On generalized Melvin solutions for Lie algebras of rank 4

We deal with generalized Melvin-like solutions associated with Lie algebras of rank 4 (A₄, B₄, C₄, D₄, F₄). Any solution has static cylindrically symmetric metric in D dimensions in the presence of four Abelian two-form and four scalar fields. The solution is governed by four moduli functions H_s(z) (s= 1, …, 4) of squared radial coordinate z= ρ² obeying four differential equations of the Toda chain type. These functions are polynomials of powers (n₁, n₂, n₃, n₄) = (4, 6, 6, 4), (8, 14, 18, 10), (7, 12, 15, 16), (6, 10, 6, 6), (22, 42, 30, 16) for Lie algebras A₄, B₄, C₄, D₄, F₄, respectively. The asymptotic behaviour for the polynomials at large z is governed by an integer-valued 4 × 4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A₄ case) the matrix representing a generator of the Z₂-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are studied. We also present two-form flux integrals over a two-dimensional submanifold. Dilatonic black hole analogs of the obtained Melvin-type solutions, e.g. “phantom” ones, are also considered. The phantom black holes are described by fluxbrane polynomials under consideration.