On multidimensional analogs of Melvin's solution for classical series of Lie algebras

TL;DR

This paper generalizes Melvin's solution to higher dimensions for any simple Lie algebra, introducing a set of differential equations and proposing a method to compute associated polynomial solutions for classical Lie algebra series.

Contribution

It presents a multidimensional extension of Melvin's solution for arbitrary simple Lie algebras and develops a computational program for fluxbrane polynomials, including explicit results for D_4.

Findings

Derived differential equations governing the solutions.

Provided explicit fluxbrane polynomials for D_4.

Suggested reduction formulas for classical Lie algebra series.

Abstract

A multidimensional generalization of Melvin's solution for an arbitrary simple Lie algebra $\cal G$ is presented. The gravitational model contains n 2-forms and $l \geq n$ scalar fields, wheren is the rank of $\cal G$. The solution is governed by a set of n functions obeying n ordinary differential equations with certain boundary conditions. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). A program (in Maple) for calculating of these polynomials for classical series of Lie algebras is suggested (see Appendix). The polynomials corresponding to the Lie algebra D_4 are obtained. It is conjectured that the polynomials for A_n-, B_n- and C_n-series may be obtained from polynomials for D_{n+1}-series by using certain reduction formulas.