# Dynamics on nilpotent character varieties

Sean Lawton  Department of Mathematical Sciences, George Mason University,
4400 University Drive, Fairfax, Virginia 22030, USA<EMAIL_ADDRESS>and
Jean-Philippe Burelle  Département de mathématiques, Université de Sherbrooke,
Sherbrooke, QC J1K 2R1, Canada<EMAIL_ADDRESS>
###### Abstract.

Let $\mathsf{Hom}^{0}(\Gamma,G)$ be the connected component of the identity of
the variety of representations of a finitely generated nilpotent group
$\Gamma$ into a connected compact Lie group $G$, and let
$\mathsf{X}^{0}(\Gamma,G)$ be the corresponding moduli space. We show that
there exists a natural $\mathsf{Out}(\Gamma)$-invariant measure on
$\mathsf{X}^{0}(\Gamma,G)$ and that whenever $\mathsf{Out}(\Gamma)$ has at
least one hyperbolic element, the action of $\mathsf{Out}(\Gamma)$ on
$\mathsf{X}^{0}(\Gamma,G)$ is mixing with respect to this measure.

###### Key words and phrases:

character variety, ergodicity, nilpotent group

###### 2010 Mathematics Subject Classification:

Primary 22D40, 20F18, 22F50; Secondary 14L30, 37A25

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Abelian Groups are Goldman
  4. 4 Examples of Nilpotent Groups that are Goldman
  5. 5 Exotic Components in the Abelian Case
    1. 5.1 Action on commutators
  6. 6 Forni-Goldman Type Theorem for $\mathbb{Z}^{n}$

##  1\. Introduction

Let $\Gamma$ be a finitely presentable group and let $G$ be a connected
compact Lie group. Then, the set of group homomorphisms
$\mathsf{Hom}(\Gamma,G)$ is a topological space, independent of a choice of
presentation of $\Gamma$. One way to see this is to give $\Gamma$ the discrete
topology and consider the compact-open topology on the set of continuous
functions $C^{0}(\Gamma,G)$. This endows $\mathsf{Hom}(\Gamma,G)\subset
C^{0}(\Gamma,G)$ with the subspace topology. The group $\mathsf{Aut}(\Gamma)$
acts on $\mathsf{Hom}(\Gamma,G)$ via $\alpha\cdot\rho=\rho\circ\alpha^{-1}$.
This action descends to an $\mathsf{Out}(\Gamma)$-action on the conjugation
orbit space $\mathsf{X}(\Gamma,G):=\mathsf{Hom}(\Gamma,G)/G$ known as the
$G$-character variety of $\Gamma$.

When $\Gamma$ is the fundamental group of a closed orientable surface of genus
$g\geq 2$, there is an $\mathsf{Out}(\Gamma)$-invariant symplectic measure on
$\mathsf{X}(\Gamma,G)$ by [Gol84]. In [Gol97, PX02] it was shown that
$\mathsf{Out}(\Gamma)$ acts ergodically with respect to this measure.111Recent
work has been announced that the Torelli subgroup is sufficient to establish
ergodicity [Bou20]. This theorem was generalized to most “relative character
varieties” [PX03, GLX20], and for some $G$, it also holds for non-orientable
surface groups [Pal11]. When $\Gamma$ is a free group of rank 3 or greater,
again there is an invariant measure where $\mathsf{Out}(\Gamma)$ is ergodic
[Gol07, Gel08]. Given the above examples, it is natural to ask:

######  Question 1.

For what $\Gamma$ does there exist an $\mathsf{Out}(\Gamma)$-invariant measure
with full support on $\mathsf{X}(\Gamma,G)$? If so, does
$\mathsf{Out}(\Gamma)$ act ergodically for all connected compact Lie groups
$G$?

In [MP02] (see also [Pal11]) the first question is explored given a
presentation of $\Gamma$ satisfying certain conditions. In general,
$\mathsf{X}(\Gamma,G)$ may have many connected components, and
$\mathsf{Out}(\Gamma)$ permutes them, fixing the component containing the
trivial representation. In the case of hyperbolic surface groups and free
groups, the moduli space $\mathsf{X}(\Gamma,G)$ is connected and as mentioned
above $\mathsf{Out}(\Gamma)$ acts ergodically with respect to measures first
considered by Goldman. As such, we call any $\Gamma$ goldman if
$\mathsf{Out}(\Gamma)$ acts ergodically on the connected component of the
identity $\mathsf{X}^{0}(\Gamma,G)$ with respect to an invariant measure (for
any connected compact Lie group $G$). From this point of view, hyperbolic
surface groups and free groups of rank greater than 2 are goldman.

It is the purpose of this short note to show that finitely generated abelian
groups are goldman, and to prove many but not all finitely generated nilpotent
groups are goldman as well (we note that finitely generated nilpotent groups
are finitely presentable). Precisely, we prove:

######  Theorem 1.

Let $\Gamma$ be a finitely generated group and $G$ a compact connected Lie
group. If $\Gamma$ is nilpotent, then there exists a finite
$\mathsf{Out}(\Gamma)$-invariant measure on $\mathsf{X}^{0}(\Gamma,G)$ with
full support. If $\mathsf{Aut}(\Gamma)$ has a hyperbolic element $($in
particular, if $\Gamma$ is abelian$)$, then the $\mathsf{Out}(\Gamma)$ action
is strongly mixing.

Since strong mixing implies ergodicity, this immediately implies that the two
classes of groups in Theorem 1 are goldman if the conditions of the theorem
are satisfied. We also give examples of non-abelian nilpotent groups that do
in fact satisfy this condition, and some that do not.

We emphasize that Theorem 1 only requires the existence of a single hyperbolic
element 222In fact, we only require it has no eigenvalue a root of unity, see
Theorem 3. in $\mathsf{Aut}(\Gamma)$ whereas in the case of hyperbolic surface
groups, a single hyperbolic element is not sufficient to obtain ergodicity
[Bro96].

When character varieties of nilpotent groups have other connected components
than the one containing the trivial representation, the group
$\mathsf{Out}(\Gamma)$ acts by some permutation on these components. In
Section 5, we investigate the $\mathsf{Out}(\mathbb{Z}^{n})$ action on a
specific class of examples where the _exotic components_ have been completely
classified by Adem, Cohen, and Gómez [ACG13]. The components in these examples
are all isolated points, and we prove that the action of
$\mathsf{Out}(\mathbb{Z}^{n})$ is transitive.

We also show that the action of $\mathsf{Out}(\Gamma)$ does not generally fix
other components in the character variety by explicitly describing the action
in some special cases. Lastly, as a corollary of our main theorem, letting
$\phi_{t}$ denote the flow associated to any noncompact $1$-parameter subgroup
of $\mathsf{SL}(n,\mathbb{R})$, we define an associated flow
$\widehat{\phi}_{t}$ on the space

$F_{K}=(\mathsf{SL}(n,\mathbb{R})\times\mathsf{X}^{0}(\mathbb{Z}^{n},K))/\mathsf{SL}(n,\mathbb{Z}),$

and prove that $\widehat{\phi}_{t}$ is strongly mixing. This result should be
compared to the main theorem in [FG18].

### Acknowledgments

Burelle thanks the Institute des Hautes Études Scientifiques for hosting him
in 2019 when this work was started. Lawton is partially supported by a
Collaboration grant from the Simons Foundation, and also thanks the Institute
des Hautes Études Scientifiques for hosting him in 2019 and again in 2021 when
this work was advanced. We thank Lior Silberman for a very helpful
conversation about nilpotent automorphisms which allowed us to expand Section
4. We also thank David Fisher, Giovanni Forni, Ben McReynolds, and Mark Sapir
for helpful references and/or discussions.

##  2\. Preliminaries

The following definitions and lemmas are standard (see [Qua12] for a quick
summary).

######  Definition 1.

A measure-preserving transformation $T$ is a map from a measure space
$(X,\mu)$ to itself such that for each measurable subset $U$ of the space,
$\mu(T^{-1}(U))=\mu(U)$.

######  Definition 2.

A measure-preserving transformation $T$ is strong-mixing if, for any
measurable sets $A,B$ one has

$\lim_{n\to\infty}\mu(A\cap T^{-n}B)=\mu(A)\mu(B).$

The following two lemmas follow directly from definitions.

######  Lemma 1.

Let $T$ be strong-mixing on two finite measure spaces $X,Y$. Then the diagonal
action of $T$ on the product measure space $X\times Y$ is strong-mixing.

###### Proof.

Since the $\sigma$-algebra of measurable sets on $X\times Y$ is generated by
measurable sets of the form $A\times B$ for measurable sets $A\subset X$ and
$B\subset Y$, it suffices to check the definition for $A\times B$. Indeed:

$\displaystyle\lim_{n\to\infty}(\mu_{X}\times\mu_{Y})((A_{1}\times A_{2})\cap
T^{-n}(B_{1}\times B_{2}))$
$\displaystyle=\lim_{n\to\infty}(\mu_{X}\times\mu_{Y})((A_{1}\times
A_{2})\cap(T^{-n}B_{1}\times T^{-n}B_{2}))$
$\displaystyle=\lim_{n\to\infty}(\mu_{X}\times\mu_{Y})((A_{1}\cap
T^{-n}B_{1})\times(A_{2}\cap T^{-n}B_{2}))$
$\displaystyle=\lim_{n\to\infty}\mu_{X}(A_{1}\cap
T^{-n}B_{1})\mu_{Y}(A_{2}\cap T^{-n}B_{2})$
$\displaystyle=\mu_{X}(A_{1})\mu_{Y}(B_{1})\mu_{X}(A_{2})\mu_{Y}(B_{2})$
$\displaystyle=\mu_{X}(A_{1}\times B_{1})\mu_{Y}(A_{2}\times B_{2}).\qed$

######  Lemma 2.

Let $X$ be a finite measure space, $\pi:X\to Y$ is a measurable function where
$Y$ has the push-forward measure. Let $T_{X},T_{Y}$ be measure-preserving maps
on $X,Y$ respectively, such that $\pi\circ T_{X}=T_{Y}\circ\pi$. If $T_{X}$ is
strong-mixing on $X$, then $T_{Y}$ is strong-mixing on $Y$.

###### Proof.

Let $\mu_{X}$ be the measure on $X$ and $\mu_{Y}$ be the measure on $Y$. By
assumption, we have that $\mu_{Y}=\mu_{X}\circ\pi^{-1}$ and $\pi\circ
T_{X}=T_{Y}\circ\pi$. Then:

$\displaystyle\lim_{n\to\infty}\mu_{Y}(A\cap T_{Y}^{-n}B)$
$\displaystyle=\lim_{n\to\infty}\mu_{X}(\pi^{-1}(A\cap T_{Y}^{-n}B))$
$\displaystyle=\lim_{n\to\infty}\mu_{X}(\pi^{-1}A\cap\pi^{-1}T_{Y}^{-n}B)$
$\displaystyle=\lim_{n\to\infty}\mu_{X}(\pi^{-1}A\cap T_{X}^{-n}\pi^{-1}B)$
$\displaystyle=\mu_{X}(\pi^{-1}A)\mu_{X}(\pi^{-1}B)$
$\displaystyle=\mu_{Y}(A)\mu_{Y}(B).\qed$

######  Definition 3.

The transformation $T:X\rightarrow X$ is said to be ergodic if for any
invariant set $U\subset X$, either $\mu(U)=0$ or $\mu(X\setminus U)=0$.

