Bootstrapping $\mathcal{N}=3$ superconformal theories

TL;DR

This work launches a concrete bootstrap program for four-dimensional ${ m N}=3$ SCFTs by combining a protected 2d chiral algebra sector with crossing symmetry of half-BPS Coulomb-branch operators. It introduces a new 2d chiral algebra with super Virasoro symmetry that depends on a central charge and derives the corresponding superconformal blocks and crossing equations for the Coulomb-branch operators, focusing on dimensions two and three. Using both analytical chiral-algebra data and numerical SDPB bounds, the authors constrain 4d CFT data, fix certain Schur-sector OPEs, and demonstrate how to exclude ${ m N}=4$ solutions to zoom in on a minimal ${ m N}=3$ theory with known central charge and OPE coefficients. The results illustrate how protected subsectors can be bootstrapped to produce universal constraints, and how chiral-algebra inputs can sharpen central-charge bounds and OPE data, paving the way for a more detailed map of the ${ m N}=3$ landscape and its higher-rank generalizations.

Abstract

We initiate the bootstrap program for $\mathcal{N}=3$ superconformal field theories (SCFTs) in four dimensions. The problem is considered from two fronts: the protected subsector described by a $2d$ chiral algebra, and crossing symmetry for half-BPS operators whose superconformal primaries parametrize the Coulomb branch of $\mathcal{N}=3$ theories. With the goal of describing a protected subsector of a family of $\mathcal{N}=3$ SCFTs, we propose a new $2d$ chiral algebra with super Virasoro symmetry that depends on an arbitrary parameter, identified with the central charge of the theory. Turning to the crossing equations, we work out the superconformal block expansion and apply standard numerical bootstrap techniques in order to constrain the CFT data. We obtain bounds valid for any theory but also, thanks to input from the chiral algebra results, we are able to exclude solutions with $\mathcal{N}=4$ supersymmetry, allowing us to zoom in on a specific $\mathcal{N}=3$ SCFT.

Figures (7)

• Figure 1: Numerically minimum allowed central charge for the $\hat{{\mathcal{B}}}_{[2,0]}$, $\hat{{\mathcal{B}}}_{[0,2]}$ four-point function as a function of the inverse of the number of derivatives $\Lambda$. The dashed horizontal line marks the central charge of the $\mathrm{U}(1)$$\mathcal{N}=4$ SYM theory. The middle orange line shows a linear fit to all the data points, while the top and bottom blue lines show fits to different subsets of the points.
• Figure 2: Minimum allowed central charge from the correlation function of $\hat{{\mathcal{B}}}_{[3,0]}$ and its conjugate, as a function of the inverse of the number of derivatives $\Lambda$. The dashed horizontal line marks the central charge of the $\mathrm{U}(1)$$\mathcal{N}=4$ SYM theory. The two blue lines show linear fits to different subsets of points, in order to give very rough idea of where the bound is converging to with $\Lambda \to \infty$.
• Figure 3: Upper bound on the OPE coefficient squared of $\hat{{\mathcal{B}}}_{[3,3]}$ versus the inverse central charge $1/c$. The shaded region is excluded and the number of derivatives is increased from 10 to 24 in steps of two. The two green curves show the possible value of the OPE coefficient computed by the chiral algebra in section \ref{['sec:fixingB33']}, while the green dot shows the expected value for the ${\mathcal{N}}=3$ theory of $1/c=0.8$, extracted from the chiral algebra of Nishinaka:2016hbw. The red line and dots corresponds to the solution of ${\mathcal{N}}=4$ SYM theories. The two dashed lines correspond to the minimum central charges for an interacting ${\mathcal{N}}=2$Liendo:2015ofa and ${\mathcal{N}}=3$ SCFTs Cornagliotto:2016 ($c^{-1}=\tfrac{30}{11}\approx 2.73$ and $c^{-1}=\tfrac{24}{13}\approx 1.84$ respectively).
• Figure 4: Upper bound on the OPE coefficient squared of $\bar{{\mathcal{B}}}_{[2,2]}$ ($|\tilde{\lambda}_{\bar{{\mathcal{B}}}_{[2,2]}}|^2$, depicted on the left) and of $\bar{{\mathcal{C}}}_{[0,2],(\tfrac{1}{2},0)}$ ($|\tilde{\lambda}_{\bar{{\mathcal{C}}}_{[0,2],(\tfrac{1}{2},0)}}|^2$, shown on the right) versus the inverse central charge $1/c$. The first vertical dashed line marks $c=\tfrac{13}{24}$ and the second $c=\tfrac{11}{30}$ (the minimal central charges for ${\mathcal{N}}=3$ and ${\mathcal{N}}=2$ interacting theories respectively Liendo:2015ofaCornagliotto:2016). The number of derivatives $\Lambda$ is increased from 10 to 24 in steps of two. The red dots mark the value of this OPE coefficient for generalized free field theory and $\mathrm{U}(1)$${\mathcal{N}}=4$ SYM, while the green line marks the central charge $c=\tfrac{15}{12}$ of the simplest known ${\mathcal{N}}=3$ SCFT, with the green dot providing an upper bound for the OPE coefficients of this theory.
• Figure 5: Upper bound on the dimensions of long multiplets ${\mathcal{A}}_{[0,0],0}^{\Delta>2}$ (left) and ${\mathcal{A}}_{[1,1],0}^{\Delta> 4}$ (right) for different values of the inverse of the central charge $c$. The maximum number of derivatives is $\Lambda=24$, and the weaker bounds correspond to decreasing the number of derivatives by two. The red dots mark the dimension of the first long operators for generalized free field theory and $\mathrm{U}(1)$${\mathcal{N}}=4$ SYM, while the green line marks the central charge $c=\tfrac{15}{12}$ of the simplest known ${\mathcal{N}}=3$ SCFT, with the green dot providing an upper bound for this theory. The two dashed lines correspond to the minimum central charges for an interacting ${\mathcal{N}}=2$Liendo:2015ofa and ${\mathcal{N}}=3$ SCFTs Cornagliotto:2016.

Theorems & Definitions (2)

• Remark 1
• Remark 2