On Brane Solutions with Intersection Rules Related to Lie Algebras

Abstract

The review is devoted to exact solutions with hidden symmetries arising in a multidimensional gravitational model containing scalar fields and antisymmetric forms. These solutions are defined on a manifold of the form $M=M_0\times M_1\times \dots \times M_n$, where all $M_i$ with $i\ge 1$ are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, $M_0$ is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on $M_0$. The monomial forms (of the electric or magnetic type) appear in the so-called composite brane ansatz for fields of forms. Under certain restrictions on branes, the sigma-model approach for the solutions to field equations was derived in earlier publications with V.N.Melnikov. The sigma model is defined on the manifold $M_0$ of dimension $d_0\ne 2$. By using the sigma-model approach, several classes of exact solutions, e.g., solutions with harmonic functions, S-brane, black brane and fluxbrane solutions, are obtained. For $d_0=1$, the solutions are governed by moduli functions that obey Toda-like equations. For certain brane intersections related to Lie algebras of finite rank—non-singular Kac–Moody (KM) algebras—the moduli functions are governed by Toda equations corresponding to these algebras. For finite-dimensional semi-simple Lie algebras, the Toda equations are integrable, and for black brane and fluxbrane configurations, they give rise to polynomial moduli functions. Some examples of solutions, e.g., corresponding to finite dimensional semi-simple Lie algebras, hyperbolic KM algebras: $H_2(q,q)$, $A{E}_3$, $H{A}_2^{(1)}$, $E_{10}$ and Lorentzian KM algebra $P_{10}$, are presented.

1. Introduction

In this review, we deal with certain aspects of multidimensional models of gravity, which are rather popular at present time. As is widely known, the history of the multidimensional approach begins with the well-known papers of Kaluza and Klein on five-dimensional theories, which were continued then by Jordan in his works. These papers were in some sense a source of inspiration for Brans and Dicke in their well-known work on a scalar-tensor gravitational theory. After their work, many investigations were performed in models with fundamental scalar fields, both conformal and non-conformal; see [1] and the references therein.

A revival of the ideas of many dimensions started in the 1970s and continues now, mainly due to the development of unified theories. In the 1970s, interest in multidimensional gravitational models was stimulated mainly by: (i) the ideas of gauge theories leading to a non-Abelian generalization of the Kaluza–Klein approach and (ii) by supergravitational theories [2,3].

In the 1980s, the supergravitational theories “gave the baton” to the superstring theory [4] and in the 1990s to the so-called M-theory [5,6].

Usually, having a multidimensional model, one can obtain a four-dimensional one by a dimensional reduction based on the decomposition of the manifold:

$$M=M^4\times M_{\mathrm{int}},$$

(1)

where $M^4$ is a four-dimensional manifold and $M_{\mathrm{int}}$ is some internal manifold (widely considered to be compact).

Here, we overview a sigma-model approach and several classes of exact solutions for the multidimensional gravitational model governed by the Lagrangian:

$$\mathcal{L}=R[g]-2\mathsf{\Lambda }-h_{\alpha \beta }{g}^{MN}{\partial }_M{\phi }^{\alpha }{\partial }_N{\phi }^{\beta }-\sum _a\frac{1}{n_a!}exp(2{\lambda }_{a\alpha }{\phi }^{\alpha }){(F^a)}^2,$$

(2)

where g is metric, $F^a=d{A}^a$ are forms (of ranks $n_a$), ${\phi }^{\alpha }$ are scalar fields, ${\lambda }_a=({\lambda }_{a\alpha })$ are certain vectors and $\mathsf{\Lambda }$ is a cosmological constant (the matrix $(h_{\alpha \beta })$ is invertible). The Lagrangians of such a type may describe (pure bosonic) solutions in supergravitational models [7], when the so-called Chern–Simons terms are zero.

We deal with the solutions having a block-diagonal (warped-product) metrics g defined on the D-dimensional product manifold M, i.e.,

$$g=e^{2\gamma }{g}^0+\sum _{i=1}^n{e}^{2{\varphi }^i}{g}^i,M=M_0\times M_1\times \dots \times M_n,$$

(3)

where $g^0$ is a metric on $M_0$ and $g^i$ are fixed Einstein (or Ricci-flat) metrics on $M_i$, $i>0$. The moduli $\gamma ,{\varphi }^i$ and scalar fields ${\phi }^{\alpha }$ are functions on $M_0$, and fields of forms are also governed by several scalar functions on $M_0$. Any $F^a$ is supposed to be a sum of (linear independent) monoms, corresponding to electric or magnetic p-branes (p-dimensional analogues of membranes), i.e., the so-called composite p-brane ansatz is considered.

Here, we are interested in brane solutions, which have intersection rules related to a certain subclass of Lie algebras, namely non-singular Kac–Moody (KM) algebras. Kac–Moody (KM) Lie algebras [8,9,10] play a rather important role in different areas of mathematical physics (see [10,11,12,13] and the references therein).

We recall that KM Lie algebra is a Lie algebra generated by the relations [10]:

$$\begin{array}{c}\hfill [h_i,h_j]=0,[e_i,f_j]={\delta }_{ij}{h}_j,\end{array}$$

(4)

$$\begin{array}{c}\hfill [h_i,e_j]=A_{ij}{e}_j,[h_i,f_j]=-A_{ij}{f}_j,\end{array}$$

(5)

$$\begin{array}{c}\hfill {(\mathrm{ad}{e}_i)}^{1-A_{ij}}(e_j)=0(i\ne j),\end{array}$$

(6)

$$\begin{array}{c}\hfill {(\mathrm{ad}{f}_i)}^{1-A_{ij}}(f_j)=0(i\ne j).\end{array}$$

(7)

Here, $A=(A_{ij})$ is a generalized Cartan matrix, $i,j=1,\dots ,r$, and r is the rank of the KM algebra. This means that all $A_{ii}=2$; $A_{ij}$ for $i\ne j$ are non-positive integers, and $A_{ij}=0$ implies $A_{ji}=0$.

In what follows, the matrix A is restricted to be non-degenerate (i.e., $detA\ne 0$) and symmetrizable i.e., $A=B\mathcal{D}$, where B is a symmetric matrix and $\mathcal{D}$ is an invertible diagonal matrix ($\mathcal{D}$ may be chosen in such way that all of its entries ${\mathcal{D}}_{ii}$ are positive rational numbers [10]). Here, we do not consider singular KM algebras with $detA=0$, e.g., affine ones. Recall that affine KM algebras are of much interest for conformal field theories, superstring theories, etc. [4,11].

In the case when A is positive definite (the Euclidean case), we get ordinary finite dimensional Lie algebras [10,11]. For non-Euclidean signatures of A, all KM algebras are infinite-dimensional. Among these, the Lorentzian KM algebras with pseudo-Euclidean signatures $(-,+,\dots ,+)$ 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 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 [14,15,16] and have rank $2\le r\le 10$. For $r\ge 3$, there is a finite number of hyperbolic algebras. For rank 10, there are four algebras, known as $E_{10}$, $B{E}_{10}$, $C{E}_{10}$ and $D{E}_{10}$. Hyperbolic KM algebras appeared in ordinary gravity [17] (${\mathcal{F}}_3=A{E}_3=H_3$), supergravity: [18,19] ($E_{10}$), [20] (${\mathcal{F}}_3$), strings [21], etc.

The growth of interest in hyperbolic algebras in theoretical and mathematical physics appeared in 2001 after the publication of Damour and Henneaux [22] devoted to a description of chaotic (BKL-type [23]) behavior near the singularity in string-inspired low energy models (e.g., supergravitational ones) [24] (see also [25]). 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 [26]) elaborated in our papers [27,28,29,30,31] (see also [32,33,34]).

The description of oscillating behavior near the singularity in $D=11$ supergravity [2] (which is related to M-theory [5,6]) in terms of motion of a point-like particle in a nine-dimensional billiard (of finite volume) corresponding to the Weyl chamber of the hyperbolic KM algebra $E_{10}$ inspired another description of $D=11$ supergravity in [35]: a formal “small tension” expansion of $D=11$ supergravity near a space-like singularity was shown to be equivalent (at least up to 30th order in height) to a null geodesic motion in the infinite dimensional coset space ${\mathcal{E}}_{10}/K({\mathcal{E}}_{10})$ (here, $K({\mathcal{E}}_{10})$ is the maximal compact subgroup of the hyperbolic Kac–Moody group ${\mathcal{E}}_{10}(R)$).

Recall that $E_{10}$ KM algebra is an over-extension of the finite dimensional Lie algebra $E_8$, i.e., $E_{10}=E_8^{++}$. However, there is another extension of $E_8$—the so-called very extended Kac–Moody algebra of the $E_8$ algebra—called $E_{11}=E_8^{+++}$ (to get an understanding of very extended algebras and some of their properties, see [36] and the references therein). It has been proposed by P.West that the Lorentzian (non-hyperbolic) KM algebra $E_{11}$ is responsible for a hidden algebraic structure characterizing 11-dimensional supergravity [37]. The same very extended algebra occurs in $IIA$ [37] and $IIB$ supergravities [38]. Moreover, it was conjectured that an analogous hidden structure is realized in the effective action of the bosonic string (with the KM algebra $k_{27}=D_{24}^{+++}$) [37] and also for pure D dimensional gravity (with the KM algebra $A_{D-3}^{+++}$ [39]). It has been suggested in [40] that all of the so-called maximally-oxidized theories (see also [13]) possess the very extension $G^{+++}$ of the simple Lie algebra G. It was shown in [41] that the BPSsolutions of the oxidized theory of a simply laced group $\mathcal{G}$ form representations of a subgroup of the Weyl transformations of the algebra $G^{+++}$. For other aspects of very-extended Kac–Moody algebras (e.g., $E_{11}$), see also [42,43,44,45] and the references therein.

In this paper, we briefly review another possibility for utilizing non-singular (e.g., hyperbolic) KM algebras suggested in three of our papers [46,47,48]. This possibility (implicitly assumed also in [49,50,51,52,53,54]) 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 $M=M_0\times M_1\times \dots \times M_n$, where $M_i$ are Ricci-flat spaces ($i\ge 1$). 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 [55]. We deal here with generalized Cartan matrices of the form:

$$A_{s{s}^{\prime }}\equiv \frac{2(U^s,U^{s^{\prime }})}{(U^{s^{\prime }},U^{s^{\prime }})},$$

(8)

$s,s^{\prime }\in S$, with $(U^s,U^s)\ne 0$, for all $s\in S$ (S is a finite set). Here, $U^s$ are the so-called brane (co-)vectors. They are linear functions on ${\mathbb{R}}^N$, where $N=n+l$, and l is the number of scalar fields. The indefinite scalar product $(.,.)$ [56] is defined on ${({\mathbb{R}}^N)}^{*}$ and has the signature $(-1,+1,\dots ,+1)$ 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 ${\alpha }_s$ of the generalized KM algebra after a suitable normalizing. For indecomposable A (when the KM algebra is simple), the matrices $((U^s,U^{s^{\prime }}))$ and $(({\alpha }_s|{\alpha }_{s^{\prime }}))$ are proportional to each other. Here, $(.|.)$ is a non-degenerate bilinear symmetric form on a root space obeying $({\alpha }_s|{\alpha }_s)>0$ for all simple roots ${\alpha }_s$ [10].

We note that the minisuperspace bilinear form $(.,.)$ coming from the multidimensional gravitational model [56] should not coincide with the bilinear form $(.|.)$ from [10], and hence, there exist physical examples where all $(U^s,U^s)$ are negative. Some examples of this are given below (see Section 5). For $D=11$ supergravity and ten-dimensional $IIA$, $IIB$ supergravities, all $(U^s,U^s)=2$ [44,55] and corresponding KM algebras are simply laced. It was shown in our papers [29,30,31] that the inequality $(U^s,U^s)>0$ is a necessary condition for the formation of the billiard wall (ifone 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 [56] (for the electric case, see also [57,58,59]):

$$(U^s,U^{s^{\prime }})=d_{s{s}^{\prime }}+\frac{d_s{d}_{s^{\prime }}}{2-D}+{\chi }_s{\chi }_{s^{\prime }}<{\lambda }_s,{\lambda }_{s^{\prime }}>,$$

(9)

where $d_s$ and $d_{s^{\prime }}$ are dimensions of the brane world volumes corresponding to branes s and $s^{\prime }$, respectively, $d_{s{s}^{\prime }}$ is the dimension of intersection of the brane world volumes, D is the total space-time dimension, ${\chi }_s=+1,-1$ for electric or magnetic brane, respectively, and $<{\lambda }_s,{\lambda }_{s^{\prime }}>$ is the non-degenerate scalar product of the l-dimensional dilatonic coupling vectors ${\lambda }_s$ and ${\lambda }_{s^{\prime }}$ corresponding to branes s and $s^{\prime }$.

