Squashed Toric Manifolds and Higher Depth Mock Modular Forms

TL;DR

This paper demonstrates that the elliptic genera of squashed toric Calabi-Yau manifolds are higher-depth mock modular forms, revealing their automorphic properties and connecting them to indefinite theta series.

Contribution

It identifies the automorphic nature of these elliptic genera as higher-depth mock modular forms and generalizes their computation for additional toric charges.

Findings

Elliptic genera are higher-depth mock modular forms.

Explicit connection to indefinite theta series.

Generalized elliptic genera for additional charges.

Abstract

Squashed toric sigma models are a class of sigma models whose target space is a toric manifold in which the torus fibration is squashed away from the fixed points so as to produce a neck-like region. The elliptic genera of squashed toric-Calabi-Yau manifolds are known to obey the modular transformation property of holomorphic Jacobi forms, but have an explicit non-holomorphic dependence on the modular parameter. The elliptic genus of the simplest one-dimensional example is known to be a mixed mock Jacobi form, but the precise automorphic nature for the general case remained to be understood. We show that these elliptic genera fall precisely into a class of functions called higher-depth mock modular forms that have been formulated recently in terms of indefinite theta series. We also compute a generalization of the elliptic genera of these models corresponding to an additional set of charges corresponding to the toric symmetries. Finally we speculate on some relations of the elliptic genera of squashed toric models with the Vafa-Witten partition functions of $\mathcal{N}=4$ SYM theory on $\mathbb{CP}^2$.