Monodromy Data Research Articles

Irregular conformal blocks and connection formulae for Painlevé V functions

We prove a Fredholm determinant and short-distance series representation of the Painlevé V tau function τt associated with generic monodromy data. Using a relation of τt 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∞ are conjectured. Explicit evaluations of the connection constants relating the tau function asymptotics as t → 0, +∞, i∞ are obtained. We also show that irregular conformal blocks of rank 1, for arbitrary central charge, are obtained as confluent limits of the regular conformal blocks.

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.

The uniformization of the moduli space of principally polarized abelian 6-folds

Abstract Starting from a beautiful idea of Kanev, we construct a uniformization of the moduli space 𝒜 6 \mathcal{A}_{6} of principally polarized abelian 6-folds in terms of curves and monodromy data. We show that the general principally polarized abelian variety of dimension 6 is a Prym–Tyurin variety corresponding to a degree 27 cover of the projective line having monodromy the Weyl group of the E 6 E_{6} lattice. Along the way, we establish numerous facts concerning the geometry of the Hurwitz space of such E 6 E_{6} -covers, including: (1) a proof that the canonical class of the Hurwitz space is big, (2) a concrete geometric description of the Hodge–Hurwitz eigenbundles with respect to the Kanev correspondence and (3) a description of the ramification divisor of the Prym–Tyurin map from the Hurwitz space to 𝒜 6 \mathcal{A}_{6} in the terms of syzygies of the Abel–Prym–Tyurin curve.

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.

Real-root property of the spectral polynomial of the Treibich–Verdier potential and related problems

We study the spectral polynomial of the Treibich–Verdier potential. Such spectral polynomial, which is a generalization of the classical Lamé polynomial, plays fundamental roles in both the finite-gap theory and the ODE theory of Heun's equation. In this paper, we prove that all the roots of such spectral polynomial are real and distinct under some assumptions. The proof uses the classical concept of Sturm sequence and isomonodromic theories. We also prove an analogous result for a polynomial associated with a generalized Lamé equation, where we apply a new approach based on the viewpoint of the monodromy data.

A Lie-theoretic Description of the Solution Space of the tt*-Toda Equations

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized by monodromy data of a certain flat $SL_{n+1}\mathbb{R}$-connection. Using Boalch's Lie-theoretic description of Stokes data, and Steinberg's description of regular conjugacy classes of a linear algebraic group, we express this monodromy data as a convex subset of a Weyl alcove of $SU_{n+1}$.

Analytic geometry of semisimple coalescent Frobenius structures

We present some results of a joint paper with Dubrovin (see references), as exposed at the Workshop “Asymptotic and Computational Aspects of Complex Differential Equations” at the CRM in Pisa, in February 2017. The analytical description of semisimple Frobenius manifolds is extended at semisimple coalescence points, namely points with some coalescing canonical coordinates although the corresponding Frobenius algebra is semisimple. After summarizing and revisiting the theory of the monodromy local invariants of semisimple Frobenius manifolds, as introduced by Dubrovin, it is shown how the definition of monodromy data can be extended also at semisimple coalescence points. Furthermore, a local Isomonodromy theorem at semisimple coalescence points is presented. Some examples of computation are taken from the quantum cohomologies of complex Grassmannians.

On reducible monodromy representations of some generalized Lamé equation

In this note, we compute the explicit formula of the monodromy data for a generalized Lam\'{e} equation when its monodromy is reducible but not completely reducible. We also solve the corresponding Riemman-Hilbert problem.

Transition asymptotics for the Painlevé II transcendent

We consider real-valued solutions u=u(x|s), x∈R, of the second Painlevé equation uxx=xu+2u3 which are parameterized in terms of the monodromy data s≡(s1,s2,s3)⊂C3 of the associated Flaschka–Newell system of rational differential equations. Our analysis describes the transition, as x→−∞, between the oscillatory power-like decay asymptotics for |s1|<1 (Ablowitz–Segur) to the power-like growth behavior for |s1|=1 (Hastings–McLeod) and from the latter to the singular oscillatory power-like growth for |s1|>1 (Kapaev). It is shown that the transition asymptotics are of Boutroux type; that is, they are expressed in terms of Jacobi elliptic functions. As applications of our results we obtain asymptotics for the Airy kernel determinant det(I−γKAi)|L2(x,∞) in a double scaling limit x→−∞, γ↑1, as well as asymptotics for the spectrum of KAi.

On the existence of monodromies for the Rabi model

We discuss the existence of monodromies associated with the singular points of the eigenvalue problem for the Rabi model. The complete control of the full monodromy data requires the taming of the Stokes phenomenon associated with the unique irregular singular point. The monodromy data, in particular, the composite monodromy, are written in terms of the parameters of the model via the isomonodromy method and the τ function of the Painlevé V. These data provide a systematic way to obtain the quantized spectrum of the Rabi model.

Exact conformal blocks for the W-algebras, twist fields and isomonodromic deformations

We consider the conformal blocks in the theories with extended conformal W-symmetry for the integer Virasoro central charges. We show that these blocks for the generalized twist fields on sphere can be computed exactly in terms of the free field theory on the covering Riemann surface, even for a non-abelian monodromy group. The generalized twist fields are identified with particular primary fields of the W-algebra, and we propose a straightforward way to compute their W-charges. We demonstrate how these exact conformal blocks can be effectively computed using the technique arisen from the gauge theory/CFT correspondence. We discuss also their direct relation with the isomonodromic tau-function for the quasipermutation monodromy data, which can be an encouraging step on the way of definition of generic conformal blocks for W-algebra using the isomonodromy/CFT correspondence.

