Duality identities for moduli functions of generalized Melvin solutions related to classical Lie algebras of rank 4
Summary: We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank \(4\) (namely, \(A_4\), \(B_4\), \(C_4\), and \(D_4\)) corresponding to certain internal symmetries of the solutions. The system under consideration is a static cylindrically symmetric gravitational configuration in \(D\) dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions \(H_s(z)\) (\(s = 1, \ldots, 4\)) of squared radial coordinate \(z = \rho^2\) obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers \((n_1, n_2, n_3, n_4) = (4,6, 6,4),(8,14,18,10),(7,12,15,16),(6,10,6, 6)\) for Lie algebras \(A_4\), \(B_4\), \(C_4\), and \(D_4\), respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued \(4 \times 4\) matrix \(\nu\) connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in \(A_4\) case) the matrix representing a generator of the \(\mathbb{Z}_2\)-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances. We also calculate 2-form flux integrals over \(2\)-dimensional discs and corresponding Wilson loop factors over their boundaries.
scientific article; zbMATH DE number 6992025
scientific article