duality in physics
TRANSCRIPT
Duality in Physics
Dominic Barker and Leon FordSupervisor: Dr Paul Saffin
School of Physics and Astronomy
University of Nottingham
UK
31.05.2016
Abstract
This report discusses some of the important dualities in physics, noting that they arise in a multitude of different
areas. One such duality is the AdS/CFT duality, which has influenced research across physics. Dualities can
offer insight into the fundamental behaviour of the Universe, connecting different formalisms of the same theory.
This report examines theories which can produce topological defects and Q-balls and notes their key individual
properties. We analyse the Kibble mechanism which can produce topological defects, these are believed to arise
in areas of physics such as cosmology and condensed matter. We then explain some key properties of topological
defects, arriving at a description of the complex kink. Q-balls are then similarly analysed, these are believed to
exist in some theories of baryogenesis and dark matter. The report then proceeds to prove that there is a particular
classical duality between the static complex kink and the 1 + 1 dimensional stationary Q-ball, with an interchange
of topological charge and Noether charge. The proven duality is also explored numerically and the computational
limitations are discussed.
Contents
1 Introduction 2
1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Applications of Q-balls and Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Kibble Mechanism and the Monopole Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 Baryogenesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Dualities in Physics 5
2.1 Early Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 String Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 T-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.2 S-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 AdS/CFT Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Thirring Model and the Sine-Gordon Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3 Topological Defects 10
3.1 The Kink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.1.1 Static Kink Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1.2 Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Higher Dimensional Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.1 The Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2.2 Derrick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.1 Vacuum Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3.2 Homotopy Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3.3 Winding Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Complex Kink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4.1 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4.2 Topological Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Q-balls 24
4.1 Noether’s Theorem and Conserved Charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Q-ball Action and Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.3 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5 Duality 28
5.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Duality Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 Proving the Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Specifying the Duality to the Stationary Q-ball and Static Kink 31
6.1 Dual Q-ball Analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.2 Dual Complex Kink Analytics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.3 Comparing the Complex Kink and the Dual Q-ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
6.4 Numerical Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Q-ball Dynamics 42
8 Conclusions and Summary 42
1
1 Introduction
The term duality, in physics, describes an equivalence between two seemingly different theories which are, in fact,
two different prescriptions of the same underlying theory. There are many dualities in physics, the most famous
being the dualities within string theory that link the variations together. The duality between the Sine-Gordon
Model and the Thirring model in 1+ 1 dimensions emphasise the fact that the theories can be completely different:
The Sine-Gordon Model is a bosonic model whilst, the Thirring model describes a fermionic field[1]. Dual theories
are related by a set of Duality Transformations, which map observables in one theory to the observables in the
respective dual theory. In the Sine-Gordon/Thirring Duality one transformation relates the coupling constants.
This, in turn, allows for observables to be calculated in non-perturbative strongly coupled theory by considering
perturbative techniques in the weakly coupled theory. The dualities discussed above will be considered in more
detail in the second section to familiarise the reader with the concept of duality. This will be aided by a historical
account of dualities in physics. However, the primary purpose of this project is to prove one duality in particular:
the duality between the 1+ 1 dimensional static complex kink and the 1+ 1 dimensional stationary complex Q-ball.
The duality between the kink and the Q-ball will be proven by considering a set of duality transformations between
the two theories and by showing a relationship between the equations of motion of both theories. Firstly however,
the properties of both the kink (section 3) and the Q-ball (section 4) will be discussed. To aid proceedings, simple
cases of both will be reviewed before considering the dual theories. In addition, an introduction to topological
defects as a whole is given in order to understand the concepts behind the complex kink in the duality. Initially
however, theories where topological defects and Q-balls exist will be discussed.
1.1 Conventions
We have chosen the convention for the Minkowski metric to be (+1,-1,-1,-1)
Ô⇒ ∂µ∂µ♠ = ∂2
0♠ − (∂21♠ + ∂2
2♠ + ∂23♠) (1)
and have also chosen to denote topological defect fields and Q-ball fields by ψ and φ respectively. Furthermore
natural units h = c = 1 have been implemented along with Einstein notation.
1.2 Applications of Q-balls and Topological Defects
As stated above, the primary aim of this project was to prove the duality of the static one-dimensional complex
kink and the one-dimensional stationary Q-ball. The theory of both Q-balls and topological defects are discussed
in more detail in the subsequent sections below, however, before they are introduced fully, here is a brief discussion
of their existence within the Universe.
There are many areas of physics where topological defects have been identified as possible phenomena but only in
some cases have they been discovered. Topological defects are solitons, i.e. non dispersive wave solutions within
a field whose existence arises due to the initial boundary conditions. Their existence has been proven in condense
matter systems, however, they are yet to be discovered in cosmology[2]. Although topological defects have not been
observed in cosmology, a variety of types have been theorised, such as cosmic strings and monopoles[3, 4].
Q-balls on the other hand, are non-topological solitons[5]. Their existence arises from the conservation of Noether
charge (see section 4.3[6].) It is assumed they exist, particularly in dark matter theories and in baryogenesis[7, 8].
Below is a more in depth discussion of these objects. Firstly, a feasible theory of how topological defects could form
via the Kibble mechanism is discussed along with the monopole problem[9]. Secondly, the theory in which Q-balls
exist known as the Affleck-Dine mechanism: a theory which attempts to explains baryogenesis, is considered[7].
2
(a) V = a∣φ∣2 + b∣φ∣4 (b) V = −a∣φ∣2 + b∣φ∣4
Figure 1: The above is the form of potential before (left) and after (right) Tc, in the complex plane. Initially,
the Universe is hot the potential has the form of (a) and as the universe cools the potential changes to (b). The
minimum in figure (a) becomes a maximum and the field rolls down into the new vacuum in (b) under fluctuations.
Note both potentials have a U(1) symmetry and the vacuum manifold in (b) is a circle.
1.2.1 Kibble Mechanism and the Monopole Problem
The Kibble mechanism explains how a theory could produce topological defects in the Universe. As explained later
in the report (see section 3.3.1), existence of a topological defect requires the symmetry of a system to be broken.
Research has lead to the following temperature dependent Lagrangian for a complex φ4 theory[10, 11]:
L = 1
2∣∂µφ∣2 − V (∣φ∣) = 1
2∣∂µφ∣2 +m2
0(1 − ( TTc
)2
)∣φ∣2 − λ
4!∣φ∣4. (2)
Consider this as a toy model to explain the Kibble mechanism. The only difference to the temperature independent
Lagrangian is the fact that the mass term has a temperature dependent coupling constant: m2 = m20(1 − ( T
Tc)
2
).
The Lagrangian has a U(1) symmetry and so the field has the following solution: φ = ∣φ∣eiθ.
Initially, the Universe was hot and T > Tc. From equation (2), a potential of the form V = a∣φ∣2 + b∣φ∣4, where
a and b are positive constants (see figure 1a) is implied. In the vacuum solution the potential is minimised and for
this potential it is observed that ∣φ∣ = 0. Hence U(1) symmetry of the theory is not broken because if ∣φ∣ = 0, then
any rotation by angle θ leaves the vacuum solution invariant.
As time progresses however, the Universe cools and eventually the symmetry of the system is broken when T < Tcand the potential has a non-zero expectation value. The extrema at φ = 0 now becomes a maximum, as a becomes
negative and the minimum potential lies at ∣φ∣ =√
4!m2
2λ, hence the non-zero expectation value (see figure 1b).
Therefore the field, under slight fluctuations, “rolls down” the potential to the the minimum value into the vacuum
state, breaking the U(1) symmetry. However, now the vacuum manifold lies on a circle, as φvac =√
4!m2
2λeiθ, and
there is an infinite choice of vacuum states for the field to roll down into. Therefore, for non-causally connected
regions there is no necessity for the field to be in the same vacuum state and there can be various regions in the
Universe with different values for φ. It is at the boundary of these regions that topological defects arise. This, in
essence, is the Kibble mechanism, although the actual mechanism has other subtleties. One example is to consider
regions meeting at a point in space, see figure 8 for the 2D equivalent, with varying φvac. This is known as the
hedgehog formation and resembles the topological defect, known as the monopole, in three-dimensions[4].
Using the Kibble mechanism, one can calculate the density of monopoles today: ρm,0.[4] Also, using the fact
3
that monopoles are massive and thus non-relativistic, the density of monopoles is: ρm = mnm,Tc , at temperatures
where the Kibble mechanism is initiated. Using the Kibble mechanism, the mass density of the monopoles at the
time of symmetry breaking can be calculated using the following assumptions:
• One monopole exists per region and a region size is the Hubble volume l3 ∼H−3, i.e. the volume of a causally
connected region.
• A radiation dominated Universe at the time of symmetry breaking, so that the Friedmann equation is: H2 =8πG
3ρ
• The mean energy density of the Universe is the mean energy density of a Fermi-Dirac distribution, ρ = π2
30g∗T 4,
where g∗ is the effective degree of freedom. This accounts for all particles.
Combining the second and third assumption results in H =√
8π3g∗90
T 2
Mpl. Hence, combining this with the first
assumption, one can calculate the density of monopoles at the critical temperature to be:
nm,Tc ∼ l−3 ∼√
8π3g∗90
T 2c
M3pl
. (3)
Also note that one can use the average energy and pressure density of the Fermi-Dirac distribution, along with the
first law of thermodynamics, to calculate the entropy density:
s = 4ρ
3T= 2π2
45g∗T 3. (4)
Finally, this can be used along with the conservation of entropy and monopole number, neglecting annihilation, to
calculate the density of monopoles today[4]:
ρm,0 ∼√g∗m
T 6c
M3pl
g∗,0T 30
T 3c
∼ 1012 m
1016GeV( Tc
1016GeV)
3√g∗102
GeV cm−3. (5)
The GeV scale has been chosen due to grand unified theory predictions. However, this is clearly not observed, and
is therefore known as the monopole problem.
1.2.2 Baryogenesis
The role of baryogensis appears to explain the asymmetry in the Universe between baryonic matter and anti-matter.
