# Notes on the Dynamics of Noncommutative U(2) and Commutative SU(3)
Instantons

Douglas J. Smith  Calum J. Robson  Joseph A. Farrow

###### Abstract

We examine the dynamics of noncommutative instantons of instanton number $2$
and commutative instantons of instanton number $3$ in 5d Super Yang Mills
theory. We begin by detailing the construction of the 1/4-BPS instanton
solutions, their moduli space, and the moduli space potential using an
explicit parametrisation of the moduli space coordinates in terms of the
biquaternions. We then go on to numerically analyse the dynamics on the moduli
spaces we have constructed, discussing some of the numerical issues which
arose and describing the code we developed to solve them

##  1 Introduction

The aim of this paper is to present new solutions for the moduli space
dynamics of 2- and 3- instantons in 5d SYM theory. Instantons are a specific
example of topological solitons, which are nonlinear solutions to certain
PDEs. Because the properties of these solutions are tied to topological
invariants of the spaces they are defined upon, they are very stable – no
continuous transformation (including time evolution) of the solutions can
cause these properties to change. Originally discovered in [1], research into
their properties took off after the discovery of the ADHM method for
constructing them in [2]. Whilst instantons were originally constructed as
solutions in 4 Euclidean dimensions, we can also define them in 5d SYM. Here
the instantons are static solutions in every slice of the 4 Euclidean
dimensions, and the 5th timelike dimension is seen as describing their
evolution. In the context of string theory this arises since the instantons
appear as 1/2-BPS states corresponding to an interacting system of D0-branes
and D4-branes [3] [4].
In this system, there are have 5 $SU(N)$ scalar fields which describe the
transverse positions of the D4-branes. Separating the branes gives at least
one of these scalars a non-zero expectation value. In the low energy limit
this corresponds to introducing a non-zero scalar field on top of the
instanton equations. This configuration would usually be an unstable field
configuration due to interactions with the Higgs field however introduction of
the scalar field gives the instantons an electric charge which balances the
scalar Higgs field, producting a stable solution.[5][6]. These charged
instantons are known as dyonic instantons, and are 1/4-BPS rather than
1/2-BPS. They are the low energy limits of a bound state of fundamental
strings and D0-branes.
This system has been the subject of particular interest as the low energy
limit of M5 branes in M-Theory. The low energy dynamics of these objects is
described using the so-called (2,0) theory, a 6d Superconformal field theory.
It has been shown that if we dimensionally reduce this theory, we get 5d SYM
[7][8]. Another way of reducing the number of dimensions is to compactify one
of the 6 dimensions into an $S^{1}$. Somewhat surprisingly, it turns out that
this also gives 5d SYM theory. In this case, the instanton sector, with
separate solutions labelled by the integer instanton number $k$, has been
shown to agree for low $k$ with the Kaluza-Klein modes arising from the
compactifaction, which are labelled by an integer winding number. This raises
the possibility that if these sectors are in fact identical, then 5d SYM with
all instantons included corresponds to including all the Kaluza-Klein modes of
the compactified (2,0) theory. This would imply that the 6d (2,0) theory is
the UV fixed point of 5d SYM, even though 5d SYM is pertubatively non-
renormalisable[9][10].
Directly calculating the dynamic behaviour of instanton solutions is both
analytically and computationally expensive. The moduli space method developed
by Manton [11] simplifies things by treating the free parameters as
coordinates on a manifold, called the moduli space. Evolution of an instanton
solution is approximated for slow motion by geodesic motion on the moduli
space. This moduli space contains singularities when the size of the instanton
shrinks to zero size. They correspond in the string theory to a transition
between the Coloumb and Higgs branches of the D0 theory [4].
This is the motivation for introducing noncommutativity, corresponding to
adding a Fayet-Illiopoulos term to the string theory. Defining the instantons
on a noncommutative spacetime has the effect of introducing a minimum size for
the instantons. This resolves the moduli space singularities as the instantons
can only shrink to a finite size, so cannot reach the singularity whee their
size vanishes.
We begin by introducing instantons themselves, and the noncommutative
spacetimes we will be studying them on. This includes a brief discussion of
the biquaternions – the algebra $\mathbb{C}\times\mathbb{H}$. The calculation
of instanton solutions uses the ADHM construction first developed in [2]. We
briefly review the use of this method in the $SU(2)$ Yang Mills case.
We then introduce the instanton moduli space [11]. We recap how this
construction can be extended to dyonic instantons via introducing a potential
on the Moduli space, following the presentation in [12]. A practical method
for calculating the moduli space metric and potential for noncommutative
$U(N)$ instantons is presented in Appendix C . This generalises that presented
for $SU(2)$ commutative instantons in [13].

In the second part of the paper, we look for solutions to the equations we
have derived. The noncommutative two-instanton case was first studied in [14]
however we found an error in this result. We were unable to find the exact
result for the full moduli space but we present a solution defined on a
subspace of the full moduli space taking values in the
$\mathbb{C}\times\mathbb{C}$ subgroup of $\mathbb{C}\times\mathbb{H}$. This is
a geodesic submanifold of the full moduli space. After finding this solution,
we use it to derive the metric and potential on this subspace. We then to
numerically evaluate scattering in this subspace. We compare these results to
the results for the commutative two-instanton in [13].
Finally, we look at the commutative three-instanton case. Again we present a
solution for the complex subspace $\mathbb{C}\times\mathbb{C}$ and calculate
the moduli space metric and potential for that subspace. Numerical scattering
calculations proved to be very computationally expensive, however we were able
to plot scalar field and topological charge density profiles, which allowed us
to make some comparison to the two instanton case in the appropriate limits.

We use two different numerical algorithms to integrate the moduli space
equation of motion, depending on the algebraic complexity of the moduli space
metrics and potentials considered. For two non-commutative instantons with an
orthogonal gauge embedding and 4 dimensional moduli space, we find the system
sufficiently simple to solve with the numerical algorithm developed in [13].
Increasing to the full 6 dimensional gauge embedding results in significantly
more complex metrics and potentials on the moduli space, and we find that
these cases are no longer tractable to the algorithm from [13]. To overcome
this problem we develop a new numerical algorithm, which we describe in
Appendix G. We will attach a Mathematica notebook containing the methods
discussed in this appendix to a future arXiv submission of the paper, which
can be accessed by following the links ‘Other formats’ and then ‘Download
Source’.

##  2 Background Material

We begin by defining our notations and conventions for the biquaternions and
for noncommutative spacetime. These will be used throughout the rest of the
paper.

###  2.1 Quaternions and Biquaternions

The group $\mathbb{C}\times\mathbb{H}$, known as Complex Quaternions,
Biquaternions and even Tessarions has a long history [15]. To avoid confusion
we will refer to the group as Biquaternions in the rest of the paper. As
discussed in, e.g. [16], the algebra is equipped with three notions of
conjugation. We write a general element of the group as

$q=q_{R}+iq_{I}=q_{R0}+\textbf{q}_{R}+i(q_{I0}+i\textbf{q}_{I})$ (2.1)

Where $q_{R},q_{I}\in\mathbb{H}$, and correspondingly
$q_{R0},q_{I0}\in\mathbb{R}$ and $\textbf{q}_{R},\textbf{q}_{I}$ belong to the
quaternion imaginary part of $\mathbb{H}$. Then we have a complex conjugation
$q^{\star}$, which takes

$q_{R}+iq_{I}\rightarrow q_{R}-iq_{I}$ (2.2)

We also have a quaternion conjugation $\bar{q}$

$q_{R}+iq_{I}\rightarrow\bar{q}_{R}+i\bar{q}_{I}=q_{R0}-\textbf{q}_{R}+i(q_{i0}-\textbf{q}_{I})$
(2.3)

Finally we have a total conjugation $q^{\dagger}$ which applies both these
operations simultaneously

$q_{R}+iq_{I}\rightarrow\bar{q}_{R}-i\bar{q}_{I}=q_{R0}-\textbf{q}_{R}-i(q_{i0}-\textbf{q}_{I})$
(2.4)

To clarify our notation, we use the basis $\sigma_{n}$ for the quaternions,
with $\sigma_{n}=(\mathbb{1}_{2},i\tau_{i})$, where the $\tau_{i}$ are the
standard Pauli matrices. We also define
$\bar{\sigma}_{n}=(\mathbb{1}_{2},-i\tau_{i})$. Further, we define the
selfdual object

$\sigma_{mn}=\frac{1}{4}\Big{(}\sigma_{m}\bar{\sigma}_{n}-\sigma_{n}\bar{\sigma}_{m}\Big{)}$
(2.5)

and the anti- selfdual

$\bar{\sigma}_{mn}=\frac{1}{4}\Big{(}\bar{\sigma}_{m}\sigma_{n}-\bar{\sigma}_{n}\sigma_{m}\Big{)}$
(2.6)

Here, we define a selfdual matrix as one for which $A_{nm}^{\star}=A_{mn}$,
and an anti-selfdual one as $A_{nm}^{\star}=-A_{mn}$. With these definitions,
we have

$\sigma_{0}=\begin{bmatrix}1&0\\\ 0&1\end{bmatrix},\ \
\sigma_{1}=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix}\ \
\sigma_{2}=\begin{bmatrix}0&i\\\ i&0\end{bmatrix}\ \
\sigma_{3}=\begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}$ (2.7)

Finally, the fact that the biquaternions have multiple notions of conjugation
means that there are multiple notions of the imaginary part. We will use
$\text{Im}_{\mathbb{H}}$ to denote the quaternion imaginary part, defined for
$q=q_{0}+\textbf{q}\in\mathbb{H}$, with $q_{0}\in\mathbb{R}$ and q a purely
imaginary quaternion, as

$\text{Im}_{\mathbb{H}}(q)=\textbf{q}$ (2.8)

We also have the complex imaginary part, defined for $z\in\mathbb{C}$,
$z=x+iy$ as

$\text{Im}_{\mathbb{C}}(z)=y$ (2.9)

Note that $\text{Im}_{\mathbb{C}}$ doesn’t include the factor $i$ which we
must add in by hand where it is required – this is done to match with the
usual definition of Im in the complex case, however it does mean some care has
to be taken when restricting from $\mathbb{H}$ to $\mathbb{C}$, as $i$ then
corresponds to the imaginary quaternion basis vector, which is included in
$\text{Im}_{\mathbb{H}}$ but not in $\text{Im}_{\mathbb{C}}$.
In the case of a biquaternion $q=q_{R}+iq_{I}$ we have

$\text{Im}_{\mathbb{C}}(q)=q_{I}\ ;\
\text{Im}_{\mathbb{H}}(q)=\textbf{q}_{R}+i\textbf{q}_{I}$ (2.10)

where $\textbf{q}_{R}$ and $\textbf{q}_{I}$ are the quaternion imaginary parts
of $q_{R}$ and $q_{I}$ respectively. We similarly define
$\text{Re}_{\mathbb{H}}$ and $\text{Re}_{\mathbb{C}}$.

###  2.2 Noncommutativity

It is convenient to introduce the study of noncommutative spacetimes into the
study of instantons. It is convenient because it allows us to resolve
singularities on the instanton moduli space (see the next section). It was
first show in [17] that this was possible, and since then many examples have
been constructed (see e.g. [18][19][20] for a selection).
To construct a noncommutative version of $\mathbb{R}^{4}$, we simply impose an
anticommutation relation on the spacetime coordinates

$\big{[}x^{m},x^{n}\big{]}=\theta^{mn}$ (2.11)

Here $m,n$ are the Euclidean Lorentz indices, and $\theta^{mn}$ is a real,
antisymmetric, constant matrix. We can always rotate it into the form

$\theta^{mn}=\begin{bmatrix}0&\theta^{12}&0&0\\\ -\theta^{12}&0&0&0\\\
0&0&0&\theta^{34}\\\ 0&0&-\theta^{34}&0\end{bmatrix}$ (2.12)

There are several interesting subcases of this matrix [20]; in this paper we
consider the selfdual (SD) case, where $\theta^{12}=\theta^{34}=2\zeta$.
The noncommutativity of the spacetime coordinates forces us to modify our
notion of the multiplication of functions. Rather than the usual
multiplication, we use the Moyal Star Product [21]. This is defined as

$f(x)\star
g(x)=\exp\Big{(}\frac{i}{2}\theta^{ij}\partial_{i}\partial^{\prime}_{j}\Big{)}f(x)g(x^{\prime})|_{x=x^{\prime}}$
(2.13)

This gives the following expansion on powers of $\theta^{ij}$

$f(x)\star
g(x)=f(x)g(x)+\frac{i}{2}\theta^{ij}\partial_{i}f(x)\partial_{j}g(x)+\mathcal{O}(\theta^{2})$
(2.14)

Using this, the gauge potential and field strength become

$A_{i}\rightarrow g^{-1}\star A_{i}\star g+g^{-1}\star\partial_{i}g$ (2.15)

and

$F_{ij}=\partial_{[i}A_{j]}-i\big{[}A_{i},A_{j}\big{]}_{\star}$ (2.16)

where

$\big{[}A_{i},A_{j}\big{]}_{\star}=A_{i}\star A_{j}-A_{j}\star A_{i}$ (2.17)

This has two effects on our instanton solutions. First of all, it allows us to
find solutions with no commutative equivalent, since the additional length
scale $[\zeta]=[length]^{2}$ and the fact we are not in Euclidean flat space
means Derrick’s theorem does not apply.
Secondly, and less positively, in theory it implies we have an infinite number
of terms to calculate. However we can avoid this thanks to an isomorphism
between the algebra of functions with the $\star$-product, and certain
operators over Hilbert space. This is more fully discussed in [22].

##  3 Dyonic Instantons

Now we have discussed these notational conventions we define what is meant by
dyonic instantons. Following the presentation in [13] we start with the action

$S_{YM}=\int
d^{5}x\frac{1}{4}F_{\mu\nu}^{a}F_{a}^{\mu\nu}+\frac{1}{2}D_{\mu}\phi
D^{\mu}\phi$ (3.1)

We consider static solutions, for which the integral is taken over the four
spatial dimensions. These have energy, topological charge, and electric charge
given respectively by

$\displaystyle E=$ $\displaystyle\int
d^{4}x\text{Tr}\big{(}\frac{1}{2}F_{i0}F_{i0}+\frac{1}{4}F_{ij}F_{ij}+\frac{1}{2}D_{0}\phi
D_{0}\phi+\frac{1}{2}D_{i}\phi D_{i}\phi\big{)},$ $\displaystyle k=$
$\displaystyle-\frac{1}{16\pi^{2}}\int
d^{4}x\epsilon_{ijkl}\text{Tr}\big{(}F_{ij}F_{kl}\big{)},$ $\displaystyle
Q_{E}=$ $\displaystyle\int d^{4}x\text{Tr}\big{(}D_{i}\phi F_{i0}\big{)}=\int
d^{4}x\text{Tr}\big{(}D_{i}\phi\big{)}^{2}$ (3.2)

We use a Bogomolny argument of the type [23] to give us a bound on the energy,
by completing the square

$\displaystyle E=\ $ $\displaystyle\int
d^{4}x\text{Tr}\Big{(}\frac{1}{8}\big{(}F_{ij}\pm\frac{1}{2}\epsilon_{ijkl}F_{kl}\big{)}^{2}+\frac{1}{2}\big{(}F_{i0}\pm
D_{i}\phi\big{)}^{2}+\frac{1}{2}D_{i}\phi D_{i}\phi$ (3.3)
$\displaystyle+\frac{1}{8}\epsilon_{ijkl}F_{ij}F_{kl}\mp
F_{i0}D_{i}\phi\Big{)}$

So we get

$E\geq 2\pi^{2}\lvert k\rvert+\lvert Q_{E}\rvert$ (3.4)

where

$\displaystyle k=-\frac{1}{16\pi^{2}}\int d^{4}x\
\epsilon_{ijkl}\text{Tr}(F_{ij}F_{kl}),$ (3.5) $\displaystyle Q_{E}=\int
d^{4}x\text{Tr}(D_{i}\phi F_{i0})=\int d^{4}x\ \text{Tr}(D_{i}\phi)^{2}$

The conditions for this bound to be saturated are

$\displaystyle F_{ij}=\frac{1}{2}\epsilon_{ijkl}F_{kl}$ (3.6) $\displaystyle
F_{i0}=D_{i}\phi$ $\displaystyle D_{0}\phi=0$

With the boundary condition on $\phi$ that it goes to the vev
$\phi_{0}=i\textbf{q}$ at infinity [5] where q is an arbitrary imaginary
quaternion. The second and third of these are satisfied provided the fields
are static and $A_{0}=\phi$. Requiring our solutions to obey Gauss’ law,
$D_{i}E_{i}=ig\big{[}\phi,D_{0}\phi\big{]}$. imposes the further equation

$D^{2}\phi=0$ (3.7)

So, to find a dyonic instanton solution, we can use the ADHM method to
calculate a self dual field strength $F_{\mu\nu}$, then additionally calculate
the background scalar field using equation (3.7). An important corollary is
the observation that when $\phi=0$, the solution reduces to precisely that of
a pure instanton, and such solutions are in one-to-one correspondence with the
pure instanton solutions discussed above.

