PhD Thesis - Dr. Luchini

Universidade de S˜
ao Paulo
Instituto de F´ısica de S˜
ao Carlos

Gabriel Luchini Martins

Hidden symmetries in gauge theories &
quasi-integrablility

S˜ao Carlos
2013

Gabriel Luchini Martins

Hidden symmetries in gauge theories &
quasi-integrablility

Tese apresentada ao Programa de P´osGradua¸c˜ao em F´ısica do Instituto de F´ısica de
S˜ao Carlos da Universidade de S˜ao Paulo, para
obten¸c˜ao do t´ıtulo de doutor em Ciˆencias.
´
Area
de Concentra¸c˜ao: F´ısica B´asica
Orientador: Prof. Dr. Luiz Agostinho Ferreira

Vers˜ao Corrigida
(vers˜ao original dispon´ıvel na Unidade que aloja o Programa)

S˜ao Carlos
2013

AUTORIZO A REPRODUÇÃO E DIVULGAÇÃO TOTAL OU PARCIAL DESTE
TRABALHO, POR QUALQUER MEIO CONVENCIONAL OU ELETRÔNICO PARA
FINS DE ESTUDO E PESQUISA, DESDE QUE CITADA A FONTE.

Ficha catalográfica elaborada pelo Serviço de Biblioteca e Informação do IFSC,
com os dados fornecidos pelo(a) autor(a)
Luchini, Gabriel
Hidden symmetries in gauge theories & quasiintegrabiity / Gabriel Luchini; orientador Luiz
Agostinho Ferreira - versão corrigida -- São Carlos,
2013.
113 p.
Tese (Doutorado - Programa de Pós-Graduação em
Física Básica) -- Instituto de Física de São Carlos,
Universidade de São Paulo, 2013.
1. Solitons. 2. Formulação de curvatura nula. 3.
Simetrias escondidas. 4. Espaço dos Laços. 5. Cargas
Conservadas. I. Agostinho Ferreira, Luiz, orient.
II. Título.

ACKNOWLEDGEMENTS

I would like to express my deep gratitude to Clisthenis P. Constantinidis for his companionship since the beginning of my studies in physics; from the first year of my graduation as
a professor, as my supervisor during my masters and as a very good friend always. I finish my
PhD studies in S˜ao Carlos thanks to his many good advises, including the one stating I should
come to work with Luiz Agostinho.
I wish to thank my friends in Vit´oria, Z´e, Ulysses dS, Ivanzito (and all respective ladies) for
standing by me in every new step I make. I also want to thank Massayuki for having listened
to me during these 4 years. I could say more, but I think I’ve said enough. I would also like
to thank Ritinha for her friendship, and Yvoninha and Mariana for all the help they gave me
since I came to S˜ao Carlos. I wish to express my gratitude for the staff in the Institute and in
particular to Silvio, who can still be very patient with my requests!
This work is a consequence of hundreds knocks on Luiz’s door, who luckily moved to
another room a little bit more distant from my office during my second year as a student. I
learned with him many valuable lessons, but two of them are very special: first, that an example
is much better than a thousand theorems, and the other one is that research is something that
must be done for yourself, with honesty and not as a proof of your abilities for the others. I
am deeply thankful for the faith that he seems to have in me.
Part of the content of this thesis (half of it) is due to a collaboration with Wojciech
Zakrzewski from the Department of Mathematical Science at Durham University. My gratitude
for him is very big. The opportunity he gave me to not just work with him but also to go
to Durham and participate in that magnificent non-perturbative environment was of a major
relevance for my growth. I extend my gratitude for all the people I met there, and somehow
contributed to all that. In particular, Laura da Costa and Karen Blundell in Grey College.
Also, I must mention how lucky I was in meeting David Tapp, who helped me with everything
I needed and was (and is) a true friend, that I really hope to see again.
During almost my entire PhD studies time I was the only student in the group which made

my life even harder. In this last year, however, Vinicius Aurichio joined us and this was great
for me. I am very glad for his companionship as my office mate. Also the presence of David
Foster as a pos-doc gave a much more enthusiastic feeling to the place and definitely our
daily discussions about math, physics and women gave me much more motivations to work. I
learned a lot with him, and for that I am very grateful.
It is not even necessary to say that I could only get to this point thanks to Arlete, Marina,
Nat´alia and Mercedes. Although very far away their love for me made them always very close.
Harder than handle a PhD, is to do it and take care of Lays... but what does not kill us
makes us stronger, and I am deeply grateful to have met her and for her being sharing all this
with me. I found in her the hidden symmetry that makes my happiness conserved. I wish also
to thank her family that gave me a safe place to be every time a needed.
A very special thanks to Thiago Mosqueiro, who developed this amazing Latex template
that makes my thesis looks more important than it is.

’...when you have eliminated all which is impossible, then whatever remains, however improbable, must be the truth.’
Sherlock Holmes Quote - The Blanched Soldier

RESUMO

LUCHINI, G. Simetrias escondidas em teorias de calibre & quasi-integrabilidade. 2013. 113 p.
Tese (Doutorado em F´ısica B´asica) – Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao
Paulo, S˜ao Carlos, 2013.

Essa tese discute algumas extens˜oes de id´eias e t´ecnicas usadas em teorias de campos integr´aveis para tratar teorias que n˜ao s˜ao integr´aveis. Sua apresenta¸c˜ao ´e feita em duas partes.
A primeira tem como tema teorias de calibre em 3 e 4 dimens˜oes; propomos o que chamamos
de equa¸c˜ao integral para uma tal teoria, o que nos permite de maneira natural a constru¸c˜ao
de suas cargas invariantes de calibre, e independentes da parametriza¸c˜ao do espa¸co-tempo. A
defini¸c˜ao de cargas conservadas in variantes de calibre em teorias n˜ao-Abelianas ainda ´e um
assunto em aberto e acreditamos que a nossa solu¸c˜ao pode ser um primeiro passo em seu
entendimento. A formula¸c˜ao integral mostra uma conex˜ao profunda entre diferentes teorias
de calibre: elas compartilham da mesma estrutura b´asica quando formuladas no espa¸co dos
la¸cos. Mais ainda, em nossa constru¸c˜ao os argumentos que levam `a conserva¸c˜ao das cargas
s˜ao dinˆamicos e independentes de qualquer solu¸c˜ao particular. Na segunda parte discutimos
o recentemente introduzido conceito de quasi-integrabilidade: em (1 + 1) dimens˜oes existem
modelos n˜ao integr´aveis que admitem solu¸c˜oes solitonicas com propriedades similares `aquelas
de teorias integr´aveis. Estudamos o caso de um modelo que consiste de uma deforma¸c˜ao
(n˜ao-integr´avel) da equa¸c˜ao de Schr¨odinger n˜ao-linear (NLS), proveniente de um potencial
mais geral, obtido a partir do caso integr´avel. O que se busca ´e desenvolver uma abordagem
matem´atica sistem´atica para tratar teorias mais realistas (e portanto n˜ao integr´aveis), algo
bastante relevante do ponto de vista de aplica¸c˜oes; o modelo NLS aparece em diversas ´areas da
f´ısica, especialmente no contexto de fibra ´otica e condensa¸c˜ao de Bose-Einstein. O problema
foi tratado de maneira anal´ıtica e num´erica, e os resultados se mostram interessantes. De fato,
sendo a teoria n˜ao integr´avel n˜ao ´e encontrado um conjunto com infinitas cargas conservadas,
mas, pode-se encontrar um conjunto com infinitas cargas assintoticamente conservadas, i.e.,
quando dois solitons colidem as cargas que eles tinham antes tem os seus valores alterados,

mas ap´os a colis˜ao, os valores inicias, de antes do espalhamento, s˜ao recobrados.

Palavras-chave: Solitons. Formula¸c˜ao de curvature nula. Simetrias escondidas. Espa¸co
de la¸cos. Cargas conservadas.

ABSTRACT

LUCHINI, G. Hidden symmetries in gauge theories & quasi-integrablility. 2013. 113 p. Tese
(Doutorado em F´ısica B´asica) – Instituto de F´ısica de S˜ao Carlos, Universidade de S˜ao Paulo,
S˜ao Carlos, 2013.

This thesis is about some extensions of the ideas and techniques used in integrable field
theories to deal with non-integrable theories. It is presented in two parts. The first part
deals with gauge theories in 3 and 4 dimensional space-time; we propose what we call the
integral formulation of them, which at the end give us a natural way of defining the conserved
charges that are gauge invariant and do not depend on the parametrisation of space-time.
The definition of gauge invariant conserved charges in non-Abelian gauge theories is an open
issue in physics and we think our solution might be a first step into its full understanding.
The integral formulation shows a deeper connection between different gauge theories: they
share the same basic structure when written in the loop space. Moreover, in our construction
the arguments leading to the conservation of the charges are dynamical and independent of
the particular solution. In the second part we discuss the recently introduced concept called
quasi-integrability: one observes soliton-like configurations evolving through non-integrable
equations having properties similar to those expected for integrable theories. We study the
case of a model which is a deformation of the non-linear Schr¨odinger equation consisting of a
more general potential, connected in a way with the integrable one. The idea is to develop a
mathematical approach to treat more realistic theories, which is in particular very important
from the point of view of applications; the NLS model appears in many branches of physics,
specially in optical fibres and Bose-Einstein condensation. The problem was treated analytically
and numerically, and the results are interesting. Indeed, due to the fact that the model is not
integrable one does not find an infinite number of conserved charges but, instead, a set of
infinitely many charges that are asymptotically conserved, i.e., when two solitons undergo a
scattering process the charges they carry before the collision change, but after the collision
their values are recovered.

Keywords:

Solitons. Zero curvature formulation. Hidden symmetries. Loop space.

Conserved charges.

LIST OF FIGURES

1.1 A 1-soliton solution propagates through the string of pendula. The energy is
not dissipated, so, after the pendulum flips 180 degrees it starts to decelerate,
and stops at the bottom, without wiggling. . . . . . . . . . . . . . . . . .
2.1 The

1
n!

p. 16

factor appears due to the symmetry relating the n! integrations in

the path-ordered product. . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 31

2.2 One can use a family of homotopically equivalent loops to scan a 2-dimensional
surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 34

2.3 The zero curvature implies that the Wilson line is independent of the path.
This leads to a conservation law. . . . . . . . . . . . . . . . . . . . . . .

p. 35

2.4 On the left, a surface Σ in M is scanned with loops based at xR . On the
right, this surface is represented in LM, where each loop in M corresponds
to a point and the surface from xR to the boundary ∂Σ is a path. A variation
of this surface, leaving the boundary fixed, is also represented in LM. . . .

p. 37

2.5 The border of the surface is kept fix while performing the variation. When
the surfaces are closed, the border is contracted to xR and the initial surface
(ζ = 0) becomes the closed infinitesimal surface ΣR while the final surface
(ζ = 2π) becomes the boundary of a volume. . . . . . . . . . . . . . . . .

p. 38

3.1 The surface independence of V means that it can be calculated from the
infinitesimal loop around xR (the initial point in loop space) to the boundary
1
loop S∞
(the final point in loop space) using any of the two surfaces (paths

in loop space) presented here. . . . . . . . . . . . . . . . . . . . . . . . .

p. 44

4.1 When the volume Ω becomes the infinitesimal cube the integral equations
imply the differential Yang-Mills equations. The big arrows on the bottom
and top surfaces indicate the sign of

dxµ
.
dτ

. . . . . . . . . . . . . . . . . .

p. 49

5.1 Plot of | ψ |2 against x for the one-soliton solution of the unperturbed NLS
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 87

5.2 Trajectories of two Solitons at v = 0.4 (ǫ = 0) . . . . . . . . . . . . . . .

p. 88

5.3 Trajectories of two solitons at rest (ǫ = 0) . . . . . . . . . . . . . . . . .

p. 88

5.4 Heights of the solitons originaly at rest (ǫ = 0) . . . . . . . . . . . . . . .

p. 89

5.5 Trajectories of two solitons at v = 0.4 (ǫ = 0.06) . . . . . . . . . . . . . .

p. 90

5.6 Trajectories (and the energy) of two solitons at rest (ǫ = 0.06) . . . . . .

p. 91

5.7 Trajectories (and the energy) of two solitons at rest (ǫ = −0.06) . . . . .

p. 92

5.8 Heights of the two solitons observed in their scattering at rest (ǫ = −0.06
c = 0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 92

5.9 Time integrated anomaly of two solitons sent at v = 0.4 (ǫ = 0.06) . . . .

p. 93

5.10 Time integrated anomaly of two solitons at rest (ǫ = 0.06) . . . . . . . . .

p. 94

5.11 Time integrated anomaly of two solitons sent at v = 0.4 (ǫ = −0.06) . . .

p. 94

5.12 Time integrated anomaly of two solitons at rest (ǫ = −0.06) . . . . . . . .

p. 94

A.1 The regularisation of the Wilson line operator is done by replacing the path
that passes through to the origin by a path going around it. . . . . . . . .

p. 101

SUMMARY

1 Introduction

p. 15

2 Hidden Symmetries, Stokes theorem and conservation laws

p. 29

2.1 The standard non-Abelian Stokes theorem . . . . . . . . . . . . . . . . .

p. 30

2.2 Generalisation of the Stokes theorem . . . . . . . . . . . . . . . . . . . .

p. 36

3 Integral formulation of theories in 2 + 1 dimensions

p. 41

3.1 The integral equations of Chern-Simons theory . . . . . . . . . . . . . . .

p. 41

3.2 The integral equations of (2 + 1)-dimensional Yang-Mills theory . . . . . .

p. 45

4 The integral Yang-Mills equation in 3 + 1 dimensions

p. 47

4.1 The full Yang-Mills integral equation . . . . . . . . . . . . . . . . . . . .

p. 47

4.2 The self-dual sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 51

4.3 Monopoles and dyons . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 53

4.4 Instantons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 60

4.5 Merons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 63

5 Quasi-integrable deformation of the non-linear Schr¨
odinger equation
5.1 Definition of the model . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1

p. 67
p. 67

On the parity symmetry . . . . . . . . . . . . . . . . . . . . . . .

p. 76

5.2 Dynamics versus parity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 78

5.2.1

Deformations of the NLS theory . . . . . . . . . . . . . . . . . . .

p. 79

5.3 The parity properties of NLS solitons . . . . . . . . . . . . . . . . . . . .

p. 83

5.3.1

The one-soliton solutions . . . . . . . . . . . . . . . . . . . . . .

p. 83

5.3.2

The two-soliton solutions . . . . . . . . . . . . . . . . . . . . . .

p. 84

5.4 Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 87

5.4.1

The NLS model . . . . . . . . . . . . . . . . . . . . . . . . . . .

p. 87

5.4.2

The modified model with ε 6= 0 . . . . . . . . . . . . . . . . . . .

p. 90

6 Final Comments

p. 95

REFERENCES

p. 97

Appendix A -- Regularisation of Wilson lines

p. 101

Appendix B -- Explicity quantities involved in equation (5.1.25)

p. 107

Appendix C -- The Hirota solutions

p. 111

15

CHAPTER 1

Introduction

All mentors have a way of seeing more
of our faults than we would like.
Padm´e Amidala

Great achievements in physics were done from attempts to put together apparently conflicting theories. For instance, the incompatibility between Maxwell’s electromagnetism and
the Galilean covariance led to the development of special relativity; the loss of energy of the
orbiting electron predicted by classical electrodynamics and the stability of the atom (an empirical fact), among others, led to the construction of the laws of quantum mechanics; the
difficulty in introducing Newton’s gravity into the principles of special relativity gave birth to
the general theory of relativity.
Our main approach to understand Nature within its different aspects, as in the solution of
the problems mentioned above, is through symmetry principles. Three of the four interactions,
namely the weak, strong and electromagnetic, are based on the so called gauge principle while
the gravitational interaction relies on the equivalence principle.
The symmetries are not just fundamental as a basic ingredient in the construction of the
theories, letting the physical degrees of freedom be identified, but are also important in the
development of systematic methods leading to solutions and/or observables.
In particular, symmetries referred to as “hidden” play a major role in the understanding
of non-linear field theories in 2-dimensional space-time and in the development of exact/nonperturbative methods to treat them. Such theories are certainly important in many branches of
condensed matter and, in higher energy physics, are used as toy-models for realistic scenarios.
The existence of soliton-like solutions in 3 and 4 dimensions (and the lack of them) leads to
the quest of improvement and/or development of powerful tools such as the zero curvature
representation used for integrable theories in 2 dimensions(1, 2).

1 Introduction

16

Figure 1.1 –

A 1-soliton solution propagates through the string of pendula. The energy is not
dissipated, so, after the pendulum flips 180 degrees it starts to decelerate, and stops at
the bottom, without wiggling.

To be concrete let us consider the example of the well-known sine-Gordon equation describing the dynamics of the real scalar field ϕ(t, x):
∂t2 ϕ − ∂x2 ϕ +

m2
sin (βϕ) = 0.
β

(1.0.1)

This equation appears in diverse phenomena in physics, and in particular as the continuous
version of the mechanical model presented in figure (1.1): a set of pendula attached to a
rubber band. In that case ϕ stands for the angle of the straight rod holding the mass blob to
the rubber band, and m and β are some combinations of the value of gravity, the length of the
rod, the separation between two pendula and the torsion of the rubber band. For a small angle
we can consider sin (βϕ) ≈ βϕ, and (1.0.1) becomes the linear Klein-Gordon equation whose

solutions are ordinary (linear) propagating waves. This is the perturbative sector of the theory.
Perturbative methods are very well developed in physics and roughly speaking everything in
quantum electrodynamics is done using them; the standard model, one of the cornerstones of
modern science, is a consequence of the success of that approach. On the other hand this
string of pendula presents very interesting configurations in the non-perturbative sector. In
figure (1.1) the 1-soliton solution ϕ(t, x) =

4
β

arctan e

√m

1−v 2

(x−vt)

is sketched. In opposition

to the linear waves a soliton do not admit the superposition principle. It propagates with
constant velocity without changing its shape or dissipating energy, and when two of them
undergo a scattering process the only effect they feel is a shift from the position they would
have if they were propagating freely. These features lead to the interpretation of solitons as
particles. Besides, generally the coupling of solitons is inversely proportional to the coupling
constant of fundamental particles, so that they tend to be free in the strong coupling regime,
and this is certainly interesting when (as often happens) there is a duality between solitons and
particles(3) involving the weak and strong coupling sectors. Their stability and the behaviour

17

1 Introduction

just described arise∗ from the existence of infinitely many conserved quantities (often called
charges) that can eventually be obtained when one recast the dynamical equations of the
theory as a zero curvature equation, Gtx ≡ ∂t Cx − ∂x Ct + [Ct , Cx ] = 0, i.e., the vanishing of

the curvature of the Lie algebra valued 1-form connection C = Ct dt + Cx dx, a functional of

the fields and its derivatives, implies the equations of motion of the theory and vice versa.

and

In the case of sine-Gordon theory if we take the components of the connection as



iβ ∂ϕ
1 −i β ϕ
iβϕ
2 +
2
e
e
m ∂x
λ

m 


Ct =

4 


ϕ
iβ
∂ϕ
iβϕ
−i
β
e 2 +λe 2
− m ∂x
Cx =



m 

4 

ϕ

iβ ∂ϕ
m ∂t

ϕ

ϕ

−ei β 2 + λ e−i β 2

then the curvature becomes

im
Gtx =
4



∂t2 ϕ

−

∂x2 ϕ

ϕ

ei β 2 − λ1 e−i β 2



∂ϕ
− iβ
m ∂t





m2
+
sin (βϕ) 

β

and we clearly see that Gtx = 0 ⇔ ∂t2 ϕ − ∂x2 ϕ +

m2
β




1

0

0 −1


,





,


sin (βϕ) = 0.

There is no recipe to get a flat connection like this for a given theory. Its existence
is related to the integrability of the theory(5). In fact, the curvature Gtx comes from the
compatibility condition† of the associated linear problem described by the set of two equations
(∂µ + Cµ ) Ψ = 0, with µ = 0, 1 corresponding to the t and x components. The quantity
Ψ is an element of the group G. This equation can be solved if the connection is flat:
Cµ = −∂µ Ψ Ψ−1 . A gauge transformation Cµ → Cµ′ ≡ hCµ h−1 − ∂µ h h−1 with h in the

gauge group G implies that Gµν → hGµν h−1 , and therefore does not affect the zero curvature

representation; the curvature remains zero. However, the connection changes and due to the
non-homogeneity of this transformation one can produce non-trivial solutions for Ψ from very
simple ones which is very powerful. For instance, in the string of pendula we can then start
from the vacuum configuration, where every pendula are at rest at the bottom, and with such
a gauge transformation get a highly non-trivial configuration, which of course is a solution of
the equation of motion since the curvature of this gauged connection is also zero.
∗

In the case of soliton-like solutions in d + 1-dimensions, with d > 1, the stability of the so called topological
solitons (4) is related to their topological charges.
†
The system is said compatible if [∂t + Ct , ∂x + Cx ] = 0.

1 Introduction

18

Notice that the parameter λ was introduced in the definition of the connection in a way
that it does not appear in the equation of motion. It is called the spectral parameter and is
crucial in the obtention of an infinite number of conserved charges, which, as said before, is
responsible for the stability of the soliton. It must be noticed that these charges are not, a priori,
related to the Noether’s charges, i.e., with the symmetries of the equation of motion. Indeed,
equation (1.0.1) has just the 2-dimensional Poincar`e invariance and a discrete symmetry under
ϕ→

2πn
β

+ ϕ. This is certainly far from being a set of infinitely many symmetries generating

that infinite number of conserved charges. Instead, such symmetries giving the stability of the
solitons are related to the fact that the charge operator undergoes an iso-spectral evolution in
time, and therefore, its eigenvalues (the charges) are conserved; that time evolution is then a
symmetry, but this is only revealed when the zero curvature representation of the equations
of motion of the theory is found, and that is why one refers to it as a hidden symmetry. This
charge operator can be naturally obtained with the use of the non-Abelian Stokes theorem:
P1 e−

H

∂Σ

R

Cµ dxµ

= P2 e

Σ

W −1 Gµν W dxµ dxν

.

The l.h.s of the equation above is the path-ordered integral of the connection C along a curve,
which is the boundary of the 2-dimensional surface Σ. That P1 refers to this ordering. Let us
be more precise. This quantity is obtained from the following equation
dxµ
dW
+ Cµ
W =0
dσ
dσ

(1.0.2)

which defines the Wilson line W (in a finite representation of C, a matrix) along a curve
parametrised by σ. The solution of it is given by an infinite series
W (σ) = 1l −

Z

0

σ

dxµ
Cµ (σ ) ′ +
dσ
′

Z

σ

dxµ
Cµ (σ ) ′
dσ
′

0

Z

σ′

Cν (σ ′′ )
0

dxν
W (σ ′′ )dσ ′′ dσ ′
dσ ′′

up to a multiplication by a constant element from the right, and σ ≥ σ ′ ≥ σ ′′ , etc. This
series can be formally written as an exponential. Due to the non-Abelian character of C, the

order it appears in the products matters, and to guarantee that this is respected we introduce
P1 . The r.h.s of the non-Abelian Stokes theorem is the ordered integral of the curvature of
C, G = dC + C ∧ C, conjugated with W , on the surface Σ. As we discuss in chapter 2 this

surface is scanned with loops based at xR , a point on its border we call reference point. Every
point on Σ belongs to a unique loop, and every loop can be obtained by smooth variations
from the point-loop around xR , until we reach the border, which is the final loop. So, the
Wilson line appearing inside the integral on the r.h.s of the theorem above is calculated along
each such loop from the reference point. Then, we can get W on the curve ∂Σ by considering
that this curve is the result of variations from the point-loop. This point of view was presented

19

1 Introduction

first in (6) and we reproduce it in chapter 2. One finds that it is possible to calculate W using
the equation

Z
dW
∂xµ ∂xν
−1
− W dσW Gµν W
=0
(1.0.3)
dτ
∂σ ∂τ
where τ parametrises the variation from one loop to another, and the integration appearing
here is performed along the entire loop. The solution of it gives exactly the r.h.s of the
non-Abelian Stokes theorem, where P2 stands for the ordering with respect to τ , that we call
surface-ordering.
Once a zero curvature representation for the equations of motion is found, this theorem
implies that the Wilson line along any of the loops Γc scanning the surface Σ is the same, i.e.,
WΓc = P1 e−

H

C

= 1l.

This leads to a very important property of the Wilson line: it is path independent. This
is not difficult to see. Consider that the loop Γc is made of the composition of two paths:
Γc = Γ2 ◦ Γ1 . Then, we use the fact that the Wilson line follows such a decomposition,

becoming WΓc = WΓ2 · WΓ1 . Next, we take the reverse order of the path Γ2 , and using the

= WΓ−1
and that WΓc = 1l, we get WΓ1 = WΓ2 ; the Wilson line calculated
fact that WΓ−1
2
2
from xR to the point Γ1 ∩ Γ2 is the same, independently of the path.
Now, the path independence is what leads to the conserved charges. We split space-time

into space and time, and take each of these paths Γ1 and Γ2 to be composed by two other
paths. The path Γ1 is made of a path that goes from the reference point to the spatial
boundary, at constant time, say t = 0. This path is called Γ0 to stress the fact that it takes
the whole space at time zero. Then the second path forming Γ1 is the “time evolution” of
the spatial boundary, that goes from t = 0 to some given t > 0. This path is called Γ∞ to
emphasize that it is just the point on the border, that we will take to be at infinity, that “goes
up in time” until we reach the point Γ1 ∩ Γ2 , at spatial infinity and time t > 0. Now the

path Γ−1
2 starts with the “time evolution” of the reference point, from time zero to that time
t > 0. This path is the Γ−∞ . Then we compose it with the path that goes from the reference
point at time t > 0 to the border of space at this time, i.e., this path, called Γt , is the one
that sweeps the whole space, like Γ0 , but at time t > 0. At the end we are going from xR
to the spatial border at time t > 0 using two different paths. We use again the fact that the
Wilson line follows the decomposition of the path to get WΓ−1
· WΓ−1
· WΓ∞ · WΓ0 = 1l. With
t
−∞

appropriate boundary conditions we can make the Wilson lines corresponding to the “time
evolution” of the borders coincide (and we rename them in a very suggestive way as U(t))

1 Introduction

20

and get the aforementioned iso-spectral evolution
WΓt = U(t) · WΓ0 · U −1 (t).
Then it becomes clear that the eigenvalues of W calculated over the whole space at a certain
time slice is the same for any time. This is a conservation law, and it appeared here because
we had a flat connection C, so that dC + C ∧ C = 0, which through the Stokes theorem gave

the path independence of the Wilson line.

In 1998 Luiz Agostinho Ferreira, Joaquin Sanchez-Guill´en and Orlando Alvarez proposed
that maybe a first step into the construction of the concept of integrability for theories in a
higher (or any) dimensional space-time could be encoded in a generalisation of this charge
operator. If the space-time M is now d + 1-dimensional, a generalisation of W would involve a connection which is a differential form of higher degree; a d-form. Remarkably, the
demonstration they give for the non-Abelian Stokes theorem (that first appeared in (6), then
in (7) and can also be found in details in (8)) gives a systematic method to generalise it for
connections of higher degree and also (or consequently) to define the objects that generalise
W . In fact, what they noticed is that the natural environment to extend the zero curvature
representation and therefore the path-independence of the Wilson line, is the so called loop
space. For a d + 1-dimensional space-time M the loop space LM consists of the set of maps
from the (d−1)-sphere S d−1 to M, keeping the image of one point, for instance the north-pole
of S d−1 , fixed as the reference point xR in M. For d = 2, for instance, the image of these
maps are loops based on xR . A loop in M corresponds to a point in LM. So, if we scan a
2-dimensional surface Σ with loops, the initial loop will be a point in loop space and the final
loop, the boundary of the surface, another point. The bulk of the surface consists basically
of a set of loops that are continuously deformed from the initial one, the point-loop around
xR ; this corresponds to a path in loop space. Notice now that for d = 1 the loop space is the
set of mappings from the sphere S 0 to the 2-dimensional space-time. This sphere is in fact
made of two points. One has its image fixed in xR by construction, so the map is made from
a point, the other one in S 0 to another point in space-time. So, the loop space coincides with
space-time. That is a great motivation to understand how the loop space is the natural place
to generalise the Stokes theorem and consequently the zero-curvature representation.
The above construction in loop space for the d = 2 case, where the surface is scanned with
loops, is exactly the way (1.0.3) was introduced. Indeed, this equation can be used to generalise
R
µ ∂xν
is a connection
the Wilson line, once, as presented in (7), the quantity dσW −1 Gµν W ∂x
∂σ ∂τ

in loop space. Then, the generalisation of this term is done by replacing G by a general 2-form
Rσ
µ
B, so it readsA = 0 dσ ′ W −1 (σ ′ )Bµν (σ ′ )W (σ ′ ) ∂x
δxν . Thus, following what we said before
∂σ′

21

1 Introduction

the idea is to search for this connection in loop space such that F = δA + A ∧ A = 0,

i.e., its curvature in loop space vanishes. This would lead to a generalisation of the above
mentioned path-independence, which is the property one seeks for the construction of the
conserved charges. Considerable progress was done in this direction; the vanishing of such a
curvature can be seen as a guide for integrability in higher dimensions. Some very interesting
well-known models were studied under this perspective, and original results appear thanks
to this zero curvature formulation in loop space of these models (besides the applications
presented in (7) see for instance (9)).
Regardless the success of that construction in this thesis we follow a different approach.
Instead of looking for a flat connection in loop space we propose that the differential equations
of motion of the theory have an integral version, which is based on the standard and/or
generalised Stokes theorem, whose general form in a space-time M of dimension d + 1 is
R

Pd−1 e

∂Ω

A

R

= Pd e

Ω

F

,

a relation between the quantities A and F , constructed from (d − 1) and d-forms respectively.

