Contents

# Contents

## Idea

Studies of holography have been hampered by the lack of a simple solvable model that can capture features of Einstein gravity. The simplest model, which is a single matrix quantum mechanics, does not appear to lead to black holes. $\mathcal{N} = 4$ super Yang Mills at strong ’t Hooft coupling certainly leads to black holes, and exact results are known at large N for many anomalous dimensions and some vacuum correlation functions, but at finite temperature the theory is difficult to study.

A system that reproduces some of the dynamics of black holes should be interacting, but we might hope for a model with interactions that are simple enough that it is still reasonable solvable.

Kitaev has proposed to study a quantum mechanical model of $N$ Majorana fermions interacting with random interactions (Kitaev 15). It is a simple variant of a model introduced by Sachdev and Ye (Sachdev-Ye 93), which was first discussed in relation to holography in (Sachdev 10).

From Maldacena 18:

The SYK model gives us a glimpse into the interior of an extremal black hole… That’s the feature of SYK that I find most interesting… It is a feature this model has, that I think no other model has

## Definition

Let $\mathcal{J}_{ijkl}$ be random variables with expectation values $E[\mathcal{J}_{ijkl}]=0$ and $E[\mathcal{J}_{ijkl}^2]=\frac{6J^2}{N^3}$.

The Lagrangian density definign the SYK model is this:

$L = \frac{1}{2} \sum_{i=1}^N \chi_i \partial_t \chi_i - \frac{1}{4!} \sum_{i,j,k,l=1}^N \mathcal{J}_{ijkl} \chi_i \chi_j \chi_k \chi_l$

## References

### General

Review includes

The model is due to

with further discussion in

Further developments in

• Biao Lian, S. L. Sondhi, Zhenbin Yang, The chiral SYK model (arXiv:1906.03308)