Universal Hodge Bundle and Mumford Isomorphisms on the Abelian Inductive Limit of Teichm\"uller Spaces

Abstract

Let $\Mg$ denote the moduli space of compact Riemann surfaces of genus $g$. Mumford had proved that, for each fixed genus $g$, there are isomorphisms asserting that certain higher $DET$ bundles over $\Mg$ are certain fixed (genus-independent) tensor powers of the Hodge line bundle on $\Mg$. Here we obtain a coherent, genus-independent description of the Mumford isomorphisms over certain infinite-dimensional ``universal'' parameter spaces of compact Riemann surfaces (with or without marked points). We work with an inductive limit of Teichm\"uller spaces comprising complex structures on a certain ``solenoidal Riemann surface'', $H_{\infty,abel}$. The main result constructs these universal $DET$ and Hodge line bundles on this direct limit, equips them with their Quillen metrics, and proves that they are related by the appropriate Mumford isomorphisms. Our work can be viewed as a contribution towards a non-perturbative formulation of the bosonic string theory, since it provides an interpretation of the Polyakov measure in a genus-independent fashion.