Yang–Mills equations
In physics and mathematics, and especially differential geometry and gauge theory, the Yang–Mills equations are a system of partial differential equations for a connection on a vector bundle or principal bundle. They arise in physics as the Euler–Lagrange equations of the Yang–Mills action functional. They have also found significant use in mathematics. Solutions of the equations are called Yang–Mills connections or instantons. The moduli space of instantons was used by Simon Donaldson to prove Donaldson's theorem. Motivation. Physics. In their foundational paper on the topic of gauge theories, Robert Mills and Chen -Ning Yang developed (essentially independent of the mathematical literature) the theory of principal bundles and connections in order to explain the concept of "gauge symmetry" and "gauge invariance" as it applies to physical theories. The gauge theories Yang and Mills discovered, now called "Yang–Mills theories", generalised the classical work of James Maxwell on Maxwell's equations, which had been phrased in the language of a formula_1 gauge theory by Wolfgang Pauli and others. The novelty of the work of Yang and Mills was to define gauge theories for an arbitrary choice of Lie group formula_2, called the "structure group  (or in physics the gauge group", see Gauge group (mathematics) for more details). This group could be non-Abelian as opposed to the case formula_3 corresponding to electromagnetism, and the right framework to discuss such objects is the theory of principal bundles. The essential points of the work of Yang and Mills are as follows. One assumes that the fundamental description of a physical model is through the use of "fields", and derives that under a "local gauge transformation" (change of local trivialisation of principal bundle), these physical fields must transform in precisely the way that a connection formula_4 (in  physics, a "gauge field") on a principal bundle transforms. The "gauge field strength" is the curvature formula_5 of the connection, and the energy of the gauge field is given (up to a constant) by the Yang–Mills action functional The principle of least action dictates that the correct equations of motion for this physical theory should be given by the Euler–Lagrange equations of this functional, which are the Yang–Mills equations derived below: Mathematics. In addition to the physical origins of the theory, the Yang–Mills equations are of important geometric interest. There is in general no natural choice  of connection on a vector bundle or principal bundle. In the special case where this bundle is the tangent bundle to a Riemannian manifold, there is such a natural choice, the Levi-Civita connection, but in general there is an infinite-dimensional space of possible choices. A Yang–Mills connection gives some kind of natural choice of a connection for a general fibre bundle, as we now describe. A connection is defined by its local forms formula_8 for a trivialising open cover formula_9 for the bundle formula_10. The first attempt at choosing a canonical connection might be to demand that  these forms vanish. However, this is not possible unless the trivialisation is flat, in the sense that the transition functions formula_11 are constant functions. Not every bundle is flat, so this is not possible in general. Instead one might ask that the local connection forms formula_12 are themselves constant. On a principal bundle the correct way to phrase this condition is that the curvature formula_13 vanishes. However, by Chern–Weil theory if the curvature formula_5 vanishes (that is to say, formula_4 is a flat connection), then the underlying principal bundle must have trivial Chern classes, which is a topological obstruction  to the existence of flat connections: not every principal bundle can have a flat connection. The best one can hope for is then to ask that instead of vanishing curvature, the bundle has curvature "as small as possible". The Yang–Mills action functional described above is precisely (the square of) the formula_16-norm of the curvature, and its Euler–Lagrange equations describe the critical points of this functional, either the absolute minima or local minima. That is to say, Yang–Mills connections are precisely those that minimize their curvature. In this sense they are the natural choice of connection on  a principal or vector bundle over a manifold from a mathematical point of view. Definition. Let formula_17 be a compact, oriented, Riemannian manifold. The Yang–Mills equations can be phrased for a connection on a vector bundle or principal formula_2-bundle over formula_17, for some compact Lie group formula_2. Here the latter convention is presented. Let