Abstract
A non-commutative Feynman graph is a ribbon graph and
can be drawn on a genus g 2-surface with a boundary. We
formulate a general convergence theorem for the non-commutative
Feynman graphs in topological terms and prove it for some classes
of diagrams in the scalar field theories. We propose a
non-commutative analog of Bogoliubov-Parasiuk's recursive
subtraction formula and show that the subtracted graphs from a
class Ωd satisfy the conditions of the convergence
theorem. For a generic scalar non-commutative quantum field theory
on 
d, the class Ωd is smaller than the class
of all diagrams in the theory. This leaves open the question of
perturbative renormalizability of non-commutative field theories.
We comment on how the supersymmetry can improve the situation and
suggest that a non-commutative analog of Wess-Zumino
model is renormalizable.