Vertical Versus Diagonal Dimensional Reduction for p-branes

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 spacetime 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. † Research supported in part by DOE Grant DE-FG05-91-ER40633 ⋆ Research supported in part by the Commission of the European Communities under contract SCI*-