The Bootstrap Program for Boundary CFT_d

TL;DR

The paper extends the conformal bootstrap to BCFTs, formulating boundary crossing constraints for scalar and tensor two-point functions and showing both analytic (ε-expansion) and numerical (linear programming) routes to constrain operator spectra and OPE data. Analytically, it derives exact BCFT solutions at free theory and one-loop order in ε, and numerically, it obtains universal bounds on bulk gaps and boundary operator dimensions, with detailed Ising-model benchmarks. The work highlights the BCFT framework as a tractable arena for bounding bulk dynamics via boundary data, especially for tensor operators, while also underscoring positivity assumptions and the need for further theoretical justification of bulk-block positivity. Overall, the study opens quantitative avenues to connect boundary phenomena with bulk CFT data across dimensions, including the Ising model near boundary transitions.

Abstract

We study the constraints of crossing symmetry and unitarity for conformal field theories in the presence of a boundary, with a focus on the Ising model in various dimensions. We show that an analytic approach to the bootstrap is feasible for free-field theory and at one loop in the epsilon expansion, but more generally one has to resort to numerical methods. Using the recently developed linear programming techniques we find several interesting bounds for operator dimensions and OPE coefficients and comment on their physical relevance. We also show that the "boundary bootstrap" can be easily applied to correlation functions of tensorial operators and study the stress tensor as an example. In the appendices we present conformal block decompositions of a variety of physically interesting correlation functions.

Figures (12)

• Figure 1: Two-point function crossing symmetry in boundary CFT.
• Figure 2: Phase diagram for the surface critical behavior of the Ising model in dimension $2 < d <4$. Temperature is plotted on the horizontal axis and the (relative) surface interaction strength on the vertical axis. The extraordinary transition disappears for $d=4$, while the special transition is absent in $d=2$.
• Figure 3: Upper bound for the first boundary operator in the special transition.
• Figure 4: Improved bound for the first boundary operator in the special transition. The bulk spectrum is assumed to satisfy $\Delta_\text{bulk} \ge 2 \Delta_{\text{ext}}$.
• Figure 5: Upper bound for the dimension of the second boundary operator in $\langle \sigma \sigma \rangle$ as a function of the dimension of the first boundary operator.