Geometric Properties of Classical Yang-Mills Theory on Differentiable Manifolds

Abstract

Gauge theories make up a class of physical theories that attempt to describe the physics of particles at a fundamental level. The purpose here is to study Yang-Mills theory at the classical level in terms of the geometry of fiber bundles and differentiable manifolds. It is shown how fundamental particles of bosonic and fermionic nature can be described mathematically. The Lagrangian for the basic interactions is presented and then put together in a unified form. Finally, some basic theorems are proved for a Yang-Mills on compact four-dimensional manifolds.

1. Introduction

In 1954 C. N. Yang and R. Mills proposed a classical field theory that incorporates Lie groups at a fundamental level [1]. Since then, great progress has been made in the area of subatomic physics by realizing that physics which is described by non-abelian Lie groups can display many novel features and play a major role in the kinds of physical theories they describe [2, 3, 4, 5, 6, 7]. These features are alluded to having no classical analogu. When formulated using rigorous mathematics, Yang-Mills theories as well as gauge theories make elegant use of complicated structures called fiber bundles and associated vector bundles. These are indispensable in physics where spacetime, the base manifold has a non-trivial topology. This occurs in string theory for example spacetime is usually assumed to be a product ${\mathbb{M}}^4\times K$ of Minkowski spacetime with a compact Riemannian manifold. If Euclidean spacetime ${\mathbb{R}}^4$ is compactified to the $4$-sphere $S^4$, a similar situation applies [8, 9, 10, 11, 12, 13, 14]. Fields in spacetime often cannot be described simply by a map to a fixed vector space but as sections of a non-trivial vector bundle. In these cases, fields on spacetime often cannot be described simply by a map to a fixed vector space, but rather as sections of a nontrivial vector bundle [15, 16, 17, 18, 19, 20].

The Lagrangian and action of a field theory should be invariant under the action of certain symmetry groups such as the Lorentz group, gauge symmetry, and conformal symmetry [21, 22, 23, 24, 25, 26]. This means the Lagrangian for the fields, hence the laws of physics, are invariant under symmetry transformations. For spontaneously broken gauge theories, the Lagrangian is invariant under gauge transformations that originate in a Lie group. The Higgs condensate yields a vacuum configuration invariant only under a subgroup of $G$, and at the same time provides a mechanism for giving mass to particles. Although the quantum versions of theories are not discussed here, it is important to state that symmetries of the classical theory, such as gauge symmetries, do not necessarily hold in the quantized theory. The main reason is the quantization method, such as the path integral measure may not be invariant under the symmetry [27, 28, 29, 30, 31]. In this event, the theory is said to be anomalous.

In mathematical terms, suppose $\pi :E\to M$ is a surjective differentiable map between smooth manifolds. If $x\in M$ is an arbitrary point, the nonempty subset $E_x={\pi }^{-1}\left(x\right)\subset E$ is called the fiber of $\pi$ over $x$. For a subset $U\subset M$ we set $E_U={\pi }^{-1}\left(U\right)\subset E$, the part of $E$ above $U$, and it is the union of all fibers $E_x$, where $x\in U$. A differentiable map $s:M\to E$ such that $\pi \circ s=I_M$ where $I_M$ is the identity map is called a global section of $\pi$. A differentiable map $s:U\to E$, defined on some open subset $U\subset M$ satisfying $\pi \circ s=I_U$ is called a local section. A differentiable map $s:U\to E$ is a local section of $\pi :E\to U$ if and only if $s\left(x\right)\in E_x$, for all $x\in U$. Fiber bundles are an important generalization of products $E=M\times F$ and can be understood as twisted products. The fibers are still embedded submanifolds and are all diffeomorphic. The fibration in general is only locally trivial, so locally a product which is not global.

Definition 1.1 Let, $E$, $F$, $M$ be manifolds and $\pi :E\to M$ a surjective differentiable map. Then, $\left(E,\pi ,M;F\right)$ is called a fiber bundle if: for every $x\in M$, there exists an open $U\subset M$ around $x$ such that $\pi$ restricted to $E_U$ can be trivialized, so there is a diffeomorphism ${\varphi }_U:E_U\to U\times F$ such that ${\mathrm{pr}}_1\circ {\varphi }_U=\pi$. Denote a fiber bundle as $F\to E\to M$, $E$ is called the total space, $M$ the base manifold, $F$ the general fiber, $\pi$ the projection and $\left(U,{\varphi }_U\right)$ a bundle chart.

Using a bundle chart, $\left(U,{\varphi }_U\right)$, the fiber $E_x={\pi }^{-1}\left(x\right)$ is seen to be an embedded submanifold of the total space $E$ for every $x\in M$, and ${\left.{\varphi }_{U_2}={\mathrm{pr}}_2\circ {\varphi }_U\right|}_{E_x}:E_x\to F$ is a diffeomorphism between the fiber over $x\in U$ and the general fiber. For physical reasons, it is essential to include pseudo-Riemannian metrics in the picture. Let $M$ be a smooth manifold. A pseudo-Riemannian metric $g$ of signature $\left(s,t\right)$ where $\left(+,\cdots +,-,\cdots ,-\right)$ is a section that defines at each $x\in M$ a non-degenerate symmetric bilinear form $g_x:T_{x}M\times T_{x}M\to \mathbb{R}$ of signature $\left(s,t\right)$.

Principal fiber bundles are a combination of the concepts of fiber bundle and group action; that is, fiber bundles have a Lie group action such that both structures can be made compatible. Let $G\to P\to M$ be a fiber bundle with general fiber a Lie group $G$ and a smooth action $P\times G\to P$ on the right. For a principal $G$-bundle, the action of $G$ preserves the fibers of $\pi$ and is simply transitive on them. The orbit map $G\to P$ such that $g\to p\cdot g$ is a bijection for all $x\in M$, $p\in P_x$. There exists a bundle atlas of $G$- equivariant bundle charts ${\varphi }_i:P_{U_i}\to U_i\times G$ satisfying ${\varphi }_i\left(p\cdot g\right)={\varphi }_i\left(p\right)\cdot g$, for all $p\in P_{U_i}$, $g\in G$, where on the right $G$ acts on pairs $\left(a,x\right)\in U_i\times G$ by $\left(x,a\right)\cdot g=\left(x,\mathit{ag}\right)$. The group $G$ is called the structure group $P$. Two features distinguish a principal bundle $P\to M$ from a standard fiber bundle whose general fiber is a Lie group $G$: there exists a right $G$-action on $P$ simply transitive on each fiber $P_x$, $x\in M$ and bundle $P$ has a principal bundle atlas. If $P\to M$ is a principal $G$-bundle, $p\in P$, $g\in G$, ${\tau }_g$ denotes the right translation $p\to p\cdot g$. The fiber $P_x$ is a submanifold of the total space $P$ for every $x\in M$ and the orbit map $g\to p\cdot g$ is an embedding for all $p\in P_x$.

