Диссертация (1137342), страница 17
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It is not clear to us how to extract from them irregular (longdistance) asymptotic expansions. Let us mention a recent work [Nag] whichrelates such expansions to irregular conformal blocks of a different type.4. Given a matrix K ∈ CX×X indexed by elements of a discrete set X, it is almosta tautology to say that the principal minors det KY∈2X define a determinantalpoint process on X and a probability measure on 2X . Fredholm determinant representations and combinatorial expansions of tau functions thus generalize in anatural way various families of measures of random matrix or representationtheoretic origin, such as Z- and ZW -measures [BO05, BO01] (the former correspond to the scalar case N = 1 with n = 4 regular singular points, and thelatter are related to hypergeometric kernel considered in the last subsection).We believe that novel probabilistic models coming from isomonodromy deservefurther investigation.5.
Perhaps the most intriguing perspective is to extend our setup to q-isomonodromyproblems, in particular q-difference Painlevé equations, presumably related tothe deformed Virasoro algebra [SKAO] and 5D gauge theories. Among the results pointing in this direction, let us mention a study of the connection problemfor q-Painlevé VI [Ma] based on asymptotic factorization of the associated linearproblem into two systems solved by the Heine basic hypergeometric series 2 ϕ1 ,and critical expansions for solultions of q-P (A1 ) equation recently obtained in[JR].Tau functions as Fredholm determinantsRiemann-Hilbert setupThe classical setting of the Riemann-Hilbert problem (RHP) involves two basic ingredients:744.2.
Tau functions as Fredholm determinants• A contour Γ on a Riemann surface Σ of genus g consisting of a finite set of smooth oriented arcs that canintersect transversally. Orientation of the arcs definespositive and negative side Γ± of the contour in theusual way, see Fig. 4.5.• A jump matrix J : Γ → GL (N, C) that satisfies suitable smoothness requirements.+-Figure 4.5:entationlabeling ofof ΓOriandsidesThe RHP defined by the pair (Γ, J) consists in finding an analytic invertible matrixfunction Ψ : Σ\Γ → GL (N, C) whose boundary values Ψ± on Γ± are related by Ψ+ =JΨ− . Uniqueness of the solution is ensured by adding an appropriate normalizationcondition.In the present work we are mainly interested in the genus 0 case: Σ = P1 . Let usfix a collectiona := (a0 = 0, a1 , .
. . , an−3 , an−2 = 1, an−1 = ∞)of n distinct points on P1 satisfying the condition of radial ordering 0 < |a1 | < . . . <|an−3 | < 1. To reduce the amount of fuss below, it is convenient to assume thata1 , . . . , an−3 ∈ R>0 . The contour Γ will then be chosen as a collection[n−1 [n−2 Γ=γk ∪`kk=0k=0of counter-clockwise oriented circles γk of sufficiently small radii centered at ak , andthe segments `k ⊂ R joining the circles γk and γk+1 , see Fig.
4.6.g4g0l0a0g1a1g2l1a2g3l2Figure 4.6: Contour Γ for n = 5The jumps will be defined by the following data:75a3l34. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functions• An n-tuple of diagonal N ×N matrices Θk = diag {θk,1 P, . . . , θk,N } ∈ CN ×N (withk = 0, . . . , n − 1) satisfying Fuchs consistency relation n−1k=0 Tr Θk = 0 and having non-resonant spectra. The latter condition means that θk,α − θk,β ∈/ Z\ {0}.• A collection of 2n matrices Ck,± ∈ GL (N, C) subject to the constraints−1−1= Ck+1,− Ck+1,+,M0→k := Ck,− e2πiΘk Ck,+k = 0, .
. . , n − 3,−1−1,= Cn−1,− e−2πiΘn−1 Cn−1,+M0→n−2 := Cn−2,− e2πiΘn−2 Cn−2,+M0→n−1 := 1 =−1Cn−1,− Cn−1,+=(4.11)−1C0,− C0,+,which are simultaneously viewed as the definition of M0→k ∈ GL (N, C). Onlyn of the initial matrices (for example, Ck,+ ) are therefore independent.The jump matrix J that we are going to consider is then given by−1J (z) = M0→k,k = 0, . .
. , n − 2,`k−1J (z) = (ak − z)−Θk Ck,±,=z ≷ 0, k = 0, . . . , n − 2,γk−1J (z) = (−z)Θn−1 Cn−1,±,=z ≷ 0.(4.12)γn−1Throughout this chapter, complex powers will always be understood as z θ = eθ ln z ,the logarithm being defined on the principal branch. The subscripts ± of Ck,± aresometimes omitted to lighten the notation.A major incentive to study the above RHP comes from its direct connection tosystems of linear ODEs with rational coefficients.
Indeed, define a new matrix Φ byz outside γ0...n−1 ,Ψ (z) ,Θk(4.13)Φ (z) = Ck (ak − z) Ψ (z) ,z inside γk , k = 0, . . . , n − 2,−Θn−1Cn−1 (−z)Ψ (z) ,z inside γn−1 .−1It has only piecewise constant jumps JΦ (z) ]a ,a [ = M0→kon the positive real axis.k k+1The matrix A (z) := Φ−1 ∂z Φ is therefore meromorphic on P1 with poles only possibleat a0 , . . . , an−1 . It follows immediately that∂z Φ = ΦA (z) ,A (z) =n−2Xk=0Ak,z − ak(4.14)with Ak = Ψ (ak )−1 Θk Ψ (ak ). Thus Φ (z) is a fundamental matrix solution for a classof Fuchsian systems related by constant gauge transformations. It has prescribedmonodromy and singular behavior that are encoded in the connection matrices Ckand local monodromy exponents Θk . The freedom in the choice of the gauge reflectsthe dependence on the normalization of Ψ.The monodromy representation ρ : π1 (P1 \a) → GL (N, C) associated to Φ isuniquely determined by the jumps.
