On the cobordism classification of symmetry protected topological phases

TL;DR

This paper proves that symmetry protected topological (SPT) phases without Hall effects are classified by cobordism invariants within the framework of topological quantum field theory, providing a direct and constructive approach.

Contribution

It offers a direct proof linking SPT phases to cobordism invariants and constructs corresponding topological field theories from these invariants.

Findings

Partition functions of SPT phases are cobordism invariants

Constructed unitary topological field theories from cobordism invariants

Theories with the same cobordism invariant are isomorphic

Abstract

In the framework of Atiyah's axioms of topological quantum field theory with unitarity, we give a direct proof of the fact that symmetry protected topological (SPT) phases without Hall effects are classified by cobordism invariants. We first show that the partition functions of those theories are cobordism invariants after a tuning of the Euler term. Conversely, for a given cobordism invariant, we construct a unitary topological field theory whose partition function is given by the cobordism invariant. Two theories having the same cobordism invariant partition functions are isomorphic.