On generalized Melvin solutions for Lie algebras of rank 4

TL;DR

This paper constructs and analyzes generalized Melvin solutions linked to rank 4 Lie algebras, revealing polynomial moduli functions governed by Toda chain equations, and explores their asymptotic behavior, symmetries, dualities, and black hole analogs.

Contribution

It introduces explicit polynomial solutions for generalized Melvin configurations associated with rank 4 Lie algebras and studies their properties and physical implications.

Findings

Polynomial moduli functions for each Lie algebra case are explicitly derived.

Asymptotic behavior is characterized by a matrix related to the inverse Cartan matrix.

Black hole analogs and flux integrals are also analyzed.

Abstract

We deal with generalized Melvin-like solutions associated with Lie algebras of rank $4$ ($A_4$, $B_4$, $C_4$, $D_4$, $F_4$). Any solution has static cylindrically-symmetric metric in $D$ dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions $H_s(z)$ ($s = 1,...,4$) of squared radial coordinate $z=\rho^2$ obeying four differential equations of the Toda chain type. These functions are polynomials of powers $(n_1,n_2, n_3, n_4) = (4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6), (22,42,30,16)$ for Lie algebras $A_4$, $B_4$, $C_4$, $D_4$, $F_4$, respectively. The asymptotic behaviour for the polynomials at large $z$ is governed by an integer-valued $4 \times 4$ matrix $\nu$ connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in $A_4$ case) the matrix representing a generator of the $\mathbb{Z}_2$-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are studied. We also present 2-form flux integrals over a $2$-dimensional submanifold. Dilatonic black hole analogs of the obtained Melvin-type solutions, e.g. "fantom" ones, are also considered. The phantom black holes are described by fluxbrane polynomials under consideration.