Aspects of the Hypermultiplet Moduli Space in String Duality

TL;DR

The paper investigates how the hypermultiplet moduli spaces of type IIA (or F-theory) on a Calabi–Yau threefold and heterotic strings on a K3 surface times a torus map into one another, focusing on the integral structures given by $H^3(X,\mathbb{Z})$ and $H^2(S_H,\mathbb{Z})$ and bundle data. By performing a stable degeneration, it decomposes the degeneration of the Calabi–Yau into two pieces whose Ramond–Ramond moduli split into three tori, corresponding to the heterotic K3 data and the two $E_8$-bundle sectors, with a transcendental lattice $M$ encoding the B-field. The bundle moduli are then described in terms of the Mordell–Weil group and spectral data on rational elliptic surfaces, yielding concrete realizations for SU(2) and $G_2$ structure groups via spectral curves and cameral covers, including Prym constructions. The authors argue that the boundary map equates the threefold and heterotic moduli in a precise lattice–Hodge–structure sense, providing a richer analogue of mirror symmetry and offering a framework to probe nonperturbative effects in both theories. This leads to a refined picture in which the Calabi–Yau and bundle data are encoded in lower-dimensional geometric objects, clarifying how duality acts at the boundary of the moduli space and guiding further explorations of nonperturbative string dynamics.

Abstract

A type IIA string (or F-theory) compactified on a Calabi-Yau threefold is believed to be dual to a heterotic string on a K3 surface times a 2-torus (or on a K3 surface). We consider how the resulting moduli space of hypermultiplets is identified between these two pictures in the case of the E8xE8 heterotic string. As examples we discuss SU(2)-bundles and G2-bundles on the K3 surface and the case of point-like instantons. We are lead to a rather beautiful identification between the integral cohomology of the Calabi-Yau threefold and some integral structures on the heterotic side somewhat reminiscent of mirror symmetry. We discuss the consequences for probing nonperturbative effects in the both the type IIA string and the heterotic string.

Figures (6)

• Figure 1: The stable degeneration for $X$.
• Figure 2: The stable degeneration $X^\sharp$.
• Figure 3: The discriminant for an SU(2)-bundle.
• Figure 4: The discriminant for a $G_2$-bundle.
• Figure 5: The blow-up for a generic point-like $E_8$ instanton.

Theorems & Definitions (1)

• Theorem 1