Matrix model, supersymmetirc gauge theory, and discrete Painlev - PowerPoint PPT Presentation

Matrix model, supersymmetirc gauge theory, and discrete Painlevé equation Takeshi Oota (NITEP and OCAMI) December 13, 2018 International Symposium in Honor of Professor Nambu for the 10th Anniversary of his Nobel Prize in Physics, Media Center, Osaka City Univeristy Joint work with Hiroshi Itoyama and Katsuya Yano Ref: arXiv: 1805.05057, arXiv:1812.00811

Introduction

gauge theories has been known for long time and investigated from various viewpoints. AGT relation [Alday-Gaiotto-Tachikawa, 2009] supersymmetric gauge theories correspondence 3 • The correspondence between (large N ) matrix models and 2d conformal blocks = Nekrasov partition function of N = 2 = ⇒ Refined correspondence between (finite N ) matrix models and ( N = 2 ) gauge theories [Example] ( β -deformed) matrix model having the potential with three logarithmic terms ← → N = 2 SU ( 2 ) with N f = 4 • (Motivation) Renewed interest in Painlevé/Gauge • We concentrate on one of irregular limits to N f = 2

Painlevé/Gauge correspondence PI [Bonelli-Lisovyy-Maruyoshi-Sciarappa-Tanzini, 1612.06235 [hep-th]] [Kajiwara-Masuda-Noumi-Ohta-Yamada, nlin/0403009] theories Argyres-Douglas Painlevé equations 4 PII PIV PIII 3 PIII 2 PIII 1 PV PVI N = 2 supersymmetric SU ( 2 ) gauge theories with N f flavors N f = 4 N f = 3 N f = 2 N f = 1 N f = 0 H 2 AD H 1 AD H 0 AD

[Dzhamay-Takenawa, 1408.3778 [math-ph]] PII alt d-PI d-PII alt d-PII Painlevé and associated discrete Painlevé (dP) d-PV PI d-PIII, d-PIV PIV PIII 3 PIII 2 PIII 1 PV PVI (alt=alternate) 5 N f = 4 N f = 3 N f = 2 N f = 1 N f = 0 H 2 AD H 1 AD H 0 AD

In this talk, we concentrate on d-PV from the viewpoint of matrix models. alt d-PI PVI alt d-PII d-PIII, d-PIV d-PII PI PII PIV PIII 3 PIII 2 PIII 1 PV 6 N f = 4 N f = 3 N f = 2 N f = 1 N f = 0 H 2 AD H 1 AD H 0 AD

Unitary matrix model

Generalized Gross-Witten-Wadia model 2 g i d z i Let us consider the following unitary matrix model This reduces to multiple integrals over eigenvalues z 8 1 ∫ ( ) Z U ( N ) ( M ) := [ d U ] exp Tr W U ( U ) , vol ( U ( N )) where U is an N × N unitary matrix model, with the potential ( ) W U ( z ) = − 1 z + 1 + M log z , ( M ∈ Z ). ( N ) ( N ) � ∏ ∑ ∆( z )∆( z − 1 ) exp Z U ( N ) = 1 W U ( z i ) , N ! 2 π i z i i = 1 i = 1 ∏ ∏ ∆( z − 1 ) = ( z − 1 − z − 1 ∆( z ) = ( z i − z j ), j ). 1 ≤ i < j ≤ N 1 ≤ i < j ≤ N

Properties The free energy where the planar contribution is 9 (1) When M = 0 , this is the famous Gross-Witten-Wadia (GWW) model. In the large N , this model exhibits the third order phase transition at S = 1 where S := Ng . (2) For M ̸ = 0 with M finite, the generalized GWW model also exhibits the third order phase transition at S = 1 in the large N . ∞ ∑ F k ( S ) g 2 k − 2 , F = log Z U ( N ) ( M ) = k = 0  ( S ≥ 1 ), ( 1 / 4 ),  F 0 ( S ) = ( 1 / 2 ) S 2 ( log S − ( 3 / 2 )) + S , ( 0 ≤ S ≤ 1 ).  F ′′′ 0 is discontinuous at S = 1 .

