मुख्य Magnetic monopoles

Magnetic monopoles

Письма в ЭЧАЯ. 2011. Т. 8, №7(170). С. 1242-1251
These are notes of the frst part of the lectures given at the JINR-ISU Baikal Summer School on Physics of Elementary Particles and Astrophysics (July 2010). I review classical monopole solutions of the SU(2) Yang-Mills-Higgs theory, providing a pedagogical introduction into to the theory of non-Abelian monopoles both in the BPS limit and beyond of it. I brieаfly discuss monopole dynamics, the idea of the moduli space and some of the basic properties which are connected with the feld theoretical aspects of these classical solutions.
Это - первая часть лекций, прочитанных на Байкальской школе по физике элементарных частиц и астрофизике, организованной ОЯИ и ИГУ (июль 2010). Представлен обзор решений с монополями SU(2) Янга-Милса. Дается педагогическое введение в неабелеву теория пространства монополей в пределе BPS и за его рамками. Кратко обсуждаются динамика монополей идея пространства модулей и некоторые из основных идей, связанных с теоретическими аспектами этих классических решений.
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¨¸Ó³ ¢ —Ÿ. 2011. ’. 8, º 7(170). ‘. 1242Ä1251

”ˆ‡ˆŠ ‹…Œ…’›• —‘’ˆ– ˆ ’Œƒ Ÿ„. ’…ˆŸ

Ya. Shnir1
Department of Mathematical Sciences, Durham University, Durham, UK
These are notes of the ˇrst part of the lectures given at the JINRÄISU Baikal Summer School on
the SU (2) YangÄMillsÄHiggs theory, providing a pedagogical introduction into to the theory of nonAbelian monopoles both in the BPS limit and beyond of it. I brie y discuss monopole dynamics, the
idea of the moduli space and some of the basic properties which are connected with the ˇeld theoretical
aspects of these classical solutions.
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PACS: 12.39.Dc; 12.39.Fe

We consider non-Abelian classical Lagrangian of a YangÄMillsÄHiggs theory with the
gauge group SU (N ), which describes coupled gauge and Higgs ˇelds:
1 a aμν 1 μ a
L = − Tr Fμν F μν + Tr Dμ φDμ φ − V (φ) = − Fμν
+ (D φ )(Dμ φa ) − V (φ). (1)
T a , φ = φa T a and
Here, Aμ = Aaμ T a is an SU (N ) connection with ˇeld strength Fμν = Fμν
we use standard normalization of the Hermitian generators of the gauge group: Tr (T a T b ) =
1/2δab, a, b = 1, 2, 3, which satisfy the Lie algebra [T a , T b ] = iεabc T c. The scalar ˇeld
φ = φa T a transforms in the adjoint representation of SU (N ) with the covariant derivative
deˇned by Dμ φ = ∂μ φ + ie[Aμ , φ]. The nonzero vacuum expectation value of the scalar ˇeld
corresponds to the symmetry-breaking Higgs potential V (φ):

V (φ) = λ(|φ|2 − v 2 )2 ,
where the group norm of the scalar ˇeld is deˇned as |φ|2 = 2 Tr φ2 = φa φa .
1 E-mail:



Magnetic Monopoles 1243

In the simplest nontrivial case of SU (2) group we have T a = σ a /2 and V (φ) = λ(φa φa −
v ) . The energy of the conˇguration is minimal if the following conditions are satisˇed:
2 2

φa φa = v 2 ,

= 0,

Dn φa = 0.


