Geometric applications of dualities

对偶性的几何应用

• 批准号: 0700446
• 负责人: Tony Pantev
• 金额: $ 13.71万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0700446&HistoricalAwards=false
• 关键词: Geometric; applications; dualities

This is a research in the field of algebraic geometry - a classical subject studying the solutions to systems of polynomial equations. The project addresses four problems providing novel interfaces between complex geometry and string theory and quantum physics. The first one outlines a new way to extract Hodge theoretic invariants directly from the sheaf theory of commutative or noncommutative spaces. The formal structure of these linear entities will be studied and through the physical notion of quantum mirror symmetry used to produce new invariants of symplectic manifolds. In the second project a new method is proposed for proving the K-equivalence conjecture in birational geometry. The third project concerns the problem of deformation quantization of geometric dualities and symmetries in the complex analytic context. The fourth project analyzes the supersymmetry constraints for D-branes with non-trivial fluxes wrapping algebraic cycles in Calabi-Yau threefolds. The large N quantization of the branes is probed in the context of algebraically completely integrable systems.The understanding of these questions is essential for unifying variouslinearization procedures in algebraic geometry, symplectic topology,theoretical and mathematical physics. The project sets the stage forunderstanding the basic structure of algebraic varieties in a waysuitable for pragmatic use in a broad spectrum of applications. Theproject outlines concrete interdisciplinary applications to matrixquantum mechanics, string dualities and topological black holes.This project also aims to organize a concentrated effort on enhancingthe geometric arsenal of techniques used in high energy physics andcondensed matter theory. This will be achieved by training a group ofyoung researchers, and graduate and undergraduate students inmathematics and physics, and by a curriculum development of courses onHodge theory, non-commutative geometry and mirror symmetry, on thegraduate and undergraduate level.

这是代数几何领域的一项研究——研究多项式方程组解的经典学科。该项目解决了四个问题，在复杂几何、弦理论和量子物理学之间提供了新颖的接口。 第一个概述了一种直接从交换或非交换空间的层理论中提取霍奇理论不变量的新方法。 将研究这些线性实体的形式结构，并通过量子镜对称的物理概念用于产生新的辛流形不变量。在第二个项目中，提出了一种新方法来证明双有理几何中的 K 等价猜想。第三个项目涉及复杂分析背景下几何二元性和对称性的变形量化问题。第四个项目分析了 D 膜的超对称约束，其中非平凡通量包裹了 Calabi-Yau 三重代数环。 膜的大 N 量子化是在代数完全可积系统的背景下进行探讨的。理解这些问题对于统一代数几何、辛拓扑、理论和数学物理中的各种线性化过程至关重要。该项目为理解代数簇的基本结构奠定了基础，以适合在广泛应用中实际使用的方式。该项目概述了矩阵量子力学、弦对偶性和拓扑黑洞的具体跨学科应用。该项目还旨在集中精力增强高能物理和凝聚态理论中使用的几何技术。这将通过在数学和物理学领域培训一批年轻研究人员、研究生和本科生，以及在研究生和本科生层面开发霍奇理论、非交换几何和镜像对称课程来实现。