Exact microscopic wave function for a topological quantum membrane.

The higher dimensional quantum Hall liquid constructed recently supports stable topological membrane excitations. Here we introduce a microscopic interacting Hamiltonian and present its exact ground state wave function. We show that this microscopic ground state wave function describes a topological quantum membrane. We also construct variational wave functions for excited states using the noncommutative algebra on the four sphere. Our approach introduces a nonperturbative method to quantize topological membranes.