**77**: 653

https://doi.org/10.1140/epjc/s10052-017-5235-5

Regular Article - Theoretical Physics

## On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra

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

^{2}
Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Str., Moscow, 117198, Russia

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

Received:
27
June
2017

Accepted:
12
September
2017

Published online:
30
September
2017

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra is considered. The solution contains a metric, *n* Abelian 2-forms and *n* scalar fields, where *n* is the rank of . It is governed by a set of *n* moduli 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. These polynomials depend upon integration constants , . In the case when the conjecture on the polynomial structure for the Lie algebra is satisfied, it is proved that 2-form flux integrals over a proper 2*d* submanifold are finite and obey the relations , where the are certain constants (related to dilatonic coupling vectors) and the are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, . The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra . Examples of polynomials and fluxes for the Lie algebras , , , , and are presented.

*© The Author(s), 2017*