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

Abstract

Abstract We focus on 4D N $$ \mathcal{N} $$ = 2 string vacua described both by perturbative Heterotic theory and by Type IIA theory; a Calabi-Yau three-fold X IIA 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 X IIA. 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 X IIA 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.

Keywords

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