[Note: Originally posted on CV on 4th November 2005. 25 comments on it here.
Feel free to add new ones here.]
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[Warning! This is an unusually technical post.]
Ok, so last time, I told you a bit about the motivations for what I’ve been up to. Now I want to simply show you some of the product. I’m going to use pictures, words, and equations. I will lose some of you, and for that I’m sorry. But I hope that the words will still give you the gist of the thing. I’ll answer some of your questions in the comments.
Consider the following equation (first found and studied in this context in about 1991/1992 and reported e.g. here, and here, and here):
where
The leading boundary conditions for the solutions we wish to consider are:
This non-linear differential equation actually contains a lot of string theory information, and it is packaged in a way that is just the sort of thing we dream about in several other parts of string- (and M-) theory: It does not refer to strings by worldsheets, or any way that relies on thinking of the string as a string. We’ve learned from this context and several other studies that when you can identify a string unambiguously in your description, it means more often than not that you are stuck in perturbation theory and so missing a huge amount of the story. So what you look for are ways of defining string theories (or whatever they are since they are about more than strings) without starting with strings.
So how do I find strings? Well, the free energy and partition function (i.e. extremely important defining quantities) for the physical model is given by
We can develop corrections to the leading behaviour above by just iterating. Actually, you can do this yourself…. substitute in u=z + correction, where “correction” is of order nu (nu is the Greek letter that looks like a curly v above), and then, neglecting anything that is higher order than that, you’ll get a simple equation for the correction, which you can solve. Then you can solve for the next order in the same way, and so on…. You can do this separately for either the large positive z or the large negative z regimes.
The result for positive z regime is
and so integrating twice and dumping the constant (which turns out to be non-universal physics) we get the free energy:
I’ve written it in terms of the natural dimensionless combination of parameters which keeps showing up at each term:
This is the string coupling! In fact, each term is a term in the “world-sheet” expansion of a string theory…. Have a look (for the technically observant, I’ve not put the sphere term, as it turns out to be non-universal in this example):
So these “world-sheets” are two dimensional surfaces that strings sweep out as they move. A particle sweeps out a line as it moves, a string sweeps out a sheet. To use language from field theory, say, these are the “Feynmann diagrams” for the string theory. Quantum mechanics (yes, this is a quantum theory, and nu plays the role of hbar, Planck’s constant) tells us that we must sum over all paths the strings can take (for a given process), and this is what we see here.
Notice that the innocent-looking parameter Gamma appears in a special way. Every time there is a boundary on the string world-sheet (so it is an “open string”), there is a factor of Gamma. This actually counts the number of a certain type of “D-brane” that is in the background in which the string is moving. (They are extended objects defined as places (dynamical objects) on which string endpoints live. See the picture on the right, showing a snapshot of the strings at one instant, so they are not sweeping out sheets.)
Background? Ah, so these simple string theories have a quite simple spacetime (when it can be identified), which is one reason they are called “minimal” strings. but on the other hand it is a complicated background. This is because there is only one continuous dimension in the target space, but the strength of the string coupling varies from point to point. In fact it grows arbitrarily strong as you move to one end. This is in fact the end that the background D-branes (called “ZZ” branes (link) in this context) are located There is another type of D-brane in these models called “FZZT” branes (link, link) which stretch along the target space. I might talk about those some other time, since their story is a nice one too.
You might ask whether we have to force Gamma to be positive and an integer by hand, since the equation surely does not care about our stringy interpretation. Turns out that it does. Amazingly, the properties of the equation and its solutions are such that Gamma positive and integer are a very special sector, without you having to impose this! This is a cute result of a study we (James Carlisle, cvj, and Jeff Pennington) did around this time last year, and written up here. That Gamma is positive and integer might remind you of something else that can be counted discretely too. That story is really cute too, and I’ll talk about that in a later post, perhaps. (If you can’t wait, you can read ahead about it from the recent paper we posted on the arXiv on Wednesday.)
So this is all rather nice, I hope you agree. We recover a string theory -with Gamma D-branes- in one perturbative regime of the equation…. we keep expanding and get stringy Feynman diagrams at whatever order we like.
But the really great thing is that we’ve got more than just the perturbation theory! We’ve got every thing else. At this point, we leave most of the field of string theory in the dust, because most of what we can do with strings, as I’ve talked about in other posts, is based upon string perturbation theory. We need to know more about strings, and in particular, we need information to all orders in perturbation theory and we need information about stuff that cannot be described in perturbation theory at all.
