\documentclass{ws-procs975x65}
\usepackage{psfrag}
%\documentclass[12pt]{article}
%\input epsf.sty
%\topmargin -.5cm
%\textheight 21cm
%\oddsidemargin -.125cm
%\textwidth 16cm
 
\def\ZZZ{{\hbox{ Z\kern-1.6mm Z}}}
\newcommand{\rrho}{r}
\newcommand{\bA}{{\bf A}}
\newcommand{\bG}{{\bf G}}
\newcommand{\bF}{{\bf F}}
\newcommand{\bbb}{{\bar b}}
\newcommand{\gam}{\tau}
\newcommand{\eps}{\epsilon}
\newcommand{\ra}{\rangle}
\newcommand{\la}{\langle}
\newcommand{\T}{\chi_{T}(k)}
\newcommand{\Tm}{\chi_{T}(k')}
\newcommand{\Cn}{{\cal C}_n}
\newcommand{\vp}{\varphi}
\newcommand{\ve}{\varepsilon}
\newcommand{\tl}{\wt\lambda}
\newcommand{\dt}{(\vec \nabla T)^2}
\newcommand{\hp}{{\wh\Phi}}
\newcommand{\hq}{{\wh Q_B}}
\newcommand{\he}{{\wh\eta_0}}
\newcommand{\ha}{{\wh{A}}}
\newcommand{\lllb}{\Bigl\langle\Bigl\langle}
\newcommand{\rrrb}{\Bigr\rangle\Bigr\rangle}
\newcommand{\tf}{\wt f}
 
\newcommand{\vv} {\bar v}
\newcommand{\uu} {\bar u}
\newcommand{\K}{{\rm K_1}}
\newcommand{\Kt}{{\rm \widetilde K_1}}
 
\newcommand{\B}{b'}
\newcommand{\C}{c'}
\newcommand{\bB}{\bar b'}
\newcommand{\Bu}{B_{\vec u}}
\newcommand{\VV}{{\cal V}}
\newcommand{\BB}{{\cal B}}
\newcommand{\II}{{\cal I}}
\newcommand{\AAA}{{\bf a}}
\newcommand{\GG}{{\cal G}}
%\newcommand{\ggg}{{\bf g}}
\newcommand{\fff}{{\bf f}}
\newcommand{\ccc}{{\bf c}}
\newcommand{\FF}{{\cal F}}
\newcommand{\HH}{{\cal H}}
\newcommand{\MM}{{\cal M}}
\newcommand{\CC}{{\cal C}}
\newcommand{\bC}{{\bf C}}
\newcommand{\OO}{{\cal O}}
\newcommand{\QQ}{{\cal Q}}
\newcommand{\PP}{{\cal P}}
\newcommand{\EE}{{\cal E}}
\newcommand{\LL}{{\cal L}}
%\newcommand{\lll}{\langle\langle}
\newcommand{\rrr}{\rangle\rangle}
%\newcommand{\square}{\Box}
\newcommand{\half}{{1\over 2}}
\newcommand{\wt}{\widetilde}
\newcommand{\wh}{\widehat}
\newcommand{\wc}{\check}
\newcommand{\wb}{\bar}
\newcommand{\bd}{\bar{\rm D}}
\newcommand{\RR}{{\cal R}}
\newcommand{\NN}{{\cal N}}
\newcommand{\TT}{{\cal T}}
\newcommand{\bg}{\bar g}
\newcommand{\bb}{\bar b}
\newcommand{\bT}{\bar \Theta}
\newcommand{\SSS}{{\cal S}}
\newcommand{\tlx}{\left(\tilde \lambda ; X^0(0) \right)}
\newcommand{\al}{\alpha}
 
 
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\newcommand{\ben}{\begin{eqnarray}\displaystyle}
\newcommand{\een}{\end{eqnarray}}
\newcommand{\refb}[1]{(\ref{#1})}
\newcommand{\p}{\partial}
\newcommand{\sectiono}[1]{\section{#1}\setcounter{equation}{0}}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\theequation}{\arabic{equation}}
\def\one{{\hbox{ 1\kern-.8mm l}}}
\def\zero{{\hbox{ 0\kern-1.5mm 0}}}

\markboth{\quad \small \sl Ashoke Sen \hfill}{\hfill\small \sl 
 Tachyons in String Theory \quad }
\setcounter{page}{2035}
\begin{document}
 
\title{Tachyons in String Theory}

%\vspace*{4.0ex}
 
\author{%
Ashoke Sen}
 
\address{%
Harish-Chandra Research Institute\\
Chhatnag Road, Jhusi,\\
Allahabad 211019, India}
 
%\centerline{E-mail: ashoke.sen@cern.ch,
%sen@mri.ernet.in}
 
\maketitle
 
\abstracts{%
We give a non-technical review of some of the recent developments in our
understanding of the tachyon in string theory. We also illustrate the
conjecture that open string theory provides a complete description of the
dynamics of unstable D-branes.
}

%\bigskip ~ \bigskip

\newpage

In this brief review I plan to give a general overview of tachyons and 
their significance in open string theory. I shall try to address the 
following questions: 
\begin{enumerate}
\item What are tachyons?
\item How do we deal with tachyons in quantum field theory?
\item How do tachyons appear in string theory?
\item How do we make sense of tachyons in string theory?
\end{enumerate}
I shall end the review by describing an open string completeness
conjecture that arises out of the study of open string tachyons.



Let us begin with the question:
What are tachyons?
Historically tachyons were described as particles which travel faster than
light. If we use the relativistic equation
\be \label{ere1}
v = {p c\over \sqrt{p^2 + m^2 c^2}}\, ,
\ee
relating the velocity $v$ of a particle to its momentum $p$, mass $m$ and 
the velocity 
of light $c$, then we see that tachyons can also be regarded as particles 
with  negative mass$^2$,
{i.e.} imaginary mass.
Both descriptions sound equally bizarre.
On the other hand tachyons have been known to exist in string theory 
almost since
its birth, 
and hence
we need to make sense of them.

Actually tachyons do appear in conventional quantum field theories as
well. 
Consider, for example, a classical scalar field $\phi$ with potential 
$V(\phi)$. To simplify notation I shall from now on set the Planck's 
constant $\hbar$ and the velocity of light $c$ to 1.
In $p$-space and 1-time dimension labelled by the time coordinate $x^0$
and space coordinates $x^i$ ($1\le i\le p$) the lagrangian of the scalar
field is
\be \label{e1}
L={1\over 2} \, \int d^{p} x [(\p_0\phi)^2 - \p_i\phi \p_i\phi - V(\phi)] 
\, .
\ee
Normally we choose the origin of $\phi$ so that the potential $V(\phi)$
has a minimum at $\phi=0$.
In this case
quantization of $\phi$ gives a scalar particle of
mass$^2=V''(\phi)|_{\phi=0}$.
This gives a positive mass$^2$ particle.
But now suppose the potential has a maximum at $\phi=0$.
Then $V''(\phi)|_{\phi=0}$ is negative.
Naive quantization will give a particle of negative mass$^2$.
Thus we have 
a tachyon!

