0.70004pt}^{3}\gamma_{sp\hskip 0.70004pt}^{2}\gamma_{sp\hskip
0.70004pt}^{1}\right)\gamma_{sp\hskip 0.70004pt}^{0}$
$\displaystyle\Rightarrow\bar{\psi}^{\pm}=\mp\bar{\psi}^{\pm}{\Gamma_{{sp\hskip
0.70004pt}}}.$ (95)

Here, (78) is used. Therefore, it is provided that
$\bar{\psi}\in{\mathcal{}S}_{sp\hskip 0.70004pt}^{+}$ for
$\psi\in{\mathcal{}S}_{sp\hskip 0.70004pt}^{-}$. Terms $i\hskip
1.00006pt\bar{\psi}(\iota_{{\gamma}_{sp\hskip 0.50003pt}}d)\psi$ and
$\bar{\psi}\psi$ are $SO(1,3)$ invariant as follows:

$\displaystyle{{G}_{SO}}\left(i\hskip
1.00006pt\bar{\psi}(\iota_{{\gamma}_{sp\hskip 0.50003pt}}d)\psi\right)$
$\displaystyle=i\hskip
1.00006pt\psi^{\dagger}S({\mbox{\boldmath$\Lambda$}})^{-1}\left(S({\mbox{\boldmath$\Lambda$}})\gamma_{sp\hskip
0.70004pt}^{0}S({\mbox{\boldmath$\Lambda$}})_{~{}}^{-1}\right)\left(S({\mbox{\boldmath$\Lambda$}})(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d)S({\mbox{\boldmath$\Lambda$}})^{-1}\right)S({\mbox{\boldmath$\Lambda$}})\psi,$
$\displaystyle=i\hskip 1.00006pt\bar{\psi}(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d)\psi,$

where ${{G}_{SO}}\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d\right)=S({\mbox{\boldmath$\Lambda$}})(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d)S({\mbox{\boldmath$\Lambda$}})^{-1}$ is used. Similar
calculations provide the $SO(1,3)$ invariance for $\bar{\psi}\psi$.

######  Remark 3.5.

There exists super algebra in $End\left({\mathcal{}S}_{sp\hskip
0.70004pt}\right)=End^{+}\left({\mathcal{}S}_{sp\hskip 0.70004pt}\right)\oplus
End^{-}\left({\mathcal{}S}_{sp\hskip 0.70004pt}\right)$.

###### Proof.

Suppose operator $\mathcal{O}_{0}$ ($\mathcal{O}_{1}$) includes an even (odd)
number of $\gamma_{sp\hskip 0.70004pt}$’s. For
$\psi^{\pm}\in{\mathcal{}S}_{sp\hskip 0.70004pt}^{\pm}$,

$\displaystyle{\Gamma_{{sp\hskip 0.70004pt}}}\psi^{\pm}=\pm\psi^{\pm},~{}~{}$
$\displaystyle\textrm{and}~{}~{}\left\\{\begin{array}[]{rrl}{\Gamma_{{sp\hskip
0.70004pt}}}\left(\mathcal{O}_{0}\psi^{\pm}\right)=&\mathcal{O}_{0}{\Gamma_{{sp\hskip
0.70004pt}}}\psi^{\pm}=&\pm\left(\mathcal{O}_{0}\psi^{\pm}\right),\\\
{\Gamma_{{sp\hskip
0.70004pt}}}\left(\mathcal{O}_{1}\psi^{\pm}\right)=&-\mathcal{O}_{1}{\Gamma_{{sp\hskip
0.70004pt}}}\psi^{\pm}=&\mp\left(\mathcal{O}_{1}\psi^{\pm}\right)\end{array}\right.$

owing to (79). Thus, we obtain
$\mathcal{O}_{0}\in{End}^{+}\left({\mathcal{}S}_{sp\hskip 0.70004pt}\right)$
and $\mathcal{O}_{1}\in{End}^{-}\left({\mathcal{}S}_{sp\hskip
0.70004pt}\right)$. Moreover, it is obvious that

$\displaystyle\mathcal{O}_{0}\mathcal{O}_{0},\mathcal{O}_{1}\mathcal{O}_{1}\in{End}^{+}\left({\mathcal{}S}_{sp\hskip
0.70004pt}\right)~{}~{}$
$\displaystyle\textrm{and}~{}~{}\mathcal{O}_{0}\mathcal{O}_{1},\mathcal{O}_{1}\mathcal{O}_{0}\in{End}^{-}\left({\mathcal{}S}_{sp\hskip
0.70004pt}\right).$

Therefore, the remark is maintained. ∎

The Dirac operator ${\not\\!d^{~{}}_{{sp\hskip 0.70004pt}}}$ belongs to
$\mathcal{O}_{1}$ and flips a parity of $\psi^{\pm}$. We note that

$\displaystyle\begin{array}[]{ccccc}\bar{\psi}\left(i\hskip
1.00006pt{\not\\!d^{~{}}_{{sp\hskip
0.70004pt}}}\right)\psi&=&\left(\bar{\psi}{\Gamma_{{sp\hskip
0.70004pt}}}\right)\left({\Gamma_{{sp\hskip 0.70004pt}}}\left(i\hskip
1.00006pt{\not\\!d^{~{}}_{{sp\hskip
0.70004pt}}}\right)\psi\right)&=&\bar{\psi}^{+}\left(i\hskip
1.00006pt{\not\\!d^{~{}}_{{sp\hskip
0.70004pt}}}\right)\psi^{+}+\bar{\psi}^{-}\left(i\hskip
1.00006pt{\not\\!d^{~{}}_{{sp\hskip 0.70004pt}}}\right)\psi^{-},\\\
\bar{\psi}\psi&=&\left(\bar{\psi}{\Gamma_{{sp\hskip
0.70004pt}}}\right)\left({\Gamma_{{sp\hskip
0.70004pt}}}\psi\right)&=&\bar{\psi}^{+}\psi^{-}+\bar{\psi}^{-}\psi^{+}.\end{array}$

####  3.2.2 Gauge bundle

This section treats a principal bundle with a compact Lie group in a Lorentz
manifold, referred to as the principal gauge bundle. We exploit the $SU(N)$
group as a structure group, namely the gauge group. A scalar function (field)
appearing in the gauge bundle as a section is called the Higgs field in
physics.

######  Definition 3.6 (Principal gauge bundle).

A principal gauge bundle is defined as a tuple such that:

$\displaystyle\left({T^{*}{\mathcal{}M}}^{\otimes
N},\pi_{{SU}},{\mathcal{}M},SO(1,3)\otimes{SU(N)}\right),~{}$
$\displaystyle\textrm{where}~{}{T^{*}{\mathcal{}M}}^{\otimes
N}:=\overbrace{{T^{*}{\mathcal{}M}}\otimes\cdots\otimes{T^{*}{\mathcal{}M}}}^{N}$

A space of complex-valued scalar fields, namely the Higgs field,
${\mbox{\boldmath$\phi$}}=(\phi^{1},\phi^{2},\cdots,\phi^{N})^{T}$ is
introduced as a section
${\mbox{\boldmath$\phi$}}\in{\mbox{\boldmath$\Phi$}}=\Gamma\left({\mathcal{}M},\Omega^{0}({T^{*}{\mathcal{}M}}^{\otimes
N})\otimes{SU(N)}\right)$ belonging to the primary representation of the
$SU(N)$ symmetry.

$SU(N)$ group operator ${{G}_{SU}}$ acts on the section as

$\displaystyle{{G}_{SU}}:{\mbox{\boldmath$\Phi$}}\rightarrow{\mbox{\boldmath$\Phi$}}:{\mbox{\boldmath$\phi$}}\mapsto{{G}_{SU}}\left({\mbox{\boldmath$\phi$}}\right):={\mbox{\boldmath$g$}}_{SU}^{~{}}\hskip
1.00006pt{\mbox{\boldmath$\phi$}},$

that has an $(N\times N)$-matrix representation such as

$\displaystyle\left[{{G}_{SU}}\left({\mbox{\boldmath$\phi$}}\right)\right]^{I}=\left[{\mbox{\boldmath$g$}}_{SU}^{~{}}\right]^{I}_{J}\phi^{J},$

where ${\mbox{\boldmath$g$}}_{SU}^{~{}}$ is a unitary matrix with
det$[{\mbox{\boldmath$g$}}_{SU}^{~{}}]=1$. Lie algebra of $SU(N)$ group is
represented as

$\displaystyle\left[\tau^{~{}}_{I},\tau^{~{}}_{J}\right]:=i\sum_{K}\hskip
1.00006ptf^{~{}}_{IJK}\hskip 1.99997pt\tau^{~{}}_{K},$

where $f_{{\bullet\bullet}\bullet}$ is a structure constant of $SU(N)$, and
${\mbox{\boldmath$\tau$}}=(\tau_{1},\tau_{2},\cdots,\tau_{N^{2}-1})\in{\mathfrak{s}}{\mathfrak{u}}(N)$.
A gauge connection ${\mathfrak{A}}^{~{}}_{{SU}}:={\mathfrak{A}}_{SU}^{I}\hskip
1.99997pt\tau^{~{}}_{I}\in\Omega^{1}\otimes{\mathfrak{s}}{\mathfrak{u}}(N)$ is
a Lie-algebra valued one-form object. Covariant differential
$d^{~{}}_{{SU}\otimes{SO}}$ on $p$-form object
${\mathfrak{a}}\in\Omega^{p}({T^{*}{\mathcal{}M}})$ concerning $SU(N)$ and
$SO(3,1)$ is defined as

$\displaystyle d^{~{}}_{{SU}\otimes{SO}}\hskip 1.99997pt{\mathfrak{a}}\hskip
1.00006pt:={\mbox{\boldmath$1$}}_{SU}\hskip
1.00006ptd_{\mathfrak{w}}{\mathfrak{a}}-i\hskip
1.00006pt{{c^{~{}}_{SU}}}\hskip
1.00006pt[{\mathfrak{A}}^{~{}}_{SU},{\mathfrak{a}}]_{\wedge}.$ (96)

Real constant ${c^{~{}}_{SU}}\in\mathbb{R}$ is a coupling constant of the
gauge interaction in physics. Connection ${\mathfrak{A}}^{~{}}_{SU}$ belongs
to an adjoint representation of the gauge group as

$\displaystyle{{G}_{SU}}\left({\mathfrak{A}}^{~{}}_{SU}\right)$
$\displaystyle={\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.99997pt{\mathfrak{A}}^{~{}}_{SU}\hskip
1.99997pt{\mbox{\boldmath$g$}}_{SU}^{~{}}+i\hskip
1.00006pt{c^{-1}_{SU}}{\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006ptd{\mbox{\boldmath$g$}}_{SU}^{~{}},$ (97)

which ensures the $SU(N)$ covariance of the covariant differential. At the
same time, ${\mathfrak{A}}^{~{}}_{SU}$ is a vector in ${T^{*}{\mathcal{}M}}$,
which is Lorentz-transformed as

$\displaystyle{{G}_{SO}}:{\mathfrak{A}}^{~{}}_{SU}\mapsto{{G}_{SO}}({\mathfrak{A}}^{~{}}_{SU})$
$\displaystyle={\mbox{\boldmath$\Lambda$}}{\mathfrak{A}}^{~{}}_{SU}.$ (98)

Corresponding gauge curvature two-form $\mathfrak{F}_{SU}$ is defined through
a structure equation such that:

$\displaystyle\mathfrak{F}_{SU}=\mathfrak{F}_{SU}^{I}\hskip
1.99997pt\tau^{~{}}_{I}$
$\displaystyle:=d_{\mathfrak{w}}{\mathfrak{A}}^{~{}}_{SU}-i\hskip
1.00006pt{{c^{~{}}_{SU}}}\hskip
1.00006pt{\mathfrak{A}}^{~{}}_{SU}\wedge{\mathfrak{A}}^{~{}}_{SU},$
$\displaystyle=\left(d{\mathfrak{A}}_{SU}^{I}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mathfrak{w}}\wedge{\mathfrak{A}}_{SU}^{I}+\frac{{c^{~{}}_{SU}}}{2}f^{I}_{~{}JK}\hskip
1.00006pt{\mathfrak{A}}_{SU}^{J}\wedge{\mathfrak{A}}_{SU}^{K}\right)\tau^{~{}}_{I},$
(99)

where $f^{I}_{~{}JK}=f^{~{}}_{IJK}$. Curvature $\mathfrak{F}^{~{}}_{SU}$ is
$SO(1,3)$ covariant owing to the Lorentz-covariant differential in (99) and is
$SU(N)$-transformed as an adjoint representation such that:

$\displaystyle{{G}_{SU}}\left(\mathfrak{F}^{I}_{{SU}}\right)$
$\displaystyle={\mbox{\boldmath$g$}}^{~{}}_{SU}\hskip
1.00006pt\mathfrak{F}^{I}_{{SU}}\hskip
1.00006pt{\mbox{\boldmath$g$}}_{SU}^{-1}+i\hskip
1.00006pt{c^{-1}_{SU}}{\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006ptd{\mbox{\boldmath$g$}}_{SU}^{~{}}.$

The first and second Bianchi identities are provided as

$\displaystyle d_{{{SU}\otimes{SO}}}^{2}\hskip
1.99997pt{\mathfrak{a}}^{a}={c^{~{}}_{gr}}\eta_{{\circ\circ}}\mathfrak{R}^{a{\circ}}\wedge{\mathfrak{a}}^{\circ}-i{c^{~{}}_{SU}}\hskip
1.00006pt\mathfrak{F}_{{SU}}\wedge{\mathfrak{a}}^{a},~{}~{}\textrm{and}~{}~{}d^{~{}}_{{{SU}\otimes{SO}}}\hskip
1.99997pt\mathfrak{F}_{{SU}}=0,$ (100)

where ${\mathfrak{a}}\in\Omega^{p}({T^{*}{\mathcal{}M}})$. The gauge
connection and curvature are represented using the trivial bases in
${T^{*}{\mathcal{}M}}$, respectively, as

$\displaystyle{\mathfrak{A}}^{~{}}_{SU}$
$\displaystyle={\mathfrak{A}}_{SU}^{I}\hskip
1.99997pt\tau^{~{}}_{I}=:{\mathscr{A}}^{I}_{a}{\mathfrak{e}}^{a}\hskip
1.99997pt\tau^{~{}}_{I},~{}~{}\textrm{and}~{}~{}\mathfrak{F}_{SU}=\mathfrak{F}_{SU}^{I}\hskip
1.99997pt\tau^{~{}}_{I}=:\frac{1}{2}\mathscr{F}^{I}_{ab}\hskip
1.00006pt{\mathfrak{e}}^{a}\wedge{\mathfrak{e}}^{b}\hskip
1.99997pt\tau^{~{}}_{I};$ (101)

and the second Bianchi identity is represented as

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

In this expression, tensor coefficients of the gauge curvature is provided
using those of the gauge connection such that:

$\displaystyle\mathscr{F}^{I}_{ab}$
$\displaystyle=\partial_{a}{\mathscr{A}}^{I}_{b}-\partial_{b}{\mathscr{A}}^{I}_{a}+{{c^{~{}}_{SU}}}\hskip
1.00006ptf^{I}_{~{}JK}{\mathscr{A}}^{J}_{a}{\mathscr{A}}^{K}_{b}+{\mathscr{A}}^{I}_{\circ}{\mathcal{T}}^{\circ}_{~{}ab},$
(102)

where ${\mathcal{T}}$ is a tensor coefficient of the torsion form defined as
${\mathfrak{T}}^{\bullet}=:{\mathcal{T}}^{\bullet}_{\hskip
3.5pt{\circ\circ}}{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}/2$. When
the space-time manifold is torsion-less, the gauge curvature is the same as it
in the flat space-time.

####  3.2.3 Spinor-gauge bundle

The gauge group introduced in the previous section also acts on the spinor
section (field), which also has the $SU(N)$ symmetry. The spinor field creates
matter particles from the quantum mechanical point of view; the gauge
curvature acts on them as forces among matters, and the gauge connection
mediates forces between the spinor fields. A spinor-gauge bundle is a Whitney
sum of Spinor- and gauge bundles.

######  Definition 3.7 (Principal spinor-gauge bundle).

A principal spinor-gauge bundle is a tuple such as

$\displaystyle\left({\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes
N},\pi_{{sp\hskip
0.70004pt}}\oplus\pi_{{SU}},{\mathcal{}M},Spin(1,3)\otimes{SU(N)}\right)~{}$
$\displaystyle\textrm{where}~{}{\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes
N}:=\overbrace{{\mathcal{}S}_{sp\hskip
0.70004pt}\otimes\cdots\otimes{\mathcal{}S}_{sp\hskip 0.70004pt}}^{N}.$

The total space of the gauge bundle is lifted to the spin manifold.

A connection and curvature are provided, respectively, as

$\displaystyle{\mathfrak{A}}_{sg}={\mathfrak{A}}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes{\mathfrak{A}}^{~{}}_{SU},~{}~{}\textrm{and}~{}~{}\mathfrak{F}_{sg}=\mathfrak{F}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes\mathfrak{F}_{SU},$ (103)

Corresponding Bianchi identities are obtained by direct calculations as

$\displaystyle\left({d}_{sg}\right)^{2}={c^{~{}}_{gr}}\hskip
1.00006pt\mathfrak{F}_{sp\hskip 0.70004pt}\cdot{{\mathscr{S}}}\otimes\hskip
1.99997pt{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes\left(-i{{c^{~{}}_{SU}}}\hskip 1.00006pt\hskip
1.00006pt\mathfrak{F}_{SU}\right),~{}{d}_{sp\hskip
0.70004pt}\mathfrak{F}_{sp\hskip 0.70004pt}={d}_{SU}\mathfrak{F}_{SU}=0.$
(104)

We introduce $N$ spinors such that
${\mbox{\boldmath$\psi$}}=(\psi^{1},\cdots,\psi^{N})^{T}$ with the $SU(N)$
symmetry. A covariant differential on
${\mbox{\boldmath$\psi$}}\in{\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes N}$ in
the spinor-gauge bundle is provided as

$\displaystyle d_{sg}{\mbox{\boldmath$\psi$}}$
$\displaystyle:={\mbox{\boldmath$1$}}_{SU}\otimes{d}{\mbox{\boldmath$\psi$}}+{c^{~{}}_{gr}}{\mbox{\boldmath$1$}}_{SU}\otimes{\mathfrak{A}}_{sp\hskip
0.70004pt}{\mbox{\boldmath$\psi$}}-i\hskip 1.00006pt{{c^{~{}}_{SU}}}\hskip
1.00006pt{\mathfrak{A}}^{~{}}_{SU}\otimes{\mbox{\boldmath$\psi$}}.$

A Dirac operator concerning the spinor-gauge bundle is provided as

$\displaystyle{\not\\!d}_{{sg}}$
$\displaystyle:\Gamma\left({\mathcal{}M},{\Omega}^{0}\left({\mathcal{}S}_{sp\hskip
0.70004pt}^{\otimes{N}}\otimes{SU(N)}\right)\right)\rightarrow\Gamma\left({\mathcal{}M},{\Omega}^{0}\left({\mathcal{}S}_{sp\hskip
0.70004pt}^{\otimes{N}}\otimes{SU(N)}\right)\right)$
$\displaystyle:{\mbox{\boldmath$\psi$}}\mapsto{\not\\!d}_{{sg}}{\mbox{\boldmath$\psi$}}:=\iota_{{\gamma}_{sp\hskip
0.50003pt}}(d+i{c^{~{}}_{gr}}\hskip
1.00006pt{{\mathfrak{w}}\cdot{\mathscr{S}}})\hskip
1.99997pt{\mbox{\boldmath$\psi$}}\otimes{\mbox{\boldmath$1$}}_{SU}-{i}{c^{~{}}_{SU}}\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\mathfrak{A}}_{{SU}}\otimes{\mbox{\boldmath$\psi$}},$
$\displaystyle\hskip 55.00008pt=\left(\gamma_{sp\hskip
0.70004pt}^{\circ}\partial_{\circ}-\frac{1}{2}{c^{~{}}_{gr}}\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}_{\circ}^{\mu}\omega_{\mu}^{~{}{\star\star}}{{\mathscr{S}}}_{\star\star}\right){\mbox{\boldmath$\psi$}}-i{c^{~{}}_{SU}}\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathscr{A}}^{I}_{\circ}\tau^{~{}}_{I}\hskip
1.99997pt{\mbox{\boldmath$\psi$}}.$

####  3.2.4 Hodge-dual connection and curvature

This section introduces dual connections and curvatures concerning the Hodge-
dual operator in the gauge-, spinor- and spinor-gauge-bundles. Current and the
following sections explicitly express the metric tensor’s determinant for
later convenience;
${{\sigma^{~{}}_{\eta}}}:=$det$[{\mbox{\boldmath$\eta$}}]=-1$ for the
Minkowski space. Formulae in the $\theta$-metric space are provided by simply
replacing $\eta$ to ${\mbox{\boldmath$\eta$}}_{\theta}$.

