Non-Abelian discrete gauge symmetries in F-theory

The presence of non-abelian discrete gauge symmetries in four-dimensional f-theory compactifications is investigated. such symmetries are shown to arise from seven-brane configurations in genuine f-theory settings without a weak string coupling description. gauge fields on mutually non-local seven-branes are argued to gauge both r-r and ns-ns two-form bulk axions. the gauging is completed into a generalisation of the heisenberg group with either additional seven-brane gauge fields or r-r bulk gauge fields. the former case relies on having seven-brane fluxes, while the latter case requires torsion cohomology and is analysed in detail through the m-theory dual. remarkably, the m-theory reduction yields an abelian theory that becomes non-abelian when translated into the correct duality frame to perform the f-theory limit. the reduction shows that the gauge coupling function depends on the gauged scalars and transforms non-trivially as required for the groups encountered. this field dependence agrees with the expectations for the kinetic mixing of seven-branes and is unchanged if the gaugings are absent.