We construct the stress-energy tensor correlation functions in probabilistic Liouville conformal field theory (LCFT) on the two-dimensional sphere $${\mathbb {S}}^2$$ S 2 by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric on $${\mathbb {S}}^2$$ S 2 . In particular we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In Kupiainen et al. (Commun Math Phys 371:1005–1069, 2019) the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in Oikarinen (Ann Henri Poincaré 20(7):2377–2406, 2019) allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.

中文翻译：

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刘维尔共形场理论中的应力能量

我们通过研究 LCFT 相关函数的变化，在二维球面 $${\mathbb {S}}^2$$ S 2 上构建了概率刘维尔共形场理论 (LCFT) 中的应力-能量张量相关函数到 $${\mathbb {S}}^2$$ S 2 上的平滑黎曼度量。特别地，我们为这些相关函数导出了保形病房标识。这构成了在 LCFT 的规范希尔伯特空间上构建 Virasoro 代数表示的基础。在 Kupiainen 等人。(Commun Math Phys 371:1005–1069, 2019) 使用不同的应力-能量张量定义和分部高斯积分，推导出了一个和两个应力-能量张量插入的保形 Ward 恒等式。