Conformally covariant boundary correlation functions with a quantum group

TL;DR

This paper introduces a correspondence linking boundary correlation functions in conformal field theory to quantum group representations, aiding in solving problems related to Schramm-Loewner evolutions.

Contribution

It establishes a fundamental correspondence between Coulomb gas integrals and quantum group tensor product vectors, enabling explicit solutions for boundary correlation functions.

Findings

Established a correspondence between Coulomb gas integrals and quantum group representations.

Proved that properties of boundary correlation functions follow from representation theory.

Provided a framework for solving problems related to SLE using quantum groups.

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] for explicitly solving 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 sl2. We show that desired properties of the functions are guaranteed by natural representation theoretical properties of the vectors.