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  {\Large\cite{#1}}\hspace{-0.1mm}}

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\markboth{\quad \small \em Yaron Oz\hfill}{\hfill \small \em 
Aspects of tachyon condensation in string theory\quad}

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\begin{document}
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\title{ ASPECTS OF TACHYON CONDENSATION\\[1mm] IN STRING THEORY}
\author{YARON OZ}
\address{Theory Division, CERN, CH-1211, Geneva  23, Switzerland\\
and\\
Raymond and Beverly Sackler Faculty of Exact Sciences,\\
School of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv
69978, Israel}

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% You may repeat \author \address as often as necessary      %
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\maketitle\abstracts{
  We suggest a correspondence between brane-antibrane systems and
  stable triples $(E_1,E_2,T)$, where $E_1,E_2$ are holomorphic vector
  bundles and the tachyon $T$ is a map between them.  Under the
  assumption of holomorphicity, the brane-antibrane field equations
  reduce to a set of vortex equations, which are equivalent to the
  mathematical notion of stability of the triple. We discuss some
  examples and show that the theory of stable triples suggests a new
  notion of BPS bound states and stability.  We then propose a form of
  the effective action of the tachyon and gauge fields for
  brane-antibrane systems, written in terms of the supercurvature.
  Kink and vortex solutions with constant infinite gauge field
  strength reproduce the exact tensions of the lower-dimensional
  D-branes.  }

%\vspace{0.5cm}


{\small\tableofcontents}

\newpage


\section{Introduction}


Systems of non-BPS brane configurations have been extensively
studied recently (for a review see Ref.\,\protect\citebk{review}).
A basic non-BPS system is the coincident brane-antibrane
configuration, which is not stable. It has a tachyon on the
world-volume of the branes that arises from the open string
stretched between the branes and the antibranes, and it is charged
under the world-volume gauge groups.  The decay of the system can
be seen by the tachyon rolling down to the minimum of its
potential. Upon tachyon condensation one can end up with lower
dimensional BPS branes, if the original brane-antibrane system
contained the corresponding charges.

Most of the analysis of branes-antibranes systems is carried out in
flat space.
It is clearly of interest to extend the available methods
to  consider curved closed string backgrounds.
In the following we will discuss some aspects of such systems in
curved spaces.
First,
we will
propose a correspondence between brane-antibrane systems and stable
triples $(E_1,E_2,T)$, where $E_1,E_2$ are holomorphic vector bundles
and the tachyon, $T$, is a map between them. Under the assumption of holomorphicity, the
brane-antibrane field equations reduce to a set of vortex
equations. The latter are equivalent to the topological notion of
stability of the triple $(E_1,E_2,T)$. 
This is quite analogous to the case of a single vector bundle where
solutions of Hermitian Yang-Mills equations correspond to stable
holomorphic bundles. We discuss some examples and
show that the theory of stable triples suggests a new notion of BPS
bound states upon tachyon condensation.

Second, we will consider the higher order terms in the tachyon and
gauge fields effective action.
We 
 will use the notion of superconnections, which
when 
considering the branes-antibranes system 
appears naturally via the Chan-Paton factors. We will
make the assumption that the  effective action of
tachyon and gauge fields 
for the $Dp-\bar{Dp}$-branes system can be written
in a Quillen-like framework
in terms of the supercurvature.
The tachyon potential that arises in this framework is exponential in the tachyon field.
We will 
suggest a form of the  effective action and use it to
study the process of tachyon
condensation. 
Kink solutions that we will find, with infinite constant value of the gauge field
strength,
reproduce the exact tensions  of the lower-dimensional
D-branes at the minimum of the tachyon potential. 

This article  is organized as follows. In Section 2 we will introduce
the triples, the vortex equations and propose the
correspondence.
We will then consider some examples.
In Section 3 we will relate our description of BPS D-branes as stable
triples to existing descriptions of BPS states. In particular, we will
show that the theory of stable triples suggests a new notion of BPS
bound states and stability. 
In Section 4 we will introduce the notion of superconnections and
supercurvatures,
propose an effective action of
the tachyon and gauge fields for branes-antibranes systems, 
study the 
kink solutions and derive
the exact tensions  of the lower-dimensional
Dp-branes.


This article is mainly based on Refs.\,\protect\citebk{Oz:2000bs} and
\protect\citebk{Alishahiha:2000du}. I omitted many relevant works,
which can be found in the reference lists of these papers.







\section{Branes-antibranes systems}

In this section we will propose a correspondence between systems of  
coinciding branes and antibranes wrapping a manifold $X$
and stable triples $(E_1,E_2,T)$. $E_1$ and $E_2$ are holomorphic
vector bundles on $X$. Physically, they correspond to the branes and
antibranes respectively. $T$ is a homomorphism between the vector
bundles $T:E_2 \rightarrow E_1$. It is the tachyon field
that arises from the open string stretched between the branes and the
antibranes.
With a holomorphic ansatz, we
will recast the field equations of the brane-antibrane system as a set
of vortex equations. The solutions of the vortex equations
represent BPS configurations. Such solutions correspond to a
mathematical construction of stable triples on $X$.\,\cite{Brad} With
this correspondence the analysis of tachyon condensation leading to
BPS branes will be replaced by a stability analysis. 


\subsection{Tachyons and triples}

Let us  briefly review the construction of D-branes from
coincident brane-antibrane configurations. We will
consider Type IIA string theory compactified on
Calabi--Yau manifold $X$ of complex dimension $d$. 

A configuration of $n$ branes wrapping the Calabi--Yau manifold is
described by a $U(n)$ vector bundle $E$ on $X$. This carries charges
for various RR fields  
%
\begin{equation}
   Q = \ch(E)\sqrt{\Ahat(X)} = \ch(E)\sqrt{Td(X)}\ ,
\label{ve}
\end{equation}
%
which is an element of the cohomology $H^*(X,Z)$ known as the Mukai
vector. For the equality in (\ref{ve}) we used the fact that
$Td(X)$ on a Calabi--Yau manifold is equal to the
A-roof genus $\Ahat(X)$. 
In this expression $\ch(E)$ is the Chern character of the
vector bundle $E$  
%
\begin{equation}
   \ch(E) = Tr \exp\left[\frac{F}{2\pi}\right] \ ,
\label{che}
\end{equation}
%
where $F$ is the field strength of the gauge field on the brane.
It has an expansion in terms of the
Chern classes  
%
\begin{equation}
   \ch(E) = n + c_1(E) + \frac{1}{2}(c_1^2(E) - 2c_2(E)) 
      + \dots \ . 
\label{ee}
\end{equation}
%
The 
A-roof genus $\Ahat(X)$ and has an expansion in terms of the
Pontrjagin classes 
%
\begin{equation}
   \Ahat(X) = 1 - \frac{p_1(X)}{24} + \dots \ .
\label{eA}
\end{equation}

Consider a configuration of $n_1$ branes wrapped on $X$,
together with $n_2$ antibranes. In general, the configuration 
is described by specifying a $U(n_1)$ vector bundle $E_1$ on $X$ for
the branes together with a second $U(n_2)$ bundle $E_2$ for the
antibranes. The net D-brane charge is then the difference of the Mukai
vectors for the two bundles
%
\begin{equation}
   Q = Q_1 - Q_2 =
   \left(\ch(E_1)-\ch(E_2)\right)\sqrt{\Ahat(X)} \ .
\label{eq:Q}
\end{equation}
%
In general, we would expect the antibranes to annihilate against the
branes. However, if the bundles are different then there is a net
D-brane charge and the branes cannot completely annihilate and still
conserve charge.  

