# On spinning strings in I-brane background

Sagar Biswas
Department of Physics, Ramakrishna Mission Vidyamandira,
Belur Math, Howrah 711 202, India Email<EMAIL_ADDRESS>
###### Abstract:

We revisit semiclassical strings, in particular we focus on rigidly rotating
strings, in the near horizon geometry of two orthogonal stacks of NS5-branes
(I-branes) using the string sigma model. We determine the conserved charges
for the probe string moving in the resulting $\mathbb{R}_{t}\times
S^{3}_{\theta_{1}}\times S^{3}_{\theta_{2}}$ background supported by NS-NS
two-forms and find a regularized dispersion relation for different values of
integration constants. Using configurations that move simultaneously on both
spheres, we obtain the giant magnon like dispersion relation for one
particular set of parameters, while other consistent set of values gives us a
dispersion relation reminiscent of the single spike.

Bosonic strings, AdS/CFT correspondence

##  1 Introduction

$AdS/CFT$ correspondence [2, 3, 4], in its most famous avatar, relies on the
duality between type IIB string states in $AdS_{5}\times S^{5}$ and certain
operators in a four dimensional Conformal Field Theory (CFT) living on the
boundary of $AdS_{5}$. The exact matching of observables from both sides of
the duality has turned out to be one of the most powerful tools of modern
theoretical physics. Moreover, the appearance of Integrability as a key
feature in AdS/CFT correspondence makes it possible to find exact solutions of
both string theory in AdS space and gauge theory on its boundary. Using the
language of integrable systems, which naturally occur with elegant
mathematical techniques, one can explicitly solve the spectral problem of
$AdS/CFT$, i.e. the matching between string energies in the bulk with
anomalous dimensions of boundary CFT operators. Integrability techniques were
first developed in the SU(2) sector of planar $\mathcal{N}=4$ SYM, as the
action of the Dilatation operator in this sector can be reduced to an
integrable Heisenberg spin chain [5, 6, 7]. Followed by this, there has been
number of studies put forth to establish the appearance of integrability in
other lower dimensional counterparts of AdS/CFT duality [8]-[16].

In conjunction to the advent of integrability techniques in the last two
decades, study of the rigidly rotating strings in the semiclassical
approximation has turned out one of the interesting areas of research. In this
connection the so called Hofman-Maldacena (HM) limit [17], which can be seen
as a large angular momentum limit, simplifies considerably the problem of
finding out the spectrum on both sides of the duality. In this particular
instance, the spectrum consists of an elementary gauge theory excitation known
as magnon which propagates with a conserved momentum $p$ along the associated
spin chain, and this is dual to a certain rotating closed string in the
$S^{3}\in AdS_{5}$. Furthermore, a more general class of rotating string
solution in $AdS_{5}$ is the spiky string which describes the higher twist
operators from dual field theory view point [18] and magnon solutions can be
thought of as a subspace of these spike solutions. In fact it was futher
determined in [19] that if one solves the most general form of equations of
motion for a rigidly rotating string on a sphere one encounters two set of
solutions in different realms of the parameter space, corresponding precisely
to the giant magnon and single spike solutions.

To understand the AdS/CFT like dualities in more general backgrounds, it is
instructive to study rigidly rotating string in the gravity side which can
give us invaluable information about the the corresponding gauge theory
operators in the boundary side. Various other explorations in different exact
string backgrounds has been undertaken by many authors, and for a non-
exhaustive list one can see [20]-[44]. This intrigue forms the basis of our
current investigation where we study such a novel brane background. The so-
called I-brane background [45] rises as the 1+1 dimensional intersection of
two orthogonal stacks of NS5-branes, with one set of branes lying along
$(x^{0},x^{1},\cdots,x^{5})$, and other set lying along
$(x^{0},x^{6},\cdots,x^{9})$ directions. Holographically the near horizon
geometry of the stacks of parallel NS5 branes accomodates the non-local Little
String Theory (LST) in its worldvolume [46]. When all five branes are
coincident, in the S-dual picture, the near horizon geometry is given by

$R^{2,1}\times R_{\phi}\times SU(2)_{k_{1}}\times SU(2)_{k_{2}}$

where, $R_{\phi}$ is one combination of the radial directions away from the
two sets of NS5-branes, and the coordinates of $R^{2,1}$ are $x^{0},x^{1}$ and
another combination of two radial directions. The two $SU(2)$s with levels
$k_{1,2}$ describe the angular three-spheres corresponding to $(R^{4})_{2345}$
and $(R^{4})_{6789}$. As mentioned in [45] this background exhibits a higher
Poincare symmetry, ISO(2,1), than the expected ISO(1,1) and twice as many
supercharges one might expect. Various aspects of such intersecting brane
solutions in string theory has been explored in the literature [47, 48, 49,
50, 51, 52, 53, 54, 55, 56, 57, 58, 59].

Semiclassical strings on I-brane have been studied by probing the geometry
with both fundamental strings and D1-strings in [56]-[57], and the dual single
spike and giant magnon solutions have been constructed by studying rigidly
rotating fundamental strings using Nambu-Goto action in [60]. A similar class
of semiclassical solution was obtained in its S-dual picture i.e., by studying
a D1-string on two orthogonal stack of D5-branes in [38], which mimics the
behavior of F1 strings in stacks of NS5 branes.111For related spinning string
solutions from Neumann-Rosochatius like integrable systems, one can refer to
[61]. But, the analysis in the above two cases had a caveat, as in both the
cases one could not decouple the two spheres in the near horizon limit and
could only study the string dynamics under some restrictions imposed on the
equations of motion. For example, in [60] we could only study (i) the string
rotating in one of the sphere $(\theta_{1}=\theta)$ by setting the other zero
to $(\theta_{2}=0)$ and (ii) the string rotating around both the spheres by
setting $(\theta_{1}=\theta_{2}=\theta)$. In case (i) we obtain the dispersion
relations as,

$\displaystyle\tilde{J}$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{K}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{\pi}{2}-\theta_{0}\Big{)}}\
,~{}~{}~{}\text{single spike}$ $\displaystyle\tilde{E}-J$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{K}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\frac{\Delta\phi}{2}}\
,~{}~{}~{}~{}~{}~{}~{}~{}\text{giant magnon}$ (1.1)

which are the dispersion relations associated to strings in single NS5-brane
backgrounds as obtained in [43]. Here $\theta_{0}$ is related to the initial
conditions on the string dynamics. In case (ii) the dispersion relations found
were,

$\displaystyle\tilde{J}$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{K}^{2}+h\frac{\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{\pi}{2}-\theta_{0}\Big{)}}\
,~{}~{}~{}\text{single spike}$ $\displaystyle\tilde{E}-J$ $\displaystyle=$
$\displaystyle\sqrt{\tilde{K}^{2}+g\frac{\lambda}{\pi^{2}}\sin^{2}\frac{\Delta\phi}{2}}\
,~{}~{}~{}~{}~{}~{}~{}~{}\text{giant magnon}$ (1.2)

where $h$ ard $g$ are complicated combinations of different winding numbers.
This indicates that these classical configurations are unable to access both
$SU(2)$s distinctly in the gauge theory picture. In the current paper we will
remedy the situation, by focussing on the same background but using the gauge
fixed sigma model for the dynamics of the probe string. As Polyakov action of
string possess more symmetries than Nambu-Goto action of string this will
allow us to decouple the spheres and we could study the rotating string
simultaneously on both the spheres without any such restrictions.

The rest of the paper is organized as follows. In section (2), we will review
the near-horizon geometry of I-brane and find the equations of motion for the
rigidly rotating strings so that those are consistent with Virasoro
constraints. We will also completely classify possible configurations of the
string and find the associated profiles by integrating the equations of
motion. In section (3), we write down the conserved charges for the string
motion, and describe how the diverging charges can be regulated. Then, using
different consistent values of our parameters, we arrive at regularized
dispersion relations for these charges, which can be directly compared with
the dispersion relation obtained in NS5-branes. In section (4) we summarize
and comment on further investigations.

##  2 Probe strings on I-brane

As mentioned in the introduction, I-branes originate from $1+1$ dimensional
intersections of fivebranes that intersect on a $\mathbb{R}^{1,1}$. In what
follows, we will give a brief introduction to the target space geometry
associated to this background, and discuss fundamental string probes using
rigidly rotating embeddings.

###  2.1 Near-horizon geometry

The geometry of I-brane arises when $k_{1}$ number of NS5-branes lying along
$(0,1,\cdots,5)$ intersect $k_{2}$ number of NS5-branes lying along
$(0,1,6,\cdots,9)$ directions in (0,1)-plane. If the branes are coincident,
then the type IIB supergravity solution is given by [45],

$\displaystyle
ds^{2}=-(dx^{0})^{2}+(dx^{1})^{2}+H_{1}(y)\sum_{\alpha=2}^{5}(dy^{\alpha})^{2}+H_{2}(z)\sum_{p=6}^{9}(dz^{p})^{2}\
,$ $\displaystyle e^{2\phi}$ $\displaystyle=$ $\displaystyle H_{1}(y)H_{2}(z)\
,~{}~{}~{}H_{\alpha\beta\gamma}=-\epsilon_{\alpha\beta\gamma\delta}\partial^{\delta}H_{1}(y)\
,~{}~{}~{}H_{mnp}=-\epsilon_{mnpq}\partial^{q}H_{2}(z)\ ,$ (2.3)

where the harmonic functions are $H_{1}=1+\frac{k_{1}l_{s}^{2}}{y^{2}}$,
$H_{2}=1+\frac{k_{2}l_{s}^{2}}{z^{2}}$ and
$y=\sqrt{\sum_{\alpha=2}^{5}(y^{\alpha})^{2}}$ and
$z=\sqrt{\sum_{p=6}^{9}(z^{p})^{2}}$. In the near horizon limit
($\frac{k_{1}l_{s}^{2}}{y^{2}}>>1$ and $\frac{k_{2}l_{s}^{2}}{z^{2}}>>1$), the
metric and NS-NS fields are given by,

$\displaystyle
ds^{2}=-(dx^{0})^{2}+(dx^{1})^{2}+k_{1}l_{s}^{2}\frac{dr_{1}^{2}}{r_{1}^{2}}+k_{1}l_{s}^{2}d\Omega_{1}^{2}+k_{2}l_{s}^{2}\frac{dr_{2}^{2}}{r_{2}^{2}}+k_{2}l_{s}^{2}d\Omega_{2}^{2}\
,$ $\displaystyle b_{\phi_{1}\psi_{1}}=2k_{1}l_{s}^{2}\sin^{2}\theta_{1}\
,~{}~{}~{}b_{\phi_{2}\psi_{2}}=2k_{2}l_{s}^{2}\sin^{2}\theta_{2}\ ,$ (2.4)

where the three spheres are given by:

