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. © 2021, The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature.