Classifying Spaces, Virasoro Equivariant Bundles, Elliptic Cohomology and Moonshine

This work explores some connections between the elliptic cohomology of classifying spaces for finite groups, Virasoro equivariant bundles over their loop spaces and Moonshine for finite groups. Our motivation is as follows: up to homotopy we can replace the loop group LBG by the disjoint union ⨿ [γ]BCG(γ) of classifying spaces of centralizers of elements γ representing conjugacy classes of elements in G. An elliptic object over LBG then becomes a compatible family of graded infinite dimensional representations of the subgroups CG(γ), which in turn defines an element in J. Devoto’s equivariant elliptic cohomology ring EllG. Up to localization (inversion of the order of G) and completion with respect to powers of the kernel of the homomorphism EllG −→ Ell{1}, this ring is isomorphic to Ell ∗(BG), see [5, 6]. Moreover, elliptic objects of this kind are already known: for example the 2-variable Thompson series forming part of the Moonshine associated with the simple groups M24 and the Monster M. Indeed the compatibility condition between the characters of the representations of CG(γ) mentioned above was originally formulated by S. Norton in an Appendix to [21], independently of the work on elliptic genera by various authors leading to the definition of elliptic cohomology (see the various contributions to [17]). In a slightly different direction, J-L. Brylinski [2] introduced the group of Virasoro equivariant bundles over the loop space LM of a simply connected manifold M as part of his investigation of a Dirac operator on LM with coefficients in a suitable vector bundle. Our suggested definition of an elliptic object (= equivariant bundle over a not necessarily simply-connected space) starts from this, and builds in additional structure suggested by Moonshine. We propose it as only provisional, for while it fits in well with Devoto’s construction, the localization which this requires suggests that, even in the very special case of X = BG, further refinement will be necessary to obtain a geometric definition of an elliptic-like cohomology theory. The following example may help the reader to follow our general construction. Accepting for the moment that up to completion and localization a class in Ell(BG) is represented by an infinite dimensional bundle over the loop space LBG, restriction to the subspace of constant loops defines a map Ell∗(BG) −→ K∗(BG)((q)). The image of the representation ring R(G) in the coefficients of the power series ring on the right is dense, giving a privileged position to representations whose characters satisfy a modularity condition. The 1-dimensional Thompson series of [4, 21, 32] are certainly of this type. For the Mathieu group M24 we have a particularly