On the Heterotic/F-Theory Duality in Eight Dimensions

TL;DR

The paper tests the heterotic string on $T^2$ against F-theory on $K3$ in eight dimensions by matching exact one-loop threshold couplings $\Delta_{\rm eff}(T,U)$ computed on the heterotic side with geometric data from elliptically fibered $K3$ in F-theory. It develops a detailed framework linking modular integrals, elliptic genera, and holomorphic prepotentials to Chern–Simons couplings on Kodaira 7-planes, Green’s functions on $\mathbb{Z}_N$ curves, and mirror-map structures between open and closed string sectors, aided by Picard–Fuchs equations and their symmetric-square extensions. The results reproduce the expected singularities at gauge enhancements and establish a quantitative duality between the two descriptions, while revealing the role of global brane geometry and potentially new geometric objects (such as Sym$^2(K3)$) in encoding non-perturbative interactions. This strengthens the eight-dimensional duality and provides a blueprint for extracting exact non-perturbative information from the geometry of elliptic fibrations.

Abstract

We review quantitative tests on the duality between the heterotic string on T^2 and F-theory on K3. On the heterotic side, certain threshold corrections to the effective action can be exactly computed at one-loop order, and the issue is to reproduce these from geometric quantities pertaining to the K3 surface. In doing so we learn about certain non-perturbative interactions of 7-branes.

Figures (10)

• Figure 1: Encircling the location of a 7-brane in the $z$-plane leads to a jump of the perceived type IIB string coupling, $\tau_{IIB}\to\tau_{IIB}+1$.
• Figure 2: Fibration of an elliptic curve over ${\rm I P}^1$, which in total makes a $K3$ surface. At 24 points the fibers and therefore the string coupling become singular, and this is where the 7-branes are located.
• Figure 3: Enlargement of non-abelian gauge symmetries: a) $U(n)$ is generated by open strings stretched between $n$$D7$-branes of type $(0,1)$; b) if we add a pair of $(1,1)$ and $(1,-1)$ branes, then the extra "indirect" trajectories extend $U(n)$ to $SO(2n)$; c) $E_8$ is generated by colliding ${\cal E}_6$ and ${\cal H}_2$ Kodaira 7-planes.
• Figure 4: One the left we show three singular fibers of type $A_0$. The open string trajectories are projections of 2-cycles of non-zero volume in the $K3$. On the right we have collided the singular fibers to form a singularity of type $A_2$, which is associated with the simultaneous vanishing of several intersecting 2-cycles in the $K3$.
• Figure 5: 7-plane configuration of the first three models in (1.15), which describes $K3$ surfaces with elliptic $({\cal E}_{8-k}\times {\cal H}_k)^2$ singularities. We have indicated the monodromies given by $\omega=e^{2\pi i/N}$, and also exhibited multiplets of (mutually non-local) open strings that run between the ${\cal H}_k$ planes. For $\lambda(\tau)\rightarrow 1$ the ${\cal H}_k$ planes merge into a single plane, the strings between them then giving rise to massless charged gauge fields that enhance the non-abelian gauge group.