Extended field theories are local and have classifying spaces

TL;DR

This paper proves that all extended functorial field theories are local, constructs a classifying space for geometric data, and develops a geometric theory of power operations, addressing a conjecture and advancing the understanding of field theories.

Contribution

It establishes the locality of extended functorial field theories and constructs a classifying space for their geometric data, solving a conjecture and enabling new geometric power operations.

Findings

All extended functorial field theories are local.

Constructed a classifying space for geometric data of field theories.

Developed a geometric theory of power operations.

Abstract

We show that all extended functorial field theories, both topological and nontopological, are local. We define the smooth (infinity,d)-category of bordisms with geometric data, such as Riemannian metrics or geometric string structures, and prove that it satisfies codescent with respect to the target S, which implies the locality property. We apply this result to construct a classifying space for concordance classes of functorial field theories with geometric data, solving a conjecture of Stolz and Teichner about the existence of such a space. We use our classifying space construction to develop a geometric theory of power operations, following the recent work of Barthel, Berwick-Evans, and Stapleton.