Sakharov suggested that microphysical laws could account for this asymmetry, rather than the initial conditions of
the Universe. He suggested three conditions that must be met for baryon asymmetry to be produced[8]:
1. Baryon number conservation must be violated in the fundamental laws.
2. CP violation.
3. Thermal inequilibrium.
There are many baryogenesis theories which satisfy the above rules and a comprehensive introduction can be found
in reference [8]. One which is of particular interest is the Affleck-Dine mechanism[7]. In the Affleck-Dine mechanism,
one introduces an additional scalar field which carries a baryon number. Such a scalar field could have the following
Lagrangian:
L = ∣∂µφ∣2 −m2∣φ∣2. (6)
This has a corresponding conserved Noether charge (See section 4.3 for more detail on Noether current)[8]:
nB = jµ = i(φ∂µφ − φ∂µφ). (7)
4
The current can be considered as the baryon number, nB . One can consider adding baryon violating interaction
terms,
Lint = λ∣φ∣4 + εφ3φ + δφ4. (8)
These also violate the CP symmetry of the theory, thus satisfying two of the three conditions. By assuming the
coupling constants to be small and the field as initially entirely real, one can approximately solve the imaginary
part of equation of motion in a Roberson-Walker background:
φimag + 3Hφimag +m2φimag ≈ Im(ε + δ)φ3re. (9)
This can be solved to find the form of φimag in a matter or radiation dominated Universe. Then, combining φimagwith φreal, one can calculate the non-zero baryon number by substituting these into (7) (see reference [8]). Such a
set up can occur naturally in supersymmetric models and so is a favoured candidate for baryogenesis.
Considering the stability of the field, one can calculate small perturbations in the field by considering the equation
of motion in spherical coordinates and by writing the field as φ = ρeiΩ. Using perturbation theory in a Robertson-
Walker Universe, one can find equations relating the perturbations in both directions of the field. Using numerical
methods to solve the perturbed equation of motion, one can see that Q-ball form from small perturbations[8].
Now that the physical implications of Q-balls and topological defects have been discussed, dulities in physics
will be considered. This is in order to establish the methodology behind dualities, before the duality between the
complex kink and the 1 + 1 stationary Q-ball is proven.
2 Dualities in Physics
2.1 Early Dualities
One of the first major dualities in physics to be discovered is known as the Kramers-Wannier Duality, which describes
the duality between the high and low temperature states of the two-dimensional Ising model: a simple mathematical
model describing the behaviour of charges in a ferromagnetic system. In 1941, Kramers and Wannier discovered
this duality and used it to deduce the exact critical temperature in the Ising Model.[12, 13]. By considering a matrix
duality transformation to a low temperature Ising model, one can create an entirely new system that is still an Ising
model but at high temperature, and vice versa. The particular operation shown in Kramers’ and Wannier’s work
results in the following relations[12]:
e2K∗
= coth (K), (10)
Where K ∶= J2kbT
. Relation (10) implies two things: firstly, as the temperature increases in the initial Ising Model
(described by K), the new Ising Model (described by K∗ ) must decrease in temperature. This is more clearly
observed by rearranging (10) to the following:
sinh (2K) sinh (2K∗) = 1. (11)
Therefore, there is a relationship between the description of a 2D Ising model at high temperature and at low
temperature. Thus, quantities such as the free energy can be calculated by approximating the Boltzmann factor
in the high temperature model then, by using the duality, one can then calculate the low temperature limit[14].
The Ising model has only one first order phase transition and since the duality relates two temperatures, one can
identify the critical point, where K = K∗ = Kc. Solving equation (10) at Kc results in finding Kc = 0.44 which
matches Onsager’s exact result[15], confirming the duality.
In 1931 Dirac suggested the existence of a magnetic monopole with a quantised charge [16]
g = 1
2n, (12)
5
where n is an integer. This was in order to produce a symmetry between electricity and magnetism. Dirac, in fact,
found an alternate solution to the classical electromagnetic theory by considering the topology of the scenario[17, 18].
Dirac’s theory showed that the strength of a magnetic monopole would be very large and therefore the energy re-
quired to create it would also be large, hence the lack of observation.
In classical electromagnetism, a duality exists.[19]. If ρe = Je = 0, where ρ is the charge density and J is cur-
rent density, the equations are invariant under the following duality transformations: E → B and B → −E. This
transformation can be repeated to show a charge conjugation duality[19]. The duality still holds with the inclusion
of a source by introducing monopoles, hence changing the second and fourth Maxwell relations to:
∇ ⋅B = ρg,
∇×E + ∂B∂t
= −J [10]g .
(13)
A duality exists when transforming ρe,Je → ρg,Jg and ρg.Jg → −ρe,−Je. This, in essence equates, to the following
transformation of (12)
e→ g = 2πn
eand g → −e = −2πn
g. (14)
This is a self-duality as both theories are described by Maxwell’s equations. In both theories the electric or magnetic
field has a strong coupling, from relation (14). This is therefore a strong-strong coupled duality[20]. Furthermore,
considering the invariance of the Maxwell action in 1+ 3 spacetime dimensions, one can find a duality between two
electromagnetic theories related by e′ = 2πe
.
In 1977, Montonen and Olive gave the conjecture that there is a strong-weak duality within electromagnetism[21].
They suggested that, in the framework where magnetic charges are solitons in a field where electric charges are
Noether charges, the dual quantum field is still an electromagnetic theory with the same framework. In this dual
theory the monopole fields are the heavy gauge particles and so the magnetic charge is the Noether charge. Hence,
just as above, the electric and magnetic fields have interchanged to create a dual theory that is invariant to the
original one. Evidence to suggest that this is the case is described in reference [20].
The following sections will outline some of the more recent and explored examples of duality in physics, start-
ing with string theory.
2.2 String Theory
String theory is built on the assumption that the building blocks of the Universe are strings, rather than point
particles. There are multiple string theories containing different types of string with different properties and un-
derlying symmetry groups.
Despite the various differences in the string theories, in 1995, Hull, Townsend and Witten related them. There
are five known super string theories in ten dimensions: Type I, Type IIA, Type IIB, Heterotic SO(32) and Het-
erotic E8 × E8, each with different underlying formalisms. They argued that using a series of different duality
transformations, these five theories could be linked. Witten suggested that this implied an underlying theory,
known as M-theory, which describes the framework of string theory. He hypothesised that the five theories are
in fact different perturbative limits of this fundamental theory[22]. The dualities also pointed to the existence of
eleven-dimensional supergravity as another limit of M-theory and hence figure 2[23].
In string theory, different perturbative limits can be mapped to one another via various duality transformations.
These duality transformations are categorised by either T-Duality or S-Duality and therefore, it is useful to briefly
discuss both of these different general duality types.
6
Figure 2: M-Theory is usually illustrated as the above: where the five different types of String theory and Super-
gravity are limits of the overall theory[24].
2.2.1 T-Duality
As previously stated, physicists noticed that the five superstring theories were related by highly non-trivial dualities.
T duality refers to a duality relating a theory in which strings propagate on a circle of some radius R, to a theory
in which strings propagate on a circle of radius 1R
.
In T-duality, there is a compactification of extra spatial dimensions in a ten-dimensional superstring theory. Con-
sider the x9 direction in flat, ten-dimensional spacetime, then compactify it into a circle of radius R, such that
x9 ≈ x9 + 2πR, (15)
where now a dimension is compactified into a circle of radius R, so the space becomes cylindrical.
A string traveling around this circle will have its momentum quantised in integer multiples of 1R
, or momen-
tum modes. So a string in the nth quantised momentum state will contribute to the total mass squared of the
particle as,
m2n =
n2
R2. (16)
It is also possible for a closed string to wind around the cylinder. The number of times a closed string winds around
this cylinder is called the winding number, denoted by w.
Tension is defined as energy per unit length, thus a closed string has energy from the tension of being stretched
around the new circular dimension. The winding contribution Ew to the string’s total energy is therefore given by
Ew = 2πwR × T = wRα, (17)
where T is the tension of the string and α relates to the length scale of our string theory.
The total mass squared for each mode of a closed string is,
m2 = n2
R2+ w
2R2
α2+ 2
α(nL + nR − 2). (18)
The first term represents some energy from the momentum mode, i.e. if R decreases, the frequency of string oscil-
lation increases. The second term represents some energy from the winding mode, i.e. if R increases, the tension in
a closed string will increase. The final term includes contributions from the number of oscillation modes, nL and
nR, excited on a closed string in the right-moving and left-moving directions respectively around the string.
7
If one were to wrap one string around a cylindrical space of radius R and another around a cylindrical space
of radius αR
, then there would be a duality between the theories with an interchange of winding number, w, and
momentum number, n[25].
Due to it’s dependence on the winding property, T-duality is unique to string theory. Point particles cannot
have winding modes. If string theory is correct, this would imply that on a deep level, the separation between large
and small distance scales in physics is somewhat interchangeable.
2.2.2 S-Duality
Strong-weak duality relates two theories with different coupling constants, such that in one theory a coupling of
g is equivalent to the dual theory with a coupling constant of 1g, for example the previously discussed duality in
electromagnetism.[26] Another notable S-Duality is the sine-Gordon/Thirring model duality, relating a theory of
bosons to a theory of fermions, this duality will be discussed in more detail in section 2.4.
In an S-duality, a strong coupling in one theory can relate to a weak coupling in another theory. In one the-
ory, the strings break apart and join to other strings with ease, this results in a sea of perpetually interacting
strings. However, this is not the case in the other theory.
S-Duality can be very useful in perturbation theory since theories with a strong coupling are hard to analyse.
However, if a dual theory exists in which the coupling is weak, perturbative calculations can be performed in the
weak theory and mapped back to the strong theory to yield results. This is explained in more in depth in section
2.4 of this report, with the example of the sine-Gordon/Thirring model duality.
2.3 AdS/CFT Duality
The AdS/CFT (anti-de Sitter/Conformal Field Theory) duality is perhaps one of the most important examples of
duality regarding practical applications of duality in physics[27]. It is a conjectured idea originating from super-
string theory. As previously discussed, superstring theory is the prime candidate for the unified theory of the four
fundamental forces in nature.
AdS/CFT correspondence in its original formulation claims that a strongly-coupled four-dimensional gauge theory
is equivalent to a gravitational theory in five-dimensional anti-de Sitter spacetime. This duality relates different
dimensional theories, and is often referred to as a type of a more general theory, called a holographic theory. A
holographic theory encodes a higher dimensional theory to a lower one[28].
The AdS/CFT duality is also referred to as a gauge/gravity correspondence. The theoretical foundation behind the
standard model, excluding gravity, is understood by gauge theories. Unfortunately though, it is not an easy task to
analyse a gauge theory at strong coupling. The AdS/CFT duality claims that one can compute a strongly coupled
gauge theory using a curved spacetime - the AdS spacetime. de Sitter found a solution of the Einstein equation
with a constant positive curvature (de Sitter spacetime). The AdS spacetime has a constant negative curvature
instead, hence anti.
Typically, a duality states the equivalence between two theories which, at first, seem different. In the AdS/CFT
duality, the gauge theory and the gravitational theory look very different. As discussed above, if the gauge the-
ory is strongly-coupled, one can use the weakly-coupled gravitational theory instead, which makes analysis much
easier. The above relation is a particular case, the zero temperature case. At finite temperature, it is replaced by
strongly-coupled gauge theory which is equivalent to a gravitational theory of AdS black holes[29].
8
Figure 3: The original AdS/CFT duality paper has been cited in all of these areas of physics in arXiv[29]
The AdS/CFT duality originated from string theory and so, at first, was only discussed in that context, but in
recent years, it has been considered in a wide range of areas in physics. It has become a powerful tool in so called
“practical” areas of physics. Examples are condensed matter physics, quantum chromodynamics (QCD)[30], nuclear
physics and non-equilibrium physics. In fact, the original AdS/CFT paper[27] has been cited in all physics arXivs,
see figure 3.
One specific example is the quark-gluon plasma (QGP), which is believed to be a high energy phase in the early
universe. At high enough temperatures, quarks and gluons are deconfined and form a quark-gluon plasma state,
which according to QGP experiments, behaves like a fluid with a very small shear viscosity. This implies that
QGP is strongly coupled, which makes theoretical analysis difficult. However the viscosity value implied by the
experiments is close to matching the value predicted by the AdS/CFT duality, using black holes. This has triggered
AdS/CFT duality research beyond its string theory origin.
Other applications AdS/CFT duality are not limited to QCD: Strongly coupled systems often arise in condensed-
matter physics such as high critical temperature superconductivity. Partly inspired by the success of AdS/CFT
use in QCD, researchers attempted to apply the duality results to condensed matter physics. As the AdS/CFT
duality started to be used in other areas beyond particle physics, it’s established a “cross-cultural” character, where
researchers in other fields often initiate new applications.
2.4 Thirring Model and the Sine-Gordon Model
In 1975, Coleman produced a paper showing the duality between the (bosonic) quantum sine-Gordon model and
the massive (fermionic) Thirring model[1]. This duality is one of the first to show a weak/strong duality, also known
as the aforementioned S-duality. The sine-Gordon Lagrangian is as follows:
L = 1
2∂µφ∂
µφ + α0
β2cos (βφ) + γ0. (19)
This Lagrangian describes a scalar field in (1+1) dimensions and α0, β, γ0 are real parameters. If β < 8π this theory
corresponds to the massve Thirring model with zero charge which has the following Lagrangian:
L = ψiγµψ −1
2gψγµψψγµψ −m′Zψψ. (20)
Notice that the sine-Gordon equation describes a bosonic scalar field (φ), whereas the Thirring model describes a
fermionic Dirac field (ψ). g is a free parameter and Z is a cutoff-dependent constant.
Using perturbative techniques to produce a series in powers of m′ for the Thirring Model and powers of α0 for
9
the sine-Gordon model, the following relations between the two theories can be established[1]:
4π
β2= 1 + g
π,
− β
2πεµν∂νφ = ψγµψ,
α0
β2cos (βφ) = −m′σ.
(21)
The duality between the two indicates that bosons are in fact fermions. There is a caveat to this however: bosons
are fermions if both theories are massless and if the universe is (1+1) dimensional. Coleman also notes that neither
of the two theories are the underlying fundamental theory[6]. Another important property to note from (21): as the
coupling constant β of the sine-Gordon model increases, the coupling constant g of the Thirring model decreases
and visa versa. This strong-weak coupling characteristic is useful for perturbative techniques[31]. For example,
scattering amplitudes given by:
A = limt±∞
⟨f ∣U(t+, t−) ∣i⟩ , (22)
describe the probability of an interaction, and can be calculated to a good approximation, as the expectation value
of the first order term in Dyson’s formula[32],
U(t, t0) = 1 − i∫t
t0dt
′
Hint(t′
) + (−i)2 ∫t
t0∫
t′
t0dt
′′
Hint(t′
)Hint(t′′
) + ... (23)
All higher order terms would have to be considered for a strong coupling regime. Hence if one can calculate am-
plitudes in a dual theory weakly coupled regime, results could be obtained for the strongly coupled regime of the
other theory, where perturbative techniques are not applicable.
Now that some particular dualities in physics have been discussed, we will now discuss the properties of Q-balls
and topological in more detail.
3 Topological Defects
The prerequisite material behind the theories of our project’s duality is now explained. We start with topological
defects, which have been discussed briefly in the context of the real universe in section 1.
A topological defect (sometimes called a topological soliton) is a stable solution to a set of partial differential
equations and is homotopically distinct from the vacuum. It is a stable object which forms due to the boundary
conditions of a scalar field in a symmetry breaking potential.
The continuous nature of the space requires smooth transitions from one field value to another throughout space.
This condition predicts many different types of topological defect, some of which are reviewed in this report in order
to accumulate the required background information needed to understand the complex kink in the duality of this
report.
3.1 The Kink
A kink (or 1D domain wall) is a type of topological defect in one spatial dimension. Consider a real scalar field,
ψ(x, t), with the relativistically invariant Lagrangian density[33],
L = 1
2ηµν∂µψ∂νψ − V (ψ) (24)
where ηµν is the Minkowski metric with the convention (+1,-1,-1,-1) and V is a discrete symmetry breaking potential:
V = λ4(ψ2 − 1)2, (25)
10
Figure 4: Plot of potential in equation (25).
meaning that there are a discrete number, in this case 2, of distinct vacua.
Consider the boundary conditions of a scenario where at x→ −∞, ψ = −1 and x→ +∞, ψ = +1.
Figure 5: Plot of the kink’s field, as a function of position.
At some arbitrary point, x0, which we can choose to be at the origin, between −∞ and +∞ the field must be outside
the vacuum state, which is the potential minima, and thus the topological kink contains energy.
3.1.1 Static Kink Equation of Motion
One can consider the case of the static kink, where ψ = 0, and calculate the equations of motion from equations
(15) and (16),∂2ψ
∂x2= λψ(ψ2 − 1). (26)
11
After some mathematical manipulation including a separation of variables, one arrives at the spatial profile function,
of the field, for the static kink.
ψ(x) = ±tanh[√
λ
2(x − x0)], (27)
which one can see from figure 4 above.
3.1.2 Energy
In order to deduce the energy, consider the Hamiltonian of the system,
H = ∫ dx[1
2(dψdx
)2
+ V ], (28)
and note that this Hamiltonian can be rewritten in a more useful form:
H = ∫ dx
⎡⎢⎢⎢⎢⎣
1
2(dψdx
±√
2V )2
± (√
2Vdψ
dx)⎤⎥⎥⎥⎥⎦. (29)
If we now introduce a super potential ω such that dωdψ
=√
2V , and note that the Hamiltonian is extremised, then
after integrating by parts,
H = ∓[ω]∞−∞ + ∫ dx1
2[dψdx
±√
2V ]2
. (30)
Finally, after considering the boundary conditions for ω and noting that the integral in (21) must be 0 due to the
Bogomol’nyi argument[34], the energy of the static kink can be seen as:
E = 1√2
4
3. (31)
3.2 Higher Dimensional Topological Defects
The kink described in section 3.1 is the simplest type of topological defect, with one spatial dimension and one
real scalar field. The following section explains more complex topological defects, which can contain more than one
field, Ψ = Ψ(ψ1, ψ2, ..., ψn), and more than one spatial dimension, X = X(x0, x1, ..., xm), where m does not have to
be equal to n.
3.2.1 The Vortex
Vortices are seen in condensed matter systems, and are also related to cosmic strings: a topological defect believed
to exist by cosmologists.[2]
Vortices are characterised by an axial or cylindrical symmetry in two dimensional space. This U(1) type sym-
metry implies a two component field Ψ = (ψ1, ψ2), or equivalently a complex field, Ψ = ψ1 + iψ2 which has a defined
value everywhere, when mapped to two-dimensional real space. The associated symmetry breaking potential for
the vortex is of the form V = V (∣Ψ∣). One can consider the potential of the same form as that of the real kink, but
now there is a complex field, thus the potential takes the form,
V (∣Ψ∣) = λ4(∣Ψ∣2 − 1)2. (32)
There are an infinite number of potential minima, lying on a circle. This is equivalent to stating that the vac-
uum manifold is a circle (S1). By the definition of a topological defect stated at the beginning of the section,
a vortex is a solution which is distinct from the S1 vacuum manifold, discussed in more detail later in this sec-
tion. At spatial infinity, the field has stabilised in the vacuum somewhere on the circle, breaking the U(1) symmetry.
12
Figure 6: The left plot shows a mexican hat potential, where a system with a complex field has a circle of degenerate
vacua. The right graph simply shows the potential against the modulus of the field.
Vortices arise when the topological boundary conditions require the field to leave the vacuuum at some point
in space. This occurs when a rotation around the defect in real space maps to an integer number of rotations of the
field in field space, as can be seen in figure 7, this integer is known as the winding number, N , which is discussed
in more detail in section 3.3.3.[11]
Figure 7: (a) shows the y and x axis of real space, with the vortex centered at the origin, the different coloured
circles at each point correspond to a mapping to field space, (b). One can see that for (i), one rotation around the
origin in real space corresponds to one full rotation around the manifold in field space; hence there is a winding
number, N = 1. For (ii) we have a similar scenario, but one rotation through real space maps to two full rotations
in field space, hence this corresponds to a vortex of winding number, N = 2.
The nature of the U(1) symmetry breaking potential implies an ansatz for the field, Ψ, of the form,
Ψ = f(r)eiθ, (33)
where f(r) is a radial function and θ is an angle. Now if one consider the Lagrangian,
L = ∂µΨ∂µΨ − V (∣Ψ∣) (34)
By using the principle of least action and considering the ansatz in equation (33), one can obtain the equations of
motion in terms of the newly defined variables r and θ to yield, for the static case,
f ′′ + 1
rf ′ − 1
r2f − f dV
d∣Ψ∣2 = 0. (35)
13
Figure 8: Three diagrams of vortices with winding number, N = 1. All three configurations, (a), (b) and (c), are
topologically equivalent as they can be continuously deformed into one another.
This equation cannot be solved analytically like for the 1D real kink in section 3.1.1, but it can be solved numerically.
