  For a ring $R$, we construct a universal $K_R$-torsor $\mathcal{T}_R\to
K_{Tate(R)}$ on the $K$-theory space of Tate $R$-modules. This torsor is
closely related to canonical central extensions of loop groups. Just like
classical loop group theory has features of $K$-theory (e.g. determinant
bundles, tame symbol cocycle for Kac-Moody extension), the $K$-theory torsor
relates higher loop groups with higher $K$-theory. We study the classifying
"index" map of this torsor in detail. We explain how it arises in analogy with
the classical index map of Fredholm operators, and we relate the $K$-theory
torsor to previously studied dimension and determinant torsors.