Relations (2) and (9) define the brane intersection rules [55]:

$$\begin{array}{c}\hfill d_{s{s}^{\prime }}=d_{s{s}^{\prime }}^o+\frac{1}{2}{K}_{s^{\prime }}{A}_{s{s}^{\prime }},\end{array}$$

(10)

$s\ne s^{\prime }$, where $K_s=(U^s,U^s)$ and:

$$d_{s{s}^{\prime }}^o=\frac{d_s{d}_{s^{\prime }}}{D-2}-{\chi }_s{\chi }_{s^{\prime }}<{\lambda }_s,{\lambda }_{s^{\prime }}>$$

(11)

are the dimensions of the so-called orthogonal (or $A_1\oplus \dots \oplus A_1$-) intersections of branes following from the orthogonality conditions [56]:

$$(U^s,U^{s^{\prime }})=0,$$

(12)

$s\ne s^{\prime }$. The orthogonality relations (12) for brane intersections in the non-composite electric case were suggested in [57,58] and for the composite electric case in [59].

Relations (9) and (11) were derived in [56] 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), Ricci-flat factor spaces $M_i$ had arbitrary dimensions $d_i$ and signatures. The intersection rules (11) appeared earlier (in different notations) in [60,61,62] when all $d_i=1$ ($i>0$), and $<.,.>$ was positive definite (in [60,61], $l=1$, and total space-time had a pseudo-Euclidean signature). The intersection rules (11) were also used in [55,63,64,65] in the context of intersecting black brane solutions.

It was proven in [66] 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: $\mathsf{\nabla }Riem=0$) if and only if:

$$(U^s-U^{s^{\prime }})(U^s,U^{s^{\prime }})=0$$

(13)

for all s and $s^{\prime }$, i.e., when any two brane vectors $U^s$ and $U^{s^{\prime }}$, $s\ne s^{\prime }$, are either coinciding $U^s=U^{s^{\prime }}$ or orthogonal $(U^s,U^{s^{\prime }})=0$ (for two electric branes and $l=1$, see also [67]).

Now, relations for brane vectors $U^s$ (2) and (9) (with $U^s$ being identified with roots ${\alpha }_s$) are widely used in the $G^{+++}$-approach [13,41]. The orthogonality condition (12) describing the intersection of branes [56,57,58,59] was rediscovered in [44] (for some particular intersecting configurations of M-theory, it was done in [68]). 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., $({\alpha }_s|{\alpha }_{s^{\prime }})=0$, $s\ne s^{\prime }$, where ${\alpha }_s$ should be real positive roots (“real” means that $({\alpha }_s|{\alpha }_s)>0$). In [44], another condition on brane, root vectors was found: ${\alpha }_s+{\alpha }_{s^{\prime }}$ should not be a root, $s\ne s^{\prime }$. 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 do not take place due to non-zero contributions of Chern-Simons terms.

It should be noted that the orthogonality relations for brane intersections (12) appeared in 1996–1997. The standard intersection rules (11) gave back the well-known zero binding energy configurations preserving some supersymmetries. These brane configurations were originally derived from supersymmetry and duality arguments (see for example [69,70,71] and the reference therein) or by using a no-force condition (suggested for M-branes in [72]).

2. The Model

2.1. The Action

We consider the model governed by action:

$$\begin{array}{cc}\hfill S=& \frac{1}{2{\kappa }^2}{\int }_M{d}^{D}z\sqrt{\left|g\right|}\{R[g]-2\mathsf{\Lambda }-h_{\alpha \beta }{g}^{MN}{\partial }_M{\phi }^{\alpha }{\partial }_N{\phi }^{\beta }\hfill \\ & -\sum _{a\in \Delta }\frac{{\theta }_a}{n_a!}exp[2{\lambda }_a(\phi )]{(F^a)}_g^2\}+S_{GH},\hfill \end{array}$$

(14)

where $g=g_{MN}d{z}^M\otimes d{z}^N$ is the metric on the manifold M, $dimM=D$, $\phi =({\phi }^{\alpha })\in {\mathbb{R}}^l$ is a vector of dilatonic scalar fields, $(h_{\alpha \beta })$ is a non-degenerate symmetric $l\times l$ matrix ($l\in \mathbb{N}$), ${\theta }_a\ne 0$,

$$F^a=d{A}^a=\frac{1}{n_a!}{F}_{M_1\dots M_{n_a}}^{a}d{z}^{M_1}\wedge \dots \wedge d{z}^{M_{n_a}}$$

is an $n_a$-form ($n_a\ge 2$) on a D-dimensional manifold M, $\mathsf{\Lambda }$ is a cosmological constant and ${\lambda }_a$ is a one-form on ${\mathbb{R}}^l$: ${\lambda }_a(\phi )={\lambda }_{a\alpha }{\phi }^{\alpha }$, $a\in \Delta$, $\alpha =1,\dots ,l$. In (14), we denote $\left|g\right|=|det(g_{MN})|$, ${(F^a)}_g^2=F_{M_1\dots M_{n_a}}^a{F}_{N_1\dots N_{n_a}}^a{g}^{M_1{N}_1}\dots g^{M_{n_a}{N}_{n_a}},$ $a\in \Delta$, where $\Delta$ is some finite set (for example, of positive integers), and $S_{\mathrm{GH}}$ is the standard Gibbons–Hawking boundary term [73]. In models with one time, all ${\theta }_a=1$ when the signature of the metric is $(-1,+1,\dots ,+1)$. ${\kappa }^2$ is the multidimensional gravitational constant.

2.2. Ansatz for Composite Branes

Let us consider the manifold:

$$M=M_0\times M_1\times \dots \times M_n,$$

(15)

with the metric:

$$g=e^{2\gamma (x)}{\widehat{g}}^0+\sum _{i=1}^n{e}^{2{\varphi }^i(x)}{\widehat{g}}^i,$$

(16)

where $g^0=g_{\mu \nu }^0(x)d{x}^{\mu }\otimes d{x}^{\nu }$ is an arbitrary metric with any signature on the manifold $M_0$ and $g^i=g_{m_i{n}_i}^i(y_i)d{y}_i^{m_i}\otimes d{y}_i^{n_i}$ is a metric on $M_i$ satisfying the equation:

$$R_{m_i{n}_i}[g^i]={\xi }_i{g}_{m_i{n}_i}^i,$$

(17)

$m_i,n_i=1,\dots ,d_i$; ${\xi }_i=\mathrm{const}$, $i=1,\dots ,n$. Here, ${\widehat{g}}^i=p_i^{*}{g}^i$ is the pullback of the metric $g^i$ to the manifold M by the canonical projection: $p_i:M\to M_i$, $i=0,\dots ,n$. Thus, $(M_i,g^i)$ are Einstein spaces, $i=1,\dots ,n$. The functions $\gamma ,{\varphi }^i:M_0\to \mathbb{R}$ are smooth. We denote $d_{\nu }=\dim M_{\nu }$; $\nu =0,\dots ,n$; $D={\sum }_{\nu =0}^n{d}_{\nu }$. We put any manifold $M_{\nu }$, $\nu =0,\dots ,n$, to be oriented and connected. Then, the volume $d_i$-form:

$${\tau }_i\equiv \sqrt{|g^i(y_i)|}d{y}_i^1\wedge \dots \wedge d{y}_i^{d_i},$$

(18)

and signature parameter:

$$\epsilon (i)\equiv \mathrm{sign}(det(g_{m_i{n}_i}^i))=\pm 1$$

(19)

are correctly defined for all $i=1,\dots ,n$.

Let $\mathsf{\Omega }=\mathsf{\Omega }(n)$ be a set of all non-empty subsets of $\{1,\dots ,n\}$. The number of elements in $\mathsf{\Omega }$ is $\left|\mathsf{\Omega }\right|=2^n-1$. For any $I=\{i_1,\dots ,i_k\}\in \mathsf{\Omega }$, $i_1<\dots <i_k$, we denote:

$$\begin{array}{c}\hfill \tau (I)\equiv {\widehat{\tau }}_{i_1}\wedge \dots \wedge {\widehat{\tau }}_{i_k},\end{array}$$

(20)

$$\begin{array}{c}\hfill \epsilon (I)\equiv \epsilon (i_1)\dots \epsilon (i_k),\end{array}$$

(21)

$$\begin{array}{c}\hfill M_I\equiv M_{i_1}\times \dots \times M_{i_k},\end{array}$$

(22)

$$\begin{array}{c}\hfill d(I)\equiv \sum _{i\in I}{d}_i.\end{array}$$

(23)

Here, ${\widehat{\tau }}_i=p_i^{*}{\widehat{\tau }}_i$ is the pullback of the form ${\tau }_i$ to the manifold M by the canonical projection: $p_i:M\to M_i$, $i=1,\dots ,n$. We also put $\tau (\varnothing )=\epsilon (\varnothing )=1$ and $d(\varnothing )=0$.

For fields of forms, we consider the following composite electromagnetic ansatz:

$$\begin{array}{c}\hfill F^a=\sum _{I\in {\mathsf{\Omega }}_{a,e}}{\mathcal{F}}^{(a,e,I)}+\sum _{J\in {\mathsf{\Omega }}_{a,m}}{\mathcal{F}}^{(a,m,J)}\end{array}$$

(24)

where:

$$\begin{array}{c}\hfill {\mathcal{F}}^{(a,e,I)}=d{\mathsf{\Phi }}^{(a,e,I)}\wedge \tau (I),\end{array}$$

(25)

$$\begin{array}{c}\hfill {\mathcal{F}}^{(a,m,J)}=e^{-2{\lambda }_a(\phi )}*>(d{\mathsf{\Phi }}^{(a,m,J)}\wedge \tau (J))\end{array}$$

(26)

are elementary forms of electric and magnetic types, respectively, $a\in \Delta$, $I\in {\mathsf{\Omega }}_{a,e}$, $J\in {\mathsf{\Omega }}_{a,m}$ and ${\mathsf{\Omega }}_{a,v}\subset \mathsf{\Omega }$, $v=e,m$. In (26), $*=*[g]$ is the Hodge operator on $(M,g)$:

$${(*\omega )}_{M_1\dots M_{D-k}}=\frac{{\left|g\right|}^{1/2}}{k!}{\epsilon }_{M_1\dots M_{D-k}{N}_1\dots N_k}{\omega }^{N_1\dots N_k},$$

where $\mathrm{rank}\omega =k$.

For scalar functions, we put:

$$\begin{array}{c}\hfill {\phi }^{\alpha }={\phi }^{\alpha }(x),{\mathsf{\Phi }}^s={\mathsf{\Phi }}^s(x),\end{array}$$

(27)

$s\in S$. Thus, ${\phi }^{\alpha }$ and ${\mathsf{\Phi }}^s$ are functions on $M_0$.

Here and below:

$$\begin{array}{c}\hfill S=S_e\bigsqcup S_m,S_v={\bigsqcup }_{a\in \Delta }\{a\}\times \{v\}\times {\mathsf{\Omega }}_{a,v},\end{array}$$

(28)

$v=e,m$. Here and in what follows, ⊔ means the union of non-intersecting sets. The set S consists of elements $s=(a_s,v_s,I_s)$, where $a_s\in \Delta$ is the color index, $v_s=e,m$ is the electro-magnetic index and set $I_s\in {\mathsf{\Omega }}_{a_s,v_s}$ describes the location of brane.

Due to (25) and (26):

$$\begin{array}{c}\hfill d(I)=n_a-1,d(J)=D-n_a-1,\end{array}$$

(29)

for $I\in {\mathsf{\Omega }}_{a,e}$ and $J\in {\mathsf{\Omega }}_{a,m}$ (i.e., in the electric and magnetic case, respectively).

