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 $M = M_0 \times M_1 \times \cdots \times M_n$, where $M_i$ are Einstein spaces ($i \geq 1$). 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: $H_2(q,q)$, $AE_3$, $HA_2^{(1)}$, $E_{10}$ and Lorentzian KM algebra $P_{10}$ are presented

Multitemporal generalization of Schwarzschild solution

The $n$-time generalization of Schwarzschild solution is considered. The equations of geodesics for the metric are integrated. The multitemporal analogues of Newton laws for the extended objects described by the solution are suggested. The scalar-vacuum generalization of the solution is also presented.Comment: 7 page

Cosmological solutions in multidimensional model with multiple exponential potential

A family of cosmological solutions with $(n+1)$ Ricci-flat spaces in the theory with several scalar fields and multiple exponential potential is obtained when coupling vectors in exponents obey certain relations. Two subclasses of solutions with power-law and exponential behaviour of scale factors are singled out. It is proved that power-law solutions may take place only when coupling vectors are linearly independent and exponential dependence occurs for linearly dependent set of coupling vectors. A subfamily of solutions with accelerated expansion is singled out. A generalized isotropization behaviours of certain classes of general solutions are found. In quantum case exact solutions to Wheeler-DeWitt equation are obtained and special "ground state" wave functions are considered.Comment: 22 pages, 1 figur

Multitemporal generalization of the Tangherlini solution

The n-time generalization of the Tangherlini solution [1] is considered. The equations of geodesics for the metric are integrated. For $n = 2$ it is shown that the naked singularity is absent only for two sets of parameters, corresponding to the trivial extensions of the Tangherlini solution. The motion of a relativistic particle in the multitemporal background is considered. This motion is governed by the gravitational mass tensor. Some generalizations of the solution, including the multitemporal analogue of the Myers-Perry charged black hole solution, are obtained.Comment: 14 pages. RGA-CSVR-005/9

On Brane Solutions with Intersection Rules Related to Lie Algebras

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 = M0 x M1 x . . . x Mn , where all Mi with i >= 1 are fixed Einstein (e.g., Ricci-flat) spaces. We consider a warped product metric on M. Here, M0 is a base manifold, and all scale factors (of the warped product), scalar fields and potentials for monomial forms are functions on M0 . 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 M0 of dimension d0 ≠ 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 d0 = 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: H2(q, q) , AE3, HA(1)2, E10 and Lorentzian KM algebra P10 , are presented

Stable Exponential Cosmological Type Solutions with Three Factor Spaces in EGB Model with a Λ-Term

We study a D-dimensional Einstein–Gauss–Bonnet model which includes the Gauss–Bonnet term, the cosmological term Λ and two non-zero constants: α1 and α2. Under imposing the metric to be diagonal one, we find cosmological type solutions with exponential dependence of three scale factors in a variable u, governed by three non-coinciding Hubble-like parameters: H≠0, h1 and h2, obeying mH+k1h1+k2h2≠0, corresponding to factor spaces of dimensions m>1, k1>1 and k2>1, respectively, and depending upon sign parameter ε=±1, where ε=1 corresponds to cosmological case and ε=−1—to static one). We deal with two cases: (i) mk1k2 and (ii) 1k1=k2=k, k≠m. We show that in both cases the solutions exist if εα=εα2/α1>0 and αΛ>0 satisfy certain (upper and lower) bounds. The solutions are defined up to solutions of a certain polynomial master equation of order four (or less), which may be solved in radicals. In case (ii), explicit solutions are presented. In both cases we single out stable and non-stable solutions as u→±∞. The case H=0 is also considered

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored