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 method, 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 crossing-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 ϕ4 Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule ϕ1 × ϕ1 = I + ϕ2 + T , where ϕ1, ϕ2 are scalar operators, I is the identity operator and T is the stress tensor.

Highlights

  • Factors Cijk called OPE coefficients or structure constants of the operator algebra.1 The normalized two-point functions of the primary operators are fixed by conformal symmetry, while the three-point functions are determined by conformal invariance up to the OPE coefficients

  • We propose a novel approach to study conformal field theories (CFTs) in general dimensions

  • It is important that the infinite number of subleading operators are irrelevant to the achieved precision. This indicates that the CFT data of these low-lying operators already provide a successful approximation of the 3d Ising CFT which is consistent with crossing symmetry

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Summary

Inverse bootstrapping method

In the quantum inverse scattering method [83], one begins with the solutions of a nontrivial consistency condition, i.e. the Yang-Baxter equation [84, 85]. In the inverse bootstrapping method, we will start from the solutions of a non-trivial consistency condition, namely the crossing equation (2.1) to be defined below.. The CFT data can be directly deduced from a given crossing symmetric function. By working at the level of the solution space, the inverse perspective provides us with a natural truncation ansatz obeying the crossing equation (2.1)

Crossing symmetric functions
Truncating the fusion rules
Operator decoupling
CFTs in various dimensions
Canonical free scalar theory
Discussion
Findings
A Series expansion of conformal blocks
Full Text
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