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\topmatter
\title A Symmetric Family of Yang-Mills Fields
\endtitle
\author Lorenzo Sadun \endauthor
\address{Department of Mathematics, University of Texas, Austin, Texas 78712}
\endaddress
\email sadun\@math.utexas.edu \endemail
\thanks {This work was partially supported by an NSF Mathematical
Sciences Postdoctoral Fellowship}
\endthanks
\subjclass{58E15, also 34B15, 53C07, 65L15, 81T13}
\endsubjclass
\abstract{We examine a family of finite energy $SO(3)$
Yang-Mills connections over $S^4$, indexed by two real parameters.
This family includes both smooth connections (when both parameters are
odd integers), and connections
with a holonomy singularity around 1 or 2 copies of $RP^2$. These singular
YM connections interpolate between the smooth solutions. Depending on the
parameters, the curvature may be self-dual, anti-self-dual, or neither.
For the (anti)self-dual connections, we compute the formal dimension of
the moduli space. For the non-self-dual connections we examine the second
variation of the Yang-Mills functional, and count the negative and zero
eigenvalues. Each component of the
non-self-dual moduli space appears to consist only of
conformal copies of a single solution.}
\endabstract
\endtopmatter
\baselineskip=15pt
\document
\heading {\bf 1. Introduction and Statement of Results} \endheading
\noindent{\it 1.1 Main Results}
Until recently, the phrase ``Yang-Mills theory in four dimensions''
essentially meant the study of smooth solutions to the
(anti)self-duality equations
$$ *F = \pm F, \tag 1.1 $$
where $F$ is the curvature of a connection $A$, usually with gauge group
$SU(2)$ or $SO(3)$, on a bundle over a Riemannian 4-manifold $M$, which
may or may not have a boundary. The moduli space of such solutions, up
to gauge invariance, gives topological information about $M$, a fact
which was exploited by Donaldson and others to make tremendous
progress in the topology of 4-manifolds (see [DK] for an overview).
In recent years the field has expanded in two directions. First, there is
the study of non-self-dual Yang-Mills connections. These are
solutions to the full Yang-Mills equations,
$$ d_A^* F = 0, \tag 1.2 $$
whose curvature is neither self-dual nor anti-self-dual. It was
generally assumed that, for gauge group $SU(2)$ or $SO(3)$, such
solutions did not exist on bundles over
the 4-sphere, until Sibner, Sibner, and
Uhlenbeck [SSU] constructed such a solution on the trivial $SU(2)$
bundle over $S^4$ in 1988.
The other, and more important, extension has been to consider
finite-energy (anti)self-dual connections with a holonomy
singularity. The first example was found by Forgacs,
Horvath, and Palla [FHP1, FHP2]. The first general results were
the regularity theorems of Sibner and Sibner [SiSi1, SiSi2]. The subject
gained attention when
Kronheimer and Mrowka [K, KM] used instantons with holonomy
to study embeddings of 2-manifolds in 4-manifolds. Since then others
have tried to advance the general theory of such instantons
[R1, R2], but there are many tricky questions that are still
not well understood.
In this paper we consider a family of solutions to
the full $SO(3)$ Yang-Mills equations (1.2)
on $X=S^4\bs \{S_+ \cup S_-\}$, where $S_+$ and $S_-$ are linked embedded
copies of $RP^2$. This family contains solutions both with and without
holonomy, and whose curvature is self-dual, anti-self-dual,
and non-self-dual. Although only a few of these solutions have been
written in closed form [BoSe], these solutions can all be well-approximated
numerically, and their asymptotic behavior as the
indexing parameters get large is well-understood [SS3].
It is hoped that these examples will help researches build an intuition
for how singular and non-self-dual YM connections behave.
The solutions are indexed by two positive real
numbers $(r,t)$, which
reflect the holonomy of the connection around $S_+$ and $S_-$, respectively.
When $r$ and $t$ are odd integers, the holonomy about $S_\pm$ is trivial,
and the solution can be smoothly extended to all of $S^4$. These
smooth solutions
were previously discussed, for gauge group $SU(2)$,
in [SS1, SS2, SS3, BoMo, Bor].
The solutions we consider are all symmetric with respect to an
$SO(3)$ action on $S^4$. By considering only symmetric connections,
we reduce the Yang-Mills
equations and the self-duality equations to a system of ODEs, which we
call the reduced YM (or self-duality) equations. Using ODE methods, we
then prove:
\proclaim {Theorem 1.1} {For each pair of non-negative real numbers $(r,t)$
there exists a Yang-Mills connection on the trivial bundle over $X$ with
the following properties:
\item {i.} The holonomy around $S_+$ is (conjugate to) $\exp(i \pi (r+1))$.
The holonomy around $S_-$ is (conjugate to) $\exp(i \pi (t+1))$.
\item {ii.} The integral over $X$ of the Chern-Weil form
(a.k.a. the fractional Chern number) is $(r^2 - t^2)/8$.
\item {iii.} If $r$ and $t$ are both greater than 1, or both strictly between 0
and 1, then the solution has non-self-dual curvature.
\item {iv.} If $r \ge 1 \ge t$, or if $t=0$,
then the solution has anti-self-dual curvature. \hfil \break
If $t \ge 1 \ge r$, or if $r=0$,
then the solution has self-dual curvature.}
\endproclaim
Some of these results are not new. The dimensional reduction from 4 to 1
dimensions was developed by Urakawa [U], and was applied by Bor and
Montgomery [BoMo, Bor] to this particular symmetry. In [SS1, SS2] this
method was used to prove the existence of non-self-dual YM
connections with $r>1, t>1$. The case
$r=1, t=$ odd has been studied by Bor and Segert [BoSe] using
an equivariant ADHM construction [ADHM].
By the Peter-Weyl theorem, deformations of the solutions can be
decomposed into irreducible representations of the symmetry group $SO(3)$.
The linearized Yang-Mills equations for these deformations
reduce to a countable collection
of ODE systems, one for each representation. By counting the solutions
to these ODEs, with appropriate boundary conditions, we deduce
\proclaim {Theorem 1.2} {If $r\ge 3$ and $t=1$, then the number of
linearly independent regular solutions
to the linearized anti-self-duality equations (i.e. the formal dimension
of the moduli space) generically equals
$$ \left \{ r \right \}^2 -4,
\tag 1.3 $$
where $\{ x \}$ denotes the greatest {\it odd} integer less than or equal to $x$. If $r \ge 1 \ge t$ and either $r < 3$ or $t < 1$, then the dimension
generically equals zero. }
\endproclaim
For $t=1$, this result concurs with the general results of Kronheimer and
Mrowka [KM], who studied anti-self-dual connections with orientable
singular sets. As in [KM], the discontinuities in the dimension of
the moduli space all occur when the ($SO(3)$)
holonomy is trivial. In our case we have the further result that
the dimension is continuous from above.
Direct comparison with [KM] is made difficult by the fact that our
singular set $S_+$ is non-orientable. The $C^2$ bundle associated
to our principal bundle
does not split into a sum of line bundles near $S_+$. It splits
locally, but parallel transport along a generator of $\pi_1(S_+)$
interchanges the two factors. This makes the ``monopole number'' $l$
ill defined. (One could of course try lifting to a double cover
of $S_+$, in which case the bundle does split, but then the
monopole number is always zero). The holonomy parameter $\alpha$
is easier to identify.
If we let $\rho$ denote the fractional part of $(r+1)/4$, then
$\alpha$ is the smaller of $\rho$ and $1-\rho$.
The mechanism by which the dimension of the moduli space jumps
is extremely simple. Let $d$ be
the distance from a point to the singular set $S_+$.
The natural boundary conditions at $S_+$ are that
certain components of the connection remain bounded as $d \to 0$ if the
holonomy around $S_+$ is trivial, and go to zero as
$d \to 0$ if the holonomy is nontrivial.
Solving the linearized anti-self-duality equations in a particular
representation of $SO(3)$
gives solutions that behave like $d^{(r-M)/2}$, where $M$ is an odd integer
that depends on the representation. If $r>M$ the solution
goes to zero as $d \to 0$, and so satisfies the boundary conditions.
If $r=M$ then the solution approaches a finite limit at $d=0$, and so is
still admissible. However, if $rFrom $F$ we compute the Yang-Mills functional, with the result
$$ \split {\Cal Y \Cal M}(A) = S(a)=
{\pi^2} & \int_0^{\pi/3} d\theta \Big [
(a_1')^2 G_1 +
(a_2')^2 G_2 +
(a_3')^2 G_3 \\ & +
{(a_1+a_2a_3)^2 \over G_1} +
{(a_2+a_1a_3)^2 \over G_2} +
{(a_3+a_1a_2)^2 \over G_3} \Big ]. \endsplit \tag 2.21
$$
For equivariant connections, the Yang-Mills equations $d_A^*F=0$
are equivalent to the Euler-Lagrange equations for the one-dimensional
functional $S(a)$ [SS2].
Since $G_3(\theta)$ is bounded away from zero near $\theta=0$,
a finite-action reduced
connection must have $a_3'$ integrable near $\theta=0$, so the
boundary value $r\equiv a_3(0)$ is well-defined.
Similarly, $t=a_2(\pi/3)$ is well-defined. Also, since $G_1$ and $G_2$ have
zeroes at $\theta=0$, finite-action reduced connections must have
$$
\lim_{\theta \to 0}[a_1(\theta)+ a_2(\theta) a_3(\theta)] =
\lim_{\theta \to 0}[a_2(\theta)+ a_1(\theta) a_3(\theta)]
= 0. \tag 2.22
$$
If $r \ne \pm 1$, these conditions imply that both $a_1(0)$
and $a_2(0)$ exist and equal zero. Similarly, if $t \ne \pm 1$ then
$a_1(\pi/3)=a_3(\pi/3)=0$. We call a finite-action reduced connection
with $a_3(0)=r$ and $a_2(\pi/3)=t$ a {\it reduced (r,t) connection}.
The boundary values $r$ and $t$ are related to the holonomy of the
connection $A$ around $S_+$ and $S_-$, respectively.
For fixed $\theta \in I$, we consider the path
$(\theta, \exp(\tau K_3))$ on $Y$, where
$\tau$ runs from $0$ to $\pi$.
This path projects to a closed loop on $X$, from $Q_\theta$ to itself.
The tangent vector $d/d\tau$ along the path is $l_3$, so parallel
transport is given by integrating $a_3$ along the path. The
point $(\theta,1,1)$ in $P_Y$ is transported to $(\theta, \exp(\pi K_3),
\exp(\pi a_3(\theta)K_3))$. Under the identification (2.10), this is
equivalent to $(\theta,1,\exp( (1 + a_3(\theta)) \pi K_3))$. Taking the
limit as $\theta \to 0$, we see that the holonomy around $S_+$ is
$\exp((r+1)\pi K_3)$, which is trivial if and only if $r$ is an odd
integer. Similarly, the holonomy around $S_-$ is $\exp((t+1)\pi K_2)$,
which is trivial if and only if $t$ is an odd
integer.
Finally, we compute $C_2$, the integral of the second Chern-Weil form:
$$
C_2 \equiv \int_X { Tr ( F \wedge F) \over 8 \pi^2}.
\tag 2.23
$$
%
Since $X$ is an open manifold, $C_2$ need not be an integer, depending on
the holonomy around the two-dimensional singular sets [FHP1, FHP2]. From equation (2.19) we immediately get
$$ \eqalign{
Tr(F \wedge F) & = (a_1' (a_1 + a_2 a_3) + cyclic) \;
d\theta \wedge \beta^1 \wedge \beta^2 \wedge \beta^3 \cr
& = {1 \over 2 } \ d (a_1^2 + a_2^2 + a_3^2 + 2a_1 a_2 a_3)
\wedge \beta^1 \wedge \beta^2 \wedge \beta^3. }\tag 2.24
$$
Integrating first over the symmetry group and then over $I$, we get
$$ C_2 = \int_I {-1 \over 8 } \ d (a_1^2 + a_2^2 + a_3^2 + 2a_1 a_2 a_3)
= CS(0)-CS(\pi/3),\tag 2.25
$$
where
$$
CS(\theta)= (a_1(\theta)^2 + a_2(\theta)^2 + a_3(\theta)^2 +
2a_1(\theta) a_2(\theta) a_3(\theta))/8 \tag 2.26
$$
is the {\it reduced Chern-Simons function}.
If $r \ne \pm 1$, then $a_1(0)=a_2(0)=0$, and $CS(0)= r^2/8$. If $r=\pm 1$,
then $CS(0)= r^2/8 + \lim_{\theta \to 0}
(a_1(\theta) + r a_2(\theta))^2/8$. This second term is zero
(eq. 2.22), so we still have $CS(0)=r^2/8$.
Similarly, $CS(\pi/3)=t^2/8$, and so $C_2=(r^2-t^2)/8$.
The reduced self-duality equations (2.20), the Yang-Mills functional (2.21),
and the reduced Chern-Simons functional (2.26) are left unchanged if
we flip the signs of two of the three functions $(a_1, a_2, a_3)$. This is
a consequence of gauge invariance, since global gauge transformations by
$k_1$, $k_2$, or $k_3$ flip the signs of
$a_2$ and $a_3$, $a_1$ and $a_3$, or $a_1$ and $a_2$, respectively.
Without loss of generality, we can therefore restrict our attention to
non-negative $r$ and $t$.
\noindent{\it 2.4 Classifying deformations}
Once we have established the existence of equivariant Yang-Mills
connections, we will wish to consider infinitesimal deformations
of these solutions.
These deformations need not be equivariant, and can take the general form
$$
\delta A = \sum_{i,j} \alpha_{ij}(\theta,g) \beta^i \otimes l_j +
\sum_{i} \gamma_{i}(\theta,g) d \theta \otimes l_i.\tag 2.27
$$
Since the metric on $X$ and original solution $A$ are invariant
under the action of $G$, the solutions of the linearized self-duality
equations
$$ *\delta F = \pm \delta F \tag 2.28 $$
and the eigenspaces of the Yang-Mills Hessian can be decomposed into
irreducible representations of $SO(3)$, and in particular can be chosen
to be eigenfunctions of the Laplacian.