In this case however it is clear what we are doing wrong.
When we identify $V''(0)$ as the mass$^2$ of the particle, we are making
an approximation.
We expand $V(\phi)$ in a Taylor series expansion in $\phi$, and treat the
cubic and higher order terms as small corrections to the quadratic term.
This is true only if the quantum fluctuations of $\phi$ around $\phi=0$
are small.
But if $V(\phi)$ has  a maximum at $\phi=0$, then $\phi=0$ is a
classically unstable point.
Hence
we cannot expect the fluctuations of $\phi$ to be small.
The remedy to this difficulty is to
find the minimum $\phi_0$ of the potential $V(\phi)$, and quantize the 
system around this point. More precisely this means that we can 
expand the potential around $\phi=\phi_0$, and treat the
cubic and higher order terms in the expansion to be small.
The mass$^2$ of the particle now can be identified as $V''(\phi_0)$.
This is positive since $V(\phi)$ has a minimum at $\phi=\phi_0$.
Hence the theory does not have tachyons.

Thus we see that the existence of a tachyon in a scalar field theory 
implies that the potential for 
the scalar field has a local maximum at the origin. In order for the 
theory to be sensible at least in perturbation theory, the potential must 
also have a (local) minimum. Typically whenever the potential in a scalar 
field theory has more 
than one extremum we can construct non-trivial (possibly unstable) 
classical 
solutions which depend on one or more spatial directions.
For example consider a potential of the form $V(\phi)\propto 
(\phi^2-\phi_0^2)^2$. In this case $V(\phi)$ has a maximum at $\phi=0$ and 
a pair of minima at $\phi=\pm\phi_0$. In this theory we can construct a 
domain wall solution where
1) $\phi$ depends on one spatial coordinate $x^1$, 
2) as $x^1\to\infty$, $\phi\to\phi_0$, 
3) as $x^1\to-\infty$, $\phi\to -\phi_0$, and
4) the total energy is minimized subject to these constraints.
For this configuration the energy density is concentrated around 
$x^1\simeq 0$.
This gives rise to 
a `codimension 1 defect'.
For more complicated cases we can have more complicated defects (of higher
codimension). Examples of such defects are
vortices which are codimension 2 defects, 
't Hooft--Polyakov monopoles which are codimension 3 defects, etc.
In general for a 
codimension $k$ defect the energy density is localized around a subspace 
of
dimension $(p-k)$.


The lessons learned from the field theory examples may be summarized as 
follows:
\begin{itemize}
\item Existence of tachyons in the spectrum tells us that we are expanding
the potential around its maximum rather than its minimum.
\item The correct procedure to deal with such a situation is to find the 
(global) minimum of the potential and expand the potential around the 
minimum. The resulting theory has a positive mass$^2$ scalar particle 
instead of a tachyon.
\item Associated with the existence of tachyons we often have non-trivial 
space dependent classical solutions (defects).
\end{itemize}

We now turn to the discussion of tachyons in
string theory.
The conventional description of string theory is based on `first
quantized' formalism rather than a field theory.
We take a string (closed or open) and quantize it maintaining Lorentz
invariance.
This gives infinite number of states characterized by momentum $\vec p$ 
and
other
discrete quantum numbers $n$. It turns out that the energy of the $n$th 
state carrying momentum $\vec p$ is given by
$E_n = \sqrt{\vec p^2 + m_n^2}$, where $m_n$ is some constant. This 
state clearly has the
interpretation of being a particle of mass $m_n$.
Thus string theory contains infinite number of single `particle' states,
as if it is a field theory with infinite number of fields.

Quantization of some closed or open strings gives rise to
states with negative $m_n^2$ for some $n$.
This corresponds to a
tachyon!
For example, the
original bosonic string theory formulated in (25+1) dimensions has a
tachyon in the spectrum of closed strings.
This theory is
thought to be inconsistent due to this reason.

Superstring theories are free from tachyons in the spectrum of closed 
string.
But for certain boundary conditions, there can
be tachyon in the spectrum of
open strings even in superstring theories.
Thus the question is:
Does the existence of tachyons make the theory inconsistent?
Or does it simply indicate that we are quantizing the theory around
the wrong point?
The 
problem in analyzing this question stems from the fact that
unlike the example in a scalar field theory, the tachyon in string theory
does not originally come from quantization of a scalar field. Thus
in order to understand the tachyon, we have to reconstruct the scalar
field and its potential from the known results in string theory, and then
analyze if the potential has a minimum.

It turns out that
for open string tachyons we now know the answer in many cases.
On the other hand, closed string tachyons are only beginning to be 
explored.
Hence
we shall focus mainly on open string tachyons in this review.

There are five consistent, apparently different, superstring theories in
9-space and 1-time dimension.
We shall focus on two of them, known as type IIA and type IIB string
theories.
Elementary excitations in this theory come from quantum states of the 
closed strings.
But besides these elementary excitations these theories also contain
`composite' objects known as D-branes or more explicitly Dirichlet
$p$-branes.

A D$p$-brane is a $p$-dimensional object. Thus for example
D0-brane corresponds to a particle like object, a D1-brane corresponds to 
a string-like object, a 
D2-brane corresponds to a
membrane like object and so on.
But unlike the kinks and other defects in field theory which are 
associated with classical solutions of the equations of motion of the 
fields, 
D-branes are defined by saying what happens in their presence rather than
by saying what they are.
Consider, for example, a static flat D$p$-brane in flat space-time, lying 
along a
$p$-dimensional subspace. The
definition of a D$p$-brane is simply that fundamental strings can end on 
the $p$-dimensional hypersurface along which the D-brane lies.
This has been illustrated in Fig.\,\ref{f1}.

\begin{figure}
\centerline{\includegraphics[height=3cm]{fig1}}
%\begin{center} 
%\epsfysize=3cm 
%\epsfbox{fig1.eps} 
%\end{center}
\caption{Fundamental strings (shown by dashed line) ending on a D-brane 
(shown by solid line).} 
\label{f1}
\end{figure}
Quantum states of a fundamental open string with ends on a D-brane
represent
quantum excitation modes of the D-brane. D-branes need to satisfy various 
consistency requirements, and as a result D-branes for different $p$ have 
different properties. For type IIA string theory, these properties are 
summarized as follows:
\begin{enumerate}
\item For even $p$, D$p$-branes are oriented  and are known as BPS 
D-branes due to some special properties which they possess. For these 
branes, the
mass per unit $p$-volume, also known as the tension $\TT_p$ of the brane, 
is given by 
\be \label{e3}
\TT_p = {1\over (2\pi)^p g_s}\, ,
\ee
in a unit in which the tension of the fundamental string is  ${1\over
2\pi}$. We shall use this unit throughout this review.
$g_s$ is a dimensionless constant known as the string coupling constant. 
We shall do most of our analysis to lowest order in the perturbation 
expansion 
in $g_s$.

