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 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.
Highlights
Introduction and overviewRecent developments in the context of the Weak Gravity Conjecture [1], reviewed in [2, 3], have revived interest in string dualities, which underlie the emergence of tensionless strings [4,5,6,7,8,9] at infinite distance boundaries of moduli space.the previous work [6] initiated a study of the emergence of asymptotically tensionless heterotic strings in N = 1 supersymmetric string compactifications in d = 4 dimensions
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
In this work we have investigated the rich interplay between the modular properties of elliptic genera, the enumerative geometry of genus zero relative BPS invariants on elliptic fourfolds with background fluxes, and the structure of U (1) anomalies for effective field theories in four dimensions
Summary
Recent developments in the context of the Weak Gravity Conjecture [1], reviewed in [2, 3], have revived interest in string dualities, which underlie the emergence of tensionless strings [4,5,6,7,8,9] at infinite distance boundaries of moduli space. The previous work [6] initiated a study of the emergence of asymptotically tensionless heterotic strings in N = 1 supersymmetric string compactifications in d = 4 dimensions These strings arise as solitonic objects in certain flux compactifications on Calabi-Yau fourfolds in F-theory. In this context we observe an intriguing relation between partition functions associated with transversal and nontransversal fluxes, and this turns out to be a manifestation of the elliptic holomorphic anomaly equation of [25].
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