On generalized Melvin solutions for Lie algebras of rank 3

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra G is considered. The gravitational model contains n 2-forms and l > n scalar fields, where n is the rank of G. The solution is governed by a set of n functions H_s (z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials H_s(z), s = 1,…, 6, corresponding to the Lie algebra E ₆ are obtained. They depend upon integration constants Q_s, s = 1,…, 6 . The polynomials obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances which are presented in the paper. The power-law asymptotic relations for E ₆ - polynomials at large z are governed by integer-valued matrix v = A ⁻¹ (I + P), where A ⁻¹ is inverse Cartan matrix, I is identity matrix and P is permutation matrix, corresponding to a generator of the Z ₂-group of symmetry of the Dynkin diagram. The 2-form fluxes Φ^s are calculated, s = 1, …, 6. (paper)