John Huerta - Research Statement
نویسنده
چکیده
My research is an investigation into the mathematical foundations of supersymmetry, especially the theories of superstrings and supergravity. Over the past 30 years, string theory and related ideas have proven to be connected to many beautiful notions in differential geometry, such as mirror symmetry [Kontsevich] and gerbes [Brylinski, Freed and Witten]. The overarching goal of my work is to extend the tools of differential geometry, formulating a kind of “categorified differential geometry” suitable for superstring theory, M theory and supergravity. Categorification is the process of replacing set-based mathematics with mathematics based on categories—it has the effect of boosting up dimensions so that mathematics used for the description of point particles becomes appropriate for extended objects: strings and membranes. Through categorification, the tools of gauge theory, which we can loosely think of as the mathematics for describing the parallel transport of point particles, become higher gauge theory, which we can loosely think of as the mathematics for describing the parallel transport of strings, membranes and higher [Baez and Schreiber]. Into this categorification process, one can introduce supersymmetry, which is mathematics with a Z2-grading throughout. Discovered in the 1970s by physicists, it is an early example of noncommutative geometry. Mathematicians have known how to make the basic objects of supersymmetry (supermanifolds and superalgebra) precise for years [Deligne and Morgan, Kostant, Varadarajan]. Yet the mathematical study of the most fundamental supersymmetric theories in contemporary physics (super-Yang–Mills theory, superstrings, supermembranes, and supergravity) is a rich and active topic. In my work, I exploit the mysterious connection between supersymmetry and division algebras (noted by [Duff 1988], among others) as a tool to categorify supergeometry, as well as explain the strange dimensions in which supersymmetry works. To say what I have done, a good place to begin is with these strange dimensions. At its start, my research began by puzzling over this pattern:
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