Bulk from Bi-locals in Thermo Field CFT

We study the Large $N$ dynamics of the $O(N)$ field theory in the Thermo field dynamics approach. The question of recovering the high temperature phase and the corresponding $O(N)$ gauging is clarified. Through the associated bi-local representation we discuss the emergent bulk space-time and construction of (Higher spin) fields. We note the presence of `evanescent' modes in this construction and also the mixing of spins at finite temperature.Comment: 24 page. v2: references added, minor corrections. v3: typo corrected, clarification added, conclusion revised, version to appear in JHE

SYK-like Tensor Models on the Lattice

We study large $N$ tensor models on the lattice without disorder. We introduce techniques which can be applied to a wide class of models, and illustrate it by studying some specific rank-3 tensor models. In particular, we study Klebanov-Tarnopolsky model on lattice, Gurau-Witten model (by treating it as a tensor model on four sites) and also a new model which interpolates between these two models. In each model, we evaluate various four point functions at large $N$ and strong coupling, and discuss their spectrum and long time behaviors. We find similarities as well as differences from SYK model. We also generalize our analysis to rank-$D$ tensor models where we obtain analogous results as $D=3$ case for the four point functions which we computed. For $D>5$, we are able to compute the next-to-subleading ${1 \over N}$ corrections for a specific four point function.Comment: 46 pages, 29 figures; v2:typos corrected, reference added; v3:minor revisions, to be published in JHE

SYK Models and SYK-like Tensor Models with Global Symmetry

In this paper, we study an SYK model and an SYK-like tensor model with global symmetry. First, we study the large $N$ expansion of the bi-local collective action for the SYK model with manifest global symmetry. We show that the global symmetry is enhanced to a local symmetry at strong coupling limit, and the corresponding symmetry algebra is the Kac-Moody algebra. The emergent local symmetry together with the emergent reparametrization is spontaneously and explicit broken. This leads to a low energy effective action. We evaluate four point functions, and obtain spectrum of our model. We derive the low energy effective action and analyze the chaotic behavior of the four point functions. We also consider the recent 3D gravity conjecture for our model. We also introduce an SYK-like tensor model with global symmetry. We first study chaotic behavior of four point functions in various channels for the rank-3 case, and generalize this into a rank-$(q-1)$ tensor model.Comment: 61 pages, 8 figures; v2: typos corrected, references added, appendix D revised; v3: typos corrected, 1 figure added for clarification, substantial revisions and clarifications (conclusions unchanged

On the Chaos Bound in Rotating Black Holes

We study out-of-time-order correlators (OTOCs) of rotating BTZ black holes using two different approaches: the elastic eikonal gravity approximation, and the Chern-Simons formulations of 3-dimensional gravity. Within both methods the OTOC is given as a sum of two contributions, corresponding to left and right moving modes. The contributions have different Lyapunov exponents, $\lambda_L^{\pm}=\frac{2\pi}{\beta}\frac{1}{1\mp \ell \Omega}$, where $\Omega$ is the angular velocity and $\ell$ is the AdS radius. Since $\lambda_L^{-} \leq \frac{2\pi}{\beta} \leq \lambda_L^{+}$, there is an apparent contradiction with the chaos bound. We discuss how the result can be made consistent with the chaos bound if one views $\beta_{\pm}=\beta(1\mp \ell \Omega)$ as the effective inverse temperatures of the left and right moving modes.Comment: 35 pages, 2 figures. v2: references added, typos corrected, and clarifications added to the discussion sectio