One year in Edinburgh, string amplitudes and the KLT relations

My second semester at the MSc in Mathematical Physics at the University of Edinburgh is almost done (2 weeks) and I'm getting ready for examinations. The assessment scheme is a pretty tough one: I've attended twelve courses since September -some of which ended up being much harder and workload intense than I thought (e.g. linear analysis, advanced QFT, QCD, Lie groups/algebras, standard model)- and 9 of my examinations will count between 80% and 100% towards my final mark; furthermore, I need an average of above 50% with no more than 3 marks below 50% in order to get the degree.

So yes, it's crazy. And I shouldn't be wasting time writing here I guess, but at least I've started by (kind of) blocking my access to facebook, which is a huge distraction.

It's a bit ironic, but throughout the year I became uninspired to follow the path of mathematical physics and even the academic one (completely the opposite to what I expected). The reasons are varied but I guess I can just put it simply as a feeling of not belonging. It's not that I suddenly found that I don't like mathematical physics, I obviously love it, but considering it a full time job in Academia is a somewhat different matter. I'm still considering the pursuit of a PhD in Quantum Information, though, but I think I'll narrow my options to the U.S/Canada and Mexico, if I decide otherwise, God knows (whatever that means) what I'll end up doing.

In other news, one of the courses I'm taking concerns giving two 25 minute talks related to the dissertation topic that one has chosen. My dissertation will be about the so-called colour-kinematics duality: to design and implement Mathematica code which creates the relevant (loop-level) diagrams, determines the Jacobi relations among them, and allows for these objects to be manipulated.

There is a whole other world -unknown to me until now- about this and the most surprising aspect of this colour-kinematics duality is that it enables to obtain a gravity theory scattering amplitudes very easily as (prop to) the square of a gauge theory amplitudes. This then connects with another set of relations between scattering amplitudes in string theory known as the KLT-relations, which so happen to reduce to this gravity = gauge theory squared relations in the low-energy (point-like) limit.

There are two very nice popular-science level stories about this:

• Gravity is Yang-Mills Squared at the blog 4 gravitons
• Betting on the Future of Quantum Gravity at Quanta Magazine

What I did for my first presentation was to talk about string amplitudes and KLT: what does these things mean and how to interpret what the KLT relations are saying. It was a bit of a detour because I won't really be working with string theory, however I wanted to find out at least the basics of where this thing come from. For anyone who knows string theory well, I didn't say anything new and most of the slides are contained merely in the introduction of Superstring Theory by Witten, Green and Schwarz.

For anyone really serious about the topic, this paper looks like a really good review:

Perturbative Gravity and Gauge Theory Relations -- A Review
Thomas Sondergaard
arXiv:1106.0033 [hep-th]

In my second talk I'll look in a bit of a detail (as time allows me to prepare) to colour-kinematics (BCJ duality, the Kleiss-Kuijf relations, the double copy and other such words) and probably I'll share something about it here.

These are my slides about string amplitudes and the KLT relations (handout mode):

The Dirac action for p-branes from the Polyakov-type action

I still have around half a year off until I start attending my Master's courses (at least if everything, e.g. funding, etc... comes ok), so now I'm trying to follow the String Theory lectures by David Tong. As with the QFT ones, these too are great as an introductory reading at a graduate level or for self-study. Solutions to the example sheets are available from a PhD student named Chris Blair; in particular, here for the first one. What I'm concerned about is problem 2.

The p-brane of me got the joke a little late
(hurdles of a non-native speaker)

In part b, one is asked to

Show that the Dirac action
\begin{equation}S_\text{Dirac}=-T\int{d}^{p+1}\sigma\sqrt{-\det\gamma}\end{equation} where $\sigma^\alpha$, $\alpha=0,\ldots,p$ are coordinates on the brane world-volume and $\gamma_{\alpha\beta}$ is the pullback of the Minkowski metric onto the brane,
\begin{equation}\gamma_{\alpha\beta}=\frac{\p{X}^\mu}{\p\sigma^\alpha}\frac{\p{X}^\nu}{\p\sigma^\beta}\eta_{\mu\nu}\end{equation} is equivalent to the Polyakov-type action with dynamical world-volume metric $g_{\alpha\beta}$,
\begin{equation}S=-\frac{T}{2}\int{d}^{p+1}\sigma\sqrt{-g}\left(g^{\alpha\beta}\p_\alpha{X}^\mu\p_\beta{X}^\nu\eta_{\mu\nu}-(p-1)\right)\label{poly1}\end{equation}

Now, the thing is that in the solution by Blair, he starts off with (\ref{poly1}) replacing the $g$'s by $\gamma$'s, which doesn't seem to be what is asked for. This confused me badly, but finally I managed to see what's going on.

