W1+∞ algebra as a symmetry behind AGT relation High Energy Accelerator Research Organization (KEK) Institute of Particle and Nuclear Studies (IPNS) Shotaro Shiba 2011/12/06 Reference: S. Kanno, Y. Matsuo and S. S., Phys. Rev. D84 (2011) 026007 S. S., arXiv: 1111.1899 [hep-th] (submitted to JHEP)
Situation in 1994-1997 M5-brane system 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]
Situation in 1994-1997 M5-brane system [Witten ’97] 4, 5 0, 1, 2, 3 6, 10 Planck scale
Situation in 1994-1997 M5-brane system [Witten ’97] (deformation) 4, 5 2-dim Riemann surface 0, 1, 2, 3 6, 10 Planck scale
Situation in 1994-1997 M5-brane system [Witten ’97] D4/NS5 4-dim N=2 super Yang-Mills 4, 5 2-dim Riemann surface 0, 1, 2, 3 × 6, 10 Planck scale
Situation in 1994-1997 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]
Situation in 2009-2011 M5-brane system [Alday-Benini-Tachikawa ’09] 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 symmetry [Gaiotto ’09] Correlation function Partition function
Contents 1. Introduction (finished) 2. 4-dim N=2 super Yang-Mills 3. 2-dim CFT (=Liouville/Toda) 4. AGT relation 5. W1+∞algebra 6. Current results & future directions
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)
Partition function (1-loop part) gauge antifund. bifund. fund. mass mass mass VEV Each factor is defined as where : double Gamma function
Partition function (instanton part) gauge antifund. Young tableaux bifund. fund. where , : coupling constant and (A) (L) fraction of polynomials
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
2-dim CFT(=Liouville/Toda) 4-dim SYM SU(N) quiver • 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 instanton number expansion Partition function level expansion 1-loop part … x x x x x x instantonpart primary field exists at each pole
On CFT up to spin N (for SU(N) Toda) spin 2 spin 3 Generators: Commutation relation: where Primary fields are defined as highest weight states: nonlinear terms! (N-component) Toda field
Correlation function position fraction of polynomials instanton part? propagator = inverse Shapovalov matrix where the 3-point function is 1-loop part primary fields double Gamma function degenerate field
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]
Correspondence of parameters • For a general descending quiver group • with , • we can show the correspondence of 1-loop and primary part, if we set • ‘full’ puncture : • ‘simple’ puncture : • general puncture : • where , , • , , [Kanno-Matsuo-SS-Tachikawa ’09] [Drukker-Passerini ’10] [SS ’11]
It’s an interesting result! But… Why do they correspond to each other? Our question at this time: Where does “U(1) factor” come from? … Larger symmetry exists behind AGT relation!? We also hope to check the remaining correspondence of instantonpart and descendant part in terms of this larger symmetry.
W1+∞algebra • Lie algebra of differential operators on a circle • generators: where • W1+∞algebra central extension
W1+∞ algebra contains the following generators: • U(1) generator • Virasoro generator • W3 generator • WN generator (N<∞) complicated nonlinear terms … The central charge for each generator can be shifted. Our conjecture: This U(1) generator corresponds to “U(1) factor”, i.e. W1+∞algebra exists behind AGT relation!?
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.
AGT relation in terms of W1+∞algebra [Kanno-Matsuo-SS ’11] • We consider the chain vector (3-point function with a level-n descendant field) • whose inner product is 4-point function. (a projector onto level-n state) • The chain vector of W1+∞algebra must be obtainedas • get the chain vectors of WN and U(1) algebra from recursion relation • use the relation of W1+∞ algebra and WN algebra (in the previous slide) • The bases can be defined by products of Schur polynomials • then the coefficients agree with Nekrasov’sinstanton partition function. corresponding to free bosons [Belavin-Belavin ’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]
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.