formula_21 denote a principal formula_2-bundle over formula_17. Then a connection on formula_21 may be specified by a Lie algebra-valued differential form formula_4 on the total space of the principal bundle. This connection has a curvature form formula_5, which is a two-form  on formula_17 with values in the adjoint bundle formula_28 of formula_21. Associated to the connection formula_4 is an exterior covariant derivative formula_31, defined on the adjoint bundle. Additionally, since formula_2 is compact, its associated compact Lie algebra admits an invariant inner product under the adjoint representation. Since formula_17 is Riemannian, there is an inner product on the cotangent bundle, and combined with the invariant inner product on formula_28 there is an inner product on the bundle formula_35 of formula_28-valued two-forms on formula_17. Since formula_17 is oriented, there is an formula_16-inner product on the sections of this bundle . Namely, where inside the integral the bundle-wise inner product is being used, and formula_41 is the Riemannian volume form of formula_17. Using this formula_16-inner product, the formal adjoint operator of formula_31 is defined by Explicitly this is given by formula_46 where formula_47 is the Hodge star operator acting on two-forms. Assuming the above set up, the Yang–Mills equations are a system of (in general non-linear) partial differential equations given by Since the Hodge star is an isomorphism, by the explicit formula for formula_48 the Yang–Mills equations can equivalently be written A connection satisfying () or  () is called a Yang–Mills connection. Every connection automatically satisfies the Bianchi identity formula_49, so Yang–Mills connections can be seen as a non-linear analogue of harmonic differential forms, which satisfy In this sense the search for Yang–Mills connections can be compared to Hodge theory, which seeks a harmonic representative in the de Rham cohomology class of a differential form. The analogy being that a Yang–Mills connection is like a harmonic representative in the set of all possible connections on a principal bundle. Derivation. The Yang–Mills equations are the Euler–Lagrange equations of the Yang–Mills  functional, defined by To derive the equations from the functional, recall that the space formula_51 of all connections on formula_21 is an affine space modelled on the vector space formula_53. Given a small deformation formula_54 of a connection formula_4 in this affine space, the curvatures are related by To determine the critical points of (), compute The connection formula_4 is a critical point of the Yang–Mills functional if and only if this vanishes for every formula_59, and this occurs precisely when () is satisfied. Moduli space of Yang–Mills connections. The Yang–Mills equations are gauge invariant. Mathematically, a gauge transformation  is an automorphism formula_60 of the principal bundle formula_21, and since the inner product on formula_28 is invariant, the Yang–Mills functional satisfies and so if formula_4 satisfies (), so does formula_65. There is a moduli space of Yang–Mills connections modulo gauge transformations. Denote by formula_66 the gauge group of automorphisms of formula_21. The set formula_68 classifies all connections modulo gauge transformations, and the moduli space formula_69 of Yang–Mills connections is a subset. In general neither formula_70 or formula_69 is Hausdorff or a smooth manifold. However, by restricting to irreducible connections, that is, connections formula_4 whose holonomy group is  given by all of formula_2, one does obtain Hausdorff spaces. The space of irreducible connections is denoted formula_74, and so the moduli spaces are denoted formula_75 and formula_76. Moduli spaces of Yang–Mills connections have been intensively studied in specific circumstances. Michael Atiyah and Raoul Bott studied the Yang–Mills equations for bundles over compact Riemann surfaces. There the moduli space obtains an alternative description as a moduli space of holomorphic vector bundles. This is the Narasimhan–Seshadri theorem, which was proved in this form relating Yang–Mills connections to holomorphic vector bundles by Donaldson. In this setting the moduli  space has the structure of a compact Kähler manifold. Moduli of Yang–Mills connections have been most studied when the dimension of the base manifold formula_17 is four. Here the Yang–Mills equations admit a simplification from a second-order PDE to a first-order PDE, the anti-self-duality equations. Anti-self-duality equations. When the dimension of the base manifold formula_17 is four, a coincidence occurs. The Hodge star operator takes differential formula_79-forms to differential formula_80-forms, where