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arxiv: 2605.19552 · v2 · pith:SQGMTFAJnew · submitted 2026-05-19 · ✦ hep-th · math.AG

Large Order Enumerative Geometry, Black Holes and Black Rings

Sergei Alexandrov , Albrecht Klemm , Boris Pioline This is my paper

Pith reviewed 2026-06-30 18:31 UTC · model grok-4.3

classification ✦ hep-th math.AG
keywords Gopakumar-Vafa invariants5D black holesblack ringsstable pair invariantsDonaldson-Thomas invariantsCalabi-Yau threefoldsenumerative geometryentropy
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The pith

The 5D index matches the entropy of rotating black holes below a critical angular momentum and is dominated by black rings above it.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper numerically analyzes the large charge asymptotics of 5D indices and related invariants using newly available high-genus Gopakumar-Vafa data for hypergeometric Calabi-Yau threefolds. It establishes that the 5D index reproduces the Bekenstein-Hawking-Wald entropy of BMPV black holes, including four-derivative corrections, when the angular momentum parameter m stays below a critical threshold. Beyond that threshold the index instead follows the entropy of black rings with minimal dipole charge. The analysis also uncovers multiple phase transitions in stable pair and Donaldson-Thomas invariants, each with distinct growth behaviors, while confirming an earlier conjecture on the asymptotics of topological string free energies.

Core claim

Exploiting high-genus Gopakumar-Vafa invariants, the growth of the 5D index Ω_5D(d,m) at large d agrees perfectly with the entropy of rotating 5D BMPV black holes for m below a critical value, including subleading four-derivative terms, but switches to being dominated by black rings with the smallest dipole charge when m exceeds that value. The stable pair invariant PT(d,m) shows a similar black ring-hole transition at negative m together with two additional transitions at positive m leading first to a plateau and then to polynomial growth proportional to m to the power 2d-1. The rank one DT invariant DT(d,m) follows the same pattern at negative m before entering a D0-brane dominated phase w

What carries the argument

The 5D index Ω_5D(d,m) extracted from Gopakumar-Vafa invariants, which serves as the generating function whose large-charge asymptotics are compared to black hole and black ring entropies.

If this is right

  • The subleading entropy correction from four-derivative interactions is reproduced by the index below the critical m.
  • For m above critical, the smallest dipole charge black rings determine the index.
  • PT invariants exhibit a plateau phase and then polynomial growth ~ m^{2d-1} after the initial transition.
  • DT invariants transition to a phase with entropy of order m^{2/3} dominated by D0-branes.
  • The fixed-genus large-degree behavior of GV invariants is determined, including the g-dependent constant, and extended approximately to large g.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The observed phase transitions likely mark points where different BPS configurations exchange dominance in the index.
  • If the numerical agreement persists at even larger charges, it would support the idea that enumerative invariants fully encode the quantum corrections to black hole entropy.
  • The noted effectiveness of the PT/MSW relation may point to a deeper algebraic identity between these invariants.

Load-bearing premise

The high-genus Gopakumar-Vafa invariant data accurately captures the true mathematical values and that the extraction of large-charge asymptotics is free from truncation or convergence artifacts.

What would settle it

A mismatch between the numerically computed 5D index at a charge above the critical m and the entropy formula for the black ring with minimal dipole charge would falsify the dominance claim.

Figures

Figures reproduced from arXiv: 2605.19552 by Albrecht Klemm, Boris Pioline, Sergei Alexandrov.

