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\begin{document}



\begin{center}

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics

String Theory (8.821) -- Prof. J. McGreevy -- Fall 2007


\vskip.3in
\begin{large}
{ \bf Problem Set 1}

{\it Classical Worldsheet Dynamics}
\end{large}
\vskip.2in
{\bf Reading:} GSW \S2.1, Polchinski \S1.2-1.4.  Try \S3.2-3.3.

\vskip.2in
{\bf Due:} Friday, September 21, 2007 at 3 PM, in the 8.821 lockbox.


\end{center}


\begin{enumerate}




\item {\bf Weyl symmetry.}\footnote{Problems 1-4 are due to 
Marty Halpern.}
are 
A (global) Weyl transformation acts on a field theory coupled to gravity 
by 
$$F(x) \mapsto \lambda^{-W(F)} F(x) $$
where $F$ is a field, and $W(F)$ is a number called
the {\it Weyl weight} of $F$.  
$W$ is normalized so that $W(g_{\mu\nu}) = 2 $.

(a) Find $W(\varphi)$ so that 
$$ S = \int d^D x \sqrt g \half g^{\mu\nu} \del_\mu \varphi \del_\nu \varphi $$
is Weyl invariant.

(b) Consider coordinate transformations of the form
$ x \mapsto x' = \lambda x ,$
which can be called Einstein-scale transformations.
How do $\varphi$ and $g_{\mu\nu}$ transform?

(c) Write down the transformation of $\varphi$ and $g_{\mu\nu}$ 
under simultaneous Weyl and Einstein-scale transformations.
How does the result for $\varphi$ relate to 
the engineering dimension you would assign $\varphi$
in order to make $S$ dimensionless?


\item {\bf Local Weyl Invariance.}
Consider a gravity plus matter system in $D$ dimensions
$$ 
S = S_{grav}[g_{\mu\nu}] + S_{matter}[F, g_{\mu\nu}]
$$
where $S_{matter}$ is invariant under the {\it local} Weyl transformation
$$
F(x) \mapsto \lambda(x)^{-W[F]}F(x), ~~~g_{\mu\nu}(x) 
\mapsto \lambda^{-2}(x) g_{\mu\nu}(x).
$$
Using the equations of motion, show that 
the matter stress tensor
$$ T^{\mu\nu} = -{2\over  \sqrt{g}} {\delta S_{matter} \over \delta
g_{\mu\nu}} $$
is traceless:
$$T^{\mu\nu}g_{\mu\nu} = 0 .$$

%In the flat-space limit, the matter stress tensor is traceless.


\item
{\bf Symmetries of the Polyakov action.}
Recall the Polyakov action
$$ S_{P} = - {1\over 4 \pi \alpha'} \int d^2 \sigma 
\sqrt{-\gamma} \gamma^{ab} \del_a X^\mu \del_b X_\mu. $$
Think of this as the action for a system of matter ($X$) coupled
to 2d gravity ($\gamma_{ab}$).

(a) Show that the Polyakov action is locally Weyl-invariant.

(b) Obtain an expression for the stress tensor
$$ T_{ab} \equiv -{2\over  \sqrt{\gamma}} {\delta S_P \over \delta \gamma^{ab}} 
$$
and check that it is traceless.
[You may want to use the relation  
$$ \delta ( \det A)= \det A ~ \tr (A^{-1} \delta A ) ,$$
where $A$ is any nonsingular matrix, and $\delta A$ is any small deformation.]


(c) Use the equation of motion for $\gamma_{ab}$ 
to show that when evaluated on a solution
$$ S_P[ \gamma|_{EOM}, X] = S_{Nambu-Goto}[X]. $$

\item {\bf Virasoro algebra (no central extension yet).}
Consider the classical mechanics of $D$ two-dimensional 
free bosons $X^\mu$,
which enjoy the canonical Poisson bracket
$$ \{ X^\mu(\sigma) , P^\mu(\sigma' ) \} = 
\eta^{\mu\nu} \delta(\sigma - \sigma')$$
with their canonical momenta $P^\mu$.
Show that the objects
%$$ L_\pm \equiv \pm {\pi \alpha^\prime \over 2} W_{\pm}^2, 
$$ T_{\pm\pm} \equiv \pm {\pi \alpha^\prime \over 2} W_{\pm}^2, 
~~ W_\pm^\mu \equiv
%\left( 
P^\mu \pm { \del_\sigma X^\mu \over 2\pi \alpha^\prime} 
%\right)
$$
satisfy two commuting copies of the Virasoro algebra
%$$ \{ L_\pm(\sigma) , L_\pm (\sigma' \} = ( L_\pm(\sigma) 
%+ L_\pm (\sigma') ) \del_\sigma \delta(\sigma - \sigma') $$
%$$ \{ L_\pm(\sigma), L_\mp (\sigma') \}  = 0 . $$
$$ \{ T_{\pm\pm}(\sigma) , T_{\pm\pm} (\sigma' \} = ( T_{\pm\pm}(\sigma) 
+ T_{\pm\pm} (\sigma') ) \del_\sigma \delta(\sigma - \sigma') $$
$$ \{ T_{\pm\pm}(\sigma), T_{\mp\mp} (\sigma') \}  = 0 . $$
You may use the identity
$$ f(\sigma) f(\sigma') \del_\sigma  \delta(\sigma - \sigma') 
= \half(f^2 (\sigma) + f^2(\sigma') ) \del_\sigma \delta(\sigma -\sigma') , 
\forall f .$$
[This problem may be an opportunity to apply the Tedious Problem Rule
discussed in class.]



