We prove that the isomonodromic tau function on a torus with Fuchsian singularities and generic monodromies in $GL(N,\mathbb{C})$ can be written in terms of a Fredholm determinant of Cauchy-Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a series labeled by charged partitions. As an example, we show that in the case of $SL(2,\mathbb{C})$ this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function, or equivalently of a free fermion conformal block on the torus. Based on these results, we also propose a definition of the tau function of the Riemann-Hilbert problem on a torus with generic jump on the A-cycle.
We extend the approach to $τ$-functions as Widom constants developed by Cafasso, Gavrylenko and Lisovyy to orthogonal loop group Drinfeld-Sokolov hierarchies and isomonodromic deformations systems. The combinatorial expansion of the $τ$-function as a sum of correlators, each expressed as products of finite determinants, follows from using multicomponent fermionic vacuum expectation values of certain dressing operators encoding the initial conditions and the dependence on the flow (or deformation) parameters. When reduced to the orthogonal case, these correlators become finite Pfaffians and the…
We derive Fredholm determinant representation for isomonodromic tau functions of Fuchsian systems with $n$ regular singular points on the Riemann sphere and generic monodromy in $\mathrm{GL}(N,\mathbb C)$. The corresponding operator acts in the direct sum of $N(n-3)$ copies of $L^2(S^1)$. Its kernel has a block integrable form and is expressed in terms of fundamental solutions of $n-2$ elementary 3-point Fuchsian systems whose monodromy 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 building blocks ha…
For one-matrix models with polynomial potentials, the explicit relationship between the partition function and the isomonodromic tau function for the 2x2 polynomial differential systems satisfied by the associated orthogonal polynomials is derived.
It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $τ$-functions.
The Brezin-Gross-Witten tau function is a tau function of the KdV hierarchy which arises in the weak coupling phase of the Brezin-Gross-Witten model. It falls within the family of generalized Kontsevich matrix integrals, and its algebro--geometric interpretation has been unveiled in recent works of Norbury. We prove that a suitably generalized Brezin-Gross-Witten tau function is the isomonodromic tau function of a $2\times 2$ isomonodromic system and consequently present a study of this tau function purely by means of this isomonodromic interpretation. Within this approach we derive effective …
The Fredholm determinants of a special class of integral operators K supported on the union of m curve segments in the complex plane are shown to be the tau-functions of an isomonodromic family of meromorphic covariant derivative operators D_l. These have regular singular points at the 2m endpoints of the curve segments and a singular point of Poincare index 1 at infinity. The rank r of the vector bundle over the Riemann sphere on which they act equals the number of distinct terms in the exponential sums entering in the numerator of the integral kernels. The deformation equations may be viewed…
The isomonodromic tau-function for the Hurwitz spaces of branched coverings of genus zero and one are constructed explicitly. Such spaces may be equipped with the structure of a Frobenius manifold and this introduces a flat coordinate system on the manifold. The isomonodromic tau-function, and in particular the associated $G$-function, are rewritten in these coordinates and an interpretation in terms of the caustics (where the multiplication is not semisimple) is given.
We identify the Kontsevich-Penner matrix integral, for finite size $n$, with the isomonodromic tau function of a $3\times 3$ rational connection on the Riemann sphere with $n$ Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann-Hilbert boundary value problem, we can pass to the limit $n\to\infty$ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of the KP hierarchy. This allows to derive the…
The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarith…
The goal of this note is to show that the Riemann-Hilbert problem to find multivalued analytic functions with $SL(2,\mathbb{C})$-valued monodromy on Riemann surfaces of genus zero with $n$ punctures can be solved by taking suitable linear combinations of the conformal blocks of Liouville theory at $c=1$. This implies a similar representation for the isomonodromic tau-function. In the case $n=4$ we thereby get a proof of the relation between tau-functions and conformal blocks discovered in \cite{GIL}. We briefly discuss a possible application of our results to the study of relations between cer…
In this work we find the isomonodromic (Jimbo-Miwa) tau-function corresponding to Frobenius manifold structures on Hurwitz spaces. We discuss several applications of this result. First, we get an explicit expression for the G-function (solution of Getzler's equation) of the Hurwitz Frobenius manifolds. Second, in terms of this tau-function we compute the genus one correction to the free energy of hermitian two-matrix model. Third, we find the Jimbo-Miwa tau-function of an arbitrary Riemann-Hilbert problem with quasi-permutation monodromy matrices. Finally, we get a new expression (analog of ge…
The Hamiltonian approach to the theory of dual isomonodromic deformations is developed within the framework of rational classical R-matrix structures on loop algebras. Particular solutions to the isomonodromic deformation equations appearing in the computation of correlation functions in integrable quantum field theory models are constructed through the Riemann-Hilbert problem method. The corresponding $τ$-functions are shown to be given by the Fredholm determinant of a special class of integral operators.
