N = 2 Dualities and 2D TQFT Space of Complex Structures Σ = Parameter Space of the 4D Theory

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

= 2 Dualities and 2d TQFT N

Abhijit Gadde

with L. Rastelli, S. Razamat and W. Yan arXiv:0910.2225 arXiv:1003.4244 arXiv:1104.3850 arXiv:1110:3740 ...

California Institute of Technology

Indian Israeli String Meeting 2012

Abhijit Gadde N = 2 Dualities and 2d TQFT Space of complex structures Σ = parameter space of the 4d theory. Mapping class group of Σ = (generalized) 4d S-duality Punctures: SU(2) → SU(N) = Flavor symmetry: Commutant We will focus on "maximal" puncture: with SU(N) ﬂavor symmetry Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled ﬁxed points Vast generalization of “N = 4 S-duality as modular group of T 2”.

6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4)

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . . . ] N

Compactify of the 6d (2, 0) AN−1 theory on a 2d surface Σ, with punctures. =⇒ N = 2 superconformal theories in four dimensions.

Abhijit Gadde N = 2 Dualities and 2d TQFT We will focus on "maximal" puncture: with SU(N) ﬂavor symmetry Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled ﬁxed points Vast generalization of “N = 4 S-duality as modular group of T 2”.

6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4)

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . . . ] N

Abhijit Gadde N = 2 Dualities and 2d TQFT Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled ﬁxed points Vast generalization of “N = 4 S-duality as modular group of T 2”.

6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4)

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . . . ] N

Abhijit Gadde N = 2 Dualities and 2d TQFT 6=4+2: beautiful and unexpected 4d/2d connections. For ex., Correlators of Liouville/Toda on Σ compute the 4d partition functions (on S4)

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . . . ] N

Compactify of the 6d (2, 0) AN−1 theory on a 2d surface Σ, with punctures. =⇒ N = 2 superconformal theories in four dimensions. Space of complex structures Σ = parameter space of the 4d theory. Mapping class group of Σ = (generalized) 4d S-duality Punctures: SU(2) → SU(N) = Flavor symmetry: Commutant We will focus on "maximal" puncture: with SU(N) ﬂavor symmetry Sphere with 3 punctures = Theories without parameters Free hypermultiplets Strongly coupled ﬁxed points Vast generalization of “N = 4 S-duality as modular group of T 2”.

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications A new paradigm for 4d = 2 SCFTs [Gaiotto, . . . ] 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)

Abhijit Gadde N = 2 Dualities and 2d TQFT Superconformal Index = twisted partition function on S3 × S1 = Witten index in radial quantization Independent of the gauge theory coupling and invariant under S-duality. Independent of coupling = Independent of complex structure on Σ =⇒ 2d Topological QFT It encodes the protected spectrum of the 4d theory. Useful tool to understand physics. Our aim: To compute the superconformal index of these N = 2 theories (even the strongly coupled ones) by exploiting TQFT structure.

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

In this talk we will discuss the implications of another interesting connection: A protected 4d quantity, the superconformal index, is computed by topological QFT on Σ.

Abhijit Gadde N = 2 Dualities and 2d TQFT Our aim: To compute the superconformal index of these N = 2 theories (even the strongly coupled ones) by exploiting TQFT structure.

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

In this talk we will discuss the implications of another interesting connection: A protected 4d quantity, the superconformal index, is computed by topological QFT on Σ. Superconformal Index = twisted partition function on S3 × S1 = Witten index in radial quantization Independent of the gauge theory coupling and invariant under S-duality. Independent of coupling = Independent of complex structure on Σ =⇒ 2d Topological QFT It encodes the protected spectrum of the 4d theory. Useful tool to understand physics.

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT 3 † Consider a 4d SCFT. On S × R (radial quantization), Q = S. ¯ The superconformal algebra implies (taking Q = Q1−˙ )

2 {S, Q} = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 .

where ∆ 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.

I (p, q , t , . . . ) = Tr(−1)F pj1+j2−rq−j1+j2−rtR+r ....

F j1+j2−r −j1+j2−r R+r F1 Fi I (p, q, t; x1, . . . , xi) = Tr(−1) p q t x1 . . . xi ≡ I (p, q, t; x).

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005]

The SC index is the Witten index I = T r(−1)F e−βH+M 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. The SC Index counts (with signs) the (semi)short multiplets, up to equivalence relations that sets to zero ⊕i Shorti =Long.

Abhijit Gadde N = 2 Dualities and 2d TQFT F j1+j2−r −j1+j2−r R+r F1 Fi I (p, q, t; x1, . . . , xi) = Tr(−1) p q t x1 . . . xi ≡ I (p, q, t; x).

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005]

2 {S, Q} = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 .

where ∆ 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.

I (p, q , t , . . . ) = Tr(−1)F pj1+j2−rq−j1+j2−rtR+r ....

Abhijit Gadde N = 2 Dualities and 2d TQFT ≡ I (p, q, t; x).

Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005]

2 {S, Q} = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 .

where ∆ 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.

I (p, q , t , . . . ) = Tr(−1)F pj1+j2−rq−j1+j2−rtR+r ....

F j1+j2−r −j1+j2−r R+r F1 Fi I (p, q, t; x1, . . . , xi) = Tr(−1) p q t x1 . . . xi

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications

The Superconformal Index[Romelsberger; Kinney, Maldacena,Minwalla, Raju 2005]

2 {S, Q} = ∆ − 2 j2 − 2 R + r ≡ H ≥ 0 .

where ∆ 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.

I (p, q , t , . . . ) = Tr(−1)F pj1+j2−rq−j1+j2−rtR+r ....

F j1+j2−r −j1+j2−r R+r F1 Fi I (p, q, t; x1, . . . , xi) = Tr(−1) p q t x1 . . . xi ≡ I (p, q, t; x).

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index 2d TQFT and orthogonal polynomials Results and Applications 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 (p, q, t),

where Rj labels the representation. Then the index is computed by enumerating the gauge-invariant words,   Z ∞ 1 X X Rj n n n n n I(p, q, t, V) = [dU] exp  f (p , q , t ) · χR (U , V ) , n j n=1 j

Here U is the matrix of the gauge group, V the matrix of the flavor group and Rj label representations of the fields under the flavor and gauge groups.

χRj (U) is the character of the group element in representation Rj . The measure of integration [d U] is the invariant Haar measure. n Z Y [dU] χRj (U) = #of singlets in R1 ⊗ · · · ⊗ Rn . j=1

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Index is independent of the pair of pants decomposition ⇔ 2d TQFT

a2 a1

V : I(a1, a2, a3) : η(a1, a2)=∆(a1)I (a1)δ(a1, a2)

a1 a2

4 punctured sphere: H I(a1, a2, a3, a4) = [da][db] I(a1, a2, a) η(a, b) I(b, a3, a4)

S duality ⇒ I(a1, a2, a3, a4) is invariant under permutations of ai.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications

Any punctured Riemann surface can be obtained by gluing pair of pants in more than one way.

3 Building blocks: 3-punctured sphere ⇔ 4d SCFT TN with SU(N) ﬂavor symmetry

T2: Free hypermultiplet T3: Strongly coupled Minahan-Nemeschensky theory with E6 symmetry TN : Strongly coupled theories Gluing pair of pants ⇔ Gauging the common SU(N) with vector multiplet.

Abhijit Gadde N = 2 Dualities and 2d TQFT a2 a1

V : I(a1, a2, a3) : η(a1, a2)=∆(a1)I (a1)δ(a1, a2)

a1 a2

4 punctured sphere: H I(a1, a2, a3, a4) = [da][db] I(a1, a2, a) η(a, b) I(b, a3, a4)

S duality ⇒ I(a1, a2, a3, a4) is invariant under permutations of ai.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications

Any punctured Riemann surface can be obtained by gluing pair of pants in more than one way.

3 Building blocks: 3-punctured sphere ⇔ 4d SCFT TN with SU(N) ﬂavor symmetry

T2: Free hypermultiplet T3: Strongly coupled Minahan-Nemeschensky theory with E6 symmetry TN : Strongly coupled theories Gluing pair of pants ⇔ Gauging the common SU(N) with vector multiplet. Index is independent of the pair of pants decomposition ⇔ 2d TQFT

Abhijit Gadde N = 2 Dualities and 2d TQFT 4 punctured sphere: H I(a1, a2, a3, a4) = [da][db] I(a1, a2, a) η(a, b) I(b, a3, a4)

S duality ⇒ I(a1, a2, a3, a4) is invariant under permutations of ai.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications

Any punctured Riemann surface can be obtained by gluing pair of pants in more than one way.

3 Building blocks: 3-punctured sphere ⇔ 4d SCFT TN with SU(N) ﬂavor symmetry

a2 a1

V : I(a1, a2, a3) : η(a1, a2)=∆(a1)I (a1)δ(a1, a2)

a1 a2

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications

Any punctured Riemann surface can be obtained by gluing pair of pants in more than one way.

3 Building blocks: 3-punctured sphere ⇔ 4d SCFT TN with SU(N) ﬂavor symmetry

a2 a1

V : I(a1, a2, a3) : η(a1, a2)=∆(a1)I (a1)δ(a1, a2)

a1 a2

4 punctured sphere: H I(a1, a2, a3, a4) = [da][db] I(a1, a2, a) η(a, b) I(b, a3, a4)

S duality ⇒ I(a1, a2, a3, a4) is invariant under permutations of ai.

Abhijit Gadde N = 2 Dualities and 2d TQFT S duality =⇒ Associativity: CαβC γδ = CαγC βδ 2 pt function:

X αβ α β η(a, b) = η f (a)f (b) α,β

Choose f α(a) to be orthonormal =⇒ ηαβ = δαβ One can still perform orthogonal transformations on f α(a) and diagonalize Cαβγ → Cααα

γ Cαβγ ≡ [Nα]β Associativity ⇒ [Nα,Nβ ] = 0.

Simultaneously diagonalize Nα!

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

3 pt function:

X α β γ I(a, b, c) = Cαβγ f (a)f (b)f (c) α,β,γ

Abhijit Gadde N = 2 Dualities and 2d TQFT 2 pt function:

X αβ α β η(a, b) = η f (a)f (b) α,β

Choose f α(a) to be orthonormal =⇒ ηαβ = δαβ One can still perform orthogonal transformations on f α(a) and diagonalize Cαβγ → Cααα

γ Cαβγ ≡ [Nα]β Associativity ⇒ [Nα,Nβ ] = 0.

Simultaneously diagonalize Nα!

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

3 pt function:

X α β γ I(a, b, c) = Cαβγ f (a)f (b)f (c) α,β,γ S duality =⇒ Associativity: CαβC γδ = CαγC βδ

Abhijit Gadde N = 2 Dualities and 2d TQFT One can still perform orthogonal transformations on f α(a) and diagonalize Cαβγ → Cααα

γ Cαβγ ≡ [Nα]β Associativity ⇒ [Nα,Nβ ] = 0.

Simultaneously diagonalize Nα!

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

3 pt function:

X α β γ I(a, b, c) = Cαβγ f (a)f (b)f (c) α,β,γ S duality =⇒ Associativity: CαβC γδ = CαγC βδ 2 pt function:

X αβ α β η(a, b) = η f (a)f (b) α,β

Choose f α(a) to be orthonormal =⇒ ηαβ = δαβ

Abhijit Gadde N = 2 Dualities and 2d TQFT γ Cαβγ ≡ [Nα]β Associativity ⇒ [Nα,Nβ ] = 0.

Simultaneously diagonalize Nα!

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

3 pt function:

X α β γ I(a, b, c) = Cαβγ f (a)f (b)f (c) α,β,γ S duality =⇒ Associativity: CαβC γδ = CαγC βδ 2 pt function:

X αβ α β η(a, b) = η f (a)f (b) α,β

Choose f α(a) to be orthonormal =⇒ ηαβ = δαβ One can still perform orthogonal transformations on f α(a) and diagonalize Cαβγ → Cααα

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

3 pt function:

X α β γ I(a, b, c) = Cαβγ f (a)f (b)f (c) α,β,γ S duality =⇒ Associativity: CαβC γδ = CαγC βδ 2 pt function:

X αβ α β η(a, b) = η f (a)f (b) α,β

Choose f α(a) to be orthonormal =⇒ ηαβ = δαβ One can still perform orthogonal transformations on f α(a) and diagonalize Cαβγ → Cααα

γ Cαβγ ≡ [Nα]β Associativity ⇒ [Nα,Nβ ] = 0.

Simultaneously diagonalize Nα!

Abhijit Gadde N = 2 Dualities and 2d TQFT Problem: We don’t know I(a, b, c) except for SU(2)

Solution: Lift the SU(2) result to SU(N)!

α α f (a)SU(2) f (a)SU(N) X −→ X −→ α,β,γ∈SU(2)reps. α,β,γ∈SU(N)reps.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

Conclusion: Choose f α(a) s.t.

ηαβ = δαβ

Cαβγ = Cααα δαβδβγ !

Abhijit Gadde N = 2 Dualities and 2d TQFT Solution: Lift the SU(2) result to SU(N)!

α α f (a)SU(2) f (a)SU(N) X −→ X −→ α,β,γ∈SU(2)reps. α,β,γ∈SU(N)reps.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

Conclusion: Choose f α(a) s.t.

ηαβ = δαβ

Cαβγ = Cααα δαβδβγ !

Problem: We don’t know I(a, b, c) except for SU(2)

Abhijit Gadde N = 2 Dualities and 2d TQFT α α f (a)SU(2) f (a)SU(N) X −→ X −→ α,β,γ∈SU(2)reps. α,β,γ∈SU(N)reps.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

Conclusion: Choose f α(a) s.t.

ηαβ = δαβ

Cαβγ = Cααα δαβδβγ !

Problem: We don’t know I(a, b, c) except for SU(2)

Solution: Lift the SU(2) result to SU(N)!

Abhijit Gadde N = 2 Dualities and 2d TQFT X X −→ α,β,γ∈SU(2)reps. α,β,γ∈SU(N)reps.

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Discrete basis

Conclusion: Choose f α(a) s.t.

ηαβ = δαβ

Cαβγ = Cααα δαβδβγ !

Problem: We don’t know I(a, b, c) except for SU(2)

Solution: Lift the SU(2) result to SU(N)!

f α(a) f α(a) SU(2) −→ SU(N)

Conclusion: Choose f α(a) s.t.

ηαβ = δαβ

Cαβγ = Cααα δαβδβγ !

Problem: We don’t know I(a, b, c) except for SU(2)

Solution: Lift the SU(2) result to SU(N)!

α α f (a)SU(2) f (a)SU(N) X −→ X −→ α,β,γ∈SU(2)reps. α,β,γ∈SU(N)reps.

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Immediate nontrivial result: SU(2) −→ SU(3)

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Example: Hall-Littlewood polynomial p = 0, q = 0

SU(2) SU(2) : Free hypermultiplet

SU(2)

∞ X 1 n n 1 n 1 n 1 I(a , a , a ) = exp t (a + )(a + )(a + ) 1 2 3 SU(2) n 1 an 2 an 3 an n=1 1 2 3 ∞ 3 X 1 Y HL −1 ∼ 1 1 Pλ (ai, ai |t) HL 2 − 2 λ=0 Pλ (t , t |t) i=1 ∞ 3 X Y λ Cλλλ f (ai) λ=0 i=1

HL where, Pλ (a) ∼ χλ(a) − tχλ−2(a)

SU(2) SU(2) : Free hypermultiplet

SU(2)

HL where, Pλ (a) ∼ χλ(a) − tχλ−2(a)

Immediate nontrivial result: SU(2) −→ SU(3)

Abhijit Gadde N = 2 Dualities and 2d TQFT 3 3 SU(3) ﬂavor symmetry is enhanced: SU(3) → E6!

This expression agrees with the index of E6 theory obtained from Argyres-Seiberg duality [AG,Rastelli,Razamat,Yan]

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Example: Hall-Littlewood polynomial p = 0, q = 0

SU(3) SU(3) : Minahan-Nemeschansky Theory SU(3)3 E → 6 SU(3)

3 X 1 Y HL I(a , a , a ) ∼ P (a |t) 1 2 3 SU(3) P HL(t, t−1, 1|t) λ i λ∈SU(3) rep. λ i=1

HL X λ1 λk Y xi − txj Pλ (x1, . . . , xk)U(k) ∼ x1 . . . xk xi − xj σ∈Sk i

Abhijit Gadde N = 2 Dualities and 2d TQFT This expression agrees with the index of E6 theory obtained from Argyres-Seiberg duality [AG,Rastelli,Razamat,Yan]

Short review of superconformal index TQFT structure 2d TQFT and orthogonal polynomials Example: Hall-Littlewood polynomials Results and Applications Example: Hall-Littlewood polynomial p = 0, q = 0

SU(3) SU(3) : Minahan-Nemeschansky Theory SU(3)3 E → 6 SU(3)

3 X 1 Y HL I(a , a , a ) ∼ P (a |t) 1 2 3 SU(3) P HL(t, t−1, 1|t) λ i λ∈SU(3) rep. λ i=1

HL X λ1 λk Y xi − txj Pλ (x1, . . . , xk)U(k) ∼ x1 . . . xk xi − xj σ∈Sk i

3 3 SU(3) ﬂavor symmetry is enhanced: SU(3) → E6!

SU(3) SU(3) : Minahan-Nemeschansky Theory SU(3)3 E → 6 SU(3)

3 X 1 Y HL I(a , a , a ) ∼ P (a |t) 1 2 3 SU(3) P HL(t, t−1, 1|t) λ i λ∈SU(3) rep. λ i=1

HL X λ1 λk Y xi − txj Pλ (x1, . . . , xk)U(k) ∼ x1 . . . xk xi − xj σ∈Sk i

3 3 SU(3) ﬂavor symmetry is enhanced: SU(3) → E6!

This expression agrees with the index of E6 theory obtained from Argyres-Seiberg duality [AG,Rastelli,Razamat,Yan]

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary of results

I(p,q,t)

I(0,q,t)

f α(a) : Macdonald polynomial

I(0, 0, t) I(0, t, t)

f α(a) : Hall-Littlewood polynomial f α(a) : Schur polynomial

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

Abhijit Gadde N = 2 Dualities and 2d TQFT Large class of N = 2 theories: Large N limit of the index of the 4d theory corresponding to the genus g surface:

∞ N→∞ Y j g−1 Ig = (1 − t ) j=2

Index of all the N = 2 theories is also independent of N in the large N limit Puzzle: what is the reason for this general mysterious cancellation?

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Large N limit

N = 4 SYM: 1/16 BPS states in N = 4 SYM ⇔ gravitons, giant gravitons (D 5 branes), black holes [Gutowski,Reall] in AdS5 × S Black hole states grow as N 2 but the index is independent of N in the large N limit [Kinney,Maldacena,Minwalla,Raju] Mysterious cancellations between bosonic and fermionic black hole microstates?

