# ”Discrete” vacuum geometry as a tool for Dirac fundamental quantization of
Minkowskian Higgs model.

L. D. Lantsman
Wissenschaftliche Gesellschaft bei J$\rm\ddot{u}$dische Gemeinde zu Rostock,
Augusten Strasse, 20,
18055, Rostock, Germany;
Tel. 049-0381-799-07-24,
<EMAIL_ADDRESS>
###### Abstract

We demonstrate that assuming the ”discrete” vacuum geometry in the Minkowskian
Higgs model with vacuum BPS monopole solutions can justify the Dirac
fundamental quantization of that model.

The important constituent of this quantization is getting various rotary
effects, including collective solid rotations inside the physical BPS monopole
vacuum, and just assuming the ”discrete” vacuum geometry seems to be that
thing able to justify these rotary effects.

More precisely, assuming the ”discrete” geometry for the appropriate vacuum
manifold implies the presence of thread topological defects (side by side with
point hedgehog topological defects and walls between different topological
domains) inside this manifold in the shape of specific (rectilinear) threads:
gauge and Higgs fields located in the spatial region intimately near the axis
$z$ of the chosen (rest) reference frame.

This serves as the source of collective solid rotations proceeding inside the
BPS monopole vacuum suffered the Dirac fundamental quantization.

It will be argued that indeed the first-order phase transition occurs in the
Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac.

This comes to the coexistence of two thermodynamic phases inside the
appropriate BPS monopole vacuum.

There are the thermodynamic phases of collective solid rotations and
superfluid potential motions.

PACS: 12.38.Aw, 14.80.Bn, 14.80.Hv, 67.40.-w, 67.40.Bz, 68.18.Jk, 74.20.De,
74.25.-q.
Keywords: Non-Abelian Theory, BPS Monopole, Minkowski Space, Topological
Defects, Phase transitions, Superfluidity, Superconductivity.

###### Contents

  1. 1 Introduction.
  2. 2 Statement of problem.
  3. 3 How one can introduce ”discrete” geometry for degeneration spaces in Minkowskian Higgs models.
  4. 4 How topologically nontrivial threads arise inside the discrete non-Abelian vacuum manifold.
  5. 5  Specific of Higgs phase in Dirac quantisation scheme.

##  1 Introduction.

In the recent studies [1, 2, 3], the Gauss-shell reduction of the Minkowskian
Higgs model with vacuum BPS monopole solutions was discussed.

Such Gauss-shell reduction is, indeed, the particular case of the Dirac
fundamental quantization [4] of gauge theories, when such (Minkowskian) gauge
model is written down in terms of topological Dirac variables [5] $A_{i}^{D}$
($i=1,2$): transverse and gauge invariant (i.e. automatically physical)
functionals of gauge fields.

As a result of this Gauss-shell reduction, the picture of the BPS monopole
vacuum as a medium possessing manifest superfluid properties and various
rotary effects was given.

More exactly, superfluid properties of the BPS monopole vacuum suffered the
Dirac fundamental quantization [4] come to its potentiality, described
correctly by the Bogomol’nyi [1, 2, 6, 7, 8, 9, 10, 11],

${\bf B}=\pm D\Phi,$ (1.1)

and Gribov ambiguity [2, 7, 8, 12],

$[D^{2}_{i}(\Phi_{k}^{(0)})]^{ab}\Phi_{(0)b}=0,$ (1.2)

equations.

These Eqs. implicate the vacuum ”magnetic” field $\bf B$ and Higgs vacuum BPS
monopole modes $\Phi$ (in particular, topologically trivial modes
$\Phi_{(0)a}$).

$\Phi_{k}^{(0)}$ are Yang-Mills (YM) vacuum BPS monopole modes, and $D$ is the
(covariant) derivative.

The connection between the Bogomol’nyi and Gribov ambiguity equations is
realised via the Bianchi identity $D~{}B=0$ (the latter identity is equivalent
to say that the vacuum ”magnetic” field $\bf B$ is ”transverse”).

In Ref. [6] it was pointed out to the transparent analogy between the BPS
monopole vacuum and a liquid helium II specimen, possessing the manifest
superfluidity.

This analogy comes, in particular, to the mathematical resemblance between the
Bogomol’nyi equation (1.1) and the expression [6, 13]

${\bf v}_{0}=\frac{\hbar}{m}\nabla\Phi(t,{\bf r}),$ (1.3)

for the critical velocity ${\bf v}_{0}$ of superfluid potential motions inside
a liquid helium II specimen 111$v_{0}={\rm min}~{}(\epsilon/p)$ for the ratio
of the energy $\epsilon$ and momentum $p$ for quantum excitations spectrum in
the given liquid helium II specimen. Herewith [14] at velocities of the liquid
exceeding a critical velocity $v_{0}={\rm min}~{}(\epsilon/p)$ for the ratio
of the energy $\epsilon$ and momentum $p$ for quantum excitations spectrum in
the liquid helium II, the dissipation of the liquid helium energy occurs via
arising excitation quanta with momenta $\bf p$ directed antiparallel to the
velocity vector $\bf v$. Such dissipation of the liquid helium energy becomes
advantageous [15] just at  $\epsilon+{\bf
p~{}v}<0\Longrightarrow\epsilon-p~{}v<0.$ .

Eq. (1.3) implicates the phase $\Phi(t,{\bf r})$ of the complex-value helium
Bose condensate wave function $\Xi(t,{\bf r})\in C$, given as [6, 13]

$\Xi(t,{\bf r})=\sqrt{n_{0}(t,{\bf r})}~{}e^{i\Phi(t,{\bf r})},$ (1.4)

with $n_{0}(t,{\bf r})$ being the number of particles in the ground energy
state $\epsilon=0$.

$m$ is the mass of a helium atom.

On the other hand, for any potential motion, the condition [6]

${\rm rot}~{}{\rm grad}~{}{\Phi}=0$ (1.5)

is always fulfilled (to within a constant) for a scalar field $\Phi$.

In particular, the potentiality condition (1.5) is fulfilled for the crucial
velocity ${\bf v}_{0}$, (1.3), of superfluid potential motions inside a liquid
helium II, i.e.

${\rm rot}~{}{\bf v}_{0}=0.$

This implies the absence of vortices and friction forces inside the superfluid
component in a liquid helium II specimen.

The similar conclusion may be drawn also about the vacuum ”magnetic” field
$\bf B$ in the Minkowskian Higgs model with vacuum BPS monopole solutions,
described correctly by the Bogomol’nyi equation (1.1) [1, 2, 6, 7, 8, 9, 10,
11] (with the correction that in the latter case the operator ${\rm grad}$
would be replaced with the [covariant] derivative $D$).

Additionally, the Gribov ambiguity equation (Gribov equation) (1.2) the
Minkowskian BPS monopole vacuum as an incompressible medium.

This fact was discussed in Ref. [2]; it comes to the presence of typical
topological invariants in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac [4].

There are the magnetic charge ${\bf m}$ and the degree of the map referring to
the $U(1)\subset SU(2)$ embedding.

At resolving the YM Gauss law constraint [1, 2, 3, 7, 8, 12]

$\frac{\delta W}{\delta
A^{a}_{0}}=0\Longleftrightarrow[D^{2}(A)]^{ac}A_{0c}=D^{ac}_{i}(A)\partial_{0}A_{c}^{i}$
(1.6)

in terms of topological Dirac variables $A^{D}_{i}$, satisfying the gauge [5]

$D_{i}^{ab}(\hat{A}^{D})\partial_{0}(\hat{A}_{i}^{D})\equiv 0,$ (1.7)

the right-hand side of this constraint vanishes, and it becomes the
homogeneous equation [3, 7, 8, 12]

$[D^{2}_{i}(\Phi^{(0)})]^{ac}A_{0c}=0,$ (1.8)

permitting, in the Minkowskian Higgs model with vacuum BPS monopoles quantized
by Dirac [4], the family of zero mode solutions [12, 16],

$A_{0}^{c}(t,{\bf x})={\dot{N}}(t)\Phi_{(0)}^{c}({\bf x})\equiv Z^{c},$ (1.9)

implicating the topological variable $\dot{N}(t)$ and Higgs (topologically
trivial) vacuum Higgs BPS monopole modes $\Phi_{0}^{a}({\bf x})$.

The topological variables $\dot{N}(t)$ (respectively, $N(t)$), may be
specified [3, 7, 8, 12] via the relation

$\displaystyle\nu[A_{0},\Phi^{(0)}]$ $\displaystyle=$
$\displaystyle\frac{g^{2}}{16\pi^{2}}\int\limits_{t_{\rm in}}^{t_{\rm
out}}dt\int
d^{3}xF^{a}_{\mu\nu}\widetilde{F}^{a\mu\nu}=\frac{\alpha_{s}}{2\pi}\int
d^{3}xF^{a}_{i0}B_{i}^{a}(\Phi^{(0)})[N(t_{\rm out})-N(t_{\rm in})]$ (1.10)
$\displaystyle=N(t_{\rm out})-N(t_{\rm in})=\int\limits_{t_{\rm in}}^{t_{\rm
out}}dt\dot{N}(t),$

taking account of the natural duality between the tensors $F^{a}_{i0}$ and
$F^{a}_{ij}$.

Herewith $\nu[A_{0},\Phi^{(0)}]$ is referred to as the vacuum Chern-Simons
functional, implicating the asymptotical states ”in” and ”out” taking in the
time instants $t_{\rm in}$ and $t_{\rm out}$, respectively.

As it was discussed in Ref. [3] (repeating the arguments [17]), it may be set

$t_{\rm in}\to-\infty;\quad t_{\rm out}\to\infty.$

The nontrivial topological dynamics inherent in the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac comes to the specific Josephson
effect [17] in the enumerated model, i.e. to the existence of collective solid
rotations inside the physical BPS monopole vacuum.

These collective solid rotations are described correctly by the free rotator
action functional

$W_{N}=\int d^{4}x\frac{1}{2}(F_{0i}^{c})^{2}=\int dt\frac{{\dot{N}}^{2}I}{2}$
(1.11)

involving the rotary momentum [5]

$I=\int_{V}d^{3}x(D_{i}^{ac}(\Phi_{k}^{0})\Phi_{0c})^{2}=\frac{4\pi^{2}\epsilon}{\alpha_{s}}=\frac{4\pi^{2}}{\alpha_{s}^{2}}\frac{1}{V<B^{2}>}.$
(1.12)

The YM coupling constant

$\alpha_{s}=\frac{g^{2}}{4\pi(\hbar c)^{2}}$

enters this expression for $I$ together with the typical size $\epsilon$ [7,
8] of BPS monopoles and the vacuum expectation value $<B^{2}>$ of the
”magnetic” field.

In Ref. [3] there was argued that $\dot{N}(t)$ play the role of angular
velocities of collective solid rotations inside the Minkowskian Higgs vacuum
suffered the Dirac fundamental quantization [4].

Meanwhile, the action functional (1.11) contains vacuum ”electric” monopoles
(in the terminology [7, 8])

$F^{a}_{i0}={\dot{N}}(t)D^{ac}_{i}(\Phi_{k}^{(0)})\Phi_{(0)c}({\bf x})$ (1.13)

and is associated with the real spectrum

$P_{N}={\dot{N}}I=2\pi k+\theta;\quad\theta\in[-\pi,\pi];$ (1.14)

of the topological momentum $P_{N}$.

The one of manifestations of the Josephson effect [17] occurring inside the
Minkowskian Higgs vacuum suffered the Dirac fundamental quantization are the
existence of never vanishing (excluding the value $\theta=0$ of the
$\theta$-angle) vacuum ”electric” fields (”electric” monopoles) (1.13) in the
shape [5]

$F^{a}_{i0}\equiv
E_{i}^{a}=\dot{N}(t)~{}(D_{i}(\Phi_{k}^{(0)})~{}\Phi_{(0)})^{a}=P_{N}\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)})=(2\pi
k+\theta)\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a}(\Phi_{(0)}).$ (1.15)

Any vacuum ”electric” field (1.15) achieves its minimum in the zero
topological sector of the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac:

$(E_{i}^{a})_{\rm
min}=\theta\frac{\alpha_{s}}{4\pi^{2}\epsilon}B_{i}^{a};\quad-\pi\leq\theta\leq\pi.$
(1.16)

Such is, briefly, the picture of the (topologically nontrivial) dynamics
inherent in the Minkowskian Higgs model with vacuum BPS monopoles quantized by
Dirac [4].

This picture seems to be correct at least at the absolute zero temperature
$T=0$, when collective solid rotations inside the BPS monopole vacuum proceed
in the ”non-stop” regime [17] and ”friction forces” between this BPS monopole
vacuum and its surroundings are absent.

In Ref. [3], there was asserted that ”geometrically”, collective solid
rotations inside the BPS monopole vacuum suffered the Dirac fundamental
quantization [4] are, indeed, rotations around the infinitely narrow cylinder
of the effective diameter $\sim\epsilon$ (with $\epsilon$ being the typical
size of BPS monopoles) along the axis $z$ of the chosen (rest) reference frame
222 For the first time, the idea of ”cylinder topologies” in (non-Abelian)
gauge theories was proposed, probably in the paper [18] (where also the
topological dynamical variable $N(t)$ was introduced)..

In the present study we attempt to ground this assertion.

This will be associated immediately with assuming the ”discrete vacuum
geometry” [1, 3]

$R_{\rm YM}\equiv SU(2)/U(1)\simeq{\bf Z}\otimes G_{0}/U_{0},$

with

$\pi_{1}(U_{0})=\pi_{1}(G_{0})=0$

and

$SU(2)\equiv G;\quad U(1)\equiv U,$

for the degeneration space (vacuum manifold) in the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac, involving [9] the existence of
(rectilinear, topologically nontrivial) threads inside the mentioned vacuum
manifold, just located intimately near the axis $z$ of the chosen (rest)
reference frame.

Vortices generated by such threads, similar to those [15] one can observe in a
(rested) liquid helium II specimen, may be identified naturally with
collective solid rotations inside the BPS monopole vacuum suffered the Dirac
fundamental quantization [4].

Unlike the spatial region near the axis $z$, far off this axis (including the
region $|{\bf x}|\to\infty$), superfluid potential motions inside the BPS
monopole vacuum occur.

These motions are described correctly by the Bogomol’nyi, (1.1), and Gribov,
(1.2), equations.

The coexistence of (delimited somewhat in the space) collective solid
rotations and superfluid potential motions inside the BPS monopole vacuum
suffered the Dirac fundamental quantization is thus on hand [3].

It is the sign of the first-order phase transition occurring in the
Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac.

This phase transition is additional to the second-order one associated with
the spontaneous breakdown of the initial $SU(2)$ gauge symmetry down to the
$U(1)$ one.

In the present study we reveal the role of the discrete geometry assumed for
the vacuum manifold $R_{\rm YM}$ in this first-order phase transition just as
in the series of physical effects (including collective solid vacuum
rotations) taking place in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac.

##  2 Statement of problem.

The ensuing exposition is organized as follows.

In Section 3 (repeating partially the theses of Ref. [1]) we, utilizing the
general theory of topological defects (see e.g. §$\Phi$1 in [9]), construct
explicitly the degeneration space (vacuum manifold) $R_{YM}$ for the
Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac [4].

As we have seen already, it is possible to postulate the look [1] for this
manifold, implicating the discrete multiplier ${\bf Z}$.

Such look of the YM vacuum manifold $R_{YM}$ is destined by the discrete
factorisation [1] of the initial, $SU(2)$, and residual, $U(1)$, gauge
symmetries groups in the enumerated model:

$G\equiv SU(2)\simeq G_{0}\otimes{\bf Z};\quad H\equiv U(1)\simeq
U_{0}\otimes{\bf Z}.$

In Section 2, repeating the arguments [9], we discover three kinds of
topological defects inside the vacuum manifold $R_{YM}$. There are thread and
point hedgehog (stable) defects and domain walls between different topological
sectors of $R_{YM}$.

Herewith thread topological defects are of a great importance for us as those
determined physical rotary effects and the first-order phase transition taking
place in the Minkowskian Higgs model with vacuum BPS monopoles quantized by
Dirac. We shall make sure in this soon.

As it is well known [9], the sufficient condition for thread topological
defects to exist in a gauge theory involving the spontaneous breakdown of the
initial symmetry group is the isomorphism

$\pi_{1}(R)=\pi_{0}(H)\neq 0$ (2.1)

between the appropriate homotopical groups of the degeneration space $R$ and
residual symmetry group $H$ in the considered gauge theory.

