On Brane Solutions Related to Non-Singular Kac-Moody Algebras

A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M0 × M1 × ... × Mn, where Mi are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g. solutions with...

Full description

Saved in:

Bibliographic Details
Published in:	Symmetry, Integrability and Geometry: Methods and Applications
Date:	2009
Main Authors:	Ivashchuk, V.D., Melnikov, V.N.
Format:	Article
Language:	English
Published:	Інститут математики НАН України 2009
Online Access:	https://nasplib.isofts.kiev.ua/handle/123456789/149113
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Cite this:	On Brane Solutions Related to Non-Singular Kac-Moody Algebras / V.D. Ivashchuk, V.N. Melnikov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 111 назв. — англ.

Institution

Digital Library of Periodicals of National Academy of Sciences of Ukraine

_version_	1860190377820028928
author	Ivashchuk, V.D. Melnikov, V.N.
author_facet	Ivashchuk, V.D. Melnikov, V.N.
citation_txt	On Brane Solutions Related to Non-Singular Kac-Moody Algebras / V.D. Ivashchuk, V.N. Melnikov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 111 назв. — англ.
collection	DSpace DC
container_title	Symmetry, Integrability and Geometry: Methods and Applications
description	A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M0 × M1 × ... × Mn, where Mi are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g. solutions with harmonic functions, S-brane and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are reviewed. Some examples of solutions, e.g. corresponding to hyperbolic KM algebras: H2(q,q), AE3, HA2(1), E10 and Lorentzian KM algebra P10 are presented.
first_indexed	2025-12-07T18:05:36Z
format	Article
fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 070, 34 pages On Brane Solutions Related to Non-Singular Kac–Moody Algebras Vladimir D. IVASHCHUK and Vitaly N. MELNIKOV Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya Str., Moscow 119361, Russia Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia, 6 Miklukho-Maklaya Str., Moscow 117198, Russia E-mail: ivashchuk@mail.ru, melnikov@phys.msu.ru Received October 01, 2008, in final form June 15, 2009; Published online July 07, 2009 doi:10.3842/SIGMA.2009.070 Abstract. A multidimensional gravitational model containing scalar fields and antisym- metric forms is considered. The manifold is chosen in the form M = M0 ×M1 × · · · ×Mn, where Mi are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g. solutions with harmonic functions, S-brane and black brane ones) with intersection rules related to non-singular Kac–Moody (KM) algebras (e.g. hyperbolic ones) are reviewed. Some examples of solutions, e.g. corresponding to hyperbolic KM algebras: H2(q, q), AE3, HA (1) 2 , E10 and Lorentzian KM algebra P10 are presented. Key words: Kac–Moody algebras; S-branes; black branes; sigma-model; Toda chains 2000 Mathematics Subject Classification: 17B67; 17B81; 83E15; 83E50; 83F05; 81T30 1 Introduction Kac–Moody (KM) Lie algebras [1, 2, 3] play a rather important role in different areas of mathe- matical physics (see [3, 4, 5, 6] and references therein). We recall that KM Lie algebra is a Lie algebra generated by the relations [3] [hi, hj ] = 0, [ei, fj ] = δijhj , [hi, ej ] = Aijej , [hi, fj ] = −Aijfj , (adei)1−Aij (ej) = 0 (i 6= j), (adfi)1−Aij (fj) = 0 (i 6= j). Here A = (Aij) is a generalized Cartan matrix, i, j = 1, . . . , r, and r is the rank of the KM algebra. It means that all Aii = 2; Aij for i 6= j are non-positive integers and Aij = 0 implies Aji = 0. In what follows the matrix A is restricted to be non-degenerate (i.e. det A 6= 0) and sym- metrizable i.e. A = BD, where B is a symmetric matrix and D is an invertible diagonal matrix (D may be chosen in such way that all its entries Dii are positive rational numbers [3]). Here we do not consider singular KM algebras with det A = 0, e.g. affine ones. Recall that affine KM algebras are of much interest for conformal field theories, superstring theories etc. [4, 7]. In the case when A is positive definite (the Euclidean case) we get ordinary finite dimensional Lie algebras [3, 4]. For non-Euclidean signatures of A all KM algebras are infinite-dimensional. Among these the Lorentzian KM algebras with pseudo-Euclidean signatures (−,+, . . . ,+) for the Cartan matrix A are of current interest, since they contain a subclass of the so-called hyperbolic KM algebras widely used in modern mathematical physics. Hyperbolic KM algebras mailto:ivashchuk@mail.ru mailto:melnikov@phys.msu.ru http://dx.doi.org/10.3842/SIGMA.2009.070 2 V.D. Ivashchuk and V.N. Melnikov are by definition Lorentzian Kac–Moody algebras with the property that removing any node from their Dynkin diagram leaves one with a Dynkin diagram of the affine or finite type. The hyperbolic KM algebras can be completely classified [8, 9] and have rank 2 ≤ r ≤ 10. For r ≥ 3 there is a finite number of hyperbolic algebras. For rank 10, there are four algebras, known as E10, BE10, CE10 and DE10. Hyperbolic KM algebras appeared in ordinary gravity [10] (F3 = AE3 = H3), supergravity: [11, 12] (E10), [13] (F3), strings [14] etc. The growth of interest in hyperbolic algebras in theoretical and mathematical physics ap- peared in 2001 after the publication of Damour and Henneaux [15] devoted to a description of chaotic (BKL-type [16]) behaviour near the singularity in string inspired low energy models (e.g. supergravitational ones) [17] (see also [18]). It should be noted that these results were based on a billiard approach in multidimensional cosmology with different matter sources (for D = 4 suggested by Chitre [19]) elaborated in our papers [20, 21, 22, 23, 24] (for a review see also [25, 26]). The description of oscillating behaviour near the singularity in D = 11 supergravity [27] (which is related to M -theory [28, 29]) in terms of motion of a point-like particle in a 9- dimensional billiard (of finite volume) corresponding to the Weyl chamber of the hyperbolic KM algebra E10 inspired another description of D = 11 supergravity in [30]: a formal “small tension” expansion of D = 11 supergravity near a space-like singularity was shown to be equi- valent (at least up to 30th order in height) to a null geodesic motion in the infinite dimensional coset space E10/K(E10) (here K(E10) is the maximal compact subgroup of the hyperbolic Kac– Moody group E10(R)). Recall that E10 KM algebra is an over-extension of the finite dimensional Lie algebra E8, i.e. E10 = E++ 8 . But there is another extension of E8 – the so-called the very extended Kac– Moody algebra of the E8 algebra – called E11 = E+++ 8 . (To get an understanding of very extended algebras and some of their properties see [31] and references therein). It has been proposed by P. West that the Lorentzian (non-hyperbolic) KM algebra E11 is responsible for a hidden algebraic structure characterizing 11-dimensional supergravity [32]. The same very extended algebra occurs in IIA [32] and IIB supergravities [33]. Moreover, it was conjectured that an analogous hidden structure is realized in the effective action of the bosonic string (with the KM algebra k27 = D+++ 24 ) [32] and also for pure D dimensional gravity (with the KM algebra A+++ D−3 [34]). It has been suggested in [35] that all the so-called maximally oxidised theories (see also [6]), possess the very extension G+++ of the simple Lie algebra G. It was shown in [36] that the BPS solutions of the oxidised theory of a simply laced group G form representations of a subgroup of the Weyl transformations of the algebra G+++. In this paper we briefly review another possibility for utilizing non-singular (e.g. hyperbolic) KM algebras suggested in three of our papers [37, 38, 39]. This possibility (implicitly assumed also in [40, 41, 42, 43, 44, 45, 46]) is related to certain classes of exact solutions describing intersecting composite branes in a multidimensional gravitational model containing scalar fields and antisymmetric forms defined on (warped) product manifolds M = M0 × M1 × · · · × Mn, where Einstein factor spaces Mi (i ≥ 1) are Ricci-flat (at least) for i ≥ 2. From a pure mathematical point of view these solutions may be obtained from sigma-models or Toda chains corresponding to certain non-singular KM algebras. The information about the (hidden) KM algebra is encoded in intersection rules which relate the dimensions of brane intersections with non-diagonal components of the generalized Cartan matrix A [47]. We deal here with generalized Cartan matrices of the form Ass′ ≡ 2(U s, U s′ ) (U s′ , U s′) , s, s′ ∈ S, (1.1) with (U s, U s) 6= 0, for all s ∈ S (S is a finite set). Here U s are the so-called brane (co-)vectors. They are linear functions on RN , where N = n + l and l is the number of scalar fields. The On Brane Solutions Related to Non-Singular Kac–Moody Algebras 3 indefinite scalar product (·, ·) [48] is defined on (RN )∗ and has the signature (−1,+1, . . . ,+1) when all scalar fields have positive kinetic terms, i.e. there are no phantoms (or ghosts). The matrix A is symmetrizable. U s-vectors may be put in one-to-one correspondence with simple roots αs of the generalized KM algebra after a suitable normalizing. For indecomposable A (when the KM algebra is simple) the matrices ((U s, U s′ )) and ((αs\|αs′)) are proportional to each other. Here (·\|·) is a non-degenerate bilinear symmetric form on a root space obeying (αs\|αs) > 0 for all simple roots αs [3]. We note that the minisuperspace bilinear form (·, ·) coming from multidimensional gravi- tational model [48] “does not know” about the definition of (·\|·) in [3] and hence there exist physical examples where all (U s, U s) are negative. Some examples of this are given below in Sec- tion 5. For D = 11 supergravity and ten dimensional IIA, IIB supergravities all (U s, U s) = 2 [47, 49] and corresponding KM algebras are simply laced. It was shown in our papers [22, 23, 24] that the inequality (U s, U s) > 0 is a necessary condition for the formation of the billiard wall (in one approaches the singularity) by the s-th matter source (e.g. a fluid component or a brane). The scalar products for brane vectors U s were found in [48] (for the electric case see also [50, 51, 52])( U s, U s′) = dss′ + dsds′ 2−D + χsχs′〈λs, λs′〉, (1.2) where ds and ds′ are dimensions of the brane worldvolumes corresponding to branes s and s′ respectively, dss′ is the dimension of intersection of the brane worldvolumes, D is the total space-time dimension, χs = +1,−1 for electric or magnetic brane respectively, and 〈λs, λs′〉 is the non-degenerate scalar product of the l-dimensional dilatonic coupling vectors λs and λs′ corresponding to branes s and s′. Relations (1.1), (1.2) define the brane intersection rules [47] dss′ = do ss′ + 1 2 Ks′Ass′ , s 6= s′, where Ks = (U s, U s) and do ss′ = dsds′ D − 2 − χsχs′〈λs, λs′〉 (1.3) is the dimension of the so-called orthogonal (or (A1⊕A1)-) intersection of branes following from the orthogonality condition [48]( U s, U s′) = 0, (1.4) s 6= s′. The orthogonality relations (1.4) for brane intersections in the non-composite electric case were suggested in [50, 51] and for the composite electric case in [52]. Relations (1.2) and (1.3) were derived in [48] for rather general assumptions: the branes were composite, the number of scalar fields l was arbitrary as well as the signature of the bilinear form 〈·, ·〉 (or, equivalently, the signature of the kinetic term for scalar fields), (Einstein) factor spaces Mi had arbitrary dimensions di and signatures. The intersection rules (1.3) appeared