# On Yang–Mills fields from anti-de Sitter spaces

Kaushlendra Kumar  Institute of Physics, Humboldt University, Zum Großen
Windkanal 2, 12489 Berlin<EMAIL_ADDRESS>
###### Abstract

Motivated by some recent progress involving a non-compact gauge group, we
obtain classical gauge fields using non-compact foliations of anti-de Sitter
space in $4$ dimensions (required dimensionality for conformal invariance of
Yang–Mills theory) and transfer these to Minkowski spacetime using a series of
conformal maps. This construction also builds upon some previous works
involving $\textrm{SU}(2)$ gauge group in that we now use its non-compact
counterpart $\textrm{SU}(1,1)$ here. We note down gauge fields in both Abelian
as well as non-Abelian settings and find them to be divergent at some
hyperboloid, which is a hypersurface of co-dimension $1$ inside the conformal
boundary of $\textrm{AdS}_{4}$. In spite of this hurdle we find a physically
relevant field configuration in the Abelian case, reproducing a known result.

##  1 Introduction

Vacuum Yang–Mills theory in $4$ dimensions is conformally invariant—a fact
that can be used to map their solutions to any conformally related
spacetime111It is a known fact that all three variants of FLRW spacetimes viz.
Minkowski $\mathds{R}^{1,3}$, de Sitter $\textrm{dS}_{4}$ and anti-de Sitter
$\textrm{AdS}_{4}$ are covered this way. after obtaining them on a more
suitable one equipped with symmetric-foliations, and therefore admitting a
natural gauge group. This exercise was carried out for $\textrm{dS}_{4}$ whose
foliation submanifold $S^{3}$ arises as group manifold of $\textrm{SU}(2)$;
the resulting $\textrm{SU}(2)$-equivariant connection ansatz reproduced [4] a
well known Yang–Mills solutions obtained in 1977 by Lüscher [8]. This was
achieved by following set of conformal correspondences: ${\cal I}\times
S^{3}\xleftarrow{\text{conformal}}\textrm{dS}_{4}\xrightarrow{\text{conformal}}\mathds{R}^{1,3}$
where the cylinder domain ${\cal
I}=(-{\textstyle\frac{\pi}{2}},{\textstyle\frac{\pi}{2}})$ needs to be doubled
to facilitate a gluing of two $\textrm{dS}_{4}$ copies so as to cover entire
Minkowski spacetime. Furthermore, solutions for its Abelian counterpart
$U(1)\subset\textrm{SU}(2)$ (in a non-symmetric setting) produces a family of
electromagnetic knotted fields [4, 5]. These EM knots have rather intriguing
physical features, many of which have been explored recently [6, 7]. Another
motivation for such an exploration of classical gauge fields is that only a
few of them are known in mathematical physics literature (see e.g. [1, 2, 3]
for reviews). To this end, novel results were obtained on de Sitter and anti-
de Sitter spaces in [10, 11] for $\textrm{SU}(2)$ as well as some higher-
dimensional generalizations (in de Sitter case) for $\textrm{SO}(4)$ in [12].

There has been some recent progress [13] towards obtaining gauge fields via
$H^{3}$ and $\textrm{dS}_{3}$ foliations of Minkowski space regions with non-
compact gauge group $\textrm{SO}(1,3)$, as opposed to compact ones discussed
above. Inspired by this success, we considered $\textrm{AdS}_{3}$ foliations
of $\textrm{AdS}_{4}$ since the former arises as group manifold of
$\textrm{SU}(1,1)$. Classical gauge fields were obtained here for
$\textrm{SU}(1,1)$-symmetric gauge configuration [14] (of both Abelian and
non-Abelian type) by employing following strategy: ${\cal
I}\times\textrm{AdS}_{3}\xleftarrow{\text{conformal}}\textrm{AdS}_{4}\xrightarrow{\text{conformal}}{\cal
I}\times S^{3}_{+}$ followed by the previous ${\cal I}\times
S^{3}\xrightarrow{\text{conformal}}\mathds{R}^{1,3}$ after gluing two
$\textrm{AdS}_{4}$ copies to recover full $S^{3}$. However, unlike the
previous compact case, this gluing is not smooth and leads to a singular
conformal boundary; this feature propagates to the corresponding gauge fields
but in a milder fashion. We present such field configurations in a compact
form and compute their stress energy tensor. We also make use of several plots
to explain various features of these conformal maps and resulting field
configurations (such as orientation preserving gluing).

##  2 Geometrical toolkit for anti-de Sitter space $\mathrm{AdS}_{4}$

We start with the following isometric embedding of $\mathrm{AdS}_{4}$—endowed
with coordinates $(x^{1},x^{2},x^{3},x^{4},x^{5})$ and global radius
$R$—inside $\mathds{R}^{2,3}$ via

$\displaystyle-(x^{1})^{2}-(x^{2})^{2}+(x^{3})^{2}+(x^{4})^{2}+(x^{5})^{2}\ =\
-R^{2}\ .$ (1)

###  2.1 $\mathrm{AdS}_{4}$-foliations of the form ${\cal
I}\times\mathcal{M}_{3}$

First, we have $\mathcal{M}_{3}=\mathrm{AdS}_{3}$ foliation with a spacelike
parameter $\psi\in{\cal I}=(-\pi/2,\pi/2)$. This
$\mathrm{AdS}_{3}\hookrightarrow\mathds{R}^{2,2}$ can be described using
embedding coordinates $\alpha^{i}(\rho,\tau,\phi)$ for $i=1,\ldots,4$ with
spatial parameters $\rho\in\mathds{R}_{+}$, $\phi\in[0,2\pi]$ and temporal
parameter $\tau\in[0,2\pi]$ as

$\begin{aligned} \alpha^{1}&=\cosh{\rho}\,\cos{\tau},\
\alpha^{2}=\cosh{\rho}\,\sin{\tau},\\\ \alpha^{3}&=\sinh{\rho}\,\cos{\phi},\
\alpha^{4}=\sinh{\rho}\,\sin{\phi},\end{aligned}\quad\implies\quad-(\alpha^{1})^{2}-(\alpha^{2})^{2}+(\alpha^{3})^{2}+(\alpha^{4})^{2}=-1\
.$ (2)

The following global $\textrm{AdS}_{4}$ embedding coordinates,

$x^{i}\ =\ R\sec\chi\,\alpha^{i},\qquad x^{5}\ =\ R\tan\chi\ ,$ (3)

then yields its induced metric (arising from flat $\mathds{R}^{2,3}$ metric)
in terms of $\textrm{AdS}_{3}$ metric $\mathrm{d}\Omega^{2}_{1,2}$ as

$\displaystyle\mathrm{d}s^{2}\ =\
\frac{R^{2}}{\cos^{2}\\!{\psi}}\bigl{(}\mathrm{d}\psi^{2}-\cosh^{2}\\!\rho\,\mathrm{d}\tau^{2}+\mathrm{d}\rho^{2}+\sinh^{2}\\!\rho\,\mathrm{d}\phi^{2}\bigr{)}\
=\
\frac{R^{2}}{\cos^{2}\\!{\psi}}\bigl{(}\mathrm{d}\psi^{2}+\mathrm{d}\Omega^{2}_{1,2}\bigr{)}\
.$ (4)

Next, we have ${\cal M}_{3}{=}S^{3}_{+}$ (upper hemisphere) foliations,
embedded inside $\mathds{R}^{2,3}$ using

$\displaystyle x^{1}=R\sec\chi\cos\tau,\ x^{2}=R\sec\chi\sin\tau,\
x^{3}=R\tan\chi\,\beta^{1},\ x^{4}=R\tan\chi\,\beta^{2},\
x^{5}=R\tan\chi\,\beta^{3}\ ,$ (5)

where $\chi\in[0,\pi/2)$ for the half-sphere and canonical $S^{2}$ coordinates
$\beta^{i}(\theta,\phi)$ with $\theta\in[0,\pi]$ are

$\displaystyle\beta^{1}\ =\ \sin\theta\,\cos\phi\ ,\qquad\beta^{2}\ =\
\sin\theta\,\sin\phi\ ,\qquad\beta^{3}\ =\ \cos\theta\ .$ (6)

These are related to the coordinates ($\rho,\psi$) above (3) as follows,

$\tanh\rho\ =\ \sin\theta\,\sin\chi\quad\quad\textrm{and}\quad\quad\tan\psi\
=\ -\cos\theta\,\tan\chi\ .$ (7)

In this case, the induced metric demonstrates a $S^{3}_{+}$-cylinder structure
with round metric $\mathrm{d}\Omega^{2}_{3+}$; the latter can be expressed
using $S^{2}$ round metric $\mathrm{d}\Omega^{2}_{2}$ as

$\displaystyle\mathrm{d}s^{2}\ =\
\frac{R^{2}}{\cos^{2}\\!{\chi}}\bigl{(}-\mathrm{d}{\tau}^{2}+\mathrm{d}\chi^{2}+\sin^{2}\\!\chi\,\mathrm{d}\Omega^{2}_{2}\bigr{)}\
=\
\frac{R^{2}}{\cos^{2}\\!{\chi}}\bigl{(}-\mathrm{d}{\tau}^{2}+\mathrm{d}\Omega^{2}_{3+}\bigr{)}\
.$ (8)

This temporal parameter $\tau$ can be extended to full $\mathds{R}$ by going
to the universal cover $\widetilde{\textrm{AdS}_{4}}$.

