Twisting to Abelian BF/Chern-Simons Theories

TL;DR

This paper demonstrates how a specific supersymmetric theory in three dimensions can be twisted to produce abelian BF and Chern-Simons topological quantum field theories, and relates them to Donaldson-Witten theories, with a similar construction in two dimensions.

Contribution

It introduces a novel twist of a 3D N=4 supersymmetric matter theory to derive abelian BF and Chern-Simons theories and connects them to Donaldson-Witten TQFTs, extending to 2D with N=2 supersymmetry.

Findings

Twist maps supersymmetric theory to abelian BF theory on curved 3-manifolds.

Adding surface terms relates the theory to Chern-Simons actions.

The construction links BF and Chern-Simons theories to Donaldson-Witten TQFTs.

Abstract

Starting from a $D=3$, $N=4$ supersymmetric theory for matter fields, a twist with a Grassmann parity change is defined which maps the theory into a gauge fixed, abelian $BF$ theory on curved 3-manifolds. After adding surface terms to this theory, the twist is seen to map the resulting supersymmetric action to two uncoupled copies of the gauge fixed Chern-Simons action. In addition, we give a map which takes the $BF$ and Chern-Simons theories into Donaldson-Witten TQFT's. A similar construction, but with $N=2$ supersymmetry, is given in two dimensions.