Weigand, Timo (2006): Compactifications of the Heterotic String with Unitary Bundles. Dissertation, LMU München: Fakultät für Physik
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### Abstract

In this thesis we investigate a large new class of four-dimensional supersymmetric string vacua defined as compactifications of the $E_8 \times E_8$ and the $SO(32)$ heterotic string on smooth Calabi-Yau threefolds with unitary gauge bundles and heterotic five-branes. This opens up the way for phenomenologically interesting string compactifications on simply connected manifolds in that the conventional gauge symmetry breaking via Wilson lines is replaced by the embedding of non-flat line bundles into the ten-dimensional gauge group. The first part of the thesis discusses the implementation of this idea into the $E_8 \times E_8$ heterotic string. After specifying a large class of group theoretic embeddings featuring unitary bundles, we analyse the effective four-dimensional ${\cal N}=1$ supergravity upon compactification. The simultaneous presence of five-branes and abelian gauge groups requires the introduction of new anomaly cancelling counter terms into the effective action. These are also derived by an M-theory computation. The full set of Green-Schwarz terms allows for the extraction of the threshold corrections. From the gauge invariant K\"ahler potential for the moduli fields we derive a modification of the Fayet-Iliopoulos D-terms arising at one-loop in string perturbation theory. From this we conjecture a one-loop deformation of the Hermitian Yang-Mills equation and introduce the idea of $\lambda$-stability as the perturbatively correct stability concept generalising the notion of Mumford stability valid at tree-level. We then proceed to a definition of $SO(32)$ heterotic vacua with unitary gauge bundles in the presence of heterotic five-branes and find agreement of the resulting spectrum with the S-dual framework of Type I/Type IIB orientifolds. A similar analysis of the effective four-dimensional supergravity is performed. Further evidence for the proposed one-loop correction to the stability condition is found by identifying the heterotic corrections as the S-dual of the perturbative part of $\Pi$-stability as the correct stability concept in Type IIB theory. After reviewing the construction of holomorphic stable vector bundles on elliptically fibered Calabi-Yau manifolds via spectral covers, we provide semi-realistic examples for $SO(32)$ heterotic vacua with Pati-Salam and MSSM-like gauge sectors. These can be viewed, by S-duality, as the generalisation of toroidal magnetized $D9$-branes to non-abelian braneworlds on genuine Calabi-Yau manifolds. We finally discuss the construction of realistic vacua with flipped $SU(5)$ GUT and MSSM gauge group within the $E_8 \times E_8$ framework, based on the embedding of line bundles into both $E_8$ factors. Some of the appealing phenomenological properties of this stringy realisation of flipped $SU(5)$ models, in particular stability of the proton, are discussed. MSSM-like gauge coupling unification is possible for the threshold corrected gauge couplings. We explicitly construct a couple of supersymmetric string vacua in both setups with precisely the three observed chiral matter generations and without any exotic chiral states.