pith. sign in

arxiv: 2606.19479 · v1 · pith:PC5OGK4Pnew · submitted 2026-06-17 · ✦ hep-th · math.NT

Generating Function of single-centered Black Hole Index in CHL Models

Ranveer Kumar Singh This is my paper

Pith reviewed 2026-06-26 19:24 UTC · model grok-4.3

classification ✦ hep-th math.NT
keywords CHL modelssingle-centered black hole indexquarter BPS dyonstwo-centered black holesSiegel modular formsbound state metamorphosisgenerating functionsZ_N symmetry
0
0 comments X

The pith

The generating function for single-centered black hole indices in Z_N CHL models is obtained by subtracting the two-centered contribution from the quarter BPS dyon index using bound state metamorphosis.

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

This paper constructs the generating function for the index of single-centered black holes in general Z_N CHL models. It starts from the known index of quarter BPS dyons, given by a meromorphic Siegel modular form, and subtracts the generating function that counts two-centered black holes. Black hole bound state metamorphosis supplies the two-centered piece required for the subtraction. A reader would care because the single-centered index isolates the microscopic degeneracy of individual black holes without multi-center contamination. The resulting series is shown to converge for the cases N=2 and N=3.

Core claim

We present the construction of the generating function of single-centered black hole index in general Z_N CHL models. This is done by subtracting from the index of quarter BPS dyons, described by a meromorphic Siegel modular form, the generating function for the index of two-centered black holes. We use black hole bound state metamorphosis in CHL models for the construction of the generating function of two-centered black hole index. We prove the convergence of the generating function for the cases N=2,3.

What carries the argument

Subtraction of the two-centered black hole index (built via black hole bound state metamorphosis) from the quarter BPS dyon index (a meromorphic Siegel modular form).

If this is right

  • The single-centered index generating function is now defined for arbitrary N.
  • Convergence is established at least for N=2 and N=3.
  • The construction isolates single-centered contributions within the full dyon spectrum.
  • The method uses the known meromorphic Siegel form as the starting point for all Z_N cases.

Where Pith is reading between the lines

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

  • The same subtraction procedure could be applied to other orbifold or twisted compactifications where multi-centered states are present.
  • Explicit expansion of the resulting function for small N might reveal closed-form expressions or additional modular identities.
  • Comparison with microscopic state counting in the single-center limit could test whether the subtracted index matches expected entropy formulas.
  • The approach might extend to higher-derivative corrections or other BPS indices beyond the quarter-BPS sector.

Load-bearing premise

Black hole bound state metamorphosis applies in CHL models and correctly supplies the two-centered black hole index for the subtraction.

What would settle it

An explicit mismatch between the subtracted series and an independent count of single-centered states for low charges when N=2 would show the construction fails.

read the original abstract

We present the construction of the generating function of single-centered black hole index in general $\mathbb{Z}_N$ CHL models. This is done by subtracting from the index of quarter BPS dyons, described by a meromorphic Siegel modular form, the generating function for the index of two-centered black holes. We use black hole bound state metamorphosis in CHL models for the construction of the generating function of two-centered black hole index. We prove the convergence of the generating function for the cases $N=2,3$.

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 / 0 minor

Summary. The manuscript constructs the generating function of the single-centered black hole index in general Z_N CHL models by subtracting the generating function for the two-centered black hole index (built via black hole bound state metamorphosis) from the meromorphic Siegel modular form for the quarter-BPS dyon index. Convergence of the resulting series is proven only for N=2 and N=3.

Significance. If the construction is valid, it isolates single-centered contributions in CHL models, which is relevant for microscopic black hole entropy counting in orbifold compactifications. The approach extends prior work on dyon indices and modular forms, but its scope is limited by the partial convergence result.

major comments (1)
  1. [Abstract] Abstract: the central construction for general N>3 subtracts a two-centered index obtained from black hole bound state metamorphosis, yet convergence is established only for N=2,3. Without a proof that the subtracted object equals the single-centered index for N>3 (or that metamorphosis supplies the complete two-centered piece without surviving CHL wall-crossing terms), the claim that the result is the single-centered generating function does not hold for arbitrary N.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading of our manuscript and the constructive comment. We respond to the major comment below and indicate the revision we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central construction for general N>3 subtracts a two-centered index obtained from black hole bound state metamorphosis, yet convergence is established only for N=2,3. Without a proof that the subtracted object equals the single-centered index for N>3 (or that metamorphosis supplies the complete two-centered piece without surviving CHL wall-crossing terms), the claim that the result is the single-centered generating function does not hold for arbitrary N.