A fiber bundle $V\to E\to M$ is called a real or complex vector bundle of rank $m$ if: The general fiber $V$ and every fiber $E_x$ for $x\in M$, are $m$-dimensional vector space over $\mathbb{K}=\mathbb{R}$ or $\mathrm{ℂ}$, and there exists a bundle atlas ${\left\{\left(U_i,{\varphi }_i\right)\right\}}_{i\in I}$ for $E$ such that induced maps ${\varphi }_{\mathit{ix}}:E_x\to V$ are vector space isomorphisms for all $x\in U_i$. Such an atlas is called a vector bundle atlas for $E$, and the chart a vector bundle chart. There are two features that distinguish a vector bundle $E\to M$ from a standard fiber bundle; the vector space structure on each fiber $E_x$, $x\in M$ and the bundle $E$ has a vector bundle atlas. An example of this is the tangent bundle of a smooth manifold which is canonically a smooth real vector bundle [32, 33].

Definition 1.2 Let $G$ be a Lie group and $M$ a Manifold. Suppose that $M\times G\to M$ is a right action. For $X\in g_L$ we define the associated fundamental vector field $\stackrel{\sim }{X}$ on $M$ by

$${\left.{\stackrel{\sim }{X}}_p=\frac{d}{\mathit{dt}}\right|}_{t=0}(p\cdot exp\left(\mathit{tX}\right).$$E1

If we denote by ${\varphi }_p$, the orbit map for the right action, ${\varphi }_p:G\to M$, $g\to p\cdot g$, then

$${\stackrel{\sim }{X}}_p=\left(D_e{\varphi }_p\right)\left(X_p\right).$$E2

Similarly, suppose that $G\times M\to M$ is a left action. Then we define the fundamental vector field by

$${\left.{\stackrel{\sim }{X}}_p=\frac{d}{\mathit{dt}}\right|}_{t=0}\left(exp\left(-\mathit{tX}\right)\cdot p\right),$$E3

for $p\in M$. If we denote by ${\varphi }_{p^{\prime }}$ the following orbit map for the left action, ${\varphi }_p^{\prime }:G\to M$, $g\to g^{-1}\cdot p$, then

$${\stackrel{\sim }{X}}_p=\left(D_e{\varphi }_p^{\prime }\right)\left(X_e\right).$$E4

The fundamental vector field will also be denoted $X_f$ when the presentation requires.

It is shown here that a physical theory can be constructed based on the idea of a differentiable manifold along with many other associated mathematical structures that can be defined on it. The result is a theory which can be used to describe fundamental interactions of elementary particles at the classical level. This also permits the introduction of other ideas which can have a physical influence such as topological invariants. There is no discussion with regard to quantization of gauge theories. These interactions include the strong and weak forces. Physically, Yang-Mills fields represent forces or carriers of force. The first half of the paper introduces most of the mathematical concepts needed to describe particles of both fermionic and bosonic nature. The last part specializes to Yang-Mills in four dimensions. It is discussed how the field equations can be obtained from a variational principle and how the theory of partial differential equations plays a role in their study.

2. Matter fields and couplings

Lie groups appear in principal bundles in gauge theories. These are associated to vector bundles which describe particles and where representations on vector spaces are built into gauge theories. Connections are associated with gauge fields and give rise to covariant derivatives representing interactions.

Definition 2.1 A connection one-form on a principal $G$ bundle $\pi :P\to M$ is a one-form $A\in {\mathrm{\Omega }}^1\left(P,g\right)$ on the total space $P$ so that $r_g^{\ast }A={\mathit{Ad}}_{g^{-1}}A$ for all $g\in G$ and $A\left(\stackrel{\sim }{X}\right)=X$ for all $X\in g_L$, where $\stackrel{\sim }{X}$ is the fundamental vector field associated to $X$ and $g_L$ the Lie algebra of $G$. This is called a gauge field in physics. $\square$.

At $p\in P$, a connection one-form is a linear map $A_p:T_{p}P\to g_L$ and ${\mathit{Ad}}_{g^{-1}}$ is a linear isomorphism of the Lie algebra to itself. There is a correspondence between Ehresmann connections and connection one-forms. Physically we want certain objects to be gauge invariant. A global gauge transformation is a bundle automorphism of $P$ or a diffeomorphism $f:P\to P$ which preserve the fibers of $P$ and is $G$-equivariant

$$\pi \circ f=\pi ,f\left(p\cdot g\right)=f\left(p\right)\cdot g,p\in P,g\in G.$$E5

Under composition of diffeomorphisms, the set of all gauge transformations forms a group $\mathrm{Aut}\left(P\right)$. A local gauge transformation is a bundle automorphism denoted $\mathrm{Aut}\left(P\right)$. In physics, gauge transformations are often defined as maps on the base manifold $M$ to the structure group $G$ even for non-abelian Lie groups.

Let $\pi :P\to M$ be a principal $G$-bundle. A physical gauge transformation is a smooth map $\pi :U\to G$ defined on an open set $U\subset M$. The set of all physical gauge transformations forms a group ${\mathcal{C}}^{\infty }\left(U,G\right)$ with pointwise multiplication.

If $s:U\to P$ is a local gauge of the principal bundle on an open subset $U\subset M$, the local connection one-form or local gauge field $A_s\in {\mathrm{\Omega }}^1\left(U,g_L\right)$ determined by $s$ is defined as

$$A_s=A\circ Ds=s^{\ast }A.$$E6

Suppose we have a manifold chart on $U$ and ${\left\{{\partial }_{\mu }\right\}}_{\mu =1,\cdots ,n}$ are the local coordinate vector fields on $U$. Set $A_{\mu }=A_s\left({\partial }_{\mu }\right)$ and choose a basis $\left\{e_a\right\}$ for the Lie algebra $g_L$ and then expand $A_{\mu }$ over that basis

$$A_{\mu }=\sum _{a=1}^{dim{g}_L}{A}_{\mu }^a{e}_a.$$E7

The corresponding real-valued fields $A_{\mu }^a\in {\mathcal{C}}^{\infty }\left(U,\mathbb{R}\right)$ and one forms $A_s^a$ are called local gauge boson fields in physics.

A principal bundle can have many gauges and it is of interest to determine how the local connection one-forms transform as we change the local gauge. Let $s_i:U_i\to P$ and $s_j:U_j\to P$ be local gauges with $U_i\cap U_j\ne \varnothing$, then there exists $g_{\mathit{ij}}\left(x\right):U_i\cap U_j\to G$ such that

$$s_{\mathit{ij}}\left(x\right)=s_j\left(x\right)\cdot g_{\mathit{ji}}\left(x\right),x\in U_i\cap U_j.$$E8

In (8), $g_{\mathit{ij}}$ is the smooth transition function between local trivializations. There are local connection $1$-forms $A_i\in A_{s_i}\in {\mathrm{\Omega }}^1\left(U_i,g_L\right)$ and $A_j\in A_{s_j}\in {\mathrm{\Omega }}^1\left(U_j,g_L\right)$ and it is desired to obtain the relationship between $A_i$ and $A_j$. If ${\mu }_G\in {\mathrm{\Omega }}^1\left(G,p\right)$ is the Mauer-Cartan form defined as ${\mu }_G\left(v\right)=D_g{L}_{g^{-1}}\left(v\right)$ for $v\in T_{g}G$, set ${\mu }_{\mathit{ji}}\in g_{\mathit{ji}}^{\ast }{\mu }_G\in {\mathrm{\Omega }}^1\left(U_i\cap U_j\right)$. The theorem which follows accounts for the transformation of local gauge fields.

Theorem 2.1 The local connection one-form transforms as

$$A_i={\mathit{Ad}}_{g^{-1}}\circ A_j+{\mu }_{\mathit{ji}}$$E9

on $U_i\cap U_j$. If $G\subset \mathbb{G}L\left(n,\mathbb{K}\right)$ is a matrix Lie group then

$$A_i=g_{\mathit{ji}}^{-1}\cdot A_j\cdot g_{\mathit{ji}}+g_{\mathit{ji}}^{-1}\cdot {\mathit{dg}}_{\mathit{ji}}$$E10

where $\cdot$ denotes matrix multiplication, $g_{\mathit{ji}}^{-1}$ the inverse of $g_{\mathit{ji}}$ in $G$ and ${\mathit{dg}}_{\mathit{ji}}$ the differential of each component of the function $g_{\mathit{ji}}:U_i\cap U_j\to G\subset {\mathbb{K}}^{n\times n}$. If $G$ is abelian, then $A_i=A_j+{\mu }_{\mathit{ji}}=A_j+g_{\mathit{ji}}^{-1}\cdot {\mathit{dg}}_{\mathit{ji}}$.

Proof: Let $s\in U_i\cap U_j$ and $Z\in T_{x}M$ and set

$$X=D_x{s}_j\left(Z\right)\in T_{s_j\left(x\right)}P,Y\in D_x{g}_{\mathit{ji}}\left(Z\right)\in T_{g_{\mathit{ji}}\left(x\right)}G.$$E11

with group action $\mathrm{\Phi }:P\times G\to P$ given as $\left(p,g\right)\to \mathit{pg}$, we calculate using the differential of map $\mathrm{\Phi }$$\left(X,Y\right)\to \left(D_X{r}_g\right)\left(X\right)+{\mu }_G{\left(Y\right)}_{\mathit{xg}}$, where $r_g$ is right translation ${\mu }_G$ is the Mauer-Cartan form, and the chain rule

$$D_x{s}_i\left(Z\right)=D_x\left(\mathrm{\Phi }\circ \left(s_j,g_{\mathit{ji}}\right)\right)\left(Z\right)=D_{s_j\left(x\right)}{r}_{g_{\mathit{ji}}}\left(X\right)+{\mu }_G{\left(Y\right)}_{s_j\left(x\right)}{\left|{}_f=D_{s_j\left(x\right)}{r}_{g_{\mathit{ji}}\left(x\right)}\left(X\right)+{\mu }_{\mathit{ji}}{\left(Z\right)}_{s_i\left(x\right)}\right|}_f.$$E12

By the defining properties of the connection form $A$, we have

$$\begin{array}{c}{\left.A_i\left(Z\right)=A\left(D_x{s}_i\left(Z\right)\right)=A(D_{s_j\left(x\right)}{r}_{g_{\mathit{ji}}\left(x\right)}\left(X\right)+{\mu }_{\mathit{ji}}{\left(Z\right)}_{s_i\left(x\right)}\right|}_f)=\left(r_{g_{j}i}^{\ast }\left(x\right)A\right)\left(X\right)+{\mu }_{\mathit{ji}}\left(Z\right)\end{array}$$

$$={\mathit{Ad}}_{g_{\mathit{ji}}\left(x\right)}^{-1}\circ A_j\left(Z\right)+{\mu }_{\mathit{ji}}\left(Z\right).$$E13

The second claim follows by recalling that for a matrix Lie group ${\mathit{Ad}}_{g^{-1}}\cdot a\cdot g$ for all $g\in G$, $a\in g_L$ and ${\mu }_G\left(v\right)=g^{-1}v$ for $v\in T_{g}G$

$${\mu }_{\mathit{ji}}\left(Z\right)={\mu }_G\left(D_x{g}_{\mathit{ji}}\left(Z\right)\right)=g_{\mathit{ji}}^{-1}\cdot {\mathit{dg}}_{\mathit{ji}}\left(Z\right).$$E14

Theorem 2.2 Let $P\to M$ be a principal bundle and $A\in {\mathrm{\Omega }}^1\left(P,g_L\right)$ a connection one-form on $P$. Suppose that $f\in \mathcal{G}\left(P\right)$ is a global bundle isomorphism. Then $f^{\ast }A$ is a connection one-form on $P$

$$f^{\ast }A=\mathit{Ad}{\sigma }_{f^{-1}}\circ A+{\sigma }_f^{\ast }{\mu }_G.$$E15

Proof: This follows from the definition of a connection $1$-form and the previous Theorem.

Let $H$ be the associated horizontal vector bundle defined as the kernel of $A$. Then $\mathit{TP}=V\oplus H$ and we set ${\pi }_H:\mathit{TP}\to H$ for the projection onto the horizontal vector bundle.

Definition 2.2 The two-form $F\in {\mathrm{\Omega }}^2\left(P,g_L\right)$ defined by

$$F\left(X,Y\right)=dA\left({\pi }_H\left(X\right),{\pi }_H\left(Y\right)\right),X,Y\in T_{p}P,p\in P$$E16

is called the curvature two form of $A$. Sometimes $F^A$ is written to emphasize the dependence on $A$. $\square$

Definition 2.3 Let $P$ be a manifold and $g_L$ a Lie algebra. For $\eta \in {\mathrm{\Omega }}^1\left(P,g_L\right)$ and $\varphi \in {\mathrm{\Omega }}^1\left(P,g_L\right)$, define $\left[\eta ,\varphi \right]\in {\mathrm{\Omega }}^{k+l}\left(P,g_L\right)$ to be

$$\left[\eta ,\varphi \right]\left(X_1,\dots ,X_{k+l}\right)=\frac{1}{k!l!}\sum _{\sigma \in S_{k+l}}sgn\left(\sigma \right)\left[\eta \left(X_{\sigma \left(1\right)},\dots ,X_{\sigma \left(k\right)}\right),\varphi \left(X_{k+1},\dots ,X_n\right)\right],$$E17

where the commutators on the right are the commutator in the Lie algebra $g_L$. This is often written $\eta \wedge \varphi$ as well.

It is useful to recall that if $X=\stackrel{\sim }{V}$ be a fundamental vector field and $Y$ a horizontal vector field on $P$, then the commutator $\left[X,Y\right]$ is horizontal.

