The anomaly line bundle of the self-dual field theory

TL;DR

This paper explicitly determines the anomaly line bundle for abelian self-dual fields, revealing its structure over the space of metrics and its implications for global gravitational anomalies.

Contribution

It provides an explicit description of the anomaly line bundle, including its torsion part, and proposes a non-covariant action principle for self-dual fields on Riemannian manifolds.

Findings

The anomaly bundle differs from the Dirac determinant bundle by a non-trivial flat bundle.

Holonomies of the flat bundle are explicitly determined.

The results are relevant for computing the global gravitational anomaly of the self-dual field.

Abstract

In this work, we determine explicitly the anomaly line bundle of the abelian self-dual field theory over the space of metrics modulo diffeomorphisms, including its torsion part. Inspired by the work of Belov and Moore, we propose a non-covariant action principle for a pair of Euclidean self-dual fields on a generic oriented Riemannian manifold. The corresponding path integral allows to study the global properties of the partition function over the space of metrics modulo diffeomorphisms. We show that the anomaly bundle for a pair of self-dual fields differs from the determinant bundle of the Dirac operator coupled to chiral spinors by a flat bundle that is not trivial if the underlying manifold has middle-degree cohomology, and whose holonomies are determined explicitly. We briefly sketch the relevance of this result for the computation of the global gravitational anomaly of the self-dual field theory, that will appear in another paper.