二维CFT不同数学定义的比较
Comparing Different Mathematical Definitions of 2D CFT
摘要：
第一部分介绍了维拉索罗代数的表示理论、Wightman公理和顶点代数。第二部分是对上述定义的比较。我们给出了Luescher和Mack的一个证明，即一个具有能量动量张量的伸缩不变的二维QFT产生了维拉索罗代数的两个可交换的酉表示。在最后一章中，我们比较了Wightman QFT和顶点代数。我们提出了卡茨的证明，每一个Wightman Moebius CFT（一个二维Wightman QFT包含准素场）产生两个交换强生成的正能量Moebius共形顶点代数。如果每个共形权的生成域的个数是有限的，则这些顶点代数是酉拟顶点算子代数。作为一个推论使用Luescher-Mack定理，我们得到一个Wightman CFT（Wightman Moebius CFT与能量动量张量）产生一个共形顶点代数，进而成为一个酉顶点算子代数，如果每个共形权的生成域的数量是有限的。我们扭转Kac的论点，并得到一个逆证明，两个酉（拟）顶点算子代数可以结合起来，给一个Wightman（Moebius）CFT。
英文摘要：
We give introductions into the representation theory of the Virasoro algebra, Wightman axioms and vertex algebras in the first part. In the second part, we compare the above definitions. We give a proof of Luescher and Mack that a dilation invariant 2D QFT with an energy-momentum tensor gives rise to two commuting unitary representations of the Virasoro algebra. In the last chapter we compare Wightman QFTs to vertex algebras. We present Kac's proof that every Wightman Moebius CFT (a 2D Wightman QFT containing quasiprimary fields) gives rise to two commuting strongly-generated positive-energy Moebius conformal vertex algebras. If the number of generating fields of each conformal weight is finite, then these vertex algebras are unitary quasi-vertex operator algebras. As a corollary using Luescher--Mack's Theorem we obtain that a Wightman CFT (a Wightman Moebius CFT with an energy-momentum tensor) gives rise to a conformal vertex algebra which furthermore becomes a unitary vertex operator algebra, if the number of generating fields of each conformal weight is finite. We reverse Kac's arguments and get a converse proof that two unitary (quasi)-vertex operator algebras can be combined to give a Wightman (Moebius) CFT.
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