On generalized Melvin solutions for Lie algebras of rank 3

Abstract

Generalized Melvin solutions for rank-[Formula: see text] Lie algebras [Formula: see text], [Formula: see text] and [Formula: see text] are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions [Formula: see text] ([Formula: see text] and [Formula: see text] is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers [Formula: see text] for Lie algebras [Formula: see text], [Formula: see text], [Formula: see text], respectively. The solutions depend upon integration constants [Formula: see text]. The power-law asymptotic relations for polynomials at large [Formula: see text] are governed by integer-valued [Formula: see text] matrix [Formula: see text], which coincides with twice the inverse Cartan matrix [Formula: see text] for Lie algebras [Formula: see text] and [Formula: see text], while in the [Formula: see text]-case [Formula: see text], where [Formula: see text] is the identity matrix and [Formula: see text] is a permutation matrix, corresponding to a generator of the [Formula: see text]-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. Two-form flux integrals over a two-dimensional disc of radius [Formula: see text] and corresponding Wilson loop factors over a circle of radius [Formula: see text] are presented.