Geometric quantization seeks to associate to a classical geometric BohrSommerfeld Lagrangian submanifold a holomorphic wave function, i.e., a section
of a complex line bundle over a K¨ahler manifold. There is a general question of how different bases of such sections are related, which is equivalent to computing
inner products of such sections in the (quantum) Hilbert space inner product on the space of sections. The explicit form of this problem we study is the case of
two sections related by an isometric biholomorphism of the structure: can one calculate the asysmptotics of these inner products geometrically from underlying
Bohr-Sommerfeld Lagrangians. Examples of such are known in the literature if the cycles intersect nicely. We study the case where the cycles do not intersect in
the original manifold M, sometimes called in the physics literature “the classically forbidden” case. We study the extension of the problem into the complex domain
where we can in a few interesting cases derive a geometric formula for these inner products (as ℏ → 0), which depends on a kind of intersection product between
complexified Bohr-Sommerfeld cycles. This depends on global properties of the Bergman kernel function and at times, global algebraic geometry. The proofs are,
up to now, dependent on difficult computations with global kernels. This is joint work in progress with Alejandro Uribe and Tony Yau.
