Unraveling the Bott spiral

TL;DR

This paper develops a homotopy-theoretic model for the Bott spiral in symmetry-protected topological phases, connecting free and interacting fermionic SPTs through advanced mathematical tools and answering key questions about dimensional reduction.

Contribution

It introduces a twisted generalization of the Atiyah--Bott--Shapiro orientation and computes spiral maps of invertible field theories, advancing understanding of free-to-interacting transitions in SPTs.

Findings

IFTs are more sensitive to symmetry data than K-theory.

Bott periodicity in interactions relies on an isomorphism of extraspecial groups.

A novel 4-periodic description of twisted ko-homology was used.

Abstract

We construct and compute a homotopy-theoretic model for the Bott spiral of symmetry-protected topological phases (SPTs) studied by Queiroz--Khalaf--Stern. We model free and interacting fermionic SPTs using K-theory and reflection-positive invertible field theories (IFTs), resp., and define a twisted generalization of the Atiyah--Bott--Shapiro orientation to produce a free-to-interacting map. We also define and compute spiral maps of IFTs to model dimensional reduction in this context, answering a question of Hason--Komargodski--Thorngren. Our analysis highlights two general aspects of homotopical free-to-interacting maps. First, IFTs are more sensitive than K-theory is to the input symmetry data; in particular, the specification of an Altland--Zirnbauer class is insufficient information to define symmetry type for an IFT. Second, the remnant of Bott periodicity on the interacting side relies on an isomorphism of two extraspecial groups of order 32. Our computations use a novel 4-periodic description of a sector of the twisted ko-homology of elementary abelian 2-groups.