Deformation Quantization via Categorical Factorization Homology

TL;DR

This paper introduces a categorical approach to deformation quantization using factorization homology, connecting various existing quantizations and computing related homological invariants.

Contribution

It develops a framework linking local coefficient quantizations to manifold invariants, introducing shifted Poisson and BD categories, and relating different quantization methods.

Findings

Reproduces deformations by Li-Bland and evera using the Drinfeld category.

Establishes a relation between their quantization and those by Alekseev, Grosse, and Schomerus.

Computes factorization homology with values in a ribbon category enriched over bar-modules.

Abstract

This paper develops an approach to categorical deformation quantization via factorization homology. We show that a quantization of the local coefficients for factorization homology is equivalent to consistent quantizations of its value on manifolds. To formulate our results we introduce the concepts of shifted almost Poisson and BD categories. Our main example is the character stack of flat principal bundles for a reductive algebraic group $G$, where we show that applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and \v{S}evera. As a direct consequence, we can conclude a precise relation between their quantization and those introduced by Alekseev, Grosse, and Schomerus. To arrive at our results we compute factorization homology with values in a ribbon category enriched over complete $\mathbb{C}[[\hbar]]$-modules. More generally, we define enriched skein categories which compute factorization homology for ribbon categories enriched over a general closed symmetric monoidal category $\mathcal{V}$.