##### Dual curvature in gauge bundle:

Dual curvature in the gauge bundle are defined as follows:

######  Definition 3.8 (Hodge-dual geuge-curvature).

A dual curvature of the $SU(N)$ gauge-curvature is defined owing to the Hodge-
dual operator as

$\displaystyle{\widehat{\mathrm{H}}}:\mathfrak{F}^{I}_{SU}\mapsto\hat{\mathfrak{F}}^{I}_{{SU}}:=\frac{1}{2}\hat{\mathscr{F}}^{I}_{\circ\circ}\hskip
1.00006pt{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ},~{}~{}$
$\displaystyle\textrm{where}~{}~{}\hat{\mathscr{F}}^{I}_{ab}:=\frac{\sqrt{{{\sigma^{~{}}_{\eta}}}}}{2}\epsilon_{ab{\circ\circ}}\eta^{{\circ}\star}\eta^{{\circ}\star}\mathscr{F}^{I}_{{\star\star}}.$
(105)

For the dual curvature, two-group operators, ${G}^{~{}}_{SO}$ and
${G}^{~{}}_{SU}$, are homomorphic to each other:

######  Remark 3.9.

A following commutative diagram is maintained:

###### Proof.

The Lorentz transformation of a dual curvature is given as

$\displaystyle{{G}_{SO}}\left(\hat{\mathfrak{F}}^{I}_{{SU}}\right)$
$\displaystyle={\mbox{\boldmath$\Lambda$}}\hat{\mathfrak{F}}^{I}_{{SU}}{\mbox{\boldmath$\Lambda$}}^{-1}={\mbox{\boldmath$\Lambda$}}{\mbox{\boldmath$\epsilon$}}{\mbox{\boldmath$\eta$}}^{-1}\hskip
1.00006pt\mathfrak{F}^{I}_{{SU}}\hskip
1.00006pt{\mbox{\boldmath$\eta$}}^{-1}{\mbox{\boldmath$\Lambda$}}^{-1},$
$\displaystyle=\left({\mbox{\boldmath$\Lambda$}}{\mbox{\boldmath$\epsilon$}}{\mbox{\boldmath$\Lambda$}}^{-1}\right)\left({\mbox{\boldmath$\Lambda$}}{\mbox{\boldmath$\eta$}}^{-1}{\mbox{\boldmath$\Lambda$}}^{-1}\right)\left({\mbox{\boldmath$\Lambda$}}\mathfrak{F}^{I}_{{SU}}{\mbox{\boldmath$\Lambda$}}^{-1}\right)\left({\mbox{\boldmath$\Lambda$}}{\mbox{\boldmath$\eta$}}^{-1}{\mbox{\boldmath$\Lambda$}}^{-1}\right),$
$\displaystyle={\mbox{\boldmath$\epsilon$}}{\mbox{\boldmath$\eta$}}^{-1}\left({\mbox{\boldmath$\Lambda$}}\mathfrak{F}^{I}_{{SU}}{\mbox{\boldmath$\Lambda$}}^{-1}\right){\mbox{\boldmath$\eta$}}^{-1}.$

Therefore, it is maintained that

${{G}_{SO}}\circ{\widehat{\mathrm{H}}}\left(\mathfrak{F}^{I}_{{SU}}\right)={\widehat{\mathrm{H}}}\circ{{G}_{SO}}\left(\mathfrak{F}^{I}_{{SU}}\right).$

Similar calculations show that a dual curvature is transformed under the
$SU(N)$ transformation as

${{G}_{SU}}(\hat{\mathfrak{F}}^{I}_{{SU}})={\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006pt\hat{\mathfrak{F}}^{I}_{{SU}}\hskip
1.00006pt{\mbox{\boldmath$g$}}_{SU}^{~{}}+i\hskip 1.00006ptc^{-1}_{SU}\hskip
1.00006pt{\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006ptd{\mbox{\boldmath$g$}}_{SU}^{~{}}.$

Therefore, the remark is maintained. ∎

When the dual gauge-curvature is given, the dual gauge-connection is provided
as a solution of the structure equation. Though the existence of the dual
connection is not trivial, we assume it in this section. Appendix A discusses
a necessary condition to exist the dual connection.

Dual gauge-connection $\hat{\mathfrak{A}}^{I}_{{SU}}$ is defined as a solution
of the structure equation:

$\displaystyle\hat{\mathfrak{F}}^{~{}}_{{SU}}$
$\displaystyle=d_{\mathfrak{w}}\hat{\mathfrak{A}}^{~{}}_{{SU}}-i\hskip
1.00006pt{{c}^{~{}}_{SU}}\hskip
1.00006pt\hat{\mathfrak{A}}^{~{}}_{{SU}}\wedge\hat{\mathfrak{A}}^{~{}}_{{SU}},$
(106)

which consists of $6N$ independent first-order differential equations. We
write the dual gauge connection by means of expression (75), which is
compatible with the definition of the dual connection as

$\displaystyle{\widehat{\mathrm{H}}}\left(\hat{\mathfrak{F}}^{I}_{{SU}}\right)$
$\displaystyle={\widehat{\mathrm{H}}}\left(d_{\mathfrak{w}}\hat{\mathfrak{A}}^{I}_{{SU}}+\frac{{c}^{~{}}_{SU}}{2}f^{I}_{~{}JK}\hskip
1.00006pt\hat{\mathfrak{A}}^{J}_{{SU}}\wedge\hat{\mathfrak{A}}^{K}_{{SU}}\right)=\mathfrak{F}^{I}_{{SU}}.$

A solution of equations (106) for given $\hat{\mathfrak{F}}^{~{}}_{{SU}}$ is
Lie algebra-valued one-form object such that
$\hat{\mathfrak{A}}^{~{}}_{{SU}}=\hat{\mathfrak{A}}^{I}_{{SU}}\hskip
1.99997pt\tau^{~{}}_{I}=\hat{\mathscr{A}}^{I}_{\circ}{\mathfrak{e}}^{\circ}\hskip
1.99997pt\tau^{~{}}_{I}\in\Omega^{1}\otimes{\mathfrak{s}}{\mathfrak{u}}(N)$. A
following remark ensures that $\hat{\mathfrak{A}}^{~{}}_{{SU}}$ is the $SU(N)$
connection in the local Lorentz manifold ${\mathcal{}M}$.

######  Remark 3.10.

One-form $\hat{\mathfrak{A}}^{~{}}_{{SU}}$ is transformed under the structure
group as

$\displaystyle{{G}_{SU}}(\hat{\mathfrak{A}}^{I}_{{SU}})$
$\displaystyle={\mbox{\boldmath$g$}}^{~{}}_{SU}\hskip
1.99997pt\hat{\mathfrak{A}}^{~{}}_{{SU}}\hskip
1.99997pt{\mbox{\boldmath$g$}}_{SU}^{-1}+i\hskip 1.00006pt{c}^{-1}_{SU}\hskip
1.00006pt{\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006ptd{\mbox{\boldmath$g$}}_{SU}^{~{}};$

thus, it is a connection of the dual gauge bundle.

###### Proof.

Tensor coefficients of the dual curvature defined in (105) are transformed as

${{G}_{SU}}(\hat{\mathscr{F}}^{I}_{ab})={\mbox{\boldmath$g$}}^{~{}}_{SU}\hskip
1.00006pt\hat{\mathscr{F}}^{I}_{ab}\hskip
1.99997pt{\mbox{\boldmath$g$}}_{SU}^{-1}+i\hskip 1.00006pt{c}^{-1}_{SU}\hskip
1.00006pt{\mbox{\boldmath$g$}}^{-1}_{SU}\hskip
1.00006ptd{\mbox{\boldmath$g$}}_{SU}^{~{}},$

owing to its definition. One-form object $\hat{\mathfrak{A}}^{I}_{{SU}}$ is
defined as a solution of equation (106). Hence,
$\hat{\mathfrak{A}}^{I}_{{SU}}$ is a connection due to Remark 2.3. ∎

A dual map and its adjoint maps on a $SU(N)$ connection are introduced through
structure equations and dual maps in the curvature as follows:

We formally represent
${\widehat{\mathrm{H}}}({\mathfrak{A}}^{~{}}_{SU})=\hat{\mathfrak{A}}^{~{}}_{SU}$
and
${\widehat{\mathrm{H}}}{\circ}{\widehat{\mathrm{H}}}({\mathfrak{A}}^{~{}}_{SU})={\mathfrak{A}}^{~{}}_{SU}$.
A covariant differential in the dual bundle is provided as

$\displaystyle\hat{d}^{~{}}_{SU}{\mathfrak{a}}$
$\displaystyle:={\mbox{\boldmath$1$}}^{~{}}_{SU}\hskip
1.00006ptd{\mathfrak{a}}-i\hskip 1.00006pt{{{c}^{~{}}_{SU}}}\hskip
1.00006pt[\hat{\mathfrak{A}}^{~{}}_{{SU}}{\mathfrak{a}}]_{\wedge}\hskip
1.00006pt,$

for ${\mathfrak{a}}\in\Omega^{p}$, and Bianchi identities in the dual bundle
are provided as

$\displaystyle\left(\hat{d}^{~{}}_{{SU}}\right)^{2}=-i{{c}^{~{}}_{SU}}\hskip
1.00006pt\hat{\mathfrak{F}}^{~{}}_{{SU}},~{}~{}\textrm{and}~{}~{}\hat{d}^{~{}}_{{SU}}\hat{\mathfrak{F}}^{~{}}_{{SU}}=0.$
(107)

##### Dual curvature in spnior bundle:

The dual Lorentz curvature and the Lorentz connection are defined in section
3.1.2; the dual spinor-connection is obtained using the dual Lorentz
connection and the $Spin(3,1)$ generator. The dual spinor connection is
defined as

$\displaystyle\hat{\mathfrak{A}}_{sp\hskip 0.70004pt}:=-\frac{1}{2}\hskip
1.00006pt\eta_{\circ\circ}\eta_{\star\star}[\hat{\mathfrak{w}}]^{{\circ}\star}{\mathscr{S}}^{{\circ}\star}.$

The dual spinor-curvature is provided for given $\hat{\mathfrak{A}}_{sp\hskip
0.70004pt}$ using the structure equation as

$\displaystyle\hat{\mathfrak{F}}_{sp\hskip 0.70004pt}^{~{}}$
$\displaystyle:=d\hat{\mathfrak{A}}_{sp\hskip
0.70004pt}^{~{}}+{{c}^{~{}}_{gr}}\hskip 1.00006pt\hat{\mathfrak{A}}_{sp\hskip
0.70004pt}^{~{}}\wedge\hat{\mathfrak{A}}_{sp\hskip 0.70004pt}^{~{}},$

where the dual Lorentz curvature fulfils
${\widehat{\mathrm{H}}}(\mathfrak{F}_{sp\hskip
0.70004pt}^{~{}})=\hat{\mathfrak{F}}_{sp\hskip 0.70004pt}^{~{}}$.

Spinor valued section
$\hat{\psi}\in\Gamma({\mathcal{}M},{\mathcal{}S}_{sp\hskip
0.70004pt}\otimes{SU(N)})$ is introduced in the dual space. The dual operator
and its adjoint act on sections as

$\displaystyle{\widehat{\mathrm{H}}}({\mbox{\boldmath$\psi$}})={{\mbox{\boldmath$\hat{\psi}$}}}~{}~{}\textrm{and}~{}~{}{\widehat{\mathrm{H}}}({{\mbox{\boldmath$\hat{\psi}$}}})={{\mbox{\boldmath$\psi$}}}.$

Remark 3.5 is maintained also for $\hat{\psi}$. The Dirac conjugate of $\psi$
and $\hat{\psi}$ is denoted as $\bar{\hat{\psi}}$ and $\bar{\hat{\psi}}$,
whose components are a Dirac conjugate of each element and is homomorphic with
respect to dual and its adjoint operators as follows:

$\displaystyle{\widehat{\mathrm{H}}}({\mbox{\boldmath$\bar{\hat{\psi}}$}})=\overline{{\widehat{\mathrm{H}}}({{\mbox{\boldmath$\psi$}}})}={\mbox{\boldmath$\bar{\hat{\psi}}$}},~{}~{}\textrm{and}~{}~{}{\widehat{\mathrm{H}}}({\mbox{\boldmath$\bar{\hat{\psi}}$}})=\overline{{\widehat{\mathrm{H}}}({\mbox{\boldmath$\hat{\psi}$}})}={{\sigma^{~{}}_{\eta}}}{\mbox{\boldmath$\bar{\psi}$}}.$

Kramers pairs for the spinor sections are

$\displaystyle{KP}\left(\psi\right):=\left(\psi,\hat{\psi}\right)~{}~{}\textrm{and}~{}~{}{KP}\left(\bar{\psi}\right):=\left(\bar{\psi},\bar{\hat{\psi}}\right).$

##### Dual curvature in spnior-gauge bundle:

The spinor- and gauge connections and their dual connections belong to the
same bundle as remark 2.9; the spinor-gauge connection is provided as

$\displaystyle\hat{\mathfrak{A}}_{sg}:=\hat{\mathfrak{A}}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes\hat{\mathfrak{A}}_{{SU}},~{}~{}\textrm{and}~{}~{}\hat{\mathfrak{F}}_{sg}:=\hat{\mathfrak{F}}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes\hat{\mathfrak{F}}_{{SU}},$

and construct Kramers pairs ${KP}({\mathfrak{A}}_{sg}^{~{}})$ and
${KP}(\mathfrak{F}_{sg}^{~{}})$.

The covariant differential and the dual Dirac operator are provided as

$\displaystyle{\not\\!d^{~{}}_{{sg}}}{\mathfrak{a}}$
$\displaystyle:=d{\mathfrak{a}}\hskip 1.99997pt{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}+{{c}^{~{}}_{gr}}\left[\hat{\mathfrak{A}}_{sp\hskip
0.70004pt},{\mathfrak{a}}\right]_{\wedge}\otimes{\mbox{\boldmath$1$}}_{SU}+{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes\left(-i\hskip 1.00006pt{{c}^{~{}}_{SU}}\hskip
1.00006pt[\hat{\mathfrak{A}}^{~{}}_{{SU}},{\mathfrak{a}}]_{\wedge}\right),$
$\displaystyle{\not\\!\hat{d}^{~{}}_{{sg}}}{{\mbox{\boldmath$\hat{\psi}$}}}$
$\displaystyle:=\iota_{{\gamma}_{sp\hskip
0.50003pt}}(d+i{{c}^{~{}}_{gr}}\hskip
1.00006pt\hat{\mathfrak{w}}\cdot{{\mathscr{S}}}){{\mbox{\boldmath$\hat{\psi}$}}}\otimes{\mbox{\boldmath$1$}}_{SU}-{i}{{c}^{~{}}_{SU}}\hskip
1.00006pt\iota^{~{}}_{\gamma_{sp\hskip
0.50003pt}}\hat{\mathfrak{A}}^{~{}}_{{SU}}\otimes{{\mbox{\boldmath$\hat{\psi}$}}},$

where
${{\mbox{\boldmath$\hat{\psi}$}}}=(\hat{\psi}^{1},\hat{\psi}^{2},\cdots,\hat{\psi}^{N})^{T}$,
and ${\mathfrak{a}}\in\Omega^{p}$ and
${{\mbox{\boldmath$\hat{\psi}$}}}\in{\mathcal{}S}_{sp\hskip
0.70004pt}^{\otimes N}\otimes{SU(N)}$. The dual- and adjoint operators are
homomorphic as follows:

$\begin{array}[]{cclccl}{\widehat{\mathrm{H}}}\left({{\not\\!d^{~{}}_{{sg}}}}{\mbox{\boldmath$\psi$}}\right)&={\not\\!\hat{d}^{~{}}_{{sg}}}{{\mbox{\boldmath$\hat{\psi}$}}},&{\widehat{\mathrm{H}}}\left({\not\\!\hat{d}^{~{}}_{{sg}}}{{\mbox{\boldmath$\hat{\psi}$}}}\right)&={\not\\!d^{~{}}_{{sg}}}{\mbox{\boldmath$\psi$}}.\end{array}$

The existence of the dual spinor that provides the dual curvature as a
solution of the equation of motion is not trivial and is discussed in Appendix
A.

###  3.3 Yang–Mills bundle

The gauge-spinor bundle is referred to as the Yang–Mills bundle when it is
represented using the $({2\times 2})$-matrix. The connection and curvature of
the Yang–Mills bundle are defined in the secondary superspace as

$\displaystyle{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{\textrm{Y}\textrm{M}}:={\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{sp\hskip
0.70004pt}+{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{SU},~{}~{}\textrm{where}~{}~{}{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{sp\hskip
0.70004pt}:=\left(\begin{array}[]{cc}{\mathfrak{A}}^{~{}}_{{sp\hskip
0.70004pt}}&{0}\\\ {0}&\hat{\mathfrak{A}}^{~{}}_{{sp\hskip
0.70004pt}}\end{array}\right),~{}~{}{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{SU}:=\left(\begin{array}[]{cc}{\mathfrak{A}}^{~{}}_{{SU}}&{0}\\\
{0}&\hat{\mathfrak{A}}^{~{}}_{{SU}}\end{array}\right).$

The Yang–Mills curvature is provided using the structure equation such that:

$\displaystyle{\mbox{\boldmath$\mathfrak{F}$}}^{~{}}_{\textrm{Y}\textrm{M}}$
$\displaystyle:=d{\mathfrak{A}}^{~{}}_{\textrm{Y}\textrm{M}}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{{sp\hskip
0.70004pt}}\wedge{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{{sp\hskip
0.70004pt}}\otimes{\mbox{\boldmath$1$}}_{{SU}}+{\mbox{\boldmath$1$}}_{{sp\hskip
0.70004pt}}\otimes(-ic^{~{}}_{SU}\hskip
1.00006pt{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{{SU}}\wedge{\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{{SU}}),$
$\displaystyle=\left(\begin{array}[]{cc}\mathfrak{F}^{~{}}_{sg}&0\\\
0&\hat{\mathfrak{F}}^{~{}}_{sg}\end{array}\right).$ (110)

The dual operator in the Yang–Mills bundle is introduced as

$\displaystyle{\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}={{\mbox{\boldmath$1$}}_{SU}\otimes{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}}\hskip
1.00006pt\Gamma\gamma{\widehat{\mathrm{H}}}={{\mbox{\boldmath$1$}}_{SU}\otimes{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}}\left(\begin{array}[]{cc}0&-{\widehat{\mathrm{H}}}\\\
{\widehat{\mathrm{H}}}&0\end{array}\right)=-{\widehat{\mathrm{H}}}^{-1}_{{\textrm{Y}\textrm{M}}},$

that acts on the curvature as

$\displaystyle{\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}\left(\mathfrak{F}^{~{}}_{\textrm{Y}\textrm{M}}\right):={\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}\hskip
1.00006pt\mathfrak{F}^{~{}}_{\textrm{Y}\textrm{M}}\hskip
1.00006pt{\widehat{\mathrm{H}}}^{-1}_{{\textrm{Y}\textrm{M}}}=\left(\begin{array}[]{cc}\hat{\mathfrak{F}}^{~{}}_{{sg}}&0\\\
0&\mathfrak{F}^{~{}}_{{sg}}\\\ \end{array}\right).$

Dual operator ${\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}$ acts on
the Yang–Mills connection formally as defined in section 2.4.2. A structure
equation are homomorphic such that:

that is apparent owing to relations

$\displaystyle{\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}\hskip
1.00006pt{\mbox{\boldmath$\mathfrak{X}$}}\hskip
1.00006pt{\widehat{\mathrm{H}}}^{-1}_{{\textrm{Y}\textrm{M}}}={\mbox{\boldmath$\mathfrak{X}$}}\bigl{|}_{\bullet\leftrightarrow\hskip
0.70004pt\hat{\bullet}},$
$\displaystyle~{}~{}\textrm{for}~{}~{}{\mbox{\boldmath$\mathfrak{X}$}}={\mbox{\boldmath${\mathfrak{A}}$}}^{~{}}_{\textrm{Y}\textrm{M}}~{}\textrm{or}~{}~{}{\mbox{\boldmath$\mathfrak{F}$}}^{~{}}_{\textrm{Y}\textrm{M}}.$

A covariant differential in the Yang–Mills bundle is provided as

$\displaystyle{{\mathcal{}D}}^{~{}}_{\textrm{Y}\textrm{M}}$
$\displaystyle:=\left(\begin{array}[]{cc}{\not\\!d^{~{}}_{{sg}}}&0\\\
0&{\not\\!\hat{d}^{~{}}_{{sg}}}\\\ \end{array}\right).$

The covariant differential is compatible to the dual operator such as

${\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}({{\mathcal{}D}}^{~{}}_{\textrm{Y}\textrm{M}})={{\widehat{\mathrm{H}}}^{~{}}_{{\textrm{Y}\textrm{M}}}}\hskip
1.00006pt{{\mathcal{}D}}^{~{}}_{\textrm{Y}\textrm{M}}\hskip
1.00006pt{{\widehat{\mathrm{H}}}^{-1}_{{\textrm{Y}\textrm{M}}}}={{\mathcal{}D}}^{~{}}_{\textrm{Y}\textrm{M}}\bigl{|}_{d_{sg}\leftrightarrow\hat{d}_{{sg}}},$

and the dual operator preserves the Bianchi identities.

