The global anomaly of the self-dual field in general backgrounds
Samuel Monnier
TL;DR
This work develops a general formula for the global gravitational anomaly of the self-dual field on backgrounds with abelian gauge data, formulating the anomaly as a holonomy of the anomaly line bundle over the space of background fields. Central to the construction are differential cocycles shifted by the Wu class, the adiabatic limit, and mapping-torus techniques which express holonomies in terms of the Hirzebruch L-genus and the background curvature via hol(c) = exp( (1/8) ∫_W (L(TW) - 4F_W^2) ). The theta characteristic tilde{η} is fixed by the adiabatic limit and shown to live in (1/2)Λ, independent of the metric, resolving prior puzzles about metric dependence and torsion anomalies. The formalism is then applied to the cohomological version of Type IIB supergravity, showing pure gravitational anomaly cancellation but a potential mixed gauge-gravitational anomaly that vanishes only when a certain spin cobordism group disappears. Overall, the results provide a rigorous global-anomaly framework for self-dual fields in general backgrounds and deliver concrete consistency checks for higher-dimensional supergravity theories.
Abstract
We prove a formula for the global gravitational anomaly of the self-dual field theory in the presence of background gauge fields, assuming the results of arXiv:1110.4639. Along the way, we also clarify various points about the self-dual field theory. In particular, we give a general definition of the theta characteristic entering its partition function and settle the issue of its possible metric dependence. We treat the cohomological version of type IIB supergravity as an example of the formalism: a mixed gauge-gravitational global anomaly, occurring when the B-field and Ramond-Ramond 2-form gauge fields have non-trivial Wilson lines, cancels provided a certain cobordism group vanishes.