2.3. The Sigma Model

Let $d_0\ne 2$ and:

$$\begin{array}{c}\hfill \gamma ={\gamma }_0(\varphi )\equiv \frac{1}{2-d_0}\sum _{j=1}^n{d}_j{\varphi }^j,\end{array}$$

(30)

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):

$$(\mathbf{R}\mathbf{1})d(I\cap J)\le d(I)-2,$$

(31)

for any $I,J\in {\mathsf{\Omega }}_{a,v}$, $a\in \Delta$, $v=e,m$ (here $d(I)=d(J)$) and

$$(\mathbf{R}\mathbf{2})d(I\cap J)\ne 0for{d}_0=1,d(I\cap J)\ne 1for{d}_0=3.$$

(32)

It was proven in [56] that equations of motion for the model (14) and the Bianchi identities:

$$\begin{array}{c}\hfill d{\mathcal{F}}^s=0,\end{array}$$

(33)

$s\in S_m$, for fields from (16), (24)–(27), when Restrictions (31) and (32) are imposed, are equivalent to the equations of motion for the $\sigma$-model governed by the action:

$$\begin{array}{cc}\hfill S_{\sigma 0}=& \frac{1}{2{\kappa }_0^2}\int d^{d_0}x\sqrt{|g^0|}\{R[g^0]-{\widehat{G}}_{AB}{g}^{0\mu \nu }{\partial }_{\mu }{\sigma }^A{\partial }_{\nu }{\sigma }^B\hfill \\ & -\sum _{s\in S}{\epsilon }_{s}exp(-2U_A^s{\sigma }^A)g^{0\mu \nu }{\partial }_{\mu }{\mathsf{\Phi }}^s{\partial }_{\nu }{\mathsf{\Phi }}^s-2V\},\hfill \end{array}$$

(34)

where $({\sigma }^A)=({\varphi }^i,{\phi }^{\alpha })$, $k_0\ne 0$, the index set S is defined in (28),

$$V=V(\varphi )=\mathsf{\Lambda }{e}^{2{\gamma }_0(\varphi )}-\frac{1}{2}\sum _{i=1}^n{\xi }_i{d}_i{e}^{-2{\varphi }^i+2{\gamma }_0(\varphi )}$$

(35)

is the potential,

$$\begin{array}{c}\hfill ({\widehat{G}}_{AB})=\left(\begin{array}{cc}{G}_{ij}& 0\\ 0& h_{\alpha \beta }\end{array}\right)\end{array}$$

(36)

is the target space metric with:

$$\begin{array}{c}\hfill G_{ij}=d_i{\delta }_{ij}+\frac{d_i{d}_j}{d_0-2}\end{array}$$

(37)

and co-vectors:

$$\begin{array}{c}\hfill U_A^s=U_A^s{\sigma }^A=\sum _{i\in I_s}{d}_i{\varphi }^i-{\chi }_s{\lambda }_{a_s}(\phi ),(U_A^s)=(d_i{\delta }_{i{I}_s},-{\chi }_s{\lambda }_{a_s\alpha }),\end{array}$$

(38)

$s=(a_s,v_s,I_s)$. Here, ${\chi }_e=+1$ and ${\chi }_m=-1$;

$$\begin{array}{c}\hfill {\delta }_{iI}=\sum _{j\in I}{\delta }_{ij}\end{array}$$

(39)

is an indicator of i belonging to I: ${\delta }_{iI}=1$ for $i\in I$ and ${\delta }_{iI}=0$ otherwise; and:

$$\begin{array}{c}\hfill {\epsilon }_s={(-\epsilon [g])}^{(1-{\chi }_s)/2}\epsilon (I_s){\theta }_{a_s},\end{array}$$

(40)

$s\in S$, $\epsilon [g]\equiv \mathrm{sign}det(g_{MN})$. More explicitly, (40) reads:

$${\epsilon }_s=\epsilon (I_s){\theta }_{a_s}\mathrm{for}{v}_s=e;{\epsilon }_s=-\epsilon [g]\epsilon (I_s){\theta }_{a_s},\mathrm{for}{v}_s=m.$$

(41)

For finite internal space volumes $V_i$ (e.g., compact $M_i$) and electric p-branes (i.e., all ${\mathsf{\Omega }}_{a,m}=\varnothing$), the action (34) coincides with the action (14) when ${\kappa }^2={\kappa }_0^2{\prod }_{i=1}^n{V}_i$.

3. Solutions Governed by Harmonic Functions

3.1. Solutions with a Block-Orthogonal 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., ${\xi }_i=\mathsf{\Lambda }=0$, $i=1,\dots ,n$. In certain situations, these solutions describe extremal black branes charged by fields of forms.

The solutions crucially depend on scalar products of U^s-vectors $(U^s,U^{s^{\prime }})$; $s,s^{\prime }\in S$, where:

$$\begin{array}{c}\hfill (U,U^{\prime })={\widehat{G}}^{AB}{U}_A{U}_B^{\prime },\end{array}$$

(42)

for $U=(U_A),U^{\prime }=(U_A^{\prime })\in {\mathbb{R}}^N$, $N=n+l$ and:

$$({\widehat{G}}^{AB})=\left(\begin{array}{cc}{G}^{ij}& 0\\ 0& h^{\alpha \beta }\end{array}\right)$$

(43)

is the inverse matrix to the matrix (36). Here, as in [74], we have:

$$G^{ij}=\frac{{\delta }^{ij}}{d_i}+\frac{1}{2-D},$$

(44)

$i,j=1,\cdots ,n$.

The scalar products (42) for vectors $U^s$ were calculated in [56] and are given by:

$$\begin{array}{c}\hfill (U^s,U^{s^{\prime }})=d(I_s\cap I_{s^{\prime }})+\frac{d(I_s)d(I_{s^{\prime }})}{2-D}+{\chi }_s{\chi }_{s^{\prime }}{\lambda }_{a_s\alpha }{\lambda }_{a_{s^{\prime }}\beta }{h}^{\alpha \beta },\end{array}$$

(45)

where $(h^{\alpha \beta })={(h_{\alpha \beta })}^{-1}$, and $s=(a_s,v_s,I_s)$, $s^{\prime }=(a_{s^{\prime }},v_{s^{\prime }},I_{s^{\prime }})$ belong to the index set S defined in (28). This relation is a very important one since it encodes brane data (e.g., intersections) via the scalar products of U-vectors.

Let:

$$S=S_1\bigsqcup \cdots \bigsqcup S_k,$$

(46)

$S_i\ne \varnothing$, $i=1,\cdots ,k$, and:

$$(U^s,U^{s^{\prime }})=0$$

(47)

for all $s\in S_i$, $s^{\prime }\in S_j$, $i\ne j$; $i,j=1,\cdots ,k$. Relation (46) means that the set S is a union of k non-intersecting (non-empty) subsets $S_1,\cdots ,S_k$. According to (47), the set of vectors $(U^s,s\in S)$ has a block-orthogonal structure with respect to the scalar product (42), i.e., it splits into k mutually orthogonal blocks $(U^s,s\in S_i)$, $i=1,\cdots ,k$.

Here, we consider exact solutions in the model (14), when vectors $(U^s,s\in S)$ obey the block-orthogonal decomposition (46) and (47) with scalar products defined in (45) [46]. These solutions were obtained from the corresponding solutions to the $\sigma$-model equations of motion [46].

Proposition 1.

Let $(M_0,g^0)$ be Ricci-flat: $R_{\mu \nu }[g^0]=0$. Then, the field configuration:

$$\begin{array}{c}\hfill g^0,{\sigma }^A=\sum _{s\in S}{\epsilon }_s{U}^{sA}{\nu }_s^{2}ln{H}_s,{\mathsf{\Phi }}^s=\frac{{\nu }_s}{H_s},\end{array}$$

(48)

$s\in S$, satisfies the field equations corresponding to the action (34) with $V=0$ if the real numbers ${\nu }_s$ obey the relations:

$$\begin{array}{c}\hfill \sum _{s^{\prime }\in S}(U^s,U^{s^{\prime }}){\epsilon }_{s^{\prime }}{\nu }_{s^{\prime }}^2=-1\end{array}$$

(49)

$s\in S$, the functions $H_s>0$ are harmonic, i.e., $\Delta [g^0]H_s=0$, $s\in S$, and $H_s$ are coinciding inside blocks: $H_s=H_{s^{\prime }}$ for $s,s^{\prime }\in S_i$, $i=1,\cdots ,k$.

Using the sigma-model solution from Proposition 1 and the relations for contra-variant components [56]:

$$\begin{array}{c}\hfill U^{si}={\delta }_{i{I}_s}-\frac{d(I_s)}{D-2},U^{s\alpha }=-{\chi }_s{\lambda }_{a_s}^{\alpha },\end{array}$$

(50)

$s=(a_s,v_s,I_s)$, we get [46]:

$$\begin{array}{c}\hfill g={\left(\prod _{s\in S}{H}_s^{2d(I_s){\epsilon }_s{\nu }_s^2}\right)}^{1/(2-D)}\left\{{\widehat{g}}^0+\sum _{i=1}^n\left(\prod _{s\in S}{H}_s^{2{\epsilon }_s{\nu }_s^2{\delta }_{i{I}_s}}\right){\widehat{g}}^i\right\},\end{array}$$

(51)

$$\begin{array}{c}\hfill {\phi }^{\alpha }=-\sum _{s\in S}{\lambda }_{a_s}^{\alpha }{\chi }_s{\epsilon }_s{\nu }_s^{2}ln{H}_s,\end{array}$$

(52)

$$\begin{array}{c}\hfill F^a=\sum _{s\in S}{\mathcal{F}}^s{\delta }_{a_s}^a,\end{array}$$

(53)

where $i=1,\cdots ,n$, $\alpha =1,\cdots ,l$, $a\in \Delta$ and:

$$\begin{array}{c}\hfill {\mathcal{F}}^s={\nu }_{s}d{H}_s^{-1}\wedge \tau (I_s),\mathrm{for}{v}_s=e,\end{array}$$

(54)

$$\begin{array}{c}\hfill {\mathcal{F}}^s={\nu }_s({*}_{0}d{H}_s)\wedge \tau ({\overline{I}}_s),\mathrm{for}{v}_s=m,\end{array}$$

(55)

$H_s$ are harmonic functions on $(M_0,g^0)$, which coincide inside blocks (i.e., $H_s=H_{s^{\prime }}$ for $s,s^{\prime }\in S_i$, $i=1,\cdots ,k$), and the relations (49) on the parameters ${\nu }_s$ are imposed. Here, the matrix $((U^s,U^{s^{\prime }}))$ and parameters ${\epsilon }_s$, $s\in S$, are defined in (45) and (40), respectively; ${\lambda }_a^{\alpha }=h^{\alpha \beta }{\lambda }_{a\beta }$, ${*}_0=*[g^0]$ is the Hodge operator on $(M_0,g^0)$, and:

$$\begin{array}{c}\hfill \overline{I}=\{1,\cdots ,n\}\backslash I\end{array}$$

(56)

is the dual set (in (55), we redefined the sign of the ${\nu }_s$-parameter).

3.2. Solutions Related to Non-Singular KM Algebras

Now, we study the solutions (51)–(55) in more detail and show that some of them may be related to non-singular KM algebras. We put:

$$\begin{array}{c}\hfill K_s\equiv (U^s,U^s)\ne 0\end{array}$$

(57)

for all $s\in S$ and introduce the matrix $A=(A_{s{s}^{\prime }})$:

$$\begin{array}{c}\hfill A_{s{s}^{\prime }}\equiv \frac{2(U^s,U^{s^{\prime }})}{(U^{s^{\prime }},U^{s^{\prime }})},\end{array}$$

(58)

$s,s^{\prime }\in S$. Here, some ordering in S is assumed.

Using this definition and (45), we obtain the intersection rules [55]:

$$\begin{array}{c}\hfill d(I_s\cap I_{s^{\prime }})=\Delta (s,s^{\prime })+\frac{1}{2}{K}_{s^{\prime }}{A}_{s{s}^{\prime }},\end{array}$$

(59)

where $s\ne s^{\prime }$, and:

$$\begin{array}{c}\hfill \Delta (s,s^{\prime })=\frac{d(I_s)d(I_{s^{\prime }})}{D-2}-{\chi }_s{\chi }_{s^{\prime }}{\lambda }_{a_s\alpha }{\lambda }_{a_{s^{\prime }}\beta }{h}^{\alpha \beta }\end{array}$$

(60)

defines the so-called “orthogonal” intersection rules [56] (see also [60,61,62] for $d_i=1$).

In $D=11$ and $D=10$ ($IIA$ and $IIB$) supergravity models, all $K_s=2$, and hence, the intersection rules (59) in this case have a simpler form [55]:

$$\begin{array}{c}\hfill d(I_s\cap I_{s^{\prime }})=\Delta (s,s^{\prime })+A_{s{s}^{\prime }},\end{array}$$

(61)

where $s\ne s^{\prime }$, implying $A_{s{s}^{\prime }}=A_{s^{\prime }s}$. The corresponding KM algebra is simply-laced in this case.

For $detA\ne 0$, Relation (49) may be rewritten in the equivalent form:

$$\begin{array}{c}\hfill -{\epsilon }_s{\nu }_s^2(U^s,U^s)=2\sum _{s^{\prime }\in S}{A}^{s{s}^{\prime }}\equiv b_s,\end{array}$$

(62)

where $s\in S$, and $(A^{s{s}^{\prime }})=A^{-1}$. Thus, Equation (49) may be resolved in terms of ${\nu }_s$ for certain ${\epsilon }_s=\pm 1$, $s\in S$. We note that due to (47), the matrix A has a block-diagonal structure, and hence, for any i-th block, the set of parameters $({\nu }_s,s\in S_i)$ depends on the matrix inverse to the matrix $(A_{s{s}^{\prime }};s,s^{\prime }\in S_i)$.

Now, we consider the one-block case when the brane intersections are related to some non-singular KM algebras.

3.2.1. Finite-Dimensional Lie Algebras [47]

Let A be a Cartan matrix of a simple finite-dimensional Lie algebra. In this case, $A_{s{s}^{\prime }}\in \{0,-1,-2,-3\}$, $s\ne s^{\prime }$. The elements of inverse matrix $A^{-1}$ are positive (see Ch. 7 in [11]), and hence, we get from (62) the same signature relation as in the orthogonal case [56]:

$$\begin{array}{c}\hfill {\epsilon }_s(U^s,U^s)<0,\end{array}$$

(63)

$s\in S$.

When all $(U^s,U^s)>0$, we get ${\epsilon }_s<0$, $s\in S$.

Moreover, all $b_s$ are natural numbers:

$$\begin{array}{c}\hfill b_s=n_s\in \mathbb{N},\end{array}$$

(64)

$s\in S$.

The integers $n_s$ coincide with the components of the twice dual Weyl vector in the basis of simple co-roots (see Ch. 3.1.7 in [11]).

