Decoding a Three-Dimensional Conformal Manifold

TL;DR

The paper analyzes a three-field ${\\cal N}=2$ Wess-Zumino model in 3d, revealing a one-complex-dimensional conformal manifold realized as ${\\bf CP}^1/{\\mathcal S}_4$ with three special self-dual points (XYZ, cWZ${}^3$, and ${\\mathbb Z}_2\times{\\mathbb Z}_2$). It combines a four-loop-accurate $4-\\varepsilon$ expansion with a nonperturbative numerical conformal bootstrap to map operator dimensions and OPE coefficients along the manifold, demonstrating remarkable agreement between the two approaches. The authors also derive exact localization results for $C_T$ and $C_J$, and analyze the chiral ring and Zamolodchikov metric, showing how duality acts monodromically on the spectrum and fixes critical points of certain dimensions. Overall, the work provides a detailed, cross-validated view of a concrete 3d SCFT with an exactly marginal coupling, illustrating how bootstrap and perturbation theory can be synergistic in higher dimensions and offering a template for studying other cubic or multi-field conformal manifolds.

Abstract

We study the one-dimensional complex conformal manifold that controls the infrared dynamics of a three-dimensional $\mathcal{N}=2$ supersymmetric theory of three chiral superfields with a cubic superpotential. Two special points on this conformal manifold are the well-known XYZ model and three decoupled copies of the critical Wess-Zumino model. The conformal manifold enjoys a discrete duality group isomorphic to $S_4$ and can be thought of as an orbifold of $\mathbf{CP}^1$. We use the $4-\varepsilon$ expansion and the numerical conformal bootstrap to calculate the spectrum of conformal dimensions of low-lying operators and their OPE coefficients, and find a very good quantitative agreement between the two approaches.

Figures (13)

• Figure 1: The black lines denote the images of the real line ${\rm Im}\,\tau = 0$ under the various dualities, and therefore each of them is invariant under an appropriate $\mathbb{Z}_2$ reflection subgroup of the duality group. Each cell defines a fundamental domain, and the orange shading is the domain we chose in \ref{['fun']}. The triangles, circles, and crosses are dual to the XYZ, $\text{cWZ}^3$, and $\mathbb{Z}_2\times\mathbb{Z}_2$ theories with $\tau=0,1$ and $(1-\sqrt{3})\omega^2$, which are self-dual under the $S_3$, $S_3$, and $\mathbb{Z}_2\times \mathbb{Z}_2$ subgroups of $S_4$ defined in \ref{['special']}. There is an extra circle at $\tau=\infty$. The plot on the right is focused on a particular fundamental domain and shows additional degeneracies in the spectrum of quadratic operators \ref{['bilinears']} along the boundaries of the fundamental domain.
• Figure 2: The 3-loop Padé[1,2] resummed $4-\varepsilon$-expansion values for the singlet. The cross, circle, and triangle denote values of $\tau$ that correspond to the $\mathbb{Z}_2\times\mathbb{Z}_2$, cWZ$^3$, and XYZ models, respectively.
• Figure 3: The 3-loop Padé[1,2] resummed $4-\varepsilon$-expansion values for the doublets. The cross, circle, and triangle denote values of $\tau$ that correspond to the $\mathbb{Z}_2\times\mathbb{Z}_2$, cWZ$^3$, and XYZ models, respectively.
• Figure 4: Left: Bounds on the unprotected scaling dimensions $\Delta_{\bold6,0}$ and $\Delta_{\bold1,0}$ for the XYZ model in $d=3$, computed using $G\rtimes S_3$ flavor symmetry crossing equations. Right: Bounds on the unprotected scaling dimensions $\Delta_{\bold1,0}$, $\Delta_{\bold4,0}$, and $\Delta_{\bold4',0}$ for the $\mathbb{Z}_2\times\mathbb{Z}_2$ model in $d=3$, computed using $G\rtimes (\mathbb{Z}_2\times\mathbb{Z}_2)$ flavor symmetry crossing equations. In terms of the two-dimensional irreps of $G$, we have $\Delta_{\bold6,0}=\Delta_{\bold2',0}=\Delta_{\bold2",0}=\Delta_{\bold2",0}$, $\Delta_{\bold4,0}=\Delta_{\bold2,0}=\Delta_{\bold2',0}$, and $\Delta_{\bold4',0}=\Delta_{\bold2",0}=\Delta_{\bold2"',0}$. In both plots the orange denotes the allowed region, and we conjecture that the theory lives at the corner. These bounds were computed with $\Lambda=27$.
• Figure 5: Upper bounds on the unprotected scaling dimensions of the scalar singlet and doublets for real $1-\sqrt{3}\leq\tau\leq1+\sqrt{3}$, computed using the $G\rtimes \mathbb{Z}_2$ flavor symmetry crossing equations. The cross, circle, and triangle denote the results from the previous sections for the enhanced symmetry points $\tau=1\pm \sqrt{3}, 1,0$ for the $\mathbb{Z}_2\times\mathbb{Z}_2$, cWZ$^3$, and XYZ models respectively. (For the XYZ model, the top and bottom triangles correspond to the doublets while the middle one corresponds to the singlet. See also Table \ref{['finalResults']} in the Discussion section.) The dotted lines show the 3-loop resummed $4-\varepsilon$-expansion results. These bounds were computed with $\Lambda=19$.