Bootstrapping boundary-localized interactions

TL;DR

This work investigates conformal boundary conditions for a free real scalar using the conformal bootstrap. The bulk equation of motion enforces a shadow-pair structure for boundary operators, with dimensions \\widehat{\\Delta}_1=\\frac{d}{2}-1 and \\widehat{\\Delta}_2=\\frac{d}{2} where the sum equals the boundary dimension, and yields shadow-consistent three-point data. The authors derive a complete set of exact relations connecting boundary OPE data and bulk-to-boundary couplings, and formulate crossing equations for a mixed system of boundary modes. Numerical bootstrap in 4d/3d reveals a large allowed region but uncovers a sharp kink near a_{\\phi^2} \\approx 0.215, corresponding to a spin-2 operator with \\widehat{\\Delta}_{\\tau} \\approx 3.966 and a small boundary central charge C_D \\approx 0.0050, suggesting a potential new local conformal boundary condition for the free scalar. These results tightly constrain the BCFT landscape for the free scalar and hint at richer boundary dynamics beyond Dirichlet and Neumann, meriting further analytic and numerical exploration.

Abstract

We study conformal boundary conditions for the theory of a single real scalar to investigate whether the known Dirichlet and Neumann conditions are the only possibilities. For this free bulk theory there are strong restrictions on the possible boundary dynamics. In particular, we find that the bulk-to-boundary operator expansion of the bulk field involves at most a `shadow pair' of boundary fields, irrespective of the conformal boundary condition. We numerically analyze the four-point crossing equations for this shadow pair in the case of a three-dimensional boundary (so a four-dimensional scalar field) and find that large ranges of parameter space are excluded. However a `kink' in the numerical bounds obeys all our consistency checks and might be an indication of a new conformal boundary condition.

Figures (5)

• Figure 1: A plot showing the upper bound on the dimension of ${\widehat{\varepsilon}}$, the first scalar, other than the identity, seen by any of the OPEs in our correlator system. The unshaded region is the one that follows from a single correlator $\left < \widehat{O}_1\widehat{O}_1\widehat{O}_1\widehat{O}_1 \right >$. The pink region, which is more restrictive, uses the multi-correlator system but the only inputs it uses from the exact relations are the odd-spin operator dimensions given in table \ref{['tab:selprottower']}. The blue region, more restrictive again, follows from a genuine use of the exact relations. Since these depend on $a_{\phi^2}$, we have extremized $\widehat{\Delta}_{\widehat{\varepsilon}}$ over a third axis which is not shown.
• Figure 2: Bounds on the dimension of the leading spin 2 operator $\widehat{\tau}$ over the range $-\frac{1}{4} < a_{\phi^2} < \frac{1}{4}$ with our best estimate for the allowed region shaded in blue. The curves have $n_{\mathrm{max}} = 5, 6, 7, 8$ in the notation of Nakayama:2016Cappelli:2019. As for the number of derivative components being kept in each crossing equation, these correspond to $21, 28, 36, 45$ respectively. The dotted line shows the maximum possible value for $\widehat{\Delta}_{\widehat{\tau}}$ from leading order conformal perturbation theory under the assumption that the Ising model is the 3d CFT with the lowest central charge.
• Figure 3: Six allowed regions for the OPE-space vector of the unit-normalized displacement. The dotted line shows the physical locus for $\hat{\lambda}_{11\text{D}}$ and $\hat{\lambda}_{22\text{D}}$, i.e. \ref{['displRel']} divided by $\sqrt{C_{\text{D}}}$. When this line becomes vertical (defining a unique $\hat{\lambda}_{11\text{D}}$ in order for $\hat{\lambda}_{22\text{D}}$ to be finite), it saturates our bound. This does not quite happen in the opposite limit of the line becoming horizontal. Note that in the GFF example there are two candidates for the displacement. Both $[\widehat{O}_1 \widehat{O}_1]_{1, 0}$ and $[\widehat{O}_2 \widehat{O}_2]_{0, 0}$ are compatible with these bounds if we treat them as different operators that satisfy $\hat{\lambda}_{11\text{D}}\hat{\lambda}_{22\text{D}} = 0$.
• Figure 4: An asymmetric plot showing the minimum and maximum $C_{\text{D}}$ as a function of $a_{\phi^2}$. In the blue region, all dimension 4 scalars not singled out by \ref{['addd']} are constrained to satisfy $b_{T\text{D}^\prime} = 0$. No such constraint is made in the pink region which leads to weaker bounds. The dotted lines give the predictions of conformal perturbation theory which are model-independent at leading order. A slight kink in the upper right corner looks well positioned to be identified with the kink in figure \ref{['TvsA_4d']}.
• Figure 5: The maximum possible $\widehat{\Delta}_{\widehat{\tau}}$ for several points in the most interesting region of figure \ref{['CvsA_4d']}. Planes are inserted below points with the same value of $a_{\phi^2}$ for visibility. The red point has its spectrum shown in the left columns in table \ref{['CDTab']}.