Диссертация (1137342), страница 15
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. , an−3 . In the case N = 2, Schlesinger equations reduce to the Garnier systemGn−3 , see for example [IKSY, Chapter 3] for the details. Setting further n = 4, we areleft with only one time t ≡ a1 and the latter system becomes equivalent to a nonlinear2nd order ODE — the Painlevé VI equation.The main object of our interest is the isomonodromic tau function of Jimbo-MiwaUeno [JMU].
It is defined as an exponentiated primitive of the 1-formn−2da ln τJMU1Xresz=ak Tr A2 (z) dak .:=2 k=0(4.3)The definition is consistent since the 1-form on the right is closed on solutions of thedeformation equations (4.2). It generates the hamiltonians of the Schlesinger system.Dealing with the Garnier system, we will assume the standard gauge where Tr A (z) =0 and denote the eigenvalues of Ak by ±θk with k = 0, . . . , n − 1.
In the Painlevé VIcase, it is convenient to modify this notation as (θ0 , θ1 , θ2 , θ3 ) 7→ (θ0 , θt , θ1 , θ∞ ). Thelogarithmic derivative ζ (t) := t (t − 1) dtd ln τVI (t) then satisfies the σ-form of PainlevéVI,22θ02tζ 0 − ζζ 0 + θ02 + θt2 + θ12 − θ∞2.tζ 0 − ζ2θt2(t − 1)ζ 0 − ζt(t−1)ζ 00 = −2 det 0222202ζ + θ0 + θt + θ1 − θ∞ (t − 1)ζ − ζ2θ1(4.4)664.1.
IntroductionMonodromy of the associated linear problems provides a complete set of conservedquantities for Painlevé VI, the Garnier system and Schlesinger equations. By thegeneral solution of deformation equations we mean the solution corresponding togeneric monodromy data. The precise genericity conditions will be specified in themain body of the text.In [Pal90], Palmer (developing earlier results of Malgrange [Mal]) interpreted theJimbo-Miwa-Ueno tau function (4.3) as a determinant of a singular Cauchy-Riemannoperator acting on functions with prescribed monodromy. The main idea of [Pal90]is to isolate the singular points a0 , . .
. , an−1 inside a circle C ⊂ P1 and represent theFuchsian system (4.1) by a boundary space of functions on C that can be analyticallycontinued inside with specified branching. The variation of positions of singularitiesgives rise to a trajectory of this space in an infinite Grassmannian. The tau functionis obtained by comparing two sections of an associated determinant bundle.The construction suggested in the present chapter is essentially a refinement ofPalmer’s approach, translated into the Riemann-Hilbert framework.
A single circle Cis replaced by the boundaries of n − 3 annuli which cut the n-punctured sphere P1 \ainto trinions (pairs of pants), see e.g. Fig. 4.2a below. To each trinion is assigned aFuchsian system with 3 regular singular points whose monodromy is determined bymonodromy of the original system. We show that the isomonodromic tau function isproportional to a Fredholm determinant:τJMU (a) = Υ (a) · det (1 − K) ,(4.5)where the prefactor Υ (a) is a known elementary function.
The integral operator Kacts on holomorphic vector functions on the union of annuli and involves projectionson certain boundary spaces.The pay-off of a more complicated Grassmannian model is that the kernel of Kmay be written explicitly in terms of 3-point solutions1 . In particular, for N = 2(i.e. for the Garnier system) the latter have hypergeometric expressions. The n = 4specialization of our result is as follows.Theorem 4.1. Let the independent variable t of Painlevé VI equation vary inside thereal interval ]0, 1[ and let C = {z ∈ C : |z| = R, t < R < 1} be a counter-clockwiseoriented circle. Let σ, η be a pair of complex parameters satisfying the conditionsθ0 ± θt + σ ∈/ Z,1|<σ| ≤ ,2θ0 ± θt − σ ∈/ Z,1σ 6= 0, ± ,2θ1 ± θ∞ + σ ∈/ Z,θ1 ± θ∞ − σ ∈/ Z.General solution of the Painlevé VI equation (4.4) admits the following Fredholm1We would like to note that somewhat similar refined construction emerged in the analysis ofmassive Dirac equation with U (1) branching on the Euclidean plane [Pal93].
