Integrable E-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

Abstract

We construct the actions of a very broad family of 2d integrable σ-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d fields h and L, with L depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for L in terms of h this produces a 2d integrable field theory for the 2d field h whose Lax connection is given by L(h). We construct a general class of solutions to this constraint and show that the resulting 2d integrable field theories can all naturally be described as E-models.

O(D,D)-covariant two-loop β-functions and Poisson-Lie T-duality

“…Recently,[23] constructed E-models[24] for a large class of integrable σ-models and thereby makes their PL symmetry manifest.…”

Modave Lectures on Classical Integrability in 2d Field Theories

“…In the context of classical field theories, however, great development is being made, see e.g. [13][14][15].…”

“…The Belavin-Drinfeld results indicate that there is an intimate relationship between the poles of the -matrix and the integrable field theory. For recent interesting works exploring such relations from other viewpoints see for instance [13][14][15].…”

Modave Lectures on Classical Integrability in $2d$ Field Theories

“…Contrary to the finite-dimensional case, which involved an "if-and-only-if" statement, here there are in principle several possibilities for the form of the Poisson brackets {L σ,1 (σ; z), L σ,2 (σ , z }. 15 There are however only two main cases studied in the literature, as we will discuss. Note, nevertheless, that the existence of other structures are not necessarily excluded.…”