Theorem 2.3 (Structure Equations) The curvature form $F$ of a connection form $A$ satisfies

$$F=dA+\frac{1}{2}\left[A,A\right].$$E18

Proof: Eq. (18) can be checked by inserting $X,Y\in T_{p}P$ on both sides. Suppose $X,Y$ are both vertical. Then $X,Y$ are fundamental vectors $X={\stackrel{\sim }{V}}_p$, $Y={\stackrel{\sim }{W}}_p$ forcertain elements $V,W\in G_L$,

$$F\left(X,Y\right)=\mathit{dA}\left({\pi }_H\left(X\right),{\pi }_H\left(Y\right)\right)=0,\frac{1}{2}\left[A,A\right]\left(X,Y\right)=\left[A\left(X\right),A\left(Y\right)\right]=\left[V,W\right].$$E19

The differential of a one-form $A$ is given by

$$\mathit{dA}\left(X,Y\right)=L_X\left(A\left(Y\right)\right)-L_Y\left(A\left(X\right)\right)-A\left(\left[X,Y\right]\right),$$E20

where vectors $X,Y$ are extended to vector fields in a neighborhood of $p$. If the extension is chosen by the fundamental vector fields $\stackrel{\sim }{V}$ and $\stackrel{\sim }{W}$, then $\mathit{dA}\left(X,Y\right)=L_X\left(W\right)-L_Y\left(V\right)-\left[V,W\right]=-\left[V,W\right]$, since $V,W$ are constant maps from $P$ to $g_L$.

If both $X$ and $Y$ are horizontal $F\left(X,Y\right)=\mathit{dA}\left(X,Y\right)$ and $\frac{1}{2}\left[A,A\right]\left(X,Y\right)=\left[A\left(X\right),A\left(Y\right)\right]=0$.

If $X$ is vertical and $Y$ is horizontal, then $X={\stackrel{\sim }{V}}_p$ for some $V\in g_L$, and we have

$$F\left(X,Y\right)=\mathit{dA}\left({\pi }_H\left(X\right),{\pi }_H\left(Y\right)\right)=\mathit{dA}\left(0,Y\right)=0,\frac{1}{2}\left[A,A\right]\left(X,Y\right)=\left[A\left(X\right),A\left(Y\right)\right]=\left[V,0\right]=0.$$E21

Thus since $\stackrel{\sim }{V},Y]$ is horizontal

$$\mathit{dA}\left(X,Y\right)=L_{\stackrel{\sim }{V}}\left(A\left(Y\right)\right)-L_Y\left(V\right)-A\left(\left[\stackrel{\sim }{V},Y\right]\right)=-A\left(\left[\stackrel{\sim }{V},Y\right]\right)=0.$$E22

Connections define an important idea in geometry: that of parallel transport in principal and associated vector bundles and leads to the concept of covariant derivative on an associated vector bundle. An interesting result is that if $X=\stackrel{\sim }{V}$ be a fundamental vector field and $Y$ a horizontal vector field on $P$, then the commutator $\left[X,Y\right]$ is horizontal. In a similar way, $F$ can be written locally as was done for the local section. If we have a manifold chart on $U$ and $\left\{{\partial }_i\right\}$ are local coordinate basis vector fields on $U$, then $F_{\mathit{\mu \nu }}=F_s\left({\partial }_{\mu },{\partial }_{\nu }\right)$ and $F_{\mathit{\mu \nu }}={\sum }_{a=1}^{dim{g}_L}{F}_{\mathit{\mu \nu }}^a{e}_a$ and locally the structure equations take the form $F_{\mathit{\mu \nu }}={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+\left[A_{\mu },A_{\nu }\right]$.

Definition 2.4 Let $\gamma :\left[a,b\right]\to M$ be a curve in $M$. The map

$${\mathrm{\Pi }}_g^A:P_{\gamma \left(a\right)}\to P_{\gamma \left(b\right)},p\to {\gamma }_p^{\ast }\left(b\right),$$E23

is called parallel transport in the principal bundle $P$ along $\gamma$ with respect to the connection $A$.

Similarly for a curve $\gamma :\left[0,1\right]\to M$ the map ${\mathrm{\Pi }}_{\gamma }^{E,A}:E_{\gamma \left(0\right)}\to E_{\gamma \left(1\right)}$, given by $\left[p,v\right]\to \left[{\mathrm{\Pi }}_{\gamma }^A\left(p\right),v\right]$ is a well-defined and linear isomorphism, called parallel transport in the associated vector bundle $E$ along the curve $\gamma$ with respect to $A$. Let $\mathrm{\Phi }$ be a section of $E$, $x\in M$ and $X\in T_{x}M$ a tangent vector. A covariant derivative is to be defined by choosing an arbitrary curve $\gamma :\left(-\epsilon ,\epsilon \right)\to M$ with $\gamma \left(0\right)=x$, $\dot{\gamma }\left(0\right)=X$. For each $u\in \left(-\epsilon ,\epsilon \right)$ parallel transport the vector $\mathrm{\Phi }\left(\gamma \left(u\right)\right)\in E_{\gamma \left(u\right)}$ back to $E_x$ along $\gamma$. Take the derivative at $u=0$ of the curve which results in $E_x$ giving an element in $E_x$. Formally, set

$${\left.D\left(\mathrm{\Phi },\gamma ,x,A\right)=\frac{d}{\mathit{du}}\right|}_{u=0}{\left({\mathrm{\Pi }}_{{\gamma }_u}^{E,A}\right)}^{-1}(\mathrm{\Phi }\left(\gamma \left(u\right)\right)\in E_x.$$E24

The restriction of the curve $\gamma$ starting at $0$ and ending at time $u$ for $u\in \left(-\epsilon ,\epsilon \right)$ is denoted ${\gamma }_u$. Parallel transport ${\mathrm{\Pi }}_{\gamma }^A$ is a smooth map between the fibers $P_{\gamma \left(a\right)}$ and $P_{\gamma \left(b\right)}$ and does not depend on the parametrization of the curve. Let $\gamma$ be a curve in $M$ from $x$ to $y$ and ${\gamma }^{\prime }$ a curve from $y$ to $z$. Denote $\gamma$ acting followed by ${\gamma }^{\prime }$ by ${\gamma }^{\ast }{\gamma }^{\prime }$, where $\gamma$ comes first, then ${\mathrm{\Pi }}_{\gamma \circ {\gamma }^{\prime }}^A={\mathrm{\Pi }}_{{\gamma }^{\prime }}^A\circ {\mathrm{\Pi }}_{\gamma }^A$.

