The 1/N Expansion of Multi-Orientable Random Tensor Models

Abstract

Multi-orientable group field theory (GFT) has been introduced in [1], as a quantum field theoretical simplification of GFT, which retains a larger class of tensor graphs than the colored one. In this paper we define the associated multi-orientable identically independent distributed multi-orientable tensor model and we derive its 1/N expansion. In order to obtain this result, a partial classification of general tensor graphs is performed and the combinatorial notion of jacket is extended to the m.o. graphs. We prove that the leading sector is given, as in the case of colored models, by the so-called melon graphs.The interface between combinatorics and theoretical physics is rapidly growing. This interface has multiple aspects, such as the interplay of combinatorics (algebraic, analytic and so on) with quantum field theory (QFT) [2] (a typical example being the elegant Connes-Kreimer algebraic reformulation of the combinatorics of renormalization [3]), or the interplay of combinatorics (enumerative, bijective and so on) with statistical physics and integrable systems [4]. This paper is situated at the interface between combinatorics and random tensor models, which naturally generalize random matrix models. These models are interesting candidates for a fundamental theory of quantum gravity [5], in relation to many other quantum gravity approaches, such as dynamical triangulations [6], loop quantum gravity [7], non-commutative geometry [8] or even string theory [9].Nevertheless, the combinatorics and the topology of these tensor models is rather involved. Progress came from a simplified colored version of these models [10], first proposed within the framework of group field theory (GFT) [11] and then as an even more general theory of random tensors [12,13,14]. Colored models discard a significant class of the initial graphs considered in earlier tensor models but retain triangulations of all piecewise linear manifolds in any dimension [15]. They admit a 1/N expansion when the size N of the 1

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