A renormalized Riemann-Roch formula and the Thom isomorphism for the free loop space (Q2765018)

This paper is motivated by Witten's results relating applications of fixed-point formulae on the free loop space \(LX\) of a compact smooth manifold \(X\) to the topology of \(X\). Starting with ordinary equivariant cohomology on \(LX\), one obtains K-theoretic objects on \(X\), while starting with equivariant K-theory on \(LX\) one obtains quantities related to elliptic cohomology. In view of the chromatic program, these three cohomology theories detect the first three layers of stable homotopy types, so one hopes that Witten's construction in general relates theories detecting the \(n\)th layer to quantities associated to theories detecting the \((n+1)\)st layer. In this paper, the authors consider a general even periodic complex oriented cohomology theory \(E\) with formal group law \(F\). Their first result is that the fixed point formula for an appropriate equivariant extension of \(E\) is identical to the Riemann-Roch formula for the quotient of \(F\) by a suitable free cyclic subgroup \((\hat q)\). This quotient is studied using a formal group scheme \(\text{Tate}(F)\), whose torsion part is used as an approximation of \(F/(\hat q)\). Locally over a prime \(p\), the formal group scheme \(\text{Tate}(F)\) can be characterized as a universal extension. Finally, the authors show that their results fit nicely into the framework of Thom prospectra.NEWLINENEWLINEFor the entire collection see [Zbl 0972.00054].