3.2.2. 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:

$$\begin{array}{c}\hfill {\epsilon }_s(U^s,U^s)>0,\end{array}$$

(65)

since all $b_s$ are negative in the hyperbolic case [36]:

$$\begin{array}{c}\hfill b_s<0,\end{array}$$

(66)

where $s\in S$.

For $(U^s,U^s)>0$, we get ${\epsilon }_s>0$, $s\in S$. If ${\theta }_{a_s}>0$ for all $s\in S$, then:

$$\epsilon (I_s)=1\mathrm{for}{v}_s=e;\epsilon (I_s)=-\epsilon [g]\mathrm{for}{v}_s=m.$$

(67)

For a pseudo-Euclidean metric g, all $\epsilon (I_s)=1$, and hence, all branes are Euclidean or should contain an even number of time directions: $2,4,\dots$. For $\epsilon [g]=1$, only magnetic branes may be pseudo-Euclidean.

Remark 1.

The inequalities (66) guarantee the existence of a principal (real) $so(1,2)$ subalgebra for any hyperbolic Kac–Moody algebra [36,75]. Similarly, the inequalities (64) imply the existence of a principal $so(3)$ subalgebra for any finite dimensional (semi-)simple Lie algebra. It was shown in [36] that this property is not just restricted to hyperbolic algebras, but holds for a wider class of Lorentzian algebras obeying the inequalities $b_s\le 0$ for all s.

Example 1.

${\mathcal{F}}_3=A{E}_3$ algebra [48].

Now, we consider an example of the solution corresponding to the hyperbolic KM algebra ${\mathcal{F}}_3$ with the Cartan matrix:

$$\begin{array}{c}\hfill A=\left(\begin{array}{ccc}2& -2& 0\\ -2& 2& -1\\ 0& -1& 2\end{array}\right),\end{array}$$

(68)

${\mathcal{F}}_3$ contains ${\mathbf{A}}_{\mathbf{1}}^{(\mathbf{1})}$ affine subalgebra (it corresponds to the Geroch group) and ${\mathbf{A}}_{\mathbf{2}}$ subalgebra. There exists an example of the solution with the A-matrix (68) for 11-dimensional model governed by the action:

$$\begin{array}{c}\hfill S=\int d^{11}z\sqrt{\left|g\right|}\left\{R[g]-\frac{1}{4!}{(F^4)}^2-\frac{1}{4!}{(F^{4*})}^2\right\},\end{array}$$

(69)

where $\mathrm{rank}{F}^4=\mathrm{rank}{F}^{4*}=4$. Here, $\Delta =\{4,4*\}$. We consider a configuration with two magnetic five-branes corresponding to the form $F^4$ and one electric two-brane corresponding to the form $F^{4*}$. We denote $S=\{s_1,s_2,s_3\}$, $a_{s_1}=a_{s_3}=4$, $a_{s_2}=4*$ and $v_{s_1}=v_{s_3}=m$, $v_{s_2}=e$, where $d(I_{s_1})=d(I_{s_3})=6$ and $d(I_{s_2})=3$.

The intersection rules (59) read:

$$\begin{array}{c}\hfill d(I_{s_1}\cap I_{s_2})=0,d(I_{s_2}\cap I_{s_3})=1,d(I_{s_1}\cap I_{s_3})=4.\end{array}$$

(70)

For the manifold (15), we put $n=5$ and $d_1=2$, $d_2=4$, $d_3=d_4=1$, $d_5=2$. The corresponding brane sets are the following: $I_{s_1}=\{1,2\}$, $I_{s_2}=\{4,5\}$, $I_{s_3}=\{2,3,4\}$.

The solution reads:

$$\begin{array}{c}\hfill g=H^{-12}\left\{-dt\otimes dt+H^9{\widehat{g}}^1+H^{13}{\widehat{g}}^2+H^4{\widehat{g}}^3+H^{14}{\widehat{g}}^4+H^{10}{\widehat{g}}^5\right\},\end{array}$$

(71)

$$\begin{array}{c}\hfill F^4=\frac{dH}{dt}\left\{{\nu }_{s_1}{\widehat{\tau }}_3\wedge {\tau }_4\wedge {\widehat{\tau }}_5+{\nu }_{s_3}{\widehat{\tau }}_1\wedge {\widehat{\tau }}_5\right\},\end{array}$$

(72)

$$\begin{array}{c}\hfill F^{4*}=\frac{dH}{dt}\frac{{\nu }_{s_2}}{H^2}dt\wedge {\widehat{\tau }}_4\wedge {\widehat{\tau }}_5,\end{array}$$

(73)

where ${\nu }_{s_1}^2=\frac{9}{2}$, ${\nu }_{s_2}^2=5$ and ${\nu }_{s_3}^2=2$ (see (62)).

All metrics $g^i$ are Ricci-flat ($i=1,\dots ,5$) and have Euclidean signatures (this agrees with Relations (65) and (40)), and $H=ht+h_0>0$, where $h,h_0$ are constants. The metric (71) may be also rewritten using the synchronous time variable $t_s$:

$$\begin{array}{c}\hfill g=-d{t}_s\otimes d{t}_s+f^{3/5}{\widehat{g}}^1+f^{-1/5}{\widehat{g}}^2+f^{8/5}{\widehat{g}}^3+f^{-2/5}{\widehat{g}}^4+f^{2/5}{\widehat{g}}^5,\end{array}$$

(74)

where $f=5h{t}_s=H^{-5}>0$, $h>0$ and $t_s>0$. The metric describes the power-law “inflation” in $D=11$. It is singular for $t_s\to +0$.

In the next example, we consider a chain of the so-called $B_D$-models ($D\ge 11$) suggested in [55]. For $D=11$, the $B_D$-model coincides with the truncated (i.e., without the Chern–Simons term) bosonic sector of $D=11$ supergravity [2], which is related to M-theory. For $D=12$, it coincides with the truncated 12-dimensional model from [76], which may be related to F-theory [77].

${\mathit{B}}_{\mathit{D}}$-models: The $B_D$-model has the action [55]:

$$\begin{array}{c}\hfill S_D=\int d^{D}z\sqrt{\left|g\right|}\left\{R[g]+g^{MN}{\partial }_M\overrightarrow{\phi }{\partial }_N\overrightarrow{\phi }-\sum _{a=4}^{D-7}\frac{1}{a!}exp[2{\overrightarrow{\lambda }}_a\overrightarrow{\phi }]{(F^a)}^2\right\},\end{array}$$

(75)

where $\overrightarrow{\phi }=({\phi }^1,\cdots ,{\phi }^l)\in {\mathbb{R}}^l$, ${\overrightarrow{\lambda }}_a=({\lambda }_{a1},\cdots ,{\lambda }_{al})\in {\mathbb{R}}^l$, $l=D-11$, $\mathrm{rank}{F}^a=a$, $a=4,\cdots ,D-7$. Here, vectors ${\overrightarrow{\lambda }}_a$ satisfy the relations:

$$\begin{array}{c}\hfill {\overrightarrow{\lambda }}_a{\overrightarrow{\lambda }}_b=N(a,b)-\frac{(a-1)(b-1)}{D-2}={\mathsf{\Lambda }}_{ab},\end{array}$$

(76)

$$\begin{array}{c}\hfill N(a,b)=min(a,b)-3,\end{array}$$

(77)

$a,b=4,\cdots ,D-7$ and ${\overrightarrow{\lambda }}_{D-7}=-2{\overrightarrow{\lambda }}_4$. For $D>11$, vectors ${\overrightarrow{\lambda }}_4,\cdots ,{\overrightarrow{\lambda }}_{D-8}$ are linearly independent (it may be verified that matrix $({\mathsf{\Lambda }}_{ab})$ is positive definite, and hence, the set of vectors obeying (76) does exist).

The model (75) contains l scalar fields with a negative kinetic term (i.e., $h_{\alpha \beta }=-{\delta }_{\alpha \beta }$ in (14)) coupled to several forms (the number of forms is (l + 1)).

For the brane world volumes, we have the following dimensions (see (29)):

$$\begin{array}{c}\hfill d(I)=3,\cdots ,D-8,I\in {\mathsf{\Omega }}_{a,e},\end{array}$$

(78)

$$\begin{array}{c}\hfill d(I)=D-5,\cdots ,6,I\in {\mathsf{\Omega }}_{a,m}.\end{array}$$

(79)

Thus, there are $(l+1)$ electric and $(l+1)$ magnetic p-branes, $p=d(I)-1$. In the $B_D$-model, all $K_s=2$.

Example 2.

$H_2(q_1,q_2)$ algebra [46].

Let

$$\begin{array}{c}\hfill A=\left(\begin{array}{cc}2& -q_1\\ -q_2& 2\end{array}\right),q_1{q}_2>4,\end{array}$$

(80)

$q_1,q_2\in \mathbb{N}$. This is the Cartan matrix for the hyperbolic KM algebra $H_2(q_1,q_2)$ [10]. From (62), we get:

$$\begin{array}{c}\hfill {\epsilon }_s{\nu }_s^2(U^s,U^s)(q_1{q}_2-4)=2q_s+4,\end{array}$$

(81)

$s\in \{1,2\}=S$. An example of the $H_2(q,q)$-solution for the $B_D$-model with two electric p-branes ($p=d_1,d_2$), corresponding to $F^a$ and $F^b$ fields and intersecting on time manifold, is the following [46]:

$$\begin{array}{c}\hfill g=H^{-2/(q-2)}{\widehat{g}}^0-H^{2/(q-2)}dt\otimes dt+{\widehat{g}}^1+{\widehat{g}}^2,\end{array}$$

(82)

$$\begin{array}{c}\hfill F^a={\nu }_{1}d{H}^{-1}\wedge dt\wedge {\widehat{\tau }}_1,\end{array}$$

(83)

$$\begin{array}{c}\hfill F^b={\nu }_{2}d{H}^{-1}\wedge dt\wedge {\widehat{\tau }}_2,\end{array}$$

(84)

$$\begin{array}{c}\hfill \overrightarrow{\phi }=-({\overrightarrow{\lambda }}_a+{\overrightarrow{\lambda }}_b){(q-2)}^{-1}lnH\end{array}$$

(85)

where $d_0=3$, $d_1=a-2$, $a=q+4$, $a<b$, $d_2=b-2$, $D=a+b$, and ${\nu }_1^2={\nu }_2^2={(q-2)}^{-1}$. The signature restrictions are: ${\epsilon }_1={\epsilon }_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 $a=7$, $b=8$, $d_1=5$, $d_2=6$ and $q=$ 3.

Remark 2 (Affine Lie algebras).

We note that affine KM algebras (with det $A=0$) do not appear in the solutions (51)–(55). Indeed, any affine Cartan matrix satisfies the relations:

$$\sum _{s^{\prime }\in S}{a}_{s^{\prime }}{A}_{s^{\prime }s}=0$$

(86)

with $a_s>0$ called Coxeter labels [11], $s\in S$. This relation makes impossible the existence of the solution to Equation (49), since the latter is incompatible with Equations (48) and (58).

3.2.3. Generalized Majumdar–Papapetrou Solutions

Now, we return to a “multi-block” solution (51)–(55). Let $M_0={\mathbb{R}}^{d_0}$, $d_0>2$, $g^0={\delta }_{\mu \nu }d{x}^{\mu }\otimes d{x}^{\nu }$, $d_1=1$ and $g^1=-dt\otimes dt$. For:

$$\begin{array}{c}\hfill H_s=1+\sum _{b\in X_s}\frac{q_{sb}}{{|x-b|}^{d_0-2}},\end{array}$$

(87)

where $X_s$ is the finite non-empty subset $X_s\subset M_0$, $s\in S$, all $q_{sb}>0$, and $X_s=X_{s^{\prime }}$, $q_{sb}=q_{s^{\prime }b}$ for $b\in X_s=X_{s^{\prime }}$, $s,s^{\prime }\in S_j$, $j=1,\dots ,k$. The harmonic functions (87) are defined in domain $M_0\setminus X$, $X={\bigcup }_{s\in S}{X}_s$, and generate the solutions (51)–(55).

Denote $S(b)\equiv \{s\in S|b\in X_s\}$, $b\in X$ (in the one-block case, when $k=1$, all $X_s=X$ and $S(b)=S$). We have a horizon at point b w.r.t. time t, when $x\to b\in X$, if and only if:

$$\begin{array}{c}\hfill {\xi }_1(b)\equiv \sum _{s\in S(b)}(-{\epsilon }_s){\nu }_s^2{\delta }_{1I_s}-\frac{1}{d_0-2}\ge 0.\end{array}$$

(88)

This relation follows just from the requirement of the infinite propagation time of light to $b\in X$.

Majumdar–Papapetrou solution: Recall that the well-known four-dimensional Majumdar–Papapetrou (MP) solution [78,79] corresponding to the Lie algebra ${\mathbf{A}}_{\mathbf{1}}$ in our notation reads:

$$\begin{array}{c}\hfill g=H^2{\widehat{g}}^0-H^{-2}dt\otimes dt,\end{array}$$

(89)

$$\begin{array}{c}\hfill F=\nu d{H}^{-1}\wedge dt,\end{array}$$

(90)

where ${\nu }^2=2$, $g^0={\sum }_{i=1}^{3}d{x}^i\otimes d{x}^i$ and H is a harmonic function. We have one electric zero-brane (point) “attached” to the time manifold; $d(I_s)=1$, ${\epsilon }_s=-1$ and $(U^s,U^s)=1/2$. In this case (e.g., for the extremal Reissner–Nordström black hole), we get ${\xi }_1(b)=1$, $b\in X$. Points b are the points of (regular) horizon.

