On generalized Melvin solution for the Lie algebra $$E_6$$ E6

Abstract

Abstract A multidimensional generalization of Melvin’s solution for an arbitrary simple Lie algebra $${\mathcal {G}}$$ G is considered. The gravitational model in D dimensions, $$D \ge 4$$ D≥4 , contains n 2-forms and $$l \ge n$$ l≥n scalar fields, where n is the rank of $${\mathcal {G}}$$ G . The solution is governed by a set of n functions $$H_s(z)$$ 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 $$H_s(z)$$ Hs(z) , $$s = 1,\ldots ,6$$ s=1,…,6 , for the Lie algebra $$E_6$$ E6 are obtained and a corresponding solution for $$l = n = 6$$ l=n=6 is presented. The polynomials depend upon integration constants $$Q_s$$ Qs , $$s = 1,\ldots ,6$$ 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 $$E_6$$ E6 -polynomials at large z are governed by the integer-valued matrix $$\nu = A^{-1} (I + P)$$ ν=A-1(I+P) , where $$A^{-1}$$ A-1 is the inverse Cartan matrix, I is the identity matrix and P is a permutation matrix, corresponding to a generator of the $$Z_2$$ Z2 -group of symmetry of the Dynkin diagram. The 2-form fluxes $$\Phi ^s$$ Φs , $$s = 1,\ldots ,6$$ s=1,…,6 , are calculated.