$d\Omega_{1}^{2}=d\theta_{1}^{2}+\sin^{2}\theta_{1}d\phi_{1}^{2}+\cos^{2}\theta_{1}d\psi_{1}^{2}\
,~{}~{}~{}d\Omega_{2}^{2}=d\theta_{2}^{2}+\sin^{2}\theta_{2}d\phi_{2}^{2}+\cos^{2}\theta_{2}d\psi_{2}^{2}\
.$

These $d\Omega_{1,2}$ are the volume elements on the sphere along
$(y^{2},\cdots,y^{5})$ and $(z^{6},\cdots,z^{9})$ directions respectively. To
proceed further we make the following change of variables (we choose
$l_{s}=1,k_{1}=k_{2}=N$)

$\displaystyle\rho_{1}=\ln\frac{r_{1}}{\sqrt{N}}\
,~{}~{}\rho_{2}=\ln\frac{r_{2}}{\sqrt{N}}\ ,~{}~{}x^{0}=\sqrt{N}t\
,~{}~{}x^{1}=\sqrt{N}x\ .$ (2.5)

The final form of the metric and the background NS-NS field are given by

$\displaystyle
ds^{2}=N(-dt^{2}+dx^{2}+d\rho_{1}^{2}+d\theta_{1}^{2}+\sin^{2}\theta_{1}d\phi_{1}^{2}+\cos^{2}\theta_{1}d\psi_{1}^{2}+d\rho_{2}^{2}+d\theta_{2}^{2}$
(2.6)
$\displaystyle+\sin^{2}\theta_{2}d\phi_{2}^{2}+\cos^{2}\theta_{2}d\psi_{2}^{2})\
,~{}~{}b_{\phi_{1}\psi_{1}}=2N\sin^{2}\theta_{1}\
,~{}~{}b_{\phi_{2}\psi_{2}}=2N\sin^{2}\theta_{2}\ .$

In the rest of the paper, we will be mainly concerned with the above
background in our computations.

###  2.2 String equations of motion

In the near horizon geometry (2.6), to study a fundamental string coupled to
NS-NS B-field, we use the Polyakov action with a WZ term,

$\displaystyle S=-\frac{\sqrt{\lambda}}{4\pi}\int d\sigma
d\tau[\sqrt{-\gamma}\gamma^{\alpha\beta}g_{MN}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}-\epsilon^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}b_{MN}]\
,$ (2.7)

where $\lambda$ is the ’t Hooft coupling, $\gamma^{\alpha\beta}$ is the
worldsheet metric and $\epsilon^{\alpha\beta}$ is the antisymmetric tensor
defined as $\epsilon^{\tau\sigma}=-\epsilon^{\sigma\tau}=1$. Variation of the
action with respect to $X^{M}$ gives us the following equations of motion

$\displaystyle
2\partial_{\alpha}(\eta^{\alpha\beta}\partial_{\beta}X^{N}g_{KN})$
$\displaystyle-$
$\displaystyle\eta^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\partial_{K}g_{MN}-2\partial_{\alpha}(\epsilon^{\alpha\beta}\partial_{\beta}X^{N}b_{KN})$
(2.8) $\displaystyle+$
$\displaystyle\epsilon^{\alpha\beta}\partial_{\alpha}X^{M}\partial_{\beta}X^{N}\partial_{K}b_{MN}=0\
,$

and variation with respect to the metric gives the two Virasoro constraints,

$\displaystyle
g_{MN}(\partial_{\tau}X^{M}\partial_{\tau}X^{N}+\partial_{\sigma}X^{M}\partial_{\sigma}X^{N})$
$\displaystyle=$ $\displaystyle 0\ ,$ (2.9) $\displaystyle
g_{MN}(\partial_{\tau}X^{M}\partial_{\sigma}X^{N})$ $\displaystyle=$
$\displaystyle 0\ .$ (2.10)

We use the conformal gauge (i.e.
$\sqrt{-\gamma}\gamma^{\alpha\beta}=\eta^{\alpha\beta}$) with
$\eta^{\tau\tau}=-1$, $\eta^{\sigma\sigma}=1$ and
$\eta^{\tau\sigma}=\eta^{\sigma\tau}=0$) to solve the equations of motion. For
studying a generic class of rotating strings we use the ansatz,

$\displaystyle t=\kappa\tau\ ,~{}~{}~{}x=\mu\tau\
,~{}~{}~{}\rho_{1}=m_{1}\tau\ ,~{}~{}~{}\rho_{2}=m_{2}\tau\
,~{}~{}~{}\theta_{1}=\theta_{1}(y)\ ,~{}~{}~{}\theta_{2}=\theta_{2}(y)\ ,$
$\displaystyle\phi_{1}=\nu_{1}(\tau+f_{1}(y))\
,~{}~{}~{}\phi_{2}=\nu_{2}(\tau+f_{2}(y))\
,~{}~{}~{}\psi_{1}=\omega_{1}(\tau+g_{1}(y))\
,~{}~{}~{}\psi_{2}=\omega_{2}(\tau+g_{2}(y))\ ,$

where the generalised variable $y=\sigma-v\tau$, which takes worldsheet
coordinates from $(\sigma,\tau)$ to $(y,\tau)$. Solving $\phi_{1}$ and
$\psi_{1}$ equations using the above set of embeddings, we get,

$\displaystyle\frac{\partial f_{1}}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})}\Big{[}\frac{c_{1}}{\nu_{1}\sin^{2}\theta_{1}}-\frac{2\omega_{1}}{\nu_{1}}-v\Big{]}\
,$ $\displaystyle\frac{\partial g_{1}}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})}\Big{[}\frac{c_{2}}{\omega_{1}\cos^{2}\theta_{1}}+\frac{2\nu_{1}\sin^{2}\theta_{1}}{\omega_{1}\cos^{2}\theta_{1}}-v\Big{]}\
.$ (2.12)

Again solving $\phi_{2}$ and $\psi_{2}$ equations we get,

$\displaystyle\frac{\partial f_{2}}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})}\Big{[}\frac{c_{3}}{\nu_{2}\sin^{2}\theta_{2}}-\frac{2\omega_{2}}{\nu_{2}}-v\Big{]}\
,$ $\displaystyle\frac{\partial g_{2}}{\partial y}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})}\Big{[}\frac{c_{4}}{\omega_{2}\cos^{2}\theta_{2}}+\frac{2\nu_{2}\sin^{2}\theta_{2}}{\omega_{2}\cos^{2}\theta_{2}}-v\Big{]}\
,$ (2.13)

where $c_{1,2,3,4}$ are constants here. Finally, solving for $\theta_{1}$ and
$\theta_{2}$ equations we get,

$\displaystyle\frac{\partial^{2}\theta_{1}}{\partial y^{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})^{2}}\Big{[}\frac{c_{1}^{2}}{\sin^{4}\theta_{1}}-\frac{(c_{2}+2\nu_{1})^{2}}{\cos^{4}\theta_{1}}+3(\nu_{1}^{2}-\omega_{1}^{2})\Big{]}\sin\theta_{1}\cos\theta_{1}\
,$ $\displaystyle\frac{\partial^{2}\theta_{2}}{\partial y^{2}}$
$\displaystyle=$
$\displaystyle\frac{1}{(1-v^{2})^{2}}\Big{[}\frac{c_{3}^{2}}{\sin^{4}\theta_{2}}-\frac{(c_{4}+2\nu_{2})^{2}}{\cos^{4}\theta_{2}}+3(\nu_{2}^{2}-\omega_{2}^{2})\Big{]}\sin\theta_{2}\cos\theta_{2}\
.$ (2.14)

Integrating them we get,

$\displaystyle\Big{(}\frac{\partial\theta_{1}}{\partial y}\Big{)}^{2}$
$\displaystyle=$ $\displaystyle
c_{5}-\frac{1}{(1-v^{2})^{2}}\Big{[}\frac{c_{1}^{2}}{\sin^{2}\theta_{1}}+\frac{(c_{2}+2\nu_{1})^{2}}{\cos^{2}\theta_{1}}-3(\nu_{1}^{2}-\omega_{1}^{2})\sin^{2}\theta_{1}\Big{]}\
,$ (2.15) $\displaystyle\Big{(}\frac{\partial\theta_{2}}{\partial
y}\Big{)}^{2}$ $\displaystyle=$ $\displaystyle
c_{6}-\frac{1}{(1-v^{2})^{2}}\Big{[}\frac{c_{3}^{2}}{\sin^{2}\theta_{2}}+\frac{(c_{4}+2\nu_{2})^{2}}{\cos^{2}\theta_{2}}-3(\nu_{2}^{2}-\omega_{2}^{2})\sin^{2}\theta_{2}\Big{]}\
,$ (2.16)

where $c_{5}$ and $c_{6}$ are again the integration constants.

