# New Trends in Lagrangian and Hamiltonian Aspects of Integrable Systems

## Leeds Workshop 13-14 May 2022

This workshop is part of the series of workshops in the LMS supported network Classical and Quantum Integrability. It is a hybrid event with in-person and online speakers and participants. There will be a conference dinner.

## Registration

Please use the Registration form to indicate your interest in taking part in person or virtually and to indicate whether you would like to attend the dinner on Friday evening. We will go to a very nice French influenced restaurant called Sous-le-nez. This is primarily to help us get an idea of numbers. Thanks!

## Programme

Here is the event programme.

## Titles and Abstracts

**Anastasia Doikou** "*Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation*"

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones subject to certain extra constraints dictated by the q-deformation.

**Clare Dunning** "*Rational solutions of the Painlevé equations*"

Rational solutions of the Painlevé equations and their higher-order analogues are often written in the form of a logarithmic derivative of a ratio of particular polynomials. These polynomials have a wide range of applications, including in classical and quantum integrable systems, random matrix theory, supersymmetric quantum mechanics, orthogonal polynomials. The polynomials may be expressed as certain Wronskian determinants labelled by partitions. Combinatorial aspects of the partitions turn out to play intriguing roles in various aspects of the polynomials including their coefficients, distribution of zeros in the complex plane and the discriminant.

**Sylvain Lacroix** "*Integrable sigma-models at RG fixed points: quantisation as affine Gaudin models***"**

In this talk, I will present first steps towards the quantisation of integrable sigma-models using the formalism of affine Gaudin models, approaching these theories through their conformal limits. The talk will mostly focus on a specific example called the Klimcik model. After recalling the relation between this theory and affine Gaudin models at the classical level, I will explain how its integrable structure splits into two decoupled chiral parts in the conformal limit, built respectively from left-moving and right-moving degrees of freedom. Finally, I will briefly sketch how the quantisation of these chiral integrable structures can be studied using the language of affine Gaudin models and vertex operator algebras. This is based on joint work with G. Kotousov and J. Teschner.

**Linyu Peng **"*The difference variational bicomplex and discrete integrable systems*"

We define the difference variational bicomplex and examine its exactness after introducing the prolongation structure for finite difference equations, a discrete counterpart of the jet structure for differential equations. The difference variational bicomplex provides a convenient setting for the study of discrete variational calculus, inverse problems, symmetry analysis, etc. In particular, we will show its connection with the recent development of discrete integrable systems with the closure relation of Lagrangian multiforms. This is based on joint work with Peter Hydon and Frank Nijhoff.

**Mats Vermeeren **"*Lagrangian multiforms and the Toda hierarchy*"

The first part of this talk will be an introduction to Lagrangian multiform theory, which is a variational formulation of discrete and continuous integrable systems. I will highlight some nice features of this approach and some connections to more common notions of integrability. In the second part I will present the newly developed semi-discrete version of Lagrangian multiforms. The main example will be the Toda lattice and the hierarchy of ODEs which it is part of. There is more to this hierarchy than meets the eye: by constructing a semi-discrete Lagrangian multiform for it we will find a hierarchy of PDEs hiding within.

**Jennifer Winstone** "*3-dimensional mixed BF theory and Hitchin’s integrable system*"

The affine Gaudin model, associated with an untwisted affine Kac-Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological-holomorphic Chern-Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework. Based on joint work arXiv:2201.07300 with B. Vicedo.

*Local organisers: Farrokh Atai, Vincent Caudrelier, Allan Fordy, Oleg Chalykh*