††institutetext: ICTP South American Institute for Fundamental Research
Instituto de Física Teórica, UNESP - Univ. Estadual Paulista
Rua Dr. Bento Teobaldo Ferraz 271, 01140-070, São Paulo, SP, Brazil

# 4D Chern-Simons and the pure spinor $\ads$ superstring

Nathan Berkovits,  Rodrigo S. Pitombo<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Four-dimensional Chern-Simons (4DCS) theory is useful for understanding
integrable sigma-models and constructing new ones. In this paper, we show how
to derive the complete pure spinor $\ads$ superstring sigma-model from 4DCS
theory with defects. The matter sector of this sigma model was previously
derived by Costello and Yamazaki, and we propose here that the pure spinor
ghosts come from gauge-fixing meromorphic transformations of 4DCS which lead
to the usual pure spinor Lax connection including the ghost contribution.

##  1 Introduction

An important step in fully understanding the AdS/CFT duality in its original
formulation maldacena_large-n_1999  is to study the behavior of superstrings
propagating in the $\ads$ background. In this context, the pure spinor
formalism for the superstring berkovits_super-poincare_2000  plays an
essential role since it can describe Type IIB backgrounds with Ramond-Ramond
flux berkovits_ten-dimensional_2002  and its worldsheet action in an $\ads$
background is quantizable berkovits_quantum_2005 . Sigma models that describe
superstrings in this background are special because they are integrable (at
least classically). This means that the equations of motion for these models
are equivalent to the flatness of a Lax connection $\mathcal{L}(z)$:

$d\mathcal{L}(z)+\mathcal{L}(z)\wedge\mathcal{L}(z)=0\;\;\;\;\forall
z\in\mathds{C}$ (1)

where $z$ is called the spectral parameter. The Green-Schwarz bena_hidden_2004
, pure spinor Brenno_Carlini_Vallilo_2004 ; berkovits_brst_2005  and the
recently introduced B-RNS-GSS chandia_b-rns-gss_2023  formalisms have this
property. Moreover, integrable structures also appear in the field theory side
of the duality minahan_bethe-ansatz_2003 ; beisert_n4_2003 .

Discovering that a model has a Lax connection satisfying (1) usually involves
guesswork. However, in a series of papers costello_gauge_2018-1 ;
costello_gauge_2018 ; costello_gauge_2019  Costello, Witten and Yamazaki have
shown that one can use a four-dimensional version of Chern-Simons theory with
defects to construct sigma models in such a way that their integrability is
guaranteed, as the flatness equation is implied by one of the equations of
motion. For our purposes, the theory lives in
$\mathds{R}^{2}\times\mathds{CP}^{1}$ and the spectral parameter on which
$\mathcal{L}$ depends is reinterpreted as a holomorphic coordinate on the
Riemann sphere.

Using 4D Chern-Simons theory (4DCS), several integrable field theories have
been shown to come from different defect configurations in the 4D manifold
such as the principal chiral model, the WZW model, the Yang-Baxter model
costello_gauge_2019 ; lacroix_four-dimensional_2022 , and others
fukushima_non-abelian_2022 ; fukushima_faddeev-reshetikhin_2021 ;
liniado_integrable_2023 ; caudrelier_zakharovmikhailov_2021 ;
lacroix_integrable_2021 ; vicedo_4d_2021 . Thus, a natural question is whether
one can construct the superstring sigma model on $\ads$ with this framework.
In costello_gauge_2019  Costello and Yamazaki constructed the matter sector
(20) of the pure spinor superstring on $\ads$, and some integrable
deformations fukushima_yang-baxter_2020 ; tian_ensuremathlambda-deformed_2021
as well as the Green-Schwarz sigma model costello_chern-simons_2020  on $\ads$
were also constructed using this framework. Nevertheless, none of the
constructions included the ghost sector of the worldsheet action, which plays
an essential role in the pure spinor formalism.

The purpose of this work is to show how the 4DCS framework with gauge
supergroup $\mathrm{PSU}(2,2|4)$ can be used to describe the complete $\ads$
pure spinor superstring including the ghost sector. In the construction of
costello_gauge_2019 , there is some arbitrariness in the choice of pole
structure for the Lax connection. In particular, one imposes by hand which
components of $\mathcal{L}$ have poles at given points, giving rise to the
chiral and anti-chiral defects. Here, we propose that this arbitrary choice
should be interpreted as a gauge-fixing condition. We then use a set of
$\mathrm{PSU}(2,2|4)$-valued meromorphic gauge transformations to restrict the
pole structure of $\mathcal{L}$.

Fixing the meromorphic gauge symmetry will lead to a set of 32 bosonic and 10
fermionic ghosts localized in two-dimensional subspaces of
$\mathds{R}^{2}\times\mathds{CP}^{1}$. This characterizes the ghost system as
a so-called “order defect”, and defects of this type were thoroughly analyzed
in costello_gauge_2019 . Adapting a cohomology argument of aisaka_new_2003  to
$\ads$, we argue that these ghosts should be equivalent to the usual pair of
left and right-moving pure spinor bosonic ghosts, and the system composed by
the 4DCS gauge field coupled to the ghosts yields the complete pure spinor
$\ads$ superstring. Thus, 4DCS not only gives a complete description of the
formalism but also furnishes a natural origin for the pure spinor ghosts in
terms of gauge-fixing. Note that in berkovits_simplifying_2009 , the 22
bosonic pure spinor ghosts in an $\ads$ background were similarly derived from
32 bosonic and 10 fermionic ghosts by gauge-fixing $\mathrm{PSU}(2,2|4)$ local
symmetries.

The paper is organized as follows: In the following subsection, we review the
pure spinor formalism in $\ads$. In section 2 we introduce 4D Chern-Simons and
the defects that lead to coset sigma models. In section 3 we discuss gauge
invariance and argue that for a specific gauge-fixing, the Faddeev-Popov
procedure leads to a ghost system and BRST charge which are equivalent to
those of the pure spinor formalism. And in section 4, we show how the 4D setup
of the defects and ghosts leads to the 2D action and Lax connection of the
pure spinor superstring in $\ads$.

###  1.1 Review of the pure spinor formalism in $\ads$

We are interested in describing superstrings propagating in $\ads$. The
associated superspace can be described in terms of the supercoset

$\frac{\mathrm{PSU}(2,2|4)}{\mathrm{SO}(4,1)\times\mathrm{SO}(5)}.$ (2)

The $\mathfrak{psu}(2,2|4)$ algebra is given by

$\displaystyle\left\\{Q_{\alpha},Q_{\beta}\right\\}=\gamma_{\alpha\beta}^{a}P_{a},\quad\left\\{\hat{Q}_{\hat{\alpha}},\hat{Q}_{\hat{\beta}}\right\\}=\gamma^{a}_{\hat{\alpha}\hat{\beta}}P_{a},$
(3)
$\displaystyle{\left[P_{a},\hat{Q}_{\hat{\beta}}\right]=-\frac{1}{2}\left(\eta\gamma_{a}\right)^{\alpha}_{\hat{\beta}}Q_{\alpha},}$
(4)
$\displaystyle{\left[P_{a},Q_{\alpha}\right]=\frac{1}{2}\left(\gamma_{a}\eta\right)_{\alpha}^{\hat{\beta}}\hat{Q}_{\hat{\beta}},}$
(5) $\displaystyle\left[P_{a},P_{b}\right]=-L_{ab},$ (6)
$\displaystyle\left\\{Q_{\alpha},\hat{Q}_{\hat{\beta}}\right\\}=\frac{1}{2}\left(\gamma^{ab}\eta\right)_{\alpha\hat{\beta}}L_{ab}$
(7)

and the non-vanishing supertraces are

$\displaystyle\mathrm{sTr}\left(P_{a}P_{b}\right)=\eta_{ab},\quad\mathrm{sTr}\left(Q_{\alpha}\hat{Q}_{\hat{\beta}}\right)=-2\eta_{\alpha\hat{\beta}},$
(8) $\displaystyle\mathrm{sTr}\left(L_{ab}L_{cd}\right)=\eta_{a[b}\eta_{c]d.}$
(9)

This supercoset describes a “generalized symmetric space” because the
superalgebra $\mathfrak{psu}(2,2|4)$ has an automorphism $\rho$ which induces
a $\mathds{Z}^{4}$ grading

$\mathfrak{g}=\mathfrak{g}^{(0)}\oplus\mathfrak{g}^{(1)}\oplus\mathfrak{g}^{(2)}\oplus\mathfrak{g}^{(3)}$
(10)

such that

$\rho\cdot\mathfrak{g}^{(k)}=i^{k}\mathfrak{g}^{(k)}$ (11)

where

$\displaystyle\mathfrak{g}^{(0)}=\mathrm{span}(L_{ab})\cong\mathfrak{so}(4,1)\oplus\mathfrak{so}(5)$
(12) $\displaystyle\mathfrak{g}^{(1)}=\mathrm{span}(Q_{\alpha})$ (13)
$\displaystyle\mathfrak{g}^{(2)}=\mathrm{span}(P_{a})$ (14)
$\displaystyle\mathfrak{g}^{(3)}=\mathrm{span}(\hat{Q}_{\hat{\alpha}})$ (15)

Therefore, the superstring in this space can be described by the supergroup-
valued degree of freedom

$g(w,\bar{w})\in\mathrm{PSU}(2,2|4)$ (16)

with the equivalence relation

$g(w,\bar{w})\sim
g(w,\bar{w})h(w,\bar{w})\;\;\;\;\;h\in\mathrm{SO}(4,1)\times\mathrm{SO}(5)$
(17)

The matter sector of the action is constructed out of the
$\mathfrak{psu}(2,2|4)$-valued left-invariant currents

$J:=g^{-1}dg$ (18)

which can be decomposed in terms of the $\mathds{Z}^{4}$ grading of
$\mathfrak{psu}(2,2|4)$ as

$J=J^{(0)}+J^{(1)}+J^{(2)}+J^{(3)}$ (19)

