UC Santa Barbara

In F-theory, a branch of string theory, the geometry of an elliptically fibered variety determines important features of the associated physical theory. In particular, the singular fibers of the elliptic fibration give rise to Lie-theoretic data that determine the nonabelian part of the gauge group. This connection between singular fibers and Lie theory suggests the possibility of a similar construction in arithmetic geometry.Motivated by this analogy, we introduce arithmetic gauge algebras and arithmetic gauge groups associated to an elliptic curve E/K, where K is the fraction field of a Dedekind domain R. These objects are constructed from the Mordell–Weil group of E together with the geometric special fibers of a minimal proper regular model. The action of the absolute Galois group on the irreducible components of the geometric special fibers gives rise to arithmetic dual graphs and associated abstract Cartan matrices, from which semisimple Lie algebras and groups of Lie type are constructed. This provides an arithmetic analogue of the gauge group construction appearing in F-theory.