  We look at connection Laplacians L,g defined by a field h:G to K, where G is
a finite set of sets and K is a normed division ring which does not need to be
commutative, nor associative but has a conjugation leading to the norm as the
square root of h^* h. The target space K can be a normed real division algebra
like the quaternions or an algebraic number field like a quadratic field. For
parts of the results we can even assume K to be a Banach algebra like an
operator algebra on a Hilbert space. The K-valued function h on G then defines
connection matrices L,g in which the entries are in K. We show that the
Dieudonne determinants of L and g are both equal to the abelianization of the
product of all the field values on G. If G is a simplicial complex and h takes
values in the units U of K, then g^* is the inverse of L and the sum of the
energy values is equal to the sum of the Green function entries g(x,y). If K is
the field C of complex numbers, we can study the spectrum of L(G,h) in
dependence of the field h. The set of matrices with simple spectrum defines a
|G|-dimensional non-compact Kaehler manifold that is disconnected in general
and for which we can compute the fundamental group of each connected component.
