Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
2006-11-14
Class.Quant.Grav.24:1833-1840,2007
Physics
High Energy Physics
High Energy Physics - Theory
10 pages, unnecessary details removed, typos fixed, references added
Scientific paper
10.1088/0264-9381/24/7/009
There are two natural Chern-Simons theories associated with the embedding of a three-dimensional surface in Euclidean space; one is constructed using the induced metric connection -- it involves only the intrinsic geometry, the other is extrinsic and uses the connection associated with the gauging of normal rotations. As such, the two theories appear to describe very different aspects of the surface geometry. Remarkably, at a classical level, they are equivalent. In particular, it will be shown that their stress tensors differ only by a null contribution. Their Euler-Lagrange equations provide identical constraints on the normal curvature. A new identity for the Cotton tensor is associated with the triviality of the Chern-Simons theory for embedded hypersurfaces implied by this equivalence. The corresponding null surface stress capturing this information will be constructed explicitly.
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