On multidimensional analogs of Melvin’s solution for classical series of Lie algebras

2017

Symmetry

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 × M 1 × . . . × M n , where all M i with i ≥ 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 = 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) , AE 3 , H A (1) 2 , E 10 and Lorentzian KM algebra P 10 , are presented.

Section: Generalized Melvin Solution With Several Two-formsmentioning

confidence: 99%

Section: Generalized Melvin Solution With Several Two-formsmentioning

confidence: 99%

See 1 more Smart Citation

On Brane Solutions with Intersection Rules Related to Lie Algebras

2017

Symmetry

“…The D-dimensional warped product solution from Ref. [2] comprises two factor spaces: 1-dimensional subspace M 1 and a (D − 2)-dimensional Ricci-flat subspace M 2 . Here M 1 is either R or S 1 .…”

Section: Introductionmentioning

confidence: 99%

“…[2] the electro-vacuum Melvin solution was generalized for the D-dimensional model which contains metric g, n 2-form fields F s = d A s and l scalar fields ϕ α . The model also includes n dilatonic coupling vectors belonging to R l .…”

Section: Introductionmentioning

confidence: 99%

On generalized Melvin solution for the Lie algebra $$E_6$$ E 6

Bolokhov

2017

Eur. Phys. J. C

Self Cite

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is considered. The gravitational model in D dimensions, D ≥ 4, contains n 2-forms and l ≥ n scalar fields, where n is the rank of G. The solution is governed by a set of n functions H s (z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials H s (z), s = 1, . . . , 6, for the Lie algebra E 6 are obtained and a corresponding solution for l = n = 6 is presented. The polynomials depend upon integration constants Q s , s = 1, . . . , 6. They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for E 6 -polynomials at large z are governed by the integer-valued matrix ν = A −1 (I + P), where A −1 is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the Z 2 -group of symmetry of the Dynkin diagram. The 2-form fluxes s , s = 1, . . . , 6, are calculated.

Black brane solutions governed by fluxbrane polynomials

Journal of Geometry and Physics

2014

A family of composite black brane solutions in the model with scalar fields and fields of forms is presented. The metric of any solution is defined on a manifold which contains a product of several Ricci-flat "internal" spaces. The solutions are governed by moduli functions H_s (s = 1, ..., m) obeying non-linear differential equations with certain boundary conditions imposed. These master equations are equivalent to Toda-like equations and depend upon the non-degenerate (m x m) matrix A. It was conjectured earlier that the functions H_s should be polynomials if A is a Cartan matrix for some semisimple finite-dimensional Lie algebra (of rank m). It is shown that the solutions to master equations may be found by using so-called fluxbrane polynomials which can be calculated (in principle) for any semisimple finite-dimensional Lie algebra. Examples of dilatonic charged black hole (0-brane) solutions related to Lie algebras A_1, A_2, C_2 and G_2 are considered.Comment: 16 pages, Latex, no figures, several typos are correcte