Classical Yang-Baxter equation, Lagrangian multiforms and ultralocal integrable hierarchies

Abstract: I will first review the main ideas of Lagrangian multiform theory which provides a variational framework for integrable systems, focusing on 1+1 dimensional models. The key idea is to encode variationally the compatibility of the flows in an integrable hierarchy by introducing a generalisation of standard Lagrangians and action functionals, and a corresponding generalisation of the principle of least action.

I will then present a systematic construction of Lagrangian multiforms for hierarchies of ultralocal field theories. The ''multitime'' Euler-Lagrange equations produce the infinite collection of flatness equations for the Lax connection. It is based on a few key (algebraic) ingredients, the main one being the classical r-matrix and the classical Yang-Baxter equation. In particular, the construction casts the classical Yang-Baxter in a variational context for the first time. For simplicity of exposition, I will focus on the simplest example of the so-called Ablowitz-Kaup-Newell-Segur hierarchy which already contains all the essential features (classical r-matrix, generating formalism). I will then explain briefly how these features generalise to produce Lagrangian multiforms for many other (known and new) hierarchies : e.g. Zakharov-Mikhailov, Faddeev-Reshetikhin model, deformed sigma/Gross-Neveu models. Time permitting, I will explain how one can easily couple different hierarchies together to form new ones and obtain the corresponding Lagrangian multiform.

HEP - theorymathematical physics

Audience: researchers in the topic

Nagoya Math-Phys Seminar

Series comments: Zoom link will be shown at the seminar website 2 hours before the start of the seminar. (In the case that the seminar website is dead, it will be displayed here.)

Slides and videos of the talks will be uploaded at the seminar website.