Matching conditions for a brane of arbitrary codimension

TL;DR

This paper derives matching conditions for distributional branes of arbitrary codimension in Lovelock gravity using a differential-form formalism and generalized Gauss-Codazzi equations. It shows that odd codimension yields Israel-like junctions, while even codimension produces induced brane gravity through dimensional reduction of bulk Euler densities, with the topological defect parameter $\beta$ setting the on-brane Planck scale. The authors construct explicit maximally symmetric 1-brane solutions in codimensions 3 and 4, illustrating how Gauss-Bonnet (and higher Lovelock) terms generate brane charges and deficit topologies, and how the bulk dynamics reduce to a brane action in favorable cases. The results connect with prior codimension-2 analyses and imply that Lovelock theories can support higher-codimension defects with on-brane gravity, albeit with caveats about bulk regularity and the role of $\beta$ as a dynamical or radiative degree of freedom.

Abstract

We present matching conditions for distributional sources of arbitrary codimension in the context of Lovelock gravity. Then we give examples, treating maximally symmetric distributional p-branes, embedded in flat, de Sitter and anti-de Sitter spacetime. Unlike Einstein theory, distributional defects of locally smooth geometry and codimension greater than 2 are demonstrated to exist in Lovelock theories. The form of the matching conditions depends on the parity of the brane codimension. For odd codimension, the matching conditions involve discontinuities of Chern-Simons forms and are thus similar to junction conditions for hypersurfaces. For even codimension, the bulk Lovelock densities induce intrinsic Lovelock densities on the brane. In particular, this results in the appearance of the induced Einstein tensor for p>2. For the matching conditions we present, the effect of the bulk is reduced to an overall topological solid angle defect which sets the Planck scale on the brane and to extrinsic curvature terms. Moreover, for topological matching conditions and constant solid angle deficit, we find that the equations of motion are obtained from an exact p+1 dimensional action, which reduces to an induced Lovelock theory for large codimension. In essence, this signifies that the distributional part of the Lovelock bulk equations can naturally give rise to induced gravity terms on a brane of even co-dimension. We relate our findings to recent results on codimension 2 branes.