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

Abstract

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 ≥ 1 . 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 µ > 0 . In the extremal case µ = 0 such polynomials were considered previously by H. Lü, 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 M 2 and M 5 branes intersecting at a point and Kaluza-Klein dyon, are considered.

Composite fluxbranes with general intersections

“…It may be verified that the constraints (2.38) and (2.40) are satisfied identically, due to conditions (3.1), (3.2). Relations (3.7) and (3.8) look similar to analogous relations for black branes [37,38,34] which appear from the horizon condition. These relations play a key role in this paper.…”

“…This conjecture is valid for Lie algebras A m and C m+1 , m ≥ 1 , at may be verified with a little modification of the proof for the black brane case [37,38].…”

“…We remind that a slightly different equations occur for black brane solutions from [36,37,38] where µ > 0 is extremality parameter, z ∈ (0, (2µ) −1 ) and allB s = 0 , (in most interesting casesB s < 0 ). For black brane solutions the finite limits (on a horizon) H s ((2µ) −1 − 0) = H s0 > 0 exist.…”

Black-brane solution for A3 algebra

“…This conjecture was verified for solutions with A m and C m series of Lie algebras [2,3] and also confirmed by special black-brane "block-orthogonal" solutions considered earlier in [15,16]. It should be noted that an analogue of this conjecture for extremal black holes was considered earlier in [11].…”

“…In this paper we continue investigation of the black-brane solutions with arbitrary intersection rules [1,2,3,4]. These solutions are governed by functions H s = H s (R) obeying a set of second order non-linear differential equations (equivalent to Toda-type ones) with certain boundary conditions imposed.…”