The Clifford algebra and Dirac operator are introduced to the Yang–Mills
bundle as follows: One-dimensional Clifford algebra represented as

$\displaystyle\gamma^{~{}}_{\textrm{Y}\textrm{M}}:={\mbox{\boldmath$1$}}_{\textrm{Y}\textrm{M}}\hskip
1.99997pt\gamma,~{}~{}$
$\displaystyle\textrm{where}~{}~{}{\mbox{\boldmath$1$}}_{\textrm{Y}\textrm{M}}:={\mbox{\boldmath$1$}}_{{sp\hskip
0.70004pt}}\otimes{\mbox{\boldmath$1$}}_{{SU}}.$

and corresponding chiral and projection operators are, respectively, provided
as

$\displaystyle\Gamma^{~{}}_{\textrm{Y}\textrm{M}}:={\mbox{\boldmath$1$}}_{\textrm{Y}\textrm{M}}\hskip
1.99997pt\Gamma~{}~{}\textrm{and}~{}~{}P^{\pm}_{\textrm{Y}\textrm{M}}:=\frac{{\mbox{\boldmath$1$}}^{~{}}_{\textrm{Y}\textrm{M}}\pm\gamma^{~{}}_{\textrm{Y}\textrm{M}}}{2}.$
(111)

They fulfilled relations
$\gamma^{~{}}_{\textrm{Y}\textrm{M}}\Gamma^{~{}}_{\textrm{Y}\textrm{M}}+\Gamma^{~{}}_{\textrm{Y}\textrm{M}}\gamma^{~{}}_{\textrm{Y}\textrm{M}}=0$
and
$\Gamma^{2}_{\textrm{Y}\textrm{M}}=\gamma^{2}_{\textrm{Y}\textrm{M}}={\mbox{\boldmath$1$}}_{\textrm{Y}\textrm{M}}$.
The Yang–Mills form is defined owing to the Clifford algebra as follows:

######  Definition 3.11 (Yang–Mills form).

$\displaystyle\mathfrak{ym}$
$\displaystyle:=\gamma^{~{}}_{\textrm{Y}\textrm{M}}\hskip
1.99997pt\mathfrak{F}^{~{}}_{\textrm{Y}\textrm{M}}=\left(\begin{array}[]{cc}0&\hat{\mathfrak{F}}_{{sg}}^{~{}}\\\
\mathfrak{F}_{{sg}}^{~{}}&0\end{array}\right)\otimes{\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}.$

The secondary superspace is induced on a vector space
$\\{\mathfrak{F}^{~{}}_{{SU}}\\}/G_{SU}\otimes\\{\hat{\mathfrak{F}}^{~{}}_{{SU}}\\}/G_{SU}$
owing to chiral operator $\Gamma^{~{}}_{\textrm{Y}\textrm{M}}$. Clifford
algebra $\gamma^{~{}}_{\textrm{Y}\textrm{M}}$ indices the Dirac operator such
that:

$\displaystyle{\not\\!\mathcal{}D}^{~{}}_{\textrm{Y}\textrm{M}}$
$\displaystyle:=\gamma^{~{}}_{\textrm{Y}\textrm{M}}{{\mathcal{}D}}^{~{}}_{\textrm{Y}\textrm{M}}=\left(\begin{array}[]{cc}0&{\not\\!d^{~{}}_{{sg}}}\\\
{\not\\!\hat{d}^{~{}}_{{sg}}}&0\\\ \end{array}\right).$

The Dirac spinor defined over the Yang–Mills bundle is defined as

$\displaystyle{\mbox{\boldmath$\Psi$}}^{~{}}_{\textrm{Y}\textrm{M}}$
$\displaystyle:=\left(\begin{array}[]{c}{\mbox{\boldmath$\psi$}}\\\
{{\mbox{\boldmath$\hat{\psi}$}}}\end{array}\right).$

The dual operator acts on the spinor field as

$\displaystyle{\widehat{\mathrm{H}}}\left({\mbox{\boldmath$\Psi$}}^{~{}}_{\textrm{Y}\textrm{M}}\right)=\left(\begin{array}[]{r}-{{\mbox{\boldmath$\hat{\psi}$}}}\\\
{\mbox{\boldmath$\psi$}}\end{array}\right).$

The Dirac conjugate of ${\mbox{\boldmath$\Psi$}}^{~{}}_{\textrm{Y}\textrm{M}}$
and its dual in this representation are provided as

$\displaystyle\overline{{\mbox{\boldmath$\Psi$}}}^{~{}}_{\textrm{Y}\textrm{M}}~{}:=~{}{\mbox{\boldmath$\Psi$}}^{\dagger}_{\textrm{Y}\textrm{M}}\left(\gamma^{0}_{sp\hskip
0.70004pt}\gamma^{~{}}_{\textrm{Y}\textrm{M}}\right)$
$\displaystyle=\left({\mbox{\boldmath$\psi$}}^{\dagger},{{\mbox{\boldmath$\hat{\psi}$}}}^{\dagger}\right)\left(\begin{array}[]{cc}{\mbox{\boldmath$0$}}&\gamma^{0}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{{SU}}\\\ \gamma^{0}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{{SU}}&{\mbox{\boldmath$0$}}\\\
\end{array}\right)=\left(\bar{{{{\mbox{\boldmath$\hat{\psi}$}}}}},\bar{{{\mbox{\boldmath$\psi$}}}}\right),$

and

$\displaystyle{\widehat{\mathrm{H}}}\left(\overline{{\mbox{\boldmath$\Psi$}}}^{~{}}_{\textrm{Y}\textrm{M}}\right)=\left(\bar{{{{\mbox{\boldmath$\hat{\psi}$}}}}},-\bar{{{\mbox{\boldmath$\psi$}}}}\right).$

##  4 Lagrangian and equation of motion

In classical mechanics, equations of motion are extracted from the action
integral using the variational method with appropriate boundary conditions.
The Lagrangian is a four-form object consisting of connections, curvatures,
sections, and invariant differentials in principal bundles. This section
introduces three fundamental Lagrangians, i.e., Einstein–Hilbert Lagrangian
for gravity, matter-field and gauge-filed Lagrangians for the Yang–Mills
theory, then extracts equations of motion from the Lagrangian.

###  4.1 Einstein–Hilbert action

The gravitational Lagrangian four-form consists of curvature $\mathfrak{R}$
and section ${\mathfrak{e}}$ in the co-Poincaré bundle; the definition is
given as follows[33, 34]:

######  Definition 4.1 (Gravitational Lagrngian).

The gravitational Lagrangian and the corresponding action integral are defined
as

$\displaystyle{\mathfrak{L}}_{Gr}$
$\displaystyle:=\frac{1}{\hbar\kappa}\left(\frac{1}{2}Str\left[{\mbox{\boldmath$\mathfrak{F}$}}_{Gr}\wedge{\mbox{\boldmath$\mathfrak{F}$}}_{Gr}\right]-\Lambda_{\mathrm{c}}{\widehat{\mathrm{H}}}(1)\right),$
(112)

where $\Lambda_{\mathrm{c}}$ is the cosmological constant. Integration of the
Lagrangian

$\mathscr{I}_{Gr}:=\int_{\Sigma_{4}}{\mathfrak{L}}_{Gr}$

is referred to as the gravitational action, where
$\Sigma_{4}\subseteq{\mathcal{}M}$ is appropriate closed and oriented subset
of the Lorentz manifold.

The cosmological constant has a physical dimension of length inverse squared
$[\Lambda_{\mathrm{c}}]=L^{-2}$ in our units. This study sets the Lagrangian
form and action integral to null-dimensional object
$[{\mathfrak{L}}]=[\mathscr{I}]=1$. Here, fundamental constants $\hbar\kappa$
appear associated with the gravitational Lagrangian owing to keeping a
physical dimension of the Lagrangian to be null. Although the Planck constant
appears in the Lagrangian, the theory is still classical.

######  Remark 4.2.

The gravitational Lagrangian defined by Definition 4.1 is equivalent to the
standard Einstein–Hilbert Lagrangian

$\displaystyle{\mathfrak{L}}_{\textrm{EH}}:=\frac{1}{\hbar\kappa}\left(\frac{1}{2}\mathfrak{R}^{\circ\circ}\wedge{\mathfrak{S}}_{\circ\circ}-\Lambda_{\mathrm{c}}\hskip
1.00006pt{\widehat{\mathrm{H}}}(1)\right).$

###### Proof.

The supertrace of the gravitational form squared is provided as

$\displaystyle Str[\mathfrak{F}^{~{}}_{Gr}\wedge\mathfrak{F}^{~{}}_{Gr}]$
$\displaystyle={\mathrm{Tr}}_{\hskip
0.70004pt{T{\mathcal{}M}}}\left[\hat{\mathfrak{F}}^{~{}}_{{c\mathrm{P}}}\wedge\mathfrak{F}^{~{}}_{{c\mathrm{P}}}\right]-{\mathrm{Tr}}_{\hskip
0.70004pt{T{\mathcal{}M}}}\left[\mathfrak{F}^{~{}}_{{c\mathrm{P}}}\wedge\hat{\mathfrak{F}}^{~{}}_{{c\mathrm{P}}}\right]=-2\hskip
1.00006pt{\mathrm{Tr}}_{\hskip
0.70004pt{T{\mathcal{}M}}}\left[\mathfrak{F}^{~{}}_{{c\mathrm{P}}}\wedge\hat{\mathfrak{F}}^{~{}}_{{c\mathrm{P}}}\right].$

Theorem 4.2 in Ref.[33] and Remark 2.1 in Ref.[34] show that

$\displaystyle{\mathrm{Tr}}_{\hskip
0.70004pt{T{\mathcal{}M}}}\left[\mathfrak{F}^{~{}}_{{c\mathrm{P}}}\wedge\hat{\mathfrak{F}}^{~{}}_{{c\mathrm{P}}}\right]=\|\mathfrak{F}^{~{}}_{{c\mathrm{P}}}\|^{2}=\frac{1}{2}\mathfrak{R}^{{\circ}\star}\wedge{\mathfrak{S}}_{\star{\circ}}=-\frac{1}{2}\mathfrak{R}^{{\circ}\star}\wedge{\mathfrak{S}}_{{\circ}\star},$

Therefore, we obtain

$\displaystyle\frac{1}{2}Str\left[\mathfrak{F}_{Gr}\wedge\mathfrak{F}_{Gr}\right]=\frac{1}{2}\mathfrak{R}^{\circ\circ}\wedge{\mathfrak{S}}_{\circ\circ};$

thus, ${\mathfrak{L}}_{Gr}={\mathfrak{L}}_{\textrm{EH}}$ is maintained. ∎

A space of the two-form objects is split into two subspaces owing to the
Hodge-dual operator as mentioned in section 2.4.2; thus, four-form object
${\mathrm{Tr}}_{T{\mathcal{}M}}[\mathfrak{R}\wedge{\mathfrak{S}}]$ is split
into two parts such as

$\displaystyle{\mathrm{Tr}}_{T{\mathcal{}M}}[\mathfrak{R}\wedge{\mathfrak{S}}]={\mathrm{Tr}}[\mathfrak{R}^{+}\wedge{\mathfrak{S}}^{+}]+{\mathrm{Tr}}_{T{\mathcal{}M}}[\mathfrak{R}^{-}\wedge{\mathfrak{S}}^{-}].$

More precisely, it is provided that

$\displaystyle{\mathrm{Tr}}_{T{\mathcal{}M}}[\mathfrak{R}^{\pm}\wedge{\mathfrak{S}}^{\pm}]$
$\displaystyle=\frac{1}{2}\left(R\pm i\epsilon^{\hskip
4.89998pt{\star\star}}_{{\circ\circ}}R^{{\circ\circ}}_{\hskip
4.89998pt{\star\star}}\right){\widehat{\mathrm{H}}}(1),$
$\displaystyle{\mathrm{Tr}}_{T{\mathcal{}M}}[\mathfrak{R}\wedge{\mathfrak{S}}]$
$\displaystyle=R\hskip 1.99997pt{\widehat{\mathrm{H}}}(1),$

where $R$ is the scalar curvature defined in (5). Coordinate expressions using
the trivial basis are

$\displaystyle\left[\mathfrak{R}\wedge{\mathfrak{S}}\right]^{ab}$
$\displaystyle=\frac{1}{4}\eta_{\star\star}\hskip
1.00006ptR^{a\star}_{~{}~{}{\circ\circ}}\hskip 1.00006ptS^{\star
b}_{~{}~{}{\circ\circ}}\hskip
1.99997pt{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ},$
$\displaystyle=\frac{1}{4}\epsilon^{\circ\circ\circ\circ}\eta^{~{}}_{{\star\star}}\hskip
1.99997ptR^{a\star}_{~{}~{}{\circ\circ}}\hskip 1.00006ptS^{\star
b}_{~{}~{}{\circ\circ}}\hskip 1.99997pt{\widehat{\mathrm{H}}}(1).$ (113)

The torsion-less Riemann-curvature tensor has additional symmetry such that:

$\displaystyle\eta_{a{\circ}}\eta_{b{\circ}}R^{{\circ\circ}}_{~{}~{}cd}=\eta_{c{\circ}}\eta_{d{\circ}}R^{{\circ\circ}}_{~{}~{}ab},$
$\displaystyle\textrm{when}~{}~{}{\mathfrak{T}}^{a}=0,$

and thus, one of $\mathfrak{R}^{\pm}$ yields the scalar curvature like

$\displaystyle\eta_{\circ\circ}[\mathfrak{R}^{+}\wedge{\mathfrak{S}}]^{\circ\circ}$
$\displaystyle=\eta_{\circ\circ}[\mathfrak{R}^{-}\wedge{\mathfrak{S}}]^{\circ\circ}=\frac{1}{2}R\hskip
1.00006pt{\widehat{\mathrm{H}}}(1),$
$\displaystyle\textrm{when}~{}~{}{\mathfrak{T}}^{a}=0.$

The Euler-Lagrange equation of motion requires a torsion-less condition due to
the Einstein–Hilbert Lagrangian; thus, the Einstein–Hilbert Lagrangian can be
written using only one of the subspaces $V^{\pm}(\Omega^{2})$. When we choose
SD curvature $\mathfrak{R}^{+}$ (accordingly with ${\mathfrak{w}}^{+}$) to
construct the Lagrangian, the gravitational Lagrangian is represented as

$\displaystyle{\mathfrak{L}}_{Gr}$
$\displaystyle=\frac{1}{\hbar\kappa}\left(\mathfrak{R}^{+}\wedge{\mathfrak{S}}^{+}-\Lambda_{\mathrm{c}}{\widehat{\mathrm{H}}}(1)\right),$
$\displaystyle\textrm{when}~{}~{}{\mathfrak{T}}^{a}=0.$

We note that the volume form can be written using one of
${\mathfrak{S}}^{\pm}$ such that:

$\displaystyle\frac{1}{3!}{\widehat{\mathrm{H}}}(1)$
$\displaystyle=-\eta_{\circ\circ}\left[{\mathfrak{S}}^{+}\wedge{\mathfrak{S}}^{+}\right]^{\circ\circ}=\eta_{\circ\circ}\left[{\mathfrak{S}}^{-}\wedge{\mathfrak{S}}^{-}\right]^{\circ\circ}.$

( It is true for any four-form objects that
$\Omega^{4}\ni{\mathfrak{a}}\propto{\mathfrak{b}}^{\pm}\wedge{\mathfrak{b}}^{\pm}$
for ${\mathfrak{b}}\in\Omega^{2}$ since
${\mathfrak{b}}^{\pm}\wedge{\mathfrak{b}}^{\mp}=0$. )

The gravitational Lagrangian is invariant under the general coordinate
transformation and the co-Poincaré transformation[33]. In the local Lorentzian
manifold, the number of independent components of the Lagrangian form is 10.
Six components are from the curvature two-form, and four are from the vierbein
form. These degrees of freedom correspond to the total degree of freedom for
the Poincaré group. In reality, the Einstein–Hilbert Lagrangian is given using
only one of the SD- or ASD curvature; thus, only three components of the
Lorentz connection are utilised in the Lagrangian. The Lagrangian form is now
treated as a function with two independent forms
$({\mathfrak{w}}^{ab},{\mathfrak{e}}^{c})$ and is applied to a variational
operator independently. This method is commonly known as the Palatini
method[40, 41].

Equations of motion in a vacuum can be obtained by requiring a stationary
condition of the action for the spin- and vierbein forms separately. From the
variation concerning connection form ${\mathfrak{w}}$, an equation of motion
is provided as

$\displaystyle{\mathfrak{T}}^{a}\wedge{\mathfrak{e}}^{b}$
$\displaystyle=\left(d{\mathfrak{e}}^{a}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mathfrak{w}}^{a}_{~{}{\circ}}\wedge{\mathfrak{e}}^{\circ}\right)\wedge{\mathfrak{e}}^{b}=0,$
(114)

which is referred to as the torsion-less equation. The torsion-less property
is obtained from the solution of the equation of motion rather than being an
independent constraint. This equation of motion includes six independent
equations, which is the same as the number of the independent components of
${\mathfrak{w}}$; thus, the Lorentz connection form is uniquely determined
from (114) when the vierbein form is given.

Next, taking the variation with respect to the vierbein form, one can obtain
an equation of motion as

$\displaystyle\frac{1}{2}\epsilon_{a{\circ\circ}{\circ}}\mathfrak{R}^{\circ\circ}\wedge{\mathfrak{e}}^{\circ}-\Lambda_{\mathrm{c}}\mathfrak{V}_{a}$
$\displaystyle=0,$ (115)

where
$\mathfrak{V}_{a}:=\epsilon_{a{\circ}{\circ\circ}}{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ}/3!$
is a three-dimensional volume form. Equation (115) is referred to as the
Einstein equation in a vacuum (those with the Yang–Mills fields are provided
in section 4.5). The curvature- and vierbein-forms can be uniquely determined
up to $GL(4,\mathbb{R})$ symmetry by solving equations (114) and (115)
simultaneously.

We note that component representations of (113) and (115) are provided in the
$\theta$-metric space as

$\displaystyle(\ref{RSv})$
$\displaystyle\rightarrow\frac{\sqrt{{{\sigma^{~{}}_{\theta}}}}}{4}\left[{\mbox{\boldmath$\epsilon$}}_{\theta}\right]^{\circ\circ\circ\circ}\eta_{\theta\hskip
0.70004pt{\star\star}}\hskip 1.99997ptR^{a\star}_{~{}~{}{\circ\circ}}\hskip
1.00006ptS^{\star b}_{~{}~{}{\circ\circ}}\hskip
1.99997pt{\widehat{\mathrm{H}}}(1),$ $\displaystyle(\ref{EoM2})$
$\displaystyle\rightarrow\left(R_{ab}-\frac{1}{2}R\hskip
1.99997pt\eta_{\theta\hskip 0.70004ptab}+\Lambda_{\mathrm{c}}\hskip
1.99997pt\eta_{\theta\hskip 0.70004ptab}\right){\widehat{\mathrm{H}}}(1).$

###  4.2 Yang–Mills action

The Yang–Mills Lagrangian consists of spinor field $\psi$, gauge connection
${\mathfrak{A}}^{~{}}_{SU}$, and dual curvature
$\hat{\mathfrak{F}}^{~{}}_{{SU}}$. The spinor field represents a charge
distribution of fermionic matter fields for classical mechanics. The gauge
field represents a potential function as a source of force fields such as
electromagnetic, weak, or strong forces. Their equations of motion govern the
dynamics of fields. Gauge and matter fields contribute to the structure of
space-time through their stress-energy tensor.

####  4.2.1 Matter field

The matter-field Lagrangian-form is defined in the spinor-gauge bundle
introduced in section 3.3 using the Dirac operator given in (94). A mass term
is introduced in the secondary superpsace of the Yang–Mills bundle as follows:

$\displaystyle{\mbox{\boldmath$\mu$}}^{~{}}_{{\textrm{Y}\textrm{M}}}$
$\displaystyle:=\gamma\left(\begin{array}[]{cc}\mu&0\\\
0&\hat{\mu}\end{array}\right){\mbox{\boldmath$1$}}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}.$

Real constants $0\leq\mu,\hat{\mu}\in\mathbb{R}$ are particle masses in
physics, and they have a mass dimension $[\mu]=[\hat{\mu}]=M$. The matter-
field Lagrangian density is defined as

$\displaystyle{\mathcal{}L}_{M}$
$\displaystyle:=\overline{{\mbox{\boldmath$\Psi$}}}_{{\textrm{Y}\textrm{M}}}\left(i\hskip
1.00006pt{\not\\!\mathcal{}D}^{~{}}_{\textrm{Y}\textrm{M}}-\frac{1}{\hbar}{\mbox{\boldmath$\mu$}}^{~{}}_{{\textrm{Y}\textrm{M}}}\right){\mbox{\boldmath$\Psi$}}_{{\textrm{Y}\textrm{M}}}.$
(116)

Physical constant $\hbar$ is inserted to keep a physical dimension of
Lagrangian densities to $[{\mathcal{}L}]=L^{-4}$.

######  Definition 4.3 (Matter-field Lagrangian and action).

The Lagrangian and action integral for the matter field are defined as

$\displaystyle{\mathfrak{L}}_{M}:={\mathcal{}L}_{M}\hskip
1.00006pt{\widehat{\mathrm{H}}}(1)~{}~{}\mathrm{and}~{}~{}\mathscr{I}_{M}:=\int_{\Sigma_{4}}{\mathfrak{L}}_{M}.$

The matter-field Lagrangian is Hermitian self-adjoint.

####  4.2.2 Gauge field

The gauge-field Lagrangian is defined by means of a super trace as follows:

######  Definition 4.4 (Gauge-field Lagrangian and action).

The gauge-field Lagrangian and corresponding gauge action are defined as
follows:

$\displaystyle{\mathfrak{L}}_{{\textrm{Y}\textrm{M}}}:=-\frac{1}{2}Str[\mathfrak{ym}\wedge\mathfrak{ym}],$
$\displaystyle~{}~{}~{}\mathscr{I}_{\textrm{Y}\textrm{M}}:=\int_{\Sigma_{4}}{\mathfrak{L}}_{{\mathrm{YM}}},$
(117)

which is equivalent to the standard definition of the Yang-Mills Lagrangian.

