Title:
A renormalized Riemann-Roch formula and the Thom isomorphism for the
free loop space
Authors:
Matthew Ando
University of Illinois at Urbana-Champaign
mando@math.uiuc.edu
Jack Morava
Johns Hopkins University
jack@math.jhu.edu
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 $X$, may be viewed as 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 we show that its torsion subgroup is. In the
case that $E$ is the Borel theory associated to the Lubin-Tate theory
$E_n$, this leads to a description of the functor represented by
$E_n[[q]], analogous to the relationship between the Tate curve and
$K$-theory. For a more general equivariant $E$, we show that the
formal products which arise in this discussion may be naturally
viewed as Thom classes for Thom prospectra as considered by
Cohen-Jones-Segal. These prospectra seem to define interesting models
for the physicists' space of `small' loops on $X$.