Isomonodromic τ-functions and W N conformal blocks

We study the solution of the Schlesinger system for the 4-point $\mathfrak{sl}_N$ isomonodromy problem and conjecture an expression for the isomonodromic $\tau$-function in terms of 2d conformal field theory beyond the known $N=2$ Painlev\'e VI case. We show that this relation can be used as an alternative definition of conformal blocks for the $W_N$ algebra and argue that the infinite number of arbitrary constants arising in the algebraic construction of $W_N$ conformal block can be expressed in terms of only a finite set of parameters of the monodromy data of rank $N$ Fuchsian system with three regular singular points. We check this definition explicitly for the known conformal blocks of the $W_3$ algebra and demonstrate its consistency with the conjectured form of the structure constants.

Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa I. Stokes Data

We describe all smooth solutions of the two-function tt*-Toda equations (a version of the tt* equations, or equations for harmonic maps into SL(n,R)/SO(n)) in terms of (i) asymptotic data, (ii) holomorphic data, and (iii) monodromy data. This allows us to find all solutions with integral Stokes data. These include solutions associated to nonlinear sigma models (quantum cohomology) or Landau-Ginzburg models (unfoldings of singularities), as conjectured by Cecotti and Vafa.

Type IIB flux vacua from G-theory I

We construct non-perturbatively exact four-dimensional Minkowski vacua of type IIB string theory with non-trivial fluxes. These solutions are found by gluing together, consistently with U-duality, local solutions of type IIB supergravity on $$ {T}^4\times \mathbb{C} $$ with the metric, dilaton and flux potentials varying along $$ \mathbb{C} $$ and the flux potentials oriented along T 4. We focus on solutions locally related via U-duality to non-compact Ricci-flat geometries. More general solutions and a complete analysis of the supersymmetry equations are presented in the companion paper [1]. We build a precise dictionary between fluxes in the global solutions and the geometry of an auxiliary K3 surface fibered over $$ \mathbb{C}{\mathrm{\mathbb{P}}}^1 $$ . In the spirit of F-theory, the flux potentials are expressed in terms of locally holomorphic functions that parametrize the complex structure moduli space of the K3 fiber in the auxiliary geometry. The brane content is inferred from the monodromy data around the degeneration points of the fiber.

Isomonodromy Aspects of the tt* Equations of Cecotti and Vafa II: Riemann–Hilbert Problem

In Guest et al. (arXiv:1209.2045) (part I) we computed the Stokes data for the smooth solutions of the tt*-Toda equations whose existence we had previously established by p.d.e. methods. Here we formulate the existence problem as a Riemann–Hilbert problem, based on this Stokes data, and solve it under certain conditions (Theorem 5.4). In the process, we compute the connection matrix for all smooth solutions, thus completing the computation of the monodromy data (Theorem 5.5). We also give connection formulae relating the asymptotics at zero and infinity of all smooth solutions (Theorem 4.1), clarifying the region of validity of the formulae established earlier by Tracy and Widom. Finally, we resolve some conjectures of Cecotti and Vafa concerning the positivity of S + St (where S is the Stokes matrix) and the unimodularity of the eigenvalues of the monodromy matrix (Theorem 5.6). In particular, we show that “unitarity implies regularity” for the tt*-Toda equations.

Connection Problem for the Sine-Gordon/Painlevé III Tau Function and Irregular Conformal Blocks: Fig. 1.

The short-distance expansion of the tau function of the radial sine-Gordon/Painlev\'e III equation is given by a convergent series which involves irregular $c=1$ conformal blocks and possesses certain periodicity properties with respect to monodromy data. The long-distance irregular expansion exhibits a similar periodicity with respect to a different pair of coordinates on the monodromy manifold. This observation is used to conjecture an exact expression for the connection constant providing relative normalization of the two series. Up to an elementary prefactor, it is given by the generating function of the canonical transformation between the two sets of coordinates.

Exact WKB analysis and cluster algebras

We develop the mutation theory in the exact WKB analysis using the framework of cluster algebras. Under a continuous deformation of the potential of the Schrödinger equation on a compact Riemann surface, the Stokes graph may change the topology. We call this phenomenon the mutation of Stokes graphs. Along the mutation of Stokes graphs, the Voros symbols, which are monodromy data of the equation, also mutate due to the Stokes phenomenon. We show that the Voros symbols mutate as variables of a cluster algebra with surface realization. As an application, we obtain the identities of Stokes automorphisms associated with periods of cluster algebras. The paper also includes an extensive introduction of the exact WKB analysis and the surface realization of cluster algebras for nonexperts.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Cluster algebras in mathematical physics’.

Geometric Aspects of the Painlevé Equations PIII(D6) and PIII(D7)

The Riemann-Hilbert approach for the equations ${\rm PIII(D_6)}$ and ${\rm PIII(D_7)}$ is studied in detail, involving moduli spaces for connections and monodromy data, Okamoto-Painlev\'e varieties, the Painlev\'e property, special solutions and explicit B\"acklund transformations.

A Riemann–Hilbert approach to Painlevé IV

The methods of [vdP-Sa, vdP1, vdP2] are applied to the fourth Painlevé equation. One obtains a Riemann–Hilbert correspondence between moduli spaces of rank two connections on ℙ1 and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto–Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on ℙ1 are introduced, inspired by the symmetric form for PIV, studied by M. Noumi and Y. Yamada.

Black hole monodromy and conformal field theory

The analytic structure of solutions to the Klein-Gordon equation in a black hole background, as represented by monodromy data, is intimately related to black hole thermodynamics. It encodes the ``hidden conformal symmetry'' of a nonextremal black hole, and it explains why features of the inner event horizon appear in scattering data such as greybody factors. This indicates that hidden conformal symmetry is generic within a universality class of black holes.