Supersymmetric Charge Constraints on AdS Black Holes from Free Fields

Abstract: Supersymmetric AdS black hole solutions exist only when their angular momenta and charges satisfy a certain constraint that depends on the dimension. We show that these nonlinear relations on the conserved charges agree with a computation in the dual supersymmetric CFT in its free limit, with interactions entering only through a uniform rescaling of all charges. Our computations apply to the highly non-trivial charge constraints for AdS$_4$, AdS$_5$ and AdS$_7$ black holes, and generalize an earlier one for the analogous constraint in AdS$_3$. Our results suggest a microscopic understanding of AdS black holes beyond the scope of supersymmetric indices.

• The paper shows that supersymmetric charge constraints on AdS black holes emerge microscopically at the free field level in dual SCFTs.
• It employs fully refined grand canonical partition functions to compute statistical averages of macroscopic charges across diverse AdS dimensions.
• The findings match gravitational charge constraints through symmetry and combinatorial arguments, highlighting uniform rescaling from interactions.

Supersymmetric Charge Constraints on AdS Black Holes from Free Fields

Introduction and Motivation

The paper addresses the intricate class of nonlinear constraints satisfied by the conserved charges (angular momenta and R-charges) of supersymmetric black holes in asymptotically AdS spacetimes. These constraints are universally observed in explicit supergravity solutions across dimensions (AdS$_4$, AdS$_5$, AdS$_7$), but their microscopic origin in the dual SCFTs has remained obscure. Notably, these constraints involve nonlinear relations among the charges, extending far beyond the linear BPS bounds and lying outside the reach of protected supersymmetric indices, which by construction average over directions associated with preserved supercharges and become independent of corresponding chemical potentials.

The central objective of this work is to demonstrate that these nonlinear charge constraints can be microscopically realized already at the free field level in the dual SCFTs by considering appropriately defined supersymmetric ensembles. The authors show that the precise functional forms of these constraints match the corresponding gravitational constraints up to a coupling-dependent overall rescaling, with interactions only uniformly rescaling the charges. The analysis extends earlier results from AdS$_3$ black holes to higher dimensions, providing generic formulas and explicit combinatorial arguments for AdS$_4$, AdS$_5$, and AdS$_7$. The approach goes beyond the supersymmetric index formalism and employs fully refined grand canonical partition functions for BPS letters in free superconformal multiplets.

Framework and Methodology

The bulk-side existence of supersymmetric AdS black holes is limited by nonlinear constraints on their charges, derivable from demanding regularity and causality of the geometry (e.g., absence of closed timelike curves and smooth horizons). Previous field-theoretic approaches, relying primarily on supersymmetric indices, cannot recover these constraints, as the index becomes insensitive in the direction conjugate to the preserved supercharge. Specifically, by fixing chemical potentials so that microstates and their images under the preserved supercharge are counted with equal (opposite) sign, the index erases all dependence on the corresponding charge.

To overcome this limitation, the authors define a supersymmetric grand canonical ensemble—distinct from the index—where microstates along the direction of the preserved supercharge ${\cal Q}$ are given equal statistical weight. This is achieved by imposing a linear constraint among the chemical potentials (fugacities): states differing only by the quantum numbers of ${\cal Q}$ are assigned equal weights. As a result, the statistical averages for macroscopic charges are over ensembles restricted to charge configurations that saturate the supercharge constraint.

The detailed prescription is as follows:

• Identify the superconformal algebra, multiplet structure, and BPS letter content in the free field theory. List all free fields, their quantum numbers, BPS descendants (created by derivatives), and equations of motion that impose relations among fields.
• Construct the fully refined single-particle grand canonical partition function $Z_\text{sp}$, keeping all chemical potentials, including those invisible to the supersymmetric index.
• Obtain the full ensemble partition function by exponentiation, corresponding to the large-$N$ (classical statistics) limit valid for black hole configurations.
• Compute macroscopic charges as statistical averages, i.e., derivatives of $\log Z$ with respect to appropriate chemical potentials.
• Impose the supersymmetric ensemble constraint (a linear relation among fugacities/chemical potentials set by the quantum numbers of ${\cal Q}$).
• Extract the resulting nonlinear constraint satisfied by the ensemble-averaged charges, and compare its structure with the gravitational result.

This analysis is performed explicitly in detail for AdS$_3$/CFT$_2$, AdS$_4$/CFT$_3$, AdS$_5$/CFT$_4$, and AdS$_7$/CFT$_6$.

