Topological Field Theory Coupled to Parameter Spaces

Topological quantum field theories (TQFTs) describe the IR fixed points of wide classes of gapped theories and are useful for studying many-body quantum phases of matter.
In this talk, I will talk about TQFTs coupled to parameter spaces. I first explain the motivation for studying such TQFTs with parameter spaces from two perspectives: generalizing the description of the partition function with background gauge fields, and generalizing to invariants of many-body gapped phases over parameter spaces, known as the Berry phase.
Then I will explain how these two are related by showing two physically motivated maps that connect them. The construction of these maps provides physical evidence for the Cobordism Hypothesis. I also discuss other related topics, such as the bulk-boundary correspondence.
The talk is based on my ongoing work with Ryan Thorngren (UCLA).