Extended quantum field theory, index theory and the parity anomaly

TL;DR

This paper develops a geometric and algebraic framework using functorial quantum field theory to describe the parity anomaly in fermionic systems, connecting index theory, cobordism bicategories, and topological phases.

Contribution

It introduces a new geometric cobordism bicategory incorporating background fields and constructs invertible extended quantum field theories capturing anomalies.

Findings

Explicit construction of a geometric cobordism bicategory with background fields

Derivation of a Hamiltonian perspective on the parity anomaly using index theory

Computation of the 2-cocycle for the gauge symmetry projective representation

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.