This isomorphism is correct, in particular, for the vacuum manifold $R_{YM}$.

We ground this utilizing the holonomies group arguments (developed in Ref.
[7]; see also [9]).

The existence of point hedgehog topological defects inside the same
degeneration space $R$ is controlled by the isomorphism [6, 9]

$\pi_{2}(R)=\pi_{1}(H)\neq 0.$ (2.2)

These topological defects are always associated, in (Minkowskian) Higgs non-
Abelian models, with various monopole solutions (typical of which will be us
discussed below). Geometrically, they are located at the origin of
coordinates.

Besides two said types of topological defects, the YM vacuum manifold $R_{YM}$
possesses the some more kind of topological defects. There are domain walls
between different topological sectors of this manifold.

The cause of said domain walls is in the manifest ”discrete” geometry of the
Minkowskian YM vacuum manifold $R_{YM}$ [9]:

$\pi_{0}(R_{YM})={\bf Z}.$

Indeed, the width of a domain wall is roughly proportional to the inverse of
the lowest mass of all the physical particles being present in the (gauge)
model considered [19].

In the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac it
is just the typical size $\epsilon\sim(m/\sqrt{\lambda})^{-1}$ of BPS
monopoles.

Thus in the Bogomol’nyi-Prasad-Sommerfeld (BPS) limit [6, 7, 8, 9, 10]

$m\to 0;\quad\lambda\to 0$ (2.3)

for the Higgs mass $m$ and selfinteraction constant $\lambda$, widths of
domain walls inside $R_{YM}$ can take arbitrary values, including infinite
ones.

Actually [5, 7, 8], $\epsilon$ is the function of the spatial volume $V\sim
r^{3}$:

$\epsilon(V)\sim V^{-1}\propto r^{-3}.$

Thus walls between topological domains become in fact infinite (of the typical
width $\epsilon(0)\to\infty$) in the spatial region $r=0$ (including the
infinite narrow cylinder of the effective diameter $\epsilon(\infty)\to 0$
with its axis symmetry $z$).

Vice verse, as $|{\bf x}|\to\infty$, domain walls become infinitely thin (of
the typical width $\epsilon(\infty)\to 0$).

In Ref. [3], it was argued (implicating the general QFT arguments) that
asymptotical (vacuum) states ”in” and ”out” (cf. (1.10)) in the Minkowskian
Higgs model with vacuum BPS monopoles quantized by Dirac would be separated by
the infinite time interval $T\to\infty$ [17].

Assuming infinite widths for domain walls inside the vacuum manifold $R_{YM}$
in the spatial region intimately near the axis $z$ of the chosen (rest)
reference frame is in a good agreement with this setting $T\to\infty$.

Section 4 of the present study we devote to the discussion about
(topologically nontrivial) rectilinear threads containing inevitably inside
the vacuum manifold $R_{YM}$.

It will be shown, repeating the arguments [9], that there are YM fields

$A_{\theta}(\rho,\theta,z)\equiv A_{\mu}\partial x^{\mu}/\partial\theta$

These fields may be always represented in the $z$-invariant shape as [9]

$A_{\theta}(\rho,\theta,z)=\exp(iM\theta)A_{\theta}(\rho)\exp(-iM\theta),$
(2.4)

with $M$ being the generator of the group $G_{1}$ of rigid rotations
compensating changes in the vacuum (Higgs-YM) configuration
$(\Phi^{a},A_{\mu}^{a})$ at rotations around the axis $z$ of the chosen (rest)
reference frame.

In (2.4),

$A_{\theta}(\rho)=M+\beta(\rho),$

where the function $\beta(\rho)$ approaches zero as $\rho\to\infty$.

The elements of $G_{1}$ can be set as [9]

$g_{\theta}=\exp(iM\theta).$ (2.5)

YM fields $A_{\theta}$ are manifestly invariant with respect to shifts along
the axis $z$.

It is important that rectilinear threads $A_{\theta}$ don’t coincide with
vacuum YM BPS monopole solutions, and, on the contrary, there are, indeed,
gaps between directions of ”magnetic” tensions vectors: ${\bf B}_{1}$,

$|{\bf B}_{1}|\sim\partial_{\rho}A_{\theta}(\rho,\theta,z),$ (2.6)

and $\bf B$, given by the Bogomol’nyi equation (1.1) and diverging as $r^{-2}$
at the origin of coordinates [10].

These gaps just testify in favour the above discussed first-order phase
transition occurring in the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac [4].

In the Higgs sector of that model, there exist [9] also z-invariant (vacuum)
Higgs solutions in a (small) neighbourhood of the origin of coordinates:

$\Phi^{(n)}(\rho,\theta,z)=\exp(M\theta)~{}\phi(\rho)\quad(n\in{\bf
Z}),\quad\nabla_{\mu}\phi(\rho)\leq{\rm
const}~{}\rho^{-1+\delta};\quad\delta>0;$ (2.7)

that can join (in a smooth wise) vacuum Higgs BPS monopoles, belonging to the
same topology $n$ and disappearing [20] at the origin of coordinates.

Herewith, speaking ”in a smooth wise”, we imply that the covariant derivative
$D\Phi$ of any vacuum Higgs field $\Phi_{a}^{(n)}$ merges with the covariant
derivative of such a vacuum Higgs BPS monopole solution.

Following [9], the vacuum $z$-invariant, i.e. axially symmetric, (Higgs-YM)
configuration $(\Phi^{a},A_{\mu}^{a})$ (given via Eqs. (2.4) and (2.7),
possessing, as it can be demonstrated [9], a finite linear energy density and
obeying the appropriate equations of motions) can be treated as a rectilinear
thread solution (called also the rectilinear thread vortex or the rectilinear
thread).

The same terminology ”rectilinear threads” is applicable (as we have seen this
above with the example of Eq. (2.4) [9]) to z-invariant (axially symmetric) YM
fields $A_{\theta}$.

Obvious locating (2.4) of (topologically nontrivial) threads $A_{\theta}$ at
the origin of coordinates (actually, in the spatial region $\rho\to 0$; the
same is correctly also for Higgs thread modes $\Phi^{a}$ due to (2.7)) permits
the natural (somewhat alternative) geometrical interpretation of
(topologically nontrivial) threads as infinitely narrow tubes around the axis
$z$ over which the family of vortex solutions $(\Phi^{a},A_{\mu}^{a})$ is
specified actually (disappearing rapidly outside this spatial region).

The requirement for vacuum Higgs fields $\Phi_{a}^{(n)}$ to be smooth is quite
natural if the goal is pursued, in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac [4], to justify various rotary effects inherent
in this model.

In particular, vacuum ”electric” monopoles (1.15) [5] are directly
proportional to $D_{i}(\Phi_{k}^{(0)})~{}\Phi_{(0)}$.

These vacuum ”electric” monopoles, in turn, enter explicitly the action
functional (1.11), describing, in the Dirac fundamental quantization scheme
[4], collective solid rotations inside the Minkowskian BPS monopole vacuum.

Such (smooth) sawing together appropriate vacuum Higgs modes $\Phi^{(n)}$
(which are [9] specific thread rectilinear vortices) and BPS monopoles is
intended to remove the seeming contradiction between the manifest superfluid
properties of the Minkowskian BPS monopole vacuum (suffered the Dirac
fundamental quantization [4]), setting by the Bogomolny’i, (1.1), and Gribov
ambiguity, (1.2), equations.

Moreover, one can assert [16] that

$D~{}B\sim D~{}E=0$ (2.8)

for vacuum ”magnetic” and ”electric” tensions in the quested Minkowskian Higgs
model, i.e. that these tensions are, indeed, ”transverse” vectors colinear
each other.

This just implies the potential nature of the ”electric” tension $\bf E$, that
can be perceived as the above contradiction, on the face of it. Note that Eq.
(1.15) [5] just reflects this colinearity.

Going out from this contradiction seems to be just in locating (topologically
nontrivial) threads in the infinitely narrow cylinder of the effective
diameter $\epsilon$ around the axis $z$ and in joining (in a smooth wise)
vacuum Higgs fields $\Phi_{a}^{(n)}$ and BPS monopole solutions.

In this case collective solid rotations (vortices) inside the Minkowskian BPS
monopole vacuum, occurring actually in that spatial region around the axis $z$
and described correctly by the action functional (1.11), become quite
”legitimate”, and simultaneously, the Gauss law constraint (1.8) is satisfied
outward this region with smooth vacuum ”electric” monopoles $E_{i}^{a}$ [5],
(1.15). Latter ones can be treated as thread solutions since vacuum Higgs
fields $\Phi_{a}^{(n)}$ are such (the same assertion is correct, of course,
for Higgs vacuum BPS monopoles).

On the other hand, in the region of thread topological defects inside the
discrete vacuum manifold $R_{\rm YM}$, Eq. (2.8) is violated since the vacuum
”magnetic” field $\bf B$ suffers a break in this region according to the
arguments [10] (the vacuum ”magnetic” field $\bf B$ set for BPS monopole
solutions via the Bogomol’nyi equation (1.1) [9, 10, 11] diverges as $r^{-2}$
at the origin of coordinates).

In other words, the plot of $D\Phi$ behaves as $r^{-2}$ at the origin of
coordinates for vacuum Higgs BPS monopole modes $\Phi$ (whence $DB\sim
D^{2}\Phi=O(r^{-3})$ therein), while the plot of $D\Phi$ for Higgs thread
solutions $\Phi$ [9] joins the plot for the transverse vacuum “electric” field
$E\sim D\Phi$ (2.8) [16] (in the latter case $\Phi$ are vacuum BPS monopole
solutions, specified via the Bogomol’nyi equation (1.1)). Such is the
geometrical picture connected with Eq. (2.8).

The said is the one more argument in favour of the first-order phase
transition occurring in the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac and coming therein to the coexistence of two thermodynamic
phases: the thermodynamic phases of collective solid rotations and superfluid
potential motions, inside the appropriate Minkowskian BPS monopole vacuum.

Herewith the enough clear-cut picture can be observed how the enumerated
thermodynamic phases are distributed inside the discrete vacuum manifold
$R_{\rm YM}$.

Thread topological defects (vortices), associated with rectilinear threads
$A_{\theta}$, are located intimately near the axis $z$ of the chosen (rest)
reference frame. Actually, they refer to the cylinder of the effective
diameter $\epsilon$ with $z$ serving its symmetry axis.

Simultaneously, superfluid potential motions refer to the spatial region out
of this cylinder, including the spatial region $|{\bf x}|\to\infty$
(corresponding to the infrared region of the momentum space).

The important consequence of the presence of rectilinear threads $A_{\theta}$
in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac
[4] and involving the ”discrete” vacuum geometry for the vacuum manifold
$R_{\rm YM}$ is the effect [9] of annihilating two equal magnetic charges
${\bf m}_{1}={\bf m}_{2}={\bf m}(n)\neq 0$ ($n\in{\bf Z}$) colliding at
crossing a rectilinear topologically nontrivial thread $A_{\theta}(n)$.

This effect takes place actually within a fixed topological domain inside the
discrete vacuum manifold $R_{\rm YM}$, possessing the topological number $n$.

Colliding magnetic charges with different topological numbers are suppressed
in the spatial region near the axis $z$ (of the chosen rest reference frame)
by large widths of domain walls in this spatial region.

The said means the possible annihilation of all the topologically nontrivial
YM vacuum BPS monopole modes and excitations over this BPS monopole vacuum
(suffered the Dirac fundamental quantization [4]) during a definite time.

As a consequence of such possible annihilation, Higgs (BPS monopole) modes
become free electric fields (at least in the region out of vortices cores,
where superfluid potential motions occur inside the vacuum manifold $R_{\rm
YM}$): their electric charges $e$ are dual (due to the Dirac quantization [21]
of electric and magnetic charges) to zero magnetic charges can only survive
upon the above-described annihilation.

Such situation when Higgs modes possess arbitrary electric charges $e$, while
magnetic charges ${\bf m}\neq 0$ are confined is referred to as the Higgs
phase in modern physical literature (see e.g. [22]).

If quarks are incorporated in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac [4], disappearing topologically nontrivial YM
modes via the ”colliding” mechanism [9] can cause the possibility to observe
free ”coloured” quarks in the spatial region near the axis $z$.

The said can serve as a (perhaps, enough rough) representation for the
asymptotical freedom of quarks in that model.

Vice verse, in the spatial region $|{\bf x}|\to\infty$, where walls between
different topological domains inside $R_{\rm YM}$ become highly thin, the
infrared topological and physical confinement of quarks (in the spirit [12,
23]).

Such confinement can be ensured, for instance, by surviving, upon colliding
[9] at threads $A_{\theta}$, topologically trivial and, perhaps, topologically
nontrivial, YM modes.

In detail, the specific of the Higgs phase in the non-Abelian model (with BPS
monopoles) quantized by Dirac [4] will be us discussed in Section 5.

Also we discuss important consequences for QCD (based on the Minkowskian Higgs
model with vacuum BPS monopoles quartzes by Dirac) of assuming the ”discrete”
geometry.

The thing is that the Bogomol’nyi equation (1.1) is fulfilled to within the
sign before the (covariant) derivative $D$ [11].

Herewith it can be shown (repeating the arguments [9]) that this duality in
specifying the sign of the vacuum ”magnetic” field $\bf B$ becomes unavoidable
when there exists the (already mentioned) group $G_{1}$ of global rotations,
compensating changes in the vacuum (YM-Higgs) configuration
$(\Phi^{a},A_{\mu}^{a})$ at rotations around the axis $z$ of the chosen (rest)
reference frame.

This causes inverting the sign before the generator [9] $h\equiv
h(\Phi)\equiv\Phi/a$ (with $\Phi$ being a Higgs BPS monopole mode and
$a^{2}\equiv m/\sqrt{\lambda}$) of the residual $U(1)$ gauge symmetry group in
the quested Minkowskian Higgs model.

More exactly, to within an isomorphism, the elements [9]

$\alpha(n)=(P\exp(-\int\limits_{0}^{2\pi}A_{\theta}d\theta))^{-1}$ (2.9)

(with $P$ standing for parallel transports along appropriate curves
[integration ways]) of the holonomies group [7, 9] $H\simeq U(1)$ may be
constructed involving (topologically nontrivial) threads $A_{\theta}$.

In this case there exist gauge transformations [9]

$\tilde{h}(\Phi)=\alpha(2k)h(\Phi)\alpha(2k)^{-1}=-h(\Phi)$ (2.10)

mapping the topological domain with a topological number $k$ in the discrete
group space $H$ (and by that the appropriate topological domain inside the
degeneration space $R_{\rm YM}$) into the topological domain with the
topological number $-k$.

Herewith the change in the sign of $\bf B$ in the Bogomol’nyi equation (1.1)
occurs automatically at such mapping, accompanied also by changing the signs
of magnetic charges $\bf m$.

Thus vacuum ”magnetic” fields $\bf B$ and $-\bf B$ can be identified by means
of gauge transformations (2.10), implicating (topologically nontrivial)
threads $A_{\theta}$, and this implies, regarding the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac, the manifest invariance of the
appropriate vacuum ”magnetic” energy, squared by $|\bf B|$, with respect to
changes in the sign of the vacuum ”magnetic” field $\bf B$.

The same gauge transformations (2.10) influence also the ”electric” energy of
the Minkowskian BPS monopole vacuum (suffered the Dirac fundamental
quantization [4]), squared by the topological momentum $P_{N}$ and given by
the action functional (1.11).

As a result, the ”electric” energy of this vacuum and the action functional
(1.11) corresponding to this ”electric” energy prove to be also invariant with
respect to changes in signs of topological charges $k$.

Just this reasoning about the vacuum ”magnetic” and ”electric” energies of the
Minkowskian BPS monopole vacuum allows to draw the conclusion that the look of
the initial and residual gauge symmetries groups in the Minkowskian Higgs
model with vacuum BPS monopoles quantized by Dirac would be modified:
respectively,

$G_{M}=G_{0}\otimes{\bf Z}/{\mathbb{Z}}_{2};\quad U_{1}=U_{0}\otimes{\bf
Z}/{\mathbb{Z}}_{2}.$

This implies the modification

$R^{\prime}_{YM}=({\bf Z}/{\mathbb{Z}}_{2})\otimes G_{0}/U_{0}$

in the look of the appropriate degeneration space (vacuum manifold) $R_{\rm
YM}$.

Section 5 of the present study we devote to the specific of the Higgs phase in
the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac.

Above, this specific was us characterized; so we shall not repeat ourselves.
On the other hand, note the following.

