T对偶环面背景下闭玻色弦的非对易性与非结合性
Noncommutativity and nonassociativity of closed bosonic string on T-dual toroidal backgrounds
AI总结：
本文通过 Buscher 与广义 T对偶研究闭玻色弦，证明 $Q$ 通量理论对易，而 $R$ 通量理论因线积分变量导致非局域，并呈现非对易、非结合性。
摘要：
本文考虑在常态度规和仅有一个非零分量的 Kalb-Ramond 场（$B_{xy}=Hz$，其中场强 $H$ 为无穷小）存在下的闭玻色弦。利用 Buscher T对偶程序，我们沿 $x$ 和 $y$ 方向进行对偶化，并采用广义 T对偶程序沿 $z$ 方向对偶化，同时施加平凡绕数条件。经过前两次 T对偶化后，我们得到仅在局域上良定义的 $Q$ 通量理论；而在全部三次 T对偶化之后，得到非局域的 $R$ 通量理论。非局域性的来源是定义为线积分的变量 $ΔV$，它作为背景场的自变量出现。将 T对偶变换定律改写为正则形式并使用标准 Poisson 代数，我们得到 $Q$ 通量理论是对易的，而 $R$ 通量理论是非对易且非结合的。因此，非局域性与闭弦的非对易性和非结合性之间存在关联。
英文摘要：
In this article we consider closed bosonic string in the presence of constant metric and Kalb-Ramond field with one non-zero component, $B_{xy}=Hz$, where field strength $H$ is infinitesimal. Using Buscher T-duality procedure we dualize along $x$ and $y$ directions and using generalized T-duality procedure along $z$ direction imposing trivial winding conditions. After first two T-dualizations we obtain $Q$ flux theory which is just locally well defined, while after all three T-dualizations we obtain nonlocal $R$ flux theory. Origin of non-locality is variable $ΔV$ defined as line integral, which appears as an argument of the background fields. Rewriting T-dual transformation laws in the canonical form and using standard Poisson algebra, we obtained that $Q$ flux theory is commutative one and the $R$ flux theory is noncommutative and nonassociative one. Consequently, there is a correlation between non-locality and closed string noncommutativity and nonassociativity.
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