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Integral torus partition functions just above c=1 have no fixed spectral gap

Primarily AI-generated textHuman understanding: some partsmath-ph — Mathematical Physics

Contributed by Tzu Chen Huang ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

We consider modular-invariant torus partition functions of central charge c>1c>1 that have a unique vacuum, an expansion in non-degenerate Virasoro characters with positive integer multiplicities, integer spins, no nonvacuum state at the unitarity bound, and parity symmetry. We prove that for every g>0g>0 there is ϵ(g)>0\epsilon(g)>0 such that no such partition function with 1<c<1+ϵ(g)1<c<1+\epsilon(g) has all nonvacuum primary dimensions at least gg. Thus the spectral gap closes as c→1+c\to 1^+ in this class, as suggested by the numerical bootstrap study of Benjamin, Langhoff, Li and Thaler. The proof is by contradiction and compactness. Using the crystalline-measure reduction of Eberhardt and Levine, we show that every limit at c=1c=1 is an equal-weight average of two compact-boson partition functions, at least one at a half-integer radius. Positive Gaussian test functions adapted to the two circle lattices then force every state of a nearby partition function, through a logarithmically growing energy window, to lie within a power of c−1c-1 of a lattice point. On the other hand, the central-charge shift of the vacuum contributes a second-order term at zero momentum that no distribution supported on circle lattices and compatible with modular invariance can carry; a finite family of Gaussian tests detects this. Limits with two half-integer radii, where lattice points collide, require a separate analysis. The threshold ϵ(g)\epsilon(g) is not effective; we describe the additional estimates that would make it so.

Provenance statement

The proof presented here was developed by GPT-6 Astra under the direction of the author. The writeup was examined and edited by Claude Fable 5.2: five independent referee passes, one for each part of the proof, checked the arguments line by line, and a further pass adjudicated their findings.

References

  1. N. Benjamin, K. Langhoff, W. Li and J. Thaler, Constraining the spectral gap of 2d CFTs just above c=1c=1 , arXiv:2611.07011 (https://arxiv.org/abs/2611.07011).arXiv
  2. L. Eberhardt and N. Levine, Classification of c=1c=1 CFTs , arXiv:2609.16111 (https://arxiv.org/abs/2609.16111).arXiv
  3. S. Hellerman, A universal inequality for CFT and quantum gravity , J. High Energy Phys. 08 (2011) 130, arXiv:0902.2790 (https://arxiv.org/abs/0902.2790).arXiv
  4. D. Friedan and C. A. Keller, Constraints on 2d CFT partition functions , J. High Energy Phys. 10 (2013) 180, arXiv:1307.6562 (https://arxiv.org/abs/1307.6562).arXiv
  5. S. Collier, Y.-H. Lin and X. Yin, Modular bootstrap revisited , J. High Energy Phys. 09 (2018) 061, arXiv:1608.06241 (https://arxiv.org/abs/1608.06241).arXiv
  6. T. Hartman, D. Mazáč and L. Rastelli, Sphere packing and quantum gravity , J. High Energy Phys. 12 (2019) 048, arXiv:1905.01319 (https://arxiv.org/abs/1905.01319).arXiv
  7. J. Kaidi and E. Perlmutter, Discreteness and integrality in conformal field theory , J. High Energy Phys. 02 (2021) 064, arXiv:2008.02190 (https://arxiv.org/abs/2008.02190).arXiv
  8. N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality , J. High Energy Phys. 01 (2021) 130, arXiv:2006.04839 (https://arxiv.org/abs/2006.04839).arXiv
  9. P. J. Cohen, On a conjecture of Littlewood and idempotent measures , Amer. J. Math. 82 (1960), 191--212.

Version history

  1. v1 · 109 pagesSubmitted by Tzu Chen HuangInitial depositCurrentNov 10, 2026