FuSeGC · First Steps in Mirror Symmetry for Generalized Complex Geometry

First Steps in Mirror Symmetry for Generalized Complex Geometry

FuSeGC is a research project in pure mathematics, specifically in geometry. Mathematics is an important tool in theoretical physics: It is the language in which physical theories are written, and large parts of modern fundamental physics are written in the language of geometry in particular. In order to formulate theories about the physics of the universe at all scales, physicists need a well-developed geometric language. Ideas from physics frequently inspire pure mathematical research, but also vice versa: Understanding in fundamental physics can often only advance once a suitable mathematical language to formulate theories has been developed. Developing the backbone of a part of this language is what this research project is about: Just as applied science and engineering need to rest on a solid foundation of fundamental and theoretical science to be successful, fundamental physics needs mathematical tools to open up new ways of thinking about and formulating theories. This research project is inspired by (theoretical) discoveries in string theory in particular: String theory is one proposal for a unified theory encompassing both quantum physics and gravity, but it is less one theory than a whole toolbox for building many different theories. (It is an open problem to find the one theory that describes our particular universe, and it might of course well turn out that none does.) At the core of string theory lies the idea that the macroscopic physics we observe could at very small scale arise from tiny vibrating strings (the fundamental objects of string theory) moving through a curved spacetime (the concept first introduced by Einstein in his general theory of relativity). The geometry of this spacetime heavily influences the physics observed at larger scales, but it turns out that two completely different spacetime geometries can result in the same physics: This is referred to as mirror symmetry, and the two spacetimes are said to be mirror partners. The precise mathematical formulation of mirror symmetry still contains many open problems, and is one of the most active research areas in modern geometry – which spaces have mirror partners, and how to we find them? How general a phenomenon is mirror symmetry, and what insights can we gain about the geometry on either side of the duality by translating from one to the other? These questions are interesting to mathematicians independently of any physical applications because mirror symmetry relates two different types of geometry, which allows us to translate problems about one geometry to the other, where they might be much easier to solve. The answer can then be translated back into the original context. With FuSeGC, my goal is to extend what is known about mirror symmetry so far to a new class of geometries. Generalized complex (GC) structures constitute a relatively new type of geometry that include both types of geometry involved in mirror symmetry as examples; more precisely as the two endpoints of the spectrum of GC structures. Thus there is a natural question: Is mirror symmetry fundamentally a generalized complex duality? Since modern mirror symmetry is a theory involving complex mathematical technology, so in order to begin to answer this question, much of this technology needs to be adapted and expanded to GC geometry. This is what FuSeGC does, for a selection of carefully chosen contexts.

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