Comments on N=1 Heterotic String Vacua

TL;DR

The paper addresses how to characterize moduli, stability, and chiral matter in four-dimensional $N=1$ heterotic vacua on elliptically fibered Calabi–Yau threefolds. It leverages the spectral-cover description and Fourier–Mukai transforms to relate bundle data on $X$ to spectral data on the base, derives stability-preserving properties under FM, and computes moduli counts for vertical five-brane transitions via index theory and localization on the intersection curve $S$; it also connects these results to F-theory/M-theory dual pictures. A key contribution is an explicit framework to count transition moduli, using Parseval-type relations and vanishing arguments on $S$, and a criterion to ensure vanishing of certain chiral matter sectors, with implications for Yukawa couplings. Overall, the work clarifies how the geometry of spectral covers, stability, and brane transitions control the low-energy content of these heterotic vacua and their dual descriptions.

Abstract

We analyze three aspects of N=1 heterotic string compactifications on elliptically fibered Calabi-Yau threefolds: stability of vector bundles, five-brane instanton transitions and chiral matter. First we show that relative Fourier-Mukai transformation preserves absolute stability. This is relevant for vector bundles whose spectral cover is reducible. Then we derive an explicit formula for the number of moduli which occur in (vertical) five-brane instanton transitions provided a certain vanishing argument applies. Such transitions increase the holonomy of the heterotic vector bundle and cause gauge changing phase transitions. In an M-theory description the transitions are associated with collisions of bulk five-branes with one of the boundary fixed planes. In F-theory they correspond to three-brane instanton transitions. Our derivation relies on an index computation with data localized along the curve which is related to the existence of chiral matter in this class of heterotic vacua. Finally, we show how to compute the number of chiral matter multiplets for this class of vacua allowing to discuss the associated Yukawa couplings.