On the l.h.s we have the ordered integration of A on the boundary of the hyper-volume Ω, a

sub-manifold of M, and on the r.h.s the ordered integration of F in the bulk of Ω.

The idea follows more or less what we have for electromagnetism, i.e., the differential
equations known as the Maxwell’s equations can be integrated and with the use of Stokes
theorem one gets the laws for the fluxes of electric and magnetic fields. Indeed, here we
show how such an integral formulation is done for gauge theories. The integral formulation of
electromagnetism in terms of fluxes (done by Faraday with his invention of the lines of fields)
precedes Maxwell’s differential equations and were and are very important for the understanding
of the phenomena. However, non-Abelian gauge theories did not follow the same historical
route, and as far as we know, the integral version of Yang-Mills theories were never presented
before, so apparently this is the first time it is done.
We promote the Stokes theorem from a mathematical identity to a physical equation,
which we call the integral equation. The physical fields constitute A and F above, more or
less like Cµ is written in terms of the physical fields in the zero curvature formulation. Then
we write the integral equation in a way that when Ω is considered infinitesimal the differential
equations are recovered.
The key point in this formulation is that the integral equation gives the possibility of finding
the path independence property, which is the important thing in the obtention of the conserved
charges, without going through the need of a zero curvature. Lets see how it happens for the

1 Introduction

22

case of theories in (1 + 1)-dimensional space-time, i.e., let us reformulate the ideas presented
before from another perspective. Take Ω a 1-dimensional sub-manifold of M, which is a path
going from xR to xf . Take a field g(x), and element of the group G. We then construct the
quantity g(xf ) · g −1 (xR ), i.e., a quantity made of the field g(x) on the border of Ω. Then,
take also the field Cµ (x) and define the quantity P1 e−

R

Ω

C

, on the bulk of Ω. Finally we claim

a relation between g(x) and Cµ (x) given by the the integral equation
g(xf ) · g −1(xR ) = P1 e−

R

Ω

C

.

Now, if the border of Ω is fixed then the l.h.s of the above equation is fixed, for any path
linking the points xR and xf , so, as a consequence of the integral equation the quantity on the
r.h.s P1 e−

R

Ω

C

, which is simply the Wilson line, obtained from (1.0.2), is path independent,

which is the property we want. We achieved the path independence without having to talk
about a zero curvature.
If Ω is an infinitesimal path then g(x) ≈ g(xR ) + ∂µ g(xR )δxµ and the l.h.s becomes

1l + ∂µ g(xR ) · g −1 (xR )δxµ , while the r.h.s 1l − Cµ (xR )δxµ , which gives the differential equation

Cµ = −∂µ gg −1. This is exactly saying that the connection is flat, which is the condition for

the solution of the associated linear problem (∂µ + Cµ ) Ψ = 0. Of course, this leads to a zero

curvature, but from this point of view it is just a consequence, and not something we need
from the beginning. That might seems just another point of view in this case, with no further
implications, but it is crucial to the construction in higher dimensions.
Remarkably our integral formulation of gauge theories enable us to define naturally what
the conserved charges are. This discussion in gauge theories is not closed: there are many
attempts (see for instance (10–14)) to define charges that are conserved and also invariant
under gauge transformations, a fundamental property for any physical quantity.
In the standard literature‡ the charges associated to the Yang-Mills theory are constructed
as follows. From the Yang-Mills equations Dν F νµ = J µ , Dν Feνµ = 0, where Jµ stands for the

matter current, Feµν = 12 ǫµνρλ F ρλ is the Hodge dual of the field strength, and the covariant
derivative is Dµ ⋆ = ∂µ + ie [Aµ , ⋆], one defines the quantity j µ ≡ ∂ν F νµ = J µ − ie [Aν , F νµ ]
i
h
νµ
νµ
µ
e
e
e
and its Hodge dual j ≡ ∂ν F = −ie Aν , F , which are locally conserved due to the
antisymmetry of the field strength tensor. With appropriate boundary conditions the charges
coming from j µ and jeµ are written as
Z
Z
eYM =
QYM =
dS · E
Q
dS · B
2
S∞

‡

2
S∞

Basically this can be found in any good book about the subject. One example is (15).

23

1 Introduction

where Ei = F0i and Bi = − 21 ǫijk Fjk are the non-Abelian electric and magnetic fields. Under

a gauge transformation h ∈ G these charges become
Z
Z
−1
eYM →
QYM →
dS · hEg
Q
2
S∞

2
S∞

dS · hBg −1,

and the eigenvalues of them remain invariant only under gauge transformations that go to a
eYM → h∞ Q
eYM h−1 , and not under a general
constant h∞ at infinity, QYM → h∞ QYM h−1 , Q

gauge transformation.

∞

∞

We present here what we think may be the starting point to find a solution to this problem.
We borrow the idea used to build the charges responsible for the stability of the solitons in
integrable field theories to get the conserved gauge invariant charges in gauge theories. The
integral equations are formulated for theories in (2 + 1)-dimensional space-time, namely the
Chern-Simons theory and Yang-Mills, and in (3 + 1) dimensions we describe the Yang-Mills
theory and its self-dual sector. The charge operator is found in all the cases, and we give the
charges explicitly for some configurations of the (3 + 1)-dimensional Yang-Mills. Our results
look very promising. In some cases the integral formulation leads naturally to the quantisation
condition of the charges. Moreover it puts different gauge theories in the same status, i.e.,
apparently the gauge theories can be written in loop space under the same structure, as an
equation for fluxes. Also, this links gauge theories and integrability, although we still do not
understand how to get an infinite number of conserved charges (a crucial feature in integrable
theories), if any, or to use this to construct the solutions. This remains to be investigated
together with some other points we present along the thesis.
It is important to emphasize that there is a quite vast literature on integral and loop
space formulations of gauge theories (see for instance (16–24)). Our approach differs in many
aspects of those formulations even though it shares some of the ideas and insights permeating
them.
This is the content of the first part of this thesis. All the results presented here were
recently published in two articles (8) and (25) by Luiz Agostinho Ferreira and me.
The second part is about a concept called quasi-integrability introduced recently (26) by
Luiz A. Ferreira and Wojciech Zakrzewski. Performing simulations with soliton-like solutions
evolving through non-integrable theories they observed a behaviour very similar to that expected for integrable theories (which explains the name “quasi-integrable”). They were able to
associate this with the existence of a set of infinitely many quantities that are asymptotically
conserved, so giving them the name of quasi-conserved charges.

1 Introduction

24

As proposed in (26) a (1 + 1)-dimensional theory is said to be quasi-integrable if even
without a zero curvature representation of its dynamical equations it presents soliton like
solutions that preserve their basic physical properties like mass, topological charges, etc. when
they undergo a scattering process. Also, this theory must have an infinity number of what
they call quasi-conservation laws: the corresponding charges are conserved when evaluated on
the one-soliton solutions, and are asymptotically conserved in the scattering of these solitons.
Summarising: during the scattering of the solitons their charges can vary in time, but when
the solitons are well separated after the collision they regain the values they had before the
scattering. The quasi-conservation law is of the form
d Q(n)
= βn (t)
dt

(1.0.4)

with n an integer. The asymptotically conservation is expressed as
Z ∞
(n)
(n)
Q (t → ∞) − Q (t → −∞) =
dt βn = 0.

(1.0.5)

−∞

In (26) they considered some modifications of the sin-Gordon equation following (27)
whose equations of motion were written as what is now called an anomalous zero curvature
representation. Basically, the curvature is not zero for any other case but when the potential of
the theory is exactly the integrable one, i.e., the potential of the integrable theory is modified in
a way that it can be recovered (for instance, by fixing some parameter), so, this anomalous zero
curvature becomes the zero curvature when this happens. Using the techniques already known
in integrable field theories the quasi-conserved charges were found and employing analytical
and numerical methods the scattering of solitons was studied and it was verified that for some
special solutions the charges are indeed asymptotically conserved. The key observation of (26)
was based on the fact that the two-soliton solutions satisfying (1.0.5) had the property that
their fields were eigenstates of a very special space-time parity transformation
P :





x˜, t˜ → −˜
x, −t˜

with

x˜ = x − x∆

t˜ = t − t∆ .

(1.0.6)

where the point (x∆ , t∆ ) in space-time, depends upon the parameters of the solution. Since
R∞
(n)
the charges are obtained from some densities, i.e., Q(n) = −∞ dx j0 , so are the functions
R∞
R∞
R∞
βn = −∞ dx γn , called integrated anomaly. Therefore, the vanishing of −∞ dt −∞ dx γn ,

follows from the properties of γn under (1.0.6). An important remark is that the solutions for

which the fields are eigenstates of the parity (1.0.6) cannot be selected by choosing appropriate
initial boundary conditions; the boundary conditions are set at a given initial time and the
transformation (1.0.6) relates the past and the future of the solutions. In other words, boundary
conditions are kinematical statements, and the fact that a field is an eigenstate under (1.0.6)

25

1 Introduction

is a dynamical statement. The physical mechanism that guarantees that such special solutions
have the required parity properties is not clear yet. That is the main motivation of this thesis:
it is crucial to look at other models, with different symmetries and physical content, that are
also deformations of integrable models, and analyse if the quasi-integrability phenomenon also
happens.
Hence in this thesis we look at the non-linear Schr¨odinger (NLS) model and its perturbations. The NLS is an integrable theory. It differs from the sine-Gordon in the sense
that its soliton solutions are not topological and it is non-relativistic. It worth to mention
that this model appears in several branches of science, from condensed matter to biology,
being extremely expressive in the context of optical fibres. Hence the understanding of quasiintegrability for this model would have very important implications. The modifications of the
NLS model considered here have equations of motion of the form
i ∂t ψ = −∂x2 ψ +

∂V
ψ,
∂ | ψ |2

(1.0.7)

where ψ is a complex scalar field and V is a potential dependent only on the modulus of ψ.
The NLS equation corresponds to V ∼| ψ |4 . The analysis of such models start by writing

the equations of motion (1.0.7) as an anomalous zero curvature equation of the form
∂t Ax − ∂x At + [Ax , At ] = X ,

(1.0.8)

where the connection Aµ is a functional of ψ and its derivatives, and takes values in the SL(2)
loop algebra (Kac-Moody algebra with vanishing central element), and X is the anomaly that
vanishes when V is the NLS potential.

Then we discuss how to construct the infinite set of quasi-conserved charges by employing
the standard techniques of integrable field theories known as Drinfeld-Sokolov reduction (28),
or abelianisation procedure (2, 29, 30). With them we gauge transform the Ax component of
the connection into an infinite dimensional Abelian sub-algebra of the loop algebra, generated
by T3n ≡ λn T3 . Even though the anomaly X prevents the gauge transformation to rotate

the At component into the same Abelian sub-algebra, the component of the transformed
curvature (1.0.8) in that sub-algebra leads to a set with infinitely many quasi-conservation laws,
R∞
R∞
(n)
(n)
∂ µ jµ = γn , or equivalently leads to (1.0.4) with Q(n) = −∞ dx j0 and βn = −∞ dx γn .

Next a more refined technique, involving two ZZ2 transformations, is used to understand

the conditions for the vanishing of the integrated anomalies. The first ZZ2 is an order two
automorphism of the SL(2) loop algebra and the second is the parity transformation (1.0.6).

1 Introduction

26

For the solutions for which the field ψ transforms under (1.0.6) as
ψ → ei α ψ ∗

with α constant

(1.0.9)

R x˜
dt −˜x0 0 dx γn = 0, where t˜0 and x˜0 are any given fixed values of the
space-time coordinates t˜ and x˜, respectively, introduced in (1.0.6). This leads to

it is shown that

R ˜t0

−t˜0





Q(n) t = t˜0 + t∆ = Q(n) t = −t˜0 + t∆

(1.0.10)

which is a type of a mirror symmetry for the charges. Therefore, for a two-soliton solution
satisfying (1.0.9), the asymptotic conservation of the charges (1.0.5) follows from this stronger
conclusion.
Such results certainly unravel important structures responsible for the phenomena called
quasi-integrability. They involve an anomalous zero curvature equation, internal and external
ZZ2 symmetries, and algebraic techniques borrowed from integrable field theories. However,
they rely on the assumption (1.0.9) which is a dynamical statement since it relates the past
and the future of the solutions. In order to shed more light on this issue the relation between
(1.0.9) and the dynamics defined by (1.0.7) is studied.
It is easier to work with the modulus and phase of ψ, and so the fields are parametrised
√
ϕ
as ψ = R ei 2 , with R and ϕ being real scalars fields. They are separated into their eigencomponents under the parity (1.0.6), as R = R(+) + R(−) , and ϕ = ϕ(+) + ϕ(−) . The


assumption (1.0.9) implies that the solution should contain only the components R(+) , ϕ(−) ,


and nothing of the pair R(−) , ϕ(+) . By splitting the equations of motion (1.0.7) into their
even and odd components under (1.0.6), we show that there cannot exist non-trivial solutions


carrying only the pair R(−) , ϕ(+) . In addition, if the potential V in (1.0.7) is a deformation
of the NLS potential, in the sense that we can expand it as

V = VNLS + ε V1 + ε2 V2 + . . .

(1.0.11)

with ε being a deformation parameter, then we can make even stronger statements. In such
a case we expand the equations of motion and the solutions into power series in ε, as
(±)

R(±) = R0

(±)

+ ε R1

(±)

+ ε2 R2

+ ... ;

(±)

ϕ(±) = ϕ0

(±)

+ ε ϕ1

(±)

+ ε 2 ϕ2

+ ...

(1.0.12)
If we select a zero order solution, i.e., a solution of the NLS equation, satisfying (1.0.9), carrying


(+)
(−)
only the pair R0 , ϕ0 , then the equations for the first order fields, which are obviously


(+)
(−)
satisfies inhomogeneous equations, while
linear in them, are such that the pair R1 , ϕ1

27

1 Introduction





(−)
(+)
(−)
(+)
the pair R1 , ϕ1 , satisfies homogeneous ones. Therefore, R1 , ϕ1
= (0, const.), is


(+)
(−)
= (0, const.), is not. By selecting the
a solution of the equations of motion, but R1 , ϕ1


(−)
(+)
first order solution such that the pair R1 , ϕ1
is absent, we see that the same happens in


(+)
(−)
second order, i.e., that the pair R2 , ϕ2
also satisfies inhomogeneous equations, and the


(−)
(+)
the homogeneous ones. By repeating this procedure, order by order, one
pair R2 , ϕ2
can build a perturbative solution which satisfies (1.0.9), and so has charges satisfying (1.0.10).

Note that the converse could not be done, i.e., we cannot construct a solution involving only


the pair R(−) , ϕ(+) . So, the dynamics dictated by (1.0.7) favours solutions of the type
(1.0.9).

Finally we discuss the conditions for the soliton solutions of the NLS equation to satisfy
the parity property (1.0.9). As it is well known there are two basic types of NLS soliton
solutions: the bright solitons for η < 0, and dark solitons for η > 0, where η is the coupling
constant of the NLS potential given by VN LS = η | ψ |4 . The names originate from the
fact that the values of | ψ |2 increase (decrease) as one approaches the core of a bright
(dark) soliton. For a more detailed discussion about NLS bright/dark solitons see (31–34)

and references therein. We shall show that the one-bright-soliton and the one-dark-soliton
solutions of the NLS equation satisfy the condition (1.0.9), and that not all two-bright-soliton
solutions satisfy it. However, one can choose the parameters of the general solution so that
the corresponding two-bright-soliton solutions do satisfy (1.0.9). This involves a choice of the
relative phase between the two one-bright-solitons forming the two-soliton solution. Therefore,
our perturbative expansion explained above can be used to build a sub-sector of two-brightsoliton solutions of (1.0.7) that obeys (1.0.9) and so has charges satisfying (1.0.10). This
would constitute our quasi-integrable sub-model of (1.0.7). We do not analyse in this thesis the
two-dark soliton solutions of the NLS equation basically for conciseness. The construction of
the general two-dark soliton solution requires a modification of the Hirota’s method described
in appendix C for the case of bright solitons. In addition, our numerical code would have to
be altered to deal with dark solitons.


(±)
(±)
Despite the fact that the equations of motion satisfied by the n-order fields Rn , ϕn

are linear, the coefficients are highly non-linear in the lower order fields and so, unfortunately,
these equations are not easy to solve. We then use numerical methods to study the properties

of our solutions. In addition, such numerical analysis can clarify possible convergence issues of
our perturbative expansions. We chose to perform our numerical simulations for a potential
of the form
V =


2+ε
2η
| ψ |2
2+ε

η<0

(1.0.13)

1 Introduction

28

The choice of such potential is rather arbitrary. It possesses a property however which might
be relevant, i.e. it does not shift the vacuum of the NLS potential. Other choices like those
shown in (5.2.4) may introduce additional vacua besides ψ = 0.
With the huge contribution of Wojciech Zakrzewski from Durham University several simulations were done using the 4th order Runge Kutta method of simulating the time evolution.
These simulations involved the NLS case with the two bright solitons sent towards each other
with different values of velocity (including v = 0) and for various values of the relative phase.
We then repeated that for the modified models. We looked at various values of ε and have
found that the numerical results were reliable for only a small range of ε around 0. For very
small values we saw no difference from the results for the NLS model but for |ε| ∼ 0.1 or

∼ 0.2 the results of the simulations became less reliable. Hence, we are quite confident of

our results for |ε| < 0.1 and in the numerical section we present the results for ε = ±0.06.

Also the results for the anomaly as seen in the simulations are presented and they confirm our
expectations.
The results about the quasi-integrable deformations of the NLS theory presented here were
published by Luiz Ferreira, Wojciech Zakrzewski and me in (35).

29

CHAPTER 2

Hidden Symmetries, Stokes theorem
and conservation laws
Non-linear field theories are ubiquitous in Nature. Such theories, in opposition to the
linear ones (e.g. electromagnetism) do not admit the superposition principle and in general
the solutions are not only harder to be found but also they behave differently when compared to
linear waves. In particular the soliton solutions(? ) propagate with constant velocity without
changing their shape or dissipating energy, and when two of them undergo a scattering process
the only effect they feel is a shift from the position they would have if there was no scattering.
These features lead to the interpretation of solitons as particles. Their stability and the
behaviour just described arise from the existence of infinitely many conserved quantities that
can eventually be obtained when one recast the dynamical equations of the theory as a zero
curvature equation(1), Gtx ≡ ∂t Cx − ∂x Ct + [Ct , Cx ] = 0, i.e., the vanishing of the curvature

of the Lie algebra valued 1-form connection C = Ct dt+Cx dx, a functional of the fields and its
derivatives, implies the equations of motion of the theory and vice versa. There is no recipe to
build such a connection, and also the set of theories that can be described in this way (known
as integrable) is not very big.
One can immediately notice that the gauge invariance of the zero curvature equation
reveals a new (hidden) symmetry of the theory. We now want to discuss how to build up
the conserved charges. For theories in 2-dimensions this is done through the (standard∗ ) nonAbelian Stokes theorem, and in the next section we shall prove it following the approach in
(6).

∗

The label “standard” here is because we intend to generalise this theorem later, remaining at the end with
the generalised and the standard non-Abelian Stokes theorems.

2 Hidden Symmetries, Stokes theorem and conservation laws

30

2.1

The standard non-Abelian Stokes theorem

Let M be the 2-dimensional space-time manifold. We introduce the Wilson line W , a
non-local object† defined by the integration of the first order linear equation
dxµ
dW
+ Cµ
W = 0,
dσ
dσ

(2.1.1)

along a curve Γ, a 1-dimensional sub-manifold of M. The points of Γ are parametrised by
σ ∈ [0, 2π], so, xµ = xµ (σ); the initial point (here called the reference point) is xµ (0) ≡ xR ,
and the final point, xµ (2π) ≡ xf . Together with the equation we set the constant element

WR being the initial condition , i.e., the value of W at the reference point. This equation is
solved iteratively: one integrates it from the reference point to some arbitrary point xµ (σ) in
Γ
WΓ [σ, 0] = WR −

Z

σ

Cµ (σ ′ )

0

dxµ
WΓ [σ ′ , 0]dσ ′
dσ ′

and use this result, but for WΓ [σ ′ , 0], inside the Wilson line in the integrand above, which
gives
WΓ [σ, 0] = WR −

Z

0

σ

dxµ
Cµ (σ ) ′ WR +
dσ
′

Z

0

σ

dxµ
Cµ (σ ) ′
dσ
′

Z

σ′

Cν (σ ′′ )

0

dxν
WΓ [σ ′′ , 0]dσ ′′ dσ ′
dσ ′′

and so on, indefinitely. Notice that the product of connections appears in a certain order: the
rightmost term in Cµ (σ ′ )Cν (σ ′′ ) . . . Cρ (σ ′...′ ) is the first one, from the reference point to the
final point (the direction the path is oriented) since σ ≥ σ ′ ≥ σ ′′ ≥ · · · ≥ 0. It is then useful

to introduce the path-ordered product as
Z

P1
Z Z0

σ

Z

σ′

dxµ dxν ′′ ′
dσ dσ =
dσ ′ dσ ′′
0
Z Z
µ
ν
′
′′ dx dx
′′
′
Cµ (σ )Cν (σ ) ′ ′′ dσ dσ +
dσ dσ
σ≥σ′ ≥σ′′ ≥0
Cµ (σ ′ )Cν (σ ′′ )

Cν (σ ′′ )Cµ (σ ′ )
σ≥σ′′ ≥σ′ ≥0

dxν dxµ ′ ′′
dσ dσ
dσ ′′ dσ ′

to guarantee that the rightmost term always comes first. Considering a 2-dimensional plane
with vertical axis σ ′ and horizontal axis σ ′′ , the first term on the r.h.s above corresponds to the
integration on the top triangle in figure (2.1), while the second term, on the bottom triangle.
Due to the evident symmetry σ ′ ↔ σ ′′ between these terms the result of the integrations must

†

Taking the connection in a finite representation of the Lie algebra, it becomes a matrix, and W is a matrix
as well.

31

2.1 The standard non-Abelian Stokes theorem

Figure 2.1 –

1
factor appears due to the symmetry relating the n! integrations in the pathThe n!
ordered product.

be the same and therefore
Z σ Z σ′
Z σ Z σ′
µ
ν
ν
µ
1
′′
′
′
′′ dx dx
′
′′ dx dx
P1
Cµ (σ )Cν (σ ) ′ ′′ dσ dσ =
Cµ (σ )Cν (σ ) ′ ′′ dσ ′′ dσ ′
dσ dσ
2
dσ dσ
0
0
0 σ≥σ′ ≥σ′′ ≥0
0
2
Z σ
µ
1
dx
≡
P1
Cµ (σ ′ ) ′ dσ ′ .
2
dσ
0
The generalisation of the path-ordering for products involving more terms is straightforward:






P1 (Cµ (σ1 )Cν (σ2 ) . . . Cρ (σn )) = Cµ σπ(1) Cν σπ(2) . . . Cρ σπ(n) where π stands for the




permutation such that σπ(1) ≥ · · · ≥ σπ(n) . The integrations appearing in this series
are defined on simplexes. A n-simplex is, roughly speaking, the generalisation of triangles:

they are geometrical objects with flat sides that form the convex set of their n + 1 vertices.
The very first term of the series of W (after the integration constant) is the integration on a
1-simplex, which is a line. The second term, on a triangle, which is a 2-simplex. The third,
on a tetrahedron, and so on. The symmetry pattern appearing in the case discussed above of
the 2-simplex persists and for a n-dimensional cube one has n! simplexes, so, a factor n!1 will
Rσ

n
appear in front of the path-ordered integral for the nth term: n!1 P1 0 Cµ dxµ . Finally the

iterative solution of (2.1.1) can be recognised as an infinite series

!
Z σ
Z σ
2
3
1
1
µ
µ
WΓ [σ, 0] =
1l − P1
Cµ dx + P1
Cµ dx
Cµ dx
− P1
+ . . . · WR
2!
3!
0
0
0
n
Z σ
∞
X
(−1)n
µ
P1
Cµ dx
=
· WR
n!
0
n=0
Z

σ

µ

which can be formally written as the path-ordered exponential

 Z σ
dxµ ′
WΓ [σ, 0] = P1 exp −
Cµ ′ dσ · WR .
dσ
0

(2.1.2)

2 Hidden Symmetries, Stokes theorem and conservation laws

32

Although maybe obvious it is important to point out that M must be arc-connected, so that we
can define the path Γ linking two points. Now, this path may be formed by the composition
of several paths, for instance Γ may be the union of a path Γ1 with another, Γ2 , with an
intersection point Γ1 ∩ Γ2 . We denote this by Γ = Γ2 ◦ Γ1 , following the path-ordering idea

that the rightmost part comes first. It is not difficult to see that under such a decomposition
the Wilson line is also decomposed as WΓ = WΓ2 · WΓ1 , the product being the group product‡

once W belongs to the group G if C is in its Lie algebra.

Consider a gauge transformation of the connection Cµ → Cµ′ ≡ hCµ h−1 − ∂µ h h−1

with h in the gauge group G. We suppose that the Wilson line is transformed to W ′ , such
that the equation

dW ′
dσ

µ

+ Cµ′ dx
W ′ = 0 holds. Multiplying the first term of this equation
dσ

by h · h−1 and with some simple manipulations, putting an h term in evidence on the left,

we end up with

d
dσ

µ

(h−1 · W ′ ) + Cµ dx
(h−1 · W ′ ) = 0, which clearly requires W ′ = h · W .
dσ

However, this equation (and equation (2.1.1), more generally) defines the Wilson line up to

a constant term on the right, i.e., if h · W is a solution of this equation, then h · W · k is

also a solution, with k a constant element of the group. Let us take then W ′ = h · W · k.

Suppose Γ = Γ2 ◦ Γ1 , and denote the intersection point Γ1 ∩ Γ2 by x˜. Then WΓ will be

decomposed as discussed above and the gauge transformation will change the first part as
x), where we
WΓ1 → h(˜
x) · WΓ1 · k(xR ), while the second part as WΓ2 → h(xf ) · WΓ2 · k(˜

used a different constant element for Γ2 . The constant elements can be calculated anywhere
(since they are constant), and we set this to be done in the initial point of each curve. Then
we have W ′ = h(xf ) · WΓ2 · k(˜
x) · h(˜
x) · WΓ1 · k(xR ), which by consistency must be equal to

h(xf ) · W · k(xR ), so k(˜
x) = h−1 (˜
x), and since the intersection point is arbitrary, it holds for
the entire curve. Finally, we conclude that the Wilson line calculated on a given curve Γ with

boundary ∂Γ = {xR ∪ xf } transforms under a gauge transformation of the connection as
WΓ → h(xf ) · WΓ · h−1 (xR ).