Figure 1
Figure 1. Figure 1: Asymptotics of genus zero GV invariants for the quintic X5. On the left, we plot the N-th iteration of the depth 1 logarithmic Richardson transform r (N) 1 := (L1) N [r (0) d ], for N = 0, 1, 2, 3. This is identical to the plot in [2], but extended up to degree d = 1780. On the right, we plot the N-th iteration of the depth 2 logarithmic Richardson transform r (N) 2 := (L2) N [r (0) d ], for N = 0, 1, 2. T… view at source ↗
Figure 2
Figure 2. Figure 2: Ratio of the exact GW invariant and asymptotic estimate for X5 (left) and X4,2 (right), for genus 0 up to 10, as a function of the order N of the logarithmic Richardson transform, using degree d ≤ 1600. 2.4 Phenomenology of GV invariants at fixed degree                                                                              … view at source ↗
Figure 3
Figure 3. Figure 3: log | GV(g) d | as a function of g for X5 (left) and X4,2 (right). Different degrees correspond to different colors. In both cases the maximal shown degree is 44. Positive GV invariants are shown by dots, while negative ones by crosses. Gaps correspond to unknown invariants. Let us now fix the degree d and consider the dependence of GV invariants GV(g) d on the genus g, bounded by gmax(d). In [PITH_FULL_I… view at source ↗
Figure 4
Figure 4. Figure 4: The values of gkink(d) for X5 (left) and X4,2 (right). At the bottom, gkink(d) is normalized by 2(wd) 3/2 . • Finally, unfortunately, we do not have enough data to guess the form of GV invariants after the kink, i.e. for gkink(d) < g ≤ gmax(d). It is tempting to claim that they are again captured by a Gaussian, but we do not have a sufficient set of constraints to determine its parameters. These observatio… view at source ↗
Figure 5
Figure 5. Figure 5: log |Ω5D(d, m)| as a function of m for X5 (left) and X4,2 (right). Different degrees cor￾respond to different colors. The maximal shown degrees are 26 and 31, respectively. Positive BPS indices are shown by dots, while negative ones by crosses. Thus, one expects that at large enough d, keeping the ratio m/d3/2 fixed, the logarithm of the 5D index (3.3) should reproduce the black hole entropy (3.7) for |m| … view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The EN transforms of the series (3.19) for X4,2 with N = 0, . . . , 4. The sets of functions used in the transform are {d −1 , d−3/2 log(d), d−3/2 , d−2} (left) and {d −1/2 , d−1 , d−3/2 , d−2} (right). 22It is worth noting that the log-term is not necessary to include to achieve this remarkable convergence. – 24 – [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The Richardson and EN transforms of slog(d) for X4,2 with N = 0, . . . , 7. The set of functions used in the E-algorithm is {d 1−k/ log(d)}k≥1. Given the uncertainty on the value of α, it seems pointless to try and investigate further subleading terms. However, one can also try to check the prediction (3.15) for the difference of the logarithmic corrections in two ensembles. In the static case, the angular… view at source ↗
Figure 9
Figure 9. Figure 9: The Richardson and EN transforms of sd log(d) for X4,2 with N = 0, . . . , 7. The set of functions used in the E-algorithm is {d 1−k/ log(d)}k≥1. 3.3.2 Slow-spinning black holes Let us now consider 5D index Ω5D(d, m) at large d keeping m non-vanishing but fixed. Fol￾lowing [29], we define ρ(d, m) = d 3/2 m2 log [PITH_FULL_IMAGE:figures/full_fig_p027_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: The Richardson transforms of ˜s0(d) for X5 and X4,2 with N = 0, . . . , 4. The results of several Richardson transforms of ˜s0(d) are shown in [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The Richardson transforms of ˜s1(d) for X5 and X4,2 with N = 0, . . . , 4 (top) and the values of (RN [˜s1])d for maximally available d for N up to 10 and the same manifolds (bottom). 3.3.3 Fast-spinning black holes Now we turn to spinning black holes in the regime where the ratio ω = J/Q3/2 is kept finite while Q is scaled to infinity, but ω remains smaller than the critical value given by the position –… view at source ↗
Figure 12
Figure 12. Figure 12: Tests of the quantum correction to the entropy of rotating black holes. Different columns correspond to three different ways to check whether the tested function represents the correction. Different rows correspond to four tested functions: g(ω) = 1 + 1 3 ω 2 , 1 1−ω2 , 1 and 1− 4 3 ω 2 1−ω2 . All functions are plotted for X4,2 and d ranging from 11 (orange) to 31 (blue). Due to symmetry, we restrict to p… view at source ↗
Figure 13
Figure 13. Figure 13: 5D index (blue) vs. Bekenstein-Hawking-Wald entropy of spinning BMPV black holes, for X5 (left) and X4,2 (right). The red curve shows the classical Bekenstein-Hawking entropy (3.5), the brown curve includes the 4-derivative correction at linear order ((3.7) with g(ω) given in (3.8d)), and the green curve includes it at non-linear order as in (3.27). Given the last observation, it is natural to reconsider … view at source ↗
Figure 14
Figure 14. Figure 14: Left: log |Ω5D| (blue) and the black ring entropy S br for r = 1, . . . , 6 with n = m. Right: log |Ω5D| (blue), PT1(d, n) (red) and S br for r = 1 (green) with n = m + 2. The purple dashed curve is the quantum corrected black hole entropy (3.27). All graphs are made for X4,2 and the maximal available degree d = 31. In fact, the match can be further improved if one takes into account that the D0-brane cha… view at source ↗
