Generalised Complex Geometry and the Planck Cone

Abstract

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised 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 letter we analyse the role played by generalised complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalised 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 viceversa. 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 viceversa.