AGT on the S-duality Wall

Kazuo Hosomichi 4th Taiwan String Workshop 16th Dec. 2011 @ NCTU

M5-brane

A fundamental DOF in M-theory, (5+1)-dim object.

When many of them are put together, they support a mysterious theory --- (2,0)-theory --- on the worldvolume.

Studying various compactifications of (2,0) theory

* gives new insights to low-dim SUSY gauge theories

* helps us understand M5-branes better

1. AGT relation

A correspondence between 4D N=2 gauge theories and 2D CFTs

Gaiotto's gauge theories

( Σ (τ) { ⋯ }) CFT N , g,n , Y 1, ,Y n Families of 4D N = 2 SCFTs describing N M5-branes wrapped on punctured Riemann surfaces.

Σ (τ) g ,n : genus g surfaces with n punctures

τ : 3g-3+n complex structure moduli

⋯ Y 1, Y 2, ,Y n : Young diagram with N boxes, height less than N

N＝２ case : SU(2) generalized quivers

Pants decomposition gives a Lagrangian. Pants curve ( 3g-3+n of them )

SU (2)

SU (2) Pair of pants = 4 hypermultiplets

SU (2)

Pants curve

Re(τ) Tubular region = SU(2) gauge multiplet τ : moduli = gauge coupling Im(τ)

Moore-Seiberg graph

m1 a2 a6 a10 a4 a a1 8

a5 a7 a9 a3 m2 m3 m4

： ai Coulomb branch parameter ： mi matter mass

To a graph with parameters, one can associate (ϵ ϵ τ) Nekrasov partition function ZNek 1 , 2 ,m,a ,

ϵ ϵ 1 , 2 : parameters of Ω-deformation

Basic examples

1. “ Nf = 4 ” theory

m1 m4 SU(2) SQCD a with 4 fund. hypers

m 4-punctured sphere 2 m3

2. “ N = 2* ” theory

SU(2) SQCD m with 1 adjoint hyper a 1-punctured torus

Partition function on 4-sphere

Pestun 2007 : (τ) (τ) - N=2 SUSY theories can be put on 4-sphere. Z Nek Z Nek - Partition function can be obtained by an explicit path integral using localization principle.

where

： radius of 4-sphere

Instantons localize to N- and S-poles.

Riemann surface

m1 a2 a6 a10 Moore-Seiberg graph a4 a a1 8

a a7 a9 a3 5 m2 Holomorphic function m3 m4

m1 a2 a6 2 Integral of a10 a a 4 a8 | Hol. Function |^2 ∫ da 1 a5 a7 a9 a3 m2 Partition function m3 m4

. . . looks like constructing 2D CFT correlators.

Liouville CFT

Operator : V m2 V m 1 w m : Liouville momentum z

OPE :

m1 Correlator : m 2 m3 m4

Holomorphic diff. eqn. in τ Virasoro × Virasoro Ward identity τ Holomorphic diff. eqn. in Conformal block functions

Solutions to Virasoro Ward identity.

(Holomorphic functions of τ. )

m1

Moore-Seiberg graph m 2 m3 m4 = A complete set of solutions m1 a2 a6 a10 a4 a a1 8 mi : momentum of external insertions a5 a7 a9 a3 m2 a : momentum of intermediate states m3 m4 i

Correlator

AGT conjecture (Alday-Gaiotto-Tachikawa, 2009)

A correspondence between SU(2) generalized quiver theory and Liouville CFT

SU(2) generalized quiver Liouville theory

Ω-deformation parameter Central charge

Nekrasov partition function Conformal block

Partition function on 4-sphere Correlator

Some follow-ups

* Is the correspondence true?

Nekrasov partition functions and conformal blocks satisfy the same recursion relation (Fateev-Litvinov '0912, . . . )

* Generalizations?

- SU(N) quiver ↔ Toda CFT （Wyllard, '0907）

WN algebra

- Inclusion of Wilson / 't Hooft loop operators Okuda-san's talk (AGGTV '0909, DGOT '0909)

- Non-conformal gauge theories (Gaiotto '0908, . . . )

An Open Problem

* Why are they related?

A direct explanation would be to show

(2,0) theory on 4-sphere = Liouville / Toda CFT

Another way -- use the equivalence of conformal blocks with

- Dotsenko-Fateev integral? ( Dijkgraav-Vafa's large N matrix model )

- wave function of quantum Teichmueller theory?

- SL(2,R) Chern-Simons path integral?

Another Open Problem

Partition function on round 4-sphere Liouville correlator

ϵ =ϵ = Round 4-sphere corresponds to 1 2 , b 1.

≠ How to deform the round 4-sphere to get b 1?

2. 3d version of AGT

S-duality

Different pants decompositions of the same surface

[Gauge theory] ＝ Different Lagragians for the same SCFT

[CFT] ＝ Defferent choices for complete sets of solutions to Virasoro Ward Id.

Example: Liouville torus 1-point [ SU(2) N=2* ]

τ m −1/ τ m a a'

They satisfy the same diff eqn.

: “S-duality Kernel”

S-duality Kernel

(Teschner '03)

Double-sine function:

3D AGT correspondence

Drukker-Gaiotto-Gomis '09, KH-Lee-Park '10

Liouville “S-duality kernel” = partition function on 3-sphere of the gauge theory on “S-duality wall”

S-duality of SU(2) N=2* SYM

Different pants decompositions of 1-punctured torus

τ − /τ τ N=2* SYM with couplings and 1 are equivalent. −1/ τ But dynamical variables are related via electric-magnetic duality. 0 1

Janus and S-duality walls (Gaiotto-Witten '08)

S-dualize the left half

[Janus wall] [S-duality wall] Coupling τ jumps here. Fields are connected via S-duality

coupling coupling

Partition function on the wall (Drukker-Gaiotto-Gomis '09)

[AGT]

The couplings at the two poles do not need to be the same [with Janus wall]

τ =− / τ Set ' 1 and use Teschner's formula [with S-duality wall]

N-pole Wall S-pole

Wall partition function = S-duality Kernel S-duality wall of N=4 SYM

Wall theory = T[G] L G G (3+1)-dim L Isometry of Coulomb branch G worldvolume Higgs branch G T[G]

T[G] coupled to bulk G gauge field

G G GL G G G

T[G] T[G] D J S(D) S(D)

S-dual of “Dirichlet boundary for G^L SYM” Brane construction of T[SU(N)]

Dirichlet boundary T[SU(N)] coupled to bulk SU(N)

S-dual N D3(0126) N D3(0126)

N D5 (012789) N NS5(012345)

Ungauging SU(N) T[SU(N)]

N N−1 3 2 1

D=3 N=4 Quiver theory N D5(012789) N NS5(012345) T[SU(2)] and its mass-deform

3D N=4 U(1) SQED with 2 electron hypers. Fields (in 3D N=2 terms)

U(1) vector ∼ ( σ) V A m , ϕ Neutral chiral D3(0126) Electron chirals q1 , q2

Positron chirals ̃ 1 ̃ 2 D5(012789) NS5(012345) q , q

To get S-duality wall of N=2* SU(2) SYM between vacua (a,a'),

± Separate D5's along x3 Electrons get masses a

Separate NS5's along x7 U(1) FI parameter a' ∼ Deform N=4 to N=2* Chiral matters get masses m

Partition function of the Wall S^3

(FI) (ϕ)

( ) ( ) q1 q2

(q̃ 1 ) (q̃ 2)

3D AGT relation (Drukker-Gaiotto-Gomis '09, KH-Lee-Park '10)

Wall partition function S-duality Kernel

Generalization : replace S^3 by ellipsoid 3. Generalization of 3D AGT （ 6=3+3 ）

Teichmuller space

Teichmuller space { n-punctured, genus g Riemann surfaces } { Diff_0 }

Moduli space

{ Modular group }

cf) string theory = 2D Quantum Gravity

~ Teichmuller theory ~ Liouville theory

Quantum Teichmuller theory

Geometric quantization of

m1 a2 a6 a10 a4 a a1 8

： a5 a7 a9 Complete basis of Hilbert space a3 m2 m3 m 1. ： eigenstates of hol coordinates 4

2. ：eigenstates of lengths along punctures / pants curves

Note : arbitrary closed Riemann surface can be written as Σ=( )/Γ UHP Γ ： discrete subgroup of SL(2,R)

2cosh(2π bl) = Tr γ (γ∈Γ)

H. Verlinde, '90

1. Conformal block = wave function in quantum Teichmuller theory

2. Quantum Teichmuller theory = SL(2,R) Chern-Simons theory

Example: 1-punctured Torus

1. Triangulate.

B 2. Assign Fock coordinates to edges.

A

3. Calculate the cycle lengths.

z B zC

z A

S-duality Kernel =

Re-phrasing

Wall partition function S-duality Kernel

Overlap of Teichmuller wave functions

SL(2,R) Chern-Simons

a m m

a '

I Generalization of 3D AGT （Terashima-Yamazaki, Dimofte-Gukov, ・・・）

SL(2,R) CS theory (level k) N=2 SUSY gauge theory T[M] 3 on 3-manifold M on ellipsoid S b

Outlook

1. 3D “extension” of AGT correspondence will relate wider range of physics and mathematics.

2. Further study of AGT will lead to - inventions of new useful observables (such as partition function on sphere) - better understanding of various non-local operators