Диссертация (Isomonodromic deformations and quantum field theory), страница 14

. ., where all other factors do not depend at all on z, so that allmonodromies comes from a single kernel K.Now it is easy to proveTheorem 3.5. Solution of the linear problem with n marked points is given by(z − w)Kαβ (z, w) withhθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )ψ̃αθ0 (z)ψβθ0 (w)|θ 0 iKαβ (z, w) =hθ ∞ |Vθn−2 (tn−2 ) .

. . Vθ1 (t1 )|θ 0 i(3.126)whereas its isomonodromic tau-function is defined byτ (t1 , . . . , tn−2 ) = hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 i(3.127)Proof: First, insert resolutions of unity between each two (radially-ordered) vertexoperators, e.g.Xτ · Kαβ (z, w) =hθ ∞ |Vθn−2 (tn−2 )|Y n−3 , mn−3 , σ n−3 ihY n−3 , mn−3 , σ n−3 |×{Y 1 ,mi }× . . . × hY 2 , m2 , σ 2 |Vθ2 (t2 )|Y 1 , m1 , σ 1 ihY 1 , m1 , σ 1 |Vθ1 (t1 )ψ̃αθ0 (z)ψβθ0 (w)|θ 0 i(3.128)for 0 < |z|, |w| < |t1 | and similarly in the other regions. Due to Lemma 3.4 themonodromies of the fermionic fields do not depend on the intermediate states, but onlyon the vertex operators and the set of charges σ’s 3 , therefore it is enough to reduce theproblem of computation of all monodromies to the collection of corresponding threepoint problems with different vertex operators Vθj (tj ) inserted. So we have proven that3In addition to (n−3) time parameters ({t1 , .

. . , tn } modulo Möbius transformation, which alwaysallow to fix three of them to 0, 1, ∞) and n sets of W-charges {θ j } the isomonodromic tau-functiondepends upon the charges {σ k } ∈ (R/Z)N −1 , k = 1, . . . , n − 3 in the intermediate channels andtheir duals {β k }, which we had already discussed in the context of ambiguity in normalization ofthe vertex operators and their matrix elements.603.5. Isomonodromic tau-functions and Fredholm determinants(z − w)Kαβ (z, w) = [Φ(z)Φ−1 (w)]αβ (to cancel extra singularity in (3.126)), actuallygives a solution to the multi-point Riemann-Hilbert problem.In order to prove (3.127) considerXttNψ̃α (z + )ψα (z − ) =+ J(z) + tU2 (z) + . .

.(3.129)22tαso thatttttt TrK(z + , z − ) = Tr Φ(z + )Φ(z − )−1 =2222hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )J(z)|θ 0 ihθ|V(t) . . . Vθ1 (t1 )U2 (z)|θ 0 i∞θn−2n−2=N +t+ t2+ ...hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 ihθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 i(3.130)where from (3.72) and the conformal Ward identitieshθ ∞ |Vθn−2 (tn−2 ) . .

. Vθ1 (t1 )U2 (z)|θ 0 i=hθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 i=nXi=1!∂i+loghθ ∞ |Vθn−2 (tn−2 ) . . . Vθ1 (t1 )|θ 0 iz − ti z − ti1 2θ2 iwhere we have extended this formula to include t1 = 0Pand tn = ∞.Aiwe getNow solving the linear system (3.88) with A(z) = i z−titt −1tt22Φ(z + )Φ(z − ) = Φ(z) 1 + A(z) + (∂A(z) + A(z) ) + . . .

×22282tt2× 1 + A(z) + (−∂A(z) + A(z) ) + . . . Φ(z)−1 =28t22= Φ(z) 1 + tA(z) + A(z) + . . . Φ(z)−12(3.131)(3.132)Therefore, due to the definition of the tau-function!∂i+log τ (t1 , . . . , tn ) + . . .(z − ti )2 z − ti(3.133)Comparing this formula with (3.131) completes the proof.ntt2 XtTr Φ(z + )Φ(z − )−1 =222 i=11 2θ2 iFredholm determinantConsider now the isomonodromic tau-function τ (t) = hθ ∞ |Vν 1 (1)Vν t (t)|θ 0 i, corresponding to the problem on sphere with four marked points at z = 0, t, 1, ∞.

Insertingthe resolution of unity one can writeXτ (t) = hθ ∞ |Vν 1 (1)Vν t (t)|θ 0 i =hθ ∞ |Vν 1 (1)|Y, m; σihY, m; σ|Vν t (t)|θ 0 i =Y,m=Xhθ ∞ |Vν 1 (1)|{pα,i }, {qα,i }; σih{qα,i }, {pα,i }; σ|Vν t (t)|θ 0 i{{pα,i },{qα,i }}(3.134)613. Free fermions, W-algebras and isomonodromic deformationsHere we have used first just a particular case of the expansion (3.128), applying itto the simplest nontrivial isomonodromic tau-function. However,now it is useful toLσσnotice, that summation over the basis in total space H =Hm can be performedm∈ZNin Frobenius coordinates just forgetting restriction #pα = #qα for the states (3.56)σ, hence there is no restriction in summation range in the r.h.s.