An orthonormal basis for $L^2(SO(3))$ is given by the functions
$$
\Psi_{l,m_r,m_l}(g), \qquad l=0,1,2,\ldots, \qquad m_r=-l,1-l,\ldots,l
\qquad m_l=-l,1-l,\ldots,l, \tag 2.29
$$
where $\Psi_{l,m_r,m_l}$ is an eigenfunction of the Laplacian with
eigenvalue $l(l+1)$, of $-i r_3$ with eigenvalue $m_r$, and of $-i l_3$
with eigenvalue $m_l$. Since right-translations and left-translations
commute, $l$ and $m_l$ are preserved by left-translations, while $l$ and
$m_r$ are preserved by right-translations.
As a result, we may fix $l$ and $m_l$ while looking for
eigenvalues of the Hessian and solutions of the linearized self-duality
equations. Moreover, for fixed $l$ either a solution exists for all
$m_l$ or for none. We can therefore do our calculations for a
single value of $m_l$, and then multiply the multiplicity by $2l+1$ to
account for the other values. We choose a useful basis as follows:
\proclaim {Proposition 2.1} There exist real functions $\Psi^\pm_{l,m}$,
$m = 1,2,\ldots,l$ and $\Psi_{l,0}$, which, together with their
left translates, span the $l$-th eigenspace of the Laplacian on $SO(3)$,
and have the following properties:
\item 1 If $m$ is odd, then $\Psi^+_{l,m}$ is an x-type
function and $\Psi^-_{l,m}$ is y-type.
\item 2 If $m$ is even, then $\Psi^+_{l,m}$ is a z-type
function and $\Psi^-_{l,m}$ is invariant under $\Gamma$.
\item 3 $\Psi_{l,0}$ is z-type if $l$ is odd and invariant if $l$ is even.
\item 4
$$
l_3 \Psi^\pm_{l,m} = \pm m \Psi^\mp_{l,m} . \tag 2.30
$$
\endproclaim
\noindent {\it Proof: \quad} We start with the functions $\Psi_{l,0,m}$.
Their span, which we
call $W$, is a $2l+1$-dimensional
representation of the right-action of $SO(3)$. Together with their
left-translates, the $\Psi_{l,0,m}$'s span the $l$-th eigenspace
of the Laplacian.
Since $\Psi_{l,0,m}$ is an eigenfunction of $l_3$ with eigenvalue $im$,
we have $\Psi_{l,0,m}(g \exp(\pi K_3))= (-1)^m \Psi_{l,0,m}(g)$. Thus
for $m$ odd
$\Psi_{l,0,m}$ must be a linear combination of x-type and y-type functions,
and for $m$ even $\Psi_{l,0,m}$ must be a linear combination of z-type and
invariant functions.
Because of its own transformation properties under $\Gamma$, $l_3$ maps
x-type and y-type functions to each other (e.g. if
$\psi$ is x-type then $l_3\psi$ is y-type) and maps z-type and invariant
functions to each other. Since $\Psi_{l,m}$ is an eigenfunction
of $l_3$, $l_3$ must map the x-type (z-type) and y-type (invariant)
parts of $\Psi^\pm_{l,0,m}$ to multiples of each other. This also
shows that none of these parts are zero. The projection $P_+$ onto
the x-type and z-type parts of a function can be written in terms of
right-translations,
$$ (P_+ \psi)(g) = (\psi(g) - \psi (g k_2))/2, \tag 2.31 $$
so for any $\psi \in W$, $P_+ \psi \in W$.
For $m>0$, let $\Psi^+_{l,m}= P_+ \Psi_{l,0,m}$,
rescaled to have unit norm,
and let $\Psi^-_{l,m} = l_3 \Psi^+_{l,m}/m$. Since $l_3^2$ commutes with
$P_+$, $l_3^2 \Psi^+_{l,m} = -m^2 \Psi^+_{l,m}$, and
equation (2.30) follows. It also follows that $\Psi^-_{l,m}$ has unit norm.
To complete our basis we take $\Psi_{l,0}= \Psi_{l,0,m}$.
Any two of these functions either correspond to different eigenvalues
of $l_3^2$ or to different tranformation properties under $\Gamma$, and
so must be orthogonal. Our collection, which contains $2l+1$ elements,
is therefore an orthonormal basis for $W$.
$\Psi_{l,0}$ must be either z-type or invariant. It cannot be a combination
of both, or else we could decompose it into $P^\pm_{l,0}$
and end up with $2l+2$ linearly
independent functions in $W$.
To see which type $\Psi_{l,0}$ is, we use a simple counting argument.
The dimension of the x-type subspace of $W$ equals the dimension of
the y-type subspace, since $l_3$ maps x-type and z-type functions
to each other. This is also the dimension of the z-type subspace,
as $l_2$ maps x-type and z-type functions to each other. Of the
basis elements $\Psi^\pm_{l,m}$, there are
$[l/2]$ z-type functions ($m$ even) and $[(l+1)/2]$ x-type functions
($m$ odd), where $[x]$ denotes the integer part of $x$.
To keep the total number of x-type and z-type basis elements
equal, $\Psi_{l,0}$ must be z-type if $l$ is odd, and invariant if $l$ is even.
\qed
\noindent{\it 2.5 $SO(3)$ vs. $SU(2)$}
We have done our construction for symmetry group $G=SO(3)$ and gauge
group $H=SO(3)$. However, the construction was first carried out for
$SU(2)$, and is nearly identical.
In the $SU(2)$ case, we take symmetry and gauge groups $\tilde G=
\tilde H = SU(2)$, and think of $SU(2)$ as the set of unit quaternions.
Since $\tilde G$ is the double cover of $G$, the action of $G$ on $S^4$
lifts to an action of $\tilde G$. The isotropy group
of $Q_\theta$ is now the 8 element group
$\tilde \Gamma = \{ \pm 1, \pm i, \pm j, \pm k\}$. To get a principal
bundle over $X$ we define the bundle $P_{\tilde Y} = I \times \tilde G
\times \tilde H$ and mod out by the equivalence relation (2.10), where
now we think of $g,h,\gamma$ as arbitrary elements of $\tilde G$,
$\tilde H$, and $\tilde \Gamma$ respectively.
The dimensional reduction procedure is identical to that of section 2.3,
with equations (2.17) through (2.25) remaining true. The only difference
is in the interpretations of the two numbers $(r,t)$. For $SU(2)$, the
holonomy around $S_+$ is $\exp(\pi (r+1) k/2)$, rather than
$\exp(\pi (r+1) K_3)$, and the holonomy around $S_-$ is
$\exp(\pi (t+1) j/2)$.
Note that the holonomy for $r$ is {\it not} conjugate to that for
$-r$, so an $(r,t)$ $SU(2)$ connection cannot be gauge-equivalent
to a $(-r,t)$
connection, an $(r,-t)$ connection, or a $(-r,-t)$ connection.
Although the (4-dimensional) connections are not gauge-equivalent, the
one-dimensional equations are still invariant under a pair of sign
flips.
If $a=(a_1,a_2,a_3)$ is a reduced connection,
then the four reduced connections
$(a_1,a_2,a_3)$, $(a_1,-a_2,-a_3)$, $(-a_1,a_2,-a_3)$ and $(-a_1,-a_2,a_3)$
all have the same curvature (up to sign), have the same Chern number,
are all self-dual if any one is, and are all Yang-Mills if any one is.
As $SO(3)$ connections they are all gauge-equivalent, but as $SU(2)$
connections they are all distinct.
Finally we consider deformations of $SU(2)$ connections.
Since $-1 \in \tilde \Gamma$ acts
trivially on $l_i$ and $\beta^j$, our most general deformation is
composed of even functions on $\tilde G$. However, even functions on
$\tilde G$ are in 1-1 correspondence with arbitrary functions on
$\tilde G / Z_2 = G$, so the classification of deformations of $SU(2)$ connections is identical to that of $SO(3)$ connections.
\heading {\bf 3. Existence and classification of Yang-Mills Solutions}
\endheading
In this section we prove the existence of a family of equivariant
Yang-Mills solutions, parametrized by $r$ and $t$. Specifically,
\proclaim {Theorem 3.1} {For each pair of non-negative real numbers $(r,t)$
there exists a Yang-Mills $(r,t)$-connection on $P_X$. Furthermore:
\item {i.} If $r$ and $t$ are both greater than 1, or both strictly between 0
and 1, then the solution has non-self-dual curvature.
\item {ii.} If $r \ge 1 \ge t$, or if $t=0$,
then the solution has anti-self-dual curvature.
If $t \ge 1 \ge r$, or if $r=0$,
then the solution has self-dual curvature.}
\endproclaim
This, together with the general properties of $(r,t)$ connections shown
in section 2, gives theorem 1.1. Two theorems were proven in [SS2] which,
taken together, establish the case $r>1,t>1$.
\proclaim {Theorem 3.2} {{\rm [SS2]} For each pair of non-negative
real numbers $(r,t)$
with $r \ne 1$, $t \ne 1$, there exists a Yang-Mills
$(r,t)$-connection on $P_X$. }
\endproclaim
\proclaim {Theorem 3.3} {{\rm [SS2]} There do not exist any
finite-action self-dual
$(r,t)$-connections with $r>1$. There do not exist any
finite-action anti-self-dual
$(r,t)$-connections with $t>1$.}
\endproclaim
To complete part (i) of theorem 3.1, we must prove an analog of theorem 3.3
for $0 \epsilon$ on $N$, the third term
is dominated by the second on some smaller neighborhood $M$ of zero.
Thus $T$ is non-increasing on $M$. However, $T(0)=0$,
so $T$ must be exactly zero on $M$, so $a_1=a_2=0$ on $M$.
If $a_1=a_2=0$ at any point in $(0, \pi/3)$, then they are zero on all of
$(0,\pi/3)$, as the equations (3.1) are linear in $a_{1,2}$. So $t=
\lim_{\theta \to \pi/3}a_2(\theta)=0$. \qed
There {\it is} an anti-self-dual $(r,0)$ solution for any $r$.
It is
$$a_1=a_2=0, \qquad a_3(\theta) = r \exp\left(- \int_0^\theta
{dy \over G_3(y)} \right ). \tag 3.4
$$
The integral of $1/G_3$ diverges at $\theta=\pi/3$, so $a_3(\pi/3)=0$, as it
should. There is also a self-dual $(0,t)$ solution for any $t$, namely
$$a_1=a_3=0, \qquad a_2(\theta) = t \exp\left( -\int^{\pi/3}_\theta
{dy \over G_2(y)} \right ). \tag 3.5
$$
\noindent{\it 3.2. Anti-self-dual Yang-Mills connections: $r \ge 1 \ge t$}
In this section we prove
\proclaim {Theorem 3.5} {For each pair of real numbers $(r,t)$
with $r \ge 1 \ge t \ge 0$, there exists a
finite-energy solution to the reduced ASD equations with
$a_3(0)=r$ and $a_2(\pithree)=t$. For each $t \ge 1 \ge r \ge 0$ there
exists a finite-energy solution to the reduced SD equations.}
\endproclaim
We prove only the first statment. The proof of the
second is completely analogous.
We begin with local analysis near $\theta=0$.
\proclaim {Lemma 3.6} Suppose $r \ge 1$. Then for any constant $c$
there exists a solution to the reduced ASD equations on a neighborhood
of $\theta=0$, of the form
$$
\eqalign{ a_1(\theta) = & -c \theta^{(r-1)/2} + O(\theta^{(r+1)/2}), \cr
a_2(\theta) = & \phantom{-} c \theta^{(r-1)/2} + O(\theta^{(r+1)/2}), \cr
a_3(\theta) = & \phantom{-}r + O(\theta^2).} \tag 3.6
$$
\endproclaim
\noindent{\it Proof: } We solve the ASD equations by iteration.
Let $\phi_3(\theta)=\exp(- \int_0^\theta {dy \over G_3(y)})$
and let
$a_{1,2}=\phi^1_{1,2}(\theta)$, and $a_{1,2}=\phi^2_{1,2}(\theta)$ be
solutions to the linear ODE system
$$
a_1' = - {(a_1+r a_2) \over G_1}, \qquad
a_2' = - {(a_2+r a_1) \over G_2}. \tag 3.7 $$
We can choose our normalizations such that
$$
\eqalign{
\phi_1^1(\theta) = & -\theta^{(r-1)/2} + O(\theta^{(r+1)/2}), \cr
\phi_2^1(\theta) = & \theta^{(r-1)/2} + O(\theta^{(r+1)/2}), \cr
\phi_1^2(\theta) = & \theta^{(-r-1)/2} + O(\theta^{(-r+1)/2}), \cr
\phi_2^2(\theta) = & \theta^{(-r-1)/2} + O(\theta^{(-r+1)/2}).} \tag 3.8
$$
Let $a_{1,2}^{(0)}=c \phi^1_{1,2}$, $a_3^{(0)} = r\phi_3$, and let
$a_i^{(k+1)}$ be solutions, of the form (3.6), to the differential equations
$$ \eqalign{
a_1^{(k+1)}{}' = & - {(a_1^{(k+1)}+ r a_2^{(k+1)}) / G_1}
+ [r-a_3^{(k)}] a_2^{(k)}/G_1, \cr
a_2^{(k+1)}{}' = & - {(a_2^{(k+1)}+r a_1^{(k+1)}) / G_2}
+ [r-a_3^{(k)}] a_1^{(k)}/G_2, \cr
a_3^{(k+1)}{}' = & - {(a_3^{(k+1)}+a_1^{(k)}a_2^{(k)}) / G_3}.}
\tag 3.9 $$
These solutions can be written explicitly, using the method
of variation of parameters:
$$ \eqalign{
a_1^{(k+1)}(\theta) = & (c+ h_1^{(k)}(\theta)) \phi^1_1(\theta) +
h_2^{(k)}(\theta)\phi^2_1(\theta) \cr
a_2^{(k+1)}(\theta) = & (c+ h_1^{(k)}(\theta)) \phi^1_2(\theta) +
h_2^{(k)}(\theta)\phi^2_2(\theta) \cr
a_3^{(k+1)}(\theta) = & (r + h_3^{(k)}(\theta)) \phi_3(\theta),}
\tag 3.10 $$
where
$$ \eqalign{
h_1^{(k)}(\theta)= & \int_0^\theta W(y)
(r-a_3^{(k)}(y)) \left [{a_2^{(k)}(y)\phi_2^2(y)\over G_1(y)}-
{a_1^{(k)}(y)\phi_1^2(y) \over G_2(y)} \right ] dy, \cr
h_2^{(k)}(\theta)= & \int_0^\theta W(y)
(r-a_3^{(k)}(y)) \left [{-a_2^{(k)}(y)\phi_2^1(y)\over G_1(y)} +
{ a_1^{(k)}(y)\phi_1^1(y) \over G_2(y)}\right ] dy, \cr
h_3^{(k)}(\theta)= & -\int_0^\theta { a_1^{(k)}(y)a_2^{(k)}(y))
\over \phi_3(y) G_3(y)} dy,}
\tag 3.11 $$
and
$W(y)=1/(\phi_1^1(y)\phi_2^2(y)-\phi_1^2(y)\phi_2^1(y))=-\theta/2+O(\theta^2)$.
