3d Conformal Field Theories via Fuzzy Sphere Algebra

TL;DR

This paper investigates the algebraic structure of fuzzy sphere models that aim to realize 3d conformal field theories, analyzing their limits, symmetries, and how they approximate known algebraic structures in different regimes.

Contribution

It provides a detailed algebraic analysis of fuzzy sphere models, including their limits, symmetries, and a representation of the conformal algebra, clarifying their connection to 3d CFTs.

Findings

Density mode algebra satisfies Jacobi identity

High-angular-momentum modes recover Girvin-MacDonald-Platzman algebra

Explicit $so(3,2)$ conformal algebra representation in minimal systems

Abstract

Fuzzy sphere models conjecturally realize 3d CFTs in small systems of spinful fermions, but why they work so well is still not fully understood. Their Hamiltonians are built from electron density operators projected to the lowest Landau level. We analyze the algebra of the density modes and verify that it satisfies the Jacobi identity. The fuzzy sphere geometry admits two thermodynamic limits: a local planar limit yielding the fuzzy plane, and a commutative limit yielding an ordinary sphere. In the planar limit, high-angular-momentum modes recover the Girvin-MacDonald-Platzman algebra, whereas in the commutative limit, the low-angular-momentum modes become semiclassical. Upon further restricting to a subspace with few spin flips above the paramagnetic reference state, they behave approximately as harmonic oscillators. We also find an explicit representation of the conformal algebra $so(3,2)$ in the minimal two-electron system and extend it to larger systems via an $so(3)$ equivariant coproduct. Because the coproduct splits one $so(3)$ representation into a tensor product, it is structurally mismatched with the thermodynamic limit of critical fuzzy sphere models.