###  2.2 Gluing of two $\mathrm{AdS}_{4}$ copies

We now proceed to glue two copies of $\mathrm{AdS}_{4}$ to recover full round
$3$-sphere, i.e. $S^{3}=S^{3}_{+}\cup S^{2}\cup S^{3}_{-}$ in order to apply
below-mentioned conformal map. To this end, we note down the following map
that glues northern copy $S^{3}_{+}$ with southern one $S^{3}_{-}$ along the
boundary $S^{2}$ at $\chi{=}\frac{\pi}{2}$:

$\displaystyle\tanh{\rho}\ =\
\varepsilon\,\sin{\theta}\,\sin{\chi}\quad\textrm{and}\quad\tan{\psi}\ =\
-\varepsilon\,\cos{\theta}\,\tan{\chi}\ ,\quad\textrm{with}\quad$ (9)
$\displaystyle\varepsilon|_{S^{3}_{+}}$ $\displaystyle=+1:\
\rho\in\mathds{R}_{+}\Leftrightarrow\chi\in[0,\tfrac{\pi}{2})\quad\textrm{and}\quad\varepsilon|_{S^{3}_{-}}=-1:\
\rho\in\mathds{R}_{-}\Leftrightarrow\chi\in(\tfrac{\pi}{2},\pi]\ .$

This map preserves the orientation along the gluing boundary $\partial
S^{3}_{\pm}{=}S^{2}$. To see this, we note down some key points in table 1 for
both coordinate systems and then plot these in spherical coordinates
$(\chi,\theta)$222This could also be demonstrated in other coordinate system
as well; see [14] for details. individually in figures 3 and 3 as well as in
combined fashion in figure 3. Notice in the figures that the open boundary of
the AdS space (for fixed $\tau$) is depicted with dashed lines while some
points (like the north pole) are identified as is clear from table 1. Finally,
the gluing happens by identifying the points $(P_{2},\ldots,P_{5})$ with
$(P_{2}^{\prime},\ldots,P_{5}^{\prime})$ pairwise so that same-colored
segments are coincide and the orientation remains preserved. Another point to
note here is that this gluing is not smooth, but consists of a singularity
arising from the conformal boundary $\chi=\pi/2$ as is clear from above metric
(8).

Table 1:  Key points on northern hemisphere $(\varepsilon{=}+1;P)$ and southern hemisphere $(\varepsilon{=}-1;P^{\prime})$ in two coordinate systems for some infinitesimal $\epsilon>0$. $S^{3}_{+}$ | $(\rho,\psi)_{+}$ | $(\chi,\theta)_{-}$ | $S^{3}_{-}$ | $(\rho,\psi)_{-}$ | $(\chi,\theta)_{-}$
---|---|---|---|---|---
$\mathrm{P}_{1}$ | $(0,0{-}\epsilon)$ | $(0,0)$ | $\mathrm{P}^{\prime}_{1}$ | $(0,0{-}\epsilon)$ | $(\pi,0)$
$\mathrm{P}_{2}$ | $(0,-\frac{\pi}{2})$ | $(\frac{\pi}{2},0)$ | $\mathrm{P}^{\prime}_{2}$ | $(0,-\frac{\pi}{2})$ | $(\frac{\pi}{2},0)$
$\mathrm{P}_{3}$ | $(\infty,-\frac{\pi}{2})$ | $(\frac{\pi}{2},\frac{\pi}{2}{-}\epsilon)$ | $\mathrm{P}^{\prime}_{3}$ | $(-\infty,-\frac{\pi}{2})$ | $(\frac{\pi}{2},\frac{\pi}{2}{-}\epsilon)$
$\mathrm{P}_{4}$ | $(\infty,\frac{\pi}{2})$ | $(\frac{\pi}{2},\frac{\pi}{2}{+}\epsilon)$ | $\mathrm{P}^{\prime}_{4}$ | $(-\infty,\frac{\pi}{2})$ | $(\frac{\pi}{2},\frac{\pi}{2}{+}\epsilon)$
$\mathrm{P}_{5}$ | $(0,\frac{\pi}{2})$ | $(\frac{\pi}{2},\pi)$ | $\mathrm{P}^{\prime}_{5}$ | $(0,\frac{\pi}{2})$ | $(\frac{\pi}{2},\pi)$
$\mathrm{P}_{6}$ | $(0,0{+}\epsilon)$ | $(0,\pi)$ | $\mathrm{P}^{\prime}_{6}$ | $(0,0{+}\epsilon)$ | $(\pi,\pi)$
|  |  |  |  |

Figure 1: Depiction of $S^{3}_{+}$-boundary with colored segments between key
points.

Figure 2: Depiction of $S^{3}_{-}$-boundary with colored segments between key
points.

Figure 3: Depiction of the gluing across $S^{2}$ boundary while preserving
orientation.

###  2.3 Conformal mapping to Minkowski space $\mathds{R}^{1,3}$

So far, we have seen an effective comformal map between
$\textrm{AdS}_{3}$-cylinder (4) and $S^{3}$-cylinder (8,9):
$(\rho,\psi)\rightarrow(\chi,\theta)$ while keeping $\tau$ and $\phi$ fixed.
We can now map the $S^{3}$-cylinder (post-gluing) to Minkowski space
$\mathds{R}^{1,3}$ equipped with polar coordinates $(t,r,\theta,\phi)$333Let
us recall Minkowski polar coordinates:
$(t,x,y,z)=(t,r\sin\theta\cos\phi,r\sin\theta\sin\phi,r\cos\theta)$.:
$(\tau,\chi)\rightarrow(t,r)$ via

$\displaystyle\sin\tau\ =\ \gamma\frac{t}{R}\quad\textrm{and}\quad\sin\chi\ =\
\gamma\frac{r}{R}\quad\textrm{with}\quad\gamma\ =\
\frac{2R^{2}}{\sqrt{4R^{2}t^{2}+(r^{2}-t^{2}+R^{2})^{2}}}\ ,$ (10)

where the $S^{2}$ coordinates $(\theta,\phi)$ are identified. A key features
of this map is that the full Minkowski space gets embedded inside half of the
doubled AdS domain with null-boundary given by $\chi{=}|\tau|$; this is due to
the following inequality:

$\displaystyle\gamma\ =\ \cos{\tau}-\cos{\chi}\
>0\qquad\implies\qquad\chi>\lvert\tau\rvert\ .$ (11)

This feature is clearly exemplified through $(\tau,\chi)$ Penrose diagram in
figure 5 demonstrating the flat spacetime embedding inside half of the (glued)
$\textrm{AdS}_{4}$ space with future and past null boundaries. The $(t,r)$
plot in figure 5 further illustrates the gluing-boundary inside Minkowski
spacetime. It should be noted here that every point inside shaded regions of
these plots hides a $2$-sphere and that such regions coming from different
$\textrm{AdS}_{4}$ copies are color-coded (yellow and orange shades).

Figure 4: Penrose diagram inside AdS4 spaces joined together (dotted black
line) with constant (blue lines) $t$\- and (brown lines) $r$-hypersurfaces and
the lightcone at the origin (red lines).

Figure 5: Minkwski spacetime in polar coordinates $(t,r)$ with constant
$\tau$\- & $\chi$-slices shown in blue and brown lines respectively while the
broken brown line shows the gluing-boundary.

The conformal metric (8) in these Minkowski coordinates take the following
form:

$\mathrm{d}s^{2}\ =\
\frac{4\,R^{4}}{(r^{2}{-}t^{2}{-}R^{2})^{2}}\,\bigl{(}-\mathrm{d}t^{2}+\mathrm{d}r^{2}+r^{2}\mathrm{d}\Omega_{2}^{2}\bigr{)}\
,$ (12)

which shows that the singularity at the gluing-boundary is a one-sheeted
hyperboloid $H^{1,2}_{R}$:

$\bigl{\\{}\chi{=}\tfrac{\pi}{2}\bigr{\\}}\qquad\Longleftrightarrow\qquad\bigl{\\{}r^{2}{-}t^{2}{=}R^{2}\bigr{\\}}\
=:\ H^{1,2}_{R}\ .$ (13)

##  3 $\textrm{SU}(1,1)$-symmetric gauge theory on $\textrm{AdS}_{4}$

Let us review some algebraic results required to construct the relevant
connection one-form. To this end, we start with the group manifold of
$\textrm{SU}(1,1)$ which is $\textrm{AdS}_{3}$, easily seen by the map

$\displaystyle
g:\;\mathrm{AdS_{3}}\,\rightarrow\,\mathrm{SU(1,1)}\qquad\mathrm{via}\qquad(\alpha^{1},\alpha^{2},\alpha^{3},\alpha^{4})\,\mapsto\,\begin{pmatrix}\alpha^{1}{-}\mathrm{i}\alpha^{2}&\alpha^{3}{-}\mathrm{i}\alpha^{4}\\\
\alpha^{3}{+}\mathrm{i}\alpha^{4}&\alpha^{1}{+}\mathrm{i}\alpha^{2}\end{pmatrix}\
.$ (14)

We use this $g$ to obtain left-invariant one-forms $e^{\alpha},\ \alpha=0,1,2$
via Maurer–Cartan method:

$\displaystyle\Omega_{L}(g)\ =\ g^{-1}\mathrm{d}g\ =\ e^{\alpha}\,I_{\alpha}$
(15)

where $I_{\alpha}$ are $\mathfrak{sl}(2,\mathds{R})$ generators satisfying

$\displaystyle[I_{\alpha},I_{\beta}]\ =\ 2\,f_{\
\alpha\beta}^{\gamma}\,I_{\gamma}\quad\quad\textrm{and}\quad\quad\mathrm{tr}(I_{\alpha}\,I_{\beta})\
=\ 2\,\eta_{\alpha\beta}\ ,$ (16)

with $f^{2}_{\ 01}=f^{1}_{\ 20}=-f^{0}_{\ 12}=1$ and
$(\eta_{\alpha\beta})=\mathrm{diag}(-1,1,1)$. The resulting one-forms look
like,

$\displaystyle e^{0}$ $\displaystyle\ =\
\cosh^{2}\\!{\rho}\;\mathrm{d}\tau+\sinh^{2}\\!{\rho}\;\mathrm{d}\phi\ ,$ (17)
$\displaystyle e^{1}$ $\displaystyle\ =\
\cos{(\tau{-}\phi)}\;\mathrm{d}\rho+\sinh{\rho}\;\cosh{\rho}\;\sin{(\tau{-}\phi)}\;\mathrm{d}{(\tau{+}\phi)}\
,$ $\displaystyle e^{2}$ $\displaystyle\ =\
-\sin{(\tau{-}\phi)}\;\mathrm{d}\rho+\sinh{\rho}\;\cosh{\rho}\;\cos{(\tau{-}\phi)}\;\mathrm{d}{(\tau{+}\phi)}\
.$

