Conformally covariant boundary correlation functions with a quantum group

Abstract

Particular boundary correlation functions of conformal field theory are needed to answer some questions related to random conformally invariant curves known as Schramm–Loewner evolutions (SLE). In this article, we introduce a correspondence and establish its fundamental properties, which are used in the companion articles [JJK16, KP16] to explicitly solve two such problems. The correspondence associates Coulomb gas type integrals to vectors in a tensor product representation of a quantum group, a $q$-deformation of the Lie algebra ${\mathfrak{sl}}_2$. We show that the desired properties of the functions are guaranteed by natural representation-theoretical properties of the vectors.