Chern-Simons theory and three-dimensional surfaces

TL;DR

This paper explores two different Chern-Simons theories related to three-dimensional surface embeddings, revealing their classical equivalence and deriving new geometric identities and stress tensors.

Contribution

It demonstrates the classical equivalence of intrinsic and extrinsic Chern-Simons theories for surfaces and introduces a new identity for the Cotton tensor.

Findings

The two Chern-Simons theories are classically equivalent.

Their stress tensors differ only by a null contribution.

A new identity for the Cotton tensor is derived.

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.