Theorem 2.4 Let $s:U\to P$ be a local gauge $A_s=s^{\ast }A$ and $\varphi :U\to V$ the map with $\mathrm{\Phi }=\left[s,\varphi \right]$. Then the vector $D\left(\mathrm{\Phi },\gamma ,x,A\right)\in E_x$ is given by

$$D\left(\mathrm{\Phi },\gamma ,x,A\right)=\left[s\left(x\right),\mathit{d\varphi }\left(X\right)+{\rho }_{\ast }\left(A_s\left(X\right)\right)\varphi \left(x\right)\right].$$E25

Proof: It holds that

$${\left({\mathrm{\Pi }}_{{\gamma }_t}^{E,A}\right)}^{-1}\left(\mathrm{\Phi }\left(\gamma \left(t\right)\right)\right)=\left[{\left({\mathrm{\Pi }}_{{\gamma }_t}^A\right)}^{-1}\right(s\left(\gamma \left(t\right)\right),\varphi \left(\gamma \left(H\right)\right)].$$E26

Let $q\left(t\right)$ be the unique smooth curve determined in the fiber $P_x$ such that ${\mathrm{\Pi }}_{\gamma \left(t\right)}^A\left(q\left(t\right)\right)=s\left(\gamma \left(t\right)\right)$. Write $q\left(t\right)=s\left(x\right)\cdot g\left(t\right)$ and $g\left(t\right)$ is a uniquely determined smooth curve in $G$

$${\left({\mathrm{\Pi }}_{{\gamma }_i}^{E,A}\right)}^{-1}\left(\mathrm{\Phi }\left(\gamma \left(t\right)\right)\right)=\left[q\left(t\right),\mathrm{\Phi }\left(\gamma \left(t\right)\right)\right]=\left[s\left(x\right),\rho \left(g\left(t\right)\right)\varphi \left(\gamma \left(t\right)\right)\right].$$E27

For $t=0$, we have

$$s\left(x\right)=s\left(\gamma \left(0\right)\right)={\mathrm{\Pi }}_{{\gamma }_0}^A\left(q\left(0\right)\right)=q\left(0\right),g\left(0\right)=a\in G.$$E28

Consequently, $\dot{g}\left(0\right)\in g_L$, and it follows that

$${\left.D\left(\mathrm{\Phi },\gamma ,x,A\right)=\frac{d}{\mathit{dt}}\right|}_{t=0}\left[s\left(x\right),\rho \left(g\left(t\right)\right)\varphi \left(\gamma \left(t\right)\right)\right]=\left[s\left(x\right),{\rho }_x\left(\dot{g}\left(0\right)\right)\varphi \left(x\right)+\mathit{d\varphi }\left(X\right)\right].$$E29

To finish, ${\rho }_r\left(\dot{g}\left(0\right)\right)$ is calculated,

$${\left.\frac{d}{\mathit{dt}}\right|}_{t=0}s\left(\gamma \left(t\right)\right)=\mathit{ds}\left(X\right),\frac{d}{\mathit{dt}}{\left|{}_{t=0}{\mathrm{\Pi }}_{{\gamma }_i}^A\left(g\left(t\right)\right)=\dot{q}\left(0\right)+\frac{d}{\mathit{dt}}\right|}_{t=0}{\mathrm{\Pi }}_{{\gamma }_i}^A\left(s\left(x\right)\right).$$E30

Since the curve ${\mathrm{\Pi }}_{{\gamma }_t}^A\left(s\left(x\right)\right)$ is horizontal, with respect to $A$, we obtain $A_s\left(x\right)=A\left(\mathit{ds}\left(X\right)\right)=A\left(\dot{q}\left(0\right)\right)$. Since $\dot{q}\left(0\right)$ and $\dot{g}{\left(0\right)}_{s\left(x\right)}$ are related by ${\varphi }_{\ast }$, the map that associates to a Lie algebra element the corresponding vector field on $M$ is a homeomorphism, hence, $A\left(\dot{q}\left(0\right)\right)=\dot{g}\left(0\right)$ by definition of connection one-form. It follows that

$${\rho }_{\ast }\left(\dot{g}\left(0\right)\right)={\rho }_{\ast }\left(A_s\left(X\right)\right),$$E31

and so the claim.

In fact, the theorem implies that $D\left(\mathrm{\Phi },\gamma ,x,A\right)$ depends only on the tangent vector $X$ not on the curve $\gamma$ itself. Now we are in a position to define the covariant derivative.

Definition 2.5 Let $\mathrm{\Phi }$ be a section of an associated vector bundle $E$ and $X\in \mathcal{X}\left(M\right)$ a vector field on $M$. The covariant derivative ${\nabla }_X^A\mathrm{\Phi }$ of the section of $E$ defined by

$$\left({\nabla }_X^A\mathrm{\Phi }\right)\left(x\right)=D\left(\mathrm{\Phi },\gamma ,x,A\right),$$E32

where $\gamma$ is any vector through $X_x$ tangent to $\gamma$. The covariant derivative is a map ${\nabla }^A:\mathrm{\Gamma }\left(E\right)\to {\mathrm{\Omega }}^1\left(M,E\right)$.

The fact that ${\nabla }^A\mathrm{\Phi }$ is a smooth one-form in ${\mathrm{\Omega }}^1\left(M,E\right)$ for every $\mathrm{\Phi }\in \mathrm{\Gamma }\left(E\right)$ is clear from the local formula. In physics the covariant derivative in a local gauge $s:U\to P$ with $\mathrm{\Phi }=\left[s,\varphi \right]$ is given as

$${\nabla }_X^A\mathrm{\Phi }=\left[s,{\nabla }_X^A\varphi \right]{\nabla }_X^A\varphi \left(x\right)=\mathit{d\varphi }\left(X_x\right)+{\rho }_{\ast }\left(A_i\left(X_x\right)\right)\varphi \left(x\right).$$E33

The map ${\nabla }^A$ is $\mathbb{K}$-linear in both entries and satisfies ${\nabla }_{\mathit{fX}}^A\mathrm{\Phi }=f{\nabla }_X^A\mathrm{\Phi }$ for all smooth functions $f\in {\mathcal{C}}^{\infty }\left(M,\mathbb{R}\right)$. The Leibnitz rule ${\nabla }_X^A\left(\lambda \mathrm{\Phi }\right)=\left(L_X\lambda \right)\mathrm{\Phi }+\lambda {\nabla }_X^A\mathrm{\Phi }$ holds for all smooth functions $\lambda \in {\mathcal{C}}^{\infty }\left(M,\mathbb{K}\right)$.

Suppose $\gamma :\left[0,1\right]\to M$ is a closed curve in $M$, $\gamma \left(0\right)=\gamma \left(1\right)=x$, a loop. Then parallel transport ${\mathrm{\Pi }}_{\gamma }^{E,A}$ is a linear isomorphism of the fiber $E_x$ to itself. This isomorphism is called the holonomy ${\mathit{Hol}}_{\gamma ,x}^E$ of the loop $\gamma$ in the basepoint $x$ with respect to the connection $A$. The Wilson loop is the map $W_{\gamma }^E$ that associates to a connection $A$ and loop $\gamma$ the number $W_{\gamma }^E\left(A\right)=\mathrm{Tr}\left({\mathit{Hol}}_{\gamma ,x}^E\left(A\right)\right)$.

