Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model

We study a quartic matrix model with partition function Z=∫dMexpTr(-ΔM2-λ4M4). The integral is over the space of Hermitian (Λ+ 1) × (Λ+ 1) matrices, the matrix Δ , which is not a multiple of the identity matrix, encodes the dynamics and λ> 0 is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series and hence is a well-defined analytic function of the coupling constant in certain analytic domain of λ, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual ϕ⁴ theory on the 2-dimensional Moyal space also called the 2-dimensional Grosse–Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.