Consider the static Hamiltonian for the vortex,
H = ∫ dx[∂iΨ∂iΨ + V (∣Ψ∣)]. (36)
If one were to use the ansatz for the vortex field then, in polar coordinates, the Hamiltonian can be written:
HV ortex = ∫R
0drdθ ⋅ r(f ′2 + f
2
r2+ λ
4(f2 − 1)2). (37)
Now as we take the limit r →∞, then the form of f becomes[11]
f ∣r→∞ → 1 − n
r2+ ... (38)
and hence, the Hamiltonian approximates to
HV ortex ∝ ∫R
0drdθr ⋅ (n1
r6+ 1
r2+ n2
r4+O(r−8)), (39)
where n1 and n2 are constants. For the large r limit, the Hamiltonian is dominated by the 1r2
term, giving
HV ortex ∼ ElnR, (40)
where E is a constant. We can therefore see that this energy diverges at large radius and is not therefore a finite
energy solution.
Although this conclusion for the energy of a vortex should imply that they are not feasible in the real Universe,
the infinite energy assumes an individual vortex which does not interact with any other object throughout space.
14
Figure 9: Two vortex configurations of different winding number. The left diagram shows a scenario in which one
clockwise rotation around the defect, denoted by the shaded circle in the middle, correspond to a full anticlockwise
rotation of the field and thus has a winding number, N = −1. The right diagram shows a scenario in which one
clockwise rotation about the defect’s centre corresponds to two full clockwise rotations in field space and thus this
configuration has a winding number of N = 2.
However, the concept of winding number can play an important role in suppressing the divergent energy of such a
system. For example, if a vortex with a winding number of N = +1 is near a vortex of winding number of N = −1,
these infinite energies may cancel.
3.2.2 Derrick’s Theorem
Derrick’s Theorem states: “for a wide class of non-linear wave equations there exists no stable time-independent
solutions of finite energy.”[35] Hence, for spatial dimensions higher than one, singular topological defects cannot exist.
If we consider a single real field, ψ, in d spatial dimensions, then the Lagrangian density,
L = 1
2∂µψ∂
µψ − V (ψ), (41)
gives rise to a Hamiltonian, which for the static case, is given by
H = ∫ [1
2∂iψ∂
iψ + V (ψ)]ddx, (42)
which is extremised. This has both kinetic and potential components, and i represents the derivative with respect
to the spatial dimensions xi, where i = 1, ..., d.
Imagine we have a solution, ψ(x), such that we can define the following:
IK = ∫ ddx1
2(∂ψ(x)
∂x)
2
, (43)
IV = ∫ ddxV (ψ(x)). (44)
This gives the extremised Hamiltonian the form:
Hψ = IK + IV . (45)
15
Figure 10: Numerically solving the equation of motion for the vortex, (35), gives the profile of the field, ψ, as a
function of position.
One can also define a set of equations, ψλ(x) = ψ(λx), where λ is a scaling variable. The case of λ = 1 corresponds
to ψ(x), which is the solution that extremises the Hamiltonian. The general Hamiltonian can now be written as.
Hλ = ∫ ddx⎛⎝
1
2[ ∂∂xψ(λx)]
2
+ V (ψ(λx))⎞⎠. (46)
If we note that λ is simply a constant, and use a change of variables, X = λx then one can show that the Hamiltonian
can be written,
Hλ =⎛⎝λ2−d ∫ ddX
1
2[ ψ(X)dX
]2⎞⎠+⎛⎝λ−d ∫ ddXV (ψ(X))
⎞⎠
(47)
Ô⇒ Hλ = λ2−dIK + λ−dIV . (48)
So we notice that equation (45) and equation (48) are of the same form but with a difference in factors of the scaling
variable λ in each integral.
This result can be used to check the stability of static topological defect solutions in different dimensional sys-
tems:
• For example, d = 1 gives the following equation:
Hλ = λIK + 1
λIV . (49)
One can see that, at λ = 1, we have a static solution:
∂Hλ
∂λ∣λ=1 = 0 Ô⇒ IK = IV . (50)
• Now d = 2 gives the following equation:
Hλ = IK + 1
λ2IV . (51)
16
Figure 11: For d=1, the Hamiltonian is plotted as a function of scaling variable λ, the blue line represents the full
Hamiltonian, and the black and red dashed lines represent the contibution from IK and IV terms respectively.
Which, one can see, has no static point:
∂Hλ
∂λ= −2I2
λ≠ 0 ∀ λ ∈ R. (52)
By this argument, there exists no stable solution for a soliton of dimensions above one. However, it was shown
in section 3.2.1 that the two-dimensional vortex does have a stable solution. However, thus far it is assumed the
vortex has a finite energy. This is not necessarily true: Equation (52) disproves the stability of a single static vortex.
However, in the real Universe, equation (52) does not disprove the existence of configurations including multiple
vortices and anti-vortices, or vortices confined to a finite spatial region, where energies could be finite.
3.3 Topology
Previously in this section, some key properties of topological defects have been outlined. This introduced concepts
such as vacuum manifolds and winding numbers in a non-technical, but intuitive way. This section aims to more
rigorously and comprehensively explain these concepts. This is necessary when attempting to deduce information
about systems which may not be diagrammatically or intuitively comprehensible as for the kink and vortex.
3.3.1 Vacuum Manifolds
The idea of a vacuum manifold has already loosely been introduced. It describes the nature and shape of the vacuum
with regards to the potential of the theory. For the vortex example, the vacuum was at the bottom of a mexican
hat potential and formed a circle of degenerate minima. Mathematically, a circular vacuum manifold is denoted by
S1, where the S comes from the spherical nature of the manifold, and the number one comes from the fact that
the two-dimensional circle has one degree of freedom. In higher dimensional topological defects, for example in the
monopole, three fields are present and hence the vacuum manifold is the surface of a three-dimensional sphere. This
vacuum manifold is called S2 as there is still spherical symmetry, but the number two refers to the two degrees of
freedom.
If one labels the vacuum manifold as M, then with a theory of symmetry group G, the symmetry of the sys-
tem is broken when ψ chooses a particular vacuum state. The symmetry group G is then replaced by a new
17
Figure 12: For d=2, the Hamiltonian is plotted as a function of scaling variable λ, the blue line represents the full
Hamiltonian, and the black and red dashed lines represent the contribution from IK and IV terms respectively.
symmetry group, H, which is a subgroup of G. One can retrieve the following important result about the vacuum
manifold of the theory:
M = G/H. (53)
This is closely related to Goldstone’s theorem describing the number of massless bosons in a theory by considering
symmetry breaking.[36] The operation in the equation is not division, but refers to the space of cosets of H in G,
i.e. one must consider the elements of G with the elements of H factored out.
This stems from group theory. The groups we are considering are compact Lie groups: which are groups that
are a differentiable manifold and are thus useful in systems relevant to this project. We previously discussed the
concept and role of symmetry breaking in the formation of topological defects (see section 1.2.1). More mathemati-
cally, a compact Lie group G is spontaneously broken into a smaller subgroup, H, with a vacuum expectation value
of ψ0 ∈M, we can generate the remainder of M by transformations of the form,
ψ =D(g)ψ0, g ∈ G, (54)
and g ∉H since ψ0 =D(h)ψ0 for h ∈H [31][11].
Equation (53) relates the symmetry groups of a theory, before and after symmetry breaking, to the nature of
the vacuum manifold. It can be used in a wide variety of circumstances that extend beyond the basic examples we
have discussed thus far in this report, which have been intuitively understandable.
3.3.2 Homotopy Theory
As stated at the beginning of section 3, the definition of a topological defect is a solution to a set of partial differen-
tial equations and is homotopically distinct from the vacuum manifold. Homotopy refers to the topological nature
of functions in space. Two functions are homotopic if one can be continuously deformed into the other without
leaving the manifold in which the function exists.
Consider a manifold, M, formed by a two dimensional plane, but containing a circular hole, as shown in fig-
ure 13. The figure shows three closed paths which begin and end at point x0: loops (a), (b) and (c). Loops (a) and
18
(b) are homotopic, since one can be continuously deformed to give the other. Loop (c), however, is not homotopic
with respect to loop (a) or (b), since the circle, which does not exist inM, forbids continuous deformation to either
loop (a) or (b).
Figure 13: A two dimensional plane containing a hole outside the manifold, as indicated by the shaded area. Three
loops in the manifold are shown about the point x0.
More mathematically, if one writes a loop in figure 13 as a continuous function, f(α) ∈M, with 0 ⩽ α ⩽ 1 such that
f(0) = f(1) = x0. Then f(α) can be smoothly deformed into another continuous function, g(α), only if there is
another continuous function, k(β,α), with 0 ⩽ β,α ⩽ 1 such that,
k(0, α) = f(α), k(1, α) = g(α), k(β,0) = k(β,1) = x0. (55)
One can therefore think of k as representing all the loops between f and g, which start and end at x0. f and g are
homotopic to each other about the point x0, and the set of paths in k is known as a homotopy[31].
Paths which are homotopic to f satisfy the conditions for a group, such that:
• There exists an identity, f(α) = x0, which is a trivial path which remains at x0 throughout the path.
• There is a well defined product with the following conditions:
(f g) =⎧⎪⎪⎨⎪⎪⎩
f(2α), 0 ⩽ α ⩽ 12
g(2α − 1) 12⩽ α ⩽ 1.
(56)
Which means that the product of two paths joins the end point of one, to the beginning of the other.
• There is an inverse given by
f−1(α) = f(1 − α), (57)
i.e. the inverse of any path about a point, is the same path, but with a reversed direction.
This described group is referred to as the fundamental group of M, or π1(M), where one should note that the
choice of x0 is arbitrary, since any point on the manifold behaves equivalently.
For example, if one chooses our manifold as the R2 plane, then any loop on the manifold can be shrunk to a
point. We write this mathematically as
π1(R2) = 0. (58)
19
Thus, if one considers figure 13, it relates to the vortices example from section 3.2.1. The manifold,M, in the case
of the vortex, is the vacuum manifold, which is a circle (S1). One can understand this by imagining a manifold as
shown in figure 14, which is related to figure 13. Again loops (a) and (b) are homotopic, but (c) is not homotopic
to either (a) or (b). However, now notice that loops (a) and (b) can be shrunk to an infinitely small loop about the
point x0. Also, the manifold can be shrunk such that the thickness of the manifold is infinitesimal. In this limit,
the manifold, M, is S1 (as for the vortex), and the choice of points x0 exist anywhere on that circle. Now loops
(a) and (b) both correspond to vacuum solutions, and loop (c) corresponds to a vortex with a winding number of
N = 1. There can be any integer number for the winding number of a vortex, all of which give unique behaviour.
