Equal-charge projection of the $\mathcal{N}=4$ index: exact large-$N$ formula and finite-rank $U(3)$ coefficients

Abstract: The equal-charge branch of supersymmetric rotating AdS$<em>5$ black holes has $Q_1=Q_2=Q_3$. The corresponding microcanonical sector of the $\mathcal{N}=4$ superconformal index is obtained by projecting to equal charges, or equivalently by extracting the constant term in the two charge-difference fugacities. We prove that for the large-$N$ multigraviton sector the projected index factorizes exactly as [ \mathcal{I}^{\rm eqQ}\infty(x,p) =\prod_{k\ge1}(1-p^kx^{3k})(1-p^{-k}x^{3k}) \sum_{n\ge0}\mathsf{p}(n)^3x^{6n}, ] where $\mathsf{p}(n)$ is the partition function. This factorization gives, for every spin sector, an explicit onset energy below which the large-$N$ coefficient is zero. Exact $U(3)$ computations show that finite-rank coefficients can nevertheless appear at energies where the large-$N$ coefficient vanishes, including beyond the classical $U(3)$ black-hole bound. We also determine the full line $j=6J_R+6$. In particular, with $j<sup>*(J_R)$ denoting this large-$N$ onset energy, [ d_3^{\rm eqQ}(87,\tfrac{27}{2})=1, \qquad j^*(\tfrac{27}{2})-87=1554, ] and the first giant-graviton sector already contributes one unit at this point. All coefficients are coefficients of the $(-1)<sup>F$-graded index, not positive degeneracies. The main conclusion is that the high-spin tail survives the exact equal-charge projection.

• The paper presents an exact large-N factorization of the equal-charge projected index, identifying vanishing intervals and onset energies for multigraviton coefficients.
• It systematically computes finite-N corrections for U(3), revealing nonzero tail contributions far below the classical black hole bound.
• The work demonstrates that finite-rank effects, including giant graviton sectors, contribute nontrivially to supersymmetric black hole microstate counting.

Equal-Charge Projection of the $\mathcal{N}=4$ Index: Large-$N$ Exact Formula and Finite-$N$ $U(3)$ Tail Structure

Overview and Motivation

This work provides an exact analytic and computational treatment of the equal-charge projection of the superconformal index for $\mathcal{N}=4$ $U(N)$ SYM. The projection corresponds to enforcing the physical microcanonical sector relevant for supersymmetric rotating $\mathrm{AdS}_5$ black holes with $Q_1=Q_2=Q_3$. The focus is on two central aspects: (1) an exact large-$N$ factorization of the projected multigraviton index, leading to explicit vanishing intervals for the large-$N$ coefficients, and (2) a systematic finite-rank analysis for $N$0, revealing how finite-$N$1 corrections generate nonzero coefficients even in sectors where the large-$N$2 index vanishes, including well beyond the classical black hole bound.

Superconformal Index and Equal-Charge Projection

The superconformal index $N$3 of $N$4 $N$5 SYM is a protected, $N$6-graded count of $N$7-BPS operators, refined by energy, angular momentum, and three $N$8-charges. In previous AdS/CFT studies, interest has focused on large-charge limits and saddle points reconstructing Bekenstein-Hawking entropy for supersymmetric $N$9 black holes. The standard "equal-fugacity" index, specializing $N$0, maintains a sum over all charge assignments, but does not restrict to the physically relevant microcanonical sector of equal charges.

This paper instead imposes the exact microcanonical equal-charge projection by extracting the constant term in the two charge-difference fugacities. This produces an index counting only states with $N$1 for each $N$2, with all other charges suppressed.

Exact Large-$N$3 Factorization and the Onset Phenomenon

A key analytic result is the exact factorization of the equal-charge projected index in the large-$N$4 (multigraviton) sector:

$N$5

where $N$6 is the partition function. The structure of the prefactor (Euler pentagonal numbers) allows for an explicit analytic determination, for each $N$7, of the unique minimal energy $N$8 at which the large-$N$9 coefficient becomes nonzero (“onset”). For energies $U(3)$0 in the interval $U(3)$1, the corresponding projected multigraviton coefficient vanishes exactly. The onset can be explicitly calculated using a divisor minimization exploiting generalized pentagonal numbers, and an exact formula for the coefficient at onset is given. Analytically, the support of the multigraviton sector is completely understood, with computational confirmation for all relevant grades.

For all $U(3)$2 in the vanishing interval, any nonzero finite-$U(3)$3 coefficient is a pure finite-rank effect and cannot originate from the multigraviton index.