###  3.1 The ADHM Construction

To calculate an instanton solution, we use the ADHM construction. We follow an
ansatz-based construction outlined in [20]. A good discussion of this can be
found in [24].
The main ingredient in the ADHM construction is the ADHM Data $\Delta$. This
is an $(N+2k)\times 2k$ matrix, where $N$ is the degree of the gauge group
$SU(N)$ or $U(N)$, and $k$ is the instanton number, or topological degree. In
the commutative case the entries are usually taken to be real, whereas in the
noncommutative case they are taken as being complex. However, they can be
taken to be complex in the commutative case too, with the real solution
recovered using the symmetries due to the additional redundancy. Therefore we
will treat the entries as being complex in the remainder of this paper unless
otherwise stated. With this in mind, we have

$\Delta=\begin{bmatrix}\Lambda\\\ \Omega\end{bmatrix}$ (3.8)

where $\Lambda$ is an $N\times 2k$ complex matrix and $\Omega$ is a $2k\times
2k$ hermitian matrix. It is often useful for the purpose of performing
calculations to treat these as being instead biquaternion-valued matrices (or
quaternion-valued, for real matrices). The matrix $\Omega$ can always be
treated as a $k\times k$ matrix of biquaterions, but $\Lambda$ is not as
straightforward. For some values of $N$ and $k$ there is a similar
identification – for example, for the $U(2)$ instantons we consider in this
paper, we can always write $\Lambda$ as a row of $N$ (bi)quaternions. However,
in general, this is not possible.
This difficulty is mitigated by the fact that we will always end up
considering $\Delta^{\dagger}\Delta$ in any practical calculation, and as we
shall see below, this can always be written in (bi)quaternion form.
The commutative ADHM method involves solving the equation

$\Delta^{\dagger}\Delta=\mathbb{1_{2}}\otimes f^{-1}$ (3.9)

where $f$ is an invertible $k\times k$ matrix, and we can think of
$\mathbb{1}_{2}$ as the quaternion identity. This also means we can look at
the ADHM equation above as

$\text{Im}_{\mathbb{H}}\big{(}\Delta^{\dagger}\Delta\big{)}_{ij}=0$ (3.10)

where $\text{Im}_{\mathbb{H}}$ takes the quaternion imaginary part. In the
noncommutative case we must modify the ADHM equation (3.9) to be

$(\Delta^{\dagger}\Delta)_{ij}=\mathbb{1_{2}}\otimes
f^{-1}_{ij}-4\zeta\sigma_{3}\delta_{ij}$ (3.11)

which we can view as

$\text{Im}_{\mathbb{H}}\big{(}\Delta^{\dagger}\Delta\big{)}_{ij}=-4\zeta\sigma_{3}\delta_{ij}$
(3.12)

This solution has some residual freedom – we can transform any solution
$\Delta$ to

$\Delta\rightarrow\ \begin{bmatrix}1&0\\\ 0&R^{T}\end{bmatrix}\Delta\ R$
(3.13)

to obtain a new solution, where $R$ lies in $O(k)$ if we are using real
quaternions, or $SU(k)$ if we are using biquaternions. This additional freedom
in the biquaternion case cancels out the additional degrees of freedom from
the complexified ADHM parameters. This freedom is also important in obtaining
the moduli space metric (see appendix C). Once we have solved equation (3.9),
or its noncommutative analogue, we can use it to calculate the gauge potential
and field strength.
To do this, we need to find a zero eigenvector $U(x)$ of $\Delta^{\dagger}$,
normalised so that $U^{\dagger}U=1$. There are $N$ such vectors, spanning the
nullspace of $\Delta$, satisfying

$U^{\dagger}\Delta=\Delta^{\dagger}U=0$ (3.14)

This implies that $U$ has dimension $N+2k$ as a complex vector. Once we have
this $U$ we can use it to define the gauge potential as

$A_{\mu}=U^{\dagger}\partial_{\mu}U$ (3.15)

and the field strength as

$F_{\mu\nu}=-4U^{\dagger}bf\sigma_{mn}b^{\dagger}U$ (3.16)

where $b$ is a $(N+2k)\times 2k$ matrix whose top $N\times 2k$ part is $0$ and
whose bottom $2k\times 2k$ part is the identity, and $\sigma_{mn}$ is as given
in (2.6). This procedure works for both the commutative and the noncommutative
cases. An additional subtlety in the noncommutative case comes from the
assumption implicit in the above construction that we can factorise the
projection operator

$1-U\bar{U}$ (3.17)

as

$1-\mathcal{P}\equiv\delta_{\lambda\kappa}\delta\indices{{}_{\alpha}^{\beta}}-\mathcal{P}\indices{{}_{\lambda\kappa}{}_{\alpha}^{\beta}}=\Delta
f\bar{\Delta}$ (3.18)

This is called the, ‘Completeness relation’. It is automatically satisfied in
the commutative case, however there are some complications in the
noncommutative case. The issue is that, whereas the normalisation of $U(x)$ is
straightforward in the commutative case, there is are subtleties in the case
where $x$ is itself an operator. We must therefore be careful to pick a good
definition for this normalisation. These issues were first discussed in [20].
Checking this is highly non trivial, however as it only affects the value of
$U$, and not the validity of the remainder of the solution, it is only
necessary if one is constructing an explicit expression for the gauge
potential, which we are not. The only point we use $U$ is in Appendix D, and
here we take it in the limit $x\rightarrow\infty$ in which any noncommutative
effects (which go as $\frac{\zeta}{x^{r}}$ for some positive integer $r$) are
automatically neglected. This does not affect our results therefore, though a
full investigation might be a fruitful topic for future research.

###  3.2 The Moduli Space

The Moduli Space of instanton solutions is a the space of inequivalent
solutions to the self dual Yang Mills equations, (3.9). This was first
introduced for instantons in [11]. Calculating the dynamics of an individual
instanton solution over a period of time is difficult. However for
sufficiently slow velocities we can approximate such a solution by a slow
transition between different instanton solutions with marginally different
initial conditions. This corresponds to motion on the Moduli Space. For a
review of Techniques and applications see e.g. [12] [13][25][6]. The Moduli
Space is parameterised by the $4kN-4$ free ADHM parameters, which are called
the Collective Coordinates. To define small velocities, we must introduce a
moduli space metric. To do this we look at small fluctuations $A_{m}(x)+\delta
A_{m}(x)$. If this is also to be a solution to the equations (and hence lie in
the moduli space), the $\delta A_{m}$ must satisfy the linearised self duality
equation

$\mathcal{D}_{m}\delta A_{n}-\mathcal{D}_{n}\delta
A_{m}=\epsilon_{mnkl}\mathcal{D}_{k}\delta A_{l}$ (3.19)

In addition, it must not be related to $A_{n}(x)$ by a local gauge
transformation. We therefore require the solutions to be orthogonal to gauge
transformations. To define this orthogonality, we take the natural metric on
the space of all solutions

$g\big{(}\delta A_{m}(x),\delta A^{\prime}_{m}(x)\big{)}=\int
d^{4}x\text{Tr}\big{(}\delta A_{m}(x)\delta A^{\prime}_{m}(x)\big{)}$ (3.20)

and then use this to induce a metric on the moduli space after quotienting out
the gauge-equivalent solutions. We then require that under this metric, zero
modes $\delta A_{i}(x)$ are orthogonal to all gauge transformations
$D_{i}\Lambda$. This is equivalent to satisfying

$D_{i}\delta A_{i}=0$ (3.21)

For small pertubations, we get the following action on the moduli space
[26][12]

$S=\frac{1}{2}\int d^{5}x\text{Tr}\big{(}F_{i0}F_{i0}-D_{i}\phi
D_{i}\phi+D_{0}\phi D_{0}\phi\big{)}$ (3.22)

If we neglect terms of order $\dot{z}^{2}\lvert q\rvert^{2}$, where $z(t)$
refers to any of the collective coordinates on the moduli space, we get the
effective action

$S=\frac{1}{2}\int dt\Big{(}g_{rs}\dot{y}^{r}\dot{y}^{s}-\lvert
q\rvert^{2}g_{rs}K^{r}K^{s}\Big{)}$ (3.23)

Where the $K_{r}$ are Killing vectors of the moduli space and satisfy
$D_{m}\phi=\lvert q\rvert K^{r}\delta_{r}A_{m}$. Then equation (3.23) is the
sum of a free instanton and the potential

$V=\frac{1}{2}\int d^{5}x\text{Tr}(D_{i}\phi D_{i}\phi)=\frac{\lvert
q\rvert^{2}}{2}\int dtg_{rs}K^{r}K^{s}$ (3.24)

This solution is valid in the limit

$\dot{z}^{2},\lvert q\rvert\leq 1;$ (3.25)

where we can ignore terms of order $\dot{z}^{2}\lvert q\rvert^{2}$ and higher.
The geometric interpretation of this is both that the kinetic energy of the
instanton solution is sufficiently small, and that the potential evaluated on
the instanton solutions, which lie on the moduli space, is shallow compared to
the potential on non-BPS solutions evaluated off the moduli space. This allows
us to imagine our approximate solution as lying in a steep valley given by the
locally small potential around the moduli space solutions, where the small
kinetic energy prevents our dynamics from, ‘climbing away’ from the moduli
space.

###  3.3 The Complex Subspace

There is one final technical point to discuss, which applies to both Pure and
Dyonic Instantons. The Moduli space has several subspaces, which are preserved
under the geodesic motion. This means a geodesic beginning in one of these
subspaces (i.e. whose initial tangent vector lies in that subspace) will
remain in it throughout its motion.
The subspace we are interested in is as follows. The moduli space is a
manifold over the collective coordinates z. However, as we saw in section 3.1
these collective coordinates are elements of the ADHM matrix $\Delta$ and
therefore are parametrised by the (bi)quaternions. That algebra can be thought
of as $\mathbb{C}\times\mathbb{C}\times\mathbb{H}$. Therefore, in the same way
that the quaternions contain an invariant complex subspace $\mathbb{C}$, the
biquaternions have the subspace $\mathbb{C}\times\mathbb{C}$. We can therefore
restrict from the full moduli space where the collective coordinates are
biquaternions, to a submanifold where they lie in
$\mathbb{C}\times\mathbb{C}$.
This corresponds to conjugating all the moduli space coordinates by a unit
quaternion $q$, e.g. $\tau\rightarrow q\tau\bar{q}$. This corresponds
geometrically to leaving the real quaternion part fixed, whilst the imaginary
quaternion is rotated around an axis in $S^{3}$ represented by $q$. Imposing
invariance under such rotations corresponds to requiring our solutions to be
fixed points under this rotation, which constrains them to lie in a two
dimensional plane within the four dimensional quaternions. Because
$\bar{q}q=1$, if we multiply two (bi)quaternions together and apply this
rotation to each of them then the result is that the entire product is rotated
– e.g.

$\Delta^{\dagger}\Delta\rightarrow
q\Delta^{\dagger}\bar{q}q\Delta\bar{q}=q\Delta^{\dagger}\Delta\bar{q}$ (3.26)

We can therefore think of all our equations and objects (for example the
scalar field and potential) as being rotated in the same overall way. In the
commutative space, for pure instantons we can see the invariance of this
subspace automatically, since the elements of $\mathbb{C}\in\mathbb{H}$
automatically commute with all other elements, meaning that they form an ideal
within that group (and ideals are invariant subspaces) [27]. For dyonic
instantons, we must choose the imaginary direction to be the same as the vev
in SU(2), since otherwise the vev will not be preserved. The result is that
the transformation maps solutions to solutions, and so the fixed point
manifold thus generated is also a geodesic submanifold of our moduli space. In
the noncommutative case, the presence of the noncommutative parameter means
that the spacetime coordinates do not automatically commute. The only complex
subspace which is preserved in this case is the complex subspace spanned by
$\\{\mathbb{1},\sigma_{3}\\}$, as, since $\sigma_{3}$ is the direction
associated with the noncommutativity, it is preserved under rotations of the
space, and $\mathbb{1}$ commutes with everything. We must then align the plane
within $\mathbb{H}$ that we are preserving with this direction. Hence we see,
as would be expected, the presence of a noncommutative parameter reduces the
symmetries of the theory.
Calculations on the full quaternion moduli space are very computationally
expensive, and this subspace is often much easier to run simulations on. In
addition, the fact that elements in this subspace commute makes solving the
ADHM equations on this restricted part of the theory much easier.

##  4 The Two Instanton Solution

Now we have discussed the technical background, we present our work on the
dynamics of the noncommutative U(2) 2-instanton. First, we review the single
U(2) instanton as presented in [18]. This is necessary to test the two
instanton solution in the appropriate limits.
Next we derive the ADHM equations for the two Instanton case, using
biquaterion coordinates. We were unable to find a solution for the full moduli
space, however we were able to find a solution for the geodesic submanifold
discussed in section 3.3. This corrects the solution in [14]
After finding this solution we use it to derive the metric and potential for
the relevant moduli space, and show that the metric and potential behave
suitably in the commutative limit and in the limit of the instantons being far
separated.

###  4.1 The Single U(2) Instanton

First, we state the solution for a single U(2) instanton in noncommutative
space. We will follow [18], however we rederive their solution in our
notation. One key difference is that the relation between their
$\zeta^{\prime}$ and our $\zeta$ is $\zeta^{\prime}=2\zeta$ Other discussions
of the solution can be found in [20] and [19]. First, for the single
noncommutative U(2) instanton, the ADHM data has the form

$\begin{bmatrix}v_{R}+v_{I}\sigma_{3}\\\ X-x^{\prime}\end{bmatrix};\
v_{R},v_{I},X,x^{\prime}\in\mathbb{H}$ (4.1)

As discussed in [18], we can set $X-x^{\prime}=0$ by an symmetry
transformation related to the centre of mass. Then we can solve for $v_{I}$ in
terms of the free parameter $v_{R}$.

$v_{I}=-\frac{2\zeta v_{R}}{\lvert v_{R}\rvert^{2}};\ \ v_{R}\in\mathbb{H}$
(4.2)

This gives us 4 ADHM coordinates, as required. Now we have this, we can
calculate the metric and potential using the method discussed in appendices B
and D. The potential is equal to

$\mathcal{V}=8\pi^{2}\lvert\textbf{q}\rvert^{2}\Big{(}\rho^{2}-\frac{16\zeta^{2}}{\rho^{2}}cos^{2}(\theta)\Big{)}$
(4.3)

and the metric is

$ds^{2}=8\pi^{2}\bigg{(}dv_{R}^{2}+dv_{I}^{2}-\frac{\big{(}v_{R}dv_{I}-\bar{v}_{I}dv_{R}\big{)}^{2}}{\lvert
v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}}\bigg{)}$ (4.4)

Here $\lvert\textbf{q}\rvert$ is the vev of the scalar field $\phi,\
v_{R}=\rho\ \text{cos}(\theta)$ and $v_{I}=\rho\ \text{sin}(\theta)$.

###  4.2 Two U(2) Instantons

We now move on to the case of two U(2) instantons. In the commutative case, a
solution was found for the real quaternions (and with gauge group SU(2)) in
[13]. A solution for the noncommutative case was postulated in [14], however
this is not in fact correct, and an alternative solution is therefore
presented here. We were unable to find a solution for the full moduli space,
however we obtained a solution for the subspace defined in section 3.3. For
the case of two U(2) instantons, the ADHM data has the form

$\Delta=a-bx;\ a=\begin{bmatrix}\Lambda\\\ \Omega\end{bmatrix}=\
\begin{bmatrix}v&w\\\ \tau&\sigma^{\star}\\\ \sigma&-\tau\end{bmatrix},\
b=\begin{bmatrix}0&0\\\ 1&0\\\ 0&1\end{bmatrix}$ (4.5)

Note that here $\Omega$ is constrained to be hermitian (under complex
conjugation) rather than symmetric, as in the real ADHM construction. $v,w$
and $\sigma$ lie in the biquaternions, however due to the requirement that
$\Omega$ be hermitian, $\tau$ remains a member of $\mathbb{H}$. Proceeding as
in section 3.1, equation (3.9) gives the following. First, the diagonal
equations

$\displaystyle
v^{\dagger}v+\lvert\tau\rvert^{2}+\sigma^{\dagger}\sigma=f^{-1}_{11}\mathbb{1}+2\zeta\sigma_{3}$
$\displaystyle
w^{\dagger}w+\lvert\tau\rvert^{2}+(\sigma^{\dagger}\sigma)^{\star}=f^{-1}_{11}\mathbb{1}-2\zeta\sigma_{3}$
(4.6)

Next, the off diagonal constraints are given by

$\displaystyle
v^{\dagger}w+\bar{\tau}\sigma^{\star}-\sigma^{\dagger}\tau=f^{-1}_{12}\mathbb{1}$
$\displaystyle
w^{\dagger}v+(\sigma^{\dagger})^{\star}\tau-\bar{\tau}\sigma=f^{-1\star}_{12}\mathbb{1}$
(4.7)

This gives a total of four equations for the complex ADHM constraints. For
completeness we list them here

$\displaystyle
2\text{Im}_{\mathbb{H}}(\bar{\sigma}_{R}\sigma_{I})-\text{Im}_{\mathbb{H}}(\bar{w}_{R}w_{I})+\text{Im}_{\mathbb{H}}(\bar{v}_{R}v_{I})=0$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{w}_{R}w_{I})+\text{Im}_{\mathbb{H}}(\bar{v}_{R}v_{I})=-4\zeta\sigma_{3}$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{\tau}\sigma_{I})=\frac{\text{Im}_{\mathbb{H}}(\bar{w}_{R}v_{I}+\bar{v}_{R}w_{I})}{2}\equiv\frac{\Upsilon}{2}$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{\tau}\sigma_{R})=\frac{\text{Im}_{\mathbb{H}}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})}{2}\equiv\frac{\Lambda}{2}$
(4.8)

As a check, if we assume that our solutions to the ADHM equations are entirely
real and that $\zeta=0$, we have the complex imaginary parts of all our
variables being 0, and we have only the one equation which is not trivially
satisfied (just as in [13])

$\text{Im}_{\mathbb{H}}(\bar{\tau}\sigma_{R})=\frac{\text{Im}_{\mathbb{H}}(\bar{w}_{R}v_{R})}{2}$
(4.9)

It should be noted that no new degrees of freedom are introduced compared to
the real ADHM equations. Complexifying $v,w$ and $\sigma$ adds twelve degrees
of freedom. However, each of the 3 new equations affecting the imaginary part
of an expression adds 3 constraints, giving 9. Recall that we have a residual
O(2) symmetry on our solutions to the ADHM equation in the quaternion case,
which is promoted to a U(2) symmetry in the biquaternion case allows us to
remove a further three degrees of freedom. This gives a total of 12 degrees of
freedom removed, cancelling the number of new parameters and showing that
there are no new solutions. We have checked this explicitly by constructing
the transformation taking a commutative ADHM solution with biquaterion
coordinates to the standard solution parameterised by quaternions, however
this is time consuming and not especially illuminating.

