# Yang–Mills–Scalar–Matter Fields in the Quantum Hopf Fibration

Gustavo Amilcar Saldaña Moncada  Gustavo Amilcar Saldaña Moncada
Instituto de Matemáticas UNAM<EMAIL_ADDRESS>
###### Abstract.

In this paper we present solutions to the non–commutative geometrical version
of the Yang–Mills–Scalar–Matter theory in the Hopf fibration
$S^{1}\hookrightarrow S^{3}\rightarrow S^{2}$ using the $3D$–calculus.

MSC 2010: 46L87, 58B99.

Keywords: Fields, quantum Hopf fibration, canonical quantum principal
connection

##  1\. Introduction

In Differential Geometry, the Hopf fibration is perhaps maybe one of the most
basic and well–established examples of principal bundles. This bundle is
particularly important in Physics since its connections can describe magnetic
monopoles. Even more, gauge theory allows us to study matter fields in the
presence of magnetic monopoles. In Non–Commutative Geometry there is an analog
concept known as the quantum Hopf fibration or the $q$–deformed Dirac monopole
bundle [BM].

This paper aims to show in a concrete example that the theory presented in
[Sa1] and [Sa2] is non–trivial, i.e., we are going to present a
non–commutative geometrical version of magnetic monopoles and its interaction
with space–time scalar matter fields. Unlike examples shown in [Sa2] and [Sa3]
where we used trivial quantum principal bundles; here we will use the quantum
Hopf fibration which relates the structure group and the base space in a
non–trivial way. The importance of this work lies in the fact it provides a
better support of the general theory showing explicit and interesting
solutions of the field equations.

To accomplish our purpose, we will consider the quantum Hopf fibration
together with its $3D$–calculus, which represent the non–commutative
counterpart of the classical differential calculus, but of course, it is
possible to use others, for example, the $4D$–calculus like in [LZ]. It is
worth remarking (again) that the theory presented here is just an application
of the general theory ([Sa1], [Sa2]) not an ad hoc theory created for this
space; and this work follows the research line of M. Durdevich shown in [D1],
[D2]and [D3], among other papers. If the reader wants to check another
concrete example of the general theory, we recommend [Sa3].

The paper is organized into five sections. In the second section we are going
to develop our study of the quantum Hopf fibration using the $3D$ $\ast$–first
order differential calculus ($\ast$–FODC) to creat the differential calculus
in the quantum Hopf fibration shown in [D3]. This differential calculus arises
from the classical differential geometry and like in that case, one can define
a particular quantum principal connection (qpc) which will play the role of
the canonical principal connection on the Hopf fibration. In the third section
we will talk about the associated quantum vector bundles (associated qvb) as
well as all the necessary conditions to work with Yang–Mills fields and
space–time scalar matter fields [Sa1], [Sa2]. In the fourth section we are
going to present solutions to the fields equations in this space as well as
the spectrum of the left and right Laplace-de Rham operator for qvbs [Sa2].
The last section is about some concluding comments. It is worth mentioning
that this paper is not self–contained, so we strongly recommended to read
[D3], [Sa1], [Sa2] before this paper.

##  2\. The quantum Hopf Fibration

Let us take the compact matrix quantum group (cmqg) $\mathcal{SU}_{q}(2)$ (the
quantum $SU(2)$ group [W2]) with $q$ $\in$ $(-1,1)-\\{0\\}$. The dense
$\ast$–algebra $SU_{q}(2)$ is generated by two letters $\\{\alpha,\gamma\\}$
and they satisfy

$\displaystyle\alpha^{\ast}\alpha+\gamma^{\ast}\gamma=\mathbbm{1},\qquad\alpha\alpha^{\ast}+q^{2}\gamma\gamma^{\ast}=\mathbbm{1},$
$\displaystyle\gamma\gamma^{\ast}=\gamma^{\ast}\gamma,\quad
q\gamma\alpha=\alpha\gamma,\quad q\gamma^{\ast}\alpha=\alpha\gamma^{\ast},$

and its $\ast$–Hopf algebra structure is given by

$\phi(\alpha)=\alpha\otimes\alpha-q\gamma^{\ast}\otimes\gamma,\qquad\phi(\gamma)=\gamma\otimes\alpha+\alpha^{\ast}\otimes\gamma,$
$\epsilon(\alpha)=1,\qquad\epsilon(\gamma)=0,$
$\kappa(\alpha)=\alpha^{\ast},\quad\kappa(\alpha^{\ast})=\alpha,\quad\kappa(\gamma)=-q\gamma,\quad\kappa(\gamma^{\ast})=-q^{-1}\gamma^{\ast}.$

In an abuse of notation, we will identify the group $U(1)$ with the Laurent
polynomial algebra, i.e.,

$U(1):=\mathbb{C}[z,z^{\ast}]=\mathbb{C}[z,z^{-1}]$

and in this way, its $\ast$–Hopf algebra structure is defined by

$\phi^{\prime}(z)=z\otimes
z,\qquad\epsilon^{\prime}(z)=1,\qquad\kappa^{\prime}(z)=z^{\ast},\qquad\kappa^{\prime}(z^{\ast})=z.$

This algebra is commutative and $\kappa^{\prime}$ is a $\ast$–algebra
morphism.