In particular, the matter part is mazzucato_superstrings_2012

$S_{m}=\int
d^{2}w\;\mathrm{sTr}\big{(}\frac{1}{2}J_{\bar{w}}^{(2)}J_{w}^{(2)}+\frac{3}{4}J_{w}^{(3)}J_{\bar{w}}^{(1)}+\frac{1}{4}J_{w}^{(1)}J_{\bar{w}}^{(3)}\big{)}$
(20)

We also have ghosts which play a fundamental role in cancelling the conformal
anomaly and constructing physical vertex operators. As in 10D flat space, they
are bosonic fields with 10D Weyl spinor indices ($\lambda^{\alpha}$,
$\hat{\lambda}^{\hat{\alpha}}$) that satisfy the pure spinor constraints

$\displaystyle\lambda^{\alpha}\gamma^{a}_{\alpha\beta}\lambda^{\beta}=0$ (21)
$\displaystyle\hat{\lambda}^{\hat{\alpha}}\gamma^{a}_{\hat{\alpha}\hat{\beta}}\hat{\lambda}^{\hat{\beta}}=0$
(22)

In the $\ads$ model, they can be conveniently written in terms of
$\mathfrak{g}$-valued ghosts as

$\displaystyle\lambda:=\lambda^{\alpha}Q_{\alpha}\in\mathfrak{g}^{(1)}$ (23)
$\displaystyle\hat{\lambda}:=\hat{\lambda}^{\hat{\alpha}}\hat{Q}_{\hat{\alpha}}\in\mathfrak{g}^{(3)}$
(24)

From the structure constants in (3), it is direct to see that the pure spinor
constraints can be rewritten as

$\\{\lambda,\lambda\\}=\\{\hat{\lambda},\hat{\lambda}\\}=0$ (25)

The model also has anti-ghosts
$(y,\hat{y})\in(\mathfrak{g}^{(3)},\mathfrak{g}^{(1)})$ which can be used to
construct the ghost-number zero objects

$N=-\\{y,\lambda\\}\;\;\;\;\;\hat{N}=-\\{\hat{y},\hat{\lambda}\\}\;\;\in\mathfrak{g}^{(0)}$
(26)

The ghost contribution to the action is then

$S_{gh}=\int
d^{2}w\;\mathrm{sTr}\big{(}y\nabla_{\bar{w}}^{(0)}\lambda+\hat{y}\nabla^{(0)}_{w}\hat{\lambda}-N\hat{N}\big{)},$
(27)

where we defined the $\mathfrak{g}^{(0)}$-covariant derivative

$\nabla^{(0)}=d+[J^{(0)},\cdot]$ (28)

The complete action is just the sum

$S=S_{m}+S_{gh.}$ (29)

This action is invariant under the BRST transformations

$\displaystyle\;\;\;\;\;\;\;Q\cdot g=g(\lambda+\hat{\lambda})$ (30)
$\displaystyle Q\cdot y=-J^{(3)}_{w}\;\;\;\;Q\cdot\hat{y}=-J^{(1)}_{\bar{w}}$
(31) $\displaystyle\;\;\;\;\;\;Q\cdot\lambda=Q\cdot\hat{\lambda}=0$ (32)

and the Noether procedure yields the BRST charge

$\displaystyle Q=\oint dw\;\mathrm{sTr}\big{(}\lambda
J^{(3)}_{\bar{w}}\big{)}-\oint
d\bar{w}\;\mathrm{sTr}\big{(}\hat{\lambda}J^{(1)}_{w}\big{)}$ (33)

In this sigma model, the equations of motion are equivalent to the flatness of
the Lax connection

$\displaystyle\mathcal{L}_{w}=J_{w}^{(0)}+\frac{1}{u}J_{w}^{(1)}+\frac{1}{u^{2}}J_{w}^{(2)}+\frac{1}{u^{3}}J_{w}^{(3)}-(1-\frac{1}{u^{4}})N$
(34)
$\displaystyle\mathcal{L}_{\bar{w}}=J_{\bar{w}}^{(0)}+uJ_{\bar{w}}^{(3)}+u^{2}J_{\bar{w}}^{(2)}+u^{3}J_{\bar{w}}^{(1)}-(1-u^{4})N.$
(35)

Moreover, one can show mikhailov_perturbative_2007  that, under BRST
transformations, the Lax connection transforms as

$Q\cdot\mathcal{L}=d\tilde{\lambda}+[\mathcal{L},\tilde{\lambda}]$ (36)

where

$\tilde{\lambda}=\frac{1}{u}\lambda+u\hat{\lambda}.$ (37)

Note that (36) has exactly the form of the gauge transformation in Chern-
Simons theory. Thus, it seems reasonable that the ghosts of the formalism can
be reinterpreted as Faddeev-Popov ghosts in a gauge-fixing of 4D Chern-Simons
theory. Moreover, from the form of (37) it is natural to expect that they are
related to meromorphic gauge transformations. This will be shown below by
starting with the 4DCS action of costello_gauge_2019  for the $\ads$
superstring and gauge-fixing the local symmetries.

##  2 4D Chern-Simons

Our starting point is the 4D Chern-Simons (4DCS) action given by
costello_gauge_2018 ; costello_gauge_2018-1 ; costello_gauge_2019

$S=\frac{1}{2\pi i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}A\wedge
dA+\frac{1}{3}A\wedge\big{[}A\wedge A\big{]}\big{)}$ (38)

with a gauge (super)group $G=\mathrm{PSU}(2,2|4)$ and
$\mathcal{M}=\mathds{R}^{2}\times\mathds{CP}^{1}$. Throughout the paper,
($z,\bar{z}$) are coordinates on $\mathds{CP}^{1}$ and $(w,\bar{w})$ are
coordinates on $\mathds{R}^{2}$. Moreover, $\omega(z)$ is a meromorphic 1-form
on $\mathds{CP}^{1}$

$\omega:=\phi(z)dz.$ (39)

which can have poles and zeroes at special points. These special points are
called disorder defects costello_gauge_2019  and play an important role in the
relation between 4DCS and 2D integrable field theories. One can construct a
plethora of known integrable models by choosing different functions $\phi(z)$
and gauge groups $G$.

The equation of motion for $A$ is

$\displaystyle\omega\wedge F(A)=0$ (40)

where the field-strength $F$ is defined as

$F(A):=dA+A\wedge A$ (41)

###  2.1 Coset models and $\ads$

To describe sigma-models with target space $\ads$, we need to introduce
another kind of defect costello_gauge_2019  which is a line with ends at $z=0$
and $z=1$ such that, when the line defect is crossed, the $\mathbb{Z}^{4}$
automorphism $\rho$ of the superalgebra is applied on the gauge field, as
shown in fig. 1.

$z=0$$z=1$$z$-plane$A$$\rho(A)$$u=0$$u$-plane$A$$\rho(A)$$\rho^{2}(A)$$\rho^{3}(A)$
Figure 1: $z$-plane with line defect and change of coordinates to 4-fold
cover. The line defect correspond to the 4 lines connecting $u=0$ and
$u=\infty$ in the second diagram.

It is easier to understand the implications of this defect in a 4-fold cover
of $\mathbb{CP}^{1}$ defined by the change of coordinates.

$z=-\frac{1}{4(u^{4}-1)}$ (42)

The holomorphic 1-form $dz$ is then

$dz=\frac{u^{3}}{(u^{4}-1)^{2}}du=:\omega(u)$ (43)

The 1-form has zeroes at $u=0$ and $u=\infty$ (the pre-images of the ends of
the line defect) and second-order poles at $\\{p_{k}\\}=\\{1,i,-1,-i\\}$. In
addition to understanding the behavior of the theory at these poles and
zeroes, we also need to deal with the line defect. In particular, we shall
impose that the fields are single-valued on the $z$-plane. In the $u$-plane,
this implies that field configurations should be invariant under the
simultaneous permutation of the 4 “slices” of the cover and application of
$\rho$, as can be seen in the second diagram of fig. 1. This is true if, and
only if,

$\displaystyle A(u)=\rho\cdot A(iu),$ (44)

and we shall call (44) the $\rho$-constraint.

Such a configuration of defects was used to construct the matter sector of the
$\ads$ pure spinor superstring in costello_gauge_2019 . As we will see in the
following sections, the equations of motion imply that the gauge field along
$\Sigma$ can have poles either at $u=0$ or $u=\infty$. In the construction of
costello_gauge_2019 , one imposes by hand that $A_{w}$ only has poles at $u=0$
and that $A_{\bar{w}}$ only has poles at $u=\infty$. However, there is no a
priori reason for the Lax connection to satisfy this property, and we will
argue below that this should be interpreted as a gauge-fixing condition. We
will see in section 3 that after appropriately imposing this gauge-fixing, the
usual left- and right-moving pure spinor ghosts emerge as Faddeev-Popov
ghosts.

###  2.2 Boundary conditions

In this theory, it is convenient to consider small contours around the poles
$p_{k}=\\{1,i,-1,-i\\}$ of $\omega(u)$ as boundaries. We shall impose
Dirichlet boundary conditions on the gauge field $A_{\Sigma}(p_{k})$ at these
contours so that the boundary terms in the variation of the action vanish.
Note that by varying the action, we get

$\delta S=\frac{1}{2\pi
i}\sum_{k}\int_{\Sigma}\oint_{\mathcal{C}_{k}}\omega(u)\wedge\mathrm{sTr}\big{(}A_{\Sigma}\wedge\delta
A_{\Sigma}\big{)}+\frac{1}{\pi
i}\int_{\Sigma\times\mathds{CP}^{1}}\omega\wedge\mathrm{sTr}\big{(}\delta
A\wedge F(A)\big{)}$ (45)

where the index $k$ goes over the boundaries. So the boundary variation is

$\displaystyle\delta S_{\\{p_{k}\\}}=\frac{1}{2\pi
i}\sum_{\\{p_{k}\\}}\int_{\Sigma}\oint_{\mathcal{C}_{p_{k}}}\frac{du}{(u-p_{k})^{2}}\frac{u^{3}}{\prod_{j\neq
k}(u-p_{j})^{2}}\mathrm{sTr}\big{(}\delta A_{\Sigma}\wedge A_{\Sigma}\big{)},$
(46)

which vanishes if we impose the Dirichlet boundary condition

$A_{\Sigma}(p_{k})=0.$ (47)

##  3 Gauge invariance and gauge fixing

The action is invariant under the gauge symmetry

$A\mapsto hAh^{-1}-dhh^{-1}\;\;\;\;h\in\mathrm{PSU}(2,2|4)$ (48)

provided that condition (47) is preserved at $u^{4}=1$. This means that the
$\bar{u}$ component of $A$ can be gauged to zero everywhere except near the
points $u^{4}=1$, which leads to $G$-valued degrees of freedom at the vicinity
of $u^{4}=1$

$\displaystyle A_{\bar{u}}=-\partial_{\bar{u}}\hat{g}\hat{g}^{-1}$ (49)
$\displaystyle
A_{\Sigma}=\hat{g}\mathcal{L}\hat{g}^{-1}-d_{\Sigma}\hat{g}\hat{g}^{-1}$ (50)

where $\mathcal{L}$ is a 1-form with components only along
$\Sigma\equiv\mathds{R}^{2}$.

The Dirichlet boundary conditions on $A_{\Sigma}$ translate to

$\displaystyle\mathcal{L}(p_{k})=\hat{g}^{-1}d\hat{g}|_{u=p_{k}}$ (51)

In terms of ($\mathcal{L},\hat{g}$) the 4DCS action (38) is

$\displaystyle S(\mathcal{L},\hat{g})=$ $\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge
d\mathcal{L}\big{)}+\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\hat{g}^{-1}d\hat{g}\big{)}+$
$\displaystyle+\frac{1}{6\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\big{)}.$
(52)

To get an $A_{\bar{u}}$ that satisfies the desired gauge-fixing conditions, we
use (48) to fix $\hat{g}$ to be archipelago-likedelduc_unifying_2020 . It will
then be equal to $\mathds{1}$ outside small disks around the poles
$\\{p_{k}\\}$. Inside the disks, there is a smaller disk in which
$\hat{g}=g_{k}(w,\bar{w})$ (for the disk around $p_{k}$) and an outer annulus
in which $\hat{g}$ smoothly interpolates between $g_{k}$ and $\mathds{1}$
depending only on $|u-p_{k}|$ and $(w,\bar{w})$. These conditions are
schematically depicted in Fig. 2. Note that the $\rho$-constraint fixes
$g_{k}=\rho^{k}\cdot g$ where we defined $g_{0}\equiv g$.

$\hat{g}=g_{k}(w,\bar{w})$$p_{k}$$\hat{g}(|u-p_{k}|,w,\bar{w})$$\hat{g}=\mathds{1}$
Figure 2: Schematic depiction of archipelago gauge conditions.

###  3.1 Residual gauge transformations and ghosts

In archipelago gauge, the equation of motion (40) implies

$\displaystyle\phi(u)\partial_{\bar{u}}\mathcal{L}=0$ (53)
$\displaystyle\phi(u)(d\mathcal{L}+\mathcal{L}\wedge\mathcal{L})=0$ (54)

The residual gauge transformations should be consistent with these equations
and leave $A_{\bar{u}}$ unchanged. Since (53) implies that $\mathcal{L}$ only
has poles up to third-order, the residual transformations are

$\displaystyle\delta\mathcal{L}=\nabla_{\Sigma}\tilde{\Lambda}$ (55)

where

$\nabla_{\Sigma}:=d_{\Sigma}+[\mathcal{L},\cdot],$ (56)

and $\tilde{\Lambda}$ should be defined such that $\delta\mathcal{L}$ only has
poles up to third order at $u=0$ or $u=\infty$ and should vanish at $u^{4}=1$.
The precise form of these parameters will depend on the specific value of
$\mathcal{L}$, but the general structure is

$\displaystyle\tilde{\Lambda}=\big{(}1/u-u^{3}\big{)}\Lambda_{1}(w,\bar{w})+\big{(}1/u^{2}-u^{2}\big{)}\Lambda_{2}(w,\bar{w})+\big{(}u-1/u^{3}\big{)}\Lambda_{3}(w,\bar{w})+\mathrm{higher}\;\mathrm{poles}$
(57)

where $\Lambda_{k}\in\mathfrak{g}^{(k)}$. Here, ($\Lambda_{1}$, $\Lambda_{2}$,
$\Lambda_{3}$) are independent parameters and the higher poles are determined
in terms of ($\Lambda_{1}$, $\Lambda_{2}$, $\Lambda_{3}$) by the requirement
that $\nabla_{\Sigma}\tilde{\Lambda}$ only has poles up to third order. We
therefore have 32 fermionic ($\Lambda_{1}$ and $\Lambda_{3}$) and 10 bosonic
($\Lambda_{2}$) gauge parameters. So a naive counting indicates that the ghost
system obtained from gauge-fixing this symmetry is equivalent to $32-10=22$
bosonic ghosts, which is precisely the number of degrees of freedom of a pair
of pure spinors.

To further understand this, let’s use $(\Lambda_{1},\Lambda_{2},\Lambda_{3})$
to impose gauge-fixing conditions on $\mathcal{L}$. Let

$\mathcal{L}=\sum_{k}u^{k}\mathcal{L}^{k}.$ (58)

We can first use $\Lambda_{1}$ and $\Lambda_{3}$ to impose

$\mathcal{L}_{w}^{1}=\mathcal{L}_{\bar{w}}^{-1}=0,$ (59)

which leads to a pair of ghosts
$(Z,\hat{Z})\in(\mathfrak{g}^{(1)},\mathfrak{g}^{(3)})$. We can then use
$\Lambda_{2}$ to gauge away 5 components of $\mathcal{L}_{w}^{2}$ and
$\mathcal{L}_{\bar{w}}^{-2}$. To choose which components are gauged away, we
follow the procedure of berkovits_simplifying_2009  by defining the matrices
($\mathcal{N}$, $\bar{\mathcal{N}}$) as in Appendix A and imposing

$\mathcal{N}(Z)\cdot\mathcal{L}^{2}_{w}=\bar{\mathcal{N}}(\hat{Z})\cdot\mathcal{L}_{\bar{w}}^{-2}=0$
(60)

Given these conditions, one can derive further restrictions from the flatness
equation. The vanishing of fourth- and fifth-order poles at $u=0$ of the
curvature implies

$\displaystyle[\mathcal{L}_{w}^{-1},\mathcal{L}_{\bar{w}}^{-3}]+[\mathcal{L}_{w}^{-2},\mathcal{L}_{\bar{w}}^{-2}]=0$
(61)
$\displaystyle[\mathcal{L}_{w}^{-2},\mathcal{L}_{\bar{w}}^{-3}]+[\mathcal{L}_{w}^{-3},\mathcal{L}_{\bar{w}}^{-2}]=0$
(62)

which in components gives

$\displaystyle
L_{ab}[(\mathcal{L}^{-1}_{w})^{\alpha}(\mathcal{L}_{\bar{w}}^{-3})^{\hat{\beta}}(\gamma^{ab}\eta)_{\alpha\hat{\beta}}+(\mathcal{L}_{w}^{-2})^{a}\eta^{bc}\mathcal{N}^{I}_{c}\bar{\mathcal{L}}_{I}]=0$
(63) $\displaystyle
Q_{\alpha}[(\mathcal{L}^{-2}_{w})^{a}(\eta\gamma_{a})_{\hat{\beta}}^{\alpha}(\mathcal{L}_{\bar{w}}^{-3})^{\hat{\beta}}+\eta^{ab}\mathcal{N}^{I}_{b}\bar{\mathcal{L}}_{I}(\eta\gamma_{a})^{\alpha}_{\hat{\beta}}(\mathcal{L}_{w}^{-3})^{\hat{\beta}}]=0$
(64)

where we used condition (60) to write

$(\mathcal{L}_{\bar{w}}^{-2})^{a}=\eta^{ab}\mathcal{N}^{I}_{b}\bar{\mathcal{L}}_{I}$
(65)

Assuming that $\mathcal{L}_{w}^{-1}$ and half of $\mathcal{L}_{w}^{-2}$ are
non-zero and generic (as restricting their values would correspond to other
gauge-fixing conditions), equations (63) and (64) imply

$\mathcal{L}_{\bar{w}}^{-3}=\bar{\mathcal{L}}_{I}=0.$ (66)

So we have shown that in this gauge,
$\mathcal{L}_{\bar{w}}^{-1}=\mathcal{L}_{\bar{w}}^{-2}=\mathcal{L}_{\bar{w}}^{-3}=0$,
i.e. $\mathcal{L}_{\bar{w}}$ has no poles at $u=0$. An analogous argument
considering the poles of $F(\mathcal{L})$ at $u=\infty$ implies that
$\mathcal{L}_{w}$ has no poles at $u=\infty$.

With this gauge-fixing, the ghost system can be conveniently expressed as

$\displaystyle\tilde{C}=\big{(}1/u-u^{3}\big{)}Z(w,\bar{w})+\big{(}1/u^{2}-u^{2}\big{)}c(w,\bar{w})+\big{(}1/u^{3}-u\big{)}\hat{Z}(w,\bar{w})+\mathrm{higher}\;\mathrm{poles},$
(67)

where the higher poles are complicated functions of ($Z$, $c$, $\hat{Z}$).
Analogously, the anti-ghosts ($y_{\alpha}$, $\hat{y}_{\alpha}$, $b_{I}$,
$\bar{b}_{I}$) and Lagrange multipliers ($l_{\alpha}$,
$\hat{l}_{\hat{\alpha}}$, $f_{I}$, $\bar{f}_{I}$) can be compactly written as

$\displaystyle B=y+ub\cdot\bar{\mathcal{N}}$ (68)
$\displaystyle\bar{B}=\hat{y}+\frac{1}{u}\hat{b}\cdot{\mathcal{N}}$ (69)
$\displaystyle\mathcal{F}=l+uf\cdot\bar{\mathcal{N}}$ (70)
$\displaystyle\bar{\mathcal{F}}=\hat{l}+\frac{1}{u}\bar{f}\cdot\mathcal{N}$
(71)

The Faddeev-Popov action can then succinctly be written as

$\displaystyle
S_{FP}=\int_{\Sigma}d^{2}w\bigg{\\{}\oint_{\mathcal{C}_{0}}du\;\mathrm{sTr}\big{(}B\nabla_{\bar{w}}\tilde{C}+\mathcal{F}\mathcal{L}_{\bar{w}}\big{)}+\oint_{\mathcal{C}_{\infty}}du\;\mathrm{sTr}\big{(}\bar{B}\nabla_{w}\tilde{C}+\bar{\mathcal{F}}\mathcal{L}_{w}\big{)}\bigg{\\}}$
(73)

or in components

$\displaystyle S_{FP}=\int_{\Sigma}$ $\displaystyle
d^{2}w\;\mathrm{sTr}\big{\\{}y\big{(}\nabla^{0}_{\bar{w}}Z+[\mathcal{L}_{\bar{w}}^{1},c]-[\mathcal{L}_{\bar{w}}^{2},\hat{Z}]+[\mathcal{L}_{\bar{w}}^{3},\tilde{\Lambda}_{-4}(Z,\hat{Z},c)]\big{)}+$
$\displaystyle+b\cdot\bar{\mathcal{N}}\big{(}\nabla_{\bar{w}}^{0}c-[\mathcal{L}_{\bar{w}}^{1},\hat{Z}]+[\mathcal{L}_{\bar{w}}^{2},{\tilde{\Lambda}_{-4}}(Z,\hat{Z},c)]+[\mathcal{L}_{\bar{w}}^{3},\tilde{\Lambda}_{-5}(Z,\hat{Z},c)]\big{)}+$
$\displaystyle+l\mathcal{L}_{\bar{w}}^{-1}+f\cdot\bar{\mathcal{N}}\mathcal{L}_{\bar{w}}^{-2}+\mathrm{right-
movers}\big{\\}}.$ (74)

The BRST currents can be easily derived through the Noether procedure, and the
left-moving component is

$\displaystyle j_{w}^{B}=$
$\displaystyle\oint_{\mathcal{C}_{0}}du\;\mathrm{sTr}\big{(}\mathcal{F}\tilde{C}-\frac{1}{2}B\\{\tilde{C},\tilde{C}\\}\big{)}$
$\displaystyle=$ $\displaystyle
l_{\alpha}Z^{\alpha}+f_{I}\mathcal{N}^{I}_{a}c^{a}-\frac{1}{2}y_{\alpha}\\{\tilde{C},\tilde{C}\\}_{-1}^{\alpha}-\frac{1}{2}b_{I}\bar{\mathcal{N}}^{I}_{a}\\{\tilde{C},\tilde{C}\\}^{a}_{-2}$
(75)

where $\\{\tilde{C},\tilde{C}\\}_{k}$ denotes the $k$-th Laurent coefficient
of $\\{\tilde{C},\tilde{C}\\}$. Note that

$\displaystyle\\{\tilde{C},\tilde{C}\\}_{-1}=-\\{c,\hat{Z}\\}+\;\mathrm{higher}\;\mathrm{pole}\;\mathrm{contributions}$
(76)
$\displaystyle\\{\tilde{C},\tilde{C}\\}_{-2}=-\frac{1}{2}\\{Z,Z\\}-\frac{1}{2}\\{\hat{Z},\hat{Z}\\}+\;\mathrm{higher}\;\mathrm{pole}\;\mathrm{contributions}$
(77)

The left-moving BRST current is then

$\displaystyle
j_{w}^{B}=l_{\alpha}Z^{\alpha}+f_{I}\mathcal{N}^{I}_{a}c^{a}+\frac{1}{4}y_{\alpha}(\eta\gamma_{a})^{\alpha}_{\hat{\beta}}c^{a}\hat{Z}^{\hat{\beta}}-\frac{1}{2}b_{I}$
$\displaystyle\bar{\mathcal{N}}^{I}_{a}\eta^{ab}\big{(}\mathcal{N}^{J}_{b}\Phi_{J}(Z)+\bar{\mathcal{N}}^{J}_{b}\Phi_{J}(\hat{Z})\big{)}$
$\displaystyle+\;\mathrm{higher}\;\mathrm{pole}\;\mathrm{contributions}.$ (78)

Then using (116) for $\bar{\mathcal{N}}$ yields

$\displaystyle j_{w}^{B}=l_{\alpha}Z^{\alpha}+f_{I}\mathcal{N}^{I}_{a}c^{a}+$
$\displaystyle\frac{1}{4}y_{\alpha}(\eta\gamma_{a})^{\alpha}_{\hat{\beta}}c^{a}\hat{Z}^{\hat{\beta}}-\frac{1}{2}b_{I}\bar{\mathcal{N}}^{I}_{a}\eta^{ab}\mathcal{N}^{J}_{b}\Phi_{J}(Z)+\;\mathrm{higher}\;\mathrm{pole}\;\mathrm{contributions}.$
(79)

This is the BRST current of berkovits_simplifying_2009  plus contributions
from higher-order poles in $\tilde{C}$.

Following berkovits_simplifying_2009 , this BRST charge is an $\ads$
generalization of the one in aisaka_new_2003 . Thus, using similar arguments
to those of aisaka_new_2003 , it should be possible to show that its
cohomology is equivalent to the cohomology of the charge

$\displaystyle Q_{\lambda,\hat{\lambda}}=\oint
dw\;l_{\alpha}\lambda^{\alpha}+\oint
d\bar{w}\hat{l}_{\hat{\alpha}}\hat{\lambda}^{\hat{\alpha}},$ (80)

where the ghost system is now composed of a pair of pure spinors
($\lambda$,$\hat{\lambda}$) with the associated Faddeev-Popov action

$S_{\lambda,\hat{\lambda}}=\oint_{\mathcal{C}_{0}}du\int_{\Sigma}d^{2}w\;Q\cdot\mathrm{sTr}\big{(}B\mathcal{L}_{\bar{w}}\big{)}+\oint_{\mathcal{C}_{\infty}}du\int_{\Sigma}d^{2}w\;Q\cdot\mathrm{sTr}\big{(}\bar{B}\mathcal{L}_{w}\big{)}$
(81)

With this new BRST charge, the BRST transformation of the Lax connection is

$Q\cdot\mathcal{L}=\nabla_{\Sigma}\big{(}\frac{1}{u}\lambda+u\hat{\lambda}\big{)}$
(82)

which can be verified a posteriori through the Noether procedure. Thus, the
action for the pure spinor ghosts is

$S_{\lambda,\hat{\lambda}}=-\int_{\Sigma}d^{2}w\;\mathrm{sTr}\big{(}y\nabla^{0}_{\bar{w}}\lambda+\hat{y}\nabla_{w}^{0}\hat{\lambda}-l\mathcal{L}_{\bar{w}}^{-1}-\hat{l}\mathcal{L}_{w}^{1}\big{)}$
(83)

where we already used the multipliers ($f_{I}$,$\bar{f}_{I}$) to impose the
conditions (60). Moreover, note that since the BRST transformation of
$\mathcal{L}$ no longer vanishes at $u^{4}=1$, we also need to define $Q\cdot
g=g(\lambda+\hat{\lambda})$ to respect the boundary conditions of (47).

##  4 The 2D sigma model from 4D action

The gauge-fixed action is then obtained by the sum of (52) with the Faddeev-
Popov ghosts and the contributions from the Lagrangian multipliers

$\displaystyle S=$ $\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}+\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\hat{g}^{-1}d\hat{g}\big{)}+$
$\displaystyle+\frac{1}{6\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\big{)}+$
$\displaystyle+\int_{\Sigma}d^{2}w\;\mathrm{sTr}\big{(}y\nabla^{0}_{\bar{w}}\lambda+\hat{y}\nabla_{w}^{0}\hat{\lambda}+l\mathcal{L}_{\bar{w}}^{-1}+\hat{l}\mathcal{L}^{1}_{w}\big{)}$
(84)

where we redefined $\lambda\to-\lambda$ and $\hat{\lambda}\to-\hat{\lambda}$.

We shall now vary the action with respect to $\mathcal{L}$ and use the
resulting equations of motion, gauge fixing conditions and boundary conditions
$\mathcal{L}(p_{k})=\rho^{k}\cdot J$ to solve for $\mathcal{L}$ in terms of
$J$ and the ghosts. We then plug $\mathcal{L}$ back in (84) to obtain the
action for the associated 2D integrable field theory. As we will see, the
resulting action and Lax connection describe the pure spinor superstring in
$\ads$. The resulting model is BRST invariant and we show that the BRST charge
(80) coincides with the usual one in the pure spinor formalism by writing the
Lagrange multipliers ($l_{\alpha},\hat{l}_{\hat{\alpha}}$) in terms of
physical fields.

###  4.1 Equations of motion and Lax connection

First of all, note that the WZ-term in the second line of (84) vanishes for
the 1-form (43) since for archipelago-like $\hat{g}$

$\displaystyle\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\wedge\hat{g}^{-1}d\hat{g}\big{)}\propto\sum_{k}\mathrm{Res}{(\phi,p_{k})}\int_{\Sigma\times\mathcal{A}_{k}}\mathrm{sTr}\big{(}J\wedge
J\wedge J\big{)}$ (85)

where $\mathcal{A}_{k}$ denotes the small annulus around $p_{k}$. As $\phi(u)$
only has double poles, all terms on the right-hand side of (85) are zero.
Moreover, the second term in (84) becomes

$\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\hat{g}^{-1}d\hat{g}\big{)}=\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\rho^{k}\cdot(g^{-1}dg)\big{)}.$
(86)

A special feature of (84) is that the ghosts only couple to $\mathcal{L}$ at
the points $u=\\{0,\infty\\}$, which means that they “source” higher poles for
the Lax connection at these points. To see this, let’s vary the action with
respect to $\mathcal{L}$

$\displaystyle\delta S=$ $\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{\\{}\delta\mathcal{L}\wedge\bar{d}\mathcal{L}+\mathcal{L}\wedge\bar{d}(\delta\mathcal{L})\big{\\}}+\int_{\mathcal{M}}d^{2}wd^{2}u\;\mathrm{sTr}\big{(}\delta_{u,0}N\delta\mathcal{L}_{\bar{w}}^{(0)}+\delta_{u,\infty}\hat{N}\delta\mathcal{L}^{(0)}_{w}\big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\delta\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}+\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\delta\mathcal{L}\big{)}+$
$\displaystyle+\int_{\mathcal{M}}d^{2}wd^{2}u\;\mathrm{sTr}\big{(}\delta_{u,0}N\delta\mathcal{L}_{\bar{w}}^{(0)}+\delta_{u,\infty}\hat{N}\delta\mathcal{L}^{(0)}_{w}\big{)}$
$\displaystyle=$ $\displaystyle\frac{1}{\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\delta\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}+\int_{\mathcal{M}}d^{2}wd^{2}u\;\mathrm{sTr}\big{(}\delta_{u,0}N\delta\mathcal{L}_{\bar{w}}^{(0)}+\delta_{u,\infty}\hat{N}\delta\mathcal{L}^{(0)}_{w}\big{)}.$
(87)

The boundary terms cancel due to the boundary condition
$\mathcal{L}(p_{k})=\rho^{k}\cdot J$ and $N$ and $\hat{N}$ are defined in
(26). Therefore, the equations of motion for $\mathcal{L}$ are

$\displaystyle\frac{1}{\pi
i}\phi(u)\partial_{\bar{u}}\mathcal{L}_{w}=-N\delta^{2}(u)$ (88)
$\displaystyle\frac{1}{\pi
i}\phi(\xi)\partial_{\bar{\xi}}\mathcal{L}_{\bar{w}}=-\hat{N}\delta^{2}(\xi)$
(89)

where we used the coordinates $\xi=\frac{1}{u}$ centered at infinity to write
the e.o.m. for $\mathcal{L}_{\bar{w}}$. The e.o.m. imply that the fourth-order
poles in $\mathcal{L}$ have the coefficients

$\displaystyle\mathcal{L}_{w}^{-4}=\frac{N}{2}$ (90)
$\displaystyle\mathcal{L}_{\bar{w}}^{4}=\frac{\hat{N}}{2}$ (91)

Now, joining the e.o.m., the boundary conditions, the gauge fixing conditions
and the $\rho$-constraint, $\mathcal{L}$ is

$\displaystyle\mathcal{L}_{w}=J_{w}^{(0)}+u^{-1}J_{w}^{(1)}+u^{-2}J_{w}^{(2)}+u^{-3}J_{w}^{(3)}-(1-u^{-4})\frac{N}{2}$
(92)
$\displaystyle\mathcal{L}_{\bar{w}}=J_{\bar{w}}^{(0)}+uJ_{\bar{w}}^{(3)}+u^{2}J_{\bar{w}}^{(2)}+u^{3}J_{\bar{w}}^{(1)}-(1-u^{4})\frac{\hat{N}}{2},$
(93)

This is, indeed, the Lax connection for the pure spinor formalism in $\ads$
magro_review_2012 . Note that after gauge-fixing, this is the unique solution
with poles up to third order in the Lax connection (not counting the fourth-
order pole sourced by the ghosts).

###  4.2 2D Action

Let us now show that, as expected, the 2D action associated with the solution
(92, 93) is the usual pure spinor superstring action. To do this, just plug
(92) and (93) in (84). Let’s do this term by term. Using the identity
$\partial_{\bar{u}}\big{(}\frac{1}{u^{n}}\big{)}=(-1)^{n}\frac{2\pi
i}{(n-1)!}\partial_{u}^{(n-1)}\delta(u)$, the first term yields

$\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}=$
$\displaystyle\frac{1}{2\pi i}\bigg{\\{}\int_{\mathcal{M}}d^{2}\xi
d^{2}w\phi(\xi)\;\mathrm{sTr}\bigg{[}\big{(}J_{w}^{(0)}-\frac{N}{2}\big{)}\partial_{\bar{\xi}}(\xi^{-4})\frac{\hat{N}}{2}\bigg{]}+$
$\displaystyle+\int_{\mathcal{M}}d^{2}ud^{2}w\phi(u)\;\mathrm{sTr}\bigg{[}\big{(}J_{\bar{w}}^{(0)}-\frac{\hat{N}}{2}\big{)}\partial_{\bar{u}}(u^{-4})\frac{N}{2}\bigg{]}\bigg{\\}}$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\bigg{\\{}\int_{\mathcal{M}}d^{2}\xi
d^{2}w\phi(\xi)\;\mathrm{sTr}\bigg{[}\big{(}J_{w}^{(0)}-\frac{N}{2}\big{)}\frac{\pi
i}{3}\partial^{3}_{\xi}\delta(\xi)\frac{\hat{N}}{2}\bigg{]}+$
$\displaystyle+\int_{\mathcal{M}}d^{2}ud^{2}w\phi(u)\;\mathrm{sTr}\bigg{[}\big{(}J_{\bar{w}}^{(0)}-\frac{\hat{N}}{2}\big{)}\frac{\pi
i}{3}\partial^{3}_{u}\delta(u)\frac{N}{2}\bigg{]}\bigg{\\}}$ $\displaystyle=$
$\displaystyle-\frac{1}{6}\bigg{\\{}\int_{\mathcal{M}}d^{2}\xi
d^{2}w\partial^{3}_{\xi}\phi(\xi)\;\mathrm{sTr}\bigg{[}\big{(}J_{w}^{(0)}-\frac{N}{2}\big{)}\delta(u)\frac{\hat{N}}{2}\bigg{]}+$
$\displaystyle+\int_{\mathcal{M}}d^{2}ud^{2}w\partial^{3}_{u}\phi(u)\;\mathrm{sTr}\bigg{[}\big{(}J_{\bar{w}}^{(0)}-\frac{\hat{N}}{2}\big{)}\delta(u)\frac{N}{2}\bigg{]}\bigg{\\}}$
(94)

Integrating along $\mathds{CP}^{1}$ with the delta functions gives

$\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}=-\int_{\Sigma}\;d^{2}w\mathrm{sTr}\big{(}\frac{J^{(0)}_{w}\hat{N}}{2}+\frac{J^{(0)}_{\bar{w}}N}{2}-\frac{N\hat{N}}{2}\big{)}$
(95)

The second term can be directly computed to yield

$\displaystyle\frac{1}{2\pi
i}\sum_{k}\oint_{\mathcal{C}_{k}}\omega(u)\int_{\Sigma}\mathrm{sTr}\big{(}\mathcal{L}\wedge\rho^{k}\cdot
J\big{)}=$ $\displaystyle\frac{1}{2\pi
i}\sum_{k}\sum_{r=0}^{4}\oint_{\mathcal{C}_{k}}\omega(u)e^{-\frac{ikr\pi}{2}}\bigg{\\{}\frac{1}{u^{r}}\int
d^{2}w\;\mathrm{sTr}\big{(}\mathcal{L}_{w}^{-r}J^{(4-r)}_{\bar{w}}\big{)}-u^{4-r}\int
d^{2}w\;\mathrm{sTr}\big{(}\mathcal{L}_{\bar{w}}^{4-r}J^{(4-r)}_{w}\big{)}\bigg{\\}}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\int
d^{2}w\;\mathrm{sTr}\big{(}J_{\bar{w}}^{(2)}J_{w}^{(2)}+\frac{3}{2}J_{w}^{(3)}J_{\bar{w}}^{(1)}+\frac{1}{2}J_{w}^{(1)}J_{\bar{w}}^{(3)}+\frac{\hat{N}J^{(0)}_{w}}{2}+\frac{NJ^{(0)}_{\bar{w}}}{2}\big{)}.$
(96)

The third term only depends on the regular part of the Lax connection and is

$\displaystyle\int d^{2}w\;$
$\displaystyle\mathrm{sTr}\big{(}y\partial_{\bar{w}}\lambda+Y\commutator{J_{\bar{w}}^{(0)}-\frac{\hat{N}}{2}}{\lambda}+\hat{y}\partial_{w}\hat{\lambda}+\hat{y}\commutator{J_{w}^{(0)}-\frac{N}{2}}{\hat{\lambda}}\big{)}=$
(97) $\displaystyle=$ $\displaystyle\int
d^{2}w\;\mathrm{sTr}\big{(}y\nabla_{\bar{w}}^{(0)}\lambda+\hat{y}\nabla_{w}^{(0)}\hat{\lambda}-N\hat{N}\big{)}$
(98)

where

$\displaystyle\nabla^{(0)}=d+\commutator{J^{(0)}}{\cdot}$ (99)

Summing all contributions, the action is

$\displaystyle\int
d^{2}w\;\mathrm{sTr}\big{(}J_{\bar{w}}^{(2)}J_{w}^{(2)}+\frac{3}{2}J_{w}^{(3)}J_{\bar{w}}^{(1)}+\frac{1}{2}J_{w}^{(1)}J_{\bar{w}}^{(3)}+y\nabla^{(0)}_{\bar{w}}\lambda+\hat{y}\nabla^{(0)}_{w}\hat{\lambda}-\frac{N\hat{N}}{2}\big{)}$
(100)

which is the action for the $\ads$ pure spinor superstring (up to an
irrelevant overall factor) if we redefine $\lambda\mapsto 2\lambda$ and
$\hat{\lambda}\mapsto 2\hat{\lambda}$.