We were unable to solve these equations for the full biquaternion valued
space, however we were able to calculate solutions restricted to the complex
subspace defined in 3.3. After some calculation (see appendix A), we get:

$\displaystyle v_{I}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert v_{R}\rvert^{2}}$
$\displaystyle w_{I}=\frac{-2\zeta w_{R}\sigma_{3}}{\lvert w_{R}\rvert^{2}}$
$\displaystyle\sigma_{R}=\frac{\tau\text{Im}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})}{2\lvert\tau\rvert^{2}}=\frac{(\lvert
v_{R}\rvert^{2}\lvert w_{R}\rvert^{2}+4\zeta^{2})}{2\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert
w_{R}\rvert^{2}}\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{R})\sigma_{3}$
$\displaystyle\sigma_{I}=\frac{\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{I}+\bar{v}_{R}w_{I})}{2\lvert\tau\rvert^{2}}=-\frac{\zeta(\lvert
w_{R}\rvert^{2}+\lvert v_{R}\rvert^{2})}{\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert
w_{R}\rvert^{2}}\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{R}\sigma_{3})\sigma_{3}$
(4.10)

We can check our assumption about the symmetries by checking both that our
solution really does solve the ADHM equations, and that there are no residual
symmetries remaining. To show there are no residual symmetries, we consider a
general U(2) transformation

$\Delta\mapsto\begin{bmatrix}1&0\\\ 0&R^{\dagger}\end{bmatrix}\Delta R;\
R=\begin{bmatrix}a&b\\\ -\bar{b}&\bar{a}\end{bmatrix};\ a,b\in\mathbb{C},\
\lvert a\rvert^{2}+\lvert b\rvert^{2}=1$ (4.11)

This generates the transformation

$\begin{bmatrix}v&w\\\ \tau&\sigma^{\star}\\\
\sigma&-\tau\end{bmatrix}\mapsto\begin{bmatrix}v^{\prime}&w^{\prime}\\\
\tau^{\prime}&\sigma^{\prime\star}\\\
\sigma^{\prime}&-\tau^{\prime}\end{bmatrix}\ =\
\begin{bmatrix}av-\bar{b}w&bv+\bar{a}w\\\ (\lvert a\rvert^{2}-\lvert
b\rvert^{2})\tau-
ab\sigma-\bar{a}\bar{b}\sigma^{\star}&2\bar{a}b\tau-b^{2}\sigma+\bar{a}^{2}\sigma^{\star}\\\
2a\bar{b}\tau+a^{2}\sigma-\bar{b}^{2}\sigma^{\star}&-(\lvert
a\rvert^{2}-\lvert
b\rvert^{2})\tau+ab\sigma+\bar{a}\bar{b}\sigma^{\star}\end{bmatrix}$ (4.12)

Now, in appendix A we use the residual symmetry to do two things. First, we
require that $\text{Re}_{\mathbb{C}}(\bar{\tau}\sigma)=0$. This imposes the
condition that $\sigma$ has no component proportional to $\tau$. Second, we
require that $\bar{w}_{R}w_{I}=\bar{v}_{R}v_{I}$. We need to work out the form
of the transformation in equation (4.12) so that the new variables
$v^{\prime},\tau^{\prime}$ etc, also satisfy these conditions. If the form of
the transformation is completely determined by this, then we know we have no
remaining symmetries to consider.
To aid in this, we write

$\displaystyle a$
$\displaystyle=\text{cos}\chi(\text{cos}\theta+i\text{sin}\theta)$
$\displaystyle b$
$\displaystyle=\text{sin}\chi(\text{cos}\phi+i\text{sin}\phi)$ (4.13)

The first of these conditions,
$\text{Re}_{\mathbb{C}}(\bar{\tau}^{\prime}\sigma^{\prime})=0$ requires that
$\lvert a\rvert^{2}-\lvert b\rvert^{2}=0$, and hence that both
$\chi=\frac{n\pi}{4}$ for $n$ from 1 to 7, and also $\theta=-\phi$. This
condition on $\chi$ gives the dihedral group of order 16 as a group of
discrete rotations. The implications of this are discussed in [13].
Now, we look at the second part of our symmetry, which is unique to the
noncommutative case. Keeping
$\bar{w}^{\prime}_{R}w^{\prime}_{I}=-\bar{v}^{\prime}_{R}v^{\prime}_{I}$
requires $\cos(2\theta)+i\sin(2\theta)=0$. This leads to

$\theta=\frac{n\pi}{2};\ \text{e.g.}\
\theta=0,\frac{\pi}{2},\pi,\frac{3\pi}{2}...$ (4.14)

Now, $\theta=\frac{n\pi}{2}$ multiplies $a,b$ by $\pm 1$ and so does not
change the symmetries in [13]. If we set $\theta=0,\pi$ we multiply $a$ and
$b$ by $\pm i$ and $\mp i$ respectively. This doesn’t change $\tau$, but sends
$\sigma\rightarrow-\sigma$. It also interchanges the complex real and
imaginary parts of $v^{\prime}$ and $w^{\prime}$. Between them these two
conditions fully fix the form of the transformation in equation 4.12 and
therefore there is no residual symmetry.
The next step is to calculate the scalar field, potential and metric. The
calculations and results are long and not particularly illuminating, and so
are given in appendices B and C. Now we will go on to investigate the dynamics
on the moduli space via numerical methods.

##  5 Two Instanton Dynamics

In this section we discuss the dynamics of the instantons on the
noncommutative two instanton moduli space we have constructed, solving for
motion on the moduli space numerically. Where possible we use the same
numerical algorithms developed in [13] to produce the following figures, and
for when the scattering problem becomes more complex we use a new numerical
algorithm which we outline in Appendix G.

###  5.1 The Setup

$y$$x$$b$$\omega$$\chi$$\tilde{\rho}_{2}$ Figure 1: The setup of the
Instantons. The instantons are located at $\pm(x,b)=\
\big{(}\omega\text{cos}(\chi),\omega\text{sin}(\chi)\big{)}$. They have size
$\tilde{\rho}_{i}=\sqrt{\rho_{i}^{2}+\frac{4\zeta^{2}}{\rho_{i}^{2}}}$

The ADHM data are in terms of
$v=v_{R}+iv_{I},w_{R}+iw_{I},\sigma=\sigma_{R}+i\sigma_{I}$ and $\tau$. We
showed in appendix that $v_{I},w_{I}$ and $\sigma$ depend on the collective
coordinates $v_{R},w_{R}$ and $\tau$. As stated in section 3.3, we are working
on the subspace of the total moduli space with the collective coordinates in
$\mathbb{C}\times\mathbb{C}$ rather than $\mathbb{C}\times\mathbb{H}$. This
means that $v$ and $w$ are in $\mathbb{C}\times\mathbb{C}$, whilst $\tau$ is
in $\mathbb{C}$ (since it lies on the diagonal, and $\Delta$ is hermitian with
respect to the complex structure).
As is standard [23], we interpret $v,w$ as describing the embedding of the
instantons into the gauge group, with $\tilde{\rho_{1}}=\lvert v\rvert$ and
$\tilde{\rho_{2}}=\lvert w\rvert$ giving the physical size; and $\tau$ giving
the position. This is shown in figure 1. We also rewrite our independent ADHM
coordinates in polar form as

$\displaystyle v_{R}=\rho_{1}\big{(}\cos(\theta_{1})+i\sin(\theta_{1})\big{)}$
$\displaystyle w_{R}=\rho_{2}\big{(}\cos(\theta_{2})+i\sin(\theta_{2})\big{)}$
$\displaystyle\tau=\omega\big{(}\cos(\chi)+i\sin(\chi)\big{)}$ (5.1)

It is convenient to describe the initial conditions in cartesian coordinates
$b$ and $x$ as indicated in figure 1. The relation between the parameters
$b,x$ and $\omega,\chi$ is

$\displaystyle x=\omega\cos(\chi)$ $\displaystyle b=\omega\sin(\chi)$
$\displaystyle\omega=\sqrt{b^{2}+x^{2}}$ $\displaystyle\chi=\arctan(b/x)$
(5.2)

Now, using the equations

$\displaystyle v_{I}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert v_{R}\rvert^{2}}$
$\displaystyle w_{I}=\frac{-2\zeta w_{R}\sigma_{3}}{\lvert w_{R}\rvert^{2}}$
(5.3)

It follows that in the noncommutative case, the total instanton size
$\rho_{i}$ is defined as

$\tilde{\rho}_{i}=\sqrt{\rho_{i}^{2}+\frac{4\zeta}{\rho_{i}^{2}}}$ (5.4)

where the index $i$ in $\rho_{i}$ is either 1 or 2, referring to the magnitude
of $v$ or $w$ respectively. Calculating the scattering with general $\rho_{1}$
and $\rho_{2}$, and with general gauge embedding is very computationally
expensive for the noncommutative case. Therefore we did a lot of the
simulations in the, ‘Orthogonal’ case where $\rho_{1}=\rho_{2}$ and the
relative gauge angle between the two instantons is
$\theta_{1}-\theta_{2}=\pi/2$. There are several technical issues which
emerged. First of all, at the collision point there are two ingoing and two
outgoing paths. Where the instanton positions nearly coincide, it is difficult
to work out which ingoing and outgoing paths ought to be connected. The
plotting programme we used appears to choose sensibly except in a few cases
where it appears to connect the wrong pairs, as evidenced by a discontinuity
in the path near the origin. This can be seen on several of the graphs, e.g.
in figure 6.
The second issue is with the parameterisation of the instanton position in
terms of $\tau$. The position of the instanton is given by the eigenvalues of
the submatrix

$\begin{bmatrix}\tau&\sigma^{\star}\\\ \sigma&-\tau\end{bmatrix}$ (5.5)

of the ADHM data [13]. Recall that

$\displaystyle\sigma_{R}$
$\displaystyle=\frac{\text{Im}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})}{2}=\frac{(\lvert
v_{R}\rvert^{2}\lvert w_{R}\rvert^{2}+4\zeta^{2})}{2\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert w_{R}\rvert^{2}}\tau\text{Im}(\bar{w}_{R}v_{R})$
$\displaystyle\sigma_{I}$
$\displaystyle=\frac{\text{Im}(\bar{w}_{R}v_{I}+\bar{v}_{R}w_{I})}{2}=-\frac{\zeta(\lvert
w_{R}\rvert^{2}+\lvert v_{R}\rvert^{2})}{\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert
w_{R}\rvert^{2}}\tau\text{Im}(\bar{w}_{R}v_{R}\sigma_{3})$ (5.6)

In the subspace under discussion, and in the coordinates we are using, this
becomes

$\displaystyle\sigma_{R}$
$\displaystyle=\frac{i\big{(}\rho_{1}^{2}\rho_{2}^{2}+4\zeta^{2}\big{)}\big{(}\cos(\chi)+i\sin(\chi)\big{)}\sin(\theta_{1}-\theta_{2})}{2\rho_{1}\rho_{2}\omega}$
$\displaystyle\sigma_{I}$
$\displaystyle=\frac{-i\zeta\big{(}\rho_{1}^{2}+\rho_{2}^{2}\big{)}\big{(}\cos(\chi)+i\sin(\chi)\big{)}\cos(\theta_{1}-\theta_{2})}{\rho_{1}\rho_{2}\omega}$
(5.7)

At large $\tau$, the matrix is effectively diagonal, and so the positions of
the two instantons can be approximated by $\tau$ and $-\tau$ respectively. At
small values of $\tau$, however, $\sigma$ becomes very large and therefore
$\pm\tau$ is no longer a good description. A better approach is to diagonalise
the full matrix which gives the parametrisation
$\pm\sqrt{\tau^{2}+\sigma^{2}}$ for the position (note that this is in general
a complex number), though this can give a discontinuity at the origin due to
the presence of the square root, with both positive and negative values. In
the commutative case this is the true position of the instanton. In the
noncommutative case, the noncommutativity of the underlying space means that
the meaning of, ‘true position’ is not clear, however it still makes more
sense to use the $\sqrt{\tau^{2}+\sigma^{2}}$ parametrisation, as the presence
of $\zeta$ in $\sigma$ means that this becomes more important (as shown in
figure 2).

Figure 2: Scattering of Dyonic instantons with $b=0.5$ and $\zeta=1.15$. Both
plots are solutions of the same initial set up, with different choices of
parameterisation of the position The left plot shows the $\lvert\tau\rvert$
parametrisation, the right shows $\pm\sqrt{\tau^{2}+\sigma^{2}}$. The radii of
the instantons are not shown. In this case the $\sigma$ behaviour dominates
and after the interaction the position of the instantons goes as
$\frac{1}{\lvert\tau\rvert^{2}}$ Note that after 2400 time steps the
$\sqrt{\tau^{2}+\sigma^{2}}$ had left the plot region, whereas the $\tau$ case
has been run for 50000 time steps and still has not left the plotted area.
This is because the size of the instantons (hence $v$ and $w$) becomes very
large and so $\sigma$ dominates $\tau$ in the definition of the instanton
position, so that merely plotting $\tau$ is a very inaccurate approximation to
the position.

We now move on to looking at the graphs.

###  5.2  Pure Instantons

We start with the four parameter orthogonal instantons. There are two
procedures we can use to investigate the moduli space dynamics when
noncommutativity is introduced. The first is to start off with a particular
example of commutative scattering and see how turning on the noncommutativity
affects this. The second relies on the fact that choosing a value for the
noncommutative parameter $\zeta$ sets an overall scale. We can therefore
‘scan’ the parameter space for interesting behaviour by varying one of the
other parameters at a time for a fixed value of $\zeta$. It should be noted
that this scanning process is not in itself sensitive to periodic ambiguities
in the scattering angles – e.g. instantons moving parallel and not interacting
and instantons reflecting directly off each other would both register a
scattering angle of zero. Therefore we must supplement this scanning by
looking at individual plots to check the interpretation of the scattering
angles we have found.
Both these methods have their uses, and we will use each in turn. With the
first method, we will see how a typical example of scattering changes with the
parameter $\zeta$. We will also investigate what happens to the orthogonal
scattering in the noncommutative case. We will then use the second method to
see if there is interesting systematic behaviour associated with the other
ADHM parameters.
First, we take a typical example of scattering in the commutative case, and
see what happens when we add in noncommutativity. In this case, the parameters
$\\{\rho,\theta,b,x\\}$ take the values $\\{1,0,0.5,50\\}$ and their initial
derivatives are $\\{0,0,0,-0.03\\}$. The change in scattering angle as we
change the value of $\zeta$ from 0 to 5 is shown in figure 4. The most obvious
feature is the presence of a peak. This is a general feature of scattering as
we change $\zeta$ – see figure 3.

Figure 3: Plot of scattering angle vs. impact parameter for different values
of $\zeta$. From left to right we have $\zeta=\\{0.65,1.5,3\\}$

The peak is hard to resolve numerically – there seems to be a discontinuity.
To analyse this we can zoom in on that section of the graph (figure 6). Part
of the issue seems to be a numerical error based on the code jumping between
$\pm\theta$ at $\pi/4$ and $\pi/2$. This cannot completely explain the
phenomenon however. Looking at the graphs in 6, the instantons appear to merge
then divide again in a way that, ‘swaps over’ the ingoing and outgoing tracks.
This may indicate the presence of a bound state at the cusp, which the
numerics cannot fully resolve. Another possibility is that there is suddenly a
second scattering at right angles. It is not clear why there should be such a
sharp transition, but perhaps it happens when the instantons become large
enough to overlap during the scattering process. This would be a good topic
for future work with more powerful numerical methods.
In general, the effect of the noncommutativity is to increase the repulsion
between the instantons. Initially, the instanton scattering angle seems to
rotate anticlockwise, going from glancing off each other, to moving parallel,
to crossing over. This occurs rapidly as $\zeta$ changes from 0.85 until about
0.88 (figure 5).
At the first apparent discontinuity, the instantons change from moving across
each others paths, to repelling and turning back on themselves, so that their
paths form a loop near the interaction point. This change happens somewhere
between $\zeta=0.8818695$ and $\zeta=0.88187$. The beginnings of the looping
behaviour can be seen in the first graph in figure 6, however as can be seen
in the second graph there is no way of assigning the trajectories to different
instantons. This is a general feature of instanton dynamics- you cannot
distinctly seperate instantons when they get too close [23].
Some of the issue with correctly defining the angle can be seen from the
bottom two graphs in figure 6, where the loops are joined in two different
ways. Comparing the last graph in 6, the last graph in figure 4 and the first
graph in figure 7 indicates that the particles seem to loop back on
themselves. This is further evidence for the fact that discontinuities in the
scattering angle graph 6 are based on a breakdown of the notion of the
instanton positions as they begin to intersect.
After the peak (figure 7), the scattering angle appears to rotate clockwise –
the loop at the interaction point is, ‘unwound’. This leads to them then
repelling entirely before the angle widens to about $\pi/4$, with the
instantons repelling rather than glancing off each other as they did at the
start.

Figure 4: Change of scattering angle (left) with noncommutative parameter
$\zeta$ for $b=0.5$, with the other parameters as discussed in the main body
of the text. On the right is shown the area around the discontinuity, which
seems to be a region where the numerics have confused $\pm\theta$.

Figure 5: Scattering for two instantons with $b=0.5$, and, moving in each row
from left to right, $\zeta=\\{0.1,0.86,0.87,0.88\\}$; This corresponds to the
region to the left of and around the peak in figure 4. Where the sizes are not
shown this is in order to make the trajectories clearer. Note that the
instantons go from glancing off one another, to moving parallel, to crossing
over, and then deflecting

Figure 6: Graphs of the interaction around the peak, zoomed in at the center,
with $\zeta=\\{0.8187,0.89,0.9\\}$. The vertical lines on the graph are the
results of confusion about which parts of the trajectory belong to which
instanton, and can be ignored. More detailed interpretation of the graphs is
given in the main text.

Figure 7: Scattering for two instatons with $b=0.5$, and
$\zeta={0.9,1,1.15,2}$. This corresponds to the right side of the peak. Note
that the instanton angle begins to turn back on itself, until the scattering
becomes a direct repulsion $\zeta=1.15$, then opens to about $\pi/4$

We observed this behaviour for a number of different set ups, including where
the instantons began too far apart to originally interact. An overall feature
of all this graphs is that whereas in the commutative case the instantons
shrink through zero size then expand again, in the noncommutative case, as we
would expect, they shrink to a finite size before expanding; since due to the
noncommutativity the zero size point cannot be reached.

Plotting the scattering angle for differing values of the impact parameter $b$
shows the same distinctive spike for a particular value of $b$ (figure 3). As
a numerical check, if we interchange the roles of $b$ and $\zeta$ by plotting
the scattering angle for varying $\zeta$ whilst keeping $b$ fixed, the spike
appears at the same $(b,\zeta)$ coordinates. If we plot the graphs of
scattering angle vs. impact parameter for different values of $\zeta$ we see
that the overall behaviour stays the same, however the position of the peak
moves to the right as $\zeta$ increases as shown in figure 3. This behaviour
only appears when $\zeta\neq 0$, and therefore seems to be unique to the
noncommutative case. This would be another topic for further investigation.
The next case we will look at is the case where the scattering is orthogonal
in the commutative case (so $b=0$). This remains consistently orthogonal in
the noncommutative case – see figure 8. Overall the scattering keeps its
perpendicular character. We can explain this analytically in a similar way as
in the commutative case in [13]. As discussed above, the location of the
instantons is described by a combination of $\tau$ and $\sigma$. Because
$\sigma$ behaves like $1/\lvert\tau\rvert$, the change in which parameter
dominates happens when $\lvert\tau\rvert=\lvert\sigma\rvert$. Recall the
definitions of $\tau$ and $\sigma$ in equation (5.1). For the case of
orthogonal scattering, $\chi=0$. Therefore $\tau=\omega$, and so lies entirely
on the $x$-axis. On the contrary, $\sigma$ is proportional to $i$ and so lies
on the $y$-axis. Therefore, as the dominant parameter in the position changes
between $\tau$ and $\sigma$, the instanton motion changes from the $x$-axis to
the $y$-axis and so they scatter orthogonally.

Figure 8: Examples of scattering behaviour for the orthogonal scattering with
$b=0$ for $\zeta=\\{0,2\\}$.

