New Calabi-Yau Geometries in String Theory and Supersymmetry

自第一次超弦革命以来，弦理论见证了思想从物理学到数学的自由流动。 它的未来取决于扩展这个界面，将想法和人员聚集在一起。 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。 理论粒子物理和数学接口的研究通常分为两类，一类侧重于特定的物理问题，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学、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-丘几何所发挥的关键作用，将这些群体聚集在一起。 许多重要的弦对偶性是根据卡拉比-丘几何上的纤维结构来表述的。 在过去的六年里，我的研究项目在周期积分和纤维卡拉比-丘流形的代数子流形之间建立了一本字典。 通过证明著名的纤维霍奇猜想的有效形式，我提供了一种全新的方法来“从内到外”理解纤维卡拉比-丘流形及其模空间。 我的研究涉及弦理论物理学的很大一部分，包括：源自杂质/F 理论二元性中的 Clingher-Doran-Malmendier-Morrison 计划的“非几何”杂质紧化；通过镜像 Calabi-Yau 纤维和 Tyurin 简并获得的 Calabi-Yau 流形和 Landau-Ginzburg 模型的统一；通过描述 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 orientifold 理论进行分类；以及光滑 K3 表面纤维 Calabi-Yau 三重上垂直 D4-D2-D0 束缚态生成函数的模块化特性的表征。 我最近发现了超对称物理学和卡拉比-丘几何之间的联系。超多重态向世界线的降维，即去除了它们的空间维度，由称为 Adinkra 的彩色二分图进行编码。我的工作表明，每个 Adinkra 都有一个自然关联的超级黎曼曲面，因此 Adinkra 图作为二聚体模型嵌入到曲面中。超对称表示理论的“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个研究方向，这使得我被任命为马里兰大学第一位坎波巴西客座物理学教授。

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.