NOTE: This material is a first draft for Section 21.11 (Four-Dimensional Manifolds) of Zeidler’s outline forVolume IV. Much of the required background material will need to be supplied in the earlier Chapters 16-19of Part IV (The Qualitative Behavior of Physical Fields (Topology)).

CONTENTS

1. FOUR-DIMENSIONAL MANIFOLDS 11.1. Yang-Mills Equations and BPST Instantons 11.1.1. The Quaternionic Hopf Bundle 21.1.2. Yang-Mills Connections on the Hopf Bundle 31.1.3. The Moduli Space of Instantons on the Hopf Bundle 81.2. Donaldson’s Theorem on Intersection Forms 91.2.1. Topological Background 91.2.2. Donaldson’s 1983 Theorem 121.3. Exotic Differentiable Structures 161.3.1. Freedman’s Theorem 161.3.2. An Exotic R4 181.4. Donaldson Invariants 201.4.1. Moduli Spaces of Connections 201.4.2. The Uhlenbeck Compactification 251.4.3. The Invariants 281.5. The Mathai-Quillen Formalism and Witten’s TQFT Partition Function 331.5.1. The Euler Number 341.5.2. The Universal Thom Form 401.5.3. Equivariant Cohomology 441.5.4. Gaussian Representatives of the Thom Class 481.5.5. Witten’s Partition Function 521.6. The Seiberg-Witten Invariants for Spinc-Manifolds 641.6.1. Real and Complex Clifford Algebras of R4 651.6.2. Spin and Spinc Structures 731.6.3. Seiberg-Witten Equations 771.6.4. Seiberg-Witten Moduli Space 821.6.5. Zero-Dimensional Seiberg-Witten Invariant and Witten’s Conjecture 86References 88Index 91

1. FOUR-DIMENSIONAL MANIFOLDS

1.1. Yang-Mills Equations and BPST Instantons. In this section we will introduce the Yang-Mills equa-tions and the anti-self-dual equations on R4 and then describe the simplest example of an instanton. This is

1

2

the so-called BPST instanton, or pseudoparticle, first introduced in [BPST]. We will do this in some detailin the hope of motivating the deep work of Donaldson [Don1], [Don2] on the application of gauge theory tothe topology of smooth 4-manifolds. We will turn to the work of Donaldson in Sections 1.2 and 1.4.

1.1.1. The Quaternionic Hopf Bundle. Generally we will identify R4 as a real vector space and as a C∞-manifold with the division algebra H of quaternions. Sp(1) will denote the Lie group of unit quaternions(those g ∈ H with ‖g‖ = 1). As a submanifold of H, Sp(1) is diffeomorphic to the 3-sphere S3. It is alsoisomorphic to the Lie group SU(2) of 2 × 2 complex matrices U that are unitary (U−1 = U

T) and satisfy

det U = 1. Indeed, every such U can be written uniquely in the form U =

a b−b a

, where a = a1 + a2i and

b = b1 + b2i are complex numbers satisfying |a|2 + |b|2 = 1. Then the map a b−b a

7→ a + bj = a1 + a2i + (b1 + b2i)j = a1 + a2i + b1j + b2k

is a Lie group isomorphism. We will allow ourselves the luxury of adopting whichever view of this Liegroup is most convenient in any given situation. We can therefore view its Lie algebra in one of twoways. It is isomorphic to the Lie algebra ImH of pure imaginary quaternions with the commutator bracket[x, y] = xy − yx = 2 Im (xy) determined by quaternion multiplication and also to the Lie algebra su(2) of2 × 2 complex matrices A that are skew-Hermitian (A

T= −A) and tracefree (tr A = 0) with the matrix

commutator bracket [A, B] = AB − BA. A convenient basis for su(2) consists of T j = − 12 iσ j, j = 1, 2, 3,

where σ1 =

0 11 0

, σ2 =

0 −ii 0

and σ3 =

1 00 −1

are the Pauli spin matrices. T1,T2,T3 is an

orthonormal basis relative to the inner product 〈A, B〉 = −2 tr (AB) on su(2).Now we begin our investigation by recalling the construction of the quaternionic Hopf bundle

Sp(1) → S7 π→ HP1.

We identify the 7-sphere S7 with the subset ofH2 consisting of all pairs p = (q1, q2) of quaternions satisfying‖q1‖2 + ‖q2‖2 = 1. Then we can define a smooth right action σ : S7 × Sp(1) → S7 of Sp(1) on S7 byσ(p, g) = p · g = (q1, q2) · g = (q1g, q2g). This action is free and the orbits p · Sp(1) are submanifolds ofS7 diffeomorphic to S3. Define an equivalence relation on S7 that identifies two points if and only if theyare on the same orbit and denote the equivalence class of (q1, q2) ∈ S7 by [q1, q2]. The orbit space S7/Sp(1)consisting of all of these equivalence classes is, by definition, the quaternionic projective line HP1. We letπ : S7 → HP1 be the natural projection π(q1, q2) = [q1, q2] and provide HP1 with the quotient topologydetermined by π. HP1 is provided with a differentiable structure determined by the atlas consisting of twocharts (Uk, ϕk), k = 1, 2, defined as follows.

Uk =[q1, q2] ∈ HP1 : qk , 0

, k = 1, 2

ϕk : Uk → H, k = 1, 2

ϕ1([q1, q2]) = q2(q1)−1

ϕ2([q1, q2]) = q1(q2)−1

3

Notice that each Uk covers all but one point of HP1, but ϕk maps Uk onto H. Relative to this differentiablestructure the map π : S7 → HP1 is smooth. Moreover, each of the maps

Ψk : π−1(Uk)→ Uk × Sp(1), k = 1, 2,

Ψk(p) = (π(p), ψk(p)), k = 1, 2,

where

ψk(p) = ψk(q1, q2) = qk/‖qk‖, k = 1, 2,

is a local trivialization. Specifically, each is a diffeomorphism that is equivariant with respect to the givenright action of Sp(1) on S7 and the natural right action of Sp(1) on Uk × Sp(1), that is,

Ψk(p · g) = (π(p), ψk(p)g), k = 1, 2.

Consequently, S7 is a principal Sp(1)-bundle overHP1 (see Chapter I, Section 5, page 50, of [KN1]) whichwe denote symbolically by Sp(1) → S7 π

→ HP1. It is called the quaternionic Hopf bundle.

Remark 1.1. HP1 is diffeomorphic to the 4-sphere S4. One can see this in the following way. Let (US , ϕS )and (UN , ϕN) be the charts on S4 corresponding to stereographic projection from the north and south polesof S4, respectively. If ϕ1 denotes the map ϕ1([q1, q2]) = ϕ1([q1, q2]), then ϕ−1

S ϕ2 and ϕ−1N ϕ1 are

diffeomorphisms of HP1 minus a point to S4 minus a point. On the intersection of their domains theyagree so they determine a global diffeomorphism of HP1 onto S4. Composing with π : S7 → HP1 gives aprincipal Sp(1)-bundle

Sp(1) → S7 π1→ S4

over S4 which is also often called the quaternionic Hopf bundle. Some care should be exercised, however,since reversing the roles of ϕ1 and ϕ2 gives another identification of HP1 with S4, but the correspondingSp(1)-bundle over S4 is not equivalent to the one we just described. This is most readily shown by computingtheir Chern numbers which turn out to be 1 in the first case and -1 in the second.

1.1.2. Yang-Mills Connections on the Hopf Bundle. Now fix a point p = (q1, q2) ∈ S7 ⊆ H2. The orbit ofour Sp(1)-action containing p (that is, the fiber of π : S7 → HP1 above π(p) = [q1, q2]) is a submanifoldof S7 diffeomorphic to S3. The subset of the tangent space Tp(S7) to S7 at p consisting of tangent vectorsto this fiber is called the vertical space at p and is denoted Vertp(S7). It is a 3-dimensional linear subspaceof Tp(S7) which, in turn, can be identified with a linear subspace of Tp(H2) Tp(R8) R8. Thus, relativeto the usual Euclidean inner product on R8, Vertp(S7) has an orthogonal complement and we will call theintersection of this orthogonal complement with Tp(S7) the (natural) horizontal space at p and denote itHorp(S7). At each p ∈ S7 we therefore have a natural orthogonal decomposition

Tp(S7) Vertp(S7) ⊕ Horp(S7).

If one fixes g ∈ Sp(1) and explicitly computes the derivative at p of the diffeomorphism σg : S7 → S7 givenby σg(p) = p · g one finds that

(σg)∗p(Horp(S7)) = Horp·g(S7).

4

One also checks that the distribution p 7→ Horp(S7) is C∞ in the sense that, if X is any smooth vector fieldon S7 and if one decomposes each Xp = X(p) into vertical and horizontal parts

Xp = Vert(Xp) + Hor(Xp) ∀p ∈ S7,

then the vector fields Vert(X) and Hor(X) defined by Vert(X)p = Vert(Xp) and Hor(X)p = Hor(Xp) arealso smooth. Consequently, the distribution p 7→ Horp of 4-dimensional spaces is a connection on thequaternionic Hopf bundle (see Chapter II, Section 1, page 63, of [KN1]). We call this the natural connectionon Sp(1) → S7 π

→ HP1.Any connection on a principal G-bundle arises from a connection 1-form, that is, a g-valued 1-form on the

bundle space whose kernel at each point is the horizontal space at that point (see Proposition 1.1, Chapter II,Section 1, page 64, of [KN1]). Identifying the Lie algebra of Sp(1) with ImH and defining an ImH-valued1-form ω onH2 by

ω = Im ( q1dq1 + q2dq2 )

one can show that the connection 1-form ω for the natural connection on the quaternionic Hopf bundle isthe restriction of ω to S7, that is,

ω = ι∗ω,

where ι : S7 → H2 is the inclusion map (see Section 6.1, pages 334-335, of [Nab4]).In the physics literature it is more common to describe connections locally on the base manifold by

pulling back the connection 1-form by a section of the bundle corresponding to some trivialization (thesepullbacks are called gauge potentials). For the trivializations (Uk,Ψk), k = 1, 2, of the Hopf bundle each Uk

covers all but one point of HP1 so that the connection is uniquely determined by either one of the corre-sponding pullbacks. For example, the inverse of Ψ1 : π−1(U1) → U1 × Sp(1) is given by Ψ−1

1 ([q1, q2], g) =

(‖q1‖ g, q2(q1/‖q1‖)−1g) ∈ S7 ⊆ H2 so the associated section s1 : U1 → π−1(U1) is

s1([q1, q2]) = Ψ−11 ([q1, q2], 1) =

(‖q1‖, q2(q1/‖q1‖)−1 )

.

Since U1 is also the domain of the standard chart (U1, ϕ1) on HP1 we can write s∗1ω in terms of thesecoordinates on HP1. More precisely, we pull s∗1ω back to H by ϕ−1

1 to obtain an ImH-valued 1-form(s1 ϕ

−11 )∗ω on H. For reasons that will become clear shortly we will denote this 1-form A1,0. Computing

the pullback explicitly gives

A1,0 = Im( q

1 + ‖q‖2dq

)on H (see Chapter 5, Section 9, pages 297-301, of [Nab4]). Regarded simply as an ImH-valued 1-formon H R4, A1,0 first appeared in a quite different context in the physics literature [BPST] where it wasinitially referred to as a pseudoparticle. We will have more to say about this shortly.

Thus far we know only one connection on the Hopf bundle and we would now like to produce some more.For this we recall that an automorphism of the principal bundle Sp(1) → S7 π

→ HP1 is a diffeomorphismf : S7 → S7 of S7 onto itself that respects the action of Sp(1) on S7 ( f (p · g) = f (p) · g). Each suchautomorphism induces a diffeomorphism fHP1 : HP1 → HP1 ofHP1 onto itself by π f = fHP1π. If fHP1

happens to be the identity map, then f is called a (global) gauge transformation of Sp(1) → S7 π→ HP1.

Now, if f is any automorphism and ω is any connection 1-form, then the pullback f ∗ω is also a connection

5

1-form. If f is a gauge transformation, the the connections ω and f ∗ω are said to be gauge equivalent. Onecan show (see Chapter 6, Section 1, pages 336-341, of [Nab4]) that, by judiciously choosing automorphismsof the Hopf bundle by which to pull back the natural connection ω, one can produce new connections ωλ,nfor (λ, n) ∈ (0,∞) ×H that are uniquely determined by the ImH-valued 1-forms

Aλ,n = Im( q − nλ2 + ‖q − n‖2

dq)

onH. For reasons that we will discuss shortly, Aλ,n is called the BPST instanton with center n and spread λ.Although all of the ωλ,n differ from the natural connection ω = ω1,0 by an automorphism, distinct pairs (λ, n)give rise to connections that are not gauge equivalent so the ωλ,n all represent distinct gauge equivalenceclasses of connections on the Hopf bundle (see Chapter 6, Section 3, page 357, of [Nab4]). We will see thata great deal more is true.

Any connection ω on any principal bundle has a curvature Ω that can be calculated from the CartanStructure Equation Ω = dω + 1

2 [ω,ω] and is uniquely determined by a family of pullbacks F = s∗Ω, calledgauge field strengths, by sections corresponding to some trivializing cover (see Chapter II, Theorem 5.2,page 77, of [KN1]. For the connection ωλ,n on the Hopf bundle the curvature Ωλ,n is uniquely determinedby the single gauge field strength Fλ,n = dAλ,n + 1

2 [Aλ,n,Aλ,n]. A rather tedious, but routine calculation (seeChapter 5, Section 11, pages 324-328, of [Nab4]) gives

Fλ,n =λ2

(λ2 + ‖q − n‖2)2 dq ∧ dq

=2λ2

(λ2 + ‖q − n‖2)2

[(dx1 ∧ dx2 − dx3 ∧ dx4) i

+ (dx1 ∧ dx3 + dx2 ∧ dx4) j + (dx1 ∧ dx4 − dx2 ∧ dx3) k],

where x1, x2, x3 and x4 are standard coordinates on R4.The ImH-valued 2-forms Fλ,n on H R4 have a number of properties that are crucial to our story. We

will denote by ∗ν the Hodge dual of the p-form ν, 0 ≤ p ≤ 4, on R4 corresponding to the usual orientationand inner product on R4. Extending ∗ to ImH-valued forms componentwise one finds that each Fλ,n isanti-self-dual in the sense that

∗Fλ,n = −Fλ,n.

The Hodge star operator ∗ on R4 also defines a pointwise inner product 〈µ, ν〉 on the real valued p-forms onR4 ( µ ∧∗ ν = 〈µ, ν〉volR4 ). Combined with an inner product on the Lie algebra this gives a pointwise innerproduct on Lie algebra-valued 1-forms. The most useful way to describe this is to identify ImH with su(2)and use the inner product 〈A, B〉 = −2 tr(AB) on su(2). Then the squared norm ‖F‖2 of a gauge field strengthis given by

−tr (F ∧∗ F) =12‖F‖2 volR4 ,

6

where the wedge product is computed as a matrix product with the entries multiplied by the ordinary wedgeproduct. For Fλ,n this gives

‖Fλ,n‖2 =

96λ4

(λ2 + ‖q − n‖2)4 .

Notice that ‖Fλ,n‖2 has a maximum value of 96/λ4 at q = n and that, for a fixed n, its variation with λ is suchthat the total field strength

12

∫R4‖Fλ,n‖

2 volR4 =

∫R4

48λ4

(λ2 + ‖q − n‖2)4 d4q = 8π2

remains constant at 8π2. Thus, the BPST gauge potential Aλ,n on R4 has field strength that is “centered” atn in R4 with a “spread” that is determined by λ (hence the terminology introduced earlier) and a total fieldstrength that is independent of n and λ (see Figure 1). One can think of this in the following way. Fix thecenter n. Then, as a function of λ, the maximum value 96/λ4 of ‖Fλ,n‖2 approaches infinity as λ → 0 insuch a way as to maintain a constant value for the total field strength. Thus, as λ → 0, the field strengthconcentrates more and more at n. We will see that the reason for this as well as its consequences are deeperthan one might expect.

FIGURE 1. BPST Field Strengths ‖Fλ,n‖2

Now we would like to temporarily suppress from our minds where the potentials Aλ,n came from, that is,the Hopf bundle, and regard them simply as Lie algebra-valued 1-forms onR4. Any such Lie algebra-valued1-form A can be thought of as a gauge potential for a connection on the trivial Sp(1)-bundle over R4 and sohas a gauge field strength F = dA + 1

2 [A,A]. We define the Yang-Mills action YM(A) of A by

YM(A) =

∫R4−tr (F ∧∗ F) =

12

∫R4‖F‖2 volR4 . (1)

The motivation for the terminology comes from the attempts of Yang and Mills in [YM] to construct a non-Abelian generalization of classical electromagnetic theory to describe the isotopic spin of a nucleon. Theintegral in (1) might well be infinite, but if it is not we will say that A has finite action and then think ofYM(A) as the total field strength of the gauge potential A.

7

The Euler-Lagrange equations for the Yang-Mills action YM(A) under variations of A are

dA∗F = d∗F + [A, ∗F] = 0, (2)

where dA∗F is the covariant exterior derivative of ∗F associated with A. These are the Yang-Mills Equationson R4. Quite independently of YM, any gauge field strength F, being the pullback of a curvature, satisfies apurely geometrical condition called the Bianchi Identity

dAF = dF + [A,F] = 0. (3)

(see Chapter II, Theorem 5.4, page 78, of [KN1]. Now notice that, if the gauge field strength F of A

happens to be anti-self-dual (∗F = −F), then the Bianchi identity implies that the Yang-Mills equations areautomatically satisfied. A finite action gauge potential A on R4 with anti-self-dual gauge field strength F

is called an instanton on R4. In particular, an instanton on R4 is a solution to the Yang-Mills equationsand so is a critical point of the Yang-Mills action. In fact, however, instantons are absolute minima for theYang-Mills action (see Chapter 6, Section 3, page 361, of [Nab4]).

We have described a family of instantons Aλ,n parametrized by (λ, n) ∈ (0,∞) ×R4 and have lately beenthinking of them simply as ImH-valued 1-forms on R4 and forgetting where they came from. In orderto unearth the motivation for Donaldson’s work we will now want to identify HP1 with S4 in the mannerdescribed in Remark 1.1 and view the Aλ,n as gauge potentials corresponding to connections ωλ,n on

Sp(1) → S7 π1→ S4.

Stated otherwise, the Aλ,n can be viewed as connection 1-forms on the trivial Sp(1)-bundle over R4

Sp(1) → R4 × Sp(1)→ R4

that “come from” connection 1-formsωλ,n on the nontrivial Hopf bundle over S4. Turning matters about, onemight say that these connections on the trivial bundle overR4 “extend to S4” in the sense that S4 = R4∪∞

is the 1-point compactification of R4 and, due to their asymptotic behavior as ‖x‖ → ∞ in R4 (they havefinite action), the connections extend to the point at infinity. Notice, however, that the extension processinvolves not only the connections, but the bundle on which they are defined as well. A consequence of thefamous Removable Singularities Theorem of Karen Uhlenbeck [Uhl] asserts that this interpretation is not asfanciful as it might seem. Specifically, Uhlenbeck proved the following. Let A be an ImH-valued 1-formon R4 that satisfies the Yang-Mills equations (2) and has finite Yang-Mills action YM(A) < ∞. Then thereexists a unique (up to equivalence) Sp(1)-bundle Sp(1) → P

π→ S4 over S4, a connection 1-form ω on P and

a section s : S4−N → π−1(S4−N) such that A = (sϕ−1S )∗ω, where ϕS : S4−N → R4 is stereographic

projection from the north pole N.Thus, finite action Yang-Mills potentials on R4 extend to connections on some principal Sp(1)-bundle

over S4. But a principal Sp(1)-bundle Sp(1) → Pπ→ S4 over S4 is uniquely determined up to equivalence

by its second Chern number c2(P)[S4] (see Chapter 6, Section 4, pages 328-334, of [Nab5]) which can becalculated from

c2(P)[S4] =1

8π2

∫R4

tr (F ∧ F),

8

where F = dA+ 12 [A,A]. Now suppose that A is an instanton so that ∗F = −F. Then −tr (F∧∗F) = tr (F∧F)

so

c2(P)[S4] =1

8π2 YM(A).

Thus, the Yang-Mills action of an instanton A on R4 directly encodes the topology of the bundle over S4 towhich A extends. But YM(A) is determined by the asymptotic behavior of the field strength F on R4 so itis this physical characteristic of the field that is represented by the Chern number. Physicists call −c2(P)[S4]the instanton number or topological charge of A. In particular, each Aλ,n has instanton number -1. The“reason” that all of the BPST instantons Aλ,n have the same total field strength YM(Aλ,n) is now clear; theyall extend to (that is, come from) the same Sp(1)-bundle over S4, that is, the Hopf bundle. Notice alsothat, because the Chern number is always an integer (see Chapter 6, Section 4, pages 328-334, of [Nab5]),the topological charge of an instanton on R4 is “quantized”. In particular, its total field strength, being anintegral multiple of 8π2, cannot be altered by a continuous variation of the field and so is “conserved”, butfor purely topological reasons, unlike the more common Noether conserved quantities.

1.1.3. The Moduli Space of Instantons on the Hopf Bundle. The motivation for Donaldson’s approach tothe topology of smooth 4-manifolds comes not from the instantons themselves, but rather from their gaugeequivalence classes. If A is an instanton on R4 with topological charge -1 we will denote by [A] its gaugeequivalence class and by M the collection of all such gauge equivalence classes. The BPST instantonsAλ,n each determine a point [Aλ,n] in M and, moreover, [Aλ,n] = [Aλ′,n′] if and only if (λ, n) = (λ′, n′).Much deeper is the fact, proved by Atiyah, Hitchin and Singer in [AHS], that every instanton on R4 withtopological charge -1 is gauge equivalent to some BPST instanton Aλ,n so that the [Aλ,n] exhaust all of M.

M =

[Aλ,n] : (λ, n) ∈ (0,∞) ×R4

M is called the moduli space of instantons on R4 with topological charge -1. One can identify this with themoduli space of connections with anti-self-dual curvature on the Sp(1)-bundle over S4 with Chern number1, that is, on the quaternionic Hopf bundle Sp(1) → S7 π1

→ S4. The map (λ, n) ∈ (0,∞) ×R4 7→ [Aλ,n] ∈Mis a bijection. Points that are “nearby” in the usual topology of (0,∞) × R4 ⊆ R5 give rise to connectionswhose gauge potentials are “close” in the sense that their field strengths are centered at nearby points andhave approximately the same scale so it would seem appropriate to identify M topologically with the opensubspace (0,∞)×R4 ofR5. We will see in Section 1.3 that any two differentiable structures onR5 (which ishomeomorphic to (0,∞)×R4) are necessarily diffeomorphic so there is no ambiguity about the appropriatedifferentiable structure for M. We therefore identify M as a smooth manifold with (0,∞)×R4. A still moreinstructive picture is obtained by noting that there is an orientation preserving, conformal diffeomorphismof (0,∞) × R4 onto the open 5-dimensional ball B5 in R5 and that, with this, one can supply M with“spherical coordinates” (see Chapter 6, Section 5, pages 372-375, of [Nab4]). The picture of M that emergesfrom this is as follows. M is viewed as the open 5-ball B5 with the gauge equivalence class [A1,0] of thenatural connection at the center. Proceeding radially outward from [A1,0] toward a point on ∂B5 = S4 oneencounters potentials all of which have the same center n, but become more and more concentrated, that is,for which the spread λ → 0. This boundary point in S4 can then, at least intuitively, be identified with aconnection/gauge potential concentrated entirely at n, that is, one whose curvature/field strength is a sort of

9

Dirac delta. A particularly pleasing aspect of this picture is that the base manifold S4 of the bundle to whichthe connections Aλ,n extend emerges as the boundary of the moduli space in some compactification of M(M B5 → B

5= B5 ∪ S4 ) and that a point in it can be viewed as a sort of singular connection. The moral

is that the topologies of the underlying 4-manifold S4 and the moduli space M are inextricably linked. Inthe next section we will begin to see how Donaldson exploited this idea.

1.2. Donaldson’s Theorem on Intersection Forms. We will now provide a very broad sketch of the pro-found generalization, due to Donaldson [Don1], of the scenario suggested by the example in the precedingsection. The idea is to study the topology of a smooth 4-manifold X by identifying X with the boundary ina compactification of a certain moduli space of connections on some principal bundle over X.

1.2.1. Topological Background. Throughout this section X will denote a compact, connected, simply con-nected, oriented, smooth 4-manifold.

Remark 1.2. A simply connected 4-manifold is always orientable, but much of what we will have to sayis very sensitive to the specific orientation chosen so we will tend to emphasize this by specifying that ourmanifolds are oriented.

Typical examples include the 4-sphere S4, the product S2 × S2 of two 2-spheres, the complex projectiveplane CP2 with its natural orientation as a complex manifold, the same manifold CP

2with the opposite ori-

entation, the connected sum CP2#CP2

of CP2 and CP2

and the Kummer surface K3 which can be thoughtof as the submanifold (complex algebraic surface) in CP3 whose homogeneous coordinates x1, x2, x3, x4

satisfy x41 + x4

2 + x43 + x4

4 = 0.We begin by enumerating a few classical results on the topology of such manifolds. We will be partic-

ularly interested in the homology of X. Since X is assumed connected, H0(X;Z) Z (see Theorem 3.2.3,page 171, of [Nab4]). Being simply connected, π1(X) is trivial so the Hurewicz Theorem (see Corollary12.2, page 48, of [Green]) implies that H1(X;Z) 0 as well and this implies that H2(X;Z) is torsion free(see Proposition E.1, page 217, of [FU1]). Poincare Duality (see Theorem 26.6, page 164, of [Green]) thenimplies that H4(X;Z) Z and H3(X;Z) 0. Thus, manifolds of the sort we are considering can differhomologically only in H2(X;Z) which is a finitely generated, free Abelian group (Z-module) and thereforecompletely determined by the second Betti number b2(X), that is, the rank of H2(X;Z). Since there is notorsion in H2(X;Z),

H2(X;Z) Hom (H2(X;Z),Z) H2(X;Z)

(see Corollary 23.14, page 134, of [Green]).Another very special property of this type of manifold is that every homology class α in H2(X;Z) can

be represented by a smoothly embedded, compact, oriented surface (2-manifold) Σ in X. This means thatthere exists a smooth embedding ι : Σ → X of Σ in X such that ι∗([Σ]) = α, where [Σ] is the fundamentalclass of Σ in H2(Σ;Z) and ι∗ : H2(Σ;Z) → H2(X;Z) is the map induced by ι. (see Proposition E.9,page 224, of [FU1] or Theorem 1.1, page 20, of [Kirby]). A given α ∈ H2(X;Z) can be represented by a

10

variety of surfaces of different genus. The elements of H2(X;Z) can also be represented by complex linebundles L→ X over X because these line bundles are determined up to equivalence by their 1st Chern classc1(L) ∈ H2(X;Z). We point out as well that, if X1 and X2 are two manifolds of the type under considerationhere, the Mayer-Vietoris sequence (see Part II, Section 17, page 74, of [Green]) implies that H2( · ;Z) isadditive on connected sums, that is,

H2(X1#X2;Z) H2(X1;Z) ⊕ H2(X2;Z).

The intersection form on X is a certain integer-valued, symmetric, bilinear form QX : H2(X;Z) ×H2(X;Z) → Z on H2(X;Z) (see Chapters 3 and 4 of [Scor]). This can be defined in a variety of ways,but we will opt for the following. If H2(X;Z) 0, then QX takes the value 0 at the only point inH2(X;Z) × H2(X;Z); this is generally called the empty form and denoted ∅. This is the case, for exam-ple, when X = S4. Otherwise, suppose α1 and α2 are two homology classes in H2(X;Z). Select smoothlyembedded, compact, oriented surfaces Σ1 and Σ2 in X representing them. By the Transversality Theorem(see Chapter 3, Section 2, of [Hirsch]), one can assume that these surfaces intersect transversally, that is,the tangent spaces to Σ1 and Σ2 span the tangent space to X at each point in the intersection Σ1 ∩ Σ2, so thatΣ1 ∩ Σ2 is a finite set of isolated points in X. A point p ∈ Σ1 ∩ Σ2 is assigned the value 1 if an orientedbasis for Tp(Σ1) together with an oriented basis for Tp(Σ2) gives an oriented basis for Tp(X); otherwise it isassigned the value −1. Then QX(α1, α2) is the sum of these values over all of the intersection points.

Remark 1.3. Identifying the second homology H2(X;Z) with the second cohomology H2(X;Z) by Poincareduality, the intersection form QX : H2(X;Z) × H2(X;Z) → Z is given by pairing the cup product with thefundamental class of X

QX(α1, α2) = 〈α1 ^ α2, [X] 〉

and this serves to define the intersection form if X is merely a topological rather than a smooth 4-manifold.

If H2(X;Z) , 0, then QX is unimodular in the sense that its matrix relative to any basis for H2(X;Z) hasdeterminant ±1 (see Chapter 3, Section 2, page 116, of [Scor]). QX is said to be even if QX(α, α) is even forall α ∈ H2(X;Z) and odd otherwise. QX is positive definite (respectively, negative definite) if QX(α, α) > 0(respectively, QX(α, α) < 0) for every nonzero α ∈ H2(X;Z). QX is definite if it is either positive or negativedefinite and indefinite otherwise. If H2(X;Z) , 0 then the maximal rank of a Z-submodule of H2(X;Z) onwhich QX is positive (respectively, negative) definite is denoted b+

2 (X) (respectively, b−2 (X)). If H2(X;Z) = 0,then both of these are taken to be zero. Then the second Betti number of X is b2(X) = b+

2 (X) + b−2 (X). Thesignature of X (or of QX) is σ(X) = σ(QX) = b+

2 (X) − b−2 (X). The signature is additive for both disjointunions and connected sums. Two intersection forms QX1 and QX2 are said to be equivalent if and only ifthere exist bases for H2(X1;Z) and H2(X2;Z) relative to which QX1 and QX2 have the same matrix. Theintersection form is additive on connected sums, that is,

QX1#X2 = QX1 ⊕ QX2

(see Chapter 3, Section 2, page 118, of [Scor]).

11

Example 1.1. In each of the following examples an unspecified basis is chosen for the Z-module H2(X;Z)and QX is given as a matrix relative to this basis (see Chapter 1, Section 1, pages 3-5, of [DK]).

X H2(X;Z) QX b+2 (X) σ(X)

S4 0 ∅ 0 0CP2 Z (1) 1 1

CP2

Z (−1) 0 −1

S2 × S2 Z ⊕Z

0 11 0

1 0

CP2#CP2

Z ⊕Z

1 00 −1

1 0

K3 22Z 2(−E8) ⊕ 30 11 0

3 −16

where E8 is the even, positive definite, unimodular matrix given by

E8 =

2 −1 0 0 0 0 0 0−1 2 −1 0 0 0 0 00 −1 2 −1 0 0 0 00 0 −1 2 −1 0 0 00 0 0 −1 2 −1 0 −10 0 0 0 −1 2 −1 00 0 0 0 0 −1 2 00 0 0 0 −1 0 0 2

The following classical result of Rokhlin [Rokh] will be useful in the next section when we discuss the

existence of exotic differentiable structures.

Theorem 1.1. (Rokhlin) Let X be a compact, connected, simply connected, oriented, smooth 4-manifoldwith even intersection form. Then the signature σ(X) is divisible by 16.

Remark 1.4. Rokhlin’s Theorem is generally stated for spin 4-manifolds which always have even intersec-tion form. For simply connected 4-manifolds the converse is also true, that is, an even intersection formimplies the existence of a spin structure. For a proof of Rokhlin’s Theorem 1.1 see Chapter X, Section 1,pages 64-65, of [Kirby].

It has been known for some time that the intersection form is a basic invariant of compact 4-manifolds.In 1949, J.H.C. Whitehead [White] proved the following.

Theorem 1.2. (Whitehead) Two compact, connected, simply connected, topological 4-manifolds X1 and X2

have the same homotopy type if and only if their intersection forms are equivalent.

