We study a quartic matrix model with partition function $$Z=\int d\ M\exp \mathrm{Tr}\ (-\Delta M^2-\frac{\lambda }{4}M^4)$$ Z = ∫ d M exp Tr ( - Δ M 2 - λ 4 M 4 ) . The integral is over the space of Hermitian $$(\varLambda +1)\times (\varLambda +1)$$ ( Λ + 1 ) × ( Λ + 1 ) matrices, the matrix $$\Delta $$ Δ , which is not a multiple of the identity matrix, encodes the dynamics and $$\lambda >0$$ 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 $$\lambda $$ 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 $$\phi ^4$$ ϕ 4 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.
Annales Henri Poincaré – Springer Journals
Published: Jun 2, 2018
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