Bilinear equations on Painlevé τ-functions from CFT

Transcription

1 Bilinear equations on Painlevé τ-functions from CFT M. Bershtein, A. Shchechkin 19 June 2014 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

2 Painleve equations Painlevé VI is the most general equation of type q = F (t, q, q ) without movable critical points except poles. d 2 q dt 2 = 1 ( 1 2 q + 1 q ) ( ) 2 ( dq 1 q t dt t + 1 t ) dq q t dt + ( 2q(q 1)(q t) ( + t 2 (t 1) 2 θ 1 ) 2 θ 0 2t ( ) ) 2 q 2 + θ2 1 (t 1) θ 2 (q 1) 2 t 1 4 t(t 1) (q t) 2. 2nd order, 4 parameters θ = (θ 0, θ t, θ 1, θ ) M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

3 Painleve equations Painlevé VI is the most general equation of type q = F (t, q, q ) without movable critical points except poles. d 2 q dt 2 = 1 ( 1 2 q + 1 q ) ( ) 2 ( dq 1 q t dt t + 1 t ) dq q t dt + ( 2q(q 1)(q t) ( + t 2 (t 1) 2 θ 1 ) 2 θ 0 2t ( ) ) 2 q 2 + θ2 1 (t 1) θ 2 (q 1) 2 t 1 4 t(t 1) (q t) 2. 2nd order, 4 parameters θ = (θ 0, θ t, θ 1, θ ) Confluence PVI PV PIII 1 PIII 2 PIII 3 PIV PII PI M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

4 Painleve equations Painlevé VI is the most general equation of type q = F (t, q, q ) without movable critical points except poles. d 2 q dt 2 = 1 ( 1 2 q + 1 q ) ( ) 2 ( dq 1 q t dt t + 1 t ) dq q t dt + ( 2q(q 1)(q t) ( + t 2 (t 1) 2 θ 1 ) 2 θ 0 2t ( ) ) 2 q 2 + θ2 1 (t 1) θ 2 (q 1) 2 t 1 4 t(t 1) (q t) 2. 2nd order, 4 parameters θ = (θ 0, θ t, θ 1, θ ) Confluence PVI PV PIII 1 PIII 2 PIII 3 PIV PII PI Painlevé III 3 d 2 q dt 2 = 1 q ( dq dt ) 2 1 dq t dt + 2q2 t t, M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

5 Isomonodromic deformations Linear system of rank N with n regular singularities a = {a 1,..., a n } on CP 1 : n A ν z Φ = A(z)Φ, A(z) =, z a ν n ν=1 ν=1 A ν sl N, Aν = 0; A ν = G νt νgν 1, where T ν = diag {λ ν,1,..., λ ν,n }; Non-resonance condition: λ ν,j λ ν,k / Z. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

6 Isomonodromic deformations Linear system of rank N with n regular singularities a = {a 1,..., a n } on CP 1 : n A ν z Φ = A(z)Φ, A(z) =, z a ν n ν=1 A ν sl N, Aν = 0; A ν = G νt νgν 1, where T ν = diag {λ ν,1,..., λ ν,n }; Non-resonance condition: λ ν,j λ ν,k / Z. Let us now vary the positions of singularities a ν simultaneously evolving A ν s in such way that the monodromy is preserved. Schlesinger deformation equations: aµ A ν = z 0 a ν z 0 a µ [A µ, A ν ] a µ a ν, µ ν, ν=1 aν A ν = µ ν [A µ, A ν ] a µ a ν. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

7 Isomonodromic deformations Linear system of rank N with n regular singularities a = {a 1,..., a n } on CP 1 : n A ν z Φ = A(z)Φ, A(z) =, z a ν n ν=1 A ν sl N, Aν = 0; A ν = G νt νgν 1, where T ν = diag {λ ν,1,..., λ ν,n }; Non-resonance condition: λ ν,j λ ν,k / Z. Let us now vary the positions of singularities a ν simultaneously evolving A ν s in such way that the monodromy is preserved. Schlesinger deformation equations: ν=1 aµ A ν = z 0 a ν [A µ, A ν ], µ ν, aν A ν = z 0 a µ a µ a ν µ ν ) Corollary: d( µ<ν Tr A µa ν d ln (a µ a ν ) = 0. [A µ, A ν ] a µ a ν. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

