GENERALIZED COMPLEX GEOMETRY AND THE PLANCK CONE

Abstract

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalized complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this paper we analyze the role played by generalized complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalized complex geometry as an appropriate geometrical setup for dualities. The latter are interpreted as transformations connecting points in the interior of the Planck cone with points in the exterior, and vice versa. The Planck cone bears some resemblance with the relativistic light-cone. However the latter cannot be traversed by physical particles, while dualities do connect the region outside the Planck cone with the region inside, and vice versa.