It is known that many Calabi-Yau manifolds form a connected web. The question of whether all Calabi-Yau manifolds form a single web depends on the degree of singularity that is permitted for the varieties that connect the distinct families of smooth manifolds. If only conifolds are allowed then, since shrinking two-spheres and three-spheres to points cannot affect the fundamental group, manifolds with different fundamental groups will form disconnected webs. We examine these webs for the tip of the distribution of Calabi-Yau manifolds where the Hodge numbers (h 11 , h 21 ) are both small. In the tip of the distribution the quotient manifolds play an important role. We generate via conifold transitions from these quotients a number of new manifolds. These include a manifold with χ = −6, that is an analogue of the χ = −6 manifold found by Yau, and manifolds with an attractive structure that may prove of interest for string phenomenology. We also examine the relation of some of these manifolds to the remarkable Gross-Popescu manifolds that have Euler number zero.

We construct supersymmetric compactifications of E 8 ×E 8 heterotic string theory which realise exactly the massless spectrum of the Minimal Supersymmetric Standard Model (MSSM) at low energies. The starting point is the standard embedding on a Calabi-Yau threefold which has Hodge numbers (h 1,1 , h 2,1 ) = (1, 4) and fundamental group Z 12 , which gives an E 6 grand unified theory with three net chiral generations. The gauge symmetry is then broken to that of the standard model by a combination of discrete Wilson lines and continuous deformation of the gauge bundle. On eight distinct branches of the moduli space, we find stable bundles with appropriate cohomology groups to give exactly the massless spectrum of the MSSM.

We propose a 4 + 1-dimensional action that is a candidate for realising a standard-model-like effective theory for fields dynamically localised to a domain-wall brane. Our construction is in part based on the conjecture that the Dvali-Shifman mechanism for dynamically localising gauge bosons to a domain wall works correctly in 4 + 1-d. Assuming this to be so, we require the gauge symmetry to be SU(5) in the bulk, spontaneously breaking to SU(3)⊗SU(2)⊗U(1) inside the domain wall, thus dynamically localising the standard model gauge bosons provided that the SU(5) theory in the bulk exhibits confinement. The wall is created jointly by a real singlet-Higgs field η configured as a kink, and an SU(5) adjoint-Higgs field χ that takes nonzero values inside the wall. Chiral 3 + 1dimensional quarks and leptons are confined and split along the bulk direction via their Yukawa couplings to η and χ. The Higgs doublet and its colour triplet SU(5) partner are similarly localised and split. The splittings can suppress coloured-Higgs-induced proton decay and, because of the different localisation profiles, the usual SU(5) mass relation me = m d does not arise. Localised gravity is generated via the Randall-Sundrum alternative to compactification.

We present a complete intersection Calabi-Yau manifold Y that has Euler number −72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z12 and the non-Abelian dicyclic group Dic3. The quotient manifolds have χ = −6 and Hodge numbers (h 11 , h 21 ) = (1, 4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h 11 , h 21 ) = (2, 2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y . The manifold Y may also be realised as a hypersurface in a toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.

We present a supersymmetric extension of the Standard Model in which only one electroweak doublet acquires a vacuum expectation value and gives mass to Standard Model fermions. As well as the novel accommodation of a Standard Model Higgs within a supersymmetric framework, this leads to a very predictive model, with some advantages over the MSSM. In particular, problems with proton decay, flavour changing neutral currents and large CP violation are ameliorated, primarily due to the presence of an anomalyfree R-symmetry. Since supersymmetry must be broken at a low scale, gravity-mediated effects which break the R-symmetry are naturally small. The R-symmetry requires the presence of adjoint chiral superfields, to give Dirac masses to the gauginos; these adjoints are the only non-MSSM fields in the visible sector. The LSP is a very light neutralino, which is mostly bino. Such a light neutralino is not in conflict with experiment, and is a striking prediction of the minimal model. Additional scenarios to raise the mass of this neutralino to the weak scale are also outlined. Prospects for discovery at the LHC are briefly discussed, along with viable scenarios for achieving gauge-coupling unification.