On Bohr-Sommerfeld bases

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of Borthwick-Paul-Uribe [3], in which every vector of a BS basis is determined by some half-weight Legendrian distribution coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying symplectic manifold. The advantage of BS bases (compared to the bases of theta functions in [23]) is that we can use the powerful methods of asymptotic analysis of quantum states. This shows that Bohr-Sommerfeld bases are quasiclassically unitary. Thus we can apply these bases to compare the Hitchin connection [11] and the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in the combinatorial theory (see, for example, [14] and [15]).