Bootstrapping the Minimal 3D SCFT

Abstract

We study the conformal bootstrap constraints for 3D conformal field theories with a ${\mathbb{Z}}_2$ or parity symmetry, assuming a single relevant scalar operator $\epsilon$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $\sigma$, we map out the allowed space of dimensions and three-point couplings of such "Ising-like" CFTs. If we allow a second relevant odd scalar ${\sigma }^{\prime }$, we identify a feature in the allowed space compatible with 3D $\mathcal{N}=1$ superconformal symmetry and conjecture that it corresponds to the minimal $\mathcal{N}=1$ supersymmetric extension of the Ising CFT. This model has appeared in previous numerical bootstrap studies, as well as in proposals for emergent supersymmetry on the boundaries of topological phases of matter. Adding further constraints from 3D $\mathcal{N}=1$ superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions ${\Delta }_{\sigma }={\Delta }_{\epsilon }-1=.58444(22)$ and three-point couplings ${\lambda }_{\sigma \sigma \epsilon }=1.0721(2)$ and ${\lambda }_{\epsilon \epsilon \epsilon }=1.67(1)$. We additionally place bounds on the central charge and use the extremal functional method to estimate the dimensions of the next several operators in the spectrum. Based on our results we observe the possible exact relation ${\lambda }_{\epsilon \epsilon \epsilon }/{\lambda }_{\sigma \sigma \epsilon }=\tan (1)$.

arXiv comment16 pages, 6 figures; V2: references added

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• Carving out OPE space and precise $O(2)$ model critical exponents
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• Bootstrapping Mixed Correlators in Three-Dimensional Cubic Theories
• The M-theory Archipelago
• Bootstrapping boundary-localized interactions
• New Developments in the Numerical Conformal Bootstrap
• Analytic and Numerical Bootstrap of CFTs with $O(m)\times O(n)$ Global Symmetry in 3D
• Non-Wilson-Fisher kinks of $O(N)$ numerical bootstrap: from the deconfined phase transition to a putative new family of CFTs
• Higher Spin ANEC and the Space of CFTs
• Bootstrapping Monodromy Defects in the Wess-Zumino Model
• Conformal bootstrap bounds for the $U(1)$ Dirac spin liquid and $N=7$ Stiefel liquid
• Precision Bootstrap for the $\mathcal{N}=1$ Super-Ising Model
• Seeking SUSY fixed points in the $4-\epsilon$ expansion
• The 3d $\mathcal{N}=6$ Bootstrap: From Higher Spins to Strings to Membranes
• Spectrum continuity and level repulsion: the Ising CFT from infinitesimal to finite $\mathbf{\epsilon }$
• A roadmap for bootstrapping critical gauge theories: decoupling operators of conformal field theories in $d>2$ dimensions
• Bootstrapping Coulomb and Higgs branch operators
• Bootstrapping Mixed MN Correlators in 3D
• Analytic and numerical bootstrap for the long-range Ising model
• Is there supersymmetric Lee-Yang fixed point in three dimensions?
• Spin-models, dynamics and criticality with atoms in tilted optical superlattices
• Superconformal invariants and spinning correlators in 3d $\mathcal{N}=2$ SCFTs
• More Analytic Bootstrap: Nonperturbative Effects and Fermions
• The Inversion Formula and 6j Symbol for 3d Fermions
• Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods
• New AdS${}_4$ Vacua in Dyonic ISO(7) Gauged Supergravity
• Factorized lightcone expansion of conformal blocks
• Free and Interacting Fermionic Conformal Field Theories on the Fuzzy Sphere
• Exact 3D Conformal Blocks from Fractional Calculus