lectures on nonperturbative aspects of supersymmetric gauge theories

INSTITUTE OF PHYSICS PUBLISHING CLASSICAL AND QUANTUM GRAVITY

Class. Quantum Grav. 22 (2005) S77–S105 doi:10.1088/0264-9381/22/8/003

Lectures on nonperturbative aspects ofsupersymmetric gauge theories*

Nikita A Nekrasov1

Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France

E-mail: nikita@ihes.fr

Received 9 December 2004Published 29 March 2005Online at stacks.iop.org/CQG/22/S77

AbstractAn introduction to (supersymmetric) gauge theories and their applications isfollowed by a discussion of the recent solution of N = 2 theories by meansof direct instanton counting. The gauge/string duality in the context of BPSstates and topological strings is reviewed, and the quantum gravitational foampicture emerging out of gauge-theoretic considerations is explained.

Introduction

Gauge theory accurately describes the physics of elementary particles over a wide energyscale. The perturbative expansion is adequate for non-Abelian gauge theories at highenergies. For low-energy physics the effective gauge coupling grows strong in theabsence of gauge symmetry breaking. As a consequence, non-perturbative effects becomeimportant. In particular, low-energy physics is affected by the gauge theory instantons,whose careful treatment is necessary to get non-singular low-energy effective theory. In therealistic gauge theories, such as SU(3) gauge theory, the instanton effects are mixed withthose of anti-instantons, and also are dressed by various perturbative corrections. In thesupersymmetric gauge theories the situation is simpler, as the perturbation expansion aboutinstanton background is essentially trivial, and there are interesting correlation functions whereinstantons and anti-instantons do not mix.

In these lectures we shall explain the recent solution of the N = 2 gauge theory bymeans of direct instanton counting. We shall also discuss the possible applications to the lesssymmetric N = 1 case.

We shall also discuss gauge theory/string theory duality in the context of the theoriesabove. It turns out that many of the exactly calculable on the gauge theory side quantities havea topological string interpretation.

* Lectures given at the RTN Winter School on Strings, Supergravity and Gauge Fields, which took place in Barcelona,on 12–16 January 2004.1 On leave of absence from Institute for Theoretical and Experimental Physics, Moscow, 117259, Russia.

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We shall conclude with a quantum gravitational foam picture which emerges out of thegauge theory considerations.

1. Supersymmetric gauge theories

In the first lecture we shall briefly recall the necessity of gauge theories, motivate theintroduction of the supersymmetry (susy), discuss extended susy, and also recall a few advancedtopics, such as twisting.

1.1. Why gauge theory

We shall not give an extensive apology for gauge theories. Suffice it to say that gauge theoriesare the most up-to-date accurate descriptions of all the known fundamental interactions exceptfor gravity. For more details, including renormalizability, asymptotic freedom and otherimportant aspects, we refer the reader to [27, 31, 65].

1.2. Perturbative expansion versus non-perturbative effects

In quantum field theory with coupling constant g, where the Lagrangian can be written as

L = Lfree + gLint (1)

with quadratic non-degenerate term Lfree, the correlation functions can be expanded in g:

〈O1(x1) . . .On(xn)〉 = 〈O1(x1) . . .On(xn)〉free + g

∫ddy〈O1(x1) . . .On(xn)Lint(y)〉free

∫ ∫ddy1 ddy2〈O1(x1) . . .On(xn)Lint(y1)Lint(y2)〉free + · · · . (2)

However, this expansion is formal, and in fact does not make sense, as generally the freetheory correlation functions have divergences as y → xi, i = 1, . . . , n, and also as y1 → y2,etc. These divergences are taken care of by allowing the operators Oi and the interactionLagrangian Lint to depend on g in a non-trivial way. The dependence on g involves theultraviolet cut-off, which also enters the proper definition of the integrals in equation (2). Theprocedure is known as regularization and renormalization.

If the theory is regularized and renormalized the terms of fixed order in the couplingconstant, e.g. ≈gn, are finite. They depend, however, on the details of the renormalization,i.e. on the normalization point µ.

In addition to these perturbative contributions to the correlation function, one has variousnon-perturbative effects. They come from the fact that the action S = Lfree + gLint mayhave several critical points, and in general all of these contribute to the path integral. Thecontribution of a non-trivial critical point ϕ∗ goes like

Zϕ∗ = e− 1

g2 S(ϕ∗) Pfaff(∂2SF )√Det ∂2SB

(1 + A2(ϕ∗)g2 + · · ·) (3)

where ∂2SF is the fermionic quadratic part of the action expanded about ϕ∗, and ∂2SB is thebosonic one.

In the case of pure gauge theory, with the action

4π2g2

∫TrF ∧ �F (4)

Lectures on nonperturbative aspects of supersymmetric gauge theories S79

by rescaling the gauge field A �→ g · A one brings the Lagrangian to the form

L = ‖dA‖2 + g Tr dA ∧ �[A,A] + g2 Tr [A,A] ∧ �[A,A] (5)

which can be used (after gauge fixing) to write the perturbative amplitudes. However, theaction (4) possesses non-trivial solutions of equations of motion with finite action, and thesesolutions (instantons and anti-instantons) lead to extra terms in the low-energy effective action.To the standard Yang–Mills action (4) one can add a topological term, which does not affectequations of motion, nor the perturbation theory, which measures the instanton charge:

∫TrF ∧ F. (6)

This term depends on the extra parameter, �, also known as theta, or the instanton angle.The physics is periodic with respect to the shifts � → � + 2π , as (6) enters gauge theoryamplitudes only in the exponentiated form, while 1

∫TrF ∧ F is quantized2.

As was shown by Callan, Dashen and Gross [13], and also by ’t Hooft [32], the effects ofthe instantons can be systematically taken into account, at least for the single instanton. Thecorrection to the effective action, which comes from the single instanton, has the form of anintegral over the moduli space of centred instantons. The remaining integral over the positionof the instanton is the integral which converts the Lagrangian into the action.

So, on general grounds, one can write3

Leff = Lcl + g2L1-loop + g4L2-loop + · · · + e− 8π2

g2 cos�L1-inst(g). (7)

1.3. Wilsonian effective action

There are several ways of packaging the information in quantum field theory. One is theso-called effective action, the generating function �(ϕ,µ) of all 1PI diagrams, which is theLegendre transform of the free energy W [J,µ] as a functional of sources J . Both depend onthe normalization point µ which is used to define the renormalized vertices. The momentain the loop integrals defining W [J,µ] are integrated over from zero to the ultraviolet cut-off�UV, which is then taken to infinity, while at the same time adjusting the bare couplingconstants.

This object is almost impossible to calculate, and it contains too much information.Another object is the Wilsonian effective action. It is defined, roughly, by integrating all thefields of momenta of magnitude higher than a given value (and less the ultraviolet cut-off ).This value, µ, is the parameter of the Wilsonian action. The Wilsonian action Seff[ϕslow, µ] isthe functional of the remaining, unintegrated fields. By construction, they are slowly varying.For the theories with massive particles only the Wilsonian action and the 1PI effective actionsare quite similar, for µ much less than the mass scale. However, in the presence of masslessparticles the Wilsonian action is much better defined; it does not suffer from the infrareddivergences present in �[ϕ,µ]. These divergences lead to various unwanted effects.

The Wilsonian action can be expanded in 1µ2 ∂

2, i.e. in the derivatives of the fields.

1.4. Supersymmetry

Conventional gauge theory is defined on flat Minkowski space. Lorentz invariance allows us todefine it on a general curved manifold. However, one can define it on more exotic manifolds,

2 We use the standard notation for simple Lie groups: Tr = 1h∨ Tradj.

3 cos�L1-inst should really read 12 (e

i�L1-inst + e−i�L1-anti-inst).

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namely on noncommutative spaces. The simplest non-trivial example of such a manifold isthe superspace.

Gauge theory on the superspace is the supersymmetric (susy) gauge theory. The fields ofthe theory are functions of the ordinary, bosonic, four spacetime coordinates xµ, and the odd,fermionic, coordinates ϑα, α = 1, 2 (for minimal susy), and their conjugates ϑ α . The minimalsupersymmetry in four dimensions is generated by two complex supercharges Qα, α = 1, 2,which form the usual Poincare superalgebra:

{Qα, Qα} = −2iσµ

αα∂µ, {Q1,Q2} = 0. (8)

The superspace meaning of the supercharges (8) is the superanalogue of the ordinarytranslations ∂µ = ∂

∂xµ :

Qα = ∂

∂ϑα− iσµ

ααϑα ∂

∂xµ, Qα = ∂

∂ϑα− iσµ

αβϑβ ∂

∂xµ. (9)

To write down supersymmetric Lagrangians one makes use of the superspace covariantderivatives:

Dα = ∂

∂ϑα+ iσµ

ααϑα ∂

∂xµ, Dα = ∂

∂ϑα+ iσµ

αβϑβ ∂

∂xµ. (10)

The remarkable property of the supersymmetric theories is the cancellation of various quantumcorrections thanks to the competition between the bosonic and fermionic loops. In particular,vacuum energy (if the susy is not unbroken) is exactly zero.

