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]).