Free-to-Interacting Maps and the Bott Spiral

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. Time permitting, I will also discuss how to model dimensional reduction and symmetry breaking for the Bott spiral in terms of homotopy theory.

 

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