Brane World Sum Rules

TL;DR

The paper derives sum rules from the D-dimensional Einstein equations for warped brane-worlds with a spatially periodic internal space, showing that the sum of brane tensions and the non-negative integral of bulk scalar gradient energy is tied to the four-dimensional curvature $R_g$, vanishing for flat branes. The authors apply these constraints to several models, finding that RSI satisfies the condition, but smooth domain-wall generalizations are forbidden; Goldberger-Wise stabilization requires backreaction and a tuned cosmological constant to meet the rules; the DeWolfe-Freedman-Gubser-Karch construction can satisfy the constraints with an explicit superpotential, and supersymmetric BKP branes automatically comply due to BPS relations. Overall, the sum rules provide a practical diagnostic tool for assessing brane-world constructions and underscore the special consistency of supersymmetric setups in singular spaces.

Abstract

A set of consistency conditions is derived from Einstein equations for brane world scenarios with a spatially periodic internal space. In particular, the sum of the total tension of the flat branes and the non-negative integral of the gradient energy of the bulk scalars must vanish. This constraint allows us to make a simple consistency check of several models. We show that the two-brane Randall-Sundrum model satisfies this constraint, but it does not allow a generalization with smooth branes (domain walls), independently of the issue of supersymmetry. The Goldberger-Wise model of brane stabilization has to include the backreaction on the metric and the fine tuning of the cosmological constant to satisfy the constraints. We check that this is achieved in the DeWolfe-Freedman-Gubser-Karch scenario. Our constraints are automatically satisfied in supersymmetric brane world models.