It is a classical result attributed to Weil that the moduli space of G-bundles on a curve over a finite field admits a description in terms of arithmetic quotients of the adelic points of the group G. Replacing G by the centrally extended loop group LG^, we show how the corresponding loop arithmetic quotients parametrize G-bundles on certain algebraic surfaces together with the information of a relative second Chern class. The main ingredient in our construction is a certain adelic Riemann-Roch type formula linking central extensions of loop groups with second Chern classes. This is joint work with Howard Garland.