Isomonodromic Tau Functions Research Articles

Expansions for semiclassical conformal blocks

We propose a relation the expansions of regular and irregular semiclassical conformal blocks at different branch points making use of the connection between the accessory parameters of the BPZ decoupling equations to the logarithm derivative of isomonodromic tau functions. We give support for these relations by considering two eigenvalue problems for the confluent Heun equations obtained from the linearized perturbation theory of black holes. We first derive the large frequency expansion of the spheroidal equations, and then compare numerically the excited quasi-normal mode spectrum for the Schwarzschild case obtained from the large frequency expansion to the one obtained from the low frequency expansion and with the literature, indicating that the relations hold generically in the complex modulus plane.

Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes

We study linear scalar perturbations of slowly accelerating Kerr-Newman-anti-de Sitter black holes using the method of isomonodromic deformations. The conformally coupled Klein-Gordon equation separates into two second-order ordinary differential equations with five singularities. Nevertheless, the angular equation can be transformed into a Heun equation, for which we provide an asymptotic expansion for the angular eigenvalues in the small acceleration and rotation limit. In the radial case, we recast the boundary value problem in terms of a set of initial conditions for the isomonodromic tau function of Fuchsian systems with five regular singular points. For the sake of illustration, we compute the quasi-normal modes frequencies.

Monodromy dependence and symplectic geometry of isomonodromic tau functions on the torus

We compute the monodromy dependence of the isomonodromic tau function on a torus with n Fuchsian singularities and SL(N) residue matrices by using its explicit Fredholm determinant representation. We show that the exterior logarithmic derivative of the tau function defines a closed one-form on the space of monodromies and times, and identify it with the generating function of the monodromy symplectomorphism. As an illustrative example, we discuss the simplest case of the one-punctured torus in detail. Finally, we show that previous results obtained in the genus zero case can be recovered in a straightforward manner using the techniques presented here.

Isomonodromic Tau Functions on a Torus as Fredholm Determinants, and Charged Partitions

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 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.

From Quantum Curves to Topological String Partition 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 of generalised theta series type from which one can extract the topological string partition functions for each chamber.

Painlevé/CFT correspondence on a torus

This Review details the relationship between the isomonodromic tau-function and conformal blocks on a torus with one simple pole. It is based on the author’s talk at ICMP 2021.

Quantum Spectral Problems and Isomonodromic Deformations

We develop a self-consistent approach to study the spectral properties of a class of quantum mechanical operators by using the knowledge about monodromies of 2times 2 linear systems (Riemann–Hilbert correspondence). Our technique applies to a variety of problems, though in this paper we only analyse in detail two examples. First we review the case of the (modified) Mathieu operator, which corresponds to a certain linear system on the sphere and makes contact with the Painlevé mathrm {III}_3 equation. Then we extend the analysis to the 2-particle elliptic Calogero–Moser operator, which corresponds to a linear system on the torus. By using the Kyiv formula for the isomonodromic tau functions, we obtain the spectrum of such operators in terms of self-dual Nekrasov functions (epsilon _1+epsilon _2=0). Through blowup relations, we also find Nekrasov–Shatashvili type of quantizations (epsilon _2=0). In the case of the torus with one regular singularity we obtain certain results which are interesting by themselves. Namely, we derive blowup equations (filling some gaps in the literature) and we relate them to the bilinear form of the isomonodromic deformation equations. In addition, we extract the epsilon _2rightarrow 0 limit of the blowup relations from the regularized action functional and CFT arguments.

Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

The prevertices of the conformal map between a generic, n-vertex, simply connected, polycircular arc domain and the upper half plane are determined by finding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of five vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko and Lisovyy (2018 Commun. Math. Phys. 363 1–58) is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.

Tau-functions à la Dubédat and probabilities of cylindrical events for double-dimers and CLE(4)

Building upon recent results of Dubédat [7] on the convergence of topological correlators in the double-dimer model considered on Temperleyan approximations \Omega^\delta to a simply connected domain \Omega\subset\mathbb C we prove the convergence of probabilities of cylindrical events for the double-dimer loop ensembles on \Omega^\delta as \delta\to 0 . More precisely, let \lambda_1,\dots,\lambda_n\in\Omega and L be a macroscopic lamination on \Omega\setminus\{\lambda_1,\dots,\lambda_n\} , i.e., a collection of disjoint simple loops surrounding at least two punctures considered up to homotopies. We show that the probabilities P_L^\delta that one obtains L after withdrawing all loops surrounding no more than one puncture from a double-dimer loop ensemble on \Omega^\delta converge to a conformally invariant limit P_L as \delta \to 0 , for each L . Though our primary motivation comes from 2D statistical mechanics and probability, the proofs are of a purely analytic nature. The key techniques are the analysis of entire functions on the representation variety Hom (\pi_1(\Omega\setminus\{\lambda_1,\dots,\lambda_n\})\to\mathrm{SL}_2(\mathbb C) and on its (non-smooth) subvariety of locally unipotent representations. In particular, we do not use any RSW-type arguments for double-dimers. The limits P_L of the probabilities P_L^\delta are defined as coefficients of the isomonodromic tau-function studied in [7] with respect to the Fock–Goncharov lamination basis on the representation variety. The fact that P_L coincides with the probability of obtaining L from a sample of the nested CLE(4) in \Omega requires a small additional input, namely a mild crossing estimate for this nested conformal loop ensemble.

