The $τ$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant
Harini Desiraju








 


The τ-function of the Ablowitz-Segur family of solutions to Painlevé  II as a Widom constant
     H. Desiraju[harini.desiraju@sissa.it]
  

  Scuola Internazionale Superiore di Studi Avanzati,   

  Via Bonomea, 265, 34136 Trieste, Italy.
    

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 τ-functions of certain Painlevé equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. 
 
 This paper is a step towards understanding whether the τ-function of Painlevé II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlevé II and the corresponding τ-function is known to be the Fredholm determinant of the Airy Kernel. We develop a formalism for open contour in parallel to the one formulated in <cit.> in terms of the Widom constant and verify that the Widom constant for Ablowitz-Segur family of solutions is indeed the determinant of the Airy Kernel. Finally, we construct a suitable basis and obtain the minor expansion of the Ablowitz-Segur τ-function.
 

 



§ INTRODUCTION

Painlevé equations are nonlinear second order ODEs  whose solutions are widely recognized  as important special functions with a broad range of applications. The integrability property of these equations was obtained by representing them as an isomonodromic system  of ordinary differential equations. The   Riemann-Hilbert  (RH)  method then proved to be a powerful technique to study  solutions and their properties. An important object related to the solutions is the so called τ-function.
 
In the theory of isomonodromic deformations, the τ-function (τ_JMU) was introduced by the Kyoto school  and it is  constructed starting from a certain 1-form ω_JMU on the space of the deformation parameters <cit.>. If  the parameters are of isomonodromic type, then the form ω_JMU is closed  with respect to differentiation with respect to the parameters. The corresponding τ_JMU function is defined  locally as 
 
    dlogτ_JMU=ω_JMU

 where d denotes total differentiation with respect to the parameters.
 A notable example  of  τ-function of the Painlevé II equation is the Tracy-Widom distribution  <cit.>. 
  Such τ- function has the property of being expressed as a Fredholm determinant of the Airy kernel. Many relevant solutions of the Painlevé equations that appear in various branches of mathematics  turn out to be expressed as a Fredholm determinant of some integrable operator, as defined by Its, Izergin, Korepin, and Slavnov <cit.>.
  For example the gap probability distribution in random matrices is  Fredholm determinant with the   sine kernel (Painlevé V)  <cit.>, the  correlation function   of  stochastic point processes on a one-dimensional lattice originated from  representations of the infinite symmetric group is a Fredholm determinant with hypergeometric kernel (Painlevé VI) <cit.>,<cit.>. 
  
  It is natural to inquire whether  all solutions of Painlevé equations can be expressed as a Fredholm determinant of some integrable operator.
In a series of recent papers Cafasso, Gavrylenko, Lisovyy <cit.>, <cit.>  showed  that the  generic  τ-function  of the  PVI, PV, PIII equation can be expressed as a Fredholm determinant.
The key feature of this construction is to reduce the Riemann-Hilbert problem  (RHP) associated   to the isomonodromic system to a RHP on the circle  for a jump  matrix G.


Then one can define a  Toeplitz operator T_G=Π_+G where G is the jump (called symbol in the literature of Toeplitz determinants) of the RHP and Π_+ the  projection operator to analytic functions in the interior of the circle.
It has been shown in <cit.>,<cit.> that the  Fredholm determinant 

    τ[G]= ( T_G^-1∘ T_G ),
   coincides  up to a factor  with the isomonodromic τ function (<ref>).
The above τ function is also called Widom constant  since such quantity was obtained by Widom in the description of the asymptotic behaviour of Toeplitz determinants <cit.>, <cit.>, when the size of the matrix tends to infinity, as a refinement of the strong Szegö theorem.




This approach  cannot be directly implemented to  the cases where the RHP is formulated on a contour that is not a circle as is the case for the  Painlevé equations PI, PII and  PIV. 
There are several examples of  τ-functions expressed   as a  Fredholm determinant  like the  Tracy-Widom distribution related to  Painlevé II <cit.>,  or like the example obtained in <cit.> related to 
Painlevé IV. 
However the generic τ-function  of the Painlevé I, II and IV equations does not seem to have a Fredholm determinant representation.
 The main obstacle to develop the procedure implemented in <cit.> and <cit.> is   the impossibility to reduce the RHP problem of the Painlevé I, II and IV equations to a RHP on the circle.
However it is expected that the generic RHP for these equations could be reduced to a RHP on the line for a jump matrix G.


Then one considers the  projection operator Π_+  to holomorphic functions on the semi-plane  and define the operator  T_G=Π_+G.
For the case considered  in this manuscript,  this operator    is trace class and  therefore the  Fredholm determinant (<ref>)
is well defined.
In this manuscript  we develop this machinery for the Painlevé II equation by considering  the Ablowitz-Segur family of solutions for PII as a toy model.
This example  serves as a starting point to obtain the generic Painlevé II τ-function.
 


 This paper is structured as follows. We will first setup the machinery to extend the formalism in <cit.>,<cit.>  to a line contour and show in <ref> that the τ-function can be written as a Widom constant.  Next we show in <ref> that the Widom constantcoincides with the isomonodromic τ-function (<ref>).
 Finally in <ref> we construct an appropriate basis and study the minor expansion of the Widom constant.





§ SETUP

Let J(z,t): iℝ→ SL(2,ℂ) be a smooth matrix function of z depending analytically on the complex parameter t in some domain. We assume that ||J(z)- 1|| = 𝒪(|z|^-1) as z →±  i∞. We shall refer to J as the jump matrix. In association with the data of the contour (iℝ) and jump matrix J one can introduce two Riemann Hilbert problems, also known as factorization problems. They consist of two 2× 2 matrices Θ(z,t) and Ψ(z,t) such that:


  
  * Θ(z,t), Ψ(z,t) are analytic in z ∈ℂ/iℝ and admit continuous boundary values from the left (+) and right (-) sides of iℝ.
  