For sufficiently small $\theta$,
$a^{(k+1)}-a^{(k)}$ is small compared to $a^{(k)}-a^{(k-1)}$, so the
iteration converges. Letting $a_i=\lim_{k \to \infty}
a_i^{(k)}$ we get our desired solution, of form (3.6), to the ASD
equations (3.1). \qed
Note that, if $c$ is positive, then for $\theta$ sufficiently small
$a_3(\theta)$ and $a_2(\theta)$ are positive and $a_1(\theta)$ is
negative. We next prove that these signs persist for all $\theta$.
\proclaim {Lemma 3.7} Suppose $a(\theta)$ is a solution of the
reduced ASD equations, and that at some point $\theta_0 \in (0,\pi/3)$,
$a_{2,3}(\theta_0) > 0 > a_1(\theta_0)$. Then for all $\theta \in
(\theta_0,\pi/3)$, $a_{2,3}(\theta) > 0 > a_1(\theta)$.
\endproclaim
\noindent{\it Proof: } Suppose the conclusion is false. Then there is a
smallest value of $\theta$, call it $\theta_1$, where one (or more) of
the $a_i$'s is equal to zero. If two or three of the functions are zero at
$\theta_1$, then the ASD equations imply that they are zero for all
$\theta$, and in particular at $\theta_0$, contradicting the assumption.
So all we need rule out is the possiblity that exactly one of the $a_i$'s
is zero at $\theta_1$.
Suppose $a_3(\theta_1)=0$ (the other two cases are similar). Since
$a_3(\theta)$ is positive for all $\theta \in (\theta_0, \theta_1)$,
$a_3'(\theta_1)$ cannot be positive. However, $a_3'(\theta_1) =
-a_1(\theta_1)a_2(\theta_1)/G_3(\theta_1)$ {\it is} positive.
Contradiction. \qed
Combining lemmas 3.6 and 3.7, we see that there is a 1-parameter
family of solutions to the ASD equations, with definite sign properties.
However, it is
not immediately clear whether a given solution is defined on all of
$[0,\pi/3]$ or just on a neighborhood of zero, and whether it has
finite or infinite energy. The finite vs. infinite energy question
is resolved by the following lemmas.
\proclaim {Lemma 3.8}
Suppose $a(\theta)$ is a solution of the
reduced ASD equations with $r>0$, and $CS(\theta)$ is the reduced
Chern-Simons functional. The action $S(a)$, restricted to any
subinterval $(\theta_0,\theta_1) \subset (0,\pi/3)$ is
$8\pi^2(CS(\theta_1) - CS(\theta_0))$. Furthermore, either
\item {1.} $a$ is a finite-action reduced connection
defined on all of $[0,\pi/3]$,
$CS(\theta)$ is positive for all $\theta \in (0,\pi/3)$,
and $S(a) \le \pi^2 r^2$, or
\item {2.} There is a point $\theta_0 \in (0,\pi/3)$ such that
$CS(\theta_0)=0$, and $a$ has infinite action. ($a$ may or may
not be defined on all of $[0,\pi/3]$).
\endproclaim
\noindent{\it Proof: } If the curvature $F_A$ is anti-self-dual, then
the Chern-Weil form equals $1/8 \pi^2$ times the action density.
Integrating the Chern-Weil form first over the symmetry group
and then over $(\theta_0,\theta_1)$ we get $CS(\theta_1)-CS(\theta_0)$.
Integrating the action density we get the action.
\noindent 1) Now suppose $a$ is a finite-action ASD connection.
We first show that
$a$ is defined on all of $[0,\pi/3)$. We already know that $a$ is defined
on some neighborhood $[0,\delta]$. Since $G_3$ is bounded below on
$[\delta,\pi/6]$, finite action implies that $a_3$ is of finite variation
on $[\delta,\pi/6]$ (see [SS2] for the precise estimates), and in particular is
bounded. Given bounded $a_3$, the equations for $a_{1,2}$ are linear with
bounded coefficients, and so $a_{1,2}$ cannot blow up on $[\delta,\pi/6]$.
Similarly, $G_2$ is bounded below on $[\pi/6,\pi/3]$, so $a_2$ is bounded,
so the equations for $a_{1,3}$ cannot give blowup at any point prior to
$\pi/3$. Thus $a$ is defined on all of $[0,\pithree)$. The finite-action
boundary conditions then give finite limits {\it at} $\pithree$.
The finite-action
boundary conditions at $\pi/3$ also imply that $CS(\pi/3)=t^2 \ge 0$.
For any point $\theta_0 \in (0,\pi/3)$, $CS(\theta_0)=t^2 +
\hbox{(action on $(\theta,\pi/3)$)}/8\pi^2 >0$.
$S(a) = 8\pi^2(CS(0)-CS(\pithree)) = \pi^2(r^2-t^2) \le \pi^2 r^2$.
\noindent 2) If the action is unbounded, there is a point $\theta_0$ where
the action on $[0,\theta_0]$ equals $\pi^2 r^2$. Since $CS(0)=r^2/8$,
$CS(\theta_0)$ must equal zero. \qed
Note that $CS(\theta)$ is a non-increasing function of $\theta$.
If we can show that $CS(\theta)$ is positive in a neighborhood of
$\pi/3$, then it must be positive everywhere and $a$ has finite action.
\proclaim {Lemma 3.9} If $00$.
\endproclaim
\noindent{\it Proof: } $$ \eqalign{
CS(\theta) = & (a_2^2 + a_1^2 + a_3^2 + 2 a_1 a_2 a_3)/8 \cr
= & (a_2^2 + a_2(a_1 + a_3)^2 + (1-a_2)(a_1^2 + a_3^2))/8} \tag 3.12
$$
As long as $0 1/(4y)$. As a result,
$$ \eqalign{
\big (a_3(\theta_1) -a_3(\theta_2)\big )^2 \le &
{\delta \over \theta_2 -\theta_1} \big (a_3(\theta_1) -a_3(\theta_2)\big )^2
\le \delta \int_{\theta_1}^{\theta_2} (a_3'(y))^2 dy \cr
\le & 4 \delta^2 \int_{\theta_1}^{\theta_2} G_3(y) (a_3'(y))^2 dy
\le 4 M^2 \delta^2.} \tag 3.13
$$
The bound on $|r-a_3(\theta_2)|$ then follows from taking $\theta_1=0$ and
$\delta=\theta_2$.
The proof of statement 2 is similar, as $G_2(y) = G_3(\pi/3-y)$.
\proclaim {Proposition 3.11} Given $r>0$ and $\epsilon \in (0,1)$,
there exists a $\delta$ such that
\item 1 Every ASD reduced connection $a$ defined on $[0,\pi/3-\delta]$
with $a_3(0)=r$ and $0 < a_2(\pi/3-\delta) < 1-\epsilon$ can be
extended to be a finite-action ASD connection on all of $[0,\pi/3]$ with $|a_2(\pi/3)-a_2(\pi/3-\delta)|
< \epsilon$.
\item 2 No ASD reduced connection defined on $[0,\pi/3\!-\!\delta]$
with $a_3(0)=r$ and with $a_2(\pi/3\!-\!\delta) > 1+\epsilon$
can be extended to a finite-action ASD solution on $[0, \pi/3]$.
\endproclaim
\noindent{\it Proof: } Let $\delta = \min(1/10, \epsilon/(2r))$.
\noindent 1) Suppose $00$, so
$\theta_2 > \theta_1$. Also by lemma 3.9, $a_2(\theta_2) > 1$, so
$|a_2(\theta_2)-a_2(\theta_1)| > \epsilon$.
However, the energy between $\theta_1$ and $\theta_2$ cannot
be greater than the total energy between $0$ and
$\theta_2$, which by lemma 3.8 is $\pi^2 r^2$.
So by lemma 3.10,
$|a_2(\theta_2)-a_2(\theta_1)| \le \epsilon$, which is a contradiction.
The bound on $a_2(\pi/3)$ also follows from lemma 3.10.
\noindent 2) Now suppose $a_2(\pi/3-\delta) > 1+\epsilon$.
If $a$ can be extended to a finite-action solution on $[0,\pi/3]$, then
the energy between $\pi/3-\delta$ and $\pi/3$ must be less than $\pi^2 r^2$
(by lemma 3.8). By lemma 3.10,
$t \ge a_2(\pi/3-\delta) - \epsilon >1$.
But by theorem 3.3 there are no finite-action ASD
connections with $t>1$. \qed
Proposition 3.11 gives a fairly coherent description of the family of ASD
solutions given by lemma 3.6. For $c=0$ we get the explicit solution
of equation (3.4). As $c$ increases, $a_2(\theta)$ increases for any fixed
$\theta$, and in particular $t$ increases. As long as $t$ remains less
than 1, the solution has finite energy.
Eventually $c$ hits a critical value, which we call $c_1$, for
which $t=1$. For $c$ slightly greater than $c_1$ we get infinite energy
solutions on $(0,\pi/3)$. The limit $t=a_2(\pi/3)$ still exists (and is
greater than 1), but $a_1$ and $a_3$ diverge as $\theta$ approaches $\pi/3$.
Eventually $c$ reaches another critical value, which we call $c_2$, after
which the solution blows up prior to $\pi/3$. To complete the proof of
theorem 3.5 we must show that $c_1$ is neither zero nor infinite, and that,
for $c \in [0,c_1]$, $t$ varies continuously with $c$.
The critical value $c_2$ is not needed for the proof of theorem 3.5.
We merely remark, without proof, that it corresponds to $t=2$.
\proclaim {Lemma 3.12} Let $r\ge 1$ be fixed and let $a^c$
denote the ASD solution of lemma 3.6 with constant $c$.
Let $c_1 = \inf\{c | a^c \hbox{has infinite action} \}$.
Then
\item {1.} If $c_1 \in (0,\infty)$, then $a^{c_1}$ is a finite
action connection with $t=1$.
\item {2.} The boundary value $t$ is a continuous function
of $c$ for $c \in [0,c_1]$.
\item {3.} $c_1 \in (0,\infty)$.
\endproclaim
\noindent{\it Proof: } 1) For small $\theta$, it is clear from equation (3.6)
that $a^c(\theta)$ depends continuously on $c$. Since solutions to ODE's
depend continuously on their initial conditions, $a^c(\theta)$ must
depend continuously on $c$ for any $\theta \in (0, \pi/3)$ such that $a^c(\theta)$ is defined.
Let $S_\delta(a)$ be the energy of the reduced
connection $a$ between $\theta=\delta$ and $\theta=\pithree-\delta$. For ASD
connections, the derivative and non-derivative terms are equal, so
$$
S_\delta(a) = 2 \pi^2 \int_\delta^{\pithree-\delta}
\Big [
{(a_1+a_2a_3)^2 \over G_1} +
{(a_2+a_1a_3)^2 \over G_2} +
{(a_3+a_1a_2)^2 \over G_3} \Big ]. \tag 3.14
$$
As $c \to c_1$ from below,
$a^c$ approaches $a^{c_1}$
pointwise. Furthermore, all of the $a^c$'s with $c 1 - 2\epsilon$. But then for all
$c$ sufficiently close to $c_1$, $a^c_2(\pithree-\delta) < 1-\epsilon$, so
by proposition 3.11 all such $a^c$'s have finite action.
This contradicts the definition of $c_1$.
\noindent 2) By lemma 3.10, the family $a_2^c$ is uniformly
equicontinuous near
$\theta = \pithree$ for $c \le c_1$. This, combined with the continuous
dependence of $a^c_2(\theta)$ on $c$ for fixed $\theta<\pithree$, gives
continuous dependence of $t=a^c_2(\pithree)$ on $c$.
\noindent 3) To show that $c_1>0$ we pick $\epsilon =1/2$ and
let $\delta=\min(1/10,\epsilon/(2r))$. Since
$a^0_2(\pithree-\delta)=0$, for all sufficiently small
$c$ we have $a^c_2(\pithree-\delta)<1 - \epsilon$, so $a^c$ has finite action.
Thus $c_1 > 0$.
To show that $c_1<\infty$, we show that for sufficiently large
$c$, $a^c$ has action greater than $\pi^2 r^2$, and so by lemma 3.8 has
infinite action. We do this by choosing a $c$ so big that
either there is energy greater than
$\pi^2 r^2$ in a small interval $(0,\delta)$, or
$a_2(\delta)$ is so big that it takes energy greater than
$\pi^2 r^2$ on $(\delta, \pithree)$ to bring $a_2$ back down.
First we bound the variation of $a_3$. By lemma 3.10,
if the action between 0 and $\theta$ is bounded by $\pi^2 r^2$, then
$$|a_3(\theta) - r| \le 2 r \theta. \tag 3.16 $$
Next we look at the growth of $|a_1-a_2|$. By equation (3.2),
$$ \eqalign{ d\log|a_1 \! - \! a_2|/d\theta =&
(a_3 \! - \! 1)(G_1^{-1} \! + \! G_2^{-1})/2
- (a_3 \! + \! 1) (G_1^{-1} \! - \! G_2^{-1})
(a_1 \! + \! a_2)/[2(a_1 \! - \! a_2)] \cr
\ge & (a_3 \! - \! 1)(G_1^{-1} \! + \! G_2^{-1})/2 -
(a_3 \! + \! 1) |G_1^{-1} \! - \! G_2^{-1}|/2,
} \tag 3.17
$$
where we have used the fact that $a_1$ and $a_2$ have different signs
(lemma 3.7), so $|a_1-a_2| > |a_1+a_2|$.
