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Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

S-duality, the 4d Superconformal Index and 2d Topological QFT

Leonardo Rastelli

with A. Gadde, E. Pomoni, S. Razamat and W. Yan arXiv:0910.2225,arXiv:1003.4244, and in progress

Yang Institute for Theoretical , Stony Brook

Rutgers, November 9 2010 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . . . )

Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . . . )

Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. Moore-Seiberg groupoid of Σ = (generalized) 4d S-duality • Vast generalization of “ =4 S-duality as of T 2”. N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

A new paradigm for 4d =2 susy gauge theories N (Gaiotto, . . . )

Compactification of the (2, 0) 6d theory on a 2d surface Σ, with punctures. = =2 superconformal⇒ theories in four dimensions. N Space of complex structures Σ = parameter space of the 4d • theory. Moore-Seiberg groupoid of Σ = (generalized) 4d S-duality • Vast generalization of “ =4 S-duality as modular group of T 2”. N 6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition • functions (on S4) Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

In this talk we will uncover another surprising connection: A protected 4d quantity, the superconformal index, is computed • by topological QFT on Σ. A “microscopic” 2d definition of the TQFT still lacking. We will define it in terms of its abstract operator algebra. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

In this talk we will uncover another surprising connection: A protected 4d quantity, the superconformal index, is computed • by topological QFT on Σ. A “microscopic” 2d definition of the TQFT still lacking. We will define it in terms of its abstract operator algebra. Index = twisted partition function on S3 S1. Independent of the moduli and invariant under× S-duality. It encodes the protected spectrum of the 4d theory. Useful tool. Computing the index in different duality frames gives very • non-trivial checks of Gaiotto’s dualities. Conversely, assuming S-duality we will explicitly compute the • index of 4d theories lacking a Lagrangian description. Surprising connection with elliptic hypergeometric function, an • active area of mathematical research. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005] The SC Index counts (with signs) the (semi)short multiplets, up to equivalence relations that sets to zero i Shorti =Long. ⊕ F 2(∆+j ) 2 j (r+R) (t, v,y,... )= Tr( 1) t 2 y 1 v− ... . I − Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005] The SC Index counts (with signs) the (semi)short multiplets, up to equivalence relations that sets to zero i Shorti =Long. ⊕ F 2(∆+j ) 2 j (r+R) (t, v,y,... )= Tr( 1) t 2 y 1 v− ... . I − 3 Consider a 4d SCFT. On S R (radial quantization), Q† = S. × The implies (taking Q = Q¯ ) • 2+

2 S, Q =∆ 2 j2 2 R + r H 0 . { } − − ≡ ≥ where E is the conformal dimension, (j1, j2) the SU(2)1 SU(2)2 ⊗ Lorentz spins, and (R,r) the quantum numbers under the SU(2)R U(1)r R-symmetry. ⊗ The SC index is the Witten index • F βH+M = Tr( 1) e− I − Here M is a generic combination of charges (weighted by chemical potentials) which commutes with S and Q. States with H > 0 come in pairs, boson + fermion, and cancel out,so • I is β-independent. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The Index as a Matrix Integral If the theory has Lagrangian description there is a simple recipe to compute the index. One defines a single-letter partition function as the index evaluated on the • set of the basic objects (letters) in the theory with H = 0 and in a definite representation of the gauge and flavor groups:

f Rj (t, y, v), where labels the representation. Rj Then the index is computed by enumerating the gauge-invariant words, •

∞ 1 n n n n n (t,y,v, V) = [dU] exp f Rj (t ,y , v ) χ (U , V ) , I 0 n · Rj 1 Z nX=1 Xj @ A Here U is the matrix of the gauge group, V the matrix of the flavor group and label representations of the fields under the flavor and gauge groups. Rj χ (U) is the character of the group element in representation j . • Rj R The measure of integration [d U] is the invariant Haar measure. • n [dU] χ (U)=#of singlets in 1 n . Rj R ⊗···⊗R Z jY=1 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