We then used the scanning method to look for interesting behaviour amongst the
other parameters. Fixing $\zeta$ fixes the length scale of the system,
therefore we investigated the behaviour of the system keeping $\zeta$ at a
constant value of 1, and looking at how the scattering angle of the instantons
depends on the other parameters. The variables for which there was notable
behaviour were $\dot{\rho}$ and $\dot{\theta}$. These showed similarly
interesting behaviour in both cases, and so I will discuss them together. As
can be seen in figure 9, in both cases, a small pertubation in $\dot{\rho}$
and $\dot{\theta}$ causes almost orthogonal scattering, no matter what the
initial scattering angle. The difference is that, as shown in figure
LABEL:fig:thprsca1, there seems to be jump of the scattering angle from
positive to negative, for positive and negative $\dot{\theta}$, which is not
present for $\dot{\rho}$. However it is unclear even from individual
scattering graphs if this is a misidentification of incoming and outgoing
particles. Even if there is no scattering in the, ‘base’ case where both are
zero, as in figure 8, we still get the same orthogonal scattering behaviour,
however there is not the same jump in scattering angle at the origin. The
reason for this behaviour seems to be that changing either of these parameters
from zero makes the instanton size very large, causing a high degree of
interaction (and hence orthogonal scattering) no matter what the initial
separation is. There is a subtlety in that $\dot{\rho}$ is not the variation
in the actual size, but only in the parameter $\rho$. The variation in the
actual size is give by

$\dot{\tilde{\rho}}=\frac{\dot{\rho}\rho-\frac{4\zeta^{2}\dot{\rho}}{\rho^{3}}}{\sqrt{\rho^{2}+\frac{4\zeta^{2}}{\rho^{2}}}}$
(5.8)

Therefore $\dot{\rho}$ is a good approximation when $2\zeta/\rho$ is small. We
have been careful to only consider such cases

Figure 9: (left) Graph showing variation of scattering angle vs. gauge angle
$\theta$ for $b=0.5$ and $\zeta=1$. Note the jump in scattering angle between
positive and negative $\dot{\theta}$. (right) Graph showing variation of
scattering angle vs. $\dot{\rho}$ for $b=0.5$ and $\zeta=0.1$. Note that here
there is no jump in the scattering angle, unlike in the case of $\dot{\theta}$

We now move on to dyonic instantons. We follow the same method as before.
First we look at how changing $\zeta$ changes a specific case of scattering.
Then we use the scanning technique to look for interesting behaviour linked to
varying the ADHM parameters. In the dyonic case, the length scale is still set
by $\zeta$, however the scale of the time dimension is no longer arbitrary,
but is set by $\lvert q\rvert$. Therefore we must consider the consequences of
varying both.
Looking at specific scattering examples, we again see that the noncommutative
parameter initially introduces a repulsive effect (eg. figure 10). Here, the
basic values of the parameters are as in the pure case, except that we give
$\theta$ a small initial velocity of $0.1$ in order to avoid numerical issues.
Any changes to these parameters will be discussed in the captions to the
graphs. As discussed in [13], dyonic instantons oscillate along their motion,
and this effect is much more observable with the noncommutativity turned on.
Scanning along the scattering angle in both the commutative and noncommutative
case gave very noisy graphs from which it is hard to deduce any global
behaviour. However we found some interesting examples of orbiting behaviour,
especially for small $q$ relative to the other parameters – a particularly
impressive example is figure 11.

Figure 10: Plot of dyonic instanton scattering for $b=-1,\ \lvert
q\rvert=0.1,\ \zeta=0$ (above), $\zeta=1$ (below). Note the visible
oscillations on the right hand graph. Figure 11: Graph of scattering with
orbiting behaviour, with $\zeta=0.5,\ b=0.5,\ q=0.00438$

We can then systematically look for interesting behaviour amongst the
remaining parameters. A recurring feature was the presence of stable
combinations of $\zeta$ and $q$ for which there was a clear pattern of
behaviour, with no observable pattern outside of these regions.
We started by looking at varying $\theta$, but this did not yield any
interesting systematic behaviour – only random noise. We then looked at
$\rho$. Here there seemed to be a window where the behaviour matched the
commutative case, e.g. with $\zeta=0.1$ and $q=0.1$, as shown in figure 12.
Exploring around that point showed that the behaviour persisted with roughly
$\zeta<1$ and with $q>0.08$
We then moved on to looking at $b$. Here there was similar behaviour. At low
$\zeta$ there did not seem to be any overall pattern, however increasing
$\zeta$ led to graphs having a linear pattern, as in figure 12. This seems to
be two examples of the same phenomenon, with very different scaling caused by
the differences in the relative values of $\zeta$. The magnitude of $q$ did
not seem to have a major effect on whether any linear pattern in the behaviour
was observed past a certain point, but continuing to make $\zeta$ larger
caused the non linear behaviour to return.

Figure 12: Left: Scattering angle vs. parameter $\rho$ for $\zeta=q=0.1$. As
discussed in the text, we expect $\rho$ to be a good approximation to the true
initial size after the peak Right: Scattering angle vs. Impact parameter for
$\zeta=q=0.5$

. The case b=0 is discussed in figure 8, and not included in this graph.

Finally, we did not find any discernible patterns for $\dot{\theta}$ or
$\dot{\rho}$ either. This difference as compared to the pure case is probably
because the potential prevents the instanton size from growing large in the
dyonic case, and therefore the transition to orthogonal scattering cannot
occur.

###  5.3 The Six Parameter Space

We now move on to look at the full six parameter space, where the instantons
are free to have different sizes and to vary in their gauge angle. The
additional parameters add greatly to the complexity of the numerics, which we
describe in Appendix G. Due to this it is no longer possible to use the
numerical algorithm from [13] as we had in the four parameter case and so we
had to use a new algorithm, outlined in Appendix G to produce the figures in
this section.
There are two parameters to examine here. These are the relative gauge angle
$\phi$ and the relative sizes of the instantons. Unless otherwise stated, the
initial conditions for $\\{\rho_{1},\rho_{2},\theta,x\\}$ take the values
$\\{1,1,0,50\\}$, and the initial derivatives of all parameters are zero,
except $\dot{x}=-0.03$. In the noncommutative case it is tricky to
systematically explore the latter as the instanton sizes are nonlinear
functions of $\zeta$ and the $\rho_{i}$. Therefore we chose to keep $\zeta$
fixed to set the overall length scale, and to vary the impact parameter rather
than the instanton size, looking at cases where the separation was much
smaller than, larger than and of the same order as the sizes of the
instantons. Initially we kept the instantons the same size. We then checked
the behaviour in three cases $\rho_{1}<b<\rho_{2}$, $b<\rho_{1}<\rho_{2}$ and
$\rho_{1}<\rho_{2}<b$.
In the commutative case, varying the gauge angle produces a clear sinusoidal
variation (figure 13). This pattern held for different values of the impact
parameter, however when the impact parameter was small compared to the
instanton size, the varation takes on more of a, ‘square’ shape (figure 13).
As can be seen both from these two figures and from the scattering angles in
figure 16, at $\phi=n\pi$, where the instantons are parallel in the gauge
group, the interaction between the instantons disappears and they just move
past one another. Conversely, the instantons interact most strongly at
$\phi=n\pi+\pi/2$, where they are orthogonal in the gauge group. Changing the
relative sizes of the instantons did not seem to affect this sinusoidal
behaviour, but it did change the strength of the interaction, with the
scattering angle decreasing when the Instantons were different sizes, with
smaller sizes making the scattering angle smaller (figure 14).

Figure 13: Left: Graph of varying scattering angle $\phi$ with $\zeta=0$,
$\rho_{1}=\rho_{2}=1$ and $b=0.5$. Right: Graph of varying scattering angle
$\phi$ with $\zeta=0$, $\rho_{1}=\rho_{2}=1$ and $b=0.1$.

Figure 14: Graph showing variation of scattering with gauge angle $\phi$,
where $\zeta=0$ and $b=0.5$. In both graphs $\rho_{1}=1$. In the left graph,
$\rho_{2}=5$, and in the right graph $\rho_{2}=0.1$. Note that the scattering
angle is much smaller in this case.

The behaviour in the noncommutative case is not so simple. The outline of the
sinusoidal pattern is still present, but it is significantly disrupted, as in
figure 15. Increasing the impact parameter somewhat restores the behaviour
(figure 15). There is therefore much less variation in the scattering angle
for the noncommutative case. The Instantons also no longer stop interacting
when they are parallel in the gauge group, instead oscillating between minimum
and maximum scattering angles, as in figure 15. Making one of the instantons
smaller than the other and the impact parameter did not seem to have too much
of an effect, however, making one larger than the impact parameter further
disrupted the sinusoidal pattern, as in figure 17.

Figure 15: Left: Graph showing variation of scattering angle $\phi$ with
$\zeta=1$, $\rho_{1}=\rho_{2}=1$ and $b=0.5$. The true instanton size is
therefore $\sqrt{2}$, and so is roughly comparable to the separation. The
splitting of the left peak appears to be a numerical error. Right: Graph
showing variation of scattering angle $\phi$ with $\zeta=1$,
$\rho_{1}=\rho_{2}=1$ and $b=4$. The true instanton size is therefore
$\sqrt{2}$, and so is much smaller than the separation. Note that the
sinusoidal form is much more preserved, but now oscillates around zero rather
than away from it

Figure 16: Graph showing scattering examples from figure 15, with $\phi=\pi$
left and $\phi=3\pi/2$ right. Note that the scales are different on the two
graphs, and that the behaviour is extremely different.  Figure 17: Graph
showing varying gauge angle $\phi$ with $\zeta=1$, $\rho_{1}=1;\rho_{2}=5$ and
$b=0.5$. The true instanton sizes are $\sqrt{2}$ and just over 25
respectively.

###  5.4 Conclusions

We end this section by reviewing the main results. We looked at both the full
six parameter space, and also a four parameter subspace where the instantons
were orthogonally embedded in the gauge group. This was necessary to analyse
the dyonic case. Overall, increasing the noncommutative parameter $\zeta$
increases the repulsion between the instantons. The form this takes is not
straightforward, and in the pure instanton case involves a peak with strange
behaviour which requires a future, more detailed analysis with more
sophisticated simulations. However in general even if the instantons begin by
not interacting, they move from glancing off each other, to reflecting
entirely as the parameter $\zeta$ increases.
We also found that orthogonal scattering was present in the noncommutative
case as well as the commutative case. Systematically looking at the other
parameters, we saw that, as expected, increasing $\rho$ strengthens the
repulsive effect of the scattering, and increasing the separation $b$
decreases it. Further interesting behaviour was observed seeing how the
scattering changed when the quantities $\dot{\rho}$ and $\dot{\theta}$ were
varied. For any nonzero value of these initial velocities, the scattering
rapidly became almost orthogonal. This seems to be because making these
parameters nonzero causes a rapid increase in the instanton size, and hence a
very strong interaction.
This behaviour is not found in the dyonic case; probably because the presence
of the potential suppresses the instanton size. In the dyonic case there was
the additional feature of orbiting behaviour, some of a high winding number
and great complexity.
Finally, we were able to use the six parameter pure instanton case to analyse
changing the gauge embedding. In general we found that the scattering
oscillated with the gauge angle, but that this was suppressed as $\zeta$ was
increased.

##  6 Three Instantons

We now move on to the case of three instantons in SU(2) Yang Mills. Here we
only consider a commutative background, not a noncommutative one. We also use
the usual version of the commutative ADHM construction with the quaternions
rather than the biquaternion construction outlined above. As before, we begin
by solving the ADHM constraints. We then calculate the scalar field for the
dyonic case and use this to calculate the moduli space potential. Finally, we
calculate the moduli space metric. The results in this section are original.
There is, however, some related work in [28] and its related papers. There,
some three monopole solutions are found, using two methods involving writing
the solution as a reduction of the ADHM equations. The first method is to use
the JNR ansatz. This corresponds to taking $\Omega$ to be diagonal in our
notation. The second is to calculate Axial monopoles using ADHM data which has
axial symmetry imposed on it via the Manton- Sutcliffe method. In our
notation, there is a non-diagonal but specific form for $\Omega$, and the
instanton size $v$ is chosen to be zero. These specific symmetries do not seem
to match the ones we have chosen, and hence it it not immediately clear how
the results in that work relate to those presented here, but it would be
interesting and worthwhile to pursue this in future.
Since we are in the commutative case, we need to solve the equation
$\Delta^{\dagger}\Delta=0$. For three instantons in SU(2) Yang Mills, the ADHM
data $\Delta$ is

$\begin{bmatrix}\Lambda\\\ \Omega\end{bmatrix}=\ \begin{bmatrix}u&v&w\\\
\tau_{1}&\sigma_{1}&\sigma_{2}\\\ \sigma_{1}&\tau_{2}&\sigma_{3}\\\
\sigma_{2}&\sigma_{3}&\tau_{3}\end{bmatrix}$ (6.1)

where the entries of $\Delta$ all lie in $\mathbb{H}$. With $\Delta$ as given
above, we have three equations, one for each basis vector of $o(3)$. These are

$\displaystyle\text{Im}_{\mathbb{H}}\big{(}\bar{u}v+(\bar{\tau}_{1}-\bar{\tau}_{2})\sigma_{1}+\bar{\sigma}_{2}\sigma_{3}\big{)}=0$
$\displaystyle\text{Im}_{\mathbb{H}}\big{(}\bar{u}w+(\bar{\tau}_{1}-\bar{\tau}_{3})\sigma_{2}+\bar{\sigma}_{1}\sigma_{3}\big{)}=0$
$\displaystyle\text{Im}_{\mathbb{H}}\big{(}\bar{v}w+(\bar{\tau}_{2}-\bar{\tau}_{3})\sigma_{3}+\bar{\sigma}_{1}\sigma_{2}\big{)}=0$
(6.2)

Note that these are now nonlinear in the ADHM data. Again, we were unable to
find a solution on the full quaternion moduli space; however, if we restrict
to the complex subspace as in the noncommutative 2 instanton case, and use the
three residual symmetries to set the real parts of the $\sigma_{i}$ to zero,
the terms in $\text{Im}(\bar{\sigma}_{i}\sigma_{j})$ vanish, and we can solve
as

$\displaystyle\sigma_{1}=\frac{\tau_{1}-\tau_{2}}{\lvert\tau_{1}-\tau_{2}\lvert^{2}}\bigg{(}\alpha-\text{Im}_{\mathbb{C}}(\bar{u}v)\bigg{)}$
$\displaystyle\sigma_{2}=\frac{\tau_{1}-\tau_{3}}{\lvert\tau_{1}-\tau_{3}\lvert^{2}}\bigg{(}\beta-\text{Im}_{\mathbb{C}}(\bar{u}w)\bigg{)}$
$\displaystyle\sigma_{1}=\frac{\tau_{2}-\tau_{3}}{\lvert\tau_{2}-\tau_{3}\lvert^{2}}\bigg{(}\gamma-\text{Im}_{\mathbb{C}}(\bar{v}w)\bigg{)}$
(6.3)

For constants $\alpha,\beta,\gamma\in\mathbb{R}$. These are then constrained
by the condition $\text{Re}_{\mathbb{C}}(\sigma_{i})=0$ to be

$\displaystyle\alpha=-\frac{\text{Im}_{\mathbb{C}}(\tau_{1}-\tau_{2})\text{Im}_{\mathbb{C}}(\bar{u}v)}{\text{Re}(\tau_{1}-\tau_{2})}$
$\displaystyle\beta=-\frac{\text{Im}_{\mathbb{C}}(\tau_{1}-\tau_{3})\text{Im}_{\mathbb{C}}(\bar{u}w)}{\text{Re}(\tau_{1}-\tau_{3})}$
$\displaystyle\gamma=-\frac{\text{Im}_{\mathbb{C}}(\tau_{2}-\tau_{3})\text{Im}_{\mathbb{C}}(\bar{v}w)}{\text{Re}(\tau_{2}-\tau_{3})}$
(6.4)

The minus sign comes from the fact that each $\text{Im}_{\mathbb{C}}$ comes
with a $\sigma_{3}$, which multiply together to give $-1$. The above equations
give a solution for the complex subspace. As with the two instanton case, the
solutions for the scalar field, metric and potential are long and are given in
Appendix F

###  6.1 3 Instanton Dynamics

The next step is to analyse the dynamics numerically, as was done in the case
of two instantons. Unfortunately we were unable to generate enough simulations
to carry out a full analysis, however were able to observe some particular
behaviours by plotting the scalar field profiles. When the instantons are far
separated, this gives three peaks at the positions of each instanton, with the
position defined as $\tau_{i}$ (figure 18). This confirms the interpretation
of that parameter. If we move one instanton far away from the others (off to
the right of the plot, in fact) then we see two peaks which look very similar
to the graphs for two commutative instantons found in [13]. The splitting in
the right peak increases the closer the third instanton gets. In the graph in
question, the two instantons shown are at $(\pm 1,0)$ and the third is at
$(0,40)$. Finally, we were able to approximate some aspects of the scattering
by plotting the Topological charge Density (figure 19). Here, if one instanton
is kept, ‘stationary’ at the origin, and the other instantons are plotted at
successively closer values of $\tau_{i}$, there appears to be the kind of
right angled scattering that is a familiar part of Soliton Dynamics. The fact
that the instantons are moving away at very small values of $\tau_{i}$ is a
function of the fact that the position depends both upon $\tau_{i}$ and
$\sigma_{i}$, as in the two instanton case.

Figure 18: (left) Plot of the scalar field profile for three separated
instantons. (right)Plot of the scalar field profile for two instatons, with
the third far separated off to the right

Figure 19: Plot of the topological charge density with one instanton at the
origin and the other two at decreasing values of $\tau_{i}$. Note the apparent
right angled scattering

##  7 Conclusion

We have presented a new notation and method for working with noncommutative
ADHM instantons, by writing them explicitly in terms of biquaternion
components. After deriving the form of the ADHM equation, as well as the
Moduli Space and Potential, for general SU(2) instanton number in this
notation, we attempted to solve them for the two-instanton case.
First, we rederived the commutative solution which was found in [13], but
using biquaternions rather than quaternions. We wereunable to find a solution
on the full subspace, however we were able to find one for the subspace of the
moduli space spanned by the $\mathbb{C}\times\mathbb{C}$ subalgebra of
$\mathbb{C}\times\mathbb{H}$. We used this solution to calculate the metric
and potential for that subspace, and checked its behaviours in various limits.
Once we had these two solutions, we investigated the dynamics on the
noncommutative moduli space numerically. In general, increasing the value of
the noncommutative parameter $\zeta$ increased the repulsion between the
instantons, however the addition of the potential suppressed the repulsive
force, particularly for large $\zeta$.
Finally, we looked at the case of three $U(2)$ instantons. Again, we were able
to find a solution on the submanifold of the moduli space spanned by the
$\mathbb{C}$ subgroup of $\mathbb{H}$. This solution once more allowed us to
calculate the metric and potential on that submanifold, and to numerically
graph the scalar field profiles. These solutions were very challenging
numerically – however the limited results we got indicated the presence of
right angled scattering, and the appropriate behaviour of the solution in
various limits. In terms of further work, the most obvious thing to do is to
try and improve the efficiency of the numerical evaluations so that we can
explore the dyonic six parameter case for the noncommutative two instantons,
and to access more of the three particle scattering in the three instanton
case. Analytically, we could try and extend our ADHM solutions from the
subspaces of the moduli spaces to the full moduli spaces.