It seems to be a very useful thing, to compare (as it will be done in Section
5) the view about the Higgs phase at applying the Dirac fundamental
quantization method [4] to the Minkowskian Higgs model (with vacuum BPS
monopoles) with that associated with the so-called dual superconducting
picture of confinement for quark and magnetic monopoles [22].

This conception of the dual superconducting picture, enveloping QCD based on
the Minkowskian Higgs model with violating the ”initial” $SU(2)$ gauge
symmetry, includes the ”Mandelstam” linearly increasing potential $O(Kr)$
(with $K={\rm const}$ being the string tension [24]) and fixing the maximal
Abelian gauge (MAG), coming to the maximal diaganalization of the $SU(2)$
gauge matrices.

MAG can be achieved in the Yang-Mills (YM) theory by means maximizing the
integral

$\int d^{4}x\left[(A_{\mu}^{1})^{2}+(A_{\mu}^{2})^{2}\right];\quad
A_{\mu}=A_{\mu}^{a}\tau_{a}.$

It is equivalent to the fact (see e.g. Ref. [25]) of maximization of the
quantity

$\sum_{x}\sum_{\mu=1}^{4}\mbox{Tr}[\tau_{3}U_{\mu}(x)\tau_{3}U^{\dagger}_{\mu}(x)],$

where $U_{\mu}(x)$ are $SU(2)$ gauge matrices.

MAG can be got [26] by imposing the ”Lorentz” gauge

$D_{\mu}^{ab}A_{\mu}^{b}=0.$

The dual superconducting picture of confinement includes the Higgs phase, in
which Higgs modes are considered [22] as ”free” particles with nonzero
electric charges while all the magnetic charges are confined, and also the
confinement phase [22], dual to the Higgs phase, when Higgs modes become free
and purely ”magnetic” objects while gluons, monopoles and (incorporated)
quarks are confined by (infinitely narrow) Meissner flux tubes [24], of the
hadronic size $\sim$ 1 fm ($\sim$ 5 GeV-1), similar to ones in a
superconductor.

Herewith the role of the Mandelstam linearly increasing potential $O(Kr)$ [24]
between (electric, magnetic) charges comes in this confinement mechanism just
to breaking Meissner flux tubes (strings) exceeding the hadronic size $\sim$ 1
fm. Such breaking is advantageous from the energetic standpoint.

It is transparent, in the light of the said above, that this dual
superconducting picture of confinement foresees violating the ”initial”
$SU(2)$ gauge symmetry group down to its $U(1)$ subgroup, that implicates the
appearance of Higgs modes. In this case MAG fixing seems to be quite correct.

Indeed, it is not enough to violate the ”initial” $SU(2)$ gauge symmetry group
down to its $U(1)$ subgroup in order to achieve the dual superconducting
picture of confinement.

On the contrary, this Abelian $U(1)$ gauge symmetry, with Higgs modes
appearing upon MAG fixing, implying the breakdown of the initial $SU(2)$ gauge
symmetry, would be violated farther, and this is the thing necessary for
justifying the dual superconducting picture of confinement [22] in the YM
theory.

Really, explaining the dual superconducting picture of confinement [22] can be
found in the framework of the Nielsen-Olesen (NO) Abelian Higgs model [27],
involving NO vortices.

This model serves as a highly correct representation for the Abelian Higgs
theory, in which the initial $U(1)$ gauge symmetry group is violated.

On the other hand, just MAG fixing can reduce the initial non-Abelian Higgs
model to the NO one.

In this case there exist specific solutions to the equations of motions, so-
called NO vortices [27].

Herewith there is a highly transparent analogy between NO vortices and above-
mentioned Meissner flux tubes.

Discussing briefly the NO model [27] in Section 5, we demonstrate that NO
vortices are the particular case of thread topological defects inside the
appropriate vacuum manifold in this model swept by the nonzero value
$|\phi|\neq 0$ for the Higgs field $\phi$ in the ”asymmetric” phase.

The important task in this situation is to ascertain the look of this
manifold.

This task is connected closely with the thermodynamic specific of the Abelian
Higgs model [27], involving NO vortices.

It turns out that the first-order phase transition takes place in this model,
accompanied by gradual violating the $U(1)$ gauge symmetry (contrary to
”momentary” violating gauge symmetries groups in theories involving second-
order phase transitions [6, 28]).

The typical picture of first-order phase transitions in gauge theories is
following.

At first-order phase transitions in such theories, ”bubbles” [28] of
”asymmetric” thermodynamic phases are ”conceived” inside ”symmetric”
thermodynamic phases at temperatures below a Curie point $T_{c}$.

There are always prolonged processes at which ”symmetric” thermodynamic phases
become metastable, coexisting during a definite time with stable
(”asymmetric”) thermodynamic phases.

On the other hand, order parameters (a $\phi(T)$) appear (disappear) rapidly
at first-order phase transitions, and this implies the finite difference
$\Delta F$ of the free energy. This difference is called the latent heat of
the given first-order phase transition [29]. Such situation coexisting two
thermodynamic phases below $T=T_{c}$ is referred to as supercooling in modern
literature (e.g. [29]).

When the temperature $T$ decreases further, down to some its value
$T_{b}<T_{c}$ (or when the system of fields is perturbed by either external or
internal fluctuations), it drops rapidly in the ”true vacuum” corresponding to
the stable (”asymmetric”) thermodynamic phase with $\phi(T)=\phi_{0}$
(herewith the system leaves the ”false vacuum” $\phi(T)=0$ corresponding to
the metastable, ”symmetric” thermodynamic phase).

As a result, the metastable thermodynamic phase disappears entirely, and the
appropriate latent heat is liberated in this dropping process. This phenomenon
is referred to as reheating  (see e.g. [29]) 333The above discussed dropping
the system of fields in the ”true vacuum” $\phi(T)=\phi_{0}$ in gauge theories
implicating first-order phase transitions can be explained [29] by thermal
fluctuations or by various tunnelling effects shifting this system of fields
over the potential barrier (discontinuity) between $\phi(T)=0$ and
$\phi(T)=\phi_{0}$..

The Abelian Higgs model [27], involving NO vortices, is the particular case of
gauge theories involving first-order phase transitions.

Thus all the us discussed above remains valid also for the model [27].

The specific only in choosing the vacuum expectation value $<|\phi|^{2}>$ as
the order parameter in that model, with all ensuing consequences.

From the geometrical standpoint, the first-order phase transition in the
Abelian Higgs model [27] can be explained by assuming the gradual destruction
of the continuous $U(1)\simeq S^{1}$ group space in that model to a discrete
set, can be obtained at cutting off this group space by its topological
domains (there are, indeed, the countable number ${\bf Z}=\pi_{1}S^{1}$ of
such topological domains).

Then at the first-order phase transition taking place in the NO model [27],
the ”bubble” of the ”asymmetric” (stable) thermodynamic phase inside the
”symmetric” (metastable) one is represented just by the above discrete set.

In turn, the metastable thermodynamic phase in the NO model [27] can be
represented by the continuous set consisting of a part of topological domains
remained still ”gluing together”.

In the temperature interval $[T_{b};T_{c}]$, corresponding to the supercooling
situation [29] in the Abelian Higgs model [27], these (continuous and
discrete) sets coexist.

But when the temperature descends below $T_{b}$, the metastable thermodynamic
phase, represented by the topological domains ”gluing together”, disappears
entirely. Herewith only the discrete set, corresponding to the stable
thermodynamic phase and ”true” vacuum $|\phi|=\phi_{0}$, survives.

Such is the outline of the first-order phase transition occurring in the
Abelian Higgs model [27] with NO vortices.

Assuming this scheme, involving the ”discrete” vacuum geometry (it can be
shown that the vacuum manifold in this case is just isomorphic to the above
discussed discrete set, we shall denote as $\tilde{U}$), for the first-order
phase transition can serve for justifying namely NO vortices, as a particular
case of thread topological defects, in the model [27].

Geometrically, NO vortices are located inside the appropriate vacuum manifold
that is isomorphic to $\tilde{U}$.

The reasoning here similar to that in the Minkowskian Higgs model with vacuum
BPS monopoles quantized by Dirac, that is the basic topic of the present
study.

The distinction only in some (although greatly important) details regarding
the nature of vortices in the enumerated models.

For understanding the role of the Abelian Higgs model [27] with NO vortices in
the dual superconducting picture of confinement in QCD, the transparent
analogy between physical processes occurring in the former model and in Type
II superconductors proves to be very helpful.

Type II superconductors are described correctly in the framework of the
Ginzburg-Landau model [30].

The key points in describing Type II superconductors are the following.

Firstly, the Ginzburg-Landau model [30] implicates the initial $U(1)$ gauge
symmetry group, like the NO model [27]. Thus both the models are patterns of
Abelian theories.

In this case the Ginzburg-Landau model [30] can be got formally from the NO
model [27] (as a typical pattern of Abelian Higgs theories) by identifying a
Cooper pair with the Higgs field squared $|\phi|^{2}$.

Secondly, it is the general property of superconductors (of Type I as well of
Type II), exponential falling down [30] the magnetic field $\bf B$ at moving
deep into a superconducting region:

${\bf B}(x)={\bf B}_{0}e^{-x/\lambda_{L}},$

with

$\lambda_{L}=\frac{1}{M_{\gamma}}$

being the penetration or London depth, depending on the ”photon mass”

$M_{\gamma}=\sqrt{2}~{}ea,\quad a\equiv|\phi|,$

where $\phi$ is the wave function of the studied superconductor.

As in the NO model [27], the value $a^{2}$ plays the role of the order
parameter in the Ginzburg-Landau superconductivity model [30]. $a^{2}=0$ in
the symmetric (normal) thermodynamic phase, while $a^{2}\neq 0$ in the
asymmetric (superconducting) thermodynamic phase.

Thirdly, there is the following specific [13] of Type II superconductors.

Let us place a superconductor in an external magnetic field $\bf H$.

Then, in definition [13], Type II superconductors are those in which the
Ginzburg-Landau parameter [30]

$\kappa=\frac{\lambda_{L}}{\xi}<1/\sqrt{2}.$

In this Eq. $\xi$ is the coherence length, whose physical sense is [13] the
correlations radius of fluctuations in the appropriate order parameter
$<|\phi|^{2}>$. Indeed, $\kappa$ is a function of the temperature $T$:
$\kappa\equiv\kappa(T)$.

It turns out [13, 30] that the inequality $\kappa<1/\sqrt{2}$ proper to Type
II superconductors is associated with the negative surface tension

$\alpha_{ns}=\int\limits_{-\infty}^{\infty}(f_{s}-f_{n})~{}dx<0$

between the normal (n) and superconductor (s) phases.

In the latter case it becomes advantageous, from the thermodynamic standpoint,
to compensate increasing the volume energy by this negative surface energy
$\alpha_{ns}<0$.

This implies arising germs of the n-phase inside the s one at the values $H$
of the background magnetic field $H$ exceeding a field $H_{c_{1}}$, called
[13] the lower critical field. Vice verse, germs of the s-phase inside the n
one arise at the values $H$ don’t exceeding a field $H_{c_{2}}$, called [13]
the upper critical field.

Thus in the interval $H_{c_{1}}<H<H_{c_{2}}$, one can observe [13] the mixed
state in Type II superconductors (there are definite alloyed metals and
combinations of metals). In this case a Type II superconductor abides
simultaneously in the n and s states.

At $H\leq H_{c_{1}}$, a specimen is purely in the s-state, while it is purely
in the n-state at $H\geq H_{c_{2}}$.

It is easy to see the transparent parallel between the mixed state in Type II
superconductors and the coexistence of bubbles of the metastable and stable
thermodynamic phases in the temperature interval $[T_{b};T_{c}]$ in the NO
model [27] (indeed, one would replace temperatures $T$ with magnetic tensions
$H$ in the case of Type II superconductors). The mixed state in Type II
superconductors is also the supercooling case [29].

Ascertaining the explicit dependence $H(T)$ (as it was performed in Ref. [13])
seems to be helpful for understanding this parallel.

Thus the first-order phase transition occurs in Type II superconductors. As a
result, the conclusion can be drawn that the case of Type II superconductors
can be treated as a nonrelativistic limit of the Abelian Higgs NO model [27].

As in the NO model [27], the nontrivial topological content of the Ginzburg-
Landau superconductivity theory [30] can be discovered. We demonstrate that it
is asociated with including the electromagnetic vector potential $\bf A$ in
the expressionin [13]

${\bf j}_{s}=\frac{\hbar e}{2m}n_{s}(\nabla\phi-\frac{2e}{\hbar c}{\bf A})$

for the gauge ($U(1)$) invariant superconductivity current ${\bf j}_{s}$
(involving the density $n_{s}$ of superconductive electrons).

In this case

${\rm rot}~{}{\bf j}_{s}\sim{\rm rot}~{}{\bf A}\neq 0,$

and this is the sign of nontrivial topologies in the Ginzburg-Landau
superconductivity theory [30] according the arguments similar to those
explained topologically nontrivial vortices [15] in a liquid helium II
specimen (at rest).

More exactly, since

$\oint_{C}{\bf A}d{\bf l}=\int{\rm rot}~{}{\bf A}\cdot d{\bf f}=\int{\bf
B}d{\bf f}\equiv{\Phi},$

where in this context, $\Phi$ is the flux of the magnetic field $\bf B$,

$\oint_{C}{\bf A}d{\bf l}=\frac{\hbar c}{2e}\oint_{C}\nabla\Phi\cdot d{\bf
l}=\frac{\hbar c}{2e}\delta\Phi$

for an integration way $C$ lying inside a superconductor far from its surface.

However from the claim for the superconductor wave function to be finite, it
follows that this change of the phase $\Phi$ can take only values $2\pi n$
($n\in{\bf Z}$).

One comes thus to the result [13]

$\Phi=n\Phi_{0},\quad\Phi_{0}=\frac{\pi\hbar c}{|e|}=2\cdot 10^{-7}~{}{\rm
G~{}cm^{2}}$

for the phase $\Phi$. The value $\Phi_{0}$ is called the elementary quant of
the magnetic flux [13].

The variation $\delta\Phi$ is indeed infinitesimal, that corresponds to the
fact that ${\rm rot}~{}\nabla\Phi=0$. Thus one can observe actual disappearing
nontrivial topologies deep into a superconductor.

Indeed, in Type II superconductors the background magnetic field $H$
penetrates into the superconducting specimen along the parallel each other
lines, so-called Abrikosov threads, forming a lattice. These serve as embryos
of the normal phase in the specimen, generating simultaneously circular
superconducting currents inside it [13]. Note that these Abrikosov threads
belonging to the “continuous” thermodynamical phase n are additional to those
would exist inside the discrete vacuum manifold $\tilde{U}(1)$.

In the present study Abrikosov threads will be analyzed repeating the
arguments [13]. In particular, it will be argued that the first order phase
transition occurring in Type II superconductors can be reduced to the non-
smooth behaviour of the order parameter $|\phi(r)|^{2}$ in the points locating
Abrikosov threads.

The geometric (topological) specific of that first order phase transition will
be also discussed and the series of hypotheses concerning this specific will
be suggested.

In the light of the said about the Abelian Higgs NO model [27] involving NO
vortices and a parallel between this and the Ginzburg-Landau theory [30],
especially in that part of the latter one concerning Type II superconductors,
the dual superconducting picture of confinement in QCD can be explained enough
good as that corresponding to the stable thermodynamic phase, below the
temperature $T_{b}$ (when the metastable phase disappears entirely and the
system fall down in the ”true” vacuum $|\phi|=\phi_{0}$), in the NO model
[27].

In this stable thermodynamic phase, an external magnetic field $\bf B$, being
present, is concentrated [31] in cores of vortices and will not extend into
the superconducting region.

As a result, the interaction energy of magnetic monopoles grows linearly with
their separation $d$ [31]:

$V=\frac{1}{2}\int d^{3}x\,{\bf B}^{2}\propto\frac{4\pi
d}{e^{2}\lambda_{L}^{2}},$

as it would be in the dual superconducting picture [22] of confinement.

In the modern literature the above discussed (phenomenological) picture of
confinement based on MAG fixing and the dual superconducting picture (similar
to that one discovers in the Ginzburg-Landau model [30] concerning Type II
superconductors and implicating [22] the Higgs and confinement phases) is
referred to as the Abelian dominance [32]).

The situation changes rather in a radical way, in comparison with the Abelian
dominance [32] picture of confinement, in the Minkowskian Higgs model with
vacuum BPS monopoles quantized by Dirac [4].

