

\input harvmac.tex


\input epsf
\input tables.tex
\noblackbox
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hep-th/0302159.}
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for D-branes with Traveling Waves,'' hep-th/0303214.}
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Nucl. Phys. {\bf B277} (1986) 593\semi D. Garfinkle and T. Vachaspati,
``Cosmic String Travelling Waves,'' Phys. Rev. {\bf D42} (1990) 1960.}
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in String Theory,'' Phys. Rev. {\bf D51} (1995) 2896, hep-th/9409021
\semi C. Callan, 
J. Maldacena, and A. Peet, ``Extremal Black Holes 
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J.Gauntlett, J.Harvey, D. Waldram,
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of the multiply wound rotating string,'' Nucl. Phys.
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{\bf B538} (2002) 366,
hep-th/0204062.}
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JHEP 0212 (2002) 061, hep-th/0211042.}
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Nucl. Phys. {\bf B646} (2002) 524, hep-th/0207016.}
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string radiation from moving D-branes,'' Nucl. Phys.
{\bf B517} (1998) 92, hep-th/9710049.}
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{\bf B50} (1972) 222\semi M. Ademollo, A. D'Adda, R. D'Auria, F. Gliozzi,
E. Napolitano, S. Sciuto, and P. Di Vecchia, Nucl. Phys.
{\bf B77} (1974) 89\semi Nucl. Phys. {\bf B94} (1975).}
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infinite strings,'' Phys. Rev. {\bf D42} (1990) 354.}
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and Products,'' Academic Press (1965).}






\def\tx{\vbox{\sl\centerline{Physics Department}% 
\centerline{University of Texas at Austin}%
\centerline{Austin, TX 78712 USA}}}




\Title{\vbox{\baselineskip12pt
\hbox{UTTG-01-03}\hbox{hep-th/0304173}}}
{\vbox{\centerline{Gravitational Radiation from Travelling Waves on D-Strings}}}


{\bigskip
\centerline{Julie D. Blum}
\bigskip
\tx

\bigskip
\medskip
\centerline{\bf Abstract}
Boundary states that preserve supersymmetry are constructed for 
fractional D-strings with travelling waves on a ${\bf C}^3/
{{\bf Z}_2\times {\bf Z}_2}$ orbifold.  The gravitational
radiation emitted between two D-strings with antiparallel travelling waves
is calculated.  
}

\Date{4/03}

\newsec{Introduction}


Topological defects such as cosmic strings are not thought to be the
primary source of large scale structure in the universe while evidence
for some sort of inflationary scenario is compelling.  Nevertheless, they may 
play a lesser role in structure formation and other high energy
phenomena like baryogenesis, very energetic cosmic rays, and gamma ray
bursts.  These defects are predicted by string theory and possible
observable effects are worth considering.

Recent work has studied unidirectional travelling waves on 
Dirichlet(D)-strings as an exactly solvable two-dimensional conformal
field theory \one .  Purely left or right moving waves of arbitrary
profile on D-strings preserve one-quarter of the supersymmetry
in supersymmetric string theory.  These D-strings are dual under the
action of type IIB $SL(2,{\bf Z})$ S-duality to fundamental string
states with only left movers or right movers excited.  Here, we will
calculate the gravitational radiation emitted from the interaction of
two fractional D-strings with pulses moving in opposite directions on an $N=2$ 
supersymmetric ${\bf C}^3/{\bf Z}_2\times {\bf Z}_2$ orbifold of type IIB
string theory.  The boundary states in this calculation are 
supersymmetric although the interaction breaks supersymmetry and
allows for nontrivial radiation.  The interaction of two D-strings with 
pulses moving in the same direction preserves supersymmetry and does not 
radiate.  In this model the D-string does
not produce a conical deficit in the geometry due to the noncompact extra
dimensions.  Treating more ``realistic'' four-dimensional models seems
difficult, at least in type IIB, because the supergravity with
travelling wave requires that the Minkowski dimensions be 
asymptotically flat far from the D-strings.

In the next section we show that the supergravity travelling wave solution
can be applied to the case of a regular (unwrapped) D-string on the conifold.
A supergravity solution of purely fractional branes on the conifold that 
is asymptotically flat in the Minkowski dimensions has yet to be 
constructed but should exist.  Our calculation considers fractional
branes in order to restrict movement in the extra dimensions.  In
section three we construct the relevant boundary states and calculate
amplitudes with no insertions, in section four we calculate the 
gravitational radiation from the nonsupersymmetric interaction, and
in section five we conclude.  
While this work was progressing there was
a paper \two\ discussing various T-dual phenomena related to the 
travelling wave.  As my work was nearing completion a second related
paper \three\ appeared which has some overlap with section three.  This paper
constructs boundary states for D-strings with travelling waves in the
bosonic string and calculates the amplitude between strings with oppositely
directed pulses.  


\newsec{Supergravity Solution of a Wiggly D-string on the Conifold}

Exact solutions in nonsupersymmetric gravitational theories of travelling
waves on cosmic strings were constructed in \four .  These solutions and
generalizations could also be constructed in string theory or supergravity
\five .  Recent work has analyzed interesting configurations related
by various dualities to the travelling wave \six\seven .

In this paper we will calculate the amplitude of interactions of 
fractional branes on the ${\bf C}^3/{\bf Z}_2\times {\bf Z}_2$ orbifold.
The resolution of this orbifold contains three ${\bf P}^1$'s that
each look locally like the conifold.  By tuning two of the Kahler
parameters to be large, we can approximate the geometry of the conifold.
In the resolved orbifold the fractional branes are equivalent to bound
states of D3-branes wrapped on a conifold and a D1-brane.  Of course,
the string theory calculation can only be done in the orbifold limit.
Far from the branes the geometry should be little affected by whether
the branes are fractional or regular and whether the orbifold is 
resolved.  Thus, asymptotically the geometry is the conifold (or any
noncompact Calabi-Yau threefold) and Minkowski space.  

We consider a Dirichlet onebrane parallel to the lightcone directions 
$x^{\pm}=
{t\pm x^1\over \sqrt{2}}$.  The two transverse directions of Minkowski 
space we will denote by $\vec{y}$.  The string is localized at $\vec{y}=0$
and at the origin of the conifold ($r_{con}=0$) where $r_{con}$ is a radial
coordinate on the resolved conifold.  Adding the travelling wave amounts
to the $x^-$ dependent translation $\vec{y}=\vec{Y}(x^-)$ and requiring
the metric to be asymptotically flat in the Minkowski directions.  
The solution takes the form
%
\eqn\metric{\eqalign{ds^2 &=e^{-{\phi\over 2}}g_{\mu\nu}dx^{\mu}dx^{\nu}\cr 
&=-2 
e^{-2\phi}dx^+ dx^- 
-(e^{-2\phi}-1)
\|\dot{\vec{Y}}\|^2 {dx^-}^2 +2(e^{-2\phi}-1) 
\dot{\vec{Y}}\cdot d{\vec y} dx^- +{\| d\vec{y}\|}^2 +ds^2_
{con}\cr B_{+-}^{RR}&=1-e^{-2\phi}\cr B_{-i}^{RR}&=
\sqrt{2}\dot{Y}_i(e^{-2\phi}-1).\cr }}
%
Here, $\vec{Y}=(Y^2 , Y^3)$ depends only on $x^-$, the dot denotes
differentiation with respect to $x^-$, 
${1\over\sqrt{g}}\partial_{\mu}(\sqrt{g} g^{\mu\nu}\partial_{\nu} 
e^{-2\phi})\sim\delta^{(8)}({\vec x}-{\vec Y})$
where $\phi(\vec{y}',r_{con})$ is
the dilaton.  The
coordinate $\vec{y}'=\vec{y}-\vec{Y}(x^-)$.  For large distances in
the variable $R=\sqrt{\|\vec{y}'\|^2 +r^2_{con}}$ and no angular momentum,
$e^{-2\phi}\sim 1+{c\over R^6}$ where $c$ is constant.
Supersymmetry is broken to $1\over 16$ by the conditions 
%
\eqn\susy{\eqalign{\gamma^{+-}\epsilon&=\epsilon^*\cr \gamma^-\epsilon&=0\cr
D_{\mu}^{(conifold)}\epsilon&=0\cr .}}
%
Here, $\epsilon$ is a supersymmetric spinor of type IIB supergravity.  
The effective theory has two
supercharges of negative chirality.  There is a T-dual configuration
with three fivebranes intersecting on a string (two fivebranes
intersect in a threebrane while the third fivebrane intersects one
of the two in a threebrane and the other in a string) with momentum in
eleven dimensional supergravity.  This triple intersection is different
from the one analyzed in \eight\ as it is nonchiral. \foot{Note that the
conjectured theories in \eight\ with $(0,2)$ supersymmetry obtained by wrapping
fivebranes on fourcycles in Calabi-Yau fourfolds have anomalies that
cannot be cancelled and are therefore inconsistent.}

\newsec{Boundary States and Interactions for Travelling Waves}

In this section we construct boundary states for the fractional D-string 
with travelling wave on the orbifold ${\bf C}^3/{\bf Z}_2\times {\bf Z}_2$.
Most of the ingredients needed for our calculation in this section and
the next can be found in \nine .  The boundary state formalism was
introduced in \ten\ as an expression of the duality between one loop 
open string and tree closed string amplitudes and developed in
\eleven\ as a solution of a boundary condition on left and right movers.

