In describing the intersection of one D3-brane with $N$ D1-branes,
one has the option of starting from the dynamics of the D3-brane
and trying to derive the D1-branes (this was the approach of the
previous section), or of starting from the nonabelian dynamics of 
multiple D1-branes and trying to derive the D3-brane. The latter 
approach has been applied in \cite{ncbion} to the case in which there
is no background $B$ field. In this section we will review that work
and show how to generalize it to the case where $B\ne 0$ and the
bion is tilted.

We begin by reviewing the results from \cite{ncbion} on the $B=0$ case, 
while introducing some notation which will be useful later.
We consider the nonabelian Born--Infeld action of 
equation (\ref{eqn:SBI}) specialized to the case of N coincident D1-branes,
flat background spacetime ($G_{\mu\nu}=\eta_{\mu\nu}$), vanishing
$B$ field, vanishing worldvolume gauge field and constant dilaton. 
The action then depends only on the $N\times N$ matrix transverse 
scalar fields $\Phi^i$'s. In general, $i=1,\ldots,8$, but since we are
interested in studying the D1/D3-brane intersection, we will allow
only three transverse coordinate fields to be active ($i=1,2,3$). 
The explicit reduction of the static gauge action ($X^0 = \tau$ and 
$X^9=\sigma$) is then
\be
\label{eqn:SBI1}
S_{BI} = -T_1 \int d\sigma d\tau STr 
 \sqrt{ -\det(\eta_{ab}+\lambda^2\partial_a\Phi^i 
   Q^{-1}_{ij}\partial_b\Phi^j) det(Q^{ij}) } ~,
\ee 
where
\be 
Q^{ij} = \delta^{ij} + i\lambda[\Phi^i,\Phi^j]~.
\ee 
Since the dilaton is constant, we incorporate it in the 
tension $T_1$ as a factor of $g^{-1}$. 
 
We look for static solutions ({\it{i.e.}}, $\Phi = \Phi(\sigma)$
only). Since we have no hope of finding a general static solution of 
these nonlinear matrix equations, we make some simplifying 
assumptions which have a chance of being valid on the restricted 
class of BPS solutions. The action (\ref{eqn:SBI1}) depends only 
on the two matrix structures $\partial_a\Phi^i$ and
$W_i \equiv \half i \epsilon_{ijk}[\Phi^j,\Phi^k]$ and, because of 
the nature of the $STr$ instruction, they may be treated as commuting 
quantities until the final step of doing the gauge trace to evaluate
the action. This allows us to evaluate the determinants in the 
definition of the action (\ref{eqn:SBI1}) and to convert the energy
functional to the following form:
\be 
U_{B=0} = \int d \sigma STr \sqrt {1 + \lambda^2(\partial \Phi^i)^2 
+ \lambda^2(W_i)^2 + \lambda^4 (\partial \Phi^i W_i)^2 } ~.
\label{eqn:S0} 
\ee 
Continuing to treat $\Phi^i$ and $W^i$ as commuting objects, we see that
this energy functional can be written as a sum of two squares
\be 
U_{B=0} = \int d \sigma STr \sqrt { 
(1 \pm \lambda^2 \partial \Phi^i W_i)^2 
+ \lambda^2 (\partial \Phi^i \mp W_i)^2} ~,
\ee 
and is minimized by a displacement field satisfying the 
first-order BPS-like equation $\partial \Phi^i = \pm W_i$. 
This equation, written more explicitly as
\be 
\partial \Phi_i = \pm \half i \epsilon_{ijk}[\Phi^j,\Phi^k] ~,
\label{eqn:BPS0} 
\ee 
is known as the Nahm equation \cite{nahm0}. The $\pm$ ultimately 
corresponds to the choice between a bundle of D1- or 
$\overline {\textrm{D1}}$-branes. The Nahm equation is a very 
plausible candidate for the exact equation to be satisfied by a BPS 
solution of this system and the fact that the Myers action 
(\ref{eqn:SBI1}) implies it in the BPS limit is very satisfactory.  

