Diagonal boundary conditions in critical loop models

In critical loop models, we define diagonal boundaries as boundaries that couple to diagonal fields only. Using analytic bootstrap methods, we show that diagonal boundaries are characterised by one complex parameter, analogous to the boundary cosmological constant in Liouville theory. We determine disc 1-point functions, and write an explicit formula for disc 2-point functions as infinite combinations of conformal blocks. For a discrete subset of values of the boundary parameter, the boundary spectrum becomes discrete, and made of degenerate representations. In such cases, we check our results by numerically bootstrapping disc 2-point functions. We sketch the interpretation of diagonal boundaries in lattice loop models. In particular, a loop can neither end on a diagonal boundary, nor change weight when it touches it. In bulk-to-boundary OPEs, numbers of legs can be conserved, or increase by even numbers.

💡 Research Summary

In this work the authors investigate boundary conformal field theory (BCFT) for critical loop models, a class of two‑dimensional statistical systems whose continuum limit is described by a CFT. They introduce the notion of diagonal boundary conditions, defined by the requirement that the boundary couples only to diagonal bulk fields (V_{P}) (with momentum (P\in\mathbb{C})) and to the degenerate fields (\psi^{\rm d}{\langle1,s\rangle}). Non‑diagonal boundaries would also allow non‑diagonal bulk fields (V{(r,0)}) to have non‑zero one‑point functions.

The analysis proceeds by applying the analytic bootstrap method, i.e. solving crossing symmetry equations that involve the simplest degenerate bulk field (V^{\rm d}{\langle1,2\rangle}). The bulk‑to‑boundary operator product expansion (OPE) of this field contains two boundary fields with momenta (P{(1,1)}) and (P_{(1,3)}). Imposing the condition that the coefficient (\mu) of the non‑degenerate boundary field vanishes yields the diagonal boundary. This leads to a shift equation for the disc one‑point function (\langle V_{P}\rangle_{\sigma}) which is solved by

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