Squashed toric manifolds and higher depth mock modular forms

Rajesh Kumar Gupta*, Sameer Murthy, Caner Nazaroglu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)
142 Downloads (Pure)


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 N = 4 SYM theory on ℂℙ 2 .

Original languageEnglish
Article number64
Number of pages44
JournalJournal of High Energy Physics
Issue number2
Publication statusPublished - 12 Feb 2019


  • Anomalies in Field and String Theories
  • Conformal Field Theory
  • Sigma Models


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