Holomorphic geometry of non-perturbative superstring theories

Abstract

It is shown that the natural complex structure over the loop superspace, Ω M d|d NS, associated to the Neveu-Schwarz superstring, is invariant not only under pure rotations (S 1) but also under the less evident symmetry group OSp(1|2) ⊂ Superdiff S 1. [Moreover, it is proved that there is a unique Lorentz and OSp (1|2) invariant complex structure on Ω M d|d NS.] This result implies that the superspace of all admissible complex structures over Ω M d|d is isomorphic to the homogeneous Kähler supermanifold Superdiff S 1/OSp(1|2) rather than to Superdiff S 1/ S 1 as was stated by Harari et al. [Nucl. Phys. B 294 (1987) 556] and Zhao et al. [Phys. Lett. B 199 (1987) 37]. The Ricci curvature for Superdiff S 1/OSp(1|2) is calculated. Applying the method of geometric quantization to the Neveu-Schwarz superstring, we construct a representation of superstring vacua in terms of antiholomorphic and horizontal sections of a certain vector bundle over Superdiff S 1/OSp(1|2); it is proved that such sections exist only in dimension 10. We also perform a geometric quantization of the Ramond superstring theory. Again our conclusions do not match with those of Harari et al., because we use a different complex structure on the loop superspace Ω M d|d R associated with Ramond superstrings; this complex structure is responsible for the fermionic nature of the corresponding vacuum states.