Comparing this to the growth of $\theta^{(r-1)/2}$ we see that
$$ \eqalign{
\log \left ( |a_1 - a_2| \over 2 c \theta^{(r-1)/2} \right ) \ge &
\int_0^\theta dy (a_3(y)-r)(G_1^{-1}(y)+G_2^{-1}(y))/2 \cr + &
(r-1) (G_1^{-1}(y) \! + \! G_2^{-1}(y) \! - \! y^{-1})/2
- (a_3(y) \! + \! 1)|G_1^{-1} \! - \! G_2^{-1}|/2} \tag 3.18
$$
The integrals on the right hand side can all be bounded, independent of $c$,
using (3.16).
As a result, for any small $\delta$, by choosing $c$ large enough we can
force $|a_1(\delta)-a_2(\delta)|$ to be as large as we wish. If we can
prove the bound $|a_1(\delta)+a_2(\delta)|<|a_1(\delta)-a_2(\delta)|/2$, independently of
$c$, then we will have forced $a_2(\delta)$ to be arbitrarily large, and
we will be done.
However, by equations (3.2),
$|a_1 + a_2|' \le |(a_3-1)(G_1^{-1}-G_2^{-1})(a_1-a_2)|/2$.
Integrating this from $0$ to $\theta$, using (3.16) and the
fact that $|a_1-a_2|$ is an increasing function, gives our desired bound
on $|a_1 + a_2|$.
\qed
\noindent{\it Proof of theorem 3.5: } For any fixed $r \ge 1$,
consider the family of functions $a^c$ with $c \in [0,c_1]$. These are
all finite-energy solutions to the ASD equations. Since $t$ is a depends
continuously on $c$, $c=0$ implies $t=0$, and $c=c_1$ implies $t=1$.
Therefore as $c$ increases from 0 to $c_1$, $t$ must take on all values
between 0 and 1. \qed
Theorems 3.2, 3.3, 3.4, and 3.5, together with the explicit solutions of
equations (3.4) and (3.5), cover all the cases of theorem 3.1.
\heading {\bf 4. Anti-self-dual connections}
\endheading
In this section we take $r \ge 1 \ge t$ and look
at deformations of the anti-self-dual YM solutions of theorem 3.5.
The goal is
to prove theorems 1.2 and 1.3, which give the formal
dimensions of the moduli spaces of anti-self-dual connections with
appropriate boundary conditions.
These proofs appear at the end of section 4.4.
In section 4.1 we derive the linearized (anti)self-duality equations for
deformations of the connection. These form a countable collection of
linear ODE systems, one for each representation of the symmetry group $SO(3)$.
We also write down an appropriate gauge condition. Finally, we compute
the second variation of the Yang-Mills functional (i.e. the Hessian).
In section 4.2 we derive appropriate boundary conditions for the ODEs of
section 4.1. The boundary conditions differ, depending on whether $r$ is
an odd integer or not. When $r$ is an odd integer the boundary conditions
at $\theta=0$ follow from smoothness on $S^4$. When $r$ is not an odd
integer we have a choice of boundary conditions, depending on how we
define our space of connections and our gauge group. We consider both
the strongest reasonable boundary conditions, which we term ``regularity'',
and the weakest reasonable conditions, which we term ``weak regularity''.
In section 4.3 we find approximate solutions to the linearized ASD
equations, and show that these solutions have the same asymptotic
behavior near $\theta=0$ as the exact solutions.
Finally, in section 4.4 we
compute the dimension of the moduli space. For each
representation of $SO(3)$, we compute the dimension of the stable manifold
of the ASD equations near $\theta=0$ and $\theta=\pi/3$.
For generic metrics these manifolds
intersect transversally, so we can compute the dimension of the space of
admissible solutions to the ASD equations. Summing over all representations
we get the dimension of the moduli space. We get two sets of answers, one
for regular solutions and one for weakly regular solutions.
\noindent{\it 4.1 The ASD equations}
Let $a=(a_1,a_2,a_3)$ be a reduced $(r,t)$ connection, corresponding to
an equivariant Yang-Mills connection $A$. We wish to consider deformations
of $A$. As discussed in section 2.4, these take the general form
$$
\delta A = \sum_{i,j} \alpha_{ij}(\theta,g) \beta^i \otimes l_j +
\sum_{i} \gamma_{i}(\theta,g) d \theta \otimes l_i.\tag 4.1
$$
One easily establishes the following four propositions. Propositions 4.1,
4.3, and 4.4 are
direct computations, while proposition 4.2 follows from
proposition 4.1 and equations (2.6).
\proclaim {Proposition 4.1} The first variation of the curvature is
given by
$$ \eqalign{ \delta F = & d_A(\delta A) \cr
= & (\alpha'_{11} \!- \!l_1 \gamma_1) (011) +
(\alpha'_{12}\! -\! l_1 \gamma_2\!-\!a_1 \gamma_3) (012) +
(\alpha'_{13}\! -\! l_1 \gamma_3 \!+\!a_1 \gamma_2) (013) \cr
% & + (\alpha'_{21} - l_2 \gamma_1 +a_2 \gamma_3) (021)
% + (\alpha'_{22} - l_2 \gamma_2) (022) +
%(\alpha'_{23} - l_2 \gamma_3 -a_2 \gamma_1) (023) \cr
%& +(\alpha'_{31} - l_3 \gamma_1 -a_3 \gamma_2) (031)
%+(\alpha'_{32} - l_3 \gamma_2 +a_3 \gamma_1) (032)
%+ (\alpha'_{33} - l_3 \gamma_3) (033) \cr
&+(l_2 \alpha_{31} - l_3 \alpha_{21} - \alpha_{11} - a_3 \alpha_{22}
- a_2 \alpha_{33}) (231) \cr &
+(l_2 \alpha_{32} - l_3 \alpha_{22} - \alpha_{12} + a_3 \alpha_{21}) (232)\cr &
+(l_2 \alpha_{33} - l_3 \alpha_{23} - \alpha_{13} + a_2 \alpha_{31}) (233)
%&+(l_3 \alpha_{11} - l_1 \alpha_{31} - \alpha_{21} + a_3 \alpha_{12})
%(311)\cr &
%+(l_3 \alpha_{12} - l_1 \alpha_{32} - \alpha_{22} - a_1 \alpha_{33}
%- a_3 \alpha_{11}) (312)\cr &
%+(l_3 \alpha_{13} - l_1 \alpha_{33} - \alpha_{23} + a_1 \alpha_{32}) (313) \cr
%&+(l_1 \alpha_{12} - l_2 \alpha_{11} - \alpha_{31} + a_2 \alpha_{13})
%(121)\cr &
%+(l_1 \alpha_{22} - l_2 \alpha_{12} - \alpha_{32} + a_1 \alpha_{23}) (122)\cr %& +(l_1 \alpha_{23} - l_2 \alpha_{13} - \alpha_{33} - a_1 \alpha_{22}
%- a_2 \alpha_{11}) (123)
+ \hbox{cyclic}, } \tag 4.2
$$
where $(0jk)$ and $(ijk)$ are shorthand for $d\theta \wedge \beta^j \otimes
l_k$ and $\beta^i \wedge \beta^j \otimes l_k$, respectively, and ${}'$
denotes $\partial/\partial\theta$.
\endproclaim
\proclaim {Proposition 4.2} The linearized anti-self-duality equations
$* \delta F =-\delta F$ are
$$\eqalign{
G_1 (\alpha'_{11} - l_1 \gamma_1) = &
(l_2 \alpha_{31} - l_3 \alpha_{21} -
\alpha_{11} - a_3 \alpha_{22} - a_2 \alpha_{33}) \cr
%
G_1 (\alpha'_{12} - l_1 \gamma_2 -a_1 \gamma_3)= &
(l_2 \alpha_{32} - l_3 \alpha_{22} - \alpha_{12} + a_3 \alpha_{21}) \cr
%
G_1 (\alpha'_{13} - l_1 \gamma_3 +a_1 \gamma_2) = &
(l_2 \alpha_{33} - l_3 \alpha_{23} - \alpha_{13} + a_2 \alpha_{31}) \cr
%
G_2 (\alpha'_{21} - l_2 \gamma_1 +a_2 \gamma_3)= &
(l_3 \alpha_{11} - l_1 \alpha_{31} - \alpha_{21} + a_3 \alpha_{12}) \cr
%
G_2 (\alpha'_{22} - l_2 \gamma_2)= &
(l_3 \alpha_{12} - l_1 \alpha_{32} - \alpha_{22} - a_1 \alpha_{33}
- a_3 \alpha_{11}) \cr
%
G_2 (\alpha'_{23} - l_2 \gamma_3 -a_2 \gamma_1) = &
(l_3 \alpha_{13} - l_1 \alpha_{33} - \alpha_{23} + a_1 \alpha_{32}) \cr
%
G_3 (\alpha'_{31} - l_3 \gamma_1 -a_3 \gamma_2) = &
(l_1 \alpha_{12} - l_2 \alpha_{11} - \alpha_{31} + a_2 \alpha_{13}) \cr
%
G_3 (\alpha'_{32} - l_3 \gamma_2 +a_3 \gamma_1) =&
(l_1 \alpha_{22} - l_2 \alpha_{12} - \alpha_{32} + a_1 \alpha_{23}) \cr
%
G_3 (\alpha'_{33} - l_3 \gamma_3) = &
(l_1 \alpha_{23} - l_2 \alpha_{13} - \alpha_{33} - a_1 \alpha_{22}
- a_2 \alpha_{11})
} \tag 4.3
$$
\endproclaim
\proclaim {Proposition 4.3} The covariant divergence of $\delta A$ is
$$ \eqalign{d_A^*(\delta A) = &
\left [ {l_1 \alpha_{11} \over f_1^2} +
{l_2 \alpha_{21} - a_2 \alpha_{23} \over f_2^2} +
{l_3 \alpha_{31} + a_3 \alpha_{32} \over f_3^2} +
{(f_1f_2f_3\gamma_1)' \over f_1f_2f_3}
\right ] l_1 \cr
&+ \left [ {l_1 \alpha_{12} + a_1 \alpha_{13} \over f_1^2} +
{l_2 \alpha_{22} \over f_2^2} +
{l_3 \alpha_{32} - a_3 \alpha_{31} \over f_3^2} +
{(f_1f_2f_3\gamma_2)' \over f_1f_2f_3} \right ] l_2 \cr
&+ \left [ {l_1 \alpha_{13} - a_1 \alpha_{12} \over f_1^2} +
{l_2 \alpha_{23} + a_2 \alpha_{21} \over f_2^2} +
{l_3 \alpha_{33} \over f_3^2} + {(f_1f_2f_3\gamma_3)' \over f_1f_2f_3}
\right ] l_3 } \tag 4.4
$$
\endproclaim
\proclaim {Proposition 4.4} The Hessian is given by
$$ \delta^2 S = \langle \delta F,\delta F \rangle + 2 \langle F,
\delta^2 F \rangle, \tag 4.5 $$
where the second variation of the curvature is
$$ \eqalign{\delta^2 F = & [\delta A, \delta A] \cr = &
(\gamma_2 \alpha_{13} - \gamma_3 \alpha_{12}) (011) +
(\gamma_3 \alpha_{11} - \gamma_1 \alpha_{13}) (012) +
(\gamma_1 \alpha_{12} - \gamma_2 \alpha_{11}) (013) \cr
& + (\alpha_{12} \alpha_{23} \! - \!\alpha_{22} \alpha_{13}) (121)
\!+\! (\alpha_{13} \alpha_{21}\! - \!\alpha_{23} \alpha_{11}) (122)
\!+\! (\alpha_{11} \alpha_{22}\! -\! \alpha_{21} \alpha_{12}) (123) \cr
& + \hbox{cyclic }
} \tag 4.6
$$
\endproclaim
\noindent{\it 4.2 Boundary values}
We wish to study the linearized anti-self-duality equations (4.3), together
with the gauge-fixing condition $d_A^* (\delta A) =0$. We restrict
our attention to a
particular representation of $SO(3)$, and decompose the functions
$\alpha_{ij}(\theta,g)$ and
$\gamma_i(\theta,g)$ along the basis of proposition 2.1. That is, we write
$$\alpha_{13}(\theta,g) = \sum_{m>0, \hbox{ odd}} \alpha^m_{13}(\theta)
\Psi_{l,m}^-(g), \tag 4.7 $$
with similar expansions for the other components of $\alpha$ and $\gamma$.
The ASD and gauge-fixing equations then become a system of ODEs for the
functions $\alpha_{ij}^m$ and $\gamma_i^m$ on the interval $[0,\pi/3]$.
The question is what boundary conditions to impose at $0$ and $\pi/3$.
The answer depends on $r$ and $t$.
\proclaim {Proposition 4.5} Let $r$ be a positive odd integer, let $A$
be an equivariant connection that is smooth on a neighborhood of $Q_0$,
and let $\delta A$ be a deformation of $A$ that is smooth near $Q_0$.
Then, with the exception of the 6 modes listed below, all components of
$\alpha$ and $\gamma$ are zero at $\theta=0$,
and $\alpha_{3j}^m{}'(0)=0$ for all $j,m$.
\noindent If $r \ge 3$, then the 6 exceptional modes are as follows:
\item {1.} $\alpha_{23}^1(0) = - \alpha_{13}^1(0)$ may be nonzero.
\item {2.} $\alpha_{33}^2{}'(0) = 2 \gamma_3^2(0)$ may be nonzero.
\item {3.} $-\alpha_{22}^{r-1}(0) = \alpha_{11}^{r-1}(0) =
\alpha_{12}^{r-1}(0) = \alpha_{21}^{r-1}(0)$ may be nonzero.
