A Renormalized Riemann-roch Formula and the Thom Isomorphism for the Free Loop Space

Abstract. We show that the fixed-point formula in an equivariant complex-oriented cohomology theory E, applied to the free loop space of a manifold X, defines a (renormalized) Riemann-Roch formula for the quotient of the group law of E by a free cyclic subgroup. If E is K-theory, this explains how the elliptic genus associated to the Tate elliptic curve emerges from Witten’s analysis of the fixed-point formula in K-theory. [In general this quotient is not representable, but by using the theory of p-divisible groups, we show that its torsion subgroup is.] The equivariant Borel extensions of the cohomology theories associated to Lubin-Tate lifts provide a large class of new examples. For a general equivariant E, we show that the formal Weierstrass products defined by these quotients have a natural interpretation as Thom classes for prospectra similar to those considered by Cohen,Jones, and Segal. These prospectra seem to define interesting models for the physicists’ space of ‘small’ loops on a manifold. [For some earlier work along these lines, see Ando, Morava, and Sadofsky in Geometry and Topology 2 (1998).]