  * The boundary values Θ_±, Ψ_± satisfy the jump conditions
  
    J(z,t)= Θ_-^-1(z,t)Θ_+(z,t) = Ψ_+^-1(z,t)Ψ_-(z,t)     ;    z∈ iℝ

  
  * The functions Θ(z,t) and Ψ(z,t) are normalized at infinity.
  
    lim_z →∞Θ(z,t) = I  ; lim_z →∞Ψ(z,t) = I

  where the limit is intended as limit in any proper subsector of the left/right half-planes.
  

The two solutions (if they exist) are the two Birkhoff factorizations; we stipulate to call Θ(z,t) the direct Riemann-Hilbert problem and Ψ(z,t) the dual. The contour iℝ divides the complex plane  into the right half (negative side) and the left half (positive side). The space L^2( iℝ,|dz| )⊗ℂ^2 can be split as the direct sum of two closed subspaces (Hardy spaces):
 
    ℋ = L^2(iℝ, ℂ^2) = ℋ_+⊕ℋ_-
 the functions on ℋ are all column vectors. The two subspaces consist of (vector valued) functions in L^2(iℝ) that are boundary values from the left(+)/right(-) of analytic functions that tend to 0 at infinity. Notice that this splitting is orthogonal. On these spaces, one can define projection operators Π_± such that 


    Π_+ : ℋ→ℋ_+   ;   Π_-: ℋ→ℋ_-

explicitly, Π_± are just the Cauchy transforms

    Π_+ f(z) = ∫_iℝdw/2π if(w)/w-z     z<0 
    Π_- f(z) = -∫_iℝdw/2π if(w)/w-z     z>0

 with the equality Π_+ + Π_-≡𝕀. To define the τ-function, we  first  define the operator T_J^-1:ℋ→ℋ_+ for the symbol J^-1 by multiplication followed by projection

    T_J^-1 (f) = Π_+( J^-1 f)

T_J is similarly defined 

    T_J (f) = Π_+( J f ).



With the operators T_J and T_J^-1, we define the τ-function along the same lines as the Widom constant.  

We define the Widom constant with respect to the operators T_J, T_J^-1

    τ [J] = ( T_J^-1∘ T_J)



It is a Fredholm determinant for the case of Ablowitz-Segur solutions as will be shown later. 


 τ[J] as defined in (<ref>) admits an equivalent representation as the determinant 

    τ[J] = ℋ[1 + U]

where 1 denotes the identity operator on ℋ, U: ℋ→ℋ is an operator represented in the splitting ℋ_± as
U = (  [ 0 a; b 0 ]) and a:ℋ_-→ℋ_+; b: ℋ_+→ℋ_- are given by,

    a = Θ_+Π_+Θ_+^-1 -Π_+  ;    b = Π_-- Θ_-Π_-Θ_-^-1.
 





Substituting (<ref>) in (<ref>) and manipulating the terms gives the familiar form of the determinant representation of τ-function in <cit.>. 

    τ [J] = ℋ_+[ T_J^-1∘ T_J ] = ℋ_+ [Π_+ J^-1Π_+ J] 
    
    =   ℋ_+[Π_+Θ_+^-1Θ_-Π_+Θ_-^-1Θ_+ ] 
    
    = ℋ_+[Θ_+Π_+Θ_+^-1Θ_-Π_+Θ_-^-1  ] 
    
     =ℋ_+[Θ_+Π_+Θ_+^-1Θ_- (1- Π_-) Θ_-^-1  ] 
    
     = ℋ_+[ 1- (Θ_+Π_+Θ_+^-1) ( Θ_-Π_-Θ_-^-1)  ] 
    
     = ℋ_+[ 1- (Θ_+Π_+Θ_+^-1 - Π_+) (Π_-- Θ_-Π_-Θ_-^-1)  ] 
    
    = ℋ[1 + U]

where 

    U = (  [ 0 a; b 0 ])

and 

    a = Θ_+Π_+Θ_+^-1 -Π_+  ;    b = Π_--Θ_-Π_-Θ_-^-1.
 
Notice that 
    a: ℋ_-→ℋ_+   ;     b: ℋ_+→ℋ_-.
 

Now, one can repeat the same computation as above in terms of the dual RHP Ψ_± and get the following. 
 
    a = Ψ_+Π_+Ψ_+^-1 -Π_+   b = Π_-- Ψ_-Π_-Ψ_-^-1.












§ TOY MODEL: ABLOWITZ-SEGUR SOLUTION
 

The Ablowitz-Segur family <cit.>
 of solutions  of  the  Painlevé II   equation 
    ∂_s^2 u = su + 2u^3

are specified uniquely by the boundary condition 

    u(s) ≃κ Ai(s);    s→ +∞,   κ∈ℂ .

It is well known that the solution of a Painlevé equation can be obtained by solving a Riemann-Hilbert problem for a matrix valued  function  <cit.> 
Γ(z,s) analytic in the complex z domain ℂ minus some contours.  
For the Ablowitz-Segur family of solutions,  the contours  are shown in figure <ref>,
 and the  Riemann-Hilbert problem satisfied by Γ(z,s) is as follows
 
 
  * Γ(z,s) is  is analytic in ℂ\Σ with  Σ≡γ_1∪γ_2;
 
  * the  boundary values Γ_±(z,s) on the oriented contours γ_1 and γ_2  satisfying the following jump conditions

    Γ_+(z,s) = G_1(z,s)Γ_-(z,s)     z∈γ_1
      Γ_+(z,s) = G_2(z,s)Γ_-(z,s)    z ∈γ_2


  * the asymptotic behaviour at infinity is specified by 

    Γ(z,s) =   1 + Γ^(1)(s)z+O(z^-2),    |z|→∞ .