\item {4.} $\alpha_{22}^{r+1}(0) = \alpha_{11}^{r+1}(0) =
-\alpha_{21}^{r+1}(0) = \alpha_{12}^{r+1}(0)$ may be nonzero.
\item {5.} $\alpha_{31}^{r-2}{}'(0) = \alpha_{32}^{r-2}{}'(0) =
2 \gamma_2^{r-2}(0) = -2 \gamma_1^{r-2}(0)$ may be nonzero.
\item {6.} $\alpha_{31}^{r+2}{}'(0) = \alpha_{32}^{r+2}{}'(0) =
2 \gamma_1^{r+2}(0) = -2 \gamma_2^{r+2}(0)$ may be nonzero.
\noindent If $r=1$, then the exceptional modes 1, 2, 4 and 6 are as before.
However, in place of modes 3 and 5 we have
\item {$3.'$} If $l$ is odd then $\alpha_{12}^0(0) = \alpha_{21}^0(0)$ may
be nonzero. \hfill \break
If $l$ is even then $\alpha_{11}^0(0) = -\alpha_{22}^0(0)$
may be nonzero.
\item{$5.'$} $\alpha_{31}^{1}{}'(0) = -\alpha_{32}^{1}{}'(0) =
2 \gamma_2^{1}(0) = 2 \gamma_1^{1}(0)$ may be nonzero.
\endproclaim
\noindent Note that if $r>l-2$, then some of the exceptional modes may
not exist, as $m$ cannot be greater than $l$.
\noindent{\it Proof:} We choose appropriate nonsingular coordinates
near $Q_0$ and write down an arbitrary smooth connection form $\delta A$
in these coordinates, relative to the section $\delta$ of formula
(2.16). We then write $\delta A_\delta$ a different way, by starting with
the expansion (4.1), applying the transition function (2.15) and doing a
change of coordinates. Comparing the two expressions shows that the only possible nonzero coefficients at zero
are those of modes 1--6.
An arbitrary element $Q \in S^4 \subset V$ can be written
in terms of the basis (2.2), as
$$ Q = \sum_{i=0}^4 x_i Q_i. \tag 4.8 $$
We use $(x_1,x_2,x_3,x_4)$ as coordinates for $S^4$ near $Q_0$.
Our connection form near $Q_0$, relative to the section $\delta$ is then
$$ \delta A_\delta = \sum_{i=1}^4 \sum_{j=1}^3 \delta A_i^j dx_i
\otimes l_j. \tag 4.9 $$
Converting between the $\{ x \}$ and $\{\theta,y\}$
coordinates is easy. We apply
the group element $g=\exp(y_1K_1 + y_2K_2)\exp(y_3 K_3)$ to $Q_\theta$, and
decompose relative to the basis (4.8). To first order in $y_1, y_2, \theta$
we have
$$ (x_1,x_2,x_3,x_4)= (y_2 \sqrt{3}/2, - y_1 \sqrt{3}/2,
\theta \cos(2 y_3), \theta \sin(2 y_3)). \tag 4.10
$$
Our standard vector fields on $S^4$ are, to lowest order, given by
$$\eqalign{
\partial_\theta = & \cos(2y_3) \partial/\partial x_3 +
\sin(2y_3) \partial/\partial x_4 \cr
l_1 = & ad_g(K_1)(Q_\theta) = \sqrt{3} \sin(y_3) \partial/\partial x_1 -
\sqrt{3} \cos(y_3) \partial/\partial x_2 \cr
l_2 = & ad_g(K_2)(Q_\theta) = \sqrt{3} \cos(y_3) \partial/\partial x_1 +
\sqrt{3} \sin(y_3) \partial/\partial x_2 \cr
l_3 = & \partial/\partial y_3 = 2 \theta( - \sin(2y_3) \partial/\partial x_3 +
\cos(2y_3) \partial/\partial x_4),} \tag 4.11
$$
and so their duals are
$$ \eqalign{
d \theta = & \cos(2 y_3) dx_3 + \sin(2y_3) dx_4 \cr
\beta^1 = &(\sin(y_3) dx_1 - \cos(y_3) dx_2)/\sqrt{3} \cr
\beta^2 = &(\cos(y_3) dx_1 + \sin(y_3) dx_2)/\sqrt{3} \cr
\beta^3 = & (\cos(2y_3) dx_4 - \sin(2y_3) dx_3)/(2 \theta)}
\tag 4.12
$$
We now take the expansion (4.1), which gives $\delta A_\kappa$, and apply
the transition function (2.15). This transforms the lie algebra of
$H=SO(3)$ as follows:
$$ \eqalign{
l_1 \mapsto & \cos(r y_3) l_1 - \sin(r y_3) l_2 \equiv \tilde l_1 \cr
l_2 \mapsto & \sin(r y_3) l_1 + \cos(r y_3) l_2 \equiv \tilde l_2\cr
l_3 \mapsto & l_3 \equiv \tilde l_3.} \tag 4.13
$$
As a result,
$$ \delta A_\delta = \sum_{i,j} \alpha_{ij}(\theta,g) \beta^i \otimes
\tilde l_j +
\sum_{i} \gamma_{i}(\theta,g) d \theta \otimes \tilde l_i.\tag 4.14 $$
Making the substitutions (4.12) for the 1-forms and (4.13) for the lie
algebra, we have an expansion of $\delta A_\delta$ in terms of $dx_i
\otimes l_j$. Comparing to (4.9) gives $\delta A_i^j$ in terms of
$\alpha$ and $\gamma$. Inverting this relationship we get that
$$ \eqalign {
\alpha_{23} =& \sqrt{3} \left [ \delta A_1^3 \cos(y_3) +
\delta A_2^3 \sin(y_3) \right ] +O(\theta) \cr
%
\alpha_{13} =& \sqrt{3} \left [ \delta A_1^3 \sin(y_3) -
\delta A_2^3 \cos(y_3) \right ] +O(\theta) \cr
%
\alpha_{33}/(2\theta) =& \delta A_4^3 \cos(2 y_3) -
\delta A_3^3 \sin(2 y_3) +O(\theta) \cr
%
\gamma_{3} =& \delta A_3^3 \cos(2 y_3) +
\delta A_4^3 \sin(2 y_3) +O(\theta) \cr
%
\alpha_{22} - \alpha_{11} = &\sqrt{3} \left [
(\delta A_1^2 + \delta A_2^1) \cos((r-1)y_3) +
(\delta A_1^1 - \delta A_2^2) \sin((r-1)y_3) \right ] +O(\theta) \cr
%
\alpha_{12} + \alpha_{21} = &\sqrt{3} \left [
(\delta A_1^1 - \delta A_2^2) \cos((r-1)y_3) -
(\delta A_1^2 + \delta A_2^1) \sin((r-1)y_3) \right ] +O(\theta) \cr
%
\alpha_{22} + \alpha_{11} = &\sqrt{3} \left [
(\delta A_1^2 - \delta A_2^1) \cos((r+1)y_3) +
(\delta A_1^1 + \delta A_2^2) \sin((r+1)y_3) \right ] +O(\theta) \cr
%
\alpha_{12} - \alpha_{21} = &\sqrt{3} \left [
-(\delta A_1^1 + \delta A_2^2) \cos((r+1)y_3) +
(\delta A_1^2 - \delta A_2^1) \sin((r+1)y_3) \right ] +O(\theta) \cr
%
\alpha_{31}/(2\theta) + \gamma_2 = &
(\delta A_4^1 + \delta A_3^2) \cos((r-2)y_3) -
(\delta A_4^2 - \delta A_3^1) \sin((r-2)y_3) +O(\theta) \cr
%
\alpha_{32}/(2\theta) - \gamma_1 = &
(\delta A_4^2 - \delta A_3^1) \cos((r-2)y_3) +
(\delta A_4^1 + \delta A_3^2) \sin((r-2)y_3) +O(\theta) \cr
%
\alpha_{31}/(2\theta) - \gamma_2 = &
(\delta A_4^1 - \delta A_3^2) \cos((r+2)y_3) -
(\delta A_4^2 + \delta A_3^1) \sin((r+2)y_3) +O(\theta) \cr
%
\alpha_{32}/(2\theta) + \gamma_1 = &
(\delta A_4^2 + \delta A_3^1) \cos((r+2)y_3) +
(\delta A_4^1 - \delta A_3^2) \sin((r+2)y_3) +O(\theta). } \tag 4.15
$$
The first two equations give rise to mode 1, the next two to mode 2, and so on.
\qed
We next wish to study what boundary conditions to apply to deformations of
$(r,t)$ YM connections for which $r$ is not an odd integer. Modes 3--6
are clearly impossible, so the natural choice of boundary conditions is to
require that, with the exception of the modes 1 and 2,
all components of $\alpha$ and $\gamma$ be zero at $\theta=0$,
and $\alpha_{3j}^m{}'(0)=0$ for all $j,m$. We call a deformation that
meets these conditions {\it regular at $0$}. If in addition all components
$\alpha$, $\gamma$ and their linear combinations grow or decay as a power
of $\theta$ near $\theta=0$, then we say the deformation is {\it
power-law regular}.
Power-law regularity is very similar to a condition that Johan R\aa de
recently proved ([R1], Theorem 2). R\aa de showed that, locally,
a finite action YM connection with
a non-removable holonomy singularity is gauge-equivalent to a connection
for which the ``direct'' components of the connection (in our notation
$(r-a_3)/f_3$, $\alpha_{13}$, $\alpha_{23}$, $\gamma_3$, and
$\alpha_{33}/f_3$) are at most $O(\theta^0)$, while the
remaining components are a positive power of $\theta$ smaller.
The R\aa de estimates are slightly weaker than power-law regularity, since
they allow all components of $\alpha_{i3}$ and $\gamma_3$ to be
$O(\theta^{\hbox{0 or 1}})$,
not just the particular components of exceptional modes
1 and 2. The other difference is that R\aa de's gauge condition is
slightly different from $d_A^*(\delta A)=0$.
A weaker, but still reasonable, set of boundary conditions is to
allow components of $\gamma$ to have integrable singularities at
$\theta=0$, while requiring all components of $\alpha$
(excepting mode 1) to go to zero at $\theta=0$. We call a deformation
that satisfies these conditions {\it weakly regular}, and if it also
exhibits power-law growth we call it {\it weakly power-law regular}.
\proclaim {Proposition 4.6} If $A$ is a finite-action $(r,t)$ YM connection
and if $\delta A$ is power-law regular, then for all real $\tau$,
$A+\tau \delta A$ has square-integrable curvature near $\theta=0$.
\endproclaim
\noindent {\it Proof:} Curvature is quadratic in the connection, so
$F_{A + \tau \delta A} = F_A + \tau \delta F + \tau^2 \delta^2 F$.
Since $A$ has finite action, $F_A$ is in $L^2$. So $F_{A+ \tau \delta A}$
is in $L^2$ for all $\tau$ if $\delta F$ and $\delta^2 F$ are.
If $\delta A$ is power-law regular, then there
exists a constant $s > 0$ such that all components of $\alpha_{1j}$,
$\alpha_{2j}$, $\gamma_j$ and $\alpha_{3j}/\theta$ are $O(\theta^s)$,
except for those of modes 1 and 2, which are constant $+ O(\theta^s)$.
This makes all the (03i), (23i), and (31i) components of $\delta F$
be $O(\theta^s)$ (the $O(1)$ components from modes 1 and 2 cancel). Since
$1/G_1$, $1/G_2$, and $G_3$ are $O(\theta^{-1})$, the expressions
$$
|F_{23i}|^2/G_1, \qquad |F_{31i}|^2/G_2, \qquad |F_{03i}|^2 G_3
\tag 4.16
$$
are $O(\theta^{2 s - 1})$, and so are integrable near zero. Similarly, all
the (01i), (02i), and (12i) components of $\delta F$ are $O(\theta^{s-1})$.
When squared and multiplied by an $O(\theta)$ metric factor, this again
gives $O(\theta^{2 s - 1})$, which is integrable. Thus $\delta F$ is
square-integrable near $\theta=0$.
$\delta^2 F$ is even easier. Since both special modes involve $l_3$, each
term in $\delta^2 F$ has at least one factor that is not from a special
mode, and so is $O(\theta^s)$. Thus $\delta^2 F$ is square-integrable. \qed
Unfortunately, the converse of proposition 4.6 is false. One can find
finite-energy deformations $\delta A$ that are arbitrarily singular,
simply by applying an arbitrarily singular infinitesimal
gauge transformation to $A$. This problem did not arise in the smooth
case (proposition 4.5), as a gauge transformation there would have to take
on a definite (and finite) value at $\theta=0$. However, if $r$ is not
an odd integer we work on the open manifold $X$ that does not contain
$Q_0$, so we have no a priori
control on gauge transformations near $\theta=0$.
Even applying a gauge-fixing condition such as $d^*_A(\delta A)=0$ does
not fix the problem, since the equation $d^*_A d_A \phi =0$, where $\phi$
is an infinitesimal gauge tranformation on $P_X$, is not elliptic.
To control the problem we have to limit the behavior of $\phi$ near
$\theta=0$ by hand. Since for gauge transformations $\gamma=\phi'$,
this translates into controlling $\gamma$. Requiring $\phi$ to have
a definite limit as $\theta \to 0$ corresponds to requiring $\gamma$
to be integrable, which leads to weak regularity. If we further
require $\gamma$ to be $O(1)$ we get the stronger notion of regularity.
\proclaim {Proposition 4.7} Suppose $r\ge 0$ is not an odd integer. Let $A$
be a finite-action equivariant $(r,t)$ connection that is a solution of
the Yang-Mills equations,
and let $\delta A$ be a deformation of $A$ such that $d_A^*(\delta A)=0$,
such that $\delta F$ is square-integrable on $X$,
and such that $\delta A$ has power-law growth near $\theta=0$. Then,
if the $\gamma$
components of $\delta A$ are integrable near $\theta=0$,
$\delta A$ is weakly power-law regular near $\theta=0$. If
the $\gamma$
components of $\delta A$ are $O(1)$,
$\delta A$ is power-law regular near $\theta=0$.
\endproclaim
\noindent {\it Proof:} We sequentially prove the following six
statements. The first two are mild regularity results, which we
then use to prove four stronger results, which imply the theorem.