By defining the $\ast$–Hopf algebra epimorphism

$j:SU_{q}(2)\longrightarrow U(1)$

such that $j(\alpha)=z,$ $j(\gamma)=0$, we can consider

${{}_{SU_{q}(2)}}\Phi:=(\mathrm{id}_{SU_{q}(2)}\otimes
j)\circ\phi:SU_{q}(2)\longrightarrow SU_{q}(2)\otimes U(1).$

Now we define the quantum $2$–sphere as (the quantum space whose
$\ast$–algebra of $\mathbb{C}$–valued functions is given by) the
$\ast$–subalgebra of $SU_{q}(2)$

$(\mathbb{S}^{2}_{q},\cdot,\mathbbm{1},\ast),$

where

$\mathbb{S}^{2}_{q}:=\\{x\in
SU_{q}(2)\mid{{}_{SU_{q}(2)}}\Phi(x)=x\otimes\mathbbm{1}\\}$

which can be viewed as the $\ast$–algebra generated by
$\\{\alpha\alpha^{\ast},\alpha\gamma^{\ast}\\}$ and it is a quantum sphere in
the sense of [P]. In this way, the quantum principal $U(1)$–bundle over
$\mathbb{S}^{2}_{q}$ given by

(1) $\zeta_{HF}=(SU_{q}(2),\mathbb{S}^{2}_{q},{{}_{SU_{q}(2)}}\Phi)$

is usually called the quantum Hopf fibration [BM], [D2].

Now to accomplish our purpose we will take the differential calculus on
$\zeta_{HF}$ shown [D3]; however, we are going to use the notation presented
in [Sa1], [Sa2]. The bicovariant $\ast$–FODC of $U(1)$ will be denoted by
$(\Gamma,d)$ and we shall take

(2) $\beta^{\prime}:=\\{\varsigma:=\pi^{\prime}(z-z^{\ast})\\}$

as a Hamel basis of ${{}_{\mathrm{inv}}}\Gamma$. In this case, the quantum
germs map and the adjoint corepresentation will be denoted by
$\pi^{\prime}:U(1)\longrightarrow{{}_{\mathrm{inv}}}\Gamma$ and
$\mathrm{ad}^{\prime}:{{}_{\mathrm{inv}}}\Gamma\longrightarrow{{}_{\mathrm{inv}}}\Gamma\otimes
U(1)$, respectively. For this space, the universal differential envelope
$\ast$–calculus of $(\Gamma,d)$ ([D1]), $(\Gamma^{\wedge},d,\ast)$, satisfies

$\Gamma^{\wedge k}=\\{0\\}\;\;\mbox{ for }\;\;k\geq 2;$

however, it differs from the classical differential calculus on $U(1)$ [D3].

######  Definition 2.1.

With the previous differential calculus, the linear map

$\displaystyle\omega^{\mathrm{c}}:{{}_{\mathrm{inv}}}\Gamma$
$\displaystyle\longrightarrow\Omega^{1}(SU_{q}(2))$ $\displaystyle\theta$
$\displaystyle\longmapsto\mathbbm{1}\otimes\theta$

is a real, regular and multiplicative quantum principal connection (qpc) and
it is called the canonical quantum principal connection in $\zeta_{HF}$.

######  Proposition 2.2.

The only regular qpc is $\omega^{\mathrm{c}}$.

###### Proof.

According to the general theory of qpcs we know that every regular qpc has to
have the form [So]

$\omega^{\mathrm{c}}+\lambda$

such that
$\varphi\,\lambda(\theta)=(-1)^{k}\,\lambda(\theta\circ{\kappa^{\prime}}^{-1}(\varphi^{(1)}))\varphi^{(0)}$
for all $\varphi$ $\in$ $\mathrm{Hor}^{k}SU_{q}(2)$, $\theta$ $\in$
${{}_{\mathrm{inv}}}\Gamma$ with
${{}_{\mathrm{H}}}\Phi(\varphi)=\varphi^{(0)}\otimes\varphi^{(1)}$ (in
Sweedler’s notation). We are going to prove that $\lambda=0$. A direct
calculation shows that

${{}_{\mathrm{H}}}\Phi(\lambda(\varsigma))=({{}_{\mathrm{H}}}\Phi\otimes\mathrm{id}_{U(1)})ad^{\prime}(\varsigma)\;\Longleftrightarrow\;\lambda(\varsigma)\,\in\,\Omega^{1}(\mathbb{S}^{2}_{q});$

so $\lambda(\varsigma)=x\eta_{-}+y\eta_{+}$ with
$x=\displaystyle\sum_{m+k-l=2}\lambda_{mkl}\,\alpha^{m}\gamma^{k}\gamma^{\ast
l}$,
$y=\displaystyle\sum_{p+q-r=-2}\,\mu_{pqr}\alpha^{p}\gamma^{q}\gamma^{\ast
r},$ $\lambda_{mkl}$, $\mu_{pqr}$ $\in$ $\mathbb{C}$ (these elements form a
Hamel basis). Due to the fact that $\lambda$ has to satisfies
$\lambda(\varsigma)\,\alpha=\alpha\,\lambda(\varsigma\circ
z)=q^{-2}\alpha\,\lambda(\varsigma),$ we get

$x\eta_{-}\alpha=q^{-2}\alpha
x\eta_{-}\;\Longrightarrow\;x=\sum_{m+k-l=2}\lambda_{mkl}\,\alpha^{m}\gamma^{k}\gamma^{\ast
l}\,\mbox{ with }\,k+l=1,\;m\geq 0.$

Applying the same process to $\gamma$ we find that $m=-1$ which is a
contradiction, so $x=0$. A similar calculation shows $y=0$ and hence
$\lambda=0$. ∎

A quick calculation shows that the operator $D$ presented in [D3] is the
covariant derivative of $\omega^{\mathrm{c}}$ and its curvature fulfills

(3) $R^{\omega^{\mathrm{c}}}(\varsigma)=(1+q^{2})q\,\eta_{-}\eta_{+}.$

It is worth mentioning that for the form of the differential calculus, the
only possible embedded differential ([D2], [So]) is $\delta=0$ and for $q=1$,
$\omega^{\mathrm{c}}$ is the principal connection associated to the
Levi–Civitta connection.

##  3\. Associated Quantum Vector Bundles and the Quantum Hodge Operator

In accordance with the general theory of associated qvbs, we need to check
that Equations 34 and 35 of [Sa1] hold.

