Quantum black hole cohomologies

Technical Summary: Quantum Black Hole Cohomologies

Problem Statement
The microstates of BPS black holes in $AdS_5 \times S^5$ have been extensively studied using the classical cohomologies of $\mathcal{N}=4$ super-Yang-Mills (SYM) theory, specifically focusing on $1/16$-BPS operators. A prevailing assumption in this program is that the spectrum of classical cohomologies (defined by the tree-level supercharge $Q_0$) accurately represents the full quantum spectrum of strictly protected states. However, this conjecture lacks rigorous verification. Recent work in the $SO(7)$ gauge theory demonstrated that certain classical cohomologies acquire anomalous dimensions at the 1-loop level, effectively "lifting" them from the protected spectrum. This paper investigates whether such quantum lifts occur in the simpler $SU(2)$ gauge theory, aiming to determine the extent to which classical cohomologies overestimate the true quantum BPS spectrum.

Methodology
The authors analyze the perturbative quantum corrections to the supercharge cohomologies in $SU(2)$ $\mathcal{N}=4$ SYM. The supercharge is expanded as $Q = Q_0 + g_{YM}^2 Q_1 + \dots$, where $Q_0$ is the classical supercharge and $Q_1$ represents the 1-loop correction.

A classical cohomology $O_0$ (satisfying $Q_0 O_0 = 0$) lifts at the 1-loop level if it fails to remain closed under the full quantum supercharge. This occurs via two mechanisms:

1. Non-exactness: $Q_1 O_0$ is not $Q_0$-exact (i.e., $Q_1 O_0 \neq Q_0 \Lambda$ for any $\Lambda$).
2. Exactness: $O_0$ becomes $Q$-exact at 1-loop, meaning $O_0 = Q_1 \Lambda_0 + Q_0 \Lambda_1$ for some classical cohomology $\Lambda_0$.

The authors focus on the first mechanism, which is sufficient for operators in the "BMN subsector" (where a specific quantum number $d_{BMN} = 0$). To determine if $Q_1 O_0$ is $Q_0$-exact, the authors employ a rigorous computational strategy:

• Basis Construction: They construct a basis for the space of potential pre-images $\Lambda$ that satisfy the required charge constraints (R-charges, angular momenta, and field number $Y$). They utilize two methods: matching only Cartan charges and matching full $SU(3)_R$ representation singlets.
• Linear Algebra Formulation: The problem of checking if $Q_1 O_0$ lies in the image of $Q_0$ is converted into a linear algebra problem ($A\vec{x} = \vec{b}$).
• Finite Field Arithmetic: To handle the massive size of the coefficient matrices and avoid floating-point errors or integer overflow, the authors perform Gaussian elimination over finite fields $\mathbb{Z}_p$ using large primes ($2^{31}-1$, etc.). They cross-check results using multiple primes and rational reconstruction to ensure accuracy.

Key Results
The study examines specific classical cohomologies in the $SU(2)$ theory, categorized into "threshold," "hairy," and "core" fortuitous cohomologies:

1. Threshold and Hairy Cohomologies (Unlifted):

   • The threshold fortuitous cohomology $O_0$ (the lightest black hole state) is found to be unlifted at 1-loop. $Q_1 O_0$ is explicitly shown to be $Q_0$-exact.
   • "Hairy" black hole cohomologies, constructed as products of $O_0$ with graviton operators (e.g., $w_2 O_0$, $O'_1$, $O''_1$), are also found to be unlifted at 1-loop.
   • The authors note that $O_0$ is unique in its charge sector ($j=24, Y=7$) within the $SU(2)$ theory, preventing it from pairing with a graviton state (which requires an even number of fields in $SU(2)$), unlike the pairing observed in $SO(7)$.

2. Heavier Core Cohomologies (Lifted):

   • The heavier "core" fortuitous cohomologies $O_n$ (for $n=1, 2, 3$) are found to be lifted at 1-loop.
   • For these states, $Q_1 O_n$ is not $Q_0$-exact. Consequently, they acquire anomalous dimensions and pair with other classical cohomologies to form massive supermultiplets.
   • A specific analysis of $O_1$ reveals that its quantum superpartner $Q_1 O_1$ splits into two distinct cohomologies: $O''_1$ (which is unlifted) and a new state $\tilde{O}_1$. This results in a partial cancellation in the index at $j=36$, where only a subset of the classical states remain protected.

3. Entropy Estimation:

   • The authors compare the Cardy limit entropy of the classical cohomologies (refined by a bonus $U(1)$ symmetry counting the number of fields $Y$) against the indicial entropy of strictly protected states.
   • They find that the maximum entropy of the refined classical index exceeds the protected index entropy by approximately 1.2%.
   • This implies that while the classical cohomologies overestimate the true BPS spectrum, the overestimation is macroscopically small but non-zero. Specifically, the fraction of surviving classical cohomologies at large charges is estimated to be $\sim e^{-0.012 S}$.

Significance and Claims
The paper claims to provide the first explicit demonstration of quantum lifts in the $SU(2)$ theory, confirming that the assumption of classical cohomologies representing the full quantum spectrum is not universally true.

• Protection Mechanism: The results suggest a hierarchy of protection: the lightest fortuitous cohomology ($O_0$) and its "hairy" descendants are robust against 1-loop corrections, while heavier "core" states are susceptible to lifting.
• Entropy Correction: The calculation of the $\approx 1.2\%$ entropy discrepancy provides a quantitative lower bound on the overestimation of the BPS spectrum by classical methods. The authors argue that this small discrepancy suggests that most classical cohomologies at large charges are lifted, leaving only a small fraction of the classical states as true quantum BPS microstates.
• Methodological Contribution: The work establishes a rigorous computational framework using finite field linear algebra to resolve the $Q_0$-exactness problem, a task previously considered a bottleneck in the field.

The authors remain modest regarding higher-order effects, noting that further lifts are possible at higher loops, non-perturbative levels, or finite coupling, but their 1-loop analysis establishes that the classical spectrum is not identical to the quantum one.