Nonperturbative Mellin Amplitudes: Existence, Properties, Applications

Abstract

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories.

Figures (23)

• Figure 1: We divide the region $u, v >0$ into $4$ regions coloured in blue, pink, red and grey. The regions are separated by the curves $v=(1-\sqrt{u})^2$ and $v=(1+\sqrt{u})^2$ or, equiavelntly, $z = \bar{z}$. In the grey region $z$ and $\bar{z}$ are the complex conjugate of each other. In the colored regions $z$ and $\bar{z}$ are real and independent variables. In the red region we have that $z, \bar{z} \in (-\infty, 0)$. In the blue and pink regions we have that $z, \bar{z} \in (0,1)$ and $z, \bar{z} \in (1,\infty)$ respectively.
• Figure 2: Cylinder picture of points $1$, $3$ and $4$. The point at $(t=0, \phi=\pi)$ should be identified with the point at $(t=0, \phi=-\pi)$. The colored area signifies regions where point $2$ is spacelike separated from three other points and no light-cones has been crossed. Note that the colored region is a double cover in the cross ratio space. Indeed changing $t_2 \to - t_2$ does not change the cross ratios (\ref{['crossratiosspecial']}).
• Figure 3: Different regions in the $(u,v)$ plane that we will find convenient to consider. Different regions are mapped into each other by crossing.
• Figure 4:
• Figure 5: We use red, blue and grey to colour the regions where the ${\cal O}(x_1) \times {\cal O}(x_2)$, ${\cal O}(x_1) \times {\cal O}(x_4)$ and ${\cal O}(x_1) \times {\cal O}(x_3)$ OPE channels converge respectively.