发表机构
Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文将Krauel等人关于仿射顶点算子代数L(k,0)环面1点函数的结果推广到n点函数，针对\udb2a alg{sl}(2)完成分类、构造显式形式族及相关映射类群表示，其S矩阵显式形式仍待解决。
AI 中文摘要
Krauel、Shafiq和Wood实现了由\udb2alg{sl}(2)构造的简单仿射顶点算子代数（VOA）\udb2a L(k,0)的环面1点函数，将其作为与循环R-模结构及模张量范畴\udb2a rep{L(k,0)}相关的向量值模形式。我们将这一结果推广到环面n点函数，其定义为绕τ-圈的n个缠结算子链的迹。在对角层附近的局部（Laurent或Puiseux）展开的每个系数均为向量值模形式（VVMF），只要满足温和的最低权假设，就属于\udb2a KSW的R-模。在环面素性假设（下文针对\udb2a alg{sl}(2)已明确验证）下，算子积展开将短程主导行为归约为已分类的1点理论。针对\udb2a alg{sl}(2)，我们对融合链共形块空间进行分类，确定环面n点本原向量，并详细处理n=2的情况，得到显式的(k+1)维向量值雅可比型形式族，推广了\udb2a KSW的k级形式\udb2a η^{3k/2}。我们还将\udb2a KSW的范畴图景（由模张量范畴构造的\udb2a B_3 = Mod(1,1)的表示）推广到n穿孔环面的映射类子群在融合链块空间上的表示，该子群由S、T及相邻编织生成，具有Verlinde型维数公式和范畴S、T算子。除该抽象构造外，\udb2a alg{sl}(2)的显式S矩阵形式仍待解决。
英文摘要
Krauel, Shafiq and Wood realized torus \(1\)-point functions for the simple affine \voa{}s \(L(k,0)\) built from \(\alg{sl}(2)\) as vector-valued modular forms attached to a cyclic \(R\)-module structure and to the modular tensor category \(\rep{L(k,0)}\). We extend this to torus \(n\)-point functions, defined as traces of chains of \(n\) intertwining operators winding once around the \(τ\)-cycle. Every coefficient of their local (Laurent, or Puiseux) expansion about a diagonal stratum is again a \vvmf{}, lying, whenever a mild lowest-weight hypothesis holds, in the very \(R\)-modules of \cite{KSW}. Granted a torus-primarity hypothesis verified explicitly below for \(\alg{sl}(2)\), the operator product expansion then reduces leading short-distance behaviour to the classified \(1\)-point theory. For \(\alg{sl}(2)\) we classify the fusion-chain conformal block spaces, identify torus \(n\)-point primary vectors, and treat \(n=2\) in detail, obtaining an explicit \((k+1)\)-dimensional family of vector-valued Jacobi-type forms generalizing the level-\(k\) forms \(η^{3k/2}\) of \cite{KSW}. We also extend \KSW's{} categorical picture, representations of \(\Bthree=\PMod(\surf{1}{1})\) built from a modular tensor category, to representations, on fusion-chain block spaces, of a mapping-class subgroup of the \(n\)-punctured torus generated by \(S\), \(T\), and adjacent braidings, with a Verlinde-type dimension formula and categorical \(S\)-, \(T\)-operators. The explicit \(\alg{sl}(2)\) matrix form of \(S\) beyond this abstract construction remains open.