We discuss an extension of the Jimbo-Miwa-Ueno differential 1-form to a form closed on the full space of extended monodromy data of systems of linear ordinary differential equations with rational coefficients. This extension is based on the results of M. Bertola generalizing a previous construction by B. Malgrange. We show how this 1-form can be used to solve a long-standing problem of evaluation of the connection formulae for the isomonodromic tau functions which would include an explicit computation of the relevant constant factors. We explain how this scheme works for Fuchsian systems and, …
We construct the general solution of a class of Fuchsian systems of rank $N$ as well as the associated isomonodromic tau functions in terms of semi-degenerate conformal blocks of $W_N$-algebra with central charge $c=N-1$. The simplest example is given by the tau function of the Fuji-Suzuki-Tsuda system, expressed as a Fourier transform of the 4-point conformal block with respect to intermediate weight. Along the way, we generalize the result of Bowcock and Watts on the minimal set of matrix elements of vertex operators of the $W_N$-algebra for generic central charge and prove several propertie…
We consider the theory of multicomponent free massless fermions in two dimensions and use it for construction of representations of W-algebras at integer Virasoro central charges. We define the vertex operators in this theory in terms of solutions of the corresponding isomonodromy problem. We use this construction to get some new insights on tau-functions of the multicomponent Toda type hierarchies for the class of solutions, given by the isomonodromy vertex operators and get useful representation for the tau-function of isomonodromic deformations.
In this paper, we show that it is always possible to deform a differential equation $\partial_x Ψ(x) = L(x) Ψ(x)$ with $L(x) \in \mathfrak{sl}_2(\mathbb{C})(x)$ by introducing a small formal parameter $\hbar$ in such a way that it satisfies the Topological Type properties of Bergère, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce $\hbar$. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expr…
[Note: important Corrigendum now available at arXiv:1601.04790] The isomonodromic tau function defined by Jimbo-Miwa-Ueno vanishes on the Malgrange's divisor of generalized monodromy data for which a vector bundle is nontrivial, or, which is the same, a certain Riemann-Hilbert problem has no solution. In their original work, Jimbo, Miwa, Ueno did not derive the dependence on the (generalized) monodromy data (i.e. monodromy representation and Stokes' parameters). We fill the gap by providing a (simpler and more general) description in which all the parameters of the problem (monodromy-changing …
We construct a fundamental system of a $q$-difference Lax pair of rank $N$ in terms of 5d Nekrasov functions with $q=t$. Our fundamental system degenerates by the limit $q\to 1$ to a fundamental system of a differential Lax pair, which yields the Fuji-Suzuki-Tsuda system. We introduce tau functions of our system as Fourier transforms of 5d Nekrasov functions. Using asymptotic expansions of the fundamental system at $0$ and $\infty$, we obtain several determinantal identities of the tau functions.
Here we review some recent developments in the theory of isomonodromic deformations on Riemann sphere and elliptic curve. For both cases we show how to derive Schlesinger transformations together with their action on tau-function, and construct classes of solutions in terms of multi-dimensional theta-functions.
The authors show that a wide class of Fredholm determinants arising in the representation theory of "big" groups such as the infinite-dimensional unitary group, solve Painleve equations. Their methods are based on the theory of integrable operators and the theory of Riemann-Hilbert problems.