Supertrace of the Yang–mills form squared is provided as

$\displaystyle Str[\mathfrak{ym}\wedge\mathfrak{ym}]$
$\displaystyle={\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\hat{\mathfrak{F}}^{~{}}_{sg}\wedge\mathfrak{F}^{~{}}_{sg}\right]-{\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\mathfrak{F}^{~{}}_{sg}\wedge\hat{\mathfrak{F}}^{~{}}_{sg}\right]=-2\hskip
1.00006pt{\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\mathfrak{F}^{~{}}_{sg}\wedge\hat{\mathfrak{F}}^{~{}}_{sg}\right],$
$\displaystyle=-2\|\mathfrak{F}^{~{}}_{sg}\|^{2};$

thus, the gauge-field Lagrangian is represented using the trivial basis as

$\displaystyle{\mathfrak{L}}_{{\textrm{Y}\textrm{M}}}=\|\mathfrak{F}^{~{}}_{sg}\|^{2}=\frac{1}{4}\hskip
1.00006pt\left(\sum_{I=1}^{N}\eta^{{\circ}\star}\eta^{{\circ}\star}\mathscr{F}^{I}_{\circ\circ}\mathscr{F}^{I}_{\star\star}\hskip
1.99997pt{\mathrm{Tr}}\left[\tau^{I}_{SU}\cdot\tau^{I}_{SU}\right]\right){\widehat{\mathrm{H}}}(1),$

which can be verified by direct calculations from the definition (105). This
Lagrangian is the same as the standard definition of the Yang–Mills
Lagrangian[42, 43].

######  Remark 4.5.

The Yang–Mills action with the ${SU(N)}$ principal group in the Euclidean
metric space has extremal at the SD- or ASD curvature.

###### Proof.

The Yang–Mills action has the representation owing to SD and ASD curvatures as

$\displaystyle\mathscr{I}_{\textrm{Y}\textrm{M}}$
$\displaystyle=\int_{\Sigma_{4}}\|\mathfrak{F}^{~{}}_{sg}\|^{2}=\frac{1}{2}\int_{\Sigma_{4}}\left(\|\mathfrak{F}^{+}_{sg}\|^{2}+\|\mathfrak{F}^{-}_{sg}\|^{2}\right).$

Here, the trace concerning $SU(N)$ indices is not explicitly written for
simplicity. On the other hand, the second Chern class is provided as

$\displaystyle c^{~{}}_{2}\left(\mathfrak{F}^{~{}}_{sg}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}\left({\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\mathfrak{F}^{~{}}_{sg}\wedge\mathfrak{F}^{~{}}_{sg}\right]-{\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\mathfrak{F}^{~{}}_{sg}\right]^{2}\right),$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{\Sigma_{4}}\left(\|\mathfrak{F}^{+}_{sg}\|^{2}-\|\mathfrak{F}^{-}_{sg}\|^{2}\right).$

From the first line to the second line, we use ${\mathrm{Tr}}_{\hskip
0.70004pt{SU}}\left[\mathfrak{F}^{~{}}_{sg}\right]=0$ due to
${\mathrm{Tr}}\left[\tau_{SU}^{I}\right]=0$ for a $SU(N)$ gauge group. When
$\|\mathfrak{F}\|^{2}$ is positive definite, equivalently, when the space-time
manifold has the Euclidean metric, they yield
$\mathscr{I}_{\textrm{Y}\textrm{M}}\geq
4\pi^{2}\left|{c}_{2}\left(\mathfrak{F}^{~{}}_{sg}\right)\right|$; thus, the
remark is maintained. ∎

In the $\theta$-metric space, it is true only when $\theta=1$. We note that
there are solutions to the Yang–Mills equation other than the SD- nor ASD
connection. Sibner, Sibner and Uhlenbeck[44] reported the $SU(2)$ Yang–Mills
connections which are not the SD- nor ASD connection in $S^{4}$.

###  4.3 Full Lagrangian and equation of motion in Yang–Mills theory

In summary, the Yang–Mills Lagrangian and action integral is obtained as

$\displaystyle{\mathfrak{L}}_{\mathrm{MYM}}:={\mathfrak{L}}_{M}+{\mathfrak{L}}_{\textrm{Y}\textrm{M}}~{}~{}\textrm{and}~{}~{}\mathscr{I}_{\mathrm{MYM}}:=\int{\mathfrak{L}}_{\mathrm{MYM}},$

and it is expressed using the trivial basis as

$\displaystyle{\mathfrak{L}}_{\mathrm{MYM}}$
$\displaystyle=\left\\{\bar{\psi}\left(i\hskip 1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}\partial_{\circ}-\frac{1}{2}{c^{~{}}_{gr}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}_{\circ}^{\mu}\omega_{\mu}^{~{}{\star\star}}{\mathscr{S}}_{\star\star}+{c^{~{}}_{SU}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}^{\mu}_{\circ}{\mathscr{A}}^{I}_{\mu}\tau^{~{}}_{I}\right)\psi-\frac{1}{\hbar}{\mu}\bar{\psi}\psi\right\\}{\widehat{\mathrm{H}}}(1)$
$\displaystyle~{}+\left\\{(\psi,{c^{~{}}_{SU}},\mu)\rightarrow(\hat{\psi},{{c}^{~{}}_{SU}},\hat{\mu})\right\\}{\widehat{\mathrm{H}}}(1)+\frac{1}{4}\mathscr{F}^{I}_{\circ\circ}\mathscr{F}_{I}^{\circ\circ}\hskip
1.00006pt{\widehat{\mathrm{H}}}(1).$ (118)

Terms with dual field $\hat{\psi}$ in formula (118) represents the magnetic
monopole in physics, which is not confirmed experimentally to date. This
section treats only principal filed $\psi$. Section 5.2 discusses a magnetic
monopole in detail.

######  Remark 4.6 (Yang–Mills equations of motion).

Two equations of motion is obtained from the Yang–Mills action such that

$\displaystyle\left(i{\not\\!d}_{{sg}}-\frac{\mu}{\hbar}\right)\psi$
$\displaystyle=0,$ (119) $\displaystyle
d_{sg}\hat{\mathfrak{F}}^{~{}}_{{SU}}-{c^{~{}}_{SU}}\hskip
1.00006pt\bar{\psi}\gamma_{sp\hskip 0.70004pt}^{\circ}\psi\hskip
1.99997pt\mathfrak{V}_{\circ}$ $\displaystyle=0.$ (120)

The first equation is known as the Dirac equation, and the second equation is
the Yang–Mills equation.

###### Proof.

The Dirac equation can be easily obtained by taking a variational operation on
the Yang–Mills Lagrangian concerning conjugate field $\bar{\psi}$ as

$\displaystyle\hbar\hskip
1.00006pt\delta_{\bar{\psi}}\mathscr{I}_{\mathrm{YM}}=\int_{\Sigma_{4}}\delta\bar{\psi}\left(i\hbar\hskip
1.00006pt{\not\\!d^{~{}}_{{sg}}}\psi-{\mu}\psi\right)\hskip
1.00006pt{\widehat{\mathrm{H}}}(1)=0$ $\displaystyle\Rightarrow i\hbar\hskip
1.00006pt{\not\\!d^{~{}}_{{sg}}}\psi-{\mu}\psi=0.$

For gauge-field ${\mathfrak{A}}^{~{}}_{SU}$, an equation of motion is provided
owing to a variational operation concerning connection
${\mathfrak{A}}^{~{}}_{SU}$ such that:

$\displaystyle\delta_{{\mathfrak{A}}^{~{}}_{SU}}\mathscr{I}_{\mathrm{YM}}$
$\displaystyle=-\frac{1}{2}\int_{\Sigma_{4}}\sum_{I}\left((d\hat{\mathfrak{F}}^{I}_{{SU}}-{c^{~{}}_{SU}}\hskip
1.00006pt\bar{\psi}\hskip 1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\widehat{\mathrm{H}}}(1)\psi)\delta_{K}^{I}+{c^{~{}}_{SU}}f^{I}_{~{}JK}\hat{\mathfrak{F}}^{I}_{{SU}}\wedge{\mathfrak{A}}_{SU}^{J}\right)\wedge\delta{\mathfrak{A}}_{SU}^{K}$
$\displaystyle~{}-\frac{1}{2}\int_{\partial\Sigma_{4}}{\mathrm{Tr}}\left[\hat{\mathfrak{F}}^{~{}}_{{SU}}\wedge\delta{\mathfrak{A}}^{~{}}_{SU}\right]=0,$
$\displaystyle\Rightarrow
d_{sg}\hat{\mathfrak{F}}^{~{}}_{{SU}}={c^{~{}}_{SU}}\hskip
1.00006pt\bar{\psi}\gamma_{sp\hskip 0.70004pt}^{\circ}\psi\hskip
1.99997pt\mathfrak{V}_{\circ},$

where $\partial\Sigma_{4}$ is the integration boundary in which variation
$\delta{\mathfrak{A}}^{~{}}_{SU}$ vanishes. In above calculations, a relation

$\displaystyle\delta_{{\mathfrak{A}}^{~{}}_{SU}}\left(\bar{\psi}\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip 0.50003pt}}{\mathfrak{A}}_{SU}^{I}\hskip
1.00006pt\psi\hskip 1.00006pt{\widehat{\mathrm{H}}}(1)\right)$
$\displaystyle=\frac{\delta}{\delta{\mathfrak{A}}_{SU}^{K}}\left(\bar{\psi}\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\mathfrak{A}}_{SU}^{I}\psi\hskip
1.00006pt{\widehat{\mathrm{H}}}(1)\right)\delta{\mathfrak{A}}_{SU}^{K},$
$\displaystyle=\bar{\psi}\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\widehat{\mathrm{H}}}(1)\right)\psi\hskip
1.00006pt\delta{\mathfrak{A}}_{SU}^{I}$

with

$0=\iota_{{\gamma}_{sp\hskip
0.50003pt}}({\mathfrak{A}}_{SU}^{I}\wedge{\widehat{\mathrm{H}}}(1))=(\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\mathfrak{A}}_{SU}^{I})\hskip
1.99997pt{\widehat{\mathrm{H}}}(1)-{\mathfrak{A}}_{SU}^{I}\wedge\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}{\widehat{\mathrm{H}}}(1)\right)$

is used. ∎

An expression of equations (120) using the trivial basis in ${T{\mathcal{}M}}$
is provided as

$\displaystyle~{}\hbar\left(i\gamma_{sp\hskip
0.70004pt}^{\circ}\partial_{\circ}-\frac{1}{2}{c^{~{}}_{gr}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}_{\circ}^{\mu}\omega_{\mu}^{~{}{\star\star}}{{\mathscr{S}}}_{\star\star}+{c^{~{}}_{SU}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}^{\mu}_{\circ}{\mathscr{A}}^{I}_{\mu}t_{I}\right)\psi={\mu}\hskip
1.00006pt\psi,$ (121)
$\displaystyle~{}\left(\eta^{{\circ\circ}}\partial_{\circ}\mathscr{F}^{I}_{~{}{\circ}a}+{c^{~{}}_{SU}}\hskip
1.00006pt\eta^{{\circ\circ}}f^{I}_{~{}JK}{\mathscr{A}}^{J}_{~{}{\circ}}\mathscr{F}^{K}_{~{}{\circ}a}\right)\tau^{~{}}_{I}=-{c^{~{}}_{SU}}\eta_{a{\circ}}\hskip
1.00006pt\bar{\psi}\gamma_{sp\hskip 0.70004pt}^{\circ}\psi.$ (122)

In equation (122), information of the curved space-time is included in the
component representation of $\mathscr{F}^{I}$ provided in (102) through a
torsion of the space-time manifold.

The Noether’s theorem ensures vanishing divergence such that:

$\displaystyle d\left(\bar{\psi}\hskip 1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}\psi\right)=\partial_{\circ}\left(\bar{\psi}\gamma^{\circ}\psi\right)=0,$

which is referred to as a current conservation in physics.

###  4.4 Scalar action

This section provides a geometrical construction of the Higgs field. A scalar
field belonging to the fundamental representation of the gauge field is
referred to as the Higgs filed in physics. Higgs field
${\mbox{\boldmath$\phi$}}\in C^{\infty}({T{\mathcal{}M}})$ is introduced as a
complex-valued section in the gauge bundle. The equation of motion for the
Higgs field is known as the Klein–Gordon equation and is provided from an
action integral using the trivial basis in ${\mathcal{}M}$ such as

$\displaystyle\tilde{\mathscr{I}}_{\textrm{Higgs}}:=-\frac{1}{4}\int_{\Sigma_{4}}\left[\eta^{{\circ\circ}}\nabla_{\circ}{\mbox{\boldmath$\phi$}}^{\dagger}\nabla_{\circ}{\mbox{\boldmath$\phi$}}-V\left(\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}\right)\right]{\widehat{\mathrm{H}}}(1),$

where
$\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}={\mbox{\boldmath$\phi$}}^{\dagger}{\mbox{\boldmath$\phi$}}$,
$\nabla_{\mu}=\partial_{\mu}-i\hskip
1.00006ptc^{~{}}_{SU}{\mathscr{A}}^{I}_{\mu}t_{I}$ is a $SU(N)$-covariant
differential in ${T\mathscr{M}}$, and
$V\left(\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}\right)$ is a
$SU(N)$-invariant potential term.

The Dirac equation was originated from a genius idea by Dirac to realize a
square-root of Laplace operator
$\Box:=\eta^{{\circ\circ}}\partial_{\circ}\partial_{\circ}$. In this study,
the Laplacian is defined as a square of the Dirac operator; thus, an action
integral of the Higgs field is defined as

$\displaystyle\mathscr{I}_{\textrm{Higgs}}$
$\displaystyle:=-\frac{1}{4}\int_{\Sigma_{4}}{\mathrm{Tr}}\left[({\not\\!d^{~{}}_{{sg}}}{\mbox{\boldmath$\phi$}}^{\dagger}){\not\\!d^{~{}}_{{sg}}}{\mbox{\boldmath$\phi$}}-{{\mbox{\boldmath$1$}}}_{sp\hskip
0.70004pt}V\left(\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}\right)\right]{\widehat{\mathrm{H}}}(1).$

######  Remark 4.7.

An action integral of the Higgs field is expressed using the trivial basis as

$\displaystyle\mathscr{I}_{\textrm{Higgs}}$
$\displaystyle=\int_{\Sigma_{4}}{\widehat{\mathrm{H}}}(1)\hskip
1.00006pt{\mbox{\boldmath$\phi$}}^{\dagger}\left\\{\Box+\frac{R}{4}+c_{s}\partial^{\circ}{\mathscr{A}}_{\circ}+c_{s}^{2}{\mathscr{A}}^{\circ}{\mathscr{A}}_{\circ}+\frac{1}{2}\left(\partial_{\circ}{\mathcal{E}}^{\mu}_{\circ}+c_{s}A_{\circ}{\mathcal{E}}^{\mu}_{\circ}\right)\omega_{\mu}^{~{}{\circ\circ}}\right\\}{\mbox{\boldmath$\phi$}}$
$\displaystyle\hskip 5.0pt+\int_{\Sigma_{4}}{\widehat{\mathrm{H}}}(1)\hskip
1.00006ptV\left(\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}\right)-\frac{1}{4}\int_{\Sigma_{4}}{\widehat{\mathrm{H}}}(1)\hskip
1.00006pt\partial_{\circ}\left({\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{\circ}{\not\\!d^{~{}}_{{sg}}}\right]\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}\right).$

An Euler–Lagrange equation of motion is provided by requiring a stationary
condition on a variational operation concerning the field
${\mbox{\boldmath$\phi$}}^{\dagger}$. The last term does not contribute to an
equation of motion owing to the boundary condition.

###### Proof.

Direct calculations provide the result similar with the Lichnerowicz
formula444See, e.g., chapter 1.3 of Ref.[45]. such that:

$\displaystyle{\mathrm{Tr}}\left[({\not\\!d^{~{}}_{{sg}}}{\mbox{\boldmath$\phi$}}^{\dagger})({\not\\!d^{~{}}_{{sg}}}{\mbox{\boldmath$\phi$}})\right]$
$\displaystyle={\mathrm{Tr}}\left[\left(\gamma_{sp\hskip
0.70004pt}^{a}\partial_{a}{\mbox{\boldmath$\phi$}}^{\dagger}+i\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}{\mbox{\boldmath$\phi$}}^{\dagger}\right)\left(\gamma_{sp\hskip
0.70004pt}^{b}\partial_{b}{\mbox{\boldmath$\phi$}}+i\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}{\mbox{\boldmath$\phi$}}\right)\right],$
$\displaystyle=-{\mathrm{Tr}}\left[{\mbox{\boldmath$\phi$}}^{\dagger}\gamma_{sp\hskip
0.70004pt}^{a}\left(\partial_{a}{\mbox{\boldmath$\phi$}}\gamma_{sp\hskip
0.70004pt}^{b}\partial_{b}{\mbox{\boldmath$\phi$}}\right)+i\hskip
1.00006pt{\mbox{\boldmath$\phi$}}^{\dagger}\gamma_{sp\hskip
0.70004pt}^{a}\left(\partial_{a}\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}{\mbox{\boldmath$\phi$}}\right)\right]$
$\displaystyle~{}+{\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{a}\partial_{a}{\mbox{\boldmath$\phi$}}^{\dagger}\left(\gamma_{sp\hskip
0.70004pt}^{b}\partial_{b}{\mbox{\boldmath$\phi$}}+i\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}{\mbox{\boldmath$\phi$}}\right)\right]$
$\displaystyle~{}+{\mathrm{Tr}}\left[i\hskip
1.00006pt\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}{\mbox{\boldmath$\phi$}}^{\dagger}\gamma_{sp\hskip
0.70004pt}^{a}\partial_{a}{\mbox{\boldmath$\phi$}}\right]-{\mathrm{Tr}}\left[\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}\right)^{2}\right],$
$\displaystyle=-4{\mbox{\boldmath$\phi$}}^{\dagger}\partial^{a}\partial_{a}{\mbox{\boldmath$\phi$}}-i{\mbox{\boldmath$\phi$}}^{\dagger}{\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{a}\left(\partial_{a}\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}\right){\mbox{\boldmath$\phi$}}\right]$
$\displaystyle~{}-{\mathrm{Tr}}\left[\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}\right)^{2}\right]\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}+\partial_{\circ}\left({\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{\circ}{\not\\!d^{~{}}_{{sg}}}\right]\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}\right).$

The second and third terms, respectively, give formulae such that:

$\displaystyle{\mbox{\boldmath$\phi$}}^{\dagger}{\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{\circ}\left(\partial^{~{}}_{\circ}\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}\right){\mbox{\boldmath$\phi$}}\right]$
$\displaystyle=\frac{1}{2}{\mathrm{Tr}}\left[\gamma_{sp\hskip
0.70004pt}^{\diamond}\gamma_{sp\hskip
0.70004pt}^{\diamond}{\mathscr{S}}^{\circ\circ}\right]\eta_{{\circ}\star}\eta_{{\circ}\star}\left(\partial^{~{}}_{\diamond}{\mathcal{E}}^{\mu}_{\diamond}\omega_{\mu}^{~{}{\star\star}}\right)\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}+4i\left(c_{s}\partial^{~{}}_{\circ}{\mathscr{A}}^{\circ}\right)\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2},$
$\displaystyle=-2i\left(\partial^{~{}}_{\circ}{\mathcal{E}}^{\mu}_{\circ}\omega_{\mu}^{~{}{\circ\circ}}\right)\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}+4ic_{s}\left(\partial^{~{}}_{\circ}{\mathscr{A}}^{\circ}\right)\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2},$

and

$\displaystyle-{\mathrm{Tr}}\left[\left(\iota_{{\gamma}_{sp\hskip
0.50003pt}}\mathfrak{O}\right)^{2}\right]\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2}=-2\left({\mathcal{E}}^{\mu}_{\circ}\omega_{\mu~{}\star}^{~{}{\circ}}\hskip
1.99997pt{\mathcal{E}}^{\nu}_{\diamond}\omega_{\nu}^{~{}\star\diamond}+c^{~{}}_{s}{\mathscr{A}}^{~{}}_{\circ}{\mathcal{E}}^{\mu}_{\circ}\omega_{\mu}^{~{}{\circ\circ}}+2c_{s}^{2}{\mathscr{A}}^{~{}}_{\circ}{\mathscr{A}}^{\circ}\right)\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}^{2},$

where a $SU(N)$ index of ${\mathscr{A}}$ is omitted for simplicity. On the
other hand, from the definition of the Lorentz curvature,

$\displaystyle\mathfrak{R}^{ab}$
$\displaystyle=d{\mathfrak{w}}^{ab}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mathfrak{w}}^{a}_{\circ}\wedge{\mathfrak{w}}^{{\circ}b},$
$\displaystyle=\left(\partial_{\mu}\omega_{\nu}^{~{}ab}+{c^{~{}}_{gr}}\hskip
1.00006pt\omega_{\mu~{}{\circ}}^{~{}a}\hskip
1.99997pt\omega_{\nu}^{~{}{\circ}b}\right)dx^{\mu}\wedge dx^{\nu},$
$\displaystyle=\left\\{\left(\partial_{\circ}{\mathcal{E}}^{\mu}_{\circ}\omega_{\mu}^{~{}ab}\right)-\left(\partial_{\circ}{\mathcal{E}}^{\mu}_{\circ}\right)\omega_{\mu}^{~{}ab}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mathcal{E}}^{\mu}_{\circ}\omega_{\mu~{}\star}^{~{}a}\hskip
1.99997pt{\mathcal{E}}^{\nu}_{\circ}\omega_{\nu}^{~{}\star
b}\right\\}{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ},$
$\displaystyle=:\frac{1}{2}R^{ab}_{\hskip 4.20004pt{\circ\circ}}\hskip
1.99997pt{\mathfrak{e}}^{\circ}\wedge{\mathfrak{e}}^{\circ},$
$\displaystyle\Longrightarrow R:=R^{{\circ}\star}_{\hskip
4.20004pt{\circ}\star}=2\left(\partial^{~{}}_{\circ}\omega_{\circ}^{~{}{\circ\circ}}+{c^{~{}}_{gr}}\hskip
1.00006pt\omega_{\star}^{~{}\star{\circ}}\omega_{\diamond}^{~{}{\circ}\diamond}\eta^{~{}}_{{\circ\circ}}-\left(\partial^{~{}}_{\circ}{\mathcal{E}}^{\mu}_{\circ}\right)\omega_{\mu}^{~{}{\circ\circ}}\right),$

where $\omega_{a}^{~{}bc}:={\mathcal{E}}^{\mu}_{a}\omega_{\mu}^{~{}bc}$.
Finally, by gathering up above results into $\mathscr{I}_{\textrm{Higgs}}$,
the remark is maintained. ∎

Consequently, the equation of motion for the Higgs field is provided as the
Euler–Lagrange equation such that:

$\displaystyle\left(\Box+\frac{R}{4}+\frac{\partial
V\left(\bigl{|}{\mbox{\boldmath$\phi$}}\bigr{|}\right)}{\partial{\mbox{\boldmath$\phi$}}}\right){\mbox{\boldmath$\phi$}}$
$\displaystyle+\eta^{\circ\circ}_{~{}}\sum_{I}\left(c^{~{}}_{s}\partial^{~{}}_{\circ}{\mathscr{A}}^{I}_{\circ}+c_{s}^{2}f^{I}_{~{}JK}{\mathscr{A}}^{J}_{\circ}{\mathscr{A}}^{K}_{\circ}\right){\mbox{\boldmath$\phi$}}$
$\displaystyle\hskip
5.0pt+\frac{{c^{~{}}_{gr}}}{2}\left(\partial^{~{}}_{\circ}{\mathcal{E}}^{\mu}_{\circ}+c^{~{}}_{s}\sum_{I}{\mathscr{A}}^{I}_{\circ}{\mathcal{E}}^{\mu}_{\circ}\right)\omega_{\mu}^{~{}{\circ\circ}}{\mbox{\boldmath$\phi$}}=0,$

where $SU(N)$ indices are indicated explicitly. It is clear that this equation
is $GL(4,\mathbb{R})$ and $SU(N)$ invariant owing to its construction.

