(Image above was used in an earlier post. It is of a pool at the Getty Villa.)

(Somewhat technical post follows.)

I was sitting in a random city somewhere watching the rain and it reminded me that I did not get to mentioning the completion of my other project. I reported on it in a paper with two students, Ram and Jeff, and the title was (perhaps intriguingly?) “String Theory and Water Waves”. You can see it here.

The work follows on from some projects I did a while back, and concerns attempts to gain understanding of string theory at a (perhaps) deeper level by working with various simple models of string theory that are complicated enough to capture many of the theory’s remarkable features while being simple enough to actually solve completely (in a sense). Actually, I laid a lot of the groundwork for this post by writing three earlier blog posts in the “News…” series, and you can find them here, here and here.

Here are some words from the introduction to get you excited. What you may have gathered from the previous posts was that I’d been playing with some of these simple string theories already, and that they have lots of exciting features that teach us interesting non-perturbative (i.e. strongly coupled, roughly speaking (but see footnote**)) string physics. Two of them are called the type 0A and the type 0B theories, and they are somewhat different in that they are governed by somewhat different underlying structures (defining equations and so forth). Well, look at this:

We have found that there is a larger framework into which the type 0A and type 0B string theories can be naturally embedded and within which they are connected as parts of a larger theory. We found further that the two string theories are merely special points in a much larger tapestry of possibilities. When perturbation theory is examined, other special points suggest themselves, and they turn out to be just as “stringy” as the original type 0 theories, deserving to be thought of as string theories as well. We begin the program of trying to identify some of these theories, with some success. We also find that the larger framework provides natural definitions of regimes of the type 0 string theories that are hard to define using perturbation theory, and we will report more fully on non–perturbative aspects in a follow–up paper.

In this sense, we have a precise analogue of M–theory. We have a larger physical framework that is not itself a string theory, but that can be readily specialized to yield string theories as special limits. We can move between different theories in a quite natural way, which is nonetheless outside the framework of any of its daughter string theories. We find this encouraging and exciting.

Note the last point about M-theory. This is the name the field gave to whatever the full theory is (we still don’t know what it is) underlies string theory when you go beyond perturbation theory (i.e. weak coupling). All we know is that string theory is not really the best description, and other physical degrees of freedom are better suited to do the job. Now, in many cases (and the reason we’re able to make so much progress while not knowing what M-theory is) one can proceed because you sometimes find that at strong coupling a new regime takes over that is described in terms of degrees of freedom that are, while not the strings you started with, still familiar (in surprising ways – this is the “duality” that you hear people talking about in the last 15 or so years a lot in the field). These might be other types of string theories, or instead maybe some sort of field theory. However it still leaves the issue open of what the big framework is, within which we are clearly just tinkering around the edges of. That’s the thing called “M-Theory”. I’ll leave you to dig a bit more into the history of the idea by hunting more on the web, starting with the references in my earlier posts.

My point here is that in some ways we’ve found a framework that acts just like M-theory, but for this simple string (“minimal” string, as they are often called) setting. What is it? Turns out that the governing equations all come from a single family of equations derived from what is known as the dispersive water wave hierarchy. Yes, certain model equations that come from studying the physics of water waves (of a certain sort) are the things we’ve discovered are relevant. You don’t just take the simple water wave equation, but a whole infinite family of equations that share certain properties with it and so form a sort of ladder of equations that get increasingly more involved as you go up the ladder. Each rung on the ladder gives a model, and if you take the model and bend it one way, you get type 0A string theory. Bending another way pops out type 0B, and bending yet other ways pop out new string theories that we spend a bit of time trying to understand. It’s all rather exciting.

Another really exciting thing about our project is mentioned at the end of the abstract, for example:

Notably, we discover that the Painlevé IV equation plays a key role in organizing the string theory physics, joining its siblings, Painlevé I and II, whose roles have previously been identified in this minimal string context

Now, I’m going to stop here, and not unpack that last bit, but hope that those of you who are interested in those remarkable beasts, the six Painlevé equations, reported on in 1897 by Paul Painlevé, the Mathematician and Politician might be intrigued.

-cvj

**(Footnote added in update) To equate “non-perturbative” and “strongly coupled” is not quite right, since there are physical effects that contribute in ways that are simply invisible in perturbation theory since they are *smaller* than any term in the expansion. Instanton terms, for example. “Strongly coupled” usually refers to physics that is inaccessible to perturbation theory since the expansion parameter (a dimensionless coupling) gets too large to use to organize a sensible expansion. So it is “non-perturbative”, but using “strongly coupled” misses out other ways of being non-perturbative. (There are actually ways of connecting all of these terms further, such as examining the spectrum of masses of objects in the theory as a function of coupling, etc, but I won’t go into it here.) I was using the more colloquial term in the discussion above instead of “non-perturbative” so as not to scare away some of the readers who are not technically experienced in this area. Thanks, Oliver Rosten, for asking me to clarify that these things need not be the same. (Actually, for the technically interested, there’s an excellent book by Jean Zinn-Justin that discusses a lot of this sort of thing. “Quantum Field Theory and Critical Phenomena”. I recommend it.)

Sounds very exciting!

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Congratulations, excellent work Clifford.