equation is equivalent to the closedness of the curvature; thus, the existence
of the dual connection is stated as the following theorem:

######  Theorem A.2.

Suppose Lie algebra valued two-form object $\mathfrak{F}$ with an abelian
gauge group in the flat space-time is given in submanifold
$\Sigma_{\mathfrak{F}}\subseteq{\mathcal{}M}$. Following two statements are
equivalent to each other:

  1. 1.

Lie-algebra valued one-form object ${\mathfrak{A}}$ exists in compact manifold
$\Sigma_{\mathfrak{A}}\subseteq\Sigma_{\mathfrak{F}}$ as a solution of
structure equation $\mathfrak{F}=d{\mathfrak{A}}$, and the principal bundle is
constructed using connection ${\mathfrak{A}}$ and curvature $\mathfrak{F}$.

  2. 2.

The second homology is vanished in $\Sigma_{\mathfrak{A}}$;
$H_{2}(\Sigma_{{\mathfrak{A}}},\partial\Sigma_{{\mathfrak{A}}};\mathbb{R})=0$.

###### Proof.

Vanishing second homology is equivalent to existence of connection
${\mathfrak{A}}$ owing to Remark A.1. When connection ${\mathfrak{A}}$ is
provided as a solution of structure equation $\mathfrak{F}=d{\mathfrak{A}}$
from the given $\mathfrak{F}$, it is Lie algebra valued one-form belonging to
an adjoint representation as shown in Remark 2.2. ∎

######  Example A.3.

Coulomb-type potential

A domain of a connection for thr Coulomb-type potential is
$\Sigma_{{\mathscr{A}}}:={T^{*}{\mathcal{}M}}{\setminus}\\{r=0\\}$, as given
in Example 5.1, and corresponding curvature $(\ref{FU1})$ has the same domain.
For dual curvature $\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}$ given by (134),
tensor coefficient $\hat{\mathscr{F}}^{~{}}_{U_{\textit{1}}}$ is also defined
in ${T{\mathcal{}M}}$ except $r=0$ and is static (no time dependence); thus,
the domain of $\hat{\mathscr{F}}^{~{}}_{U_{\textit{1}}}$ is
$\Sigma_{\hat{\mathscr{F}}}={T{\mathcal{}M}}{\setminus}\\{r=0\\}\cong\mathbb{R}\otimes(\mathbb{R}^{3}{\setminus}\\{r=0\\})$.
The second homology of a compact submanifold of the domain;
$\overline{\Sigma}_{\hat{\mathscr{F}}}:=D^{1}{\otimes}(D^{3}{{\setminus}}\\{0\\})$
is calculated as $H_{2}(\overline{\Sigma}_{\hat{\mathscr{F}}};\mathbb{R})\cong
H_{2}(D^{1},S^{0};\mathbb{R}){\oplus}H_{2}(D^{3},S^{2}{\oplus}S^{2};\mathbb{R})=0$;
thus, there exists dual connection $\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$
in $\Sigma_{\hat{\mathscr{A}}}\subseteq\Sigma_{\hat{\mathscr{F}}}$.

######  Example A.4.

Linear electric current

A linear electric current makes a stationary magnetic field around the current
in a flat space-time manifold, and it has a $U(1)$ connection such that:

$\displaystyle{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=\frac{1}{2}\log{\left(x^{2}+y^{2}\right)}\hskip
1.00006ptdz\in\Omega^{1}\left({T^{*}{\mathcal{}M}}{\setminus}\\{x=0\wedge
y=0\\}\right).$

Corresponding curvature and dual curvature are respectively provided as

$\displaystyle\mathfrak{F}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=\frac{x}{x^{2}+y^{2}}\hskip 1.00006ptdx\wedge
dz+\frac{y}{x^{2}+y^{2}}\hskip 1.00006ptdy\wedge dz$
$\displaystyle{\longrightarrow}^{{\widehat{\mathrm{H}}}}\hskip
11.99998pt\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=i\frac{y}{x^{2}+y^{2}}\hskip
1.00006ptdt{\wedge}dx-i\frac{x}{x^{2}+y^{2}}\hskip 1.00006ptdt{\wedge}dy,$

The dual connection is provided as a solution of structure equation (106) such
that

$\displaystyle\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=i\tan^{-1}{\frac{y}{x}}\hskip 1.99997ptdt,$

which is indeterminate at $x=y=0$ with any $z$; thus, its domain is
$\Sigma_{\hat{\mathscr{A}}}\cong\mathbb{R}\otimes\left(\mathbb{R}^{3}{\setminus}\mathbb{R}\right)$.
The second homology of $\Sigma_{\hat{\mathscr{A}}}$ is the same as that of
Example 5.1 such that
$H_{2}(\overline{\Sigma}_{\hat{\mathscr{A}}};\mathbb{R}){\cong}H_{2}(D^{1};\mathbb{R}){\oplus}H_{2}(D^{1}{\otimes}S^{1},T^{2};\mathbb{R})=0$;
thus, the dual connection exists in the same domain. A domain of the dual
connection is the same as that of the dual curvature.

The second Chern class for ${\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$ and
$\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$ is provided as

$c^{~{}}_{2}({\mathfrak{A}}^{~{}}_{U_{\textit{1}}})=c^{~{}}_{2}(\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}})=\left((4\pi(x^{2}+y^{2})\right)^{-2}.$

######  Example A.5.

Plane wave

A plane wave solution in globally flat $\mathscr{M}$ is considered. A $U(1)$
connection and corresponding curvature are, respectively, provided using the
Cartesian coordinate system as

$\displaystyle{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=\sin{(t-z)}\left(dx+dy\right),$
$\displaystyle\mathfrak{F}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=\cos{(t-z)}\left(dt\wedge dx+dt\wedge dy+dx\wedge dz+dy\wedge
dz\right),$

which represent plane wave lying in a $(x,y)$-plane propagating to $z$
direction with speed $c=1$, and its dual curvature is obtained as follows:

$\displaystyle\mathfrak{F}^{~{}}_{U_{\textit{1}}}=\cos{(t-z)}\left(dt\wedge
dx-dt\wedge dy+dx\wedge dz-dy\wedge dz\right).$

The domain of $\mathfrak{F}^{~{}}_{U_{\textit{1}}}$ is the entire space of
${T^{*}{\mathcal{}M}}{\cong}\mathbb{R}^{4}$, which provides the vanishing
second homology; $H_{2}(\mathbb{R}^{4},\not\\!0;\mathbb{R})=0$. The dual
connection is provided as follows:

$\displaystyle\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}=i\sin{(t-z)}\left(dx-
dy\right),$

whose domain is also entire ${T^{*}{\mathcal{}M}}$.

###  A.2 $SU(N)$ gauge group

Next, we treat a non-abelian gauge with $SU(N)$ group in the flat space-time.
In this case, exactness and closedness are corresponding to the structure
equation and Bianchi identity, respectively, such that:

$\displaystyle\textit{Covariant-exactness}\hskip 3.00003pt:$
$\displaystyle\hskip 11.00008pt\hat{\mathfrak{F}}^{~{}}_{{SU}}\hskip
3.99994pt=d\hat{\mathfrak{A}}^{~{}}_{{SU}}-i\hskip
1.00006pt{{c}^{~{}}_{SU}}\hskip
1.00006pt\hat{\mathfrak{A}}^{~{}}_{{SU}}\wedge\hat{\mathfrak{A}}^{~{}}_{{SU}}$
$\displaystyle\cdots\cdots(\ref{eq1})^{\prime},$
$\displaystyle\textit{Covariant-closedness}:$ $\displaystyle\hskip
1.00006pt\hat{d}_{{SU}}\hat{\mathfrak{F}}^{~{}}_{{SU}}=d\hat{\mathfrak{F}}^{~{}}_{{SU}}-i\hskip
1.00006pt{{c}^{~{}}_{SU}}\hskip
1.00006pt[\hat{\mathfrak{A}}^{~{}}_{{SU}},\hat{\mathfrak{F}}^{~{}}_{{SU}}]_{\wedge}=0$
$\displaystyle\cdots\cdots(\ref{BianchihSU})^{\prime},$

where Lorentz-covariant differential $d_{\mathfrak{w}}$ in (106) and in (107)
is replaced by the external derivative owing to the flat space-time. As is the
case of an abelian gauge, when the dual curvature is covariant-exact (given by
the structure equation $(\ref{eq1})^{\prime}$), it is trivially covariant-
closed owing to the Bianchi identity $(\ref{BianchihSU})^{\prime}$. The
question here is that “Does dual curvature $\hat{\mathfrak{F}}^{~{}}_{{SU}}$
satisfying $(\ref{BianchihSU})^{\prime}$ have a solution for
$(\ref{eq1})^{\prime}$ ?”. A key to treat this question is the Chern–Simons
form, which is introduced in the spinor-gauge bundle as follows:

######  Definition A.6 (Chern–Simons form).

The Chern–Simons forms in a gauge bundle is defined as a functional of $\hskip
1.00006pt{\mathfrak{A}}_{SU}^{\\#}$ in the total space such that:

$\displaystyle{{\mathfrak{c}}{\mathfrak{s}}}_{SU}({\mathfrak{A}}^{\\#}_{SU}):={{\mathfrak{c}}{\mathfrak{s}}}^{I}_{SU}({\mathfrak{A}}^{\\#}_{SU})\hskip
1.99997ptt_{I}:=-i\left[{\mathfrak{A}}^{\\#}_{SU},d{\mathfrak{A}}_{SU}^{\\#}-i\hskip
1.00006pt{{c^{~{}}_{SU}}}\hskip
1.00006pt{\mathfrak{A}}_{SU}^{\\#}\wedge{\mathfrak{A}}_{SU}^{\\#}\right]_{\wedge}.$
(140)

A superscript “#” represents a pull-buck of forms from the base manifold to
the total manifold with respect to the bundle map.

Hereafter, “#” and a subscript “${SU}$” are omitted for simplicity unless
otherwise denoted. The Chern–Simons form of a dual connection is defined by
$(\ref{CSform})$ with replacing ${\mathfrak{A}}^{~{}}_{SU}$ to
$\hat{\mathfrak{A}}^{~{}}_{{SU}}$. Discussions below are common for both
principal- and dual connections.

We introduce a map

$\displaystyle\pi_{\mathfrak{a}}:\Omega^{p}\rightarrow\Omega^{p-1}:\left[{\mathfrak{a}},{\mathfrak{b}}\right]_{\wedge}\mapsto
i{\mathfrak{b}},$

for ${\mathfrak{a}}\in\Omega^{1}$ and ${\mathfrak{b}}\in\Omega^{p-1}$. An
anti-derivative operator
$\delta_{\mathfrak{a}}:\Omega^{p}\rightarrow\Omega^{p-1}$ is defined in the
total space of the principal bundle as follows:

$\displaystyle\delta_{\mathfrak{a}}{\mathfrak{W}}({\mathfrak{A}})$
$\displaystyle=\pi^{~{}}_{\mathfrak{a}}\left(\lim_{\varepsilon\to
0}\frac{{\mathfrak{W}}\left({\mathfrak{A}}+\varepsilon{\mathfrak{a}}\right)-{\mathfrak{W}}\left({\mathfrak{A}}\right)}{\varepsilon}\right),$
(141)

where ${\mathfrak{W}}({\mathfrak{A}})$ is an invariant polynomial in Weil
algebra. de Rham cohomology of Weil algebra with respect to anti-derivative
operator $\delta_{\mathfrak{a}}$ is trivial such that:

$\displaystyle
H_{{dR}}^{k}(W({\mathfrak{A}});\delta_{\mathfrak{a}})=\begin{cases}\mathbb{R}&(k=0)\\\
0&(k>0),\end{cases}$ (142)

where $W({\mathfrak{A}})\in{\mathfrak{W}}({\mathfrak{A}})$ is Weil algebra
appearing in the gauge bundle.

######  Remark A.7.

The principal curvature form in the total space is an exact form by means of
the Chern–Weil theory, and there exists the Chern–Simons form which gives the
curvature form such that
$\mathfrak{F}=\delta_{\mathfrak{a}}{{\mathfrak{c}}{\mathfrak{s}}}$.

