Black-Hole States and Heavy Operators

A high-energy boundary state admits a semiclassical black-hole description only for a specified class of observables and under dynamical assumptions stronger than matching its energy. Universal density-of-states growth can establish black-hole thermodynamics for an ensemble; heavy-light correlators can reproduce propagation on a black-hole background in controlled limits. Neither result proves that every individual heavy state has a unique smooth interior. This assessment uses evidence available through 2026-08-20.

Required background. Gauge Theory and String-Theory Preparation supplies large-N counting, operator dimensions, and factorization used below.

Helpful background. Quantum Gravity frames semiclassical geometry, Quantum Information supplies typicality and distinguishability, Black-Hole Geometry and Mechanics identifies the candidate bulk solutions, and Quantum Fields in Curved Spacetime clarifies state-dependent correlators.

Energy, dimension, and the semiclassical window

On the cylinder, a CFT state created by a primary operator of dimension $\Delta_H$ has energy set by $\Delta_H$ relative to the vacuum Casimir energy. For a parity-invariant AdS$_3$/CFT$_2$ pair with $c_L=c_R=c$, the massless nonrotating BTZ threshold is $\Delta_H=c/12$. This is distinct from the order-$c$ Hawking–Page/Cardy window: a nonrotating horizon with radius of order the AdS radius has $\Delta_H-c/12=O(c)$, and sparse-spectrum large-$c$ universality extends the Cardy density down to $\Delta_H\ge c/6$ Hartman, Keller, and Stoica 2014, §§ 2–3. In that regime,

$$S(E) \simeq 2\pi\sqrt{\frac{c}{6}\left(L_0-\frac{c}{24}\right)} +2\pi\sqrt{\frac{c}{6}\left(\bar L_0-\frac{c}{24}\right)}.$$

Here $L_0$ and $\bar L_0$ denote eigenvalues $h,\bar h$, and $c$ is the central charge of each chiral sector, not their sum. For a boundary circle of circumference $2\pi L$, the nonrotating BTZ map makes the two thresholds concrete:

$$EL=\Delta_H-\frac{c}{12} =\frac{c}{12}\frac{r_+^2}{L^2}, \qquad c=\frac{3L}{2G_3}.$$

The massless geometry $r_+=0$ gives $\Delta_H=c/12$, whereas the saddle crossing $r_+=L$ gives $\Delta_H=c/6$. A real horizon and universal dominance in the canonical ensemble are different requirements. The displayed Cardy formula counts states in an energy window; a rotating or fixed-spin application requires its corresponding spectral hypotheses. It does not assign a geometry to each vector. A sparse light spectrum, large central charge, and a window broad enough to contain many states can make canonical and microcanonical predictions agree for sufficiently simple observables. Special BPS states, coherent superpositions, descendants with macroscopic occupation, or states carrying additional conserved charges can violate that typical behavior at the same total energy.

For a light operator $\mathcal O_L$ in a heavy state $|H\rangle$, a useful diagnostic is

$$G_H(t)=\langle H|\mathcal O_L(t)\mathcal O_L(0)|H\rangle.$$

If the heavy-light conformal block exponentiates and the vacuum block dominates in a controlled time and cross-ratio domain, $G_H$ can display an effective thermal scale. That is an observable-level statement. Corrections grow at late times, for sufficiently complex operators, or when nonvacuum blocks are not suppressed.

Results and limitations through 2026-08-20

Evidence or calculation	Controlled result	Limitation or adverse case
Hartman, Keller, and Stoica 2014, §§2–4, pp. 5–24	Modular invariance plus a sufficiently sparse light spectrum gives universal large-c thermodynamics matching thermal AdS and BTZ over declared energy ranges.	The theorem constrains densities and canonical thermodynamics; it does not provide a state-by-state smooth geometry or interior algebra.
Fitzpatrick, Kaplan, and Walters 2015, §§2–4, pp. 4–25	In a semiclassical heavy-light limit, Virasoro vacuum blocks reproduce thermal-form correlators associated with a classical background.	Vacuum-block dominance, operator lightness, cross-ratio domain, and perturbative large-c order are assumptions; special heavy states can retain nonthermal data.
Kraus, Sivaramakrishnan, and Snively 2017, §§2–5, pp. 4–31	With sparseness and large-c factorization, modular constraints make thermal correlators of light operators universal up to controlled corrections.	The result concerns thermal correlators and specified light data, not arbitrary individual eigenstates, complex probes, or nonperturbative interior observables.

The established evidence is strongest for ensemble thermodynamics and restricted correlators in large-c two-dimensional CFTs. It supports a controlled exterior black-hole saddle for those observables. The broader claim that generic holographic CFTs, individual heavy eigenstates, and all interior probes share one universal description remains conjectural and model-dependent.

Competing positions and discriminating tests

One position emphasizes eigenstate typicality: a typical state in a narrow high-energy band should reproduce thermal expectation values for simple observables, with deviations exponentially small in entropy. A stronger position treats the heavy operator itself as defining a smooth geometry, sometimes including an interior. The competing account stresses that code subspaces, observable complexity, conserved charges, and nonperturbative accuracy matter; agreement on a few correlators does not fix an entire bulk spacetime.

The sharp discriminator is not whether $\Delta_H$ is large, but whether a declared observable family satisfies factorization and universality uniformly over the state class. Compare one-point functions and heavy-light correlators at fixed energy. If two orthogonal states have the same energy and simple exterior data but differ for another low-complexity probe, energy alone is insufficient. If differences appear only in exponentially complex or exponentially late probes, a coarse semiclassical description may still be valid within its error tolerance.

An explicit adverse family can be built from heavy states with additional charge or large coherent occupation. Their stress tensor may match an average energy while other one-point functions remain order one. Such states need not be typical in the microcanonical measure and need not resemble an uncharged stationary black hole.

Exterior geometry is not an interior theorem

Bulk reconstruction is approximate on a regulated code subspace. A heavy-state correlator that has quasinormal decay at intermediate times supplies evidence for an exterior horizon response, but finite entropy requires recurrences and a discrete spectrum at sufficiently late times. Inferring a state-independent interior operator also raises complementary-recovery and code-subspace questions treated in the later interior reconstruction page.

This separation resolves an apparent tension. Semiclassical geometry can be an accurate emergent description of a restricted algebra without being an exact microscopic identity. The error must be stated in $1/c$, inverse coupling, exponential entropy, time, and operator complexity. Without those qualifiers, “the heavy state is a black hole” hides the actual claim.

What remains unresolved

It remains open how broad the universal state class is in higher-dimensional interacting holographic CFTs, which low-complexity observables diagnose atypicality most efficiently, and how nonperturbative accuracy changes the state-to-geometry map. Evidence does not establish a unique smooth interior for every state in a Cardy-sized energy band. Nor is there a theorem that energy, large N, and a sparse low spectrum are sufficient for all black-hole observables.

What could change these conclusions

The conclusions could change if a theorem proves heavy-state universality for a substantially larger operator algebra, if an independent nonperturbative calculation reconstructs or rules out a common interior at controlled error, or if an explicit counterexample produces order-one exterior differences within a state class previously claimed to be typical. A new result matters only after its central-charge limit, energy window, conserved sectors, factorization assumptions, and observable complexity are compared with the matrix above.

Exercises

Explain why Cardy growth can establish black-hole entropy scaling without assigning the same smooth geometry to every counted state.

Solution

Cardy growth constrains the number of states in an energy range through modular invariance, so its logarithm can match the BTZ entropy. It does not constrain every matrix element of every light operator in each state. Therefore distinct states can share the same energy and density-of-states contribution while carrying different charges or one-point functions. As a check, an ensemble average can be thermal even when atypical members are not; an additional factorization or typicality argument is needed before inferring common correlators.

References

• Fitzpatrick, A. Liam, Jared Kaplan, and Matthew T. Walters. “Virasoro Conformal Blocks and Thermality from Classical Background Fields.” Journal of High Energy Physics 11 (2015): 200. DOI.
• Hartman, Thomas, Christoph A. Keller, and Bogdan Stoica. “Universal Spectrum of 2d Conformal Field Theory in the Large c Limit.” Journal of High Energy Physics 09 (2014): 118. DOI.
• Kraus, Per, Allic Sivaramakrishnan, and River Snively. “Black Holes from CFT: Universality of Correlators at Large c.” Journal of High Energy Physics 09 (2017): 106. DOI.