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 bulkboundary correspondence in a large class of time-reversal invariant gauge-gravity symmetryprotected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3 + 1 dimensions. • Dai-Freed theories describing chiral anomalies in odd dimensions d have been sketched in [Mon15]. They are an extended version of the field theories constructed in [DF95].• Wess-Zumino theories describing the anomaly in self-dual field theories have been constructed in [Mon15].• The theory describing the anomaly of the worldvolume theory of M5-branes has been constructed as an unextended quantum field theory in [Mon17].

Parallel transport of higher flat gerbes as an extended homotopy quantum field theory

“…The only non-trivial information contained in such a 2-functor are the coherence isomorphisms, which amount to a complex number α(g 1 , g 2 ) ∈ C × for every pair of composable morphisms g 1 and g 2 in Γ . These numbers form a groupoid 2-cocycle, see [Kir04] for the case of equivariant linear categories, [SW18-P] for general 2-vector bundles and [MS18a] for a detailed discussion of 2-line bundles. Given a 2-line bundle we can construct the corresponding 2-coycle by first choosing trivializations and then calculating the number corresponding to the coherence isomorphisms.…”

“…By chasing through the definition of parallel sections of a 2-vector bundle in [SW18-P] and using (3.15) and (3.16) in [MS18a], we can observe the following:…”

’t Hooft Anomalies of Discrete Gauge Theories and Non-abelian Group Cohomology

“…There are different definitions for natural symmetric monoidal 2-transformations corresponding to different levels of strictness. Here we use [MS18,Definition B.13], which seems to be best suited for physical applications.…”

“…are part of the structure corresponding to a symmetric monoidal 2-functor [MS18, Definition B.12]. There is a long list of coherence and compatibility conditions that these morphisms have to satisfy, see [MS18,Proposition 3.14] for details.…”

“…A straightforward but tedious calculation shows that these definitions satisfy [MS18,eqs. (3.18)-(3.22)].…”