Comparing the two Virasoro constraints, we get the following relation between
various constants,

$c_{1}\nu_{1}+c_{2}\omega_{1}+c_{3}\nu_{2}+c_{4}\omega_{2}=\alpha^{2}v\ ,$
(2.17)

where $\alpha=\sqrt{\kappa^{2}-\mu^{2}-m_{1}^{2}-m_{2}^{2}}$. Also, to make
sure the Virasoro constraints are explicitly consistent with the equations of
motion, we require:

$c_{5}+c_{6}=\frac{1}{(1-v^{2})^{2}}[4\nu_{1}^{2}+4\nu_{2}^{2}-\omega_{1}^{2}-\omega_{2}^{2}+4(c_{1}\omega_{1}+c_{2}\nu_{1}+c_{3}\omega_{2}+c_{4}\nu_{2})+\alpha^{2}(1+v^{2})]\
.$ (2.18)

###  2.3 Solving for the string profile

Using the boundary condition in the limit, $\frac{\partial\theta_{1}}{\partial
y}\rightarrow 0$ as $\theta_{1}\rightarrow\frac{\pi}{2}$, we can get from
(2.15) the relation between the constants,

$\displaystyle c_{2}+2\nu_{1}=0\
,~{}~{}~{}c_{5}=\frac{1}{(1-v^{2})^{2}}[c_{1}^{2}+3(\omega_{1}^{2}-\nu_{1}^{2})]$
(2.19)

Putting in the conditions, we get the string profile on the sphere
$\theta_{1}$ from (2.15),

$\frac{\partial\theta_{1}}{\partial
y}=\pm\frac{\sqrt{3(\omega_{1}^{2}-\nu_{1}^{2})}}{(1-v^{2})}\cot\theta_{1}\sqrt{\sin^{2}\theta_{1}-\alpha_{1}^{2}}\
,$ (2.20)

where the constant

$\alpha_{1}=\frac{c_{1}}{\sqrt{3(\omega_{1}^{2}-\nu_{1}^{2})}}$

Similarly using the other boundary condition in the limit,
$\frac{\partial\theta_{2}}{\partial y}\rightarrow 0$ as
$\theta_{2}\rightarrow\frac{\pi}{2}$ in (2.16) we get,

$\displaystyle c_{4}+2\nu_{2}=0\
,~{}~{}~{}c_{6}=\frac{1}{(1-v^{2})^{2}}[c_{3}^{2}+3(\omega_{2}^{2}-\nu_{2}^{2})]$
(2.21)

Putting in the conditions, we get from (2.16),

$\frac{\partial\theta_{2}}{\partial
y}=\pm\frac{\sqrt{3(\omega_{2}^{2}-\nu_{2}^{2})}}{(1-v^{2})}\cot\theta_{2}\sqrt{\sin^{2}\theta_{2}-\alpha_{2}^{2}}\
,$ (2.22)

where

$\alpha_{2}=\frac{c_{3}}{\sqrt{3(\omega_{2}^{2}-\nu_{2}^{2})}}.$

Now solving (2.20) and (2.22) we get,

$\displaystyle y$ $\displaystyle=$
$\displaystyle\mp\frac{(1-v^{2})}{\sqrt{3(\omega_{1}^{2}-\nu_{1}^{2})}\sqrt{1-\alpha_{1}^{2}}}\cosh^{-1}\Big{(}\frac{\sqrt{1-\alpha_{1}^{2}}}{\cos\theta_{1}}\Big{)}\
,$ (2.23) $\displaystyle y$ $\displaystyle=$
$\displaystyle\mp\frac{(1-v^{2})}{\sqrt{3(\omega_{2}^{2}-\nu_{2}^{2})}\sqrt{1-\alpha_{2}^{2}}}\cosh^{-1}\Big{(}\frac{\sqrt{1-\alpha_{2}^{2}}}{\cos\theta_{2}}\Big{)}\
.$ (2.24)

We now move on to discuss the various classes of strings that occur here for
different values of the parameters introduced. Using conditions (2.19) and
(2.21) in (2.17) and (2.18) we get,

$\displaystyle(c_{1}-2\omega_{1})\nu_{1}+(c_{3}-2\omega_{2})\nu_{2}$
$\displaystyle=$ $\displaystyle\alpha^{2}v\ ,$ (2.25)
$\displaystyle(c_{1}-2\omega_{1})^{2}+(c_{3}-2\omega_{2})^{2}+\nu_{1}^{2}+\nu_{2}^{2}$
$\displaystyle=$ $\displaystyle\alpha^{2}(1+v^{2})\ .$ (2.26)

Solving (2.25) and (2.26) we get,

$c_{1}-2\omega_{1}=\frac{\alpha^{2}\nu_{1}v\pm\nu_{2}\sqrt{(\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2})(\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2})}}{\nu_{1}^{2}+\nu_{2}^{2}}\
.$ (2.27)

From (2.27) we can see $c_{1}-2\omega_{1}$ have equal roots for
$\alpha^{2}=\nu_{1}^{2}+\nu_{2}^{2}$ and
$\nu_{1}^{2}+\nu_{2}^{2}=\alpha^{2}v^{2}$. As we will show in the following,
these two roots are responsible for two classes of dynamics.

We can easily inspect out the following cases for the parameter space spanned
by the constants:

i) $\nu_{1}^{2}+\nu_{2}^{2}=\alpha^{2}$: $c_{1}-2\omega_{1}=\nu_{1}v$ and
$c_{3}-2\omega_{2}=\nu_{2}v$.

ii) $\nu_{1}^{2}+\nu_{2}^{2}=\alpha^{2}v^{2}$:
$c_{1}-2\omega_{1}=\frac{\nu_{1}}{v}$ and
$c_{3}-2\omega_{2}=\frac{\nu_{2}}{v}$.

Again for $\alpha^{2}>\nu_{1}^{2}+\nu_{2}^{2}>\alpha^{2}v^{2}$ we have real
roots for $c_{1}-2\omega_{1}$.

iii) If we take

$c_{1}-2\omega_{1}=\frac{\alpha^{2}\nu_{1}v+\nu_{2}\sqrt{(\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2})(\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2})}}{\nu_{1}^{2}+\nu_{2}^{2}}$
(2.28)

then it leads us to

$c_{3}-2\omega_{2}=\frac{\alpha^{2}\nu_{2}v-\nu_{1}\sqrt{(\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2})(\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2})}}{\nu_{1}^{2}+\nu_{2}^{2}}$
(2.29)

iv) If we take

$c_{1}-2\omega_{1}=\frac{\alpha^{2}\nu_{1}v-\nu_{2}\sqrt{(\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2})(\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2})}}{\nu_{1}^{2}+\nu_{2}^{2}}$
(2.30)

then we have

$c_{3}-2\omega_{2}=\frac{\alpha^{2}\nu_{2}v+\nu_{1}\sqrt{(\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2})(\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2})}}{\nu_{1}^{2}+\nu_{2}^{2}}$
(2.31)

##  3 Conserved Charges and Regularized Dispersion Relations

To determine the conserved charges associated to the string motion, we start
from the full form of the sigma model action in background (2.6),

$\displaystyle S$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\lambda}}{4\pi}\int d\tau
d\sigma\Big{[}-[(\partial_{\sigma}t)^{2}-(\partial_{\tau}t)^{2}]+[(\partial_{\sigma}x)^{2}-(\partial_{\tau}x)^{2}]+[(\partial_{\sigma}\rho_{1})^{2}-(\partial_{\tau}\rho_{1})^{2}]$
(3.32)
$\displaystyle+[(\partial_{\sigma}\theta_{1})^{2}-(\partial_{\tau}\theta_{1})^{2}]+\sin^{2}\theta_{1}[(\partial_{\sigma}\phi_{1})^{2}-(\partial_{\tau}\phi_{1})^{2}]+\cos^{2}\theta_{1}[(\partial_{\sigma}\psi_{1})^{2}-(\partial_{\tau}\psi_{1})^{2}]$
$\displaystyle+[(\partial_{\sigma}\rho_{2})^{2}-(\partial_{\tau}\rho_{2})^{2}]+[(\partial_{\sigma}\theta_{2})^{2}-(\partial_{\tau}\theta_{2})^{2}]+\sin^{2}\theta_{2}[(\partial_{\sigma}\phi_{2})^{2}-(\partial_{\tau}\phi_{2})^{2}]$
$\displaystyle+\cos^{2}\theta_{2}[(\partial_{\sigma}\psi_{2})^{2}-(\partial_{\tau}\psi_{2})^{2}]+4\sin^{2}\theta_{1}(\partial_{\sigma}\phi_{1}\partial_{\tau}\psi_{1}-\partial_{\tau}\phi_{1}\partial_{\sigma}\psi_{1})$
$\displaystyle+4\sin^{2}\theta_{2}(\partial_{\sigma}\phi_{2}\partial_{\tau}\psi_{2}-\partial_{\tau}\phi_{2}\partial_{\sigma}\psi_{2})\Big{]}\
.$

From this action we can easily determine the conserved energy and spins using
the Noether procedure, and they can be written as following,

$\displaystyle E$ $\displaystyle=$
$\displaystyle-\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}t)}~{}dy=\frac{\sqrt{\lambda}\kappa}{2\pi}\int_{-L/2}^{L/2}~{}dy=\frac{\sqrt{\lambda}\kappa
L}{2\pi}\ ,$ $\displaystyle D$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}x)}~{}dy=\frac{\sqrt{\lambda}\mu}{2\pi}\int_{-L/2}^{L/2}~{}dy=\frac{\sqrt{\lambda}\mu
L}{2\pi}\ ,$ $\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\rho_{1})}~{}dy=\frac{\sqrt{\lambda}m_{1}}{2\pi}\int_{-L/2}^{L/2}~{}dy=\frac{\sqrt{\lambda}m_{1}L}{2\pi}\
,$ $\displaystyle P_{2}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\rho_{2})}~{}dy=\frac{\sqrt{\lambda}m_{2}}{2\pi}\int_{-L/2}^{L/2}~{}dy=\frac{\sqrt{\lambda}m_{2}L}{2\pi}\
,$ $\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\phi_{1})}~{}dy=-\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}[3\nu_{1}\sin^{2}\theta_{1}+c_{1}v]~{}dy\
,$ $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\phi_{2})}~{}dy=-\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}[3\nu_{2}\sin^{2}\theta_{2}+c_{3}v]~{}dy\
,$ $\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\psi_{1})}~{}dy=\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}[\omega_{1}-2c_{1}+2\nu_{1}v+3\omega_{1}\sin^{2}\theta_{1}]dy\
,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle\int\frac{\partial\mathcal{L}}{\partial(\partial_{\tau}\psi_{2})}~{}dy=\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}[\omega_{2}-2c_{3}+2\nu_{2}v+3\omega_{2}\sin^{2}\theta_{2}]dy\
.$ (3.33)

Here $E,D,P_{1}$ and $P_{2}$ diverges as $L\rightarrow\infty$. To evaluate
$J_{1},J_{2},K_{1}$ and $K_{2}$ we need to specify the values of constants
$c_{1,2,3,4}$. Also the angle deficit on the two different spheres are given
by,

$\displaystyle\Delta\phi_{1}$ $\displaystyle=$
$\displaystyle\nu_{1}\int\frac{\partial f_{1}}{\partial
y}~{}dy=\frac{1}{1-v^{2}}\int_{-L/2}^{L/2}\Big{[}\frac{c_{1}}{\sin^{2}\theta_{1}}-2\omega_{1}-\nu_{1}v\Big{]}~{}dy\
,$ $\displaystyle\Delta\phi_{2}$ $\displaystyle=$
$\displaystyle\nu_{2}\int\frac{\partial f_{2}}{\partial
y}~{}dy=\frac{1}{1-v^{2}}\int_{-L/2}^{L/2}\Big{[}\frac{c_{3}}{\sin^{2}\theta_{2}}-2\omega_{2}-\nu_{2}v\Big{]}~{}dy\
.$ (3.34)

In what follows, we will discuss about the fate of these charges for different
regime of the parameter space as we discussed before and find the dispersion
relation among them.

