Toda p-brane black holes and polynomials related to Liealgebras

Abstract

Black hole generalized p-brane solutions fora wide class of intersection rules are obtained.The solutions are defined on a manifold that contains a productof n-1 Ricci-flat `internal' spaces.They are defined up to a set of functions Hs obeyingnon-linear differential equationsequivalent to Toda-type equations withcertain boundary conditions imposed. A conjecture on polynomialstructure of governing functions Hs for intersections related tosemisimple Lie algebras is suggested. This conjecture is proved forthe Lie algebras: Am, Cm + 1, m⩾1.For simple Lie algebrasthe powers of polynomials coincide with the components of the dual Weylvector in the basis of simple roots. The coefficients of polynomialsdepend upon the extremality parameter µ>0. In the extremalcase µ = 0 such polynomials were considered previously byH Lü, J Maharana, S Mukherji and C N Pope.Explicit formulae for the A2-solution are obtained. Two examples ofA2-dyon solutions, i.e., dyon in D = 11 supergravitywith M2 and M5 branes intersecting at a point and the Kaluza-Kleindyon, are considered.