We write this mathematically as,
π1(S1) = Z. (59)
Figure 14: This figure is analogous to figure 10, but the manifold, M, is now a doughnut, rather than an infinite
2D plane with a circular hole in it. The figure includes a path, (c), of winding number N = 1, and two paths, (a)
and (b), of winding number N = 0.
3.3.3 Winding Numbers
The concept of a winding number has been briefly discussed in the string theory section, 2.2.1, and in the vortex
section, 3.2.1. In the vortex section, the winding number was defined as the number of times the field travels around
the vacuum manifold per one rotation in real space, as shown in figure 7. However, a more mathematical definition
of topological degree (winding number in a given dimension) is:
degf = ∫xfω(ω)∫y ω
, (60)
where fω(ω) is the pull back in real space, which is the Jacobian between field space and real space, multiplied by
the real space volume element. ∫y ω is the integral of the field space volume element[37].
For the case of the vortex this can be written,
N = 1
2π∫
2π
0dθdα
dθ, (61)
where θ is the angle in field space and α(θ) is the angle to which θ maps to on the circle vacuum manifold. One
notices that, for the familiar example of a vortex with a winding number of 1, α(θ) = θ, so
20
Figure 15: The left figure shows two paths which are topologically equivalent, both with a winding number N = 1.
The right figure shows a path with a winding number N = 2.
N = 1
2π∫
2π
0dθ = 1
2π2π = 1. (62)
One can also consider more complex cases, where the vacuum manifold is not of the vortex form of an S1 (circle)
vacuum manifold. For example, the monopole is in three spatial dimensions, R3, and has three real fields associated
with it, with an S2 sphere shell vacuum manifold. The analogue equation for winding number in this scenario can
be given in spherical coordinates by,
N = 1
4π∫ d2Ω
sinα
sin θ(dαdθ
dβ
dφ− dβdθ
dα
dφ). (63)
As mentioned previously, the spherical symmetry in the three dimensional monopole requires two angular compo-
nents in both real space, θ and φ, and field space, α and β.
3.4 Complex Kink
The one-dimensional real kink and two-dimensional vortex have been discussed, it is therefore appropriate to intro-
duce the one-dimensional complex kink, which is a theory of this report’s duality.
The complex kink is characterised by a complex field (2 field space dimensions) and one real space dimension.
The potential is of the same form as for the vortex,
V (∣ψ∣) = λ4(∣ψ∣2 − 1)2, (64)
i.e. a mexican hat potential as previously displayed. Unlike the vortex however, the complex kink has only one
spatial dimension. As for the real kink, at x → −∞ and x → +∞ the field must be in the vacuum, but in different
vacua to one another. For the real kink case, one has a discrete number, 2, of possible vacua. But now, due to the
continuous vacuum manifold of the complex field potential, one has an infinite number of possible vacua, making a
circle vacuum manifold, as in the vortex.
So long as at the two spatial infinities stabilise in a different position on the vacuum manifold, one can have a
kink solution, i.e. one which is required to leave the vacuum manifold (see figure 16).
21
Figure 16: Some examples of a complex kink on an argand diagram, where the two spatial infinities, x → −∞ and
x → +∞, are the points on the vacuum manifold, and leave the vacuum manifold in between. Notice there are an
infinite number of possible defects, due to the arbitrary choice of end points.
3.4.1 Stability
If one assume a Lagrangian of the form:
L = ∂µψ∂µψ − V (∣ψ∣), (65)
then unfortunately the complex kink is unstable (see figure 17). The continuous nature of the vacuum manifold
allows a shrinking of the defect to a more energetically favourable point on the vacuum manifold, which is no longer
a defect.
Figure 17: For a complex kink with the Langragian in equation (65), the left configuration is unstable under small
perturbations, hence, the defect is dissipative and results in the far right configuration, which is simply a vacuum
state throughout all of space.
To make a complex kink stable, one must introduce a new term into the Lagrangian to pin the field at spatial
infinities, in order to stop the kink from dissipating.
One can do this by introducing a new conformally flat field space metric, G(∣Ψ∣). Conformally flat field space
refers to the space being proportional to the flat field metric. Now one can write the Lagrangian as:
L = 1
G∂µψ∂
µψ − W , (66)
22
where we have defined W = V (∣ψ∣)G
, which is a new term related to the potential.
Now if one considers the complex sine-Gordon model,
G = 1 − ∣ψ∣2, (67)
which implies a complex kink Lagrangian of the form,
L = ∂µψ∂µψ
1 − ∣ψ∣2 − λ4(1 − ∣ψ∣2). (68)
Now as the field approaches the vacuum at spatial infinity, ∣ψ∣→ 1, thus to keep the Lagrangian from being divergent,
the kinetic terms in the numerator must → 0 which amounts to requiring that the spatial infinity vacua are pinned
and hence, the complex kink cannot dissipate unlike in the case of the previous Lagrangian.
3.4.2 Topological Charge
In section 3.4.1, the existence of the complex kink was argued by introducing a conformally flat field space metric
which is of the form of the complex sine-Gordon model. Now the properties of the stable complex kink are investi-
gated.
One can construct a conserved current for the kink:
Jµ = κεµν∂νθ, (69)
where θ is a real field, and κ is a constant. Also εµν = −ενµ for µ ≠ ν, and εµν = 0 for µ = ν with ε01 = 1. One can
simply show that this current is conserved using the relations for ε stated above.
A natural choice for the field θ can be seen from considering the following ansatz:
Figure 18: A diagramatic representation of θ at an arbitrary point on the vacuum manifold in the form of an argand
diagram with ψIm vs ψRe.
ψ = e−iθ ∣ψ∣. (70)
We defined the topological charge as[38]
Q = ∫xJ0 = κ∫
x∂xθ = κ(θx=∞ − θx=−∞), (71)
23
where κ is a constant. Since we already argued the existence of the static complex kink, we know that the topolog-
ical charge must therefore be a conserved quantity.
Now that the theory of topological defects has been discussed, arriving at the complex kink, we now turn to
the theory of a type of non topological solition: the Q-ball.
4 Q-balls
This section considers the non-topological defect known as the Q-ball, whose stability arises from the associated
conserved Noether charge. This can be understood by considering the Lagrangian mechanics of the system. The
following section refamiliarises the reader with Noether’s theorem, before considering a theory where Q-balls are a
minimum energy solution. Once the ansatz for the field is proposed, the equation of motion is stated generally and
derived using the principle of least action. Once the appropriate relations are derived, the characteristics of the
Q-ball are considered.
4.1 Noether’s Theorem and Conserved Charge
Noether’s theorem states that: every continuous symmetry of the Lagrangian gives rise to a conserved current jµ(x)such that[39]
∂µjµ = 0. (72)
This theorem can be derived by considering the action under variations of the field. Such a derivation can be found
in any field theory textbook[39]. For the case of the Q-ball, the Lagrangian is invariant under a U(1) transformation.
The field is complex and thus can be treated as two separate real fields. By considering a transformation, such as
eiα, one can deduce the Noether current. This is done by considering an infinitesimal transformation,
φ→ φ + iαφ = φ + δφ, (73)
and by using the definition of the Noether current,
jµ ∶= ∂L∂(∂µφa)
δφa, (74)
where the subscript, a, indicates that all fields in the theory are considered. Noether’s theorem is the result of:
∂µjµ = δL − δS
δφ(x)aδφ(x)a. (75)
When the equations of motion are satisfied, the second term on the right hand side of the equation is equal to zero
and the current is only conserved if the Lagrangian has a continuous symmetry.
For an arbitrary U(1) symmetric complex action of the form,
S[φ] = ∫1
G(∣φ∣)∂µφ∂µφ −W (∣φ∣), (76)
the Noether current is[39]
j′µ = αjµ =i
G(∣φ∣) (φ∂µφ − φ.∂µφ). (77)
This can be found by considering equation (74). By convention one can rescale the current so that αjµ is the new
definition of the Noether current, thus for the rest of this discussion, αjµ → jµ.
One can split equation (72) into its temporal and spatial constituents. Then, using Stokes theorem and the fact
that no current flows out of an infinite manifold, one comes to the final conclusion
d
dtQ ∶= ∫
x∂0j0 = ∫
x∂iji = 0, (78)
24
where Q is defined as the charge and j0 is defined as the charge density, which we will label q. These two variables
are observables and, as discussed previously, dualities in physics are proven by linking equivalent observables in the
dual theories. Charge, in this case, is no different and there is a duality transformation that exists between the
charge of the one dimensional stationary Q-ball and the topological charge of the static complex kink. This issue
is addressed later, however the definition in (78) is also useful for determining the mathematical form of the Q-ball
field. This is done by considering the Lagrangian and making use of Lagrange multipliers. This prescription is
considered in the next section.
4.2 Q-ball Action and Hamiltonian
Q-balls are a solution of the following action[38]:
S[φ] = ∫1
G(∣φ∣)∂µφ∂µφ −W (∣φ∣). (79)
G(∣φ∣) and W (∣φ∣) are specific expressions that allow for Q-ball mechanics. Using the definition of the Lagrangian,
one can calculate the Hamiltonian of the system and introduce a Lagrange multiplier:[40]
H =∫x
[(∣φ∣2 + ∣φ′∣2)G(∣φ∣) +W (∣φ∣)] + ω(Q − ∫
xi(φ∂µφ − φ∂µφ)
G(∣φ∣) ). (80)
Note that the constraint found in the ω term is equal to zero due to equation (78). One can rearrange the above
into the following form
H =∫x
[ ∣φ − iωφ∣2
G(∣φ∣) − ω2∣φ∣2G(∣φ∣) +
φ′φ′
G(∣φ∣) +W] + ωQ. (81)
Now minimising the Hamiltonian with respect to φ, one finds δH = 2G(∣φ∣) ∣φ−iωφ∣δφ = 0. Therefore, to ensure δH = 0
for any arbitrary variation in the derivative of the field, one must set φ− iωφ = 0. Solving the differential equation,
one finds the form of the field to be
φ = ∣φ(x)∣eiωt. (82)
This result can be used to simplify the Lagrangian to
L = − 1
G(∣φ∣)∂i∣φ∣∂i∣φ∣ −W (φ) + ω2∣φ∣2
G(∣φ∣) . (83)
4.3 Equations of Motion
Firstly, we considered a simple one-dimensional Q-ball theory, setting G(∣φ∣) = 1 allowing W (∣φ∣) = V , the potential.