###### Proof.

Suppose the curvature is provided as a solution of the structure equation. The
Chern–Simons form is expressed using ${\mathfrak{A}}$ as

$\displaystyle{{\mathfrak{c}}{\mathfrak{s}}}^{I}\left({\mathfrak{A}}\right)$
$\displaystyle=f^{I}_{~{}JK}\hskip
1.99997pt{\mathfrak{A}}^{J}\wedge\left(d{\mathfrak{A}}^{K}+\frac{{c}}{2}f^{K}_{~{}LM}{\mathfrak{A}}^{L}\wedge{\mathfrak{A}}^{M}\right),$
$\displaystyle=\frac{1}{2}\left\\{f^{I}_{~{}JK}\hskip
1.99997ptd\left({\mathfrak{A}}^{J}\wedge{\mathfrak{A}}^{K}\right)+{c}\hskip
1.00006ptf^{I}_{~{}JK}\hskip 1.00006ptf^{K}_{~{}LM}\hskip
1.99997pt{\mathfrak{A}}^{J}\wedge{\mathfrak{A}}^{L}\wedge{\mathfrak{A}}^{M}\right\\}.$

Anti-derivative operator is applied on the Chern–Simons form as follows:

$\displaystyle\delta_{\mathfrak{a}}{{\mathfrak{c}}{\mathfrak{s}}}=\pi^{~{}}_{\mathfrak{a}}\left(f^{I}_{~{}JK}\hskip
1.99997pt{\mathfrak{a}}^{J}\wedge\left(d{\mathfrak{A}}^{K}+\frac{{c}}{2}f^{K}_{~{}LM}\hskip
1.99997pt{\mathfrak{A}}^{L}\wedge{\mathfrak{A}}^{M}\right)t_{I}\right)=-i\pi^{~{}}_{\mathfrak{a}}\left(\left[{\mathfrak{a}},\mathfrak{F}\right]_{\wedge}\right)=\mathfrak{F},$

where the Jacobi identity $f_{ABE}\hskip 3.00003ptf_{ECD}+f_{CBE}\hskip
3.00003ptf_{AED}+f_{DBE}\hskip 3.00003ptf_{ACE}=0$ is used. Vector space
$\Omega^{2}$ is interpreted as a dual space of $\Omega^{1}$ with respect to a
bilinear map
$\Omega^{1}\otimes\Omega^{2}\rightarrow\mathbb{R}[{\mathfrak{s}}{\mathfrak{u}}_{N}]$
such that

$\displaystyle\langle{\mathfrak{a}},\mathfrak{F}\rangle_{\wedge}$
$\displaystyle:=\int_{\Sigma}[{\mathfrak{a}},\mathfrak{F}]_{\wedge}\in\mathbb{R}[{\mathfrak{s}}{\mathfrak{u}}_{N}],$

where $\Sigma\subset{\mathcal{}M}$ is a three-dimensional oriented and compact
submanifold of ${\mathcal{}M}$, and
$\mathbb{R}[{\mathfrak{s}}{\mathfrak{u}}_{N}]$ is polynomial algebra of Lie
algebra ${\mathfrak{s}}{\mathfrak{u}}_{N}$ with real coefficients. This
bilinear map is non-degenerated and induces homomorphism
$\Omega^{2}\cong{\Omega^{1}}^{\\#}$; thus, $\mathfrak{F}$ is recognized as a
one-form object in the space of connections. Therefore, the existence of the
Chern–Simons form is maintained owing to its de Rham cohomology
$H_{{dR}}^{1}(\Sigma_{{\mathfrak{c}}{\mathfrak{s}}};\delta_{\mathfrak{A}})=0$
and the de Rham theorem, where $\Sigma_{{\mathfrak{c}}{\mathfrak{s}}}$ is a
domain of the Chern–Simons form. ∎

######  Remark A.8.

The Chern–Simons form is a closed form with respect to the external derivative
such that $d{{\mathfrak{c}}{\mathfrak{s}}}=0$, and it is consistent with the
structure equation $(\ref{eq1})^{\prime}$.

###### Proof.

The Bianchi identity $(\ref{BianchihSU})^{\prime}$ is represented using the
Chern–Simons form as $d\mathfrak{F}+{{\mathfrak{c}}{\mathfrak{s}}}=0$, when
$\mathfrak{F}$ is defined using the structure equation $(\ref{eq1})^{\prime}$.
Therefore, the Chern–Simons form is closed such that
$dd\mathfrak{F}+d{{\mathfrak{c}}{\mathfrak{s}}}=d{{\mathfrak{c}}{\mathfrak{s}}}=0$
with respect to the external derivative in ${T^{*}{\mathcal{}M}}$. The
external derivative of the Chern–Simons form is directly calculated as

$\displaystyle d{{\mathfrak{c}}{\mathfrak{s}}}^{I}$
$\displaystyle=f^{I}_{~{}JK}\left(d{\mathfrak{A}}^{J}\wedge\mathfrak{F}^{K}-{\mathfrak{A}}^{J}\wedge
d\mathfrak{F}^{K}\right),$
$\displaystyle=f^{I}_{~{}JK}\left(d{\mathfrak{A}}^{J}\wedge\mathfrak{F}^{K}+{c}f^{K}_{~{}LM}{\mathfrak{A}}^{J}\wedge{\mathfrak{A}}^{L}\wedge\mathfrak{F}^{M}\right),$
$\displaystyle=f^{I}_{~{}JK}\left(d{\mathfrak{A}}^{J}+\frac{{c}}{2}f^{J}_{~{}ML}{\mathfrak{A}}^{M}\wedge{\mathfrak{A}}^{L}\right)\wedge\mathfrak{F}^{K}=f^{I}_{~{}JK}\hskip
1.99997pt\widetilde{\mathfrak{F}}^{J}\wedge\mathfrak{F}^{K},$

where
$\widetilde{\mathfrak{F}}^{J}:=d{\mathfrak{A}}^{J}+{c}f^{J}_{~{}ML}{\mathfrak{A}}^{M}\wedge{\mathfrak{A}}^{L}/2$.
In this calculaiotn, the Bianchi identity $(\ref{BianchihSU})^{\prime}$ and
the Jacobi identity are used. We note that the structure equation
$(\ref{eq1})^{\prime}$ is not assumed here. When
$\widetilde{\mathfrak{F}}^{J}\wedge\mathfrak{F}^{K}$ is symmetric in exchange
$J$ for $K$, relation $d{{\mathfrak{c}}{\mathfrak{s}}}^{I}=0$ is maintained.
Therefore, closedness of the Chern–Simon form is consistent with
$\widetilde{\mathfrak{F}}^{I}=\mathfrak{F}^{I}=d{\mathfrak{A}}^{I}-i\hskip
1.00006pt{{c}}\hskip
3.00003pt\left[{\mathfrak{A}}\wedge{\mathfrak{A}}\right]^{I}$. ∎

Whereas $(\ref{eq1})^{\prime}\Rightarrow(d{{\mathfrak{c}}{\mathfrak{s}}}=0)$
is trivial, inverse statement
$(d{{\mathfrak{c}}{\mathfrak{s}}}=0)\Rightarrow(\ref{eq1})^{\prime}$ for given
$\mathfrak{F}$ is not always true. When non-singular two-form objects
$\mathfrak{F}^{I}$ are given, the existence of one-form objects
${\mathfrak{A}}^{I}$ fulfilling both $(\ref{BianchihSU})^{\prime}$ and
$(\ref{eq1})^{\prime}$ is discussed here. The Bianchi identity
$(\ref{BianchihSU})^{\prime}$ is represented using trivial bases in
${T^{*}{\mathcal{}M}}$ such that

$\displaystyle
t_{I}\left(\partial_{\circ}\mathscr{F}^{I}_{\circ\circ}+{c}\hskip
1.00006ptf^{I}_{\hskip
0.70004ptJK}{\mathscr{A}}^{J}_{\circ}\mathscr{F}^{K}_{\circ\circ}\right){\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}$
$\displaystyle=0,$

which is equivalent to

$\displaystyle\epsilon^{a{\circ}{\circ\circ}}\left(\partial_{\circ}\mathscr{F}^{I}_{\circ\circ}+{c}\hskip
1.00006ptf^{I}_{\hskip
0.70004ptJK}{\mathscr{A}}^{J}_{\circ}\mathscr{F}^{K}_{\circ\circ}\right)$
$\displaystyle=0.$ (143)

We note that the flat space-time yields $d{\mathfrak{e}}=0$. Equation (143) is
represented using the dual curvature as

$\displaystyle\partial_{\circ}[\hat{\mathscr{F}}^{I}]^{a{\circ}}+{c}\hskip
1.00006ptf^{I}_{\hskip
0.70004ptJK}[\hat{\mathscr{F}}^{K}]^{a{\circ}}{\mathscr{A}}^{J}_{\circ}=0.$

Unknown function ${\mathscr{A}}^{I}_{a}$ has
$N_{\mathscr{A}}:=N_{\textrm{ST}}\times N_{SU}$ independent components, where
$N_{\textrm{ST}}$ is a dimension of the space-time manifold and
$N_{SU}=N^{2}-1$ is total number of freedom for the $SU(N)$ symmetry. Thus,
equation (143) has $(N_{\mathscr{A}}\times N_{\mathscr{A}})$-matrix
representation such that

$\displaystyle\left[\partial\hat{\mathscr{F}}\right]^{(I;a)}+{c}\left[f\hat{\mathscr{F}}\right]^{(I;a)}_{(J;{\circ})}\left[{\mathscr{A}}\right]^{(J;{\circ})}=0,$
(144)

where

$\displaystyle\left[\partial\hat{\mathscr{F}}\right]^{(I;a)}$
$\displaystyle:=\eta^{a\star}\eta^{{\circ\circ}}\partial_{\circ}\hat{\mathscr{F}}^{I}_{\star{\circ}},\hskip
10.00002pt\left[f\hat{\mathscr{F}}\right]^{(I;a)}_{(J;b)}:=\eta^{a{\circ}}f^{I}_{\hskip
0.70004ptJK}\hskip 1.00006pt\hat{\mathscr{F}}^{K}_{{\circ}b},$
$\displaystyle\left[{\mathscr{A}}\right]^{(J;a)}$
$\displaystyle:=\eta^{a{\circ}}{\mathscr{A}}^{J}_{\circ},$

and $(I;a):=N_{SU}\times(I-1)+a$ runs from 1 to $N_{\mathscr{A}}$. Suppose all
$\mathscr{F}^{I}_{\bullet\bullet}$ (and thus,
$\partial_{\bullet}\mathscr{F}^{I}_{\bullet\bullet}$ too) are given. If matrix
$[\hat{\mathscr{F}}]$ is invertible, $[f\hat{\mathscr{F}}]$ is also
invertible; thus ${\mathfrak{A}}$ is uniquely provided from given
$\mathfrak{F}$. It is proven for the $SU(2)$ Yang–Mills theory:

######  Example A.9.

Solution of equation $(\ref{bicomp3})$ for SU(2)

Matrix $\left[f\hat{\mathscr{F}}\right]$ has a block representation such that

$\displaystyle\left[f\hat{\mathscr{F}}\right]=\left(\begin{array}[]{ccc}\hskip
10.00002pt{\mbox{\boldmath$0$}}&\hskip
10.00002pt\left[\hat{\mathscr{F}}^{3}\right]&-\left[\hat{\mathscr{F}}^{2}\right]\\\
-\left[\hat{\mathscr{F}}^{3}\right]&\hskip
10.00002pt{\mbox{\boldmath$0$}}&\hskip
10.00002pt\left[\hat{\mathscr{F}}^{1}\right]\\\ \hskip
10.00002pt\left[\hat{\mathscr{F}}^{2}\right]&-\left[\hat{\mathscr{F}}^{1}\right]&\hskip
10.00002pt{\mbox{\boldmath$0$}}\end{array}\right),$

where $0$ is a $(4\times 4)$ zero-matrix. Here, matrix

$\displaystyle\tilde{{\mbox{\boldmath$F$}}}:=\left(\begin{array}[]{ccc}{\mbox{\boldmath$1$}}&\left[\hat{\mathscr{F}}^{2}\right]\left[\hat{\mathscr{F}}^{1}\right]^{-1}&\left[\hat{\mathscr{F}}^{3}\right]\left[\hat{\mathscr{F}}^{1}\right]^{-1}\\\
\left[\hat{\mathscr{F}}^{1}\right]\left[\hat{\mathscr{F}}^{2}\right]^{-1}&{\mbox{\boldmath$1$}}&\left[\hat{\mathscr{F}}^{3}\right]\left[\hat{\mathscr{F}}^{2}\right]^{-1}\\\
\left[\hat{\mathscr{F}}^{1}\right]\left[\hat{\mathscr{F}}^{3}\right]^{-1}&\left[\hat{\mathscr{F}}^{2}\right]\left[\hat{\mathscr{F}}^{3}\right]^{-1}&{\mbox{\boldmath$1$}}\end{array}\right),$

are introduced, where $1$ is $(4\times 4)$ unit-matrix and $[\bullet]^{-1}$ is
inverse of matrix $\bullet$. Notably, all $[\mathscr{F}^{\bullet}]$ are
assumed to be invertible. Matrix $\left[f\hat{\mathscr{F}}\right]$ is block-
diagonalized using $\tilde{{\mbox{\boldmath$F$}}}$ as

$\displaystyle\tilde{{\mbox{\boldmath$F$}}}.\left[f\hat{\mathscr{F}}\right]$
$\displaystyle=\left(\begin{array}[]{ccc}\left[\hat{\mathscr{F}}^{3}\right]\left[\hat{\mathscr{F}}^{1}\right]^{-1}\left[\hat{\mathscr{F}}^{2}\right]&0&0\\\
0&\left[\hat{\mathscr{F}}^{1}\right]\left[\hat{\mathscr{F}}^{2}\right]^{-1}\left[\hat{\mathscr{F}}^{3}\right]&0\\\
0&0&\left[\hat{\mathscr{F}}^{2}\right]\left[\hat{\mathscr{F}}^{3}\right]^{-1}\left[\hat{\mathscr{F}}^{1}\right]\end{array}\right)$
$\displaystyle~{}~{}-\left(\begin{array}[]{ccc}(2\leftrightarrow 3)&0&0\\\
0&(3\leftrightarrow 1)&0\\\ 0&0&(1\leftrightarrow 2)\end{array}\right)$

Matrix $[\hat{\mathscr{F}}^{I}]$ are anti-symmetric as
$[\hat{\mathscr{F}}^{I}]^{T}=-[\hat{\mathscr{F}}^{I}]$ and
$([\hat{\mathscr{F}}^{I}]^{-1})^{T}=-[\hat{\mathscr{F}}^{I}]^{-1}$; thus,
diagonal block-matrices are symmetric such that

$\displaystyle\tilde{{\mbox{\boldmath$F$}}}_{J}$
$\displaystyle:=[\hat{\mathscr{F}}^{I}][\hat{\mathscr{F}}^{J}]^{-1}[\hat{\mathscr{F}}^{K}]-(K\leftrightarrow
I),$ $\displaystyle\tilde{{\mbox{\boldmath$F$}}}_{J}^{T}$
$\displaystyle=[\hat{\mathscr{F}}^{K}]^{T}([\hat{\mathscr{F}}^{J}]^{-1})^{T}[\hat{\mathscr{F}}^{I}]^{T}-(K\leftrightarrow
I),$
$\displaystyle=-[\hat{\mathscr{F}}^{K}][\hat{\mathscr{F}}^{J}]^{-1}[\hat{\mathscr{F}}^{I}]+(K\leftrightarrow
I)=\tilde{{\mbox{\boldmath$F$}}}_{J}.$

Therefore, diagonal blocks of
$\tilde{{\mbox{\boldmath$F$}}}.[f\hat{\mathscr{F}}]$ are invertible matrices;
thus, $[{\mathscr{A}}]$ has a unique solution.

Field matrices $\mathscr{F}^{I}$ are anti-symmetric $(N_{\textrm{ST}}\times
N_{\textrm{ST}})$-matrices; thus, they have $N_{SU}\times
N_{\textrm{ST}}\times(N_{\textrm{ST}}-1)/2$ independent components in total.
On the other hand, ${\mathscr{A}}^{I}$ has $(N_{SU}\times
N_{\textrm{ST}})$-independent components. If curvature $\mathfrak{F}$ is
provided from connection ${\mathfrak{A}}$ through the structure equation, all
component of $\mathscr{F}^{I}$ are not independent; there are $N_{SU}\times
N_{\textrm{ST}}\times(N_{\textrm{ST}}-3)/2$ constraints. In this case,
algebraic equation $(\ref{bicomp3})$ has a unique solution that fulfils the
structure equation. When arbitrary anti-symmetric matrices $\mathscr{F}^{I}$
are given without specifying ${\mathscr{A}}^{I}$, the solution of algebraic
equation $(\ref{bicomp3})$ does not fulfil $(\ref{eq1})^{\prime}$ in general.
A necessary condition to realize a relation
$(\ref{BianchihSU})^{\prime}\Rightarrow(\ref{eq1})^{\prime}$ is provided
below.

######  Theorem A.10.

Suppose $SU(N)$ Lie-algebra valused non-singular two-form object
$\mathfrak{F}$ given in $\Sigma_{\mathscr{F}}\subseteq{T^{*}{\mathcal{}M}}$ is
an adjoint representation of $SU(N)$. Following two statements are equivalent:

  1. 1.

$\mathfrak{F}$ is an $SU(N)$-curvature associated with $SU(N)$-connection
${\mathfrak{A}}$.

  2. 2.

Solution of algebraic equation $d\mathfrak{F}-i\hskip 1.00006pt{{c}}\hskip
1.00006pt[{\mathfrak{A}},\mathfrak{F}]_{\wedge}=0$ with respect to
${\mathfrak{A}}$ in compact manifold $\Sigma_{\mathfrak{A}}$ has the vanishing
first homology such that;
$H_{1}(\Sigma_{\mathfrak{A}},\partial\Sigma_{\mathfrak{A}};\mathbb{R})=0$.

###### Proof.

1. $\Rightarrow$ 2.: When $\mathfrak{F}$ is a curvature, it is represented using structure equation $\mathfrak{F}=d{\mathfrak{A}}-i\hskip 1.00006pt{{c}}\hskip 1.00006pt{\mathfrak{A}}\wedge{\mathfrak{A}}$ and fulfils the Bianchi identity $d\mathfrak{F}-i\hskip 1.00006pt{{c}}\hskip 1.00006pt[{\mathfrak{A}},\mathfrak{F}]_{\wedge}=0$.
2. $\Rightarrow$ 1.: When non-singular tensors $\mathscr{F}^{I}_{ab}$ for $\mathfrak{F}=\mathscr{F}^{I}_{\circ\circ}({\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}){t_{I}}$ are provided, equation $d\mathfrak{F}-i\hskip 1.00006pt{{c}}\hskip 1.00006pt[{\mathfrak{A}},\mathfrak{F}]_{\wedge}=0$ has a unique solution for ${\mathscr{A}}^{I}_{a}$. When $\Sigma_{\mathfrak{A}}$ has the vanished first homology; $H_{1}(\Sigma_{\mathfrak{A}},\partial\Sigma_{\mathfrak{A}};\mathbb{R})=0$, the statement 2. equivalently asserts that the exact form $\widetilde{{\mathfrak{c}}{\mathfrak{s}}}:=-i\hskip 1.00006pt{{c}}\hskip 1.00006pt[{\mathfrak{A}},\mathfrak{F}]_{\wedge}$ exists in oriented closed $\Sigma_{\widetilde{{\mathfrak{c}}{\mathfrak{s}}}}$ corresponding to given $\mathfrak{F}$ such that $\widetilde{{\mathfrak{c}}{\mathfrak{s}}}=-d\mathfrak{F}$. Therefore, the first-homology of $\widetilde{{\mathfrak{c}}{\mathfrak{s}}}$ is vanished and $\widetilde{{\mathfrak{c}}{\mathfrak{s}}}$ trivially induces exact form $d\mathfrak{F}$ owing to Remark A.1. On the other hand, statement 1. also uniquely induces the Chern–Simons form, which fulfils the Bianchi identity $d\mathfrak{F}+{{\mathfrak{c}}{\mathfrak{s}}}=0$; thus, $\widetilde{{\mathfrak{c}}{\mathfrak{s}}}={{\mathfrak{c}}{\mathfrak{s}}}$ is fulfilled. Therefore, the theorem is maintained. ∎

Figure 6:  Components of the dual $SU(2)$ connection
${\hat{\mathscr{A}}}^{1}_{y}=-\mathrm{cn}(t-z,-1)/\mathrm{sn}(t-z,-1)$ and
${\hat{\mathscr{A}}}^{2}_{x}=\mathrm{sn}(t-z,-1)/\mathrm{cn}(t-z,-1)$ with
respect to a $z$-coordinate at $t=0$ are shown in the figure.

######  Example A.11.

$SU(2)$ connection I.

A following $SU(2)$ connection provided using the Jacobi elliptic functions is
considered:

$\displaystyle{c^{~{}}_{SU}}\hskip 1.00006pt{\mathfrak{A}}^{I}$
$\displaystyle:=\left\\{\begin{array}[]{cl}-i\hskip
1.99997pt\mathrm{cn}(t-z,-1)/\mathrm{sn}(t-z,-1)\hskip 1.00006ptdx&(I=1),\\\
-i\hskip 1.99997pt\mathrm{sn}(t-z,-1)/\mathrm{cn}(t-z,-1)\hskip
1.00006ptdy&(I=2),\\\ -i\left(z\hskip 1.00006ptdt+z\hskip
1.00006ptdz\right),&(I=3)\end{array}\right..$ (148)

Here, we use coordinate vector $\xi=(t,x,y,z)$ with null physical dimension.
Connection (148) has singularities aligning on lines $t-z=2l\hskip
1.00006ptK_{-1}$ and $t-z=(2l+1)\hskip 1.00006ptK_{-1}$, respectively, for
${\mathscr{A}}^{1}_{x}$ and ${\mathscr{A}}^{2}_{y}$, where $l\in\mathbb{Z}$
and $K_{-1}=1.31103\cdots$ is the complete elliptic integral of the first
kind; thus, a domain of the $SU(2)$ connection is provided such that
$\Sigma_{\mathfrak{A}}=(\mathbb{R}^{2}/(l\mathbb{R}))\otimes\mathbb{R}^{2}$.
The first and the second $\mathbb{R}^{2}$ in the domain correspond to $t-z$
and $x-y$ planes, respectively. The behavior of functions
${\mathscr{A}}^{1}_{x}$ and ${\mathscr{A}}^{2}_{y}$ with respect to a
$z$-coordinate at $t=0$ is drawn in Figure 6. Domain $\Sigma_{\mathfrak{A}}$
is isomorphic to
$\mathbb{R}^{3}\otimes(\mathbb{R}/\\{\tau\in\mathbb{R}\mid\tau=lK_{-1}\\})\cong(\mathbb{R}^{4}\oplus\mathbb{R}^{4}\oplus\cdots)$
using new coordinates $\xi^{\prime}:=(\tau:=t-z,x,y,t+z)$. Therefore, it is
obtained that
$H^{3}_{{dR}}(\Sigma_{\mathfrak{A}}){\cong}H_{1}(D^{4},\not\\!0;\mathbb{R}){\oplus}H_{1}(D^{4},\not\\!0;\mathbb{R})\oplus\cdots=0$.

This connection induces the $SU(2)$-curvature (with $c^{~{}}_{SU}=1$) such
that;

$\displaystyle~{}{c^{~{}}_{SU}}\hskip 1.00006pt\mathfrak{F}_{~{}}^{I}=$
$\displaystyle~{}\left\\{\begin{array}[]{ccl}i\frac{\mathrm{dn}(t-z,-1)}{\mathrm{sn}^{2}(t-z,-1)}\hskip
1.00006pt\left(dt\wedge dx-dz\wedge
dx\right)+z\frac{\mathrm{sn}(t-z,-1)}{\mathrm{cn}(t-z,-1)}\hskip
1.00006pt\left(dt\wedge dy-dy\wedge dz\right),&(I=1),\\\
-z\frac{\mathrm{cn}(t-z,-1)}{\mathrm{sn}(t-z,-1)}\hskip
1.00006pt\left(dt\wedge dx+dz\wedge
dx\right)-i\frac{\mathrm{dn}(t-z,-1)}{\mathrm{cn}^{2}(t-z,-1)}\hskip
1.00006pt\left(dt\wedge dy+dy\wedge dz\right),&(I=2),\\\ i\hskip
1.99997ptdt\wedge dz-dx\wedge dy,&~{}(I=3),\end{array}\right.$

through the structure equation. Then, dual curvature
$\hat{\mathfrak{F}}_{~{}}^{I}={\widehat{\mathrm{H}}}\left(\mathfrak{F}^{I}_{~{}}\right)$
is calculated from $\mathfrak{F}^{I}_{~{}}$. Both curvatures have the same
homology class as that of the connection.

