标题： 一个关于非半单TQFT的量子场论 显示英文标题

标题： A QFT for non-semisimple TQFT

摘要： 我们构造了一类三维量子场论家族$\mathcal T_{n,k}^A$，据推测它们提供了对基于量子群$U_q(\mathfrak{sl}_n)$在偶数单位根$q=\text{exp}(i\pi/k)$下模的非半单数学TQFT的物理实现及其推广。 这些理论$\mathcal T_{n,k}^A$被定义为某些三维$\mathcal N=4$Chern-Simons-物质理论的拓扑扭转，这些理论也具有弦/M理论的实现。 它们可以被看作是$SU(n)_{k-n}$Chern-Simons理论，耦合到一个扭曲的$\mathcal N=4$物质 sector（非半单性的来源）。 我们证明了$\mathcal T_{n,k}^A$支持两种不同的对数型顶点算子代数的全纯边界条件，其中一种是$\mathfrak{sl}_n$型的 Feigin-Tipunin 代数；并且我们推测这两个顶点算子代数之间存在一种新的对数型等级-秩对偶关系。（我们进行了详细的计算以支持这一推测。） 因此，我们将$\mathcal T_{n,k}^A$中的线算符范畴与边界 Feigin-Tipunin 代数的导出模范畴联系起来，并且——利用对于$n=2$已经建立并且对一般情形下的$n$预期成立的类似对数型 Kazhdan-Lusztig 对应关系——与$U_q(\mathfrak{sl}_n)$模的导出范畴联系起来。 我们从量子群和VOA的角度分析了$\mathcal T_{n,k}^A$的许多其他关键特征，并将其与之匹配，包括通过平坦的$PSL(n,\mathbb C)$连接的变形、单形式对称性和（导出的）亏格为$g$的状态空间的指标。 显示英文摘要

摘要： 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 $PSL(n,\mathbb C)$ connections, one-form symmetries, and indices of (derived) genus-$g$ state spaces.