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\define\khat{{\hat k}}
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\topmatter
\title Constructing Non-Self-Dual Yang-Mills Connections on $S^4$ \\
with Arbitrary Chern Number
\endtitle
\author Lorenzo Sadun \\ and \\ Jan Segert \endauthor
\address{Courant Institute of Mathematical Sciences,
New York University,
251 Mercer Street,
New York, NY 10012 }
\endaddress
\address{Department of Mathematics, University of Missouri, Columbia,
Missouri 65211}
\endaddress
\address{{\it L.S.'s permanent address:} Department of Mathematics,
University of Texas, Austin, TX 78712}
\endaddress
\thanks{The first author was partially supported by
NSF Grant DMS-8806731. \newline The second author was
partially supported by a Bantrell Fellowship and
NSF Grant DMS-8801918.}
\endthanks
\subjclass{81E13, also 34B15, 53C05, 58E30}
\endsubjclass
\endtopmatter
\baselineskip=12.4pt
\document
A connection $A$ on a principal bundle over a 4-manifold $M$
is called {\it Yang-Mills\/} if it is a critical point of the
Yang-Mills (YM) action
$$ S(A) = \int_{M} |F_A|^2 dVol = \int_{M} - Tr(*F_A\wedge F_A),
\tag 1 $$
where $F_A=dA+[A,A]$ is the curvature of the connection $A$ and $*$
is the Hodge dual. Equivalently, Yang-Mills connections are solutions
of the {\it Yang-Mills equations\/},
$$d_A*F_A=0, \tag 2$$
where $d_A$ denotes the
covariant exterior derivative. These are the variational equations of
the YM action, and constitute a system of second-order PDE's in $A$.
For a given second Chern number $C_2$, the YM action is bounded below
by $8\pi^2 |C_2|$. To see this, let
$$ F_\pm = {1 \over 2} \left ( F_A \pm *F_A\right )$$
be the self-dual and anti-self-dual parts of the curvature. We can then
express the action and the Chern number as
$$ S(A) = \int_M |F_A|^2 = \int_M |F_+|^2 + |F_-|^2 $$
$$ C_2 = {-1 \over 8\pi^2} \int_M Tr(F_A\and F_A)={1\over 8\pi^2}\int_M
|F_+|^2 - |F_-|^2. $$
The absolute minima of the action, connections
with action attaining the topological bound
$8\pi^2|C_2|$, are thus characterized as having either self-dual
curvature ($*F_A=F_A$, hence $F_-=0$) or anti-self-dual curvature
($*F_A=-F_A$).
These (anti)\,self-dual connections have been well-understood for some
time. The first non-trivial example was the self-dual $SU(2)$ instanton
on $S^4$, discovered in 1975 [BPST]. Three years later, all self-dual
connections on $S^4$ were classified [ADHM], not only for $SU(2)$ but
for all classical groups. The study of self-dual $SU(2)$ connections
over arbitrary 4-manifolds led to spectacular progress in topology,
including the discovery of exotic differentiable structures on
$\real^4$ (see [FU] for an
overview).
A natural question is whether any {\it non-\/}self-dual
(NSD) Yang-Mills connections exist.
Several classes of NSD YM connections on four-manifolds
are known.
One class, due to Itoh [I], consists of homogeneous connections on bundles
over $S^4$ with several large structure groups, including $SU(4)$.
Other solutions on $S^4$ have been constructed by `twistor' methods.
Buchdal [Bu] has
produced solutions with the noncompact structure group
$SL(2,\comp)$, and Manin [Ma] has produced solutions with various
compact groups of
very high dimension (as well as with supergroups).
Parker [P] has constructed a solution on $S^4$ with a nonstandard
Riemannian metric and structure group $SU(2)$.
Some solutions are also known on other four-manifolds,
namely $S^2 \times S^2$ [Ur], and $S^1 \times S^3$ [P, Ur].
Until recently it appeared that for $SU(2)$, NSD YM connections
over the standard 4-sphere $S^4$ might not exist.
Indeed, analogies with harmonic maps from $S^2$ to $S^2$
appeared to indicate that none exist [AJ], and
NSD YM connections with certain simple symmetries were ruled out [T2,\,JT].
Moreover, it was shown that
no {\it local\/} minima of the YM action
exist [BLS,\,T1]; YM connections are either global minima (hence
(anti)\,self-dual) or saddle points.