The solution Γ(z,s) of the  above  Riemann Hilbert problem  (when it exists)   determines the Painlevé transcendent  via the relation

    u(s) = 2 Γ^(1)_12(s).

From the above data, the isomonodromic τ-function is defined by <cit.>

    ∂_slogτ_JMU(s) = - Res_z=∞[ Γ(z,s)' Γ^-1(z,s) (zσ_3) ]

and we have  the relation

    u^2(s)=-d^2ds^2logτ_JMU(s).

Instead of solving the Riemann Hilbert problem for Γ we factorize it into two separate Riemann-Hilbert problems, one for the function Θ_1(z,s)  analytic in  z∈ℂ\γ_1 and 
one for the function Θ_2(z,s)  analytic in  z∈ℂ\γ_2  with boundary values

    Θ_1+ (z,s) = G_1(z,s)Θ_1-(z,s)    z∈γ_1
      Θ_2+ (z,s) = G_2(z,s) Θ_2- (z,s)    z∈γ_2

and 

    Θ_1(z,s) =   1 + 𝒪(z^-1)  ,  Θ_2(z,s) =   1 + 𝒪(z^-1)   .


It is straightforward to solve the Riemann Hilbert problems (<ref>), (<ref>). The solutions are given by the Cauchy transforms of the respective jumps G_1, G_2

    Θ_1 (z, s)= [ [                            1                                                         0; κ∫_γ_1e^-ν(λ,s)/λ - zdλ/2π i                                                         1 ]] 
    Θ_2(z,s) = [  [                           1                             κ∫_γ_2e^ν(λ,s)/λ - zdλ/2π i;                           0                                                       1 ]].

Next we define the matrix valued function Θ(z,s) such that

    Θ(z,s) = Θ_2(z,s)    z<0 
    Θ_1(z,s)    z≥ 0
 
Clearly the matrix function  Θ(z,s) is analytic in ℂ\  iℝ  and the boundary values Θ_± on iℝ satisfy   the  jump condition

    Θ_-(z,s)^-1Θ_+(z,s)= J(z,s)= [[                             1                                 κ∫_γ_2e^ν(λ,s)/λ - zdλ/2π i; -κ∫_γ_1e^-ν(λ,s)/λ - zdλ/2π i                                                           1 ]]   .

and 

    lim_z→∞Θ(z,s) =  1.




 §.§ Computing the τ-function

In this section we want to make sense of the quantity (T_J^-1∘ T_J)   introduced in (<ref>)   when the matrix J is as in  (<ref>).
In Proposition <ref> we show that 

    τ[J] =(T_J^-1∘ T_J)= ℋ[  1 + U ]

where 1 denotes the identity operator on ℋ,  and 
U = (  [ 0 a; b 0 ])  with  a:ℋ_-→ℋ_+; b: ℋ_+→ℋ_-  given by,

    a = Θ_+Π_+Θ_+^-1 -Π_+  ;    b = Π_-- Θ_-Π_-Θ_-^-1

with Θ_± the boundary values of the matrix Θ  defined in (<ref>).

 We want to show that the  quantity  (<ref>)  is a Fredholm determinant and coincides with the τ-function defined in  <cit.>.





We remind the reader of the result in <cit.> where  the τ-function of  the Ablowitz-Segur family of solutions for Painlevé II is given by the following  Fredholm determinant[γ_± are γ_1,2 rotated by π/2 and this is also the source of the factor of i in the exponential in <cit.>.]


    τ(s) = [ Id_L^2(γ_+∪γ_-) - κ[ [ 0 ℱ; 𝒢 0 ]] ] = [ Id_L^2(γ_+) - κ^2 ℱ∘𝒢]

with 

    ℱ : L^2(γ_-) → L^2(γ_+)   𝒢 : L^2(γ_+) → L^2(γ_-)


and 

    (ℱg)(z) = e^-i/2ν(z,s)∫_ℝ-icdw/2π ie^i/2ν(w,s) g(w)/w-z
    
    (𝒢g)(z) = e^i/2ν(z,s)∫_ℝ+icdw/2π ie^-i/2ν(w,s) g(w)/w-z


 
 
 The τ-function  (<ref>) of the  Ablowitz-Segur family of solutions of the Painlevé II equation is the Widom constant defined in  (<ref>).
 





The  Widom constant can be obtained from (<ref>) by computing   the operators a, b explicity. Let  f(z)∈ℋ_-, h(z)∈ℋ_+  be  vector valued
functions

    f ≡( [ f_1; f_2 ]);    h ≡( [ h_1; h_2 ]),

then 

    af(z) =  ∫_iℝdw/2π iΘ_2(z) Θ_2^-1 (w)-1 /w-z f(w)  
    
    bh(z) =  ∫_iℝdw/2π i1 -Θ_1(z) Θ_1^-1 (w)  /w-z h(w),

with Θ_1 and Θ_2 are as in (<ref>).









