Monodromy dependence of Painlev tau functions Oleg Lisovyy - PowerPoint PPT Presentation

Monodromy dependence of Painlevé tau functions Oleg Lisovyy Institut Denis-Poisson, Université de Tours, France CIRM, 08/04/2019 collaborations with A. Its, A. Prokhorov, M. Cafasso, P. Gavrylenko

Example 1: Sine kernel Introduce sin x − y � � � 2 τ ( t ) = det 1 − K , K ( x , y ) = π ( x − y ) . � ( 0 , t ) ◮ τ ( t ) is a Painlevé V tau function: ζ ( t ) = t d dt ln τ ( t ) satisfies t ζ ′′ � 2 + t ζ ′ − ζ t ζ ′ − ζ + 4 ζ ′ 2 � � � � � = 0 . ( ζ -PV) ◮ Asymptotics: t 4 τ ( t → 0 ) = 1 − t t 6 � � 2 π + 576 π 2 + O , � 1 t − 4 �� 4 e − t 2 τ ( t → ∞ ) = τ sine · t − 1 � 1 + 2 t 2 + O 32

Conjecture [Dyson, ’76]: √ 12 e 3 ζ ′ ( − 1 ) = � 1 � 3 7 � � τ sine = 2 2 G G . 2 2 ◮ Barnes G -function is essentially defined by the recurrence relation G ( z + 1 ) = Γ ( z ) G ( z ) ; it has integral and product representations, etc ◮ proved in [Ehrhardt, ’04; Krasovsky, ’04; Deift, Its, Krasovsky, Zhou, ’06]

Example 2: 2D Ising model ◮ Nearest-neighbor interaction � H [ σ ] = − J σ i , j ( σ i + 1 , j + σ i , j + 1 ) , i , j σ 3 , 2 of spin variables σ i , j = ± 1. ◮ Spin-spin correlation function: σ 0 , 0 [ σ ] σ 0 , 0 σ r x , r y e − β H [ σ ] � � � σ 0 , 0 σ r x , r y = � [ σ ] e − β H [ σ ] ◮ Phase transition at s ≡ sinh 2 β J = 1 � σ � ◮ Spontaneous magnetization [Yang, ’52]: 1 �� 1 − s − 4 � 1 8 , s > 1 , � σ � = 0 , s < 1 , ◮ Correlation length Λ ∼ 2 − 1 2 | s − 1 | − 1 T c T

Scaling limit of the 2-point functions is described by R � T → T c , Λ → ∞ , r 2 x + r 2 y → ∞ , Λ → t , R = → Λ − 1 3 4 2 8 τ ± ( t ) , � � σ 0 , 0 σ r x , r y T ≷ T c , Scaled correlations can be written in terms of Fredholm determinants & related to Painlevé functions [McCoy, Tracy, ’73; Wu, McCoy, Tracy, Barouch, ’76] (PV, PIII( D 6 ), PIII( D 8 )). In particular, both ζ ± = t d dt ln τ ± ( t ) satisfy t ζ ′′ � 2 = 4 ζ − t ζ ′ � 2 + 4 ζ ′ � 2 � ζ ′ � 2 ζ − t ζ ′ � � � � � + ( ζ -PV) ◮ Long distances (form factor expansions): e − t √ τ + ( t → ∞ ) ∼ , τ − ( t → ∞ ) ∼ 1 . 2 π t ◮ Short distances (conformal limit): τ ± ( t → 0 ) ∼ τ Ising · ( 2 t ) − 1 4 .

Theorem [Tracy, ’91]: � 1 � 3 12 e 3 ζ ′ ( − 1 ) = G 1 � � τ Ising = 2 G . 2 2 ◮ alternative proof in [Bothner, ’17] ◮ constant factors in the asymptotics of tau functions (connection constants) were computed for many other (special!) Fredholm determinant solutions of Painlevé equations: - correlator of twist fields in sine-Gordon field theory at the free-fermion point [Basor, Tracy, ’91] - Airy kernel [Tracy, Widom, ’92; Deift, Its, Krasovsky, ’06] - Bessel kernel [Tracy, Widom, ’93] - confluent hypergeometric kernel [Deift, Krasovsky, Vasilevska, ’10] - hypergeometric kernel [Lisovyy, ’09] - . . .

Summary : ◮ Ising scaled correlator = specific PV tau function ◮ it has Fredholm determinant representation ◮ its asymptotics at one singular point ( t → ∞ ) is “easy” ◮ the asymptotics at the other singular point ( t → 0) is difficult (connection constant) Questions : ◮ Can the general solutions of Painlevé equations be written as Fredholm determinants? ◮ How to compute the relevant connection constants? In this talk, I will mainly focus on the Painlevé VI case.

Isomonodromic tau function [Jimbo, Miwa, Ueno, ’79] Consider a system of linear ODEs with rational coefficients d Φ dz = A ( z ) Φ , A , Φ ∈ Mat N × N ◮ Laurent expansions of A ( z ) at singularities  A ν ( z − a ν ) − r ν � � ( z − a ν ) r ν + 1 + O as z → a ν ,   A ( z ) = − z r ∞ − 1 A ∞ + O � z r ∞ − 2 � as z → ∞ ,   where r 1 , . . . , r n , r ∞ ∈ Z ≥ 0 . ◮ assume A ν are diagonalizable, A ν = G ν Θ ν, − r ν G − 1 ν , Θ ν, − r ν = diag { θ ν, 1 , . . . , θ ν, N } . and non-resonant ( θ ν, k are distinct whenever r ν = 0).

At each singular point, there is a unique formal solution ∞ Φ ( ν ) ( z ) e Θ ν ( z ) , Φ ( ν ) ( z ) = 1 + g ν, k ( z − a ν ) k , Φ ( ν ) form ( z ) = G ν ˆ ˆ � k = 1 where Θ ν ( z ) are diagonal and have the form − 1 Θ ν, k ( z − a ν ) k + Θ ν, 0 ln ( z − a ν ) . � Θ ν ( z ) = k k = − r ν Isomonodromic times: ◮ positions of singularities a ν ◮ diagonal elements Θ ν, k � = 0 Monodromy data: ◮ Stokes matrices relating canonical solutions in different sectors at a ν ◮ formal monodromy exponents Θ ν, 0 ◮ connection matrices relating canonical solutions at different singularities

Theorem [Jimbo, Miwa, Ueno, ’79]: Let us collectively denote the times by T . The 1-form Φ ( ν ) ( z ) − 1 ∂ z ˆ Φ ( ν ) ( z ) d T Θ ν ( z ) � � � ˆ ω JMU = − res z = a ν Tr ν = 1 ,..., n , ∞ is closed when restricted to isomonodromic family of A ( z ) . It thus defines the isomonodromic tau function by d T ln τ JMU = ω JMU Example . For A ( z ) = A 0 A t A 1 z + z − t + z − 1 (4 simple poles 0 , t , 1 , ∞ ) ∂ t ln τ JMU = Tr A 0 A t + Tr A t A 1 t − 1 . t For 2 × 2 matrices, this is Painlevé VI tau function. Aim : Extend Jimbo-Miwa-Ueno differential to the space of monodromy data (the space of parameters and initial conditions for Painlevé).

Isomonodromic deformations tau functions [Gamayun, Iorgov, OL, ’12-13] [Iorgov, OL, Teschner, ’14] [Bershtein, Shchechkin, ’14] [Alday, Gaiotto, Tachikawa, ’09] 2D CFT 4D SUSY YM conformal blocks partition functions θ 1 θ t σ + n � e in η τ VI ( t ) = N 0 n ∈ Z θ ∞ θ 0 θ ) t ( σ + n ) 2 − θ 2 0 − θ 2 = N 0 ( 1 − t ) 2 θ t θ 1 � e in η � B λ,µ ( σ + n , � t + | λ | + | µ | n ∈ Z λ,µ ∈ Y = N 0 t σ 2 − θ 2 0 − θ 2 t det ( 1 + K ) ← Task 1 ◮ explicit integrable (2 × 2 or 4 × 4) matrix kernel K involving 2 F 1 functions; acts on vector-valued functions on a circle (and not on an interval!)

