Infinitesimal 2-braidings from 2-shifted Poisson structures
Cameron Kemp, Robert Laugwitz, Alexander Schenkel
TL;DR
The paper develops a concrete first-order deformation framework linking $2$-shifted Poisson structures on a finitely generated semi-free CDGA $A$ to infinitesimal braidings on the homotopy $2$-category of $A$-dg-modules, thereby realizing a braided monoidal deformation at order $\hbar$. It uses the differential-graded algebraic geometry toolkit—cochain complexes, semi-free CDGAs, and completed $n$-shifted polyvectors—to formulate $2$-shifted Poisson structures via Maurer–Cartan equations and then constructs a $\gamma$-equivariant infinitesimal $2$-braiding on ${}_A extbf{C}$ from the bivector data $\pi^{(2)}$. The work analyzes prospects for lifting to all orders in $\hbar$, showing that strict coherence (Drinfeld associators) is insufficient in general and that higher coherence data (pentagonators/hexagonators) are needed; it provides explicit examples from Lie $N$-algebras, including ordinary Lie algebras and string Lie $2$-algebras, illustrating how these deformations might model higher quantum groups. Overall, the results connect shifted Poisson geometry with higher-categorical representation theory, suggesting a pathway to representation categories of higher quantum groups and enriching the landscape of 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, Toën, Vaquié 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'.