We begin by computing (<ref>)

    Θ_2(z) Θ_2^-1(w) -1 = [ [                                                 0                                                                                                     κ∫_γ_2(  e^ν(λ,s)/λ - z - e^ν(λ,s)/λ - w) dλ/2π i;                                                 0                                                                                                                                                     0 ]]

substituting (<ref>) in (<ref>) and focusing on the only non-zero entry a_12,


    a_12 f_2(z) = -κ∫_i ℝdw/2π i∫_γ_2dλ/2π ie^ν(λ,s)/(λ - z)(λ - w) f_2(w)

integrating over λ,

    a_12 f_2(z) = -κ∫_i ℝ - ϵdw/2π ie^ν(w,s)/w- z f_2(w)

A similar computation for b gives that the only non-zero entry is b_21 that reads

    b_21 h_1(z) =  κ∫_i ℝ∫_γ_1e^-ν(λ,s)/(λ - z)(λ - w) h_1(w) dλ/2π idw/2π i

integrating over λ

    b_21 h_1(z) = κ∫_i ℝ+ϵe^-ν(w,s)/(w - z) h_1(w)  dw/2π i

Substituing a and b back in (<ref>), we get the following


    τ(s) = [ Id_L^2(iℝ)⊗ℂ^2 -  [ [ 0 a; b 0 ]] ] 
    = [ Id_L^2(iℝ)⊗ℂ^2  -  [
     [    0    0    0 a_12;    0    0    0    0;    0    0    0    0; b_21    0    0    0 ]] 
     ] 
    
     =[ Id_L^2(iℝ)  -  [
     [    0 a_12; b_21    0 ]] 
     ].


 Further, it is straightforward to see that the operator [ [ 0 a; b 0 ]] is trace class. In other words, a_12 and b_21 are Hilbert-Schmidt

    |a_12|^2 = -κ^2 ∫_iℝ +ϵ |dz| ∫_iℝ-ϵ |dw| e^ν(w,s)+ν(w̅)/|w-z|^2 = -κ^2 ∫_iℝ +ϵ |dz| ∫_iℝ-ϵ |dw| e^2Reν(w,s)/|w-z|^2 < +∞
    
    |b_21|^2 = -κ^2 ∫_iℝ -ϵ |dz| ∫_iℝ+ϵ |dw| e^-ν(w,s)-ν(w̅)/|w-z|^2 = -κ^2 ∫_iℝ -ϵ |dz| ∫_iℝ+ϵ |dw| e^-2Reν(w,s)/|w-z|^2 < +∞
 
(<ref>) and (<ref>) are clearly convergent, implying that a_21 and b_12 are Hilbert-Schmidt operators. Therefore, the determinant _ℋ [1+U] is Fredholm and coincides with the τ-function in (<ref>).

 Further, it is shown in <cit.> that 

    [ Id_L^2(γ_L∪γ_R) + U ] = [ Id_L^2([s,∞) ) - κ^2 K_Ai|_[s,∞)]

where K_Ai is the Airy kernel, which implies the τ-function (<ref>) is the determinant of the Airy Kernel. It is a well known result <cit.> that the solution of (<ref>) is related to the Airy Kernel as 

    u(s)^2 = -d^2/ds^2log[  1 - κ^2 K_Ai|_[s,∞)]






 §.§ Relation to the JMU tau-function

The logarithmic derivative of the Widom constant  (<ref>) can be shown to coincide with the logarithmic derivative of the isomonodromic τ-function (<ref>). To this  end, we begin by defining a matrix valued function Y(z,s) as a ratio of Θ(z,s) and Γ(z,s).

    Y(z,s) =Θ_2^-1(z,s) Γ(z,s)    z<0 
    Θ_1^-1(z,s)Γ(z,s)    z ≥ 0


Y(z,s) has a jump only on iℝ. Its boundary values satisify the following relation

    Y_+(z,s) = J^-1(z,s)Y_-(z,s)    z∈ iℝ

and Y(z,s) → 1 as z→∞.

Notation: ' ≡∂/∂ z and ≡∂/∂ s. All functions depend on z,s unless stated otherwise. 




The logarithmic derivative of Widom constant  in (<ref>) is

    ∂_slogτ[J] = ∫_iℝdz/2π i{ J^-1J̇[ -Y_+' Y_+^-1 + Θ_2^-1Θ_2'  ]  } .



Begin with the Fredholm determinant [This computation follows from Theorem 2.3 in <cit.>. The difference being the choice of factorisation.]

    τ[J] = [  T_J^-1∘ T_J]= [PQ]

where P=T_J^-1 = Π_+ J^-1 and Q= T_J = Π_+J. Inverses are P^-1 = Y_-Π_+ Y_+^-1 and Q^-1 = Θ_+^-1Π_+Θ_-. Computing the logarithmic derivative

    ∂_tlog[ PQ ] = [ ∂_t P P^-1 + Q^-1∂_t Q ] 
    
    = [ -Π_+ J^-1∂_t J J^-1 Y_-Π_+ Y_+^-1 + Θ_+^-1Π_+Θ_-Π_+ ∂_t J ] 
    
    =[ -Π_+ J^-1∂_t J    Y_+ Y_-^-1 Y_-Π_+ Y_+^-1 + Θ_+^-1Π_+Θ_-(1-Π_- )∂_t J ] 
    
    =[ -Π_+ J^-1∂_t J    Y_+Π_+ Y_+^-1 + Θ_+^-1Π_+Θ_-∂_t J ] 
    
    =[ -Π_+ J^-1∂_t J    Y_+Π_+ Y_+^-1 + Θ_+^-1Π_+Θ_+ J^-1∂_t J ] 
    
    =∫_iℝdz/2π i{ J^-1∂_t J [ - ∂_z Y_+ Y_+^-1 + Θ_+^-1∂_zΘ_+] }

to obtain the last expression  we use the fact that J^-1∂_t J is a multiplication operator and only the diagonal parts of Y_+Π_+ Y_+^-1 and Θ_+^-1Π_+Θ_+ contribute to the expression. 


