Introduction to the Yang-Mills Equations Final Project for Math 581 11 May 2012 S´ebastien Picard [email protected] Contents 1. Introduction 1 2. Connections and Curvature 2 3. The Hodge Star Operator 9 4. Statement of the Yang-Mills Equation 13 5. Short Excursion into Hyperbolic Yang-Mills: The Maxwell Equations 15 6. Hodge Theory 19 7. Variational Formulation 28 8. Manifolds of Dimension d =4 31 9. ASD Connections and Gauge Fixing 36 References 41 1. Introduction We discuss some topics on the Yang-Mills equations. We first develop some of the geomet- ric background needed to understand the statement of the equation; namely, the notions of a connection, curvature, and the Hodge star operator. We then state the equation and briefly discuss gauge transformations. As an aside, which is interesting from a physics standpoint, we derive the Maxwell’s equations from the Yang-Mills framework. Since Yang-Mills theory is a generalization of Hodge theory, we spend some time carefully proving the Hodge theorem. We conclude by investigating the Yang-Mills equations over a compact four-manifold and discuss selfdual and antiselfdual instantons. The majority of these notes were written while following Jost’s Riemannian Geometry and Geometric Analysis [1], with some inspiration from more physics-oriented texts such as [2]. The last section was written while following [4]. The prerequisites are graduate level courses in smooth manifolds and partial differential equations. The Einstein summation convention is always assumed. 1 2 2. Connections and Curvature In this section, we explore connections on vector bundles. Let π : E → M to be a vector bundle of rank n over a smooth manifold M. We use the notation Γ(E) to denote the set of smooth sections σ : M → E of the vector bundle E. In the case E = T M, we have that σ ∈ Γ(T M) is a smooth vector field on M. A connection on a vector bundle in a generalization of the directional derivative in Rd. The precise definition is the following: Definition 2.1. Let M be a smooth manifold, and π : E → M a vector bundle. A connection on E is a map D : Γ(T M) × Γ(E) → Γ(E), (X, σ) 7→ DX σ such that for any τ, σ ∈ Γ(E), X,Y ∈ Γ(T M), and f ∈ C∞(M), the following four properties hold: (1) DX+Y σ = DX σ + DY σ, (2) DfX σ = fDX σ, (3) DX (σ + τ)= DX σ + DX τ, (4) DX (fσ)= X(f)σ + fDX σ. Rule (4) is called the Leibniz rule. We start by investigating the local properties of D. Given any point q ∈ M, by the definition of a vector bundle, there exists a neighbourhood U containing q and a diffeomorphism Φ : π−1(U) → U ×Rn that acts linearly on the fibers. (Φ is called a local n trivialization.) If {e1,...,en} is the standard basis of R , we obtain local sections εi : M → E, −1 εi(p) = Φ (p, ei) such that at each p ∈ U, {εi(p)} forms a basis of the fiber Ep. We can thus write any section σ : M → E in the form i σ(p)= σ (p)εi(p), where the σi are functions on U. We define the trivial connection D0 as follows: 0 i DX (σ) := X(σ )εi. It is easy to check that D0 indeed defines a connection. Now, take any given connection D. Locally, we can define the difference 3 0 A(X) := DX − DX , A : Γ(T M) → C∞(M, End(Rn)), X 7→ A(X). Hence for each smooth vector field X, we obtain A(X), which is a matrix-valued function on i M. Said in another way, given p ∈ M, (A(X))(p) acts on a vector σ (p)εi(p). We can check ∞ i i that A is C -linear. Indeed, we have A(X)(σ εi)= σ A(X)εi, since the derivative terms in the Leibniz rule cancel: i i i i i 0 A(X)(σ εi)= X(σ )εi − X(σ )εi − σ DX εi = σ (DX − DX )εi. Conversely, any map A : Γ(T M) → C∞(M, End(Rn)) that is C∞-linear defines a connection 0 DX = DX + A(X). Therefore, the maps A classify connections on a trivial bundle. We thus obtain the following local decomposition for any connection D: 0 DX = DX + A(X) (1) . Written out explicitly, under this decomposition we have i j DX σ = X(σ )εi + Aij(X)σ εi We now define the curvature operator corresponding to a connection D. Definition 2.2. Let D be a connection on the vector bundle E → M. Given vector fields V, W ∈ Γ(T M), we define the curvature F (V, W ):Γ(E) → Γ(E) in the following way: F (V, W )σ = DV DW σ − DW DV − D[V,W ]σ. i We investigate the form in F in local coordinates. The first claim is that if we have V = V ∂i, i j k W = W ∂i, we can write F (V, W )= V W Fjk, where Fjk := F (∂j, ∂k). 