The map ${\nabla }^A$ can be regarded as a generalization of the differential $d:{\mathcal{C}}^{\infty }\left(M\right)\to {\mathrm{\Omega }}^1\left(M\right)$. The differential $d$ can be identified with the covariant derivative on the trivial line bundle over $M$. The differential can be uniquely be extended in the standard way to an exterior derivative $d:{\mathrm{\Omega }}^k\left(M\right)\to {\mathrm{\Omega }}^{k+1}\left(M\right)$ by demanding $\mathit{dd}f=0$ for all $f\in {\mathcal{C}}^{\infty }\left(M\right)$ and $d\left(\alpha \wedge \beta \right)=\mathit{d\alpha }\wedge \beta +{\left(-1\right)}^k\alpha \wedge \mathit{d\beta }$ for $\alpha \in {\mathrm{\Omega }}^k\left(M\right)$ and $\beta \in {\mathrm{\Omega }}^l\left(M\right)$. This differential satisfies $d\circ d=0$ on all forms, and so the de Rham cohomology $H_{\mathbb{R}}^k\left(M\right)$ is well-defined for all $k$.

It is useful to show the covariant derivative can be extended similarly to an exterior covariant derivative

$$d_A:{\mathrm{\Omega }}^k\left(M,E\right)\to {\mathrm{\Omega }}^{k+1}\left(M,E\right).$$E34

This exterior covariant derivative, however, in general does not satisfy $d_A\circ d_A=0$. There is a well-defined vector product $\wedge :{\mathrm{\Omega }}^k\left(M\right)\times {\mathrm{\Omega }}^1\left(M\right)\to {\mathrm{\Omega }}^{k+1}\left(M,E\right)$ between standard differential forms, with values in $\mathbb{K}$, and differential forms with values in $E$. Here we get the product between a scalar in $\mathbb{K}$ and a vector in $E$, which is well-defined. Let $\omega$ be an element of ${\mathrm{\Omega }}^k\left(M,E\right)$ and choose a local basis $e_1,\dots ,e_r$ of $E$ over an open set $U\subset M$, then $\omega$ can be written

$$\omega =\sum _{i=1}^r{\omega }_i\otimes e_i,$$E35

with uniquely defined $k$-forms ${\omega }_i\in {\mathrm{\Omega }}^k\left(U\right)$.

It should be stated that the definition of forms can be extended by defining $\mathcal{C}\left(M,W\right)$ as the set of all smooth maps from $M$ into the vector space $W$, which has a canonical structure of a manifold, so that smooth maps are defined. A one-form on $M$ with values in $W$ is an alternating ${\mathcal{C}}^{\infty }\left(M\right)$ nonlinear map $\omega :\chi \left(M\right)\times \cdots \times \chi \left(M\right)\to {\mathcal{C}}^{\infty }\left(M,W\right)$. The set of all $k$-forms on $M$, and values in $W$ can be identified with ${\mathrm{\Omega }}^k\left(M,W\right)={\mathrm{\Omega }}^k\left(M\right){\otimes }_{\mathbb{R}}W$. It is said forms in ${\mathrm{\Omega }}^1\left(M,W\right)$ are twisted with $W$. Scalar product of twisted forms can be defined by choosing a local frame for $E$ over $U\subset M$ and expand $k$-forms $F$, $G$ twisted with $E$ as $F={\sum }_{i=1}^r{F}_i\otimes e_i$, $G={\sum }_{i=1}^r{G}_i\otimes e_i$ with $F_i,G_i\in {\mathrm{\Omega }}^k\left(U,\mathbb{R}\right)$. Set

$${⟨F,G⟩}_E=\sum _{i,j=1}^r⟨F_i,G_j⟩{⟨e_i,e_j⟩}_E$$E36

with Hodge star operator ${}^{\ast }:{\mathrm{\Omega }}^k\left(M,E\right)\to {\mathrm{\Omega }}^{n-k}\left(M,E\right)$ by ${}^{\ast }F={\sum }_{i=1}^r\left(\ast F_i\right)\otimes e_i$, and codifferential $d^{\ast }={\left(-1\right)}^{t+\mathit{nk}+1\ast }{d}^{\ast }$.

Definition 2.6 Let $\nabla$ be a covariant derivative on a vector bundle $E$. Define the exterior covariant derivative or differential $d_{\nabla }:{\mathrm{\Omega }}^k\left(M,E\right)\to {\mathrm{\Omega }}^{k+1}\left(M,E\right)$ by

$$d_{\nabla }\omega =\sum _{i=1}^r\left(d{\omega }_i\otimes e_i+{\left(-1\right)}^k{\omega }_i\wedge \nabla e_i\right).$$E37

If $\nabla ={\nabla }^A$ is the covariant derivative on an associated vector bundle determined by connection $A$ on a principal bundle, write $d_A=d_{\nabla }$. $\square$.

Theorem 2.5 The definition of $d_{\nabla }$ is independent of the choice of local basis $\left\{e_i\right\}$ for $E$.

Proof: Let $\left\{e_{i^{\prime }}\right\}$ be another local basis of $E$ over $U$. Then there exist unique functions $C_{\mathit{ji}}\in \mathcal{C}\left(U,\mathbb{K}\right)$ with $e_i^{\prime }={\sum }_{i=1}^r{C}_{\mathit{ji}}{e}_i$. The matrix $C$ with entries $C_{\mathit{ji}}$ is invertible. Let $C^{-1}$ be the inverse matrix with entries $C_{\mathit{ij}}^{-1}$ and define

$${\omega }_j^{\prime }=\sum _{l=1}^r{C}_{\mathit{lj}}^{-1}{\omega }_l.$$E38

Then $\omega ={\sum }_{i=1}^r{\omega }_i\otimes e_i={\sum }_{j=1}^r{\omega }_j^{\prime }\otimes e_j^{\prime }$. Now let us calculate

$$\begin{array}{c}\sum _{j=1}^r\left(d{\omega }_j^{\prime }\otimes e_j^{\prime }+{\left(-1\right)}^k{\omega }_j^{\prime }\otimes \nabla e_j^{\prime }\right)=\sum _{i,j,l=1}^r(d\left(C_{\mathit{lj}}^{-1}\right)\wedge C_{\mathit{ji}}{\omega }_l\otimes e_i+C_{\mathit{lj}}^{-1}{C}_{\mathit{ji}}d{\omega }_l\otimes e_i\end{array}$$

$$\begin{array}{c}+{\left(-1\right)}^k{C}_{\mathit{lj}}^{-1}{\omega }_l\wedge {\mathit{dC}}_{\mathit{ji}}\otimes e_i+{\left(-1\right)}^k{C}_{\mathit{lj}}^{-1}{C}_{\mathit{ji}}{\omega }_l\wedge \nabla e_i)\\ \sum _{i=1}^r\left(d{\omega }_i\otimes e_i+{\left(-1\right)}^k{\omega }_i\otimes e_i\right)+\sum _{i,j,l=1}^r(d\left(C_{\mathit{lj}}^{-1}{C}_{\mathit{ji}}+C_{\mathit{ij}}^{-1}{\mathit{dC}}_{\mathit{ji}}+C_{\mathit{lj}}^{-1}{\mathit{dC}}_{\mathit{ji}}\right)\wedge {\omega }_l\otimes e_i.\end{array}$$E39