There are two important ”points of distinction” between the ”Abelian
dominance” [32] sight on confinement and that sight based upon the Dirac
fundamental quantization method [4] applied to (Minkowskian) QCD and coming to
the Gauss-shell reduction of the appropriate Hamiltonian.

Firstly, at constructing Minkowskian Gauss-shell QCD (underlying by vacuum BPS
monopole solutions) the specific gauge, different from MAG, can be fixed in
the more general case when the initial $SU(3)_{\rm col}$ gauge symmetry group
inherent in Minkowskian Gauss-shell QCD is violated in the

$SU(3)_{\rm col}\to SU(2)_{\rm col}\to U(1)$ (2.11)

wise.

Then interesting physical effects can be achieved at choosing the
antisymmetric Gell-Mann matrices $\lambda_{2}$, $\lambda_{5}$, $\lambda_{7}$
as generators for $SU(2)_{\rm col}$ [1, 5, 12].

Among such consequences fixing the (”antisymmetric”) ”gauge”, one can mention
fermionic rotary (axial) degrees of freedom ${\bf v}_{1}={\bf r}\times{\bf K}$
(with $\bf K$ being the polar colour vector, $SU(2)_{\rm col}$ triplet).

These fermionic rotary degrees of freedom ${\bf v}_{1}$ somewhat similar to
rotary terms one can detect in molecules.

On the other hand, to explain the appearance of above fermionic rotary
degrees, assuming the ”discrete” group geometry for $SU(2)_{\rm col}$ and
$U(1)$ in Eq. (2.11) (then also the appropriate vacuum manifold $SU(2)_{\rm
col}/U(1)$ possesses a ”discrete” geometry) seems to be plausible.

Secondly, the Gauss-shell reduction of Minkowskian QCD in the BPS monopole
background implies [1, 5, 7, 8, 12, 20] the Green function of the Gauss law
constraint can be expand by the Coulomb and ”golden section” potentials.

Thus the role of the Mandelstam linearly increasing potential $\sim Kr$ (and
the dual superconducting picture [22, 31] together with) in Minkowskian
constraint-shell QCD is underestimated considerably, at least at low
temperatures $T\to 0$.

And only at enough high temperatures $T\neq 0$, a chance appears to
incorporate the Mandelstam linearly increasing potential in Minkowskian
constraint-shell QCD with vacuum BPS monopoles.

Such possibility will be also us discussed in Section 5.

##  3 How one can introduce ”discrete” geometry for degeneration spaces in
Minkowskian Higgs models.

Recall again the recent paper [6]. In this paper some modern Minkowskian Higgs
models with stationary vacuum monopole solutions were analysed.

Besides already mentioned BPS monopole model [7, 8, 9, 10, 11], one can
mention the ’t Hooft-Polyakov monopole model [33, 34].

As it was discussed in [6], such (vacuum) monopole solutions are compatible
with the ”continuous” vacuum geometry

$R\equiv SU(2)/U(1)\simeq S^{2}.$ (3.1)

Also it was argued that Minkowskian Higgs models with stationary vacuum
monopole solutions, involving the ”continuous” vacuum geometry (3.1), can be
described quite correctly in the framework of the Faddeev-Popov (FP)
”heuristic” quantization scheme [35], coming [1, 6] to actual fixing the Weyl
gauge $A_{0}=0$ (for instance via the multiplier $\delta(A_{0})$ in
appropriate FP path integrals).

From the topological viewpoint, the continuous vacuum manifold $R$
444Following [9], $R$ may be called the degeneration space.It is quite correct
herewith to interpret the initial symmetry group $G$ in a Minkowskian Higgs
model as that does not change the energy functional (Hamiltonian) of that
model, while the residual symmetry group $H\subset G$ as that consisting of
transformations that keep invariant a fixed equilibrium state. All these
states (at a fixed temperature $T<T_{c}$, with $T_{c}$ being the appropriate
Curie point in which the initial symmetry $G$ is violated and the second-order
phase transition occurs) just form the appropriate degeneration space $R=G/H$.
The natural claim to this space is herewith to be topological. The structure
of a degeneration space may be investigated with the aid of the Landau theory
of second-order phase transitions. An equilibrium state is determined by the
condition for the free energy of the given system to be minimal. In the Landau
theory of second-order phase transitions (the pattern of which is the
Minkowskian Higgs model) one supposes that an equilibrium state may be found
at minimizing the free energy of the given system by the set of states
specified by a finite number of parameters (called order parameters), but not
by the set of all the states. , (3.1), possesses the one kind of topological
defects, namely the point hedgehog topological defects.

These topological defects are associated with vacuum monopole solutions in
appropriate Minkowskian Higgs models.

Point hedgehog topological defects inside $R$ are determined by the
topological chain [6, 7]

$\pi_{2}(R)\equiv\pi_{2}S^{2}=\pi_{1}(H)\equiv\pi_{1}U(1)=\bf Z.$ (3.2)

From the thermodynamic points of view, all the kinds of topological defects
may be explained (repeating the arguments [9]) by violating the thermodynamic
equilibrium over a region in the appropriate coordinate (in particular,
Minkowski) space.

For example, point hedgehog topological defects inside $R$ are associated with
violating the thermodynamic equilibrium in an infinitesimal neighbourhood of
the origin of coordinates. Such neighbourhoods are topologically equivalent to
the two-sphere $S^{2}$.

In the present study we attempt to demonstrate our readers that going out from
the FP ”heuristic” quantization scheme [35] to the Dirac ”fundamental” one [4]
in the Minkowskian Higgs model with vacuum BPS monopole solutions claims the
in principle new geometrical approach to constructing the appropriate vacuum
manifold (in comparison with assuming the ”continuous” $\sim S^{2}$ vacuum
geometry in the former case).

This ”new geometrical approach” (whose outlines were contemplated in the
recent paper [1]) comes to assuming the ”discrete” geometry for the vacuum
manifold $SU(2)/U(1)$.

As we shall make sure soon, this assuming explains enough good various vacuum
rotary effects [3] inherent in the physical BPS monopole vacuum suffered the
Dirac fundamental quantization [4].

So, let us represent [1] the initial, $SU(2)\equiv G$, and residual,
$U(1)\equiv H$, gauge symmetries groups in the Minkowskian Higgs model with
vacuum BPS monopoles quantized by Dirac in the shape of discrete spaces

$SU(2)\simeq G_{0}\otimes{\bf Z};\quad U(1)\simeq U_{0}\otimes{\bf Z},$ (3.3)

respectively 555Such representation for gauge groups spaces was proposed for
the first time in the paper [16]..

From the topological viewpoint, the discrete representation (3.3) for the
gauge groups $G$ and $H$ extracts ”small” (topologically trivial) and ”large”
(corresponding to topological numbers $n\neq 0$) gauge transformations in the
complete set of appropriate gauge transformations (the idea of such
subdividing for gauge transformations was suggested in Ref. [18]).

According to the terminology [18], the complete groups $G_{0}$ and $H_{0}$
just contain ”small” gauge transformations, that implies

$\pi_{n}G_{0}=\pi_{n}H_{0}=0$ (3.4)

for loops in the group spaces $G_{0}$ and $H_{0}$ in all the dimensions $n\geq
1$.

Simultaneously, in definition,

$\pi_{0}G_{0}=\pi_{0}H_{0}=0,$ (3.5)

i.e. $G_{0}$ and $H_{0}$ are maximal connected components [9] in their gauge
groups (respectively, $G$ and $H$).

Later Eq. implies [9] that

$\pi_{0}[G_{0}\otimes{\bf Z}]=\pi_{0}[G_{0}\otimes{\bf Z}]=\pi_{0}({\bf
Z})={\bf Z}.$ (3.6)

The latter relation indicates transparently the discrete nature of the
appropriate group spaces 666The origin of Eq. (3.6) was revealed in the
monograph [9], in §T20. It is the particular case of the more general relation
$\pi_{i}(K)=\pi_{i}(L_{1})+\dots+\pi_{i}(L_{r})$ for a group $K$ which is the
product of the groups $L_{1}\dots L_{r}$ at a fixed $i$ (it is correctly for
the Lie groups of the series $SU$, $U$ and $SO$, with which modern theoretical
physics deals). .

To construct the degeneration space (vacuum manifold) $R_{\rm YM}\equiv G/H$
corresponding to the discrete representation (3.3) for the gauge groups $G$
and $H$, let us multiply Eq. [6, 36]

$G=H\oplus G/H$ (3.7)

left by $\bf Z$, substituting simultaneously in (3.7) the ”discrete”
representations (3.3) for the gauge symmetries groups:

${\bf Z}\otimes G_{0}={\bf Z}\otimes U_{0}+{\bf Z}\otimes G_{0}/U_{0}.$ (3.8)

From latter Eq. we learn that the degeneration space in the Minkowskian Higgs
model with vacuum BPS monopoles quantized by Dirac [4] is

$R_{\rm YM}={\bf Z}\otimes G_{0}/U_{0}.$ (3.9)

Again $\pi_{1}(R_{\rm YM})={\bf Z}$ since $U_{0}\otimes{\bf Z}$ is multi-
connected.

It is correctly because the isomorphism (2.1) (the proof of this isomorphism
was given in Ref. [9], in §T20, we recommend our readers for understanding the
matter).

Further, it becomes transparent from Eq. (3.9) that the ”small” coset
$G_{0}/U_{0}$ is one-connected:

$\pi_{1}(G_{0}/U_{0})=0.$

Really, the coset $G_{0}/U_{0}$ is treated as the space of $U_{0}$-orbits on
$G_{0}$; the latter space is one-connected.

One can see also the topological equivalence between $G_{0}/U_{0}$ and the
subset of one-dimensional ways on $R_{\rm YM}$ can be contracted into a point.

The vacuum manifold $R_{\rm YM}$ is transparently multi-connected:

$\pi_{0}(R_{YM})={\bf Z}.$ (3.10)

This Eq. may be explained in the framework of our above remark [9] about
homotopical groups of products of groups.

Latter Eq. has very important physical sense.

It implies [9] that domain walls exist between different topological sectors
in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac.

The origin of said domain walls is in the ”discrete” factorisation (3.3) of
the residual gauge symmetry group $U(1)$ 777Indeed, the above-mentioned
”discrete” factorisation (3.3) of the residual gauge symmetry group $U(1)$ can
be referred to as a “discrete-continuous” in contrast to purely ”discrete”
symmetries. The typical look of latter symmetries is [37] $\exp(2\pi i/p)$
($p\in{\bf Z}$). If, for instance, a single complex scalar field $\varsigma$
is given, the appropriate low energy theory will be invariant under the
transformation  $\varsigma\to\exp(2\pi i/p)~{}\varsigma.$ On the other hand,
as it was noted by Krauss and Wilczek [37], the most important consequence of
continuous local (gauge) symmetries is the existence of gauge fields. Such
fields are introduced in order to formulate covariant derivatives. These are
necessary since invariant interactions involving gradients can be formed;
these interactions, in turn, are necessary in order that charged fields can
propagate. In the case of a discrete symmetry there is no similar need to
introduce a gauge potential because the ordinary derivative already transforms
simply. Thus one can assert that the distinction between the original gauge
low energy theory and the ”discrete” model is in the absence of the
interaction items in the latter case. The on principle another situation is in
the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac, the
subject of the present discussion. In this model the factorisation (3.3)
(calling for justifying the Dirac quantization scheme [4] applying here), the
presence of the “continuous” factors $U_{0}$ and $G_{0}$ in Eqs. (3.3) and
(3.9) comes off the Krauss- Wilczek prohibition [37] for interactions. It is
so since now in each topological domain of the vacuum manifold $R_{\rm YM}$,
(3.9), two “large” (stationary) gauge matrices corresponding to a fixed
topological number $n$ can be always connected by a “small” gauge
transformation. This can be written down as [5] $v^{(n)}({\bf x})\to
v^{(n)^{\prime}}({\bf x})=v^{(n)}({\bf x})u_{0}^{-1};\quad u_{0}\in U_{0}.$
Appropriately, gauge fields setting “large” gauge matrices $v^{(n)}({\bf x})$
are transformed according to the law  $\frac{\sigma\cdot{\bf
A}^{(n)}_{\mu}}{2}\to\frac{\sigma\cdot{\bf
A}^{(n)^{\prime}}_{\mu}}{2}=u_{0}(\frac{\sigma\cdot{\bf
A}^{(n)}_{\mu}}{2})u_{0}^{-1}-\frac{i}{g}[\partial_{\mu}u_{0}^{-1}]u_{0};$
with $\sigma$ being Pauli matrices and $g$ being the (YM) coupling constant.
This transformation law remains correct, in particular, for topological Dirac
variables $A_{i}^{D}$ ($i=1,2$) [5, 20] (see below), which, by their
construction [12] are gauge invariants.Thus in the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac, with the “discrete-continuous”
vacuum geometry (3.9) been assumed, interactions “have vested right to
existence”. This permission for interactions has a crucial importance for the
mentioned model. For example, in the BPS (Wu-Yang) monopole background there
exist the such solution for the Gauss law constraint permitting the expansion
[5, 7, 8, 12, 20] in terms of the Coulob and “golden section” potentials.
This, in turn, serves as base for describing the four-fermionic instantaneou
interaction. In the quark sector of the Minkowskian Higgs model with vacuum
BPS monopoles quantized by Dirac this is the initial step in constracting
quarkonium ($q\Gamma\bar{q}$ with $\Gamma$ being a combination of Dirac
matrices) bound states. The algorithm constracting such bound states utilizing
the Markov-Yukawa prescription [38] (see e.g. §3.1 in the revue [12]).And
moreover, the mentioned “golden section” instantaneous potential can lead [12]
to rearrangement of the naive perturbation series of the spontaneous chiral
symmetry breaking type (especially, in the infrared momentum region). .

As it is well known (see e.g. §7.2 in [28] or the paper [19]), the width of a
domain (or Bloch, in the terminology [19]) wall is roughly proportional to the
inverse of the lowest mass of all the physical particles in the (gauge) model
considered.

In Minkowskian Higgs models the typical such scale is the (effective) Higgs
mass $m/\sqrt{\lambda}$.

In particular, in the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac [4], $m/\sqrt{\lambda}$ is the only mass scale different
from zero.

This fact was explained in Ref. [2]. This was argued by zero masses of YM
fields represented by topological Dirac variables $A_{i}^{D}$ ($i=1,2$).

These variables become massless at the Dirac removal [2, 5]

$U(t,{\bf x})(A_{0}^{(0)}+\partial_{0})U^{-1}(t,{\bf x})=0.$ (3.11)

of temporal YM components $A_{0}$, that are, indeed, nondynamical fields
possessing zero canonical momenta

$E_{0}\equiv\partial{\cal L}/\partial(\partial_{0}A_{0})=0.$

In this context [7, 8],

$U(t,{\bf x})=v({\bf
x})T\exp\\{\int\limits_{t_{0}}^{t}[\frac{1}{D^{2}(\Phi^{\rm
BPS})}\partial_{0}D_{k}(\Phi^{\rm BPS}){\hat{A}}^{k}]~{}d\bar{t}~{}\\},$
(3.12)

with $v({\bf x})$ being stationary Gribov topological multipliers (depending
indeed on topological numbers $n$).

If, additionally, the transverse gauge (1.7) [5] is fixed for topological
Dirac variables $A_{i}^{D}$, this just implies the masslessness of YM fields.
Here is a transparent analogy with the photon case in QED. In that case one
gets plane waves by fixing the Lorentz gauge

$\phi=0;\quad\partial_{i}A^{i}=0~{}~{}(i=1,2,3)\Leftrightarrow\partial_{\mu}A^{\mu}=0$

(with $A_{0}\equiv\phi$ being the scalar potential).

As it was argued in the monograph [39], such fixing the (relativistic
invariant) Lorentz gauge implies the equation of motion (referred to as the
D’alambert  or wave equation)

$\Delta{\bf A}-\frac{\partial^{2}{\bf A}}{\partial t^{2}}=0$ (3.13)

(at setting $c=1$ for the light velocity). Latter Eq. shows transparently the
absence of the (rest) mass for a photon.

As it was stressed in Ref. [3] and in Section 2 of the present work, in the
Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac [4], the
typical value of the length dimension inversely proportional to
$m/\sqrt{\lambda}$ is the size $\epsilon$ of BPS monopoles.

It may be given as [5, 7, 8]

$\frac{1}{\epsilon}=\frac{gm}{\sqrt{\lambda}}\sim\frac{g^{2}<B^{2}>V}{4\pi}.$
(3.14)

Thus $\epsilon$ is, in turn, inversely proportional to the spatial volume
$V\sim r^{3}$ occupied by the appropriate (YM-Higgs) field configuration.

The said allows to assert that $\epsilon$ disappears in the infinite spatial
volume limit $V\to\infty$, while it is maximal at the origin of coordinates
(herewith it can be set $\epsilon(0)\to\infty$).

This means, due to the above reasoning [19], that walls between topological
domains inside $R_{\rm YM}$ become infinitely wide, $O(\epsilon(0))\to\infty$,
at the origin of coordinates.

The fact $\epsilon(\infty)\to 0$ is also meaningful. This implies actual
merging of topological domains inside the vacuum manifold $R_{\rm YM}$, (3.9),
at the spatial infinity.

This promotes the infrared topological confinement (destructive interference)
of Gribov ”large” multipliers $v^{(n)}({\bf x})$ in gluonic and quark Green
functions in all the orders of the perturbation theory.

The latter fact was demonstrated utilizing the strict mathematical language in
Ref. [23] (partially these arguments [23] were reproduced in [1]).

The existence of domain walls between different topological sectors of the
vacuum manifold $R_{YM}$, (3.9), permits their treatment as quite separated
sets.

The notion of quite separated sets [40] means in this case that always there
exists such a function $f:R_{YM}\to I$ ($I\equiv[0,1]$) for two homotopical
classes $A$ and $B$ inside $R_{YM}$ that $f(x)=0$ as $x\in A$, while $f(x)=1$
as $x\in B$. Here $x$ is a (one-dimensional) way.

One speaks in this case that $f$ separates the sets $A$ and $B$.

The nontrivial isomorphism (2.1) [9] satisfying (as we have discussed above)
for the vacuum manifold $R_{YM}$, (3.9), implies the presence of thread
topological defects inside this manifold.

On the level of phenomenology this implies [9] violating the thermodynamic
equilibrium along definite lines in the given vacuum manifold.

In the simple case when such lines, threads, are straight, the thermodynamic
equilibrium is violated in an infinitely narrow tube around each of these
threads. Thus regions of thread topological defects have the topology of a
cylinder (topologically equivalent to the circle $S^{1}$). But it is well
known that $\pi_{1}S^{1}={\bf Z}$.

In our researches about the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac [4] we assume that the region of thread topological defects
inside the vacuum manifold $R_{YM}$, (3.9), is the infinitely narrow tube of
the infinite length around the axis $z$ in the chosen rest reference frame.
The effective ”diameter” of this cylinder of the infinite length is just
$O[\epsilon(\infty)]$, that is, indeed, an infinitely small value.

In the next section we shall ground the above assumption investigating the
properties [9] of rectilinear threads $A_{\theta}$ inside $R_{YM}$.

Furthermore, it may be shown that

$\pi_{2}(R_{YM})={\bf Z}=\pi_{1}H.$ (3.15)

The isomorphism $\pi_{1}(H)={\bf Z}$ follows from the arguments of the
holonomies group, stated in Refs. [7, 9].

Actually, these come to the existence of one-dimensional loops in the $H$
group space, forming holonomies elements

$b_{\gamma}=P\exp(-\oint_{\Sigma}{\bf T}\cdot A_{\mu}dx^{\mu})$ (3.16)

over one-dimensional cycles $\Sigma$.

The holonomies elements $b_{\gamma}$, belonging to the $U(1)\subset SU(2)$
embedding, make the complete holonomies group (we shall denote alls as $H$)
isomorphic to $U(1)$ [7].

Eq. (3.16) makes the isomorphism $\pi_{1}(H)={\bf Z}$ highly transparent (as
that coming to $\pi_{1}S^{1}={\bf Z}$).

The relation

$\pi_{2}(R_{\rm YM})=\pi_{1}H,$

is the particular case of more general Eq. (2.2).

The good proof of the isomorphism (2.2) was demonstrated in the monograph [9],
in §T20, and we recommend our readers this monograph to acquaint with it.

The nontrivial isomorphism (3.15) testifies in favour of point hedgehog
topological defects inside the vacuum manifold $R_{\rm YM}$ in the Minkowskian
Higgs model with vacuum BPS monopoles quantized by Dirac [4].

These point hedgehog topological defects and vacuum BPS monopole solutions are
indivisible.

As in the case of (vacuum) ’t Hooft-Polyakov monopole solutions [33, 34],
associated with the ”continuous” vacuum $\sim S^{2}$ geometry (3.1) and the FP
”heuristic” quantization scheme [35], point hedgehog topological defects
inside the discrete vacuum manifold $R_{\rm YM}$, (3.9), come to violating the
thermodynamic equilibrium in an infinitesimal neighbourhood of the origin of
coordinates.

In particular, this violating implies the singularity of the appropriate
vacuum ”magnetic” fields $\bf B$, serving as the order parameters in the
mentioned Higgs models, at the origin of coordinates.

In spite of definite distinctions between the ’t Hooft-Polyakov and BPS Higgs
models (these distinctions were pointed out, for instance, in the recent paper
[6]; the principal of them is the absence of superfluid properties in the ’t
Hooft-Polyakov model [33, 34]) the vacuum ”magnetic” field $\bf B$ diverges as
$r^{-2}$ at the origin of coordinates in both models 888For the ’t Hooft-
Polyakov model [33, 34] this was demonstrated in Ref. [36], while for the BPS
Higgs model, in the original papers [10].:

$B_{k}\sim\frac{r_{k}}{gr^{3}}.$ (3.17)

Moreover, there is a likeness between Eqs. (3.15) and (3.2) as the particular
cases of the general relation (2.2) [9], inducing point topological defects in
(gauge) theories.

On the other hand, there are no thread topological defects inside the
continuous vacuum manifold $R$, (3.1), since $\pi_{1}S^{2}=0$ and moreover
when the ”continuous” geometry is assumed for the residual $U(1)$ gauge
symmetry in the (Minkowskian) Higgs model, also $\pi_{0}~{}U(1)=0$.

In the next section it will be argued that the first-order phase transition
occurs indeed in the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac [4].

To facilitate this job, now we should like discuss an useful analogy, us give
the liquid helium II model.

Just in the latter model, possessing manifest superfluid properties [14], also
the first-order phase transition happens.

It is associated with (spontaneous) violating the superfluidity [14] in a
(resting) liquid helium II specimen along definite (rectilinear) lines,
threads.

This implies [15] the presence of thread (rectilinear) vortices in that
specimen.

The similar situation with arising (rectilinear) vortices takes place when
this specimen turns together with the cylindrical vessel in which it is
contained.

The phenomenology of the latter case was stated enough good in the monograph
[13], in §29. We don’t intend, in the present study, to retail all the said in
the monograph [13] about this subject; however we, for all that, would like to
elucidate some theses of the helium turning model. This will be very helpful
for us in the next section.

A good argument in favour of the first-order phase transition occurring in the
liquid helium II turning model is the nontrivial contribution $\Delta E$ [13]
in the total helium (He4) Hamiltonian 999 This has the typical look [41]
$\displaystyle\widehat{\cal H}$ $\displaystyle=$ $\displaystyle\int
d^{3}r~{}\hat{a}^{+}({\bf r},t)\left({-\hbar^{2}\over
2m}\;\nabla^{2}\right)\hat{a}({\bf r},t)$ $\displaystyle\quad+$
$\displaystyle{1\over 2}\int d^{3}r\int d^{3}r^{\prime}\hat{a}^{+}({\bf
r},t)~{}\hat{a}^{+}({\bf r}^{\prime},t)V(|{\bf r}-{\bf r}|)~{}\hat{a}({\bf
r}^{\prime},t)~{}\hat{a}({\bf r},t).$ Boson creation and annihilation
operators: $a^{+}({\bf r},t)$ and $a^{+}({\bf r},t)$, respectively, can be
expressed in terms of the phase $\Phi(t,{\bf r})$ of the Bose condensate wave
function (1.4) [13]. Note that such expressing boson creation and annihilation
operators in terms of the one phase function $\Phi(t,{\bf r})$ is in a good
agreement with the quantum-mechanical calculations about the liquid helium II
proposed by Bogolubov and co-authors [42], in particular, with the Bogolubov
transformations [42, 43, 44] $\hat{b}^{+}_{{\bf p}}=u_{{\bf
p}}\hat{\xi}^{+}_{{\bf p}}+v_{{\bf p}}\hat{\xi}_{{\bf-p}},$ with  $u_{{\bf
p}}^{2}-v_{{\bf p}}^{2}=1$ and the creation (respectively, annihilation)
operators  $\hat{b}^{+}_{{\bf p}}=\frac{\hat{a}_{0}\hat{a}^{+}_{{\bf
p}}}{\sqrt{n_{0}}},\quad\hat{b}_{{\bf p}}=\frac{\hat{a}^{+}_{0}\hat{a}_{{\bf
p}}}{\sqrt{n_{0}}},$ expressed through the ”initial” creation (annihilation)
operators entering actually the Bogolubov model Hamilton operator, can be
represented as [42, 43]
$\hat{H}=-\sum\limits_{a=1}^{N}\frac{\hbar^{2}}{2m}\Delta_{a}+\frac{1}{2}V(|{\bf
r}_{a}-{\bf r}_{b}|),$ (with $V(|{\bf r}_{a}-{\bf r}_{b}|)$ being the
interaction energy between particles $a$ and $b$ and $N$ being the complete
number of particles in the considered system), brought in the diagonal form
$\hat{H}=\hat{H}_{0}+\sum\limits_{{\bf p}\neq 0}\epsilon({\bf
p})\hat{\xi}^{+}_{{\bf p}}\hat{\xi}_{{\bf p}}$ via the above Bogolubov
transformations. Herewith the creation (respectively, annihilation) operators
$\hat{a}^{+}_{0}$ and $\hat{a}_{0}$ corresponds to the zero momenta ${\bf
p}=0$ and the number $n_{0}$ of helium atoms possessing these zero momenta.
from the global solid potential rotations of the helium specimen turning
together with the cylindrical vessel where it is contained.

This contribution is given with Eq.

$\Delta E\sim L\pi\rho_{s}\frac{\hbar^{2}}{m^{2}}\ln\frac{R}{a}:$ (3.18)

with $L$ being the length of the vessel, $\rho_{s}$ being the density of the
superfluid component in the helium II specimen, $m$ being the mass of a helium
atom; at last $R$ and $a$ are, respectively, the radius of the vessel and an
arbitrary distance of atomic scales.

The shortcoming of this Eq. is its logarithmic divergence at $R/a\to\infty$.

The important peculiarity of the liquid helium II turning model [13] is the
appearance of rectilinear threads (accompanied by appropriate vortices),
parallel to the rotation axis (altitude) of the cylindrical vessel (herewith
chosen to coincide with the axis $z$ of the given rest reference frame).

The liquid helium II turning model [13] possesses initially the manifest
$U(1)~{}\otimes~{}O(2)$ symmetry.

In this model $O(2)$ is the group of global (rigid) potential rotations around
the axis $z$ of the liquid helium II specimen and the cylindrical vessel where
it is contained.

Unlike $O(2)$, $U(1)$ is the gauge symmetry group.

In the ”general” liquid helium II theory [14, 42], the latter group can be
represented by matrices

$v_{n}({\bf r})=\exp(i~{}n\lambda({\bf r}))$ (3.19)

maintaining the Bogolubov (dioganalized) Hamiltonian $\widehat{\cal H}$ [41,
42, 43].

Herewith the phase $\Phi(t,{\bf r})$ of the complex-value helium Bose
condensate wave function $\Xi(t,{\bf r})$ [13], (1.4), is suffered the gauge
transformations

$\Phi(t,{\bf r})\to\Phi(t,{\bf r})+n\lambda({\bf r}),\quad n\in{\bf Z}.$
(3.20)

And again, ”small” and ”large” gauge transformations [18] can be picked out
among (3.19), (3.20).

As it is well known, superfluidity phenomena [14] in a liquid helium II
specimen are associated with violating the initial $U(1)$ gauge symmetry of
the Bogolubov Hamiltonian $\widehat{\cal H}$. This second-order phase
transition occurs in the fixed Curie point $T_{c}\to 0$.

To explain in this case rotary effects (coming to appearance of thread
vortices) in the liquid helium II turning model [13] (as well as in the liquid
helium II at rest theory [15]), it is worth to assume that violating the
initial $U(1)$ gauge symmetry in the liquid helium II theory in such a wise
that the $U(1)\simeq S^{1}$ group space turns into the discrete quite
separated set [40].

More exactly, it may be assumed the spontaneous breakdown

$U(1)\simeq U_{0}\otimes{\bf Z}\longrightarrow{\bf
Z}\simeq\tilde{U}(1);\quad\pi_{0}U_{0}=\pi_{1}U_{0}=0.$ (3.21)

Geometrically, the discrete (quite separated) set $\tilde{U}(1)$ may be seen
as the circle $S^{1}$ cut off by its topological domains (vice verse, to get
the continuous $U(1)$ group space, these topological domains would be again
pasted together).

On the other hand, in the liquid helium II turning model [13], the appropriate
group $O(2)$ of rigid rotations remains exact.

This implies the (spontaneous) breakdown

$O(2)\otimes U(1)\simeq O(2)\otimes U_{0}\otimes{\bf Z}\longrightarrow
O(2)\otimes{\bf Z}\simeq O(2)\otimes\tilde{U}(1)\equiv H$ (3.22)

of the initial symmetry group $O(2)\otimes U(1)$ inherent in the liquid helium
II turning model [13].

Upon some mathematical manipulations, resembling somehow ones [36] led to Eq.
(3.9) in the Minkowskian Higgs model with vacuum BPS monopoles quantized by
Dirac, the degeneration space in the liquid helium II turning model [13] may
be founded. It proves to be

$[{\bf Z}\otimes O(2)]\otimes U_{0}\equiv R,$ (3.23)

and it is topologically equivalent to $S^{1}$.

The latter fact may be explaining by reasoning that the group $O(2)$ of
$2\times 2$ orthogonal matrices with determinants $\pm 1$ always may be
represented as

$SO(2)\otimes{\mathbb{Z}}_{2},$

with $SO(2)\simeq U(1)$ being the group of orthogonal matrices with
determinants $+1$, while ${\mathbb{Z}}_{2}\subset\bf Z$.

It is easy to see that $\pi_{1}R={\bf Z}$.

In detail, this may be argued by the above ascertained topological equivalence
between the liquid helium II turning degeneration space $R$ and the circle
$S^{1}$.

On the other hand, also $\pi_{0}H={\bf Z}$ due to the manifest presence of the
discrete multiplier $\tilde{U}(1)$ in Eq. (3.22).

Thus the general isomorphism (2.1) [9] is satisfied in the case of the liquid
helium II turning model [13].

In this concrete case it implies the presence of (rectilinear) thread
topological defects (vortices) in the quested model.

These rectilinear vortices contribute with the item $\Delta E$ [13], (3.18),
additional to the Bogolubov helium (diagonalized) Hamiltonian $\widehat{\cal
H}$ [41].

Just this increasing $\Delta E$ the helium energy (in comparison with that
given by $\widehat{\cal H}$ and referring to the helium specimen at rest) is
the sign of the first order phase transition occurring.

Of course, there are walls between different topological domains inside the
discrete liquid helium II turning degeneration space $R$, (3.23).

On the other hand, there are no point topological defects inside $R$ since
$\pi_{2}S^{1}=0$.

The case when a liquid helium II is at rest is somewhat simpler than the case
[13] of the liquid helium II turning together with the cylindrical vessel
where it is contained.

In the former case, to justify the spontaneous appearance of rectilinear
vortices 101010As it was explained in [15] (see also [3]), the appearance of
rectilinear vortices in a liquid helium II specimen is set by Eq.
$n=\frac{m}{2\pi\hbar}\oint\limits_{\Gamma}{\bf v}^{(n)}d{\bf l};\quad
n\in{\bf Z}.$ This Eq. implicates the helium mass $m$ and the tangential
velocity ${\bf v}^{(n)}$ of a rectilinear vortex. In this case, trivial
topologies $n=0$ correspond to disappearing the cyclic integral on the right-
hand side of latter Eq. It is just the case of superfluid potential motions in
the given liquid helium II specimen with ${\rm rot}~{}{\bf v}^{(0)}=0$. In
Ref. [3], the explanation of vortices [15] in a (rested) liquid helium II
specimen as a particular case of the Josephson effect [17] (coming to circular
persistent motions of material points) was given. , it may be presumed that
the initial (continuous) $U(1)$ gauge symmetry group of the Bogolubov helium
Hamiltonian $\widehat{\cal H}$ [41] is violated in the

$U(1)\to\tilde{U}(1)$ (3.24)

wise.