It is easy to see that strong mixing implies ergodicity. Indeed, without loss
of generality assume $\mu$ is a probability measure. If $A$ is invariant, then
$\mu(A)=\lim_{n\to\infty}\mu(A\cap T^{-n}A)=\mu(A)^{2}$, and so $\mu(A)=0$ or
$\mu(A)=1=\mu(X)$.

##  3\. Abelian Groups are Goldman

Let $\Gamma$ be a finitely generated nilpotent group and let $K$ be a compact,
connected Lie group.

The quotient map $\Gamma\rightarrow\Gamma/[\Gamma,\Gamma]$ induces an
inclusion of representation spaces

$\iota:\mathsf{Hom}(\Gamma/[\Gamma,\Gamma],K)\rightarrow\mathsf{Hom}(\Gamma,K).$

Bergeron and Silberman showed that this inclusion restricts to a homeomorphism
between the connected components containing the trivial representation.

######  Theorem 2 ([BS16]).

$\iota:\mathsf{Hom}^{0}(\Gamma/[\Gamma,\Gamma],K)\rightarrow\mathsf{Hom}^{0}(\Gamma,K)$

is a homeomorphism.

Moreover, writing $\Gamma/[\Gamma,\Gamma]\cong\mathbb{Z}^{r}\oplus A$ where
$r=\mathrm{rank}(H^{1}(\Gamma,\mathbb{Z}))$ and $A$ is a finite abelian group,
any continuous deformation of the trivial representation must map $A$ to the
identity since Lie groups do not have small subgroups. Therefore,
$\mathsf{Hom}^{0}(\Gamma/[\Gamma,\Gamma],K)\cong\mathsf{Hom}^{0}(\mathbb{Z}^{r},K)$.

Since the Lie group $K$ is compact, the identity component of the character
variety $\mathsf{X}^{0}(\mathbb{Z}^{r},K)$ has a simple description (see
[Bai07, Remark 4]): for any representation $\rho:\mathbb{Z}^{r}\rightarrow K$,
the images of the generators

$\rho(e_{1}),\dots,\rho(e_{r})$

are simultaneously diagonalizable, and so

$\mathsf{X}^{0}(\mathbb{Z}^{r},K)\cong(U(1)^{k})^{r}/W,$

where $k=\mathrm{rank}(K)$ and $W$ is the Weyl group of $K$ acting diagonally
on $r$ copies of the maximal torus $U(1)^{k}$.

Consider an automorphism $F\in\mathsf{Aut}(\Gamma)$.

######  Definition 4.

We say $F$ is _hyperbolic_ if the induced automorphism
$\Gamma/[\Gamma,\Gamma]\rightarrow\Gamma/[\Gamma,\Gamma]$, also denoted by
$F$, is a hyperbolic automorphism of the free abelian part $\mathbb{Z}^{r}$.
Hyperbolic automorphisms of free abelian groups are those which have no
eigenvalue on the unit circle.

The automorphism $F$ induces an automorphism on the character variety by pre-
composition which we denote by
$\widehat{F}\in\mathsf{Aut}(\mathsf{X}(\Gamma,K))$. By the above observations,
$\mathsf{X}^{0}(\Gamma,K)\cong(U(1)^{k})^{r}/W$. We can write an element of
$(U(1)^{k})^{r}$ as a $k\times r$ matrix

$\Theta_{\rho}:=\begin{pmatrix}\theta_{1,1}&\dots&\theta_{1,r}\\\
\vdots&\ddots&\vdots\\\ \theta_{k,1}&\dots&\theta_{k,r}\end{pmatrix},$

where
$\theta_{i,j}=\frac{\arg(\rho(e_{j})_{i})}{2\pi}\in\mathbb{R}/\mathbb{Z}$.
Then, if the automorphism $F$ acts on the free part $\mathbb{Z}^{r}$ via the
matrix $M_{F}$, then $\widehat{F}$ acts by on the character variety by
$\Theta_{\rho}\mapsto\Theta_{\rho}M_{F}$ preserving $W$-orbits.

By assumption, $M_{F}$ has no eigenvalues of modulus $1$ and so
$\Theta_{\rho}\mapsto\Theta_{\rho}M_{F}$ is the diagonal action of a
_hyperbolic toral automorphism_ on $r$ copies of the $k$-torus $U(1)^{k}$.

The following lemma is well-known to dynamicists, and often left as an
exercise (see for example [KH95]). We include a proof here for the convenience
of the reader (who may not be a dynamicist).

######  Lemma 3.

An automorphism of the $n$-torus $T^{n}$ is strong mixing if and only if it
does not have an eigenvalue that is a root of unity.

###### Proof.

Let $M:T^{n}\rightarrow T^{n}$ be a toral automorphism with no eigenvalue a
root of unity. Let $f,g\in L^{2}(T^{n})$, it is sufficient to show that

$\int_{T^{n}}(f\circ
M^{k})g~{}d\mu\xrightarrow{k\rightarrow\infty}\int_{T^{n}}fd\mu\int_{T^{n}}g~{}d\mu,$

or equivalently, denoting the Hilbert inner product by $\langle\ ,\ \rangle$,

$\langle f\circ M^{k},g\rangle\xrightarrow{k\rightarrow\infty}\langle
f,1\rangle\langle g,1\rangle.$

This is because specializing $f,g$ to indicator functions of measurable sets
gives exactly the definition of strong mixing.

Denote by $e_{\vec{n}}(\vec{x})=e^{2\pi i\langle\vec{n},\vec{x}\rangle}$ the
standard Fourier Hilbert basis, where $\vec{n}\in\mathbb{Z}^{n}$.
Precomposition by $M^{k}$ permutes these basis vectors:

$e_{\vec{n}}(M^{k}\vec{x})=e^{2\pi
i\langle\vec{n},M^{k}\vec{x}\rangle}=e^{2\pi
i\langle(M^{\dagger})^{k}\vec{n},\vec{x}\rangle}=e_{(M^{\dagger})^{k}\vec{n}}(\vec{x}),$

where $\dagger$ is transpose.

Moreover, the $M$ orbit of every non-zero integer vector $\vec{n}$ is
unbounded, since if it were bounded then it would be finite and hence $M^{k}$
would have a fixed vector for some $k$, but by assumption $M$ has no
eigenvalue that is a root of unity.

Therefore, for non-constant basis vectors we have

$\langle e_{\vec{n}}\circ
T^{k},e_{\vec{m}}\rangle\xrightarrow{k\rightarrow\infty}0.$

This implies that for finite linear combinations of basis vectors, the
statement holds. Now, since the Fourier series for $f$ and $g$ converge in
$L^{2}$, if we denote the partial sums by $f_{n}$ and $g_{n}$ we have:

$\displaystyle|\langle f\circ M^{k},g\rangle-\langle f_{n}\circ
M^{k},g_{n}\rangle|$ $\displaystyle=|\langle(f-f_{n})\circ
M^{k},g\rangle+\langle f_{n}\circ M^{k},g-g_{n}\rangle|$
$\displaystyle\leq\left\lVert f-f_{n}\right\rVert\left\lVert
g\right\rVert+\left\lVert f_{n}\right\rVert\left\lVert
g-g_{n}\right\rVert\xrightarrow{n\rightarrow\infty}0$

and therefore, $\langle f\circ
M^{k},g\rangle\xrightarrow{k\rightarrow\infty}\langle f,1\rangle\langle
g,1\rangle$ as desired.

For the converse, if an automorphism $M:T^{n}\rightarrow T^{n}$ has an
eigenvalue which is a root of unity, then $M^{\dagger}$ also has an eigenvalue
which is a root of unity ($M$ and $M^{\dagger}$ have the same spectrum). Thus,
there exists $k\geq 1$ and non-zero $\vec{v}$ so that
$(M^{\dagger})^{k}\vec{v}=\vec{v}$. Hence $M^{\dagger}\vec{u}=\vec{u}$ for
$\vec{u}:=\sum_{i=0}^{k-1}(M^{\dagger})^{i}\vec{v}$. Then from our above
calculations:
$e_{\vec{u}}(M\vec{x})=e_{(M^{\dagger})\vec{u}}(\vec{x})=e_{\vec{u}}(\vec{x}).$
Thus, there is a non-constant invariant measurable function. Hence, the action
cannot be ergodic (since if it was ergodic all invariant measurable functions
would be constant), let alone strong mixing. ∎

We can now prove Theorem 1 from the Introduction, which follows from the
following slightly more general theorem:

######  Theorem 3.

Let $\Gamma$ be a finitely generated nilpotent group, and $G$ a compact
connected Lie group. Then $\mathsf{Out}(\Gamma)$ is strongly mixing on
$\mathsf{X}^{0}(\Gamma,G)$ whenever there exists an element in
$\mathsf{Aut}(\Gamma)$ that does not have any eigenvalue that is a root of
unity.

###### Proof.

First, if the free part of $\Gamma/[\Gamma,\Gamma]$ is empty, then
$\mathsf{X}^{0}(\Gamma,G)$ is a point (the identity representation), and the
theorem holds trivially.

Now assume that the free part of $\Gamma/[\Gamma,\Gamma]$ has rank $r\geq 1$.
Then by Theorem 2 and [Bai07, Remark 4]),
$\mathsf{X}^{0}(\Gamma,K)\cong(U(1)^{k})^{r}/W$.

Since a toral automorphism with no eigenvalue a root of unity is strong mixing
with respect to Lebesgue measure (Lemma 3), its diagonal action on a product
of tori is also strong mixing with respect to the product measure (Lemma 1).

The Weyl group acts on $(U(1)^{k})^{r}$ by Euclidean reflections, which
preserve Lebesgue measure, and so the $\mathsf{Out}(\Gamma)$-action on

$\mathsf{X}^{0}(\Gamma,K)\cong(U(1)^{k})^{r}/W$

has an invariant measure, induced by the product of Lebesgue measures on
$(U(1)^{k})^{r}$. Therefore, the action of $\widehat{F}$ on
$\mathsf{X}^{0}(\Gamma,K)\cong(U(1)^{k})^{r}/W$ is strong mixing with respect
to this measure by Lemma 2. ∎

######  Remark 1.

Whenever $\Gamma$ is abelian with non-trivial free part,
$\mathsf{Out}(\Gamma)$ contains a hyperbolic element.

######  Remark 2.

The “only if” part of Lemma 3 implies that if the group of automorphisms of
$\Gamma$ is unipotent then a single automorphism cannot be mixing nor ergodic
on $\mathsf{X}^{0}(\Gamma,G)$.

######  Remark 3.

It is known that if $\Gamma$ is finitely generated and torsion-free, then
$\mathsf{Out}(\Gamma)$ is non-trivial [Ree56]. It appears to be an open
problem whether there exists a _finitely generated_ nilpotent group with
trivial outer automorphism group (which does occur if $\Gamma$ is allowed to
be infinitely generated [Zal72]).

In the next section we provide large classes of examples of non-abelian
nilpotent groups that admit hyperbolic outer automorphisms, and some that do
not.

##  4\. Examples of Nilpotent Groups that are Goldman

If a subset $S$ of a group $G$ projects onto a generating set of the
abelianization $G/[G,G]$, then $S$ is said to weakly generate $G$. Every
generating set of $G$ is a weak generating set, but the converse is not always
true. For nilpotent groups, however, the converse holds :

######  Lemma 4.

[MKS04, Lemma 5.9] Let $S$ be a subset of a nilpotent group $G$. Then $S$
weakly generates $G$ if and only if $S$ generates $G$.

