# Global and Local Multiple SLEs for $\kappa \leq 4$ and Connection Probabilities for Level Lines of GFF

## TL;DR

This paper constructs and verifies pure partition functions for multiple SLE$(ppa)$ with ppa  (0,4], connecting them to crossing probabilities in statistical mechanics and GFF level lines, and proves the equivalence of different SLE constructions.

## Contribution

It provides explicit formulas for multiple SLE partition functions, confirms their relation to extremal measures, and links these functions to GFF level line connection probabilities.

## Key findings

- Constructed pure partition functions for multiple SLE(ppa) with ppa  (0,4]
- Verified the equivalence of global and local SLE constructions
- Connected partition functions to GFF level line connection probabilities

## Abstract

This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple SLE$(\kappa)$ with $\kappa \in (0,4]$ and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from [BBK05, KP16]. We prove that the two approaches to construct multiple SLEs - the global, configurational construction of [KL07, Law09a] and the local, growth process construction of [BBK05, Dub07, Gra07, KP16] - agree. The pure partition functions are closely related to crossing probabilities in critical statistical mechanics models. With explicit formulas in the special case of $\kappa = 4$, we show that these functions give the connection probabilities for the level lines of the Gaussian free field (GFF) with alternating boundary data. We also show that certain functions, known as conformal blocks, give rise to multiple SLE(4) that can be naturally coupled with the GFF with appropriate boundary data.