10 0 Painlevé II (alt dPII) equations (with different parameters). If we set (3) The partition function Z U ( N ) ( M ) is a τ -function of Painlevé III ′ equation. (4) The partition function Z U ( N ) ( M ) can be written as N − 1 ) N − j ( ∏ Z U ( N ) ( M ) = h N 1 − A j ( M ) B j ( M ) . j = 1 X n ( M ) := A n + 1 ( M ) Y n ( M ) := B n + 1 ( M ) A n ( M ) , B n ( M ) , then X n and Y n respectively satisfies the alternate discrete

11 g s (Irregular limit of AGT relation). hypermultiplets. g s g s 1 identifications: (5) We expect that this partition function Z U ( N ) ( M ) is closely related to the instanton partition function of the 4d N = 2 supersymmetric SU ( 2 ) gauge theory with N f = 2 hypermultiplets in the self-dual Ω background via parameter g = Λ 2 N = − ( m 1 + m 2 ) M = ( m 2 − m 1 ) , , . Here Λ 2 is dynamical mass scale and m i are the mass of the • The planar loop equation of the matrix model can be identified with the Seiberg-Witten curve of N f = 2 model.

(6) In the double scaling limit, the alt dPII equation goes to the Painlevé II equation. The loop equation in the double scaling limit goes to the Seiberg-Witten curve of the Argyres-Douglas (AD) model of type H 1 . 12 4 4 8 N f = 2 (first realization) H 1 AD (first realization)

Some Details

14 4 q d 2 q s q with a Hamiltonian d s d p d q s is a Hamiltonian system d s (3) The τ -function of PIII ′ The Painlevé III ′ equation ( q = q ( s ) ) ( d q ) 2 ( d q ) 4 s 2 ( γ q + α ) + β 4 s + δ d s 2 = 1 − 1 + q 2 d s = ∂ H III ′ d s = − ∂ H III ′ ∂ p , ∂ q [ ] q 2 p 2 − ( q 2 + v 2 q − s ) p + 1 H III ′ ( s ) = 1 2 ( v 1 + v 2 ) q . The parameters α , β , γ , and δ of PIII ′ are fixed as α = − 4 v 1 , β = 4 ( v 2 + 1 ), γ = 4 , δ = − 4 .

15 1 g s Let g s with • The τ -function of PIII ′ is defined by H III ′ ( s ) = d d s log τ ( s ). σ ( s ) := sH III ′ ( s ) = q 2 p 2 − ( q 2 + v 2 q − s ) p + 1 2 ( v 1 + v 2 ) = s d d s log τ ( s ). This function satisfies the σ -form of PIII ′ : ( ) 2 ( s σ ′′ ) 2 − 4 ( σ − s σ ′ ) σ ′ ( σ ′ − 1 ) − v 2 σ ′ − 1 2 ( v 1 + v 2 ) = 0 . Using [Forrester-Witte, math-ph/0201051], we can see that τ ( s ) = s ( 1 / 2 ) MN Z U ( N ) ( M ), s = 4 g 2 , v 1 = M + N = − 2 m 1 v 2 = − M + N = − 2 m 2 , .

(4) alt dPII satisfies the discrete Painlevé II equation (dPII equation) n 16 • The unitary matrix model Z U ( N ) ( M ) can be solved by the method of orthogonal polynomials. When M = 0 , the GWW model is studied by this method in [Periwal-Shevits, 1990]. • When M = 0 , N − 1 ∏ j ) N − j . ( 1 − R 2 Z U ( N ) ( 0 ) = I 0 ( 1 / g ) N j = 1 Here I ν ( z ) is the modified Bessel function of the first kind. R n R n + 1 + R n − 1 = 2 n g R n . 1 − R 2