The more supersymmetry has the theory, the softer are its ultraviolet properties. Themaximal rigid supersymmetry in four dimensions, N = 4, is so constrained that the theoryhas no parameters except for the choice of the gauge group G, and the (complexified) gaugecoupling for each simple factor of G. Moreover, these couplings do not get renormalized.

Less constrained is the N = 2 supersymmetry. In this case one has two sets of N = 1supercharges. The superalgebra is generated by Qi

α, i = 1, 2, and Qαj, j = 1, 2. Theindices i, j label the two components of the two-dimensional representation of the globalsymmetry group SU(2)I which rotates two N = 1 subalgebras of N = 2 superalgebra amongthemselves: {

Qiα, Qαj

} = −2iδijσ

αα∂µ,{Qi

α,Qjβ

} = εijεαβZ (11)

where Z is the central element.

1.4.1. Representations of the supersymmetry algebra. It is straightforward to work out thelow-spin representations of the algebras (11), (8). In the minimal susy case the representationscrucially depend on whether the particle it describes is massive or massless.

In the massive case we can choose the Lorentz frame where P0 = M,P1 = P2 = P3 = 0.Writing Qα = √

2Mbα, Qα = √2Mb†α we get

{bα, b

} = δαα, {b1, b2} = 0. In other words,we have two fermionic oscillators. The minimal representation has four helicity states: thevacuum |↓〉, which has spacetime helicity − 1

2 , two zero helicity states b†1|↓〉 and b

†2|↓〉, and

one + 12 helicity state b

†2|↓〉.

In the massless case we can choose the Lorentz frame with P0 = P3 = E,P1 = P2 = 0.In this case we should introduce the oscillators as follows: Q1 = √

2Eb, Q1 = √2Eb†, with

{b, b†} = 1 and Q2, Q2 anticommuting with everything. The minimal representation wouldbe two dimensional, with the states |↓〉 and b†|↓〉. The CPT invariant multiplet will have twosuch conjugate representations in it. Depending on the lowest helicity of the state |↓〉 in eachof them we would get the chiral multiplet, with helicities − 1

2 , 0, 0,+ 12 , or the vector multiplet,

with helicities −1,− 12 ,+ 1

2 ,+1.

1.4.2. Superfields. The chiral superfield is a compact way of packaging the chiral multipletrepresentation. Introducing the chiral coordinates yµ = xµ + iϑσµϑ , we write

�(y, ϑ) = φ(y) +√

2ϑψ(y) − ϑϑf. (12)

In the physical space coordinates the components of the chiral superfield are

�(y, ϑ) = φ(x) +√

2ϑψ + iϑσµϑ∂µφ(x) − ϑϑf

− i√2ϑϑ∂µψ(x)σµϑ − 1

4ϑ2ϑ2∂2φ(x). (13)

The vector superfield is longer. In the so-called Wess–Zumino gauge it can be expanded as

V = ϑσµϑAµ(x) + iϑ2ϑ λ(x) − iϑ2ϑλ(x) + 12ϑ

2ϑ2D(x). (14)

The vector superfield is real: V † = V . This field transforms non-trivially under gaugetransformations: V �→ V + ϕ + ϕ†, which preserve the Wess–Zumino gauge: ϕ = ϑσµϑ∂µα,for real α. Its non-Abelian generalization has all components in the Lie algebra g of the gaugegroup G. The gauge transformations look as follows:

e2V �→ eiϕ†e2V e−iϕ, e−2V �→ eiϕ e−2V e−iϕ†

. (15)

To write the gauge-invariant Lagrangian one uses the chiral-looking gauge-covariantsuperfield:

Wα = − 14 D

2(e−2VDα e2V ), Wα = 14D

2(e2V Dα e−2V ), (16)

which expands as follows:

Wα = −2iλα + 2ϑαD(y) + 2i (σµνθ)α Fµν + ϑ2(σµDµλ(y))α. (17)

1.4.3. Susy Lagrangians. The simplest supersymmetric theory is the N = 1 susy sigmamodel. It describes maps of R4|4 into some Kahler manifold X. The Lagrangian is written interms of the following geometric data: the Kahler metric on X, which locally can be describedusing Kahler potential K(z, z), where zI are local holomorphic coordinates on X, and theholomorphic function W(z) on X, called the superpotential. Here is the Lagrangian:

LN=1σ-model =∫

d2ϑ d2ϑK(�,�†) +∫

d2ϑW(�) +∫

d2ϑW (�†). (18)

The simplest supersymmetric gauge theory Lagrangian has the following structure:

LN=1YM = Im

(∫d2ϑτ TrWαW

τ = �

is the complexified gauge coupling g being the ordinary Yang–Mills gauge coupling and �

the instanton angle.One can also add some charged matter, with coupling to the Yang–Mills multiplet. The

setup is the following: the Lie group G acts on X by isometries. Its complexification GC actsby holomorphic transformations:

z �→ g · z. (20)

The compact subgroup G preserves the Kahler metric and, in particular, the Kahler formω = ∂∂K . In this context one has the so-called moment map:

µ : X → g∗ (21)

which plays an important role in symplectic geometry.

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The gauge-invariant σ -model Lagrangian:

Lmatter =∫

d2ϑ d2ϑK(�†, (e2V · �)) +

(∫d2ϑW(�) + c.c.

where the Kahler potential is analytically continued to the holomorphic function on X × X

(actually, to the formal neighbourhood of the diagonal). The superpotential W should beGC-invariant.

Renormalizable field theories are obtained for X a vector space, which is somerepresentation of G. This representation has to be real, X∗ ≈ X, to avoid gauge anomaly.

Effective field theories for supersymmetric theories are described by supersymmetricLagrangians. The sigma model part of such a Lagrangian may very well correspond tononlinear X, which would nevertheless have to be Kahler (for N = 1), hyper-Kahler or specialKahler (for N = 2) or flat manifold (or orbifold) (for N = 4).

1.4.4. Extended susy. Extended supersymmetry multiplets behave in an interesting way.The representations of N = 2 superalgebra are obtained by combining the representations

of N = 1 algebra.In the absence of central extension (Z = 0) they contain 16 helicity states. If the central

charge of the representation is non-zero, then one can get shorter multiplets for massiveparticles. This happens precisely when the so-called BPS bound is saturated:

M = 12 |Z|.

The short multiplets have four helicity states. There are two such multiplets: vector andhypermultiplet.

N = 2 vector multiplet consists of N = 1 vector and chiral multiplet in the adjointrepresentation, i.e. X = gC. N = 2 hypermultiplet consists of two N = 1 chiral multiplets,transforming in dual representations of the gauge group.

The vector multiplet can be written compactly in terms of the N = 2 superfield� = �(y, ϑ1) + ϑα

2 Wα + · · · . In section 2 we describe the special geometry [21, 71] ofthe space of scalars in the N = 2 vector multiplets.

1.4.5. Twisted susy. The SU(2)I R-symmetry of the N = 2 gauge theory allows us to twistthe susy algebra. One declares the Lorentz group SU(2)L × SU(2)R to be the subgroupSU(2)L × SU(2)� of the original Lorentz ×SU(2)I global symmetry group, where SU(2)�is the diagonal subgroup in SU(2)R × SU(2)I . Practically this means that the i index of theR-symmetry doublet becomes the α index of SU(2)R . Consequently, the supercharges of theN = 2 susy become

Qiα, Qαj −→ Qµ, Q, Qµν (23)

i.e. 1-form, a scalar, and self-dual bi-vector, which anticommute as

Q2 = Z, {Q,Qµ} = ∂µ, {Q, Qµν} = 0, {Qµ,Qν} = gµνZ

{Qµν, Qκλ} = (gµ[κgλ]ν + εµνκλ

)Z {Qµ, Q

νλ} = (δ[νµ gλ]κ + gµαε

ανλκ)∂κ .

1.4.6. Topological gauge theory. Twisted theory can be formulated on any four-manifoldX, while preserving at least one fermionic symmetry, namely Q. In this way the observablesannihilated by Q become distinguished. For one thing, their correlation functions areindependent of the metric on X and provide the smooth structure invariants of X. ForG = SO(3), and the observables constructed out of Tr�2 these are the celebrated Donaldson

invariants. For G = U(1) with a single hypermultiplet the corresponding correlators are theso-called Seiberg–Witten monopole invariants.

If X has extra structure the twisted gauge theory might have more fermionic symmetries.If X is Kahler, then one gets one more supercharge: Qω = ωµνQ

µν . If X has isometries, saygenerated by the vector fields Va = V

µa ∂µ, then one gets extra supercharges: Qa = V

µa Qµ.

2. Prepotential of the N = 2 gauge theory

In this section we discuss a convenient way of parametrizing the effective action of thefour-dimensional N = 2 gauge theory.

