Bootstrapping Boundaries and Branes

TL;DR

This work develops a systematic bootstrap for two-dimensional BCFTs by analyzing the annulus partition function with identical boundaries, deriving universal and model-specific bounds on brane tension ($g$-function) under spectral gaps. It combines linear programming and analytic functionals, recasting the annulus problem as a twist-field four-point function and revealing ties to sphere packing and modular bootstrap. The authors obtain universal upper bounds on stable branes for $1\le c<25$, universal lower bounds tied to bulk scalar gaps, and demonstrate exact saturations at $c=8$ and $c=24$ with explicit extremal spectra; they also connect these bounds to end-of-the-world branes in AdS$_3$ gravity and to RCFTs such as $SU(2)_1$, $SU(3)_1$, $(G_2)_1$, $(Spin(8))_1$, $(E_8)_1$, and Monster-based theories. The results illuminate when D-branes are stable, provide diagnostic tools for brane decays, and hint at deep links between BCFT, topological defects, and holography in the large central-charge limit.

Abstract

The study of conformal boundary conditions for two-dimensional conformal field theories (CFTs) has a long history, ranging from the description of impurities in one-dimensional quantum chains to the formulation of D-branes in string theory. Nevertheless, the landscape of conformal boundaries is largely unknown, including in rational CFTs, where the local operator data is completely determined. We initiate a systematic bootstrap study of conformal boundaries in 2d CFTs by investigating the bootstrap equation that arises from the open-closed consistency condition of the annulus partition function with identical boundaries. We find that this deceivingly simple bootstrap equation, when combined with unitarity, leads to surprisingly strong constraints on admissible boundary states. In particular, we derive universal bounds on the tension (boundary entropy) of stable boundary conditions, which provide a rigorous diagnostic for potential D-brane decays. We also find unique solutions to the bootstrap problem of stable branes in a number of rational CFTs. Along the way, we observe a curious connection between the annulus bootstrap and the sphere packing problem, which is a natural extension of previous work on the modular bootstrap. We also derive bounds on the boundary entropy at large central charge. These potentially have implications for end-of-the-world branes in pure gravity on AdS$_3$.

Figures (15)

• Figure 1: The bootstrap constraint \ref{['eq:CardyLewellen']} arising from equating the boundary and bulk OPE of the bulk two-point function $\langle \phi_i(z_1)\phi_j(z_2)\rangle_{\alpha}$.
• Figure 2: The complex plane with intervals $[0,z]$ and $[1,\infty)$ removed is biholomorphic to the open annulus of modulus $t$, where $z=\lambda(i t)$. This allows us to think of the cylinder partition function as a four-point function of twist fields in the plane.
• Figure 3: The overlap of the t-channel initial state with the state $|{\mathcal{P}}_i\rangle\otimes|{\mathcal{P}}_j\rangle\in{\mathcal{H}}_{\alpha\beta}\otimes{\mathcal{H}}_{\alpha\beta}$ is equal to the boundary two-point function $\langle\beta\,{\mathcal{P}}_i\,\alpha\,{\mathcal{P}}_j\,\beta\rangle\sim\delta_{ij}$.
• Figure 4: Upper (red) and lower (black) bounds on the $g$-function for stable boundary conditions in $c=1$ CFTs as a function of the gap in the bulk scalar sector. The dashed blue and magenta lines denote the curves $\Delta_{\rm gap}^{1/4}$ and $2^{-1/2}\Delta_{\rm gap}^{1/4}$, corresponding to the $g$-functions of the Dirichlet boundary conditions on the circle and orbifold branches, respectively. The latter truncates at $\Delta_{\rm gap} = {1\over 8}$ due to the moduli-independent gap in the twisted sector. The green points denote the $g$-function of the stable rational branes in the exceptional orbifold theories.
• Figure 5: Upper (red) and lower (black) bounds on the $g$-function for stable boundary conditions in $c=2$ CFTs as a function of the gap in the bulk scalar sector.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2