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

Domenico Fiorenza, Alessandro Valentino

TL;DR

This work develops a unified, higher-categorical framework linking boundary conditions, anomalies, and modular structures in topological quantum field theories (TQFTs). By introducing TQFTs with moduli level $m$ and treating anomalies as invertible TQFTs of moduli level $1$, it shows that anomalous $n$-dimensional TQFTs arise as homotopy fixed points for $n$-characters on $ ext{∞}$-groups and can be related to boundary data via dimensional reduction. The authors prove that, in the fully extended setting, anomalous theories correspond to boundary conditions for the anomaly theory, providing a precise mechanism to derive projective representations of mapping class groups and linking boundary phenomena to higher-categorical invariants. These results offer a rigorous bridge between boundary conditions, anomalies, and fully extended TQFTs, with potential implications for RT/TV theories and Crane–Yetter constructions, and they suggest avenues for applying this framework to the quantization of classical field theories and higher defects.

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.

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

TL;DR

This work develops a unified, higher-categorical framework linking boundary conditions, anomalies, and modular structures in topological quantum field theories (TQFTs). By introducing TQFTs with moduli level and treating anomalies as invertible TQFTs of moduli level , it shows that anomalous -dimensional TQFTs arise as homotopy fixed points for -characters on -groups and can be related to boundary data via dimensional reduction. The authors prove that, in the fully extended setting, anomalous theories correspond to boundary conditions for the anomaly theory, providing a precise mechanism to derive projective representations of mapping class groups and linking boundary phenomena to higher-categorical invariants. These results offer a rigorous bridge between boundary conditions, anomalies, and fully extended TQFTs, with potential implications for RT/TV theories and Crane–Yetter constructions, and they suggest avenues for applying this framework to the quantization of classical field theories and higher defects.

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 , 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 -characters on -groups. By using dimensional reduction on manifolds with boundaries, we show how boundary conditions for -dimensional TQFTs produce -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.

Paper Structure

This paper contains 18 sections, 11 theorems, 60 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary notions on higher category theory
  3. Cobordism $(\infty,k)$-categories
  4. Topological Quantum Field Theories
  5. TQFTs with moduli level
  6. Fully extended TQFTs
  7. Anomalies in Topological Quantum Field Theories
  8. Invertible TQFTs
  9. Anomalies
  10. $n$-characters and projective representations
  11. Discrete groups
  12. 2-characters for $\infty$-groups
  13. Projective representations from TQFTs
  14. Boundary conditions for TQFTs
  15. Boundary conditions
  16. ...and 3 more sections

Key Result

Lemma 2.6

The formation of the homotopy $n$-category is the adjoint $\infty$-functor to the inclusion of $n$-discrete categories into $(\infty,n)$-categories, i.e., if $\mathcal{C}$ and $\mathcal{D}$ are $(\infty,n)$-categories, with $\mathcal{D}$ discrete, then one has a natural equivalence of $\infty$-categ

Figures (1)

  • Figure :

Theorems & Definitions (78)

  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Remark 2.5
  • Lemma 2.6
  • Example 2.7
  • Lemma 2.8
  • proof
  • Lemma 2.9
  • ...and 68 more