###  4.3 Lagrange multipliers and the BRST charge

The BRST charge obtained in section 3 is

$\displaystyle Q_{\lambda,\hat{\lambda}}=\oint
dw\;\mathrm{sTr}\big{(}l\lambda\big{)}+\oint
d\bar{w}\;\mathrm{sTr}(\hat{l}\hat{\lambda})$ (101)

Since we only need $f_{0}$ and $\bar{f}_{0}$, use the multipliers with $k>0$
to impose the associated gauge fixing conditions. Then, the relevant terms in
(84) for the computation of ($f_{0},\;\bar{f}_{0}$) are

$\displaystyle\frac{1}{2\pi
i}\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}+\int_{\Sigma}d^{2}w\mathrm{sTr}\big{(}l\mathcal{L}_{\bar{w}}^{-1}+\
\hat{l}\mathcal{L}_{w}^{1}\big{)}$ (102)

Since the multipliers couple to Laurent modes of $\mathcal{L}$, they are
related to other modes through equations of motion. As $\mathcal{L}$ is
meromorphic on-shell, let’s perform a Laurent expansion on $\mathcal{L}$ in
the first term of (102). In particular, we are interested in the terms for
which either $\mathcal{L}^{-1}_{\bar{w}}$ or $\mathcal{L}_{w}^{1}$ appear.
They are

$\displaystyle\int_{\mathcal{M}}\omega\wedge\mathrm{sTr}\big{(}\mathcal{L}\wedge\bar{d}\mathcal{L}\big{)}$
$\displaystyle=-4\pi
i\int_{\mathcal{M}}d^{2}ud^{2}w\;\mathrm{sTr}\big{(}\delta_{u,0}\mathcal{L}_{w}^{-3}\mathcal{L}_{\bar{w}}^{-1}-\delta_{u,\infty}\mathcal{L}_{\bar{w}}^{3}\mathcal{L}_{w}^{1}+\ldots\big{)}$
$\displaystyle=-4\pi
i\int_{\Sigma}d^{2}w\;\mathrm{sTr}\big{(}\mathcal{L}_{w}^{-3}\mathcal{L}_{\bar{w}}^{-1}-\mathcal{L}_{\bar{w}}^{3}\mathcal{L}_{w}^{1}+\ldots\big{)}$
(103)

And so varying with respect to $\mathcal{L}_{\bar{w}}^{-1}$ and
$\mathcal{L}_{w}^{1}$ in (102) gives

$\displaystyle f_{0}=\frac{1}{2}\mathcal{L}_{w}^{-3}$ (104)
$\displaystyle\bar{f}_{0}=-\frac{1}{2}\mathcal{L}_{\bar{w}}^{3}.$ (105)

Then, from the solution (92, 93) $\mathcal{L}^{-3}_{w}=J_{w}^{(3)}$ and
$\mathcal{L}^{3}_{\bar{w}}=J_{\bar{w}}^{(1)}$. Thus, the BRST charge is

$\displaystyle Q_{\lambda,\hat{\lambda}}=\frac{1}{2}\oint
dw\;\mathrm{sTr}\big{(}J^{(3)}_{w}\lambda\big{)}-\frac{1}{2}\oint
d\bar{w}\;\mathrm{sTr}\big{(}J_{\bar{w}}^{(1)}\hat{\lambda}\big{)},$ (106)

Performing again the redefinition $\lambda\mapsto 2\lambda$ and
$\hat{\lambda}\mapsto 2\hat{\lambda}$ we obtain the correct BRST charge for
the pure spinor superstring in $\ads$ (33).

###### Acknowledgements.

NB would like to thank Kevin Costello and Masahito Yamazaki for useful
discussions, and FAPESP grants 2021/14335-0, 2019/21281-4, 2019/24277-8 and
CNPq grant 311434/2020-7 for partial financial support. RP would like to thank
João Gomide, Eggon Viana, and Lucas N.S. Martins for useful discussions, and
FAPESP grant 2022/05236-1 for partial financial support.

##  Appendix A Pure spinors and the matrices ($\mathcal{N}$,
$\bar{\mathcal{N}}$)

A ten-dimensional pure spinor $\lambda^{\alpha}$ is defined to satisfy

$\lambda^{\alpha}\gamma^{a}_{\alpha\beta}\lambda^{\beta}=0$ (107)

Another useful way to write (107) is by defining the $\mathfrak{g}$-valued
object $\lambda:=\lambda^{\alpha}Q_{\alpha}$. Then, from (3), the pure spinor
constraint can also be written as

$\anticommutator{\lambda}{\lambda}=0$ (108)

Although not apparent, there are only 5 independent constraints in (107). This
can be seen by breaking the $\mathrm{SO}(10)$ chiral spinor in terms of
representations of $\mathrm{U}(5)$ as

$\displaystyle\mathbf{16}\to(\mathbf{1},\mathbf{10},\mathbf{\bar{5}})$ (109)
$\displaystyle\lambda^{\alpha}\to(\lambda_{+},\lambda_{[ab]},\lambda^{a})\;\;\;a=1,\ldots,5.$
(110)

In this language, the pure spinor constraint reads

$\lambda_{+}\lambda^{a}-\frac{1}{8}\epsilon^{abcde}\lambda_{bc}\lambda_{de}=0$
(111)

which proves that there are only 5 independent constraints. Now, decomposing
$(Z\gamma^{a}Z)$ in terms of $\mathrm{SO}(4,1)\times\mathrm{SO}(5)$ yields

$\Phi_{I}(Z):=(Z\gamma_{I}Z)\;\;\;\;\Psi_{\tilde{I}}:=(Z\gamma_{\tilde{I}}Z)\;\;\;I=0,\ldots
4\;\;\;\;\tilde{I}=5\ldots{9}$ (112)

Since the pure spinor constraint has 5 independent components

$\Phi_{I}(Z)=0\iff\Psi_{\tilde{I}}(Z)=0$ (113)

and $Z^{\alpha}$ is a pure spinor if $\Phi_{I}(Z)=0$. Moreover, (113) implies
that there exists an invertible matrix $M_{\tilde{J}}^{I}$ such that

$\Psi_{\tilde{J}}(Z)=M_{\tilde{J}}^{I}(Z)\Phi_{I}(Z).$ (114)

We can then define the matrix $\mathcal{N}_{I}^{a}(Z)$ which satisfies

$Z\gamma_{a}Z=\mathcal{N}^{I}_{a}(Z)\Phi_{I}(Z)$ (115)

where $\mathcal{N}_{a}^{I}=\delta_{a}^{I}$ for $a=0,\ldots,4$ and
$\mathcal{N}_{a}^{I}=M_{a}^{I}$ for $a=5,\ldots,9$. Since
$(Z\gamma^{a}Z)(Z\gamma_{a}Z)=0$, $\mathcal{N}$ satisfies

$\eta^{ab}\mathcal{N}^{I}_{a}\mathcal{N}^{J}_{b}=0.$ (116)

This discussion also holds for right-moving spinors $\hat{Z}^{\hat{\alpha}}$,
which allows us to define $\bar{\mathcal{N}}^{I}_{a}(\hat{Z})$ satisfying
analogous identities. Assuming that the $10\times 10$ matrix obtained by
joining ($\mathcal{N}_{a}^{I}$, $\bar{\mathcal{N}_{b}^{J}}$) is invertible, we
can decompose a generic $\mathrm{SO}(1,9)$ vector $v_{a}$ as

$v_{a}=\mathcal{N}^{I}_{a}\upsilon_{I}+\bar{\mathcal{N}}^{I}_{a}\bar{\upsilon}_{I}$
(117)

where ($\upsilon^{I}$,$\bar{\upsilon}^{I}$) is a pair of $\mathrm{SO}(5)$
vectors. In particular, note that due to (116)

$\displaystyle\mathcal{N}_{a}^{I}v^{a}=0\implies
v_{a}=\mathcal{N}^{I}_{a}\upsilon_{I}$ (118)
$\displaystyle\bar{\mathcal{N}}^{I}_{a}v^{a}=0\implies
v_{a}=\bar{\mathcal{N}}^{I}_{a}\bar{\upsilon}_{I}$ (119)

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# DTW at Qur’an QA 2022: Utilising Transfer Learning with Transformers for
Question Answering in a Low-resource Domain

###### Abstract

The task of machine reading comprehension (MRC) is a useful benchmark to
evaluate the natural language understanding of machines. It has gained
popularity in the natural language processing (NLP) field mainly due to the
large number of datasets released for many languages. However, the research in
MRC has been understudied in several domains, including religious texts. The
goal of the Qur’an QA 2022 shared task is to fill this gap by producing state-
of-the-art question answering and reading comprehension research on Qur’an.
This paper describes the DTW entry to the Quran QA 2022 shared task. Our
methodology uses transfer learning to take advantage of available Arabic MRC
data. We further improve the results using various ensemble learning
strategies. Our approach provided a partial Reciprocal Rank (pRR) score of
0.49 on the test set, proving its strong performance on the task.

Keywords: Machine Reading Comprehension, Transformers, Transfer Learning,
Ensemble Learning, Qur’an

languageresourceLanguage Resources

DTW at Qur’an QA 2022: Utilising Transfer Learning with Transformers for
Question Answering in a Low-resource Domain

Damith Premasiri1, Tharindu Ranasinghe1, Wajdi Zaghouani2, Ruslan Mitkov1
---
1University of Wolverhampton, UK
2Hamad Bin Khalifa University, Qatar
{damith.premasiri, tharindu.ranasinghe<EMAIL_ADDRESS>
<EMAIL_ADDRESS>

Abstract content

## 1\. Introduction

Machine Reading Comprehension (MRC) is a challenging Natural Language
Processing (NLP) application [Baradaran et al., 2022]. The concept of MRC is
similar to how humans are evaluated in examinations where a person should
understand the text and answer questions based on the text. Similarly, the
goal of a typical MRC task requires a machine to read a set of text passages
and then answer questions about the passages. MRC systems could be widely
applied in many NLP systems such as search engines and dialogue systems.
Therefore, the NLP community has shown a great interest in MRC tasks over
recent years.