earlier (in different notations) in [53, 54, 55] when all di = 1 (i > 0) and 〈·, ·〉 was positive definite (in [53, 54] l = 1 and total space-time had a pseudo-Euclidean signature). The intersection rules (1.3) were also used in [56, 57, 58, 47] in the context of intersecting black brane solutions. It was proved in [59] that the target space of the sigma model describing composite electro- magnetic brane configurations on the product of several Ricci-flat spaces is a homogeneous (coset) space G/H. It is locally symmetric (i.e. the Riemann tensor is covariantly constant: ∇Riem = 0) if and only if( U s − U s′)( U s, U s′) = 0 4 V.D. Ivashchuk and V.N. Melnikov for all s and s′, i.e. when any two brane vectors U s and U s′ , s 6= s′, are either coinciding U s = U s′ or orthogonal (U s, U s′ ) = 0 (for two electric branes and l = 1 see also [60]). Now relations for brane vectors U s (1.1) and (1.2) (with U s being identified with roots αs) are widely used in the G+++-approach [36, 6]. The orthogonality condition (1.4) describing the intersection of branes [50, 51, 52, 48] was rediscovered in [49] (for some particular intersecting configurations of M-theory it was done in [61]). It was found in the context of G+++-algebras that the intersection rules for extremal branes are encoded in orthogonality conditions between the various roots from which the branes arise, i.e. (αs\|αs′) = 0, s 6= s′, where αs should be real positive roots (“real” means that (αs\|αs) > 0). In [49] another condition on brane, root vectors was found: αs + αs′ should not be a root, s 6= s′. The last condition is trivial for M-theory and for IIA and IIB supergravities, but for low energy effective actions of heterotic strings it forbids certain intersections that does not take place due to non-zero contributions of Chern–Simons terms. It should be noted that the orthogonality relations for brane intersections (1.4) which ap- peared in 1996–97, were not well understood by the superstring community at that time. The standard intersection rules (1.3) gave back the well-known zero binding energy configurations preserving some supersymmetries. These brane configurations were originally derived from su- persymmetry and duality arguments (see for example [62, 63, 64] and reference therein) or by using a no-force condition (suggested for M-branes in [65]). 2 The model 2.1 The action We consider the model governed by action S = 1 2κ2 ∫ M dDz √ \|g\| { R[g]− 2Λ− hαβgMN∂Mϕα∂Nϕβ − ∑ a∈∆ θa na! exp[2λa(ϕ)](F a)2g } + SGH , (2.1) where g = gMNdzM ⊗ dzN is the metric on the manifold M , dim M = D, ϕ = (ϕα) ∈ Rl is a vector of dilatonic scalar fields, (hαβ) is a non-degenerate symmetric l × l matrix (l ∈ N), θa 6= 0, F a = dAa = 1 na! F a M1...Mna dzM1 ∧ · · · ∧ dzMna is an na-form (na ≥ 2) on a D-dimensional manifold M , Λ is a cosmological constant and λa is a 1-form on Rl : λa(ϕ) = λaαϕα, a ∈ ∆, α = 1, . . . , l. In (2.1) we denote \|g\| = \|det(gMN )\|, (F a)2g = F a M1...Mna F a N1...Nna gM1N1 · · · gMnaNna , a ∈ ∆, where ∆ is some finite set (for example, of positive integers), and SGH is the standard Gibbons–Hawking boundary term [66]. In models with one time all θa = 1 when the signature of the metric is (−1,+1, . . . ,+1). κ2 is the multidimensional gravitational constant. 2.2 Ansatz for composite branes Let us consider the manifold M = M0 ×M1 × · · · ×Mn, (2.2) On Brane Solutions Related to Non-Singular Kac–Moody Algebras 5 with the metric g = e2γ(x)ĝ0 + n∑ i=1 e2φi(x)ĝi, (2.3) where g0 = g0 µν(x)dxµ ⊗ dxν is an arbitrary metric with any signature on the manifold M0 and gi = gi mini (yi)dymi i ⊗ dyni i is a metric on Mi satisfying the equation Rmini [g i] = ξig i mini , (2.4) mi, ni = 1, . . . , di; ξi = const, i = 1, . . . , n. Here ĝi = p∗i g i is the pullback of the metric gi to the manifold M by the canonical projection: pi : M → Mi, i = 0, . . . , n. Thus, (Mi, g i) are Einstein spaces, i = 1, . . . , n. The functions γ, φi : M0 → R are smooth. We denote dν = dimMν ; ν = 0, . . . , n; D = ∑n ν=0 dν . We put any manifold Mν , ν = 0, . . . , n, to be oriented and connected. Then the volume di-form τi ≡ √ \|gi(yi)\| dy1 i ∧ · · · ∧ dydi i , and signature parameter ε(i) ≡ sign(det(gi mini )) = ±1 are correctly defined for all i = 1, . . . , n. Let Ω = Ω(n) be a set of all non-empty subsets of {1, . . . , n}. The number of elements in Ω is \|Ω\| = 2n − 1. For any I = {i1, . . . , ik} ∈ Ω, i1 < . . . < ik, we denote τ(I) ≡ τ̂i1 ∧ · · · ∧ τ̂ik , ε(I) ≡ ε(i1) · · · ε(ik), MI ≡ Mi1 × · · · ×Mik , d(I) ≡ ∑ i∈I di. Here τ̂i = p∗i τ̂i is the pullback of the form τi to the manifold M by the canonical projection: pi : M → Mi, i = 1, . . . , n. We also put τ(∅) = ε(∅) = 1 and d(∅) = 0. For fields of forms we consider the following composite electromagnetic ansatz F a = ∑ I∈Ωa,e F (a,e,I) + ∑ J∈Ωa,m F (a,m,J), (2.5) where F (a,e,I) = dΦ(a,e,I) ∧ τ(I), (2.6) F (a,m,J) = e−2λa(ϕ) ∗ (dΦ(a,m,J) ∧ τ(J)) (2.7) are elementary forms of electric and magnetic types respectively, a ∈ ∆, I ∈ Ωa,e, J ∈ Ωa,m and Ωa,v ⊂ Ω, v = e,m. In (2.7) ∗ = ∗[g] is the Hodge operator on (M, g). For scalar functions we put ϕα = ϕα(x), Φs = Φs(x), s ∈ S. (2.8) Thus, ϕα and Φs are functions on M0. Here and below S = Se t Sm, Sv = ta∈∆{a} × {v} × Ωa,v, v = e,m. (2.9) 6 V.D. Ivashchuk and V.N. Melnikov Here and in what follows t means the union of non-intersecting sets. The set S consists of elements s = (as, vs, Is), where as ∈ ∆ is color index, vs = e,m is electro-magnetic index and set Is ∈ Ωas,vs describes the location of brane. Due to (2.6) and (2.7) d(I) = na − 1, d(J) = D − na − 1, (2.10) for I ∈ Ωa,e and J ∈ Ωa,m (i.e. in the electric and magnetic case, respectively). 2.3 The sigma model Let d0 6= 2 and γ = γ0(φ) ≡ 1 2− d0 n∑ j=1 djφ j , i.e. the generalized harmonic gauge (frame) is used. Here we put two restrictions on sets of branes that guarantee the block-diagonal form of the energy-momentum tensor and the existence of the sigma-model representation (without additional constraints): (R1) d(I ∩ J) ≤ d(I)− 2, (2.11) for any I, J ∈ Ωa,v, a ∈ ∆, v = e,m (here d(I) = d(J)) and (R2) d(I ∩ J) 6= 0 for d0 = 1, d(I ∩ J) 6= 1 for d0 = 3. (2.12) It was proved in [48] that equations of motion for the model (2.1) and the Bianchi identities: dFs = 0, s ∈ Sm, for fields from (2.3), (2.5)–(2.8), when Restrictions (2.11) and (2.12) are imposed, are equivalent to the equations of motion for the σ-model governed by the action Sσ0 = 1 2κ2 0 ∫ dd0x √ \|g0\| { R[g0]− ĜABg0µν∂µσA∂νσ B − ∑ s∈S εs exp (−2U s AσA)g0µν∂µΦs∂νΦs − 2V } , (2.13) where (σA) = (φi, ϕα), k0 6= 0, the index set S is defined in (2.9), V = V (φ) = Λe2γ0(φ) − 1 2 n∑ i=1 ξidie −2φi+2γ0(φ) is the potential, (ĜAB) = ( Gij 0 0 hαβ ) (2.14) is the target space metric with Gij = diδij + didj d0 − 2 On Brane Solutions Related to Non-Singular Kac–Moody Algebras 7 and co-vectors U s A = U s AσA = ∑ i∈Is diφ i − χsλas(ϕ), (U s A) = (diδiIs ,−χsλasα), s = (as, vs, Is). (2.15) Here χs = +1 for vs = e and χs = −1 for vs = m; δiI = ∑ j∈I δij is an indicator of i belonging to I: δiI = 1 for i ∈ I and δiI = 0 otherwise; and εs = (−ε[g])(1−χs)/2ε(Is)θas , s ∈ S, ε[g] ≡ sign det(gMN ). (2.16) More explicitly (2.16) reads εs = ε(Is)θas for vs = e, εs = −ε[g]ε(Is)θas for vs = m. For finite internal space volumes Vi (e.g. compact Mi) and electric p-branes (i.e. all Ωa,m = ∅) the action (2.13) coincides with the action (2.1) when κ2 = κ2 0 ∏n i=1 Vi. 3 Solutions governed by harmonic functions 3.1 Solutions with block-orthogonal set of U s and Ricci-flat factor-spaces Here we consider a special class of solutions to equations of motion governed by several harmonic functions when all factor spaces are Ricci-flat and the cosmological constant is zero, i.e. ξi = Λ = 0, i = 1, . . . , n. In certain situations these solutions describe extremal black branes charged by fields of forms. The solutions crucially depend upon scalar products of U s-vectors (U s, U s′ ); s, s′ ∈ S, where (U,U ′) = ĜABUAU ′ B, (3.1) for U = (UA), U ′ = (U ′ A) ∈ RN , N = n + l and (ĜAB) = ( Gij 0 0 hαβ ) is the inverse matrix to the matrix (2.14). Here as in [67] we have Gij = δij di + 1 2−D , i, j = 1, . . . , n. The scalar products (3.1) for vectors U s were calculated in [48] and are given by (U s, U s′ ) = d(Is ∩ Is′) + d(Is)d(Is′) 2−D + χsχs′λasαλas′βhαβ , (3.2) where (hαβ) = (hαβ)−1, and s = (as, vs, Is), s′ = (as′ , vs′ , Is′) belong to S. This relation is a very important one since it encodes brane data (e.g. intersections) via the scalar products of U -vectors. Let S = S1 t · · · t Sk, Si 6= ∅, i = 1, . . . , k, (3.3) 8 V.D. Ivashchuk and V.N. Melnikov and (U s, U s′ ) = 0 (3.4) for all s ∈ Si, s′ ∈ Sj , i 6= j; i, j = 1, . . . , k. Relation (3.3) means that the set S is a union of k non-intersecting (non-empty) subsets S1, . . . , Sk. According to (3.4) the set of vectors (U s, s ∈ S) has a block-orthogonal structure with respect to the scalar product (3.1), i.e. it splits into k mutually orthogonal blocks (U s, s ∈ Si), i = 1, . . . , k. Here we consider exact solutions in the model (2.1), when vectors (U s, s ∈ S) obey the block- orthogonal decomposition (3.3), (3.4) with scalar products defined in (3.2) [37]. These solutions were obtained from the corresponding solutions to the σ-model equations of motion [37]. Proposition 3.1. Let (M0, g 0) be Ricci-flat: Rµν [g0] = 0. Then the field configuration g0, σA = ∑ s∈S εsU sAν2 s lnHs, Φs = νs Hs , s ∈ S, satisfies the field equations corresponding to the action (2.13) with V = 0 if the real numbers νs obey the relations∑ s′∈S ( U s, U s′) εs′ν2 s′ = −1, s ∈ S, (3.5) the functions Hs > 0 are harmonic, i.e. ∆[g0]Hs = 0, s ∈ S, and Hs are coinciding inside blocks: Hs = Hs′ for s, s′ ∈ Si, i = 1, . . . , k. Using the sigma-model solution from Proposition 3.1 and the relations for contra-variant components [48]: U si = δiIs − d(Is) D − 2 , U sα = −χsλ α as , s = (as, vs, Is), we get [37]: g = (∏ s∈S H2d(Is)εsν2 s s )1/(2−D){ ĝ0 + n∑ i=1 (∏ s∈S H 2εsν2 s δiIs s ) ĝi } , (3.6) ϕα = − ∑ s∈S λα as χsεsν 2 s lnHs, (3.7) F a = ∑ s∈S Fsδa as , (3.8) where i = 1, . . . , n, α = 1, . . . , l, a ∈ ∆ and Fs = νsdH−1 s ∧ τ(Is) for vs = e, (3.9) Fs = νs(∗0dHs) ∧ τ(Īs) for vs = m, (3.10) Hs are harmonic functions on (M0, g 0) which coincide inside blocks (i.e. Hs = Hs′ for s, s′ ∈ Si, i = 1, . . . , k) and the relations (3.5) on the parameters νs are imposed. Here the matrix ((U s, U s′ )) and parameters εs, s ∈ S, are defined in (3.2) and (2.16), respectively; λα a = hαβλaβ , ∗0 = ∗[g0] is the Hodge operator on (M0, g 0) and Ī = {1, . . . , n} \ I is the dual set. (In (3.10) we redefined the sign of νs-parameter.) On Brane Solutions Related to Non-Singular Kac–Moody Algebras 9 3.2 Solutions related to non-singular KM algebras Now we study the solutions (3.6)–(3.10) in more detail and show that some of them may be related to non-singular KM algebras. We put Ks ≡ (U s, U s) 6= 0 for all s ∈ S and introduce the matrix A = (Ass′): Ass′ ≡ 2(U s, U s′ ) (U s′ , U s′) , s, s′ ∈ S. (3.11) Here some ordering in S is assumed. Using this definition and (3.2) we obtain the intersection rules [47] d(Is ∩ Is′) = ∆(s, s′) + 1 2 Ks′Ass′ , (3.12) where s 6= s′, and ∆(s, s′) = d(Is)d(Is′) D − 2 − χsχs′λasαλas′βhαβ (3.13) defines the so-called “orthogonal” intersection rules [48] (see also [53, 54, 55] for di = 1). In D = 11 and D = 10 (IIA and IIB ) supergravity models all Ks = 2 and hence the intersection rules (3.12) in this case have a simpler form [47]: d(Is ∩ Is′) = ∆(s, s′) + Ass′ , (3.14) where s 6= s′, implying Ass′ = As′s. The corresponding KM algebra is simply-laced in this case. For detA 6= 0 relation (3.5) may be rewritten in the equivalent form −εsν 2 s (U s, U s) = 2 ∑ s′∈S Ass′ ≡ bs, (3.15) where s ∈ S and (Ass′ ) = A−1. Thus, equation (3.5) may be resolved in terms of νs for certain εs = ±1, s ∈ S. We note that due to (3.4) the matrix A has a block-diagonal structure and, hence, for any i-th block the set of parameters (νs, s ∈ Si) depends upon the matrix inverse to the matrix (Ass′ ; s, s′ ∈ Si). Now we consider one-block case when the brane intersections are related to some non-singular KM algebras. Finite-dimensional Lie algebras [38]. Let A be a Cartan matrix of a simple finite- dimensional Lie algebra. In this case Ass′ ∈ {0,−1,−2,−3}, s 6= s′. The elements of inverse matrix A−1 are positive (see Chapter 7 in [4]) and hence we get from (3.15) the same signature relation as in the orthogonal case [48]: εs ( U s, U s ) < 0, s ∈ S. When all (U s, U s) > 0 we get εs < 0, s ∈ S. Moreover, all bs are natural numbers: bs = ns ∈ N, s ∈ S. (3.16) The integers ns coincide with the components of the twice dual Weyl vector in the basis of simple co-roots (see Chather 3.1.7 in [4]). 10 V.D. Ivashchuk and V.N. Melnikov Hyperbolic KM algebras. Let A be a generalized Cartan matrix corresponding to a simple hyperbolic KM algebra. For the hyperbolic algebras the following relations are satisfied εs(U s, U s) > 0, (3.17) since all bs are negative in the hyperbolic case [31]: bs < 0, where s ∈ S. (3.18) For (U s, U s) > 0 we get εs > 0, s ∈ S. If θas > 0 for all s ∈ S, then ε(Is) = 1 for vs = e, ε(Is) = −ε[g] for vs = m. For a metric g of pseudo-Euclidean signature all ε(Is) = 1 and, hence, all branes are Euclidean or should contain even number of time directions: 2, 4, . . .. For ε[g] = 1 only magnetic branes may be pseudo-Euclidean. Remark 3.1. The inequalities (3.18) guarantee the existence of a principal (real) so(1, 2) subal- gebra for any hyperbolic Kac–Moody algebra [68, 31]. Similarly the inequalities (3.16) imply the existence of a principal so(3) subalgebra for any finite dimensional (semi-)simple Lie algebra. It was shown in [31] that this property is not just restricted to hyperbolic algebras, but holds for a wider class of Lorentzian algebras obeying the inequalities bs ≤ 0 for all s. Example 3.1. F3 = AE3 algebra [39]. Now we consider an example of the solution corre- sponding to the hyperbolic KM algebra F3 with the Cartan matrix A =  2 −2 0 −2 2 −1 0 −1 2  , (3.19) F3 contains A(1) 1 affine subalgebra (it corresponds to the Geroch group) and A2 subalgebra. There exists an example of the solution with the A-matrix (3.19) for 11-dimensional model governed by the action S = ∫ d11z √ \|g\| { R[g]− 1 4! ( F 4 )2 − 1 4! ( F 4∗)2} , where rankF 4 = rankF 4∗ = 4. Here ∆ = {4, 4∗}. We consider a configuration with two magnetic 5-branes corresponding to the form F 4 and one electric 2-brane corresponding to the form F 4∗. We denote S = {s1, s2, s3}, as1 = as3 = 4, as2 = 4∗ and vs1 = vs3 = m, vs2 = e, where d(Is1) = d(Is3) = 6 and d(Is2) = 3. The intersection rules (3.12) read d(Is1 ∩ Is2) = 0, d(Is2 ∩ Is3) = 1, d(Is1 ∩ Is3) = 4. For the manifold (2.2) we put n = 5 and d1 = 2, d2 = 4, d3 = d4 = 1, d5 = 2. The corresponding brane sets are the following: Is1 = {1, 2}, Is2 = {4, 5}, Is3 = {2, 3, 4}. The solution reads g = H−12 { −dt⊗ dt + H9ĝ1 + H13ĝ2 + H4ĝ3 + H14ĝ4 + H10ĝ5 } , (3.20) F 4 = dH dt {νs1 τ̂3 ∧ τ̂4 ∧ τ̂5 + νs3 τ̂1 ∧ τ̂5} , On Brane Solutions Related to Non-Singular Kac–Moody Algebras 11 F 4∗ = dH dt νs2 H2 dt ∧ τ̂4 ∧ τ̂5, where ν2 s1 = 9 2 , ν2 s2 = 5 and ν2 s3 = 2 (see (3.15)). All metrics gi are Ricci-flat (i = 1, . . . , 5) and have Euclidean signatures (this agrees with relations (3.17) and (2.16)), and H = ht + h0 > 0, where h, h0 are constants. The metric (3.20) may be also rewritten using the synchronous time variable ts g = −dts ⊗ dts + f3/5ĝ1 + f−1/5ĝ2 + f8/5ĝ3 + f−2/5ĝ4 + f2/5ĝ5, where f = 5hts = H−5 > 0, h > 0 and ts > 0. The metric describes the power-law “inflation” in D = 11. It is singular for ts → +0. In the next example we consider a chain of the so-called BD-models (D ≥ 11) suggested in [47]. For D = 11 the BD-model coincides with the truncated (i.e. without Chern–Simons term) bosonic sector of D = 11 supergravity [27] which is related to M -theory. For D = 12 it coincides with truncated 12-dimensional model from [69] which may be related to F -theory [70]. BD-models. The BD-model has the action [47] SD = ∫ dDz √ \|g\| { R[g] + gMN∂M ~ϕ∂N ~ϕ− D−7∑ a=4 1 a! exp[2~λa~ϕ](F a)2 } , (3.21) where ~ϕ = (ϕ1, . . . , ϕl) ∈ Rl, ~λa = (λa1, . . . , λal) ∈ Rl, l = D−11, rankF a = a, a = 4, . . . , D−7. Here vectors ~λa satisfy the relations ~λa ~λb = N(a, b)− (a− 1)(b− 1) D − 2 = Λab, N(a, b) = min(a, b)− 3, (3.22) a, b = 4, . . . , D−7 and ~λD−7 = −2~λ4. For D > 11 vectors ~λ4, . . . , ~λD−8 are linearly independent. (It may be verified that matrix (Λab) is positive definite and hence the set of vectors obeying (3.22) does exist.) The model (3.21) contains l scalar fields with a negative kinetic term (i.e. hαβ = −δαβ in (2.1)) coupled to several forms (the number of forms is (l + 1)) . For the brane worldvolumes we have the following dimensions (see (2.10)) d(I) = 3, . . . , D − 8, I ∈ Ωa,e, d(I) = D − 5, . . . , 6, I ∈ Ωa,m. Thus, there are (l + 1) electric and (l + 1) magnetic p-branes, p = d(I)− 1. In BD-model all Ks = 2. Example 3.2. H2(q1, q2) algebra [37]. Let A = ( 2 −q1 −q2 2 ) , q1q2 > 4, q1, q2 ∈ N. This is the Cartan matrix for the hyperbolic KM algebra H2(q1, q2) [3]. From (3.15) we get εsν 2 s (U s, U s)(q1q2 − 4) = 2qs + 4, s ∈ {1, 2} = S. An example of the H2(q, q)-solution for BD-model with two electric p-branes (p = d1, d2), corresponding to F a and F b fields and intersecting on time manifold, is the following [37]: g = H−2/(q−2)ĝ0 −H2/(q−2)dt⊗ dt + ĝ1 + ĝ2, 12 V.D. Ivashchuk and V.N. Melnikov F a = ν1dH−1 ∧ dt ∧ τ̂1, F b = ν2dH−1 ∧ dt ∧ τ̂2, ~ϕ = ( ~λa + ~λb ) (q − 2)−1 lnH, where d0 = 3, d1 = a − 2, a = q + 4, a < b, d2 = b − 2, D = a + b, and ν2 1 = ν2 2 = (q − 2)−1. The signature restrictions are : ε(1) = ε(2) = −1. Thus, the space-time (M, g) should contain at least three time directions. The minimal D is 15. For D = 15 we get a = 7, b = 8, d1 = 5, d2 = 6 and q = 3. (Here we have eliminated a typo in a sign of scalar fields that was originally in [37].) Remark 3.2. We note that affine KM algebras (with det A = 0) do not appear in the solutions (3.6)–(3.10). Indeed, any affine Cartan matrix satisfy the relations∑ s∈S asAss′ = 0 with as > 0 called Coxeter labels [4], s ∈ S. This relation makes impossible the existence of the solution to equation (3.5) (see (3.11)). Generalized Majumdar–Papapetrou solutions. Now we return to a “multi-block” so- lution (3.6)–(3.10). Let M0 = Rd0 , d0 > 2, g0 = δµνdxµ ⊗ dxν , d1 = 1 and g1 = −dt ⊗ dt. For Hs = 1 + ∑ b∈Xs qsb \|x− b\|d0−2 , (3.23) where Xs is finite non-empty subset Xs ⊂ M0, s ∈ S, all qsb > 0, and Xs = Xs′ , qsb = qs′b for b ∈ Xs = Xs′ , s, s′ ∈ Sj , j = 1, . . . , k. The harmonic functions (3.23) are defined in domain M0 \X, X = ⋃ s∈S Xs, and generate the solutions (3.6)–(3.10). Denote S(b) ≡ {s ∈ S\| b ∈ Xs}, b ∈ X. (In the one-block case, when k = 1, all Xs = X and S(b) = S.) We have a horizon at point b w.r.t. time t, when x → b ∈ X, if and only if ξ1(b) ≡ ∑ s∈S(b) (−εs)ν2 s δ1Is − 1 d0 − 2 ≥ 0. This relation follows just from the requirement of the infinite propagation time of light to b ∈ X. Majumdar–Papapetrou solution. Recall that the well-known 4-dimensional Majumdar– Papapetrou (MP) solution [71] corresponding to the Lie algebra A1 in our notations reads g = H2ĝ0 −H−2dt⊗ dt, F = νdH−1 ∧ dt, where ν2 = 2, g0 = ∑3 i=1 dxi ⊗ dxi and H is a harmonic function. We have one electric 0-brane (point) “attached” to the time manifold; d(Is) = 1, εs = −1 and (U s, U s) = 1/2. In this case (e.g. for the extremal Reissner–Nordström black hole) we get ξ1(b) = 1, b ∈ X. Points b are the points of (regular) horizon. For certain examples of finite-dimensional semisimple Lie algebras (e.g. for A1 ⊕ · · · ⊕A1, A2 etc.) the poles b in Hs correspond to (regular) horizons and the solution under consideration describes a collection of k blocks of extremal charged black branes (in equilibrium) [37]. Hyperbolic KM algebras. Let us consider a generalized one-block (k = 1) MP solution corresponding to a hyperbolic KM algebra such that (U s, U s) > 0 for all s ∈ S. In this case all εs > 0, s ∈ S, and hence ξ1(b) < 0. Thus, any point b ∈ X is not a point of the horizon. (It may be checked using the analysis carried out in [37] that any non-exceptional point b is a singular one). As a consequence, the generalized MP solution corresponding to any hyperbolic KM algebra does not describe a collection of extremal charged black branes (in equilibrium) when all (U s, U s) > 0. On Brane Solutions Related to Non-Singular Kac–Moody Algebras 13 3.3 Toda-like solutions 3.3.1 Toda-like Lagrangian Action (2.13) may be also written in the form Sσ0 = 1 2κ2 0 ∫ dd0x √ \|g0\|{R[g0]− GÂB̂(X)g0µν∂µXÂ∂νX B̂ − 2V }, (3.24) where X = (XÂ) = (φi, ϕα,Φs) ∈ RN , and the minisupermetric G = GÂB̂(X)dXÂ ⊗ dXB̂ on the minisuperspace M = RN , N = n + l + \|S\| (\|S\| is the number of elements in S) is defined by the relation (GÂB̂(X)) =  Gij 0 0 0 hαβ 0 0 0 εs exp(−2U s(σ))δss′  . (3.25) Here we consider exact solutions to field equations corresponding to the action (3.24) Rµν [g0] = GÂB̂(X)∂µXÂ∂νX B̂ + 2V d0 − 2 g0 µν , (3.26) 1√ \|g0\| ∂µ[ √ \|g0\|GĈB̂(X)g0µν∂νX B̂]− 1 2 GÂB̂,Ĉ(X)g0,µν∂µXÂ∂νX B̂ = V,Ĉ , (3.27) where s ∈ S. Here V,Ĉ = ∂V/∂XĈ . We put XÂ(x) = F Â(H(x)), where F : (u−, u+) → RN is a smooth function, H : M0 → R is a harmonic function on M0 (i.e. ∆[g0]H = 0), satisfying u− < H(x) < u+ for all x ∈ M0. We take all factor spaces as Ricci-flat and the cosmological constant is set to zero, i.e. the relations ξi = 0 and Λ = 0 are satisfied. In this case the potential is zero : V = 0. It may be verified that the field equations (3.26) and (3.27) are satisfied identically if F = F (u) obeys the Lagrange equations for the Lagrangian L = 1 2 GÂB̂(F )Ḟ ÂḞ B̂ (3.28) with the zero-energy constraint E = 1 2 GÂB̂(F )Ḟ ÂḞ B̂ = 0. (3.29) This means that F : (u−, u+) → RN is a null-geodesic map for the minisupermetric G. Thus, we are led to the Lagrange system (3.28) with the minisupermetric G defined in (3.25). The problem of integrability will be simplified if we integrate the Lagrange equations corre- sponding to Φs (i.e. the Maxwell-type equations for s ∈ Se and Bianchi identities for s ∈ Sm): d du ( exp(−2U s(σ))Φ̇s ) = 0 ⇐⇒ Φ̇s = Qs exp(2U s(σ)), (3.30) where Qs are constants, and s ∈ S. Here (F Â) = (σA,Φs). We put Qs 6= 0 for all s ∈ S. For fixed Q = (Qs, s ∈ S) the Lagrange equations for the Lagrangian (3.28) corresponding to (σA) = (φi, ϕα), when equations (3.30) are substituted, are equivalent to the Lagrange equations for the Lagrangian LQ = 1 2 ĜABσ̇Aσ̇B − VQ, (3.31) 14 V.D. Ivashchuk and V.N. Melnikov where VQ = 1 2 ∑ s∈S εsQ 2 s exp[2U s(σ)], (3.32) the matrix (ĜAB) is defined in (2.14). The zero-energy constraint (3.29) reads EQ = 1 2 ĜABσ̇Aσ̇B + VQ = 0. (3.33) 3.3.2 The solutions Here, as above we are interested in exact solutions for a special case when Ks = (U s, U s) 6= 0, for all s ∈ S, and the generalized Cartan matrix (3.11) is non-degenerate. It follows from the non-degeneracy of the matrix (3.11) that vectors U s, s ∈ S, are linearly independent. Hence, the number of vectors U s should not exceed the dimension of Rn+l, i.e. \|S\| ≤ n + l. The exact solutions were obtained in [40] and are g = (∏ s∈S f2d(Is)hs/(D−2) s ){ exp(2c0H + 2c̄0)ĝ0 + n∑ i=1 (∏ s∈S f −2hsδiIs s ) exp(2ciH + 2c̄i)ĝi } , (3.34) exp(ϕα) = (∏ s∈S f hsχsλα as s ) exp(cαH + c̄α), α = 1, . . . , l, (3.35) and F a = ∑ s∈S Fsδa as with Fs = Qs (∏ s′∈S f −Ass′ s′ ) dH ∧ τ(Is), s ∈ Se, Fs = Qs(∗0dH) ∧ τ(Īs), s ∈ Sm, where ∗0 = ∗[g0] is the Hodge operator on (M0, g 0). Here fs = fs(H) = exp(−qs(H)), (3.36) where qs(u) is a solution to the Toda-like equations q̈s = −Bs exp (∑ s′∈S Ass′qs′ ) (3.37) with Bs = KsεsQ 2 s, s ∈ S, and H = H(x) (x ∈ M0) is a harmonic function on (M0, g 0). Vectors c = (cA) and c̄ = (c̄A) satisfy the linear constraints U s(c) = ∑ i∈Is dic i − χsλasαcα = 0, U s(c̄) = 0, s ∈ S, (3.38) and c0 = 1 2− d0 n∑ j=1 djc j , c̄0 = 1 2− d0 n∑ j=1 dj c̄ j . (3.39) On Brane Solutions Related to Non-Singular Kac–Moody Algebras 15 The zero-energy constraint reads 2ET + hαβcαcβ + n∑ i=1 di(ci)2 + 1 d0 − 2 ( n∑ i=1 dic i )2 = 0, (3.40) where ET = 1 4 ∑ s,s′∈S hsAss′ q̇s ˙qs′ + ∑ s∈S As exp (∑ s′∈S Ass′qs′ ) (3.41) is an integration constant (energy) for the solutions from (3.37) and As = 1 2εsQ 2 s. We note that equations (3.37) correspond to the Lagrangian LT = 1 4 ∑ s,s′∈S hsAss′ q̇s ˙qs′ − ∑ s∈S As exp (∑ s′∈S Ass′qs′ ) , where hs = K−1 s . Thus, the solution is given by relations (3.34)–(3.36) with the functions qs being defined in (3.37) and with relations on the parameters of solutions cA, c̄A (A = i, α, 0), imposed by (3.38), (3.39), (3.40). 