Line bundles in equivariant elliptic cohomology

Given a compact Lie group G acting on a space X, the G-equivariant elliptic cohomology of X is naturally a scheme Ell_G(X) (with a map down to the moduli space of G-bundles on elliptic curves). Given a G-equivariant vector bundle V on X, one obtains an interesting line bundle Thom(V) on Ell_G(X). Both topologists and string theorists have predicted that given two vector bundles V_1, V_2 whose first Chern classes both vanish and whose second Chern classes agree, the resulting line bundles Thom(V_1) and Thom(V_2) should agree in Pic(Ell_G(X)). I'll describe how the theory of pushforwards in topology gives rise to this subtle question in algebraic geometry, and I hope to indicate in broad strokes the proof of this conjecture. This is joint work with D. Berwick-Evans.

The synchronous discussion for Arnav Tripathy’s talk is taking place not in zoom-chat, but at https://tinyurl.com/2021-10-01-at  (and will be deleted after ~3-7 days).