Journal of High Energy Physics

, 2019:64 | Cite as

Squashed toric manifolds and higher depth mock modular forms

  • Rajesh Kumar GuptaEmail author
  • Sameer Murthy
  • Caner Nazaroglu
Open Access
Regular Article - Theoretical Physics


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 ℂℙ2.


Anomalies in Field and String Theories Conformal Field Theory Sigma Models 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Rajesh Kumar Gupta
    • 1
    Email author
  • Sameer Murthy
    • 1
  • Caner Nazaroglu
    • 2
  1. 1.Department of MathematicsKing’s College LondonLondonU.K.
  2. 2.Mathematical InstituteUniversity of CologneCologneGermany

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