A Tau function for $q$-Painlevé VI as a Fredholm determinant
Mathematical Institute, Andrew Wiles Building, Woodstock Rd, Oxford OX2 6GG, United Kingdom（数学研究所）；School of Mathematics and Statistics, University of Sydney, Camperdown, NSW 2006, Australia（悉尼大学数学与统计学院）
Riemann-Hilbert problems, Fredholm determinants, explicit combinatorial expansions, and connection formulas for the general $q$-Painlevé III$_3$ tau functions
This paper describes the reconstruction of the topological string partition function for certain local Calabi-Yau (CY) manifolds from the quantum curve, an ordinary differential equation obtained by quantising their defining equations. Quantum curves are characterised as solutions to a Riemann-Hilbert problem. The isomonodromic tau-functions associated to these Riemann-Hilbert problems admit a family of natural normalisations labelled by the chambers in the extended Kähler moduli space of the local CY under consideration. The corresponding isomonodromic tau-functions admit a series expansion o…
We show that the Kontsevich integral on $n\times n$ matrices ($n< \infty$) is the isomonodromic tau function associated to a $2\times 2$ Riemann--Hilbert problem. The approach allows us to gain control of the analysis of the convergence as $n\to\infty$. By an appropriate choice of the external source matrix in Kontsevich's integral, we show that the limit produces the isomonodromic tau function of a special tronquée solution of the first Painlevé hierarchy, and we identify the solution in terms of the Stokes' data of the associated linear problem. We also show that there are several tau functi…
We prove that the topological recursion formalism can be used to compute the WKB expansion of solutions of second order differential operators obtained by quantization of any hyper-elliptic curve. We express this quantum curve in terms of spectral Darboux coordinates on the moduli space of meromorphic $\mathfrak{sl}_2$-connections on $\mathbb{P}^1$ and argue that the topological recursion produces a $2g$-parameter family of associated tau functions, where $2g$ is the dimension of the moduli space considered. We apply this procedure to the 6 Painlevé equations which correspond to $g=1$ and cons…
The prepotentials for the quiver supersymmetric gauge theories are defined as quasiclassical tau-functions, depending on two different sets of variables: the parameters of the UV gauge theory or the bare compexified couplings, and the vacuum condensates of the theory in IR. The bare couplings are introduced as periods on the UV base curve, and the consistency of corresponding gradient formulas for the tau-functions is proven using the Riemann bilinear relations. It is shown, that dependence of generalised prepotentials for the quiver gauge theories upon the bare couplings turns to coincide wit…
We define a tau function for a generic Riemann-Hilbert problem posed on a union of non-intersecting smooth closed curves with jump matrices analytic in their neighborhood. The tau function depends on parameters of the jumps and is expressed as the Fredholm determinant of an integral operator with block integrable kernel constructed in terms of elementary parametrices. Its logarithmic derivatives with respect to parameters are given by contour integrals involving these parametrices and the solution of the Riemann-Hilbert problem. In the case of one circle, the tau function coincides with Widom'…
Several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by Jimbo, Miwa and Ueno (JMU) in the early 1980s \cite{JMU}. Recent advances in the theory of tau functions \cite{ILP}, based on earlier works of B. Malgrange and M. Bertola, have allowed to extend the original Jimbo-Miwa-Ueno differential form to a 1-form closed on the full space of extended monodromy data of the underlying Lax pairs. This in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions, including…
Inspired by recent formulæ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov-Witten invariants of $\mathbb{P}^1$ as an isomonodromic tau function associated with a difference equation. As a byproduct we obtain an analogue of the Kontsevich matrix model for this tau function. A connection with the Charlier ensemble is also considered.
