Inverse Bootstrapping Conformal Field Theories

Abstract

We propose a novel approach to study conformal field theories (CFTs) in general dimensions. In the conformal bootstrap program, one usually searches for consistent CFT data that satisfy crossing symmetry. In the new approach, we reverse the logic and interpret manifestly crossing-symmetric functions as generating functions of conformal data. Physical CFTs can be obtained by scanning the space of symmetric functions. By truncating the fusion rules, we are able to concentrate on the low-lying operators and derive some approximate relations for their conformal data. It turns out that the free scalar theory, the 2d minimal model CFTs, the ${\varphi }^4$ Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule ${\varphi }_1\times {\varphi }_1=I+{\varphi }_2+T$, where ${\varphi }_1,{\varphi }_2$ are scalar operators, $I$ is the identity operator and $T$ is the stress tensor.

38 pages, 2 figures. v2: 39 pages, 2 figures, typos fixed, references added, names of cutoffs changed, emphasis added about the convergence of the truncation ansatz. v3: typos fixed, final version

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Cited by in corpus (21)

• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Spheres, Charges, Instantons, and Bootstrap: A Five-Dimensional Odyssey
• Simplifying large spin bootstrap in Mellin space
• Rigorous bounds on irrelevant operators in the 3d Ising model CFT
• Scalar CFTs and Their Large N Limits
• Conformal manifolds: ODEs from OPEs
• Null bootstrap for non-Hermitian Hamiltonians
• Towards a Bootstrap approach to higher orders of epsilon expansion
• New method for the conformal bootstrap with OPE truncations
• Improving the five-point bootstrap
• Anomalous dimensions of spinning operators from conformal symmetry
• Ising model close to $d=2$
• Lightcone expansions of conformal blocks in closed form
• Principle of minimal singularity for Green's functions
• Easy bootstrap for the 3D Ising model: a hybrid approach of the lightcone bootstrap and error minimization methods
• Conformal Invariance and Multifractality at Anderson Transitions in Arbitrary Dimensions
• ${\varphi }^n$ trajectory bootstrap
• Factorized lightcone expansion of conformal blocks
• Anomalous dimensions from conformal field theory: Generalized ${\varphi }^{2n+1}$ theories
• Boundary anomalous dimensions from BCFT: O($N$)-symmetric ${\varphi }^{2n}$ theories with a boundary and higher-derivative generalizations
• Analytic Studies of Fermions in the Conformal Bootstrap