###  3.1 Case I: $c_{1}=2\omega_{1}+\nu_{1}v$ and $c_{3}=2\omega_{2}+\nu_{2}v$

Using the above values of constants we find,

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\nu_{1}\int_{-L/2}^{L/2}\cos^{2}\theta_{1}~{}dy-(3\nu_{1}+2\omega_{1}v+\nu_{1}v^{2})\int_{-L/2}^{L/2}~{}dy\Big{]}\
,$ $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\nu_{2}\int_{-L/2}^{L/2}\cos^{2}\theta_{2}~{}dy-(3\nu_{2}+2\omega_{2}v+\nu_{2}v^{2})\int_{-L/2}^{L/2}~{}dy\Big{]}\
,$ $\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle-\frac{3\omega_{1}\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}\cos^{2}\theta_{1}~{}dy\
,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle-\frac{3\omega_{2}\sqrt{\lambda}}{2\pi(1-v^{2})}\int_{-L/2}^{L/2}\cos^{2}\theta_{2}~{}dy\
.$ (3.35)

As $L\rightarrow\infty$, the charges $J_{1}$ and $J_{2}$ diverges like
$E,D,P_{1}$ and $P_{2}$. Combining these diverging quantities as,

$\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=\frac{\sqrt{\lambda}\alpha L}{2\pi}\
,$ (3.36)

and combining them with $J_{1}$ and $J_{2}$, we can find finite quantities.
Rescaling, $\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}$ as,

$\displaystyle\tilde{E_{1}}$ $\displaystyle=$
$\displaystyle\frac{3\nu_{1}+2\omega_{1}v+\nu_{1}v^{2}}{\alpha(1-v^{2})}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}\
,$ $\displaystyle\tilde{E_{2}}$ $\displaystyle=$
$\displaystyle\frac{3\nu_{2}+2\omega_{2}v+\nu_{2}v^{2}}{\alpha(1-v^{2})}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}\
.$ (3.37)

we find,

$\displaystyle\tilde{E_{1}}+J_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3\lambda}\nu_{1}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\sqrt{1-\alpha_{1}^{2}}\
,$ $\displaystyle\tilde{E_{2}}+J_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3\lambda}\nu_{2}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\sqrt{1-\alpha_{2}^{2}}\
.$ (3.38)

On the other hand the charges $K_{1}$ and $K_{2}$ are finite,

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3\lambda}\omega_{1}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\sqrt{1-\alpha_{1}^{2}}\
,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3\lambda}\omega_{2}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\sqrt{1-\alpha_{2}^{2}}\
.$ (3.39)

In this case the angle deficits are finite,

$\displaystyle\Delta\phi_{1}=2\cos^{-1}(\alpha_{1})\
,~{}~{}~{}\Delta\phi_{2}=2\cos^{-1}(\alpha_{2})\ .$ (3.40)

which implies

$\displaystyle\alpha_{1}=\cos\frac{\Delta\phi_{1}}{2}\
,~{}~{}~{}\alpha_{2}=\cos\frac{\Delta\phi_{2}}{2}\ ,$ (3.41)

Now these conserved charges hold the relationship

$\displaystyle\tilde{E_{1}}+J_{1}$ $\displaystyle=$
$\displaystyle\sqrt{K_{1}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{\Delta\phi_{1}}{2}\Big{)}}\
,$ $\displaystyle\tilde{E_{2}}+J_{2}$ $\displaystyle=$
$\displaystyle\sqrt{K_{2}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{\Delta\phi_{2}}{2}\Big{)}}\
.$ (3.42)

These results looks like the Giant Magnon solution obtained by studying
F-strings in NS5-brane [43]. But, here we see two independent magnon-like
solution occurs in two orthogonal stacks of NS5-branes.

###  3.2 Case II: $c_{1}=2\omega_{1}+\frac{\nu_{1}}{v}$ and
$c_{3}=2\omega_{2}+\frac{\nu_{2}}{v}$

Again, in this case $E,D,P_{1}$ and $P_{2}$ diverges as $L\rightarrow\infty$.
We will use the same combination of these diverging quantities as,

$\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=\frac{\sqrt{\lambda}\alpha L}{2\pi}\
.$ (3.43)

For $c_{1}=2\omega_{1}+\frac{\nu_{1}}{v}$ and
$c_{3}=2\omega_{2}+\frac{\nu_{2}}{v}$, the other conserved charges becomes,

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\nu_{1}\int_{-L/2}^{L/2}\cos^{2}\theta_{1}~{}dy-2(2\nu_{1}+\omega_{1}v)\int_{-L/2}^{L/2}~{}dy\Big{]}\
,$ $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\nu_{2}\int_{-L/2}^{L/2}\cos^{2}\theta_{2}~{}dy-2(2\nu_{2}+\omega_{2}v)\int_{-L/2}^{L/2}~{}dy\Big{]}\
,$ $\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\omega_{1}\int_{-L/2}^{L/2}\cos^{2}\theta_{1}~{}dy+\frac{2\nu_{1}}{v}(1-v^{2})\int_{-L/2}^{L/2}~{}dy\Big{]}\
,$ $\displaystyle K_{2}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{\lambda}}{2\pi(1-v^{2})}\Big{[}3\omega_{2}\int_{-L/2}^{L/2}\cos^{2}\theta_{2}~{}dy+\frac{2\nu_{2}}{v}(1-v^{2})\int_{-L/2}^{L/2}~{}dy\Big{]}\
.$ (3.44)

As $L\rightarrow\infty$, the charges $J_{1},J_{2},K_{1}$ and $K_{2}$ all
diverges. Subtracting the diverging piece we can find the finite piece as,

$\displaystyle(J_{1})_{reg}$ $\displaystyle=$ $\displaystyle
J_{1}+\frac{2(2\nu_{1}+\omega_{1}v)}{\alpha(1-v^{2})}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=\frac{\sqrt{3\lambda}\nu_{1}\sqrt{1-\alpha_{1}^{2}}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\
,$ $\displaystyle(J_{2})_{reg}$ $\displaystyle=$ $\displaystyle
J_{2}+\frac{2(2\nu_{2}+\omega_{2}v)}{\alpha(1-v^{2})}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=\frac{\sqrt{3\lambda}\nu_{2}\sqrt{1-\alpha_{2}^{2}}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\
,$ $\displaystyle(K_{1})_{reg}$ $\displaystyle=$ $\displaystyle
K_{1}+\frac{2\nu_{1}}{\alpha
v}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=-\frac{\sqrt{3\lambda}\omega_{1}\sqrt{1-\alpha_{1}^{2}}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\
,$ $\displaystyle(K_{2})_{reg}$ $\displaystyle=$ $\displaystyle
K_{2}+\frac{2\nu_{2}}{\alpha
v}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=-\frac{\sqrt{3\lambda}\omega_{2}\sqrt{1-\alpha_{2}^{2}}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\
.$ (3.45)

In this case the angle deficits are also diverging, again subtracting the
diverging piece, we obtain,

$\displaystyle(\Delta\phi_{1})_{reg}$ $\displaystyle=$
$\displaystyle\Delta\phi_{1}-\frac{2\pi\nu_{1}}{\sqrt{\lambda}\alpha
v}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=2\cos^{-1}(\alpha_{1})\ ,$
$\displaystyle(\Delta\phi_{2})_{reg}$ $\displaystyle=$
$\displaystyle\Delta\phi_{2}-\frac{2\pi\nu_{2}}{\sqrt{\lambda}\alpha
v}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}=2\cos^{-1}(\alpha_{2})\ ,$ (3.46)

which implies

$\displaystyle\alpha_{1}=\cos\frac{(\Delta\phi_{1})_{reg}}{2}\
,~{}~{}~{}\alpha_{2}=\cos\frac{(\Delta\phi_{2})_{reg}}{2}\ .$ (3.47)

Now these conserved charges hold the relationship

$\displaystyle(J_{1})_{reg}$ $\displaystyle=$
$\displaystyle\sqrt{(K_{1})_{reg}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{(\Delta\phi_{1})_{reg}}{2}\Big{)}}\
,$ $\displaystyle(J_{2})_{reg}$ $\displaystyle=$
$\displaystyle\sqrt{(K_{2})_{reg}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{(\Delta\phi_{2})_{reg}}{2}\Big{)}}\
.$ (3.48)

This result looks like the single spike solution obtained by studying
F-strings in NS5-brane [43]. Again here we obtain two independent single-spike
relations on two orthogonal spheres. Two independent single-spike relations
were also obtained while studying rigidily rotating strings in curved
NS5-branes [63] which arises from the near horizon-geometry of
NS1-NS1’-NS5-NS5’ brane [62].

