An extension of the $\mathrm{U}\!\left(1\right)$ BF theory, Turaev-Viro invariant and Drinfeld center construction. Part I: Quantum fields, quantum currents and Pontryagin duality

TL;DR

This paper introduces a mathematical framework extending U(1) BF theory, Turaev-Viro invariants, and Drinfeld center constructions to broader contexts, focusing on quantum fields, currents, and Pontryagin duality on closed manifolds.

Contribution

It develops foundational mathematical tools for extending abelian topological quantum field theories and invariants to more general settings.

Findings

Establishes a mathematical background for the extensions.

Connects quantum fields and currents with Pontryagin duality.

Lays groundwork for subsequent articles in the series.

Abstract

In this first of a series of articles dedicated to natural extensions of the U(1) BF theory, abelian Turaev-Viro (TV) construction and corresponding Drinfeld center construction for any closed oriented smooth manifolds, we present the mathematical background that will be used.