Conformal field theory
The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of "c". Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
Geometric quantization
The next step in the process of geometric quantization is the choice of a polarization. A polarization is a choice at each point in formula_6 a Lagrangian subspace of the complexified tangent space of formula_6. The subspaces should form an integrable distribution, meaning that the commutator of two vector fields lying in the subspace at each point should also lie in the vector field at each point. The "quantum" (as opposed to prequantum) Hilbert space is the space of sections of formula_14 that are covariantly constant in the direction of the polarization. The idea is that in the quantum Hilbert space, the sections should be functions of only formula_27 variables on the formula_28-dimensional classical phase space.