Диссертация (1137342), страница 19

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Let L ∈ End (H+ ) be the operator defined by (4.28). We define thetau function associated to the Riemann-Hilbert problem for Ψ as(4.29)τ (a) := det L−1 .In order to demonstrate the relation of (4.29) to conventional definition [JMU] ofthe isomonodromic tau function and its extension [ILP], let us compute the logarithmic derivatives of τ with respect to isomonodromic times a1 , . .

. , an−3 . At this pointit is convenient to introduce the notation∆k =1Tr Θ2k ,2¯ k = 1 Tr S2 .∆k2(4.30)¯ 0 ≡ ∆0 and ∆¯ n−2 ≡ ∆n−1 .Recall that ∆Theorem 4.11. We haveτ (a) = Υ (a)−1 τJMU (a) ,(4.31)where τJMU (a) is defined up to a constant independent of a byXda ln τJMU =Tr Ak Al d ln (ak − al ) ,(4.32)0≤k<l≤n−2and the prefactor Υ (a) is given byΥ (a) =n−3Y¯ −∆¯ k−1 −∆k∆ak k.(4.33)k=1Proof. We will proceed in several steps.Step 1. Choose a collection of points a0 close to a in the sense that the same annulican be used to define the tau function τ (a0 ). The collection a0 will be considered fixedwhereas a varies.

Let us compute the logarithmic derivatives of the ratio τ (a) /τ (a0 ).First of all we can writeτ (a)−10 −10PΣ,+ a PΣ,+ (a) P⊕,+ (a)(4.34)= det P⊕,+ aτ (a0 )Note that since PΣ,+ (a) : H+ → HT (a) can be viewed as a projection of elements ofH along HA , the compositionPa0 →a := PΣ,+ (a) PΣ,+ a0−1: HT a0 → HT (a)is also a projection along HA . It therefore coincides with the restriction PΣ H (a0 ) .TOne similarly shows that−1Fa0 →a := P⊕,+ (a) P⊕,+ a0= P⊕ H (a0 ) .T844.2.

Tau functions as Fredholm determinantsThe exterior logarithmic derivative of (4.34) can now be written asτ (a)= − TrHT (a0 ) da (Fa→a0 Pa0 →a ) · Pa→a0 Fa0 →a =da lnτ (a0 )= − TrHT (a0 ) Fa→a0 · da Pa0 →a · Pa→a0 Fa0 →a =00= − TrH P⊕ a · da PΣ (a) · PΣ a P⊕ (a) .(4.35)The possibility to extend operator domains as to have the second equality is a consequence of (4.23).

Furthermore, using once again the projection properties, one showsthatP⊕ (a) 1 − P⊕ a0 = 0.PΣ (a) 1 − PΣ a0 = 0,which reduces the equation (4.35) ton−2X[k]da ln τ (a) = − TrH P⊕ da PΣ = −TrH[k] P⊕ da PΣ .(4.36)k=1Step 2. Let us now proceed to calculation of the right side of (4.36). Computationsof the same type have already been used in the proofs of Lemmata 4.6 and 4.7. Theidea is that Ψ[k] and Ψ̂ have the same jumps on the contour Γ[k] which reduces theintegrals in (4.19), (4.22) to residue computation.

In particular, for f [k] ∈ H[k] withk = 2, . . . , n − 3 we have[k]P⊕ da PΣ f [k] (z) =1(2πi)2‹[k][k]−1Ψ+ (z) Ψ+ (z 0 )[k]−1da Ψ̂+ (z 0 ) Ψ̂+ (z 00 )f [k] (z 00 ) dz 0 dz 00.(z − z 0 ) (z 0 − z 00 )[k]Cin ∪Cout(4.37)The integrals are computed with the prescription that z is located inside the contourof z 0 , itself located inside the contourof z 00 , and then passing to boundary values. But000 −1000 −1since the function (z − z ) da Ψ̂+ (z ) Ψ̂+ (z )has no singularity at z 00 = z 0 , thecontours of z 0 and z 00 can be moved through each other.

This identifies the trace ofthe integral operator on the right of (4.37) with[k]Tr P⊕ da PΣ =no−1−1[k][k]00‹Tr Ψ+ (z) Ψ+ (z ) da Ψ̂+ (z ) Ψ̂+ (z)dz dz 01=−=[k](2πi)2 Cin[k] ∪Cout(z − z 0 )2on−1−1 [k][k]00‹Tr Ψ+ (z ) da Ψ̂+ (z ) · Ψ̂+ (z) Ψ+ (z) dz dz 01=−[k](2πi)2 Cin[k] ∪Cout(z − z 0 )2n o−1−1[k][k]00‹Tr da Ψ̂+ (z)· Ψ+ (z) Ψ+ (z ) Ψ̂+ (z ) dz dz 01−,[k](2πi)2 Cin[k] ∪Cout(z − z 0 )2where z is considered to be inside the contour of z 0 . The first term vanishes since the[k][k]contours Cin and Cout in the integral with respect to z can be merged.

