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 (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.