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 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≠ 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.
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BibTeX
@misc{chen2026quasiexponentials,
title = {{Quasi-exponentials and semi-classical limits for the chordal BPZ system}},
author = {Mo Chen and Hao Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.38192},
eprint = {2609.38192},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38192},
}
RIS
TY - UNPB
TI - Quasi-exponentials and semi-classical limits for the chordal BPZ system
AU - Mo Chen
AU - Hao Wu
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2609.38192
UR - https://arxiv.org/abs/2609.38192
AB - 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≠ 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.
ER -