Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors

TL;DR

This paper explores boundary conditions in extended topological quantum field theories, linking anomalies, homotopy fixed points, and dimensional reduction to understand how boundary conditions relate to anomalies and projective modular functors.

Contribution

It introduces the concept of TQFTs with moduli level m and connects anomalous theories to homotopy fixed points, providing a new framework for understanding boundary conditions in extended TQFTs.

Findings

Boundary conditions for extended TQFTs relate to topological anomalies.

Anomalous theories correspond to homotopy fixed points for n-characters.

Dimensional reduction links boundary conditions to lower-dimensional anomalous theories.

Abstract

We study boundary conditions for extended topological quantum field theories (TQFTs) and their relation to topological anomalies. We introduce the notion of TQFTs with moduli level $m$, and describe extended anomalous theories as natural transformations of invertible field theories of this type. We show how in such a framework anomalous theories give rise naturally to homotopy fixed points for $n$-characters on $\infty$-groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for $n+1$-dimensional TQFTs produce $n$-dimensional anomalous field theories. Finally, we analyse the case of fully extended TQFTs, and show that any fully extended anomalous theory produces a suitable boundary condition for the anomaly field theory.