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.
Local Identities Involving Jacobi Elliptic Functions
本文导出了一些涉及Jacobi椭圆函数的任意秩的局部恒等式，并利用它们得到了一些新的结果。首先，我们提出了一个替代的，更简单的推导我们最近发现的循环恒等式，随着扩展到几个新的循环恒等式的任意秩。其次，我们得到了循环恒等式的一个推广，其中连续项具有乘法相位因子exp（2 i π/s），其中s是任意整数。第三，我们通过推导四个类似于我们先前讨论的循环和的主恒等式的局部“主恒等式”来系统化局部恒等式。第四，我们指出，许多地方的身份可以被认为是精确离散的标准非线性微分方程所满足的雅可比椭圆函数。最后，我们得到了一些明确的答案，一些明确的积分和更简单的形式，不定积分涉及雅可比椭圆函数。
