# A Second Course in String Theory

I’ve been lucky enough to have funding from SEPnet to create a new lecture course recently, following on from David Tong’s famous Part III course on string theory. The notes are intended for the beginning PhD student, bridging the gap between Masters level and the daunting initial encounters with academic seminars. If you’re a more experienced journeyman, I hope they’ll provide a useful reminder of certain terminology. Remember what twist is? What does a D9-brane couple to? Curious about the scattering equations? Now you can satisfy your inner quizmaster!

Here’s a link to the notes. Comments, questions and corrections are more than welcome.

Thanks are particularly due to Andy O’Bannon for his advice and support throughout the project.

# 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!