On generalized Melvin solutions for Lie algebras of rank 3

Abstract

Abstract 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 Hs (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 Hs (z), s = 1,…, 6, corresponding to the Lie algebra E 6 are obtained. They depend upon integration constants Qs , 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.