Constraints on Conformal Field Theories in Diverse Dimensions from the Bootstrap Mechanism

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.

Unitary subsector of generalized minimal models

“…This gives us a more straightforward opportunity to use the techniques in [21,22]. The explicit solution for hσσσσi along this line was found in [23], which focused on its special role in the nonunitary (severe truncation) bootstrap of [24][25][26]. 5 This solution exhibits Virasoro symmetry with a central charge given by…”

N=4Superconformal Bootstrap

“…Altogether, one ultimately finds the following expression for the S 4 partition function, log Z S 4 (q) = log where the functions f (τ ) are Kähler transformations that drop out in the computation of the curvature, and G(z) is Barnes' G-function. 43 The function H(a, q) has been defined in [126] by means of a somewhat intricate recursion relation that we will not review here. It is a building block of the Virasoro four-point conformal block with c = 25, all four external dimensions equal to one, and internal dimension equal to 1 + a 2 .…”

𝜖-expansion in critical ϕ3-theory on real projective space from conformal field theory

“…The last quantity C φ φφ A φ may be computed in the numerical truncated conformal bootstrap on the real projective space as studied in [14] with the help of the bulk data as also numerically studied [23][24] by using the numerical truncated conformal bootstrap. We have checked that at ǫ = 0.05, the truncated bootstrap only with the first two-terms…”