Enforced symmetry breaking by invertible topological order

It is well known that two-dimensional fermionic systems with a nonzero Chern number must break the time reversal symmetry, manifested by the appearance of chiral edge modes on an open boundary. Such an incompatibility between topology and symmetry can occur more generally. We will refer to this phenomenon as enforced symmetry breaking by topological orders. In this work, we systematically study enforced breaking of a general finite group $G_f$ by a class of topological orders, namely 0D, 1D and 2D fermionic invertible topological orders. Mathematically, the symmetry group $G_f$ is a central extension of a bosonic group $G$ by the fermion parity group $Z_2^f$, characterized by a 2-cocycle $\lambda$ $\in H^2(G,Z_2)$. With some minor assumptions and for given $G$ and $\lambda$, we are able to obtain a series of criteria on the existence or non-existence of enforced symmetry breaking by the fermionic invertible topological orders. Using these criteria, we discover many examples that are not known previously. For 2D systems, we define the physical quantities to describe symmetry-enriched invertible topological orders and derive some obstruction functions using both fermionic and bosonic languages. In the latter case which is done via gauging the fermion parity, we find that some obstruction functions are consequences of conditional anomalies of the bosonic symmetry-enriched topological states, with the conditions inherited from the original fermionic system. We also study enforced breaking of the continuous group $SU_f(N)$ by 2D invertible topological orders through a different argument.