2.1. N = 2 rigid special geometry

The four-dimensional N = 2 supersymmetric action can be written as an integral over thesuperspace:

LN=2 =∫

d4ϑ F(�) +∫

d4ϑ F(�). (25)

The classical action has the form (25) with F(�) = τ04π i Tr�2, where � is the superfield

taking values in the adjoint representation of the Lie algebra g. The effective Wilsonian actionis an infinite expansion in derivatives. However, the leading term, with two derivatives andfour fermions, at most has the form (25) with another F , and with � = (Ai ), i = 1, . . . , r ,in the Cartan subalgebra t ⊂ g, r = dim t = rkg. The superfield A can be expanded, in thiscase, as

Ai = ai + ϑψi + ϑϑF−,i + · · · (26)

and the Lagrangian becomes

LeffN=2 = τij (a)F

−,i ∧ F−,j + τij (a)F+,i ∧ F +,j + Im τij dai ∧ � daj

τij = ∂2F∂ai∂aj

The effective Abelian theory has a very special target space M, which is the moduli spaceof vacua of the effective theory. In fact, its geometry is special Kahler. Geometrically,M = (t ⊗ C) /W , where W is the Weyl group of G. The metric on M can be read off theLagrangian (27):

gij = Im τij .

The special feature of M is the integral affine structure it carries. What it means is that ai arespecial coordinates, defined up to some discrete transformations. On an ordinary manifold,with no special structure on it, the local coordinates are defined up to arbitrary smooth changesof coordinates. On the complex manifold we only allow holomorphic changes of coordinates.Our friend M has coordinates defined up to Sp(2r,Z) transformations. It is convenient to addto ai extra r ‘dual’ coordinates aD,i , which are related to ai via

aD,i = ∂F∂ai

. (28)

Then Sp(2r,Z) acts linearly on the vector

(ai aD,i)t .

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2.2. Higher dimensional point of view

Theories with extended supersymmetry can be viewed as reductions of higher dimensionalminimally supersymmetric field theories. This point of view can be useful, especially if oneconsiders compactification instead of reduction. One should take care of the regularization ofthe higher dimensional theory. As we shall discuss later in these lectures all the constructionswe employ have string theory realization, which makes them well defined in the ultraviolet.

2.2.1. Lift to five dimensions. The N = 2 gauge theory in four dimensions is a dimensionalreduction of the N = 1 five-dimensional theory. The latter theory needs an ultravioletcompletion to be well defined. However, some features of its low-energy behaviour arerobust [67].

In particular, the effective gauge coupling runs because of the vacuum polarization by thecharged particles. These particles are W-bosons (for non-Abelian theory), four-dimensionalinstantons, viewed as solitons in five-dimensional theory, and the bound states thereof.

We see from (27) that the effective gauge couplings are determined by the prepotential.We conclude, therefore, that the prepotential can be viewed as some sort of generating functionof the beta function contributions of the charged particles in the theory spectrum. Moreover,one can show that only BPS particles make a nonvanishing contribution to the prepotential,the long multplets giving total contribution zero [53].

To calculate the effective couplings we need to know the multiplicities, the masses, thecharges and the spins of the BPS particles present in the spectrum of the theory [24, 53]. Thiscan be done, in principle, by careful quantization of the moduli space of collective coordinatesof the soliton solutions (which are four-dimensional gauge instantons). Now suppose thetheory is compactified on a circle. Then the one-loop effect of a given particle consists of abulk term, present in the five-dimensional theory, and a new finite-size effect, having to dowith the loops wrapping the circle in spacetime [53].

If in addition the noncompact part of the spacetime in going around the circle is rotated,as we shall arrange below, then the loops wrapping the circle would have to be localized nearthe origin in the spacetime. Physically, the multiplicities of the BPS states are accounted forby the supersymmetric character-valued index [24]:∑

solitons

TrH(−)F e−rP5 er�·M erA·I (29)

where P5 is the momentum in the fifth direction, M is the generator of the Lorentz rotations,I is the generator of the R-symmetry rotations and r is the circumference of the fifth circle.Under certain conditions on � and A this trace has some supersymmetry which allows us toevaluate it. In the process one gets some integrals over the instanton collective coordinates, asin [11, 49, 69, 70]. As in [49], these integrals are exactly calculable, thanks to the equivariantlocalization.

2.2.2. Lift to six dimensions and twisted compactification. Consider lifting the N = 2 four-dimensional theory to the N = (1, 0) six-dimensional theory. This is done in a unique fashion.Vector multiplets lift to vector multiplets, and hypers lift to hypers.

Now compactify the N = (1, 0) theory on a 2-torus with the twisted boundary conditions(along both A and B cycles). As we go around a non-contractible loop � ∼ nA + mB, thespacetime and the fields of the gauge theory charged under the R-symmetry group SU(2)Iare rotated by the element (ei(na1+mb1)σ3 , ei(na2+mb2)σ3 , ei(na2+mb2)σ3) ∈ SU(2)L × SU(2)R ×SU(2)I = Spin(4) × SU(2)I . In other words, we compactify the six-dimensional N = 1

susy gauge theory on the manifold with the topology T2 × R4 with the metric and the R-symmetry gauge field Wilson line:

ds2 = r2 dz dz + (dxµ + �µν x

ν dz + �µν x

ν dz)2,

Aa = (�µν dz + �µν dz)ηaµν, µ = 1, 2, 3, 4, a = 1, 2, 3

where η is the anti-self-dual ’t Hooft symbol. It is convenient to combine a1,2 and b1,2 intotwo complex parameters ε1,2:

ε1 − ε2 = 2(a1 + ib1), ε1 + ε2 = 2(a2 + ib2). (31)

The antisymmetric matrices �, � are given by

� = ε1 + ε2

3 +ε1 − ε2

3 , � = ε1 + ε2

3 +ε1 − ε2

3 (32)

where we have used the decomposition SU(2)L × SU(2)R of the rotation group Spin(4).Clearly, [�, �] = 0. In the limit r → 0 we get four-dimensional gauge theory. We could alsotake the limit to the five-dimensional theory by considering the degenerate torus T2. We notein passing that the complex structure of the torus T2 could be kept finite. The resulting four-dimensional theory (for gauge group SU(2)) is related to the theory of the so-called E-strings[22, 46]. The instanton contributions to the correlation functions of the chiral operators in thistheory are related to the elliptic genera of the instanton moduli space [7] and could be summedup, giving rise to the Seiberg–Witten curves for these theories. We discuss neither elliptic nortrigonometric limits, even though they lead to interesting integrable systems [44].

2.2.3. �-background. The prepotential of the low-energy effective action can be madeobservable. To this end one should consider the theory in the so-called �-background. It isobtained in the r → 0 limit of the compactification on the torus with the metric (30).

The action of the four-dimensional theory in the limit r → 0 is not that of the puresupersymmetric Yang–Mills theory on R4. Rather, it is a deformation of the latter by the�, �-dependent terms. We shall write down here only the terms with bosonic fields (forsimplicity, we have set the bare theta angle to zero):

S(�)bos = − 1

µν +(Dµ� − �ν

λxλFµν

)(Dµ� − �ν

λxλFµν

)D = [�, �] + �ν

λxλDν� − �ν

λxλDν�.

We call the theory (33) an N = 2 theory in the �-background. By construction it is thedeformation of the ordinary N = 2 super-Yang–Mills theory. This deformation violatesPoincare invariance. It also violates supersymmetry. However, one can construct certainfermionic symmetries of the theory, which will commute to the spacetime rotation, insteadof translation. In this way the �-background supersymmetry is similar to the anti-de Sittersupersymmetry.

It will prove useful that �-deformation can be described as a superspace-dependent barecoupling τ0:

τ0(x, θ;�UV) = τ0 (�UV) + �−abϑ

b2 + �µν�µλx

νxλ. (34)

It is worth mentioning at this point that the (super)spacetime-dependent bare coupling issimilar, in spirit, to the spacetime-dependent regulator mass, which was found by ’t Hooft tobe quite useful in analysing the one-loop corrections to the effective action in the instantonbackground [32].

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2.3. Chiral observables and higher times

We are going to study the correlation functions of chiral observables. These observablesare gauge-invariant holomorphic functions of the superfield Ψ. Viewed as a function on thesuperspace, every such observable O can be decomposed:

O[�(x, θ)] = O(0) + O(1)ϑ + · · · + O(4)ϑ21ϑ

22 . (35)

The component O(4) can be used to deform the action of the theory; this deformation isequivalent to the addition of O to the bare prepotential. Thus ultimately the prepotential ofthe effective theory will depend on the infinite set of couplings �t , called higher times, whichare in one-to-one correspondence with the gauge-invariant polynomials on the Lie algebra ofthe gauge group:

F0(�a,�t, �).

We shall discuss them more thoroughly in the next lecture.

2.4. The prepotential comes out of the closet

The low-energy effective action depends on the bare gauge coupling, via dimensionaltransmutation. Moreover, the axial anomaly makes it also bare theta angle dependent. N = 2susy relates these two dependences, by making the prepotential a holomorphic function of thecomplex combination τ0 of the bare gauge coupling and the theta angle.

So the low-energy effective action has the form (β1 is the one-loop beta functioncoefficient)

Slow-energy =∫

d4x d4ϑF0( �A,�) + c.c.

� = �UV exp

(2π i τ0(�UV)

) (36)

where the difference with (25) is that we made the � dependence explicit. The significanceof this is that expression (36) also makes sense when � is promoted to the superfield. Infact, from the string theory perspective log� should be viewed as the scalar component of yetanother vector multiplet.

