(A somewhat more technical post follows.)
Escher’s “Circle Limit III”, nicely illustrating (Euclidean) AdS_2 for us.
Well, I think I promised to say a bit more about what I’ve been up to in that work that resulted in the paper I talked about in an earlier post
. The title of my paper, “Non-perturbative JT gravity”
has JT (Jackiw-Teitelbiom) gravity in it, so I could start there.
JT gravity is a model of two dimensional gravity that goes back many decades. But there’s not much going on in 2D (or (1+1) dimensions, one space and one time), I hear you cry! Well, nevertheless this turns out to be a very useful arena for studying very important quantum key properties of black holes in more realistic dimensions. Let me take a step back to unpack that.
There is a long tradition (which you might know about) of studying models of black holes in string/M-theory, in different limits and approaches, each with their own advantages and disadvantages. The simplest models are often too simple to capture some of the important features of black holes, and some models get nicely at certain aspects while having little to say about others. And of course, some of the more realistic (but still simple) black hole models are still too complicated to be directly solvable in order to reliably explore phenomena of interest.
I’d say that a certain class of models that has been discussed a lot recently has something new to offer. The prototype is something called the SYK model (Sachdev-Ye-Kitaev). It’s an even more crazy sounding starting point than 2D gravity, as it is a 1D model. There’s no space at all, just time: It is a model in (0+1) dimensions, if you like. It’s a special model of quantum mechanics, actually a bunch of \(\tilde N\) fermions (where \(\tilde N\) is large), with certain random couplings between them. It was noticed that this model, at temperature \(T\), as a simple thermal with quantum chaos, scrambles quantum information at a particular rate (measured by the “Lyapunov exponent” \(\lambda_L\)) that was associated with the scrambling rate of black holes: \(\lambda_L=2\pi k_B T/\hbar\). So it became of interest to study it as a simple, solvable model of this sort of behaviour. (Kitaev made a convincing case in a series of talks at the KITP back in 2015 that if you find a model with black-hole-like scrambling, it has a secret dual gravitational character to it, in a sense we’ve discussed here on the blog before. – I recommend looking at the talks (here and here), and also a nice paper of Maldacena, Shenker, and Stanford (also from Spring 2015) that discusses the gravitationally-motivated bounds on chaos that the SYK model saturates.).
Well, I heard people mention the SYK model increasingly over 2015 but I was thinking about too many other things (and was mostly dealing with being mostly sleep-deprived as a new dad), and so did not really pay much attention. Bandwidth issues, we’d say these days. I was happy to declare “we’ve got some of our best people on it”, and continued chipping away on other things where I could. Then in December 2017 I found myself sitting next to David Gross on a flight to the East coast (long story I forgot to blog about), and he mentioned that he’d been really excited by the SYK model and maybe I should have a look, since there’s a lot there that I’d probably like. And *still* some time went by without me clearing up bandwidth to look at any of it.
Catch up began in the Fall of last year (2019), but only after similarly missing following what was going on in a related area: JT gravity. One way of thinking about how JT gravity enters the story is to simply state that it is a holographic dual of the SYK model, in the sense of AdS/CFT, where you have a gravity theory on one side, and it is dual to a (conformally invariant) non-gravitational field theory on the other side. The key thing is that the gravitational theory has one dimension more than the field theory. I’ve spoken of such things a lot here on this blog so I won’t review, but instead let you dig a bit and find things to read in the archives. (You could put AdS/CFT in the search window, or maybe start here.)
A ball showing the AdS/CFT setup. The interior has gravity on an anti-de Sitter spacetime, the the boundary has a dual (conformal) field theory.
Simply/glibly put, the duality says that some complicated (strongly coupled) dynamics of a field theory can be re-written as a simpler dynamics involving gravity, and the “re-writing” uses an additional spatial dimension. Most famously you have (super) Yang-Mills theory in 4D (which is conformally invariant, so let’s write it as CFT_4) being equivalent to a theory of gravity in 5D anti-de Sitter (AdS_5). So AdS_5/CFT_4. Many examples are known in various dimensions that fill out the pattern AdS_(d+1)/CFT_d. The picture I like to draw is of a ball (see above right), the interior of which is the AdS space where gravity is operating (I’ve cut a hole for you to see the inside), and the boundary of which has the (one dimension fewer) conformal field theory. You often hear people refer to the “boundary” and the “bulk” – well, that’s the picture.
Important technical note: In the setup I drew above, the metric which measures spatial distances makes spacings smaller and smaller as you move out from the Click to continue reading this post