  In this paper we construct a non-commutative geometry over a configuration
space of gauge connections and show that it gives rise to a candidate for an
interacting, non-perturbative quantum gauge theory coupled to a fermionic field
on a curved background. The non-commutative geometry is given by an
infinite-dimensional Bott-Dirac type operator, whose square gives the Hamilton
operator, and which interacts with an algebra generated by
holonomy-diffeomorphisms. The Bott-Dirac operator and the associated Hilbert
space relies on a metric on the configuration space of connections, which
effectively works as a covariant ultra-violet regulator. We show that the
construction coincides with perturbative quantum field theory in a local limit.
Questions concerning Lorentz invariance and the fermionic sector as well as the
issue of existence are left open.
