标题： $c=0$ 三维共形场论中的对数算符 显示英文标题

标题： Logarithmic operators in $c=0$ bulk CFTs

摘要： 我们研究了渗流和自避走（SAW）的bulk共形场论中的Kac算符（例如，能量算符），其中心荷为$c=0$。通过要求团簇模型和环路模型CFT中的三重常数有限且实值，可以推导出这些算符在一般$c$下的适当归一化。在$c=0$处，Kac算符成为零模态态和对数多重态的底场，并且与$c<1$的Liouville CFT比较表明可能存在任意高秩的Jordan块。我们基于Kac算符给出了对数算符的一般构造，并专注于能量算符与hull算符混合的秩-2对。通过取$c\to 0$极限，我们计算了它们的一些共形数据，并用此来研究$c=0$处的算符代数。根据团簇分解，我们发现与先前的观点相反，在$c=0$处，bulk能量算符的四点关联函数并不为零，而与秩-3 Jordan块相关的第二能量算符的耦合起到了关键作用。 这揭示了零范数算符以复杂对数结构在 $c=0$ 体 CFT 中构建长程高点关联的迷人方式。 显示英文摘要

摘要： We study Kac operators (e.g. energy operator) in percolation and self-avoiding walk bulk CFTs with central charge $c=0$. The proper normalizations of these operators can be deduced at generic $c$ by requiring the finiteness and reality of the three-point constants in cluster and loop model CFTs. At $c=0$, Kac operators become zero-norm states and the bottom fields of logarithmic multiplets, and comparison with $c<1$ Liouville CFT suggests the potential existence of arbitrarily high rank Jordan blocks. We give a generic construction of logarithmic operators based on Kac operators and focus on the rank-2 pair of the energy operator mixing with the hull operator. By taking the $c\to 0$ limit, we compute some of their conformal data and use this to investigate the operator algebra at $c=0$. Based on cluster decomposition, we find that, contrary to previous belief, the four-point correlation function of the bulk energy operator does not vanish at $c=0$, and a crucial role is played by its coupling to the rank-3 Jordan block associated with the second energy operator. This reveals the intriguing way zero-norm operators build long-range higher-point correlations through the intricate logarithmic structures in $c=0$ bulk CFTs.