Constructing the Infrared Conformal Generators on the Fuzzy Sphere

TL;DR

This paper develops methods to construct and analyze conformal generators on the fuzzy sphere, enabling the identification of primary operators and understanding symmetry breaking in a numerical CFT setting.

Contribution

It introduces a numerical approach to extract conformal generators from the energy momentum tensor on the fuzzy sphere, accounting for finite-size corrections and enabling operator classification.

Findings

Generators match CFT predictions with sub-percent accuracy for low-lying states.

Methods effectively distinguish primary and descendant operators in dense spectra.

Accurate construction of conformal generators at finite N with controlled corrections.

Abstract

We investigate the conformal algebra on the fuzzy sphere, and in particular the generators of translations and special conformal transformations which are emergent symmetries in the infinite IR but are broken along the RG flow. We show how to extract these generators using the energy momentum tensor, which is complicated by the fact that one does not have a priori access to the energy momentum tensor of the CFT limit but rather must construct it numerically. We discuss and quantitatively analyze the main sources of corrections to the conformal generators due to the breaking of scale-invariance at finite energy, and develop efficient methods for removing these corrections. The resulting generators have matrix elements that match CFT predictions with accuracy varying from sub-percent level for the lowest-lying states up to several percent accuracy for states with dimension $\sim 5$ with $N=16$ fermions. We show that the generators can be used to accurately identify primary operators vs descendant operators in energy ranges where the spectrum is too dense to do the identification solely based on the approximate integer spacing within conformal multiplets.