Hosotani breaking of E6 to a subgroup of rank five

Abstract

On multiply connected manifolds, it is possible to construct vacuum gauge configurations with nontrivial holonomy groups. This is the basis of the Hosotani mechanism. This naturally suggests a ‘‘Hosotani inverse problem’’: If we wish to break a gauge group G to a subgroup H, what are the possible finite holonomy groups having this effect, and what can one say about the fundamental groups of the underlying manifolds? Usually, this problem is too difficult to solve, but we show that, for G=E6 and H locally isomorphic to the rank five group SU(3)×SU(2)×U(1)×U(1), a complete solution is possible. It is hoped that the results will aid a search for examples of Calabi–Yau manifolds leading to a low-energy gauge group of rank five.