(2.1.3)

The Wilson line is defined along a curve linking two points. An important question to be
understood concerns the dependence of W on the curve, i.e., will the result of the integration
of (2.1.1) from xR to xf depend on the chosen path? In order to answer that we analyse
the behaviour of W under variations of the type xµ → xµ + δxµ . The simplest case is when

we change the speed of the curve by changing σ → σ ′ (σ). This produces an overall factor
dσ′
dσ

in (2.1.1), which does not change anything. So, we must consider variations that are

not tangent to the curve. The reference point will remain fixed, and on the next point of
the curve, infinitesimally close, we define a vector T µ , orthogonal to S µ ≡
‡

In (2.1.2) we have the exponential map of a Lie algebra element.

dxµ
,
dσ

the tangent

33

2.1 The standard non-Abelian Stokes theorem

vector. That point will suffer a variation in the direction of this normal vector. To keep track
of that a new parameter τ ∈ [0, 2π] is introduced. For τ = 0 the point is on the curve Γ,

and for any other value it is some place else. So, the normal vector can be written as the
velocity T µ ≡

dxµ
.
dτ

We want the next point, infinitesimally close to this first one in Γ to do the

same, and so on. It is possible to vary every point of the curve in the same way because the
vector T µ is parallel transported along Γ, as one can check by calculating its Lie derivative:
£S T µ = S ν ∂ν T µ − T ν ∂ν S µ = 0. For that reason once a variation (i.e., a direction of the

normal vector, etc.) is defined the whole curve changes smoothly to another curve Γ + δΓ,
and this process continues until τ = 2π defining a 2-dimensional surface.
In order to calculate the variation of the Wilson line when the curve is changed we take

the variation of equation (2.1.1). Multiplying it by W −1 from the left, after simple manip−1

µ

µ

d
= −Cµ dx
W , and dWdσ = W −1 Cµ dx
, one gets dσ
(W −1 δW ) +
ulations using that dW
dσ
dσ
dσ
Rσ
µ

µ

W −1 δ Cµ dx
W = 0, which can be integrated: δW = −W 0 dσ ′ W −1 δ Cµ dx
W.
dσ
dσ′
R
R σ ′ −1
µ
µ
σ
+ 0 dσ ′ W −1 Cµ dδx
W and we use
The integral on the r.h.s becomes 0 dσ W δCµ W ∂x
∂σ′
dσ′

δCµ = ∂ν Cµ δxν in the first term while the last one is integrated by parts, where we again make
use of the equations for W and W −1 , and also that

dCµ
dσ

ν

= ∂ν Cµ ∂x
. After all appropriate
∂σ

substitutions the result is
µ

δW = −Cµ δx W + W

Z

σ

dσ ′ W −1Gµν W
0

∂xµ ν
δx
∂σ ′

where Gµν = ∂µ Cν − ∂ν Cµ + [Cµ , Cν ]. We are omitting a more precise notation for the sake

of clearness, so, lets remark that in the above equation all the Wilson lines outside the integral
are defined along the curve Γ, from σ = 0 to σ 6= 0. The integral defines something which

is on that curve as well, so the Wilson lines appearing in the integrand, and the curvature
components Gµν , are defined inside this curve at some point parametrised by σ ′ ∈ [0, σ]. So,
these Wilson lines are calculated using equation (2.1.1) from the reference point to xµ (σ ′ ).

If both the reference point and the final point remain fixed then the first term on the r.h.s
above vanishes. Moreover, since the variation is orthogonal to the curve, δxµ = xµ (τ + δτ ) −
xµ (τ ) =

∂xµ
∂τ

and δW = W (τ + δτ ) − W (τ ) = dW
δτ , so
dτ
Z σ
∂xµ ∂xν
dW
−W
=0
dσ ′ W −1 Gµν W ′
dτ
∂σ ∂τ
0

(2.1.4)

defines how the Wilson line changes in τ , i.e., when we vary the curve. In fact, it shows more.
This equation shows that there are two different, but equivalent ways to get W along a curve.
Consider Γ to be the curve at τ = 2π, resulting from the continuous deformation of the curve
in τ = 0. We can get W on Γ by integration of (2.1.1) directly, knowing WR , but also, we

34

Figure 2.2 –

2 Hidden Symmetries, Stokes theorem and conservation laws

One can use a family of homotopically equivalent loops to scan a 2-dimensional surface.

can get W on Γ by integrating (2.1.4), knowing W at τ = 0. In particular let us consider the
case where xR = xf . The curve τ = 0 becomes the infinitesimal loop around the reference
point (see figure (2.2)), so that the Wilson line there coincides with WR . When this curve
is varied we produce a new loop, and this process goes on until the loop labelled by τ = 2π
is reached. This final loop encloses an area Σ, so we might refer to it as the boundary ∂Σ.
Because of the nature of the variations performed, as discusses previously, each point inside Σ
belongs to a single loop, i.e., the loops do not intersect. We figure out that given a reference
point in the boundary of a (simply-connected) 2-dimensional surface, we can scan this surface
using a homotopic family of loops, based at this reference point. The Wilson line calculated
from (2.1.1) reads, in this case,
W∂Σ = P1 exp

Z

2π
0

dxµ
dσ Cµ
dσ



· WR .

(2.1.5)

The integration of (2.1.4) can be done similarly to that of (2.1.1). For convenience we
R 2π
µ ∂xν
, so that (2.1.4) can be written as
define C ≡ 0 dσW −1 Gµν W ∂x
∂σ ∂τ

dW
− W C = 0.
(2.1.6)
dτ
Rτ
R τ R τ′
Integrating it iteratively one gets W (τ ) = WR +WR 0 C(τ ′ )dτ ′ + 0 0 W (τ ′′ )C(τ ′′ )C(τ ′ )dτ ′′ dτ ′ ,

etc. and we notice the need to keep track of the ordering in the products inside the integrand.
However, this time the rightmost terms are the ones that appear later in the scanning of the

surface, since τ ≥ τ ′ ≥ τ ′′ ≥ · · · ≥ 0. Then, the solution can be formally written as

Z 2π

Z 2π Z 2π
∂xµ ∂xν
−1
W∂Σ = WR · P2 exp
C dτ = WR · P2 exp
dτ dσ W Gµν W
∂σ ∂τ
0
0
0
(2.1.7)
where P2 stands for the ordering in τ , referred to as surface-ordering for obvious reasons.

35

2.1 The standard non-Abelian Stokes theorem

Figure 2.3 –

The zero curvature implies that the Wilson line is independent of the path. This leads
to a conservation law.

Since the results obtained in (2.1.5) and (2.1.7) must be the same, we have the identity


Z
 I
−1
µ
ν
µ
(2.1.8)
W Gµν W dx dx
P1 exp −
Cµ dx · WR = WR · P2 exp
Σ

∂Σ

which is the non-Abelian Stokes theorem.
Now, if we have a zero curvature representation of the equations of motion, we have
that under a path deformation, keeping the border fixed, δW = 0, i.e., the Wilson line is
independent of the path. In particular, for a closed curve Γc , the above relation holds, and
plugging Gµν = 0 in the r.h.s, we get W∂Σ = WR , or, multiplying by WR−1 from the right,

 I
µ
(2.1.9)
P1 exp −
Cµ dx = 1l.
∂Σ

In order to construct the conserved charges we start by splitting space-time into space and time.
We consider that M is flat, with a Minkowski metric. In curved space-time this separation is
not trivial to be done. Then we take the closed path as Γc = Γ−1
2 ◦ Γ1 , where Γ1 = Γ∞ ◦ Γ0 ,

linking the reference point to a final one, and Γ2 = Γt ◦ Γ−∞ , a different way to link the same

two points, according to the figure (2.3). For simplicity we could consider WR = 1l, or in the
centre of the group Z(G), so that it may be dropped from (2.1.8). Or, we could just take
WΓ = QΓ ·WR . Equation (2.1.9) implies that the operator QΓ (or equivalently the Wilson line)

is independent of the path linking the reference point to the final point: QΓc = Q−1
Γ2 · QΓ1 = 1l.

−1
−1
Then, QΓc = Q−1
Γ−∞ · QΓt · QΓ∞ · QΓ0 = 1l, thus QΓt = QΓ∞ · QΓ0 · QΓ−∞ . What do we have?

Notice that the quantity WΓ0 is the Wilson line calculated over the entire space, at time zero,

and WΓt is the Wilson line calculated over the entire space, at some later time. The paths
Γ−∞ and Γ∞ are simply the “evolution in time” of the spatial boundary. Taking as boundary
conditions Ct (t, ∞) = Ct (t, −∞) we have QΓ∞ = QΓ−∞ ≡ U(t), and we get an iso-spectral

2 Hidden Symmetries, Stokes theorem and conservation laws

36

evolution for
QΓt = P1 e−

R

Γt

Cx dx

,

which implies that its eigenvalues are conserved in time, or equivalently Tr(QnΓt ). Also, it is
clear that these charges (the eigenvalues) are not affected by gauge transformations.

2.2

Generalisation of the Stokes theorem

In the previous section it was shown how the (standard) non-Abelian Stokes theorem can
be used, with the fact that the connection is flat, to build up conserved charges. These charges
do not come from Noether’s symmetries, being only revealed once the equations of motion
are written as a zero curvature equation. The vanishing of the curvature is equivalent to the
path independence of the Wilson line, and considering a loop as in figure (2.3), where we
clearly separate the space and time, one can use this independence, with appropriate boundary
conditions to get an iso-spectral evolution for the “spatial Wilson line” (i.e., the Wilson line
calculated over the entire space, at certain time slice), whose eigenvalues are recognised as
the conserved charges.
This construction works pretty well in the (1+1)-dimensional space-time, but this is not the
case for higher dimensions, where neither the concept of integrability is understood. An idea to
generalise this zero curvature formulation for (d + 1)-dimensional theories was presented(6) in
1997 by Luiz Agostinho Ferreira, Joaquin S´anchez-Guill´en and Orlando Alvarez. They noticed
that the loop space looks like the adequate place to follow the steps mentioned above to get
the conserved charges. Given a (d + 1) dimensional space-time M, the loop space LM is
defined by the set of mappings from the (d − 1) sphere (S d−1 ) to M, fixing the image of a
point of the sphere, say, the north-pole, as the reference point xR in M. The images of these

maps are (d − 1) closed hyper-surfaces in M based at xR , and each of them corresponds to a

point in LM, while the hyper-volume, in between two such surfaces, corresponds to a path in

LM. It is not difficult to see that in the case d = 1 the loop space coincides with space-time.
For d = 2, the loops based at xR in M are points in LM, and the area of the surface between
the infinitesimal loop around xR and the loop forming the boundary at τ = 2π is a path in
LM. For d = 3 the surfaces in M correspond to points in LM and the 3-dimensional volumes
in space-time to paths in the loop space.
So, following the approach in (6) the first step is to look for a charge operator that
generalises WΓt , the “spatial Wilson line”. If the dimensionality of space increases then we
expect the generalisation of the Wilson line to be related to a connection which is a differential

37

2.2 Generalisation of the Stokes theorem

Figure 2.4 –

On the left, a surface Σ in M is scanned with loops based at xR . On the right, this
surface is represented in LM , where each loop in M corresponds to a point and the
surface from xR to the boundary ∂Σ is a path. A variation of this surface, leaving the
boundary fixed, is also represented in LM .

form with higher degree. Notice that WΓt is the path-ordered exponential of a 1-form, which
is something that we integrate over a 1-dimensional manifold, the space. For a space with 2
dimensions, we might expect a 2-form, and so on. For a theory in (d + 1) dimensions their
idea was to introduce a d-form field. So, one needs to find a way to define a generalisation of
the Wilson line but now for a surface, and then for a volume, etc. which, in the loop space,
correspond always to a path. Well, in fact this problem is more or less solved. The equation
we are looking for is a generalisation of (2.1.6), where instead of W we write V (and call it
the Wilson surface), and instead of Gµν , we introduce an anti-symmetric tensor§ Bµν :
dV
−V
dτ

Z

2π

dσW −1Bµν W

0

∂xµ ∂xν
= 0,
∂σ ∂τ

(2.2.1)

with the initial condition being the constant VR , calculated on the infinitesimal surface around
the reference point. This equation, when integrated in τ , defines the Wilson surface V , in a
2-dimensional surface. How to do that was already discussed, and the result is formally written
as
VΣ = VR · P2 exp
where C ≡

R 2π
0

µ

dσW −1 Bµν W ∂x
∂σ

∂xν
.
∂τ ′

Z

0

τ

Cdτ

′



(2.2.2)

As discussed in (7), the quantity C appearing here can be understood as a connection in

the loop space, so that the above V is a direct generalisation of the Wilson line, indeed. As

before, one could consider variations of the surface, and analyse how V changes. Following
the pattern for W , this variation will produce a new equation for V , such that if we consider
the surface to be closed and the boundary of a volume then V can be obtained by integrating
(2.2.1) on that surface directly or by integrating this new equation along the volume from
the infinitesimal closed surface around xR . The result from this new equation defines V as
a volume-ordered exponential of something that can be identified as the curvature of C in
§

The 2-form B = 12 Bµν dxµ ∧ dxν is not necessarily exact.

2 Hidden Symmetries, Stokes theorem and conservation laws

38

Figure 2.5 –

The border of the surface is kept fix while performing the variation. When the surfaces
are closed, the border is contracted to xR and the initial surface (ζ = 0) becomes the
closed infinitesimal surface ΣR while the final surface (ζ = 2π) becomes the boundary
of a volume.

loop space, and the equivalence of the two ways of finding V is the generalisation of Stokes
theorem, as we shall discuss in a while.
Then, in (6) it is explained how to obtain local conditions that would make this curvature
in loop space vanishes, and the Wilson surface becomes path independent, and we get a guide
to define integrability for theories in (2 + 1)-dimensional space-time, and a way to calculate
the conserved charges. Then, for theories in (3 + 1) dimensions we generalise what was done
here for V , to, say, a Wilson volume, and so on.
In this thesis we shall use basically the same ideas of (6) and (7), but instead of looking
for a zero curvature in loop space we propose an integral equation of motion for the theory
which is a consequence of the standard or generalised non-Abelian Stokes theorem and of the
differential equations of the physical fields. Now that the goal was explained, let us discuss
the calculations to generalise the non-Abelian Stokes theorem involving a 2-form connection.
As mentioned before, consider variations of the surface: Σ → Σ + δΣ. Following the

same reasoning used for the variation of the path Γ in the case of the Wilson line, we take
these variations to be in the direction perpendicular to the surface, and parametrise them with
ζ ∈ [0, 2π], so defining a velocity

dxµ
.
dζ

If we consider the surface to be closed, becoming the

border of the volume Ω, we notice that by continuously varying from the infinitesimal closed
surface around the reference point ΣR , where the Wilson surface is VR , we scan Ω (see figure
(2.5)). The surface labelled by ζ = 2π corresponds to the boundary ∂Ω. Each of this surfaces,
in turn, are scanned with loops, based at xR . The calculation is long but straightforward and
the result for the variation of the Wilson surface reads¶

¶

Although we shall be in most of the discussion interested in variations perpendicular to the surface, equation
(2.2.3) holds for any kind of variation, including those tangent to the surface.

39

2.2 Generalisation of the Stokes theorem

Z

δV =
−

Z

σ
0

2π

dτ
0

Z



2π

dσ V (τ ) W −1 (Dλ Bµν + Dµ Bνλ + Dν Bλµ ) W

0

 W ′

′
W
dσ ′ Bκλ
(σ ) − GW
(σ
),
B
(σ)
×
κλ
µν

∂xκ ∂xµ
× ′
∂σ ∂σ



∂xλ ′ ν
∂xν
(σ )δx (σ) − δxλ (σ ′ )
(σ)
∂τ
∂τ



V −1 (τ )

!

∂xµ ∂xν λ
(2.2.3)
δx
∂σ ∂τ

·V

where Dµ ⋆ = ∂µ ⋆ + [Cµ , ⋆], and we use the notation X W ≡ W −1 XW . Again, since the

variation is with respect to the parameter ζ, this equation can be written as an equation in
that parameter:

dV
− KV = 0,
dζ

(2.2.4)

where
K =

Z

0

2π

dτ
Z

σ

Z

2π



dσ V (τ ) W −1 (Dλ Bµν + Dµ Bνλ + Dν Bλµ ) W
0

∂xµ ∂xν ∂xλ
∂σ ∂τ ∂ζ

 W ′

′
W
dσ ′ Bκλ
(σ ) − GW
κλ (σ ), Bµν (σ) ×
0


∂xκ ∂xµ ∂xλ ′ ∂xν
∂xλ ′ ∂xν
× ′
(σ )
(σ) −
(σ )
(σ)
V −1 (τ ).
∂σ ∂σ
∂τ
∂ζ
∂ζ
∂τ

−

The quantities W and V inside K are calculated from (2.1.1) and (2.2.1) respectively. W is

integrated along the loops scanning each surface, while V is calculated on each of the surfaces

scanning the volume. The integration of (2.2.4) is done similarly as those explained before.
Now we have a volume-ordering which is represented by the P3 . The result can be written
formally as
VΣc = P3 exp

Z

0

2π

Kdζ



· VR .

(2.2.5)

The equality between the Wilson surface obtained in (2.2.5) and that in (2.2.2) (for Σc = ∂Ω)
is the statement of the generalised non-Abelian Stokes theorem:


Z
I
Kdζ · VR .
Cdτ = P3 exp
VR · P2 exp

(2.2.6)

Ω

∂Ω

The idea now is to set the quantities Bµν and Cµ in terms of physical fields, whose
dynamics are governed by differential equations, and we state that the Stokes theorem is the
integral equation of motion, in the sense that when the hyper-volume becomes infinitesimal
one recovers the differential equations. We shall formulate the integral equations using (2.1.8)
and (2.2.6). In fact, the general form of the integral equation in the (d + 1)-dimensional
space-time M is

R

Pd−1 e

∂Ω

A

R

= Pd e

Ω

F

(2.2.7)

2 Hidden Symmetries, Stokes theorem and conservation laws

40

where Ω is a sub-manifold of M with dimension d, and ∂Ω its boundary. The quantities A

and F are defined from (d − 1)-forms and d-forms respectively, in terms of the physical fields.
R

Now, if we consider Ω to be closed, the l.h.s becomes 1l and we get Pd e

Ωc

F

= 1l. This

relation will lead us to a conservation law, in a similar way of that we saw appearing from
the zero curvature formulation in (1 + 1) dimensions, in terms of the Wilson line: splitting
the space-time into space and time and with appropriate boundary conditions we find an isoR

spectral evolution for the operator Pd e

Ωspace

F

calculated over the space, at some time slice.

This will give us the conserved charges.
The main difficulty in this approach, as in the construction of the zero curvature representation in 2-dimensions, is to find the appropriate way to write Cµ and Bµν in terms of
the physical fields, in a way that the integral equations will imply the differential equations
when Ω becomes infinitesimal. In this thesis we show how this can be done for gauge theories:
Chern-Simons and Yang-Mills in (2 + 1) dimensions and Yang-Mills in (3 + 1) dimensions. We
explored some configurations of the latter case such as the monopoles, dyons and also in the
euclidean sector with the merons and instantons, calculating their charges, which arise from
our construction.

41

CHAPTER 3

Integral formulation of theories in
2 + 1 dimensions
In this chapter we show how the standard non-Abelian Stokes theorem (2.1.8) provides
an integral version of Chern-Simons and Yang-Mills equations in three dimensions. These
equations lead us to some conserved quantities in a very natural way. The construction is very
similar to what is usually done in the 2-dimensional space-time, but now the paths in figure
(2.3) are replaced by surfaces, which, due to the construction in loop space, are scanned by
loops. The physics of the problem show us the boundary conditions, which are in fact precisely
what is needed to obtain an iso-spectral evolution for the charge operator that generalises the
“spatial Wilson surface” as described above.

3.1

The integral equations of Chern-Simons theory

We consider a 3-dimensional Minkowski space-time M. The field Cµ (µ = 0, 1, 2) in
(2.1.8) is written in terms of the gauge field Aµ (containing the physical degrees of freedom)
as Cµ = ieAµ ; e is the gauge coupling constant. So, the curvature of C reads Gµν =
∂µ Cν − ∂ν Cµ + [Cµ , Cν ] = ieFµν , where Fµν is the field strength. The differential Chern-

Simons equation is given by Fµν =

1
ǫ J ρ,
κ µνρ

coupling constant. Using that we obtain Gµν

where J µ is the matter current, and κ is a
= ieκ Jeµν , where Jeµν is the Hodge dual of J µ .

Finally, we have the l.h.s and the r.h.s of (2.1.8) in terms of the physical fields: the first is
related to the gauge field Aµ while the second, to the matter field Jµ . Then, the integral
equation reads

I
P1 exp −ie

µ

Aµ dx
∂Σ



= P2 exp



ie
κ

Z

Σ

W

−1


µ
ν
e
Jµν W dx dx .

(3.1.1)

One notice that the integration constants WR do not appear in the above equation. This is
because the Stokes theorem is now promoted from a mathematical identity to a physical equa-

3 Integral formulation of theories in 2 + 1 dimensions

42

tion, and therefore we require its gauge covariance, which in turn implies that the integration
constants must be in the centre of the group, as we now show, and therefore they become
irrelevant or factorisable from (3.1.1).
The r.h.s of the above equation is V , calculated from (2.2.1), where in C we use Bµν =

ie e
J .
κ µν

This quantity, C, is integrated along the loop, starting and ending at the same point

xR . Under a gauge transformation Aµ → hAµ h−1 + ei ∂µ h h−1 the field strength transforms

as Fµν → hFµν h−1 , and similarly the dual of the current (by consistency of the Chern-Simons

equation). For a loop, the Wilson line transforms as W → h(xR ) · W · h−1 (xR ) and it

is easy to see then that C → C ′ = hCh−1 . Now, we suppose that V defined by (2.2.1)

becomes V ′ , satisfying

dV ′
dτ

V ′ = h(xR ) · V h−1 (xR ).

− V ′ C ′ = 0. Then, one finds that under a gauge transformation

Now, equation (2.2.1) defines V up to a constant group element on the left, i.e., if V is
a solution of (2.2.1), then V k = k · V is a solution too. So, under a gauge transformation
V k transforms like V k → k · h · V · h−1 , which, on the other hand, is V k → h · k · V · h−1 .

Consistency implies that k · h = h · k, so k belongs to the centre of the group. Equivalently,
the l.h.s. of (3.1.1) is W , a solution of (2.1.1) with Cµ = ieAµ . This is defined up to a

constant element on the right, so if W is a solution, W q = W · q is a solution as well. A

completely equivalent reasoning as that given for V k holds for W q . Then, we conclude that

the same construction applies to the case where k and q correspond to WR , and that is how
it can be ignored in the integral equation.
The integral equation is defined naturally on the loop space LΣ; a loop based at xR on
the border ∂Σ corresponds to a point in LΣ, and the surface Σ corresponds to a path. If we
change the reference point or de scanning the integral equation will transform “covariantly”,
i.e., both sides will change accordingly. A change on the scanning of the surface with loops
(i.e., if one chooses a new way of constructing the loops scanning Σ) will change the path in
LΣ but the physical surface remains the same.
Now, we verify that (3.1.1) is indeed the integral equation of Chern-Simons theory, by
showing that when Σ is an infinitesimal surface, the differential equation is recovered. So, let
Σ be an infinitesimal rectangle of sides∗ δx δy, the reference point being the origin of the
Cartesian system. Lets expand the l.h.s around this point. We remember that the quantity


H
P1 exp −ie ∂Σ Aµ dxµ is a solution of equation (2.1.1), with Cµ = ieAµ , along the curve

∂Σ. So, we can consider the “infinitesimal version” of this equation as W (σ + δσ) = W (σ) −

Cµ (σ)δxµ W (σ) and calculate it iteratively along the rectangle. For instance, when we go from
∗

Regardless the labels x and y, it does not mean we are in space; on the contrary, the surface is in space-time.

43

3.1 The integral equations of Chern-Simons theory

xR to δx we get W (xR +δx) = WR −Cµ (xR )δxµ WR . Next, W (xR +δx+δy) = W (xR +δx)−

Cµ (xR + δx)δy µ W (xR + δx), and we use the previous result and taylor expand the connection;

everything up to second order. Doing that, and paying attention to the signs that change
when we go in the negative direction of the axis, the result appears with not much difficulty:
H


P1 exp −ie ∂Σ Aµ dxµ ≈ 1l + ieFµν δxµ δy ν , no sums in µ, ν. For the r.h.s, it is direct, if we
 R

keep things up to second order: P2 exp ieκ Σ W −1 Jeµν W dxµ dxν ≈ 1l + ieκ Jeµν δxµ δy ν . This
came from the “infinitesimal version” of (2.2.1), V (τ + δτ ) = V (τ ) + V (τ )C(τ )δτ , using the
fact that we only need it at the reference point since if we go any further a higher order term

will appear in the Taylor expansion† . Of course, due to the infinitesimal character of Σ the
integration becomes the integrand times the area. Finally, we clearly see that the equality of
these expansions implies the Chern-Simons equation.
Our next step is to build up the conserved charges. In order to do so we consider the
surface to be closed, Σc , so that it has no border and as a consequence of the integral equation
we get
VΣc = P2 exp



ie
κ

Z

Σc

W

−1

Jeµν W dxµ dxν



= 1l,

(3.1.2)

where we emphasize the fact that this quantity is a solution of (2.2.1), with Bµν =

ie e
J .
κ µν

The surface Σc can be considered as formed by two other surfaces (see figure (3.1)),
Σc = Σ1 ◦ Σ−1
2 , and the intersection of them is a certain loop Γ. Then the above equation

implies VΣ1 = VΣ2 . This equation is expressing the fact that the Wilson surface V linking

the infinitesimal loop around xR to the loop Γ is surface independent, or, in loop space, path
independent. As we did in the 2-dimensional case we split space-time into space and time.
(0)

The surface Σ1 is composed by a purely spatial part at t = 0, the disk D∞ that extends from
the reference point to the whole space; and there is a second part which is the the boundary
(0)

1
1
∂D∞ = S∞
(a point in LM, and a loop in M) moving forward until a time t > 0: S∞
× R.

Thus, VΣ1 = VD∞
(0) · VS 1 ×R . The surface Σ2 is also composed by two parts. The first part is
∞

the infinitesimal loop around xR moving forward in time, SR1 × R, and the second part is the
(t)

whole space, D∞ . Thus VΣ2 = VSR1 ×R · VD∞
(t) . Then, surface independence implies
V D∞
(0) · VS 1 ×R = VS 1 ×R · V (t) .
∞
D∞
R

These quantities are calculated from (2.2.1), on each surface, scanned by loops based at xR .
(0)

Consider the scanning of Σ1 . We start by scanning D∞ with the loops staring and ending
1
at xR . Then, we have to scan the cylinder S∞
× R, which is at spatial infinity. Basically we

1
keep going around the same spatial circle S∞
, but at each lap we move forward in time, so
†

We use W ≈ (1l − Cµ (xR )δxµ ), its inverse and J ρ (xR + δx) ≈ J ρ (xR ) + ∂µ J ρ (xR )δxµ

3 Integral formulation of theories in 2 + 1 dimensions

44

Figure 3.1 –

The surface independence of V means that it can be calculated from the infinitesimal
1
loop around xR (the initial point in loop space) to the boundary loop S∞
(the final point
in loop space) using any of the two surfaces (paths in loop space) presented here.

there is always a leg linking xR with the circle above. The integration of (2.2.1) depends on
H
µ ∂xν
in τ , on the spatial surface, so it will depend
the integration of C = ieκ dσW −1Jeµν W ∂x
∂σ ∂τ
on the integration of J0 over the area of the disk. Then, if we set that at spatial infinity
p
1
1 ×R = VR . Also,
J0 ∼ r2+δ
, with δ > 0 and r = x21 + x22 , this integration will vanish, and VS∞

due to the fact that SR1 has an infinitesimal radius (it is just a point, basically), we have that
VSR1 ×R = VR . So, remembering that VR is in the centre of the group one gets
V D∞
(t) = V (0) ,
D∞
which relates the “spatial Wilson surfaces” at different times. But, that is not all. One has to
keep in mind that all Wilson surfaces are calculated by scanning the surface with a family of
loops based at the fixed reference point xR , in the border of space at t = 0. If we want a real

conservation law we need to be able to calculate the “spatial Wilson surface” from the reference
point at any given time slice, independently. So, we consider the reference point at time t, xtR ,
(t)

in the border of D∞ . Then, we can decompose every loop we use to scan Σ2 into three parts:
the “leg” going forward in time from xR to xtR , then the part that goes around the disk, and the
“leg” coming back from xtR to xR ; consequently the Wilson line will be decomposed as well. In
particular, when we are somewhere in the loop going around the disk, W (σ, xR ) = W (σ, xtR ) ·
R 2π
W (xtR , xR ), and we have C = 0 dσ W −1 (xtR , xR ) · W −1 (σ, xR )Bµν W (σ, xR ) · W (xtR , xR ) =
W −1 (xtR , xR )CW (xtR , xR ), where W (xtR , xR ) is calculated using (2.1.1). So, lets call now V t (t)
D∞

the Wilson surface on the disk
the Wilson surface on the disk

(t)
D∞

(t)
D∞

obtained from the reference point

xtR ,

and similarly, VD∞
(t)

obtained from the reference point xR . Then we see that

3.2 The integral equations of (2 + 1)-dimensional Yang-Mills theory

45

−1 t
integration of (2.2.4) gives, using C as above, V t (t) = W (xtR , xR )·VD∞
(xR , xR ). Finally,
(t) ·W
D∞

the Wilson surface independence implies in the iso-spectral evolution
−1 t
VDt (t) = W (xtR , xR ) · VD∞
(xR , xR ),
(0) · W

(3.1.3)

∞

and therefore the eigenvalues of the operator
 Z



I
ie
t
µ
−1 e
µ
ν
VD(t) = P2 exp
Aµ dx
W Jµν W dx dx = P1 exp −ie
1,(t)
(t)
∞
κ D∞
S∞

(3.1.4)

are the conserved charges.

