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

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 $L_{\mathrm{\infty }}$-algebra that can be used to construct a classical integrable model. Another application of this class of $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-algebra that can serve as correlation functions.