Friday, July 14, 2023

2:00 p.m. (sharp)

Zoom Web Conference

PhD Candidate: Yucong Jiang

Supervisor: Marco Gualtieri

Thesis title: Integration of generalized Kähler structures

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Generalized Kähler (GK) geometry was first discovered in 1984 by Gates, Hull, and Roček in their study of N = (2,2) supersymmetric σ-models, which extended Zumino’s work on the relationship between Kähler geometry and N = (2, 2) supersymmetry. In this thesis we develop a new approach to the field of generalized Kähler (GK) geometry by addressing the integration problem of GK structures.

To tackle this problem, we first rephrase the definition of GK structures in terms of holomorphic Manin triples. In this way, we have 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 equivalence of symplectic double groupoids. This notion allows us to access the underlying holomorphic structures associated with GK structures. Additionally, we propose the concept of multiplicative Lagrangian branes as a mean to access the underlying smooth data, such as GK metrics.

We then employ techniques from Poisson Geometry, such as gauge transformations, and IM 2-forms to solve the integration problem and obtain a reconstruction theorem. As applications, we provide a definition of generalized Kähler classes and present a Hamiltonian flow construction of GK metrics.

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The draft of the thesis can be found here: thesis

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