Conformal blocks, $q$-combinatorics, and quantum group symmetry

TL;DR

This paper introduces a $q$-analogue of Fomin's formulas linking conformal block functions in conformal field theory to multiple SLE partition functions, using quantum group methods to reveal their algebraic structure.

Contribution

It develops a $q$-deformation of Fomin's formulas and constructs conformal block functions via quantum group representation theory, connecting combinatorics, probability, and algebra.

Findings

Established a $q$-analogue of Fomin's formulas

Connected conformal blocks to multiple SLE partition functions

Provided a quantum group-based construction of conformal blocks

Abstract

In this article, we find a $q$-analogue for Fomin's formulas. The original Fomin's formulas relate determinants of random walk excursion kernels to loop-erased random walk partition functions, and our formulas analogously relate conformal block functions of conformal field theories to pure partition functions of multiple SLE random curves. We also provide a construction of the conformal block functions by a method based on a quantum group, the $q$-deformation of $\mathfrak{sl}_2$. The construction both highlights the representation theoretic origin of conformal block functions and explains the appearance of $q$-combinatorial formulas.