The last term is zero since

$$0=d{\delta }_{\mathit{li}}=d\left(\sum _{j=1}^r{C}_{\mathit{lj}}^{-1}{C}_{\mathit{ji}}\right)=\sum _{j=1}^r\left(d\left(C_{\mathit{lj}}^{-1}\right)C_{\mathit{ji}}+C_{\mathit{lj}}^{-1}{\mathit{dC}}_{\mathit{ji}}\right).$$E40

The derivative $d_{\nabla }$ also satisfies

$$d_{\nabla }\left(\omega +{\omega }^{\prime }\right)=d_{\nabla }\omega +d_{\nabla }{\omega }^{\prime },d_{\nabla }\left(\sigma \otimes e\right)=\mathit{d\sigma }+{\left(-1\right)}^k\sigma \wedge \nabla e,$$E41

as well as the Leibnitz formula for exterior covariant derivative. Unlike the case of the standard exterior derivative $d$, it can be shown that $d_{\nabla }$ in general has square $d_{\nabla }\circ d_{\nabla }\ne 0$, a fact related to the curvature $F^{\nabla }$ of the covariant derivative $\nabla$.

3. Yang-Mills Lagrangians

In physics, the Lagrangians that are used are restricted out of an infinite set of possible Lagrangians by various principles. The Lagrangian or action of a field theory should be invariant under certain transformations of the fields by symmetry groups. The laws of physics have to be invariant as well, a second meaning of symmetry is invariance of the actual field configurations. In spontaneously broken gauge theories, the Lagrangian is invariant under gauge transformations with values in a given Lie group $G$. However, due to the Higgs field, the vacuum is invariant under a subgroup $H\subset G$ of transformations. The purpose of the Higgs is to give mass to the particles that appear in the Lagrangians without at the same time breaking gauge invariance. A quantum field theory associated to the Lagrangian should be renormalizable so after the renormalization of parameters, finite results that can be compared with experiment are obtained.

The scalar product of forms is given as

$$⟨\omega ,\eta ⟩=\sum _{{\mu }_1<\cdots <{\mu }_k}{\omega }_{{\mu }_1\cdots {\mu }_k}{\eta }^{{\mu }_1\cdots {\mu }_k}=\frac{1}{k!}{\omega }_{{\mu }_1\cdots {\mu }_k}{\eta }^{{\mu }_1\cdots {\mu }_k},{\left|\omega \right|}^2=⟨\omega ,\omega ⟩.$$E42

To write the Yang-Mills equations, the Hodge star operator written as ${}^{\ast }{\mathrm{\Omega }}^k\left(M,\mathbb{K}\right)\to {\mathrm{\Omega }}^{n-k}\left(M,\mathbb{K}\right)$ is the linear map on real-valued forms so that if ${\mathit{dv}}_g$ is the volume element on $M$,

$$\omega \wedge \ast \eta =⟨\omega ,\eta ⟩{\mathit{dv}}_g,\omega ,\eta \in {\mathrm{\Omega }}^k\left(M,\mathbb{R}\right).$$E43

The $L^2$-scalar product of forms ${⟨\cdot ,\cdot ⟩}_{L^2}:{\mathrm{\Omega }}_0^k\left(M,\mathbb{K}\right)\times {\mathrm{\Omega }}_0^1\left(M,\mathbb{K}\right)\to \mathbb{K}$ is defined by

$${⟨\omega ,\eta ⟩}_{L^2}={\int }_M⟨\omega ,\eta ⟩{\mathit{dv}}_g.$$E44

To obtain a finite integral, it is usual to work with forms of compact support. The codifferential $d^{\ast }{\mathrm{\Omega }}^{k+1}\to {\mathrm{\Omega }}^k\left(M\right)$ is

$$d^{\ast }={\left(-1\right)}^{t+\mathit{nk}+1\ast }{d}^{\ast }.$$E45

Theorem 3.1 Let $M$ be a manifold without boundary. Then the codifferential $d^{\ast }$ is the formal adjoint of the differential $d$ with respect to the $L^2$ scalar product on forms of compact support ${⟨\mathit{d\omega },\eta ⟩}_{L^2}={⟨\omega ,d^{\ast }\eta ⟩}_{L^2}$ for all $\omega \in {\mathrm{\Omega }}_0^k\left(M\right)$, $\eta \in {\mathrm{\Omega }}_0^{k+1}\left(M\right)$.

Proof: The difference $⟨\mathit{d\omega },\eta ⟩-⟨\omega ,d^{\ast }\eta ⟩$ with respect to the pointwise scalar product of the forms. Applying ${}^{\ast }$ twice gives a map $\ast \ast :{\mathrm{\Omega }}^{n-k}\left(M\right)\to {\mathrm{\Omega }}^{n-k}\left(M\right)$ is given by

$$\ast \ast ={\left(-1\right)}^{t+\left(n-k\right)k}.$$E46

Therefore, we have

$$\begin{array}{c}⟨\mathit{d\omega },\eta ⟩-⟨\omega ,d^{\ast }\eta ⟩{\mathit{dv}}_g=\left(\mathit{d\omega }\right)\wedge {}^{\ast }\eta -\omega \wedge \ast \left(d^{\ast }\eta \right)=\left(\mathit{d\omega }\right)\wedge \ast \eta +{\left(-1\right)}^k\omega \wedge \left(d^{\ast }\eta \right)\\ =d\left(\omega \wedge \ast \eta \right).\end{array}$$E47

Stokes’ Theorem applied here implies the result.

This knowledge allows us to define the covariant codifferential $d_{\nabla }^{\ast }:{\mathrm{\Omega }}^{k+1}\left(M,E\right)\to {\mathrm{\Omega }}^k\left(M,E\right)$ by

$$d_{\nabla }^{\ast }={\left(-1\right)}^{t+\mathit{nk}+1\ast }{d_{\nabla }}^{\ast }.$$E48

To define the Yang-Mills Lagrangian and the associated Yang-Mills equations, procced as follows. To do so, we use an $n$-dimensional, oriented, psuedo-Riemannian manifold $\left(M,g\right)$, with signature $\left(s,t\right)$ a principal $G$-bundle $P\to M$ with compact structure group $G$ of dimension $r$, a scalar product on $g_L$, which is $\mathit{Ad}$, invariant and an orthonormal vector space basis $T_i$ for $g_L$.

