Диссертация (1137342), страница 20

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To avoid possible confusionof the reader, we explicitly indicate the contours and jump matrices for RHPs for Ψ̃[L]894. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsand Ψ[R] in Fig. 4.11; note the independence of jumps on t. In particular, inside thedisk around ∞ we have Φ̃[L] (z) = (−z)S Ψ̃[L] (z). Since the annulus A belongs to thedisk around ∞ in the RHP for Ψ[L] , the formula (4.51) yields the following expressionfor Ψ[L] inside A:!∞z X[L]Ψ[L] (z) = (−z)−S Φ[L] (z) = t−S Ψ̃[L]= t−S 1 +gk tk z −k G[L]∞ , (4.52a)tAk=1[L]where the N × N matrix coefficients gk are independent of t.

Analogous expressionfor Ψ[R] (z) inside A does not contain t at all:!∞X[R][R]Ψ[R] (z) = 1 +gk z k G0 .(4.52b)Ak=1The formulae (4.52) allow to extract from the determinant representation (4.48) theasymptotics of 4-point Jimbo-Miwa-Ueno tau function τJMU (t) as t → 0 to any desiredorder.

We are now going to explain the details of this procedure.Rewrite the integral kernel d (z, z 0 ) as0−Sd (z, z ) = t1 − Ψ̃[L]ztΨ̃[L]z − z0z 0 −1Stt .The block matrix elements of d in the Fourier basis are therefore given byd−pq = t−S d̃−pq tS · tp+q ,p, q ∈ Z0+ ,(4.53)where N × N matrix coefficients d̃−pq are independent of t. They can be extractedfrom the Fourier series−11 − Ψ̃[L] (z) Ψ̃[L] (z 0 )z−=z01X1d̃−pq z − 2 −p z 0− 2 −q ,(4.54)p,q∈Z0+and are therefore expressed in terms of the coefficients of local expansion of the 3point solution Φ̃[L] (z) around z = ∞ by straightforward algebra. For instance, thefirst few coefficients are given by− 12d̃− 21d̃− 21d̃52[L][L] [L]12− 32[L] 2[L]32[L]= g1 ,= g2 − g1 ,[L] [L][L] 3= g3 − g2 g1 − g1 g2 + g1 ,......d̃− 32d̃......3212[L]= g2 ,[L][L] [L]= g3 − g2 g1 ,− 52d̃12[L]= g3 ,......Different lines above contain the coefficients of fixed degree p + q ∈ Z>0 which appearsin the power of t in (4.53). Very similar formulas are also valid for matrix elementsof a:113[R][R][R] 2[R]a− 12 = g1 ,a− 32 = g2 − g1 ,a− 12 = g2 ,...222904.3.

Fourier basis and combinatoricsThe crucial point for the asymptotic analysis of τ (t) is that for small t the operatord becomes effectively finite rank. Indeed, fix a positive integer Q. To obtain a uniformapproximation of d (z, z 0 ) up to order O tQ , it suffices to take into account its Fouriercoefficients d−pq with p+q ≤ Q; recall that the eigenvalues of S are chosen as to satisfy(4.45). Since here p, q ∈ Z0+ , the total number of relevant coefficients is finite and equalto Q (Q − 1) /2.

It follows that the only terms in the Fourierexpansion of a (z, z 0 )that contribute to the determinant (4.48) to order O tQ correspond to monomials11z p− 2 z 0q− 2 with p + q ≤ Q. This is summarized inTheorem 4.13. Let Q ∈ Z>0 . The 4-point tau function τJMU (t) has the followingasymptotics as t → 0:hi10 aQTr(S2 −Θ20 −Θ2t )Q2,UQ =. (4.55)det (1 − UQ ) + O tτJMU (t) ' tdQ 0Here UQ denotes a 2N Q × 2N Q finite matrix whose N Q × N Q-dimensional blocks aQand dQ are themselves block lower and block upper triangular matrices of the form −1Q− 21−1−1a −10···0d̃Q−21 tQ · · · d̃ 23 t2d̃ 21 t2222 .... Q− 32− 32 2 .. .·.a.0·d̃t3 S1−2−S t ,2aQ = ,d=t33Q......−Q Q a 122.0·.· d̃ 3 t. −221111−QQ220···0d̃ 1 ta− 12 a− 32 · · · a 1 −Q2222where a−qp , d̃−pq are determined by (4.49a), (4.50), (4.54), and the conjugation by tS inthe expression for dQ is understood to act on each N × N block of the interior matrix.Moreover, strengthening the condition (4.45) to strict inequality |< (σα − σβ )| < 1improves the error estimate in (4.55) to o tQ .Remark 4.14.

