Determinant Bundles, Quillen Metrics, and Mumford Isomorphisms Over the Universal Commensurability Teichmüller Space

Abstract

There exists on each Teichmüller space $T_g$ (comprising compact Riemann surfaces of genus $g$), a natural sequence of determinant (of cohomology) line bundles, $DET_n$, related to each other via certain ``Mumford isomorphisms''. There is a remarkable connection, (Belavin-Knizhnik), between the Mumford isomorphisms and the existence of the Polyakov string measure on the Teichmüller space. This suggests the question of finding a genus-independent formulation of these bundles and their isomorphisms. In this paper we combine a Grothendieck-Riemann-Roch lemma with a new concept of $C^{*} \otimes Q$ bundles to construct such an universal version. Our universal objects exist over the universal space, $T_\infty$, which is the direct limit of the $T_g$ as the genus varies over the tower of all unbranched coverings of any base surface. The bundles and the connecting isomorphisms are equivariant with respect to the natural action of the universal commensurability modular group.

ACTA MATHEMATICA (to appear); finalised version with a note of clarification regarding the connection of the commensurability modular group with the virtual automorphism group of the fundamental group of a closed Riemann surface; 25 pages. LATEX