\item {1.} All components of all the $\alpha_{ij}$'s are $O(1)$.
\item {2.} All components of all the $\alpha_{3i}$'s are $O(\theta^s)$
for some positive constant $s$.
\item {3.} All components of
$\alpha_{11}$, $\alpha_{22}$, $\alpha_{12}$ and $\alpha_{21}$ are $o(1)$.
\item {4.} If $\gamma$ is $O(1)$ then all components of $\alpha_{31}$,
$\alpha_{32}$, $\theta \gamma_1$ and $\theta \gamma_2$ are $o(\theta)$.
\item {5.} All components of
$\alpha_{33}$ and $\theta \gamma_3$
are $o(\theta)$, with the exception of mode 2, which may be $O(\theta)$.
\item {6.} All components of $\alpha_{13}$ and $\alpha_{23}$
are $o(1)$, with the possible exception of mode 1, which may be $O(1)$.
Note that, if $\gamma$ is integrable and exhibits power-law behavior,
it must go as $\theta^{s-1}$ for some positive $s$.
Since $G_3$, $1/G_1$ and $1/G_2$ all have simple poles at
$\theta=0$, the square-integrability of $\delta F$ implies that
all the $(03i)$, $(23i)$, and $(31i)$ components of $\delta F$
are $o(1)$, and that the remaining components are $o(\theta^{-1})$.
Furthermore, the three
components of $d_A^*(\delta A)$ are identically zero.
We will control the various components of $\alpha$ and $\gamma$ by
the corresponding components of $\delta F$ and $d_A^*(\delta A)$.
Since all the $(0ij)$ components of $\delta F$ are at worst $o(\theta^{-1})$,
and since $\gamma$ is $o(\theta^{-1})$, all components of
$\alpha_{ij}'$ are at
worst $o(\theta^{-1})$, which implies statement 1.
Next we look at the components of $d_A^*(\delta A)$. All the terms
involving $\alpha_{1i}$ and $\alpha_{2i}$ are $O(1)$, and the terms
involving $\gamma_i$ are $O(\theta^{s-2})$, so the remaining terms must
also be $O(\theta^{s-2})$.
This implies that for any odd $m$,
$ -m \alpha_{31}^m + r \alpha_{32}^m$ and $ m \alpha_{32}^m - r \alpha_{31}$
are $O(\theta^s)$,
and for any even $m$,
$m \alpha_{33}^m$ is $O(\theta^s)$. Since $r$ is not an odd integer, this
proves that all components of $\alpha_{3i}$ are $O(\theta^s)$, with the
possible exception of $\alpha_{33}^0$.
To control $\alpha_{33}^0$ we look at the (033) component of
$\delta F$, namely $\alpha_{33}'-l_3\gamma_3$. Since this is $o(1)$,
and since $l_3 \Psi_{l,0}=0$,
$\alpha_{33}^0{}'$ must be $o(1)$, so $\alpha_{33}^0$ is a constant
plus $o(\theta)$. However, by the Sibners' theorem, the holonomy around
$S_+$ must be constant, so $\alpha_{33}^0(0) \Psi_{l0}$
cannot depend on $y_1$ or $y_2$.
If $l>0$, this implies that the constant part of $\alpha_{33}^0$ vanishes.
If $l=0$, a deformation $\alpha_{33}(0) \ne 0$ {\it can} have finite energy,
but it is a change in the value of $r$. For fixed $r$, this is
inadmissible.
We next prove statement 3.
We look at the $\Psi_{l,m}$ components of the (231), (232), (311) and (312)
components of $\delta F$. These components are equal to
$$ -m \alpha_{21}^m \!-\! \alpha_{11}^m \!-\! r \alpha_{22}^m, \qquad
m \alpha_{22}^m \!-\! \alpha_{12}^m \!+\! r \alpha_{21}^m, \qquad
-m \alpha_{11}^m \!-\! \alpha_{21}^m \!+\! r \alpha_{12}^m, \qquad
m \alpha_{12}^m \!-\! \alpha_{22}^m \!-\! r \alpha_{11}^m, \tag 4.17
$$
plus terms that, by statements 1 and 2, are $o(1)$.
Since $m$ is even and $r$ is
not odd, these four expressions are linearly independent, so the only way
for these four components of $\delta F$ to be $o(1)$ is
for $\alpha_{11,12,21,22}^m$ to all be $o(1)$.
Statement 4 is similar. We look at the $l_1$ and $l_2$
components of $d_A^*(\delta A)$ and the (031) and (032)
components of $\delta F$. To order $O(\theta)$ we have that
$$\alpha_{31}^m \!-\! m \theta\gamma_1^m \!-\! r \theta\gamma_2^m, \qquad
\alpha_{32}^m \!+\! m \theta\gamma_2^m \!+\! r \theta\gamma_1^m, \qquad
4 \theta\gamma_1^m \!-\! m \alpha_{31}^m \!+\! r \alpha_{32}^m, \qquad
4\theta\gamma_2^m \!+\! m \alpha_{32}^m \!-\! r \alpha_{31}^m \tag 4.18
$$
are all zero.
Since $r$ is not an odd integer, these four expressions are linearly
independent, and so all components of $\theta \gamma_1$, $\theta \gamma_1$,
$\alpha_{31}^m$, and $\alpha_{32}^m$ must be $o(\theta)$.
To prove statement 5 we look at the $l_3$ component of
$d_A^*(\delta A)$ and the (033)
component of $\delta F$. To order $O(\theta)$ we have
$$\alpha_{33}^m - m \theta\gamma_3^m =
4 \theta\gamma_3^m - m \alpha_{33}^m=0. \tag 4.19
$$
If $m \ne 2$ these are linearly independent, implying that
$\alpha_{33}^m$ and $\theta\gamma_3^m$ are $o(\theta)$.
If $m=2$ the two equations are
linearly dependent, and we find
$2\gamma_3^2(0)= \alpha_{33}^2{}'(0)$ may be nonzero. This is mode 2.
To prove statement 6 we look at the (233) and (313) components of $\delta F$
to order $O(1)$. For $m \ne 1$ the expressions are linearly independent,
so all components are $o(1)$.
For $m=1$ the two expressions are linearly
dependent, and we get mode 1.
\qed
\noindent{\it 4.3 Solutions to the anti-self-duality equations}
In this section we look for solutions to the linearized
(anti)self-duality equations (4.3), restricted to a particular representation
of $SO(3)$, near $\theta=0$ and $\theta=\pi/3$. We make repeated use of
the following standard fact about solutions to an ODE system near a
regular singular point.
\proclaim {Proposition 4.8} Let $x$ be a real variable, $y(x)$ an $n$-vector,
and $M$ a fixed $n \times n$ matrix.
Then the $n$-dimensional
space of solutions to the ODE
$$ x \; dy/dx = M y \tag 4.20 $$
on $(0,\epsilon)$ is spanned by functions of the form
$$ y(x) = \xi x^{\lambda}, \tag 4.21 $$
where $\xi$ is an eigenvector of $M$ with eigenvalue $\lambda$.
If we add a forcing term to the right-hand side,
$$ x dy/dx = M y + \xi x^s, \tag 4.22 $$
then a particular solution is $y= \xi x^{s}/(s-\lambda)$ if $s \ne \lambda$
or $y = \xi x^s \log(x)$ if $s=\lambda$.
\endproclaim
The number of solutions to (4.20) that are regular near $x=0$
depends only on the eigenvalues of $M$. Each positive eigenvalue gives a
solution that vanishes at $x=0$, each zero eigenvalue gives a solution that
is nonzero but finite at $x=0$, and each negative eigenvalue gives a solution
that diverges at $x=0$. If $s>0$ in (4.22), then the particular solution
vanishes at $x=0$.
We can find the dimension of the space of solutions to
the anti-self-duality equations near $\theta=0$ by studying the
leading-order terms.
The leading-order parts of
the ASD equations (4.3) and the gauge-fixing equations are
$$ \eqalign{
%
\theta \alpha'_{11} = &
( - l_3 \alpha_{21} -
\alpha_{11} - r \alpha_{22} ) / 2 \cr
%
\theta\alpha'_{12} = &
( - l_3 \alpha_{22} - \alpha_{12} + r \alpha_{21}
) / 2 \cr
%
\theta\alpha'_{13} = &
( - l_3 \alpha_{23} - \alpha_{13} ) / 2 \cr
%
\theta\alpha'_{21}= &
(l_3 \alpha_{11} - \alpha_{21} + r \alpha_{12}) / 2 \cr
%
\theta\alpha'_{22}= &
(l_3 \alpha_{12} - \alpha_{22}
- r \alpha_{11}) / 2 \cr
%
\theta\alpha'_{23} = &
(l_3 \alpha_{13} - \alpha_{23}) / 2 \cr
%
\theta\alpha'_{31} = & (l_3 \Gamma_1 +r \Gamma_2)/ 2 \cr
%
\theta \alpha'_{32} = & (l_3 \Gamma_2 -r \Gamma_1)/ 2 \cr
%
\theta\alpha'_{33} = & l_3 \Gamma_3 / 2 \cr
%
\theta\Gamma_1' = & (-l_3 \alpha_{31} - r \alpha_{32}) / 2 \cr
%
\theta\Gamma_2' = &
(-l_3 \alpha_{32} + r \alpha_{31}) / 2 \cr
%
\theta\Gamma_3' = & -l_3 \alpha_{33} / 2 ,
} \tag 4.23
$$
where $\Gamma_i \equiv f_3 \gamma_i$.
The exact equations differ from (4.23)
as follows. There are $O(1) \alpha_{3i}$, $O(1) \Gamma_i$,
$O(\theta) \alpha_{1i}$ and $O(\theta) \alpha_{2i}$ corrections to the
expressions for $\theta \alpha_{1i}'$ and $\theta \alpha_{2i}'$. There
are $O(\theta^2) \alpha_{3i}$, $O(\theta^2) \Gamma_i$,
$O(\theta^2) \alpha_{1i}$ and $O(\theta^2) \alpha_{2i}$
corrections to the expressions for $\theta \alpha_{3i}'$ and
$\theta \Gamma_i'$. Some of these errors come from our having
replaced
$$ \eqalign {
a_3 = & r + O(\theta^2) \cr
f_3 = & 2 \theta + O(\theta^3) \cr
f_1f_2 = & 3 + O(\theta^2) \cr
f_1/f_2 = & 1 + O(\theta)} \tag 4.24
$$
with their leading order terms. Others come from our having neglected the
$\alpha_{3i}$ and $\Gamma_i$ contributions to the derivatives of
$\alpha_{1i}$ or $\alpha_{2i}$, and vice-versa. We shall see that these
higher-order corrections make no essential difference.
The ODE system (4.23) decomposes into 4 uncoupled subsystems, one for
$\alpha_{33}$ and $\Gamma_3$, one for $\alpha_{31}$, $\alpha_{32}$,
$\Gamma_1$ and $\Gamma_2$, one for $\alpha_{13}$ and $\alpha_{23}$, and
one for $\alpha_{11}$, $\alpha_{21}$, $\alpha_{12}$ and $\alpha_{22}$.
Moreover, $l_1$ and $l_2$ do not appear, so we can solve the equations
separately for each value of $m$. Using proposition 4.8 and equation
(2.30), we get
\proclaim {Proposition 4.9} The solutions to the equations (4.23) are
spanned by the following:
\item {1.} $
\alpha_{33}^m = \Gamma_3^m = \theta^{m/2}, \qquad m>0 $ even,
\item {2.} $
\alpha_{33}^m = -\Gamma_3^m = \theta^{-m/2}, \qquad m>0 $ even,
\item {3.} $
\alpha_{31}^m = \alpha_{32}^m = \Gamma_1^m = -\Gamma_2^m =
\theta^{(m-r)/2}, \qquad m>0 $ odd,
\item {4.} $
\alpha_{31}^m = \alpha_{32}^m = -\Gamma_1^m = \Gamma_2^m =
\theta^{(r-m)/2}, \qquad m>0 $ odd,
\item {5.} $
\alpha_{31}^m = -\alpha_{32}^m = \Gamma_1^m = \Gamma_2^m =
\theta^{(r+m)/2}, \qquad m>0 $ odd,
\item {6.} $
\alpha_{31}^m = -\alpha_{32}^m = -\Gamma_1^m = -\Gamma_2^m =
\theta^{-(r+m)/2}, \qquad m>0 $ odd,
\item {7.} $
\alpha_{13}^m = \alpha_{23}^m =
\theta^{-(m+1)/2}, \qquad m>0 $ odd,
\item {8.} $
\alpha_{13}^m = -\alpha_{23}^m =
\theta^{(m-1)/2}, \qquad m>0 $ odd,
\item {9.} $
\alpha_{11}^m = \alpha_{12}^m = -\alpha_{21}^m = \alpha_{22}^m =
\theta^{(m-r-1)/2}, \qquad m>0 $ even,
\item {10.} $
\alpha_{11}^m = -\alpha_{12}^m = \alpha_{21}^m = \alpha_{22}^m =
\theta^{-(m+r+1)/2}, \qquad m>0 $ even,
\item {11.} $
\alpha_{11}^m = \alpha_{12}^m = \alpha_{21}^m = -\alpha_{22}^m =
\theta^{(r-m-1)/2}, \qquad m>0 $ even,
\item {12.} $
-\alpha_{11}^m = \alpha_{12}^m = \alpha_{21}^m = \alpha_{22}^m =
\theta^{(r+m-1)/2}, \qquad m>0 $ even,
\noindent If $l$ is even there are the solutions
\item {13.} $
\alpha_{33}^0 = 1,$
\item {14.} $
\alpha_{11}^0 = \alpha_{22}^0 = \theta^{-(r+1)/2},$
\item {15.} $
\alpha_{11}^0 = -\alpha_{22}^0 = \theta^{(r-1)/2},$
\noindent while if $l$ is odd we have
\item {13${}'$.} $
\Gamma_3^0 = 1,$
\item {14${}'$.} $
\alpha_{12}^0 = \alpha_{21}^0 = \theta^{(r-1)/2},$
\item {15${}'$.} $
\alpha_{12}^0 = -\alpha_{21}^0 = \theta^{-(r+1)/2}.$
\endproclaim
We are also interested in solutions to the ASD equations near $\theta=\pi/3$
for $t\le 1$. These are equivalent to solutions of the self-duality
equations near $\theta=0$ for $r\le 1$.