###  3.3 Case III:
$c_{1}=2\omega_{1}+\frac{\alpha^{2}\nu_{1}v\pm\nu_{2}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}$
and
$c_{3}=2\omega_{2}+\frac{\alpha^{2}\nu_{2}v\mp\nu_{1}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}$

Let $\beta_{1}^{2}=\alpha^{2}-\nu_{1}^{2}-\nu_{2}^{2}$ and
$\beta_{2}^{2}=\nu_{1}^{2}+\nu_{2}^{2}-\alpha^{2}v^{2}$. Unlike previous two
cases where we use $\beta_{1}^{2}=0$ and $\beta_{2}^{2}=0$, we now consider
$\beta_{1}^{2}$ and $\beta_{2}^{2}$ as some positive quantities, i.e.,
$\alpha^{2}>\nu_{1}^{2}+\nu_{2}^{2}$ and
$\nu_{1}^{2}+\nu_{2}^{2}>\alpha^{2}v^{2}$, and combining them we get,

$\alpha^{2}>\nu_{1}^{2}+\nu_{2}^{2}>\alpha^{2}v^{2}\ ,$ (3.49)

which implies $v^{2}<1$. One can easily check in this case all the conserved
charges diverges. However, one can find the finite quantities by subtracting
the diverging piece as,

$\displaystyle(J_{1})_{reg}$ $\displaystyle=$ $\displaystyle
J_{1}+\frac{1}{\alpha(1-v^{2})}\Big{[}3\nu_{1}+2\omega_{1}v+\frac{\alpha^{2}\nu_{1}v^{2}\pm\nu_{2}v\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}$
$\displaystyle=\frac{\sqrt{3\lambda}\nu_{1}\sqrt{1-\alpha_{1}^{2}}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\
,$ $\displaystyle(J_{2})_{reg}$ $\displaystyle=$ $\displaystyle
J_{2}+\frac{1}{\alpha(1-v^{2})}\Big{[}3\nu_{2}+2\omega_{2}v+\frac{\alpha^{2}\nu_{2}v^{2}\mp\nu_{1}v\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}$
$\displaystyle=\frac{\sqrt{3\lambda}\nu_{2}\sqrt{1-\alpha_{2}^{2}}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\
,$ $\displaystyle(K_{1})_{reg}$ $\displaystyle=$ $\displaystyle
K_{1}-\frac{2}{\alpha(1-v^{2})}\Big{[}\nu_{1}v-\frac{\alpha^{2}\nu_{1}v\pm\nu_{2}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}$
$\displaystyle=-\frac{\sqrt{3\lambda}\omega_{1}\sqrt{1-\alpha_{1}^{2}}}{\pi\sqrt{\omega_{1}^{2}-\nu_{1}^{2}}}\
,$ $\displaystyle(K_{2})_{reg}$ $\displaystyle=$ $\displaystyle
K_{2}-\frac{2}{\alpha(1-v^{2})}\Big{[}\nu_{2}v-\frac{\alpha^{2}\nu_{2}v\mp\nu_{1}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}$
(3.50)
$\displaystyle=-\frac{\sqrt{3\lambda}\omega_{2}\sqrt{1-\alpha_{2}^{2}}}{\pi\sqrt{\omega_{2}^{2}-\nu_{2}^{2}}}\
.$

In this case the angle deficits are also diverging, again subtracting the
diverging piece, we obtain,

$\displaystyle(\Delta\phi_{1})_{reg}$ $\displaystyle=$
$\displaystyle\Delta\phi_{1}+\frac{2\pi}{\sqrt{\lambda}\alpha(1-v^{2})}\Big{[}\nu_{1}v-\frac{\alpha^{2}\nu_{1}v\pm\nu_{2}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}\
,$ $\displaystyle(\Delta\phi_{2})_{reg}$ $\displaystyle=$
$\displaystyle\Delta\phi_{2}+\frac{2\pi}{\sqrt{\lambda}\alpha(1-v^{2})}\Big{[}\nu_{2}v-\frac{\alpha^{2}\nu_{2}v\mp\nu_{1}\beta_{1}\beta_{2}}{\nu_{1}^{2}+\nu_{2}^{2}}\Big{]}\sqrt{E^{2}-D^{2}-P_{1}^{2}-P_{2}^{2}}\
.$

Now these conserved charges hold the relationship

$\displaystyle(J_{1})_{reg}$ $\displaystyle=$
$\displaystyle\sqrt{(K_{1})_{reg}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{(\Delta\phi_{1})_{reg}}{2}\Big{)}}\
,$ $\displaystyle(J_{2})_{reg}$ $\displaystyle=$
$\displaystyle\sqrt{(K_{2})_{reg}^{2}-\frac{3\lambda}{\pi^{2}}\sin^{2}\Big{(}\frac{(\Delta\phi_{2})_{reg}}{2}\Big{)}}\
.$ (3.52)

Like previous section here also we obtain two independent single spike
relations in the two orthogonal spheres of I-brane and each single spike is
exactly similar as obtained by studying F-strings in NS5-brane.

##  4 Summary and conclusion

In this short paper, we revisit the rigidly rotating probe strings in I-brane
with Polyakov action of strings. Our aim was to study the rotating string
solution over the whole space of I-brane without any restriction. Unlike
Nambu-Goto action, Polyakov action of strings allows us decouple the two
orthogonal spheres that arises from the near horizon limit of orthogonal
stacks of NS5-branes. This might be because Polyakov action of string is more
symmetric than Nambu-Goto action of string and this extra symmetry allows us
to decouple the spheres. This allows us to study the rotating strings in both
the spheres simultaneously, however the motion is visibly complicated in
nature. In this case we consider three different cases in the parameter space
of integration constants and obtain the giant magnon like dispersion relation
in one case and single spike for all other cases. This observation leads us to
affirm that single spike or spiky string are more generalized solution of
rotating strings while giant magnon is a special case of spiky string
solutions.

We also obtain two independent giant magnon and single spike relations in two
orthogonal spheres, we have seen this type of behaviour while studying
rotating string in curved NS5-branes in [62, 63]. This type of behaviour was
not seen while studying rotating string using Nambu-Goto action under the
restriction $\theta_{1}=\theta_{2}=\theta$. This might be because under these
restriction we were studying rigidly rotating string on an effective sphere of
the form
$d\Omega^{2}=2d\theta^{2}+\sin^{2}\theta(d\phi_{1}^{2}+d\phi_{2}^{2})+\cos^{2}\theta(d\psi_{1}^{2}+d\psi_{2}^{2})$.
With the identification $d\phi_{1}^{2}+d\phi_{2}^{2}=2d\phi^{2}$ and
$d\psi_{1}^{2}+d\psi_{2}^{2}=2d\psi^{2}$ and rescaling $d\Omega^{2}\rightarrow
d\Omega^{2}/2$ we effectively studied rotating strings on
$\mathbb{R}_{t}\times S^{3}$ and hence we obtain one giant magnon or single
spike relation. Whereas as Polyakov action of string didn’t require such
restriction we could study the rotating strings on $\mathbb{R}_{t}\times
S^{3}\times S^{3}$ and hence we obtain two independent giant magnon and single
spike relations for two orthogonal spheres.

We have mentioned similar kind of difficulty was faced while studying rotating
D1-strings on two orthogonal stacks of D5-branes [38], where we determined the
string equations of motion using DBI-action. This scenario is explicitly
S-dual to the present case, i.e. F1 strings in stacks of NS5-branes. Like
Nambu-Goto action DBI action is of square root form and this action does not
bear enough symmetries to decouple the spheres and we had to study the
rotating D1-string under the same restriction of
$\theta_{1}=\theta_{2}=\theta$. This restriction can be avoided if one use an
action for D1-string which is of the form of Polyakov action. Another
interesting arena that we could explore in this regard could be the
oscillating circular string, also known as “Pulsating Strings” [23] in the
literature. The energy-spin dispersion relation for such states may give us
further insight into the dual field theory operators. We hope to focus on
these investigations in the near future.

## Acknowledgements

It is pleasure to thank Aritra Banerjee for some useful discussions which
helps to gain deep insights and improves the manuscript.

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# Learning with Noise: Enhance Distantly Supervised Relation Extraction with
Dynamic Transition Matrix

Bingfeng Luo1, Yansong Feng∗1, Zheng Wang2, Zhanxing Zhu3,
Songfang Huang4, Rui Yan1 and Dongyan Zhao1
1ICST, Peking University, China
2School of Computing and Communications, Lancaster University, UK
3Peking University, China
4IBM China Research Lab, China
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

###### Abstract

Distant supervision significantly reduces human efforts in building training
data for many classification tasks. While promising, this technique often
introduces noise to the generated training data, which can severely affect the
model performance. In this paper, we take a deep look at the application of
distant supervision in relation extraction. We show that the dynamic
transition matrix can effectively characterize the noise in the training data
built by distant supervision. The transition matrix can be effectively trained
using a novel curriculum learning based method without any direct supervision
about the noise. We thoroughly evaluate our approach under a wide range of
extraction scenarios. Experimental results show that our approach consistently
improves the extraction results and outperforms the state-of-the-art in
various evaluation scenarios.

##  1 Introduction

Distant supervision (DS) is rapidly emerging as a viable means for supporting
various classification tasks – from relation extraction Mintz et al. (2009)
and sentiment classification Go et al. (2009) to cross-lingual semantic
analysis Fang and Cohn (2016). By using knowledge learned from seed examples
to label data, DS automatically prepares large scale training data for these
tasks.

While promising, DS does not guarantee perfect results and often introduces
noise to the generated data. In the context of relation extraction, DS works
by considering sentences containing both the subject and object of a $<$_subj_
, rel, _obj_ $>$ triple as its supports. However, the generated data are not
always perfect. For instance, DS could match the knowledge base (KB) triple,
$<$_Donald Trump_ , born-in, _New York_ $>$ in _false positive_ contexts like
_Donald Trump worked in New York City_. Prior works Takamatsu et al. (2012);
Ritter et al. (2013) show that DS often mistakenly labels real positive
instances as negative (_false negative_) or versa vice (_false positive_), and
there could be confusions among positive labels as well. These noises can
severely affect training and lead to poorly-performing models.

Tackling the noisy data problem of DS is non-trivial, since there usually
lacks of explicit supervision to capture the noise. Previous works have tried
to remove sentences containing unreliable syntactic patterns Takamatsu et al.
(2012), design new models to capture certain types of noise or aggregate
multiple predictions under the at-least-one assumption that at least one of
the aligned sentences supports the triple in KB Riedel et al. (2010); Surdeanu
et al. (2012); Ritter et al. (2013); Min et al. (2013). These approaches
represent a substantial leap forward towards making DS more practical.
however, are either tightly couple to certain types of noise, or have to rely
on manual rules to filter noise, thus unable to scale. Recent breakthrough in
neural networks provides a new way to reduce the influence of incorrectly
labeled data by aggregating multiple training instances attentively for
relation classification, without explicitly characterizing the inherent noise
Lin et al. (2016); Zeng et al. (2015). Although promising, modeling noise
within neural network architectures is still in its early stage and much
remains to be done.

In this paper, we aim to enhance DS noise modeling by providing the capability
to explicitly characterize the noise in the DS-style training data within
neural networks architectures. We show that while noise is inevitable, it is
possible to characterize the noise pattern in a unified framework along with
its original classification objective. Our key insight is that the DS-style
training data typically contain useful clues about the noise pattern. For
example, we can infer that since some people work in their birthplaces, DS
could wrongly label a training sentence describing a working place as a born-
in relation. Our novel approach to noisy modeling is to use a dynamically-
generated transition matrix for each training instance to (1) characterize the
possibility that the DS labeled relation is confused and (2) indicate its
noise pattern. To tackle the challenge of no direct guidance over the noise
pattern, we employ a curriculum learning based training method to gradually
model the noise pattern over time, and utilize trace regularization to control
the behavior of the transition matrix during training. Our approach is
flexible – while it does not make any assumptions about the data quality, the
algorithm can make effective use of the data-quality prior knowledge to guide
the learning procedure when such clues are available.

