Verlinde 线、任意子置换与 $E_{8,1}$ CFT 中的共轭对
Verlinde lines, anyon permutations and commutant pairs inside $E_{8,1}$ CFT
AI总结：
本文提出赤道投影框架，通过非负整数矩阵编码缺陷耦合，揭示$E_{8,1}$CFT中拓扑缺陷线的模自洽性，并扩展到三特征共轭对，产生新的非meromorphic理论。
摘要：
我们开发了一种缺陷理论的细化方法，用于完善二维meromorphic CFTs。通常情况下，普通的环面分区函数——往往仅仅是真空特征函数——并不揭示状态如何在对称线下组织。我们的核心提议是一种“赤道投影”框架：从将共轭分解为互易的有理chiral代数的分解中，我们通过非负整数矩阵M来编码单环耦合，该矩阵将特征函数配对并满足模性 intertwining 关系。可逆的拓扑缺陷线直接作用于这种胶合数据（Verlinde线通过S矩阵特征值对角线作用，任意子置换线通过braided-autoequivalence置换），使缺陷幅度的模性自洽性自动成立，并明确区分产生真正模不变量的插入与定义一致非holomorphic界面的插入。我们进一步表明，Hegde等人在2021年提出的“替换规则”作为缺陷作用的赤道投影出现，我们通过系统地处理$E_{8,1}$理论中的三特征共轭对，将这些构造扩展到两特征示例之外。唯一的$c=8$meromorphic CFT $E_{8,1}$作为通用测试场，产生新的缺陷分区函数，并澄清了$\mathrm{Pic}(\mathcal{C})$和$\mathrm{Aut}^{\mathrm{br}}(\mathcal{C})$的作用。最后，我们概述了向更高中心电荷（例如$c=32,40$）的扩展，产生了超越$c=24$ Schellekens景观的非meromorphic理论作为缺陷/界面后代。
英文摘要：
We develop a defect-theoretic refinement of meromorphic 2d CFTs in which the ordinary torus partition function -- often just the vacuum character -- does not reveal how states organize under symmetry lines. Our central proposal is an \emph{equatorial projection} framework: from a commutant decomposition into commuting rational chiral algebras with categories $\mathcal{C}$ and $\widetilde{\mathcal{C}}$, we encode genus-one couplings by a non-negative integer matrix $M$ pairing characters and satisfying modular intertwiner relations. Invertible topological defect lines act directly on this gluing data (Verlinde lines diagonally via $S$-matrix eigenvalues, and anyon-permuting lines by braided-autoequivalence permutations), making modular covariance of defect amplitudes automatic and sharply distinguishing insertions that yield genuine modular invariants from those defining consistent non-holomorphic interfaces. We further show that the \emph{replacement rules} of \cite{Hegde:2021sdm, Lin:2019hks} arise as equatorial projections of defect actions, and we extend these constructions beyond two-character examples by systematically treating three-character commutant pairs in the $E_{8,1}$ theory. The unique $c=8$ meromorphic CFT $E_{8,1}$ serves as a universal testbed, producing new defect partition functions and clarifying the roles of $\mathrm{Pic}(\mathcal{C})$ and $\mathrm{Aut}^{\mathrm{br}}(\mathcal{C})$. Finally, we outline extensions to higher central charges (e.g.\ $c=32,40$), yielding modular-invariant non-meromorphic theories beyond the $c=24$ Schellekens landscape \cite{Schellekens:1992db} as defect/interface descendants of meromorphic parents.
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