As in the liquid helium II turning case [13], $\pi_{0}~{}\tilde{U}(1)={\bf
Z}$.

Herewith the set $\tilde{U}(1)$ (as well as the set $\tilde{U}(1)\otimes O(1)$
in the case [13] of a liquid helium II specimen turning together with the
cylindrical vessel where it is contained) can be treated as the residual
symmetry group in the discussed helium models.

On the other hand, it can be demonstrated (applying the manipulations similar
to those necessary in the liquid helium II turning case [13] to ascertain the
look (3.23) for the appropriate degeneration space $R$) that the degeneration
space $\tilde{R}$ in the liquid helium II at rest theory coincides (to within
an isomorphism) 111111Indeed, $\tilde{R}\simeq S^{1}/{\bf Z}$. In this case,
as it is easy to see,  $\pi_{1}\tilde{R}=\pi_{1}S^{1}={\bf Z}.$ with the
residual gauge symmetry group

$\tilde{U}(1)\simeq U_{0}\otimes{\bf Z}.$

Thus, formally,

$\pi_{1}\tilde{R}=\pi_{0}~{}\tilde{U}(1)={\bf Z},$

and this just implies the existence of thread topological defects (rectilinear
vortices) in the liquid helium II at rest theory.

The alone fact [3, 15] referring vortices in a liquid helium II at rest to
nontrivial topologies $n\neq 0$, while superfluid potential motions are
referred to trivial topologies $n=0$, is very remarkable.

This allows to associate various superfluidity effects in a liquid helium II
at rest specimen with ”small” $U(1)$ gauge matrices while ”large” ones with
(rectilinear) vortices therein.

All these gauge matrices are given (to within the natural isomorphism
$U(1)\simeq\tilde{U}(1)$) with Eq. (3.19).

Thus the connection between the superfluid and rotary effects in the liquid
helium II model and the topological degeneration of appropriate data (the
phase $\Phi(t,{\bf x})$ of the helium Bose condensate [13], in the first
place) may be observed.

Finally, it may be argued, as in the liquid helium II turning model [13], that
the first-order phase transition occurs also in the liquid helium II at rest
model [15].

Returning again to the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac [4], note the very transparent analogy between the rotary
item $\Delta E$, (3.18), in the liquid helium II turning model [13] and the
free rotator action functional $W_{N}$, (1.11), in the former case.

The important distinction between the both theories is, however, in the
logarithmic divergence of $\Delta E$, (3.18), as $R/a\to\infty$, while the
free rotator action functional $W_{N}$, (1.11), is suppressed actually by the
value of $\epsilon(\infty)\to 0$.

The just performed brief analysis of the liquid helium II turning model [13]
(as well as of the liquid helium II at rest model) suggests, by analogy, the
idea to link the various rotary properties [3, 5, 7, 8, 12] of the Minkowskian
Higgs model to vacuum BPS monopoles quantized by Dirac (described correctly by
the free rotator action functional (1.11)) with the discrete vacuum geometry
(3.9), including thread topological defects inside the appropriate vacuum
manifold $R_{\rm YM}$ (in the similar way, has been outlined above, in which
such topological defects induce rotary effects in the liquid helium II
theory).

This will be the topic of the next section, where also some arguments in
favour of the first-order phase transition occurring in the Minkowskian Higgs
model with vacuum BPS monopole solutions quantized by Dirac and similar to
that taking place in the liquid helium II theory (these arguments are
connected with the ”electric” and ”magnetic” contributions to the total vacuum
energy).

##  4 How topologically nontrivial threads arise inside the discrete non-
Abelian vacuum manifold.

The outlines of this topic were us projected in Introduction.

It should be begun from the fact that the Minkowskian Higgs model [9, 10, 11]
with vacuum BPS monopoles is the specific model where the duality in
specifying the sign of the vacuum ”magnetic” field $\bf B$ takes place. This
duality is just set by the Bogonol’nyi equation (1.1) [11].

This duality implies automatically the duality in specifying the signs of
magnetic charges $\bf m$, given by the general formula [9]

${\bf m}=\frac{1}{4\pi}\int\limits_{\Gamma}d{\bf S}~{}{\bf
B}=\frac{1}{8\pi}\int
d^{3}x~{}\partial_{i}\\{\epsilon^{ijk}<F_{jk}^{b},\Phi_{b}>a^{-1}\\}.$ (4.1)

Note that latter Eq. is irrelatively correct, actually, to the concrete choice
of the vacuum geometry: either ”continuous” or ”discrete” one, in the quested
Minkowskian Higgs model.

Really, due to the Bianchi identity $D~{}B=0$, the definition (4.1) of
magnetic charges $\bf m$ may be recast formally to the look

${\bf m}=\frac{1}{4\pi}\int\limits_{\Gamma}d{\bf S}~{}{\bf
B}\sim\frac{1}{4\pi}\int\limits_{V}{\rm div}~{}{\bf B}~{}dV=0,$

and this integral does not depend on the choice of the space-like surface in
the (Minkowski) space.

However, the actual divergence of the ”magnetic” tension ${\bf B}$ at the
origin of coordinates in all the Minkowskian Higgs models (for instance, [10,
33, 34]) implies that the latter integral is different from zero as the origin
of coordinates lies inside the chosen space-like surface $\Gamma$.

Furthermore, the above duality [11] in specifying the sign of the vacuum
”magnetic” field $\bf B$ affects also the generator $h(\Phi)$ [9] of the
residual $U(1)$ gauge group in the studied Minkowskian Higgs model.

It is so since $h(\Phi)$ may be read easily from Eq. (4.1).

Now let us return to the simpler case of ”continuous” ($\sim S^{2}$) vacuum
geometries in Minkowskian Higgs models.

In this case the appropriate degeneration space: say, $R$, is one-connected:

$\pi_{0}(H)=\pi_{1}(R)=0,$

and one can always choice, in the unique way, a continuous branch of the
function $h(\Phi)$ on $R$.

Another situation is when the degeneration space $R$ is multi-connected
(”discrete”), i.e. when

$\pi_{0}(H)=\pi_{1}(R)\neq 0,$

and therefore there exist thread topological defects in the gauge theory in
question (in particular, in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac [4]).

On the one hand (as we have seen this above), the nontrivial isomorphism (2.1)
[9] implies the existence of thread topological defects inside the discrete
vacuum manifold $R$.

On the other hand, it turns out (see §$\Phi$13 in [9]) that the duality in the
choose of the sign of $h(\phi)$ (and therefore also the duality [11] in the
choose of the direction of $\bf B$) may become unavoidable.

This occurs, in particular, when one can invert the function $h(\Phi)$ via the
gauge transformations

$\tilde{h}(\Phi)=\alpha h(\Phi)\alpha^{-1}=-h(\Phi),$ (4.2)

with

$\alpha=\exp(2\pi M)$ (4.3)

and the element $M$ belonging to the Lee algebra of the group $G_{1}$ of rigid
transformations compensating changes in the vacuum (YM-Higgs) thread
configuration $(\Phi^{a},A_{\mu}^{a})$ at rotations around the axis $z$ of the
chosen reference frame.

The general look of an element $g_{\theta}\in G_{1}$ may be set by Eq. (2.5)
[9].

Herewith our above discussion about thread topological defects inside
degeneration spaces in the liquid helium II models (turning [13] and at rest
[15]) prompts us the idea to associate the rectilinear threads (as infinite
thin tubes of infinite lengths) with the above axial symmetry.

In the monograph [9] (in §$\Phi$12) the strict proof was given that the
unitary group $G_{1}$ is a possible symmetry for alike threads (by analogy
with the group $O(2)$ of orthogonal $2\times 2$ matrices with determinants
$\pm 1$, one utilizes in the liquid helium II turning model [13]).

Omitting details, note that the proof of this statement comes [9] to the proof
that in each topological class, rectilinear threads exist giving a finite
contribution to the appropriate energy integrals (due to first-order phase
transitions occurred) and satisfying the appropriate equations of motion.

The topological type of a thread defect is determined by the group $G_{1}$:
more exactly by its generator $M$.

To prove this statement, we betake to the argument of the holonomies group [7,
9].

Let us rewrite an element (3.16) of the holonomies group $H\simeq U(1)$ in the
shape

$\alpha=(P\exp(-\oint A_{\mu}dx^{\mu}))^{-1}.$ (4.4)

Due to the standard Pontryagin degree of a map theory, elements $\alpha$ of
the holonomies group $H$ depend on integers $n$: $\alpha\equiv\alpha(n)$
($n\in\bf Z$).

Going over then to the cylindrical coordinates, we come to Eq. (2.9) [9],
implicating rectilinear (topologically nontrivial) threads $A_{\theta}$, may
be represented as (2.4).

Alternatively, $A_{\theta}$ can be represented also as [9]

$A_{\theta}(\rho)=M+\beta(\rho),$ (4.5)

where the function $\beta(\rho)$ approaches zero as $\rho\to\infty$.

This also ensures [9] a finite energies densities in Minkowskian Higgs models
involving (topologically nontrivial) threads $A_{\theta}(\rho)$ and testifies
to their location in the region $\rho\to 0$, i.e. (intimately) near the origin
of coordinates.

Thus at the spatial infinity elements $\alpha(n)\in H$ acquire the quested
look 121212In Ref. [9] the following general proof of Eq. (4.2) was
given.Firstly, there was assumed that the elements $\alpha(n)$, (4.4), of the
holonomies group $H\simeq U(1)$ belong to its adjoint representation.
Secondly, the ”generalized case” when $U(1)$ is the subgroup of an invariant
subgroup $\tilde{H}=K\otimes U(1)$ in the initial gauge symmetry group $G$
(with $K$ locally isomorphic to a product of simple non-Abelian Lee groups)
was discussed in Ref. [9]. A continuous family $\alpha_{t}$ of elements of
$\tilde{H}\subset G$ was considered connecting the element $\alpha(n)$, (4.4),
with the unit element of $G$ (it was put $0\leq t\leq 1$, $\alpha_{1}=\alpha$,
$\alpha_{0}=1$, $\alpha h(\Phi_{0})\alpha^{-1}=-h(\Phi_{0})$; with $\Phi_{0}$
being the fixed point in the vacuum manifold). At assuming [9] that the
elements $\alpha(n)$, (4.4), belong to the adjoint representation of the group
$G$, the generator of the group $U(1)\subset\tilde{H}(T(\alpha_{t})\Phi_{0})$
(with $T(g)$ being the transformation corresponding to the group element $g\in
G$) can be taken as
$\tilde{H}(T(\alpha_{t})\Phi_{0})=\alpha_{t}h(\Phi_{0})\alpha_{t}^{-1}.$ Of
course, another choice is possible, with this generator of the opposite sign,
but if the purpose is to have a continuous one-valued branch of the function
$h(\Phi)$, the sign of the generator $h(\alpha_{t}(\Phi_{0}))$ is no longer
arbitrary. Then at $t=1$ one has  $h(\Phi_{0})=h(\alpha(\Phi_{0}))=\alpha
h(\Phi_{0})\alpha^{-1}=-h(\Phi_{0}).$ In latter Eq. the fact [9] that all the
subgroups $\tilde{H}(\Phi)$ in $G$ are mutually conjugate (i.e. that there
exist an intrinsic automorphism in $G$ mapping $\tilde{H}(\Phi)$ into a
$\tilde{H}(\Phi^{\prime})$) was taken into account. Thus it becomes obvious
that the assumed existence of a continuous one-valued branch for $h(\Phi)$ is
inconsistent.This proof [9] of Eq. (4.2) can be generalized easy for
nontrivial topologies $n\neq 0$. To do this, one would replace $\alpha_{0}=1$
(that corresponds to trivial topologies $n=0$) with an arbitrary $\alpha_{0}$.

$\alpha(n)=\exp(2\pi M).$ (4.6)

We see that the dependence of gauge fields $A$ on the angle $\theta$
disappears at the spatial infinity.

It is equivalent to damping, at the spatial infinity, various rotary effects,
associated in Minkowskian non-Abelian theories with thread topological
defects.

In particular, it is correctly for the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac [4] and involving the ”discrete” vacuum geometry
(3.9).

On the other hand, the function $\beta(\rho)$ should be different from zero
effectively only in the $\epsilon$-neighbourhood of the origin of coordinates
to ensure in this infinitesimal region the pointed above collective rotary
phenomena in the appropriate Minkowskian YM vacuum via the nontrivial
dependence of $A_{\theta}(\rho,\theta,z)$ on $\theta$.

In the $\epsilon$-neighbourhood of the origin of coordinates (i.e. at $\rho\to
0$), the function $\beta(\rho)$ would be singular. It is the necessary thing
[45] to ensure gauge transformations (4.2) [9], changing the sign of the
$U(1)$ generator $h(\Phi)$ (that implies changing the sign of the magnetic
charge $\bf m$: the resulting flip of the magnetic charge $\bf m$ is $2m$,
from $-m$ to $m$).

Due to the ”total” conservation law for the magnetic charge $\bf m$, YM thread
solutions $A_{\theta}(\rho,\theta,z)$ [9] would bear the ”compensating”
magnetic charge $-2m$ [45].

The distinctive property of such ”compensating” magnetic charges $-2{\bf m}$,
in comparison with ”ordinary” magnetic charges $\bf m$ (associated with vacuum
BPS monopole solutions), is [37] their ”discrete” origin, the presence of
(topologically nontrivial) threads in the Minkowskian Higgs model quantized by
Dirac [4] under consideration. Finally, this is associated with the ”discrete”
vacuum geometry (3.9) us assumed to justify the Dirac fundamental quantization
of the above model.

The ”discrete” magnetic charges (let us denote them as ${\bf m}_{\bf Z}$)
connected with (topologically nontrivial) threads are not shielded by the
condensate of the Higgs field.

More exactly [37], imagining a particle of the magnetic charge ${m}e$ (with
$e$ being the unit magnetic charge), the operator

$\Im_{V}=\exp(2\pi i{\bf M}_{V}/{\bf m}_{\bf Z}),$ (4.7)

with ${\bf M}_{V}$ being the magnetic charge distributed in the spatial volume
$V$ (specefied via the general formula (4.1) [9]), can be constructed as a
long range observable.

Note [45] that there is no gauge invariant way to pinpoint when the abov
charge transfer has occured. The field configuration $(\Phi^{a},A_{\mu}^{a})$
is $U(1)$-invariant, but the direction of the arrow on the $A_{\mu}^{a}$ field
line is gauge dependent. One can move the cut by performing a singular gauge
transformation (4.2); this alters the apparent time of the charge transfer
without actually changing the physics of the process.

Indeed, above described (vacuum) YM thread modes $A_{\theta}(\rho,\theta,z)$
[9] would be different from vacuum YM BPS monopoles (for instance, in the
Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac).

It is associated with the obvious divergence as $r^{-2}$ [10] (c.f. (3.17)
[36]) of the vacuum ”magnetic” field $\bf B$ at the origin of coordinates in
Minkowskian non-Abelian models involving vacuum BPS monopole solutions.

In this case there is impossible to link YM fields
$A_{\theta}(\rho,\theta,z)$, representing topologically nontrivial
(rectilinear) threads in the Minkowskian Higgs model, to vacuum BPS monopoles
quantized by Dirac, with vacuum YM BPS monopole solutions (with same
topological numbers), at least in a smooth wise (that is connected with the
$\sim\partial_{\mu}A_{\nu}$ items always entering the ”magnetic” field $\bf B$
in YM theories).

On the other hand, the value $<B^{2}>$, the vacuum expectation value of the
”magnetic” field $\bf B$ squared, can serve [8] as the order parameter in
Minkowskian Higgs models with YM fields.

Herewith its nonzero value, $<B^{2}>\neq 0$, is the sign violating the initial
$SU(2)$ gauge symmetry group down to its $U(1)$ subgroup, and it is always the
second-order phase transition [6, 28].

If the Minkowskian Higgs model with YM fields implies the continuous $\sim
S^{2}$ vacuum geometry, the possible (as in the BPS or ’t Hooft-Polyakov
models) divergence of $\bf B$ at the origin of coordinates testifies [6] in
favour of point hedgehog topological defects located just at the origin of
coordinates.

The origin of such topological defects is in the nontrivial isomorphism (2.2).

This singularity of (vacuum) ”magnetic” fields can be removed by regularising
appropriate energy integrals.

