The Anomaly Line Bundle of the Self-Dual Field Theory

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 one 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.

The Global Anomaly of the Self-Dual Field in General Backgrounds

“…In [3] (with some refined arguments presented in Appendix A), we showed that the anomaly line bundle A of the self-dual field theory is given by…”

“…While the global gravitational anomalies of chiral fermionic field theories were understood a long time ago [1], a formula for the global anomaly of the self-dual field theory on generic manifolds was derived only recently [3,4]. A self-dual 2ℓ-form field on a 4ℓ`2-dimensional manifold can be naturally sourced by a 2ℓ`1-form abelian gauge field.…”

“…The derivation in [3,4] was not completely rigorous. Several consistency checks of the proposed anomaly formula were presented, one of them being the cancellation of global gravitational anomalies in the cohomological version of type IIB supergravity (see Section 5).…”

Global gravitational anomaly cancellation for five-branes

“…A formula for the global anomaly of the self-dual field was determined in [1], but only in the case when the self-dual field has no zero modes, or equivalently when the middle-degree cohomology of the underlying manifold vanishes. A general formula for the gravitational anomaly was determined in [5], building on [6,27]. The formula below, which includes the coupling to a background abelian gauge field, was proven in [7].…”

“…Previous checks of global anomaly cancellation in dimension higher than 2 include type I supergravity [1,2], 11-dimensional supergravity [3,4] and type IIB supergravity [5]. In the present paper, we use recent results concerning the global gravitational anomaly of the self-dual field [6,5,7] in order to check the cancellation of global mixed gaugegravitational anomalies in M-theory backgrounds containing M5-branes. The cancellation of local gravitational anomalies in such backgrounds was checked in [8,9,10,11].…”