An infinity-categorical TQFT from instantons

Abstract

In this paper, we lift the instanton TQFT from ordinary categories to the $\infty$-categorical setting by constructing a functor $CI:{\mathrm{BI}}^{\partial }⟶\mathsf{D}$ from an $\infty$-cobordism category ${\mathrm{BI}}^{\partial }$ for instantons to an $\infty$-derived category $\mathsf{D}$ of $2$-periodic chain complexes. The construction of ${\mathrm{BI}}^{\partial }$ is a modification of the $\infty$-cobordism category ${\mathrm{Bord}}_4$ introduced by Lurie and by Calaque--Scheimbauer using complete Segal spaces. The target $\infty$-category $\mathsf{D}$ is obtained via the dg nerve of the dg category of $2$-periodic chain complexes of finitely presented projective $\mathbb{Z}$-modules. The information encoded by the functor $CI$ was already developed by Kronheimer--Mrowka using families of metrics on cobordisms. Our reinterpretation in terms of $\infty$-categories conceptually simplifies the construction of the hypercube complexes underlying the link spectral sequence. This viewpoint also clarifies hyperbox complexes and compression in Manolescu--Ozsváth's work, and possible $A_{\infty }$-structures, especially their signs and higher coherence. In addition, we lift the generalized cap product $\mu$-operators in instanton Floer homology to the chain level and construct explicit homotopies and higher homotopies establishing the commutativity of multiple $\mu$-operators in even degrees.

arXiv comment130 pages, 6 figures. Many details expanded relative to v1. Appendix C by Longke Tang