# Induced parity-odd effective action for a Dirac field on $S^2 \times S^1$

## TL;DR

This paper computes the parity-odd effective action for massive Dirac fermions on a spherical manifold with a circle, analyzing how large gauge fields influence the fermionic determinant in this geometric setting.

## Contribution

It provides a detailed evaluation of the parity-odd effective action for Dirac fermions on $S^2 imes S^1$, focusing on large gauge field configurations.

## Key findings

- Explicit form of the parity-odd effective action derived.
- Insights into the behavior of fermionic determinants under large gauge fields.
- Analysis specific to the geometry of $S^2 imes S^1$.

## Abstract

We evaluate the parity-odd part of the effective action due to massive Dirac fermions on a $S^2 \times S^1$ manifold, minimally coupled to an external Abelian gauge field. We do that for a special class of gauge-field configurations, which is specially suitable to the study of the behaviour of the fermionic determinant under large gauge field configurations, which are allowed by the space-time geometry.