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arXiv 1706.07856hep-th
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简单有限维李代数广义Melvin解的通量积分

On flux integrals for generalized Melvin solution related to simple finite-dimensional Lie algebra

V. D. Ivashchuk

备注：
10 pages, Latex, no figures, prepared for a talk at RUSGRAV-16 conference, 2nd revised version, several typos (mainly grammar ones) are eliminated. arXiv admin note: text overlap with arXiv:1706.06621
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摘要：

考虑了有限维单李代数$\cal G$的广义Melvin解.该解包含一个度量、$n$阿贝尔2-形式和$n$标量场，其中$n$是$\cal G$的秩。它是由一组$n$模函数$H_s(z)$服从$n$常微分方程与一定的边界条件施加。前面已经指出，这些函数应该是多项式--所谓的通量膜多项式。这些多项式取决于积分常数$q_s$、$s = 1,\dots,n$。在李代数$\cal G$的多项式结构猜想成立的情况下，证明了在适当的$2d$子流形上的2-形式通量积分$Φ^s$是有限的，并且满足以下关系：$q_s Φ^s = 4 πn_s h_s$，其中$h_s > 0$是某些常数（与声子耦合矢量相关）和$n_s$是多项式的幂，它们是在单（共）根$s = 1,\dots,n$的基础上的二次对偶Weyl向量的分量。本文的主要关系式对有限维半单李代数$\cal G$的解是成立的。给出了李代数$A_1$、$A_2$、$A_3$、$C_2$、$G_2$和$A_1 + A_1$的多项式和通量的例子。

英文摘要：

A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra $\cal G$ is considered. The solution contains a metric, $n$ Abelian 2-forms and $n$ scalar fields, where $n$ is the rank of $\cal G$. It is governed by a set of $n$ moduli functions $H_s(z)$ obeying $n$ ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials - the so-called fluxbrane polynomials. These polynomials depend upon integration constants $q_s$, $s = 1,\dots,n$. In the case when the conjecture on the polynomial structure for the Lie algebra $\cal G$ is satisfied, it is proved that 2-form flux integrals $Φ^s$ over a proper $2d$ submanifold are finite and obey the relations: $q_s Φ^s = 4 πn_s h_s$, where $h_s > 0$ are certain constants (related to dilatonic coupling vectors) and $n_s$ are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, $s = 1,\dots,n$. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra $\cal G$. Examples of polynomials and fluxes for the Lie algebras $A_1$, $A_2$, $A_3$, $C_2$, $G_2$ and $A_1 + A_1$ are presented.

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