12

1.2.2. Donaldson’s 1983 Theorem. In 1982, the situation took a dramatic turn when Freedman [Freed]proved that every unimodular, symmetric,Z-valued bilinear form on a finitely generated, free Abelian groupis the intersection form of at least one (and at most two) compact, connected, simply connected, orientedtopological 4-manifolds, thereby giving a complete classification of such manifolds and, in particular, prov-ing the 4-dimensional Poincare conjecture (see Section 1.3.1). This is, in particular, true of the vast, impen-etrable maze of definite forms. If you do not believe that this is a vast, impenetrable maze, consider the factthat when the rank is 40 there are at least 1051 equivalence classes of positive definite forms (see Chapter II,Section 6, page 28, of [MH]). In 1983, however, Donaldson [Don1] showed that the differential topologistneed not venture into this maze because there is only one positive/negative definite unimodular, symmetric,Z-valued bilinear form that can arise as the intersection form of a compact, connected, simply connected,oriented smooth 4-manifold. In particular, there are enormous quantities of topological 4-manifolds that canadmit no smooth structure at all.

Theorem 1.3. (Donaldson’s Theorem on Intersection Forms) Let X be a compact, connected, simply con-nected, oriented, smooth 4-manifold with positive (respectively, negative) definite intersection form QX .Then there is a basis for H2(X;Z) relative to which the matrix of QX is the identity matrix (respectively,minus the identity matrix).

Donaldson’s Theorem is remarkable, but even more remarkable is its proof which is a byproduct of theanalysis of an instanton moduli space quite like the BPST instanton moduli space discussed in Section 1.1.3.The proof is an extremely delicate blend of topology, geometry, and analysis (see [Mor2], [FU1], [Law],or Chapter 8 of [DK]). We will offer only a minimalist sketch of the ideas that lie behind the proof in thenegative definite case. Thus, we assume that our compact, connected, simply connected, oriented, smooth4-manifold X has

b+2 (X) = 0,

but b2(X) , 0. Denote by

Sp(1) → Pπ→ X

the principal Sp(1)-bundle over X with second Chern number 1. Note that when X = S4 this is just thequaternionic Hopf bundle. Now choose a Riemannian metric g for X. With g and the given orientation of Xone can define a Hodge star operator ∗ on the differential forms defined on X. Since dim X = 4 the Hodgedual of a 2-form is a 2-form.

Now let ω be a connection on Sp(1) → Pπ→ X. The curvature Ω of ω is a Lie algebra-valued 1-form

on P, not on X. However, pulling Ω back by the local sections corresponding to some trivializing cover ofX gives a family of Lie algebra-valued 2-forms on X that are related by the adjoint action of Sp(1) on itsLie algebra. As a result these locally defined 2-forms on X with values in the Lie algebra piece togetherinto a globally defined 2-form Fω on X with values, not in the Lie algebra, but in the adjoint bundle ad P(the vector bundle associated to Sp(1) → P

π→ X by the adjoint representation). This 2-form Fω is also

13

often referred to as the curvature of ω. The Hodge star extends to such ad P-valued forms so ∗Fω is another2-form on X with values in the adjoint bundle. We will say that the connection ω is g-anti-self-dual if

∗Fω = −Fω.

Now, it is not at all clear that any such 2-forms exist on Sp(1) → Pπ→ X. It is, in fact, a deep theorem of

Taubes [Taub1] that, because b+2 (X) = 0, there do exist g-anti-self-dual connections on the (Chern number

1) bundle Sp(1) → Pπ→ X for any choice of g. However, if X = S2 × S2, where b+

2 (X) = 1, with itsstandard metric and orientation there are no such connections on the Sp(1)-bundle with Chern number 1.The same is true ofCP2, but, oddly enough, not ofCP

2(Donaldson theory is highly orientation dependent).

A g-anti-self-dual connection on Sp(1) → Pπ→ X is called an instanton on Sp(1) → P

π→ X.

A (global) gauge transformation of our bundle is a diffeomorphism f : P → P of P onto itself thatrespects the right action of Sp(1) on P ( f (p · g) = f (p) · g for all p ∈ P and all g ∈ Sp(1)) and coversthe identity map on X (π f = π). These form a group under composition, called the gauge group ofSp(1) → P

π→ X and denoted G(P). The gauge group acts on connections by pullback (ω · f = f ∗ω) and

two connectionsω andω′ are said to be gauge equivalent if there exists an f ∈ G(P) such thatω′ = f ∗ω. Thecollection of all gauge equivalence classes of g-anti-self-dual connections on Sp(1) → P

π→ X is denoted

M = M(P, g)

and called the moduli space of g-anti-self-dual connections on Sp(1) → Pπ→ X.

Remark 1.5. We are about to describe what purports to be a picture of the moduli space of g-anti-self-dualconnections on Sp(1) → P

π→ X. We must preface this by saying that the amount of technical work that

must be done in order to produce this picture is prodigious and cannot really be carried out in the smoothcontext we have just described. In order to obtain analytically tractable spaces in which to work one mustintroduce various Sobolev completions of the space of connections and the gauge group. There are standardprocedures for doing this, but for our very modest sketch we will be content to suppress these completionsand simply recommend a few sources for the material we omit . The constructions are outlined in Section 1,pages 33-49, of [Nab1]; more thorough discussions are available on pages 85-107 of [Mor2] and in ChapterII, Section 7, of [Law].

We have chosen the Riemannian metric g and it turns out that if we choose badly the moduli spaceM(P, g) might well have no decent mathematical structure at all. However, Donaldson has shown that, fora “generic” choice of g, M(P, g) has quite a nice structure that we will now describe (for a sketch of howthis structure comes about see Section B.2 of [Nab4] and for all of the details see [Mor2], [FU1], [Law], or[DK]). We will simply describe the structure. Thus, we let g denote some generic metric and denote themoduli space M(P, g) simply as M. Then, in a word, M looks like Figure 2. More precisely, Donaldsonproves all of the following.

(1) If m is one-half the number of α ∈ H2(X;Z) with QX(α, α) = −1, then there exist pointsp1, . . . , pm

in M such that M −

p1, . . . , pm

is a smooth, orientable, 5-dimensional manifold.

14

FIGURE 2. Moduli Space

(2) Each pi, i = 1, . . . ,m, has a neighborhood in M that is homeomorphic to a cone over CP2 withvertex at pi and these neighborhoods are smooth away from pi (the cone over CP2 is obtained fromthe cylinder CP2 × [0, 1] by identifying CP2 × 1 to a point, which is then called the vertex of thecone).

(3) There is a compact subspace K of M such that M − K is an open submanifold of M −p1, . . . , pm

diffeomorphic to the cylinder X × (0, 1).

Now construct from M another space M0 by cutting off the open top half of each cone and the openbottom half of the cylinder (see Figure 3). Then M0 is a compact manifold with boundary that inherits anorientation from M. The boundary is the disjoint union of a copy of X and m copies of CP2 or CP

2. Let us

suppose that there are p copies of CP2 and q copies of CP2. Thus M0 is a cobordism between X and the

disjoint union

CP2 tp· · · tCP2 t CP

2t

q· · · tCP

2= pCP2 t qCP

2

(see Chapter 1, Section 1, page 28, of [Scor]). As it happens, the signature of the intersection form isa cobordism invariant (see Chapter 3, Section 2, page 123 of [Scor]). Since the signature is additive ondisjoint unions and b+

2 (X) = 0, b+2 (X) − b−2 (X) = p − q gives b2(X) = b−2 (X) = q − p and therefore

b2(X) ≤ q + p = m.

15

FIGURE 3. Cobordism

Next we will show that b2(X) ≥ m as well. Select some α1 ∈ H2(X;Z) with QX(α1, α1) = −1. Note thatthere must be at least one such since b+

2 (X) = 0, but b2(X) , 0. Then there is a QX-orthogonal decomposition

H2(X;Z) = Zα1 ⊕G1.

Suppose first that there is no α2 ∈ H2(X;Z) with QX(α2, α2) = −1 and α2 , ±α1. Then G1 = ∅ soH2(X;Z) = Zα1 and α1 is a basis for H2(X;Z) relative to which the matrix of QX is (−1). In this case weare done. Now suppose there is such an α2. The Schwartz Inequality gives

(QX(α1, α2))2 < QX(α1, α1)QX(α2, α2) = 1.

But QX(α1, α2) is an integer so QX(α1, α2) = 0 and therefore α2 ∈ G1. Now repeat the argument inside G1

and continue inductively until you run out of α ∈ H2(X;Z) with QX(α, α) = −1. There are 2m such α andthey occur in pairs (α,−α) so the result is a QX-orthogonal decomposition

H2(X;Z) = Zα1 ⊕ · · · ⊕Zαm ⊕G,

where G is either empty or the QX-orthogonal complement of Zα1 ⊕ · · · ⊕Zαm. In particular, m ≤ b2(X) so

b2(X) = m = p + q.

But H2(X;Z) is a finitely-generated, free Abelian group so G must be empty and

H2(X;Z) = Zα1 ⊕ · · · ⊕Zαm.

Since QX(αi, αi) = −1 for every i = 1, . . . ,m, the matrix of QX relative to the basis α1, . . . , αm is minus theidentity and this is Donaldson’s Theorem.

16

1.3. Exotic Differentiable Structures. A differentiable structure (or smooth structure) on a (Hausdorff,second countable) topological space X is a maximal atlas of C∞-related charts. It is a simple matter todefine different differentiable structures on the same space X. For example, if X = R, then the standarddifferentiable structure is the unique maximal atlas determined by the single chart (R, idR). The chart (R, ϕ),where ϕ : R→ R is given by ϕ(x) = x3 is not in this standard differentiable structure because ϕ−1(x) = x1/3

is not differentiable with respect to the standard structure at x = 0. Consequently, the chart (R, ϕ) onR determines another differentiable structure that is not the same as the standard one. Nevertheless, thedifferentiable manifolds determined by (R, idR) and (R, ϕ), call them R and R′, respectively, are not toodifferent. Specifically, the map ϕ : R′ → R is a diffeomorphism. More generally, one can show that anytwo differentiable structures on the topological space R are necessarily diffeomorphic. Still more generally,one has the following result (see the Appendix in [Miln3]).

Theorem 1.4. Any smooth, connected, 1-dimensional manifold (without boundary) is diffeomorphic eitherto the circle S1 if it is compact, or to R if it is not compact.

Up to diffeomorphism, R has a unique differentiable structure. Rado [Rado] proved, in 1925, that thesame is true of R2 and, in 1952, Moise [Moi] proved the same thing for R3. Eventually, the combined workof a number of people showed that, in fact, the same is true of Rn for n = 1, 2, 3, 5, 6, 7, . . .. It will not haveescaped your attention that n = 4 is conspicuously absent from this list. Indeed, our objective in this sectionis to very briefly sketch how Donaldson’s Theorem 1.3 on intersection forms, when combined with deepresults of Freedman [Freed] on topological 4-manifolds, implies the existence of smooth 4-manifolds thatare homeomorphic to R4, but not diffeomorphic to R4 with its standard differentiable structure. These arecalled exotic or fake R4s.

Remark 1.6. We will be concerned only with the case of R4, but we point out that, in 1956, Milnor [Miln1]proved the existence of smooth 7-manifolds that are homeomorphic, but not diffeomorphic to the 7-sphereS7. In fact, there are 28 such exotic differentiable structures on S7.

1.3.1. Freedman’s Theorem. We will need to borrow a number of Freedman’s results in order to outline theconstruction of an exoticR4 so we might as well begin with the major result of [Freed] (see also [FQ]). It isa complete classification of all compact, connected, simply connected topological 4-manifolds.

Theorem 1.5. (Freedman) Let Q be anyZ-valued, symmetric, unimodular bilinear form on a finitely gener-ated, free Abelian group. Then there exists a compact, connected, simply connected topological 4-manifoldwhose intersection form is equivalent to Q.

(1) If Q is even, there is exactly one such manifold XQ up to homeomorphism.(2) If Q is odd, there are exactly two such manifolds up to homeomorphism, at least one of which does

not admit a differentiable structure.

17

Example 1.2. The form Q = E8 is even so Freedman’s Theorem 1.5 guarantees the existence of a uniquetopological 4-manifold XE8 whose intersection form is E8. The signature of E8 is 8 so Rokhlin’s Theorem1.1 implies that XE8 cannot admit a differentiable structure. On the other hand, the form E8 ⊕ E8 is evenand has signature 16 so Rokhlin’s Theorem 1.1 does not prohibit Freedman’s topological 4-manifold XE8⊕E8

from admitting a differentiable structure. Nevertheless, since E8 ⊕ E8 is positive definite, but not equivalentto the identity form, Donaldson’s Theorem 1.3 implies that XE8⊕E8 cannot admit a smooth structure.

The combination of Freedman’s Theorem 1.5 and Donaldson’s Theorem 1.3 yields a huge number ofcompact, connected, simply connected, topological 4-manifolds that can admit no smooth structure at all(one for every definite form Q other than ±id). One should contrast this with the following result of FrankQuinn (see Chapter 8, Section 2, of [FQ] ) which implies, in particular, that removing a single point fromany of these examples results in a 4-manifold that does admit a smooth structure.

Theorem 1.6. (Quinn) Any connected, noncompact 4-manifold admits a differentiable structure.

Freedman’s Theorem is quite remarkable and it has many equally remarkable consequences of which wewill record just a few. Suppose first that X1 and X2 are two compact, connected, simply connected, smooth4-manifolds with the same intersection form Q, that is, QX1 and QX2 are both equivalent to Q. Whitehead’sTheorem 1.2 implies that X1 and X2 have the same homotopy type, but now we see that much more istrue. Indeed, Freedman’s Theorem 1.5 implies that, up to homeomorphism, there is exactly one topological4-manifold with this intersection form. Consequently, we have the following.

Theorem 1.7. Let X1 and X2 be two compact, connected, simply connected, smooth 4-manifolds with equiv-alent intersection forms. Then X1 and X2 are homeomorphic.

Combining this with the Whitehead Theorem 1.2 and applying it to the 4-sphere S4 one obtains the4-Dimensional Poincare Conjecture.

Corollary 1.8. If a topological 4-manifold X is homotopy equivalent to the 4-sphere S4, then it is homeo-morphic to S4.

Example 1.3. According to Donaldson’s Theorem, if X is a compact, connected, simply connected, smooth4-manifold and the intersection form QX is definite, then QX is either the identity or minus the identity,that is, either (1)⊕ b(X)

· · · ⊕ (1) or (−1)⊕ b(X)· · · ⊕ (−1). We know that the first of these is the intersection form

of CP2 # b(X)· · · #CP2 and the second is the intersection form of CP

2# b(X)· · · #CP

2so Freedman’s Theorem

implies that X is homeomorphic to the first of these of QX is positive definite and to the second if QX is

negative definite. If QX is indefinite and odd, then it is equivalent to (1)⊕ b+(X)· · · ⊕ (1) ⊕ (−1)⊕ b−(X)

· · · ⊕ (−1)

(see Chapter II, Section 4, of [MH]) and so X is homeomorphic to CP2 # b+(X)· · · #CP2 #CP

2# b−(X)· · · #CP

2.

18

If QX is indefinite and even, then it is equivalent to ±(E8 ⊕2m· · · ⊕ E8)⊕ n

0 11 0

for integers m ≥ 0 and n > 0

(see Chapter II, Section 5, of [MH]). For example, −(E8 ⊕ E8) ⊕ 30 11 0

is represented by the K3-surface.

Finally we point out that homeomorphism can generally not be strengthened to diffeomorphism here. Indeed,Donaldson [Don2] has shown that CP2# 9CP

2itself admits nonstandard differentiable structures so that a

smooth 4-manifold with intersection form (1) ⊕ 9(−1) need not be diffeomorphic to CP2# 9CP2.

1.3.2. An ExoticR4. Now we will sketch one construction of an exoticR4, that is, a 4-dimensional, smoothmanifold that is homeomorphic, but not diffeomorphic to R4 with its usual differentiable structure. Thereare other constructions (see Chapter 9, Sections 3 and 4, of [GS]) and many other examples (uncountablymany according to a theorem of Taubes [Taub2]). The constructions are all quite involved and we must becontent with a very superficial sketch of just one of them relying on a great deal of heavy artillery.

Once we have a 4-manifold that is homeomorphic toR4 we will need some criterion to guarantee that it isnot diffeomorphic to R4 with its standard smooth structure. This much at least is easy. In standard R4 thereare arbitrarily large smoothly embedded 3-spheres so every compact set is contained inside some smoothlyembedded 3-sphere.

Lemma 1.9. Let R be a smooth 4-manifold that is homeomorphic to R4. If R is diffeomorphic to R4 withits standard smooth structure, then every compact set in R is contained inside some smoothly embedded3-sphere in R. Consequently, if there exists a compact set C in R that is not contained inside any smoothlyembedded 3-sphere in R, then R is an exotic R4, that is, R is homeomorphic, but not diffeomorphic to R4.

We begin the construction with X = CP2# 9CP2. This is a 4-manifold and, as we pointed out above,

Donaldson [Don2] has shown that X itself admits non-standard differentiable structures. Nevertheless, wewill consider only the standard smooth structure on X coming from CP2 and CP

2. The intersection form

of X can be written as QX = (1) ⊕ 9 (−1) = (1) ⊕ 8 (−1) ⊕ (−1) relative to some basis e0, e1, . . . , e9 forH2(X;Z), where QX(e0, e0) = 1,QX(e j, e j) = −1 for j = 1, . . . , 9, and QX(e j, ek) = 0 for j , k. Now,consider the element

α = 3e0 + e1 + · · · + e8 ∈ Span e0, . . . , e8 ⊆ H2(X;Z).

Then

QX(α, α) = 1

and, for any β = x0e0 + x1e1 + · · · + x8e8 + x9e9 ∈ H2(X;Z),

QX(α, β) = 3x0 − x1 − · · · − x8.

Consequently, β is in the QX-orthogonal complement of α if and only if

3x0 = x1 + · · · + x8 and x9 ∈ Z is arbitrary.

19

Write 〈α〉⊥ for the QX-orthogonal complement of α in H2(X;Z). The eight elements

e2 − e1, e2 − e3, e4 − e3, e4 − e5, e6 − e5, e6 − e7, e8 − e7, e0 + e6 + e7 + e8

of Span e0, . . . , e8 are easily seen to form a basis for 〈α〉⊥ ∩ Spane0, . . . , e8. Computing QX of its variouspairs of elements one finds that, relative to this basis, the matrix of the restricted intersection form QX is just−E8. The restriction of QX to all of 〈α〉⊥ is therefore −E8 ⊕ (−1). Consequently, QX can be written as

QX = −E8 ⊕ (−1) ⊕ (1).

According to Freedman’s Theorem 1.5 this corresponds to a topological splitting

X = N #CP2

of X into the connected sum of a topological 4-manifold N with intersection form QN = −E8 ⊕ (−1) andCP2. Notice, however, that this cannot be a smooth splitting since QN is negative definite, but not equivalentto minus the identity so that, by Donaldson’s Theorem 1.3, N cannot not admit a smooth structure at all.

The form −E8 ⊕ (−1) is defined on 〈α〉⊥ = H2(N;Z) so α generates the homology of the CP2-summandin X = N #CP2. We have already pointed out that every homology class in H2(X;Z) can be represented bya smoothly embedded, compact, oriented surface in X. However, the genus of such a representing surface isnot unique and determining the minimal such genus is generally difficult. In the case of our α ∈ H2(X;Z)we would like to argue that this minimal genus is greater than zero, that is, that α cannot be represented bya smoothly embedded 2-sphere in X. We will provide only a brief sketch of the argument. Let us supposeto the contrary that α is represented by some smooth embedding of a 2-sphere Σs in X. Identify Σs with asmooth submanifold of X diffeomorphic to S2. Then Σs has a tubular neighborhood in X and any two suchare isotopic (see Theorems 5.2 and 5.3 of [Hirsch]). More specifically, we construct a neighborhood of Σs

in X as follows. Fix some Riemannian metric g on X. Then for ε > 0 sufficiently small, we define

Ng(ε) =

x ∈ X : distg(x,Σs) < ε

and

Ng(ε) =

x ∈ X : distg(x,Σs) ≤ ε.

Define a projection map

πN : Ng(ε)→ Σs

by letting πN(x) be the point in Σs closest to x for every x ∈ Ng(ε). Then Ng(ε) is an open neighborhoodof Σs in X and (Ng(ε), πN) is a disc bundle over Σs. Each such disc bundle over Σs is characterized by aninteger, generally called the Euler number, which can be computed in a variety of ways, one of which is theself-intersection QX(α, α) of the homology class α represented by Σs (see Chapter 4, Section 1, page 103,and Exercise 1.4.11 (b), page 28, of [GS]). In our case QX(α, α) = 1 so the Euler number of (Ng(ε), πN) is 1.Now we would like to compare this with an analogous situation in CP2. The homology H2(CP2;Z) of CP2

is isomorphic to Z and is generated by the fundamental class [CP1] of any copy of CP1 S2 in CP2. Theself-intersection QCP2([CP1], [CP1]) is 1 so the disc bundle over CP1 constructed as above for Σs also hasEuler number 1. It follows, in particular, that the neighborhood Ng(ε) of Σs in X is diffeomorphic to a smoothtubular neighborhood of CP1 in CP2 which has boundary diffeomorphic to S3 (because the boundary of adisc bundle is a circle bundle and Euler number 1 implies that this must be the Hopf bundle) and complement

20

diffeomorphic to a standard 4-dimensional closed ball. From this it follows that we can cut out Σs and itstubular neighborhood from X and smoothly glue in that 4-ball instead. Since the homology of this 4-ballis trivial, this gives a new compact, connected, simply connected 4-manifold in which the homology in Xthat was generated by α has been killed. As a result, this new smooth 4-manifold has intersection form−E8 ⊕ (−1) and we have already seen that this contradicts Donaldson’s Theorem 1.3.

Thus, α cannot be represented by a smoothly embedded 2-sphere in X. On the other hand, a deep result ofFreedman on Casson handles (Theorem 1.1 of [Freed]) implies that α can be represented by a topologicallyembedded 2-sphere Σt in X and that Σt is has an open neighborhood U in X that is homeomorphic to a discbundle over S2 with Euler number 1. As above, U is then homeomorphic to an open subset of CP2. Usinga topological embedding of U in CP2 we can transfer the topological sphere Σt to CP2. We will use thesame symbols U and Σt for these subsets of CP2. Freedman showed that the complement CP2 − Σt of Σt

in CP2 is homeomorphic to an open 4-dimensional ball and therefore homeomorphic to R4. This open ballCP2 − Σt inherits a smooth structure from CP2 and we would like to argue that, with this smooth structure,CP2 − Σt cannot be diffeomorphic to R4 with its standard smooth structure, that is, that the submanifoldCP2 − Σt of CP4 is an exotic R4. The idea is to apply Lemma 1.9 to CP2 − Σt.

We will argue that the compact subsetCP2−U ofCP2−Σt cannot be enclosed by any smoothly embedded3-sphere in CP2 − Σt. Suppose to the contrary that CP2 − U is contained inside the smoothly embedded3-sphere S in CP2 − Σt. Then S is contained in U. Now return to X = CP2# 9CP

2. The 3-sphere S

also embeds smoothly in X. Thus, Σt is enclosed in X by a smoothly embedded 3-sphere. Now we canproceed exactly as in the case of Σs by cutting the 3-sphere out of X and smoothly gluing in a 4-dimensionalball, thereby killing the homology in X generated by α. The result is again a compact, connected, simplyconnected 4-manifold with intersection form −E8 ⊕ (−1) and this violates Donaldson’s Theorem 1.3. Thiscontradiction implies that CP2 − Σt is not diffeomorphic to R4 and is therefore an exotic R4.

1.4. Donaldson Invariants. In Section 1.2 we witnessed the emergence of the Yang-Mills equations andgauge-theoretic techniques in differential topology. Donaldson [Don1] proved his Theorem 1.3 on the in-tersection form of a compact, connected, simply connected, oriented, smooth 4-manifold X with b+

2 (X) = 0(or b−2 (X) = 0) by studying the moduli space of gauge equivalence classes of anti-self-dual connections onthe Sp(1)-bundle over X with Chern number 1. Here we will briefly describe the program carried out byDonaldson [Don3] to refine these techniques in order to produce a family of very sensitive invariants forsmooth 4-manifolds. The technical work required to do this is prodigious and, in a sense, the invariantsthemselves were superseded with the appearance of Seiberg-Witten theory (Section 1.6) so we will not be-labor the technical issues. A more detailed outline and an account of many of the details is to be found in[Mor2] with still more details in Chapter III of [FM]; for the full story, one should consult [DK].

1.4.1. Moduli Spaces of Connections. Throughout this section X will denote a compact, connected, simplyconnected, oriented, smooth 4-manifold. When the need arises we will also specify assumptions concerningb+

2 (X). We will write

Sp(1) → Pkπk→ X

for the principal Sp(1)-bundle over X with Chern number k.

21

Remark 1.7. The Lie group Sp(1) is isomorphic to SU(2) which, in turn, is the universal double cover of therotation group SO(3). The entire program we intend to describe can be carried out equally well with SO(3)-bundles over X. Locally these two approaches are entirely equivalent, but there can be important globaldifferences. The reason is that, whereas SU(2)-bundles over X are characterized by the second Chern classc2, SO(3)-bundles over X are characterized by the first Pontrjagin class p1 and the second Stiefel-Whitneyclass w2. The flexibility of being able to choose w2 can be quite useful (see Section II (vi) of [Don3],Sections 9.1.1, 9.1.2 and 9.1.3 of [DK] and also [FS1]). We will consider only the Sp(1) case here, but [DK]treats both in detail.

A(Pk) will denote the set of all smooth connection 1-forms on Pk and G(Pk) is the group of all difeomor-phisms f of Pk onto itself that respect the right action of Sp(1) on Pk ( f (p · g) = f (p) · g for all p ∈ Pk andall g ∈ Sp(1)) and cover the identity map on X (πk f = πk). G(Pk) is called the gauge group, its elementsare called (global) gauge transformations, and it acts on A(Pk) on the right by pullback, that is, for everyω ∈ A(Pk), ω · f = f ∗ω ∈ A(Pk). Two connections ω,ω′ ∈ A(Pk) are said to be gauge equivalent if thereexists an f ∈ G(Pk) for which ω′ = f ∗ω. The set of all gauge equivalence classes [ω] for ω ∈ A(Pk) is calledthe moduli space of connections on Sp(1) → Pk

πk→ X and is written

B(Pk) = A(Pk)/G(PK).

Remark 1.8. As we mentioned in Remark 1.5 such moduli spaces generally have no reasonable mathematicalstructure due to the assumed smoothness of the connections and gauge transformations. To obtain objectsthat one can study analytically it is necessary to introduce various Sobolev completions. As we did earlierwe will suppress these completions here and simply send those interested in seeing how it is done to thereferences provided in Remark 1.5. We should point out, however, that our next two Propositions are thekeys to defining Sobolev completions of A(Pk) and G(Pk).

For each integer i ≥ 0 we will denote by Ωi(Pk, sp(1)) the real vector space of all i-forms on Pk withvalues in the Lie algebra sp(1) of Sp(1). Ωi

ad(Pk, sp(1)) will denote the linear subspace of Ωi(Pk, sp(1))consisting of those elements ϕ of Ωi(Pk, sp(1)) that are tensorial of type ad, that is, that satisfy the followingtwo conditions (see Chapter II, Section 5, page 75, of [KN1]).

(1) ϕ vanishes if one of its arguments is vertical (that is, tangent to a fiber of Pk).(2) If g ∈ Sp(1) and σg : Pk → Pk is the diffeomorphism given by σg(p) = p · g, then σ∗gϕ = g−1ϕg.

Next let Ωi(X, ad Pk) denote the vector space of all i-forms on Pk with values in the adjoint bundle ad Pk (thevector bundle associated to Pk by the adjoint (conjugation) action of Sp(1) on its Lie algebra sp(1)). This iseasily seen to be isomorphic to Ωi

ad(X, sp(1)). If ω and ω0 are two connection 1-forms on Pk, then ω−ω0 istensorial of type ad so we have the following.

Proposition 1.10. A(Pk) is an affine space modeled on the vector space Ω1ad(X, sp(1)) Ω1(X, ad Pk). In

particular, if ω0 is any fixed element of A(Pk), then

A(Pk) =ω0 + ϕ : ϕ ∈ Ω1

ad(Pk, sp(1).

22

Similarly, one can consider the nonlinear adjoint bundle Ad Pk. This is the fiber bundle associated to Pk

by the adjoint (conjugation) action of Sp(1) on itself. Let Ω0(X,Ad Pk) be the set of smooth sections ofAd Pk. This is a group under pointwise multiplication in the fibers and is easily seen to be isomorphic to thegroup Ω0

Ad(Pk,Sp(1)) of smooth maps ψ : Pk → Sp(1) that satisfy ψ(p · g) = g−1ψ(p)g for all p ∈ Pk and allg ∈ Sp(1). One can then prove the following (see Lemma 4.1.2 of [Mor2]).

Proposition 1.11. G(Pk) Ω0Ad(Pk, Sp(1)) Ω0(X,Ad Pk)

If ω is in A(Pk), then the stabilizer Stab(ω) of ω is the subgroup of G(Pk) that leaves ω fixed.

Stab(ω) =

f ∈ G(Pk) : ω · f = ω

Any such stabilizer contains the subgroup Z2 of G(Pk) generated by the constant sections through ±1 inSp(1). If Stab(ω) = Z2, then ω is said to be irreducible; otherwise, ω is reducible. The reason for theterminology is that, when ω is reducible, Stab(ω)/Z2 U(1) and ω is induced from a connection on aU(1)-bundle (see Corollary 4.3.5 of [Mor2]). We will see that reducible and irreducible connections playvery different roles in the analysis of the moduli spaces. We will write A(Pk) for the open subset of A(Pk)consisting of the irreducible connections and will introduce, in addition to B(Pk) = A(Pk)/G(Pk) the modulispace of irreducible connections

B(Pk) = A(Pk)/G(Pk).

The objects of real interest in Donaldson theory are certain subspaces of B(Pk) and B(Pk). One describesthem by first choosing a Riemannian metric g on X. Together with the orientation of X this gives a Hodgestar operator ∗ on forms defined on X. We will say that a connection ω on Pk is g-anti-self dual (g-ASD) ifthe curvature form Fω ∈ Ω2(X, ad Pk) satisfies ∗Fω = −Fω. Note, however, that such connections can existonly if the Chern number is non-negative. Indeed,

k =1

8π2

∫X

tr (Fω ∧ Fω) =1

8π2

∫X

(|F−ω|

2 − |F+ω|

2 )vol, (4)

where F±ω = 12 (Fω ±

∗Fω) are the self-dual and anti-self-dual parts of Fω. Thus, F+ω = 0 if and only if k ≥ 0.

We will therefore restrict our attention henceforth to bundles Pk with

k ≥ 0.

For any such k we let AAS D(Pk, g) denote the subset of A(Pk) consisting of all g-ASD connections andAAS D(Pk, g) the subset of AAS D(Pk, g) consisting of the irreducible elements. The corresponding modulispaces of gauge equivalence classes are

M(Pk, g) = AAS D(Pk, g)/G(Pk)

and

M(Pk, g) = AAS D(Pk, g)/G(Pk).

For a given X, g, and k ≥ 0, AAS D(Pk, g) (and therefore M(Pk, g)) might well be empty. This is the case,for example, when k = 1 if X is either CP2 or S2 × S2 and g is the standard metric (Fubini-Study in the caseof CP2), but not if k = 2 (see Examples 4.1.3 and 4.1.5 of [DK]). In Section 1.1 we have described M(Pk, g)

23

when X = S4, k = 1, and g is the standard metric on S4. One can show that, in this case, every connection inAAS D(P1, g) is irreducible so M(P1, g) = M(P1, g).

For any X and any g the k = 0 bundle is trivial and it follows from (4) that any g-ASD connection isnecessarily flat. Conversely, a flat connection is certainly g-ASD. Since flat connections exist on any trivialbundle, the moduli space M(P0, g) is nonempty. However, since we have assumed X is simply connected,any two flat connections on P0 are gauge equivalent so M(P0, g) consists of a single point (see Chapter II,Section 9, of [KN1]). For this reason we will henceforth restrict our attention to

k > 0.

