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A supersymmetric index has zeros inside the unit disk exactly when its arithmetic coefficients grow exponentially, at a rate fixed by the nearest zero.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-07-31 16:29 UTC pith:KNH4FML7

load-bearing objection Clean analytic iff between interior zeros and exponential growth of δ(ν), with usable IR diagnostics and a solid giant-graviton check; worth engaging.

arxiv 2607.28143 v1 pith:KNH4FML7 submitted 2026-07-30 hep-th

Interior zeros of supersymmetric indices

Yu Nakayama , Tadashi Okazaki This is my paper
classification hep-th
keywords supersymmetric indexinterior zerosarithmetic coefficientsplethystic logarithms-confinementgiant graviton expansionSchur indexsupersymmetric zeta function
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Supersymmetric indices count protected states with signs, so unlike ordinary partition functions they need not vanish anywhere inside the unit disk. This paper proves they do vanish inside |q|<1 if and only if a sequence of arithmetic coefficients δ(ν), built from the plethystic logarithm by Möbius inversion, grows exponentially; the growth rate is exactly the inverse distance to the nearest zero. Each coefficient is fixed by finitely many terms of the q-series, so the test needs only a truncated expansion, not a closed form. Exponential growth rules out free or s-confining infrared descriptions, which the authors demonstrate for SU(2) SQCD with four and five flavors while recovering the known meson content at three flavors. When a giant-graviton expansion is available, the same zeros are finite-N effects sitting where one giant graviton cancels the vacuum, a law checked on the N=4 U(N) Schur index.

Core claim

A supersymmetric index I(q) has zeros in the open unit disk if and only if its arithmetic coefficients δ(ν) satisfy lim sup |δ(ν)|^{1/ν} = ρ^{-1} > 1, where ρ is the modulus of the nearest zero. The coefficients are obtained from finitely many Taylor coefficients of log I by Möbius inversion, so interior zeros are readable from the series alone and obstruct any free or s-confining infrared description.

What carries the argument

The arithmetic coefficients δ(ν) of the supersymmetric zeta function: Möbius transforms of the Taylor coefficients of log I. Their exponential growth rate equals the inverse zero distance, converting the zero problem into a finite computation on the q-expansion.

Load-bearing premise

The protected state counts grow slower than any exponential, so the index is holomorphic inside the unit disk and the only singularities of its logarithm are zeros.

What would settle it

Compute high-order arithmetic coefficients for SU(2) SQCD with Nf=3,4,5 or for the U(N) Schur index; check whether exponential growth appears exactly when independent root-finding finds zeros inside |q|<1, and whether the measured growth rate matches the modulus of the nearest zero.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Free and s-confining duals are excluded whenever δ(ν) grows exponentially, giving an infrared diagnostic that needs only a truncated index expansion.
  • For indices with a giant-graviton expansion the nearest zero lies near the cancellation of the one-brane term against the vacuum, with 1−ρ ~ (log E)/E set by the brane energy E.
  • Zero cardinality inside any radius r<1 is readable from the circle average of log|I| via Jensen’s formula, again from the series alone.
  • The Dirichlet series for the supersymmetric zeta function diverges precisely when interior zeros are present, forcing a return to the full Mellin representation.

Where Pith is reading between the lines

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

  • The same growth test should classify 3d and 6d superconformal indices and defect indices once their q-expansions are known to moderate order.
  • Tracking zero distance across continuous families of theories could link finite-N zeros to large-N complex phase transitions of the index.
  • Indices whose giant-graviton sectors reduce to deformed exponentials may inherit classical zero-distribution conjectures, offering a bridge between holographic expansions and special-function theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves that a supersymmetric index I(q) has zeros inside the open unit disk if and only if the arithmetic coefficients δ(ν) extracted from its plethystic logarithm grow exponentially, with lim sup |δ(ν)|^{1/ν} equal to the inverse modulus ρ^{-1} of the nearest zero. Each δ(ν) is obtained from finitely many Taylor coefficients of log I via Möbius inversion, so the criterion is accessible from a q-series expansion alone. Under the standard subexponential bound on BPS degeneracies, Appendix A supplies a short complex-analysis argument (Cauchy–Hadamard plus divisor-count suppression) establishing the equivalence. The authors apply the diagnostic to 4d N=1 SU(2) SQCD, ruling out free/s-confining IR descriptions for N_f=4,5 while recovering the free-meson content at N_f=3, and to the N=4 U(N) Schur index, where interior zeros are shown to be a finite-N effect located by the energy of a single giant graviton, with the large-N approach rate (14) verified numerically up to N=10^4.

