β-deformed matrix models and Nekrasov Partition Functions

Transcription

1 β-deformed matrix models and Nekrasov Partition Functions Takeshi Oota (OCAMI) Joint Work with H. Itoyama(Osaka City U. & OCAMI) Progress in Quantum Field Theory and String Theory April 4,

2 1. Introduction (Random) matrix models 0-dim. field theory Various types of matrix models: choice of ensemble (type of matrix) choice of the action W various usage in 2d gravity (non-critical string), IKKT model, Dijkgraaf-Vafa (DV) theory, 2/26

3 (Example) Hermitian one-matrix model matrix M: an NxN Hermitian matrix Large N limit matrix model curve DV theory( 02): gauge theory/matrix model correspondence identify this curve with the Seiberg-Witten curves of the N=2 gauge theories Alday-Gaiotto-Tachikawa (AGT) conjecture ([ ]) revisit this gauge theory/matrix model correspondence 3/26

4 AGT conjecture from the view point of gauge theory/matrix model correspondence renewed interests in β-deformed matrix models Itoyama-T.O.-Maruyoshi [ ], Eguchi-Maruyoshi [ ], Sulkowski [ ], Mironov-Morozov-Shakirov [ ], Itoyama-T.O. [ ], Mironov-Morozov-Morozov [ ], Awata-Yamada [ ], Eguchi-Maruyoshi [ ], Itoyama-T.O.-Yonezawa [ ], Maruyoshi-Yagi [ ], Mironov-Morozov-Shakirov [ ], Marshakov-Mironov-Morozov [ ], Mironov-Morozov-Shakirov [ ; ], Mironov-Morozov-Popolitov-Shakirov [ ], Itoyama-Yonezawa [ ], Bonelli-Maruyoshi-Tanzini [ ], Itoyama-T.O. [ ], Nishinaka-Rim [ ], Morozov [ ], Mironov-Morozov-Zakirova [ ],.. Many variants of β-deformed matrix models Here we concentrate on a type which has close connection with the Selberg integral. 4/26

5 4d/2d connection through 0d matrices (DV,[ ]) 4d N=2 SU(2) with N f =4 instanton part of Nekrasov part. fn. AGT 2d CFT four-point conformal block DF multiple integral 0d β-deformed matrix model of Selberg type Mironov-Morozov-Shakirov [ ], Itoyama-T.O [ ] 5/26

6 Matrix model technology Saddle point evaluation in the large N limit Orthogonal polynomials Recursion relations message: (a generalization of) the Selberg integral also plays important role in the β-deformed matrix models, which have connection with N=2 gauge theories Kadell integral (1997) (Macdonald s ex-conjecture (1987)) 6/26

7 Euler beta function Selberg integral (1944) (multiple generalization of the beta fn.) Atle Selberg, Bemerkinger om et multipelt integral (Remarks on a multiple integral), Norsk Matematisk Tidsskrift 26 (1944), (in Norwegian) 7/26

8 Macdonald-Kadell formula (Kadell integral) :a partition : the Jack symmetric polynomial 8/26

9 Plan of this talk 1. Introduction 2. β-deformed matrix model 3. AGT(4d/2d connection) through 0d matrices 4. On a basis of q-expansion 5. Summary 9/26

10 2. β-deformed matrix model Hermitian One Matrix Model: 10/26

11 (generalized) β-deformed MM (for some contour ) β-deformed MM of Selberg Type If Selberg integral 11/26

12 MM of Selberg Type: average w.r.t. the Selberg integral Exactly calculable: Itoyama-T.O. [ ] expansion in Jack polynomial basis and by using Kadell integral 12/26

13 13/26

14 14/26

15 The Macdonald-Kadell formula: arm length leg length coarm length coleg length 15/26

16 3. AGT(4d/2d connection) through 0d matrices 4d: instanton part of Nekrasov part. fn. 2d: conformal block 0d: average in matrix model 16/26

17 4d: Instanton part of Nekrasov partition function for N=2 SU(2) gauge theory with N f =4 (antifundamental matters) 17/26

18 2d: conformal block model independent (representation theoretic) quantity Free field representation of the conformal block screening charges 18/26

19 0d: β-deformed matrix model of Selberg type 19/26

20 20/26

21 4. On a basis of q-expansion There is a bais of q-expansion such that Factorization (conjecture) Itoyama-T.O. [ ] 21/26

22 Explicit forms 22/26

23 Some properties of M (1) In general, they are inhomogeneous polynomials (2) (3) They are greatly simplified at β=1 23/26

24 Cauchy identity 24/26

25 q-expansion in Schur polynomial basis But for generic β, Only for β=1, Mironov-Morozov-Shakirov [ ] (β=1 for SU(N): Zhang-Matsuo [ ]) 25/26

26 5. Summary β-deformed matrix model of Selberg type calculation method by using the Jack polynomials and Kadell integral Application to q-expansion of the Nekrasov partition function Characterization of and for generic β? generalization to other gauge theories? 26/26