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Deformation Quantization via Categorical Factorization Homology

Eilind Karlsson, Corina Keller, Lukas Müller, Ján Pulmann

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 Ševera. 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 $\mathcal{V}$.

Deformation Quantization via Categorical Factorization Homology

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 , where we show that applying the general framework to the Drinfeld category reproduces deformations previously introduced by Li-Bland and Ševera. 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 -modules. More generally, we define enriched skein categories which compute factorization homology for ribbon categories enriched over a general closed symmetric monoidal category .

Paper Structure

This paper contains 51 sections, 36 theorems, 181 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Part I: Enriched skein theory.
  3. Part II: Categorical deformation quantization.
  4. Enriched skein categories
  5. Background on oriented factorization homology
  6. Basic definitions
  7. Excision
  8. Enriched category theory
  9. Conventions and notation.
  10. Enriched coends
  11. The 2-category V-Cat
  12. Change of enrichment.
  13. Tensor product of $\mathcal{V}$-categories.
  14. Cocompleteness of $\mathcal{V}-\mathsf{Cat}$.
  15. The 2-category V-Pres
  16. ...and 36 more sections

Key Result

Lemma 2.7

AFfh Let $M = M_- \bigcup_{M_0} M_+$ be a collar-gluing of oriented $n$-manifolds and let $\mathcal{A}$ be a framed $\mathsf{E}_n$-algebra in $\mathcal{C}$. There is an equivalence of categories where on the right hand side the relative tensor product is computed by the colimit of the 2-sided bar construction \begin{tikzcd}[column sep = small] \dots \arrow[r,yshift=1.5ex] \arrow[r,yshift=0.5ex]

Figures (12)

  • Figure 1: An example of a $\mathcal{C}$-colored ribbon q-graph $\Gamma$; $m_i$ are objects of the ribbon category $\mathcal{C}$.
  • Figure 2: Left-to-right: Disk embeddings, or isotopies thereof, in $\mathsf{Disk}_2^{\operatorname{or}}$ that give rise to the multiplication $m$ and the braiding $\beta$ in $\mathcal{A} = \mathcal{A}(\mathbb{D}^2)$ ($\sigma$ denotes the symmetry in $\mathcal{C}$). Loop in the space of disk embeddings coming from rotating the disk by $2\pi$. This gives rise to a natural isomorphism $\theta$ satisfying certain properties, and is called a twist on $\mathcal{A}$.
  • Figure 3: Example of a collar-gluing of the genus 2 surface.
  • Figure 4: The map which induces the right $\int_{M_0} \mathcal{A}$-module structure on $\int_{M_-} \mathcal{A}$. Here, the green collar depicts the product manifold $N \times (-1, 1)$.
  • Figure 5: An example of a $\mathcal{C}$-colored ribbon q-graph $\Gamma$.
  • ...and 7 more figures

Theorems & Definitions (138)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Lemma 2.7
  • Theorem 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 128 more