On generalized Melvin solution for the Lie algebra $E_6$

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\cal G$ is considered. The gravitational model in $D$ dimensions, $D \geq 4$, contains $n$ 2-forms and $l \geq n$ scalar fields, where $n$ is the rank of $\cal G$. The solution is governed by a set of $n$ 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). The polynomials $H_s(z)$, $s = 1,\dots,6$, for the Lie algebra $E_6$ are obtained and a corresponding solution for $l = n = 6$ is presented. The polynomials depend upon integration constants $Q_s$, $s = 1,\dots,6$. 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 $E_6$-polynomials at large $z$ are governed by integer-valued matrix $\nu = A^{-1} (I + P)$, where $A^{-1}$ is the inverse Cartan matrix, $I$ is the identity matrix and $P$ is permutation matrix, corresponding to a generator of the $Z_2$-group of symmetry of the Dynkin diagram. The 2-form fluxes $\Phi^s$, $s = 1,\dots,6$, are calculated.


Introduction
In this paper we deal with a multidimensional generalization of the Melvin solution [1] which was considered earlier in ref. [2]. This solution is governed by a simple finitedimensional Lie algebra. It is a special case of the so-called generalized fluxbrane solutions from [3]. For generalizations of the Melvin solution, fluxbrane solutions and their applications, see refs. [4]- [33] and the references therein.
We remind the reader that Melvin's original solution in 4d space-time describes the gravitational field of a magnetic flux tube. The multidimensional analog of such a flux tube, supported by a certain configuration of fields of forms, is referred to as a fluxbrane (a "thickened brane" of magnetic flux). The appearance of fluxbrane solutions was motivated by superstring/M-theory models. A physical interest in such solutions is that they supply an appropriate background geometry for studying various processes involving branes, instantons, Kaluza-Klein monopoles, pair production of magnetically charged black holes and other configurations which can be studied via a special kind of Kaluza-Klein reduction ("modding technique") of a certain multidimensional model in the presence of U(1) isometry subgroup.
In ref. [2] the electro-vacuum Melvin solution was generalized for the D -dimensional model which contains metric g , n 2 -form fields F s = dA s and l scalar fields ϕ α . The model also includes n dilatonic coupling vectors belonging to R l . The D -dimensional warped product solution from ref. [2] comprises two factor spaces: 1 -dimensional subspace M 1 and a (D − 2) -dimensional Ricci-flat subspace M 2 . Here M 1 is either R or S 1 . For M 1 = S 1 we have a cylindrically symmetric solution with the isometry group U(1) × Isom(M 2 ) , where Isom(M 2 ) is the isometry group of M 2 .
The generalized fluxbrane solutions from ref. [2] are governed by functions H s (z) > 0 defined on the interval (0, +∞) which obey the non-linear differential equations s = 1, ..., n , where P s > 0 for all s . Parameters P s are proportional to Q 2 s , where Q s are integration constants and z = ρ 2 , where ρ is a radial parameter. The boundary condition (1.2) guarantees the absence of a conic singularity (in the metric) for ρ = +0 . The integration constants Q s are coinciding up to a sign with values of magnetic fields on the axis of the symmetry.
In this paper we assume that (A ss ′ ) is a Cartan matrix for some simple finite-dimensional Lie algebra G of rank n ( A ss = 2 for all s ).
According to a conjecture suggested in [3], the solutions to Eqs. (1.1), (1.2) governed by the Cartan matrix (A ss ′ ) are polynomials: where P where we denote (A ss ′ ) = (A ss ′ ) −1 . Integers n s are components of a twice dual Weyl vector in the basis of simple co-roots [35]. The set of fluxbrane polynomials H s defines a special solution to open Toda chain equations [36,37] corresponding to a simple finite-dimensional Lie algebra G ; see ref. [38]. In refs. [2,39] a program (in Maple) for calculation of these polynomials for classical series of Lie algebras (of A -, B -, C -and D -series) was suggested.
It should be noted that the open Toda chain corresponding to the Lie algebra G has a hidden symmetry group G T = exp(G) . The solution from ref. [2] corresponding to this group is a special case of solutions from [3]. It may be obtained by using an 1dimensional sigma-model [40,41,42] with (2 + l + n) -dimensional target space. The isometry group of this target space G sm (related to the sigma model) was studied in detail in [43]. For another more general setup with non-diagonal metrics (which is valid for flat M 2 ) see also [9]. The group G sm is another hidden symmetry group related to our model. Here the Toda Lagrangian L T may be obtained from the sigma-model one after integrating the Maxwell-type equations corresponding to potentials Φ s (u) = A s φ (u) , where u is a radial variable and φ is a coordinate on M 1 ( 0 < φ < 2π for M 1 = S 1 ), and obtaining integration constants Q s . The Toda Lagrangian L T = L T (x,ẋ, Q) (ẋ = dx du ) is responsible for equations of motion for 2 scale factors and l scalar fields described by x = (x a ) for fixed Q = (Q s ) .
We note also that there are several multidimensional aspects of generalized Melvin solution from ref. [2]: (1) the space-time dimension D (for Melvin's solution D = 4 ), (2) the rank of the Toda group G T which is equal to n (in Melvin's case n = 1 ) and (3) the dimension of the target space of the corresponding sigma-model which is equal to N = n + l + 2 (in Melvin's case N = 3 ).
Here we verify the conjecture from ref. [3] for the Lie algebra E 6 . In Section 2 the generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G is considered. The exact solution for the Lie algebra E 6 is presented in Section 3, while the fluxbrane polynomials are listed in the Appendix. Here duality relations for the polynomials H s (z) and asymptotic formulas for z → +∞ are presented, as well as the asymptotics for the solutions at large distances and a calculation of flux integrals. We find that any flux Φ s depends upon the integration constant Q s and does not depend upon the other constants Q s ′ , s ′ = s . The flux Φ s is proportional to n s Q −1 s , where n s are integer numbers (1.4): n s = 16, 30, 42, 30, 16, 22 for s = 1, 2, 3, 4, 5, 6 , respectively.
Here we consider a family of exact solutions to the field equations corresponding to the action (2.1) and depending on one variable ρ . The solutions are defined on the manifold where M 1 is a one-dimensional manifold (say S 1 or R ) and M 2 is a (D-2)-dimensional Ricci-flat manifold. The solution reads [2] g = The parameters h s satisfy the relations where is the Cartan matrix for a simple Lie algebra G of rank n . It may be shown that if the matrix (h αβ ) has an Euclidean signature and l ≥ n , there exists a set of co-vectors λ 1 , . . . , λ n obeying (2.9). Thus the solution is valid at least when l ≥ n and the matrix (h αβ ) is positive-definite.
If w = +1 and the (Ricci-flat) metric g 2 has a pseudo-Euclidean signature, we get a multidimensional generalization of Melvin's solution [1].
Melvin's solution (without scalar field) corresponds to D = 4 , n = 1 , For w = −1 and g 2 of Euclidean signature we obtain a cosmological solution with a horizon (as 3 The solution for the Lie algebra E 6 Here we deal with the solution for n = l = 6 , w = +1 and M 1 = S 1 , which corresponds to the Lie algebra E 6 . We put here h αβ = δ αβ and denote (λ sa ) = (λ a s ) = λ s , s = 1, . . . , 6 .
The matrix A = (A ss ′ ) is coincides with the Cartan matrix for the exceptional Lie algebra E 6 This matrix is graphically depicted at Fig. 1 by the Dynkin diagram.
For the Lie algebra E 6 we find the set of six fluxbrane polynomials, which are listed in the appendix. Here as in [38] we parametrize the polynomials by using other parameters (here denoted B s ) instead of P s :

4)
s = 1, . . . , 6 . This is necessary to avoid huge denominators in monomials of H s . The polynomials have the following structure: The powers of polynomials are in agreement with the relation (3.3). In what follows we denote Due to (3.5) the polynomials have the following asymptotical behavior The matrix (3.8) is related to the inverse Cartan matrix as follows: where I is 6 × 6 identity matrix and is permutation matrix. This matrix corresponds to the permutation σ ∈ S 6 ( S 6 is the symmetric group) σ : (1, 2, 3, 4, 5, 6) → (5, 4, 3, 2, 1, 6), (3.11) by the relation P = (P i j ) = (δ i σ(j) ) . Here σ is the generator of the group G = {σ, id} which is the symmetry group of the Dynkin diagram. G is isomorphic to the group Z 2 . σ is a composition of two transpositions: (1 5) and (2 4) .
Let us denoteB i = B σ(i) , i = 1, . . . , 6 . We call the ordered set (B i ) a dual one to the ordered set (B i ) . By using the relations for polynomials from the appendix we are led to the following two identities which are verified with the aid of Mathematica.

20)
s, s ′ = 1, . . . , 6 . For large enough K there exist vectors λ s of equal length which obey relations (3.20). Indeed, the matrix (Γ ss ′ ) is positive-definite for K > K 0 , where K 0 is some positive number. Hence there exists a matrix Λ , such that Λ T Λ = Γ . We put (Λ as ) = (λ a s ) and get the set of vectors obeying (3.20). Remark. Let us put h αβ = −δ αβ . It may be shown (along a line as was done for h αβ = δ αβ ) that, for K < K 0 , where K 0 is some negative number, there exist vectors λ s of equal length which obey relations following from (2.8) and (2.9). Thus, for both choices of signatures h αβ = ±δ αβ we get the same algebra (in our case E 6 ) and the same hidden group G T . So, the properties of the matrix (h αβ ) are not a priori known from the properties of the group G T . In the case of phantom scalar fields, when h αβ = −δ αβ , we get solutions which are defined for ρ < ρ 0 , where ρ 0 > 0 . The cosmological analogs of such solutions with phantom scalar fields where considered for Lie algebras of rank 2 and 3 in refs. [44] and [45], respectively. We note that another (sigma model) hidden group G sm (see Introduction) depends upon the choice of the matrix (h αβ ) [43].
for ρ → +∞ , and the equality 6 1 A sl n l = 2 (following from (1.4)), we get as ρ → +∞ , where (3.25) s = 1, . . . , 6 . Due to (3.9) we get Aν = I + P It is remarkable that any flux Φ s depends only upon n s and the integration constant Q s , which for D = 4 and g 2 = −dt ⊗ dt + dx ⊗ dx is coinciding up to a sign with the value of the x -component of the magnetic field on the axis of symmetry.
Analogous relations were found recently in ref. [46] for solutions corresponding to Lie algebras of rank 2 ; see also ref. [47].
The asymptotic relations for the solution under consideration for ρ → +∞ read

Conclusions
Here we have obtained a multidimensional generalization of Melvin's solution for the Lie algebra E 6 . The solution is governed by a set of six fluxbrane polynomials H s (z) , s = 1, . . . , 6 , which are presented in the appendix. These polynomials define special solutions to open Toda chain equations corresponding to the Lie algebra E 6 . The polynomials H s (z) depend also upon parameters Q s , which are coinciding for D = 4 (up to a sign) with the values of colored magnetic fields on the axis of symmetry. The symmetry and duality identities for polynomials were verified. The duality identities may be used in deriving (1/ρ) -expansion for solutions at large distances ρ , e.g. for asymptotic relations, which are presented in the paper. The power-law asymptotic relations for E 6 -polynomials at large z are governed by integer-valued matrix ν . This matrix is related to the inverse Cartan matrix A −1 by the formula ν = A −1 (I + P ) , where I is identity matrix and P is permutation matrix. The matrix P corresponds to a permutation σ ∈ S 6 , which is the generator of the Z 2 -group of symmetry of the Dynkin diagram.
We have also calculated 2d flux integrals Φ s , s = 1, . . . , 6 . Any flux Φ s depends only upon one parameter Q s , while the integrand F s depends upon all parameters Q 1 , . . . , Q 6 . An open question is how to apply the approach of this paper to other finitedimensional simple Lie algebras.

Acknowledgments
This work was supported in part by the Russian Foundation for Basic Research Grant no. 16-02-00602 and by the Ministry of Education of the Russian Federation (the agreement number 02.a03.21.0008 of 24 June 2016).

Appendix
In this appendix we present polynomials corresponding to the Lie algebra E 6 . The polynomials were calculated by using a certain program in Mathematica. We denote the variable z in bold and capital inside the polynomials for better readability: