论文标题
四个维度
Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions
论文作者
论文摘要
我们研究了与四维弦理论中的磁通量和椭圆形属模块化的卡拉比四倍的枚举几何形状之间的相互作用。我们认为，对椭圆属的某些贡献是由模块化或准模块化形式的衍生物给出的，这些衍生物编码嵌入给定的四倍的bps calabi-yau或非calabi-yau的三倍。结果，椭圆属只是一种准雅各比形式，而不是通常的意义上的模块化或准模块化形式。这表现为光谱流对称性的全态异常，并且在椭圆形的全态异常方程中，该方程在不同的通量扇区之间映射。我们通过详细的示例研究（包括四个维度的非关键字符串）来支持我们的一般考虑。对于临界杂质弦，我们解释了由于衍生部门的特性，如何恢复消除异常。本质上，尽管椭圆属的模块化部门负责取消涉及通用B场的异常，但准雅各比的一个可以占用可能存在的其他B场。因此，再次，多种数学成分，即四倍的代数几何形状，相对gromow-witten与通量背景有关的理论以及（Quasi-）jacobi形式的模块化特性，并以严格的一致性为需要以一种有趣的方式来巩固。
We investigate the interplay between the enumerative geometry of Calabi-Yau fourfolds with fluxes and the modularity of elliptic genera in four-dimensional string theories. We argue that certain contributions to the elliptic genus are given by derivatives of modular or quasi-modular forms, which encode BPS invariants of Calabi-Yau or non-Calabi-Yau threefolds that are embedded in the given fourfold. As a result, the elliptic genus is only a quasi-Jacobi form, rather than a modular or quasi-modular one in the usual sense. This manifests itself as a holomorphic anomaly of the spectral flow symmetry, and in an elliptic holomorphic anomaly equation that maps between different flux sectors. We support our general considerations by a detailed study of examples, including non-critical strings in four dimensions. For the critical heterotic string, we explain how anomaly cancellation is restored due to the properties of the derivative sector. Essentially, while the modular sector of the elliptic genus takes care of anomaly cancellation involving the universal B-field, the quasi-Jacobi one accounts for additional B-fields that can be present. Thus once again, diverse mathematical ingredients, namely here the algebraic geometry of fourfolds, relative Gromow-Witten theory pertaining to flux backgrounds, and the modular properties of (quasi-)Jacobi forms, conspire in an intriguing manner precisely as required by stringy consistency.
