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.
Highlights
The duality between the Heterotic string theory and the Type IIA string theory has been known for a long time
We focus on 4D 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
For a given X, choose any Higgs cascade attached to the branch of moduli space64 of the Type IIA compactification of X
Summary
The duality between the Heterotic string theory and the Type IIA string theory has been known for a long time. A lattice pair ΛS ⊕ΛT that fits into II4,20 is assigned for those branches [7–10], and a modular form Φ of certain type that depends on ΛS is assigned [11, 12] It is known, that there are physically distinct branches of vacua that cannot be distinguished by the triple of invariants (ΛS, ΛT , Φ). Section 3.1.1 includes the basic definition of Ψ, which follows from the idea of [11] and subsequent works This modular form appears in the integrand of the 1-loop gauge threshold correction in the Heterotic string.
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