Later, we also consider algebras of local operators $A$ that are more general than the tensor product algebras discussed so far and continue to call homomorphisms as in Eq. (5) $G$-reps.

If $G$ is continuous (e.g., a Lie group), one should require this homomorphism to be continuous, but this would require defining a sensible topology on $\text{LAut}(A)$, which we do not attempt to do in this paper.

Specifically, this follows from the classification of fermionic QCAs in one dimension [59]. The classifying group contains no elements of finite order, so the generator of a ${\mathbb{Z}}_2$ symmetry must have trivial QCA index. However, this implies that the ${\mathbb{Z}}_2$ symmetry cannot permute the trivial and nontrivial one-dimensional fermionic topological phases, which is the key property of the QFT anomaly.

In particular, any local algebra $A$ defines a local net of algebras in the sense of Ref. [73] by assigning $A\{S\}$ to each bounded set $S$, except that, in our definitions, we do not assume a norm on $A$, so, in particular, we cannot say it is a $C^{*}$-algebra (although we do indeed expect that, in all cases of physical interest, $A$ can be endowed with a norm such that the norm closure of $A$ is a $C^{*}$-algebra). Conversely, given any local net of algebras in a $C^{*}$-algebra $A$, we can define a local algebra $A^{\prime }={\cup }_{S\subseteq X\text{bounded}}A\{S\}$ with $a⊲S\iff$ ($\exists$ bounded $S^{\prime }\subseteq S$ with $a\in A\{S^{\prime }\}$), except that the axioms of Ref. [73] do not necessarily imply the locally finite-dimensional condition.

In a finite system, by “boundary states” we mean the projected states defined in Sec. 7b, in the case where the projected algebra is a boundary algebra. One can show (also in infinite systems, using the methods of Appendix pp7) that an automorphism of the boundary algebra induces an automorphism of boundary states.

Of course, there is no real spontaneous symmetry breaking in $d=0$, but the criterion in terms of the symmetry transformation of the LIOMs can be applied equally well in $d=0$.

In a general local algebra, $a\in A$ is called positive if $a={\gamma }^{†}\gamma$ for some $\gamma \in A$. For invertible local algebras, this is equivalent to saying that $a\otimes \mathbb{I}$ maps to a positive operator under the local isomorphism from $A\otimes \overline{A}$ to an on-site algebra.

One subtlety is that it is conceivable that, in order for the edge map from $I_{d+1}^G\to {\text{Anom}}_d^{\mathrm{QFT}}(G)$ to be an isomorphism, one should consider equivalences of invertible states under a generalization of finite-depth circuits allowing for rapidly decaying tails rather than equivalences by strictly local circuits. However, in the rest of the paper, we do not adopt such a definition of $I_{d+1}^G$.

The final step in Appendix pp6 was left implicit, namely, showing that if we have a local isomorphism between two on-site algebras, then those algebras can be taken to be the same. The symmetry-enriched analog is to show that if two on-site $G$-reps are conjugate, then they can be taken to be the same after rearranging sites. This follows from a generalization of Appendix A in Ref. [32], replacing the matching of on-site Hilbert space dimensions with a matching of symmetry representations.

One could also allow the $h_i$’s to get permuted among themselves by the symmetry. For internal symmetries, this would not affect the conclusions. However, in the case of noninternal symmetries—such as translation symmetry—where the orbits of the permutation are not supported on bounded regions, the lattice anomaly of the permuting symmetry need not be preserved under the projection.

Start with a loop-condensate model for the toric code with the constraint that there must be an odd number of occupied edges incident on each vertex (in order to realize the required translation symmetry fractionalization) and decorate occupied edges with SO(3) SPTs.

The original Walker-Wang construction was only in three spatial dimensions. However, presumably there should be generalizations in any dimension.

$\mathrm{LAut}(*)$ can be constructed as a direct limit of the algebras $\mathrm{LAut}(M)$, where $M$ grows by stacking ancillas.

The conjecture Eq. (113) must be modified to include twisted generalized cohomology groups for antiunitary symmetries or for fermionic systems when the group $G$ involves a nontrivial extension of a bosonic symmetry by fermion parity. The same is true for lattice anomalies in Eq. (114). We do not describe these cases, beyond noting that they can be dealt with.

For infinite groups, one may need to allow for $N=N_g$ to depend on the group element, with different $N_g$ all related by stacking or removing ancillas. Any two unitaries ${\tilde{\gamma }}_g(t)$ and ${\tilde{\gamma }}_h(t)$ can still be multiplied by considering them to act on a common Hilbert space with all the ancillas from both $N_g$ and $N_h$.

For elements $a,b\in A$, $a\le b$ means that $b-a$ is positive.