## Network Improvements

The desire to have glitch-free online teaching and business meetings at home has driven me to do some infrastructure improvements I should have done years ago: extending the Ethernet backbone of the home network. Connecting the jacks (ethernet connectors) is a tad fiddly (but trivial), but the results are worthwhile!

In particular, it is far better (than being connected to WiFi) to have the computer connected via ethernet (had to get the appropriate dongle for my mac) when teaching via Zoom. It gives a more robust setup than you can get WiFi (unless you’re right on top of one of your WiFi transponders). Since I’ve switched to ethernet for such sessions I’ve never had even the hint of a loss of quality in my zoom session, which is important when trying to focus on the teaching subject matter, and not technology issues. I even did a radio/podcast interview the other day (online), reasonably confident that the quality was strong throughout.

There’s an ethernet connected repeater station far from the main base station, so that when it now broadcasts the network wirelessly, I there is stronger network coverage all over the house than when it was just connected over WiFi. Should have done this years ago.

-cvj

## Online Teaching Methods

Sharing my live virtual chalkboard while online teaching using Zoom (the cable for the iPad is for power only).

It is an interesting time for all of us right now, whatever our walk of life. For those of us who make our living by standing up in front of people and talking and/or leading discussion (as is the case for teachers, lecturers, and professors of various sorts), there has been a lot of rapid learning of new techniques and workflows as we scramble to keep doing that while also not gathering in groups in classrooms and seminar rooms. I started thinking about this last week (the week of 2nd March), prompted by colleagues in the physics department here at USC, and then tested it out last Friday (6th) live with students from my general relativity class (22 students). But they were in the room so that we could iron out any issues, and get a feel for what worked best. Since then, I gave an online research seminar to the combined Harvard/MIT/USC theoretical physics groups on Wednesday (cancelling my original trip to fly to the East Coast to give it in person), and that worked pretty well.

But the big test was this morning. Giving a two hour lecture to my General Relativity class where we were really not all in the same room, but scattered over the campus and city (and maybe beyond), while being able to maintain a live play-by-play working environment on the board, as opposed to just showing slides. Showing slides (by doing screen-sharing) is great, but for the kind of physics techniques I’m teaching, you need to be able to show how to calculate, and bring the material to life – the old “chalk and talk” that people in other fields tend to frown upon, but which is so essential to learning how to actually *think* and navigate the language of physics, which is in large part the diagrams and equations. This is the big challenge lots of people are worried about with regards going online – how do I do that? (Besides, making a full set of slides for every single lecture you might want to do For the next month or more seems to me like a mammoth task – I’d not want to do that.)

So I’ve arrived at a system that works for me, and I thought I’d share it with those of you who might not yet have found your own solution. Many of the things I will say may well be specific to me and my institution (USC) at some level of detail, but aspects of it will generalize to other situations. Adapt as applies to you.

Here goes:

## Talk Prep

How do I prepare my research talks? I usually just sit down with a pencil, some paper and a cup of something warm, and I just draw/map out the story. Each box is a beat of the narrative, and ends up corresponding to one or two slides (if I’m doing slides). Then I’m more or less done.

(The facility of this old school approach is that drawing it out keeps it visual, less heavy with equations. Too many (if any) slides or long periods laden with equations (at least in physics) just aren’t so great. Also, it allows me to move these thumbnails/pages/sketches around if I need to, to sculpt the narrative. I can sit back and see if it’s all there at the end.)

(For this Harvard/MIT seminar, scheduled for Wednesday, I don’t yet know if I am going to get to give it. Wisdom about travel and gatherings is a bit uncertain right now, and things are changing as I type. Decisions on Monday. Update:- we changed it to a remote talk.)

–cvj

## Custard

When looking for an excuse to have some custard, simply whip up an upside down cake built on a bit of fruit, spices, and whatnot you might have (here: apples cooked in butter, basking on caramel, with toasted walnuts, cinnamon…etc).

–cvj

## New and Improved!

Updated with lots of new results! For your reading pleasure as 1912.03637 on the arXiv. (And still a swift read at 14 pages.)
New results include a fascinating differential equation for spectral densities, and some lovely E=0 results.

#physics #blackholes #gravity

–cvj

## Synchronicity?

Here’s a striking coincidence. Last Friday I was preparing to deliver a lecture on special relativity to my undergrad General Relativity class with this Hobbity thought experiment (that helps one discover Lorentz-Fitzgerald contraction), when I heard that Christopher Tolkien (the boy the Hobbit was originally written for) had died. (RIP. And thanks for the maps, the Silmarillion and so much more.)

I took the opportunity in class to pay brief tribute to him and to encourage a new generation to delve into the books, the world, and more, as expanded and illuminated for us by Tolkien the younger.

-cvj

## Black Holes and a Return to 2D Gravity! – Part II

(A somewhat more technical post follows.)

