On the geometric quantization of bosonic string

Abstract

It is shown that the natural complex structure over the space of based loops, Omega Rd-1,1, in the Minkowski space Rd-1,1, which was used by Bowick and Rajeev (1987) in their geometric formulation of string field theories, is invariant not only under pure rotations (S1) but also under less evident symmetry group SL(2,R)Diff S1 (moreover, it is proved that there is a unique Lorentz and SL(2,R) invariant complex structures on Omega Rd-1,1). This implies that the space of all admissible complex structures over Omega Rd-1,1 is isomorphic to the manifold Diff S1/SL(2,R) rather than to Diff S1/S1 as was claimed by Bowick and Rajeev. The author shows that the method of geometric quantization, when applied to the open bosonic string theory along the lines suggested by Bowick and Rajeev, provides a representation of all reparametrization invariant string vacuum states in terms of antiholomorphic and horizontal sections of certain antiholomorphic vector bundles over Diff S1/SL(2,R); it is also shown that such sections exist only in dimension d=26.