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In this paper we construct the noncommutative Grosse-Wulkenhaar model on2-dimensional Moyal plane with the method of loop vertex expansion. We treatrenormalization with this new tool, adapt Nelson's argument and prove Borelsummability of the perturbation series. This is the first non-commutativequantum field theory model to be built in a non-perturbative sense.

Constructive Renormalization of the 2-dimensional Grosse-Wulkenhaar Model with Multiscale Loop Vertex Expansions