3d Conformal Field Theories via Fuzzy Sphere Algebra

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.