For our immediate purposes it suffices to make � superspace dependent, by substituting(34) into (36).

In this background one can completely integrate the vector multiplets remaining in theWilsonian action, the only parameter left being the value of the Higgs vev �a. Indeed, forconstant � the Wilsonian action contains the vector multiplets because they are massless, andintegrating them produces the usual infrared divergences. But in the �-background (34) thereare no infrared divergences left, and the non-trivial finite value of the integral

Seff(�a) =∫

d4x d4ϑF0(�a,�(x, ϑ))

is a good indication that (in the ordinary, � = const, background this integral would give∞ × 0):

Seff(�a,�) = F0(�a,�)

ε1ε2+ · · · (37)

where � = �(0, 0), and · · · stands for less singular, as ε → 0, terms, which come fromthe higher derivative couplings in the effective action. Similarly, the partition function of the

theory in the �-background would behave as

Z(�a,�, ε1, ε2) = exp

ε1ε2F0(�a,�)

+ε1 + ε2

ε1ε2F 1

2(�a,�) + F1(�a,�) +

(ε1 + ε2)2

ε1ε2G1(�a,�) + · · ·

). (38)

2.5. Instanton moduli space integration

The nice property of the chiral observables is the independence of their correlation functionsof the anti-chiral deformations of the theory, in particular of τ0.4 We can, therefore, considerthe limit τ0 → ∞. In this limit the term

τ0‖F +‖2

in the action localizes the path integral onto the instanton configurations. In the fixed instantonsector the index (29) can be, therefore, represented by the finite-dimensional integral over themoduli space of instantons.

2.5.1. Localization. The vev of the Higgs field shrinks these instantons to pointlike ones.This phenomenon is of course well known [32].

In addition, the �-background further localizes the measure on the instantons, invariantunder rotations. This is a relatively new idea. As a result, all integrations are eliminated,reducing them to the single sum over the pointlike-invariant instantons. One then has todesingularize and partially compactify the instanton moduli spaces to make these invariantconfigurations non-degenerate. This can be done explicitly for G = U(N) and implicitlyfor all classical gauge groups, using ADHM construction [6], and for all Lie groups usingAtiyah’s construction.

2.5.2. Factorization. Finally, the instanton moduli space has the following factorizationproperty. The instantons A1 and A2 of charges k1 and k2 respectively, which are widelyseparated, can be superposed in a sense that by a small perturbation A1 + A2 can be made tosolve the instanton equations, thus giving rise to the gauge field of the instanton charge k1 +k2.In this sense the moduli spaces of instantons allow some sort of multiplication (which gives asemi-group structure at the level of homotopy) [73]:

Mk1 × Mkp ↪→ Mk1+···+kp . (39)

In the sense of this multiplication the universal moduli space of instantons is an exponent

M = �∞k=0M′

k =′ exp[R4 × �∞

k=0Mcentrek

where Mcentrek is the quotient of the ordinary instanton moduli space by the action of the group

of translations. This exponentiation is nothing but the physically well-known phenomenon ofthe cluster decomposition. When applied to the partition function Z(�a, ε1, ε2,�) the clusterproperty (40) implies (38).

To recapitulate, by appropriately deforming the theory (in a controllable way) we achievethat the path integral has isolated saddle points and, thanks to the supersymmetry, is exactlygiven by the WKB approximation. The final answer is then the sum over these critical pointsof the ratio of bosonic and fermionic determinants. This sum is shown to be equal to thepartition function of an auxiliary statistical model, describing the random growth of the Young

4 However, beware of the holomorphic anomaly.

S88 N A Nekrasov

diagrams (for the gauge groups which are the products of unitary groups5). We describe thismodel in detail in the next lecture.

3. Random partitions and limit shapes

The partition function of the four-dimensional N = 2 gauge theory in the �-background canbe reduced to the sum of equivariant integrals over instanton moduli spaces:

Z(�a,�, ε1, ε2) =∞∑k=0

�2khZk(�a, ε1, ε2). (41)

For G = SU(N) these integrals can be evaluated explicitly [57]. In the limit ε1, ε2 → 0 wherethe �-background approaches flat space the partition function Z has the essential singularity:

Z(�a,�) = exp1

ε1ε2F0(�a,�) + less singular terms (42)

where F0 is the sought-for prepotential of the effective theory.

3.1. Instanton integrals

The trace (29) is given by the integral over the moduli space of instantonsMk , which calculatesthe trace of the element of the symmetry group ofMk on the space of holomorphic functions onMk . The moduli spaceMk is acted upon by the group G of constant gauge transformations, andby the group of isometries of R4. Actually the group SO(4, 1) of conformal transformations ofS4 acts there. However, the trace is only non-trivial when restricted to the compact subgroupof the global symmetry group. There, it is sufficient to consider only the elements of themaximal torus, which is the product of the maximal torus T of G and U(1) × U(1) whichcomes from the group SO(4) of rotations:

Z5dk (�a, ε1, ε2;β) = TrHk

(eβ�a × (eβε1 , eβε2)) (43)

where �a ∈ t = LieT. To get the four-dimensional theory we should take the limit β → 0. Inthis limit the trace (43) scales as

Z5dk (�a, ε1, ε2;β) ≈ 1

β2khZ4d

k (�a, ε1, ε2). (44)

For the classical gauge groups the moduli space Mk is the quotient of an affine algebraicvariety Nk by an algebraic group GD

k , where GD stands for the Corrigan–Goddard dual group(U(k) for G = SU(N), SO for Sp and Sp for SO) [14]. This allows us to represent thetrace in (43) as the integral over the maximal compact subgroup of the trace on the space ofholomorphic functions on the affine algebraic variety, see [59] for details.

3.2. Statistical mechanics of the instanton gaz: random partitions

In the case of G = SU(N) the integrals can be evaluated using residues. The poles arein one-to-one correspondence with N-tuples of partitions, �λ = (λ1, . . . , λN). The instantoncharge k is equal to the sum of the sizes of all partitions:

k = |�λ| ≡N∑l=1

|λl|. (45)

5 It seems that one gets the sums over Young diagrams for all classical gauge groups, but at the time of writing thishas not been shown to complete satisfaction, see [48, 59] for the current state of affairs.

The partition λl can be identified with the set of pairs of positive integers (i, j), 1 � j � λl,i ,where the non-negative integers λl,1 � λl,2 � λl,3 · · · represent the partition λl of the number

|λl| =∑i

λl,i .

To describe the residue corresponding to �λ we need further notation. Let

W =N∑l=1

eal , V�λ =N∑l=1

∑(i,j)∈λl

eε1(i−1)+ε2(j−1),

E�λ = W − V�λ(1 − eε1)(1 − eε2) M�λ = WW ∗ − E�λE∗�λ

(1 − e−ε1)(1 − e−ε2)

where we use the notation for X = ∑α exα , X∗ = ∑

α e−xα . Then

Zinstk (a, ε1, ε2) =

∑�λ,|�λ|=k

µλ(�a, ε1, ε2)

µλ(�a, ε1, ε2) =∏α

m�λ,α

M�λ =∑α

em�λ,α . (48)

The full partition function is obtained by summing (47) and then multiplying it by

Zpert(a, ε1, ε2;�) = exp

∑l,n

γ (al − an; ε1, ε2)

which can be viewed as the regularized contribution of the WW ∗/(1 − e−ε1)(1 − e−ε2) partin (46):

Z(a, ε1, ε2;�) = Zpert(a, ε1, ε2;�) ×∞∑k=0

�2kNZinstk (a, ε1, ε2). (50)

3.3. Limit shape and Seiberg–Witten curves

For small ε1, ε2 the sum over �λ in equation (50) has a saddle point [55]. The partitions whichcontribute most to the partition function are of large size,

|λl| ∼ 1

ε1ε2

for which the profiles of their Young diagrams can be approximated by smooth curves. Thesecurves can be glued together to a single real analytic curve. Here are the formulae:

f (x) =∑l

|x − al| +∑l,i

|x − al − ε1(i − 1) − ε2λl,i | − |x − al − ε1(i − 1)|

− |x − al − ε1i − ε2λl,i | + |x − al − ε1i|. (51)

The maximizing configuration f∗(x) obeys a certain equation [55]:∫dy(y − x)f ′′

∗ (y)log

(x − y

)= σ ′(f ′

∗(x)). (52)

S90 N A Nekrasov

where e = 2.718 28 . . . , and σ(y) is a concave, piecewise-linear function on [−N,N ] suchthat

σ ′(y) = ξl, y ∈ [−N + 2(l − 1),−N + 2l],

σ(−N) = −σ(N) = −∑l

Finally, the numbers ξ1 > · · · > ξN are the Lagrange multipliers, which should be arrangedso that the profile f∗(x) corresponds to the eigenvalues al of the Higgs field.

