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 Hs(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.