An example of particular interest is CP2

with its standard metric and k = 1 (see Example 4.1.2 of [DK]).In this case, one can write out explicit formulas for representatives of each gauge equivalence class of g-ASDconnections very much as we did for S4 in Section 1.1.2. The moduli space M(P1, g) turns out to be an opencone on CP2. The base of the closed cone is a copy of CP

2each point of which corresponds to a sequence

of irreducible g-ASD connections becoming ever more concentrated at a point. The vertex of the closedcone corresponds to a reducible g-ASD connection. Reducible connections always give rise to such a conesingularity in the moduli space. By adding the base and the vertex one obtains a natural compactification ofM(P1, g). One should compare this example with both the discussion of the 4-sphere in Section 1.1.3 andthe scenario described in Section 1.2.2 for Donaldson’s Theorem 1.3 (see Figure 2).

Sorting out the mathematical structure of the moduli spaces M(Pk, g) and M(Pk, g) when k > 0 requiresan enormous amount of very delicate technical work. The path one must follow is outlined in [Nab1] (pages44-49), while a more thorough discussion can be found in [Mor2] (pages 95-107) and the whole story is inChapters 4 and 5 of [DK]. We must be content with a simple statement of the results we need. For these werequire one more preliminary result.

We will denote by R(X) the space of all Riemannian metrics on X. This is a space of sections of a fiberbundle over X and can be given the structure of a pathwise connected Banach manifold (see Lemma 5.6.1of [Mor2]). With this structure one can prove the following (see Theorem 5.3.1 of [Mor2]).

Theorem 1.12. Let X be a compact, connected, simply connected, oriented, smooth 4-manifold with b+2 (X) >

0 and let k > 0 be an integer. Then there is a dense Gδ-set RG(X) in R(X), the elements of which are calledgeneric Riemannian metrics on X, such that for every generic g ∈ RG(X), M(Pk, g) = M(Pk, g) is a smoothmanifold of dimension

8k − 3(1 + b+2 (X)).

Very roughly, here is how one might go about showing that M(Pk, g) is a smooth manifold for a genericchoice of g. Define

M(Pk,R(X)) =([ω], g) ∈ B(Pk) × R(X) : ω is g-ASD

.

This is an infinite-dimensional smooth submanifold of B(Pk) × R(X). One shows that the projection map

M(Pk,R(X))→ R(X)

24

is smooth with Fredholm derivative at each point. Then the Banach space version of Sard’s Theorem (seeTheorem 3.6.12 of [AMR]) implies that the set of regular values of the projection map is a dense Gδ-set inR(X). But then, for any g in this set, the inverse image of g(

B(Pk) × g)∩ M(Pk,R(X)) = M(Pk, g)

under the projection map is a smooth submanifold of M(Pk,R(X)). Calculating the dimension of this mani-fold is much more subtle. One associates with each g-ASD ω a certain elliptic complex, computes its indexfrom the Atiyah-Singer Index Theorem, and shows that, under the assumptions we have made, this is justthe dimension of M(Pk, g) (see pages 44-47 of [Nab1] for a sketch and Section 5.2, pages 96-98, of [Mor2]for more details).

A few remarks are in order. If 8k−3(1+b+2 (X)) is negative, then the moduli space is generically empty. The

restriction on b+2 (X) arises because the subset of R(X) consisting of those g for which reducible g-ASD con-

nections on Pk exist is a countable union of smooth submanifolds of codimension b+2 (X). Thus, if b+

2 (X) = 0,reducibles are generically unavoidable and one expects cone singularities in the moduli space M(Pk, g). Aswe saw in Section 1.2.2, this is a good, not a bad thing since the singularities lead to Donaldson’s Theorem1.3.

Crudely put, the idea behind defining the Donaldson invariants is to regard the moduli space M(Pk, g)as a cycle in B(Pk) over which to integrate certain differential forms on B(Pk, g). Somewhat less crudely,the invariants are obtained by pairing the cohomology of B(Pk, g) with the fundamental class [M(Pk, g)] ofM(Pk, g). In order to carry out such a program a great many questions need to be answered: What are thesecohomology classes of B(Pk, g)? Does M(Pk, g) have a fundamental class? At very least, M(Pk, g) must beorientable and, if the result is to be a differential topological invariant, the integrals must be independent ofthe choice of generic metric g. The following result of Donaldson addresses the first of these issues.

Theorem 1.13. Let X be a compact, connected, simply connected, oriented, smooth 4-manifold with b+2 (X) >

0. Let g ∈ RG(X) be a generic Riemannian metric on X and k > 0 an integer. Then the smooth (8k − 3(1 +

b+2 (X))-dimensional manifold M(Pk, g) is orientable. An orientation for M(Pk, g) is determined by the given

orientation of X and a choice of orientation for the vector space H2+(X;R) of real-valued, self-dual 2-forms

on X.

The proof amounts to constructing an explicit model of the determinant line bundle (top exterior power)of the tangent bundle of M(Pk, g) from a family of differential operators on X and exhibiting a nonzerosection (see Theorem 5.5.1 of [Mor2]).

Regarding the metric independence one proceeds in the following way. Let g0 and g1 be two elementsof RG(X). Since R(X) is a pathwise connected Banach manifold we can join g0 and g1 with a smooth pathgt : 0 ≤ t ≤ 1

in R(X). Donaldson has shown that, if b+

2 (X) > 1, then an arbitrarily small perturbation ofthe path

gt : 0 ≤ t ≤ 1

will encounter no reducible connections, that is, for a generic path from g0 to g1,

M(Pk, gt) = M(Pk, gt) for each t ∈ [0, 1]. We can then define the corresponding parametrized moduli spaceM(Pk, gt) to be the subspace of B(Pk) × [0, 1] given by

M(Pk, gt) =

( [ω], t ) ∈ B(Pk) × [0, 1] : [ω] ∈ M(Pk, gt)

25

The proof of the following is another application of the infinite-dimensional version of Sard’s Theorem (seeTheorems 5.6.2 and 5.6.3 of [Mor2]).

Theorem 1.14. Let X be a compact, connected, simply connected, oriented smooth 4-manifold with b+2 (X) >

1 and let k > 0 be an integer. Let g0, g1 ∈ RG(X) be generic Riemannian metrics on X. Then, for a genericpath

gt : 0 ≤ t ≤ 1

joining g0 and g1 in R(X), the parametrized moduli space M(Pk, gt) is a smooth,

orientable manifold with boundary. A choice of orientation O for H2+(X;R) determines an orientation for

M(Pk, gt). The oriented boundary of M(Pk, gt) is the disjoint union of M(Pk, g1) with the orientationinduced by O and M(Pk, g0) with the orientation opposite to the one induced by O.

A rough, intuitive interpretation of this result is that, if b+2 (X) > 1, then a generic variation of g varies

M(Pk, g) within a single homology class of B(Pk) so pairing either M(Pk, g0) or M(Pk, g1) with a coho-mology class of B(Pk) will give the same result. Since g0, g1 ∈ RG(X) were arbitrary it will follow that theDonaldon invariants (once they are defined in the manner suggested above) are independent of the choice ofgeneric Riemannian metric on X. For this reason we will assume henceforth that

b+2 (X) > 1.

1.4.2. The Uhlenbeck Compactification. There is much that remains to be done in order to carry out theprogram we have described for constructing the Donaldson invariants. Since the idea is to pair certaincohomology classes of B(Pk) with a homology class determined by the M(Pk, g) for generic g one mustisolate

(1) appropriate cohomology classes of B(Pk), and(2) an appropriate homology class of B(Pk) determined by M(Pk, g).

An initial step toward the first of these issues is, in Donaldson’s view, “an exercise in algebraic topology”(see page 9 of [Don4]) in which he constructs maps

µ : H2(X;Z)→ H2(B(Pk);Z) (5)

and

µ : H0(X;Z)→ H4(B(Pk);Z) (6)

and, by restriction, maps

µ : H2(X;Z)→ H2(M(Pk, g);Z) (7)

and

µ : H0(X;Z)→ H4(M(Pk, g);Z) (8)

for which it is customary to use the same symbol. Any of these is likely to be referred to as the Donaldsonµ-map (see pages 59-61 of [Nab1] for an informal sketch of the construction, Section 7.2 of [Mor2] orSections 3.1.5 - 3.1.10 of [FM] for more details, and Sections 5.1 and 5.2 of [DK] for the whole story). Forany α ∈ H2(X;Z) we then have a 2-dimensional cohomology class µ(α) ∈ H2(M(Pk, g);Z) and, roughly

26

speaking, the idea is to form a sufficiently large product µ(α) ^ · · ·^ µ(α) that can be paired with M(Pk, g).In order for this to succeed, of course, the dimension 8k − 3(1 + b+

2 (X)) of the moduli space must be evensince every µ(α) has degree two. This is the case if and only if b+

2 (X) is odd so, from this point on, we willassume

b+2 (X) is odd and b+

2 (X) ≥ 3.

Under these assumptions it will be convenient to write

2dk = 8k − 3(1 + b+2 (X)) (9)

so that the cohomology classes we would like to pair with M(Pk, g) are µ(α) ^ dk· · ·^ µ(α) for α ∈ H2(X;Z).

For reasons that we will discuss next the µ-map is not quite up to the task we have in mind for it and we willneed to make some adjustments in the program we have been describing.

Dealing with the second issue is not by any means an “exercise”. If we think, intuitively, of a secondhomology class as a surface Σ and a second cohomology class as a 2-form, then we would ideally like to“integrate µ(Σ)∧ dk

· · · ∧ µ(Σ) over M(Pk, g)”. We have seen that we can assume M(Pk, g) is a smooth, finite-dimensional manifold. If M(Pk, g) were compact we could integrate over it. More precisely, if M(Pk, g)were compact it would carry a fundamental class [M(Pk, g)] which we could pair with any µ(α) ^ dk

· · ·^

µ(α) for α ∈ H2(X;Z). . However, as we have seen, the manifolds M(Pk, g) are generally not compactso such a fundamental class need not exist. To rectify the situation Donaldson employed deep analyticalresults of Uhlenbeck to introduce what has come to be known as the Uhlenbeck compactification M(Pk, g)of M(Pk, g). We will have a very brief look at how this is done.

Since we have assumed that b+2 (X) ≥ 3 and g is generic, M(Pk, g) = M(Pk, g). To construct a com-

pactification one must understand how a sequence in M(Pk, g) can fail to have a convergent subsequencein M(Pk, g) so we consider a sequence

ω j

of irreducible g-ASD connections on Pk that gives rise to a

sequence[ω j]

in M(Pk, g) without a convergent subsequence.

Remark 1.9. If our hypotheses had not excluded the presence of reducible g-ASD connections, then thesequence

ω j

of irreducible g-ASD connections could converge to a reducible g-ASD connection ω. Then

ω would correspond to a point [ω] in M(Pk, g) at which there is a cone singularity of the sort we saw inDonaldson’s Theorem 1.3 (see Figure 2). In our present context this cannot occur.

The Compactness Theorem of Karen Uhlenbeck (see Theorem 6.1.1 of [Mor2]) implies that, after passingto a subsequence, the sequence

Fω j

of curvatures must have pointwise norms ‖Fω j(x) ‖ which, like the

BPST instantons of Section 1.1.2, become increasingly concentrated at a point (or finite set of points) in X.Intuitively, these connections “converge to a connection with curvature supported on a finite set of isolatedpoints in X” which, because of its singular nature, does not correspond to a point in the moduli spaceM(Pk, g). The compactification adds these points and this essentially amounts to adding a copy of X. Forexample, it will add the S4 boundary of the moduli space of BPST instantons in Section 1.1.3 and the copyof X at the bottom of Figure 2.

To describe the Uhlenbeck compactification M(Pk, g) of M(Pk, g) more precisely we recall that, for anytopological space X and any integer t ≥ 1, the tth symmetric product of X is the topological quotient st(X) of

27

X × t· · · × X by the action of the symmetric group on t letters that permutes the coordinates. It is therefore

the space of unordered and possibly non-distinct t-tuples of points in X. Intuitively, one should think ofthese as the points at which the connections can concentrate. In particular, s1(X) is just X. Now consider thedisjoint union

M(Pk, g) t M(Pk−1, g) × s1(X) t M(Pk−2, g) × s2(X) t · · · t M(P0, g) × sk(X). (10)

The elements of M(Pk−t, g) × st(X) for any t = 1, 2, . . . , k are called ideal g-ASD connections on X. For anypoint of the form

([ω], x1, . . . , xt ) in M(Pk−t, g)× st(X), ω is called a background connection. Uhlenbeck’s

Compactness Theorem (Theorem 6.1.1 of [Mor2]) motivates the appropriate way of defining sequentialconvergence in the disjoint union (10). For any point

([ω], x1, . . . , xt ) in M(Pk−t, g) × st(X), for some

t = 1, 2, . . . , k, we introduce an associated measure on X called the curvature density and defined by

‖Fω(x) ‖2 d volg +

t∑l=1

8π2δxl ,

where volg is the volume form on X determined by the metric g and δxl is the point measure at xl of mass 1.We will say that a sequence

([ωn], xn

1, . . . , xnt(n)

), n = 1, 2, . . ., of points in the disjoint union (10) converges

to(

[ω], x1, . . . , xt)∈M(Pk−t, g) × st(X) if the following two conditions are satisfied.

(1) The curvature densities for the(

[ωn], xn1, . . . , x

nt(n)

)converge weakly to the curvature density for(

[ω], x1, . . . , xt)

as n→ ∞, that is, for any continuous real-valued function ϕ on X,∫X‖Fωn(x) ‖2ϕ(x) d volg(x) +

t(n)∑l=1

8π2ϕ(xnl )→

∫X‖Fω(x) ‖2ϕ(x) d volg(x) +

t∑l=1

8π2ϕ(xl).

(2) For every compact set K ⊆ X − x1, . . . , xt the restrictions of the connections ωn to K converge, upto gauge equivalence, to the restriction of ω to K.

Entirely analogous definitions cover the cases in which some (or all) of the terms of the sequence are inM(Pk, g) (drop the corresponding sums to the left of the arrow) or the limit is in M(Pk, g) (drop the sum tothe right of the arrow). These notions of convergence supply the disjoint union (10) with a topology and onecan show that this topology is Hausdorff, second countable and metrizable. Moreover, M(Pk, g) and eachof the “strata” M(Pk−t, g) × st(X) inherits its usual topology as a subspace. The Uhlenbeck compactificationM(Pk, g) of M(Pk, g) is defined to be the closure of M(Pk, g) in this topology on the disjoint union (10). Adetailed proof of Theorem 1.15 is available in Chapter 4, Section 4, pages 158-170, of [DK].

Theorem 1.15. Let X be a compact, connected, simply connected, oriented, smooth 4-manifold with b+2 (X) >

1, g a generic Riemannian metric on X, and Sp(1) → Pkπk→ X the principal Sp(1)-bundle over X with Chern

number k. Then M(Pk, g) is compact and contains M(Pk, g) as a dense, open subset.

Remark 1.10. We should point out that our references [Mor2] and [FM] take the Uhlenbeck compactificationto be the entire disjoint union (10) with the topology we have described. This is also compact, but M(Pk, g)is not dense unless k is sufficiently large so it may not really qualify as a “compactification” of M(Pk, g) inthe usual sense.

28

The Uhlenbeck compactification M(Pk, g) is therefore a separable, compact, metrizable topological space.It generally does not admit a manifold structure, however, so we are not a priori guaranteed the existenceof a fundamental class. However, being separable and metrizable, M(Pk, g) has a topological dimensionthat can be described in a number of equivalent ways (see Chapter 9, Section 2, pages 356-357, of [DK]where it is defined as the covering dimension of the topological space M(Pk, g)). Similarly, each “stratum”M(Pk, g) × st(X), t ≥ 1, has a topological dimension and one can deduce from general results in algebraictopology that if each M(Pk, g)× st(X), t ≥ 1, has codimension at least two in M(Pk, g), then M(Pk, g) carriesa fundamental class

[M(Pk, g)] ∈ H2dk (M(Pk, g);Z).

Writing out all of the relevant dimensions one finds that this codimension condition is satisfied only if theChern number k is in the so-called stable range of X, that is,

k ≥14

(5 + 3b+

2 (X)). (11)

This is equivalent to

dk ≥12

(7 + 3b+

2 (X))

(12)

so one can equally well say that the codimension condition is satisfied if dk is in the stable range for X givenby (12).

Even for k in the stable range there is one more obstacle to overcome before one can define the Donaldsoninvariants. The µ-map µ : H2(X;Z) → H2(M(Pk, g);Z) carries each α ∈ H2(X;Z) to a cohomology classµ(α) of M(Pk, g), not M(Pk, g), so one cannot pair µ(α) ^ dk

· · ·^ µ(α) with [M(Pk, g)]. There are a numberof ways to deal with this (see Section 9.2 of [DK]), but we will simply note that one can construct a map

µ : H2(X;Z)→ H2(M(Pk, g);Z)

which, when followed by the restriction map,

H2(M(Pk, g);Z)→ H2(M(Pk, g);Z)

is equal to µ. The construction of this extension of the µ-map proceeds one stratum M(Pk−t, g) × st(X) ata time and requires a detailed understanding of the Taubes [Taub1] gluing procedure (see Section 7.4, page127, of [Mor2] for a sketch).

1.4.3. The Invariants. With this one can finally begin introducing the Donaldson invariants. Lest theybe lost in the turmoil of the preceding pages we will begin by laying down all of our hypotheses. LetX denote a compact, connected, simply connected, oriented, smooth 4-manifold with b+

2 (X) odd and atleast 3. Let g ∈ RG(X) be a generic Riemannian metric for X and fix an orientation of H2

+(X;R). Letk ≥ 1

4(

5 + 3b+2 (X)

)be an integer in the stable range of X and write the dimension of the smooth moduli

space M(Pk, g) = M(Pk, g) as 2dk = 8k − 3(1 + b+2 (X)). With all of these assumptions in place we will

define a Donaldson invariant γdk (X) of X. This can be thought of either as a dk-multilinear, Z-valued map

29

on H2(X;Z) or, equivalently, as a Z-valued, homogeneous polynomial of degree dk on H2(X;Z) and it iscustomary to use the same symbol γdk (X) for both. As a multilinear map

γdk (X) : H2(X;Z)× dk· · · ×H2(X;Z)→ Z

it is given by

γdk (X)(α1, . . . , αdk ) =⟨µ(α1) ^ · · ·^ µ(αdk ), [M(Pk, g)]

⟩.

The corresponding homogeneous polynomial

γdk (X) : H2(X;Z)→ Z

is defined by

γdk (X)(α) =⟨µ(α) ^ dk

· · ·^ µ(α), [M(Pk, g)]⟩.

This is not quite the end, however. It will soon be important to have the definition of γdk (X) extended tooperate on H0(X;Z) as well as H2(X;Z). One begins by returning to our, as yet unused, µ-map (8).

µ : H0(X;Z)→ H4(M(Pk, g);Z)

Unfortunately, unlike µ : H2(X;Z) → H2(M(Pk, g);Z), this map does not admit an extension to the Uh-lenbeck compactification. More precisely, if we denote by x the generator of H0(X;Z) Z, the classµ(x) ∈ H4(M(Pk, g);Z) does not extend to a cohomology class on M(Pk, g). However, µ(x) does extend toa class µ(x) on the complement of the set of points in M(Pk; g) with trivial background connection and thisis enough, under certain restrictions, to extend our definition of the Donaldson invariants to include the 4-dimensional classes. Basically, the restriction is that there must be “enough” 2-dimensional classes as well.Specifically, if a and b are non-negative integers with dk = 4k− 3

2 (1+b+2 (X)) = a+2b and a ≥ 1

4 (5+3b+2 (X)),

then, for α1, . . . , αa ∈ H2(X;Z), the class

µ(α1) ^ · · ·^ µ(αa) ^ µ(x)b = µ(α1) ^ · · ·^ µ(αa) ^ µ(x) ^ b· · ·^ µ(x)

can be paired with the fundamental class [M(Pk, g)] and we can define

γdk (X) : H2(X;Z)× a· · · ×H2(X;Z) × H0(X;Z)× b

· · · ×H0(X;Z)→ Z

by

γdk (X) (α1, . . . , αa, n1x, . . . , nbx) = n1 · · · nb⟨µ(α1) ^ · · ·^ µ(αa) ^ µ(x)b, [M(Pk, g)]

⟩.

For any non-negative integers a and b with dk = 4k − 32 (1 + b+

2 (X)) = a + 2b and a ≥ 14 (5 + 3b+

2 (X)) theelements of

H2(X;Z)× a· · · ×H2(X;Z) × H0(X;Z)× b

· · · ×H0(X;Z)

are called k-stable for X and γdk (X) is defined on any k-stable element.At this point we have defined a Donaldson invariant γd(X) for any d ≡ −3

2 (1+b+2 (X)) mod 4 and d = a+2b

with a and b non-negative integers and a ≥ 14 (5 + 3b+

2 (X)). These are called the stable range Donaldsoninvariants. Before proceeding with the definitions we will take a moment to list a few properties of theseγd(X) that remain true for all of those we still have to define (see Chapter III of [FM] and Chapter 9 of[DK]).

30

(1) γd(X) does not depend on the choice of the generic metric g.(2) The γd(X) are invariant under orientation preserving diffeomorphisms of X that also preserve the

chosen orientation for H2+(X;R).

(3) Reversing the orientation of H2+(X;R) reverses the sign of γd(X).

(4) Reversing the orientation of X can have a much more dramatic effect on the Donaldson invariants.In many examples, the invariants all vanish for one of the two possible orientations for X.

Now we will briefly address the problem of extending the definition of γd(X) to include still more valuesof d. Doing so depends on what are called blowup formulas. By definition, the blowup of a 4-manifold X isthe 4-manifold X#CP

2obtained by forming the connected sum of X and CP

2. For any non-negative integer

n, the n-fold blowup of X is

X#CP2# n· · · #CP

2= X# nCP

2.

Note that, since b+2 (CP

2) = 0, b+

2 (X) = b+2 (X# nCP

2) so the stable ranges of X and X# nCP

2are the

same. Moreover, since QCP

2 is negative definite, there are no self-dual harmonic 2-forms on CP2, so

there is a natural identification of H2+(X;R) and H2

+(X# nCP2;R). In particular, orienting H2

+(X;R) orients

H2+(X# nCP

2;R) as well. The simplest of the blowup formulas (n = 1) can be stated in the following way. If

dk is in the stable range of X, then dk+1 = dk +4 is in the stable range of X#CP2. The blowup formula asserts

that, if e ∈ H2(X#CP2;Z) is a generator for H2(CP

2;Z) ⊆ H2(X#CP

2;Z) H2(X;Z) ⊕ H2(CP

2;Z), then

γdk+1(X#CP2)(α1, . . . , αdk , e, e, e, e) = −2γdk (X)(α1, . . . , αdk ) (13)

(see Chapter III, Theorem 8.1, of [FM]). The point is that it is entirely possible for dk+1 to be in the stablerange of X#CP

2even if dk is not in the stable range of X and in this case we can define γdk (X) by (13), that

is,

γdk (X)(α1, . . . , αdk ) = −12γdk+1(X#CP

2)(α1, . . . , αdk , e, e, e, e). (14)

Notice, however, that with this definition γdk (X) is no longer Z-valued.The general result is as follows (see Chapter III, Theorem 8.3, of [FM]). Let

s = (α1, . . . , αa, n1x, . . . , nbx)

be k-stable for X. Then γdk (X)(s) is defined. Consider, for some n ≥ 1, the n-fold blowup X# nCP2

of X.

For each i = 1, . . . , n, let ei ∈ H2(X# nCP2;Z) be the generator of the ith CP

2-summand in X# nCP

2. Since

dk+n = dk + 4n = (a + 4n) + 2b,

(s, e1, e1, e1, e1, . . . , en, en, en, en)

is (k + n)-stable for X# nCP2

so

γdk+n(X# nCP2)(s, e1, e1, e1, e1, . . . , en, en, en, en)

is defined. The blowup formula of Donaldson asserts that

γdk (X)(s) =

(−

12

)nγdk+n(X# nCP

2)(s, e1, e1, e1, e1, . . . , en, en, en, en). (15)

31

Once again, the point is that, even if s is not k-stable for X because a < 14 (5+3b+

2 (X)) one can certainly choose

n sufficiently large that a+4n ≥ 14 (5+3b+

2 (X)) = 14 (5+3b+

2 (X# nCP2)) and so (s, e1, e1, e1, e1, . . . , en, en, en, en)

is (k + n)-stable for X# nCP2. Checking that the value of γdk+n(X# nCP

2)(s, e1, e1, e1, e1, . . . , en, en, en, en)

does not depend on the choice of such a sufficiently large n one can then define γdk (X)(s) by the right-handside of (15). Notice that, with this definition, the Donaldson invariants γdk (X) take their values in Z[ 1

2 ].We have now defined γd(X) for any d satisfying d ≡ −3

2 (1 + b+2 (X)) mod 4. There is one final trick for

extending the range of d-values for which we can define γd(X). Suppose d satisfies

d ≡ −32

(1 + b+2 (X)) mod 2, (16)

but

d . −32

(1 + b+2 (X)) mod 4. (17)

Then

d = 2k −32

(1 + b+2 (X))

for some odd integer k. Then k + 1 is even and we can write

d + 2 = 4(k + 1

2

)−

32

(1 + b+2 (X)) = d(k+1)/2 ≡ −

32

(1 + b+2 (X)) mod 4.

Consequently, γd+2 is defined. Since d(k+1)/2 = a+2b, where a = d and b = 1, it is defined on (α1, . . . , αd, x),where α1, . . . , αd ∈ H2(X;Z) and x is the generator of H0(X;Z). When d satisfies (16) and (17) we define

γd(X)(α1, . . . , αd) =12γd+2(X)(α1, . . . , αd, x).

Remark 1.11. We would like to make an observation that will be of use to us in the next section. Thedimension of the moduli space M(Pk, g) is given by dimM(Pk, g) = 8k − 3(1 + b+

2 (X)). By choosing b+2 (X)

and k appropriately one can arrange that dimM(Pk, g) = 0 so that M(Pk, g) is a set of isolated points. Thiscannot occur for k in the stable range since k ≥ 1

4 (5 + 3b+2 (X)) implies dimM(Pk, g) ≥ 7 + 3b+

2 (X) > 0.On the other hand, if we take X to be a manifold with b+

2 (X) = 7 and then consider the SU(2)-bundle overX with Chern number k = 3 we do, indeed, have dimM(P3, g) = 0. Such 4-manifolds are easy enough tofind since b+

2 (pCP2# qCP2) = p − q so, in particular, b+

2 (7CP2) = 7. In this case it turns out that all ofthe Donaldson invariants of 7CP2 are identically zero (see Proposition 9.3.1 of [DK]). More interestingexamples of 0-dimensional moduli spaces (for SO(3)-bundles) can be found in Section 9.1 of [DK].

This completes our list of the Donaldson invariants. There are infinitely many such invariants and, asone might expect, computing any one of them for a given X is generally a Herculean task (see [O’G], forexample). As it happens, however, many of the most significant applications of Donaldson theory arise fromgeneral results on the vanishing or non-vanishing of the invariants. These results are themselves deep andtechnically very demanding, but have the advantage of being applicable to a wide range of problems. Wewill illustrate this with just one example. The first result we need is called the Connected Sum Theorem(see Theorem 8.1.1 of [Mor2] and Theorem 9.3.4 of [DK] for sketches and Theorem 4.9 of [Don3] for acomplete proof).

32

Theorem 1.16. (Connected Sum Theorem) Let X be a compact, connected, simply connected, oriented,smooth 4-manifold with b+

2 (X) odd and suppose that X is orientation preserving diffeomorphic to X1#X2

where b+2 (X1) > 0 and b+

2 (X2) > 0. Then each γd(X) is identically zero.

Next we recall that a complex surface is a complex manifold of complex dimension two and therefore realdimension four. These all have natural orientations as complex manifolds. A complex algebraic surface isa complex surface that is defined by homogeneous polynomial equations in some complex projective spaceCPn. The K3 surface is an example, but there are a great many others (see [Beau]). The following is aNon-Vanishing Theorem for complex algebraic surfaces (see Theorem 8.1.2 of [Mor2] for a sketch, Chapter10 of [DK] for a more detailed discussion and [Don3] for a complete proof).

Theorem 1.17. (Non-Vanishing Theorem) Let X be a compact, connected, simply connected, complex alge-braic surface with b+

2 (X) odd and greater than 1. Then, for all sufficiently large d, γd(X) is not identicallyzero.

As an application we will consider the K3 surface. We know that b+2 (K3) = 3 and that the intersection

form of K3 is 2(−E8) ⊕ 30 11 0

. Note that0 11 0

is the intersection form of S2 × S2 and b+2 (S2 × S2) = 1.

Consequently, the intersection form of 3(S2 ×S2) is 30 11 0

and b+2 (S2 ×S2) = 3. According to Freedman’s

Theorem 1.5, K3 is homeomorphic to 2X−E8 # 3(S2 × S2). Topologically, K3 decomposes into a connectedsum, one summand of which is 3(S2 × S2). We will show that K3 cannot be written as a smooth connectedsum, one summand of which is 3(S2 ×S2), unless the other summand is has trivial second homology (which2X−E8 clearly does not). In fact, we will deduce from Theorems 1.3, 1.16 and 1.17 that this, and muchmore, is true not only for K3, but for any compact, connected, simply connected, complex algebraic surfacewith even intersection form and b+

2 (X) odd and greater than 1. Results such as these were quite inaccessiblebefore the advent of Donaldson theory.

Corollary 1.18. Let X be a compact, connected, simply connected, complex algebraic surface with evenintersection form and b+

2 (X) odd and greater than 1. Suppose that X is orientation preserving diffeomorphicto X1#X2, where b+

2 (X1) > 0 and b+2 (X2) > 0. Then either X1 or X2 has trivial second homology.

Proof. Since X is compact, connected, and simply connected, so are X1 and X2. By Theorem 1.17, theDonaldson invariants γd(X) are not identically zero for sufficiently large d. But then, by Theorem 1.16,either b+

2 (X1) = 0 or b+2 (X2) = 0. Without loss of generality we can assume that b+

2 (X1) = 0. Thus, eitherb2(X) = 0 or X1 has negative definite intersection form. Suppose b2(X) , 0. Then, by Donaldson’s Theorem1.3, the intersection form of X1 is equivalent to the diagonal form with all diagonal entries −1. But, since Xhas even intersection form, so does X1 and there can be no α ∈ H2(X1;Z) with QX1(α, α) = −1 ≡ 1 mod 2and this is a contradiction. Thus, b2(X) = 0 and H2(X;Z) is trivial.

33

Between 1983 and 1994 the differential topology of 4-manifolds was dominated by the ideas of Donald-son and, after 1990, this amounted to the calculation of Donaldson invariants. Much progress was made,but the computations were enormously complicated, there were infinitely many invariants to be computed,and there were no known relations among the invariants to lighten the load. This all changed in 1994, but intwo quite different ways. The first event was a remarkable breakthrough in the computation of Donaldsoninvariants by Kronheimer and Mrowka [KM1]. Their procedure was to consolidate all of the γd(X) intoa single formal power series, called the Donaldson series DX(α), on the second homology. This is to bethought of as the generating function for the Donaldson invariants. Next they introduced a condition calledKM-simple type, satisfied by all known examples of the type of 4-manifold we are considering here, whichguaranteed that the invariants themselves could be retrieved from a knowledge of the Donaldson series. Allof this would be rather uninteresting, of course, if it were not possible to obtain information about DX(α)independently of actually calculating the Donaldson invariants. The Structure Theorem of Kronheimer andMrowka (proved in [KM1]) asserts that, in fact, the entire Donaldson series is completely determined by afinite set of data. The following statement of the Structure Theorem is taken from [FS2] which contains asimplification of the proof in [KM1].