It turns out that
all open string states on a BPS D-brane have mass$^2\ge 0$. Hence there 
are no tachyons in the spectrum.
 
\item For odd $p$, the D$p$-branes are unoriented (non-BPS).
The 
tension $\wt\TT_p$ of a non-BPS D$p$-brane is given by
\be \label{e5}
\wt\TT_p = {\sqrt{2}\over 
(2\pi)^p g_s}\, .
\ee
Each such D-brane has one open string mode with mass$^2=-{1\over 2}$. In 
other words, there is a
tachyonic mode on each of these non-BPS D$p$-branes.
\end{enumerate}

For type IIB string theory the situation is reversed. There are now
oriented (BPS) D$p$-branes for odd $p$ and unoriented (non-BPS)
D$p$-branes for even $p$. The results that we shall discuss will be valid 
both for type IIA and type IIB string theory. Whether we are talking about 
type IIA theory or type IIB theory should be understood from the context. 
For example, if we are referring to a non-BPS D$p$-brane, then it should 
be 
understood that we are talking about type IIA theory if $p$ is odd, and 
type IIB theory if $p$ is even.

For oriented D-branes we define an
anti-D-brane ($\bd$-brane) to be a D-brane with opposite orientation.
It turns out that 
a coincident BPS D$p$-brane $\bd p$-brane pair has two tachyonic
modes, each of mass$^2=-{1\over 2}$, from the open strings with one end on
the brane and one end on the antibrane (sectors (c) and (d) in 
Fig.\,\ref{f2}). 

\begin{figure}
\centerline{\includegraphics[height=4cm]{fig2}}
%\begin{center} 
%\epsfysize=4cm 
%\epsfbox{fig2.eps} 
%\end{center}
\caption{The tachyon on a D$p$-$\bd p$-brane pair comes from open 
strings whose two ends lie on two different branes.} \label{f2}
\end{figure}

Since string theory is formulated in a way that is different from a field
theory, the method of analysis in string theory is very different from
that in a field theory.
Nevertheless it is useful to use the language of field theory to describe 
various situations in string theory. In particular,
if we
use the analogy with field theory origin of tachyons, then 
for a non-BPS D$p$-brane, the dynamics of the single tachyonic mode
should be
described
by a real scalar field $T$ with negative mass$^2$ in $p$-space and one
time dimensions.
We shall refer to
$T$ as tachyon field.
For the D$p$-$\bd p$ system, the dynamics of the pair of tachyonic
modes
should be
described
by a complex scalar field $T$ with negative mass$^2$.
Various results in string theory can be stated {\it as
if} the dynamics of the tachyon $T$ is described by an effective potential
$V_{eff}(T)$ or more generally an effective action $S_{eff}(T)$. We shall 
first state the main results in this language and then briefly mention the 
various stringy techniques which are used to derive these results.

Let us begin by reviewing the
properties of $S_{eff}(T)$ and $V_{eff}(T)$ which follow from simple 
considerations.
First of all it is known that $S_{eff}(T)$ has simple symmetry properties. 
For example,
for a non-BPS
D$p$-brane $S_{eff}(-T)=S_{eff}(T)$. 
On the other hand for a D-$\bd$ system, $S_{eff}(e^{i\phi}T) = 
S_{eff}(T)$. 
The other property of that is obvious is that
$V_{eff}(T)$ has a maximum at $T=0$, since the field $T$ is tachyonic.
\psfrag{Ep}{\Large$\EE_p$}
\psfrag{T0}{\Large$T_0$}
\psfrag{T}{\Large$T$}
\psfrag{V}{\Large$V$}
\begin{figure}
\centerline{\includegraphics[height=6cm]{fig3G}}
%\begin{center} 
%\epsfysize=6cm 
%\epsfbox{fig3.eps} 
%\end{center}
\vskip -2cm
\caption{The tachyon potential on a non-BPS D$p$-brane. The tachyon 
potential on a brane-antibrane system can be obtained by revolving this 
diagram around the vertical axis, so that we get a mexican hat potential.} 
\label{f3}
\end{figure}

The interesting questions the answers to which are not obvious
are:
\begin{enumerate}
\item Does $V_{eff}(T)$ have a minimum? 
\item If it does have a minimum, then what kind of mass spectrum do we get
by quantizing the theory around the minimum?
\item Do we get topological defects involving the tachyon?
\end{enumerate}
etc. It turns out that the answers to many of these questions are now 
known. These results can be summarized as follows:
\begin{enumerate}
\item $V_{eff}(T)$ does have a minimum at some value $|T|=T_0$. 
Furthermore, at this 
minimum\cite{9805170,9911116}
\be \label{e6} 
V_{eff}(T_0) + \EE_p
=0\, ,
\ee
where 
$\EE_p$ denotes the total energy density of the original system. 
Thus
$\EE_p$ = $\wt\TT_p$ for a non-BPS D$p$ brane, and 
$\EE_p$ = $2 \TT_p$ for D$p$ -- $\bd p$ system. 
Thus at $|T|=T_0$ the total energy density vanishes identically. 
This situation has been illustrated in Fig.\,\ref{f3}.

\item $|T|=T_0$ configuration describes the closed string vacuum without
any D-brane\cite{9805170,9911116}. 
Thus around this minimum there are no physical open string excitations.
This is natural from the point of view of string theory, since the total 
energy vanishes at $T=T_0$, and hence we can identify this configuration 
as vacuum without any D-brane. Since there is no D-brane, there should be 
no open strings in the spectrum. However,  
this result is very surprising from the point of view of a normal field 
theory.
Shifting the point around which we expand the potential can make a
negative
mass$^2$ state into a positive mass$^2$ state, but we do not normally 
eliminate the
state altogether.
On the other hand here expanding the action around the minimum of the
potential not only gets rid of the original tachyon state, but also gets
rid of the infinite number of other open string states which were present.
\psfrag{-T0}{\Large$-T_0$}
\psfrag{Xp}{\Large$x^p$}
\begin{figure}
\centerline{\includegraphics[height=5cm]{fig4G}}
%\begin{center} 
%\epsfysize=5cm 
%\epsfbox{fig4.eps} 
%\end{center}
\caption{Tachyonic kink solution representing a BPS D$(p-1)$-brane.} 
\label{f4}
\end{figure}

\item There are classical solutions of the equations of motion of $T$,
representing lower dimensional 
D-branes\cite{9805019,9808141,9810188,9812031,9812135,9904207}. 
For example, 
on a non-BPS D$p$-brane a kink as shown in Fig.\,\ref{f4} represents a 
D$(p-1)$-brane. 
For this solution the
energy density is localized around a codimension 1 subspace ($x^p=0$) 
This looks like an ordinary kink solution in a field theory, but there is 
an important difference.
In a conventional field theory, a defect lives on the space in which
the field theory lives.
Here, at the bottom of the potential, the object (original D$p$-brane) 
whose
dynamics the field theory describes disappears altogether.
Nevertheless defects in the field can survive and describe non-trivial
objects in the (9+1)-dimensional space-time in which full string
theory lives.

There are also other more complicated examples of `tachyonic defects'.
For example, a
vortex solution on a D$p$-$\bd p$ pair describes a BPS 
D$(p-2)$-brane\cite{9808141}. On 
the other hand, a
't Hooft--Polyakov monopole on a pair of coincident non-BPS D$p$-branes
describes a non-BPS D$(p-3)$-brane\cite{9812135}.
In this way all D-branes can be regarded as defects in the tachyon field
living on D-branes of maximal dimension.
This gives a more conventional description of D-branes as defects in the
tachyon field. 
But more importantly this description gives a way to classify D$p$-branes
based on a branch of mathematics known as K-theory\cite{9810188,9812135}.
Several new stable D-branes in various string theories have been
discovered using this general scheme.
\end{enumerate}

So far we have only described the properties of static solutions of the 
tachyon 
effective field theory. Let us now turn to dynamics, namely time dependent 
solutions of the equations of motion.
In the case of  a conventional scalar field, if we displace the field from
its maximum and let it roll down the potential, the scalar field will
oscillate about its minimum.
Energy-momentum tensor $T_{\mu\nu}$ for this solution will have the form
\be \label{e7}
T_{00} = \EE\,, \qquad T_{ij} = - p(x^0) \delta_{ij}\,, \qquad 
T_{i0}=0\,,
\ee
where $i,j$ refer to the spatial directions.
Here
$\EE$ denotes the energy density, and  remains constant due to energy 
conservation.
$p$ denotes the
pressure, and will typically oscillate about an average value (0 for a
conventional scalar field) as the scalar field oscillates about its 
minimum.
We can now ask:
What happens if we displace the tachyon field on  a
D-$\bd$ pair (or a non-BPS D-brane) and let it roll down the hill?
It turns out that in this case the 
energy density remains constant as usual by energy conservation, 
but the pressure goes to zero asymptotically instead of oscillating about
0.
Thus the final state is a
gas of non-zero energy density and zero pressure\cite{0203211,0203265}. 
Another important quantity that characterizes the system is the dilaton 
charge that measures the coupling of the dilaton to the system. It turns 
out that the final system also has zero dilaton charge.

Let me now say a few words about the various 
techniques which are used to derive the various results that we have 
mentioned so far. We begin by 
clarifying 
a point that may have been somewhat misleading.
In stating the various results 
we have represented the tachyon by a single scalar field.
But in reality
it is inconsistent to deal only with the tachyon and not take into
account its coupling with infinite number of other fields 
representing the massive
string states.
Thus in order to study the classical dynamics of the tachyon field, we
actually have to solve infinite number of coupled equations involving 
infinite
number of fields.

There are various approaches to this problem, but I shall discuss only 
two of them in some detail. We can use an
indirect approach where
we use the fact that there is a one to one correspondence between
solutions of equations of motion in string theory and two dimensional
conformal field theories.
In this approach we directly try to get a solution of the equations of
motion (describing the defect solutions or the time dependent rolling 
tachyon solution 
for example) by constructing the
corresponding conformal field theory in two dimensions.
This avoids the need to find the tachyon potential or its
coupling to other fields. 
This procedure has been used to derive analytical results both for static
and dynamical properties\cite{9808141,9904207,0203211,0203265}.
In the direct approach (based on string field 
theory)\cite{kostsam,9605088,9912249,0001084,0002237,0002211,0005036}
we take into account the coupling of the tachyon to
all
the other fields and try to solve the coupled equations for all the fields
using some approximation scheme, known as level truncation.
In this scheme, we include only fields below a certain fixed mass 
(say $M$). 
This 
gives
a finite number of fields, and the
corresponding equations can be solved (numerically).
Then we include more fields, with mass below $M'$ ($M'>M$) and repeat 
the procedure.
If the procedure converges as we go to larger and larger cut-off on the 
mass, then we are on the right track.
So far this procedure has been used to study only the static properties of
the tachyon. In these applications
the results converge rapidly to the conjectured answers.

There are various other approaches all of which has been successful to
various extents in studying the properties of the tachyon. I shall only 
list them here without giving any details:
\begin{enumerate}
\item Renormalization group flow\cite{0003101}
\item Non-commutative geometry\cite{0005031,0005006}
\item Boundary string field theory\cite{0009103,0009148,0010108}
\end{enumerate}
etc.



Whatever be the method used, some of the results stated earlier 
({e.g.} disappearance of open string states at the bottom of 
the tachyon potential, asymptotic vanishing of pressure for the 
rolling tachyon solution etc.) certainly 
seem very strange from the point of view of a conventional 
scalar field theory in which the action is given by the sum of a kinetic 
and a potential term. We can now ask if it is possible to write down an 
(unconventional) scalar field theory that can describe this 
apparently strange dynamics of the tachyon. It turns out that the 
qualitative aspects of the tachyon 
dynamics near the minimum of 
the potential is 
describable in terms of a non-standard action for the tachyon field 
$T$\cite{0003122,0003221,0204143},
\be \label{e8}
-\int d^{p+1} x \,  V(T) \, \sqrt{1 + 
\eta^{\mu\nu} \p_\mu T
\p_\nu T}\, ,
\ee
where 
$\eta$ is the diagonal matrix with eigenvalues 
$(-1, 1, 1, \ldots 1)$, and $V(T)$ is the tachyon effective potential 
which in 
this parametrization has its minima at $T=\pm\infty$ and its maximum at 
$T=0$. 

Various features of this theory makes this an attractive model for 
describing open string tachyon dynamics. First of all it has kink 
solutions with zero thickness but finite tension describing a BPS 
D$(p-1)$-brane, whose world-volume 
theory is given by the Dirac--Born--Infeld 
action\cite{0011226,0208217,0104218,0102174,0012080,0303057}. This 
is the expected result in full string theory.
The time dependent solutions describing the rolling of the tachyon is best 
described in the Hamiltonian 
formalism. The Hamiltonian for this system is given 
by\cite{0009061,0204143}
\be \label{e9}
H = \int d^p x \sqrt{\Pi^2 + (V(T))^2} \, \sqrt{1 + (\vec\nabla T)^2}\, ,
\ee
where $\Pi$ is the momentum conjugate to $T$. As the tachyon rolls 
down the potential hill, 
$V(T)\to 0$. Thus at late time we can ignore the $V(T)$ term in the 
Hamiltonian. It can be shown that in this limit the equations of motion 
derived from the Hamiltonian \refb{e9} are identical to the equations of 
motion of a pressureless non-interacting fluid (dust) with the 
identification that $|\Pi|\sqrt{1 + (\vec\nabla T)^2}$ is interpreted as 
the energy density $\rho$ of the dust, and $-\p_\mu T$ is interpreted as 
the local $(p+1)$-velocity $u_\mu$ of the dust particle\cite{0204143}.
Thus
at late time the classical solutions in this field theory are in one to 
one 
correspondence with the 
configurations of a system of non-interacting dust. Since dust particles at 
rest correspond to a pressureless fluid, this automatically explains the 
result as to why the solutions describing a homogeneous rolling tachyon 
evolve into a system of zero pressure. It is also easy to explain the 
vanishing of the dilaton charge. For a D-brane the dilaton charge is 
proportional to the lagrangian. At late time $V(T)\to 0$. Furthermore, 
equations of motion derived from the Hamiltonian \refb{e9} gives 
$\eta^{\mu\nu}\p_\mu T \p_\nu T = -1$.  {}From \refb{e8} we see that for 
such a configuration the lagrangian and hence the dilaton charge vanishes.
Finally, since a dust does not support propagating plane waves (a 
compressed dust 
remains compressed for example), we expect that quantization of this 
theory around the minimum of the potential does not lead to particle like 
excitations. This is what happens for the open string theory around the 
tachyon vacuum. 

Although the qualitative features of the tachyon effective action given 
above do not depend on the specific choice of the potential 
$V(T)$ as long as it is even, has a maximum at $T=0$ and approaches 
zero as $T\to\pm\infty$, it turns out that the choice $V(T)\propto {\rm 
sech}(T/\sqrt 2)$ also reproduces some (but not all) of the quantitative 
aspects of tachyon dynamics on a non-BPS 
D-brane\cite{0301076,0303139,0304197}. Although 
originally the action \refb{e9} was proposed in order to reproduce the 
results involving the open string tachyon without any first principle 
derivation, a partial justification of the 
validity of this action has been given in \cite{0304045}.

So far we have discussed the dynamics of the open string tachyon at the 
purely classical level, and have ignored the coupling of the 
D-brane to closed strings. Since D-branes act as source for various closed 
string fields, a time dependent open string field 
configuration such as the rolling tachyon solution acts as a time 
dependent source for closed string fields, and produces closed string 
radiation. This can be computed using the standard techniques. For 
unstable Dp-branes with all $p$ directions wrapped on circles, we get
the following result\cite{0303139,0304192}:

\begin{enumerate}
\item Total amount of energy carried by any single closed string mode is a 
negligible fraction ($\sim g_s$) of the total energy of the D-brane.

\item Total amount of energy carried by all the closed string modes is 
infinite.

\item For an emitted closed string of mass $M$, the typical momentum 
carried by the string along directions transverse to the D-brane is of 
order $\sqrt{M}$, and the typical winding charge carried by the string 
along directions tangential to the D-brane is of order $\sqrt{M}$.

\end{enumerate}

Clearly since the D-brane has finite mass, the total amount of energy 
radiated by the D-brane cannot be infinite. The most naive way to 
regularize the divergence is to put an explicit cut-off on the energy of 
the emitted closed string, the natural cut-off being of the order of the 
mass of the original D-brane. Since this is of order $1/g_s$, the 
conclusions 1-3 above get modified to:
\begin{enumerate}
\item All the energy of the D-brane is radiated away into closed strings 
even though any single closed string mode carries a small ($\sim g_s$) 
fraction of the D-brane energy.

\item Most of the energy is carried by closed strings of mass $\sim 
1/g_s$.

\item The typical momentum 
carried by these closed strings along directions transverse to the 
D-brane is of 
order $\sqrt{1/g_s}$, and the typical winding charge carried by these 
strings
along directions tangential to the D-brane is also of order 
$\sqrt{1/g_s}$.

\end{enumerate}

{}From this one would tend to conclude that the effect of closed string 
emission should invalidate the classical open string results on the 
rolling tachyon system discussed earlier. There are however some 
surprising coincidences:
\begin{enumerate}
\item The classical open string results tell us that the final system 
associated with the rolling tachyon configuration has 
zero pressure. On the other hand closed string emission results tell us 
that the final closed strings have momentum/mass and winding/mass ratio of 
order $\sqrt{g_s}$ and hence pressure/ energy density ratio of order 
$g_s$. In the $g_s\to 0$ 
limit this vanishes. Thus it appears that the classical open string 
analysis correctly predicts the equation of state of the final system of 
closed strings into which the system decays.

\item The open string analysis tells us that the final system has zero 
dilaton charge. By analyzing the properties of the closed string radiation 
produced by the decaying D-brane one finds that these closed strings also 
carry zero dilaton charge. Thus the classical open string analysis 
correctly captures the properties of the final state closed strings 
produced during the D-brane decay.

\end{enumerate}

These results suggest that the classical open string theory already 
knows about the properties of the final state closed strings produced by 
the decay of the D-brane\cite{0305011,0306137}. This can be formally 
stated as an open string completeness
conjecture according to which the complete dynamics 
of a 
D-brane is captured by the open string theory without any need to 
explicitly consider the coupling of the system to closed 
strings.\footnote{\,Previously this was called the open-closed string 
duality conjecture\cite{0306137}. However since there are many different 
kinds of open-closed string duality conjecture, we find the name open 
string completeness conjecture more appropriate. In fact the proposed 
conjecture is not a statement of equivalence between the open and closed 
string description since the closed string theory could have many more 
states which are not accessible to the open string theory.} Closed strings 
provide 
a dual description of the system. This does not imply that any arbitrary 
state in string theory can be described in terms of open string theory on 
an unstable D-brane, but that all the quantum states required to 
describe the dynamics of a given D-brane 
are contained in this open string theory.

