关于秩为4的李代数的广义Melvin解
On generalized Melvin solutions for Lie algebras of rank 4
摘要：
研究秩为$4$（$A_4$，$B_4$，$C_4$，$D_4$，$F_4$）的李代数的广义Melvin型解.在四个阿贝尔2-形式和四个标量场的存在下，任何解在$D$维数上都具有静态柱对称度规。该解由四个模函数$H_s(z)$（$s = 1,...,4$）的平方径向坐标$z=ρ^2$服从四个微分方程的户田链型。这些函数分别是李代数$A_4$，$B_4$，$C_4$，$D_4$，$F_4$的幂多项式$(n_1,n_2, n_3, n_4) = (4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6), (22,42,30,16)$.多项式的渐近行为在大$z$是由一个整数值$4 \times 4$矩阵$ν$连接在一定的方式与逆Cartan矩阵的李代数和（在$A_4$的情况下）的矩阵表示生成器的$\mathbb{Z}_2$ -群对称的Dynkin图。研究了多项式的对称性和对偶恒等式。我们也给出了$2$维子流形上的2-形式通量积分.膨胀黑洞类似物的Melvin型的解决方案，例如“fantom”的，也被认为是。幻影黑洞是由通量膜多项式所描述的。
英文摘要：
We deal with generalized Melvin-like solutions associated with Lie algebras of rank $4$ ($A_4$, $B_4$, $C_4$, $D_4$, $F_4$). Any solution has static cylindrically-symmetric metric 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 are polynomials of powers $(n_1,n_2, n_3, n_4) = (4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6), (22,42,30,16)$ for Lie algebras $A_4$, $B_4$, $C_4$, $D_4$, $F_4$, respectively. The asymptotic behaviour for the polynomials at large $z$ is governed by an integer-valued $4 \times 4$ matrix $ν$ 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 studied. We also present 2-form flux integrals over a $2$-dimensional submanifold. Dilatonic black hole analogs of the obtained Melvin-type solutions, e.g. "fantom" ones, are also considered. The phantom black holes are described by fluxbrane polynomials under consideration.
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