Weak topological phases in the presence of interactions

We investigate the stability of weak symmetry-protected topological phases (SPTs) in the presence of short-range interactions, focusing on the tenfold way classification. Using Atiyah's Real $\mathit{KR}$-theory and Anderson-dualized bordism, we classify free and interacting weak phases across all Altland-Zirnbauer symmetry classes in low dimensions. Extending the free-to-interacting map of Freed-Hopkins, we mathematically compute how the behavior of free weak SPTs changes when interactions are introduced as well as predict intrinsically-interacting weak phases in certain classes. Our mathematical techniques involve T-duality and the James splitting of the torus. Our results provide a mathematical framework for understanding the persistence of weak SPTs under interactions, with potential implications for experimental and theoretical studies of these phases.