In the second854. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsterm the integral with respect to z 0 is determined by the residue at z 0 = z, whichyields˛n o−1−11[k][k][k]Tr P⊕ da PΣ =Tr da Ψ̂+ (z)· Ψ+ (z) · ∂z Ψ+ (z) Ψ̂+ (z) dz.[k]2πi Cin[k] ∪Cout[k]Recall that Ψ̂+ , Ψ+ are related to fundamental matrix solutions Φ, Φ[k] of n-pointand 3-point Fuchsian systems by−Sk−1−1Φ (z) ,Ψ̂+ (z) [k] = Sk−1 (−z)−Sk Φ (z) ,Ψ̂+ (z) [k] = Sk−1 (−z)CC out in[k][k]−S−1Ψ+ (z) [k] = Sk−1 (−z)−Sk Φ[k] (z) .Ψ+ (z) [k] = Sk−1(−z) k−1 Φ[k] (z) ,CoutCinThis leads to˛on1−1[k]Tr P⊕ da PΣ =Tr da Φ−1 · Φ[k] · ∂z Φ[k] Φ dz =[k]2πi Cin[k] ∪Coutno−1= resz=ak Tr da Φ · Φ−1 ∂z Φ · Φ−1 − ∂z Φ[k] · Φ[k].(4.38)The contributions of the subspaces H[1] and H[n−2] to the trace (4.36) can be computed[k][k]in a similar fashion.

The only difference is that instead of merging Cin with Cout one[1][n−2]should now shrink the contour Cout to 0 and Cinto ∞. The result is given by thesame formula (4.38).Step 3. To complete the proof, it now remains to compute the residues in (4.38).Note that near the regular singularity z = ak the fundamental matrices Φ, Φ[k] arecharacterized by the behavior!∞XΦ (z → ak ) = Ck (ak − z)Θk 1 +gk,l (z − ak )l Gk ,(4.39a)Φ[k] (z → ak ) = Ck (ak − z)Θk1+l=1∞X![k]g1,l (z − ak )l[k]G1 .(4.39b)l=1The coinciding leftmost factors ensure the same local monodromy properties.

The[k]rightmost coefficients appear in the n-point and 3-point RHPs as Gk = Ψ (ak ), G1 =Ψ[k] (ak ). It becomes straightforward to verify that as z → ak , one hashi−1[k]∂z Φ · Φ−1 − ∂z Φ[k] · Φ[k] = Ck (ak − z)Θk gk,1 − g1,1 + O (z − ak ) (ak − z)−Θk Ck−1 ,Θk dakΘk−1da Φ · Φ = Ck (ak − z)−+ O (1) (ak − z)−Θk Ck−1 .z − akIn combination with (4.36), (4.38), this in turn implies thatda ln τ (a) =n−3X[k]Tr Θk gk,1 − g1,1 dak .k=186(4.40)4.2. Tau functions as Fredholm determinantsSubstituting local expansion (4.39a) into the Fuchsian system (4.14), we may recursively determine the coefficients gk,l . In particular, the first coefficient gk,1 satisfies!n−2XAlgk,1 + [Θk , gk,1 ] = G−1Gk ,(4.41)ka−akll=0,l6=kso thatn−3Xk=1Tr (Θk gk,1 ) dak =n−3 Xn−2XTr Ak Aldak = da ln τJMU .ak − alk=1 l=0,l6=k(4.42)The 3-point analog of the relation (4.41) is[k]hi[k][k][k] A[k] −1g1,1 + Θk , g1,1 = G1 0 G1 ,akwhich gives[k] 2[k] 2[k] 2 Tr A[k] A[k]Tr A∞ − A0 − A1¯k − ∆¯ k−1 − ∆k∆[k]01Tr Θk g1,1 ===.

(4.43)ak2akakCombining (4.40) with (4.42) and (4.43) finally yields the statement of the theorem.Corollary 4.12. Jimbo-Miwa-Ueno isomonodromic tau function τJMU (a) admits ablock Fredholm determinant representationτJMU (a) = Υ (a) · det (1 − K) ,(4.44)where the operator K is defined by (4.26). Its N × N subblocks (4.20) are expressed interms of solutions Ψ[k] of RHPs associated to 3-point Fuchsian systems with prescribedmonodromy.Example: 4-point tau functionIn order to illustrate the developments of the previous subsection, let us considerthe simplest nontrivial case of Fuchsian systems with n = 4 regular singular points.Three of them have already been fixed at a0 = 0, a2 = 1, a3 = ∞.

