Infinitesimal 2-braidings from 2-shifted Poisson structures

TL;DR

This paper demonstrates how 2-shifted Poisson structures induce explicit infinitesimal 2-braidings in derived categories, linking derived algebraic geometry with higher quantum groups.

Contribution

It provides a concrete first-order deformation quantization construction from 2-shifted Poisson structures in derived algebraic geometry.

Findings

Explicit infinitesimal 2-braiding from 2-shifted Poisson structures

Connection to higher quantum groups via Chevalley-Eilenberg algebras

First-order deformation quantization in derived settings

Abstract

It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal dg-category of finitely generated semi-free $A$-dg-modules. This provides a concrete realization, to first order in the deformation parameter $\hbar$, of the abstract deformation quantization results in derived algebraic geometry due to Calaque, Pantev, To\"en, Vaqui\'e and Vezzosi. Of particular interest is the case when $A$ is the Chevalley-Eilenberg algebra of a Lie $N$-algebra, where the braided monoidal deformations developed in this paper may be interpreted as candidates for representation categories of `higher quantum groups'.