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

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.

arXiv comment26 pages, 6 figures. Exposition improved, bibliography updated. Final version, to appear in Comm. Math. Phys

References in corpus (9)

• Higher-dimensional Algebra and Topological Quantum Field Theory
• Topological boundary conditions in abelian Chern-Simons theory
• Field theory representation of gauge-gravity symmetry-protected topological invariants, group cohomology and beyond
• Bicategories for boundary conditions and for surface defects in 3-d TFT
• On the structure of open-closed topological field theory in two dimensions
• Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras
• Cohomological Twisting of 2-Linearization and Extended TQFT
• A geometric approach to boundaries and surface defects in Dijkgraaf-Witten theories
• AQFT from n-functorial QFT

Cited by in corpus (15)

• Notes on generalized global symmetries in QFT
• A theory of 2+1D bosonic topological orders
• Symmetry as a shadow of topological order and a derivation of topological holographic principle
• Non-invertible anomalies and mapping-class-group transformation of anomalous partition functions
• Holographic theory for continuous phase transitions -- the emergence and symmetry protection of gaplessness
• (Op)lax natural transformations, twisted quantum field theories, and "even higher" Morita categories
• A Non-Perturbative Definition of the Standard Models
• Remarks on the Green-Schwarz terms of six-dimensional supergravity theories
• Categories of quantum liquids I
• On dualizability of braided tensor categories
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Gapped domain walls between 2+1D topologically ordered states
• Extended quantum field theory, index theory and the parity anomaly
• Metallic states beyond Tomonaga-Luttinger liquids in one dimension
• Supersymmetric field theories from twisted vector bundles