手征CFT中的局部能量界与强定域性
Local energy bounds and strong locality in chiral CFT
AI总结：
本文针对共形权d>2的手征共形场无法用线性能量界证明强定域性的问题，提出基于d-1次能量界推局部能量界、结合微分同胚对称的新方法，证明W₃代数酉真空表示的顶点算子代数强定域性，构造c>2的新共形网并证明其非完全有理。
摘要：
若一族量子场能生成冯·诺依曼代数的局部网，则称其具有强定域性。目前可直接证明量子场强定域性的方法较少，其中线性能量界应用最广，但共形权$d>2$的手征共形场无法满足线性能量界。本文提出一种证明二维共形场论中强定域性的新直接方法：我们证明，若手征共形场满足次数为$d-1$的能量界，则它也满足某种局部形式的能量界，而这一局部能量界反过来可推出强定域性，微分同胚对称性在证明中发挥了核心作用。作为具体应用，我们证明由$\u202fW_3$-代数的酉真空表示给出的顶点算子代数具有强定域性；当中心荷$c>2$时，这一结果构造出了一类新的共形网。我们进一步证明这些网不满足强可加性，因此不是完全有理的。
英文摘要：
A family of quantum fields is said to be strongly local if it generates a local net of von Neumann algebras. There are few methods of showing directly strong locality of a quantum field. Among them, linear energy bounds are the most widely used, yet a chiral conformal field of conformal weight $d>2$ cannot admit linear energy bounds. In this paper we give a new direct method to prove strong locality in two-dimensional conformal field theory. We prove that if a chiral conformal field satisfies an energy bound of degree $d-1$, then it also satisfies a certain local version of the energy bound, and this in turn implies strong locality. A central role in our proof is played by diffeomorphism symmetry. As a concrete application, we show that the vertex operator algebra given by a unitary vacuum representation of the $\mathcal{W}_3$-algebra is strongly local. For central charge $c > 2$, this yields a new conformal net. We further prove that these nets do not satisfy strong additivity, and hence are not completely rational.
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