We apply our method to the relation extraction task and evaluate under various
scenarios on two benchmark datasets. Experimental results show that our
approach consistently improves both extraction settings, outperforming the
state-of-the-art models in different settings.

Our work offers an effective way for tackling the noisy data problem of DS,
making DS more practical at scale. Our main contributions are to (1) design a
_dynamic_ transition matrix structure to characterize the noise introduced by
DS, and (2) design a curriculum learning based framework to adaptively guide
the training procedure to learn with noise.

##  2 Problem Definition

The task of distantly supervised relation extraction is to extract knowledge
triples, $<$_subj_ , rel, _obj_ $>$, from free text with the training data
constructed by aligning existing KB triples with a large corpus. Specifically,
given a triple in KB, DS works by first retrieving all the sentences
containing both _subj_ and _obj_ of the triple, and then constructing the
training data by considering these sentences as support to the existence of
the triple. This task can be conducted in both the sentence and the bag
levels. The former takes a sentence $s$ containing both $subj$ and $obj$ as
input, and outputs the relation expressed by the sentence between $subj$ and
$obj$. The latter setting alleviates the noisy data problem by using the at-
least-one assumption that at least one of the retrieved sentences containing
both $subj$ and $obj$ supports the $<$_subj_ , rel, _obj_ $>$ triple. It takes
a bag of sentences $S$ as input where each sentence $s\in S$ contains both
$subj$ and $obj$, and outputs the relation between $subj$ and $obj$ expressed
by this bag.

Figure 1: Overview of our approach

##  3 Our approach

In order to deal with the noisy training data obtained through DS, our
approach follows four steps as depicted in Figure 1. First, each input
sentence is fed to a sentence encoder to generate an embedding vector. Our
model then takes the sentence embeddings as input and produce a predicted
relation distribution, $\mathbf{p}$, for the input sentence (or the input
sentence bag). At the same time, our model dynamically produces a transition
matrix, $\mathbf{T}$, which is used to characterize the noise pattern of
sentence (or the bag). Finally, the predicted distribution is multiplied by
the transition matrix to produce the observed relation distribution,
$\mathbf{o}$, which is used to match the noisy relation labels assigned by DS
while the predicted relation distribution $\mathbf{p}$ serves as output of our
model during testing. One of the key challenges of our approach is on
determining the element values of the transition matrix, which will be
described in Section 4.

###  3.1 Sentence-level Modeling

#### Sentence Embedding and Prediction

In this work, we use a piecewise convolutional neural network Zeng et al.
(2015) for sentence encoding, but other sentence embedding models can also be
used. We feed the sentence embedding to a full connection layer, and use
_softmax_ to generate the predicted relation distribution, $\mathbf{p}$.

#### Noise Modeling

First, each sentence embedding $\mathbf{x}$, generated b sentence encoder, is
passed to a full connection layer as a non-linearity to obtain the sentence
embedding $\mathbf{x}_{n}$ used specifically for noise modeling. We then use
_softmax_ to calculate the transition matrix $\mathbf{T}$, for each sentence:

$T_{ij}=\frac{exp({\mathbf{w}_{ij}^{T}\mathbf{x}_{n}+b})}{\sum_{j=1}^{|\mathbb{C}|}{exp({\mathbf{w}_{ij}^{T}\mathbf{x}_{n}+b}})}$
(1)

where $T_{ij}$ is the conditional probability for the input sentence to be
labeled as relation $j$ by DS, given $i$ as the true relation, $b$ is a scalar
bias, $|\mathbb{C}|$ is the number of relations, $\mathbf{w}_{ij}$ is the
weight vector characterizing the confusion between $i$ and $j$.

Here, we dynamically produce a transition matrix, $\mathbf{T}$, specifically
for each sentence, but with the parameters ($\mathbf{w}_{ij}$) shared across
the dataset. By doing so, we are able to adaptively characterize the noise
pattern for each sentence, with a few parameters only. In contrast, one could
also produce a global transition matrix for all sentences, with much less
computation, where one need not to compute $\mathbf{T}$ on the fly (see
Section 6.1).

#### Observed Distribution

When we characterize the noise in a sentence with a transition matrix
$\mathbf{T}$, if its true relation is $i$, we can assume that $i$ might be
erroneously labeled as relation $j$ by DS with probability $T_{ij}$. We can
therefore capture the observed relation distribution, $\mathbf{o}$, by
multiplying $\mathbf{T}$ and the predicted relation distribution,
$\mathbf{p}$:

$\mathbf{o}=\mathbf{T}^{T}\bm{\cdot}\mathbf{p}$ (2)

where $\mathbf{o}$ is then normalized to ensure $\sum_{i}{o_{i}}=1$.

Rather than using the predicted distribution $\mathbf{p}$ to directly match
the relation labeled by DS Zeng et al. (2015); Lin et al. (2016), here we
utilize $\mathbf{o}$ to match the noisy labels during training and still use
$\mathbf{p}$ as output during testing, which actually captures the procedure
of how the noisy label is produced and thus protects $\mathbf{p}$ from the
noise.

###  3.2 Bag Level Modeling

#### Bag Embedding and Prediction

One of the key challenges for bag level model is how to aggregate the
embeddings of individual sentences into the bag level. In this work, we
experiment two methods, namely average and attention aggregation Lin et al.
(2016). The former calculates the bag embedding, $\mathbf{s}$, by averaging
the embeddings of each sentence, and then feed it to a _softmax_ classifier
for relation classification.

The attention aggregation calculates an attention value, $a_{ij}$, for each
sentence $i$ in the bag with respect to each relation $j$, and aggregates to
the bag level as $\mathbf{s}_{j}$, by the following equations111While Lin et
al. (2016) use bilinear function to calculate $a_{ij}$, we simply use dot
product since we find these two functions perform similarly in our
experiments.:

$\displaystyle\mathbf{s}_{j}=\sum_{i}^{n}{a_{ij}\mathbf{x}_{i}};\mbox{
}a_{ij}=\frac{exp(\mathbf{x}_{i}^{T}\mathbf{r}_{j})}{\sum_{i^{\prime}}^{n}{exp(\mathbf{x}_{i^{\prime}}^{T}\mathbf{r}_{j})}}$
(3)

where $\mathbf{x}_{i}$ is the embedding of sentence $i$, $n$ the number of
sentences in the bag, and $\mathbf{r}_{j}$ is the randomly initialized
embedding for relation $j$. In similar spirit to Lin et al. (2016), the
resulting bag embedding $\mathbf{s}_{j}$ is fed to a _softmax_ classifier to
predict the probability of relation $j$ for the given bag.

#### Noise Modeling

Since the transition matrix addresses the transition probability with respect
to each true relation, the attention mechanism appears to be a natural fit for
calculating the transition matrix in bag level. Similar to attention
aggregation above, we calculate the bag embedding with respect to each
relation using Equation 3, but with a separate set of relation embeddings
$\mathbf{r^{\prime}}_{j}$. We then calculate the transition matrix,
$\mathbf{T}$, by:

$T_{ij}=\frac{exp({\mathbf{s}_{i}^{T}\mathbf{r^{\prime}}_{j}+b_{i}})}{\sum_{j=1}^{|\mathbb{C}|}{exp(\mathbf{s}_{i}^{T}\mathbf{r^{\prime}}_{j}+b_{i}})}$
(4)

where $\mathbf{s}_{i}$ is the bag embedding regarding relation $i$, and
$\mathbf{r^{\prime}}_{j}$ is the embedding for relation $j$.

##  4 Curriculum Learning based Training

One of the key challenges of this work is on how to train and produce the
transition matrix to model the noise in the training data without any direct
guidance and human involvement. A straightforward solution is to directly
align the observed distribution, $\mathbf{o}$, with respect to the noisy
labels by minimizing the sum of the two terms:
$CrossEntropy(\mathbf{o})+Regularization$. However, doing so does not
guarantee that the prediction distribution, $\mathbf{p}$, will match the true
relation distribution. The problem is at the beginning of the training, we
have no prior knowledge about the noise pattern, thus, both $\mathbf{T}$ and
$\mathbf{p}$ are less reliable, making the training procedure be likely to
trap into some poor local optimum. Therefore, we require a technique to guide
our model to gradually adapt to the noisy training data, e.g., learning
something simple first, and then trying to deal with noises.

Fortunately, this is exactly what curriculum learning can do. The idea of
curriculum learning Bengio et al. (2009) is simple: starting with the easiest
aspect of a task, and leveling up the difficulty gradually, which fits well to
our problem. We thus employ a curriculum learning framework to guide our model
to gradually learn how to characterize the noise. Another advantage is to
avoid falling into poor local optimum.

With curriculum learning, our approach provides the flexibility to combine
prior knowledge of noise, e.g., splitting a dataset into reliable and less
reliable subsets, to improve the effectiveness of the transition matrix and
better model the noise.

###  4.1 Trace Regularization

Before proceeding to training details, we first discuss how we characterize
the noise level of the data by controlling the trace of its transition matrix.
Intuitively, if the noise is small, the transition matrix $\mathbf{T}$ will
tend to become an identity matrix, i.e., given a set of annotated training
sentences, the observed relations and their true relations are almost
identical. Since each row of $\mathbf{T}$ sums to 1, the similarity between
the transition matrix and the identity matrix can be represented by its trace,
$trace(\mathbf{T})$. The larger the $trace(\mathbf{T})$ is, the larger the
diagonal elements are, and the more similar the transition matrix $\mathbf{T}$
is to the identity matrix, indicating a lower level of noise. Therefore, we
can characterize the noise pattern by controlling the expected value of
$trace(\mathbf{T})$ in the form of regularization. For example, we will expect
a larger $trace(\mathbf{T})$ for reliable data, but a smaller
$trace(\mathbf{T})$ for less reliable data. Another advantage of employing
trace regularization is that it could help reduce the model complexity and
avoid overfitting.