Figure 15
Figure 15. Figure 15: Positions of the kink ωkink(d) for the 5D index for X4,2, its analytic expression (3.32) (red) and the approximation (3.33) (purple dashed). In [PITH_FULL_IMAGE:figures/full_fig_p032_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: At the top: log | GV(g) d | (blue) and its approximation (3.46) (red) for X5 (left) and X4,2 (right) for degrees from 34 to 44. The approximation is computed using (3.44) with α = β = 0. At the bottom: the same but now for degrees from 16 to 26 for X5 (left) and from 21 to 31 for X4,2 (right), and the approximation is computed using (3.43) with the actual values of the 5D index. – 34 – [PITH_FULL_IMAGE:f… view at source ↗
Figure 17
Figure 17. Figure 17: log |PT(d, m)| as a function of m for 10 maximal available degrees for X5 (left) and X4,2 (right). Different degrees correspond to different colors. Positive invariants are shown by dots, while negative ones by crosses. Using the GV/PT relation, one can compute PT invariants up to the same degree for which GV invariants are known at all genera (see the column dmod in [PITH_FULL_IMAGE:figures/full_fig_p03… view at source ↗
Figure 18
Figure 18. Figure 18: Left: the ratio log | PT(d,m)| log |P T(d,0)| as a function of J/Q3/2 , for 10 maximal available degrees for X4,2. The colors change from orange (d = 22) to blue (d = 31). Right: log |PT(d, m)| for X4,2 and d = 31 calculated in three ways: using (4.5) (blue), restricting to g = 0 in this formula and to k = 1 in (4.6) (red), and restricting to g ≥ 1 and k = 1, respectively (green). Below we analyze each of… view at source ↗
Figure 19
Figure 19. Figure 19: Left: log |PT(d, m)| for X4,2 and d = 31 (blue), its approximation obtained by restricting to k = 1 in (4.6) and to g ≥ 1 in (4.5) (green), and one more approximation obtained by additional restriction to d = 31 in (4.5) (red crosses). Right: log |S1(d, m + 2)| (blue) and log |S−1(d, m)| (red) for X4,2 and d = 31. Furthermore, the function S1 in the r.h.s. of (4.10) is closely related to the 5D index (3.3… view at source ↗
Figure 20
Figure 20. Figure 20: The ratio between the approximation PT1(d, m) (4.16) and the exact PT invariant PT(d, m) for X5, d = 26 (left) and X4,2, d = 31 (right). 4.3 Plateau The plateau phase of PT invariants is the only regime where it is important to take into account the plethystic exponential in the MNOP formula. Indeed, acting on the genus zero contribution in (4.5), it produces factors 1/(1 − (−q)k ) 2 . Let us take k = 2. … view at source ↗
Figure 21
Figure 21. Figure 21: The plots of log |PT(d, m)| for X4,2, d = 31 (blue) and its approximations (red). On the left, the approximation is obtained from MNOP formula where one takes into account only the contribution of GV(g) 29 invariants with g ≥ 2 with the plethystic exponential replaced by the usual one and GV(0) 1 with the plethystic exponential replaced by its k = 2 contribution. On the right, it is the logarithm of (4.19… view at source ↗
Figure 22
Figure 22. Figure 22: The plots of log |PT(d, m)| for X4,2, d = 31 and its approximations: the actual PT invariants (blue), the contribution of genus 0 GV invariants with the plethystic exponential replaced by the usual one (red), additional restriction to degrees d ≤ 20 (brown), d ≤ 10 (green) and d = 1 (magenta). Besides, the purple dashed curve is the approximation (4.23). 5. Growth of DT invariants The Donaldson-Thomas inv… view at source ↗
Figure 23
Figure 23. Figure 23: Left: log | DT(d, m)| as a function of m for 10 maximal available degrees for X4,2. Different degrees correspond to different colors. Positive invariants are shown by dots, while negative ones by crosses. Right: log | DT(d, m)| (blue), log |PT(d, m)| (green) and log |Ω5D| (red) for the maximal available degree d = 31. In fact, the behavior of DT invariants at large m is dominated by the MacMahon factor M(… view at source ↗
Figure 24
Figure 24. Figure 24: The plots of log | DT(d, m)| for X4,2, d = 31 (blue) and its approximations given by (5.2) with either exact Fourier coefficients ωχ (red) or replaced by their asymptotics (D.9) (green). Besides, the purple dashed curve is the approximation (5.4) multiplied by DT(d, 0). While the discussion above assumes that the degree d is fixed, a natural question is to study the behavior of DT(d, m) as m2/d3 is held f… view at source ↗
Figure 25
Figure 25. Figure 25: All graphs are made for X4,2 and λ = 5 + 25i (left), 5 + 5i (middle) and 5 + 2i (right). Top: log |Fd| (blue) and two contributions, (C.4) (red) and (C.9) (green). Down: gmax(d) (red) and g∗(d) (green). Although on the left the green curve is dominating for d > 20, it does not contribute since d∗ > gmax for d > 18. In contrast, on the right it provides the correct approximation as d∗ remains always less t… view at source ↗
Figure 26
Figure 26. Figure 26: Logarithm of the Fourier coefficients of the χ-th power of the MacMahon function (in red), compared to the asymptotic formula (in blue) (D.8) for χ = 1 (left) or (D.9) for χ = −1 (right). 38This result differs from the one quoted in [30, (6.2)], and was obtained by Charles Cosnier-Horeau and the last-named author in May 2015. – 55 – [PITH_FULL_IMAGE:figures/full_fig_p056_26.png] view at source ↗
read the original abstract