8 Isomonodromic deformations Linear system of rank N with n regular singularities a = {a 1,..., a n } on CP 1 : n A ν z Φ = A(z)Φ, A(z) =, z a ν n ν=1 A ν sl N, Aν = 0; A ν = G νt νgν 1, where T ν = diag {λ ν,1,..., λ ν,n }; Non-resonance condition: λ ν,j λ ν,k / Z. Let us now vary the positions of singularities a ν simultaneously evolving A ν s in such way that the monodromy is preserved. Schlesinger deformation equations: aµ A ν = z 0 a ν [A µ, A ν ], µ ν, aν A ν = [A µ, A ν ]. z 0 a µ a µ a ν a µ a ν µ ν ) Corollary: d( µ<ν Tr A µa ν d ln (a µ a ν ) = 0. Isomonodromic τ is defined on universal covering of C n \ {a i a j } by ν=1 i j d ln τ(a 1,..., a n ) = µ<ν Tr A µ A ν d ln (a µ a ν ). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

9 M 0,4,θ Painlevé VI correspond to the rank 2 linear systems on CP 1 with 4 regular singular points 0, t, 1,. z Φ = A(z)Φ, A(z) = A 0 z + A t z t + A 1 z 1. A = A 0 A t A 1. Eigenvalues of A ν are (θ ν, θ ν ), ν = 0, 1, t, M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

10 M 0,4,θ Painlevé VI correspond to the rank 2 linear systems on CP 1 with 4 regular singular points 0, t, 1,. z Φ = A(z)Φ, A(z) = A 0 z + A t z t + A 1 z 1. A = A 0 A t A 1. Eigenvalues of A ν are (θ ν, θ ν ), ν = 0, 1, t, Monodromy matrices M 0, M t, M 1, M SL 2 (C). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

11 M 0,4,θ Painlevé VI correspond to the rank 2 linear systems on CP 1 with 4 regular singular points 0, t, 1,. z Φ = A(z)Φ, A(z) = A 0 z + A t z t + A 1 z 1. A = A 0 A t A 1. Eigenvalues of A ν are (θ ν, θ ν ), ν = 0, 1, t, Monodromy matrices M 0, M t, M 1, M SL 2 (C). The module space of flat rank 2 connections: { } / trm ν = 2 cos(2πθ ν ), M 0,4,θ = M ν SL 2 M M 1 M t M 0 = 1 dim M 0,4,θ = 2 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

12 M 0,4,θ Painlevé VI correspond to the rank 2 linear systems on CP 1 with 4 regular singular points 0, t, 1,. z Φ = A(z)Φ, A(z) = A 0 z + A t z t + A 1 z 1. A = A 0 A t A 1. Eigenvalues of A ν are (θ ν, θ ν ), ν = 0, 1, t, Monodromy matrices M 0, M t, M 1, M SL 2 (C). The module space of flat rank 2 connections: { } / trm ν = 2 cos(2πθ ν ), M 0,4,θ = M ν SL 2 M M 1 M t M 0 = 1 dim M 0,4,θ = 2 As coordinates on M 0,4,θ one can use σ µν defined by trm µ M ν = 2 cos(2πσ µν ), µ, ν = 0, t, 1. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

13 Painlevé VI τ function Jimbo relation: m ν = trm ν = 2 cos(2πθ ν ), p µν = trm µ M ν = 2 cos(2πσ ν ), p 0t p 1t p 01 + p0t 2 + p1t 2 + p01 2 p 0t (m 0 m t + m 1 m ) p 1t (m 1 m t + m 0 m ) p 01 (m 0 m 1 + m t m ) + (m0 2 + mt 2 + m1 2 + m 2 + m 0 m t m 1 m 4) = 0 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

14 Painlevé VI τ function Jimbo relation: m ν = trm ν = 2 cos(2πθ ν ), p µν = trm µ M ν = 2 cos(2πσ ν ), p 0t p 1t p 01 + p0t 2 + p1t 2 + p01 2 p 0t (m 0 m t + m 1 m ) p 1t (m 1 m t + m 0 m ) p 01 (m 0 m 1 + m t m ) + (m0 2 + mt 2 + m1 2 + m 2 + m 0 m t m 1 m 4) = 0 s ±1 0t We will use coordinates σ 0t, s 0t : = (m tm 1 +m 0 m ± ip 01 sin 2πσ 0t ) (m 0 m 1 +m t m ip 1t sin 2πσ 0t ) e ±2πiσ0t (cos 2π(θ t σ 0t ) m 0 ) (cos 2π(θ 1 σ 0t ) m ) This coordinates are are also closely related to the Nekrasov, Rosly Shatashvili coordinates α, β.. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

