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.
In part b, one is asked to
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.
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| 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.
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