The four fractional branes satisfying open-closed duality constraints on
this orbifold take the form
%
\eqn\frbrane{|B_{\alpha}>=|F_0>+\sum_j\epsilon_{\alpha j}|F_j>}
%
where $\prod_j{\epsilon_{\alpha j}}=1$, 
$\epsilon_{\alpha j}=\pm 1$, $\alpha\in\{0,1,2,3\}$,
$j\in\{1,2,3\}$, $F_0$ corresponds to a onebrane in flat ten
dimensions, and $F_i$ corresponds to a fractional onebrane at one of the three
possible ${\bf C}^2/{\bf Z}_2$ orbifolds.  The three $|B_{\alpha}>$ with
$\alpha\neq 0$ are equivalent in the resolution of the orbifold to a
bound state of two threebranes wrapped on one of the three conifold
${\bf P}^1$'s, a threebrane and antithreebrane on the other two
conifold ${\bf P}^1$'s, plus a onebrane.  The state $|B_0>$ has two
threebranes wrapped on each conifold plus a onebrane.

The interaction of two boundary states is given by a cylinder amplitude 
in which a closed string is exchanged between the two boundaries.  The
length of the cylinder is $l$, and the amplitude takes the form
%
\eqn\amp{{\cal A}^{ab}_{\alpha\beta}=
\int dx^+ dx^-\int_0^\infty dl <B_{\alpha},
\vec{Y}_{1a}|e^{-lH}|B_{\beta},\vec{Y}_{2b}>}
%
where $H$ is the closed string Hamiltonian and $\vec{Y}_{ia}$ 
are the amplitudes
for the travelling waves on the D-strings.  $\vec{Y}_{ia}=\vec{Y}_i(x^a)$
where $a=\pm$.  To construct the boundary
states, let us write the expansions for closed string bosonic and
fermionic fields as a function of $z=\sigma +i\tau$ where $\sigma$ is
periodic with $0\leq\sigma\leq 1$, $0\leq\tau\leq l$, and 
$\alpha'={1\over 2\pi}$.
%
\eqn\expansion{\eqalign{X^{\mu}(z)&={{\hat x^{\mu}}\over 2} -{z\over 2}
{\hat p^{\mu}}+{i\over\sqrt{4\pi}}\sum_{n\neq 0}{1\over n}a^{\mu}_{n}
e^{2\pi inz}\cr {\tilde X}^{\mu}({\bar z})&={{\hat x^{\mu}}\over 2} +{{\bar z}
\over 2}
{\hat p^{\mu}}+{i\over\sqrt{4\pi}}\sum_{n\neq 0}{1\over n}{\tilde a}^{\mu}_{n}
e^{-2\pi in{\bar z}}\cr Z^i (z)&={i\over\sqrt{4\pi}}\sum_r{1\over r}
b^i_{r}e^{2\pi irz}\cr {\tilde Z}^i (z)&={i\over\sqrt{4\pi}}\sum_r{1\over r}
{\tilde b}^i_{r}e^{-2\pi ir{\bar z}}\cr
\Psi^{\mu}(z)&=\sum_p \psi^{\mu}_p e^{2\pi ipz}\cr
{\tilde \Psi}^{\mu}({\bar z})&=\sum_p {\tilde\psi}^{\mu}_p 
e^{-2\pi ip{\bar z}}\cr \Lambda^i (z)&=\sum_p\lambda^i_p e^{2\pi ipz}\cr
{\tilde\Lambda}^i (z)&=\sum_p{\tilde\lambda}^i_p e^{-2\pi ip{\bar z}}\cr}}
%
Here, $X^{\mu}(\Psi^{\mu})$ are bosonic (fermionic) coordinates in the
untwisted directions while $Z^i(\Lambda^i)$ are coordinates in the directions
twisted by a ${\bf Z}_2$ action, and $n\in {\bf Z}$, $r\in {\bf Z}+{1\over 2}$,
while $p\in {\bf Z}$ or ${\bf Z}+{1\over 2}$ depending on whether the 
fermion sector is Ramond (R) or Neveu-Schwarz (NS).  The commutation
relations are 
%
\eqn\commutation{\eqalign{[a^{\mu}_{n},a^{\nu}_{m}]&=n 
\eta^{\mu\nu}\delta_{n+m} \cr \{\psi^{\mu}_p,\psi^{\nu}_q\}&=\eta^{\mu\nu}
\delta_{p+q}\cr [b^i_r,{\bar b}_s^j]&=r\delta^{ij}\delta_{r+s},\cr}}
%
etc. where $\eta^{00}=-1$ and $\eta^{ij}=\delta^{ij}$ for
$i,j\neq 0$.

The boundary conditions for oscillator modes at $\tau=0$ are the following:
%
\eqn\bc{\eqalign{(a_n^{\pm}+{\tilde a}_{-n}^{\pm})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(a_n^{\mu}-{\tilde a}_{-n}^{\mu})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(b_r^{i}-{\tilde b}_{-r}^{i})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\psi_p^{\pm}+i\eta{\tilde \psi}_{-p}^{\pm})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\psi_p^{\mu}-i\eta{\tilde \psi}_{-p}^{\mu})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\psi_p^{\mu}-i\eta{\tilde \psi}_{-p}^{\mu})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\lambda_p^{i}-i\eta{\tilde \lambda}_{-p}^{i})|F_1 ,{\vec Y}_a,\eta>&=0\cr}}
%
for $\mu\in\{2,3,4,5\}$, $i\in\{2,3\}$, $n,p\neq 0$, and $\eta=\pm 1$.
For zero modes in the RR sector we have the conditions:
%
\eqn\bczm{\eqalign{{\hat p}^a|F_1 ,{\vec Y}_a,\eta>&=0\cr
({\hat x}^{\mu}-Y^{i}_a\delta_i^{\mu})|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\psi^{\pm}_0+i\eta{\tilde \psi}^{\pm}_0)|F_1 ,{\vec Y}_a,\eta>&=0\cr
(\psi_0^{\mu}-i\eta{\tilde \psi}_{0}^{\mu})|F_1 ,{\vec Y}_a,\eta>&=0\cr}}
%
Note that translational invariance in one of the light cone directions
is broken.  The boundary state satisfying the above constraints can be
written as
%
\eqn\bostate{|F_1,{\vec Y}_a,\eta>=|F_1,\eta>_{osc}\otimes
|F_1,{\vec Y}_a,\eta>_0\otimes |ghost>}
%
\eqn\boosc{\eqalign{|F_1,\eta>_{osc}&=exp[\sum_{n>0}{1\over n}(a_{-n}^-
{\tilde a}^+_{-n}+a_{-n}^+
{\tilde a}^-_{-n}+a_{-n}^{\mu}\cdot
{\tilde a}^{\mu}_{-n})\cr &+\sum_{r>0}{1\over r}(b_{-r}^{i}\cdot
{\tilde{\bar b}}^{i}_{-r} +{\tilde b}^{i}_{-r}\cdot{\bar b}^{i}_{-r})\cr &+
i\eta\sum_{p>0}(\psi_{-p}^-
{\tilde \psi}^+_{-p}+\psi_{-p}^+
{\tilde \psi}^-_{-p}+\psi_{-p}^{\mu}\cdot
{\tilde \psi}^{\mu}_{-p}\cr &+\lambda_{-p}^{i}\cdot
{\tilde{\bar \lambda}}^{i}_{-p} +{\tilde \lambda}^{i}_{-p}\cdot
{\bar \lambda}^{i}_{-p})]|0>\cr}}
%
\eqn\bozer{\eqalign{&|F_1,{\vec Y}(x^+),\eta>_0^{RR}
=\cr &\int {d^4 q\over(2\pi)^4}
e^{-i{\vec Y}(x^+)\cdot {\vec q}}\int{dq^-\over 2\pi} e^{iq^- x^+}
exp[i\eta(\psi_{0}^-
{\tilde \psi}^+_{0}+{\bar\psi}^2{\tilde\psi}^2+{\bar\psi}^4{\tilde\psi}^4)]
(|0>\otimes|{\tilde 0}>)_{q^- ,{\vec q}}.\cr}}
%
The oscillator vacuum is annihilated by modes with $n,p,r>0$.  In the 
NS-NS sector there are additional fermion zero modes for the $\lambda^i$, but
the $\psi^{\mu}$ zero modes are not present.  Here, $\psi^{\mu}=
{-i \psi^{\mu}_0 +\psi^{\mu +1}_0\over\sqrt{2}}$ for $\mu\in\{2,4\}$.
The zero mode vacuum in the RR sector is defined by the conditions:
%
\eqn\zmv{\eqalign{{\hat p}^+(|0>\otimes|{\tilde 0}>)_{q^- ,{\vec q}}&=0\cr
{\hat p}^-(|0>\otimes|{\tilde 0}>)_{q^- ,{\vec q}}&=q^-\cr
{\hat {\vec p}}(|0>\otimes|{\tilde 0}>)_{q^- ,{\vec q}}&={\vec q}\cr
\psi^+_0|0>&=0\cr
{\tilde\psi}^-_0|{\tilde 0}>&=0\cr
\psi^{\mu}| 0>&=0\cr
{\tilde{\bar \psi}}^{\mu}|{\tilde 0}>&=0\cr}}
%
Interchanging $+$ and $-$ give the boundary conditions for the ${\vec Y}(x^-)$
pulse.  We will not write the ghost contribution explicitly.  The ghosts
will cancel the contribution to the partition function from two untwisted
directions.  In odd spin structures of the RR sector ($\eta_1\eta_2=-1$)
the ghost contribution does not vanish.

