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 Δ1 ≲ c/9.1.
Highlights
Tonni, and Vichi [3]
We provide numerical evidence that the asymptotic bound on the spectral gap from spinless modular bootstrap, at large central charge c, is ∆1 c/9.1
The method relies on finding extremal functionals, often numerically, which bound the space of CFTs, and in certain cases encode the spectrum of the target theory [7, 8]
Summary
The functional bootstrap is an optimization problem, and as such, it has a primal and dual formulation. The primal bootstrap refers to the crossing equations themselves. (assuming the Fi are sufficiently well behaved to bring the derivatives inside the sum) We will restrict our attention to bootstrap problems without spin dependence In this case we set z = z (or τ = −τin the modular bootstrap), and drop the spin label l. This class of problems includes spinless modular bootstrap, 1d correlators, the sl(2) subsector of correlator bootstrap in higher dimensions
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