ICMP lecture on Heterotic/F-theory duality

TL;DR

The work surveys the conjectured isomorphism between heterotic and F‑theory moduli across eight, six, and four dimensions, highlighting how cameral/spectral covers and del Pezzo fibrations encode heterotic data while Deligne cohomology and period maps govern F‑theory moduli. It shows that, in the geometric limit, the heterotic Prym fibers correspond to abelian components of F‑theory intermediate Jacobians, yielding a precise matching of the continuous moduli and a finite‑index embedding for discrete data, with exact results in several special cases (notably for base $B=\mathbb{P}^1$ and certain gauge groups). The analysis integrates intricate algebraic geometry (spectral/cameral covers, del Pezzo fibrations, Prym varieties) with holomorphic integrable systems, contributing to a rigorous understanding of duality in elliptically fibered compactifications. The findings advance the mathematical framework of string dualities and illuminate how integrable system structures bridge heterotic and F‑theory descriptions.

Abstract

The heterotic string compactified on an (n-1)-dimensional elliptically fibered Calabi-Yau Z-->B is conjectured to be dual to F-theory compactified on an n-dimensional Calabi-Yau X-->B, fibered over the same base with elliptic K3 fibers. In particular, the moduli of the two theories should be isomorphic. The cases most relevant to the physics are n=2, 3, 4, i.e. the compactification is to dimensions d=8, 6 or 4 respectively. Mathematically, the richest picture seems to emerge for n=3, where the moduli space involves an analytically integrable system whose fibers admit rather different descriptions in the two theories. The purpose of this talk is to review some of what is known and what is not yet known about this conjectural isomorphism. Some of the underlying mathematics of principal bundles on elliptic fibrations is reviewed in the accompanying Taniguchi talk (hep-th/9802094).