The full boundary state $|F_{\alpha}, {\vec Y}_+>$ satisfying a GSO projection
can be written as
%
\eqn\fullbo{\eqalign{|F_{i}, {\vec{Y}}_+>&=c_1(|F_{i}, {\vec{Y}}_+ ,+>_{NS-NS}+
|F_{i}, {\vec{Y}}_+ ,->_{NS-NS}+|F_{i}, {\vec{Y}}_+ ,+>_{RR}+
|F_{i}, {\vec{Y}}_+ ,->_{RR})\cr
|F_{0}, {\vec{Y}}_+>&=c_2(|F_{0}, {\vec{Y}}_+ ,+>_{NS-NS}-
|F_{0}, {\vec{Y}}_+ ,->_{NS-NS}+|F_{0}, {\vec{Y}}_+ ,+>_{RR}+
|F_{0}, {\vec{Y}}_+ ,->_{RR})\cr}}
%
up to overall normalizations $c_1$, $c_2$.  The closed string Hamiltonian is 
%
\eqn\ham{\eqalign{H_{\alpha}&=
{{\hat p}^2\over 2}+2\pi[\sum_{n>0}(-a^+_{-n} a^-_n-
a^-_{-n} a^+_n+a^{\mu}_{-n}\cdot a^{\mu}_{n}- 
{\tilde a}^+_{-n} {\tilde a}^-_n-{\tilde a}^-_{-n} 
{\tilde a}^+_n+{\tilde a}^{\mu}_{-n}\cdot {\tilde a}^{\mu}_{n})\cr
&+\sum_{r>0}(b^i_{-r}\cdot b^i_{r}+{\tilde b}^i_{-r}\cdot {\tilde b}^i_{r})\cr
&+\sum_{p>0} p(-\psi^+_{-p} \psi^-_p-
\psi^-_{-p} \psi^+_p+\psi^{\mu}_{-p}\cdot \psi^{\mu}_{p}- 
{\tilde \psi}^+_{-p} {\tilde \psi}^-_p-{\tilde \psi}^-_{-p} 
{\tilde \psi}^+_p +{\tilde \psi}^{\mu}_{-p}\cdot {\tilde \psi}^{\mu}_{p}
+\lambda^i_{-p}\cdot \lambda^i_{p}+{\tilde \lambda}^i_{-p}\cdot 
{\tilde \lambda}^i_{p})]\cr &+a_{\alpha}^{s}+H_{ghost}\cr}}
%
where the index $s=1$ denotes the NS-NS sector while $s=2$ indicates the 
RR sector, $a_0^{1}=-1$ and the other $a_i^{s}=0$.

Partition function amplitudes where both pulses are a function of $x^+$
or $x^-$ are supersymmetric, and one obtains the usual result up to
the zero mode contribution--these amplitudes vanish.  When we take pulses
${\vec Y}_1 (x^-)$ and ${\vec Y}_2 (x^+)$, the fermion zero mode 
contribution in the even spin structure for the lightcone directions
is halved.  Note that terms linear in the spin structure index $\eta$
vanish in the sum.  The odd spin structures still
vanish because of zero mode fermions in other directions,  but the
RR sector contribution is one-half the NS-NS sector because of the 
lightcone zero modes.  This effect is similar to a brane-antibrane
interaction but weaker as the sign of the RR interaction does
not change.  The amplitudes are as follows:
%
\eqn\ampnonsusy{\eqalign{\int dx^+ dx^-\int_0 ^{\infty} dl 
<F_1 ,{\vec Y}_{1-}|e^{-lH}|F_1 ,{\vec Y}_{2+}>&=
{c_1^2\over 8\pi^2}\int dx^+ dx^-\int_0 ^{\infty} {dl\over l^2}
e^{-b^2 (x^+ ,x^-)\over 2l} {f_3^4(q)f_2^4(q)\over f_1^4(q)f_4^4(q)}\cr
\int dx^+ dx^-\int_0^{\infty} dl 
<F_0 ,{\vec Y}_{1-}|e^{-lH}|F_0 ,{\vec Y}_{2+}>&=
{c_2^2\over 32\pi^4}\int dx^+ dx^-\int_0 ^{\infty} {dl\over l^4}
e^{-b^2 (x^+ ,x^- )\over 2l} {f_2^8 (q)\over f_1^8 (q)}\cr}}
%
where $q=e^{-2\pi l}$ and 
%
\eqn\fdefs{\eqalign{b(x^+ ,x^-)&=|{\vec Y}_{2+}-{\vec Y}_{1-}|\cr
f_1(q)&=q^{1\over 12}\prod_{n=1}^{\infty}(1-q^{2n})\cr
f_2(q)&=\sqrt{2}q^{1\over 12}\prod_{n=1}^{\infty}(1+q^{2n})\cr
f_3(q)&=q^{-1\over 24}\prod_{n=1}^{\infty}(1+q^{2n-1})\cr
f_4(q)&=q^{-1\over 24}\prod_{n=1}^{\infty}(1-q^{2n-1}).\cr}}
%
The constants $c_1$ and $c_2$ can be determined from the supersymmetric
case by comparing open and closed cylinders.  We find that
$c_2={\pi\over 2}c_1$.

