An infinity-categorical TQFT from instantons

Abstract

In this paper, we upgrade the instanton TQFT from ordinary categories to a functor $CI$ from an $\infty$-cobordism category $\mathrm{BI}$ for instantons to an $\infty$-derived category $\mathsf{D}$ of $2$-periodic chain complexes and sums of homogeneous chain maps. The construction of $\mathrm{BI}$ is a modification of the $\infty$-cobordism category $\mathrm{Bord}_4$ constructed by Lurie and Calaque--Scheimbauer via complete Segal spaces. The construction of $\mathsf{D}$ follows from the dg-nerve of a dg-category of $2$-periodic chain complexes over finitely generated projective modules over $\mathbb{Z}$. The information encoded in the functor $CI$ was already developed by Kronheimer--Mrowka using families of metrics on cobordisms, but our reinterpretation through $\infty$-categories simplifies the construction of the hypercube of chain complexes for the link spectral sequence. In addition, we upgrade the generalized cap product $\mu$-operators in instanton Floer homology to the chain level and construct explicit homotopies and higher homotopies for commutativity of multiple $\mu$-operators in even degrees.