Equation (52) should be satisfied for any point x, for which f ′∗(x) is a point of continuity

of σ ′. When

f ′∗(x) ∈ {−N + 2l | l = 1, · · · , N − 1}

then considering the left and right derivatives separately one obtains the inequalities

Xf∗(x) ∈ (σ ′(f ′∗(x) − 0), σ ′(f ′

∗(x) + 0)), (53)

where, by definition,

](x) =

∫−y �=x

dy(y − x)

∣∣∣∣∣y − x

∣∣∣∣∣ − 1

)f ′′(y). (54)

The transform Xf is closely related to the standard Hilbert transform

[Hg] (x) = 1

∫−y �=x

dyg(y)

y − x. (55)

Indeed,

[Xf ]′′ = πH(f ′′).

3.3.1. Turning on the higher times. As we discussed before, the N = 2 gauge theories have arich set of exactly calculable correlation functions. Namely, the correlators of the observables(35) are saturated by instantons, and are given exactly by the one-loop approximation aroundinstanton solutions.

In what follows ν, λl denote partitions, l = 1, . . . , N, �λ = (λ1, . . . , λN). With eachpartition ν one associates the gauge theory operator Oν , which is simply the Schur functionsν evaluated on the Higgs field φ eigenvalues,

Z(�a; �t, ε1, ε2) =∑

�λµ�λ(�a, ε1, ε2) expO[φ�λ; �t] (56)

where O[X; �t] is a symmetric function of the eigenvalues of the matrix X of indefinite size,and �t are some coordinates on the space of all symmetric functions. For example, one couldtake �t = (tν),

O[X; �t] =∑ν

tνsν(X)

with sν being Schur functions. Or, large N gauge theory suggests expansion in multi-traceoperators: �t = (tk; tkl; tklm; . . .),

O[X; �t] =∑k

tk TrXk +∑k,l

tkl TrXk TrXl +∑k,l,m

tklm TrXk TrXl TrXm.

Finally, the matrix φλ is characterized by the Chern character:

Tr eβφ�λ =∑l

eβal +∑l,i

eβ(al+ε1(i−1))(eβε2λl,i − 1)(1 − eβε1) (57)

Z(�a; �t, ε1, ε2) = exp

ε1ε2F0(�a,�t) + · · ·

). (58)

The limit shape changes as a function of times �t . In fact, the limit shape should be viewednot as a curve, but rather as a (non-compact) calibrated Calabi–Yau manifold, i.e. a pair(curve, meromorphic differential). Krichever has associated the so-called Whitham integrablehierarchy with these data, and the natural expectation is that �t prove to be the times of thishierarchy [19, 26, 42, 47, 55]. However, this has not been established so far.

3.4. Extended symmetry point

It turns out that the partition function Z(�a,�t, ε1, ε2) has some magnificent properties whenε1 = −ε2 = h. We therefore introduce a special notation:

Z(�a,�t, h) = Z(�a,�t, h,−h). (59)

3.4.1. Dual partition function. The first special property is the relation to free fermions andintegrable hierarchies of Toda type: define

D(�ξ ; �t, h) =∑

�M∈ZN ,∑

Ml=p modN

e2π ih

�M·�ξZ(h( �M + �ρ); �t, h), (60)

which is the ‘quantum’ electro-magnetic dual partition function. The dual partition functionZD can be rewritten in terms of the N-tuples of partitions, as the original Z . statistical sumcould. However, and this is quite remarkable, one can also repackage these N-tuples ofpartitions into a single one, and then use the fermion representation of the Plancherel measureon the partitions to write ZD as a free fermion correlator [55]:

D(�ξ ; �t, h) = 〈p| eJ1h e�t · �W e�ξ · �J 0 e− J−1

h |p〉 (61)

where �J 0 stands for the zero modes of the u1(N) current algebra, J−1 represents the uN (1)current modes and �W stands for the W generators, corresponding to the symmetric functions.

3.4.2. Relation to topological strings. The second miracle that happens at the pointε1 + ε2 = 0 is the stringy nature of the ε-expansion. We shall of course discuss it inmore detail in the next lectures, but here we simply mention that in the expansion

Z(�a,�t, h) = exp

∞∑

h2g−2Fg(�a,�t) (62)

the terms Fg turn out to be the genus g topological string amplitudes. Moreover, there areseveral topological string theories which are behind (62). There are topological strings onlocal Calabi–Yau threefolds [37], which engineer the N = 2 gauge theory on R4. These showup most naturally in the considerations of Z5d—the five-dimensional analogue of Z . The roleof higher times �t is not understood in this case.

There are also topological strings on non-Calabi–Yau manifolds, such as P1, where thehigher times are mapped to the gravitational descendents [41], thanks to the free fermionrepresentation found in [62].

S92 N A Nekrasov

3.5. More general N = 2 theories: hypermultiplets and higher dimensions

3.5.1. Theories with matter. TheN = 2 theory with hypermultiplets in some representation Rof the gauge group can be treated similarly to the pure theory case. Consider the decompositionof R into the sum of irreducible representations:

R =⊕f

rf ⊗ Mf (63)

where Mf is the multiplicity space6. In the representation ring of U(N) the representation Rcan be expressed as a polynomial in the fundamental representation N and its conjugate N:

R = ⊕l,lmll ⊗ N⊗l ⊗ N⊗l (64)

where mll are the multiplicity spaces (they can be related to Mf , of course). The partitionfunction of the N = 2U(N) gauge theory with the matter hypermultiplet in the representation(64) is given by

Z(�a; m,�, h) =∑

�λ�|�λ|µ�λ(�a, h)ν�λ(�a,m, h)

ν�λ(�a,m, h) =∏�

�(�a, �λ,m)

∑l,l

Trmllem

E�λE∗�λ

(1 − e−h)(1 − eh)=

∑�

e�(�a,�λ,m)

and everything is understood with the regularization as in the case of pure gauge theory (i.e.with the functions γ in place of the infinite products and so on).

Formula (65) is a straightforward consequence of the fact that the matter contribution tothe instanton integral is the equivariant Euler class of the bundle over the moduli space ofinstantons, whose sections are the solutions of the Dirac equation in the instanton background,where the spinor transforms in the matter representation.

3.5.2. Quiver theories. One can similarly treat quiver theories. We consider the simplestcase: the U(N) × U(N) theory with bi-fundamental matter: (N, N). The generalization toother cases is straightforward. The partition function is given by

Z(�a1, �a2;�1,�2; h) =∑�λ1,�λ2

�|�λ1|1 �

|�λ2|2 µ�λ1

(�a1, h)µ�λ2(�a2, h)ν�λ1,�λ2

(�a1, �a2, h)

ν�λ1,�λ2(�a1, �a2, h) =

∏�

�(�λ1, �λ2; �a1, �a2, h)

E�λ1E∗

�λ2

(1 − e−h)(1 − eh)=

∑�

e�(�λ1,�λ2;�a1,�a2,h).

Note that the Z2 ‘orbifold’ of the N = 4 theory [36, 40], which corresponds to the theoryon the regular D3 branes at the C2/Z2 singularity, has the gauge group U(N) × U(N) andthe matter in the representations (N, N) × (N,N). For that theory in (66) one should useE�λ1

E∗�λ2

+ E�λ2E∗

�λ1instead of E�λ1

E∗�λ2

. The perturbative part of the partition function of thistheory is independent of the regulator mass if �a1 = �a2. This is in agreement with the well-known fact that the perturbative beta function in this theory vanishes for the coupling whichis the descendent of the N = 4 one. But we see that even for �a1 = �a2 instantons producenon-trivial non-perturbative corrections to the effective couplings.

6 Note that from the N = 1 susy point of view the matter chiral multiplets transform in the representation R ⊕ R∗.

3.5.3. Higher dimensional theories. The five-dimensional N = 1 supersymmetric theorycompactified on a circle, and the six-dimensional N = 1 theory compactified on a torus can beregularized by embedding in the string theory, as we describe below. The low-energy physicsis four dimensional, but the prepotential will depend on the radii of the compactification.See [55] for the five-dimensional partition function, [53] for the original integrable systemdescription of the prepotentials and [35] for the six-dimensional theory description.

3.6. Breaking susy down to N = 1: Dijkgraaf–Vafa proposal

N = 2 supersymmetry can be softly broken down to N = 1 by adding a superpotential W0(�)

to the action

LN=1 = LN=2 +∫

d2ϑ W0(�) + c.c. (67)

The choice of two out of four chiral thetas corresponds to the choice of N = 1 subalgebrawithin N = 2 superalgebra.

In their series of papers, originated in paper [17], Dijkgraaf and Vafa made a proposalfor the superpotential of the effective low-energy N = 1 theory. It basically consists of thefollowing: take the original superpotential W0 and consider the matrix model with the singleN × N matrix � with the action

IN = 1

VolU(N)

∫D� e− 1

gsTr W0(�)

. (68)

The effective superpotential is a function of glueball superfields S1, . . . , Sp, which correspondto the gauge symmetry breaking pattern: U(N) → U(N1) × · · · × U(Np), according to theminima of the tree level superpotential W0.

4. String theory

In this lecture we discuss the following string-theory-related issues: realizations of N = 2four-dimensional theories as low-energy sectors of string theory compactifications or D-branearrangements; �-background realizations; topological strings and their relation to the gaugetheory F-terms. In the next lecture we shall discuss the topological string calculations onR6 in more detail, relate them to the quantum spacetime foam picture and also present itssix-dimensional gauge theory realization.

