Integrable Models From Non-Commutative Geometry With Applications to 3D Dualities

TL;DR

This paper introduces a new class of strong homotopy algebras derived from inner deformations, linking them to integrable models and 3D dualities, especially in the context of Chern--Simons theories and deformation quantization.

Contribution

It constructs a novel class of $L_$-algebras from associative algebra deformations, applying them to integrable models and 3D bosonization duality, extending deformation quantization to Poisson Orbifolds.

Findings

New class of strong homotopy algebras from inner deformations

Application to classical integrable models via $L_$-algebras

Connection to 3D bosonization duality and higher spin symmetry

Abstract

We discuss a new class of strong homotopy algebras constructed via inner deformations. Such deformations have a number of remarkable properties. In the simplest case, every one-parameter family of associative algebras leads to an $L_\infty$-algebra that can be used to construct a classical integrable model. Another application of this class of $L_\infty$-algebras is related with the three-dimensional bosonization duality in Chern--Simons vector models, where it implements the idea of the slightly-broken higher spin symmetry. One large class of associative algebras originates from Deformation Quantization of Poisson Manifolds. Applications to the $3d$-bosonization duality require, however, an extension to deformation quantization of Poisson Orbifolds, which is an open problem. The $3d$-bosonization duality can be proven by showing that there is a unique class of invariants of the $L_\infty$-algebra that can serve as correlation functions.