Explicit de Sitter flux vacua for global string models with chiral matter

We address the open question of performing an explicit stabilisation of all closed string moduli (including dilaton, complex structure and Kähler moduli) in fluxed type IIB Calabi-Yau compactifications with chiral matter. Using toric geometry we construct Calabi-Yau manifolds with del Pezzo singularities. D-branes located at such singularities can support the Standard Model gauge group and matter content or some close extensions. In order to control complex structure moduli stabilisation we consider Calabi-Yau manifolds which exhibit a discrete symmetry that reduces the effective number of complex structure moduli. We calculate the corresponding periods in the symplectic basis of invariant three-cycles and find explicit flux vacua for concrete examples. We compute the values of the flux superpotential and the string coupling at these vacua. Starting from these explicit complex structure solutions, we obtain AdS and dS minima where the Kähler moduli are stabilised by a mixture of D-terms, non-perturbative and perturbative α ′ corrections as in the LARGE Volume Scenario. In the considered example the visible sector lives at a dP6 singularity which can be higgsed to the phenomenologically interesting class of models at the dP3 singularity