The ${\mathcal N}=2$ superconformal bootstrap

TL;DR

This work launches a systematic conformal bootstrap program for four-dimensional ${ emph{N}=2}$ SCFTs by adopting an operator-algebraic framework that covers both Lagrangian and non-Lagrangian theories. It develops the superconformal block structure and Ward identities for two key four-point functions—the moment-map correlator and the ${ hspace{-0.2pt}{ obreak ext{E}}_r}$ chiral correlator—and uses semidefinite programming to extract universal bounds on central charges, operator dimensions, and OPE coefficients. The study provides explicit analytic inputs for short multiplets via chiral algebra data (e.g., flavor central charges $k$ and conformal anomalies $c$) and presents extensive numerical bounds for ${rak{su}}(2)$ and ${rak{e}}_6$ flavor symmetries, connecting them to known rank-one theories and defect SCFTs. It demonstrates the potential to chart the ${ emph{N}=2}$ SCFT landscape and to constrain non-Lagrangian theories that evade traditional Lagrangian descriptions, with implications for class ${ m S}$ constructions and Argyres–Douglas points.

Abstract

In this work we initiate the conformal bootstrap program for ${\mathcal N}=2$ superconformal field theories in four dimensions. We promote an abstract operator-algebraic viewpoint in order to unify the description of Lagrangian and non-Lagrangian theories, and formulate various conjectures concerning the landscape of theories. We analyze in detail the four-point functions of flavor symmetry current multiplets and of ${\mathcal N}=2$ chiral operators. For both correlation functions we review the solution of the superconformal Ward identities and describe their superconformal block decompositions. This provides the foundation for an extensive numerical analysis discussed in the second half of the paper. We find a large number of constraints for operator dimensions, OPE coefficients, and central charges that must hold for any ${\mathcal N}=2$ superconformal field theory.

Figures (27)

• Figure 1: The two inequivalent OPE channels for the $\mathcal{E}_{r}$ four-point function.
• Figure 2: The $(c,k)$ plane for theories with an ${\mathfrak{su}}(2)$ flavor symmetry. The red region on the right is excluded by analytic unitarity bounds, whereas we are guaranteed to have valid solutions to the crossing symmetry constraints in the blue region. The curves connect points corresponding to theories related to F-theory singularities of different rank, which increases with $c$. We show the $(c,k)$ values corresponding the ${\mathfrak{su}}(2)_L$ flavor symmetry of all F-theory singularities with rank $N \geqslant 2$, and also to ${\mathfrak{su}}(2)$ flavor symmetry of the rank $N \geqslant 1$$H_1$ theory. We also show a curve connecting points corresponding to the $(c,k)$ values of $\mathcal{N}=4$ SYM with gauge group $SU(N)$. The "new" rank one theory is one of the theories obtained in Argyres:2007tq. It has a product flavor symmetry with one factor being ${\mathfrak{su}}(2)$, which is the one whose value of $k$ is shown in the plot. The vertical dotted line corresponds to the value of $k$ for the codimension two defect of the six-dimensional $(2,0)$ theory of type $A_1$, which effectively has $c\to\infty$.
• Figure 3: Bounds for the central charge $c$ of a theory with ${\mathfrak{su}}(2)$ flavor symmetry as a function of the flavor central charge $k$. These bounds are a consequence of crossing symmetry for the $\hat{\mathcal{B}}_1$ four-point function. The red regions on the right are excluded by the analytic bounds \ref{['analytic_bound']}, and the gray region at the bottom is the numerically excluded region. The gray and black lines correspond to the numerical bounds, shown for $\Lambda=10,14,\ldots,30$, with the strongest bound (black line) corresponding to $\Lambda = 30$. The curves are interpolations through the data points shown in the figure. The red dot denotes the free hypermultiplet theory.
• Figure 4: Minimum allowed value of $c$ for a theory with ${\mathfrak{su}}(2)$ flavor symmetry and $k=1$ as a function of the (inverse of) the maximum number of derivatives. The red dots are our data points, and the blue curves are possible extrapolations to infinite $\Lambda$ intended to guide the eye. The dashed line corresponds to the central charge of the free hypermultiplet $c=\frac{1}{12}$.
• Figure 5: Upper bounds for the dimension of the first unprotected singlet scalar operator in theories with ${\mathfrak{su}}(2)$ flavor symmetry, as a function of $1/k$ and $1/c$. The cutoff used for this plot was $\Lambda=18$. The two- and a three-dimensional plots are generated with the same data set. The gray and light red surfaces in the figure are the excluded regions from Fig. \ref{['Fig:SU2-cbound']}, and the vertical red wall is added help visualize the constraints imposed by the analytic bounds. The black dot is the generalized field theory solution to crossing.

Theorems & Definitions (5)

• Conjecture 1: CFT Noether "theorem"
• Conjecture 2: Free generation of the Coulomb chiral ring
• Conjecture 3: Geometrization of the Coulomb chiral ring
• Conjecture 4: Decomposability
• Conjecture 5: Geometrization of the Higgs chiral ring