Stringy Mathematics

在过去的几年里，弦理论和数学之间存在着显著的相互作用。例如，超对称场论的进步揭示了四个流形不变量，而代数几何的进步阐明了镜像对称性的起源。这个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。该项目的更广泛影响集中在通过在学校和讲习班上的讲座以及通过直接合作促进跨学科联系的改善。第一个研究目标是为杂交链开发一种最近发现的镜像对称性的模拟。传统的镜像对称性适用于具有(2；2)超对称性的理论。然而，这些紧化构成了仅具有(0；2)超对称性的更一般的杂化弦紧化的一个特殊子类。(2；2)理论的许多有趣的结构，如量子上同调(或手征)环，都推广到这种更丰富的环境。使用双重描述，在许多例子中可以准确地确定手性环，从而导致关于杂化弦瞬子修正的预测。(0；2)对偶性的研究还处于起步阶段，有许多方向需要探索：例如，S对偶性将杂性世界薄片瞬子映射成第一类开弦理论的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着迷。焦点的第二个区域涉及熔剂的压实作用。在磁通的存在下，弦目标空间不必是Ricci平坦的。近年来发现了仅涉及NS-NS通量的这类紧凑例子。这些都是有扭转的紧实度。显然，对于这类真空应该有双重描述(在镜像对称的意义上)，但目前对这些双重真空知之甚少。由于一般的字符串紧凑涉及到通量，为这些情况构造对偶描述可能既增强了我们对字符串模空间的理解，又导致了数学上的新问题。第三个方向围绕着矩阵积分和模形式之间的关系。通过计算一个复杂的矩阵积分，计算了十维IIB型D瞬子的扭曲配分函数。然而，这些矩阵积分是以特定的模形式编码的，这出现在IIB类型字符串的有效动作中。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费解的：为什么它是真的？它是否推广到U-对偶群较大的低维？它是否延伸到其他孤子，如单极子？有诱人的迹象表明，最后两个问题的答案是肯定的，但仍有许多有待理解的地方。

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.