Andreas, B.; Ruipérez, D. Hernández Comments on \(N=1\) heterotic string vacua. (English) Zbl 1069.81560 Adv. Theor. Math. Phys. 7, No. 5, 751-786 (2003). Summary: 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 a 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 associated Yukawa couplings. Cited in 7 Documents MSC: 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory 83E30 String and superstring theories in gravitational theory 83E15 Kaluza-Klein and other higher-dimensional theories 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 81V22 Unified quantum theories 81-02 Research exposition (monographs, survey articles) pertaining to quantum theory 81T13 Yang-Mills and other gauge theories in quantum field theory 81T60 Supersymmetric field theories in quantum mechanics PDFBibTeX XMLCite \textit{B. Andreas} and \textit{D. H. Ruipérez}, Adv. Theor. Math. Phys. 7, No. 5, 751--786 (2003; Zbl 1069.81560) Full Text: DOI arXiv