The above theorem gives the asymptotics of τJMU (t) to arbitrary finiteorder Q in terms of solutions Φ[R] (z), Φ̃[L] (z) of two 3-point Fuchsian systems withprescribed monodromy around regular singular points 0, 1, ∞. For Q = 1 and underassumption |< (σα − σβ )| < 1, its statement may be rewritten ashi1[R] 1−S [L] STr(S2 −Θ20 −Θ2t )2det 1 − g1 tg1 t + o (t) .(4.56)τJMU (t) ' tA result equivalent to this last formula has been recently obtained in [ILP, Proposition3.9] by a rather involved asymptotic analysis based on the conventional RiemannHilbert approach. For N = 2, the leading term in the expansion of the determinantappearing in (4.56) gives Jimbo asymptotic formula [Jimbo] for Painlevé VI.Fourier basis and combinatoricsStructure of matrix elementsLet us return to the general case of n regular singular points on P1 .

We have alreadyseen in the previous subsection certain advantages of writing the operators which914. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsappear in the Fredholm determinant representation (4.44) of the tau function in theFourier basis. This motivates us to introduce the following notation for the integralkernels of the 3-point projection operators a[k] , b[k] , c[k] , d[k] from (4.20):[k]−1[k]−1Ψ (z) Ψ+ (z 0 )a (z, z ) := +z − z00[k]=X11[k]p − 2 +p 0− 2 +qz,a[k]−q zz, z 0 ∈ Cin ,p,q∈Z0+(4.57a)[k]b[k] (z, z 0 ) :=−[k]Ψ+ (z) Ψ+ (z 0 )z − z0−1=X1b[k]pq z − 2 +p z 0− 21 −q[k][k]z ∈ Cin , z 0 ∈ Cout ,,p,q∈Z0+(4.57b)[k][k]Ψ+ (z) Ψ+ (z 0 )z − z00[k]c (z, z ) :=−1=X11− 2 −p 0− 2 +qz,c[k] −p−q z[k][k]z ∈ Cout , z 0 ∈ Cin ,p,q∈Z0+(4.57c)0[k]d (z, z ) :=[k]01 − Ψ[k]+ (z) Ψ+ (z )z−−1z0=X11[k]d[k]−pq z − 2 −p z 0− 2 −q ,z, z 0 ∈ Cout .p,q∈Z0+(4.57d)Just as before in (4.49b), the overall minus signs in the expressions for b[k] (z, z 0 ) and[k]d[k] (z, z 0 ) are introduced to absorb the negative orientation of Cout .Our task in this subsection is to understand the dependence of matrix elementsp[k]p[k] −p[k]−p0a[k]−q , b q , c −q , dq on their indices p, q ∈ Z+ .

To this end recall that (cf (4.15))[k]Ψ+(−1(−z)−Sk−1 Sk−1Φ[k] (z) ,(z) =(−z)−Sk Sk−1 Φ[k] (z) ,[k]z ∈ Cin ,[k]z ∈ Cout .(4.58)where Φ[k] (z) denotes the fundamental solution of the 3-point Fuchsian system (4.16).[k]Theorem 4.15. Denote by r[k] the rank of the matrix A1 which appears in the Fuch[k] [k]sian system (4.16). Let ur , vr ∈ CN with r = 1, . . . , r[k] be the column and rowvectors giving the decomposition[k][k]ak A1=−rX[k]u[k]r ⊗ vr .(4.59)r=1[k] pLet ψr[k] p,, ψ̄r[k] −pϕr[k] −p, ϕ̄r∈ CN be the coefficients of the Fourier expansions924.3.

