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Multiple backward Schramm–Loewner evolution and coupling with Gaussian free field
Letters in Mathematical Physics ( IF 1.2 ) Pub Date : 2021-03-05 , DOI: 10.1007/s11005-021-01374-5
Shinji Koshida

It is known that a backward Schramm–Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition function, which is determined by a system of Belavin–Polyakov–Zamolodchikov-like equations so that these Loewner chains are commutative. After this observation, we define a multiple backward SLE as a tuple of mutually commutative backward Loewner chains. It immediately follows that each backward Loewner chain in a multiple backward SLE is obtained as a Girsanov transform of a backward SLE. We also discuss coupling of a multiple backward SLE with a GFF with boundary perturbation and find that a partition function and a boundary perturbation are uniquely determined so that they are coupled with each other.



中文翻译:

多重后向Schramm-Loewner演化以及与高斯自由场的耦合

众所周知,向后的Schramm-Loewner演化(SLE)与具有边界扰动的自由边界高斯自由场(GFF)耦合,可以进行量子表面的保形焊接。通过对具有多个标记边界点的量子表面进行共形焊接的推广,我们提出了多个向后SLE的概念。为此,我们研究了两个反向Loewner链之间的换向关系,因此,我们发现每个反向Loewner链的驱动过程必须具有由分配函数的对数导数给定的漂移项,该漂移项由系统确定类Belavin–Polyakov–Zamolodchikov方程组,以便这些Loewner链是可交换的。经过这一观察,我们将多重后向SLE定义为相互可交换的后向Loewner链的元组。紧随其后的是,获得了多个反向SLE中的每个反向Loewner链,作为反向SLE的Girsanov变换。我们还讨论了多向后SLE与具有边界扰动的GFF的耦合,发现分配函数和边界扰动是唯一确定的,因此它们彼此耦合。

更新日期:2021-03-07
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