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REVIEW 3 major objections 4 minor 37 references

Properties of scalar partition functions of 2d CFTs

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The scalars of any 2d CFT obey a crossing equation whose high-temperature corrections are governed by the nontrivial zeros of the Riemann zeta function.

desk verdict Solid generalization of the Narain scalar bootstrap with a real but openly admitted unproven spectral-density assumption; worth refereeing, but the 'any 2d CFT' claim is conditional. read the letter →

arxiv 2505.18314 v1 pith:KEH5CI3R submitted 2025-05-23 hep-th

Nathan Benjamin , Cyuan-Han Chang , A. Liam Fitzpatrick , Tobi Ramella This is my paper
classification hep-th MSC 11M0611F7281T40
keywords 2dconformalfieldtheoryscalarprimaryoperatorsmodularcrossingequationRiemannzetazerosMWKpartitionfunctionbootstrapintegralgap
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper establishes that the scalar Virasoro primary operators of any two-dimensional conformal field theory satisfy a nontrivial modular crossing equation on their own, without needing the spinning part of the spectrum. The equation equates a sum over scalar states against a fixed kernel to a constant $\varepsilon$, which is the regulated integral of the full primary partition function over the modular fundamental domain. The paper's second, stronger result is an exact identity for the scalar partition function at high temperature: after subtracting the MWK (Poincaré-sum) gravity spectrum of light states, the remaining difference is $\varepsilon$ plus an infinite series of powers $y^{(1+z_k)/2}$, where $z_k$ are the nontrivial zeros of the Riemann zeta function, plus terms fixed by the light spectrum and nonperturbatively small corrections. Under the Riemann hypothesis, the zeta-zero series has an overall envelope $y^{3/4}$. Besides giving a new route to compute regulated modular integrals, the first crossing equation yields rigorous bootstrap bounds on the scalar gap for central charge $c<7$.

What carries the argument

The carrying object is the MWK-subtracted primary partition function $\hat Z_p = Z_p - Z^{\mathrm{gravity}}$, where $Z^{\mathrm{gravity}}$ is the Poincaré-sum completion of all light primary states. Its scalar part is decomposed spectrally into the constant (the modular integral), a continuum of Eisenstein series, and Maass cusp forms; the cusp forms drop out of the scalar sector. The overlap $(\hat Z_p,E_s)$ is evaluated by unfolding, turning it into an integral of the scalar density against $\Delta^{1/2-s}$; the assumption (3.26) justifies exchanging integrals and deforming the $s$-contour. The factor $\zeta(2s)/\zeta(2s-1)$ in the functional identity places poles at $s=(1+z_k)/2$, which produce the zeta-zero terms, while the residue at $s=1/2$ fixes the light-state contribution. Functionals that annihilate powers $t^{1-z_k}$ convert the second crossing equation into the first one, Eq. (1.6), which is sign-definite and suitable for bootstrap bounds.

What would settle it

Compute the MWK-subtracted scalar density $\rho_p^{\mathrm{scalars}}(\Delta) - \sum \rho_{\mathrm{MWK}}^{\mathrm{scalars}}(\Delta)$ for a CFT with a known spectrum (e.g., a compactified free boson at irrational radius) and fit the large-$\Delta$ decay: if the falloff is slower than $O(\Delta^{-1/2})$ — say $O(\Delta^{-1/4})$ — or has oscillations of that amplitude, the integral exchange behind (3.33) breaks down. Alternatively, evaluate both sides of (3.33) at two small values of $y$ with the zeta-zero sum truncated; any $y$-dependent discrepancy larger than the exponentially small tail would falsify the identity.

Watch

Extended reading notes

Core claim

The central discovery is that the scalar sector of a 2d CFT is self-consistent: the scalar Virasoro primaries obey Eq. (1.6), a crossing equation with no reference to spinning operators, and the more powerful Eq. (3.33). In the second equation, $Z_p^{\mathrm{scalars}}(y)$ minus the scalar part of the MWK gravity partition function equals $\frac{3}{\pi}(\hat Z_p,1) + \sum_k \mathrm{Re}(\delta_k y^{(1+z_k)/2})$ plus perturbative terms fixed by the light spectrum and a nonperturbative sum over heavy scalar states. The coefficients $\delta_k$ are theory-dependent residues of the spectral overlap of the subtracted partition function with Eisenstein series, evaluated at the points $s=(1+z_k)/2$ selected by zeros of $\zeta(2s-1)$. This shows that the high-temperature (small $y$) behavior of any modular-invariant primary partition function knows about the nontrivial zeros of the Riemann zeta function, and that the scalar spectrum alone determines the regulated modular integral.

