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It is well known that (1+1)-D bosonic symmetry-protected topological (SPT)phases with symmetry group $G$ can be identified by the projectiverepresentation of the symmetry at the edge. Here, we generalize this result tohigher dimensions. We assume that the representation of the symmetry on thespatial edge of a ($d+1$)-D SPT is /local/ but not necessarily /on-site/, suchthat there is an obstruction to its implementation on a region with boundary.We show that such obstructions are classified by the cohomology group$H^{d+1}(G, U(1))$, in agreement with the classification of bosonic SPT phasesproposed in [Chen et al, Science 338, 1604 (2012)]. Our analysis allows for astraightforward calculation of the element of $H^{d+1}(G, U(1))$ correspondingto physically meaningful models such as non-linear sigma models with a thetaterm in the action. SPT phases outside the classification of Chen et al arethose in which the symmetry cannot be represented locally on the edge. Withsome modifications, our framework can also be applied to fermionic systems in(2+1)-D.

Classifying symmetry-protected topological phases through the anomalous action of the symmetry on the edge