What needs to be done is similar to what is done in part a of the problem. As with the string case, $g_{\alpha\beta}$ is now a new field that is fixed by its own equations of motion. First, let's write (\ref{poly1}) as
\begin{equation}S=-\frac{T}{2}\int{d}^{p+1}\sigma\sqrt{-g}\left(g^{\alpha\beta}\gamma_{\alpha\beta}-(p-1)\right)\label{poly12}\end{equation} so that it can be handled more easily (also, from here it's evident that Blair's approach does not solves the problem). To get the equations of motion for $g_{\alpha\beta}$, we should vary the action, where it's needed the fact that
\begin{equation}\delta\sqrt{-g}=-\frac{1}{2}\sqrt{-g}\,g_{\alpha\beta}\delta{g}^{\alpha\beta}\end{equation} which can be obtained through
\begin{equation}\delta\det{M}=\det{M}\,\text{Tr}\left(M^{-1}\delta{M}\right)\end{equation} namely, Jacobi's formula for invertible $M$ (which here is $g_{\alpha\beta}$, i.e. $\delta{g}=g\,g^{\alpha\beta}\delta{g}_{\alpha\beta}$). Using this, and stationary action $\delta{S}=0$, the equations of motion are
\begin{equation}-\frac{1}{2}g_{\alpha\beta}\left(g^{\rho\sigma}\gamma_{\rho\sigma}-(p-1)\right)+\gamma_{\alpha\beta}=0\end{equation} so that $g_{\alpha\beta}$ is related to $\gamma_{\alpha\beta}$ as
\begin{equation}g_{\alpha\beta}=\frac{2\gamma_{\alpha\beta}}{g^{\rho\sigma}\gamma_{\rho\sigma}-(p-1)}\label{poly2}\end{equation} which is basically eq. (1.25) in the lecture notes for $p=1$, as expected. Now, to finally recover the Dirac action from here, just take the determinant on both sides of (\ref{poly2}), so that
\begin{equation}\sqrt{-g}=\frac{2\sqrt{-\gamma}}{g^{\rho\sigma}\gamma_{\rho\sigma}-(p-1)}\end{equation} and as the $\rho$ and $\sigma$ indices in the denominator are dummy, inserting this in (\ref{poly12}),
\begin{equation}S\bigg|_{\text{EOM}}=-T\int{d}^{p+1}\sigma\sqrt{-\det\gamma}=S_\text{Dirac}\end{equation} which is the actual desired result. Now, of course if $g_{\mu\nu}=\gamma_{\mu\nu}$ the result follows as the conformal factor is simply equal to one and it can be readily used that $\gamma^{\mu\nu}\gamma_{\mu\nu}=p+1$ on (\ref{poly12}) to get the result; however, that apparent solution can get misleading because one can think that the Dirac action is recovered only when $g_{\mu\nu}=\gamma_{\mu\nu}$. The relevant thing of this example is to see that once this new field satisfy it's equations of motion, the original Dirac (or Nambu-Goto) action is recovered. Also one should be able to see what gauge symmetry this new field brings upon, as the equations of motion for $X^\mu$ are the same: for the string it is Weyl invariance, however, I couldn't find the case for the $p$-brane but I guess it should be an analogous conformal symmetry (for now I can only say in page 58 it's stated that a $U(1)$ gauge field $A_\mu$ lives on a D-brane, however, the group language still seems a bit foreign to me ;-)

It seems funny, but in the end all of (super)string theory boils down (the history is quite interesting, from scattering amplitudes to the superstring revolutions to today) to the naive step from relativistic points to strings to branes / worldline to worldsheet to worldvolume.