%From: "Eric D'Hoker"
%Date: Thu, 23 Jan 1997 15:31:15 -0500 (EST)
%Subject: Strings problem set 1
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%%%% STRING THEORY : Problem Set 1, January 23, 1997
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\centerline{{\bf STRING THEORY}}
\centerline{ Problem Set \# 1}
\centerline{January 23, 1997}
\bigskip
\bigskip
Consider the quantum field theory of a single real scalar field
$x$, on a 2-dimensional surface $\Sigma$ with Riemannian metric $g$
and action
$$
S[x,g] = {1 \over 4 \pi} \int _{\Sigma} d \mu _g \biggl \{
\12 g^{mn} \partial _m x \partial _n x + Q R_g x \biggr \}.
$$
Here, $Q \in {\bf C}$ and $R_g$ is the Gaussian curvature of $g$.
Introduce local complex coordinates $z,\ \bar z$ in which $g=2g_{z \bar z}
|dz|^2$.
\bigskip
\noindent
{\bf 1.)} \ \ \ \ For $Q=0$, define the renormalized stress tensor by
$$
T(z) = - \12 \lim _{w \to z} \bigl ( \partial _z x \partial _w x
+ { 1 \over (z-w)^2} \bigr ).
$$
a) Using the OPE of $\partial _z x $ and $\partial _w x$, derive
all singular terms in the OPE of $T(z)$ and $T(w)$, and confirm
the form derived on general grounds
$$
T(z) T(w) \sim {c/2 \over (z-w)^4} + {2 \over (z-w)^2} T(w)
+ {1 \over z-w} \partial _w T(w).
$$
b) Show, by explicit conformal reparametrization $z \to z' = f(z)$,
that $T(z)$ transforms as a projective connection. \hfill\break
c) Use the result of b) to relate the stress tensor $T(z)$ and
the Virasoro generators $L_m$ on the cylinder and on the annulus.
\bigskip
\noindent
{\bf 2.)} \ \ \ \ For $Q\not=0$, consider the flat metric $g=2|dz|^2$.
\noindent
a) Derive the OPE of $\partial _z x$ and $\partial _wx$.
\hfill\break
b) Compute the stress tensor
and show that $T= T_{zz}$ is complex analytic at the classical level.
Define a suitable renormalized stress tensor. \hfill\break
c) Use the result of b) to obtain the central charge of the theory.
\hfill\break
d) Show that $: \exp \{ \beta x(z,\bar z) \} : $ is a primary field
for any value of $\beta \in {\bf C}$, and compute its conformal
weight ($h, \bar h$). \hfill\break
e) Is the field $\partial _z x$ primary ? Derive its transformation
law under conformal reparametrizations $z \to z' = f(z)$.
\end