Quasi-exponentials and semi-classical limits for the chordal BPZ system

Abstract

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 $\lambda$. We determine the number of global real-valued solutions modulo additive constants. When $\lambda =0$, 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 $\lambda \ne 0$ in this article. For $\lambda >0$, 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 $\lambda <0$.

v2: Added Appendix B comparing the quantum and semi-classical BPZ systems. Main results unchanged. 32 pages, 0 figures