黑洞、贝特 ansatz 与椭圆卡洛杰罗-莫塞系统
Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser Systems
AI总结：
本文定义了四维N=4超杨-米尔斯BAE解到对应椭圆卡洛杰罗-莫塞系统势极值与极点的映射，证明其关联两系统对称性，提出该映射为双射的猜想，用秩-2代数的实例为猜想提供证据。
AI中文摘要：
我们定义了一个映射，将具有任意半单规范代数$\frak{g}$的四维$\boldsymbol{\textit{N}=4}$超杨-米尔斯理论的贝特 ansatz 方程（BAEs）的解，映射到$\frak{g}$型非扭曲椭圆卡洛杰罗-莫塞系统势的极值和极点。我们猜想该映射在卡洛杰罗-莫塞极值的原像上是双射，并证明它使两个系统的对称性相互关联，包括规范对称性（环面、外尔和中心不变性）以及$\boldsymbol{\textit{PSL}(2,\boldsymbol{\textit{Z}})}$作用，因此两侧的解都组织成轨道，每个BAE轨道映射到单个卡洛杰罗-莫塞极值或极点的轨道。所得到的系统为非扭曲系统这一事实，带来一个结果：BAE解与$\boldsymbol{\textit{N}=4}$理论在$\boldsymbol{\textit{R}^{3,1}}$上的$\boldsymbol{\textit{N}=1^\boldsymbol{\textit{*}}}$形变的真空之间的猜想对应关系（后者是扭曲系统的极值），无法扩展到非单链$\frak{g}$。该对应关系在单链情况下也不成立，$\boldsymbol{\textit{su}(N)}$除外：我们展示了一个$\boldsymbol{\textit{so}(8)}$解，它流向卡洛杰罗-莫塞势的极点而非极值，因此没有$\boldsymbol{\textit{N}=1^\boldsymbol{\textit{*}}}$对应物。我们针对所有秩-2的$\frak{g}$（包括经典和例外类型）详细说明了该映射，并将这些情况作为该猜想的证据。
英文摘要：
We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang--Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero--Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero--Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{PSL}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero--Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^\ast$ deformation of $\mathcal{N}=4$ on $\mathbb{R}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{su}(N)$: we exhibit an $\mathfrak{so}(8)$ solution that flows to a pole of the Calogero--Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^\ast$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.