S-duality for =2 SU(2) SYM with Nf =4 N

Im τ S-duality τ 1 is accompanied by an • →− τ SO(8) triality transformation 2 2¯ and thus we have eight = 1 χsf • ∼ N in fundamental of SU(2). Generalized quivers: internal edges = • gauge groups; external edges = flavour Re τ groups; vertices = Tri-Fundamental χsf. 0 1 Triality permutes the four SU(2) flavor • factors. On the diagrams this is implemented as a b channel crossing.

c d Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Generalized SU(2) quivers Some examples:

(a) (b)

The generalized quivers in (a) arise from different pairs-of-paint decomposition of the same Riemann surface. The corresponding 4d theories are related by S-dualities. They must have the same superconformal index. The same applies to (b). Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The index for the A1 theories The index is read off from the quiver

NG 1 n n n n I Edges n∞=1 n fadj (t ,y , v ) χadj (UI ) = d UI e ∈ I P P Z "I=1 # Y 1 n n n n n n I,J,K V ertices n∞=1 n f3 fund(t ,y , v ) χ3 fund(UI ,UJ ,UK ) e { }∈ − − P P Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The index for the A1 theories The index is read off from the quiver

NG 1 n n n n I Edges n∞=1 n fadj (t ,y , v ) χadj (UI ) = d UI e ∈ I P P Z "I=1 # Y 1 n n n n n n I,J,K V ertices n∞=1 n f3 fund(t ,y , v ) χ3 fund(UI ,UJ ,UK ) e { }∈ − − P P Define a “metric” and “structure constants”

1 n n n n n n n∞=1 n f3 fund(t ,y , v ,..) χ3 fund(UI ,UJ ,UK ) CUI UJ UK = e − − , P 1 n n n n UI UJ n∞=1 fadj (t ,y , v ,..) χadj (U ) η = e n I δˆ(UI , UJ ). P so that the index can be written as

UM UN = CU U U η , I I J K I,J,K M,N { Y}∈V { Y}∈G where indices are contracted by integration over the Haar measure.

“NF –point correlator, with the quiver as a Feynman diagram”. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

TQFT interpretation

α | i

β α | i | i

γ β | i | i (a) (b)

TQFT interpretation of the structure constants Cαβγ and of the metric ηαβ Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

TQFT interpretation

α | i

β α | i | i

γ β | i | i (a) (b)

TQFT interpretation of the structure constants Cαβγ and of the metric ηαβ The structure constants and the metric have to satisfy a set of axioms, which • guarantee independence of correlators from the way one decomposes the Riemann surface into “pairs of pants”. Most of the axioms are simply verified, they reduce to the statement that the • indices are lowered/raised with the metric.

α | i

= γ α h | | i α h |

β γ h | h | (a) (b) Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The one non-trivial condition is associativity of the algebra

δ ǫ δ ǫ Cαβ Cδγ = Cβγ Cδα

α α | i | i

ǫ β = ǫ β h | | i h | | i

γ γ | i | i Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The one non-trivial condition is associativity of the algebra

δ ǫ δ ǫ Cαβ Cδγ = Cβγ Cδα

α α | i | i

ǫ β = ǫ β h | | i h | | i

γ γ | i | i Associativity of the algebra is equivalent to invariance of the index under channel crossing of the graph and thus is implied by S-duality

α γ

α γ

θ = θ

β δ β δ

Crucially depends on the field content. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Index of a chiral superfield = elliptic Gamma function

Mathematicians have a name for the index of the chiral superfield: • elliptic Gamma function

1 j+1 k+1 1 z− p q Γ(z; p, q) − . ≡ 1 zpj qk j,k 0 Y≥ −

The index of a χsf is (Dolan and Osborn - 2008) • 2 ∞ 1 chi k k k t 3 3 1 exp f t , v , y =Γ ; p, q , p = t y, q = t y− . " k # √v Xk=1 “ ” „ « Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Index of a chiral superfield = elliptic Gamma function

Mathematicians have a name for the index of the chiral superfield: • elliptic Gamma function

