Topological constraints in geometric deformation quantization on domains with multiple boundary components

A topological constraint on the possible values of the universal quantization parameter is revealed in the case of geometric quantization on (boundary) curves diffeomorphic to $S^1$, analytically extended on a bounded domain in $\mathbb{C}$, with $n \ge 2$ boundary components. Unlike the case of one boundary component (such as the canonical Berezin quantization of the Poincar\'e upper-half plane or the case of conformally-invariant 2D systems), the more general case considered here leads to a strictly positive minimum value for the quantization parameter, which depends on the geometrical data of the domain (specifically, the total area and total perimeter in the smooth case). It is proven that if the lower bound is attained, then $n=2$ and the domain must be annular, with a direct interpretation in terms of the global monodromy.