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

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, 𝐴 4 , 𝐵 4 , 𝐶 4


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 = = 4, governed by a 4 × 4 Cartan matrix ( ) for simple nonexceptional Lie algebras of rank 4: 4 , 4 , 4 , and 4 . The solutions from [2] are special case of the so-called generalized fluxbrane solutions from [3].
Here, as in [2], we assume that the matrix ( ) is a Cartan matrix for some simple finite-dimensional Lie algebra G of rank ( = 2 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 we denote ( ) = ( ) −1 . 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 finitedimensional Lie algebra G [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 → 1/ , 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 4 and 4 ) or isomorphic to the group Z 2 (for Lie algebra 4 ) or isomorphic to the group 3 which is the group of permutation of 3 elements (for Lie algebra 4 ). These duality identities may be used in deriving a 1/ -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: 2 , 2 = 2 , 2 in [27], and for rank-3 algebras 3 , 3 , and 3 in [28]. Also, in [29] the conjecture from [3] was verified for the Lie algebra 6 and certain duality relations for six 6 -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 4 , 4 , 4 , and 4 . We find symmetry properties and duality relations for polynomials and present asymptotic relations for the solutions. We also calculate 2-form flux integrals Φ ( ) = ∫ 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].

The Setup and Generalized Melvin Solutions
Let us consider the following product manifold: where 1 = 1 and 2 is a ( − 2)-dimensional Ricci-flat manifold.
For further convenience, let us denote = 2 . As it was shown in earlier works, the functions ( ) > 0 obey the set of master equations with the boundary conditions = 1, . . . , 4. The boundary condition (9) guarantees the absence of a conic singularity (for the metric (5)) for = +0.
There are some relations for the parameters ℎ : where , = 1, . . . , 4. 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 G of rank 4 then there exists a set of vectors → 1 , . . . , (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 = 4, one (electromagnetic) 2-form, 1

Solutions Related to Simple Classical Rank-4 Lie Algebras
In this section we consider the solutions associated with the simple nonexceptional Lie algebras G of rank 4. This means than the matrix = ( ) coincides with one of the Cartan matrices for G = 4 , 4 , 4 , 4 , respectively.
Each of these matrices can be graphically described by drawing the Dynkin diagrams pictured on Figure 1 for these four Lie algebras.
One can prove this conjecture by solving the system of nonlinear algebraic equations for the coefficients of these polynomials following from master equations (8). Below we present a list of the polynomials obtained by using appropriate MATHEMATICA algorithm. For convenience, we use the rescaled variables (as in [25]): for Lie algebras 4 , 4 , 4 , 4 , respectively. In these four cases there is a simple property Note that for Lie algebras 4 , 4 , and 4 we have where −1 is inverse Cartan matrix, whereas in the 4 -case the matrix ] is related to the inverse Cartan matrix as follows: Here is 4 × 4 identity matrix and is a permutation matrix corresponding to the permutation ∈ 4 ( 4 is symmetric group) : (1, 2, 3, 4) → (4, 3, 2, 1) , by the following relation = ( ) = ( ( ) ). Here is the generator of the group = { , id} which is the group of symmetry of the Dynkin diagram for 4 . is isomorphic to the group Z 2 .
In case of 4 the group of symmetry of the Dynkin diagram is isomorphic to the symmetric group 3 acting on the set of three vertices {1, 3, 4} of the Dynkin diagram via their permutations. The existence of the above symmetry groups ≅ Z 2 and ≅ 3 implies certain identity properties for the fluxbrane polynomials ( ).
Let us denotê= ( ) for the 4 case and̂= for 7 action (trivial or nontrivial) of the group Z 2 on vertices of the Dynkin diagrams for above algebras.
Then we obtain the following identities which were directly verified by using MATHEMATICA algorithms.
We call relations (46) as duality ones.