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

TL;DR

This paper explores the modular properties of elliptic genera in four-dimensional string theories, revealing their quasi-Jacobi form nature and the associated holomorphic anomalies, with implications for Calabi-Yau fourfolds and flux backgrounds.

Contribution

It introduces the concept that elliptic genera in 4D string theories are quasi-Jacobi forms with holomorphic anomalies, linking algebraic geometry, fluxes, and modularity in a novel way.

Findings

Elliptic genera are quasi-Jacobi forms with holomorphic anomalies.

Derivative of modular forms encode BPS invariants of embedded threefolds.

Anomaly cancellation involves both modular and quasi-Jacobi sectors.

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.