Also there are no contributions from point hedgehog topological defects into
these energies integrals unlike the case of first-order phase transitions when
such contributions appear (as, for instance, $\Delta E$ [13], (3.18), in the
case of a liquid helium II specimen turning together with the cylindrical
vessel where it is contained).

Unlike the former case, gaps in the directions of $\bf B$ and ${\bf B}_{1}$,
(2.6), in the Minkowskian Higgs model with vacuum BPS monopoles quantized by
Dirac and involving the ”discrete” vacuum geometry (3.9) according to our
assumption are evidences in favour of the first-order phase transition
occurring in this model.

This is in a good agreement with the general theory of first-order phase
transitions (discussed, for instance, in Ref. [28], in §3.1; see also [6]).

The discontinuous behaviour of plots for order parameters is just the sign of
first-order phase transitions occurring in physical theories 131313The
discussed gaps in the directions of $\bf B$ and ${\bf B}_{1}$ discovered in
the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac is a
specific manifestation of singularities associated with vortices presenting in
that model. The existence of alike singularities in gnon-Abelian gauge
theories quantized by Dirac was predicted as far back in Ref. [16] (in the
English version of that paper). .

The above discussed gaps do not influence however the ”right” of gauge
matrices $\alpha(n)$, (2.9) (taking account of the ”discrete” vacuum geometry
(3.9), involving [topologically nontrivial] threads), to represent the
holonomies group $H$ coinciding with the residual gauge symmetry group $U(1)$
in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac
[4].

This may be explained by the natural isomorphism between exponential
multipliers $\alpha(n)$, referring to (topologically nontrivial) threads
arising because of the ”discrete” vacuum geometry (3.9), and Gribov
topological multipliers $v^{(n)}({\bf x})$, referring to superfluid potential
motions in the above model.

Note that the dependence of the theory considered on nontrivial topologies $n$
also disappears at the spatial infinity, i.e. in the infrared region of
transferred momenta.

It is the next in turn evidence in favour of the infrared topological
confinement (in the spirit [1, 23]) occurring in the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac.

This ”confluence” of different topologies $n$ in the infrared region of the
momentum space is closely related with the tendency of domain walls between
different topological sectors of the vacuum manifold $R_{YM}$ to become more
”thin” as the spatial volume $V$ approaches infinity.

In this limit, the typical width of a domain wall, $\epsilon~{}(V)$, goes,
indeed, to zero, as it was discussed above (see (3.14)).

The above discussed examples of liquid helium II at rest [15] and turning
together with the cylindrical vessel where it is contained [13] give us
pattern of physical systems in which first-order phase transitions occur.

The ”centre of gravity” of the enumerated models is in the appearance of
(rectilinear) threads generated appropriate vortices, i. e. thread topological
defects.

On the other hand, it was demonstrated above (repeating the arguments [13]),
in the case of a liquid helium II specimen turning together with the
cylindrical vessel where it is contained, the ”rotary” contribution $\Delta
E$, (3.18), supplements the typical (diagonalized) Bogolubov Hamiltonian
$\widehat{\cal H}$ [41].

It is the sign of the first-order phase transition taking place in the liquid
helium II turning model [13].

Herewith according to the modern terminology [6], this ”rotary” contribution
$\Delta E$, (3.18), is the latent heat, while the thermodynamic phase in whose
framework collective solid potential rotations [13, 15] in a liquid helium II
specimen turning together with the cylindrical vessel where it is contained
coexist with superfluid motions setting by the Hamiltonian $\widehat{\cal H}$
is characterized by the supercooling phenomenon.

In this case superfluid motions inside a liquid helium II specimen represent
the stable state corresponding to the second-order phase transition occurred
therein beforehand and coming to violating the $U(1)$ gauge symmetry of the
helium Hamiltonian $\widehat{\cal H}$ [41] 141414The transparent evidence in
favour of the second-order phase transition occurring in a helium: actually,
it is the transition between the liquid helium I high-temperature and liquid
helium II modifications (the latter one possesses manifest superfluid
properties described in the Landau-Bogolubov theory [14, 42]) is the
discontinuity [46] in the plot of the helium heat capacity in the Curie point
$T_{c}\to 0$.For instance [41], in the case of superfluid helium films, the
experimental data result  $C_{p}(T)\equiv T(\frac{\partial S}{\partial
T})\approx c_{1}+c_{2}\exp[c_{3}/|T-T_{c}|^{1/2}]$ for the helium heat
capacity $C_{p}(T)$, with $S$ being the appropriate entropy; $c_{i}$
($i=1,2,3$) are constants.  (in the (3.22) wise in the liquid helium II
turning case [13]).

In turn, collective solid potential rotations in this specimen are referred to
the metastable state (in the terminology [28]).

There are concrete computations, given in Ref. [41] with the example of a
rested liquid helium II specimen, demonstrating discontinuities in the plot of
the vacuum expectation value $<|\Xi|^{2}(t,{\bf r})>$ for the helium Bose
condensate wave function $\Xi$, (1.4), near locations of vortices cores.

Herewith the value $<|\Xi|^{2}(t,{\bf r})>$ can serve quite naturally as the
order parameter in the Landau-Bogolubov helium theory [14, 42].

So $<|\Xi|^{2}(t,{\bf r})>=0$ for the helium I modification [46],
corresponding to the initial $U(1)$ gauge symmetry of the helium Hamiltonian
$\widehat{\cal H}$ [41], while $<|\Xi|^{2}(t,{\bf r})>\neq 0$ upon this
symmetry is violated in the (3.21) wise, that corresponds to the helium II
modification.

Then, utilizing Eq. (1.4) [6, 13], the correlation (Green) function

$G(r)\equiv<\Xi(t,{\bf r}),\Xi^{*}(t,{0})>$

(it is different from zero for the liquid helium II modification) can be
recast to the look [41]

$\displaystyle G(r)$ $\displaystyle=$ $\displaystyle\langle\Xi(t,{\bf
r}),\Xi^{*}(t,0)\rangle$ (4.8) $\displaystyle=$
$\displaystyle\Xi_{0}^{2}\exp\left[-{1\over 2}\langle[\Phi(t,{\bf
r})-\Phi(t,0)]^{2}\rangle\right],$

where the last line is a general property of Gaussian fluctuations.

To a first approximation, one can neglect fluctuations in the amplitude of
$\Xi$, setting it to be a $\Xi_{0}$. Then the vacuum expectation value
$<|\Xi|^{2}(t,{\bf r})>$ of the helium Bose condensate wave function $\Xi$
will equal to $G(0)=(\Xi_{0})^{2}$.

On the other hand, the above discussed expression [15] for topological indices
$n\in{\bf Z}$ via circular velocities ${\bf v}^{(n)}$ is mathematically
equivalent to [41]

$\frac{m}{2\pi\hbar}\oint_{\Gamma}\nabla\Phi~{}d{\bf
l}=\int_{\Omega}d^{2}r\;n_{v}({\bf r}),$ (4.9)

with the vortex ”charge density” given as

$n_{v}({\bf r})=\sum_{\alpha=1}^{N}\;n_{\alpha}\delta({\bf r}-{\bf
r}_{\alpha})$ (4.10)

for a collection $N$ of vortices located at positions $\\{{\bf r}_{\alpha}\\}$
with integer charges $\\{n_{\alpha}\\}$. Without loss of generality, it may be
set $z=0$ in all the calculations [41] following. In (4.9), Eq. (1.3) [6, 13],
with appropriate replacing ${\bf v}_{0}\to{\bf v}^{(n)}$, was utilized.

Applying then the Stokes formula to (4.9), one find upon integrating:

$\displaystyle\epsilon_{ij}\partial_{i}\partial_{j}\Phi(t,{\bf r})$
$\displaystyle=$
$\displaystyle\partial_{x}\partial_{y}\Phi-\partial_{y}\partial_{x}\Phi$
(4.11) $\displaystyle\approx$ $\displaystyle n_{v}({\bf r}).$

To cast this Eq. in a more familiar form, one can introduce [41] the value
$\tilde{\Phi}(t,{\bf r})$ dual to $\Phi(t,{\bf r})$:

$\partial_{i}\Phi(t,{\bf r})=\epsilon_{ij}\partial_{j}\tilde{\Phi}(t,{\bf
r}).$ (4.12)

Then

$\nabla^{2}\tilde{\Phi}(t,{\bf r})=n_{v}({\bf r}).$ (4.13)

In particular,

$\nabla^{2}\tilde{\Phi}(t,{\bf r})=0$ (4.14)

in the case when vortices are absent, i.e. in the case of purely superfluid
motions inside a liquid helium II specimen.

Eq. (4.13) is the particular case of the Poisson equation. Thus it permits its

$\tilde{\Phi}(t,{\bf r})=\sum_{\alpha}\;n_{\alpha}G({\bf r},{\bf r}_{\alpha})$
(4.15)

solution, where the Green function $G({\bf r},{\bf r}_{\alpha})$ (defined as
on the left-hand side of Eq. (4.8)) satisfies

$\nabla^{2}G({\bf r},{\bf r}_{\alpha})=\delta({\bf r}-{\bf r}_{\alpha}).$
(4.16)

For $|{\bf r}-{\bf r}_{\alpha}|$ large enough and both the points far from any
boundaries, the Green function $G({\bf r},{\bf r}_{\alpha})$ has the look (see
§11.8 in [47])

$G({\bf r},{\bf r}_{j})\approx{1\over 2\pi}\ln\left({|{\bf r}-{\bf
r}_{j}|\over\xi_{0}}\right)+C$ (4.17)

on the $z=0$ plane, where $C$ is a constant which contributing to the vortex
core energy.

The origin of the parameter $\xi_{0}$ entering Eq. (4.17) is following [41].
It is the characteristic length related to the coefficients of the first two
terms of Eq. [13]

${F\over T}=\int\;d^{2}r\left[{1\over 2}\;A|\nabla~{}\Xi|^{2}+{1\over
2}\;a|\Xi|^{2}+b|\Xi|^{4}+\cdots\right]$ (4.18)

for the helium free energy $F$.

In this case $\xi_{0}$ can be defined as [41]

$\xi_{0}=\sqrt{A/|a|},$ (4.19)

with $a$ given as $a=a^{\prime}(T-T_{c})$.

As it follows from (4.17), at setting $C=0$, the Green function $G({\bf
r},{\bf r}_{j})$ diverges logarithmically when $|{\bf r}-{\bf
r}_{j}|\to\infty$, while it approaches zero when

$|{\bf r}-{\bf r}_{j}|\to\xi_{0}.$

Additionally, $G({\bf r},{\bf r}_{j})$ diverges logarithmically also at $|{\bf
r}-{\bf r}_{j}|\to 0$. The latter property of $G({\bf r},{\bf r}_{j})$, as we
shall se soon, is very important.

The just performed analysis of $G({\bf r},{\bf r}_{j})$ allows to draw the
series important conclusions about the phase $\tilde{\Phi}(t,{\bf r})$ of the
helium wave function $\Xi$ (taking over the complete collection $N$ of
topologies: including the trivial one $s_{\alpha}=0$, corresponding to
superfluid motions [14, 15], in a liquid helium II specimen at rest).

So, due to (4.8),

$<|(\Xi)|^{2}(t,{\bf r})>\equiv<\Xi(t,{\bf r}),\Xi^{*}(t,{\bf r})>\approx
G(0)=1/N\sum\limits_{\alpha}<|\Xi^{2}|_{\alpha}(t,{\bf r})>,$ (4.20)

serving the order parameter in the helium theory, diverges logarithmically at
each index $\alpha(n)$ ($n\in{\bf Z}$) in the same (4.17) [41] sense that the
Green function $G({\bf r}_{\alpha},{\bf r}_{\alpha})\approx G(0)$, i.e. in the
points $\\{{\bf r}_{\alpha}\\}$ where quantum vortices are located in a liquid
helium II specimen at rest 151515In (4.20) the Green function $G(0)$ is
different from that given via Eq. (4.8) and involving the constant amplitude
$\Xi_{0}$. Now the amplitude of $\Xi(t,{\bf r})$ wouldn’t be constant, and
this implies that $G(0)$ has the nontrivial look $G({\bf r}_{\alpha},{\bf
r}_{\alpha})$ [41], (4.17)..

This is associated immediately with (rectilinear) quantum vortices arising
spontaneously [15] in a liquid helium II specimen at rest. Herewith also the
phase $\tilde{\Phi}(t,{\bf r})$ of the helium wave function $\Xi$ possesses
the same behaviour that the Green function $G({\bf r},{\bf r}_{j})$ (it is
correctly due to (4.15)).

The said is true for nontrivial vortices topologies $n\neq 0$ due to (4.15).
However for $n=0$, i.e. in the ”superfluid case”, $\tilde{\Phi}(t,{\bf r})$
becomes an uncertain value when $|{\bf r}-{\bf r}_{j}|\to\infty$ (or when
$|{\bf r}-{\bf r}_{j}|\to 0$).

On the other hand, the phase $\tilde{\Phi}(t,{\bf r})$ (or $\Phi(t,{\bf r})$)
of the helium wave function $\Xi$, (1.4), disappears from Eq. (4.20).

Summarizing up, now we can assert that in the presence of (rectilinear)
quantum vortices [15] in a liquid helium II specimen, the order parameter
$<|\Xi|^{2}(t,{\bf r})>$ in the helium model suffers discontinuities of the
logarithmic nature in the points $\\{{\bf r}_{\alpha}\\}$ in which these
(rectilinear) quantum vortices are located (the long distances logarithmic
singularities of $G(0)$ are not associated, probably, with quantum vortices).

The discovered discontinuity (of the logarithmic nature) in the plot of the
vacuum expectation value $<|\Xi|^{2}(t,{\bf r})>$ for the helium Bose
condensate wave function $\Xi$, serving the order parameter in the helium
theory, just testifies in favour of the first-order phase transition occurring
in a liquid helium II rested specimen with arising therein quantum vortices.

There are, indeed, lot of distinctions between the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac [4] and the liquid helium II at
rest theory [15] (the same concerns also the liquid helium II turning model
[13], involving rigid $O(2)$ rotations).

But, in spite of these distinctions, the discontinuities in the appropriate
order parameters: $<|\Xi|^{2}(t,{\bf r})>$ in the helium theory and $<B^{2}>$
in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac,
can be considered as the common trait of the both models associated with
rotary effects (including the ”discrete” vacuum geometry and thread
topological defects).

Moreover, drawing a parallel between the Minkowskian Higgs model with vacuum
BPS monopoles quantized by Dirac and the liquid helium II at rest theory [15],
one can refer collective solid rotations inside the appropriate BPS monopole
vacuum to the metastable thermodynamic phase, characterised by the latent heat
equal to the ”electric” energy $\sim E^{2}$. This ”electric” energy is
described correctly by the free rotator action functional (1.11).

Simultaneously, superfluid potential motions inside this vacuum, involving the
”magnetic” energy $\sim B^{2}$, may be referred to the stable thermodynamic
phase.

It is again the supecooling situation.

As it was discussed in Section 2, on the face of it, there is a contradiction
between the rotary effects in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac (these effects, including collective vacuum
rotations, are determined by the action functional (1.11)) and the manifest
potential nature of the Minkowskian BPS monopole vacuum.

More exactly, this potential nature of the Minkowskian BPS monopole vacuum
(suffered the Dirac fundamental quantization [4]) comes, in particular, to the
colinearity (2.8) [16] of vectors $\bf B$ and $\bf E$ (vacuum ”magnetic” and
”electric” fields, respectively).

This is associated with the explicit look (1.13), (1.15) [5, 7, 8, 12] for
vacuum ”electric” monopoles $\bf E$, can be expressed through vacuum Higgs BPS
monopole modes $\Phi_{(0)}({\bf x})$.

Also the vacuum ”magnetic” field can be expressed through vacuum Higgs BPS
monopole modes $\Phi_{(0)}({\bf x})$, now through the Bogomol’nyi equation
(1.1).

There is however a way to solve the above contradiction.

It turns out that, side by side with (topologically nontrivial) threads
$A_{\theta}$, one can construct also specific vacuum Higgs $z$-invariant
solutions [9], having the look (2.7) in the cylindrical coordinates, such that
the vacuum Higgs field $\Phi_{a}^{(n)}$ specified by (2.7) merges with the
appropriate Higgs vacuum BPS monopole solution in the Minkowskian Higgs model
with vacuum BPS monopoles quantized by Dirac [4], herewith belonging to the
same topology $n$.

Additionally, we should also claim that the covariant derivative $D\Phi$ of
any vacuum Higgs field $\Phi_{a}^{(n)}$ specified via (2.7) merges with the
covariant derivative of such a vacuum Higgs BPS monopole solution.

