Extended Functorial Field Theories and Anomalies in Quantum Field Theories
arXiv:2003.08217
Abstract
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. 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.
187 pages, PhD Thesis based on arXiv:1709.03860, arXiv:1802.10455 and arXiv:1811.05446
References in corpus (35)
- Topological Insulators
- Topological insulators and superconductors
- Generalized Global Symmetries
- Symmetry protected topological orders and the group cohomology of their symmetry group
- Braiding statistics approach to symmetry-protected topological phases
- Coupling a QFT to a TQFT and Duality
- Fermion Path Integrals And Topological Phases
- Electromagnetic and gravitational responses and anomalies in topological insulators and superconductors
- Twisted equivariant matter
- Classifying gauge anomalies through SPT orders and classifying gravitational anomalies through topological orders
- Interacting topological phases and modular invariance
- Chern-Simons Theory with Finite Gauge Group
- The "Parity" Anomaly On An Unorientable Manifold
- Dai-Freed anomalies in particle physics
- The Classification of Two-Dimensional Extended Topological Field Theories
- Setting the quantum integrand of M-theory
- Non-Abelian String and Particle Braiding in Topological Order: Modular SL(3,Z) Representation and 3+1D Twisted Gauge Theory
- Anomalies of discrete symmetries in various dimensions and group cohomology
- On Gauging Symmetry of Modular Categories
- Constructing symmetric monoidal bicategories
- Bundle Gerbes Applied to Quantum Field Theory
- On time-reversal anomaly of 2+1d topological phases
- Short-range entanglement and invertible field theories
- Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter
- Orbifolds of n-dimensional defect TQFTs
- 3-dimensional defect TQFTs and their tricategories
- Cohomological Twisting of 2-Linearization and Extended TQFT
- Eilenberg-Watts calculus for finite categories and a bimodule Radford theorem
- Lecture Notes on Mathematical Methods of Classical Physics
- Double Bicategories and Double Cospans
- Lecture notes on 2-dimensional defect TQFT
- A note on the parity anomaly from the Hamiltonian point of view
- Orientation twisted homotopy field theories and twisted unoriented Dijkgraaf-Witten theory
- Mathematical remarks on the cohomology of gauge groups and anomalies
- Symmetries and defects in three-dimensional topological field theory