W 1+ ∞ algebra as a symmetry behind AGT relation

1. W1+∞ algebra as a symmetry behind AGT relation High Energy Accelerator Research Organization (KEK) Institute of Particle and Nuclear Studies (IPNS) ShotaroShiba 2011. 09. 17. Reference: S. Kanno, Y. Matsuo and S. S., Phys. Rev. D84 (2011) 026007

2. What is AGT relation? M5-brane system after reduction to the system of superstring can be interpreted as the extra dimensions [Witten ’97] N=2 as a moduli space 4-dim super Yang-Mills theory 2-dim Riemann surface with poles [Seiberg-Witten’94]

3. What is AGT relation? M5-brane system correspondence of physical quantities! [Alday-Gaiotto-Tachikawa’09] more concrete correspondence AGT relation CFT on it N=2 4-dim super Yang-Mills theory 2-dim Riemann surface with poles Correlation function Partition function

4. 4-dim N=2 super Yang-Mills • Field contents • gauge fields (adjoint) • matter fields (fundamental/antifund./bifund.) • fermionicsuperpartners • Partition function • classical part • 1-loop part • instanton part N1 N2 for SU(N) quiver instanton number expansion (just a phase) The higher loop corrections vanish because of N=2 supersymmetry! (as a nonperturbative corrections)

5. 2-dim CFT(=Liouville/Toda) • Field contents • primary fields • descendant fields • : Virasoro generator , : W3 generator , … • Correlation function • primary field part • descendant field part • … of propagators for SU(N) Toda / Liouville=SU(2) Toda level expansion … x x x x x x primary field exists at each pole

6. AGT relation • 4-dim super Yang-Mills theory : Partition function • 2-dim Toda theory : Correlation function YM coupling mass slightly different equivalent! position momentum add invariant under the flip of Weyl group of SU(N) sym. : “U(1) factor” • SU(2) quiver [Alba-Morozov ’09] • SU(3) quiver [Kanno-Matsuo-SS ’10] • [Drukker-Passerini ’10] • SU(2) [Alday-Gaiotto-Tachikawa ’09] • SU(3) [Mironov-Morozov ’09]

7. It’s an interesting result! But… Whydo they correspond to each other? Our question at this time: Where does “U(1) factor” come from? … Larger symmetry exists behind AGT relation!?

8. W1+∞algebra • Lie algebra of differential operators on a circle • generators: where • W1+∞algebra central extension

9. W1+∞ algebra contains the following generators: • U(1) generator • Virasoro generator • W3 generator • WN generator (N<∞) complicated nonlinear terms … Our conjecture: This U(1) generator corresponds to “U(1) factor”, i.e. W1+∞algebra exists behind AGT relation!?

10. On representation highest weight state • General representation • Quasi-finite representation • Unitary representation • Free fermion representation ⊃ ⊃ central charge ⊃ Relation of W1+∞generator and WN generator [Kanno-Matsuo-SS ’11] • Bosonization • Split of U(1) part already well known U(1) field Toda field The relation includes extra complicated nonlinear terms. This explains the nonlinearity of WN algebra, although W1+∞algebrais linear.

11. Current results Our expectation:We want to show that Correlation function of 2-dim CFT with W1+∞algebra = Correlation function for 2-dim CFT with WN algebra + “U(1) factor” = Partition function for 4-dim SU(N) quiver gauge theory 4pt 4pt We showed that 4pt correlation functions plus U(1) factor in the case of N=2,3 with a specific central charge c=N-1 can be written in terms of W1+∞algebra. [Kanno-Matsuo-SS ’11]

12. Future directions • Case of general central charge (difficult) • Both functions become more complicated form. • Case of quiver gauge theory • More than 4-point correlation functions must be • calculated. • Case of N>3 • The way of embedding WN in W1+∞must be clarified. • In N=∞ case, AdS/CFT for M5 system can be discussed.