Load-bearing premise

The argument assumes that after subtracting the reference spectrum built from light states, the scalar primary density falls off at least as $O(\Delta^{-1/2})$ with the scaling dimension $\Delta$, so integrals and contour deformations can be exchanged; the paper checks this numerically for solvable theories but does not prove it for all 2d CFTs.

Editorial extensions

If this is right

  • For c<7, crossing equation (1.6) gives a positive sum rule; using it with semidefinite programming yields rigorous upper bounds on the scalar gap for 1<c<7, reported in Table 1.
  • Identity (3.33) computes regulated modular integrals from scalar spectra, including cases with c>7 where (1.6) diverges; the paper demonstrates this for the (E8)_1 and ((E8)_1)^4 WZW models and for meromorphic modular functions.
  • If the Riemann hypothesis is true, the nonperturbative zeta-zero series in (3.33) has an envelope of size y^{3/4} at small y; the paper notes the converse would follow if every zeta zero has at least one CFT with nonzero delta_k.
  • The same subtraction strategy gives a formal crossing equation for each spin sector, Eq. (8.3), with Maass cusp-form contributions; for the c=1 free boson it reproduces the known spin-j spectrum.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct test of the assumed O(Delta^{-1/2}) falloff in non-rational exactly solvable theories (for instance, free bosons at generic radius) would tell how close this is to a theorem.
  • Analytically continuing y to y+it in (3.33) gives a spectral form factor whose late-time behavior might exhibit statistics inherited from the zeta zeros; the paper does not perform this continuation, but the structure invites it.
  • If the residues delta_k can be extracted reliably for individual CFTs, their pattern could serve as a measure of how 'chaotic' a spectrum is: rational CFTs with large degeneracies should make the MWK subtraction nearly exact, while chaotic theories might display larger zeta-zero oscillations.
  • The r-plane pole structure suggests a possible analytic continuation of (1.6) beyond c=7; if found, it would extend scalar-gap bounds into the holographic regime.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies the scalar primary sector of the torus partition function of an arbitrary 2d CFT. Its first main result is Eq. (1.6), a crossing equation for the scalar Virasoro primary density with an r-independent constant epsilon, obtained by unfolding a Petersson inner product against a c=1 free-boson seed. Its second and more elaborate result is Eq. (3.33), an exact high-temperature expansion that expresses the difference between the scalar partition function and the MWK/gravity completion of the light spectrum as the sum of a modular integral term, an infinite series controlled by nontrivial zeros of the Riemann zeta function, terms fixed by the light spectrum, and exponentially small corrections from heavy scalars. The derivation uses harmonic analysis on the modular surface, subtraction of MWK Poincare sums, contour deformation in the Eisenstein spectral parameter, and linear functionals. The paper also contains analytic checks for Liouville theory, MWK and Rademacher sums, numerical fits for WZW models including c>7 cases, and semidefinite-programming bounds on the scalar gap for c<7.

Significance. If the main theorem were established for all 2d CFTs, it would be a substantial extension of the scalar modular bootstrap: it moves from Narain lattices to generic Virasoro primaries and produces a striking structural connection between the high-temperature asymptotics of scalar partition functions and the nontrivial zeros of the Riemann zeta function. The analytic machinery, including the explicit MWK scalar densities and the derivation of sum rules from spectral decomposition, is valuable independent of the final theorem. The paper also demonstrates a practical route to computing regulated modular integrals, with explicit analytic checks and several nontrivial WZW examples. However, the central universal claim is conditional on an unproven spectral falloff assumption, and the numerical evidence does not close that gap; this limits the current significance to a well-developed but conditional framework.

