来自量子群表示的导出范畴的三维拓扑量子场论
3-dimensional TQFTs from derived categories of quantum group representations
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中文总结 AI 辅助
该研究证明有限模张量范畴A的导出∞-范畴D(A)支持三维拓扑量子场论，其无标记版本产生射影映射类群作用，相关导出场论可作为N=4超对称QFT拓扑A扭曲的数学形式化。
中文摘要 AI 辅助
对于任意有限模张量范畴A，我们证明其关联的导出∞-范畴D(A)支持三维拓扑量子场论（TQFT）。该TQFT呈现为从带D(A)中对象标记的曲面及适当配边构成的∞-范畴到dg向量空间∞-范畴的对称幺半函子。我们证明该理论中的态空间可自然等同于D(A)的线性化映射空间。尽管我们的TQFT需要导出∞-范畴的标记，但我们证明在同伦截断后可移除所有标记。得到的无标记TQFT在二维上产生映射类群的射影作用，特别是在霍奇上同调上产生射影SL₂(ℤ)作用。我们预期这些映射类群作用可复现Lentner等人arXiv:2003.06527与Schweigert-Woike arXiv:2004.14343的结果。在三维中，我们得到幂级数值的纽结不变量和模张量范畴的幂级数值不变量，还得到闭三维流形的幂级数不变量，尽管这些已可在阿贝尔层面计算。我们的导出场论被提出作为某些N=4超对称量子场论的拓扑A扭曲的数学形式化，遵循物理原理，我们讨论了通过朗兰兹对偶群沿量子群表示的类似（猜想）变形，沿局部系统对TQFT进行变形的可能性。
英文摘要
For any finite modular tensor category A, we show that the associated derived $\infty$-category D(A) supports a topological quantum field theory in dimension 3. This TQFT takes the form of a symmetric monoidal functor from an $\infty$-category of surfaces with markings by objects in D(A), and appropriately decorated bordisms, to the $\infty$-category of dg vector spaces. We show that the state spaces in this theory are naturally identified with linearized mapping spaces for D(A). Though our TQFT requires markings from the derived $\infty$-category, we show that all markings can be removed after taking a homotopy truncation. The resulting unmarked TQFT produces projective mapping class group actions on cohomology in dimension 2, and in particular a projective SL_2(Z)-action on Hochschild cohomology. We expect these mapping class group actions to recover those of Lentner et al. arxiv:2003.06527 and Schweigert-Woike arxiv:2004.14343. In dimension 3 we obtain power-series valued knot invariants, and power-series valued invariants for modular tensor categories. We also obtain power-series invariants for closed 3-manifold, though these can already be calculated at the abelian level. Our derived field theories are proposed as mathematical formalizations for topological A-twists of certain N=4 supersymmetric QFTs, in dimension 3. Following physical principles, we discuss the possibility of deforming our TQFTs along local systems via an analogous (conjectural) deformation of quantum group representations along the Langlands dual group.
