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

TL;DR

This paper develops an algebraic cochain-based framework for classifying invertible field theories with higher-form symmetries, generalizing group cohomology and connecting it to classifying space cohomology.

Contribution

It extends the algebraic approach to finite higher-form symmetries with trivial higher-group structure, linking it to classifying space cohomology via simplicial homotopy theory.

Findings

Generalization of algebraic cochains to higher-form symmetries

Establishment of isomorphism with classifying space cohomology

Explicit construction of Eilenberg-MacLane spaces

Abstract

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. We explore the alternative algebraic approach based on cochains that directly characterize the lattice lagrangian of invertible field theories and the anomalous phase factor of topological operator rearrangements. In the current literature, the algebraic approach has been systematically described for only finite 0-form symmetries. In this initial work, we generalize it to finite higher-form symmetries with trivial higher-group structure. We carefully analyze the algebraic cochains and abstract a purely algebraic structure that naturally generalizes group cohomology. Using techniques from simplicial homotopy theory, we show its isomorphism to the cohomology of classifying spaces. The proof is based on an explicit construction of Eilenberg-MacLane spaces and their products.