| Summary: | <p>In this thesis we study string compactifications on manifolds equipped with a G-structure, placing a special emphasis on the interplay between geometry and physics. We follow two complementary approaches.</p>
<p>In the first part of the thesis we adopt a sigma model perspective and focus on the worldsheet superconformal field theory. We consider compactifications on 7-dimensional Extra Twisted Connected Sum (ETCS) G2 manifolds as well as 8-dimensional Generalized Connected Sum (GCS) Spin(7) manifolds. These are special holonomy manifolds obtained by gluing together two open manifolds along isomorphic asymptotic ends in a particular fashion. We find that this geometric construction is reproduced in the worldsheet algebra via a diamond of algebra inclusions. A study of the automorphisms of these algebras leads us to conjecture new mirror maps for GCS manifolds. Finally, we explore—making use of the worldsheet algebras—whether these constructions produce manifolds of generic special holonomy.</p>
<p>In the second part of the thesis we change gears and consider string compactifications from a supergravity point of view. In particular, we focus on compactifications of the heterotic string down to three spacetime dimensions preserving minimal supersymmetry N = 1. These are described by the heterotic G2 system, which is the 7-dimensional analogue of the Hull–Strominger system. Finding solutions to this system involves the study of 7-dimensional manifolds with integrable G2-structures and the construction of G2-instantons on bundles over them. In addition, the anomaly cancellation condition mixes the different degrees of freedom of the theory in a highly non-trivial way. As a result, explicit background solutions are difficult to obtain. We construct new families of solutions to the system on homogeneous 3-Sasakian manifolds—that means either the 7-sphere or the Aloff-Wallach space N(1,1)—equipped with squashed metrics. Our solutions present constant dilaton and AdS3 spacetime. Some of these families can be regarded as finite deformations from a given solution, providing an explicit description of a particular direction in the moduli space.</p>
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