A three‐generation Calabi‐Yau manifold with small Hodge numbers

Abstract

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.

Stabilizing all Kähler moduli in type IIB orientifolds

“…There is a free Z 2 ×Z 2 ×Z 2 -action on X [21,23,24,28,57]. We will always divide out this group action and define…”

Groups acting freely on Calabi–Yau threefolds embedded in a product of del Pezzo surfaces

“…The solid dots correspond to quotients found in this paper. The blue rings represent the ones known until now (with respect to the data collected in [2][3][4]6]). The black rings are quotients by groups whose order is maximal.…”

“…From the picture below, we can summarize our results as follows. The dots (3,5), (2,7) and (5,13) represent NEW CalabiYau threefolds. There exists a Calabi-Yau manifold corresponding to the pair (1, 5) with non-abelian fundamental group; see [4].…”

“…The two cubic Fermat surfaces are examples of degree 3 del Pezzo surfaces, i.e., smooth surfaces with ample anticanonical divisor which can be obtained as the blow-up of P 2 C at six points in general position. A first generalization in this direction was given by Braun, Candelas and Davies in [3]. In that paper, they discover a new Calabi-Yau manifold with Euler characteristic −6 and small Hodge numbers.…”

“…In general, special attention is given to those Calabi-Yau manifolds that have Euler characteristic 6 in absolute value since they correspond to three-generation families (see, for instance, [3]). …”