The Bulk Dual of SYK: Cubic Couplings

The SYK model, a quantum mechanical model of $N \gg 1$ Majorana fermions $\chi_i$, with a $q$-body, random interaction, is a novel realization of holography. It is known that the AdS$_2$ dual contains a tower of massive particles, yet there is at present no proposal for the bulk theory. As SYK is solvable in the $1/N$ expansion, one can systematically derive the bulk. We initiate such a program, by analyzing the fermion two, four and six-point functions, from which we extract the tower of singlet, large $N$ dominant, operators, their dimensions, and their three-point correlation functions. These determine the masses of the bulk fields and their cubic couplings. We present these couplings, analyze their structure and discuss the simplifications that arise for large $q$.Comment: 39 pages, v2: Evaluation of integral in Sec. 3.3.2 correcte

A line of CFTs: from generalized free fields to SYK

We point out that there is a simple variant of the SYK model, which we call cSYK, that is $SL(2,R)$ invariant for all values of the coupling. The modification consists of replacing the UV part of the SYK action with a quadratic bilocal term. The corresponding bulk dual is a non-gravitational theory in a rigid AdS$_2$ background. At weak coupling cSYK is a generalized free field theory; at strong coupling, it approaches the infrared of SYK. The existence of this line of fixed points explains the previously found connection between the three-point function of bilinears in these two theories at large $q$.Comment: 26 pages, v

All point correlation functions in SYK

Large $N$ melonic theories are characterized by two-point function Feynman diagrams built exclusively out of melons. This leads to conformal invariance at strong coupling, four-point function diagrams that are exclusively ladders, and higher-point functions that are built out of four-point functions joined together. We uncover an incredibly useful property of these theories: the six-point function, or equivalently, the three-point function of the primary $O(N)$ invariant bilinears, regarded as an analytic function of the operator dimensions, fully determines all correlation functions, to leading nontrivial order in $1/N$, through simple Feynman-like rules. The result is applicable to any theory, not necessarily melonic, in which higher-point correlators are built out of four-point functions. We explicitly calculate the bilinear three-point function for $q$-body SYK, at any $q$. This leads to the bilinear four-point function, as well as all higher-point functions, expressed in terms of higher-point conformal blocks, which we discuss. We find universality of correlators of operators of large dimension, which we simplify through a saddle point analysis. We comment on the implications for the AdS dual of SYK.Comment: 67 pages, v

Laura Chinellato: L’ara di Ratchis a Cividale

A good mosaic should have a clear-cut frame, a central panel proposing the main subject, and subsidiary ones amplifying the theme. Laura Chinellato’s L’ara di Ratchis a Cividale is indeed such a successful mosaic of words and ideas. Its frame is made up of a prelude and postlude by two distinguished medieval art scholars, Valentino Pace and Hjalmar Torp; its centerpiece consists of the author’s extensive and enlightened formal, iconographic, epigraphic and material analysis and is further amplified by experts in related areas – Stefano Gasparri (history), Laris della Pietra (liturgy), Maria Teresa Constantini, (conservation, restoration, reconstruction of polychromy), and Alessandro Princivalle and Davide Manzato (scientific analysis and measurements). All in all, this book is a model enterprise creating a material and spiritual ID, indeed a biography, of a key work of Pre-Romanesque figurative arts. As emphasized by Valentino Pace in his Preface, the Altar of Ratchis is “among the most important monuments of the 8th century”, one in which “epigraphy, figured images, signs, material and color converge to communicate a message of faith and prestige, which this book helps us understand”. But it is also a station on a way to the future, for, as stated by Hjalmar Torp in his concluding remarks, this is a work “based on twelve years of research which includes a detailed analysis of 300 years of scholarship constitutes … a firm point of continuous research.

Dimensionally Reduced SYM_4 as Solvable Matrix Quantum Mechanics

We study the quantum mechanical model obtained as a dimensional reduction of N=1 super Yang-Mills theory to a periodic light-cone "time". After mapping the theory to a cohomological field theory, the partition function (with periodic boundary conditions) regularized by a massive term appears to be equal to the partition function of the twisted matrix oscillator. We show that this partition function perturbed by the operator of the holonomy around the time circle is a tau function of Toda hierarchy. We solve the model in the large N limit and study the universal properties of the solution in the scaling limit of vanishing perturbation. We find in this limit a phase transition of Gross-Witten type.Comment: 29 pages, harvmac, 1 figure, formulas in appendices B and C correcte

Field Theory as a Matrix Model

A new formulation of four dimensional quantum field theories, such as scalar field theory, is proposed as a large N limit of a special NxN matrix model. Our reduction scheme works beyond planar approximation and applies for QFT with finite number of fields. It uses quenched coordinates instead of quenched momenta of the old Eguchi-Kawai reduction known to yield correctly only the planar sector of quantum field theory. Fermions can be also included.Comment: 16 pages, 3 figure

Two-Matrix model with ABAB interaction

Using recently developed methods of character expansions we solve exactly in the large N limit a new two-matrix model of hermitean matrices A and B with the action S={1\over 2}(\tr A^2+\tr B^2)-{\alpha\over 4}(\tr A^4+\tr B^4) -{\beta\over 2} \tr(AB)^2. This model can be mapped onto a special case of the 8-vertex model on dynamical planar graphs. The solution is parametrized in terms of elliptic functions. A phase transition is found: the critical point is a conformal field theory with central charge c=1 coupled to 2D quantum gravity.Comment: harvmac, 24 pages, 5 figures (1 color figure