Anomaly cascade in (2+1)D fermionic topological phases

Abstract

We develop a theory of anomalies of fermionic topological phases of matter in (2+1)D with a general fermionic symmetry group $G_f$. In general, $G_f$ can be a non-trivial central extension of the bosonic symmetry group $G_b$ by fermion parity $(-1{)}^F$. We encounter four layers of obstructions to gauging the $G_f$ symmetry, which we dub the anomaly cascade: (i) An ${\mathcal{H}}^1(G_b,{\mathbb{Z}}_{\mathbf{T}})$ obstruction to extending the symmetry permutations on the anyons to the fermion parity gauged theory, (ii) An ${\mathcal{H}}^2(G_b,\ker r)$ obstruction to extending the $G_b$ group structure of the symmetry permutations to the fermion parity gauged theory, where $r$ is a map that restricts symmetries of the fermion parity gauged theory to the anyon theory, (iii) An ${\mathcal{H}}^3(G_b,{\mathbb{Z}}_2)$ obstruction to extending the symmetry fractionalization class to the fermion parity gauged theory, and (iv) the well-known ${\mathcal{H}}^4(G_b,U(1))$ obstruction to developing a consistent theory of $G_b$ symmetry defects for the fermion parity gauged theory. We describe how the ${\mathcal{H}}^2$ obstruction can be canceled by anomaly inflow from a bulk (3+1)D symmetry-protected topological state (SPT) and also its relation to the Arf invariant of spin structures on a torus. If any anomaly in the above sequence is non-trivial, the subsequent ones become relative anomalies. A number of conjectures regarding symmetry actions on super-modular categories, guided by general expectations of anomalies in physics, are also presented.

arXiv comment30+9 pages, 1+1 figures. v2: revised Summary of Results (Section I.A) and Section VI for clarity, Conjecture VI.4 changed to Eq. 80 and now proven, removed now-irrelevant Conjecture VI.3, fixed typos, updated acknowledgments and references. v3: updated and refined conjectures, various corrections and reorganization, added Appendix D

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• Non-perturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1)D
• Fractionalizing Global Symmetry on Looplike Topological Excitations
• Rotational Symmetry Protected Edge and Corner States in Abelian topological phases
• Enforced symmetry breaking by invertible topological order
• Fractionalization as an alternate to charge ordering in electronic insulators
• Towards a complete classification of non-chiral topological phases in 2D fermion systems
• Gapped boundary of (4+1)d beyond-cohomology bosonic SPT phase
• Anomaly of $(2+1)$-Dimensional Symmetry-Enriched Topological Order from $(3+1)$-Dimensional Topological Quantum Field Theory
• Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
• Ordering the topological order in the fractional quantum Hall effect
• Systematic Construction of Interfaces and Anomalous Boundaries for Fermionic Symmetry-Protected Topological Phases
• Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry
• Boundary topological orders of (4+1)d fermionic ${\mathbb{Z}}_{2N}^{\mathrm{F}}$ SPT states
• Exactly solvable models for fermionic symmetry-enriched topological phases and fermionic 't Hooft anomaly