Using the above proposition we can identify the Widom constant with the isomonodromic  τ function.

 
The logarithmic derivative of the  Widom  constant (<ref>) coincides exactly with the logarithmic derivative of the (isomonodromic) JMU τ- function (<ref>) for Ablowitz-Segur family of solutions namely :

    ∂_slogτ[J] = ∂_slogτ_JMU = - Res_z=∞[ Γ' Γ^-1( z σ_3) ] .



  We prove the statement by simplifying the expression of ∂_slogτ[J] in  (<ref>):


    { J^-1J̇[  -Y_+' Y_+^-1 + Θ_2^-1Θ_2' ]  }.

We first perform algebraic manipulation on   Y_+' Y_+^-1 using  (<ref>)  






    Y_+' Y_+^-1 = (Θ_2^-1Γ)' (Γ^-1Θ_2) 
    
    = (-Θ_2^-1Θ_2' Θ_2^-1Γ + Θ_2^-1Γ' ) (Γ^-1Θ_2) 
    
    =  Θ_2^-1(- Θ_2' Θ_2^-1 +  Γ' Γ^-1)Θ_2

and expressing J in terms of Θ_1 and Θ_2 we obtain 

    J^-1J̇  = (Θ_2^-1Θ_1) (̇Θ̇_̇1̇^̇-̇1̇Θ̇_̇2̇)̇
    
    = (Θ_2^-1Θ_1) (-Θ_1^-1Θ̇_1Θ_1^-1Θ_2 + Θ_1^-1Θ̇_2) 
    
    = Θ_2^-1  (-Θ̇_1Θ_1^-1  +  Θ̇_2Θ_2^-1 ) Θ_2
    
    = Θ_2^-1Δ(Θ̇Θ^-1   ) Θ_2

where 

    Δ (Θ̇Θ^-1) = Θ̇_2Θ_2^-1 -Θ̇_1Θ_1^-1.

Substituting (<ref>) and (<ref>) in (<ref>) and using cyclicity of trace,

    { J^-1J̇   Y_+' Y_+^-1} = {Δ( Θ̇Θ^-1)  ( Γ' Γ^-1  + Θ_2' Θ_2^-1)  }

The term -Δ( Θ̇Θ^-1)Θ_2' Θ_2^-1 is explicit and cancels the term J^-1J̇Θ_2' Θ_2^-1. After the simplification, (<ref>) is 

    -∫_iℝd z/2π i{Δ( Θ̇Θ^-1)  Γ' Γ^-1}

since Γ has no jump on iℝ, (<ref>) can be further simplified

    -∫_iℝd z/2π i{Δ( Θ̇Θ^-1)  Γ' Γ^-1} = -∫_iℝd z/2π iΔ{Θ̇Θ^-1Γ' Γ^-1}
    
     = ∫_Σdz/2π iΔ{Θ̇Θ^-1Γ' Γ^-1}
    
     = ∫_γ_1dz/2π iΔ{Θ̇Θ^-1Γ' Γ^-1} + ∫_γ_2dz/2π iΔ{Θ̇Θ^-1Γ' Γ^-1}

Let us begin by computing the integral on γ_1 in (<ref>)

    Δ{Θ̇Θ^-1Γ' Γ^-1} = {Θ̇_1+Θ^-1_1+Γ_+' Γ_+^-1  - Θ̇_1-Θ^-1_1-Γ_-' Γ_-^-1}

computing (<ref>) term by term by substituting (<ref>) for Γ_+

    Γ_+'Γ_+^-1  = (G_1Γ_- )' (Γ_-^-1 G_1^-1) 
    
     = G_1[ G_1^-1 G_1'  +  Γ_-' Γ_-^-1]G_1^-1

and (<ref>) for Θ_1+

    Θ̇_1+Θ_1+^-1 = (̇ ̇Ġ_̇1̇Θ̇_̇1̇-̇)̇Θ_1-^-1 G_1^-1
    
    = G_1[ G_1^-1Ġ_1 + Θ̇_1-Θ_1-^-1]G_1^-1.

Substituting (<ref>), (<ref>) in (<ref>) and using cyclicity

    {( Θ̇_1+Θ_1+^-1) Γ_+' Γ_+^-1} = [ ( G_1^-1Ġ_1 + Θ̇_1-Θ_1-^-1) ( G_1^-1 G_1'  +  Γ_-' Γ_-^-1)  ]

In (<ref>), notice that the term 
 (G_1^-1Ġ_1  +  Θ̇_1-Θ_1-^-1) ( G_1^-1 G_1' ) is traceless. Furthermore, we have the following identity 
 2 G_1^-1Ġ_1 = - z G_1^-1σ_3 G_1 +  z σ_3. The terms Θ̇_1-Θ_1-^-1Γ_-^-1Γ_-' in (<ref>) and (<ref>)  cancel each other out.



So, all that is left to compute on the contour γ_1 is the following

    ∫_γ_1dz/2π i[ G_1^-1Ġ_1Γ_-' Γ_-^-1] = 1/2∫_γ_1d z/2π i[ (  z G_1^-1σ_3 G_1 + z σ_3 ) Γ_-' Γ_-^-1]
 
We begin by computing the following term

    ( G_1^-1σ_3 G_1Γ_-' Γ_-^-1)

the term Γ_-'  Γ_-^-1 can be simplified by substituting (<ref>)

    Γ_-'  Γ_-^-1 = ( G_1^-1Γ_+)' (  Γ_+^-1 G_1) 
    
