Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-12-14
Physics
High Energy Physics
High Energy Physics - Theory
37 pages, LATEX
Scientific paper
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.
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