This would allow us to use the moduli space to calculate the number of BPS
states in this topological sector, by calculating a specific Witten index [29]
on the moduli space. As argued in [29] the index is invariant under changes of
the parameters of a theory, provided that these changes can be expressed as a
conjugation of the supersymmetric variables in the theory, as such a
conjugation leaves the supersymmetry properties of a theory unchanged. The
moduli space potential is one of these parameters.
In [30] it is argued that the Dirac operator on the moduli space goes as
$\text{exp}(-q)$, where $q$ is the absolute value of the moduli space
potential, outside of small regions around the zeros of that potential. It is
shown that rescaling the potential does not change the Witten index, so the
number of BPS states remains the same in the limit that $q\rightarrow\infty$.
In this limit the moduli space dynamics are exponentially suppressed outside
of the small regions around the zeros. On the commutative moduli space this
procedure cannot be carried out due to the presence of singularities, but
because these singularities are removed by the multiplicity, in this case
these regions should be describable by supersymmetric harmonic oscillators, as
was found to be the case in [18] for single U(2) instantons. Calculating the
Witten index, and hence the number of BPS states, would then be fairly
straightforward. Therefore a key goal for future research would be to find the
full moduli space and take the potential to infinity. This would suppress the
dynamics everywhere except from regions of a small radius around the zeros of
the potential, where it would be described by a supersymmetric quantum
mechanics, for which we could calculate the Witten Index, and hence the
partition function. Once we have calculated this function, we can compare it
to the result directly calculated from the field theory in [31]. This would
provide a check for the $k=2$ case of the hypothesis mentioned in the
introduction, where the (2,0) theory describing the interaction of multiple M5
Branes is the strong coupling completion of 5d Super Yang Mills. If this is
true, then the instantons of topological charge $k$ should match with states
of Kaluza Klein momentum $k$ around the circle of compactification, as
explained in [32]. The $k=2$ section of the partition function for Kaluza
Klein states was calculated in [31], and so comparing this to the partition
function on the instanton moduli space would be an important next step in
verifying this conjecture.

##  Appendix A Solving the ADHM equations

As stated in equation 4.2, the noncommutative ADHM equations are:

$\displaystyle
2\text{Im}_{\mathbb{H}}(\bar{\sigma}_{R}\sigma_{I})-\text{Im}_{\mathbb{H}}(\bar{w}_{R}w_{I})\text{Im}_{\mathbb{H}}(\bar{v}_{R}v_{I})=0$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{w}_{R}w_{I})+\text{Im}_{\mathbb{H}}(\bar{v}_{R}v_{I})=-4\zeta\sigma_{3}$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{\tau}\sigma_{I})=\frac{\text{Im}_{\mathbb{H}}(\bar{w}_{R}v_{I}+\bar{v}_{R}w_{I})}{2}\equiv\frac{\Upsilon}{2}$
$\displaystyle\text{Im}_{\mathbb{H}}(\bar{\tau}\sigma_{R})=\frac{\text{Im}_{\mathbb{H}}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})}{2}\equiv\frac{\Lambda}{2}$
(A.1)

where $\sigma_{3}$ is the quaternion basis element

$\begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}$ (A.2)

We can solve the third and fourth equations as

$\sigma_{R}=\frac{\tau}{\lvert\tau\rvert^{2}}\Big{(}\alpha+\frac{\Lambda}{2}\Big{)}$
(A.3)

and

$\sigma_{I}=\frac{\tau}{\lvert\tau\rvert^{2}}\Big{(}\gamma+\frac{\Upsilon}{2}\Big{)}$
(A.4)

We can use the second equation to deduce that

$\bar{w}_{R}w_{I}=\beta-\bar{v}_{R}v_{I}-4i\zeta\sigma_{3}$ (A.5)

and so

$w_{I}=\frac{w_{R}\beta-w_{R}\bar{v}_{R}v_{I}-4i\zeta w_{R}\sigma_{3}}{\lvert
w_{R}\rvert^{2}}$ (A.6)

We want 12 independent real parameters, or three independent quaternion ones.
We therefore aimed to solve for the other parameters in terms of $v_{R},w_{R}$
and $\tau$. We were unable to find a solution to these equations for the full
biquaterion moduli space. However were able to find a solution on a complex
valued geodesic submanifold. This subspace comes from restricting the
quaternions to the subspace consisting of elements $z\in\mathbb{C}$ written as
$x+y\sigma_{3}$, for $x,y\in\mathbb{R}$ and $\sigma_{3}$ is given by

$\sigma_{3}=\begin{bmatrix}i&0\\\ 0&-i\end{bmatrix}$ (A.7)

Note that $\sigma_{3}^{2}=-\mathbb{1}_{\mathbb{H}}$, and therefore
$\sigma_{3}$ can play the role of the imaginary unit. We start with the second
ADHM equation, now for complex variables

$\text{Im}_{\mathbb{C}}(\bar{v}_{R}v_{I})+\text{Im}_{\mathbb{C}}(\bar{w}_{R}w_{I})=-4\zeta\sigma_{3}$
(A.8)

Recall that we are using the notation $\text{Im}_{\mathbb{H}}$ to mean the
imaginary quaternion part of an element of $\mathbb{H}$; e.g. for
$q=q_{0}+\textbf{q}\in\mathbb{H}$,

$\text{Im}_{\mathbb{H}}(q)=\textbf{q}$ (A.9)

On the other hand, $\text{Im}_{\mathbb{C}}$ takes the imaginary component of
an element of $\mathbb{C}$. If $z\in\mathbb{C};\ z=a+ib$

$\text{Im}_{\mathbb{C}}(z)=b$ (A.10)

With these definitions in mind, we can solve this for $w_{I}$ and $v_{I}$ in
terms of the other variables by finding a particular solution, then by adding
the null space, found by solving

$\text{Im}_{\mathbb{C}}(\bar{v}_{R}v_{I})+\text{Im}_{\mathbb{C}}(\bar{w}_{R}w_{I})=0$
(A.11)

A particular solution is given by

$v_{Ip}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert v_{R}\rvert^{2}};\
w_{Ip}=\frac{-2\zeta w_{R}\sigma_{3}}{\lvert w_{R}\rvert^{2}}$ (A.12)

We already know the solution to the null equation; it is

$\tilde{v}_{I}=\frac{v_{R}}{\lvert
v_{R}\rvert^{2}}\Big{(}\beta-\bar{w}_{R}\tilde{w}_{I}\Big{)}$ (A.13)

For arbitrary real $\beta$ and arbitrary quaternion $\tilde{w}_{I}$. Therefore
we have the general solution

$\displaystyle v_{I}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert
v_{R}\rvert^{2}}+\frac{v_{R}}{\lvert
v_{R}\rvert^{2}}\Big{(}\beta-\bar{w}_{R}\tilde{w}_{I}\Big{)}$ (A.14)
$\displaystyle w_{I}=\frac{-2\zeta w_{R}\sigma_{3}}{\lvert
w_{R}\rvert^{2}}+\tilde{w}_{I}$

To complete this general solution we need to solve for $\tilde{w}_{I}$. This
is done by solving the first ADHM equation

$\text{Im}_{\mathbb{C}}(\bar{\sigma}_{R}\sigma_{I})=\text{Im}_{\mathbb{C}}(\bar{v}_{R}v_{I})-\text{Im}_{\mathbb{C}}(\bar{w}_{R}w_{I})$
(A.15)

We can use two of the symmetries in equation (4.11) to set
$\text{Re}(\bar{\tau}\sigma)=0$, by analogy to [13]. This corresponds to
removing any component proportional to $\tau$ from $\sigma$. If we do this,
then the equation becomes

$-\frac{\text{Im}_{\mathbb{C}}(\Lambda\Upsilon)}{\lvert\tau\rvert^{2}}=\text{Im}_{\mathbb{C}}(\bar{v}_{R}v_{I})-\text{Im}_{\mathbb{C}}(\bar{w}_{R}w_{I})$
(A.16)

If we now restrict to the complex plane spanned by $\mathbb{1}$ and
$\sigma_{3}$ the LHS becomes zero, since $\Lambda$ and $\Upsilon$ are both
proportional to $\sigma_{3}$, and hence their product is real and so
$\text{Im}_{\mathbb{C}}(\Lambda\Upsilon)=0$. Putting the solutions in (A.14)
into the RHS we get

$\text{Im}_{\mathbb{C}}(\bar{w}_{R}\tilde{w}_{I})=0$ (A.17)

This leads to the solution

$\displaystyle v_{I}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert
v_{R}\rvert^{2}}+Bv_{R}$ (A.18) $\displaystyle w_{I}=\frac{-2\zeta
w_{R}\sigma_{3}}{\lvert w_{R}\rvert^{2}}+Aw_{R}$

for $A,B\in\mathbb{R}$. We can then use the remaining two symmetries to set
$A$ and $B$ above to zero – see the discussion below equation (4.11). Then we
get the full solution for the complex subspace

$\displaystyle v_{I}=\frac{-2\zeta v_{R}\sigma_{3}}{\lvert v_{R}\rvert^{2}}$
$\displaystyle w_{I}=\frac{-2\zeta w_{R}\sigma_{3}}{\lvert w_{R}\rvert^{2}}$
$\displaystyle\sigma_{R}=\frac{\tau\text{Im}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})}{2\lvert\tau\rvert^{2}}=\frac{(\lvert
v_{R}\rvert^{2}\lvert w_{R}\rvert^{2}+4\zeta^{2})}{2\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert
w_{R}\rvert^{2}}\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{R})\sigma_{3}$
$\displaystyle\sigma_{I}=\frac{\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{I}+\bar{v}_{R}w_{I})}{2\lvert\tau\rvert^{2}}=-\frac{\zeta(\lvert
w_{R}\rvert^{2}+\lvert v_{R}\rvert^{2})}{\lvert\tau\rvert^{2}\lvert
v_{R}\rvert^{2}\lvert
w_{R}\rvert^{2}}\tau\text{Im}_{\mathbb{C}}(\bar{w}_{R}v_{R}\sigma_{3})\sigma_{3}$
(A.19)

##  Appendix B Solving the Scalar Field

The method outlined here is mainly based on Appendix 1 in [13], generalised to
the case of arbitrary non commutative instantons. That method is in turn based
on [24]. It begins with the ansatz

$\phi=iU^{\dagger}\mathcal{A}U;\ \mathcal{A}=\begin{bmatrix}q&0\\\
0&P\end{bmatrix}$ (B.1)

where $\phi$ is the scalar field we are trying to calculate and $U$ is an
element of the null space of the ADHM Matrix $\Delta$. Further, $q\in u(N)$,
where $N$ is the degree of the instaton gauge group, and $P\in u(k)$, where
$k$ is the instanton number. In fact, $iq$ is the vev of the scalar field. For
the real ADHM construction, we can use $o(k)$ rather than $u(k)$. In the
biquaternion case in theory there is an additional $u(1)$, promoting the
symmetry group to $u(2)$. In what follows we use our freedom to choose the vev
so that it lies in the $su(2)$ part of this overall $u(2)$.
Note that the equation for $\phi$ has the form of a rotation of $\mathcal{A}$
by $U$. We can think of this as follows. The matrix $\mathcal{A}$ belongs in
$u(N)\times u(k)$. We can imagine it being defined on a $u(N)\times u(k)$
bundle over $\mathbb{R}^{4}$. However we know the ADHM construction breaks
the, ‘gauge group’ $u(N)\times u(k)$ down to $u(N)$. We can therefore see the
rotation as rotating $\mathcal{A}$ into the $u(N)$ subspace picked out by the
ADHM constraints. This interpretation can be confirmed by the straightforward
observation that $U^{\dagger}(\mathbb{1}-UU^{\dagger})\mathcal{A}U=0$. A long
and algebraic justification for the ansatz is given in [24]. Regardless of the
justification for the ansatz, once we have it, the problem of solving for
$\phi$ becomes the problem of solving for $P$ above. It is shown in [24] that
the equation of motion for $\phi$

$D^{2}\phi=0$ (B.2)

expands as

$D^{2}\phi=-4iU^{\dagger}\\{bfb^{\dagger},\mathcal{A}\\}U+4iU^{\dagger}bf\text{Tr}_{2}(\Delta^{\dagger}\mathcal{A}\Delta)fb^{\dagger}U=0$
(B.3)

Here, $\text{Tr}_{2}$ refers to the quaternion trace on each element of a
matrix, not to the trace of the matrix itself, which is written Tr. Hence,
applied to a (complex/ real) quaternion valued matrix, $\text{Tr}_{2}$ will
give a complex/ real valued matrix, whereas Tr will give a (complex/real)
quaternion.
With $\mathcal{A}$ written as above, the first term is
$-4iU^{\dagger}\\{f,p\\}U$. For the second term, we recall that $\Delta$ can
be written as

$\Delta=\begin{bmatrix}\Lambda\\\ \Omega-\mathbb{1}x\end{bmatrix}$ (B.4)

Writing $\Omega^{\prime}=\Omega-\mathbb{1}x$, recalling that $\Omega$ and
$\Omega^{\prime}$ are hermitian, and using the ADHM constraint
$\Delta^{\dagger}\Delta=\Lambda^{\dagger}\Lambda+\Omega^{\prime\dagger}\Omega^{\prime}=f^{-1}$
we can see

$\displaystyle\text{Tr}_{2}(\Delta^{\dagger}\mathcal{A}\Delta)$
$\displaystyle=\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\text{Tr}_{2}(\Omega^{\prime\dagger}\mathcal{A}\Omega^{\prime})$
$\displaystyle=\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\frac{1}{2}\text{Tr}_{2}\big{(}[\Omega^{\prime\dagger},P]\Omega^{\prime}-\omega^{\prime\dagger}[\Omega^{\prime},P]+\\{P,\Omega^{\prime\dagger}\Omega^{\prime}\\}\big{)}$
$\displaystyle=\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\frac{1}{2}\big{(}[\Omega^{\prime\dagger},P]\Omega^{\prime}-\Omega^{\prime\dagger}[\Omega^{\prime},P]+\\{P,f^{-1}\\}-\\{P,\Lambda^{\dagger}\Lambda\\}\big{)}$
$\displaystyle=\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\frac{1}{2}\big{(}2\Omega^{\prime\dagger}P\Omega^{\prime}-\\{\Omega^{\prime\dagger}\Omega^{\prime},P\\}+\\{P,f^{-1}\\}-\\{P,\Lambda^{\dagger}\Lambda\\}\big{)}$
(B.5)

Now, note that $x$ in the above expression is always the coefficient of
$\mathbb{1}$. Therefore the terms involving $x$ in the above expression
cancel, and we can everywhere replace $\Omega^{\prime}$ by $\Omega$ (this is
most easily seen from the third line above).
We can use these to rewrite (B.3) as

$D^{2}\phi=-4i\bigg{(}U^{\dagger}\\{f,P-\frac{1}{2}\text{Tr}_{2}(P)\\}U+U^{\dagger}bf\Big{(}\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\frac{1}{2}\big{(}2\Omega^{\prime\dagger}P\Omega^{\prime}-\\{\Omega^{\prime\dagger}\Omega^{\prime},P\\}-\\{P,\Lambda^{\dagger}\Lambda\\}\big{)}\Big{)}\bigg{)}$
(B.6)

Since $P$ has complex components, not quaternion valued ones,
$\text{Tr}_{2}(P)=P$ and the first term vanishes. Hence the e.o.m.
$D^{2}\phi=0$ is equivalent to

$\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\frac{1}{2}\big{(}2\Omega^{\prime\dagger}P\Omega^{\prime}-\\{\Omega^{\prime\dagger}\Omega^{\prime},P\\}-\\{P,\Lambda^{\dagger}\Lambda\\}\big{)}=0$
(B.7)

This gives one equation for each component of $P$, allowing us to solve for
$P$ and hence, by extension, for $\phi$.

