# Conformal amplitude hierarchy and the Poincare disk

## TL;DR

This paper explores the complex structure of amplitudes in the 2d $O(n)$ model's conformal field theory, revealing a hierarchical pattern of zeros and poles on the Poincaré disk linked to rational angles and modular symmetries.

## Contribution

It uncovers a hierarchical amplitude structure in the 2d $O(n)$ model, connecting zeros and poles to rational angles and modular group symmetries, with implications for operator algebras and logarithmic functions.

## Key findings

- Amplitude zeros and poles occur at rational angles forming a hierarchical tree.
- The structure is related to the Poincaré disk and Farey paths.
- Symmetry of the subgroup $ ext{Γ}(2)$ arises from rational angle classes.

## Abstract

The amplitude for the singlet channels in the 4-point function of the fundamental field in the conformal field theory of the 2d $O(n)$ model is studied as a function of $n$. For a generic value of $n$, the 4-point function has infinitely many amplitudes, whose landscape can be very spiky as the higher amplitude changes its sign many times at the simple poles, which generalize the unique pole of the energy operator amplitude at $n=0$. In the stadard parameterization of $n$ by angle in unit of $\pi$, we find that the zeros and poles happen at the rational angles, forming a hierarchical tree structure inherent in the Poincar\'{e} disk. Some relation between the amplitude and the Farey path, a piecewise geodesic that visits these zeros and poles, is suggested. In this hierarchy, the symmetry of the congruence subgroup $\Gamma(2)$ of $SL(2,\mathbb{Z})$ naturally arises from the two clearly distinct even/odd classes of the rational angles, in which one respectively gets the truncated operator algebras and the logarithmic 4-point functions.