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

Understand

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.