    Authors: The construction proceeds by subtracting the two-centered index, obtained via black hole bound state metamorphosis, from the meromorphic Siegel modular form encoding the quarter-BPS dyon index; this subtraction is formulated uniformly for arbitrary N. The metamorphosis procedure is applied precisely to remove the contributions from two-centered configurations, including those associated with wall-crossing in the CHL models. We agree that a complete proof that the resulting object coincides with the single-centered index (i.e., that no residual CHL wall-crossing terms survive) is supplied only for N=2 and N=3, where convergence of the series is established. For N>3 the same formal subtraction is performed, but the corresponding convergence statement and the verification that the subtraction exhausts the two-centered piece are not proven. We will therefore revise the abstract to state explicitly that the generating function is constructed for general N by this subtraction, while convergence is proven only for N=2 and N=3. This change removes any overstatement of the result for arbitrary N. revision: yes

Circularity Check

0 steps flagged

No circularity: subtraction construction uses external modular form and prior metamorphosis concept without self-reduction.

full rationale

The derivation subtracts a two-centered index (constructed via black hole bound state metamorphosis) from the known meromorphic Siegel modular form for the quarter-BPS dyon index. The abstract states the steps explicitly but contains no equations that define the output in terms of itself, no fitted parameters renamed as predictions, and no load-bearing self-citation that reduces the central claim to an unverified prior result by the same authors. Convergence is separately proved for N=2,3, supplying independent content. This is self-contained against the external modular-form benchmark and matches the default non-circular outcome.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; limited visibility into parameters or entities. The work assumes standard modular form properties and metamorphosis applicability without introducing new fitted parameters or entities in the summary.

axioms (2)
  • domain assumption The index of quarter BPS dyons is described by a meromorphic Siegel modular form.
    Stated directly in abstract as the base for subtraction.
  • domain assumption Black hole bound state metamorphosis applies in CHL models for two-centered index construction.
    Invoked explicitly for building the two-centered generating function.

pith-pipeline@v0.9.1-grok · 5600 in / 1147 out tokens · 24887 ms · 2026-06-26T19:24:34.744978+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 32 linked inside Pith

  1. [1]

    Strominger and C

    A. Strominger and C. Vafa,Microscopic origin of the Bekenstein-Hawking entropy,Phys. Lett. B379(1996) 99–104, [hep-th/9601029]

    Pith/arXiv arXiv 1996

  2. [2]

    Sen,Extremal black holes and elementary string states,Mod

    A. Sen,Extremal black holes and elementary string states,Mod. Phys. Lett. A10(1995) 2081–2094, [hep-th/9504147]

    Pith/arXiv arXiv 1995

  3. [3]

    Dijkgraaf, E

    R. Dijkgraaf, E. P. Verlinde, and H. L. Verlinde,Counting dyons in N=4 string theory,Nucl. Phys. B484(1997) 543–561, [hep-th/9607026]

    Pith/arXiv arXiv 1997

  4. [4]

    Dijkgraaf, G

    R. Dijkgraaf, G. W. Moore, E. P. Verlinde, and H. L. Verlinde,Elliptic genera of symmetric products and second quantized strings,Commun. Math. Phys.185(1997) 197–209, [hep-th/9608096]

    Pith/arXiv arXiv 1997

  5. [5]

    H. L. Verlinde,Counting dyons in four-dimensional N=4 string theory,Nucl. Phys. B Proc. Suppl.58(1997) 141–148

    1997

  6. [6]

    Lopes Cardoso, B

    G. Lopes Cardoso, B. de Wit, J. Kappeli, and T. Mohaupt,Asymptotic degeneracy of dyonic N = 4 string states and black hole entropy,JHEP12(2004) 075, [hep-th/0412287]

    Pith/arXiv arXiv 2004

  7. [7]

    D. Shih, A. Strominger, and X. Yin,Recounting Dyons in N=4 string theory,JHEP10 (2006) 087, [hep-th/0505094]

    Pith/arXiv arXiv 2006

  8. [8]

    Sen,Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen

    A. Sen,Black Hole Entropy Function, Attractors and Precision Counting of Microstates, Gen. Rel. Grav.40(2008) 2249–2431, [arXiv:0708.1270]

    Pith/arXiv arXiv 2008

  9. [9]