###  4.2 Training

To tackle the challenge of no direct guidance over the noise patterns, we
implement a curriculum learning based training method to first train the model
without considerations for noise. In other words, we first focus on the loss
from the prediction distribution $\mathbf{p}$ , and then take the noise
modeling into account gradually along the training process, i.e., gradually
increasing the importance of the loss from the observed distribution
$\mathbf{o}$ while decreasing the importance of $\mathbf{p}$. In this way, the
prediction branch is roughly trained before the model managing to characterize
the noise, thus avoids being stuck into poor local optimum. We thus design to
minimize the following loss function:

$\displaystyle L$
$\displaystyle=\sum_{i=1}^{N}{-((1-\alpha)log(o_{iy_{i}})+\alpha
log(p_{iy_{i}}))}$ (5) $\displaystyle-\beta trace(\mathbf{T}^{i})$

where 0$<$$\alpha$$\leq$1 and $\beta$$>$0 are two weighting parameters,
$y_{i}$ is the relation assigned by DS for the $i$-th instance, $N$ the total
number of training instances, $o_{iy_{i}}$ is the probability that the
observed relation for the $i$-th instance is $y_{i}$, and $p_{iy_{i}}$ is the
probability to predict relation $y_{i}$ for the $i$-th instance.

Initially, we set $\alpha$=1, and train our model completely by minimizing the
loss from the prediction distribution $\mathbf{p}$. That is, we do not expect
to model the noise, but focus on the prediction branch at this time. As the
training progresses, the prediction branch gradually learns the basic
prediction ability. We then decrease $\alpha$ and $\beta$ by 0$<$$\rho$$<$1
($\alpha^{*}$=$\rho$$\alpha$ and $\beta^{*}$=$\rho$$\beta$) every $\tau$
epochs, i.e., learning more about the noise from the observed distribution
$\mathbf{o}$ and allowing a relatively smaller $trace(\mathbf{T})$ to
accommodate more noise. The motivation behind is to put more and more effort
on learning the noise pattern as the training proceeds, with the essence of
curriculum learning. This gradually learning paradigm significantly
distinguishes from prior work on noise modeling for DS seen to date. Moreover,
as such a method does not rely on any extra assumptions, it can serve as our
default training method for $\mathbf{T}$.

#### With Prior Knowledge of Data Quality

On the other hand, if we happen to have prior knowledge about which part of
the training data is more reliable and which is less reliable, we can utilize
this knowledge as guidance to design the curriculum. Specifically, we can
build a curriculum by first training the prediction branch on the reliable
data for several epochs, and then adding the less reliable data to train the
full model. In this way, the prediction branch is roughly trained before
exposed to more noisy data, thus is less likely to fall into poor local
optimum.

Furthermore, we can take better control of the training procedure with trace
regularization, e.g., encouraging larger $trace(\mathbf{T})$ for reliable
subset and smaller $trace(\mathbf{T})$ for less relaibale ones. Specifically,
we propose to minimize:

$\displaystyle
L=\sum_{m=1}^{M}{\sum_{i=1}^{N_{m}}{-log(o_{mi,y_{mi}})}}-\beta_{m}trace(\mathbf{T}^{mi})$
(6)

where $\beta_{m}$ is the regularization weight for the $m$-th data subset, $M$
is the total number of subsets, $N_{m}$ the number of instances in $m$-th
subset, and $\mathbf{T}^{mi}$, $y_{mi}$ and $o_{mi,y_{mi}}$ are the transition
matrix, the relation labeled by DS and the observed probability of this
relation for the $i$-th training instance in the $m$-th subset, respectively.
Note that different from Equation 5, this loss function does not need to
initiate training by minimizing the loss regarding the prediction distribution
$\mathbf{p}$, since one can easily start by learning from the most reliable
split first.

We also use trace regularization for the most reliable subset, since there are
still some noise annotations inevitably appearing in this split. Specifically,
we expect its $trace(\mathbf{T})$ to be large (using a positive $\beta$) so
that the elements of $\mathbf{T}$ will be centralized to the diagonal and
$\mathbf{T}$ will be more similar to the identity matrix. As for the less
reliable subset, we expect the $trace(\mathbf{T})$ to be small (using a
negative $\beta$) so that the elements of the transition matrix will be
diffusive and $\mathbf{T}$ will be less similar to the identity matrix. In
other words, the transition matrix is encouraged to characterize the noise.

Note that this loss function only works for sentence level models. For bag
level models, since reliable and less reliable sentences are all aggregated
into a sentence bag, we can not determine which bag is reliable and which is
not. However, bag level models can still build a curriculum by changing the
content of a bag, e.g., keeping reliable sentences in the bag first, then
gradually adding less reliable ones, and training with Equation 5, which could
benefit from the prior knowledge of data quality as well.

##  5 Evaluation Methodology

Our experiments aim to answer two main questions: (1) is it possible to model
the noise in the training data generated through DS, even when there is no
prior knowledge to guide us? and (2) whether the prior knowledge of data
quality can help our approach better handle the noise.

We apply our approach to both sentence level and bag level extraction models,
and evaluate in the situations where we do not have prior knowledge of the
data quality as well as where such prior knowledge is available.

###  5.1 Datasets

We evaluate our approach on two datasets.

####  TimeRE

We build TimeRE by using DS to align time-related Wikidata Vrandečić and
Krötzsch (2014) KB triples to Wikipedia text. It contains 278,141 sentences
with 12 types of relations between an entity mention and a time expression. We
choose to use time-related relations because time expressions speak for
themselves in terms of reliability. That is, given a KB triple $<$$e$, rel,
$t$$>$ and its aligned sentences, the finer-grained the time expression $t$
appears in the sentence, the more likely the sentence supports the existence
of this triple. For example, a sentence containing both _Alphabet_ and
_October-2-2015_ is very likely to express the inception-time of _Alphabet_ ,
while a sentence containing both _Alphabet_ and _2015_ could instead talk
about many events, e.g., releasing financial report of 2015, hiring a new CEO,
etc. Using this heuristics, we can split the dataset into 3 subsets according
to different granularities of the time expressions involved, indicating
different levels of reliability. Our criteria for determining the reliability
are as follows. Instances with full date expressions, i.e., Year-Month-Day,
can be seen as the most reliable data, while those with partial date
expressions, e.g., Month-Year and Year-Only, are considered as less reliable.
Negative data are constructed heuristically that any _entity-time_ pairs in a
sentence without corresponding triples in Wikidata are treated as negative
data. During training, we can access 184,579 negative and 77,777 positive
sentences, including 22,214 reliable, 2,094 and 53,469 less reliable ones. The
validation set and test set are randomly sampled from the reliable (full-date)
data for relatively fair evaluations and contains 2,776, 2,771 positive
sentences and 5,143, 5,095 negative sentences, respectively.

####  EntityRE

is a widely-used entity relation extraction dataset, built by aligning triples
in Freebase to the New York Times (NYT) corpus Riedel et al. (2010). It
contains 52 relations, 136,947 positive and 385,664 negative sentences for
training, and 6,444 positive and 166,004 negative sentences for testing.
Unlike TimeRE, this dataset does not contain any prior knowledge about the
data quality. Since the sentence level annotations in EntityRE are too noisy
to serve as gold standard, we only evaluate bag-level models on EntityRE, a
standard practice in previous works Surdeanu et al. (2012); Zeng et al.
(2015); Lin et al. (2016).

###  5.2 Experimental Setup

#### Hyper-parameters

We use 200 convolution kernels with widow size 3. During training, we use
stochastic gradient descend (SGD) with batch size 20. The learning rates for
sentence-level and bag-level models are 0.1 and 0.01, respectively.

Sentence level experiments are performed on TimeRE, using 100-d word
embeddings pre-trained using GloVe Pennington et al. (2014) on Wikipedia and
Gigaword Parker et al. (2011), and 20-d vectors for distance embeddings. Each
of the three subsets of TimeRE is added after the previous phase has run for
15 epochs. The trace regularization weights are $\beta_{1}=0.01$,
$\beta_{2}=-0.01$ and $\beta_{3}=-0.1$, respectively, from the reliable to the
most unreliable, with the ratio of $\beta_{3}$ and $\beta_{2}$ fixed to 10 or
5 when tuning.

Bag level experiments are performed on both TimeRE and EntityRE. For TimeRE,
we use the same parameters as above. For EntityRE, we use 50-d word embeddings
pre-trained on the NYT corpus using word2vec Mikolov et al. (2013), and 5-d
vectors for distance embedding. For both datasets, $\alpha$ and $\beta$ in Eq.
5 are initialized to 1 and 0.1, respectively. We tried various decay rates,
{0.95, 0.9, 0.8}, and steps, {3, 5, 8}. We found that using a decay rate of
0.9 with step of 5 gives best performance in most cases.

#### Evaluation Metric

The performance is reported using the precision-recall (PR) curve, which is a
standard evaluation metric in relation extraction. Specifically, the
extraction results are first ranked decreasingly by their confidence scores,
then the precision and recall are calculated by setting the threshold to be
the score of each extraction result one by one.

#### Naming Conventions

We evaluate our approach under a wide range of settings for sentence level
(sent_) and bag level (bag_) models: (1) _mix: trained on all three subsets of
TimeRE mixed together; (2) _reliable: trained using the reliable subset of
TimeRE only; (3) _PR: trained with prior knowledge of annotation quality,
i.e., starting from the reliable data and then adding the unreliable data; (4)
_TM: trained with dynamic transition matrix; (5) _GTM: trained with a global
transition matrix. In bag level, we also investigate the performance of
average aggregation (_avg) and attention aggregation (_att).

##  6 Experimental Results

###  6.1 Performance on TimeRE

Figure 2: Sentence Level Results on TimeRE

#### Sentence Level Models

The results of sentence level models on TimeRE are shown in Figure 2. We can
see that mixing all subsets together (sent_mix) gives the worst performance,
significantly worse than using the reliable subset only (sent_reliable). This
suggests the noisy nature of the training data obtained through DS and
properly dealing with the noise is the key for DS for a wider range of
applications. When getting help from our dynamic transition matrix, the model
(sent_mix_TM) significantly improves sent_mix, delivering the same level of
performance as sent_reliable in most cases. This suggests that our transition
matrix can help to mitigate the bad influence of noisy training instances.

