Harmonic Maps and Self-Dual Equations for Immersed Surfaces

Details Harmonic Maps and Self-Dual Equations for Immersed Surfaces Harmonic Maps and Self-Dual Equations for Immersed Surfaces

1993-10-18

arxiv.org/abs/hep-th/9310107v1

Physics

High Energy Physics

High Energy Physics - Theory

V5A 1S6,26pages, LaTeX, IMSc/93-44 and SFU.HEP.109/1993

Scientific paper

The immersion of the string world sheet, regarded as a Riemann surface, in $R^3$ and $R^4$ is described by the generalized Gauss map. When the Gauss map is harmonic or equivalently for surfaces of constant mean curvature, we obtain Hitchin's self-dual equations, by using $SO(3)$ and $SO(4)$ gauge fields constructed in our earlier studies. This complements 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. The immersion in $R^n, \ n>4$ is described by the generalized Gauss map. It is shown that when the Gauss map is harmonic, the mean curvature of the immersed surface is constant. $SO(n)$ gauge fields are constructed from the geometry of the surface and expressed in terms of the Gauss map. It is found Hitchin's self- duality relations for the gauge group $SO(2)\times SO(n-2)$.

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