###  4.5 Torsion and stress-energy forms of Yang–Mills theory

Yang–Mills field is a source of torsion and curvature of space-time. The
right-hand side of the torsion equation (114) and the Einstein equation (115)
is given by torsion and a stress-energy tensor of the Yang–Mills field,
respectively. They are defined as follows:

######  Definition 4.8 (torsion and stress-energy forms).

Torsion two-form ${\mathfrak{T}}_{{\mathrm{YM}}}$ and stress-energy three-form
$\mathfrak{E}_{{\mathrm{YM}}}$ of the Yang–Mills Lagrangian are respectively
defined as

$\displaystyle{\mathfrak{T}}_{{\mathrm{YM}}}\wedge{\mathfrak{e}}:=\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta{\mathfrak{w}}}-d\left(\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta(d{\mathfrak{w}})}\right)~{}~{}\mathrm{and}~{}~{}\mathfrak{E}_{{\mathrm{YM}}}:=\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta{\mathfrak{e}}}-d\left(\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta(d{\mathfrak{e}})}\right).$

######  Remark 4.9 (torsion and stress-energy forms of Yang–Mills Lagrangian).

For the Yang–Mills Lagrangian, a torsion form is provided as

$\displaystyle{\mathfrak{T}}_{{\mathrm{YM}}}^{a}$
$\displaystyle=\left\\{K^{a}_{~{}b_{1}b_{2}}+\frac{1}{2}\left(\delta^{a}_{b_{1}}K^{\circ}_{~{}b_{2}{\circ}}-\delta^{a}_{b_{2}}K^{\circ}_{~{}b_{1}{\circ}}\right)\right\\}{\mathfrak{e}}^{b_{1}}\wedge{\mathfrak{e}}^{b_{2}},$
(123)

where

$\displaystyle K^{a}_{~{}bc}$
$\displaystyle:=\hbar\left(\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{a}{{\mathscr{S}}}^{~{}}_{bc}\psi\right),$ (124)

and the stress-energy form is

$\displaystyle[\mathfrak{E}_{\mathrm{YM}}]_{a}$
$\displaystyle=\left\\{\hbar\bar{\psi}\left(i\hskip 1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}\partial_{\circ}-\frac{1}{2}{c^{~{}}_{gr}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}_{\circ}^{\mu}\omega_{\mu}^{~{}{\star\star}}{{\mathscr{S}}}^{~{}}_{\star\star}+{c^{~{}}_{SU}}\hskip
1.00006pt\gamma_{sp\hskip
0.70004pt}^{\circ}{\mathcal{E}}^{\mu}_{\circ}{\mathscr{A}}^{I}_{\mu}t_{I}\right)\psi-{m_{e}}\bar{\psi}\psi\right.$
$\displaystyle\hskip
20.00003pt\left.-\frac{\hbar}{4}\mathscr{F}^{I}_{\circ\circ}\mathscr{F}_{I}^{\circ\circ}\right\\}\mathfrak{V}_{a}.$
(125)

###### Proof.

For the torsion form, a tensor $K$ is introduced as

$\displaystyle\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta{\mathfrak{w}}^{ab}}$
$\displaystyle=:K^{\circ}_{~{}ab}\mathfrak{V}_{\circ}.$

Coefficients of tensor $K$ can be obtained by direct calculations from (118)
as

$\displaystyle\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta{\mathfrak{w}}^{ab}}=\frac{\delta{\mathfrak{L}}_{{\mathrm{YM}}}}{\delta\omega_{\mu}^{~{}ab}}\partial_{\mu}$
$\displaystyle=\hbar\hskip
1.00006pt{c^{~{}}_{gr}}\left(\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mathscr{S}}}^{~{}}_{ab}\psi\right)\frac{1}{3!}\epsilon_{{\circ}\star{\star\star}}{\mathfrak{e}}^{\star}\wedge{\mathfrak{e}}^{\star}\wedge{\mathfrak{e}}^{\star},$
$\displaystyle=\hbar\hskip
1.00006pt{c^{~{}}_{gr}}\left(\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mathscr{S}}}^{~{}}_{ab}\psi\right)\mathfrak{V}_{\circ},$

and thus, (124) is provided. Torsion form (123) can be obtained by solving
algebraic equation

$\epsilon_{ab{\circ\circ}}\hskip
1.00006pt{\mathfrak{T}}_{{\mathrm{YM}}}^{\circ}\wedge{\mathfrak{e}}^{\circ}=K^{\circ}_{~{}ab}\hskip
1.00006pt\mathfrak{V}_{\circ}.$

Energy-stress form $\mathfrak{E}_{{\mathrm{YM}}}$ can be obtained by direct
calculations from (118). ∎

In summary, the Einstein equation and the torsion equation with the Yang–Mills
field are respectively provided using the trivial basis in ${T{\mathcal{}M}}$
as

$\displaystyle\frac{1}{\kappa}\left(\frac{1}{2}\epsilon_{a{\circ\circ}{\circ}}\mathfrak{R}^{\circ\circ}\wedge{\mathfrak{e}}^{\circ}-\Lambda_{\mathrm{c}}\mathfrak{V}_{a}\right)$
$\displaystyle=-[\mathfrak{E}_{{\mathrm{YM}}}]_{a}~{}\Leftarrow~{}(\ref{esform}),$
(126)

and

$\displaystyle\frac{1}{\kappa}\left(d{\mathfrak{e}}^{a}+{c^{~{}}_{gr}}\hskip
1.00006pt{\mathfrak{w}}^{a}_{~{}{\circ}}\wedge{\mathfrak{e}}^{\circ}\right)=\hbar\left(\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{a}{{\mathscr{S}}}^{~{}}_{{\circ}\star}\psi+2\delta^{a}_{\hskip
1.39998pt[{\circ}}{\eta}^{~{}}_{\star]\diamond}\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{\diamond}\psi\right){\mathfrak{e}}^{{\circ}}\wedge{\mathfrak{e}}^{\star}.$

The stress-energy due to the gauge field tensor (corresponding to the last
term of the right-hand side of (125)) is provided as
$T^{ab}_{SU}:=-{\hbar}\hskip
1.00006pt\eta^{ab}\mathscr{F}^{I}_{\circ\circ}\mathscr{F}_{I}^{\circ\circ}/4$.
Whereas the standard form of the stress-energy tensor is traceless, its trace
is not zero. The standard traceless stress-energy tensor can be obtained by
adding total derivative term $\hbar\hskip
1.00006ptd({\mathfrak{A}}^{~{}}_{SU}\wedge\mathfrak{F}_{SU})$ to Lagrangian
${\mathfrak{L}}_{{\textrm{Y}\textrm{M}}}$. This additional term gives term
$\hbar\hskip
1.00006pt\eta_{\circ\circ}\mathscr{F}_{I}^{a{\circ}}\mathscr{F}_{I}^{b{\circ}}$
in the stress-energy tensor, and the standard representation of the stress-
energy tensor for the gauge field is obtained as

$\displaystyle T^{\overline{ab}}_{SU}$ $\displaystyle:=\hbar\hskip
1.00006pt\sum_{I}\left(\eta_{\circ\circ}\mathscr{F}_{I}^{a{\circ}}\mathscr{F}_{I}^{b{\circ}}-\frac{1}{4}\eta^{ab}\mathscr{F}^{I}_{\circ\circ}\mathscr{F}_{I}^{\circ\circ}\right),$
(127)

which is traceless such as $\eta_{{\circ\circ}}T^{{\circ\circ}}_{SU}=0$.

##  5 Index theorem in general relativistic Yang–Mills theory

Both Lagrangians of general relativity and the Yang–Mills theory are defined
using the supertrace in the secondary superspace. According to methods given
in section 2.5.1 and 2.6, this section discussed a topological index in the
general relativistic Yang–Mills bundle with the Euclidean- and Lorentzian-
metrics.

###  5.1 Chern index of space-time manifold

This section treats the characteristic class of the Lorentz manifold. The
characteristic class is an element of the cohomology group of the classifying
space of the principal bundle. We utilize the Chren characteristic to classify
the Lorentz manifold. The first Chern class of the Lorentz manifold is zero
since the Lorentz curvature $\mathfrak{R}$ is traceless concerning tensor
coefficients in ${T{\mathcal{}M}}$. In the four-dimensional manifold, only the
second Chern class can have a non-zero value provided such that:

$\displaystyle
c^{~{}}_{2}\left(\mathfrak{R}\right)=p_{1}\left(\mathfrak{R}\right)=\frac{{\mathrm{Tr}}_{{T{\mathcal{}M}}}\left[\mathfrak{R}\wedge\mathfrak{R}\right]}{8\pi^{2}}\in
H^{4}({\mathcal{}M},\mathbb{Z})$

where $\mathfrak{R}$ is provided as a solution of the Einstein equation and
the torsion-less condition. A flat Lorentz-manifold has no singularity, and
its domain is the whole ${\mathcal{}M}{\cong}\mathbb{R}^{4}$. After the one-
point compactification of ${\mathcal{}M}$, the Poincaré duality theorem
insists that
$H^{4}({\mathcal{}M}{\cong}S^{4},\mathbb{Z}){\cong}H_{0}(S^{4},\mathbb{Z}){\cong}\mathbb{Z}$.
The de Sitter space-time, as well as the Friedmann–Lemaître–Robertson–Walker
space-time, has topology $M_{\textrm{dS}}\cong\mathbb{R}{\otimes}S^{3}$, and
after the one-point compactification of the time coordinate, it has cohomology
such that:

$\displaystyle H^{4}(S^{1}{\otimes}S^{3},\mathbb{Z})$
$\displaystyle{\cong}\sum_{i+j=4}H^{i}(S^{1},\mathbb{Z}){\otimes}H^{j}(S^{3},\mathbb{Z}){\cong}H_{0}(S^{1},\mathbb{Z}){\otimes}H_{0}(S^{3},\mathbb{Z}){\cong}\mathbb{Z}\times\mathbb{Z}.$

Here, we use the Künneth theorem and the Poincaré duality theorem.

On the other hand, black hole solutions have a singularity, e.g., the
Schwarzschild solution is defined in
$\mathbb{R}^{1}{\otimes}\mathbb{R}^{3}\setminus\\{0\\}$ and the Kerr–Newman
solution has a domain as
$\mathbb{R}^{1}{\otimes}\mathbb{R}^{3}{\setminus}S^{1}$. The singularity-ring
$S^{1}$ in the Kerr–Newman solution is homotopically contractible; thus, its
homology is the same as the Schwarzschild solution’s one. Therefore, it is
enough to consider the cohomology of the Schwarzschild solution. An oriented
and closed domain of the Schwarzschild solution, denoted as
${\mathcal{}M}_{\textrm{Schw}}$, is constructed owing to the one-point
compactification of ${\mathcal{}M}$ as follows: Time variable $t$ is one-point
compactified as $\mathbb{R}{\mapsto}S^{1}$. A three-dimensional spacial
manifold without the origin is the closed manifold such that
$\mathbb{R}^{3}\setminus\\{0\\}{\cong}S^{3}$; thus, the domain of the
Schwarzschild solution is a closed manifold
${\mathcal{}M}_{\textrm{Schw}}{\cong}S^{1}{\otimes}S^{3}$. The cohomology of a
domain of the Schwarzschild solution is provided as
$H^{4}({\mathcal{}M}_{\textrm{Schw}},\mathbb{Z}){\cong}\mathbb{Z}\times\mathbb{Z}$.

In summary, the second Chern class of the known solutions of the Einstein
equation in the Lorentz bundle has integral cohomology. In reality, direct
calculations of
${\mathrm{Tr}}_{{T{\mathcal{}M}}}[\mathfrak{R}\wedge\mathfrak{R}]$ with the
$\theta$-metric show

$\displaystyle c^{~{}}_{2}\left(\mathfrak{R}\right)$
$\displaystyle\propto\eta_{\theta\hskip 0.70004pt{\circ\circ}}\hskip
1.99997pt\eta_{\theta\hskip
0.70004pt{\star\star}}\mathfrak{R}^{{\circ}\star}\wedge\mathfrak{R}^{\star{\circ}}=0,$

for all solutions listed above. Therefore, the Chern characteristic is also
zero for the space-time manifold of the known solution of the Einstein
equation.

###  5.2 Index in spinor-gauge bundle

The $\mathbb{Z}_{2}$-grading superspace of gauge fields consisting of
${\mathfrak{A}}_{{SU}}$ and $\hat{\mathfrak{A}}^{~{}}_{{SU}}$ induces the
corresponding superspace of spinor fields $\psi$ and $\hat{\psi}$,
respectively. Whereas a spinor concerning $\hat{\mathfrak{A}}^{~{}}_{{SU}}$
(magnetic monopole) has not been observed experimentally, a dual spinor is
introduced as a section in the spinor-gauge bundle. The second term of
Lagrangian (118) is kept in theory in this section. This section treats the
flat space-time with ${\mathfrak{w}}=0$.

The magnetic-monopole Lagrangian is provided as

$\displaystyle{\mathfrak{L}}_{\textrm{MP}}$
$\displaystyle:=\left\\{{\mbox{\boldmath$\bar{\hat{\psi}}$}}\left(i\hbar\hskip
1.00006pt\hat{d}_{{SU}}-\hat{\mu}{\mbox{\boldmath$1$}}_{SU}\right){{\mbox{\boldmath$\hat{\psi}$}}}\right\\}{\widehat{\mathrm{H}}}(1).$

The second Bianchi identities of gauge and its dual fields are provided as

$\displaystyle
d^{~{}}_{SU}\mathfrak{F}^{~{}}_{{SU}}=\hat{d}^{~{}}_{{SU}}\hat{\mathfrak{F}}^{~{}}_{{SU}}=0.$
(128)

Equations of motion for the gauge- and spinor-fields are provided as

$\displaystyle d_{SU}\hat{\mathfrak{F}}^{~{}}_{{SU}}={c^{~{}}_{SU}}\hskip
1.00006pt{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mbox{\boldmath$\psi$}}}\hskip
1.99997pt\mathfrak{V}_{\circ},$
$\displaystyle\hat{d}_{{SU}}\mathfrak{F}_{SU}={{c}^{~{}}_{SU}}\hskip
1.00006pt{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mbox{\boldmath$\hat{\psi}$}}}\hskip
1.99997pt\mathfrak{V}_{\circ},$ (129) $\displaystyle\left(i\hbar\hskip
1.00006pt{\not\\!d^{~{}}_{{sg}}}-\mu{\mbox{\boldmath$1$}}_{SU}\right){\mbox{\boldmath$\psi$}}=0,$
$\displaystyle\left(i\hbar\hskip
1.00006pt{\not\\!\hat{d}^{~{}}_{{sg}}}-\hat{\mu}{\mbox{\boldmath$1$}}_{SU}\right){{\mbox{\boldmath$\hat{\psi}$}}}=0.$
(130)

The equation of motion for $\hat{\psi}$ is provided owing to a variational
operation on $\mathscr{I}_{\mathrm{YM}}$ with respect to
$\hat{\mathfrak{A}}_{{SU}}$. The Bianchi identities (128) and equations of
motion (129) are generally independent to each other. The equation of motion
for the spinor section has an expression using tensor coefficients such that:

$\displaystyle-\sqrt{{{\sigma^{~{}}_{\theta}}}}\left(\delta^{I}_{J}\partial_{\circ}+c^{~{}}_{SU}f^{I}_{~{}JK}[{\mathscr{A}}^{~{}}_{\theta}]^{K}_{\circ}\right)\left[\mathscr{F}_{\theta}^{J}\right]^{{\circ}a}\tau^{~{}}_{I}=\left[j^{~{}}_{\theta}\right]^{a},$
(131)

where

$\displaystyle\left[\mathscr{F}_{\theta}^{J}\right]^{ab}:=\eta_{\theta}^{a{\circ}}\eta_{\theta}^{b\star}\left[\mathscr{F}_{\theta}^{J}\right]_{{\circ}\star}~{}~{}\textrm{and}~{}~{}j^{a}:={c^{~{}}_{SU}}\hskip
1.00006pt{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{\theta}^{a}{{\mbox{\boldmath$\psi$}}}.$

We note that both $\hat{\mathfrak{F}}^{~{}}_{SU}$ and $\mathfrak{V}_{\bullet}$
are defined using the Levi-Civita tensor; thus, it can be factored out after
rearranging indices. Therefore, the equation of motion for the spinor section
is provided with the connection and curvature of the spinor-gauge bundle.

For the $U(1)$ gauge, the Bianchi identities correspond to the (so-called)
first set of the Maxwell equations, and they are not equations but identities.
The second set of the Maxwell equations corresponds to (129) and govern the
dynamics of electric- and magnetic fields. We note that the dual field
$\hat{\mathfrak{F}}^{~{}}_{{SU}}$ is the electromagnetic field created by an
electron field $\psi$ through the electromagnetic potential
${\mathfrak{A}}_{{SU}}$ in this study.

######  Example 5.1.

electric and magnetic poles

This example treats a topological index of the spinor-gauge bundle with the
$U(1)$-gauge group. Spinor fields $\psi$ and $\hat{\psi}$, conventionally
called an electron and a magnetic monopole, are referred to as electric pole
and magnetic pole in this study, respectively. We start a discussion from a
Coulomb-type electric-potential of an electric pole in the flat space-time
manifold with the $\theta$-metric. When a Coulomb-type potential in globally
flat $\mathscr{M}$ is provided as

$\displaystyle V(r)=\frac{e^{i\pi\theta/2}}{r}$

using a local three-dimensional polar coordinate
$\xi^{a}=(t,r,\theta_{1},\theta_{2})$. a connection of the gauge bundle is
obtained as

$\displaystyle{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle:=V(r)\hskip
1.99997ptdt\in\Omega^{1}\left({T^{*}{\mathcal{}M}}{\setminus}\\{r=0\\}\right).$
(132)

A curvature is obtained owing to the structure equation of an abelian gauge-
group such that:

$\displaystyle\mathfrak{F}^{~{}}_{U_{\textit{1}}}:=d{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}=\left(\partial_{\circ}V(r)\right)d\xi^{\circ}{\wedge}dt=\frac{e^{i\pi\theta/2}}{r^{2}}\hskip
1.99997ptdt\wedge dr.$ (133)

Thus, the dual curvature is provided as

$\displaystyle\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle=\frac{e^{i\pi\theta/2}}{\sqrt{-e^{i\pi\theta}}}\frac{1}{r^{2}}(rd\theta_{1})\wedge(r\sin{\theta_{1}}d\theta_{2}).$
(134)

We note that

$\displaystyle\left.\frac{e^{i\pi\theta/2}}{\sqrt{-e^{i\pi\theta}}}\right|_{\theta\rightarrow+0}=\left.\frac{e^{i\pi\theta/2}}{\sqrt{-e^{i\pi\theta}}}\right|_{\theta\rightarrow
1}=i.$

We treat the $\theta$-metric space with $0\leq\theta\leq 1$; thus, we set
$e^{i\pi\theta/2}/\sqrt{-e^{i\pi\theta}}=i$, hereafter.

