On generalized Melvin solution for the Lie algebra E 6

• Published in: The European physical journal. C, Particles and fields Particles and fields, 2017-10, Vol.77 (10), p.1-16
• Main Authors: Bolokhov, S V, Ivashchuk, V D
• Format: Article
• Language: English
• Subjects: Algebra; Differential equations; Fluxes; Functions (mathematics); Lie groups; Matrix methods; Ordinary differential equations; Permutations; Polynomials; Quantum theory; Symmetry
• ISSN: 1434-6044; 1434-6052
• DOI: 10.1140/epjc/s10052-017-5234-6

A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra G is considered. The gravitational model in D dimensions, D≥4, contains n 2-forms and l≥n scalar fields, where n is the rank of G. The solution is governed by a set of n functions Hs(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials (the so-called fluxbrane polynomials). The polynomials Hs(z), s=1,…,6, for the Lie algebra E6 are obtained and a corresponding solution for l=n=6 is presented. The polynomials depend upon integration constants Qs, s=1,…,6. They obey symmetry and duality identities. The latter ones are used in deriving asymptotic relations for solutions at large distances. The power-law asymptotic relations for E6-polynomials at large z are governed by the integer-valued matrix ν=A-1(I+P), where A-1 is the inverse Cartan matrix, 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 2-form fluxes Φs, s=1,…,6, are calculated.