W gravity, N=2 strings, and 2+2 SU*(∞) Yang–Mills instantons

Abstract

It is conjectured that W gravity can be interpreted as the gauge theory of φ diffeomorphisms in the space of dimensionally reduced D=2+2 SU*(∞) Yang–Mills instantons. These φ diffeomorphisms preserve a volume-three form and are those that furnish the correspondence between the dimensionally reduced Plebanski equation and the KP equation in (1+2) dimensions. A supersymmetric extension furnishes super-W gravity. The super-Plebanski equation generates self-dual complexified super gravitational backgrounds (SDSG) in terms of the super-Plebanski second heavenly form. Since the latter equation yields N=1, D=4 SDSG complexified backgrounds associated with the complexified-cotangent space of the Riemannian surface, (T*Σ)c, required in the formulation of SU*(∞) complexified Self-Dual Yang-Mills theory, (SDYM); it naturally follows that the recently constructed D=2+2 N=4 SDSYM theory, as the consistent background of the open N=2 superstring, can be embedded into the N=1 SU*(∞) complexified self-dual-super-Yang–Mills (SDSYM) in D=3+3 dimensions. This is achieved after using a generalization of self-duality for D≳4. Finally, the plausible relationship between the geometry of N=2 strings and the moduli of SU*(∞) complexified SDSYM in 3+3 dimensions is indicated.