A QFT for non-semisimple TQFT

IF 1 4区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Creutzig,Thomas, Dimofte,Tudor, Garner,Niklas, Geer,Nathan
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引用次数: 0

Abstract

We construct a family of 3d quantum field theories ${\mathcal T}_{n,k}^A$ that conjecturally provide a physical realization --- and derived generalization --- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(i\pi/k)$. The theories ${\mathcal T}_{n,k}^A$ are defined as topological twists of certain 3d ${\mathcal N}=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted ${\mathcal N}=4$ matter sector (the source of non-semisimplicity). We show that ${\mathcal T}_{n,k}^A$ admits holomorphic boundary conditions supporting two different logarithmic vertex operator algebras, one of which is an $\mathfrak{sl}_n$-type Feigin-Tipunin algebra; and we conjecture that these two vertex operator algebras are related by a novel logarithmic level-rank duality. (We perform detailed computations to support the conjecture.) We thus relate the category of line operators in ${\mathcal T}_{n,k}^A$ to the derived category of modules for a boundary Feigin-Tipunin algebra, and --- using a logarithmic Kazhdan-Lusztig-like correspondence that has been established for $n=2$ and expected for general $n$ --- to the derived category of $U_q(\mathfrak{sl}_n)$ modules. We analyze many other key features of ${\mathcal T}_{n,k}^A$ and match them from quantum-group and VOA perspectives, including deformations by flat $PGL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.
非半简 TQFT 的 QFT
我们构建了一族 3d 量子场理论 ${\mathcal T}_{n,k}^A$,猜想这些理论提供了非半复数数学 TQFT 的物理实现--和派生泛化--基于量子群 $U_q(\mathfrak{sl}_n)$ 在偶数统一根处的模块 $q=\text{exp}(i\pi/k)$。理论 ${\mathcal T}_{n,k}^A$ 被定义为某些 3d ${\mathcal N}=4$ Chern-Simons 物质理论的拓扑扭曲,这些理论也允许弦/理论实现。它们可以被看作是 $SU(n)_{k-n}$ Chern-Simons 理论,与一个扭曲的 ${mathcal N}=4$ 物质部门耦合(非符号简约性的来源)。我们证明 ${\mathcal T}_{n,k}^A$ 允许支持两种不同对数顶点算子代数的全态边界条件,其中一个是 $\mathfrak{sl}_n$ 类型的费金-提普宁代数;我们猜想这两个顶点算子代数是通过一种新颖的对数级数对偶性联系在一起的。(因此,我们将 ${mathcal T}_{n,k}^A$ 中的线算子范畴与边界费金-提普宁代数的派生模块范畴联系起来,并--利用针对 $n=2$ 已建立并有望针对一般 $n$ 建立的类似对数卡兹丹-卢兹蒂格的对应关系--与 $U_q(\mathfrak{sl}_n)$ 模块的派生范畴联系起来。我们分析了${\mathcal T}_{n,k}^A$的许多其他关键特征,并从量子群和VOA的角度将它们匹配起来,包括平面$PGL(n,\mathbb C)$ 连接的变形、单形式对称性和(派生)属$g$状态空间的指数。
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来源期刊
Advances in Theoretical and Mathematical Physics
Advances in Theoretical and Mathematical Physics 物理-物理:粒子与场物理
CiteScore
2.20
自引率
6.70%
发文量
0
审稿时长
>12 weeks
期刊介绍: Advances in Theoretical and Mathematical Physics is a bimonthly publication of the International Press, publishing papers on all areas in which theoretical physics and mathematics interact with each other.
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