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The spinless modular bootstrap at large central charge

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

Contributed by Anthropic

Version 1 / Nov 07, 2026 / CC BY 4.0

Abstract

The spinless modular bootstrap bounds the dimension Δ1\Delta_1 of the lightest primary other than the vacuum in a two-dimensional conformal field theory, using only modular SS invariance of the torus partition function at zero angular potential. For unitary, compact theories with central charge c>1c>1 and no conserved currents beyond the Virasoro algebra, these bounds read Δ1≤c/Λ+o(c)\Delta_1\le c/\Lambda+o(c) at large cc. The best slope Λ\Lambda was known only to lie between 8.5038.503 and 1212, where the bound would reach the BTZ black hole threshold. We show that it is exactly ΛV=12π2/(π2+4 arcsinh21)≈9.1266\Lambda_V=12\pi^2/(\pi^2+4\,\mathrm{arcsinh}^2 1)\approx9.1266. No linear functional in a class that contains all finite combinations of derivatives improves on c/ΛVc/\Lambda_V by more than O(c1/3log⁡2/3c)O(c^{1/3}\log^{2/3}c), and an explicit functional gives Δ1<c/ΛV+0.483 c1/3+O(1)\Delta_1<c/\Lambda_V+0.483\,c^{1/3}+O(1) for all sufficiently large cc. The value has an electrostatic origin: as a function of a complexified dimension, c−1log⁡∣g∣c^{-1}\log\lvert g\rvert of a functional gg is a two-dimensional potential with positive charges at the zeros of gg. Positivity forces gg to fall as fast as the Cardy density of states grows, and the maximum principle bounds the potential by an explicit harmonic function, which fixes ΛV\Lambda_V. The field of this potential is the spectral curve of a mean-field theory of the optimal functionals, and its charge density is the distribution that the zeros of any functionals attaining ΛV\Lambda_V must follow. We also conjecture the coefficient of the c1/3c^{1/3} correction to the optimal bound, supported by numerics up to c=2048c=2048.

Provenance statement

The main result was first obtained autonomously by an internal Claude research model. The publicly available Claude Opus 5.5 later obtained the same result independently. Claude then wrote this paper, following brief guidance from a physicist, who also checked every main result.

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  1. v1 · 131 pagesDeposited by AnthropicInitial depositCurrentNov 07, 2026