A String Theory Primer

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%Kiyoshi Shiraishi Sept 11, 1997
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\title{A String Theory Primer}
\author{Kiyoshi Shiraishi}
\begin{document}

\maketitle
\begin{abstract}
We focus our attention on bosonic closed string theory.
\end{abstract}

\section{Classical action}

\noindent {\bf Polyakov action}:
\be
I=-\frac{T}{2}\int d^2\sigma\sqrt{-\gamma}\gamma^{ab}
\partial_aX^\mu\partial_bX^\nu \eta_{\mu\nu},
\ee
where
$a,b=0,1$,$\mu,\nu=0,\dots,D-1$, and $\eta_{\mu\nu}=diag.(-1,1,\dots,1)$.
$\tau=\sigma^0, \sigma=\sigma^1$. $T=1/(2\pi\alpha')$.

Taking the following variations, we get:
\bea
\frac{\delta I}{\delta X^\mu}=0 &\rightarrow&
\nabla^a\nabla_a X^\mu=0 \\
\frac{\delta I}{\delta \gamma_{ab}}=0 &\rightarrow&
T_{ab}=0, 
\eea
where
\be
T_{ab}=\partial_aX^\mu\partial_bX_\mu-
\frac{1}{2}\gamma_{ab}\gamma^{cd}\partial_cX^\mu\partial_dX_\mu .
\ee

Note that $T^{a}_{a}=T_{ab}\gamma^{ab}$ identically vanishes
($\leftrightarrow$ Weyl invariance).


\vspace{1cm}
\noindent {\bf Invariances}:

\begin{itemize}

\item global Poincar\'e invariance
\bea
\delta X^{\mu}&=&a^{\mu}{}_{\nu} X^{\nu}+b^{\mu}~~~~~a_{\mu\nu}=-a_{\nu\mu} \\
\delta \gamma_{ab}&=&0 
\eea

\item local reparametrization invariance
\bea
\delta X^{\mu}&=&\xi^a \partial_a X^{\mu} \\
\delta \gamma_{ab}&=&\nabla_a \xi_b + \nabla_b \xi_a
\eea

\item local Weyl invariance
\bea
\delta X^{\mu}&=&0 \\
\delta \gamma_{ab}&=&2 \Lambda(\sigma) \gamma_{ab}
\eea

\end{itemize}


\vspace{1cm}
\noindent {\bf Conformal gauge}

Using these invariances, we can take
\be
\gamma_{ab}=e^{2 \phi}\eta_{ab} .~~~~~(conformal~gauge)
\ee
Then the equation of motion for $X^{\mu}$ reads
\be
\left(\frac{\partial^2}{\partial \tau^2}-
\frac{\partial^2}{\partial \sigma^2}\right) X^{\mu}(\tau,\sigma)=0 .
\ee
This is the 2-dimensional wave equation. Thus $X^{\mu}$ can be considered
as free fields in 2-dimensions.

\vspace{1cm}
\noindent {\bf Boundary conditions}

By using the reparametrization invariance, we can take:

\begin{itemize}

\item for closed string
\be
X^{\mu}(\tau,\sigma+\pi)=X^{\mu}(\tau,\sigma) ,
\ee

\item for open string
\be
\left.\frac{\partial X^{\mu}}{\partial \sigma}\right|_{\sigma=0}=
\left.\frac{\partial X^{\mu}}{\partial \sigma}\right|_{\sigma=\pi}=0 .
\ee

\end{itemize}

\vspace{1cm}
\noindent {\bf Components of $T_{ab}$}

In conformal gauge:

\bea
T_{00}&=&T_{11}=\frac{1}{2}\left(\dot{X}^2+{X'}^2\right) \\
T_{01}&=&T_{10}=\dot{X}\cdot X' ,
\eea
where $\dot{X}^\mu=\frac{\partial X^{\mu}}{\partial\tau}$ and 
${X^{\mu}}'=\frac{\partial X^{\mu}}{\partial\sigma}$.


Thus
\be
T_{ab}=0 \Leftrightarrow \left(\dot{X}\pm X'\right)^2=0 .
\ee

\vspace{1cm}
\noindent {\bf Canonical conjugate of $X^{\mu}$}

In conformal gauge
\be
I=-\frac{T}{2}\int d\tau d\sigma \left[-\dot{X}^2+{X'}^2\right] .
\ee
Then the canonical momentum is
\be
\Pi_{\mu}=T \dot{X}^{\mu} .
\ee

\section{Quantization of closed strings}

\subsection{Covariant approach (global Poincar\'e inv. is manifest)}

$X^{\mu}$ can be expanded as the equation of motion and the boundary
condition satisfied:
\be
X^{\mu}(\tau,\sigma)=x^{\mu}+2\alpha' p^{\mu} \tau+
\frac{i}{2}\sqrt{2\alpha'}\sum^{\infty}_{n=-\infty,n\neq 0}
\frac{1}{n}\left(\alpha^{\mu}_{n} e^{-2in(\tau-\sigma)}
+\tilde{\alpha}^{\mu}_{n} e^{-2in(\tau+\sigma)}\right) .
\ee
Here $\alpha^{\mu}_{-n}=\alpha^{\mu}_{n}\dag$ and
$\tilde{\alpha}^{\mu}_{-n}=\tilde{\alpha}^{\mu}_{n}\dag$.

We propose the equal-time($\tau$) commutation relations:
\be
\left[X^{\mu}(\tau,\sigma), \Pi^{\nu}(\tau,\sigma')\right]=
\left[X^{\mu}(\tau,\sigma), T \dot{X}^{\nu}(\tau,\sigma')\right]=
i \eta^{\mu\nu} \delta (\sigma-\sigma'), etc.
\ee
Then we get
\be
[x^{\mu},p^{\nu}]=i \eta^{\mu\nu}, 
[\alpha^{\mu}_{m},\alpha^{\nu}_{n}]=m \delta_{m+n,0} \eta^{\mu\nu}
[\tilde{\alpha}^{\mu}_{m},\tilde{\alpha}^{\nu}_{n}]=
m \delta_{m+n,0} \eta^{\mu\nu},
[\alpha^{\mu}_{m},\tilde{\alpha}^{\nu}_{n}]=0 .
\ee


\vspace{1cm}
\noindent {\bf Virasoro operators}

\bea
L_{m}=\frac{T}{2}\int^{\pi}_{0}d\sigma e^{+2im(\tau-\sigma)}
\frac{1}{2}\left(T_{00}-T_{01}\right)=
\frac{1}{2}\sum^{\infty}_{n=-\infty}\alpha_{m-n}\cdot\alpha_{n} \\
\tilde{L}_{m}=\frac{T}{2}\int^{\pi}_{0}d\sigma e^{+2im(\tau+\sigma)}
\frac{1}{2}\left(T_{00}+T_{01}\right)=
\frac{1}{2}\sum^{\infty}_{n=-\infty}\tilde{\alpha}_{m-n}\cdot
\tilde{\alpha}_{n} 
\eea
for $m\neq 0$, and
\bea
L_{0}=
\frac{\alpha'}{4}p^2+\sum^{\infty}_{n=1}\alpha_{-n}\cdot\alpha_{n} \\
\tilde{L}_{0}=
\frac{\alpha'}{4}p^2+\sum^{\infty}_{n=1}\tilde{\alpha}_{-n}\cdot
\tilde{\alpha}_{n} 
\eea
for $m=0$.

Note that $L_{-n}=L_{n}\dag$ and $\tilde{L}_{-n}=\tilde{L}_{n}\dag$.


They satisfies the Virasoro algebla:
\bea
\left[L_{m},L_{n}\right]&=&(m-n)L_{m+n}+\frac{D}{12}(m^3-m)\delta_{m+n,0} \\
\left[\tilde{L}_{m},\tilde{L}_{n}\right]&=&(m-n)\tilde{L}_{m+n}+
\frac{D}{12}(m^3-m)\delta_{m+n,0} \\
\left[L_{m},\tilde{L}_{n}\right]&=&0
\eea

\vspace{1cm}
\noindent {\bf Virasoro condition for physical states}
\be
L_{n}|phys\rangle=0~~~and~~~\tilde{L}_{n}|phys\rangle=0
\ee
for $n>0$.
\be
\langle phys'|L_{-n}|phys\rangle=\langle phys'|L_{n} \dag |phys\rangle=
(\langle phys|L_{n}|phys'\rangle)\dag=0
\ee
for $n>0$. Thus $\langle phys'|T_{ab}|phys\rangle=0$.