Every branch pointwas isolated there in a separate strip, which ultimately allowed to derive an explicit Fredholmdeterminant representation for the tau function of appropriate Dirac operator [SMJ]. In physicalterms, the determinant corresponds to a resummed form factor expansion of a correlation functionof U (1) twist fields in the massive Dirac theory. The paper [Pal93] was an important source ofinspiration for the present work, although it took us more than 10 years to realize that the stripsshould be replaced by pairs of pants in the chiral problem.674.
Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsdeterminant representation:σ 2 −θ02 −θt2τVI (t) = const · t−2θt θ1(1 − t)det (1 − U ) ,U=0 ad 0,(4.6)g+where the operators a, d ∈ End (C ⊗ L (C)) act on g =with g± ∈ L2 (C) asg−˛˛11000(ag) (z) =a (z, z ) g (z ) dz ,(dg) (z) =d (z, z 0 ) g (z 0 ) dz 0 , (4.7)2πi C2πi C22and their kernels are explicitly given byK++ (z) K+− (z)K−− (z 0 ) −K+− (z 0 )0 2θ1−1(1 − z )K−+ (z) K−− (z)−K−+ (z 0 ) K++ (z 0 )0,a (z, z ) =z − z0 K̄++ (z) K̄+− (z)K̄−− (z 0 ) −K̄+− (z 0 )t 2θt1 − 1 − z0K̄−+ (z) K̄−− (z)−K̄−+ (z 0 ) K̄++ (z 0 )0,d (z, z ) =z − z0(4.8)withθ1 + θ∞ ± σ, θ1 − θ∞ ± σK±± (z) = 2 F1;z ,±2σ2− (θ1 ± σ)2θ∞1 + θ1 + θ∞ ± σ, 1 + θ1 − θ∞ ± σK±∓ (z) = ±z 2 F1;z ,2 ± 2σ2σ (1 ± 2σ)θt + θ0 ∓ σ, θt − θ0 ∓ σ tK̄±± (z) = 2 F1; ,∓2σz221 + θt + θ0 ∓ σ, 1 + θt − θ0 ∓ σ t∓2σ ∓iη θ0 − (θt ∓ σ) tK̄±∓ (z) = ∓ t e; .2 F12 ∓ 2σ2σ (1 ∓ 2σ) zz(4.9)Moreover, we demonstrate that for a special choice of monodromy in the PainlevéVI case, U becomes equivalent to the hypergeometric kernel of [BO05] and therebyreproduces previously known family of Fredholm determinant solutions [BD].
Thehypergeometric kernel is known to produce other random matrix integrable kernelsin confluent limits.Another part of our motivation comes from isomonodromy/CFT/gauge theorycorrespondence. It was conjectured in [GIL12] that the tau function associated tothe general Painlevé VI solution coincides with a Fourier transform of 4-point c = 1Virasoro conformal block with respect to its intermediate momentum.
Two independent derivations of this conjecture have been already proposed in [ILTe] and [BShch].The first approach [ILTe] also extends the initial statement to the Garnier system.Its main idea is to consider the operator-valued monodromy of conformal blocks withadditional level 2 degenerate insertions. At c = 1, Fourier transform of such conformal blocks reduces their “quantum” monodromy to ordinary 2 × 2 matrices.
It cantherefore be used to construct the fundamental matrix solution of a Fuchsian system684.1. Introductionwith prescribed SL (2, C) monodromy. The second approach [BShch] uses an embedding of two copies of the Virasoro algebra into super-Virasoro algebra extended byMajorana fermions to prove certain bilinear differential-difference relations for 4-pointconformal blocks, equivalent to Painlevé VI equation. An interesting feature of thismethod is that bilinear relations admit a deformation to generic values of Virasorocentral charge.Among other developments, let us mention the papers [GIL13, ILT14, Nag] whereasymptotic expansions of Painlevé V, IV and III tau functions were identified withFourier transforms of irregular conformal blocks of different types. The study ofrelations between isomonodromy problems in higher rank and conformal blocks ofWN algebras has been initiated in [Gav, GMtw, GMfer].The AGT conjecture [AGT] (proved in [AFLT]) identifies Virasoro conformalblocks with partition functions of N = 2 4D supersymmetric gauge theories.