These conditions deˇne the vacuum. Note that the very deˇnition (3) forces the classical
vacuum of the SU (2) YangÄMillsÄHiggs theory to be degenerated. Indeed, the condition
V (φ) = 0 means that |φ| = v, i.e., the set of vacuum values of the Higgs ˇeld forms a sphere
. All the points on this sphere are equivalent because there is a well-deˇned SU (2) gauge
transformation which connects them. If v 2 = 0 the SU (2) symmetry is spontaneously broken
to U (1).
Thus, the solutions of the classical ˇeld equations map the vacuum manifold M = Svac
onto the boundary of 3-dimensional space, which is also a sphere S 2 . These maps are
characterized by a winding number n = 0, ±1, ±2 . . . which is the number of times Svac
covered by a single turn around the spatial boundary S 2 . The celebrated 't HooftÄPolyakov
solution [3, 4] corresponds to the ®hedgehog¯ asymptotic of the scalar ˇeld:
r→∞ r
Such a behavior obviously mixes the spatial and isotopic indices and deˇnes a single mapping
of the vacuum M onto the spatial asymptotic.
As was mentioned by 't Hooft [3], the conˇgurations which are characterized by different
winding numbers cannot be continuously deformed into each other. Indeed, the gauge transformation of the form U = ei(σk ϕ̂k )θ/2 rotates the isovector to the third axis. However, if we
try to ®comb the hedgehog¯, that is, to rotate the scalar ˇeld everywhere in space to a given
direction (so-called unitary or Abelian gauge), the singularity of the gauge transformation on
the south pole does not allow us to do it globally. Thus, there is no well-deˇned global
gauge transformation which connects the trivial and the ®hedgehog¯ conˇgurations and this
singularity results in the inˇnite barrier separating them.
Note that the electric charge of the massive vector bosons A± is given by the unbroken
U (1) subgroup. In general, this is a subgroup H of the gauge group G, the action of which
leaves the Higgs vacuum invariant. Obviously, that is a little group of the rotation in isospace
about the direction given by the vector φa . The generator of it, (φa T a )/a, must be identiˇed
with the operator of electric charge Q. Thus, the expression for the covariant derivative can
be written in the form
Dμ = ∂μ + ieAaμ T a = ∂μ + iQAem
φa −→

that allows us to deˇne the ®electromagnetic projection¯ of the gauge potential
1 a a
Aμ φ ,
Q = e φa T a .
Taking into account the deˇnition of the generators T a of the gauge group, we can easily see
that the minimal allowed eigenvalues of the electric charge operator now are q = ±e/2.
Thus, it would be rather natural to introduce the electromagnetic potential as a projection
of the SU (2) gauge potential Aaμ onto that direction, see Eq. (6). Furthermore [5], a general
solution of the equation Dμ φa = 0, for φa φa = a2 , can be written as
μ =

Aaμ =

εabc φb ∂μ φc + φa Λμ ,
a2 e


1244 Shnir Ya.

where Λμ is an arbitrary 4-vector. It can be identiˇed with the electromagnetic potential
because Eq. (7) yields for φa φa = a2
φa a
A = Λμ ≡ Aem
μ .
a μ
Inserting Eq. (7) into the deˇnition of the ˇeld strength tensor yields
= Fμν


where Fμν = ∂μ Aν − ∂ν Aμ +

εabc φa ∂μ φb ∂ν φc .
a3 e


This gauge-invariant deˇnition of the electromagnetic ˇeld strength tensor Fμν suggested
by 't Hooft [3] has a very deep meaning. It is rather obvious that in the topologically trivial
sector the last term in Eq. (8) vanishes and then we have
Fμν = ∂μ Aν − ∂ν Aμ .
This is precisely the case of standard Maxwell electrodynamics. Of course, in this sector
there is no place for a monopole because the Bianchi identities are satisˇed: ∂ μ Fμν ≡ 0.
However, for the conˇguration with nontrivial boundary conditions (4), (14), also the
Higgs ˇeld gives a nonvanishing contribution to the electromagnetic ˇeld strength tensor (8).
Then, the second pair of Maxwell equations becomes modiˇed:
∂ μ Fμν = kν .


Note that, if the electromagnetic potential Aem
μ is regular, the magnetic, or topological current
kμ is expressed via the scalar ˇeld alone:
kμ =

εμνρσ ∂ ν F ρσ = 3 εμνρσ εabc ∂ ν φa ∂ ρ φb ∂ σ φc .
2a e


At ˇrst glance this current is independent of any property of the gauge ˇeld. It is conserved
by its very deˇnition:
∂μ kμ ≡ 0,
unlike a Noether current which is conserved because of some symmetry.
The static regular solution of the corresponding ˇeld equations was constructed numerically
by employing the spherically symmetric ansatz [3, 4] for the gauge and the Higgs ˇelds,
φa = 2 H(ξ), Aan = εamn 2 [1 − K(ξ)], Aa0 = 0,
where H(ξ) and K(ξ) are functions of the dimensionless variable ξ = ver.
The condition of vanishing covariant derivative of the scalar ˇeld on the spatial asymptotic (3), together with the choice of the nontrivial hedgehog conˇguration, implies that
at r → ∞
 a
− eεabc Abn = 0.
The simple transformation
 a
rc rk
rc rk
r2 δan − ra rn
= (δan δck − δak δnc ) 2 = −εabc εbnk 3

Magnetic Monopoles 1245

then provides an asymptotic form of the gauge potential
Aak (r) −→


εank 2 .


This corresponds to the non-Abelian magnetic ˇeld
Bna −→


ra rn


Therefore, the boundary conditions (4), (14) are compatible with the existence of a long-range
gauge ˇeld associated with an Abelian subgroup which is unbroken in the vacuum. Since
this ˇeld decays like 1/r2 , which is typical behavior of the Coulomb-like ˇeld of a point-like
charge, and since the electric components of the ˇeld strength tensor vanish, we can recognize
a monopole in such a ®hedgehog¯ conˇguration with a ˇnite energy.
The explicit forms of the shape functions of the scalar and gauge ˇeld can be found
numerically. It turns out that the functions H(ξ) and K(ξ) approach rather fast the asymptotic
values. Thus, there is almost vacuum outside of some region of the order of the characteristic
scale Rc , which is called the core of the monopole. One could estimate this size by simple
arguments [7]. The total energy of the monopole conˇguration consists of two components:
the energy of the Abelian magnetic ˇeld outside the core and the energy of the scalar ˇeld
inside the core:
E = Emag + Es ∼ 4πg 2 Rc−1 + 4πv 2 Rc ∼

4π  −1
Rc + m2v Rc .

This sum is minimal if Rc ∼ m−1
v . In other words, inside the core at distances shorter
∼ (ve)−1 , the original SU (2) symmetry is
than the wavelength of the vector boson m−1
restored. However, outside the core this symmetry is spontaneously broken down to the
Abelian electromagnetic subgroup.
Note that the solution given by the 't HooftÄPolyakov ansatz (12) corresponds to the
condition Aa0 = 0. One could consider a more general case, where this time component of
the vector potential is not equal to zero, but is also a function of the spatial coordinates [8]:
Aa0 =



This ˇeld conˇguration corresponds to the non-Abelian dyon, which has both magnetic and
electric charges. The electric charge of the system of the ˇelds can be deˇned as
a a
dSn En =
dSn En φ =
d3 x Ena Dn φa .
Here, we invoked the ˇeld equations, according to which Dn Ena = 0, and made use of the
relation εabc φb D0 φc = 0, which is valid for the ansatz under consideration. The asymptotic
behavior of the proˇle function J(r) is very similar to that of the scalar ˇeld:
J(r) → 0 as

r → 0,

J(r) → Cr

as r → ∞.


The arbitrary constant C is connected with the electric charge of the dyon (17) [8]. The
charge vanishes if C = 0.

1246 Shnir Ya.

Indeed, substituting the ansatz (12) into the integral (17), after some algebra we obtain

= Cg.


However, on the classical level there is no reason for the electric charge (17), unlike the
magnetic charge, to be quantized and the constant C in (19) remains an arbitrary parameter.
Finally, we note that the time component of the vector-potential (16) is in isospace parallel
to the direction of the Higgs ˇeld. Moreover, one can consider it as an additional triplet of
the scalar ˇelds. This is the so-called JuliaÄZee correspondence φa  Aa0 .

Unfortunately, the system of nonlinear coupled differential equations on the functions
H(ξ) and K(ξ) in general has no analytical solution. The only known exception is the very
special case of vanishing scalar potential V = 0 [9Ä11]. This is the so-called Bogomol'nyiÄ
PrasadÄSommerˇeld (BPS) limit.
In the BPS limit of vanishing Higgs potential the scalar ˇeld also becomes massless and
the energy of the static ˇeld conˇguration is taking the form
 
Tr (εijk Fij ± Dk φ) ∓ εijk Tr (Fij Dk φ) d3 r.
Thus, the absolute minimum of the energy corresponds to the static conˇgurations which
satisfy the ˇrst-order Bogomol'nyi equations:
εijk Fij = ±Dk φ,


which are solved by

sinh ξ

H = ξ coth ξ − 1.


Deˇnitely, the solution to the ˇrst-order BPS equation (21) automatically satisˇes the system
of ˇeld equations of the second order.
BPS monopoles are very remarkable objects. Let us brie y recapitulate the properties of
these solutions:
• The BPS equation, together with the Bianchi identity, means that Dn Dn φa = 0. Therefore, the condition Dn φa Dn φa = (1/2)∂n ∂n (φa φa ) holds and the energy of the monopole
conˇguration in the BPS limit is independent of the properties of the gauge ˇeld:
= gv.
d3 x∂n ∂n (φa φa ) =
If the conˇguration has both electric and magnetic charges, the monopole mass becomes
M =v

g2 + q2 .

This yields the so-called Bogomol'nyi bound on the monopole mass.


Magnetic Monopoles 1247

• In comparison with the 't HooftÄPolyakov solution, the behavior of the Higgs ˇeld of
the monopole in the BPS limit has changed drastically: as we can see from (22), alongside
with the exponentially decaying component it also obtains a long-distance Coulomb tail
φa → vr̂a −



r → ∞.


The reason for this is that in the limit V (φ) = 0, the scalar ˇeld becomes massless.
• The long-range monopoleÄmonopole interaction is composed of two parts originating
from the long-range scalar force and the standard electromagnetic interaction, which could
be either attractive or repulsive [12]. Mutual compensation of both contributions leaves the
pair of BPS monopoles static but the monopole and antimonopole would interact with double
• The Bogomol'nyi equation may be treated as a three-dimensional reduction of the
integrable self-duality equations. Indeed, the JuliaÄZee correspondence means that
Dn φa  Dn Aa0 ≡ F0n
Bna = Dn φa  F0n = F0n


Therefore, if we suppose that all the ˇelds are static, the Euclidean equations of self-duality
= Fμν
reduce to Eqs. (21) and the monopole solutions in the Bogomol'nyi limit could
be considered as a special class of self-dual ˇelds.
• The analogy between the Euclidean YangÄMills theory and the BPS equations can be
traced up to the solutions. It was shown [13, 14] that the solutions of these equations are
exactly equal to an inˇnite chain of instantons directed along the Euclidean time axis t in
d = 4. More precisely, the BPS monopole is equivalent to an inˇnite chain of instantons
having identical orientation in isospace and separated by an interval τ0 = 2π. An alternative
conˇguration is a chain of correlated instantonÄantiinstanton pairs, which corresponds to an
inˇnite monopole loop.

In the BPS limit the JuliaÄZee dyonic solutions have a very interesting interpretation [15,
16]. First, we note that for the static ansatz (12), (16) and the choice A0 = 0, the kinetic
energy of the conˇguration
T = d3 x Tr (En En + D0 φD0 φ)
is equal to zero. Moreover, in this case the Gauss law
Dn En − ie [φ, D0 φ] = 0


can be satisˇed trivially, with En = D0 φ = 0.
Let us now consider time-dependent ˇelds An (r, t), φa (r, t), but suppose that their time
dependence arises as a result of a gauge transformation of the original static conˇguration:
An (r, t) = U (r, t)An (r, 0)U −1 (r, t) −

U (r, t)∂n U −1 (r, t) .


1248 Shnir Ya.

Here, U (r, t) = eieωt with ω(r) a parameter of the transformation. If the time interval δt is
very small, we can expand
U (r, δt) ≈ 1 + ieωδt + . . .
Now it follows from (29) that
An (r, δt) ≈ An (r) + (ie[ω, An (r)] − ∂n ω) δt,


∂0 An = ie[ω, An (r)] − ∂n ω = −Dn ω.


and we have
In a similar way we obtain for the time dependence of the scalar ˇeld:
∂0 φ = ie [ω, φ] .


These gauge transformations simultaneously affect the time component of the gauge potential, which for the monopole conˇgurations (12), (16) is a pure gauge:
A0 (r, t) = − U (r, t) ∂0 U −1 (r, t) = −ω.


Since the gauge transformations (32) and (33) do not change the potential energy of the
conˇguration, the parameter ω can be identiˇed with the gauge zero mode. This is one of four
collective coordinates (they are also called moduli) of the one-monopole conˇguration [16].
The other three specify the position of the monopole in space. Their appearance re ects an
obvious breaking of translational invariance of the original Lagrangian (1) by the monopole
conˇguration: the position of the monopole in R3 can be chosen arbitrarily.
However, deˇned in this way, the gauge zero mode is not physical, since the gauge
transformations (32) and (33) do not affect the non-Abelian electric ˇeld:
Ena = ∂0 An − Dn A0 = −Dn ω + Dn ω ≡ 0,
D0 φ = ∂0 φ + ie[A0 , φ] = ie [ω, φ] − ie [ω, φ] ≡ 0.


