*(A relatively technical post follows.)*

So imagine the following:

*You’re walking along the street, minding your own business, and somebody walks up to you and tries to sell you a string theory. So you stop and examine the goods, since you’re in the market for string theories, on the lookout for any that might be novel, useful, bright, or shiny, etc. You never know when one or other property might come in useful. *

Question: How do you know that it is a string theory? Let me be sure to point out that it comes with a lot of the defining path integral done for you. In other words, you don’t have to do the integral over string world sheet metrics and world sheet fields. This was done in the factory for you. What you have access to are parameters such as the coefficients of the operators in the theory, and you can also adjust the value of the string coupling.

So a lot of the stuff you would recognize as a string theory in your typical string theorist’s notebook have been cleaned up. They’ve been integrated over. The observable physics actually never cared about them (the technical details of summing over metrics – slicing up the moduli space of inequivalent metrics properly at each genus, etc etc…. all done), assuming you’ve done the integrals properly. The factory did it all for you.

So what criteria do you use to decide that it is a string theory at all? Actually, this is not an idle question. Think about the issue in the context of trying to understand some phenomenon or phenomena in Nature. How would you know you had a string theory description underlying the physics?

Well, what we might start doing at this point is start listing various things we’ve learned about strings that we think are rather spiffy about the theory that make them different from what we’ve seen before. I’m sure you have your favourites.

A word or more of caution though. From my new paper:

With a few notable (and highly instructive) examples in D â‰¤ 2, string theory still lacks a satisfactory and wellâ€“understood nonâ€“perturbative definition. It is fair to say that while strings have marvellous properties that may prove a great boon for studying Nature, we have not been learning about these properties systematically, but instead by following the theory into regimes which have become accessible to us by various techniques. As a result, it is not clear what the big picture is â€”certainly not clear is the complete list of phenomena we should expect from string theory.

My point? Do we really know enough about what string thory is to decide when to rule out something as being stringy or not. How do you know when to hand over the cash to the person on the street trying to sell you one?

Ok, so you’re thinking: What’s he getting at? What’s in this paper?

Well, one of the things of which we are all very proud -that we show off at theorist parties to all our theorist friends from other fields, you’ll admit- are branes, right? We’ve spent a huge amount of effort on them in the last decade or so expecially, and we’ve done a lot with them… everything from understanding the microphysics of black holes, understanding profound conenctions between gravity and gauge theory, modeling important phenomena such as confinement in strongly coupled gauge theory, and even constructing potential phenomenological scenarios where our *entire universe* is a brane living in higher dimensions. Branes are just great. In fact, since branes were properly understood, we know that string theory is not really a theory of strings at all.

So you’d probably list branes high up on the list of objects you’d want to see in a string theory. If you’re really tring to impress, you’d probably quote the classic paper of Shenker’s that connects the presence of branes to the behaviour of string perturbation theory. Perturbation theory is an asymptotic series, and there are corrections that are exponentially small in the inverse string coupling, g, that are smaller than any order in a perturbative expansion in small g. Shenker showed how to connect those exp(-1/g) effects to the growth of string perturbation theory. We now see that those effects are associated to D-branes that have tension/action proportional to 1/g, and so they would seem to be pretty darned fundamental things in what we might call string theories.

In the paper, entitled “Strings Without Branes” (I know the title will raise giggles if it turns out to be wrong, but fortune favours the brave) I ask:

In the light of this, one may ask the simple question: Is it possible that one can have a string theory without these features? What kind of properties would the theory have to possess, and if it has those properties, does it still qualify to be considered as a string theory?

Specifically, what if the perturbative expansion was not an asymptotic expansion? What if you could resum it exactly and get all the information about the physics? Would you still call it a string theory?

Well, here’s part of the abstract:

We present a class of solvable models which resemble string theories in many respects but have a strikingly different nonâ€“perturbative sector. In particular, there are no exponentially small contributions, which normally are associated with branes and related objects. Perturbation theory is no longer an asymptotic expansion, and so can be completely resummed to yield all the non-perturbative physics.

We examine a number of other properties of the theories, for example constructing and examining the physics of loop operators, which can be computed exactly in all models, and gain considerable understanding of the difference between these new theories and the more familiar ones, including the possibility of how to interpolate between the two types.

I began wondering about this issue (i.e., How do you recognise a string theory if it walked up and shook you by the hand?) a while ago, off and on, and then again this year in earnest. Two weeks in the monastery I referred to previously helped me crystallize some of the thoughts into a specific narrative, with the specific examples presented, and I’ve been chipping away at some of these thoughts and computations ever since then (including at the other monastery), resulting in the (long overdue) paper appearing tonight.

Before reading the paper -if interested- you can quickly refresh your memory of some of the context and technology I use by reading this post, and the papers referenced within. Why? Because:

The theories are defined naturally by starting with models that yield wellâ€“understood string theories and allowing a fluxâ€“like background parameter to take halfâ€“integer rather than integer values.

