On generalized Melvin solutions for Lie algebras of rank 3
Abstract
Generalized Melvin solutions for rank-3 Lie algebras A3, B3 and C3 are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions H1(z),H2(z),H3(z) (z=ρ2 and ρ is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers (n1,n2,n3)=(3,4,3),(6,10,6),(5,8,9) for Lie algebras A3, B3, C3, respectively. The solutions depend upon integration constants q1,q2,q3≠0. The power-law asymptotic relations for polynomials at large z are governed by integer-valued 3×3 matrix ν, which coincides with twice the inverse Cartan matrix 2A−1 for Lie algebras B3 and C3, while in the A3-case ν=A−1(I+P), where I is the identity matrix and P is a permutation matrix, corresponding to a generator of the ℤ2-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 R and corresponding Wilson loop factors over a circle of radius R are presented.