Exploiting newly available data on Gopakumar-Vafa invariants at high genus for one-parameter hypergeometric Calabi-Yau threefolds, we study numerically the growth of the 5D indices, stable pair (PT) invariants and rank one Donaldson-Thomas (DT) invariants at large charges. For the 5D index $\Omega_{5D}(d,m)$, below a critical value of the angular momentum $m$, we find perfect agreement with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes, including the subleading correction from four-derivative interactions. When $m$ exceeds the critical value, the 5D index is instead dominated by black rings with the smallest possible dipole charge. The stable pair invariant $PT(d,m)$, which is determined by 5D indices, has a similar black ring/hole transition at negative $m$ (now interpreted as the D0-brane charge) but surprisingly exhibits two other phase transitions at positive $m$: first, to a plateau and then to a polynomial growth $\sim m^{2d-1}$. In each phase, we derive an approximate expression for the invariant. Finally, the rank one DT invariant $DT(d,m)$ is similar to $PT(d,m)$ at negative $m$, and then transitions to a phase dominated by D0-branes, with entropy of order $m^{2/3}$. Along the way, we determine the fixed genus, large degree behavior of GV invariants (including the overall $g$-dependent constant), extend it to an approximate formula valid also for large $g$, point out the unreasonable effectiveness of a simple PT/MSW relation, and study the growth of topological free energies at fixed degree, confirming a conjecture of Mari\~no.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper uses newly available high-genus Gopakumar-Vafa invariant data for one-parameter hypergeometric Calabi-Yau threefolds to numerically study the large-charge growth of the 5D index Ω_5D(d,m), stable pair invariants PT(d,m), and rank one DT invariants DT(d,m). It reports perfect agreement below a critical m with the Bekenstein-Hawking-Wald entropy of rotating 5D BMPV black holes including four-derivative corrections, transitioning to black ring domination above it. Similar phase transitions are identified for PT and DT, with approximate expressions derived in each phase, and a conjecture of Mariño on topological free energies is confirmed. The fixed genus large degree behavior of GV invariants is determined, including g-dependent constant, and extended to large g.

