  In 1994, Grojnowski gave a construction of an equivariant elliptic cohomology
theory associated to an elliptic curve over the complex numbers. Grojnowski's
construction has seen numerous applications in algebraic topology and geometric
representation theory, however the construction is somewhat ad hoc and there
has been significant interest in the question of its geometric interpretation.
We show that there are two global models for Grojnowski's theory, which shed
light on its geometric meaning. The first model is constructed as the
Borel-equivariant cohomology of a double free loop space, and is a holomorphic
version of a construction of Rezk from 2016. The second model is constructed as
the loop group-equivariant K-theory of a free loop space, and is a slight
modification of a construction given in 2014 by Kitchloo, motivated by ideas in
conformal field theory. We investigate the properties of each model and show
that each is isomorphic to Grojnowski's theory, for any given elliptic curve.