##  Appendix C Constructing the Moduli Space Metric and Potential

This section is based on Appendix 2 in [13], which is itself based on the
method of [33] for calculating the metric determinant. This technique was
adapted in [6] for the moduli space metric of two instantons, which they
calculated to order $\lvert\tau\rvert^{-2}$. In [13] this is extended to the
full metric for 2 commutative U(2) instantons. We present the argument for
arbitrary gauge group and topological charge.
As in section 3.2, the metric on the moduli space is defined as

$g_{rs}=\int d^{4}x\
\text{Tr}^{\star}\big{(}\delta_{r}A_{i}\delta_{s}A_{i}\big{)}$ (C.1)

where

$\delta_{r}A_{i}=\partial_{r}A_{i}-D_{i}\epsilon_{r}$ (C.2)

and

$\text{Tr}^{\star}(q)=\text{Tr}_{2}\big{(}\text{Tr}(q)\big{)}$ (C.3)

There is one mode for each of the $8k$ moduli space coordinates, labelled here
by the indices $r$ and $s$. The index $i$ refers to spacetime coordinates.
Recall these zero modes are orthogonal to gauge transformations by definition

$D_{i}(\delta_{r}A_{i})=0$ (C.4)

We can use this fact to find an explicit expression for the metric. First, we
need an expression for $\partial_{r}A_{i}|_{z=z_{0}}$ in terms of the ADHM
data. To do this, we recall that $A_{i}=U^{\dagger}\partial_{i}U$, and use the
identity $U=PU$ with $U$ the projection operator $\mathbb{1}-\Delta
f\Delta^{\dagger}$ to derive

$\partial_{r}U=-\Delta f\partial_{r}\Delta^{\dagger}U+P\partial_{r}U$ (C.5)

Using this result, the definition $A_{i}=U^{\dagger}\partial_{i}U$, and the
product rule allows us to get the necessary result

$\partial_{r}A_{i}|_{z=z_{0}}=-iU^{\dagger}\partial_{r}\Delta
f\bar{e}_{i}b^{\dagger}U+iU^{\dagger}be_{i}f\partial_{r}\Delta^{\dagger}U+D_{i}(iU^{\dagger}\partial_{r}U)$
(C.6)

around an arbitrary point of the moduli space $z_{0}$. The zero mode is then
this expression with the gauge dependent part removed. The third term above is
explicitly a gauge transformation, however we also need to ensure that there
is no gauge part implicit in the first two terms. To do this, we use the
residual transformations described in equation (LABEL:symms2) to transform the
ADHM data as

$\Delta\rightarrow q\Delta R,\ U\rightarrow QU,\ Q(z_{0})=\mathbb{1},\
R(z_{0})=\mathbb{1}$ (C.7)

It can be seen that this transformation leaves $A_{i}$ invariant, and that

$\partial_{r}A_{i}|_{z=z_{0}}=-iU^{\dagger}C_{r}f\bar{e}_{i}b^{\dagger}U+iU^{\dagger}be_{i}fC^{\dagger}_{r}U+D(iU^{\dagger}\partial_{r}(Q^{\dagger}U))$
(C.8)

with

$C_{r}=\partial_{r}\Delta+\partial_{r}Q\Delta+\Delta\partial_{r}R$ (C.9)

It turns out we can choose $C_{r}$ so that the first two terms of
$\delta_{r}A_{i}$ have no gauge part – i.e. they are a zero mode. To do so we
must prove the following

######  Lemma C.1.

If we choose $C_{r}$ to be independent of $x$ with

$\Delta^{\dagger}C_{r}=(\Delta^{\dagger}C_{r})^{T\star}$ (C.10)

the expression

$\partial_{r}A_{i}=-iU^{\dagger}C_{r}f\bar{e}_{i}b^{\dagger}U+iU^{\dagger}be_{i}fC^{\dagger}_{r}U$
(C.11)

will be a zero mode

To do this, we first note that the condition (C.10) is equivalent to the two
conditions

$a^{\dagger}C_{r}=(a^{\dagger}C_{r})^{T\star};\
b^{\dagger}C_{r}=(b^{\dagger}C_{r})^{T\star}$ (C.12)

and then consider the expression (forming part of $\delta_{r}A_{i}$ above)

$a_{i}:=U^{\dagger}bfe_{i}$ (C.13)

We can then calculate

$D_{i}a_{j}=\partial_{i}a_{j}-iA_{i}a_{j}\
=U^{\dagger}e_{i}bf\Delta^{\dagger}bfe_{j}+U^{\dagger}bf(\bar{e}_{i}b^{\dagger}\Delta+\Delta^{\dagger}be_{i})$
(C.14)

We then write $\Delta^{\dagger}b$ in terms of its quaternion components as
$c_{k}\bar{e}_{k}$, where the $c_{k}$ are complex valued matrices. It is
important to note that since $\Delta^{\dagger}b=\Omega$, the bottom $2k\times
2k$ part of the ADHM data, the $c_{k}$ are hermitian, since $\Omega$ is
hermitian by construction. Keeping this fact in mind, we can write (C.14) as

$D_{i}a_{j}=U^{\dagger}bfc_{k}f(e_{i}\bar{e}_{k}e_{j}+\bar{e}_{i}e_{k}e_{j}+\bar{e}_{k}e_{i}e_{j})$
(C.15)

Now, we use the identity $\bar{e}_{i}e_{j}=-\bar{e}_{j}e_{i}+2\delta_{ij}$ to
get

$D_{i}a_{j}=-U^{\dagger}bfc_{k}(e_{i}\bar{e}_{j}e_{k}-2\delta_{jk}e_{i}-2\delta_{ik}e_{j})$
(C.16)

Then we can see $a_{j}$ satisfies both the linear selfdual field equation

$D_{[i}a_{j]}=\frac{1}{2}\epsilon_{ijkl}a_{k}a_{l}$ (C.17)

and the zero mode condition $D_{i}a_{i}=0$. What does this say about the full
mode $\delta_{r}A_{i}$? We calculate

$\displaystyle D_{i}(\delta_{r}A_{j})$
$\displaystyle=-iD_{i}U^{\dagger}C_{r}D_{i}a_{j}^{\dagger}+ia_{j}C_{r}^{\dagger}D_{i}U-iU^{\dagger}C_{r}(D_{i}a_{j})^{\dagger}+iD_{i}a_{j}C_{r}^{\dagger}U$
$\displaystyle=-iU^{\dagger}bf\big{(}e_{i}\Delta^{\dagger}C_{r}\bar{e}_{j}-e_{j}C_{r}^{\dagger}\Delta\bar{e}_{i}\big{)}fb^{\dagger}U-iU^{\dagger}C_{r}D_{i}a_{j}^{\dagger}+iD_{i}a_{j}C_{r}^{\dagger}U$
$\displaystyle-
iU^{\dagger}C_{r}D_{i}a_{j}^{\dagger}+iD_{i}a_{j}C_{r}^{\dagger}U$ (C.18)

Here we have used the fact that

$D_{i}U^{\dagger}-iA_{i}U^{\dagger}=U^{\dagger}e_{i}bf\Delta^{\dagger}$ (C.19)

The discussion above of $D_{i}a_{j}$ shows that the last two terms of (C) are
a zero mode. We must therefore check the first two terms. The only parts of
these which depend on the moduli space coordinates are

$e_{i}\Delta^{\dagger}C_{r}\bar{e}_{j}-e_{j}C_{r}^{\dagger}\Delta\bar{e}_{i}\equiv
K_{ij}$ (C.20)

So the first two terms being a zero mode are equivalent to

$K_{[ij]}=\frac{1}{2}\epsilon_{ijkl}K_{kl}\ ;\ K_{ii}=0$ (C.21)

and these are satisfied iff
$\Delta^{\dagger}C_{r}=(\Delta^{\dagger}C_{r})^{T\star}$. This proves the
above lemma. To use this result, we must see what this condition says about
the form of $C_{r}$. First we define

$\displaystyle C_{r}$
$\displaystyle=\partial_{r}\Delta+\partial_{r}Q\Delta+\Delta\partial_{r}R$
(C.22) $\displaystyle Q$ $\displaystyle=\begin{bmatrix}q&0\\\
0&R^{-1}\end{bmatrix}$ (C.23)

Note that we can write $C_{r}$ as

$\partial_{r}a+\partial_{r}Qa+a\partial_{r}R+\Big{(}\partial_{r}b+\partial_{r}Qb+b\partial_{r}R\Big{)}x$
(C.24)

Next we set $q=1$. This means it does not contribute to the variation of $Q$,
which means

$\partial_{r}Q=-b\partial_{r}Rb^{\dagger}$ (C.25)

(The conjugation by $b$ is necessary to give $\partial_{r}Q$ the correct
dimensions). Then we see that the part of $C_{r}$ proportional to $x$ is zero,
since $\partial_{r}b$ is zero as $b$ is a constant matrix and the other two
terms cancel. This leaves us with

$C_{r}=\partial_{r}a+\partial_{r}Qa+a\partial_{r}R$ (C.26)

With this form, and the fact that $R^{T\star}=-R$, since $R$ is antiunitary,
it is straightforward that $b^{\dagger}C_{r}=(b^{\dagger}C_{r})^{T\star}$. The
second condition, $a^{\dagger}C_{r}=(a^{\dagger}C_{r})^{T\star}$, is satisfied
iff

$a^{\dagger}\partial_{r}a-(a^{\dagger}\partial_{r}a)^{T\star}-a^{\dagger}b\partial_{r}Rb^{\dagger}a-(a^{\dagger}b\partial_{r}Rb^{\dagger}a)^{T\star}+a^{\dagger}a\partial_{r}R-(a^{\dagger}b\partial_{r}Rb^{\dagger}a)^{T\star}=0$
(C.27)

We have therefore reduced the problem of finding the zero modes to solving the
above equation.
The metric is then derived from the inner product of two zero modes. To find
this, we use the following result from ([33])

$\text{Tr}^{\star}(\delta_{r}A_{i}\delta_{s}A_{i})=-\frac{1}{2}\partial^{2}\text{Tr}^{\star}\big{(}C_{r}^{\dagger}PC_{s}f+fC_{r}^{\dagger}C_{s}\big{)}$
(C.28)

where $P=\mathbb{1}-\Delta f\Delta^{\dagger}$. We can then use Stoke’s Theorem
to find the metric

$\displaystyle g_{rs}$ $\displaystyle=-\frac{1}{2}\int_{\mathcal{M}}\
\partial^{2}\text{Tr}^{\star}\big{(}C_{r}^{\dagger}PC_{s}f+fC_{r}^{\dagger}C_{s}\big{)}$
$\displaystyle=\int_{\partial\mathcal{M}}\
\text{Tr}^{\star}\big{(}C^{\dagger}_{r}P_{\infty}C_{s}+C^{\dagger}_{r}C_{s}\big{)}_{ij}$
$\displaystyle=2\pi^{2}\text{Tr}^{\star}\big{(}C^{\dagger}_{r}P_{\infty}C_{s}+C^{\dagger}_{r}C_{s}\big{)}_{ij}$
$\displaystyle=2\pi^{2}\text{Tr}^{\star}\Big{(}\partial_{r}a^{\dagger}(1+P_{\infty})\partial_{s}a-\big{(}a^{\dagger}\partial_{r}a-(a^{\dagger}\partial_{r}a)^{T}\big{)}_{ij}\partial_{s}R\Big{)}$
(C.29)

Here

$\displaystyle P_{\infty}$
$\displaystyle=lim_{x\rightarrow\infty}P=\mathbb{1}_{n+2k\times
n+2k}-bb^{\dagger}$ $\displaystyle=$
$\displaystyle\begin{bmatrix}\mathbb{1}_{n/2\times n/2}&0\\\ 0&0_{k\times
k}\end{bmatrix}$ (C.30)

remembering

$\Delta(x)=\begin{bmatrix}\Lambda\\\ \Omega+\tilde{\rho}\mathbb{1}_{k\times
k}\end{bmatrix}-x\begin{bmatrix}0\\\ \mathbb{1}_{k\times k}\end{bmatrix}$
(C.31)

(Note – the term in $\tilde{\rho}$ gives the center of mass, and is usually
absorbed into the $x$ component by a suitable choice of coordinates, but it is
there, and therefore we consider it here – albeit briefly). The first term
above then gives

$2\pi^{2}\text{Tr}^{\star}\big{(}da^{\dagger}(1+P_{\infty})da\big{)}=2\pi^{2}\text{Tr}\big{(}2\Lambda^{\dagger}\Lambda+\Omega^{\dagger}\Omega+2d\tilde{\rho}^{\dagger}d\tilde{\rho}\big{)}$
(C.32)

The $d\tilde{\rho}^{\dagger}d\tilde{\rho}$ directions are flat and decouple
from the rest of the metic and so we ignore them (They correspond to the
position of the centre of mass). This gives the first part of the metric

$ds_{1}^{2}=2\pi^{2}\text{Tr}^{\star}\big{(}da^{\dagger}(1+P_{\infty})da\big{)}=2\pi^{2}\text{Tr}^{\star}\big{(}2\Lambda^{\dagger}\Lambda+\Omega^{\dagger}\Omega\big{)}$
(C.33)

Now for the second part of the metric

$ds_{2}^{2}=2\pi^{2}\text{Tr}^{\star}\Big{(}\big{(}a^{\dagger}da-(a^{\dagger}da)^{T\star}\big{)}dR\Big{)}$
(C.34)

To find an explicit expression here, we write $dR$ in terms of its components
considered as a $U(k)$ matrix, and solve for them using (C.27). We get one
equation for each component, and solving them gives $dR$ in terms of the ADHM
parameters. We will see this explicitly in the specific cases below. Once we
have done this, we can put all these parts together to get the full metric

$\displaystyle ds^{2}=ds_{1}^{2}+\ ds_{2}^{2}$
$\displaystyle=2\pi^{2}\Big{(}\text{Tr}^{\star}\big{(}da^{\dagger}(1+P_{\infty})da\big{)}$
$\displaystyle=2\pi^{2}\bigg{(}\text{Tr}^{\star}\big{(}2d\Lambda^{\dagger}d\Lambda+d\Omega^{\dagger}d\Omega\big{)}+\text{Tr}^{\star}\Big{(}\big{(}a^{\dagger}da-(a^{\dagger}da)^{T\star}\big{)}dR\Big{)}\bigg{)}$
(C.35)

##  Appendix D Constructing the Potential

We can use the Metric to calculate the potential for the dyonic instanton
moduli space. This also makes use of the solution for the scalar field in
Appendix B. Recall the definition of the potential

$\mathcal{V}=\int d^{4}x\ \text{Tr}(D_{i}\phi D_{i}\phi)$ (D.1)

Integrating by parts and using the fact that $D^{2}\phi=0$ via its equation of
motion we get

$\mathcal{V}=\text{lim}_{R\rightarrow\infty}\int_{\lvert
x\rvert=R}dS^{3}\hat{x}_{i}\text{Tr}(\phi D_{i}\phi)$ (D.2)

Using the facts that $\phi=U^{\dagger}\mathcal{A}U,D_{i}=\partial_{i}-iA_{i}$
and $A_{i}=U^{\dagger}\partial_{i}U$, a moderately long calculation [13] gives

$D_{i}\phi=iU^{\dagger}e_{i}bf\Delta^{\dagger}\mathcal{A}U+iU^{\dagger}\mathcal{A}\Delta
f\bar{e}_{i}b^{\dagger}U$ (D.3)

To fully evaluate this integral, we need an expression for $U$. In general
this would be rather complicated, however we only need the value of $U$ on the
boundary, in the limit $R\rightarrow\infty$. For a general ADHM matrix

$\begin{bmatrix}v_{1}&v_{2}&v_{3}&\dots&v_{k}\\\
\tau_{1}-x&\sigma^{\star}_{1}&\sigma^{\star}_{2}&\dots&\sigma^{\star}_{k-1}\\\
\sigma_{1}&\tau_{2}-x&\sigma^{\star}_{k}&\dots&\sigma^{\star}_{2k-3}\\\
\vdots&&\ddots&&\vdots\\\
\sigma_{k-1}&\sigma_{2k-3}&\sigma_{3k-4}&\dots&\tau_{k}-x\end{bmatrix}$ (D.4)

the condition $\Delta^{\dagger}U=0$ is solved to leading order in $\lvert
x\rvert$ by

$U_{1}\mapsto 1;\ U_{i}\mapsto\frac{x}{\lvert
x\rvert^{2}}v^{\dagger}_{i-1},i\neq 1$ (D.5)

We might worry here about the issue discussed in section 3.1, where $U$ may or
may not satisfy the completeness relation (3.18). In general we would need to
worry about this, however if we expand in powers of $\zeta$, any terms
including a correction of order $\zeta^{n}$ would, by dimensional analysis,
also have to go as $\lvert x\rvert^{-2n}$, and are therefore neglected in this
limit. We also need these two results for the behaviour of other quantities in
this limit

$\displaystyle\Delta\mapsto\begin{bmatrix}\Lambda\\\
-x\mathbb{1}_{k}\end{bmatrix}$ $\displaystyle f\mapsto\frac{1}{\lvert
x\rvert^{2}}\mathbb{1}_{k}$ (D.6)

We can use these to expand equation (D.3), and then multiplying by
$\hat{x}_{i}$ we get, to leading order

$\hat{x}_{i}D_{i}\phi=\frac{2i}{\lvert
x\rvert^{3}}\bigg{(}q\Lambda\Lambda^{\dagger}-\Lambda
P\Lambda^{\dagger}\bigg{)}+\mathcal{O}\Big{(}\frac{1}{\lvert
x\rvert^{4}}\Big{)}$ (D.7)

Remembering that $\phi=iq$ on the boundary, we can then write, to leading
order

$\displaystyle\mathcal{V}$
$\displaystyle=\text{lim}_{R\rightarrow\infty}\int_{\lvert
x\rvert=R}dS^{3}\hat{x}_{i}\text{Tr}(\phi D_{i}\phi)$
$\displaystyle=-2\text{lim}_{R\rightarrow\infty}\int_{\lvert
x\rvert=R}dS^{3}\frac{1}{\lvert
x\rvert^{3}}\bigg{(}q^{2}\Lambda\Lambda^{\dagger}-q\Lambda
P\Lambda^{\dagger}\bigg{)}+\mathcal{O}\Big{(}\frac{1}{\lvert
x\rvert^{4}}\Big{)}$
$\displaystyle=-4\pi^{2}\text{Tr}\Big{(}q^{2}\Lambda\Lambda^{\dagger}-q\Lambda
P\Lambda^{\dagger}\Big{)}$ (D.8)

Now we have these general expressions and methods for the ADHM solutions,
moduli space metric, and potential, the following appendices will provide
particular solutions for the cases discussed in the main text

##  Appendix E Scalar Field, Metric and Potential in the Two Instanton case

###  E.1 The Scalar Field

We begin with the Scalar Field. Following the method in Appendix B, we have
the Ansatz:

$\phi=U^{\dagger}\mathcal{A}U;\ \mathcal{A}=\begin{bmatrix}q&0\\\
0&P\end{bmatrix}$ (E.1)

Here $q$ is in the odd graded part of $\mathbb{C}\times\mathbb{H}$; i.e.
$q=iq_{0}+\textbf{q}$, where $q_{0}\in\mathbb{R}$ and
$\textbf{q}\in\text{Im}_{\mathbb{H}}\mathbb{H}$. The matrix $P$ is
antihermitian, and so can be parameterised by

$\begin{bmatrix}ai&ci-b\\\ ci+b&di\end{bmatrix}$ (E.2)

The equation for the scalar field is

$2\text{Tr}_{2}(\Lambda^{\dagger}q\Lambda)+\text{Tr}_{2}([\Omega^{\dagger},P]\Omega-\Omega^{\dagger}[\Omega,P])-\text{Tr}_{2}(\\{P,\Lambda^{\dagger}\Lambda\\})=0$
(E.3)

Solving this equation is a lengthy calculation, which gives:

$\displaystyle a=$
$\displaystyle-\frac{1}{\Theta}\bigg{(}A(3)\big{(}g^{2}N_{AI}-f^{2}N_{AR}\big{)}+A(2)w(4gP-
fN_{AR})+A(1)w(4fP-gN_{AI})$
$\displaystyle-\big{(}(16P^{2}-N_{AR}N_{AI}\big{)}\Big{(}A(3)(2s+w)+wA(4)\Big{)}\bigg{)}$
$\displaystyle b=$
$\displaystyle\frac{1}{2\Theta}\bigg{(}A(1)\left(f^{2}(v+w)-2N_{AI}(\text{sv}+\text{sw}+\text{vw})\right)+A(2)(fg(v+w)+8P(\text{sv}+\text{sw}+\text{vw}))$
$\displaystyle+(4fP+gN_{AI})\Big{(}A(3)(v-w)-A(4)(v+w)\Big{)}\bigg{)}$
$\displaystyle c=$
$\displaystyle\frac{1}{2\Theta}\bigg{(}A(1)\Big{(}fg(v+w)-8P(sv+sw+vw)\Big{)}+A(2)\Big{(}g^{2}(v+w)+2N_{AR}(sv+sw+vw)\Big{)}$
$\displaystyle+(fN_{AR}+4gP)\Big{(}A(3)(v-w)+A(4)(v+w)\Big{)}\bigg{)}$
$\displaystyle d=$
$\displaystyle-\frac{1}{\Theta}\bigg{(}A(3)\big{(}f^{2}N_{AR}-g^{2}N_{AI}\big{)}+A(2)v(4gP-
fN_{AR})+A(1)v(4fP-gN_{AI})$
$\displaystyle+(A(3)(2s+v)-A(4)v)\big{(}16P^{2}-XY\big{)}\bigg{)}$ (E.4)

where

$\displaystyle A(1)$
$\displaystyle=4q_{0}\text{Re}_{\mathbb{H}}(\bar{v}_{R}w_{I}-\bar{v}_{I}w_{R})-4\text{Re}_{\mathbb{H}}(\bar{v}_{R}\textbf{q}w_{R}+\bar{v}_{I}\textbf{q}w_{I})$
$\displaystyle A(2)$
$\displaystyle=4q_{0}\text{Re}_{\mathbb{H}}(\bar{v}_{R}w_{R}+\bar{v}_{I}w_{I})+4\text{Re}_{\mathbb{H}}(\bar{v}_{R}\textbf{q}w_{I}-\bar{v}_{I}\textbf{q}w_{R})$
$\displaystyle A(3)$ $\displaystyle=q_{0}\big{(}\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}-\lvert w_{R}\rvert^{2}-\lvert
w_{I}\rvert^{2}\big{)}+2\text{Re}_{\mathbb{H}}(\bar{v}_{R}qv_{I}-\bar{w}_{R}qw_{I})$
$\displaystyle A(4)$ $\displaystyle=q_{0}\big{(}\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}+\lvert w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2}\big{)}+2\text{Re}_{\mathbb{H}}(\bar{v}_{R}v_{I}+\bar{w}_{R}qw_{I})$
$\displaystyle f$
$\displaystyle=\text{Re}_{\mathbb{H}}(\bar{w}_{R}v_{R}+\bar{w}_{I}v_{I})$
$\displaystyle g$
$\displaystyle=\text{Re}_{\mathbb{H}}(\bar{w}_{I}v_{R}-\bar{v}_{I}w_{R})$
$\displaystyle x$ $\displaystyle=\lvert\sigma_{R}\rvert^{2}$ $\displaystyle y$
$\displaystyle=\lvert\sigma_{I}\rvert^{2}$ $\displaystyle P$
$\displaystyle=\text{Re}_{\mathbb{H}}(\bar{\sigma}_{R}\sigma_{I})$
$\displaystyle v$ $\displaystyle=\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}$ $\displaystyle w$ $\displaystyle=\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}$ $\displaystyle N_{AR}$
$\displaystyle=\lvert v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}+\lvert
w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2}+4(\lvert\tau\rvert^{2}+\lvert\sigma_{R}\rvert^{2})$
$\displaystyle N_{AI}$ $\displaystyle=\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}+\lvert w_{R}\rvert^{2}+\lvert
w_{I}\rvert+4(\lvert\tau\rvert^{2}+\lvert\sigma_{I}\rvert^{2})$
$\displaystyle\Theta$
$\displaystyle=(v+w)\big{(}f^{2}N_{AR}-g^{2}N_{AI}\big{)}+2\big{(}16P^{2}-N_{AR}N_{AI}\big{)}(sv+\text{sw}+\text{vw})$
(E.5)

In the commutative limit from [13], that is, $\zeta=0$ and the imaginary
quaternion parts $q_{I}$ set to zero, this becomes

$b=\frac{-2\text{Re}_{\mathbb{H}}(\bar{v}\textbf{q}w)}{\Sigma_{+}+4(\lvert\tau\rvert^{2}+\lvert\sigma_{R}\rvert^{2})};\
a,b,d=0$ (E.6)

which is precisely the result in that paper.
Another useful limit is that in which $\lvert\tau\rvert\mapsto\infty$. In this
case

$\displaystyle a=\frac{q_{0}(\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2})+2\text{Re}(\bar{v}_{R}\textbf{q}v_{I})}{\lvert
v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}}$ $\displaystyle d=\frac{q_{0}(\lvert
w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2})+2\text{Re}(\bar{w}_{R}\textbf{q}w_{I})}{\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}}$ (E.7)

With $b,c=0$. This corresponds to the two instantons being far separated. In
this case we would expect them to look like two single U(2) instantons, and we
see that we do in fact have two copies of the solution for a single instanton
given in (4.1). The next step is to use this to explicitly calculate the
potential.

###  E.2 The Potential

Recall from Appendix D that the potential is given by

$V=\int d^{4}x\text{Tr}(D_{i}\phi D_{i}\phi)$ (E.8)

Integrating by parts, and using the equation of motion for $\phi$

$D^{2}\phi=0$ (E.9)

we get

$V=\text{lim}_{R\mapsto\infty}\int_{\lvert
x\rvert=R}dS^{3}\hat{x}_{i}\text{Tr}\big{(}\phi D_{i}\phi\big{)}$ (E.10)

We know that the vector $U$, being a null vector of $\Delta$, must solve

$\displaystyle
v^{\dagger}U_{1}+(\tau^{\dagger}-x^{\dagger})U_{2}+\sigma^{\dagger}U_{3}=0$
$\displaystyle
w^{\dagger}U_{1}+\sigma^{\dagger}U_{2}-(\tau^{\dagger}+x^{\dagger})U_{3}=0$
(E.11)

This is solved on the boundary by

$\displaystyle U_{1}\mapsto\ 1$ $\displaystyle U_{2}\mapsto\ \frac{x}{\lvert
x\rvert^{2}}v^{\dagger}$ $\displaystyle U_{3}\mapsto\ \frac{x}{\lvert
x\rvert^{2}}w^{\dagger}$ (E.12)

Solving these equations following the method in Appendix D is long, however
eventually we get:

$\displaystyle\mathcal{V}=$ $\displaystyle 8\pi^{2}\Bigg{(}\lvert
q\rvert^{2}\big{(}\lvert v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}+\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}\big{)}$
$\displaystyle+4q_{0}\text{Re}_{\mathbb{H}}(\bar{v}_{R}\vec{q}v_{I}+\bar{w}_{R}\vec{q}w_{I})-a\big{(}q_{0}(\lvert
v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}+2\text{Re}_{\mathbb{H}}(\bar{v}_{R}\textbf{q}v_{I})\big{)}$
$\displaystyle-d\big{(}q_{0}(\lvert w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2})+2\text{Re}_{\mathbb{H}}(\bar{w}_{R}\vec{q}w_{I})\big{)}+2b\text{Re}_{\mathbb{H}}(\bar{v}_{R}\vec{q}w_{R}+\bar{v}_{I}\vec{q}w_{I})$
$\displaystyle-2bq_{0}\text{Re}_{\mathbb{H}}(w_{I}\bar{v}_{R}-w_{R}\bar{v}_{I})-2cq_{0}\text{Re}_{\mathbb{H}}(w_{R}\bar{v}_{R}+w_{I}\bar{v}_{I})-2c\text{Re}_{\mathbb{H}}(\bar{v}_{R}\vec{q}w_{I}-\bar{v}_{I}\vec{q}w_{R})\Bigg{)}$
(E.13)

We can choose the $q_{0}$ to be zero by requiring the vev to lie in $SU(2)$.
This simplifies our solution to

$\displaystyle\mathcal{V}=\ $ $\displaystyle 8\pi^{2}\Bigg{(}\lvert
q\rvert^{2}\big{(}\lvert v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}+\lvert
w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2}\big{)}-a\big{(}2\text{Re}(\bar{v}_{R}\textbf{q}v_{I})\big{)}-d\big{(}2\text{Re}(\bar{w}_{R}\vec{q}w_{I})\big{)}$
$\displaystyle+2b\text{Re}(\bar{v}_{R}\vec{q}w_{R}+\bar{v}_{I}\vec{q}w_{I})-2c\text{Re}(\bar{v}_{R}\vec{q}w_{I}-\bar{v}_{I}\vec{q}w_{R})\Bigg{)}$
(E.14)

Where $a,b,c,d$ are given above.
If instead we go back to the large $\tau$ limit, using (E.1)

$\displaystyle\mathcal{V}=8\pi^{2}\Bigg{(}\lvert q\rvert^{2}\big{(}\lvert
v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}+\lvert w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2}\big{)}+4q_{0}\text{Re}(\bar{v}_{R}\vec{q}v_{I}+\bar{w}_{R}\vec{q}w_{I})$
$\displaystyle-\frac{\big{(}q_{0}(\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}+2\text{Re}(\bar{v}_{R}\textbf{q}v_{I})\big{)}^{2}}{\lvert
v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}}-\frac{\big{(}q_{0}(\lvert
w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2})+2\text{Re}(\bar{w}_{R}\vec{q}w_{I})\big{)}^{2}}{\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}}\Bigg{)}$ (E.15)

In this case, the $q_{0}$ parts cancel explicitly, and we get

$\mathcal{V}=8\pi^{2}\lvert\textbf{q}\rvert^{2}\Bigg{(}\hat{\textbf{q}}\Big{(}\lvert
v_{R}\rvert^{2}+\lvert v_{I}\rvert^{2}+\lvert w_{R}\rvert^{2}+\lvert
w_{I}\rvert^{2}\big{)}-\frac{4\text{Re}^{2}(\bar{v}_{R}\hat{\textbf{q}}v_{I})}{\lvert
v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}}-\frac{4\text{Re}^{2}(\bar{w}_{R}\hat{\textbf{q}}w_{I})}{\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}}\Bigg{)}$ (E.16)

We would expect this is the potential for two copies of the single U(1)
instanton, and if we compare to the result in section 4.1 we can easily see
that this is the case.

###  E.3 The Metric

As in Appendix C, we begin by calculating $a^{\dagger}\delta C_{r}$, and
impose the condition

$a^{\dagger}\delta C_{r}=\big{(}a^{\dagger}\delta C_{r}\big{)}^{T\star}$
(E.17)

Once again, we carefully note that $T$ involves taking the transpose
considered as a $2\times 2$ matrix of biquaternions. It does not affect the
quaternions themselves. The operation $\star$ takes the complex conjugate of
each element, which again does not affect the quaternions but only their
complex coefficients.
We can expand $\delta R$ in the $u(2)$ basis as

$\delta R=\begin{bmatrix}id\phi&id\psi-d\theta\\\
id\psi+d\theta&id\chi\end{bmatrix}$ (E.18)

this gives 3 simultaneous equations for the derivations in the different gauge
directions. As discussed below equation (C.34), we can use these along with
equation (C.27) to find $\delta R$ in full:

$\displaystyle d\phi=$ $\displaystyle\frac{1}{\Phi}\bigg{(}-2B(1)(2s+w)(4fP-
gN_{AI})+2B(2)((2s+w)(4gP-fN_{AR})$
$\displaystyle-(B(3)(2s+w)+B(4)w)\left(16P^{2}-N_{AR}N_{AI}\right)-2B(4)\left(f^{2}N_{AR}-8fgP+g^{2}N_{AI}\right)\bigg{)}$
$\displaystyle d\theta=$
$\displaystyle\frac{1}{\Phi}\bigg{(}2B(1)\left(f^{2}(4s+v+w)-N_{AI}(sv+sw+vw)\right)$
$\displaystyle+2B(2)(fg(4s+v+w)-4P(sv+sw+vw))$
$\displaystyle+(B(3)(4s+v+w)-B(4)(v-w))(4fP-gN_{AI})\bigg{)}$ $\displaystyle
d\psi=$ $\displaystyle\frac{1}{\Phi}\bigg{(}-2B(1)(fg(4s+v+w)-4P(sv+sw+vw))$
$\displaystyle-2B(2)\left(g^{2}(4s+v+w)-2N_{AR}(sv+sw+vw)\right)$
$\displaystyle+(B(3)(4s+v+w)-B(4)(v-w))(4gP-fN_{AR})\bigg{)}$ $\displaystyle
d\chi=$ $\displaystyle\frac{1}{\Phi}\bigg{(}-2B(1)(2s+v)(4fP-
gN_{AI})+2B(2)(2s+v)(4gP-fN_{AR})$
$\displaystyle-\left(16P^{2}-N_{AR}N_{AI}\right)(B(3)(2s+v)-B(4)v)+2B(4)\left(f^{2}N_{AR}-8fgP+g^{2}N_{AI}\right)\bigg{)}$
(E.19)

where the terms are defined in (E.1) with the addition of

$\displaystyle B(1)=$
$\displaystyle\bar{v}_{R}dw_{R}+\bar{v}_{I}dw_{I}-\bar{w}_{R}dv_{R}-w_{I}dv_{I}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau)$
$\displaystyle B(2)=$
$\displaystyle\bar{v}_{R}dw_{I}-\bar{v}_{I}dw_{R}+\bar{w}_{R}dv_{I}-\bar{w}_{I}dv_{R}+2(\bar{\sigma}_{I}d\tau-\bar{\tau}d\sigma_{I})$
$\displaystyle B(3)=$
$\displaystyle\bar{v}_{R}dv_{I}-\bar{v}_{I}dv_{R}+\bar{w}_{R}dw_{I}-\bar{w}_{I}dw_{R}$
$\displaystyle B(4)=$
$\displaystyle\bar{v}_{R}dv_{I}-\bar{v}_{I}dv_{R}-\bar{w}_{R}dw_{I}+\bar{w}_{I}dw_{R}+2(\bar{\sigma}_{R}d\sigma_{I}-\bar{\sigma}_{I}d\sigma_{R})$
$\displaystyle\Phi=$ $\displaystyle
4\left((4s+v+w)\left(f^{2}X-8fgP+g^{2}Y\right)+\left(16P^{2}-\text{XY}\right)(sv+sw+vw)\right)$
(E.20)

We can now use this in our formula (C)

$ds^{2}=ds_{1}^{2}+ds_{2}^{2}=2\pi^{2}\bigg{(}\text{Tr}^{\star}\big{(}2d\Lambda^{\dagger}d\Lambda+d\Omega^{\dagger}d\Omega\big{)}+\text{Tr}^{\star}\Big{(}\big{(}a^{\dagger}da-(a^{\dagger}da)^{T\star}\big{)}dR\Big{)}\bigg{)}$
(E.21)

First we have that $a^{\dagger}da-(a^{\dagger}da)^{T\star}$ is

$\displaystyle\begin{bmatrix}0&\bar{v}_{R}dw_{R}+\bar{v}_{I}dw_{I}-\bar{w}_{R}dv_{R}-\bar{w}_{I}dv_{I}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau)\\\
-\big{(}\bar{v}_{R}dw_{R}+\bar{v}_{I}dw_{I}-\bar{w}_{R}dv_{R}-\bar{w}_{I}dv_{I}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau)\big{)}&0\end{bmatrix}$
$\displaystyle+i\begin{bmatrix}2\big{(}\bar{v}_{R}dv_{I}-\bar{v}_{I}dv_{R}+\bar{\sigma}_{R}d\sigma_{I}-\bar{\sigma}_{I}d\sigma_{R}\big{)}&\bar{v}_{R}dw_{I}-\bar{v}_{I}dw_{R}+\bar{w}_{R}dv_{I}-\bar{w}_{I}dv_{R}+2\big{(}\bar{\sigma}_{I}d\tau-\bar{\tau}d\sigma_{I}\big{)}\\\
\bar{v}_{R}dw_{I}-\bar{v}_{I}dw_{R}+\bar{w}_{R}dv_{I}-\bar{w}_{I}dv_{R}+2\big{(}\bar{\sigma}_{I}d\tau-\bar{\tau}d\sigma_{I}\big{)}&2\big{(}\bar{w}_{R}dw_{I}-\bar{w}_{I}dw_{R}-\bar{\sigma}_{R}d\sigma_{I}+\bar{\sigma}_{I}d\sigma_{R}\big{)}\end{bmatrix}$
(E.22)

Once we have this it is fairly straightforward to calculate the metric as

$\displaystyle ds^{2}=$ $\displaystyle
8\pi^{2}\bigg{(}d^{2}v_{R}+d^{2}v_{I}+d^{2}w_{R}+d^{2}w_{I}+d^{2}\tau+d^{2}\sigma_{R}+d^{2}\sigma_{I}$
$\displaystyle-\text{Re}_{\mathbb{H}}\Big{(}\big{(}\bar{v}_{R}dv_{I}-\bar{v}_{I}dv_{R}+\bar{\sigma}_{R}d\sigma_{I}-\bar{\sigma}_{I}d\sigma_{R}\big{)}d\phi+\big{(}\bar{w}_{R}dw_{I}-\bar{w}_{I}dw_{R}-\bar{\sigma}_{R}d\sigma_{I}+\bar{\sigma}_{I}d\sigma_{R}\big{)}d\chi$
$\displaystyle+\big{(}\bar{v}_{R}dw_{R}+\bar{v}_{I}dw_{I}-\bar{w}_{R}dv_{R}-\bar{w}_{I}dv_{I}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau))\big{)}d\theta$
$\displaystyle+\big{(}\bar{v}_{R}dw_{I}-\bar{v}_{I}dw_{R}+\bar{w}_{R}dv_{I}-\bar{w}_{I}dv_{R}+2\big{(}\bar{\sigma}_{I}d\tau-\bar{\tau}d\sigma_{I}\big{)}\big{)}d\psi\Big{)}$
(E.23)

We can check the behaviour of this solution in various limits. First of all,
the commutative real limit, where the various imaginary quaternion parts
$q_{I}$ and the noncommutative parameter $\zeta$ are set to zero. In this
limit we have

$d\phi=d\psi=d\chi=0;d\theta=\frac{\bar{v}_{R}dw_{R}-\bar{w}_{R}dv_{R}-\bar{w}_{R}dv_{R}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau)}{\lvert
v_{R}\rvert^{2}+\lvert
w_{R}\rvert^{2}+4(\lvert\tau\rvert^{2}+\lvert\sigma_{R}\rvert^{2})}$ (E.24)

This allows us to calculate the metric to be

$ds^{2}=8\pi^{2}\Big{(}d^{2}v_{R}+d^{2}w_{R}+d^{2}\tau+d^{2}\sigma_{R}-\frac{dk^{2}}{N_{A}}\Big{)}$
(E.25)

with

$\displaystyle N_{A}$ $\displaystyle=\lvert v_{R}\rvert^{2}+\lvert
w_{R}\rvert^{2}+4(\lvert\tau\rvert^{2}+\lvert\sigma_{R}\rvert^{2})$ (E.26)
$\displaystyle dk$
$\displaystyle=\bar{v}_{R}dw_{R}-\bar{w}_{R}dv_{R}-\bar{w}_{R}dv_{R}+2(\bar{\tau}d\sigma_{R}-\bar{\sigma}_{R}d\tau)$

exactly as in [13]. The second limit we can check is the limit in which
$\lvert\tau\rvert\mapsto\infty$. Since this corresponds to the two instantons
becoming far separated, in this limit, we would expect to get two copies of
the solution for a single $U(2)$ instanton. We in fact get

$d\phi=\frac{v_{R}dv_{I}-\bar{v}_{I}dv_{R}}{\lvert v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}};\ d\chi=\frac{w_{R}dw_{I}-\bar{w}_{I}dw_{R}}{\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}}$ (E.27)

This gives the metric

$\displaystyle
ds^{2}=8\pi^{2}\bigg{(}d^{2}v_{R}+d^{2}v_{I}+d^{2}w_{R}+d^{2}w_{I}-\frac{\big{(}v_{R}dv_{I}-\bar{v}_{I}dv_{R}\big{)}^{2}}{\lvert
v_{R}\rvert^{2}+\lvert
v_{I}\rvert^{2}}-\frac{\big{(}w_{R}dw_{I}-\bar{w}_{I}dw_{R}\big{)}^{2}}{\lvert
w_{R}\rvert^{2}+\lvert w_{I}\rvert^{2}}\bigg{)}$ (E.28)

This is precisely the sum of two copies of the form in section 4.1 above,
equation (4.4).