For $n=0$,
\be
(L_{0}-\tilde{L}_{0})|phys\rangle=0,~~~
(L_{0}-a)|phys\rangle=0,~~~
(\tilde{L}_{0}-a)|phys\rangle=0
\ee


\vspace{1cm}
\noindent {\bf $(mass)^2$ for physical states}

\bea
M^2=-p^{\mu}p_{\mu}=\frac{2}{\alpha'}(N+\tilde{N}-2a), \\
N-\tilde{N}=0 ,
\eea
where
\be
N=\sum^{\infty}_{n=1}\alpha_{-n}\cdot\alpha_{n}, ~~~~~~~
\tilde{N}=\sum^{\infty}_{n=1}\tilde{\alpha}_{-n}\cdot\tilde{\alpha}_{n}. 
\ee

Note that
\be
[N,\alpha^{\mu}_{-n}]=n \alpha^{\mu}_{n}, 
[\tilde{N},\tilde{\alpha}^{\mu}_{-n}]=n \tilde{\alpha}^{\mu}_{n}. 
\ee

\vspace{1cm}
\noindent {\bf spurious states}
\be
L_{-n}|*\rangle
\ee
for $n>0$.
\be
|phys\rangle\sim |phys\rangle+L_{-n}|*\rangle
\ee

\subsection{Light cone quantization (global Poincar\'e inv. is not manifest)}

\be
X^{\pm}=\frac{1}{\sqrt{2}}(X^0\pm X^1)
\ee

By using local reparametrization inv., we can take
\be
X^{+}(\tau)=x^{+}+2\alpha' p^{+} \tau .
\ee

There are commutation relations for $\alpha^{i}_{n}$ and
$\tilde{\alpha}^{i}_{n}$, similar to the previous approach, 
but $i=2,\dots,D-1$ in the present case.

\vspace{1cm}
\noindent {\bf $(mass)^2$}

\bea
\dot{X}^2+{X'}^2=0 &\Rightarrow& 2\alpha' p^{+}\dot{X}^{-}=
\frac{1}{2}(\dot{X}^{i}{}^2+{{X^{i}}'}^2) \\
\dot{X}\cdot{X'}=0 &\Rightarrow& 2\alpha' p^{+}{X^{-}}'=
{\dot{X}^{i}}\cdot{{X^{i}}'}
\eea

{}From these

\bea
M^2=-p^{\mu}p_{\mu}=2 p^{+}p^{-}-p^{i}p^{i}=
\frac{1}{\alpha'}\sum^{\infty}_{n=-\infty,n\neq 0}
(\alpha^{i}_{-n}\alpha^{i}_{n}+
\tilde{\alpha}^{i}_{-n}\tilde{\alpha}^{i}_{n}), \\
\sum^{\infty}_{n=-\infty,n\neq 0}(\alpha^{i}_{-n}\alpha^{i}_{n}-
\tilde{\alpha}^{i}_{-n}\tilde{\alpha}^{i}_{n})
=0 .
\eea

\bea
\sum^{\infty}_{n=-\infty,n\neq 0}
\alpha^{i}_{-n}\alpha^{i}_{n}&=&
\sum^{\infty}_{n=1}\alpha^{i}_{-n}\alpha^{i}_{n}+
\sum^{\infty}_{n=1}\alpha^{i}_{n}\alpha^{i}_{-n} \nn
&=&2 \sum^{\infty}_{n=1}\alpha^{i}_{-n}\alpha^{i}_{n}+
\sum^{D-1}_{i=2}\sum^{\infty}_{n=1}[\alpha^{i}_{n},\alpha^{i}_{-n}] \nn
&=&2 \sum^{\infty}_{n=1}\alpha^{i}_{-n}\alpha^{i}_{n}+
(D-2)\sum^{\infty}_{n=1} n .
\eea

Here, for Riemann zeta fn. $\zeta(s)=\sum^{\infty}_{n=1}n^{-s}$,
$\zeta(-1)=-\frac{1}{12}$, we replace $\sum^{\infty}_{n=1} n \Rightarrow
-\frac{1}{12}$.


\bea
M^2=\frac{2}{\alpha'}(N+\tilde{N}-2a), \\
N-\tilde{N}=0,
\eea
where
\be
N=\sum^{\infty}_{n=1}\alpha^{i}_{-n}\alpha^{i}_{n}, ~~~~~~~
\tilde{N}=\sum^{\infty}_{n=1}\tilde{\alpha}^{i}_{-n}\tilde{\alpha}^{i}_{n}. 
\ee


By checking the algebla of generators of
Lorentz transformation $M_{\mu\nu}$ ($\leftrightarrow$ global
Lorentz inv.), we must take $D=26$ and $a=1$.