As we discussed, under a gauge transformation this operator changes as V t (t) → h(xR ) ·
D∞

V t (t) h−1 (xR ), and therefore the charges, which are the eigenvalues, remain invariant. Also,
D∞

if one decides to change the scanning, i.e., the parametrisation of the surface, the path
independence in loop space (3.1.2) shows that the charge operator above remains the same.
Moreover, by changing the reference point xR to some other point on the border of the disk,
the charge operator will change by conjugation with the Wilson line joining the two points,
and clearly, the charges remain the same. Since the reference point is at the border of the
disk, at infinity, our boundary condition says that Fµν ∼ 0, and therefore Aµ becomes flat, so

W joining the two different reference points is independent of the curve we choose.

3.2

The integral equations of (2 + 1)-dimensional YangMills theory

We start by choosing the connection Cµ in terms of the physical fields, in this case, the


Yang-Mills field Aµ : Cµ = ie Aµ + β Feµ , where Feµ = 12 ǫµνρ F νρ is the Hodge dual of the field
strength Fµν = ∂µ Aν − ∂ν Aµ + ie [Aµ , Aν ], and β ∈ C is an arbitrary parameter. Notice that

this choice is compatible with the fact that Cµ is a connection, i.e., given that under a gauge
transformation Aµ → hAµ h−1 + ei ∂µ h h−1 , and Fµν → hFµν h−1 , the combination above

implies that Cµ → hCµ h−1 − ∂µ h h−1 , as expected. Now, in the r.h.s of the Stokes theorem

we have the curvature of Cµ , Gµν . The differential equation of Yang-Mills read Dν F νµ = J µ
and Dµ Feµ = 0, where Dµ ⋆ = ∂µ ⋆ + [Aµ , ⋆] and Jµ is the matter current. Thus, calculating
h

i
Gµν and using the differential equations one gets Gµν = ie Fµν − β Jeµν + ieβ 2 Feµ , Feν ,
and finally the integral equation of Yang-Mills theory reads
P1 e−ie

H

∂Σ

(Aµ +β Feµ )dxµ = P eie
2

R

Σ

W −1 (Fµν −β Jeµν +ieβ 2 [Feµ ,Feν ])W dxµ dxν

.

(3.2.1)

3 Integral formulation of theories in 2 + 1 dimensions

46

Now we need to check if when Σ is considered to be infinitesimal, the differential equations are recovered. Using the same reasoning as before we consider Σ to be an infinitesimal rectangular surface and expand each side of the integral equation on it, which gives:
h
i



H
µ
e
P1 e−ie ∂Σ (Aµ +β Fµ )dx ≈ 1l + ie Fµν + β Dµ Feν − Dν Feµ + ieβ 2 Feµ , Feν δxµ δy ν for the

h
i
R
µ
ν
2 e e
−1
e
l.h.s and, P2 eie Σ W (Fµν −β Jµν +ieβ [Fµ ,Fν ])W dx dx ≈ 1l + ie Fµν − β Jeµν + ieβ 2 Feµ , Feν
for the r.h.s. Clearly, equating both terms we get Dν F νµ = J µ .

The conserved charges are constructed in an analogous way as we did for the Chern-Simons
theory. For a closed surface we get that
VΣc = P2 eie

R

Σc

W −1 (Fµν −β Jeµν +ieβ 2 [Feµ ,Feν ])W dxµ dxν

= 1l,

(3.2.2)

expressing the surface independence of the Wilson surface joining the infinitesimal loop around
xR and SR1 , if we take Σc as before. The next step is to look for the iso-spectral evolution of
the “spatial Wilson surface”. The difference is that now the boundary conditions are defined
not just in terms of the matter current, but also in terms of the Yang-Mills field strength,
since in VΣc above both quantities appear. So, we consider J0 ∼

1
r 2+δ

with δ and ε positive, at r → ∞, then one gets the charge operator
ie

VDt (t) = P2 e
∞

R

(t)
D∞

W −1 (Fµν −β Jeµν +ieβ 2 [Feµ ,Feν ])W dxµ dxν

−ie

= P1 e

H

1,(t)
S∞

and also Fµν ∼

(Aµ +β Feµ )dxµ

1
,
r 2+ε

(3.2.3)

whose eigenvalues are the conserved charges, with the same nice features under gauge transformation and re-parametrisation, as discussed before.

47

CHAPTER 4

The integral Yang-Mills equation in
3 + 1 dimensions
Here we present the construction of the integral equations of Yang-Mills theory in 4dimensional space-time, and derive the conserved charges from our construction in loop space.
The generalised Stokes theorem is used, and also the standard one, in the case of the self-dual
sector. Moreover the standard Stokes theorem can appear as an identity given some conditions,
originating another operator whose eigenvalues are also conserved charges. We calculate
the charges for some configurations of the theory namely monopoles, dyons, instantons and
merons.

4.1

The full Yang-Mills integral equation

Let M be the 4-dimensional, simply-connected, Minkowski space-time manifold. Suppose
Ω to be a 3-dimensional sub-manifold in M, i.e., a volume. According to the ideas we discussed
before, the loop space we are interested in is that of the maps from the 2-sphere into M, so
that a volume such as Ω in M becomes a path in loop space and each point in this path in
loop space is a surface in M. In order to proceed like this we define a reference point xR
on the border of Ω, and there we build an infinitesimal closed surface ΣR , which is changed
continuously (picture someone blowing a balloon from xR ) until reaches the border of Ω.
Each of these surfaces is scanned by a family of loops, also based at xR . On the surface
we calculate the Wilson surface defined by (2.2.1), and the Wilson lines along the loops
are calculated from (2.1.1).We shall construct the integral equations of Yang-Mills from the
generalised Stokes theorem (2.2.6) and the differential equations Dν F νµ = J µ , Dν Feνµ = 0,

where Jµ stands for the matter current and Feµν = 21 ǫµνρλ F ρλ is the Hodge dual of the field
R 2π
strength. For convenience we introduce the notation K = 0 dτ V J V −1 , in K of (2.2.4).

The first step is to define C and J in terms of the Yang-Mills field. We set Bµν in (2.2.1) as
Bµν = αFµν + β Feµν , where α and β are arbitrary constants in C. Then it is not difficult to

4 The integral Yang-Mills equation in 3 + 1 dimensions

48

see that this choice and the differential equations give us

Z 2π
 ∂xµ ∂xν ∂xλ

J =
dσ W −1 ieβ Jeµνλ W
∂σ ∂τ ∂ζ
0
Z σ
h

i
W
W
W
W
+
dσ ′ (α − 1) Fκλ
(σ ′ ) + β Feκλ
(σ ′ ), αFµν
(σ) + β Feµν
(σ) ×
0


∂xλ ′ ∂xν
∂xκ ∂xµ ∂xλ ′ ∂xν
(σ )
(σ) −
(σ )
(σ)
.
×
∂σ ′ ∂σ
∂τ
∂ζ
∂ζ
∂τ
where J ρ =

1 ρµνλ e
ǫ
Jµνλ .
3!

Now we can plug that Bµν in C, and get the l.h.s of Stokes theorem

(2.2.6), and that J into K to get the r.h.s. With the integration constants in the centre of the

group, following the arguments given before, we write down the integral Yang-Mills equation:


Z
 Z


−1
µ
ν
−1
e
. (4.1.1)
dζ dτ V J V
αFµν + β Fµν W dx dx = P3 exp
P2 exp ie
W
Ω

∂Ω

Our next step is to show that when Ω is an infinitesimal cube of sides δx, δy and δz, the

differential equations are recovered. We start by letting the reference point to be at one of the
corners of the cube (see figure (4.1)), such that the sides form a Cartesian frame. We need to
evaluate each side of the integral equation above on that cube, up to the first non-trivial order.
The l.h.s of the integral equation is defined on the surface of the (closed) cube. This surface
is scanned with loops, based at xR . The bottom plane of the cube is “scanned” with one loop,
going around the border and coming back to xR . The signs of

dxµ
dσ

and

dxµ
dτ

are positive. Of

course, the cube being infinitesimal so that only one loop is required, makes this harder to see.
So, for the sake of visualisation we imagine this cube as a finite surface. Then,
means that we start from the reference point going to the border, while for

dxµ
dτ

dxµ
dτ

positive

negative we

start from the border, going to the reference point. After “scanning” the infinitesimal bottom
surface we move along a straight line in the δz direction, until reach the top surface. Then,
µ

, going
we continue moving around the border of that surface, but now with a negative dx
 R

 dτ

−1
µ
ν
e
back to the reference point. So, we evaluate P2 exp ie ∂Ω W
αFµν + β Fµν W dx dx

in two parts: at the bottom surface and at the top, and then, we repeat this calculation for the

other 2 pairs of surfaces of the cube. At the bottom surface we calculate the integrand at the
reference point. If we take it at any other point then the contributions will be of higher order;

notice that W (xR +δx) ∼ 1l−ieAµ (xR )δxµ , and Fµν (xR +δx) ∼ F (xR )+∂ρ Fµν (xR )δxρ , so, at
o




n 
≈ ie αFµν + β Feµν δxµ δy ν . For the
the bottom ie W −1 αFµν + β Feµν W δxµ δxν
xR

top surface we have to use these Taylor expansions for W and Fµν , in the direction δz µ . This
o


n

≈
calculation is straightforward and we get −ie W −1 αFµν + β Feµν W δxµ δxν
xR +δz




−ie αFµν + β Feµν δxµ δy ν − ie αDλ Fµν + βDλ Feµν δxµ δy ν δz λ . Thus, summing these
contributions the terms involving the area cancel each other and then summing the three

49

4.1 The full Yang-Mills integral equation

Figure 4.1 –

When the volume Ω becomes the infinitesimal cube the integral equations imply the
differential Yang-Mills equations. The big arrows on the bottom and top surfaces indicate
µ
the sign of dx
dτ .




e
pairs of surfaces we get −ie αDλFµν + βDλ Fµν + cyclic permutations δxµ δy ν δz λ . This is
the l.h.s of Stokes theorem. The r.h.s is trivial: 1l + iβ Jeµνλ δxµ δy ν δz λ . Clearly, the equality of

the results from each side of the equation implies the differential Yang-Mills equations. Now

we proceed to construct the conserved charges. Let Ω be a closed volume, so that ∂Ω = ∅,
and the integral equation (3.2.1) becomes
Z
VΩc = P3 exp

Ωc

dζ dτ V J V

−1



= 1l,

(4.1.2)

which express the fact that the operator VΩc is independent of the paths in loop space, which
2,(0)

in space-time are the volumes joining the infinitesimal closed surface SR

and the 2-sphere

2,(t)

S∞ . We now proceed to explain how to decompose Ωc getting the surfaces joining these
2-spheres such that the generalisation of the “spatial Wilson surface” can be obtained. Take
Ωc = Ω−1
2 ◦ Ω1 , where Ω1 and Ω2 are also composed by the union of two other volumes each,

as we now explain. The volume Ω1 links two surfaces (its borders): the infinitesimal 2-sphere
2,(0)

around the reference point SR

2,(t)

, and the 2-sphere at spatial infinity at time t > 0, S∞ .

We then have Ω1 = Ω∞ ◦ Ω0 , where Ω0 is the volume (or path in loop space) at t = 0
2,(0)

that consists of the whole space from SR
2,(0)

border, so it goes from S∞

2,(t)

2,(0)

to S∞ , and Ω∞ is the “time evolution” of the

to S∞ . The volume Ω2 is made of Ωt ◦ Ω−∞ , where Ω−∞ is
2,(0)

the “time evolution” of the border S0

, the 2-sphere around the reference point, from t = 0

2,(t)

2,(t)

to t > 0, where we call it SR , and Ωt stands for the volume linking S0
2,(t)

to the border

S∞ . The Wilson volume VΩ is calculated on each of these volumes from (2.2.4). Using the
decomposition explained above equation (4.1.2) is written as
VΩ−1
· VΩ−1
· VΩ∞ · VΩ0 = 1l
t
−∞

4 The integral Yang-Mills equation in 3 + 1 dimensions

50

and now our first task is to find appropriate boundary conditions that make this an iso-spectral
evolution equation for the “spatial Wilson volume” VΩt . Although it must be clear by now,
we remember that as in the previously discussed cases the idea is to find conditions over
the “connection” (Cµ in 2-dimensions, C in 3-dimensions and K now) such that the Wilson

operators related to the boundaries can be made equal. Here we must analyse VΩ±∞ . The
R 2π
“connection” appearing in (2.2.4) is K = 0 dτ V J V −1 , and it is calculated on the surfaces
scanning the volumes. The surfaces for both borders, i.e., around the reference point and at
spatial infinity, can be described topologically as cylinders; the first being SR2 × R while the

2
second S∞
× R. In order to scan the first surface one starts at the reference point and go

around the sphere SR2 , with a set of loops, until scan it all. Then, move forward in time to the

next point in R, and scan the sphere again, so that when it is done the opposite vertical path
in R is taken to come back to the reference point. This goes on until a certain time t. Since
the tiny sphere in the first surface has an infinitesimal radius it gives no contribution to K. In

fact we could split this quantity into K = K↑ + K◦ + K↓ , where K↑ and K↓ correspond to the
integrations in τ when we go up and down along R, while K◦ is the “connection” calculated

on the sphere. Thus because of the tiny radius of the sphere K◦ = 0 and we get K = K↑ + K↓ .
Since the sign of

dxµ
dτ

in K↑ is the opposite of that in K↓ , but the integrands are the same, then

K vanishes and the Wilson volume is the integration constant VR ∈ Z(G). Now for the other

border we scan in the same way, except that now the sphere is huge, with an infinite radius.
It is not difficult to see that the same things happen for K↑ and K↓ . It is now our job to find
the appropriate boundary conditions to get K◦ = 0 so that the Wilson volumes on the borders

become the same. Indeed, keeping in mind that K◦ is calculated on the sphere at spatial infinity,

if we assume that at spatial infinity Jµ ∼

1
R2+δ

and

Fµν ∼

1

3

R 2 +δ

with δ, δ ′ > 0 one

′

−1
2 ×R , so V (t) = VS 2 ×R ·V (0) ·V 2
.
gets exactly the desired behaviour for K◦ and VSR2 ×R = VS∞
∞
S ×R
Ω∞
Ω∞
R

As before we have to find a way to calculate the “spatial Wilson volume” at each slice of
time independently; in the above equation everything is calculated from the reference point at
time t = 0. Following the same argumentations already given it is not very hard to see that K at

the reference point at t > 0 is related to that at t = 0 by KxtR = W (xtR , xR )KxR W −1 (xtR , xR ),
·
and therefore the Wilson volume satisfies an identical relation: V t (t) = W (xtR , xR ) · VΩ(t)
∞
Ω∞

W −1 (xtR , xR ). Finally we find the desired iso-spectral evolution
VΩt (t) = U(t) · VΩ(0)
· U −1 (t)
∞

(4.1.3)

∞

with U(t) = W (xtR , xR ) · VSR2 ×R . So, we have found that the eigenvalues of the operator
Q ≡ P2 eie

R

∂S

W −1 (αFµν +β Feµν )W dxµ dxν

R

= P3 e

S

dζdτ V J V −1

,

(4.1.4)

51

4.2 The self-dual sector

S being the spatial surface at a given time slice, are the conserved charges, satisfying all good
properties discussed before: they are invariant under gauge transformations, re-parametrisation
and change of the reference point.
Once the volume independence equation (4.1.2) is obtained we proceed to split the closed
volume Ωc into two volumes, sharing the same boundaries. Basically, we look at that in loop
2,(0)

space as two paths that can be deformed into each other linking two fixed points, SR
2,(t)
S∞ .

and

That these points are fixed means that not just the physical surfaces remain the same

but also if one changes their parametrisation, nothing will happen in the loop space. The
2,(0)

scanning of SR

2,(t)

is trivial, so there is nothing to worry about. For S∞ , we must analyse

what happens with the Wilson surface, defined by (2.2.1). Equation (2.2.3) gives how it
changes under a general variation of the space-time points. A re-parametrisation consists
of variations that are parallel to the surface, then the first term, involving the 3-form with
components (Dλ Bµν + cyclic parmutations) on a 2-dimensional surface vanishes. In order to
guarantee the re-parametrisation independence one needs
Z 2π
Z σ
h

i
W
W
W
W
dσ
dσ ′ (α − 1) Fκλ
(σ ′ ) + β Feκλ
(σ ′ ), αFµν
(σ) + β Feµν
(σ) ×
0
0


ν
∂xκ ∂xµ ∂xλ ′ ν
λ
′ ∂x
(σ )δx (σ) − δx (σ )
(σ) = 0.
× ′
∂σ ∂σ
∂τ
∂τ
This integration is performed over a loop at spatial infinity, so that the accomplishment of
this condition is determined by the behaviour of the field strength ar r → ∞. There are at

least two sufficient conditions that make it possible to get re-parametrisation invariance: (i)
for r → ∞ the field strength behaves like Fµν →

1
r2

and (ii) the commutator appearing above

W
vanishes, i.e., the quantities Fµν
lie in an Abelian sub-algebra. As we discuss later, the first

condition is exactly the case of instantons solutions while monopoles dyons and merons fulfil
the second condition.
An analogous argumentation holds for the the charges in 3-dimensional theories. However,
1
there it is trivial since the boundary is a circle S∞
and the re-parametrisations (variations

tangent to the circle) imply an overall factor multiplying the equation (2.1.1), which changes
nothing.

4.2

The self-dual sector

We are interested here in the Euclidean space-time where the Yang-Mills field strength
satisfies the so called self-duality equation Fµν = κFeµν , with κ = ±1. We now show that

using the standard non-Abelian Stokes theorem (2.1.8) we formulate the integral equation

4 The integral Yang-Mills equation in 3 + 1 dimensions

52

of self-dual Yang-Mills. Let Cµ = ieAµ , where Aµ is the Yang-Mills field. Consider the


e
quantity Hµν = ie αFµν + κ (1 − α) Fµν , with α an arbitrary parameter. If one writes it
 


as Hµν = ie α Fµν − κFeµν + κFeµν then it becomes evident that when the self-duality

equation holds, Hµν = ieFµν , coinciding with Gµν = ieFµν , the curvature of Cµ given above.
This also happens trivially in the case α = 1, when we get just an identity. The integral

self-dual Yang-Mills equation is (the integration constants are in the centre of the group)


Z

I


µ
−1
µ
ν
e
P1 exp −ie
Aµ dx = P2 exp
W
αFµν + κ (1 − α) Fµν W dx dx , (4.2.1)
∂Σ

Σ

for when Σ is an infinitesimal rectangle with sides δx, δy one gets (the calculations are


H
done identically to those we discussed before) P1 exp −ie ∂Σ Aµ dxµ ≈ 1l + ieFµν δxµ δy ν



R


and P2 exp Σ W −1 αFµν + κ (1 − α) Feµν W dxµ dxν ≈ 1l + ie αFµν + κ (1 − α) Feµν ,
which implies the differential equations, when α = 1 is excluded. The 2-dimensional surface

Σ is a sub-manifold of the 4-dimensional Euclidean space-time M, and every calculation here
is identical to what was done in the Chern-Simons or 3-dimensional Yang-Mills case. Also,

the construction
of the conserved charges is similar. The surface independence equation reads
R
−1 αF
( µν +κ(1−α)Feµν )W dxµ dxν
(t) W
= 1l, obtained from the integral equation (4.2.1)
V Σc = P 2 e D ∞
1
for r → ∞. Finally the
when Σ is closed. Now the boundary condition is Fµν = κFeµν ∼ 2+δ
r

charge operator reads

R

VDt (t) = P2 e

(t)
D∞

∞

W −1 (αFµν +κ(1−α)Feµν )W dxµ dxν

−ie

= P1 e

H

1,(t)
S∞

Aµ dxµ

.

(4.2.2)

There is more to it. If we consider Σc to be the boundary of a volume Ω, then we can
R

use equation (2.2.6) to get P3 e

Ω

dζdτ V J V −1
R

= 1l. This can be simplified once the self-duality

W V −1 dxµ dxν dxρ
ieκ(1−α) Ω V Jeµνρ

equation holds, giving P3 e

= 1l, which is valid for any volume Ω,

and therefore one concludes that Jµ vanishes. Well, this is expected when the self-duality
equations are introduced in the Yang-Mills equations.
Suppose Bµν = ieλFµν in the generalised non-Abelian Stokes theorem, where F is the
curvature of the 1-form Lie algebra-valued connection A. Then the first term of K in the r.h.s
of the theorem vanishes, since it becomes just the Bianchi identity for A. What remains in
P2 eieλ

R

∂Ω

W dxµ dxν
Fµν

R

= P3 e

Ω

dζdτ V J V −1

is
J

Z

2π

Z

σ
 W ′

W
= e λ (λ − 1)
dσ
dσ ′ Fκρ
(σ ), Fµν
(σ) ×
0
0

∂xκ ∂xµ ∂xρ ′ ∂xν
∂xρ ′ ∂xν
× ′
(σ )
(σ) −
(σ )
(σ) ,
∂σ ∂σ
∂τ
∂ζ
∂ζ
∂τ
2

(4.2.3)

53

4.3 Monopoles and dyons

which is quite non-trivial for λ 6= 0, 1; it says that the “flux” of the rescaled field strength (on
the l.h.s) depends on that commutator term, and therefore is apparently non-zero even when
there is no matter current. This fact deserves further investigation.
As a final remark we notice that with the self-duality condition the full Yang-Mills integral
equation (4.1.1) becomes the integral Bianchi identity (4.2.3) with λ = α + κβ.

4.3

Monopoles and dyons

We now want to evaluate the operator (4.1.4) for the monopole field given by
1
n
ˆj
1 i
Ai = − ǫijk
Tk =
∂i g g −1 ;
e
r
2e
nk
1
ǫijk 2 n
ˆ·T ;
Fij =
e
r
where ni =

xi
r

A0 = 0
F0i = 0

(4.3.1)

is unit vector in the radial direction, r 2 = x21 + x22 + x23 defines the radial

distance in the Cartesian system, Ti are the generators of the SU(2) Lie algebra satisfying
[Ti , Tj ] = i ǫijk Tk

i, j, k = 1, 2, 3

(4.3.2)

and g is the group element g = exp (i π n
ˆ · T ). The field (4.3.1) corresponds to the exact

Wu-Yang (36) configuration and also to the ’tHooft-Polyakov (37) configuration∗ at r → ∞.
Lets first consider the operator (4.1.4) as the Wilson surface, defined by the surface-

ordered exponential of the field strength and its dual. This quantity is then calculated on the
surface of the 2-sphere at spatial infinity (and that is why our calculations hold equally for the
Wu-Yang and ’tHooft-Polyakov case), by scanning it with loops based at the reference point
xR . The Wilson line W , appearing inside the integral, is calculated from (2.1.1) integrating
from the reference point to a given point on a certain loop. One then notice that the term
n
ˆ · T appearing in the definition of the field strength is covariantly constant, i.e., Di (ˆ
n · T) =

∂i (ˆ
n · T ) + ie [Ai , (ˆ
n · T )] = 0, which implies that

d
dσ

(W −1 n
ˆ · T W ) = 0, and therefore

W −1 n
ˆ · T W is constant along any loop, and since the sphere is covered with loops, this is
2
constant everywhere on S∞
. So, we can choose any point to perform the integration. Lets take

it to be done at xR . The “conjugate field strength” there is then W −1 Fij W ≡

1
ǫ nk
e ijk r 2

TR ,

where we denote n
ˆ · T at xR by TR . Notice that since we bring everything to the reference

point the surface-ordering becomes irrelevant; in fact, the “conjugate field strength” is now
in the Abelian sub-algebra of U(1) generated by TR . Introducing the Abelian magnetic field
BiR = − 12 ǫijk W −1 Fjk W = − 1e nr2i TR it is easy to see that the charge operator becomes the
∗

The Higgs field behaviour is not presented since it has no relevance for the following discussion.

4 The integral Yang-Mills equation in 3 + 1 dimensions

54

standard exponential
−ieα

R

dS·BR

= ei 4παTR .
(4.3.3)
R
It is more convenient to define the quantity GR ≡ S 2 dS · BR = − 4π
T . Thus, according
e R
Q=e

2
S∞

∞

to our construction, the eigenvalues of GR are the conserved magnetic charges. Choosing a

finite dimensional representation of SU(2) or SO(3) they are integers or half-integers. So, it
follows that the magnetic charge must be quantised as integer multiples of

2π
.
e

Now we can calculate the operator Q in (4.1.4) as a volume-ordered exponential
R

e−ieαGR = P3 e

space

dζdτ V Jmonopole V −1

,

(4.3.4)

where†
Z

2π

Z

σ

 W ′
W
Jmonopole = e α (α − 1)
dσ
(σ ), Fµν
(σ) ×
dσ ′ Fκρ
0
0

∂xρ ′ ∂xν
∂xκ ∂xµ ∂xρ ′ ∂xν
(σ )
(σ) −
(σ )
(σ) ,
× ′
∂σ ∂σ
∂τ
∂ζ
∂ζ
∂τ
2

(4.3.5)

The ’tHooft-Poliyakov monopole has a core, and inside it the result presented before for the
magnetic charge does not hold. We shall not discuss this here. We remark though that the
Higgs field does not appear in our formula for the magnetic charge.
For the Wu-Yang monopole we can evaluate that integral over the entire space. In doing
that we inevitably face the problem of passing through a singularity point of the gauge field.
This is solved by a regularisation process of the Wilson line. For the sake of the continuation of
the discussion we leave the calculations to the appendix A; the result is that Jmonopole vanishes
everywhere and therefore

R

e−i e α GR = P3 e

space

dζdτ V Jmonopole V −1

= 1l.

(4.3.6)

This result implies that the magnetic charge for the Wu-Yang monopole is quantised as
eigenvalues of GR =

2πn
eα

n = 0, ±1, ±2 . . .

(4.3.7)

If the parameter α is indeed arbitrary, and there is no physical condition to fix it, then the
only acceptable value for the integer n is n = 0, and so the magnetic charge of the Wu-Yang
monopole should vanish. Perhaps we have to go to the quantum theory to settle that issue.
It might happen that quantum conditions restrict the allowed values of α. That is one of the
important points of our construction to be further investigated.
†

In the case of the ’tHooft-Poliyakov monopole, the static solution of the Higgs field requires A0 = 0, which
implies J0 = 0, and that is why no current appears in the formula of J also in this case; the Wu-Yang
monopole has no current by definition.