For certain examples of finite-dimensional semisimple Lie algebras (e.g., for ${\mathbf{A}}_{\mathbf{1}}\oplus \dots \oplus {\mathbf{A}}_{\mathbf{1}}$, ${\mathbf{A}}_{\mathbf{2}}$, etc.), the poles b in $H_s$ correspond to (regular) horizons, and the solution under consideration describes a collection of k blocks of extremal charged black branes (in equilibrium) [46].

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 $s\in S$. In this case, all ${\epsilon }_s>0$, $s\in S$, and hence, ${\xi }_1(b)<0$. Thus, any point $b\in X$ is not a point of the horizon (it may be checked using the analysis carried out in [46] 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 $(U^s,U^s)>0$.

Remark 3.

Here, we do not consider the solutions in supergravity models (e.g., in $D=11$ supergravity) with partially preserving supersymmetries. For supersymmetric solutions, see [71,80,81,82,83,84,85,86,87,88] and the references therein.

3.3. Toda-Like Solutions

3.3.1. Toda-Like Lagrangian

Action (34) may be also written in the form:

$$S_{\sigma 0}=\frac{1}{2{\kappa }_0^2}\int d^{d_0}x\sqrt{|g^0|}\{R[g^0]-{\mathcal{G}}_{\widehat{A}\widehat{B}}(X)g^{0\mu \nu }{\partial }_{\mu }{X}^{\widehat{A}}{\partial }_{\nu }{X}^{\widehat{B}}-2V\}$$

(91)

where $X=(X^{\widehat{A}})=({\varphi }^i,{\phi }^{\alpha },{\mathsf{\Phi }}^s)\in {\mathbf{R}}^N$, and the mini-supermetric

$\mathcal{G}={\mathcal{G}}_{\widehat{A}\widehat{B}}(X)d{X}^{\widehat{A}}\otimes d{X}^{\widehat{B}}$ on the minisuperspace $\mathcal{M}={\mathbf{R}}^N$, $N=n+l+\left|S\right|$ ($\left|S\right|$ is the number of elements in S) is defined by the relation:

$$({\mathcal{G}}_{\widehat{A}\widehat{B}}(X))=\left(\begin{array}{ccc}{G}_{ij}& 0& 0\\ 0& h_{\alpha \beta }& 0\\ 0& 0& {\epsilon }_{s}exp(-2U^s(\sigma )){\delta }_{s{s}^{\prime }}\end{array}\right).$$

(92)

Here, we consider exact solutions to field equations corresponding to the action (91):

$$\begin{array}{c}\hfill R_{\mu \nu }[g^0]={\mathcal{G}}_{\widehat{A}\widehat{B}}(X){\partial }_{\mu }{X}^{\widehat{A}}{\partial }_{\nu }{X}^{\widehat{B}}+\frac{2V}{d_0-2}{g}_{\mu \nu }^0,\end{array}$$

(93)

$$\begin{array}{c}\hfill \frac{1}{\sqrt{|g^0|}}{\partial }_{\mu }[\sqrt{|g^0|}{\mathcal{G}}_{\widehat{C}\widehat{B}}(X)g^{0\mu \nu }{\partial }_{\nu }{X}^{\widehat{B}}]-\frac{1}{2}{\mathcal{G}}_{\widehat{A}\widehat{B},\widehat{C}}(X)g^{0,\mu \nu }{\partial }_{\mu }{X}^{\widehat{A}}{\partial }_{\nu }{X}^{\widehat{B}}=V_{,\widehat{C}},\end{array}$$

(94)

where $s\in S$. Here, $V_{,\widehat{C}}=\partial V/\partial X^{\widehat{C}}$.

We put:

$$\begin{array}{c}\hfill X^{\widehat{A}}(x)=F^{\widehat{A}}(H(x)),\end{array}$$

(95)

where $F:(u_{-},u_{+})\to {\mathbb{R}}^N$ ($u\mapsto F(u)$) is a smooth function, $H:M_0\to \mathbb{R}$ is a harmonic function on $M_0$ (i.e., $\Delta [g^0]H=0$), satisfying $u_{-}<H(x)<u_{+}$ for all $x\in M_0$. We take all factor spaces as Ricci-flat, and the cosmological constant is set to zero, i.e., the relations ${\xi }_i=0$ and $\mathsf{\Lambda }=0$ are satisfied.

In this case, the potential is zero: $V=0$. It may be verified that the field Equations (93) and (94) are satisfied identically if the map F obeys the Lagrange equations for the Lagrangian:

$$L=\frac{1}{2}{\mathcal{G}}_{\widehat{A}\widehat{B}}(F){\dot{F}}^{\widehat{A}}{\dot{F}}^{\widehat{B}}$$

(96)

with the zero-energy constraint:

$$E=\frac{1}{2}{\mathcal{G}}_{\widehat{A}\widehat{B}}(F){\dot{F}}^{\widehat{A}}{\dot{F}}^{\widehat{B}}=0.$$

(97)

Here, ${\dot{F}}^{\widehat{A}}=\frac{d{F}^{\widehat{A}}}{du}$. This means that $F:(u_{-},u_{+})\to {\mathbb{R}}^N$ is a null-geodesic map for the minisupermetric $\mathcal{G}$. Thus, we are led to the Lagrange system (96) with the minisupermetric $\mathcal{G}$ defined in (92).

The problem of integrability will be simplified if we integrate the Lagrange equations corresponding to ${\mathsf{\Phi }}^s$ (i.e., the Maxwell-type equations for $s\in S_e$ and Bianchi identities for $s\in S_m$):

$$\begin{array}{c}\hfill \frac{d}{du}\left(exp(-2U^s(\sigma )){\dot{\mathsf{\Phi }}}^s\right)=0⟺{\dot{\mathsf{\Phi }}}^s=Q_{s}exp(2U^s(\sigma )),\end{array}$$

(98)

where $Q_s$ are constants and $s\in S$. Here, $(F^{\widehat{A}})=({\sigma }^A,{\mathsf{\Phi }}^s)$. We put $Q_s\ne 0$ for all $s\in S$.

For fixed $Q=(Q_s,s\in S)$, the Lagrange equations for the Lagrangian (96) corresponding to $({\sigma }^A)=({\varphi }^i,{\phi }^{\alpha })$, when Equation (98) is substituted, are equivalent to the Lagrange equations for the Lagrangian:

$$L_Q=\frac{1}{2}{\widehat{G}}_{AB}{\dot{\sigma }}^A{\dot{\sigma }}^B-V_Q,$$

(99)

where:

$$V_Q=\frac{1}{2}\sum _{s\in S}{\epsilon }_s{Q}_s^{2}exp[2U^s(\sigma )],$$

(100)

and the matrix $({\widehat{G}}_{AB})$ is defined in (36). The zero-energy constraint (97) reads:

$$E_Q=\frac{1}{2}{\widehat{G}}_{AB}{\dot{\sigma }}^A{\dot{\sigma }}^B+V_Q=0.$$

(101)

3.3.2. The Solutions

Here, as above, we are interested in exact solutions for a special case when $K_s=(U^s,U^s)\ne 0$, for all $s\in S$, and the generalized Cartan matrix (58) is non-degenerate. It follows from the non-degeneracy of the matrix (58) that vectors $U^s,s\in S,$ are linearly independent. Hence, the number of vectors $U^s$ should not exceed the dimension of ${\mathbb{R}}^{n+l}$, i.e., $\left|S\right|\le n+l$.

The exact solutions were obtained in [49] and are:

$$\begin{array}{c}\hfill g=\left(\prod _{s\in S}{f}_s^{2d(I_s)h_s/(D-2)}\right)\{exp(2c^{0}H+2{\overline{c}}^0){\widehat{g}}^0\\ \hfill +\sum _{i=1}^n\left(\prod _{s\in S}{f}_s^{-2h_s{\delta }_{i{I}_s}}\right)exp(2c^{i}H+2{\overline{c}}^i){\widehat{g}}^i\},\end{array}$$

(102)

$$\begin{array}{c}\hfill exp({\phi }^{\alpha })=\left(\prod _{s\in S}{f}_s^{h_s{\chi }_s{\lambda }_{a_s}^{\alpha }}\right)exp(c^{\alpha }H+{\overline{c}}^{\alpha }),\end{array}$$

(103)

$\alpha =1,\cdots ,l$ and $F^a={\sum }_{s\in S}{\mathcal{F}}^s{\delta }_{a_s}^a$ with:

$$\begin{array}{c}\hfill {\mathcal{F}}^s=Q_s\left(\prod _{s^{\prime }\in S}{f}_{s^{\prime }}^{-A_{s{s}^{\prime }}}\right)dH\wedge \tau (I_s),s\in S_e,\end{array}$$

(104)

$$\begin{array}{c}\hfill {\mathcal{F}}^s=Q_s({*}_{0}dH)\wedge \tau ({\overline{I}}_s),s\in S_m,\end{array}$$

(105)

where ${*}_0=*[g^0]$ is the Hodge operator on $(M_0,g^0)$. Here,

$$f_s=f_s(H)=exp(-q^s(H)),$$

(106)

where $q^s(u)$ is a solution to the Toda-like equations:

$$\ddot{q^s}=-B_{s}exp(\sum _{s^{\prime }\in S}{A}_{s{s}^{\prime }}{q}^{s^{\prime }})$$

(107)

with:

$$B_s=K_s{\epsilon }_s{Q}_s^2.$$

(108)

$s\in S$, and $H=H(x)$ ($x\in M_0$) is a harmonic function on $(M_0,g^0)$. Vectors $c=(c^A)$ and $\overline{c}=({\overline{c}}^A)$ satisfy the linear constraints:

$$\begin{array}{c}\hfill U^s(c)=U_A^s{c}^A=\sum _{i\in I_s}{d}_i{c}^i-{\chi }_s{\lambda }_{a_s\alpha }{c}^{\alpha }=0,U^s(\overline{c})=0,\end{array}$$

(109)

$s\in S$, and:

$$c^0=\frac{1}{2-d_0}\sum _{j=1}^n{d}_j{c}^j,{\overline{c}}^0=\frac{1}{2-d_0}\sum _{j=1}^n{d}_j{\overline{c}}^j.$$

(110)

The zero-energy constraint reads:

$$\begin{array}{c}\hfill 2E_T+h_{\alpha \beta }{c}^{\alpha }{c}^{\beta }+\sum _{i=1}^n{d}_i{(c^i)}^2+\frac{1}{d_0-2}{\left(\sum _{i=1}^n{d}_i{c}^i\right)}^2=0,\end{array}$$

(111)

where:

$$E_T=\frac{1}{4}\sum _{s,s^{\prime }\in S}{h}_s{A}_{s{s}^{\prime }}\dot{q^s}\dot{q^{s^{\prime }}}+\sum _{s\in S}{A}_{s}exp(\sum _{s^{\prime }\in S}{A}_{s{s}^{\prime }}{q}^{s^{\prime }})$$

(112)

is an integration constant (energy) for the solutions from (107) and $A_s=\frac{1}{2}{\epsilon }_s{Q}_s^2$.

We note that Equation (107) correspond to the Lagrangian:

$$L_T=\frac{1}{4}\sum _{s,s^{\prime }\in S}{h}_s{A}_{s{s}^{\prime }}\dot{q^s}\dot{q^{s^{\prime }}}-\sum _{s\in S}{A}_{s}exp(\sum _{s^{\prime }\in S}{A}_{s{s}^{\prime }}{q}^{s^{\prime }}),$$

(113)

where $h_s=K_s^{-1}$.

Thus, the solution is given by Relations (102)–(106) with the functions $q^s$ being defined in (107) and with relations on the parameters of solutions $c^A$, ${\overline{c}}^A$ $(A=i,\alpha ,0)$, imposed by (109)–(111).

4. Cosmological-Type, e.g., S-Brane, Solutions

Now, we consider the case $d_0=1$, $M_0=\mathbb{R}$, i.e., we are interested in applications to the sector with dependence on a single variable. We consider the manifold:

$$M=(u_{-},u_{+})\times M_1\times \dots \times M_n$$

(114)

with a metric:

$$g=w{\mathrm{e}}^{2\gamma (u)}du\otimes du+\sum _{i=1}^n{\mathrm{e}}^{2{\varphi }^i(u)}{\widehat{g}}^i,$$

(115)

where $w=\pm 1$, u is a distinguished coordinate, which by convention, will be called “time”; $(M_i,g^i)$ are oriented and connected Einstein spaces (see (17)), $i=1,\cdots ,n$. The functions $\gamma ,{\varphi }^i$: $(u_{-},u_{+})\to \mathbb{R}$ are smooth.

Here, we adopt the brane ansatz from Section 2, putting $g^0=wdu\otimes du$.