Sibner, Sibner and Uhlenback [SSU] recently showed that NSD YM
connections on the trivial
bundle $S^4 \times SU(2)$ do exist.
As Lesley Sibner explained in this meeting, their construction involves
using minmax theory to generate monopoles on hyperholic space
${\Bbb H}^3$, which correspond to Yang-Mills connections on $\real^4$
with a certain $U(1)$ symmetry.
\medskip
In this talk we would like to explain an alternate and
somewhat simpler method for constructing NSD YM connections.
This work is described in the papers [SS],
and is based extensively on the work of Urakawa [Ur] and the work of Bor
and Montgomery [BoMo].
Our construction produces examples
not only on the trivial bundle,
but on all $SU(2)$ bundles over $S^4$, except those with
second Chern number equal to
$\pm 1$. We still do not know whether any NSD YM connections exist
with Chern number $\pm 1$.
The strategy is as follows:
\noindent 1) \qquad Pick a symmetry on $S^4$. This reduces all
calculations from a 4-dimensional space, $S^4$, to a much smaller
space, the space of group orbits. In our case the symmetry
group is the rotation group $SO(3)$ (and its cover $SU(2)$),
and the space of orbits is isomorphic to
the interval $[0,\pi/3]\subset \real$.
\noindent 2) \qquad Consider $SU(2)$ connections on $S^4$ that are
equivariant under this symmetry.
Since connections live on bundles,
we need to lift the symmetry group action from the base manifold
$S^4$ to $SU(2)$ bundles over $S^4$.
The equivariant connections fall into distinct classes,
corresponding to different lifts of the group action.
In our case these classes
are indexed by two positive odd integers $n_\pm$.
The bundle corresponding to
$(n_+, n_-)$ has second Chern number $C_2= (n_+^2 - n_-^2)/8$.
\noindent 3) \qquad In each class of equivariant
connections, look for {\it minima\/} of the
action. By the principle of symmetric criticality [Pal], such
minima must be stationary points of the action in the space of
all connections, i.e. must by Yang-Mills. However, they need not
be minima in the space of all connections, as the second variation in
some non-equivariant directions may be negative.
In our case, we show that minima exist
for all classes with $n_+ \ne 1$, $n_- \ne 1$.
\noindent 4) \qquad Finally, show that some classes do not contain
(anti)\,self-dual connections. In our case we find that self-dual
connections can only exist for $n_-=1$ and anti-self-dual connections
only exist for $n_+ = 1$.
These results give NSD YM connections in every class $(n_+, n_-)$
with $n_\pm \ge 3$. Since every integer $N$ except $\pm 1$ can be written
as $N=(n_+^2-n_-^2)/8$ with $n_\pm \ge 3$ in at least one way, this
gives examples with every Chern number except $\pm 1$. For some
Chern numbers we get several solutions (e.g. $5=(7^2-3^2)/8=
(11^2-9^2)/8$), and for the trivial bundle
we have a countably infinite number of solutions (just take $n_+=n_-$).
\heading 1. The Symmetry \endheading
Let $V \simeq \real^5$ be the space of symmetric, traceless, real
$3 \times 3$ matrices $Q$, with inner product $\langle Q,Q' \rangle
= \half Tr(QQ')$. It's convenient to work with an explicit
orthonormal basis
$$ Q_0 = {1 \over \sqrt{3}} \pmatrix -1 & 0 & 0 \cr 0 & -1 & 0 \cr
0 & 0 & 2 \endpmatrix ;
\qquad Q_1 = \pmatrix 0 & 0 & 1 \cr 0 & 0 & 0 \cr 1 & 0 & 0 \endpmatrix ;
\qquad Q_2 = \pmatrix 0 & 0 & 0 \cr 0 & 0 & 1 \cr 0 & 1 & 0
\endpmatrix ; $$
$$ Q_3 = \pmatrix -1 & 0 & 0 \cr 0 & 1 & 0 \cr 0 & 0 & 0 \endpmatrix;
\qquad Q_4 = \pmatrix 0 & 1 & 0 \cr 1 & 0 & 0 \cr 0 & 0 & 0
\endpmatrix . $$
We let $SO(3)$ act on $V$ by conjugation,
$g(Q)=gQg^{-1}$. Restricting ourselves to the unit sphere in $V$,
this gives an action of $SO(3)$ on $S^4$. Since $SU(2)$ is the
double cover of $SO(3)$, this also gives an action of $SU(2)$ on
$S^4$.
