Some Lower Dimensional Quantum Field Theories Reduced from Chern-Simons Gauge Theories

TL;DR

This paper explores how symmetry reductions of Euclidean Chern-Simons gauge theories lead to lower-dimensional quantum field theories, revealing connections to boundary conformal field theories and supersymmetric quantum mechanics.

Contribution

It establishes relations between pure Chern-Simons theories and reduced quantum field theories, including the derivation of 2D $BF$-theory and insights into supersymmetric structures.

Findings

Relations between Chern-Simons theories on $S^3$ and reduced QFTs

Reduction to $BF$-theory on the disk and boundary CFTs

Reduced actions related to supersymmetric quantum mechanics

Abstract

We study symmetry reductions in the context of Euclidean Chern-Simons gauge theories to obtain lower dimensional field theories. Symmetry reduction in certain gauge theories is a common tool for obtaining explicit soliton solutions. Although pure Chern-Simons theories do not admit solitonic solutions, symmetry reduction still leads to interesting results. We establish relations at the semiclassical regime between pure Chern-Simons theories on $S^3$ and the reduced Quantum Field Theories, based on actions obtained by the symmetry reduction of the Chern-Simons action, spherical symmetry being the prominent one. We also discuss symmetry reductions of Chern-Simons theories on the disk, yielding $BF$-theory in two dimensions, which signals a curious relationship between symmetry reductions and the boundary conformal field theories. Finally, we study the Chern-Simons-Higgs instantons and show that under certain circumstances, the reduced action can formally be viewed as the action of a supersymmetric quantum mechanical model. We discuss the extent to which the reduced actions have a fermionic nature at the level of the partition function.