Since identical bundles can annihilate, adding
the same bundle to $E_1$ and $E_2$ gives the same physical
configuration. That is we should identify $(E_1\oplus V,E_2\oplus V)$ with
$(E_1,E_2)$, which is the equivalence class identification made in
K-theory. In K-theory one identifies the Chern class of
the pair $(E_1,E_2)$ with the difference of Chern classes
$\ch(E_1)-\ch(E_2)$, thus providing the map from K-theory to the
Mukai charge (\ref{eq:Q}). In fact, a D-brane charge is more accurately
measured by the K-theory class rather than the cohomological Mukai
charge. In particular, K-theory includes more information than the
Chern classes themselves. For instance, Chern classes miss torsion.

Physically, the annihilation happens because there is a tachyonic mode
$T$ in the open string connecting the branes and antibranes. The
tachyon potential has a minimum away from zero. However, if the
bundles $E_1$ and $E_2$ are different, there is a topological
obstruction to the tachyon being at the minimum everywhere on $X$. The
tachyon $T$ transforms in the fundamental representation of each
bundle $({\bf n_2, \bar{n}_1})$, and it must respect the
twisting of each bundle. In general, even if $n_1=n_2$ if the bundles
are different, it cannot do so and remain everywhere at the minimum of
the tachyon potential. Instead it must be zero on some sub-manifold
$C$ of $X$. There is a vortex solution representing a
lower-dimensional brane localized on $C$. In particular, all the lower
dimensional branes can be built out of D9-branes in this
way.\cite{wittenk}

To specify a general D-brane configuration we need to
specify the bundles $E_1$ and $E_2$ together with the condensed
tachyon $T$. Since $T$ is in the bi-fundamental representation, it
represents a map between the bundles. Thus the full information is the
triple $(E_1,E_2,T)$ giving the complex 
%
\begin{equation}
   E_2 \stackrel{T}{\longrightarrow} E_1 \ .
\label{eq:triple}
\end{equation}
%
As we have noted, the D-brane charges are characterized by the
K-theory class of the pair $(E_1,E_2)$. However for a given D-brane
charge there is generically a moduli space of different D-brane
states. It is natural to ask what characterizes these distinct
D-brane states. In general, this should be some equivalence
class of triples $(E_1,E_2,T)$, giving a finer classification than
simply the K-theory class. In the following, we will consider BPS
configurations. We will see that this implies that the bundles and
maps are holomorphic. A possible equivalence class 
is that for holomorphic
bundles $E_1$ and $E_2$, we should identify triples in the same
derived category, which essentially means considering
complexes of bundles of the form (\ref{eq:triple}), modulo exact
sequences.  

\subsection{The vortex equations}

Consider the low-energy effective action of the
world-volume theory of a  configuration of coinciding $n_1$
branes and $n_2$ antibranes wrapping a manifold $X$
\begin{equation}
   S = \int_X \left[ \frac{1}{4}\tr_1 F_1^2 + \frac{1}{4}\tr_2 F_2^2 
          + (DT)^a_{\bar{b}}(DT^*)^{\bar{b}}_{a}
          + \l \left(T^a_{\bar{b}}T^{*\bar{b}}_b-\a^2\d^{a}_{b}\right)^2 \right] \ .
\label{eq:action}
\end{equation}
%
Typically, we will take $n_1=n_2$, so only lower-dimensional branes
remain after condensation. There are higher order corrections to
(\ref{eq:action}), and in general one also expects
the kinetic terms of the tachyon and the
gauge fields to depend on the tachyon background. Such corrections
modify the field equations and the precise description
of the tachyon rolling to the minimum of its potential.
We expect, however, that it should not matter
for the topological construction of the lower-dimensional
branes upon the condensation of the tachyon. Here we think about the
lower-dimensional branes as the BPS branes.
Quantitative properties of the lower-dimensional
branes such as the size of vortex solutions will be modified, upon the
inclusion of the corrections.

We further assume the same gauge coupling for the two gauge groups
and rescaled the gauge and tachyon fields in  (\ref{eq:action}).
In (\ref{eq:action}), $a$ is the index of the fundamental
representation of $E_1$ and $\bar{a}$ the anti-fundamental of $E_2$.
The parameter $\a^2$ in the tachyon potential is related to the value
of the tachyon field at the minimum of the potential
%
\begin{equation}
   \a^2 = \frac{1}{n_1}\tr(TT^*)|_{minimum} \ .
\label{a2}
\end{equation} 
%
Since $T$ has charge $\pm e$ under the gauge groups its covariant
derivative is
%
\begin{equation}
   D_M T^a_{\bar{b}} = \partial_M T^a_{\bar{b}} 
        + ie(A^1_M)^a_{b} T^b_{\bar{b}}
        - ieT^a_{\bar{a}}(A^2_M)^{\bar{a}}_{\bar{b}} \ .
\label{eq:D}
\end{equation}
%
From now on we will suppress the indices and write, for instance
$DT=dT+ieA_1T-ieTA_2$. 

The equations of motion  read
%
\begin{eqnarray}
   D_1^M F^1_{MN} &=& ie\left[
      T\left(D_NT^*\right) - \left(D_NT\right)T^* \right] \ , \nonumber\\
   D_2^M F^2_{MN} &=& ie\left[
      T^*\left(D_NT\right) - \left(D_NT^*\right)T \right] \ , \nonumber\\
   D^2 T &=& 2\l \left(TT^*T - \a^2T\right) \ ,
\label{eq:eom}
\end{eqnarray}
%
where $D_1=d+ieA_1$ and $D_2=d+ieA_2$. 

