Fast Conformal Bootstrap and Constraints on 3d Gravity

TL;DR

The paper introduces a fast modular bootstrap algorithm built on truncating the crossing equations to a finite polynomial system, and demonstrates that such truncations yield rigorous bounds on the spectra of unitary CFTs, connecting primal solutions to extremal functionals via a duality framework. It extends modular bootstrap into the semiclassical, large-$c$ regime, achieving bounds like $\Delta_1 \lesssim c/9.1$ up to $c\sim1800$ and obtaining high-precision spectral data across thousands of operators, including a striking link to the modular $j$-function at $c=12$. The method shows orders-of-magnitude speedups over traditional semidefinite programming, with robust numerical stability and a practical spectrum-guess generator that seeds Newton iterations, enabling large derivative orders $P$. While the current focus is on spinless modular bootstrap, the results offer a path toward spinning modular bootstrap and new insights into 3d gravity, BTZ black holes, and related holographic questions.

Abstract

The crossing equations of a conformal field theory can be systematically truncated to a finite, closed system of polynomial equations. In certain cases, solutions of the truncated equations place strict bounds on the space of all unitary CFTs. We describe the conditions under which this holds, and use the results to develop a fast algorithm for modular bootstrap in 2d CFT. We then apply it to compute spectral gaps to very high precision, find scaling dimensions for over a thousand operators, and extend the numerical bootstrap to the regime of large central charge, relevant to holography. This leads to new bounds on the spectrum of black holes in three-dimensional gravity. We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge $c$, is $Δ_1 \lesssim c/9.1$.

Figures (5)

• Figure 1: Example of an extremal functional for modular bootstrap, with $c=12$ and truncation order $P=6$. There are single zeroes at $\Delta_0 \approx 0$ and $\Delta_1 \approx 2.13$, and double zeroes at $\Delta_{2}\approx 3.43$ and $\Delta_3 \approx 5.13$. The additional single zero near $\Delta \approx 1$ plays no role in the discussion. (The root near the origin is slightly shifted due to null state contributions described in section \ref{['ss:modularreview']}, but this too small to be visible in the plot.)
• Figure 2: Spectrum of the $c=12$ solution at truncation order $P=110$ (left) and $P=200$ (right). The horizontal axis is the state number, $\mu = 1, 2,\dots,P/2$. Note that the curves are approximately the same shape. This observation is used to generate very accurate guesses for the initial point in Newton's method.
• Figure 3: Upper bound on $\Delta_1/c$, as a function of $c$. Dots are numerical data for truncation at $P =$ 270, 310, 350, 390, 470, 550, 630, 710, 870, 1030, 1350, 1670, 1990, 2310, from top to bottom. The solid red line is the extrapolation to $P = \infty$. The dashed blue line is the asymptotic estimate at large $c$, $1/9.08$.
• Figure 4: The numerical spectrum $\Delta_\mu$ for $c=500$ and $P=2310$. In the larger plot we show every 15th scaling dimension to reduce clutter. The smaller inset figure shows the first 30 scaling dimensions with deviation away from linearity. In upcoming, the linear regime of the spectrum with $\Delta_\mu = \frac{c-4}{8} + \mu$ (dashed line) will be explained analytically.
• Figure 5: Runtime of SDPB and our Newton-based algorithm for modular bootstrap at c=12, vs the number of polynomials, $P$. Larger $c$ requires more precision, so runtimes are somewhat longer.