Bimodules and branes in deformation quantization

TL;DR

This paper extends Kontsevich's formality theorem to subspaces (branes) in a vector space, showing their deformation quantizations are Koszul dual, thus resolving an open question in deformation quantization theory.

Contribution

It proves a version of Kontsevich's formality theorem for branes, establishing Koszul duality between certain deformation quantizations, advancing understanding of dualities in deformation quantization.

Findings

Deformation quantizations of S(X*) and wedge(X) are Koszul dual.

Provides a new proof of a formality theorem for subspaces in vector spaces.

Answers an open question on Koszul duality in deformation quantization.

Abstract

We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space $X$. The result implies in particular that the Kontsevich deformation quantizations of $\mathrm{S}(X^*)$ and $\wedge(X)$ associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.