Squashed Toric Manifolds and Higher Depth Mock Modular Forms

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 ${\mathbb{CP}}^2$.

arXiv comment43 pages; v2: Typos corrected, brief comments and clarifications added, accepted for publication in JHEP

References in corpus (23)

• Phases of $N=2$ Theories In Two Dimensions
• A Strong Coupling Test of S-Duality
• Linear Dilatons, NS5-branes and Holography
• Elliptic genera of 2d N=2 gauge theories
• Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups
• Duality of the Fermionic 2d Black Hole and N=2 Liouville Theory as Mirror Symmetry
• Holography for Non-Critical Superstrings
• E-Strings and N=4 Topological Yang-Mills Theories
• The non-compact elliptic genus: mock or modular
• Notes on Non-Critical Superstrings in Various Dimensions
• A Twisted Non-compact Elliptic Genus
• Non-holomorphic Modular Forms and SL(2,R)/U(1) Superconformal Field Theory
• Multiple D3-instantons and mock modular forms I
• A holomorphic anomaly in the elliptic genus
• $R^3$ index for four-dimensional $N=2$ field theories
• Multiple D3-instantons and mock modular forms II
• Wall-crossing made smooth
• Moonshine in Fivebrane Spacetimes
• $r$-Tuple Error Functions and Indefinite Theta Series of Higher-Depth
• Donaldson-Witten theory and indefinite theta functions
• Elliptic genera and real Jacobi forms
• Higher Poles and Crossing Phenomena from Twisted Genera
• Mock modularity from black hole scattering states

Cited by in corpus (4)

• Black holes and higher depth mock modular forms
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• Mock modularity at work, or black holes in a forest
• Traces of reciprocal singular moduli