We point out that the noncommutative selfdual $phi^3$ model can be mapped to the Kontsevich model, for a suitable choice of the eigenvalues in the latter. This allows to apply known results for the Kontsevich model to the quantization of the field theory, in particular the KdV flows and Virasoro constraints. The 2- and 4-dimensional cases are worked out explicitly. We obtain nonperturbative expressions for the genus expansion of the free energy and some n-point functions. The full renormalization for finite coupling is found, and all contributions in a genus expansion of any n-point function are finite after renormalization. A critical coupling is determined beyond which the model is unstable. This provides a nontrivial interacting NC field theory in 4 dimensions.

University of Vienna