Field theories, stable homotopy theory and Khovanov homology

TL;DR

This paper explores converting 1+1-dimensional topological quantum field theories into stable homotopical data and applies this framework to recent refinements of Khovanov homology, including module structures and stable homotopy types.

Contribution

It introduces a method to translate TQFTs into stable homotopical data and applies it to recent Khovanov homology refinements, connecting quantum field theories with homotopy theory.

Findings

Conversion of TQFTs into stable homotopy data using Elmendorf-Mandell machinery

Refinement of Khovanov homology into a module over connective K-theory spectrum

Refinement of Khovanov homology into a stable homotopy type by Lipshitz and Sarkar

Abstract

In this paper, we discuss two topics: first, we show how to convert 1+1-topological quantum field theories valued in symmetric bimonoidal categories into stable homotopical data, using a machinery by Elmendorf and Mandell. Then, we discuss, in this framework, two recent results (independent of each other) on refinements of Khovanov homology: our refinement into a module over the connective k-theory spectrum and a stronger result by Lipshitz and Sarkar refining Khovanov homology into a stable homotopy type.