发表机构
Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究弦向BPZ系统半经典极限的Hamilton-Jacobi方程，确定全局实值解的数量，利用Schwarzian对应与拟指数联系，并证明$\lambda>0$时解的存在性与$\lambda<0$时解的不存在性。
AI 中文摘要
我们研究了弦向Belavin--Polyakov--Zamolodchikov方程在共同特征值参数$\lambda$下的形式半经典极限所导致的非线性Hamilton--Jacobi系统。我们确定了模加法常数下全局实值解的数量。当$\lambda=0$时，解的数量可以通过Eremenko提出的解与有理函数之间的联系来推导。本文重点关注$\lambda\neq 0$的情形。对于$\lambda>0$，Schwarzian对应关系将梯度的代数方程与具有给定临界点的拟指数联系起来。Mukhin、Tarasov和Varchenko的枚举与横截性结果给出了光滑实分支，而Poisson括号恒等式确立了它们的精确性。能量界排除了$\lambda<0$时的全局实值解。
英文摘要
We study the nonlinear Hamilton--Jacobi system arising as the formal semi-classical limit of the chordal Belavin--Polyakov--Zamolodchikov equations with a common eigenvalue parameter $λ$. We determine the number of global real-valued solutions modulo additive constants. When $λ=0$, the number of solutions can be derived using the connection between the solutions and rational functions, due to Eremenko. We focus on the case when $λ\neq 0$ in this article. For $λ>0$, a Schwarzian correspondence relates the algebraic equations for the gradients to quasi-exponentials with prescribed critical points. Enumeration and transversality results of Mukhin, Tarasov, and Varchenko yield smooth real branches, and a Poisson bracket identity establishes their exactness. An energy bound rules out global real-valued solutions for $λ<0$.
Commentsv2: Added Appendix B comparing the quantum and semi-classical BPZ systems. Main results unchanged. 32 pages, 0 figures