Core Results and Strong Numerical Claims

Applying the above prescription, the authors establish:

• For AdS$_5$/CFT$_4$ (4d $\mathcal{N}=4$ SYM), the supersymmetric black hole charge constraint

$$\left( Q_1 Q_2 Q_3 + \frac{N^2}{2} J_1 J_2 \right) - \left(Q_1+Q_2+Q_3+\frac{N^2}{2}\right) \left(Q_1 Q_2 + Q_2 Q_3 + Q_3 Q_1 - \frac{N^2}{2}(J_1+J_2)\right)=0$$

is exactly reproduced by consideration of the statistical ensemble of BPS letters in the free vector multiplet, after imposing the supersymmetric ensemble constraint on the fugacities [(2405.17648), eq. (5dcc),(5dccmicro)]. All functional dependence on charges is matched, with only a uniform normalization discrepancy, interpretable as the effective number of accessible degrees of freedom at strong coupling.

• For AdS$_4$/CFT$_3$ (3d $\mathcal{N}=8$), the analogous highly nonlinear constraint among four charges and angular momentum is recovered from statistical counting in the free hypermultiplet, again matching the gravity result modulo a global rescaling [(2405.17648), eq. (4dcc), (4dccmicro)].
• For AdS$_7$/CFT$_6$ ($6d$ $(2,0)$ theory), the equally complex charge constraint is reproduced by the free field prescription involving the tensor multiplet [(2405.17648), eq. (7dcc),(7dccmicro)].
• Importantly, in all cases the nonlinear functional form of the charge constraint is already manifest in the free theory, with interactions entering only as a uniform rescaling that sets the overall number of effective degrees of freedom relative to Newton's constant.
• In AdS$_3$/CFT$_2$, the result is exact, with no normalization ambiguity, due to the additional rigidity imposed by the super-Virasoro algebra and spectral flow [(2405.17648), §3]. The supersymmetric ensemble is naturally motivated by character theory.

These results suggest the supersymmetric charge constraints are consequences of symmetry and representation structure, not strong coupling dynamics or gravitational interactions per se. The analysis sidesteps the blind spot of the supersymmetric index and accesses data invisible to the index (e.g., dependence on chemical potentials conjugate to all conserved charges).

Implications and Future Directions

From a holographic standpoint, these findings are significant for several reasons:

• The charge constraints, previously regarded as emergent gravitational phenomena, are demonstrated to have simple combinatorial and group-theoretic origins in dual CFTs, with nontrivial functional forms.
• The matching determines Newton's constant or, equivalently, the effective number of free fields required, up to a uniform factor. The observed reduction in the number of effective degrees of freedom at strong coupling compared to the free theory is consistent with established results (e.g., $3/4$ factor in D3-brane entropy [Gubser:1996de]).
• The work uncovers a route to understanding black hole microstate counting beyond the limitations of supersymmetric indices, which could be crucial for ongoing efforts to compute fine-grained spectra and subleading corrections (quantum entropy, index saddles).
• It opens future investigations into the coupling dependence of these ensembles, the role of gauge singlet constraints (absent in the free theory), and the precise implementation at finite $N$ and finite coupling, potentially through refined matrix model techniques.
• The approach also lays groundwork for extending such results to less supersymmetric (microstate-rich) black holes in higher-dimensional AdS, as well as to the interplay of macroscopic geometric constraints and microscopic ensemble properties in holography.

Conclusion

The paper provides a detailed prescription whereby the nonlinear charge constraints satisfied by supersymmetric AdS black holes arise naturally in the dual SCFT, already at the level of free field representations. The procedure relies on going beyond supersymmetric indices to fully refined grand canonical partition functions and constructing supersymmetric ensembles that reflect the charge structure imposed by the preserved supercharge. This reveals that the charge constraints are consequences of BPS combinatorics and the superconformal algebra rather than strong coupling, with interactions affecting only the overall scale of physical charges. The results offer a new perspective on the microscopic origin of black hole properties in AdS/CFT, and motivate further research in non-index and non-BPS directions in quantum gravity and field theory dualities.

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References:

• "Supersymmetric Charge Constraints on AdS Black Holes from Free Fields" (2405.17648)
• Gubser, S. S., Klebanov, I. R., & Peet, A. W., "Entropy and temperature of black 3-branes" [hep-th/9602135]