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到数学再到数学的自由流动。它的未来取决于扩大这种接口，将思想和人员聚集在一起。我的工作为新出现的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的研究一般分为两类，一类专注于具体的物理问题，包括弦现象学和超对称应用于对撞机物理。第二类涉及受到理论粒子物理学、其结构和方法的强烈启发的数学家，特别是弦理论和超对称场论。我的研究通过研究弦对偶和超对称性，以及Calabi-Yau几何学所扮演的关键角色，将这些群体聚集在一起。 用Calabi-Yau几何上的纤维结构表示了一些重要的弦对偶。在过去的六年里，我的研究项目已经建立了一本关于周期积分和纤维Calabi-Yau流形的代数子流形的词典。通过证明著名的Hodge猜想对纤维的有效形式，我提供了一种全新的方法来理解纤维Calabi-Yau流形及其模空间。 我的研究涉及弦理论物理学的一大部分，包括：由Clingher-Doran-Malmendier-Morison程序在异质/F-理论对偶中产生的“非几何”异质紧化；通过镜像Calabi-Yau纤颤和Tyurin简并得到的Calabi-Yau流形和Landau-Ginzburg模型的统一；通过描述椭圆曲线和椭圆纤维Calabi-Yau流形的新的拓扑KR-理论的新变体来分类它们的定向理论；以及描述光滑K3表面上垂直D4-D2-D0束缚态的生成函数的模块化性质。 我最近发现了超对称性物理学和卡拉比-尤几何学之间的联系。超多重到世界线的降维，剥离了它们的空间维度，由一个被称为Adinkra的彩色二部图编码。我的工作表明，每个Adinkra都有一个自然关联的超黎曼曲面，使得Adinkra图作为二聚体模型嵌入到曲面中。这种超对称表示理论的“几何化”提供了超多重态和镜像对称性之间的基本联系，这是理论物理学的一个研究方向，使我被任命为马里兰大学有史以来第一位坎波巴西物理学客座教授。

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.