We will see this fact is very useful to understand which nilpotent groups
admit hyperbolic automorphisms.

######  Example 1.

For a group $\Gamma$, the lower central series is defined by
$\ell_{0}(\Gamma)=\Gamma$ and $\ell_{i+1}(\Gamma)=[\Gamma,\ell_{i}(\Gamma)]$
where $[X,Y]$ is the commutator subgroup generated by $X,Y\leq\Gamma$. A group
is nilpotent, by definition, if the series

$\Gamma=\ell_{0}(\Gamma)\geq\ell_{1}(\Gamma)\geq\ell_{2}(\Gamma)\geq\cdots$

terminates, that is, for some $s$, $\ell_{s}(\Gamma)$ is the trivial group
$1$. When $s$ is the minimal such integer we say $\Gamma$ is nilpotent of step
size $s$. The free nilpotent group of step size $s$ and of rank $r$ is the
quotient group $N_{s,r}:=F_{r}/\ell_{s+1}(F_{r})$ where $F_{r}$ is a free
group of rank $r$.

The abelianization homomorphism $N_{s,r}\to
N_{r,s}/[N_{s,r},N_{s,r}]\cong\mathbb{Z}^{r}$ induces a homomorphism
$\varphi:\mathsf{Out}(N_{s,r})\to\mathsf{GL}(r,\mathbb{Z})$. Lemma 4 implies
$\varphi$ is surjective, and hence there are hyperbolic automorphisms of
$N_{s,r}$ for all $s\geq 1$ and $r\geq 1$.

######  Example 2.

The $(2n+1)$-dimensional discrete Heisenberg group is the group is
$H_{2n+1}(\mathbb{Z}):=\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}$
with the group law:

$(\mathbf{a}_{1},\mathbf{b}_{1},c_{1})*(\mathbf{a}_{2},\mathbf{b}_{2},c_{2})=(\mathbf{a}_{1}+\mathbf{a}_{2},\mathbf{b}_{1}+\mathbf{b}_{2},c_{1}+c_{2}+(\mathbf{a}_{1}\cdot\mathbf{b}_{2}-\mathbf{a}_{2}\cdot\mathbf{b}_{1})).$

Since $\\{(\mathbf{0},\mathbf{0},c)\in H_{2n+1}(\mathbb{Z})\ |\
c\in\mathbb{Z}\\}$ is central in $H_{2n+1}(\mathbb{Z})$, and
$\mathbf{a}_{1}\cdot\mathbf{b}_{2}-\mathbf{a}_{2}\cdot\mathbf{b}_{1}$ defines
a symplectic form any automorphism must preserve, the abelianization
homomorphism

$H_{2n+1}(\mathbb{Z})\to
H_{2n+1}(\mathbb{Z})/[H_{2n+1}(\mathbb{Z}),H_{2n+1}(\mathbb{Z})]$

induces a homomorphism
$\mathsf{Out}(H_{2n+1}(\mathbb{Z}))\to\mathsf{Sp}(2n,\mathbb{Z}).$ Every
$M\in\mathsf{Sp}(2n,\mathbb{Z})$ defines an automorphism of
$H_{2n+1}(\mathbb{Z})$ by

$M(\mathbf{a},\mathbf{b},c):=(M(\mathbf{a},\mathbf{b}),c)$

since it preserves the group law. Thus
$\mathsf{Out}(H_{2n+1}(\mathbb{Z}))\to\mathsf{Sp}(2n,\mathbb{Z})$ is an
epimorphism and we conclude there exists hyperbolic elements in
$\mathsf{Out}(H_{2n+1}(\mathbb{Z}))$.

######  Remark 4.

In the case that $n=1$ we can say more. In [Osi15], it is shown that
$\mathsf{Out}(H_{3}(\mathbb{Z}))\cong\mathsf{GL}(2,\mathbb{Z})\cong\mathsf{SL}(2,\mathbb{Z})\ltimes(\mathbb{Z}/2\mathbb{Z})$.
This is consistent with our prior observations since
$\mathsf{SL}(2,\mathbb{Z})\cong\mathsf{Sp}(2,\mathbb{Z})$ and
$H_{3}(\mathbb{Z})\cong N_{2,2}$.

######  Corollary 1.

The action of $\mathsf{Out}(\Gamma)$ on $\mathsf{X}^{0}(\Gamma,K)$ is ergodic
with respect to Lebesgue measure whenever $\Gamma$ is a free nilpotent group,
or a discrete Heisenberg group.

###### Proof.

Theorem 3 and Examples 1 and 2 establish the corollary. ∎

Moreover, there exists finitely generated nilpotent groups that do not admit
hyperbolic automorphisms as the next example shows.

######  Example 3.

Dyer [Dye70] shows there exists a nilpotent Lie algebra whose automorphism
group is strictly unipotent. As Dyer says in his introduction “for connected
simply connected nilpotent Lie groups the choice of group or algebra language
is a matter of taste.” Consequently, by taking $\mathbb{Z}$-points of the
corresponding Lie group that Dyer considers, we have a finitely generated
nilpotent $\Gamma$ whose automorphism group is strictly unipotent, hence there
can be no hyperbolic automorphism in that case.

######  Remark 5.

Dyer [Dye70] proves that all automorphisms in his example are unipotent by
showing they are trivial on the abelianization. Thus, Dyer’s example shows
that there are examples where the corresponding action on
$\mathsf{X}^{0}(\Gamma,G)$ is not ergodic and hence we know that the class of
finitely generated nilpotent groups are not all goldman.

######  Example 4.

The following example was communicated to us by Lior Silberman. Let $G$ be a
connected reductive affine algebraic group defined over $\mathbb{Z}$, and
$P<G$ a parabolic subgroup. Then $P=LU$ where $L$ is the Levi factor of $P$,
and $U$ is the unipotent radical of $P$. Since $P\cong L\ltimes U$, $L$ acts
on $U$ by conjugation $($and hence are automorphisms$)$. In general, unipotent
Lie groups $($having all elements unipotent$)$ are nilpotent groups.

The set of $\mathbb{Z}$-points of $U$ forms a finitely generated nilpotent
group, and the set of $\mathbb{Z}$-points of $L$ acts on $U$ by $($non-
inner$)$ automorphisms. Since $L$ is reductive, its locus of
$\mathbb{Z}$-points generally contains hyperbolic elements.

######  Example 5.

As pointed out to us by David Fisher, Nilmanifolds with Anosov diffeomorphisms
generally give examples too. See for example [LW09] and the references
therein.

The fact that there exist examples of finitely generated nilpotent $\Gamma$
where $\mathsf{Out}(\Gamma)$ does not satisfy the conditions of Theorem 3 does
not a priori rule out the possibility that the action on
$\mathsf{X}^{0}(\Gamma,G)$ is ergodic. So we pose:

######  Question 2.

Is there an example of a finitely generated nilpotent group $\Gamma$ such that
$\mathsf{Out}(\Gamma)$ is unipotent and the action on
$\mathsf{X}^{0}(\Gamma,G)$ is ergodic?

Given the example by Dyer, we suspect that answer is no, and so formulate:

######  Conjecture 1.

Let $\Gamma$ be a finitely generated nilpotent group, and $G$ a connected
compact Lie group. Then $\mathsf{Out}(\Gamma)$ acts ergodically on
$\mathsf{X}^{0}(\Gamma,G)$ with respect to Lebesgue measure if and only if
$\mathsf{Out}(\Gamma)$ contains an element with no eigenvalue a root of unity.

##  5\. Exotic Components in the Abelian Case

In this section we describe the action on the non-identity components in
special cases where they can be explicitly computed, as investigated in
[ACG13]. Let $p$ be prime and $\mathbb{Z}_{p}$ be the cyclic group of order
$p$. We will use the same notation for center of $\mathsf{SU}(p)$ which is
realized as the subgroup of scalar matrices with values $p$-th roots of unity.
Let $\Delta(p)$ be the diagonal embedding of $\mathbb{Z}_{p}$ in
$(\mathbb{Z}_{p})^{m}=Z(\mathsf{SU}(p)^{m})$. Let
$G_{m,p}:=\mathsf{SU}(p)^{m}/\Delta(p)$. For example,
$G_{1,p}=\mathrm{PU}(p)$.

The main theorems in [ACG13] explicitly describe the components in
$\mathsf{Hom}(\mathbb{Z}^{r},G_{m,p})$ and
$\mathsf{X}(\mathbb{Z}^{r},G_{m,p})$. In particular,
$\mathsf{X}(\mathbb{Z}^{r},G_{m,p})$ consists of the identity component

$\mathsf{X}^{0}(\mathbb{Z}^{r},G_{m,p})\cong
T^{r}/W\cong(S^{1})^{(p-1)rm}/(\Sigma_{p})^{m}$

where $\Sigma_{p}$ is the symmetric group on $p$ letters, and

$\frac{p^{(m-1)(r-2)}(p^{r}-1)(p^{r-1}-1)}{p^{2}-1}$

discrete points.

A priori, the only component we can generally say is fixed by

$\mathsf{Out}(\mathbb{Z}^{r})\cong\mathsf{GL}(r,\mathbb{Z})$

is the identity component. Each isolated point in
$\mathsf{X}(\mathbb{Z},G_{m,p})$ corresponds to a path-component in
$\mathsf{Hom}(\mathbb{Z}^{r},G_{m,p})$; all of which are isomorphic to the
homogeneous space $\mathsf{SU}(p)^{m}/(\mathbb{Z}_{p}^{m-1}\times E_{p})$
where $E_{p}\subset\mathsf{SU}(p)$ is isomorphic to the quaternion group
$Q_{8}$ if $p$ is even and the group of triangular $3\times 3$ matrices over
the $\mathbb{Z}_{p}$, with 1’s on the diagonal when $p$ is odd (either way it
is “extra-special” of order $p^{3}$).

The action of $\mathsf{GL}(r,\mathbb{Z})$ on the character variety lifts to an
action on $\mathsf{Hom}(\mathbb{Z}^{r},G_{m,p})$ and so we study this to
understand the dynamics.

###  5.1. Action on commutators

Pairs of elements $x,y\in\mathsf{SU}(p)^{m}$ such that the commutator
$[x,y]=xyx^{-1}y^{-1}$ belongs to the central group
$\Delta(p)\cong\mathbb{Z}_{p}$ are called _almost-commuting_ elements. An
$r$-tuple of pairwise almost-commuting elements
$x_{1},\dots,x_{r}\in\mathsf{SU}(p)^{m}$ commutes in the quotient $G_{m,p}$
and therefore defines a representation of $\mathbb{Z}^{r}$.

The two results from [ACG13] that we will use are the following:

######  Lemma 5 (Lemma 8 of [ACG13]).

Let $x,y\in\mathsf{SU}(p)^{m}$ be a pair of almost-commuting elements with
$[x,y]=c\neq 1$. Assume $z$ almost-commutes with $x$ and $y$, and let
$[x,z]=c^{b}$ and $[y,z]=c^{a}$ for integers $0\leq a,b<p$. Then, there exists
$w\in Z(SU(p)^{m})$ such that $z=wx^{-a}y^{b}$.