To leading order, the self-duality and gauge-fixing equations for $\alpha_{3i}'$ and $\Gamma_i'$
are the same as the ASD and gauge-fixing equations. The leading order
SD expressions for
$\alpha_{1i}'$ and $\alpha_{2i}'$ are minus those of (4.23). The solutions
to the leading-order SD equations
are therefore similar to those given by proposition 4.9, the difference
being that the exponents in solutions 7--15 (or 7--15${}'$) flip sign.
\proclaim {Proposition 4.10} The regular (or weakly regular) solutions
to the linearized anti-self-duality equations (4.3) with the gauge
condition $d_A^*(\delta A)=0$ near $\theta=0$ are in 1-1 correspondence
with regular (or weakly regular) solutions to (4.23).
\endproclaim
\noindent {\it Proof:} We must show that the higher-order terms neglected
in equation (4.23) do not affect regularity. By the general
theory of ODEs, the behavior of any linear ODE system near
a regular singular point is controlled by the leading-order terms in
the equation. One solves the leading-order equation, then uses the
solution to compute the correction terms, then solves the equation again
using the correction terms as a source, and continues iterating. The
iterative procedure converges, giving a solution to the full system that
has the same growth behavior near the singular point as the first solution
to the leading-order equation.
The only potential trouble in our case
comes from the $O(1)\alpha_{33}$ and
$O(1) \Gamma_i$ corrections to $\theta\alpha_{1i}'$ and $\theta\alpha_{2i}'$
and the $O(\theta^2)\alpha_{1i}$ and $O(\theta^2)\alpha_{2i}$ corrections to
$\theta\alpha_{33}'$ and $\theta\Gamma_i'$. It is not immediately obvious
that these terms are really lower order.
However, if a solution to (4.23) is
regular, then $\alpha_{33}$ and $\Gamma_i$ are at most $O(\theta)$.
Treating these as source terms for the equations for
$\alpha_{1i}'$ and $\alpha_{2i}'$, we get (by proposition 4.8) that
$\alpha_{1i}$ and $\alpha_{2i}$ change by at most $O(\theta\log(\theta))$,
which does not affect their regularity.
Similarly, if a solution to (4.23) is weakly regular, then
$\alpha_{33}$ and $\Gamma_i$ are at most $O(\theta^s)$ for some $s>0$,
so $\alpha_{1i}$ and $\alpha_{2i}$ change by at most $O(\theta^s\log(\theta))$.
Also if a solution is
regular or weakly regular, then $a_{1i}$ and $a_{2i}$ are at most $O(1)$, so
their contribution to $\alpha_{33}$ and $\Gamma_i$ is at most
$O(\theta^2\log(\theta))$, which again does not affect regularity or
weak regularity.
\qed
\noindent{\it 4.4 Dimensions of solution spaces}
For a given representation of $SO(3)$, there are $6l+3$ linearly independent
solutions to the ASD and gauge-fixing equations, since these constitute
a 1-st order system of ODEs in the $6l+3$ variables $\alpha_{ij}^m$ and
$\gamma_i^m$. We will compute the dimension $N_+$ ($N_+^w$)
of the space of solutions
that are regular (weakly regular)
at $\theta=0$, and the dimension $N_-$ ($N_-^w$) of solutions that
satisfy the boundary conditions at $\theta=\pi/3$.
Generically, the dimension $N(l)$ of the
space of solutions that
meet the boundary conditions at both endpoints will then be
$N_+ + N_- - 6l - 3$
(or 0 dimensional if $N_+ + N_- < 6l+3$). We can then sum this number over
representations of $SO(3)$ to get the dimension of the space of allowable
infinitesimal deformations. Generically, this equals the dimension of
the moduli space.\footnote{We cannot be certain that the
round metric on $S^4$ gives generic behavior.
We may have to vary our metric functions $f_1$, $f_2$, and $f_3$ in a
generic way away from the endpoints.}
We begin by computing $N_+$.
\proclaim {Proposition 4.11} Suppose $l \ge 1$, and let $M=l+1$ if $l$ is
even and let $M=l+2$ if $l$ is odd. Then the number of regular
solutions is $N_+=3l+1$ for $r \ge M$, and
$N_+ < 3l+1$ for $1 \le r < M$.
\endproclaim
\noindent{\it Proof:} We first consider $l \ge 2$ even, and evaluate solutions
according to the classification of proposition 4.9. There are of course
$l/2$ positive even values of $m$, the largest being $l$, and there are
$l/2$ positive odd values of $m$, the largest being $l-1$. So there
are $l/2$ solutions of each type 1--12, and 1 solution of each type 13--15.
For $r>l+1$, all the type 1, 4, 5, 8, 11, and 12 solutions are regular at
$\theta=0$, as all have sufficiently large powers of $\theta$
for all values of $m$. None of the type 2, 3, 6, 7, 9, and 10 solutions
are regular, as they all carry negative powers of $\theta$. In addition,
solution 15 is regular, while 13 and 14 are not, giving a total of $3l+1$
regular solutions.
When $r=l+1$ the counting is the same. The $m=l-1$ case of type 4 goes as
$\theta^1$, but this is allowed by exceptional mode 5 of proposition 4.5.
Similarly, the $m=l$ case of type 11 goes as $\theta^0$, but this is
exceptional mode 3. However, when $r$ drops below $l+1$, these two modes
become singular, and $N_+$ drops down to $3l-1$.
Decreasing $r$ further, $N_+$ remains $3l-1$ until $r$ hits $l-1$. At that
point the $m=l$ case of type 9 becomes regular (exceptional mode 4), and
$N_+=3l$. However, once $r$ drops below $l-1$ the $m=l-3$ case of type 4
and the $m=l-2$ case of type 11 become singular, and $N_+$ drops to $3l-2$.
As $r$ decreases further, the pattern repeats itself. Whenever $r$ hits an
odd integer $\le l-3$, two modes become regular. One is of type 3
and corresponds to exceptional mode 6, while the other is of type 9 and
corresponds to exceptional mode 4. $N_+$ thus increases to $3l$. However,
once $r$ drops below the odd integer two other modes, one of type 4 and
one of type 11, become singular, reducing $N_+$ back down to $3l-2$.
In any case, $N_+$ is only bigger than $3l$ when $r \ge M = l+1$.
We next consider $l \ge 3$ odd. There are $(l+1)/2$ positive odd values of $m$
and $(l-1)/2$ positive even values of $m$.
For $r \ge l+2$, all the type 1, 4, 5, 8, 11, and 12 solutions are
regular at $\theta=0$, while none of the type 2, 3, 6, 7, 9, and 10
solutions are regular. In addition, solution 14${}'$ is regular while
13${}'$ and 15${}'$ are not. This makes for a total of $N_+=3l+1$.
When $r$ drops below $l+2$, the $m=l$ case of type 4 becomes singular,
reducing $N_+$ to $3l$, where it remains through $r=l$. When $r$ drops
below $l$, the $m=l-2$ case of type 4 and the $m=l-1$ case of type 11
become singular, reducing $N_+$ to $3l-2$. When $r$ hits $l-2$, and
at every odd integer thereafter, a type 3 solution and a type 9 solution
become regular, increasing $N_+$ to $3l$. However, once $r$ drops below
the odd integer, a type 4 solution and a type 11 solution become
singular, reducing $N_+$ to $3l-2$ again.
$N_+$ is only bigger than $3l$ when $r \ge M = l+2$.
Finally we consider $l=1$. For $l=1$, solutions of type 1, 2, or 9--12 do
not exist. The type 5, 8, and 14${}'$ solutions are always regular,
the type 4 solution is regular for $r \ge 3$, and the type 3, 6, 7, 13${}'$,
and 15${}'$ solutions are never regular. So $N_+=4=3l+1$ for $r \ge M=3$
and $N_+ = 3 < 3l+1$ for $r M^w$, and
$N_+^w = 3l$ for $1 < r < M^w$.
\endproclaim
\noindent{\it Proof:} The counting is the same as in the proof of
proposition 4.11, with the following exceptions. For $l$ even,
when $l-1 < r < l+1$ the $m=l-1$ type 4 solution is weakly regular
but not regular, which increases $N_+^w$ from $3l-1$ to $3l$. When
$rl+1$ and $N_+^w=3l$
when $r1$, let $M^w=l+2$ for $l$ odd and
$l+1$ for $l$ even.
For generic metrics and weakly regular boundary conditions, and for $r$
not an odd integer,
$N^w(l)=1$ if $l \ge 1$ and $r > M^w$, and
$N^w(l)=0$ otherwise. For $l=1$, $N^w(1)=1$ is $t<1$ and $N^w(1)=0$ is $t=1$.
\endproclaim
\noindent {\it Proof of Theorem 1.2:} Let $\{ r \}$ be the greatest odd
integer less than or equal to $r$. By theorem 4.14, $N(l)=1$ for all $l$
between 2 and $\{ r\}-1$, and equals 0 for all other values of $l$.
By the discussion before proposition 2.1, the spin-$l$ representation appears $2l+1$ times in the decomposition of
$L^2(SO(3))$, and the anti-self-duality equations are the same for each
appearance. Thus the total number of regular solutions to the linearized
anti-self-duality equations is
$$ \sum_{l=0}^\infty (2l+1)N(l) = \sum_{l=2}^{\{ r \} -1} (2l+1)=
\{ r \}^2-4. \tag 4.25 $$
For a generic metric this equals the dimension of the moduli space.
\qed
\noindent {\it Proof of Theorem 1.3:} If $t=1$, the total number of
weakly regular solutions to the linearized
anti-self-duality equations is
$$ \sum_{l=0}^\infty (2l+1)N^w(l) = \sum_{l=2}^{\{ r \} } (2l+1)=
(\{ r \}+1)^2-4. \tag 4.26 $$
If $t<1$ we must add on the contribution of the $l=1$ representation to
get $(\{ r \}+1)^2-1$.
\qed
\heading {\bf 5. Non-self-dual connections}
\endheading
In this section we consider equivariant
non-self-dual connections without holonomy.
That is, we consider the case where $r$ and $t$ are odd integers $\ge 3$.
These are non-minimal critical points of the Yang-Mills functional, and we
investigate the index and the nullity of the Hessian at these points. The
nullity gives information about the moduli space of non-self-dual YM
solutions, while the index gives topological information about the space
of all connections, modulo gauge transformations.
Our investigation is numerical. We treat the Hessian one representation
of $SO(3)$ at a time, and numerically diagonalize the Hessian in that
representation, using a finite mode approximation to the space of
deformations. We have results for $r$ and $t$ up to $13$, and $l$ up to $5$.
We also apply the method to (anti)self-dual connections $(r,1)$ and $(1,t)$,
where the results are already known [BoSe], as a test of our method.
The numerical method is detailed in section 5.1. The results are presented in section 5.2. In section 5.3 we discuss their significance.
\noindent{\it 5.1 The numerical method.}
Bor and Montgomery [BoMo] showed that, for a general smooth equivariant
connection, the reduced connection $(a_1,a_2,a_3)$, which is naturally
defined only on $[0,\pi/3]$, can be extended to be a function on the
entire circle. These functions have the following properties:
$$a_3(\theta) = a_3(-\theta), \qquad a_2(\theta) = \pm a_3(\theta + 2\pi/3),
\qquad a_1(\theta) = \pm a_3(\theta - 2\pi/3). \tag 5.1 $$
The signs in the second and third equations depend on whether $r$ and $t$ are
congruent to $1 \pmod 4$ or $3 \pmod 4$.
The relations (5.1) essentially follow from the fact that, on $S^4$, the
points $Q_{\theta}$, $Q_{-\theta}$ and $Q_{\theta \pm 2 \pi/3}$ lie on
the same orbit of the symmetry group $G=SO(3)$. The connection form at
$Q_{-\theta}$ (or $Q_{\theta \pm 2 \pi/3}$) can either be written in terms
of $a_i(-\theta)$, or as rotated versions of the connection form at $Q_\theta$.
So instead of working with three functions on the interval $[0,\pi/3]$, we
can work with the single function $a_3$ on the entire circle. Since this
function is smooth, its Fourier coefficients decrease rapidly, and the function
can be well-approximated with a finite Fourier series. This fact was used
in [SS3] to get numerical approximations to the YM solutions for various
values of $r$ and $t$. We minimized the YM functional in the space of
functions whose Fourier expansions vanished after a fixed number of
terms. Taking 10 terms gave remarkably good results in most cases,
and taking 20
or 30 terms always gave the minimizing action to within one part in $10^{10}$.
The same trick of combining several functions on $[0,\pi/3]$ into one
function on the circle works with deformations. Since the coordinates
$(\theta,g)$ and $(-\theta, g \exp(\pi K_3/2))$ describe the same point on
the sphere, we can decompose the connection at this point either in terms
of $\alpha_{ij}^m(\theta)$ and $\gamma_j^m(\theta)$ or
in terms of $\alpha_{ij}^m(-\theta)$ and $\gamma_j^m(-\theta)$. This,
and similar equivalences between $\theta$ and $\theta \pm 2\pi/3$, gives
relations similar to (5.1).
For example, for $l=1$ a deformation can be written in terms of the 9
functions $\alpha_{32}^1$, $\alpha_{31}^1$, $\alpha_{23}^1$, $\alpha_{13}^1$,
$\alpha_{12}^0$, $\alpha_{21}^0$, $\gamma_1^1$, $\gamma_2^1$, and
$\gamma_3^0$ on $[0, \pi/3]$. Alternatively, it can be written in terms
of two functions, $\alpha$ and $\gamma$ on the circle. If $r \equiv t
\equiv 3 \pmod 4$, then
$$ \eqalign{
\alpha_{32}^1(\theta) = \alpha(\theta), \qquad &
\alpha_{21}^0(\theta) = \alpha(\theta + 2\pi/3), \qquad \qquad \! \!
