Bootstrapping the $(A_1,A_2)$ Argyres-Douglas theory

TL;DR

This work advances the numerical bootstrap study of the ${\cal N}=2$ Argyres-Douglas theory $(A_1,A_2)$ by concentrating on the four-point function of the Coulomb-branch chiral ring generator. By fixing $\Delta_{\phi}=\frac{6}{5}$ and leveraging the known central charge bound $c_{4d}\!=\frac{11}{30}$, the authors extract rigorous upper and lower bounds on OPE coefficients for a family of semi-short multiplets across spins up to $\ell=20$, and supplement these with large-spin estimates obtained via the Lorentzian inversion formula. The analysis also yields a numerical range for the OPE coefficient of the next chiral-ring operator and estimates for the dimension of the first unprotected, $R$-neutral long multiplet, illustrating how cross-channel data and inversion techniques jointly constrain the spectrum. The results, though not yet as precise as the 3d Ising model, indicate a consistent large-spin structure and suggest a promising hybrid program that combines numerical bootstrap with analytic inversion to “solve” the $(A_1,A_2)$ theory in the near future.

Abstract

We apply bootstrap techniques in order to constrain the CFT data of the $(A_1,A_2)$ Argyres-Douglas theory, which is arguably the simplest of the Argyres-Douglas models. We study the four-point function of its single Coulomb branch chiral ring generator and put numerical bounds on the low-lying spectrum of the theory. Of particular interest is an infinite family of semi-short multiplets labeled by the spin $\ell$. Although the conformal dimensions of these multiplets are protected, their three-point functions are not. Using the numerical bootstrap we impose rigorous upper and lower bounds on their values for spins up to $\ell=20$. Through a recently obtained inversion formula, we also estimate them for sufficiently large $\ell$, and the comparison of both approaches shows consistent results. We also give a rigorous numerical range for the OPE coefficient of the next operator in the chiral ring, and estimates for the dimension of the first R-symmetry neutral non-protected multiplet for small spin.

Figures (7)

• Figure 1: Numerical lower bound (black dots) on the central charge of theories with an ${\mathcal{N}}=2$ chiral operator of dimension $r_{0}=\frac{6}{5}$ as a function of the inverse cutoff $\Lambda$. The lines correspond to various extrapolations to infinitely many derivatives, and the horizontal dashed line marks $c=\tfrac{11}{30}$ -- the central charge of the $(A_1,A_2)$ SCFT.
• Figure 2: Numerical upper bound on the OPE coefficient squared of the operator ${\mathcal{B}}_{1,\frac{7}{5}(0,0)}$ appearing in the chiral channel for $\Lambda=26,\ldots,40$, and external dimension $r_0=\frac{6}{5}$. Left: Upper bound on the OPE coefficient for different values of the central charge, with the strongest bound corresponding to $\Lambda=40$; the dashed lines mark the minimum central charge as extracted from figure \ref{['Fig:c_min']} for each cutoff $\Lambda$, and the solid line marks $c=\tfrac{11}{30}$. Right: Bound on the OPE coefficient for $c=\tfrac{11}{30}$ as a function of the inverse cutoff $\Lambda$, together with various extrapolations to infinitely many derivatives.
• Figure 3: Numerical upper and lower bounds on the OPE coefficient squared of the chiral operator ${\mathcal{E}}_{\frac{12}{5}}$ for increasing number of derivatives and external dimension $r_0=\frac{6}{5}$. Left: Bounds on the OPE coefficient for different values of the central charge, with cutoffs $\Lambda=26,\ldots,40$, the vertical line marks $c=\tfrac{11}{30}$. Right: Various extrapolations of the lower and upper bounds at $c=\tfrac{11}{30}$ for infinite $\Lambda$.
• Figure 4: Numerical upper and lower bounds on the OPE coefficient squared of the chiral channel multiplet $\mathcal{C}_{0,\frac{7}{5}\left(\frac{\ell }{2}-1,\frac{\ell }{2}\right)}$, for $\ell=2,4$, and external dimension $r_0=\frac{6}{5}$. The bounds were obtained for cutoffs $\Lambda=26,\ldots,34$ and the vertical line marks $c=\tfrac{11}{30}$. The dashed line corresponds to the value obtained from the Lorentzian inversion formula of Caron-Huot:2017vep applied to the chiral channel and using as input only the exchange of the identity and stress-tensor superblocks in the non-chiral channel, and thus valid for sufficiently large $\ell$ (see section \ref{['sec:chiral_inv']} for more details).
• Figure 5: Numerical estimates for the first scalar long operator in the non-chiral (a) and chiral (b) channels obtained from the functionals of figures \ref{['Fig:c_min']}, \ref{['Fig:Bbound']}, \ref{['Fig:Ebound']}, and \ref{['Fig:Cl24']}. The data points are color-coded according to the bound the extremal functional was extracted from, and in the cases where the bounds are plotted as a function of $c$, the functional for $c=\tfrac{11}{30}$ was used. The lines give an estimate of the extrapolation to infinitely many derivatives.