Elliptic loop spaces

TL;DR

This paper introduces an elliptic analogue of loop spaces in derived algebraic geometry, expanding the classical rational and trigonometric cases to include elliptic objects, and explores their categorical and cohomological properties.

Contribution

It defines an elliptic loop space as a stack of maps from an exotic elliptic curve avatar, completing the classical trichotomy and linking to equivariant elliptic Hodge cohomology.

Findings

Defines the elliptic loop space in derived algebraic geometry.

Establishes a connection to equivariant elliptic Hodge cohomology.

Completes the trichotomy of rational, trigonometric, and elliptic objects.

Abstract

We introduce an elliptic avatar of loop spaces in derived algebraic geometry, completing the familiar trichotomoy of rational, trigonometric and elliptic objects. Heuristically, the elliptic loop space of $\mathcal{Y}$ is the stack of maps to $\mathcal{Y}$ from a certain exotic avatar $\mathcal{S}_{E}$ of the elliptic curve $E$, such that the category of quasi-coherent sheaves on $\mathcal{S}_{E}$ is the convolution category of zero-dimensionally supported coherent sheaves on $E$. For quotient stacks, the structure sheaf of the elliptic loop space gives rise to a theory of equivariant elliptic Hodge cohomology.