  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 Teichm\"uller space of Riemann surfaces of each fixed genus. The
description of $d(Poly)$ via the Mumford isomorphisms for determinant bundles
over Teichm\"uller 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 Teichm\"uller 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.

Algebraic Geometry;High Energy Physics - Theory;Algebraic Geometry