Aspects of geometric quantization: Hamiltonian flows in imaginary time

May 18, 2026

In chapter 1 we introduce concepts from symplectic geometry, with a review on symplectic forms, symplectomorphisms, tautological and canonical forms and relevant Lagrangian submanifolds. Going on to chapter 2, we discuss complex manifolds and their structures, Dolbeault theory and Kähler manifolds. The chapter ends with a few examples of Kähler manifolds. Chapter 3 consists of a series of results following the works [1] and [2]. We study the action on complex structures and a subset of complexified flows relevant for the process of quantization, the complexified Hamiltonian flows. We introduce the action of these flows on Kähler manifolds and show that the path of Kähler metrics obtained through this action is a geodesic relative to the Mabuchi Metric. In chapter 4, there is a motivation for geometric quantization, following the Groenewold-Van Hove theorem. Concepts relevant for the pre-quantization process such as complex line bundles, parallel transport and curvature that lead to the integrality condition and Kostant-Soriau quantization. Polarizations are introduced, followed by the core of geometric quantization: Polarized sections and Bohr-Sommerfeld conditions. Chapter 5 provides the insightful example of quantization of the Riemann sphere S^2 . We consider the Fubini-Study form derived from a Kähler potential ϕ and the line bundle O(N ). Following the works [3] and [4], we study a family of holomorphic polarizations P s , s ≥ 0 and find the polarized sections in terms of the N -tensor product of O(1) and identify the Bohr-Sommerfeld cycles for the s → ∞ limit polarization.

Authors in the community:

• 

  S

  Simão José Gaspar dos Santos

  ist1105723

Supervisors of this institution:

• 

  J

  João Luís Pimentel Nunes

  ist12894

mathematics - Mathematics

por - Portuguese

Instituto Superior Técnico