Absolute anomalies in (2+1)D symmetry-enriched topological states and exact (3+1)D constructions

Daniel Bulmash, Maissam Barkeshli

Certain patterns of symmetry fractionalization in (2+1)D topologically
ordered phases of matter can be anomalous, which means that they possess an
obstruction to being realized in purely (2+1)D. In this paper we demonstrate
how to compute the anomaly for symmetry-enriched topological (SET) states of
bosons in complete generality. We demonstrate how, given any unitary modular
tensor category (UMTC) and symmetry fractionalization class for a global
symmetry group $G$, one can define a (3+1)D topologically invariant path
integral in terms of a state sum for a $G$ symmetry-protected topological (SPT)
state. We present an exactly solvable Hamiltonian for the system and
demonstrate explicitly a (2+1)D $G$ symmetric surface termination that hosts
deconfined anyon excitations described by the given UMTC and symmetry
fractionalization class. We present concrete algorithms that can be used to
compute anomaly indicators in general. Our approach applies to general symmetry
groups, including anyon-permuting and anti-unitary symmetries. In addition to
providing a general way to compute the anomaly, our result also shows, by
explicit construction, that every symmetry fractionalization class for any UMTC
can be realized at the surface of a (3+1)D SPT state. As a byproduct, this
construction also provides a way of explicitly seeing how the algebraic data
that defines symmetry fractionalization in general arises in the context of
exactly solvable models. In the case of unitary orientation-preserving
symmetries, our results can also be viewed as providing a method to compute the
$\mathcal{H}^4(G, U(1))$ obstruction that arises in the theory of $G$-crossed
braided tensor categories, for which no general method has been presented to
date.