At the level of critical string theory one cannot prove this conjecture. 
However it turns out that this conjecture has a simple realization in a 
non-critical two dimensional string theory. This theory has two equivalent 
descriptions: 1) as a regular string theory in a somewhat complicated 
background\cite{DAVID,DK} in which the world-sheet dynamics of the 
fundamental string is described by the direct sum of a free scalar field 
theory and 
the Liouville theory with central charge 25, and 2) as a theory of free 
non-relativistic 
fermions moving under a shifted inverted harmonic oscillator 
potential $-{1\over 2} \, q^2 + {1\over g_s}$\cite{GROMIL,BKZ,GINZIN}. 
Although in the free fermion description the potential is unbounded from 
below, the ground state of the system has all the negative energy states 
filled, and hence the second quantized theory is well defined. The map 
between these two theories is also known. In particular the closed string 
states
in the first
description are related to the quanta of the scalar field obtained by 
bosonizing the 
second quantized fermion field in the second 
description\cite{DASJEV,SENWAD,GROKLE}.

In the regular string theory description the theory also has an unstable 
D0-brane with a tachyonic mode\cite{0101152}. The classical properties of 
this tachyon 
are identical to those discussed earlier in this review in the context of 
critical string theory. In particular one 
can construct time dependent solution describing the rolling of the 
tachyon away from the maximum of the potential. Upon taking into account 
possible closed string emission effects one finds that as in the case of 
critical string theory, the D0-brane decays completely into closed 
strings\cite{0305159}.

By examining the coherent closed string field configuration produced in
the D0-brane decay, and translating this into the fermionic description
using the known relation between the closed string fields and the
bosonized fermion, one discovers that the radiation produced by `D0--brane
decay' precisely corresponds to a single fermion excitation in the theory.  
This suggests that the D0-brane should be identified as the single fermion
excitation in the matrix model\cite{0304224,0305159,0305194}. Thus its 
dynamics is described by that of
a single particle moving under the inverted harmonic oscillator potential 
with a lower-cutoff on the energy at the fermi level due to Pauli 
exclusion principle. 

Given that the dynamics of a D0-brane is described by an open string 
theory, 
we see from this analysis that in this theory, the single particle
mechanics with inverted harmonic oscillator potential corresponds to the
open string theory describing the dynamics of the D0-brane. 
A consistency check of this proposal is that the 
second derivative of the inverted harmonic oscillator potential at the 
maximum precisely matches the negative mass$^2$ of the open string tachyon 
living on the D0-brane.
This `open string theory' clearly
has the ability to describe the complete dynamics of a single D0-brane,
which corresponds to single fermion excitations in the fermionic 
description. It is possible but not necessary to describe the system in 
terms of the scalar field obtained by bosonizing the second quantized 
fermion field. This corresponds to the closed string description of the 
system. This result is completely consistent with the open string 
completeness
conjecture proposed earlier in the context of critical string theory.


We conclude by mentioning briefly the status of closed string 
tachyon condensation. Clearly we would like to know if we can
make sense of closed string tachyon that appears in the original
bosonic string theory.
Existence of the tachyon is the only thing wrong with this theory, and
hence by making sense of this tachyon we may make the theory consistent.
For this we
need to
1) establish the existence of the minimum of the potential,
and
2) find an interpretation of the physics around this minimum.
This is still an unsolved problem.
However some progress has been made in understanding other kind of closed
string
tachyons which appear in superstring theories in non-trivial 
background\cite{0108075,0405064}.
In each case that has been understood, the minimum of the tachyon
potential always corresponds to some kind of stable background.
Thus
the tachyon reflects the instability of the original background to
decay
into the new background.
Success of this analysis raises hope that perhaps the tachyon in
(25+1) dimensional bosonic string theory may also be understood in a
similar manner.

\begin{thebibliography}{99}

\bibitem{9805170}
A.~Sen,
%``Tachyon condensation on the brane antibrane system,''
JHEP {\bf 9808}, 012 (1998)
[hep-th/9805170].
%%CITATION = HEP-TH 9805170;%%

\bibitem{9911116}
A.~Sen,
%``Universality of the tachyon potential,''
JHEP {\bf 9912}, 027 (1999)
[hep-th/9911116].
%%CITATION = HEP-TH 9911116;%%

\bibitem{9805019}
A.~Sen,
%``Stable non-BPS bound states of BPS D-branes,''
JHEP {\bf 9808}, 010 (1998)
[hep-th/9805019].
%%CITATION = HEP-TH 9805019;%%

\bibitem{9808141}
A.~Sen,
%``SO(32) spinors of type I and other solitons on brane-antibrane pair,''
JHEP {\bf 9809}, 023 (1998)
[hep-th/9808141].
%%CITATION = HEP-TH 9808141;%%

\bibitem{9810188}
E.~Witten,
%``D-branes and K-theory,''
JHEP {\bf 9812}, 019 (1998)
[hep-th/9810188].
%%CITATION = HEP-TH 9810188;%%

\bibitem{9812031}
A.~Sen,
%``BPS D-branes on non-supersymmetric cycles,''
JHEP {\bf 9812}, 021 (1998)
[hep-th/9812031].
%%CITATION = HEP-TH 9812031;%%

\bibitem{9812135}
P.~Horava,
%``Type IIA D-branes, K-theory, and matrix theory,''
Adv.\ Theor.\ Math.\ Phys.\  {\bf 2}, 1373 (1999)
[hep-th/9812135].
%%CITATION = HEP-TH 9812135;%%

\bibitem{9904207}
A.~Sen,
{\em Non-BPS states and branes in string theory}, hep-th/9904207.
%%CITATION = HEP-TH 9904207;%%

\bibitem{0203211}
A.~Sen,
%``Rolling tachyon,''
JHEP {\bf 0204}, 048 (2002)
[hep-th/0203211].
%%CITATION = HEP-TH 0203211;%%

\bibitem{0203265}
A.~Sen,
%``Tachyon matter,''
JHEP {\bf 0207}, 065 (2002)
[hep-th/0203265].
%%CITATION = HEP-TH 0203265;%%

\bibitem{kostsam}
V.~A.~Kostelecky and S.~Samuel,
%``On A Nonperturbative Vacuum For The Open Bosonic String,''
Nucl.\ Phys.\ B {\bf 336}, 263 (1990).
%%CITATION = NUPHA,B336,263;%%

\bibitem{9605088}
V.~A.~Kostelecky and R.~Potting,
%``Expectation Values, Lorentz Invariance, and CPT in the Open Bosonic 
%String,''
Phys.\ Lett.\ B {\bf 381}, 89 (1996)
[hep-th/9605088].
%%CITATION = HEP-TH 9605088;%%

\bibitem{9912249}
A.~Sen and B.~Zwiebach,
%``Tachyon condensation in string field theory,''
JHEP {\bf 0003}, 002 (2000)
[hep-th/9912249].
%%CITATION = HEP-TH 9912249;%%