\alpha_{13}^1(\theta) = \alpha(\theta- 2\pi/3)), \cr
%
\alpha_{31}^1(\theta) = -\alpha(-\theta), \qquad &
\alpha_{23}^1(\theta) = -\alpha(-\theta - 2\pi/3), \qquad \!
\alpha_{12}^0(\theta) = -\alpha(-\theta+ 2\pi/3), \cr
%
\gamma_3^0(\theta) = \gamma(\theta)=-\gamma(-\theta), \qquad &
\gamma_2^1(\theta) = \gamma(\theta+2\pi/3), \qquad \qquad
\gamma_1^1(\theta) = \gamma(\theta-2\pi/3). } \tag 5.2
$$
If $r \equiv 1 \pmod 4$ or $t \equiv 1 \pmod 4$ then similar relations,
with some signs changed, apply.
These functions of the circle can be Fourier decomposed. Since $\gamma$ is
an odd function, it can be decomposed into sine functions, while $\alpha$
decomposes into both cosines and sines. For each frequency, we thus
have 3 modes, two sines and a cosine, to consider. For larger values of
$l$ we need more than 2 functions on the circle to describe $\delta A$,
and in general we have $2l+1$ modes associated to each frequency.
These Fourier modes form a natural basis for the space of deformations
for each representation of $SO(3)$. We cut off this basis at a maximum
frequency $N$ to get a finite-dimensional space.
We compute the matrix elements of the Hessian (4.5) and
of $\int [d_A^*(\delta A)]^2$ with respect to this basis, and then
numerically diagonalize the matrix of $\delta^2 S +
\int (d_A^*(\delta A))^2$.
There are three potential sources of error in this procedure, none of
which actually cause problems. Round-off error
in the computer gives errors of order of magnitude $10^{-14}$.
The second source of error is our finite-mode approximation in calculating
the reduced YM connection $(a_1, a_2, a_3)$. By taking enough Fourier
components
we limit this error to less than 1 part in $10^6$. This
means that a zero eigenvalue may appear slightly negative (or positive),
but will still be a factor of $10^6$ smaller than the other eigenvalues.
Finally, there is our finite-mode approximation for the deformations. By
restricting ourselves to a subspace of the space of deformations, we raise
all the eigenvalues. In principle, this could mean that negative or zero
modes could actually appear positive.
In practice, however, this does not seem to occur. The positive and
negative eigenvalues change only
slightly when the highest allowed frequency, which we denote $N$,
is increased, say from 20 to 30. The eigenvalues
close to zero shrink quickly as $N$ is increased, as is to be expected
of true zero modes.
Table 1 shows the ten smallest eigenvalues, for $l=2$ and $N=20$, for two
different critical points. One critical point has self-dual
curvature, with $(r,t)=(1,5)$,
while the second is non-self-dual, with $(r,t)=(5,3)$.
The results are completely
unambiguous. (1,5) has one zero mode and no negative modes. (5,3) has one zero mode and one negative mode.
\vfill\eject
\centerline{\bf Table 1. Lowest Eigenvalues for $l=2$, $N=20$}
\smallskip
$$ \vbox{\offinterlineskip\hrule
\halign{\vrule#&\strut\quad\hfil#\hfil\quad&\vrule#&\qquad#\hfil\quad&
\vrule# \cr
& $(r,t)=(1,5)$ & & $(r,t)=(5,3)$ & \cr \hline
& 206.274560510209
& & 171.127481544641 & \cr \hline
& 162.883311525700
& & 135.206418912372 & \cr \hline
& 125.836315992737
& & 122.690117492145 & \cr \hline
& 120.723046719095
& & 91.3946372008735 & \cr \hline
& 115.713667988154
& & 88.8817452481566 & \cr \hline
& 84.6803226260188
& & 80.0597577062249 & \cr \hline
& 75.2720093738385
& & 44.6778661554892 & \cr \hline
& 47.6005044927976
& & 38.3540729641350 & \cr \hline
& 35.6509304053161
& & 2.64380290969914$\times 10^{-10}$ & \cr \hline
& 2.06865281842209$\times 10^{-12}$
& & $\!\!\!\!\!-$457.162248939746 & \cr}
\hrule} $$
\noindent{\it 5.2 Results.}
Let $h_-(l,r,t)$ be the dimension of the negative eigenspace of the Hessian
for the $(r,t)$ YM connection, restricted to the spin-$l$ representation of
$SO(3)$. Let $h_0(l,r,t)$ be the dimension of the zero eigenspace. Of course,
for (anti)self-dual connections $h_-(l,r,t)=0$ for all $l$.
The results for $h_-$, for $3 \le r,t \le 13$ are as follows:
\item {$1$:} $h_-(1,r,t)=1$ .
\item {$2$:} $h_-(2,r,t)=1$.
\item {$3$:} $h_-(3,r,t)=2$ when $r$ and $t$ are both greater than 3.
$h_-(3,r,t)=1$ when $r>t=3$ or $t>r=3$. $h_-(3,3,3)=0$.
\item {$4$:} $h_-(4,r,t)=h_-(3,r,t)$.
\item {$5$:} $h_-(5,r,t)=3$ when $r$ and $t$ are both greater than 5.
$h_-(5,r,t)=2$ when $r > t=5$ or $t>r=5$. $h_-(5,r,t)=1$ when
$r>5$ and $t=3$ or when $t>5$ and $r=3$ or when $r=t=5$.
$h_-(5,5,3)=h_-(5,3,5)=h_-(5,3,3)=0$.
\noindent These results suggest the following conjecture:
\proclaim {Conjecture 1} For any positive odd integers $r$ and $t$ and
any positive integer $k$,
$$h_-(2k,r,t)=h_-(2k-1,r,t)=
P \left ( {2k- P[2k+1-r] - P[2k+1-t] \over 2} \right ), \tag 5.3
$$
where $P(x)=x$ when $x>0$ and 0 otherwise.
\endproclaim
In other words, $h_-=k$ if both $r$ and $t$ are greater or equal to $2k+1$,
is reduced by one
if $r=2k-1$, is reduced by two if $r=2k-3$, and so on, with similar reductions
for the value of $t$.
Note that, in addition to matching our numerical results for
non-self-dual connections,
formula (5.3) gives the correct dimension for the (anti)self-dual cases,
namely zero for all $l$.
>From conjecture 1 we derive the total index of the Hessian.
$$ \eqalign{
\hbox{Index of Hessian of} (r,t) \hbox{connection} & = \sum_{l=0}^\infty
(2l+1) h_-(l,r,t) \cr
& = \sum_{k=1}^\infty 8 k P \left ( {2k- P[2k+1-r] - P[2k+1-t]} \right )/2 \cr
& = (r-1)(t-1)(r+t-2)/2.} \tag 5.4
$$
This index is always a multiple of 8, and grows very quickly with
increasing $r$ and $t$. The six smallest indices are
8, 24, 24, 48, 48 and 64, corresponding to the (3,3), (5,3), (3,5), (7,3),
(3,7), and (5,5) connections, respectively.
Taubes has shown [T1] that a NSD YM connection of second Chern number $C_2$
over $S^4$ must have Morse index at least $2|C_2|+2$.
Conjecture 1 implies that equivariant NSD YM connections far exceed that
lower bound. If $t=3$, formula (5.4) gives an
index of $r^2-1$, four times the Taubes bound. Larger values of $t$
given an even greater discrepancy, since increasing $t$ increases the
index but decreases the Chern number. In general, if $r\ge t$,
the index is at least $2(t-1)$ times larger than the Taubes bound.
We next turn to the nullity, where our results are extremely simple.
For all the non-self-dual cases,
$h_0(2,r,t)=h_0(3,r,t)=1$ and $h_0(1,r,t)=h_0(4,r,t)=h_0(5,r,t)=0$. Since
the $l=2$ and $l=3$ zero modes (and no others) are required by conformal
symmetry, this suggests the following:
\proclaim {Conjecture 2} For any positive odd integers $r\ge 3$ and $t\ge 3$,
and for any $l$, $h_0(l,r,t)$ equals 1 if $l=2$ or 3 and is zero otherwise.
\endproclaim
\noindent{\it 5.3 Discussion and Open Problems.}
Let ${\Cal N_k}$ be the moduli space of all YM connections on an $SO(3)$
bundles over $S^4$ with second Chern number $-k$, modulo gauge
transformations, and let ${\Cal M}_k \subset {\Cal N}_k$ be the
moduli space of (anti)self-dual
connections. Conjecture 2 implies that
the component of ${\Cal N_k}$ containing a non-self-dual $G$-equivariant
YM connection consists only of conformal copies of that connection.
Topologically, this component
is the quotient of the conformal group $SO(4,1)$ by the subgroup $G=SO(3)$
that fixes the equivariant connection, and does not
in any way depend on the values of $r$ and $t$.
There is a natural extension of conjecture 2 to non-equivariant connections,
namely
\vfill\eject
\proclaim {Conjecture 3} Let $A_1$ and $A_2$ be non-self-dual YM connections
over $S^4$ with second Chern number $-k$. Then either $A_1$ and $A_2$ lie in
different components of ${\Cal N_k}$, or $A_1$ is gauge-equivalent to
a conformal copy of $A_2$.
\endproclaim
At first glance, conjecture 3 is a disappointment.
The
non-self-dual components of the moduli space lack the rich structure
of the (anti)self-dual component.
However, conjecture 3 also says that non-self-dual YM connections are the
closest thing possible to nondegenerate critical points of the YM functional.
This makes it much easier to identify their topological role.
If we allow ourselves to vary the metric functions $f_i(\theta)$, we can
even find true non-degenerate critical points. The proof of theorem 3.1 uses
only the asymptotic properties of the $f_i$'s, so the construction of
NSD YM connections proceeds just as well with altered metric. The only
difference is that, for a generic set of functions $f_i$,
the conformal group is reduced from $SO(4,1)$ to
$G=SO(3)$. By definition, each equivariant connection is left unchanged
by $G$, so there are no conformal copies of a given equivariant NSD YM connection. We would then expect these NSD YM connections to be
non-degenerate critical points of the YM functional.
We can get the same effect without changing the metric if we take $r$ or
$t$ to be other than an odd integer. We then would be working on $S^4 -
S_\pm$ rather than on $S^4$. Since the only conformal transformations of
$S^4$ that leave $S_+$ (or $S_-$) fixed are in $G$, the conformal group of
$S^4 -S_\pm$ is just $G$.
In either case, we would have a non-degenerate critical point. Such points
are stable under small perturbations of the functional, and hence under small
changes in the metric. These changes {\it need not be equivariant}.
The $(r,t)$ NSD YM connections persist even when all symmetry
has been broken. They have topological significance, and are not
mere flukes of symmetry.
We now return to smooth bundles over $S^4$ with the round metric. Let
${\Cal A}_k$ be the space of smooth $SO(3)$ connections with second
Chern number $-k$, and let ${\Cal G}$ be group of gauge transformations.
The Yang-Mills functional is gauge-invariant, and so is a functional on
$B_k= {\Cal A}_k/{\Cal G}$. We are interested in describing
the topology of $B_k$, via Morse theory, by the critical
points of the YM functional, i.e. by ${\Cal N_k}$.
Of course, Morse theory need not be exact, since the Yang-Mills functional
in 4 dimensions does not satisfy the Palais-Smale condition.
In the case of $B_0$, critical points
of index 1 do not exist, as $1<2|k|+2=2$, yet Taubes
explicitly found a non-contractible loop. This loop corresponds to
what Taubes calls a {\it critical end} of $B_k$ [T2], not to a critical point.
In general, the topology of $B_k$ is described
partly by the minima of the YM functional (${\Cal M}_k$), partly by
the higher critical points (the rest of ${\Cal N}_k$), and partly by
critical ends. The obvious question is how much of the topology
comes from each of these three contributions.
Atiyah and Jones [AJ] conjectured that ${\Cal M}_k$ is homotopy
equivalent to $B_k$ up through a range. The Atiyah-Jones
conjecture was recently proven by Boyer et al [BHMM1, BHMM2],
who showed that $B_k$ and
${\Cal M}_k$ are homotopy equivalent at least through dimension $[k/2]-2$.
Boyer et al also
found that, above the range of equivalence, $H_*({\Cal M}_k)$ has torsion
elements, while $H_*(B_k)$ is free. The higher critical points and
the critical ends must not only provide the topology of $B_k$ that is not
in ${\Cal M}_k$, but must also cancel those elements of $H_*({\Cal M}_k)$
that are not in $H_*(B_k)$.
The space $B_k$ does not depend in any way on the metric, and
the topology of ${\Cal M}_k$ is also metric-invariant. However,
the higher critical points and the critical ends very much depend on
the metric. A change in metric can, by breaking conformal
symmetry, convert a critical surface into a discrete set of critical
points. A change in metric can also change a critical end into a
critical point.
Parker [Pa] showed how, by changing the metric on $S^4$ an
arbitrarily small amount, one can form a NSD YM connection with Chern
number zero and with energy arbitrarily close to two instanton units.
Parker's solution consists of a very small instanton and a very small
anti-instanton, centered at antipodal points. If the metric were round,
both the instanton and anti-instanton would bubble off under gradient
flow. However, with the slightly altered metric, bubbling off is
energetically unfavorable and the instanton and anti-instanton balance.
Parker in effect raised the energy of Taubes' critical end and turned
it into a critical point.
Presumably, the process can be repeated for other critical ends.
It is unclear, however, whether there exist critical ends that
cannot be so removed. We close with some open questions:
\proclaim {Questions} Given integers $i$, $q$ and $k$, do there exist
metrics on $S^4$, arbitrarily close to the round metric in the $C^q$
norm, such that $B_k$ contains no critical ends with index less than $i$?
If so, how large is the space of such metrics? Do there exist
metrics that eliminate all critical ends?
\endproclaim
\medskip
\noindent {\it Acknowledgements:} I thank Gil Bor, Jalal Shatah, Karen
Uhlenbeck, and especially Jan Segert for helpful discussions. Many of
the ideas about decomposing deformations of the connection into representations
of $SO(3)$ are due to Segert. This work was partially supported by
an NSF Mathematical Sciences Postdoctoral Fellowship.
\vfill\eject
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\enddocument
\bye
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