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

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$(κ)$ 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 $κ\in (8/3,8)$, while they may have zeroes for $κ\le 8/3$. 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 $κ= 8/3$ and $κ= 8$, in accordance with logarithmic minimal models $M(2,1)$ and $M(2,3)$, respectively. Second, we construct $\SLE_κ$ pure partition functions and show that they are real-analytic in $κ\in (0,8)$ and decay to zero as a polynomial of $(8-κ)$ as $κ\to 8$. 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$(κ)$ measures ($κ\in (4,8)$) uniquely determined by their re-sampling property.