Tannakian QFT: from spark algebras to quantum groups

TL;DR

This paper develops a nonperturbative framework for constructing Hopf algebras representing categories of line operators in 3D topological quantum field theories, connecting Tannakian formalism with quantum groups and boundary conditions.

Contribution

It introduces a novel nonperturbative method to build Hopf algebras from semi-extended operators in TQFT, applying Tannakian formalism to quantum field theories with new insights into quantum groups.

Findings

Constructed Hopf algebras from spark algebras in 3D TQFTs

Defined R-matrices, ribbon twists, and Drinfeld doubles topologically

Applied framework to finite-group gauge theories and 3d supersymmetric theories

Abstract

We propose a nonperturbative construction of Hopf algebras that represent categories of line operators in topological quantum field theory, in terms of semi-extended operators (spark algebras) on pairs of transverse topological boundary conditions. The construction is a direct implementation of Tannakian formalism in QFT. Focusing on d=3 dimensional theories, we find topological definitions of R-matrices, ribbon twists, and the Drinfeld double construction for generalized quantum groups. We illustrate our construction in finite-group gauge theory, and apply it to obtain new results for B-twisted 3d $\mathcal{N}=4$ gauge theories, a.k.a. equivariant Rozansky-Witten theory, or supergroup BF theory (including ordinary BF theory with compact gauge group). We reformulate our construction mathematically in terms of abelian and dg tensor categories, and discuss connections with Koszul duality.