These obey Cartan structure equation and provide orthonormal-frame on
$\textrm{AdS}_{3}$-cylinder (4):

$\mathrm{d}e^{\alpha}+f^{\alpha}_{\ \beta\gamma}\;e^{\beta}\wedge
e^{\gamma}=0\quad\quad\textrm{and}\quad\quad\mathrm{d}s^{2}_{\mathrm{cyl}}\ =\
\mathrm{d}\psi^{2}+\eta_{\alpha\beta}\,e^{\alpha}e^{\beta}\ .$ (18)

Now a generic gauge field ${\cal A}$ in this frame can be made
$\text{SU}(1,1)$-symmetric by,

$\displaystyle{\cal A}\ =\
\mathcal{A}_{\psi}\,e^{\psi}+\mathcal{A}_{\alpha}\,e^{\alpha}\qquad\xrightarrow{{\cal
A}_{\psi}=0}\qquad\mathcal{A}\ =\ X_{\alpha}(\psi)\;e^{\alpha}$ (19)
$\displaystyle\implies{\cal F}\ =\ \mathrm{d}{\cal A}+{\cal A}\wedge{\cal A}\
=\ X_{\alpha}^{{}^{\prime}}\;e^{\psi}\wedge
e^{\alpha}+\tfrac{1}{2}\bigl{(}-2f_{\
\beta\gamma}^{\alpha}X_{\alpha}+[X_{\beta},X_{\gamma}]\bigr{)}\;e^{\beta}\wedge
e^{\gamma}\ ,$

where $X^{\prime}_{\alpha}$ in the field strength expression correspond to
$\mathrm{d}X_{\alpha}/\mathrm{d}\psi$. Next, we impose the Gauss-law
constraint $[X_{\alpha},X^{{}^{\prime}}_{\alpha}]=0$ arising from the eom
$*\mathrm{d}*{\cal F}=0$ by following choice of components,

$\displaystyle X_{0}=\Theta_{0}(\psi)\,{I_{0}}\ ,\quad
X_{1}=\Theta_{1}(\psi)\,{I_{1}}\quad\textrm{and}\quad
X_{2}=\Theta_{2}(\psi)\,{I_{2}}\ .$ (20)

The Yang–Mills Langrangian
$\mathcal{L}={\textstyle\frac{1}{4}}\mathrm{tr}({\cal F}\wedge*{\cal F})$ can
then be readily computed to get,

$\displaystyle\mathcal{L}$ $\displaystyle\ =\
\tfrac{1}{4}\mathrm{tr}\mathcal{F}_{\psi\alpha}\mathcal{F}^{\psi\alpha}+\tfrac{1}{8}\mathrm{tr}\mathcal{F}_{\beta\gamma}\mathcal{F}^{\beta\gamma}$
(21) $\displaystyle\ =\
\tfrac{1}{2}\sum_{\alpha}(\Theta^{\prime}_{\alpha})^{2}-2\bigl{\\{}(\Theta_{1}{-}\Theta_{2}\Theta_{0})^{2}+(\Theta_{2}{-}\Theta_{0}\Theta_{1})^{2}+(\Theta_{0}{-}\Theta_{1}\Theta_{2})^{2}\bigr{\\}}\
.$

This enjoys discrete symmetry of the permutation group $S_{4}$, acting by
permuting $\\{\Theta_{\alpha}\\}$ and flipping the sign of any two
$\Theta^{\prime}s$. Its maximal normal subgroups $S_{3}$ and $D_{8}$ yields
exact solutions:

  * •

Non-equivariant Abelian ansätze:

$\Theta_{0}\ \textrm{or}\ \Theta_{1}\ \textrm{or}\ \Theta_{2}\ =\
\mathrm{h}(\psi)\quad\mathrm{while}\quad\textrm{rest}\ =\
0\quad\implies\quad\mathrm{h}^{\prime\prime}=-4\,\mathrm{h}\ ,$ (22)

resulting into a harmonic equation whose solutions are well-known
trigonometric functions.

  * •

Equivariant non-Ablian ansatz:

$\Theta_{0}=\Theta_{1}=\Theta_{2}\ =:\
\tfrac{1}{2}\bigl{(}1+\Phi(\psi)\bigr{)}\quad\implies\quad\Phi^{{}^{\prime\prime}}\
=\ 2\,\Phi\,(1-\Phi^{2})\ =\ -{\textstyle\frac{\partial V}{\partial\Phi}}\ ,$
(23)

where the double-well potential
$V(\Phi){=}{\textstyle\frac{1}{2}}(\Phi^{2}-1)^{2}$ has been plotted in figure
8. The resultant anharmonic equation here also admits analytic solutions in
terms of Jacobi elliptic functions.

Figure 6: Plot of the potential $V(\Phi)$.

Figure 7: Color-graded density plot for the energy density
$\rho\propto\frac{1}{(\lambda^{2}{+}4z^{2})^{2}}|_{t=0}$ emphasizing its
maxima near $z{=}0$.

Figure 8: Electric (red) and magnetic (green) field lines for $\widetilde{{\bf
S}}^{(0)}$ at $t{=}10$ and $\psi_{0}{=}{\textstyle\frac{\pi}{2}}$ inside the
sphere of radius $r{=}\sqrt{101}$.

##  4 $\textrm{AdS}_{4}$ gauge fields on Minkowski space

We have already seen a series of conformal maps
$(\tau,\rho,\psi)\rightarrow(\tau,\chi,\theta)\rightarrow(t,r,\theta)$ defined
via (9) and (10) above. We can use these with $R{=}1$ along with abbreviations
$x\cdot\mathrm{d}{x}:=x_{\mu}\mathrm{d}{x}^{\mu}$,
$\varepsilon^{1}=\varepsilon^{2}:=\varepsilon$ and $\varepsilon^{0}:=1$ to
write the $\textrm{AdS}_{3}$-cylinder one-forms $e^{\alpha}$ (17) and
$e^{\psi}:=\mathrm{d}\psi$ in Minkowski coordinates as,

$\displaystyle e^{\alpha}$ $\displaystyle\ =\
{\textstyle\frac{\varepsilon^{\alpha}}{\lambda^{2}{+}4z^{2}}}\,\big{(}2(\lambda{+}2)\,\mathrm{d}{x}^{\alpha}-4x^{\alpha}\,x\cdot\mathrm{d}{x}-4\,f^{\alpha}_{\
\beta\gamma}\,x^{\beta}\mathrm{d}{x}^{\gamma}\big{)}\ ,$ (24) $\displaystyle
e^{\psi}$ $\displaystyle\ =\
{\textstyle\frac{\varepsilon}{\lambda^{2}{+}4z^{2}}}\,\big{(}{-}2\lambda\,\mathrm{d}{z}+4z\,x\cdot\mathrm{d}{x}\big{)}\
,\qquad\textrm{where}\quad\lambda\ :=\ r^{2}-t^{2}-1\ .$

This allows one to read-off various vierbein components, viz.
$e^{\alpha}_{\mu}$ and $e^{\psi}_{\mu}$ to be used below.

###  4.1 Nonabelian fields

For the nonabelian case we use the the ansatz (23) for gauge field ${\cal A}$
(19). The corresponding field strength $F$ of $\mathcal{A}\equiv
A=\tfrac{1}{2}\Big{(}1+\Phi\bigl{(}\psi(x)\bigr{)}\Big{)}\,I_{\alpha}\;e^{\alpha}_{\
\mu}\;\mathrm{d}x^{\mu}$ then computes to,

$F\ =\
\tfrac{1}{2}\Big{(}\Phi^{\prime}\bigl{(}\psi(x)\bigr{)}\,I_{\alpha}\,e^{\psi}_{\
\mu}e^{\alpha}_{\ \nu}\ -\
\tfrac{1}{2}\bigl{(}1{-}\Phi\bigl{(}\psi(x)\bigr{)}^{2}\bigr{)}\,I_{\alpha}\,f^{\alpha}_{\
\beta\gamma}\,e^{\beta}_{\ \mu}e^{\gamma}_{\
\nu}\Big{)}\,\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\ .$ (25)

One can easily extract the color-electromagnetic fields from this $F$ as
follows: $E_{a}{:=}F_{a0}$ and $B_{a}{:=}\frac{1}{2}\epsilon_{abc}F_{bc}$. We
can express the color EM fields thus obtained rather succinctly in terms of a
Riemann–Silberstein vector ${\bf S}:={\bf E}+\mathrm{i}{\bf B}$ as,

$\displaystyle S_{x}$
$\displaystyle=-{\textstyle\frac{2(\mathrm{i}\varepsilon\Phi^{\prime}+\Phi^{2}{-}1)}{(\lambda{-}2\mathrm{i}z)(\lambda{+}2\mathrm{i}z)^{2}}}\,\Bigl{\\{}2\bigl{[}ty{+}\mathrm{i}x(z{+}\mathrm{i})\bigr{]}I_{0}+2\varepsilon\bigl{[}xy{+}\mathrm{i}t(z{+}\mathrm{i})\bigr{]}I_{1}+\varepsilon\bigl{[}t^{2}{-}x^{2}{+}y^{2}{+}(z{+}\mathrm{i})^{2}\bigr{]}I_{2}\Bigr{\\}}\
,$ (26) $\displaystyle S_{y}$
$\displaystyle={\textstyle\frac{2(\mathrm{i}\varepsilon\Phi^{\prime}+\Phi^{2}{-}1)}{(\lambda{-}2\mathrm{i}z)(\lambda{+}2\mathrm{i}z)^{2}}}\,\Bigl{\\{}2\bigl{[}tx{-}\mathrm{i}y(z{+}\mathrm{i})\bigr{]}I_{0}+\varepsilon\bigl{[}t^{2}{+}x^{2}{-}y^{2}{+}(z{+}\mathrm{i})^{2}\bigr{]}I_{1}+2\varepsilon\bigl{[}xy{-}\mathrm{i}t(z{+}\mathrm{i})\bigr{]}I_{2}\Bigr{\\}}\
,$ (27) $\displaystyle S_{z}$
$\displaystyle={\textstyle\frac{2(\mathrm{i}\varepsilon\Phi^{\prime}+\Phi^{2}{-}1)}{(\lambda{-}2\mathrm{i}z)(\lambda{+}2\mathrm{i}z)^{2}}}\,\Bigl{\\{}\mathrm{i}\bigl{[}t^{2}{+}x^{2}{+}y^{2}{-}(z{+}\mathrm{i})^{2}\bigr{]}I_{0}+2\varepsilon\bigl{[}\mathrm{i}tx{-}y(z{+}\mathrm{i})\bigr{]}I_{1}+2\varepsilon\bigl{[}\mathrm{i}ty{+}x(z{+}\mathrm{i})\bigr{]}I_{2}\Bigr{\\}}\
.$ (28)

Interestingly, any explicit solution $\Phi$ do not couple to color components
of these fields ${\bf S}$; this fact reflects below in that the corresponding
physical quantities (arising from the stress-energy tensor) depends only on a
conserved parameter—rather than an explicit form—of such solutions.

