Hamiltonian monodromy via geometric quantization and theta functions

In this paper, Hamiltonian monodromy is addressed from the point of view of geometric quantization, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections. In the case of completely integrable Hamiltonian systems with two degrees of freedom, a link is established between monodromy and (two-level) theta functions, by resorting to the by now classical differential geometric interpretation of the latter as covariantly constant sections of a flat connection, via the heat equation. Furthermore, it is shown that monodromy is tied to the braiding of the Weierstraß roots pertaining to a Lagrangian torus, when endowed with a natural complex structure (making it an elliptic curve) manufactured from a natural basis of cycles thereon. Finally, a new derivation of the monodromy of the spherical pendulum is provided.