**77**: 664

https://doi.org/10.1140/epjc/s10052-017-5234-6

Regular Article - Theoretical Physics

## On generalized Melvin solution for the Lie algebra

^{1}
Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St, Moscow, 119361, Russia

^{2}
Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russia

^{*} e-mail: ivashchuk@mail.ru

Received:
22
June
2017

Accepted:
14
September
2017

Published online:
5
October
2017

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra is considered. The gravitational model in *D* dimensions, , contains *n* 2-forms and scalar fields, where *n* is the rank of . The solution is governed by a set of *n* functions 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 , , for the Lie algebra are obtained and a corresponding solution for is presented. The polynomials depend upon integration constants , . 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 -polynomials at large *z* are governed by the integer-valued matrix , where is the inverse Cartan matrix, *I* is the identity matrix and *P* is a permutation matrix, corresponding to a generator of the -group of symmetry of the Dynkin diagram. The 2-form fluxes , , are calculated.

*© The Author(s), 2017*