Abstract
We show that non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Specifically, we focus on Calabi-Yau three-folds constructed as complete intersections in products of projective spaces, although we expect similar methods to apply to other constructions and also to Calabi-Yau four-folds. Among the wide range of possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications, on all complete intersection Calabi-Yau manifolds. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form.
Highlights
Of CY compactifications, in particular the spectrum of massless particles and holomorphic quantities in the low-energy theory, can be extracted with methods of algebraic geometry without any recourse to the metric
Whether a given manifold with SU(3) structure does provide a solution to string theory depends on the pattern of torsion classes which have to match constraints arising from the chosen flux
We have explored the relation between these two approaches by constructing non-trivial SU(3) structures on Calabi-Yau three-folds and by analysing their possible role in string compactifications
Summary
We discuss possible non-trivial SU(3) structures on the quintic CY, defined as the anti-canonical hypersurface in the ambient space A = P4. This is a warm-up example for the section, where this discussion will be generalised to all CICY manifolds. We begin with some general background and notation for the projective space P4
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