We develop a general framework for the description of anomalies using extended
functorial field theories extending previous work by Freed and Monnier. In this
framework, anomalies are described by invertible field theories in one dimension
higher and anomalous field theories live on their boundaries.
We provide precise mathematical definitions for all concepts involved using the
language of symmetric monoidal bicategories. In particular, field theories with
anomalies will be described by symmetric monoidal transformations. The use of
higher categorical concepts is necessary to capture the Hamiltonian picture of anomalies. The relation to the path integral and the Hamiltonian description of anomalies
will be explained in detail. Furthermore, we discuss anomaly inflow in detail.
We apply the general framework to the parity anomaly in fermionic systems coupled to background gauge and gravitational fields on odd-dimensional spacetimes.
We use the extension of the Atiyah-Patodi-Singer index theorem to manifolds with
corners due to Loya and Melrose to explicitly construct an extended invertible field
theory encoding the anomaly. This allows us to 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.
As a second application, we study discrete symmetries of Dijkgraaf-Witten theories and their gauging. Non-abelian group cohomology is used to describe discrete
symmetries and we derive concrete conditions for such a symmetry to admit ’t Hooft
anomalies in terms of the Lyndon-Hochschild-Serre spectral sequence. We give an
explicit realization of a discrete gauge theory with ’t Hooft anomaly as a state on the
boundary of a higher-dimensional Dijkgraaf-Witten theory using a relative version
of the pushforward construction of Schweigert and Woike.Doctoral Training Grant ST/N509099/1 UK Science and Technology Facilities Council (STFC)
