Diagonal fields in critical loop models

Abstract

In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\le 1$. We study their $N$-point functions, which depend on the $2^{N-1}$ weights of combinatorially inequivalent loops on a sphere with $N$ punctures. Using a numerical conformal bootstrap approach, we find that $4$-point functions decompose into infinite but discrete linear combinations of conformal blocks. We conclude that diagonal fields belong to an extension of the $O(n)$ model.

9 pages, v3: corrected factorization results

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