4 Cosmological-type solutions Now we consider the case d0 = 1, M0 = R, i.e. we are interested in applications to the sector with dependence on a single variable. We consider the manifold M = (u−, u+)×M1 × · · · ×Mn with a metric g = we2γ(u)du⊗ du + n∑ i=1 e2φi(u)ĝi, where w = ±1, u is a distinguished coordinate which, by convention, will be called “time”; (Mi, g i) are oriented and connected Einstein spaces (see (2.4)), i = 1, . . . , n. The functions γ, φi: (u−, u+) → R are smooth. Here we adopt the brane ansatz from Section 2 putting g0 = wdu⊗ du. 4.1 Lagrange dynamics It follows from Subsection 2.3 that the equations of motion and the Bianchi identities for the field configuration under consideration (with the restrictions from Subsection 2.3 imposed) are equivalent to equations of motion for 1-dimensional σ-model with the action Sσ = µ 2 ∫ duN { Gijφ̇ iφ̇j + hαβϕ̇αϕ̇β + ∑ s∈S εs exp[−2U s(φ, ϕ)](Φ̇s)2 − 2N−2Vw(φ) } , (4.1) where ẋ ≡ dx/du, Vw = −wV = −wΛe2γ0(φ) + w 2 n∑ i=1 ξidie −2φi+2γ0(φ) (4.2) 16 V.D. Ivashchuk and V.N. Melnikov is the potential with γ0(φ) ≡ ∑n i=1 diφ i, and N = exp(γ0−γ) > 0 is the modified lapse function, U s = U s(φ, ϕ) are defined in (2.15), εs are defined in (2.16) for s = (as, vs, Is) ∈ S, and Gij = diδij − didj are components of “pure cosmological” minisupermetric, i, j = 1, . . . , n [67]. In the electric case (F (a,m,I) = 0) for finite internal space volumes Vi the action (4.1) coincides with the action (2.1) if µ = −w/κ2 0, κ2 = κ2 0V1 · · ·Vn. Action (4.1) may be also written in the form Sσ = µ 2 ∫ duN { GÂB̂(X)ẊÂẊB̂ − 2N−2Vw } , (4.3) where X = (XÂ) = (φi, ϕα,Φs) ∈ RN , N = n + l + \|S\|, and minisupermetric G is defined in (3.25). Scalar products. The minisuperspace metric (3.25) may be also written in the form G = Ĝ + ∑ s∈S εse −2Us(σ)dΦs ⊗ dΦs, where σ = (σA) = (φi, ϕα), Ĝ = ĜABdσA ⊗ dσB = Gijdφi ⊗ dφj + hαβdϕα ⊗ dϕβ, is the truncated minisupermetric and U s(σ) = U s AσA is defined in (2.15). The potential (4.2) now reads Vw = (−wΛ)e2UΛ(σ) + n∑ j=1 w 2 ξjdje 2Uj(σ), where U j(σ) = U j AσA = −φj + γ0(φ), (U j A) = (−δj i + di, 0), (4.4) UΛ(σ) = UΛ AσA = γ0(φ), (UΛ A) = (di, 0). The integrability of the Lagrange system (4.3) crucially depends upon the scalar products of co-vectors UΛ, U j , U s (see (3.1)). These products are defined by (3.2) and the following relations [48] ( U i, U j ) = δij dj − 1, (4.5) ( U i, UΛ ) = −1, ( UΛ, UΛ ) = −D − 1 D − 2 ,( U s, U i ) = −δiIs , ( U s, UΛ ) = d(Is) 2−D , (4.6) where s = (as, vs, Is) ∈ S; i, j = 1, . . . , n. Toda-like representation. We put γ = γ0(φ), i.e. the harmonic time gauge is considered. Integrating the Lagrange equations corresponding to Φs (see (3.30)) we are led to the Lagrangian from (3.31) and the zero-energy constraint (3.33) with the modified potential VQ = Vw + 1 2 ∑ s∈S εsQ 2 s exp[2U s(σ)], (4.7) where Vw is defined in (4.2). 4.2 Solutions with Λ = 0 Here we consider solutions with Λ = 0. On Brane Solutions Related to Non-Singular Kac–Moody Algebras 17 4.2.1 Solutions with Ricci-f lat factor-spaces Let all spaces be Ricci-flat, i.e. ξ1 = · · · = ξn = 0. Since H(u) = u is a harmonic function on (M0, g 0) with g0 = wdu⊗ du we get for the metric and scalar fields from (3.34), (3.35) [40] g = (∏ s∈S f2d(Is)hs/(D−2) s ){ exp(2c0u + 2c̄0)wdu⊗ du + n∑ i=1 (∏ s∈S f −2hsδiIs s ) exp(2ciu + 2c̄i)ĝi } , (4.8) exp(ϕα) = (∏ s∈S f hsχsλα as s ) exp(cαu + c̄α), α = 1, . . . , l, (4.9) and F a = ∑ s∈S δa as Fs with Fs = Qs (∏ s′∈S f −Ass′ s′ ) du ∧ τ(Is), s ∈ Se, (4.10) Fs = Qsτ(Īs), s ∈ Sm, (4.11) Qs 6= 0, s ∈ S. Here fs = fs(u) = exp(−qs(u)) and qs(u) obey Toda-like equations (3.37). Relations (3.39) and (3.40) take the form c0 = n∑ j=1 djc j , c̄0 = n∑ j=1 dj c̄ j , 2ET + hαβcαcβ + n∑ i=1 di(ci)2 − ( n∑ i=1 dic i )2 = 0, with ET from (3.41) and all other relations (e.g. constraints (3.38)) are unchanged. This solution in the special case of an Am Toda chain, was obtained earlier in [72] (see also [73]). Some special configurations were considered earlier in [74, 75, 76]. Currently, the cosmological solutions with branes are considered often in a context of S- brane terminology [77]. S-branes were originally space-like analogues of D-branes, see also [78, 79, 80, 81, 45, 82, 83] and references therein. 4.2.2 Solutions with one curved factor-space The cosmological solution with Ricci-flat spaces may be also modified to the following case: ξ1 6= 0, ξ2 = · · · = ξn = 0, i.e. one space is curved and others are Ricci-flat and 1 /∈ Is, s ∈ S, i.e. all “brane” submanifolds do not contain M1. The potential (3.32) is modified for ξ1 6= 0 as follows (see (4.7)) VQ = 1 2 ∑ s∈S εsQ 2 s exp[2U s(σ)] + 1 2 wξ1d1 exp[2U1(σ)], where U1(σ) is defined in (4.4) (d1 > 1). For the scalar products we get from (4.5) and (4.6) (U1, U1) = 1 d1 − 1 < 0, (U1, U s) = 0 (4.12) for all s ∈ S. 18 V.D. Ivashchuk and V.N. Melnikov The solution in the case under consideration may be obtained by a little modification of the solution from the previous section (using (4.12), relations U1i = −δi 1/d1, U1α = 0) [40] g = (∏ s∈S [fs(u)]2d(Is)hs/(D−2) ){ [f1(u)]2d1/(1−d1) exp(2c1u + 2c̄1) × [wdu⊗ du + f2 1 (u)ĝ1] + n∑ i=2 (∏ s∈S [fs(u)]−2hsδiIs ) exp(2ciu + 2c̄i)ĝi } , (4.13) exp(ϕα) = (∏ s∈S f hsχsλα as s ) exp(cαu + c̄α), (4.14) F a = ∑ s∈S δa as Fs, (4.15) with forms Fs defined in (4.10) and (4.11). Here fs = fs(u) = exp(−qs(u)) where qs(u) obey Toda-like equations (3.37) and f1(u) =  R sinh( √ C1(u− u1)), C1 > 0, ξ1w > 0; R sin( √ \|C1\|(u− u1)), C1 < 0, ξ1w > 0; R cosh( √ C1(u− u1)), C1 > 0, ξ1w < 0; \|ξ1(d1 − 1)\|1/2 , C1 = 0, ξ1w > 0, u1, C1 are constants and R = \|ξ1(d1 − 1)/C1\|1/2. The vectors c = (cA) and c̄ = (c̄A) satisfy the linear constraints U r(c) = U r(c̄) = 0, r = s, 1, (for r = s see (3.38)) and the zero-energy constraint C1 d1 d1 − 1 = 2ET + hαβcαcβ + n∑ i=2 di(ci)2 + 1 d1 − 1 ( n∑ i=2 dic i )2 . 4.2.3 Special solutions for block-orthogonal set of vectors Us Let us consider block-orthogonal case: (3.3), (3.4). In this case we get fs = (f̄s)bs where bs = 2 ∑ s′∈S Ass′ , (Ass′ ) = (Ass′)−1 and f̄s(u) =  Rs sinh( √ Cs(u− us)), Cs > 0, ηsεs < 0; Rs sin( √ \|Cs\|(u− us)), Cs < 0, ηsεs < 0; Rs cosh( √ Cs(u− us)), Cs > 0, ηsεs > 0; \|Qs\| \|νs\| (u− us), Cs = 0, ηsεs < 0, where Rs = \|Qs\|/(\|νs\|\|Cs\|1/2), ηsν 2 s = bshs, ηs = ±1, Cs, us are constants, s ∈ S. The constants Cs, us are coinciding inside the blocks: us = us′ , Cs = Cs′ , s, s′ ∈ Si, i = 1, . . . , k. The ratios εsQ 2 s/(bshs) also coincide inside the blocks, or, equivalently, εsQ 2 s bshs = εs′Q2 s′ bs′hs′ , s, s′ ∈ Si, i = 1, . . . , k. (4.16) On Brane Solutions Related to Non-Singular Kac–Moody Algebras 19 For energy integration constant (3.41) we get ET = 1 2 ∑ s∈S Csbshs. (4.17) The solution (4.13)–(4.15) with a block-orthogonal set of U s-vectors was obtained in [85, 86] (for non-composite case see also earlier paper by K.A. Bronnikov [84]). The generalized KM algebra corresponding to the generalized Cartan matrix A in this case is semisimple. In the special orthogonal (or A1⊕· · ·⊕A1) case when: \|S1\| = · · · = \|Sk\| = 1, the solution was obtained in [47]. Thus, here we presented a large class of exact solutions for invertible generalized Cartan matrices (e.g. corresponding to hyperbolic KM algebras). These solutions are governed by Toda-type equations. They are integrable in quadratures for finite-dimensional semisimple Lie algebras [87, 88, 89, 90, 91] in agreement with Adler–van Moerbeke criterion [91] (see also [92]). The problem of integrability of Toda-chains related to Lorentzian (e.g. hyperbolic) KM al- gebras is much more complicated than in the Euclidean case. This is supported by the result from [93] (based on calculation of the Kovalevskaya exponents) where it was shown that the known cases of algebraic integrability for Euclidean Toda chains have no direct analogues in the case of spaces with pseudo-Euclidean metrics because the full-parameter expansions of the general solution contain complex powers of the independent variable. A similar result, using the Painlevé property, was obtained earlier for 2-dimensional Toda chains related to hyperbolic KM algebras [94]. Remark 4.1. It was shown in [95] that all supergravity billiards corresponding to sigma-models on any U/H non compact-symmetric space and obtained by compactifying supergravity to D = 3 are fully integrable. As far as we know this result could not be reformulated in terms of integrability of Toda-chains corresponding to certain Lorentzian (e.g. hyperbolic) KM algebras. 4.3 Examples of S-brane solutions Example 4.1. S-brane solution governed by E10 Toda chain Let us consider the B16-model in 16-dimensional pseudo-Euclidean space of signature (−,+, . . . , +) with six forms F 4, . . . , F 9 and five scalar fields ϕ1, . . . , ϕ5, see (3.21). Recall that for two branes corresponding to F a and F b forms the orthogonal (or (A1 + A1)-) intersection rules read [47, 46]: (a− 1)e ∩o (b− 1)e = N(a, b) = min(a, b)− 3, (a− 1)e ∩o (D − b− 1)m = a− 1−N(a, b), where dv ∩o d′v′ denotes the dimension of orthogonal intersection for two branes with the di- mensions of their worldvolumes being d and d′. dv ∩o d′v′ coincides with the symbol ∆(s, s′) from (3.13)1. The subscripts v, v′ = e,m here indicate whether the brane is electric (e) or mag- netic (m) one. In what follows we will be interested in the following orthogonal intersections: 4e ∩o 4e = 2, 4e ∩o 5e = 2, 4e ∩o 11m = 3, 5e ∩o 11m = 4. Here we deal with 10 (S-)branes: eight electric branes s1, s2, s3, s4, s5, s6, s8, s9 corresponding to 5-form F 5, one electric brane s7 corresponding to 6-form F 6 and one magnetic brane s10 corresponding to 4-form F 4. The brane sets are as follows: I1 = {3, 4, 10, 12}, I2 = {1, 6, 7, 12}, I3 = {8, 9, 10, 12}, I4 = {1, 2, 3, 12}, I5 = {5, 6, 10, 12}, I6 = {1, 4, 8, 12}, I7 = {2, 7, 10, 12, 13}, I8 = {3, 6, 8, 12}, I9 = {1, 10, 11, 12}, I10 = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}. It may be verified that these sets do obey E10 intersection rules following from the relations (3.14) (with Isi = Ii) and the Dynkin diagram from Fig. 1. 