There remains asingle time variable a1 ≡ t. To be able to apply previous results, it is assumed that0 < t < 1.The monodromy data are given by 4 diagonal matrices Θ0,t,1,∞ of local monodromyexponents and connection matrices C0 , Ct,± , C1,± , C∞ satisfying the relations−1M0 ≡ C0 e2πiΘ0 C0−1 = Ct,− Ct,+,−1−1e2πiS = Ct,− e2πiΘt Ct,+= C1,− C1,+Observe that, in the hope to make the notation more intuitive, it has been slightlychanged as compared to the general case. The indices 0, 1, 2, 3 are replaced by0, t, 1, ∞.

Also, for n = 4 there is only one nontrivial matrix M0→k (namely, withk = 1). Therefore it becomes convenient to work from the very beginning in a874. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions-s( )-z-s( )-zt+8-18-z Q( ) C-z-Q-1( ) 0C0t01e -2pis-1M1t-z)-Q C-1t,-(-z-Q(1 ) C1,+8t-z)-Q C-1t,e -2pisM0(-1t-z-Q1-1(1 ) C1,-Figure 4.10: Contour Γ̂ and jump matrices Jˆ for the 4-punctured spheredistinguished basis where M0→1 is given by a diagonal matrix e2πiS with Tr S =Tr (Θ0 + Θt ) = − Tr (Θ1 + Θ∞ ). In terms of the previous notation, this correspondsto setting S1 = S and S1 = 1.

The eigenvalues of S will be denoted by σ1 , . . . , σN .Recall (cf Assumption 4.3) that S is chosen so that these eigenvalues satisfy|< (σα − σβ )| ≤ 1,σα − σβ 6= ±1.(4.45)The 4-punctured sphere is decomposed into two pairs of pants T [L] , T [R] by oneannulus A as shown in Fig. 4.10. The space H is a sum[L]H = H+ ⊕ H− ,[R]H± = Hout,± ⊕ Hin,∓ .(4.46)Both subspaces H± may thus be identified with the space HC := CN ⊗L2 (C) of vectorvalued square integrable functions on a circle C centered at the origin and belongingto the annulus A. It will be very convenient for us to represent the elements of HCby their Laurent series inside A,X1f (z) =f p z − 2 +p ,f p ∈ CN .(4.47)p∈Z0In particular, the first and second component of H+ in (4.46) consist of functionswith vanishing negative and positive Fourier coefficients, respectively, i.e.

they maybe identified with Π+ HC and Π− HC . At this point the use of half-integer indices p ∈ Z0for Fourier modes may seem redundant, but its convenience will quickly become clear.When n = 4, the representation (4.44) reduces to12 −Θ2 −Θ20aTrS()t det (1 − U ) ,0τJMU (t) = t 2U=∈ End (HC ) ,(4.48)d 0884.2. Tau functions as Fredholm determinants-s( )-z-1e-1( ) C0-z0-z0e-z -Q-11M-1( ) C1-1M0Q-2pis1-z-Q18-z-Q08-s( )-z-Q(1 ) C1,+-2pis8-z -Q(1 ) t Ct,+-1(1 ) C1,--1(1 ) t Ct,-Figure 4.11: Contours and jump matrices for Ψ̃[L] (left) and Ψ[R] (right)where the operators a ≡ a[R] ≡ a[2] : Π− HC → Π+ HC and d ≡ d[L] ≡ d[1] : Π+ HC →Π− HC are given by1(ag) (z) =2πi(dg) (z) =12πi˛−100Ψ[R] (z) Ψ[R] (z 0 )a (z, z ) =z − z000a (z, z ) g (z ) dz ,C−1,(4.49a)˛1 − Ψ[L] (z) Ψ[L] (z 0 )−1d (z, z 0 ) g (z 0 ) dz 0 ,d (z, z 0 ) =Cz − z0.(4.49b)The contour C is oriented counterclockwise, which is the origin of sign difference in theexpression for d as compared to (4.20d).

In the Fourier basis (4.47), the operators aand d are given by semi-infinite matrices whose N ×N blocks a−qp , d−pq are detereminedbyXX1111a (z, z 0 ) =a−qp z − 2 +p z 0− 2 +q ,d (z, z 0 ) =d−pq z − 2 −p z 0− 2 −q .(4.50)p,q∈Z0+p,q∈Z0+It should be emphasized that the indices of a−qp and d−pq belong to different ranges,since in both cases p, q are positive half-integers.The matrix functions Ψ[L] (z), Ψ[R] (z) appearing in the integral kernels of a andd solve the 3-point RHPs associated to Fuchsian systems with regular singularities at0, t, ∞ and 0, 1, ∞, respectively. In order to understand the dependence of the 4-pointtau function on the time variable t, let us rescale the fundamental solution of the firstsystem by settingz .(4.51)Φ[L] (z) = Φ̃[L]tThe rescaled matrix Φ̃[L] (z) solves a Fuchsian system characterized by the same monodromy as Φ[L] (z) but the corresponding singular points are located at 0, 1, ∞. Denoteby Ψ̃[L] (z) the solution of the RHP associated to Φ̃[L] (z).

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