Tensor coefficient $\hat{\mathscr{F}}$ is defined in ${T^{*}{\mathcal{}M}}$
except $r=0$ and is static; thus, the domain of $\hat{\mathscr{F}}$ is
$\Sigma_{\hat{\mathscr{F}}}={T^{*}{\mathcal{}M}}{\setminus}\\{r=0\\}\cong\mathbb{R}\otimes(\mathbb{R}^{3}{\setminus}\\{r=0\\})$.
An equation of motion in the spinor-gauge bundle is provided as

$\displaystyle d\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}=i\hskip
1.00006ptd\left(\frac{1}{r^{2}}\hskip
1.99997pt(rd\theta_{1})\wedge(r\sin{\theta_{1}}d\theta_{2})\right)={c^{~{}}_{U_{\textit{1}}}}\hskip
1.00006pt{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mbox{\boldmath$\psi$}}}\hskip
1.99997pt\mathfrak{V}_{\circ}.$ (135)

This integration in three-dimensional volume $D^{3}$ is performed as

$\displaystyle\int_{D^{3}}d\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}=\int_{\partial
D^{3}=S^{2}}\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}=i\int_{S^{2}}\sin{\theta_{1}}\hskip
1.00006ptd\theta_{1}\wedge d\theta_{2}=4\pi i.$ (136)

The first equality is just a formal relation because
$d\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}=0$ in reality; this is special for
the $U(1)$-gauge bundle. We note that the covariant differential is just the
external derivative for an abelian gauge group; thus, the equation of motion
is equivalent to the Bianchi identity.

For the right-hand side of (135), we obtain a stationary electric-pole
solution at the origin. An integration is performed as

$\displaystyle{c^{~{}}_{U_{\textit{1}}}}\int_{D^{3}}{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{\circ}{{\mbox{\boldmath$\psi$}}}\hskip
1.99997pt\mathfrak{V}_{\circ}=\int_{0}^{1}dr\int_{0}^{\pi}rd\theta_{1}\int_{0}^{2\pi}r\sin{\theta_{1}}d\theta_{2}\hskip
3.00003ptj^{0}=4\pi{i},$

where $j^{a}$ is a static electron current given as

$\displaystyle~{}j^{0}=4\pi{i}\hskip
1.00006pt\delta^{3}({\mbox{\boldmath$r$}})=4\pi{i}\frac{\delta(r)\delta(\theta_{1})\delta(\theta_{2})}{r^{2}\sin{\theta_{1}}},$
$\displaystyle~{}j^{1}=j^{2}=j^{3}=0;$

thus, it is consistent with (136). A topological invariant of the electric-
pole current is provided as the Chern characteristic such that:

$\displaystyle ch\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)$
$\displaystyle=dim\left(U(1)\right)+c_{1}\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)+\frac{1}{2}\left(c_{1}\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)^{2}-2c^{~{}}_{2}\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)\right),$

where

$\displaystyle dim\left(U(1)\right)$ $\displaystyle=1,$ $\displaystyle
c_{1}\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)$
$\displaystyle=\frac{i}{2\pi}\int_{S^{2}}\textrm{Tr}_{U_{\textit{1}}}[\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}]=-2,$
$\displaystyle
c^{~{}}_{2}\left(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\right)$
$\displaystyle=\frac{1}{8\pi^{2}}\int_{D^{4}}\left(\textrm{Tr}_{U_{\textit{1}}}[\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}\wedge\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}]-\textrm{Tr}_{U_{\textit{1}}}[\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}]\wedge{\mathrm{Tr}}[\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}]\right)=0.$

We note that
$\textrm{Tr}_{U_{\textit{1}}}[\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}]=\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}$
for the $U(1)$ gauge group. Therefore, we obtain the Chern characteristic of
the electric pole as $ch(\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}})=1$ for any
$\theta\in[0,1]$.

Figure 3:  Arrows show three-dimensional vector representation of the dual
connection for the Coulomb type static potential. Vectors do not defined on a
$z$-axis in this example.

The dual-connection is provided as a solution of the structure equation for
the given dual-curvature such that:

$\displaystyle
d\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}=\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}}=\frac{i}{r^{2}}(rd\theta_{1})\wedge(r\sin{\theta_{1}}d\theta_{2}).$

This equation has a unique solution with boundary condition of
$\lim_{r\rightarrow\infty}\hat{\mathscr{A}}=0$ such as

$\displaystyle\hat{\mathfrak{A}}^{~{}}_{{U_{\textit{1}}}}=-\frac{i}{r}\cot{\theta_{1}}\hskip
1.99997pt(r\sin{\theta_{1}}d\theta_{2})=-i\cos{\theta_{1}}d\theta_{2},$

and it is represented using Cartesian coordinate $\xi^{a}=(t,x,y,z)$ as

$\displaystyle\hat{\mathfrak{A}}^{~{}}_{{U_{\textit{1}}}}=i\hskip
1.00006pt\frac{y\hskip 1.00006ptdx-x\hskip 1.00006ptdy}{x^{2}+y^{2}}\hskip
1.00006pt\frac{z}{r}\in\Omega^{1}\left({T^{*}{\mathcal{}M}}{\setminus}\\{x=0\wedge
y=0\\}\right).$

This solution is known as the Dirac monopole, and a one-dimensional manifold
subtracted from $\mathbb{R}^{3}$ is known as the Dirac string. Three-
dimensional vectors of connection
$({\mathscr{A}}_{x},{\mathscr{A}}_{y},{\mathscr{A}}_{z})$ at $t=0$ are
visualized in Figure 3. We note that the one-dimensional singularity along the
Dirac string is removable, and an essential singularity is a point at
$x=y=z=0$. In reality, the same solution is represented using a cylindrical
coordinate $\xi_{\textrm{cyl}}:=(t,\rho,\phi,z)$ as

$\displaystyle\hat{\mathfrak{A}}^{~{}}_{{U_{\textit{1}}}}=\frac{iz}{\sqrt{\rho^{2}+z^{2}}}\hskip
1.00006ptd\phi;~{}$ $\displaystyle\textrm{thus,}~{}~{}\lim_{\rho\rightarrow
0}\hat{\mathfrak{A}}^{~{}}_{{U_{\textit{1}}}}\big{|}_{z\neq
0}\rightarrow{i\hskip 1.00006pt\textrm{sign}(z)d\phi}.$

Before calculating the topological index of the dual spinor-gauge bundle, we
consider a relation between the spin-gauge connection and its dual; Figure 4
shows a schematic diagram of its structure. We start from gauge-connection
${\mathfrak{A}}^{~{}}_{SU}$; then, the structure equation concerning
${\mathfrak{A}}^{~{}}_{SU}$ gives gauge curvature $\mathfrak{F}^{~{}}_{SU}$.
The Hodge-dual of the gauge field (curvature) gives the equation of motion in
the spinor-gauge bundle such as the left equation in (129). We note that dual-
curvature $\hat{\mathfrak{F}}^{~{}}_{SU}$ is not the curvature corresponding
the spinor-gauge connection, and spinor filed $\psi$ interacts with gauge-
connection ${\mathfrak{A}}^{~{}}_{SU}$. When we start from the spinor current
$j(\psi)$, a solution of equations (129), (130), and the Bianchi identity
(128) provides a set of fields
$({\mathfrak{A}}^{~{}}_{SU},\hat{\mathfrak{F}}^{~{}}_{SU})$; then, the gauge
curvature $\mathfrak{F}^{~{}}_{SU}$ is provided as the Hodge-dual of
$\hat{\mathfrak{F}}^{~{}}_{SU}$ that fulfils the structure equation with dual
connection $\hat{\mathfrak{A}}^{~{}}_{SU}$. The equation of motion of the
(dual-)spinor field has a tensor-coefficient representation owing to the
(dual-)connection and (dual-)curvature, respectively, e.g., as shown in (131).
Dual connection $\hat{\mathfrak{A}}^{~{}}_{SU}$ is provided as a solution of
the structure equation owing to the given dual curvature
$\hat{\mathfrak{F}}^{~{}}_{SU}$, and the dual-spinor current
$\hat{j}(\hat{\psi})$ is given as a solution of the equation of motion and
interacts with the dual connection.

Figure 4:  A diagram shows a relation between spin-gauge bundle and its dual
bundle. Abbreviations “str. eq.” and “E.o.M.” mean the structure function and
the equation of motion, respectively.

In our case, a topological index of the Dirac monopole is provided owing to
gauge-connection ${\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$ given as (132). A
direct calculation of the first Chern class concerning
${\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$ results indeterminate integration;
thus, the indirect method is necessary. We note that
${\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$ is homotopically equivalent to
$\hat{\mathfrak{A}}^{~{}}_{U_{\textit{1}}}$. When two connection are embedded
in the five-dimensional smooth manifold such that:

$\displaystyle\overline{{\mathfrak{A}}}^{~{}}_{U_{\textit{1}}}$
$\displaystyle:=s\hskip
1.00006pt{{\mathfrak{A}}}^{~{}}_{U_{\textit{1}}}+(1-s)\hskip
1.00006pt{\hat{\mathfrak{A}}}^{~{}}_{U_{\textit{1}}}\in(\Sigma\amalg\hat{\Sigma})\otimes[0,1],$

where $\Sigma\cong\hat{\Sigma}\cong{S^{2}}$. Moreover,
$\overline{{\mathfrak{A}}}^{~{}}_{U_{\textit{1}}}$ is the gauge connection at
any $s\in[0,1]$ because the gauge group is abelian and the structure equation
is a linear equation. Therefore, two connections have the same Chern index as
$ch(\mathfrak{F}^{~{}}_{U_{\textit{1}}})=ch(\hat{\mathfrak{F}}^{~{}}_{U_{\textit{1}}})=1$.
It is true for any values of $\theta\in[0,1]$.

The stationary condition of the Yang–Mills action provides the equation of
motion (129); thus, solutions are SD or ASD when the gauge group is $SU(N)$,
and the metric is Euclidean. In this example, the curvature is not SD nor ASD
independent of the value of T because the gauge group is $U(1)$.

###  5.3 Index in gauge field

The four-dimensional Euclidean space-time has the SD- or ASD curvature as a
solution of the equation of motion obtained as the stationary point of the
Yang–Mills action as shown in Section 4.2.2. The instanton solution[46], is an
example of the (A)SD solution, and the instanton number is a topological
invariant in the four-dimensional Euclidean space-time. This section treats
the instanton solution in a flat space-time (${\mathfrak{w}}=0$) with a
$(4\times 4)$-matrix representation of the $\theta$-metric space. The
instanton number in the Minkowski space-time is also discussed in this
section.

Coupling constant ${c^{~{}}_{SU}}$ is set to unity for simplicity. The $SU(2)$
generator is extended to a four dimensional vector by adding the zeroth
component such that
${{\mbox{\boldmath$\tau$}}}\rightarrow{{\mbox{\boldmath$\tau$}}}_{4}:=({\mbox{\boldmath$1$}}_{2}/2,-i\tau_{1},-i\tau_{2},-i\tau_{3})$,
where $\tau_{i}=\sigma_{i}/2$ is the $SU(2)$ generator. A gauge-group element
at $\xi\in{\mathcal{}M}_{\theta}$ has a representation such that:

$\displaystyle g(\xi):=v(\xi)\hskip
1.00006pt{\mbox{\boldmath$\gamma$}}_{\theta}=\sum_{a=0}^{3}v^{a}(\xi)\gamma_{\theta}^{a},$

that yields $\textrm{det}[g(\xi)]=\left(\eta^{~{}}_{\theta\hskip
0.70004pt{\circ\circ}}\hskip 1.00006pt\xi^{\circ}\xi^{\circ}\right)^{2}$,
where $v^{\bullet}(\xi)$ is a position vector in ${T{\mathcal{}M}}_{\theta}$.
A generator of the Lorentz transformation in the $\theta$-metric space is
provided as
$\Xi:=C_{{\circ\circ}}\gamma^{\circ}_{\theta}\gamma^{\circ}_{\theta}$, where
$C_{\bullet\bullet}\in\mathbb{R}$ is a boost or rotation parameter. The group
element square is an invariant under the Lorentz transformation as

$\displaystyle g^{\prime}(\xi)g^{\prime}(\xi)$
$\displaystyle=\Xi^{-1}g(\xi)\Xi^{-1}\Xi
g(\xi)\Xi=\Xi^{-1}g(\xi)g(\xi)\Xi=g(\xi)g(\xi),$
$\displaystyle=\eta^{~{}}_{\theta\hskip 0.70004pt{\circ\circ}}\hskip
1.00006pt\xi^{\circ}\xi^{\circ}{\mbox{\boldmath$1$}}_{sp\hskip 0.70004pt}.$

An instanton connection at $\xi^{a}=(t,x,y,z)$ is provided as

$\displaystyle{\mathscr{A}}_{\theta}$
$\displaystyle=\sum_{I=1}^{4}{\mathscr{A}}_{\theta}^{I}\left(\begin{array}[]{cc}[{{\mbox{\boldmath$\tau$}}}_{4}]_{I}&0\\\
0&[{{\mbox{\boldmath$\tau$}}}_{4}]_{I}\end{array}\right)=\frac{r_{\theta}^{2}}{r_{\theta}^{2}-\lambda^{2}}g(\xi)dg^{-1}(\xi)),$

where

$\displaystyle
g(\xi):=\frac{1}{r^{~{}}_{\theta}}\sum_{a=0}^{3}\xi^{a}\gamma_{\theta}^{a},$
$\displaystyle~{}~{}\textrm{and}~{}~{}r_{\theta}^{2}:=\eta^{~{}}_{\theta\hskip
0.70004pt{\circ\circ}}\hskip
1.00006pt\xi^{\circ}\xi^{\circ}=e^{i\pi\theta}t^{2}-x^{2}-y^{2}-z^{2}.$

An instanton connection is provided using a chiral representation of
$\gamma^{i}_{\theta}$ as

$\displaystyle[{\mathscr{A}}_{\theta}]_{\circ}{\mathfrak{e}}_{\theta}^{\circ}$
$\displaystyle=\frac{1}{r^{2}_{\theta}-\lambda^{2}}\left(\begin{array}[]{cc}[{{\mbox{\boldmath$a$}}}_{\theta}]_{\circ}{\mathfrak{e}}_{\theta}^{\circ}&0\\\
0&[\tilde{{\mbox{\boldmath$a$}}}_{\theta}]_{\circ}{\mathfrak{e}}_{\theta}^{\circ}\end{array}\right),$

where

$\displaystyle~{}{\mathfrak{e}}^{a}_{\theta}:=(dt_{\theta},dx,dy,dz),~{}~{}~{}t_{\theta}:=ie^{i\pi\theta/2}\hskip
1.00006ptt,$
$\displaystyle~{}\begin{array}[]{lcclcc}[{{\mbox{\boldmath$a$}}}_{\theta}]_{0}&=i\left(\begin{array}[]{cc}z&x-iy\\\
x+iy&-z\end{array}\right),&[\tilde{{\mbox{\boldmath$a$}}}_{\theta}]_{0}&=-i\left(\begin{array}[]{cc}z&x-iy\\\
x+iy&-z\end{array}\right),\\\
{[{{\mbox{\boldmath$a$}}}_{\theta}]}_{1}&=\left(\begin{array}[]{cc}-iy&-it_{\theta}+z\\\
-it_{\theta}-z&iy\end{array}\right),&[\tilde{{\mbox{\boldmath$a$}}}_{\theta}]_{1}&=\left(\begin{array}[]{cc}-iy&it_{\theta}+z\\\
it_{\theta}-z&iy\end{array}\right),\\\
{[{{\mbox{\boldmath$a$}}}_{\theta}]}_{2}&=i\left(\begin{array}[]{cc}x&it_{\theta}-z\\\
-it_{\theta}-z&-x\end{array}\right),&[\tilde{{\mbox{\boldmath$a$}}}_{\theta}]_{2}&=i\left(\begin{array}[]{cc}x&-it_{\theta}-z\\\
it_{\theta}-z&-x\end{array}\right),\\\
{[{{\mbox{\boldmath$a$}}}_{\theta}]}_{3}&=\left(\begin{array}[]{cc}-it_{\theta}&-x+iy\\\
x+iy&it_{\theta}\end{array}\right),&[\tilde{{\mbox{\boldmath$a$}}}_{\theta}]_{3}&=\left(\begin{array}[]{cc}it_{\theta}&-x+iy\\\
x+iy&-it_{\theta}\end{array}\right).\end{array}$

The instanton has two sets of $SU(2)$ connections according to
${{\mbox{\boldmath$a$}}}_{\theta}$ and
$\tilde{{\mbox{\boldmath$a$}}}_{\theta}$ such that:

${\mathfrak{A}}_{\theta}^{I}=\left[{\mathscr{A}}_{\theta}^{I}\right]_{\circ}dx^{\circ}=\left(\frac{2}{r_{\theta}^{2}-\lambda^{2}}\right)\times\left\\{\begin{array}[]{cl}0,&(I=0),\\\
-x\hskip 1.00006ptdt_{\theta}+t_{\theta}\hskip 1.00006ptdx+z\hskip
1.00006ptdy-y\hskip 1.00006ptdz,&(I=1),\\\ -y\hskip
1.00006ptdt_{\theta}-z\hskip 1.00006ptdx+t_{\theta}\hskip 1.00006ptdy+x\hskip
1.00006ptdz,&(I=2),\\\ -z\hskip 1.00006ptdt_{\theta}+y\hskip
1.00006ptdx-x\hskip 1.00006ptdy+t_{\theta}\hskip
1.00006ptdz,&(I=3),\end{array}\right.$

and

$\tilde{\mathfrak{A}}_{\theta}^{I}=\left[\tilde{\mathscr{A}}_{\theta}^{I}\right]_{\circ}dx^{\circ}=\left(\frac{2}{r_{\theta}^{2}-\lambda^{2}}\right)\times\left\\{\begin{array}[]{cl}0,&(I=0),\\\
x\hskip 1.00006ptdt_{\theta}-t_{\theta}\hskip 1.00006ptdx+z\hskip
1.00006ptdy-y\hskip 1.00006ptdz,&(I=1),\\\ y\hskip
1.00006ptdt_{\theta}-z\hskip 1.00006ptdx-t_{\theta}\hskip 1.00006ptdy+x\hskip
1.00006ptdz,&(I=2),\\\ z\hskip 1.00006ptdt_{\theta}+y\hskip
1.00006ptdx-x\hskip 1.00006ptdy-t_{\theta}\hskip
1.00006ptdz,&(I=3),\end{array}\right.$

The instanton connection in the Euclidean space is provided as
${\mathscr{A}}_{E}:={\mathscr{A}}_{\theta}\big{|}_{\theta\rightarrow 1}$.

Instanton curvatures, $\mathfrak{F}_{\theta}$ and
$\tilde{\mathfrak{F}}_{\theta}$, are induced owing to the structure equation
with ${\mathfrak{A}}_{\theta}$ and $\tilde{\mathfrak{A}}_{\theta}$,
respectively; Euclidean and Minkowskian curvatures are provided from instanton
curvature in the $\theta$-metric space such as

$\displaystyle\mathfrak{F}^{+}_{E}$
$\displaystyle=\mathfrak{F}^{~{}}_{\theta}\big{|}_{\theta\rightarrow 1},\hskip
13.00005pt\mathfrak{F}^{-}_{E}~{}=~{}\tilde{\mathfrak{F}}_{\theta}\big{|}_{\theta\rightarrow
1},$ $\displaystyle\mathfrak{F}^{+}_{M}$
$\displaystyle=\mathfrak{F}^{~{}}_{\theta}\big{|}_{\theta\rightarrow+0},\hskip
5.0pt\mathfrak{F}^{-}_{M}~{}=~{}\tilde{\mathfrak{F}}^{~{}}_{\theta}\big{|}_{\theta\rightarrow+0}.$

They yield $\hat{\mathfrak{F}}^{\pm}_{E}=\pm\mathfrak{F}^{\pm}_{E}$ and
$\hat{\mathfrak{F}}^{\pm}_{M}=\pm\mathfrak{F}^{\pm}_{M}$; thus, instanton
solutions in the $\theta$-metric space provide the SD and ASD solutions
simultaneously for the both Euclidean and Minkowskian limit. Moreover, it is
true for any $\theta\in[0,1]$ that

$\displaystyle{\widehat{\mathrm{H}}}(\mathfrak{F}^{~{}}_{\theta})=+\mathfrak{F}^{~{}}_{\theta}~{}~{}\textrm{and}~{}~{}{\widehat{\mathrm{H}}}(\tilde{\mathfrak{F}}^{~{}}_{\theta})=-\tilde{\mathfrak{F}}^{~{}}_{\theta}.$
(137)

For the Euclidean-metric space, according to a sign of the second Chern class,
the SD- or ASD curvature gives a solution of the Yang–Mills equation as shown
in Remark 4.5. On the other hand, in the Minkowski space-time, the SD- and ASD
curvature do not always provide the extremal of the Yang–Mills action. We
confirm that instanton curvature ${\mathfrak{F}}^{\pm}_{M}$ in the Minkowski
space-time is a solution of the Yang–Mills equation: The Yang–Mills equation
in the $\theta$-metric space has a tensor component representation as (131).
For the SD curvature, the equation can be written using a half of components
as

$\displaystyle[\textrm{Y-M eqs}]_{\theta}:=$
$\displaystyle\left\\{\begin{array}[]{rcl}\left(\nabla_{1;I}\mathscr{F}^{I}_{01}+\nabla_{2;I}\mathscr{F}^{I}_{02}+\nabla_{3;I}\mathscr{F}^{I}_{03}\right){{\sigma^{~{}}_{\theta}}}^{-\frac{1}{2}}dt&=&{\mbox{\boldmath$\bar{\hat{\psi}}$}}\left(e^{i\pi\theta/2}\gamma_{sp\hskip
0.70004pt}^{0}\right){{\mbox{\boldmath$\psi$}}}\hskip
1.00006pt{{\sigma^{~{}}_{\theta}}}^{-\frac{1}{2}}\hskip 1.99997ptdt,\\\
\left(-{{\sigma^{~{}}_{\theta}}}^{-\frac{1}{2}}\hskip
1.00006pt\nabla_{0;I}\mathscr{F}^{I}_{01}-\nabla_{3;I}\mathscr{F}^{I}_{02}+\nabla_{2;I}\mathscr{F}^{I}_{03}\right)dx&=&{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{1}{{\mbox{\boldmath$\psi$}}}\hskip 1.99997ptdx,\\\
\left(\nabla_{3;I}\mathscr{F}^{I}_{01}-{{\sigma^{~{}}_{\theta}}}^{-\frac{1}{2}}\hskip
1.00006pt\nabla_{0;I}\mathscr{F}^{I}_{02}-\nabla_{1;I}\mathscr{F}^{I}_{03}\right)dy&=&{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{2}{{\mbox{\boldmath$\psi$}}}\hskip 1.99997ptdy,\\\
\left(-\nabla_{2;I}\mathscr{F}^{I}_{01}+\nabla_{1;I}\mathscr{F}^{I}_{02}-{{\sigma^{~{}}_{\theta}}}^{-\frac{1}{2}}\hskip
1.00006pt\nabla_{0;I}\mathscr{F}^{I}_{03}\right)dz&=&{\mbox{\boldmath$\bar{\hat{\psi}}$}}\gamma_{sp\hskip
0.70004pt}^{3}{{\mbox{\boldmath$\psi$}}}\hskip 1.99997ptdz,\end{array}\right.$

where

$\displaystyle\nabla_{a;J}:=\left(\delta^{I}_{J}\partial_{a}+f^{I}_{~{}JK}[{\mathscr{A}}^{~{}}_{\theta}]^{K}_{a}\right)\tau^{~{}}_{I}.$

The SD Yang–Mills equations in the Euclidean- and Lorentzian metrics are
provided as

$\displaystyle[\textrm{Y-M eqs}]_{E}:=\lim_{\theta\rightarrow 1-0}[\textrm{Y-M
eqs}]_{\theta},~{}~{}$ $\displaystyle\textrm{and}~{}~{}[\textrm{Y-M
eqs}]_{M}:=\lim_{\theta\rightarrow+0}[\textrm{Y-M eqs}]_{\theta}.$

Accordingly, equations have a simple relation to each other such that:

$\displaystyle t\mapsto-it:[\textrm{Y-M eqs}]_{E}\mapsto[\textrm{Y-M
eqs}]_{M};$

thus, the connection with the Lorentzian metrics is provided from the
Euclidean metric as

$\displaystyle t\mapsto-
it:{\mathfrak{A}}_{E}^{I}\mapsto{\mathfrak{A}}_{M}^{I}.$

Therefore, the Lorentzian metric’s SD connection is a solution to the Yang-
Mills equation in the Minkowski space-time. It is also true for the ASD
connection.

