The geometric cobordism hypothesis

TL;DR

This paper generalizes the cobordism hypothesis to include bordisms with various geometric structures, extending the classification of topological field theories to non-topological cases with additional geometric data.

Contribution

It extends the cobordism hypothesis to arbitrary geometric structures and generalizes the classification of invertible field theories beyond topological cases.

Findings

Proves a generalized cobordism hypothesis for geometric structures.

Reduces the problem to geometrically framed bordism categories.

Classifies non-topological field theories with geometric data.

Abstract

We prove a generalization of the cobordism hypothesis of Baez--Dolan and Hopkins--Lurie for bordisms with arbitrary geometric structures, such as Riemannian metrics, complex and symplectic structures, principal bundles with connections, or geometric string structures. Our methods rely on the locality property for fully extended functorial field theories established in arXiv:2011.01208, reducing the problem to the special case of geometrically framed bordism categories. As an application, we upgrade the classification of invertible fully extended topological field theories by B\"okstedt--Madsen and Schommer-Pries to nontopological field theories, generalizing the work of Galatius--Madsen--Tillmann--Weiss to arbitrary geometric structures.