  We investigate the field dependence of the gauge coupling functions of
four-dimensional Type IIB orientifold and F-theory compactifications with
space-time filling seven-branes. In particular, we analyze the constraints
imposed by holomorphicity and covariance under shift-symmetries of the bulk and
brane axions. This requires introducing quantum corrections that necessarily
contain Riemann theta functions on the complex torus spanned by the D7-brane
Wilson line moduli. Our findings hint towards a new underlying geometric
structure for gauge coupling functions in string compactifications. We
generalize this discussion to a genuine F-theory compactification on an
elliptically fibered Calabi-Yau fourfold. We perform the first general
dimensional reduction of eleven-dimensional supergravity and dualization to the
F-theory frame. The resulting effective action is compared with the circle
reduction of a four-dimensional N=1 supergravity theory. The F-theory geometry
elegantly unifies bulk and brane degrees of freedom and allows us to infer
non-trivial results about holomorphicity and shift-symmetries. For instance, we
gain new insight into kinetic mixing of bulk and brane gauge fields.
