与共形网的真空区相关的零亏格函子型共形场论
A genus zero functorial CFT associated to the vacuum sector of a conformal net
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中文总结 AI 辅助
本文从任意共形网出发，构造零亏格函子型共形场论并证明共形网满足迹类条件，其$L_0$本征空间为有限维。
中文摘要 AI 辅助
从任意共形网出发，我们构造了一个零亏格函子型共形场论，其希尔伯特空间为该共形网的真空区。具体而言，我们针对小圆盘的 operad 和共形嵌入构造了一个代数，取值于希尔伯特空间与有界线性映射构成的范畴。我们采用闭圆盘，且所构造的共形嵌入明确允许入射圆盘的像与出射圆盘的边界重叠。作为应用，我们证明所有共形网均满足迹类条件：若$L_0$为该共形网的共形哈密顿量，则当$0 \le r < 1$时，算子$r^{L_0}$是迹类的；特别地，共形网的$L_0$本征空间自动为有限维。
英文摘要
Starting from an arbitrary conformal net, we construct a genus zero functorial conformal field theory whose Hilbert space is the vacuum sector of the net. Specifically, we construct an algebra for the operad of little discs and conformal embeddings with values in the category of Hilbert spaces and bounded linear maps. We work with closed discs, and our conformal embeddings explicitly allow the image of an incoming disc to overlap with the boundary of the outgoing disc. As an application, we show that all conformal nets satisfy the trace class condition: if $L_0$ is the conformal Hamiltonian of the net, then the operators $r^{L_0}$ are trace class whenever $0 \le r < 1$. In particular, the $L_0$-eigenspaces of a conformal net are automatically finite-dimensional.
