Microscopic Black-Hole Entropy: Claim and Ensemble Contract

A microscopic entropy comparison is meaningful only after the counted quantity, charge sector, ensemble, moduli chamber, supersymmetry, coupling regime, and asymptotic order are fixed. Equality of leading exponentials can be a major result, but it does not imply equality of absolute degeneracies or of subleading corrections.

Required background. Black-Hole Thermodynamics at the QFT Interface supplies the macroscopic entropy and ensemble distinctions; Holographic Duality: Claims, Dictionaries, and Regimes supplies the correspondence criteria.

Helpful background. Thermodynamic Limits, Phases, and Ensemble Equivalence explains when ensemble transforms are controlled; The Witten Index, Vacuum Counting, and Its Failure Modes distinguishes an index from a count.

The quantities being compared

For fixed conserved charges $\Gamma$, an absolute degeneracy $d(\Gamma)$ defines

$$S_{\rm micro}(\Gamma)=\log d(\Gamma).$$

A protected helicity trace instead has signs and insertions,

$$\Omega_k(\Gamma) =\operatorname{Tr}_{\mathcal H_\Gamma} (-1)^{2J_3}(2J_3)^k ,$$

so $\log\lvert\Omega_k\rvert$ equals $\log d$ only when cancellations and multiplet factors are controlled. A grand-canonical generating function

$$Z(\phi)=\sum_\Gamma d(\Gamma)e^{-\phi\cdot\Gamma}$$

requires an inverse transform, including its contour and measure, before it can be compared with a fixed-charge black hole.

Macroscopically, the leading two-derivative entropy is $S_{\rm BH}=A_H/(4G)$ Bekenstein 1973, with its temperature fixed by Hawking radiation Hawking 1975. Higher-derivative theories require the appropriate Noether-charge entropy, and quantum fields add determinant, zero-mode, and ensemble contributions. The comparison must use the same charges and boundary conditions on both sides.

First application: state a reproducible entropy comparison

Before comparing a microscopic formula with a black hole, record:

• the electric, magnetic, angular-momentum, and brane charges $\Gamma$, including their lattice normalization;
• whether the microscopic object is $d$, an index $\Omega_k$, or a partition function;
• microcanonical, canonical, or mixed ensemble and the transform between them;
• the chamber in moduli space and whether single- and multicenter states are separated;
• the protected supersymmetry sector and the interpolation from weak to strong coupling;
• the simultaneous large-charge scaling;
• the retained terms, for example $S_0(\Gamma)+a\log\Lambda+O(\Lambda^{-1})$.

For a scale $\Gamma=\Lambda\hat\Gamma$, a clean claim has the form

$$\log\lvert\Omega_k(\Lambda\hat\Gamma)\rvert =S_{\rm macro}(\Lambda\hat\Gamma) +o\!\left(S_{\rm macro}\right), \qquad \Lambda\to\infty,$$

with the same $\hat\Gamma$, chamber, and ensemble on both sides. Sen’s quantum entropy function gives a fixed-charge AdS₂ path-integral framework for refining the macroscopic side Sen 2005.

Adversarial control: preserve the exponential, change the claim

Suppose $d_+=e^{S_0}(1+\varepsilon)$ and $d_-=e^{S_0}(1-\varepsilon)$ are bosonic and fermionic counts. The absolute degeneracy is $d_++d_-\sim2e^{S_0}$, while the index is $d_+-d_-\sim2\varepsilon e^{S_0}$. For exponentially small $\varepsilon$, their leading entropies differ by an amount of order $S_0$. Similarly, a Gaussian inverse-ensemble transform produces logarithmic determinants that change the coefficient of $\log\Lambda$ without changing the leading area term.

This control shows why leading Cardy or saddle agreement licenses leading asymptotic entropy in a protected sector, not automatic equality of signs, chambers, ensembles, or corrections. A non-BPS or finite-charge extrapolation requires new evidence rather than continuity alone.

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

• Bekenstein, Jacob D. “Black Holes and Entropy.” Physical Review D 7, 2333–2346 (1973). DOI.
• Hawking, Stephen W. “Particle Creation by Black Holes.” Communications in Mathematical Physics 43, 199–220 (1975). DOI.
• Sen, Ashoke. “Black Hole Entropy Function and the Attractor Mechanism in Higher Derivative Gravity.” Journal of High Energy Physics 2005, 9 (2005): 038. DOI. Open PDF.