Multiple chordal SLE($κ$) and quantum Calogero-Moser system
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| Format: | Preprint |
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2025
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| _version_ | 1866916782629978112 |
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| author | Zhang, Jiaxin |
| author_facet | Zhang, Jiaxin |
| contents | We study multiple chordal SLE$(κ)$ systems in a simply connected domain $Ω$, where $z_1, \ldots, z_n \in \partial Ω$ are boundary starting points and $q \in \partial Ω$ is an additional marked boundary point.
As a consequence of the domain Markov property and conformal invariance, we show that the presence of the marked boundary point $q$ gives rise to a natural equivalence relation on partition functions. While these functions are not necessarily conformally covariant, each equivalence class contains a conformally covariant representative.
Building on the framework introduced in \cite{Dub07}, we demonstrate that in the $\mathbb{H}$-uniformization with $q = \infty$, the partition functions satisfy both the null vector equations and a dilatation equation with scaling exponent $d$.
Using techniques from the Coulomb gas formalism in conformal field theory, we construct two distinct families of solutions, each indexed by a topological link pattern of type $(n, m)$ with $2m \leq n$.
In the special case $Ω= \mathbb{H}$ and $q = \infty$, we further show that these partition functions correspond to eigenstates of the quantum Calogero-Moser system, thereby extending the known correspondence beyond the standard $(2n, n)$ setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16093 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiple chordal SLE($κ$) and quantum Calogero-Moser system Zhang, Jiaxin Probability Mathematical Physics Complex Variables We study multiple chordal SLE$(κ)$ systems in a simply connected domain $Ω$, where $z_1, \ldots, z_n \in \partial Ω$ are boundary starting points and $q \in \partial Ω$ is an additional marked boundary point. As a consequence of the domain Markov property and conformal invariance, we show that the presence of the marked boundary point $q$ gives rise to a natural equivalence relation on partition functions. While these functions are not necessarily conformally covariant, each equivalence class contains a conformally covariant representative. Building on the framework introduced in \cite{Dub07}, we demonstrate that in the $\mathbb{H}$-uniformization with $q = \infty$, the partition functions satisfy both the null vector equations and a dilatation equation with scaling exponent $d$. Using techniques from the Coulomb gas formalism in conformal field theory, we construct two distinct families of solutions, each indexed by a topological link pattern of type $(n, m)$ with $2m \leq n$. In the special case $Ω= \mathbb{H}$ and $q = \infty$, we further show that these partition functions correspond to eigenstates of the quantum Calogero-Moser system, thereby extending the known correspondence beyond the standard $(2n, n)$ setting. |
| title | Multiple chordal SLE($κ$) and quantum Calogero-Moser system |
| topic | Probability Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2505.16093 |