Even-periodic cohomology theories for twisted parametrized spectra

TL;DR

This paper explores conditions under which twisted parametrized spectra become untwisted over even-periodic $E_2$-rings, with implications for equivariant theories and gauge theoretic applications.

Contribution

It provides a sufficient condition for untwisting twisted parametrized spectra over even-periodic $E_2$-rings and investigates equivariant generalizations within $ abla$-categories.

Findings

Sufficient condition for untwisting spectra established

Application to gauge theoretic settings suggested

Framework for future generalizations of Floer homotopy theory

Abstract

We recall the notion of twisted parametrized spectra defined by Douglas and provide a sufficient condition for an $\infty$-category of twisted parametrized module spectra to be untwisted over an even-periodic $E_2$-ring. It is an easy consequence of the universal property of Thom spectra. We also investigate a genuine equivariant generalization based on the theory of $\infty$-categories internal to the $\infty$-topos of $G$-spaces for a compact Lie group $G$. We expect that our sufficient condition is satisfied in a number of gauge theoretic settings. This article is intended as a warm-up for a generalization of certain Seiberg-Witten Floer stable homotopy theory, which we look forward to.