Geometry and supersymmetry in type II string theory
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Abstract
My thesis is associated with the field of D6-brane model building in type IIA string theory, where the fractional D6-brane stacks are wrapped on special Lagrangian cycles in backgrounds of factorisable toroidal orientifolds T6/(Z2xZ2MxΩR) with 2M=2,6,6' and with discrete torsion. I develop an explicit formalism for complex structure deformations of the Z2 orbifold singularities and observe how the volumes of the special Lagrangian cycles change under deformations. I show that, depending on the concrete model, this procedure can be used to stabilise several or all twisted complex structure moduli, or that, alternatively, the sizes of gauge couplings can be varied.
As a starting point, I introduce the orientifold T6/(Z2xZ2xΩR) on square torus lattices, now expressed as hypersurface in a complex weighted projective space. I demonstrate the process of concrete deformations and discuss how explicit volumes of fractional cycles, bulk cycles, and exceptional cycles can be computed. These can be used to determine physical quantities like gauge couplings. My investigations also show if the cycles keep their special Lagrangian property under deformations. After that, I present the orientifold T6/(Z2xZ6'xΩR) with underlying hexagonal tori, which is very interesting for model building with intersecting D6-brane stacks. For the concrete construction, I start again with T6/(Z2xZ2xΩR) and mod out an additional Z3 symmetry by hand, which implies that the deformation parameters are organised in Z3 and ΩR invariant orbits. In this setup, new technical difficulties arise. In addition, I show first results for the phenomenologically very appealing orientifold T6/(Z2xZ6xΩR) on one rectangular and two hexagonal tori, whose underlying structure is much more complicated than in the previous examples. Hence, so far only local descriptions are found.
In concrete models, there exist stacks of N coincident D6-branes, wrapped on fractional cycles, which carry gauge groups SO(2N), USp(2N), and/or U(N). I show that, depending on the model, the impact of the deformations can be divided into three different cases. Firstly, the brane stack does not couple to a certain deformation, which gives rise to a flat direction in the moduli space. Secondly, the branes couple only to the orientifold-even part of the deformed exceptional cycle, but not to the orientifold-odd part, in which case one can adjust the corresponding gauge couplings by changing the volume of the exceptional cycle. Thirdly, if the brane stack couples to the orientifold-odd part, one finds that the respective complex structure modulus is stabilised, which is a desirable property to find a unique vacuum. For D6-branes with SO(2N) or USp(2N) gauge group, by construction all orientifold-odd contributions are zero and no moduli are stabilised. On the other hand, in a Pati-Salam model with three particle generations and only U(N) gauge groups all three cases can be observed, and actually ten out of 15 twisted complex structure moduli can be stabilised at the orbifold point.