major comments (3)
  1. [Sec. 3.2.2, Eq. (3.26)] The load-bearing premise of the derivation is stated as an assumption and is not proven, and as written it is not even well-defined pointwise. For a theory with infinitely many scalar primaries, the left-hand side of Eq. (3.26) is a sum of delta functions with degeneracies growing like the Cardy density, so it cannot be O(Delta^{-1/2}) pointwise as Delta tends to infinity; it must be interpreted in some smeared or integrated sense, but no such regularization is defined. This matters because the exchange of integrals leading to Eq. (3.27), and hence the contour deformation leading to Eq. (3.33), rely exactly on this falloff. The numerical evidence in Sec. 6 involves rational WZW models and checks the final crossing equation rather than Eq. (3.26) directly, and footnote 8, which weakens the assumption to 'not superpolynomial,' is likewise unproven. The universal claim for arbitrary 2d CFTs is therefore not established.
  2. [Introduction and Sec. 5] The paper states that Eq. (1.6) can be defined for c>7 by analytic continuation in r, but no precise analytic-continuation procedure is supplied. Section 5 analyzes pole locations in the complex r plane but concludes only that it would be 'extremely interesting' to extend convergence beyond c=7; it does not provide the continuation. Since the abstract and introduction claim validity for any 2d CFT without a c restriction, the treatment of the c>7 regime is currently not justified. This is not merely a cosmetic issue, because the numerical uses of Eq. (3.33) in Sec. 6 are presented as going beyond the c<7 convergence window of Eq. (1.6).
  3. [Sec. 6, Eqs. (6.4), (6.13), (6.16)] The numerical extractions of epsilon and delta_k are presented without error bars, fit ranges, or systematic convergence checks with respect to the cutoff in the zeta-zero sum and the neglected nonperturbative terms. For instance, Eq. (6.4) quotes Re(epsilon) ~ -15.07 to two decimals, and the fits in Figs. 1-4 use different cutoffs (5, 5, 20) without demonstrating stability in the cutoff or quantifying the contribution of omitted zeros. Since these numbers are offered both as evidence for Eq. (3.33) and as definite predictions for modular integrals, the numerical support is weaker than stated.
minor comments (4)
  1. [Table 1] The c=1.3 row is non-monotonic in nmax: the bound is 0.55203 at nmax=30 but 0.57043 at nmax=40. Please check the numerics or explain why the bound can increase with the number of derivatives.
  2. [Abstract] The phrase 'and an infinite series terms directly related to the nontrivial zeros' should read 'and an infinite series of terms directly related to the nontrivial zeros.'
  3. [Figs. 1-4] The vertical axes of Figs. 1-4 are unlabeled, and the horizontal axis is written as 'log(T)' without specifying the base; please label the plotted quantity explicitly and state the base.
  4. [Sec. 4.4] The sentence 'We have checked that their scalar Virasoro primaries numerically obey our crossing equation to arbitrarily high precision' is not quantified; please specify the numerical precision and the range of parameters checked.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the scalar crossing equations are derived from modular invariance, unfolding, and spectral decomposition; the central unproven input (3.26) is an explicit assumption, not a fitted or self-referential one.

full rationale

The derivation chain is self-contained in the relevant sense. Equation (3.21) follows from the spectral decomposition of the subtracted partition function \hat Z_p = Z_p - Z_gravity, with the harmonic-analysis input (3.2)/(3.10) cited to the textbook [14] and to prior work [15] that is itself an externally checkable technique, not the target result. The quantities \epsilon and \delta_k are introduced as integrals/residues: \epsilon via (1.11) and (3.44), and \delta_k via the residue definition after (3.28), so their numerical extraction in Sec. 6 is an output or check, not a fitted input renamed as a prediction. The only fragile step, Eq. (3.26), is explicitly labeled an assumption with numerical evidence in Sec. 6 and is even weakened in footnote 8; an unproven assumption is a correctness or rigor risk, not circularity, because the paper does not define (3.26) in terms of the conclusion (3.33). External anchors also exist: the SU(2)_1 modular integral agrees with [12] (Fig. 2), and the meromorphic-function integral (6.20) agrees with [12, 22]. Minor self-citations to [1] and [15] are to prior methods, and per the review rules these are independent support rather than load-bearing self-reference. I therefore find no step that reduces, by construction, to its own input.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The derivation relies on standard harmonic analysis and an explicit but unproven convergence assumption. No new physical entities are introduced.

