An algebraic model for rational $T^2$-equivariant elliptic cohomology

TL;DR

This paper develops a rational $T^2$-equivariant elliptic cohomology theory based on algebraic models derived from an elliptic curve, extending prior circle-based constructions to the 2-torus.

Contribution

It constructs an algebraic model for rational $T^2$-equivariant elliptic cohomology, generalizing the circle case to the 2-torus using geometric and algebraic tools.

Findings

Constructed an object $EC_{T^2}$ in the algebraic model category $dA(T^2)$.

Established Quillen equivalence to rational $T^2$-spectra.

Extended elliptic cohomology models from circle to 2-torus.

Abstract

We construct a rational $T^2$-equivariant elliptic cohomology theory for the 2-torus $T^2$, starting from an elliptic curve C over the complex numbers and a coordinate data around the identity. The theory is defined by constructing an object $EC_{T^2}$ in the algebraic model category $dA(T^2)$, which by Greenlees and Shipley is Quillen-equivalent to rational $T^2$-spectra. This result is a generalization to the 2-torus of the construction [Gre05] for the circle. The object $EC_{T^2}$ is directly built using geometric inputs coming from the Cousin complex of the structure sheaf of the surface CxC.