Compactifications of the Heterotic String with Unitary Bundles

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.