4.1. Lagrange Dynamics

It follows from Section 2.3 that the equations of motion and the Bianchi identities for the field configuration under consideration (with the restrictions from Section 2.3 imposed) are equivalent to equations of motion for the one-dimensional $\sigma$-model with the action:

$$S_{\sigma }=\frac{\mu }{2}\int du\mathcal{N}\left\{G_{ij}{\dot{\varphi }}^i{\dot{\varphi }}^j+h_{\alpha \beta }{\dot{\phi }}^{\alpha }{\dot{\phi }}^{\beta }+\sum _{s\in S}{\epsilon }_{s}exp[-2U^s(\varphi ,\phi )]{({\dot{\mathsf{\Phi }}}^s)}^2-2{\mathcal{N}}^{-2}{V}_w(\varphi )\right\},$$

(116)

where $\dot{x}\equiv dx/du$,

$$V_w=-wV=-w\mathsf{\Lambda }{\mathrm{e}}^{2{\gamma }_0(\varphi )}+\frac{w}{2}\sum _{i=1}^n{\xi }_i{d}_i{\mathrm{e}}^{-2{\varphi }^i+2{\gamma }_0(\varphi )}$$

(117)

is the potential with ${\gamma }_0(\varphi )\equiv {\sum }_{i=1}^n{d}_i{\varphi }^i$, $\mathcal{N}=exp({\gamma }_0-\gamma )>0$ is the lapse function, $U^s=U^s(\varphi ,\phi )$ are defined in (38), ${\epsilon }_s$ are defined in (40) for $s=(a_s,v_s,I_s)\in S$, $G_{ij}=d_i{\delta }_{ij}-d_i{d}_j$ are components of “pure cosmological” minisupermetric, $i,j=1,\cdots ,n$, and matrix $(G_{ij})$ has pseudo-Euclidean signature $(-,+,\cdots ,+)$ [74,89].

In the electric case $({\mathcal{F}}^{(a,m,I)}=0)$ for finite internal space volumes $V_i$, the action (116) coincides with the action (14) if $\mu =-w/{\kappa }_0^2$, ${\kappa }^2={\kappa }_0^2{V}_1\dots V_n$.

Action (116) may be also written in the form:

$$S_{\sigma }=\frac{\mu }{2}\int du\mathcal{N}\left\{{\mathcal{G}}_{\widehat{A}\widehat{B}}(X){\dot{X}}^{\widehat{A}}{\dot{X}}^{\widehat{B}}-2{\mathcal{N}}^{-2}{V}_w\right\},$$

(118)

where $X=(X^{\widehat{A}})=({\varphi }^i,{\phi }^{\alpha },{\mathsf{\Phi }}^s)\in {\mathbb{R}}^N$, $N=n+l+\left|S\right|$, and minisupermetric $\mathcal{G}$ is defined in (92).

Scalar products: The minisuperspace metric (92) may be also written in the form $\mathcal{G}=\widehat{G}+{\sum }_{s\in S}{\epsilon }_s{\mathrm{e}}^{-2U^s(\sigma )}d{\mathsf{\Phi }}^s\otimes d{\mathsf{\Phi }}^s$, where $\sigma =({\sigma }^A)=({\varphi }^i,{\phi }^{\alpha })$,

$$\begin{array}{c}\hfill \widehat{G}={\widehat{G}}_{AB}d{\sigma }^A\otimes d{\sigma }^B=G_{ij}d{\varphi }^i\otimes d{\varphi }^j+h_{\alpha \beta }d{\phi }^{\alpha }\otimes d{\phi }^{\beta },\end{array}$$

(119)

is the truncated minisupermetric, and $U^s(\sigma )=U_A^s{\sigma }^A$ is defined in (38). The potential (117) now reads:

$$V_w=(-w\mathsf{\Lambda }){\mathrm{e}}^{2U^{\mathsf{\Lambda }}(\sigma )}+\sum _{j=1}^n\frac{w}{2}{\xi }_j{d}_j{\mathrm{e}}^{2U^j(\sigma )},$$

(120)

where

$$\begin{array}{c}\hfill U^j(\sigma )=U_A^j{\sigma }^A=-{\varphi }^j+{\gamma }_0(\varphi ),(U_A^j)=(-{\delta }_i^j+d_i,0),\end{array}$$

(121)

$$\begin{array}{c}\hfill U^{\mathsf{\Lambda }}(\sigma )=U_A^{\mathsf{\Lambda }}{\sigma }^A={\gamma }_0(\varphi ),(U_A^{\mathsf{\Lambda }})=(d_i,0).\end{array}$$

(122)

The integrability of the Lagrange system (118) crucially depends on the scalar products of co-vectors $U^{\mathsf{\Lambda }}$, $U^j$, $U^s$ (see (42)). These products are defined by (45) and the following relations [56]:

$$\begin{array}{c}\hfill (U^i,U^j)=\frac{{\delta }_{ij}}{d_j}-1,\end{array}$$

(123)

$$\begin{array}{c}\hfill (U^i,U^{\mathsf{\Lambda }})=-1,(U^{\mathsf{\Lambda }},U^{\mathsf{\Lambda }})=-\frac{D-1}{D-2},\end{array}$$

(124)

$$\begin{array}{c}\hfill (U^s,U^i)=-{\delta }_{i{I}_s},(U^s,U^{\mathsf{\Lambda }})=\frac{d(I_s)}{2-D},\end{array}$$

(125)

where $s=(a_s,v_s,I_s)\in S$; $i,j=1,\cdots ,n$.

Toda-like representation: We put $\gamma ={\gamma }_0(\varphi )$, i.e., the harmonic time gauge is considered. Integrating the Lagrange equations corresponding to ${\mathsf{\Phi }}^s$ (see (98)), we are led to the Lagrangian from (99) and the zero-energy constraint (101) with the modified potential:

$$V_Q=V_w+\frac{1}{2}\sum _{s\in S}{\epsilon }_s{Q}_s^{2}exp[2U^s(\sigma )],$$

(126)

where $V_w$ is defined in (117).

4.2. Solutions with $\mathsf{\Lambda }=0$

Here, we consider solutions with $\mathsf{\Lambda }=0$.

4.2.1. Solutions with Ricci-Flat Factor-Spaces

Let all spaces be Ricci-flat, i.e., ${\xi }_1=\cdots ={\xi }_n=0$.

Since $H(u)=u$ is a harmonic function on $(M_0,g^0)$ with $g^0=wdu\otimes du$, we get for the metric and scalar fields from (102) and (103) [49]:

$$\begin{array}{c}\hfill g=\left(\prod _{s\in S}{f}_s^{2d(I_s)h_s/(D-2)}\right)\{exp(2c^{0}u+2{\overline{c}}^0)wdu\otimes du\\ \hfill +\sum _{i=1}^n\left(\prod _{s\in S}{f}_s^{-2h_s{\delta }_{i{I}_s}}\right)exp(2c^{i}u+2{\overline{c}}^i){\widehat{g}}^i\},\end{array}$$

(127)

$$\begin{array}{c}\hfill exp({\phi }^{\alpha })=\left(\prod _{s\in S}{f}_s^{h_s{\chi }_s{\lambda }_{a_s}^{\alpha }}\right)exp(c^{\alpha }u+{\overline{c}}^{\alpha }),\end{array}$$

(128)

$\alpha =1,\cdots ,l$, and $F^a={\sum }_{s\in S}{\delta }_{a_s}^a{\mathcal{F}}^s$ with:

$$\begin{array}{c}\hfill {\mathcal{F}}^s=Q_s\left(\prod _{s^{\prime }\in S}{f}_{s^{\prime }}^{-A_{s{s}^{\prime }}}\right)du\wedge \tau (I_s),s\in S_e,\end{array}$$

(129)

$$\begin{array}{c}\hfill {\mathcal{F}}^s=Q_s\tau ({\overline{I}}_s),s\in S_m\end{array}$$

(130)

$Q_s\ne 0$, $s\in S$.

Here, $f_s=f_s(u)=exp(-q^s(u))$ and $q^s(u)$ obey Toda-like Equation (107).

Relations (110) and (111) take the form:

$$\begin{array}{c}\hfill c^0=\sum _{j=1}^n{d}_j{c}^j,{\overline{c}}^0=\sum _{j=1}^n{d}_j{\overline{c}}^j,\end{array}$$

(131)

$$\begin{array}{c}\hfill 2E_T+h_{\alpha \beta }{c}^{\alpha }{c}^{\beta }+\sum _{i=1}^n{d}_i{(c^i)}^2-{\left(\sum _{i=1}^n{d}_i{c}^i\right)}^2=0,\end{array}$$

(132)

with $E_T$ from (112), and all other relations (e.g., Constraints (109)) are unchanged.

This solution in the special case of an ${\mathbf{A}}_{\mathbf{m}}$ Toda chain was obtained earlier in [90] (see also [91]). Some special configurations were considered earlier in [92,93,94].

Currently, the cosmological solutions with branes are considered often in a context of S-brane terminology [95]. S-branes were originally space-like analogues of D-branes; see also [53,96,97,98,99,100,101,102,103] and the references therein.

Remark 4.

The solutions of this subsection could be readily extended to the case when the Toda-like potential for scalar fields is added (into the action) [104,105].

4.2.2. Solutions with One Curved Factor-Space

The cosmological solution with Ricci-flat spaces may be also modified to the following case: ${\xi }_1\ne 0,{\xi }_2=\dots ={\xi }_n=0$, i.e., one space is curved, the others are Ricci-flat and $1\notin I_s$, $s\in S$, i.e., all “brane” submanifolds do not contain $M_1$.

The potential (117) is modified for ${\xi }_1\ne 0$ as follows (see (126)):

$$V_Q=\frac{1}{2}\sum _{s\in S}{\epsilon }_s{Q}_s^{2}exp[2U^s(\sigma )]+\frac{1}{2}w{\xi }_1{d}_{1}exp[2U^1(\sigma )],$$

(133)

where $U^1(\sigma )$ is defined in (121) ($d_1>1$).

For the scalar products, we get from (123) and (125):

$$\begin{array}{c}\hfill (U^1,U^1)=\frac{1}{d_1}-1<0,(U^1,U^s)=0\end{array}$$

(134)

for all $s\in S$.

The solution in the case under consideration may be obtained by a small modification of the solution from the previous section (using (134), relations $U^{1i}=-{\delta }_1^i/d_1$, $U^{1\alpha }=0$) [49]:

$$\begin{array}{c}\hfill g=\left(\prod _{s\in S}{[f_s(u)]}^{2d(I_s)h_s/(D-2)}\right)\{{[f_1(u)]}^{2d_1/(1-d_1)}exp(2c^{1}u+2{\overline{c}}^1)\\ \hfill \times [wdu\otimes du+f_1^2(u){\widehat{g}}^1]+\sum _{i=2}^n\left(\prod _{s\in S}{[f_s(u)]}^{-2h_s{\delta }_{i{I}_s}}\right)exp(2c^{i}u+2{\overline{c}}^i){\widehat{g}}^i\}.\end{array}$$

(135)

$$\begin{array}{c}\hfill exp({\phi }^{\alpha })=\left(\prod _{s\in S}{f}_s^{h_s{\chi }_s{\lambda }_{a_s}^{\alpha }}\right)exp(c^{\alpha }u+{\overline{c}}^{\alpha }),\end{array}$$

(136)

and $F^a={\sum }_{s\in S}{\delta }_{a_s}^a{\mathcal{F}}^s$ with forms ${\mathcal{F}}^s$ defined in (129) and (130).

Here, $f_s=f_s(u)=exp(-q^s(u))$, where $q^s(u)$ obey Toda-like Equation (107) and:

$$\begin{array}{c}\hfill f_1(u)=Rsinh(\sqrt{C_1}(u-u_1)),C_1>0,{\xi }_{1}w>0;\end{array}$$

(137)

$$\begin{array}{c}\hfill Rsin(\sqrt{|C_1|}(u-u_1)),C_1<0,{\xi }_{1}w>0;\end{array}$$

(138)

$$\begin{array}{c}\hfill Rcosh(\sqrt{C_1}(u-u_1)),C_1>0,{\xi }_{1}w<0;\end{array}$$

(139)

$$\begin{array}{c}\hfill {\left|{\xi }_1(d_1-1)\right|}^{1/2},C_1=0,{\xi }_{1}w>0,\end{array}$$

(140)

$u_1,C_1$ are constants, and $R=|{\xi }_1(d_1-1)/C_1{|}^{1/2}$.

