New Calabi-Yau Geometries in String Theory and Supersymmetry

自从第一次超弦革命以来，弦理论见证了从物理学到数学再到物理学的思想自由流动。它的未来取决于扩展这个界面，将思想和人聚集在一起。我的工作为新兴物理学建立了一个关键框架，采用高度结构化的物理系统，并使用几何语言来约束、分类和发展物理理论。理论粒子物理与数学相结合的研究通常分为两类，一类关注具体的物理问题，包括弦现象学和超对称在对撞机物理中的应用。第二类涉及受到理论粒子物理及其结构和方法，特别是弦理论和超对称场论强烈启发的数学家。我的研究通过研究弦对偶性和超对称性，以及卡拉比-丘几何所起的关键作用，将这些群体聚集在一起。

Since the first superstring revolution, string theory has seen a free flow of ideas from physics to mathematics and back again. Its future depends on expanding this interface, bringing together ideas as well as people. My work sets up a key framework for new and emerging physics, taking highly structured physical systems and using the language of geometry to constrain, classify, and evolve physical theories. Research at the interface of theoretical particle physics and mathematics generally falls into two categories, one whose focus is on specific physical questions, including string phenomenology and supersymmetry applications to collider physics. The second category involves mathematicians strongly inspired by theoretical particle physics, its structures and methods, and especially string theory and supersymmetric field theory. My research brings these groups together by studying string dualities and supersymmetry, and the critical role played by Calabi-Yau geometry. A number of important string dualities are formulated in terms of fibration structures on Calabi-Yau geometries. Over the past six years, my research program has built a dictionary between period integrals and algebraic submanifolds of fibered Calabi-Yau manifolds. By proving effective forms of the famous Hodge Conjecture for the fibers, I provide an entirely new approach to understanding both fibered Calabi-Yau manifolds and their moduli spaces "from the inside out." My research cuts across a large swath of the physics of string theory, including: the "non-geometric" Heterotic compactifications stemming from the Clingher-Doran-Malmendier-Morrison program in Heterotic/F-theory duality; unification of Calabi-Yau manifolds and Landau-Ginzburg models obtained by mirroring Calabi-Yau fibrations and Tyurin degenerations; the classification of orientifold theories on elliptic curves and elliptic fibered Calabi-Yau manifolds via new variants of topological KR-theory describing their BPS spectra; and the characterization of the modularity properties of the generating functions of vertical D4-D2-D0 bound states on smooth K3 surface fibered Calabi-Yau threefolds. I have recently uncovered a link between the physics of supersymmetry and Calabi-Yau geometry. The dimensional reduction of supermultiplets to the world-line, which strips away their spatial dimensions, is encoded by a colored bipartite graph known as an Adinkra. My work shows that there is a super Riemann surface naturally associated with each Adinkra such that the Adinkra graph is embedded into the surface as a dimer model. This “geometrization” of supersymmetric representation theory provides a fundamental connection between supermultiplets and mirror symmetry, a line of research in theoretical physics which led to my appointment as the first ever Visiting Campobassi Professor of Physics at the University of Maryland.