Since all matrices in $V$ are diagonalizable, it's not hard to
check that
\proclaim{Proposition 1} Every $Q \in S^4$ is related by the
group action to a unique $Q_\theta = \cos(\theta)Q_0 +
\sin(\theta)Q_3$ with
$0 \le \theta \le \pi/3$. The orbits of $Q_0$ and $Q_{\pi/3}$
are two-dimensional, while all other orbits are three-dimensional.
\endproclaim
As a result, equivariant connections forms are determined by their
values on the path $Q_\theta$, $0 \le \theta \le \pi/3$.
We put
coordinates $(\theta, y^1, y^2, y^3)$ on a neighborhood of
this path by
$$ (\theta, \vec y) \mapsto \exp(\vec y \cdot \vec \sigma) \left (
Q_\theta \right ),$$
where $\sigma_1$, $\sigma_2$, and $\sigma_3$ are the usual
(antihermitian) generators of $SU(2)$, and the action of
$SU(2)$ on $Q_\theta$ is as above.
On the path the tangent vectors
$\partial_\theta \equiv \partial / \partial \theta$ and $\partial_i
\equiv \partial / \partial y^i$ are orthogonal but not orthonormal.
The vector $\partial_\theta$ is normalized, but the length of the vector
$\partial_i$ at $Q_\theta$ is $f_i(\theta)$, where
$$ f_1(\theta) = 2 \sin(\pi/3 + \theta); \qquad
f_2(\theta) = 2 \sin(\pi/3 - \theta); \qquad
f_3(\theta) = 2 \sin(\theta). $$
Note that $f_3$ vanishes at $\theta = 0$, as $Q_0$ is invariant
under rotations about the $z$-axis. Similarly, $f_2$ vanishes at
$\pi/3$.
>From this it is easy to see how the Hodge dual operator $*$ acts on
2-forms:
$$ *(d\theta \wedge dy^i) = {f_j f_k \over f_i} dy^j \wedge dy^k,
\tag 3 $$
where $(i,j,k)$ are cyclic permutations of (1,\,2,\,3). To
simplify the notation we define the functions
$$G_1 = {f_2 f_3 \over f_1}; \qquad G_2={f_1 f_3 \over f_2}; \qquad
G_3 = {f_1 f_2 \over f_3}. $$
$G_1$ and $G_2$ have zeroes at $\theta=0$, while $G_3$
has a pole. Similarly, $G_1$ and $G_3$ have zeroes at $\theta=\pi/3$,
while $G_2$ has a pole.
\medskip
\heading 2. Equivariant Connections \endheading
We next look at $SU(2)$ connections on $S^4$ that are equivariant
under the above action of $SU(2)$. Such equivariant connections
appeared in a study on non-Abelian Berry's phase [ASSS], and were
classified by
Bor and Montgomery [BoMo], who took Urakawa's general theory of
equivariant connections with one-dimensional orbit spaces [Ur] and
applied it to this particular symmetry. Much of this section is
due to [BoMo].
An equivariant connection is of course determined by its values on
the path $\{ Q_\theta \}$. The most general Lie-algebra valued
one-form on the path is
$$ A = - \sum_{i,j=1}^3 \alpha_{ij}(\theta) dy^i \otimes \sigma^j
- \sum_{i=1}^3 \beta_i(\theta) d\theta \otimes \sigma^i, $$
where $\alpha_{ij}$ and $\beta_i$ are real-valued functions.
However, while rotations by
$180^\circ$ about the $x$, $y$, or $z$ axes send $Q_\theta$ to itself,
they do not preserve a general $A$ of this form.