The vectors $c=(c^A)$ and $\overline{c}=({\overline{c}}^A)$ satisfy the linear constraints:

$$\begin{array}{c}\hfill U^r(c)=U^r(\overline{c})=0,r=s,1,\end{array}$$

(141)

(for $r=s$, see (109)) and the zero-energy constraint:

$$C_1\frac{d_1}{d_1-1}=2E_T+h_{\alpha \beta }{c}^{\alpha }{c}^{\beta }+\sum _{i=2}^n{d}_i{(c^i)}^2+\frac{1}{d_1-1}{\left(\sum _{i=2}^n{d}_i{c}^i\right)}^2.$$

(142)

4.2.3. Special Solutions for Block-Orthogonal Set of Vectors $U^s$

Let us consider block-orthogonal case: (46) and (47). In this case, we get:

$$f_s={({\overline{f}}_s)}^{b_s}$$

(143)

where $b_s=2{\sum }_{s^{\prime }\in S}{A}^{s{s}^{\prime }}$, $(A^{s{s}^{\prime }})={(A_{s{s}^{\prime }})}^{-1}$ and:

$$\begin{array}{c}\hfill {\overline{f}}_s(u)=R_{s}sinh(\sqrt{C_s}(u-u_s)),C_s>0,{\eta }_s{\epsilon }_s<0;\end{array}$$

(144)

$$\begin{array}{c}\hfill R_{s}sin(\sqrt{|C_s|}(u-u_s)),C_s<0,{\eta }_s{\epsilon }_s<0;\end{array}$$

(145)

$$\begin{array}{c}\hfill R_{s}cosh(\sqrt{C_s}(u-u_s)),C_s>0,{\eta }_s{\epsilon }_s>0;\end{array}$$

(146)

$$\begin{array}{c}\hfill \frac{|Q^s|}{|{\nu }_s|}(u-u_s),C_s=0,{\eta }_s{\epsilon }_s<0,\end{array}$$

(147)

where $R_s=|Q_s|/(|{\nu }_s\left|\right|C_s{|}^{1/2})$,

$${\eta }_s{\nu }_s^2=b_s{h}_s,$$

(148)

${\eta }_s=\pm 1$, $C_s$, $u_s$ are constants, $s\in S$. The constants $C_s$, $u_s$ are coinciding inside the blocks: $u_s=u_{s^{\prime }}$, $C_s=C_{s^{\prime }}$, $s,s^{\prime }\in S_i$, $i=1,\dots ,k$. The ratios ${\epsilon }_s{Q}_s^2/(b_s{h}_s)$ also coincide inside the blocks, or equivalently,

$$\begin{array}{c}\hfill \frac{{\epsilon }_s{Q}_s^2}{b_s{h}_s}=\frac{{\epsilon }_{s^{\prime }}{Q}_{s^{\prime }}^2}{b_{s^{\prime }}{h}_{s^{\prime }}},\end{array}$$

(149)

$s,s^{\prime }\in S_i$, $i=1,\dots ,k$.

For the energy integration constant (112), we get:

$$E_T=\frac{1}{2}\sum _{s\in S}{C}_s{b}_s{h}_s.$$

(150)

The solution (135)–(130) with a block-orthogonal set of $U^s$-vectors was obtained in [106,107] (for the non-composite case, see also [108]). The generalized KM algebra corresponding to the generalized Cartan matrix A in this case is semisimple. In the special orthogonal (or $A_1\oplus ...\oplus A_1$) case when: $|S_1|=\dots =|S_k|=1$, the solution was obtained in [55].

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 ([109,110,111,112,113]) in agreement with the Adler–van Moerbeke criterion [113] (see also [114]).

The problem of integrability of Toda-chains related to Lorentzian (e.g., hyperbolic) KM algebras is much more complicated than in the Euclidean case. This is supported by the result from [115] (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 Painleve property, was obtained earlier for two-dimensional Toda chains related to hyperbolic KM algebras [116].

4.3. Examples of S-Brane Solutions

Example 3.

S-brane solution governed by the $E_{10}$ Toda chain.

Let us consider the $B_{16}$-model in 16-dimensional pseudo-Euclidean space of signature $(-,+,\dots ,+)$ with six forms $F^4,\dots ,F^9$ and five scalar fields ${\phi }^1,\dots ,{\phi }^5$; see (75). Recall that for two branes corresponding to the $F^a$ and $F^b$ forms, the orthogonal (or $(A_1+A_1)$-) intersection rules read [54,55]:

$$\begin{array}{c}\hfill {(a-1)}_e{\cap }_o{(b-1)}_e=N(a,b)=\min (a,b)-3,\end{array}$$

(151)

$$\begin{array}{c}\hfill {(a-1)}_e{\cap }_o{(D-b-1)}_m=a-1-N(a,b)\end{array}$$

(152)

where $d_v{\cap }_o{d}_{v^{\prime }}^{\prime }$ denotes the dimension of orthogonal intersection for two branes with the dimensions of their world volumes being d and $d^{\prime }$. $d_v{\cap }_o{d}_{v^{\prime }}^{\prime }$ coincides with the symbol $\Delta (s,s^{\prime })$ from (60) (Here, as in [54], our notations differ from those adopted in string theory. For example, for the intersection of M2- and M5-branes, we write $3{\cap }_{o}6$ = 2 instead of $2\cap 5$ = 1.). The subscripts $v,v^{\prime }=e,m$ here indicate whether the brane is an electric (e) or a magnetic (m) one. In what follows, we will be interested in the following orthogonal intersections: $4_e{\cap }_o{4}_e=2$, $4_e{\cap }_o{5}_e=2$, $4_e{\cap }_o{11}_m=3$, $5_e{\cap }_o{11}_m=4$.

Here, we deal with 10 (S-)branes: eight electric branes $s_1,s_2,s_3,s_4,s_5,s_6,s_8,s_9$ corresponding to five-form $F^5$, one electric brane $s_7$ corresponding to six-form $F^6$ and one magnetic brane $s_{10}$ corresponding to four-form $F^4$. The brane sets are as follows: $I_1=\{3,4,10,12\}$, $I_2=\{1,6,7,12\}$, $I_3=\{8,9,10,12\}$, $I_4=\{1,2,3,12\}$, $I_5=\{5,6,10,12\}$, $I_6=\{1,4,8,12\}$, $I_7=\{2,7,10,12,13\}$, $I_8=\{3,6,8,12\}$, $I_9=\{1,10,11,12\}$, $I_{10}=\{1,2,3,4,5,6,7,8,9,10,11\}$.

It may be verified that these sets do obey $E_{10}$ intersection rules following from the relations (61) (with $I_{s_i}=I_i$) and the Dynkin diagram from Figure 1.

Now, we present a cosmological S-brane solution from Section 4.2.1 for the configuration of ten branes under consideration. In what follows, the relations ${\epsilon }_s=+1$ and $h_s=1/2$, $s\in S$, are used.

The metric (127) reads:

$$\begin{array}{c}\hfill g={\left[{(\prod _{s\ne 7,10}{f}_s)}^4{f}_7^5{f}_{10}^{11}\right]}^{1/14}\{-e^{2c^{0}t+2{\overline{c}}^0}dt\otimes dt+{(f_2{f}_4{f}_6{f}_9{f}_{10})}^{-1}{e}^{2c^{1}t+2{\overline{c}}^1}d{x}^1\otimes d{x}^1\\ \hfill +{(f_4{f}_7{f}_{10})}^{-1}{e}^{2c^{2}t+2{\overline{c}}^2}d{x}^2\otimes d{x}^2+{(f_1{f}_4{f}_8{f}_{10})}^{-1}{e}^{2c^{3}t+2{\overline{c}}^3}d{x}^3\otimes d{x}^3\\ \hfill +{(f_1{f}_6{f}_{10})}^{-1}{e}^{2c^{4}t+2{\overline{c}}^4}d{x}^4\otimes d{x}^4+{(f_5{f}_{10})}^{-1}{e}^{2c^{5}t+2{\overline{c}}^5}d{x}^5\otimes d{x}^5\\ \hfill +{(f_2{f}_5{f}_8{f}_{10})}^{-1}{e}^{2c^{6}t+2{\overline{c}}^6}d{x}^6\otimes d{x}^6+{(f_2{f}_7{f}_{10})}^{-1}{e}^{2c^{7}t+2{\overline{c}}^7}d{x}^7\otimes d{x}^7\\ \hfill +{(f_3{f}_6{f}_8{f}_{10})}^{-1}{e}^{2c^{8}t+2{\overline{c}}^8}d{x}^8\otimes d{x}^8+{(f_3{f}_{10})}^{-1}{e}^{2c^{9}t+2{\overline{c}}^9}d{x}^9\otimes d{x}^9\\ \hfill +{(f_1{f}_3{f}_5{f}_7{f}_9{f}_{10})}^{-1}{e}^{2c^{10}t+2{\overline{c}}^{10}}d{x}^{10}\otimes d{x}^{10}+{(f_9{f}_{10})}^{-1}{e}^{2c^{11}t+2{\overline{c}}^{11}}d{x}^{11}\otimes d{x}^{11}\\ \hfill +{(\prod _{s=1}^9{f}_s)}^{-1}{e}^{2c^{12}t+2{\overline{c}}^{12}}d{x}^{12}\otimes d{x}^{12}+f_7^{-1}{e}^{2c^{13}t+2{\overline{c}}^{13}}d{x}^{13}\otimes d{x}^{13}\\ \hfill +e^{2c^{14}t+2{\overline{c}}^{14}}d{x}^{14}\otimes d{x}^{14}+e^{2c^{15}t+2{\overline{c}}^{15}}d{x}^{15}\otimes d{x}^{15}\}.\end{array}$$

(153)

For scalar fields (128), we get:

$$\begin{array}{c}\hfill {\phi }^{\alpha }=\frac{1}{2}[-{\lambda }_{5\alpha }(\sum _{s\ne 7,10}ln{f}_s)-{\lambda }_{6\alpha }ln{f}_7+{\lambda }_{4\alpha }ln{f}_{10}]+c_{\phi }^{\alpha }t+{\overline{c}}_{\phi }^{\alpha },\end{array}$$

(154)

$\alpha =1,\cdots ,5$ (here, we used the relations ${\lambda }_a^{\alpha }=-{\lambda }_{a\alpha }$).

The form fields (see (129) and (130)) are as follows:

$$\begin{array}{c}\hfill F^4=Q_{10}d{x}^{12}\wedge d{x}^{13}\wedge d{x}^{14}\wedge d{x}^{15},\end{array}$$

(155)

$$\begin{array}{c}\hfill F^5=Q_1{f}_1^{-2}{f}_{2}dt\wedge d{x}^3\wedge d{x}^4\wedge d{x}^{10}\wedge d{x}^{12}+Q_2{f}_1{f}_2^{-2}{f}_{3}dt\wedge d{x}^1\wedge d{x}^6\wedge d{x}^7\wedge d{x}^{12}\\ \hfill +Q_3{f}_2{f}_3^{-2}{f}_{4}dt\wedge d{x}^8\wedge d{x}^9\wedge d{x}^{10}\wedge d{x}^{12}+Q_4{f}_3{f}_4^{-2}{f}_{5}dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d{x}^{12}\\ \hfill +Q_5{f}_4{f}_5^{-2}{f}_{6}dt\wedge d{x}^5\wedge d{x}^6\wedge d{x}^{10}\wedge d{x}^{12}+Q_6{f}_5{f}_6^{-2}{f}_{7}dt\wedge d{x}^1\wedge d{x}^4\wedge d{x}^8\wedge d{x}^{12}\\ \hfill +Q_8{f}_7{f}_8^{-2}{f}_{9}dt\wedge d{x}^3\wedge d{x}^6\wedge d{x}^8\wedge d{x}^{12}+Q_9{f}_8{f}_9^{-2}dt\wedge d{x}^1\wedge d{x}^{10}\wedge d{x}^{11}\wedge d{x}^{12},\end{array}$$

(156)

$$\begin{array}{c}\hfill F^6=Q_7{f}_6{f}_7^{-2}{f}_8{f}_{10}dt\wedge d{x}^2\wedge d{x}^7\wedge d{x}^{10}\wedge d{x}^{12}\wedge d{x}^{13},\end{array}$$

(157)

where $Q_s\ne 0$, $s=1,\dots ,10$. Here:

$$c^0=\sum _{j=1}^{15}{c}^j,{\overline{c}}^0=\sum _{j=1}^{15}{\overline{c}}^j,$$

(158)

$f_s=exp(-q^s(t))$ and $q^s(t)$ obey Toda-type equations:

$$\ddot{q^s}=-2Q_s^{2}exp(\sum _{s^{\prime }=1}^{10}{A}_{s{s}^{\prime }}{q}^{s^{\prime }}),$$

(159)

$s=1,\dots ,10$, where $(A_{s{s}^{\prime }})$ is the Cartan matrix for the KM algebra $E_{10}$ (with the Dynkin diagram from Figure 1), and the energy integration constant:

$$E_T=\frac{1}{8}\sum _{s,s^{\prime }=1}^{10}{A}_{s{s}^{\prime }}\dot{q^s}\dot{q^{s^{\prime }}}+\frac{1}{2}\sum _{s=1}^{10}{Q}_s^{2}exp(\sum _{s^{\prime }=1}^{10}{A}_{s{s}^{\prime }}{q}^{s^{\prime }}),$$

(160)

obeys the constraint:

$$2E_T-\sum _{\alpha =1}^5{(c_{\phi }^{\alpha })}^2+\sum _{i=1}^{15}{(c^i)}^2-{\left(\sum _{i=1}^{15}{c}^i\right)}^2=0.$$

(161)

The brane constraints (109) are in our case:

$$\begin{array}{c}\hfill U^1(c)=c^3+c^4+c^{10}+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^1(\overline{c})=0,\\ \hfill U^2(c)=c^1+c^6+c^7+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^2(\overline{c})=0,\\ \hfill U^3(c)=c^8+c^9+c^{10}+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^3(\overline{c})=0,\\ \hfill U^4(c)=c^1+c^2+c^3+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^4(\overline{c})=0,\\ \hfill U^5(c)=c^5+c^6+c^{10}+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^5(\overline{c})=0,\\ \hfill U^6(c)=c^1+c^4+c^8+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^6(\overline{c})=0,\\ \hfill U^7(c)=c^2+c^7+c^{10}+c^{12}+c^{13}-\sum _{\alpha =1}^5{\lambda }_{6\alpha }{c}_{\phi }^{\alpha }=0,U^7(\overline{c})=0,\\ \hfill U^8(c)=c^3+c^6+c^8+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^8(\overline{c})=0,\\ \hfill U^9(c)=c^1+c^{10}+c^{11}+c^{12}-\sum _{\alpha =1}^5{\lambda }_{5\alpha }{c}_{\phi }^{\alpha }=0,U^9(\overline{c})=0,\\ \hfill U^{10}(c)=\sum _{i=1}^{11}{c}^i+\sum _{\alpha =1}^5{\lambda }_{4\alpha }{c}_{\phi }^{\alpha }=0,U^{10}(\overline{c})=0.\\ \hfill \end{array}$$

(162)

Remark 5.