15 Painlevé VI τ function Jimbo relation: m ν = trm ν = 2 cos(2πθ ν ), p µν = trm µ M ν = 2 cos(2πσ ν ), p 0t p 1t p 01 + p 2 0t + p 2 1t + p 2 01 p 0t (m 0 m t + m 1 m ) p 1t (m 1 m t + m 0 m ) p 01 (m 0 m 1 + m t m ) + (m m 2 t + m m 2 + m 0 m t m 1 m 4) = 0 s ±1 0t We will use coordinates σ 0t, s 0t : = (m tm 1 +m 0 m ± ip 01 sin 2πσ 0t ) (m 0 m 1 +m t m ip 1t sin 2πσ 0t ) e ±2πiσ0t (cos 2π(θ t σ 0t ) m 0 ) (cos 2π(θ 1 σ 0t ) m ) This coordinates are are also closely related to the Nekrasov, Rosly Shatashvili coordinates α, β. For the Painlevé VI equations σ, s are the integration constants. τ-form of Painlevé VI: D VI (τ(t), τ(t)) = 0, where D VI is certain bilinear differential operator (combination of Hirota operators).. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

16 t = 0 expansion Denote ν = θ 2 ν, σ = σ 2, s = s 0t, σ = σ 0t. Jimbo formula (1982) in Re σ < 1 2 : τ( θ, s, σ; t) = const ( t σ 0 t + C ±1 t σ±1 0 t ), where C ±1 = s ±1 [Gamma functions]. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

17 t = 0 expansion Denote ν = θ 2 ν, σ = σ 2, s = s 0t, σ = σ 0t. Jimbo formula (1982) in Re σ < 1 2 : τ( θ, s, σ; t) = const ( t σ 0 t + C ±1 t σ±1 0 t ), where C ±1 = s ±1 [Gamma functions]. Gamayun, Iorgov, Lisovyy (2012) conjecture Theorem The expansion Painlevé VI τ function near t = 0 can be written as τ( θ, s, σ; t) = n Z C(σ + n, θ ) s n F(, (σ + n) 2 t) F(, (σ + n) 2, t) 4-point CP 1 conformal block for central charge c = 1. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

18 t = 0 expansion Denote ν = θ 2 ν, σ = σ 2, s = s 0t, σ = σ 0t. Jimbo formula (1982) in Re σ < 1 2 : τ( θ, s, σ; t) = const ( t σ 0 t + C ±1 t σ±1 0 t ), where C ±1 = s ±1 [Gamma functions]. Gamayun, Iorgov, Lisovyy (2012) conjecture Theorem The expansion Painlevé VI τ function near t = 0 can be written as τ( θ, s, σ; t) = n Z C(σ + n, θ ) s n F(, (σ + n) 2 t) F(, (σ + n) 2, t) 4-point CP 1 conformal block for central charge c = 1. The coefficients C(σ, θ ) are defined by use of Barnes G-function G(z + 1) = Γ(z)G(z): C(σ, ɛ,ɛ θ ) = =±1 G(1 + θ t + ɛθ 0 + ɛ σ)g(1 + θ 1 + ɛθ + ɛ σ) ɛ=±1 G(1 + 2ɛσ). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

19 Applications: expansions and connection. Another proof: by Iorgov, Lisovyy, Teschner M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18 t = 0 expansion Denote ν = θ 2 ν, σ = σ 2, s = s 0t, σ = σ 0t. Jimbo formula (1982) in Re σ < 1 2 : τ( θ, s, σ; t) = const ( t σ 0 t + C ±1 t σ±1 0 t ), where C ±1 = s ±1 [Gamma functions]. Gamayun, Iorgov, Lisovyy (2012) conjecture Theorem The expansion Painlevé VI τ function near t = 0 can be written as τ( θ, s, σ; t) = n Z C(σ + n, θ ) s n F(, (σ + n) 2 t) F(, (σ + n) 2, t) 4-point CP 1 conformal block for central charge c = 1. The coefficients C(σ, θ ) are defined by use of Barnes G-function G(z + 1) = Γ(z)G(z): C(σ, ɛ,ɛ θ ) = =±1 G(1 + θ t + ɛθ 0 + ɛ σ)g(1 + θ 1 + ɛθ + ɛ σ) ɛ=±1 G(1 + 2ɛσ).

