Anomaly and invertible field theory with higher-form symmetry: Extended group cohomology

In the realm of invertible symmetry, the topological approach based on classifying spaces dominates the classification of 't Hooft anomalies and symmetry protected topological phases. In contrast, except for discrete 0-form symmetry, a systematic algebraic approach based on cochains remains poorly explored. In this work, we investigate the systematic algebraic approach for discrete higher-form symmetry with a trivial higher-group structure. Studying the discrete formulation of invertible field theories in arbitrary dimension, we extract a purely algebraic structure that we call extended group cohomology, which directly characterizes and classifies the lattice lagrangian of invertible field theories and the behavior of anomalous topological operators. Using techniques from simplicial homotopy theory, we prove the isomorphism between extended group cohomology and cohomology of classifying spaces. The proof is based on an explicit construction of Eilenberg-MacLane spaces and their products. Our findings also clarify the discrete formulation of a class of generalized Dijkgraaf-Witten-Yetter models.