\bibitem{0001084}
N.~Berkovits,
%``The tachyon potential in open Neveu-Schwarz string field theory,''
JHEP {\bf 0004}, 022 (2000)
[hep-th/0001084].
%%CITATION = HEP-TH 0001084;%%

\bibitem{0002237}
N.~Moeller and W.~Taylor,
%``Level truncation and the tachyon in open bosonic string field theory,''
Nucl.\ Phys.\ B {\bf 583}, 105 (2000)
[hep-th/0002237].
%%CITATION = HEP-TH 0002237;%%

\bibitem{0002211}
N.~Berkovits, A.~Sen and B.~Zwiebach,
%``Tachyon condensation in superstring field theory,''
Nucl.\ Phys.\ B {\bf 587}, 147 (2000)\\{}
[hep-th/0002211].
%%CITATION = HEP-TH 0002211;%%

\bibitem{0005036}
N.~Moeller, A.~Sen and B.~Zwiebach,
%``D-branes as tachyon lumps in string field theory,''
JHEP {\bf 0008}, 039 (2000)
[hep-th/0005036].
%%CITATION = HEP-TH 0005036;%%

\bibitem{0003101}
J.~A.~Harvey, D.~Kutasov and E.~J.~Martinec,
{\em On the relevance of tachyons},\\
hep-th/0003101.
%%CITATION = HEP-TH 0003101;%%

\bibitem{0005031}
J.~A.~Harvey, P.~Kraus, F.~Larsen and E.~J.~Martinec,
%``D-branes and strings as non-commutative solitons,''
JHEP {\bf 0007}, 042 (2000)\\{}
[hep-th/0005031].
%%CITATION = HEP-TH 0005031;%%

\bibitem{0005006}
K.~Dasgupta, S.~Mukhi and G.~Rajesh,
%``Noncommutative tachyons,''
JHEP {\bf 0006}, 022 (2000)
[hep-th/0005006].
%%CITATION = HEP-TH 0005006;%%

\bibitem{0009103}
A.~A.~Gerasimov and S.~L.~Shatashvili,
%``On exact tachyon potential in open string field theory,''
JHEP {\bf 0010}, 034 (2000)
[hep-th/0009103].
%%CITATION = HEP-TH 0009103;%%

\bibitem{0009148}
D.~Kutasov, M.~Marino and G.~W.~Moore,
%``Some exact results on tachyon condensation in string field theory,''
JHEP {\bf 0010}, 045 (2000)
[hep-th/0009148].
%%CITATION = HEP-TH 0009148;%%

\bibitem{0010108}
D.~Kutasov, M.~Marino and G.~W.~Moore,
{\em Remarks on tachyon condensation in superstring field theory},
hep-th/0010108.
%%CITATION = HEP-TH 0010108;%%

\newpage

\bibitem{0003122}
M.~R.~Garousi,
%``Tachyon couplings on non-BPS D-branes and Dirac-Born-Infeld action,''
Nucl.\ Phys.\ B {\bf 584}, 284 (2000)
[hep-th/0003122];
%%CITATION = HEP-TH 0003122;%%
%``On-shell S-matrix and tachyonic effective actions,''
Nucl.\ Phys.\ B {\bf 647}, 117 (2002)
[hep-th/0209068].
%%CITATION = HEP-TH 0209068;%%
 
\bibitem{0003221}
E.~A.~Bergshoeff, M.~de Roo, T.~C.~de Wit, E.~Eyras and S.~Panda,
%``T-duality and actions for non-BPS D-branes,''
JHEP {\bf 0005}, 009 (2000)
[hep-th/0003221].
%%CITATION = HEP-TH 0003221;%%


\bibitem{0204143}
A.~Sen,
%``Field theory of tachyon matter,''
Mod.\ Phys.\ Lett.\ A {\bf 17}, 1797 (2002)
[hep-th/0204143].
%%CITATION = HEP-TH 0204143;%%

\bibitem{0011226}
J.~A.~Minahan and B.~Zwiebach,
%``Gauge fields and fermions in tachyon effective field theories,''
JHEP {\bf 0102}, 034 (2001)
[hep-th/0011226].
%%CITATION = HEP-TH 0011226;%%

\bibitem{0208217}
N.~D.~Lambert and I.~Sachs,
%``Tachyon dynamics and the effective action approximation,''
Phys.\ Rev.\ D {\bf 67}, 026005 (2003)
[hep-th/0208217].
%%CITATION = HEP-TH 0208217;%%

\bibitem{0104218}
N.~D.~Lambert and I.~Sachs,
%``On higher derivative terms in tachyon effective actions,''
JHEP {\bf 0106}, 060 (2001)
[hep-th/0104218].
%%CITATION = HEP-TH 0104218;%%

\bibitem{0102174}
K.~Hashimoto and S.~Hirano,
%``Metamorphosis of tachyon profile in unstable D9-branes,''
Phys.\ Rev.\ D {\bf 65}, 026006 (2002)
[hep-th/0102174].
%%CITATION = HEP-TH 0102174;%%

\bibitem{0012080}
G.~Arutyunov, S.~Frolov, S.~Theisen and A.~A.~Tseytlin,
%``Tachyon condensation and universality of DBI action,''
JHEP {\bf 0102}, 002 (2001)
[hep-th/0012080].
%%CITATION = HEP-TH 0012080;%%

\bibitem{0303057}
A.~Sen,
%``Dirac-Born-Infeld action on the tachyon kink and vortex,''
Phys.\ Rev.\ D {\bf 68}, 066008 (2003)
[hep-th/0303057].
%%CITATION = HEP-TH 0303057;%%

\bibitem{0009061}
G.~W.~Gibbons, K.~Hori and P.~Yi,
%``String fluid from unstable D-branes,''
Nucl.\ Phys.\ B {\bf 596}, 136 (2001)
[hep-th/0009061].
%%CITATION = HEP-TH 0009061;%%

\bibitem{0301076}
C.~j.~Kim, H.~B.~Kim, Y.~b.~Kim and O.~K.~Kwon,
%``Electromagnetic string fluid in rolling tachyon,''
JHEP {\bf 0303}, 008 (2003)\\{}
[hep-th/0301076].
%%CITATION = HEP-TH 0301076;%%

\bibitem{0303139}
N.~Lambert, H.~Liu and J.~Maldacena,
{\em Closed strings from decaying D-branes},\\
hep-th/0303139.
%%CITATION = HEP-TH 0303139;%%

\bibitem{0304197}
P.~Brax, J.~Mourad and D.~A.~Steer,
%``Tachyon kinks on non BPS D-branes,''
Phys.\ Lett.\ B {\bf 575}, 115 (2003)
[hep-th/0304197].
%%CITATION = HEP-TH 0304197;%%

