Perturbative Prepotential and Monodromies in N=2 Heterotic Superstring

TL;DR

The paper analyzes the perturbative monodromies of the one-loop prepotential $F$ in $N=2$ heterotic string models, showing that logarithmic singularities from massless states at enhanced-symmetry loci modify the duality group from a product of modular groups to a fundamental-group (braid-like) structure. By computing the one-loop correction in the two-moduli $(T,U)$ case and recasting the theory in a linear symplectic basis, the authors identify a normal abelian subgroup $H$ of monodromies with $G/H \cong O(2,2;\mathbb{Z})$, and extend the construction to general $(4,4)$ and $(4,0)$ compactifications, illustrating how Weyl reflections and logarithmic terms govern the quantum monodromies. They provide explicit braid-group representations in concrete models (including a two-modulus $(4,0)$ example) and outline how the perturbative structure constrains, and is constrained by, non-perturbative physics, laying groundwork for understanding monodromies within $Sp(2r+4,\mathbb{Z})$ in a controlled setting. The work thus connects perturbative string computations, special geometry, and braid-group mathematics to illuminate the perturbed duality structure of four-dimensional $N=2$ string vacua.

Abstract

We discuss the prepotential describing the effective field theory of N=2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory.