On a complexification of the moduli space of Bohr - Sommerfeld lagrangian cycles

Abstract

In the previous papers we present a construction of the set ${\mathcal{U}}_{SBS}$ in the direct product ${\mathcal{B}}_S\times \mathbb{P}\Gamma (M,L)$ of the moduli space of Bohr - Sommerfeld lagrangian submanifolds of fixed topological type and the projectivized space of smooth sections of the prequantization bundle $L\to M$ over a given compact simply connected symplectic manifold $M$. Canonical projections $p:{\mathcal{U}}_{SBS}\to \mathbb{P}\Gamma (M,L)$ and $q:{\mathcal{U}}_{SBS}\to {\mathcal{B}}_S$ are studied in the present text: first, we show that the differential $\mathrm{d}p$ at a given point is an isomorphism, which implies that a natural complex structure can be defined on ${\mathcal{U}}_{SBS}$; second, the projection $q:{\mathcal{U}}_{SBS}\to {\mathcal{B}}_S$ splits as the combination ${\mathcal{U}}_{SBS}\to T{\mathcal{B}}_S\to {\mathcal{B}}_S$ such that the fibers of the first map are complex subsets in ${\mathcal{U}}_{SBS}$. This implies that an appropriate section of the first map should define a complex structure on $T{\mathcal{B}}_S$; therefore it can be seen as a complexification of the moduli space ${\mathcal{B}}_S$. The construction can be exploited in the Lagrangian approach to Geometric Quantization

arXiv commentThis work was done as an extended comment to a short lecture course given at the VII school on Geometry and Physics in Bialowieza (Poland). The author cordially thanks the organizers and the participants of the shool for hospitality, valuable discussions and remarks