4 Since partial derivatives commute, we have [∂j , ∂k] = 0, and hence Fjk = DjDk − DkDj, (2) where for convenience, we denote Di := D∂i . Before launching into the computation, we note how to write the Lie bracket in coordinates: i k i k [V, W ]=(V ∂iW − W ∂iV )∂k. We now compute i k i k k i i k i k F (V ∂i, W ∂k)σ = DV ∂i DW ∂k σ − DW ∂k DV ∂i σ − D(V ∂iW −W ∂iV )∂k σ k i i k i k i k = DV ∂i W Dkσ − DW ∂k V Diσ − (V ∂iW )Dkσ − (W ∂iV )Dkσ i k i k k i k i = V ∂iW Dkσ + V W DiDkσ − W ∂kV Diσ − W V DkDiσ i k i k − V ∂iW Dkσ − W ∂iV Dkσ i k = V W (DiDkσ − DkDiσ) i k = V W F (∂i, ∂k)σ. j k This shows that F (V, W )= V W Fjk. i The next step is to write Fjk in terms of A. As usual, locally we write σ = σ εi. Using the decomposition from (1), and the notation Ai = A(∂i), we compute i Fjkσ =(DjDk − DkDj)σ εi 0 0 i 0 0 i =(Dj + Aj)(Dk + Ak)σ εi − (Dk + Ak)(Dj + Aj)σ εi 0 i i 0 i i =(Dj + Aj)(∂kσ εi + Akσ εi) − (Dk + Ak)(∂jσ εi + Ajσ εi) i i i i i i i i = ∂j∂kσ εi + ∂j(Akσ )εi + Aj∂kσ εi + AjAkσ εi − ∂k∂jσ εi − ∂k(Ajσ )εi − Ak∂jσ εi − AkAjσ εi i i i i i i i i =(∂jAk)σ εi + Ak∂jσ εi + Aj∂kσ εi + AjAkσ εi − (∂kAj)σ εi − Aj∂kσ εi − Ak∂jσ εi − AkAjσ εi i =(∂jAk − ∂kAj + AjAk − AkAj)σ εi. We have thus derived the following useful local expression: i Fjkσ =(∂jAk − ∂kAj + [Aj, Ak])σ εi. (3) 5 Written this way, it is easy to see the skew-symmetry of the curvature operator: Fjk = ∂jAk − ∂kAj + [Aj, Ak] = −(∂kAj − ∂jAk − [Aj, Ak]) = −(∂kAj − ∂jAk + [Ak, Aj]) = −Fkj. There is a different way to look at the curvature operator F . The ultimate goal is to gener- alize the definition of the connection D in a way akin to the exterior derivative d. The rest of this section introduces new notation that will be used throughout the document. We start by rewriting the curvature operator, and then move on to generalize the connection D. We rewrite the curvature operator in the following way: j k F = Fjk ⊗ dx ∧ dx . (4) We use the so-called “Alt convention” for the wedge product; for covectors ωi ∈ Γ(T M ∗) and vector fields Vi ∈ Γ(T M), we have 1 ω1 ∧···∧ ωk(V ,...,V )= det(ωi(V )). i k k! j To justify (4), we compute for vector fields V, W j k i l F (V, W )= Fjk ⊗ dx ∧ dx (V ∂i ⊗ W ∂l) F = jk (V jW k − W jV k) 2 1 = (V jW kF − V kW jF ) 2 jk jk 1 = (V jW kF + V kW jF ) 2 jk kj j k = V W Fjk. Now that we have tweaked the definition of the curvature F , we revisit the definition of a connection D. Given a vector bundle E → M of rank n, we can construct the bundle End(E). If Ep is 6 the fiber over p ∈ M, we can look at the space of linear transformations on the vector space Ep. The fibers of End(E) are given by this space of linear transformations, and allows us to the define the vector bundle End(E) → M of rank n2. We want to define extend our definition of a connection D to allow it to act on sections of End(E). The motivation for the definition is to have a Leibniz rule; if T ∈ Γ(End(E)), σ ∈ Γ(E), and V is a smooth vector field, then we should have DV (T σ)=(DV T )σ + T (DV σ). With this motivation in mind, we define (DV T )σ := DV (T σ) − T (DV σ). (5) Locally, if we write D in its decomposition D = D0 + A, we obtain 0 0 (DjT )σ =(Dj + Aj)(T σ) − T ((Dj + Aj)σ)=(∂jT )σ + T ∂jσ + AjT σ − T ∂jσ − T Ajσ. After cancellation, we see that DjT = ∂jT + [Aj, T ].