Quasi-Jacobi forms, elliptic genera and strings in four dimensions

Journal of High Energy Physics

Wolfgang Lerche¹  Guglielmo Lockhart¹  Seung-Joo Lee²  Timo Weigand³ 

[1] CERN, Theory Department, 1 Esplanade des Particules, CH-1211, Geneva 23, Switzerland;Center for Theoretical Physics of the Universe, Institute for Basic Science, 34051, Daejeon, South Korea;PRISMA Cluster of Excellence and Mainz Institute for Theoretical Physics, Johannes Gutenberg-Universität, 55099, Mainz, Germany;

关键词: F-Theory;    String Duality;    Superstrings and Heterotic Strings;    Topological Strings;   

DOI  :  10.1007/JHEP01(2021)162

来源: Springer

【 摘 要 】

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 may 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.

【 授权许可】

CC BY