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Large N limit

∞ N→∞ Y j g−1 Ig = (1 − t ) j=2

Index of all the N = 2 theories is also independent of N in the large N limit Puzzle: what is the reason for this general mysterious cancellation?

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Outline

1 Short review of superconformal index

2 2d TQFT and orthogonal polynomials TQFT structure Example: Hall-Littlewood polynomials

3 Results and Applications Large N limit Instanton partition function

x4,x5,x6,x7

x8,x9 E6 singularity

k D3 k D3

N D7

Higgs branch of k D3 = Instanton moduli space of k instantons

Index of rank k E6 theory = k E6 instanton partition function

Index of T3 theory = partition function of single E6 instanton

Abhijit Gadde N = 2 Dualities and 2d TQFT Reduce on S1 Partition function of 3d theories on S3 → [AG,Yan;Dolan,Spiridonov,Vartanov;Imamura] 3 Reduce on Hopf ﬁber of S The index of 3d theories [Kim,. . . ]: 1 →2 partition function on S S [Benini,Nishioka,Yamazaki] × I expect one can play similar games in other dimensions and with other exact observables

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Reductions to 3d

4d Superconformal Index: Partition function on S1 S3 ×

Abhijit Gadde N = 2 Dualities and 2d TQFT 3 Reduce on Hopf ﬁber of S The index of 3d theories [Kim,. . . ]: 1 →2 partition function on S S [Benini,Nishioka,Yamazaki] × I expect one can play similar games in other dimensions and with other exact observables

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Reductions to 3d

4d Superconformal Index: Partition function on S1 S3 × Reduce on S1 Partition function of 3d theories on S3 → [AG,Yan;Dolan,Spiridonov,Vartanov;Imamura]

Abhijit Gadde N = 2 Dualities and 2d TQFT I expect one can play similar games in other dimensions and with other exact observables

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Reductions to 3d

4d Superconformal Index: Partition function on S1 S3 × Reduce on S1 Partition function of 3d theories on S3 → [AG,Yan;Dolan,Spiridonov,Vartanov;Imamura] 3 Reduce on Hopf ﬁber of S The index of 3d theories [Kim,. . . ]: 1 →2 partition function on S S [Benini,Nishioka,Yamazaki] ×

Abhijit Gadde N = 2 Dualities and 2d TQFT Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Reductions to 3d

Abhijit Gadde N = 2 Dualities and 2d TQFT What about three parameter? Possible to add surface and line operators Relation to AGT corrrespondence? 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.

Thank You!!

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary and Outlook

We have computed the superconformal index (two parameter) for ALL Gaiotto theories.

Abhijit Gadde N = 2 Dualities and 2d TQFT Possible to add surface and line operators Relation to AGT corrrespondence? 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.

Thank You!!

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary and Outlook

We have computed the superconformal index (two parameter) for ALL Gaiotto theories. What about three parameter?

Abhijit Gadde N = 2 Dualities and 2d TQFT Relation to AGT corrrespondence? 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.

Thank You!!

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary and Outlook

We have computed the superconformal index (two parameter) for ALL Gaiotto theories. What about three parameter? Possible to add surface and line operators

Abhijit Gadde N = 2 Dualities and 2d TQFT 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.

Thank You!!

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary and Outlook

We have computed the superconformal index (two parameter) for ALL Gaiotto theories. What about three parameter? Possible to add surface and line operators Relation to AGT corrrespondence?

Abhijit Gadde N = 2 Dualities and 2d TQFT Thank You!!

Short review of superconformal index Large N limit 2d TQFT and orthogonal polynomials Instanton partition function Results and Applications Summary and Outlook

We have computed the superconformal index (two parameter) for ALL Gaiotto theories. What about three parameter? Possible to add surface and line operators Relation to AGT corrrespondence? 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.

Thank You!!

Abhijit Gadde N = 2 Dualities and 2d TQFT