Anomaly cascade in (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mn>2</mml:mn><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>)-dimensional 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 nontrivial central extension of the bosonic symmetry group ${G}_{b}$ by fermion parity ${(\ensuremath{-}1)}^{F}$. 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 ${\mathcal{H}}^{2}({G}_{b},kerr)$ 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 nontrivial, 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.

Topics & Concepts