In their study of the representation theory of loop groups, Pressley and Segal introduced a determinant line bundle over an infinite dimensional Grassmann manifold. Mickelsson and Rajeev subsequently generalized the work of Pressely and Segal to obtain representations of the groups Map(M, G) where M is an odd dimensional spin manifold. In the course of their work, Mickelsson and Rajeev introduc...

Any singularity free vector field X defined on an open set in a three-dimensional Euclidean space with curl X = 0 admits a complex line bundle Fa with a fibre-wise defined symplectic structure, a principal bundle Pa and a Heisenberg group bundle Ga . For the non-vanishing constant vector field X the geometry of Pa defines for each frequency a Schrödinger representation of any fibre of the Heise...

We show that holomorphic bundles on O(−k) for k ≥ 0 are algebraic. We also show that such bundles are trivial outside the zero section. 1 Preliminaires The line bundle on P given by transition function z is usually denoted O(−k). Since we will be studying bundles over this space, we will denote O(−k) by Mk when we want to view this space as the base of a bundle. We give Mk the following charts ...

In this paper, by using the regulator map of Beilinson-Deligne on a curve, we show that the quantization condition posed by Gukov is true for the SL2(C) character variety of the hyperbolic knot in S. Furthermore, we prove that the corresponding C∗valued closed 1-form is a secondary characteristic class (Chern-Simons) arising from the vanishing first Chern class of the flat line bundle over the ...