Duality Identities for Moduli Functions of Generalized Melvin Solutions Related to Classical Lie Algebras of Rank 4

摘要

We consider generalized Melvin-like solutions associated with nonexceptional Lie algebras of rank 4 (namely, A4, B4, C4, and D4) 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, . . . ,4) of squared radial coordinate z = ρ~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 4×4 matrix v connected in a certain waywith the inverse Cartanmatrix of the Lie algebra and (in A_4 case) thematrix representing a generator of the 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 correspondingWilson loop factors over their boundaries.