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

Abstract

We study a quartic matrix model with partition function $Z=\int dM\exp \mathrm{Tr}(-\Delta M^2-\frac{\lambda }{4}{M}^4)$. 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 ${\varphi }^4$ 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.

arXiv comment53 pages, essentially overlapping with arXiv:1205.0196; Accepted for publication by Annales Henri poincare

References in corpus (26)

• String Theory and Noncommutative Geometry
• Noncommutative Geometry and Matrix Theory: Compactification on Tori
• Noncommutative Perturbative Dynamics
• D-branes and Deformation Quantization
• Renormalisation of ϕ^4-theory on noncommutative R^4 in the matrix base
• Algebras of distributions suitable for phase-space quantum mechanics. I
• Duality in Scalar Field Theory on Noncommutative Phase Spaces
• Renormalisation of ϕ^4-theory on noncommutative R^2 in the matrix base
• Vanishing of Beta Function of Non Commutative ${\Phi }_4^4$ Theory to all orders
• Moyal Planes are Spectral Triples
• Exact Solution of Quantum Field Theory on Noncommutative Phase Spaces
• Power-counting theorem for non-local matrix models and renormalisation
• The β-function in duality-covariant noncommutative ϕ^4-theory
• Renormalization of noncommutative phi 4-theory by multi-scale analysis
• Scaling behaviour of three-dimensional group field theory
• The 1/N Expansion of Tensor Models Beyond Perturbation Theory
• How to Resum Feynman Graphs
• Constructive ${\varphi }^4$ field theory without tears
• Loop Vertex Expansion for Phi^2k Theory in Zero Dimension
• Two and Three Loops Beta Function of Non Commutative ${\Phi }_4^4$ Theory
• Constructive Matrix Theory
• Self-dual noncommutative ϕ^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
• Renormalization of the noncommutative phi^3 model through the Kontsevich model
• Renormalization of the Orientable Non-commutative Gross-Neveu Model
• Quantum field theory on the degenerate Moyal space
• Renormalization of Orientable Non-Commutative Complex ${\Phi }_3^6$ Model

Cited by in corpus (3)

• Detecting scaling in phase transitions on the truncated Heisenberg algebra
• Renormalization footprints in the phase diagram of the Grosse-Wulkenhaar model
• Constructive expansion for quartic vector fields theories. I. Low dimensions