Algebraic equation
$d\hat{\mathfrak{F}}-i{c^{~{}}_{SU}}[\hat{\mathfrak{A}},\hat{\mathfrak{F}}]_{\wedge}=0$
for given $\hat{\mathfrak{F}}$ has a unique solution such that;

${c^{~{}}_{SU}}\hskip
1.00006pt\hat{{\mathfrak{A}}}^{I}:=\left\\{\begin{array}[]{cl}-\mathrm{cn}(t-z,-1)/\mathrm{sn}(t-z,-1)\hskip
1.00006ptdy,&(I=1),\\\ \mathrm{sn}(t-z,-1)/\mathrm{cn}(t-z,-1)\hskip
1.00006ptdx,&(I=2),\\\ i\left(z\hskip 1.00006ptdt+z\hskip
1.00006ptdz\right),&(I=3),\end{array}\right.$

which fulfils structure equation
$\hat{\mathfrak{F}}=d\hat{\mathfrak{A}}-i{c^{~{}}_{SU}}\hat{\mathfrak{A}}\wedge\hat{\mathfrak{A}}$.
A domain of the dual connection has the same first homology such that
$H_{1}(\Sigma_{\mathfrak{A}};\mathbb{R})=H_{1}(\Sigma_{\hat{{\mathfrak{A}}}};\mathbb{R})=0$.
Behavior of the dual $SU(2)$ connection is drawn in Figure 6, and the dual
textitelectric fields defined as
${\hat{E}}^{I}_{a}:=\hat{\mathfrak{F}}^{I}_{0a}$ are shown in Figure 7.

Here, we obtained the Kramers pairs of curvatures and connections owing to
exact solutions of the classical Yang–Mills equation with the $SU(2)$ gauge
group. Connections and curvatures fulfil covariant-exactness
$(\ref{eq1})^{\prime}$ and covariant-closedness $(\ref{BianchihSU})^{\prime}$.
The equation of motion in this system is provided as

$\displaystyle{c}^{~{}}_{SU}\left(d\hat{\mathfrak{F}}^{I}-i{c^{~{}}_{SU}}\left[\hat{\mathfrak{F}},{\mathfrak{A}}\right]^{I}_{\wedge}\right)$
$\displaystyle=\left\\{\begin{array}[]{ccl}\frac{\mathrm{sn}}{\mathrm{cn}}\hskip
1.99997pt\mathfrak{V}^{1}+i\frac{\mathrm{sn}\hskip
0.70004pt\mathrm{cn}-4z\hskip 0.70004pt\mathrm{dn}}{\mathrm{cn}^{2}}\hskip
1.99997pt\mathfrak{V}^{2}&(I=1)\\\ \frac{\mathrm{cn}}{\mathrm{sn}}\hskip
1.99997pt\mathfrak{V}^{2}-i\frac{\mathrm{sn}\hskip
0.70004pt\mathrm{cn}+4z\hskip 0.70004pt\mathrm{dn}}{\mathrm{sn}^{2}}\hskip
1.99997pt\mathfrak{V}^{1}&(I=2)\\\
z\frac{\mathrm{sn}^{4}+\mathrm{cn}^{4}}{\mathrm{sn}^{2}\mathrm{cn}^{2}}\left(\mathfrak{V}^{0}+\mathfrak{V}^{3}\right)&(I=3)\end{array}\right.,$
$\displaystyle=j^{I}_{\circ}\mathfrak{V}^{\circ},$

where the argument of the Jacobi elliptic function is common and it is omitted
in the above formulae.

Figure 7: The Electric fields of $I=1$ are shown in figures. In the left
figure, solid (dashed) lines show $\hat{E}_{y}$ ($i\hat{E}_{x}$),
respectively.

###  A.3 $SU(N)$ gauge group in curved space-time

Finally, non-abelian gauge of $SU(N)$ group in the curved space-time is
considered. Exactness and closedness are represented, respectively, such that:

$\displaystyle\textit{Covariant\hskip 5.0ptexactness}\hskip 3.00003pt:$
$\displaystyle\hskip
13.00005pt\mathfrak{F}=d{\mathfrak{A}}+{c^{~{}}_{gr}}{\mathfrak{w}}\wedge{\mathfrak{A}}-i\hskip
1.00006pt{{c}}\hskip 1.99997pt{\mathfrak{A}}\wedge{\mathfrak{A}},\vspace{1mm}$
(149) $\displaystyle\textit{Covariant\hskip 5.0ptclosedness}:$
$\displaystyle\hskip
1.00006ptd_{G}\mathfrak{F}=d\mathfrak{F}+{c^{~{}}_{gr}}[{\mathfrak{w}},\mathfrak{F}]_{\wedge}-i\hskip
1.00006pt{{c}}\hskip 1.00006pt[{\mathfrak{A}},\mathfrak{F}]_{\wedge}=0.$ (150)

A component representation of, e.g., ${\mathfrak{w}}\wedge{\mathfrak{A}}$,
using the trivial bases in ${T^{*}{\mathcal{}M}}$ is provided as

$\displaystyle{\mathfrak{w}}\wedge{\mathfrak{A}}$
$\displaystyle=[{\mathfrak{w}}\wedge{\mathfrak{A}}]_{ab}\hskip
1.00006pt{\mathfrak{e}}^{a}\wedge{\mathfrak{e}}^{b}=\left(\eta_{b{\circ}}\hskip
1.00006pt\omega_{\mu}^{\hskip
2.10002pt{\circ}\star}{\mathscr{A}}^{I}_{\star}{\mathcal{E}}^{\mu}_{a}\right)\hskip
1.00006pt{\mathfrak{e}}^{a}\wedge{\mathfrak{e}}^{b}\hskip 1.99997ptt_{I}.$

Here, an approximation that spin form ${\mathfrak{w}}$ is a external field
that has constant values independent of ${\mathfrak{A}}$ and $\mathfrak{F}$,
is exploited. In this case, the Chern–Simons form is defied as

$\displaystyle{{\mathfrak{c}}{\mathfrak{s}}}\left({\mathfrak{w}},{\mathfrak{A}}\right):=[{c^{~{}}_{gr}}{\mathfrak{w}}-i\hskip
1.00006pt{{c}}\hskip 1.00006pt{\mathfrak{A}},\mathfrak{F}]_{\wedge},$

and thus, its closedness with respect to the external derivative is
maintained;
$d{{\mathfrak{c}}{\mathfrak{s}}}\left({\mathfrak{w}},{\mathfrak{A}}\right)=0$.
Anti-derivative operator $\delta_{{\mathfrak{w}}\otimes{\mathfrak{A}}}$ is
defined similarly as (141), and the exactness with respect to the anti-
derivative operator $\delta_{{\mathfrak{w}}\otimes{\mathfrak{A}}}\hskip
1.00006pt{{\mathfrak{c}}{\mathfrak{s}}}\left({\mathfrak{w}},{\mathfrak{A}}\right)=\mathfrak{F}$
is maintained. Theorem A.10 is also maintained in this case because spin form
${\mathfrak{w}}$ is assumed to be constant valued.

###  A.4 Existence of dual spinor

It is known that existence of a spin structure in a bundle is distinguished
owing to the first and second Stiefel–Whitney classes[51]:

######  Remark A.12.

Suppose $w^{i}(M)\in H^{i}(M,\mathbb{Z}_{2})$ is $i^{th}$ Stiefel–Whitney
class of orientable manifold $M$.

  1. 1.

Manifold $M$ has a spin structure, if and only if

$\displaystyle w^{2}(M)\in H^{2}(M,\mathbb{Z}_{2})\cong
Hom(H_{2}(M;\mathbb{Z}_{2}),\mathbb{Z}_{2})$

vanishes; $w^{2}(M)=0$.

  2. 2.

When $M$ has a spin structure, each spin structure has one-to-one
correspondence to

$\displaystyle H^{1}(M,\mathbb{Z}_{2})\cong Hom(\pi_{1}(M),\mathbb{Z}_{2}),$

where $\pi_{1}(M)$ is fundamental group of $M$.

###### Proof.

For a proof of the remark, see, e.g., chapter 4, section 5 in Ref.[52] ∎

For a three-dimensional manifold
${\mathcal{}M}^{3}_{\overline{r=0}}:=\mathbb{R}^{3}{\setminus}\\{r=0\\}$, any
closed surface
$\Sigma^{2}_{\overline{r=0}}\subset{\mathcal{}M}^{3}_{\overline{r=0}}$ such
that $[\Sigma^{2}_{\overline{r=0}}]\in
H_{2}({\mathcal{}M}^{3}_{\overline{r=0}},\mathbb{Z}_{2})$ is homologous to
$S^{2}$, and any vector bundle in $S^{2}$ is isomorphic to $\pi_{1}(SO(3))$.
Therefore, both first and second Stiefel–Whitney classes vanish, and only one
spin structure exists in ${\mathcal{}M}^{3}_{\overline{r=0}}$. As shown in
Example 5.1, the Coulomb type potential is defined in
$\mathbb{R}\otimes{\mathcal{}M}^{3}_{\overline{r=0}}$; thus, there exists a
spinorial section corresponding to an electric monopole in the Yang–Mills
theory with $U(1)$-gauge group (an electron in electromagnetic theory). On the
other hand, a manifold
${\mathcal{}M}^{3}_{\overline{x=y=0}}:=\mathbb{R}^{3}{\setminus}\\{x=y=0\\}$
has a non-trivial two-dimensional surface such as a torus winding around the
Dirac string. Therefore, the second Stiefel–Whitney class does not vanish;
$w^{2}({\mathcal{}M}^{3}_{\overline{x=y=0}})\cong\mathbb{Z}$; thus, the spin
structure does not exist in a dual-bundle. Therefore, in a space with an
isolated magnetic monopole, an electron does not exist. If a pair of magnetic
monopoles with opposite charges makes a magnetic dipole, the second
Stiefel–Whitney class is trivial; thus, a spin structure exists.

## References

  * [1] Yang, C.N., Mills, R.L.: Conservation of isotopic spin and isotopic gauge invariance. Phys. Rev. 96, 191–195 (1954). doi:10.1103/PhysRev.96.191
  * [2] Atiyah, M.F., Singer, I.M.: The Index of elliptic operators. 1. Annals Math. 87, 484–530 (1968). doi:10.2307/1970715
  * [3] Atiyah, M.F., Segal, G.B.: The Index of elliptic operators. 2. Annals Math. 87, 531–545 (1968). doi:10.2307/1970716
  * [4] Atiyah, M.F., Singer, I.M.: The Index of elliptic operators. 3. Annals Math. 87, 546–604 (1968). doi:10.2307/1970717
  * [5] Atiyah, M.F., Singer, I.M.: The Index of elliptic operators. 5. Annals Math. 93, 139–149 (1971). doi:10.2307/1970757
  * [6] Atiyah, M.F., Singer, I.M.: The Index of elliptic operators. 4. Annals Math. 93, 119–138 (1971). doi:10.2307/1970756
  * [7] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. I. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 77, pp. 43–69 (1975). Cambridge University Press
  * [8] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. II. Mathematical Proceedings of the Cambridge Philosophical Society 78(3), 405–432 (1975). doi:10.1017/S0305004100051872
  * [9] Atiyah, M.F., Patodi, V.K., Singer, I.M.: Spectral asymmetry and Riemannian geometry. III. Mathematical Proceedings of the Cambridge Philosophical Society 79(1), 71–99 (1976). doi:10.1017/S0305004100052105
  * [10] Schwarz, A.S.: On regular solutions of Euclidean Yang–Mills equations. Physics Letters B 67(2), 172–174 (1977). doi:10.1016/0370-2693(77)90095-8
  * [11] Witten, E.: Monopoles and four manifolds. Math. Res. Lett. 1, 769–796 (1994). doi:10.4310/MRL.1994.v1.n6.a13. hep-th/9411102
  * [12] Donaldson, S.K.: An application of gauge theory to four-dimensional topology. J. Differential Geom. 18(2), 279–315 (1983). doi:10.4310/jdg/1214437665
  * [13] Fujikawa, K.: Path-integral measure for gauge-invariant fermion theories. Phys. Rev. Lett. 42, 1195–1198 (1979). doi:10.1103/PhysRevLett.42.1195
  * [14] Alvarez-Gaumé, L., Ginsparg, P.: The topological meaning of non-abelian anomalies. Nuclear Physics B 243(3), 449–474 (1984). doi:10.1016/0550-3213(84)90487-5
  * [15] Alvarez-Gaumé, L., Ginsparg, P.H.: The Structure of Gauge and Gravitational Anomalies. Annals Phys. 161, 423 (1985). doi:10.1016/0003-4916(85)90087-9. [Erratum: Annals Phys. 171, 233 (1986)]
  * [16] Berline, N., Getzler, E., Vergne, M.: Heat Kernels and Dirac Operators. Grundlehren Text Editions, (2003). https://www.springer.com/gp/book/9783540200628
  * [17] Atiya, M., Hitchin, N., Singer, I.: Self-duality in four-dimensional riemannian geometry. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 362(1711), 425–461 (1978). doi:10.1098/rspa.1978.0143
  * [18] Woit, P.: Euclidean Twistor Unification (2021). 2104.05099
  * [19] Fukaya, H., Onogi, T., Yamaguchi, S.: Atiyah-patodi-singer index from the domain-wall fermion dirac operator. Phys. Rev. D 96, 125004 (2017). doi:10.1103/PhysRevD.96.125004
  * [20] Fukaya, H., Furuta, M., Matsuo, S., Onogi, T., Yamaguchi, S., Yamashita, M.: A physicist-friendly reformulation of the Atiyah-Patodi-Singer index and its mathematical justification. PoS LATTICE2019, 061 (2019). doi:10.22323/1.363.0061. 2001.01428
  * [21] Fukaya, H., Furuta, M., Matsuo, S., Onogi, T., Yamaguchi, S., Yamashita, M.: The Atiyah–Patodi–Singer Index and Domain-Wall Fermion Dirac Operators. Commun. Math. Phys. 380(3), 1295–1311 (2020). doi:10.1007/s00220-020-03806-0. 1910.01987
  * [22] Bär, C., Strohmaier, A.: An index theorem for Lorentzian manifolds with compact spacelike Cauchy boundary. American Journal of Mathematics (2017). © 2017. This is an author produced version of a paper accepted for publication in American Journal of Mathematics. Uploaded in accordance with the publisher’s self-archiving policy.
  * [23] Bär, C., Hannes, S.: Boundary Value Problems for the Lorentzian Dirac Operator. Geometry and Physics: Volume I, (2018). doi:10.1093/oso/9780198802013.003.0001
  * [24] Bertlmann, R.A.: Anomalies in Quantum Field Theory. International Series of Monographs on Physics, (2000). doi:10.1093/acprof:oso/9780198507628.001.0001
  * [25] McKean, H.P.J., Singer, I.M.: Curvature and the eigenvalues of the Laplacian. Journal of Differential Geometry 1(1-2), 43–69 (1967). doi:10.4310/jdg/1214427880
  * [26] Geroch, R.P.: Topology in general relativity. Journal of Mathematical Physics 8(4), 782–786 (1967). doi:10.1063/1.1705276. https://doi.org/10.1063/1.1705276
  * [27] Thom, René F.: Veriétés différentiables cobordantes. C.R. Acad.Sic. Paris 78, 1733–1735 (1953)
  * [28] Thom, René F.: Quelques propriétés globales des variétés diffrentiables. Comment. Math. Helv. 28, 17–86 (1954)
  * [29] Hirsch, M.W.: Differential Topology. Graduate texts in mathematics, (1976). doi:10.1007/978-1-4684-9449-5
  * [30] Gibbons, G.W.: Cosmological fermion-number non-conservation. Physics Letters B 84(4), 431–434 (1979). doi:10.1016/0370-2693(79)91232-2
  * [31] Gibbons, G.W.: Spectral asymmetry and quantum field theory in curved spacetime. Annals of Physics 125(1), 98–116 (1980). doi:10.1016/0003-4916(80)90120-7
  * [32] Bär, C., Strohmaier, A.: A Rigorous Geometric Derivation of the Chiral Anomaly in Curved Backgrounds. Communications in Mathematical Physics 347(3), 703–721 (2016). doi:10.1007/s00220-016-2664-1
  * [33] Kurihara, Y.: Characteristic classes in general relativity on a modified Poincaré curvature bundle. Journal of Mathematical Physics 58(9), 092502 (2017). doi:10.1063/1.4990708
  * [34] Kurihara, Y.: Symplectic structure for general relativity and Einstein–Brillouin–Keller quantization. Classical and Quantum Gravity 37(23), 235003 (2020). doi:10.1088/1361-6382/abbc44
  * [35] Zanelli, J.: Chern–Simons forms in gravitation theories. Classical and Quantum Gravity 29(13), 133001 (2012)
  * [36] Frè, P.G.: Gravity, a Geometrical Course vol. 1: Development of the Theory and Basic Physical Applications, (2013). doi:10.1007/978-94-007-5361-7
  * [37] Krasnov, K.: Formulations of General Relativity: Gravity, Spinors and Differential Forms. Cambridge Monographs on Mathematical Physics, (2020)
  * [38] Moore, J.D.: Lectures on Seiberg-Witten Invariants. Lecture notes in mathematics Berlin: Mathematical biosciences subseries, vol. 1629 (1996). doi:10.1007/BFb0092948
  * [39] Nakajima, S.: Application of covariant analytic mechanics with differential forms to gravity with Dirac field. Electron. J. Theor. Phys. 13, 95 (2016). 1510.09048
  * [40] Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali dal principio di hamilton. Rendiconti del Circolo Matematico di Palermo (1884-1940) 43(1), 203–212 (2008). doi:10.1007/BF03014670
  * [41] Ferraris, M., Francaviglia, M., Reina, C.: Variational formulation of general relativity from 1915 to 1925 “Palatini’s method” discovered by Einstein in 1925. General Relativity and Gravitation 14(3), 243–254 (1982). doi:10.1007/BF00756060
  * [42] Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry vol. 1, (1996)
  * [43] Donaldson, S.K., Kronheimer, P.B.: The Geometry of Four-manifolds. Oxford mathematical monographs. Oxford University Press, Oxford, England (1990)
  * [44] Sibner, L.M., Sibner, R.J., Uhlenbeck, K.: Solutions to Yang–Mills equations that are not self-dual. Proceedings of the National Academy of Sciences 86(22), 8610–8613 (1989). doi:10.1073/pnas.86.22.8610
  * [45] Markvorsen, S., Min-Oo, M.: Global Riemannian Geometry: Curvature and Topology. Advanced Courses in Mathematics - CRM Barcelona, (2003)
  * [46] Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G., Manin, Y.I.: Construction of instantons. Physics Letters A 65(3), 185–187 (1978). doi:10.1016/0375-9601(78)90141-X
  * [47] Thouless, D.J.: Quantization of particle transport. Phys. Rev. B 27, 6083–6087 (1983). doi:10.1103/PhysRevB.27.6083
  * [48] Fu, L., Kane, C.L.: Time reversal polarization and a ${Z}_{2}$ adiabatic spin pump. Phys. Rev. B 74, 195312 (2006). doi:10.1103/PhysRevB.74.195312
  * [49] Fu, L., Kane, C.L.: Topological insulators with inversion symmetry. Phys. Rev. B 76, 045302 (2007). doi:10.1103/PhysRevB.76.045302
  * [50] Warner, F.W.: Foundations of Differentiable Manifolds and Lie Groups. Graduate Texts in Mathematics, (2013). https://link.springer.com/book/10.1007/978-1-4757-1799-0
  * [51] Milnor, J.W., Stasheff, J.D., Stasheff, J.: Characteristic Classes. Annals of mathematics studies, (1974). https://www.maths.ed.ac.uk/~v1ranick/papers/milnstas.pdf
  * [52] Cohen, R.L.: The Topology of Fibre Bundles Lecture Notes, (1998). http://math.stanford.edu/~ralph/fiber.pdf


# Evidence of stellar oscillations in the post-common envelop binary candidate
ASASSN-V J205543.90+240033.5

J..Takata  Department of Astronomy, School of Physics, Huazhong University of
Science and Technology, Wuhan 430074, People’s Republic of China  A.K.H. Kong
Institute of Astronomy, National Tsing Hua University, Hsinchu 30013, Taiwan
X.F. Wang  Department of Astronomy, School of Physics, Huazhong University of
Science and Technology, Wuhan 430074, People’s Republic of China  F.F. Song
Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan
Province, People’s Republic of China  Key Laboratory for the Structure and
Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming,
People’s Republic of China  University of Chinese Academy of Sciences, 100049
Beijing, People’s Republic of China  J. Mao  Yunnan Observatories, Chinese
Academy of Sciences, 650011 Kunming, Yunnan Province, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, 100012 Beijing, People’s Republic China  Key
Laboratory for the Structure and Evolution of Celestial Objects, Chinese
Academy of Sciences, 650011 Kunming, People’s Republic of China  X. Hou
Yunnan Observatories, Chinese Academy of Sciences, 650011 Kunming, Yunnan
Province, People’s Republic of China  Key Laboratory for the Structure and
Evolution of Celestial Objects, Chinese Academy of Sciences, 650011 Kunming,
People’s Republic of China  C.-P. Hu  Department of Physics, National Changhua
University of Education, Changhua 50007, Taiwan  L. C.-C. Lin  Department of
Physics, National Cheng Kung University, Tainan 701401, Taiwan  K.L. Li
Department of Physics, National Cheng Kung University, Tainan 701401, Taiwan
C.Y. Hui  Department of Astronomy and Space Science, Chungnam National
University, Daejeon 305-764, Korea<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

ASASSN-V J205543.90+240033.5 (ASJ2055) is a possible post-common envelope
binary system. Its optical photometric data shows an orbital variation about
$0.52$ days and a fast period modulation of $P_{0}\sim 9.77$ minute, whose
origin is unknown. In this Letter, we report an evidence of the stellar
oscillation of the companion star as the origin of the fast period modulation.
We analyze the photometric data taken by TESS, Liverpool telescope, and Lulin
One-meter Telescope. It is found that the period of the 9.77-minute signal
measured in 2022 August is significantly shorter than that in 2021
July/August, and the magnitude of the change is of the order of $|\triangle
P_{0}|/P_{0}\sim 0.0008(4)$. Such a large variation will be incompatible with
the scenario of the white dwarf spin as the origin of the 9.77-minute periodic
modulation. We suggest that the fast periodic signal is related to the
emission from the irradiated companion star rather than that of the white
dwarf. Using existing photometric data covering a wide wavelength range, we
estimate that the hot white dwarf in ASJ2055 has a temperature of $T_{eff}\sim
80000$ K and is heating the oscillating M-type main-sequence star with
$T_{eff}\sim 3500$ K on its un-irradiated surface. The stellar oscillation of
M-type main-sequence star has been predicted in theoretical studies, but no
observational confirmation has been done. ASJ2055, therefore, has a potential
to be a unique laboratory to investigate the stellar oscillation of a M-type
main-sequence star and the heating effect on the stellar oscillation.

††software:
Ximage
https://heasarc.gsfc.nasa.gov/xanadu/ximage/ximage.html
IRAF
https://iraf-community.github.io, Tody 1993

##  1 Introduction

Figure 1: Left: Orbital modulation of ASJ2055 observed ZTF ($\lambda=750$ nm).
The solid line is the model light curve (section 3.3). Right: Spectrum of
ASJ2055. The solid line and dashed line are the Koester’s WD model (Koester,
2010) and the Planck function, respectively, with the effective temperature of
$T_{eff,wd}=80000$ K, where the radius of WD and the distance to the target
from Earth are assumed to be $R_{WD}=1.7\times 10^{9}$ cm and $d=1.7$ kpc,
respectively. The dashed-dotted lines represent theoretical spectra of the
heated companion star (section 3.3) at the optical peak ($\phi_{orb}=0$) and
minimum ($\phi_{orb}=0.5$), respectively. For the observed data, the
interstellar dust extinction is calculated using the model of Fitzpatrick &
Massa (2007) ($R_{V}=3.1$) implemented in extinction of Python-code (Barbary,
2016); the extinction in the $V$ bands, $A_{V}=0.5$, is estimated from the
relation $N_{H}/A_{V}\sim 1.8\times 10^{21}~{}{\rm cm^{2}mag^{-1}}$ with
$N_{H}\sim 9\times 10^{20}~{}{\rm cm^{2}}$ inferred from the sky position of
ASJ2055 (https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl). Error bar
of each observational point is smaller than the size of each symbol.

