Eta-invariants and anomalies in U(1)-Chern-Simons theory

TL;DR

This paper explores the geometric and analytical aspects of U(1)-Chern-Simons theory on Seifert three-manifolds, proposing a new formulation of its partition function and establishing conjectural equivalences using regularization techniques.

Contribution

It introduces an alternative formulation of the U(1)-Chern-Simons partition function via stationary phase approximation and regularization, linking it to eta invariants and hypoelliptic operators.

Findings

New formulation of the U(1)-Chern-Simons partition function

Rigorous proof of conjectural equivalence with traditional theory

Insights into eta invariants and hypoelliptic operators

Abstract

This paper studies U(1)-Chern-Simons theory and its relation to a construction of Chris Beasley and Edward Witten. The natural geometric setup here is that of a three-manifold with a Seifert structure. Based on a suggestion of Edward Witten we are led to study the stationary phase approximation of the path integral for U(1)-Chern-Simons theory after one of the three components of the gauge field is decoupled. This gives an alternative formulation of the partition function for U(1)-Chern-Simons theory that is conjecturally equivalent to the usual U(1)-Chern-Simons theory. The goal of this paper is to establish this conjectural equivalence rigorously through appropriate regularization techniques. This approach leads to some rather surprising results and opens the door to studying hypoelliptic operators and their associated eta invariants in a new light.