Dimensional Reduction and Topological Invariants of Symmetry-Protected Topological Phases
Nathanan Tantivasadakarn
TL;DR
The paper develops a partition-function approach to distinguishing SPT phases for bosonic systems with finite abelian symmetry and fermionic Gu-Wen phases, by evaluating cocycles on closed manifolds with flat connections and, for fermions, on spin manifolds. It introduces a general dimensional reduction framework via slant products that enables a complete set of invariants in higher dimensions (notably 4+1D) and connects these invariants to braiding statistics of gauged theories, proving exact correspondences in the bosonic case and providing substantial evidence in the fermionic Gu-Wen setting. The main technical advance is constructing invariants such as $\\mathcal{I}_{ij}$, $\\mathcal{I}_{ijk}$ for bosonic SPTs and $\\mathcal{Z}$ for Gu-Wen fermions, and showing how dimensional reduction preserves the topological information necessary to distinguish all phases within the respective classifications. Together, these results deepen the link between cohomological classifications, manifold-based invariants, and braiding data, with implications for higher-dimensional SPTs and potential extensions to non-Abelian or non-unitary symmetries.
Abstract
We review the dimensional reduction procedure in the group cohomology classification of bosonic SPT phases with finite abelian unitary symmetry group. We then extend this to include general reductions of arbitrary dimensions and also extend the procedure to fermionic SPT phases described by the Gu-Wen super-cohomology model. We then show that we can define topological invariants as partition functions on certain closed orientable/spin manifolds equipped with a flat connection. The invariants are able to distinguish all phases described within the respective models. Finally, we establish a connection to invariants obtained from braiding statistics of the corresponding gauged theories.