It is well–know that a complete set of mutually inequivalent irreducible
unitary finite dimensional (smooth) $U(1)$–corepresentations $\mathcal{T}$ is
in biyection with $\mathbb{Z}$. These corepresentations are unitary with the
canonical inner product of $\mathbb{C}$ and it is worth mentioning that for
all $n$ $\in$ $\mathbb{Z}$, the matrix of the canonical corepresentation
morphism between the corepresentation and its double contragradiant is the
identity matrix (see Equation 35 of [Sa1]).

######  Proposition 3.1.

For every $n$ $\in$ $\mathbb{Z}$, Equations 34 and 35 of [Sa1] are satisfied.

###### Proof.

By taking

$\displaystyle T^{\mathrm{triv}}:\mathbb{C}$
$\displaystyle\longrightarrow\mathbb{S}^{2}_{q}$ $\displaystyle w$
$\displaystyle\longmapsto w\mathbbm{1}$

it follows that the statement is true for $n=0$. Now let us take $n$ $\in$
$\mathbb{N}$ and consider the linear maps

$T^{n}_{k+1}:\mathbb{C}\longrightarrow\mathrm{SU}_{q}(2)$

defined by

$T^{n}_{k+1}(1)=\left[\begin{array}[]{lcr}n\\\ k\\\
\end{array}\right]^{\frac{1}{2}}_{q^{-2}}\alpha^{n-k}\gamma^{k}=:x^{n}_{k+1\,1}$

with $k=0,...,n$, where $\displaystyle\left[\begin{array}[]{lcr}n\\\ k\\\
\end{array}\right]_{q^{-2}}$ is the Gaussian binomial coefficient also known
as the $q$–binomial coefficient [KS]. Due to the fact that
${{}_{\mathrm{SU}_{q}(2)}}\Phi(\alpha)=\alpha\otimes z,$
${{}_{\mathrm{SU}_{q}(2)}}\Phi(\gamma)=\gamma\otimes z$ we get

$T^{n}_{k}\;\in\;\textsc{Mor}(n,{{}_{\mathrm{SU}_{q}(2)}}\Phi).$

According to [KS], these elements form the first column of the
$\mathcal{SU}_{q}$–representation matrix for spin $l=\dfrac{n}{2}$, $u^{l}$.
Since $u^{l\,\dagger}u^{l}=\mathrm{Id}_{n+1}$ $\in$
$M_{n+1}(\mathrm{SU}_{q}(2))$ (here $\dagger$ is denoting the transpose
conjugate matrix) we get that Equation 34 holds. Taking

$Z^{n}=(q^{2(i-1)}\delta_{ij})\;\in\;M_{n+1}(\mathbb{C})$

where $\delta_{ij}$ is the Kronecker delta, Equation 35 holds since in this
case

$W^{n\,\mathrm{T}}X^{n\,\ast}=\mathrm{Id}_{n_{\alpha}}\quad\mbox{ with }\quad
W^{n}=(w^{n}_{ij})=Z^{n}X^{n},\;X^{n}=(x^{n}_{k+1\,1})$

is the the $(1,1)$–entry of $u^{l}u^{l\dagger}=\mathrm{Id}_{n+1}$ [KS]. For
negative integers $n$ it is enough to take the last column of $u^{l}$ with
$l=\dfrac{|n|}{2}$ to ensure that the Equation 34 is holds and taking

$Z^{n}=(q^{-2(|n|+1-i)}\delta_{ij})\;\in\;M_{|n|+1}(\mathbb{C})$

we get that Equation 35 holds since in this case
$W^{n\,\mathrm{T}}X^{n\,\ast}=\mathrm{Id}_{n_{\alpha}}$ will be the
$(|n|+1,|n|+1)$–entry of $u^{l}u^{l\dagger}=\mathrm{Id}_{|n|+1}$ [KS]. ∎

We have to remark that Equations 36 of [Sa1] can be viewed in terms of
Hopf–Galois extension’s theory [HM]. However, an advantage of having proven
Proposition 3.1 is that it provides us with the left and right generators of
the associated qvbs [Sa1], [Sa2]. In this way, it is possible to take left and
right associated quantum vector bundles (qvbs) and induced quantum linear
connections (qlc) for all $n$ $\in$ $\mathbb{Z}$. The left/right qvb
associated to the corepresentation $n$ will be denoted by
$\zeta^{\mathrm{L}}_{n}:=(\Gamma^{\mathrm{L}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q}),+,\cdot)$,
$\zeta^{\mathrm{R}}_{n}:=(\Gamma^{\mathrm{R}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q}),+,\cdot)$,
respectively. Moreover, the induced qlc by any $\omega$ will be denote by
$\nabla^{\omega}_{n}$, $\widehat{\nabla}^{\omega}_{n}$, respectively.

Now we have to verify if $\zeta_{HF}$ satisfies all the conditions written in
Definition 2.1 and Remark 2.3 of [Sa2]. In order to do this, we need the
following lemma.

