\input amstex \documentstyle{amsppt} \magnification=\magstep1 \parskip=5pt \pagewidth{5.0in} \pageheight{7.0in} \CenteredTagsOnSplits \TagsOnRight \define\Tr{\text{Tr}} \define\half{{1 \over 2}} \define\real{{\Bbb R}} \define\comp{{\Bbb C}} \define\ihat{{\hat \imath}} \define\jhat{{\hat \jmath}} \define\pithree{\pi/3} \define\khat{{\hat k}} \predefine\oldand{\and} \redefine\and{{\wedge}} \define\npnp{(n_+,n_-)} \define\inflim{\varliminf} \define\hil{{\Cal H}} \define\wto{\rightharpoondown} \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. \Refs \ref\key [ADHM] \by M.F. Atiyah, V.G. Drinfeld, N.J. HItchin, Y.I. Manin \paper Construction of Instantons \jour Phys. Lett. \vol 65A \page 185 \yr 1978 \endref \ref\key [AJ] \by M.F. Atiyah, J.D.S. 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