20 Painlevé III τ function Gamayun, Iorgov, Lisovyy conjecture Theorem The expansion Painlevé III 3 τ function near t = 0 can be written as τ(σ, s; t) = C(σ + n)s n F((σ + n) 2 t), n Z F(σ 2 t) irregular (Whittaker) limit of Virasoro (Vir) ( conformal block for ) central charge c = 1. The coefficients C(σ) = 1/ G(1 2σ)G(1 + 2σ). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

21 Painlevé III τ function Gamayun, Iorgov, Lisovyy conjecture Theorem The expansion Painlevé III 3 τ function near t = 0 can be written as τ(σ, s; t) = C(σ + n)s n F((σ + n) 2 t), n Z F(σ 2 t) irregular (Whittaker) limit of Virasoro (Vir) ( conformal block for ) central charge c = 1. The coefficients C(σ) = 1/ G(1 2σ)G(1 + 2σ). Our proof is based on the bilinear equation on τ functions : D III (τ(t), τ(t)) = 0, where D III = 1 2 D4 [log t] t d dt D2 [log t] +1 2 D2 [log t] +2tD0 [log t], Here D[log k t] is a Hirota differential operator defined by: f (e α t)g(e α t) = + k=0 D[log k t] (f (t), g(t))αk k! M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

22 Bilinear conformal blocks relations We substitute τ(t) to the equation and collect terms with the same powers of s. The vanishing condition of s m coefficient have the form: ( ) C(σ + n + m)c(σ n) D III F((σ + n + m) 2 t), F((σ n) 2 t = 0, n Z M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

23 Bilinear conformal blocks relations We substitute τ(t) to the equation and collect terms with the same powers of s. The vanishing condition of s m coefficient have the form: ( ) C(σ + n + m)c(σ n) D III F((σ + n + m) 2 t), F((σ n) 2 t = 0, n Z The s m term coincide with of s m+2 term after the shift σ σ + 1. Therefore it is sufficient to prove the vanishing of s 0 and s 1 terms: ( ) C(σ + n)c(σ n) D III F((σ + n) 2 t), F((σ n) 2 t) = 0, n Z n Z+ 1 2 ( ) C(σ + n)c(σ n) D III F((σ + n) 2 t), F((σ n) 2 t) = 0 (we made the shift σ σ + 1 2, n n in s1 term). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

24 Bilinear conformal blocks relations We substitute τ(t) to the equation and collect terms with the same powers of s. The vanishing condition of s m coefficient have the form: ( ) C(σ + n + m)c(σ n) D III F((σ + n + m) 2 t), F((σ n) 2 t = 0, n Z The s m term coincide with of s m+2 term after the shift σ σ + 1. Therefore it is sufficient to prove the vanishing of s 0 and s 1 terms: ( ) C(σ + n)c(σ n) D III F((σ + n) 2 t), F((σ n) 2 t) = 0, n Z n Z+ 1 2 ( ) C(σ + n)c(σ n) D III F((σ + n) 2 t), F((σ n) 2 t) = 0 (we made the shift σ σ + 1 2, n n in s1 term). Each summand Vir Vir. The whole sum Vir Vir F NSR, where F is a Majorana fermion algebra and NSR is Neveu Schwarz Ramond algebra, N = 1 superanalogue of the Virasoro algebra. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

25 Instanton counting Let M(C 2 ; r, N) denotes the moduli space of instantons on C 2 (of rank r, c 2 = N). By Z(ɛ 1, ɛ 2, a; q) we denote Nekrasov instanton partition function for the pure U(2) gauge theory on C 2 : Z(ɛ 1, ɛ 2, a; q) = N=0 qn 1. M(C 2 ;r,n) Due to AGT relation Z(ɛ 1, ɛ 2, a; q) correspond to irregular confromal block with c = (ɛ1+ɛ2)2 ɛ 1ɛ 2. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