17 (string equations): dPII equation. 2 n g B n to 2 n g A n • When M ∈ Z , the results of [Periwal-Shevits] are generalized N − 1 ∏ ( 1 − A j B j ) N − j . Z U ( N ) ( M ) = ( − 1 ) MN I M ( 1 / g ) N j = 1 A n and B n satisfy the following system of recursion relations A n + 1 + A n − 1 = , B n + 1 + B n − 1 = , 1 − A n B n 1 − A n B n A n B n + 1 − A n + 1 B n = 2 M g . When M ̸ = 0 , A n ̸ = B n . But when M = 0 , we can set A n = B n = R n and these recursion relations reduces to the

Let n Y n 2 g n 2 g X n 18 Then these variables satisfy the alt dPII equations: X n ( M ) = A n + 1 ( M ) Y n ( M ) = B n + 1 ( M ) ( n ≥ 0 ). A n ( M ) , B n ( M ) , ( ) ( n + 1 ) + = 1 − X n + 1 + n − M , 1 + X n X n + 1 1 + X n X n − 1 ( n + 1 ) ( ) + = 1 − Y n + 1 + n + M . 1 + Y n Y n + 1 1 + Y n Y n − 1

(6) Double scaling limit turns into a continuous function Let N 19 1 N • The double scaling limit: N → ∞ , S = Ng → 1 with κ ≡ 1 ( 1 − S ) ( 1 / 2 )( 2 − γ st ) , ( γ st = − 1 ) kept finite. Here γ st is the susceptibility of the system. • In the N → ∞ limit, we assume that a discrete variable f n ( n ) f n → f ≡ f ( x ), x ≡ n N , ( 0 ≤ x ≤ 1 ). a 3 ≡ 1 ( c = κ − 2 / 3 ), N , S ≡ N g = 1 − c a 2 , n g = n NNg = S x = 1 − 1 2 a 2 t , A n B n = a 2 u ( t ).

Then the string equations turns into the PII equation: [Flaschka-Newell, 1980] 2 p 2 with the Hamiltonian This is a Hamiltonian system 20 u ′′ = ( u ′ ) 2 2 u + y 2 − 1 2 t u − M 2 2 u . Let p u ≡ − u ′ / u . Then, this is equivalent to u ′ = − p u u , p ′ u = 1 u − u + 1 2 t + M 2 2 u 2 . 2 p 2 u ′ = ∂ H II u = − ∂ H II p ′ , ∂ u , ∂ p u 2 u 2 − 1 H II = − 1 u u + 1 2 t u + M 2 2 u .

By a canonical transformation the Hamiltonian becomes By rescaling y and t , we get the standard form of PII equation: Then This leads to 21 p y ( u , p u ) → ( y , p y ); u = − p y , p u = y + M , 2 ( y 2 + t ) p y + M y . H II = 1 y + 1 2 p 2 y ′ = p y + 1 2 y 2 + 1 p ′ 2 t , y = − p y y − M . ( 1 ) y ′′ = 1 2 y 3 + 1 2 − M 2 t y + . y ′′ = 2 y 3 + t y + α , 2 − M . α = 1

22 p n p y dPI): • Bäcklund transformations for PII are generated by s 1 and π : s 1 ( y ) = y + 2 M , π ( y ) = − y , π ( p y ) = − p y − y 2 − t , s 1 ( p y ) = p y , s 1 ( M ) = − M , π ( M ) = 1 − M . • The translation T = s 1 π . T n ( M ) = M + n , ( n ∈ Z ). Let y n ( t ) = T n ( y ( t )) , p n ( t ) = T n ( p y ( t )) , ( n ∈ Z ) . Then they obey y n + 1 + y n = − 2 ( M + n ) p n + p n − 1 = − y 2 n − t . , These leads to the alternate discrete Painlevé I equation (alt + 2 ( M + n − 1 ) 2 ( M + n ) = y 2 n + t . y n + 1 + y n y n + y n − 1

PVI d-PV alt d-PI PV alt d-PII d-PIII, d-PIV d-PII PI PII PIV PIII 3 PIII 2 PIII 1 23 N f = 4 N f = 3 N f = 2 N f = 1 N f = 0 H 2 AD H 1 AD H 0 AD