Significance. If correct, the result supplies a practical, closed-form-independent infrared diagnostic for supersymmetric QFTs and a quantitative link between finite-N index zeros and giant-graviton energetics. The central equivalence is elementary, self-contained, and falsifiable from series coefficients; the ancillary Mathematica notebook and high-order expansions (matrix integrals to m≤150, Schur indices to m=1000) make the numerical claims reproducible. The obstruction to infinite-product/s-confining representations and the connection to the deformed-exponential/Sokal zero problem are of genuine interest to both the hep-th and special-functions communities. Strengths include the clean Appendix A proof, the parameter-free growth-rate formula, and the explicit large-N test of (14).

minor comments (5)
  1. [§II] §II and footnote [20]: the assumption of rational grading (ν ∈ (1/a)Z) is stated, but a one-sentence remark on how the diagnostic would be adapted for irrational R-charges after a-maximization would help readers who encounter generic fixed points.
  2. [Fig. 1] Fig. 1 caption and End Matter: the fitted form |δ| ∝ ρ_Q^{-m}/m is used, yet the main text only quotes the lim-sup relation (11). A brief cross-reference to the 1/ν prefactor discussion in Appendix B would make the dashed-line fits self-explanatory.
  3. [Appendix C] Eq. (14) and Appendix C: the constant w_∞ ≈ 0.7378 is obtained from the deformed exponential Φ(w,θ). Citing the numerical precision (or the minimization method) used for min_θ |w_*(θ)| would strengthen reproducibility of the asymptotic law.
  4. [Table I] Table I: the column header ρ^{1/2}_δ is slightly ambiguous (it is the estimate of |Q_*|). Renaming to ρ_Q^δ or “estimated |Q_*|” would avoid a momentary misreading.
  5. [References] References: the supersymmetric-zeta paper [7] is listed as JHEP 06 (2026) 003; confirm the final bibliographic data before publication so that the arXiv–journal mapping is stable.

Circularity Check

0 steps flagged

No significant circularity: the zeros↔growth equivalence is a self-contained complex-analysis argument, not forced by its inputs.

full rationale

The load-bearing claim is the iff between interior zeros of I and exponential growth of the arithmetic coefficients δ(ν) at rate ρ^{-1} (Eqs. 9–11). Appendix A derives this from Cauchy–Hadamard applied to log I plus a standard divisor bound showing that the finite Möbius sum (7) preserves lim sup |L_m|^{1/m}. The only external hypothesis is the subexponential bound (2) on BPS degeneracies, stated explicitly to guarantee holomorphy of I in |Q|<1; it is not fitted from the zeros being diagnosed. Each δ(ν) is computed from finitely many series coefficients of the same index whose zeros are under test, then the growth rate is cross-checked against independent root-finding (matrix-integral truncations for SU(2) SQCD; closed-form S_N and Padé/Newton for N=4 Schur). The giant-graviton location law (14) is read off from the leading wrapping term of the known expansion and verified a posteriori up to N=10^4; it is not inserted by fitting ρ. Self-citation to the authors’ prior zeta-function paper [7] only supplies nomenclature and motivation for δ(ν); the growth–zero theorem and its proofs do not rely on any uniqueness or existence result from [7]. No step reduces a claimed prediction to a fitted input or to a definitional identity.

