On generalized Melvin solutions for Lie algebras of rank 3

TL;DR

This paper explores generalized Melvin solutions for rank-3 Lie algebras, deriving polynomial moduli functions, asymptotic behaviors, and flux integrals, extending known solutions to new algebraic structures with specific boundary conditions.

Contribution

It introduces explicit polynomial solutions and asymptotic relations for generalized Melvin solutions associated with rank-3 Lie algebras $A_3$, $B_3$, and $C_3$, including duality identities and flux calculations.

Findings

Polynomial moduli functions with specific powers for each Lie algebra.

Asymptotic relations governed by the inverse Cartan matrix and symmetry considerations.

Explicit flux integrals and Wilson loop factors for the solutions.

Abstract

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 $q_1, q_2, q_3 \neq 0$. 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 $2 A^{-1}$ 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. 2-form flux integrals over a $2$-dimensional disc of radius $R$ and corresponding Wilson loop factors over a circle of radius $R$ are presented.