Symmetry Protected Topological Phases, Anomalies, and Cobordisms: Beyond Group Cohomology

TL;DR

This paper introduces a cobordism-based classification for symmetry protected topological phases with finite symmetry groups, explaining recent discoveries beyond traditional group cohomology methods and linking to anomalies.

Contribution

It proposes a novel cobordism framework for classifying SPT phases, extending beyond group cohomology and connecting to gauge and gravitational anomalies.

Findings

Classifies SPT phases using cobordism groups of classifying spaces.

Explains bosonic SPT phases not captured by group cohomology.

Links SPT classification to gauge and gravitational anomalies.

Abstract

We propose that Symmetry Protected Topological Phases with a finite symmetry group G are classified by cobordism groups of the classifying space of G. This provides an explanation for the recent discovery of bosonic SPT phases which do not fit into the group cohomology classification. We discuss the connection of the cobordism classification of SPT phases to gauge and gravitational anomalies in various dimensions.