The most common way of dealing with MRC tasks is to train a machine learning
model on an annotated dataset. Over the years, researchers have experimented
with different machine learning approaches ranging from traditional algorithms
such as support vector machines [Suzuki et al., 2002, Yen et al., 2013] to
embedding based neural approaches such as transformers, with the latter
providing state-of-the-art results in many datasets. We discuss them
thoroughly in Section 2. However, an annotated dataset is an essential
requirement for these machine learning models. Identifying this, the NLP
community has developed several datasets in recent years. The most popular MRC
dataset is the Stanford Question Answering Dataset (SQuAD), which contains
more than 100,000 annotated examples [Rajpurkar et al., 2016]. SQuAD dataset
has been extended to several languages including Arabic [Mozannar et al.,
2019], Dutch [Rouws et al., 2022], Persian [Abadani et al., 2021] and Sinhala
[Jayakody et al., 2016]. However, MRC datasets have been limited to common
domains such as Wikipedia and MRC in low-resource domains, including religious
books, have not been explored widely by the community [Baradaran et al.,
2022]. Moreover, most researchers focus on a few popular MRC datasets, while
most other MRC datasets are not widely known and studied by the community
[Zeng et al., 2020].

Qur’an QA 2022 shared task[Malhas et al., 2022] has been organised to address
these gaps in MRC research. The goal of the shared task is to trigger state-
of-the-art question answering and reading comprehension research on a book
that is sacredly held by more than 1.8 billion people across the world. The
shared task relies on a recently released dataset of 1,337 question-passage-
answer triplets extracted from the holy Qur’an [Malhas and Elsayed, 2020].
Despite the novelty, the dataset poses several challenges. Firstly, since the
dataset contains texts from Qur’an, modern embedding models would have
problems encoding them. Therefore, we experimented with different pre-
processing techniques to handle the texts from Qur’an. Secondly, the dataset
is small compared to other MRC datasets such as SQuAD [Rajpurkar et al.,
2016], and it would be difficult to fine-tune the state-of-the-art neural
models. We experiment with different techniques such as transfer learning and
ensemble learning to overcome this. We show that state-of-the-art neural
models can be applied in smaller MRC datasets utilising the above methods.

We address two research questions in this paper:

RQ1: Do ensemble models provide better results compared to single models?

RQ2: Can other Arabic MRC resources such as SOQAL [Mozannar et al., 2019] be
used to improve the results for Qur’an MRC?

The code of the experiments has been released as an open-source Github
project111The Github project is available on
https://github.com/DamithDR/QuestionAnswering. The project has been released
as a Python package222The Python package is available on
https://pypi.org/project/quesans/ and the pre-trained machine learning models
are freely available to download in HuggingFace model hub333The pre-trained
models are available on https://huggingface.co/Damith/AraELECTRA-
discriminator-SOQAL and https://huggingface.co/Damith/AraELECTRA-
discriminator-QuranQA. Furthermore, we have created a docker image of the
experiments adhering to the ACL reproducibility criteria444The docker image is
available on https://hub.docker.com/r/damithpremasiri/question-answering-
quran.

The rest of the paper is structured as follows. Section 2 presents an overview
of MRC datasets and machine learning models. Section 3 describes the data we
used in the experiments. In Section 4 we explain the experiments carried out.
Section 5 discusses the results answering the research questions. Finally, the
paper outlines future works and provides conclusions.

## 2\. Related Work

Machine reading comprehension is not newly proposed. The earliest known MRC
system dates back to 1977 when [Lehnert, 1977] developed a question answering
program called the QUALM. In 1999 [Hirschman et al., 1999] constructed a
reading comprehension system exploiting a corpus of 60 development and 60 test
stories of 3rd to 6th-grade material. Due to the lack of high-quality MRC
datasets and the poor performance of MRC models, this research field was
understudied until the early 2010s. However, with the creation of large MRC
datasets and with the success of word embedding based neural models in the NLP
field, research in MRC has been popular in recent years. We present the
related work in MRC in two broad categories; datasets and models.

#### Datasets

In 2013, [Richardson et al., 2013] created the MCTest dataset which contained
500 stories and 2000 questions. This dataset can be considered the first big
MRC dataset. A breakthrough in MRC was achieved in 2015 when [Hermann et al.,
2015] defined a new dataset generation method that provides large-scale
supervised reading comprehension datasets. This was followed by the creation
of large scale MRC datasets such as SQuAD[Rajpurkar et al., 2016]. Later the
SQuAD dataset has been expanded to many languages including Arabic [Mozannar
et al., 2019], Dutch [Rouws et al., 2022], French [d’Hoffschmidt et al., 2020]
and Russian [Efimov et al., 2020]. Furthermore, SQuAD has been extended to
low-resource languages such as Persian [Abadani et al., 2021] and Sinhala
[Jayakody et al., 2016] proving that SQuAD has been an important benchmark in
MRC research. MRC datasets have been compiled on different domains such as
news [Trischler et al., 2017], publications [Dasigi et al., 2021] and natural
sciences [Welbl et al., 2017]. As far as we know, Qur’an Reading Comprehension
Dataset used in this shared task is the first dataset created on religious
texts [Malhas and Elsayed, 2020].

#### Methods

Most MRC systems in the early 2000s were rule-based or statistical models
[Riloff and Thelen, 2000, Charniak et al., 2000]. These models do not provide
good results compared to the neural methods introduced in recent years
[Baradaran et al., 2022]. [Hermann et al., 2015] developed a class of
attention based deep neural networks that learn to read real documents and
answer complex questions with minimal prior knowledge of language structure.
Since 2015, with the emergence of various large scale, supervised datasets,
neural network models have shown state-of-the-art results in MRC tasks. The
recently introduced transformer models such as BERT [Devlin et al., 2019] have
already exceeded human performance over the related MRC benchmark datasets
[Zeng et al., 2020]. A critical contribution of the SQuAD benchmark is that it
provides a system to submit the MRC models and a leaderboard to display the
top results555SQuAD leaderboard is available on
https://rajpurkar.github.io/SQuAD-explorer/. This has enabled the NLP
community to keep track of the state-of-the-art MRC systems. Other languages
have also followed this approach666Korean MRC leaderboard is available on
https://korquad.github.io/. However, the NLP community has focused mainly on
improving system performance on popular benchmarks such as SQuAD and has not
focused on improving results on benchmarks with limited coverage, which we
address in this research paper.

## 3\. Data

MRC tasks are usually divided into four categories: cloze style, multiple-
choice, span prediction, and free form [Liu et al., 2019]. The Qur’an QA 2022
shared task777More information on the Qur’an QA 2022 shared task is available
on https://sites.google.com/view/quran-qa-2022/ belongs to the span prediction
category where the MRC system needs to select the correct beginning and end of
the answer text from the context. The event organisers provided the QRCD
(Quran Reading Comprehension Dataset), which contained 1,093 tuples of
question-passage pairs that are coupled with their extracted answers to
constitute 1,337 question-passage-answer triplets. QRCD is a JSON Lines
(JSONL) file; each line is a JSON object that comprises a question-passage
pair and its answers extracted from the accompanying passage. Figure 1 shows a
sample training tuple. The distribution of the dataset into training,
development and test sets is shown in Table 1.

Figure 1: Sample Json object from the QRCD dataset [Malhas and Elsayed, 2020]

Dataset | % | Q-P Pairs | Q-P-A Triplets
---|---|---|---
Training | 65% | 710 | 861
Development | 10% | 109 | 128
Test | 25% | 274 | 348
All | 100% | 1,093 | 1,337

Table 1: Shared Task Data Composition. Column Q-P Pairs shows the number of
Question Passage pairs, Column Q-P-A Triplets shows the number of Question
Passage Answer triplets in the dataset

#### SOQAL

contains two Arabic MRC datasets; Arabic Reading Comprehension Dataset (ARCD)
[Mozannar et al., 2019], composed of 1,395 questions posed by crowdworkers on
Wikipedia articles, and a machine translation of the SQuAD [Mozannar et al.,
2019] containing 48,344 questions. SQuAD is widely used as the standard
dataset in English MRC tasks, therefore using the machine translation of the
same dataset will be helpful for the learning process. Compared to QRCD, SOQAL
is a large dataset and both of these datasets belong to the span prediction
MRC category. Therefore, they can be used to perform transfer learning which
we describe in Section 4.

## 4\. Methodology

With the introduction of BERT [Devlin et al., 2019], transformer models have
achieved state-of-the-art results in different NLP applications such as text
classification [Ranasinghe and Hettiarachchi, 2020], information extraction
[Plum et al., 2022] and event detection [Giorgi et al., 2021]. Furthermore,
the transformer architectures have shown promising results in SQuAD dataset
[Zhang et al., 2021, Zhang et al., 2020, Yamada et al., 2020, Lan et al.,
2020]. In view of this, we use transformers as the basis of our methodology.
Transformer architectures have been trained on general tasks like language
modelling and then can be fine-tuned for MRC tasks. [Devlin et al., 2019]. For
the MRC task, transformer models take an input of a single sequence that
contains the question and paragraph separated by a [SEP] token. Then the model
introduces a start vector and an end vector. The probability of each word
being the start-word is calculated by taking a dot product between the final
embedding of the word and the start vector, followed by a softmax over all the
words. The word with the highest probability value is considered. The
architecture of transformer-based MRC model is shown in Figure 2.

Figure 2: Transformer Architecture for MRC

We experimented with seven popular pre-trained transformer models that
supports Arabic; camelbert-mix [Inoue et al., 2021], camelbert-ca [Inoue et
al., 2021], mbert-cased [Devlin et al., 2019] , mbert-uncased [Devlin et al.,
2019], AraELECTRA-generator [Antoun et al., 2021], AraELECTRA-discriminator
[Antoun et al., 2021] and AraBERTv2 [Antoun et al., 2020]. These models are
available in HuggingFace model hub [Wolf et al., 2020]. For all the
experiments we used a batch-size of eight, Adam optimiser with learning rate
$2\mathrm{e}{-5}$, and a linear learning rate warm-up over 10% of the training
data. During the training process, the parameters of the transformer model, as
well as the parameters of the subsequent layers, were updated. The models were
trained using only training data. All the models were trained for five epochs.
For some of the experiments, we used an Nvidia GeForce RTX 2070 GPU, whilst
for others we used a GeForce RTX 3090 GPU. This was purely based on the
availability of the hardware and it was not a methodological decision.

