The global gravitational anomaly of the self-dual field theory

TL;DR

This work derives a comprehensive formula for the global gravitational anomaly of the self-dual field on 4ℓ+2 dimensional manifolds, uncovering deep links between determinant line bundles, Siegel theta functions, and the Hopkins-Singer differential-cohomology framework. The method hinges on relating the anomaly bundle to fourth roots of determinant/flat bundles, expressing holonomies via eta and Arf invariants of mapping tori, and matching with the Hopkins-Singer construction. A key outcome is that, in the naive differential-cohomology picture for type IIB supergravity, the global gravitational anomaly cancels on all 10D spin manifolds, illustrating the consistency of the theory under these assumptions and offering a path toward broader physical applications. The work also lays mathematical groundwork on intermediate Jacobians, modular geometry, and quadratic refinements, and proposes conjectures about the Hopkins-Singer bundle’s role in encoding global anomalies across families. Overall, it strengthens the bridge between high-energy physics anomalies and refined differential-geometric/topological invariants with practical implications for string/M-theory backgrounds.

Abstract

We derive a formula for the global gravitational anomaly of the self-dual field theory on an arbitrary compact oriented Riemannian manifold. Along the way, we uncover interesting links between the theory of determinant line bundles of Dirac operators, Siegel theta functions and a functor constructed by Hopkins and Singer. We apply our result to type IIB supergravity and show that in the naive approximation where the Ramond-Ramond fields are treated as differential cohomology classes, the global gravitational anomaly vanishes on all 10-dimensional spin manifolds. We sketch a few other important physical applications.