26 Instanton counting Let M(C 2 ; r, N) denotes the moduli space of instantons on C 2 (of rank r, c 2 = N). By Z(ɛ 1, ɛ 2, a; q) we denote Nekrasov instanton partition function for the pure U(2) gauge theory on C 2 : Z(ɛ 1, ɛ 2, a; q) = N=0 qn 1. M(C 2 ;r,n) Due to AGT relation Z(ɛ 1, ɛ 2, a; q) correspond to irregular confromal block with c = (ɛ1+ɛ2)2 ɛ 1ɛ 2. Nakajima-Yoshioka blow-up equations (2003): q k2 Z(ɛ 1, ɛ 2, a; q) = l n(a, ɛ 1, ɛ 2 ) Z(ɛ 1, ɛ 2 ɛ 1, a+nɛ 1 ; q) Z(ɛ 1 ɛ 2, ɛ 2, a+nɛ 2 ; q), n Z Impossible to make c (1) = c (1) = 1. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

27 Instanton counting Let M(C 2 ; r, N) denotes the moduli space of instantons on C 2 (of rank r, c 2 = N). By Z(ɛ 1, ɛ 2, a; q) we denote Nekrasov instanton partition function for the pure U(2) gauge theory on C 2 : Z(ɛ 1, ɛ 2, a; q) = N=0 qn 1. M(C 2 ;r,n) Due to AGT relation Z(ɛ 1, ɛ 2, a; q) correspond to irregular confromal block with c = (ɛ1+ɛ2)2 ɛ 1ɛ 2. Nakajima-Yoshioka blow-up equations (2003): q k2 Z(ɛ 1, ɛ 2, a; q) = l n(a, ɛ 1, ɛ 2 ) Z(ɛ 1, ɛ 2 ɛ 1, a+nɛ 1 ; q) Z(ɛ 1 ɛ 2, ɛ 2, a+nɛ 2 ; q), n Z Impossible to make c (1) = c (1) = 1. X 2 minimal resolution of C 2 /Z 2. Bonelli, Maruyoshi, Tanzini (2012): q 2n2 Z X2 (ɛ 1, ɛ 2, a; q) = l n(a, ɛ 1, ɛ ) Z(2ɛ 1, ɛ 2 ɛ 2, a+2nɛ 1 ; q)z(ɛ 1 ɛ 2, 2ɛ 2, a+2nɛ 2 ; q) 2 2n Z If ɛ 1 + ɛ 2 = 0 then c (1) = c (2) = 1. The coefficients 1/l n(a, ɛ 1, ɛ 2 ) are proportional to C(σ + n)c(σ n) C(σ) 2. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

28 The F NSR algebra The F NSR algebra is a direct sum of the free-fermion algebra F with generators f r (r Z ) and NSR (Neveu-Schwarz-Ramond or Super Virasoro) algebra with generators L n, G r (n Z, r Z ). {f r, f s } = δ r+s,0, [L n, L m ] = (n m)l n+m + c NSR 8 (n3 n)δ n+m, ( ) 1 [L n, G r ] = 2 n r G n+r, {G r, G s } = 2L r+s + c NSR 2 (r )δ r+s,0. The central charge c NSR is parameterized as: c NSR = 1 + 2(b + b 1 ) 2. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

29 The F NSR algebra The F NSR algebra is a direct sum of the free-fermion algebra F with generators f r (r Z ) and NSR (Neveu-Schwarz-Ramond or Super Virasoro) algebra with generators L n, G r (n Z, r Z ). {f r, f s } = δ r+s,0, [L n, L m ] = (n m)l n+m + c NSR 8 (n3 n)δ n+m, ( ) 1 [L n, G r ] = 2 n r G n+r, {G r, G s } = 2L r+s + c NSR 2 (r )δ r+s,0. The central charge c NSR is parameterized as: c NSR = 1 + 2(b + b 1 ) 2. Two commuting Virasoro (Crnkovic et al 1989; Lashkevich 1993 ) L (1) n = 1 1 b 2 L n 1 + 2b2 2(1 b 2 r : f n r f r : + b ) 1 b 2 f n r G r, r= r= 1 L (2) n = 1 b 2 L n 1 + 2b 2 2(1 b 2 r : f n r f r : + b 1 ) 1 b 2 f n r G r, r= The central charges: c (1) = 1 + 3(b+b 1 ) 2 (b 2 1), c (2) = 1 + 3(b+b 1 ) 2 (b 2 1). r= M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

30 Verma modules Denote Verma module of Vir as πvir. This module has highest weight vector, defined by properties: L 0 =, L n = 0, n > 0, We denote π NS F NSR Verma module of F NSR, NS = 1 8 (b + b 1 ) P2. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