We further used following fine-tuning strategies to improve the performance.

### 4.1. Ensemble Learning

Ensemble learning is a popular technique in machine learning, where different
machine learning models contribute to a single solution. As different machine
learning algorithms tend to learn differently, the final predictions each one
of them provides can be slightly different. However, they have the potential
to contribute to the final output with ensemble learning. Usually, ensemble
learning provides better results compared to single models [Sagi and Rokach,
2018].

Transformer models that we used as the base model are prone to the random seed
[Hettiarachchi and Ranasinghe, 2020]. The same architecture can provide
different results for different random seeds [Uyangodage et al., 2021]. To
avoid the impact of this, we performed self ensemble. We trained the same
architecture using five different random seeds and ensembled the output files
using Algorithm 1.

Algorithm 1 Ensemble Learning Algorithm for MRC

$R\leftarrow\textit{all results files}$

$r_{i}\leftarrow\textit{i(th) result file}$

$Q\leftarrow\textit{all questions}$

$q_{j}\leftarrow\textit{j(th) question}$

$A\leftarrow\textit{all answers}$

$a_{i,j}\leftarrow\textit{answer for question j in files file i}$

$a_{j}\leftarrow\textit{all unique answers for question j in all files}$

$a_{j,k}\leftarrow\textit{answer k from unique answers for question j}$

$a_{i,j,m}\leftarrow\textit{answer m from file i to question j}$

$answer_{j,k}\leftarrow\textit{temporary score}$

repeat

for each $a_{j,k}\in a_{j}$ do

repeat

for each $a_{i,j,m}\in r_{i}$ do

if $a_{i,j,m}=a_{j,k}$ then

$score_{j,k}\leftarrow average(score_{a_{i,j,m}},score_{a_{j,k}})$

$answer_{j,k}\leftarrow a_{j,k},score_{j,k}$

end if

end for

until all items iterated in R

$\textit{final\\_answers}\leftarrow answer_{j,k}$

end for

until all unique answers iterated in for question j

$\textbf{answers}\leftarrow\textbf{sort(answers)}$

repeat

for each $q_{j}\in Q$ do

repeat

for each $answer_{j,k}\in final_{a}nswers$ do

$rank_{j,k}\leftarrow\textit{assign rank}$

end for

until iterate all answers for question j

end for

until iterate all questions iterated in Q

### 4.2. Transfer Learning

One limitation of the QRCD dataset is that training set only contains 710
annotated QnA pairs and as a result transformer models would find it difficult
to properly fine-tune their weights in the training process. A common practice
to overcome this is to utilise transfer learning. The main idea of transfer
learning is that we train a machine learning model on a resource rich setting,
save the weights of the model and when we initialise the training process for
a lower resource setting, start with the saved weights from the resource rich
setting . Transfer learning has improved results for many NLP tasks such as
offensive language identification [Ranasinghe and Zampieri, 2020], machine
translation [Nguyen and Chiang, 2017] and named entity recognition [Lee et
al., 2018].

For this task, we first trained a transformer-based MRC model on SOQAL dataset
which contained more training data compared to the QRCD dataset as mentioned
in Section 3. Then when we started the training for QRCD dataset we started
from the saved weights from the SOQAL dataset.

## 5\. Results and Discussion

In this section, we report the experiments we conducted and their results. As
advised by the task organisers, we used partial Reciprocal Rank (pRR) score to
measure the model performance. It is a variant of the traditional Reciprocal
Rank evaluation metric that considers partial matching. We also report Exact
Match (EM), and F1@1 in the results tables, which are evaluation metrics
applied only to the top predicted answer. The EM metric is a binary measure
that rewards a system only if the top predicted answer matches exactly one of
the gold answers. In comparison, the F1@1 metric measures the token overlap
between the top predicted answer and the best matching gold answer. The
reported results are for the dev set.

As can be seen in Table 2, camelbert-mix model produced the best results with
0.549 pRR value. This was closely followed by camelbert-ca and AraELECTRA-
discriminator. Transformer models built specifically for Arabic generally
outperformed multilingual models.

Model | pRR | EM | F1@1
---|---|---|---
AraELECTRA-discriminator | 0.516 | 0.303 | 0.495
AraELECTRA-generator | 0.355 | 0.339 | 0.324
camelbert-mix | 0.549 | 0.193 | 0.529
camelbert-ca | 0.535 | 0.119 | 0.516
mbert-cased | 0.425 | 0.321 | 0.405
mbert-uncased | 0.440 | 0.220 | 0.424
AraBERTv2 | 0.501 | 0.294 | 0.472

Table 2: Results of different transformer models without ensemble learning or
transfer-learning. Column pRR shows the partial Reciprocal Rank score, Column
EM shows results for exact match and Column F1@1 shows F1@1 score. The top
three results are highlighted in Bold.

To answer our RQ1, we performed self ensemble learning. Table 3 shows the
results of different models with results ensemble. Even though there was a
slight improvement in AraELECTRA-discriminator, the overall impact for the
results from the ensemble was very low. And we noticed that some of the models
had performed less when using ensemble. However, the results were stable
compared to the single models. Therefore, we used self ensemble learning even
though it did not contribute to improving the results. With these findings, we
answer our RQ1, ensemble models do not provide better results compared to
single models; however, they provide more consistent results.

Model | pRR | EM | F1@1
---|---|---|---
AraELECTRA-discriminator | 0.528 | 0.321 | 0.500
AraELECTRA-generator | 0.364 | 0.128 | 0.335
camelbert-mix | 0.520 | 0.303 | 0.497
camelbert-ca | 0.495 | 0.239 | 0.467
mbert-cased | 0.438 | 0.220 | 0.417
mbert-uncased | 0.424 | 0.220 | 0.399
AraBERTv2 | 0.475 | 0.239 | 0.436

Table 3: Results of different transformer models with self ensemble learning.
Column pRR shows the partial Reciprocal Rank score, Column EM shows results
for exact match and Column F1@1 shows F1@1 score. The top three results are
highlighted in Bold.

To answer our RQ2, we performed transfer learning from SOQAL [Mozannar et al.,
2019] to QRCD dataset as mentioned in Section 4. We only conducted the
experiments for the best model from the self ensemble setting. As can be seen
in the results in Table 4, transfer learning improved the results for
AraELECTRA-discriminator. Without transfer learning, AraELECTRA-discriminator
scored only 0.528 pRR, while with transfer learning, it provided 0.616 pRR. We
did not observe improvements in other transformer models. However, the 0.616
pRR we got with performing transfer learning with AraELECTRA-discriminator was
the best result for the dev set. With this, we answer our RQ2, other Arabic
MRC resources such as SOQAL [Mozannar et al., 2019] can be used to improve the
results for Qur’an MRC. We believe that this finding will be important to the
researchers working on low-resource MRC datasets.

Model | pRR | EM | F1@1
---|---|---|---
AraELECTRA-discriminator | 0.616 | 0.394 | 0.609
camelbert-mix | 0.520 | 0.284 | 0.494
AraBERTv2 | 0.430 | 0.138 | 0.412

Table 4: Results of different transformer models after transfer learning.
Column pRR shows the partial Reciprocal Rank score, Column EM shows results
for exact match and Column F1@1 shows F1@1 score.

Based on the results of the dev set, we selected three models for the final
submission; camelbert-mix with ensemble learning but without transfer
learning, camelbert-mix with transfer learning and ensemble learning and
AraELECTRA-discriminator with transfer learning and ensemble learning. Table 5
shows the results that the organisers provided on the test set for our
submitted models.

Model | TL | EN | pRR | EM | F1@1
---|---|---|---|---|---
camelbert-mix | ✗ | ✓ | 0.290 | 0.084 | 0.258
camelbert-mix | ✓ | ✓ | 0.408 | 0.138 | 0.390
AraELECTRA-discriminator | ✓ | ✓ | 0.495 | 0.226 | 0.476

Table 5: Results of different transformer models on the test set. Column TL
implies whether we performed transfer learning or not and the Column EN shows
whether we performed ensemble learning. Column pRR shows the partial
Reciprocal Rank score, Column EM shows results for exact match and Column F1@1
shows F1@1 score.

AraELECTRA-discriminator performed best in the test set too. The camelbert-mix
mode without transfer learning has decreased its performance from 0.549 to
0.290, which is a 47% decrease. However, the models with transfer learning
have performed comparatively high, confirming our answer to the RQ2.

## 6\. Conclusion

In this paper, we have presented the system submitted by the DTW team to the
Qur’an QA 2022 shared task in the 5th Workshop on Open-Source Arabic Corpora
and Processing Tools. We have shown that AraELECTRA-discriminator with
transfer learning from an Arabic MRC dataset is the most successful
transformer model from several transformer models we experimented with. Our
best system scored 0.495 pRR in the test set. With our RQ1, we showed that
transformer models based on self ensemble provided stable results than single
models in Qur’an QA task. Revisiting our RQ2, we showed that transfer learning
could be used to improve the MRC results of the Qur’an. We believe that this
finding would pave the way to enhance MRC in many low-resource domains. Our
code, software and the pre-trained models have been made available freely to
the researchers working on similar problems.

In future work, we would like to explore more to transfer learning. We will be
exploring cross-lingual transfer learning with larger English MRC datasets
such as SQuAD, as cross-lingual transfer learning has shown splendid results
in many NLP tasks [Ranasinghe et al., 2021]. Furthermore we will be exploring
zero-shot and few-shot learning, which could benefit a multitude of low-
resource languages.

## 7\. Acknowledgements

This project was partially funded by the University of Wolverhampton’s RIF4
Research Investment Funding provided for the Responsible Digital Humanities
lab (RIGHT).

We would like to thank the Qur’an QA 2022 shared task organisers for running
this interesting shared task and for replying promptly to all our inquiries.
Furthermore, we thank the anonymous OSACT 2022 reviewers for their insightful
feedback.

## 8\. References

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