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

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.

arXiv comment24 pages, 7 figures. v3: minor improvements. Accepted for publication in Annales de l'Institut Henri Poincaré D

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• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Commutation relations for SLE
• Boundary Partitions in Trees and Dimers
• SLE martingales and the Virasoro algebra
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Cited by in corpus (7)

• Boundary correlations in planar LERW and UST
• Large deviations of multichordal SLE${}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• Bijections on Dyck tilings: DTS/DTR bijections, Dyck tableaux and tree-like tableaux
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA
• Dyck tilings of type $D$
• Planar UST Branches and $c=-2$ Degenerate Boundary Correlations