Bootstrapping SCFTs with Four Supercharges

TL;DR

The paper develops a dimensionally continuous framework for four-supercharge superconformal algebras and constructs corresponding superconformal blocks across 2≤d≤4 using a Casimir-based approach. It then applies numerical bootstrap to bound the leading unprotected scalar in the chiral–antichiral OPE, revealing three kinks that interpolate between the d=2 𝒩=(2,2) minimal model (c=1) and the d=4 free chiral multiplet via the critical Wess-Zumino model, with strong support that the first kink corresponds to the cWZ fixed point in all dimensions. The work also analyzes OPE data and the central charge C_T, benchmarking against known 2d minimal models and providing detailed d-dependent predictions with ε-expansion consistency, supported by a companion 3d study. Overall, the results illustrate a coherent, dimension-agnostic bootstrap landscape for SCFTs with four supercharges and identify promising directions for disentangling the remaining kinks and their physical realizations.

Abstract

We study the constraints imposed by superconformal symmetry, crossing symmetry, and unitarity for theories with four supercharges in spacetime dimension $2\leq d\leq 4$. We show how superconformal algebras with four Poincaré supercharges can be treated in a formalism applicable to any, in principle continuous, value of $d$ and use this to construct the superconformal blocks for any $d\leq 4$. We then use numerical bootstrap techniques to derive upper bounds on the conformal dimension of the first unprotected operator appearing in the OPE of a chiral and an anti-chiral superconformal primary. We obtain an intriguing structure of three distinct kinks. We argue that one of the kinks smoothly interpolates between the $d=2$, $\mathcal N=(2,2)$ minimal model with central charge $c=1$ and the theory of a free chiral multiplet in $d=4$, passing through the critical Wess-Zumino model with cubic superpotential in intermediate dimensions.

Figures (13)

• Figure 1: Upper bound on the lowest-dimension neutral scalar operator, $[\Phi \bar{\Phi}]$, appearing in the $\Phi \times \bar{\Phi}$ OPE. The dashed vertical lines correspond to $\Delta_\Phi=\frac{d-1}{3}$, the protected dimension of $\Phi$ in the cWZ model in dimension $d$. The value of $d$ associated to a line is indicated by its color, which matches the corresponding bound plot.
• Figure 2: A close-up of the bounds in Figure \ref{['fig:epsbound']}. Note that the first kink in every dimension corresponds to $\Delta_\Phi=\frac{d-1}{3}$ (the locations of the vertical lines).
• Figure 3: The central charge, $C_T$, of the boundary solution, i.e. when $\Delta_{[\Phi \bar{\Phi}]}$ saturates the bounds given in Figure \ref{['fig:epsbound']}. The crosses denote the value of $C_T$ for a free chiral multiplet in dimension $d$. The dashed vertical lines lie at $\Delta_\Phi=\frac{d-1}{3}$, corresponding to the chiral primary field of the cWZ model in dimension $d$.
• Figure 4: A close-up of the curves in Figure \ref{['fig:CT']}. The minimum in every dimension exactly corresponds to $\Delta_\Phi=\frac{d-1}{3}$ (the locations of the vertical lines). Note that $C_T$ in $d=2$ lies precisely at 2, corresponding to the known value $c=\bar{c} = 1$ of the lowest $\mathcal{N}=2$ minimal model (see Section \ref{['sec:2d_minmodels']}).
• Figure 5: An extended view of the upper bound on $\Delta_{[\Phi \bar{\Phi}]}$ in $d=2$ (with $n_{max}=9$). The blue crosses mark the exact dimensions of operators from various superconformal minimal models. The cross at $(\frac{1}{3}, 2)$ corresponds to the super-Ising model (i.e. the $k=1$ super-Virasoro minimal model). The dashed green like corresponds to $\Delta_{[\Phi\bar{\Phi}]}=2 \Delta_\Phi$, the expected value in mean field theory.