Now let us consider the PR scenario where one can build a curriculum by first
training on the reliable subset, then gradually moving to both reliable and
less reliable data. We can see that, this simple curriculum learning based
model (sent_PR) further outperforms sent_reliable significantly, indicating
that the curriculum learning framework not only reduces the effect of noise,
but also helps the model learn from noisy data. When applying the transition
matrix approach into this curriculum learning framework using one reliable
subset and one unreliable subset generated by mixing our two less reliable
subsets, our model (sent_PR_seg2_TM) further improves sent_PR by utilizing the
dynamic transition matrix to model the noise. It is not surprising that when
we use all three subsets separately, our model (sent_PR_TM) significantly
outperforms all other models by a large margin.

(a) Attention Aggregation

(b) Average Aggregation

Figure 3: Bag Level Results on TimeRE

#### Bag Level Models

In this setting, we first look at the performance of the bag level models with
attention aggregation. The results are shown in Figure 3(a). Consider the
comparison between the model trained on the reliable subset only
(bag_att_reliable) and the one trained on the mixed dataset (bag_att_mix). In
contrast to the sentence level, bag_att_mix outperforms bag_att_reliable by a
large margin, because bag_att_mix has taken the at-least-one assumption into
consideration through the attention aggregation mechanism (Eq. 3), which can
be seen as a denoising step within the bag. This may also be the reason that
when we introduce either our dynamic transition matrix (bag_att_mix_TM) or the
curriculum of using prior knowledge of data quality (bag_att_PR) into the bag
level models, the improvement regarding bag_att_mix is not as significant as
in the sentence level.

However, when we apply our dynamic transition matrix into the curriculum built
upon prior knowledge of data quality (bag_att_PR_TM), the performance gets
further improved. This happens especially in the high precision part compared
to bag_att_PR. We also note that the bag level’s at-least-one assumption does
not always hold, and there are still false negative and false positive
problems. Therefore, using our transition matrix approach with or without
prior knowledge of data quality, i.e., bag_att_mix_TM and bag_att_PR_TM, both
improve the performance, and bag_att_PR_TM performs slightly better.

The results of bag level models with average aggregation are shown in Figure
3(b), where the relative ranking of various settings is similar to those with
attention aggregation. A notable difference is that both bag_avg_PR and
bag_avg_mix_TM improve bag_avg_mix by a larger margin compared to that in the
attention aggregation setting. The reason may be that the average aggregation
mechanism is not as good as the attention aggregation in denoising within the
bag, which leaves more space for our transition matrix approach or curriculum
learning with prior knowledge to improve. Also note that bag_avg_reliable
performs best in the very-low-recall region but worst in general. This is
because that it ranks higher the sentences expressing either birth-date or
death-date, the simplest but the most common relations in the dataset, but
fails to learn other relations with limited or noisy training instances, given
its relatively simple aggregation strategy.

#### Global v.s. Dynamic Transition Matrix

Figure 4: Global TM v.s. Dynamic TM

We also compare our dynamic transition matrix method with the global
transition matrix method, which maintains only one transition matrix for all
training instances. Specifically, instead of dynamically generating a
transition matrix for each datum, we first initialize an identity matrix
$\mathbf{T}^{\prime}\in\mathbb{R}^{|\mathbb{C}|\times|\mathbb{C}|}$, where
$|\mathbb{C}|$ is the number of relations (including no-relation). Then the
global transition matrix $\mathbf{T}$ is built by applying _softmax_ to each
row of $\mathbf{T}^{\prime}$ so that $\sum_{j}{\mathbf{T}_{ij}}=1$:

$T_{ij}=\frac{e^{T^{\prime}_{ij}}}{\sum_{j=1}^{|\mathbb{C}|}{e^{T^{\prime}_{ij}}}}$
(7)

where $T_{ij}$ and $T^{\prime}_{ij}$ are the elements in the $i^{th}$ row and
$j^{th}$ column of $\mathbf{T}$ and $\mathbf{T}^{\prime}$. The element values
of matrix $\mathbf{T}^{\prime}$ are also updated via backpropagation during
training. As shown in Figure 4, using one global transition matrix (_GTM) is
also beneficial and improves both the sentence level (sent_PR) and bag level
(bag_att_PR) models. However, since the global transition matrix only captures
the global noise pattern, it fails to characterize individuals with subtle
differences, resulting in a performance drop compared to the dynamic one
(_TM).

#### Case Study

We find our transition matrix method tends to obtain more significant
improvement on noisier relations. For example, time_of_spacecraft_landing is
noisier than time_of_spacecraft_launch since compared to the launching of a
spacecraft, there are fewer sentences containing the landing time of a
spacecraft that talks directly about the landing. Instead, many of these
sentences tend to talk about the activities of the crew. Our sent_PR_TM model
improves the F1 of time_of_spacecraft_landing and time_of_spacecraft_launch
over sent_PR by 9.09% and 2.78%, respectively. The transition matrix makes
more significant improvement on time_of_spacecraft_landing since there are
more noisy sentences for our method to handle, which results in more
significant improvement on the quality of the training data.

###  6.2 Performance on EntityRE

We evaluate our bag level models on EntityRE. As shown in Figure 5, it is not
surprising that the basic model with attention aggregation (att) significantly
outperforms the average one (avg), where att in our bag embedding is similar
in spirit to Lin et al. (2016), which has reported the-state-of-the-art
performance on EntityRE. When injected with our transition matrix approach,
both att_TM and avg_TM clearly outperform their basic versions.

Similar to the situations in TimeRE, since att has taken the at-least-one
assumption into account through its attention-based bag embedding mechanism,
thus the improvement made by att_TM is not as large as by avg_TM.

Figure 5: Results on EntityRE Method | P@R_10 | P@R_20 | P@R_30
---|---|---|---
Mintz | 39.88 | 28.55 | 16.81
MultiR | 60.94 | 36.41 | -
MIML | 60.75 | 33.82 | -
avg | 58.04 | 51.25 | 42.45
avg_TM | 58.56 | 52.35 | 43.59
att | 61.51 | 56.36 | 45.63
att_TM | 67.24 | 57.61 | 44.90
Table 1: Comparison with feature-based methods. P@R_10/20/30 refers to the
precision when recall equals 10%, 20% and 30%.

We also include the comparison with three feature-based methods: Mintz Mintz
et al. (2009) is a multiclass logistic regression model; MultiR Hoffmann et
al. (2011) is a probabilistic graphical model that can handle overlapping
relations; MIML Surdeanu et al. (2012) is also a probabilistic graphical model
but operates in the multi-instance multi-label paradigm. As shown in Table 1,
although traditional feature-based methods have reasonable results in the low
recall region, their performances drop quickly as the recall goes up, and
MultiR and MIML did not even reach the 30% recall. This indicates that, while
human-designed featurs can effectively capture certain relation patterns,
their coverage is relatively low. On the other hand, neural network models
have more stable performance across different recalls, and att_TM performs
generally better than other models, indicating again the effectiveness of our
transition matrix method.

##  7 Related Work

In addition to relation extraction, distant supervision (DS) is shown to be
effective in generating training data for various NLP tasks, e.g., tweet
sentiment classification Go et al. (2009), tweet named entity classifying
Ritter et al. (2011), etc. However, these early applications of DS do not well
address the issue of data noise.

In relation extraction (RE), recent works have been proposed to reduce the
influence of wrongly labeled data. The work presented by Takamatsu et al.
(2012) removes potential noisy sentences by identifying bad syntactic patterns
at the pre-processing stage. Xu et al. (2013) use pseudo-relevance feedback to
find possible false negative data. Riedel et al. (2010) make the _at-least-one
assumption_ and propose to alleviate the noise problem by considering RE as a
multi-instance classification problem. Following this assumption, people
further improves the original paradigm using probabilistic graphic models
Hoffmann et al. (2011); Surdeanu et al. (2012), and neural network methods
Zeng et al. (2015). Recently, Lin et al. (2016) propose to use attention
mechanism to reduce the noise within a sentence bag. Instead of characterizing
the noise, these approaches only aim to alleviate the effect of noise.

The _at-least-one assumption_ is often too strong in practice, and there are
still chances that the sentence bag may be false positive or false negative.
Thus it is important to model the noise pattern to guide the learning
procedure. Ritter et al. (2013) and Min et al. (2013) try to employ a set of
latent variables to represent the true relation. Our approach differs from
them in two aspects. We target noise modeling in neutral networks while they
target probabilistic graphic models. We further advance their models by
providing the capability to model the fine-grained transition from the true
relation to the observed, and the flexibility to combine indirect guidance.

Outside of NLP, various methods have been proposed in computer vision to model
the data noise using neural networks. Sukhbaatar et al. (2015) utilize a
global transition matrix with weight decay to transform the true label
distribution to the observed. Reed et al. (2014) use a hidden layer to
represent the true label distribution but try to force it to predict both the
noisy label and the input. Chen and Gupta (2015); Xiao et al. (2015) first
estimate the transition matrix on a clean dataset and apply to the noisy data.
Our model shares similar spirit with Misra et al. (2016) in that we all
dynamically generate a transition matrix for each training instance, but,
instead of using vanilla SGD, we train our model with a novel curriculum
learning training framework with trace regularization to control the behavior
of transition matrix. In NLP, the only work in neural-network-based noise
modeling is to use one single global transition matrix to model the noise
introduced by cross-lingual projection of training data Fang and Cohn (2016).
Our work advances them through generating a transition matrix dynamically for
each instance, to avoid using one single component to characterize both
reliable and unreliable data.

##  8 Conclusions

In this paper, we investigate the noise problem inherent in the DS-style
training data. We argue that the data speak for themselves by providing useful
clues to reveal their noise patterns. We thus propose a novel transition
matrix based method to dynamically characterize the noise underlying such
training data in a unified framework along the original prediction objective.
One of our key innovations is to exploit a curriculum learning based training
method to gradually learn to model the underlying noise pattern without direct
guidance, and to provide the flexibility to exploit any prior knowledge of the
data quality to further improve the effectiveness of the transition matrix. We
evaluate our approach in two learning settings of the distantly supervised
relation extraction. The experimental results show that the proposed method
can better characterize the underlying noise and consistently outperform
start-of-the-art extraction models under various scenarios.

## Acknowledgement

This work is supported by the National High Technology R&D Program of China
(2015AA015403); the National Natural Science Foundation of China (61672057,
61672058); KLSTSPI Key Lab. of Intelligent Press Media Technology; the UK
Engineering and Physical Sciences Research Council under grants EP/M01567X/1
(SANDeRs) and EP/M015793/1 (DIVIDEND); and the Royal Society International
Collaboration Grant (IE161012).

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