55

4.3 Monopoles and dyons

Let us now consider the case of dyon solutions. For the Wu-Yang and the ’tHooft-Polyakov
case, as calculated by Julia and Zee (38), the space components of the gauge potential and
field tensor, namely Ai and Fij , i, j = 1, 2, 3, are the same as those in (4.3.1), and the time
components, at spatial infinity, are replaced by
A0 =

M
γ n
ˆ·T
1
n
ˆ·T +
+ O( 2 ) ;
e
e r
r

F0i =

γ ni
1
n
ˆ · T + O( 3 ) ;
2
e r
r

r→∞

(4.3.8)

with M and γ being parameters of the solution. In the case of the Wu-Yang dyon, i.e. when
there is no Higgs field and no symmetry breaking, the formulas (4.3.8), as well as (4.3.1), are
true everywhere and not only at spatial infinity. In other words, there are no terms of order r −2
and r −3 in A0 and F0i respectively. Concerning the quantities entering in the charge operator
here we have

nk
γ
W −1 Feij W → − εijk 2 TR
e
r

r→∞

(4.3.9)

So, W −1 Feij W also belongs to the abelian subalgebra U(1) generated by TR , and it is in fact

proportional to W −1 Fij W . Therefore, the surface ordering is not relevant in the evaluation
of the operator (4.1.4), and we get in the dyon case that
h R
i
R
~ B
~ R +β
~ E
~R
−i e α S 2 dΣ·
dΣ·
S2

QS = e

∞

∞

= e−i e [α GR +β KR ] = ei 4 π [α−β γ] TR

where we have introduced the abelian electric field EiR = W −1 F0i W =

γ ni
e r2

are the same as before, and using Gauss law we have defined
Z
4πγ
~ ·E
~ R = KR
TR
dΣ
and so
KR =
e
2
S∞

(4.3.10)

~ R and GR
TR , B

(4.3.11)

According to (4.1.4) the eigenvalues of QS are constant in time, and so we conclude from
(4.3.10) that the eigenvalues of (α GR + β KR ) are constants. But if we assume that the
parameters α and β are arbitrary it follows that the eigenvalues of GR and KR are independently
constant in time. We have seen that, by evaluating the eigenvalues of TR on finite dimensional
representations of the gauge group SU(2) or SO(3), where they are integers or half-integers,
the magnetic charges, eigenvalues of GR , are quantised. Under the same assumptions it follows
that

2πγ n
n = 0, ±1, ±2, . . .
(4.3.12)
e
Again from (4.1.4) we can express the charges in terms of volume ordered integrals:
eigenvalues of KR =

R

e−i e [α GR +β KR ] = P3 e

space

dζdτ V Jdyon V −1

(4.3.13)

4 The integral Yang-Mills equation in 3 + 1 dimensions

56

with
Jdyon ≡

Z

2π
0

Z



dxi dxj dxk
dσ dτ dζ

W
dσ ieβ Jeijk




i
W
′
W
W
e
e
+ e
dσ
(α −
+ β Fij (σ ) , αFkl + β Fkl (σ)
0


d xi d xk d xj (σ ′ ) d xl (σ) d xj (σ ′ ) d xl (σ)
×
(4.3.14)
−
d σ′ d σ
dτ
dζ
dζ
dτ
2

σ

′

h

1) FijW

In the case of the Wu-Yang dyon we have Je123 = J0 = 0, since there are no sources. However,

for the Julia-Zee dyon we have that the Higgs field contributes to the current Jµ . One can

extract the operators GR and KR from (4.3.13), by setting β = 0 and α = 0 respectively.
Then, the Higgs field contributes to the electric charge only.
In the case of the Wu-Yang dyon it is possible to evaluate (4.3.14) after a regularisation of
the Wilson line operator passing through the singularity of the gauge potential (4.3.1). That
calculation is shown in the appendix A and it was found that Jdyon vanishes in all loops for

the Wu-Yang dyon. Therefore

R

e−i e [α GR +β KR ] = P3 e

space

dζdτ V Jdyon V −1

= 1l

(4.3.15)

which implies that
eigenvalues of [α GR + β KR ] =

2πn
e

n = 0, ±1, ±2, . . .

(4.3.16)

Again, if the parameters α and β are indeed arbitrary, then it follows from (4.3.16) that by
taking β = 0, the eigenvalues of GR should obey (4.3.7). On the other hand by taking α = 0,
one concludes that (4.3.16) implies that
2πn
eβ

eigenvalues of KR =

n = 0, ±1, ±2 . . .

(4.3.17)

Now, if (4.3.7) and (4.3.17) should hold true for arbitrary values of α and β respectively, then
the only acceptable value of the integer n in both equations is n = 0, and consequently the
electric and magnetic charges of the Wu-Yang dyon should vanish. As discussed below (4.3.7),
we have perhaps to consider of the quantum theory to settle that issue, since there could be
quantum conditions restricting the values of α and β.
Suppose now that we consider the standard non-Abelian Stokes theorem (2.1.8) where Σ
is a closed surface, boundary of the volume Ω. Then, we have the identity (with Cµ = ieAµ )
P2 eie

R

∂Ω

W −1 Fµν W dxµ dxν

= 1l.

57

4.3 Monopoles and dyons

For any configuration satisfying Fµν ∼

1
,
r 2+δ

with δ > 0 at r → ∞ one can build up, using

this identity, the iso-spectral evolution of the operator
Vx′(t)
R

(t)
D∞
(t)




ie

= P2 e

R

(t)
D∞

µ dxν
dτ

dσdτ W −1 Fµν W dx
dσ

−ie

= P1 e

H

1,(t)
S∞

µ

dσ Aµ dx
dσ

.

(4.3.18)

Note Rthat if D∞ is a spatial surface then the surface ordered integral in (4.3.18), namely
ie

P2 e

(t)
D∞

µ dxν
dτ

dσdτ W −1 Fµν W dx
dσ

, corresponds to the flux of the non-abelian magnetic field (Bi ≡
(t)

− 21 εijk Fjk ) through that surface. On the other hand, if D∞ has a time component then that

integral corresponds to the flux of the non-abelian electric field (Ei ≡ F0i ) through such spatialtemporal surface. Note that the conservation of those fluxes can be intuitively understood by
(t)

1,(t)

the fact that the border of D∞ is the circle S∞

of infinite radius. Therefore, if the field
1,(t)

configuration is localized in a region at a finite distance to the plane containing S∞

the

solid angle defined by that circle is 2 π spheroradians. If that field configuration evolves in the
time t changing its distance to that plane by a finite amount, the solid angle will remain the
same, and so should the flux of the magnetic or electric fields. Of course, that is an intuitive
view, and so not precise, of the conservation of the charge, but we will show that it stands
reasonable in the examples we discuss here.
It is worth evaluating the conserved charges associated to the operator (4.3.18) in the
case of the monopole and dyon solutions. For simplicity we shall take the circle of infinite
1,(t)

radius S∞

to lie on the plane x1 x2 , for some constant values of x3 and x0 . The calculation

for any other plane is similar and leads, as we shall see, to similar results. We use polar
coordinates on the plane, with the polar angle being the parameter σ parametrising the circle,
2

i.e. x1 = ρ cos σ, x2 = ρ sin σ, and r 2 = ρ2 + (x3 ) , with x3 constant, and ρ → ∞.

Therefore, for both the monopole and dyon solutions, we get from (4.3.1) that on the circle
of infinite radius we have Aµ

dxµ
dσ

= 1e T3 , since on that circle ρ ∼ r → ∞ and the unit vector

n
ˆ , on that limit, has components only on the plane x1 x2 . Therefore, from (4.3.18), we have
Vx′(t)
R

(t)
D∞




−ie

= P1 e

H

1,(t)
S∞

µ

dσ Aµ dx
dσ

= e−i 2 π T3

(4.3.19)

The eigenvalues of (4.3.18) are conserved in the time t which in this case can be any
linear combination of x0 and x3 . But that is equivalent to say that the eigenvalues of T3 are
conserved in t. Since those are integers or half integers in a finite dimensional representation


(t)
of SU(2), it follows that the operator V ′(t) D∞ is either 1l or −1l, i.e. an element of the
xR

centre of SU(2). As pointed out below (4.3.18) such conserved charge can be interpreted as
(t)

1,(t)

the non-abelian magnetic flux through the surface D∞ , which border is S∞ . Indeed, we see
that the argument of the exponential in (4.3.19) is half of the argument of the exponential

4 The integral Yang-Mills equation in 3 + 1 dimensions

58

in (4.3.3), if one takes α = 1, and considers that TR and T3 have the same norm and
so the same eigenvalues. Remember TR is the value of n
ˆ · T at the reference point xR

and so Tr (TR )2 = Tr (Ti Tj ) ni nj = λ, where Tr (Ti Tj ) = λ δij , and λ depends upon the
representation used. Since the argument of the exponential in (4.3.3) corresponds to the total
(t)

2
flux of the magnetic field through S∞
, we see that it is the double of the flux through D∞ .

Due to the spherical symmetry of the solution that is compatible with interpretation, given
2
corresponds to a solid angle of 4 π spheroidal as seen from the centre
below (4.3.18), since S∞
(t)

of the solution and D∞ corresponds to only 2π steradians.

There are several important comments regarding our charges for monopoles and dyons.
First of all we recall that usually the charges associated to the Yang-Mills theory are defined
from the current j µ = ∂ν F νµ and its dual, and with appropriate boundary conditions are writes
as
QYM =

Z

2
S∞

eYM =
Q

dS · E

Z

2
S∞

dS · B

(4.3.20)

where Ei = F0i and Bi = − 12 ǫijk Fjk are the non-Abelian electric and magnetic fields.
The charges we constructed are different from those given by (4.3.20). Indeed, from
(4.3.1) and (4.3.8) we have that the magnetic and electric fields at spatial infinity for the
Wu-Yang and ’tHooft-Polyakov cases are given by
Bi → −

1 ni
n
ˆ·T ;
e r2

Ei →

γ ni
n
ˆ·T ;
e r2

r→∞

(4.3.21)

So, they do not lie on an abelian U(1) subalgebra like BiR and EiR given above, and when
integrated on the two-sphere at infinity lead to the vanishing of the charges (4.3.20), i.e.
monopole/dyon

QY M

monopole/dyon

e
=Q
YM

= 0.

(4.3.22)

Note that even though the evaluation of the charges (4.3.3) and (4.3.10) rely on the choice
of a reference point xR , which leads to a particular generator TR , the charges do not depend
upon that reference point. Indeed, if one changes the reference point from xR to x
eR , then

the operator QS changes as QS → W −1 (e
xR , xR ) · QS · W (e
xR , xR ) where W (e
xR , xR ) is the
holonomy from the old reference point xR to the new one x
eR . Therefore the charges, which
are the eigenvalues of QS , do not change.

Also, under gauge transformations g ∈ G the charges defined by (4.3.20) become
Z
Z
−1
e
QYM →
dS · gEg
QYM →
dS · gBg −1,
2
S∞

2
S∞

and the eigenvalues of them remain invariant only under gauge transformations that go to a

4.3 Monopoles and dyons

59

−1 e
eYM g −1. The charges we constructed
constant g∞ at infinity: QYM → g∞ QYM g∞
, QYM → g∞ Q
∞

here are invariant under general gauge transformations instead, and not just to these restricted
ones.

Note that since the charges are the eigenvalues of the operator (4.1.4), the number of
charges is equal to the rank of the gauge group G. However, since the field tensor and its
Hodge dual come multiplied by the arbitrary parameters α and β respectively, the number
of charges is in fact twice the rank of the gauge group. So, we have rank of G magnetic
charges and rank of G electric charges. In this sense the number of charges does not pay
attention to the pattern of symmetry breaking. Indeed, our calculations have shown that the
electric and magnetic charges are the same for the Wu-Yang case, which is a solution of the
pure Yang-Mills theory, and for the ’tHooft-Polyakov case which has a Higgs field breaking
the gauge symmetry from SO(3) down to SO(2). In fact, as we have shown above the Higgs
field does not play any role in the evaluation of the charges. In addition the conservation of
the charges is dynamical, i.e. it follows directly from the integral form of the equations of
motion (4.1.1). That contrasts to the conservation of the magnetic charge of the ’tHooftPolyakov monopole which follows from topological (homotopy) considerations related to the
mapping of the Higgs field from the spatial infinity to the Higgs vacua. Another point relates
to the quantisation of the magnetic charge, which in the case of ’tHooft-Polyakov monopole
comes from the topology again, i.e. the charge is determined by second homotopy group of the
Higgs vacua. In our case, the quantization of the charges comes from the integral equations of
motion themselves, without any reference to the Higgs field since it works equally well for the
Wu-Yang and ’tHooft-Polyakov monopoles. It is worth pointing out that the magnetic charges
of monopoles of ’tHooft-Polyakov type have already been expressed in the literature, as surface
ordered integral using the ordinary non-abelian Stokes theorem. See for instance section 5 of
Goddard and Olive’s review paper (39). However, that construction is totally based on the
properties of the Higgs vacua, since the fact that the Higgs field must be covariantly constant
at spatial infinity leads to an equation for it similar to (2.1.1) for the Wilson line W . In addition
the argument for the conservation of the magnetic charge is particular to that type of solution
since it is based on topology considerations of the solution. The generalised non-Abelian
Stokes theorem (2.2.6), and consequently the integral Yang-Mills equations (4.1.1) were not
known by that time. We believe that the role played by the integral Yang-Mills equation in
monopole and dyon solutions deserves further study specially in the quantum theory. It might
connect to the so-called Abelian projection and arguments for confinement.

4 The integral Yang-Mills equation in 3 + 1 dimensions

60

4.4

Instantons

The instantons are euclidean self-dual solutions where the gauge potentials become of the
pure gauge form at infinity, i.e. Aµ →

1
e

∂µ g g −1 , for s → ∞, with s2 = x21 + x22 + x23 + x24 ,

where xµ , µ = 1, 2, 3, 4, being the Cartesian coordinates in the Euclidean space-time.

Let us consider the case where the gauge group is SU(2) and take the one-instanton
solution (4) given by
2
xν − aν
Aµ = − σµν
;
e
(xρ − aρ )2 + λ2

Fµν = κ Feµν =

4 λ2
σµν

 (4.4.1)
e (xρ − aρ )2 + λ2 2

with κ = ±1, and where aµ , µ = 1, 2, 3, 4, and λ are parameters of the solution, and
σi4 = −κ Ti ;

σij = εijk Tk ;

[Ti , Tj ] = i εijk Tk

(4.4.2)

with i, j, k = 1, 2, 3, Ti being the generators of the SU(2) Lie algebra, and the quantities σµν
satisfy 12 εµναβ σαβ = κ σµν .
2
If one considers a 2-sphere S∞
of infinite radius surrounding the instanton then we have

that the integrand in the surface ordered integral in (4.1.4) behaves as
(α + κ β) Fµν

∂xµ ∂xν
1
→ 2
∂σ ∂τ
r

as

r→∞

(4.4.3)

2
2
= 1l, and
where r is the radius of S∞
. Therefore, the operator (4.1.4) is unity, i.e. QS∞

that unity comes from the exponentiation of the trivial element of the Lie algebra. So, the
one-instanton solution have vanishing charges associated to (4.1.4) .

Let us now evaluated the charges associated to the operator (4.2.2). Without any loss of
1,(t)

generality take the circle S∞

of infinite radius to lie on the plane x1 x2 , at some constant

values of x3 and x4 . Due to the symmetries of the one-instanton solution the calculation on
any other plane is very similar. We shall use polar coordinates x1 = ρ cos σ, x2 = ρ sin σ,
2

2

with s2 = ρ2 + (x3 ) + (x4 ) , and ρ → ∞, where we have taken the polar angle σ to be the
1,(t)

same as the parameter of the circle S∞ . Therefore, using (4.4.1), the integrand of the path

ordered integral in (4.2.2) becomes
Aµ

(xν − aν )
dxµ
.
= −(2/e) ρ (−σ1ν sin σ + σ2ν cos σ)
dσ
(xµ − aµ )2 − λ2

(4.4.4)

As ρ → ∞, the only non-vanishing terms are those where xν is one of the coordinates of the

61

4.4 Instantons

plane, i.e. x1 or x2 . Then Aµ

dxµ
dσ
−ie

V D∞ = P1 e

→ (2/e) σ12 , and so (4.2.2) becomes
H

µ

1,(t)
S∞

dσ Aµ dx
dσ

1,(t)

where D∞ is the infinity disk with border S∞

= e−i 2

R 2π
0

dθ σ12

= e−i 4 π T3

(4.4.5)

on the plane x1 x2 . From (4.2.2) this operator

should correspond‡ to the (euclidean) magnetic flux Φ of Bi = − 12 ǫijk Fjk through D∞ , i.e.

VD∞ = e−i e Φ(B,D∞ ) . If one takes finite dimensional representations of SU(2) or SO(3), the
eigenvalues of T3 are integers or half integers and so VD∞ = 1l, which is compatible with the

fact the connection Aµ for the one-instanton is flat in the limit s → ∞. However, that fact

also implies that the flux should be quantised as
Φ (B, D∞ ) =

2πn
e

n = 0, ±1, ±2, . . .

(4.4.6)

However, following the same reasoning, the charges coming from (4.1.4) should also be associated, in such self-dual case solution, to the (euclidean) magnetic flux through the closed
2
sphere S∞
. But as we have shown below (4.4.3) that flux must vanish. Therefore, the only

compatible value of n in (4.4.6) seems to be n = 0.

Let us now consider the case of the two-instanton solution. A closed form for the regular
(non-singular) form of that solution is not easy. However we need only its asymptotic form
to calculate the charges and that is provided by Giambiagi and Rothe (40). Consider a twoinstanton regular solutions where the position four-vector of each instanton is given by aµ1 and
aµ2 . Then the asymptotic form of the connection is given by (40)
Aµ →

4
[(x · a) σµλ bλ + bµ xν σνλ bλ ]
e a2 s2

as

s→∞

(4.4.7)

where s2 = x21 + x22 + x23 + x24 , σµν is the same as in (4.4.2), aµ is the difference between the
two position four-vectors, and bµ is the reflection of aµ through the hyperplane perpendicular
to xµ , i.e.
aµ ≡ aµ1 − aµ2

bµ ≡ aµ − 2

(x · a)
xµ
x2

(4.4.8)

and so b2 = a2 .
The leading term of the connection in (4.4.7) is flat, falling as 1/s as s → ∞. The

leading term of the field tensor which would fall as 1/s2 vanishes, and therefore Fµν falls at
least as 1/s3 . Consequently, the integrand of the surface ordered integral in (4.1.4), namely
Fµν

dxµ dxν
,
dσ dτ

falls faster than 1/s, so it vanishes in the limit s → ∞. We then conclude that,

similarly to the one-instanton case, the charges associated to the operator (4.1.4) vanish when
evaluated on the two-instanton solution.
‡

Use the self-duality condition in the integrand on the l.h.s of (4.2.2).

4 The integral Yang-Mills equation in 3 + 1 dimensions

62

We now evaluate the charges associated to the operator (4.2.2) for the two-instanton
1,(t)

solution. Given an infinite plane (disk) D∞ with border being the circle S∞
1

of infinite radius

2

we can choose, without loss of generality, the axis x and x to lie on that plane. We then split
the vector aµ in its perpendicular and parallel parts with respect to the plane, i.e. aµ = aµ⊥ +aµk ,

and take the axis x1 to lie along aµk , and the axis x3 to lie along aµ⊥ . In addition we take

polar coordinates on the plane x1 x2 , such that x1 = ρ cos σ, and x2 = ρ sin σ, with the polar
1,(t)

angle σ being the same as the parameter in (4.2.2), parametrising S∞ . Then the integrand
1,(t)

of the path ordered integral in (4.2.2) along the infinite circle S∞

for the connection (4.4.7)

becomes (ρ → ∞)



| ak | | a⊥ |
4 | ak |2
dxµ
→
T3 +
[cos (2σ) T1 + sin (2σ) T2 ]
Aµ
dσ
e
a2
a2

(4.4.9)

We now perform a gauge transformation Aµ → (g2 g1 ) Aµ (g2 g1 )−1 + ei ∂µ (g2 g1 ) (g2 g1 )−1 ,
with g1 = ei 2 σ T3 and g2 = ei 2 ϕ T2 . The angle ϕ is defined as follows: since a2 =| ak |2 + |

a⊥ |2 , we parametrise it as | ak |=| a | cos ϕ, and | a⊥ |=| a | sin ϕ, with 0 ≤ ϕ ≤ π2 . So,
since the vector aµ was chosen to lie on the plane x1 x3 , ϕ is the angle between aµ and the

plane x1 x2 measured along the plane x1 x3 . Under such a gauge transformation one gets that
Aµ

dxµ
2
dxµ
→ A′µ
= T3
dσ
dσ
e

(4.4.10)

and so
−ie

P1 e

H

µ

1
S∞

dσ Aµ dx
dσ

−ie

→ g1 (σ = 2 π)−1 g2−1 P1 e

H

1
S∞

µ

dσ A′µ dx
dσ

g2 g1 (σ = 0)

(4.4.11)

Finally, the operator (4.2.2) becomes
−ie

V (D∞ ) = P1 e

H

1
S∞

µ

dσ Aµ dx
dσ

= e−i 4 π T3 e−i 2 ϕ T2 e−i 4 π T3 ei 2 ϕ T2

(4.4.12)

We can try to interpret that result in terms of the (euclidean) magnetic flux Φ through the
infinite disk D∞ . Since we are dealing with a non-abelian gauge theory one should not expect
a linear superposition of the fluxes of each instanton. We have seen in (4.4.5) that the
exponentiated flux of a single instanton is e−i 4 π T3 . In addition, since ϕ is the angle between
the line passing through the centres of the instantons and the disc D∞ , the result (4.4.12)
could give a hint on how the fluxes compose. That is certainly a point which deserves further
study. Again, as in any finite dimensional representation of SU(2) we have that e−i 4 π T3 = 1l,
and so VD∞ = 1l, which is compatible with the fact that Aµ is flat at the leading order we
have performed the calculation. Using the flux interpretation of the charges we can write
VD∞ = e−i e Φ2−inst. (D∞ ) , and so the two-instanton flux Φ2−inst. (D∞ ) is quantised as in (4.4.6).

63

4.5 Merons

4.5

Merons

Merons are singular euclidean solutions not self-dual with one-half unit of topological
charge (see (41–43)). We shall work here with such solutions in the Coulomb gauge since is is
more suitable for the evaluation of the charges and it also connects with monopole solutions.
The solution for a one-meron located at the origin is given by (41, 44)
!
1
x4
nj
Ai = − εijk
1− p 2
Tk
A4 = 0
e
r
x4 + r 2

(4.5.1)

with r 2 = x21 + x22 + x23 , i, j, k = 1, 2, 3, and Tk are the generators of the SU(2) Lie algebra.
Note that for x4 = 0 the connection (4.5.1) coincides with that for the Wu-Yang monopole

given in (4.3.1). In addition, it interpolates between two vacuum configurations, i.e. for
x4 → ∞ the connection (4.5.1) vanishes, and for x4 → −∞ it becomes of a pure gauge form

Ai = ei ∂i g g −1 , with g = exp (i π n
ˆ · T ).

In order to evaluate the charges (4.1.4) we need the field tensor at infinity, which is given
by
Fij →

nk
1
εijk 2 n
ˆ·T
e
r

F4i →

1
nj
εijk 2 Tk
e
r

r→∞

(4.5.2)

Note that when taking the limit r → ∞ we have kept x4 finite. The double limit r → ∞ and
x4 → ±∞, is not well defined. The asymptotic form of the space components of the dual

tensor is (ε1234 = 1)

1 1
Feij → − 2 [ni Tj − nj Ti ]
er

r → ∞.

(4.5.3)

2
of infinite radius and centred
If we evaluate the operator (4.1.4) on a spatial 2-sphere S∞

2
at the origin, it turns out that n
ˆ is perpendicular to S∞
and the derivatives

dxi
dσ

and

dxi
,
dτ
2
S∞
.

with σ parametrising the loops scanning the sphere and τ labelling them, are parallel to
i ∂xj
Therefore, we have that Feij ∂x
= 0. Consequently, the calculation of the operator (4.1.4)
∂σ ∂τ

for the one-meron solution is identical to that for the monopole (see calculation leading to
(4.3.3)). So, we have that
ie

QS = P2 e

R

2
S∞

dσ dτ W −1 [α Fij +β Feij ]W

∂xi ∂xj
∂σ ∂τ

−i e α

=e

R

2
S∞

~ B
~R
dΣ·

= e−i e α GR = ei 4 π α TR(4.5.4)

where we have introduced a (euclidean) magnetic field in a way similar to that in (4.3.3),
i.e. BiR ≡ − 21 ǫijk W −1 Fjk W = − 1e

ni
r2

TR , with TR being the value of n
ˆ · T at the reference

point xR used in the scanning of the sphere . Using the same arguments as in the case of
the monopole we conclude that the magnetic charges, eigenvalues of GR are quantised as in

4 The integral Yang-Mills equation in 3 + 1 dimensions

64

(4.3.7).
We then conclude that the one-meron solution has a magnetic charge conserved in the
euclidean time x4 , and it is quantised in units of

2π
.
e

What it is not clear is what happens to

that charge in the limit x4 → ±∞, since as we have seen, the connection (4.5.1) becomes

flat in that limit, and so the charge should disappear. One of the difficulties in answering that
is the fact that the double limit r → ∞ and x4 → ±∞, of the connection (4.5.1) is not well

defined.

The evaluation of the charges from the operator (4.3.18) for the one-meron solution is
1,(t)

also identical to that of the monopole. Indeed, if we consider the circle S∞

of infinite radius

to lie on spatial planes, then only the components Ai , i = 1, 2, 3, of the connection matters.
But in the limit r → ∞ the connection (4.5.1) becomes identical to that for the monopole

(4.3.1). Therefore one gets that

H
i


−ie 1,(t) dσ Ai dx
dσ
(t)
S∞
= e−i 2 π T3
Vx′(t) D∞
= P1 e

(4.5.5)

R

The eigenvalues of that operator are conserved in the euclidean time x4 , and their interpretation, given below (4.3.19), in terms of the magnetic flux through the surface which border
1,(t)

is S∞

remains valid. Again, we do not know what happens to those charges in the limit

x4 → ±∞, for the same reasons given above in the case of the one-meron magnetic charge.
The two-meron solution in the Coulomb gauge corresponding to one meron siting at the
position xµ = aµ = (0, 0, 0, a) and the other at xµ = bµ = (0, 0, 0, b) is given by (41, 44)


2
1
r + (x4 − a) (x4 − b) 
nj 
Ai = − ǫijk
1+ q
Tk
A4 = 0.
(4.5.6)
e
r
(x − a)2 (x − b)2
Expanding it in powers of

1
r

one gets

i
1 (a − b)2
nj
Ai = ∂i g g −1 +
ǫijk 3 Tk + O
e
e
2
r



1
r5



(4.5.7)

with g = exp (i π n
ˆ · T ), and so the leading term is of pure gauge form, i.e. it is flat. Therefore,
we have that

Fij ∼ O



1
r4



F0i ∼ O



1
r5



i ∂xj
Consequently, the integrand in (4.1.4), namely (αFij + β Feij ) ∂x
, behaves as O
∂σ ∂τ

limit r → ∞. Therefore, the charges associated to (4.1.4) vanish, i.e. QS = 1l.

(4.5.8)
1
r2




in the

Note that in the limit r → ∞ the spatial component of the connection (4.5.6) for the

two-meron solution is twice that of the one-meron solution (4.5.1). Therefore, the evaluation

65

4.5 Merons

of the charges associated to the operator (4.3.18) is very similar to that leading to (4.5.5) and
gives
Vx′(t)
R

(t)
D∞




−ie

= P1 e

H

i

1,(t)
S∞

dσ Ai dx
dσ

= e−i 4 π T3 = 1l

(4.5.9)

where the last equality follows from the fact that the leading term of Ai is flat. The interpretation for such conserved charges, given below (4.3.18), holds true, i.e. they correspond to the
1,(t)

magnetic flux through the surface which border is S∞ , and such fluxes are also quantized.
The meron-antimeron solution in the Coulomb gauge, corresponding to a meron and an
anti-meron located at xµ = −aµ and xµ = aµ respectively, with aµ = (0, 0, 0, a), is given by

(41, 44)



nj 
1
1− q
Ai = − ǫijk
e
r

Again expanding in powers of

2

2

x −a
2

(x + a) (x − a)

1
r

2



 Tk

A4 = 0

(4.5.10)

one gets

2 a2
nj
Ai = −
ǫijk 3 Tk + O
e
r



1
r5

Then, similarly to the two-meron case one has that Fij ∼ O



(4.5.11)
1
r4




and F0i ∼ O

1
r5




, and so

the charges coming from (4.1.4) are trivial, i.e. QS = 1l. In addition, since the connection falls
faster than

1
r

−ie

the integrand in the operator (4.3.18) vanishes, i.e. P1 e

The corresponding charges are also trivial in this case.