A lot of people working in strings today are not familiar with the context, being trained in the more fashionable issues of the day. If you’re one, don’t let that put you off: I don’t write things so that you need to know any of the details of the background from where this technology arises. I explain pretty much all you need, going quite slowly and carefully through the most unfamiliar bits, I hope. It is all pretty simple, so don’t assume you need to swot up on prehistoric matters from the late 80s and early 90s to get the essence of the story.

(For the mathematically minded among you, a lot of rather tasty structures that show up, in the context of exact solutions of non-linear diffential equations. Properties of the Painleve equations (I and II) are involved, and natural generalizations of structures that might be familiar from the classic literature on these objects. And there’s more…)

One interesting point here is that the models I construct are very closely related to models that are known to be stringy, and so I argue that they deserve to be thought of as string theories too. They might just be rather curious toys that have no application, but I’m not so sure. I think that these simple non-critical strings will have applications in a wide variety of applications -perhaps as effective models of subsectors of physics of a larger scenario. (See for example a recent paper of Caporaso et. al., in a topological string context.) And more directly, since I’ve noticed that you can phrase the difference between the “traditional” stringy models and the “non-traditional” ones in terms of the presence, at least formally, of a background field, what if our more “grown up” string theories can be driven to a phase where they have this sort of behaviour? Perhaps we ought to be on the lookout for this type of phenomenon all over string theory.

-cvj

### On this day on Asymptotia...

- I Dare! - 2014
- Fail Lab 6: Pyrotechnics! - 2013
- Bus Moment - 2011
- Another Friday, Another President - 2010
- Sketches while Zonked - 2009
- Summer Reading: Fresh Air From Pollan - 2008

Cool! And completely over my head, but let’s see if I got this much:

1. Because they involve branes, grown-up string theories have series that can’t be summed, and therefore problems with no closed-form solutions?

2. Your baby theories are similar to string theories, but have no branes (

giggle), and therefore guarantee closed-form solutions to everything?Hi Aaron… Kind of… but be careful: Having an asymptotic series in perturbation theory is not a problem. It just means you have to do more than perturbation theory, generally. It tells you that your theory is quite rich, in fact. This is true in field theory too, generically. It is also common in even some of the simplest quantum mechanical systems one can think of.

Cheers,

-cvj

I saw this paper on the ArXiv this morning and am looking forward to reading it. Hopefully I am not alone in recognizing the irony here—the man who taught us about D branes now says that they’re not altogether necessary. Thank you for a very readable post Dr. Johnson.

Hmm… so if asymptotic series aren’t all that bad, and don’t prevent you from coming up with usable answers, what’s the attraction of being able to calculate things exactly?

p.s. When you say “exactly,” I’ve been assuming you mean in closed form, with no series to sum or anything. Is that how most people actually use the term?

It is not an attraction per se….. I make an observation that it is possible, and that it means something about the physics. It does not mean that not being able to do it is bad. Far from it… it is very good. Consider the degenerate two-well system in QM. Then do more wells…. Then do an infinite number.

Thanks,

-cvj

Actually, do all the papers on string theory agree nowadays to the need of having “inside the box” a Lagrangian corresponding to a relativistic string as physics understands it? Sometimes, specially with membranes and with topology stuff, I am under the impresion that any, say, “theory of one dimensional geometric segments” is considered to be a theory of strings.

Hello Clifford,

I like the fact that scientists sense the emergence of response necessary, for actions initiated from their fellow scientists(Lee Smolin).

This is the kind of debate that has been long hoped for as I watched and looked for such situations to occur, for they were most enlightening factors too, “what use this subject” even to the lay person.

So one had to indeed look for such processes “in physics” to compare any “model assumption” to help push perspective forward?

Let me get this straight, or at least twisted in a knot. If a string theory hasn’t got a brane, then it’s a straw-man argument.

[...] The first equation I shall try is the following (for more on unpacking this equation and its meaning, see this post and links therein): [...]

\left[ \begin {array}{c} - 1.0\, \left( - 1.0\,\beta+{\frac {m \left( m\beta+\sqrt { 2.0\,{m}^{2}\alpha\,J+ 2.0\,\alpha\,{J}^{2}- 1.0\,J{\beta}^{2}} \right) }{{m}^{2}+J}} \right) {J}^{-1}\\\noalign{\medskip}- 1.0\, \left( - 1.0\,\beta+{\frac {m \left( m\beta- 1.0\,\sqrt { 2.0\,{m}^{2}\alpha\,J+ 2.0\,\alpha\,{J}^{2}- 1.0\,J{\beta}^{2}} \right) }{{m}^{2}+J}} \right) {J}^{-1}\end {array} \right]