For example, rotation by $180^\circ$ about the
$z$-axis flips the signs of $dy^1$, $dy^2$, $\sigma^1$, and $\sigma^2$,
but not the signs of $dy^3$, $d\theta$, and $\sigma^3$. As a
result, for an equivariant connection the coefficients $\alpha_{13}$,
$\alpha_{23}$, $\alpha_{31}$, $\alpha_{32}$, $\beta_1$, and $\beta_2$
must be identically zero. Similarly, invariance under $180^\circ$
rotations about the $x$-axis forces $\alpha_{12}$, $\alpha_{21}$ and
$\beta_3$ to be zero. Thus equivariant connections may be described
by only three real-valued functions, $a_i = \alpha_{ii}$, and we write
$$ A = - \sum_{i=1}^3 a_i(\theta) dy^i \otimes \sigma^i. $$
We call such a triplet of functions $a = (a_1, a_2, a_3)$ a
{\it reduced connection}.
Given an equivariant connection $A$, the curvature $F_A$ is easily computed:
$$ F_A = \big ( (a_1 + a_2 a_3) dy^2 \wedge dy^3 - a_1' d\theta \wedge
dy^1 \big ) \otimes \sigma^1 + (cyclic),$$
where ${}'$ denotes $d/d\theta$, and $(cyclic)$ denotes the other
cyclic permutations of the indices (1,\,2,\,3). By equation (3) the
(anti)\,self-duality equations are then
$$
-a_1' = \pm {(a_1+a_2a_3) \over G_1}, \qquad
-a_2' = \pm {(a_2+a_1a_3) \over G_2}, \qquad
-a_3' = \pm {(a_3+a_1a_2) \over G_3}, \tag 4 $$
where $+$ denotes self-duality and $-$ denotes anti-self-duality.
>From $F_A$ and $*$ we compute the action (1). The result is
$$ \split S(A) = {\pi^2} \int_0^{\pi/3} d\theta & \Big [
(a_1')^2 G_1 +
(a_2')^2 G_2 +
(a_3')^2 G_3 \\ & +
{(a_1+a_2a_3)^2 \over G_1} +
{(a_2+a_1a_3)^2 \over G_2} +
{(a_3+a_1a_2)^2 \over G_3} \Big ]. \endsplit \tag 5
$$
Finite action connections must have well-defined boundary values
$r=a_3(0)$ and $t=a_2(\pi/3)$. Also, since $G_1$ and $G_2$ have
zeroes at $\theta=0$, they must have
$$a_1(0)+ a_2(0) a_3(0) = a_2(0) + a_1(0) a_3(0) = 0. $$
If $r \ne \pm 1$, then these conditions imply that both $a_1(0)$
and $a_2(0)$ equal zero. Similarly, if $t \ne \pm 1$ then
$a_1(\pi/3)=a_2(\pi/3)=0$.
Not all finite-action reduced connections correspond to
connections on all of $S^4$. A holonomy condition for infinitesimal
paths around $Q_0$ forces $r \equiv -1 \pmod 4$, and a similar condition
at $Q_{\pi/3}$
forces $t \equiv -1 \pmod 4$. If these conditions are met we define
the positive odd integers $n_+ = |r|$, $n_-=|t|$.
(If the holonomy conditions are not met, then our reduced connection corresponds
to a singular connection with non-integer Chern
number [FHP,\,SiSi], with the singularities occurring
at the orbits of $Q_0$ and $Q_{\pi/3}$.)
Finally, we compute the Chern number of a connection.
On our path the Chern form is
$$ \split {-1 \over 8 \pi^2} Tr(F_A \wedge F_A) = & {-1 \over 8\pi^2}
(a_1'(a_2+a_2a_3)+ cyclic) d\theta\wedge dy^1 \wedge dy^2 \wedge dy^3 \\
= & {-1 \over 16 \pi^2} d(a_1^2+a_2^2+a_3^2+2a_1a_2a_3)\wedge dy^1
\wedge dy^2 \wedge dy^3, \endsplit $$
which we integrate, first over the group and then over
$[0,\pi/3]$, to get a Chern number of $(r^2-t^2)/8= (n_+^2-n_-^2)/8$.
\medskip
\heading 3. Nonexistence of Self-Dual Connections \endheading
Before showing that Yang-Mills connections do exist, we would like to
prove that in certain classes (anti)\,self-dual connections do
not exist. We prove this not only for the non-singular classes
$\npnp$ with $n_\pm \ge 3$, but also for a large number of
singular classes $(r,t)$. Specifically,
\proclaim {Theorem 2} There are no finite-action self-dual
reduced connections with $|t|>1$. There are no finite-action
anti-self-dual reduced connections with $|r|>1$. \endproclaim
We prove the second statement, the first being similar. An
anti-self-dual connection has non-positive Chern number, so
$|t| \ge |r| > 1$. Since $|r|$ and $|t|$ both differ from 1,
finite action implies that
$a_1(0)=a_2(0)=a_1(\pi/3)=a_3(\pi/3)=0$. We will show that a
solution to the anti-self-dual equations with $a_1(0)=a_2(0)=0$
must have $a_3(\pi/3) \ne 0$, contradicting this.
