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Integrability of Liouville theory: proof of the DOZZ formula
Annals of Mathematics ( IF 5.7 ) Pub Date : 2020-01-01 , DOI: 10.4007/annals.2020.191.1.2
Antti Kupiainen 1 , Rémi Rhodes 2 , Vincent Vargas 3
Affiliation  

Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the 3 point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e. to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first rigorous integrability result on Gaussian Multiplicative Chaos measures.

中文翻译:

Liouville 理论的可积性:DOZZ 公式的证明

Dorn 和 Otto (1994) 以及独立的 Zamolodchikov 和 Zamolodchikov (1996) 提出了一个显着的显式表达式,即所谓的 DOZZ 公式,用于 Liouville 共形场理论 (LCFT) 的 3 点结构常数,预期描述标度极限正确嵌入黎曼球体的大型平面图。在本文中,我们根据 F. David 和作者早先给出的高斯乘法混沌,基于 LCFT 的严格概率构造给出了 DOZZ 公式的证明。这一结果是证明 LCFT 可积性的路径中的一个基本步骤,即从数学上证明物理学家使用的保形自举方法的合理性。从纯概率的角度来看,我们的证明构成了高斯乘法混沌测度的第一个严格的可积性结果。
更新日期:2020-01-01
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