Localization and Non-Renormalization in Chern-Simons Theory

TL;DR

This paper uses localization and supersymmetry to clarify renormalization properties in Chern-Simons theory, showing that one-loop shifts are nonperturbative effects explained by a non-renormalization theorem, with applications to various manifolds.

Contribution

It introduces a localization-based approach to understand renormalization in Chern-Simons theory, unifying perturbative and nonperturbative insights.

Findings

One-loop shifts are nonperturbative effects explained by a non-renormalization theorem.

The approach applies to flat space, Seifert manifolds, and solid tori.

Provides a unified understanding of Wilson line renormalization.

Abstract

We revisit and clarify some aspects of perturbative renormalization in pure Chern-Simons theory by means of a localization principle associated with an underlying supersymmetry. This perspective allows the otherwise perturbative one-loop shifts to be interpreted as nonperturbative consequences of a non-renormalization theorem, while providing a unified understanding of their origin (particularly in the case of Wilson lines). We illustrate this approach explicitly for $SU(2)$ Chern-Simons theory in flat space, on Seifert manifolds, and on a solid torus.