    Chaudhuri, G

    S. Chaudhuri, G. Hockney, and J. D. Lykken,Maximally supersymmetric string theories in D <10,Phys. Rev. Lett.75(1995) 2264–2267, [hep-th/9505054]

    Pith/arXiv arXiv 1995

  10. [10]

    Chaudhuri and J

    S. Chaudhuri and J. Polchinski,Moduli space of CHL strings,Phys. Rev. D52(1995) 7168–7173, [hep-th/9506048]

    Pith/arXiv arXiv 1995

  11. [11]

    Sen,Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories,JHEP05(2007) 039, [hep-th/0702141]

    A. Sen,Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories,JHEP05(2007) 039, [hep-th/0702141]

    Pith/arXiv arXiv 2007

  12. [12]

    Sen,Two centered black holes and N=4 dyon spectrum,JHEP09(2007) 045, [arXiv:0705.3874]

    A. Sen,Two centered black holes and N=4 dyon spectrum,JHEP09(2007) 045, [arXiv:0705.3874]

    Pith/arXiv arXiv 2007

  13. [13]

    M. C. N. Cheng and E. Verlinde,Dying Dyons Don’t Count,JHEP09(2007) 070, [arXiv:0706.2363]

    Pith/arXiv arXiv 2007

  14. [14]

    Bachas and E

    C. Bachas and E. Kiritsis,F(4) terms in N=4 string vacua,Nucl. Phys. B Proc. Suppl.55 (1997) 194–199, [hep-th/9611205]. – 59 –

    Pith/arXiv arXiv 1997

  15. [15]

    Gregori, E

    A. Gregori, E. Kiritsis, C. Kounnas, N. A. Obers, P. M. Petropoulos, and B. Pioline,R**2 corrections and nonperturbative dualities of N=4 string ground states,Nucl. Phys. B510 (1998) 423–476, [hep-th/9708062]

    Pith/arXiv arXiv 1998

  16. [16]

    Sen,Arithmetic of Quantum Entropy Function,JHEP08(2009) 068, [arXiv:0903.1477]

    A. Sen,Arithmetic of Quantum Entropy Function,JHEP08(2009) 068, [arXiv:0903.1477]

    Pith/arXiv arXiv 2009

  17. [17]

    D. P. Jatkar and A. Sen,Dyon spectrum in CHL models,JHEP04(2006) 018, [hep-th/0510147]

    Pith/arXiv arXiv 2006

  18. [18]

    J. R. David, D. P. Jatkar, and A. Sen,Product representation of Dyon partition function in CHL models,JHEP06(2006) 064, [hep-th/0602254]

    Pith/arXiv arXiv 2006

  19. [19]

    J. R. David, D. P. Jatkar, and A. Sen,Dyon Spectrum in N=4 Supersymmetric Type II String Theories,JHEP11(2006) 073, [hep-th/0607155]

    Pith/arXiv arXiv 2006

  20. [20]

    J. R. David, D. P. Jatkar, and A. Sen,Dyon spectrum in generic N=4 supersymmetric Z(N) orbifolds,JHEP01(2007) 016, [hep-th/0609109]

    Pith/arXiv arXiv 2007

  21. [21]

    J. R. David and A. Sen,CHL Dyons and Statistical Entropy Function from D1-D5 System, JHEP11(2006) 072, [hep-th/0605210]

    Pith/arXiv arXiv 2006

  22. [22]

    Dabholkar, D

    A. Dabholkar, D. Gaiotto, and S. Nampuri,Comments on the spectrum of CHL dyons,JHEP 01(2008) 023, [hep-th/0702150]

    Pith/arXiv arXiv 2008

  23. [23]

    Banerjee and A

    S. Banerjee and A. Sen,Duality orbits, dyon spectrum and gauge theory limit of heterotic string theory on T**6,JHEP03(2008) 022, [arXiv:0712.0043]

    Pith/arXiv arXiv 2008

  24. [24]

    Bhand, A

    A. Bhand, A. Sen, and R. K. Singh,Generating Function of Single Centered Black Hole Index from the Igusa Cusp Form,arXiv:2510.05219

    arXiv

  25. [25]

    Sen,How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?,Gen

    A. Sen,How Do Black Holes Predict the Sign of the Fourier Coefficients of Siegel Modular Forms?,Gen. Rel. Grav.43(2011) 2171–2183, [arXiv:1008.4209]

    Pith/arXiv arXiv 2011

  26. [26]