${\mathcal{}M}_{1}$ is constructed owing to the one-point compactification of
${\mathcal{}M}_{E}$ as follows: For Euclidean manifold ${\mathcal{}M}_{E}$,
variable transformations $r_{E}\mapsto\lambda\rho$ for $0<r_{E}\leq\lambda$
and $r_{E}\mapsto\lambda/\rho$ for $\lambda<r_{E}<\infty$ are applied; thus,
two compact manifolds $D^{4}$ are obtained, namely $D^{4}_{\textrm{in}}$ and
$D^{4}_{\textrm{out}}$, respectively. Then, surfaces of $D^{4}_{\textrm{in}}$
and $D^{4}_{\textrm{out}}$ are equated to each other with keeping three
angular variables; thus, ${\mathcal{}M}_{E}$ is compacitified into union of
two compact manifolds such that
${\mathcal{}M}_{E}\mapsto{\mathcal{}M}_{1}=D_{\textrm{in}}^{4}\cup
D_{\textrm{out}}^{4}$ with $D^{4}_{\textrm{in}}\cap
D^{4}_{\textrm{out}}=\partial{D^{4}_{\textrm{in}}}=-\partial{D^{4}_{\textrm{out}}}=S^{3}$
and $\partial(D^{4}_{\textrm{in}}{\cup}D^{4}_{\textrm{out}})=\emptyset$. The
first Pontrjagin number is obtained as an integration of the Pontrjagin class
in compactified volumes such that:

$\displaystyle
p_{1}({\mathcal{}M}_{1})=-\left(\int_{D^{4}_{\textrm{in}}}\left.\frac{{\mathrm{Tr}}_{SU}\left[\mathfrak{F}^{~{}}_{1}\wedge\mathfrak{F}^{~{}}_{1}\right]}{8\pi^{2}}\right|_{r_{E}\rightarrow\lambda\rho}+\int_{D^{4}_{\textrm{out}}}\left.\frac{{\mathrm{Tr}}_{SU}\left[\mathfrak{F}^{~{}}_{1}\wedge\mathfrak{F}^{~{}}_{1}\right]}{8\pi^{2}}\right|_{r_{E}\rightarrow\lambda/\rho}\right),$

where

$\displaystyle\mathfrak{F}^{~{}}_{1}=\mathfrak{F}^{~{}}_{E}$
$\displaystyle=:\left.\frac{1}{2}\left[\mathscr{F}_{E}^{~{}}\right]_{{\circ\circ}}{\mathfrak{e}}_{\theta}^{\circ}\wedge{\mathfrak{e}}_{\theta}^{\circ}\right|_{\theta\rightarrow
1}.$

A trace of an instanton curvature squared is obtained such that:

$\displaystyle{\mathrm{Tr}}_{SU}\left[\mathfrak{F}^{~{}}_{1}\wedge\mathfrak{F}^{~{}}_{1}\right]$
$\displaystyle=\sum_{I=1}^{4}\epsilon_{E}^{{\circ\circ}{\circ\circ}}\frac{1}{2}\left[\mathscr{F}_{E}^{I}\right]_{{\circ\circ}}\frac{1}{2}\left[\mathscr{F}_{E}^{I}\right]_{{\circ\circ}}\hskip
1.99997pt{\mathrm{Tr}}_{2\times
2}\left[\left[\textbf{t}_{4}\right]_{I}.\left[\textbf{t}_{4}\right]_{I}\right]{\widehat{\mathrm{H}}}(1),$
(138)
$\displaystyle=-48\left(\frac{\lambda}{r_{E}^{2}+\lambda^{2}}\right)^{4}{\widehat{\mathrm{H}}}(1),$

thus, $p_{1}({\mathcal{}M}_{1})$ is calculated as

$\displaystyle p_{1}({\mathcal{}M}_{1})$
$\displaystyle=\frac{6}{\pi^{2}}\int_{0}^{1}\frac{\rho^{3}}{(\rho^{2}+1)^{4}}d\rho\hskip
1.99997pt\int_{S^{3}}d\Omega_{4}~{}=~{}1,$

where $d\Omega_{4}$ is an integration measure for an angular integration on
$S^{3}$, which gives $\int_{S^{3}}d\Omega_{4}=2\pi^{2}$. The completely anti-
symmetric tensor in the Euclidean space is provided as
$[{\mbox{\boldmath$\epsilon$}}_{E}]^{0123}=[{\mbox{\boldmath$\epsilon$}}_{E}]_{0123}=1$.

On intersection $D^{4}_{\textrm{in}}\cap D^{4}_{\textrm{out}}=S^{3}$, angular
variables coincide as

$\displaystyle\partial
D_{\textrm{in}}\ni\theta_{i}^{\textrm{in}}=\theta_{i}^{\textrm{out}}\in-\partial
D_{\textrm{out}},$

for $i=1,2,3$ with keeping an orientation on their surfaces. When
$\theta_{3}^{\textrm{in}}$ maps to $\theta_{3}^{\textrm{out}}$ as
$n\times\hskip 1.99997pt\theta_{3}^{\textrm{in}}=\theta_{3}^{\textrm{out}}$
with $0<n\in\mathbb{Z}$, an instanton number is given as
$p_{1}({\mathcal{}M}^{\mathrm{c}}_{1})=n$ due to the $\theta_{3}$ integration
from $0$ to $2n\pi$, which corresponds to the winding number around infinity.

The first Pontrjagin number is provided in the entire space of
$W_{\theta}:={{\mathcal{}M}_{\theta}}\otimes(\theta\in[0,1])$. After the same
compactification as in a Euclidean space in $W_{\theta}$, a four-dimensional
polar coordinate is introduced as

$\displaystyle\begin{array}[]{ll}\hskip
1.99997ptt:=\lambda\rho\cos{\theta_{1}},&x:=\lambda\rho\sin{\theta_{1}}\cos{\theta_{2}},\\\
y:=\lambda\rho\sin{\theta_{1}}\sin{\theta_{2}}\cos{\theta_{3}},&z:=\lambda\rho\sin{\theta_{1}}\sin{\theta_{2}}\sin{\theta_{3}}.\end{array}$

The first Pontrjagin number is provided by integrations such that:

$\displaystyle p_{1}({\mathcal{}M}_{\theta})$
$\displaystyle=-i\frac{6}{\pi^{2}}e^{i\pi\theta/2}\int_{0}^{1}d\rho\int_{0}^{\pi}\rho^{2}\sin^{2}{\theta_{1}}\hskip
1.00006ptd\theta_{1}\int_{0}^{\pi}\rho\sin{\theta_{2}}\hskip
1.99997ptd\theta_{2}\int_{0}^{2\pi}d\theta_{3}$
$\displaystyle\left[\left\\{1+\rho^{2}\left(1-\left(1+e^{i\pi\theta}\right)\cos^{2}{\theta_{1}}\right)\right\\}^{-4}+\left\\{\left(1+e^{i\pi\theta}\right)\cos^{2}{\theta_{1}}-\left(1+\rho^{2}\right)\right\\}^{-4}\right],$
$\displaystyle=1,$

for $0<\theta\leq 1$. We note that $\epsilon_{E}^{{\circ\circ}{\circ\circ}}$
in (138) is replaced by ${{\sigma^{~{}}_{\theta}}}\hskip
1.00006pt\epsilon_{\theta}^{{\circ\circ}{\circ\circ}}$ in the $\theta$-metric
space. The Pontrjagin class with the Minkowski metric is a divergent integral;
thus, it is not well-defined. A domain of the instanton solution in the
Lorentz bundle with the Lorentzian metric is not the entire space of
${\mathcal{}M}_{0}$ but is
${\mathcal{}M}_{0}\setminus{\mathcal{}M}_{\lambda}$, where
${\mathcal{}M}_{\lambda}:=\\{\xi\in{\mathcal{}M}_{0}\mid{\eta_{\circ\circ}\xi^{\circ}\xi^{\circ}-\lambda^{2}=0}\\}$;
thus, manifold ${\mathcal{}M}_{0}$ is not closed as a principal bundle with
the instanton connection. Taking the closure of manifold ${\mathcal{}M}_{0}$
with respect to $\theta$ from the right ($\theta=+0$), ${\mathcal{}M}_{0}$ has
the first Pontrjagin number as
$p_{1}({\mathcal{}M}_{\theta})|_{\theta\rightarrow+0}=1$. The instanton
solution in the Lorentzian metric is an example of Theorem 2.18: manifolds
${\mathcal{}M}_{0}$ and ${\mathcal{}M}_{1}$ are cobordant to each other, and
the first Pontrjagin number of ${\mathcal{}M}_{0}$ is consistently defined
owing to that of ${\mathcal{}M}_{1}$.

The SD- and ASD-instanton-curvatures fulfil the Bianchi identity as

$\displaystyle
d\mathfrak{F}_{\theta}^{\pm}+\left[\mathfrak{F}_{\theta}^{\pm},{\mathfrak{A}}_{\theta}^{\pm}\right]_{\wedge}=0.$

When we take the SD- or ASD solution as the principal curvature of the gauge
bundle, we have

$\displaystyle\mathfrak{F}_{SU}^{~{}}=\mathfrak{F}_{\theta}^{\pm}=\pm{\widehat{\mathrm{H}}}(\mathfrak{F}_{\theta}^{\pm})=\pm\hat{\mathfrak{F}}_{SU}^{~{}};$

thus, the equation of motion is provided from (120) as

$\displaystyle{d}\hat{\mathfrak{F}}_{SU}^{~{}}+\left[\hat{\mathfrak{F}}_{SU}^{~{}},{\mathfrak{A}}_{SU}^{~{}}\right]_{\wedge}=\pm{d}\mathfrak{F}_{SU}^{~{}}\pm\left[\mathfrak{F}_{SU}^{~{}},{\mathfrak{A}}_{SU}^{~{}}\right]_{\wedge}=\pm\bar{\psi}_{\textrm{inst}}\hskip
1.00006pt\gamma_{sp\hskip 0.70004pt}^{\circ}\psi_{\textrm{inst}}\hskip
1.99997pt\mathfrak{V}_{\circ}=0,$

where $\psi_{\textrm{inst}}$ is an instanton spinor-field. Last equality is
due to the Bianchi identity. This is true for any $\theta\in(0,1]$. Therefore,
the source field does not exist in a domain of the instanton curvature.

Figure 5:  This figure shows regions where $\|\mathfrak{F}^{~{}}_{SU}\|>1$ for
the Euclidean metric (a circle at the origin) and for the Lorentzian metric (a
mashed region along diagonal lines). Here, we set $\lambda=1$ in
$\mathfrak{F}^{~{}}_{SU}$.

A physical interpretation of an instanton in the Minkowski-bundle is as
follows: The SD- and ASD-connections domain with the Lorentzian metric is
${\mathcal{}M}_{0}\setminus{\mathcal{}M}_{\lambda}$; thus the domain of
$\psi_{\textrm{inst}}$ is ${\mathcal{}M}_{\lambda}$, which is on the light-
cone at far from the origin, as shown in Figure 5. After quantization of
spinor field $\psi_{\textrm{inst}}$, a quantity

$\displaystyle
p_{\textrm{inst}}\left(\Sigma_{3}\right):=\int_{\Sigma_{3}}\bar{\psi}_{\textrm{inst}}\hskip
1.00006pt\gamma_{sp\hskip 0.70004pt}^{\circ}\psi_{\textrm{inst}}\hskip
1.99997pt\mathfrak{V}_{\circ}$

may give a provability to observe an instanton in three-dimensional volume
$\Sigma_{3}\in{\mathcal{}M}_{\lambda}$ after appropriate normalization; thus,
an instanton is localized in the light-cone and is interpreted as a massless
particle with an $SU(2)$-charge. Conservation of the Lorentzian-instanton
number is ensured as the topological invariant (the first Pontrjagin number)
owing to cobordism.

###  5.4 Index in chiral representation

This section discusses the topological index of the Yang–Mills bundle using a
new $2N$-dimensional vector representation, namely a chiral representation. A
spinor-gauge bundle has a $\mathbb{Z}_{2}$-grading total space as
${\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes N}=({{\mathcal{}S}_{sp\hskip
0.70004pt}^{+}})^{\otimes N}\oplus({{\mathcal{}S}_{sp\hskip
0.70004pt}^{-}})^{\otimes N}$ owing to projection operator $P^{\pm}_{sp\hskip
0.70004pt}$ as discussed in section 3.2.1. The chiral representation of
spinors and covariant differential operator are introduced as

$\displaystyle{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{ch}$
$\displaystyle:=\gamma_{\textrm{Y}\textrm{M}}\hskip
1.00006pt{\not\\!d}_{sg}\left(\begin{array}[]{cc}P^{+}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}&0\\\ 0&P^{-}_{sp\hskip
0.70004pt}\otimes{\mbox{\boldmath$1$}}_{SU}\end{array}\right)=\left(\begin{array}[]{cc}0&{\not\\!d^{-}_{{sg}}}\otimes{\mbox{\boldmath$1$}}_{SU}\\\
{\not\\!d^{+}_{{sg}}}\otimes{\mbox{\boldmath$1$}}_{SU}&0\end{array}\right),$
$\displaystyle{\mbox{\boldmath$\Psi$}}_{ch}$
$\displaystyle:=\left(\begin{array}[]{cc}{\mbox{\boldmath$\psi$}}^{+}\\\
{\mbox{\boldmath$\psi$}}^{-}\end{array}\right),$

where
${\not\\!d^{\pm}_{{sg}}}:={\not\\!d^{~{}}_{{sg}}}{{\circ}}P^{\pm}_{sp\hskip
0.70004pt}$. A Dirac conjugate of ${\mbox{\boldmath$\Psi$}}_{ch}$ is provided
as

$\displaystyle{\mbox{\boldmath$\overline{\Psi}$}}_{ch}:={\mbox{\boldmath$\Psi$}}^{\dagger}_{ch}\gamma^{0}{\mbox{\boldmath$\gamma$}}_{{\textrm{Y}\textrm{M}}}=\left({\mbox{\boldmath$\bar{\hat{\psi}}$}}^{-},{\mbox{\boldmath$\bar{\hat{\psi}}$}}^{+}\right).$

In this representation, the Lagrangian of the spinor field is provided as

$\displaystyle{\mathfrak{L}}_{ch}:=-\left(\hskip
1.00006pt\overline{{\mbox{\boldmath$\Psi$}}}_{ch}\hskip
1.99997pti{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{ch}{\mbox{\boldmath$\Psi$}}_{ch}+\frac{\mu}{\hbar}\hskip
1.99997pt\overline{{\mbox{\boldmath$\Psi$}}}_{ch}{\mbox{\boldmath$\Psi$}}_{ch}\right){\widehat{\mathrm{H}}}(1),$

which provides the same equation of motion as those from the matter-field
Lagrangian in the Yang–Mills bundle.

A kinetic term of the matter Lagrangian denoted by $K_{sg}:=i\bar{\psi}\hskip
1.00006pt{\not\\!d}\psi$ is Hermitian self-conjugate. For the Dirac operator
given in (94), its Hermitian conjugate is provided that

$\displaystyle K_{sg}^{\dagger}$ $\displaystyle=\left(i\hskip
1.00006pt\bar{\psi}(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d)\psi\right)^{\dagger}=-i\hskip
1.00006pt\left(\partial_{\star}\psi^{\dagger}\right)(\gamma_{sp\hskip
0.70004pt}^{\star})^{\dagger}(\gamma_{sp\hskip
0.70004pt}^{0})^{\dagger}\psi=-i\hskip
1.00006pt\left(\partial_{\star}\bar{\psi}\right)\gamma_{sp\hskip
0.70004pt}^{\star}\psi,$ (139) $\displaystyle=i\hskip
1.00006pt\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{\star}(\partial_{\star}\psi)-i\hskip
1.00006pt\left(\partial_{\star}\bar{\psi}\gamma_{sp\hskip
0.70004pt}^{\star}\psi\right),$

and the last term is vanished owing to the current conservation; thus,
$K_{sg}$ is a Hermitian self-conjugate as $K_{sg}=K_{sg}^{\dagger}$. Moreover,
$\mathrm{Ker}({\not\\!d^{-}_{{sg}}})^{\dagger}\cong\mathrm{Coker}({\not\\!d^{+}_{{sg}}})={\mathcal{}S}^{-}/\mathrm{Im}({\not\\!\mathcal{}D}^{+})$
is provided owing to $(\ref{Ksp})$ and

$\displaystyle i\hskip 1.00006pt\bar{\psi}^{+}(\iota_{{\gamma}_{sp\hskip
0.50003pt}}d)\psi^{-}=-i\hskip
1.00006pt\left(\partial_{\star}\bar{\psi}^{+}\right)\gamma_{sp\hskip
0.70004pt}^{\star}\psi^{-}=-i\hskip
1.00006pt\bar{\psi}^{+}(\overleftarrow{\iota_{{\gamma}_{sp\hskip
0.50003pt}}d})\psi^{-},$

where $(\overleftarrow{\bullet})$ is a differential operator acting from the
left. Operator ${\not\\!d^{~{}}_{{sg}}}$ is the Dirac operator; thus the
Atyah–Singer index theorem (Remark 2.14) is maintained in the spinor-gauge
bundle after appropriate compactification of base manifold ${\mathcal{}M}_{E}$
with the Euclidean metric.

The topological index with the Lorentzian metric is defined owing to cobordism
through the $\theta$-metric space. The Dirac equation is provided in the
$\theta$-metric space as

$\displaystyle{{\not\\!\mathcal{}D}}_{\theta}{\mbox{\boldmath$\psi$}}$
$\displaystyle:=\iota_{{\mbox{\boldmath$\gamma$}}_{\theta}}(d+i{c^{~{}}_{gr}}\hskip
1.00006pt{{\mathfrak{w}}\cdot{\mathscr{S}}}_{\theta}){\mbox{\boldmath$\psi$}}\otimes{\mbox{\boldmath$1$}}_{SU}-{i}{c^{~{}}_{SU}}\hskip
1.00006pt\iota_{{\mbox{\boldmath$\gamma$}}_{\theta}}{\mathfrak{A}}^{~{}}_{SU}\hskip
1.99997pt{\mbox{\boldmath$\psi$}},$

where

$\displaystyle{{\mathfrak{w}}\cdot{\mathscr{S}}}_{\theta}:=\frac{i}{2}\eta^{~{}}_{\theta\hskip
0.70004pt{\circ\circ}}\eta^{~{}}_{\theta\hskip
0.70004pt{\star\star}}{\mathcal{E}}^{\mu}_{a}\omega^{~{}{\circ}\star}_{\mu}\hskip
1.00006pt{{\mathscr{S}}}^{{\circ}\star}_{\theta}~{}~{}\textrm{and}~{}~{}{{\mathscr{S}}}_{\theta}^{ab}:=\frac{i}{4}[\gamma_{\theta}^{a},\gamma_{\theta}^{b}].$

Spinor bundle is formally extended to the $\theta$-metric space as
$\left({\mathcal{}S}^{\theta}_{sp\hskip 0.70004pt},\pi^{\theta}_{sp\hskip
0.70004pt},{\mathcal{}M}_{\theta},Spin_{\theta}\right)$, where
${\mathcal{}M}_{\theta}$ is a Riemannian manifold with metric tensor
${\mbox{\boldmath$\eta$}}^{~{}}_{\theta}$ defined in (16) and
${\mathcal{}S}^{\theta}_{sp\hskip 0.70004pt}$ is a Clifford module with the
Clifford algebra ${\mbox{\boldmath$\gamma$}}^{~{}}_{\theta}$ defined in (85).
At $\theta=+0$ and $\theta=1$, the $\theta$-metric gives Minkowski and Euclid
spinor bundles with $Spin_{\theta}|_{\theta\rightarrow+0}=Spin(1,3)$ and
$Spin_{\theta}|_{\theta\rightarrow 1}=Spin(4)$, respectively. Dirac spinor
$\psi^{~{}}_{\theta}$ is presented as a section in the $\theta$-metric spinor
bundle; thus, the chiral representation introduced above is extended in the
$\theta$-metric spinor bundle.

