Bounds on $\mathcal{N}=1$ Superconformal Theories with Global Symmetries

TL;DR

The paper develops a non-Abelian extension of the conformal bootstrap for four-dimensional N=1 SCFTs with SU(N) global symmetry by analyzing the 4-point function of the current multiplet top component J^a. It derives the full set of SU(N) adjoint scalar sum rules, constructs non-Abelian superconformal blocks, and computes how SUSY constrains the J^a×J^b OPE, including new operators beyond previous Abelian analyses. Using these results, it obtains a rigorous lower bound on the current central charge τ, with τ growing with N and constraints sharpened by potential spectral gaps; the Abelian case is also explored to bound OPE coefficients. The findings illuminate the minimal amount of charged content in any such SCFT and connect to holographic interpretations via a bound on bulk gauge coupling parameters. Overall, the work extends the conformal bootstrap toolkit to non-Abelian supersymmetric currents and provides quantitative bounds on current data in a broad class of theories.

Abstract

Recently, the conformal-bootstrap has been successfully used to obtain generic bounds on the spectrum and OPE coefficients of unitary conformal field theories. In practice, these bounds are obtained by assuming the existence of a scalar operator in the theory and analyzing the crossing-symmetry constraints of its 4-point function. In $\mathcal{N}=1$ superconformal theories with a global symmetry there is always a scalar primary operator, which is the top of the current-multiplet. In this paper we analyze the crossing-symmetry constraints of the 4-point function of this operator for $\mathcal{N}=1$ theories with $SU(N)$ global symmetry. We analyze the current-current OPE, and derive the superconformal blocks, generalizing the work of Fortin, Intrilligator and Stergiou to the non-Abelian case and finding new superconformal blocks which appear in the Abelian case. We then use these results to obtain bounds on the coefficient of the current 2-point function.

Figures (2)

• Figure 1: Lower bounds on $\tau$ for different gauge groups $SU(N)$, obtained for different search space sizes $k$.
• Figure 2: Lower bounds on $\tau$ for different gauge groups $SU(N)$ with high values of $N$, obtained for different search space sizes $k$.