H

1,(t)
S∞

i

dσ Ai dx
dσ

= 1l.

66

4 The integral Yang-Mills equation in 3 + 1 dimensions

67

CHAPTER 5

Quasi-integrable deformation of the
non-linear Schr¨
odinger equation
In this chapter we analyse the phenomenon of quasi-integrability through some deformations of the non-linear Schr¨odinger equation. In section 5.1 the models we study are described
in details, the anomalous zero-curvature is constructed, the quasi-conserved charges are calculated and we establish the conditions that have to be satisfied by the solutions for the integrated
anomalies to vanish. We also give an argument, valid in a space-time of any dimension, for a
field theory to possess charges satisfying symmetries of the type given in (1.0.10). In section
5.2 we discuss how the dynamics of the model favours solutions satisfying (1.0.9). We also
discuss further the relation between the dynamics and parity for the case when the potential is
a deformation of the NLS potential. In section 5.3 we discuss the parity properties of the one
and two-soliton solutions of the NLS theory and show how to select those that satisfy (1.0.9).
We then present, in section 5.4, the results of our numerical simulations which support the
analytical results discussed in the previous sections. The numerical simulations were a major
contribution made by Wojciech Zakrzewski from the Durham University.In the appendix B we
present the details of the calculation used in section 5.1, and in appendix C we use the Hirota
method to construct one and two-bright-soliton solutions of the NLS theory.

5.1

Definition of the model

We consider a non-relativistic complex scalar field in (1 + 1) dimensions whose dynamics
is governed by the Lagrangian
L=





i
ψ ∂t ψ − ψ ∂t ψ − ∂x ψ ∂x ψ − V | ψ |2 ,
2

(5.1.1)

where ψ is the complex conjugate of ψ. The equations of motion are
i ∂t ψ = −∂x2 ψ +

∂V
ψ
∂ | ψ |2

(5.1.2)

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

68

together with its complex conjugate. The corresponding “Hamiltonian” reads


H =| ∂x ψ |2 +V | ψ |2 .

(5.1.3)

We shall consider solutions of (5.1.2) satisfying the following boundary conditions
| ψ |x=−∞ =| ψ |x=∞ ;

∂x ψ → 0

for

x → ±∞.

(5.1.4)

It is easy to check that the energy E, momentum P and normalisation N of the solutions of
the equations of motion(5.1.2) satisfying (5.1.4), as defined below, are conserved in time:
Z ∞


E =
dx | ∂x ψ |2 +V ,
(5.1.5)
−∞
Z ∞


P = i
(5.1.6)
dx ψ ∂x ψ − ψ ∂x ψ ,
−∞
Z ∞
N =
(5.1.7)
dx | ψ |2 .
−∞

In fact, these conserved quantities correspond to the Noether charges of the model. The
energy E is connected with the invariance of (5.1.1) under time translations, the momentum
P under the space translations, and N is related to the U(1) symmetry of the Lagrangian
(5.1.1)
ψ → ei α ψ

α ≡ const.

(5.1.8)

The integrable Non-Linear Schr¨odinger theory (NLS) corresponds to the potential
VNLS = η | ψ0 |4 ,

(5.1.9)

i ∂t ψ0 = −∂x2 ψ0 + 2 η | ψ0 |2 ψ0 .

(5.1.10)

which leads to the NLS equation

The sign of the parameter η plays an important role in the physical properties of the
solutions, and we refer to (31–34) for a more detailed discussion of this point. Indeed, for
η < 0 we have the so-called bright soliton solutions given by
i

h

i
2
ρ2 − v4 t+ v2 x

e
|ρ|
ψ0 = p
| η | cosh [ρ (x − v t − x0 )]

(5.1.11)

with ρ, v and x0 being real parameters of the solution. For η > 0 we have the dark soliton

69

5.1 Definition of the model

solution given by
|ρ|
i
ψ0 = √ tanh [ρ (x − v t − x0 )] e
η

h

v
2


 i
2
x− 2 ρ2 + v4 t

.

(5.1.12)

Note, that the solutions are defined up to an overall constant phase due to the symmetry
(5.1.8).
The equation (5.1.2) admits an anomalous zero curvature representation (Lax-ZakharovShabat equation) with the connection given by∗

Ax = −i T31 + γ¯ ψ T+0 + γ ψ T−0 ,
(5.1.13)




δV
T30 − γ¯ ψ T+1 + γ ψ T−1 − i γ¯ ∂x ψ T+0 − γ ∂x ψ T−0 ,
At = i T32 + i
2
δ |ψ|

where the generators Tin , i = 3, +, −, and n integer, satisfy the so-called SL(2) loop algebra

commutation relations



T3m , T±n = ±T±m+n ;




T+m , T−n = 2 T3m+n ,

(5.1.14)

which can be realized in terms of the finite SL(2) algebra generators as Tin ≡ λn Ti , with λ

an arbitrary complex parameter. The curvature of the connection (5.1.13) is given by


δV
2
0
∂t Ax − ∂x At + [Ax , At ] = X T3 + i γ¯ −i ∂t ψ + ∂x ψ − ψ
T+0
δ | ψ |2


δV
2
T−0
(5.1.15)
− i γ i ∂t ψ + ∂x ψ − ψ
δ | ψ |2

with
X ≡ −i ∂x



δV
− 2 γ γ¯ | ψ |2
δ | ψ |2



(5.1.16)

In consequence, when the equations of motion (5.1.2) are imposed, the terms on the r.h.s. of
(5.1.15), proportional to T+0 and T−0 , vanish. Note also that by taking
η ≡ γ γ¯ ,

(5.1.17)

the anomaly X, given in (5.1.16), vanishes for the NLS potential (5.1.9), and so the curvature
(5.1.15) vanishes, which makes the NLS equation integrable.
In this thesis we discuss a generalisation of this theory (i.e., deformations of the NLS
potential) which makes the resultant theory non-integrable (the anomaly (5.1.16) does not
vanish), but, as we will show, it exhibit properties very similar to the integrable theory, like
∗

As it will become clear later, γ¯ is not necessarily the complex conjugate of γ.

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

70

the solitons preserving their shapes after the scattering etc. In addition, we will show, using
the algebraic techniques borrowed from integrable field theories, that the anomalous LaxZakharov-Shabat equation (5.1.15) leads to an infinite number of quasi-conservation laws and
we will find that, under some special circumstances, the corresponding charges are conserved
asymptotically in the scattering of soliton type solutions of the deformed theory.
In order, to employ the algebraic techniques mentioned above it is more convenient to
work with a new basis of the SL(2) loop algebra and a new parametrisation of the fields. We
shall use the modulus R of ψ 2 and its phase ϕ, defined by
ψ=

√

ϕ

R ei 2 .

(5.1.18)

In addition, the complex parameters γ and γ¯ , appearing in the connection (5.1.13), are
written as
γ=i

p
| η | eiφ ,

γ¯ = −i σ

p
| η | e−iφ ,

γ γ¯ = η,

σ = sign η.
(5.1.19)

The new basis of the SL(2) loop algebraic is then defined as
bn = T3n ,
which satisfy
[bm , bn ] = 0 ;

F1n =



1
σ T+n − T−n ,
2

[bn , F1m ] = F2n+m ;

F2n =



1
σ T+n + T−n ,
2

[bn , F2m ] = F1n+m ;

(5.1.20)

[F1n , F2m ] = σ bn+m . (5.1.21)

As usual we perform the gauge transformation
Aµ → A˜µ ≡ g˜ Aµ g˜−1 + ∂µ g˜ g˜−1 ;

with

g˜ = ei( 2 +φ) b0
ϕ

(5.1.22)

and find that the connection (5.1.13) has now become
p
√
i
A˜x = −i b1 + ∂x ϕ b0 − 2 i | η | R F10 ,
2
p
√
δV
i
b0 + 2 i | η | R F11
A˜t = i b2 + ∂t ϕ b0 + i
2
δR


p
√
∂x R 0
0
+
F + i ∂x ϕ F1 .
|η| R −
R 2

(5.1.23)

For the fields which satisfy the equations of motion (5.1.2) the curvature becomes


h
i
δ
V
− 2ηR .
F˜tx = ∂t A˜x − ∂x A˜t + A˜x , A˜t = X b0 ;
with
X ≡ −i ∂x
δR
(5.1.24)

71

5.1 Definition of the model

To go further we carry out the usual abelianisation technique of the integrable field theories
(2, 28–30); i.e., we perform a further gauge transformation
A˜µ → aµ = g A˜µ g −1 + ∂µ g g −1

(5.1.25)

with
P∞

g=e

n=1

F (−n)

;

(−n)

The parameters ζi

(−n)

F (−n) ≡ ζ1

where

(−n)

F1−n + ζ2

F2−n .

(5.1.26)

are chosen, as we will explain below, so that the ax component of the

transformed connection lies in the infinite abelian sub-algebra spanned by the generators bn .
An important role in our construction is played by the grading operator d defined as
d≡λ

d
,
dλ

[d, bn ] = n bn ,

[d, Fin ] = n Fin .

(5.1.27)

The A˜x component of the connection (5.1.23) has generators of grade 0 and 1. Since the
group element (5.1.26) is an exponentiation of negative grade generators, the ax component
of the transformed connection has generators of grades ranging from 1 to −∞. Splitting the

transformed potential (5.1.25) into its eigen-sub-spaces under the grading operator (5.1.27),
P
(n)
i.e., ax = ∞
n=1 ax , we find that
a(1)
= −i b1 ,
x


a(0)
= i b1 , F (−1) + A˜(0)
x
x
h
i


i  (−1)  (−1) 
(−2)
(−1) ˜(0)
a(−1)
=
i
b
,
F
+
F
,
A
−
F
, F
, b1
1
x
x
2!
+ ∂x F (−1) ,
i


 h (−2)
i  (−2)  (−1)
(−3)
(0)
˜
F
, F
, b1
a(−2)
=
i
b
,
F
+
F
,
A
−
1
x
x
2!
ii
  1 h (−1) h (−1)
i  (−1)  (−2)
F
, F
, b1 +
F
, F
, A˜(0)
−
x
2!
2!

i  (−1)  (−1)  (−1)
F
, F
, F
, b1
−
3!

1  (−1)
+ ∂x F (−2) +
F
, ∂x F (−1) ,
2!
..
.

(0)
where we have denoted A˜x = 2i ∂x ϕ b0 − 2 i

(5.1.28)

p
√
| η | R F10 (see (5.1.25)).

An important ingredient of this construction is the observation that the generator b1 is a
semi-simple element (in fact any bn is) in the sense that it splits the SL(2) loop algebra G

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

72

into the kernel (Ker) and image (Im) of its adjoint action, i.e.
G = Ker + Im ;

with

[ b1 , Ker ] = 0 ;

Im = [ b1 , G ] . (5.1.29)

The Ker and Im sub-spaces do not have common elements, i.e. any element of G commuting

with b1 cannot be written as a commutator of b1 with some other element of G. One notes

from (5.1.21) that bn constitute a basis of Ker, and Fin , i = 1, 2, a basis of Im. In addition,
one notes from (5.1.28) that the first time that F (−n) appears in the expansion of ax , is in


the component ax−n+1 of grade −n + 1, and it appears in the form b1 , F (−n) . Therefore,
one can choose the parameters in F (−n) so that they cancel the image component of ax−n+1 .

This can be done recursively starting at the component of grade 0 and working downwards.

It is then clear that the gauge transformation (5.1.26) can rotate the ax component of the
connection into the Abelian sub-algebra generated by the bn ’s, i.e.,
ax = −i b1 +

∞
X
n=0

a(3,−n)
b−n .
x

(5.1.30)

Note from (5.1.23) that A˜x depends on the real fields R and ∂x ϕ. Thus, the components
(3,n)

ax

are polynomials in these fields and their x-derivatives, and they do not depend on the

potential V . In consequence, the ax component of the connection is the same for any choice
of the potential. In the appendix (B) we give explicit expressions for the first few components
of ax .
On the other hand the A˜t component of the connection (5.1.23) depends on the choice
of the potential V . In fact, for the case of the NLS potential (5.1.9) we note that the gauge
transformation (5.1.25), with the group element (5.1.26) fixed as above, does rotate at into an
Abelian sub-algebra generated by the bn ’s, when the equations of motion (5.1.10) are satisfied.
For other choices of V this does not take place even when the equations of motion (5.1.2) are
imposed. Thus, we find that
∞ h
i
X
(3,−n)
(1,−n) −n
(2,−n) −n
at = i b2 +
at
b−n + at
F1 + at
F2 .

(5.1.31)

n=0

Next we note that at does not have the grade 1 component due to the fact that the
coefficient of F10 in A˜x , and the coefficient of F11 in A˜t , are the same up to a sign (see (5.1.23)).
Under the gauge transformation (5.1.25) the curvature Ftx transforms to Ftx → g Ftx g −1 ,
and so from (5.1.24) we see that

∂t ax − ∂x at + [ at , ax ] = X g b0 g −1 .

(5.1.32)

73

5.1 Definition of the model

Since ax lies in the kernel of b1 it follows that [ at , ax ] has components only in the image of
b1 . Thus, denoting
g b0 g −1 =

∞
X
 (3,−n)

α
b−n + α(1,−n) F1−n + α(2,−n) F2−n

(5.1.33)

n=0

we find that
(3,−n)

∂t a(3,−n)
− ∂x at
x

= X α(3,−n) ;

n = 0, 1, 2, . . . .

(5.1.34)

The explicit expressions for the first few α(i,−n) , i = 1, 2, 3, are given in appendix
B. Note that if the time component of the connection satisfies the boundary condition
(3,−n)

at

(3,−n)

(x = ∞) = at

(x = −∞), which is the case in the example we consider, then

we have anomalous conservation laws
d Q(n)
= βn ;
dt

with

(n)

Q

=

Z

∞
−∞

dx a(3,−n)
x

;

where

βn =

Z

∞
−∞

dx X α(3,−n) .
(5.1.35)

Of course, in the case of the NLS theory we get an infinite number of conserved quantities
since the anomaly X, given in (5.1.16) or (5.1.24), vanishes for the NLS potential (5.1.9).
We now use a more refined algebraic technique to explore the structure of the anomalies
βn . The key ingredients are the two ZZ2 transformations, one in the internal space of the loop
algebra and the other in space-time. The first ZZ2 transformation is an order 2 automorphism
of the SL(2) loop algebra (5.1.21) given by
Σ (bn ) = −bn ,

Σ (F1n ) = −F1n ,

Σ (F2n ) = F2n .

(5.1.36)

The second ZZ2 transformation is a space-time reflection around a given point (x∆ , t∆ ), i.e.,
P :





x˜, t˜ → −˜
x, −t˜

with

x˜ = x − x∆

t˜ = t − t∆ .

(5.1.37)

Consider now solutions of the equations of motion (5.1.2) of the theory (5.1.1) such that,
in addition, they satisfy the following property under the parity (5.1.37) (see (5.1.18))
P :

R→R;

ϕ → −ϕ + constant.

(5.1.38)

Then, the x-component of the connection (5.1.23) transforms as
 
Σ A˜x = −A˜x

 
P A˜x = A˜x

(5.1.39)

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

74

and so it is odd under the joint action of the two ZZ2 transformations:



˜
Ω Ax = −A˜x ,

Ω ≡ Σ P.

(5.1.40)

In fact, this property is valid for every individual component of A˜x . Thus we see that we






have Ω b1 , F (−n) = − b1 , Ω F (−n) , and so
(1 + Ω)



b1 , F (−n)






= b1 , (1 − Ω) F (−n)

(5.1.41)

(0)
Since A˜x is odd under Ω, it follows from the second equation of (5.1.28) that



(−1)
(1 + Ω) a(0)
.
x = i b1 , (1 − Ω) F

(5.1.42)

The r.h.s. of (5.1.42) is clearly in the image of the adjoint action, and we have chosen the
F (−n) to rotate ax into the kernel of that same adjoint action. Therefore, the only possibility
for (5.1.42) to hold is that both sides vanish, i.e., that
(1 + Ω) a(0)
x = 0,

(1 − Ω) F (−1) = 0

(5.1.43)

and so that F (−1) is even under Ω. Using this fact we see from the third equation in (5.1.28)

that



(−2)
(1 + Ω) a(−1)
=
i
b
,
(1
−
Ω)
F
.
1
x

(5.1.44)

Furthermore, using same arguments we conclude also that
(1 + Ω) a(−1)
= 0,
x

(1 − Ω) F (−2) = 0

(5.1.45)

and so that F (−2) is even under Ω as well. Again, from the fourth equation in (5.1.28) we


(−2)
see that (1 + Ω) ax = i b1 , (1 − Ω) F (−3) , and so by the same arguments as before we
conclude that

(1 + Ω) a(−2)
= 0,
x

(1 − Ω) F (−3) = 0.

(5.1.46)

Repeating this reasoning we reach the conclusion that all F (−n) are even under Ω. So,

the group element g, given in (5.1.26), is even under Ω
Ω (g) = g.

(5.1.47)

To go further we note that since A˜x and ∂x are odd under Ω, and since g is even (5.1.25)
demonstrates that ax has to be odd under Ω. One can verify all these claims by inspecting
(−n)

the explicit expressions for the parameters ζi

given in appendix B. Since the F (−n) are even

75

5.1 Definition of the model



(−n)
under Ω, and since the generators satisfy (5.1.36), it follows from (5.1.26) that P ζ1
=


(−n)
(−n)
(−n)
= ζ2 .
−ζ1
and P ζ2
Next we use the Killing form of the SL(2) loop algebra given by
Tr (bn bm ) =

1
δn+m,0 ;
2

Tr (bn Fim ) = 0 ;

which can be realized by Tr (⋆) ≡

1
2πi

H

dλ
tr (⋆),
λ

i = 1, 2,

(5.1.48)

with tr being the ordinary finite matrix

trace, and Tin = λ Ti , i = 3, ±. In this case we see from (5.1.33) that





α(3,−n) = 2 Tr g b0 g −1 bn = 2 Tr Σ (g) b0 Σ g −1 bn ,

(5.1.49)

where in the last equality we have used the fact that the Killing form is invariant under Σ,
and that all the bn ’s are odd under it. Thus, using (5.1.47) we have that







P α(3,−n) = 2 Tr Ω (g) b0 Ω g −1 bn = 2 Tr g b0 g −1 bn = α(3,n)

(5.1.50)

and so we see that all the α(3,−n) ’s are even under P . Note that X, given in (5.1.24), is an
x-derivative of a functional of R. Since we have assumed that R is even under P , we see from
(5.1.38) that X is odd, i.e., that P (X) = −X and so that
Z

t˜0

dt
−t˜0

Z

x
˜0

dx X α(3,−n) = 0,

(5.1.51)

−˜
x0

where t˜0 and x˜0 are given fixed values of the shifted time t˜ and space coordinate x˜ respectively,
introduced in (5.1.37). Therefore, by taking x˜0 → ∞, we conclude that the non-conserved
charges (5.1.35) satisfy the following mirror time-symmetry around the point: t∆ .




Q(n) t = t˜0 + t∆ = Q(n) t = −t˜0 + t∆ .

(5.1.52)

In consequence, even though the charges Q(n) vary in time, they are symmetric with
respect to t = t∆ . Note that we have derived this property for any potential V which depends
only on the modulus of ψ. The only assumption we have made is that we are considering
fields ψ which satisfy (5.1.38).
In the next sections we will show that such solutions are very plausible and that, in fact,
the one and two-soliton solutions of the theories (5.1.1) can always be chosen to satisfy
(5.1.38). This fact has far reaching consequences for the properties of the theories (5.1.1).
For instance, by taking t˜0 → ∞ one concludes that the scattering of two-soliton solutions

presents an infinite number of charges which are asymptotically conserved. Since the S-matrix
relies only on asymptotic states, it is quite plausible that the theories (5.1.1) share a lot of

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

76

interesting properties with integrable theories (but which have been believed to be only true
for integrable field theories).

5.1.1

On the parity symmetry

The properties leading to charges satisfying (5.1.52) can be realised in a much wider
context. Indeed, consider a field theory in a space-time of (d + 1) dimensions with fields
labelled by ϕa , a = 1, 2, . . . n. These fields can be scalars, vectors, spinors, etc., and the
indices a just label their components. Consider a fixed point xµ∆ in space-time, and a reflection
P around it, i.e.,
x˜µ → − x˜µ

P :

with

x˜µ = xµ − xµ∆

µ = 0, 1, 2 . . . d.

(5.1.53)

Suppose that a such field theory possesses a classical solution ϕsa such that the fields evaluated
on it are eigenvectors of P up to constants, i.e. that
P (ϕsa ) = εa ϕsa + ca ,

εa = ±1 ;

ca = const.

(5.1.54)

Consider now a functional of the fields and of their derivatives F = F (ϕa , ∂µ ϕa , ∂µ ∂ν ϕa , . . .),
that is even under P when evaluated on a particular solution, i.e.
P [F (ϕsa , ∂µ ϕsa , ∂µ ∂ν ϕsa , . . .)] = F (ϕsa , ∂µ ϕsa , ∂µ ∂ν ϕsa , . . .) .

(5.1.55)

Next, look at a rectangular spatial volume V bounded by hyperplanes crossing the axes

of the space coordinates at the points ±˜
xi0 , i = 1, 2, . . . d, corresponding to fixed values of

the shifted space coordinates x˜i introduced in (5.1.53), i.e. such that the point xi∆ lies in the
very centre of V. The integral of this functional over V
Z
Q=
dd x F

(5.1.56)

V

satisfies
dQ
=
d x0

Z

dF
d x
=
d x0
V
d

Z


δF
δF
δF
d x
∂0 ϕa +
∂0 ∂µ ϕa +
∂0 ∂µ ∂ν ϕa + . . . .
δϕa
δ∂µ ϕa
δ∂µ ∂ν ϕa
V
(5.1.57)
d



When evaluated on the solution ϕsa each term in the integrand in (5.1.57) is odd under P .
The reasons for this are simple: any derivative of the form ∂0 ∂µ1 . . . ∂µm ϕa , when evaluated on
ϕsa , has an eigenvalue of P equal to εa (−1)m+1 . Since F evaluated on ϕsa is even under P , it
follows that any derivative of the form

δF
δ∂µ1 ...∂µm ϕa

has an eigenvalue of P equal to εa (−1)m ,

when evaluated on ϕsa . Therefore, when evaluated on ϕsa each term of the integrand on the

77

5.1 Definition of the model

r.h.s. of (5.1.57) is odd under P . Consequently, one finds that
s

0

Q x˜




s

0

− Q −˜
x
=

Z

x
˜0

−˜
x0

0

dx




=

Z

x
˜0

dx0

−˜
x0

d Qs
d x0

(5.1.58)


δ Fs
δ Fs
δ Fs
s
s
s
∂0 ϕa +
∂0 ∂µ ϕa +
∂0 ∂µ ∂ν ϕa + . . . = 0,
d x
δϕsa
δ∂µ ϕsa
δ∂µ ∂ν ϕsa
V

Z

d



where the superscript s denotes that Q is evaluated on the solution ϕsa , and x˜0 is a given fixed
value of the shifted time introduced in (5.1.53).
Summarizing: if one has a solution of the theory such that all the fields evaluated on this
solution are eigenstates of P , i.e. they satisfy (5.1.54), then any even functional of these fields
and their derivatives leads to charges that satisfy a mirror time-symmetry like (5.1.58).
In the case studied in this thesis we have shown that the x-component of the connection,
ax , is odd under the transformation Ω = Σ P , i.e. (1 + Ω) ax = 0. Since ax lies in the
abelian subalgebra generated by the bn ’s (see (5.1.30)), which are odd under Σ (see (5.1.36)),
(3,−n)

it follows that the charge densities ax

are even under P . Therefore, the charges Q(n)

introduced in (5.1.35) are in the class of charges (5.1.56) discussed in this subsection. So, the
assumption of the existence of a solution satisfying (5.1.38) has much deeper consequences. It
implies not only that the charges (5.1.35) satisfy the mirror time-symmetry (5.1.52), but also
that any charge built out of a density that is even under P when evaluated on this solution,
also satisfies (5.1.52). The fact that a solution satisfies (5.1.38) implies that its past and
future w.r.t. to the point in time t∆ , are strongly linked and, in consequence, so are many of
its properties. Indeed, the mirror time-symmetry (5.1.52) is a direct consequence of such a
link between the past and the future. The non-linear phenomena behind the quasi-integrability
properties we are discussing are certainly driven by the parity property (5.1.38). However, we
still have to understand the basic physical processes guarantee that a given solution satisfies
(5.1.38). This is one of the great challenges for our techniques to understand. In the next
section, we argue that for the theories (5.1.1) for which the potential V is a deformation of
the NLS potential (5.1.9), the solutions satisfying (5.1.38) are favoured by the dynamics if
the corresponding undeformed solution of the integrable NLS theory also satisfies (5.1.38).

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

78

5.2

Dynamics versus parity

In terms of fields R and ϕ introduced in (5.1.18), the equations of motion (5.1.2) become
∂t R = −∂x (R ∂x ϕ) ,
R2
1
∂V
−R2 ∂t ϕ = −R ∂x2 R +
(∂x ϕ)2 + (∂x R)2 + 2 R2
.
2
2
∂R

(5.2.1)

Let us analyze what type of solutions these equations admit if we assume that the fields
of these solutions are eigenstates of the of parity transformation P introduced in (5.1.37). We
split the fields as
R = R(+) + R(−) ;
where



P R(±) = ±R(±) ,

ϕ = ϕ(+) + ϕ(−) ,


P ϕ(±) = ±ϕ(±) + constant.

(5.2.2)

(5.2.3)

Let us now assume that we have a solution for which R(+) = ϕ(−) = 0. Then, the l.h.s.
of the first equation in (5.2.1) is even under P , and its r.h.s. is odd. Thus, ∂t R(−) = 0


and ∂x R(−) ∂x ϕ(+) = 0. In addition, the second equation in (5.2.1) implies that ∂t ϕ(+) =
 (−)
−2 ∂∂RV
.
Note also that if we have a solution for which R(−) = ϕ(−) = 0 we get very similar results,


 (−)
.
namely that ∂t R(+) = 0, ∂x R(+) ∂x ϕ(+) = 0, and that ∂t ϕ(+) = −2 ∂∂RV

In a similar way, if we assume that our solution satisfies R(+) = ϕ(+) = 0, then the second
 (−)
= 0. This condition, however, excludes potentials
equation in (5.2.1) implies that ∂∂RV

that are even functions of R, like the integrable NLS potential (5.1.9). Thus, we would not
expect interesting non-trivial solutions, like a two-soliton solution, with one of these three
classes of cases in which the fields are eigenstates of P .
The only remaining case is the one we assumed in (5.1.38), namely, that R(−) = ϕ(+) = 0.
One can easily check that the equations (5.2.1) do not impose any restrictions on the solutions
of this type. Indeed,

∂V
∂R

is always even under P for any V , if R is even under P .

Consequently, we would expect most of the interesting non-trivial results for solutions of
the theories (5.1.1), for which the fields evaluated on them are eigenstates of P , to fall into
the class (5.1.38). Of course, there can also exist classes of non-trivial solutions for which
the parity components are mixed and the above arguments do not apply. However, this does
not mean that the results of these arguments are necessarily incorrect. Sometimes they may
still hold even though one has to work harder to prove them. In the next section we present

79

5.2 Dynamics versus parity

a detailed analysis of the case in which the potential V is a deformation of the NLS potential
(5.1.9). Our analysis shows that the mixed solutions can always be “gauged away”, order by
order, in the perturbation expansion around the NLS theory.

5.2.1

Deformations of the NLS theory

We now consider the theories (5.1.1) for which the potential V is a deformation of the
NLS potential (5.1.9). The deformation is introduced through a parameter ε such that for
ε = 0, V corresponds to (5.1.9). We will not consider here the deformations for which the
potential depends upon the phase of ψ. Examples of such potentials are
V (1) = η R2+ε ;

V (2) = η R2 + ε R3 ;

V (3) = η R2 e−ε R ,

(5.2.4)

where R =| ψ |2 was introduced in (5.1.18).
We start our analysis by expanding the solutions of the corresponding equations of motion
in powers of ε around the solution of the integrable NLS theory as
R = R0 + ε R1 + ε2 R2 + . . . ,

ϕ = ϕ0 + ε ϕ1 + ε 2 ϕ2 + . . .

