Delocalized equivariant elliptic cohomology
by Ian Grojnowski
This is an old paper, which has been circulating quietly for almost a
decade.
It contains a definition of an equivariant elliptic cohomology theory
for compact connected Lie groups and reasonable topological spaces.
The theory is defined over Q, i.e. neglects torsion completely, and
yet was still interesting.
This is because of the well known heuristic identifying elliptic
cohomology with something like the K-theory of the loop space. The
functor of "loops into" is not local---there is no Mayer-Vietoris
style patching. Yet elliptic cohomology has such a property.
However the equivariant elliptic cohomology defined here does not
satisfy such a naive locality propery. Instead, the elliptic
cohomology of a space is a non-trivial bundle on the canonical abelian
variety associated to the group.
The crudest invariant of such a bundle is its first Chern class.
This is a combinatorial shadow of the failure of locality.
These same obstruction invariants occur in the study of semi-infinite
D-modules on the infinitesimal neighbourhood of formal loops in the
loop space of an algebraic variety; just as one would expect.
--
Since this paper was written there have been several developments.
Rosu and Ando used this theory to give a new proof of Witten rigidity,
and Greenlees constructed a model for that part of rational
equivariant S^1 homotopy that is seen by an elliptic cohomology
theory.
(There has also been the extraordinary work of Hopkins et al on tmf).