The UV/IR mixing problem makes renormalization difficult when considering quantum field theories on noncommutative spaces. To avoid the UV/IR mixing, Grosse and Steinacker adjusted the action of scalar Phi cubic theories on Moyal spaces by adding harmonic oscillator potentials and showed it renormalizable. A similar model for the Phi 4-th model, which is also renormalizable, is called Grosse-Wulkenhaar model. These models can be described as matrix models. In practice, the partition function of Phi cubic model basically coincides with Kontsevich model. In this seminar, we will show all multipoint correlation functions of the Phi cubic type model in the large N limit can be computed by solving Schwinger-Dyson equations. For the finite N case, we will see that we can also calculate all multipoint functions, exactly. Recently, a hybrid model of Phi cubic and Phi 4th potential is studied as a finite matrix model. This is also related to an integrable system, too. This hybrid model will also be explained.

https://indico.math.cnrs.fr/event/9969/