Abstract
The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial “swapping” property, allowing to swap infinite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving finite sums of derivatives. However, it is far from obvious for “cut-touching” functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazáč in his work on analytic derivation of optimal bootstrap bounds. We derive general swapping criteria for the cut-touching functionals, and check in a few explicit examples that Mazáč’s functionals pass our criteria.
Highlights
The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data
Swapping is easy to justify for the popular functionals involving finite sums of derivatives. It is far from obvious for “cut-touching” functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazac in his work on analytic derivation of optimal bootstrap bounds
A comment is in order concerning the origin of the positivity property of the conformal blocks and of their power series coefficients, which played an important role in the above proof
Summary
The function G(z), while originally defined on the interval 0 < z < 1, allows an analytic continuation into the complex plane of z with cuts along (−∞, 0) and (1, +∞) (“cut plane”). The subdisks |ρ| 1− are mapped onto the subregions of the z plane shown in figure 1 In any such subregion the series (1.5) converges uniformly and exponentially fast. The argument proves that the function G(z) can be analytically extended through this cut, and one can circle around the origin through a second, third etc sheet. The same is true for the cut (1, +∞) because the function G(z) is crossing symmetric, eq (1.3) (or because we can equivalently run the argument around z = 1) In this way one can explore the full domain of analyticity of G(z), which is an infinitely-sheeted Riemann surface if ∆φ is an irrational number. A comment is in order concerning the origin of the positivity property of the conformal blocks and of their power series coefficients, which played an important role in the above proof. This explains why all terms in the power series expansion of G(ρ) have to be positive [14]
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