######  Lemma 3.2.

Let us consider the linear functional

(4) $\displaystyle\int_{\mathbb{S}^{2}_{q}}:\Omega^{2}(\mathbb{S}^{2}_{q})$
$\displaystyle\longrightarrow\mathbb{C}$ $\displaystyle p\,\eta_{-}\eta_{+}$
$\displaystyle\longmapsto h_{q}(p),$

where $h_{q}$ is the quantum Haar measure of $\mathcal{SU}_{q}$ [W1]. Then

$d(\Omega^{1}(\mathbb{S}^{2}_{q}))\subseteq\mathrm{Ker}\left(\int_{\mathbb{S}^{2}_{q}}\right).$

###### Proof.

Let us start by remembering the definition of $D$ for degree zero:

$D(a)=a^{(0)}(\pi_{-}(a^{(1)})+\pi_{+}(a^{(1)})),$

where $\pi_{\pm}:=\rho_{\pm}\circ\pi$ with
$\pi:\mathrm{SU}_{q}(2)\longrightarrow{{}_{\mathrm{inv}}}\Xi$ the quantum
germs map and
$\rho_{\pm}:{{}_{\mathrm{inv}}}\Xi\longrightarrow\mathbb{C}\eta_{\pm}$ the
canonical projection. In this way, we define the linear functional

$\lambda_{-}:\mathrm{SU}_{q}(2)\longrightarrow\mathbb{C}$

such that $\pi_{-}(a)=\lambda_{-}(a)\eta_{-}$. Notice that $\mathbbm{1}$ $\in$
$\mathrm{Ker}(\lambda_{-})$.

Consider $y\eta_{+}$ $\in$ $\Omega^{1}(\mathbb{S}^{2}_{q})$. Hence

$\displaystyle\int_{\mathbb{S}^{2}_{q}}d(y\eta_{+})\,=\,h_{q}(y^{(1)})\,\lambda_{-}(y^{(2)})\,=\,\lambda_{-}(h_{q}(y^{(1)})y^{(2)})$
$\displaystyle=$ $\displaystyle\lambda_{-}(h_{q}\ast y)$ $\displaystyle=$
$\displaystyle\lambda_{-}(h_{q}(y)\mathbbm{1})\,=\,\lambda_{-}(\mathbbm{1})h_{q}(y)=0.$

In an analogous way it can be proved that

$\int_{\mathbb{S}^{2}_{q}}d(x\eta_{-})=0$

and therefore the Lemma follows. ∎

######  Proposition 3.3.

The quantum $2$–sphere satisfies all the conditions written in Definition 2.1
and Remark 2.3 of [Sa2] with respect to the graded differential $\ast$–algebra
of base forms.

###### Proof.

First of all let us observe that $\mathbb{S}^{2}_{q}$ is (obviously)
$C^{\ast}$–closeable.

  1. (1)

$\mathbb{S}^{2}_{q}$ is oriented since for $k>2$,
$\Omega^{k}(\mathbb{S}^{2}_{q})=0$ and

$\mathrm{dvol}:=\eta_{-}\eta_{+}$

is a quantum $2$–volume form.

  2. (2)

A direct calculation shows that a lqrm can be defined on $\mathbb{S}^{2}_{q}$
by means of

$\displaystyle\langle-,-\rangle:\mathbb{S}^{2}_{q}\times\mathbb{S}^{2}_{q}$
$\displaystyle\longrightarrow\mathbb{S}^{2}_{q}$
$\displaystyle(\;\hat{p}\;,\;p\;)$
$\displaystyle\longmapsto\hat{p}\,p^{\ast},$
$\displaystyle\langle-,-\rangle\;\;:\;\;\Omega^{1}(\mathbb{S}^{2}_{q})\;\quad\times\;\quad\Omega^{1}(\mathbb{S}^{2}_{q})$
$\displaystyle\longrightarrow\;\mathbb{S}^{2}_{q}$
$\displaystyle((\hat{x}\eta_{-}+\hat{y}\eta_{+}),(x\eta_{-}+y\eta_{+}))$
$\displaystyle\longmapsto\dfrac{1}{2}\left(q^{2}\hat{x}x^{\ast}+\hat{y}y^{\ast}\right)$

and finally

$\displaystyle\langle-,-\rangle:\Omega^{2}(\mathbb{S}^{2}_{q})\;\times\;\Omega^{2}(\mathbb{S}^{2}_{q})$
$\displaystyle\longrightarrow\mathbb{S}^{2}_{q}$
$\displaystyle(\;\hat{p}\,\mathrm{dvol}\;,\;p\,\mathrm{dvol}\;)$
$\displaystyle\longmapsto\hat{p}\,p^{\ast}.$

With this lqrm, $\mathrm{dvol}$ is actually a lqr $2$–form. Taking into
account Remark LABEL:a.2.2, we get a rqrm with a rqr $2$–form.

  3. (3)

According to [W1], $h_{q}$ is a faithful state on $\mathrm{SU}_{q}(2)$ and
hence the linear functional of Equation 4 is actually a quantum integral. In
this way, by the previous lemma we conclude that
$(\mathbb{S}^{2}_{q},\cdot,\mathbbm{1},\ast)$ is a quantum space without
boundary.

  4. (4)

A direct calculation shows

$\star_{\mathrm{L}}\,p=p^{\ast}\,\mathrm{dvol}$

for all $p$ $\in$ $\mathbb{S}^{2}_{q}$;

$\star_{\mathrm{L}}(p\,\mathrm{dvol})=p^{\ast}$

for all $p\,\mathrm{dvol}$ $\in$ $\Omega^{2}(\mathbb{S}^{2}_{q})$ and finally

$\star_{\mathrm{L}}\,\mu=\dfrac{1}{2}\left(-y^{\ast}\eta_{-}+x^{\ast}\eta_{+}\right),$

for all $\mu=x\eta_{-}+y\eta_{+}$ $\in$ $\Omega^{1}(\mathbb{S}^{2}_{q}).$ To
define $\star_{\mathrm{R}}$ we can use Remark 2.2 of [Sa2].

∎

It is worth mentioning that for this differential calculus, the only possible
embedded differential is $\delta=0$ [D2], [Sa2].

##  4\. Yang–Mills–Scalar–Matter Fields

In this section we are going to show solutions of the field equations of the
Yang–Mills–Scalar–Matter theory ([Sa2]) using
$(\zeta_{HF},\omega^{\mathrm{c}})$ and all the structures that we have just
defined.

