This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the $${{\mathcal {W}}}$$ W -algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra $${\mathfrak {s}}{\mathfrak {l}}_3$$ s l 3 on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat $$\mathrm {SL}(3)$$ SL ( 3 ) -connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel–Nielsen coordinates from Teichmüller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.

中文翻译：

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Toda 共形块、量子群和平面连接

本文研究了户田共形场论、量子群论和平面连接模空间的量子化之间的关系。我们使用 $${{\mathcal {W}}}$$ W 代数的自由场表示来定义与李代数 $${\mathfrak { 相关的 Toda 共形场论的共形块空间的自然基s}}{\mathfrak {l}}_3$$ sl 3 在三个穿孔的球体上，具有与三个穿孔相关联的通用类型的表示。简并域的算子值单体可用于描述平面 $$\mathrm {SL}(3)$$ SL ( 3 ) -连接的模空间的量化。结果表明，单项矩阵的矩阵元素可以表示为更多初等算子的Laurent多项式，这些算子在自由场表示中有一个简单的定义。这些算子被确定为来自 Teichmüller 理论的 Fenchel-Nielsen 坐标的自然更高阶类似物的量化对应物。简要概述了对非拉格朗日 SUSY 场论研究的可能应用。