Tau functions, Fredholm determinants and combinatorics Oleg Lisovyy - PowerPoint PPT Presentation

Tau functions, Fredholm determinants and combinatorics Oleg Lisovyy Institut Denis-Poisson, Université de Tours, France Montréal, 27/07/2018 joint work with M. Cafasso & P. Gavrylenko 1712.08546 [math-ph]

Motivation: Painlevé equations Painlevé III (massive scaling limit) [Wu,McCoy,Tracy,Barouch,’76] 2D Ising correlation Painlevé VI (lattice) [Jimbo,Miwa,’81] Painlevé V (sine kernel) impenetrable Bose gas [Jimbo,Miwa,Mori,Sato,’80] Painlevé II–VI (Airy kernel, etc) random matrix theory [Tracy,Widom,’92; ...]

Painlevé equations describe simplest cases of monodromy preserving deformations of linear ODEs with rational coefficients. E.g. Painlevé VI corresponds to rank 2 Fuchsian system with 4 regular singularities at 0 , t , 1 , ∞ : A ( z ) = A 0 A t A 1 ∂ z Φ = Φ A ( z ) , z + z − t + z − 1 Isomonodromy equations are dA 0 = [ A 0 , A t ] dA 1 = [ A 1 , A t ] , t − 1 , A ∞ = const dt t dt For A 0 , t , 1 and A ∞ := − A 0 − A t − A 1 traceless 2 × 2 matrices, with eigenvalues ± θ 0 , t , 1 , ∞ , these equations are equivalent to Painlevé VI.

Painlevé VI : t ζ ′ − ζ ζ ′ + θ 2   2 θ 2 0 + θ 2 t + θ 2 1 − θ 2 0 ∞ � t ( t − 1 ) ζ ′′ � 2 t ζ ′ − ζ ( t − 1 ) ζ ′ − ζ 2 θ 2 = − 2 det   t ζ ′ + θ 2 ( t − 1 ) ζ ′ − ζ 0 + θ 2 t + θ 2 1 − θ 2 2 θ 2 ∞ 1 ◮ ζ ( t ) = ( t − 1 ) Tr A 0 A t + t Tr A 1 A t = t ( t − 1 ) d dt ln τ ◮ τ ( t ) is the Painlevé VI tau function

Geometric confluence diagram [Chekhov, Mazzocco, Rubtsov, ’15]: V deg V VI III III 3 III III 2 3 u ′′ + u ′ t = sin u III III 1 II FN q ′′ = 6 q 2 + t I IV II JM

Painlevé project : ◮ develop a general approach that would allow to derive systematically (asymptotic) series for PI-PV functions ◮ explicit expressions for coefficients of the series + connection formulas (in terms of monodromy of the associated linear problem)

Painlevé project : ◮ develop a general approach that would allow to derive systematically (asymptotic) series for PI-PV functions ◮ explicit expressions for coefficients of the series + connection formulas (in terms of monodromy of the associated linear problem) All classical “linear” special functions admit explicit representations. The Painlevé transcendents do not. A. Fokas, A. Its, A. Kapaev, V. Novokshenov, Painlevé transcendents. The Riemann-Hilbert approach , (2006)

General�solutions�of Painlevé�equations Fredholm�determinants Series�representations ◮ block integrable kernels ◮ summation over partitions/Young diagrams ◮ Widom’s constants

General solution of PVI [Gamayun, Iorgov, OL, ’12]: PVI tau function is a Fourier transform of c = 1 Virasoro conformal block: θ 1 θ t � � e in η B ( � e in η τ ( t ) = θ, σ + n , t ) = ( t ) σ + n θ 0 θ ∞ n ∈ Z n ∈ Z θ, σ, t ) = t σ 2 − θ 2 0 − θ 2 t � ∞ ◮ B ( � k = 0 B k ( � θ, σ ) t k ◮ B k determined by commutation relations of Vir ◮ AGT correspondence [Alday, Gaiotto, Tachikawa, ’09]: sum over pairs B ( t ) = Z inst ( t ) = [Nekrasov, ’04] of Young diagrams Series representation for PVI tau function (proof in [Gavrylenko, OL, ’16]) � e in η � θ, σ + n ) t ( σ + n ) 2 + | λ | + | µ | B λ,µ ( � τ ( t ) = n ∈ Z λ,µ ∈ Y

V deg III 3 III 3 V III III 2 VI III 1 III II FN IV I II JM ◮ PVI, PV, PIII 1 , 2 , 3 surfaces may be cut into solvable pieces Gauss Whittaker Bessel ◮ More surprisingly, Fourier transform also appears in “irregular type” expansions for PI–PV at t = ∞ .

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 ◮ τ [ J ] appears in the large size asymptotics of Toeplitz determinants with symbol J and is called Widom’s constant in this context

If the direct RHP is solvable, then τ [ J ] may also be written as � � 0 a + − τ [ J ] = det H ( 1 + K ) , K = , a − + 0 where a ±∓ = Ψ ± Π ± Ψ − 1 ± − Π ± : H ∓ → H ± are integral operators � 1 � z , z ′ � � z ′ � dz ′ , ( a ±∓ f ) ( z ) = a ±∓ f 2 π i C with block integrable kernels = ± 1 − Ψ ± ( z ) Ψ ± ( z ′ ) − 1 � z , z ′ � a ±∓ . z − z ′ In applications to Painlevé: ◮ Ψ ± (direct factorization) are given and define the jump J = Ψ − − 1 Ψ + ◮ Ψ ± are expressed via classical special functions (Gauss, Kummer & Bessel for PVI, PV, PIII’s) ◮ dual factorization ( ¯ Ψ ± in J = ¯ Ψ + ¯ Ψ − 1 − ) is the problem to be solved

Differentiation formula Theorem : Let ( z , t ) �→ J ( z , t ) be a smooth family of GL ( N , C ) -loops which depend on an extra parameter t and admit direct & dual factorization. Then � 1 � � �� J − 1 ∂ t J ∂ z ¯ Ψ − ¯ Ψ − 1 − + Ψ − 1 ∂ t ln τ [ J ] = Tr + ∂ z Ψ + dz . 2 π i C ◮ proof in [Widom, ’74]; rediscovered by [Its, Jin, Korepin, ’06] ◮ related results in the study of dependence of isomonodromic tau functions on monodromy [Bertola, ’09] Corollary : in isomonodromic RHPs, Widom’s constant τ [ J ] ≃ Jimbo-Miwa-Ueno tau function

Dual RHP 1 for ˜ Ψ 1 t 0 � G ( ν ) ( z ) , z ∈ D ν , ˜ Ψ ( z ) = ∈ R ≥ 0 ∪ ¯ D 0 ∪ ¯ D t ∪ ¯ D 1 ∪ ¯ Φ ( z ) , z / D ∞ .

Dual RHP 1 for ˜ Ψ 1 t 0 C out C in � ( − z ) − S ˜ Ψ ( z ) , z ∈ A , ˆ Ψ ( z ) = ˜ ∈ ¯ Ψ ( z ) , z / A .

Dual RHP 2 for ˆ Ψ 1 t 0 C out C in

Dual RHP 2 for ˆ Ψ C 1 t 0 C out C in � Ψ + ( z ) − 1 ˆ Ψ ( z ) , outside C , ¯ Ψ ( z ) = Ψ − ( z ) − 1 ˆ Ψ ( z ) , inside C .

Dual RHP 3 for ¯ Ψ C � Ψ + ( z ) − 1 ˆ Ψ ( z ) , outside C , ¯ Ψ ( z ) = Ψ − ( z ) − 1 ˆ Ψ ( z ) , inside C . ◮ contour C (single circle !), smooth jump J : C → GL ( N , C ) given by − 1 J ( z ) = Ψ − ( z ) − 1 Ψ + ( z ) = ¯ Ψ + ( z ) ¯ Ψ − ( z ) ◮ we are in the previously described setup!

Widom’s differentiation formula implies that − Tr A + 0 A + ∂ t ln τ [ J ] = Tr A 0 A t + Tr A t A 1 t , t t − 1 t � �� � ∂ t ln τ JMU ( t ) so that in turn 2 Tr ( S 2 − Θ 2 0 − Θ 2 t ) τ [ J ] . 1 τ JMU ( t ) = t ◮ Recall that � � 0 a + − τ [ J ] = det ( 1 + K ) , K = , a − + 0 = ± 1 − Ψ ± ( z ) Ψ ± ( z ′ ) − 1 � z , z ′ � a ±∓ . z − z ′ ◮ τ JMU ( t ) for 4-point system written via auxiliary 3-point solutions ◮ hypergeometric representations for N = 2 = ⇒ PVI tau function !

Schematically, � t 1 � = τ JMU 0 8 � 1 �   1 a + − t � � � 1 � 0 8 det τ JMU τ JMU  t  � � 0 8 0 8 a − + 1 0 8

Similarly, for a linear system with 2 irregular singularities � � τ JMU = 0 8  � �  1 a + − � � � � 0 8   τ JMU τ JMU det  � �  0 8 0 8   a − + 1 0 8

Series representations Given K ∈ C X × X , we can expand Fredholm determinant � � K mm + 1 � � � K mm K mn � � det ( 1 + K ) = det K Y = 1 + � + . . . � � 2 ! K nm K nn � Y ∈ 2 X m ∈ X m , n ∈ X � � 0 a + − ◮ our case: K = a − + 0 ◮ in the Fourier basis, � � z , z ′ � ∓ q z − 1 2 ± p z ′− 1 a ± p 2 ± q , a ±∓ = p , q ∈ Z ′ + with a ± p ∓ q ∈ Mat N × N ( C ) . � 0 � a p ◮ multi-indices m , n , . . . of principal minors det K Y = det h a h 0 p incorporate color indices α = 1 , . . . N and (half-)integer Fourier indices

N = 1 case : { p { h { a 9 0 2 p 5 2 + - 3 a - + { 0 2 h 11 2 9 5 3 11 2 2 2 2 ◮ combinatorial expansion � ( − 1 ) | p | det a p h det a h det ( 1 + K ) = p , ( p , h ) with balance condition | p | = | h |

NW E N = Q = ... 9 7 5 3 1 1 3 5 7 9 ... 2 - - - - - 2 2 2 2 2 2 2 2 2 ◮ A Maya diagram is a map m : Z ′ → {− 1 , 1 } subject to the condition m ( p ) = ± 1 for all but finitely many p ∈ Z ′ ± (positions of particles and holes) ◮ Maya diagram = charged partition/Young diagram ◮ charge ( m ) = ♯ (particles) − ♯ (holes) ◮ for N = 1 principal minors are labeled by partitions