Tensor models and embedded Riemann surfaces

Abstract

Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/Nexpansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces.Group field theories [1] and tensor models are higher dimensional analogues of matrix models [2]. Matrix integrals have been shown to provide a natural framework within which to frame a multitude of physical and mathematical questions ranging through the fields of statistical and condensed matter physics all the way to the more abstract enumeration of virtual knots and tangles. This highlights how apparently disparate physical and mathematical phenomena in fact share certain universal features.One particular facet that sparked considerable interest was the realization that matrix models could give a non-perturbative definition of 2d quantum gravity [3]. One considers a statistical ensemble of N ×N (oftentimes hermitian) matrices. The Feynman graphs arising in the perturbative expansion of the free energy describe discrete Riemann surfaces. Remarkably, the expansion can be ordered in powers of 1/N labelled by their topological invariant, the Euler characteristic. In the large-N limit, the 2-sphere dominates and moreover, one can tune the coupling constant so that in a double scaling limit one describes a continuum theory of 2d quantum gravity.Tensor models hope to reproduce the same successes that matrix models have enjoyed, with the ultimate aim of being viable candidates for quantum theories of gravity in higher dimensions. A stumbling block seemed to be that they generated a plethora of unwanted structures; not only simplicial manifolds, but simplicial pseudomanifolds [4]. Recently, after much work [5,11], a promising step has been made in that direction with the construction of a 1/N -topological expansion [6][7][8] for the so-called coloured tensor models [9,10]. In that context, it was shown that for arbitrary dimension d, only graphs corresponding to d-spheres arise at leading order in the 1/N -expansion. Central to this construction were the ribbon graphs associated to the Feynman graphs of the tensor model. These ribbon graphs are algebraic objects that capture the topological properties of the Feynman graphs. They contain two classes of subgraphs, known as bubbles [9] and jackets [11], which are of particular significance. While bubbles are easily identified as Riemann surfaces embedded in the dual triangulation, the topological properties of the jac...

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