On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra

Abstract

Abstract A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra $$\mathcal G$$ G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of $$\mathcal G$$ G . It is governed by a set of n moduli 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. These polynomials depend upon integration constants $$q_s$$ qs , $$s = 1,\dots ,n$$ s=1,⋯,n . In the case when the conjecture on the polynomial structure for the Lie algebra $$\mathcal G$$ G is satisfied, it is proved that 2-form flux integrals $$\Phi ^s$$ Φs over a proper 2d submanifold are finite and obey the relations $$q_s \Phi ^s = 4 \pi n_s h_s$$ qsΦs=4πnshs , where the $$h_s > 0$$ hs>0 are certain constants (related to dilatonic coupling vectors) and the $$n_s$$ ns are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, $$s = 1,\dots ,n$$ s=1,⋯,n . The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra $$\mathcal G$$ G . Examples of polynomials and fluxes for the Lie algebras $$A_1$$ A1 , $$A_2$$ A2 , $$A_3$$ A3 , $$C_2$$ C2 , $$G_2$$ G2 and $$A_1 + A_1$$ A1+A1 are presented.