In other words, vacuum Higgs solutions (2.7), corresponding to the above
described axial symmetry $G_{1}$, would merge with appropriate Higgs vacuum
BPS monopole solutions actually in a smooth wise.

All this can be achieved by the appropriate choice of the function
$\phi(\rho)$ in (2.7) (the latter one satisfies the spatial asymptotic (2.7)
[9]).

By this the goal is pursued to link $D\Phi_{a}^{(0)}$ to vacuum ”electric”
monopoles (1.13) in the Minkowskian Higgs model with vacuum BPS monopoles
quantized by Dirac 161616As it was explained above, the plot of
$D\Phi_{a}^{(0)}$ in the thread topological defects region inside the vacuum
manifold $R_{\rm YM}$ does not coincide with the plot of the vacuum ”magnetic”
field $\bf B$, given by the Bogomol’nyi equation (1.1), i.e. via vacuum Higgs
BPS monopole solutions. This vacuum ”magnetic” field $\bf B$ diverges as
$1/r^{2}$ [10] at the origin of coordinates..

Indeed, this confluence of vacuum Higgs solutions would occur already at the
notably lesser scale of distances: at the hadronic distances scales, $\sim$1
fm., to ensure the correct infrared behaviour [23] of quark Green functions in
the Minkowskian (constraint-shell) QCD.

Actually, the following picture takes place.

As it was noted in Ref. [20], Higgs BPS monopole solutions disappear at the
origin of coordinates (in its infinitesimal neighbourhood of the effective
radius $\epsilon$).

In this situation the continuous and smooth prolongation of Higgs vacuum BPS
monopole solutions with appropriate Higgs modes $\Phi_{a}^{(n)}$ [9], (2.7),
seems to be quite natural. It is controlled by the proper choice of
$\phi(\rho)$.

It is just the way solving the above discussed problem how to co-ordinate the
”colinearity condition” (2.8) [16] (reflecting the superfluid nature of the
Minkowskian BPS monopole vacuum suffered the Dirac fundamental quantization
[4]) with collective solid rotations inside this vacuum.

Really, Higgs (vacuum) modes $\Phi_{a}^{(n)}$, (2.7), refer to the rigid axial
symmetries group $G_{1}$ [9].

This group, in turn, compensates local rotations of the vacuum (YM-Higgs)
configuration $(\Phi_{a}^{(n)},A_{\mu}^{a})$ around the axis $z$ (just such
rotations are apprehended as rectilinear vortices).

On the other hand, this vacuum (YM-Higgs) configuration can be linked, in a
continuous and smooth wise in its ”Higgs” part, to Higgs vacuum BPS monopoles,
disappearing at the origin of coordinates [20].

By that two things are ensured.

Firstly, the continuity (and smoothness) achieved now for Higgs vacuum
solutions, ”legalizes” collective solid rotations inside the Minkowskian Higgs
model with vacuum BPS monopoles quantized by Dirac [4] (as a sign of thread
topological defects inside the appropriate vacuum manifold $R_{\rm YM}$) over
the infinitely narrow cylinder of the effective diameter $\epsilon$ along the
axis $z$. These rotations are described correctly by the action functional
(1.11) [5].

Secondly, simultaneously, the Gauss law constraint (1.8) and the ”colinearity
condition” (2.8) are satisfied outward of this infinitesimal region with
smooth vacuum ”electric” monopoles (1.13).

On the other hand, being a continuous and smooth value now, $<\Phi^{(0)}>^{2}$
(with all its Gribov copies $<\Phi^{(n)}>^{2}$; see e.g. [2]) cannot be
considered as an order parameter in the Minkowskian Higgs model with vacuum
BPS monopoles quantized by Dirac since alike order parameter would suffer
singularities in coordinate points where vortices (thread topological defects)
are located inside the vacuum manifold $R_{\rm YM}$ (cf. our above discussion
[41] about quantum vortices in a liquid helium II specimen).

Thus $<B^{2}>\sim<D~{}\Phi^{(n)}>^{2}$ [9] rather than $<\Phi^{(n)}>^{2}$
serves as an order parameter in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac. It is the specific of the first-order phase
transition occuring in that model and involving the discrete vacuum geometry
(3.9). This distinguishes it from the Minkowskian Higgs models involving the
continuous $\sim S^{2}$ vacuum geometry, where just the “Higgs” vacuum
expectation value $<\Phi>^{2}$ serves as the order parameter.

Note that also in the “classical”Minkowskian (YM-Higgs) model with vacuum BPS
monopoles [6, 9, 10, 11], involving the “continuous” $\sim S^{2}$ vacuum
geometry and in which only the second-order phase transition occurs, $<B^{2}>$
can serve as an order parameter, but now side by side with $<\Phi^{2}>$.

When we view very attentively Eq. (1.12) [5] for the ”rotary momentum” $I$ of
the BPS monopole vacuum, we can note that the integration there is carried out
over the infinite spatial volume $V\to\infty$.

At neglecting the infinitesimally small interval $[0,\epsilon(\infty)]$ in
this integral, one get nevertheless the same result.

More exactly, the integral (1.12) may be evaluated in the following way [5]:

$I\simeq\frac{4\pi^{2}}{\alpha_{s}}\int\limits_{\epsilon}^{R}dr~{}\frac{d}{dr}(r^{2}\frac{d}{dr}f_{0}^{BPS}(r));\quad
R\to\infty$ (4.21)

It involves the ”Higgs” BPS ansatz [7, 8]

$f_{0}^{BPS}(r)=\left[\frac{1}{\epsilon\tanh(r/\epsilon)}-\frac{1}{r}\right],$

disappearing [20] at the origin of coordinates.

We see that the extrapolation of Higgs vacuum BPS monopoles inward the
interval $[0,\epsilon]$, where the potential superfluid nature of the
Minkowskian BPS monopole vacuum suffered the Dirac fundamental quantization
[4] is violated, does not affects the integrals (1.12), (4.21) so long as the
integration limits in Eq. (4.21) are $[\epsilon,\infty]$.

This is just the region where the potential superfluid nature of that vacuum
takes place.

It is determined, in the Dirac fundamental quantization scheme [4], by the
Bogomol’nyi equation (1.1), YM Gauss law constraint (1.8) and Gribov ambiguity
equation (1.2) as the logical consequence [2, 7, 8] of the Bogomol’nyi
equation (1.1).

As to the interval $[0,\epsilon]$, integrating inside this interval comes to
integrating over the base of the infinitely narrow cylinder along the axis $z$
(in the chosen rest reference frame), so long as the Higgs BPS ansatz
$f_{0}^{BPS}(r)$ is a function of the distance $r$ only. Thus the interval
$[0,\epsilon]$ would result a vestigial contribution to the ”rotary momentum”
$I$, (1.12), and one can neglect it actually.

On hand thus a delimitation between superfluid potential motions and
collective solid rotations inside the Minkowskian BPS monopole vacuum suffered
the Dirac fundamental quantization [4].

Superfluid potential motions therein (controlled by the Bogomol’nyi and Gribov
ambiguity equations) are referred actually to the spatial interval
$r\in[\epsilon(\infty);\infty]$, while collective solid rotations to the
spatial interval $r\in[0,\epsilon(\infty)]$.

This delimitation is the next in turn evidence in favour of the first-order
phase transition occurring in the Minkowskian Higgs model with vacuum BPS
monopoles quantized by Dirac and coming to the coexistence of two
thermodynamic phases inside the Minkowskian BPS monopole vacuum suffered the
Dirac fundamental quantization. The thermodynamic phases of superfluid
potential motions and collective solid rotations are just such phases.

To this stage of investigations about the Minkowskian Higgs model with vacuum
BPS monopoles quantized by Dirac, we cannot conclude regarding the concrete
nature of the boundary between the mentioned thermodynamic phases as well as
regarding various surface effects.

At the same time, we should like note that the transparent way to verify that
the first-order phase transition occurs in any physical model is to compute
the appropriate surface tension integral [13]

$\alpha_{12}=\int\limits_{-\infty}^{\infty}(f_{1}-f_{2})~{}dx,$ (4.22)

where $f_{1}$ and $f_{2}$ are the free energies densities referring to two
thermodynamic phases coexisting at the first-order phase transition occurring
in the given physical model.

Meanwhile, already now some conclusions may be drawn concerning to the domain
structure of the Minkowskian vacuum manifold $R_{\rm YM}$, (3.9). It is just
associated with the $\epsilon(V)\sim V^{-1}$ dependence [7, 8].

So intimately to the axis $z$ of the chosen rest reference frame, domain walls
become very wide (the typical wide of a domain wall ibid of the
$\sim\epsilon(0)\to\infty$ order, while alone topological domains become
microscopically thin.

One can assume herewith that in the mentioned spatial region, topological
domains inside $R_{\rm YM}$ (at neglecting their thickness) possess
approximately the geometry of planes (perpendicular to the axis $z$). This
implies also concentrating rectilinear threads $A_{\theta}$ inside these
highly thin topological domains.

Vice verse, at the spatial infinity topological domains merge (as it was noted
above), while walls between them become infinitely thin (of the typical wide
$\epsilon(\infty)\to 0$) and this promotes the infrared topological
confinement [23] of ”large” multipliers $v^{(n)}({\bf x})$ in the (quark,
gluonic) Green functions in all the orders of the perturbation theory.

Actually, since the confluence of topological domains would occurs already at
the hadronic distances scales, $\sim$1 fm., one can speak that only in the
spatial region intimately near the axis $z$ of the chosen rest reference frame
the ”discrete” vacuum geometry $R_{\rm YM}$, (3.9), in the Minkowskian Higgs
model with vacuum BPS monopoles quantized by Dirac [4] is different from the
continuous $\sim S^{2}$ vacuum geometry, (3.1), proper to the ”classical”
Minkowskian Higgs model with BPS monopoles [9, 10, 11], involving the
”heuristic” FP quantization scheme [35], associated with fixing the $A_{0}=0$
gauge [6].

Nevertheless, this distinction in vacuum geometries plays the crucial role, as
we see from the current discussion, for understanding the Minkowskian Higgs
model with vacuum BPS monopoles quantized by Dirac.

The important consequence of the presence of topologically nontrivial threads
in the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac
[4] is annihilating [9] two equal magnetic charges ${\bf m}_{1}={\bf
m}_{2}={\bf m}(n)\neq 0$ ($n\in{\bf Z}$) colliding at crossing a rectilinear
topologically nontrivial thread $A_{\theta}(n)$: (2.4), (4.5).

This effect is the particular case of changes in the relative sign of two
magnetic charges as the integration region $\Gamma$ in (4.1) intersects a
(rectilinear) thread.

Suppose that the magnetic charge $|{\bf m}|=m_{1}$ is concentrated to the left
of the thread $A_{\theta}(n)$, while the magnetic charge $|{\bf m}|=m_{2}$ is
concentrated to the right of this thread. Herewith we concentrate these
magnetic charges in points.

Concentrating magnetic charges in points does not change the matter, despite
magnetic charges are distributed volume actually due to (4.1).

Due to (4.2), the both magnetic charges are specified to within their signs.

One can draw a closed curve connected the both magnetic charges (see Fig. 15
in [9]). It consists of two lines, $\gamma_{1}$ and $\gamma_{2}$, between the
charges, one rounds in the opposite directions.

This implies that the vacuum ”magnetic” tension $\bf B$ (specifying via the
Bogomol’nyi equation (1.1)) changes its sign depending on the ”direction”:
either $\vec{\gamma}_{1}$ or $\vec{\gamma}_{2}$, in which one crosses the
thread.

Really, we should always consider the relative sign of the charges ${\bf
m}_{1}$ and ${\bf m}_{2}$: more precisely, our interest is the difference
$m_{1}-m_{2}$ as we move in the ”direction” $\vec{\gamma}_{1}$; contrariwise,
our interest is the difference $m_{2}-m_{1}$ as we move in the opposite
”direction” $\vec{\gamma}_{2}$.

Taking account of the duality (4.2) in the definition of a magnetic charge, we
encounter few ”alternatives” for the values of the relative magnetic charge in
both the cases.

The most interesting case is when $m_{1}=m_{2}$. In this case some two vacuum
BPS monopoles belonging to the one fixed topological class $n\in\bf Z$
annihilated mutually at approaching each other crossing the topologically
nontrivial thread $A_{\theta}$ lying in the Minkowskian YM vacuum manifold
$R_{YM}$, (3.9)(the same is correctly also for excitations over the BPS
monopole vacuum).

The above reasoning demonstrates that magnetic charges are not maintained in
the Minkowskian Higgs model with vacuum BPS monopoles quantized by Dirac.

Magnetic charges cease then to be motion integrals; at the same time they are
still topological invariants: continuous deformations do not change the
difference $m_{2}-m_{1}$.

In the monograph [9], in §$\Phi 13$, there was proposed the following, more
general, definition of a magnetic charge, taking account of rectilinear
(topologically nontrivial) threads in appropriate non-Abelian gauge models.

One considers a membrane pulling on the rectilinear thread (now treated as a
geometrical object in the manner us discussed above) lying along the axis $z$
of the chosen coordinate system. Then one removes this membrane from the
Minkowskian space. This results a one-connected set, on which one can choice a
fixed continuous branch of the magnetic tension $\bf B$.

In fact, this ”trick” comes to the restriction of the integration region in
Eq. (4.1) for a magnetic charge by the ”left” (respectively, ”right”) semi-
space of the Minkowskian space with respect to the axis $z$. This ensures that
magnetic charges are maintained in each of these semi-spaces taken separately.

In the present study we don’t have for our object deriving the concrete
contribution from the above discussed annihilating effect [9] for equal
topologically nontrivial magnetic charges colliding at crossing a thread
$A_{\theta}$ in the total action functional (Hamiltonian) of the quested
Minkowskian Higgs model with vacuum BPS monopoles, although such contribution
exists undoubtedly.

Nevertheless, now we should like make some remarks may be helpful for finding
such contribution.

For instance, in the paper [48] possible interactions between (YM) magnetic
monopoles and antimonopoles were studied involving forming so-called monopole
molecules.

Herewith only two types of monopole molecules are possible [48], reflecting
ordinary Feynman rules for gauge non-Abelian fields (see e.g. §9.1 in [24]).

There are, firstly, confined colourless magnetic monopoles $M$ and $\tilde{M}$
(with the symbol $M$ referring to a monopole with the magnetic charge $\bf m$
and the symbol $\tilde{M}$ referring to an antimonopole with the magnetic
charge $-\bf m$, respectively). These monopole molecules were also referred to
as monopole-antimonopole mesons in Ref. [48]. The well-known monopole-
antimonopole confinement and annihilation mechanism via Abrikosov threads
(strings) of the ”magnetic” field [9, 36, 49] is implicated here (see also
§6.2 in [28]).

On the phenomenological level, such Abrikosov threads [9, 36, 49] can be
describes with the aid of the Mandelstam linearly increasing potential $E\sim
Kr$ [24] (with $K$ being the tension of the Abrikosov string).

Secondly [48, 24], confined colour triplet and antitriplet of monopoles are
also possible in the nature. Herewith $MMM$ (respectively,
$\tilde{M}\tilde{M}\tilde{M}$) states were referred to as baryons in [48].

The presence of topologically nontrivial threads in a gauge non-Abelian theory
(for instance, rectilinear threads (2.4) [9]), implying the annihilation of
monopole pairs with equal magnetic charges colliding at crossing such a
(rectilinear) topologically nontrivial thread, changes substantially the
matter in comparison with the case of the Abrikosov mechanism [28, 36, 48] of
the monopole-antimonopole confinement.

The former case is rather similar to the case of confined colour triplet and
antitriplet of monopoles (where the role of the one field in such triplets is
played by the rectilinear thread $A_{\theta}$). The only important specific is
that the rectilinear thread $A_{\theta}$ would bear always the magnetic charge
${\bf m}_{\bf Z}=-2m$ [45, 37] due to (above discussed) reasoning maintaining
the magnetic charge.

Note that this “baryonic” way annihilating equal magnetic charges colliding
with the rectilinear thread $A_{\theta}$ is somewhat alternative to one
discussed in Ref. [9]. In the latter case the $\bf m$ and $-\bf m$ pairs
forming upon crossing by the one of the magnetic charges the thread
$A_{\theta}$ annihilate rather in the “mesonic” $\tilde{M}M$ way as it was
described in [48].

The serious objection against this “mesonic” annihilation way is its
temperature dependence [28, 48]. More precisely [48], the annihilation of