发表机构
Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay; Aix-Marseille University, CNRS, Institut de Mathématiques de Marseille (I2M), and Institut Universitaire de France (IUF); Beijing Institute of Technology（巴黎萨克雷大学; 艾克斯-马赛大学; 北京理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过散射方法建立半环半群的谱分解，将边界态空间分解为Virasoro不可约表示，并利用Ward恒等式将关联函数表示为谱参数积分，完成带边界Liouville CFT的共形自举。
AI 中文摘要
本文是带边界的紧致曲面上Liouville共形场论共形自举证明的第二部分，致力于半环半群的谱理论及由此产生的自举公式。基于第一部分~\cite{GRW1}中建立的Segal公理和粘合性质，我们将半环半群的生成元识别为边界哈密顿量，并通过散射方法建立其谱分解。这导致边界态空间关于边界谱测度的不可约Virasoro表示的直接积分分解。将此分解与边界变形的Ward恒等式相结合，我们将一般关联函数表示为对切割曲线上的谱参数的积分，其被积函数由体态和边界结构常数及相应的共形块乘积给出。这些结果也支撑了共形块的解析性和对称性~\cite{GRSSbloc}，并有助于在共形块空间上构造映射类群表示~\cite{PapierBlocs}。
英文摘要
This paper is the second part of the proof of the conformal bootstrap for Liouville conformal field theory on compact surfaces with boundary. It is devoted to the spectral theory of the half-annulus semigroup and to the resulting bootstrap formula. Building on the Segal axioms and gluing properties established in Part~1~\cite{GRW1}, we identify the generator of the half-annulus semigroup with the boundary Hamiltonian and establish its spectral decomposition by scattering methods. This yields a direct-integral decomposition of the boundary state space into irreducible Virasoro representations, governed by the boundary spectral measure. Combining this decomposition with Ward identities for boundary deformations, we express general correlation functions as integrals over spectral parameters attached to cutting curves, with integrands given by products of bulk and boundary structure constants and the corresponding conformal blocks. These results also underpin the analyticity and symmetry properties of conformal blocks~\cite{GRSSbloc}, and contribute to the construction of mapping class group representations on spaces of conformal blocks~\cite{PapierBlocs}.