Theorem 1.19. (Structure Theorem) Let X be a compact, connected, simply connected, oriented, smooth4-manifold with b+

2 (X) > 1 and odd. Suppose that X is of simple type and some Donaldson invariant ofX is not identically zero. Then there exist finitely many cohomology classes K1, . . . ,Ks ∈ H2(X;Z), calledKM-basic classes, and nonzero rational numbers a1, . . . , as, called KM-coefficients, such that

DX(α) = eQX(α,α)/2s∑

i=1

aieKi(α).

With the appearance of this result in early 1994 Donaldson theory seemed to have turned a corner. Longand complex calculations of an apparently infinite set of independent invariants were suddenly replacedby the (certainly nontrivial, but at least finite) problem of determining the basic classes and coefficients.Initially, this resulted in very considerable progress (see [Stern] for the state of the art as of July, 1994). Asfate would have it, however, the fall of 1994 witnessed another event which effectively rendered this triumphof Kronheimer and Mrowka moot. Edward Witten, at the end of a lecture at M.I.T. on N = 2 supersymmetricYang-Mills theory, made a conjecture which, within weeks, brought about the demise of Donaldson theoryand initiated an entirely new approach to the study of smooth 4-manifolds. The story of this event and theensuing frenzy is told by one who was there in [Taub3]. In Section 1.5 we will take just one, ever so modest,step toward understanding how one might go about finding the Donaldson invariants in quantum field theoryand then, in Section 1.6, we will have a look at the revolution to which this led.

1.5. The Mathai-Quillen Formalism and Witten’s TQFT Partition Function. In 1988, Edward Witten[Witt2], prompted by Atiyah, constructed a quantum field theory in which the Donaldson invariants appearedas expectation values of certain observables and thus ushered in the study of what has come to be known astopological quantum field theory (TQFT). This construction of Witten’s was a remarkable achievement andprovided the most direct sort of link between topology and physics. Topologists were (and still are) eager

34

to understand and exploit the insights that gave rise to such a radically new view of smooth 4-manifoldinvariants. These insights, however, sprang from the deepest regions of theoretical physics and were largelyinaccessible to mathematicians. In this section we will not presume to make Witten’s ideas palatable to thosetrained in modern mathematics, but will only provide a brief hint as to how the simplest of the Donaldsoninvariants might fit into the path integral framework of quantum field theory. That this is possible at all is, inlarge measure, the result of a beautiful paper of Atiyah and Jeffrey [AJ] that we will discuss in due course.

1.5.1. The Euler Number. We begin with a few general remarks. We consider an oriented, real vectorbundle ξ = (πE : E → X) of even rank (fiber dimension) n = 2k over a compact, oriented, smooth manifoldX of dimension n. Then E is an oriented manifold of dimension 2n. Examples include the tangent andcotangent bundles of an oriented, even dimensional manifold. We will denote by V the typical fiber of ξ andwill assume that V comes equipped with a positive definite inner product coming from a fiber metric on ξ.The exterior algebra of V will be denoted

∧V = ⊕p ∧p V =

(⊕p ∧

2pV)⊕

(⊕p ∧

2p+1V)

= V0 ⊕ V1

where Z2 = 0, 1. Select some oriented, orthonormal basis ψ1, . . . , ψn for V . These basis elements arethen odd generators for ∧V and we think of the elements of ∧V as polynomials with real coefficients in theanti-commuting variables ψ1, . . . , ψn . The corresponding volume form for V is defined by vol = ψ1 · · ·ψn

(we suppress the customary wedge ∧ and write the product in ∧V by juxtaposition). Notice that one candefine the exponential map on ∧V by the usual power series, noting that the series necessarily terminates forany element of ∧V due to the anti-commutativity of exterior multiplication.

Remark 1.12. Soon it will be useful to adopt some terminology that arose in supersymmetric physics sowe will pause to formulate a few definitions. A superalgebra is a real or complex vector space A with aZ2-grading A = A0 ⊕ A1 and an associative, bilinear multiplication A × A → A that satisfies AiAj ⊆ Ai+j

and which contains an element 1 ∈ A that satisfies 1a = a1 = a for all a ∈ A. The elements of A0 are saidto be even and have degree 0 ∈ Z2, while the elements of A1 are odd and have degree 1 ∈ Z2. A is saidto be supercommutative if, for all a1, a2 ∈ A, a1a2 = (−1)d1d2a2a1, where d1 = deg a1 and d2 = deg a2

(moving an odd thing past an odd thing costs a minus sign). In particular, 1 ∈ A0. Thus, ∧V , with exteriormultiplication, is a supercommutative superalgebra. Other examples include the algebra Ω∗(V) = ∧V∗ offorms on V or, more generally, the algebra of forms Ω∗(X) on a smooth manifold X. Another class ofexamples that will play an important role soon is constructed in the following way. Let G be some compact,connected, matrix Lie group with Lie algebra g (for example, SO(V) and so(V)). The vector space dual ofg will be written g∗ and Sym(g∗) will denote the symmetric algebra of g∗. We will identify Sym(g∗) withthe algebra of polynomial functions with real coefficients on g. More precisely, we let ζ1, . . . , ζn denote abasis for g and x1, . . . , xn the corresponding dual basis for g∗. Each x j is then a real-valued linear functionon g and Sym(g∗) is isomorphic to the polynomial algebra R[x1, . . . , xn] generated by x1, . . . , xn. Then

C[g] = Sym(g∗) ⊗ C

is identified with the algebra of polynomial functions on gwith complex coefficients. Each element P ofC[g]has an algebraic degree deg P as a polynomial, but, since C[g] is commutative , it will be more convenient

35

to regard C[g] as a supercommutative superalgebra in which all of the elements are even by doubling thedegrees. Thus,

C[g] = C[g]0 ⊕ C[g]1,

where C[g]0 = C[g] and C[g]1 is the trivial subspace of C[g].If A and B are two supercommutative superalgebras, then their supertensor product, denoted A⊗B has as

its underlying vector space the ordinary tensor product A ⊗ B with the Z2-grading

(A ⊗ B)0 = A0 ⊗ B0 ⊕ A1 ⊗ B1

(A ⊗ B)1 = A0 ⊗ B1 ⊕ A1 ⊗ B0

and with multiplication defined, not in the usual way ((a1⊗b1)(a2⊗b2) = (a1b1)⊗ (a2b2)) for homogeneouselements), but rather by

(a1 ⊗ b1)(a2 ⊗ b2) = (−1)deg b1 deg a2(a1b1) ⊗ (a2b2).

With this structure the maps a 7→ a⊗1 and b 7→ 1⊗b are algebra monomorphisms preserving theZ2-gradingand such that

(a ⊗ 1)(1 ⊗ b) = a ⊗ b, and (1 ⊗ b)(a ⊗ 1) = (−1)deg b deg aa ⊗ b.

Using these embeddings to identify A and B with subalgebras of A⊗B and writing ab rather than a ⊗ b thisgives

ba = (−1)deg b deg aab.

Moreover, if A and B are supercommutative, then so is A⊗B. An example with which we will deal shortly is

C[g] ⊗Ω∗(X).

Notice that, since everything in C[g] is even, this coincides with the usual tensor product C[g] ⊗ Ω∗(X)except that it is Z2-graded by (

C[g] ⊗Ω∗(X))0 = C[g] ⊗ Ω(X)0(

C[g] ⊗Ω∗(X))1 = C[g] ⊗ Ω(X)1

The elements of C[g] ⊗Ω∗(X) are sums of homogeneous terms of the form P ⊗ ϕ, where P ∈ C[g] andϕ ∈ Ω∗(X) and should be thought of as Ω∗(X)-valued polynomial functions on g. For example, (P ⊗ ϕ)(ζ) =

P(ζ)ϕ for every ζ ∈ g.

Now we return to the general development. The vector bundle ξ has associated with it an Euler number

χ(ξ)

that can be defined in a variety of ways, of which we will begin with two.

36

(1) The Euler number χ(ξ) of ξ is the intersection number of any generic section of ξ with the zero-section of ξ. We should say what all of this means (see Theorems 21.9 and 21.11 of [MT]). Lets0 : X → E be the zero section of ξ (s0(x) = 0x, where 0x is the zero element of the fiber π−1

E (x)). Lets : X → E be another section of ξ. Both s0 and s are diffeomorphisms of X onto closed submanifoldss0(X) and s(X) of E. Now suppose that these submanifolds intersect, that is, s(x) = s0(x) = 0x forsome x ∈ X. Then s is said to be transverse to s0 at x if

(s0)∗x(Tx(X)) ⊕ s∗x(Tx(X)) = Ts(x)(E), (18)

where (s0)∗x and s∗x are the derivatives of s0 and s at x, respectively. By the Transversality Theorem(see Chapter 3, Section 2, of [Hirsch]), there is a dense set of sections s with the property that theyare transverse to s0 at every x ∈ X for which s(x) = 0x. These are called generic sections. For anygeneric section s the submanifolds s0(X) and s(X) of E intersect transversally so s0(X) ∩ s(X) is asubmanifold. Since dim s0(X) = dim s(X) = n and dim E = 2n, the intersection has dimensionzero and is therefore a set of isolated points. Since s0(X) ∩ s(X) is compact, it is a finite set ofisolated points. Next notice that the diffeomorphisms s0 and s and the given orientation of X provides0(X) and s(X) with orientations. Because of (18) we can then assign an index Ind (s; p) to everyp = s(x) ∈ s0(X) ∩ s(X) as follows. We take Ind (s; p) = 1 if an oriented basis for Tp(s0(X)) =

(s0)∗x(Tx(X)) together with an oriented basis for Tp(s(X)) = s∗x(Tx(X)) gives an oriented basisfor Tp(E); otherwise, Ind (s; p) = −1. The sum of these indices over all p ∈ s0(X) ∩ s(X) is theintersection number of s with s0 when s0(X)∩ s(X) , ∅ and the sum is independent of the choice ofgeneric section s. In this case the Euler number of ξ is given by

χ(ξ) =∑

p∈s0(X)∩s(X)

Ind (s; p). (19)

If s0(X) ∩ s(X) = ∅, then the intersection number of s with s0 is taken to be 0, as is χ(ξ).

(2) The Euler number of ξ is

χ(ξ) =

∫X

e(ξ), (20)

where e(ξ) ∈ Hn(X;R) is the Euler class of ξ. Again, we should elaborate just a bit. The Euler classe(ξ) can be defined in a number of ways. We will first describe how it can be viewed as a Chern-Weil characteristic class (see Chapter XII, Section 5, of [KN2]). The Chern-Weil construction ofcharacteristic classes requires two ingredients. One needs a principal bundle G → P

πP→ X and a

polynomial P on the Lie algebra g that is ad G-invariant (P(gAg−1) = P(A) for all A ∈ g and allg ∈ G). One chooses a connection ω on G → P

πP→ X, computes the curvature Ω and evaluates the

polynomial on the curvature to obtain a differential form P(Ω) on P. This form is then the pullbackby πP of a form P(Ω) on X which can be shown to be closed and whose cohomology class [P(Ω)]does not depend on the choice of ω. Making various choices for P gives rise to various characteristicclasses for G → P

πP→ X. One obtains characteristic classes for vector bundles by regarding them as

associated to some principal bundle G → PπP→ X by a representation of G.

37

Now, to define the Euler class of the oriented, real vector bundle ξ = (πE : E → X) one choosesa fiber metric on ξ and considers the corresponding oriented, orthonormal frame bundle

SO(2k) → FSO(ξ)πSO→ X. (21)

Then ξ is the vector bundle associated to the principal bundle (21) by the defining action of SO(2k)on V R2k. For the ad SO(2k)-invariant polynomial one chooses the Pfaffian Pf : so(2k) → R

defined as follows. For each skew-symmetric matrix A = (Ai j)i, j=1,...,2k ∈ so(2k) we associate anelement of ∧2V ⊆ ∧V given by ∑

i< j

Ai jψiψ j =

12ψT Aψ,

where ψ =

ψ1

...

ψ2k

and “T” means transpose.. Then ( 12ψ

T Aψ)k is in ∧2kV , which is 1-dimensional, so

it is a multiple of vol = ψ1 · · ·ψ2k. We define Pf (A) by1k!

(12ψT Aψ)k = Pf (A) vol. (22)

By expanding ( 12ψ

T Aψ)k one finds that

Pf (A) =1

2kk!

∑σ∈S 2k

sgn(σ) Aσ(1)σ(2) · · · Aσ(2k−1)σ(2k),

where S 2k is the group of permutations of 1, 2, . . . , 2k − 1, 2k and sgn(σ) is the sign of the permu-tation σ. Then Pf is, indeed, ad SO(2k)-invariant and satisfies (Pf (A))2 = det A (see Appendix Bof [MT]). Now we choose a connection ω on the SO(2k)-bundle (21) with curvature Ω. Then the2k-form on FSO(ξ) given by

Pf (−1

2πΩ) =

(−1)k

2kπk Pf (Ω) =(−1)k

22kπkk!

∑σ∈S 2k

sgn(σ) Ωσ(1)σ(2) · · ·Ωσ(2k−1)σ(2k)

descends to 2k-form Pf (− 12πΩ) on X whose cohomology class is the Euler class of ξ.

e(ξ) =[Pf (−

12π

Ω)]. (23)

Remark 1.13. It is not obvious, but is nevertheless true, that e(ξ) does not depend on the initial choice of afiber metric for ξ (see Theorem 5.1, page 318, of [KN2]). We should point out also that, if s is any localsection of the principal bundle (21), then the form Pf (− 1

2πΩ) can be described locally by

(−1)k

2kπk Pf (s∗Ω) = (2π)−kPf (−s∗Ω). (24)

Furthermore, under a change of local section the forms s∗Ω transform by conjugation and the Pfaffian isinvariant under conjugation so (24) is also invariant and these local descriptions piece together into globaldescription of Pf(− 1

2πΩ) and so to a representative of the Euler class. It is customary to notationally suppressthe “piecing together” aspect of this and refer to (24) as a representative of e(ξ).

38

Example 1.4. Consider the 2-sphere S2 with its usual orientation and Riemannian metric. Let π : TS2 →

S2 be the tangent bundle of S2. The corresponding oriented, orthonormal frame bundle is SO(2) →

FSO(TS2)πSO→ S2. If θ and φ are the usual angular coordinates on S2, then e1, e2 = ∂∂φ ,

1sin φ

∂∂θ is an

oriented, orthonormal frame field on S2, that is, a section s of FSO(TS2). The dual oriented, orthonormalfield of 1-forms is e1, e2 = dφ, sin φ dθ. Thus, the metric volume form is e1 ∧ e2 = sin φ dφ dθ. Onecomputes de1 = 0 = 0 (e1 ∧ e2) and de2 = cos φ dφ dθ = cot φ (e1 ∧ e2). Thus, the Levi-Civita connection ωon the frame bundle is determined by

s∗ω =

0 cos φ dθ−cos φ dθ 0

.Since SO(2) is Abelian, the curvature of ω is Ω = dω and so

s∗Ω =

0 −sin φ dφ dθsin φ dφ dθ 0

.A representative of the Euler class is therefore

(2π)−1Pf (−s∗Ω) =1

2πsin φ dφ dθ.

Consequently, the Euler number of the tangent bundle of S2 is

χ(TS2) =1

2π

∫S2

sin φ dφ dθ = 2.

Notice that this coincides with the Euler characteristic χ(S2) of S2.

The vector bundle ξ = (πE : E → X) gives rise to another approach to the Euler class that will soonbe important to us. We will say that a smooth p-form α on E has compact vertical support if, for everyx ∈ X, the restriction α|π−1

E (x) of α to the fiber π−1E (x) has compact support. The space of all such is denoted

ΩpCV (E;R). Then d : Ω

pCV (E;R) → Ω

p+1CV (E;R) and the resulting co-complex is denoted Ω∗CV (E;R).

The cohomology of this co-complex is denoted H∗CV (E;R) and called the compact vertical cohomology ofE. According to the Thom Isomorphism Theorem (see Theorem 6.17 of [BT]) there is an isomorphismT : H∗(X;R) → H∗+2k

CV (E;R) and this carries the generator of H0(X;R) onto a cohomology class U(E) ∈H2k

CV (E;R) whose integral over each fiber is 1 (see Proposition 6.18 of [BT]). This is called the Thom classand it has the property that, if s is any section of ξ, then

e(ξ) = s∗U(ξ) (25)

(see Proposition 6.41 of [BT]).Before proceeding any further with the technical issues we will try to see where this is going. Consider

the special case in which X is a compact, oriented manifold and ξ is its tangent bundle πT X : T X → X. In thiscase the Euler number χ(ξ) of ξ is just the Euler characteristic χ(X) of X, usually defined as the alternatingsum of the dimensions of the cohomology groups Hp(X;R) (see Proposition 11.24 of [BT]). A section ofξ is then just a vector field on X and the statement that χ(X) is the intersection number of a generic vectorfield with the vector field that is identically zero is called the Poincare-Hopf Theorem. On the other hand,the Chern-Gauss-Bonnet Theorem is the statement that χ(X) can be expressed as the integral over X of the

39

Euler class of the tangent bundle. The point is that χ(X) is a topological invariant and can be calculatedeither as a sum of signed points or as the integral of a cohomology class.

To see what this has to do with Donaldson invariants we would like to adopt a somewhat different per-spective on the simplest of these invariants (see Remark 1.11). For this and for all of the subsequent dis-cussions we will adopt a somewhat more economical notation, writing P for Pk, M for M(Pk, g), G forG(Pk), and so on. The gauge group G does not act freely on the space A if irreducible connections sinceeven irreducible connections have a Z2 stabilizer. However, G = G/Z2 does act freely on A so we have aninfinite-dimensional principal bundle

G → A→ A/G = B

over the Banach manifold B (note that the orbits in A of the G-action are the same as those of the G-actionso the quotient is still B). We will build a vector bundle associated to this principal bundle as follows.Consider the (infinite-dimensional) vector space Ω2

+(X, ad P) of self-dual 2-forms on X with values in theadjoint bundle ad P. We claim that there is a smooth left action of G on Ω2

+(X, ad P). To see this we firstthink of G as the group of sections of the nonlinear adjoint bundle Ad P under pointwise multiplication.Since the elements of Ω2

+(X, ad P) take values in the sp(1)-fibers of ad P and the elements of G take values inthe Sp(1)-fibers of Ad P, G acts on Ω2

+(X, ad P) by pointwise conjugation. Moreover, conjugation takes thesame value at ± f ∈ G so this G-action on Ω2

+(X, ad P) descends to a G-action. Thus, we have an associatedvector bundle

A ×G Ω2+(X, ad P), (26)

the elements of which are equivalence classes [ω, α] = [ω f , f −1 α] with ω ∈ A, α ∈ Ω2+(X, ad P), and

f ∈ G. Now recall that sections of associated vector bundles can be identified with equivariant maps fromthe principal bundle space into the typical fiber of the vector bundle. In our case we have an obvious map

F+ : A→ Ω2+(X, ad P)

of A into Ω2+(X, ad P) that sends a connection to the self-dual part of its curvature.

F+(ω) = F+ω =

12

(Fω + ∗Fω)

Since the action of G on A is by conjugation and curvatures transform by conjugation under a gauge trans-formation, F+ is equivariant so F+ can be identified with a section

s+ : B→ A ×G Ω2+(X, ad P),

given explicitly by

s+([ω]) = [ω, F+ω].

Now notice that the moduli space M of irreducible anti-self-dual connections is precisely the zero set of thesection s+. Thus, if

s0 : B→ A ×G Ω2+(X, ad P),

40

is the zero section, M can be identified with

M = s0(B) ∩ s+(B).

Now suppose that M is 0-dimensional and compact, that is, a finite set of points each equipped witha sign (for concrete examples, see Section 9.1 of [DK]). Then the corresponding Donaldson invariant isthe sum of these signs and one sees quite clearly the sense in which this invariant can be regarded as aninfinite-dimensional analogue of the Poincare-Hopf version of the Euler characteristic. Taking this anal-ogy seriously would suggest the possibility of an integral representation of the 0-dimensional Donaldsoninvariant analogous to the Gauss-Bonnet-Chern Theorem. Notice, however, that such an “integral” wouldnecessarily be over the infinite-dimensional moduli space B and such integrals are notoriously difficult todefine rigorously. But, as Nigel Hitchin [Hitch] has phrased it,

“This is such stuff as quantum field theory is made of.”

Indeed, it was Edward Witten who first produced such an integral representation, not directly, but as thepartition function for the quantum field theory introduced in [Witt2]. The integrals are purely formal and thearguments leading to the partition function and its identification with the Donaldson invariant are physicaland require a deep understanding of the methods of (supersymmetric) quantum field theory. A still deeperunderstanding of physics is required to see how this quantum field-theoretic interpretation of the Donaldsoninvariants eventually pointed toward technically much simpler invariants that were destined to supplantDonaldson theory. We will not presume to describe the physics behind all of this, but will follow insteada path forged by Atiyah and Jeffrey [AJ]. The arguments are still formal in that they apply rigorous finite-dimensional theorems due to Mathai and Quillen [MQ] in an infinite-dimensional context, but they are atleast geometrical. These finite-dimensional theorems are quite nontrivial and live most naturally in thecontext of equivariant cohomology, while the computation required to implement the beautiful idea in [AJ]is relatively straightforward, but lengthy. In both cases we merely sketch what is involved and refer toSection 5 of [Nab2] for a more detailed discussion; additional sources of background information include[BGV], [Blau], [BT], and [GuSt].

1.5.2. The Universal Thom Form. We will begin the discussion by introducing what may seem at first glanceto be a rather strange notational device. We consider again the oriented, real vector bundle ξ = (πE : E → X)of rank n = 2k with typical fiber V . Motivated by the definition (22) of the Pfaffian we define the Berezin(or fermionic) integral of any f ∈ ∧V to be the real coefficient fvol of vol = ψ1 · · ·ψ2k in the polynomial fand write this as ∫

f Dψ = fvol.

Here Dψ is just a formal reminder of the name we have given to the odd generators of ∧V . With this (22)can be written ∫

e12ψ

T AψDψ = Pf (A) (27)

41

for any A ∈ so(2k). In particular, the representative (24) of the Euler class can now be written

(2π)−k∫

e12ψ

T (−s∗Ω)ψDψ.

We will also need to extend this notion of Berezin integration in the following way. Let A be any su-percommutative superalgebra and consider the supertensor product A ⊗ ∧ V (see Remark 1.12). Regard theelements of A ⊗ ∧ V as polynomials in the odd variables ψ1, . . . , ψ2k with coefficients in A and define theBerezin integral of F ∈ A ⊗ ∧ V to be the coefficient in A of vol = ψ1 · · ·ψ2k.∫

F Dψ = Fvol

Example 1.5. Introduce coordinate functions u1, . . . , u2k on V corresponding to the basis ψ1, . . . , ψ2k. Thus,u1, . . . , u2k is the basis for V∗ dual to ψ1, . . . , ψ2k. Let A = Ω∗(V) be the algebra of complex-valueddifferential forms on V . Then each du j, j = 1, . . . , 2k, is in Ω1(V) and −idu jψ

j = iψ jdu j (sum overj = 1, . . . , 2k) is in Ω∗(V) ⊗ ∧ V . Writing du for the column vector (du1 · · · du2k)T we will show that∫

eiψT du Dψ =

∫eiψ jdu j Dψ = du1 · · · du2k,

where we again suppress the customary ∧. To see this we compute as follows.∫eiψT du Dψ =

∫ei(ψ1u1+···+ψ2ku2k) Dψ =

∫eiψ1u1 · · · eiψ2ku2k Dψ

since the elements ψ ju j are all even in Ω∗(V) ⊗ ∧ V and therefore commute. Next, using the (terminating)exponential series and discarding terms that do not contribute to the coefficient of vol = ψ1 · · ·ψ2k, we obtain∫

eiψT du Dψ =

∫(1 + iψ1u1) · · · (1 + iψ2ku2k)Dψ =

∫(iψ1u1) · · · (iψ2ku2k)Dψ

= i2k∫

(−1)12 (2k)(2k+1)du1 · · · du2k ψ

1 · · ·ψ2k Dψ

= du1 · · · du2k.

Notice also that, if we write ‖u‖2 = u21 + · · · + u2

2k ∈ Ω0(V) and identify this with the even element ‖u‖2 ⊗ 1in Ω∗(V) ⊗ ∧ V , then

(2π)−k∫

e−12 ‖u‖

2+iψT du Dψ = (2π)−ke−12 ‖u‖

2du1 · · · du2k (28)

and this is a form that integrates to 1 over V .

The Berezin integral (2π)−k∫

e−12 ‖u‖

2+iψT du Dψ in (28) is often referred to as a Gaussian representativeof the Thom class of V , thought of as a vector bundle over a point. It does not have compact support,but is rapidly decreasing and we will find that this is enough for our purposes. Our objective is constructsimilar Gaussian representatives of the Thom class for the vector bundle ξ = (πE : E → X) and from themrepresentatives of the Euler class (see (25)).

42

Remark 1.14. We will make no particular use of the fact, but we should point out that a Gaussian represen-tative of the Thom class is easily converted into an actual representative of the Thom class, that is, into aform with compact support (see Section 10.3, page 156, of [GuSt]).

The first step is the construction of what is called a universal Thom form. Once again we let V denotethe typical fiber of ξ. As in the previous example we choose an oriented, orthonormal (with respect to thefiber metric) basis ψ1, . . . , ψ2k for V with coordinate functions u1, . . . , u2k. The basis identifies SO(V)with SO(2k) and therefore so(V) with so(2k). Choose a basis ζ1, . . . , ζn, n = k(2k − 1), for so(2k) and letx1, . . . , xn be the dual basis for so(2k)∗. Now let

A = C[so(2k)] ⊗Ω∗(V)

(see Remark 1.12). Every element of A is a sum of terms of the form P ⊗ ϕ, where P is a polynomial inx1, . . . , xn and ϕ is a form on V . We regard these elements as polynomial functions on so(2k) with values inΩ∗(V). One should keep in mind that we have doubled the degrees of the polynomial parts so that C[so(2k)]is purely even. Now consider the supercommutative superalgebra

A ⊗ ∧ V = C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V.

The elements are polynomials in the odd variables ψ1, . . . , ψ2k with coefficients in A. Consequently, Berezinintegration of an element of C[so(2k)] ⊗Ω∗(V) ⊗ ∧V yields an element of C[so(2k)] ⊗Ω∗(V). Our universalThom form will be obtained in this way.

Notice first that Ω∗(V) ⊗ ∧ V can be identified with the subalgebra of C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V for whichthe polynomial part P is 1 ∈ C[so(2k)]. In particular, the exponent in (28) is in C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V .

−12‖u‖2 + iψT du ∈ C[so(2k)] ⊗Ω∗V) ⊗ ∧ V (29)

We intend to add just one more term and then exponentiate. For this extra term we observe the following.Each ζ ∈ so(2k) is represented on V by a linear transformation Mζ whose matrix relative to the basisψ1, . . . , ψ2k is skew-symmetric. Specifically,

Mζ(v) =ddt

[(exp(−tζ)(v)

] ∣∣∣t=0

for each v ∈ V , where exp : so(2k) → SO(2k) is the exponential map on so(2k). We will write (Mζ) for thematrix of Mζ relative to ψ1, . . . , ψ2k . Since this is skew-symmetric,

12ψT (Mζ)ψ = −

12

2k∑j=1

ψ jMζψj =

12

2k∑j=1

(Mζψj)ψ j. (30)

Writing Ma for Mζa , a = 1, . . . , n, and ζ = xa(ζ)ζa (summation convention over a = 1, . . . , n), we haveMζ = xa(ζ)Ma since ζ 7→ Mζ is an isomorphism. Thus,

−12

2k∑j=1

ψ jMζψj = −

12

xa(ζ)2k∑j=1

ψ jMaψj (31)

43

Now notice that − 12 xa ∈ C[so(2k)] and

∑2kj=1 ψ

jMaψj ∈ ∧V . Thus,

−12

xa ⊗

( 2k∑j=1

ψ jMaψj)∈ C[so(2k)] ⊗ ∧ V

or, better yet,

−12

xa ⊗ 1 ⊗( 2k∑

j=1

ψ jMaψj)∈ C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V.

It is customary to write this even element of the algebra simply as

−12

2k∑j=1

ψ jxaMaψj ∈ C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V. (32)

Now one can exponentiate to obtain

e−12∑2k

j=1 ψj xa Maψ

j∈ C[so(2k)] ⊗Ω∗(V) ⊗ ∧ V

and then perform a Berezin integration to get∫e−

12∑2k

j=1 ψj xa Maψ

jDψ ∈ C[so(2k)] ⊗Ω∗(V).

Since the Ω∗(V)-part is 1, this is essentially a C-valued polynomial on so(2k). Indeed, by (27), (30) and(31), its value at any ζ ∈ so(2k) is given by

(2π)−k∫

e−12∑2k

j=1 ψj xa(ζ)Maψ

jDψ = Pf (Mζ).

Now we combine (29) and (32) to obtain

−12‖u‖2 + iψT du −

12

2k∑j=1

ψ jxaMaψj ∈ C[so(2k)] ⊗Ω∗V ⊗ ∧ V (33)

Exponentiating, computing the Berezin integral and then multiplying by (2π)−k gives an element ofC[so(2k)] ⊗Ω∗(V) denoted ν and called the (Mathai-Quillen) universal Thom form of V .

ν = (2π)−k∫

e−12 ‖u‖

2+iψT du− 12∑2k

j=1 ψj xa Maψ

jDψ

(34)

= (2π)−ke−12 ‖u‖

2∫

exp(iψT du −

12

2k∑j=1

ψ jxaMaψj)Dψ

We will see the ν is basically a machine for producing representatives of the Thom class of any vector bundleξ with typical fiber V .

44

Remark 1.15. It is an instructive exercise to compute ν explicitly when V = R2 with its usual orientation

and inner product. Letting ψ1, ψ2 be the standard basis for R2 and taking ζ1 =

0 1−1 0

as the basis for

so(2), the result is the following non-homogeneous element of C[so(2)] ⊗Ω∗(R2).

ν = (2π)−1 e−12 (u2

1+u22) du1du2 + (2π)−1 x1 e−

12 (u2

1+u22). (35)

Notice that each term has degree 2 in C[so(2)] ⊗Ω∗(R2) and the first term integrates to 1 over R2.

1.5.3. Equivariant Cohomology. The construction of the universal Thom form no doubt appears rather adhoc. As it happens, ν lives quite naturally in the context of equivariant cohomology and we would like to sayjust a few words about this; the standard reference for all of this material is [GuSt]. Equivariant cohomologyarose from attempts to understand the topology of the orbit space X/G of a topological space X on whichsome topological group G acts. We will be concerned only with the case in which X is a smooth manifoldand G is a compact, connected, matrix Lie group acting smoothly on X on the left. For this action we willwrite

σ : G × X → X

σ(g, x) = g · x = σg(x) = σx(g).

We denote by g the Lie algebra of G, by C[g] the graded algebra of complex-valued polynomial functionson g, and by Ω∗(X) the graded algebra of complex-valued, smooth differential forms on X. We considerthe supertensor product C[g] ⊗Ω∗(X), every element of which is a sum of terms of the form P ⊗ ϕ, whereP ∈ C[g] and ϕ ∈ Ω∗(X). These are best thought of as Ω∗(X)-valued polynomials on g.

(P ⊗ ϕ)(ζ) = P(ζ)ϕ

The degree of a homogeneous element α = P ⊗ ϕ is taken to be

deg α = deg (P ⊗ ϕ) = 2 deg P + deg ϕ,

where deg P is the polynomial degree of P and deg ϕ is the cohomological degree of the form ϕ. Thus,

C[g] ⊗Ω∗(X) =⊕

k=2i+ j

Ci[g] ⊗Ω j(X).

