  We present quantum holonomy theory, which is a non-perturbative theory of
quantum gravity coupled to fermionic degrees of freedom. The theory is based on
a C*-algebra that involves holonomy-diffeomorphisms on a 3-dimensional manifold
and which encodes the canonical commutation relations of canonical quantum
gravity formulated in terms of Ashtekar variables. Employing a Dirac type
operator on the configuration space of Ashtekar connections we obtain a
semi-classical state and a kinematical Hilbert space via its GNS construction.
We use the Dirac type operator, which provides a metric structure over the
space of Ashtekar connections, to define a scalar curvature operator, from
which we obtain a candidate for a Hamilton operator. We show that the classical
Hamilton constraint of general relativity emerges from this in a semi-classical
limit and we then compute the operator constraint algebra. Also, we find states
in the kinematical Hilbert space on which the expectation value of the Dirac
type operator gives the Dirac Hamiltonian in a semi-classical limit and thus
provides a connection to fermionic quantum field theory. Finally, an
almost-commutative algebra emerges from the holonomy-diffeomorphism algebra in
the same limit.