Thereexist combinatorial representations of the latter objects [Nek], expressing them assums over tuples of Young diagrams. This fact is of crucial importance for isomonodromy theory, since it gives (contradicting to an established folklore) explicit seriesrepresentations for the Painlevé VI and Garnier tau functions. Since the very firstpaper [GIL12] on the subject, there has been a puzzle to understand combinatorialtau function expansions directly within the isomonodromic framework. There havealso been attempts to sum up these series to determinant expressions; for example, in[Bal] truncated infinite series for c = 1 conformal blocks were shown to coincide withpartition functions of certain discrete matrix models.In this work, we show that combinatorial series correspond to the principal minorexpansion of the Fredholm determinant (4.5), written in the Fourier basis of thespace of functions on annuli of the pants decomposition.
Fourier modes which labelthe choice of rows for the principal minor are related to Frobenius coordinates ofYoung diagrams. It should be emphasized that this combinatorial structure is validalso for N > 2 where CFT/gauge theory counterparts of the tau functions have yetto be defined and understood.We prove in particular the following result, originally conjectured in [GIL12] (thedetails of notation concerning Young diagrams are explained in the next subsection):Theorem 4.2.
General solution of the Painlevé VI equation (4.4) can be written asXinη 0~e B θ; σ + n; t ,(4.10)τVI (t) = const ·n∈Z~ σ; t) is a double sum over Young diagrams,where B(θ,X~ σ; t = N θ1 N θt tσ2 −θ02 −θt2 (1 − t)2θt θ1~Bθ,σt|λ|+|µ| ,B θ,λ,µθ∞ ,σ σ,θ0λ,µ∈YBλ,µ22(θ+σ+i−j)−θ1∞~ σ =θ,×220h(i,j)λ−i+µ−j+1+2σijλ(i,j)∈λ2Y (θt − σ + i − j)2 − θ02 (θ1 − σ + i − j)2 − θ∞×,2 (i, j) µ0 − i + λ − j + 1 − 2σ 2hiµj(i,j)∈µY (θt + σ + i − j)2 − θ02694. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsQNθθ32,θ1 ==±G (1 + θ3 + (θ1 + θ2 )) G (1 − θ3 + (θ1 − θ2 )).G(1 − 2θ1 )G(1 − 2θ2 )G(1 + 2θ3 )Here σ ∈/ Z/2, η 0 are two arbitrary complex parameters, and G (z) denotes the BarnesG-function.The parameters σ play exactly the same role in the Fredholm determinant (4.6)and the series representation (4.10), whereas η and η 0 are related by a simple transformation.
An obvious quasiperiodicity of the second representation with respect tointeger shifts of σ is by no means manifest in the Fredholm determinant.NotationThe monodromy matrices of Fuchsian systems and the jumps of associated RiemannHilbert problems appear on the left of solutions. These somewhat unusual conventionsare adopted to avoid even more confusing right action of integral and infinite matrixoperators. The indices corresponding to the matrix structure of rank N RiemannHilbert problem are referred to as color indices and are denoted by Greek letters, suchas α, β ∈ {1, .
. . , N }. Upper indices in square brackets, e.g. [k] in T [k] , label differenttrinions in the pants decomposition of a punctured Riemann sphere. We denote byZ0 := Z + 21 the half-integer lattice, and by Z0± = {p ∈ Z0 | p ≷ 0} its positive andnegative parts. The elements of Z0 , Z0± will be generally denoted by the letters p andq.jil'3= 3|l|= 17l2= 5Figure 4.1:{6, 5, 4, 2}.=(2,3)al( )=2ll( )=1hl( )=4Young diagram associated to the partition λ =The set of all partitions identified with Young diagrams is denoted by Y.
Forλ ∈ Y, we write λ0 for the transposed diagram, λi and λ0j for the number of boxesin the ith row and jth column of λ, and |λ| for the total number of boxes in λ. Let = (i, j) be the box in the ith row and jth column of λ ∈ Y (see Fig. 4.1). Itsarm-length aλ () and leg-length lλ () denote the number of boxes on the right andbelow. This definition is extended to the case where the box lies outside λ by theformulae aλ () = λi − j and lλ () = λ0j − i. The hook length of the box ∈ λ isdefined as hλ () = aλ () + lλ () + 1.Outline of the chapterThe chapter is organized as follows.
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