Stringy Mathematics

在过去的几年里，弦理论和数学之间有了显著的相互作用。例如，超对称场论的进步揭示了四种流形不变量，而代数几何的进步阐明了镜像对称的起源。该项目的目的是通过集中研究数学家和物理学家都感兴趣的三个领域来进一步加强这种相互作用。本项目更广泛的影响集中在通过在学校和讲习班的讲座以及通过直接合作来改善跨学科关系。第一个研究目标是开发一种最近发现的异质弦的镜像对称模拟。传统的镜像对称适用于（2;2）超对称理论。然而，这些紧化构成了只有（0;2）超对称的更一般的杂散弦紧化的一个特殊子类。（2; 2）理论中许多有趣的结构，如量子上同调（或手性）环，可以推广到这种更丰富的环境中。利用对偶描述，可以在许多例子中精确地确定手性环，从而预测异质弦的瞬态修正。对（0;2）对偶的研究是一个刚刚起步的课题，有许多方向可以探索：例如，s对偶将异质性世界表瞬子映射为I型开放弦理论的d-瞬子。这表明了开弦瞬子和闭弦瞬子之间的关系，这可能对物理学家和几何学家都很有吸引力。第二个重点领域涉及与通量的紧化。在通量存在的情况下，弦目标空间不一定是里奇平坦的。近年来发现了这类仅涉及NS-NS通量的紧凑例子。这些是有扭转的紧化。很明显，这种真空应该有对偶描述（在镜像对称的意义上），但目前对这些对偶知之甚少。由于一般的弦紧化涉及到通量，为这些情况构建对偶描述既可能增强我们对弦模空间的理解，又可能导致新的数学问题。第三个方向围绕着矩阵积分和模形式之间的关系。通过计算一个复杂矩阵积分，计算了十维IIB型d -实例的扭曲配分函数。然而，这些矩阵积分以特定的模形式编码，这种形式出现在类型IIB字符串的有效作用中。这种模形式完全由超对称决定。u对偶群SL（2；Z）和矩阵积分之间的联系既有趣又令人困惑：为什么它是真的？它能推广到更低的维度吗在那里u对偶群更大？它会扩展到其他孤子，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。

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.