    =  G_1^-1(  -G_1' G_1^-1  + Γ_+' Γ_+^-1) G_1

substituting (<ref>) in (<ref>) and using the cyclic property of the trace

    (  G_1^-1σ_3 G_1Γ_-^-1Γ_-') = [ σ_3 Γ_+' Γ_+^-1  - G_1' G_1^-1σ_3 ]

note that G_1' G_1^-1σ_3 is traceless. Substituting (<ref>) in (<ref>) we have

    1/2∫_γ_1dz/2π i[(  z G_1^-1σ_3 G_1 -  z σ_3 ) Γ_-' Γ_-^-1] = 1/2∫_γ_1dz/2π i[-  z σ_3 ( Γ_+' Γ_+^-1 - Γ_-' Γ_-^-1) ]

repeating this exercise and computing the integral on γ_2 in (<ref>), we get exactly the same expression. Putting all together

    ∂_s lnτ[J] = ∫_Σd z/2π i[ -z σ_3  ( Γ_+' Γ_+^-1 - Γ_-' Γ_-^-1)    ] 
    
    = -Res_z=∞(  z σ_3 Γ' Γ^-1)





§ MINOR EXPANSION

The Hilbert space L^2(S^1) admits a natural orthonormal basis of Fourier modes (i.e. the monomials z^n,   n ∈ℤ).  The minor expansion of the Fredholm determinant (<ref>) in this particular basis gives rise to interesting combinatorics. In the case of Painlevé VI, V, III the combinatorics correspond to certain Nekrasov Partition functions of certain Gauge theories <cit.>. 

In this spirit, we would like to propose, at least, a reasonable expansion of the Fredholm determinant of our operator in a similar guise. In our case the underlying Hilbert space L^2(iℝ)  does not immediately suggest a natural discrete orthonormal basis. Here below we want to propose a very natural such basis: the main guiding principle is that of identifying the Hardy space ℋ_+ with the Hardy space of the interior of the disk, and pulling back the monomial basis.  


The Fredholm determinant of the τ-function in (<ref>) can be expanded, on an appropriate basis, in terms of minors, that can be labelled by Maya diagrams (m_X)

    τ [s] = ∑_m_X ∈𝕄;  | p | = | h |α_p_X^h_Xβ_h_X^p_X

where the coefficients α_m^n, β_n^m are as follows 

    β_0^0 = α_0^0 = (4∂^2/∂ s^2 - 1  )  ( 1- ∂/∂ s)^-1 Ai(s)

where Ai(s) is the Airy function, and

    α_m^n = (-1)^m+n/(m!)^2 n! (m+n+1)!( D)^m+nα_0^0
    β_n^m = (-1)^m+n/(n!)^2 m! (m+n+1)!( D)^m+nβ_0^0

with D= 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s )









Recall that, 

    τ(s) = [ Id_L^2(iℝ) -  [ [ 0 a; b 0 ]] ]
    =[ Id_L^2(iℝ)⊗ℂ^2  -  [
     [    0    0    0 a_12;    0    0    0    0;    0    0    0    0; b_21    0    0    0 ]] 
     ]

of which a_12 and b_21 are the only non zero entries. Therefore, the  determinant of the 4 × 4 block operator can be reduced to a determinant of a 2× 2 block operator. Let us denote a_12≡α, b_21≡β

    τ[s] = [ Id_L^2(iℝ)- ( [    0 a_12; b_21    0 ])  ] = [ Id_L^2(iℝ)- ( [ 0 α; β 0 ])  ]

We remind the reader here that the block decomposition is due to the splitting L^2(iℝ) = ℋ = ℋ_+⊕ℋ_-.

The first step to obtain the minor expansion is constructing a suitable basis to expand α(z,w) and β(z,w),

    α(z,w): ℋ_-→ℋ_+   ;   β(z,w) : ℋ_+→ℋ_-.




 §.§ Basis construction
 
The spaces ℋ_± are Hardy spaces of functions analytic on the left and right half of the complex planes respectively. To construct the bases of ℋ_±, we employ the Paley-Weiner theorem which identifies ℋ_+ as the image under Fourier transform of functions supported on a half–line. Specifically, let  ℂ_+ = { z: z=x+iy , y>0  }, 

    H^2(ℂ_+) = { f:f is analytic in  ℂ_+  andsup_0<y<+∞∫_-∞^+∞| f(z) |^2 dx < ∞}.

By definition, the boundary values of f∈ H^2(ℂ_+) on ℝ define a function in L^2(ℝ) and we can think of H^2(ℂ_+) as a (closed) subspace of L^2(ℝ). With this understanding, the Paley–Wiener theorem can be stated as the following identity:

    ℱ H^2(ℂ_+) = L^2[0,∞) .

The space ℋ_+  can be isometrically mapped to H^2(ℂ_+) by a variable change z → iz. 

We have that Laguerre functions  (L_n^λ(z) z^λ e^-z) provide a basis of L^2(ℝ_+). Using the Paley–Wiener theorem, upon inverse Fourier transform, they yield a basis for H^2(ℂ_+) and an innocent change of variable z → iz gives a basis on ℋ_+. We can comfortably restrict ourselves to λ =0. Following <cit.>, the Fourier transform (ℓ̂_n^λ(t)) of the Laguerre functions L_n^λ(x) e^-x/2 x^λ/2 for λ=0, and using the notation and L_n^0≡ L_n, is 

    L_n(x)  e^-x/2 =  e^-x/2/n!∑_k=0^n (-n)_k x^k/k!
    ℓ̂_n (t) = -2/n!( 1+2it/2it-1)^n 1/2it-1

ℓ̂_n forms a complete basis on H^2(ℂ_+, dt). With the change of variable 2it =z, (<ref>) reads

