Note on explicit construction of conformal generators on the fuzzy sphere

TL;DR

This paper presents an explicit method to construct conformal generators on the fuzzy sphere, aiding the analysis of three-dimensional conformal field theories with improved primary state identification.

Contribution

We explicitly construct conformal generators on the fuzzy sphere from the microscopic Hamiltonian, enabling better primary state separation in regularized conformal field theories.

Findings

Successfully captures all primaries with spin < 4 and dimension < 7

Distinguishes primary states from others with high precision

Provides a practical tool for analyzing fuzzy sphere regularized CFTs

Abstract

The lowest Landau level on the sphere was recently proposed as a continuum regularization of the three-dimensional conformal field theories, the so-called fuzzy sphere regularization. In this note, we propose an explicit construction of the conformal generators on the fuzzy sphere in terms of the microscopic Hamiltonian. Specifically, we construct the generators for the translation and special conformal transformation, which are used in defining the conformal primary states and thus are of special interest. We apply our method to a concrete example, the fuzzy sphere regularized three-dimensional Ising conformal field theory. We show that it can help capture all primaries with spin $\ell < 4$ and scaling dimension $\Delta < 7$. In particular, our method can clearly separate the primary from other states that differ in scaling dimension by $1\%$, making it hard otherwise based solely on using the conformal tower associated with the primaries.