Chern–Simons theory and three-dimensional surfaces

DOI10.1088/0264-9381/24/7/009zbMATH Open1112.83040arXivhep-th/0611144OpenAlexW3100827240MaRDI QIDQ3435179

Publication date: 25 April 2007

Published in: Classical and Quantum Gravity (Search for Journal in Brave)

Abstract: 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.

Full work available at URL: https://arxiv.org/abs/hep-th/0611144

Mathematics Subject Classification ID

Relativistic gravitational theories other than Einstein's, including asymmetric field theories (83D05) Analogues of general relativity in lower dimensions (83C80)