We can now compute the corresponding stress-energy tensor $T_{\mu\nu}$ given
by,

$\displaystyle T_{\mu\nu}\ =\ -\tfrac{1}{2g^{2}}\,\bigl{(}\delta_{\mu}^{\
\rho}\delta_{\nu}^{\
\lambda}\eta^{\sigma\tau}-\tfrac{1}{4}\eta_{\mu\nu}\eta^{\rho\lambda}\eta^{\sigma\tau}\bigr{)}\,\mathrm{tr}\big{(}F_{\rho\sigma}F_{\lambda\tau}\bigr{)}\
,$ (29)

and express them, using mechanical energy
$\epsilon:=-\tfrac{1}{4}\bigl{(}(\Phi^{\prime})^{2}+(1{-}\Phi^{2})^{2}\bigr{)}$,
compactly as follows:

$\begin{split}\bigl{(}T_{\mu\nu}\bigr{)}\ =\
\frac{8}{g^{2}}\,\frac{\epsilon}{(\lambda^{2}{+}4z^{2})^{3}}\begin{pmatrix}\mathfrak{t}_{\alpha\beta}&\mathfrak{t}_{\alpha
3}\\\
\mathfrak{t}_{3\alpha}&\mathfrak{t}_{33}\end{pmatrix}\end{split}\quad\textrm{with}\quad\left\\{\
\begin{split}\mathfrak{t}_{\alpha\beta}&\ =\
-\eta_{\alpha\beta}(\lambda^{2}{+}4z^{2})+16x_{\alpha}x_{\beta}z^{2}\,,\\\
\mathfrak{t}_{3\alpha}&\ =\ \mathfrak{t}_{\alpha 3}\ =\
-8x_{\alpha}z\,(\lambda{-}3z^{2})\ ,\\\ \mathfrak{t}_{33}&\ =\
3\lambda^{2}-4z^{2}(1+4\lambda-4z^{2})\ .\end{split}\right.$ (30)

We find that these fields ${\bf E},{\bf B}$ and their stress-energy tensor
$T_{\mu\nu}$ are not singular at the full hyperboloid
$H^{1,2}\equiv\lambda{=}0$ but on a hypersurface given by the intersection

$\\{\lambda{=}0\\}\cap\\{z{=}0\\}\Leftrightarrow x^{2}+y^{2}-t^{2}\ =\ 1\ .$
(31)

We have plotted the energy density $\rho=\mathfrak{t}_{00}$ at $t{=}0$
highlighting the role of $xy$-plane in figure 8.

###  4.2 Electromagnetic fields

We consider following harmonic functions as solutions to the Abelian eom (22)
such that they would have same structural form across the gluing surface
(mediated by $\varepsilon$):

$\displaystyle\widetilde{A}^{(0)}$ $\displaystyle\ =\ -\tfrac{1}{2}\,\cos
2\bigl{(}\psi(x){+}\varepsilon\psi_{0}\bigr{)}\,e^{0}_{\
\mu}\;\mathrm{d}x^{\mu}\ ,$ (32) $\displaystyle\widetilde{A}^{(1)}$
$\displaystyle\ =\ \tfrac{1}{2}\,\sin
2\bigl{(}\psi(x){+}\varepsilon\psi_{0}\bigr{)}\,e^{1}_{\
\mu}\;\mathrm{d}x^{\mu}\ ,$ $\displaystyle\widetilde{A}^{(2)}$ $\displaystyle\
=\ \tfrac{1}{2}\,\sin 2\bigl{(}\psi(x){+}\varepsilon\psi_{0}\bigr{)}\,e^{2}_{\
\mu}\;\mathrm{d}x^{\mu}\ .$

As before, we can go ahead and compute the corresponding field strengths
$\widetilde{F}$, e.g. this one

$\displaystyle\widetilde{F}^{(0)}\ =\ \bigl{\\{}\sin
2\bigl{(}\psi(x){+}\varepsilon\psi_{0}\bigr{)}\,e^{\psi}_{\ \mu}e^{0}_{\
\nu}-\cos 2\bigl{(}\psi(x){+}\varepsilon\psi_{0}\bigr{)}\,e^{1}_{\
\mu}e^{2}_{\ \nu}\big{\\}}\;\mathrm{d}x^{\mu}\wedge\mathrm{d}x^{\nu}\ ,$ (33)

and then extract the electric $\widetilde{{\bf E}}$ and magnetic
$\widetilde{{\bf B}}$ fields. These can again be casted into a nice compact
form using Riemann–Silberstein vector $\widetilde{{\bf S}}:=\widetilde{{\bf
E}}+\mathrm{i}\widetilde{{\bf B}}$ as follows,

$\displaystyle\widetilde{{\bf S}}^{(0)}$ $\displaystyle\ =\
\frac{4\mathrm{e}^{2\mathrm{i}\psi_{0}}}{(\lambda{+}2\mathrm{i}z)^{3}}\begin{pmatrix}-2\,\bigl{(}t\,y+\mathrm{i}x\,(z{+}\mathrm{i})\bigr{)}\\\\[4.0pt]
2\,\bigl{(}t\,x-\mathrm{i}y\,(z{+}\mathrm{i})\bigr{)}\\\\[4.0pt]
\mathrm{i}\,\bigl{(}t^{2}+x^{2}+y^{2}-(z{+}\mathrm{i})^{2}\bigr{)}\end{pmatrix}\
,$ (34) $\displaystyle\widetilde{{\bf S}}^{(1)}$ $\displaystyle\ =\
-\frac{4\mathrm{e}^{2\mathrm{i}\psi_{0}}}{(\lambda{+}2\mathrm{i}z)^{3}}\begin{pmatrix}2\mathrm{i}\,\bigl{(}x\,y+\mathrm{i}t\,(z{+}\mathrm{i})\bigr{)}\\\\[4.0pt]
-\mathrm{i}\,\bigl{(}t^{2}+x^{2}-y^{2}+(z{+}\mathrm{i})^{2}\bigr{)}\\\\[4.0pt]
2\,\bigl{(}t\,x+\mathrm{i}y\,(z{+}\mathrm{i})\bigr{)}\end{pmatrix}\ ,$ (35)
$\displaystyle\widetilde{{\bf S}}^{(2)}$ $\displaystyle\ =\
\frac{4\mathrm{e}^{2\mathrm{i}\psi_{0}}}{(\lambda{+}2\mathrm{i}z)^{3}}\begin{pmatrix}-\mathrm{i}\,\bigl{(}t^{2}-x^{2}+y^{2}+(z{+}\mathrm{i})^{2}\bigr{)}\\\\[4.0pt]
2\mathrm{i}\,\bigl{(}x\,y-\mathrm{i}t\,(z{+}\mathrm{i})\bigr{)}\\\\[4.0pt]
-2\,\bigl{(}t\,y-\mathrm{i}x\,(z{+}\mathrm{i})\bigr{)}\end{pmatrix}\ .$ (36)

Like before, these fields are also singular on the hypersurface (31). We
demonstrate this in figure 8 by plotting typical field lines for
$\widetilde{{\bf E}}^{(0)}$ and $\widetilde{{\bf B}}^{(0)}$ (34); these
accumulate and intersect at the singular boundary (denoted with black circle).
We omit here the expressions for the stress-energy components (due to their
bulky form) and refer the reader to [14], where such expressions corresponding
to fields in (34) have been noted.

Incidentally, the above Riemann–Silberstein vectors are reminiscent of the
Hopf–Ranãda (HR) electromagnetic knots [4] and suggests their interpretation
as a non-compact cousin of the HR knot. Upon further exploration we find that
a special case of the field configuration (34):

$\psi_{0}=z=0\quad\implies\quad\frac{4}{(x^{2}{+}y^{2}{-}t^{2}{-}1)^{3}}\begin{pmatrix}2(x-t\,y)\\\\[4.0pt]
2(y+t\,x)\\\\[4.0pt] t^{2}{+}x^{2}{+}y^{2}{+}1\end{pmatrix}\ ,$ (37)

reproduces the magnetic fields444In our convention, these gauge fields are
interpreted as two electric fields and one magnetic field. of a recently
constructed magnetic vortex [15, eq. (5.15)] under the following
identification of the $\textrm{AdS}_{3}$ coordinates $(x^{0},x^{1},x^{2})$
used in [15],

$x^{0}\ =\ -t\ ,\quad x^{1}\ =\ -y\quad\textrm{and}\quad x^{2}\ =\ x\ .$ (38)

##  5 Summary

  * •

First, we employed the $\text{AdS}_{3}$-slicing of $\text{AdS}_{4}$ and the
group manifold structure of $\text{SU}(1,1)$ to find Yang–Mills solutions on
${\cal I}\times\text{AdS}_{3}$.

  * •

Next, we used a series of conformal maps to transfer these solutions to
Minkoowsi space, since Yang–Mills theory is conformally invariant in
$4$-dimensions.

  * •

These gauge fields are singular on a $2$-dimensional hyperboloid
$x^{2}{+}y^{2}{-}t^{2}=1$, but this singularity is milder than the one we
started with, namely $r^{2}{-}t^{2}{-}1=0$.

  * •

Due to this singularity the total energy diverges for both kinds of gauge
fields (see [14] for details), thereby limiting their physical usefulness.

