Abstract
We present a general method for computing Hodge numbers for Calabi-Yau manifolds realised as discrete quotients of complete intersections in products of projective spaces. The method relies on the computation of equivariant cohomologies and is illustrated for several explicit examples. In this way, we compute the Hodge numbers for all discrete quotients obtained in Braun’s classification [1].
Highlights
Showing that there is a tension between directly breaking E8 to the Standard Model group and obtaining a particle spectrum free from exotics
A systematic study of discrete symmetry groups, G, on CICY manifolds X was initiated by Candelas and Davies in ref. [43], and completed by Braun in ref. [1], who classified, through an automated scan, all finite group actions that descend from linear automorphisms of the ambient space, given by a product of projective spaces, to free actions on the CICY manifold
Our results are summarised in figure 1. This figure represents the tip of the Hodge plot of all Calabi-Yau manifolds presently known, highlighting CICY quotients and the new results obtained in the present paper
Summary
We explain the general method to compute Hodge numbers of CICY quotients. We being with a general set-up of CICY manifolds X and first review how to compute the “upstairs” cohomology H2,1(X) of these manifolds. The “downstairs” cohomology H2,1(X ) of this quotient is given by the G-invariant part of the upstairs cohomology H2,1(X) and we explain in detail how to calculate this G-invariant part. In this way, we can obtain the Hodge number h2,1(X ) of the quotient. Since the Euler number of the quotient is obtained from its upstairs counterpart by dividing by the group order |G|, this fixes h1,1(X ) as well
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