The action of G on X together with the adjoint action of G on g give a natural action of G on C[g] ⊗Ω∗(X)defined as follows. If α = P ⊗ ϕ is a homogeneous element of C[g] ⊗Ω∗(X) and g ∈ G, then g · α is theelement of C[g] ⊗Ω∗(X) whose value at any ζ ∈ g is given by

(g · α)(ζ) = (g · (P ⊗ ϕ))(ζ) = P(g−1ζg)σ∗g−1ϕ

An element α of C[g] ⊗Ω∗(X) is said to be G-invariant if g · α = α for every g ∈ G or, equivalently,

α(gζg−1) = σ∗g−1α(ζ)

for every g ∈ G and every ζ ∈ g. The graded algebra of all G-invariant elements of C[g] ⊗Ω∗(X) is denoted

Ω∗G(X) = [C[g] ⊗Ω∗(X) ]G

45

and its elements are called G-equivariant differential forms on X. Notice that C[g]G (those P in C[g] satis-fying P(gζg−1) = P(ζ) for all g ∈ G and all ζ ∈ g) and Ω∗(X)G (those ϕ ∈ Ω∗(X) satisfying σ∗

g−1ϕ = ϕ for allg ∈ G) can both be identified with subalgebras of Ω∗G(X). Our grading of C[g] ⊗Ω∗(X) gives

Ω∗G(X) =

∞⊕k=0

ΩkG(X) =

⊕k=2i+ j

[Ci[g] ⊗Ω j(X)

]G.Note that the universal Thom form (35) forR2 is clearly SO(2)-equivariant since the adjoint action of SO(2)on so(2) is trivial (SO(2) is Abelian) and the exponential factor e−

12 (u2

1+u22) is rotationally invariant. It is not

quite so obvious, but the analogous statement is true of every universal Thom form, that is,

ν = (2π)−k∫

e−12 ‖u‖

2+iψT du− 12∑2k

j=1 ψj xa Maψ

jDψ ∈

[C[so(V)] ⊗Ω∗(V)

]SO(V).

To build a cohomology from Ω∗G(X) we define the G-equivariant exterior derivative dG on Ω∗G(X) asfollows. For any α ∈ Ω∗G(X) the value of dGα at ζ ∈ g is given by

(dGα)(ζ) = d(α(ζ)) − ιζ#(α(ζ)),

where ζ# is the vector field on X defined at each x ∈ X by

ζ#(x) =ddt

(exp(−tζ) · x)∣∣∣t=0

and ιζ# denotes interior multiplication (contraction) by ζ#.

Example 1.6. We will show that dSO(2)ν = 0, where ν is the universal Thom form of R2 in (35). Anyζ ∈ so(2) can be written

ζ =

0 λ

−λ 0

= λζ1.

Then x1(ζ) = λ so

ν(ζ) =1

2πe−

12 (u2

1+u22)du1du2 +

12π

λ e−12 (u2

1+u22).

Thus,

d(ν(ζ)) = 0 + (2π)−1λ d(e−12 (u2

1+u22)) = (2π)−1e−

12 (u2

1+u22)λ(−u1du1 − u2du2).

Next we compute

ιζ#(ν(ζ)) = (2π)−1e−12 (u2

1+u22)ιζ#(du1du2) + 0

= (2π)−1e−(u21+u2

2)( ιζ#(du1)du2 − du1ιζ#(du2)).

The proof will be complete if we show

ιζ#(du1) = u1(ζ#) = −λu2 and ιζ#(du2) = u2(ζ#) = λu1.

46

Let v = v1ψ1 + v2ψ

2 ∈ R2 be arbitrary. Then

ζ#(v) =ddt

(exp(−tζ) · v

) ∣∣∣t=0

=ddt

((1 − tζ +

12

t2ζ2 − · · · ) · v) ∣∣∣

t=0

= −ζv = −λ

0 1−1 0

v1

v2

=

−λv2

λv1

=

−λu2(v)λu1(v)

as required.

An element α ∈ Ω∗G(X) with dGα = 0 is said to be G-equivariantly closed and the previous examplegeneralizes to show that this is true of any universal Thom form (see pages 86-87 of [GuSt]).

dSO(V)ν = 0

One can show also that, for any α ∈ Ω∗G(X),

dG dG(α) = 0

so d2G = 0 on Ω∗G(X) and, due to the grading we have introduced on Ω∗G(X),

dG : ΩkG(X)→ Ωk+1

G (X).

Consequently,

(Ω∗G(X), dG)

is a co-chain complex. The cohomology of this co-chain complex is called the Cartan model of the G-equivariant cohomology of X and is denoted

H∗G(X).

Every G-equivariantly closed form α ∈ Ω∗G(X), such as the universal Thom form ν, determines a G-equivariant cohomology class that is denoted [α]. If α = dGβ for some β ∈ Ω∗G(X), then α is said to beG-equivariantly exact and in this case the cohomology class [α] is trivial.

Remark 1.16. Let S3 =(z1, z2) ∈ C2 : |z1|2 + |z2|2 = 1

be the 3-sphere and S1 =

eiθ : θ ∈ R

the Lie group

of complex numbers of modulus one. Define a free left action of S1 on S3 by

eiθ · (z1, z2) = (eiθz1, eiθz2).

The orbit space S3/S1 is, by definition, the complex projective line CP1, which is diffeomorphic to the2-sphere S2 (see (1.2.8), page 53, of [Nab4]). The S1-equivariant cohomology H∗

S1(S3) of S3 is computedon pages 90-93 of [Nab2] and is found to coincide with the de Rham cohomology of the orbit space S2.According to the following theorem of Cartan, this is no accident (see Theorem 2.5.1 of [GuSt]).

47

Theorem 1.20. (Cartan) Let X be a smooth manifold and G a compact, connected Lie group. Suppose thereis a smooth, free, left action of G on X. Then the orbit space X/G is a smooth manifold and the G-equivariantcohomology H∗G(X) of X is isomorphic to the de Rham cohomology H∗deRham(X/G;C) of X/G with complexcoefficients.

Finally we will need to introduce a notion of integration for equivariant differential forms and cohomol-ogy classes. For this we will now assume that X is compact and oriented and that the G-action preserves theorientation of X in the sense that each of the diffeomorphisms σg : X → X is orientation preserving. Let αbe an element of Ω∗G(X). For each ζ ∈ g we write α(ζ) as

α(ζ) = α(ζ)[0] + α(ζ)[1] + · · · + α(ζ)[n],

where α(ζ)[k] ∈ Ωk(X) and n = dim X. Now we define an element∫Xα ∈ C[g]G

by ( ∫Xα)

(ζ) =

∫Xα(ζ)

de f=

∫Xα(ζ)[n].

Notice that∫

X α really is G-invariant since( ∫Xα)

(gζg−1) =

∫Xα(gζg−1)[n] =

∫Xσ∗g−1(α(ζ)[n]) =

∫Xα(ζ)[n] =

( ∫Xα)

(ζ).

Thus, ∫X

: Ω∗G(X)→ C[g]G. (36)

Example 1.7. Let ν be the universal Thom form for R2 given by (35). Then, for every ζ ∈ so(2),( ∫R2ν)

(ζ) =

∫R2ν(ζ)[2] =

∫R2

(2π)−1e−12 (u2

1+u22)du1du2 = 1

so ∫R2ν = 1 (the constant function in C[so(2)]SO(2)).

The result in the previous Example generalizes to any universal Thom form (see pages 87-88 of [GuSt]).

ν = (2π)−k∫

e−12 ‖u‖

2+iψT du− 12∑2k

j=1 ψj xa Maψ

jDψ⇒

∫Vν = 1

Finally, note that, if α = dGβ is G-equivariantly exact, then, for each ζ ∈ g, α(ζ)[n] = d(β(ζ)[n−1]) so, byStokes’ Theorem,

∫X α = 0 ∈ C[g]G. Consequently, the integration map (36) descends to cohomology.∫

X: H∗G(X)→ C[g]G

48

1.5.4. Gaussian Representatives of the Thom Class. With this digression on equivariant cohomology behindus we will return to the general development. Recall that our interest in the universal Thom class [ν] residesin the fact, asserted earlier, that its representative ν can be used to manufacture Gaussian representativesof the Thom class of any oriented, real vector bundle with typical fiber V . We will now briefly sketch theprocedure.

We begin by once again recalling our basic assumptions. X will denote a compact, oriented, smoothmanifold of even dimension n = 2k and we will consider an oriented, real vector bundle over X whosetypical fiber V has dimension n = 2k and that has been equipped with a positive definite fiber metric. Wewill now use the fact that any such vector bundle can be viewed as the vector bundle ξρ = (πρ : P×ρ V → X)associated to some principal bundle

G → PπP→ X

by a representation

ρ : G → SO(V) (ρ(g)(v) = g · v)

of G on V . We briefly recall the construction of ξρ. The actions of G on P and on V combine to give anaction of G on P × V defined by

(p, v) · g = (p · g, g−1 · v).

The action is free so the orbit space P ×ρ V is a smooth manifold and the projection

P × VQ→ P ×ρ V

Q(p, v) = [p, v]

defines a principal G-bundle. Let π1 : P × V → P be the projection onto the first factor and define πρ by thecommutativity of the following diagram.

P × Vπ1

−−−−−→ P

Qy yπP

P ×ρ Vπρ

−−−−−→ X

We are trying to produce representatives of the Thom class on P ×ρ V . Notice that the right-hand side ofthe diagram is the arena in which the Chern-Weil procedure produces characteristic classes. As motivationfor what is to come we will briefly recall how this goes. Choose a connection ω on P with curvature Ω anda P ∈ C[g]G. Evaluate P(Ω) to obtain a form on P. P(Ω) is basic, that is, G-invariant and horizontal withrespect to ω, so it descends to a form on X. This form on X is closed and its cohomology class does notdepend on the choice of ω so it represents a characteristic class of G → P

πP→ X. The entire process can be

encapsulated in the existence of a map, called the Chern-Weil map

CWω :C[g]G → Ω∗(P)BAS IC Ω∗(X)

CWω(P) = P(Ω)

The idea is to carry out a similar procedure on the left-hand side G → P × VQ→ P ×ρ V of the diagram.

49

Again choose a connection ω on P with curvature Ω. Then π∗1ω is a connection on P × V with curvatureπ∗1Ω. Identify T(p,v)(P × V) with Tp(P) ⊕ Tv(V). Since π∗1ω and π∗1Ω depend only on the Tp(P)-summandsand there agree with ω and Ω, respectively, we will use the same symbols to denote the forms on P×V . Theω-horizontal spaces in P × V are given by Hor(p,v)(ω) = Horp(ω) ⊕ Tv(V). The decomposition

T(p,v)(P × V) =(

Horp(ω) ⊕ Tv(V))⊕ Vert(p,v)(P × V)

determines a projection πHor(ω) of forms on P×V to ω-horizontal forms on P×V (evaluate on ω-horizontalparts of the tangent vectors). Now let α = P ⊗ ϕ be a homogeneous element of Ω∗G(V) = [C[g] ⊗Ω∗(V)]G.One can evaluate the polynomial part on the curvature Ω as in the ordinary Chern-Weil procedure to obtainP(Ω)⊗ ϕ in Ω∗(P)⊗Ω∗(V). This one can identify with the form P(Ω)∧ ϕ on P×V which, because P⊗ ϕ isG-invariant, is in Ω∗(P × V)G. It is generally not ω-horizontal, however, so to obtain a basic form on P × Vone must compose with the horizontal projection πHor(ω). This gives a map

α = P ⊗ ϕ ∈ Ω∗G(V) 7→ πHor(ω)[P(Ω) ∧ ϕ

]∈ Ω∗(P × V)BASIC Ω∗(P ×ρ V)

and extending by linearity gives the generalized Chern-Weil map

CWω : Ω∗G(V)→ Ω∗(P × V)BASIC Ω∗(P ×ρ V).

One can show that CWω is a co-chain map

d CWω = CWω dG

so it carries G-equivariantly closed forms on V to ordinary closed forms on P × V which then descend toclosed forms on P ×ρ V .

Notice that the representation ρ : G → SO(V) induces a Lie algebra homomorphism ρ∗ : g→ so(V) (justthe derivative of ρ at the identity in G). With this any P⊗ϕ in Ω∗SO(V)(V) gives rise to an element (Pρ∗)⊗ϕin Ω∗G(V) and therefore, by linearity, so does every element of Ω∗SO(V)(V). In particular, the universal Thomform gives rise to

νG = (2π)−k∫

e−12 ‖u‖

2+iψT du− 12∑2k

j=1 ψj[xaρ∗]Maψ

jDψ

in Ω∗G(V). Applying the map CWω to νG gives a basic, closed form on P×V that is the horizontal projectionof

(2π)−k∫

e−12 ‖u‖

2+iψT du− 12∑2k

j=1 ψj(xa[ρ∗Ω])Maψ

jDψ.

It will often be more convenient to work with a more compact form of this that we obtain as follows. Wewill write (ρ∗Ω) for the skew-symmetric matrix image of the curvature under ρ∗. In somewhat more detail,the values of the 2-form Ω are in g and ρ∗ carries each of these to so(V). Each of these values corresponds,via the action of so(V) on V , to a linear transformation xa[ρ∗Ω]Ma on V . This we can then identify with askew-symmetric matrix

(xa[ρ∗Ω]Ma

). In this way we think of (ρ∗Ω) as a matrix of 2-forms on P with values

in so(V). Now use the fact that, for any skew-symmetric matrix S , ψT Sψ = −∑2k

j=1 ψjSψ j. Moreover, we

can implement the horizontal projection by insisting that the form be evaluated only on the horizontal partsof tangent vectors to P × V . With this we define a basic, closed form U on P × V by

U = (2π)−ke−12 ‖u‖

2∫

exp(iψT du +

12ψT (ρ∗Ω)ψ

)Dψ (evaluated on horizontal parts). (37)

50

Then U descends to a closed form U on P ×ρ V (U = Q∗U). This is, in fact, a Gaussian representative ofthe Thom class of P ×ρ V (see Chapter 10 and, in particular, Section 10.4 of [GuSt])

Remark 1.17. One can write out an explicit, albeit rather cumbersome, formula for the ω-horizontal projec-tion of any form ϕ. Writing ω = ωaζa and ιa = ιζ#

aone has

πHor(ω)(ϕ) = ϕ − ω1 ∧ ι1ϕ − ω2 ∧ ι2ϕ − · · · − ω

n ∧ ιnϕ

+∑

1≤a1<···<ar≤n

(−1)r(r+1)/2ωa1 ∧ · · · ∧ ωar ∧ (ιa1 · · · ιar )(ϕ) (38)

We will see an instance of this in the next example. However, it will generally be most useful for ourpurposes to view this as evaluating on horizontal parts. For example, if we recall the Cartan formula Ω =

dω + 12 [ω,ω] and notice that the second term vanishes on ω-horizontal parts by definition, the right-hand

side of (37), when evaluated on horizontal parts, gives the same result whether or not 12 [ω,ω] is present. We

can therefore simplify (37) and write

U = (2π)−ke−12 ‖u‖

2∫

exp(iψT du +

12ψT (ρ∗(dω))ψ

)Dψ (evaluated on horizontal parts). (39)

We will have considerably more cosmetic surgery to perform on U and U as we proceed.

Since our primary concern is with representatives of the Euler class, we will want to pull U back by asection of the vector bundle πρ : P ×ρ V → X. Now, any section of the associated bundle P ×ρ V can bewritten as

x ∈ X(s,S s)−→ (s(x), S (s(x))) ∈ P × V

Q−→ [s(x), S (s(x))] ∈ P ×ρ V,

where s is a section of G → PπP→ X and S : P → V is an equivariant map, that is, S (p · g) = ρ(g−1)(s(p))

for all p ∈ P and all g ∈ G (see Section 6.8, page 384, of [Nab4]). Thus,

(Q (s, S s))∗U = (s, S s)∗(Q∗U) = ((1, S ) s)∗U = s∗((1, S )∗U)

so to pull U back by a section of P×ρV we compute (1, S )∗U, which simply pulls the V-factors of U back bythe equivariant map S to get a form on P which we then pull back to X by a section s of the principal bundleG → P

πP→ X. We never need to explicitly compute U since U contains all of the necessary information. The

essential fact, particularly when we later mimic these ideas in the infinite-dimensional context of quantumfield theory, is that S and s are entirely arbitrary and one can obtain very different looking representatives ofthe Euler class by making different choices for S and s. We will illustrate all of this with an example.

Example 1.8. We begin with the universal Thom form forR2 using the notation established in Remark 1.15,Example 1.6, and Example 1.7.

ν = (2π)−1e−12 (u2

1+u22) (du1du2 + x1)

51

As the vector bundle with typical fiberR2 we take the tangent bundle TS2 of the 2-sphere S2, which we needto regard as a vector bundle associated to some principal bundle. We provide S2 with its usual orientationand Riemannian metric and consider the oriented, orthonormal frame bundle

SO(2) → FSO(S2)πSO−→ S2.

If ρ : SO(2)→ SO(2) is the identity representation of SO(2) on SO(2) (ρ = idSO(2)), then

FSO(S2) ×ρ R2 TS2.

In this case, ρ∗ : so(2) → so(2) is also the identity so x1 ρ∗ = x1. Now let ω = ω1ζ1 be a connectionon the frame bundle with curvature Ω = Ω1ζ1 (soon we will be more specific and take ω to be the Levi-Civita connection, but for the moment we will leave it arbitrary). Then x1(ρ∗Ω) = Ω1 so CWω(νSO) is thehorizontal projection of

ϕ = (2π)−1e−12 (u2

1+u22) (du1du2 + Ω1)

on FSO(S2) × R2 with respect to the connection ω on FSO(S2) × R2 (keep in mind that this really meansthe pullback to FSO(S2) ×R2 of the chosen connection on FSO(S2) by the projection π1 : FSO(S2) ×R2 →

FSO(S2) onto the first factor). We need to compute the ω-horizontal projection of ϕ. The explicit formula(38) in this special case is

πHor(ω)(ϕ) = ϕ − ω1 ∧ ι1ϕ.

Here ι1 = ιζ#1, where ζ#

1 is the vector field on FSO(S2) ×R2 determined by

ζ#1 (p, v) =

ddt

(p · exp(tζ1), exp(−tζ1) · v

) ∣∣∣t=0

=ddt

( p · exp(tζ1) )∣∣∣t=0 ⊕

ddt

( exp(−tζ1) · v )∣∣∣t=0

= ζ#1 (p) ⊕ ζ#

1 (v),

where we identify T(p,v)(FSO(S2) ×R2) = Tp(FSO(S2)) ⊕ Tv(R2). As in Example 1.6 we have

ιζ1(du1du2) = −u1du1 − u2du2.

Moreover, ι1Ω1 = 0 since ζ#1 is vertical and Ω is dω evaluated on horizontal parts. Thus,

CWω(νSO) = (2π)−1e−12 (u2

1+u22)( Ω1 + du1du2 + ω1 ∧ (u1du1 + u2du2)

). (40)

Next, we have seen that to get a representative of the Euler class of the tangent bundle of S2 we mustchoose a section s of the frame bundle and an equivariant map S of FSO(S2) toR2. As a section of the framebundle we choose the oriented, orthonormal frame field corresponding to the spherical coordinate chart, thatis,

s(φ, θ) =

(φ, θ, ∂φ,

1sin φ

∂θ

).

We choose an equivariant map FSO(S2)S→ R2 by beginning with a vector field on S2 (that is, a section of

TS2). This can be chosen arbitrarily and we will take V = γ sin θ ∂φ+γ cos θ cot φ ∂θ, where γ is an arbitraryreal parameter (when γ = 1 this is the infinitesimal generator for rotations about the x-axis). Relative to the

52

frame field s the components of V are γ sin θ and γ cos θ cos φ. The map S : FSO(S2) → R2 is defined onthe image of s by

S s(φ, θ) = (sin θ, cos θ cos φ)

and elsewhere by equivariance. We have seen that to pull back CWω(νSO) by the section x 7→ [s(x), S (s(x))]one proceeds in two steps. First pull back the R2-parts by S , that is, substitute u1 = γ sin θ and u2 =

γ cos θ cos φ, to obtain a form on FSO(S2) and then pull this form back to S2 by s. To make matters quiteconcrete we will now choose a specific connection on the frame bundle. The most natural choice is theLevi-Civita connection which is given in terms of spherical coordinates by 0 −cos φ dθ

cos φ dθ 0

= (−cos φ dθ) ζ1

so that s∗ω1 = −cos φ dθ and therefore s∗Ω1 = sin φ dφ dθ. The result of all of these substitutions and a bitof trigonometry is the following representative of the Euler class of the tangent bundle of S2.

(2π)−1e−12γ

2(sin2θ+cos2θ cos2φ)sin φ (1 + γ2cos2θ sin2φ) dφ dθ

In particular, since the Euler characteristic of S2 is 2, we obtain the following not altogether obvious integralformula.

12π

∫ 2π

0

∫ π

0e−

12γ

2(sin2θ+cos2θ cos2φ)sin φ (1 + γ2cos2θ sin2φ) dφ dθ = 2

1.5.5. Witten’s Partition Function. Now recall that our interest in the Mathai-Quillen formalism stems fromthe fact that Atiyah and Jeffrey [AJ] have shown how it can be adapted and formally applied to the infinite-dimensional vector bundle A×GΩ2

+(X, ad P) to yield an integral representation of the 0-dimensional Donald-son invariant and which coincides with the partition function of Witten’s topological quantum field theory[Witt2]. We will now sketch the appropriate adaptation of the formula (39), still working in the finite-dimensional context, and will then move on to A×G Ω2

+(X, ad P). There is quite a lot to be verified along theway, but we will be content to illustrate a few of the computations and refer to [AJ] and Section 5 of [Nab2]for more details.

We will begin augmenting the assumptions we have made thus far and making some specific choicesof the various ingredients we require for the construction. We will assume that the principal bundle G →

PπP→ X has bundle space P that is oriented and that the action of G on P preserves the orientation in the

sense that each of the diffeomorphisms σg : P → P is orientation preserving. Next we recall that for anyaction of a compact Lie group G on a manifold P it is always possible to choose a Riemannian metric onP that is G-invariant, that is, for which the diffeomorphisms σg are all isometries (see Chapter IV, Example1.4, page 155, of [KN1]). We fix, once and for all, a G-invariant Riemannian metric on P, denoted simply〈 , 〉. Similarly, we fix an ad-invariant inner product ( , ) on the Lie algebra g. This, in turn, determines bytranslation a bi-invariant Riemannian metric on G which can be normalized so that the volume of G withrespect to the corresponding metric volume form is one.

At each p ∈ P the Riemannian metric 〈 , 〉 defines an orthogonal complement to the vertical subspaceof Tp(P) (the tangent space to the G-orbit through p) and, since G acts by isometries, these orthogonal

53

complements are invariant under the action of G and so determine a connection ω on G → PπP→ X (see

Chapter II, Section 1, page 63, of [KN1]). Henceforth we will use this connection exclusively. In particular,the pullback connection π∗1ω on P × V , which we continue to denote by the same symbol ω, has

Hor(p,v)(ω) = Tp(p ·G)⊥ ⊕ Tv(V).

at each point.Now define, for each p ∈ P, a linear map

Cp : g→ Vertp(P) ⊆ Tp(P)

by

Cp(ζ) = ζ#(p) =ddt

(p · exp (tζ))∣∣∣t=0.

This is an isomorphism onto Vertp(P), but we wish to regard it as a map into Tp(P). Relative to the invariantinner products on g and Tp(P), Cp has an adjoint C∗p : Tp(P)→ g. The map

Rp = C∗p Cp : g→ g

is self-adjoint with trivial kernel so there is an inverse

R−1p : g→ g.

Moreover, as a map onto Vertp(P), Cp has an inverse

C−1p : Vertp(P)→ g.

Note that, on Vertp(P), C−1p agrees with ωp, that is,

C−1p (w) = ωp(w), w ∈ Vertp(P).

Indeed, if w ∈ Vertp(P), then w = η#(p) = Cp(η) for a unique η ∈ g and, since ω is a connection, ωp(η#(p)) =

η for every η ∈ g. Thus, ωp(w) = η = C−1p (w). Similarly, one checks that

C∗p(v) = Rp(ωp(v)), v ∈ Tp(P)

(see page 76 of [Nab2]) so C∗ can be thought of as a g-valued 1-form on P. Selecting bases we can identifyeach Rp with an invertible matrix and ω with a matrix of real-valued 1-forms on P so this becomes a matrixequation C∗ = Rω and we will write it as

ω = R−1C∗.

From this we compute

dω = dR−1 ∧C∗ + R−1dC∗.

Noting that the first term vanishes on horizontal vectors we find that, in the expression (39), we can replacedω by R−1dC∗ to obtain

U = (2π)−ke−12 ‖u‖

2∫

exp(iψT du +

12ψT (ρ∗(R−1dC∗))ψ

)Dψ (evaluated on horizontal parts). (41)

The next objective is to remove the explicit appearance of the inverse in (41) by using the Fourier inversionformula. We will follow the conventions of [RS2]. Thus, we let W be an oriented, real vector with inner

54

product ( , ) and orthonormal coordinates w = (w1, . . . ,wn). If f (w) = f (w1, . . . ,wn) is in the Schwartzspace S(W) of rapidly decreasing functions of w1, . . . ,wn, then the Fourier transform F[ f ] = f of f is afunction of another set of coordinates y = (y1, . . . , yn) in W defined by

F[ f ](y) = f (y) = (2π)−n/2∫

We−i (w,y) f (w) dw.

The Fourier Inversion Formula then asserts that f is a Schwartz function of y1, . . . , yn and

f (w) = (2π)−n/2∫

Wei (w,y) f (y) dy.

Combining these two gives

f (w) = (2π)−n∫

W

∫W

ei (w,y)e−i (z,y) f (z) dz dy.

If R is a self-adjoint matrix with positive determinant, then one can use this formula to compute f (R−1w).To get an integral that does not depend explicitly on R−1, however, we also make the change of variabley 7→ Ry. Then (R−1w,Ry) = (w, y) and d(Ry) = det R dy so

f (R−1w) = (2π)−n∫

W

∫W

ei(w,y)e−i(z,Ry) f (z) det R dz dy. (42)

Now return to the expression (41) for U. We take W to be the Lie algebra g with the chosen ad-invariantinner product ( , ). Letting ζ = (ζ1, . . . , ζn) denote orthonormal coordinates on g we consider the function ong defined by

f (ζ) = (2π)−ke−12 ‖u‖

2∫

exp(iψT du +

12ψT (ρ∗(ζ))ψ

)Dψ.

The values of f are forms on V whose components relative to du1, . . . , du2k are polynomials in ζ1, . . . , ζn.These polynomials are not in the Schwartz space S(g), but we nevertheless propose to apply the Fourierformula (42) componentwise to f (ζ). This is rather sloppy, of course, and could be made more precise byinserting a rapidly decaying factor e−ε(ζ,ζ) and taking the limit as ε → 0. However, since our objective is aformula to be applied formally to an infinite-dimensional situation in which even k is infinite and completerigor is (for the time being at least) out of the question anyway, we will not be scrupulous about such matters.Thus, we let λ = (λ1, . . . , λn) be another Lie algebra variable in g and apply (42) with w = dC∗, that is, withw = dC∗(χ1, χ2) for each fixed pair (χ1, χ2) of tangent vectors to P. Evaluating this on horizontal parts thengives, by (41), our next expression for U. The result is as follows.

U = (2π)−n(2π)−ke−12 ‖u‖

2∫g

∫g

∫exp

(iψT du +

12ψT (ρ∗(ζ))ψ + i(dC∗, λ) − i(ζ,Rλ)

)det RDψ dλ dζ

(evaluated on horizontal parts) (43)

Notice that this expression contains one Berezin (“fermionic”) and two ordinary (“bosonic”) integrations.Next we would like to include the parenthetical remark “evaluated on horizontal parts” in (43) directly

into the integral expression for U. For this we need the notion of a normalized vertical volume form for a

principal bundle, which is essentially an analogue of the Thom form for a vector bundle. Let G → QπQ→ M

be a principal G-bundle with M compact and orientable and dim G = n. We assume that the bundle isorientable in the sense that there exists a smooth n-form Ψ on Q such that, if m ∈ M and ιm : π−1

Q (m) → Q

55

is the inclusion map of the fiber above m into Q, then ι∗mΨ is an orientation for the submanifold π−1Q (m) G

of Q. It follows that Q is orientable and one can assume that the action of G on Q is orientation preserving.Then a normalized vertical volume form is an n-form Θ on Q such that the integral of Θ over each fiber isone, that is, if m ∈ M and ιm : π−1

Q (m) → Q is the inclusion map, then∫π−1

Q (m)ι∗mΘ = 1.

If β is any top rank form on M, then ∫Mβ =

∫Qπ∗Q β ∧ Θ. (44)

One can produce a normalized vertical volume form from a connection ω on the bundle as follows.Assume that ζ1, . . . , ζn is an orthonormal basis for g relative to the normalized, ad-invariant inner product( , ) and consistent with the orientation G inherits as a fiber in Q. Write ω = ω jζ j, where ω j ∈ Ω1(Q) forj = 1, . . . , n. Then we claim that

Θ = ω1 ∧ · · · ∧ ωn (45)

is a normalized vertical volume form on Q. To see this note that, for each m ∈ M,

ι∗mΘ = (ι∗mω1) ∧ · · · ∧ (ι∗mω

n).

Now notice that, on π−1Q (m) G, the form ι∗mω

j is dual to the vector field ζ#j for j = 1, . . . , n. Indeed, a

defining property of a connection is that ω(A#) = A for every A ∈ g so, on π−1Q (m), (ι∗mω)(ζ#

k ) = ζk for eachk = 1, . . . , n. Thus,

ζk = (ι∗mω)(ζ#k ) = (ι∗mω

j)(ζ#k ) ζ j

so (ι∗mωj)(ζ#

k ) = δ jk = (ζ#j , ζ

#k ). Consequently, the integral of ι∗mΘ over π−1

Q (m) agrees with the volume of Gwith respect to the normalized, bi-invariant Riemannian metric on G corresponding to ( , ) and this is one.

Now, notice that wedging the normalized vertical volume form Θ given by (45) with the horizontal pro-jection πHor(ω)(ϕ) given by (38) gives

πHor(ω)(ϕ) ∧ Θ = ϕ ∧ Θ. (46)

Integrating the left-hand side over a fiber gives

πHor(ω)(ϕ)∫π−1

Q (m)ι∗mΘ = πHor(ω)(ϕ).

We conclude that computing the ω-horizontal projection of a form ϕ on Q can be accomplished by wedgingϕ with the normalized vertical volume form Θ and then integrating over a fiber.

We would like to apply this to the form U in (43), where the principal bundle is G → P × VQ→ P ×ρ V .

However, we would like to include the procedure as part of the integration formula so we first note that Θ

can be written as a Berezin integral. For this we let η1, . . . , ηn denote either the oriented, orthonormal basisζ1, . . . , ζn if n(n − 1)/2 is even or an odd permutation of ζ1, . . . , ζn if n(n − 1)/2 is odd. Regard these asodd generators of the exterior algebra

∧g and consider the following element of Ω∗(Q) ⊗

∧g.

e∑n

a=1 ωaηa = eω1η1 · · · eωnηn = (1 + ω1η1) · · · (1 + ωnηn)

56

Performing the Berezin integration gives∫e∑n

a=1 ωaηaDη = (−1)n(n−1)/2∫

ω1 · · ·ωnη1 · · · ηnDη = ω1 · · ·ωn = ω1 ∧ · · · ∧ ωn = Θ. (47)

Now we would like to express the Berezin integral representation (47) of Θ without explicit referenceto the connection forms ωa. For this we assume, as we did earlier for G → P

πP→ X, that Q has an

invariant Riemannian metric and that ω is the connection on Q whose horizontal spaces are the orthogonalcomplements of the G-orbits. Then we have available the maps C, C∗, and R. Now consider the element ofC1[g] ⊗Ω1(Q) whose value at any η ∈ g is the 1-form on Q defined by

(C∗, η)(χ) = (C∗χ, η) = (χ,Cη)

for any vector field χ on Q. One can exponentiate (C∗, η) in C[g] ⊗Ω∗(Q) and compute the Berezin integralof the result. If one writes out everything explicitly and has a little patience one finds that∫

e(C∗,η) Dη = det R∫

e∑n

a=1 ωaηa Dη = (det R) Θ

(see pages 106-109 of [Nab2]). Thus,

Θ = (det R)−1∫

e(C∗,η) Dη. (48)

We can now ensure the horizontal projection in (43) by multiplying through by the normalized vertical

volume form Θ of G → P × VQ→ P ×ρ V as given by (48), thus adding one more term in the exponential

and canceling the determinant.