31 Verma modules Denote Verma module of Vir as πvir. This module has highest weight vector, defined by properties: L 0 =, L n = 0, n > 0, We denote π NS F NSR Verma module of F NSR, NS = 1 8 (b + b 1 ) P2. Proposition (BBFLT) For generic NS the space πf NSR NS is isomorphic to Vir Vir module: πf NSR NS = πvir Vir n 2n Z M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

32 Verma modules Denote Verma module of Vir as πvir. This module has highest weight vector, defined by properties: L 0 =, L n = 0, n > 0, We denote π NS F NSR Verma module of F NSR, NS = 1 8 (b + b 1 ) P2. Proposition (BBFLT) For generic NS the space πf NSR NS is isomorphic to Vir Vir module: πf NSR NS = πvir Vir n 2n Z In other words, there exist vectors P, n : L (η) 0 P, n = (η) n P, n, L (η) P, n = 0, k > 0, 2n Z, η = 1, 2. k We can write that P, n = (1) n (2) n. The highest weights: (1) n = (b + b 1 ) 2 (P + 2nb)2 8(b 2 + 1) 2(b 2 1), (2) n = (b + b 1 ) 2 8(b 2 1) + (P + 2nb 1 ) 2 2(b 2 1), M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

33 Whittaker vector Whittaker vector for Vir defined by: W (q) = N=0 q +N 2 N L 0 N = ( + N) N, L 1 N = N 1, N > 0, L k N = 0, k > 1 This relation can be rewritten as L 1 W (q) = q 1/2 W (q), L k W (q) = 0, k > 1. First examples are: 0 =, 1 = 1 2 L 1. The Whittaker limit of the four point conformal block defined by: F c ( ; q) = W (q), W (q) = N, N q +N N=0 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

34 Whittaker vector Whittaker vector for Vir defined by: W (q) = N=0 q +N 2 N L 0 N = ( + N) N, L 1 N = N 1, N > 0, L k N = 0, k > 1 This relation can be rewritten as L 1 W (q) = q 1/2 W (q), L k W (q) = 0, k > 1. First examples are: 0 =, 1 = 1 2 L 1. The Whittaker limit of the four point conformal block defined by: F c ( ; q) = W (q), W (q) = N, N q +N N=0 Whittaker vector for NSR defined by: W (q) NSR = N=0 q +N 2 N NS G 1/2 W (q) NSR = q 1/4 W NSR, G s W (q) NSR = 0, s > 1 2. Whittaker vector for F NSR defined as the tensor product W (q) F NSR = 1 W (q) NSR, where 1 is a vacuum: f r 1 = 0, r > 0. The Whittaker limit of the four point conformal block defined by F cns ( NS q) = W F NSR (q) W F NSR (q) = W NSR (q) W NSR (q). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

35 Whittaker vector decomposition Decomposition of F NSR representation provides decomposition Whittaker vector: W F NSR (q) = v(q) n, 2n Z where v(q) n πvir Vir n. It turns out that v(q) n is Whittaker vector for algebra Vir Vir: M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

36 Whittaker vector decomposition Decomposition of F NSR representation provides decomposition Whittaker vector: W F NSR (q) = v(q) n, 2n Z where v(q) n πvir Vir n. It turns out that v(q) n is Whittaker vector for algebra Vir Vir: Proposition W F NSR (q) = 2n Z ( ) l n (P, b) W (1) n (β (1) q) W (2) n (β (2) q). Here W (1) n W (2) n tensor product of Whittaker vectors, ( ) 2 ( ) 2 β (1) b = 1 b 1 b, β (2) b = b b. 1 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

37 Whittaker vector decomposition Decomposition of F NSR representation provides decomposition Whittaker vector: W F NSR (q) = v(q) n, 2n Z where v(q) n πvir Vir n. It turns out that v(q) n is Whittaker vector for algebra Vir Vir: Proposition W F NSR (q) = 2n Z ( ) l n (P, b) W (1) n (β (1) q) W (2) n (β (2) q). Here W (1) n W (2) n tensor product of Whittaker vectors, ( ) 2 ( ) 2 β (1) b = 1 b 1 b, β (2) b = b b. 1 Taking the scalar square of we have: F c NS( NS q) = 2n Z l 2 n (P, b)f (1) n F (2) n, where F n (1) = F c (1)( (1) n β (1) q), F n (2) = F c (2)( (2) n β (2) q). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

