Vertical versus diagonal dimensional reduction for p-branes
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In addition to the double dimensional reduction procedure that employs world-volume Killing symmetries of p-brane supergravity solutions and acts diagonally on a plot of p versus space-time dimension D, there exists a second procedure of "vertical" reduction. This reduces the transverse-space dimension via an integral that superposes solutions to the underlying Laplace equation. We show that vertical reduction is also closely related to the recently introduced notion of intersecting p-branes. We illustrate this with examples, and also construct a new D = 11 solution describing four intersecting membranes, which preserves 1/16 of the supersymmetry. Given the two reduction schemes plus duality transformations at special points of the scalar modulus space, one may relate most of the p-brane solutions of relevance to superstring theory. We argue that the maximum classifying duality symmetry for this purpose is the Weyl group of the corresponding Cremmer-Julia supergravity symmetry Er(+r). We also discuss a separate class of duality-invariant p-branes with p = D - 3.