Lie Theory in Generalized Kähler Geometry

Generalized Kähler (GK) geometry was discovered in the study of N = (2, 2) supersymmetric σ-models. In this thesis we develop a new approach to GK geometry by addressing the integration problem of GK structures. To tackle this problem, we first reformulate the definition of GK structures in terms of holomorphic Manin triples. Throughout this approach, we discovered an intimate connection between GK geometry and double structures invented by Ehresmann and further developed by Mackenzie in the fields of Poisson geometry and Lie theory.We introduce and develop the concept of holomorphic Morita equivalences of symplectic double groupoids, 2-morphisms between Morita equivalences, and multiplicative LS bisections in order to access the underlying holomorphic data, and smooth data associated with GK structures. We then employ techniques from Poisson Geometry, such as gauge transformations, and IM 2-forms to solve the integration problem and prove a differentiation theorem. As applications, we provide a definition of generalized Kähler classes and present a Hamiltonian flow construction of GK metrics. In the last chapter, we introduce and develop the notion of groupoid equivariant gerbes, double groupoid equivariant gerbes, Morita equivalences between equivariant gerbes. We make a connection between equivariant gerbes and shifted symplectic structures, and determine the gerbe prequantization of pluriclosed 2-forms.

Department dc:contributor.department

Mathematics

Year dc:date.issued

2023

Author dc:creator

• Jiang, Yucong

Advisor dc:contributor.advisor

• Gualtieri, Marco

• Generalized geometry
• Lie theory
• Manin triples
• Mathematical physics
• Poisson geometry
• Symplectic double groupoids

Handle dc:identifier.uri

http://hdl.handle.net/1807/129916

OAI identifier oai:identifier

oai:utoronto.scholaris.ca:1807/129916