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Annales de l’Institut Henri Poincaré D

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Volume 6, Issue 3, 2019, pp. 449–487
DOI: 10.4171/AIHPD/88

Published online: 2019-04-09

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

Alex Karrila[1], Kalle Kytölä[2] and Eveliina Peltola[3]

(1) Aalto University, Finland
(2) Aalto University, Finland
(3) Université de Genève, Switzerland

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.

Keywords: Conformal blocks, conformal field theory (CFT), Dyck tilings, multiple SLEs, quantum group representations, $q$-combinatorics

Karrila Alex, Kytölä Kalle, Peltola Eveliina: Conformal blocks, $q$-combinatorics, and quantum group symmetry. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), 449-487. doi: 10.4171/AIHPD/88