The total amplitude in the field theory ($l\rightarrow\infty$) limit is
%
\eqn\ampllarge{\eqalign{\lim_{l\rightarrow\infty}
{\cal A}_{\alpha\beta}^{-+}&\sim
{c_1^2\over 2\pi^2}(-1+4\delta_{\alpha\beta})\int dx^+ dx^- 
(\int_l ^{\infty} {dl'\over {l'}^2}
e^{-b^2 (x^+ ,x^-)\over 2l'})\cr
&+{c_1^2\over 8\pi^2}\int dx^+ dx^- (\int_l ^{\infty} {dl'\over {l'}^4}
e^{-b^2 (x^+ ,x^- )\over 2l'} ).\cr}}
%
The potential is positive for $\alpha=\beta$ and 
decreases with separation of the D-strings, while it is negative
for $\alpha\neq\beta$ and increases with separation.  For small $l$ a modular
transformation $l\rightarrow{1\over 2t}$ reveals a tachyon in the
open string spectrum for $b(x^+ ,x^-)<b_{crit}=\sqrt{2\pi^2\alpha'}$.
This tachyon turns out not to be present for the $\alpha\neq\beta$
case.  
For sufficiently
large separation $b(x^+ ,x^-)$ there is no divergence and the potentials
for large t are
%
\eqn\amptlargeaa{\lim_{t\rightarrow\infty}{\cal A}_{\alpha\alpha}^{-+}\sim
{c_1^2\over 4\pi^2} 
\int dx^+ dx^- (\int_t^{\infty}{dt'\over t'^2}
e^{-t'[b^{2}(x^+ ,x^-)-b_{crit}^2]})}
%
\eqn\amptlargeab{\lim_{t\rightarrow\infty}{\cal A}_{\alpha\beta}^{-+}\sim
{-c_1^2\over 2\pi^2}\int dx^+ dx^- (\int_t^{\infty}{dt'\over t'^2}
e^{-t' b^{2}(x^+ ,x^-)})}
%
where the second amplitude has $\alpha\neq\beta$.  The behavior of the 
large $t$ region as a function of $b$ in the $\alpha=\beta$ case
is the same as that for the large $l$ region provided that $b>b_{crit}$.
There is a runaway repulsive potential pushing the stable vacuum to
$b\rightarrow\infty$. We expect a tachyonic phase transition as 
the average $b\rightarrow b_{crit}$
in which the system decays to a supersymmetric set of D-strings with
a unidirectional travelling wave.  What would be interesting to
understand are subcritical separations that occur on small pieces of
the  D-strings.  For $\alpha\neq\beta$ there is a minimum with diverging,
negative two-dimensional vacuum energy at $b=0$ while there
is a maximum with zero vacuum energy as $b\rightarrow\infty$.
Since these D-strings with waves break supersymmetry, there will be
large corrections.  Nevertheless, for the $\alpha=\beta$ case with
$b>>b_{crit}$ and $g_s=\lim_{R\rightarrow\infty}e^{\phi}<<1$, 
the radiation calculation of the next
section should be approximately valid.  There is less control over the
string coupling for the $\alpha\neq\beta$ case with negative 
vacuum energy, and the coupling will become strong as $b\rightarrow 0$.
To first order the system has no tachyon for $b>0$, and we will
assume that the radiation calculation has some validity for 
$b>>b_{crit}$.  



\newsec{Gravitational Radiation}

Calculation of gravitational radiation from a cosmic string with
travelling waves was performed in \twelve .
In this section we calculate the massless NS-NS closed string emission from
the interaction of two D-strings with antiparallel travelling waves.  A
similar calculation of radiation from moving D-particle interactions is
extensively discussed in \nine , and many of the results there will
be useful in our calculation.  We want to determine the following
amplitude:
%
\eqn\amprad{{\cal A}^{-+}_{\alpha\beta}(p)=
\int dx^+ dx^-\int_0^\infty dl \int_0^l d\tau <B_{\alpha},
\vec{Y}_{1-}|e^{-lH}V_p(\sigma ,\tau)|B_{\beta},\vec{Y}_{2+}>}
%
where
%
\eqn\vertexop{V_p(z,{\bar z})=e_{\mu\nu}(\partial X^{\mu}-{1\over 2}
p\cdot\Psi\Psi^{\mu})({\bar\partial} X^{\mu}+{1\over 2}
p\cdot{\tilde{\Psi}}{\tilde\Psi}^{\nu})e^{ip\cdot x}}
%
and $2p^+ p^-=\|{\vec p}\|^2\equiv p^2$.  
We restrict the index $\mu\in\{+,-,2,3\}$.  Here, $X^{\mu}=X^{\mu}(z)+
{\tilde X}^{\mu}({\bar z})$.
As explained in \nine , the axionic amplitude $e_{\mu\nu}=B_{\mu\nu}$
comes from the odd spin structure of the RR sector.  Because there are
extra fermion zero modes for the ${\bf Z}_2\times {\bf Z}_2$ orbifold,
the odd spin structure vanishes unless the $V_p$ insertion 
has these zero modes.  If we consider polarizations in the extra dimensions,
then there will be a term with four zero modes.  We will not discuss this
case further here.

Our gauge choice is the following.
The dilaton has a polarization tensor with nonvanishing components
$e_{++}={(p^-)^2\over p^2}$, $e_{--}={(p^+)^2\over p^2}$, $e_{+-}={-1\over 2}$,
and $e_{ij}=\delta_{ij}-{p_i p_j\over p^2}$ for $i,j\in\{2,3\}$.
The graviton has components $e_{++}=h_{++}^p$, $e_{--}=h_{--}^p$,
$e_{i+}=h_{i+}^p$, and
$e_{i-}=h_{i-}^p$ with the others vanishing and satisfies 
$\eta^{\mu\nu}h_{\mu\nu}=0$.  Both the dilaton and graviton satisfy
$p^{\mu}e_{\mu\nu}=0$, and the dilaton has $\eta^{\mu\nu}e_{\mu\nu}=2$.
The index $p$ indicates that these components are a Fourier transform that
will be defined more precisely near the finish of this calculation.
This gauge is consistent with working in the rescaled metric \metric .

The amplitude can be split into a zero mode and oscillator part.  
Defining $l'=l-\tau$ so that $l'=0$ at the boundary with pulse
${\vec Y}_1(x^-)$ and $l'=l$ at the boundary with ${\vec Y}_2(x^+)$,
the amplitude between two $F_1$ D-strings can be written as
(see \nine\ for details)
%
\eqn\maineq{\eqalign{{\cal A}_{F_1}(p)=&\int dx^+ dx^-(e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr &(\int_0^{\infty}d\tau
\int_0^{\infty}dl'\int {d^4k\over(2\pi)^4} e^{i{\vec k}\cdot{\vec b}(x^+ ,x^-)}
e^{-l'k^2/2} e^{-\tau q^2/2}<e^{ip\cdot x}>_{osc}{\cal N}_{F_1}).\cr}}
%
Note that $k^+ =p^+$, $q^- =-p^-$, and ${\vec k}={\vec q}+{\vec p}$
with $k^- =q^+ =0$.  Also, ${\vec b}(x^+ ,x^-)
={\vec Y}_1(x^-)-{\vec Y}_2(x^+)$.  
The factor
%
\eqn\Namp{{\cal N}_{F_1}\equiv\sum_{s,a}Z_{F_1}^{s,a}{\cal M}^{s,a}
\epsilon^{s,a}_{F_1}}
%
where 
%
\eqn\zpart{Z_{F_1}^{s,a}=<F_1,\eta_1|e^{-lH}|F_1,\eta_2 >^{s,a}_{osc},}
%
$a=\pm 1$
with $\eta_1\eta_2=a$, and $\epsilon^{s,a}_{F_1}=\pm 1$ is determined by
\fullbo .  For instance,
%
\eqn\Zresults{\eqalign{Z^{1 +}_{F_1}&={-2Z^{2 +}_{F_1}\over 
1+2i\eta_1 F_0^{2+}}=
{f^4_3(q) f^4_2(q)\over f^4_1(q) f^4_4(q)}\cr
Z^{1 +}_{F_0}&={f^8_3(q)\over f_1^8(q)}\cr
Z^{1 -}_{F_0}&={f^8_4(q)\over f_1^8(q)}\cr
Z^{2 +}_{F_0}&=-{f^8_2(q)\over 2 f_1^8(q)}(1+2i\eta_1 F_0^{2+})\cr}}
%
where $F_0^{2+}$ is defined below.

In the even spin structure correlation functions are defined as
%
\eqn\defcorreven{<{\cal O}(\sigma ,\tau)>^s_{osc}\equiv{<F_i ,\eta_1|
e^{-lH}{\cal O}(\sigma ,\tau)|F_i ,\eta_2>^s_{osc}\over Z^s_{F_i}}.}
%
The correlation function
%
\eqn\expp{<e^{ip\cdot X}>_{osc}=\prod_{n=0}^{\infty}
(1-q^{2n}e^{-4\pi\tau})^{-{p^2\over 2\pi}}
(1-q^{2n}e^{-4\pi l'})^{-{p^2\over 2\pi}}.}
%
The odd spin structure will only contribute to the $F_0$ amplitudes in
the NS-NS sector and is defined as above.

The factor ${\cal M}^{sa}$ takes the form
%
\eqn\mfactor{\eqalign{{\cal M}^{sa}&=e_{\mu\nu}\{ <\partial X^{\mu}
{\bar\partial}X^{\nu}>_{osc}-<\partial X^{\mu} p\cdot X>_{osc}
<{\bar\partial}X^{\nu}p\cdot X>_{osc}\cr
&+{1\over 4}(<p\cdot\Psi p\cdot{\tilde\Psi}>^{sa}
<\Psi^{\mu}{\tilde\Psi}^{\nu}>^{sa}\cr &-<p\cdot\Psi\Psi^{\mu}>^{sa}
<p\cdot{\tilde\Psi}{\tilde\Psi}^{\nu}>^{sa}+<p\cdot{\tilde\Psi}\Psi^{\mu}>^{sa}
<p\cdot\Psi{\tilde\Psi}^{\nu}>^{sa})\cr
&+{i\over 2}(<\partial X^{\mu} p\cdot X>_{osc}
<p\cdot{\tilde\Psi}{\tilde\Psi}^{\nu}>^{sa}-
<{\bar\partial}X^{\nu}p\cdot X>_{osc}<p\cdot\Psi\Psi^{\mu}>^{sa})\cr
&-{1\over 2}k^{\mu}(i<{\bar\partial}X^{\nu}p\cdot X>_{osc}
+{1\over 2}<p\cdot{\tilde\Psi}{\tilde\Psi}^{\nu}>^{sa})\cr
&+{1\over 2}k^{\nu}(i<\partial X^{\mu} p\cdot X>_{osc}
-{1\over 2}<p\cdot\Psi\Psi^{\mu}>^{sa})-{1\over 4}k^{\mu}k^{\nu}\}.\cr}}
%
The nonzero correlators, where we need to keep track of $\eta_1$, 
are the following:
%
\eqn\corrone{<\partial X^+(z){\tilde X}^-({\bar z})>_{osc}=
-<{\bar\partial}{\tilde X}^+({\bar z})X^-(z)>_{osc}={i\over 2} K(\tau,l)}
%
\eqn\corrtwo{<\partial X^i(z){\tilde X}^j({\bar z})>_{osc}=
\delta^{ij}{i\over 2} K(\tau,l)} 
%
\eqn\corrthree{<\Psi^+(z){\tilde\Psi}^-({\bar z})>^{sa}=
<\Psi^-(z){\tilde\Psi}^+({\bar z})>^{sa}=
{F_0^{sa}\over 1+2i\eta_1 F_0^{sa}}+i\eta_1 F_{osc}^{sa}}
%
\eqn\corrfour{<\Psi^i(z){\tilde\Psi}^j({\bar z})>^{sa}=i\eta_1
\delta^{ij}
({-1\over 2}\delta^s_2\delta^a_+ +F_{osc}^{sa})}
%
\eqn\corrfive{<\Psi^+(z)\Psi^-(z)>^{sa}={-i\eta_1 F_0^{sa}\over
1+2i\eta_1 F_0^{sa}}-N_{\infty}^{osc}}
%
\eqn\corrsix{<\Psi^i(z)\Psi^j(z)>^{sa}=
\delta^{ij}({1\over 2}\delta^s_2\delta^a_+
+N_{\infty}^{osc})}
%
\eqn\defone{K(\tau , l)=-\sum_{n=0}^{\infty}({q^{2n}e^{-4\pi\tau}\over
1-q^{2n}e^{-4\pi\tau}}-{q^{2n}e^{-4\pi l'}\over
1-q^{2n}e^{-4\pi l'}})}
%
\eqn\deftwo{\eqalign {&F_0^{2 +}
=<0|\Psi^-_0|0><{\tilde 0}| {\tilde\Psi}_0^+|{\tilde 0}>\cr
&F_{osc}^{2+}=-\sum_{n=0}^{\infty}(-1)^{n}({q^{2n}e^{-4\pi\tau}\over
1-q^{2n}e^{-4\pi\tau}}+{q^{2n}e^{-4\pi l'}\over
1-q^{2n}e^{-4\pi l'}})\cr
&F_{osc}^{1\pm}=-\sum_{n=0}^{\infty}(\mp )^{n}({q^{n}e^{-2\pi\tau}\over
1-q^{2n}e^{-4\pi\tau}}\pm{q^{n}e^{-2\pi l'}\over
1-q^{2n}e^{-4\pi l'}}).\cr}}
%
Note that supersymmetry breaking allows for an arbitrarily valued
fermion condensate $F_0^{2 +}$.  The infinite constant $N_{\infty}^{osc}$
does not contribute to  ${\cal M}^{sa}$ because $p^{\mu}e_{\mu\nu}=0$.

For the dilaton and the graviton the term ${\cal M}^{sa}$ simplifies to
%
\eqn\dil{\eqalign{{\cal M}^{sa}_d&=-{p^2\over 4}({1\over 2}
\delta^s_2\delta^a_+ +{i\eta_1 F_0^{sa}\over
1+2i\eta_1 F_0^{sa}}+2F_{osc}^{sa})({1\over 2}
\delta^s_2\delta^a_+ -{i\eta_1 F_0^{sa}\over
1+2i\eta_1 F_0^{sa}})\cr &-{k^2\over 4}+{1\over 4}{({\vec k}\cdot{\vec p})^2
\over p^2}-{1\over 16}p^2\cr}}
%
\eqn\grav{{\cal M}^{sa}_g=-p^+(h_{+i}^p p^i K+{1\over 4}h_{++}^p p^+
+{1\over 2}h_{+i}^p k^i +\delta^s_2\delta^a_+( {1\over 4}h_{+i}^p p^i
+{1\over 2}h_{++}^p p^+{i\eta_1 F_0^{sa}\over
1+2i\eta_1 F_0^{sa}})).}
%

Notice that the dilaton amplitude depends on fermionic non-zeromodes
while the 
graviton receives a contribution from the bosonic oscillator correlations.
The non-zeromode piece of ${\cal M}^{sa}_d$ vanishes in the NS-NS sector.
The field theory ($l\rightarrow\infty$) limits of $K$ and $F_{osc}^{2+}$ are
%
\eqn\limits{\eqalign{K&=-(f(\tau)-f(l'))\cr
F_{osc}^{2+}&=-(f(\tau)+f(l'))\cr}}
%
where $f(x)={e^{-4\pi x}\over 1+e^{-4\pi x}}$.  Integration by parts
has been used to calculate ${\cal M}^{sa}$ with
%
\eqn\intbyparts{\int_0^{\infty}d\tau\int_0^{\infty}dl'({\partial}_{\tau}
-{\partial}_{l'})\{ e^{-{q^2+m^2\over 2}\tau}e^{-{k^2+m^2\over 2}l'}
<e^{ip\cdot X}>_{osc}\}=0}
%
where the infinite surface terms have been set to zero 
by an analytic continuation from $p^2<0$.
This integration by parts in the $l\rightarrow\infty$ limit allows
one to replace $f(\tau)$ by ${-1\over 4}{q^2+m^2\over p^2}$ and 
$f(l')$ by ${-1\over 4}{k^2+m^2\over p^2}$ where $m^2$ is the momentum in the
zeromode directions orthogonal to the four Minkowski dimensions.  
We also make use of the
relation
%
\eqn\rel{\int_0^{\infty}dx e^{-xa^2\over 2}(1-e^{-4\pi x})^{-b^2\over 2\pi}
={1\over 4\pi}B({a^2\over 8\pi} ,1-{b^2\over 2\pi})
={1\over 4\pi}{\Gamma({a^2\over 8\pi})\Gamma(1-{b^2\over 2\pi})\over
\Gamma({a^2\over 8\pi}+1-{b^2\over 2\pi})}.}
%
Equation \Namp\ yields 
%
\eqn\Namptwo{\eqalign{{\cal N}_{F_1}&=\sum_{\eta_1}(Z_{F_1}^{1+}
{\cal M}^{1+}+Z_{F_1}^{2+}{\cal M}^{2+})\cr
{\cal N}_{F_0}&=\sum_{\eta_1}(Z_{F_0}^{1+}
{\cal M}^{1+}-Z_{F_0}^{1-}{\cal M}^{1-}+Z_{F_0}^{2+}{\cal M}^{2+}).\cr}}
%
Calculating these terms in the $l\rightarrow\infty$ limit yields
%
\eqn\namplimod{\lim_{l\rightarrow\infty}{\cal N}^d_{F_1}=k^2
+p^2({1\over 8}+{(F_0^{2+})^2\over 1+4(F_0^{2+})^2})-{1\over 4}
{\vec k}\cdot{\vec p}+{({\vec k}\cdot{\vec p})^2\over p^2}
+{m_1^2\over 4}}
%
\eqn\namplimzd{\lim_{l\rightarrow\infty}{\cal N}^d_{F_0}=4{\cal N}^d_{F_1}
+m_0^2-m_1^2}
%
\eqn\namplimog{\lim_{l\rightarrow\infty}{\cal N}^g_{F_1}=
2p^i p^+ h_{i+}^p{{\vec k}\cdot{\vec p}\over p^2}-(p^+)^2  h_{++}^p
-2k^i p^+ h_{i+}^p}
%
\eqn\namplimzg{\lim_{l\rightarrow\infty}{\cal N}^g_{F_0}=4{\cal N}^g_{F_1}.}
%
We also utilize the relation
%
\eqn\betalim{\lim_{\alpha' p^2\rightarrow 0}B({x\over 8\pi},1-\alpha' p^2)
={8\pi\over x^2}.}
%
Finally, the amplitudes reduce to 
%
\eqn\ampdil{\eqalign{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_{\alpha\beta}^d(p)=\int dx^+ dx^- (e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&c_1^2\sum_i\epsilon^i_{\alpha\beta}\int^{\Lambda}{d^2 m_i\over4\pi^2}
\int^{\Lambda} {d^2k\over(2\pi)^2} e^{i{\vec k}\cdot{\vec b}(x^+ ,x^-)}
({k^2\over (k^2+m_i^2)(q^2+m_i^2)}\cr
&+{p^2\over (k^2+m_i^2)(q^2+m_i^2)}
({1\over 8}+{(F_0^{2+})^2\over 1+4(F_0^{2+})^2})-{1\over 4}
{{\vec k}\cdot{\vec p}\over(k^2+m_i^2)(q^2+m_i^2)}\cr
&+{({\vec k}\cdot{\vec p})^2\over p^2(k^2+m_i^2)(q^2+m_i^2)}
+{m_i^2\over 4(k^2+m_i^2)(q^2+m_i^2)})+{\cal A}_0^d(p)\cr}}
%
\eqn\ampgrav{\eqalign{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_{\alpha\beta}^g(p)=\int dx^+ dx^- (e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&c_1^2\sum_i\epsilon^i_{\alpha\beta}\int^{\Lambda}{d^2 m_i\over4\pi^2}
\int^{\Lambda} {d^2k\over(2\pi)^2} e^{i{\vec k}\cdot{\vec b}(x^+ ,x^-)}
{1\over(k^2+m_i^2)(q^2+m_i^2)}\cr 
&(2p^i p^+ h_{i+}^p{{\vec k}\cdot{\vec p}\over p^2}-(p^+)^2  h_{++}^p
-2k^i p^+ h_{i+}^p)+{\cal A}_0^g(p)\cr}}
%
\eqn\ampdilz{\eqalign{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_0^d(p)=\int dx^+ dx^- (e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&\pi^2 c_1^2\int^{\Lambda}{d^6 m_0\over (2\pi)^6}
\int^{\Lambda} {d^2k\over(2\pi)^2} e^{i{\vec k}\cdot{\vec b}(x^+ ,x^-)}
({k^2\over (k^2+m_0^2)(q^2+m_0^2)}\cr
&+{p^2\over (k^2+m_0^2)(q^2+m_0^2)}
({1\over 8}+{(F_0^{2+})^2\over 1+4(F_0^{2+})^2})-{1\over 4}
{{\vec k}\cdot{\vec p}\over(k^2+m_0^2)(q^2+m_0^2)}\cr
&+{({\vec k}\cdot{\vec p})^2\over p^2(k^2+m_0^2)(q^2+m_0^2)}
+{m_0^2\over 4(k^2+m_0^2)(q^2+m_0^2)})\cr}}
%
\eqn\ampgravz{\eqalign{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_0^g(p)=\int dx^+ dx^- (e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&\pi^2 c_1^2\int^{\Lambda}{d^6 m_0\over (2\pi)^6}
\int^{\Lambda} {d^2k\over(2\pi)^2} e^{i{\vec k}\cdot{\vec b}(x^+ ,x^-)}
{1\over(k^2+m_0^2)(q^2+m_0^2)}\cr 
&(2p^i p^+ h_{i+}^p{{\vec k}\cdot{\vec p}\over p^2}-(p^+)^2  h_{++}^p
-2k^i p^+ h_{i+}^p)\cr}}
%
where $\epsilon^i_{\alpha\beta}=\pm 1$ and can be determined from
\frbrane .  We have introduced a cutoff on momentum, 
$\Lambda\sim{1\over\sqrt{\alpha'}}$ because the large momentum region 
picks out the small $l$ region where the above calculation is no 
longer valid, but for $b>>b_{crit}$ taking $\Lambda\rightarrow\infty$
should be okay.  Again, we can do a modular transformation 
$l\rightarrow {1\over 2t}$ to show that there is no divergence so long 
as the average separation of the D-strings is sufficiently large.
We can calculate the above integrals using the formula
%
\eqn\modbess{{1\over 4\pi^2}\int d^2k {e^{i{\vec k}\cdot{\vec b}}\over
k^2+m^2}={-1\over 2\pi}K_0(mb)}
%
where $K_0$ is the modified Bessel function of order zero.
We also need the following integrals which can be found in
\thirteen .
%
\eqn\intone{\eqalign{&\int_0^{\infty} x K_0(ax)dx={\pi\over a^2}\cr
&\int_0^{\infty} x K_0(ax) K_0(bx)dx={-1\over ab(a+b)}\cr}}
%
Defining
%
\eqn\gfunction{\eqalign{g(b)&={1\over 8\pi^3}\int d^2x{1\over |{\vec x}|
|{\vec b}-{\vec x}|(|{\vec x}|+|{\vec b}-{\vec x}|)}\cr
a_F&={7\over 8}+{(F_0^{2+})^2\over 1+4(F_0^{2+})^2}\cr}}
%
where $g(b)$ is finite for $b>0$, we obtain
%
\eqn\ampdiltwo{\eqalign
{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_{\alpha\beta}^d(p)=c_1^2(-1+4\delta_{\alpha\beta})
\int dx^+ dx^- (e^{i{\vec Y}_2(x^+)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&[{1\over 16\pi b^2}+{i{\vec b}\cdot{\vec p}\over 2\pi b^4 p^2}
+e^{i{\vec b}\cdot{\vec p}}({-3\over 16\pi b^2}-
{i{\vec b}\cdot{\vec p}\over 2\pi b^4 p^2}+a_F p^2 g(b)
-{1\over 2}\bigtriangleup_b g(b)\cr
&+{i{\vec b}\cdot{\vec p}\over 4b}\partial_b g(b))]+{\cal A}_0^d(p)\cr}}
%
\eqn\ampgravtwo{\eqalign
{\lim_{\alpha' p^2\rightarrow 0 \atop l\rightarrow\infty}
&{\cal A}_{\alpha\beta}^g(p)=c_1^2(-1+4\delta_{\alpha\beta})
\int dx^+ dx^- (e^{i{\vec Y}_1(x^-)
\cdot {\vec p}-ip^+x^- -ip^- x^+})\cdot\cr
&[-2i{p^i p^+ h_{i+}^p{\vec b}\cdot{\vec p}\over b p^2}
\partial_b g(b)-(p^+)^2 h_{++}^p g(b)
+2i{b^i p^+ h_{i+}^p\over b}\partial_b g(b)]+{\cal A}_0^g(p)\cr}}
%
where $\bigtriangleup_b$ is the two-dimensional 
Laplacian with respect to ${\vec b}$.  We leave as an exercise the calculation
of ${\cal A}_0^d(p)$ and ${\cal A}_0^g(p)$ which should vanish more
quickly for large $b$.