4.1. String theory realizations of N = 2 theories

The N = 2 theory can arise as a low-energy limit of the theory on a stack of D branes in thetype II string theory. A stack of N parallel D3 branes in IIB theory in flat R1,9 carries theN = 4 supersymmetric Yang–Mills theory [78]. A stack of parallel D4 branes in IIA theoryin flat R1,9 carries N = 2 supersymmetric Yang–Mills theory in five dimensions. Uponcompactification on a circle the latter theory reduces to the former in the limit of zero radius.

Now consider the stack of N D4 branes in the geometry S1 × R1,8 with the metric

ds2 = dxµ dxµ + r2 dϕ2 + dv2 + |dZ1 + mrZ1 dϕ|2 + |dZ2 − mrZ2 dϕ|2. (69)

Here xµ denote the coordinates on the Minkowski space R1,3, ϕ is the periodic coordinateon the circle of circumference r, v is a real transverse direction, and Z1 and Z2 are theholomorphic coordinates on the remaining C2. The worldvolume of the branes is S1 × R1,3,which is located at Z1 = Z2 = 0, and v = vl, l = 1, . . . , N . Together with the Wilson loop

S94 N A Nekrasov

eigenvalues eiσ1 , . . . , eiσN around S1, vl form N complex moduli w1, . . . , wN , parametrizingthe moduli space of vacua. In the limit r → 0 the N complex moduli lose periodicity.

It is easy to check that the worldvolume theory has N = 2 susy, with the massivehypermultiplet in the adjoint representation (of mass m). This realization is T-dual to thestandard realization with the NS5 branes, as in [79]. Note that the background (30) is similarto (69). However, the D branes are differently located, a fact which leads to very interestinggeometries upon T-dualities and lifts to M-theory, providing (hopefully) another useful insight.

However, in our story we want to analyse the pure N = 2 supersymmetric Yang–Millstheory. This can be achieved by taking the m → ∞ limit, and at the same time taking the weakstring coupling limit. The resulting brane configuration can be described using two parallelNS5 branes and N D4 branes suspended between them, as in [79], or, alternatively, as a stack ofN D3 (fractional) branes stuck at the C2/Z2 singularity, as in [15]. In fact the precise form of thesingularity is irrelevant, as long as it corresponds to a discrete subgroup of SU(2), and all thefractional branes are of the same type. The relation between these two pictures is throughthe T-duality of the resolved C2/Z2 singularity. The fractional D3 branes blow up into D5branes wrapping a non-contractible 2-sphere. The resolved space T ∗CP1 has a U(1) isometry,with two fixed points (the North and South poles of the non-contractible 2-sphere). UponT-duality these turn into two NS5 branes. The D5 branes dualize to D4 branes suspendedbetween NS5s.

The instanton effects in this theory are due to the fractional D(−1) instantons, whichbind to the fractional D3 branes, in the IIB description. The ‘worldvolume’ theory on theseD(−1) instantons is the supersymmetric matrix integral, which we describe with the help ofthe ADHM construction below. In the IIA picture the instanton effects are due to EuclideanD0 branes, which ‘propagate’ between two NS5 branes.

The IIB picture with the fractional branes corresponds to the metric (before � isturned on)

ds2 = dxµ dxµ + dw dw + ds2C2/Z2

. (70)

The singularity C2/Z2 has five moduli in the IIB string theory: three parameters of thegeometric resolution of the singularity, and the fluxes of the NSNS and RR 2-forms throughthe 2-cycle, which becomes visible upon the blowup; otherwise all of these show up as thetwisted sector massless field. The NSNS and RR fluxes are responsible for the gauge couplingson the fractional D3 branes, as in [40]:

τ0 =∫

S2BRR + τIIB

BNSNS. (71)

Our conjecture is that turning on the higher Casimirs (and gravitational descendants on the dualclosed string side) corresponds to a ‘holomorphic wave’, where τ0 holomorphically dependon w. From [38] this is known to be a solution of IIB supergravity.

We shall return to the fractional brane picture later on. Right now let us mention anotherstringy effect. By turning on the constant NSNS B-field along the worldvolume of the D3branes we deform the super-Yang–Mills on R4 to the super-Yang–Mills on the noncommutativeR4

ζ , see [12, 13, 68]. On the worldvolume of the D(−1) instantons the noncommutativity actsas a Fayet–Illiopoulos term, deforming the ADHM equations, as in [1, 2, 58], and resolvingthe singularities of the instanton moduli space, as in [51]. This deformation is a technical tool,so we shall not describe it in much detail. The necessary references can be found in [68].

At this point we remark that even for N = 1 the instantons are present in the D-branepicture. They become visible in the gauge theory when noncommutativity is turned on.Remarkably, the actual value of the noncommutativity parameter ζ does not affect theexpectation values of the chiral observables, thus simplifying our life enormously.

So far we presented the D-brane realization of the N = 2 theory. There exists anotheruseful realization, via local Calabi–Yau manifolds, introduced in [37]. This realization is usefulin relating the prepotential to the topological string amplitudes. If the theory is embedded inthe IIA string on local Calabi–Yau, then the interesting physics comes from the worldsheetinstantons, wrapping some 2-cycles in the Calabi–Yau. In the mirror IIB description one gets astring without worldsheet instantons contributing to the prepotential, and effectively reducingto some field theory. This field theory is known in the case of global Calabi–Yau. But it isnot known explicitly in the case of local Calabi–Yau. As we shall show, it can sometimes beidentified with the free fermion theory on an auxiliary Riemann surface (cf [16, 55, 57]).

The relation to the geometrical engineering of [37] is also useful in making contact betweenour �-deformation and the sugra backgrounds with graviphoton field strength. Indeed, ourconstruction involved a lift to five or six dimensions. The first case embeds easily to the IIAstring theory where this corresponds to the lift to M-theory. To see the whole six-dimensionalpicture (30) one should use IIB language and the lift to F-theory (one has to set � = 0,though).

Let us consider the five-dimensional lift. We have M-theory on the 11-fold with themetric:

ds2 = (dxµ + �µ

ν xν dϕ

)2+ r2 dϕ2 + ds2

CY. (72)

Here we assume, for simplicity, that ε1 = −ε2, so that � = �− generates an SU(2) rotation,thus preserving half of susy. Now let us reduce on the circle S1 and interpret the background(72) in the type IIA string. Using [77] we arrive at the following IIA background:

gs = (r2 + ‖� · x‖2)34 (73)

Agrav = 1

r2 + ‖� · x‖2�µνx

µ dxν (74)

ds210 = 1√

r2 + ‖� · x‖2

(r2 dx2 + �µ

ν �λκ(x

2 dx2δνκδµλ − xνxκ dxµ dxλ))

+√r2 + ‖� · x‖2 ds2

where the graviphoton U(1) field is turned on. The IIA string coupling becomes strong atx → ∞. However, the effective coupling in the calculations of Fg is

h ∼ gs

√‖dAgrav‖2 ∼ (r2 + ‖� · x‖2)−

14 → 0, x → ∞. (75)

4.2. Topological strings

In this section we recall the appearance of the topological strings in the physical stringcalculations. We consider a type II superstring compactified on a Calabi–Yau threefold. Theeffective four-dimensional supergravity action contains the terms which are calculable by thetopological string.

4.2.1. Calabi–Yau compactifications. A critical superstring lives in ten dimensions. Welive in four macroscopic dimensions, so the remaining six dimensions should be unobservablysmall. One option is to have them compactified on a manifold M, which is severely constrainedby various symmetry requirements. For example, if we want to have four-dimensionalsupersymmetry, the manifold M has to be Calabi–Yau (if we want N = 1 susy we mustalso turn on some fluxes). In what follows we assume it is Calabi–Yau. It means that M is a

S96 N A Nekrasov

complex manifold, of complex dimension 3, which can be endowed with a Ricci-flat Kahlermetric. Yau’s theorem states that for a given complex structure, and the cohomology class ofthe Kahler form k, the Ricci-flat metric which has the corresponding Kahler form, exists and isunique. Existence of the Ricci-flat metric implies that c1(M) = 0, which, in turn, implies theexistence of nowhere vanishing holomorphic 3-form �, which is unique up to a multiplicationby a complex number.

4.2.2. Effective supergravity in four dimensions. In these circumstances the low-energyphysics in four dimensions is described by the N = 2 supergravity (sugra) theory. Sucha theory contains a gravity supermultiplet, whose bosonic fields are the metric gmn,m, n =1, 2, 3, 4, and the vector field bm, called a graviphoton. In addition, there are matter multiplets,vectors and hypers. Vector multiplets contain complex scalars φa, a = 1, . . . , r , and U(1)vector fields Aa

m. The hypermultiplets contain two complex scalars Qi, Qi, i = 1, . . . , s. Inthe IIA string r = h1,1(M), and s = h1,2(M). Mirror symmetry would exchange IIA withIIB and M with M—the mirror Calabi–Yau.

