Modular forms as classification invariants of 4D \( \mathcal{N} \) = 2 Heterotic-IIA dual vacua

Abstract

We focus on 4D \( \mathcal{N} \) = 2 string vacua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi-Yau three-fold XIIA in the Type IIA language is further assumed to have a regular K3-fibration. It is well-known that one can assign a modular form Φ to such a vacuum by counting perturbative BPS states in Heterotic theory or collecting Noether-Lefschetz numbers associated with the K3-fibration of XIIA. In this article, we expand the observations and ideas (using gauge threshold correction) in the literature and formulate a modular form Ψ with full generality for the class of vacua above, which can be used along with Φ for the purpose of classification of those vacua. Topological invariants of XIIA can be extracted from Φ and Ψ, and even a pair of diffeomorphic Calabi-Yau’s with different Kähler cones may be distinguished by introducing the notion of “the set of Ψ’s for Higgs cascades/for curve classes”. We illustrated these ideas by simple examples.

A preprint version of the article is available at ArXiv.

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Enoki, Y., Watari, T. Modular forms as classification invariants of 4D \( \mathcal{N} \) = 2 Heterotic-IIA dual vacua. J. High Energ. Phys. 2020, 21 (2020). https://doi.org/10.1007/JHEP06(2020)021

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Keywords

  • String Duality
  • Superstrings and Heterotic Strings
  • Differential and Algebraic Geometry
  • Topological Strings