Abstract
A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra {mathcal {G}} is considered. The gravitational model in D dimensions, D ge 4, contains n 2-forms and l ge n scalar fields, where n is the rank of {mathcal {G}}. The solution is governed by a set of n functions H_s(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 H_s(z), s = 1,ldots ,6, for the Lie algebra E_6 are obtained and a corresponding solution for l = n = 6 is presented. The polynomials depend upon integration constants Q_s, s = 1,ldots ,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 E_6-polynomials at large z are governed by the integer-valued matrix nu = 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 Z_2-group of symmetry of the Dynkin diagram. The 2-form fluxes Phi ^s, s = 1,ldots ,6, are calculated.
Highlights
In this paper we deal with a multidimensional generalization of the Melvin solution [1] which was considered earlier in Ref. [2]
For the Lie algebra E6 we find the set of six fluxbrane polynomials, which are listed in the appendix
The solution is governed by a set of six fluxbrane polynomials Hs(z), s = 1, . . . , 6, which are presented in the appendix
Summary
In this paper we deal with a multidimensional generalization of the Melvin solution [1] which was considered earlier in Ref. [2]. This solution is governed by a simple finitedimensional Lie algebra It is a special case of the so-called generalized fluxbrane solutions from [3]. The set of fluxbrane polynomials Hs defines a special solution to open Toda chain equations [36,37] corresponding to a simple finite-dimensional Lie algebra G; see Ref. The isometry group of this target space Gsm (related to the sigma model) was studied in detail in [43] For another more general setup with non-diagonal metrics (which is valid for flat M2) see [9]. 2 the generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G is considered.
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