Holomorphic and quantum aspects of generalized Kahler geometry

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Mathematical models which are used to describe the behaviour of physical systems are often based on geometrical structures on complex curved spaces of potentially large dimension. This is true when studying mechanics, the physics of elementary particles, the behaviour of superconductors, and in many other cases. The discovery of increasingly complicated physical systems in quantum field theory has led to the development of new and interesting types of geometric structure, which differ drastically from traditional geometric forms, and so are not well understood. In key areas of theoretical physics such as renormalization and quantum gravity, progress is slowed when physicists are not able to access appropriate mathematical tools for modelling the physical systems of interest. In some cases, such mathematical tools have simply not yet been invented. The goal of the proposed research is to develop a mathematical understanding of a certain class of geometric structures called generalized Kahler structures, to discover the underlying fundamental laws which govern similar structures which arise in quantum field theory, and finally to use the powerful tools of algebraic and differential geometry to leap forward by inventing new geometric structures which help pave the way toward better models in the future.