###  4.1. Non–commutative geometrical Yang–Mills Fields

We know that every single qpc $\omega$ has the form [D2]

$\omega=\omega^{\mathrm{c}}+\lambda\qquad\mbox{ with
}\qquad\lambda(\varsigma)=x\eta_{-}+y\eta_{+}\,\in\,\Omega^{1}(\mathbb{S}^{2}_{q}).$

######  Proposition 4.1.

Every YM qpc is of the form $\omega^{\mathrm{c}}+\lambda$, where
$\lambda(\varsigma)=dp$ for some $p$ $\in$ $\mathbb{S}^{2}_{q}$.

###### Proof.

First, notice that for all qpc $\omega=\omega^{\mathrm{c}}+\lambda$ (see
Equation 3)

(5) $R^{\omega}(\varsigma)=(1+q^{2})q\eta_{-}\eta_{+}+d\,\lambda(\varsigma);$

so

$\displaystyle\left.\dfrac{\partial}{\partial
z}\right|_{z=0}\mathscr{S}_{\mathrm{YM}}(\omega+z\,\lambda^{\prime})\,$
$\displaystyle=$
$\displaystyle-\dfrac{1}{4}\,\left(\langle\lambda^{\prime}(\varsigma)\,|\,d^{\star_{\mathrm{L}}}R^{\omega}(\varsigma)\rangle_{\mathrm{L}}+\langle\lambda^{\prime}(\varsigma)^{\ast}\,|\,d^{\star_{\mathrm{R}}}R^{\omega}(\varsigma)^{\ast}\rangle_{\mathrm{R}}\right)$
$\displaystyle=$ $\displaystyle-\dfrac{1}{4}\,\left(\langle
d\lambda^{\prime}(\varsigma)\,|\,R^{\omega}(\varsigma)\rangle_{\mathrm{L}}+\langle
d\lambda^{\prime}(\varsigma)^{\ast}\,|\,R^{\omega}(\varsigma)^{\ast}\rangle_{\mathrm{R}}\right)$
$\displaystyle=$ $\displaystyle-\dfrac{1}{4}\,\left(\langle
d\lambda^{\prime}(\varsigma)\,|\,d\lambda(\varsigma)\rangle_{\mathrm{L}}+\langle
d\lambda^{\prime}(\varsigma)^{\ast}\,|\,d\lambda(\varsigma)^{\ast}\rangle_{\mathrm{R}}\right)$
$\displaystyle=$ $\displaystyle-\dfrac{1}{2}\,\langle
d\lambda^{\prime}(\varsigma)\,|\,d\lambda(\varsigma)\rangle_{\mathrm{L}}.$

Since $\langle-\,|\,-\rangle_{\mathrm{L}}$ is an inner product we conclude
that every YM qpc has the form $\omega^{\mathrm{c}}+\lambda$ with
$d\lambda(\varsigma)=0$.

In accordance with [W2], the zero cohomology group of $\mathrm{SU}_{q}(2)$ is
$\mathbb{C}$; while the first cohomology group is $\\{0\\}$. Hence, since
$\lambda(\varsigma)$ $\in$ $\Omega^{1}(\mathbb{S}^{2}_{q})$ is exact, there
exists $p$ $\in$ $\mathbb{S}^{2}_{q}$ such that $\lambda(\varsigma)=dp$. ∎

In this case, the quantum gauge group (qgg) of the Lagrangian ([Sa1])
satisfies

(6)
$\mathfrak{qGG}_{\mathrm{YM}}:=\\{\mathfrak{f}\in\mathfrak{qGG}\mid\mathfrak{f}^{\circledast}\omega^{\mathrm{c}}=\omega^{\mathrm{c}}+\lambda\mbox{
with }d\lambda=0\\}.$

We can deduce that $\mathrm{U}(1)\subseteq\mathfrak{qGG}_{\mathrm{YM}}$ and
all YM qpcs are in the same orbit, just like in the classical case.

###  4.2. Non–commutative geometrical n–multiple of Space–Time Scalar Matter
Fields.

According to [Sa2], it is enough to look for eigenvectors of the left quantum
Laplace–de Rham operator $\vartriangle_{\mathrm{L}}$ with
$\mathbb{S}^{2}_{q}$–valued eigenvalues. A direct calculation shows that

$d^{\star_{\mathrm{L}}}dp=\dfrac{1}{2}(1+q^{2})^{2}\,p\quad\mbox{ with }\quad
p=\mathbbm{1}-(1+q^{2})\gamma\gamma^{\ast},\,\alpha\gamma^{\ast},\,\alpha^{\ast}\gamma.$

It is important to mention that for $q$ $\in$ $(-1,1)-\\{0\\}$, these
eigenvalues are not $0$. In this way, taking a potential such that

$V^{\prime}=\dfrac{1}{2}(1+q^{2})^{2}$

it is easy to find non–commutative geometrical space–time scalar matter
fields. Of course, there are more solutions but they depend on the form of the
potential $V$.

###  4.3. Non–commutative geometrical Yang–Mills–Scalar–Matter Equations.

Let us take $n$ $\in$ $\mathbb{Z}$. If $n=0$, YMSM fields are triplets
$(\omega,T_{1},T_{2})$ where $\omega$ is a YM qpc and $(T_{1},T_{2})$ is an
stationary point of $\mathscr{S}_{\mathrm{SM}}$.

