Descending into the Modular Bootstrap

2026-04-01

AI Summary

This work explores the landscape of two-dimensional conformal field theories (2d CFTs) in the central charge regime \(1 < c < 8/7\), where no explicit examples are currently known, with the aim of constructing candidate spectra satisfying modular invariance. By encoding modular invariance as a loss function and combining a low-dimensional operator truncation scheme with a newly introduced singular-value-based optimizer—dubbed Sven—the approach effectively navigates the hierarchical structure of the loss landscape. Additionally, a strategy for estimating truncation-induced spectral uncertainties is incorporated. The method successfully generates multiple viable partition functions, providing evidence for a continuous solution space of the modular bootstrap in this central charge interval and yielding improved bounds on the spectral gap near \(c \approx 1\), surpassing the existing constraint \(\Delta_{\text{gap}} \leq c/6 + 1/3\).

Abstract

In this paper, we attempt to explore the landscape of two-dimensional conformal field theories (2d CFTs) by efficiently searching for numerical solutions to the modular bootstrap equation using machine-learning-style optimization. The torus partition function of a 2d CFT is fixed by the spectrum of its primary operators and its chiral algebra, which we take to be the Virasoro algebra with $c>1$. We translate the requirement that this partition function is modular invariant into a loss function, which we then minimize to identify possible primary spectra. Our approach involves two technical innovations that facilitate finding reliable candidate CFTs. The first is a strategy to estimate the uncertainty associated with truncating the spectrum to the lowest dimension operators. The second is the use of a new singular-value-based optimizer (Sven) that is more effective than gradient descent at navigating the hierarchical structure of the loss landscape. We numerically construct candidate truncated CFT partition functions with central charges between 1 and $\frac{8}{7}$, a range devoid of known examples, and argue that these candidates likely come from a continuous space of modular bootstrap solutions. We also provide evidence for a more stringent constraint on the spectral gap near $c = 1$ than the existing bound of $Δ_{\rm gap} \le \frac{c}{6} + \frac{1}{3}$.

Problem

• modular bootstrap
• 2d CFT
• spectral gap
• central charge
• modular invariance

Innovation

• modular bootstrap
• machine-learning optimization
• singular-value optimizer
• truncated spectrum uncertainty
• 2d CFT