Finite-$U(3)$4 Corrections and the High-Spin Tail in $U(3)$5

The study computes all $U(3)$6 coefficients for the equal-charge projected index exactly (using checked integer and modular arithmetic), benchmarking against the large-$U(3)$7 result. All finite-rank coefficients are $U(3)$8-graded and not positive degeneracies.

Significant nonzero $U(3)$9 coefficients are found at energies and spins for which the large-$\mathcal{N}=4$0 coefficient vanishes and that lie far below the determined onset $\mathcal{N}=4$1. Notably, there is explicit construction in sectors that violate both the classical black hole existence bound and the large-$\mathcal{N}=4$2 support—demonstrating that the tail in the index is a genuine finite-rank effect not attributable to the large-$\mathcal{N}=4$3 graviton gas or to summing over unequal charges.

A canonical example is $\mathcal{N}=4$4 with $\mathcal{N}=4$5, for which $\mathcal{N}=4$6, lying $\mathcal{N}=4$7 units below the multigraviton onset. Other explicit extractions with $\mathcal{N}=4$8 confirm the persistence and sometimes substantial magnitude of this finite-rank tail.

The finite-rank corrections admit an expansion in the number of giant-graviton sectors (“wrapping” number). In selected cases, the support of the tail is confirmed to originate in the minimal ($\mathcal{N}=4$9) giant-graviton sector, with higher sectors vanishing, at least moduloprime up to numerical bounds.

Entropic Implications and Comparison to Black Hole Physics

Within the classical black hole region, the logarithm of the absolute value of the finite-$U(N)$0 projected index coefficients tracks the Bekenstein-Hawking entropy with a near-constant deficit attributable to the exact charge projection, index sign cancellations, and finite-$U(N)$1 effects. Beyond the black hole bound, the classical entropy vanishes, but the projected index tail persists, with support arising entirely from finite-rank, non-multigraviton contributions.

The large-$U(N)$2 analysis also quantifies the suppression due to equal-charge projection relative to the sum over charges, with a profile dominated by a central limit/partition-number Gaussian scaling.

Group-Theoretic and Algorithmic Aspects

The bulk of results are phrased for $U(N)$3, but the computational method is designed so that factoring out the free $U(N)$4 center allows extraction of $U(N)$5 coefficients for selected targets, confirming that the observed finite-$U(N)$6 tail is not a center artifact. For all calculated examples, the coefficients persist and often increase in magnitude in the interacting $U(N)$7 sector.

Technically, the expansion is controlled using plethystic exponentials, charge-difference profiles, and rigorous pruning bounds for efficient extraction. All coefficients are checked for consistency, charge-permutation symmetry, and modular correctness; selected high-$U(N)$8 coefficients are reconstructed with several large primes via the Chinese remainder theorem.

Theoretical and Future Implications

The analysis rigorously demonstrates that, under microcanonical equal-charge projection, the high-spin tail in the superconformal index survives and, at finite $U(N)$9, receives strong contributions well outside classical and large-$\mathrm{AdS}_5$0 multigraviton support. This invalidates the hypothesis that the tail is an artifact of summing over unequal-charge assignments and shows that finite-$\mathrm{AdS}_5$1 quantum corrections in the gauge theory are nontrivial in sectors relevant for quantum microstate counting of supersymmetric black holes.

From a broader perspective, the tools and explicit formulae developed here enable fine-grained studies of finite-$\mathrm{AdS}_5$2 and finite-charge corrections to black hole microphysics, with potential applications to the analysis of giant graviton expansions, wrapped-brane corrections, and quantum deconfinement transitions. The exact onset and vanishing intervals, as well as the analytic machinery, provide a sharp diagnostic of the interplay between group-theoretic combinatorics, protected state counting, and the supergravity/black hole spectrum in AdS/CFT.

Future work could extend the exact computations to larger $\mathrm{AdS}_5$3 and $\mathrm{AdS}_5$4, dissect the higher giant-graviton sector decomposition in the tail, and employ these data in mapping microcanonical phase diagrams for holographic RG flows. There is also scope for direct comparison with grey-galaxy and partial deconfinement phases, as well as further elucidation of the structure of positive BPS degeneracy spectra beyond the $\mathrm{AdS}_5$5-graded index environment.

Conclusion

This study delivers an exact analysis of the equal-charge projected superconformal index for $\mathrm{AdS}_5$6 SYM, exposing the arithmetic, analytic, and physical structure of its high-spin finite-rank tail. The main implication is that finite-$\mathrm{AdS}_5$7 corrections, as computed in genuinely microcanonical sectors, supply nontrivial protected index support where both multigraviton and classical black hole descriptions vanish. The analytic apparatus built around pentagonal number onsets, charge-resolved partition structures, and giant graviton expansions sets a solid foundation for further investigation into quantum aspects of holographic BPS spectra.