Toda p-brane black holes and polynomials related to Lie algebras

Black hole generalized p-brane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold that contains a product of n - 1 Ricci-flat internal spaces. They are defined up to a set of functions H_s obeying non-linear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions H_s for intersections related to semisimple Lie algebras is suggested. This conjecture is proved for Lie algebras: A_m, C_{m+1}, m > 0. For simple Lie algebras the powers of polynomials coincide with the components of twice the dual Weyl vector in the basis of simple coroots. The coefficients of polynomials depend upon the extremality parameter \mu >0. In the extremal case \mu = 0 such polynomials were considered previously by H. L\"u, J. Maharana, S. Mukherji and C.N. Pope. Explicit formulas for A_2-solution are obtained. Two examples of A_2-dyon solutions, i.e. dyon in D = 11 supergravity with M2 and M5 branes intersecting at a point and Kaluza-Klein dyon, are considered.