ASASSN-V J205543.90+240033.5 (hereafter ASJ2055) is a binary system, which is
composed of a hot white dwarf (WD) and a cool main-sequence star that is
detached from the Roche-lobe. The information of the binary nature of ASJ2055
is reported by Kato (2021) and Kato et al. (2021), who find two periodic
modulations with $\sim 0.5$ day and $\sim 9.77$ minute in the optical data
taken by the Zwicky Transient Facility (hereafter ZTF, Masci et al., 2019).
The former is thought to be the orbital period ($P_{orb}$), while the origin
of the latter ($P_{0}$) has not been understood. An interesting property of
ASJ2005 is that the orbital light curve in the optical bands shows a single
broad peak with a large amplitude of $\triangle m\sim 1.5$ magnitude (Figure
1). This orbital modulation is interpreted as a result of the irradiation on
day-side of the companion star by the WD (Kato, 2021; Wagner et al., 2021),
and a rate of energy deposited on the companion star will be of the order of
$\sim 10^{32}~{}{\rm erg~{}s^{-1}}$.

If the periodic signal with the 9.77-minute signal represents a spin period of
the WD, ASJ2055 may be a binary system similar to AR Scorpii (Marsh et al.,
2016; Pelisoli et al., 2022), as suggested by Kato (2021). AR Scorpii
comprises a WD and a low-mass (M-type) companion star, and its orbital period
is $P_{orb}\sim 3.56$ hours. It also shows a large orbital variation
($\triangle m\sim 2$ magnitude) in the optical light curve and contains a
rapidly spinning WD with a spin-period of $P_{s}\sim 118$ seconds. Signature
of the non-thermal emission in broad energy bands from radio to X-ray (Marsh
et al., 2016; Buckley et al., 2017; Takata et al., 2018; du Plessis et al.,
2022) of AR Scorpii suggests a particle acceleration process in the WD binary
system.

Although the optical properties of two binary systems are similar to each
other, the heating process of the companion star in ASJ2055 may be different
from that in AR Scorpii. For AR Scorpii, the temperature of the WD’s surface
is $\sim 11,500$ K, indicating that WD luminosity level, $L_{WD}\sim
10^{31}~{}{\rm erg~{}s^{-1}}$, is lower than the luminosity $\sim
10^{32}~{}{\rm erg~{}s^{-1}}$ of the companion star (Marsh et al., 2016;
Garnavich et al., 2021). It is, therefore, suggested that AR Scorpii contains
a fast spinning magnetized WD with a surface magnetic field of $B_{s}\sim
10^{7-8}$ Gauss, and the WD’s magnetic field or rotation will be the energy
source of the irradiation and non-thermal activities (Marsh et al., 2016;
Buckley et al., 2017; Geng et al., 2016; Takata et al., 2017; Bednarek, 2018;
Lyutikov et al., 2020). For ASJ2055, Wagner et al. (2021) measures the
spectrum in $\sim 300-1000$ nm bands, and find that the flux is rising steeply
toward the UV bands (see Figure 1). With the property of the spectrum, they
suggest that ASJ2055 is a post-common envelop binary (hereafter PCEB) and
contains a hot WD that heats up the companion star, which is probably a M-type
star. Hence, the origin of the 9.77-minute periodic modulation has not been
well understood.

In this Letter, we carry out a more detailed photometric study to probe the
origin of the 9.77-minute periodic signal. The structure of this paper is as
follows. We describe the data reduction in section 2. We present the results
of the timing analysis of the photometric data in sections 3.1 and 3.2, and
modeling for the orbital modulation of the light curve in section 3.3. In
section 4, we suggest the oscillation of the companion star is the origin of
9.77-minute periodic signal, and AJS2055 is a new type PCEB, in which a hot WD
heats up the oscillating low-mass main-sequence star.

##  2 Data reduction

We analyze photometric data taken by ZTF, Transiting Exoplanet Survey
Satellite (hereafter TESS, Ricker et al., 2014), Lulin One-meter Telescope
(hereafter LOT) in Taiwan, two-meter Liverpool telescope (hereafter LT) in
Spain and the Neil Gehrels SWIFT Observatory (hereafter SWIFT, Burrows et al.,
2005). For ZTF data, we download the light curves from the Infrared Science
Archive111https://irsa.ipac.caltech.edu, and use the data (DR8 object) in
$r$-band to determine the orbital period (Figure 1). TESS observed ASJ2055 in
120-second cadence mode (MAST Team, 2021) in 2021 July/August and 2022 August.
We download the light curves from Muikulski Archive for Space Telescopes
(MAST) Portal222https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html
and use Pre-search Data Conditioning Simpler Aperture Photometry (PDCSAP) flux
to analyze the light curve. The top two panels in Figure 2 present the
temporal evolution of PDCSAP flux, and the light curves modulate with the
orbital period of $P_{orb}\sim 0.52$ day. We find in the figure that the light
curve generated by the TESS SAP pipeline becomes negative values around the
optical minimum. We investigate the origin of the negative flux by performing
a custom aperture photometry333https://heasarc.gsfc.nasa.gov/docs/tess/Target-
Pixel-File-Tutorial.html with the target pixel files (hereafter TPFs). We
obtain the TPFs from TESS Science Processing Operations
Center444https://heasarc.gsfc.nasa.gov/docs/tess/ and analyze them using
lightkurve tool in Python555https://docs.lightkurve.org/tutorials/2-creating-
light-curves/2-1-cutting-out-tpfs.html. From the TPFs, we find that ASJ2055 is
a faint source and the differential light curve (i.e. the light curve of the
source pixels minus the background pixels) can have negative values in the
light curve. We also confirm that the shape of the differential light curve is
not significantly affected by the choice of the background region. Therefore,
we use the pipeline-generated light curve in this study.

We carried out LOT and LT observations for several nights in 2021 and 2022;
exposure length for each night is from several ten minutes to several hours.
Table 1 summarizes the information of the LOT and LT observations. With the
extracted photometric light curve, we create a Lomb-Scargle periodogram
(hereafter LS, Lomb, 1976) to search for a periodic modulation in the light
curves, and we estimate false of alarm probability (FAP) using the methods of
Baluev (2008).

SWIFT observed ASJ2055 with a total exposure of $\sim 22$ ks. We extract the
clean event files for UVOT and XRT data using `HEASoft ver.6-29`. For UVOT
data, we extract the light curve and magnitude with command `uvotevtlc` and
`uvotsource` in `HEASoft`, respectively. For XRT data, we extract image with
`Xselect` and confirm no significant emission at the source position, which is
consistent with the result reported by Garnavich (2021). We estimate
$F_{X}\sim 5\times 10^{-14}~{}{\rm erg~{}cm^{-2}s^{-1}}$ as the 3-$\sigma$
upper limit using the command `uplimit` in `XIMAGE` package (version 4.5.1).

##  3 Results

###  3.1 Stability of the 9.77-minute signal

Figure 2: Top panels: TESS-PDCSAP flux for AJ2055 taken in 2021 July/August
(top left) and 2022 August (top right) downloaded from MAST. Bottom panels:
The LS-periodogram of 2021 (left panel) and 2022 (right panel) TESS data after
subtracting the mean value of the light curve. The frequency range of
$75-250~{}{\rm day^{-1}}$ is displayed to present the signals of $f_{0}\sim
147~{}{\rm day^{-1}}$ and $f_{1}\sim 192~{}{\rm day^{-1}}$. The frequency
$f_{0}-f_{orb}$ corresponds to the beat signal.

Figure 3: LS-periodogram for 2021 July/August (right) and 2022 August TESS
data after subtracting the orbital modulation and the data taken at the
orbital phase of $\phi_{orb}=0-0.2/0.8-1$, where the periodic signal can be
confirmed in the TESS data. The orbital modulation is removed from the light
curve. The solid lines and dashed lines correspond to FAP=0.1 and 0.01
estimated by the methods of Baluev (2008), respectively.

Figure 4: The O$-$C curves for the arrival phase of the $9.77$-minute periodic
signal using the TESS data. The light curve is folded with the frequency of
$f_{0}=147.28~{}{\rm day^{-1}}$ for 2021 July/August data (left panel) and
$f_{0}=147.40~{}{\rm day^{-1}}$ for 2022 August data (right panel). A Gaussian
function is applied to fit the pulse profile. In the right panel, the solid
curve is the fitting of the O$-$C curve and indicates the first time
derivative of the period of $\dot{P}_{0}=1.2(5)\times 10^{-7}$.

Figure 2 shows the light curves of TESS-PDCAP flux for ASJ2055 taken in 2021
July/August (top left) and 2022 August (top right), and clearly indicates a
modulation with the orbital period of $P_{orb}\sim 0.523$ day, whose frequency
signal ($f_{orb}\sim 1.91~{}{\rm day^{-1}}$) dominates the LS-periodogram of
the TESS light curve. Figure 2 displays the LS-periodogram in the frequency
range of 75 $\rm{day^{-1}}$-250 $\rm{day^{-1}}$, and shows a short periodic
signal of $P_{0}\sim 9.77$ minute ($f_{0}\sim 147~{}{\rm day}^{-1}$). In
addition to the signal $f_{0}$, we can also confirm the existence of the beat
signal at $f_{0}-f_{orb}\sim 145~{}{\rm day^{-1}}$ in 2021 data and a periodic
signal at $f_{1}\sim 192~{}{\rm day^{-1}}$. The beat signal will be related to
the fact that in the current TESS data, the short periodic signal can be
confirmed during the orbital phases $\phi_{orb}=0-0.2/0.8-1$, where
$\phi_{orb}=0$ corresponds to the optical peak in Figure 1 (section 3.2).

To remove the effect of the orbital modulation from the light curve, we fit
the light curve of Figure 2 with a functional form of
$\mathcal{F}=c_{0}\sin[2\pi(ft+c_{1})]+c_{2}t+c_{3}$, where the first term is
the periodic modulation, and the second and third terms correspond to the
linear trend and the base of the light curve, respectively. In addition, $f$
is the peak frequency in LS-periodogram, and it is the orbital frequency at
the first iteration in the pre-whitening process. Then, we subtract its
contribution from the light curve and create a new LS-periodogram to find a
new peak frequency. We iterate this process until no significant signal with
$f<10~{}{\rm day^{-1}}$ appears in the LS-periodogram. Figure 3 shows the LS-
periodogram after removing the orbital modulation with TESS data taken at the
orbital phase $\phi_{orb}=0-0.2/0.8-1$. We can see that significance of
signals increases, and we find a marginal third signal at $f_{2}\sim
103~{}{\rm day^{-1}}$ in 2022 August data (right panel in Figure 3).

We find that two detected frequencies $f_{0}=147.28(4)~{}{\rm day^{-1}}$ in
2021 data and $147.40(4)~{}{\rm day^{-1}}$ in 2022 data are significantly
different from each other; the error is estimated from the Fourier width (the
inverse of total exposure). Such a large change of the periodic signal
$|\triangle P_{0}|/P_{0}\sim 0.0008(4)$ in a scale of year will not be
realized by the change of the spin period of the WD. This suggests that the
WD’s spin as the origin of $P_{0}\sim 9.77$ minutes periodic signal is
unlikely.

We create an observed-minus-computed (hereafter O$-$C) curves of the arrival
phase of the pulsed peak with the TESS data (Figure 4), and investigate the
day/month timescale stability of the signal $f_{0}$. With the TESS light curve
after removing the orbital modulation, we determine the averaged frequency and
arrival phase of the pulse of the 2021 or 2022 TESS data set. Then, we divide
2021 or 2022 data into five segments and fold each epoch with the averaged
frequency. We determine the arrival phase of the pulsed peak and calculate the
difference from the averaged one. Figure 4 shows the O$-$C curves for 2021
data (left panel) and 2022 data (right panel). We can see that the 2021 data
does not show a significant temporal variation of the arrival phase of the
pulsed peak. In 2022 August data (right panel), on the other hand, we may see
a temporal variation of the 9.77-minute periodic modulation with the first
time derivative of the period of $\dot{P}_{0}=1.2(5)\times 10^{-7}$, which can
be also confirmed in 2-dimensional LS-periodogram. If it would correspond to
the spin down of the WD, the spin down energy were
$I_{WD}(2\pi)^{2}P^{-3}_{0}\dot{P}_{0}\sim 10^{37}~{}{\rm erg~{}s^{-1}}$ with
$I_{WD}\sim 10^{51}{\rm g~{}cm^{2}}$ being the moment of inertia of the WD.
Such a unrealistically large spin down rate also indicates that the WD’s spin
is unrelated to the 9.77-minute periodic signal.