38 Operator H Let us introduce operator H: and define F k as: F NS = W F NSR e αh W F NSR = Clearly F 0 = F c NS( NS q). H = bl (1) 0 + b 1 L (2) 0, + k=0 W F NSR H k W F NSR αk + k! = k=0 F k α k k! M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

39 Operator H Let us introduce operator H: and define F k as: F NS = W F NSR e αh W F NSR = H = bl (1) 0 + b 1 L (2) 0, + k=0 W F NSR H k W F NSR αk + k! = Clearly F 0 = F c NS( NS q). We can calculate F k using Whittaker vector decomposition: k=0 F k α k k! W F NSR (q) e αh W F NSR (q) = ln 2 (P, b) 2n Z W n (1) (β (1) q) e αbl(1) 0 W n (1) (β (1) q) W n (2) (β (2) q) e αb 1 L (2) 0 W n (2) (β (2) q). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

40 Operator H Let us introduce operator H: and define F k as: F NS = W F NSR e αh W F NSR = H = bl (1) 0 + b 1 L (2) 0, + k=0 W F NSR H k W F NSR αk + k! = Clearly F 0 = F c NS( NS q). We can calculate F k using Whittaker vector decomposition: k=0 F k α k k! W F NSR (q) e αh W F NSR (q) = ln 2 (P, b) 2n Z W n (1) (β (1) q) e αbl(1) 0 W n (1) (β (1) q) W n (2) (β (2) q) e αb 1 L (2) 0 W n (2) (β (2) q). Generalized Hirota differential operators are defined by: f (e ɛ1α q)g(e ɛ2α q) = + Dɛ n 1,ɛ 2[log q] (f (q), g(q))αn n! n=0 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

41 Calculation of F k Therefore we get: F k = 2n Z ln 2 (P, b) Db,b k (1) 1 [log q] (F n, F n (2) ). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

42 Calculation of F k Therefore we get: F k = 2n Z ln 2 (P, b) Db,b k (1) 1 [log q] (F n, F n (2) ). On the other hand we can rewrite operator H in terms of F NSR generators: H = (b + b 1 ) r : f r f r : f r G r, r Z 1/2 r Z 1/2 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

43 Calculation of F k Therefore we get: F k = 2n Z ln 2 (P, b) Db,b k (1) 1 [log q] (F n, F n (2) ). On the other hand we can rewrite operator H in terms of F NSR generators: H = (b + b 1 ) r : f r f r : f r G r, We calculate r Z 1/2 r Z 1/2 H W F NSR = q 1/4 f 1/2 W F NSR, Similarly we calculate H 2 W F NSR, H 3 W F NSR, H 4 W F NSR and get: F 0 = F NS, F2 = q 1/2 F NS, F4 = q 1/2 (2q d dq F NS q 1/2 F NS ) (b+b 1 ) 2 q 1/2 F NS, where F NS denotes F c NS( NS q). M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

44 Calculation of F k Therefore we get: F k = 2n Z ln 2 (P, b) Db,b k (1) 1 [log q] (F n, F n (2) ). On the other hand we can rewrite operator H in terms of F NSR generators: H = (b + b 1 ) r : f r f r : f r G r, We calculate r Z 1/2 r Z 1/2 H W F NSR = q 1/4 f 1/2 W F NSR, Similarly we calculate H 2 W F NSR, H 3 W F NSR, H 4 W F NSR and get: F 0 = F NS, F2 = q 1/2 F NS, F4 = q 1/2 (2q d dq F NS q 1/2 F NS ) (b+b 1 ) 2 q 1/2 F NS, where F NS denotes F c NS( NS q). Therefore we get an equation on F k : F 4 + 2q d dq F 2 (1 + (b + b 1 ) 2 ) F 2 + q F 0 = 0. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

45 Conformal blocks relations Introduce operator. D III b = D 4 b,b 1 [log q] +2q d dq D2 b,b 1 [log q] (1+(b+b 1 ) 2 )D 2 b,b 1 [log q] +qd0 b,b 1 [log q] We proved that: 2n Z l 2 n (P, b) D III b ( ) F c (1)( (1) n β (1) q), F c (2)( (2) n β (2) q) = 0 M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

