The Chevalley--Eilenberg complex and deformation quantization in presence of two branes

TL;DR

This paper demonstrates that the Chevalley--Eilenberg complex for a finite-dimensional Lie algebra over a field containing complex numbers is $A_
$-quasi-isomorphic to a deformation quantization of a specific bimodule, extending formality results to a two-brane setting.

Contribution

It establishes an $A_
$-quasi-isomorphism between the Chevalley--Eilenberg complex and a deformation quantization of an $A_
$-bimodule using the Formality Theorem with two branes.

Findings

Proves the Chevalley--Eilenberg complex is a deformation quantization of the Koszul complex.

Shows the complex is $A_
$-quasi-isomorphic to a bimodule deformation quantization.

Extends the Formality Theorem to the context of two branes.

Abstract

In this note, we prove that, for a finite-dimensional Lie algebra $\mathfrak g$ over a field $\mathbb K$ of characteristic 0 which contains $\mathbb C$, the Chevalley--Eilenberg complex $\mathrm U(\mathfrak g)\otimes \wedge(\mathfrak g)$, which is in a natural way a deformation quantization of the Koszul complex of $\mathrm S(\mathfrak g)$, is $A_\infty$-quasi-isomorphic to the deformation quantization of the $A_\infty$-bimodule $K=\mathbb K$ provided by the Formality Theorem in presence of two branes (CFFR).