The potential was also given the form V = −∣φ∣2 + ∣φ∣4 − β∣φ∣6, where β is an arbitrary coefficient. Without loss
of generality, the coefficients of ∣φ∣2 and ∣φ∣4 have been absorbed by rescaling the field. Substituting the above
definitions into (83) and using the principle of least action, one can calculate the equations of motion for one spatial
dimension as:
∣φ∣′′ + ∂
∂fVeff , (84)
where Veff = 12(ω2−1)∣φ∣2+ ∣φ∣4−β∣φ∣6. One can use the analogy of a particle on a potential hill (poh) by considering
the equations of motion in both scenarios and identifying the variables in the theory as: xpoh → φ and tpoh → x.[5, 6]
Therefore, for Q-ball solutions the initial conditions required can be evaluated using the analogy above.
Consider the form of Veff (see figure 19). If one imagines a particle with initial conditions at point A, (φ0, Veff(φ0)),then the particle would pass through the origin, then reach the point, (−φ0, Veff(−φ0)), and continue to oscillate
back and forth. If the particle were to start at C, then it would undershoot, never reach the origin and oscillate.
In order for a solution to be in the vacuum at infinity, one requires the field to start at B analytically. This
25
Figure 19: The effective potential as a function of the field. If the initial condition is A, an overshoot occurs. If the
initial condition is C, an undershoot occurs. Only with initial condition B do Q-ball solutions occur.
is equivalent to Veff = 0 Ô⇒ V = ω2∣φ∣2. Therefore, the maximum and minimum value of ω can be found by
considering the the extrema values of V∣φ∣2 . For positive ω, one finds the following results for the extrema with β = 1
4
and β = 12:
β = 1
4⇒ ωmin = 0 and ωmax = 1,
β = 1
2⇒ ωmin =
1√2
and ωmax = 1.
Solving the equation of motion (84), charge (78) and the Hamiltonian (80) (without the Lagrange multiplier), one
numerically finds the following profiles: β = 0.5 (see figure 20), β = 0.25 (see figure 21).
The important properties to note of the Q-ball are its radial profile and the fact that it is a local defect. This is also
seen in the charge and energy profile diagrams. Both diagrams represent the amount of charge and energy within
a radius, r, and one can see that both are localised within a ball.
These are the properties of a Q-ball and, as shown in the next section, the dual Q-ball theory also exhibits
these properties. However, computationally the differential equations that are solved are approximations of the
analytic equation. This is discussed in more detail in section 6.4. In essence, this means that the analytical initial
conditions do not reproduce exact Q-ball solutions numerically and one requires a bisection method, in order to
find the suitable initial conditions. A bisection method was implemented in figures 20, 21, 22.
For various ω one can find the total charge and energy (see figure 22). One should therefore expect similar
phenomena to be derivable from any Q-ball theory.
26
Figure 20: A set of graphs to show the profiles of the field, charge and energy of a Q-ball for β = 12
and ω = 0.5.
Note the charge and energy at position r is the total charge and energy within the radius r.
Figure 21: A set of graphs to show the profile of the field, charge and energy of a Q-ball for β = 14
and ω = 0.25.
Note the slight lift at the end of the charge and energy, which is due to the inaccuracy of the initial conditions.
27
Figure 22: Half the total charge and energy are plotted for the appropriate range of ω.
Now that the properties of topological defects, including the complex kink, have been disccused along with the
properties of Q-balls, the specific dual theories will be considered. However, the duality between two general
theories will first be proven, then the properties of the duality will be explored.
5 Duality
In this section the duality between the static 1 + 1 dimensional complex kink and the 1 + 1 stationary Q-balls will
be proven. However, initially a more general duality will be proven between the following theories:[38]
L = 1
G(∣φ∣)∂µφ∂µφ −W (∣φ∣), L = 1
G(∣ψ∣)∂µψ∂
µψ − W (∣ψ∣). (85)
The duality states that both theories described by these Lagrangians are in fact the same theory, due to the following
duality transformations:[38]
d∣ψ∣2d∣φ∣2 = − ∣ψ∣2
G(∣φ∣) , G(∣ψ∣) = ∣φ∣2∣ψ∣2G(∣φ∣) , W (∣ψ∣) =W (∣φ∣), (86)
∂xθ =1
2q, ∂tθ =
1
2JQ, (87)
where G(∣φ∣), G(∣ψ∣), W (∣φ∣) and W (∣ψ∣) are arbitrary functions and θ, q and JQ are related to the topological and
Noether charges of both systems respectively.
A proof of the duality is discussed in section 4.3. Also, later in the next section it will be shown that these
two theories can represent a theory which describes Q-balls and a theory which describes complex kinks.
28
5.1 Equations of Motion
One can calculate the equation of motion for both theories by considering the principle of least action. For the
benefit of the inquisitive reader who may wish to complete the exact calculation, the following relations are of use
towards the end of the derivation:
∂µG(∣φ∣)∂µφG2
= ∂µ∣φ∣2∂µφG2
∂G
∂∣φ∣2 = ∂µφ∂µφ
G2
∂G
∂∣φ∣2 φ +∂µφ∂µφ
G2
∂G
∂∣φ∣2φ, (88)
∂W (∣φ∣)∂φ
= ∂W
∂∣φ∣2∂∣φ∣2∂φ
= ∂W
∂∣φ∣2φ. (89)
Using the above relations, one finds the equation of motion to be
∂µ∂µφ − ∂µφ∂
µφ
G(∣φ∣)dG(∣φ∣)d∣φ∣2 φ + φG(∣φ∣)dW (∣φ∣)
d∣φ∣2 = 0. (90)
Similarly, for the second theory described by the second Lagrangian in equations (85), one finds the equation of
motion to be
∂µ∂µψ − ∂µψ∂
µψ
G(∣ψ∣)dG(∣ψ∣)d∣ψ∣2 ψ + ψG dW
d∣ψ∣2 = 0. (91)
These equations of motion can be used to calculate the profiles of fields φ and ψ numerically. This will be considered
in section 4.4 for the specific cases of the stationary Q-ball and the static complex kink. However, for proving the
duality in section 4.3 a convenient coordinate system is light cone coordinates:
u = 1
2(t + x), v = 1
2(t − x). (92)
Expressing the equations of motion in light cone coordinates results in:
∂uvφ − ∂uφ∂vφφ
G(∣φ∣)dG(∣φ∣)d∣φ∣2 + φG(∣φ∣)dW (∣φ∣)
d∣φ∣2 , (93)
∂uvψ − ∂uψ∂vψψ
G(∣ψ∣)dG(∣ψ∣)d∣ψ∣2 + ψG(∣ψ∣)dW (∣ψ∣)
d∣ψ∣2 , (94)
where ∂u = ∂∂u
and ∂uv = ∂2
∂u∂v. These will be used in section 4.3. But first, the general duality transformations will
be discussed in more detail.
5.2 Duality Transformations
As described in section 3, the first Lagrangian in equation (85) has a conserved Noether current. As the Lagrangian
is globally U(1) symmetric, a natural variation to consider is a small deviation in the field with respect to a U(1)rotation. The resulting Noether current is
jµ = −i
G(∣φ∣) (φ∂µφ − φ∂µφ). (95)
Defining the temporal part as q = j0 and the spatial part as JQ = j1, one can relate these components to the
topological charge of the ψ theory. Consider the following definitions for the current and charge of a topological
defect in the ψ theory (see section 3.4.2),
Jµ ∶= κεµν∂vθ, (96)
Q ∶= 2(θ∞ − θ−∞) = 2∫∞
−∞∂xθdx, (97)
where θ is a real field and κ is an arbitrary constant. Note that Jµ is conserved. When considering the components
of the currents individually and setting κ = 2, the duality transformations (87) imply jµ = Jµ. Finally, by considering
equation (97), one can confirm that the charge in both theories are equivalent.
29
Q = 2(θ∞ − θ−∞) = 2∫∞
−∞∂xθdx = ∫
∞
−∞qdx ∶= Q. (98)
This is an observable and with regards to the duality between the complex kink and Q-ball, the charge can be
numerically computed in both theories and compared.
The other three duality transformations, equations (86), are definitions which are required in proving the du-
ality. The first of the equations relates the dual fields, whereas the others are relations between the arbitrary
functions in the Lagrangian. The first transformation can be rearranged to give the following:
∣ψ∣2 = exp∫∣φ∣2
0− d∣φ′∣2G(∣φ′∣) . (99)
Even though the charges are equivalent, one must prove that both the theories, described by equations (85), are in
fact different formulations of the same theory. To do this, the mechanics of one theory need to be satisfied when
considering the mechanics of the other. Therefore, in the next section, the duality will be proven with the use of
the stated duality transformations.
5.3 Proving the Duality
In this section it will be proven that both theories described by Lagrangians (85) are the same theory, so long as:
ψ = ∣ψ∣e−iθ. (100)
Differentiating ψ with respect to space or time results in
−i∂µθ =1
2
ψ∂µψ − ψ∂µψ∣ψ∣2 . (101)
These can be related to the Noether current (95) by duality transformations (87). Then writing the temporal and
spatial components explicitly in terms of light cone coordinates results in:
ψ(∂uψ − ∂vψ) − ψ(∂uψ − ∂vψ)∣ψ∣2 = φ(∂uφ + ∂vφ) − φ(∂uφ + ∂vφ)
G(∣φ∣) , (102)
ψ(∂uψ + ∂vψ) − ψ(∂uψ + ∂vψ)∣ψ∣2 = φ(∂uφ − ∂vφ) − φ(∂uφ − ∂vφ)
G(∣φ∣) . (103)
One can create similar expressions by considering the duality transformation (99) differentiated with respect to xµ.
Using the chain rule and product rule results in the the following expression:
ψ∂µψ − ψ∂µψ∣ψ∣2 = − φ∂µφ + φ∂µφ
G(∣φ∣) . (104)
Writing the temporal and spatial components explicitly in terms of light cone coordinates results in
ψ(∂uψ − ∂vψ) + ψ(∂uψ − ∂vψ)∣ψ∣2 = −φ(∂uφ − ∂vφ) + φ(∂uφ − ∂vφ)
G(∣φ∣) , (105)
ψ(∂uψ + ∂vψ) + ψ(∂uψ + ∂vψ)∣ψ∣2 = −φ(∂uφ − ∂vφ) + φ(∂uφ − ∂vφ)
G(∣φ∣) . (106)
Finally, combining equations (102) + (103) + (105) + (106) and (102) + (105) − (103) − (106) results in the following
relations respectively:ψ∂uψ
∣ψ∣2 = − φ∂uφG(∣φ∣) , (107)
ψ∂vψ
∣ψ∣2 = − φ∂vφ
G(∣φ∣) . (108)
30
Now that the above relations have been derived they can be substituted into the equation of motion (91), which
will be written, again for convenience, in light cone coordinates:
∂uvψ − ∂uψ∂vψψ
G(∣ψ∣)dG(∣ψ∣)d∣ψ∣2 + ψG(∣ψ∣)dW (∣ψ∣)
d∣ψ∣2 . (109)
One can substitute equation (107) into the first term for ∂uψ. Also, using the duality transformations and dW (∣ψ∣)d∣ψ∣2 =
dW (∣φ∣)d∣φ∣2
dW (∣ψ∣)dW (∣φ∣)
d∣φ∣2dψ2 results in the following:
ψφ
G(∣φ∣)[1
φ∂vφ∂uφ + ∂v∂uφ + φ
dW (∣φ∣)d∣φ∣2 G(∣φ∣) + ∂vψ∂uφ
ψ+ G(∣φ∣)
φψ∂uψ∂V ψ −
∂uφ∂vG(∣φ∣)G(∣φ∣) ] = 0. (110)
Now splitting the first term into two parts by considering the product rule on ∂v ∣φ∣2 and using the fact thatdG(∣φ∣)d∣φ∣2
G∣φ∣2 = 1, the equation of motion can be re-written to give
ψφ
G(∣φ∣)[ − ∂uφ∂vφdG(∣φ∣)d∣φ∣2
φ
G(∣φ∣) + ∂uφ∂v ∣φ∣2 dG(∣φ∣)d∣φ∣2
1
G(∣φ∣) + ∂v∂uφ + φdW (∣φ∣)d∣φ∣2 G(∣φ∣)
+∂vψ∂uφψ
+ G(∣φ∣)φψ
∂uψ∂vψ −∂uφ∂vG(∣φ∣)
G(∣φ∣) ] = 0.
(111)
Using (107) and (108) one can show that the second term cancels with the last term and the fourth term cancels
with the fifth term, resulting in
∂uvψ − ∂uψ∂vψψ
G(∣ψ∣)dG(∣ψ∣)d∣ψ∣2 + ψG(∣ψ∣)dW (∣ψ∣)
d∣ψ∣2 =
ψφ
G(∣φ∣)[∂uvφ − ∂uφ∂vφdG(∣φ∣)d∣φ∣2
φ
G(∣φ∣) + φdW (∣φ∣)d∣φ∣2 G(∣φ∣)].
(112)
This shows that using the duality transformations, the equation of motion of the ψ theory can be rewritten as the
equation of motion of the φ theory multiplied by a scale factor. Therefore, if the equation of motion is satisfied in
one theory, then the equation of motion of the other theory is also satisfied, thus proving the duality.
6 Specifying the Duality to the Stationary Q-ball and Static Kink
Now that the duality has been proven the following section will specify the arbitrary functions within equations
(85), such that complex kinks and Q-balls are solutions to the respective theories. Consider setting the arbitrary
functions to:
G(∣ψ∣) = 1 − ∣ψ∣2, W (∣ψ∣) = 1 − ∣ψ∣2. (113)
This ensures the first Lagrangian in (85) is of the form of the complex kink Lagrangian, equation (68) discussed in
section 2.4.1:
L = 1
1 − ∣ψ∣2 ∂µψ∂µψ − (1 − ∣ψ∣2). (114)
Using the duality transformations equations (86) one finds:
∣ψ∣2 = 1 − ∣φ∣2, G(∣φ∣) = 1 − ∣φ∣2, W (∣φ∣) = ∣φ∣2. (115)
Substituting the above into the first Lagrangian of (85) results in the following:
L = 1
1 − ∣φ∣2 ∂µφ∂µφ − ∣φ∣2. (116)
This Lagrangian can have Q-ball solutions, however, this is not initially obvious and will be discussed in the next
section.
31
6.1 Dual Q-ball Analytics
As stated previously, it is not obvious that this Lagrangian has Q-ball solutions. For Q-ball solutions, one must have
the correct initial conditions and certain requirements for the potential. As seen in figure 23, the potential must
have a maximum at ∣φ∣ = 0 in order for a Q-ball solution to exist. This is because if the initial condition is at point
A then the profile of ∣φ∣ reaches zero as x →∞, i.e. a Q-ball solution. To find the form of the potential from this
Lagrangian, one must change to an appropriate coordinate system by using the following coordinate transformation:
∣φ∣ = sinχ. Combining this transformation with the ansatz, (82), one finds the Lagrangian to have the following
form:
L = −χ′2 − (−ω2 tan2 χ + sin2 χ). (117)
Solving for the equation of motion results in χ′′ + ∂V∂χ
and hence, one finds the potential of the form:
V = −1
2(−ω2 tan2 χ + sin2 χ). (118)
This potential has the graphical representation shown in figure 23, for 0 < ω < 1, which is the form required for
Q-ball solutions. The range of ω is found by considering the fact that a maximum must exist at χ = 0, and also by
considering that for Q-ball solutions, V (0) = 0 for any ω. Therefore, setting V = 0 in equation (118) and rearranging
to find ω2 = cos2 χ, one can differentiate with respect to χ to find the minimum, ωmin = 0, and maximum, ωmax = 1,
where we have discounted the negative ω range. In addition, setting V = 0 one can find the analytical initial
conditions for which Q-ball solutions exist,
Figure 23: A graph of the potential of the dual Q-ball in χ coordinates. For a Q-ball profile one must have initial
conditions which start at A.
χ0 = cos−1 ω Ô⇒ ∣φ∣0 = sin(cos−1(ω)), (119)
∣φ∣′0 = 0. (120)
32
Figure 24: The field, charge and energy profiles of the dual Q-ball theory for ω = 0.81.
Using the general equation of motion (90) and the ansatz (82) the equation of motion for a Q-ball is
∣φ∣′′ + ω2∣φ∣ + ω2∣φ∣2 + ∣φ∣′21 − ∣φ∣2 ∣φ∣ − (1 − ∣φ∣2)∣φ∣ = 0. (121)
The initial conditions above can be used to solve such an equation.
Initial conditions can also be found for the charge and the Hamiltonian of a Q-ball by considering equations
(78) and (80). Considering the forms of G(∣φ∣) and W (φ∣) in equations (115), as well as using the ansatz (82) and
setting the final term in the Hamiltonian to be 0 results in the following forms for the charge and energy of the
dual Q-ball:
Q(x) ∶= ∫x
0q = ∫
x
0
2ω∣φ∣21 − ∣φ∣2 , (122)
H(x) = ∫x
0
ω2∣φ∣21 − ∣φ∣2 +
∣φ∣′21 − ∣φ∣2 + ∣φ∣2. (123)
Therefore the analytical initial conditions required to solve for the absolute field profile (121), charge profile (122)
and energy profile (123) for a Q-ball field are:
∣φ∣0 = sin(cos−1(ω)), ∣φ∣′0 = 0, Q(0) = 0, H(0) = 0. (124)
The numerical results are shown in figure 24. As shown, the results have the generic Q-ball profile, similar to the
results in section 4.3. However, the profiles for larger radii overshoot or undershoot, the reason is discussed in
section 6.4. For the analytical initial conditions, one would presume that such affect would not occur. Briefly put,
the discrepancy lies within the computational limitations. However, one can see the generic form of a Q-ball profile.
The modulus of the field has a defect at the origin which decays into the vacuum state as x→∞. Both the charge
and energy can be found to be concentrated in a ball about the origin. The total charge and energy can be found
numerically for a range of ω values, just as in section 4.3. The result is shown in figure 25.
These results will be compared with the dual kink in section 6.3.
33
Figure 25: The total charge and energy for dual Q-balls over the range of ω.
6.2 Dual Complex Kink Analytics
Using the above analytics and the duality, the initial conditions for the complex kink are found in this section.
As discussed in section 5.3, the solution to the dual theory is of the form ψ = ∣ψ∣e−iθ and so the following form for
the differential of the phase is
θ′ = −i2
(ψψ′ − ψψ′)∣ψ∣2 . (125)
One can then use the first duality transformation in equations (87) and the form of the charge density from (122)
to find the phase as a function of ∣φ∣:
θ(x) = 1
2∫
x
0dx
2ω∣φ∣21 − ∣φ∣2 . (126)
Therefore θ(0) = 0 is one of the initial conditions to required to solve (125). Similarly, the Hamiltonian can be
calculated from the Lagrangian (114) resulting in
H(x) = ∫x
0dx
ψ′ψ′
1 − ∣ψ∣2 + 1 − ∣ψ∣2. (127)
Therefore, H(0) = 0 is one of the initial conditions to solve the Hamiltonian. The other two initial conditions
required to solve equations (125) and (127) are also required solve the equation of motion of the kink. These initial
conditions can be found using the first duality relation (115) and the initial conditions of the Q-ball (124). Firstly,
as θ(0) = 0 then ψ(0) = ∣ψ(0)∣, then using the duality transformation:
∣ψ(0)∣2 = 1 − ∣φ(0)∣ Ô⇒ ψ(0) = ω. (128)
The second initial conditions arises by considering the differential of ψ = ∣ψ∣eiθ. In addition to this, the initial
condition of θ′ is required. This can be established by considering the differential of equation (126) and substituting
the initial conditions of the Q-ball. This results in θ′(0) = ω−1 sin(cos−1(ω)). Putting all of the above together one
finds
ψ′(0) = sin2(cos−1(ω)). (129)
34
Figure 26: The above shows the real, imaginary parts and modulus of the dual kink field. It also shows the argand
diagram of the kink. Note, the circle is of radius one to emphasis the fact the modulus of the kink does not go
beyond one. In addition, the charge and energy profiles describe the charge and energy in a radius r.
The above two initial conditions can be used to calculate the charge and the energy of the complex kink, as well as
the equation of motion. Similar to the Q-ball case above, the equation of motion can be calculated by substituting
the definitions of G(∣ψ∣) and W (∣ψ∣) into the equation of motion (91) and assuming a static field, ψ = 0, resulting
in:
ψ′′ + ψ′2ψ1 − ∣ψ∣2 − ψ(1 − ∣ψ∣2) = 0. (130)
Therefore, the initial conditions required to solve for the analytic field profile (130), charge profile (126) and energy
profile (127) for the complex kink are:
ψ(0) = ω, ψ′(0) = sin2(cos−1(ω)), θ(0) = 0, H(0) = 0. (131)
Solving the equations numerically results in the profiles in figure 26.
