  There exists on each Teichm\"uller 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
Teichm\"uller 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.
