Equivariant elliptic cohomology via loop spaces

Abstract

We construct a cohomology theory $H_A(-)$ canonically attached to any 1-dimensional commutative group scheme A over a connective $E_\infty$-ring. The additive group in characteristic zero gives 2-periodic de Rham cohomology, the multiplicative group gives topological periodic cyclic homology, and an elliptic curve gives elliptic cohomology. The construction extends to the G-equivariant setting for an algebraic group G, yielding in the elliptic case a theory of equivariant elliptic cohomology on algebraic varieties with G-action that takes values in sheaves on the stack of G-bundles. Over the complex numbers, and for G connected reductive, this lifts Grojnowski's theory from the moduli space of G-bundles to the stack. The key player is the category of perfect complexes on A with 0-dimensional support, equipped with the convolution product induced by the group law of A. It plays the role of a categorified circle and is used to define a loop space $L_A^G(X)$, whose structure sheaf gives rise to $H_A^G(X)$ after periodization. Sheaves on $L_A^G(X)$ equivariant under loop rotation provide a category of coefficients for $H_A^G(-)$ in the same way that D-modules provide coefficients for de Rham cohomology. At the point X = *, the theory also gives rise to an $E_\infty$-ring spectrum (or sheaf thereof) $K_A$, which recovers complex K-theory for $A = \mathbf{G}_m$, and the sheaf $\mathcal{O}^{top}$ of topological modular forms for the universal elliptic curve.