Suppose $r>1$ (the case $r < -1$ is similar). Then $a_3$ is
positive and greater than 1 on a neighborhood $I_\epsilon = (0, \epsilon)$.
If at some point in this neighborhood both $a_1$ and $a_2$ are non-negative,
then by the anti-self-duality equations (4) all three derivatives
will be non-negative, and the signs will persist. In particular,
$a_3(\pi/3)$ will be positive, not zero. Similarly, if at some
point in $I_\epsilon$ both $a_1$ and $a_2$ are non-positive,
then $a_1', a_2' \le 0 \le a_3'$ and again the signs persist.
Thus it suffices to find a single point $\theta \in I_\epsilon$ at
which $a_1$ and $a_2$ have the same sign (or where one is zero).
Suppose there is no such point, so $a_1$ and $a_2$
have opposite signs on all of $I_\epsilon$.
Then $|a_1 - a_2| < |a_1 + a_2|$. Equations (4) yield
$$ \split {d (a_1 - a_2)^2 \over d\theta} & = 2 (a_1-a_2)(a_1-a_2)' \\
= - ({1 \over G_1} & + {1 \over G_2} )(a_3-1)(a_1-a_2)^2
+ ({1 \over G_1} - {1 \over G_2} )(a_3+1)(a_1-a_2)(a_1+a_2). \endsplit $$
The first term dominates in a neighborhood of $\theta=0$, since
$G_1^{-1}+G_2^{-1}$ has a pole at zero, while $G_1^{-1}-G_2^{-1}$ does
not, and since $(a_3-1)$ is bounded away from zero on $I_\epsilon$.
Thus $[(a_1-a_2)^2]'$ is strictly
negative, and $(a_1-a_2)^2$ is a strictly decreasing non-negative
function. However, at $\theta=0$
we have $(a_1-a_2)^2=(0-0)^2=0$, and cannot
decrease further. We have a contradiction, and so are done.
\medskip
\heading 4. Existence of Minima \endheading
What remains is to show that in each class $\npnp$ the action achieves
its minimum. This is to be expected [BoMo], since the
symmetry should prevent any
bubbling-off phenomena, as in the equivariant
Sobolev theorems of Parker [P].
By symmetry, such bubbling would have
to occur on a complete orbit. But each orbit contains an infinite
number of points, and Uhlenbeck's theorem only allows bubbling at a
finite number of points.
This is in fact true, and
as before, we prove our result for both
singular and non-singular connections.
\proclaim {Theorem 3} On each class $(r,t)$ with $|r|>1$, $|t|>1$
the action achieves its minimum. \endproclaim
The proof, which is only sketched here (the details being
rather grungy), is by the direct method of
the calculus of variations. We first define a Hilbert space $\hil$ with norm
%
$$ \Vert a \Vert^2 =
\int_0^{\pi/3} d\theta \Big [
(a_1')^2 G_1 +
(a_2')^2 G_2 +
(a_3')^2 G_3 +
{(a_1)^2 \over G_1} +
{(a_2)^2 \over G_2} +
{(a_3)^2 \over G_3} \Big ]. $$
%
This norm resembles the action, only with the cubic and quartic terms removed.
$\hil$ is the direct sum of three weighted Sobolev spaces,
one for each $a_i$.
We next show that, for fixed $(r,t)$, sets of bounded action have
bounded norm. This implies
that any minimizing sequence lies in a finite radius ball in $\hil$,
which is weakly compact. Finally we show that the action is weakly
lower-semicontinuous, hence that the weak limit of a minimizing sequence
achieves the minimum action.
The difficulty is in showing that bounded action implies bounded norm.
Away from the boundaries we have no problem, but near $0$ and $\pi/3$
various functions $G_i$ and $G_i^{-1}$ diverge, complicating the analysis.
