55 research outputs found

### Multipoint Conformal Blocks in the Comb Channel

Conformal blocks are the building blocks for correlation functions in
conformal field theories. The four-point function is the most well-studied
case. We consider conformal blocks for $n$-point correlation functions. For
conformal field theories in dimensions $d=1$ and $d=2$, we use the shadow
formalism to compute $n$-point conformal blocks, for arbitrary $n$, in a
particular channel which we refer to as the comb channel. The result is
expressed in terms of a multivariable hypergeometric function, for which we
give series, differential, and integral representations. In general dimension
$d$ we derive the $5$-point conformal block, for external and exchanged scalar
operators.Comment: 39 page

### Entanglement Entropy: A Perturbative Calculation

We provide a framework for a perturbative evaluation of the reduced density
matrix. The method is based on a path integral in the analytically continued
spacetime. It suggests an alternative to the holographic and `standard' replica
trick calculations of entanglement entropy. We implement this method within
solvable field theory examples to evaluate leading order corrections induced by
small perturbations in the geometry of the background and entangling surface.
Our findings are in accord with Solodukhin's formula for the universal term of
entanglement entropy for four dimensional CFTs.Comment: 28 pages, 3 appendices, 5 figure

### The Spectrum in the Sachdev-Ye-Kitaev Model

The SYK model consists of $N\gg 1$ fermions in $0+1$ dimensions with a
random, all-to-all quartic interaction. Recently, Kitaev has found that the SYK
model is maximally chaotic and has proposed it as a model of holography. We
solve the Schwinger-Dyson equation and compute the spectrum of two-particle
states in SYK, finding both a continuous and discrete tower. The four-point
function is expressed as a sum over the spectrum. The sum over the discrete
tower is evaluated.Comment: 29 pages, 2 figure

### Entanglement Entropy for Relevant and Geometric Perturbations

We continue the study of entanglement entropy for a QFT through a
perturbative expansion of the path integral definition of the reduced density
matrix. The universal entanglement entropy for a CFT perturbed by a relevant
operator is calculated to second order in the coupling. We also explore the
geometric dependence of entanglement entropy for a deformed planar entangling
surface, finding surprises at second order.Comment: 18 pages + appendice

### 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

### 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

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