R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

This article reviews the present status of the spin-foam approach to the quantization of gravity. Special attention is payed to the pedagogical presentation of the recently-introduced new models for four-dimensional quantum gravity. The models are motivated by a suitable implementation of the path integral quantization of the Plebanski formulation of gravity on a simplicial regularization. The article also includes a self-contained treatment of 2+1 gravity. The simple nature of the latter provides the basis and a perspective for the analysis of both conceptual and technical issues that remain open in four dimensions.

/pdf/the-spin-foam-approach-to-quantum-gravity-4bctgipiqk.pdf

The Spin-Foam Approach to Quantum Gravity

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.

/pdf/colored-tensor-models-a-review-3sdqhl51nj.pdf

Colored Tensor Models - a Review

Certain models with rank-3 tensor degrees of freedom have been shown by Gurau and collaborators to possess a novel large $N$ limit, where ${g}^{2}{N}^{3}$ is held fixed. In this limit the perturbative expansion in the quartic coupling constant, $g$, is dominated by a special class of ``melon'' diagrams. We study ``uncolored'' models of this type, which contain a single copy of real rank-3 tensor. Its three indices are distinguishable; therefore, the models possess $O(N{)}^{3}$ symmetry with the tensor field transforming in the tri-fundamental representation. Such uncolored models also possess the large $N$ limit dominated by the melon diagrams. The quantum mechanics of a real anticommuting tensor therefore has a similar large $N$ limit to the model recently introduced by Witten as an implementation of the Sachdev-Ye-Kitaev (SYK) model which does not require disorder. Gauging the $O(N{)}^{3}$ symmetry in our quantum mechanical model removes the nonsinglet states; therefore, one can search for its well-defined gravity dual. We point out, however, that the model possesses a vast number of gauge-invariant operators involving higher powers of the tensor field, suggesting that the complete gravity dual will be intricate. We also discuss the quantum mechanics of a complex 3-index anticommuting tensor, which has $U(N{)}^{2}\ifmmode\times\else\texttimes\fi{}O(N)$ symmetry and argue that it is equivalent in the large $N$ limit to a version of SYK model with complex fermions. Finally, we discuss similar models of a commuting tensor in dimension $d$. While the quartic interaction is not positive definite, we construct the large $N$ Schwinger-Dyson equation for the two-point function and show that its solution is consistent with conformal invariance. We carry out a perturbative check of this result using the $4\ensuremath{-}\ensuremath{\epsilon}$ expansion.

Uncolored random tensors, melon diagrams, and the Sachdev-Ye-Kitaev models

Critical behavior of colored tensor models in the large N limit

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. nonsymmetric, complex tensor. Colors appear in this setting as a canonical bookkeeping 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 ${\ensuremath{\gamma}}_{m}=1\ensuremath{-}1/m$.

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

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.

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

Spin foam models for quantum gravity are derived from lattice path integrals. The setting involves variables from both lattice BF theory and Regge calculus. The action consists in a Regge action, which depends on areas, dihedral angles and includes the Immirzi parameter. In addition, a measure is inserted to ensure a consistent gluing of simplices, so that the amplitude is dominated by configurations that satisfy the parallel transport relations. We explicitly compute the path integral as a sum over spin foams for a generic measure. The Freidel-Krasnov and Engle-Pereira-Rovelli models correspond to a special choice of gluing. In this case, the equations of motion describe genuine geometries, where the constraints of area-angle Regge calculus are satisfied. Furthermore, the Immirzi parameter drops out of the on-shell action, and stationarity with respect to area variations requires spacetime geometry to be flat.

Spin foam models for quantum gravity from lattice path integrals

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.

Valentin Bonzom

Papers

Critical behavior of colored tensor models in the large N limit

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

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

Spin foam models for quantum gravity from lattice path integrals

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