Invariant Homology on Standard Model Manifolds

TL;DR

This paper computes invariant homology classes on specific Calabi-Yau threefolds with Z_2 x Z_2 symmetry, crucial for constructing anomaly-free string models with three generations.

Contribution

It explicitly determines the invariant homology classes on Z and their generators, advancing the understanding of geometric structures relevant for string phenomenology.

Findings

Explicit invariant homology classes on Z are computed.

Seven generators of H_4(Z,Z) are identified.

Results support the construction of realistic heterotic string models.

Abstract

Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces, each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2 invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H_4(Z,Z) on Z. The existence of these homology classes on Z is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory.