Narrow Results

Generalized Melvin solutions for rank-$3$ Lie algebras $A_3$, $B_3$ and $C_3$ are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions $H_1(z),H_2(z),H_3(z)$ ($z={\rho }^2$ and $\rho$ is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers $(n_1,n_2,n_3)=(3,4,3),(6,10,6),(5,8,9)$ for Lie algebras $A_3$, $B_3$, $C_3$, respectively. The solutions depend upon integration constants $q_1,q_2,q_3\ne 0$. The power-law asymptotic relations for polynomials at large $z$ are governed by integer-valued $3\times 3$ matrix $\nu$, which coincides with twice the inverse Cartan matrix $2A^{-1}$ for Lie algebras $B_3$ and $C_3$, while in the $A_3$-case $\nu =A^{-1}(I+P)$, where $I$ is the identity matrix and $P$ is a permutation matrix, corresponding to a generator of the ${\mathbb{Z}}_2$-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.