introduction in six dimensions n - brandeis...

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification

Finite subgroups ofE8

Superconformal field theoriesin six dimensions

David R. Morrison

University of California, Santa Barbara

Recent Progress in String Theory and Mirror Symmetry

Brandeis University

7 March 2015

Based on work done with M. Bertolini, M. Del Zotto,J. J. Heckman, P. Merkx, D. Park, T. Rudelius, and C. Vafa

arXiv:1312.5746, arXiv:1412.6526, arXiv:1502.05405,

arXiv:1503.?????

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Introduction

I The maximum spacetime dimension in which asuperconformal field theory is possible is six (Nahm).

I The degrees of freedom in such a theory are notdescribed by particles, but the theory is a local quantumfield theory (Seiberg and others).

I The worldvolume quantum field theory for a (stack of)M5-branes is a six dimensional superconformal fieldtheory (with maximal supersymmetry).

I Compactification of the maximally supersymmetrictheory has led to a host of interesting theories in lowerdimesions (very active area of research since 2009).

I In this talk, we will focus instead on the minimallysupersymmetric theories, that is, theories withN = (1, 0) supersymmetry.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


N = (1, 0) superconformal field theories

I The conformal symmetry of these theories is so(6, 2).

I The superconformal algebra is described with 8supersymmetry generators Qi and 8 superconformalgenerators Sj .

I The theory has an su(2) R-symmetry.

I These theories typically have nontrivial global (flavor)symmetries.

Multiplets in a 6D supersymmetric theory:

I Gravity multiplet (gµν ,B+µν , fermions)

I Tensor multiplet(s) (S ,B−µν , fermions)

I Vector multiplet(s) (Aµ, fermions)

I Hypermultiplet(s) (φR , fermions)

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Strings and their tension

I Having Bµν ’s implies that there are strings in thesetheories, and a BPS lattice Λ ⊂ R1,T .

I The scalars S1, . . .ST are naturally parameterized by aspace whose universal cover is SO(1,T )/SO(T ).

I The expectation values 〈Si 〉 control the string tensions.

I When 〈Si 〉 = 0 we get a tensionless string, and weexpect a local superconformal field theory to describe it.

Non-zero expectation values of Si parameterize the Coulombbranch of the theory, and this is where the theory is typicallystudied in detail.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Anomaly-free gauge fields

I On the Coulomb branch, the gauge fields are potentiallysubject to an anomaly.

I The Green-Schwarz-West-Sagnotti mechanism specifiesmodified Bianchi identities for gauge fields in thesetheories, which gives the possibility of an anomaly-freetheory: potential anomalies from fermions in mattermultiplets are cancelled by the gauge field anomalywhich follows from the modified Bianchi identity.

I In practice, this requirement is a severe constraint onthe gauge groups and matter representations.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Anomaly-free gauge fields

I For example, one solution of the anomaly constraintsallows an arbitrary number of copies of su(N), labeledby an integer j = 1, . . . , p, together with (N,N)bi-fundamental matter charged under su(N)j andsu(N)j+1, with an extra N fundamentals for su(N)1 andan extra N fundamentals for su(N)p.

I In fact, these theories also have a global symmetrySU(N)× SU(N) which can be thought of as occupyingthe 0th and (p+1)st positions in the chain, so that all ofthe matter sits in bi-fundamentals. (Sometimes it’sgauge-global bi-fundamentals.)

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Other anomaliesI The global symmetry group G of such a theory is

typically anomalous, and cannot be directly gauged.However, in many cases one can add additional (free)hypermultiplets charged under G to render thecombined theory anomaly-free.

I For example, in the su(N)⊕p theory described above,the global symmetry SU(N)p+1 is anomalous unless anadditional N fundamentals for this group are added tothe theory. Then the combined theory has ananomaly-free global symmetry, which can be gauged(using the GSWS Bianchi identity), extending the chainby 1.

I One can also ask if the theory can be coupled togravity, a necessary condition for which is that thegauge-gravity anomaly vanish (using the modifiedBianchi identity for the graviton as well).This was studied in some nontrivial cases in [Del Zotto,

Heckman, Morrison, Park].

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Examples

I The N = (2, 0) theories have an ADE classification.

I A stack p of M5-branes gives one model for theN = (2, 0) theory of type Ap−1.

I The heterotic e8 ⊕ e8 string compactified on a local K3surface with a point-like instanton with instantonnumber k provides a family of examples of N = (1, 0)theory.

I Using the Horava–Witten model of the heterotic string,the previous example can be viewed as p M5-branesdissolved into the boundary M9-brane, creating apoint-like instanton in real codimension 4.

I More generally, one can put p point-like instantons atthe singular point of an ALE space C2/Γ.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


1

These examples can be studied in F-theory, usingF-theory/heterotic duality [Aspinwall-Morrison ’97].

