AI 中文总结
本文以BGKR24的工作为基础，计算了Virasoro中心项对应的标量反常项，推导了圆盘等曲面的局部沃德恒等式，恢复了裤形系数的多项式分解等结果，还证明了体插入点光滑性并推导了BPZ方程。
AI 中文摘要
本文以Baverez、Guillarmou、Kupiainen和Rhodes的工作[BGKR24]为起点，该工作表明边界参数化的解析变换对Liouville振幅的作用是可微的，其导数由Virasoro算子和一个标量反常项给出。我们计算了该标量项，其全纯部分是一个施瓦茨边界积分，为Virasoro中心项提供了几何解释。随后，我们推导了圆盘、圆环和裤形曲面的局部沃德恒等式，这些恒等式给出了后代矩阵系数的有限递推关系。从这些递推关系中，我们恢复了归一化裤形系数的多项式分解；在圆环零权重极限下，恢复了Shapovalov形式；对于具有两个入射边界的裤形曲面，恢复了形式手征顶点算子系数。我们还给出了体插入点光滑性的几何证明，并推导了退化体插入的零亏格任意水平BPZ方程。
英文摘要
The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.
Comments27 pages
