  Quantum gauge theory in the connection representation uses functions of
holonomies as configuration observables. Physical observables (gauge and
diffeomorphism invariant) are represented in the Hilbert space of physical
states; physical states are gauge and diffeomorphism invariant distributions on
the space of functions of the holonomies of the edges of a certain family of
graphs. Then a family of graphs embedded in the space manifold (satisfying
certain properties) induces a representation of the algebra of physical
observables. We construct a quantum model from the set of piecewise linear
graphs on a piecewise linear manifold, and another manifestly combinatorial
model from graphs defined on a sequence of increasingly refined simplicial
complexes. Even though the two models are different at the kinematical level,
they provide unitarily equivalent representations of the algebra of physical
observables in separable Hilbert spaces of physical states (their s-knot basis
is countable). Hence, the combinatorial framework is compatible with the usual
interpretation of quantum field theory.