For large $b$ the dominant contributions are
%
\eqn\ampdildom{{\cal A}_{\alpha\beta}^d(p)\sim(-1+4\delta_{\alpha\beta})
\int dx^+ dx^- (e^{i{\vec Y}_1(x^-)
\cdot {\vec p}-ip^+x^- -ip^- x^+})p^2 g(b)}
%
\eqn\ampgravdom{{\cal A}_{\alpha\beta}^g(p)\sim(-1+4\delta_{\alpha\beta})
\int dx^+ dx^- (e^{i{\vec Y}_1(x^-)
\cdot {\vec p}-ip^+x^- -ip^- x^+})(p^+)^2 h_{++}^p g(b)}
%
Notice that the amplitudes vanish for unidirectional travelling waves.

To complete the calculation we will assume that the strings are far
enough apart that there is an approximate solution in supergravity
that is the sum of the right and left moving fluctuations or that there
is a linear approximation even though supersymmetry is broken.  The
radiation is observed at $r_{con}=0$ (see \metric ) and 
$R>>b>>\sqrt{\alpha'}$ so we will ignore the angular dependence due
to nonzero separation.  The fluctuation of the graviton and dilaton can be 
written in position space as
%
\eqn\polar{e_{\mu\nu}\sim T'_{\mu\nu}(1-e^{-2\phi})\sim{T'_{\mu\nu}\over R^6}}
%
where the prime denotes that we drop the delta function of the 
transverse space.
The energy-momentum tensor for the travelling wave on the D-string
was calculated from the action for the D-string in \one\ with the 
result to lowest order in $\alpha'$
%
\eqn\stressenergy{\eqalign{T_{+-}&=T_D(\delta^{(8)}({\vec x}-{\vec Y}_1)
+\delta^{(8)}({\vec x}-{\vec Y}_2))\cr
T_{++}&=T_D\|\partial_+{\vec Y}_2\|^2 
\delta^{(8)}({\vec x}-{\vec Y}_2)\cr
T_{--}&=T_D\|\partial_-{\vec Y}_1\|^2\delta^{(8)}({\vec x}-{\vec Y}_1)\cr
T_{+i}&=-T_D \partial_+ Y^i_2\delta^{(8)}({\vec x}-{\vec Y}_2)\cr
T_{-i}&=-T_D \partial_- Y^i_1\delta^{(8)}({\vec x}-{\vec Y}_1)\cr}}
%
where $T_D\sim{1\over g_s\alpha'}$ is the D-string tension.
In this linear approximation there is no gravitational radiation
because there are no terms in the energy-momentum tensor containing
an interaction of the two pulses, and the Fourier transform of $h_{++}$
is annihilated by $p^+$.  There will be higher order terms with 
these interactions leading to gravitational radiation,
but the calculation will break down for $b\rightarrow b_{crit}$.

To obtain the second order of the graviton fluctuation is difficult.
There are $\alpha'$ corrections to the D-string action and to the 
supergravity fields which couple to this action.  In section three
we have shown that there is a repulsive (attractive) potential
for the $\alpha=\beta$ ($\alpha\neq\beta$) interaction.  For $b>>b_{crit}$,
this interaction is of order $1\over b^2$.  To second order the
energy-momentum tensor calculated from the D-string action by itself
will not
be covariantly conserved because one has to include the energy-momentum
of the radiation in the bulk.
The gauge choice we have made for the graviton may also
be inappropriate at second order. Rather than trying to determine
precisely the second order fluctuation we will assume that a conserved
energy-momentum tensor of the full theory incorporating the nonlocal 
interaction is given where we still take $b>>b_{crit}$.
This energy-momentum tensor takes the form
%
\eqn\stress{T_{\mu\nu}=T^{(4)}_{\mu\nu}\delta^{(6)}({\vec x})}
%
where $\mu\in\{ 0,1,2,3\}$.  We choose a gauge such that the
three degrees of freedom of the graviton and dilaton satisfy
$(\bigtriangleup^{(10)}-\partial_t^2)e_{\mu\nu}=T_{\mu\nu}$ 
where $\bigtriangleup^{(10)}$
is the Laplace operator in ten dimensions.  The retarded Green function
for the ten-dimensional wave equation takes the form
%
\eqn\green{G(x, x')\sim{1\over r^7}
(\delta(\bigtriangleup_{x,x'})-r\delta'(\bigtriangleup_{x,x'})
+{2\over 5}r^2\delta''(\bigtriangleup_{x,x'})-
{1\over 15}r^3\delta'''(\bigtriangleup_{x,x'}))}
%
where $\bigtriangleup_{x,x'}=t'-t+|{\vec x}_{10}-{\vec x'}_{10}|$
and $r=|{\vec x}_{10}-{\vec x'}_{10}|$.




We follow the analysis of \twelve\ to determine the energy spectrum, but
the details are modified due to our unrealistic setup of having ten 
noncompact dimensions.   The dilaton and graviton  
satisfy 
%
\eqn\dilgrav{\eqalign{e_{\mu\nu}({\vec x},t)&\sim g_s\int d^3 x' 
{1\over |{\vec x}-{\vec x'}|^7}
(T^{(4)}_{\mu\nu}({\vec x'},t-|{\vec x}-{\vec x'}|)
-|{\vec x}-{\vec x'}|\partial_t 
T^{(4)}_{\mu\nu}({\vec x'},t-|{\vec x}-{\vec x'}|)
\cr &+{2\over 5}|{\vec x}-{\vec x'}|^2
\partial^2_t T^{(4)}_{\mu\nu}({\vec x'},t-|{\vec x}-{\vec x'}|)
-{1\over 15}|{\vec x}-{\vec x'}|^3 \partial^3_t 
T^{(4)}_{\mu\nu}({\vec x'},t-|{\vec x}-{\vec x'}|))\cr}}
%
Taking the Fourier transform of $T_{\mu\nu}$ in the lightcone directions
yields
%
\eqn\stressfourier{T^{(4)}_{\mu\nu}({\vec x},t)={1\over 2\pi}
\int_{-\infty}^{\infty}dw\int_{-\infty}^{\infty}dk\, 
T^{(4)}_{\mu\nu}(\vec{\rho} ,k,\omega)
e^{i(kz-\omega t)}}
%
where $\rho$ is the transverse distance, $\omega={p^+ +p^-\over\sqrt{2}}$,
$k={p^+ -p^-\over\sqrt{2}}$, and 
$T_{\mu\nu}^{*(4)}({\vec\rho},-k,-\omega)=
T_{\mu\nu}^{(4)}({\vec\rho},k,\omega)$.  Note that the coordinate ${\vec \rho}$
includes the extra dimensions (these are zero for $T_{\mu\nu}^{(4)}$), 
but our observations will always be
at the orbifold point.
Then
%
\eqn\gravdiltwo{\eqalign{e_{\mu\nu}({\vec \rho},z,t)&\sim {g_s\over 2\pi}
\int_{-\infty}^{\infty}dw e^{-i\omega t}
\int_{-\infty}^{\infty}dk e^{ikz}\int {dz' d\rho'\over D^7}
e^{i(\omega D+kz')}T^{(4)}_{\mu\nu}({\vec \rho'},k,\omega)\cdot\cr
&(1+i\omega D
-{2\over 5}\omega^2 D^2 +{1\over 15}i\omega^3 D^3)\cr}}
%
where $D=\sqrt{z'^2+|{\vec \rho}-{\vec \rho'}|^2}$.  In the limit
that the observation distance $\rho$ is much larger than the extent
of the source, we can do a stationary phase approximation of the $z'$
integral.  Thus, we obtain
%
\eqn\staphase{\eqalign{e_{\mu\nu}({\vec \rho},z,t)&\sim {g_s 
e^{i\pi\omega\over 4|\omega|}\over\sqrt{2\pi}
\rho^{13\over 2}\omega^6}\int_{|k|<|\omega|}
d\omega dk\, e^{i(kz-\omega t+{\vec q}\cdot
{\vec \rho})}(\omega^2-k^2)^{11\over 4}\cdot\cr &(1+
{i\omega|\omega|\rho\over\sqrt{\omega^2-k^2}}-{2\over 5}
{\omega^4\rho^2\over\omega^2-k^2}+{i\over 15}
{\omega^5|\omega|\rho^3\over(\omega^2-k^2)^{3\over 2}})\int d{\vec\rho'}
e^{-i{\vec q}\cdot{\vec\rho'}}T^{(4)}_{\mu\nu}({\vec\rho'},k,\omega)\cr}}
%
where ${\vec q}={\omega\over|\omega|}\sqrt{\omega^2-k^2}{{\vec\rho}\over\rho}$.
Obviously, there is a divergence for travelling waves that are
highly localized in the lightcone directions so for this model, we must
consider waves with compact support in $\omega$ or, alternatively,
periodic waves with discrete frequencies.  For periodic waves we
would calculate the power rather than the energy.



