/sci/ - Science & Math

>>9334941
Let [math](M,\omega)[/math] be a symplectic manifold. It is said to be prequantizable if there exists a Hermitian line bundle [math]B[/math] on [math]M[/math] with a connection [math]\nabla =
d + \omega[/math] such that [math]\omega\in H^2(M,\mathbb{Z})[/math]. This makes the space of sections of the bundle [math]B\rightarrow M[/math] a preHilbert space, so-called the space of "wavefunctions". Given a moment map [math]\Phi: M \rightarrow \mathfrak{u}(1)^*[/math] compatible with the Hamiltonian action, one imposes gauge invariance by quotienting out the space of sections by the integrable polarization along the connection on [math]B[/math] defined by [math]\omega_{U(1)}(\xi) = \omega(\xi) + \langle \Phi,\xi\rangle[/math]. Hence in order for something to be called a "wavefunction", it needs to be a section of a linear Hermitian bundle with curvature in the integral lattice [math]H^2(M,\mathbb{Z})[/math], and it needs to reside in some leaf of the foliation of an integrable polarization of the space of sections given by the [math]U(1)[/math] moment map.
I hope this clears up any confusion you might have had.