关于复Gamma函数积分
On Complex Gamma-Function Integrals
摘要：
最近观察到${\rm SL}(2,\mathbb R)$自旋链模型中某些算符的矩阵元之间的关系采用由R.A.古斯塔夫森具有${\rm SL}(2,\mathbb C)$对称群的自旋磁体和作为局部Hilbert空间的${\rm L}_2(\mathbb C)$，产生了一类新的$Γ$ -函数积分.在这项工作中，我们提出了一个直接计算两个这样的积分。我们还分析了这些积分的性质，并表明它们包括最近在文献中讨论的星三角关系。在准经典极限下，这些积分恒等式可化为Dotsenko-Fateev积分的对偶关系。
英文摘要：
It was observed recently that relations between matrix elements of certain operators in the ${\rm SL}(2,\mathbb R)$ spin chain models take the form of multidimensional integrals derived by R.A. Gustafson. The spin magnets with ${\rm SL}(2,\mathbb C)$ symmetry group and ${\rm L}_2(\mathbb C)$ as a local Hilbert space give rise to a new type of $Γ$-function integrals. In this work we present a direct calculation of two such integrals. We also analyse properties of these integrals and show that they comprise the star-triangle relations recently discussed in the literature. It is also shown that in the quasi-classical limit these integral identities are reduced to the duality relations for Dotsenko-Fateev integrals.
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