\vspace{1cm}
\noindent {\bf Low mass states}

\begin{itemize}

\item Lowest
\be
|0;k\rangle~~~~~~~~~~~M^2=-\frac{4}{\alpha'}<0~~~~tachyon
\ee

\item 1st excited
\be
\alpha^{i}_{-1}\tilde{\alpha}^{j}_{-1}|0;k\rangle~~~~~~~~~~~M^2=
0~~~~graviton, antisymmetric field, dilaton
\ee

\end{itemize}

\subsection{Path integral quantization}

\section{Effective field theory for massless fields}

\bea
I=-\frac{T}{2}\int d^2 \sigma&& \left[\frac{}{}
\sqrt{-\gamma}\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}g_{\mu\nu}(X)+
\epsilon^{ab}\partial_a X^{\mu}\partial_b X^{\nu}B_{\mu\nu}(X)\right. \nn
& &\left.-\frac{\alpha'}{2}\sqrt{-\gamma}~{}^{(2)}\!\!R \Phi(X)\right]
\eea

In the background field,
\be
2\pi T^{a}_{a}=
\beta^{g}_{\mu\nu}\sqrt{-\gamma}\gamma^{ab}\partial_a X^{\mu}\partial_b X^{\nu}+
\beta^{B}_{\mu\nu}\epsilon^{ab}\partial_a X^{\mu}\partial_b X^{\nu}+
\beta^{\Phi}\sqrt{-\gamma}~{}^{(2)}\!\!R .
\ee

Weyl inv. $\leftrightarrow$ $T^{a}_{a}=0$

\bea
\beta^{g}_{\mu\nu}&=&R_{\mu\nu}-\frac{1}{4}H_{\mu}{}^{\lambda\sigma}
H_{\nu\lambda\sigma}+2\nabla_{\mu}\nabla_{\nu}\Phi+O(\alpha')=0 \\
\beta^{B}_{\mu\nu}&=&\nabla_{\lambda}H^{\lambda}{}_{\mu\nu}-
2(\nabla_{\lambda}\Phi)H^{\lambda}{}_{\mu\nu}+O(\alpha')=0 \\
\frac{\beta^{\Phi}}{\alpha'}&=&\frac{D-26}{48\pi^{2}\alpha'}+
\frac{1}{16\pi^2}\left[4(\nabla\Phi)^2-4\nabla^{2}\Phi-R+
\frac{1}{12}H^{2}\right]+O(\alpha')=0 
\eea

These can be derived from the effective action:
\be
S=\frac{1}{2\kappa^2}\int d^D x \sqrt{-g} e^{-2\Phi}\left[
R+4(\nabla\Phi)^2-\frac{1}{12}H^{2}-\frac{D-26}{3\alpha'}\right]
+O(\alpha')
\ee


\section{Torus compactifications of closed string theory
and T duality}

\subsection{closed string on $S^1$}

\be
X^{25}(\tau,\sigma+\pi)=X^{25}(\tau,\sigma)+2\pi R \ell,~~~~~\ell\in {\bf Z}
\ee

\bea
& &X^{25}(\tau,\sigma)= \nn
& &x^{25}+2\alpha' p^{25} \tau+2R\ell\sigma+
\frac{i}{2}\sqrt{2\alpha'}\sum_{n\neq 0}
\frac{1}{n}\left(\alpha^{25}_{n} e^{-2in(\tau-\sigma)}
+\tilde{\alpha}^{25}_{n} e^{-2in(\tau+\sigma)}\right)
\eea

\be
p^{25}\rightarrow \frac{m}{R}~~~~~~~~m\in {\bf Z}
\ee


\bea
M^2&=&\left(\frac{m}{R}\right)^2+\left(\frac{R\ell}{\alpha'}\right)^2+
\frac{2}{\alpha'}(N+\tilde{N}-2) \\
&=&(KK~mode)+(winding~mode)+(oscillation~mode)
\eea

\be
N-\tilde{N}=m\ell
\ee
 
\vspace{1cm}
\noindent {\bf T duality}
\bea
R &\leftrightarrow& \frac{\alpha'}{R} \\
\ell &\leftrightarrow& m
\eea

\subsection{closed string on $T^N$}

\subsection{background antisymmetric field}

\vspace{1cm}
\noindent {\bf T duality in general}

\begin{itemize}
\item For $S^1$
\bea
R &\rightarrow& \frac{\alpha'}{R} \\
\Phi &\rightarrow& \Phi - \ln \frac{R}{\alpha'}
\eea

\bea
~~~~~~~~\heartsuit ~~&~& S\sim \int dX^{25}e^{-2\Phi}[\cdots] \\
& &R e^{-2\Phi}=R'e^{-2\Phi'}=\frac{\alpha'}{R}e^{-2\Phi'}
\eea

\item For $T^N$ ($\alpha'=1$)
\bea
g+B &\rightarrow&  (g+B)^{-1} \\
\Phi &\rightarrow& \Phi - \frac{1}{2}\ln \det (g+B)
\eea
\end{itemize}

\section{Appendix: Cosmological constant}

\end{document}
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