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

TL;DR

This work develops a cobordism-based framework for classifying bosonic SPT phases with finite symmetry, extending beyond group cohomology by using oriented cobordism groups $\Omega^{d}_{SO}(BG,U(1))$ and their twists for time-reversal. It shows that, in dimensions up to six, cobordism yields a finer, sometimes qualitatively different, classification with explicit low-dimension examples (notably $d=4$ and $d=5$ with $\mathbb{Z}_2^T$) and connects these invariants to 't Hooft anomalies via inflow. The paper provides concrete boundary theories that cancel bulk anomalies and demonstrates how higher Stiefel-Whitney classes reveal new SPT phases invisible to group cohomology. It also discusses extensions to fermionic systems via spin cobordism and touches on relations to K-theory, outlining future research directions.

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.