Why are D-Branes Non-Perturbative?

It’s frequently said that D-branes are non-pertubative objects. In other words, you can’t learn about them by doing a series expansion in the string coupling $g$. That’s because the DBI action which encodes the dynamics of D-branes couples to the dilaton field via a term $e^{-\phi}$. Now recall that the dilaton VEV yields the string coupling and Bob’s your uncle!

But there’s a more subtle point at work here. What determines the coupling term $e^{-\phi}$? For this, we must remember that D-brane dynamics may equivalently be viewed from an open string viewpoint. To get a feel for the DBI action, we can look at the low energy effective action of open strings. Lo and behold we find our promised factor of $e^{-\phi}$.

Yet erudite readers will know that the story doesn’t end there. Recall that D-brane actions are blessed with conformal invariance. In particular the theory is unchanged up to Weyl rescaling. This means that we can change frame to eliminate the pesky $e^{-\phi}$. From our new perspective, D-branes aren’t non-perturbative any more!

There’s a price to pay for this, and it’s a steep one. It turns out that this dual description turns strings into non-perturbative objects. This is quite unhelpful, since we know a fair amount about the perturbative behaviour of fundamental strings. So most people stick with the “string frame” in which D-branes are immune to the charms of perturbation theory.

Thanks to Felix Rudolph for an enlightening discussion and for bringing to my attention Tomas Ortin’s excellent volume on Gravity and Strings.

Renormalization and Super Yang Mills Theory

It’s well known that $\mathcal{N}=4$ super Yang-Mills theory is perturbatively finite. This means that there’s no need to introduce a regulating cutoff to get sensible answers for scattering amplitude computations. In particular the $\beta$ and $\gamma$ functions for the theory vanish.

Recall that the $\gamma$ function tells us about the anomalous dimensions of elementary fields. More specifically, if $\phi$ is some field appearing in the Lagrangian, it must be rescaled to $Z \phi$ during renormalization. The $\gamma$ function then satisfies

$\displaystyle \gamma(g)=\frac{1}{2}\mu\frac{d}{d\mu}\log Z(g,\mu)$

where $g$ is the coupling and $\mu$ the renormalization scale. At a fixed point $g_*$ of the renormalization group flow, it can be shown that $\gamma(g_*)$ exactly encodes the difference between the classical dimension of $\phi$ and it’s quantum scaling dimension.

Thankfully we can replace all that dense technical detail with the simple picture of a river above. This represents the space of all possible theories, and the mass scale $\mu$ takes the place of usual time evolution. An elementary field operator travelling downstream will experience a change in scaling dimension. If it happens to get drawn into the fixed point in the middle of the whirlpool(!) the anomaly will exactly be encoded by the $\gamma$ function.

For our beloved $\mathcal{N}=4$ though the river doesn’t flow at all. The theory just lives in one spot all the time, so the elementary field operators just keep their simple, classical dimensions forever!

But there’s a subtle twist in the tale, when you start considering composite operators. These are built up as products of known objects. Naively you might expect that these don’t get renormalized either, but there you would be wrong!

So what’s the problem? Well, we know that propagators have short distance singularities when their separation becomes small. To get sensible answers for the expectation value of composite operators we must regulate these. And that brings back the pesky problem of renormalization with a vengeance.

The punchline is the although $\mathcal{N}=4$ is finite, the full spectrum of primary operators does contain some with non-trivial scaling dimensions. And that’s just as well really, because otherwise the AdS/CFT correspondence wouldn’t be quite as interesting!

Physics Through the Looking Glass

I recently took part in the popular Three Minute Thesis competition. Each contestant gets just 3 minutes to explain their research to a panel of laymen. Although I didn’t make it to the national finals, it was nevertheless great fun.

Here’s an audio recording of my speech, taken live at the QMUL finals. For the experts among you, I’m giving an account of my attempts to use twistor techniques to investigate subleading soft theorems in gauge theory and gravity!

Elsevier Journals – The QMUL Figure

A few weeks ago I reblogged Tim Gowers’ post about the cost of Elsevier journals. I noticed that my own institution (QMUL) had deflected his Freedom of Information request. Curious to learn more, I did some digging.

It turns out that QMUL paid a total of £454,422.44 to Elsevier for the academic year 2013/14. Interestingly this is more than Exeter and York, who also joined the Russell Group recently. However it’s still much cheaper than the bill Cambridge, UCL, Imperial or Edinburgh face.

Unfortunately QMUL weren’t able to provide any further breakdown of the figures. Apparently this information isn’t available to the university, which seems like a very odd way of doing business. I think it likely that the vast majority of the cost is the subscription fee.

I should point out that QMUL and Cambridge certainly have differentiated access to Elsevier journals. For example QMUL Library does not have access to Science Direct papers before the early 1990s. Cambridge University Library has universal access to this material.

However with all the smoke and mirrors in this story, it’s impossible to turn this anecdotal evidence into an accurate account of Elsevier’s charging policy. There’s clearly a need for much greater transparency.

Below is a transcript of the email I received from the QMUL FOI Department. I owe a debt of gratitude for their help.

Dear Edward Hughes

Thank you for your email of 25th April requesting information about spend on Elsevier journals at Queen Mary University of London.

The total amount paid to Elsevier for 2013/14 was £545,306.93 (inclusive of £90,884.49 VAT). We do not hold any further break down.

