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Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model

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Abstract

We study a quartic matrix model with partition function \(Z=\int d\ M\exp \mathrm{Tr}\ (-\Delta M^2-\frac{\lambda }{4}M^4)\). The integral is over the space of Hermitian \((\varLambda +1)\times (\varLambda +1)\) 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 and 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.

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Authors and Affiliations

  1. Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Xidazhi Street 92, Harbin, 150006, China

    Zhituo Wang

  2. Research Center for Operator Algebras, East China Normal University, 3663 ZhongShan North Road, Shanghai, 200062, China

    Zhituo Wang

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  1. Zhituo Wang
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Correspondence to Zhituo Wang.

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Communicated by Raimar Wulkenhaar.

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Wang, Z. Constructive Renormalization of the 2-Dimensional Grosse–Wulkenhaar Model. Ann. Henri Poincaré 19, 2435–2490 (2018). https://doi.org/10.1007/s00023-018-0688-0

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