Cameron Krulewski - Free-to-Interacting Maps and the Bott Spiral

September, 17th, 2024

In this talk, I will discuss free (i.e., noninteracting) and interacting classifications for certain fermionic symmetry-protected topological phases (SPTs) and show how to define free-to-interacting maps in terms of homotopy theory. I will apply these ideas to study the phenomenon of the "Bott spiral": as shown in work of Queiroz-Khalaf-Stern using a dimensional reduction approach, the tenfold way classification of free theories (with one additional reflection symmetry) breaks down to a large 2-torsion classification in the presence of interactions. Using K-theory and (Anderson-dual) twisted spin bordism, we can compute the same interacting classification, and with the language of fermionic groups, we can interpret the "spiral" as a failure of Morita invariance on the interacting side. The key ingredient to these maps are the twisted Atiyah-Bott-Shapiro maps, which I will discuss from multiple perspectives with examples. Time permitting, I will also discuss our definition of the interacting version of the dimensional reduction map in terms of topological field theory using a modified Smith homomorphism.

This talk is based on upcoming work joint with Arun Debray, Natalia Pacheco-Tallaj, and Luuk Stehouwer.