Elliptic loop spaces

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.