1 j+1 k+1 1 z− p q Γ(z; p, q) − . ≡ 1 zpj qk j,k 0 Y≥ −

The index of a χsf is (Dolan and Osborn - 2008) • 2 ∞ 1 chi k k k t 3 3 1 exp f t , v , y =Γ ; p, q , p = t y, q = t y− . " k # √v Xk=1 “ ” „ « The Elliptic Beta integral is a generalization of the celebrated Euler • Beta integral (Spiridonov - 2001)

6 1 1 dz i=1 Γ(ti z± ; p, q) α 1 β 1 Γ(α)Γ(β) κ = Γ(titk ; p, q) → dt t − (1 − t) − = . 2πiz Γ(z 2; p, q) Γ(α + β) Q ± i

Elliptic Cookbook Recall the character of the (anti)fundamental representation of SU(n)

n n n 1 χf = ai, χf¯ = , ai = 1 . a i i=1 i=1 i=1 X X Y

The index of a chiral multiplet in fundamental of SU(n) • n 2 t 1 Γ a± ; p, q √v i i=1 Y „ « Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Elliptic Cookbook Recall the character of the (anti)fundamental representation of SU(n)

n n n 1 χf = ai, χf¯ = , ai = 1 . a i i=1 i=1 i=1 X X Y

The index of a chiral multiplet in fundamental of SU(n) • n 2 t 1 Γ a± ; p, q √v i i=1 Y „ «

When an SU(n) symmetry is gauged we add a vector multiplet and • integrate over the gauge group

2 n 1 n 1 2 2Γ(t v; p, q) κ − − Γ(t vai/aj ; p, q) dµ(ai) . . . n! T Γ(ai/aj ; p, q) ˛ ˆ ˜ n 1 i=1 i=j I − ˛ n a =1 Y Y6 ˛ i=1 i ˛ ˛Q * For brevity we will often omit the parameters p and q from the expression˛ of the Gamma function. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The index of the SU(2) generalized quivers in terms of elliptic Gamma functions

The index of Nf = 4 SU(2) gauge theory can be written as

2 2 2 2 2 dz Γ(t vz± ) t 1 1 1 t 1 1 1 κ Γ t v Γ( a± b± z± ) Γ( c± d± z± ). 2πiz Γ(z 2) √v √v I ± ` ´ Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The index of the SU(2) generalized quivers in terms of elliptic Gamma functions

The index of Nf = 4 SU(2) gauge theory can be written as

2 2 2 2 2 dz Γ(t vz± ) t 1 1 1 t 1 1 1 κ Γ t v Γ( a± b± z± ) Γ( c± d± z± ). 2πiz Γ(z 2) √v √v I ± ` ´ This integral was recently shown to be invariant under exchanging a and c (more generally, under the of ) (van de Bult 2009)

This checks associativity of the A1 TQFT, or equivalently, S-duality for the index of the A1 theories Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

A2 generalized quivers

Im τ Generalized quivers: internal edges = SU(3) gauge • groups; external edges = flavour groups, either U(1) or SU(3); vertices = hypermultiplets. Basic example N = 6 SU(3) SYM • f 1 S-duality group generated by τ τ and • τ τ + 2. →− → τ Three possible degenerations of the four-punctured Re• 0 1 sphere: different types of punctures collide (2 possibilities), or two like punctures collide. “Usual” S-duality is the equivalence of the two • degenerations when different types of punctures collide (interchange of the two flavor U(1) or SU(3) factors) x y Argyres- brings us to the frame when • two like punctures collide. The theory consists of an SU(2) vector multiplet coupled to a fundamental a b hyper and to a strongly coupled rank-one SCFT with E6 flavor symmetry. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Weakly-coupled frame

SU SU SU(3)z (3)y SU(3)z (3)y

SU(3) SU(3)

U(1)a U(1)b U(1)b U(1)a

3 3 2 1 2 t azi ± t 1 2 xi Γ Γ b yi xj ± Γ t v 2 0 √v 0 x 1 1 0 √v 1 0 x 1 2 2 2 2 dxi i=1 j=1 j “ ” i=j j = κ Γ(t v) Y Y Y6 . Ia,z;b,y T2 @ @ A A @ A @ A 3 I i=1 2πi xi xi Y Γ 0 x 1 iY=j j 6 @ A Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Weakly-coupled frame

