Gauge groups and matter fields on some models of F-theory without section

We investigate F-theory on an elliptic Calabi-Yau 4-fold without a section to the fibration. To construct an elliptic Calabi-Yau 4-fold without a section, we introduce families of elliptic K3 surfaces which do not admit a section. A product K3 x K3, with one of the K3's chosen from these families of elliptic K3 surfaces without a section, realises an elliptic Calabi-Yau 4-fold without a section. We then compactify F-theory on such K3 x K3's. We determine the gauge groups and matter fields which arise on 7-branes for these models of F-theory compactifications without a section. Since each K3 x K3 constructed does not have a section, gauge groups arising on 7-branes for F-theory models on constructed K3 x K3's do not have U(1)-part. Interestingly, exceptional gauge group E-6 appears for some cases.