Thus, as before, the Gauss law is satisˇed trivially and the kinetic energy of the monopole (27)
is still equal to zero.
Now let us suppose that the time dependence of the ˇelds again appears as a result of the
gauge transformations (32) and (33), but that the corresponding gauge zero mode (∂0 An , ∂0 φ)
satisˇes the background gauge condition:
Dn (∂0 An ) − ie [φ, (∂0 φ)] = 0.


Then the Gauss law (28) is satisˇed, if A0 = 0 and there is a nontrivial solution of
Eqs. (32), (33) and (36) [15], where ω is proportional to φ and an additional time dependence is allowed:
ω = Υ̇(t)φ,
which corresponds to the gauge transformation
U (r, t) = exp {ieΥ(t)φ(r)} ≈ 1 + ieΥ̇φδt.


Magnetic Monopoles 1249

Here Υ(t) is an arbitrary function of time. Indeed, in this case we have ∂0 An = Υ̇Dn φ
and ∂0 φ = 0, and, since in the Bogomol'nyi limit Dn Dn φ = 0, the background gauge
condition (36) is satisˇed by the ansatz (37). However, this solution corresponds to the
generation of a non-Abelian electric ˇeld
En = ∂0 An = Υ̇(t)Dn φ = Υ̇(t)Bn ,

D0 φ = 0,

so the kinetic energy of the monopole (27) is no longer zero:
T = Υ̇2 d3 xDn φa Dn φa = Υ̇2 d3 xBna Bna = 2πvg Υ̇2 = M Υ̇2 ,



where we make use of the deˇnition of the magnetic charge and take into account that the
mass of the BPS monopole is
Since the potential energy of the conˇguration is time-independent, the gauge transformations (32) and (33), supplemented with the condition A0 = 0, deˇne a physical collective
coordinate Υ(t), that is a gauge zero mode. Its excitation corresponds to the generation of an
electric charge Q = Υ̇g. Thus, such a gauge-induced time dependence of the ˇelds transforms
the monopole into a dyon.
Note that this collective coordinate is an angular variable, which is deˇned on a circle S 1 .
Indeed, the points Υ = 2πn, n ∈ Z correspond to the same gauge transformation U (r, t),
which is unity on the spatial asymptotic [15]. However, the points Υ = 0 and, for example,
Υ = 2π, correspond to different topological classes.
To sum up, the one-monopole conˇguration in the BPS limit could be characterized by
four zero modes (moduli) that form the so-called moduli space M1 . It is clear from the above
discussion that M1 = R3 × S 1 .
Note that we can come back to the JuliaÄZee description of a dyon conˇguration just by
inverting the above discussion: we could start from a system of time-dependent ˇelds and
apply the gauge transformations (32) and (33) to compensate for that dependence. The price
we would have to pay would be the appearance of a nonzero time component of the gauge
potential A0 . This corresponds to the static ansatz (16).

Now we consider the mechanism of interaction between two widely separated monopoles.
If they are close enough to each other, the cores overlap and we have quite a complicated
picture of short-range interactions mediated by the gauge and scalar ˇelds. However, if we
consider well-separated monopoles, there is some simpliˇcation. We may suppose that the
monopole core has a radius that is much smaller than the distance between the monopoles.
Moreover, outside of this core the covariant derivatives of the scalar ˇeld vanish and thus the
gauge ˇelds obey the free YangÄMills equations. This approximation is a standard assumption
in the analysis of monopole interactions.
The result of both analytical and variational calculations conˇrm a rather surprising conclusion, ˇrst observed by Manton [12]: there is no interaction between two BPS monopoles
at all, but the monopoleÄantimonopole pair attracts each other with double strength.