formula_81. Thus, in dimension four, the Hodge star operator maps two-forms to two-forms, The Hodge star operator  squares to the identity in this case, and so has eigenvalues formula_83 and formula_84. In particular there is a decomposition into the positive and negative eigenspaces of formula_47, the self-dual and anti-self-dual two-forms. If a connection formula_4 on a principal formula_2-bundle over a four-manifold formula_17 satisfies either formula_90 or formula_91, then by (), the connection is a Yang–Mills connection. These connections are called either self-dual connections or anti-self-dual connections, and the equations the self-duality (SD) equations and the anti-self-duality (ASD) equations. The spaces of self-dual and anti -self-dual connections are denoted by formula_92 and formula_93, and similarly for formula_94 and formula_95. The moduli space of ASD connections, or instantons, was most intensively studied by Donaldson in the case where formula_96 and formula_17 is simply-connected. In this setting, the principal formula_98-bundle is classified by its second Chern class, formula_99. For various choices of principal bundle, one obtains moduli spaces with interesting properties. These spaces are Hausdorff, even when allowing reducible connections, and are generically smooth. It was shown by Donaldson that the smooth part is orientable. By the Atiyah–Singer index theorem, one may compute  that the dimension of formula_100, the moduli space of ASD connections when formula_101, to be where formula_103 is the first Betti number of formula_17, and formula_105 is the dimension of the positive-definite subspace of formula_106 with respect to the intersection form on formula_17. For example, when formula_108 and formula_109, the intersection form is trivial and the moduli space has dimension formula_110. This agrees with existence of the BPST instanton, which is the unique ASD instanton on formula_111 up to a 5 parameter family defining its centre in formula_112 and its scale. Such instantons on formula_112 may be extended across  the point at infinity using Uhlenbeck's removable singularity theorem. Applications. Donaldson's theorem. The moduli space of Yang–Mills equations was used by Donaldson to prove Donaldson's theorem about the intersection form of simply-connected four-manifolds. Using analytical results of Clifford Taubes and Karen Uhlenbeck, Donaldson was able to show that in specific circumstances (when the intersection form is definite) the moduli space of ASD instantons on a smooth, compact, oriented, simply-connected four-manifold formula_17 gives a cobordism between a copy of the manifold itself, and a disjoint union of copies of the complex projective plane  formula_115. The intersection form is a cobordism invariant up to isomorphism, showing that any such smooth manifold has diagonalisable intersection form. The moduli space of ASD instantons may be used to define further invariants of four-manifolds. Donaldson defined rational numbers associated to a four-manifold arising from pairings of cohomology classes on the moduli space. This work has subsequently been surpassed by Seiberg–Witten invariants. Dimensional reduction and other moduli spaces. Through the process of dimensional reduction, the Yang–Mills equations may be used to derive other important equations in differential geometry and gauge theory. Dimensional reduction is the  process of taking the Yang–Mills equations over a four-manifold, typically formula_112, and imposing that the solutions be invariant under a symmetry group. For example: There is a duality between solutions of the dimensionally reduced ASD equations on formula_118 and formula_120 called the Nahm transform, after Werner Nahm, who first described how to construct monopoles from Nahm equation data. Hitchin showed the converse, and Donaldson proved that solutions to the Nahm equations could further be linked to moduli spaces of rational maps from the to itself. The duality observed for these solutions is theorized to hold for arbitrary dual  groups of symmetries of a four-manifold. Indeed there is a similar duality between instantons invariant under dual lattices inside formula_112, instantons on dual four-dimensional tori, and the ADHM construction can be thought of as a duality between instantons on formula_112 and dual algebraic data over a single point. Chern–Simons theory. The moduli space of Yang–Mills equations over a compact Riemann surface formula_123 can be viewed as the configuration space of Chern–Simons theory on a cylinder formula_124. In this case the moduli space admits a geometric quantization, discovered independently by Nigel Hitchin and Axelrod–Della Pietra –Witten.