SU SU SU(3)z (3)y SU(3)z (3)y

SU(3) SU(3)

U(1)a U(1)b U(1)b U(1)a

3 3 2 1 2 t azi ± t 1 2 xi Γ Γ b yi xj ± Γ t v 2 0 √v 0 x 1 1 0 √v 1 0 x 1 2 2 2 2 dxi i=1 j=1 j “ ” i=j j = κ Γ(t v) Y Y Y6 . Ia,z;b,y T2 @ @ A A @ A @ A 3 I i=1 2πi xi xi Y Γ 0 x 1 iY=j j 6 @ A S-duality implies symmetry under a b ↔ Checked perturbatively in t and analytically proved for t = v. Using (Rains 2003) Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Strongly-coupled frame

SU SU(3)z (3)y SU(3)z

U SU(3) E6 SU(3) SU(2) (1) ⊃

U U (1)a (1)b SU(3)y

The E6 SCFT has no Lagrangian description Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Strongly-coupled frame

SU SU(3)z (3)y SU(3)z

U SU(3) E6 SU(3) SU(2) (1) ⊃

U U (1)a (1)b SU(3)y

The E6 SCFT has no Lagrangian description

(E6) Let C (x, y, z) denote the index of rank one E6 SCFT. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Strongly-coupled frame

SU SU(3)z (3)y SU(3)z

U SU(3) E6 SU(3) SU(2) (1) ⊃

U U (1)a (1)b SU(3)y

The E6 SCFT has no Lagrangian description

(E6) Let C (x, y, z) denote the index of rank one E6 SCFT. In the strongly-coupled frame, the index reads • 2 2 2 2 de Γ(t ve± ) t 1 1 (E ) ˆ y z ± ± 6 y z (s,r; , )= κ Γ(t v) 2 Γ( e s ) C ((e, r), , ) . I T 2πie Γ(e ) √v I ±

Argyres-Seiberg duality implies • 3/2 1/2 ˆ (s,r; y, z)= a,z;b,y s =(a/b) , r =(ab)− . I I

where a,z;b,y is the index in the weakly-coupled frame. I Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Inverting the SU(2) integral t2 1 1 de Γ( √ e± s± ) ˆ v 2 2 (E6) (s,r; y, z) = κ 4 Γ(t ve± ) C ((e, r), y, z) . I T 2πie Γ( t )Γ(e 2) I v ± Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Inverting the SU(2) integral t2 1 1 de Γ( √ e± s± ) ˆ v 2 2 (E6) (s,r; y, z) = κ 4 Γ(t ve± ) C ((e, r), y, z) . I T 2πie Γ( t )Γ(e 2) I v ± Inversion formula: Under certain assumptions the following holds: (Spiridonov-Warnaar 2004)

1 ds t2 − de t2 fˆ(w)= κ δ s, w; f(s) f(s)= κ δ e,s; fˆ(e) . 2πis √v ⇒ T 2πie √v ICw „ « ! I „ «

The integration contour Cw is a deformation of the unit circle

w√v t2

t2w √v

t2 w√v

√v t2w Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

The index of the E6 SCFT

Using the inversion formula we obtain the index of the E6 SCFT

3 2 2 √v 1 1 2 κ Γ(t v) ds Γ( 2 w± s± ) C(E6) ((w, r), y, z) = t 3 Γ(t2vw 2) 2πis Γ( v , s 2) × ± ICw t4 ± 1 1 1 1 3 3 2 ± 2 ± t s 3 z t s− 3 y x x Γ i Γ i j Γ t2v i 2 0 √v x r 1 0 √v r 1 x dx i=1 j=1 j ! ! i=j „ j « i Y Y Y6 . × T2 2πix @ A @ xi A I i=1 i Γ Y x i=j „ j « Y6 Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Spectrum of protected operators from the index

