Circle-equivariant classifying spaces and the rational equivariant sigma genus

The circle-equivariant spectrum MStringℂ is the equivariant analogue of the cobordism spectrum MU 〈6〉 of stably almost complex manifolds with c₁ = c₂ = 0. Given a rational elliptic curve C, Greenlees (Topology 44:1213-1227, 2005) constructs a ring T-spectrum EC representing the associated T-equivariant elliptic cohomology. The core of the present paper is the construction, when C is a complex elliptic curve, of a map of ring T-spectra MStringℂ → EC which is the rational equivariant analogue of the sigma orientation of Ando et al. (Invent. Math. 146:595-687, 2001). We support this by a theory of characteristic classes for calculation, and a conceptual description in terms of algebraic geometry. In particular, we prove a conjecture the first author made in Ando (Geom. Topol. 7:91-153, 2003).