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Register a new userThe double scaling limit of random tensor models
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Tensor models generalize matrix models and generate colored triangulations of pseudo-manifolds in dimensions $Dgeq 3$. The free energies of some models have been recently shown to admit a double scaling limit, i.e. large tensor size $N$ while tuning to criticality, which turns out to be summable in dimension less than six. This double scaling limit is here extended to arbitrary models. This is done by means of the Schwinger--Dyson equations, which generalize the loop equations of random matrix models, coupled to a double scale analysis of the cumulants.
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We study the double scaling limit of the $O(N)^3$-invariant tensor model, initially introduced in Carrozza and Tanasa, Lett. Math. Phys. (2016). This model has an interacting part containing two types of quartic invariants, the tetrahedric and the pillow one. For the 2-point function, we rewrite the sum over Feynman graphs at each order in the $1/N$ expansion as a emph{finite} sum, where the summand is a function of the generating series of melons and chains (a.k.a. ladders). The graphs which are the most singular in the continuum limit are characterized at each order in the $1/N$ expansion. This leads to a double scaling limit which picks up contributions from all orders in the $1/N$ expansion. In contrast with matrix models, but similarly to previous double scaling limits in tensor models, this double scaling limit is summable. The tools used in order to prove our results are combinatorial, namely a thorough diagrammatic analysis of Feynman graphs, as well as an analysis of the singularities of the relevant generating series.
Massive and massless potentials play an essential role in the perturbative formulation of particle interactions. Many difficulties arise due to the indefinite metric in gauge theoretic approaches, or the increase with the spin of the UV dimension of massive potentials. All these problems can be evaded in one stroke: modify the potentials by suitable terms that leave unchanged the field strengths, but are not polynomial in the momenta. This feature implies a weaker localization property: the potentials are string-localized. In this setting, several old issues can be solved directly in the physical Hilbert space of the respective particles: We can control the separation of helicities in the massless limit of higher spin fields and conversely we recover massive potentials with 2s+1 degrees of freedom by a smooth deformation of the massless potentials (fattening). We construct stress-energy tensors for massless fields of any helicity (thus evading the Weinberg-Witten theorem). We arrive at a simple understanding of the van Dam-Veltman-Zakharov discontinuity concerning, e.g., the distinction between a massless or a very light graviton. Finally, the use of string-localized fields opens new perspectives for interacting quantum field theories with, e.g., vector bosons or gravitons.
Various tensor models have been recently shown to have the same properties as the celebrated Sachdev-Ye-Kitaev (SYK) model. In this paper we study in detail the diagrammatics of two such SYK-like tensor models: the multi-orientable (MO) model which has an $U(N) times O(N) times U(N)$ symmetry and a quartic $O(N)^3$-invariant model whose interaction has the tetrahedral pattern. We show that the Feynman graphs of the MO model can be seen as the Feynman graphs of the $O(N)^3$-invariant model which have an orientable jacket. We then present a diagrammatic toolbox to analyze the $O(N)^3$-invariant graphs. This toolbox allows for a simple strategy to identify all the graphs of a given order in the $1/N$ expansion. We apply it to the next-to-next-to-leading and next-to-next-to-next-to-leading orders which are the graphs of degree $1$ and $3/2$ respectively.
The box-ball systems are integrable cellular automata whose long-time behavior is characterized by the soliton solutions, and have rich connections to other integrable systems such as Korteweg-de Veris equation. In this paper, we consider multicolor box-ball system with two types of random initial configuration and obtain the scaling limit of the soliton lengths as the system size tends to infinity. Our analysis is based on modified Greene-Kleitman invariants for the box-ball systems and associated circular exclusion processes.
We compute the exact density of states and 2-point function of the $mathcal{N} =2$ super-symmetric SYK model in the large $N$ double-scaled limit, by using combinatorial tools that relate the moments of the distribution to sums over oriented chord diagrams. In particular we show how SUSY is realized on the (highly degenerate) Hilbert space of chords. We further calculate analytically the number of ground states of the model in each charge sector at finite $N$, and compare it to the results from the double-scaled limit. Our results reduce to the super-Schwarzian action in the low energy short interaction length limit. They imply that the conformal ansatz of the 2-point function is inconsistent due to the large number of ground states, and we show how to add this contribution. We also discuss the relation of the model to $SL_q(2|1)$. For completeness we present an overview of the $mathcal{N}=1$ super-symmetric SYK model in the large $N$ double-scaled limit.
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