Fractional quantum Hall fluids support gapped excitations above the ground state — neutral modes that carry energy and angular momentum but no charge. These excitations are described by the Girvin-MacDonald-Platzman (GMP) algebra, a non-commutative structure that encodes how the quantum Hall fluid responds to geometric deformations.
Golkar, Nguyen, and Son develop the microscopic theory. By deforming the metric of the topological ground state, they construct explicit wavefunctions for the higher-spin neutral modes — excitations that transform as rank-s tensors under spatial rotations. The construction reveals fundamental dualities: the same geometric deformation that produces a spin-2 graviton mode in the Laughlin state produces a spin-1 mode in the dual composite fermion description.
The duality is not a curiosity — it constrains the physical properties of the excitations. The graviton gap, the dispersion relation, and the coupling to external probes are all related by the duality transformation. Measuring one determines the other.
The microscopic wavefunctions are exact within the lowest Landau level — no projection or approximation is needed. This makes them amenable to numerical verification: exact diagonalization on finite systems can compute overlaps between the constructed wavefunctions and the actual excited states, providing quantitative tests of the geometric theory.
The excitations of a topological fluid, built from geometry. Deform the space, and the excitation is the shape of the deformation. The GMP algebra IS the algebra of geometric deformations, and the neutral modes ARE the quantized shapes.