⚛️ high-energy theory

Large gaps and BTZ entropy in modular spectra with positive integer degeneracies

This paper constructs modular-invariant torus partition functions with positive integer degeneracies and unique vacua that realize large primary dimension gaps and reproduce the Bekenstein-Hawking entropy of BTZ black holes, including all Virasoro descendants and spins, by recursively repairing the Maloney-Witten-Keller modular completion.

Original authors: Chi-Ming Chang, Reiko Liu, Wen-Jie Ma

Published 2026-10-01

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Chi-Ming Chang, Reiko Liu, Wen-Jie Ma

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quest to understand the universe, physicists often look for a bridge between two seemingly different worlds: the smooth, continuous geometry of gravity and the discrete, jittery behavior of quantum particles. In a specific corner of theoretical physics known as the study of three-dimensional gravity, this bridge is called the AdS/CFT correspondence. It suggests that a universe with gravity can be described by a simpler, gravity-free theory living on its boundary. A crucial test of this idea involves black holes. According to the laws of thermodynamics, a black hole has an entropy, a measure of how many different ways its internal parts can be arranged. For a black hole in this three-dimensional setting, known as a BTZ black hole, the entropy is predicted by a simple formula. However, for this prediction to be truly valid, the underlying quantum theory must have a very specific structure: it must contain a unique empty state, a set of distinct energy levels, and the number of ways to reach each level must be a whole, positive number.

For years, a major obstacle stood in the way of proving this connection rigorously. When physicists tried to construct the simplest possible version of this quantum theory using a standard mathematical technique, the result was flawed. The theory produced a continuous smear of energy levels rather than distinct steps, and worse, it predicted negative numbers for the count of certain states. In the physical world, you cannot have negative amounts of anything, so this result meant the theory could not describe a real, consistent universe. The mathematical tool used to generate these results was a "modular completion," a method that ensures the theory behaves correctly under certain transformations, but it left behind these unphysical artifacts. The question remained: could one repair this broken theory to create a valid, discrete spectrum of states that still matched the famous black hole entropy formula?

A team of researchers has now answered this question with a definitive construction. They have developed a new, step-by-step method to fix the broken theory, turning the continuous smear and negative counts into a clean, discrete set of energy levels where every state has a positive, whole-number count. Their work proves that it is possible to build a quantum theory that satisfies all the strict requirements of consistency while still reproducing the exact entropy of a black hole. They did not just find a single example; they created entire families of these theories. In some cases, they forced a large gap between the empty state and the first excited state, ensuring no light particles exist. In other cases, they allowed the first excited state to sit very close to the empty state. In every single case they constructed, the number of states at high energies matched the prediction for a black hole, down to the finest details of the calculation.

The researchers achieved this by treating the flawed theory like a damaged tapestry. They identified the specific regions where the fabric was torn or where the thread count was negative. Instead of trying to patch the whole thing at once, they worked on small, bounded sections of energy. For each section, they calculated exactly what changes were needed to remove the negative parts and replace the continuous smear with distinct, countable points. They then used a clever mathematical trick to ensure that fixing one small section did not ruin the rest of the tapestry. By repeating this process over and over, moving from low energies to higher ones, they gradually replaced the entire continuous spectrum with a discrete one. Crucially, they ensured that at every step, the number of states remained a positive integer, satisfying the basic rules of quantum mechanics.

One of the most striking findings is that this repair process works even in energy ranges where the black hole is not the dominant object in the universe. In the standard view, at lower energies, the universe is expected to be filled with a hot gas of radiation rather than a black hole. The researchers showed that their repaired theories still produce the same entropy formula for black holes in this regime, matching the prediction perfectly. This suggests that the relationship between the quantum states and the black hole geometry is robust and does not depend on the black hole being the only thing present. Furthermore, they found that while the overall count of states matches the black hole prediction, the specific arrangement of the very first few energy levels can vary. This means there is a hidden freedom in the microscopic details of the theory that does not affect the large-scale thermodynamic properties.

The construction also revealed what happens when a large gap is enforced between the empty state and the first excited state. In this scenario, the number of states grows differently just above the gap compared to the standard black hole prediction. There is a specific range of energies where the count of states is exponentially higher than what the simple black hole formula would suggest. This excess is due to the specific way the researchers built the theory, creating a cluster of heavy states right above the gap. However, once the energy gets high enough, the theory settles back down and matches the black hole prediction exactly. This behavior highlights that while the large-scale entropy is universal, the detailed structure of the energy levels can vary, offering a rich landscape of possible quantum theories that all look the same from a distance.

The significance of this work lies in its rigor. The researchers did not rely on approximations or simulations that might miss subtle errors. They provided a mathematical proof that such theories exist and that they possess the necessary properties: a unique vacuum, discrete energy levels, positive integer counts, and exact agreement with the black hole entropy formula. They also showed that these theories are consistent with the symmetries required by the laws of physics. This removes a long-standing doubt about whether a consistent quantum theory of gravity could exist in this setting that also reproduces the famous black hole entropy. It confirms that the black hole entropy formula is not just a lucky guess or a feature of a specific approximation, but a fundamental consequence of the underlying quantum structure, even when that structure is forced to be discrete and free of negative probabilities.

The paper also touches on the broader implications for how we understand the universe. If multiple different quantum theories can all produce the same black hole entropy, it suggests that the macroscopic world we observe might be an average of many different microscopic possibilities. This aligns with ideas that gravity might emerge from an ensemble of theories rather than a single, unique one. The researchers' construction provides a concrete set of examples to test these ideas. They have shown that the "average" behavior of these theories matches the black hole prediction, but the individual members of the family can have different internal structures. This opens the door to asking whether the universe we live in corresponds to one specific member of this family or if it is a statistical average.

In the end, this work transforms a theoretical possibility into a mathematical reality. It takes a flawed, unphysical result and repairs it, piece by piece, until it becomes a valid description of a quantum universe. The process is meticulous, relying on precise calculations to ensure that every correction preserves the delicate balance of the theory. The result is a set of theories that are as consistent as they are complex, proving that the bridge between quantum mechanics and gravity is not only possible but can be built with the strictest of materials. The researchers have not just found a solution; they have mapped out the entire space of possible solutions, showing that the path to a consistent theory of gravity is wide open and filled with diverse, valid possibilities that all lead to the same familiar destination: the entropy of a black hole.

Technical Summary: Large Gaps and BTZ Entropy in Modular Spectra with Positive Integer Degeneracies

Problem Statement
The paper addresses a fundamental challenge in the AdS $_3$ /CFT $_2$ correspondence: constructing a microscopic spectrum for pure gravity that satisfies the rigorous constraints of a unitary, compact two-dimensional conformal field theory (CFT). Specifically, the authors seek to construct genus-one torus partition functions that possess:

A unique vacuum.
Discrete energy levels.
Positive integer degeneracies.
Exact modular invariance.
Large primary dimension gaps (specifically, gaps exceeding the standard $c/12$ threshold or arbitrary fixed shifts).

The starting point for such constructions is often the Maloney–Witten–Keller (MWK) modular completion of the Virasoro vacuum. While MWK is exactly modular invariant, its spectrum is continuous, contains negative spectral densities (particularly near the threshold for large odd spins), and includes a negative scalar contribution at the threshold. These defects prevent its interpretation as a valid partition function for a unitary CFT. Previous approaches, such as conical-defect orbifolds or stringy completions, have addressed parts of this problem but often fail to simultaneously ensure discreteness, integrality, and large gaps while maintaining exact modular invariance.

Methodology
The authors develop a recursive construction that transforms the signed continuous spectrum of the MWK completion into a discrete spectrum with positive integer multiplicities. The methodology relies on three core technical components:

Exact Local Modular Repair:
The authors define a "repair" operation that modifies the spectrum within a bounded energy band $[0, B]$ without violating modular invariance. This involves:
- Inversion: Solving for a "seed" measure whose canonical modular completion produces a specific desired spectral measure $\nu$ in the low-energy band. This requires inverting the operator mapping seeds to their induced low-energy spectra, which is achieved by establishing the strict positivity of a quadratic form associated with the kernel of the modular completion.
- Threshold Anchoring: The modular completion of a seed typically induces a non-zero coefficient at the scalar threshold $(\tilde{E}, J) = (0,0)$ . The authors construct an auxiliary modular function, the "threshold anchor," which has a unit atom at the threshold and zero measure in the low-energy band. Subtracting the appropriate multiple of this anchor cancels the unwanted threshold atom while preserving the prescribed band measure.
Integer Moment Matching:
To replace the continuous measure in a specific energy cell with a discrete set of states, the authors employ a moment-matching strategy. They replace a continuous measure $\mu_{cell}$ with a finite atomic measure $Q$ (a sum of Dirac deltas) such that:
- The total weight of $Q$ is a positive integer.
- $Q$ matches a sufficient number of moments of $\mu_{cell}$ with respect to the variable $u = \sqrt{\tilde{E} - |J|}$ .
- The polynomial inequalities derived from the large- $c$ density estimates ensure that the continuous measure is "positive enough" on the cell to allow for a positive integer atomic replacement that satisfies the moment conditions.
Recursive Discretization and Convergence:
The construction proceeds recursively. Starting from the MWK completion, the authors clear a low-energy band and place the first primary level. They then iteratively select energy cells in spin sectors, replace the continuous measure in those cells with discrete atoms via moment matching, and apply the exact modular repair to correct the spectrum at higher energies.
- Protection: The repair cutoffs are chosen to be strictly higher than the current cell and all previously processed regions, ensuring that earlier constructed levels remain unchanged.
- Convergence: By carefully selecting the cell widths, moment degrees, and repair cutoffs (an "admissible schedule"), the authors prove that the sum of the spectral corrections converges absolutely in the thermal norm for any positive temperature. This guarantees the existence of a limiting modular function with a locally finite, discrete spectrum.

