| Summary: | <p>In this thesis, we study compactifiations of ten-dimensional heterotic supergravity at O(α'), focusing on the moduli of such compactifications. We begin by studying supersymmetric compactifications to four-dimensional maximally symmetric space, commonly referred to as the Strominger system. The compactifications are of the form <em>M</em><sub>10</sub> = <em>M</em><sub>4</sub> x <em>X</em>, where <em>M</em><sub>4</sub> is four-dimensional Minkowski space, and <em>X</em> is a six-dimensional manifold of what we refer to as heterotic <em>SU</em>(3)-structure. We show that this system can be put in terms of a holomorphic operator D on a bundle Q = T&ast; X ⊕ End(<em>TX</em>) ⊕ End(V ) ⊕ <em>TX</em>, defined by a series of extensions. Here <em>V</em> is the <em>E</em><sub>8</sub> x <em>E</em><sub>8</sub> gauge-bundle, and <em>TX</em> is the tangent bundle of the compact space <em>X</em>. We proceed to compute the infinitesimal deformation space of this structure, given by TM = <em>H</em><sup>(0,1)</sup>(Q), which constitutes the infinitesimal spectrum of the lower energy four-dimensional theory. In doing so, we find an over counting of moduli by <em>H</em><sup>(0,1)</sup>(End(<em>TX</em>)), which can be reinterpreted as O(α') field redefinitions.</p> <p>In the next part of the thesis, we consider non-maximally symmetric compactifications of the form <em>M</em><sub>10</sub> = <em>M</em><sub>3</sub> x <em>Y</em> , where M<sub>3</sub> is three-dimensional Minkowski space, and <em>Y</em> is a seven-dimensional non-compact manifold with a <em>G</em><sub>2</sub>-structure. We write <em>X</em> → <em>Y</em> → &reals;, where X is a six dimensional compact space of half- at <em>SU</em>(3)-structure, non-trivially fibered over &reals;. These compactifications are known as domain wall compactifications. By focusing on coset compactifications, we show that the compact space <em>X</em> can be endowed with non-trivial torsion, which can be used in a combination with %alpha;'-effects to stabilise all geometric moduli. The domain wall can further be lifted to a maximally symmetric AdS vacuum by inclusion of non-perturbative effects in a heterotic KKLT scenario. Finally, we consider domain wall compactifications where <em>X</em> is a Calabi-Yau. We show that by considering such compactifications, one can evade the usual no-go theorems for flux in Calabi-Yau compactifications, allowing flux to be used as a tool in such compactifications, even when <em>X</em> is Kähler. The ultimate success of these compactifications depends on the possibility of lifting such vacua to maximally symmetric ones by means of e.g. non-perturbative effects.</p>
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