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

S V Bolokhov,V D Ivashchuk

doi:10.1155/2018/8179570

S V Bolokhov, V D Ivashchuk

Open Access

https://doi.org/10.1155/2018/8179570

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Abstract

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 Hs(z) (s=1,…,4) of squared radial coordinate z=ρ2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1,n2,n3,n4)=(4,6,6,4),(8,14,18,10),(7,12,15,16),(6,10,6,6) for Lie algebras A4, B4, C4, and D4, respectively. The asymptotic behaviour for the polynomials at large distances is governed by some integer-valued 4×4 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A4 case) the matrix representing a generator of the Z2-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.

Highlights

In this paper, we investigate properties of multidimensional generalization of Melvin’s solution [1], which was presented earlier in [2]
We prove some symmetry properties, as well as the so-called duality relations of fluxbrane polynomials which establishes a behaviour of the solutions under the inversion transformation ρ 󳨀→ 1/ρ, which makes the model in tune with T-duality in string models and can be mathematically understood in terms of the groups of symmetry of Dynkin diagrams for the corresponding Lie algebras
The generalized multidimensional family of Melvin-type solutions was considered corresponding to simple nonexceptional finite-dimensional Lie algebras of rank 4: G = A4, B4, C4, D4

Summary

Introduction

We investigate properties of multidimensional generalization of Melvin’s solution [1], which was presented earlier in [2]. We prove some symmetry properties, as well as the so-called duality relations of fluxbrane polynomials which establishes a behaviour of the solutions under the inversion transformation ρ 󳨀→ 1/ρ, which makes the model in tune with T-duality in string models and can be mathematically understood in terms of the groups of symmetry of Dynkin diagrams for the corresponding Lie algebras. In our case these groups of symmetry are either identical ones (for Lie algebras B4 and C4) or isomorphic to the group Z2 (for Lie algebra A4) or isomorphic to the group S3 which is the group of permutation of 3 elements (for Lie algebra D4).

The Setup and Generalized Melvin Solutions

Solutions Related to Simple Classical Rank4 Lie Algebras

Conclusions