    ℓ̂_n(z) = -2/n!( 1+z/z-1)^n 1/z-1

and will form a basis on H^2(ℋ_+, -i/2 dz). In conclusion 

    e_ℋ_+^n = i/n!( 1+z/z-1)^n 1/z-1

is a basis on H^2(ℋ_+, dz). Similarly,

    e_ℋ_-^n = i/n!( z-1/z+1)^n 1/z+1

is a basis on H^2(ℋ_-, dz)



 §.§ Minor expansion
 
Expanding α(z,w) and β(z,w) in the basis e_ℋ_+ and e_ℋ_-, the τ-function (<ref>) can be expressed as a minor expansion. Starting with α(z,w),

    α_m^n = ⟨α(z,w) e_ℋ_-^n , e_ℋ_+^m(z) ⟩
    
    = ∫_iℝdz/2π ie̅_ℋ_+^m(z) ∫_iℝ-ϵdw/2π iα(z,w) e_ℋ_-^n(w) 
    
    =-κ/m!n!∫_iℝdz/2π i(z̅+1/z̅-1)^m1/(z̅-1)∫_iℝ-ϵdw/2π ie^ν(w,s)/w-z(w-1/w+1)^n1/(w+1)
    
     =κ/m!n!∫_iℝ-ϵdw/2π ie^ν(w,s)(w-1)^m+n/(w+1)^m+n+2

Similarly for β(w,z)

    β_m^n = ⟨β(z,w) e_ℋ_+^n , e_ℋ_-^m(z) ⟩
    
    = ∫_iℝdz/2π ie̅_ℋ_-^m(z) ∫_iℝ+ϵdw/2π iβ(z,w) e_ℋ_+^n(w) 
    
    =κ/m!n!∫_iℝdz/2π i(z̅-1/z̅+1)^m1/(z̅+1)∫_iℝ+ϵdw/2π ie^-ν(w,s)/w-z(w+1/w-1)^n1/(w-1)
    
     =-κ/m!n!∫_iℝ+ϵdw/2π ie^-ν(w,s)(w+1)^m+n/(w-1)^m+n+2




 §.§ Recurrence relations
 
α_m^n, β_m^n can be made explicit by noticing that the functions such as ∫dw/2π i(w+1)^m e^-θ(w) can be written as some derivatives of the Airy function. Define the function χ_m+n as  

    χ_m+n = (w-1)^m+n/(w+1)^m+n+2 e^ν(w,s).

Then 
α_m^n in terms of χ_m+n is simply

    α_m^n = κ/m!n!∫_iℝ-ϵdw/2π ie^ν(w,s)(w-1)^m+n/(w+1)^m+n+2 = κ/m! n!∫_iℝ-ϵdw/2π iχ_m+n

and β_n^m in terms of χ_m+n is

    β_n^m = -κ/m!n!∫_iℝ+ϵdw/2π ie^-ν(w,s)(w+1)^m+n/(w-1)^m+n+2
    
    = κ/m!n!∫_iℝ-ϵdw/2π ie^ν(w,s)(w-1)^m+n/(w+1)^m+n+2 
     = κ/m! n!∫_iℝ-ϵdw/2π iχ_m+n

The function
χ_m+n obeys a recurrsion relation that can be derived as follows

    ∫dw/2π iχ_m+n(w,s) = ∫dw/2π i(w-1)^m+n/(w+1)^m+n+2 e^ν(w,s)
    
     = 2/(m+n+1)∫dw/2π i e^ν(w,s)∂_w[(w-1)^m+n+1/(w+1)^m+n+1] 
    
    = -2/(m+n+1)∫dw/2π i[(w-1)^m+n+1/(w+1)^m+n+1] ∂_w e^ν(w,s)
    
    =-2/(m+n+1)∫dw/2π i(w-1)^m+n-1/(w+1)^m+n+1 (w-1)^2  (4w^2 -s) e^ν(w,s)
    
     =-2/(m+n+1)∫dw/2π i( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) χ_m+n-1(w,s)

which gives the following equation 

    ∫_iℝdw/2π i[ χ_m+n + 2/(m+n+1)( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) χ_m+n-1] =0


We define a function Ci(t)  as follows:

    Ci(t) := ∫dw/2π i1/(w+1) e^θ(w) = ( 1- ∂/∂ s)^-1 Ai(t)

Then the function
χ_0 can be computed in terms of Ci(t)

    ∫_iℝdw/2π iχ_0 = ∫dw/2π i1/(w+1)^2 e^ν(w,s) 
    = -∫dw/2π i∂_w( 1/w+1) e^ν(w,s)
    
    =∫dw/2π i( 1/w+1) ∂_w e^ν(w,s) 
    = ∫dw/2π i( 1/w+1) (4w^2 -s) e^ν(w,s)
    
    = ∫dw/2π i( 4 ∂^2/∂ s^2 - s ) ( 1/w+1) e^ν(w,s)
    
    = ( 4 ∂^2/∂ s^2 - s ) Ci(s)

Since both  α_m^n and β_n^m are integrals of χ_m+n(<ref>), (<ref>), the recurrsion relation (<ref>) and the expression for χ_0 (<ref>) give the explicit expressions for α_m^n and β_n^m. Starting with the recursion relation for α_m^n and β_n^m

    n(m+n+1)α_m^n + 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) α_m^n-1 =0 
    
    n(m+n+1)β_m^n + 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) β_m^n-1 =0

with 

    α_0^0 = (4∂^2/∂ s^2 - 1  )  ( 1- ∂/∂ s)^-1 Ai(s)

Further,

    n α_m-1^n = m α_m^n-1   ;    nβ_m-1^n = mβ_m^n-1

Notice that (<ref>) and (<ref>) are recurrsive relations in n. Using (<ref>) similar recurrsion relations in m can be obtained

    m (m+n+1) α_m^n + 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) α_m-1^n =0 
    
    m (m+n+1) β_m^n + 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) β_m-1^n =0
 
