# Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model

Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annales Henri Poincaré Springer Journals

# Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model

, Volume 19 (8) – Jun 2, 2018
56 pages

/lp/springer_journal/constructive-renormalization-of-the-2-dimensional-grosse-wulkenhaar-GCP3PyoD7o
Publisher
Springer Journals
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Elementary Particles, Quantum Field Theory
ISSN
1424-0637
eISSN
1424-0661
D.O.I.
10.1007/s00023-018-0688-0
Publisher site
See Article on Publisher Site

### Abstract

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.

### Journal

Annales Henri PoincaréSpringer Journals

Published: Jun 2, 2018

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