Of course, the deformed potential V has the expansion




∂V
∂V
V = V |ε=0 +ε
|ε=0 +
|ε=0 R1 + O ε2
∂ε
∂R

(5.2.5)

(5.2.6)

and its gradient has the expansion
 2

∂V
∂V
∂ V
∂2 V
(5.2.7)
=
|ε=0 +ε
|ε=0 +
|ε=0 R1
∂R
∂R
∂ε ∂R
∂R2
 3

ε2
∂ V
∂3 V
∂3 V
∂2 V
2
+
|ε=0 +2
|ε=0 R1 +
|ε=0 R1 + 2
|ε=0 R2 + . . .
2 ∂ε2 ∂R
∂ε∂R2
∂R3
∂R2
We also expand the equations of motion (5.2.1) into powers of ε and at the same time we
split the equations, and so the fields, into their even and odd parts under a given space-time
parity P of the type (5.1.37). At this stage the value of the point (x∆ , t∆ ) around which
we perform the reflection is not yet important. We just use the fact that the operation P
satisfies P 2 = 1l, and so it has eigenvalues ±1. Next we introduce the following notation for

the eigen-components of the fields:

⋆(±) ≡

1
(1 ± P ) ⋆ .
2

(5.2.8)

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

80

Then the zero order part of the equations of motion (5.2.1) splits under P as
(−)
∂t R0
(+)

∂t R0

= −∂x





(+)
R0

(+)
∂x ϕ0

(+)

(−)

= −∂x R0 ∂x ϕ0

+

(−)
R0

(−)
∂x ϕ0

(−)

(+)

+ R0 ∂x ϕ0



,



(5.2.9)
(5.2.10)

and


(+)
R0



(+)
R0


2 
(+) (−)
(+)
(+)
(+)
(−)
(−)
(−)
(−)
+ R0
∂t ϕ0 − 2 R0 R0 ∂t ϕ0 = −R0 ∂x2 R0 − R0 ∂x2 R0

 
2 
2 
1  (+) 2  (−) 2
(+)
(+) (−)
(+)
(−)
(−)
R0
∂x ϕ0
+ 2 R0 R0 ∂x ϕ0 ∂x ϕ0
+ R0
+ ∂x ϕ0
2

2 
2 
1 
(+)
(−)
∂x R0
+ ∂x R0
(5.2.11)
2
(+)
(−)


2 
2   ∂ V
∂V
(+) (−)
(+)
(−)
|ε=0
|ε=0
+ 4 R0 R0
2
R0
+ R0
∂R
∂R

−
+
+
+

2

and
2 

(+)
(+) (−)
(−)
(+)
(−)
(−)
(+)
(−)
−
∂t ϕ0 − 2 R0 R0 ∂t ϕ0 = −R0 ∂x2 R0 − R0 ∂x2 R0
+ R0


2 
2 
2 
2 
(+)
(+)
(+)
(−)
(+) (−)
(−)
(−)
+
R0
∂x ϕ0
∂x ϕ0 ∂x ϕ0 + R0 R0
+ R0
+ ∂x ϕ0
(+)

2

(−)

+ ∂x R0 ∂x R0

(−)
(+)

2 
2   ∂ V
∂V
(+)
(−)
(+) (−)
+ 2
R0
+ R0
|ε=0
|ε=0
+ 4 R0 R0
.
∂R
∂R

(5.2.12)

As we have shown in section 5.1, and in particular in sub-section 5.1.1, the mirror timesymmetry property of the charges, given in (5.1.52), is valid for solutions for which the components of the fields with different eigenvalues of P are not mixed. Since, we have two fields
R and ϕ we have four possibilities for non-mixing solutions. In our analysis we shall assume
that the potentials satisfy the property
∂V
|ε=0 ∼ R0 .
∂R
(+)

If one considers solutions for which R0

(5.2.13)
(+)

= 0 and ∂ϕ0

= 0 (with ∂ standing for time

and space derivatives), then the zero order equations of motion (5.2.9)-(5.2.12) impliy that
(−)

R0

(−)
∂t R0

and

(+)

= 0. In addition, if one assumes R0
= 0. Finally, if one assumes

(+)
∂t ϕ0

(−)
R0

(−)

= 0 and ∂ϕ0

= 0 and

(−)
∂ϕ0

= 0 then (5.2.9)-(5.2.12) imply
(+)

= 0 then one finds that ∂t R0

=0

= 0. Therefore, in none of those three cases one should expect to get interesting

solutions, specially two-soliton solutions. Therefore, we shall restrict our attention to the class

81

5.2 Dynamics versus parity
(−)

of solutions for which R0

(+)

= 0 and ∂t,x ϕ0

= 0, i.e. those that satisfy

R0 → R0

P :

ϕ0 → −ϕ0 + const.

Note that with R0 even under P it follows that all derivatives of the form

(5.2.14)
∂ n+m V
∂εn ∂Rm

|ε=0 are

even under P . Now, assuming (5.2.14) one gets that the first order part of the equations of
motion (5.2.1) split under P as
(−)

∂t R1

(+)

∂t R1



(+)
(+)
(−)
(−)
,
= −∂x R0 ∂x ϕ1 + R1 ∂x ϕ0


(+)
(−)
(+)
(−)
= −∂x R0 ∂x ϕ1 + R1 ∂x ϕ0
,

(5.2.15)
(5.2.16)


2
(+)
(−)
(+) (+)
(−)
(+)
(+)
(+)
(+)
− R0
(5.2.17)
∂t ϕ1
= 2R0 R1 ∂t ϕ0 − R0 ∂x2 R1 − R1 ∂x2 R0
2 

2
(−)
(+) (+)
(+)
(−)
(−)
(+)
(+)
∂x ϕ0
+ R0 R1
+ R0
∂x ϕ0 ∂x ϕ1 + ∂x R0 ∂x R1

2  ∂ 2 V

∂2 V
(+) (+) ∂ V
(+)
(+)
+ 4 R0 R1
|ε=0 +2 R0
|ε=0 +
|ε=0 R1 ,
∂R
∂ε ∂R
∂R2
2

(+)
(+)
(+) (−)
(−)
(+)
(−)
(−)
(+)
(5.2.18)
− R0
∂t ϕ1
= 2R0 R1 ∂t ϕ0 − R0 ∂x2 R1 − R1 ∂x2 R0
2 

2
(−)
(+) (−)
(+)
(−)
(+)
(+)
(−)
∂x ϕ0
+ R0 R1
+ R0
∂x ϕ0 ∂x ϕ1 + ∂x R0 ∂x R1

2 ∂ 2 V
(+)
(−)
(+) (−) ∂ V
|ε=0 +2 R0
|ε=0 R1 .
+ 4 R0 R1
∂R
∂R2
(+)

(−)

Once the zero order solutions for R0

and ϕ0

have been found, we put them into (5.2.15)-

(5.2.18) and get four coupled partial differential equations with non-constant coefficients which
(±)

are linear in the first order fields R1

(±)

and ϕ1 .
(+)

There are two important facts about (5.2.15)-(5.2.18). First they couple R1
(−)
ϕ1

and

(−)
R1

only to

(+)
ϕ1 ,

only to

i.e. the pair of equations (5.2.15) and (5.2.18) is decoupled from

the pair formed by (5.2.16) and (5.2.17). Secondly, the pair of equations (5.2.15) and (5.2.18)
is homogeneous in the first order fields, but the equation (5.2.17) is non-homogeneous due to
2 2

(+)
∂ V
| , which does not involve the first order fields. Therefore, there
the term 2 R0
∂ε ∂R ε=0
(+)

are no solutions for which R1
solutions for which

(−)
R1

(−)

= 0 and ϕ1

= 0 and

(+)
ϕ1

= constant. On the other hand we can have

= constant. In addition, if R1 and ϕ1 , are solutions
(−)

with a non-definite parity, then R1 − R1

(+)

and ϕ1 − ϕ1

are also solutions but now with a

definite parity. So, we can always choose the first order solutions to satisfy
P :

R1 → R1

ϕ1 → −ϕ1 + const.

(5.2.19)

If we now take the zero and first order solutions satisfying (5.2.14) and (5.2.19), respec-

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

82

tively, then the second order part of the equations of motion (5.2.1) splits under P as
(−)
∂t R2
(+)

∂t R2





(−)
(−)
(+)
(+)
= −∂x R2 ∂x ϕ0
− ∂x R0 ∂x ϕ2
,
(5.2.20)






(+)
(−)
(+)
(−)
(+)
(−)
− ∂x R1 ∂x ϕ1
− ∂x R0 ∂x ϕ2
= −∂x R2 ∂x ϕ0
(5.2.21)

and


2

2
(+)
(+) (+)
(−)
(+)
(−)
(+) (+)
(−)
− R0
∂t ϕ2 − 2R0 R1 ∂t ϕ1 −
R1
+ 2 R0 R2
∂t ϕ0
(+)

(+)

(+)

(+)

(+)

(+)

= −R0 ∂x2 R2 − R2 ∂x2 R0 − R1 ∂x2 R1

2
2
1  (+) 2 
(+)
(−)
(−)
(−)
(+) (+)
(−)
(−)
+ R0
∂x ϕ0 ∂x ϕ2 +
R0
∂x ϕ1
+ 2 R0 R1 ∂x ϕ0 ∂x ϕ1
2
2 
2

2 1 
(+)
(−)
(+) (+)
(−)
R1
∂x ϕ0
(5.2.22)
+ R0 R2
∂x ϕ0
+
2


2

2
∂V
1
(+)
(+)
(+)
(+)
(+) (+)
∂x R1
|ε=0
+ ∂x R0 ∂x R2 + 2 R1
+ 4 R0 R2
+
2
∂R

 2
∂2 V
∂ V
(+)
(+) (+)
|ε=0 +
|ε=0 R1
+ 4 R0 R1
∂ε ∂R
∂R2

2

2  ∂ 3 V

∂3 V
∂3 V
∂2 V
(+)
(+)
(+)
(+)
|ε=0 +2
|ε=0 R1 +
|ε=0 R1
|ε=0 R2
+2
+ R0
∂ε2 ∂R
∂ε∂R2
∂R3
∂R2

and
−
+





(+)

R0

(+)

R0

2

2

(+)

(+)

∂t ϕ2

(−)

(+)

∂x ϕ0 ∂x ϕ2
(−)

+ ∂x R0 ∂x R2
+ 2

(+)

(−)

(−)

− 2 R0 R2 ∂t ϕ0
(+)

(−)

+ R0 R2

(+)

(−)

+ 4 R0 R2

∂2 V
(−)
|ε=0 R2 .
∂R2

(+)

(−)

= −R0 ∂x2 R2

2
(−)
∂x ϕ0

(−)

(+)

− R2 ∂x2 R0

(5.2.23)

∂V
|ε=0
∂R

Again we have a structure very similar to that discussed in the case of the equations
(5.2.15)-(5.2.18). Indeed, having found the solutions for the zero and first order fields, we
put them into (5.2.20)-(5.2.23) and get four coupled partial differential equations with non(±)

constant coefficients which are linear in R2

(±)

and ϕ2 . In addition, the pair of equations
(+)

(5.2.20) and (5.2.23) is decoupled from the pair (5.2.21) and (5.2.22), i.e. R2
only to

(−)
ϕ2

and

(−)
R2

also only to

(+)
ϕ2 .

couples

Again, the pair of equations (5.2.20) and (5.2.23) is

homogeneous in the second order fields and the pair (5.2.21) and (5.2.22) is non-homogeneous.
(−)

Thus, as before, R2

(+)

= 0 and ϕ2

(+)

= constant is a solution, but R2

(−)

= 0 and ϕ2

=

constant, cannot be a solution. In addition, if R2 and ϕ2 , are solutions, with a non-definite
(−)

parity, then R2 − R2

(+)

and ϕ2 − ϕ2

are also solutions but now with a definite parity. So, we

83

5.3 The parity properties of NLS solitons

can always choose the second order solutions to satisfy
P :

R2 → R2

ϕ2 → −ϕ2 + const.

(5.2.24)

We can repeat this process, and even though we have not proved this here, this structure
repeats itself at every order of perturbation in ε. Therefore, the fields R(−) and ϕ(+) can
always be “gauged away” since they satisfy homogeneous equations (order by order), but the
fields R(+) and ϕ(−) are robust in the sense that they always have to be present in the solution.
Thus, we have shown that the solutions satisfying (5.1.38) are favoured by the dynamics when
the potential V in (5.1.1) is a deformation of the integral NLS potential (5.1.9). We point out
however, that the property of being even or odd under P is not something that can be encoded
into the initial boundary conditions at a given initial time t0 . The properties under P involve
a link between the past and the future of the solution and so, perhaps, cannot be understood
using the usual techniques (specially numerical) of investigating the coupling of the normal
modes as the systems evolves in time. We are perhaps facing a new and intriguing non-liner
phenomenon. In the next section, we go further in our analysis and study the properties under
P of the exact one and two-soliton solutions of the integral NLS theory.

5.3

The parity properties of NLS solitons

We will now analyze the one and two soliton solutions of the integrable NLS theory (5.1.10)
under the parity transformation (5.1.37). The solutions are constructed by the Hirota’s method
described in the appendix C.

5.3.1

The one-soliton solutions

In terms of the fields R and ϕ introduced in (5.1.18) the one-bright-soliton solution
(5.1.11) is given by
R0bright

1
ρ2
;
=
2
| η | cosh [ρ (x − v t − x0 )]

ϕbright
0




v
v2
2
t+ x
=2
ρ −
4
2
(5.3.1)

Analogously, the one-dark-soliton solution (5.1.12) is given by
R0dark

ρ2
tanh2 [ρ (x − v t − x0 )] ;
=
η

ϕdark
0

 

v2
v
2
t
=2
x− 2ρ +
2
4
(5.3.2)


5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

84

Then it is clear that the relevant parity transformation, in each case, is
x˜ → −˜
x

P :

t → −t

x˜ = x − x0

with

(5.3.3)

Therefore one has that
bright/dark

P :

R0

bright/dark

→ R0

bright/dark

;

ϕ0

bright/dark

→ −ϕ0

+ 2 v x0
(5.3.4)

which is agreement with (5.2.14) and (5.1.38).
If one chooses the potential in (5.1.1) as
V =

2
η R2+ε
2+ε

(5.3.5)

then the theory has a one-soliton solution given by


1
2 + ε ρ2
R=
2
2 | η | cosh [(1 + ε) ρ (x − v t − x0 )]

1
 1+ε

;




v
v2
2
t+ x
ϕ=2
ρ −
4
2
(5.3.6)

which is a deformation of the one-bright-soliton (5.3.1). Notice that under the parity (5.3.3)
it transforms as
R→R;

P :

ϕ → −ϕ + 2 v x0

(5.3.7)

Since (5.3.6) is an exact solution of the deformed NLS theory this observation supports our
claims of section 5.2, based on the perturbative series in ε, that solutions satisfying the property
(5.3.7) are favoured by the dynamics.

5.3.2

The two-soliton solutions

The two-bright-soliton solution of the NLS model can been obtained using the Hirota
method. The details are given in the appendix C. Its expression is given in (C.0.13), which
can be rewritten as
2
N
,
ψ0 = p
|η| D

(5.3.8)

where the overall phase i e−i φ , has been absorbed using the symmetry (5.1.8), and where we
have defined
z+

D = 2e



−∆

cosh z+ + e

| ρ1 | | ρ2 |
cosh z− − 16
cos (Ω1 − Ω2 − 2 δ+ )
Λ−



(5.3.9)

85

5.3 The parity properties of NLS solitons

and
h z+


z−
z
(Ω −Ω )
(Ω −Ω )
− 2 i δ−
−i 1 2 2
i 12 2
− 2−
2
N = e e e
e
e
e
| ρ1 | e
e + | ρ2 | e
e

i
z
z
z+
(Ω1 −Ω2 )
(Ω1 −Ω2 )
−
−
e 2 e−i 2δ+ + | ρ1 | e−i 2
e− 2 ei 2δ+ .
(5.3.10)
+ e 2 e−i δ− | ρ2 | ei 2
z+

−∆
2

−i

(Ω1 +Ω2 )
2

−i δ−

In this expression we use ∆ defined by
Λ−
(v1 − v2 )2 + 4 (ρ1 − ρ2 )2
=
Λ+
(v1 − v2 )2 + 4 (ρ1 + ρ2 )2

e∆ =

(5.3.11)

and the coordinates
z+ ≡ X1 + X2 + ∆

z− ≡ X1 − X2

(5.3.12)

with


Xi = ρi x − vi t −
where

(0)
xi



Ωi =



vi2
− ρ2i
4



t−

vi
x + θi + ζi
2


2 (ρ1 ± ρ2 )
.
δ± = ArcTan
(v1 − v2 )


i = 1, 2
(5.3.13)
(5.3.14)

The quantities Ωi are linear in x and t, and so in the new coordinates z± . We can therefore,
separate the homogeneous dependence on z± by writing
Ω1 − Ω2
− δ+ = Ω− + c,
2
Ω1 + Ω2
+ δ− = Ω+ + d,
2

(5.3.15)
(5.3.16)

where Ω± are homogeneous in z± , i.e. Ω± = β±+ z+ + β±− z− , with β±± being some constants
depending on vi and ρi , i = 1, 2. Note that the constants ζi appearing in the expression of
Ωi in (5.3.13) are the phases of z1 and w1 given in (C.0.10), and so depend on vi and ρi ,
i = 1, 2. However, the constants θi also appearing in (5.3.13) are the phases of a+ and b+
given in (C.0.9), and so are independent of vi and ρi . The constants c and d introduced in
(0)

(5.3.16) depend on vi , ρi and xi

(i = 1, 2) but are linear in θ1 − θ2 and θ1 + θ2 , respectively,

and so can be traded for θi , i = 1, 2, and be considered as constants independent of vi , ρi
(0)

and xi . Thus, the two-soliton solution (5.3.8) depends only on 8 free parameters; namely,
(0)

vi , ρi , xi

(i = 1, 2) c and d, and can be written as
∆

e− 2 −i Ω+
e
ψ0 = p
|η|

Nˆ
ˆ
D

(5.3.17)

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

86

where the overall phase e−i d , has been absorbed using the symmetry (5.1.8), and where we
have introduced
ˆ = cosh z+ + e−∆ cosh z− − 16 | ρ1 | | ρ2 | cos [2 (Ω− + c)]
D
Λ−

(5.3.18)

and


z−
z−
ei δ− | ρ1 | e−i (Ω− +c+δ+ ) e 2 + | ρ2 | ei (Ω− +c+δ+ ) e− 2


z
z−
z+
−i (Ω− +c−δ+ ) − 2−
−i δ−
i (Ω− +c−δ+ )
2
2
| ρ1 | e
e
+ | ρ2 | e
e
.
+ e e

Nˆ = e−

z+
2

(5.3.19)

We are now in a position to consider the parity transformation (5.1.37) relevant for the
two-bright-soliton solution, i.e.
P :

(z+ , z− ) → (−z+ , −z− )

(5.3.20)

which can be written in terms of x and t as
P :
and





x˜, t˜ → −˜
x, −t˜

with

(0)

x∆

(5.3.21)

(0)

∆(ρ1 v1 − ρ2 v2 ) + 2 ρ1 ρ2 (v2 x1 − v1 x2 )
=
(2ρ1 ρ2 (v2 − v1 ))
(0)

t∆

t˜ = t − t∆

x˜ = x − x∆

(0)

∆(ρ1 − ρ2 ) + 2ρ1 ρ2 (x1 − x2 )
=
.
(2ρ1 ρ2 (v2 − v1 ))

(5.3.22)

Note that under this parity transformation Ω± are odd since they are linear and homogeneous in z± . Therefore, if

π
c=n ,
n ∈ ZZ
(5.3.23)
2
ˆ given in (5.3.18), is invariant under the parity P . Consethe term cos [2 (Ω− + c)], in D,
ˆ is even under P
quently, D
 
ˆ = D.
ˆ
P D
(5.3.24)
In addition, when c satisfies (5.3.23), as one can check,

 
P Nˆ = (−1)n Nˆ ∗ .

(5.3.25)

Thus, the two-bright-soliton solution (5.3.17) satisfies
P (ψ0 ) = (−1)n ψ0∗

(5.3.26)

87

5.4 Numerical analysis

Figure 5.1 –

Plot of | ψ |2 against x for the one-soliton solution of the unperturbed NLS model.

In terms of the fields R and ϕ introduced in (5.1.18), i.e. for ψ0 =
P :

R0 → R0 ;

√

ϕ0

R0 ei 2 , we find that

ϕ0 → −ϕ0 + 2 π n

(5.3.27)

which is what we have assumed in (5.2.14).

5.4

Numerical analysis

In this section we present some numerical results which support the claims we have made
in the preceding sections.
Our results concern the NLS model and its deformation discussed in the last section i.e.
with the potential of the form (5.3.5). In our numerical studies we used a fixed lattice of
5001 points with time evolution calculated using the 4th order Runge-Kutta method. The
lattice step was taken to be dx=0.01 (and sometimes 0.05 or 0.1) and the time step used was
dt=0.00005. We used both fixed and absorbing boundary conditions (to avoid any reflections
from the boundaries) but as our field configurations were always very localised in the main
section of the lattice the results did not depend on the boundary conditions (we only considered
the evolution of the solitons when they were still some distance away from these boundaries.

5.4.1

The NLS model

Let us first present some of our results for the NLS model i.e., for ε = 0). The one soliton
solution, (5.1.11), for the case of v = 0, is shown in figure (5.1). In this figure we present a
plot of |ψ|2 as a function of x.
Next we have looked at several field configurations involving two solitons i.e., given by
(5.3.17). In this case we varied the values of the free parameter c. As mentioned in the last

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

88

section, when c is an integer multiple of

π
2

the two-soliton field configuration (5.3.17) is an

eigenfunction of P in the sense of (5.3.27). We have followed the field configuration given
by (5.3.17) and have used this field configuration as an initial condition for a full simulation
and the results were virtually indistinguishable from each other. This has provided a test of
our numerical procedure. In figure (5.2) (a,b and c) we present plots of the position of one
soliton as a function of time for 3 different values of c, namely c = 0, c = 0.7 and c = 1.4.
The position was determined by looking at the maxima of the energy density. The trajectory
of the other soliton is symmetric to this one and to the right. We notice a slight dependence
on the values of c.

(a) c=0
Figure 5.2 –

(c) c=1.4

Trajectories of two Solitons at v = 0.4 (ǫ = 0)

(a) c=0
Figure 5.3 –

(b) c=0.7

(b) c=0.7

(c) c=1.4

Trajectories of two solitons at rest (ǫ = 0)

The existence of multi-soliton solutions does not directly describe the forces between the
solitons. Of course, one can deduce them by analysing in detail the time dependence of their
positions etc. Another way to proceed involves putting two solitons at rest, not too close (not
to deform them) and not too far away (so that they do interact) and see what will happen.
We have performed such a study and in (5.3) we present the trajectories of the soliton “on

89

5.4 Numerical analysis

(a) c=0

(b) c=0.7

(c) c=1.4

(d) c=0.01
Figure 5.4 –

Heights of the solitons originaly at rest (ǫ = 0)

the left” with respect to time for 3 values of the relative phase between them (equivalent to c).
We see that at c = 0 the solitons attract, at c = 0.7 the forces are quite complicated resulting
in a rather complicated trajectories and for c = 1.4 they repel. However, the parameter c has
also another role and this is associated with the heights of the solitons. When c = 0 both
solitons, when they move towards each other, stay of the same size but as they come towards
each other they overlap and some appear to be taller. When c 6= 0 the situation is more

complicated. The non-zero value of c breaks the symmetry and so one soliton tends to grow
while the other, to decrease in size. For this to happen they have to interact and so be close
enough; the two effects (both of them growing and one of them growing and the other one
getting smaller) produce a more complicated pattern of their sizes and, in part, is responsible
for their repulsion and never being able to come very close to each other. Hence the effect of
them overlapping is very small. In figure (5.4). we present the time dependence of the heights
of the solitons for the cases of c = 0.3 and c = 0.01. The first two pictures (from the left)
show the time dependence of the heights of the two solitons for c = 0.3, and the other two for
c = 0.01. The extremum of height seen in plots a) and b) corresponds to the case when the
two solitons are at the closest distance from each other. In the plots c) and d) we note that
after the scattering the values of the heights are slightly different. This may appear strange
at first but the two solitons move with marginally different velocities after the scattering; this

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

90

effect is induced during the scattering by the non-zero value of c.
And what about the conserved charges? Well, the NLS model is integrable so that all
anomalies vanish (and so all charges are conserved). In the next subsection we look at the
same problems for ǫ 6= 0 i.e., when the model is not integrable.

5.4.2

The modified model with ε 6= 0

Next we have considered the ε 6= 0 cases. This time we have only one soliton solution

(5.3.6) which is a simple deformation of the one soliton of the NLS model (5.1.11). In fact,
when one plots it for small values of ε it is hard to see any difference.

As for ε 6= 0 the model is non-integrable and we do not have analytic expressions involving

two solitons. Hence we can only use two one solitons some distance apart or use the two-

soliton solutions of the NLS model i.e., the expression for ε = 0) and take them as the initial
conditions for our numerical simulations.
In figure (5.5) we present the plots of the trajectories of one soliton (similar to figure
(5.2)) for ε = 0.06 for 3 values of c.

(a) c=0
Figure 5.5 –

(b) c=0.7

(c) c=1.4

Trajectories of two solitons at v = 0.4 (ǫ = 0.06)

Looking at the trajectories and comparing them to those of the NLS model we see very
little difference. The same was observed for other values of ε. In fact these trajectories
were obtained by starting with initial configurations corresponding to the NLS model and
then evolving them with ε 6= 0. We have also looked at the effects of evolving the initial
configurations described by two ε 6= 0 solitons ‘sewn’ together. The obtained trajectories were

very similar. This is due to the fact that the solitons are well localised and all the perturbations

induced by taking non-exact expressions were very small.
Next we looked at two solitons at rest. In this case we have taken the expressions for two

91

5.4 Numerical analysis

solitons corresponding to ε 6= 0 placed next to each other. In figure (5.6) and (5.7) we present

the plots similar to those of figure (5.3) for ε = 0.06 and for ε = −0.06.

(a) c=0.

(b) Energy for c=0.

(c) c=0.7.

(d) c=1.4.
Figure 5.6 –

Trajectories (and the energy) of two solitons at rest (ǫ = 0.06)

Comparing these plots with those of figure (5.3) we see only little difference. The dependence on c is very similar although the strength of the attraction (or repulsion) does appear
to depend on ǫ. Clearly the overall attraction (at least for c = 0) increases with the increase
of ǫ. In addition, we note that for c = 0, in the NLS case, the solitons oscillate around their
point of attraction while for ǫ 6= 0 the amplitude of their oscillation decreases (see figure

(5.3(a)) and compare with figures (5.6(a)) and (5.7(a))). This suggests that for ε 6= 0 the

solitons radiate a little and so come closer and closer to each other after each oscillation. This
is indeed the case as can be seen from the expressions of the total energy (for c = 0, the

energy is effectively conserved while for c 6= 0 it decreases a little (see figures 6b) and 7b).

After a while, however, during these interactions, they gradually change their height and then
they split up, repel and move away from each other. During this last part of the motion they
move with slightly different velocities and so their sizes are also slightly different. In this their
behaviour resembles the c 6= 0 case; so we note that as ε 6= 0 the interaction between the

solitons gradually induces their behaviour as if c were not 0. In figure (5.8) we plot the heights

of the two solitons observed in the scattering in the ε = 0.06, c = 0 case. Figure (5.8(a))
corresponds to the case of the left hand one, and figure (5.8(b)) - the right one.

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

92

(a) c=0.

(b) Energy for c=0.

(c) c=0.7.

(d) c=1.4.
Figure 5.7 –

Trajectories (and the energy) of two solitons at rest (ǫ = −0.06)

(a) The left one.
Figure 5.8 –

(b) The right one.

