Chern-Simons action for zero-mode supporting gauge fields in three dimensions

TL;DR

This paper challenges the belief that the Chern-Simons action only takes quantized values for gauge fields with zero modes, by providing an explicit counter-example in three dimensions.

Contribution

The paper presents a counter-example demonstrating that the Chern-Simons action can have non-quantized values even when zero modes are present.

Findings

Counter-example disproves the quantization conjecture

Zero modes can occur with non-quantized Chern-Simons values

Challenges previous assumptions about gauge field quantization

Abstract

Recent results on zero modes of the Abelian Dirac operator in three dimensions support to some degree the conjecture that the Chern-Simons action admits only certain quantized values for gauge fields that lead to zero modes of the corresponding Dirac operator. Here we show that this conjecture is wrong by constructing an explicit counter-example.