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Quantization of conical spaces in 3D gravity

Joris Raeymaekers

TL;DR

The paper studies quantization of conical excess solutions in 2+1D AdS gravity through the Chern–Simons formulation, treating them as topological solitons carrying winding numbers. By quantizing fluctuations along exceptional Virasoro coadjoint orbits in a semiclassical $1/c$ expansion, it finds nonunitary highest-weight Virasoro representations with a level-$s$ null vector, identifying them with the degenerate $(1,s)$ representations for large central charge. The analysis reveals a gap between metric smoothness (which signals singular cones for $s>1$) and CS observables (which are smooth), and shows that the quantum corrections reproduce Kac’s predictions for degenerate representations, suggesting a gravity realization of nonunitary holography and potential links to higher-spin theories. These results illuminate how conical spaces encode nonunitary CFT data, provide a concrete bulk computation of the 1-loop energy, and offer a framework for exploring modular properties and extensions to $W$-algebras in nonunitary holographic setups.

Abstract

We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations which lower the energy. We propose a framework to quantize them in a semiclassical expansion in the inverse central charge, which we use to compute their one-loop corrected energies. Interestingly, the resulting Virasoro representations contain a null vector, thus providing an appearance of Kac's degenerate representations, which are nonunitary at large central charge, in the context of gravity. We match the computed quantum corrections in the bulk with the properties of a class of primaries in Kac's classification.

Quantization of conical spaces in 3D gravity

TL;DR

The paper studies quantization of conical excess solutions in 2+1D AdS gravity through the Chern–Simons formulation, treating them as topological solitons carrying winding numbers. By quantizing fluctuations along exceptional Virasoro coadjoint orbits in a semiclassical expansion, it finds nonunitary highest-weight Virasoro representations with a level- null vector, identifying them with the degenerate representations for large central charge. The analysis reveals a gap between metric smoothness (which signals singular cones for ) and CS observables (which are smooth), and shows that the quantum corrections reproduce Kac’s predictions for degenerate representations, suggesting a gravity realization of nonunitary holography and potential links to higher-spin theories. These results illuminate how conical spaces encode nonunitary CFT data, provide a concrete bulk computation of the 1-loop energy, and offer a framework for exploring modular properties and extensions to -algebras in nonunitary holographic setups.

Abstract

We discuss the quantization and holographic aspects of a class of conical spaces in 2+1 dimensional pure AdS gravity. These appear as topological solitons in the Chern-Simons formulation of the theory and are closely related to the recently studied conical solutions in higher spin gravity. We discuss the classical fluctuations around these solutions, which form exceptional coadjoint orbits of the asymptotic Virasoro group. We argue that the quantization of these solutions leads to nonunitary representations of the Virasoro algebra, on account of their having boundary graviton fluctuations which lower the energy. We propose a framework to quantize them in a semiclassical expansion in the inverse central charge, which we use to compute their one-loop corrected energies. Interestingly, the resulting Virasoro representations contain a null vector, thus providing an appearance of Kac's degenerate representations, which are nonunitary at large central charge, in the context of gravity. We match the computed quantum corrections in the bulk with the properties of a class of primaries in Kac's classification.

Paper Structure

This paper contains 14 sections, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Chern-Simons formulation of 3D AdS gravity
  3. Asymptotic symmetries and coadjoint orbits
  4. Conical spaces as topological Chern-Simons solitons
  5. Winding numbers
  6. Metric-like observables
  7. Holonomies
  8. Quantizing the conical spaces
  9. Symplectic form on coadjoint orbits
  10. Conical spaces and exceptional orbits
  11. Semiclassical expansion
  12. One-loop energy correction
  13. Null states and Kac's degenerate representations
  14. Discussion

Figures (1)

  • Figure 1: The conformal weights $h_{(1,s)}$ for $1 \leq s \leq 6$ as a function of the central charge $c$. The red lines correspond to $c=1$ and $c=25$, between which the conformal weights become complex.