The Nahm equation has the trivial solution $\Phi=0$ which
corresponds to an infinitely long bundle of coincident D1-branes. 
In \cite{ncbion}, a much more interesting solution was found
by starting with the following ansatz: 
\be 
\Phi^i = \hat R(\sigma) \alpha^i,\qquad\qquad 
(\alpha^1, \alpha^2, \alpha^3) \equiv {\bf{X}} ~,
\label{eqn:ansatz0} 
\ee 
where $\alpha^i$ form an $N\times N$ representation of the 
generators of an $SU(2)$ subgroup of $U(N)$, 
$[\alpha^i, \alpha^j] = 2 i\epsilon_{ijk}\alpha^k$. 
With this ansatz, both $\partial\Phi^i$ and $W^i$ are proportional
to the generator matrix $\alpha^i$.
When the ansatz is substituted into the BPS condition
(\ref{eqn:BPS0}), we obtain a simple equation for $\hat R$,  
\be 
\hat R'= \mp 2\hat R^2 ~,
\ee 
which is solved by 
\be 
\hat R = \pm \inv{2 \sigma}~.
\label{eqn:R0} 
\ee 
Substituting the ansatz (\ref{eqn:ansatz0}) into  
(\ref{eqn:S0}) leads to the following effective action
for $\hat R(\sigma)$ :
\be 
U_{B=0}[\hat R(\sigma)] = \int d \sigma STr \sqrt { 
(1 + \lambda^2(\hat R')^2 {\bf{X}}^2) 
(1 + 4\lambda^2(\hat R)^4 {\bf{X}}^2) 
}~. 
\label{eqn:S(R)0} 
\ee 
It can be shown that (\ref{eqn:R0}) satisfies the 
equations of motion following from the action (\ref{eqn:S(R)0}). 

This solution maps very nicely onto the bion solution of the 
previous section.  At a fixed point $|\sigma|$ on the D1-brane stack,
the geometry given by (\ref{eqn:ansatz0}) is that of a sphere
with the physical radius $R^2={\lambda^2\over N}Tr (\Phi^i)^2$
(the only sensible way to pass from the matrix transverse displacement
field $\lambda\Phi^i$ to a pure number describing the geometry).
For the ansatz under consideration, this gives
\be 
R(\sigma)^2 = {{\lambda^2}\over {N}} Tr (\Phi^i)^2 
= \lambda^2 \hat R(\sigma)^2 C ~,
\ee 
where $C$ is the quadratic Casimir, equal to $N^2-1$ 
for an irreducible representation of $SU(2)$. This gives 
\be 
\label{eqn:R}
R(\sigma)= {{\lambda \sqrt{N^2-1}} \over {(2 |\sigma|)}} 
\cong {{\pi\alpha' N} \over {|\sigma|}} 
\ee 
for large N, in agreement with equation (\ref{eqn:r0}). This 
completes our synopsis of the arguments given in \cite{ncbion} for
the agreement between the commutative and noncommutative approaches
to the D1/D3-brane intersection.

A simple argument can be made at this point to strengthten the
meaning of equation (\ref{eqn:R}).  At a fixed $\sigma$, the 
Fourier transform of the density of the $D1$-strings is given
by \cite{mark}
\be
\label{density}
\widetilde \rho (k) = Tr \left ( e^{i\lambda k_i \Phi^i} 
\right )~.
\ee
This is simply the operator to which the $09$-component
of the RR 2-form $C^{(2)}$, $C_{09}$, couples.
For the solution (\ref{eqn:ansatz0}), this is evaluated to
give
\be
\widetilde \rho (k) = {\sin(\lambda N \hat R |k|)
\over \sin(\lambda \hat R |k|)}~,
\ee
which for $N\gg 1$, and $k$ such that 
$(\lambda N \hat R)^{-1} < |k| \ll (\lambda \hat R)^{-1}$
(i.e., for momentum large enough to resolve the size of the
sphere, but not large enough to resolve the individual brane 
constituents) gives
\be
\widetilde \rho (k) \cong {\sin(\lambda N \hat R |k|)
\over \lambda \hat R |k|}~.
\ee
This is precisely the Fourier transform of the density
distribution representing a thin shell,
\be
\widetilde \rho (k) = 
\int d^3 x e^{i\vec k\cdot\vec x}\rho(x) \quad \mbox{for}
\quad \rho(x) = {N \over 4 \pi R^2} \delta(|x|-R)
\quad \mbox{with} \quad R = \lambda N \hat R = 
{\pi \a' N \over |\sigma|}~,
\ee
in agreement with equation (\ref{eqn:R}).

