Harmonic Gauss Maps and Self-Dual Equations in String Theory

Abstract

The string world sheet, regarded as Riemann surface, in background $R^3$ and $R^4$ is described by the generalised Gauss map. When the Gauss map is harmonic or equivalently for surfaces of constant mean scalar curvature, we obtain an Abelian self-dual system, using $SO(3)$ and $SO(4)$ gauge fields constructed in our earlier studies. This compliments our earlier result that $h\surd g\ =\ 1$ surfaces exhibit Virasaro symmetry. The self-dual system so obtained is compared with self-dual Chern-Simons system and a generalized Liouville equation involving extrinsic geometry is obtained. \vspace{0.2cm} The world sheet in background $R^n, \ n>4$ is described by the generalized Gauss map. It is first shown that when the Gauss map is harmonic, the scalar mean curvature is constant. $SO(n)$ gauge fields are constructed from the geometry of the surface and expressed in terms of the Gauss map. It is shown that the harmonic map satisfies a non-Abelian self-dual system of equations for the gauge group $SO(2)\times SO(n-2)$.