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

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.

Highlights

In this paper we start with a generalization of a Melvin solution [1], which was presented earlier in Ref. [2]
Where g = gM N (x)dx M ⊗ dx N is a metric, φ = ∈ Rl is a set of scalar fields, is a constant symmetric nondegenerate l × l matrix (l ∈ N), dx N is a 2-form, λs is a 1-form
They are related to scalar products of certain vectors U s, which belong to a certain linear space (“truncated target space”, for our problem it has dimension l + 2), i.e. Bss = (U s, U s ) and Ks = (U s, U s) [35,36,37]

Summary

Introduction

In this paper we start with a generalization of a Melvin solution [1], which was presented earlier in Ref. [2]. It appears in the model which contains a metric, n Abelian 2-forms and l ≥ n scalar fields This solution is governed by a certain nondegenerate (quasi-Cartan) matrix ( Ass ), s, s = 1, . The set of fluxbrane polynomials Hs defines a special solution to open Toda chain equations [30,31] corresponding to a simple finite-dimensional Lie algebra G [32]. [2,33] a program (in Maple) for the calculation of these polynomials for the classical series of Lie algebras (A-, B-, C- and D-series) was suggested 3 we deal with the generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra.

The solutions

Flux integrals for a simple finite-dimensional Lie algebra

Examples

Conclusions