Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Tensor Models, Formalism and Applications

The Guest Editor for this special issue is

Razvan Gurau (Center of Theoretical Physics, Ecole Polytechnique, France)

Matrix models are a very successful framework for the study of random surfaces, instrumental in string theory, quantum gravity in dimension 2, conformal field theory, integrability, etc. However,the generalization of matrix models to higher dimensions has been a thorny issue for a long time. The first generalizations of matrix models, known as tensor models and group field theories, date back to 1990. However, for a long time progress has been slow, mainly due to the lack of an equivalent of the famous $1/N$ expansion of matrix models in higher dimensions. This problem has been solved in 2010 when the $1/N$ expansion for tensor models has been found. Since then, the field underwent a tremendous transformation and numerous results on tensor models, tensor field theories and group field theories have been established.

Today, tensor models provide a well defined and analytically controlled framework for the study of random geometries in higher dimension with possible applications to statistical physics in random geometry, conformal field theory and quantum gravity. This issue will bring together a collection of up to date reviews on the various aspects of the modern theory of tensor models. It's purpose is to serve as a reference for both mathematical and theoretical physicists, covering an in depth presentation of the mathematical formalism and its applications.

The issue contains 9 papers with the total of 264 pages.

We would like to thank all the authors who have published papers in the issue, and to give our special thanks to all the referees for providing constructive reviews.

Razvan Gurau        

Papers in this Issue:

Invitation to Random Tensors
Razvan Gurau
SIGMA 12 (2016), 094, 12 pages   [ abs   pdf ]
Constructive Tensor Field Theory
Vincent Rivasseau
SIGMA 12 (2016), 085, 31 pages   [ abs   pdf ]
Quantum Cosmology from Group Field Theory Condensates: a Review
Steffen Gielen and Lorenzo Sindoni
SIGMA 12 (2016), 082, 49 pages   [ abs   pdf ]
$(D+1)$-Colored Graphs - a Review of Sundry Properties
James P. Ryan
SIGMA 12 (2016), 076, 27 pages   [ abs   pdf ]
Large $N$ Limits in Tensor Models: Towards More Universality Classes of Colored Triangulations in Dimension $d\geq 2$
Valentin Bonzom
SIGMA 12 (2016), 073, 39 pages   [ abs   pdf ]
Flowing in Group Field Theory Space: a Review
Sylvain Carrozza
SIGMA 12 (2016), 070, 30 pages   [ abs   pdf ]
Random Tensors and Quantum Gravity
Vincent Rivasseau
SIGMA 12 (2016), 069, 17 pages   [ abs   pdf ]
Exact Renormalisation Group Equations and Loop Equations for Tensor Models
Thomas Krajewski and Reiko Toriumi
SIGMA 12 (2016), 068, 36 pages   [ abs   pdf ]
The Multi-Orientable Random Tensor Model, a Review
Adrian Tanasa
SIGMA 12 (2016), 056, 23 pages   [ abs   pdf ]

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