Conical Defects, Orbifolds, and Heavy States

Conical defects interpolate between global AdS3 and the massless BTZ threshold and provide a geometric model for heavy CFT states below black-hole formation. A quotient geometry, an orbifold theory, and an individual heavy primary are related only under additional assumptions: the quotient fixes local geometry, the orbifold requires projected and twisted sectors, and a primary represents a state whose correlators need not be captured by one classical metric.

Required background. AdS3/CFT2 and the Brown–Henneaux Central Charge supplies the charge normalization, and Heavy States, Coherent States, and Semiclassical Geometries supplies the state-to-geometry criteria.

Helpful background. Duality Operations: Gauging, Quotients, and Orbifolds supplies the full orbifold operation, while BTZ Black Holes and Modular CFT Thermodynamics supplies the threshold and high-energy regime.

Deficit angle, energy, and conformal weight

A static locally AdS3 conical defect can be parameterized by $0<\alpha\le1$, with angular deficit

$$\delta=2\pi(1-\alpha).$$

In conventions where global AdS has $M=-1/(8G_3)$ and the massless BTZ solution has $M=0$, the defect family obeys

$$M=-\frac{\alpha^2}{8G_3}, \qquad \Delta-\frac{c}{12}=\ell M, \qquad \Delta=\frac{c}{12}(1-\alpha^2)$$

for a spinless state with $\Delta=h+\bar h$. Thus $\alpha=1$ is the vacuum and $\alpha\to0$ approaches the massless BTZ threshold $\Delta=c/12$. Above threshold the corresponding uniformization parameter becomes imaginary and the classical identification is of BTZ type rather than a real angular deficit.

For $\alpha=1/N$, the geometry is the quotient AdS3/$\mathbb Z_N$. But a consistent CFT orbifold is not obtained by keeping only invariant states of an untwisted sector: modular invariance requires twisted sectors as well. Those sectors can carry states with no counterpart in a naive point-particle description.

Heavy-state probes

In a heavy–light correlator, the heavy primary fixes the leading stress-tensor expectation value. The uniformizing coordinate $w=z^\alpha$ then makes a light probe behave as though it propagated in the defect background. This is a controlled leading large-$c$ statement when $h_H/c$ is fixed, $h_L/c\to0$, and the relevant vacuum block dominates.

First application. Map a heavy primary below the BTZ threshold to a conical defect and compare its probe correlator with the quotient geometry. Determine $\alpha$ from $h_H+\bar h_H$, compute the light two-point function in the uniformizing coordinate, and compare its image singularities with geodesics in the quotient. State whether the primary is assumed spinless and whether the state is one microstate, an average, or an orbifold sector.

Backreaction and completion

As the probe weight grows, its own backreaction invalidates propagation in a fixed defect. At finite $c$, stress-tensor one-point data do not determine all higher correlators, and exact heavy states with the same energy can have different OPE data. Crossing the BTZ threshold changes the classical conjugacy class, but does not prove that every above-threshold primary has all correlators of a smooth black-hole geometry.

Adversarial control. First omit twisted sectors from a $\mathbb Z_N$ orbifold and test modular invariance; the failure shows that a geometric quotient is not a complete orbifold CFT. Second continue the real-deficit formula past $\alpha=0$ without changing branches; its geometry and probe periodicity become inconsistent. These tests locate the boundary between a useful semiclassical map and a complete state-spectrum claim.

Evidence ceiling

The relation $\Delta=c(1-\alpha^2)/12$ and the heavy–light uniformization provide controlled leading dictionaries for specified states and probes. They do not identify all defect quotients with exact CFT primaries, guarantee a consistent orbifold without twisted sectors, or turn an individual heavy state into an exact thermal ensemble.

The heavy-state semiclassical block analysis used to diagnose backreacted conical or black-hole backgrounds is developed in Fitzpatrick, Kaplan, and Walters 2015; a string orbifold sector carries additional twisted-state data Dixon et al. 1985.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Bañados, Máximo, Claudio Teitelboim, and Jorge Zanelli. “The Black Hole in Three-Dimensional Spacetime.” Physical Review Letters 69 (1992): 1849–1851. DOI; Open PDF.
• Dixon, Lance, Jeffrey Harvey, Cumrun Vafa, and Edward Witten. “Strings on Orbifolds.” Nuclear Physics B 261 (1985): 678–686. DOI.
• Fitzpatrick, A. Liam, Jared Kaplan, and Matthew T. Walters. “Virasoro Conformal Blocks and Thermality from Classical Background Fields.” Journal of High Energy Physics 2015, no. 11 (2015): 200. DOI; Open PDF.