1. Magnetic Monopoles and the
Homotopy Groups

2. Solitons
 Finite-energy, non-dissipative solutions of the
classical wave equations
 Cannot arise in linear theories such as
electrodynamics, because there is dispersion
 Non linear theories allow cancelation of
dispersive effects

3. Solitons in Field Theory
 M is the set of vacuum field configurations:
 Define a topological quantum number πd-
2(G/H), where G maps one vacuum to another
and H is the isotopy group (G/H is the
manifold of the vacuum)

4. Solitons in 1+1 Dimensions
Euler—Lagrange Equations give us the
equation of motion
This has static solutions at zero and the
potential minimum
)()(
2
1 2
φφµ V−∂=L
422
2
1
)( λφφφ +−= mV
32
2
2
2
2
λφφ
φφ
−=
∂
∂
−
∂
∂
m
xt
λ
µ
φ ±= ,0

5.  Equation of motion can also be satisfied by
the soliton kink solution:
Kinky Soliton Solutions
 4  2 2 4
 1 .0
 0 .5
0 .5
1 .0

6. Topological Charge
 The scalar fields in 2D have a conserved
current, leading to a topological charge:
 For the kink solution:
Charge Topological Quantum
Number
λ
µ2
=Q 2
2
0 Z
e
Z
=





π

7.  To construct a magnetic monopole, you can
have A of the form
Magnetic Monopole with charge g
Dirac Monopoles
φ
θ
θ
π
ˆ
)sin(
))cos(1(
4
−
=
r
g
AN

φ
θ
θ
π
ˆ
)sin(
))cos(1(
4
+
=
r
g
AS

r
r
g
AB ˆ
4 2
π
=×∇=


8. Dirac Quantization Condition
 Singularities at θ=0 and θ=π correspond to
Dirac string
 Moving in a circle around string, particle wave
function picks up a phase (e-ieg
)
 This phase factor must be equal to 1 for
string to be undetectable
e-ieg
= 1 eg = 2πn

9.  From previous calculation we see that charge
is quantized
 Monopoles require a compact U(1) gauge
group
 Can be embedded in a larger compact gauge
group i.e. SU(2)
 Also can arise in Kaluza-Klein theories
Dirac Quantization Condition

10. Soliton in 2+1 Dimensions (Nielsen-Olsen
Model)
 Here Dμ=∂μ – ieAμ and Φ is a complex valued
field
 Vacuum is identified with Fμν=0, Dμϕ=0,
 The vacuum fields have the values
λ
µϕ =
)(θχ
λ
µ i
e

11. Boundary Conditions
 As r→∞, we need to have vacuum states.
Working in gauge where A0
=0 and
considering the requirements on vacuum
solution:
 Defining χ(θ)=nθ where refers to a
homotopy class we find that magnetic field
through (x,y) plane is quantized.
Ζ∈n

12.  Φa
are scalar fields in the adjoint representation of
SU(2) (a=1,2,3).
 The vacuum state manifold is a 2 sphere with
radius

‘t Hooft-Polyakov Model (3+1
dimensions)
ZSOSOSO == ))2(())2(/)3(( 12 ππ

13. Boundary Conditions
 r→∞, need vacuum state, Dμϕa
=0
 Shown by ‘t Hooft that Wμ
a
and Fμν are related
by (Φ={ϕa
}):
 Working out electric and magnetic fields
using the r→∞ limit value of Wμ
a
ijkijki
ii
r
er
FH
FE
3
0
1
2
1
0
=−=
=−=
ε

14. Quantization of Magnetic Charge
 This is same as Dirac quantization condition
(eg = 2πn) with n = 2!
 Considering all possible solutions of the
equations of motion for this theory, it can be
shown we get the Dirac quantization
condition with n even
 Agrees with result for possible homotopy
classes

15. Mass of Monopoles
 t’ Hooft showed , where MV is gauge
boson mass
 Monopoles arise in GUTs, with mass of the scale of
the symmetry breaking of the theory
 SU(5):
 GUT monopoles are too massive to be produced in
accelerators, but could have been produced in early
universe
 Monopole searches focus on accelerators, cosmic
rays, and on monopoles possibly bound in matter
 To date, nothing found
2
4
e
M
M V
mon
π
≈

16. References
 Boya L. et al. “Homotopy and Solitons.” Fortschritte der
Physik 26, 175-214 (1978).
 Coleman, S. “Magnetic Monopole Fifty Years Later.” from The
Unity of the Fundamental Interactions, A. Zichichi, ed.
European Physical Society, 1981.
 Dine, Michael. Supersymmetry and String Theory. Cambridge
University Press, 2007.
 Goddard P. and D.I. Olive, “Magnetic monopoles in gauge
field theories.” Rep. Prog. Phys., 41, 1357-1437 (1978).
 Harvey, Jeffrey A. “Magnetic Monopoles, Duality, and
Supersymmetry.” arXiv: hep-th/9603086
 Milstead, D. and E.J. Weinberg. “Magnetic Monopoles.” from
K. Nakamura et al. (Particle Data Group), JPG 37, 075021
(2010).