Similarly to the Q-ball results in section (6.1), the profiles again overshoot or undershoot for the same reason.
The generic profiles still exist and the duality between the two theories obviously holds. It is evident that there
is a topological defect at the origin, where the charge and energy are stored. The field then reaches the vacuum
expectation value as x → ∞. The profiles can be calculated for a range of ω values and plotted. Using this, the
total charge and energy can be calculated and plotted for varying ω. The result are shown in figure 27.
Clearly, the result resembles that of the Q-ball data shown in figure 25. The reason for this is discussed in the next
section.
35
Figure 27: A graph of the total charge and energy of complex kinks in the theory for a range of ω.
6.3 Comparing the Complex Kink and the Dual Q-ball
Using the Q-ball numerics for the field, charge and energy profiles, which were found by solving the equation of
motion (121), the charge (122) and the energy (123), one can use the duality along with the transformations (86)
and (87) to calculate the complex kink field, charge and energy profile. In this section, the above is implemented.
In section 5.2, the relationship between the topological charge of the complex kink and the Noether charge of
the Q-ball has already been established. They are equal and, in fact, the profiles are equal, which can be seen by
considering equation (97). Similarly, one can show that the Hamiltonian of the Q-ball is equal to the Hamiltonian
of the kink. Finally, the modulus of the field profiles are linked by the first duality transformation in (86).
Therefore, by using the Q-ball numerics, the kink profiles can be calculated using the following equations:
∣ψ∣ = (1 − ∣φ∣2) 12 , θ(x) = Q(x), H(x)kink =H(x)Q−ball. (132)
The results are plotted in figure 28 and compared with the kink numerics for the field profile, (130), the phase
profile, (126), and the energy profile, (127).
As can be seen, the profiles for both cases match, hence the duality has been numerically established. Furthermore,
the total charge and energy for range of ω values are the same for both cases. This can be seen in figure 29 for the
charge and figure 30 for the energy.
However, as discussed above, these profiles do not fit the criteria for a Q-ball or a complex kink. Therefore, the
numerical calculations were revised and the results recreated. The results are shown in the next section.
6.4 Numerical Limitations
Numerical methods do not exactly solve the specific differential equation requested of them, but instead solve an
approximation of the differential equation. One example would be, rather than solving the equation
φ′′ = +V (φ), (133)
36
Figure 28: A comparison of the kink profiles using the kink numerics (blue) and the Q-ball numerics (red). This is
with analytical initial conditions.
the program solves an approximation by considering the Taylor series of φ(x + dx) and φ(x − dx) to form the
following approximation:
φ′′ ≈ φ(x + dx) + φ(x − dx) − 2φ(x)dx2
. (134)
Then substituting the equation (134) into (133) and rearranging, results in the field value at the next spatial step:
φ(x + dx) = 2φ(x) − φ(x − dx) + V (φ(x))dx2. (135)
The results are then collected and plotted. Time evolution can also be considered in a similar way and in fact,
this is the procedure that was undertaken in order to calculate the Q-ball dynamics in the next section. One can
see, however, by combining the two equations (134) and (133), along with the the full Taylor expansion, the actual
equation being solved is
aφ′′ + bφ′′′ + cφ′′′′... = V (φ(x)), (136)
where a, b and c are numbers. This is not equal to the analytical equation (133).
Therefore, to negate this occurrence, the initial conditions can be amended to accommodate for the approximation.
This will prevent undershooting and overshooting and therefore provide exact Q-ball and kink profiles. This is
done using a bisection method. One starts with the analytical initial conditions and calculates the absolute field
profile numerically. Depending on whether overshooting or undershooting occurs, the initial conditions are altered
accordingly until the opposite situation occurs. Then using bisection, the initial conditions that provide Q-ball and
kink solutions are found. The results for the field profile, charge profile and energy profile with the new initial
conditions are displayed in figure 32 for the Q-ball and figure 31 for the complex kink.
The total charge and the energy for a range of omega can also be calculated again. Then, using the same procedure
as in section 6.3, one can compare the Q-ball numerics to the kink numerics using the duality. In figure 33, the
37
Figure 29: A comparison between the total charge of the kink via the kink numerics (blue) and the Q-ball numerics
(red) is shown for a range of ω, with analytical initial conditions.
Figure 30: A comparison between the total energy of the kink via the kink numerics (blue) and the Q-ball numerics
(red) is shown for a range of ω, with analytical initial conditions.
38
Figure 31: The profiles of field, charge and energy for the dual kink are repeated, but with initial conditions
calculated using the bisection method.
Figure 32: The profiles of field, charge and energy for the dual Q-ball are repeated, but with initial conditions
calculated using the bisection method.
39
Figure 33: A comparison between profiles of the field, charge and energy of a kink using the kink numerics (blue)
and the Q-ball numerics (red), using initial conditions found using a bisection method. There is still, however, an
overshoot when the charge was calculated
field, charge and energy profiles are compared. In figure 34, the total charge is compared and, finally, in figure 35,
the total energy is compared.
Notice the slight variation in the profiles. This is due to the fact that now the initial conditions for the kink do not
correlate with the initial conditions for the Q-ball, due to the numerical bisection method working independently
in both theories.
40
Figure 34: A comparison between the total charge of a kink using the kink numerics (blue) and the Q-ball numerics
(red). The initial conditions were found using a bisection method. This is done for a range of ω.
Figure 35: A comparison between the total energy of a kink using the kink numerics (blue) and the Q-ball numerics
(red). The initial conditions were found using a bisection method. This is done for a range of ω.
41
7 Q-ball Dynamics
As discussed in section 6.4, the time evolution of the field can be calculated by considering Taylor expansions of the
field with respect to time and space. These can be rearranged to find approximations for the second differential of
the field.
φ′′ ≈ φ(x + dx) + φ(x − dx) − 2φ(x)dx2
, φ ≈ φ(t + dt) + φ(t − dt) − 2φ(t)tx2
. (137)
This is useful when considering the equation of motion for the Q-ball (90) with G(∣φ∣) and W (∣φ∣) substituted into
the equation of motion:
φ = φ′′ − (φ2 − φ′2)φ1 − ∣φ∣2 − φ(1 − ∣φ∣2). (138)
Substituting the approximations above, along with the definitions for the differential of the field with respect to
space and time results in the following:
φ(t + dt) = 2φ(t) − φ(t − dt) + (φ(x + dx) + φ(x − dx) − 2φ(x)) dt2
dx2+
dt2( − [φ(t) − φ(t − dt)dt
]2
+ [φ(x + dx) − φ(x)dx
]2
) φ
1 − ∣φ∣2 − dt2φ(1 − ∣φ∣2).
(139)
One can calculate the value of the field at multiple discrete temporal and spatial positions, on a lattice, and plot
the data. This can be done for a system of multiple Q-balls on the lattice, as long as the Q-balls are sufficiently far
apart such that
φTot = ∣φ1∣eiω1t + ∣φ2∣eiω2t+iα, (140)
where α is the phase between the two Q-balls.
Unfortunately, due to numerical complications, we have not been successful in producing Q-ball collision data.
We attempted to use a bisection method to form initial Q-balls and then approximated the field far from the origin
of the Q-ball as zero. We then added two fields describing Q-balls together to form the total field. Furthermore,
periodic boundary conditions were implemented. Results from the paper [41] by Battye and Sutcliffe were repeated
and similar results were found for the analytical equation they suggest. We are unsure where the computational
complications have arisen for the duality Q-balls described by equation (139). We tried varying the distances dt
and dx to no avail. Also, varying approximations for the differential of φ in time and space were used, with differing
results. Therefore, numerically producing Q-ball collisions is a possible extension to the project. In addition to
this, the results from Q-ball collisions could be mapped into the kink dual theory to see how kink collisions evolve,
so long as the phenomena created after Q-ball collisions can be approximated as individual Q-balls. As kinks are
generally harder to solve, this is possibly the better alternative.
8 Conclusions and Summary
We discussed some of the important dualities in physics, noting that they arise in a multitude of different areas.
The AdS/CFT duality has particular relevance in physics, having aided research in many areas. Examining other
dualities has been a key feature in understanding the general underlying formalisms behind duality, along with its
relevance in physics, thus allowing us to progress toward proving a particular classical duality later in the report.
We analysed the Kibble mechanism, which can produce topological defects, these are believed to arise in areas
of physics such as cosmology and condensed matter. We then explained some key properties of them, in order to
arrive at a description of the complex kink: one half of this project’s duality. We then did the same for Q-balls:
the other half of the duality. Q-balls are believed to exist in some theories of baryogenesis and dark matter.
42
After sufficiently discussing the analytics of the two theories individually, we proved a duality between the static
complex kink and the 1 + 1 dimensional stationary Q-ball. We did this by showing that the equation of motion in
one theory is proportional to the equation of motion in the other theory.
This duality allows a mapping between the two theories via the duality transformations stated in section 5.2:
we may calculate results in one theory, then yield results in the other, without having to calculate explicitly in the
second theory. We discussed in the introduction and section 2 that dualities can be a very useful tool in areas of
physics such as quantum field theory and string theory. Since results can be obtained from one theory to determine
results in the other theory, results may still be obtainable due to the duality, even if perturbative calculations may
be invalid.
Though the primary duality of this report has no obvious applications, it elegantly and comprehensibly explains
the key ideas and mathematics behind duality in a more basic and intuitive way than the dualities which exist in
quantum field theory and string theory. It also demonstrates a key idea: that dualities can relate theories which
have different fundamental properties and are seemingly unrelated.
After proving the general duality in section 5.3, we set functions G, G, W and W of the duality to be func-
tions which are shown to give the form of the previously discussed complex kink and Q-ball. We then discussed
the numerics of both theories and compared the complex kink and Q-ball to conclude that the duality holds, with
slight variations arising from the computational limitations.
After establishing the duality, we realised that it may be useful to analyse systems involving multiple Q-balls.
Analysis of these systems may lead to understanding of phenomena which can occur under Q-ball collisions, as seen
in 7, as well as in reference[41]. Knowledge of Q-ball collisions may offer insight to the behaviour of kink collisions
via the duality we have proven. A possible extension of the project is therefore to analyse Q-ball collisions.
43
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