For a special choice of integration constants $c^i=0$ and $c_{\phi }^{\alpha }=0$, we get a solution governed by the $E_{10}$ Toda chain with the energy constraint $E_T=0$. According to the result from [30], 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 $B\subset H^9$. This billiard has a finite volume since $E_{10}$ is hyperbolic.

Special one-block solution: Now, we consider a special one-block solution (see Section 4.2.3). This solution is valid when a special set of charges is considered (see (149)):

$$Q_s^2=Q^2\left|b_s\right|,$$

(163)

where $Q\ne 0$ and [47]:

$$b_s=2\sum _{s^{\prime }=1}^{10}{A}^{s{s}^{\prime }}=-60,-122,-186,-252,-320,-390,-462,-306,-152,-230,$$

(164)

$s=1,\dots ,10$. Recall that $(A^{s{s}^{\prime }})={(A_{s{s}^{\prime }})}^{-1}$.

In this case, $f_s={(\overline{f})}^{b_s}$, where:

$$\begin{array}{c}\hfill \overline{f}(t)=\left|Q\right|\sqrt{2/C}sinh(\sqrt{C}(t-t_0)),C>0,\\ \hfill \left|Q\right|\sqrt{2/\left|C\right|}sin(\sqrt{\left|C\right|}(t-t_0)),C<0,\\ \hfill \left|Q\right|\sqrt{2}(t-t_0),C=0\end{array}$$

(165)

and $t_0$ is a constant.

From (150), we get:

$$E_T=-620C,$$

(166)

where relation ${\sum }_{s=1}^{10}{b}_s=-2480$ was used.

For the special solution under consideration, the electric monomials in (156) and (157) have a simpler form:

$${\mathcal{F}}^s=Q_s{\overline{f}}^{-2}dt\wedge \tau (I_s),$$

(167)

where $s=1,2,\dots ,9$.

Solution with one harmonic function: Let $C=0$ and all $c^i={\overline{c}}^i=0$, $c_{\phi }^{\alpha }={\overline{c}}_{\phi }^{\alpha }=0$. In this case, $H=\overline{f}(t)=\left|Q\right|\sqrt{2}(t-t_0)>0$ is a harmonic function on the one-dimensional manifold $((t_0,+\infty ),-dt\otimes dt)$, and our solution coincides with the one-block solution (51)–(55) (if $\mathrm{sign}{\nu }_s=-\mathrm{sign}{Q}_s$ for all s).

Example 4.

S-brane solution governed by $H{A}_2^{(1)}$ Toda chain.

Now, we consider the $B_{11}$-model in the 11-dimensional pseudo-Euclidean space of signature $(-,+,\dots ,+)$ with four-form $F^4$.

Here, we deal with four electric branes ($SM2$-branes) $s_1,s_2,s_3,s_4$ corresponding to the four-form $F^4$. The brane sets are the following ones: $I_1=\{1,2,3\}$, $I_2=\{4,5,6,\}$, $I_3=\{7,8,9\}$, $I_4=\{1,4,10\}$.

It may be verified that these sets obey the intersection rules corresponding to the hyperbolic KM algebra $H{A}_2^{(1)}$ with the following Cartan matrix:

$$\begin{array}{c}\hfill A=\left(\begin{array}{cccc}2& -1& -1& 0\\ -1& 2& -1& 0\\ -1& -1& 2& -1\\ 0& 0& -1& 2\end{array}\right),\end{array}$$

(168)

(see (61) with $I_{s_i}=I_i$).

Now, we give a cosmological S-brane solution from Section 4.2.1 for the configuration of four branes under consideration. In what follows, the relations ${\epsilon }_s=+1$ and $h_s=1/2$, $s\in S$, are used.

The metric (127) reads [117]:

$$\begin{array}{c}\hfill g={(f_1{f}_2{f}_3{f}_4)}^{1/3}\{-e^{2c^{0}t+2{\overline{c}}^0}dt\otimes dt+{(f_1{f}_4)}^{-1}{e}^{2c^{1}t+2{\overline{c}}^1}d{x}^1\otimes d{x}^1\\ \hfill +f_1^{-1}{e}^{2c^{2}t+2{\overline{c}}^2}d{x}^2\otimes d{x}^2+f_1^{-1}{e}^{2c^{3}t+2{\overline{c}}^3}d{x}^3\otimes d{x}^3\\ \hfill +{(f_2{f}_4)}^{-1}{e}^{2c^{4}t+2{\overline{c}}^4}d{x}^4\otimes d{x}^4+f_2^{-1}{e}^{2c^{5}t+2{\overline{c}}^5}d{x}^5\otimes d{x}^5\\ \hfill +f_2^{-1}{e}^{2c^{6}t+2{\overline{c}}^6}d{x}^6\otimes d{x}^6+f_3^{-1}{e}^{2c^{7}t+2{\overline{c}}^7}d{x}^7\otimes d{x}^7\\ \hfill +f_3^{-1}{e}^{2c^{8}t+2{\overline{c}}^8}d{x}^8\otimes d{x}^8+f_3^{-1}{e}^{2c^{9}t+2{\overline{c}}^9}d{x}^9\otimes d{x}^9\\ \hfill +f_4^{-1}{e}^{2c^{10}t+2{\overline{c}}^{10}}d{x}^{10}\otimes d{x}^{10}\}.\end{array}$$

(169)

The form field (see (129)) is as follows:

$$\begin{array}{c}\hfill F^4=Q_1{f}_1^{-2}{f}_2{f}_{3}dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3+Q_2{f}_1{f}_2^{-2}{f}_{3}dt\wedge d{x}^4\wedge d{x}^5\wedge d{x}^6\\ \hfill +Q_3{f}_1{f}_2{f}_3^{-2}dt\wedge d{x}^7\wedge d{x}^8\wedge d{x}^9+Q_4{f}_3{f}_4^{-2}dt\wedge d{x}^1\wedge d{x}^4\wedge d{x}^{10},\end{array}$$

(170)

where $Q_s\ne 0$, $s=1,\dots ,4$. Here:

$$c^0=\sum _{j=1}^{10}{c}^j,{\overline{c}}^0=\sum _{j=1}^{10}{\overline{c}}^j,$$

(171)

$f_s=exp(-q^s(t))$ and $q^s(t)$ obey the Toda-type equations:

$$\ddot{q^s}=-2Q_s^{2}exp(\sum _{s^{\prime }=1}^4{A}_{s{s}^{\prime }}{q}^{s^{\prime }}),$$

(172)

$s=1,\dots ,4$, where $(A_{s{s}^{\prime }})$ is the Cartan matrix (168) for the KM algebra $H{A}_2^{(1)}$, and the energy integration constant:

$$E_T=\frac{1}{8}\sum _{s,s^{\prime }=1}^4{A}_{s{s}^{\prime }}\dot{q^s}\dot{q^{s^{\prime }}}+\frac{1}{2}\sum _{s=1}^4{Q}_s^{2}exp(\sum _{s^{\prime }=1}^4{A}_{s{s}^{\prime }}{q}^{s^{\prime }}),$$

(173)

obeys the constraint:

$$2E_T+\sum _{i=1}^{10}{(c^i)}^2-{\left(\sum _{i=1}^{10}{c}^i\right)}^2=0.$$

(174)

The brane constraints (109) read in this case as follows:

$$\begin{array}{c}\hfill U^1(c)=c^1+c^2+c^3=0,U^1(\overline{c})=0,\\ \hfill U^2(c)=c^4+c^5+c^6=0,U^2(\overline{c})=0,\\ \hfill U^3(c)=c^7+c^8+c^9=0,U^3(\overline{c})=0,\\ \hfill U^4(c)=c^1+c^4+c^{10}=0,U^4(\overline{c})=0,\end{array}$$

(175)

Since $F^4\wedge F^4=0$, this solution also obeys the equations of motion of 11-dimensional supergravity.

Special one-block solution. Now, we consider a special one-block solution (see Section 4.2.3). This solution is valid when a special set of charges is considered (see (149)):

$$Q_s^2=Q^2\left|b_s\right|,$$

(176)

where $Q\ne 0$ and:

$$b_s=2\sum _{s^{\prime }=1}^4{A}^{s{s}^{\prime }}=-12,-12,-14,-6.$$

(177)

In this case, $f_s={(\overline{f})}^{b_s}$, where $\overline{f}$ is the same as in (165).

For the energy integration constant, we have:

$$E_T=-11C,$$

(178)

(see (150)).

Example 5.

S-brane solution governed by $P_{10}$ Toda chain with $E_T=0$.

Now, we consider the $B_{11}$-model in the 11-dimensional pseudo-Euclidean space of signature $(-,+,\dots ,+)$ with four-form $F^4$.

Here, we deal with ten electric branes ($SM2$-branes) $s_1,\dots ,s_{10}$ corresponding to the four-form $F^4$. The brane sets are taken from [13,118] as: $I_1=\{1,4,7\}$, $I_2=\{8,9,10\}$, $I_3=\{2,5,7\}$, $I_4=\{4,6,10\}$, $I_5=\{2,3,9\}$, $I_6=\{1,2,8\}$, $I_7=\{1,3,10\}$, $I_8=\{4,5,8\}$, $I_9=\{3,6,7\}$, $I_{10}=\{5,6,9\}$.

These sets obey the intersection rules corresponding to the Lorentzian KM algebra $P_{10}$ (we call it Petersen algebra) with the following Cartan matrix:

$$\begin{array}{c}\hfill A=\left(\begin{array}{cccccccccc}2& -1& 0& 0& -1& 0& 0& 0& 0& -1\\ -1& 2& -1& 0& 0& 0& 0& 0& -1& 0\\ 0& -1& 2& -1& 0& 0& -1& 0& 0& 0\\ 0& 0& -1& 2& -1& -1& 0& 0& 0& 0\\ -1& 0& 0& -1& 2& 0& 0& -1& 0& 0\\ 0& 0& 0& -1& 0& 2& 0& 0& -1& -1\\ 0& 0& -1& 0& 0& 0& 2& -1& 0& -1\\ 0& 0& 0& 0& -1& 0& -1& 2& -1& 0\\ 0& -1& 0& 0& 0& -1& 0& -1& 2& 0\\ -1& 0& 0& 0& 0& -1& -1& 0& 0& 2.\end{array}\right),\end{array}$$

(179)

The Dynkin diagram for this Cartan matrix could be represented by the Petersen graph (“a star inside a pentagon”). $P_{10}$ is the Lorentzian KM algebra. It is a subalgebra of $E_{10}$ [13,118].

Let us present an S-brane solution for the configuration of 10 electric branes under consideration. The metric (127) reads [117]:

$$\begin{array}{c}\hfill g={\left(\prod _{s=1}^{10}{f}_s\right)}^{1/3}\{-dt\otimes dt+{(f_1{f}_6{f}_7)}^{-1}d{x}^1\otimes d{x}^1\\ \hfill +{(f_3{f}_5{f}_6)}^{-1}d{x}^2\otimes d{x}^2+{(f_5{f}_7{f}_9)}^{-1}d{x}^3\otimes d{x}^3\\ \hfill +{(f_1{f}_4{f}_8)}^{-1}d{x}^4\otimes d{x}^4+{(f_3{f}_8{f}_{10})}^{-1}d{x}^5\otimes d{x}^5\\ \hfill +{(f_4{f}_9{f}_{10})}^{-1}d{x}^6\otimes d{x}^6+{(f_1{f}_3{f}_9)}^{-1}d{x}^7\otimes d{x}^7\\ \hfill +{(f_2{f}_6{f}_8)}^{-1}d{x}^8\otimes d{x}^8+{(f_2{f}_5{f}_{10})}^{-1}d{x}^9\otimes d{x}^9\\ \hfill +{(f_2{f}_4{}_{}}^{}\end{array}$$