Matrix models for stationary Gromov–Witten invariants of the Riemann sphere

Inspired by recent formulæ of Dubrovin, Yang, and Zagier, we interpret the tau function enumerating stationary Gromov–Witten invariants of 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.

Correction to: The Dependence on the Monodromy Data of the Isomonodromic Tau Function

The note corrects the aforementioned paper (Bertola in Commun Math Phys 294(2):539–579, 2010). The consequences of the correction are traced and the examples updated.

Riemann-Hilbert correspondence and blown up surface defects

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with Nf = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.

Schwarz–Christoffel accessory parameter for quadrilaterals via isomonodromy

We develop the recent proposal by the authors to exploit the isomonodromic tau function defined by Jimbo, Miwa and Ueno (JMU) to solve the accessory parameter problem in conformal mapping theory. We focus here on mappings of Schwarz–Christoffel type: in particular, the mapping from the upper half plane to a four-sided polygon where the sides are all straight lines. We show that one can obtain the relevant accessory parameters—the pre-image of the polygonal vertices—via a special ‘zero curvature limit’ in which the radius of curvature of some of the edges tends to zero. We apply the procedure to rectangular domains where the JMU tau function is given by a ratio of Riemann theta functions, known as the Picard solution, and take the zero curvature limit to recover the accessory parameter obtained by Nehari using quite different methods. We then turn to trapezoids, deriving new asymptotic formulas for the accessory parameters in the limit of large and small aspect ratios. Our work lends a new geometrical perspective to problems of isomonodromy that we believe provides theoretical insight, while also showing how classical problems in conformal mapping can benefit from new ideas emerging from isomonodromic deformation theory.

Higher-rank isomonodromic deformations and W-algebras

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 properties of semi-degenerate vertex operators and conformal blocks for $c=N-1$.

On the analysis of incomplete spectra in random matrix theory through an extension of the Jimbo–Miwa–Ueno differential

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 [45]. Recent advances in the theory of tau functions [41], 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 the exact computation of all relevant constant factors. We use this method to efficiently compute the tail asymptotics of soft-edge, hard-edge and bulk scaled distribution and gap functions in the complex Wishart ensemble, provided each eigenvalue particle has been removed independently with probability 1−γ∈(0,1].

The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers

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 String and Dilaton equations via a purely Riemann-Hilbert approach. The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formul\ae\ for the correlators of this matrix model in terms of the solution of the Riemann-Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.

Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions

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 have hypergeometric representations, the kernel becomes completely explicit and has Cauchy type. In this case Fredholm determinant expansion yields multivariate series representation for the tau function of the Garnier system, obtained earlier via its identification with Fourier transform of Liouville conformal block (or a dual Nekrasov-Okounkov partition function). Further specialization to $n=4$ gives a series representation of the general solution to Painlev\'e VI equation.

Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI.

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.

On Some Hamiltonian Properties of the Isomonodromic Tau Functions

We discuss some new aspects of the theory of the Jimbo–Miwa–Ueno tau function which have come to light within the recent developments in the global asymptotic analysis of the tau functions related to the Painlevé equations. Specifically, we show that up to the total differentials the logarithmic derivatives of the Painlevé tau functions coincide with the corresponding classical action differential. This fact simplifies considerably the evaluation of the constant factors in the asymptotics of tau functions, which has been a long-standing problem of the asymptotic theory of Painlevé equations. Furthermore, we believe that this observation is yet another manifestation of L. D. Faddeev’s emphasis of the key role which the Hamiltonian aspects play in the theory of integrable system.

Monodromy dependence and connection formulae for isomonodromic tau functions

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, in particular, calculate the connection constant for generic Painlev\'e VI tau function. The result proves the conjectural formula for this constant proposed in \cite{ILT13}. We also apply the method to non-Fuchsian systems and evaluate constant factors in the asymptotics of Painlev\'e II tau function.