New Calabi-Yau Geometries in String Theory and Supersymmetry

自从第一次超弦革命以来，弦理论已经看到了从物理到数学再回来的思想自由流动。 它的未来取决于扩大这种界面，将想法和人聚集在一起。 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。 理论粒子物理学和数学的研究通常福尔斯两类，一类侧重于特定的物理问题，包括弦现象学和超对称性在对撞机物理学中的应用。 第二类是数学家，他们深受理论粒子物理学及其结构和方法的启发，特别是弦理论和超对称场论。 我的研究通过研究弦的对偶性和超对称性，以及卡-丘几何所起的关键作用，将这些群体聚集在一起。 一些重要的弦对偶制定的纤维化结构的卡-丘几何。 在过去的六年里，我的研究计划已经建立了一个字典之间的周期积分和代数子流形的纤维卡-丘流形。 通过证明著名的霍奇猜想的有效形式的纤维，我提供了一个全新的方法来理解纤维卡-丘流形和他们的模空间“从内到外。" 我的研究跨越了弦理论物理学的很大一部分，包括：“非几何”杂化紧化，源于杂化/F理论对偶中的Clingher-Doran-Malmendier-Morrison程序;通过镜像Calabi-Yau纤维化和Tyurin退化获得的Calabi-Yau流形和Landau-Ginzburg模型的统一;通过描述其BPS谱的拓扑KR-理论的新变体，对椭圆曲线和椭圆纤维Calabi-Yau流形上的定向折叠理论进行分类;以及光滑K3表面纤维化Calabi-Yau三重态上垂直D_4-D_2-D_0束缚态生成函数的模性特征。 我最近发现了超对称物理学和卡-丘几何学之间的联系。将超多重态降维到世界线，去掉它们的空间维度，由一个称为Adinkra的有色二分图编码。我的工作表明，有一个超级黎曼曲面自然与每个Adinkra，这样的Adinkra图嵌入到表面作为二聚体模型。超对称表象理论的这种“几何化”提供了超多重态和镜像对称之间的基本联系，这是理论物理学的一条研究路线，使我被任命为马里兰州大学有史以来第一位访问坎波切尼物理学教授。

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.