Fluxbrane and S-brane solutions related to Lie algebras
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luxbrane and Sbrane Solutions Related to Lie Algebras1 A. A. Golubtsovab, c and V. D. Ivashchuka, b a
VNIIMS, 46 Ozyornaya Str., Moscow, 119361 Russia Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 MiklukhoMaklaya Str., Moscow, 117198 Russia c Laboratoire de Univers et Theories (LUTh), Observatoire de Paris, Place Jules Janssen 5, 92190 Meudon, France emqil: [email protected], [email protected] b
Abstract—We overview composite fluxbrane and special Sbrane solutions for a wide class of intersection rules related to semisimple Lie algebras. These solutions are defined on a product manifold R* × M1 × … M1 × … × Mn which contains n Ricciflat spaces M1, … , Mn with 1dimensional R* and M1. They are governed by a set of moduli functions Hs, which have polynomial structure. The powers of polynomials coincide with the components of the dual Weyl vector in the basis of simple coroots. DOI: 10.1134/S1063779612050139 1
1. INTRODUCTION
2. “FLUXSBRANE” SOLUTIONS
In this paper we overview fluxbrane and special Sbrane solutions related to semisimple finite dimensional (FD) Lie algebras [1, 2]. These solutions contain a subclass of (partially) supersymmetric solu tions related to Lie algebras A1 … A1 (at least for Mi = Rdi). The solutions are governed by functions Hs(z) > 0 defined on the interval (0, +∞) and obeying differential equations A 1 d ⎛ z dH s⎞ = B s H s'ss' ⎝ ⎠ 4 s' ∈ S dz H s dz
∏
(1.1)
with the boundary conditions imposed: Hs(+0) = 1,
(1.2)
s ∈ S (S is nonempty set). Here and in what follows all Bs > 0 are constants and (Ass') is the Cartan matrix (Ass = 2) of some semisimple FD Lie algebra Ᏻ. It was conjectured in [1] that the Eqs. (1.1), (1.2) have polynomial solutions. For semisimple Lie algebra the powers of polynomials coincide with the compo nents of the dual Weyl vector in the basis of simple coroots. In [1, 2] the polynomials corresponding to Lie algebras A1 … A1, A2, C2 and G2 were presented. The conjecture may be verified for any classical simple Lie algebra using the program from [3], where the polynomials corresponding to exceptional Lie alge bras F4 and E6 were found as well.
We consider a model governed by the action
∫
D
S = d x g { R g – h αβ g –
MN
α
∂M ϕ ∂N ϕ
β
(2.1)
θ a 2 a exp [ 2λ a ( ϕ ) ] ( F ) }, N ! a∈Δ a
∑
where g = gMN(x)dxM dxN is a metric, ϕ = (ϕα) ∈ Rl is a vector of scalar fields, (hαβ) is a constant symmetric nondegenerate l × l matrix (l ∈ N), θa = ±1, Fa = dAa is Naform (na ≥ 1), λa is a 1form on Rl: λa(ϕ) = λaαϕα, a ∈ Δ, α = 1, …, l. Here Δ is some finite set. Let us consider a family of exact solutions to field equations corresponding to the action (2.1) and depending on one variable ρ. These solutions are defined on the manifold M = (0, +∞) × M1 × M2 × … × Mn,
(2.2)
where M1 is onedimensional manifold. The solutions read [2]
1 The article is published in the original.
g=⎛ ⎝ +⎛ ⎝
∏H s∈S
∏H s∈S
2h s d ( I s )/ ( D – 2 )⎞ ⎧ ( wdρdp ) s ⎠⎨
⎩
– 2 h s⎞ 2 1 s ⎠ρ g
α
+
e
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