4.2.3. Vector multiplet moduli space. Let us work, for definiteness, with the IIA string. Thenthe vector fields in four dimensions come by reduction of the U(1) RR gauge field C1 in tendimensions and by reduction of the RR 3-form C3 along 2-cycles in CY M. The vector field C1

becomes b and falls into the gravity multiplet. The graviphoton field strength T = db plays aspecial role in what follows. The scalars of the vector multiplets come from the complexifiedKahler moduli of Calabi–Yau M—the real part parametrizes the Kahler class k, which fixessome of the metric moduli, while the imaginary part comes from the periods of the NSNSB-field: φa = ∫

βaω, ω = k + iB.

The geometry of the vector multiplet moduli space Mv is governed by the holomorphicfunction, the prepotential F0(φ

a). The effective Lagrangian for the vector multiplets looks asfollows:

Im τab dφa ∧ � dφb + τabF−a ∧ F−

b − τabF+a ∧ F +

τab = ∂2F0

∂φa∂φb.

The geometry in (76) is projective special Kahler. It differs from the special Kahler geometry(27) in the following respect: the special coordinates φa in supergravity are defined up torescaling, and the prepotential is the homogeneous degree 2 function of these homogeneouscoordinates. Globally the special coordinates are defined up to the Sp(2r + 2,Z) × C∗

transformations. In the rigid supersymmetry case the special coordinates are defined up toSp(2r,Z) transformations.

In addition, there are couplings between the vector multiplets and the gravity multiplet:∞∑g=1

Fg(φa)T 2g−2R ∧ R (77)

whereT 2 = T −mnT

−mn. Note that the string dilaton does not show up among the vector multiplets.

4.2.4. Hypermultiplet moduli space. The moduli space Mh of hypermultiplets hasquaternionic-Kahler geometry. The hypermultiplet scalars come from dilaton, axion (dual tothe Bmn in four dimensions), and the Calabi–Yau C3 period, corresponding to the (3, 0)-form�:

C3 ∧ �, and∫M

C3 ∧ � (this gives rise to the so-called universal hypermultiplet); andthen h1,2(M) multiplets where Qi corresponds to the complex structure deformations of M,and Qi to the remaining periods of C3.

4.2.5. Topological strings and vector multiplets. The vector multiplet couplings Fg(φ), g �0, can be calculated using a simplified version of string theory. In the IIA superstring contextit would be the A type topological string, and in the IIB context it would be, not surprisingly,the B type topological string. The topological strings were introduced by Witten [80], andtheir importance for the superstring effective theory calculations was understood in [3, 8]. Thetype A string calculates Fg by summing over the holomorphic maps of the genus g Riemannsurfaces into M:

Fg =∑

β∈H2(M,Z)

e− ∫βω

∫Mg(M,β)

1 (78)

where Mg(M, β) is the moduli space of stable maps, introduced by Kontsevich [39]. Theintegral of 1 counts these maps, if the moduli space of stable maps consists of isolated points.Its expected dimension is zero, so this is indeed the answer in the generic situation. However,it could happen that the actual dimension is positive. Then one has to integrate the Euler classof the obstruction sheaf over the real moduli space. In this way, say, every isolated rationalcurve in the primitive homology class β would contribute to F0 not just a single exponentiale− ∫

βω but the whole series of multicover contributions:

F0 =∑

β∈Hprimitive2 (M,Z)

nβLi3(

e− ∫βω). (79)

The B model definition of the Fg amplitudes is rather tricky, except for the genus zero part.Suffice it to say that it can be computed by purely classical means, by calculating the periodsof the (3, 0)-form �.

The amplitudes Fg(φ) are naturally combined into a generating function:

Z(φ; h) = exp∞∑g=0

h2g−2Fg(φ) (80)

where h is the topological string coupling constant. In terms of the original superstringproblem, h2 ∼ g2

s F2, where gs is the IIA string coupling constant and F 2 is the Lorentz-

invariant of the graviphoton field strength.The partition function Z(φ; h) is expected to obey various remarkable equations [8], and

also to have some modular properties.

4.2.6. Kodaira–Spencer theory of gravity. The great simplification of the topological stringcompared to the physical string is the reduction, for the B type topological string, of theintegrals over the moduli space of Riemann surfaces to the locus of maximally degeneratesurfaces [8]. The combinatorics of such surfaces is that of trivalent graphs. Thus the generatingfunction of the B type topological string amplitudes can be written as the sum over Feynmandiagrams of some field theory. This theory seems to be the so-called Kodaira–Spencer theoryof gravity, introduced in [8]:

LKS = 1

[A∨ 1

∂∂A∨ +

3(A ∧ A)∨ ∧ A∨

A ∈ �−1,1(M),A∨ = ιA� ∈ �2,1(M), (A ∧ A)∨ = ιA∧A� ∈ �1,2(M).

S98 N A Nekrasov

4.2.7. Kahler gravity. The mirror version of the theory (81) is much less understood.The problem is that the worldsheet instantons contribute to the effective action. However,in the large radius limit the theory looks very much like (81) modulo the standard mirrorreplacements: A∨ ↔ ω, ∂ ↔ d, ∂ ↔ dc [9]:

LKG = 1

dcdω +

3ω ∧ ω ∧ ω

]. (82)

4.3. Topological string amplitudes and N = 2 theories in �-background

Now let us return to the discussion of N = 2 gauge theories and their F-terms, i.e. the termsin the effective action, which are obtained by the integration over half of the superspace. TheF-terms have two faces. On the one hand, they are calculable within gauge theory, by doingthe instanton moduli space integration, the way we discussed in the first two sections. On theother hand, in the string theory realization of the gauge theory they are given by the topologicalstring amplitudes.

There is one non-trivial point in this identification. As we argued before, using the lift toM-theory, it is the partition function in the �-background that captures the spin-charge particlecontent of the gauge theory, and contains all the information about the effective couplings.On the other hand, the simplest way to relate the physical and topological strings proceeds viaconsideration of the constant graviphoton field strength background [3, 8].

Note that the general �-background depends on two parameters ε1, ε2, since genericSO(4) rotation is characterized by two angles. As we discussed in (30), one has to turn on theR-symmetry Wilson lines in order to preserve fermionic symmetry, if ε1 + ε2 �= 0. For generictype II string theory compactifications R-symmetry does not exist as a global symmetry. Sowe may hope to relate simply the [3, 8] calculations in string theory to our calculations in gaugetheory only for the special �-backgrounds, which do not touch R-symmetry. To understandthe general case, with two parameters ε1, ε2, one has to work harder7. Now, for the specialcase of the �-background, ε1 = −ε2 = h, using its M-theory realization and the resultingtype II sugra background (74), we can observe that one has the non-trivial graviphoton fieldstrength. Even though it is not constant, one may hope that since the effective string couplinggoes to zero at distances much larger than r

|ε| , the higher genus string does not probe these

regions of R4; it is effectively confined near the origin, where the graviphoton field strength isalmost constant, and the twisting arguments of [8] apply.

4.3.1. Localization. The most powerful method of exact calculations in the supersymmetrictheories is the localization on the Q-fixed points, where Q is the conserved global supercharge.

The localization can be applied to the calculations of Z(φ; h) for toric varieties X, usingthe worldsheet definition [39].

Imagine calculating the Kahler gravity partition function in some background. Assumingthe background has isometries, one can set up an equivariant supercharge whose fixed pointswould correspond to the geometries with the same asymptotics as the background, and inaddition invariant under the isometry group action. This is in complete parallel with theN = 2 supersymmetric gauge theory partition function calculation [55, 57].

Going back to our gravity problem, we consider toric Calabi–Yau X0 as our background.The toric CY has T3 as a group of isometries. The Kahler gravity partition function ZKG(X0),evaluated by localization with respect to T3, would be expressed as a sum over all toric X,with the same asymptotics as X0.

7 Vafa points out that such parameters are present, if at all, only for the compactifications on rigid CYs.

As we shall explain in the next lecture, this method of evaluating the partition functionleads to the quantum spacetime foam picture in the topological string theory.

5. Quantum gravitational foam

5.1. Spacetime geometry in string theory

Since the remarkable realization by Scherk and Schwarz [66] that a string spectrum containsgravitons one is looking for the stringy approach to quantum gravity. One feature of quantumgravity—strong fluctuations of spacetime topology at Planck scales, advocated by Hawking[29]—has been very elusive. The reason is that the conventional approach to string quantizationproduces a S-matrix in the form of the expansion in the string coupling constant. TheS-matrix describes, at best, the scattering of the gravitational waves in some stringybackground, most commonly the Minkowski background (however, recent advances inAdS/CFT duality allow us to hope for other homogeneous spaces as well). At finite orderof string perturbation theory one has a finite number of gravitons, so it is unlikely that thestrong fluctuation of the spacetime topology would be seen. The only hope is to resume stringperturbation theory, perhaps analytically continuing in the string coupling:

exp∞∑g=0

gs2g−2Fg(a) + O

(e− 1

∫bndry cndts∼a

DgµνD(. . .) exp − Seff(gµν, . . .) (83)

where gs denotes string coupling, and . . . stand for superpartners and massive string modes.But to do that one needs an example of all-loop string calculation. For conventional gravitonscattering on R10 no-one has been able to go beyond two loops so far [30]. However, recently,using the advances in the topological string theory, where all-loop exact string amplitudes arenot uncommon, the quantum foam picture has indeed emerged. It is the purpose of this lectureto explain how it came about.

