Multiple SLEs for $\kappa\in (0,8)$: Coulomb gas integrals and pure partition functions

TL;DR

This paper explicitly relates SLE partition functions to Coulomb gas formalism, constructs pure partition functions, and explores their properties and asymptotics across the parameter range.

Contribution

It introduces a Coulomb gas integral construction of SLE partition functions and relates them to pure partition functions, providing new insights into their properties and measures.

Findings

Partition functions are positive for   8/3, may have zeroes for   8/3.

Partition functions admit Frobenius series expansion matching CFT algebraic content.

Pure partition functions are real-analytic in   8 and decay polynomially as   8.

Abstract

In this article, we give an explicit relationship of SLE partition functions with Coulomb gas formalism of conformal field theory. We first construct a family of SLE$(\kappa)$ partition functions as Coulomb gas integrals and derive their various properties. In accordance with an interpretation as probabilistic correlations in loop $O(n)$ models, they are always positive when $\kappa \in (8/3,8)$, while they may have zeroes for $\kappa \le 8/3$. They also admit a Frobenius series expansion that matches with the algebraic content from CFT. Moreover, we check that at the first level of fusion, they have logarithmic asymptotic behavior when $\kappa = 8/3$ and $\kappa = 8$, in accordance with logarithmic minimal models $M(2,1)$ and $M(2,3)$, respectively. Second, we construct $\SLE_\kappa$ pure partition functions and show that they are real-analytic in $\kappa \in (0,8)$ and decay to zero as a polynomial of $(8-\kappa)$ as $\kappa \to 8$. We explicitly relate the Coulomb gas integrals and pure partition functions together in terms of the meander matrix. As a by-product, our results yield a construction of global non-simple multiple chordal SLE$(\kappa)$ measures ($\kappa \in (4,8)$) uniquely determined by their re-sampling property.