Razvan Gurau

Colored Group Field Theory

Random matrix models generalize to Group Field Theories (GFT) whose Feynman graphs are dual to higher dimensional topological spaces. The perturbative development of the usual GFT’s is rather involved combinatorially and plagued by topological singularities (which we discuss in great detail in this paper), thus very difficult to control and unsatisfactory.Both these problems simplify greatly for the “colored” GFT (CGFT) model we introduce in this paper. Not only this model is combinatorially simpler but also it is free from the worst topological singularities. We establish that the Feynman graphs of our model are combinatorial cellular complexes dual to manifolds or pseudomanifolds, and study their cellular homology. We also relate the amplitude of CGFT graphs to their fundamental group.

COLORED TENSOR MODELS - A REVIEW

Colored tensor models have recently burst onto the scene as a promising con- ceptual 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 al- gebra 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.

CRITICAL BEHAVIOR OF COLORED TENSOR MODELS IN THE LARGE N LIMIT

Colored tensor models have been recently shown to admit a large N expansion, whose leading order encodes a sum over a class of colored triangulations of the D-sphere. The present paper investigates in details this leading order. We show that the relevant triangula- tions proliferate like a species of colored trees. The leading order is therefore summable and exhibits a critical behavior, independent of the dimension. A continuum limit is reached by tuning the coupling constant to its critical value while inserting an infinite number of pairs of D-simplices glued together in a specific way. We argue that the dominant triangulations are branched polymers. Tensor models (1-3) and group field theories (4-7) are the natural generalization of matrix models (8, 9) implementing in a consistent way the sum over random triangulations in dimensions higher than two. They are notoriously hard to control analytically and one usually resorts to numerical simulations (10-12). Progress has recently been made in the analytic control of tensor models with the advent of the 1/N expansion (13-15) of colored (16-18) tensor models. This expansion synthetizes several alternative evaluations of graph amplitudes in tensor models (19-27) and provides a straightforward generalization of the familiar genus expansion of matrix models (28, 29) in arbitrary dimension. The coloring of the fields allows one to address previously inaccessible questions in tensor models like the implementation of the diffeomorphism symmetry (27, 30) in the Boulatov model or the identification of embedded matrix models (31). The symmetries of tensor models have recently been studied using n-ary algebras (32, 33). This paper is the first in a long series of studies of the implications of the 1/N expansion in colored tensor models. We present here a complete analysis of the leading order in the large N limit in arbitrary dimensions, indexed by graphs of spherical topology (14). To perform the study of this leading order one needs to address the following two questions • What is the combinatorics of the Feynman graphs contributing to the leading order, i.e. the higher dimensional extension of the notion of planar graphs? Unlike in matrix models, where planarity and spherical topology are trivially related, this question is non trivial in tensor models. In particular not all triangulations of the sphere contribute to the leading order. • Is the series of the leading order summable with a non zero radius of convergence? If this is the case, then, in the large N limit, the model exhibits a critical behavior whose critical exponents one needs to compute.

The Complete 1/N Expansion of Colored Tensor Models in Arbitrary Dimension

In this paper we generalize the results of Gurau (arXiv:1011. 2726 [gr-qc], 2011), Gurau and Rivasseau (arXiv:1101.4182 [gr-qc], 2011) and derive the full 1/N expansion of colored tensor models in arbitrary dimensions. We detail the expansion for the independent identically distributed model and the topological Boulatov Ooguri model.

The 1/N Expansion of Colored Tensor Models

In this paper, we perform the 1/N expansion of the colored three-dimensional Boulatov tensor model. As in matrix models, we obtain a systematic topological expansion, with increasingly complicated topologies suppressed by higher and higher powers of N. We compute the first orders of the expansion and prove that only graphs corresponding to three spheres S3 contribute to the leading order in the large N limit.

The 1/N expansion of colored tensor models in arbitrary dimension

In this paper we extend the 1/N expansion introduced in Gurau R., Ann. Henri Poincare, 12 (2011) 829, to group field theories in arbitrary dimension and prove that only graphs corresponding to spheres SD contribute to the leading order in the large-N limit.

Random tensor models in the large N limit: Uncoloring the colored tensor models

Tensor models generalize random matrix models in yielding a theory of dynamical triangulations in arbitrary dimensions. Colored tensor models have been shown to admit a 1/N expansion and a continuum limit accessible analytically. In this paper we prove that these results extend to the most general tensor model for a single generic, i.e. non-symmetric, complex tensor. Colors appear in this setting as a canonical book-keeping device and not as a fundamental feature. In the large N limit, we exhibit a set of Virasoro constraints satisfied by the free energy and an infinite family of multicritical behaviors with entropy exponents m = 1 1/m.

Universality for random tensors

We prove two universality results for random tensors of arbitrary rank D. We first prove that a random tensor whose entries are N^D independent, identically distributed, complex random variables converges in distribution in the large N limit to the same limit as the distributional limit of a Gaussian tensor model. This generalizes the universality of random matrices to random tensors. We then prove a second, stronger, universality result. Under the weaker assumption that the joint probability distribution of tensor entries is invariant, assuming that the cumulants of this invariant distribution are uniformly bounded, we prove that in the large N limit the tensor again converges in distribution to the distributional limit of a Gaussian tensor model. We emphasize that the covariance of the large N Gaussian is not universal, but depends strongly on the details of the joint distribution.

Group field theory renormalization in the 3D case: Power counting of divergences

We take the first steps in a systematic study of group field theory (GFT) renormalization, focusing on the Boulatov model for 3D quantum gravity. We define an algorithm for constructing the 2D triangulations that characterize the boundary of the 3D bubbles, where divergences are located, of an arbitrary 3D GFT Feynman diagram. We then identify a special class of graphs for which a complete contraction procedure is possible, and prove, for these, a complete power counting. These results represent important progress towards understanding the origin of the continuum and manifoldlike appearance of quantum spacetime at low energies, and of its topology, in a GFT framework.

A generalization of the Virasoro algebra to arbitrary dimensions

Abstract Colored tensor models generalize matrix models in higher dimensions. They admit a 1 / N expansion dominated by spherical topologies and exhibit a critical behavior strongly reminiscent of matrix models. In this paper we generalize the colored tensor models to colored models with generic interaction, derive the Schwinger Dyson equations in the large N limit and analyze the associated algebra of constraints satisfied at leading order by the partition function. We show that the constraints form a Lie algebra (indexed by trees) yielding a generalization of the Virasoro algebra in arbitrary dimensions.