GEOMETRIC-QUANTIZATION OF N=2 SUPERSTRING

A nonperturbative geometric formulation of the N = 2 Neveu-Schwarz superstring theory which has been recently interpreted by H. Ooguri and C. Vafa [Mod. Phys. Lett. A5, 1389 (1990)] as a consistent quantum theory of self-dual gravity in four dimensions, is constructed. It is shown that the natural complex structure over the loop superspace OMEGA-M(d/d) associated to the N = 2 Neveu-Schwarz fermionic string, is invariant under symmetry group OSp(2/2) subset-of SuperdiffS1/2. Moreover, it is proved that there is a unique Lorentz and OSp(212) invariant complex structure on OMEGA-M(d/d) . This result implies that the superspace of all admissible complex structures over OMEGA-M(d/d) is isomorphic to the homogeneous Kahler supermanifold SuperdiffS1/2/OSp(2/2). The Ricci curvature of SuperdiffS1/2/OSp(2/2) is calculated. Applying the method of geometric quantization to the N = 2 Neveu-Schwarz superstring theory along the lines suggested by M. J. Bowick and S. G. Rajeev [Nucl. Phys. B361, 469 (1991)], a representation is constructed of nonperturbative N = 2 superstring vacua in terms of antiholomorphic and horizontal sections of a certain vector bundle over SuperdiffS1/2/OSp(2/2); it is proved that such sections exist only in complex dimension d = 2.