AI 中文总结
本文求解贝特 Ansatz 方程，获得秩2规范代数下$\boldsymbol{\textit{N}=4}$超杨-米尔斯超共形指标的解，发现非A类规范代数的新解及相关现象，阐明Cardy展开有限对数修正的贝特起源。
AI 中文摘要
我们求解贝特 Ansatz 方程（BAEs），以计算四维$\boldsymbol{\textit{N}=4}$超杨-米尔斯理论的超共形指标，该理论具有秩为2的规范代数，且角动量 fugacity 满足$p=q$。$A_2$型的解是已知的，而$D_2\boldsymbol{\bigcong}A_1\boldsymbol{\bigoplus}A_1$型的解可轻易从同样已知的$A_1$型解导出。首个真正全新的情况是$B_2\boldsymbol{\bigcong}C_2$型，我们获得了其完整的闭式解集合；对于例外情况$G_2$，我们的结果为数值解，其中有一个完全有理的解以闭式形式得到。据我们所知，这是首次在非A类规范代数（针对$\boldsymbol{\textit{N}=4}$理论或任何其他$\boldsymbol{\textit{N}=1}$理论）中找到任何解析或数值解。在此过程中，我们发现了若干在A类中不存在的现象：对指标有非平凡贡献的孤立外尔固定解；最多仅有一半完整同伦具有有理系数的孤立解（与A类完全有理的Hong-Liu族形成对比）；以及当$|\boldsymbol{\textit{\boldsymbol{\u03c9}}|\to0$（其中$p=q=:e^{2\boldsymbol{\u03c0}i\boldsymbol{\u03c9}}$）时，其极限避开了Cardy类极限文献中的标准假设，落在了常规分析未捕捉到的鞍点上，适用于所有BCD类型。我们还阐明了Cardy展开有限对数修正的贝特起源：它源于BAE解在方程规范对称性下的轨道大小，而非源于理论的一形式中心对称性，该中心对称性对指标不敏感。
英文摘要
We solve the Bethe Ansatz equations (BAEs) to evaluate the superconformal index of four-dimensional $\mathcal{N}=4$ super-Yang--Mills with rank-two gauge algebra, at equal angular momentum fugacities $p=q$. The solutions for $A_2$ are known, and those for $D_2\cong A_1\oplus A_1$ follow readily from the (equally known) $A_1$ ones. The first genuinely new case is $B_2\cong C_2$, for which we obtain the complete solution set in closed form; for the exceptional case $G_2$ our results are numerical, with a single fully rational solution obtained in closed form. To the best of our knowledge, this is the first time any solution, analytic or numerical, has been found in non-$A$-type gauge algebra (for $\mathcal{N}=4$ or any other $\mathcal{N}=1$ theory). Along the way we uncover several phenomena absent in type $A$: isolated Weyl-fixed solutions contributing nontrivially to the index; isolated solutions in which at most half of the holonomies have rational coefficients (in contrast with the $A$-type fully rational Hong--Liu family); and solutions whose $|ω|\to0$ limit (with $p=q=:e^{2πiω}$) evades assumptions standardly made in the Cardy-like limit literature, landing on saddles that the usual analysis does not capture, for all types $BC\!D$. We also clarify the Bethe origin of the finite logarithmic correction to the Cardy expansion: it arises from the orbit size of a BAE solution under the gauge symmetries of the equations, rather than from the one-form center symmetry of the theory, to which the index is insensitive.
Commentsv1: 55+15 pages, 2 appendices, 10 tables, 8 figures, 1 ancillary Mathematica file with B2=C2 solutions