1250 Shnir Ya.

The reason for this unusual behavior is that the normal magnetostatic repulsion of the two
monopoles is balanced by the long-range scalar interaction: in the BPS limit the quanta of
the scalar ˇeld are also massless.
Indeed, we have already noted that there is a crucial difference between the asymptotic
behavior of the Higgs ˇeld in the non-BPS and the BPS cases: there is a long-range tail of
the BPS monopole φa → vr̂a − ra /er2 as r → ∞. The result is that, in a system of two
widely separated monopoles, the asymptotic value of the Higgs ˇeld in the region outside
the core of the ˇrst monopole is distorted due to the long-range scalar ˇeld of the other
monopole: the mass of the ˇrst monopole will decrease and the size of its core is increased.
In other words, the additional long-range force appears as a result of violation of the original
scale invariance of the model in the BPS limit λ → 0. The scalar charge of a dyon is
just QD = g 2 + q 2 [17] and the corresponding Coulomb scalar potential of interaction is
∼ − g 2 + q 2 /r (recall that the scalar interaction is always attractive). Then the relative
motion of two well-separated BPS dyons is a geodesic motion in the four-dimensional space
of collective coordinates M0 with one compact variable (moduli space). This dynamics is
governed by the TaubÄNUT (NewmanÄUntiÄTamburino) metric
 2
2g /M
2g 2
(dΥ + a · dr)2 .
ds = 1 −
dr +
1 − 2g 2 /M r
A general description of the low-energy dynamics of BPS monopoles on the moduli space is
given by the AtiyahÄHitchin metric, whose asymptotic form is the TaubÄNUT metric [18].
Let us stop our discussion at this point. Recent developments in the understanding of
the low-energy dynamics of the supersymmetric monopoles, which basically used the same
simple picture of geodesic motion on the underlying moduli space, have greatly improved
our understanding of the structure of the vacuum of supersymmetric theories. The restricted
volume of our review does not allow us to go into detail of many remarkable works. Because
of lack of time we also do not discuss here the powerful Nahm formalism, which allows us
to obtain many results in a very simple and elegant way. In this rapidly developing situation,
we direct the reader to the original works and reviews.

1. Manton N. S., Sutcliffe P. M. Topological Solitons. Cambridge Univ. Press, 2004.
2. Shnir Ya. M. Magnetic Monopoles. Berlin; Heidelberg; N. Y.: Springer, 2005.
3. 't Hooft G. Magnetic Monopoles in Uniˇed Gauge Theories // Nucl. Phys. B. 1974. V. 79. P. 276Ä
4. Polyakov A. M. Particle Spectrum in the Quantum Field Theory // Pis'ma JETP. 1974. V. 20.
P. 430Ä433.
5. Corrigan E. et al. Magnetic Monopoles in SU (3) Gauge Theories // Nucl. Phys. B. 1976. V. 106.
P. 475.
6. Arafune J., Freund P. G. O., Goebel C. J. Topology of Higgs Fields // J. Math. Phys. 1975. V. 16.
P. 433.
7. Preskill J. Magnetic Monopoles // Ann. Rev. Nucl. Part. Sci. 1984. V. 34. P. 461.
8. Julia B., Zee A. Poles with Both Magnetic and Electric Charges in Non-Abelian Gauge Theory //
Phys. Rev. D. 1975. V. 11. P. 2227Ä2232.

Magnetic Monopoles 1251

9. Bogomol'nyi E. B. The Stability of Classical Solutions // Sov. J. Nucl. Phys. 1976. V. 24. P. 449Ä454.
10. Prasad M. K., Sommerˇeld C. M. An Exact Classical Solution for the 't Hooft Monopole and the
JuliaÄZee Dyon // Phys. Rev. Lett. 1975. V. 35. P. 760Ä762.
11. Coleman S. et al. Can One Dent a Dyon? // Phys. Rev. D. 1977. V. 15. P. 544.
12. Manton N. S. The Force between 't HooftÄPolyakov Monopoles // Nucl. Phys. B. 1977. V. 126.
P. 525.
13. Rossi P. Propagation Functions in the Field of a Monopole // Nucl. Phys. B. 1979. V. 149. P. 170.
14. Rossi P. Exact Results in the Theory of Non-Abelian Magnetic Monopoles // Phys. Rep. 1982.
V. 86. P. 317Ä362.
15. Gibbons G. W., Manton N. S. Classical and Quantum Dynamics of BPS Monopoles // Nucl. Phys.
B. 1986. V. 274. P. 183.
16. Weinberg E. J. Parameter Counting for Multi-Monopole Solutions // Phys. Rev. D. 1979. V. 20.
P. 936Ä944.
17. Manton N. S. Monopole Interactions at Long Range // Phys. Lett. B. 1985. V. 154. P. 397.
18. Atiyah M. F., Hitchin N. J. The Geometry and Dynamics of Magnetic Monopoles. Princeton Univ.
Press, 1988.