We denote the K\"ahler metric on $X$ by $g_{m\bar{n}}$, where $m$ is a
holomorphic index and $\bar{n}$ an anti-holomorphic index. There is then
a set of equations which imply the equations of motion.\cite{Oz:2000bs}
They are, first, that all the fields are holomorphic, namely
%
\begin{eqnarray}
   F^1_{mn} &=& F^1_{\bar{m}\bar{n}} = 0 \ , \nonumber\\
   F^2_{mn} &=& F^2_{\bar{m}\bar{n}} = 0 \ , \nonumber\\
   D_{\bar{m}}T &=& 0 \ .
\label{eq:holo}
\end{eqnarray}
%
Then in addition we have a Hermitian condition
%
\begin{eqnarray}
   ig^{m\bar{n}}F^1_{m\bar{n}} + eTT^* &=& 2\p\t_1 I_1 \ , \nonumber\\
   ig^{m\bar{n}}F^2_{m\bar{n}} - eT^*T &=& 2\p\t_2 I_2 \ ,
\label{eq:vortex}
\end{eqnarray}
%
where $I_{1},I_{2}$ are the identity matrices for the $E_1$ and $E_2$
bundles respectively. Together we shall call equations (\ref{eq:holo})
and (\ref{eq:vortex}) the {\it vortex
equations}. The important point, as we discuss in the next subsection,
is that solutions of the vortex equations (\ref{eq:holo}) and
(\ref{eq:vortex}) are in one-to-one correspondence with the
topological notion of stability of the triple $(E_1,E_2,T)$.\,\cite{Brad}

This requires that the
parameters in the action are related so that
%
\begin{equation}
   \l = e^2 \ .
\label{eq:lm-e}
\end{equation}
%
We also get a relation between $\t_1$, $\t_2$ and $\a$
%
\begin{equation}
   e\a^2 = \p \left( \t_1 - \t_2 \right) \ .
\end{equation}
%
Note that the relation between $\l$ and $e$, together with the
assumption that the height of the tachyon potential is the tension of
the brane system ${\cal T}_p$, imply that the tachyon charge $e$ is related to
the value of the tachyon field at the minimum of its potential by
%
\begin{equation}
   e^2 \sim {\cal T}_p \left(
       \frac{1}{n_1}\left.\tr(TT^*)\right|_{minimum}
       \right)^{-2} \ .
\end{equation}



If we add that vortex equations, take a trace and
integrate over $X$, we find 
%
\begin{equation}
   \t_1 n_1 + \t_2 n_2 = \deg E_1 + \deg E_2 \ ,
\label{eq:tau2}
\end{equation}
%
where the degree of a vector bundle $\deg E$ is defined as
%
\begin{equation}
   \deg E = \frac{1}{V(d-1)!} \int c_1(E) \wedge J^{d-1} \ ,
\label{eq:degree}
\end{equation}
%
with $J$ the K\"ahler form on $X$ and $c_1(E)$ is the first Chern
class. Thus we see that $\t_1$ and $\t_2$ are completely determined by
the parameter $\a$ and the bundles $E_1$ and $E_2$. In particular
%
\bea
   2\p\t_1 &=& 2\pi\frac{\deg E_1 + \deg E_2}{n_1+n_2}
      + \frac{2n_2}{n_1+n_2}\a^2 \ , \nonumber\\
   2\p\t_2 &=& 2\pi\frac{\deg E_1 + \deg E_2}{n_1+n_2}
      - \frac{2n_1}{n_1+n_2}\a^2 \ .
\label{eq:t1t2}
\eea
%

We expect that the solutions of the vortex equations are
supersymmetric BPS states. One way to establish this is
to analyze the supersymmetry directly. Another way, which 
is to show that these solution satisfy the Bogomol'nyi 
bound.\cite{Oz:2000bs}

With the relations (\ref{eq:lm-e}) and
(\ref{eq:t1t2}) we have an interesting correspondence. Brane
configurations where all fields are holomorphic and that arise via the
process of tachyon condensation are described by solutions to the
vortex equations (\ref{eq:holo}) and (\ref{eq:vortex}). There is one
dimensionful parameter, $\a^2$, in the equations which is related to
the value of the tachyon at the minimum of the potential and so scales
as the string scale. We should emphasize that the holomorphicity conditions  
(\ref{eq:holo}) limit our discussion to tachyon condensation that
leads to BPS branes. In order to study stable non-BPS branes we would
have to relax these conditions. 


\subsection{Stable triples}

A particularly useful property of the vortex equations is that
their solutions are in one-to-one correspondence with a topological notion
of stability of the triple $(E_1,E_2,T)$. Thus we can use stability to
analyze the existence of solutions on a general $X$, rather than
looking for solutions explicitly. 

The analogy here is to the Hermitian Yang--Mills equations (HYM)
describing the supersymmetric compactification of a single gauge
bundle $E$ on a Calabi--Yau manifold. 
They read 
%
\begin{equation}
   F_{mn} = F_{\bar{m}\bar{n}} = 0 \ ,\qquad 
   ig^{m\bar{n}}F_{m\bar{n}} = 2\p\t \ ,
\label{eq:HYM}
\end{equation}
%
which are a simple subset of the vortex equations. By
the Donaldson--Uhlenbeck--Yau theorem, solutions of the HYM
equations are in one-to-one correspondence with holomorphic vector
bundles of a particular type: those that are
poly-stable. This is defined as follows. Let the slope of a bundle be
given by
%
\begin{equation}
   \m(E) = \frac{\deg E}{\rk E} \ .
\label{eq:slope}
\end{equation}
%
A bundle $E$ is stable if for any non-trivial sub-bundle $E'\subset
E$, one has $\m(E')<\m(E)$. Poly-stability means that $E$ is the
direct sum of stable bundles each with the same slope. By this
theorem, an analytic problem --- solutions of the HYM equations --- is
a equivalent to a topological problem --- listing the stable bundles
--- and the latter problem is generally much easier to solve. We
should note that the stability problem is not quite topological: it
also depends on the choice of K\"ahler form $J$ via the definition of
$\deg E$. 

It turns out there that is an analogous notion of stability for a
triple $(E_1,E_2,T)$, such that there is a solution to the vortex
equations if and only if the triple is stable.\cite{Brad} One first
needs to define what is meant by a sub-triple. We take the definition
that $(E'_1,E'_2,T')$ is a sub-triple if 
%
\begin{enumerate}
\item[(1)] $E_i'$ is a coherent sub-sheaf of $E_i$ with $i=1,2$,
\item[(2)] $T'$ is the restriction of $T$.
\end{enumerate}

 Next one needs the analog of the $\mu$-slope $\m(E)$. With
$\sigma$ a real number, one defines the $\sigma$-slope of a triple
$(E_1,E_2,T)$ by
%
\begin{equation}
   \m_{\sigma}(T) = \frac{\deg(E_1\oplus E_2)+\sigma n_2}{n_1+n_2} \ .
\label{slope}
\end{equation}
%
A triple is then called $\sigma$-stable if for all nontrivial sub-triples 
$(E'_1,E'_2,T')$ we have  
%
\begin{equation}
   \m_{\sigma}(T') < \m_{\sigma}(T) \ .
\label{stable}
\end{equation}
%

The relation between solutions to the vortex equations (\ref{eq:holo})
and (\ref{eq:vortex}), and the $\sigma$-stability of the triple is for
$\sigma = \tau_1- \tau_2$. As seen, for instance, from (\ref{eq:t1t2}), 
for the brane-antibrane system it means
%
\begin{equation}
   \sigma \equiv \frac{\alpha^2}{\pi} \ .
\label{sigma}
\end{equation}
%
The BPS branes that arise upon the condensation of
the tachyon will be $\sigma$-stable with $\sigma$ given by the
relation (\ref{sigma}).
As will be very relevant later, since $\alpha$ goes like the string
scale $M_s$, the large volume limit corresponds to large $\sigma$.
This is the regime where we can trust the vortex equations to provide
an adequate description.
Note, in particular in the large $\sigma$ limit, the stability condition
(\ref{stable}) reads
%
\begin{equation}
   n_2 n_1' - n_2'n_1 > 0 \ ,
\label{theta}
\end{equation} 
%
where $n_i' = \rk \, E_i'$, for $i=1,2$, which is similar to a stability
condition on a quiver in the orbifold limit.


\subsection{BPS branes via tachyon condensation}


As a simple illustration of solutions to the
vortex equations and some of the discussion
to follow, let us consider how we can realize a D0-brane on
$C$ via the condensation of a D2-brane and an anti-D2-brane. 

The D2-branes will be realized as $U(1)$ bundles. For finite energy,
we require that the connection is pure gauge at infinity (thus we are
effectively considering bundles on $S^2=P^1$). To ensure that we
have a zero brane we need the difference of the bundle
charges (\ref{eq:Q}) to be one: 
%
\begin{equation}
   c_1(E_1) - c_1(E_2) = 1 \ .
\label{eq:DOcharge}
\end{equation}
%
For simplicity we can take $E_2$ to be trivial, while $E_1$
has $c_1(E_1)=1$.
This means that $E_1$ has a non-trivial holonomy at
infinity. 

Now consider a solution of the vortex
equations (\ref{eq:holo}). First $T$ must be holomorphic. This
implies that 
%
\begin{equation}
   \bar{\partial}T + iA^1_{\bar{z}}T - iA^2_{\bar{z}}T = 0 \ .
\label{eq:DT}
\end{equation}
%
Equation (\ref{eq:DT}) can be solved and gives 
%
\bea
A^1_z - A^2_z &=& i \partial \ln \, T \nonumber\\
&=& \partial\chi + i\partial\ln \, f \ ,
\label{eq:Adiff}
\eea
%
where we have written the tachyon as $T=fe^{i\chi}$. Note that locally
gauge transformations can always remove $\chi$. Since fields must
be pure gauge at infinity, we have $\ln \, f \to {const}$ at
infinity. The fact that $E_2$ is trivial and $E_1$ has $c_1(E_1)=1$
means that $A_2$ can be gauged to zero at infinity while $A_1$ can be
gauged to the form 
%
\begin{equation}
   A_1|_\infty = \partial \theta \ ,
\label{eq:A1}
\end{equation}
%
where $z=re^{i\theta}$, which is locally trivial but has global
holonomy. 

From this we see that we can gauge $T$ such that 
%
\begin{equation}
   \c|_\infty = \q \ .
\end{equation}
%
There must be some point $z=z_0$ in $C$ where this non-trivial
holonomy in $T$ untwists, at which point $T=0$. We can choose this to
be the origin $z=0$. In general this means we can globally gauge $T$
to the form  %
\begin{equation}
   T = f(r)e^{i\theta}
\label{eq:chi}
\end{equation}
%
such that $f(0)=0$. 

The form of $f$ can be determined by subtracting the vortex
equations (\ref{eq:vortex}). We have
%
\bea
   ig^{z\bar{z}}F^1_{z\bar{z}} - ig^{z\bar{z}}F^2_{z\bar{z}} 
      &=& i\partial\left(A^1_{\bar{z}} - A^2_{\bar{z}}\right) 
           - i\bar{\partial}\left(A^1_z - A^2_z\right) \nonumber\\
      &=& \partial\bar{\partial} \ln \, f^2 \ ,
\label{eq:diff}
\eea
%
so the difference of the vortex equations  (\ref{eq:vortex}) reads
%
\begin{equation}
   \partial\bar{\partial} \ln \, f^2 + 2f^2 = 2\p(\t_1-\t_2) = 2\a^2 \ .
\label{eq:feq}
\end{equation}
%
Finding the solution of this equation such that $f(0)=0$ and
$f(\infty)=\a$ then completely determines the vortex solution up to
gauge transformations. In particular, the individual vortex equations
give
%
\bea
   F^1_{z\bar{z}} &=& i\left(f^2 - \a^2\right) -2\p i \ , \nonumber\\
   F^2_{z\bar{z}} &=& -i\left(f^2 - \a^2\right)
\label{eq:F1F2}
\eea
%
for the two field strengths. 

We see that at infinity $|T|=\a$ and the tachyon is at the minimum of
its potential. However, as one moves around the $S^1$ at infinity the
phase of the tachyon rotates giving a vortex. The vortex untwists at
the origin where $T=0$. This is the position of the D0-brane. In
general there is a modulus to move the D0-brane to at any point in the
complex plane. 

We can do the same analysis without referring directly to the vortex
equations. On $P^1$ we have $E_1=\mathcal{O}(1)$ and
$E_2=\mathcal{O}$ the trivial bundle. Thus we have the triple 
%
\begin{equation}
   \mathcal{O} \stackrel{T}{\longrightarrow} \mathcal{O}(1) \ .
\label{eq:complexeg}
\end{equation}
%
We can think of this as a map between holomorphic functions on $P^1$
(that is constant functions) to meromorphic functions with, at most, a
single pole (at the zero-brane). However, such maps lie in an exact
sequence 
%
\begin{equation}
   \o \longrightarrow \mathcal{O} 
       \stackrel{T}{\longrightarrow} \mathcal{O}(1)
       {\longrightarrow} \mathcal{O}_P{\longrightarrow}\o \ .
\label{eq:equiv}
\end{equation}
%
That is the kernel of $T$ is zero. The cokernel, meanwhile, is simply
the sheaf of functions localized at a point $\mathcal{O}_P$. This is
precisely the set of points where $T$ vanishes. 
But since $\mathcal{O}_P$ is localized on a point $p$ it is precisely the
description of a D0-brane on $p$. Thus after condensation, $E_1$ and
$E_2$ are effectively replaced by their cokernel, representing a
D0-brane. Depending on the particular choice of the map $T$, the
D0-brane lies at different points $p$ in $P^1$.  

It is straightforward to show that the triple (\ref{eq:complexeg})
is $\sigma$-stable in the sense of (\ref{slope}) and (\ref{stable}), since
any sub-triple has $E_2'$ zero.


We can naturally generalize the previous example to construct, via
the process of tachyon condensation, supersymmetric $(2d-2)$-branes on
a $d$ complex dimensional Calabi--Yau manifold $X$.
Such branes are described by sheaves localized on a holomorphic
hypersurface $C$ in $X$.
As above let us assume that $E_2$ is the trivial $U(1)$ bundle
$\mathcal{O}_X$. We then require $c_1(E_1)=[C]$, the class of
$C$. This can be achieved by 
taking $E_1$ to be the bundle $\mathcal{O}_X(C)$. Note that in general
this bundle also induces lower-dimensional brane charges. Then, for
any map $T$ we have the exact sequence
%
\begin{equation}
   \o \longrightarrow \mathcal{O}_X 
        \stackrel{T}{\longrightarrow} \mathcal{O}_X(C)
        \longrightarrow \mathcal{O}_C(C) \longrightarrow\o \ ,
\end{equation}
%
where $\mathcal{O}_C(C)$ is a sheaf localized on $C$. As in the
previous example, this means we can replace the triple
$E_2 \stackrel{T}{\longrightarrow} E_1$ with the sheaf
$\mathcal{O}_C(C)$. Since this represents a bundle localized on $C$,
it describes a supersymmetric $(2d-2)$-brane configuration as required. 

Again, it is easy to see that the triple is $\sigma$-stable since any
sub-triple has $E_2'$ zero. 


\section{BPS bound states as stable triples}

In this section we will discuss and illustrate the description of 
BPS branes as stable triples in relation with other
existing descriptions of BPS branes. 

\subsection{Stable sheaves}

Let us now relate our description of BPS D-branes as stable triples to
existing descriptions of BPS states. Locally, the bosonic D-brane
degrees of freedom are a bundle $E$ with gauge fields $A_m$ on the brane
together with scalars $\Phi$ living in the normal bundle to the brane
world-volume $C$ and the adjoint of $E$. If the brane is an embedding 
$C\subset X$, then  the normal bundle is generically non-trivial, and the
scalars are its sections. The conditions that
the background on the brane preserves supersymmetry then lead to a set
of first-order holomorphic differential equations on $A_m$ and
$\Phi$. For instance, for a brane wrapping a holomorphic
two-cycle in K3, one has the Hitchin equations
%
\bea
   F_{z\bar{z}} &=& [\Phi_z, \Phi_{\bar{z}}] \ , \nonumber\\
   \bar{D}_{\bar{z}} \Phi_z &=& D_z \bar{\Phi}_{\bar{z}} \ . 
\label{eq:hitchin}
\eea
%
These conditions can be reinterpreted geometrically as a generalized
stability condition on the pair $(E,\Phi)$. 

A second description of a D-brane is as a sheaf $\cS$ on $X$. More precisely,
BPS branes correspond
to ``coherent semistable'' sheaves on $X$. The coherent
condition means that $\cS$ fits into an exact sequence
%
\begin{equation}
   E_2 \longrightarrow  E_1 \longrightarrow \cS \longrightarrow \o \ , 
\label{eq:coherent}
\end{equation}
%
where $E_1$ and $E_2$ are vector bundles (or more precisely, the
sheaves of sections of vector bundles). The semi-stability condition is
the generalization to sheaves of the geometrical condition of
stability for vector bundles. However, in contrast to the case for
vector bundles, there is no differential equation on the sheaf
corresponding to the condition of stability. From this point of view,
the requirement that the sheaves are stable is a conjecture. 

The sheaf description is related to the description in terms of fields
on the embedded brane $C$ as follows. One requires that on $C$ the
sheaf $\cS$ reduces to the vector bundle $E$. Furthermore, away from
$C$ the sheaf must be zero. Mathematically this means that the support
of $\cS$ is $C$, and the restriction of $\cS$ to $C$ is $E$, 
%
\begin{equation}
   \supp(\cS) = C \ , \qquad
   \cS|_C = E \ .
\label{eq:restrict}
\end{equation}
%
In fact, these conditions do not completely determine $\cS$. One can
show that the additional information required is precisely the
twisting of the scalar fields $\Phi$. Thus, locally, $\cS$ is
equivalent to the pair $(E,\Phi)$. In the case of $C$ being a curve
in K3 one can then see a close relation between stable
sheaves $\cS$ and solutions of the local Hitchin
equations (\ref{eq:hitchin}). However, in general, there is no
explicit justification of the requirement that $\cS$ be stable. 

\subsection{Stable triples}

We are proposing a description of BPS brane states as stable
triples. How does this relate to the sheaf and local-field
descriptions? 
Consider the large volume limit where we can neglect stringy
corrections. In this limit we can derive an important
result:
%
\begin{quote}
   \textit{In the large $\sigma$ limit, any stable triple will
   necessarily be without a kernel, that is, the map $T$ will be
   injective (one-to-one).} 
\end{quote}
% 
 
To see this, suppose there is a kernel,
$\ker(T) \subset E_2$. By definition, there is then a non-trivial
sub-triple $T':\ker(T)\to 0$. Since $E_2$ is torsion-free, any
sub-sheaf of $E_2$ must be torsion-free. This implies that if
$\ker(T)$ is non-trivial, it must be supported on the whole of $X$. In
particular, we must have $\rk(\ker(T)) > 0$. Recall that for large
$\s$ the stability condition reduced to a condition on the
ranks (\ref{theta}). However, for the sub-triple $T':\ker(T)\to 0$ we
have 
%
\begin{equation}
   n_2 n'_1 - n_1 n'_2 = -n_1 n'_2 < 0 \ ,
\end{equation} 
%
since $n'_1=0$, $n'_2 > 0$ and so the stability condition (\ref{theta})
is violated. Thus any stable triple has $\ker(T)=0$. In general, it
will have a cokernel however. The fact that $\ker(T)=0$ implies that
there is always an exact sequence 
%
\begin{equation}
   \o \longrightarrow E_2 \stackrel{T}{\longrightarrow}  E_1 
     \longrightarrow \coker(T) \longrightarrow \o \ . 
\label{eq:injective}
\end{equation}
%
Comparing with (\ref{eq:coherent}), we see that, in the large~$\s$
limit, the coherent sheaf $\cS$ is simply the cokernel $\coker(T)$ of
the tachyon map. In particular, it will be supported on some
holomorphic subspace of $X$ (or $X$ itself). For example on K3, if $E_{1}$
and $E_{2}$ of the same rank, then the brane charge 
$c_1(E_{1})-c_1(E_{2})=c_1(\cS)$ must be effective, \textit{i.e.} the BPS
branes are realized as a sheaf $\coker(T)$ localized on a holomorphic
curve $C$. In particular, $c_1(E_1)-c_1(E_2)=n[C]$, where $n$ is
the rank of $\coker(T)$ on $C$. 

This appears to justify the conjecture that D-branes are described by
stable coherent sheaves. However, it turns out that the notion of
stability for an injective triple (with $T$ injective), is different
from the notion of stability of the cokernel $\cS=\coker(T)$,
considered either as a torsion sheaf on $X$ or as a vector bundle on
its support. The reason being that the $\sigma$-slope (\ref{slope}) in
the $\sigma$-stability of the triple (\ref{stable}) involves only the
ranks and the first Chern classes of $E_{1}$ and $E_{2}$. On the other
hand, the $\mu$-slope (\ref{eq:slope}) in the $\mu$-stability of the
cokernel $\cS$ involves its first Chern class. The latter is related
to the second Chern classes of $E_{1}$ and $E_{2}$, which do not enter
the $\sigma$-slope stability. 

One might expect that the vortex equations receive corrections
involving higher-order Chern classes, which could correct this
discrepancy. However, in fact, there is a basic difference between the
stability of the triple and other notions of D-branes stability.
In general, one considers the charges of a brane-antibrane system as
elements in K-theory, and searches for geometric objects that
correspond to these charges. In this paper, we suggest a representation
of the brane-antibrane charges by holomorphic triples, which satisfy
the vortex equations. For an injective triple, the K-theory charges are
then related to the Chern classes of $\coker(T)$, the difference of
$\ch(E_1)$ and $\ch(E_2)$. However, this proposal implies that the
stability condition does not depend only on the K-theory classes of a
complex and its sub-complexes, but rather on the individual
terms.
The
$\sigma$-slope is expressed in terms of the sums of the ranks and
degrees of the individual terms in the complex, rather than their
alternating differences. 

We can also consider more
general webs of vector bundles. Basically we are working with the
representation of some quiver in the category of vector bundles on
$X$. We fix the `shape' of the web of bundles and
maps and then look for stability of all configurations of bundles and
maps for that shape. Thus, for instance, for vector bundles we take the
quiver of type $A_{1}$, for stable triples we take the quiver $A_{2}$
and for a sequence of $n$ vector bundles we will take the quiver
$A_{n}$. For all such objects one can define stability, but there are 
many notions of slope and stability (depending on discrete and
continuous parameters).
All of these stability notions specialize
to the ordinary slope stability of vector bundles when working with
$A_{1}$. However already for $A_{2}$ there are many different
stabilities. For example there is one continuous family of stabilities
(depending on $\sigma$ or $\tau$) that we use. 

Finally, note that since there are corrections to the effective action
(\ref{eq:action}), we expect the vortex equations to be deformed. This
deformation is likely to influence the stability notion when the
corrections are not negligible, and in particular in the finite
$\sigma$ regime. 





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Superconnections and branes-antibranes}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we  will introduce  the notion of 
superconnections.\cite{quillen} We will
make the assumption that the  effective action of
tachyon and gauge fields 
for branes-antibranes system can be written
in a Quillen-like framework
in terms of the supercurvature, and 
propose the form of the  effective action.


\subsection{Superconnections}



The superconnections which will be relevant for us
appear in the work of Quillen\,\cite{quillen}
on the Chern character of a K-class.
Let us briefly 
review some of its elements.

One considers a pair of complex vector bundles $E_1,E_2$ over a manifold $M$ and a 
homomorphism $T:E_2 \rightarrow E_1$.
In the branes-antibranes system the vector bundles $E_1$ and $E_2$ correspond to the branes and antibranes
respectively, and the map $T$ corresponds to the tachyon arising from the open string stretched
between them. 
One can regard $E = E_1 \oplus E_2$ as a superbundle,
that is a bundle that carries
a $Z_2$-graded structure. The fiber $V = V_1 \oplus V_2$ is a vector space with $Z_2$ grading.
Denote
the involution that gives the grading by $\varepsilon$: $\varepsilon(v) = (-1)^{deg(v)}v$.  
The algebra of endomorphisms of $V$, $End(V)$, is a superalgebra with even and odd elements.
The even endomorphisms commute with  $\varepsilon$, while the odd ones anticommute with it.
The supertrace is defined by
\be
{\rm Tr}_s (X) \equiv Tr (\varepsilon X),~~~~X \in End(V) \ .
\label{supertrace}
\ee
It vanishes for odd endomorphisms, and gives the difference of the traces on $V_1$ and $V_2$
for the even ones.

When considering differential forms on $M$ there is a natural $Z$-grading corresponding to the degree
of the forms. Thus, differential forms on $M$ with values in $E$ have a $Z \times Z_2$ grading.
What will be relevant is the total $Z_2$ grading.



Let $D$ be an odd degree connection on $E$ preserving the $Z_2$ grading
\be
D =  \pmatrix{d+A^{1}  & 0 \cr &\cr
0 & d+A^{2}} \ .
\label{D}
\ee
Denote by ${\cal T}$ the odd degree endomorphism of $E$
\be
{\cal T}=  \pmatrix{0  & iT \cr &\cr
i{\bar T} & 0} \ .
\label{cT}
\ee
The superconnection ${\cal A}= D + {\cal T}$ on $E$ is 
an operator of odd degree acting on differential form on $M$ with 
values in $E$
\be
{\cal A}=  \pmatrix{d+A^{1}  & iT \cr &\cr
i{\bar T} & d+A^{2}} \ .
\label{A}
\ee

When considering the branes-antibranes system the superconnection (\ref{A})
appears naturally via the Chan-Paton factors, where
the gauge fields of the branes $A_{\mu}^1$
and antibranes $A_{\mu}^2$ are the diagonal elements and the off-diagonal elements are the
tachyon $T$ and its conjugate $\bar{T}$. 
Note, that
while the   diagonal elements are 1-forms the off-diagonal
elements are 0-forms. However, the total grading of all the matrix
elements is one.

The supercurvature ${\cal F}={\cal A}^2$ is given by 
\be
{\cal F}=  \pmatrix{F^1-T{\bar T}  & i{\cal D}T \cr &\cr
i\overline{{\cal D}T} & F^2-T{\bar T}} \ ,
\ee
where the covariant derivatives are defined by
\bea
&&{\cal D}T \equiv dx^{\mu} D_{\mu}T=dx^{\mu}(\partial_{\mu}T+A^{1}_{\mu}T-
TA^{2}_{\mu})\ , \cr
&&\cr
&&\overline{{\cal D}T} \equiv dx^{\mu}{\overline {D_{\mu}T}}=dx^{\mu}
(\partial_{\mu}{\bar T}+A^{2}_{\mu}{\bar T}-{\bar T}A^{1}_{\mu})\ .
\label{COV} 
\eea
$F^i,\ i=1,2$ are the gauge fields strength associated with the gauge potentials $A^i,\  i=1,2$.
Note, that we used the fact that in this framework
$T$ and ${\bar T}$ anti commute with $dx^{\mu}$.
The Chern character $ch(E_1) - ch(E_2)$ is represented by 
$Tr_s\; e^{{\cal F}}$.\,\cite{quillen}





\subsection{The effective action}


One can rewrite the supercurvature (\ref{COV}) using the Clifford algebra. We replace
$dx^{\mu_1}...dx^{\mu_n}\rightarrow
(1/n!) \, \gamma^{\mu_1}...\gamma^{\mu_n}$, where
$\gamma^{\mu}$ satisfy the Clifford algebra
$\{\gamma^{\mu},\gamma^{\nu}\}=2g^{\mu \nu}$.

The supercurvature reads now
\be
{\cal F}=\pmatrix{{1\over 2}\gamma^{\mu \nu}F^1_{\mu \nu}- 
(T{\bar T} -m {\bar m}) &\;\;\;i\gamma^{\mu}D_{\mu}T\cr &\;\;\;\cr
i\gamma^{\mu}{\overline {D_{\mu}T}} &\;\;\; {1\over 2}\gamma^{\mu \nu}
F^2_{\mu \nu}- (T{\bar T} -m {\bar m})} \ ,
\label{F}
\ee
where
$\gamma^{\mu \nu}=[\gamma^{\mu},\gamma^{\nu}]/2$, namely $
 dx^{\mu} \wedge dx^{\nu}\rightarrow\gamma^{\mu \nu}$. Note that in
(\ref{F}) we used the freedom to add a constant part, represented by 
$m{\bar m}$.

We denote by ${\rm tr}$ the trace taken over the Clifford algebra 
elements, e.g.
${\rm tr}(\gamma^{\mu}\gamma^{\nu}) = 2^{[(p+2)/2]}g^{\mu\nu}$ in a 
$(p+1)$-dimensional space.  
We denote by
${\rm Tr}$ the one taken
over the matrix structure of ${\cal F}$, and ${\rm Tr}_s$ is as in (\ref{supertrace}).




Since the superconnection appears naturally in the description of the branes-antibranes system 
it is natural to ask 
whether we can write the effective action in terms of the supercurvature.
The first hint is the $Dp-D{\bar p}$ effective action up 
to second order
\be
S_2\! =\! T_p\! \int\!\! d^{p+1}x  \left({1\over 4}\,F^{1 \mu\nu}F^1_{\mu\nu}
\!+\!{1\over 4}
F^{2 \mu \nu}F^2_{\mu \nu}\! -\! D^{\mu}T{\overline{D_{\mu}T}}
-( T{\bar T} -m {\bar m})^2\right),
\label{YM}
\ee
where by $T_p$ we denote the tension of a BPS Dp-brane.
This action can be written as
\begin{equation}
S_2  = - \frac{T_p}{2^{[(p+2)/2]}}  \int d^{p+1}x \; {\rm Tr}({\rm tr} \;
{\cal F}^2) \ .
\label{AC2}
\end{equation}
One may suspect then that the higher order terms in the effective action,
in the  slowly varying fields approximation, where we neglect
terms like $\partial^k F$ and $\partial^l T,\ l > 1$, could
be of the form
${\cal F}^{n},\ n > 2$.
We will work in the slowly varying fields
approximation in the following.
We will see
that this approximation is sufficient for
the analysis of some exact properties of the tachyon condensation. 

The second hint comes from the form of the Wess-Zumino (WZ) term of the  branes-antibranes system.
In the language of differential forms it reads 
\begin{equation}
S_{WZ}=
{\mu_p e^{-m{\bar m}}}\int {\cal C} \wedge {\rm Tr_s}(\;e^{{\cal F}}) \ ,
\label{ACW}
\end{equation}
where 
\be
{\cal C}=\sum {1\over n!}\gamma^{\mu_1,\cdots\mu_n}
C_{\mu_1,\cdots\mu_n} \ ,
\ee
and $C_{\mu_1,\cdots\mu_n}$ is an 
$n$-form corresponding to the RR n-form field.

This 
WZ action is expected in view of the discussion in Sec.\, 4.1
and the fact that
D-branes charge is measured by the K-theory class.


The supercurvature ${\cal F}$
can be decomposed as 
\bea
{\cal F}&=&\pmatrix{{1\over 2}\gamma^{\mu \nu}F^1_{\mu \nu}
&\;\;\;
i\gamma^{\mu}D_{\mu}T\cr &\;\;\;\cr
i\gamma^{\mu}{\overline {D_{\mu}T}} &\;\;\; {1\over 2}\gamma^{\mu \nu}
F^2_{\mu \nu}}- ( T{\bar T} -m {\bar m})\pmatrix{{ 1\kern - .4em 1} &\;\;0\cr &\;\;\cr
0&\;\; { 1\kern - .4em 1}} \nonumber \\ [5mm]
&=&{\bar {\cal F}}-( T{\bar T} -m {\bar m}){ 1\kern - .4em 1}  \ .
\label{C}
\eea

Using this form of the curvature the WZ action (\ref{ACW}) can be written as
\begin{eqnarray}
S_{WZ}={\mu_p} \int d^{p+1}x e^{-T{\bar T}}{\cal C} \wedge
{\rm Tr_s}\left(\sum_{n\leq p+1}\frac{{\bar {\cal F}}^n}{n!}
\right) \ .\label{WZ1}
\end{eqnarray}
The WZ action (\ref{WZ1}) suggests that 
the tachyon potential is 
\be
V(T,{\bar T}) \sim  e^{-T{\bar T}} \ .
\label{TACP}
\ee
This is in accord with the effective field theory, 
string field theory,
and $\sigma$-model computations.

We now turn to the non-topological part of the branes-antibranes action, which we will
denote by  DBI. We expect to get the same tachyon potential
(\ref{TACP}) in the DBI part.
We now make the assumption that
we can write it via the supercurvature. 
Since the superconnection and supercurvature appear as part
of the structure of the system via the Chan-Paton factors one may expect this to be the case.
However, it is also possible that only the topological part
of the branes-antibranes action can be written using the supercurvature.
This is related to the question whether the superbundle structure is indeed
a structure of the brane-antibrane system or only of its topological part.
We will continue with the assumption, bearing in mind that we do not have
a proof for it. 

The requirement of being able to write the DBI part using the supercurvature,
together with the requirement of getting the 
 same tachyon potential
(\ref{TACP}) in the DBI part, uniquely fixes the DBI action to
\begin{equation}
S_{DBI} = -{\tau_0}\int d^{p+1}x \; {\rm Tr} \left( {\rm tr} \; e^{{\cal F}} \right) \ .
\label{AC1}
\end{equation}
$\tau_0$ is a normalization constant given by
$T_p / 2^{[(p+1)/2]} = \tau_0 e^{m{\bar m}}$.

The order ${\cal F}^2$ of (\ref{AC1}) is precisely (\ref{YM}).
Using the form of the curvature (\ref{C}) we have
\begin{equation}
S_{DBI}=-\frac{T_p}{2^{[(p+2)/2]}}\int d^{(p+1)}x \;e^{-T{\bar T}}\;{\rm Tr}
\left({\rm tr}\;  e^{{\bar {\cal F}}}\right)\ .
\label{ACP} 
\end{equation}


Thus, the proposed effective
action of the branes-antibranes system, written in terms of the
supercurvature (\ref{C},) is  $S = S_{DBI} + S_{WZ}$, with  $S_{DBI}$ 
given by (\ref{ACP}) and  $S_{WZ}$ by (\ref{WZ1}).


\subsection{Tachyon condensation}

Consider tachyon condensation on a $Dp$-$\bar{D}p$ system
carrying a D(p-2)-brane charge.
The tachyon should form a vortex-like configuration, with the topological charge of the vortex
 encoding the $D(p-2)$ brane charge. 

We take the tachyon configuration 
$T=\alpha z,\;\bar{T}=\bar{\alpha}\bar{z}$, where $z=x^1 + i x^2$.
 Inserting into the WZ action (\ref{WZ1}) we get the coupling of RR $p$-form to the BPS-brane. It reads
\bea
S_{WZ}^{(2)}&=&\mu_p\int d^{p+1}x  \; \frac{1}{2\;p!} \; \epsilon^{\mu_0,...\mu_{p-1}\alpha\beta} \; \nonumber \\
            &\times& C_{\mu_0...\mu_{p-1}}
            \left( (F^{1}-F^2)_{\alpha\beta}+2D_{\alpha}T{\overline{D_{\beta}T}}
\right)e^{-T{\bar T}} \nonumber \\
&=&\mu_p\;(2\pi) (1+\Delta F)\int d^{p-1}x\frac{1}{p!}\epsilon^{\mu_0...\mu_{p-1}}C_{\mu_0...\mu_{p-1}} \ ,
\eea
where $\Delta F = F^1-F^2$.
Reinstalling $2\pi\alpha'$ one thus finds $\mu_{cond}=2\pi\;\mu_{p-2}(1+\Delta F)$.
Assume that only $F_{12}^i,\ i=1,2$ is different from zero.
In order to find the
exact charge  the vortex-like solution should have $F_{12}^1-F_{12}^2=0$.

In this setup the supercurvature (\ref{C}) reads 
\be
{\bar {\cal F}}=\pmatrix{\gamma^1\gamma^2 F_{12} & i(\gamma^1+i\gamma^2) \partial_{z} T\cr
&\cr i(\gamma^1+i\gamma^2) \partial_{\bar{z}}\bar{T}&\gamma^1\gamma^2 F_{12} }.
\ee
Evaluating from this the action (\ref{ACP}) we get 
\be
S=-2T_p\int d^{p+1}x \; e^{-|T|^2}\cosh\sqrt{2\partial_z T\partial_{\bar{z}}\bar{T}-F^2} \ .
\ee
The field equations read 
\bea
&&\!\!\!\!\partial_z\left[\partial_{\bar{z}}\bar{T}e^{-|T|^2}\frac{\sinh(2|\partial T|^2-F^2)^{1/2}}{(2|\partial T|^2-F^2)^{1/2}}\right]
-\bar{T}e^{-|T|^2}\cosh(\sqrt{2|\partial T|^2-F^2})=0,\nonumber\\[2mm]
%&&\cr
&&\!\!\!\!\partial_z\left[e^{-|T|^2}F_{z\bar{z}}\frac{\sinh(2|\partial
    T|^2-F^2)^{1/2}}{(2|\partial T|^2-F^2)^{1/2}}\right]
= 0 ,
\eea
where we denote $ F=F_{12}=-(i/2) \, F_{z\bar{z}}$.

To calculate the vortex tension consider the following kink profile
\be
T=\alpha z, \;\;\;\;\; \beta=\sinh(\sqrt{2|\alpha|^2-F^2}) \ .
\ee
For this profile the equations of motion read
\bea
&&\alpha z
e^{-|\alpha|^2|z|^2}\left[\frac{|\alpha|^2\beta}{{\rm arcsinh}\beta}-\sqrt{1+\beta^2}
\right]=0 \nonumber\\[2mm]
%&&\cr
&&|\alpha|^2ze^{-|\alpha|^2|z|^2}\;\frac{|\alpha|^2\beta \sqrt{|\alpha|^2-{\rm arcsinh}^{2}\beta}}{{\rm arcsinh}\beta}=0 \ .
\eea
This profile can solve the equations of motion for $\alpha=0$ and $F=0$. This solution corresponds to the top
of the potential where we have the $Dp$-$\bar{D}p$ system. 
Condensation of the $Dp$-$\bar{D}p$ system to a $D(p-2)$ brane,
 corresponds to non zero fields with the tachyon mostly sitting at the minimum of 
the potential.
This happens for $|\alpha|\rightarrow\infty$.  $F$ has to be sent to infinity
such that the $D(p-2)$ tension saturates the BPS bound.

The correct scaling for the field strength $F$ can be found from calculating the tension of the vortex.
Plugging the $\alpha\rightarrow \infty$ solution into the action, we get
\be
S|_{\rm vortex}\!=\!-2T_p\!\int\! d^{p+1}x \, e^{-|\alpha|^2|z|^2}\!\sqrt{1+\beta^2}
=\!-2\pi T_p\;\frac{\sqrt{1+\beta^2}}{|\alpha|^2}\!\int\! d^{p-1}x \ .
\ee
Scaling F such that $|\beta|\rightarrow|\alpha|^2$ the tension of the vortex is $T_{p-2,cond}=2\pi\;T_{p-2}$. After reinstalling $2\pi\alpha'$ one finds 
\begin{equation}
T_{p-2}=(2\pi)^2\alpha'\;T_{p-2} \ ,  
\end{equation}
which is 
the correct value of the D(p-2)-brane tension.


\section*{Acknowledgments}
\addcontentsline{toc}{section}{\numberline{}Acknowledgments}

This work is supported by the US-Israel Binational Science Foundation
and by Israel Science Foundation.

\section*{References}
\addcontentsline{toc}{section}{\numberline{}References}

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\bibitem{Alishahiha:2000du}
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\bibitem{Brad}  S. Bradlow and O. Garc\'ia-Prada, ``Stable triples,
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\bibitem{wittenk}  E. Witten,
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\bibitem{quillen} D. Quillen, ``Superconnections and the Chern Character,''
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\end{thebibliography}

\end{document}

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