In LS-periodogram, the second periodic signal is confirmed at $f_{1}\sim
192.77(4)~{}{\rm day^{-1}}$ ($P_{1}\sim 7.47$ minute) for 2021 data and at
$f_{1}\sim 192.75(4)~{}{\rm day^{-1}}$ for 2022 data. This second periodic
signal would not be explained by a simple harmonic of $f_{0}\sim 147~{}{\rm
day^{-1}}$ signal or the combination of $f_{0}$ and the orbital frequency
$f_{orb}$. Moreover, we can see in 2022 data that the power of $f_{1}$-signal
is comparable to that of $f_{0}$-signal. We expect that $f_{1}$-signal has a
different mode of the stellar oscillation of the low-mass companion star
(section 4).

Table 1: Journal of LOT and LT observations. The zero orbital phase is defined at MJD 58957 | Date | Orbital phase | Filter
---|---|---|---
LOT | 2021/10/04 | 0.14-0.36 | $r$
| 2021/10/05 | 0.05-0.24 | $r$
| 2022/08/14 | 0.34-0.51 | No
| 2022/08/15 | 0.30-0.43 | No
| 2022/10/01 | 0.74-0.87 | No
| 2022/10/02 | 0.43-0.51 | No
|  | 0.75-0.81 | No
| 2022/10/13 | 0.59-0.67 | No
| 2022/11/14 | 0.60-0.75 | No
| 2022/11/15 | 0.52-0.66 | No
| 2022/11/16 | 0.42-0.56 | No
| 2022/11/17 | 0.33-0.47 | No
| 2022/11/18 | 0.24-0.38 | No
LT | 2022/08/03 | 0.58-0.63 | $r$
| 2022/08/08 | 0.31-0.36 | $r$
| 2022/08/11 | 0.95-1.0 | $r$
| 2022/08/18 | 0.43-0.49 | $r$
| 2022/08/20 | 0.47-0.51 | $r$
| 2022/09/06 | 0.56-0.59 | $r$

###  3.2 Orbital variation of the 9.77-minute signal

Figure 5: LT (left) and LOT (right) light curves folded with the orbital
period for the data taken in 2022 August and November, respectively. The
orbital modulation is removed from the data. The solid line shows a sinusoidal
function with a modulation frequency of $f_{0}=147.40~{}{\rm day^{-1}}$ for LT
data and $f_{0}=149.35~{}{\rm day^{-1}}$ for LOT data. For LOT data, the O$-$C
curve of the phase of the pulsed peak is presented. The calculated amplitude
($a$) of the modulation for each data set is presented at the top of each
panel.

The existence of the beat signal, $f_{0}-f_{orb}$, indicates an orbital
variation of the amplitude of the 9.77-minute signal. In TESS observation, we
only confirm the significant periodic signal near the optical peak (the
orbital phase $\phi_{orb}\sim 0-0.2/0.8-1$ in the left panel of Figure 1). We
carried out LOT and LT observations in 2021 and 2022 to investigate the
orbital variation of the periodic signal. The left panel of Figure 5 shows the
light curves folded with the orbital period for 2022 August LT data and the
right panel shows the light curve of November LOT data. Both data sets cover
the optical minimum of the orbital light curve, and clearly indicate a
modulation with a 9.77 minute. We confirm that the 9.77-minute periodic
modulation exists for the entire orbital phase.

We investigate the orbital variation of the 9.77-minute periodic signal with
2022 August LT and 2022 November LOT observations. For LT data, since we
cannot obtain a significant periodic signal in LS periodogram, we apply
$f_{0}=147.40(4)~{}{\rm day^{-1}}$ obtained with the 2022 August TESS data.
For November LOT data, we obtain $f_{0}\sim 149.6(2)~{}{\rm day^{-1}}$ as the
best frequency in LS-periodogram. We use a sinusoidal function to determine
its amplitude and the peak phase of the modulation. In Figure 5, we present
the O$-$C values of the peak phase for LOT data and the amplitude ($a$) for
the data of each night. We can see that within the errors, the O$-$C values
and the amplitudes are consistent with constants throughout the orbital phase.
This may suggest that the non-detection of the periodic signal in the TESS
data during the orbital phase $\phi_{orb}\sim 0.2-0.8$ is due to a low signal-
to-noise ratio. With the large size of the error bars of the current LT/LOT
data, however, a deeper observation is also desired to obtain a more solid
conclusion for the orbital variation of the 9.77-minute periodic signal.

###  3.3 Modeling for orbital modulation

Table 2: Fitting parameters for ASJ2055. $f_{roche}$ represents the Roche-lobe filling factor of the companion star, and $T_{eff,0}$ is the effective temperature of the unheated surface of the companion star. $\theta_{0}$ is the observer viewing angle measured from the direction perpendicular to the orbital plane. $d$ | 1.7 kpc
---|---
$M_{WD}$ | $0.6M_{\odot}$
$R_{WD}$ | $0.024R_{\odot}$
$T_{eff,WD}$ | 80000 K
$M_{c}$ | $0.3M_{\odot}$
$f_{roche}$ | 0.5
$T_{eff,0}$ | 3500 K
$\theta_{o}$ | $37^{\circ}$

ASJ2055 shows a large amplitude ($\triangle m\sim 1.5$ magnitude) of the
orbital modulation in the optical bands. This implies that the companion star,
which is probably M-type main-sequence star, is heated up by the irradiation
of the WD. Wagner et al. (2021) measure the spectrum of ASJ2055 in 300-1000 nm
bands and find that the spectrum rises steeply toward the shorter wavelength.
They suggest that ASJ2055 is a post-common envelope binary (hereafter PCEB)
and a hot WD irradiates the day-side of the secondary companion star. We
produce a broadband spectrum in 100-10000 nm bands using new SWIFT UVOT data
and archival GALEX (Wright et al., 2010), Pan-STARRS (Chambers & et al., 2017)
and WISE data (Cutri & et al., 2012) 666https://vizier.cds.unistra.fr/viz-
bin/VizieR, and we confirm that the spectrum is continuously rising toward
shorter wavelength bands below 300 nm (Figure 1). By assuming that the fluxes
in $\lambda<300$ nm bands is dominated by the emission from the WD, we fit the
spectrum with a Planck function (dashed line in Figure 1) with an effective
temperature of $T_{eff,WD}=80000$ K and $R_{WD}=1.7\times 10^{9}$ cm, where we
apply the distance to the source of $d=1.7$ kpc measured by GAIA (Gaia
Collaboration et al., 2021; Gaia Collaboration, 2022). We also generate
Koester’s WD atmosphere model (Koester, 2010) with the gravity acceleration of
${\rm log}g=7.25$777http://svo2.cab.inta-csic.es/theory/newov2/.

We carry out a modeling for the orbital light curve of the companion star
heated by the WD. Because of the large amplitude of the orbital modulation,
the rotational motion of the companion star is likely synchronized with the
orbital modulation. Only one surface will be heated up by the irradiation from
the WD. We take into account the gravity darkening as
$T_{eff,c}=T_{eff,0}(g/g_{max})^{\beta}$, where $g_{max}$ is the maximum
gravity and $\beta\sim 0.08-0.25$ (Lucy, 1967; Espinosa Lara & Rieutord,
2012). In this study, we apply $\beta=0.15$ in our calculation. By assuming
that all irradiated flux from the WD is used for heating of the companion
star, we may calculate the temperature ($T_{new}$) for a surface segment of
the companion star as

$\sigma_{sb}T_{new}^{4}\delta A=\sigma_{sb}T_{eff,c}^{4}\delta
A+\frac{L_{WD}}{4\pi\ell^{2}}\delta A_{irr},$ (1)

where $L_{WD}$ is the WD’s luminosity, $\sigma_{sb}$ is the Stephen-Boltzmann
constant and $\delta A$ is the area of the surface segment, $\ell$ is the
distance between the WD and the surface segment, and $\delta A_{irr}$ is the
area of the surface segment measured from the WD. For each segment, we assume
a Planck function for the emission, and apply a simple limb-darkening effect
with $I(\cos\theta)\propto 1-0.5(1-\cos\theta)$ (van Hamme, 1993), where
$\theta$ is the viewing angle of the observer measured from the direction
perpendicular to the surface. Table 2 summarizes the parameters of the
fitting. Since no sign of the accretion process is observed, the companion
star will be detached from the Roche-lobe, namely, the Roche-lobe filling
factor is less than unity $f_{roche}<1$. To explain the flux level observed by
WISE, we choose the base surface temperature of $T_{eff,0}=3500$ K. In Figure
1, we compare the results of the model with the observations. We find that the
heating due to the emission from the WD surface can explain the observed light
curve and spectrum, simultaneously.

We note that the effective temperature $(T_{eff,0}$) of the companion star and
the observer viewing angle $\theta_{0}$ are degenerated in the fitting process
for the orbital modulation, and a reasonable fit can be obtained in a
parameter range of $T_{eff,0}\sim 3300-4500$ and $\theta_{o}<60^{\circ}$, for
which a smaller effective temperature corresponds to a smaller viewing angle.
To explain the flux level of the WISE data, on the other hand, a smaller
temperature is preferred. Hence, a more detailed spectroscopic information
such as a radial velocity curve and the spectral information at night-side of
the companion star will be desired to reveal the property of the companion
star.

##  4 Discussion and summary

###  4.1 Origin of the 9.77-minutes signal

It was suggested that the pulsation with a frequency $f_{0}\sim 147~{}{\rm
day^{-1}}$ ($P_{0}\sim 9.77$ minutes) is related to the spin-period of the WD
of ASJ2055 (Kato et al., 2021). As we discussed in section 3.1, however, the
photometric data indicates a significant change in the period $|\triangle
P_{0}|/P_{0}\sim 8\times 10^{-4}$ between 2021 and 2022, and such a large
change will not be explained by the temporal variation of the WD’s spin. With
the current results, therefore, we conclude that the 9.77-minute signal is
irrelevant to the WD’s spin. It is unlikely that the periodic modulation is
related to the stellar oscillation of the WD. This is because if the periodic
emission is from the surface of the WD, we expect the detection of the
periodic signal in TESS data taken at around the optical minimum, where the
contamination of the radiation from the companion star is minimum.

One possible origin of the 9.77-minute signal is the solar-type oscillation of
the main-sequence star (García & Ballot (2019) for a recent review). In
addition to the significant periodic signals of $f_{0}\sim 147.4~{}{\rm
day^{-1}}$ and $f_{1}\sim 192.8~{}{\rm day^{-1}}$, we also find a marginal
third signal at $f_{2}\sim 103~{}{\rm day^{-1}}$ in LS-periodogram (bottom
right panel of Figure 3 ). Because these three frequencies are almost equally
separated with $\triangle f=f_{0}-f_{2}\sim f_{1}-f_{0}\sim 45~{}{\rm
day^{-1}}$, we may anticipate the main frequency $f_{0}$ and the separation
$\triangle f$ as the so-called frequency of the maximum power and large
frequency separation, respectively, of the solar-type oscillation. It has been
known that the frequency of the maximum power and the large separation of the
solar-type oscillations are well scaled as (Belkacem et al., 2011; Hekker,
2020)

$f_{max}=f_{max,\odot}\left(\frac{g}{g_{\odot}}\right)\left(\frac{T_{eff,\odot}}{T_{eff}}\right)^{1/2},$
(2)

and

$\triangle f=\triangle
f_{\odot}\left(\frac{M}{M_{\odot}}\right)^{1/2}\left(\frac{R}{R_{\odot}}\right)^{-3/2},$
(3)

respectively, where $f_{\odot}=3100~{}\mu{\rm Hz}$, $\triangle
f_{\odot}=135.1~{}\mu{\rm Hz}$, $T_{eff,\odot}=5777$ K and ${\rm
log}g_{\odot}=4.4377$ are the solar values.

With $M_{c}\sim 0.3M_{\odot}$, $R_{c}\sim 0.3R_{\odot}$ and $T_{eff}\sim 3500$
K shown in Table 2, the equations (2) and (3) expect $\triangle f\sim
450~{}\mu H$ and $f_{max}\sim 4000\mu Hz$, respectively. We find that the
observed separation $\triangle f\sim 45~{}{\rm day^{-1}}\sim 520\mu{\rm Hz}$
of ASJ2055 may be consistent with the scaling law, but the observed peak
frequency $f_{0}\sim 147.4~{}{\rm day^{-1}}\sim 1700~{}\mu{\rm Hz}$ is
significantly smaller. The day-side of the companion star is heated up to a
temperature of $T_{eff}\sim 8000$ K, with which the scaling law expects
$f_{max}\sim 2640\mu{\rm Hz}$. Hence, the heating of the companion star
surface may affect to the frequency of the stellar oscillation.