U = (2π)−n(2π)−ke−12 ‖u‖

2∫g

∫g

∫ ∫exp

(iψT du +

12ψT (ρ∗(ζ))ψ + i(dC∗, λ)

− i(ζ,Rλ) + (C∗, η))DηDψ dλ dζ (49)

Notice that, in addition to the two Berezin integrations and the two ordinary integrations, there is an implicitintegration over the fibers of P× V → P×ρ V to “integrate out” the extra factor of Θ in (46); it is customaryto leave this understood rather than explicit. Then (49) gives a basic form on P × V which descends to arepresentative U of the Thom class on P ×ρ V . Pulling U back by a section of P ×ρ V gives a representativeof the Euler class of P ×ρ V which, when integrated over X, gives the Euler number. We have already seenthat pulling back by such a section amounts to first pulling the V-parts of U back by an equivariant mapS : P→ V and then pulling the resulting form on P back by a section s of the principal bundle πP : P→ X.The first of these steps simply replaces u by S ∗u = u S , that is, by the component functions of S : P→ V .We will adhere to the usual convention of simply writing this as

(2π)−n(2π)−ke−12 ‖S ‖

2∫g

∫g

∫ ∫exp

(iψT dS +

12ψT (ρ∗(ζ))ψ + i(dC∗, λ)

− i(ζ,Rλ) + (C∗, η))DηDψ dλ dζ (50)

Pulling back this form on P by the section s : X → P of πP : P→ X gives a representative of the Euler classof P ×ρ V so its integral over X is the Euler number of P ×ρ V .

57

Remark 1.18. One can look at this calculation of the Euler number from a somewhat different point of view.Notice that if e is any representative of the Euler class of P ×ρ V and Θ is the normalized vertical volumeform for the principal bundle πP : P→ X, then (44) gives∫

Xe =

∫Pπ∗Pe ∧ Θ

so the Euler number can also be computed as an integral over P.

We have one last bit of cosmetic surgery to perform. There is a common notational device in (super-symmetric) physics whereby the integral of a top rank form over a manifold is written as two successiveintegrations, one fermionic and one bosonic. As motivation let us first suppose P is a d-dimensional, ori-ented, real vector space with a positive definite inner product. Let u1, . . . , ud be coordinate functions on Pcorresponding to some oriented, orthonormal basis for P and let du = du1∧· · ·∧dud be the volume form forP. The integral of any (properly decaying) function ϕ on P is then defined to be the integral of the d-formϕ du over P. ∫

Pϕ du

Introduce generators χ1, . . . χd for the exterior algebra∧

P. Now let α = α(u j, du j) be a d-form on P writtenin terms of the coordinates u1, . . . , ud and their differentials. Formally substituting du j 7→ χ j for j = 1, . . . , dwe find that the Berezin integral ∫

α(u j, χ j)Dχ

is precisely the function one integrates next to du to get the integral of α over P.∫Pα =

∫P

∫α(u j, χ j)Dχ du

Physicists employ precisely the same notational device when P is a manifold by thinking of u1, . . . , ud aslocal coordinates and χ1, . . . , χd as generators for the exterior algebra of the tangent space. Since it is justa notational device one does not worry too much about making this precise. Applying this convention tothe expression for the Euler number obtained by integrating over P gives the final formula toward whichall of this has been leading us. This introduces one more fermionic integration Dχ and one more bosonicintegration du. Since there is now an integration over u ∈ P we will include the exponential factor e−

12 ‖S ‖

2

in the integrand. Also, since the proliferation of integral signs is becoming rather tiresome we will nowsuppress all but one of them and leave it to the “differentials” DχDηDψ dλ dζ du to indicate the integrationintended. Finally, we will, for the sake of clarity, indicate explicitly the variables on which each termdepends. The result of all this is

(2π)−n(2π)−k∫

exp(−

12‖S (u)‖2 + iψT dS u( χ) +

12ψT (ρ∗(ζ))ψ + i(dC∗u( χ), λ)

− i(ζ,Ruλ) + (C∗u χ, η))DχDηDψ dλ dζ du. (51)

58

We will refer to (51) as the Atiyah-Jeffrey formula for the Euler number of P ×ρ V . Our objective is toformally apply it to the infinite-dimensional vector bundle (26) associated with Donaldson theory. The re-sult will be, formally at least, an expression for the “Euler number” of the bundle, that is, the 0-dimensionalDonaldson invariant (see page 40) and also, as it so happens, the partition function for Witten’s topologicalquantum field theory. We should emphasize at the outset that what we intend to do is not mathematics (andcertainly not physics). We proceed by analogy, isolating formal field-theoretic analogues of the variousfermionic and bosonic variables in (51) and natural identifications of the terms in the exponent with func-tions of these variables. In the process the perfectly well-defined fermionic and bosonic integrals in (51)metamorphose into Feynman integrals over spaces of fields with all of their attendant mathematical difficul-ties. The purist will argue that this is meaningless manipulation of symbols and we can offer no credibledefense against the charge. The only mitigating circumstance is that such formal manipulations have provedextraordinarily productive for both physics and mathematics and so should be viewed with some tolerance.

Throughout the remainder of this section X will denote a compact, connected, simply connected, oriented,smooth 4-manifold with b+

2 (X) > 1 and odd, g is a generic Riemannian metric on X and we will use thenotation established on pages 39-40. It will be more convenient to deal with matrices rather than quaternionsso we will identify Sp(1) with SU(2) and consider a smooth, principal SU(2)-bundle SU(2) → P

π→ X over X

whose Chern number k has been fixed so that the dimension of the moduli space M = M(P, g) of irreducibleg-ASD connections is zero. We consider the principal G-bundle

G → A→ A/G = B,

the vector bundle (26)

A ×G Ω2+(X, ad P)

associated to it by the action of G on Ω2+(X, ad P) and the section

s+ : B→ A ×G Ω2+(X, ad P),

of it determined by the equivariant map S = F+ : A → Ω2+(X, ad P), S (ω) = F+(ω) = F+

ω = 12 (Fω +∗ Fω),

that is,

s+([ω]) = [ω, F+ω].

The application of the Atiyah-Jeffrey formula (51) requires the existence of a Riemannian metric on theprincipal bundle space A for which the group G acts by isometries. To produce such a metric we recall thatA is an open subset of A and that A is an affine space modeled on Ω1(X, ad P) so that

Tω(A) Ω1(X, ad P)

for each ω ∈ A. Now, each of the spaces Ωp(X, ad P) has a natural L2-inner product arising from the metricg on X, the corresponding Hodge star operator ∗ and an ad-invariant inner product ( , ) on the Lie algebrasu(2). Taking (A, B) = −tr (AB) for all A, B ∈ su(2), this is given by

〈α, β〉p = −

∫X

tr (α ∧∗ β).

In particular, this is true for Tω(A) and this defines a metric on A. Since ( , ) is ad-invariant, 〈 , 〉 is invariantunder the action of G (pointwise conjugation by elements of P×Ad SU(2)), so G acts by isometries on A. This

59

metric, in turn, determines a connection on G → A→ A/G = B whose horizontal spaces are the orthogonalcomplements to the gauge orbits. Using the Hodge Decomposition Theorem for elliptic complexes one canshow that, for every ω ∈ A,

Tω(A) Tω(ω · G) ⊕ Kernel (δω) = Image (dω) ⊕ Kernel (δω),

where dω : Ω0(X, ad P)→ Ω1(X, ad P) is the covariant exterior derivative and δω : Ω1(X, ad P)→ Ω0(X, ad P)is its formal adjoint with respect to the inner products we have just introduced (see pages 73-74 of [Nab2]).

Now we will begin to determine the appropriate analogue for each of the terms in the exponent of theAtiyah-Jeffrey formula (51). First consider − 1

2‖S (u)‖2. This is a function defined on A so we will write ωrather than u. The norm is computed from the inner product 〈 , 〉2 on Ω2(X, ad P) introduced above. Thus,

−12‖S (ω)‖2 = −

12‖F+

ω‖2 =

12

∫X

tr(F+ω ∧

∗F+ω

)=

12

∫X

tr(F+ω ∧ F+

ω

)Using the orthogonality of the Hodge decomposition one finds that this can be written

−12‖S (ω)‖2 =

14

∫X

tr(Fω ∧

∗Fω)

+14

∫X

tr(Fω ∧ Fω

)(52)

The first term is of the typical Yang-Mills variety for a classical gauge theory, whereas the second termWitten [Witt2] calls a topological term since it is, up to a constant, the Chern number of the underlyingSU(2)-bundle.

Remark 1.19. Witten [Witt2] employs the notation more common in physics whereby everything is writtenin such a way as to appear local. We will not attempt to translate all that we do into this language, butwill illustrate with (52). Let Ta be an orthonormal basis for su(2) relative to (A, B) = −tr (AB). WriteFω = 1

2 Fαβdxα ∧ dxβ, where Fαβ = FaαβTa and ∗Fω = 1

2 Fαβdxα ∧ dxβ, where Fαβ = FaαβTa. Raise indices

with the metric g to get Fαβ = gαα′

gββ′

Fα′β′ and Fαβ = gαα′

gββ′

Fα′β′ . Writing the volume element of g asd4x√

g one finds that 14 tr

(Fω∧

∗Fω)

= 14 tr (FαβFαβ)

√g d4x and 1

4 tr(Fω∧Fω

)= 1

4 tr (FαβFαβ)√

g d4x. Thus,

−12‖S (ω)‖2 =

∫X

d4x√

g tr( 1

4FαβFαβ +

14

FαβFαβ)

These are the first two terms in the Witten’s TQFT Lagrangian (see (2.13) and (2.41) of [Witt2]).

To proceed we must sort out the appropriate analogues, in the Donaldson theory context, of the mapsC, C∗, and R. At each point ω in the principal bundle space A, Cω is the map from the Lie algebra ofG, which can be identified with Ω0(X, ad P), to the tangent space Tω(A) Ω1(X, ad P), defined, for eachζ ∈ Ω0(X, ad P) by

Cω(ζ) =ddt

(ω · exp (tζ))∣∣∣t=0.

Computing this locally by expanding the exponential gives the covariant exterior derivative of ζ with respectto ω, that is,

Cω = dω : Ω0(X, ad P)→ Ω1(X, ad P). (53)

60

Consequently, C∗ω is the formal adjoint

C∗ω = δω : Ω1(X, ad P)→ Ω0(X, ad P) (54)

of dω relative to the inner products on the spaces of ad P-valued forms introduced above and

Rω = C∗ω Cω = ∆0ω : Ω0(X, ad P)→ Ω0(X, ad P) (55)

is the scalar Laplacian corresponding to ω.With this information in hand we consider the term −i(ζ,Rωλ) in (51). Both ζ and λ are in the Lie algebra

of the gauge group so we introduce two “bosonic” fields

ζ, λ ∈ Ω0(X, ad P)

and interpret ( , ) as the natural inner product 〈 , 〉0 on Ω0(X, ad P).

Remark 1.20. We apply the adjectives “bosonic” and “fermionic” to the fields we introduce only because ofthe type of integral these variables correspond to in (51). The terms have physical significance in QFT, butwe do not claim to have justified any such physical connotations.

Thus, the term −i(ζ,Rωλ) is interpreted as

−i(ζ,Rωλ) = −i〈ζ,∆0ωλ〉0 = i

∫X

tr (ζ ∧ ∗(∆0ωλ)) = i

∫X

tr (∗(ζ∆0ωλ)). (56)

Remark 1.21. Atiyah and Jeffrey [AJ] point out that, to conform with Witten’s notation in [Witt2], ζ andλ must be replaced by iζ and 1

2λ, respectively. In the notation used in physics the corresponding term in[Witt2] is ∫

Xd4x√

g tr( 1

2ζ DαDαλ

).

Next we consider the term (C∗ω χ, η) in (51). Since C∗ω maps Ω1(X, ad P) to Ω0(X, ad P) we will need twofermionic fields

η ∈ Ω0(X, ad P) and χ ∈ Ω1(X, ad P)

and, as above, ( , ) = 〈 , 〉0. Thus, we find that

(C∗ω χ, η) = 〈δω χ, η〉0 = 〈 χ, dωη〉1 = −

∫X

tr ( χ ∧ ∗dωη). (57)

The fermionic variable ψ in the Mathai-Quillen formalism arises from the odd generators of the exterioralgebra of the fiber V in the vector bundle. In our case this fiber is Ω2

+(X, ad P) so we introduce a fermionicfield

ψ ∈ Ω2+(X, ad P).

61

Now we consider the term iψT dSω( χ) in (51). S = F+ : A → Ω2+(X, ad P) is the self-dual curvature map.

The derivative of this map at ω ∈ A can be identified with the self-dual projection of the covariant exteriorderivative associated with ω which we will denote

dω+ : Ω1(X, ad P)→ Ω2+(X, ad P).

Thus, dSω( χ) = dω+ χ. We will interpret finite-dimensional expressions such as

AT B = (A1 · · · Ar)

B1

...

Br

= A1B1 + · · · ArBr

in terms of the appropriate field-theoretic inner products. Thus, we obtain, using the self-duality of ψ andthe orthogonality of the Hodge decomposition,

iψT dSω( χ) = i〈ψ, dω+ χ〉2 = i〈ψ, dω χ〉2 = i〈dω χ, ψ〉2 = −i∫

Xtr (dω χ ∧ ψ). (58)

Next consider the term 12ψ

T (ρ∗(ζ))ψ. We know that ζ ∈ Ω0(X, ad P) and ψ ∈ Ω2+(X, ad P). In the Mathai-

Quillen formalism, ρ corresponds to the action of G on V that gives rise to the associated vector bundle. Inour case, G, regarded as sections of the nonlinear adjoint bundle, acts on Ω2

+(X, ad P) pointwise by conjuga-tion. The infinitesimal action is therefore bracket

ρ∗(ζ)ψ = [ζ, ψ]

so 12ψ

T (ρ∗(ζ))ψ is interpreted as the inner product

12ψT (ρ∗(ζ))ψ =

12〈ψ, [ζ, ψ]〉2 =

12〈[ζ, ψ], ψ〉2 = −

12

∫X

tr ([ζ, ψ] ∧ ψ).

Now, any ad-invariant inner product ( , ) on any Lie algebra satisfies (x, [y, z]) = ([x, y], z) so we can writethis term as

12ψT (ρ∗(ζ))ψ = −

12

∫X

tr ( ζ [ψ, ψ] ) (59)

The only remaining term is i (dC∗ω( χ), λ) and this requires a bit more work. C∗ is a 1-form on A withvalues in Ω0(X, ad P). We compute dC∗ at ω ∈ A as follows. Fix χ1, χ2 ∈ TωA Ω1(X, ad P). Since A isan affine space and A is open in A, we can regard χ1 and χ2 as constant vector fields on A. Thus,

dC∗ω( χ1, χ2) = χ1(C∗χ2) − χ2(C∗χ1) −C∗([ χ1, χ2]) = χ1(C∗χ2) − χ2(C∗χ1),

where C∗χ j is the function θ 7→ C∗θ χ j on A for j = 1, 2 and [ χ1, χ2] = 0 because the vector fields areconstant. Now,

( χ1(C∗χ2))(ω) = χ1(ω)(C∗χ2) = χ1(C∗χ2) =ddt

C∗ω+tχ1( χ2)

∣∣∣t=0 =

ddtδω+tχ1( χ2)

∣∣∣t=0

Notice that, for any λ ∈ Ω0(X, ad P),

dω+tχ1(λ) = dωλ + t [ χ1, λ] = dωλ + tBχ1(λ)

where Bχ1 : Ω0(X, ad P)→ Ω1(X, ad P) is given by Bχ1(λ) = [ χ1, λ]. Thus,

δω+tχ1( χ2) = δωχ2 + tB∗χ1( χ2),

62

where B∗χ1: Ω1(X, ad P)→ Ω0(X, ad P) is the adjoint of Bχ1 . We claim that

B∗χ1( χ2) = −∗[ χ1,

∗χ2].

Indeed, for any λ ∈ Ω0(X, ad P),

〈 Bχ1(λ), χ2 〉1 = 〈 [ χ1, λ], χ2 〉1 = −

∫X

tr ( [χ1, λ] ∧ ∗χ2) =

∫X

tr ( [λ, χ1] ∧ ∗χ2)

=

∫X

tr (λ ∧ [ χ1,∗χ2]) =

∫X

tr (λ ∧ ∗∗[ χ1,∗χ2]) = −〈 λ, ∗[ χ1,

∗χ2] 〉0

= 〈 λ,−∗[ χ1,∗χ2] 〉0

which shows that −∗[ χ1,∗χ2] = B∗χ1

( χ2). Thus,

δω+tχ1( χ2) = δωχ2 − t ∗[ χ1,∗χ2 ]

so computing the t-derivative at t = 0 gives

( χ1(C∗χ2))(ω) = −∗[ χ1,∗χ2 ]

Interchanging χ1 and χ2 gives

( χ2(C∗χ1))(ω) = −∗[ χ2,∗χ1 ] = ∗[ χ1,

∗χ2 ]

Notice that these are independent of ω so χ1(C∗χ2) = −∗[ χ1,∗χ2 ] and χ2(C∗χ1) = ∗[ χ1,

∗χ2 ]. We concludethat

dC∗ω( χ1, χ2) = −2 ∗[ χ1,∗χ2 ]

for any ω ∈ A. Putting all of this together we find that

i ( dC∗ω( χ1, χ2), λ ) = 2i∫

Xtr (λ [ χ1,

∗χ2] ). (60)

Using the metric on P we identify vector fields with 1-forms and these, in turn, with elements of the exterioralgebra and thereby arrive at the following identification of the final term in (51).

i (dC∗ω( χ), λ) = 2i∫

Xtr (λ [ χ, ∗χ]) (61)

With this we have identified all of the terms in the exponent of (51). Each is the integral over X of a trace sowe can collect them altogether into∫

Xtr

( 14

Fω ∧∗Fω +

14

Fω ∧ Fω −12ζ [ψ, ψ] − i dωχ ∧ ψ + 2i λ [ χ, ∗χ] + i ∗(ζ ∆0

ωλ) − χ ∧ ∗dωη)

(62)

Now we will introduce some of the terminology used in physics. Each fixed choice of the three bosonic(ω ∈ A, ζ, λ ∈ Ω0(X, ad P)) and three fermionic (η ∈ Ω0(X, ad P), χ ∈ Ω1(X, ad P), ψ ∈ Ω2

+(X, ad P)) fieldswill be called a field configuration and will be denoted

Φ = ( χ, η, ψ, λ, ζ, ω).

We will also write

DΦ = DχDχDψ dλ dζ dω

63

Changing all of the signs in (62) we define the Donaldson-Witten action functional S DW[Φ] on the space offield configurations by

S DW[Φ] =

∫X

tr(−

14

Fω ∧∗Fω −

14

Fω ∧ Fω +12ζ [ψ, ψ] + i dωχ ∧ ψ − 2i λ [ χ, ∗χ] − i ∗(ζ ∆0

ωλ) + χ ∧ ∗dωη)

Now, the integral in (51) can be written ∫e−S DW [Φ] DΦ. (63)

Whereas S DW[Φ] is a perfectly well-defined mathematical object, (63) can only be thought of as a Feynmanintegral over the space of field configurations.

We will conclude this section with a few remarks. Notice that in (63) we have omitted any constantsthat might correspond to (2π)−n(2π)−k in (51) since both n and k are infinite in our present circumstances.Physicists generally include in the exponent in (63), or directly in the action S DW[Φ], a factor of the form1/e2, where e is a so-called coupling constant. Mathematically, one can view the inclusion of such a factor inS DW[Φ] as simply a different choice of invariant inner product on the Lie algebra su(2), which is determinedonly up to a positive constant multiple. Classically, one can rescale and give this factor any convenient value.However, upon quantization with the Feynman integral (63) the different values of the coupling constant giverise to an entire 1-parameter family of quantum field theories and the computability of the theory generally,but not always depends on this value. Since this will be relevant to some other comments we would like tomake we will record the following alternative to (63).

ZDW =

∫e−S DW [Φ]/e2

DΦ (64)

The integral (64) represents the partition function of the quantum field theory constructed by Witten in[Witt2]. It is (formally) a function of the coupling constant e. Of course, Witten arrived at the action, andtherefore the partition function, by quite a different route than the one we have followed. We began withthe 0-dimensional Donaldson invariant, regarded it as an “Euler number” and massaged the Mathai-Quillenintegral representation of the Euler number until it could be formally applied in the infinite-dimensionalcontext of Donaldson theory. Witten’s arguments in [Witt2] were physical, but the objective was to describea quantum field theory in which the Donaldson invariants appeared as expectation values of certain ob-servables and, in particular, the 0-dimensional invariant appeared as the partition function (the observableswhose expectation values give rise to other Donaldson invariants are discussed in Section 3 of [Witt2]). Hechose the field content Φ and the action in order to ensure that S DW[Φ] exhibited certain symmetries which,together with various formal manipulations of Feynman integrals, implied that the theory was independentof both the Riemannian metric g and the coupling constant e in the sense that the infinitesimal variation ofthe partition function with respect to either g or e is zero.

The g-independence has led physicists to refer to such field theories as topological quantum field theoriesand to the expectation values we have mentioned as topological invariants. One should be aware, however,that this conflicts rather strikingly with the terminology that one would encounter in mathematics. Indeed,we have seen in Section 1.4 that the Donaldson invariants are only independent of a generic choice ofRiemannian metric g and even this is true only if b+

2 (X) > 1. Moreover, even granting this, the Donaldsoninvariants are invariants, not of the topology of X, but rather of its smooth structure and this is quite adifferent thing.

64

The e-independence of the theory is particularly significant since one is then free to compute in either theweak coupling (e→ 0) or strong coupling (e→ ∞) regime. For small values of e physicists employ variousapproximation techniques such as the semi-classical approximation which, again because of the symmetriesof the action S DW[Φ], Witten was able to show (formally) must be exact. This fortuitous circumstance is ananalogue of a well-known finite-dimensional theorem on the exactness of the stationary phase approximationdue to Duistermaat and Heckman [DH]. This result is most properly understood within the context ofequivariant cohomology and the localization of certain integrals of equivariant differential forms to the fixedpoint set of the group action (see Section 7.2 of [BGV] or Section 6 of [Nab3]). For the particular casewe have under consideration, the partition function, being gauge invariant, descends to the moduli space offield configurations and localizes to a sum over the 0-dimensional moduli space of ASD connections thusyielding the 0-dimensional Donaldson invariant.

Remark 1.22. This localization is, intuitively, not unlike the Residue Theorem which localizes a contourintegral around a closed path to a sum of contributions from the singularities of the integrand inside (althoughthere are no symmetry considerations in this case). Much closer analogues are the Equivariant LocalizationTheorems of Berline and Vergne which extract and generalize the essential content of the Duistermaat-Heckman Theorem (see Section 7.2 of [BGV] or Section 6 of [Nab3]).

There is only one question remaining. Witten found the Donaldson invariants by studying his TQFT inthe weak coupling regime (e → 0) by performing a semi-classical approximation and arguing that it mustbe exact. In the strong coupling regime such arguments are unavailable and the theory can be expected tolook quite different, but, because of the e-independence, it is physically equivalent and so, one might hope,mathematically equivalent in some sense as well. What can be learned about Donaldson theory in the e→ ∞limit of Witten’s TQFT? This is the question to which we turn in the next section.

1.6. The Seiberg-Witten Invariants for Spinc-Manifolds. Remarkable as it is that the subtle differential-topological invariants of Donaldson can be recast in the quantum field-theoretic terms described in theprevious section, the most remarkable part of the story is yet to come. Witten unearthed the Donaldsoninvariants in 1988 by analyzing his TQFT perturbatively in the weak coupling regime. He was, of course,well-aware of the fact that the e-independence of the theory suggested the possibility of an alternative de-scription of the invariants that lay buried in an analysis of the strong coupling regime. However, in 1988,there were no tools available for studying the strongly coupled, nonperturbative behavior of quantum fieldtheories. This all changed in 1994 when Seiberg and Witten [SW] developed techniques for performingexact calculations in strongly coupled N = 2 supersymmetric Yang-Mills theories, of which Witten’s 1988TQFT was an example. How they managed to do this is a story that lies in very deep waters indeed and,alas, we cannot presume to offer even an exegesis (one might consult Section 5 of [Witt3], or Lectures 17,18, and 19, pages 1351-1401, of [Delig2]). The end result, however, when applied to Witten’s TQFT gaverise to a new set of 4-manifold invariants which, from the perspective of physics at least, were “dual” tothe Donaldson invariants and so, conjectured Witten, must contain the same topological information. Thesenew invariants arose in much the same way as the Donaldson invariants from a moduli space of solutions tocertain partial differential equations. These Seiberg-Witten equations, however, were vastly simpler than the

65

anti-self-dual equations so Witten’s conjecture was received with considerable skepticism. Very soon, how-ever, it became clear that the skepticism was unwarranted and that the new and much simpler Seiberg-Witteninvariants provided a much more felicitous tool for the study of smooth 4-manifolds. The story of how theWitten conjecture was sprung on the mathematical community and the pandemonium that ensued has beentold many times, but is best heard from one who was there. For this as well as a lovely introduction to whatis to come and a very pleasant afternoon’s entertainment we heartily recommend [Taub3]. We will nowsimply get on with the business of describing the equations and the invariants (references for this materialare [Mor1], [Nicol], [Sal], Appendix A of [Nab5], and Section 7 of [Nab3]).

1.6.1. Real and Complex Clifford Algebras ofR4. The price one must pay for the eventual simplicity of theSeiberg-Witten invariants is an initial expenditure of time and energy to assemble the rather considerablealgebraic machinery required to simply write down the relevant equations. This is most conveniently phrasedin the language of Clifford algebras so we will begin with a brief synopsis of just what we need and no more(the standard reference for Clifford algebras and spin geometry is [LM]).

Any finite-dimensional, real vector space V with an inner product 〈 , 〉 has a Clifford algebra Cl (V) thatcan be realized concretely as follows. If e1, . . . , en is any orthonormal basis for V , then Cl (V) is the real,associative algebra with unit 1 generated by e1, . . . , en and subject to the relations

eie j + e jei = −2〈ei, e j〉 1, i, j = 1, . . . , n. (65)

Since we will need only the Clifford algebra Cl (4) = Cl (R4) of R4 with its usual positive definite innerproduct, we will construct an explicit matrix model. The procedure will be to identify R4 with a real linearsubspace of a matrix algebra, find an orthonormal basis for this copy of R4 satisfying the defining relations(65), where the product is matrix multiplication and 1 is the identity matrix, and then form the subalgebra itgenerates. First, identify R4 with the algebra H of quaternions q = q1 + q2i + q3j + q4k. Next we embedH in the real, associative algebra H2×2 of 2 × 2 quaternionic matrices with the product given by matrixmultiplication.

H2×2 =

q11 q12

q21 q22

: qi j ∈ H, i, j = 1, 2

Specifically, we will identifyH with the real, linear subspace ofH2×2 consisting of all elements of the form

x =

0 q−q 0

, q ∈ H. (66)

Notice that det x = ‖q‖2 so defining a norm on the set of all such x by ‖x‖2 = det x and an inner product bypolarization 〈x, y〉 = 1

4 ( ‖x + y‖2 − ‖x − y‖2 ), we find that the real linear subspace ofH2×2 consisting of all xof the form (66) is isomorphic to R4 as an inner product space. One easily checks that

e1 =

0 1−1 0

, e2 =

0 ii 0

, e3 =

0 jj 0

, e4 =

0 kk 0

(67)

form an orthonormal basis and satisfy (65). The real subalgebra of H2×2 generated by e1, e2, e3, e4 isthe real Clifford algebra of R4 and will be denoted Cl (4). Writing out the matrix products of these basis

66

elements and using (65) to eliminate linear dependencies gives the following basis for Cl (4).

e0 =

1 00 1

= 1,

e1 =

0 1−1 0

, e2 =

0 ii 0

, e3 =

0 jj 0

, e4 =

0 kk 0

,e1e2 =

i 00 −i

, e1e3 =

j 00 −j

, e1e4 =

k 00 −k

,e2e3 =

k 00 k

, e2e4 =

−j 00 −j

, e3e4 =

i 00 i

, (68)

e1e2e3 =

0 k−k 0

, e1e2e4 =

0 −jj 0

,e1e3e4 =

0 i−i 0

, e2e3e4 =

0 −1−1 0

,e1e2e3e4 =

−1 00 1

Thus, dim Cl (4) = 16. Since the real dimension ofH2×2 is also 16, we conclude that

Cl (4) = H2×2.

Notice that this basis provides Cl (4) with a natural Z2-grading

Cl (4) = Cl0(4) ⊕Cl1(4),

where Cl0(4) is spanned by e0, e1e2, e1e3, e1e4, e2e3, e2e4, e1e2e3e4 and Cl1(4) is spanned bye1, e2, e3, e4, e1e2e3, e1e2e4, e1e3e4, e2e3e4. The elements of Cl0(4) are said to be even, while those ofCl1(4) are odd. The decomposition corresponds to splitting an element of Cl (4) into its diagonal and anti-diagonal parts. Since

(Cli(4)) (Clj(4)) ⊆ Cli+j(4), i, j ∈ Z2,

Cl (4) is a Z2-graded algebra, that is, a superalgebra.

Lemma 1.21. The center Z(Cl (4)) of Cl (4) is Span e0 R.

Proof. Span e0 ⊆ Z(Cl (4)) is clear. To complete the proof it will suffice to show that every

eI = ei1 · · · eik , 1 ≤ k ≤ 4, 1 ≤ i1 < · · · < ik ≤ 4

fails to commute with something in Cl (4). For k = 1 this is clear since eie j = −e jei for i , j. For k = 4,eI = e1e2e3e4 so e1eI = (e1e1)e2e3e4 = −e2e3e4, whereas eIe1 = (−1)3(e1e1)e2e3e4 = e2e3e4. Now suppose1 < k < 4. Then eIei1 = (−1)k−1ei1eI , but if el is not among ei1 , . . . , eik , then eIel = (−1)keleI so eI cannotcommute with both ei1 and el.

67

Notice that if x, y ∈ R4 ⊆ Cl (4), then it follows from (65) and the bilinearity of the multiplication inCl (4) that

xy + yx = −2〈x, y〉 1.

If x ∈ R4 ⊆ Cl (4) and ‖x‖ = 1, then 〈x, x〉 = 1 and xx + xx = −2〈x, x〉 1 imply xx = −1 so x is invertiblein Cl (4) and x−1 = −x. We will denote by Cl×(4) the multiplicative group of units (invertible elements) inCl (4) and by Pin(4) the subgroup of Cl×(4) generated by the set of all x ∈ R4 with ‖x‖ = 1, that is, by theunit sphere S3 ⊆ R4 ⊆ Cl (4). Now, notice that an x of the form (66) has ‖x‖ = 1 if and only if q is a unitquaternion, that is, if and only if q ∈ Sp(1) and we have just seen that the set of all such x is closed underinversion (x−1 = −x). Consequently, Pin(4) is just the set of products of such elements. The even elementsof Pin(4) are just its diagonal elements and these form a subgroup denoted

Spin(4) = Pin(4) ∩Cl0(4) =

u1 00 u2

: u1, u2 ∈ Sp(1) Sp(1) × Sp(1). (69)

The topology and differentiable structures that Spin (4) inherits from H2×2 H4 R16 are the same as theproduct structures from Sp(1) so Spin (4) is a compact, simply connected Lie group. Since the Lie algebraof Sp(1) can be identified with the pure imaginary quaternions, the Lie algebra of Spin(4) can be identifiedwith

spin(4) =

Q1 00 Q2

: Q1,Q2 ∈ ImH ImH ⊕ ImH. (70)

For any u ∈ Cl×(4) we define an isomorphism adu : Cl (4)→ Cl (4) by

adu(p) = upu−1.