We review the role played by tau functions of special type - called {\it Bergman} tau functions in various areas: theory of isomonodromic deformations, solutions of Einstein's equations, theory of Dubrovin-Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau functions are natural generalizations of Dedekind's eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein's metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplac…
A classical way to introduce tau functions for integrable hierarchies of solitonic equations is by means of the Sato-Segal-Wilson infinite-dimensional Grassmannian. Every point in the Grassmannian is naturally related to a Riemann-Hilbert problem on the unit circle, for which Bertola proposed a tau function that generalizes the Jimbo-Miwa-Ueno tau function for isomonodromic deformation problems. In this paper, we prove that the Sato-Segal-Wilson tau function and the (generalized) Jimbo-Miwa-Ueno isomonodromy tau function coincide under a very general setting, by identifying each of them to the…
We consider the factorization problem of matrix symbols relative to a closed contour, i.e., a Riemann-Hilbert problem, where the symbol depends analytically on parameters. We show how to define a function $τ$ which is locally analytic on the space of deformations and that is expressed as a Fredholm determinant of an operator of "integrable" type in the sense of Its-Izergin-Korepin-Slavnov. The construction is not unique and the non-uniqueness highlights the fact that the tau function is really the section of a line bundle.
We derive a new Hamiltonian formulation of Schlesinger equations in terms of the dynamical $r$-matrix structure. The corresponding symplectic form is shown to be the pullback, under the monodromy map, of a natural symplectic form on the extended monodromy manifold. We show that Fock-Goncharov coordinates are log-canonical for the symplectic form on the extended monodromy manifold. Using these coordinates we define the symplectic potential on the monodromy manifold and interpret the isomonodromic tau-function as the generating function of the monodromy map. This, in particular, solves a recent …
The level spacing distributions which arise when one rescales the Laguerre or Jacobi ensembles of hermitian matrices is studied. These distributions are expressible in terms of a Fredholm determinant of an integral operator whose kernel is expressible in terms of Bessel functions of order $α$. We derive a system of partial differential equations associated with the logarithmic derivative of this Fredholm determinant when the underlying domain is a union of intervals. In the case of a single interval this Fredholm determinant is a Painleve tau function.
The goal of the present paper is to calculate the determinant of the Dirac operator with a mass in the cylindrical geometry. The domain of this operator consists of functions that realize a unitary one-dimensional representation of the fundamental group of the cylinder with $n$ marked points. The determinant represents a version of the isomonodromic $τ$-function, itroduced by M. Sato, T. Miwa and M. Jimbo. It is calculated by comparison of two sections of the $\mathrm{det}^*$-bundle over an infinite-dimensional grassmannian. The latter is composed of the spaces of boundary values of some local…
Iorgov, Lisovyy, and Teschner established a connection between isomonodromic deformation of linear differential equations and Liouville conformal field theory at $c=1$. In this paper we present a $q$ analog of their construction. We show that the general solution of the $q$-Painlevé VI equation is a ratio of four tau functions, each of which is given by a combinatorial series arising in the AGT correspondence. We also propose conjectural bilinear equations for the tau functions.
In this paper we study the extension of Painlevé/gauge theory correspondence to circular quivers by focusing on the special case of $SU(2)$ $\mathcal{N}=2^*$ theory. We show that the Nekrasov-Okounkov partition function of this gauge theory provides an explicit combinatorial expression and a Fredholm determinant formula for the tau-function describing isomonodromic deformations of $SL_2$ flat connections on the one-punctured torus. This is achieved by reformulating the Riemann-Hilbert problem associated to the latter in terms of chiral conformal blocks of a free-fermionic algebra. This viewpoi…
We derive Fredholm determinant and series representation of the tau function of the Fuji-Suzuki-Tsuda system and its multivariate extension, thereby generalizing to higher rank the results obtained for Painlevé VI and the Garnier system. A special case of our construction gives a higher rank analog of the continuous hypergeometric kernel of Borodin and Olshanski. We also initiate the study of algebraic braid group dynamics of semi-degenerate monodromy, and obtain as a byproduct a direct isomonodromic proof of the AGT-W relation for $c=N-1$.
We define a Chern-Simons invariant for a certain class of infinite volume hyperbolic 3-manifolds. We then prove an expression relating the Bergman tau function on a cover of the Hurwitz space, to the lifting of the function $F$ defined by Zograf on Teichmüller space, and another holomorphic function on the cover of the Hurwitz space which we introduce. If the point in cover of the Hurwitz space corresponds to a Riemann surface $X$, then this function is constructed from the renormalized volume and our Chern-Simons invariant for the bounding 3-manifold of $X$ given by Schottky uniformization, t…
The partition function for a canonical ensemble of 2D Coulomb charges in a background potential (the Dyson gas) is realized as a vacuum expectation value of a group-like element constructed in terms of free fermionic operators. This representation provides an explicit identification of the partition function with a tau-function of the 2D Toda lattice hierarchy. Its dispersionless (quasiclassical) limit yields the tau-function for analytic curves encoding the integrable structure of the inverse potential problem and parametric conformal maps. A similar fermionic realization of partition functio…
Dirac hamiltonian on the Poincare disk in the presence of an Aharonov-Bohm flux and a uniform magnetic field admits a one-parameter family of self-adjoint extensions. We determine the spectrum and calculate the resolvent for each element of this family. Explicit expressions for Green functions are then used to find Fredholm determinant representations for the tau function of the Dirac operator with two branch points on the Poincare disk. Isomonodromic deformation theory for the Dirac equation relates this tau function to a one-parameter class of solutions of the Painleve VI equation with $γ=0$…
We give a general method to compute the expansion of topological tau functions for Drinfeld-Sokolov hierarchies associated to an arbitrary untwisted affine Kac-Moody algebra. Our method consists of two main steps: first these tau functions are expressed as (formal) Fredholm determinants of the type appearing in the Borodin-Okounkov formula, then the kernels for these determinants are found using a reduced form of the string equation. A number of explicit examples are given.
We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equat…
The semisimple Frobenius manifolds related to the Hurwitz spaces $H_{g,N}(k_1, ..., k_l)$ are considered. We show that the corresponding isomonodromic tau-function $τ_I$ coincides with $(-1/2)$-power of the Bergmann tau-function which was introduced in a recent work by the authors \cite{KokKor}. This enables us to calculate explicitly the $G$-function of Frobenius manifolds related to the Hurwitz spaces $H_{0, N}(k_1, ..., k_l)$ and $H_{1, N}(k_1, ..., k_l)$. As simple consequences we get formulas for the $G$-functions of the Frobenius manifolds ${\mathbb C}^N/\tilde{W}^k(A_{N-1})$ and ${\math…
We provide a simple derivation of the constant factor in the short-distance asymptotics of the tau-function associated with the $2$-point function of the two-dimensional Ising model. This factor was first computed by C. Tracy in \cite{T} via an exponential series expansion of the correlation function. Further simplifications in the analysis are due to Tracy and Widom \cite{TW} using Fredholm determinant representations of the correlation function and Wiener-Hopf approximation results for the underlying resolvent operator. Our method relies on an action integral representation of the tau-functi…
We introduce effective form factors for one-dimensional lattice fermions with arbitrary phase shifts. We study tau functions defined as series of these form factors. On the one hand we perform the exact summation and present tau functions as Fredholm determinants in the thermodynamic limit. On the other hand simple expressions of form factors allow us to present the corresponding series as integrals of elementary functions. Using this approach we re-derive the asymptotics of static correlation functions of the XY quantum chain at finite temperature.
We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function $τ(t)$ associated to generic monodromy data. Using a relation of $τ(t)$ to two different types of irregular $c=1$ Virasoro conformal blocks and the confluence from Painlevé VI equation, connection formulas between the parameters of asymptotic expansions at $0$ and $i\infty$ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as $t\to 0,+\infty,i\infty$ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary centr…
We study a Fredholm determinant of the hypergeometric kernel arising in the representation theory of the infinite-dimensional unitary group. It is shown that this determinant coincides with the Palmer-Beatty-Tracy tau function of a Dirac operator on the hyperbolic disk. Solution of the connection problem for Painleve VI equation allows to determine its asymptotic behavior up to a constant factor, for which a conjectural expression is given in terms of Barnes functions. We also present analogous asymptotic results for the Whittaker and Macdonald kernel.
