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Blobbed topological recursion for the quartic melonic tensor model. (English) Zbl 1400.81159

Summary: Random tensor models are generalizations of random matrix models which admit \(1/N\) expansions. In this article we show that the topological recursion, a modern approach to matrix models which solves the loop equations at all orders, is also satisfied in some tensor models. While it is obvious in some tensor models which are matrix models in disguise, it is far from clear that it can be applied to others. Here we focus on melonic interactions for which the models are best understood, and further restrict to the quartic case. Then Hubbard-Stratonovich transformation maps the tensor model to a multi-matrix model with multi-trace interactions. We study this matrix model and show that after subtracting the leading order, it satisfies the blobbed topological recursion. It is a new extension of the topological recursion, recently introduced by Borot and further studied by Borot and Shadrin. Here it applies straightforwardly, yet with a novelty as our model displays a disconnected spectral curve, which is the union of several spectral curves of the Gaussian unitary ensemble. Finally, we propose a way to evaluate expectations of tensorial observables using the correlation functions computed from the blobbed topological recursion.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
60B20 Random matrices (probabilistic aspects)
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
03D80 Applications of computability and recursion theory
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
81T70 Quantization in field theory; cohomological methods
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