Even-periodic cohomology theories for twisted parametrized spectra

We recall the notion of twisted parametrized spectra defined by Douglas and provide a sufficient condition for an $\mathrm{\infty }$-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 $\mathrm{\infty }$-categories internal to the $\mathrm{\infty }$-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.