Note on explicit construction of conformal generators on the fuzzy sphere

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.