Massless Spectra of Three Generation U(N) Heterotic String Vacua

TL;DR

The paper develops a comprehensive toolkit for computing the massless spectrum of heterotic string vacua built from $U(N)$ bundles on elliptically fibered Calabi–Yau threefolds. Central to the method is the Leray spectral sequence, which localizes cohomology to base curves and spectral-intersection loci, enabling precise formulas for $H^i(X,V_a\otimes V_b)$, $H^i(X,\wedge^2 V)$, and $H^i(X, S^2 V)$ in terms of line bundles on curves. It introduces $\Lambda$-stability as a one-loop corrected stability notion and demonstrates stability for non-split extensions, then applies the machinery to construct a fully consistent three-generation flipped $SU(5)$ model with MSSM matter content and a single GUT Higgs pair, highlighting both the potential and the limitations (e.g., extra EW-Higgs) of such vacua. The work offers a robust pathway to exploring MSSM-like heterotic vacua on general simply connected Calabi–Yau manifolds without relying on nontrivial fundamental groups, with implications for model-building and phenomenology.

Abstract

We provide the methods to compute the complete massless spectra of a class of recently introduced supersymmetric E8 x E8 heterotic string models which invoke vector bundles with U(N) structure group on simply connected Calabi-Yau manifolds and which yield flipped SU(5) and MSSM string vacua of potential phenomenological interest. We apply Leray spectral sequences in order to derive the localisation of the cohomology groups H^i(X,V_a \times V_b), H^i(X,\bigwedge^2 V) and H^i(X,{\bf S}^2 V) for vector bundles defined via Fourier-Mukai transforms on elliptically fibered Calabi-Yau manifolds. By the method of bundle extensions we define a stable U(4) vector bundle leading to the first flipped SU(5) model with just three generations, i.e. without any vector-like matter. Along the way, we propose the notion of Lambda-stability for heterotic bundles.