关于秩为3的李代数的广义Melvin解
On generalized Melvin solutions for Lie algebras of rank 3
AI总结：
本文研究秩3李代数$A_3$、$B_3$、$C_3$的广义Melvin解，推导其模多项式的形式、幂次、渐近行为与对偶恒等式，给出通量积分与Wilson圈因子。
摘要：
本文研究了秩为3的李代数$A_3$、$B_3$和$C_3$对应的广义Melvin解。任意解都包含度规、三个阿贝尔2-形式和三个标量场，由三个模函数$H_1(z)$、$H_2(z)$、$H_3(z)$（其中$z = \rho^2$，$\rho$为径向变量）描述，这三个函数满足带有特定边界条件的三个微分方程。这些函数是多项式，对于李代数$A_3$、$B_3$、$C_3$，其幂次分别为$(n_1,n_2, n_3) = (3,4,3)$、$(6,10,6)$、$(5,8,9)$，解依赖于非零积分常数$q_1, q_2, q_3$。\n 多项式在大$z$情形下的幂律渐近关系由整数值$3 \times 3$矩阵$\nu$决定：对于李代数$B_3$和$C_3$，$\nu$等于两倍的逆Cartan矩阵$2 A^{-1}$；而在$A_3$的情形下，$\nu= A^{-1} (I + P)$，其中$I$是单位矩阵，$P$是置换矩阵，对应Dynkin图的$\mathbb{Z}_2$对称群的一个生成元。本文得到了多项式的对偶恒等式，以及解在远距离处的渐近关系，同时给出了半径为$R$的二维圆盘上的2-形式通量积分，以及半径为$R$的圆周上对应的Wilson圈因子。
英文摘要：
Generalized Melvin solutions for rank-$3$ Lie algebras $A_3$, $B_3$ and $C_3$ are considered. Any solution contains metric, three Abelian 2-forms and three scalar fields. It is governed by three moduli functions $H_1(z),H_2(z),H_3(z)$ ($z = ρ^2$ and $ρ$ is a radial variable), obeying three differential equations with certain boundary conditions imposed. These functions are polynomials with powers $(n_1,n_2, n_3) = (3,4,3), (6,10,6), (5,8,9)$ for Lie algebras $A_3$, $B_3$, $C_3$, respectively. The solutions depend upon integration constants $q_1, q_2, q_3 \neq 0$. The power-law asymptotic relations for polynomials at large $z$ are governed by integer-valued $3 \times 3$ matrix $ν$, which coincides with twice the inverse Cartan matrix $2 A^{-1}$ for Lie algebras $B_3$ and $C_3$, while in the $A_3$ case $ν= A^{-1} (I + P)$, where $I$ is the identity matrix and $P$ is a permutation matrix, corresponding to a generator of the $\mathbb{Z}_2$-group of symmetry of the Dynkin diagram. The duality identities for polynomials and asymptotic relations for solutions at large distances are obtained. 2-form flux integrals over a $2$-dimensional disc of radius $R$ and corresponding Wilson loop factors over a circle of radius $R$ are presented.
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