Continuing from part I: Well, I set the scene there, and so after that, a number of different ideas come together nicely. Let me list them:

What “nearly” AdS_2 looks like via JT gravity. The boundary wiggles, but has fixed length 1/T.

• Exact solution of the SYK model (or dual JT model) in that low temperature limit I mentioned before gave an answer for the partition function $$Z(\beta)$$, by solving the Schwarzian dynamics for the wiggling boundary that I mentioned earlier. (The interior has a model of gravity on $$AdS_2$$, as I mentioned before, but as we’re in 2D, there’s no local dynamics associated with that part. But we’ll see in a moment that there’s very interesting stuff to take into account there too.) Anyway, the result for the Schwarzian dynamics can be written (see Stanford and Witten) in a way familiar from standard, say, statistical mechanics: $$Z_0(\beta)=\int dE \rho_0(E) \exp(-\beta E)$$, where $$\rho_0(E)\sim\sinh(2\pi\sqrt{E})$$ is the spectral density of the model. I now need to explain why everything has a subscript 0 in it in the last sentence.
• On the other hand, the JT gravity model organises itself as a very interesting topological sum that is important if we are doing quantum gravity. First, recall that we’re working in the “Euclidean” manner discussed before (i.e., time is a spatial parameter, and so 2D space can be tessellated in that nice Escher way). The point is that the Einstein-Hilbert action in 2D is a topological counting parameter (as mentioned before, there’s no dynamics!). The thing that is being counted is the Euler characteristic of the space: $$\chi=2-2g-b-c$$, where $$g,b,c$$ are the number of handles, boundaries, and crosscaps the surface has, characterising its topology. Forget about crosscaps for now (that has to do with unorientable surfaces like a möbius strip $$(g=0,b=1,c=1)$$ – we’ll stick with orientable surfaces here). The full JT gravity action therefore has just the thing one needs to keep track of the dynamics of the quantum theory, and the partition function (or other quantities that you might wish to compute) can be written as a sum of contributions from every possible topology. So one can write the JT partition function as $$Z(\beta)=\sum_{g=0}^\infty\hbar^{-(1-2g)}Z_g(\beta)$$ where the parameter $$\hbar$$ weights different genus surfaces. In that sum the weight of a surface is $$\hbar^{-\chi}$$ and $$b=1$$ since there’s a boundary of length $$\beta$$, you may recall.

The basic Schwarzian computation mentioned above therefore gives the leading piece of the partition function, i.e., $$g=0$$, and so that’s why I put the subscript 0 on it at the outset. A big question then is what is the result for JT gravity computed on all those other topologies?!

## Black Holes and a Return to 2D Gravity! – Part I

(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

## More Pie

Well, you know the saying: When life hands you several apples, some blackberries, and a chunk of pastry left over from your last pie-making… you make another apple-blackberry pie, taking the opportunity to make it even better!

(No? Never heard that saying? Huh.)

-cvj

## Chalky

I realized recently that I’ve forgotten a great deal of my drawing skills, settling back into some clunky habits, due to zero practice. But I’m going to need them back for a project, and so will start teaching myself again. Above is a (digital) chalk doodle I did yesterday.

–cvj

## Rolling…

And so it began… All this butter had to end up inside this little bit of flour/water mix. But in lots of tasty layers of deliciousness! (And I’ll do this four times, and so that’s my morning.)

More later. I love puff pastry!

–cvj

## Spectral (again)

A newly computed spectral density function for a new example…

Oh, this? I’m very pleased with it. I’ll tell you more about what it is at a later point, but this is mostly for my own entertainment. It’s a sort of big brother to the spectral curve shown in the previous post. I’d been thinking about how to get this curve (those plus marks) to come out right for a few days, and repeatedly getting it not quite right*, with puzzlement and frustration. And then while out on a run it came to me how to do it right…. Then after writing the matlab routines to make my solution happen, out it popped! Hurrah. (Update: The inset was added to show more detail. Now it is in the paper as an added extra.)

(* Update: “not quite right” means, for example, that the large E behaviour should have been E^(3/2), generalising the E^(1/2) of the previous example, but it wasn’t for a bit, because of a normalization issue. All fixed now!)

To (most of) you it’s just a bunch of dots and wiggles and that’s ok. To me is some very nice physics that connects things I’d been thinking about back some 30 years ago with things I’m thinking about now. More later. Sorry if this is annoying, I’m just very pleased with the result.

-cvj

## Spectral

Oh, this was a recent post I did on social media that I think ought to go here too. I will try to write a longer post about the physics behind this (in involves black holes, quantum gravity, string theory, quantum chaos, and a host of other things):

A spectral density function of particular interest to me and a recent project on 2D quantum gravity.

Were you looking for more reasons to read my recent paper on JT gravity? This figure from it is my favorite reason. It summarizes a lot of the abstract – identical perturbation theory (right) but better non-perturbative regime (left). Go! https://arxiv.org/abs/1912.03637

-cvj