Circle-equivariant classifying spaces and the rational equivariant sigma genus

We analyze the circle-equivariant spectrum MString_C which is the equivariant analogue of the cobordism spectrum MU<6> of stably almost complex manifolds with c_1=c_2=0. The second author has shown how to construct the ring T-spectrum EJ representing the T-equivariant elliptic cohomology associated to a rational elliptic curve J. In the case that J is a complex elliptic curve, we construct a map of ring T-spectra MString_C --> EJ which is the rational equivariant analogue of the sigma orientation of Ando-Hopkins-Strickland. Our method gives a proof of a conjecture of the first author.