Consider now $n\not=0$. It is easy to see that

$d^{\star_{\mathrm{L}}}R^{\omega^{\mathrm{c}}}(\varsigma)=0;$

so we have to look for $T_{1}$ $\in$
$\Gamma^{\mathrm{L}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})$,
$T_{2}$ $\in$
$\Gamma^{\mathrm{R}}(\mathbb{S}^{2}_{q},\mathbb{C}_{-n}\mathbb{S}^{2}_{q})$
such that

(7) $\langle\Upsilon_{n}\circ
K^{\lambda}(T_{1})\,|\,\nabla^{\omega^{\mathrm{c}}}_{n}T_{1}\rangle_{\mathrm{L}}-\langle\widetilde{\Upsilon}_{-n}\circ\widehat{K}^{\lambda}(T_{2})\,|\,\widehat{\nabla}^{\omega^{\mathrm{c}}}_{-n}T_{2}\rangle_{\mathrm{R}}=0$

for all $\lambda$ $\in$ $\overrightarrow{\mathfrak{qpc}(\zeta_{HF})}$, and

(8)
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\left(\nabla^{\omega^{\mathrm{c}}}_{n}\,T_{1}\right)-V^{\prime}_{\mathrm{L}}(T_{1})^{\ast}\,T_{1}=0\,,\qquad\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{-n}\left(\widehat{\nabla}^{\omega^{\mathrm{c}}}_{-n}\,T_{2}\right)-T_{2}\,V^{\prime}_{\mathrm{R}}(T_{2})^{\ast}=0.$

Now it is possible to explicitly find solutions. For example, for $n>0$ the
triplet $(\omega^{\mathrm{c}},T_{1},T_{2})$ such that

$T_{1}(1)=\alpha^{n}\,,\quad T_{2}(1)=\alpha^{\ast\,n}\quad\mbox{ or }\quad
T_{1}(1)=\gamma^{n}\,,\quad T_{2}(1)=\gamma^{\ast\,n}$

is a YMSM field for a potential such that

$V^{\prime}=\dfrac{1}{2}\left(\dfrac{q^{4}(1-q^{2n})}{1-q^{2}}\right).$

It is worth mentioning that $q\longrightarrow 1$ implies
$V^{\prime}\longrightarrow n$, so we recover the winding number $n$. Of
course, there are more solutions; however, they depend on the form of the
potential $V$.

The spectrums of

$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}:\Gamma^{\mathrm{L}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})\longrightarrow\Gamma^{\mathrm{L}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})$

and

$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}:\Gamma^{\mathrm{R}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})\longrightarrow\Gamma^{\mathrm{R}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})$

for all $n$ $\in$ $\mathbb{Z}$ are shown in the following tables. In the
second row of the table $1$ and the fifth row of the table $2$, $m$, $k$ $\in$
$\mathbb{N}_{0}$ (in the other cases, $m$, $k$, $l$ $\in$ $\mathbb{N}$) and
they cannot be both $0$ at the same time. On the other hand,
$p(\gamma^{k}\gamma^{\ast\,l})$, $\widehat{p}(\gamma^{k}\gamma^{\ast\,l})$ are
polynomials with coefficients in $\mathbb{C}$ such that their terms are
$\gamma^{k}\gamma^{\ast\,l}$, $\gamma^{k-1}\gamma^{\ast\,l-1}$, etc. until
$\gamma$ or $\gamma^{\ast}$ disappear. For example

$p(\gamma\gamma^{\ast})=\widehat{p}(\gamma\gamma^{\ast})=\mathbbm{1}-(1+q^{2})\gamma\gamma^{\ast}.$

Polynomials $p(\alpha^{m}\gamma^{k}\gamma^{\ast\,l})$,
$p(\alpha^{\ast\,m}\gamma^{k}\gamma^{\ast\,l})$,
$\widehat{p}(\alpha^{m}\gamma^{k}\gamma^{\ast\,l})$,
$\widehat{p}(\alpha^{\ast\,m}\gamma^{k}\gamma^{\ast\,l})$ follow an analogous
rule. For example

$p(\alpha\gamma\gamma^{\ast})=-\dfrac{(q^{6}+3q^{4}+2q^{2}+1)(q^{2}+q^{4})}{q^{6}+2q^{4}+2q^{2}+1}\alpha+(q^{6}+3q^{4}+2q^{2}+1)\alpha\gamma\gamma^{\ast}$

and

$\widehat{p}(\alpha\gamma\gamma^{\ast})=-\dfrac{(q^{4}+2q^{2}+q^{-2}+3)(1+q^{2})}{q^{4}+2q^{2}+q^{-2}+2}\alpha+(q^{4}+2q^{2}+q^{-2}+3)\alpha\gamma\gamma^{\ast}.$

In addition, let us define the the $q^{2}$–number

$[r]:=[r]_{q^{2}}=\dfrac{1-q^{2r}}{1-q^{2}}$

for all $r$ $\in$ $\mathbb{N}$. Then let us take

$\lambda_{m,k,l}:=\dfrac{1}{2}\left([m]\,[l+1]\,q^{2(2-l)}+[k]\,[l+1]\,q^{4+2m-2l}+[l]\,[m+1]\,q^{2(1-l)}+[l]\,[k]\,q^{4+2m-2l}\right),$
$\lambda_{-m,k,l}:=\dfrac{1}{2}\left([m]\,[k+1]\,q^{2(1-m)}+[l]\,[k+1]\,q^{2-2m-2l}+[k]\,[m+1]\,q^{2(2-m)}+[l]\,[k]\,q^{4-2m-2l}\right),$
$\widehat{\lambda}_{m,k,l}:=\dfrac{1}{2}\left([m]\,[l+1]\,q^{2-2m-2k}+[k]\,[l+1]\,q^{2(1-k)}+[l]\,[m+1]\,q^{4-2m-2k}+[l]\,[k]\,q^{2(3-k)}\right)$

and

$\widehat{\lambda}_{-m,k,l}:=\dfrac{1}{2}\left([m]\,[k+1]\,q^{4-2k+2l}+[l]\,[k+1]\,q^{2(2-k)}+[k]\,[m+1]\,q^{2-2k+2l}+[k]\,[l]\,q^{2(1-k)}\right).$

Values of the first columns form linear basis of $\mathrm{SU}_{q}(2)$, thus
for each $n$ $\in$ $\mathbb{Z}$, these sections form a basis of eigenvectors.

######  Proposition 4.2.

Considering
$\textsc{Mor}(n,{{}_{\mathrm{SU}_{q}(2)}}\Phi)=\Gamma^{\mathrm{L}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})=\Gamma^{\mathrm{R}}(\mathbb{S}^{2}_{q},\mathbb{C}_{n}\mathbb{S}^{2}_{q})$
just as a vector space, the operators
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}$
and
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}$
are not simultaneously diagonalizable for each $n$ $\in$ $\mathbb{Z}$.

###### Proof.

We are going to prove that these operators do not commute each other. In fact

$(\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n})(\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n})T\not=(\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n})(\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n})T,$

where $T(\mathbbm{1})=\alpha^{n}\gamma\gamma^{\ast}$ for $n>0$;
$T(\mathbbm{1})=\alpha^{\ast\,n}\gamma\gamma^{\ast}$ for $n<0$ and
$T(\mathbbm{1})=\alpha\gamma\gamma^{\ast\,2}$ for $n=0$. ∎

As we checked in the Subection $4.2.3$, the operators
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}$
and
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}$
are symmetric and non–negative.

There is a kind of $\ast$–symmetry between both operators, at least for the
first five eigenvalues presented. Moreover, the eigenvalues are not symmetric
under the change $n\longleftrightarrow-n,$ which is a difference with the
classical case [K]. In contrast and in agreement with the classical case, both
operators are not bounded. For example, let us fix $n$ and consider the
eigenvalue of the fifth row of the table $6.1$

$\dfrac{1}{2}\left([l]\,[m+1]\,q^{2(1-l)}+[m]\,[l+1]\,q^{2(2-l)}\right)=-\dfrac{q^{2}+q^{6}+2q^{2n+4}}{2(1-q^{2})^{2}}+\dfrac{q^{2}(1+q^{2})}{2(1-q^{2})^{2}}(q^{-2l}+q^{2m+2}).$

The first term in the right–hand side of the previous equality is a fixed
number, and also the term $\dfrac{q^{2}(1+q^{2})}{2(1-q^{2})^{2}}$. However,
since $m-l=n$

$q^{-2l}+q^{2m+2}=q^{2n-2m}+q^{2m+2}\;\Longrightarrow\;\lim_{m\rightarrow\infty}q^{2n-2m}+q^{2m+2}=\pm\infty,$

depending of the sign of $q$. By taking the classical limit $q\longrightarrow
1$, both operators reproduces the spectrum of the Laplacian on associated
vector bundles of the Hopf fibration [K].

$T(1)$ | $n\in\mathbb{Z}$ |  $\lambda$
---|---|---
$\mathbbm{1}$ | $0$ |  $0$
$\alpha^{m}\gamma^{k}$ | $m+k=n$ |  $\dfrac{[n]\,q^{4}}{2}$
$\alpha^{\ast\,n}$, $\gamma^{\ast\,n}$ | $n>0$ |  $\dfrac{[n]\,q^{2(1-n)}}{2}$
$\alpha^{\ast\,m}\gamma^{\ast\,l}$ | $m+l=n$ |  $-\dfrac{[-n]\,q^{2}}{2}$
$\alpha^{m}\gamma^{\ast\,l}$ | $m-l=n$ |  $\dfrac{1}{2}\left([l]\,[m+1]\,q^{2(1-l)}+[m]\,[l+1]\,q^{2(2-l)}\right)$
$\alpha^{\ast\,m}\gamma^{k}$ | $-m+k=n$ |  $\dfrac{1}{2}\left([m]\,[k+1]\,q^{2(1-m)}+[k]\,[m+1]\,q^{2(2-m)}\right)$
$p(\gamma^{k}\gamma^{\ast\,l})$ | $k-l=n$ |  $\dfrac{1}{2}\left([l]\,q^{2(1-l)}+[k]\,q^{4}+2\,[l]\,[k]\,q^{2(2-l)}\right)$
$p(\alpha^{m}\gamma^{k}\gamma^{\ast\,l})$ | $m+k-l=n$ |  $\lambda_{m,k,l}$
$p(\alpha^{\ast\,m}\gamma^{k}\gamma^{\ast\,l})$ | $-m+k-l=n$ |  $\lambda_{-m,k,l}$
Table 1. Values for $\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}\,T=\lambda\,T$. $\widehat{T}(1)$ | $n\in\mathbb{Z}$ |  $\widehat{\lambda}$
---|---|---
$\mathbbm{1}$ | $0$ |  $0$
$\alpha^{m}\gamma^{k}$ | $m+k=n$ |  $-\dfrac{[-n]\,q^{2}}{2}$
$\alpha^{n}$, $\gamma^{n}$ | $n>0$ |  $\dfrac{[n]\,q^{2(1-n)}}{2}$
$\alpha^{\ast\,m}\gamma^{\ast\,l}$ | $m+l=n$ |  $\dfrac{[n]\,q^{4}}{2}$
$\alpha^{m}\gamma^{\ast\,l}$ | $m-l=n$ |  $\dfrac{1}{2}\left([m]\,[l+1]\,q^{2(1-m)}+[l]\,[m+1]\,q^{2(2-m)}\right)$
$\alpha^{\ast\,m}\gamma^{k}$ | $-m+k=n$ |  $\dfrac{1}{2}\left([k]\,[m+1]\,q^{2(1-k)}+[m]\,[k+1]\,q^{2(2-k)}\right)$
$\widehat{p}(\gamma^{k}\gamma^{\ast\,l})$ | $k-l=n$ |  $\dfrac{1}{2}\left([l]\,q^{2(2-k)}+[k]\,q^{2(1-n)}+[l]\,[k]\,(1+q^{4})q^{2(1-k)}\right)$
$\widehat{p}(\alpha^{m}\gamma^{k}\gamma^{\ast\,l})$ | $m+k-l=n$ |  $\widehat{\lambda}_{m,k,l}$
$\widehat{p}(\alpha^{\ast\,m}\gamma^{k}\gamma^{\ast\,l})$ | $-m+k-l=n$ |  $\widehat{\lambda}_{-m,k,l}$
Table 2. Values for
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}\,\widehat{T}=\widehat{\lambda}\,\widehat{T}$.

