On the cobordism classification of symmetry protected topological phases

Understand

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.