Heights of the two solitons observed in their scattering at rest (ǫ = −0.06 c = 0)

Furthermore, in the last section we did stress that the cases of c given by (5.3.23) are
special for all ε’s as then we could use our parity arguments to claim asymptotic conservation
of further anomalous conserved quantities (5.1.52).
So we have looked at the first nontrivial anomaly. To get its form we used the expression
of our potential (5.3.5) and so calculated X from the second formula in (5.1.24). Then we
put it into the formula for α(3,−4) given in (B.0.4). In order to avoid using the explicit value
of t∆ , which for the zero order solution (expanded in ε) is given in (5.3.22), we decided to
integrate the resultant expression for β4 . Therefore, using (5.1.24), (5.1.35) and (B.0.4) we

93

5.4 Numerical analysis

introduce the quantity
Z t
Z t
Z ∞
(4)
′
′
χ (t) ≡
dt β4 =
dt
dx X α(3,−4)
(5.4.1)
−∞
−∞
−∞


Z t
Z ∞
3
3
2 2
2
2
2
′
ε
3
= −2 i η
dt
dx (R − 1) 6 η R + (∂x ϕ) R − 2 R ∂x R + (∂x R)
2
2
−∞
−∞
As at large values of t′ the integrand in (5.4.1) vanishes, we can take, in our numerical
simulations, the lower end of the t′ -integral to be large in the past but finite. It is the quantity
χ(4) given in (5.4.1) whose plots we present next.
Clearly for ε = 0 the anomaly vanishes so in fig 9 and 10 we present our results for
ε = 0.06 and in fig 11 and 12 those for ε = −0.06.
The first figures in each group show the anomaly when the solitons were sent towards
each other at v = 0.4 and the second ones (10 and 12) those started at rest. In each case the
first figure corresponds to the special value of c, i.e. c = 0, the others to c = 0.7 and c = 1.4.
Note that the scale on the vertical axis in the figures is very different. The anomaly for the
cases corresponding to c = 0 is essentially zero thus supporting our claims of the previous
section. Of course, our results are non-perturbative but they do involve also small corrections
due to the numerical errors. In any case the smallness of the corrections suggest to us that our
claims are correct and the results are stable with respect to small perturbations. For c 6= 0 we
do see some important corrections to the anomaly as expected (even though the differences
of the trajectories are not very significant).

(a) c=0.
Figure 5.9 –

(b) c=0.7.

(c) c=1.4.

Time integrated anomaly of two solitons sent at v = 0.4 (ǫ = 0.06)

5 Quasi-integrable deformation of the non-linear Schr¨odinger equation

94

(a) c=0.
Figure 5.10 –

(b) c=0.7.

(c) c=1.4.

Time integrated anomaly of two solitons sent at v = 0.4 (ǫ = −0.06)

(a) c=0.
Figure 5.12 –

(c) c=1.4.

Time integrated anomaly of two solitons at rest (ǫ = 0.06)

(a) c=0.
Figure 5.11 –

(b) c=0.7.

(b) c=0.7.

Time integrated anomaly of two solitons at rest (ǫ = −0.06)

(c) c=1.4.

95

CHAPTER 6

Final Comments

This thesis is about using methods and/or ideas from integrable field theories to investigate
non-integrable ones. We found that in gauge theories it is possible to construct charges that
are not, in principle, related to the Noether’s charges, in an analogous way one gets the hidden
charges in integrable theories in (1 + 1)-dimensional space-time, in many cases responsible for
their integrability and/or existece of solitonic solutions. The path-independence leading to
these charges is generalised to what we call surface-independence, volume-independence etc.
and this property is obtained from the integral equations we proposed, which are like a flux
equation, based on the generalised Stokes theorem introduced for the first time in (6). The
charges calculated in this way are invariant under any gauge transformation, which is not the
case for the standard charges known for non-Abelian gauge theories in the literature. Moreover,
their conservation comes from the integral equation, and not from topology or some other
features of particular solutions. In this sense, these charges are very general. As presented
in appendix A we developed a regularisation method of the Wilson lines to deal with the
singularities of the gauge connection. This is of major importance for our calculations but
also can help to treat this common problem when dealing with holonomies in other cases.
Our results open an interesting perspective in non-Abelian gauge theories: we showed how the
loop space can be a powerful tool in this context. Certainly a deeper mathematical analysis
can bring even more structures to our attention. We calculated explicitly the charges of some
configurations of Yang-Mills theory in (3 + 1)-dimensional space-time, but as we understand,
there are a finite number of them. Although the path-independence of the Wilson line that
appear in (1 + 1)-dimensional integrable theories was extended, we could not understand yet
what would play the role of the spectral parameter. This would lead to the possibility of
investigation of integrability of gauge theories. The parameters α and β appearing in the
integral equations and after that in the charge operators and finally in the charges remain to
be understood. So far we do not see how to fix them, and this fixing is crucial for the value
of the charge.

6 Final Comments

96

We also discussed here the concept, recently introduced by Luiz Ferreira and Wojciech
Zakrzewski, of quasi-integrability in the context of the deformations of the NLS model in
(1 + 1) dimensions . The unperturbed model is fully integrable and possesses multi-soliton
solutions. The perturbations destroy integrability but the perturbative models still possess
soliton solutions.
We have looked at the problem of quasi-integrability and in this case related it to the properties of specific field configurations (like those describing multi-solitons) under very specific
parity transformations. It was showed that when one considers the perturbed models which
are not integrable, the models do not possess an infinite number of conserved charges (like the
integrable ones do). However, when one restricts the attention to specific field configurations,
sometimes it is possible to say more. Namely, when the field configurations satisfy some very
specific parity conditions (which are often physical in nature) the extra charges, though not
conserved, do satisfy some interesting conditions (given in (5.1.52)). These conditions do
restrict the scattering properties of solitons and so provide the basis of our understanding of
quasi-integrability. We have also looked at the properties of the soliton field configurations
numerically and have found a good support of our claims.

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101

APPENDIX A

Regularisation of Wilson lines

In order to calculate the r.h.s. of the relations (4.3.4) and (4.3.13), which give the
conserved charges, as volume ordered integrals, for the Wu-Yang monopole and dyon solutions
respectively, we have to evaluate the Wilson line operator W , defined by (2.1.1), for the
connection

1
xj
Ai = − ǫijk 2 Tk
(A.0.1)
e
r
which has a singularity at the origin of the coordinate system. In the case of the dyon the time
component of the connection is non-zero and also present a singularity at the origin. However,
it does not play a role in the charge calculation since all the Wilson line operators are defined
on space curves with no time component. We show in this appendix how the Wilson line

operator can be regularised, when it is integrated along a purely spatial (no time) curve Γ
˜ ε ◦ Γ1 ,
passing through the origin. In order to do that we shall split Γ in three parts, Γ = Γ2 ◦ Γ

as shown in part I of the Figure (A.1).

Figure A.1 –

The regularisation of the Wilson line operator is done by replacing the path that passes
through to the origin by a path going around it.

The solution of (2.1.1) can be written as
W = WΓ2 · WΓ˜ ε · WΓ1

(A.0.2)

The quantities WΓ1 and WΓ2 do not involve the singularity and so we should not worry about

Appendix A -- Regularisation of Wilson lines

102

˜ ε infinitesimally
them. We have to evaluate WΓ˜ ε which pass through the origin. We shall take Γ
small in such a way that we can approximate it by an infinitesimal straight line of length 2 ε
containing the origin in its middle point. Note that the quantity Ai

d xi
,
dσ

appearing in (2.1.1),

is invariant under rotations, and so we can rotate the coordinate system in such a way that
˜ ε and in the direction of growing σ, i.e. in the sense of integration
the x3 -axis lies parallel to Γ
˜ ε,
of (2.1.1), as shown in part II of the Figure A.1. Along such infinitesimal straight line Γ
parametrised as x3 = σ, we have that Ai

d xi
dσ

= − 1e

1
r2

˜ ε one
[x1 T2 − x2 T1 ]. However, on Γ

has x1 = x2 = 0, and so for r 6= 0 such expression vanishes. On the other hand, for r = 0 it
diverges, and so, we have a quite ill defined quantity.

˜ ε by a semi-circle Γε of radius ε,
In order to regularise the Wilson line we shall replace Γ
with diameter being the previous straight line, and lying on the plane x1 x3 as shown in part
II of Figure A.1. We evaluate W (Γε ) on such semi-circle and then take the limit ε → 0. The

points in Γε can be parametrised as
x1 = ε sin σ

x2 = 0

x3 = −ε cos σ

0≤σ≤π

(A.0.3)

Therefore for all such points we have r = ε, and so from (A.0.1) one has
Ai

1
d xi
= ε (A1 cos σ + A3 sin σ) = − T2
dσ
e

(A.0.4)

We note that it does not depend upon ε and σ, and lies in the direction of just one generator
of SU(2). Therefore, the problem is Abelian and the path ordering is not necessary. Then
from (2.1.1) we have
W (Γε ) = ei π T2
(A.0.5)
 
˜ ε . Of course, we would obtain different
which we take as the regularised expression for W Γ
results for different choices of curves going around the origin, specially non-planar curves.

However, as we will show below, the evaluation of the r.h.s of the relations (4.3.4) and
(4.3.13) is independent of such choices, and the regularisation of those quantities is quite
unique.
Note that for the Wu-Yang monopole and dyon solutions one has
Fij =

1
nk
ǫijk 2 n
ˆ·T ;
e
r

γ
nk
Feij = − ǫijk 2 n
ˆ·T
e
r

(A.0.6)

with γ = 0 in the pure monopole case. In the evaluation of (4.3.6) and (4.3.14) we have to
deal with the conjugated quantities FijW and FeijW , and so essentially we have to worry about
the quantity W −1 n
ˆ · T W . Our prescription is to scan the volume (the whole space) with

closed surfaces based at xR , and each of those surfaces are scanned with loops based at xR .

103

Appendix A -- Regularisation of Wilson lines

The origin lies on a given surface labelled by ζ0 , and to just one loop, labelled by τ0 , on
that surface, and corresponding to the point labelled by σ0 on that loop. For the surfaces
corresponding to ζ < ζ0 there are no problems in the integration since everything is regular.
On each loop on those surfaces W −1 n
ˆ · T W is constant and equal to TR , i.e. the value of

n
ˆ · T at the reference point xR .

Therefore, the commutators in (4.3.6) and (4.3.14) vanish for ζ < ζ0 , since the conjugated
tensors F W and FeW all lie in the direction of TR on any point of any loop scanning the surfaces
ij

ij

for ζ < ζ0 . On the surface for ζ = ζ0 everything is fine until we reach the loop corresponding

to τ = τ0 . In other words, the commutators in (4.3.6) and (4.3.14) also vanish for ζ = ζ0

and τ < τ0 . Let us consider the loop corresponding to τ = τ0 . For σ < σ0 we still have the
vanishing of those commutators since the singularity has not been touched yet. After crossing
the singularity we have that the Wilson line W becomes W2 W (Γε ) W1 (see (A.0.2)), where
W1 is the result of the integration of (2.1.1) along Γ1 , i.e. the curve from the reference point
xR up to the point marked −ε on Figure A.1, along the loop corresponding to τ = τ0 , which

passes through the origin. Similarly W2 is obtained by integrating (2.1.1) along Γ2 , i.e. the

curve from the point marked ε on Figure A.1, up to some generic point beyond the origin
along that same loop. In addition, W (Γε ) is the regularised expression, given in (A.0.5), for
˜ ε.
the integration of (2.1.1) along Γ
Along the curve Γ2 the connection (A.0.1) is regular, and so
W2−1 n
ˆ · T W2 = (ˆ
n · T )Γ0

(A.0.7)

2

where (ˆ
n · T )Γ0 is the value of n
ˆ · T at the initial point of the curve Γ2 , which is the point
2

marked ε on Figure A.1. But since we have rotated the coordinate system such that the
˜ ε , we have that (ˆ
x3 -axis lies along Γ
n · T ) 0 = T3 . Now using (A.0.5), we have that
Γ2

W −1 (Γε ) W2−1 n
ˆ · T W2 W (Γε ) = e−i π T2 T3 ei π T2 = −T3 = (ˆ
n · T )Γend
1

(A.0.8)

since −T3 is the value of n
ˆ · T at the final point of the curve Γ1 , which is the point marked

−ε on Figure A.1. Along the curve Γ1 the connection (A.0.1) is regular, and we have
W1−1 W −1 (Γε ) W2−1 n
ˆ · T W2 W (Γε ) W1 = W1−1 (ˆ
n · T )Γend W1 = TR
1

(A.0.9)

where TR is the value of (ˆ
n · T ) at the reference point xR which is the initial point of Γ1 .

Therefore, the field tensor and its dual, given in (A.0.6), lie in the direction of TR when
conjugated with W2 ·W (Γε )·W1 , and so the commutators in (4.3.6) and (4.3.14) vanish when

evaluated on the loop corresponding to τ = τ0 , i.e. the one passing through the singularity of

Appendix A -- Regularisation of Wilson lines

104

(A.0.1). Of course, the quantities (4.3.6) and (4.3.14) will vanish on all loops scanning the
surfaces for ζ > ζ0 , since the potential (A.0.1) is not singular there, and W −1 n
ˆ · T W = TR ,
for W obtained by the integration of (2.1.1) on such loops.

Consequently all the commutators in (4.3.6) and (4.3.14) vanish on any loop on the
scanning of any surface on the scanning of the volume. Since the Wu-Yang solutions have
no sources we have J˜123 = J0 = 0, and so Jmonopole and Jdyon also vanishes. Therefore we
conclude that the r.h.s. of (4.3.4) and (4.3.13) are equal to unity, i.e.
R

P3 e

space

dζdτ V Jmonopole V −1

R

= 1l ;

P3 e

space

dζdτ V Jdyon V −1

= 1l

(A.0.10)

We now come to the issue of the uniqueness of the regularisation procedure. We have
˜ ε by the semi-cicle Γε . Let us now analyze what happens to the
chosen to replace the segment Γ
quantity W −1 (Γε ) (ˆ
n · T )Γend W (Γε ) = W −1 (Γε ) (−T3 ) W (Γε ), when we make arbitrary
1

infinitesimal variations on the semi-circle Γε keeping its end points fixed, i.e. the points
marked ε and −ε on Figure A.1. We have




δ W −1 (Γε ) (−T3 ) W (Γε ) = W −1 (Γε ) (−T3 ) W (Γε ) , W −1 (Γε ) δW (Γε )


(A.0.11)
= T3 , W −1 (Γε ) δW (Γε )

where in the last equality we have used (A.0.5) and (A.0.8). The variation of the Wilson line
can be easily evaluated using for instances the techniques of section 2 of (6). When the end
points of the curve Γε are kept fixed one gets
Z
−1
W (Γε )δW (Γε ) =

0

π

dσ W −1 Fij W

dxi j
δx
dσ

(A.0.12)

where Fij is the curvature, given in (A.0.6), of the connection (A.0.1), and where W in the
integrand in (A.0.12), is obtained by integrating (2.1.1) along Γε , from its initial point at
σ = 0 to the point σ = σ where the tensor Fij is evaluated. As long as the transformed curve
does not pass through the singularity of the connection (A.0.1), the relations Di (n · T ) = 0

and

d
dσ

(W −1 n · T W ) = 0 can be used to show that W −1 n
ˆ · T W = −T3 , where −T3 is the

value of n
ˆ · T at the initial point of Γε . Therefore, the integrand in (A.0.12) always lies in the
direction of T3 , and so



δ W −1 (Γε ) (−T3 ) W (Γε ) = 0

(A.0.13)

Consequently any curve Γ, with the same end points as Γε , and that can be continuously
deformed into Γε , satisfies W −1 (Γε ) (−T3 ) W (Γε ) = W −1 (Γ ) (−T3 ) W (Γ) = T3 . That
shows that our prescription for the regularisation of the Wilson line is independent of the
˜ ε.
choice of the curve replacing the segment Γ

Appendix A -- Regularisation of Wilson lines

105

Note that the special role being played by T3 is an artifact of our choice of the orientation
of the coordinate axis with respect to the curve. Note in addition that our results do not
imply that the Wilson line does not change. It is just the conjugation of T3 by the Wilson line
that remains invariant. In the cases where the variation of the curve lies on the same plane as
Γε , then the Wilson line itself is invariant. The reason is that the r.h.s of (A.0.12) measures
the magnetic flux through the infinitesimal surface spanned by the variation, and since the
magnetic field is radial it is parallel to such surface, and so δW (Γε ) = 0 in such cases.

106

Appendix A -- Regularisation of Wilson lines

107

APPENDIX B

Explicity quantities involved in
equation (5.1.25)
(−n)

We give in this appendix the first few explicit expressions for the parameters ζi

, i = 1, 2,

introduced in (5.1.26), for the components ax of the connection defined in (5.1.25), and the
quantities α(j,−n), j = 1, 2, 3, introduced in (5.1.33). On the r.h.s. of the equations below we
use the following notation: (for partial derivatives w.r.t. x and t)
⋆(n,m) ≡ ∂xn ∂tm ⋆
(−n)

The expressions for ζi
(−1)

ζ1

(−1)

ζ2

(−2)

ζ1

(−2)

ζ2

(−3)

ζ1

(−3)

ζ2

(−4)

ζ1

(−4)

ζ2

(B.0.1)

are:

= 0,
p
√
= 2 | η | R,
p
i | η |R(1,0)
√
=
,
R
p
√
=
| η |ϕ(1,0) R,
p

p
i
| η |ϕ(1,0) R(1,0) + | η |ϕ(2,0) R
√
,
=
(B.0.2)
R
p
p
p

2

2
16 | η |3/2 σR3 + 3 | η | ϕ(1,0) R2 − 6 | η |R(2,0) R + 3 | η | R(1,0)
=
,
6R3/2
h
p
p

2
i
64 | η |3/2 σR(1,0) R3 + 9 | η | ϕ(1,0) R(1,0) R2 + 18 | η |ϕ(1,0) ϕ(2,0) R3
=
5/2
12R
p
p
p

i
(3,0) 2
(1,0) (2,0)
(1,0) 3
− 12 | η |R
,
R + 18 | η |R
R
R−9 |η | R
h
p
p
1
(2,0) (1,0)
3/2
(1,0) 3
|
η
|ϕ
R
R
−
6
| η |ϕ(1,0) R(2,0) R
=
16
|
η
|
σϕ
R
−
6
3/2
4R
i
p
p

2 p

3
+ 3 | η |ϕ(1,0) R(1,0) + | η | ϕ(1,0) R2 − 4 | η |ϕ(3,0) R2 .

Appendix B -- Explicity quantities involved in equation (5.1.25)

108
(3,n)

The components ax

introduced in (??) are:

1 (1,0)
iϕ
,
2
= 2i | η | σR,

a(3,0)
=
x
a(3,−1)
x

a(3,−2)
= i | η | σϕ(1,0) R,
(B.0.3)
x






2
2
i | η | 4 | η | R3 + σ ϕ(1,0) R2 − 2σR(2,0) R + σ R(1,0)
(3,−3)
ax
=
,
2R



2
i|η|h
(3,−4)
12 | η | ϕ(1,0) R3 − 6σR ϕ(2,0) R(1,0) + ϕ(1,0) R(2,0) + 3σϕ(1,0) R(1,0)
ax
=
4R
 i


3
+ σ ϕ(1,0) − 4ϕ(3,0) R2 .

The quantities α(j,−n) , introduced in (5.1.33) are:
α(3,0) = 1,
α(3,−1) = 0,
α(3,−2) = 2 | η | σR,

(B.0.4)

α(3,−3) = 2 | η | σϕ(1,0) R,
α

(3,−4)

and



3 | η | σ R(1,0)
3
(1,0) 2
(2,0)
= 6|η| R + |η|σ ϕ
R − 2 | η | σR
+
2
2R
2

2


2

α(1,0) = 0,

p
√
α(1,−1) = −2 | η | R,
p
√
α(1,−2) = − | η |ϕ(1,0) R,

α

(1,−3)

α(1,−4)

p



(1,0) 2

p

(B.0.5)


√
|η| R
| η |R(2,0)
1p
(1,0) 2
√
,
+
= −4 | η | σR −
|η| ϕ
R−
2
2R3/2
R
p

2
p
3 | η |ϕ(1,0) R(1,0)
3 | η |ϕ(1,0) R(2,0)
3/2
(1,0) 3/2
√
= −6 | η | σϕ R −
+
4R3/2
2 R
p
p
√

3 √
3 | η |ϕ(2,0) R(1,0) 1 p
√
−
+
| η | ϕ(1,0)
R + | η |ϕ(3,0) R
4
2 R
3/2

3/2

Appendix B -- Explicity quantities involved in equation (5.1.25)

109

and
α(2,0) = 0,
α(2,−1) = 0,
α

(2,−2)

=

α(2,−3) =
α(2,−4) =
+

p
i | η |R(1,0)
√
−
,
R
p
p
√
i | η |ϕ(1,0) R(1,0)
√
− i | η |ϕ(2,0) R,
−
(B.0.6)
R
p

2
√
√
3i | η | ϕ(1,0) R(1,0) 3 p
(1,0)
3/2
√
−
− i | η |ϕ(1,0) ϕ(2,0) R
−6i | η | σ RR
2
4 R
p
p
p


3
3i | η | R(1,0)
3i | η |R(2,0) R(1,0) i | η |R(3,0)
√
−
+
.
4R5/2
2R3/2
R

110

Appendix B -- Explicity quantities involved in equation (5.1.25)

111

APPENDIX C

The Hirota solutions

Here we construct the one and two bright soliton solutions of the integrable NLS theory
(5.1.10) using the Hirota method. The one and two dark soliton solutions require a different
procedure from the one described here. We introduce the Hirota tau-functions as
ψ0 =

i τ+
;
γ τ0

i τ−
ψ¯0 = −
,
γ¯ τ0

(C.0.1)

where η = γ γ¯ . The bright soliton solutions exist for η < 0 and so we need γ¯ = −γ ∗ , and
 ∗
τ−
then τ0 = − ττ+0 . Putting (C.0.1) into the the NLS equation (5.1.10) and its complex
conjugate we get the two Hirota equations



τ02 i∂t τ+ + ∂x2 τ+ − 2τ0 ∂x τ+ ∂x τ0 − 2τ+2 τ− −


τ0 τ+ i∂t τ0 + ∂x2 τ0 + 2τ+ (∂x τ0 )2 = 0,


τ02 −i∂t τ− + ∂x2 τ− − 2τ0 ∂x τ− ∂x τ0 − 2τ−2 τ+ −


τ0 τ− −i∂t τ0 + ∂x2 τ0 + 2τ− (∂x τ0 )2 = 0.

(C.0.2)

The one-soliton solution of (C.0.2) is given by
z1 z2
eiΓ(z1 ) e−iΓ(z2 ) ,
2
(z1 − z2 )
−iΓ(z2 )
= a− z2 e
,

τ0 = 1 + a+ a−
τ+

τ− = a+ z1 eiΓ(z1 )

(C.0.3)

with a± , z1 and z2 being complex parameters and Γ (zi ) = zi2 t − zi x. We choose z2 = z1∗

and a− = −a∗+ , which implies that τ− = −τ+∗ , and τ0 is real. We then parametrize them as
r
p
p
v2
v
+ ρ2 eiζ ,
γ = i | η | eiφ ,
a± = i a e±i θ ,
z1 = + i ρ =
γ¯ = i | η | e−iφ
2
4
(C.0.4)

Appendix C -- The Hirota solutions

112

with a > 0, and v and ρ both real. We replace a by x0 defined as
q
v2
+ ρ2
4
a
= e−ρ x0
2 |ρ|

(C.0.5)

and find from (C.0.1) that
ψ0 =

−i (θ+ζ+φ)

ie

p

|η|

i

|ρ|


i
h
2
ρ2 − v4 t+ v2 x

e
.
cosh [ρ (x − v t − x0 )]

(C.0.6)

This expression, up to an overall constant phase factor (due to the symmetry (5.1.8)) is
the one-bright-soliton given in (5.1.11).
The two-soliton solution of (C.0.2) is given by
z1 z2
w1 w2
ei Γ(z1 ) e−i Γ(z2 ) + b+ b−
ei Γ(w1 ) e−i Γ(w2 )
2
2
(z1 − z2 )
(w1 − w2 )
w1 z2
z1 w2
i Γ(z1 ) −i Γ(w2 )
e
e
+ a− b+
e−i Γ(z2 ) ei Γ(w1 )
a+ b−
2
2
(z1 − w2 )
(w1 − z2 )
2
z1 z2 w1 w2 (z1 − w1 ) (z2 − w2 )2
a+ a− b+ b−
ei Γ(z1 ) e−i Γ(z2 ) ei Γ(w1 ) e−i Γ(w2 ) ,
(z1 − z2 )2 (w1 − w2 )2 (z1 − w2 )2 (w1 − z2 )2
w2 z1 z2 (w2 − z2 )2 i Γ(z1 ) −i Γ(z2 ) −i Γ(w2 )
e
e
e
a− z2 e−i Γ(z2 ) + b− w2 e−i Γ(w2 ) + a+ a− b−
(w2 − z1 )2 (z1 − z2 )2
w1 w2 z2 (w2 − z2 )2 −i Γ(z2 ) i Γ(w1 ) −i Γ(w2 )
a− b+ b−
e
e
e
,
(w1 − w2 )2 (w1 − z2 )2
w1 z1 z2 (w1 − z1 )2 i Γ(z1 ) i Γ(w1 ) −i Γ(z2 )
e
e
e
a+ z1 ei Γ(z1 ) + b+ w1 ei Γ(w1 ) + a+ a− b+
(w1 − z2 )2 (z1 − z2 )2
w1 w2 z1 (w1 − z1 )2 i Γ(z1 ) i Γ(w1 ) −i Γ(w2 )
(C.0.7)
a+ b+ b−
e
e
e
,
(w1 − w2 )2 (w2 − z1 )2

τ0 = 1 + a+ a−
+
+
τ+ =
+
τ− =
+

where a± , b± , z1 , z2 , w1 and w2 are arbitrary complex parameters, and as before, Γ (wi ) =
wi2 t − wi x. The two-bright-soliton solution of the NLS theory (5.1.10), corresponding to

η < 0, is obtained by taking τ− = −τ+∗ , and τ0 real. One way of getting this involves putting
z2 = z1∗ ,

w2 = w1∗ ,

a− = −a∗+ ,

b− = −b∗+

(C.0.8)

p
γ = i | η | eiφ ,

p
γ¯ = i | η | e−iφ

(C.0.9)

and then parametrizing them as
a± = i a1 e±i θ1 ,

b± = i a2 e±i θ2 ,

with ai > 0, i = 1, 2, and
r
v12
v1
+ i ρ1 =
+ ρ21 eiζ1 ,
z1 =
2
4

v2
w1 =
+ i ρ2 =
2

r

v22
+ ρ22 eiζ2 .
4

(C.0.10)

113

Appendix C -- The Hirota solutions

This gives us



v12
v1
2
Γ (z1 ) =
t − z1 x =
− ρ1 t − x − i ρ1 (x − v1 t) ,
4
2

 2
v2
v2
− ρ22 t − x − i ρ2 (x − v2 t) .
Γ (w1 ) = w12 t − w1 x =
4
2
z12

(C.0.11)

(0)

Finally, we replace ai by xi , i = 1, 2, defined as
q
vi2
+ ρ2i
(0)
4
ai
= e−ρi xi .
2 | ρi |

(C.0.12)

Putting all these expressions into (C.0.7) and into (C.0.1) we obtain the final form of the
two-bright-soliton solution:
i 2 e−i φ
p
×
|η|




Λ− (X1 +X2 )
X1 −i 2(δ+ +δ− )
X2 i 2(δ+ −δ− )
X1
X2
W2 e e
+ W1 e e

 W1 e + W2 e + Λ+ e
× 
,
h i2
1 + e2 X1 + e2 X2 + ΛΛ−+ e2 (X1 +X2 ) − 32 |ρ1Λ| +|ρ2 | cos (Ω1 − Ω2 − 2 δ+ ) e(X1 +X2 )

ψ0 =

(C.0.13)

where
2

2

Λ± = (v1 − v2 ) + 4 (ρ1 ± ρ2 ) ;



2 (ρ1 ± ρ2 )
δ± = ArcTan
(v1 − v2 )



(C.0.14)

and
Wi =| ρi | e−i Ωi

(C.0.15)

with
Ωi =



vi2
− ρ2i
4



vi
t − x + θi + ζi ,
2



Xi = ρi x − vi t −

(0)
xi



i = 1, 2.
(C.0.16)