Quasi-Jacobi Forms, Elliptic Genera and Strings in Four Dimensions

Abstract

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.

90 pages, 4 figures; v2: minor comments added to match published version

References in corpus (15)

• On the Geometry of the String Landscape and the Swampland
• The Swampland: Introduction and Review
• Super-Planckian Spatial Field Variations and Quantum Gravity
• F-Theory and the Mordell-Weil Group of Elliptically-Fibered Calabi-Yau Threefolds
• Heterotic GUT and Standard Model Vacua from simply connected Calabi-Yau Manifolds
• The Weak Gravity Conjecture in three dimensions
• Enumerative geometry of Calabi-Yau 4-folds
• E + E $\to$ H
• Geometric constraints in dual F-theory and heterotic string compactifications
• The Vertical, the Horizontal and the Rest: anatomy of the middle cohomology of Calabi-Yau fourfolds and F-theory applications
• Massive U(1)s and Heterotic Five-Branes on K3
• Self-dual Strings and 2D SYM
• On the Elliptic Genus of Three E-strings and Heterotic Strings
• Symmetry enhancement in 4d Spin(n) gauge theories and compactification from 6d
• Modularity from Monodromy

Cited by in corpus (6)

• Quantum Corrections in 4d N=1 Infinite Distance Limits and the Weak Gravity Conjecture
• On Higher-Spin Points and Infinite Distances in Conformal Manifolds
• Twisted 6d $(2,0)$ SCFTs on a Circle
• Weyl invariant Jacobi forms along Higgsing trees
• Mock modularity at work, or black holes in a forest
• Stability of BPS States and Weak Coupling Limits