If you are dissatisfied with this response, you may ask QMUL to conduct a review of this decision.  To do this, please contact the College in writing (including by fax, letter or email), describe the original request, explain your grounds for dissatisfaction, and include an address for correspondence.  You have 40 working days from receipt of this communication to submit a review request.  When the review process has been completed, if you are still dissatisfied, you may ask the Information Commissioner to intervene. Please see www.ico.org.uk for details.

Yours sincerely

Paul Smallcombe

Records & Information Compliance Manager

Scattering Without Scale, Or The S-Matrix In N=4

My research focuses on an unrealistic theory called massless $\mathcal{N}=4$ super Yang-Mills (SYM). This sounds pretty pointless, at least at first. But actually this model shares many features with more complete accounts of reality.  So it’s not all pie in the sky.

The reason I look at  SYM is because it contains lots of symmetry. This simplifies matters a lot. Studying SYM is like going to an adventure playground – you can still have great fun climbing and jumping, but it’s a lot safer than roaming out into a nearby forest.

Famously SYM has a conformal symmetry. Roughly speaking, this means that the theory looks the same at every length scale. (Whether conformal symmetry is equivalent to scale invariance is a hot topic, in fact)! Put another way, SYM has no real notion of length. I told you it was unrealistic.

This is a bit unfortunate for me, because I’d like to use SYM to think about particle scattering. To understand the problem, you need to know what I want to calculate. The official name for this quantity is the S-matrix.

The jargon is quite straightforward. “S” just stands for scattering. The “matrix” part tells you that this quantity encodes many possible scattering outcomes. To get an S-matrix, you have to assume you scatter particles from far away. That’s certainly the case in big particle accelerators – the LHC is huge compared to a proton!

But remember I said that SYM doesn’t have a length scale. So really you can’t get an S-matrix. And without an S-matrix, you can’t say anything about particle scattering. Things aren’t looking good.

Fortunately all is not lost. You can try to define an S-matrix using the usual techniques that worked in normal theories. All the calculations go through fine, unless there are any low energy particles around. Any of these so-called soft particles will cause your S-matrix to blow up to infinity!

But hey, we should expect our S-matrix to be badly behaved. After all, we’ve chosen a theory without a sense of scale! These irritating infinities go by the name of infrared divergences. Thankfully there’s a systematic way of eliminating them.

Remember that I said our SYM theory is massless. All the particles are like photons, constantly whizzing about that the speed of light. If you were a photon, life would be very dull. That’s because you’d move so fast through space you couldn’t move through time. This means that essentially our massless particles have no way of knowing about distances.

Viewed from this perspective it’s intuitive that this lack of mass yields the conformal symmetry. We can remove the troublesome divergences by destroying the conformal symmetry. We do this in a controlled way by giving some particles a small mass.

Technically our theory is now called Coulomb branch SYM. Who’s Coulomb, I hear you cry? He’s the bloke who developed electrostatics 250 years ago. And why’s he cropped up now? Because when we dispense with conformal symmetry, we’re left with some symmetries that match those of electromagnetism.

In Coulomb branch SYM it’s perfectly fine to define an S-matrix! You get sensible answers from all your calculations. Now imagine we try to recover our original theory by decreasing all masses to zero. Looking closely at the S-matrix, we see it split into two pieces – finite and infinite. Just ignore the infinite bit, and you’ve managed to extract useful scattering data for the original conformal theory!

You might think I’m a bit blasé in throwing away these divergences. But this is actually well-motivated physically. The reason is that such infinities cancel in any measurable quantity. You could say that they only appear in the first place because you’re doing the wrong sum!

This perspective has been formalized for the realistic theories as the KLN theorem. It may even be possible to get a rigorous version for our beloved massless $\mathcal{N}=4$ SYM.

So next time somebody tells you that you can’t do scattering in a conformal theory, you can explain why they’re wrong! Okay, I grant you, that’s an unlikely pub conversation. But stranger things have happened.

And if you’re planning to grab a pint soon, make it a scientific one!

Elsevier journals — some facts

Excellent article uncovering the problems of journal subscription.

I’ll try chatting to the relevant people here at QMUL and see whether I can dig deeper into the legal implications threatened by Elsevier. Will let you know what I find.

Originally posted on Gowers's Weblog:

A little over two years ago, the Cost of Knowledge boycott of Elsevier journals began. Initially, it seemed to be highly successful, with the number of signatories rapidly reaching 10,000 and including some very high-profile researchers, and Elsevier making a number of concessions, such as dropping support for the Research Works Act and making papers over four years old from several mathematics journals freely available online. It has also contributed to an increased awareness of the issues related to high journal prices and the locking up of articles behind paywalls.

However, it is possible to take a more pessimistic view. There were rumblings from the editorial boards of some Elsevier journals, but in the end, while a few individual members of those boards resigned, no board took the more radical step of resigning en masse and setting up with a different publisher under a new name (as some journals have…

View original 10,726 more words

The Amplituhedron – Now In Poster Form!

Recently there’s been a lot of excitement about the amplituhedron. This wacky new object might just revolutionize our understanding of particle scattering. To help you get your head round it, here’s a poster I made for an outreach competition.

Click for the full size JPEG file. If you want a hard copy to impress your friends at home, just drop me a comment! I’ve got a few spares, which I’ll happily put in the mail.

By the way, check out The Naked Scientists podcasts – great info and entertainment for the daily commute!

Musings Mathematical and Otherwise

christopherhughfrancisbaird

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Conversations About Science with Theoretical Physicist Matt Strassler

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4 gravitons

The trials and tribulations of four gravitons and a postdoc

Eventually Almost Everywhere

A blog about probability by Dominic Yeo