On generalized Melvin solution for the Lie algebra $E_6$

Abstract

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\mathcal{G}$ is considered. The gravitational model in $D$ dimensions, $D\ge 4$, contains $n$ 2-forms and $l\ge n$ scalar fields, where $n$ is the rank of $\mathcal{G}$. 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 $H_s(z)$, $s=1,\dots ,6$, for the Lie algebra $E_6$ are obtained and a corresponding solution for $l=n=6$ is presented. The polynomials depend upon integration constants $Q_s$, $s=1,\dots ,6$. 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 integer-valued matrix $\nu =A^{-1}(I+P)$, where $A^{-1}$ is the inverse Cartan matrix, $I$ is the identity matrix and $P$ is permutation matrix, corresponding to a generator of the $Z_2$-group of symmetry of the Dynkin diagram. The 2-form fluxes ${\Phi }^s$, $s=1,\dots ,6$, are calculated.

16 pages, Latex, no figures, prepared for a talk at RUSGRAV-16 conference in Kaliningrad, 2017, 2nd. revised version, several typos are eliminated

References in corpus (3)

• On multidimensional analogs of Melvin's solution for classical series of Lie algebras
• Electric S-brane solutions corresponding to rank-2 Lie algebras: acceleration and small variation of G
• On multidimensional cosmological solutions with scalar fields and 2-forms corresponding to rank-3 Lie algebras: acceleration and small variation of G