∞ C(E6) a tk . ≡ k kX=0

4 2 3 4 t E6 5 6 6 E6 6 6 3 a0 =1, a1t = a2t = a3t = 0, a4t = χ , a5t = 0, a6t = −t χ − t + t v v 78 78 7 7 7 t 1 E6 t 1 7 2 1 a7t = y + χ + y + − t v y + v y 78 v y y „ « „ « „ « 8 8 t E6 E6 8 8 a8t = χ − χ − 1 + t v + t v v2 sym2(78) 650 “ ” 9 9 1 E6 9 1 9 3 1 a9t = − t y + χ − 2t y + + t v y + y 78 y y „ « „ « „ « 10 10 10 t E6 E6 E6 t 2 1 E6 a10t = − (χ χ − χ − 1) + y +1+ χ v 78 78 650 v y2 78 „ « t10 1 2 1 2 + y + − t10v2 y + . v y y „ « „ «

The index is E6 covariant Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Spectrum of protected operators from the index F 2(E+j ) 2 j (r+R) (t, v,y,... )= Tr( 1) t 2 y 1 v− .... I −

t4 2 3 4 E6 5 6 6 E6 6 6 3 a0 =1, a1t = a2t = a3t = 0, a4t = χ78 , a5t = 0, a6t = t χ78 t +t v , v − − t7 1 t7 1 1 t8 7 E6 7 2 8 E6 E6 8 8 a7t = y + χ + y + t v y + , a8t = χ χ 1 +t v+t v, 78 − 2 sym2(78) − 650 − v y ! v y ! y ! v „ « 1 1 1 9 9 E6 9 9 3 a9t = t y + χ78 2t y + +t v y + − y ! − y ! y !

t10 t10 1 t10 1 2 1 2 10 E6 E6 E6 2 E6 10 2 a10t = (χ78 χ78 χ650 1)+ y +1+ χ78 + y + t v y + . − v − − v y2 ! v y ! − y !

7 4 6 t6v3 t7v2(y + 1 ) + t8v t6 + t (y + 1 ) + t8v t9(y + 1 ) t /v t y v y y X − , u − , T − − . → (1 t3y)(1 t3/y) → (1 t3y)(1 t3/y) → (1 t3y)(1 t3/y) − − − − − −

E r R j1 j2 X 2 0 1 0 0 u 3 3 0 0 0 − T 2 0 0 0 0

X X X X Constraints : ( ) 650 1 = 0 , u = 0 , T = 0 . ⊗ | ⊕ ⊗ ⊗ Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

SU(3) TQFT

x y x y x a

z a b

(333) (133) (113) Cx,y,z Ca,x,y Ca,b,x

(333) rank 1 : Cx,y,z: Index of E6 SCFT.

(133) rank 0 : Ca,x,y: Index of a hypermultiplet.

(113) “rank -1” : Ca,b,x : An auxiliary construct to write the Argyres-Seiberg theory . Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

S-duality checks of the E6 index

x y x y x y

a b z w z a

(a) (b) (c)

(a) Nf = 6 SU(3) theory (in either of two S-dual frames), or Argyres-Seiberg theory.

(b) Two E6 theories “joined” by gauging an SU(3) of the flavor symmetry.

(c) E6 SCFT joined to hypers by an SU(3) gauging.

We checked associativity perturbatively in t. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Higher rank

Can in principle generalize the discussion to quivers with higher rank • gauge groups. Get many intrinsically strongly coupled theories: E SCFT, T • 7 N theories ... To obtain the index of these higher rank theories have to learn to • invert the superconformal tails.

SU(2)

SU(2) SU(4) SU(2) SU(4) SU(3) SU(2) U(1) ⊃

SU(4) SU(4) Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

32 : S-duality SO(2n + 1)/Sp(n)

The index on X • 2 α 3 dzj Γ(t e ; p, q) I =4 ∼ , N 2πiz Γ(eα; p, q) j j α X I Y Y∈

e where we formally identify zi = e i . The root systems of SO(2n + 1) and • Sp(n) are

SO(2n +1) : X = {±ei, ±ei ± ej ,i

Sp(n) : X = {±2 ei, ±ei ± ej ,i

and they define dual polyhedra. n = 2 SO(5) and Sp(2) are both • squares. n = 3 SO(7) gives a cube and Sp(3) • is an octahedron. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outline

Review of superconformal index

=2 : A1 generalized quivers N S-duality for SU(2) theories The index of the A1 theories and TQFT interpretation. Index as elliptic hypergeometric integral

=2: A generalized quivers N 2 Index of E6 theory A2 TQFT

=4 index N =1 index N Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

8 Supersymmetries

Curious recipe : compute the index by counting the states in the UV, but with the IR charge assignments (Romelsberger) Can be justified interpreting the index as the Witten index of the non-conformal theory on S3 R interpolating betweeen UV and IR fixed × points. Several Seiberg-dual pairs turn out to have the same index. • (Romelsberger, Dolan Osborn, Spiridonov Vartanov) Remarkably, setting v = t in the = 2 index gives the = 1 index of • N N the SCFT obtained (in the IR) integrating out the chiral adjoints.

We consider = 1 SCFTs that have an AdS5 dual. • N Closed formulas for the index of the SCFTs dual to AdS5 Ypq. × We check toric duality of these theories • We match the index of gauge theory to gravity on AdS T 1,1 • × Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Y p,q quiver gauge theory

p,p Y is Z2p of = 4 • N Y p,0 is Z orbifold of the conifold Y 1,0 • p 4 types of field in Y p,q theory •

U(1)r Arrows U 1 1 (x + y) − 2 V 1+ 1 (x y) 2 − Z x Y y

1 2 2 2 2 yp,q = 4p + 2pq + 3q + (2p q) 4p 3q , 3q2 − − − n p o 1 2 2 2 2 xp,q = 4p 2pq + 3q + (2p + q) 4p 3q . 3q2 − − − n p o Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Toric duality

Example of toric duality •

Figure: Different quiver diagrams for Y 4,2.

Indices of toric-dual quivers are equal using Rains’ identity • Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Toric duality

Figure: Action of Seiberg duality

Figure: Example of the Y 4,2 quiver. Middle: Seiberg duality on node 1. Right: swap nodes 1 and 2. Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Index of the conifold gauge theory and AdS T 1,1 ×

In the large N limit, we can compute quiver gauge theory index by • saddle point approximation

∞ ϕ(k) = log[det(1 i(tk,yk))] ϕ(n) = Euler Phi function I − k − Xk=1 i(x), index valued adjacency matrix of the quiver • Conifold index: • 3 1 t3ab t3 a t3 b t3 1 t3y t = + b + a + ab y I 1 t3ab 1 t3 a 1 t3 b 1 t3 1 − 1 t3y − 1 t3 1 − − b − a − ab − − y a,b are potentials of SU(2)a SU(2)b global symmetry × 1,1 KK reduction on T , spectrum of scalar laplacian (Nakayama) • On gravity side, contribution from , gravitino and vector • multiplets, exactly matches with gauge theory result Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outlook

Many possible extensions to theories with 16 supercharges (higher • rank, ADE) Possible to add line and surface operators • Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outlook

Many possible extensions to theories with 16 supercharges (higher • rank, ADE) Possible to add line and surface operators • Relation to the partition function of three dimensional theories on S3 • Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outlook

Many possible extensions to theories with 16 supercharges (higher • rank, ADE) Possible to add line and surface operators • Relation to the partition function of three dimensional theories on S3 • It must be possible to obtain a “microscopic” Lagrangian description of • the 2d TQFT by reduction of the twisted 6d (2,0) theory on S3 S1. × This would give a uniform description of the index for all An theories. Relation to Liouville/Toda? • More systematic understanding of the connection with elliptic • hypergeometric mathematics? Review of superconformal index = 2 : A generalized quivers = 2: A generalized quivers = 4 index = 1 N 1 N 2 N N

Outlook

Many possible extensions to theories with 16 supercharges (higher • rank, ADE) Possible to add line and surface operators • Relation to the partition function of three dimensional theories on S3 • It must be possible to obtain a “microscopic” Lagrangian description of • the 2d TQFT by reduction of the twisted 6d (2,0) theory on S3 S1. × This would give a uniform description of the index for all An theories. Relation to Liouville/Toda? • More systematic understanding of the connection with elliptic • hypergeometric mathematics?

Thank You