SLE$(\kappa)$ describes a curve growing in time describing the interface of many statistical models that fall within conformal field theory. The research problem here is to construct a family of loops using the Beltrami equation that might be related to the SLE-loop.

In the works [1] and [2] this was achieved for small enough $\kappa\leq \kappa_{0}$.

The SLE-loop has already being constructed in various works: [3], that constructed an SLE loop measure using a two-sided whole-plane SLE but also proved a criterion for uniqueness up to a constant to unify the various constructions; Sheffield and Werner CLE measure [4], Kemppainen and Werner CLE's Mobius-invariance [5], Miller and Schoug CLE for $\kappa\in (8/3, 8) $ [6], Fields and Lawler "SLE loops rooted at an interior point" , Benoist, S., Dubédat, J.: "Building $SLE_\kappa$ loop measures for $\kappa < 4$". Also, there was the construction in [8][9] by Morris, Holden, and Sun, who showed in ([theorem 1.3])[8] that the construction of SLE loops based on the quantum zipper satisfies Zhan's uniqueness.

The difference of the above with the approach in the works [1] and [2] is that here the construction starts from proving the existence of the welding using the standard theory of Beltrami equations (see [11] and [10]). So one is still left with the task of proving a relation to SLE. However, it provides a complementary approach that is more analytic in nature and thus can provide a new perspective on many of the achievements in imaginary geometry.

The research problem is to extend the existence and uniqueness of the conformal welding over the entire range $\kappa\in [0,4]$, and thus including the critical case which will likely be the hardest.

ArticleIs an originInverse of the Gaussian multiplicative chaos: Lehto welding of Independent Quantum disks

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