  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 [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions
with applications to the non-vanishing of Poincar\'e series of large weight,
Invent. math, 122 (1995), 359-402, preprint hep-th/9406036], whereby every
vector of a BS basis is defined by some half-weighted Legendrian distribution
coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying
symplectic manifold. The advantage of BS bases (compared to bases of theta
functions in [A. Tyurin, Quantization and ``theta functions'', Jussieu preprint
216 (Apr 1999), e-print math.AG/9904046, 32pp.]) is that we can use information
from the skillful analysis of the asymptotics of quantum states. This gives
that Bohr-Sommerfeld bases are unitary quasi-classically. Thus we can apply
these bases to compare the Hitchin connection with the KZ connection defined by
the monodromy of the Knizhnik-Zamolodchikov equation in combinatorial theory
(see, for example, [T. Kohno, Topological invariants for 3-manifolds using
representations of mapping class group I, Topology 31 (1992), 203-230; II,
Contemp. math 175} (1994), 193-217]).