1Here as in [46] our notations differ from those adopted in string theory. For example for intersection of M2- and M5-branes we write 3 ∩o 6 = 2 instead of 2 ∩ 5 = 1. 20 V.D. Ivashchuk and V.N. Melnikov p p p p p p p p pp 1 2 3 4 5 6 7 8 9 10 Fig. 1. Dynkin diagram for E10 hyperbolic KM algebra Now we present a cosmological S-brane solution from Subsection 4.2.1 for the configuration of ten branes under consideration. In what follows the relations εs = +1 and hs = 1/2, s ∈ S, are used. The metric (4.8) reads: g = [( ∏ s 6=7,10 fs )4 f5 7 f11 10 ]1/14{ −e2c0t+2c̄0dt⊗ dt + (f2f4f6f9f10)−1e2c1t+2c̄1dx1 ⊗ dx1 + (f4f7f10)−1e2c2t+2c̄2dx2 ⊗ dx2 + (f1f4f8f10)−1e2c3t+2c̄3dx3 ⊗ dx3 + (f1f6f10)−1e2c4t+2c̄4dx4 ⊗ dx4 + (f5f10)−1e2c5t+2c̄5dx5 ⊗ dx5 + (f2f5f8f10)−1e2c6t+2c̄6dx6 ⊗ dx6 + (f2f7f10)−1e2c7t+2c̄7dx7 ⊗ dx7 + (f3f6f8f10)−1e2c8t+2c̄8dx8 ⊗ dx8 + (f3f10)−1e2c9t+2c̄9dx9 ⊗ dx9 + (f1f3f5f7f9f10)−1e2c10t+2c̄10dx10 ⊗ dx10 + (f9f10)−1e2c11t+2c̄11dx11 ⊗ dx11 + ( 9∏ s=1 fs )−1 e2c12t+2c̄12dx12 ⊗ dx12 + f−1 7 e2c13t+2c̄13dx13 ⊗ dx13 + e2c14t+2c̄14dx14 ⊗ dx14 + e2c15t+2c̄15dx15 ⊗ dx15 } . For scalar fields (4.9) we get ϕα = 1 2 −λ5α  ∑ s 6=7,10 ln fs − λ6α ln f7 + λ4α ln f10 + cα ϕt + c̄α ϕ, α = 1, . . . , 5. Here we used the relations λα a = −λaα. The form fields (see (4.10) and (4.11)) are as follows F 4 = Q10dx12 ∧ dx13 ∧ dx14 ∧ dx15, F 5 = Q1f −2 1 f2dt ∧ dx3 ∧ dx4 ∧ dx10 ∧ dx12 + Q2f1f −2 2 f3dt ∧ dx1 ∧ dx6 ∧ dx7 ∧ dx12 + Q3f2f −2 3 f4dt ∧ dx8 ∧ dx9 ∧ dx10 ∧ dx12 + Q4f3f −2 4 f5dt ∧ dx1 ∧ dx2 ∧ dx3 ∧ dx12 + Q5f4f −2 5 f6dt ∧ dx5 ∧ dx6 ∧ dx10 ∧ dx12 + Q6f5f −2 6 f7dt ∧ dx1 ∧ dx4 ∧ dx8 ∧ dx12 + Q8f7f −2 8 f9dt ∧ dx3 ∧ dx6 ∧ dx8 ∧ dx12 + Q9f8f −2 9 dt ∧ dx1 ∧ dx10 ∧ dx11 ∧ dx12, F 6 = Q7f6f −2 7 f8f10dt ∧ dx2 ∧ dx7 ∧ dx10 ∧ dx12 ∧ dx13, (4.18) where Qs 6= 0, s = 1, . . . , 10. Here c0 = 15∑ j=1 cj , c̄0 = 15∑ j=1 c̄j , fs = exp(−qs(t)) and qs(t) obey Toda-type equations q̈s = −2Q2 s exp ( 10∑ s′=1 Ass′qs′ ) , s = 1, . . . , 10, On Brane Solutions Related to Non-Singular Kac–Moody Algebras 21 where (Ass′) is the Cartan matrix for the KM algebra E10 (with the Dynkin diagram from Fig. 1) and the energy integration constant ET = 1 8 10∑ s,s′=1 Ass′ q̇s ˙qs′ + 1 2 10∑ s=1 Q2 s exp ( 10∑ s′=1 Ass′qs′ ) , obeys the constraint 2ET − 5∑ α=1 (cα ϕ)2 + 15∑ i=1 (ci)2 − ( 15∑ i=1 ci )2 = 0. The brane constraints (3.38) are in our case U1(c) = c3 + c4 + c10 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U1(c̄) = 0, U2(c) = c1 + c6 + c7 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U2(c̄) = 0, U3(c) = c8 + c9 + c10 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U3(c̄) = 0, U4(c) = c1 + c2 + c3 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U4(c̄) = 0, U5(c) = c5 + c6 + c10 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U5(c̄) = 0, U6(c) = c1 + c4 + c8 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U6(c̄) = 0, U7(c) = c2 + c7 + c10 + c12 + c13 − 5∑ α=1 λ6αcα ϕ = 0, U7(c̄) = 0, U8(c) = c3 + c6 + c8 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U8(c̄) = 0, U9(c) = c1 + c10 + c11 + c12 − 5∑ α=1 λ5αcα ϕ = 0, U9(c̄) = 0, U10(c) = 11∑ i=1 ci + 5∑ α=1 λ4αcα ϕ = 0, U10(c̄) = 0. Remark 4.2. For a special choice of integration constants ci = 0 and cα ϕ = 0, we get a solution governed by E10 Toda chain with the energy constraint ET = 0. According to the result from [23] we obtain a never ending asymptotical oscillating behavior of scale factors which is described by the motion of a point-like particle in a billiard B ⊂ H9. This billiard has a finite volume since E10 is hyperbolic. Special 1-block solution. Now we consider a special 1-block solution (see Subsection 4.2.3). This solution is valid when a special set of charges is considered (see (4.16)): Q2 s = Q2\|bs\|, 22 V.D. Ivashchuk and V.N. Melnikov where Q 6= 0 and [38] bs = 2 10∑ s′=1 Ass′ = −60,−122,−186,−252,−320,−390,−462,−306,−152,−230, s = 1, . . . , 10. Recall that (Ass′ ) = (Ass′)−1. In this case fs = (f̄)bs , where f̄(t) =  \|Q\| √ 2/C sinh( √ C(t− t0)), C > 0, \|Q\| √ 2/\|C\| sin( √ \|C\|(t− t0)), C < 0, \|Q\| √ 2(t− t0), C = 0 and t0 is a constant. From (4.17) we get ET = −620C, where relation ∑10 s=1 bs = −2480 was used. For the special solution under consideration the electric monomials in (4.18) have a simpler form Fs = Qsf̄ −2dt ∧ τ(Is), where s = 1, 2, . . . , 9. Solution with one harmonic function. Let C = 0 and all ci = c̄i = 0, cα ϕ = c̄α ϕ = 0. In this case H = f̄(t) = \|Q\| √ 2(t − t0) > 0 is a harmonic function on the 1-dimensional manifold ((t0,+∞),−dt⊗dt) and our solution coincides with the 1-block solution (3.6)–(3.10) (if signνs = −signQs for all s). Example 4.2. S-brane solution governed by HA (1) 2 Toda chain. Now we consider the B11-model in 11-dimensional pseudo-Euclidean space of signature (−,+, . . . , +) with 4-form F 4. Here we deal with four electric branes (SM2-branes) s1, s2, s3, s4 corresponding to the 4- form F 4. The brane sets are the following ones: I1 = {1, 2, 3}, I2 = {4, 5, 6}, I3 = {7, 8, 9}, I4 = {1, 4, 10}. It may be verified that these sets obey the intersection rules corresponding to the hyperbolic KM algebra HA (1) 2 with the following Cartan matrix A =  2 −1 −1 0 −1 2 −1 0 −1 −1 2 −1 0 0 −1 2  (4.19) (see (3.14) with Isi = Ii). Now we give a cosmological S-brane solution from Subsection 4.2.1 for the configuration of four branes under consideration. In what follows the relations εs = +1 and hs = 1/2, s ∈ S, are used. The metric (4.8) reads: g = (f1f2f3f4)1/3 { −e2c0t+2c̄0dt⊗ dt + (f1f4)−1e2c1t+2c̄1dx1 ⊗ dx1 + f−1 1 e2c2t+2c̄2dx2 ⊗ dx2 + f−1 1 e2c3t+2c̄3dx3 ⊗ dx3 + (f2f4)−1e2c4t+2c̄4dx4 ⊗ dx4 + f−1 2 e2c5t+2c̄5dx5 ⊗ dx5 On Brane Solutions Related to Non-Singular Kac–Moody Algebras 23 + f−1 2 e2c6t+2c̄6dx6 ⊗ dx6 + f−1 3 e2c7t+2c̄7dx7 ⊗ dx7 + f−1 3 e2c8t+2c̄8dx8 ⊗ dx8 + f−1 3 e2c9t+2c̄9dx9 ⊗ dx9 + f−1 4 e2c10t+2c̄10dx10 ⊗ dx10 } . The form field (see (4.10)) is as follows F 4 = Q1f −2 1 f2f3dt ∧ dx1 ∧ dx2 ∧ dx3 + Q2f1f −2 2 f3dt ∧ dx4 ∧ dx5 ∧ dx6 + Q3f1f2f −2 3 dt ∧ dx7 ∧ dx8 ∧ dx9 + Q4f3f −2 4 dt ∧ dx1 ∧ dx4 ∧ dx10, where Qs 6= 0, s = 1, . . . , 4. Here c0 = 10∑ j=1 cj , c̄0 = 10∑ j=1 c̄j , fs = exp(−qs(t)) and qs(t) obey the Toda-type equations q̈s = −2Q2 s exp ( 4∑ s′=1 Ass′qs′ ) , s = 1, . . . , 4, where (Ass′) is the Cartan matrix (4.19) for the KM algebra HA (1) 2 and the energy integration constant ET = 1 8 4∑ s,s′=1 Ass′ q̇s ˙qs′ + 1 2 4∑ s=1 Q2 s exp ( 4∑ s′=1 Ass′qs′ ) , obeys the constraint 2ET + 10∑ i=1 (ci)2 − ( 10∑ i=1 ci )2 = 0. The brane constraints (3.38) read in this case as follows U1(c) = c1 + c2 + c3 = 0, U1(c̄) = 0, U2(c) = c4 + c5 + c6 = 0, U2(c̄) = 0, U3(c) = c7 + c8 + c9 = 0, U3(c̄) = 0, U4(c) = c1 + c4 + c10 = 0, U4(c̄) = 0. Since F 4∧F 4 = 0 this solution also obeys equations of motion of 11-dimensional supergravity. Special 1-block solution. Now we consider a special 1-block solution (see subsection 4.2.3). This solution is valid when a special set of charges is considered (see (4.16)): Q2 s = Q2\|bs\|, where Q 6= 0 and bs = 2 4∑ s′=1 Ass′ = −12,−12,−14,−6. In this case fs = (f̄)bs , where f̄ is the same as in (4.1). For the energy integration constant we have ET = −11C, (see (4.17)). 24 V.D. Ivashchuk and V.N. Melnikov Example 4.3. S-brane solution governed by P10 Toda chain with ET = 0. Now we consider the B11-model in 11-dimensional pseudo-Euclidean space of signature (−,+, . . . , +) with 4-form F 4. Here we deal with ten electric branes (SM2-branes) s1, . . . , s10 corresponding to the 4- form F 4. The brane sets are taken from [96, 6] as: I1 = {1, 4, 7}, I2 = {8, 9, 10}, I3 = {2, 5, 7}, I4 = {4, 6, 10}, I5 = {2, 3, 9}, I6 = {1, 2, 8}, I7 = {1, 3, 10}, I8 = {4, 5, 8}, I9 = {3, 6, 7}, I10 = {5, 6, 9}. These sets obey the intersection rules corresponding to the Lorentzian KM algebra P10 (we call it Petersen algebra) with the following Cartan matrix A =  2 −1 0 0 −1 0 0 0 0 −1 −1 2 −1 0 0 0 0 0 −1 0 0 −1 2 −1 0 0 −1 0 0 0 0 0 −1 2 −1 −1 0 0 0 0 −1 0 0 −1 2 0 0 −1 0 0 0 0 0 −1 0 2 0 0 −1 −1 0 0 −1 0 0 0 2 −1 0 −1 0 0 0 0 −1 0 −1 2 −1 0 0 −1 0 0 0 −1 0 −1 2 0 −1 0 0 0 0 −1 −1 0 0 2  . (4.20) The Dynkin diagram for this Cartan matrix could be represented by the Petersen graph (“a star inside a pentagon”). P10 is the Lorentzian KM algebra. It is a subalgebra of E10 [96, 6]. Let us present an S-brane solution for the configuration of 10 electric branes under consid- eration. The metric (4.8) reads: g = ( 10∏ s=1 fs )1/3{ −dt⊗ dt + (f1f6f7)−1dx1 ⊗ dx1 + (f3f5f6)−1dx2 ⊗ dx2 + (f5f7f9)−1dx3 ⊗ dx3 + (f1f4f8)−1dx4 ⊗ dx4 + (f3f8f10)−1dx5 ⊗ dx5 + (f4f9f10)−1dx6 ⊗ dx6 + (f1f3f9)−1dx7 ⊗ dx7 + (f2f6f8)−1dx8 ⊗ dx8 + (f2f5f10)−1dx9 ⊗ dx9 + (f2f4f7)−1dx10 ⊗ dx10 } . The form field (see (4.10)) is the following F 4 = Q1f −2 1 f2f5f10dt ∧ dx1 ∧ dx4 ∧ dx7 + Q2f1f −2 2 f3f9dt ∧ dx8 ∧ dx9 ∧ dx10 + Q3f2f −2 3 f4f7dt ∧ dx2 ∧ dx5 ∧ dx7 + Q4f3f −2 4 f5f6dt ∧ dx4 ∧ dx6 ∧ dx10 + Q5f1f4f −2 5 f8dt ∧ dx2 ∧ dx3 ∧ dx9 + Q6f4f −2 6 f9f10dt ∧ dx1 ∧ dx2 ∧ dx8 + Q7f3f −2 7 f8f10dt ∧ dx1 ∧ dx3 ∧ dx10 + Q8f5f7f −2 8 f9dt ∧ dx4 ∧ dx5 ∧ dx8 + Q9f2f6f8f −2 9 dt ∧ dx3 ∧ dx6 ∧ dx7 + Q10f1f6f7f −2 10 dt ∧ dx5 ∧ dx6 ∧ dx9, where Qs 6= 0, s = 1, . . . , 10. Here fs = exp(−qs(t)) and qs(t) obey the Toda-type equations q̈s = −2Q2 s exp ( 10∑ s′=1 Ass′qs′ ) , where (Ass′) is the Cartan matrix (4.20) for the KM algebra P10 and the energy constraint ET = 1 8 10∑ s,s′=1 Ass′ q̇sq̇s′ + 1 2 10∑ s=1 Q2 s exp ( 10∑ s′=1 Ass′qs′ ) = 0 On Brane Solutions Related to Non-Singular Kac–Moody Algebras 25 is obeyed. Here we used the fact that the two sets of linear equations U s(c) = 0, U s(c̄) = 0, s = 1, . . . , 10, have trivial solutions: c = 0, c̄ = 0, due to the linear independence of vectors U s. Since F 4 ∧F 4 = 0, this solution also obeys the equations of motion of 11-dimensional super- gravity. Remark 4.3. As pointed out in [96] we do not obtain a never ending asymptotic oscillating behavior of the scale factors in this case since the Lorentzian KM algebra P10 is not hyperbolic and the corresponding billiard B ⊂ H9 has an infinite volume. Special 1-block solution. Now we consider a special 1-block solution. The calculations give us the following relations bs = 2 10∑ s′=1 Ass′ = −2, s = 1, . . . , 10, and hence the special solution is valid (see (4.16)), when all charges are equal Q2 s = Q2, where Q 6= 0. In this case all fs = f̄−2, where f̄(t) = \|Q\|(t− t0), and t0 is constant. The metric (4.3) may be rewritten using the synchronous time variable ts: g = −dts ⊗ dts + At2/7 s 10∑ i=1 dxi ⊗ dxi, where A > 0 and ts > 0. This metric coincides with the power-law, inflationary solution in the model with a one-component perfect fluid when the following equation of state is adopted: p = 2 5ρ, where p is pressure and ρ is the density of fluid [97, 98]. 