Gauge Theories on Four Dimensional Riemannian Manifolds

This paper develops the Riemannian geometry of classical gauge theories Yang-Mills fields coupled with scalar and spinor fields on compact four-dimensional manifolds. Some important properties of these fields are derived from elliptic theory : regularity, an "energy gap theorem", the manifold structure of the configuration space, and a bound for the supremum of the field in terms of the energy. It is then shown that finite energy solutions of the coupled field equations cannot have isolated singularities (this extends a theorem of K. Uhlenbeck). Introduction One of the major discoveries of physics in this century is the recognition that non-abelian Lie groups play a role in particle physics. For many years this was regarded as a peculiar aspect of quantum mechanics having no classical analogue. Then in 1954 C. N. Yang and R. Mills proposed a classical field theory incorporating these groups. Recently their theory has received considerable attention from both mathematicians and physicists. Yang-Mills theory is easily described in terms of modern differential geometry. One begins with a principal bundle P with compact Lie structure group G over a manifold M. The Yang-Mills field is then the curvature f2 of a connection V on P which is a critical point of the action A(V) = j" IOl z . M When G is the circle group the Yang-Mills field satisfies Maxwell's equations. Physically, Yang-Mills fields represent forces. As such they interact with a second type of field the field of a particle. This is interpreted as a section ~b of a vector bundle associated to P and the action for the system is essentially A(V, qS) = ~ If212 + {Vq~12-m2kbl2, M * The author holds an A.M.S. Postdoctoral Fellowship 0010-3616/82/0085/0563/$08.00 564 T . H . P a r k e r where m is a constant (the mass of the particle). The critical points of this action are solutions to a pair of coupled non-linear partial differential equations the "coupled Yang-Mills equations." These are invariant under the infinite dimensional "gauge group" of all fibre preserving automorphisms of P. This setup constitutes a (classical) gauge theory and is the subject of this paper. Four-dimensional compact Riemannian manifolds are the natural context for Yang-Milts theory for several reasons. First, the four-dimensional Yang-Mills action is bounded below by the characteristic number of the bundle, so the field is constrained by the topology. This is linked by invariant theory to the conformal invariance of the action. This conformal invariance occurs only in dimension four; it means that the relevant geometry lies in the conformal structure of the base manifold. The curvature is expressed in terms of the connection form co by g2 = dco + 1[co, co] and the Yang-Mills action is, essentially, S [dcor 2 + ]co[4. This is the sum of a gradient term Idcol 2 and a non-linear ("self-interaction") term jcoj4. By the Sobolev inequalities these terms are of compatible strength only in dimension four. Thus conformal invariance which dictates the Sobolev inequalities is reflected in the analytic aspects of Yang-Mills fields. To date, the main analytic result for Yang-Mills fields is Uhlenbeck's proof [19] that a Yang-Mills field on a four-dimensional space with finite energy cannot have isolated singularities. As a consequence, a field on IR 4 with finite energy extends via stereographic projection to a field in a non-trivial bundle over S 4. This theorem is striking because it shows that the topology is inherent in the field; for 1 example the quantity ~-225_2 S f~/x f2 is always an integer the characteristic number 1 ~ M of the bundle. In this sense Uhlenbeck's theorem completes the circle : the analytic properties of the Yang-Mills field imply the topology. It is natural to ask if isolated singularities can exist for coupled Yang-Mills fields. Our main result (Theorem 8.1) shows that such isolated singularities are indeed removable. The proof depends crucially on the conformal invariance of the coupled field equations. In the first three sections we develop Yang-Mills theories on compact Riemannian four-manifolds. Section one is an overview of four-dimensional Riemannian geometry and is primarily intended to introduce the (considerable) notation used in subsequent sections. We begin by discussing the special features of the linear algebra of IR 4 which stem from the isomorphism Spin(4)=SU(2) x SU(2). This algebraic structure carries over to vector bundles over four-manifolds and, when connections are introduced, leads to relationships between the curvature, topology and differential operators on these bundles. In Sect. 2 we introduce the coupled Yang-Mills equations and show that the action is naturally associated to conformal structures on oriented four-manifolds. As in physics, we consider two types of equations : the "fermion" equations based on the Dirac operator for bundle-valued spinors, and the "boson" equations based on the bundle Laptacian. The key properties of the Yang-Mitls equations their gauge and conformal invariance extend to these coupled equations. The Yang-Mitls equations are not elliptic because of gauge invariance. Section 4 contains a local slice theorem similar to those of [4, 12, 20] for the action of the gauge group on the product of the space of connections and the space Gauge Theories on Four-Manifolds 565 of fields. In Sect. 5 this slice theorem is used to construct local "gauges" (sections of the principal bundle). This breaks the gauge invariance of the equations, which are then elliptic and possess the expected regularity; for. example a bounded weak solution is C ~. The last three sections are devoted to the proof of the removability of isolated singularities for finite energy solutions of the coupled field equations. This builds on the work of Uhlenbeck [t9, 20]. The proof involves three steps: (i) gauge independent estimates, (ii) a choice of gauge and the corresponding gauge dependent estimates, and (iii) an examination of how these estimates depend on the metric within the conformal class. Together, these yield an energy growth rate, from which the theorem follows. The gauge independent estimate of Sect. 6 is perhaps of interest in other contexts : it shows that the supremum of the total field F = f2 + IVq5 + q~ is bounded by the L 2 norm (the energy) of F. One consequence of this is the fact that a solution to the coupled Yang-Mills equations is 0 ( 1 ) around an isolated singularity. This \ / growth rate is enough to establish the existence of a particularly nice gauge around the singularity using a theorem of Uhlenbeck. Estimates in this gauge are carried out in Sect. 7. These estimates go considerably beyond those of Uhlenbeck [19] by showing that the particle field ~b satisfies an inequality (Theorem 7.6) analogous to Uhlenbeck's inequality on the curvature (Theorem 7.7). The removability of singularities is proved in the last section. Note that this means that both the bundle and the field extend across the singularity. Finally, as an application, we prove an extension theorem: solutions of the coupled field equations IR 4 which decrease at infinity at a certain specified rate extend by stereographic projection to solutions over S 4. 1. Four Dimensional Riemannian Geometry Riemannian geometry in dimension 4 is distinguished by the fact that the universal cover Spin(4) of the rotation group SO(4) is not a simple group, but decomposes as Spin(4) = SU(2) x SU(2). On the group level this is seen by identifying N 4 and C 2 and with the quaternions H. We may regard SU(2) as the group of unit quaternions. For unit quaternions 9 and h, the map x-~9-1xh is an orthogonal transformation of H = N ~ with determinant 1, and hence gives a homomorphism ~ :SU(2)x SU(2)->SO(4). This map has kernel ( 1 , 1 ) , so displays SU(2)x SU(2) as the 2-fold universal covering group of SO (4). On the algebra level the isomorphism so(4) =su(2) x su(2) is a consequence of the Hodge star operation: *:A2(IR4)~AZ(IR 4) with *2=Identity, and the metric gives an identification so(n)=Ag(V). Thus Az(IR 4) decomposes into _+ 1 eigenspaces: so(4)=Ai+GAZ_. The spaces A~ are 3-dimensional spaces of skewsymmetric matrices which are isomorphic as Lie algebras to so(3)--su(2). We will distinguish the two copies of SU(2) in Spin(4) by writing Spin(4) = SU+ (2)x SU_ (2) (this labeling is determined by orientation since a change in