Bootstrapping the Three-Dimensional Supersymmetric Ising Model

TL;DR

The conformal bootstrap program for three dimensional conformal field theories with N=2 supersymmetry is implemented and universal constraints on the spectrum of operator dimensions in these theories are found.

Abstract

We implement the conformal bootstrap program for three-dimensional CFTs with $\mathcal{N}=2$ supersymmetry and find universal constraints on the spectrum of operator dimensions in these theories. By studying the bounds on the dimension of the first scalar appearing in the OPE of a chiral and an anti-chiral primary, we find a kink at the expected location of the critical three-dimensional $\mathcal{N}=2$ Wess-Zumino model, which can be thought of as a supersymmetric analog of the critical Ising model. Focusing on this kink, we determine, to high accuracy, the low-lying spectrum of operator dimensions of the theory.

Figures (6)

• Figure 1: Bound on the dimensions of the leading unprotected operator in the $\Phi\times \bar{\Phi}$ OPE at $n_{max}=9$. There can be no unitary SCFTs in the white region. $\Delta_{\Phi}=2/3$ is indicated with a red dashed line. The small shaded rectangle at $\Delta_\Phi=2/3$ indicates the field-of-view in Fig. \ref{['fig:closeup']}.
• Figure 2: Bound on the dimension of the leading unprotected operator in the $\Phi\times \bar{\Phi}$ OPE close to $\Delta_\Phi=2/3$(upper curves), and the corresponding central charge $C_T$ of the solution on the boundary (lower curves). The numbers in parenthesis next to the values of $n_{max}$ indicate the number of constraints imposed to generate the associated curve.
• Figure 3: A closer view of Fig. \ref{['fig:closeup']}. The shaded rectangles indicate our estimated error for $\Delta_{[\Phi\bar{\Phi}]}$ and $C_T$.
• Figure 4: Bound on the dimension of the subleading superconformal scalar primary, $[\Phi\bar{\Phi}]'$, in the $\Phi\times \bar{\Phi}$ OPE. The dimension is extracted from the solution that maximizes $\Delta_{[\Phi\bar{\Phi}]}$.
• Figure 5: Charged scalar spectrum in the vicinity of $\Delta_\Phi=2/3$. (Left:) the first three spin zero operators in the spectrum (colored by order of appearance). The dashed lines correspond to $2\Delta_\Phi$, $d - 2 \Delta_\Phi$ and $2(d-1)-2\Delta_\Phi$ with $d=3$ (see BEMP). (Right:) the OPE coefficients for each operator appearing on the left hand plot (with matching colors). Observe the vanishing of the $\Phi^2$ OPE coefficient at $\Delta_\Phi \simeq 2/3$. The "noisy" operators in the spectrum plots can be seen to have vanishingly small OPE coefficients and thus correspond to small numerical artefacts in the solution.