Structure in the moduli space of vacua of low dimensional supersymmetric quantum

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This Project involves the investigation of the moduli space of vacua of supersymmetricquantum eld theories. The sets of zero-energy congurations, or moduli space of vacua, of a quantumeld theory contains rich algebro-geometric structure. For a large set of supersymmetric quantumeld theories this structure is related to nilpotent varieties in Lie algebras. These nilpotent varietiesare well understood for the classical Lie algebras in the mathematical literature and investigation ofthe interpretation of this and related structures in a physical context is the main aim of this project.The supersymmetric quantum eld theories also have an interpretation as the low energy dynamics ofsuperstring theory embeddings. The interplay between these embeddings, the quantum eld theories,the geometry of the moduli spaces of vacua, and potential generalizations of all therein, is the centralarea of concern.