Abstract: |
In the study of algebro-geometric analogue of free loop spaces, certain class of infinite dimensional vector bundles, called Tate vector bundles, plays an important role. They naturally arise, for instance, as the cotangent bundles of ind-schemes with nice properties, such as free loop spaces. In contrast to a finite dimensional vector bundle that has a determinant line bundle, a Tate vector bundle defines its determinant as a gerbe. That is, in the geometry of Tate vector bundles, one should deal with torsors not over a group sheaf, but instead over a group stack. In this talk I will give a clarified perspective on this "categorification phenomenon" from a K-theoretical point of view, implementing an idea sketched by Beilinson and Drinfeld. More precisely, I will show that the determinant gerbe of a Tate vector bundle can be considered as a piece of a larger structure called a torsor over K-theory, and prove that the moduli space of such torsors is equiva lent to the K-theory of Tate vector bundles. In the formulation and proof, I will use the recently developed theory of infinity-topoi, to combine an abstract delooping in K-theory, whose original form was a conjecture of Luigi Previdi, with the geometric fact, due to Drinfeld, that the first negative K-groups vanish Nisnevich locally. |