Each adu clearly preserves the grading of Cl (4). For proofs of the following results, leading up to Theorem1.22, one can consult Theorem 7.3 of [Nab3] or Theorem A.2.3 of [Nab5]. If u ∈ Pin(4) one can showthat adu carries R4 onto itself and is, in fact, an orthogonal transformation of R4. If u ∈ Spin(4), then thedeterminant of adu is 1 so adu is a rotation of R4.

u ∈ Spin(4)⇒ adu ∈ SO (R4).

The orthonormal basis e1, e2, e3, e4 for R4 identifies SO(R4) with SO(4) so we have a map

Spin : Spin(4)→ SO(R4) SO(4)

Spin(u) = adu

This is called the spinor map. It is a surjective Lie group homomorphism with kernel Z2 so we arrive at thefollowing.

Theorem 1.22. Spin(4) is the universal double cover of SO(4).

Remark 1.23. In Section 1.6.2 we will see how globalizing these constructions leads to the notion of aSpin structure on a manifold and how this additional structure gives rise to the existence of spinor fieldson the manifold. There is, however, an obstruction to the existence of a Spin structure called the secondStiefel-Whitney class (see Section 6.5 of [Nab5]) and many interesting 4-manifolds (CP2, for example) fail

68

to admit such a structure. Since a spinor field is an essential ingredient of Seiberg-Witten theory and sincewe would like the theory to be applicable to as large a class of 4-manifolds as possible it will be necessaryto generalize these ideas. As it happens there is a very natural generalization obtained by complexifying ourprevious algebraic considerations and this is what we will turn to next.

In order to define complex analogues of the algebraic objects we have introduced we will embed Cl (4)into a complex algebra of matrices and form the complex subalgebra it generates. The basic tool we useis the usual matrix model of the quaternions. Specifically, we consider the map γ : H → C2×2 from thequaternions to the 2 × 2 complex matrices given by

γ(q) = γ(α + βj) =

α β

−β α

, (71)

where we have written q = q1 + q2i + q3j + q4k = (q1 + q2i) + (q3 + q4i)j = α + βj. One easily checks thatγ is real linear, injective, preserves products, carries q to γ(q)

Tand satisfies det (γ(q)) = ‖q‖2 so that we can

identifyH with the set of all 2 × 2 complex matrices of the form α β

−β α

. Specifically, if we let

γ(1) =

1 00 1

= 1, γ(i) =

i 00 −i

= I,

(72)

γ(j) =

0 1−1 0

= J, γ(k) =

0 ii 0

= K

then we identify q = q1 + q2i + q3j + q4k with q11 + q2I + q3J + q4K. Next we identify Cl (4) = H2×2 witha subset of C4×4. Define Γ : H2×2 → C4×4 by

Γ

q11 q12

q21 q22

=

γ(q11) γ(q12)γ(q21) γ(q22)

. (73)

This map Γ is also real linear, injective and preserves products so we can identify Cl (4) with its image

Cl (4) = Γ(H2×2) ⊆ C4×4.

The restriction of Γ to R4 ⊆ Cl (4) is

x =

0 q−q 0

7→ Γ(x) =

0 γ(q)

−γ(q)T

0

.Since det Γ(x) = det x = ‖x‖2 = ‖q‖2 we can define an inner product on the copy Γ(R4) of R4 via polariza-tion from ‖Γ(x)‖2 = det Γ(x) and then γ |R4 is an isometry. With this we fully identify R4 with Γ(R4) andobtain an orthonormal basis E j = Γ(e j), j = 1, 2, 3, 4, satisfying

EiE j + E jEi = −2〈Ei, E j〉1, i, j = 1, 2, 3, 4.

69

In this context, Cl (4) is the real subalgebra of C4×4 generated by E1, E2, E3, E4 and a basis for Cl (4) is asin (68), but with everything capitalized and 1 changed to 1. Moreover, under γ, Sp(1) is mapped to SU(2) so

Spin(4) =

U1 00 U2

: U1,U2 ∈ SU(2) SU(2) × SU(2) (74)

and

spin(4) =

A1 00 A2

: A1, A2 ∈ su(2) su(2) ⊕ su(2). (75)

Now we regard C4×4 as a complex algebra. The complex linear subspace spanned by E1, E2, E3, E4 is acopy of C4. We define the complexified Clifford algebra Cl (4) ⊗ C of R4 to be the complex subalgebra ofC4×4 generated by E1, E2, E3, E4, that is, by Cl (4). A basis for Cl (4) ⊗ C over C is given by (68) with allof the ei capitalized and 1 changed to 1. Thus, the dimension of Cl (4)⊗C over C is 16 so Cl (4)⊗C C4×4.We will let

SC = C4

with its usual Hermitian inner product (〈z,w〉 = z1w1 + z2w2 + z3w3 + z4w4) and use the basis E1, E2, E3, E4

for C4 to identify

Cl (4) ⊗ C = EndC(SC).

Thus, the elements of Cl (4) ⊗ C (and therefore also Cl (4),R4 and Spin(4)) act as endomorphisms of SC.This action is called Clifford multiplication and will be indicated with a dot · . Restricting the Clifford actionto Spin(4) ⊆ Cl (4) ⊗ C gives a group representation of Spin(4) on SC that we will denote

∆C : Spin(4)→ GLC(SC)

and call the complex spin representation This is not irreducible. Indeed, if we write

SC = S +C ⊕ S −C

z1

z2

z3

z4

=

z1

z2

00

+

00z3

z4

,then Clifford multiplication by elements of Spin(4) ⊆ Cl0(4)⊗C, because they are block diagonal, preservesS +C

and S −C

, whereas Clifford multiplication by Cl1(4)⊗C interchanges S +C

and S −C

. In particular, ∆C resolvesinto a direct sum

∆C = ∆+C ⊕ ∆−C,

where

∆±C : Spin(4)→ SU(S ±C).

These are called the positive and negative complex spin representations. Notice, however, that Cliffordmultiplication by elements of R4 ⊆ Cl (4) ⊗ C, being odd, interchanges S +

Cand S −

C.

70

Recall that Spin(4) is the set of all even elements in the subgroup of multiplicative units in Cl (4) generatedby the unit sphere in R4 ⊆ Cl (4). For the complex analogue we add to the generators the unit circle in C.More precisely, we identify U(1) with the subset

U(1) =eθ i1 : θ ∈ R

of Cl (4) ⊗ C (often dropping the 1 and thinking of eθ i as an element of Cl (4) ⊗ C). Then

Spinc(4)

is defined to be the subgroup of the group of multiplicative units in Cl (4) ⊗ C generated by Spin(4) andU(1). Since U(1) is in the center of Cl (4) ⊗ C we have

Spinc(4) =eθ iu : θ ∈ R, u ∈ Spin(4)

=

eθ i

U1 00 U2

: θ ∈ R,U1,U2 ∈ SU(2)

=

U+ 00 U−

: U+,U− ∈ U(2), det U+ = det U−.

From Lemma 1.21 and the fact that U(1) is Abelian, it follows that the center of Spinc(4) is

Z(Spinc(4)) = U(1).

There is another useful way of looking at Spinc(4). The mapping

Spin(4) × U(1)→ Spinc(4)

(u, eθ i)→ eθ iu

is a surjective homomorphism. Its kernel is the set of all (α, α−1) ∈ Spin(4) ∩ U(1). But Spin(4) intersectsthe scalars only in ±1 so the kernel is Z2 = ±(1, 1). Thus,

Spinc(4) Spin(4) × U(1)/Z2.

Now we will define a few mappings that will be required in the next section. The first is a map

δ : Spinc(4)→ U(1) (76)

defined as follows. For ξ =

U+ 00 U−

=

eθ iU1 00 eθ iU2

∈ Spinc(4),

δ(ξ) = det U+ = det U− = e2θ i.

Then δ is a surjective homomorphism with kernel Spin(4). Next define

π : Spinc(4)→ SO(4) (77)

as follows. For each u ∈ Spin(4), adu : R4 → R4 is a rotation of R4 defined by adu(x) = uxu−1 for everyx ∈ R4. Notice, however, that if ξ = eθ iu ∈ Spinc(4), then ξxξ−1 = uxu−1 = adu(x) so we can extend theadjoint action of Spin(4) on R4 to Spinc(4) by setting adξ = adu. We define

π(ξ) = adξ = adu = Spin(u),

71

where ξ = eθ iu for some θ ∈ R and some u ∈ Spin(4). Putting δ and π together we obtain

Spinc :Spinc(4)→ SO(4) × U(1)

Spinc(ξ) = Spinc(eθ iu) = (π(ξ), δ(ξ)) = (Spin(u), e2θ i).

Spinc is a surjective homomorphism with kernel Z2 = ±1 so Spinc(4) is a double cover of SO(4)×U(1) (notthe universal cover, however, since its fundamental group is Z). The Lie algebra is therefore so(4) ⊕ u(1) spin(4) ⊕ u(1) which can be identified with the following subset of Cl (4) ⊗ C.

spinc(4) =

A1 00 A2

+ t i1 00 1

: t ∈ R, A1, A2 ∈ su(2)

In particular, spinc(4) also acts by Clifford multiplication on SC, preserving both S ±C

.The complex spin representation ∆C : Spin(4) → GLC(SC) satisfies ∆C(−1) = −1 and so it extends to a

representation

∆C : Spinc(4)→ GLC(SC)

of Spinc(4). Since the elements of Spinc(4) are all block diagonal, this representation also splits into

∆C = ∆+C ⊕ ∆−C,

where

∆±C : Spinc(4)→ U(S ±C).

One of the Seiberg-Witten equations relates the self-dual part of the curvature of a U(1)-connection toa certain trace-free endomorphism of a positive spinor bundle. The last of our algebraic preliminaries de-scribes the relationship between 2-forms and endomorphisms. Note first that there is a linear isomorphismfrom the complex-valued 2-forms Ω2(R4;C) on R4 into Cl0(4) ⊗ C. Indeed, if e1, e2, e3, e4 is the basis(67) for R4 and e1, e2, e3, e4 is its dual basis and if E1, E2, E3, E4 is the image of e1, e2, e3, e4 under Γ,then we define

ρ : Ω2(R4;C)→ Cl0(4) ⊗ C (78)

by

ρ(η) = ρ(∑

i< j

ηi jei ∧ e j)

=∑i< j

ηi jEiE j

Notice that, although ρ is clearly a linear isomorphism, it is not multiplicative. For example, e1 ∧ e1 = 0,but E1E1 = −1. There is, of course, an analogous map in any rank. Also notice that, if η real-valued(respectively, ImC-valued), then ρ(η) is skew-Hermitian (respectively, Hermitian). For example, if η isreal-valued,

ρ(η)T

=∑i< j

ηi jEiE jT

=∑i< j

ηi jETj E

Ti =

∑i< j

ηi j(−E j)(−Ei) =∑i< j

ηi jE jEi =∑i< j

ηi j(−EiE j) = −ρ(η).

72

Moreover, in the definition of η, e1, e2, e3, e4 can be replaced by any oriented, orthonormal basis providedE1, E2, E3, E4 is replaced by the image of the basis under Γ. Being even, that is, block diagonal, any ρ(η)preserves the subspaces S ±

Cof SC so we obtain endomorphisms of S ±

Cby setting

ρ±(η) = ρ(η) | S ±C.

For example,

ρ+(η) = (η12 + η34) I + (η13 + η42) J + (η14 + η23) K.

Thus, we have two maps

ρ± : Ω2(R4;C)→ EndC(S ±C).

The usual orientation and inner product onR4 determine a Hodge star operator ∗ and therefore a decom-position

Ω2(R4;C) = Ω2+(R4;C) ⊕Ω2

−(R4;C)

of Ω2(R4;C) into self-dual and anti-self-dual 2-forms.

Lemma 1.23. ρ± |Ω2±(R4;C) is a complex-linear isomorphism onto the space End0(S ±

C) of trace-free endo-

morphisms of S ±C

.

Proof. We give the argument for ρ+ |Ω2+(R4;C). The ρ− |Ω2

−(R4;C) case is analogous. A simple computa-tion shows that

ρ(e1 ∧ e2 + e3 ∧ e4) = 2I 00 0

ρ(e1 ∧ e3 + e4 ∧ e2) = 2

J 00 0

ρ(e1 ∧ e4 + e2 ∧ e3) = 2

K 00 0

Since e1∧e2 + e3∧e4, e1∧e3 + e4∧e2, e1∧e4 + e2∧e3 spans the set of self-dual 2-forms onR4, it is clearthat ρ+ |Ω2

+(R4;C) is a complex-linear, injective map into the endomorphisms of S +C

. Because I, J, and Kare trace-free, so is everything in the image of ρ+ |Ω2

+(R4;C). Furthermore, every 2 × 2 complex, trace-freematrix is a complex linear combination of I, J, and K so ρ+ |Ω2

+(R4;C) maps onto End0(S +C

).

It follows, in particular, that the map ρ+ |Ω2+(R4;C) : Ω2

+(R4;C)→ End0(S +C

) has an inverse that we willdenote simply

σ+ : End0(S +C)→ Ω2

+(R4;C). (79)

We would like to describe the image under σ+ of a particular type of trace-free endomorphism of S +C

. Forthis we consider an arbitrary element ψ of S +

Cand suppress its two zero components to write

ψ =

ψ1

ψ2

.

73

Define an endomorphism of S +C

by the matrix

ψ ⊗ ψ∗ =

ψ1

ψ2

(ψ1ψ

2)=

|ψ1|2 ψ1ψ2

ψ1ψ2 |ψ2|2

.The trace-free part of this endomorphism is

(ψ ⊗ ψ∗)0 = ψ ⊗ ψ∗ −12

tr (ψ ⊗ ψ∗) 1 =

12 (|ψ1|2 − |ψ2|2) ψ1ψ

2

ψ1ψ2 1

2 (|ψ2|2 − |ψ1|2)

(80)

The following expression for σ+((ψ ⊗ ψ∗)0) can be checked by simply applying ρ+ to the right-hand side.

σ+((ψ ⊗ ψ∗)0) = −14

i[(|ψ1|2 − |ψ2|2) (e1 ∧ e2 + e3 ∧ e4)

− 2 Im (ψ1ψ2) (e1 ∧ e3 + e4 ∧ e2)

+ 2 Re (ψ1ψ2) (e1 ∧ e4 + e2 ∧ e3)

](81)

1.6.2. Spin and Spinc Structures. Our task now is to globalize the algebraic constructions of the precedingsection. Throughout this section and the next, X will denote an oriented, smooth 4-manifold.

Remark 1.24. Compactness, connectedness, and simple connectivity are generally not required for the con-structions we wish to describe in this section and the next, but only for one of the results we wish to state.The situation changes in Sections 1.6.4 and 1.6.5, where we will impose these conditions (and more) on allof the oriented, smooth 4-manifolds we consider.

Choose a Riemannian metric g on X. The orientation and the Riemannian metric determine an oriented,orthonormal frame bundle

SO(4) → FSO(X)πSO→ X.

The fiber above each point in X is a copy of SO(4). We now know that SO(4) is double covered by Spin(4).

Spin : Spin(4)→ SO(4)

What we would like to do is piece together these double covers over the points of X and build a principalSpin(4)-bundle over X. More precisely, a Spin structure for X consists of a principal Spin(4)-bundle

Spin(4) → S (X)πS→ X

over X and a smooth map

λ : S (X)→ FSO(X)

of S (X) onto FSO(X) satisfying

πSO λ = πS (82)

and

λ(p · u) = λ(p) · Spin(u) (83)

74

for each p ∈ S (X) and every u ∈ Spin(4). Condition (82) says that the fiber π−1S (x) of πS above x ∈ X is

λ−1 of the fiber π−1SO(x) of πSO above x. On each fiber π−1

S (x) we can fix p and regard λ as a function ofu ∈ Spin(4) and (83) asserts that this is just the double cover Spin : Spin(4) → SO(4) on this fiber. In thissense, λ is just the spinor map on each fiber of πS .

S (X)↓ λ

FSO(X)

πS

↓ πSO

X

Two Spin structures

Spin(4) → S (X)πS→ X

λ : S (X)→ FSO(X)

and

Spin(4) → S ′(X)πS ′→ X

λ′ : S ′(X)→ FSO(X)

for X are said to be equivalent if there exists a bundle isomorphism f : S ′(X)→ S (X) for which λ f = λ′,that is, f ∗λ = λ′. Note that it is not enough to assume that the the Spin(4)-bundles are isomorphic (see theExample, page 200, of [Miln2]).

Unlike the frame bundle FSO(X), which exists for any oriented, Riemannian manifold, there is an ob-struction to the existence of a Spin structure. We should briefly indicate how this comes about. Let Uα beany trivializing good cover for SO(4) → FSO(X)

πSO→ X so that each intersection Uα ∩Uβ is contractible (see

Theorem 3.7 of [KN1]). Denote the transition functions by ταβ : Uα ∩Uβ → SO(4). Because Spin(4) is theuniversal cover of SO(4), each of the maps ταβ lifts to a map ταβ : Uα ∩ Uβ → Spin(4) satisfying

ταβ = Spin ταβ

(see Theorem 6.1 of [Green]). If these lifts satisfy the cocycle condition (ταβτβγ = ταγ on Uα ∩ Uβ ∩ Uγ

whenever this is nonempty), then they determine a principal Spin(4)-bundle over X that is unique up toequivalence and is easily seen to give rise to a Spin structure for X (see Proposition 5.2 of [KN1] or Theorem4.3.4 of [Nab4]). It might well happen that no such family of lifts satisfying the cocycle condition existsand, in this case, no Spin structure exists. In terms of the Cech cohomology of X with coefficients in Z2 it iseasy to isolate a class w2(X) ∈ H(X;Z2), called the second Stiefel-Whitney class, the vanishing of which isequivalent to the existence of the required lifts (see Section 6.5 of [Nab5]). Thus, an oriented, Riemannianmanifold X has a Spin structure if and only if the second Stiefel-Whitney class is trivial. If X has a Spinstructure, then it is called a Spin manifold. The following result, which describes the situation of mostinterest to us, is Corollary 2.12 of [LM].

Theorem 1.24. Let X be a compact, connected, simply connected, oriented, smooth 4-manifold. Then thesecond Stiefel-Whitney class w2(X) is trivial if and only if the intersection form QX is even.

75

Given a Spin structure on X and a representation ρ of Spin(4) on some vector space V there is an associatedvector bundle S (X) ×ρ V . Sections of such a vector bundle will be referred to as spinor fields on X. Onthe space Γ(S (X) ×ρ V) of all such sections one can define a canonical first order differential operatorD : Γ(S (X) ×ρ V) → Γ(S (X) ×ρ V) called the Dirac operator (see Chapter II, Section 5, pages 112-134, of[LM]). The study of Dirac operators on Spin manifolds has led to some of the deepest mathematics of the20th century, the most renowned example being the Atiyah-Singer Index Theorem (see Chapter III, Section13, pages 243-259, of [LM]). For our purposes, however, that is, for the Seiberg-Witten equations, we willrequire a complex analogue of Spin structures and Dirac operators and it is to this that we turn now.

A Spincstructure L on X consists of a principal Spinc(4)-bundle

Spinc(4) → S c(X)πS c→ X

over X and a smooth map of S c(X) onto FSO(X)

Λ : S c(X)→ FSO(X)

satisfying

πSO Λ = πS c

and

Λ(p · ξ) = Λ(p) · π(ξ).

The picture is essentially the same as for a Spin structure.

S c(X)↓ Λ

FSO(X)

πS c

↓ πSO

X

Two Spinc structures

Spinc(4) → S c(X)πS c→ X

Λ : S c(X)→ FSO(X)

and

Spinc(4) → (S c)′(X)π(S c)′

→ X

Λ′ : S ′(X)→ FSO(X)

for X are said to be equivalent if there exists a bundle isomorphism F : (S c)′(X) → S c(X) for whichΛ F = Λ′, that is, F∗Λ = Λ′.

The reason we work with Spinc structures rather that Spin structures is that, for oriented, Riemannian4-manifolds, they always exist (see Lemma 3.1.2, page 25, of [Mor1]). Each of the representations δ :

76

Spinc(4) → U(1), ∆C : Spinc(4) → GLC(SC), and ∆±C

: Spinc(4) → U(S ±C

) gives rise to a vector bundle

associated to Spinc(4) → S c(X)πS c→ X and we will write these as follows.

L(L) = S c(X) ×δ C

S(L) = S c(X) ×∆CSC

S±(L) = S c(X) ×∆±C

S ±C

We refer to L(L) as the determinant line bundle of L, S(L) as the spinor bundle of L, and S±(L) as thepositive and negative spinor bundles of L. The algebraic decomposition ∆C = ∆+

C⊕ ∆−

Cof ∆C gives rise to

a Whitney sum decomposition

S(L) = S+(L) ⊕ S−(L)

of S(L). Sections of S(L), S+(L), and S−(L) are referred to as spinor fields, positive spinor fields andnegative spinor fields, respectively. The spaces of such sections will be denoted Γ(S(L)), Γ(S+(L)), andΓ(S−(L)), respectively.

The determinant line bundle L(L) is, in particular, a complex line bundle and so has an associated princi-pal U(1)-bundle

U(1) → L0(L)πL0→ X

that can be described as follows. Choose a Hermitian fiber metric on L(L). Then L0(L) is the unit circlebundle of L(L), that is, the corresponding orthonormal frame bundle. One can retrieve L(L) from L0(L) asthe vector bundle associated to L0(L) by complex multiplication. L0(L) has a first Chern class c1(L0(L))and one can show that the second Stiefel-Whitney class of X is the mod 2 reduction of c1(L0(L))

w2(X) = c1(L0(L)) mod 2

and that, conversely, given a U(1)-bundle L0 over X with w2(X) = c1(L0) mod 2, there is a Spinc structure L

on X with L0(L) = L0 (see Section 1.3, pages 42-43, of [Nicol]).Recall that Spinc(4) double covers Spin(4)×U(1) by the map Spinc. It follows that S c(X) double covers the

fiber product FSO(X)·× L0(L) (this is just that part of the product bundle SO(4)×U(1) → FSO(X)×L0(L)→

X × X above the diagonal in X × X with this diagonal identified with X in the obvious way). We will use thesymbol Spinc also for this double cover.

Spinc(4) → S c(X) −→ X↓ Spinc

SO(4) × U(1) → FSO(X)·× L0(L) −→ X

Locally, the map Spinc : S c(X)→ FSO(X)·× L0(L) is given by

(x, ξ) 7→ (x,Spinc(ξ)) = (x, (π(ξ), δ(ξ)) ) = (x, (Spin(u), e2θ i) ),

where ξ = eθ iu with θ ∈ R and u ∈ Spin(4).We will also require two bundles associated to the oriented, orthonormal frame bundle of X that do not

require the existence of a Spin or Spinc structure. Notice that Spin(4), being contained in Cl×(4), acts onCl (4) by conjugation and, since (−u)p(−u)−1 = upu−1, this descends to an action of SO(4)=Spin(4)/Z2 on

77

Cl (4) which clearly preserves products since u(pq)u−1 = (upu−1)(uqu−1). The Clifford bundle is the bundleof algebras with typical fiber Cl (4) over X associated to SO(4) → FSO(X)

πSO→ X by this action. It is denoted

Cl (X) = FSO(X) ×SO(4) Cl (4).

In the same way, SO(4) acts on Cl (4) ⊗ C and one defines the complexified Clifford bundle

Cl (X) ⊗ C = FSO(X) ×SO(4) (Cl (4) ⊗ C).

These decompose into even and odd summands Cl (X) = Cl0(X) ⊕ Cl1(X) and Cl (X) ⊗ C = Cl0(X) ⊗ C ⊕Cl1(X)⊗C. Moreover, pointwise multiplication provides the spaces Γ(Cl (X)) and Γ(Cl (X)⊗C) of sectionsof these bundles with algebra structures and such sections act on sections of the spinor bundles by pointwiseClifford multiplication.

Finally, we will need to globalize the isomorphism σ+ : End0(S +C

) → Ω2+(R4;C) of (84). The map

ρ : Ω2(R4;C) → Cl0(4) ⊗ C of (78) is independent of the choice of oriented, orthonormal basis for R4

so, using local, oriented, orthonormal frame fields, that is, sections of FSO(X), it gives a map from 2-forms on X to sections of Cl0(X) ⊗ C. At each point these sections, in turn, act on the spinor bundleS(L) = S+(L) ⊕ S−(L) of any Spinc structure for X. Since this action is fiberwise, the image of a self-dual2-form on X preserves S+(L) and is, at each point, a trace-free endomorphism of S +

C. Thus, a self-dual 2-

form on X gives rise to a section of the trace-free endomorphism bundle End0(S+(L)) of the positive spinorbundle. This is an isomorphism of bundles and we will continue to write the inverse as

σ+ : Γ( End0(S+(L)) )→ Ω2+(X;C). (84)

1.6.3. Seiberg-Witten Equations. To formulate the Seiberg-Witten equations for X one chooses a Riemann-ian metric g and an associated Spinc structure L.The field content of the theory consists of a connection Aon the U(1)-principal bundle L0(L), the gauge field, and a positive spinor field ψ ∈ Γ(S+(L)), the matterfield. These two fields are coupled by two equations, the first of which, called the curvature equation, wenow describe.

The positive spinor field determines a section (ψ⊗ψ∗)0 of the trace-free endomorphism bundle End0(S+(L))as in (80). This corresponds, via (84), to a self-dual 2-form on X. The first of the Seiberg-Witten equationsrequires that this coincide with the self-dual part F+

A of the curvature of A.

F+A = σ+( (ψ ⊗ ψ∗)0 ) (85)

Remark 1.25. We should be clear on what we mean by the curvature of A. This is defined to be a Liealgebra-valued 2-form on the bundle space L0(L). Since the structure group is U(1), this is a ImC-valued2-form. Pulling this curvature back to X by the sections corresponding to some trivializing cover one obtainsa family of locally defined, ImC-valued 2-forms on X. These are related by the adjoint action which, for theAbelian group U(1), is trivial. Consequently, these locally defined pullbacks agree on the intersections oftheir domains and so determine a globally defined 2-form on X. This is what we have called the curvatureand denoted FA.

78

The second Seiberg-Witten equation, called a Dirac equation, still requires a bit more preparation. Theoriented, orthonormal frame bundle SO(4) → FSO(X)

πSO→ X has a distinguished (Levi-Civita) connection

which we will denote ωLC . This can be characterized locally as follows. If e1, e2, e3, e4 is a local, oriented,orthonormal frame field on X, that is, a section of FSO(X), with dual 1-form field e1, e2, e3, e4, then ωLC isrepresented by a skew-symmetric matrix (ωi

j) of real-valued 1-forms satisfying dei = −ωij ∧ e j, i = 1, 2, 3, 4

(see Chapter IV, Section 2, of [KN1]). Now, notice that, if X has a Spin structure Spin(4) → S (X)πS→ X,

then the map λ : S (X)→ FSO(X) is a double cover that respects the group actions so ωLC automatically liftsto a connection on S (X) (think of the connection as a distribution of horizontal spaces). However, if X hasonly a Spinc structure Spinc(4) → S c(X)

πS c→ X, then the map Λ : S c(X) → FSO(X) is not a finite covering

so ωLC alone will not determine a connection on S c(X). However, Spinc : S c(X) → FSO(X)·× L0(L) is

a double cover so, if A is any connection on L0(L), then ωLC and A together determine a connection onFSO(X)

·× L0(L) which will then lift to a connection on S c(X). Specifically, if we denote by π1 and π2 the

restrictions to FSO(X)·× L0(L) of the projections of FSO(X) × L0(L) onto FSO(X) and L0(L), respectively,

then π∗1ωLC ⊕ π∗2A is a connection on the fiber product and

ωA = (Spinc)∗(π∗1ωLC ⊕ π∗2A) (86)

is a connection on S c(X). Any such ωA is called a Spincconnection on S c(X). Notice that the Levi-Civitaconnection ωLC on the frame bundle is fixed, but the U(1)-connection A will play the role of the gauge fieldin Seiberg-Witten theory and is constrained only by the Seiberg-Witten equations, that is, by (85) and theDirac equation that we are in the process of introducing.

A Spinc connectionωA determines a covariant derivative on sections of all of the associated spinor bundlesS(L), S+(L), and S−(L). We will denote all of these by ∇A. It is in terms of these covariant derivatives thatwe will now build a “Dirac operator” on spinor fields. Begin with

∇A : Γ(S(L))→ Ω1(X;C) ⊗ Γ(S(L)).

Define

/DA : Γ(S(L))→ Γ(S(L))

as follows. Let e1, e2, e3, e4 be an oriented, orthonormal frame field on U ⊆ X. Each ei can be thoughtof either as a vector field on U or as a section of the Clifford bundle which therefore acts by Cliffordmultiplication on sections of S(L) defined on U. Thus, for each Ψ ∈ Γ(S(L)) we can define /DA on Ψ by

/DAΨ =

4∑i=1

ei · ∇AΨ(ei). (87)

One shows that this is independent of the choice of e1, e2, e3, e4 and so defines /DAΨ globally (see Lemma3.3.1 of [Mor1]). Since S(L) = S+(L) ⊕ S−(L), we can restrict /DA to sections of either S+(L) or S−(L).Since Clifford multiplication by ei switches S±(L), so will these restrictions. We will write these as

/DA : Γ(S+(L))→ Γ(S−(L)) (88)

and

/D∗A : Γ(S−(L))→ Γ(S+(L)) (89)

79

The operators /DA and /D∗A are, in fact, formal adjoints of each other with respect to the L2-inner product onsections induced by the pointwise Hermitian inner product on fibers (see Lemma 3.3.3 of [Mor1]). Theyare, moreover, elliptic operators (see pages 43-44 of [Mor1]). We will refer to /DA as the Dirac operator ofSeiberg-Witten theory. The second of the Seiberg-Witten equations requires that the positive spinor field ψbe annihilated by /DA.

/DAψ = 0. (90)

We will collect together all of this information in the form of a few definitions and then try to gain somesense of what the Seiberg-Witten equations look like locally. Thus, we let X denote an oriented, smooth4-manifold. Choose a Riemannian metric g for X and a Spinc structure L for the corresponding oriented,orthonormal frame bundle SO(4) → FSO(X)

πSO→ X. The Seiberg-Witten configuration space A(L) consists

of all pairs (A, ψ), where A is a connection on the principal U(1)-bundle L0(L) and ψ ∈ Γ(S+(L)) is apositive spinor field. An (A, ψ) ∈ A(L) is called a Seiberg-Witten monopole if it satisfies the Seiberg-WittenEquations.

F+A =σ+( (ψ ⊗ ψ∗)0 )

(91)

/DAψ = 0

Example 1.9. To gain some sense of what the Seiberg-Witten equations actually look like we will write themout explicitly when X = R4 with its standard orientation and Riemannian metric. Since R4 is contractibleall of the relevant bundles over it are trivial and we will work with explicit trivializations (see Corollary11.6, page 53, of [Steen]). Thus, the oriented, orthonormal frame bundle is

SO(4) → R4 × SO(4)πSO−→ R4,

where πSO is the projection onto the first factor. We introduce a Spinc structure L as follows. Begin with thetrivial Spinc(4)-bundle over R4

Spinc(4) → R4 × Spinc(4)πS c−→ R4,

where πS c is the projection onto the first factor. Now define

Λ : R4 × Spinc(4)→ R4 × SO(4)

by

Λ(x, ξ) = (x, π(ξ)) = (x,Spin(u)),

80

where ξ = eθ iu ∈ Spinc(4). The associated spinor bundles are therefore also trivial so their sections can beidentified with globally defined functions on R4 which we will write as

Ψ =

ψ1

ψ2

ψ3

ψ4

: R4 → SC C4

ψ =

ψ1

ψ2

00

: R4 → S +C C

2 φ =

00ψ3

ψ4

: R4 → S −C C2.

For convenience we will often abuse the notation a bit and write Ψ =

ψφ

by suppressing the zero compo-

nents. We will use x1, x2, x3, x4 for the standard coordinates on R4 and write ∂i for ∂∂xi , i = 1, 2, 3, 4, these

being applied componentwise to the spinor fields. The determinant line bundle is likewise trivial, as is thecorresponding principal U(1)-bundle L0(L)

U(1) → R4 × U(1)πL0−→ R4,

where, once again, πL0 is the projection onto the first factor. A connection on this U(1)-bundle is thenuniquely determined by an ImC-valued 1-form

A = Aidxi, Ai : R4 → ImC, i = 1, 2, 3, 4.

