More constraining conformal bootstrap

Abstract

Recently an efficient numerical method has been developed to implement the constraints of crossing symmetry and unitarity on the operator dimensions and OPE coefficients of conformal field theories (CFT) in diverse space-time dimensions. It appears that the calculations can be done only for theories lying at the boundary of the allowed parameter space. Here it is pointed out that a similar method can be applied to a larger class of CFT's, whether unitary or not, and no free parameter remains, provided we know the fusion algebra of the low lying primary operators. As an example we calculate using first principles, with no phenomenological input, the lowest scaling dimensions of the local operators associated with the Yang-Lee edge singularity in three and four space dimensions. The edge exponents compare favorably with the latest numerical estimates. A consistency check of this approach on the 3d critical Ising model is also made.

Figures (3)

• Figure 1: Each curve represents the locus of vanishing of a $3\times 3$ minor of the homogeneous system (\ref{['homo']}) in the $\Delta_\varphi,\Delta_{\varphi^2}$ plane in the free scalar massless theory in $D=3$ dimensions.
• Figure 2: Plot of some $3\times3$ minors around the solution (\ref{['solution']}) as functions of $\Delta_4$. Their convergence to zero near this solution supports our estimate of the critical exponent $\sigma$ for the three-dimensional Yang-Lee edge singularity.
• Figure 3: Plot in the plane $(\Delta_\varphi,\Delta_4)$ of the zeros of some minors in the case of Yang-Lee edge singularity in four space dimensions.