Weil-Petersson geometry and determinant bundles on inductive limits of moduli spaces

1996/10/31 by Biswas, Indranil, Nag, Subhashis
#Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th)

paper · doi:10.48550/arxiv.alg-geom/9610020

Abstract

In an earlier paper [Acta Mathematica, v. 176, 1996, 145-169, alg-geom/9505024 ] the present authors and Dennis Sullivan constructed the universal direct system of the classical Teichmüller spaces of Riemann surfaces of varying genus. The direct limit, which we called the universal commensurability Teichmüller space, T_infty, was shown to carry on it a natural action of the universal commensurability mapping class group, MC_∞. In this paper we identify an interesting cofinal sub-system corresponding to the tower of finite-sheeted characteristic coverings over any fixed base surface. Utilizing a certain subgroup inside MC_∞, (associated intimately to the inverse system of characteristic coverings), we can now descend to an inductive system of moduli spaces, and construct the direct limit ind-variety M_∞. Invoking curvature properties of Quillen metrics on determinant bundles, and naturality under finite coverings of Weil-Petersson forms, we are able to construct on M_∞ the natural sequence of determinant of cohomology line bundles, as well as the Mumford isomorphisms connecting these.

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