Minimal surfaces and strings from spinors a realization of the cartan programme

Abstract

It is shown ow the old Enneper-Weierstrass integral parametrization ofminimal surfaces inR3 and the Eisenhart ones inR3,1, when expressed through bilinear spinor polynomia, may be considered as deriving from a particularlocal realization of the possibility envisaged by Cartan: to consider ordinary vectors as generated from isotropic planes in complex spaces, in the frame of the bijective Cartan map connecting pure spinor directions to totally null planes in complex spaces. In the case ofR3 the correspondingglobal realization of the Cartan map extends the Enneper-Weierstrass parametrization to the Gauss-conformal map of the minimal surface to S2, which may be identified with the Riemann celestial sphere. Forreal spinors minimal surfaces are substituted bystrings both inR2,1 andR3,1; inR2,1 strings are globally mapped to a torus (inR4). InR3,1 (and its conformal extensions) a prescription is given to obtain strings as integrals of real, bilinear spinor null vectors, from the Enneper-Weierstrass spinor representation of minimal surfaces, through the use of unitary transformations in spinor space which allows its restriction to the real (Majorana spinor-space). It is shown that the Nambu action, or the area of the world surface described by the space-time string, is minimized by the Lagrangian density expressed as a quadrilinear spinor product formally reminding Fermi and Thirring interaction Lagrangians.