## On currents in the $O(n)$ loop model

Jesper Lykke Jacobsen, Rongvoram Nivesvivat, Hubert Saleur

SciPost Phys. 16, 111 (2024) · published 24 April 2024

- doi: 10.21468/SciPostPhys.16.4.111
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### Abstract

Using methods from the conformal bootstrap, we study the properties of Noether currents in the critical $O(n)$ loop model. We confirm that they do not give rise to a Kac-Moody algebra (for $n≠ 2$), a result expected from the underlying lack of unitarity. By studying four-point functions in detail, we fully determine the current-current OPEs, and thus obtain several structure constants with physical meaning. We find in particular that the terms $:\!J\bar{J}\!:$ in the identity and adjoint channels vanish exactly, invalidating the argument made in \[Nucl. Phys. B 419, 411 (1994)] that adding orientation-dependent interactions to the model should lead to continuously varying exponents in self-avoiding walks. We also determine the residue of the identity channel in the $JJ$ two-point function, finding that it coincides both with the result of a transfer-matrix computation for an orientation-dependent correlation function in the lattice model, and with an earlier Coulomb gas computation of Cardy [Nucl. Phys. B 265, 409 (1986)]. This is, to our knowledge, one of the first instances where the Coulomb gas formalism and the bootstrap can be successfully compared.

### Authors / Affiliations: mappings to Contributors and Organizations

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^{1}^{2}^{3}Jesper Lykke Jacobsen, -
^{4}Rongvoram Nivesvivat, -
^{2}^{5}Hubert Saleur

^{1}Sorbonne Université / Sorbonne University^{2}L'Institut de physique théorique [IPhT]^{3}Laboratoire de Physique de l’École Normale Supérieure / Physics Laboratory of the École Normale Supérieure [LPENS]^{4}Yau Mathematical Sciences Center [YMSC]^{5}University of Southern California [USC]