Well, we have that here, since the point is that the equation has more than a perturbative expansion. It has a unique exact solution. I can plot it for you here:
So you can ask questions about physics not just pertaining to the extreme right of the solution, but right in the middle, if you want to, where the expansion above makes no sense. This is really a fun and exciting thing to be able to do in such a clear and simple way.
Ok, so you might ask. Hmmm, what is that region to the far left? Well away from the other perturbative regime? Well, you can do the same expansion tricks again to get:
and hence the free energy:
Turns out that this is a completely closed string theory. There are no open strings at all! Instead, Gamma appears upon the insertion of a “vertex operator” on the worldsheet, corresponding to the string moving in a background “R-R flux”, a sort of background field in the model generated by closed strings. (Turns out you need an even number of such insertions, as shown in this paper, who clarified a number of key aspects of the modern interpretation of these models.) Here is the picture of what the diagrams encoded in the expansion look like.
So what we’ve found is that there is a completely separate regime encoded by the string equation that represents an entirely closed string theory. This is remarkable, I hope you agree, and this is an example of what is called an “open-closed transition”. Such non-perturbative connections between open and closed strings were discovered in this context a long time before the terms “open-closed transition”, or “open-closed duality” was invented, so people in the field point to other more recent examples as the prototype, such as here and even the heterotic/type I example in here. That’s what I get for being several years ahead of my time, I suppose. (See e.g. the papers here, and here, and here.)
You might ask what such a study does for the field. The simple answers are (1) Proof of principle, and (2) Controlled understanding. In other words, (1) there are several hopes expressed about how non-perturbative string theory might look -including whether it exists- and whether several of the exotic properties -such as dualities, etc- that it has can really be captured in a sensible single model. This
(and its cousins) is a concrete example. Note also that we can ask physics questions for any value of the string coupling….i.e. it’s not just duality games. (2) See my comments in the previous post.
Well, that’s probably enough to be getting on with for now. More later.
-cvj
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Great historical instruction Clifford and easy reading for layman.
Thanks Clifford
Yes, I simply mean that we have very limited understanding of how to go beyond fixed backgrounds. Given a background, we can do great things with it, and describe it in many ways sometimes (using dualities)…and since gauge gravity duals (things like AdS/CFT), we can indirectly describe classes of backgrounds (that share the same asymptotic structure, say) and imagine how they go from one type to another (the formation of a black hole in AdS, for example, is dual to an ordinary thermalization process in a quantum field theory, for example)… but there we’ve fixed the asymptotic (which is a bit better)… and so we’re still not there yet. Of course, one possibility is that we never will succeed. That might be all there is. Fixed backgrounds put in by hand. It’s a limitation in some ways, but there’s still a great deal of physics that we’ve learned, and have still to learn, while remaining in the fixed background regime.
Cheers,
-cvj
Hi Clifford,
Thanks again for your response!
“What’s missing, and what stops me from completing my answer to your question, is a good description of how strings can dynamically go from one background (solution to Einstein’s equation) to a completely different one. What we do have is really good understanding of various ways of describing given backgrounds (or classes of backgrounds) in different regimes, such as weak vs string coupling.”
I’m not sure what you mean here. What do you mean by “dynamically go from one background to a completely different one.”? Do you mean how strings can be embedded in generic backgrounds? Or how they are supposed to propagate? From your last sentence, what descriptions of the background are we talking about here? Aren’t the backgrounds fixed?
Thanks very much, and sorry for the confusion!
Fred
Hi Fred,
This is a great question, and this is a very good place to ask it.
I think that the main answers to your question are not yet known. We’re still reaching for the bits that will help us fill out the full story. What we understand is that strings are very good at describing the quanta of gravity, which almost by definition involves considering a fixed background upon which they propagate. Strings also supply you with the same machinery that General Relativity does – a set of field equations for gravity that you must satisfy if you’re working properly. Remarkably, the equations are forced on you for quantum mechanical consistency. This is profound, and nobody knows why – you ask for a quantum-mechanically and relativistically consistent theory of a 1D extended object and the equations say “ok, but on condition that you solve these equations describing gravity”. Wow. So it’s a rather nice situation. Curvature and so forth are not illusions…. they are real physical things that strings know about and respond to and require. What we expect then is that these descriptions are connected. In the same sense that on the one hand there’s this nice quantum mechanical description of photons in QED, and on the other hand the field around a solenoid is morally made of the same stuff, but described in a very different way. They’re just different regimes of the same underlying phenomenon, electromagnetism, equipped with their own tools for best describing them. Same here.
