# LPTENS Summer Institute XLVII

Aug 16 – 25, 2017
LPTENS
Europe/Paris timezone

## Abstracts Miniworkshop

Frank Loray (Université de Rennes I)

Isomonodromic deformations and Painlevé equations

We introduce the notion of isomonodromic deformations of linear systems on the Riemann sphere, and explain how Painlevé equations can be derived from them. Although we will mainly talk about the Painlevé VI case, related to fuchsian systems, we will explain how other Painlevé equations arise. We will deduce the Painlevé property from the Riemann-Hilbert correspondence. We will give a geometric description of the space of initial conditions, its symmetries, and how singularities arise.

Simon Ruijsenaars (University of Leeds)

Integrable systems of Calogero-Moser type

Systems of Calogero-Moser type are integrable N-particle systems associated with root systems. They are connected to a great many subfields of pure and applied mathematics, and also find applications in various areas of physics.

In this lecture we aim to survey this class of systems associated with A_{N-1} and BC_N. We mostly deal with the highest level in a hierarchy we shall specify. This top level can be characterised by the catchwords quantum’, elliptic’ and relativistic’. The search for joint eigenfunctions of the N commuting operators has led to a novel and very promising tool, namely so-called kernel functions. We shall also discuss this notion in some detail.

Oleg Lisovyi (Université de Tours)

Painlevé functions, Fredholm determinants and combinatorics

I am going to explain the explicit construction of general solutions to isomonodromy equations, with the main focus on the Painlevé VI equation. I will start by deriving a Fredholm déterminant representation of the Painlevé VI tau function. The corresponding integral operator acts in the direct sum of two copies of $L^2(S^1)$. Its kernel is expressed in terms of hypergeometric fundamental solutions of two auxiliary 3-point Fuchsian systems whose monodromy is determined by the monodromy of the associated linear problem via a decomposition of the 4-punctured sphere into two pairs of pants. In the Fourier basis, this kernel is given by an infinite Cauchy matrix. I will explain how the principal minor expansion of the Fredholm determinant yields a combinatorial series representation for the general solution to Painlevé VI in the form of a sum over pairs of Young diagrams. The latter series coincides with the dual Nekrasov partition function of the $mathcal N=2$ $N_f=2$ $SU(2)$ gauge theory in the self-dual $Omega$-background.

Masa-Hiko Saito (University of Kobe)

Moduli spaces of connections and Higgs bundles over curves  and Geometric Theory of equations of Painlevé type

Geometric theory of equations of Painlevé type are based on moduli spaces of stable parabolic connections on a family of smooth projective curves of arbitrary genus. We will start with algebraic constructions of moduli spaces of parabolic connections and singular parabolic Higgs bundles on a smooth projective curve. Riemann-Hilbert correspondence from a family of moduli spaces of singular connections to the corresponding moduli spaces of (generalized) monodromy data induces the isomonodromic differential equations. An analysis of RH correspondence shows the geometric Painlevé property of isomonodromic differential equations associated to each type of singular connections. Next, I will investigate explicit geometric structures of moduli spaces of parabolic connections and Higgs bundles. On a Zariski dense open set of each moduli space one can define a canonical coordinate system associated to apparent singularities and their duals. The spectral curves for Higgs bundles play essential roles for this explicit geometry. If time permits, we will explain more geometric structures of moduli spaces.

Alessandro Tanzini (SISSA)

Painlevé / Gauge theory correspondence

The relation between SU(2) gauge theories with N=2 supersymmetry in four dimensions and isomonodromy problems for SL(2) flat connections on the Riemann sphere with punctures will be illustrated by making use of M-theory construction of the gauge theory and of the related Hitchin’s integrable system. It will be discussed how this relation can be used to calculate the gauge theory partition function in the strong coupling regime and to describe S-duality properties. The embedding of these results in a non-perturbative completion of topological strings will be briefly highlighted with emphasis on their uplift to five dimensional gauge theories and multiplicative q-Painlevé equations

Geometric aspects of discrete Painlevé equations

The theory of discrete Painlevé equations has made great progress in the last two decades. In this talk, we first study simple examples of discrete Painlevé equations using their autonomous limits as a clue. Then, after recapitulating the basic ideas of Sakai's geometric theory of Painlevé equations, we will give an explicit formulation of the most generic case: the elliptic difference Painlevé equation. Finally, we will derive the isomonodromic description (Lax formulation) based on the geometric method. If time permits we will also discuss the tau functions, special solutions, quantization etc. This talk is mainly based on a review with K.Kajiwara and M.Noumi arXiv:1509.08186.

Guy Casale (Université de Rennes I)

Painlevé equations and differential Galois theories

Paul Painlevé claimed that «irreducibility» of his equations is a consequence of the computation of their «rationality groups» as defined by Jules Drach in 1898. These objects were correctly defined by H. Umemura in 1996 and by B. Malgrange in 2001. In this lecture we will present the definition of the Malgrange groupoid on some examples. Then we will explain how to compute it for second order differential equations and we will use it to prove a weak «irreducibility property» for Painlevé equations.

Ovidiu Costin (Ohio State University)

Asymptotic methods for the analysis of Painlevé equations

We present new resurgence based methods for the global analysis of problems in mathematics, and models in physics such as QFT and string theory. The starting point can simply be a perturbative expansion. This approach is particularly well suited for finding the  "large-to-small coupling" connection and for calculating the monodromy at infinity. Applied to the Painlevé P1 equation, the Stokes constant is obtained in closed form simply from the Painlevé property (all movable singularities are poles); the small argument behavior of the tritronquée solution is derived from its asymptotic behavior, a crucial ingredient we used for proving of Dubrovin's conjecture. We devise convergent rational function expansions for the tritronquée which are practical, efficient and accurate throughout its domain of analyticity sought by the Painlevé program initiated by Deift & al. Work with G. Dunne, M. Huang and S. Tanveer.

Alba Grassi (ICTP Trieste)

Topological string, Spectral theory and Painlevé equations

I will explain how to embed the Painlevé/SU(2) gauge correspondence into the framework of topological string theory. Then I will explain how to use this more general framework to obtain some new results for the SU(N) case.

Simon Ruijsenaars (University of Leeds)

Relativistic Heun operators and their E_8 spectral invariance

The eigenvalue equation for the Hamiltonian defining the nonrelativistic quantum elliptic BC_1 Calogero-Moser system is equivalent to the Heun equation. This linear 4-parameter differential equation is closely connected to the nonlinear 4-parameter Painlevé VI equation, and the connection persists at lower levels of the two hierarchies.

Decades ago, van Diejen introduced an 8-parameter difference equation generalizing the Heun equation. It may be viewed as the eigenvalue equation for the Hamiltonian defining the relativistic quantum elliptic BC_1 Calogero-Moser system. We sketch our recent results concerning the E_8 spectral invariance of a Hilbert space version of this difference operator. This self-adjoint version yields a commuting self-adjoint modular partner’ with a discrete spectrum that is also invariant under the E_8 Weyl group.

Our findings are a strong indication of a connection to Sakai’s highest level elliptic difference Painlevé equation, which also has E_8 symmetry. At lower levels in the two hierarchies, recent results by Takemura have strengthened this connection. He has shown that the linear Lax equations for the Painlevé difference equations studied by Jimbo/Sakai and Yamada can be tied in with special cases of van Diejen’s relativistic Heun equation.