It is generated by the matrices Mk = ρ (ξk )764.2. Tau functions as Fredholm determinantsa0x0a1a2an-2x1x2xn-2xn-1Figure 4.7: Generators of π1 (P1 \a)assigned to counter-clockwise loops ξ0 , . . . , ξn−1 represented in Fig. 4.7. They may beexpressed asM0 = M0→0 ,Mk+1 = M0→k −1 M0→k+1 ,which means simply that M0→k = M0 . . .
Mk−1 Mk . It is a direct consequence of thedefinition (4.11) that the spectra of Mk coincide with those of e2πiΘk .Assumption 4.3. The matrices M0→k with k = 1, . . . , n − 3 are assumed to bediagonalizable:M0→k = Sk e2πiSk Sk−1 ,Sk = diag {σk,1 , . . . , σk,N } .PkIt can then be assumed without loss in generality that Tr Sk =j=0 Tr Θj and|< (σk,α − σk,β )| ≤ 1. We further impose a non-resonancy condition σk,α − σk,β 6= ±1.In order to have uniform notation, we may also identify S0 ≡ Θ0 , Sn−2 ≡ −Θn−1 .Note that any sufficiently generic monodromy representation can be realized as described above.Auxiliary 3-point RHPsConsider a decomposition of the original n-punctured sphere into n − 2 pairs of pantsT [1] , .
. . , T [n−2] by n − 3 annuli A1 , . . . , An−3 represented in Fig. 4.8. The labeling isdesigned so that two boundary components of the annulus Ak that belong to trinions[k][k+1]T [k] and T [k+1] are denoted by Cout and Cin . We are now going to associate tothe n-point RHP described above n − 2 simpler3-point RHPs assigned to different[k][k]trinions and defined by the pairs Γ , Jwith k = 1, .
. . , n − 2.[k][k]The curves Cin and Cout are represented by circles of positive and negative orientation as shown in Fig. 4.9. For k = 2, . . . , n − 3, the contour Γ[k] of the RHP[k][k]assigned to trinion T [k] consists of three circles Cin , Cout , γk associated to boundarycomponents, and two segments of the real axis. For leftmost and rightmost trinions[1][n−2]T [1] and T [n−2] , the role of Cin and Cout is played respectively by the circles γ0 andγn−1 around 0 and ∞.The jump matrix J [k] is constructed according to two basic rules:774. Fredholm determinant and Nekrasov sum representations of isomonodromic tau functionsgkg1T [1][1]CoutA1Ak-1[k]CinTgk+1[k][k][k+1]AkCoutTgn-2[k+1][k+1]CoutCin[n-2]An-3Ak+1CinT [n-2]g0gn-1Figure 4.8: Labeling of trinions, annuli and boundary curvesg0gkg1[k]Cout[k]Cinak[3]g4Cin[2]Cout[2]Cing2g3a2a3[1]Couta1Figure 4.9: Contour Γ[k] (left) and Γ̂ for n = 5 (right)• The arcs that belong to original contour give rise to the same jumps:[k]J − J Γ[k] ∩Γ = 0.[k][k+1]• The jumps on the boundary circles Cout , Cinmimic regular singularities characterized by counter-clockwise monodromy matrices M0→k :−Sk −1−S[k+1] [k] Jk = 1, .
. . , n − 3.J [k] = (−z)Sk , [k+1] = (−z) k Sk−1 ,CoutCin(4.15)[k]The solution Ψ[k] of the RHP defined by the pair Γ[k] , Jis thus related in a way[k]analogous to (4.13) to the fundamental matrix solution Φ of a Fuchsian system with3 regular singular points at 0, ak and ∞ characterized by monodromies M0→k−1 , Mk ,−1M0→k:[k][k]A0A1[k][k] [k][k]∂z Φ = Φ A (z) ,A (z) =+.(4.16)zz − ak[k][k][k][k][k]We note in passing that the spectra of A0 , A1 and A∞ := −A0 − A1 coincide withthe spectra of Sk−1 , Θk and −Sk . The non-resonancy constraint in Assumption 4.3784.2.
Tau functions as Fredholm determinantsensures the existence of solution Φ[k] with local behavior leading to the jumps (4.15)in Ψ[k] .It will be convenient to replace the n-point RHP describedin the previous subˆsection by a slightly modified one. It is defined by a pair Γ̂, J such that (cf rightpart of Fig. 4.9)n−2[[k]ˆΓ̂ =Γ ,J [k] = J [k] .(4.17)Γk=1Constructing the solution Ψ̂ of this RHP is equivalent to finding Ψ: it is plain that−Sz ∈ Ak ,(−z) k Sk−1 Ψ (z) ,n−3S(4.18)Ψ̂ (z) =Ψ (z) ,z ∈ P1 \Ak .k=1Our aim in the next subsections is to construct the isomonodromic tau function interms of 3-point solutions Φ[k] .
This construction employs in a crucialway integralSn−3Plemelj operators acting on spaces of holomorphic functions on A := k=1 Ak .Plemelj operatorsGiven a positively oriented circle C ⊂ C centered at the origin, let us denote by V (C)the space of functions holomorphic in an annulus containing C. Any f ∈ V (C) iscanonically decomposed as f = f+ + f− , where f+ and f− denote the analytic andprincipal part of f .
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