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15-17 Oct 2019 Talence (France)
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Titles and abstracts
I will present some results on spontaneous symmetry breaking in the O(N)3 tensor model with quartic interactions. In particular I will present an SO(3)-invariant solution of the field equations, for which the tensor is proportional to a Wigner 3jm symbol. Time permitting, I will discuss its stability, and its application to the 2PI effective action.
I will discuss the enumeration of real and complex tensor model observables. The formalism is based on permutation groups. In both cases, they can be considered as the base elements of an algebra which proves to be semi-simple and so admits a matrix subalgebra decomposition. I will explain the matrix decomposition for the complex case, whereas the real case is yet to be understood.
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We consider the O(N)^3 tensor model of Klebanov and Tarnopolsky \cite{Klebanov:2016xxf} in d<4 with a free covariance modified to fit the infrared conformal scaling. We study the renormalization group flow of the model using a Wilsonian approach valid in any d (notably we do not require d=4-\epsilon with small \epsilon). At large N, the tetrahedral coupling has a finite flow, hence it becomes a free parameter. The remaining flow can be parametrized by two couplings which do not mix. We show that, at leading order in 1/N but non perturbatively in the couplings, the beta functions stop at quadratic order in the pillow and double-trace couplings. We find four fixed points which depend parametrically on the tetrahedral coupling. For purely imaginary values of the latter we identify a real and \emph{infrared attractive} fixed point. We remark that an imaginary tetrahedral coupling is in fact natural from the onset as the tetrahedral invariant does not have any positivity property, and moreover in the large-N limit beta functions depend on the square of the tetrahedral coupling, thus they remain real, as long as the other couplings stay real.
This talk provides the constructive loop vertex expansion for sta- ble matrix models with (single trace) interactions of arbitrarily high even order in the Hermitian and real symmetric cases. It relies on a new and simpler method which can also be applied in the previously treated complex case. We prove analyticity in the coupling constant of the free energy for such models in a domain uniform in the size N of the matrix.
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Four dimensional random geometries can be generated by statistical models with random variables such as rank-4 tensors. Using a suitable background-independent coarse-graining approach, we discover potential candidates for a universal continuum limit in a real rank-4 tensor model. The possible relevance for the asymptotic safety scenario for quantum gravity as well as dynamical triangulations will be discussed after having introduced the functional renormalization group as a tool for uncovering possibly interesting universality classes.
In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call $\D$th-random feuilletages (denoted by $\RR{\D}$), indexed by a parameter $\D\geq 0$ and which are candidate to play the role of the Brownian map in dimension $D$. The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes. In the planar $D=2$ case, the family of discrete snakes considered coincides with some family of (random) labeled trees known to encode planar quadrangulations. Iterating snakes provides a sequence of random trees $(\bt^{(j)}, j\geq 1)$. The $D$th-random feuilletage $\RR{D}$ is built using $(\bt^{(1)},\cdots,\bt^{(D)})$: $\RR{0}$ is a deterministic circle, $\RR{1}$ is Aldous' continuum random tree, $\RR{2}$ is the Brownian map, and somehow, $\RR{D}$ is obtained by quotienting $\bt^{(D)}$ by $\RR{D-1}$. A discrete counterpart to $\RR{\D}$ is introduced and called the $\D$th random discrete feuilletage with $n+\D$ nodes ($\DR{n}{\D}$). The proof of the convergence of $\DR{n}{\D}$ to $\RR{\D}$ after appropriate rescaling in some functional space is provided (however, the convergence obtained is too weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter of $\DR{n}{\D}$ is $n^{1/2^{\D}}$. Some elements allowing to conjecture that the Hausdorff dimension of $\RR{\D}$ is $2^\D$ are given. This is a joint work with L. Lionni.
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We will present the derivation of Schwinger-Dyson equations in Tensor Field Theory, which are obtained using Ward-Takahashi identities, focusing on the 2-point function. After taking the large N limit, we will find the 2-point function of a particular model in term of Lambert's W-function.
The spiked tensor model describes the estimation of a large rank-one tensor in Gaussian noise. We consider the evolution of the rank-one tensor driven by the gradient of the spiked tensor model Hamiltonian and study the corresponding correlation functions obtained by averaging with respect to the random initial conditions and Gaussian noise. Applying the functional super-symmetric formalism we derive the saddle point equation and show that the large N limit of the model is dominated by the melon diagrams.
We compute the OPE coefficients of the bosonic tensor model of \cite{Benedetti:2019eyl} for three point functions with two fields and a bilinear with zero and non-zero spin. We find that all the OPE coefficients are real in the case of an imaginary tetrahedral coupling constant, while one of them is not real in the case of a real coupling. We also discuss the operator spectrum of the free theory based on the character decomposition of the partition function.
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