######  Corollary 2.

Let $x_{1},\dots,x_{r}\in\mathsf{SU}(p)^{m}$ be almost-commuting elements,
$Z_{ij}:=[x_{i},x_{j}]$ for all $1\leq i,j\leq r$, and denote $Z=(Z_{ij})$ the
$r\times r$ matrix of their commutators. Then, the first two rows of $Z$ can
take arbitrary values in $\Delta(p)$, and given the two first rows the rest of
the matrix $Z$ is uniquely determined.

In order to compute the action of an outer automorphism of $\mathbb{Z}^{r}$ on
representations, we first prove a general lemma about its action on
commutators of almost-commuting elements.

Let $G$ be a group, and let $A_{1},\dots,A_{r}\in G$ such that the commutators
$[A_{i},A_{j}]=z_{i,j}\in Z(G)$ are central.

If we use additive notation for $Z(G)$ since it is abelian, then the matrix
$Z:=[A,A]$, whose entry in position $(i,j)$ is $[A_{i},A_{j}]$, is skew-
symmetric.

For $M\in\mathsf{GL}(r,\mathbb{Z})$, define
$A_{i}^{M}=A_{1}^{M_{i,1}}A_{2}^{M_{i,2}}\cdots A_{r}^{M_{i,r}}$, and
$A^{M}=(A_{1}^{M},\dots,A_{r}^{M})$.

######  Lemma 6.

$[A^{M},A^{M}]=A^{t}ZA$

###### Proof.

By a direct computation,

$\displaystyle A_{i}^{M}A_{j}^{M}=A_{1}^{M_{i,1}}A_{2}^{M_{i,2}}\cdots
A_{r}^{M_{i,r}}A_{1}^{M_{j,1}}A_{2}^{M_{j,2}}\cdots A_{r}^{M_{j,r}}=$
$\displaystyle\left(\sum_{k=1}^{r-1}M_{i,r}M_{j,k}z_{r,k}\right)A_{1}^{M_{i,1}}A_{2}^{M_{i,2}}\cdots
A_{r-1}^{M_{i,r-1}}A_{1}^{M_{j,1}}A_{2}^{M_{j,2}}\cdots
A_{r}^{M_{j,r}}A_{r}^{M_{i,r}}$ $\displaystyle=\cdots$
$\displaystyle=\left(\sum_{l=1}^{r}\sum_{k=1}^{r}M_{i,l}M_{j,k}z_{l,k}\right)A_{j}^{M}A_{i}^{M}$
$\displaystyle=(MZM^{t})_{i,j}A_{j}^{M}A_{i}^{M}$

∎

Now let us assume that the center $Z(G)$ is a finite abelian group $A$. Notice
that the action of $\mathsf{GL}(r,\mathbb{Z})$ on the skew-symmetric matrix
$Z$ above is the usual action on skew-symmetric bilinear forms. Therefore, for
any such matrix $Z$ with values in $A$, there exists
$M\in\mathsf{GL}(r,\mathbb{Z})$ such that $M^{t}ZM$ has the standard form:

$M^{t}ZM=\begin{pmatrix}0&a_{1}&0&0&\dots&0\\\ -a_{1}&0&0&0&\dots&0\\\
0&0&0&a_{2}&\dots&0\\\ 0&0&-a_{2}&0&\dots&0\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&0&\dots&0\\\
\end{pmatrix}.$

For the specific case of $G=G_{m,p}$, the center is $\mathbb{Z}_{p}$ for some
prime $p$.

######  Theorem 4.

Suppose $r\geq 3$. Then,
$\mathsf{Out}(\mathbb{Z}^{r})=\mathsf{GL}(r,\mathbb{Z})$ acts transitively on
the exotic components of $\mathsf{X}(\mathbb{Z}^{r},G_{m,p})$.

###### Proof.

The main results of [ACG13] show that the exotic components are uniquely
determined by the first two rows of the matrix of commutators $Z$, and that
these two rows can take any possible values in $\mathbb{Z}_{p}$.

Let $x_{1},\dots,x_{r}\in\mathsf{SU}(p)^{m}$ be almost-commuting elements with
matrix of commutators $Z$, and assume that they do not all commute. By the
remarks above, up to the action of $\mathsf{GL}(r,\mathbb{Z})$ we may assume
that the first two rows of $Z$ are

$\begin{pmatrix}0&a_{1}&0&0&\dots&0\\\ -a_{1}&0&0&0&\dots&0\\\ \end{pmatrix},$

where $a_{1}\neq 0$.

By Lemma 5, if those are the first two rows of the matrix of commutators, then
every other entry of the matrix $Z$ must be $0$.

Define

$M=\begin{pmatrix}a_{1}^{-1}&0\dots&0&0\\\ 0&1&\dots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\dots&1&0\\\ 0&0&\dots&0&a_{1}\\\
\end{pmatrix}\in\mathsf{GL}(r,\mathbb{Z}_{p}),$

and let $\hat{M}\in\mathsf{GL}(r,\mathbb{Z})$ be a lift of $M$. Then, the
first two rows of $\hat{M}^{t}Z\hat{M}$ are

$\begin{pmatrix}0&1&0&0&\dots&0\\\ -1&0&0&0&\dots&0\\\ \end{pmatrix},$

and so every exotic component is in the orbit of the one determined by this
matrix of commutators, and the action of $\mathsf{GL}(n,\mathbb{Z})$ is
transitive.

∎

######  Remark 6.

In characteristic two, the classification of skew-symmetric bilinear forms is
different than in odd characteristic. However, in our setting, we know that
the diagonal of the matrix $Z$ is $0$ since a group element always commutes
with itself. This implies that the normal form for $Z$ given above applies to
all characteristics.

The following proposition describes the case $r=2$.

######  Proposition 1.

If $p>2$, the group $\mathsf{Out}(\mathbb{Z}^{2})=\mathsf{GL}(2,\mathbb{Z})$
permutes the exotic components $\mathsf{X}(\mathbb{Z}^{r},G_{m,p})$ in pairs.
If $p=2$, it fixes every component.

###### Proof.

In this case, there is only one non-trivial commutator $c\in\mathbb{Z}_{p}$,
and the action of an automorphism $M\in\mathsf{GL}(2,\mathbb{Z})$ maps $c$ to
$\det(M)c$. Since the only invertible elements in $\mathbb{Z}$ are $\pm 1$,
the orbit of the exotic component with commutator $c$ is made of the two
components with commutators $\pm c$, which are distinct when $p>2$ but equal
when $p=2$. ∎

##  6\. Forni-Goldman Type Theorem for $\mathbb{Z}^{n}$

In [FG18], Forni and Goldman define a bundle $\mathbb{U}\mathfrak{M}(S)$ over
the unit sphere bundle of the Riemann moduli space associated to a surface
$S$, with fibers the character variety $\mathsf{X}(\pi_{1}(S),G)$ into a Lie
group $G$. They define a flow on $\mathbb{U}\mathfrak{M}(S)$ lifting the
Teichmüller geodesic flow in order to create a continuous version of the
mapping class group dynamics on character varieties of surfaces, and prove
that this flow is ergodic.

By analogy with this construction, we consider the flow of a $1$-parameter
subgroup on $\mathsf{SL}(n,\mathbb{R})/\mathsf{SL}(n,\mathbb{Z})$. The group
$\mathsf{SL}(n,\mathbb{Z})$ is a lattice in $\mathsf{SL}(n,\mathbb{R})$. We
can therefore use Moore’s ergodicity theorem :

######  Theorem 5.

Let $\Gamma\subset G$ be an irreducible lattice. If $H\subset G$ is a closed
noncompact subgroup, then $H$ acts ergodically on $G/\Gamma$ $($with respect
to Haar measure$).$

Let $\phi_{t}$ denote the flow associated to any noncompact $1$-parameter
subgroup of $\mathsf{SL}(n,\mathbb{R})$. Then, by Theorem 5 the $\phi_{t}$
action is ergodic on $\mathsf{SL}(n,\mathbb{R})/\mathsf{SL}(n,\mathbb{Z})$.

Let $K$ be a compact connected Lie group and consider the associated space

$F_{K}:=(\mathsf{SL}(n,\mathbb{R})\times\mathsf{X}^{0}(\mathbb{Z}^{n},K))/\mathsf{SL}(n,\mathbb{Z}),$

where $\mathsf{SL}(n,\mathbb{Z})$ acts diagonally by right multiplication on
the first factor and by outer automorphisms on the second. Then, $F_{K}$ is a
flat bundle over the quotient
$\mathsf{SL}(n,\mathbb{R})/\mathsf{SL}(n,\mathbb{Z})$ with fiber
$\mathsf{X}^{0}(\mathbb{Z}^{n},K)$.

We lift the flow $\phi_{t}$ to a flow $\widehat{\phi}_{t}$ on $F_{K}$ which is
trivial on the second factor, and equal to $\phi_{t}$ on the first. This flow
is a continuous analog of the outer automorphism action on the character
variety $\mathsf{X}^{0}(\mathbb{Z}^{n},K)$.

Following [FG18] exactly we have:

######  Theorem 6.

The action of $\widehat{\phi}_{t}$ on $F_{K}$ is strong mixing.

###### Proof.

The product of the Haar measure on $\mathsf{SL}(n,\mathbb{R})$ with the
Lebesgue measure defined in Section 3 on $\mathsf{X}^{0}(\mathbb{Z}^{n},K)$
defines a $\widehat{\phi}_{t}$ invariant measure on
$\mathsf{SL}(n,\mathbb{R})\times\mathsf{X}^{0}(\mathbb{Z}^{n},K)$, which
passes down to an invariant probability measure on the quotient $F_{K}$.

The $\phi_{t}\times\mathsf{SL}(n,\mathbb{Z})$-action on
$\mathsf{SL}(n,\mathbb{R})$ is ergodic. If we act by
$\phi_{t}\times\mathsf{SL}(n,\mathbb{Z})$ on
$\mathsf{X}^{0}(\mathbb{Z}^{n},K)$ with the first factor acting trivially, we
obtain a strongly mixing action by Theorem 1. The _multiplier criterion_
[GW16] for weak mixing states the action of a group on a product $X\times Y$
of probability spaces is ergodic if the action on $X$ is ergodic and the
action on $Y$ is weakly mixing. We deduce that the
$\widehat{\phi}_{t}\times\mathsf{SL}(n,\mathbb{Z})$-action on
$\mathsf{SL}(n,\mathbb{R})\times\mathsf{X}^{0}(\mathbb{Z}^{n},K)$ is ergodic,
and hence that the $\widehat{\phi}_{t}$ action on $F_{K}$ is ergodic.

Now, the flow $\widehat{\phi}_{t}$ extends to an action of the whole Lie group
$\mathsf{SL}(n,\mathbb{R})$. As a corollary of the Howe-Moore Theorem on
vanishing of matrix coefficients, any measure-preserving ergodic action of
$\mathsf{SL}(n,\mathbb{R})$ on a probability space is strongly mixing and
restricts to a strongly mixing action on any noncompact closed subgroup (see
for instance [BM00]). We conclude that the flow $\widehat{\phi}_{t}$ is
strongly mixing.

∎

With the above definitions, we have converted an action of
$\mathsf{SL}(n,\mathbb{Z})$ into a flow which reflects its dynamical
properties. A natural invariant of a flow is its entropy, and so we ask:

######  Question 3.

Given a choice of 1-parameter subgroup, are there natural bounds for the
entropy of the flow $\widehat{\phi}_{t}$?

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11institutetext: Guangxi Key Laboratory for Relativistic Astrophysics, School
of Physical Science and Technology, Guangxi University, Nanning 530004, China;
<EMAIL_ADDRESS>22institutetext: Key Laboratory of Space Astronomy and
Technology, National Astronomical Observatories, Chinese Academy of Sciences,
Beijing 100101, China<EMAIL_ADDRESS>33institutetext: Department of
Astronomy, University of California, Berkeley, CA 94720-3411, USA
44institutetext: School of Astronomy and Space Science, University of Chinese
Academy of Sciences, Beijing, People’s Republic of China 55institutetext:
Miller Institute for Basic Research in Science, University of California,
Berkeley, CA 94720, USA

# B3 0749+460A: A New “Changing-Look” Active Galactic Nucleus Associated with
X-ray Spectral Slope Variations

J. Wang 1122 W. K. Zheng 33 D. W. Xu 2244 T. G. Brink 33 A. V. Filippenko 3355
C. Gao 112244 S. S. Sun 112244 J. Y. Wei 2244

###### Abstract

Here we report an identification of B3 0749+460A as a new double-peaked local
“changing-look” active galactic nucleus (CL-AGN) in terms of our multi-epoch
spectroscopic analysis. By comparing our new spectra taken in 2021 with the
ones taken by the Sloan Digital Sky Survey in 2004, BOSS in 2013, and MaNGA in
2016, we reveal type transitions of Seyfert (Sy)
1.9$\rightarrow$Sy1.8$\rightarrow$Sy1.9. In the transitions, the classical
broad H$\alpha$ emission fades away since 2013, and disappears in our 2021
spectrum, although the absence of broad H$\beta$ can be traced back to at
least 2016. A follow-up observation in X-rays by the Swift/XRT reveals that
(1) the X-ray emission level gradually decreases since 2005; and (2) the X-ray
spectrum is soft in the optical “turn-off” state and hard in the ‘turn-on”
state. We argue that the disappearance of the classical broad H$\alpha$
emission can be likely explained by the disk-wind broad-line-region model, in
which the CL phenomenon is sensitive to luminosity in individual AGNs.

###### keywords:

galaxies: Seyfert — galaxies: nuclei — X-rays: galaxies — quasars: emission
lines — quasars: individual (B3 0749+460A)

##  1 Introduction

Depending on the existence of broad Balmer emission lines (full width at half-
maximum intensity (FWHM) $\gtrsim 2000$ km s-1) in their optical spectra,
active galactic nuclei (AGNs) can be classified into Type 1 and Type 2. The
latter lack broad Balmer emission lines in their spectra, widely understood by
the unified model (e.g., Antonucci 1993) in which its central engine is
obscured by the dusty torus along the line of sight to an observer. This
orientation-based model has, however, recently been challenged by the rarely
discovered “changing-look” (CL) phenomenon, in which some AGNs show a spectral
transition between Type 1, intermediate, and Type 2 within a timescale of
years to decades (e.g., MacLeod et al. 2010, 2016; Shapovalova et al. 2010;
Shappee et al. 2014; LaMassa et al. 2015; McElroy et al. 2016; Ruan et al.
2016; Runnoe et al. 2016; Parker et al. 2016; Gezari et al. 2017; Sheng et al.
2017, 2020; Stern et al. 2018; Yang et al. 2018; Wang et al. 2018b, 2019,
2020a; Frederick et al. 2019; Trakhtenbrot et al. 2019; Yan et al. 2019; Guo
et al. 2019; Ai et al. 2020; Graham et al. 2020; Kollatschny et al. 2018,
2020). The widely accepted standard disk model has also been challenged by the
CL phenomenon in terms of the viscosity crisis (e.g., Lawrence 2018, and
references therein).

The physical origin of CL-AGNs is still an open question, although the
scenario involving clumpy obscuration can be almost entirely excluded by light
echos in the mid-infrared (e.g., Sheng et al. 2017) and by spectropolarimetry
(e.g., Hutsemekers et al. 2019). There is, in fact, some evidence supporting
the idea that the CL phenomenon results from a variation of the accretion
power of a supermassive black hole (SMBH, e.g., H. Feng et al. 2021), even
though the underlying physical connection is still poorly understood. On the
one hand, some previous studies argue that CL-AGNs tend to be biased toward
the low Eddington ratio end ($L/L_{\mathrm{Edd}};L_{\mathrm{Edd}}=1.26\times
10^{38}M_{\mathrm{BH}}/M_{\odot}\ \mathrm{erg\ s^{-1}}$ is the Eddington
luminosity; e.g., MacLeod et al. 2019; Wang et al. 2020b), which is roughly
consistent with the expectation of the disk-wind broad-line region (BLR) model
(e.g., Elitzur & Ho 2009; Nicastro 2000; Elitzur & Shlosman 2006).

On the other hand, Ruan et al. (2019) and Ai et al. (2020, and references
therein) argue that the CL phenomenon might be related to an accretion-state
transition similar to that occurring in X-ray binaries (XRBs), after taking
into account a “V”-shaped correlation between X-ray hardness and
$L/L_{\mathrm{Edd}}$ identified in a few CL-AGNs. This conclusion, however, is
argued against by Wang et al. (2020b), who found that the CL phenomenon
identified in UGC 3223 is related to its X-ray emission level, rather than to
its X-ray hardness ratio.

At the current stage, understanding of the CL phenomenon is greatly hampered
by the scarcity of identified cases; there are only $\sim 100$ CL-AGNs
identified by multi-epoch spectroscopy. Among these, there are just 8 objects
showing repeat type transitions (e.g., Parker et al. 2019; Wang et al. 2020a;
Marin et al. 2019, and references therein), and only a couple of cases with a
comparison study in X-rays.

Hon et al. (2020) recently reported a significant line-profile variation in
the local AGN B3 0749+460A. In this paper, we identify the object as a new
double-peaked CL-AGN thanks to our elaborate spectral analysis, and we report
a follow-up observation in X-rays, which enables us to claim an X-ray spectral
slope-dependent CL phenomenon. The paper is organized as follows. Section 2
presents our optical spectroscopic and X-ray observations, along with the data
reduction. The spectral analysis in both optical and X-rays is described in
Section 3. Sections 4 and 5 present the results and discussion, respectively.
A significant line-profile variation in this object was reported by Hon et al.
(2020), who argue that it is an unusual object; we include a comparison with
this study. A $\Lambda$CDM cosmological model with parameters
H${}_{0}=70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$, $\Omega_{\mathrm{m}}=0.3$, and
$\Omega_{\Lambda}=0.7$ is adopted throughout.

##  2 Observations and Data Reduction

B3 0749+460A (=NPM1G+46.0092, $\alpha=\mathrm{07^{h}52^{m}44^{s}.2}$,
$\delta=\mathrm{+45\degr 56\arcmin 58\arcsec}$; J2000) is a local AGN at a
redshift of $z=0.0518$. The object was classified as a Seyfert 1.9 galaxy in
the catalog of quasars and active nuclei (12th ed.; Veron-Cetty & Veron 2006)
and the NASA/IPAC Extragalactic Database
(NED)111http://ned.ipac.caltech.edu/.. It is part of a sample of CL-AGN
candidates we selected by cross-matching the local partially obscured AGNs
extracted from the Sloan Digital Sky Survey (SDSS) DR7 with the Wide-field
Infrared Survey Explorer (WISE) catalog (Wright et al. 2010).

###  2.1 Optical Spectroscopy

####  2.1.1 Observations

We performed long-slit spectroscopic observations with the 2.16 m telescope
(Fan et al. 2016) at the Xinglong Observatory of the National Astronomical
Observatories, Chinese Academy of Sciences (NAOC) on 2021 January 12 (UT dates
are used throughout this paper), and with the 3 m Shane telescope at Lick
Observatory on 2021 January 21. The first spectrum was taken with the Beijing
Faint Object Spectrograph and Camera (BFOSC) that is equipped with a back-
illuminated E2V55-30 AIMO CCD. We used a long slit of width 2 oriented in the
north–south direction. Our spectral resolution was $\sim 10$ Å and the
wavelength coverage was 3850–8200 Å. The exposure time was $2\times 2400$ s.
Wavelength calibration was carried out with spectra of iron–argon comparison
lamps. In order to minimize the effects of atmospheric dispersion (Filippenko
1982), the spectrum was obtained as close to the meridian as possible.

The second spectrum was obtained with the Kast double spectrograph (Miller &
Stone 1993) mounted on the 3 m Shane telescope. Grism 600/4310 was used on the
blue side and grating 300/7500 on the red side, providing respective
resolutions of $\sim 5$ Å and $\sim 12$ Å and a range of 3600–10,700 Å. The
long 2 slit was aligned near the parallactic angle (Filippenko 1982) to
minimize differential light losses caused by atmospheric dispersion, and the
exposure time was 1500 s.

Flux calibration of both spectra was carried out with observations of Kitt
Peak National Observatory standard stars (Massey et al. 1988).

####  2.1.2 Data Reduction

One-dimensional (1D) spectra were extracted from the raw images by using the
IRAF222IRAF is distributed by NOAO, which is operated by AURA, Inc., under
cooperative agreement with the U.S. National Science Foundation (NSF). package
and standard procedures, including bias subtraction and flat-field correction.
Both extracted 1D spectra were then calibrated in wavelength and in flux with
the corresponding comparison lamp and standard stars. The accuracy of the
wavelength calibration is better than 1 Å for the Kast spectrum and better
than 2 Å for the BFOSC spectrum. The telluric A-band (7600–7630 Å) and B-band
(around 6860 Å) absorption produced by atmospheric $\mathrm{O_{2}}$ molecules
were removed from both spectra by using the observation of the corresponding
standard star.

Each calibrated spectrum was then corrected for the Galactic extinction of
$A_{V}=0.226$ mag (Schlafly & Finkbeiner 2011) taken from the NED. The
correction was applied by assuming the $R_{V}=3.1$ extinction law of our
Galaxy (Cardelli et al. 1989). Both spectra were then transformed to the rest
frame. The two resulting spectra are displayed in Figure 1 (lower two curves).

Figure 1: A comparison between the rest-frame spectra of B3 0749+460A taken at
different epochs. The two new spectra obtained with the Xinglong 2.16 m
telescope and the Shane 3 m telescope are displayed by the lower two curves.
The three spectra extracted from the SDSS DR7, BOSS, and MaNGA spectral
database are shown by the upper three curves. The spectra are vertically
shifted by an arbitrary amount for clarity.

###  2.2 X-ray Follow-up Observation

We proposed X-ray and ultraviolet (UV) follow-up observations of the object at
the beginning of 2021 by using the Neil Gehrels Swift Observatory (Gehrels et
al. 2004) X-ray telescope (XRT) and Ultraviolet/Optical Telescope (UVOT). The
object was targeted (ObsID=00037357002) by XRT and UVOT simultaneously on 2021
February 12. The exposure times were 1400 s in XRT Photon Counting (PC) mode
and 1400 s for the UVOT. The UVOT image is, however, useless because of heavy
contamination caused by the bright star in the field of view.

The XRT data taken in the PC mode were reduced with HEASOFT version 6.27.2,
along with the corresponding CALDB version 20190910. The source spectrum was
then extracted from the image in a circular region with a radius of 20.0. An
adjacent region free of any sources was adopted to extract the background-sky
spectrum. The corresponding ancillary response file was produced by the task
xrtmkarf. The total XRT count rate in the 0.3–10 keV range was estimated to be
$(4.62\pm 0.44)\times 10^{-2}\ \mathrm{count\ s^{-1}}$.

##  3 Analysis and Results

In order to identify the CL phenomenon in B3 0749+460A and to reveal the
underlying physics, spectral analysis was performed on the optical and X-ray
spectra described in Section 2.

###  3.1 Optical Spectroscopy: A CL-AGN

Figure 1 compares the new spectra taken in 2021 with the previous spectra,
which were obtained in different epochs, extracted from the SDSS DR7, BOSS,
and Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) survey. The
MaNGA spectrum is extracted from the datacube by an aperture with a diameter
of 3. At first glance, one can see clearly a significant variation of the
H$\alpha$ line profile. The SDSS spectrum taken in 2013 shows the existence of
a very broad, double-peaked H$\alpha$ component that is, however, absent or
weak in the other four spectra. The following spectral analysis allows us to
identify the object as a local repeat CL-AGN.

####  3.1.1 Subtraction of the Starlight Component

We first remove the starlight component from each of the five spectra by
modeling the stellar features with a linear combination of the first seven
eigenspectra through $\chi^{2}$ minimization, where the eigenspectra are built
from the standard single stellar population spectral library developed by
Bruzual & Charlot (2003). The minimization is carried out over the rest-frame
wavelength range from 3700 to 8000 Å, except for the regions with strong
emission lines — Balmer lines (both narrow and broad components), [O III]
$\lambda\lambda$4959, 5007, [N II] $\lambda\lambda$6548, 6583, [S II]
$\lambda\lambda$6716, 6731, [O II] $\lambda$3727, [O III] $\lambda$4363, and
[O I] $\lambda$6300\. Intrinsic extinction due to the host galaxy described by
the Galactic extinction curve with $R_{V}=3.1$ is also involved in the
modeling. A potential featureless continuum contributed by the central AGN is
ignored in the minimization; it is almost degenerate with the spectrum of
early-type stars, we do not focus on the stellar population in this study, and
the effect caused by the featureless continuum is usually slight (e.g., Wang
2015). The starlight templates are convolved with a fixed Gaussian profile in
advance before the minimization for the Xinglong spectrum, because the
observed absorption features are dominated by the instrumental profile. The
subtraction of the starlight component is illustrated in Figure 2.

Figure 2: An illustration of the modeling and removal of the stellar continuum
for the five spectra (see the main text for the details). In each panel, the
top black curve shows the observed rest-frame spectrum, overplotted by the
best-fit continuum indicated by the red curve. The black curve underneath
shows the continuum-removed emission-line spectrum.

####  3.1.2 Line-Profile Modeling

After removing the starlight component, we model the emission-line profiles in
each of the spectra by a linear combination of a set of Gaussian profiles in
the H$\alpha$ and H$\beta$ regions by the SPECFIT task (Kriss 1994) in IRAF.
The 2021 Xinglong spectrum is not included in the profile modeling simply
because the nearly contemporaneous 2021 Shane spectrum has better spectral
resolution and higher signal-to-noise ratio (S/N). In the modeling, the line-
flux ratios of the [O III] $\lambda\lambda$4959, 5007 and the [N II]
$\lambda\lambda$6548, 6583 doublets are fixed to their theoretical values of
1:3. A sum of a narrow component and a blueshifted broad component is required
to reproduce the [O III] $\lambda\lambda$4959, 5007 line profiles in all three
spectra (e.g., Boroson 2005; Zhang et al. 2013; Harrison et al. 2014; Woo et
al. 2017; Wang et al. 2011, 2018a).

The Balmer-line models are described as follows.

  * •

2004 SDSS spectrum. The H$\alpha$ line profile can be well reproduced by a
combination of narrow (FWHM $\approx 400$ km s-1) and broad
($\mathrm{FWHM}\approx 7800\ \mathrm{km\ s^{-1}}$) components333All line
widths are not corrected for the instrumental spectral resolution throughout
of the paper.. A broad component is not needed to reproduce the profile of
H$\beta$.

  * •

2013 BOSS spectrum. In addition to a classical broad component with
$\mathrm{FWHM}\approx 3000\ \mathrm{km\ s^{-1}}$, a double-peaked broad
H$\alpha$ (e.g., Halpern, & Filippenko 1988; Eracleous & Halpern 1994;
Strateva et al. 2003; Popovic et al. 2003) line containing redshifted and
blueshifted broad components is necessary to properly reproduce the observed
H$\alpha$ profile; thus, a total of three broad components is required.
Double-peaked Balmer emission lines are expected to originate from the
rotating gas in an accretion disk (e.g., Chen et al. 1989; Popovic et al.
2004; Bon et al. 2009a,b; Storchi-Bergmann et al. 2017).

Initially, we model the H$\beta$ profile by a sum of a narrow and a broad
H$\beta$ emission component; however, this results in a residual on the blue
side of [O III]$\lambda 4959$ line. Since the intrinsic profile of the [O
III]$\lambda 4959$ line must be physically identical to that of the [O
III]$\lambda 5007$ line, the residual motivates us to reproduce the emission-
line spectrum in the H$\beta$ region by an additional broad redshifted
H$\beta$ emission component.

We argue that both H$\alpha$ and H$\beta$ show the same double-peaked broad-
line profiles in the 2013 SDSS spectrum, because of the comparable bulk
velocities of the broad redshifted components. The bulk velocities are
calculated to be $5980\pm 90\,\mathrm{km\ s^{-1}}$ and $5310\pm
140\,\mathrm{km\ s^{-1}}$ for H$\alpha$ and H$\beta$, respectively. The broad
blueshifted H$\beta$ is likely undetectable owing to its weakness.

  * •

2016 MaNGA spectrum. Except for its significant weakness, the modeling of the
H$\alpha$ line profile is similar to that of the 2013 BOSS spectrum. The
H$\beta$ emission line can, however, be modeled well by a single narrow
component.

  * •

2021 Shane spectrum. The classical H$\alpha$ broad component with a width of
$\sim 10^{3}\ \mathrm{km\ s^{-1}}$ is not necessary for reproducing the line
profile. The H$\alpha$ broad emission line can be described by a fainter
double-peaked profile, compared to the MaNGA spectrum. Again, a single narrow
Gaussian can reproduce the H$\beta$ line profile properly.

The line-profile models are displayed in Figure 3 for the H$\beta$ (left) and
H$\alpha$ (right) regions, and the results are listed in Table 1. All of the
reported errors correspond to the 1$\sigma$ significance level and include
only the uncertainties caused by the fitting, not the removal of the stellar
continuum.

To summarize, in addition to a significant variation of the broad H$\alpha$
line profile, the spectral modeling and analysis enable us to reveal weak,
broad (and most likely double-peaked) H$\beta$ components in the SDSS 2013
spectrum, but not in the other three spectra, suggesting that the object is a
CL-AGN with type transitions of Sy1.9$\rightarrow$Sy1.8$\rightarrow$Sy1.9
(Type 1.8 and 1.9 AGNs are classified according the existence and absence of a
weak broad H$\beta$ line, respectively; e.g., Osterbrock & Ferland 2006.),
although the prototypical CL refers to a transition between Type 1 and Type 2
spectra.

Figure 3: An illustration of the line-profile modeling with a linear combination of a set of Gaussian functions for H$\beta$ (left panels) and H$\alpha$ (right panels). In each panel, the modeled stellar continuum has already been removed from the original observed spectrum. The observed and modeled line profiles are plotted with the black and red solid lines, respectively. Each Gaussian function is shown by a dashed line. The subpanel underneath the line spectrum presents the residuals between the observed and modeled profiles.  Table 1: Results of Line-Profile Modeling and Analysis Parameters | 2004 Feb. 19 | 2013 Jan. 06 | 2016 Feb. 10 | 2021 Jan. 21
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
Line flux ($\mathrm{10^{-15}\ erg\ s^{-1}\ cm^{-2}}$)
$F(\mathrm{[O~{}III]}\lambda 5007)$ | $13.4\pm 1.0$ | $16.3\pm 0.4$ | $10.3\pm 0.3$ | $25.1\pm 4.1$
$F(\mathrm{H\beta_{b}})$ |  | $3.7\pm 0.4$ |  |
$F(\mathrm{H\beta_{b2}})$ |  | $6.6\pm 0.8$ |  |
$F(\mathrm{H\alpha_{b}})$ | $34.9\pm 1.3$ | $34.2\pm 0.6$ | $9.4\pm 0.5$ |
$F(\mathrm{H\alpha_{b1}})$ |  | $16.4\pm 0.4$ | $20.4\pm 0.5$ | $37.8\pm 1.3$
$F(\mathrm{H\alpha_{b2}})$ |  | $55.8\pm 0.5$ | $18.9\pm 0.3$ | $30.7\pm 0.9$
Line width ($\mathrm{km\ s^{-1}}$)
$\mathrm{FWHM(H\beta_{b})}$ |  | $3760\pm 400$ |  |
$\mathrm{FWHM(H\beta_{b2})}$ |  | $4260\pm 520$ |  |
$\mathrm{FWHM(H\alpha_{b})}$ | $7860\pm 250$ | $3500\pm 60$ | $2210\pm 80$ |
$\mathrm{FWHM(H\alpha_{b1})}$ |  | $3640\pm 80$ | $9760\pm 220$ | $7440\pm 260$
$\mathrm{FWHM(H\alpha_{b2})}$ |  | $7000\pm 70$ | $6820\pm 140$ | $6660\pm 30$
$L/L_{\mathrm{Edd}}$ | 0.005 | 0.012 | 0.009 | 0.005

####  3.1.3 Estimation of Black Hole Mass and Eddington Ratio

After the line-profile modeling, we estimate the black hole mass
($M_{\mathrm{BH}}$) and Eddington ratio ($L/L_{\mathrm{Edd}}$) in terms of the
modeled H$\alpha$ broad emission line through the traditional method described
by Wang et al. (2020a). Briefly, $M_{\mathrm{BH}}$ can be estimated by the
calibration (Greene & Ho 2007)

$M_{\mathrm{BH}}=3.0\times 10^{6}\bigg{(}\frac{L_{\mathrm{H\alpha}}}{10^{42}\
\mathrm{erg\
s^{-1}}}\bigg{)}^{0.45}\bigg{(}\frac{\mathrm{FWHM_{H\alpha}}}{1000\
\mathrm{km\ s^{-1}}}\bigg{)}^{2}\ M_{\odot}$ (1)

and $L/L_{\mathrm{Edd}}$ through a bolometric correction of
$L_{\mathrm{bol}}=9\lambda L_{\lambda}(5100~{}{\rm\AA})$ (e.g., Kaspi et al.
2000), where (Greene & Ho 2005)

$\lambda L_{\lambda}(5100\AA)=2.4\times
10^{43}\bigg{(}\frac{L_{\mathrm{H\alpha}}}{10^{42}\ \mathrm{erg\
s^{-1}}}\bigg{)}^{0.86}\ \mathrm{erg\ s^{-1}}.$ (2)

The 2004 SDSS spectrum is used as a reference in the subsequent comparison. In
deriving the broad H$\alpha$ luminosity $L_{\mathrm{H\alpha}}$, the measured
broad H$\alpha$ line fluxes in the BOSS, MaNGA, and Shane spectra are first
scaled by a factor determined by equaling the total [O III] $\lambda$5007 line
fluxes to that of the reference one (e.g., Peterson et al. 2000). Then, the
intrinsic extinction is corrected from the narrow-line flux ratio
$\mathrm{H\alpha/H\beta}$ by assuming the Balmer decrement of standard Case B
recombination and a Galactic extinction curve with $R_{V}=3.1$. The value of
$M_{\mathrm{BH}}$ is estimated to be $\sim 1.3\times 10^{8}\,M_{\odot}$ based
on the 2004 SDSS spectrum, which we use as a fiducal value because of the
regular and symmetric H$\alpha$ emission-line profile. With this value of
$M_{\mathrm{BH}}$, the lowest row in Table 1 reports the estimated
$L/L_{\mathrm{Edd}}$, where $L_{\mathrm{bol}}$ is estimated from the total
H$\alpha$ broad emission. One can see a relation between $L/L_{\mathrm{Edd}}$
and spectral types.

###  3.2 X-ray Energy Spectrum

####  3.2.1 Swift/XRT Spectra in 2008, 2016, and 2021

  * •

Swift/XRT 2021 spectrum. Because of its low count rate, we attempt to model
the X-ray energy spectrum of B3 0749+460A by XSPEC (v12.11, Arnaud 1996) with
two simple models over the 0.3–10 keV range in terms of the C-statistic (Cash
1979; Humphrey et al. 2009; Kaastra 2017). The adopted models can be expressed
as $wabs*zwabs*powerlaw$ and $wabs*zpcfabs*powerlaw$. In both cases, the
Galactic hydrogen column density is fixed to be $N_{\mathrm{H}}=5.55\times
10^{20}\ \mathrm{cm^{-2}}$ (Kalberla et al. 2005). Both models return similar
results characterized by a quite soft X-ray spectrum with a photon index of
$\Gamma=2.8\pm 0.5$.

  * •

Swift/XRT 2008 spectrum. The object was observed by the Swift/XRT in pointing
mode with an exposure time of 9000 s on 2008 February 24, and is included in
the second Swift/XRT Point Source (2SXP) Catalog (1SXPS J075244.4+455655;
Evans et al. 2020). The energy spectrum is extracted by the method described
in Section 2.2, and analyzed by the same models applied to the 2021 spectrum.
Both best-fitted models return a hard X-ray spectrum with a photon index of
$\Gamma=1.7\pm 0.1$.

  * •

Chandra 2016 spectrum. The spectrum of the object was taken by ACIS-S onboard
the Chandra satellite on 2016 December 21. We reduced the data with software
CIAO version 4.13 and the corresponding calibration database CALDB 4.9.5. The
extracted spectrum is then again modeled by the XSPEC package, which returns a
quite hard power-law spectrum with a photon index of $\Gamma=1.5\pm 0.1$. In
addition to the continuum models aforementioned, two Gaussian functions are
required to model the weak Ar and Ca K$\alpha$ emission lines at 3.0 and 3.7
keV, respectively.

The best fits and their parameters are shown in Figure 4 and listed in Table
2. In the table, all of the quoted uncertainties correspond to a 90%
significance level.

Figure 4: X-ray spectrum of B3 0749+460A taken at different epochs and the best-fit spectral model expressed as $wabs*zwabs*powerlaw$ (Model 1, red-dashed line) and $wabs*zpcfabs*powerlaw$ (Model 2, blue-solid line). The subpanel underneath the spectrum shows the deviations, in units of $\mathrm{counts\ s^{-1}\ keV^{-1}}$, of the observed data from the best-fit model.  Table 2: X-ray spectra fit parameters of B3 0749+460A. Parameter | Value | Units | Description
---|---|---|---
| 2008 | 2016 | 2021 |  |
(1) | (2) | (3) | (4) | (5) |
Model 1 - $wabs*zwabs*powerlaw$ |  |
$N_{\mathrm{H}}$ | $0.01_{-0.01}^{+0.03}$ | $<0.01$ | $0.25_{-0.09}^{+0.12}$ | $10^{22}\,\mathrm{cm^{-2}}$ | Local column density
$\Gamma$ | $1.65_{-0.11}^{+0.14}$ | $1.48_{-0.09}^{+0.10}$ | $2.75^{+0.54}_{-0.45}$ |  | Power-law index
$F$(2–10 keV) | $2.63_{-0.33}^{+0.36}\times 10^{-12}$ | $1.50_{-0.18}^{+0.17}\times 10^{-12}$ | $3.70^{+0.28}_{-0.29}\times 10^{-13}$ | $\mathrm{erg\ s^{-1}\ cm^{-2}}$ | Unabsorbed flux
Cash statistics | 53.16/50 = 1.063 | 54.18/66 = 0.821 | 2.46/4 = 0.615 |  |
Model 2 - $wabs*zpcfabs*powerlaw$ |  |
$\eta_{\mathrm{H}}$ | $0.01_{-0.01}^{+1.33}$ | $<0.01$ | $0.30_{+0.64}^{-0.26}$ | $10^{22}\,\mathrm{cm^{-2}}$ | Local equivalent column density
$f$ | 0.95 |  | 0.95 |  | Dimensionless covering fraction (fixed)
$\Gamma$ | $1.64_{-0.10}^{+0.15}$ | $1.48_{-0.09}^{+0.10}$ | $2.76^{+1.01}_{-1.08}$ |  | Power-law index
$F$(2–10 keV) | $2.65_{-0.32}^{+0.36}\times 10^{-12}$ | $1.50_{-0.18}^{+0.17}\times 10^{-12}$ | $3.84^{+0.27}_{-0.28}\times 10^{-13}$ | $\mathrm{erg\ s^{-1}\ cm^{-2}}$ | Unabsorbed flux
Cash statistics | 56.33/49 = 1.150 | 54.18/66 = 0.809 | 2.77/3 = 0.923 |  |

####  3.2.2 Spectral Slope Transition in the CL Phenomenon

Here we compare the new X-ray observation with the previous ones to reveal the
potential physical process occurring in the CL phenomenon of the object. In
addition to the 2SXP Catalog, the object is in the XMM-Newton Slew Survey
Source Catalog (XMM-SSC, XMMSL2 J075244.7+455654). There are three detections
at different epochs in the XMM-SSC catalog from 2005 to 2011.

The comparison is shown in Table 3. Columns (2) and (3) list the intrinsic
fluxes in the 0.3–10 keV ($F$(0.3-–10) keV) and the observed flux in the
0.2–12 keV ($F$(0.2-–12) keV) bands, respectively. In order to compare with
the previous values, the tabulated fluxes in 2021 are obtained from our
fitting with the model of $wabs*zwabs*powerlaw$. The XMM-Newton hardness
ratio444The XMM-Newton hardness ratio is defined as
$\mathrm{(R_{b7}-R_{b6})/(R_{b7}+R_{b6})}$, where $R_{\mathrm{b7}}$ and
$R_{\mathrm{b6}}$ are the hard (2–12 keV) and soft (0.2–2 keV) band rates,
respectively. is listed in Column (4). Columns (5) and (6) are the hardness
ratios HR1 and HR2 defined by Evans et al. (2020) for the Swift mission555The
Swift hardness ratios HR1 and HR2 are defined as $\mathrm{HR1}=(M-S)/(M+S)$
and $\mathrm{HR2}=(H-M)/(H+M)$, where $S$, $M$, and $H$ are the count rates in
the 0.3–1 keV, 1–2 keV, and 2–10 keV energy bands, respectively.. All of the
uncertainties shown in the table correspond to a 90% significance level, after
taking into account proper error propagation. For the 2008 Swift/XRT
observation, the hardness ratios obtained in the current work are highly
consistent with the values reported in the 2SXP Catalog (Evans et al. 2020).

The upper panel in Figure 5 shows the long-term variation of the X-ray flux
and hardness ratio $\mathrm{HR1_{XMM}}$ of the object. On the one hand, as
shown in the table, the X-ray flux decreases gradually by more than an order
of magnitude during the period from 2005 to 2021, which does not follow the
spectral-type transition. On the other hand, based on the hardness ratios at
different epochs, one can see from the comparison that the CL phenomenon
identified in B3 0749+460A tends to be related to the X-ray slope. The X-ray
spectrum is found to be soft ($\mathrm{HR1=-0.50\pm 0.18}$ in 2005 and
$\mathrm{HR1=-0.41\pm 0.16}$ in 2021) at the “turn-off” state with a Seyfert
1.9-like spectrum, and to be hard ($\mathrm{HR1=0.21\pm 0.35}$) in 2011 when
the object was possibly at the “turn-on” state with a Seyfert 1.8-like
spectrum.

Table 3: Comparison of X-ray emission

Year of Obs. |  $F$(0.3–10) keV |  $F$(0.2–12) keV | $\mathrm{HR1_{XMM}}$ | $\mathrm{HR1_{Swift}}$ | $\mathrm{HR2_{Swift}}$ | Sp. type | Mission | Reference |
---|---|---|---|---|---|---|---|---|---
|  ($10^{-12}\ \mathrm{erg\ s^{-1}\ cm^{-2}})$ |  |  |  |  |  |  |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) |
2005 |  | $10.3\pm 1.9$ | $-0.50\pm 0.18$ |  |  | Sy1.9 | XMM-Newton | 1 |
2006a |  | $6.7\pm 1.8$ |  |  |  |  | XMM-Newton | 1 |
2008 | $5.2\pm 0.3$ |  |  | $0.07\pm 0.04$ | $-0.09\pm 0.04$ |  | Swift | 2 |
| $4.3\pm 0.3$ | $4.3\pm 0.4$ | $-0.35\pm 0.03$ | $0.07\pm 0.04$ | $-0.08\pm 0.04$ |  | Swift | This work |
2011 |  | $6.7\pm 2.3$ | $0.21\pm 0.35$ |  |  | Sy1.8 | XMM-Newton | 1 |
2016 | $2.2\pm 0.2$ | $2.3\pm 0.2$ | $-0.43\pm 0.03$ | $-0.05\pm 0.03$ | $-0.12\pm 0.04$ | Sy1.9 | Chandra | This work |
2021 | $2.1^{+1.0}_{-0.8}$ | $0.8^{+0.6}_{-0.3}$ | $-0.41\pm 0.16$ | $-0.42\pm 0.14$ | $-0.27\pm 0.20$ | Sy1.9 | Swift | This work |

aafootnotetext: There is no reported value of HR1 in the catalog due to the
lack of the count rate in the B7 channel.

References: 1: XMM-SSC (2017); 2: Evans et al. (2020)

Figure 5: Upper panel: Evolution of the observed X-ray flux in 0.2-12keV (red
symbol) and XMM-Newton HR1 (blue symbol). Lower panel: Multi wavelength light
curves of B3 0749+460A detected by the Catalina Real-time Transient Survey
(Drake et al. 2011), Palomar Transient Factory survey (Law et al. 2009)and
WISE . The light curves are binned by averaging the measurements within one
day. The vertical dashed lines mark the epochs of optical spectra, where the
blue line denotes the “turn-on” state with a Seyfert 1.8-like spectrum having
weak broad H$\beta$ emission lines, and the red ones the “turn-off” state with
a Seyfert 1.9-like spectrum. The corresponding years are marked beside the
vertical lines.

##  4 Conclusions and Discussion

We identify B3 0749+460A as a new local CL-AGN by comparing SDSS spectra
previously taken in 2004, 2013, and 2016 with new spectra taken by us in 2021.
The object shows type transitions of
Sy1.9$\rightarrow$Sy1.8$\rightarrow$Sy1.9. In particular, the classical broad
H$\alpha$ component fades away from 2013 to 2021. A follow-up observation in
X-rays taken by the Swift/XRT enables us to reveal (1) gradually decreasing
X-ray emission since 2005, and (2) the X-ray spectrum is soft in the optical
“turn-off” state with a Seyfert 1.9-like spectrum. Also, the X-ray spectrum is
hard in the ‘turn-on” state with a Seyfert 1.8-like spectrum, although there
is a separation of about two years between the optical and X-ray observations.

###  4.1 Broad H$\beta$ Emission

Here we argue that the identified spectral-type transitions are not caused by
the removal of the stellar continuum. The left panel in Figure 6 shows the
differential spectra of the object obtained by adopting the 2004 SDSS spectrum
as the reference, after scaling the others according to the total [O III]
$\lambda$5007 line flux. One can see that, in addition to the redshifted
H$\alpha$ broad component, there is a bump at the red side of H$\beta$ in the
2013 BOSS spectrum. Three arguments support our belief that the bump
represents the redshifted H$\beta$ broad component. First, as shown in the
right panel of the figure, the two bumps exhibit quite similar profiles, after
taking into account a measured bump flux ratio of 3 that is comparable to the
average Balmer ratio of the broad Blamer lines in AGNs (Dong et al. 2006).
Second, based on simple Gaussian fitting, the two bumps show comparable bulk
redshift velocities with respect to the corresponding rest-frame wavelength.
The measured bulk velocities are $\Delta v=5895\pm 87\,\mathrm{km\ s^{-1}}$
and $5310\pm 140\,\mathrm{km\ s^{-1}}$ for H$\alpha$ and H$\beta$,
respectively. Finally, the bump in the H$\beta$ region is unlikely to result
from He I $\lambda 4922$ broad emission (e.g., Veron et al. 2002), simply
because the broad He I $\lambda$5876 emission line is not detected in the
observed and differential spectra.

Figure 6:  Left panel: A comparison of the differential spectra taken in 2013,
2016, and 2021, when the SDSS 2004 spectrum is used as a reference. Before the
subtraction, the two spectra are scaled in flux level according to their total
[O III] $\lambda 5007$ line fluxes. The two vertical lines mark the rest-frame
wavelengths of the H$\beta$ and H$\alpha$ emission lines. One can see clearly
a redshifted broad H$\beta$ component in the BOSS 2013 spectrum, but not in
the 2021 Shane spectrum. In the Shane spectrum, the features around 5400 Å are
artificial, caused by uncertainty in the low-S/N region where the blue-side
and red-side CCD spectra are combined. Right panel: A comparison of profiles
of the two redshifted bumps, after scaling them by a measured flux ratio of
the two bumps of 3 (red, H$\beta$; blue, H$\alpha$).

###  4.2 A Comparison with Hon et al.

Although our new spectrum taken in 2021 is comparable to the 2016 MaNGA
spectrum shown by Hon et al. (2020), our current study is quite improved in
the spectral analysis when compared to Hon et al. (2020). The stellar-
component subtraction, line-profile modeling, and a comparison within the
differential spectrum enable us to argue for the existence of a broad H$\beta$
line in the 2013 SDSS spectrum, which suggests that a CL phenomenon occurred
in the object. Hon et al. (2020) argued against a typical CL phenomenon in the
object, because of the short variation timescale of $\sim 1000$ d. However, a
CL timescale of $\sim 1$–3 yr has been revealed in previous investigations
(e.g., Merloni et al. 2015; Runnoe et al. 2016; Macleod et al. 2016). In
addition, the supernova (SN) explanation of the SDSS 2013 spectrum suggested
by Hon et al. (2020) can be potentially excluded by the lack of Ca II triplet
emission in the spectrum. Finally, taking into account the 2016 MaNGA
spectrum, the object experienced type transitions of
Sy1.9$\rightarrow$Sy1.8$\rightarrow$Sy1.9 within a duration of 13 yr (from
2004 to 2016), although the broad H$\alpha$ emission is found to be further
weaker in 2021 than in 2016.

###  4.3 Physical Implications

The lower panel in Figure 5 shows the multiwavelength light curves of B4
0749+460A. The mid-infrared (MIR) light curves in the $w1$ and $w2$ bands
detected by the Wide-field Infrared Survey Explorer (WISE and NEOWISE-R;
Wright et al. 2010; Mainzer et al. 2014) indicate that the brightness in both
bands decreases gradually by $\sim 0.3$ mag when the object changes from a
Seyfert 1.8 to a Seyfert 1.9 nucleus, consistent with the expectation of the
accretion-rate enhancement scenario of the CL phenomenon (e.g., Sheng et al.
2017; Stern et al. 2018; Wang et al. 2019). The possibility of a normal SN
mentioned above can be further decreased according to the observed MIR
variability with an absolute magnitude in the $w2$ band of $-23.6$ mag. This
value is marginally larger than the brightest one (i.e., $-22.8$ mag at 4.5
$\mu$m for SN 2010jl) reported by Szalai et al. (2019), who studied a sample
of MIR light curves of hundreds of SNe.

With increasing cases of CL-AGNs, a few possible models have recently been
proposed (e.g., Wang & Bon 2021). Based on the statistical studies performed
by MacLeod et al. (2019) and Wang et al. (2019, 2020a,b), there is some
evidence for the CL phenomenon being understood by the disk-wind BLR model. In
the model, a classical BLR can be sustained if $L/L_{\mathrm{Edd}}$ is above a
critical value of $\sim 10^{-6}$–$10^{-3}$, when the fiducal values of a set
of parameters of the disk are adopted (e.g., Nicastro 2000; Elitzur & Ho
2009). In B3 0749+460A, when the $L/L_{\mathrm{Edd}}$ listed in Table 1
decreases gradually from 0.012 to 0.005, the classical broad H$\alpha$
component in fact fades away from 2013 to 2021. This link between the strength
of the classical H$\alpha$ broad component and $L/L_{\mathrm{Edd}}$ can be
reinforced if we instead estimate $L_{\mathrm{bol}}$ from the X-ray emission.
Our X-ray analysis suggests a decreasing $L/L_{\mathrm{Edd}}$ from 0.017 in
2011 to 0.002 in 2021, when a bolometric correction of
$L_{\mathrm{bol}}=16L_{\mathrm{X}}$ is adopted in the calculation. In
addition, it is well known that the double-peaked broad Blamer lines observed
frequently in AGNs can originate from gas in a single accretion disk (e.g.,
Storchi-Bergmann et al. 2017, and references therein). The double-peaked broad
H$\alpha$ component is found to weaken synchronously with the classical broad
H$\alpha$ in the object, implying a disk origin for the classical broad
H$\alpha$ component. Alternatively, Pan et al. (2021) and J. Feng et al.
(2021) show that the observed repeat CL-AGNs could be explained by reducing
the disk burst period by including the effect of a large-scale magnetic field
(e.g., Dexter & Begelman 2009) in the disk-instability model suggested by
Sniegowska et al. (2020).

A transition of the accretion state, similar to that seen in XRBs, has been
proposed recently as an explanation of the CL phenomenon (e.g., Ruan et al.
2019; Ai et al. 2020), although it seems that this scenario might not be
generally applicable. The CL phenomenon in UGC 3223, a local repeat CL-AGN, is
found to be unrelated to the X-ray hardness ratio, but instead related to the
X-ray luminosity (Wang et al. 2020b). Following Ai et al. (2020), the hardness
ratio $\mathrm{CR_{soft}/(CR_{soft}+CR_{hard})}$ is plotted against
$L/L_{\mathrm{Edd}}$ in Figure 7 for B3 0749+460A, where $L/L_{\mathrm{Edd}}$
is derived from the unabsorbed X-ray 2–10 keV luminosity by assuming a
bolometric correction of $L_{\mathrm{bol}}=16L_{\mathrm{X}}$. After converting
the photon index $\Gamma$ to the hardness ratio, the
$\Gamma-L/L_{\mathrm{Edd}}$ relationships for low-luminosity AGNs (Constantin
et al. 2009) and bright AGNs (Risaliti et al. 2009) are overplotted in the
figure, which stand for an advection-dominated accretion flow (ADAF) at low
$L/L_{\mathrm{Edd}}$ and a standard accretion disk at high
$L/L_{\mathrm{Edd}}$, respectively. The $\Gamma-L/L_{\mathrm{Edd}}$
relationships can be well understood by the change of the Compton $y$
parameter with the released energy (e.g., Esin et al. 1997; Janiuk & Czerny
2000). One can see from the figure that the evolution of the object likely
follows the ADAF locus and does not follow the V-shape transitions revealed by
Ai et al. (2020). In fact, assuming the aforementioned disk-wind model, the
(dis)appearance of a classical BLR is sensitive to accretion (wind) luminosity
in individual AGNs, suggesting that a change of accretion mode is possibly a
necessary (rather than a sufficient) condition for the CL phenomenon.

Figure 7: Hardness ratio plotted against the $L/L_{\mathrm{Edd}}$ for B3
0749+460A. $\mathrm{CR_{soft}}$ and $\mathrm{CR_{hard}}$ denote the count
rates at 0.2–2 and 2–12 keV, respectively. $L/L_{\mathrm{Edd}}$ is estimated
from the unabsorbed 2–10 keV luminosity after assuming a bolometric correction
of $L_{\mathrm{bol}}=16L_{\mathrm{X}}$. The symbol size is proportional to the
broad H$\alpha$ emission. The $\Gamma-L/L_{\mathrm{Edd}}$ relationships of
low-luminosity AGNs and bright AGNs are denoted by the dotted and dashed
lines, respectively.

###### Acknowledgements.

The authors thank the anonymous referee for a careful review and helpful
suggestions that greatly improved the manuscript. This study is supported by
the National Natural Science Foundation of China under grant 11773036, and by
the Strategic Pioneer Program on Space Science, Chinese Academy of Sciences,
grants XDA15052600 and XDA15016500. The authors are grateful for support from
the National Key Research and Development Project of China (grant
2020YFE0202100). J.W. is supported by the Natural Science Foundation of
Guangxi (2020GXNSFDA238018) and by the Bagui Young Scholars Program. This
study is supported by the Open Project Program of the Key Laboratory of
Optical Astronomy, NAOC, CAS. A.V.F.’s group is supported by the Christopher
R. Redlich Fund and the Miller Institute for Basic Research in Science (in
which A.V.F. is a Senior Miller Fellow). We thank the Swift Acting PI, Brad
Cenko, for approving our target-of-opportunity request, as well as the Swift
observation team for assistance. This study uses the NASA/IPAC Extragalactic
Database (NED), which is operated by the Jet Propulsion Laboratory, California
Institute of Technology. It also uses data collected by the Wide-field
Infrared Survey Explorer (WISE), which is a joint project of the University of
California, Los Angeles, and the Jet Propulsion Laboratory/California
Institute of Technology, funded by the National Aeronautics and Space
Administration (NASA).

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