    Dabholkar, S

    A. Dabholkar, S. Murthy, and D. Zagier,Quantum Black Holes, Wall Crossing, and Mock Modular Forms,arXiv:1208.4074

    Pith/arXiv arXiv

  27. [27]

    Chattopadhyaya and J

    A. Chattopadhyaya and J. R. David,Properties of dyons inN= 4 theories at small charges, JHEP05(2019) 005, [arXiv:1810.12060]

    Pith/arXiv arXiv 2019

  28. [28]

    Bhand, A

    A. Bhand, A. Sen, and R. K. Singh,Mock modularity in CHL models,Res. Math. Sci.12 (2025), no. 1 8, [arXiv:2311.16252]

    arXiv 2025

  29. [29]

    Banerjee, V

    N. Banerjee, V. Bhutra, and R. K. Singh,Mock modularity of twisted index in CHL models, JHEP10(2025) 008, [arXiv:2505.17182]

    arXiv 2025

  30. [30]

    Bringmann, A

    K. Bringmann, A. Folsom, K. Ono, and L. Rolen,Harmonic Maass Forms and Mock Modular Forms: Theory and Applications. Colloquium Publications. American Mathematical Society, 2017

    2017

  31. [31]

    Bringmann and S

    K. Bringmann and S. Murthy,On the positivity of black hole degeneracies in string theory, Commun. Num. Theor. Phys.7(2013), no. 1 15–56, [arXiv:1208.3476]

    Pith/arXiv arXiv 2013

  32. [32]

    Rosselló,The Immortal Dyon Index Is Positive,arXiv:2403.19630

    M. Rosselló,The Immortal Dyon Index Is Positive,arXiv:2403.19630

    arXiv

  33. [33]

    Sen,Negative discriminant states in N=4 supersymmetric string theories,JHEP10 (2011) 073, [arXiv:1104.1498]

    A. Sen,Negative discriminant states in N=4 supersymmetric string theories,JHEP10 (2011) 073, [arXiv:1104.1498]

    Pith/arXiv arXiv 2011

  34. [34]

    Chowdhury, S

    A. Chowdhury, S. Lal, A. Saha, and A. Sen,Black Hole Bound State Metamorphosis,JHEP 05(2013) 020, [arXiv:1210.4385]. – 60 –

    Pith/arXiv arXiv 2013

  35. [35]

    Chowdhury, A

    A. Chowdhury, A. Kidambi, S. Murthy, V. Reys, and T. Wrase,Dyonic black hole degeneracies inN= 4string theory from Dabholkar-Harvey degeneracies,JHEP10(2020) 184, [arXiv:1912.06562]

    arXiv 2020

  36. [36]

    Lopes Cardoso, S

    G. Lopes Cardoso, S. Nampuri, and M. Rosselló,Arithmetic of decay walls through continued fractions: a new exact dyon counting solution inN= 4 CHL models,JHEP03(2021) 154, [arXiv:2007.10302]

    arXiv 2021

  37. [37]

    Persson and R

    D. Persson and R. Volpato,Fricke S-duality in CHL models,JHEP12(2015) 156, [arXiv:1504.07260]

    Pith/arXiv arXiv 2015

  38. [38]

    Bringmann, M

    K. Bringmann, M. Raum, and O. K. Richter,Kohnen’s limit process for real-analytic siegel modular forms,Advances in Mathematics231(2012), no. 2 1100–1118, [arXiv:1105.5482]

    Pith/arXiv arXiv 2012

  39. [39]

    Bringmann, O

    K. Bringmann, O. K. Richter, and M. Westerholt-Raum,Almost holomorphic poincaré series corresponding to products of harmonic siegel–maass forms,Research in the Mathematical Sciences3(2016), no. 1 30, [arXiv:1604.05105]

    Pith/arXiv arXiv 2016

  40. [40]

    M. Westerholt-Raum,Harmonic weak siegel–maaß forms i: Preimages of non-holomorphic saito-kurokawa lifts,International Mathematics Research Notices2018(12, 2016) 1442–1472, [https://academic.oup.com/imrn/article-pdf/2018/5/1442/24261414/rnw288.pdf]

    2016

  41. [41]

    Raum,Formal fourier jacobi expansions and special cycles of codimension 2, arXiv:1302.0880

    M. Raum,Formal fourier jacobi expansions and special cycles of codimension 2, arXiv:1302.0880. – 61 –

    Pith/arXiv arXiv