##  5\. Concluding Comments

There are a lot of interesting papers about the quantum Hopf fibration and its
associated qvbs, as well as a treatment of gauge theory in this space, for
example, [HM], [La], [LZ], [LRZ], [Z]. All of them follow the line of research
of S. Majid and T. Brzeziński shown in [BM]. Unlike all these papers, the work
shown here follows the line of research of M. Durdevich in which we deal with
two kind of covariant derivatives for any qpc (both agree in the classical
case); this allows us to define induced qlc in left/right associated qvb as
well as the Lagrangians and their respective field equations. For example, if
we do not consider the right structure, Equation 7 becomes into

$\langle\Upsilon_{n}\circ
K^{\lambda}(T_{1})\,|\,\nabla^{\omega^{\mathrm{c}}}_{n}T_{1}\rangle_{\mathrm{L}}=0,$

which does not have solutions for an arbitrary $n$. Furthermore, the operators
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n},$
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}$
are not the same, they do not even commute between them!. It strongly suggests
that it is important to take into account both of structures.

The general theory is, at first sight, too restrictive in the sense that too
many conditions are necessary; however this example and the others developed
show that these conditions are (relatively) easy to satisfy. This theory can
be applied perfectly to other spaces; there are a lot of illustrative and rich
examples to study.

In this work we focused in the canonical qpc, just because it is the
non–commutative counterpart of the principal connection induced by the
Levi–Civitta connection; nevertheless, there is no problem in considering
other qpcs.

In terms of a physical interpretation, this space models left space–time
scalar matter fields and right space–time scalar antimatter fields coupled to
a magnetic monopole. Since the spectrums of
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}$
and
$\widehat{\nabla}^{\omega^{\mathrm{c}}\,\star_{\mathrm{R}}}_{n}\widehat{\nabla}^{\omega^{\mathrm{c}}}_{n}$
are discrete, the eigenvalues could be interpreted as quantum numbers.

Finally, it is worth mentioning that by considering all the fixed elements of
[La] instead of ours (for example $\omega_{\pm}$ instead of $\eta_{\pm}$), the
operator
$\nabla^{\omega^{\mathrm{c}}\,\star_{\mathrm{L}}}_{n}\nabla^{\omega^{\mathrm{c}}}_{n}$
is exactly the gauge Laplacian operator shown in [La] and all its results can
be reproduced.

## References

  * [BM] Brzezinski, T & Majid, S. : Quantum Group Gauge Theory on Quantum spaces,  Commun. Math. Phys. 157, 591–638 (1993). Erratum: Commun. Math. Phys. 167–235 (1995).
  * [D1] Durdevich, M. : Geometry of Quantum Principal Bundles I,  Commun. Math. Phys 175 (3), 457-521 (1996).
  * [D2] Durdevich, M. : Geometry of Quantum Principal Bundles II,  Rev. Math. Phys. 9 (5), 531—607 (1997).
  * [D3] Durdevich, M. : Differential Structures on Quantum Principal Bundles,  Rep. Math. Phys. 41 (1), 91-115 (1998).
  * [HM] Hajac, P & Majid, S. : Projective Module Description of the $q$–monopole,  arXiv:math/9803003v2, 30 Aug 1998.
  * [KS] Klimyk, A. & Schmudgen, K. : Quantum Groups and their Representations,  Springer, 1997.
  * [K] Kuwabara,R. : On Spectra of the Laplacian on Vector Bundles,  J. Math. Tokushima Univ., 16, 1-23 (1982).
  * [La] Landi, G. : Twisted Sigma–Model Solitons on the Quantum Projective Line, Lett. Math. Phys., 108 (8), 1955–1983 (2018).
  * [LRZ] Landi, G.; Reina, C. & Zampini, A. : Gauge Laplacians on Quantum Hopf Bundles, Commum. Math. Phys., 287, 179–209 (2009).
  * [LZ] Landi, G.& Zampini, A. : Calculi, Hodge Operators And Laplacians on a Quantum Hopf Fibration,  Rev. Math. Phys. 23 (6), 575—613 (2011).
  * [P] Podles, P., Quantum Spheres, Lett Math Phys 14 193-202 (1987).
  * [Sa1] Saldaña, M, G, A. : Geometry of Associated Quantum Vector Bundles and the Quantum Gauge Group.  arXiv:2109.01550v2, 1 Dec 2021.
  * [Sa2] Saldaña, M, G, A. : Quantum Principal Bundles and Yang–Mills–Scalar–Matter Fields.  arXiv:2109.01554v2, 1 Dec 2021.
  * [Sa3] Saldaña, M, G, A. : Yang–Mills–Scalar–Matter Fields in the Two–Point Space.  arXiv:2112.00647v1, 1 Dec 2021.
  * [So] Sontz, S, B. : Principal Bundles: The Quantum Case,  Universitext, Springer, 2015.
  * [W1] Woronowicz, S, L. : Compact Matrix Pseudogroups,  Commun. Math. Phys. 111, 613-665 (1987).
  * [W2] Woronowicz, S, L. : Twisted $SU(2)$ Group. An Example of Non–Commutative Differential Calculus,  Publ. RIMS, Kyoto Univ. 23 (1987), 117–181.
  * [Z] Zampini, A. : Warped Products and Yang-Mills Equations on Noncommutative Spaces, Lett. Math. Phys., 105 (2), 221–243 (2015).