In orthonormal coordinates, the covariant exterior derivative induced by the Levi-Civita connection on R4

is just ordinary componentwise exterior differentiation. Consequently, the covariant derivative ∇A inducedby it and the U(1)-connection A takes the form

∇A = ∇i dxi = (∂i + Ai) dxi.

Thus, for any spinor field Ψ,

∇AΨ =

(∂iψ

1 + Aiψ1) dxi

(∂iψ2 + Aiψ

2) dxi

(∂iψ3 + Aiψ

3) dxi

(∂iψ4 + Aiψ

4) dxi

.Taking ei = ∂i to be the standard oriented, orthonormal frame field on R4 we have

∇AΨ(ei) = ∇iΨ = (∂i + Ai)Ψ =

∂iψ

1 + Aiψ1

∂iψ2 + Aiψ

2

∂iψ3 + Aiψ

3

∂iψ4 + Aiψ

4

.

81

The Dirac operator (87) requires that we Clifford multiply by the basis elements ei, that is, matrix multiply

by Ei = Γ(ei) ∈ Cl (4) ⊗ C, and add. Writing this out explicitly with Ψ =

ψφ

we have

/DAΨ =

4∑i=1

ei · ∇AΨ(ei) =

4∑i=1

Ei∇AΨ(ei) =

∇1φ + I∇2φ + J∇3φ + K∇4φ

−∇1ψ + I∇2ψ + J∇3ψ + K∇4ψ

.The restriction of /DA to a positive spinor field ψ is therefore given by

/DAψ = −∇1ψ + I∇2ψ + J∇3ψ + K∇4ψ.

The Seiberg-Witten Dirac equation (90) therefore becomes

∇1ψ = I∇2ψ + J∇3ψ + K∇4ψ,

or, in complete detail,−(∂1 + A1) + i (∂2 + A2) (∂3 + A3) + i (∂4 + A4)−(∂3 + A3) + i (∂4 + A4) −(∂1 + A1) − i (∂2 + A2)

ψ1

ψ2

=

00 (92)

Notice that these are first order, linear equations.For the Seiberg-Witten curvature equation (85) we will use the local expressions (81) forσ+((ψ⊗ψ∗)0) and

the following local description of F+A . If A = Aidxi, then, since U(1) is Abelian, FA = dA =

∑i< j Fi jdxi∧dx j,

where Fi j = ∂iA j − ∂ jAi. A basis for the self-dual 2-forms on R4 is given bydx1 ∧ dx2 + dx3 ∧ dx4, dx1 ∧ dx3 + dx4 ∧ dx2, dx1 ∧ dx4 + dx2 ∧ dx3

so

F+A =

12

(FA + ∗FA) =12

(F12 + F34) (dx1 ∧ dx2 + dx3 ∧ dx4)

+12

(F13 + F42) (dx1 ∧ dx3 + dx4 ∧ dx2)

+12

(F14 + F23) (dx1 ∧ dx4 + dx2 ∧ dx3).

With this and (81) the curvature equation (85) becomes the following system of three equations.

F12 + F34 = −12

i (|ψ1|2 − |ψ2|2)

F13 + F42 = i Im (ψ1ψ2)

F14 + F23 = −i Re (ψ1ψ2)

or, in complete detail,

(∂1A2 − ∂2A1) + (∂3A4 − ∂4A3) = −12

i (|ψ1|2 − |ψ2|2)

(∂1A3 − ∂3A1) + (∂4A2 − ∂2A4) = i Im (ψ1ψ2) (93)

(∂1A4 − ∂4A1) + (∂2A3 − ∂3A2) = −i Re (ψ1ψ2)

These are again first order equations, but this time (mildly) nonlinear.

82

It is perhaps worth pointing out that the Seiberg-Witten equations (92) and (93) on R4 do have nontrivialsolutions. Indeed, if ψ = 0, then the Dirac equation is satisfied identically and A can be taken to be anyanti-self-dual connection on the trivial U(1)-bundle U(1) → R4 × U(1) → R4 and these are abundant (seepages 384-385 of [Nab5] for an explicit construction).

Remark 1.26. Our discussion of the Donaldson invariants began with the Yang-Mills action functional.The corresponding Euler-Lagrange equations are the Yang-Mills equations. These are second order partialdifferential equations. The absolute minima of the Yang-Mills action are solutions to the first order anti-self-dual equations and it is the moduli space of these that give rise to the Donaldson invariants. The pattern isthe same for the Seiberg-Witten invariants. One can write down a Seiberg-Witten action functional and thecorresponding second order Euler-Lagrange equations . Once again, however, the absolute minima of thisaction functional are the objects that give rise to the Seiberg-Witten invariants and these are just the solutionsto the first order Seiberg-Witten equations (see Section 7.1 of [Sal] or Section 7.2 of [Jost]). Specifically,the Seiberg-Witten action functional/energy has the following form.

E(A, ψ) =

∫X

(| ∇Aψ |

2 +s4| φ |2 +

14| φ |4 + | FA |

2)

d volg,

where s is the scalar curvature of X. This can be rewritten in the form

E(A, ψ) =

∫X

(| /DAψ |

2 + 2 | F+A − σ

+( (ψ ⊗ ψ∗)0 ) |2)

d volg − π〈c1(L0(L))2, [X]〉, (94)

where c1(L0(L)) is the first Chern class of the U(1)-bundle L0(L) and [X] is the fundamental class of X (seeProposition 7.3 of [Sal]). From this it is clear that the absolute minima occur for configurations that satisfythe Seiberg-Witten equations. In the next two sections we will sketch how these invariants come about.

1.6.4. Seiberg-Witten Moduli Space. Throughout this section and the next X will denote a compact, con-nected, simply connected, oriented, smooth 4-manifold (simple connectivity is not really necessary here,but will streamline the exposition and facilitate comparison with Donaldson theory). Choosing a Riemann-ian metric g on X gives an oriented, orthonormal frame bundle and one can then select a Spinc structureL. Recall that an element (A, ψ) of the Seiberg-Witten configuration space A(L) is called a Seiberg-Wittenmonopole if it satisfies the Seiberg-Witten equations (91). As in the case of Donaldson theory we are in-terested in a moduli space of such Seiberg-Witten monopoles so we must begin by isolating the appropriategauge group. This can be defined in a manner entirely analogous to Donaldson theory, but in this case itadmits a much simpler formulation and we will begin with this. We let C∞(X,U(1)) denote the group of allsmooth maps of X into U(1) under pointwise multiplication in U(1). Let γ ∈ C∞(X,U(1)) and identify U(1)with a subgroup of Spinc(4). Now define the map

σγ : S c(X)→ S c(X)

by

σγ(p) = p · γ(πS c(p))

where the dot · indicates the right action of Spinc(4) on S c(X). Then each σγ is an automorphism of theSpinc-bundle, that is, a diffeomorphism of S c(X) onto itself satisfying σγ(p·ξ) = σγ(p)·ξ for each p ∈ S c(X)

83

and each ξ ∈ Spinc(4), and which covers the identity on X in the sense that πS c σγ = πS c . Moreover, eachσγ covers the identity on FSO(X) in the following sense. Let Spinc : S c(X) → FSO(X)

·× L0(L) be the

double cover of the fiber product and π1 : FSO(X)·× L0(L) → FSO(X) the restriction to FSO(X)

·× L0(L) of

the projection of FSO(X) × L0(L) onto the first factor. Then

π1 Spinc σγ = π1 Spinc.

Remark 1.27. One can show that, conversely, any automorphism of the Spinc-bundle that covers the identityon FSO(X) is of this form (see Lemma 7.5, page 55, of [Nab3]).

The group C∞(X,U(1)) under pointwise multiplication is isomorphic to the group of all σγ under com-position and to the group of automorphisms of the Spinc-bundle that cover the identity on FSO(X) undercomposition. This group is called the Seiberg-Witten gauge group and denoted G(L). We will use whicheverof these three descriptions is most convenient in any particular context.

To define an action of G(L) on the configuration space A(L) we proceed as follows. First consider thepositive spinor field ψ ∈ Γ(S+(L)). Since U(1) acts by scalar multiplication on the fibers S +

Cof S+(L) we

can define a right action of γ ∈ C∞(X,U(1)) on Γ(S+(L)) by

(ψ · γ)(x) = γ(x)−1ψ(x)

for every x ∈ X. The same formula defines a right action of C∞(X,U(1)) on Γ(S−(L)).Next we turn to the connection A on L0(L). First note that each of the automorphisms σγ : S c(X) →

S c(X) induces an automorphism σ′γ : L0(L)→ L0(L) defined by

σ′γ (π2 Spinc) = (π2 Spinc) σγ.

We define the action of G(L) on A by

A · γ = A · σγ = (σ′γ)∗A.

One checks that the Spinc-connection corresponding to A · γ is the pullback by σγ of that corresponding toA, that is,

ωA·γ = σ∗γ ωA (95)

(see page 58 of [Nab3]). It is useful to have the action on A written locally on X. Let s be a local section ofL0(L) and consider the 1-form s∗A on X. Then define

(s∗A) · γ = s∗(A · γ) = s∗( (σ′γ)∗A ) = (σ′γ s)∗A.

A computation using the local expression for Spinc gives

(s∗A) · γ = s∗A + 2γ−1dγ

(see pages 57-58 of [Nab3]).Putting the actions of G(L) on ψ and A together we arrive at a right action of G(L) on the Seiberg-Witten

configuration space A(L)

(A, ψ) · γ = (A, ψ) · σγ = ( (σ′γ)∗A, γ−1ψ )

84

or, locally on X,

(s∗A, ψ) · γ = s∗A + 2γ−1dγ.

One finds that this action carries solutions to the Seiberg-Witten equations onto other solutions to theSeiberg-Witten equations. More precisely, one has the following (see Theorem 7.6 of [Nab3]).

Theorem 1.25. Suppose (A, ψ) ∈ A(L) satisfies

/DAψ = 0

F+A = σ+( (ψ ⊗ ψ∗)0 ).

Then, for any γ ∈ C∞(X,U(1)), (A, ψ) · γ = (A · γ, ψ · γ) satisfies

/DA·γ(ψ · γ) = 0

F+A·γ = σ+( (ψ · γ) ⊗ (ψ · γ)∗)0 ).

Thus, the space of solutions to the Seiberg-Witten equations is invariant under the action of the gaugegroup G(L) and we may, as for the anti-self-dual equations, consider the moduli space M(L) of gaugeequivalence classes of Seiberg-Witten monopoles.

M(L) =(A, ψ) ∈ A(L) : /DAψ = 0, F+

A = σ+( (ψ ⊗ ψ∗)0 )/G(L)

Remark 1.28. The ensuing analysis required to manufacture a differential-topological invariant of X fromM(L) is in many ways analogous to that of Donaldson theory. In particular, this analysis cannot be carriedout in the smooth category in which we currently find ourselves and one must work with appropriate Sobolevcompletions. As we did in the anti-self-dual case (see Remarks 1.5 and 1.8) we will forgo this here andsimply record the results we will need. We refer those interested in the analytical details to [Mor1], [Nicol],and [Sal].

In Donaldson theory the reducible connections are those left fixed by some nontrivial element of thegauge group and these give rise to singularities in the moduli space. In Seiberg-Witten theory the situationis slightly different, but the reducible configurations are particularly easy to describe.

Lemma 1.26. An element (A, ψ) of A(L) is left fixed by some nontrivial element γ of G(L) if and only ifψ = 0 and, in this case, γ : X → U(1) is a constant map.

Proof. It will suffice to argue locally so we let s be a section of L0(L) and consider s∗A. Then s∗A is leftfixed by γ ∈ G(L) if and only if

(s∗A + 2γ−1dγ, γ−1ψ) = (s∗A, ψ)

and this, in turn, is the case if and only if

γ−1ψ = ψ and 2γ−1dγ = 0.

85

Since γ , 1 the first of these is true if and only if ψ = 0. The second is true if and only if dγ = 0 and, sinceX is connected, γ must be constant.

Thus, we will say that a configuration (A, ψ) ∈ A(L) is reducible if and only if γ = 0 and irreducibleotherwise. In Donaldson theory, Theorem 1.12 guarantees the existence of a dense subspace of the spaceR(X) of Riemannian metrics for X for which reducible anti-self-dual connections do not exist and the modulispace is a smooth manifold. One cannot do quite so well for the Seiberg-Witten equations. We will firstdescribe the part of this scenario that is the same and then what must be done to get the full strength of thetheorem (see Chapter 4 of [Mor1] or Sections 7.1 and 7.2 of [Sal]).

Theorem 1.27. Let X denote a compact, connected, simply connected, oriented, smooth 4-manifold withb+

2 (X) > 0. Then there is a dense Gδ-subset RS WG (X) of the space R(X) of Riemannian metrics on X with

the following property. For any g ∈ RS WG (X) and any corresponding Spinc structure L, every solution (A, ψ)

to the Seiberg-Witten equations is irreducible. If b+2 (X) > 1, then for any generic path g(t), 0 ≤ t ≤ 1, of

Riemannian metrics in R(X) there are no reducible solutions to the Seiberg-Witten equations for any Spinc

structure corresponding to any of the metrics g(t), 0 ≤ t ≤ 1.

Missing here is any statement about the manifold structure of the moduli space. For this one must perturb,not only the metric, but the equations themselves. Specifically, if we fix a Riemannian metric g and a Spinc

structure L, then we can consider, for any η ∈ Ω2+(X, ImC), the perturbed Seiberg-Witten equations

/DAψ = 0

(96)

F+A = σ+( (ψ ⊗ ψ∗)0 ) + η.

Remark 1.29. The motivation here is not so hard to discern. If we define a map

F : A(L)→ Ω2+(X, ImC) ⊕ Γ(S−(L))

by

F(A, ψ) = ( F+A − σ

+( (ψ ⊗ ψ∗)0, /DAψ ),

then (A, ψ) satisfies the Seiberg-Witten equations if and only if it is in the zero set F−1(0, 0). With theappropriate analytical structures one can show that F is smooth and this solution space is a smooth manifoldif and only if (0, 0) is a regular value of F. If it is not, then the Sard-Smale Theorem (see Section 3.6of [AMR]) suggests that a small perturbation of (0, 0) of the form (η, 0) will be a regular value so thatF(A, ψ) = (η, 0) will define a smooth manifold of configurations.

The gauge group G(L) acts on solutions to the perturbed equations in the same way so one can once againconsider the moduli space M(L, η) of gauge equivalence classes of solutions. One then has the followingGeneric Perturbations Theorem (see Theorem 7.16 of [Sal] or Corollary 6.3.2 of [Mor1]).

86

Theorem 1.28. Let X denote a compact, connected, simply connected, oriented, smooth 4-manifold withb+

2 (X) > 0. Fix a g ∈ RS WG (X) and a corresponding Spinc structure L. Then there is a dense Gδ-set

Ω2+(X, ImC)G in Ω2

+(X, ImC) with the following property. For any η ∈ Ω2+(X, ImC)G, the moduli space

M(L, η) of solutions (A, ψ) to the perturbed Seiberg-Witten equations (96) is either empty or a smoothmanifold of dimension

14

[ ⟨c1(L0(L))2, [X]

⟩− 2χ(X) − 3σ(X)

],

where c1(L0(L)) is the first Chern class of the U(1)-bundle L0(L), [X] is the fundamental class of X, χ(X) isthe Euler characteristic of X and σ(X) is its signature.

Much of the analysis of the Seiberg-Witten moduli space (orientability, cobordism, calculation of thedimension, etc.) proceeds in a manner entirely analogous to that of the anti-self-dual moduli space (seeSections 7.2 and 7.3 of [Sal]). There is one very substantial difference, however, and it accounts for therelative simplicity of Seiberg-Witten theory. There is, in this case, no need for an “Uhlenbeck-type” com-pactification since the Seiberg-Witten moduli spaces are always compact. The proof of this rests on the factthat solutions to the (perturbed) Seiberg-Witten equations satisfy certain uniform, a priori bounds. This issomething that is categorically false for the anti-self-dual equations because these are conformally invariantin dimension four. The arguments are quite long and technical and we will simply refer those interested inthe details to either Section 7.2 and Chapter 8 of [Sal] or Sections 5.3 and 6.4 of [Mor1], where the resultsproved are much more general than the special case we will now record in the final theorem of this section.

Theorem 1.29. Let X denote a compact, connected, simply connected, oriented, smooth 4-manifold withb+

2 (X) > 0. Fix a generic Riemannian metric g ∈ RS WG (X), a corresponding Spinc structure L, a generic

perturbation η ∈ Ω2+(X, ImC)G, and an orientation for H2

+(X;R). Then the moduli space M(L, η) of solu-tions (A, ψ) to the perturbed Seiberg-Witten equations (96) is either empty or a smooth, compact, orientedmanifold of dimension

14

[ ⟨c1(L0(L))2, [X]

⟩− 2χ(X) − 3σ(X)

],

where c1(L0(L)) is the first Chern class of the U(1)-bundle L0(L), [X] is the fundamental class of X, χ(X) isthe Euler characteristic of X and σ(X) is its signature.

1.6.5. Zero-Dimensional Seiberg-Witten Invariant and Witten’s Conjecture. Throughout this section X willdenote a compact, connected, simply connected, oriented, smooth 4-manifold and we will assume that b+

2 (X)is greater than 1 and odd (the reason for strengthening the assumption b+

2 (X) > 0 in Theorem 1.29 will bedescribed in a moment). We now know that, for such a 4-manifold, we have available some very nicemoduli spaces M(L, η) and one would like to use these to produce differential-topological invariants of Xa la Donaldson. The game plan is the same: integrate selected cohomology classes over the moduli spaceto produce numbers and show that orientation preserving diffeomorphisms preserve these numbers. In thecase of Donaldson theory we focused on 0-dimensional moduli spaces because this was the simplest case

87

and we will do the same for Seiberg-Witten, but now the reasons are even more compelling, as we shall seequite soon.

Thus, we fix a generic metric g and perturbation η and suppose that there exists a Spinc structure L forwhich ⟨

c1(L0(L))2, [X]⟩

= 2χ(X) + 3σ(X). (97)

This can be written as ⟨c1(L0(L))2, [X]

⟩− σ(X)

4=χ(X) + σ(X)

2= 1 + b+

2 (X)

and the index theory of the Dirac operator implies that the leftmost quantity is even. Consequently, b+2 (X)

must be odd. We assume that b+2 (X) > 1 for the cobordism result suggested in Theorem 1.27 which is

responsible for the metric independence of the invariants. Also choose an orientation for the vector spaceH2

+(X;R) and thereby orient all of the moduli spaces. Then M(L, η) is either empty or a smooth, compact,oriented manifold of dimension zero, that is, a finite set of isolated points each of which is equipped with asign ±1 (there is an explicit description of the sign on page 246 of [Sal]). The 0-dimensional Seiberg-Witteninvariant , denoted

SW0(X,L),

is defined to be zero if the moduli space is empty and, otherwise, it is the sum of these signs. This is, indeed,an invariant in the sense that it is independent of the choice of g and η and depends only on the isomorphismclass of the Spinc structure L (see Theorem 7.22, page 247, of [Sal]). Consequently, if f : X′ → X is anorientation preserving diffeomorphism for which the induced map f ∗ : H2

+(X;R) → H2+(X′;R) preserves

the chosen orientations, then the induced Spinc structure f ∗L obtained by pulling back the Spinc(4)-bundleand the map Λ by f also satisfies (97) and, moreover, SW0(X′, f ∗L) = SW0(X,L).

Seiberg-Witten invariants can be defined even when the moduli space is not 0-dimensional (see Section7.4 of [Sal] or Section 6.7 of [Mor1]), but we will not do so here and we would like to provide a few wordsof explanation. The empirical evidence, based on the calculation of specific examples, suggests that whenb+

2 (X) > 1 nonzero Seiberg-Witten invariants occur only when the moduli space is 0-dimensional. We willsay that an X with b+

2 (X) > 1 is of SW-simple type if nonzero Seiberg-Witten invariants of X occur onlyfor 0-dimensional moduli spaces. The conjecture then is that every compact, connected, simply connected,oriented, smooth 4-manifold X with b+

2 (X) > 1 and odd is of SW-simple type. Any element of H2(X;Z) thatarises as c1(L0(X)) for some Spinc structure L satisfying (97) is referred to as a SW-basic class. Thus, SW-basic classes are just the 1st Chern classes of the U(1)-bundles corresponding to Spinc structures for whichthe Seiberg-Witten moduli space is 0-dimensional. With this terminology we can describe more preciselyour motivation for restricting attention to SW0(L, η).

We have already mentioned that Witten was led to the Seiberg-Witten equations by a deep physical“duality” between the weak and strong coupling regimes of the TQFT constructed in [Witt2]. This dualitysymmetry led Witten not just to some vague sense that the Seiberg-Witten invariants should contain the sametopological information as the Donaldson invariants, but rather to a very explicit formula relating the twofor manifolds of KM-simple type (see Theorem 1.19). This conjecture, called Witten’s Magical Formula by

88

Taubes [Taub3], essentially rewrites the Donaldson series DX(α) in terms of 0-dimensional Seiberg-Witteninvariants. In the following statement of the conjecture we use the notation of Theorem 1.19.

Witten’s Conjecture: Let X be a compact, connected, simply connected, oriented, smooth 4-manifold withb+

2 (X) > 1 and odd. Then

(1) X is of SW-simple type if and only if it is of KM-simple type and, in this case,(2) KM-basic classes coincide with the SW-basic classes, and(3) for each basic class Ki, i = 1, . . . , s, the corresponding rational number ai, i = 1, . . . , s, is given by

ai = 22+ 14 (7 χ(X)+11σ(X)) SW0(Li),

where Li is a Spinc structure for which c1(L0(Li)) = Ki.

The Donaldson series can then be written

DX(α) = eQX(α,α)/2s∑

i=1

aieKi(α) = eQX(α,α)/2s∑

i=1

22+ 14 (7 χ(X)+11σ(X)) SW0(Li) ec1(L0(Li))(α).

Witten’s Conjecture has been verified for all of the examples for which both the Donaldson and Seiberg-Witten invariants are known, but a completely general proof is still lacking. A proposal for constructingsuch a proof was outlined shortly after the appearance of the conjecture by Pidstrigatch and Tyurin [PT] andthis proposal was taken up in a long series of technically very demanding papers by Feehan and Leness (thecurrent state of affairs is described in [FL2]). A special case of the conjecture proved by Feehan and Lenessin [FL1] was used by Kronheimer and Mrowka [KM2] to resolve a long-standing problem in knot theory.

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INDEX

A⊗B, 35CWω : Ω∗G(V)→ Ω∗(P × V)BASIC Ω∗(P ×ρ V), 49CWω : C[g]G → Ω∗(P)BAS IC Ω∗(X), 48C∞(X,U(1)), 82Cp : g→ Vertp(P) ⊆ Tp(P), 53C∗p : Tp(P)→ g, 53Cl (4), 65Cl (4) ⊗C, 69Cl (V), 65Cl (X), 77Cl (X) ⊗C, 77Cl×(4), 67Cl0(4), 66Cl1(4), 66E(A, ψ), 82E8, 11H∗G(X), 46H∗CV (E;R), 38K3, 9L(L), 76QX : H2(X;Z) × H2(X;Z)→ Z, 10Rp = C∗p Cp : g→ g, 53S DW [Φ], 63SC, 69S ±C

, 69U(E) ∈ H2k

CV (E;R), 38XQ, 16[Aλ,n], 8[ω], 21[M(Pk, g)], 28∆C, 69Γ(Cl (X)), 77Γ(Cl (X) ⊗C), 77Γ : H2×2 → C4×4, 68Ω0(X,Ad Pk), 22Ωi(Pk, sp(1)), 21Ωi(X, ad Pk), 21Ω2

+(X, ad P), 39Ω∗G(X) = [C[g] ⊗Ω∗(X) ]G, 44Ω∗CV (E;R), 38Ω

pCV (E;R), 38

Ωλ,n, 5Ω0

Ad(Pk,Sp(1)), 22Ωi

ad(Pk, sp(1)), 21A(Pk), 21A(L), 79

AAS D(Pk, g), 22Aλ,n, 5B(Pk), 21DX(α), 33F, 5Fλ,n, 5G(P), 13G(Pk), 21G(L), 83M, 8, 13M(P, g), 13M(Pk, g), 22M(L), 84R(X), 23RG(X), 23S(L), 76S±(L), 76YM(A), 6χ(ξ), 35C[g], 34C[g] ⊗Ω∗(X), 44C[so(2k)] ⊗Ω∗(V), 42H, 2HP1, 2, 3γ : H→ C2×2, 68γd(X), 29γdk (X) of X, 28∆C, 71∆±C

, 71A(Pk), 22A ×G Ω2

+(X, ad P), 39AAS D(Pk, g), 22B(Pk), 22G → A→ A/G = B, 39M(Pk,R(X)), 23M(Pk, gt), 24M(Pk, g), 22∫

e−S DW [Φ] DΦ, 63∫f Dψ, 40∫

X: H∗G(X)→ C[g]G, 47∫

X: Ω∗G(X)→ C[g]G, 47

spin(4), 67su(2), 2µ(α), 26∇A, 78ν, 43

91

92

νG, 49ωA, 78ωLC , 78ωλ,n, 5M(Pk, g), 26µ, 28πHor(ω), 49ρ : Ω(R4;C)→ Cl0(4) ⊗C, 71ρ± : Ω2(R4;C)→ EndC(S ±

C), 72

σ+ : Γ( End0(S+(L)) )→ Ω2+(X;C), 77

σ+ : End0(S +C

)→ Ω2+(R4;C), 72

σ1, σ2, σ3, 2/DA : Γ(S+(L))→ Γ(S−(L)), 78/D∗A : Γ(S−(L))→ Γ(S+(L)), 78Ad Pk, 22Horp(S7), 3ImH, 2Ind (s; p), 36Pf : so(2k)→ R, 37Pf (A), 37Pin (4), 67SU(2), 2SW0(X,L), 87Spin : Spin (4)→ SO (R4), 67Spin (4), 67Spinc(4), 70Sp(1), 2Sp(1) → Pk

πk→ X, 20

Sp(1) → S7 π1→ S4, 3

Sp(1) → S7 π→ HP1, 2

Stab(ω), 22Vertp(S7), 3ad Pk, 21ad P, 12adu, 67vol = ψ1 · · ·ψn, 34/DA : Γ(S(L))→ Γ(S(L)), 78∧V , 34dA∗F, 7dG, 45fvol, 40st(X), 26

anti-self-dual, 5Atiyah-Jeffrey formula, 58Atiyah-Singer Index Theorem, 75automorphism of a principal bundle, 4

background connection, 27

basic classKM, 33SW, 87

Berezin integral, 40, 41Bianchi identity, 7blowup, 30

n-fold, 30blowup formulas, 30BPST instanton, 2, 5

center, 5spread, 5

Cartan formula, 50Cartan model, 46Cartan Structure Equation, 5Cartan’s Theorem, 47Chern-Gauss-Bonnet Theorem, 38Chern-Weil map, 48

generalized, 49Clifford algebra, 65Clifford algebra ofR4

complexified, 69real, 65

Clifford bundle, 77complexified, 77

Clifford multiplication, 69compact vertical cohomology, 38compact vertical support, 38complex algebraic surface, 32complex spin representation, 69complex surface, 32complexified Clifford algebra, 69complexified Clifford bundle, 77cone over CP2, 14Connected Sum Theorem, 32connection 1-form, 4coupling constant, 63covering dimension, 28curvature density, 27curvature equation, 77

determinant line bundle, 76Dirac equation, 78Dirac operator, 75, 79Donaldson µ-map, 25Donaldson invariants, 20, 24, 25, 28

stable range, 29Donaldson series, 33, 88Donaldson’s Theorem on Intersection Forms, 12

93

Donaldson-Witten action functional, 63doubling the degrees, 35

empty form, 10equivalence of Spin structures, 74equivalence of Spinc structures, 75equivariant cohomology, 44equivariant differential form, 45

integration, 47Equivariant Localization Theorems, 64equivariantly exact, 46Euler number, 35even element, 34even elements of Cl (4), 66exoticR4, 16, 18, 20

fakeR4, 16, 18, 20fermionic integral, 40field configuration, 62finite action, 6Fourier Inversion Formula, 54Freedman’s Theorem, 16

g-anti-self-dual, 13, 22G-equivariant cohomology, 46G-equivariant exterior derivative, 45G-equivariantly closed, 46G-invariant element of C[g] ⊗Ω∗(X), 44gauge equivalent connections, 5, 13, 21gauge field

Seiberg-Witten, 77gauge field strength, 5gauge group, 13, 21, 83gauge potential, 4gauge transformation, 4, 13, 21Generic Metrics Theorem

Donaldson, 23Seiberg-Witten, 85

Generic Perturbations Theorem, 85generic Riemannian metric, 23generic section of a vector bundle, 36

Hopf bundlequaternionic, 2

horizontal space, 3

ideal g-ASD connection, 27instanton, 1, 7, 13

BPST, 2instanton number, 8

intersection form, 10definite, 10equivalent, 10even, 10indefinite, 10negative definite, 10odd, 10positive definite, 10signature, 10

irreducible configuration, 85irreducible connection, 22

k-stable, 29KM-basic classes, 33KM-coefficients, 33KM-simple type, 33Kummer surface, 9

Levi-Civita connection, 78

matter fieldSeiberg-Witten, 77

moduli space, 20, 84metric independence, 25of connections, 21of g-anti-self dual connections, 22of g-anti-self-dual connections on the Hopf bundle, 13of instantons on the Hopf bundle, 8of irreducible connections, 22orientability, 24Seiberg-Witten, 82

n-fold blowup, 30natural connection, 4negative spinor bundle, 76negative spinor field, 76Non-Vanishing Theorem, 32normalized vertical volume form, 54

odd elements of Cl (4), 66

parametrized moduli space, 24partition function, 63Pauli spin matrices, 2perturbed Seiberg-Witten equations, 85Pfaffian, 37Poincare Conjecture

4-dimensional, 17Poincare-Hopf Theorem, 38positive spinor bundle, 76

94

positive spinor field, 76pseudoparticle, 2, 4

quaternionic Hopf bundle, 2, 3quaternionic projective line, 2Quinn’s Theorem, 17

reducible configuration, 85reducible connection, 22Removable Singularities Theorem, 7Rokhlin’s Theorem, 11

second Stiefel-Whitney class, 67, 74Seiberg-Witten action functional, 82Seiberg-Witten configuration space, 79Seiberg-Witten equations, 77, 79

onR4, 79perturbed, 85

Seiberg-Witten gauge field, 77Seiberg-Witten gauge group, 83Seiberg-Witten invariant

0-dimensional, 87Seiberg-Witten invariants, 65Seiberg-Witten matter field, 77Seiberg-Witten moduli space, 84Seiberg-Witten monopole, 79signature, 10simple type

KM, 33SW, 87

Spin manifold, 74Spin structure, 67, 73

equivalence, 74Spinc connection, 78Spinc structure, 75

equivalence, 75spinor bundle, 76

negative, 76positive, 76

spinor field, 67, 75, 76negative, 76positive, 76

spinor map, 67stabilizer, 22stable range, 28strong coupling, 64Structure Theorem of Kronheimer and Mrowka, 33superalgebra, 34

supercommutative, 34

supertensor product, 35SW-basic class, 87SW-simple type, 87symmetric product, 26

tensorial of type ad, 21Thom class, 38

Gaussian representative, 41Thom isomorphism, 38topological charge, 8topological dimension, 28topological invariant, 63topological quantum field theory, 33, 63

topological invariants, 63total field strength, 6TQFT, 33

Uhlenbeck compactification, 25, 27universal Thom form, 42, 43

as an SO(V)-equivariant differential form, 45SO(V)-equivariantly closed, 46

vertical space, 3

weak coupling, 64Whitehead’s Theorem, 11Witten’s Conjecture, 88Witten’s Magical Formula, 87

Yang-Mills action, 6Yang-Mills equations, 7