Fourier basis and combinatorics [k][k]Ψ+ (z)ur z − ak=Xp 1ψr[k] z − 2 +p ,p∈Z0+= [k][k]Ψ+ (z)ur z − ak=k[k] [k]vr Ψ+X[k]ψ̄rpz− 12 +p(4.60a)[k](4.60b),p∈Z0+Xϕ[k]r−p1z − 2 −p ,p∈Z0+z ∈ Cout .−1(z)z − ak[k]z ∈ Cin ,−1[k] [k]vr Ψ+ (z)z−a=X[k]ϕ̄r−pz− 12 −p,p∈Z0+Then the operators a[k] , b[k] , c[k] , d[k] can be represented as sums of a finite number ofinfinite-dimensional Cauchy matrices with respect to the indices p, q, explicitly givenby[k]p;αa[k]−q;β=b[k] p;αq;β =c[k] −p;α−q;β =d[k] −p;αq;β =[k] [k] p;αψ̄r q;βψr,p + q + σk−1,α − σk−1,βr=1[k] [k] p;αr[k]Xϕ̄r −q;βψr,q−p−σ+σk−1,αk,βr=1[k] [k] −p;αr[k]Xψ̄r q;βϕr,q − p + σk,α − σk−1,βr=1[k] −p;α[k] r[k]Xϕrϕ̄r −q;β,p + q − σk,α + σk,βr=1rX(4.61a)(4.61b)(4.61c)(4.61d)where the color indices α, β = 1, .

. . , N correspond to internal structure of the blocksp[k]p[k] −p[k]−pa[k]−q , b q , c −q , dq.Proof. The Fuchsian system (4.16) can be used to differentiate the integral kernels(4.57) with respect to z and z 0 . Consider, for instance, the operatorL0 = z∂z + z 0 ∂z0 + 1.1It is easy to check that2 L0 z−z0 = 0. Combining this with (4.57a), (4.58) and (4.16),one obtains e.g. that[k][k]L0 az, z0[k](z∂z + z 0 ∂z 0 ) Ψ+ (z) Ψ+ (z 0 )=z − z0−1h[k]= a−1[k]i Ψ[k] (z)0[k] Ψ+ (z )+ak A1,z, z , Sk−1 −z − akz 0 − ak0[k]where z, z 0 ∈ Cin . The crucial point here is that the dependence of the second termon z and z 0 is completely factorized.

Indeed, it follows from the last identity, the2The reader with acquintance with two-dimensional conformal field theory will recognize in thisequation the dilatation Ward identity for the 2-point correlator of Dirac fermions.934. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionspform of L0 and the notation (4.60a) that N × N matrix a[k]−q from (4.57a) satisfies theequationr[k]p [k] p Xψr[k] ⊗ ψ̄r[k] q .p + q + adSk−1 a −q =r=1The formula (4.61a) is nothing but a rewrite of this identity. The proof of Cauchy typerepresentations (4.61b)–(4.61d) for the other three operators is completely analogous.Combinatorics of determinant expansionThis subsection develops a systematic approach to the computation of multivariateseries expansion of the Fredholm determinant τ (a) = det (1 − K).

Recall that, according to Theorem 4.11, the isomonodromic tau function τJMU (a) coincides withτ (a) up to an elementary explicit prefactor.Let A ∈ CX×X be a matrix indexed by a discrete and possibly infinite set X. Ourbasic tool for expanding τ (a) is the von Koch’s formula:Xdet (1 + A) =det AY ,(4.62)Y∈2Xwhere det AY denotes the |Y| × |Y| principal minor obtained by restriction of A to asubset Y ⊆ X. Of course, the series in (4.62) terminates when X is finite.In our case, the role of the matrix A is played by the operator K written in theFourier basis. The elements of X are multi-indices which encode the following data:• the positions of the blocks a[k] , b[k] , c[k] , d[k] in K defined by (4.26);• a half-integer Fourier index of the appropriate block;• a color index taking its values in the set {1, .

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