Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States
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2026
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| _version_ | 1866915756486164480 |
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| author | Krishnan, Chethan Pathak, Pradipta S. |
| author_facet | Krishnan, Chethan Pathak, Pradipta S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18775 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States Krishnan, Chethan Pathak, Pradipta S. High Energy Physics - Theory 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. |
| title | Holography with an Inner Boundary: A Smooth Horizon as a Sum over Horizonless States |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2601.18775 |