The biggest difficulty is in bounding $a_1^2/G_1 + a_2^2/G_2$ near
$\theta = 0$ (with a similar problem at $\theta = \pi/3$).
This is done using the fact that both $G_1$ and $G_2$ go as $1/\theta$,
and noting that
$$ (a_1\!+\!a_2a_3)^2 + (a_2\!+\!a_1a_3)^2= (a_3\!+\!1)^2(a_1\!+\!a_2)^2 +
(a_3\!-\!1)^2 (a_1 \!-\! a_2)^2 \ge 2(|a_3|-1)^2 (a_1^2 + a_2^2).
$$
\noindent For $|r|>1$ we can bound $|a_3|-1$ away from zero in some
neighborhood of $\theta=0$, and so can bound $a_1^2/G_1 + a_2^2/G_2$ by
a multiple of $(a_1+a_2a_3)^2/G_1 + (a_2+a_1a_3)^2/G_2$.
This whole approach breaks down for $r=\pm 1$ or $t = \pm 1$.
In those cases bounded
action does {\it not\/} imply bounded norm, and we have no proof that
the minimum is achieved.
\medskip
\heading 5. Regularity \endheading
For each pair $(r,t)$ with $|r|>1$ and $|t|>1$ we have found solutions to
the one-dimensional variational problem that do not satisfy the
(anti)\,self-duality equations. These correspond to equivariant
non-self-dual Yang-Mills connections on $S^4$ with the two exceptional
orbits removed. The only remaining question is whether these connections may
be extended smoothly across these two orbits.
For $r \not \equiv -1 \pmod 4$ they cannot be extended across the orbit
of $Q_0$ due to a holonomy obstruction, and for $t \not \equiv -1 \pmod 4$
they cannot be extended across the orbit of $Q_{\pi/3}$. For
$r$ and $t$ congruent to $-1 \pmod 4$, however, the extension is possible.
The proof is straightforward but lengthly, and we only sketch
the main ideas here.
We first choose a particular connection (call it $B$),
which is known to be smooth by the theorems of [BoMo].
We then show that the difference between our
finite-action connection $A$ and the reference connection $B$ approaches
zero as $\theta \to 0$, and hence that $A$ can be extended
{\it continuously\/} across the orbit of $Q_0$.
We then compute the $L^2_1$ norm of $A-B$ in a particular gauge and
show it to be finite. Since $B$ is smooth, this means that $A$
is in $L^2_1$. By Uhlenbeck's regularity theorem [Uh], there exists a
gauge in which $d*\tilde A=0$, where $\tilde A$ is gauge equivalent
to $A$.
The Yang-Mills equations together with this gauge condition
form an elliptic system of equations, which implies that $\tilde A$ is
smooth. Finally, since $A$ was continuous to begin with, the gauge
transformation must be $C^1$, and so cannot change the topology of
the bundle.
\medskip
We wish to thank Gil Bor,
Percy Deift, Richard Montgomery, Lesley Sibner, Robert Sibner,
Barry Simon, Cliff Taubes, and particularly Jalal Shatah for their
help with this work.
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\book Gauge Field Theory and Complex Geometry
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\paper Non-Self-Dual Yang-Mills Connections with Nonzero Chern Number
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\paper Non-Self-Dual Yang-Mills Connections with Quadrupole Symmetry
\paperinfo preprint \yr 1990 \endref
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\by L.M. Sibner, R.J. Sibner
\paper Singular Sobolev Connections with Holonomy
\jour Bull. Amer. Math. Soc. \vol 19 \page 471 \yr 1988
\moreref \paper Classification of Singular Sobolev Connections by their
Holonomy \paperinfo preprint \yr 1989 \endref
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\by L.M. Sibner, R.J. Sibner, K. Uhlenbeck
\paper Solutions to Yang-Mills Equations which are not Self-Dual
\jour Proc. Natl. Acad. Sci. USA
\vol 86 \page 8610 \yr 1989 \endref
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Stability in Yang-Mills theories
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On the Equivalence of the First and Second order
Equations for Gauge Theories
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Connections with $L^p$ Bounds on Curvature
\jour Comm. Math. Phys. \vol 83 \page 31 \yr 1982 \endref
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Equivariant Theory of Yang-Mills Connections over
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\jour Indiana Univ. Math. J. \vol 37 \page 753 \yr 1988 \endref
\enddocument