Eur. Phys. J. C (2017) 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

Abstract

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.