Operator Dimensions from Moduli

Abstract

We consider the operator spectrum of a three-dimensional $\mathcal{N}=2$ superconformal field theory with moduli spaces of one complex dimension, such as the fixed point theory with three chiral superfields $X,Y,Z$ and a superpotential $W=XYZ$. By using the existence of an effective theory on each branch of moduli space, we calculate the anomalous dimensions of certain low-lying operators carrying large $R$-charge $J$. While the lowest primary operator is a BPS scalar primary, the second-lowest scalar primary is in a semi-short representation, with dimension exactly $J+1$, a fact that cannot be seen directly from the $XYZ$ Lagrangian. The third-lowest scalar primary lies in a long multiplet with dimension $J+2-c_{-3}{J}^{-3}+O(J^{-4})$, where $c_{-3}$ is an unknown positive coefficient. The coefficient $c_{-3}$ is proportional to the leading superconformal interaction term in the effective theory on moduli space. The positivity of $c_{-3}$ does not follow from supersymmetry, but rather from unitarity of moduli scattering and the absence of superluminal signal propagation in the effective dynamics of the complex modulus. We also prove a general lemma, that scalar semi-short representations form a module over the chiral ring in a natural way, by ordinary multiplication of local operators. Combined with the existence of scalar semi-short states at large $J$, this proves the existence of scalar semi-short states at all values of $J$. Thus the combination of $\mathcal{N}=2$ superconformal symmetry with the large-$J$ expansion is more powerful than the sum of its parts.

48 pages, 8 figures, LaTeX, typos corrected

References in corpus (21)

• Strings in flat space and pp waves from $\mathcal{N}=4$ Super Yang Mills
• Causality, Analyticity and an IR Obstruction to UV Completion
• An Index for 4 dimensional Super Conformal Theories
• The Analytic Bootstrap and AdS Superhorizon Locality
• Rigid Supersymmetric Theories in Curved Superspace
• Chiral Rings and Anomalies in Supersymmetric Gauge Theory
• Convexity and Liberation at Large Spin
• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• Index for three dimensional superconformal field theories with general R-charge assignments
• Operators with large R charge in N=4 Yang-Mills theory
• Exploring Curved Superspace
• Exactly Marginal Deformations and Global Symmetries
• Deformations of Superconformal Theories
• Semiclassics, Goldstone Bosons and CFT data
• Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus
• Compensating strong coupling with large charge
• Superconformal indices of three-dimensional theories related by mirror symmetry
• Superconformal Symmetry in Three-dimensions
• String Theory of the Regge Intercept
• Higher derivative couplings and massive supergravity in three dimensions
• A Comment on Non-Chiral Operators in SQCD and its Dual

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