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 in theories of gravity, we can change the metric by altering our frame of reference. In particular we can effect a Weyl rescaling 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.

Thanks to David Berman, for pointing out my ridiculous and erroneous claim that D-branes are conformally invariant. My mistake is utterly obvious when you recall the Polyakov action for D-branes, namely

\displaystyle S[X,\gamma]= \int d^{p+1}\xi \sqrt{|\gamma|}\left(\gamma^{ij}g_{ij}+(1-p)\right)

where \xi are worldsheet coordinates, \gamma is the worldsheet metric, and g is the pull-back of the spacetime metric. Clearly conformal invariance is violated unless p=1.

From this perspective it’s even more remarkable that one gets conformally invariant \mathcal{N}=4 SYM in 4D from the low energy action of a stack of D-branes. This dimensional conspiracy is lucidly unravelled on page 192 of David Tong’s notes. But even for a 3-brane, the higher derivative operators in the \alpha' expansion ruin conformal invariance.

Incidentally, the lack of conformal invariance is a key reason why D-branes remain so mysterious. When we quantize strings, conformal invariance is enormously helpful. Without this crutch, quantization of D-branes becomes unpleasantly technical, hence our lack of knowledge!

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!