Asymptotic behaviors of τ VI : � ˜ N 0 t σ 2 − θ 2 0 − θ 2 as t → 0 , t τ VI ( t ) ≃ N 1 ( 1 − t ) ρ 2 − θ 2 1 − θ 2 ˜ as t → 1 . t ◮ σ , ρ are 2 Painlevé VI integration constants, related to monodromy of the associated 4-point Fuchsian system Task 2 → compute the connection coefficient ˜ N 1 / ˜ N 0 Remark . Tau function can be expanded in different channels (there are different Fredholm determinant representations, adapted for asymptotic analysis near different critical points): θ 1 θ t θ 1 θ t σ + n � e in η � e in µ τ VI ( t ) = N 0 = N 1 ρ + n n ∈ Z n ∈ Z θ ∞ θ 0 θ ∞ θ 0 This allows to relate the connection coefficient to the c = 1 fusion matrix, � θ 1 θ t � θ 1 θ 1 θ t � θ 0 ; ρ θ t σ = d ρ F d ρ ρ θ ∞ σ Γ θ ∞ θ 0 θ ∞ θ 0

It turns out ln N 1 N 0 coincides (up to an elementary correction) with the generating function of the canonical transformation between two pairs of Darboux coordinates on Hom ( π 1 ( C 0 , 4 ) , SL ( 2 , C )) / SL ( 2 , C ) : σ, η and ρ, µ . This in its turn coincides (again up to an elementary correction) with the complexified volume of the hyperbolic tetrahedron with dihedral angles σ , ρ , θ 0 , t , 1 , ∞ .   θ 1   ρ 8   ln N 1 G ( 1 + z k )   � N 0 ∼ Vol θ ∞ ∼ ln   G ( 1 − z k )  σ   θ t k = 1    θ 0 ◮ z k ’s are explicit elementary (though complicated) functions of monodromy parameters ◮ conjecture in [Iorgov, OL, Tykhyy, ’13] ◮ proved in [Its, OL, Prokhorov, ’16]

Riemann-Hilbert setup C − ◮ let C ⊂ C be a circle centered at the origin + ◮ pick a loop J ( z ) ∈ Hom ( C , GL N ( C )) ◮ J ( z ) continues into an annulus A ⊃ C � J k z k , J ( z ) = k ∈ Z Two Riemann-Hilbert problems: J ( z ) = Ψ − ( z ) − 1 Ψ + ( z ) direct : − 1 J ( z ) = ¯ Ψ + ( z ) ¯ dual : Ψ − ( z )

Main definition : The tau function of RHPs defined by ( C , J ) is defined as Fredholm determinant Π + J − 1 Π + J Π + � � τ [ J ] = det H + , where H = L 2 � C , C N � and Π + is the orthogonal projection on H + along H − . Properties : ◮ dual RHP is solvable iff the operator P := Π + J − 1 Π + is invertible on H + , in which case P − 1 = ¯ Ψ + Π + ¯ Ψ − 1 − Π + ◮ likewise, for direct RHP and Q := Π + J Π + , with Q − 1 = Ψ − 1 + Π + Ψ − Π + ◮ if either direct or dual RHP is not solvable, then τ [ J ] = 0 Example : scalar case ( N = 1) ◮ direct and dual factorization coincide ◮ J ( z ) = ( 1 − t 1 z ) ν 1 ( 1 − t 2 / z ) ν 2 with | z | = 1 and | t 1 | , | t 2 | < 1, then τ [ J ] = ( 1 − t 1 t 2 ) ν 1 ν 2

Remark . τ [ J ] appears [Widom, ’76] in the asymptotics of determinant of block Toeplitz matrix with symbol J , . . .  J 0 J − 1 J − K + 1  . . . J 1 J 0 J − K + 2   T K [ J ] =  . . . .  ...  . . .   . . .  . . . J K − 1 J K − 2 J 0 In this context, τ [ J ] is called Widom’s constant. ◮ strong Szegő for N = 1: τ [ J ] = exp � ∞ k = 1 k ( ln J ) k ( ln J ) − k ◮ no nice general formula for N ≥ 2