Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States

TL;DR

The paper offers a bulk CS construction for the BTZ partition function as the S-transform of the Virasoro vacuum character, realized via two boundaries: an outer AdS$_3$ boundary and an inner stretched-horizon boundary created by excising a Wilson line. Through Drinfel'd-Sokolov gauge and Alekseev-Shatashvili boundary dynamics, primaries are associated with holonomies at the inner boundary while descendants arise from boundary gravitons, yielding a primary-vs-descendant split that reproduces Cardy entropy. A key insight is that the BTZ character emerges as a sum over hyperbolic (horizonless) bulk states when the dual cycle label lies in the exceptional orbit, revealing that smooth horizons can be regarded as emergent from a bulk sum over unsmooth microstates. The approach provides a concrete bulk realization of modular transformations, with the bulk Fourier transform between holonomy bases matching the CFT modular S-kernel. While specialized to AdS$_3$/CFT$_2$, the results illuminate how inner boundary data and horizon microstructure can underpin holographic entropy and horizon smoothness, suggesting extensions to higher dimensions and richer symmetry algebras.

Abstract

The (holomorphic) partition function of the Euclidean BTZ black hole with boundary modulus $τ$, is the $S$-image of the Virasoro vacuum character, $χ_{\rm vac}(-1/τ)$. This object decomposes into primaries via the modular $S$-kernel: $χ_{\rm vac}\left(-\frac{1}τ\right)=\int_{0}^{\infty} dP S_{0P}(P,c)χ_P(τ)$. In this paper, we provide a bulk understanding of this spectral resolution using the Chern-Simons formulation of AdS$_3$ gravity with $two$ boundaries: an asymptotic torus and an excised Wilson line at the origin ("stretched horizon"). At infinity, we impose standard AdS$_3$ Drinfel'd-Sokolov (DS) gauge to obtain the Alekseev-Shatashvili (AS) boundary action for a coadjoint orbit. At the inner boundary, removing the Wilson line prepares the state at the cut as a sum over orbits of the $spatial$ cycle. Re-inserting a spatial holonomy Wilson line acts as a delta-function projector onto the corresponding primary, which together with boundary gravitons, reproduces the Virasoro character (e.g., of a conical defect). But we can also consider projectors onto the $conjugate$ basis $\tilde P$, of the dual cycle. A key observation is that this leads to $S$-kernels instead of delta functions, with the BTZ character arising when the dual cycle label is in the exceptional orbit. Our two-boundary construction provides a bulk understanding of BTZ entropy: holonomy zero modes at the horizon have an effective central charge $c_{\rm prim}=c-1$ from the kernel measure (primaries), while the universal Dedekind-$η$ in $χ_P(τ)$ contributes $c_{\rm desc}=1$ from boundary gravitons (descendants). Together, they reproduce the full Cardy entropy. While our methods are specific to AdS$_3$/CFT$_2$, they are an explicit illustration that smoothness of the (Euclidean) horizon may emerge from a $sum$ over bulk states which are manifestly unsmooth.

Figures (3)

• Figure 1: The thickened torus $T^2 \times I$, a "solid torus with a solid torus removed".
• Figure 2: The topology of the janus $T^2 \times I$ geometry. The left end is the inner boundary, and projecting on to a primary corresponds to viewing the left torus as the boundary of an inner solid torus with a (not necessarily smoothly) contractible spatial cycle.
• Figure 3: Projecting on to a conjugate cycle orbit corresponds to viewing the left torus as the boundary of an inner solid torus with a contractible conjugate cycle. The conjugate label in the exceptional orbit corresponds to the BTZ case: thermal cycle is smoothly contractible.