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 × 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 × 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 × k3 constructed does not have a section, gauge groups arising on 7-branes for f-theory models on constructed k3 × k3’s do not have u(1)-part. interestingly, exceptional gauge group e 6 appears for some cases.