Define 2 ( ∂/∂ s -1 )^2  (4 ∂^2/∂ s^2 -s ) = D. From (<ref>) 

    α_m^n = (-1)^m+n(1+m)/n! (m+n+1)!D^nα_m^0

now using (<ref>)

    α_m^0 = (-1)^m1/m! (m+1)!D^m α_0^0

In terms of α_0^0, α_m^n is explicit

    α_m^n = (-1)^m+n/(m!)^2 n! (m+n+1)!D^m+nα_0^0

Repeating the same computation for β_n^m

    β_n^m = (-1)^m+n/(n!)^2 m! (m+n+1)!D^m+nβ_0^0




 §.§ Maya diagrams
 

 The determinant of an operator A ∈ℂ^m× m can be expanded in terms of its principal minors <cit.>. For a finite m× m matrix A, the minor expansion is given by 

    ( 1+A  ) = ∑_n=0^∞∑_i_1< ... <i_n(A_i_p,i_q)_p,q=1^n

This sequence obviously terminates after n=m. Now generalise the minor exansion for infinite dimensional matrix.


  * Let K be a infinite dimensional matrix. Instead of labelling by { 1,...,m }, K is labelled by an infinite discrete set. Define a half-integer lattice ℤ' = ℤ+ 1/2. The set of all finite subsets of ℤ' is given by { 0,1  }^ℤ' and χ⊂{ 0,1  }^ℤ'. For ℤ'_∓ = ℤ'_≶ 0, define 'particles' (p_χ) to be the positions p_χ = χ∩ℤ'_+ and 'holes'(h_χ) to be the positions h_χ= χ∩ℤ'_-. (p_χ, h_χ) define point configurations on ℤ'. Furthermore, for a block matrix, we have two indices.


  * Expansion of the block determinant, given by the particles and holes (p_χ,h_χ)


  * The index within the block, which is called the colour index. 


  * Maya diagram m_χ is constructed by drawing filled circles at the points ( ℤ'_+\ p_χ)∪ h_χ and empty circles at p_χ∪( ℤ'_-\ h_χ). Set of all Maya diagrams is denoted by 𝕄 = ∪_χ m_χ.

  * For (1+K), the minors can be labelled by the half integer lattice ℤ'. Rows and coloumns will now be labelled by χ⊂{ 0,1  }^ℤ'. The minor expansion is given by 

    [1+ K ] = ∑_χ⊂{ 0,1  }^ℤ' A_χ



  * Maya diagrams can also be written as young diagrams by playing the following game. Reading the maya diagram from the left end, draw a horizontal line pointing to the right (→) for every filled circle ([black,fill=black] (0,0) circle (.5ex);) and a vertical line pointing downwards (↓) for every empty circle ([black] (0,0) circle (.5ex);). 


Let us work out a simple example. Let K be a 2× 2 block matrix. A, B are 2× 2 matrices

    K = [ [ 0 A; B 0 ]]

The labelling goes as shown in fig.<ref>. The blue and red label are the colour indices.


 In fig. <ref>, collect the blue and red terms in terms of particles and holes. It is clear from here that minor contribution for the configurations where |p|≠ |h| is 0. p_χ = ( 5/2), h_χ = ( -5/2,-1/2); p_χ = ( 5/2, 7/2), h_χ = ( -3/2). This gives the blue and red Maya diagrams.  m_χ = ( 5/2,-5/2,-1/2);  m_χ = ( 5/2,7/2,-3/2)



and the corresponding Young tableaux are
 (2,2,1)
 (4,1)

Now for the Ablowitz-Segur τ-function, 

    τ[s] = [ 1 - ([ 0 α; β 0 ]) ]

α and β can be expanded on a discrete basis  e_ℋ_±(z)

    α(z,w) = ∑_m,n ∈ℤ_+α_m^n e_ℋ_+(z)^m e_ℋ_-(w)^n ;   β(z,w) = ∑_m,n ∈ℤ_+β_m^n e_ℋ_-(z)^m e_ℋ_+(w)^n



Since α_m^n and β_n^m in (<ref>) are not matrices themselves, the corresponding Maya diagrams are "colourless". If a_m^n and b_n^m were N× N matrices themselves, the corresponding entries in the expansion would be a_m;β^n;α and b_n;α^m;β where α, β = { 1,...,N } would be the colour indices. Furthermore, given the off-diagonal structure of the matrix U, the minors with | p |≠| h | vanish. Therefore, the minor expansion reads, 

    τ [s] = ∑_m_χ∈𝕄;  | p | = | h |α_p_χ^h_χβ_h_χ^p_χ .

The proof is now complete.



It would be extremely interesting to interpret the terms in this minor expansion in a similar way to the case of Painlevé VI, V, III. However, to our knowledge, in the case of the second Painlevé transcendent, there is no direct analog  connection with some field theory. Nonetheless the computation proceeds in a  rather natural way and may prove of use in future applications.



Acknowledgements.
 
The author thanks Alexander Its, Marco Bertola, Mattia Cafasso and Tamara Grava for illuminating discussions and suggestions. The author acknowledges   the   Centre International de Rencontres Mathematiques, Luminy, for hospitality during the semester "Integrability and Randomness in mathematical Physics" and LYSM grant for financial support. Part of the work was completed during a visit to the Department of Mathematics and Statistics at Concordia University which was supported by H2020-MSCA-RISE-2017 PROJECT No. 778010 IPADEGAN, and the author is greatful for the financial support and hospitality.






















































































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