\item {\bf Spinning strings in AdS.}
[This problem might be hard.\footnote{It is taken
from a recent paper.  For privacy's sake, let's call the authors
Steve G., Igor K, and Alexandre P.  No, that's too obvious.
Uhh...
let's say
S. Gubser, I. Klebanov and A. Polyakov.}]
In this problem we are going to think about 
the behavior 
of a string propagating in 5d anti de Sitter space (AdS).  
Specifically, we are going to study 
and use some of its conserved charges.
When these conserved quantities are large,
they can be compared to their counterparts in
a dual 4d gauge theory.

(a) Write down the Nambu-Goto
action for a bosonic string propagating 
in $AdS_5$, whose metric (in so-called global coordinates) is 
$$ ds^2 = { R^2( -\cosh^2 \rho dt^2 + d\rho^2 +  \sinh^2\rho d\Omega_3^2 ) } 
\equiv
G_{ij}dX^i dX^j~~~~(1)$$
where $ d\Omega_3^2$ denotes the line element on the unit
three-sphere,
$$ d\Omega_3^2 = d\theta_1^2 + \sin^2 \theta_1 
( d\theta_2^2 + \sin^2 \theta_2 d\phi^2 ) . $$



(b) 
The AdS background has many isometries. 
We will focus on two:
shifts of $t$ (the energy)
and shifts of $\phi$ (the spin).
These isometries of the target space
are symmetries of the NLSM, 
and therefore lead to conserved charges.
Using the Noether method,
write an expression for the conserved charge $S$ which follows from
the symmetry $\phi \to \phi + \epsilon$, with $\epsilon$ a constant.
Write an expression for the conserved charge $E$ which follows from
the symmetry $t \to t + \delta$, with $\delta$ a constant.

\bigskip
We're going to consider a spinning folded string,
which spins as a rigid rod around its center,
and lies on an equator of the $S^3$ in (1),
$\theta_1 = \theta_2 = \pi/2$.
The center of the string is at the `center' of AdS, $\rho = 0$.
Go to a static gauge $t = \tau$,
with $\rho$ some function of $\sigma$.
Consider an ansatz for the azimuthal coordinate 
$$ \phi = \omega t ;$$
this describes a string spinning around the spatial sphere.

(c) Show that with these specifications the Nambu-Goto Lagrangian becomes
$$ L = - 4 
%{\sqrt \lambda \over 2 \pi}
{R^2 \over 2\pi \alpha'}
\int_0^{\rho_0} d\rho \sqrt{ \cosh^2 \rho - ( \dot \phi)^2 \sinh^2 \rho }, $$
where $\coth^2 \rho_0 = \omega^2$,
and the factor of 4 is because there are four segments of the 
string stretching from $0$ to $\rho_0$.

(d) 
Show that the energy and spin of this configuration are 
$$ E = 4 
{\sqrt \lambda \over 2 \pi}
%{R^2 \over 2 \pi \alpha'} 
\int_0^{\rho_0} d\rho
{\cosh^2 \rho \over \sqrt{ \cosh^2 \rho - \omega^2 \sinh^2 \rho}}
$$
$$ S = 4 
{\sqrt \lambda \over 2 \pi}
%{R^2 \over 2 \pi \alpha'} 
\int_0^{\rho_0} d\rho
{\omega \sinh^2 \rho \over \sqrt{ \cosh^2 \rho - \omega^2 \sinh^2 \rho}} $$
where 
$$ \lambda = {R^4 \over \alpha'^2}.$$



(e) Next we're going to reproduce this solution from
the Polyakov action, in conformal gauge:
$$ S_{P} = {1 \over 4 \pi \alpha'} \int d\tau d \sigma G_{ij}
\del_a X^i \del^a X^j. $$

In this description, a solution must also satisfy the
equations of motion following from varying the worldsheet metric, 
namely the Virasoro constraints:
$$ 0 = T_{++} = \del_+ X^i \del_+ X^j G_{ij} $$
$$ 0 = T_{--} = \del_- X^i \del_- X^j G_{ij} $$
Show that inserting
\def\ee{e}
$$ t = \ee\tau, \phi= \ee \omega \tau, \rho = \rho(\sigma) $$
into the Virasoro constraints gives
$$ (\rho')^2 = \ee^2 ( \cosh^2 \rho - \omega^2 \sinh^2 \rho ) .$$
(Here $\ee$ is a bit of slop which can be adjusted to make sure 
period of $\sigma$ is $2\pi$.)
This leads to an expression 
for $ {d\sigma \over d\rho}.$  

Show that the (target-space) energy and spin are
$$ 
E = {R^2 \over 2 \pi \alpha'} \ee \int_0^{2\pi} d\sigma \cosh^2 \rho $$
$$
S = {R^2 \over 2 \pi \alpha'} \ee \omega \int_0^{2\pi} d\sigma \sinh^2 \rho .$$
Use your expression for 
$ {d\sigma \over d\rho}$
to show that this reproduces the answers found using the Nambu-Goto action.




(f) {\bf [super-bonus challenge]}
Expand the integrals for $E$ and $S$ at large spin 
($\omega = 1 + 2 \eta, \eta \ll 1$), and 
show that in that regime
$$ E - S \sim \sqrt \lambda \ln S. $$
The fact that this expression is reminiscent of logarithmic scaling
violations in field theory is not a coincidence; the coefficient 
in front of the log is called the
'cusp anomalous dimension.'



\end{enumerate}

Some other problems you might consider doing are 

\begin{enumerate}
\item{\bf Polchinski Problem 1.2.}
Use the Virasoro constraints to show that 
the endpoints of an open string (with Neumann boundary conditions)
move at the speed of light.

\item{\bf Polchinski Problem 1.1.} 



\end{enumerate}
\end{document}



\item {\bf Flaky question of the week.}
Think up a new use for string theory.