##  Appendix F Three Instanton Metric and Potential

###  F.1 The Scalar Field

As in the 2 Instanton case the expressions derived here for the scalar field,
metric and potential are in principle valid for the full quaternion
parametrisation. However when we substitute in the solutions for the
$\sigma_{i}$ derived in equation (6), that is only valid for that particular
complex subspace. Keeping this in mind, we use the same method as before. This
time the ansatz is given by

$\mathcal{A}=\begin{bmatrix}\textbf{q}&0\\\ 0&P\end{bmatrix}$ (F.1)

where $\textbf{q}\in\ su(2)$, and $P\in\ o(3)$, parametrised as

$\begin{bmatrix}0&-a&b\\\ a&0&-c\\\ -b&c&0\end{bmatrix}$ (F.2)

The ADHM data $\Delta$ is given, in this case, by

$\begin{bmatrix}u&v&w\\\ \tau_{1}&\sigma_{1}&\sigma_{2}\\\
\sigma_{1}&\tau_{2}&\sigma_{3}\\\ \sigma_{2}&\sigma_{3}&\tau_{3}\end{bmatrix}$
(F.3)

where $\tau_{1}+\tau_{2}+\tau_{3}=0$. Now the elements are all quaternions,
not biquaternions. The equation we want to solve is still

$2\text{Tr}_{2}(\Lambda^{\dagger}\textbf{q}\Lambda)+\text{Tr}_{2}([\Omega^{\dagger},P]\Omega-\Omega^{\dagger}[\Omega,P])-\text{Tr}_{2}(\\{P,\Lambda^{\dagger}\Lambda\\})=0$
(F.4)

Proceeding as in the previous cases, we can solve for the components of
$\mathcal{A}$ as

$\displaystyle a=$
$\displaystyle\frac{1}{\Upsilon}\bigg{(}C_{3}(2\Psi_{2}M_{A2}-\Psi_{1}\Psi_{3})+C_{2}(2\Psi_{1}M_{A3}-\Psi_{2}\Psi_{3})+C_{1}\left(-\left(4M_{A2}M_{A3}-\Psi_{3}^{2}\right)\right)\bigg{)}$
$\displaystyle b=$
$\displaystyle\frac{1}{\Upsilon}\bigg{(}-C_{3}(\Psi_{1}\Psi_{2}+2M_{A1}z)+C_{1}(2\Psi_{1}M_{A3}-\Psi_{2}\Psi_{3})-C_{2}\left(4M_{A1}M_{A3}-\Psi_{2}^{2}\right)\bigg{)}$
$\displaystyle c=$
$\displaystyle\frac{1}{\Upsilon}\bigg{(}-C_{3}\left(4M_{A1}M_{A2}-\Psi_{1}^{2}\right)-C_{2}(\Psi_{1}\Psi_{2}+2M_{A1}\Psi_{3})+C_{1}(2\Psi_{2}Y-\Psi_{1}\Psi_{3})\bigg{)}$
(F.5)

Where

$\displaystyle C_{1}=$ $\displaystyle
4\text{Re}_{\mathbb{H}}(\bar{v}\textbf{q}u)$ $\displaystyle C_{2}=$
$\displaystyle 4\text{Re}_{\mathbb{H}}(\bar{u}\textbf{q}w)$ $\displaystyle
C_{3}=$ $\displaystyle 4\text{Re}_{\mathbb{H}}(\bar{w}\textbf{q}v)$
$\displaystyle M_{A1}=$ $\displaystyle\lvert u\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{1}\rvert^{2}+\Sigma^{2}+\lvert\tau_{1}-\tau_{2}\rvert^{2}$
$\displaystyle M_{A2}=$ $\displaystyle\lvert w\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{2}\rvert^{2}+\Sigma^{2}+\lvert\tau_{1}-\tau_{3}\rvert^{2}$
$\displaystyle M_{A3}=$ $\displaystyle\lvert w\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{3}\rvert^{2}+\Sigma^{2}+\lvert\tau_{2}-\tau_{3}\rvert^{2}$
$\displaystyle\Psi_{1}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{1}\sigma_{3}-\bar{\sigma}_{2}\sigma_{1})-\bar{w}v\big{)}$
$\displaystyle\Psi_{2}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{2}\sigma_{2}-\bar{\sigma}_{1}\sigma_{3})-\bar{u}w\big{)}$
$\displaystyle\Psi_{3}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{3}\sigma_{1}-\bar{\sigma}_{3}\sigma_{2})-\bar{v}u\big{)}$
$\displaystyle\Upsilon=$ $\displaystyle
2\left(\Psi_{1}^{2}M_{A3}+\Psi_{1}\Psi_{2}\Psi_{3}-4M_{A1}M_{A2}M_{A3}+M_{A1}\Psi_{3}^{2}+\Psi_{2}^{2}M_{A2}\right)$
(F.6)

###  F.2 The Potential

We can now use this to calculate the potential, using the formula

$V=\int d^{4}x\text{Tr}(D_{i}\phi D_{i}\phi)$ (F.7)

We can now follow the standard method. Integrating by parts, and using the
equation of motion for $\phi$

$D^{2}\phi=0$ (F.8)

We get

$V=\text{lim}_{R\mapsto\infty}\int_{\lvert
x\rvert=R}dS^{3}\hat{x}_{i}\text{Tr}\big{(}\phi D_{i}\phi\big{)}$ (F.9)

We know that the vector $U$, being a null vector of $\Delta$, must solve
$\Delta^{\dagger}U=0$, which gives the equations

$\displaystyle\bar{u}U_{1}+(\bar{\tau}_{1}-\bar{x})U_{2}+\bar{\sigma}_{1}U_{3}+\bar{\sigma}_{2}U_{4}=0$
$\displaystyle\bar{v}U_{1}+\bar{\sigma}_{1}U_{2}+(\bar{\tau}_{2}-\bar{x})U_{3}+\bar{\sigma}_{3}U_{4}=0$
$\displaystyle\bar{w}U_{1}+\bar{\sigma}_{2}U_{2}+\bar{\sigma}_{3}U_{3}+(\bar{\tau}_{3}-\bar{x})U_{4}=0$
(F.10)

These can be solved in the $\lvert x\rvert^{2}\mapsto\infty$ limit as

$U_{1}\mapsto 1\ ;\ U_{2}\mapsto\frac{x\bar{u}}{\lvert x\rvert^{2}}\ ;\
U_{3}\mapsto\frac{x\bar{v}}{\lvert x\rvert^{2}}\ ;\
U_{4}\mapsto\frac{x\bar{w}}{\lvert x\rvert^{2}}\ $ (F.11)

We can continue to calculate the potential as in the previous cases. Using the
method in Appendix D , analogously to the discussion for two instantons in
Appendix E.2 we get:

$\mathcal{V}=8\pi^{2}\bigg{(}\lvert\textbf{q}\rvert^{2}\big{(}\lvert
u\rvert^{2}+\lvert v\rvert^{2}+\lvert
w\rvert^{2}\big{)}-2a\text{Re}_{\mathbb{H}}(\bar{v}\textbf{q}u)-2b\text{Re}_{\mathbb{H}}(\bar{u}\textbf{q}w)-2c\text{Re}_{\mathbb{H}}(\bar{w}\textbf{q}v)\bigg{)}$
(F.12)

with $a,b,c$ given as above.

###  F.3 $O(3)$ Metric

The final thing to calculate is the metric. As in the previous case, we need
to calculate $a^{T}\delta C_{r}$, and impose the condition

$a^{T}\delta C_{r}=\big{(}a^{T}\delta C_{r}\big{)}^{T\star}$ (F.13)

Note that here we have the operation $T$ rather than $\dagger$ as we are
dealing with the usual, real quaternions rather than the complexified version.
Since in this commutative 3-instanton case, the remaining symmetry is $o(3)$,
we can write

$\delta R=\begin{bmatrix}0&-d\phi&d\theta\\\ d\phi&0&-d\psi\\\
-d\theta&d\psi&0\end{bmatrix}$ (F.14)

We should end up, analogously to the previous case, with three simultaneous
equations. We now follow the same method as before in Appendices C and E.3.
The solution is

$\displaystyle d\phi=$
$\displaystyle\frac{1}{\Xi}\bigg{(}D_{1}\left(M_{A2}M_{A3}+\Psi_{3}^{2}\right)+D_{2}(M_{A3}\Psi_{1}-\Psi_{2}\Psi_{3})-D_{3}(M_{A2}\Psi_{2}+\Psi_{1}\Psi_{3})\bigg{)}$
(F.15) $\displaystyle d\theta=$
$\displaystyle\frac{1}{\Xi}\bigg{(}-D_{1}(M_{A3}\Psi_{1}+\Psi_{2}\Psi_{3})-D_{2}\left(M_{A1}M_{A3}-\Psi_{2}^{2}\right)+D_{3}(M_{A1}\Psi_{3}+\Psi_{1}\Psi_{2})\bigg{)}$
$\displaystyle d\psi=$
$\displaystyle\frac{1}{\Xi}\bigg{(}\text{D1}(\Psi_{1}\Psi_{3}-M_{A2}\Psi_{2})+D_{2}(M_{A1}\Psi_{3}-\Psi_{1}\Psi_{2})+D_{3}\left(M_{A1}M_{A2}-\Psi_{1}^{2}\right)\bigg{)}$

where

$\displaystyle D_{1}=$
$\displaystyle\bar{u}dv-\bar{v}du+\bar{\tau}_{1}d\sigma_{1}-\bar{\sigma}_{1}d\tau_{1}+\bar{\sigma}_{1}d\tau_{2}-\bar{\tau}_{2}d\sigma_{1}+\bar{\sigma}_{2}d\sigma_{3}-\bar{\sigma}_{3}d\sigma_{2}$
$\displaystyle D_{2}=$
$\displaystyle\bar{u}dw-\bar{w}du+\bar{\tau}_{1}d\sigma_{2}-\bar{\sigma}_{2}d\tau_{1}+\bar{\sigma}_{1}d\sigma_{3}-\bar{\sigma}_{3}d\sigma_{1}+\bar{\sigma}_{2}d\tau_{3}-\bar{\tau}_{3}d\sigma_{2}$
$\displaystyle D_{3}=$
$\displaystyle\bar{v}dw-\bar{w}dv+\bar{\sigma}_{1}d\sigma_{2}-\bar{\sigma}_{2}d\sigma_{1}+\bar{\tau}_{2}d\sigma_{3}-\bar{\sigma}_{3}d\tau_{2}+\bar{\sigma}_{3}d\tau_{3}-\bar{\tau}_{3}d\sigma_{3}$
$\displaystyle M_{A1}=$ $\displaystyle\lvert u\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{1}\rvert^{2}+\Sigma^{2}+\lvert\tau_{1}-\tau_{2}\rvert^{2}$
$\displaystyle M_{A2}=$ $\displaystyle\lvert w\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{2}\rvert^{2}+\Sigma^{2}+\lvert\tau_{1}-\tau_{3}\rvert^{2}$
$\displaystyle M_{A3}=$ $\displaystyle\lvert w\rvert^{2}+\lvert
v\rvert^{2}+3\lvert\sigma_{3}\rvert^{2}+\Sigma^{2}+\lvert\tau_{2}-\tau_{3}\rvert^{2}$
$\displaystyle\Psi_{1}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{1}\sigma_{3}-\bar{\sigma}_{2}\sigma_{1})-\bar{w}v\big{)}$
$\displaystyle\Psi_{2}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{2}\sigma_{2}-\bar{\sigma}_{1}\sigma_{3})-\bar{u}w\big{)}$
$\displaystyle\Psi_{3}=$
$\displaystyle\text{Re}_{\mathbb{H}}\big{(}3(\bar{\tau}_{3}\sigma_{1}-\bar{\sigma}_{3}\sigma_{2})-\bar{v}u\big{)}$
$\displaystyle\Xi=$ $\displaystyle
M_{A1}M_{A2}M_{A3}+M_{A1}\Psi_{3}^{2}-M_{A2}\Psi_{2}^{2}-M_{A3}\Psi_{1}^{2}$
(F.16)

Once more we use our formula, modified for real quaternions

$ds^{2}=ds_{1}^{2}+ds_{2}^{2}=2\pi^{2}\bigg{(}\text{Tr}^{\star}\big{(}2d\Lambda^{\dagger}d\Lambda+d\Omega^{\dagger}d\Omega\big{)}+\text{Tr}^{\star}\Big{(}\big{(}a^{\dagger}da-(a^{\dagger}da)^{T}\big{)}dR\Big{)}\bigg{)}$
(F.17)

which enables us to calculate the metric as

$\displaystyle ds^{2}=$ $\displaystyle
8\pi^{2}\bigg{(}d^{2}u+d^{2}v+d^{2}w+d^{2}\tau_{1}+d^{2}\tau_{2}+d^{2}\tau^{3}+d^{2}\sigma_{1}+d^{2}\sigma_{2}+d^{2}\sigma_{3}$
(F.18)
$\displaystyle-\Big{(}D_{1}d\phi+D_{2}d\theta+D_{3}d\psi\Big{)}\bigg{)}^{2}$

##  Appendix G Numerical Methods

In this appendix we develop a new numerical algorithm for calculating particle
trajectories on a curved manifold with potential, which we use to calculate
numerical instanton scattering in the moduli space approximation in Section
5.3. We will attach a Mathematica notebook containing the methods discussed in
this appendix to a future arXiv submission of the paper.

###  G.1 Overview of the Numerical System

The problem of instanton scattering in the moduli space approximation with $d$
parameters is equivalent to Newtonian mechanics on a $d$ dimensional curved
manifold with metric $g_{\mu\nu}(x)$, potential $V(x)$ and Lagrangian
$\mathcal{L}(x,\dot{x})=g_{\mu\nu}(x)\dot{x}^{\mu}\dot{x}^{\nu}-V(x)$. The
corresponding equations of motion are

$\ddot{x}^{\mu}=-\Gamma^{\mu}_{\rho\sigma}(x)\dot{x}^{\rho}\dot{x}^{\sigma}-\partial^{\mu}V(x),$
(G.1)

where
$\Gamma^{\mu}_{\rho\sigma}=\frac{1}{2}g^{\mu\nu}\left(\partial_{\rho}g_{\nu\sigma}+\partial_{\sigma}g_{\nu\rho}-\partial_{\nu}g_{\rho\sigma}\right)$
are the Christoffel connection coefficients on the manifold. We then solve for
trajectories on the moduli space in the form $x(\tau;x_{0},\dot{x}_{0})$,
where $\tau$ is the parameter on the worldline and
$x_{0},\dot{x}_{0}\in\mathbb{R}^{d}$ are initial conditions specifying a given
trajectory. Note that the calculations in the moduli space approximation used
in this work return the line element $ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}$, so
we consider $x(\tau;x_{0},\dot{x}_{0})$ as depending implicity on $ds^{2}$ and
$V(x)$, and the metric $g_{\mu\nu}$ is extracted from $ds^{2}$ as an internal
step in the calculation.

The standard approach to this problem first extracts the metric from the line
element, then calculates the derivatives of metric and the potential as well
as the inverse metric, and uses these to give the right hand side of equation
(G.1) explicitly in terms of the coordinates on the moduli space and their
derivatives. In some simple cases the resulting differential equation can then
be solved analytically, but the cases we are interested in in this work are
too complex to solve analytically and so we treat (G.1) as a numerical system.
For a given set of numerical initial data the system can then be integrated
numerically for example using NDSolve in Mathematica. This method was used to
calculate commutative two instanton scattering in [13], both for a general
gauge embedding where the moduli space has 6 degrees of freedom, and a simpler
orthogonal gauge embedding with a 4 dimensional moduli space.

In Section 4 of this work we extend these results to noncommutative instatons,
both with general and aribtrary gauge embeddings. Switching on the non-
commutative parameter $\zeta$ or adding in more instatons both result in an
increase in the algebraic complexity of the metrics and potentials on the
instanton moduli space. As the complexity increases calculating the equations
of motion analytically from the metric and potential in Mathematica can become
inaccessible to a standard desktop computer due the large amount of memory
that the expressions require. For the two non-commutative instanton scattering
with 4 dimensional moduli space considered in Section 5.2 it is possible to
calculate the equations of motion explicitly, and so in this case we use the
standard approach to integrate equation (G.1) numerically.

For the scattering of two noncommutative instantons with 6 dimensional moduli
space considered in Section 5.3 we find that it is no longer possible to
calculate the equations of motion analytically due to the increased complexity
of the functions on the moduli space. To illustrate this, we use the function
LeafCount as a simple measure of the complexity of an algebraic expression in
Mathematica. The LeafCounts of line elements, metrics and potentials for the
cases considered in [13] and all cases considered in this work are shown in
Figure 20; the cases which require the new algorithm have significantly higher
LeafCounts. A method to overcome this problem by saving large algebraic
expressions to the hard disk is presented in [34] and was used to calculate
the scattering of two noncommutative instantons in [14]. As explained in
Section 4 we have found some difficulties with the results presented in [14],
and we were not able to integrate equation (G.1) using the method from [34]
for the updated moduli space metrics and potentials that we derive in this
paper. This motivates us to develop a new algorithm which integrates equation
(G.1) numerically without explicitly calculating the equations of motion.

Number of | Commutative | $d$ |  | LeafCount |  | Algorithm Used
---|---|---|---|---|---|---
Instantons |  |  | $V(x)$ | $ds^{2}$ | $\frac{1}{d^{2}}\sum_{\mu,\nu}g_{\mu\nu}(x)$ |
2 | Yes | 4 | 35 | 105 | 10.0 | Standard
2 | Yes | 6 | N/A |  |  | Standard
2 | No | 4 | 255 | 715 | 70.6 | Standard
2 | No | 6 | ? | $6.9\times 10^{5}$ | $7.4\times 10^{5}$ | New
Figure 20: Algebraic complexity of moduli space potentials and line elements