Axiom & Free-Parameter Ledger

0 free parameters · 6 axioms · 2 invented entities

The central claim rests on standard complex analysis (Cauchy-Hadamard, argument principle, Jensen) plus standard domain facts about supersymmetric indices (holomorphy in |q|<1 under subexponential BPS growth, Möbius form of the plethystic logarithm). No fitted free parameters enter the iff theorem. Definitions (zero distance, zero cardinality, arithmetic coefficients) are bookkeeping, not new physical entities. The giant-graviton asymptotic adds the modeling assumption that higher wrapping sectors remain suppressed near the first cancellation.

axioms (6)
  • domain assumption BPS degeneracies satisfy log|d(n)|=O(n^α) with α<1, hence the index power series has radius of convergence at least 1 (Eq. 2).
    Standard for local QFTs and used to guarantee holomorphy of I in |Q|<1 so that singularities of log I are only zeros.
  • standard math Cauchy-Hadamard theorem: radius of convergence R satisfies R^{-1}=lim sup |a_m|^{1/m}.
    Applied to log I and to δ to equate growth rates with zero distance (Eqs. 10–11, Appendix A).
  • domain assumption Plethystic logarithm coefficients are the Möbius transform δ(ν)=∑_{d|νa} (μ(d)/d) L_{νa/d} (Eq. 7).
    Standard in the supersymmetric-index literature; taken from the authors' prior definition of the supersymmetric zeta function.
  • domain assumption Free and s-confining IR descriptions yield indices that are finite products of matter multiplet indices and therefore admit zero-free infinite-product representations inside the unit disk.
    Used to convert exponential growth of δ into an obstruction to free/s-confining duals (§II, §IV); standard but the converse is explicitly not claimed.
  • domain assumption Spectrum is rationally graded (ν∈(1/a)Z>0) so that an ordinary power series in Q=q^{1/a} exists.
    Stated in footnote [20]; irrational R-charges after a-maximization are deferred.
  • ad hoc to paper In a giant-graviton expansion the leading n=1 correction is −C Q^E and higher wrappings remain suppressed near the first vacuum cancellation, yielding −E log ρ_Q = log E + O(1) (Eq. 14).
    Motivated by known expansions for D3/M2/M5 branes and verified for the Schur index, but not derived for a general index.
invented entities (2)
  • zero distance ρ independent evidence
    purpose: Quantitative modulus of the nearest interior zero, identified with the inverse growth rate of δ(ν).
    Definition (Eq. 9), not a new physical object; purely a named observable extracted from the series.
  • zero cardinality N(r) independent evidence
    purpose: Count of interior zeros inside radius r, via argument principle / Jensen formula.
    Definition (Eq. 12); standard complex-analytic counting renamed for the index setting.

pith-pipeline@v1.2.0-daily-grok45 · 17238 in / 3431 out tokens · 60065 ms · 2026-07-31T16:29:57.700994+00:00 · methodology

0 comments
read the original abstract

A supersymmetric index has interior zeros ($|q|<1$) if and only if the arithmetic coefficients $\delta(\nu)$ underlying the supersymmetric zeta function grow exponentially at a rate set by the nearest zero. Each $\delta(\nu)$ follows from finitely many $q$-series coefficients, so interior zeros are detectable from the expansion alone and obstruct a free or s-confining infrared description, as we show for 4d $\mathcal{N}=1$ $SU(2)$ SQCD. With a giant graviton expansion interior zeros are a finite $N$ effect, located by the energy of one giant graviton, verified for the $\mathcal{N}=4$ $U(N)$ Schur index.

Figures

Figures reproduced from arXiv: 2607.28143 by Tadashi Okazaki, Yu Nakayama.

Figure 1
Figure 1. Figure 1: FIG. 1. Arithmetic coefficients [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: shows |δ(m/2)| up to m = 100 for N ≤ 5. The free U(1) theory has bounded coefficients, δ(m/2) = 2(−1)m+1, matching the zero-free product form of its in￾dex. Every non-Abelian theory grows exponentially in￾stead, reaching |δ(m/2)| ≃ 1.2 × 10216 at m = 1000 for U(2), so each one has interior zeros. We estimate the zero distance from this growth, ρδ := lim supm |δ(m/2)| 1/m−2 , which (11) equates with ρ [PI… view at source ↗

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Reference graph

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