Asteroseismology is a powerful tool to investigate the stellar interior
structure and can provide fundamental stellar parameters with high precision
[e.g. using scaling laws of equations (2) and (3)]. The stellar oscillation of
M-type star has been theoretically predicted (Rodríguez-López et al., 2014;
Brito & Lopes, 2021), and the efforts have been made to detect the stellar
oscillation of the M-type star (Baran et al., 2011b, a; Rodríguez et al.,
2016; Berdiñas et al., 2017). However no solid confirmation of the existence
of the stellar oscillation of M-type main-sequence star has been done; one
possible signal from pre M-type main-sequence star has been reported by
Steindl et al. (2021). Hence the method of asteroseismology has not been
applied to M-type main-sequence star. If the 9.77-minute periodic signal
discussed in this study originates from the stellar oscillation, ASJ2055
provides the evidence of the stellar oscillation of the M-type star. Further
investigation will be important to obtain a solid conclusion about the origin
of the short periodic signal and to identify the stellar type of the companion
star.

###  4.2 Other PCEB

With the launch of TESS, the population of the binary system containing a
oscillating star has been increasing (Shi et al., 2022; Chen et al., 2022).
However, the population of the WD binary system that contains an oscillating
secondary is not many, and the companion star is usually subdward B star (Reed
et al., 2011; Kern et al., 2018; Jayaraman et al., 2022). If the 9.77-minute
signal is originated from the stellar oscillation, ASJ2055 will be a new type
of PCEB, in which a hot WD heats up a oscillating M-dwarf star. ASJ2055 is a
relatively young binary system after the common-envelope phase and the cooling
timescale of the WD’s surface is of the order of several million years. Such a
PCEB with a hot WD will be a unique laboratory to study the pulsating M-type
star and the effect of the heating. We, therefore, carry out a search for
other candidates of PCEB that contain a oscillating secondary. Rebassa-
Mansergas et al. (2010) provides $\sim 3300$ of PCEB candidates from SDSS
data. From the catalog, we select 25 binary systems, in which temperature of
the WD is greater than $T_{eff}>70000$ K, and we check a short periodic signal
in the archival photometric data downloaded from MAST portal. However, we
obtained null results. This means that ASJ2005 is the rare PCEB system.

We mention SDSS J082145.27+455923.4 (hereafter J0821), which is an eclipsing
binary with an orbital period of $P_{orb}\sim 0.51$ day, a WD’s surface
temperature of $T_{eff}\sim 80000$ K and M-type companion star (Parsons et
al., 2013). Since these binary parameters are similar to those of ASJ2005, the
detection of the short periodic signal is expected. However, the data taken
with 120-second cadence model of TESS does not show a significant periodic
signal shorter than the orbital period. In spite of edge-on view, the ZTF
light curve shows the amplitude of the orbital modulation is $\sim 0.7$
magnitudes, which is smaller than $\sim 1.5$ magnitudes of ASJ2005. This
indicates that the optical emission from J0821 is more dominated by the WD
emission than that from ASJ2005. A deeper observation for J0821 is desired to
search for the short periodic modulation and to study the
similarity/dissimilarity with ASJ2055.

In summary, we carried out the photometric study for ASJ2055, whose optical
emission shows (i) an orbital modulation ($P_{orb}\sim 0.52$ hours) with an
amplitude of $\triangle m\sim 1.5$ magnitude and (ii) a $9.77$-minute
modulation. With TESS data, we found that (i) the period of the short
modulation measured in 2022 August is significantly smaller ($|\triangle
P_{0}|/P_{0}\sim 8\times 10^{-4}$) than that measured in 2021 July/August and
(ii) TESS 2022 August data also evolution of the periodic signal with the a
time derivative of $\dot{P}_{0}=1.2(5)\times 10^{-7}$. This large variability
of the 9.77-minute periodic signal will be incompatible with the scenario of
the WD’s spin. Alternatively, the oscillation of M-type star is more likely as
the origin of the periodic modulation. The optical/UV spectrum and orbital
modulation suggest that ASJ2055 is a PCEB and the radiation from a hot WD
heats up the day-side of the companion star. ASJ2055 may be a new type of the
WD binary system, in which a hot WD heats up a oscillating M-type star.

We thank to referee for his/her useful comments and suggestions. We are
grateful to Swift-TOO team for arrangements of observations for our sources.
J.M. thanks the discussion with Prof. Yan Li and Xiangdong Li. J.T. and X.F.W
are supported by the National Key Research and Development Program of China
(grant No. 2020YFC2201400) and the National Natural Science Foundation of
China (grant No. 12173014). A.K.H.K. is supported by the National Science and
Technology Council of Taiwan through grant 111-2112-M-007-020. J.M. is
supported by the National Natural Science Foundation of China (grant No.
11673062). X.H. is supported by the National Natural Science Foundation of
China through grant 12041303 and Yunnan Revitalization Talent Support Program
(YunLing Scholar Award). C.-P.H. acknowledges support from the National
Science and Technology Council through MOST 109-2112-M-018-009-MY3. L.C.-C.L.
is supported by NSTC through grants 110-2112-M-006-006-MY3 and
111-2811-M-006-012. K.L.L. is supported by the National Science and Technology
Council through grant 111-2636-M-006-024, and he is also a Yushan Young Fellow
supported by the Ministry of Education of the Republic of China (Taiwan).
C.Y.H. is supported by the research fund of Chungnam National University and
by the National Research Foundation of Korea grant 2022R1F1A1073952

## Data Availability

All data and analysis products presented in this article are available upon
request. The TESS data used in this paper can be found in MAST (MAST Team,
2021). The GALEX, Pan-STARRS and WISEdata used in this papercan be found
inVizieR (https://vizier.cds.unistra.fr/viz-bin/ VizieR). The Siwft data used
in this paper can be obtain from NASA’s HEASARC Archive
(https://heasarc.gsfc.nasa.gov/docs/archive.html).

Swift(XRT), ZTF, TESS LT and LOT.

## References

  * Baluev (2008) Baluev, R. V. 2008, MNRAS, 385, 1279
  * Baran et al. (2011a) Baran, A. S., Fox-Machado, L., Lykke, J., Nielsen, M., & Telting, J. H. 2011a, Acta Astron., 61, 325
  * Baran et al. (2011b) Baran, A. S., Winiarski, M., Krzesiński, J., et al. 2011b, Acta Astron., 61, 37
  * Barbary (2016) Barbary, K. 2016, Extinction V0.3.0, Zenodo, doi:10.5281/zenodo.804967
  * Bednarek (2018) Bednarek, W. 2018, MNRAS, 476, L10
  * Belkacem et al. (2011) Belkacem, K., Goupil, M. J., Dupret, M. A., et al. 2011, A&A, 530, A142
  * Berdiñas et al. (2017) Berdiñas, Z. M., Rodríguez-López, C., Amado, P. J., et al. 2017, MNRAS, 469, 4268
  * Brito & Lopes (2021) Brito, A., & Lopes, I. 2021, MNRAS, 507, 5747
  * Buckley et al. (2017) Buckley, D. A. H., Meintjes, P. J., Potter, S. B., Marsh, T. R., & Gänsicke, B. T. 2017, Nature Astronomy, 1, 0029
  * Burrows et al. (2005) Burrows, D. N., Hill, J. E., Nousek, J. A., et al. 2005, Space Sci. Rev., 120, 165
  * Chambers & et al. (2017) Chambers, K. C., & et al. 2017, VizieR Online Data Catalog, II/349
  * Chen et al. (2022) Chen, X., Ding, X., Cheng, L., et al. 2022, ApJS, 263, 34
  * Cutri & et al. (2012) Cutri, R. M., & et al. 2012, VizieR Online Data Catalog, II/311
  * du Plessis et al. (2022) du Plessis, L., Venter, C., Wadiasingh, Z., et al. 2022, MNRAS, 510, 2998
  * Espinosa Lara & Rieutord (2012) Espinosa Lara, F., & Rieutord, M. 2012, A&A, 547, A32
  * Fitzpatrick & Massa (2007) Fitzpatrick, E. L., & Massa, D. 2007, ApJ, 663, 320
  * Gaia Collaboration (2022) Gaia Collaboration. 2022, VizieR Online Data Catalog, I/360
  * Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1
  * García & Ballot (2019) García, R. A., & Ballot, J. 2019, Living Reviews in Solar Physics, 16, 4
  * Garnavich (2021) Garnavich, P. 2021, The Astronomer’s Telegram, 14932, 1
  * Garnavich et al. (2021) Garnavich, P., Littlefield, C., Lyutikov, M., & Barkov, M. 2021, ApJ, 908, 195
  * Geng et al. (2016) Geng, J.-J., Zhang, B., & Huang, Y.-F. 2016, ApJ, 831, L10
  * Hekker (2020) Hekker, S. 2020, Frontiers in Astronomy and Space Sciences, 7, 3
  * Jayaraman et al. (2022) Jayaraman, R., Handler, G., Rappaport, S. A., et al. 2022, ApJ, 928, L14
  * Kato (2021) Kato, T. 2021, arXiv e-prints, arXiv:2108.09060
  * Kato et al. (2021) Kato, T., Hambsch, F.-J., Pavlenko, E. P., & Sosnovskij, A. A. 2021, arXiv e-prints, arXiv:2109.03979
  * Kern et al. (2018) Kern, J. W., Reed, M. D., Baran, A. S., Telting, J. H., & Østensen, R. H. 2018, MNRAS, 474, 4709
  * Koester (2010) Koester, D. 2010, Mem. Soc. Astron. Italiana, 81, 921
  * Lomb (1976) Lomb, N. R. 1976, Ap&SS, 39, 447
  * Lucy (1967) Lucy, L. B. 1967, ZAp, 65, 89
  * Lyutikov et al. (2020) Lyutikov, M., Barkov, M., Route, M., et al. 2020, arXiv e-prints, arXiv:2004.11474
  * Marsh et al. (2016) Marsh, T. R., Gänsicke, B. T., Hümmerich, S., et al. 2016, Nature, 537, 374
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, PASP, 131, 018003
  * MAST Team (2021) MAST Team. 2021, TESS Light Curves - All Sectors, doi:10.17909/T9-NMC8-F686
  * Parsons et al. (2013) Parsons, S. G., Gänsicke, B. T., Marsh, T. R., et al. 2013, MNRAS, 429, 256
  * Pelisoli et al. (2022) Pelisoli, I., Marsh, T. R., Parsons, S. G., et al. 2022, MNRAS, 516, 5052
  * Rebassa-Mansergas et al. (2010) Rebassa-Mansergas, A., Gänsicke, B. T., Schreiber, M. R., Koester, D., & Rodríguez-Gil, P. 2010, MNRAS, 402, 620
  * Reed et al. (2011) Reed, M. D., Harms, S. L., Poindexter, S., et al. 2011, MNRAS, 412, 371
  * Ricker et al. (2014) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9143, Space Telescopes and Instrumentation 2014: Optical, Infrared, and Millimeter Wave, ed. J. Oschmann, Jacobus M., M. Clampin, G. G. Fazio, & H. A. MacEwen, 914320
  * Rodríguez et al. (2016) Rodríguez, E., Rodríguez-López, C., López-González, M. J., et al. 2016, MNRAS, 457, 1851
  * Rodríguez-López et al. (2014) Rodríguez-López, C., MacDonald, J., Amado, P. J., Moya, A., & Mullan, D. 2014, MNRAS, 438, 2371
  * Shi et al. (2022) Shi, X.-d., Qian, S.-b., & Li, L.-J. 2022, ApJS, 259, 50
  * Steindl et al. (2021) Steindl, T., Zwintz, K., Barnes, T. G., Müllner, M., & Vorobyov, E. I. 2021, A&A, 654, A36
  * Takata et al. (2018) Takata, J., Hu, C. P., Lin, L. C. C., et al. 2018, ApJ, 853, 106
  * Takata et al. (2017) Takata, J., Yang, H., & Cheng, K. S. 2017, ApJ, 851, 143
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * van Hamme (1993) van Hamme, W. 1993, AJ, 106, 2096
  * Wagner et al. (2021) Wagner, R. M., Garnavich, P., Thorstensen, J. R., Littlefield, C., & Szkody, P. 2021, Research Notes of the American Astronomical Society, 5, 242
  * Wright et al. (2010) Wright, E. L., Eisenhardt, P. R. M., Mainzer, A. K., et al. 2010, AJ, 140, 1868