Colored tensor models – a review.

*(English)*Zbl 1242.05094Summary: Colored tensor models have recently burst onto the scene as a promising conceptual and computational tool in the investigation of problems of random geometry in dimension three and higher. We present a snapshot of the cutting edge in this rapidly expanding research field. Colored tensor models have been shown to share many of the properties of their direct ancestor, matrix models, which encode a theory of fluctuating two-dimensional surfaces. These features include the possession of Feynman graphs encoding topological spaces, a \(1/N\) expansion of graph amplitudes, embedded matrix models inside the tensor structure, a resumable leading order with critical behavior and a continuum large volume limit, Schwinger-Dyson equations satisfying a Lie algebra (akin to the Virasoro algebra in two dimensions), non-trivial classical solutions and so on. In this review, we give a detailed introduction of colored tensor models and pointers to current and future research directions.

##### MSC:

05C15 | Coloring of graphs and hypergraphs |

05C75 | Structural characterization of families of graphs |

81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |

81T17 | Renormalization group methods applied to problems in quantum field theory |

81T18 | Feynman diagrams |

83C27 | Lattice gravity, Regge calculus and other discrete methods in general relativity and gravitational theory |

83C45 | Quantization of the gravitational field |