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


6D SCFTs from F-theory

I F-theory is a quantum gravity theory which allows anontrivial axio-dilaton profile, exploiting the SL(2,Z)symmetry of the type IIB string.

I The data of an F-theory compactification is provided bya base space B, a complex line bundle L on B, andsections f and g of L⊗4 and L⊗6, respectively,determining a Weierstrass equation

y2 = x3 + fx + g .

I If B has complex dimension 2 and if the total space ofthe elliptic fibration can be blown up to a Calabi–Yauthreefold, we get a 6D supersymmetric theory.

I The strings arise from wrapping D3-branes alongcomplex curves in B, and their tensions are supplied bythe areas of these curves. Thus, to get a tensionlessstring limit, we need a configuration of algebraic curveswhich are contractible to a point.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


I More generally, we can study a local model B which is aneighborhood of a curve collection {Σj}. The keycondition for contractibility is that the intersectionmatrix (Σj · Σk) be negative definite. This matrix alsodefines the lattice of BPS string charges.

I The key quantity for understanding these models is thediscriminant locus

∆ := {4f 3 + 27g2 = 0}.

This typically contains some or all of the Σj ’s ascomponents with some multiplicity. The Kodairaclassification determines the type of singular fibers andalso (when supplemented by Tate’s algorithm) thegauge algebra associated with each Σj .

I There can be additional, non-compact components of∆ in B. These are associated with global symmetries.Detailed investigation: [Bertolini, Merkx, Morrison].

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Pointlike instantons on C2/Γ, as studied by [Aspinwall-Morrison

’97].

II!

II!

12 + n

12 ! nC0

C"

Figure 2: Point-like E8 instantons in the simplest case.

figure the curly line represents the locus of I1 fibres. This will be the case in all subsequentdiagrams. The overall shape of this curve is meant to be only schematic. (In particular, wehave omitted the cusps which this curve invariably has.) The important aspect is the localgeometry of the collisions between this curve and the other components of the discriminantwhich we try to represent accurately.

This is the F-theory picture of the physics discussed in [6] that each point-like instantonleads to a massless tensor in six dimensions (here represented as a blowup of the originalbase Fn). We also see that 12 ! n of the instantons are associated to one of the E8 factorsand the other 12 + n are tied to the other E8 [14]. After blowing up the base however, onemay blow down in a di!erent way to change n. Thus after blowing up, it is not a well-definedquestion to ask which E8 a given instanton is associated to.

Now consider what happens to this picture as we go to the stable degeneration. That is,what happens to the F-theory picture when the heterotic K3 surface, on which the 24 point-like instantons live, becomes very large? Along every rational fibre, f , of the Hirzebruchsurface, Fn, the process as discussed in section 3 will occur. That is, every rational fibre willbreak into two fibres. Thus our Hirzebruch surface, Fn, will break into two surfaces whichmay be viewed as a (P1 "P1)-bundle over P1. The result is shown in figure 3, where C! is thecurve along which the two irreducible components of the base now meet. We see that X hasbroken into two irreducible threefolds (“generalized Fano threefolds”) which meet along anelliptic surface with base C!,4 which is actually a K3 surface as we shall demonstrate below.

Before the degeneration, if we had restricted the elliptic fibration !F : X # Z to one ofthe rational fibres, f , of the Hirzebruch surface we would have found an elliptic K3 surface.Now when we look at this elliptic fibration restricted to one of the P1’s into which f hasbroken, we find a rational elliptic surface instead. Let us focus on the elliptic fibration, X1,over the lower component of the surface in figure 3. Given that the curve, C0, of II! fibreswas preserved in this process, this new irreducible component will still have a section of

4We will assume that C! is parallel to the two lines of II! fibres as shown in 3. If it is not, we may blowup and blow down in order to make it so. This is equivalent to reshu!ing the distribution of instantonsbetween the two E8’s.

14

4.2 The J = 0 series

We may now employ our knowledge of the stable degeneration to answer more di!cultquestions about point-like instantons. This will be a fairly involved process in the generalcase so we will start with the simplest cases. First recall that all of the Kodaira fibres canbe associated with a particular value of the J-invariant of the elliptic fibre, except for I0 andI!0 for which J may take any value (see, for example, page 159 of [33]).

In this section we are going to force a “vertical” line of bad fibres (along an f direction)into the discriminant so that it has a transverse intersection with the “horizontal” line of II!

fibres along C0 without any additional local contributions to the collision from the rest of thediscriminant. One may show [34] that such intersections of curves within the discriminantmust correspond to fibres with the same J-invariant. In this section we require J = 0 whichcorresponds to Kodaira types II, IV, I!0, IV!, and II!. In each case, the order of vanishingof ! is twice the order of vanishing of b, with a playing no significant role. Thus, to analyzethe J = 0 cases we need only concern ourselves with the geometry of the divisor B".