5 Black brane solutions In this section we consider the spherically symmetric case of the metric (4.13), i.e. we put w = 1, M1 = Sd1 , g1 = dΩ2 d1 , where dΩ2 d1 is the canonical metric on a unit sphere Sd1 , d1 ≥ 2. In this case ξ1 = d1 − 1. We put M2 = R, g2 = −dt⊗ dt, i.e. M2 is a time manifold. Let C1 ≥ 0. We consider solutions defined on some interval [u0,+∞) with a horizon at u = +∞. When the matrix (hαβ) is positive definite and 2 ∈ Is, ∀ s ∈ S, i.e. all branes have a common time direction t, the horizon condition singles out the unique solution with C1 > 0 and linear asymptotics at infinity qs = −βsu + β̄s + o(1), u → +∞, where βs, β̄s are constants, s ∈ S [42, 43]. In this case cA/µ̄ = −δA 2 + h1U 1A + ∑ s∈S hsbsU sA, βs/µ̄ = 2 ∑ s′∈S Ass′ ≡ bs, (5.1) 26 V.D. Ivashchuk and V.N. Melnikov where s ∈ S, A = (i, α), µ̄ = √ C1, the matrix (Ass′ ) is inverse of the generalized Cartan matrix (Ass′) and h1 = (U1, U1)−1 = d1/(1− d1). Let us introduce a new radial variable R = R(u) through the relations exp(−2µ̄u) = 1− 2µ Rd̄ , µ = µ̄/d̄ > 0, where u > 0, Rd̄ > 2µ, d̄ = d1 − 1. We put c̄A = 0 and qs(0) = 0, A = (i, α), s ∈ S. These relations guarantee the asymptotic flatness (for R → +∞) of the (2 + d1)-dimensional section of the metric. Let us denote Hs = fse −βsu, s ∈ S. Then, solutions (4.13)–(4.15) may be written as follows [41, 42, 43] g = (∏ s∈S H2hsd(Is)/(D−2) s ){( 1− 2µ Rd̄ )−1 dR⊗ dR + R2dΩ2 d1 − (∏ s∈S H−2hs s )( 1− 2µ Rd̄ ) dt⊗ dt + n∑ i=3 (∏ s∈S H −2hsδiIs s ) ĝi } , (5.2) exp(ϕα) = ∏ s∈S H hsχsλα as s , (5.3) where F a = ∑ s∈S δa as Fs, and Fs = − Qs Rd1 (∏ s′∈S H −Ass′ s′ ) dR ∧ τ(Is), s ∈ Se, (5.4) Fs = Qsτ(Īs), s ∈ Sm. (5.5) Here Qs 6= 0, hs = K−1 s , s ∈ S, and the generalized Cartan matrix (Ass′) is non-degenerate. Functions Hs > 0 obey the equations d dz ( (1− 2µz) Hs d dz Hs ) = B̄s ∏ s′∈S H −Ass′ s′ , (5.6) Hs((2µ)−1 − 0) = Hs0 ∈ (0,+∞), (5.7) Hs(+0) = 1, s ∈ S, (5.8) where Hs(z) > 0, µ > 0, z = R−d̄ ∈ (0, (2µ)−1) and B̄s = εsKsQ 2 s/d̄2 6= 0. There exist solutions to equations (5.6)–(5.7) of polynomial type. The simplest example occurs in orthogonal case [58, 47] (for di = 1 see also [56, 57]): (U s, U s′ ) = 0, for s 6= s′, s, s′ ∈ S. In this case (Ass′) = diag(2, . . . , 2) is a Cartan matrix for the semisimple Lie algebra A1 ⊕ · · · ⊕A1 and Hs(z) = 1 + Psz (5.9) with Ps 6= 0, satisfying Ps(Ps + 2µ) = −B̄s, s ∈ S. (For earlier supergravity solutions see [99, 100] and references therein). In [84, 86, 101] this solution was generalized to a block-orthogonal case (3.3), (3.4). In this case (5.9) is modified as follows Hs(z) = (1 + Psz)bs , (5.10) On Brane Solutions Related to Non-Singular Kac–Moody Algebras 27 where bs are defined in (5.1) and parameters Ps coincide inside blocks, i.e. Ps = Ps′ for s, s′ ∈ Si, i = 1, . . . , k. The parameters Ps 6= 0 satisfy the relations [86, 101, 46] Ps(Ps + 2µ) = −B̄s/bs, s ∈ S, (5.11) and the parameters B̄s/bs coincide inside blocks, i.e. B̄s/bs = B̄s′/bs′ for s, s′ ∈ Si, i = 1, . . . , k. Finite-dimensional Lie algebras. Let (Ass′) be a Cartan matrix for a finite-dimensional semisimple Lie algebra G. In this case all powers bs defined in (5.1) are natural numbers which coincide with the components of twice the dual Weyl vector in the basis of simple co-roots [4] and hence, all functions Hs are polynomials, s ∈ S. Conjecture 5.1. Let (Ass′) be a Cartan matrix for a semisimple finite-dimensional Lie algeb- ra G. Then the solutions to equations (5.6)–(5.8) (if exist) have a polynomial structure: Hs(z) = 1 + ns∑ k=1 P (k) s zk, where P (k) s are constants, k = 1, . . . , ns; ns = bs = 2 ∑ s′∈S Ass′ ∈ N and P (ns) s 6= 0, s ∈ S. In the extremal case (µ = +0) an analogue of this conjecture was suggested previously in [76]. Conjecture 5.1 was verified for the Am and Cm+1 Lie algebras in [42, 43]. Explicit expressions for polynomials corresponding to Lie algebras C2 and A3 were obtained in [102] and [103] respectively. Hyperbolic KM algebras. Let (Ass′) be a Cartan matrix for an infinite-dimensional hyperbolic KM algebra G. In this case all powers in (5.1) are negative numbers and hence, we have no chance to get a polynomial structure for Hs. Here we are led to an open problem of seeking solutions to the set of “master” equations (5.6) with boundary conditions (5.7) and (5.8). These solutions define special solutions to Toda-chain equations corresponding to the hyperbolic KM algebra G. Example 5.1. Black hole solutions for A1 ⊕ A1, A2 and H2(q, q) KM algebras. Let us consider the 4-dimensional model governed by the action S = ∫ M d4z √ \|g\| { R[g]− εgMN∂Mϕ∂Nϕ− 1 2 e2λϕ(F 1)2 − 1 2 e−2λϕ(F 2)2 } . Here F 1 and F 2 are 2-forms, ϕ is scalar field and ε = ±1. We consider a black brane solution defined on R∗ × S2 × R with two electric branes s1 and s2 corresponding to forms F 1 and F 2, respectively, with the sets I1 = I2 = {2}. Here R∗ is subset of R, M1 = S2, g1 = dΩ2 2, is the canonical metric on S2, M2 = R, g2 = −dt ⊗ dt and ε1 = ε2 = −1. The scalar products of U -vectors are (we identify U i = U si): (U1, U1) = (U2, U2) = 1 2 + ελ2 6= 0, (U1, U2) = 1 2 − ελ2. The matrix A from (3.11) is a generalized non-degenerate Cartan matrix if and only if 2(U1, U2) (U2, U2) = −q, or, equivalently, ελ2 = 2 + q 2(2− q) , 28 V.D. Ivashchuk and V.N. Melnikov where q = 0, 1, 3, 4, . . . . This takes place when ε = +1, q = 0, 1, ε = −1, q = 3, 4, 5, . . . and λ2 = 2 + q 2\|2− q\| . The first branch (ε = +1) corresponds to finite dimensional Lie algebras A1 ⊕ A1 (q = 0), A2 (q = 1) and the second one (ε = −1) corresponds to hyperbolic KM algebras H2(q, q), q = 3, 4, . . . . In the hyperbolic case the scalar field ϕ is a phantom (ghost). The black brane solution reads (see (5.2)–(5.4)) g = (H1H2)h {( 1− 2µ R )−1 dR⊗ dR + R2dΩ2 2 − (H1H2)−2h ( 1− 2µ R ) dt⊗ dt } , (5.12) exp(ϕ) = (H1/H2)ελh, (5.13) F s = Qs R2 H−2 s (Hs̄)qdt ∧ dR, s = 1, 2. (5.14) Here h = (2− q)/2 and s̄ = 2, 1 for s = 1, 2 respectively. The moduli functions Hs > 0 obey the equations (see (5.6)) d dz ( (1− 2µz) Hs d dz Hs ) = 2Q2 s q − 2 H−2 s (Hs̄)q, (5.15) with the boundary conditions Hs((2µ)−1 − 0) = Hs0 ∈ (0,+∞), Hs(+0) = 1, s = 1, 2, imposed. Here µ > 0, z = 1/R ∈ (0, (2µ)−1). For q = 0, 1 the solutions to equations (5.15) with the boundary conditions imposed were given in [41, 42, 43]. They are polynomials of degrees 1 and 2 for q = 0 and q = 1, respectively. For q = 3, 4, . . . the exact solutions to equations (5.15) are not known yet. Special solution with Q2 1 = Q2 2. Now we consider the special one-block solution with the functions Hs obeying (5.10) and (5.11). Since bs = 2/(2 − q) and B̄s = 2Q2 s/(q − 2) it takes place when Q2 1 = Q2 2 = Q2 > 0. The moduli functions read Hs = H2/(2−q), H = 1 + Pz, where z = 1/R and q 6= 2. These functions obey Hs(z) > 0 for z ∈ [0, (2µ)−1] if P > −2µ (µ > 0). Due to this inequality and the relation P (P +2µ) = Q2 (following from (5.11) ) we get P = −µ + √ µ2 + Q2 > 0. In this special case the solution (5.12)–(5.14) has the following form: g = H2 {( 1− 2µ R )−1 dR⊗ dR + R2dΩ2 2 −H−4 ( 1− 2µ R ) dt⊗ dt } , ϕ = 0, (5.16) F s = Qs H2R2 dt ∧ dR, s = 1, 2. Remarkably, this special solution does not depend upon q. The metric (5.16) coincides with the metric of the Reissner–Nordström solution (when the Maxwell 2-form is F = √ 2Q(HR)−2dt ∧ dR). On Brane Solutions Related to Non-Singular Kac–Moody Algebras 29 In the extremal case µ → +0 we are lead to the special case of a Majumdar–Papapetrou type solution g = H2ĝ0 −H−2dt⊗ dt, ϕ = 0, F s = νsdH−1 ∧ dt, where g0 = ∑3 i=1 dxi ⊗ dxi, H is a harmonic function on M0 = R3 and ν2 s = 1, s = 1, 2. Here νs = −Qs/Q. 6 Conclusions Here we reviewed several families of exact solutions in multidimensional gravity with a set of scalar fields and fields of forms related to non-singular (e.g. hyperbolic) KM algebras. The solutions describe composite electromagnetic branes defined on warped products of Ricci- flat, or sometimes Einstein, spaces of arbitrary dimensions and signatures. The metrics are block-diagonal and all scale factors, scalar fields and fields of forms depend on points of some manifold M0. The solutions include those depending upon harmonic functions, S-branes and spherically-symmetric solutions (e.g. black-branes). Our approach is based on the sigma-model representation obtained in [48] under the rather general assumption on intersections of composite branes (when stress-energy tensor has a diagonal structure). We were dealing with rather general intersection rules [47] governed by invertible generalized Cartan matrix corresponding to the certain generalized KM Lie algebra G. For G = A1⊕· · ·⊕A1 (r terms) we get the well-known standard (e.g. supersymmetry preserving) intersection rules [53, 54, 55, 48]. We have also considered a class of special “block-orthogonal” solutions corresponding to semisimple KM algebras and governed by several harmonic functions. Certain examples of 1-block solutions (e.g. corresponding to KM algebras H2(q, q), AE3) were considered. In the one-block case a generalization of the solutions to those governed by several functions of one harmonic function H and obeying Toda-type equations was presented. For finite-dimensional (semi-simple) Lie algebras we are led to integrable Lagrange systems while the Toda chains corresponding to infinite-dimensional (non-singular) KM algebras are not well studied yet. Some examples of S-brane solutions corresponding to Lorentzian KM algebras HA (1) 2 = A++ 2 , E10 and P10 were presented. We have also considered general classes of cosmological-type solutions (e.g. S-brane and spherically symmetric solutions) governed by Toda-type equations, containing black brane con- figurations as a special case. The “master” equations for moduli functions have polynomial solutions in the finite-dimensional case (according to our conjecture [41, 42, 43]), while in the infinite-dimensional case we have only a special family of the so-called block-orthogonal solutions corresponding to semi-simple non-singular KM algebras. Examples of 4-dimensional dilatonic black hole solutions corresponding to KM algebras A1⊕A1, A2 and H2(q, q) (q > 2) were given. We note that the problem of integrability of Toda chain equations corresponding to (non- singular) KM algebras arises also in the context of fluxbrane solutions [44] that have also a poly- nomial structure of moduli functions for finite-dimensional Lie algebras (see also [104]). (For similar S-brane solutions governed by polynomial functions and its applications in connection with cosmological problems see [105, 106, 107].) Here we have considered only the case of non-degenerate matrix A. It is an open problem to find general classes of solutions with branes for the degenerate case when detA = 0 (e.g. corresponding to affine KM algebras). Some special solutions of such type with maximal set of composite electric S-branes (e.g. when A is not obviously a generalized Cartan matrix) were 30 V.D. Ivashchuk and V.N. Melnikov found in [108, 109] and generalized in [110, 111] for arbitrary (anti-)self-dual parallel charge density form of dimension 2m defined on Ricci-flat Riemannian sub-manifold of dimension 4m. In these examples the restrictions on brane intersections (2.11) and (2.12) were replaced by more general condition on the stress-energy tensor: TM N = 0, M 6= N . It should be noted here that the solutions related to Lorentzian (e.g. hyperbolic) KM algebras considered in Examples 4.1–5.1 are the new ones. (In Example 4.3 the brane configuration from [96, 6]) was used.) Acknowledgements This work was supported in part by the Russian Foundation for Basic Research grant Nr. 07–02– 13624–ofits. We are grateful to D. Singleton for reading the manuscript and valuable comments. We are also indebted to anonymous referees whose comments have led to the improvement of the paper. References [1] Kac V.G., Simple irreducible graded Lie algebras of finite growth, Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 1323–1367. [2] Moody R.V., A new class of Lie algebras, J. Algebra 10 (1968), 211–230. [3] Kac V.G., Infinite-dimensional Lie algebras, 3rd ed., Cambridge University Press, Cambridge, 1990. [4] Fuchs J., Schweigert C., Symmetries, Lie algebras and representations. A graduate course for physicists, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997. [5] Nikulin V.V., On the classification of hyperbolic root systems of rank three, Proc. Steklov Inst. Math. 230 (2000), no. 3, 1–241, alg-geom/9711032, alg-geom/9712033, math.AG/9905150. [6] Henneaux M., Persson D., Spindel P., Spacelike singularities and hidden symmetries of gravity, Living Rev. Relativity 11 (2008), lrr-2008-1, 232 pages, arXiv:0710.1818. [7] Green M.B., Schwarz J.H., Witten E., Superstring theory, Cambridge University Press, Cambridge, 1987. [8] Saclioglu C., Dynkin diagram for hyperbolic Kac–Moody algebras, J. Phys. A: Math. Gen. 22 (18) (1989), 3753–3769. [9] de Buyl S., Schomblond C., Hyperbolic Kac–Moody algebras and Einstein biliards, J. Math. Phys. 45 (2004), 4464–4492, hep-th/0403285. [10] Feingold A., Frenkel I.B., A hyperbolic Kac–Moody Algebra and the theory of Siegel modular forms of genus 2, Math. Ann. 263 (1983), 87–144. [11] Julia B., Kac–Moody symmetry of gravitation and supergravity theories, in Applications of Group Theory in Physics and Mathematical Physics (Chicago, 1982), Lectures in Appl. Math., Vol. 21, Amer. Math. Soc., Providence, RI, 1985, 355–373. [12] Mizoguchi S., E10 symmetry in one-dimensional supergravity, Nuclear Phys. B 528 (1998), 238–264, hep-th/9703160. [13] Nicolai H., A hyperbolic Lie algebra from supergravity, Phys. Lett. B 276 (1992), 333–340. [14] Moore G., String duality, automorphic forms, and generalized Kac–Moody algebras, in String Theory, Gauge Theory and Quantum Gravity (Trieste, 1997). Nuclear Phys. Proc. Suppl. 