Advances in Mathematical Physics

Volume 2018, Article ID 8179570, 10 pages

https://doi.org/10.1155/2018/8179570

## Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

^{1}Institute of Gravitation and Cosmology, Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St., Moscow 117198, Russia^{2}Center for Gravitation and Fundamental Metrology, VNIIMS, 46 Ozyornaya St., Moscow 119361, Russia

Correspondence should be addressed to V. D. Ivashchuk; ur.liam@kuhchsavi

Received 26 September 2018; Accepted 28 October 2018; Published 7 November 2018

Academic Editor: Andrei D. Mironov

Copyright © 2018 S. V. Bolokhov and V. D. Ivashchuk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank (namely, , , , and ) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions () of squared radial coordinate obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers for Lie algebras , , , and , respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued matrix connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in case) the matrix representing a generator of the -group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over -dimensional discs and corresponding Wilson loop factors over their boundaries.

#### 1. Introduction

In this paper, we investigate properties of multidimensional generalization of Melvin’s solution [1], which was presented earlier in [2]. Originally, model from [2] contains metric, Abelian 2-forms and scalar fields. Here we consider a special solutions with , governed by a 4 × 4 Cartan matrix for simple nonexceptional Lie algebras of rank 4: , , , and . The solutions from [2] are special case of the so-called generalized fluxbrane solutions from [3].

It is well known that the original Melvin’s solution in four dimensions describes the gravitational field of a magnetic flux tube. The multidimensional analog of such a flux tube, supported by a certain configuration of form fields, is referred to as a fluxbrane (a “thickened brane” of magnetic flux). The appearance of fluxbrane solutions was originally motivated by superstring/brane models and -theory. For generalizations of the Melvin solution and fluxbrane solutions see [4–21] and references therein.

In [3] there were considered the generalized fluxbrane solutions which are described in terms of moduli functions defined on the interval , where and is a radial coordinate. Functions obey nonlinear differential master equations of Toda-like type governed by some matrix , and the following boundary conditions are imposed: , .

Here, as in [2], we assume that the matrix is a Cartan matrix for some simple finite-dimensional Lie algebra of rank ( for all ). A conjecture was suggested in [3] that in this case the solutions to master equations with the above boundary conditions are polynomials of the form: where are constants. Here and where we denote . Integers are components of the twice dual Weyl vector in the basis of simple (co)roots [22].

Therefore, the functions (which may be called “fluxbrane polynomials”) define a special solution to open Toda chain equations [23, 24] corresponding to simple finite-dimensional Lie algebra [25]. In [2, 26] a program (in Maple) for calculation of these polynomials for classical series of Lie algebras (-, -, -, and -series) was suggested. It was pointed out in [3] that the conjecture on polynomial structure of is valid for Lie algebras of - and - series.

One of the goals of this paper is to study interesting geometric properties of the solution considered for case of nonexceptional Lie algebras of rank 4. In particular, we prove some symmetry properties, as well as the so-called duality relations of fluxbrane polynomials which establishes a behaviour of the solutions under the inversion transformation , which makes the model in tune with -duality in string models and also can be mathematically understood in terms of the groups of symmetry of Dynkin diagrams for the corresponding Lie algebras. In our case these groups of symmetry are either identical ones (for Lie algebras and ) or isomorphic to the group (for Lie algebra ) or isomorphic to the group which is the group of permutation of 3 elements (for Lie algebra ). These duality identities may be used in deriving a -expansion for solutions at large distances . The corresponding asymptotic behaviour of the solutions is studied.

The analogous analysis was performed recently for the case of rank-2 Lie algebras: , , in [27], and for rank-3 algebras , , and in [28]. Also, in [29] the conjecture from [3] was verified for the Lie algebra and certain duality relations for six -polynomials were found.

The paper is organized as follows. In Section 2 we present a generalized Melvin solutions from [2] for the case of four scalar fields and four 2-forms. In Section 3 we deal with the solutions for the Lie algebras , , , and . We find symmetry properties and duality relations for polynomials and present asymptotic relations for the solutions. We also calculate 2-form flux integrals and corresponding Wilson loop factors, where are 2-forms and is 2-dimensional disc of radius . The flux integrals converge, i.e., have finite limits for [30].

#### 2. The Setup and Generalized Melvin Solutions

Let us consider the following product manifold: where and is a -dimensional Ricci-flat manifold.

On this manifold, we define the action where is a metric on , is vector of scalar fields, is a 2-form, and is dilatonic coupling vector, ; . Here we use the notations ; .

There is a family of exact cylindrically symmetric solutions to the field equations corresponding for the action (4) and depending on the radial coordinate . The solution has the form [2], where is a metric on and is a Ricci-flat metric of signature on . Here are integration constants ( in notations of [2]).

For further convenience, let us denote . As it was shown in earlier works, the functions obey the set of master equations with the boundary conditions where . The boundary condition (9) guarantees the absence of a conic singularity (for the metric (5)) for .

There are some relations for the parameters : where . In these relations, we have denoted The latter matrix is the so-called “quasi-Cartan” matrix. One can prove that if is a Cartan matrix for a certain simple Lie algebra of rank 4 then there exists a set of vectors obeying (13). See also Remark 1 in the next section.

The solution considered can be understood as a special case of the fluxbrane solutions from [3, 19].

Therefore, here we investigate a multidimensional generalization of Melvin’s solution [1] for the case of four scalar fields and four 2-forms. Note that the original Melvin’s solution without scalar field would correspond to , one (electromagnetic) 2-form, (), , and .

#### 3. Solutions Related to Simple Classical Rank-4 Lie Algebras

In this section we consider the solutions associated with the simple nonexceptional Lie algebras of rank 4. This means than the matrix coincides with one of the Cartan matrices

for , respectively.

Each of these matrices can be graphically described by drawing the Dynkin diagrams pictured on Figure 1 for these four Lie algebras.