Spectrum of End of the World Branes in Holographic BCFTs

TL;DR

The paper analyzes overlaps of regularized BCFT boundary states (Cardy states) in holographic BCFTs and shows that off‑diagonal overlaps are exponentially suppressed by the lowest open‑string energy, with the AdS$_3$/BCFT$_2$ result $h^{(min)}_{ab}=\frac{c}{24}$ implying boundary states act as random left–right symmetric microstates of a single‑sided black hole. Using open–closed duality and on‑shell gravity actions, it connects these overlaps to the open‑string spectrum and to the boundary entropies, concluding that the number of relevant boundary microstates scales as $e^{S_{BS}}$ with $S_{BS}=\frac{\pi^2 c}{3\beta}$, i.e. a square‑root subset of all black hole microstates. The work generalizes to higher dimensions where $h^{(min)}_{a,b}$ becomes tension‑dependent, reflecting the role of end‑of‑the‑world brane tensions in the open‑string spectrum. It also contrasts the AdS$_3$/BCFT results with JT gravity, highlighting that gravity calculations in AdS$_3$/BCFT can yield nonzero off‑diagonal overlaps through disconnected brane configurations, with implications for how gravity encodes CFT microstates and for island/ensemble considerations.

Abstract

We study overlaps between two regularized boundary states in conformal field theories. Regularized boundary states are dual to end of the world branes in an AdS black hole via the AdS/BCFT. Thus they can be regarded as microstates of a single sided black hole. Owing to the open-closed duality, such an overlap between two different regularized boundary states is exponentially suppressed as $\langle ψ_{a} | ψ_{b} \rangle \sim e^{-O(h^{(min)}_{ab})}$, where $h^{(min)}_{ab}$ is the lowest energy of open strings which connect two different boundaries $a$ and $b$. Our gravity dual analysis leads to $h^{(min)}_{ab} = c/24$ for a pure AdS$_3$ gravity. This shows that a holographic boundary state is a random vector among all left-right symmetric states, whose number is given by a square root of the number of all black hole microstates. We also perform a similar computation in higher dimensions, and find that $h^{( min)}_{ab}$ depends on the tensions of the branes. In our analysis of holographic boundary states, the off diagonal elements of the inner products can be computed directly from on-shell gravity actions, as opposed to earlier calculations of inner products of microstates in two dimensional gravity.

Figures (4)

• Figure 1: A sketch of cylinder amplitude from the viewpoint of closed string (left) and open string (right).
• Figure 2: A sketch of gravity duals of a CFT on a cylinder in the connected phase (left) and disconnected phase (right) at $d=2$.
• Figure 3: Gravity evaluations of the inner products of microstates $\langle \phi_i|\phi_j\rangle$ in AdS$_2$ gravity (left) and those of regularized boundary states $\langle \psi_a|\psi_b\rangle$ in AdS$_3$ gravity (right).
• Figure 4: Gravity evaluations of the averages of squares of inner products in AdS$_2$ gravity (left) and AdS$_3$ gravity (right).