\bibitem{0304045}
D.~Kutasov and V.~Niarchos,
%``Tachyon effective actions in open string theory,''
Nucl.\ Phys.\ B {\bf 666}, 56 (2003)
[hep-th/0304045].
%%CITATION = HEP-TH 0304045;%%


\bibitem{0304192}
D.~Gaiotto, N.~Itzhaki and L.~Rastelli,
%``Closed strings as imaginary D-branes,''
Nucl.\ Phys.\ B {\bf 688}, 70 (2004) [hep-th/0304192].
%%CITATION = HEP-TH 0304192;%%

\bibitem{0305011}
A.~Sen,
%``Open and closed strings from unstable D-branes,''
Phys.\ Rev.\ D {\bf 68}, 106003 (2003) [hep-th/0305011].
%%CITATION = HEP-TH 0305011;%%

\bibitem{0306137}
A.~Sen,
%``Open-closed duality at tree level,''
Phys.\ Rev.\ Lett.\  {\bf 91}, 181601 (2003)
[hep-th/0306137].
%%CITATION = HEP-TH 0306137;%%

\bibitem{DAVID}
F.~David,
%``Conformal Field Theories Coupled To 2-D Gravity In The Conformal
%Gauge,''
Mod.\ Phys.\ Lett.\ A {\bf 3}, 1651 (1988).
%%CITATION = MPLAE,A3,1651;%%

\bibitem{DK}
J.~Distler and H.~Kawai,
%``Conformal Field Theory And 2-D Quantum Gravity Or Who's Afraid Of
%Joseph
%Liouville?,''
Nucl.\ Phys.\ B {\bf 321}, 509 (1989).
%%CITATION = NUPHA,B321,509;%%

\bibitem{GROMIL}
D.~J.~Gross and N.~Miljkovic,
%``A Nonperturbative Solution Of D = 1 String Theory,''
Phys.\ Lett.\ B {\bf 238}, 217 (1990).
%%CITATION = PHLTA,B238,217;%%

\bibitem{BKZ}
E.~Brezin, V.~A.~Kazakov and A.~B.~Zamolodchikov,
%``Scaling Violation In A Field Theory Of Closed Strings In One Physical
%Dimension,''
Nucl.\ Phys.\ B {\bf 338}, 673 (1990).
%%CITATION = NUPHA,B338,673;%%

\bibitem{GINZIN}
P.~Ginsparg and J.~Zinn-Justin,
%``2-D Gravity + 1-D Matter,''
Phys.\ Lett.\ B {\bf 240}, 333 (1990).
%%CITATION = PHLTA,B240,333;%%

\bibitem{DASJEV}
S.~R.~Das and A.~Jevicki,
%``String Field Theory And Physical Interpretation Of D = 1 Strings,''
Mod.\ Phys.\ Lett.\ A {\bf 5}, 1639 (1990).
%%CITATION = MPLAE,A5,1639;%%

\bibitem{SENWAD}
A.~M.~Sengupta and S.~R.~Wadia,
%``Excitations And Interactions In D = 1 String Theory,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 6}, 1961 (1991).
%%CITATION = IMPAE,A6,1961;%%

\bibitem{GROKLE}
D.~J.~Gross and I.~R.~Klebanov,
%``Fermionic String Field Theory Of C = 1 Two-Dimensional Quantum
%Gravity,''
Nucl.\ Phys.\ B {\bf 352}, 671 (1991).
%%CITATION = NUPHA,B352,671;%%




\bibitem{0101152}
A.~B.~Zamolodchikov and A.~B.~Zamolodchikov,
{\em Liouville field theory on a pseudosphere},
hep-th/0101152.
%%CITATION = HEP-TH 0101152;%%

\bibitem{0305159}
I.~R.~Klebanov, J.~Maldacena and N.~Seiberg,
%``D-brane decay in two-dimensional string theory,''
JHEP {\bf 0307}, 045 (2003) \\{} [hep-th/0305159].
%%CITATION = HEP-TH 0305159;%%

\bibitem{0304224}
J.~McGreevy and H.~Verlinde,
{\em Strings from tachyons: The c = 1 matrix reloated},
hep-th/0304224.
%%CITATION = HEP-TH 0304224;%%

\bibitem{0305194}
J.~McGreevy, J.~Teschner and H.~Verlinde,
{\em Classical and quantum D-branes in 2D string theory},
hep-th/0305194.
%%CITATION = HEP-TH 0305194;%%

\bibitem{0108075}
A.~Adams, J.~Polchinski and E.~Silverstein,
%``Don't panic! Closed string tachyons in ALE space-times,''
JHEP {\bf 0110}, 029 (2001)
[hep-th/0108075].
%%CITATION = HEP-TH 0108075;%%


\bibitem{0405064}
M.~Headrick, S.~Minwalla and T.~Takayanagi,
%``Closed string tachyon condensation: An overview,''
Class.\ Quant.\ Grav.\  {\bf 21}, S1539 (2004)
[hep-th/0405064].
%%CITATION = HEP-TH 0405064;%%

\end{thebibliography}



\end{document}

\bye

This finishes our discussion of the formal aspects of tachyon dynamics in 
open string theory.
Let us now briefly discuss a possible application of brane-antibrane 
annihilation process to inflationary cosmology. Typically
inflation requires a (set of) scalar field(s) with reasoanbly flat
potential near the
top and a minimum where the potential vanishes.
The question is:
can we achieve this by a brane-antibrane pair or some
variation of this configuration?
The brane-antibrane pair has large energy near the top of the potential, 
but the potential is not flat near the top. Thus this system does not seem 
to be suitable for inflation.
One 
possible way to achieve a flat potential is to separate the brane and the 
antibrane
in space\cite{0105203,0105204}.
To be more specific, 
consider (9+1) dimensional string theory with 6 of the dimensions wrapped
into a compact space $K$.
Take a D3-brane along the 3 non-compact directions, placed at a given
point in $K$, and 
an $\bd$3-brane along the same direction but placed at a different
point of $K$.
The D-$\bd$ system has large energy density, but for sufficiently large
separation there is no tachyon.
They however have an attractive weak gravitational potential, and under 
its influence the brane and the anti-brane will slowly roll towards each 
other.
{}From the point of view of a (3+1) dimensional observer, the separation
between branes is interpreted as a scalar field $\phi$.
For large value of $\phi$ the potential is almost flat, but there is a 
small potential that drives $\phi$ towards smaller value.
Thus if the universe started out with a large $\phi$ which then slowly
decreased towards zero the universe will inflate during this phase.
When $\phi$ becomes smaller than a critical value the tachyon 
develops and since 
the tachyon potential is very steep, 
the system quickly rolls down towards the minimum of the 
potential.
In this scenario the tachyon dynamics is important for understanding end
of inflation / reheating process.
The details of this reheating process
are still to be explored.


