Commutation relation and BPZ equations
In 1999, O. Schramm introduced Schramm–Loewner evolution (SLE) as a non-self-crossing random curve driven by a multiple of Brownian motion using Loewner’s transform. This definition is motivated by a quest to mathematically describe the random interfaces in 2D critical lattice models, which satisfy the conformal invariance and domain Markov property. In this talk, we will consider the law of multiple curves with conformal invariance and domain Markov property, following the framework of Dub´ edat’s commutation relation.
Tosolve Dubédat’s commutation relation,one needs to classify solutions to Belavin–Polyakov–Zamolodchikov (BPZ) equations. There are various works related to the classification of solutions to BPZ equations in the chordal setting, which we will review in this talk. BPZ equations in the radial setting are less explored. We will present recent results about classification of BPZ equations in the radial setting and explain its connection to the random-cluster model and Gaussian free field.
This talk is based on the following works:
- E. Krusell, Y. Wang, H. Wu. Commutation relations for two-sided radial SLE. arXiv:2405.07082. 2024.
- Y. Feng, H. Wu. Radial BPZ equations and partition functions of FK-Ising interfaces conditional on one-arm event. arXiv:2411.16051. 2024.