What’s missing, and what stops me from completing my answer to your question, is a good description of how strings can dynamically go from one background (solution to Einstein’s equation) to a completely different one. That would be nice. What we do have is really good understanding of various ways of describing given backgrounds (or classes of backgrounds) in different regimes, such as weak vs string coupling. This has been really great, since it has allowed us to tackle hard questions like whether black holes have an underlying quantum mechanical description in terms of microstates that can be counted (accounting for their hitherto mysterious Bekenstien-Hawking entropy). It has also allowed us to describe various gravitational phenomena entirely in terms of non-gravitational phenomena, and vice-versa (see my earlier posts from last Summer about how this is useful for understanding phenomena relevant to strongly coupled nuclear matter in various experiments. Search on QCD.) Some might use the latter example as an argument that gravity is an illusion, since perhaps we can always describe it with a non-gravitational dual (if holography is to be taken seriously), but I’m not sure whether that’s anything more than a matter of definition or semantics (in the sense that you could take lessons from GR and say that by the equivalence principle gravity is clearly already an illusion anyway…).
There are limitations in our current understanding how to extract what strings tell us about gravity (especially quantum gravity at strong coupling), but we’ve learned a lot, and we’re set to learn a lot more, I’m sure.
Cheers,
-cvj
Dear Clifford,
I have a question about the relation between string theory and classical GR I was hoping you’d entertain. We can interpret classical GR as a theory describing the curvature of spacetime in the presence of matter and how curved spacetime influences the motion of test bodies. We also know that the distribution of matter throughout the universe influences the “shape” of the spacetime manifold. However, we can also interpret the linearized Einstein equations as describing the theory of a massless spin-2 field. Lorentz invariance of the theory leads to Einstein’s equations. So far, the only consist quantum theory of a massless spin-2 field comes from string theory.
Does this mean that, in string theory, there is no notion of “curved spacetime” and that the notion of spacetime curvature is just the geometric interpretation of the classical theory which string theory does not adopt? Or is it this true only for perturbative string theory? I know the strings can propagate in curved and warped backgrounds (after all, the C-Y manifolds are supposed to be curled up and, presumably, curved). So I guess my question is: Should we expect the notion of a curved and warped spacetime manifold to always hold, or is it all supposed to be just strings and branes which, on a classical level, produces the illusion of curvature?
Thanks a lot and I hope this is an okay question to ask here.
Fred
Yes… First of all read a good solid history with technical notes. Pais’ “Inward Bound”, and Pais’ “Subtle is the Lord” are two of the finest. Then get a good concept-driven book like Zee’s “Quantum Field Theory in a Nutshell”, and a good gentle geometry book Nash and Sen. Absorb those (and others like them, and bits of the books you’ll naturally be led to in order to follow up on details) and you’ll be set.
Those aren’t the only choices, of course, but a start somewhere is good. A way into the building, as it were. Once you’re there, you’ll find your own way around…. tackling one or two specific books and being disciplined about sticking with the story for a good way will open you up to ideas and techniques that will then allow you to read lots of other books on diverse topics.
Enjoy!
-cvj
Thanks for reposting this, it’s very interesting to get a look at some modern research.
I have a slightly unrelated question. I am a physics student (still undergrad) and intensely interested in theoretical physics. I have supplemented my undergrad physics education by reading the Feynman Lectures along with other material and generally trying to get a good intuition from a number of different sources.
I’m wondering how to adapt this approach to more advanced physics. I’m aware that there’s a fair amount of books for the general public about string theory and advanced quantum theory, but I’m interested in somewhat more technical literature. I certainly wouldn’t expect to understand it completely, or even very well, but I’ve found that seeking exposure to intriguing ideas in physics before I encounter them in the classroom has helped me get more out of class and not be put off by slow-moving curricula. Is there any literature out there that you would suggest off the top of your head for somewhat technical introduction to fairly advanced physics?