SU(4) Instantons on Calabi-Yau Threefolds with Z_2 x Z_2 Fundamental Group
Ron Donagi, Burt A. Ovrut, Tony Pantev, Rene Reinbacher
TL;DR
The paper tackles constructing SU(4) gauge vacua in heterotic string theory on torus-fibered Calabi–Yau threefolds with a $\mathbb{Z}_2\times\mathbb{Z}_2$ fundamental group. It develops a concrete construction of invariant, stable rank-4 holomorphic vector bundles on the simply connected cover $X$ via nontrivial extensions of invariant rank-2 bundles, with descent to the non-simply connected quotient $Z$ (twisted by a flat $B$-field when needed). By enforcing topological conditions $c_1(V)=0$, an effective anomaly-cancellation class, and $c_3(V)=24$, and proving stability for a large class of extensions, the work yields explicit discrete parameter spaces and continuous moduli for these vacua. The resulting Standard-Model-like vacua feature a low-energy $U(1)_{B-L}$ symmetry that helps suppress nucleon decay, illustrating a phenomenologically viable path within heterotic constructions and setting groundwork for broader generalizations to other fundamental groups and structure groups.
Abstract
Structure group SU(4) gauge vacua of both weakly and strongly coupled heterotic superstring theory compactified on torus-fibered Calabi-Yau threefolds Z with Z_2 x Z_2 fundamental group are presented. This is accomplished by constructing invariant, stable, holomorphic rank four vector bundles on the simply connected cover of Z. Such bundles can descend either to Hermite-Yang-Mills instantons on Z or to twisted gauge fields satisfying the Hermite-Yang-Mills equation corrected by a non-trivial flat B-field. It is shown that large families of such instantons satisfy the constraints imposed by particle physics phenomenology. The discrete parameter spaces of those families are presented, as well as a lower bound on the dimension of the continuous moduli of any such vacuum. In conjunction with Z_2 x Z_2 Wilson lines, these SU(4) gauge vacua can lead to standard-like models at low energy with an additional U(1)_{B-L} symmetry. This U(1)_{B-L} symmetry is very helpful in naturally suppressing nucleon decay.