On equivariant affineness in elliptic cohomology

Abstract

We investigate equivariant affineness in spectral algebraic geometry: the phenomenon where $\infty$-categories of quasi-coherent sheaves of a spectral stack are equivalent to $\infty$-categories of modules over an equivariant ring spectrum. In the nonequivariant case, this is known as 0-affineness, and has been studied by Gaitsgory, Mathew--Meier, amongst others. Allowing for equivariant phenomena opens up a much wider class of spectral stacks to be studied through the lens of spectra. The spectral stacks at the heart of our analysis are oriented elliptic curves, their associated Hilbert schemes, and related constructions. When working over the universal base, so the moduli stack of oriented elliptic curves, we show that these spectral stacks are $U(1)$-, $U(n)$-, and $SU(n)$-equivariantly affine, with associated $G$-ring spectrum Gepner--Meier's equivariant topological modular forms $\mathrm{TMF}_G$. Our results also encompass the groups $Sp(n)$ as well as all finite products of such groups. Our most general result is a relative statement over an arbitrary base stack. These results can be viewed as an integral and global refinement of Greenlees' algebraic model for modules over rational $U(1)$-equivariant elliptic cohomology.