Aspects of vacuum moduli in string theory

Sammanfattning: In this thesis we explore geometric aspects of the space of vacua in supersymmetric string compactifications. More specifically, we will look at heterotic strings compactified on Calabi-Yau manifolds and, separately, G2 structure manifolds. After reviewing the basic concepts underlying the thesis, specifically dimensional reduction of supergravity theories and G structures on manifolds, we begin looking at the vacua leading to supersymmetric three dimensional theories. The underlying compact manifold has a G2 structure. We investigate some structures that further refine the G2 structure: almost contact metric three-structures. We report on a relation between these structures and (co)associative cycles in the G2 structure manifold, and initiate an exploration of the topological properties of the moduli space of such structures. Next, we identify a superpotential whose critical locus is precisely the moduli space of supersymmetry preserving backgrounds, including both gauge and geometric fields.  Finally, we study the Yukawa couplings of heterotic string compactifications on Calabi-Yau manifolds. Using sheaf cohomological tools, we find widely applicable vanishing theorems, which can constrain the Yukawa couplings of the effective four dimensional theory.