The Fourier transform of the graviton and dilaton are obtained
from the above equation as
%
\eqn\fourier{e_{\mu\nu}({\vec\rho} ,k,\omega)=
{g_s (2\pi)^{9\over 2}e^{i\pi\omega\over 4|\omega|}
\over\rho^{13\over 2}\omega^6}
T^{(4)}_{\mu\nu}({\vec q},k,\omega)
(\omega^2-k^2)^{11\over 4}(1+i{\omega|\omega| \rho\over \sqrt{\omega^2-k^2}}
-{2\over 5}{\omega^4\rho^2\over(\omega^2-k^2)}+{i\over 15}
{\omega^5|\omega|\rho^3\over(\omega^2-k^2)^{3\over 2}}).}
%

We now calculate the energy radiated through a large cylinder of radius
$\rho$ surrounding the D-strings.  Not all of the radiation is in
four dimensions--there is also radiation in the extra dimensions.
To not violate energy conservation we need to consider a seven-sphere
cylinder surrounding the D-strings.  Then we set all but one of the
angles on the seven-sphere to ${\pi\over 2}$ so that we observe the 
radiation through a cylinder at the orbifold point.  
The energy flux observed in the 
four-dimensional directions is given by
%
\eqn\energy{E=\rho^7\int_0^{2\pi}d\theta\int_{|k|<|\omega|}
dk d\omega d\Omega_7^{\vec q}
\{|\omega|(\omega^2-k^2)^3\}|{\cal A}|^2}
%
where $\Omega_7={\pi^4\over 3}$ is the volume of a seven-sphere and $\cal A$ is
the amplitude from the string calculation.  Assume that we keep the gauge
with which we did this calculation through a coordinate transformation
if necessary.  Then we substitute equation \fourier\ into 
\ampdildom ,\ampgravdom ,
and \energy .  At large distances energy is conserved because there
is no $\rho$ dependence, and we obtain
%
\eqn\endil{E_{dil}\sim g_s^2\int_0^{2\pi}d\theta \int_{|k|<|\omega|}dk d\omega
d\Omega_7^{\vec q}
(\omega^2-k^2)^{15\over 2} |\omega| |A_p|^2 
|T_{+-}^{(4)}({\vec q},k,\omega)|^2}
%
\eqn\engrav{E_{grav}\sim g_s^2\int_0^{2\pi}d\theta
\int_{|k|<|\omega|}dk d\omega d\Omega_7^{\vec q}
(\omega^2-k^2)^{11\over 2}(\omega+k)^4|\omega|
|A_p|^2 |T_{++}^{(4)}({\vec q},k,\omega)|^2}
%
where $A_p=\int dx^+ dx^- (e^{i{\vec Y}_1(x^-)
\cdot {\vec p}-ip^+x^- -ip^- x^+}) g(b)$.  The factor $|A_p|^2$ should
give a suppression of order ${1\over b^2}$.  Unless the travelling waves
are extremely localized in frequency, these energies will diverge.
This restriction will be general for noncompact extra dimensions, but
reducing the number of noncompact dimensions is helpful.  For
instance, for six noncompact dimensions, ignoring Kaluza-Klein
effects, one would obtain
%
\eqn\energysix{E^{six}_{dil}\sim g_s^2\int_0^{2\pi}d\theta 
\int_{|k|<|\omega|}dk d\omega d\Omega_3^{\vec q}
(\omega^2-k^2)^{7\over 2} |\omega| |A_p|^2 |T_{+-}^{(4)}({\vec q},k,\omega)|^2}
%
Actually, we can relate the frequency factors to supersymmetry breaking
since $|\omega|-|k|$ measures the amount of supersymmetry breaking, and
the radiation vanishes for waves with $|\omega|=|k|$.




\newsec{Conclusions}

Let us summarize our results.  We have constructed boundary states
for fractional D-strings with travelling waves on a ${\bf C}^3/
{{\bf Z}_2\times {\bf Z}_2}$ orbifold.  The underlying 
orbifold with D-strings is supersymmetric.  The interaction between
two of these D-strings with waves travelling in opposite directions 
breaks supersymmetry  and was calculated.  
If the fractional D-strings were of the same type,
the interaction was repulsive.  There was a tachyonic critical
point at small separation where an extra massless open string mode
appeared.  One might expect that there would be a tachyonic 
condensation with antiparallel momentum modes annihilating.
The end result would be a BPS state with momentum in one direction.
On the other hand, if the two D-strings were different types, the
interaction was attractive, there was no critical separation,
and there was a divergent, negative two-dimensional vacuum
energy minimum at zero separation.  The supersymmetry breaking
through the interaction of momenta in opposite directions 
reduces the RR interaction relative to the NS-NS interaction
so the potentials we have calculated in string theory should
be related to decreasing the force due to the three-form RR 
field strength relative to the gravitational force.  The
relevant term in the supergravity would be of the form
$\int H^{RR}_+\cdot H^{RR}_-$ where $H^{RR}_{\pm}$ is the 
three-form RR field strength due to the D-string with a right
or left-moving pulse.

Under the assumption that for large separations of the D-strings
compared to the critical separation, perturbative string theory
was not entirely invalid, we calculated the dilaton and graviton 
radiation emitted from the interaction of antiparallel
pulses.  To first order there is dilaton but no graviton radiation.
The dilaton radiation is to be expected because the background with
these D-strings is not supersymmetric, and there is a two-dimensional
vacuum energy at any non-infinite separation of the D-strings.
At large enough separations the effective energy-momentum tensor
does not incorporate the nonlocal interaction between the two pulses,
and the graviton does not radiate.  Where the dilaton radiation
is related to the nonzero vacuum energy, the graviton radiation
is the result of left and right moving momenta annihilating
into gravitons.  This interaction deforms the waves over time
so that one must somehow obtain an energy-momentum tensor that
incorporates the D-strings as well as the bulk.  We were not
able at this time to obtain the appropriate interacting energy-momentum
tensor but assuming its existence showed that there was graviton
radiation.  There are two variables in the model governing the
intensity of the interaction.  Increasing the distance between the D-strings
decreases the interaction.  The amount of supersymmetry breaking can be
tuned to an arbitrarily small value by studying boundary states 
in which the frequency distribution has $|\omega|-|k|<<1$ so that, for
example, the
pulse on the first D-string is a very slowly varying function of $x^-$,
while the pulse on the second D-string is an arbitrary function of $x^+$.
One could perhaps have enough control over the time dependent interaction
to calculate the variation of the waves leading 
to a supersymmetric configuration.  This interaction is much less
violent than brane-antibrane annihilation so it might be more easily
understood.

There are differences between this string theory calculation and the
classical gravity calculation of cosmic string radiation as in \twelve .
We are required by supersymmetry to separate the right and left
moving pulses so that the interaction is nonlocal.  The situation
of having both type of pulses on the same string would not give a
controllable interaction in string theory.  How to obtain the
energy-momentum tensor for this kind of nonlocal interaction is
challenging.  We are also required to include noncompact extra
dimensions so that the metric is asymptotically flat at large
distances, and the string theory calculation makes sense.  As a result
the energy spectrum is unrealistic.  Perhaps an analogous calculation
could be done in four dimensions in an orientifolded theory.  The
time dependence of these backgrounds is somewhat trivial because
the wave only travels in one spatial dimension.  Having constructed
supersymmetric boundary states, studying waves in perpendicular 
directions could be a means for obtaining theories with nontrivial
three-dimensional dependence.  The calculations in this paper were
done in the closed string boundary state framework.  One could
possibly study these questions from the dual open string viewpoint
via the S-matrix formalism developed in \one .  The supersymmetry
breaking interaction could be introduced as a small perturbation.











\bigskip\centerline{\bf Acknowledgments}\nobreak
I wish to thank J. Distler, W. Fischler, and A. Loewy for 
helpful discussions.  This work was supported in part
by the National Science Foundation under Grant No. 0071512.




\listrefs

\end