46 Conformal blocks relations Introduce operator. D III b = D 4 b,b 1 [log q] +2q d dq D2 b,b 1 [log q] (1+(b+b 1 ) 2 )D 2 b,b 1 [log q] +qd0 b,b 1 [log q] We proved that: 2n Z l 2 n (P, b) D III b ( ) F c (1)( (1) n β (1) q), F c (2)( (2) n β (2) q) = 0 We want to specialize central charges to 1. We set b = i. Other parameters are specified by: q = 4t; P = 2iσ, P (η) = iσ, η = 1, 2, (1) n = (σ + n) 2, (2) n = (σ n) 2. After this specialization we have D III b 2D III. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

47 Conformal blocks relations Introduce operator. D III b = D 4 b,b 1 [log q] +2q d dq D2 b,b 1 [log q] (1+(b+b 1 ) 2 )D 2 b,b 1 [log q] +qd0 b,b 1 [log q] We proved that: 2n Z l 2 n (P, b) D III b ( ) F c (1)( (1) n β (1) q), F c (2)( (2) n β (2) q) = 0 We want to specialize central charges to 1. We set b = i. Other parameters are specified by: q = 4t; P = 2iσ, P (η) = iσ, η = 1, 2, (1) n = (σ + n) 2, (2) n = (σ n) 2. After this specialization we have Db III 2D III. We proved: ( ) l n (2iσ, i) 2 D III F((σ + n) 2 t), F((σ n) 2 t) = 0, 2n Z M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

48 Structure constants Proposition l n (P, b) = ( 1)n 2 2n2 (β (1) ) (1) n /2 (β (2) ) (2) n /2 seven (2P, 2n)s even (2P + b + b 1, 2n), where s even (x, n) = i,j 0, i+j<2n i+j 0 mod 2 (x + ib + jb 1 ), M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

49 Structure constants Proposition l n (P, b) = ( 1)n 2 2n2 (β (1) ) (1) n /2 (β (2) ) (2) n /2 seven (2P, 2n)s even (2P + b + b 1, 2n), where s even (x, n) = i,j 0, i+j<2n i+j 0 mod 2 (x + ib + jb 1 ), This proposition was conjectured in [BBFLT]. Proved by another method L. Hadasz, Z. Jasklski. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

50 Structure constants Proposition l n (P, b) = ( 1)n 2 2n2 (β (1) ) (1) n /2 (β (2) ) (2) n /2 seven (2P, 2n)s even (2P + b + b 1, 2n), where s even (x, n) = i,j 0, i+j<2n i+j 0 mod 2 (x + ib + jb 1 ), This proposition was conjectured in [BBFLT]. Proved by another method L. Hadasz, Z. Jasklski. The calculation l n (P, b) is the last step inthe proof of AGT for NSR algebra. (mostly based on the proof for Virasoro: [AFLT], and previous works on NSR V. Belavin, B. Feigin; G. Bonelli, K. Maruyoshi, A. Tanzini; [BBFLT].) M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

51 Structure constants Proposition l n (P, b) = ( 1)n 2 2n2 (β (1) ) (1) n /2 (β (2) ) (2) n /2 seven (2P, 2n)s even (2P + b + b 1, 2n), where s even (x, n) = i,j 0, i+j<2n i+j 0 mod 2 (x + ib + jb 1 ), This proposition was conjectured in [BBFLT]. Proved by another method L. Hadasz, Z. Jasklski. The calculation l n (P, b) is the last step inthe proof of AGT for NSR algebra. (mostly based on the proof for Virasoro: [AFLT], and previous works on NSR V. Belavin, B. Feigin; G. Bonelli, K. Maruyoshi, A. Tanzini; [BBFLT].) Using the recurrence equation on G(q) one can prove for 2n Z: C(σ + n)c(σ n) C(σ) 2 = 2 n 1 k=1 1 (k 2 4σ 2 ) 2(2 n k) (4σ 2 ) 2 n = 4 NS ( 1) 2n l n (2iσ, i) 2, M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

52 Concluding remarks M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

53 Concluding remarks Results Proof of Gamayun, Iorgov, Lisovyy conjecture. Byproduct: finalizing the proof of AGT conjecture of NSR algebra. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18

54 Concluding remarks Results Proof of Gamayun, Iorgov, Lisovyy conjecture. Byproduct: finalizing the proof of AGT conjecture of NSR algebra. Questions Possible generalizations: many points, higher genus, higher ranks. Geometrical meaning Deformation of Painlevé equations. Litvinov, Lukyanov, Nekrasov, Zamolodchikov connection: c = and Painlevé. M. Bershtein, A. Shchechkin Bilinear equations on Painlevé τ-functions from CFT 19 June / 18