3d Conformal Field Theories via Fuzzy Sphere Algebra

TL;DR

The paper analyzes density-mode algebras of fuzzy-sphere Hamiltonians that realize 3d CFTs in finite spinful-fermion systems projected to the lowest Landau level, establishing a genuine Lie-algebra structure and exploring two thermodynamic limits. It derives a planar GMP-like algebra for large angular momentum modes and a semiclassical limit for small modes, clarifying how conformal-like structures emerge in appropriate scaling regimes. It constructs an explicit so(3,2) representation in the minimal two-electron sector and discusses extensions via an so(3) coproduct, highlighting structural mismatches with the thermodynamic limit relevant to critical fuzzy-sphere Hamiltonians. The work lays groundwork toward a low-energy scaling limit realizing 3d CFTs in fuzzy-sphere models, while noting limitations of coproduct-based lifts for capturing the large-system limit.

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. We further 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 Hamiltonians.