[More technical post follows.] I’ve been working on this project with (UCSB postdoc) Maciej Kolanowski on and off for a while now, but only in the last couple of weeks did I have the time to hunker down and help push the writing of the results to the finish. For your Sunday reading pleasure, it is already up on the arXiv here (it came out Thursday but I’ve been too busy to pause to post about it – partly because I’ve begun work on writing up the next paper in the backlog). The title is “Extended JT supergravity and random matrix models: The power of the string equation”, and it is co-authored with Maciej Kolanowski.
In a way, it is a natural continuation of work I’ve described here from 2023 and 2024, described here and here. At a meeting at the Institute for Advanced Study in December 2023 I described in a talk (YouTube video here, look in particular from minute 35) something miraculous I’d discovered concerning capturing certain special supergravity (and black hole) behaviour using a random matrix model. The effective physics is two dimensional and the resulting model is one with N=2 extended supergravity. The spectrum of the relevant models has a continuum part associated with non-BPS states, which begins at some threshold energy E=E0. Additionally there’s a delta function at E=0 where there is some number of BPS states. Supergravity consistency
gives a precise relation between the R-charge of the multiplet in question, the value of E0, and the amount of BPS at the origin. Miraculously, I found that if you attempt to model the non-BPS continuum part as the spectrum of a multicritical matrix model (using a special ansatz for the appropriate solution of the relevant “string equation”), that the computation looks hopelessly problematic, with an infinite number of problematic terms that spoil the matching… except if the BPS part of the spectrum is precisely of a specific form – the form given by the supergravity consistency computations in the literature! Somehow the random matrix model, apparently not “knowing” anything about supergravity, R-charges, etc… comes back and says it will only work when all those things are working together consistently. I was stunned by that result, and very excited by it when I announced it at the conference.
By the way, what’s the “string equation” I keep talking about? Well, it has shown up in work described here in several guises and playing several roles… as far back as this post from 20 years ago! I think I need to write a post about it at some point, but for now I’ll leave it to the paper to explain it to you since I wrote a very careful general introduction to what the equation is, where it comes from, and how it is used to give the spectrum of the gravity model. (Foregrounding aspects that were buried in the 20 year old discussion.)
Anyway, after giving that talk, I refined the whole scheme a bit more in the following days (finding many beautiful formulae just pop out after certain miraculous resummations happened on the plane back home to California), and even began to see certain patterns that worked for N=4 supergravity (published later after expanding the whole thing into a detailed work about how matrix models can predict supergravity gaps in Summer 2024 with Misha Usatyuk, a KITP postdoc).
One of the many things we do in this new paper gain a deeper understanding of how the methods of my 2023 paper worked, getting to the bottom of what’s powering the miracle, and refining the method to work much better than the original scheme I used. Back in 2024 I also had conjectured (based on patterns seen when working with the matrix model approach) that there ought to be some N=3 variants of the (JT) supergravity supergravity and tried to find out if people know of such a thing already. Apparently they did not, and I had no luck with that until I spoke to Joaquin Turiaci, and he went off and tried to see if it could work as part of a larger scheme he’d been working on to classify such supergravities. He came back soon after and said that indeed an N=3 model should work and have some of the features I expected it to, so I was rather pleased about that. He also shared various new examples of spectra he’d been developing with collaborators that allowed us to test out our new refined methods and indeed we were able to do more complicated examples of what I’d done in 2023/24 – feed in the non-BPS features and predict the BPS features from consistency. Moreover we were able to show what kinds of spectrum should have a matrix model realisation within our scheme and what kind should not.IT immediately suggested that the main new set of spectra of Turiaci et al must break up into two subsections, each of which is a matrix model – this amounts to key statements/predictions about the statistical independence of certain gravity sectors, but read the paper for more.
Last, and certainly not least, is the fact that (as you’ve heard me say many times here) this framework very naturally lends itself to powerful non-perturbative analyses. We are able to show that large parts of the (apparently fine, perturbativley) parameter space of these models are non-perturbatively problematic in that there is no solution of the string equation that could (in the appropriate limit) produce the leading perturbative behaviour seen. The matrix model seems to be saying that there’s something inconsistent about those parts of parameter space. Marvellously, it is precisely when the condition for there to be a BPS sector is violated.
The power of the matrix model (through the string equation) to “know” all the things it does is still somewhat miraculous to me. Why are random matrix models and the intricacies of supergravity so nicely linked to each other? How does it know so precisely when a sector does not have BPS state? It is all quite remarkable. I’ve a long standing conjecture as to what this all means, but that’s for another time.
Anyway, I should stop rambling on now. Needless to say, this has been a super-fun project to work on!
Happy reading!
—cvj