assumptions (5)
  • domain assumption Spectral decomposition of modular functions with polynomial growth (Rankin-Selberg / Zagier extension).
    Used in Section 3.1 to expand Z_p_hat and derive Eq. (3.21).
  • ad hoc to paper The subtracted scalar density satisfies O(Delta^{-1/2}) falloff as Delta goes to infinity (Eq. 3.26).
    Load-bearing for unfolding and contour deformation; only numerical evidence is given, no proof.
  • domain assumption Finiteness of the light spectrum L union J (primaries with Delta <= 0).
    Needed for the MWK subtraction in Section 3.2.1; standard for compact 2d CFTs.
  • ad hoc to paper Interchange of sums and integrals in Eq. (3.24) and (3.28) for Re(s) > 1.
    Assumed under the O(Delta^{-1/2}) condition; not proven in full generality.
  • standard math Standard facts about the Riemann zeta function and the functional equation of the Eisenstein series.
    Used in Eqs. (3.6), (3.23), and (3.29).

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Cite this review

Pith. "Pith review of Properties of scalar partition functions of 2d CFTs." pith.science (2026). https://pith.science/paper/KEH5CI3R

@misc{pith2026250518314,
  author       = {Pith},
  title        = {Pith review of: Properties of scalar partition functions of 2d CFTs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KEH5CI3R}},
  note         = {Machine review of arXiv:2505.18314}
}
read the original abstract

We study the spectrum of scalar primary operators in any two-dimensional conformal field theory. We show that the scalars alone obey a nontrivial crossing equation. This extends previous work that derived a similar equation for Narain conformal field theories. Additionally, we show that at high temperature, the difference between the true scalar partition function and the one predicted from a semiclassical gravity calculation is controlled by: the modular integral of the partition function, the light states of the theory, and an infinite series terms directly related to the nontrivial zeros of the Riemann zeta function. We give several numerical examples and compute their modular integrals.

Figures

Figures reproduced from arXiv: 2505.18314 by the authors.

Figure 1
Figure 1. A plot of the scalar partition function of the Virasoro primary (E8)1 WZW partition function with the MWK spectrum subtracted out. The resulting function is fitted to the large T prediction in (3.33), namely Re(ε) − P∞ k=1 Re  δkT − 1+zk 2  (with the sum in k cut off at 5) to high precision. The fit gives Re(ε) ≈ −15.07. In [PITH_FULL_IMAGE:figures/full_fig_p027_1.png] view at source ↗
Figure 2
Figure 2. A plot of the scalar partition function of the (SU(2))1 WZW partition function with the MWK spectrum subtracted out as in the real part of (6.12). The resulting function is fitted to the large T prediction in (3.33), namely Re(ε) − P∞ k=1 Re  δkT − 1+zk 2  (with the sum in k cut off at 5) to high precision. The fit gives Re(ε) ≈ 1.50003. This is consistent with [PITH_FULL_IMAGE:figures/full_fig_p031_2.png] view at source ↗
Figure 3
Figure 3. A plot of the scalar partition function of the (E8)1 WZW partition function with the MWK spectrum subtracted out as in the real part of (6.12). The resulting function is fitted to the large T prediction in (3.33), namely Re(ε) − P∞ k=1 Re  δkT − 1+zk 2  (with the sum in k cut off at 5) to high precision. The fit gives Re(ε) ≈ −23.823. The second example we will do is the (E8)1 WZW model again. Compared to the prev… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: A plot of the scalar partition function of the ((E8)1) 4 WZW partition function with the light spectrum subtracted out as in (6.15). Unlike previous examples, this theory has states with E < 0 of nonzero spin. The resulting function is fitted to the large T prediction …
Figure 5
Figure 5. Figure 5: Plot of a bound on the scalar gap for general 2d CFTs at various c < 7. The colors blue, green, red, and purple represent the bound we get at 10, 20, · · · , 40 derivatives respectively. See [PITH_FULL_IMAGE:figures/full_fig_p037_5.png]

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