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 odddimensional 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 AtiyahPatodiSinger 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 2cocycle of the projective representation of the gauge symmetry on the quantum state space, which is defined in a paritysymmetric 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 timereversal invariant gaugegravity symmetryprotected topological phases of quantum matter with gapless charged boundary fermions, including the standard topological insulator in 3+1 dimensions.
Original language  English 

Pages (fromto)  10491109 
Number of pages  61 
Journal  Communications in Mathematical Physics 
Volume  362 
Issue number  3 
Early online date  12 Jun 2018 
DOIs  
Publication status  Published  Sep 2018 
Keywords
 hepth
 condmat.strel
 mathph
 math.CT
 math.DG
 math.MP
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Profiles

Richard Joseph Szabo
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)