5.2. Type A string on R6

The simplest, but already inspirational example comes from the Gromov–Witten theory onC3—the simplest Calabi–Yau manifold. This theory needs regularization because of thenoncompactness of C3, but this can be cured by using the torus T3 action. Then the explicitcalculation [20] gives

Fg =∫

c3g−1(H) = B2gB2g−2

2g(2g − 2)(2g − 2)!. (84)

The all-genus partition function is then

Ztop(gs) = exp∞∑g=0

Fgg2g−2s ∼ M

12 (q)

M(q) =∞∏n=1

(1 − qn)n

q = −e−igs .

To pass from Ztop(gs) to M(q) requires adding some unstable, e.g. F0 = ζ(3), as well asnon-perturbative terms, which are fixed in [24, 64].

S100 N A Nekrasov

5.3. Crystal interpretation

5.3.1. MacMahon function. The function M(q) which enters (85) is well known incombinatorics [43, 63]. It counts three-dimensional partitions:

M(q) =∑

π−3d partitions

q |π | (86)

5.3.2. Donaldson–Thomas (DT) theory. The reason three-dimensional partitions pop up inour calculation is explained in [45, 56]—they are T3-invariant ideal sheaves: the partitionπ corresponds to the ideal Iπ in C[x, y, z]: (i, j, k) ∈ Z3\π ⇔ xi−1yj−1zk−1 ∈ Iπ . Thesummation over the partitions is actually the fixed-point formula for the integral of 1 over themoduli space of ideal sheaves, weighted by qch3(Iπ ).

For more general toric X0 the DT theory produces the partition function which is

ZDT(X0) =∫

Mideal(X0)

qch3(I ) e− ∫X0

ω0∧ch2(I ) (87)

where Mideal(X0) is the moduli space of ideal sheaves I, ch1(I ) = 0, on X0.

5.3.3. From two- to three-dimensional partitions: gauge theory in higher dimensions. Idealsheaves have showed up in physics in some other problems already. In some sense theyare singular limits of U(1) instantons. In this same sense the chp of a sheaf corresponds to1p!

)∧p, where F is the curvature of the gauge field.

In the previous sections we saw that the instanton calculus in four-dimensional gaugetheories with N = 2 supersymmetry reduces (for gauge group SU(N)) to the statisticalmodel on N-tuples of partitions. In two lines the idea of the correspondence is the following:instantons correspond to holomorphic bundles, and moduli of bundles are compactified bytorsion-free sheaves; the latter differ from bundles by ideals of zero-dimensional schemes.When working equivariantly with respect to the torus action on the moduli space the integralsof interest reduce to the summation over the torus-invariant schemes. These are labelled bytwo-dimensional partitions.

For three complex dimensional manifolds the analogous considerations suggest that three-dimensional partitions should come into play as labelling the fixed points of the torus actionon the compactified moduli space of holomorphic bundles.

The corresponding field theory problem looks as follows. One studies connections A

on a bundle E over Kahler three complex dimensional manifold X. In addition, one takesϕ ∈ �0,3 ⊗ ad E , and its complex conjugate. The BPS equations are

F 0,2 + ∂†ϕ = 0, F 2,0 + ∂†ϕ = 0, F 1,1 ∧ k ∧ k + [ϕ, ϕ] = 0 (88)

where k is a Kahler form.When we work on R6 we can again turn on the �-background, which this time will depend

on three parameters ε1, ε2, ε3. Instead of the summation over two-dimensional partitions, asin (47), we would get the sum over three-dimensional ones (for U(1) theory):

Z(q, ε1, ε2, ε3) =∑π

q |π |µπ(ε1, ε2, ε3) (89)

where the three-dimensional measure on partitions, the so-called equivariant vertex [54]measure, is defined as follows.

For the partition π define the following functions of three variables (q1, q2, q3):

Vπ(q1, q2, q3) =∑

(i,j,k)∈π

qi−11 q

j−12 qk−1

Hπ(q1, q2, q3) = 1

(1 − q1)(1 − q2)(1 − q3)− Vπ

Mπ(q1, q2, q3) = −(1 − q1)(1 − q2)(1 − q3)Hπ(q1, q2, q3)Hπ

(q−1

1 , q−12 , q−1

which describe the content of the partition, its complement and the deformation complex ofthe corresponding ideal sheaf. The equivariant vertex measure is given by

µπ(ε1, ε2, ε3) =∏

�+�+(ε1, ε2, ε3)∏

�− �−(ε1, ε2, ε3)(91)

where the linear functions of three variables �± are found from

Mπ(q1, q2, q3) = q1q2q3

(1 − q1)(1 − q2)(1 − q3)−

∑�+

e�+(ε1,ε2,ε3) +∑�−

e�−(ε1,ε2,ε3)

q1 = eε1 , q2 = eε2 , q3 = eε3 .

The partition function (89) is shown in [45] to be equal to

Z(q, ε1, ε2, ε3) = M(q)− (ε1+ε2)(ε1+ε3)(ε2+ε3)

ε1ε2ε3 . (93)

In particular, if the �-background is special, in the sense that the SO(6) rotation generated by(ε1, ε2, ε3) belongs to SU(3), i.e. preserves the Calabi–Yau structure on R6, then the partitionfunction reduces to the MacMahon function (86) which, as we saw already, counts the typeA topological string amplitudes on R6 (up to an important issue of squaring the partitionfunction8).

The fully equivariant partition function (93), not very surprisingly, agrees with theequivariant Gromov–Witten (GW) partition function (the calculation on the GW side is almostidentical to (84), thanks to Mumford’s relations).

5.4. Kahler gravity interpretation

The three-dimensional partition point of view allows us to relate the topological string to thewould-be Kahler gravity calculation. The idea is to interpret ω0 − igsF as the Kahler form ωX

on the manifold X, which is obtained from C3 by the blowup along the ideal Iπ . The weighte− ∫

X0ω0∧ch2qch3 corresponds to

(− 1

ω ∧ ω ∧ ω

which is the value of the Kahler gravity action (82).

5.5. Summary of the last two lectures

We have shown that the all-genus topological string partition function can be interpreted asa Kahler gravity partition function [33], which contains the sum over various space-timetopologies, in accordance with the quantum foam expectations [29]. It has been shown in[56] that the non-perturbative completion of the type A string on Calabi–Yau manifold Mmust include the D-brane contributions which in turn can be expressed using the topological

8 To get it fair: square.

S102 N A Nekrasov

string field theory

limit shape

T-duality S-duality

Gromov-Witten

Seiberg-Witten

Donaldson-Witten

Donaldson-Thomas

Figure 1. Z-theory.

(This figure is in colour only in the electronic version)

B string. This is the manifestation of S-duality in the context of the topological string. Theexistence of such duality suggests a more fundamental theory, perhaps higher dimensional,which would explain this duality geometrically. Indeed, in the physical superstring contextthe S-duality is a geometric symmetry of M-theory.

The full topological string partition function would therefore depend on both the Kahlerand complex moduli of Calabi–Yau, in other terms, on all the metric moduli. The natural wayto combine these moduli is in the G2 moduli of the seven-dimensional manifold B = S1 ×M .Recently, Hitchin has proposed a functional on the space of closed 3-forms on B whose criticalpoints are the metrics of G2 holonomy. Its quantization is currently being investigated, see[18, 60], and also [23].

6. Summary

In these lectures we have covered in some detail the exact non-perturbative calculations infour-dimensional N = 2 systems, both from the gauge theory and the string theory points ofview.

We gave an introduction to the gauge theories, their perturbation theory, effectiveactions, non-perturbative effects, notably those coming from instantons. We then proceededwith supersymmetry, studied the minimal and extended susy, their representations, twistedsupersymmetry, discussed topological gauge theories and invariants of four-manifolds. Afterthat we introduced the �-backgrounds and studied the partition functions of the gauge theoriesin these backgrounds, including the chiral observables. We mostly talked about N = 2theories, but also mentioned briefly the superpotential calculations of Dijkgraaf and Vafa inthe N = 1 context.

We then discussed the string theory realizations of N = 2 systems and the string theorypoint of view on the same partition functions. In particular, we saw that the instantoncalculations in the gauge theory are mapped to the topological string calculations on the stringtheory side.

This is one of the manifestations of the existence of Z-theory, the framework whichencompasses all the F-term calculations in string and gauge theories [60].

We then discussed an interesting corner of Z-theory, where the Gromov–Witten theory (thetype A topological closed string) meets the Donaldson–Thomas theory (the type B topologicalopen string). The Donaldson–Thomas computation was interpreted using the Kahler theory ofgravity as realizing the summation over the space–time topologies, which one long suspectedtakes place in any quantum gravity theory. Thus, one of the outcomes of Z-theory is the stringtheory realization of the quantum space–time foam.

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