Hamiltonian monodromy via geometric quantization and theta functions

Lagrangian fibrations and theta functions

In this thesis we study asymptotic behavior of projective embeddings of abelian varieties and their amoebas. The projective embeddings are given by theta functions. It is known that a Lagrangian fibration of the abelian variety determines a basis of theta functions. After reviewing the relation from the viewpoint of geometric quantization and mirror symmetry, we prove that the Lagrangian fibration of the abelian variety can be approximated by the moment maps of natural torus action in a suitable sense. In the final section we discuss a part of the result in more general setting.

绝热极限、Theta函数与几何量子化

Adiabatic Limit, Theta Function, and Geometric Quantization

本文研究非紧拉格朗日环面纤维化上的绝热极限，构造由Bohr-Sommerfeld点索引的L^2截面正交系，证明其收敛到delta函数截面，并推广到非可积近复结构情形。

Quantization of some moduli spaces of parabolic vector bundles on CP^1

We address quantization of the natural symplectic structure on a moduli space of parabolic vector bundles of parabolic degree zero over $\mathbf{CP}^1$ with four parabolic points and parabolic weights in {0,1/2}. Identifying such parabolic bundles as vector bundles on an elliptic curve, we obtain explicit expressions for the corresponding non-abelian theta functions. These non-abelian theta functions are described in terms of certain naturally defined distributions on the compact group SU(2).