For example, let us consider the case illustrated in figure 4 in which we add a verticalline of II! fibres along the f direction. To do this, we must subtract 5f from B" whichimplies that what remains can only produce 7 ! n and 7 + n simple point-like instantons ofthe type we discussed above. It is therefore clear that, whatever else we may have done toproduce this extra line of II! fibres, we have had to “use up” ten of the instantons. Notethat B" intersects f twice, producing collisions between the I1 part of the discriminant andthe vertical line of II! fibres as shown.

C0

C#II!

II!

II!

7 + n

7 ! n

Figure 4: 10 instantons on an E8 singularity.

Now when we consider the stable degeneration of this model, we cannot avoid havingthe new line of II! fibres pass through C!. Therefore SH has an orbifold singularity of typeE8 (i.e., locally of the form C2 divided by the binary icosahedral group). We claim thatthis geometry represents 10 point-like instantons sitting on an E8 quotient singularity in SH .This is consistent with our earlier assertion that the vertical lines determine the location ofthe point-like instantons; we will discuss this point more fully in section 4.3.

16

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


II! C06 ! n

!!

!!

!!!

""

""

"""

C"II!7 + n

#

Blow up

C"II!

II! C0

II!

7 + n

6 ! n

II!

II!

Figure 5: 11 instantons on an E8 singularity.

We may follow this same method for analyzing a vertical line of IV, I!0, or IV! fibresreplacing the vertical line of II! fibres in figure 4. These produce A2 (i.e., C2/Z3), D4, or E6

singularities on the K3 surface, SH , respectively. The other J = 0 case, namely a fibre oftype II, produces no singularity on SH and no interesting nonperturbative physics.

In table 2, we show the result of allowing k point-like E8 instantons to coalesce on asingularity of type C2/G. The local contribution to the number of massless tensors and tothe gauge algebra is listed assuming a given bound on k.

4.3 Fewer instantons

In each entry in table 2 imposing the vertical line of bad fibres forced a minimum numberof instantons into the quotient singularity. How do we analyze the situation when there arefewer instantons within the singularity?

Let us begin with the E8 quotient singularity in SH again. Consider the elliptic fibrationgiven by the lower half of figure 3 after the stable degeneration has occurred. In particularwe are interested in B#, the divisor associated to b after the contribution from the line of II!

19

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


I The intersection of two E8 branes (Kodaira type II ∗) isassociated to a Weierstrass model whose blowup is aCalabi–Yau threefold X whose map to the base B has a(complex) two-dimensional fiber over the intersectionpoint.

I In other words, when we compactify on a circle to getan M-theory model, we find an infinite tower of lightstates (from wrapping an M2-brane over arbitraryalgebraic curves within the two-dimensional fiber). Thisis another signal of conformality in the parentsix-dimensional theory.

I On the other hand, by blowing up the base B, we canensure that all fibers X → B are one dimensional. Thisis the Coulomb branch of the theory.

I Aspinwall and I worked this out in detail for the collisionof two II ∗ fibers.

The elliptic fibration of figure 4 is quite singular and requires many blowups in the basebefore it becomes smooth. For example, the degrees of (a, b, !) for II! fibres are (4, 5, 10)respectively. Thus, if two such curves intersect transversely and we blow up the point ofintersection, the exceptional divisor will contain degrees (8, 10, 20). As in section 3, thisindicates a non-minimal Weierstrass model, and when passing to a minimal model, L isadjusted in a way that subtracts (4, 6, 12) from these degrees and restores KX to 0. We arethus left with an exceptional curve of degrees (4, 4, 8), which is a curve of IV! fibres. Thisnew curve will intersect the old curves of II! fibres and these points of intersection also needblowing up. Iterating this process we finally arrive at smooth model (i.e., no further blowupsneed to be done) when we have the chain

!!

!!

!!

!!"

""

"!!

!!"

""

"!!

!!"

""

"!!

!!"

""

"!!

!!

!!

!!

II!I0

IIIV

I!0II

IV!

II

I!0IV

III0

II!.

(27)

Chains of this sort were studied systematically in [34, 35].5 (Such chains produced bycollisions in the discriminant have also been discussed in the physics literature [36, 37].6)We see that in the present example, eleven blowups are required. Various of the intersectionsin the above graph produce monodromies and the usual rules of F-theory [17, 18] then dictatethat the resulting gauge algebra from this graph will be

e8 ! su(2) ! g2 ! f4 ! g2 ! su(2) ! e8. (28)

We need two of these chains from the two intersections of curves of II! fibres in figure 4.Adding this to the 16 further blowups from the B" collisions we obtain our first result.