Significance. If the numerical results hold, this work provides compelling evidence for the microscopic origin of 5D black hole and black ring entropies from enumerative invariants, including subleading corrections. It demonstrates phase transitions in the invariants corresponding to different gravitational configurations and offers new insights into the asymptotics of GV invariants. The confirmation of Mariño's conjecture and the approximate PT/MSW relation are notable strengths. The availability of high-genus data enables these large-order studies, which are rare in the field.

major comments (1)
  1. [Numerical analysis of Ω_5D(d,m) growth (abstract and results)] The claim of 'perfect agreement' with the BMPV entropy including the precise subleading four-derivative correction is load-bearing on the numerical extraction of large-(d,m) asymptotics from the finite GV table. No information is supplied on data precision, error estimation, convergence criteria, or how the critical m is determined, nor on systematic checks by varying genus cutoff or cross-validation against known GV values. This undermines confidence in whether the agreement survives more complete data.
minor comments (1)
  1. The abstract mentions determining the overall g-dependent constant in the fixed genus large degree behavior of GV invariants; it would be helpful to state this constant explicitly in the main text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for greater transparency in our numerical methodology. We address the major comment below and will revise the manuscript to incorporate additional details on data handling and validation.

read point-by-point responses

  1. Referee: [Numerical analysis of Ω_5D(d,m) growth (abstract and results)] The claim of 'perfect agreement' with the BMPV entropy including the precise subleading four-derivative correction is load-bearing on the numerical extraction of large-(d,m) asymptotics from the finite GV table. No information is supplied on data precision, error estimation, convergence criteria, or how the critical m is determined, nor on systematic checks by varying genus cutoff or cross-validation against known GV values. This undermines confidence in whether the agreement survives more complete data.

    Authors: We agree that the manuscript would benefit from explicit documentation of the numerical procedures. In the revised version we will add a new subsection (likely in Section 3 or an appendix) that: (i) states the precision and known error bounds of the input GV invariants from the available tables; (ii) describes the fitting procedure used to extract the leading and subleading coefficients of the entropy, together with the associated uncertainties; (iii) specifies the convergence criteria with respect to the genus cutoff (including plots or tables showing stability when the cutoff is varied); (iv) explains how the critical value of m is identified (by direct comparison of the extracted growth rates to the BMPV and black-ring formulae); and (v) reports cross-checks against lower-genus or lower-degree GV data where independent results are known. These additions will make the robustness of the reported agreement quantifiable. While the present data set is finite, the observed agreement within the accessible range remains robust under the checks we have performed internally; the revision will render this transparent to the reader. revision: yes

Circularity Check

0 steps flagged

No significant circularity; comparisons are to independent supergravity formulas

full rationale

The paper extracts large-charge asymptotics of 5D indices, PT invariants and DT invariants numerically from external high-genus GV data for a specific Calabi-Yau, then compares those asymptotics to the Bekenstein-Hawking-Wald entropy of BMPV black holes (including four-derivative corrections) taken from the supergravity literature. This comparison is to an externally defined physical quantity rather than to any quantity constructed from the same invariants or fitted parameters. No self-definitional steps, no fitted inputs relabeled as predictions, and no load-bearing self-citations appear in the derivation chain. The numerical procedure itself may raise separate questions of convergence, but those do not reduce the claimed agreement to a tautology by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the reliability of external high-genus GV data whose accuracy is not independently verified in the abstract and on the assumption that numerical large-charge limits faithfully capture the true asymptotic behavior.

axioms (1)
  • domain assumption Newly available high-genus Gopakumar-Vafa data are sufficiently accurate and complete for extracting large-charge asymptotics
    The entire numerical study depends on this external dataset without reported cross-checks or error bounds.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. BMPV black hole at first order in $\alpha'$

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  2. Extended Supergravity Needs String Scale Cut-off

    hep-th 2026-06 conditional novelty 5.0

    String-scale UV cut-off in the gravitational path integral removes string-coupling dependence from the BPS black hole index in extended supergravity, matching supersymmetry expectations for zero-Euler-number cases.

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