# Abelian Turaev-Virelizier theorem and $U(1)$ BF surgery formulas

## TL;DR

This paper constructs a $U(1)$ BF theory-based invariant for 3-manifolds, demonstrating its equivalence to the Turaev-Viro invariant and providing surgery formulas akin to those in abelian Chern-Simons theory.

## Contribution

It establishes the Reshetikhin-Turaev invariant from the Drinfeld Center of a spherical category related to $U(1)$ BF theory and proves its equivalence to the Turaev-Viro invariant.

## Key findings

- Invariant coincides with Turaev-Viro invariant for closed 3-manifolds
- Provides surgery formulas similar to abelian Chern-Simons theory
- Demonstrates the Turaev-Virelizier theorem in the abelian $U(1)$ context

## Abstract

In this article we construct the Reshetikhin-Turaev invariant associated with the Drinfeld Center of the spherical category arising from the $U(1)$ BF theory on a closed $3$-manifold $M$. This invariant is shown to coincide with the Turaev-Viro invariant of $M$ thus providing an example of the Turaev-Virelizier theorem. Finally we exhibit some surgery formulas for the abelian Turaev-Viro invariant which are very similar to the surgery formulas of the abelian Reshetikhin-Turaev invariant obtained in the $U(1)$ Chern-Simons context.