Geometric quantization of N=2 superstring

Abstract

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 ΩMd‖d associated to the N=2 Neveu–Schwarz fermionic string, is invariant under symmetry group OSp(2‖2)⊆SuperdiffS1‖2. Moreover, it is proved that there is a unique Lorentz and OSp(2‖2) invariant complex structure on ΩMd‖d. This result implies that the superspace of all admissible complex structures over ΩMd‖d is isomorphic to the homogeneous Kähler 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.