AI 中文总结
该研究证明局部实一维模函子的普适性质，将其与普适Liouville作用关联，为共形场论和Schramm-Loewner演化中中心荷的出现提供数学解释，还证明布朗环测度作为SLE典范限制函数的唯一性。
AI 中文摘要
我们证明了局部实一维模函子的一个普适性质，这类函子可视为（无限维）Segal模空间上缝合运算的中心扩张。这类模函子由一个实参数（即其中心荷）表征；我们的结果是复情形下Segal专著[Seg88, Seg04]中“Mumford定理”的类似物。代数上，这类模函子由曲面缝合对的实值上同调类表征。我们将圆盘-圆盘上同调类确定为普适Liouville作用，也称为环Loewner能量。典型例子是实行列式线丛，其编码了共形场论（CFT）的迹反常及Schramm-Loewner演化（SLE）环测度的限制函数。因此，我们的结果为CFT与SLE中中心荷的出现提供了数学解释，并证明了布朗环测度作为SLE的典范限制函数的猜想唯一性。
英文摘要
We prove a universal property of local real one-dimensional modular functors, which may be thought of as central extensions of the sewing operation on the (infinite-dimensional) Segal moduli spaces. Such modular functors are characterized by one real parameter: their central charge: Our result is an analogue of ``Mumford's theorem,'' that appeared in Segal's monograph [Seg88, Seg04] in the complex case. Algebraically, the modular functors are characterized by real-valued cocycles on pairs of surfaces under sewing. We identify the disk-disk cocycle as the universal Liouville action, also known as loop Loewner energy. The guiding example is the real determinant line bundle encoding the trace anomaly of conformal field theories (CFT) and the restriction function of Schramm-Loewner evolution (SLE) loop measures. Our result thus gives a mathematical explanation for the appearance of the central charge in CFT and SLE, and proves the conjectured uniqueness of Brownian loop measureas the canonical restriction function for SLE.
Comments69 pages, 3 figures, comments welcome!
