The Tensor Track, III

We provide an informal up-to-date review of the tensor track approach to quantum gravity. In a long introduction we describe in simple terms the motivations for this approach. Then the many recent advances are summarized, with emphasis on some points (Gromov-Hausdorff limit, Loop vertex expansion, Osterwalder-Schrader positivity...) which, while important for the tensor track program, are not detailed in the usual quantum gravity literature. We list open questions in the conclusion and provide a rather extended bibliography.Comment: 53 pages, 6 figure

The Tensor Track: an Update

The tensor track approach to quantum gravity is based on a new class of quantum field theories, called tensor group field theories (TGFTs). We provide a brief review of recent progress and list some desirable properties of TGFTs. In order to narrow the search for interesting models, we also propose a set of guidelines for TGFT's loosely inspired by the Osterwalder-Schrader axioms of ordinary Euclidean QFT.Comment: 12 pages, 1 figure. This paper is based on a talk given at the XXIX International Colloquium on Group-Theoretical Methods in Physics in Tian-Jin (China), very minor change

Spheres are rare

We prove that triangulations of homology spheres in any dimension grow much slower than general triangulations. Our bound states in particular that the number of triangulations of homology spheres in 3 dimensions grows at most like the power 1/3 of the number of general triangulations.Comment: 14 pages, 1 figur

Random Tensors and Quantum Gravity

We provide an informal introduction to tensor field theories and to their associated renormalization group. We focus more on the general motivations coming from quantum gravity than on the technical details. In particular we discuss how asymptotic freedom of such tensor field theories gives a concrete example of a natural "quantum relativity" postulate: physics in the deep ultraviolet regime becomes asymptotically more and more independent of any particular choice of Hilbert basis in the space of states of the universe.Comment: Section 6 is essentially reproduced from author's arXiv:1507.04190 for self-contained purpose of the revie

The Tensor Theory Space

The tensor track is a background-independent discretization of quantum gravity which includes a sum over all topologies. We discuss how to define a functional renormalization group flow and the Wetterich equation in the corresponding theory space. This space is different from the Einsteinian theory space of asymptotic safety. It includes all fixed-rank tensor-invariant interactions, hence generalizes matrix models and the (Moyal) non-commutative field theory space.Comment: This short note is intended as a complement to arXiv:1311.1461, to appear in the Proceedings of the Workshop on Noncommutative Field Theory and Gravity in Corfu September 2013, Fortshritt. Phys. 201

Loop Vertex Expansion for Higher Order Interactions

This note provides an extension of the constructive loop vertex expansion to stable interactions of arbitrarily high order, opening the way to many applications. We treat in detail the example of the $(\bar \phi \phi)^p$ field theory in zero dimension. We find that the important feature to extend the loop vertex expansion is not to use an intermediate field representation, but rather to force integration of exactly one particular field per vertex of the initial action.Comment: 16 pages, 2 figures, V2, minor modification

The two dimensional Hubbard Model at half-filling: I. Convergent Contributions

We prove analyticity theorems in the coupling constant for the Hubbard model at half-filling. The model in a single renormalization group slice of index $i$ is proved to be analytic in $\lambda$ for $|\lambda| \le c/i$ for some constant $c$, and the skeleton part of the model at temperature $T$ (the sum of all graphs without two point insertions) is proved to be analytic in $\lambda$ for $|\lambda| \le c/|\log T|^{2}$. These theorems are necessary steps towards proving that the Hubbard model at half-filling is {\it not} a Fermi liquid (in the mathematically precise sense of Salmhofer).Comment: 31 pages, 2 figure

How to Resum Feynman Graphs

In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing but the percentage of Hepp's sectors in which the tree is leading the ultraviolet analysis. We explain how they allow to reshuffle the divergent series formulated in terms of Feynman graphs into convergent series indexed by the trees that these graphs contain. The Feynman graphs to be used are not the ordinary ones but those of the intermediate field representation, and the result of the reshuffling is called the Loop Vertex Expansion.Comment: 18 pages, 6 figures; minor revisions; improves and replaces arXiv:1006.461