Non-Cartan Mordell-Weil lattices of rational elliptic surfaces and heterotic/F-theory compactifications

The Mordell-Weil lattices (MW lattices) associated to rational elliptic surfaces are classified into 74 types. Among them, there are cases in which the MW lattice is none of the weight lattices of simple Lie algebras or direct sums thereof. We study how such "non-Cartan MW lattices" are realized in the six-dimensional heterotic/F-theory compactifications. In this paper, we focus on non-Cartan MW lattices that are torsion free and whose associated singularity lattices are sublattices of $A_7$. For the heterotic string compactification, a non-Cartan MW lattice yields an instanton gauge group $H$ with one or more $U(1)$ group(s). We give a method for computing massless spectra via the index theorem and show that the $U(1)$ instanton number is limited to be a multiple of some particular non-one integer. On the F-theory side, we examine whether we can construct the corresponding threefold geometries, i.e., rational elliptic surface fibrations over $P^1$. Except for some cases, we obtain such geometries for specific distributions of instantons. All the spectrum derived from those geometries completely match with the heterotic results.