First, the topological index of the Euclidean space-time is considered in the
$\theta$ metric space at $\theta=1$. Dirac operator
${\mbox{\boldmath${\not\\!\mathcal{}D}$}}^{E}_{ch}:={\mbox{\boldmath${\not\\!\mathcal{}D}$}}^{\theta}_{ch}|_{\theta\rightarrow
1}$ has the topological index introduced in section 2.5.1, where
${\mbox{\boldmath${\not\\!\mathcal{}D}$}}^{\theta}_{ch}$ is a Dirac operator
in the $\theta$-metric extended chiral space. Euclidean base-manifold
${\mathcal{}M}_{E}$ is assumed to be compactified appropriately. Owing to the
McKean–Singer theorem (given as Remark 2.13 in section 2.5.1), the index of
Dirac operator ${\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{ch}^{E}$ is provided
using the super trace such that:

$\displaystyle
ind\left(i{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{ch}^{E}\right)$
$\displaystyle=Str\left(e^{-t\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}}\right)={\mathrm{Tr}}\left[\Gamma_{E}\hskip
1.99997pte^{-t\hskip 1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}}\right],$
$\displaystyle={\mathrm{Tr}}\left[e^{-t\hskip
1.39998pt{{{\not\\!\mathcal{}D}}_{sg}^{E}}^{-}\hskip
1.39998pt{{{\not\\!\mathcal{}D}}_{sg}^{E}}^{+}}\right]-{\mathrm{Tr}}\left[e^{-t\hskip
1.39998pt{{{\not\\!\mathcal{}D}}_{sg}^{E}}^{+}\hskip
1.39998pt{{{\not\\!\mathcal{}D}}_{sg}^{E}}^{-}}\right],$
$\displaystyle=\int_{{\mathcal{}M}_{E}}\hat{A}(\mathfrak{R})ch(\hat{\mathfrak{F}}^{~{}}_{SU}).$

where ${{\not\\!\mathcal{}D}}_{sg}^{E}$ is the Dirac operator in the spinor-
gauge bundle in the Euclidean space-time and
${{{\not\\!\mathcal{}D}}_{sg}^{E}}^{\pm}$ are its projections owing to the
projection operators $P^{\pm}_{E}$, that is introduced in section 2.6.
Operator ${{\not\\!\mathcal{}D}}_{sg}^{E}\hskip
1.99997pt{{\not\\!\mathcal{}D}}_{sg}^{E}$ is an elliptic type; thus,
$e^{-t\hskip 1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}^{E}}$ is a heat kernel. The last equality
is owing to the Atiyah–Singer index theorem given as Remark 2.14 in section
2.5.1.

Second, we discuss the topological index with the Lorentzian metric, which is
not well-defined because operator ${\not\\!\mathcal{}D}_{sg}\hskip
1.99997pt{\not\\!\mathcal{}D}_{sg}$ is a hyperbolic type, and the
Atiyah–Singer index theorem is not fulfilled. Although the index theorem is
not applied as its original shape, the topological index is defined in the
Minkowski space-time as stated as Theorem 2.18. Minkowski manifold
${\mathcal{}M}_{M}:=\left.{\mathcal{}M}_{\theta}\right|_{\theta\rightarrow+0}$
and Euclidean manifold
${\mathcal{}M}_{E}:=\left.{\mathcal{}M}_{\theta}\right|_{\theta\rightarrow 1}$
are cobordant to each other with respect to manifold
$W_{\theta}:={\mathcal{}M}_{\theta}\otimes(\theta\in[0,1])$. Therefore, the
topological index in ${\mathcal{}M}_{M}$ is well-defined as

$\displaystyle ind\left(i{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{sp\hskip
0.70004pt}\right):=ind\left(i{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{ch}^{E}\right)=\int_{{\mathcal{}M}_{E}}\hat{A}(\mathfrak{R})ch(\hat{\mathfrak{F}}^{~{}}_{SU})\Rightarrow\int_{{\mathcal{}M}_{E}}ch(\hat{\mathfrak{F}}^{~{}}_{SU}),$

owing to Theorem 2.18. We note that the second Chern class is zero for know
solutions of the Einstein equation; thus, the index is given only by curvature
$\hat{\mathfrak{F}}^{~{}}_{SU}$. The index in the Minkowski space-time is
formally denoted as

$\displaystyle ind\left(i{\mbox{\boldmath${\not\\!\mathcal{}D}$}}_{sp\hskip
0.70004pt}\right)$ $\displaystyle={\mathrm{Tr}}\left[\Gamma_{sp\hskip
0.70004pt}\hskip 1.99997pte^{-t\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}\hskip
1.39998pt{{\not\\!\mathcal{}D}}_{sg}}\right],$

where $\Gamma_{sp\hskip 0.70004pt}=\gamma^{5}$ in the standard notation, and
it is equivalent to the standard representation of the axial anomaly[13, 24].

##  6 Summary

bundle name | bundle | conn. | curv. | sect. | physical object
---|---|---|---|---|---
co-Poincaré | $\left({\mathcal{}M},\pi_{\mathrm{L}},\mathscr{M},G_{c\mathrm{P}}\right)$ | ${\mathfrak{w}}~{}~{}$ | $\mathfrak{R}~{}~{}$ | ${\mathfrak{e}}$ | gravity
Spinor | $\left({\mathcal{}S}_{sp\hskip 0.70004pt},\pi_{sp\hskip 0.70004pt},{\mathcal{}M},Spin(1,3)\right)$ | ${\mathfrak{A}}^{~{}}_{sp\hskip 0.70004pt}$ | $\mathfrak{F}^{~{}}_{sp\hskip 0.70004pt}$ | $\psi$ | matter
Gauge | $\left({T^{*}{\mathcal{}M}},\pi_{{SU}},{\mathcal{}M},SO(1,3)\otimes{SU(N)}\right)$ | ${\mathfrak{A}}^{~{}}_{SU}$ | $\mathfrak{F}^{~{}}_{SU}$ | $\phi$ | force, Higgs
SG | $\left({\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes N},\pi_{{sp\hskip 0.70004pt}},{\mathcal{}M},Spin(1,3)\otimes{SU(N)}\right)$ | ${\mathfrak{A}}^{~{}}_{sp\hskip 0.70004pt}$ | $\mathfrak{F}^{~{}}_{sp\hskip 0.70004pt}$ | $\psi$ | force, matter
Dual SG | $\left(\hat{\mathcal{}S}_{sp\hskip 0.70004pt}^{\otimes N},\hat{\pi}_{{sp\hskip 0.70004pt}},{\mathcal{}M},Spin(1,3)\otimes{SU(N)}\right)$ | $\hat{\mathfrak{A}}^{~{}}_{sp\hskip 0.70004pt}$ | $\hat{\mathfrak{F}}^{~{}}_{sp\hskip 0.70004pt}$ | $\hat{\psi}$ | force, dual-matter
Yang–Mills | tensor products of SG and dual SG bundles |  | force, matter
Table 1:  Principal bundles introduced in this study. In a bundle name, “SG”
stands for “spinor-gauge”.

This study treated the classical Yang–Mills theory and general relativity in
four-dimension. The Chern–Weil theory concerning several principal bundles
twisting to each other provides a common mathematical language to treat
physical objects appearing in the Yang–Mills theory in curved space-time. In
reality, we introduced five principal bundles listed in Table 1. The co-
Poincar’e bundle is a model of our Universe, which provides the platform of
the Yang–Mills theory. Space-time manifold $\mathscr{M}$ with, in general,
non-zero curvature is a smooth Riemannian four-dimensional manifold. At each
point of $\mathscr{M}$, flat Lorentz-manifold ${\mathcal{}M}$ with vanishing
Levi-Civita connection $\Gamma^{\lambda}_{~{}\mu\nu}$ is associated as the
inertial space owing to Einstein’s equivalent principle. Lorentz-manifold
${\mathcal{}M}$ has vierbain form ${\mathfrak{e}}$ as a section that
determines a metric of ${\mathcal{}M}$ and connection ${\mathfrak{w}}$ to
govern gravitational interaction among any energetic objects. Other bundles
provide matter and force fields on the base manifold ${\mathcal{}M}$ as
sections and connections, respectively, and are twisted with the co-Poincaré
bundle.

The Yang–Mills theory is represented in the spinor-gauge bundles on the
Lorentz-manifold with various structure groups of the gauge symmetry.
Connection and curvature of the bundle belonging to the fundamental
representation of the structure group represent a potential function and a
force field in physics, e.g., $U(1)$-gauge group as the electromagnetic force,
$SU(2)$ as the weak force, and $SU(3)$ as the strong force, in physics. The
matter field is provided owing to a spinor-valued section in total space
${\mathcal{}S}_{sp\hskip 0.70004pt}$ of the spinor-gauge bundle and
interacting with each other through the gauge connection. A scalar field in
the gauge bundle with the structure group of $SU(2)$ is called the Higgs field
in physics. The Higgs field is a section of the $SU(2)$-gauge bundle and
belongs to the fundamental representation of the $SU(2)$ symmetry. The
equation of motion for the spinor fields consists of the Dirac operator; thus,
the Atiyah–Singer index theorem provides the topological invariant in the
spinor-gauge bundle.

The Hodge-dual of curvature forms are indispensable objects to constructing
Yang-Mills theory; a force field given by curvature has a constraint owing to
the Bianchi identity $d^{~{}}_{G}\mathfrak{F}^{~{}}_{G}=0$; thus, a gauge
curvature $\mathfrak{F}^{~{}}_{G}$ does not have dynamics in the spinor-gauge
bundle. On the other hand, the dual-curvature is not a curvature in the
spinor-gauge bundle and has an equation of motion
$d^{~{}}_{G}\hat{\mathfrak{F}}^{~{}}_{G}=j^{\circ}\mathfrak{V}_{\circ}$.
Therefore, the dual-curvature has dynamics in the spinor-gauge bundle. In
four-dimensional spaces, the Hodge-dual of a two-form object is again a two-
form object; thus, curvature two-form is decomposed into the SD and ASD parts.
In the Euclidean space, a solution to the equation of motion is a pure SD- or
ASD curvature; thus, the equation of motion is equivalent to the Bianchi
identity. The gauge field cannot have dynamics in the gauge bundle.

When a $\mathbb{Z}_{2}$-grading discrete operator, such as the Hodge-dual
operator, exists for the principal curvature, duplex superspace appears in the
system. The $\mathbb{Z}_{2}$-grading operator splits a space of curvatures
into the $\mathbb{Z}_{2}$-grading superspaces $V^{\pm}(\Omega^{2})$, namely
the primary superspace, owing to eigenvalues concerning the operator. Owing to
the isomorphism of four-dimensional spaces (13), a space of one-form objects
also splits into two subspaces; SD- and ASD curvatures are obtained using one-
form objects in one of these subspaces. A pair of the principal curvature and
its dual-curvature provide the Kramers pair. The secondary
$\mathbb{Z}_{2}$-grading superspace arises using the one-dimensional Clifford
algebra in the $({2\times 2})$-matrix representation of the Kramers pair. The
Dirac operator is provided in the secondary superspace, and the Atiyah–Singer
index theorem provides the additional topological invariants in the principal
bundles. General relativity and Yang–Mills Lagrangians are represented using
supertrace in the secondary superspace. The similarity between general
relativity and Yang–Mills gauge fields, in other words, similarity among four
forces in Nature, is evident in our representation.

A procedure to extract topological invariants in the secondary superspace of
the principal bundles commonly also appears in material physics. The
topological insulator’s edge-bulk correspondence, such as the charge- and
spin-pump phenomena, is an example appearing as a topological invariant in the
secondary superspace[47, 48, 49]. In this case, the time-reversal operator
plays the role of the Hodge-dual operator in this study, and the electron wave
function and its time-reversal provide the Kramers pair of the fermionic
section. The principal bundle consists of the Brillouin zone as the base
manifold, and the vector space of the electron wave functions as the total
space; it has the $U(2)$ symmetry as the structure group and the Berry-
connection as the principal connection. The spin polarization is the
topological invariant expressed by the first Chern number.

General relativity and the Yang–Mills theory are fundamental theories of
cosmology and elementary particle physics and treat fundamental phenomena in
our universe; thus, they are defined in the four-dimensional manifold with the
Lorentzian metric and the Bianchi identity and an equation of motion are the
hyperbolic-type. On the other hand, the Atiyah–Singer index theorem holds for
the elliptic-type Dirac operator on a compact manifold. Therefore, we cannot
apply the theorem on general relativity and the Yang–Mills theory as its
original shape, even after appropriately compacting the space-time. We
introduced the $\theta$-metric, which smoothly connects the Minkowski space-
time to the Euclidean space-time. When a compact manifold with the Lorentzian
metric is isotopic to that with the Euclidean metric, two manifolds are
cobordant, and characteristic numbers are cobordism invariant. In this case, a
characteristic index in the Minkowski space-time is defined owing to the
corresponding index in the Euclidean space-time, and the index theorem ensures
the index is topological invariant. We provided several examples of
topological invariants defined in the $\theta$-metric space and demonstrated
topological invariants working in the Minkowski space-time and the Euclidean
space-time.

The method using the $\theta$-metric may have various applications on the wide
variety of physical topics.

Field | Lagrangian | #equation
---|---|---
Gravity | $\frac{1}{\hbar\kappa}\frac{1}{2}Str\left[{\mbox{\boldmath$\mathfrak{F}$}}_{Gr}\wedge{\mbox{\boldmath$\mathfrak{F}$}}_{Gr}\right]$ | (112)
Gauge field | $-\frac{1}{2}Str[\mathfrak{ym}\wedge\mathfrak{ym}]$ | (117)
Cosmological constant | $-\frac{1}{\hbar\kappa}\Lambda_{\mathrm{c}}\hskip 1.99997pt{\widehat{\mathrm{H}}}(1)$ | (112)
Matter field | $\overline{{\mbox{\boldmath$\Psi$}}}_{{\textrm{Y}\textrm{M}}}\left(i\hskip 1.00006pt{\not\\!\mathcal{}D}^{~{}}_{\textrm{Y}\textrm{M}}-\frac{1}{\hbar}{\mbox{\boldmath$\mu$}}^{~{}}_{{\textrm{Y}\textrm{M}}}\right){\mbox{\boldmath$\Psi$}}_{{\textrm{Y}\textrm{M}}}\hskip 1.99997pt{\widehat{\mathrm{H}}}(1)$ | (116)
Table 2:  A summary table of the Lagrangians appearing in the Yang–Mills
theory.

## Competing interests

The author declares that I have no competing interests.

## Acknowledgements

I would like to thank Dr. Y. Sugiyama, Prof. J. Fujimoto, Prof. T. Kaneko, and
Prof. F. Yuasa for their continuous encouragement and fruitful discussions.

##  Appendix A existence of dual connection and dual spinor

###  A.1 $U(1)$ gauge group

First, this appendix treats the most straightforward case that the $U(1)$
group in the flat space-time (${\mathfrak{w}}=0$). In this case, the dual
connection is obtained as a solution of structure equation
$\hat{\mathfrak{F}}^{~{}}_{{SU}}=d\hat{\mathfrak{A}}^{~{}}_{{SU}}$, that
satisfies the Bianchi identity
$d\hat{\mathfrak{F}}^{~{}}_{{SU}}=d(d\hat{\mathfrak{A}}^{~{}}_{{SU}})=0$. The
existence problem of a solution for this system is equivalent to the
differential-geometrical question: “Is closed-form
$\hat{\mathfrak{F}}^{~{}}_{{SU}}$ exact ?”. For a contractible manifold, the
answer to this question is positively given by the Poincaré lemma; the general
answer is provided using the de Rham theorem (See, e.g., [50]). The de Rham
theorem asserts that $k$’th de Rham cohomology of a smooth manifold $M$,
$H_{{dR}}^{k}(M)$, is isomorphic to the real cohomology $H^{k}(M,\mathbb{R})$.

The existence problem of the one-form object ${\mathfrak{a}}$ such that
$d{\mathfrak{a}}={\mathfrak{b}}$ for given two-form object ${\mathfrak{b}}$ is
answered by a following statement[50]:

######  Remark A.1.

Suppose $M$ is $n$-dimensional smooth manifold and closed $k$-form object
${\mathfrak{b}}$ exists in a compact submanifold of $M$;
$\Sigma_{\mathfrak{b}}\subseteq M$. In this case, following two statements are
equivalent to each other:

  1. 1.

$(k-1)$-form object ${\mathfrak{a}}$ such that
${\mathfrak{b}}=d{\mathfrak{a}}$ exists in
$\Sigma_{\mathfrak{a}}\subseteq\Sigma_{\mathfrak{b}}$, where
$\Sigma_{\mathfrak{a}}$ is a compact and connected manifold.

  2. 2.

The $(n-k)$’th relative homology of pair
$(\Sigma_{\mathfrak{a}},\partial\Sigma_{\mathfrak{a}})$ is vanished;
$H_{n-k}(\Sigma_{\mathfrak{a}},\partial\Sigma_{\mathfrak{a}};\mathbb{R})=0$,
where $\partial\Sigma_{\mathfrak{a}}\subset\Sigma_{\mathfrak{a}}$ is a
boundary of compact and connected manifold $\Sigma_{\mathfrak{a}}$.

###### Proof.

1. $\Rightarrow$ 2. : Closedness of two-form ${\mathfrak{b}}$ is consistent with an exactness ${\mathfrak{b}}=d{\mathfrak{a}}$ such that $d{\mathfrak{b}}=dd{\mathfrak{a}}=0$. The $k$’th de Rham cohomology of domain $\Sigma_{\mathfrak{a}}$ concerning the external-derivative operation is defined as

$\displaystyle H^{k}_{{dR}}(\Sigma_{\mathfrak{a}})$
$\displaystyle:=Z^{k}(\Sigma_{\mathfrak{a}})/B^{k}(\Sigma_{\mathfrak{a}})=0,$

where

$\displaystyle Z^{k}(\Sigma)$
$\displaystyle:=\mathrm{Ker}\left(d:\Omega^{k}(\Sigma_{\mathfrak{a}})\rightarrow\Omega^{k+1}(\Sigma_{\mathfrak{a}})\right),$
$\displaystyle B^{k}(\Sigma_{\mathfrak{a}})$
$\displaystyle:=\mathrm{Im}\left(d:\Omega^{k-1}(\Sigma_{\mathfrak{a}})\rightarrow\Omega^{k}(\Sigma_{\mathfrak{a}})\right).$

Owing to the de Rham theorem and the Poincaré duality-theorem for a compact
and connected four-dimensional manifold, there exists isomorphism:

$H^{k}_{{dR}}(\Sigma_{\mathfrak{a}})\rightarrow
H^{k}(\Sigma_{\mathfrak{a}};\mathbb{R})\rightarrow
H_{n-k}(\Sigma_{\mathfrak{a}},\partial\Sigma_{\mathfrak{a}};\mathbb{R}).$

Thus, vanishing relative homology
$H_{n-k}(\Sigma_{\mathfrak{a}},\partial\Sigma_{\mathfrak{a}};\mathbb{R})=0$ is
maintained.
2. $\Rightarrow$ 1. : When $H_{n-k}(\Sigma_{\mathfrak{a}},\partial\Sigma_{\mathfrak{a}};\mathbb{R})=0$, the $k$’th de Rham cohomology is also vanished owing to the de Rham theorem and the Poincaré duality theorem. The vanishing $k$’th cohomology induces isomorphism between $Z^{k}(\Sigma_{\mathfrak{a}})$ and $B^{k}(\Sigma_{\mathfrak{a}})$. If $B^{k}(\Sigma_{\mathfrak{a}})$ is empty, $Z^{k}(\Sigma_{\mathfrak{a}})$ is also empty, which contradicts to the assumption of the existence of ${\mathfrak{b}}\in Z^{k}(\Sigma_{\mathfrak{a}})$. Therefore, the existence of ${\mathfrak{a}}\in\Omega^{k-1}(\Sigma_{\mathfrak{a}})$ such that $B^{k}(\Sigma_{\mathfrak{a}})\cong[d{\mathfrak{a}}]\equiv[{\mathfrak{b}}]{\cong}Z^{k}(\Sigma_{\mathfrak{a}})$ is maintained. ∎

For an abelian gauge symmetry such as a $U(1)$ gauge group, the structure