Extended quantum field theory, index theory and the parity anomaly

Abstract

We use techniques from functorial quantum field theory to provide a geometric description of the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes. We give an explicit construction of a geometric cobordism bicategory which incorporates general background fields in a stack, and together with the theory of symmetric monoidal bicategories we use it to provide the concrete forms of invertible extended quantum field theories which capture anomalies in both the path integral and Hamiltonian frameworks. Specialising this situation by using the extension of the Atiyah-Patodi-Singer index theorem to manifolds with corners due to Loya and Melrose, we obtain a new Hamiltonian perspective on the parity anomaly. We compute explicitly the 2-cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a parity-symmetric way by suitably augmenting the standard chiral fermionic Fock spaces with Lagrangian subspaces of zero modes of the Dirac Hamiltonian that naturally appear in the index theorem. We describe the significance of our constructions for the bulk-boundary correspondence in a large class of time-reversal invariant gauge-gravity symmetry-protected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3+1 dimensions.

63 pages, 3 figures; v2: clarifying comments and references added; Final version to be published in Communications in Mathematical Physics

References in corpus (4)

• Boundary Conditions for Topological Quantum Field Theories, Anomalies and Projective Modular Functors
• Corners in M-theory
• Index of Dirac operators and classification of topological insulators
• Anomalies and Invertible Field Theories

Cited by in corpus (6)

• Gravitational parity anomaly with and without boundaries
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• A note on the parity anomaly from the Hamiltonian point of view
• Lieb-Schultz-Mattis Theorem and the Filling Constraint
• The Chiral Anomaly of the Free Fermion in Functorial Field Theory
• Two-Dimensional Extended Homotopy Field Theories