To distinguish the various cases involving branes and antibranes, note
that $\sigma$ can either run from $-\infty$ to $0$, in which case
the D1($\overline {\textrm{D1}}$)-branes run `towards' the 
D3($\overline {\textrm{D3}}$)-brane plane, or from $0$ to $\infty$, 
in which case the D1($\overline {\textrm{D1}}$)-branes run `away' 
the D3($\overline {\textrm{D3}}$)-brane plane.  We have thus four cases:
\begin{itemize}
\item[{\bf{A}}] :~
Stack of D1-branes expanding to a D3-brane
, $\partial \Phi^i = + W_i$ and $\sigma \in (-\infty, 0)$
\item[{\bf{B}}] :~
Stack of D1-branes expanding to a D3-brane
, $\partial \Phi^i = + W_i$ and $\sigma \in (0, \infty)$
\item[{\bf{C}}] :~
Stack of $\overline {\textrm{D1}}$-branes expanding to a $\overline {\textrm{D3}}$-brane
, $\partial \Phi^i = - W_i$ and $\sigma \in (-\infty, 0)$
\item[{\bf{D}}] :~
Stack of $\overline {\textrm{D1}}$-branes expanding to a $\overline {\textrm{D3}}$-brane
, $\partial \Phi^i = - W_i$ and $\sigma \in (0, \infty)$
\end{itemize}
\input{figure}
Cases {\bf{A}} and {\bf{D}} correspond to the D1($\overline {\textrm{D1}}$)-branes 
running `towards' (`away from') the D3($\overline {\textrm{D3}}$)-brane plane, 
and thus should represent a positive magnetic charge, while
Cases {\bf{B}} and {\bf{C}} should represent a negative magnetic charge.
We will see shortly that this is the case and that the four cases
match the four cases found in the abelian treatment of the same
problem starting from the D3-brane (see the end of section \ref{sec-D3}).

Our next step is to turn on the background $B$ field in order to study 
the tilted bion from the noncommutative D1-brane point of view.
The only difference from the previous case is that we turn on the component
$B_{12}=\mbox{const.}$ of the background $B$ field (remember that the
D1-brane worldvolume spans the $(X^0,X^9)$ plane and that only the
$i=1,2,3$ components of the matrix transverse displacement field are
allowed to be nonzero). It is still the case that the action is
a functional only of the fields $\partial\Phi^i$ and  
$\epsilon_{ijk}[\Phi^j,\Phi^k]$ and the same reasoning as
before leads us to treat these quantities as commuting objects 
inside the $STr$ instruction. In this way, the action (\ref{eqn:SBI}) 
can be reduced to something much more explicit. To get the most 
transparent results, it helps to define rescaled fields 
\be 
\varphi_1 = \sqrt{(1+\lambda^2 B^2)} \Phi_1~, \qquad 
\varphi_2 = \sqrt{(1+\lambda^2 B^2)} \Phi_2~, \qquad 
\varphi_3 = \Phi_3~,
\label{eqn:rescale} 
\ee 
and to redefine the commutator $W^i$ as 
\be 
W_i \equiv \half i \epsilon_{ijk}[\varphi^j,\varphi^k] -
\delta^3_i B ~.
\ee 

After some rather tedious algebra (made much easier by MAPLE) 
to evaluate the determinants in the definition of the action,
we get a result for the energy functional that is almost identical
to (\ref{eqn:S0}):
\bear 
U_{B\neq 0} &=& {{1}\over{\sqrt{1+\lambda^2 B^2}}} 
\int d \sigma STr \sqrt{ 
1 + \lambda^2(\partial \varphi^i)^2 
+ \lambda^2(W_i)^2 
+ \lambda^4 (\partial \varphi^i W_i)^2 
} \nn \\ &=& 
{{1}\over{\sqrt{1+\lambda^2 B^2}}} 
\int d \sigma STr \sqrt{ 
(1 \pm \lambda^2 \partial \varphi^i W_i)^2 
+ \lambda^2 (\partial \varphi^i \mp W_i)^2 
} ~.
\label{eqn:SB} 
\eear 
The action is still `linearized' by taking 
$\partial \varphi^i = \pm W_i$, which means that the
BPS condition in the presence of a background $B$ field is
\be 
\partial \varphi_i = \pm i (\half  
\epsilon_{ijk}[\varphi^j,\varphi^k] + \delta^3_i i B)~.
\label{eqn:BPSB} 
\ee 
This is precisely the generalization of the Nahm equation 
that has been derived in the context of studies of magnetic 
monopoles in noncommutative field theory \cite{nahmB}. It is  
a plausible candidate for the exact BPS condition for the nonabelian
D1-brane system and we will show that it gives a detailed account 
of the physics of the tilted bion. The fact that the generalized 
Nahm equation is implied by the Myers action is further evidence 
for the essential correctness of the latter.

In order to solve the modified equations of motion, we have to
slightly modify the ansatz (\ref{eqn:ansatz0}), expressing the
fields $\varphi$ in terms of generators of an N-dimensional 
representation of $SU(2)$ and a scalar function $\hat R(\sigma)$:
\be 
\varphi^i = \hat R(\sigma) \alpha^i - \delta^3_i 
{{B} \over {2 \hat R(\sigma)}}~ .
\label{eqn:ansatzB} 
\ee 
When this modified ansatz is substituted into the BPS 
equation (\ref{eqn:BPSB}), we obtain the same equation for 
$\hat R$ as before, namely $\hat R'= \mp 2\hat R^2$: 
solution (\ref{eqn:R0}) still holds. If we collect the generators
into a modified triplet ${\bf{X}}\equiv
(\alpha^1, \alpha^2, \alpha^3+ {{B}\over{2{\hat R}^2}})$ and use
(\ref{eqn:ansatzB}), the action (\ref{eqn:SB}) can be expressed 
as an effective action for $\hat R(\sigma)$:
\be 
U_{B\ne 0}[\hat R(\sigma)] =  
{{1}\over{\sqrt{1+\lambda^2 B^2}}} 
\int d \sigma STr \sqrt { 
(1 + \lambda^2(\hat R')^2 {\bf{X}}^2) 
(1 + 4\lambda^2(\hat R)^4 {\bf{X}}^2) } ~.
\label{eqn:S(R)B} 
\ee 
This looks the same as (\ref{eqn:S(R)0}) but is not quite 
because ${\bf{X}}$ now depends on $\hat R$. Nevertheless,
the same radial function (\ref{eqn:R0}) continues to be a solution,
further testing the compatibility of the action with the BPS 
condition.  Notice that this does not conclusively prove that 
the ansatz (\ref{eqn:ansatzB}) with (\ref{eqn:R0}) is a solution to 
the full equations of motion implied by (\ref{eqn:SB}). 
 
It is easily seen that this solution corresponds to the 
tilted bion solution discussed in the previous section. 
Equation (\ref{eqn:rescale}) matches the ratios of axes
given in (\ref{eqn:rB}). The over-all size of the spheroid 
agrees with (\ref{eqn:rB}) by an argument identical to that 
given for $B=0$. The shift of its center is given by 
\be
\Delta^i(\sigma)= {1\over N}tr(\lambda\Phi^i)=\Delta \delta^i_3
\ee
(by virtue of the fact that $tr(\alpha^i)=0$), where
\be
\Delta = - {\lambda {B}\over{2\hat R}} = \mp \lambda B \sigma = 
\left \{ \begin{array}{ll}
\tan(\alpha) |\sigma| &\textrm{for cases {\bf{A}} and {\bf{D}}.}
\\
- \tan(\alpha) |\sigma| &\textrm{for cases {\bf{B}} and {\bf{C}}.}
\end{array} \right .  
\ee
Thus, in cases {\bf{A}} and {\bf{D}}, the bion tilts in agreement 
with section \ref{sec-D3}.  In the other two cases,
it tilts in the opposite direction.  The interpretation
is that D1-branes coming `towards' the $D3$-brane
correspond to a positively charged magnetic monopole, 
while D1-branes coming `away from' the $D3$-brane
correspond to a negatively charged one.  Similarly, 
$\overline{\textrm{D1}}$-strings coming `away from' the $D3$-brane
correspond to a positively charged magnetic monopole, 
while D1-branes coming `towards' the $D3$-brane
correspond to a negatively charged one. The geometry of the
tilted bion inferred from the nonabelian dynamics of D1-branes
perfectly matches the results of the abelian D3-brane
calculation summarized in the previous section.
\begin{figure}

\begin{center}
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\put(-3.5,-1){\line(2,1){1}}
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\put(-2.25,-0.4){$\a$}
\put(-2.39,-0.92){$+$}
\put(-1.75,-.7){\small{D3}}
\put(-1.8,0){\small{D1}}

\put(-1.3,-1.3){{\bf{B}}}
\put(-1.5,-1){\line(2,1){1}}
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\put(-0.4,-1.2){$\a$}
\put(-0.31,-0.67){$-$}
\put(0.25,-.7){\small{D3}}
\put(-.6,-1.7){\small{D1}}

\put(0.7,-1.3){{\bf{C}}}
\put(0.5,-1){\line(2,1){1}}
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\put(1.62,-0.92){$-$}
\put(2.3,-.7){$\overline{\mbox{\small{D3}}}$}
\put(1.35,0){$\overline{\mbox{\small{D1}}}$}

\put(2.7,-1.3){{\bf{D}}}
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\put(3.69,-0.67){$+$}
\put(4.25,-.7){$\overline{\mbox{\small{D3}}}$}
\put(3.85,-1.7){$\overline{\mbox{\small{D1}}}$}

\end{picture}
\end{center}
\vspace{1in}
\caption{The four cases {\bf{A}}-{\bf{D}} discussed in the text.}
\label{figure}
\end{figure}\documentclass{JHEP3}
%\documentclass[12pt]{article}

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\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bear{\begin{eqnarray}}
\def\eear{\end{eqnarray}}
\def\nn{\nonumber}

\def\half{{{1\over 2}}}

\def\const{{\mbox{const\ }}}                    

\def\Re{{\rm Re\hskip0.1em}}
\def\Im{{\rm Im\hskip0.1em}}

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\newcommand\pypx[2]{{{{\partial {#1}}\over{\partial {#2}}}}}

\newcommand\inv[1]{{1\over{#1}}}

\newcommand\tr[1]{{\mbox{tr}\{{#1}\}}}          % trace
\newcommand\Tr[1]{{\mbox{Tr}\{{#1}\}}}          % trace

\newcommand\MR[1]{{{\bf R}^{#1}}}               % Real numbers
\newcommand\MC[1]{{{\bf C}^{#1}}}               % Complex numbers
\newcommand\MS[1]{{{\bf S}^{#1}}}               % Circle, sphere,...
\newcommand\MB[1]{{{\bf B}^{#1}}}               % disk, ball,...
\newcommand\MT[1]{{{\bf T}^{#1}}}               % Torus
\newcommand\CP[1]{{{\bf CP}^{#1}}}              % CP
\newcommand\MF[1]{{{\bf F}_{#1}}}               % Ruled surface F_n

\newcommand\SUSY[1]{{{\cal N}= {#1}}}           % N=? SUSY
\newcommand\SLZ[1]{{SL({#1},\BZ)}}              % SL(*,Z)
\newcommand\SLR[1]{{SL({#1},\BR)}}              % SL(*,Z)

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\def\v{{\nu}}
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\def\t{{\tau}}
\def\h{{\eta}}
\def\x{{\xi}}
\def\z{{\zeta}}
\def\tht{{\theta}}
\def\lam{{\lambda}}

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%  SPECIAL PURPOSE DEFINITIONS AND MACROES                         %
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\input{titlepg}

\begin{document}


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\section{Introduction}\label{intro}
\input{intro}

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\section{The Bion solution on a D3-brane} 
\label{sec-D3} 
\input{D3.tex}

\section{Dual treatment by nonabelian D1-branes}
\label{sec-D1} 
\input{D1.tex}

\section{Flat D$(p+r)$-brane from D$p$-branes}
\label{sec-plane} 
\input{plane}

\section{The Worldvolume Gauge Field on the Dual Bion}
\label{sec-curved} 
\input{curved}

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\section{Conclusion}
\input{conclusion}

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%  A C K N O W L E D G M N E T S                                   %
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\section*{Acknowledgments}
\input{acknow}


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\def\np#1#2#3{{\it Nucl.\ Phys.} {\bf B#1} (#2) #3}
\def\pl#1#2#3{{\it Phys.\ Lett.} {\bf B#1} (#2) #3}
\def\physrev#1#2#3{{\it Phys.\ Rev.\ Lett.} {\bf #1} (#2) #3}
\def\prd#1#2#3{{\it Phys.\ Rev.} {\bf D#1} (#2) #3}
\def\ap#1#2#3{{\it Ann.\ Phys.} {\bf #1} (#2) #3}
\def\ppt#1#2#3{{\it Phys.\ Rep.} {\bf #1} (#2) #3}
\def\rmp#1#2#3{{\it Rev.\ Mod.\ Phys.} {\bf #1} (#2) #3}
\def\cmp#1#2#3{{\it Comm.\ Math.\ Phys.} {\bf #1} (#2) #3}
\def\mpla#1#2#3{{\it Mod.\ Phys.\ Lett.} {\bf #1} (#2) #3}
\def\jhep#1#2#3{{\it JHEP} {\bf #1} (#2) #3}
\def\atmp#1#2#3{{\it Adv.\ Theor.\ Math.\ Phys.} {\bf #1} (#2) #3}
\def\jgp#1#2#3{{\it J.\ Geom.\ Phys.} {\bf #1} (#2) #3}
\def\cqg#1#2#3{{\it Class.\ Quant.\ Grav.} {\bf #1} (#2) #3}

\def\hepth#1{{\it hep-th/{#1}}}

%   e.g.:
%  \bibitem{BHO}{E. Bergshoeff, C. Hull and T. Ortin,
%    {``Dualities in the type-II superstring Effective action,''}
%    \hepth{9504081}}

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\begin{thebibliography}{99}
\input{bibliogr}  
\end{thebibliography}

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\end{document}

