Current algebras in d + 1-dimensions and determinant bundles over infinite-dimensional Grassmannians

We extend the methods of Pressley and Segal for constructing cocycle representations of the restricted general linear group in infinite-dimensions to the case of a larger linear group modeled by Schatten classes of rank 1 ≤ p < ∞. An essential ingredient is the generalization of the determinant line bundle over an infinite-dimensional Grassmannian to the case of an arbitrary Schatten rank, p≥1. The results are used to obtain highest weight representations of current algebras (with the operator Schwinger terms) in d+1-dimensions when the space dimension d is any odd number. (orig.)