Stringy Mathematics

弦数学

• 批准号: 0401814
• 负责人: Savdeep Sethi
• 金额: $ 25.4万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401814&HistoricalAwards=false
• 关键词: Stringy; Mathematics

In the past few years, there has been significant interplay between string theory and mathematics. For example, advances in supersymmetric field theory have shed light on four manifold invariants, while advances in algebraic geometry have clarified the origins of mirror symmetry. The aim of this project is to further strengthen this interplay by research focused on three areas of interest to both mathematicians and physicists. The broader impact of this project centers on improved interdisciplinary ties fostered through lectures at schools and workshops, and through direct collaboration. The first research goal is to develop a recently discovered analogue of mirror symmetry for the heterotic string. Conventional mirror symmetry applies to theories with (2; 2) super-symmetry. However, these compactifications constitute a special subclass of more general heterotic string compactifications with only (0; 2) supersymmetry. Many of the interesting structures of (2; 2) theories, like quantum cohomology (or chiral) rings, generalize to this richer setting. Using the dual description, the chiral ring can be determined exactly in many examples, leading to predictions about heterotic string instanton corrections. The study of (0; 2) duality is a topic in its infancy, and there are many directions to explore: for example, S-duality maps heterotic world-sheet instantons into D-instantons of type I open string theory. This suggests a relation between open and closed string instantons, which is likely to be fascinating both to physicists and to geometers. The second area of focus involves compactifications with flux. In the presense of flux, a string target space need not be Ricci-flat. Compact examples of this kind involving just NS-NS fluxes have been found in recent years. These are compactifications with torsion. It is clear that there should be dual descriptions for vacua of this kind (in the sense of mirror symmetry), but there is little currently known about these duals. Since generic string compactifications involve fluxes, constructing dual descriptions for these cases is likely to both enhance our understanding of the string moduli space, and lead to novel questions in mathematics. The third direction revolves around a relation between matrix integrals and modular forms. The twisted partition function for type IIB D-instantons in ten dimensions is computed by evaluating a complicated matrix integral. Yet these matrix integrals are encoded in a particular modular form, which appears in the effective action for the type IIB string. This modular form is completely determined by supersymmetry. The connection between the U-duality group, SL(2;Z), and the matrix integrals is intriguing and puzzling: why is it true? Does it generalize to lower dimensions where the U-duality group is larger? Does it extend to other solitons like monopoles? There are tantalizing hints that the answer to the last two questions is affirmative, but much remains to be understood.

在过去的几年里，弦理论和数学之间有着重要的相互作用。例如，超对称场论的进展揭示了四个流形不变量，而代数几何的进展澄清了镜像对称的起源。该项目的目的是通过研究数学家和物理学家感兴趣的三个领域来进一步加强这种相互作用。该项目的更广泛影响集中在通过学校和研讨会的讲座以及通过直接合作促进的跨学科联系上。第一个研究目标是开发一个最近发现的镜像对称的杂种优势字符串的模拟。传统的镜像对称适用于具有（2; 2）超对称性的理论。然而，这些紧化构成了一个特殊的子类更一般的杂合弦紧化只有（0; 2）超对称。（2; 2）理论的许多有趣的结构，如量子上同调（或手征）环，都可以推广到这个更丰富的环境。使用对偶描述，手征环可以在许多例子中被精确地确定，从而导致关于杂合弦瞬子修正的预测。研究了（0; 2）对偶性是一个处于萌芽阶段的话题，有许多方向可以探索：例如，S-对偶性将杂化世界片瞬子映射到I型开弦理论的D-瞬子。这表明了开弦瞬子和闭弦瞬子之间的关系，这可能会让物理学家和几何学家都感兴趣。第二个重点领域涉及通量的紧化。在通量存在的情况下，弦目标空间不必是Ricci平坦的。近年来，已经发现了这种仅涉及NS-NS通量的紧凑例子。这些是有挠紧化。很明显，这类真空应该有双重的描述（在镜像对称的意义上），但目前对这些双重的描述知之甚少。由于一般弦紧化涉及通量，因此为这些情况构建对偶描述可能既能增强我们对弦模空间的理解，又能引发数学中的新问题。第三个方向围绕矩阵积分和模形式之间的关系。通过计算一个复杂的矩阵积分，计算了10维IIB型D-瞬子的扭曲配分函数。然而，这些矩阵积分以特定的模形式编码，这出现在IIB型弦的有效作用量中。这种模形式完全由超对称性决定。U-对偶群SL（2;Z）和矩阵积分之间的联系是有趣和令人困惑的：为什么它是真的？它是否推广到U-对偶群较大的低维情况？它会延伸到其他的孤子，比如单极子吗？有一些诱人的迹象表明，最后两个问题的答案是肯定的，但仍有许多问题有待理解。