Result 1 10 point-like E8 instantons on an E8 quotient singularity produce 38 extra masslesstensors (in addition to the dilaton) and a gauge algebra

e#38 ! f#2

4 ! g#42 ! su(2)#4. (29)

Two of the above e8 terms come from the perturbative, primordial part of the heteroticstring. All of the rest is nonperturbative. The couplings of these nonperturbative componentsare controlled by particular massless tensors. Let us introduce Gloc as the nonperturbativegauge algebra produced locally by the collision of the point-like instantons with the quotient

5Some care is required in applying the results of [34] since the Calabi–Yau condition was not relevantthere; for example, a transverse intersection of a curve of II fibres and a curve of IV fibres should not be blownup since there is no singularity in the total space of the fibration. (This blowup was done for convenience in[34], but it is implicit in [35] that it need not be done.)

6Note that the analysis in [36] is erroneous in its assertion that above collision cannot be resolved byblowups to preserve KX = 0.

17

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Quivers

I The su(N)⊕p field theory examples can also be realizedby constructions in heterotic M-theory and in F-theory.

I They are given by a stack of p M5-branes at an AN−1

singularity.

I They also have a quiver description as above.

I They are realized in F-theory by a chain of −2 curveswith Kodaira fibers of type IN over each one.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


1

1

1

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Variants

I Other types of quivers can realized by variants of thisconstruction.

I For example, putting flavor branes IN and IN+pk

through an Ap−1 singularity leads to:

A(p-1)

IN

IN+pk

IN

IN+pk

...D1

D2

D(p-2)

D(p-1)

with gauge algebra⊕p−1

j=1 su(N + jk).

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Classification of 6D SCFTs from F-theoryI A key ingredient is non-Higgsable clusters [Morrison-Vafa

’96, Morrison-Taylor ’12]: curve configurations with aminimal gauge algebra. They are: 3, 4, 5, 6, 7, 8, (12),32, 232, 322.

I The classification is largely bottom up, relying on fieldtheory and anomaly cancellation. One hopes toeventually understand it purely in those terms.

I Classification result #1: contracting {Σj} leads to a Bof the form C2/G with G ⊂ U(2).

I Classification result #2: all 6D SCFTs in F-theory canbe described by quivers, whose links may themselves beSCFTs. (Nodes and links are explicitly classified.)

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


A test: finite subgroups of E8

The basic heterotic M-theory example

1

has an important variant: the instanton solution may betwisted by a Wilson line. Such Wilson lines correspond tohomomorphisms Γ→ E8 up to conjugacy.

The E8 globalsymmetry breaks to the commutant of the image of Γ.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


A test: finite subgroups of E8

The basic heterotic M-theory example

1

has an important variant: the instanton solution may betwisted by a Wilson line. Such Wilson lines correspond tohomomorphisms Γ→ E8 up to conjugacy. The E8 globalsymmetry breaks to the commutant of the image of Γ.

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


I Cyclic subgroups of E8 (up to conjugacy) were classifiedby Victor Kac. For example, there are two cases for Z2,with commutants (E7 × SU(2))/Z2 and Spin(16)/Z2,respectively.

I Certain other subgroups of E8 (up to conjugacy) wereclassified by D. D. Frey. This includes some dihedralgroups as well as SL(2,F5) and A5. All of these arerelevant for ADE subgroups of SU(2).

I On the other hand, we can use the F-theoryclassification to ask what are the infinite chains, andhow an infinite chain can end.

I There is an almost perfect match!

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


Group Order Generators Quotient groups

ΓAn−1 = Zn n ζn Zk if k | nΓDn = Dn−2 4(n−2) ζ2n−4, δ Z2, Dih2k if k | (n−2),

D` if ` | (n−2) but 2` 6 | (n−2)ΓE6 = T 24 ζ4, δ, τ Z3, A4

∼= SL(2,F3)ΓE7 = O 48 ζ8, δ, τ Z2, S3, S4

ΓE8 = I 120 −(ζ5)3, ι A5

∼= SL(2,F5)

Finite subgroups of SU(2), where

ζn ≡[

e2πi/n

e−2πi/n

], δ ≡

[1

−1

], τ ≡ 1√

2

[e−2πi/8 e−2πi/8

e10πi/8 e2πi/8

]ι ≡ 1

e4πi/5 − e6πi/5

[e2πi/5 + e−2πi/5 1

1 −e2πi/5 − e−2πi/5

].

SCFTs in 6D

David R. Morrison

Introduction

N=(1, 0) SCFTs

Strings

No anomalies

Examples

F-theory

Quivers

Classification


E7 × SU(2) cases

[E7] 1su22

su42

[SU(2)]...

su42 [SU(4)]

[E7] 1su22

su42

su62

[SU(2)]...

su62 [SU(6)]

[E7] 1su22

so73

[SU(2)]1

so94

sp11

so114

sp21

so134

sp31

so154 ...

spn−4

1 [SO(2n)]

introduction in six dimensions n - brandeis...

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