NONCOMMUTATIVE GEOMETRY, GEOMETRICAL ASPECTS OF SKYRMIONS AND STRING THEORY

Abstract

It is pointed out here that noncommutative geometry having the space–time manifold X = M4× Z2leads to the description of a massive fermion as a skyrmion when the discrete space–time is incorporated as an internal variable. The inherent anisotropic feature of the internal space gives rise to the Skyrme term as well as the Wess–Zumino term. An array of such skyrmions may be considered as a three-dimensional Ising system. It is shown that the composite system of such skyrmions may be viewed as the (3 + 1)-dimensional relative of the Polyakov string theory when the associated Liouville field corresponds to the attachment of topological fixtures like vortex lines at the end points of an open string. These two systems are found to satisfy certain duality principles through conformal duality which are similar to T duality and S duality in conventional string picture.