Constructive Renormalization of the $2$-dimensional Grosse-Wulkenhaar Model

TL;DR

This paper rigorously analyzes a 2D Grosse-Wulkenhaar matrix model, proving the partition function's logarithm is Borel summable and establishing its analytic properties, thus advancing the understanding of non-trivial quantum field theories.

Contribution

It provides the first full construction of a non-trivial, non-solvable matrix model derived from the 2D Grosse-Wulkenhaar theory, using multi-scale loop vertex expansions.

Findings

Partition function's logarithm is Borel summable.

The model's perturbation series is analytically well-defined.

Non-planar graphs are treated on equal footing with planar ones.

Abstract

We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm Tr}\ (-\Delta M^2-\frac{\lambda}{4}M^4)$. The integral is over the space of Hermitian $(\Lambda+1)\times(\Lambda+1)$ matrices, the matrix $\Delta$, which is not a multiple of the identity matrix, encodes the dynamics and $\lambda>0$ is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series, hence is a well defined analytic function of the coupling constant in certain analytic domain of $\lambda$, 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 $\phi^4$ 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.