  * •

Nevertheless, our Abelian solutions were found to match the magnetic field of
a known vortex magnetic mode on SU$(1,1)$ [15]. The status of other two
Abelian solutions in this regard remains to be explored.

## References

## References

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  * [2] Rajaraman R 1984 Solitons and instantons (Amsterdam: North-Holland)
  * [3] Manton N and Sutcliffe P 2004 Topological solitons (Cambridge: Cambridge University Press)
  * [4] Lechtenfeld O and Zhilin G 2018 Phys. Lett. A 382 1528 (Preprint 1711.11144)
  * [5] Kumar K and Lechtenfeld O 2020 Phys. Lett. A 384 126445 (Preprint 2002.01005)
  * [6] Hantzko L, Kumar K and Picanço Costa G 2022 Eur.Phys.J.Plus 137 407 (Preprint 2106.05952)
  * [7] Kumar K, Lechtenfeld O and Picanço Costa G 2022 J. Phys. A: Math. Theor. 55 315401 (Preprint 2202.00169)
  * [8] Lüscher M 1977 Phys. Lett. 70B 321
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  * [10] Ivanova T A, Lechtenfeld O and Popov A D 2017 Phys. Rev. Lett. 119 061601 (Preprint 1704.07456)
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  * [13] Kumar K, Lechtenfeld O, Picanço Costa G and Röhrig J 2022 Phys. Lett. B 835 137564 (Preprint 2206.12009)
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# Linguistic Characterization of Divisive Topics Online: Case Studies on
Contentiousness in Abortion, Climate Change, and Gun Control

Jacob Beel,1 Tong Xiang,2 Sandeep Soni,1 Diyi Yang1
1Georgia Institute of Technology, 2Georgetown University
1School of Interactive Computing, 2Department of Computer Science
1Atlanta, Georgia 30318, 2Washington, DC 20057
{<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

As public discourse continues to move and grow online, conversations about
divisive topics on social media platforms have also increased. These divisive
topics prompt both contentious and non-contentious conversations. Although
what distinguishes these conversations, often framed as what makes these
conversations contentious, is known in broad strokes, much less is known about
the linguistic signature of these conversations. Prior work has shown that
contentious content and structure can be a predictor for this task, however,
most of them have been focused on conversation in general, very specific
events, or complex structural analysis. Additionally, many models used in
prior work have lacked interpret-ability, a key factor in online moderation.
Our work fills these gaps by focusing on conversations from highly divisive
topics (abortion, climate change, and gun control), operationalizing a set of
novel linguistic and conversational characteristics and user factors, and
incorporating them to build interpretable models. We demonstrate that such
characteristics can largely improve the performance of prediction on this
task, and also enable nuanced interpretability. Our case studies on these
three contentious topics suggest that certain generic linguistic
characteristics are highly correlated with contentiousness in conversations
while others demonstrate significant contextual influences on specific
divisive topics.

## Introduction

Divisive topics – from gun control to illegal immigration, from animal cruelty
to vaccination in children – have gained another avenue for exposure in online
media. These topics evoke opinionated and polarizing conversations and though
we broadly understand the different issues that make these topics contentious,
we know very little about the structure and dynamics at the level of
conversations around these topics. For instance, what makes a conversation
about a particular topic contentious? What are the differences between
conversations which foster healthy disagreement and discussion about a
divisive topic to those which devolve into online shouting matches? What is
the extent of this effect in conversations about a particular topic on a
particular platform?

Answers to the above questions and a nuanced understanding of these
conversations could help in resource allocation on social media platforms: for
example, determining the course of a conversation early on in its lifespan
could help to guide moderator action and perform a kind of triage for online
community moderation. Additionally, the distribution of these conversations on
the platform can help in user enculturation: for instance, platforms can guide
newcomers so that they are exposed to less contentious conversations and get a
gentle introduction to the communities dedicated to discussing divisive
topics. Alternatively, measuring the extent of contentious conversations can
be used to assess platform health (?): for example, conversations that
manifest as constant inter-group conflict or lead to hateful speech or vile ad
hominem attacks can be taken as a bad sign, marking platforms as amplifiers
and manufacturers of negative discourse; in contrast, discussions that balance
between opposing viewpoints or improve information diversity can be viewed as
good signals.

Much attention has recently been paid towards predicting conversation quality,
typically framed as controversy, from the content of and structure of
conversations (?; ?). The strength of these works lies in the use of powerful
neural networks, such as convolutional neural networks (CNN) and transformers
(BERT), to predict controversy in conversations with high accuracy. However,
these predictive models are constrained by their relative limitations in
highlighting the informative conversation characteristics that distinguish
controversial conversations from non-controversial ones. Moreover, their
predictive performance has only been tested in generic conversations, making
it hard to tease out conversation topics being divisive from participation
patterns in discussions leading to controversy. This work fills this gap by
focusing exclusively on divisive topics. In this work, we aim to address the
above gaps by answering the following research questions: RQ1 Can we
accurately predict contentiousness in conversations via explainable
conversation characteristics when controlling for divisive topics? RQ2 What
specific conversation characteristics differentiate non-contentious
conversations from contentious ones between certain divisive topics?

To address RQ1 we operationalize a set of linguistically explainable
characteristics including coarse discourse acts, toxicity, sentiment, and
occurrence of specific lexicons. In addition, we propose to leverage
conversation participants’ characteristics such as their gender, location,
account age, and prolificity. Many of these features, including discourse acts
and user features have not been used in this task previously. We follow prior
work (?; ?) to define quality of conversations. We focus on conversations
identified to belong of one of the three divisive topics: abortion, climate
change, and gun control and then look to examine the contentiousness of
conversations within those divisive topics. These conversation characteristics
are then utilized to build machine learning classifiers to predict whether a
conversation is contentious. To address RQ2, we visualize the most predictive
conversation characteristics, across and within each divisive topic. Our
analyses show that certain generic linguistic characteristics are highly
correlated with contentiousness in conversations across the board: for
instance, negative discourse acts appear significantly more often in
contentious conversations across three topics; while others demonstrate
significant contextual influences such as the varying importance of user
demographic information.

To sum up, the contributions of our work are three-fold. First, we
operationalize a set of conversation characteristics such as coarse discourse
acts and toxicity, which can improve both the predictive performance and the
explainability of contentious conversation detection. Second, we demonstrate
that the contentiousness can be detected using our explainable models with
reasonable accuracy, even at an early stage of these conversations. Third, our
research uncovers both generic and topic specific conversation characteristics
associated with discussions about divisive topics online. This work can be
leveraged for several different purposes, including further research and
aiding moderation strategies of online communities.

Note that this work uses terminology which may be different from prior work
due to the structure of this research. For instance, ? refer to conversations
as “controversial”. In this work, we elect to refer to conversations as
“contentious” and to topics as “divisive” in order to ensure that the
difference between the nature of the two is as clear as possible. The topics
we study are divisive, but will not necessarily result in contentious
conversations.

## Related Work

### Controversy in Online Discussions

There has been extensive computational research about controversy in online
discussions. To quantify controversy, ? (2010) used the ratio of positive and
negative sentiment words; ? (?) and ? (?) define controversial content as that
which attracts a mix of positive and negative feedback. Other lines of
research used predefined lexicons to collect controversial discussions (?).
Prior work on predicting controversy or distinguishing controversial
discussions from non-controversial ones rely heavily on dictionary based
approaches (?; ?). Such methods benefit from being explainable but have
limited predictive power (?). Supervised neural models such as CNNs have also
been used recently for predicting controversy (?). Since it is difficult to
get ground-truth annotations of controversy, these approaches typically use
distant supervision, such as upvote-downvote ratio, to mark examples as either
controversial or non-controversial. The growing popularity of Reddit has
attracted research attention to predict and model controversy on its platform
(?). ? (2019) uses BERT to predict controversial conversations on Reddit. ?
(2019) refines controversy into disputes, disruptions, and discrepancies via
user actions and connects those user actions to sentiment and topic analysis
to explain controversy specifically within political discussion.

### Contentiousness, Divisive Topics and Controversy

Different from all the aforementioned research, we argue that controversy
should be studied at least from two perspectives: topics being divisive, and
conversations being contentious. The key intuition is that, people can still
have deliberative and good discussions regarding a very divisive topic such as
gun control. Thus, topics being divisive does not equal to conversations being
contentious, and analyzing how conversation characteristics relate to
contentious content should control for divisive topics. For the purposes of
our work, controversial and non-controversial conversations are referred to as
contentious and non-contentious, respectively. This is more accurate to our
task at hand, which is to distinguish between two types of discourse which
emerge when even discussing divisive topics rather than detect controversy
irrespective of topic. Therefore, our work controls for topics in order to
explore why certain conversations become controversial and why others do not
within and between those topics. We do this by opting to focus on linguistic
characterizations of conversations as well as user factors rather than
conversation structure and comment timing. In contrast to both ? and ?, we
focus on using linguistic characteristics and user factors to build simpler
models, rather than end-to-end neural solutions. In doing so we create a more
explainable solution, and compare these models against the transformer
architecture BERT (?).

### Abortion, Climate Change, and Gun Control

To better study conversation characterises and their influences on
conversations being contentious, we first need to control for divisive topics,
one big confounding factor behind controversy. As a case study, we focus on
discussion around three topics – abortion, climate change, and gun control.
These contentious topics have long been in the public discourse (?) but have
gained an additional dimension: views on these topics are now recorded
permanently on social media platforms making these platforms attractive venues
for research on the social aspects of conversations (?; ?; ?). Online
discussions about these contentious topics tends to be polarizing (?). For
instance, ? (?) found that debates on contentious topics, such as gun control
or abortion, on Twitter become more polarizing as a result of increased public
attention that typically follows some external event. Similarly, ? (?)
measured polarization based on the content of tweets around issues of gun
policies for a series of documented mass shooting events. However, in contrast
to these objectives of measuring polarization around these topics, our focus
is on characterizing what linguistic features are correlated with controversy.
The topic of climate change, though relatively recent compared to gun control
and abortion, has also attracted research particularly on how it is discussed
online (?; ?). ? (?) proposed a method to summarize discussions on contentious
topics such as abortion and climate change, to better understand the
controversy and these different viewpoints. Our focus in this work is also on
explaining controversy in these topics but we diverge in two ways: first, we
focus on understanding the linguistic aspects that are predictive of
controversy in conversations; second, we use high-capacity neural networks to
classify conversations as controversial or non-controversial, allowing us to
distinguish between the two for any of the contentious topics that we
considered.

## Data

Our data consist of filtered and labeled posts and comments from Reddit.
Reddit is a content-sharing and discussion social media website where users
can post conversations, which other users can comment on. The data consist of
most conversations (also known as “posts” or “threads”) and comments over the
period of time from 2007 to February 2014 (?), and are drawn from the same
source as is used by ?. Deleted conversations are not included, but
conversations and comments which were made by accounts that have since been
deleted are included. A conversation consists of a title, selftext (the body
of an original post), and comments. The title and selftext are input by the
original poster (OP) at the time of conversation creation. We label the posts
according to the scheme proposed and validated in ? (?). Under this scheme, a
conversation is considered contentious if its upvote ratio is within the
bottom quartile of conversations, i.e. it has drawn more mixed user reactions.
Only conversations which have an upvote ratio of at least $0.5$ are
considered.

Although we use the same labelling technique as ?, we control for divisive
topics. This allows us explore the manner in which contentious conversations
differ within those divisive topics, a question which ? elect not to explore.
Additionally, we explore a larger number of subreddits by virtue of our choice
to filter by topic. As a robustness check, we also compared this labelling
technique to another which is widely used in this task (?), the ratio of
positive to negative sentiment. Across all topics, this resulted in unbalanced
data sets, with non-contentious conversations making up over $85\%$ of
conversations. We found this definition of controversy to differ
significantly, and state of the art models can not perform significantly
better than random chance. These two labelling techniques clearly measure
different phenomena, and we urge future work to explore these differences for
deeper insights.

To retain only the conversations associated with particularly divisive topics,
we began with a larger list of divisive topics available at ProCon.org
(http://procon.org) and filtered down to a subset of topics that met our
selection criteria. First we aimed to maximize the number of conversations.
Second, we aimed to minimize the number of off-topic conversations through a
process of filtering, followed by manual inspection, followed by refinement of
key words. This process was repeated until the number of off-topic
conversations was minimized. Finally, we aimed for topics with a large
difference between the upvote ratios of the two label categories. Topics which
did not satisfy this criteria, such as the “ _U.S. involvement in the Middle
East_ ”, “ _free or affordable higher education_ ”, “ _healthcare_ ”, and “
_police brutality_ ”, were excluded. The topics we focus on and the number of
conversations $n$ in each topic are: abortion (n=892), climate change
(n=1070), and gun control (n=1176). These were selected due to their large
differences aside from being political and divisive.

These discussions (conversations and comments) were identified using keywords
selected from the ProCon.org pages for these topics: abortion (e.g., _“pro-
life”, “pro-choice”, “fetus”, “abortion”_), climate change (e.g., _“global
warming”, “climate change”, “greenhouse gas”, “fossil fuel”_), and gun control
(e.g., _“ar-15”, “national rifle association”, “assault rifle”, “second
amendment”_). The three selected topics had the highest number of
conversations of the topics we considered and a wide enough range of upvote
ratio to indicate the presence of different classes of conversations. These
topics also minimized the number of false positives, conversations that
matched one or more keywords but did not actually pertain to the topic.
Additionally, these topics presented ways in which they may differ. For
instance, we hypothesized that abortion would be a topic where a non-minority
presence of male users in a conversation would lead to contentiousness,
whereas a non-minority presence of female users in a conversation would not.

## Conversation Characteristics

We operationalize a set of conversation characteristics that include both
linguistic and non-linguistic features needed by subsequent models and
explainable analyses.

### Linguistic Operationalization

Discourse Acts and Discussion Patterns contentiousness in a discussion is
reflected by the expression of opposing views or disagreements. To better
understand those discourse patterns in contentious conversations, we leverage
coarse discourse acts (?) to extract different types of patterns in users’
discussions. These discourse acts consist of the following 10 acts: question,
answer, announcement, agree, appreciate, disagree, negative, elaborate, humor,
and other. We used BERT and fine-tuned it to predict these discourse acts for
each post and each comment using the Reddit annotated data from ? (?), with
$65\%$ 10-class macro accuracy at predicting these discourse acts. The entire
conversation conversation can then be represented in terms of these discourse
acts as unigram and bigram vectors.

Toxicity Previous work has shown that toxicity is linked with contentious
conversations on social media (?; ?). Prior work has leveraged it as a signal
for unhealthy interactions in online conversations (?). We design toxicity
features in order to evaluate the rudeness, disrespectfulness, or
unreasonableness of posts within a conversation. We use Google’s Perspective
API111https://www.perspectiveapi.com/ to extract the toxicity of text. The API
returns a value between $0$ and $1$ for a given input, where $0$ indicates
complete inoffensiveness and $1$ indicates absolute offensiveness. We
construct three features with the aforementioned toxicity values: title
toxicity to represent the toxicity of the title, post toxicity to represent
the toxicity of the post, and max. comment toxicity to represent the most
toxic comments in the conversation conversation.

Sentiment Sentiment is also highly related to contentiousness (?; ?; ?). We
use Valence Aware Dictionary and sEntiment Reasoner (VADER) (?) to measure the
sentiment of posts and their comments. We use the built-in VADER lexicons from
NLTK (?) to obtain a sentiment score for any given text. We measure the level
of positivity and negativity in the post (max. sentiment, min. post
sentiment), the maximum level of positivity in the comments (max. comment
sentiment), the average amount of positivity in the comments (avg. comment
positive sentiment), as well as the maximum and average amount of negativity
in the comments (min. comment sentiment, and avg. comment negative sentiment)
and use them as separate features for prediction.

LIWC The linguistic styles and word choices that speakers choose to convey
their intent can reflect how they conceptualize meaning (?). Prior studies
suggested that information gained from examining these lexical patterns can be
useful in contentiousness prediction (?; ?). To understand how linguistic
styles affect contentious conversations, we employ the Linguistic Inquiry and
Word Count (LIWC) (?; ?). We construct a set of features to help capture
lexical evidence using its $64$ dimensions, including aspects such as
attentional focus, social relationships, thinking styles, and others (?), by
extracting a count vector of occurrences of words from the LIWC lexicon.

Text Representations We choose to represent the meaning of the text of each
conversation conversation in two vector representations: TF-IDF and contextual
representation from a finetuned BERT model. The TF-IDF representation is the
sum of the tf-idf vector for the post and the average tf-idf vector of all the
comments under the post. We set a maximum document frequency of $0.95$ and
exclude English stopwords. The BERT representation is computed using bert-
base-uncased from transformer (?) and is similarly the sum of the contextual
representation for the post and the average comment representation. We use the
second-to-last hidden states as the representation vectors.

### User Factors

We also include a set of user factors that are closely connected with
contentiousness (?), which are often leveraged to understand the types of
people who are participating in these conversations. We use the gender,
location, and prolificity of Reddit users as operational features. Our choice
of restricting to these factors is partially because this information can be
extracted more easily and with high confidence from Reddit. Additionally,
gender is often associated with different roles in online conversations and
reactions to contentiousness (?). Similarly, location is informative due to
its contextualized effect in understanding contentiousness in different
communities (?). Prolificity can be used to distinguish between newcomers and
experienced members.

Gender and location are heuristically determined by textually examining a
user’s post history for noun phrases. Gender is determined by looking for
gendered nouns or pronouns at the end of a noun phrase taking the form “I
am…”, or any other form of the copula. Genders are counted, with indeterminate
and nonbinary genders being excluded from this count. Locations are extracted
in a similar fashion, and vectorized as a count vector of geocoded locations
or the raw location if the location fails to be geocoded. The distribution of
user locations differs from the distribution of user locations generally,
indicating a connection with contentiousness. Prolificity is calculated from
the number of posts a user has made in conversations on the topic of interest.
We define a user as prolific if she/he appears in her/his respective topic
more than $25$, $50$, or $100$ times in order to explore differences between
different levels of prolificity.

To ensure quality, we validated the gender and locations of users extracted
using the above steps. We find a female/male split of $47.84\%$/$52.16\%$,
$25.56\%$/$74.44\%$, and $33.32\%$/$66.68\%$ for abortion, climate change, and
gun control, respectively. The overall gender distribution closely matches
surveyed Reddit demographics.222https://www.journalism.org/2016/02/25/reddit-
news-users-more-likely-to-be-male-young-and-digital-in-their-news-preferences
We also use location to validate the corpus we collected, where we find that
the U.S. is the most common country at $44\%$, followed by Canada, India, the
U.K, Brazil, and Germany. This does not perfectly mirror reported
demographics, but comes
close.333https://www.statista.com/statistics/325144/reddit-global-active-user-
distribution

### Subreddit

The subreddit of a conversation is included as a one-hot encoded feature to
account for its potentially confounding influence. We found no statistically
significant ($p<0.05$) evidence that these distributions were different using
the Kolmogorov-Smirnov 2 sample test (?). The $p$ for abortion, climate
change, and gun control are $0.88$, $0.60$, and $0.80$ respectively.

## Prediction of Contentious Conversations

Features | Abortion | Climate change | Gun control
---|---|---|---
Acc. | Prec. | Rec. | F1 | Acc. | Prec. | Rec. | F1 | Acc. | Prec. | Rec. | F1
TF-IDF | $0.586$ | $0.578$ | $0.583$ | $0.546$ | $0.644$ | $0.614$ | $0.581$ | $0.581$ | $0.653$ | $0.640$ | $0.617$ | $0.613$
\+ Discourse act | $0.660^{*}$ | $0.628^{*}$ | $0.613$ | $0.599$ | $0.696$ | $0.649$ | $0.617$ | $0.624$ | $0.721^{*}$ | $0.694^{*}$ | $0.657$ | $0.668$
\+ Gender | $0.644^{*}$ | $0.620^{*}$ | $0.595$ | $0.585$ | $0.639$ | $0.611$ | $0.569$ | $0.575$ | $0.685$ | $0.667^{*}$ | $0.614$ | $0.630$
\+ LIWC | $0.665^{*}$ | $0.650$ | $0.617$ | $0.608$ | $0.672$ | $0.671^{*}$ | $0.554$ | $0.590$ | $0.687$ | $0.685$ | $0.660$ | $0.652$
\+ Location | $0.624$ | $0.594$ | $0.581$ | $0.567$ | $0.641$ | $0.617$ | $0.586$ | $0.582$ | $0.693$ | $0.672$ | $0.644$ | $0.647$
\+ Prolific | $0.657^{*}$ | $0.618^{*}$ | $0.606$ | $0.596$ | $0.669^{*}$ | $0.640^{*}$ | $0.605^{*}$ | $0.607^{*}$ | $0.703^{*}$ | $0.684^{*}$ | $0.630$ | $0.648$
\+ Sentiment | $0.596$ | $0.581$ | $0.596$ | $0.556$ | $0.676$ | $0.634^{*}$ | $0.625$ | $0.616$ | $0.665$ | $0.650$ | $0.625$ | $0.625$
\+ Toxicity | $0.609$ | $0.586$ | $0.608$ | $0.565$ | $0.661$ | $0.638$ | $0.614$ | $0.610$ | $0.660$ | $0.646$ | $0.618$ | $0.618$
\+ Subreddit | $0.550$ | $0.551$ | $0.547$ | $0.548$ | $0.649$ | $0.659$ | $0.607$ | $0.631$ | $0.597$ | $0.610$ | $0.541$ | $0.573$
\+ All | $0.704^{*}$ | $0.753^{*}$ | $0.605$ | $0.663$ | $0.707$ | $0.766$ | $0.593$ | $0.660$ | $0.683^{*}$ | $0.727^{*}$ | $0.573$ | $0.631$
BERT | $0.664$ | $0.652$ | $0.641$ | $0.617$ | $0.757$ | $0.729$ | $0.678$ | $0.693$ | $0.728$ | $0.701$ | $0.709$ | $0.692$
\+ Discourse | $0.702$ | $0.664$ | $0.638$ | $0.639$ | $0.772$ | $0.754$ | $0.692$ | $0.716$ | $0.746$ | $\bm{0.726}$ | $0.703$ | $0.707$
\+ Gender | $0.660$ | $0.620$ | $0.623$ | $0.605$ | $0.761$ | $0.724$ | $0.701$ | $0.702$ | $0.739$ | $0.706$ | $0.714$ | $0.702$
\+ LIWC | $0.686$ | $0.639$ | $0.668$ | $0.635$ | $0.761$ | $0.736$ | $0.686$ | $0.701$ | $0.734$ | $0.705$ | $0.695$ | $0.692$
\+ Location | $0.652$ | $0.626$ | $0.699$ | $0.630$ | $0.746$ | $0.712$ | $0.689$ | $0.689$ | $0.741$ | $0.723$ | $0.698$ | $0.701$
\+ Prolific | $0.698$ | $0.652$ | $0.688$ | $0.652$ | $\bm{0.780}$ | $0.742$ | $\bm{0.714}$ | $\bm{0.719}$ | $\bm{0.752}$ | $0.724$ | $\bm{0.716}$ | $\bm{0.711}$
\+ Sentiment | $0.657$ | $0.653$ | $0.632$ | $0.617$ | $0.768$ | $0.744$ | $0.708^{*}$ | $0.717^{*}$ | $0.732$ | $0.707$ | $0.703$ | $0.693$
\+ Toxicity | $0.648$ | $0.612$ | $0.683$ | $0.618$ | $0.760$ | $0.735$ | $0.689$ | $0.700$ | $0.728$ | $0.701$ | $0.698$ | $0.688$
\+ Subreddit | $0.558$ | $0.552$ | $0.619$ | $0.582$ | $0.666$ | $0.652$ | $0.708$ | $0.678$ | $0.594$ | $0.580$ | $0.689$ | $0.629$
\+ All | $\bm{0.715}$ | $\bm{0.686}$ | $\bm{0.712}$ | $\bm{0.678}$ | $0.774$ | $\bm{0.755}$ | $0.685$ | $0.712$ | $0.740$ | $0.723$ | $0.707$ | $0.706$
Table 1: Comparison of the performance for each feature set. We highlight the
model with the best performance in bold. ∗ denotes having statistically
significant differences from their corresponding baselines at the $p<.05$
level.

To answer RQ1, we train classifiers to predict whether a conversation is
labeled as contentious or otherwise, incorporating the linguistic
characteristics and the user factors as features along with subreddit in the
classifier. Specifically, we show the performance of the classifier in making
two types of predictions: predicting the contentiousness of the conversation
using all the comments and predicting the contentiousness at different stages
of these conversations. We experimented with two classifiers as baselines:
Logistic Regression (LR) using TF-IDF features and Multi-layer Perceptron
(MLP) using BERT features. We then augmented these baselines to include the
additional linguistic and user characteristics described in Conversation
Characteristics. All the models were implemented using the off-the-shelf
machine learning library scikit-learn. The pretrained BERT representation was
further finetuned on Reddit comments before being used as input for the MLP.
10-fold stratified cross-validation was used to evaluate all the models and
account for potentially confounding factors. The resulting scores are
displayed in Table 1.

### Conversation Level Contentiousness Prediction

As shown in Table 1, augmenting the baselines with the linguistic and the user
features boosts the prediction performance significantly across all three
topics. Specifically, in all cases, TF-IDF + All outperforms the baseline TF-
IDF. Similarly, BERT + All also displays significant improvements over its
baseline though not to the same degree as the TF-IDF baseline. A closer look
reveals that the addition of certain features improves the results across all
topics. On the other hand, some features only provide topic-specific
improvements – for instance, adding the gender features does not improve the
performance for the climate change topic. This suggests a distinction between
features that are more related to contentiousness in general, and features
that are more related to a specific divisive topic. Our modeling of
contentiousness through this combination of linguistic and user features
highlights these differences which would otherwise go unnoticed if using just
the TF-IDF or BERT representations.

((a)) Abortion

((b)) Climate Change

((c)) Gun Control

Figure 1: Accuracy and F1 score for early prediction of contentiousness using
TF-IDF + All and BERT + All models. The percentage of comments used as input
is shown on the $X$ axis.

### Early Prediction of Contentiousness

To predict contentiousness on partial conversations, we evaluated the
previously trained models on increasing percentages of each conversation’s
comments. From a practical perspective, stakeholders such as moderators could
be interested in solutions that predict contentiousness early with high
performance i.e. without seeing all the comments or seeing only a limited
number of comments in addition to the titles and posts. To this end, we test
both TF-IDF + All and BERT + All models by gradually increasing the proportion
of conversation comments in the order they were posted, beginning at $10\%$
and ending at $100\%$ and increasing in $10\%$ intervals. The results of this
experiment for each topic are shown in Figure 1.

Overall, we found that the inclusion of more comments and features increases
the accuracy and F1 score significantly for TF-IDF + All. BERT + All also
improves over time, but less substantially. TF-IDF + All approaches the
performance of BERT + All at $100\%$ of comments, but BERT + All is much
better at early stage prediction than TF-IDF + All. Specifically, models can
obtain $88\%$ of peak performance for accuracy after the first $20\%$ of
comments have been included, consistent across all three topics. The new
features that we explore in this work provide significant performance benefits
over the baseline, while remaining explain-able, both in full-prediction and
partial prediction contexts.

## Explanation of Contentious Case-Studies

This section investigates RQ2 on how different linguistic and conversational
characteristics of conversations relate to their contentiousness and how these
differ between different contentious topics. To do so, we utilize the
regression coefficients from the aforementioned explainable classifier (TF-IDF
+ All), and further convert them into odds ratios for ease of explainability.
We visualize the top 10 predictors with high odds ratios as well as the bottom
10 predictors with low odds ratios for each topic in Figure 2.

#### Negative Discourse Acts Matter

The presence of certain discourse acts, in particular disagreement, negative
reaction, and question are indicative of contentiousness across all three
topics, with odds ratios greater than $1$ for each. Disagreement and negative
reaction are discourse acts with a primarily negative connotation, so their
appearance here is to be expected. Question, however, is less so. Its
appearance here indicates that the communication between the various sides of
a divisive topic highly contributes to conversation contentiousness. Since
answer and elaboration are significantly lower in terms of odds ratio, this
would imply that communication is either one-sided or incomplete in these
contentious conversations.

The discourse act bigrams that include disagreement, negative reaction, and
question are also heavily represented in contentious conversations, with the
vast majority of them having odds ratios greater than 1. This is even the case
when paired with positive discourse acts, such as agreement elaboration, and
appreciation, the cause of which may be that negative discourse acts are
causing an effect similar to that of the negativity bias (?; ?), a
psychological phenomenon where negative events have greater psychological
effects than positive ones, in the readers of these comments.

The positive discourse acts elaboration, agreement, and appreciation, however,
have significantly lower average odds ratios, $1.002$, $1.017$, and $0.984$,
respectively. Bigrams containing these positive discourse acts similarly have
lower odds ratios, again seen in Table LABEL:table:discourse_table. Each of
these discourse acts indicates a greater level of inter-group cooperation and
communication within these divisive topics. The least controversial act
announcement can be characterized generally as comments which are repositories
of information, usually laid out in list form. These tend to be highly upvoted
comments which guide the course of the conversation.

Figure 2: Odds ratios of the top 20 categories for contentious and non-
contentious conversations across “abortion”, “climate change”, and “gun
control”. Feature names are colored based on their category. Prolificity,
negative discourse acts, and title toxicity are generally associated with
contentiousness across all three topics. Location, other discourse acts, and
sentiment vary across topics. The feature importance across topics indicates a
divide between contextual features and generally predictive ones.

#### Location Varies Across Topics

Location is, as shown by Figure 2, variable in terms of both which specific
locations are represented and the importance of them across topics. Two
classes of location can be seen in the features for abortion: locations which
are generally more accepting of abortion and locations which are generally
less accepting of abortion. We see examples of the first category in in
Germany (odds ratio of $0.993$), and in Europe ($1.005$) and examples of the
second category in in Georgia ($1.109$) and in North Texas
($1.076$)444https://www.pewforum.org/religious-landscape-study/compare/views-
about-abortion/by/state/. The posts of these users, however, have very little
to do with their place of residence. This may indicate that a wide spread of
people with complex views on abortion or on a diminished importance of
particular locations. We also notice a third category of locations, which are
places which are heavily featured in news related to abortion, with Seattle,
San Diego, and New York having odds ratios of $1.202$, $1.203$, and $1.118$,
respectively.

Climate change may present a more political view of location. California
($1.212$) and New York ($1.184$) both had high values for the topic of climate
change, and are both heavily featured in national news about climate change as
well as politics concerning climate change. Both of these states are primarily
represented by the United States’ Democratic Party whose constituents who are,
on average, significantly more concerned about climate change on average than
those of the United States’ Republican
party.555https://www.pewresearch.org/science/2015/07/01/chapter-2-climate-
change-and-energy-issues/ Gun control, however, presents a very specific view
of location and contentiousness, with locations generally associated with gun
violence tragedies being highly predictive of contentiousness. For instance,
Aurora, Colorado (which has an odds ratio of $1.196$) where, in 2012, a
shooting was carried out which killed 12 and injured
70.666https://en.wikipedia.org/wiki/2012_Aurora,_Colorado_shooting Many users
that we have identified as being from the Denver area, very near to Aurora,
have participated in contentious conversations specifically about the
shooting. Humans tend to react to events spatially close to them more
concretely and emotionally(?) which is a likely cause, along with the
particularly volatile subject matter, for these conversations to become
contentious.

The examination of locations in contentious conversations shows that location
is highly variable across topics as well as within them. For abortion,
location may only play a broader role in contentiousness prediction. In
climate change, locations appear to be related to political affiliation. In
gun control, locations may be extremely specific and related to similarly
specific events. As such, detecting contentiousness in conversations
concerning issues involving very specific, spatially close locations can be
viewed very differently from broader issues involving more spatial separation.

#### Prolificity and Account Age Are Inversely Correlated with Contentiousness

Prolificity and account age are two related user factors which relate to
contentious conversations in very different ways. Prolificity tends to
correlate with contentious conversations across all topics. This would
indicate that users that are, to a certain extent, prolific within a given
community are more likely to participate in potentially difficult
conversations due to feeling more comfortable engaging with the community on
these issues. For instance, conversations featuring high numbers of prolific
users in the topic of abortion overwhelmingly featured religion, a topic which
many are uncomfortable talking about in unfamiliar spaces. Non-contentious
conversations still involved religion, but not nearly to the same extent.
Account age, on the other hand, is correlated with a lack of contentiousness.

#### Gender Predicts Abortion

Gender has an extremely variable impact on contentiousness depending heavily
on whether or not gender is an inherent issue for the contentious topic being
discussed. This is immediately evident in our analysis: the presence of
specific genders is only appreciably different when looking at abortion, with
male having an odds ratio of $1.128$ compared to the female value of $1.061$.
The fact that gender can be so influential in the topic of abortion as to
raise the odds ratio of male so high is a testament to the gender differences
that can appear in contentious conversations (?) and that abortion is a topic
that particularly concerns women more than men. Those studying conversations
using our method in a particular community which may deal with female-centric
issues more frequently than usual should pay more attention to gender when
determining contentiousness. Notably, these results suggest that the
differences in which female and male users engage with contentious discussion
found in ? are highly variable with topic.

#### Toxicity Equals Contentiousness?

High levels of toxicity in post titles are strong predictors of
contentiousness, as demonstrated by odds ratios that are greater than $1.1$ in
all topics. It might be that some users choose contentious or provocative
titles to attract more users to participate in conversations; similar trends
have been found on Twitter (?), where the sentiment of the news headline is
strongly related to the popularity of the news. As demonstrated in Figure 2,
however, the toxicity of other parts of conversations are weighted differently
between topics. For instance, max. comment toxicity contributes to
contentiousness across all three topics while avg. comment toxicity only
contributes to contentiousness in abortion. This might be because different
topics have their own characteristics, with people involved in discussions
about climate change being more likely to use milder language in comments,
with an average toxicity level of $0.19$. On the other hand, participants in
abortion and gun control conversations are more likely to use more toxic
language in comments, with an average toxicity level of $0.26$ and $0.25$,
respectively. Selftext toxicity is relatively less predictive, with odds
ratios of less than $1$ among all topics. This might be because posts usually
serve a pragmatic purpose, either as expansions or explanations of their title
in order to provide more information about the poster’s intent and context for
discussions. It may also be that the content of the title colours commenters’
opinions more strongly than the selftext, following the human preference for
first impressions.

#### LIWC Offers Contextual View of Language Choice on Contentiousness

In general, LIWC features offer a comprehensive view of users’ linguistic
styles and word choices and their influences on controversy. For simplicity,
here we discuss two consistently informative LIWC features: you and friends.
Here the linguistic category you contains second-person pronouns such as
“you”, and “yours”. you is positively associated with contentiousness across
all topics which might communicate to us that a repetitive utilization of you
words increases social distance between conversation participants, and
relatively related to relationship quality (?). Such situations can be
observed in our cases. The impact of the category friends is more variable,
with climate change viewing it the most indicative of contentiousness ($1.13$)
and abortion viewing it the least ($0.91$). This might indicate that higher
mentions of friend words in abortion make them harder to serve as a
distinction signal in contentiousness prediction, as abortion involves large
amount of descriptions about relationships, for which people need to refer to
related words related to friends. In both contentious and non-contentious
conversations, we can easily find such discussions. Conversely, we found that
in climate change friends tended to be associated with sarcasm, or negative
experiences with friends. In general, these results are highly contextual with
regards to target and community.

## Discussion and Conclusion

This work investigated contentious discussions via case studies on three
highly divisive topics: abortion, climate change, and gun control.
Specifically, we operationalized a set of conversation characteristics —
discourse acts, toxicity, sentiment, LIWC, text representation, and user
factors — and demonstrated that these features contributed to both reasonable
predictive performances and explainability in understanding contentious in
conversations. In doing so we discovered that different linguistic and social
features have different relationships to contentiousness and the topics
themselves. Certain features — toxicity, prolificity, and discourse acts —
remained stable predictors of contentiousness regardless of the topic. Other
features — location, gender, sentiment, and LIWC — demonstrated highly
contextual information and differences in the ways in which users engage with
these contentious topics on Reddit. The findings suggest that contentiousness
in conversations is subtle and that explainable models are necessary to
identify these nuances for both moderators, users, and researchers.

### Implications

Our work has both theoretical and empirical guidelines to assist online media
in defining, measuring, understanding, and even intervening with contentious
content on social media platforms. First, our research provides conversational
characteristics of contentious conversations which enable deep, contextual
insights into the nature of these conversations. Second, our proposed generic
conversation characteristics can be easily applied to build predictive models
for understanding other similarly divisive topics beyond the three studied
here, since they are not tailored to any specific data.

Practically, this work could motivate the design of tracking/monitoring
systems that can visualize possibly contentious conversations in any specific
groups to assist community moderators with content moderation. In addition to
warning moderators early on in a post’s lifespan whether or not it is of
concern, this work has the ability to explain to moderators (and users) why a
post is considered contentious and has attracted increased attention. Since
the explanation of moderator action is a large contributor to user
satisfaction (?), the explanation portion of our research is key for ensuring
quality moderation strategies can be derived from our work. It also sheds
light on designing better features to inform new community members on what
aspects of the conversations might cause problematic experiences, allowing for
increased retention of newer users who are not exposed to contentious content
which they may not wish to engage with early on in their time on the
subreddit. Since there is some amount of subject-matter familiarity required
to properly explain the models we have proposed here, moderators which are
very familiar with the content that is posted to their communities naturally
fit this role. Our work will also help design novel tutorials to educate users
on how to engage in discussions, especially in divisive topics, in a more
civil and constructive manner. Particularly, these tutorials can be tailored
to individual topics with the use of this work and subject-matter experts.

### Generalization

Although our work focused only on three topics: abortion, climate change, and
gun control, many of our features and characteristics of these divisive topics
can generalize well to other topics. LIWC, toxicity, and sentiment have been
shown to be powerful predictors of contentiousness from previous work(?; ?;
?). We expect the patterns that we have observed regarding user factors,
excluding location, to generalize well. For gender, the strong correlation of
gender with contentiousness in the topic of abortion suggests that topics
which relate to gendered issues will hold a similar relationship to gender,
while non-gendered issues will continue to display little to no relationship
with gender. We do not expect that the same locations should emerge in a
generalized version of this work, although that locations which are relevant
to the issue being discussed will continue to appear. In a generalized version
of this work, less specific measures of location diversity (e.g., entropy) may
perform better. We also hypothesize that discourse acts will hold generally
for contentious conversations, and leave confirmation of that hypothesis to
future work.

### Limitations

Our work also has a few limitations. First, the distant-supervision method
that we use to label the contentiousness in posts relies on upvote ratio,
which may not be a perfectly accurate analogue for contentiousness, though
this has been adopted and validated by prior work (?). The Rediquette
(Reddit’s rules of conduct) specifically states that users are not to
“downvote an otherwise acceptable post because you do not personally like it.”
Although only a subset of users follow these recommendations, this indicates
that the downvote button is mainly reserved for posts that break rules or do
not belong on a particular subreddit. We partially circumvent this limitation
by controlling for topic. Future work can conduct further annotations to
validate the correlations between human judgement and our contentiousness
measures. Another important limitation of this work is its correlational
nature. We mainly characterize which features of conversations are correlated
with contentiousness but not what causes that relationship. However, without
true random-assignment experimentation we cannot assume that the relationships
described reflect any degrees of causation.

### Ethical Considerations

This research study has been approved by the Institutional Review Board (IRB)
at the researchers’ institution. In this work, we leverage no information that
was not publicly available at the time of data collection, leveraging the
public post histories of the users who participated in the conversations which
we studied. As such, user private information is not disclosed, and none of
the posts which were used to compute the gender and location features have
been saved or need to be saved during the course of this computation.

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