Formal loops III: Additive functions and the Radon transform

To any algebraic variety X and closed 2-form omega on X, we associate the "symplectic action functional" T(omega) which is a function on the formal loop space LX introduced by the authors earlier. The correspondence omega -> T(omega) can be seen as a version of the Radon transform. We give a characterization of the functions of the form T(omega) in terms of factorizability (infinitesimal analog of additivity in holomorphic pairs of pants) as well As in terms of vertex operator algebras. These results will be used in the subsequent paper which will relate the gerbe of chiral differential operators on X (whose lien is the sheaf of closed 2-forms) and the determinantal gerbe of the tangent bundle of LX (whose lien is the sheaf of invertible functions on LX). On the level of liens this relation associates to a closed 2-form to the invertible function exp T(omega). (C) 2008 Elsevier Inc. All rights reserved.