Cochain valued TQFTs from nonsemisimple modular tensor categories

TL;DR

This paper extends a vector space valued TQFT to a cochain valued topological field theory using nonsemisimple modular tensor categories, connecting it with homotopy theory and derived categories for potential future developments.

Contribution

It introduces a cochain valued TQFT based on nonsemisimple modular tensor categories, extending previous work and incorporating homotopy invariance and derived categorical structures.

Findings

Constructed a symmetric monoidal functor to linear cochains

Identified surface values with Hom complexes in cochains

Derived 3-manifold invariants as sums of Lyubashenko invariants

Abstract

We show that a vector space valued TQFT constructed in work of De Renzi et al. [DGGPR23] extends naturally to a topological field theory which takes values in the symmetric monoidal category of linear cochains. Specifically, we consider a bordism category whose objects are surfaces with markings from the category of cochains Ch(A) over a given modular tensor category (such as the category of small quantum group representations), and whose morphisms are 3-dimensional bordisms with embedded ribbon graphs traveling between such marked surfaces. We construct a symmetric monoidal functor from the aforementioned ribbon bordism category to the category of linear cochains. The values of this theory on surfaces are identified with Hom complexes for Ch(A), and the 3-manifold invariants are alternating sums of the renormalized Lyubashenko invariant from [DGGPR23]. We show that our cochain valued TQFT furthermore preserves homotopies, and hence localizes to a theory which takes values in the derived $\infty$-category of dg vector spaces. The domain for this $\infty$-categorical theory is, up to some approximation, an $\infty$-category of ribbon bordism with labels in the homotopy $\infty$-category K(A). We suggest our localized theory as a starting point for the construction of a "derived TQFT" for the $\infty$-category of derived quantum group representations.