Valentin Bonzom

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.

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.

Radiative Corrections in the Boulatov-Ooguri Tensor Model: The 2-Point Function

The Boulatov-Ooguri tensor model generates a sum over spacetime topologies for the D-dimensional BF theory. We study here the quantum corrections to the propagator of the theory. In particular, we find that the radiative corrections at the second order in the coupling constant yield a mass renormalization. They also exhibit a divergence which cannot be balanced with a counter-term in the initial action, and which usually corresponds to the wave-function renormalization.

Bubble Divergences from Cellular Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang–Mills theory, the Ponzano–Regge model of 3d quantum gravity and discrete BF theory, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined in general, because of a phenomenon called ‘bubble divergences’. A common expectation is that the degree of these divergences is given by the number of ‘bubbles’ of the 2-complex. In this note, we show that this expectation, although not realistic in general, is met in some special cases: when the 2-complex is simply connected, or when the structure group is Abelian – in both cases, the divergence degree is given by the second Betti number of the 2-complex.

Bubble Divergences: Sorting out Topology from Cell Structure

We conclude our analysis of bubble divergences in the flat spinfoam model. In Bonzom and Smerlak, Comm. Math. Phys., (submitted), we showed that the divergence degree of an arbitrary 2-complex Γ can be evaluated exactly by means of twisted cohomology. Here, we specialize this result to the case where Γ is the 2-skeleton of the cell decomposition of a pseudomanifold, and sharpen it with a careful analysis of the cellular and topological structures involved. Moreover, we explain in detail how this approach reproduces all the previous powercounting results for the Boulatov–Ooguri (colored) tensor models, and sheds light on algebraic-topological aspects of Gurau’s 1/N expansion.

Bubble Divergences from Twisted Cohomology

We consider a class of lattice topological field theories, among which are the weak-coupling limit of 2d Yang-Mills theory and 3d Riemannian quantum gravity, whose dynamical variables are flat discrete connections with compact structure group on a cell 2-complex. In these models, it is known that the path integral measure is ill-defined because of a phenomenon known as ‘bubble divergences’. In this paper, we extend recent results of the authors to the cases where these divergences cannot be understood in terms of cellular cohomology. We introduce in its place the relevant twisted cohomology, and use it to compute the divergence degree of the partition function. We also relate its dominant part to the Reidemeister torsion of the complex, thereby generalizing previous results of Barrett and Naish-Guzman. The main limitation to our approach is the presence of singularities in the representation variety of the fundamental group of the complex; we illustrate this issue in the well-known case of two-dimensional manifolds.

Diagrammatics of a colored SYK model and of an SYK-like tensor model, leading and next-to-leading orders

The Sachdev-Ye-Kitaev (SYK) model is a model of q interacting fermions. Gross and Rosenhaus have proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, those flavors are reminiscent of the colors used in random tensor theory. This gives us the opportunity to apply some modern, yet elementary, tools developed in the context of random tensors to one particular instance of such colored SYK models. We illustrate our method by identifying all diagrams which contribute to the leading and next-to-leading orders of the 2-point and 4-point functions in the large N expansion and argue that our method can be further applied if necessary. In the second part, we focus on the recently introduced Gurau-Witten tensor model and also extract the leading and next-to-leading orders of the 2-point and 4-point functions. This analysis turns out to be remarkably more involved than in the colored SYK model.

The Ising model on random lattices in arbitrary dimensions

Abstract We study analytically the Ising model coupled to random lattices in dimension three and higher. The family of random lattices we use is generated by the large N limit of a colored tensor model generalizing the two-matrix model for Ising spins on random surfaces. We show that, in the continuum limit, the spin system does not exhibit a phase transition at finite temperature, in agreement with numerical investigations. Furthermore we outline a general method to study critical behavior in colored tensor models.

The Hamiltonian constraint in 3d Riemannian loop quantum gravity

We discretize the Hamiltonian scalar constraint of three-dimensional Riemannian gravity on a graph of the loop quantum gravity phase space. This Hamiltonian has a clear interpretation in terms of discrete geometries: it computes the extrinsic curvature from dihedral angles. The Wheeler-DeWitt equation takes the form of difference equations, which are actually recursion relations satisfied by Wigner symbols. On the boundary of a tetrahedron, the Hamiltonian generates the exact recursion relation on the 6j-symbol which comes from the Biedenharn-Elliott (pentagon) identity. This fills the gap between the canonical quantization and the symmetries of the Ponzano-Regge state-sum model for 3d gravity.

Revisiting random tensor models at large N via the Schwinger-Dyson equations

A bstractThe Schwinger-Dyson Equations (SDEs) of matrix models are known to form (half) a Virasoro algebra and have become a standard tool to solve matrix models. The algebra generated by SDEs in tensor models (for random tensors in a suitable ensemble) is a specific generalization of the Virasoro algebra and it is important to show that these new symmetries determine the physical solutions. We prove this result for random tensors at large N. Compared to matrix models, tensor models have more than a single invariant at each order in the tensor entries and the SDEs make them proliferate. However, the specific combinatorics of the dominant observables allows to restrict to linear SDEs and we show that they determine a unique physical perturbative solution. This gives a new proof that tensor models are Gaussian at large N, with the covariance being the full 2-point function.