Full Circle

snapshot of paper

Yesterday I submitted (with collaborators Felipe Rosso and Andrew Svesko) a new paper to the arXiv that I’m very excited about! It came from one of those lovely moments when a warm flash of realisation splashed through my mind, and several fragments of (seemingly separate things) that had been floating around in my head for some time suddenly all fit together. The fit was so tight and compelling that I had a feeling of certainty that it just “had to be right”. It is a great feeling, when that happens. Of course, the details had to be worked out, and everything checked and properly developed, new tools made and some very nice computations done to unpack the consequences of the idea… and that’s what resulted in this paper! It is a very natural companion to the cluster of papers I wrote last year, particularly the ones in May and June.

What’s the story? It’s all about Jackiw-Teitelboim (JT) gravity, a kind of 2D gravity theory that shows up rather generically as controlling the low temperature physics of a wide class of black holes, including 4D ones in our universe. Understanding the quantum gravity of JT is a very nice step in understanding quantum properties of black holes. This is exciting stuff!

Ok, now I’ll get a bit more technical. Some background on all this (JT gravity, matrix models, etc), can be found in an earlier pair of posts. You might recall that in May last year I put out a paper where I showed how to define, fully non-perturbatively, a class of Jackiw-Teitelbiom (JT) supergravity theories that had been defined in 2019 in a massive paper by Stanford and Witten (SW). In effect, I showed how to build them as a particular combination of an infinite number of special “minimal string” models called type 0A strings. Those in turn are made using a special class of random matrix model based on ensembles of complex matrices. That paper itself had resulted from a flash of inspiration, when I realized that some key things that had been floating around in my head since my thesis work on such models (30 years ago!) spectacularly fit some observations of SW. Shortly after, I figured out how to actually extract concrete non-perturbative information numerically from my definition, and also for a definition of ordinary JT gravity I’d presented the previous December, (the defining equations are highly non-linear and formally of infinite order). I explicitly computed things like the JT spectrum and the spectral form factor (this had never been done before for a complete JT gravity model), presenting them in my June paper, and with some more examples in an August paper.

One of the obvious cluster of questions that jumps to mind is “Why the 0A theories? Why not 0B?”. The answer to the first was, for a long time, just that they have the right properties. The answer to the second was naturally then that there may be a JT gravity with just the right properties, but I had no idea what it was. Other than conjecturing that there may be more JT gravity models to be defined non-perturbatively by combining together various other kinds of minimal string model using my scheme, (a conjecture I stand by) I did not more time worrying about the whys and wherefores of this issue. There was a pandemic on, and lots of things to calculate with the existing definition.

But there were other questions. One set centered around the fact that I was very puzzled about some curious parts of the SW paper that I could not make any sense of, concerning a class of models that had a spectral density that was infinite in extent after taking the scaling limit that defines a gravity theory. This was in contrast to the earlier mentioned models where the densities classically start at zero and go off to infinity, i.e., semi-infinite. The latter arise from starting with an unscaled spectral density, which has two ends, and then taking a scaling limit where you focus on one endpoint. In SW this infinite extent class of models seemed to arise by taking a limit where both ends go off to infinity, and I could not understand how to do that, so could not apply my methods. Again, I filed this puzzle in the back of my mind and moved on to other issues.

Months later (while thinking about some other things) is when the two seemingly separate issues (along with some other technical signposts) collided in my head and exploded, creatively! The second mystery is actually resolved by the first mystery, and I do know how to get spectral densities that stretch along the whole infinite line! You see there’s a different class of matrix models that come from having two separate spectral densities run into each other and merge. If you take the scaling limit where again you focus on the endpoints, you’ll have semi-infinite spectral density running off in both directions, merging to make a whole, infinite spectral density shape. See the figure.

diagram of zooming in on endpoints

It turns out that such models had already been identified (by Klebanov et al in 2003) as a means of capturing type 0B string theories! – in the same paper where they identified the complex matrix models (that I’d worked on in the 90s) as type 0A models. The last really nice thing about all this was that there’s a special class of solution for the string equations of the type 0B models that vanish to all orders in perturbation theory, but for which there is non-trivial non-perturbative physics. This was very analogous to a class of such solutions I’d noticed for the 0A case, that translated into some remarkable features of the gravity theory that SW had noticed. So a natural conjecture that presented itself to me is that this same type of perturbatively vanishing solution for these “double-cut” models defining 0B should again reproduce some remarkable features observed in SW, now for the mystery model.

Saying it all like this makes it sound all very obvious, but believe me, it wasn’t so clear to me before. But it is lovely that this all works out as an extremely natural 0B companion story to my 0A stories from before, coming full circle, in a sense. Re-reading various parts of SW that I had not understood before, now with this new perspective, served to give support to the idea, and in fact now I see from a clever footnote in an appendix that they partially anticipated some of the general 0A/0B connections, if not seeing all the juicy details and non-perturbative structures that all flow from the connections I worked out in my papers last year, (and in the new paper with Felipe and Andy).

In any case, even if you have a nice story, it needs to be checked, and also in this case a number of the structures needed to properly make it fly needed to be discovered and explored, and it was a huge amount of fun to do that with Felipe Rosso and Andy Svesko. We collectively refined and sharpened the whole thing into a lovely story, learning a lot from each other along the way. We showed that we can, in a very elegant manner, reproduce all the perturbative features of the JT supergravity that SW had noticed, and also extract lots of new non-perturbative physics such as the spectrum in the figure (dashed line is the perturbative result).

a density function

Non-perturbative spectral density (red) of the supercharge operator in an SJT theory.

The paper will appear on the arXiv later today – only 13 pages long (plus some appendices). Enjoy!


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