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 .