Hamilton-Jacobi Structure as a 4-D Generalization of 2-D Complex Structure

Abstract

Hamilton-Jacobi structure is a key notion of Topological Geometrodynamics proposed to generalize 2-D complex structure to 4-D case. Space-time surfaces X⁴ ⊆ H= M⁴ x CP², as analogs of Bohr orbits realizing almost deterministic holography, are simultaneous extremals of both volume action and Kahler action if X⁴ is determined by the simultaneous vanishing of 2 complex valued functions of H coordinates. In the simplest situation, the functions depend on a light-like coordinate of M² ⊆ M⁴, complex coordinate of E² orthogonal to M², and 2 complex coordinates of CP². The complex structures of 2-D Riemann surface are parameterized by the moduli space (Teichmuller parameters). The same should be true in 4-D case. The proposal is that Hamilton-Jacobi structure, defining an integrable distribution of partonic 2-surfaces and string world sheets, slices X⁴ by partonic 2-surfaces X² and string world sheets Y². The complex (hypercomplex) structure for X² would depend on point of Y² (X²). Space-time would define a 4-D surface in the product of Teichmuller space for the partonic 2-surfaces and its analog for string world sheets.