Mathematics in and out of String Theory

Abstract

We first give an exposition of how the Polyakov path integral for the bosonic string produces a natural mapping class group invariant measure, $d(Poly)$, on the Teichmüller space of Riemann surfaces of each fixed genus. The description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles over Teichmüller space is also explained. We then report on our recent results with I.Biswas and D.Sullivan, where we succeed in coherently fitting together this Polyakov-Mumford construction over the universal direct limit, $T_\infty$, of all the finite genus Teichmüller spaces. The virtual automorphism group of the fundamental group of a compact surface (of arbitrary genus $g > 1$) acts by automorphisms on $T_\infty$ as the ``commensurability mapping class group''. Our entire construction is equivariant with respect to this new modular group.