发表机构
University of Alberta; Institute for Advanced Study（阿尔伯塔大学; 高等研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文严格分类了有理c=1顶点算子代数与顶点算子超代数，证实其为秩1整格格VO(S)A或其有限群自同构的orbifold，为后续完整CFT分类奠定基础。
AI 中文摘要
面向数学家：在本系列两篇论文的第一篇中，我们对（足够好的）c=1顶点算子代数（VOA）与顶点算子超代数（VOSA）给出了数学上严格的分类。我们证实了相关共识：任何此类VO(S)A要么是与秩1整格L相关的格VO(S)A V_L，要么可通过其orbifold得到，即对某个有限自同构群G，取其G不变子代数V_L^G。所有此类G均为已知，可实现对良好c=1 VO(S)A的显式枚举。我们方法的关键要素是建立良好VOA的通用判据，该判据仅需真空特征标知识，用于检验具有整数共形维数的单模是否张成一个对称融合子范畴，该子范畴与Rep(G)是辫张量等价的。在我们的姊妹篇论文中，我们计算了良好c=1 VOA表示范畴的 ribbon自等价，并利用此结果得到了良好玻色型与费米型c=1共形场论（CFT）的分类。面向物理学家：我们对可在玻色型或费米型有理c=1共形场论（CFT）全纯部分中出现的手征代数给出严格分类，该CFT的非本原初级场均具有正共形维数。将手征代数视为3维拓扑量子场论（TQFT）的无能隙边界条件，我们的结果表明，任何此类手征代数要么是U(1)_k Chern-Simons理论的全纯边界，要么可通过对其取某个有限对称群G的不变态得到。我们对这些手征代数进行了显式枚举，并讨论了它们的非可逆对称性。在姊妹篇论文中，我们利用3维TQFT研究的技术，基于这些结果对完整的玻色型与费米型有理c=1 CFT进行了分类。
英文摘要
For mathematicians: In this first in a series of two papers, we give a mathematically rigorous classification of (sufficiently nice) $c=1$ vertex operator algebras (VOAs) and vertex operator superalgebras (VOSAs). We confirm the lore that any such VO(S)A is either a lattice VO(S)A $V_L$ associated to a rank-1 integral lattice $L$, or can be obtained as an orbifold thereof, i.e. a $G$-invariant subalgebra $V_L^G$ for some finite group $G$ of automorphisms. All such $G$ are known, allowing for an explicit enumeration of nice $c=1$ VO(S)As. A key ingredient in our approach is to establish a general criterion for nice VOAs, requiring knowledge only of the vacuum character, for testing when the simple modules with integer conformal dimension span a symmetric fusion subcategory which is braided tensor equivalent to $Rep(G)$. In our companion paper, we calculate the ribbon auto-equivalences of the representation categories of the nice $c=1$ VOAs, and leverage this to obtain the classification of nice bosonic and fermionic $c=1$ full conformal field theories. For physicists: We rigorously classify the chiral algebras that can arise in the holomorphic sector of a bosonic or fermionic rational $c=1$ conformal field theory (CFT) whose non-identity primaries all have positive conformal dimension. Thinking of chiral algebras as gapless boundary conditions of 3D topological quantum field theories (TQFTs), our result says that any such chiral algebra is either a holomorphic boundary of $U(1)_k$ Chern-Simons theory, or can be obtained by passing to the $G$-invariant states thereof for some finite group $G$ of symmetries. We explicitly enumerate these chiral algebras and also discuss their non-invertible symmetries. In a companion paper, we build on these results using techniques from the study of 3D TQFTs to classify full bosonic and fermionic rational $c=1$ CFTs.
Comments36 pages + appendices