Key Contributions and Results

Existence of Gap Families:
The paper proves the existence of two families of modular-invariant partition functions with large primary gaps for sufficiently large central charge $c$ :
- Macroscopic Gap Family: For a fixed $\kappa > 0$ , the construction yields a spectrum with a primary dimension gap $\Delta_1 = (1+\kappa)c/12$ . This exceeds the standard $c/12$ bound by an extensive amount.
- Fixed Shift Family: For any fixed $\delta \ge 0$ , the construction yields a spectrum with a gap $\Delta_1 = (c-1)/12 + \delta$ . This includes the case $\delta=0$ (scalar uncensored threshold) and $\delta=1/12$ (classical massless BTZ threshold).
BTZ Entropy Matching:
For the fixed-shift family, the authors demonstrate that the smoothed density of states, $N_\phi(E)$ , matches the prediction of a single perturbative BTZ saddle through every fixed finite order in the inverse central charge $1/c$ .
- The expansion of the spectral entropy $S_\phi(E) = \log N_\phi(E)$ reproduces the Bekenstein–Hawking term, the logarithmic correction, and higher-order coefficients derived from the boundary graviton determinant.
- Crucially, this matching holds even in the intermediate energy range $0 < E < c/12$ , where the BTZ saddle is subdominant to thermal AdS in the canonical ensemble. The construction provides direct spectral control in this regime, bypassing the limitations of general sparseness assumptions (Hartman-Keller-Stoica).
Non-Matching Regimes:
For the macroscopic gap family (large $\kappa$ ), the authors show that while BTZ matching is restored at sufficiently high energies ( $E/c > \vartheta/12$ ), there exists an interval immediately above the gap where the smoothed density of states exhibits an exponential excess over the BTZ prediction. This excess arises from primary levels with exponentially large multiplicities generated by the initial discretization.
Formal Verification:
The main theorems regarding the gap families and the entropy expansion have been formalized in Lean 4, providing a machine-checked verification of the mathematical arguments.

Significance and Claims
The paper claims to provide the first rigorous construction of genus-one partition functions that simultaneously satisfy the requirements of discreteness, positive integer degeneracies, and large primary gaps, while maintaining exact modular invariance.

Resolution of MWK Defects: The work demonstrates that the negative densities and continuous spectrum of the MWK completion can be systematically repaired to yield a valid CFT spectrum without introducing light matter or violating unitarity constraints.
Microscopic Realization of BTZ Entropy: The results confirm that the BTZ state-growth relation can be realized in a discrete spectrum with integer multiplicities, including in energy regimes where the bulk saddle is not dominant. This supports the idea that the BTZ entropy is a robust feature of the theory's spectral data, independent of the specific dominance of the saddle in the path integral.
Spectral Freedom: The construction reveals a significant degree of freedom in the microscopic spectra. Different choices of first-level data and atomic replacements lead to distinct partition functions that share the same smoothed BTZ entropy expansion. This suggests that the perturbative entropy expansion does not uniquely determine the individual energy levels or multiplicities, a feature relevant to statistical descriptions of AdS $_3$ gravity and ensemble interpretations.
Limitations: The authors explicitly state that their construction addresses genus-one consistency. They do not claim to have constructed a full CFT with consistent Operator Product Expansion (OPE) coefficients or higher-genus amplitudes. They note that determining which members of their constructed families admit such extensions remains an open problem. Similarly, while the construction resolves the MWK negativity, the specific "correcting seeds" generated do not satisfy the growth assumptions required by the modular ambiguity analysis of Alday and Bae, leaving the precise bulk interpretation of these corrections as an open question.

In summary, the paper establishes the mathematical existence of a broad class of modular spectra that bridge the gap between the continuous MWK completion and the discrete, integer-degenerate spectra required for a microscopic understanding of pure gravity in AdS $_3$ .

Technical Summary: Large Gaps and BTZ Entropy in Modular Spectra with Positive Integer Degeneracies

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