67 (1998), 56–67, hep-th/9710198. [15] Damour T., Henneaux M., E10, BE10 and arithmetical chaos in superstring cosmology, Phys. Rev. Lett. 86 (2001), 4749–4752, hep-th/0012172. [16] Belinskii V.A., Lifshitz E.M., Khalatnikov I.M., Oscillating regime of approaching to peculiar point in relyativistic cosmology, Usp. Fiz. Nauk 102 (1970), 463–500 (in Russian). Belinskii V.A., Lifshitz E.M., Khalatnikov I.M., A general solution of the Einstein equations with a time singularity, Adv. Phys. 31 (1982), 639–667. [17] Damour T., Henneaux M., Chaos in superstring cosmology, Phys. Rev. Lett. 85 (2000), 920–923, hep-th/0003139. [18] Damour T., Henneaux M., Julia B., Nicolai H., Hyperbolic Kac–Moody algebras and chaos in Kaluza-Klein models, Phys. Lett. B 509 (2001), 323–330, hep-th/0103094. http://arxiv.org/abs/alg-geom/9711032 http://arxiv.org/abs/alg-geom/9712033 http://arxiv.org/abs/math.AG/9905150 http://arxiv.org/abs/0710.1818 http://arxiv.org/abs/hep-th/0403285 http://arxiv.org/abs/hep-th/9703160 http://arxiv.org/abs/hep-th/9710198 http://arxiv.org/abs/hep-th/0012172 http://arxiv.org/abs/hep-th/0003139 http://arxiv.org/abs/hep-th/0103094 On Brane Solutions Related to Non-Singular Kac–Moody Algebras 31 [19] Chitre D.M., Investigations of vanishing of a horizon for Bianchy type IX (the Mixmaster) universe, Ph.D. Thesis, University of Maryland, 1972. [20] Ivashchuk V.D., Kirillov A.A., Melnikov V.N., On stochastic properties of multidimensional cosmological models near the singular point, Izv. Vyssh. Uchebn. Zaved. Fiz. 37 (1994), no. 11, 107–111 (English transl.: Russian Phys. J. 37 (1994), 1102–1106). [21] Ivashchuk V.D., Kirillov A.A., Melnikov V.N., On stochastic behaviour of multidimensional cosmological models near the singularity, Pis’ma ZhETF 60 (1994), no. 4, 225–229 (English transl.: JETP Lett. 60 (1994), 235–239). [22] Ivashchuk V.D., Melnikov V.N., Billiard representation for multidimensional cosmology with multicompo- nent perfect fluid near the singularity, Classical Quantum Gravity 12 (1995), 809–826, gr-qc/9407028. [23] Ivashchuk V.D., Melnikov V.N., Billiard representation for pseudo-Euclidean Toda-like systems of cosmo- logical origin, Regul. Chaotic Dyn. 1 (1996), no. 2, 23–35. [24] Ivashchuk V.D., Melnikov V.N., Billiard representation for multidimensional cosmology with intersecting p-branes near the singularity, J. Math. Phys. 41 (2000), 6341–6363, hep-th/9904077. [25] Damour T., Henneaux M., Nicolai H., Cosmological billiards, Classical Quantum Gravity 20 (2003), R145– R200, hep-th/0212256. [26] Ivashchuk V.D., Melnikov V.N., On billiard approach in multidimensional cosmological models, Gravit. Cosmol. 15 (2009), 49–58, arXiv:0811.2786. [27] Cremmer E., Julia B., Scherk J., Supergravity theory in eleven-dimensions, Phys. Lett. B 76 (1978), 409–412. [28] Hull C.M., Townsend P.K., Unity of superstring dualities, Nuclear Phys. B 438 (1995), 109–137, hep-th/9410167. [29] Witten E., String theory dynamics in various dimensions, Nuclear Phys. B 443 (1995), 85–126, hep-th/9503124. [30] Damour T., Henneaux M., Nicolai H., E10 and a small tension expansion of M-theory, Phys. Rev. Lett. 89 (2002), 221601, 4 pages, hep-th/0207267. [31] Gaberdiel M.R., Olive D.I., West P.C., A class of Lorentzian Kac–Moody algebras, Nuclear Phys. B 645 (2002), 403–437, hep-th/0205068. [32] West P., E11 and M theory, Classical Quantum Gravity 18 (2001), 4443–4460, hep-th/0104081. [33] Schnakenburg I., West P., Kac–Moody symmetries of IIB supergravity, Phys. Lett. B 517 (2001), 421–428, hep-th/0107181. [34] Lambert N.D., West P.C., Coset symmetries in dimensionally reduced bosonic string theory, Nuclear Phys. B 615 (2001), 117–132, hep-th/0107209. [35] Englert F., Houart L., Taormina A., West P., The symmetry of M-theories, J. High Energy Phys. 0309 (2003), no. 9, 020, 39 pages, hep-th/0304206. [36] Englert F., Houart L., West P., Intersection rules, dynamics and symmetries, J. High Energy Phys. 2003 (2003), no. 8, 025, 26 pages, hep-th/0307024. [37] Ivashchuk V.D., Melnikov V.N., Majumdar–Papapetrou-type solutions in sigma-model and intersecting p- branes, Classical Quantum Gravity 16 (1999), 849–869, hep-th/9802121. [38] Grebeniuk M.A., Ivashchuk V.D., Sigma-model solutions and intersecting p-branes related to Lie algebras, Phys. Lett. B 442 (1998), 125–135, hep-th/9805113. [39] Ivashchuk V.D., Kim S.-W., Melnikov V.N., Hyperbolic Kac–Moody algebra from intersecting p-branes, J. Math. Phys. 40 (1998), 4072–4083, Erratum, J. Math. Phys. 42 (2001), 5493, hep-th/9803006. [40] Ivashchuk V.D., Kim S.-W., Solutions with intersecting p-branes related to Toda chains, J. Math. Phys. 41 (2000), 444–460, hep-th/9907019. [41] Ivashchuk V.D., Melnikov V.N., p-brane black holes for general intersections, Gravit. Cosmol. 5 (1999), 313–318, gr-qc/0002085. [42] Ivashchuk V.D., Melnikov V.N., Black hole p-brane solutions for general intersection rules, Gravit. Cosmol. 6 (2000), 27–40, hep-th/9910041. [43] Ivashchuk V.D., Melnikov V.N., Toda p-brane black holes and polynomials related to Lie algebras, Classical Quantum Gravity 17 (2000), 2073–2092, math-ph/0002048. [44] Ivashchuk V.D., Composite fluxbranes with general intersections, Classical Quantum Gravity 19 (2002), 3033–3047, hep-th/0202022. http://arxiv.org/abs/gr-qc/9407028 http://arxiv.org/abs/hep-th/9904077 http://arxiv.org/abs/hep-th/0212256 http://arxiv.org/abs/0811.2786 http://arxiv.org/abs/hep-th/9410167 http://arxiv.org/abs/hep-th/9503124 http://arxiv.org/abs/hep-th/0207267 http://arxiv.org/abs/hep-th/0205068 http://arxiv.org/abs/hep-th/0104081 http://arxiv.org/abs/hep-th/0107181 http://arxiv.org/abs/hep-th/0107209 http://arxiv.org/abs/hep-th/0304206 http://arxiv.org/abs/hep-th/0307024 http://arxiv.org/abs/hep-th/9802121 http://arxiv.org/abs/hep-th/9805113 http://arxiv.org/abs/hep-th/9803006 http://arxiv.org/abs/hep-th/9907019 http://arxiv.org/abs/gr-qc/0002085 http://arxiv.org/abs/hep-th/9910041 http://arxiv.org/abs/math-ph/0002048 http://arxiv.org/abs/hep-th/0202022 32 V.D. Ivashchuk and V.N. Melnikov [45] Ivashchuk V.D., Composite S-brane solutions related to Toda-type systems, Classical Quantum Gravity 20 (2003), 261–275, hep-th/0208101. [46] Ivashchuk V.D., Melnikov V.N., Exact solutions in multidimensional gravity with antisymmetric forms, Classical Quantum Gravity 18 (2001), R82–R157, hep-th/0110274. [47] Ivashchuk V.D., Melnikov V.N., Multidimensional classical and quantum cosmology with intersecting p- branes, J. Math. Phys., 39 (1998), 2866–2888, hep-th/9708157. [48] Ivashchuk V.D., Melnikov V.N., Sigma-model for the generalized composite p-branes, Classical Quan- tum Gravity 14 (1997), 3001–3029, Corrigenda, Classical Quantum Gravity 15 (1998), 3941–3942, hep-th/9705036. [49] Englert F., Houart L., G+++ invariant formulation of gravity and M-theories: exact intersecting brane solutions, J. High Energy Phys. 2004 (2004), no. 5, 059, 15 pages, hep-th/0405082. [50] Ivashchuk V.D., Melnikov V.N., Intersecting p-brane solutions in multidimensional gravity and M-theory, Gravit. Cosmol. 2 (1996), 297–305, hep-th/9612089. [51] Ivashchuk V.D., Melnikov V.N., Generalized intersecting p-brane solutions from σ-model, Phys. Lett. B 403 (1997), 23–30. [52] Ivashchuk V.D., Melnikov V.N., Rainer M., Multidimensional sigma-models with composite electric p-branes, Gravit. Cosmol. 4 (1998), 73–82, gr-qc/9705005. [53] Aref’eva I.Ya., Rytchkov O.A., Incidence matrix description of intersecting p-brane solutions, in L.D. Fad- deev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Providence, RI, 2000, 19–38, hep-th/9612236. [54] Argurio R., Englert F., Hourant L., Intersection rules for p-branes, Phys. Lett. B 398 (1997), 61–68, hep-th/9701042. [55] Aref’eva I.Ya., Ivanov M.G., Rytchkov O.A., Properties of intersecting p-branes in various dimensions, in Supersymmetry and Quantum Field Theory (Kharkov, 1997), Lecture Notes in Phys., Vol. 509, Springer, Berlin, 1998, 25–41 hep-th/9702077. [56] Aref’eva I.Ya., Ivanov M.G., Volovich I.V., Non-extremal intersecting p-branes in various dimensions, Phys. Lett. B 406 (1997), 44–48, hep-th/9702079. [57] Ohta N., Intersection rules for non-extreme p-branes, Phys. Lett. B 403 (1997), 218–224, hep-th/9702164. [58] Bronnikov K.A., Ivashchuk V.D., Melnikov V.N., The Reissner–Nordström problem for intersecting electric and magnetic p-branes, Gravit. Cosmol. 3 (1997), 203–212, gr-qc/9710054. [59] Ivashchuk V.D., On symmetries of target space for σ-model of p-brane origin, Gravit. Cosmol. 4 (1998), 217–220, hep-th/9804102. [60] Gal’tsov D.V., Rytchkov O.A., Generating branes via sigma models, Phys. Rev. D 58 (1998), 122001, 13 pages, hep-th/9801180. [61] West P., The IIA, IIB and eleven dimensional theories and their common E11 origin, Nuclear Phys. B 693 (2004), 76–102, hep-th/0402140. [62] Tseytlin A.A., Harmonic superposition of M -branes, Nuclear Phys. B 475 (1996), 149–163, hep-th/9604035. [63] Bergshoeff E., de Roo M., Eyras E., Janssen B., van der Schaar J.P., Multiple intersections of D-branes and M -branes, Nuclear Phys. B 494 (1997), 119–143, hep-th/9612095. [64] Gauntlett J.P., Intersecting branes, in Proceeding of the February 97 APCTP Winter School on Dualities of Gauge and String Theories, World Sci. Publ., River Edge, NJ, 1998, 146–193, hep-th/9705011. [65] Tseytlin A.A., No-force condition and BPS combinations of p-branes in 11 and 10 dimensions, Nuclear Phys. B 487 (1997), 141–154, hep-th/9609212. [66] Gibbons G.W., Hawking S.W., Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977), 2752–2751. [67] Ivashchuk V.D., Melnikov V.N., Zhuk A.I., On Wheeler–DeWitt equation in multidimensional cosmology, Nuovo Cimento B 104 (1989), 575–582. [68] Nicolai H., Olive D.I., The principal SO(1,2) subalgebra of a hyperbolic Kac–Moody algebra, Lett. Math. Phys. 58 (2001), 141–152, hep-th/0107146. [69] Khviengia N. , Khviengia Z., Lü H., Pope C.N., Towards a field theory of F-theory, Classical Quantum Gravity 15 (1998), 759–773, hep-th/9703012. [70] Vafa C., Evidence for F -theory, Nuclear Phys. B 469 (1996), 403–415, hep-th/9602022. http://arxiv.org/abs/hep-th/0208101 http://arxiv.org/abs/hep-th/0110274 http://arxiv.org/abs/hep-th/9708157 http://arxiv.org/abs/hep-th/9705036 http://arxiv.org/abs/hep-th/0405082 http://arxiv.org/abs/hep-th/9612089 http://arxiv.org/abs/gr-qc/9705005 http://arxiv.org/abs/hep-th/9612236 http://arxiv.org/abs/hep-th/9612236 http://arxiv.org/abs/hep-th/9702077 http://arxiv.org/abs/hep-th/9702079 http://arxiv.org/abs/hep-th/9702164 http://arxiv.org/abs/gr-qc/9710054 http://arxiv.org/abs/hep-th/9804102 http://arxiv.org/abs/hep-th/9801180 http://arxiv.org/abs/hep-th/0402140 http://arxiv.org/abs/hep-th/9604035 http://arxiv.org/abs/hep-th/9612095 http://arxiv.org/abs/hep-th/9705011 http://arxiv.org/abs/hep-th/9609212 http://arxiv.org/abs/hep-th/0107146 http://arxiv.org/abs/hep-th/9703012 http://arxiv.org/abs/hep-th/9602022 On Brane Solutions Related to Non-Singular Kac–Moody Algebras 33 [71] Majumdar S.D., A class of exact solutions of Einstein’s field equations, Phys. Rev. 72 (1947), 390–398. Papapetrou A., A static solution of the gravitational field for arbitrary charge distribution, Proc. Roy. Irish Acad. Sect. A 51 (1947), 191–204. [72] Gavrilov V.R., Melnikov V.N., Toda chains with type Am Lie algebra for multidimensional classical cosmol- ogy with intersecting p-branes, in Proceedings of the International seminar “Curent Topics in Mathematical Cosmology” (30 March – 4 April, 1998, Potsdam, Germany), Editors M. Rainer and H.-J. Schmidt, World Scientific, 1998, 310–322, hep-th/9807004. [73] Gavrilov V.R., Melnikov V.N., D-dimensional p-brane cosmological models associated with Lie algebras of type Am, Theoret. and Math. Phys. 123 (2000), 726–743. [74] Lü H., Mukherji S., Pope C.N., Xu K.-W., Cosmological solutions in string theories, Phys. Rev. D 55 (1997), 7926–7935, hep-th/9610107. [75] Lü H., Pope C.N., Xu K.W., Liouville and Toda solitons in M -theory, Modern Phys. Lett. A 11 (1996), 1785–1796, hep-th/9604058. [76] Lü H., Maharana H., Mukherji S., Pope C.N., Cosmological solutions, p-branes and the Wheeler–DeWitt equation, Phys. Rev. D 57 (1997), 2219–2229, hep-th/9707182. [77] Gutperle M., Strominger A., Spacelike branes, J. High Energy Phys. 2002 (2002), no. 4, 018, 19 pages, hep-th/0202210. [78] Chen C.M., Gal’tsov D.M., Gutperle M., S-brane solutions in supergravity theories, Phys. Rev. D 66 (2002), 024043, 6 pages, hep-th/0204071. [79] Kruczenski M., Myers R.C., Peet A.W., Supergravity S-branes, J. High Energy Phys. 2002 (2002), no. 5, 039, 23 pages, hep-th/0204144. [80] Roy S., On supergravity solutions of space-like Dp-branes, J. High Energy Phys. 2002 (2002), no. 8, 025, 13 pages, hep-th/0205198. [81] Degger N.S., Kaya A., Intersecting S-brane solutions of D = 11 supergravity, J. High Energy Phys. 2002 (2002), no. 7, 038, 14 pages, hep-th/0206057. [82] Ohta N., Intersection rules for S-branes, Phys. Lett. B 558 (2003), 213–220, hep-th/0301095. [83] Ivashchuk V.D., On composite S-brane solutions with orthogonal intersection rules, hep-th/0309027. [84] Bronnikov K.A., Block-orthogonal brane systems, black holes and wormholes, Gravit. Cosmol. 4 (1998), 49–56, hep-th/9710207. [85] Ivashchuk V.D., Melnikov V.N., Cosmological and spherically symmetric solutions with intersecting p-branes, J. Math. Phys. 40 (1999), 6558–6576. [86] Ivashchuk V.D., Melnikov V.N., Multidimensional cosmological and spherically symmetric solutions with intersecting p-branes, in Mathematical and Quantum Aspects of Relativity and Cosmology, Proceedings of the Second Samos Meeting on Cosmology, Geometry and Relativity (1998, Pythagoreon, Samos, Greece), Editors S. Cotsakis and G.W. Gibbons, Lecture Notes in Phys., Vol. 537, Springer, Berlin, 2000, 214–247, gr-qc/9901001. [87] Toda M., Waves in nonlinear lattice, Progr. Theor. Phys. Suppl. 45 (1970), 174–200. [88] Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201–209. [89] Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. in Math. 34 (1979), 195–338. [90] Olshanetsky M.A., Perelomov A.M., Explicit solutions of the periodic Toda lattices, Invent. Math. 54 (1979), 261–269. [91] Adler M., van Moerbeke P., Kowalewski’s asymptotic method, Kac–Moody Lie algebras and regularization, Comm. Math. Phys. 83 (1982), 83–106. [92] Kozlov V.V., Treshchev D.V., Polynomial integrals of Hamiltonian systems with exponential interaction, Math. USSR-Izv. 34 (1990), 555–574. [93] Emel’yanov K.V., Tsygvintsev A.V., Kovalevskaya exponents of systems with exponential interaction, Sb. Math. 191 (2000), 1459–1469. [94] Gebert R.W., Inami T., Mizoguchi S., Toda field theories associated with hyperbolic Kac–Moody algebra – Painlevé properties and W algebras, Internat. J. Modern Phys. A 11 (1996), 5479–5493, hep-th/9503176. [95] Fre P., Sorin A.S., The arrow of time and the Weyl group: all supergravity billiards are integrable, arXiv:0710.1059. http://arxiv.org/abs/hep-th/9807004 http://arxiv.org/abs/hep-th/9610107 http://arxiv.org/abs/hep-th/9604058 http://arxiv.org/abs/hep-th/9707182 http://arxiv.org/abs/hep-th/0202210 http://arxiv.org/abs/hep-th/0204071 http://arxiv.org/abs/hep-th/0204144 http://arxiv.org/abs/hep-th/0205198 http://arxiv.org/abs/hep-th/0206057 http://arxiv.org/abs/hep-th/0301095 http://arxiv.org/abs/hep-th/0309027 http://arxiv.org/abs/hep-th/9710207 http://arxiv.org/abs/gr-qc/9901001 http://arxiv.org/abs/hep-th/9503176 http://arxiv.org/abs/0710.1059 34 V.D. Ivashchuk and V.N. Melnikov [96] Henneaux M., Leston M., Persson D., Spindel P., Geometric configurations, regular subalgebras of E10 and M-theory cosmology, J. High Energy Phys. 2006 (2006), no. 10, 021, 49 pages, hep-th/0606123. [97] Ivashchuk V.D., Melnikov V.N., Multidimensional classical and quantum cosmology with perfect fluid, Gravit. Cosmol. 1 (1995), 133–148, hep-th/9503223. [98] Alimi J.-M., Ivashchuk V.D., Kononogov S.A., Melnikov V.N., Multidimensional cosmology with anisotropic fluid: acceleration and variation of G, Gravit. Cosmol. 12 (2006), 173–178, gr-qc/0611015. [99] Cvetic M., Tseytlin A.A., Nonextreme black holes from nonextreme intersecting M-branes, Nuclear Phys. B 478 (1996), 181–198, hep-th/9606033. [100] Ohta N., Shimizu T., Non-extreme black holes from intersecting M-branes, Internat. J. Modern Phys. A 13 (1998), 1305–1328, hep-th/9701095. [101] Cotsakis S., Ivashchuk V.D., Melnikov V.N., p-brane black holes and post-Newtonian approximation, Gravit. Cosmol. 5 (1999), 52–57, gr-qc/9804080. [102] Grebeniuk M.A., Ivashchuk V.D., Kim S.-W., Black-brane solutions for C2 algebra, J. Math. Phys. 43 (2002), 6016–6023, hep-th/0111219. [103] Grebeniuk M.A., Ivashchuk V.D., Melnikov V.N., Black-brane solution for A3 algebra, Phys. Lett. B 543 (2002), 98–106, hep-th/0208083. [104] Golubtsova A.A., Ivashchuk V.D., On calculation of fluxbrane polynomials corresponding to classical series of Lie algebras, arXiv:0804.0757. [105] Goncharenko I.S., Ivashchuk V.D., Melnikov V.N., Fluxbrane and S-brane solutions with polynomials re- lated to rank-2 Lie algebras, Gravit. Cosmol. 13 (2007), 262–266, math-ph/0612079. [106] Alimi J.-M., Ivashchuk V.D., Melnikov V.N., An S-brane solution with acceleration and small enough variation of G, Gravit. Cosmol. 13 (2007), 137–141, arXiv:0711.3770. [107] Ivashchuk V.D., Kononogov S.A., Melnikov V.N., Electric S-brane solutions corresponding to rank-2 Lie algebras: acceleration and small variation of G, Grav. Cosmol. 14, No. 3 (2008), 235–240, arXiv:0901.0025. [108] Ivashchuk V.D., Singleton D., Composite electric S-brane solutions with maximal number of branes, J. High Energy Phys. 2004 (2004), no. 10, 061, 18 pages, hep-th/0407224. [109] Ivashchuk V.D., Melnikov V.N., Singleton D., On avoiding cosmological oscillating behavior for S-brane solutions with diagonal metrics, Phys. Rev. D 72 (2005), 103511, 8 pages, gr-qc/0509065. [110] Ivashchuk V.D., Melnikov V.N., Singleton D., Electric S-brane solutions with parallel forms on Ricci-flat factor space, Gravit. Cosmol. 12 (2006), 315–320, hep-th/0610167. [111] Dehnen H., Ivashchuk V.D., Melnikov V.N., S-brane solutions with (anti-)self-dual parallel charge density form on a Ricci-flat submanifold, Gravit. Cosmol. 13 (2007), 23–30, hep-th/0612260. http://arxiv.org/abs/hep-th/0606123 http://arxiv.org/abs/hep-th/9503223 http://arxiv.org/abs/gr-qc/0611015 http://arxiv.org/abs/hep-th/9606033 http://arxiv.org/abs/hep-th/9701095 http://arxiv.org/abs/gr-qc/9804080 http://arxiv.org/abs/hep-th/0111219 http://arxiv.org/abs/hep-th/0208083 http://arxiv.org/abs/0804.0757 http://arxiv.org/abs/math-ph/0612079 http://arxiv.org/abs/0711.3770 http://arxiv.org/abs/0901.0025 http://arxiv.org/abs/hep-th/0407224 http://arxiv.org/abs/gr-qc/0509065 http://arxiv.org/abs/hep-th/0610167 http://arxiv.org/abs/hep-th/0612260 1 Introduction 2 The model 2.1 The action 2.2 Ansatz for composite branes 2.3 The sigma model 3 Solutions governed by harmonic functions 3.1 Solutions with block-orthogonal set of Us and Ricci-flat factor-spaces 3.2 Solutions related to non-singular KM algebras 3.3 Toda-like solutions 3.3.1 Toda-like Lagrangian 3.3.2 The solutions 4 Cosmological-type solutions 4.1 Lagrange dynamics 4.2 Solutions with = 0 4.2.1 Solutions with Ricci-flat factor-spaces 4.2.2 Solutions with one curved factor-space 4.2.3 Special solutions for block-orthogonal set of vectors Us 4.3 Examples of S-brane solutions 5 Black brane solutions 6 Conclusions References
id	nasplib_isofts_kiev_ua-123456789-149113
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
issn	1815-0659
language	English
last_indexed	2025-12-07T18:05:36Z
publishDate	2009
publisher	Інститут математики НАН України
record_format	dspace
spelling	Ivashchuk, V.D. Melnikov, V.N. 2019-02-19T17:28:28Z 2019-02-19T17:28:28Z 2009 On Brane Solutions Related to Non-Singular Kac-Moody Algebras / V.D. Ivashchuk, V.N. Melnikov // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 111 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 17B67; 17B81; 83E15; 83E50; 83F05; 81T30 https://nasplib.isofts.kiev.ua/handle/123456789/149113 A multidimensional gravitational model containing scalar fields and antisymmetric forms is considered. The manifold is chosen in the form M = M0 × M1 × ... × Mn, where Mi are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions with intersecting composite branes (e.g. solutions with harmonic functions, S-brane and black brane ones) with intersection rules related to non-singular Kac-Moody (KM) algebras (e.g. hyperbolic ones) are reviewed. Some examples of solutions, e.g. corresponding to hyperbolic KM algebras: H2(q,q), AE3, HA2(1), E10 and Lorentzian KM algebra P10 are presented. This work was supported in part by the Russian Foundation for Basic Research grant Nr. 07–02–13624–ofits. We are grateful to D. Singleton for reading the manuscript and valuable comments. We are also indebted to anonymous referees whose comments have led to the improvement of the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Brane Solutions Related to Non-Singular Kac-Moody Algebras Article published earlier
spellingShingle	On Brane Solutions Related to Non-Singular Kac-Moody Algebras Ivashchuk, V.D. Melnikov, V.N.
title	On Brane Solutions Related to Non-Singular Kac-Moody Algebras
title_full	On Brane Solutions Related to Non-Singular Kac-Moody Algebras
title_fullStr	On Brane Solutions Related to Non-Singular Kac-Moody Algebras
title_full_unstemmed	On Brane Solutions Related to Non-Singular Kac-Moody Algebras
title_short	On Brane Solutions Related to Non-Singular Kac-Moody Algebras
title_sort	on brane solutions related to non-singular kac-moody algebras
url	https://nasplib.isofts.kiev.ua/handle/123456789/149113
work_keys_str_mv	AT ivashchukvd onbranesolutionsrelatedtononsingularkacmoodyalgebras AT melnikovvn onbranesolutionsrelatedtononsingularkacmoodyalgebras

On Brane Solutions Related to Non-Singular Kac-Moody Algebras

Institution

Similar Items