Index theorem for topological excitations on R^3 * S^1 and Chern-Simons theory

TL;DR

This work derives an index theorem for the Dirac operator on ${\bf R^3 \times S^1}$ in backgrounds sourced by topological excitations such as monopoles and KK monopoles. The index splits into a surface term governed by the eta-invariant and a topological term; their non-integer parts cancel to yield an integer index, reproducing Callias’ index in the small-circle limit and connecting smoothly to the APS index in the decompactification limit. The analysis covers static and winding monopole backgrounds, SU(2) representations, and provides explicit expressions illustrating how KK modes induce spectral jumps and how the index encodes refined non-perturbative data. The paper also discusses anomalies and the emergence of Chern-Simons terms under chirally twisted boundary conditions, highlighting implications for center-stabilized gauge theories and possible topological phases in small-S^1 regimes.

Abstract

We derive an index theorem for the Dirac operator in the background of various topological excitations on an R^3 \times S^1 geometry. The index theorem provides more refined data than the APS index for an instanton on R^4 and reproduces it in decompactification limit. In the R^3 limit, it reduces to the Callias index theorem. The index is expressed in terms of topological charge and the eta-invariant associated with the boundary Dirac operator. Neither topological charge nor eta-invariant is typically an integer, however, the non-integer parts cancel to give an integer-valued index. Our derivation is based on axial current non-conservation--an exact operator identity valid on any four-manifold--and on the existence of a center symmetric, or approximately center symmetric, boundary holonomy (Wilson line). We expect the index theorem to usefully apply to many physical systems of interest, such as low temperature (large S^1, confined) phases of gauge theories, center stabilized Yang-Mills theories with vector-like or chiral matter (at S^1 of any size), and supersymmetric gauge theories with supersymmetry-preserving boundary conditions (also at any S^1). In QCD-like and chiral gauge theories, the index theorem should shed light into the nature of topological excitations responsible for chiral symmetry breaking and the generation of mass gap in the gauge sector. We also show that imposing chirally-twisted boundary condition in gauge theories with fermions induces a Chern-Simons term in the infrared. This suggests that some QCD-like gauge theories should possess components with a topological Chern-Simons phase in the small S^1 regime.