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

V. D. Ivashchuk

doi:10.1140/epjc/s10052-017-5235-5

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2023 Impact factor 4.2

Particles and Fields

Open Access

Eur. Phys. J. C (2017) 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

V. D. Ivashchuk¹^,2^*

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

Abstract

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra $\mathcal G$ is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of $\mathcal G$ . It is governed by a set of n moduli 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. These polynomials depend upon integration constants $$q_s$$ , $s = 1,\dots ,n$ . In the case when the conjecture on the polynomial structure for the Lie algebra $\mathcal G$ is satisfied, it is proved that 2-form flux integrals $\Phi ^s$ over a proper 2d submanifold are finite and obey the relations $q_s \Phi ^s = 4 \pi n_s h_s$ , where the $$h_s > 0$$ are certain constants (related to dilatonic coupling vectors) and the $$n_s$$ are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, $s = 1,\dots ,n$ . The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra $\mathcal G$ . Examples of polynomials and fluxes for the Lie algebras $$A_1$$ , $$A_2$$ , $$A_3$$ , $$C_2$$ , $$G_2$$ and $$A_1 + A_1$$ are presented.