Summary: 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. A 5, 1389-1398 (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 \(\Omega\mathbb{M}^{d| d}\) associated to the \(N=2\) Neveu- Schwarz fermionic string, is invariant under symmetry group \(OSp(2\mid 2)\subset\text{Superdiff} S^{1/2}\). Moreover, it is proved that there is a unique Lorentz and \(OSp(2\mid 2)\) invariant complex structure on \(\Omega\mathbb{M}^{d\mid d}\). This result implies that the superspace of all admissible complex structures over \(\Omega\mathbb{M}^{d\mid d}\) is isomorphic to the homogeneous Kähler supermanifold \(\text{Superdiff} S^{1\mid 2}/OSp(2\mid 2)\). The Ricci curvature of \(\text{Superdiff} S^{1\mid 2}/OSp(2\mid 2)\) 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. B 361, 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 \(\text{Superdiff} S^{1\mid 2}/OSp(2\mid 2)\); it is proved that such sections exist only in complex dimension \(d=2\).