Power operations in the Stolz--Teichner program

TL;DR

This paper develops a geometric framework for power operations in the Stolz--Teichner program, connecting geometric field theories with cohomology theories and providing computational tools for equivariant K-theory and elliptic cohomology.

Contribution

It introduces a theory of geometric power operations for geometric field theories and relates them to classical cohomology power operations, extending the Stolz--Teichner program.

Findings

Established relations between geometric and homotopical power operations.

Provided computational methods for equivariant K-theory.

Constructed power operations for equivariant elliptic cohomology.

Abstract

The Stolz--Teichner program proposes a deep connection between geometric field theories and certain cohomology theories. In this paper, we extend this connection by developing a theory of geometric power operations for geometric field theories restricted to closed bordisms. These operations satisfy relations analogous to the ones exhibited by their homotopical counterparts. We also provide computational tools to identify the geometrically defined operations with the usual power operations on complexified equivariant $K$-theory. Further, we use the geometric approach to construct power operations for complexified equivariant elliptic cohomology.