On generalized Melvin solution for the Lie algebra $$E_6$$ E 6

2023

Symmetry

This review dealt with generalized Melvin solutions for simple finite-dimensional Lie algebras. Each solution appears in a model which includes a metric and n scalar fields coupled to n Abelian 2-forms with dilatonic coupling vectors determined by simple Lie algebra of rank n. The set of n moduli functions Hs(z) comply with n non-linear (ordinary) differential equations (of second order) with certain boundary conditions set. Earlier, it was hypothesized that these moduli functions should be polynomials in z (so-called “fluxbrane” polynomials) depending upon certain parameters ps>0, s=1,…,n. Here, we presented explicit relations for the polynomials corresponding to Lie algebras of ranks n=1,2,3,4,5 and exceptional algebra E6. Certain relations for the polynomials (e.g., symmetry and duality ones) were outlined. In a general case where polynomial conjecture holds, 2-form flux integrals are finite. The use of fluxbrane polynomials to dilatonic black hole solutions was also explored.

Section: Polynomialsmentioning

confidence: 99%

“…This matrix is graphically depicted by the Dynkin diagram in Figure 5. The inverse Cartan matrix for E 6 (16,30,42,30,16,22).…”

Section: E 6 Algebramentioning

confidence: 99%

See 1 more Smart Citation

Fluxbrane Polynomials and Melvin-like Solutions for Simple Lie Algebras

2023

Symmetry

“…For the Lie algebra E 6 we find the set of six fluxbrane polynomials [45], which are listed in Appendix.…”

Section: E 6 Algebramentioning

confidence: 99%

“…For the Lie algebra A 5 ∼ = sl(6) the polynomials read C 5 -case. For the Lie algebra C 5 ∼ = sp(5) we find In this subsection we present polynomials corresponding to the Lie algebra E 6 [45].…”

Section: A Appendixmentioning

confidence: 99%

On Fluxbrane Polynomials for Generalized Melvin-like Solutions Associated with Rank 5 Lie Algebras

2022

Symmetry

We consider generalized Melvin-like solutions corresponding to Lie algebras of rank 5 (A5, B5, C5, D5). The solutions take place in a D-dimensional gravitational model with five Abelian two-forms and five scalar fields. They are governed by five moduli functions Hs(z) (s=1,...,5) of squared radial coordinates z=ρ2, which obey five differential master equations. The moduli functions are polynomials of powers (n1,n2,n3,n4,n5)=(5,8,9,8,5),(10,18,24,28,15),(9,16,21,24,25),(8,14,18,10,10) for Lie algebras A5, B5, C5, D5, respectively. The asymptotic behavior for the polynomials at large distances is governed by some integer-valued 5×5 matrix ν connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in A5 and D5 cases) with the matrix representing a generator of the Z2-group of symmetry of the Dynkin diagram. The symmetry and duality identities for polynomials are obtained, as well as asymptotic relations for solutions at large distances.

On generalized Melvin solutions for Lie algebras of rank 3

2019

J. Phys.: Conf. Ser.

We consider generalized Melvin-like solutions associated with Lie algebras of rank 4 (namely, A4 , B4 , C4 , D4 , and the exceptional algebra F4 ) 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 = ρ 2 obeying four differential equations of the Toda chain type. These functions turn out to be polynomials of powers (n1, n2, n3, n4) ) for Lie algebras A4 , B4 , C4 , D4 , F4 , 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. Dilatonic black hole analogs of the obtained Melvin-type solutions are also presented.