Long Multiplet Bootstrap

Abstract

Applications of the bootstrap program to superconformal field theories promise unique new insights into their landscape and could even lead to the discovery of new models. Most existing results of the superconformal bootstrap were obtained form correlation functions of very special fields in short (BPS) representations of the superconformal algebra. Our main goal is to initiate a superconformal bootstrap for long multiplets, one that exploits all constraints from superprimaries and their descendants. To this end, we work out the Casimir equations for four-point correlators of long multiplets of the two-dimensional global $\mathcal{N}=2$ superconformal algebra. After constructing the full set of conformal blocks we discuss two different applications. The first one concerns two-dimensional (2,0) theories. The numerical bootstrap analysis we perform serves a twofold purpose, as a feasibility study of our long multiplet bootstrap and also as an exploration of (2,0) theories. A second line of applications is directed towards four-dimensional $\mathcal{N}=3$ SCFTs. In this context, our results imply a new bound $c \geqslant \tfrac{13}{24}$ for the central charge of such models, which we argue cannot be saturated by an interacting SCFT.

Figures (4)

• Figure 1: Lower bound on the allowed right central charge $c_R$ (non-supersymmetric side) of ${\mathcal{N}}=(2,0)$ SCFTs as a function of the external dimension, $h={\bar{h}}$, after imposing different gaps on the spectrum of superprimary scalar operators $h_{\mathrm{gap}}={\bar{h}}_{\mathrm{gap}}$. The lines with dots correspond to the full set of crossing equations. The dashed black line corresponds to the bound obtained from the crossing equations of the superconformal primary alone, which matches with the bosonic bootstrap bounds, and is obtained for a single $h_{\mathrm{gap}}=1.45 h$. The red dot marks the central charge and external dimension of the known $(2,2)$ and $(2,0)$ minimal models described below eq. \ref{['hetc32']} (see text for discussion). The bounds were obtained for $\Lambda=20$ which, with the shown scale is enough to have obtained a converged plot as exemplified in figure \ref{['Fig:convergence']}.
• Figure 2: Lower bound on the allowed left central charge $c_L$ (supersymmetric side) of ${\mathcal{N}}=(2,0)$ SCFTs as a function of the external dimension, $h={\bar{h}}$, after imposing different gaps on the spectrum of superprimary scalar operators $h_{\mathrm{gap}}={\bar{h}}_{\mathrm{gap}}$. The lines with dots correspond to the full set of crossing equations. The dashed black line corresponds to the bound obtained from the crossing equations of the superconformal primary alone, which matches with the bosonic bootstrap bounds, and is shown for a single value of $h_{\mathrm{gap}}$. The red dot marks the central charge and external dimension of the known $(2,2)$ and $(2,0)$ minimal models discussed below eq. \ref{['hetc32']} which have identical four-point functions for this external operator. The bounds were obtained for $\Lambda=20$, and the rate of convergence of the numerical bounds is shown in figure \ref{['Fig:convergence']} for $h={\bar{h}}=0.5$.
• Figure 3: Lower bound on the allowed left central charge $c_L$ (supersymmetric side) of ${\mathcal{N}}=(2,0)$ SCFTs for $h={\bar{h}}=0.5$ and $h_{\mathrm{gap}}={\bar{h}}_{\mathrm{gap}}=0.8$, as a function of the inverse of the number of derivatives ($\Lambda^{-1}$) to exemplify the convergence of our numerical results with $\Lambda$.
• Figure 4: Upper bound on the dimension of the first uncharged scalar long superconformal primary that appears in the OPE, as a function of the dimension of the external operator $h={\bar{h}}$ for $\Lambda=16,18,\ldots, 22$ derivatives. The red dot marks the dimension of the known $(2,2)$ and $(2,0)$ minimal models discussed below eq. \ref{['hetc32']}. The orange line corresponds to the generalized free field theory solution $h_{\mathrm{ex}}=2 h$.