of (3.134).in HmNow, one can still apply formulas (3.91), (3.92) for the matrix elements in (3.134).It giveshθ ∞ |Vν 1 (1)|{pα,i }, {qα,i }; σi = det KxI yJh{pα,i }, {qα,i }; σ|Vν t (t)|θ 0 i = det K̃xI yJ (t)K̃pαβ(t)α ,qβpα +qβ −σα +σβ=t(3.135)K̃pαβα ,qβwhere we have used again the multi-indices ∪α {(α, pα,i )} = {xI } and ∪α {(α, qα,i )} ={yJ }. It means, that the tau-function (3.134) can be summed up into a single Fredholm determinantXτ (t) =hθ ∞ |Vν 1 (1)|{pα,i }, {qα,i }; σih{qα,i }, {pα,i }; σ|Vν t (t)|θ 0 i ={{pα,i },{qα,i }}=Xdet Kx,y · det K̃y,x (t) =n=0{x},{y}=∞X∞XXdet Kx,y · det K̃y,x (t) =|{x}|=n|{y}|=n(3.136)Tr ∧n (K K̃(t)) = det(1 + K K̃(t)) = det (1 + Rt )n=0where basically only the Wick theorem has been used. One can also present the kernelof this operator Rt = K K̃(t) explicitly by the formula˛R(x, z) =(φ(x)φ(y)−1 − xσ y −σ )(S −1 φ̃(y/t)φ̃(z/t)−1 S − y σ z −σ )dyt−1 (x − y)(y − z)(3.137)|y|=r(where S is the diagonal matrix introduced before), so that this integral operatoracts from the space of vector-valued functions f (z) = (f1 (z), .

. . , fN (z)) on the circle|z| = r, t < r < 1. These functions have the fractional Laurent expansionXfα (z) = z σαfα,n z n(3.138)n∈Zotherwise their convolution with our kernel will be ill-defined.The representation in terms of the Fredholm determinant definitely requires further careful investigation, and it could appear to be useful for practical computationswith isomonodromic tau-functions, which basically have no explicit representations.ConclusionWe have considered in this chapter the free fermion formalism, which allows to studyrepresentations of the W-algebras at least at integer values of the central charges. The623.6.

Conclusionvertex operators are defined by their two-fermion matrix elements, which are fixedby monodromies of auxiliary linear system, and can be obtained from solution of thecorresponding Riemann-Hilbert problem.This chapter is just the first step of studying this relation (apart of the well-knownand effectively used for different applications Abelian case). A natural developmentof the above ideas is only outlined in sect. 3.5.

We are going to return elsewhere tothe problem of rewriting the isomonodromic tau-functions in terms of the Fredholmdeterminants, which can be quite useful representations (though still not an explicitform) for these complicated objects. Another point, which has to be understoodbetter is the relation of class of the isomonodromic solutions to the Toda lattices,which have been defined above using the generalized Hirota bilinear relations, to theclass of solutions, obeying the Virasoro-W constraints.634Fredholm determinant and Nekrasov sumrepresentations of isomonodromic taufunctionsAbstractWe derive Fredholm determinant representation for isomonodromic tau functions ofFuchsian systems with n regular singular points on the Riemann sphere and genericmonodromy in GL (N, C).

The corresponding operator acts in the direct sum ofN (n − 3) copies of L2 (S 1 ). Its kernel has a block integrable form and is expressed interms of fundamental solutions of n − 2 elementary 3-point Fuchsian systems whosemonodromy is determined by monodromy of the relevant n-point system via a decomposition of the punctured sphere into pairs of pants. For N = 2 these buildingblocks have hypergeometric representations, the kernel becomes completely explicitand has Cauchy type.

In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earliervia its identification with Fourier transform of Liouville conformal block (or a dualNekrasov-Okounkov partition function). Further specialization to n = 4 gives a seriesrepresentation of the general solution to Painlevé VI equation.IntroductionMotivation and some resultsThe theory of monodromy preserving deformations plays a prominent role in many areas of modern nonlinear mathematical physics. The classical works [WMTB, JMMS,TW1] relate, for instance, various correlation and distribution functions of statisticalmechanics and random matrix theory models to special solutions of Painlevé equations.

The relevant Painlevé functions are usually written in terms of Fredholm orToeplitz determinants. Further study of these relations has culminated in the development by Tracy and Widom [TW2] of an algorithmic procedure of derivation ofsystems of PDEs satisfied by Fredholm determinants with integrable kernels [IIKS]restricted to a union of intervals; the isomonodromic origin of Tracy-Widom equations has been elucidated in [Pal94] and further studied in [HI].

This raises a natural654. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsquestion:? Can the general solution of isomonodromy equations be expressed in terms of aFredholm determinant?One of the goals of the present chapter is to provide a constructive answer to thisquestion in the Fuchsian setting. Let us consider a Fuchsian system with n regularsingular points a := {a0 , . .

. , an−2 , an−1 ≡ ∞} on P1 ≡ P1 (C):∂z Φ = ΦA (z) ,A (z) =n−2Xk=0Ak,z − ak(4.1)where A0 , . . . , An−2 are N × N matrices independent of z and Φ (z) is a fundamentalmatrix solution, multivalued on P1 \a. The monodromy of Φ (z) realizes a represen1tation of the fundamental groupPn−2π1 (P \a) in GL (N, C).

When the residue matricesA0 , . . . , An−2 and An−1 := − k=0 Ak are non-resonant, the isomonodromy equationsare given by the Schlesinger system,[Ai , Ak ]i 6= k,∂ai Ak = a − a ,kiX [Ai , Ak ](4.2)A=∂. ai iai − akk6=iIntegrating the flows associated to affine transformations, we may set without loss ofgenerality a0 = 0 and an−2 = 1, so that there remains n − 3 nontrivial time variablesa1 , . .