Let $A$ be a connection $1$-form on the principal bundle $P$ with curvature two-form $F^A\in {\mathrm{\Omega }}^2\left(P,g_L\right)$. The curvature defines a twisted two-form $F_M^A\in {\mathrm{\Omega }}^2\left(M,\mathit{Ad}\left(P\right)\right)$. The Yang-Mills Lagrangian is defined by

$${\mathcal{L}}_{\mathit{YM}}=-\frac{1}{2}{⟨F_M^A,F_M^A⟩}_{\mathit{Ad}\left(P\right)}.$$E49

For a fixed connection $A$, this Lagrangian is a global smooth function ${\mathcal{L}}_{\mathit{YM}}\left(A\right):M\to \mathbb{R}$. The Yang-Mills Lagrangian is gauge invariant, ${\mathcal{L}}_{\mathit{YM}}\left(f^{\ast }A\right)={\mathcal{L}}_{\mathit{YM}}\left(A\right)$, for all bundle automorphisms $f\in \mathcal{G}\left(P\right)$ and all $A$ on $P$. In a chart with coordinates $x^{\mu }$, the components of $F^A$ are $F_{\mathit{\mu \nu }}^A=F_s^A\left({\partial }_{\mu },{\partial }_{\nu }\right)$ and they can be expanded over the Lie algebra basis as

$$F_{\mathit{\mu \nu }}^A=F_{\mathit{\mu \nu }}^{\mathit{Aa}}{T}_a,$$E50

and $F_s^{\mathit{Aa}}\in {\mathrm{\Omega }}^2\left(U\right)$ are real-valued differential forms, $F_{\mathit{\mu \nu }}^{\mathit{Aa}}$ are real-valued smooth functions on $U$. Thus, expanding (49), the Yang-Mills Lagrangian is locally

$${\mathcal{L}}_{\mathit{YM}}\left(A\right)=-\frac{1}{2}⟨F_s^A,F_s^A⟩=-\frac{1}{4}{F}_{\mathit{\mu \nu }}^{\mathit{Aa}}{F}_a^{A\mathit{\mu \nu }},$$E51

where $F_{\mathit{\mu \nu }}^{\mathit{Aa}}={\partial }_{\mu }{A}_{\nu }^a-{\partial }_{\nu }{A}_{\mu }^a+f_{\mathit{bca}}{A}_{\mu }^b{A}_{\nu }^c$, and structure constant $f_{\mathit{cba}}$ for the Lie algebra.

Suppose $\left(M,g\right)$ is compact and closed. The Yang-Mills action for a principal $G$-bundle $P\to M$ is the smooth map $S_{\mathit{YM}}:\mathcal{A}\left(P\right)\to \mathbb{R}$, with $\mathcal{A}\left(P\right)$ the space of all connection one-forms $A$ on $P$ defined by

$$S_{\mathit{YM}}\left(A\right)=-\frac{1}{2}{\int }_M{⟨F_M^A,F_M^A⟩}_{\mathit{Ad}\left(P\right)}{\mathit{dv}}_g.$$E52

A connection $A$ on the principal bundle $P$ is a critical point of the Yang-Mills action if

$${\left.\frac{d}{\mathit{du}}\right|}_{u=0}{S}_{\mathit{YM}}\left(A+u\beta \right)=0,$$E53

for all such variations on $P$.

Theorem 3.2 A connection $A$ on a principal bundle $P\to M$ is a critical point of the Yang-Mills action if and only if $A$ satisfies the Yang-Mills equation

$${d_A}^{\ast }{F}_M^A=0.$$E54

Proof: Based on the structure equations, we calculate

$$F^{A+\mathit{u\beta }}=d\left(A+\mathit{u\beta }\right)+\frac{1}{2}\left[A+\mathit{u\beta },A+\mathit{u\beta }\right]=F^A+u\left(\mathit{d\beta }+\left[A,\beta \right]\right)+\frac{1}{2}{u}^2\left[\beta ,\beta \right].$$E55

Differentiating this and using the adjoint property on $M$, it follows that

$${\left.\frac{d}{\mathit{du}}\right|}_{u=0}{⟨F_M^{A+\mathit{u\beta }},F_M^{A+\mathit{u\beta }}⟩}_{\mathit{Ad}\left(P\right),L^2}=2{⟨d_A\beta ,F_M^A⟩}_{\mathit{Ad}\left(P\right),L^2}=2{⟨\beta ,d_A^{\ast }{F}_M^A⟩}_{\mathit{Ad}\left(P\right),L^2}.$$E56

The scalar product on the Lie algebra is non-degenerate, the $L^2$-scalar product is non-degenerate. It follows that $A$ is a critical point of the Lagrangian (49) if and only if (54) holds.

Any connection $A$ on $P$ has to satisfy the Bianchi identity

$$d_A{F}_M^A=0.$$E57

When the group $G=U\left(1\right)$, the local curvature forms are independent of the choice of local gauge $s$ and define a global two-form $F_A$, so the Bianchi identity and Yang-Mills equations are given by ${\mathit{dF}}_M=0$ and $d^{\ast }{F}_M=0$. These are Maxwell’s equations for a source-free electromagnetic field on a general $n$-dimensionsl oriented pseudo-Riemannian manifold.

Fields of different types can be introduced into the picture. These include matter fields that couple to the gauge field $A$_, such as scalar fields or fermionic spinor fields, and are distinguished by the statistics they obey. These two types of particle are distinguished by an intrinsic property called spin, and this has to have its own treatment.

A complex scalar field is a smooth map $\varphi :M\to \mathrm{ℂ}$. A multiplet of complex scalar fields is a smooth map $\varphi :M\to {\mathrm{ℂ}}^r$ for some $r>1$ with the standard Hermitian scalar product $⟨v,w⟩=v^{†}w$ on ${\mathrm{ℂ}}^r$. Given a principal $g$-bundle $P\to M$ with compact structure group $G$ of dimension $r$, a complex representation $\rho :G\to \mathit{GL}\left(W\right)$ with associated complex vector bundle $E$ and $G$-invariant Hermitian scalar product ${⟨\cdot ,\cdot ⟩}_W$ on $W$ and bundle metric ${⟨\cdot ,\cdot ⟩}_E$ on the vector bundle $E$. If the dimension of $V$ is one, then a smooth section of $E$ is called a multiplet of complex scalar fields and the vector space $W$ is called a multiplet space. With the covariant derivative $d_A:\mathrm{\Gamma }\left(E\right)\to {\mathrm{\Omega }}^1\left(M,E\right)$ and the scalar product ${⟨\cdot ,\cdot ⟩}_E$ on ${\mathrm{\Omega }}^1\left(M,E\right)$, the Klein-Gordon Lagrangian can be given.

Definition 3.1 The Klein-Gordon Lagrangian for a multiplet of the complex scalar field $\mathrm{\Phi }\in \mathrm{\Gamma }\left(E\right)$ of mass $m$ coupled to a gauge field $A$ is

$${\mathcal{L}}_{\mathit{KG}}\left(\mathrm{\Phi },A\right)={⟨d_A\mathrm{\Phi },d_A\mathrm{\Phi }⟩}_E-m^2{⟨\mathrm{\Phi },\mathrm{\Phi }⟩}_E.$$E58

For given fields $\mathrm{\Phi }$ and $A$, the Klein-Gordon Lagrangian is a smooth function ${\mathcal{L}}_{\mathit{KG}}\left(\mathrm{\Phi },A\right):M\to \mathbb{R}$.

The associated action $S_{\mathit{KG}}\left(\mathrm{\Phi },A\right)$ is the integral over the Klein-Gordon Lagrangian on the closed manifold $M$. In local coordinates on $M$, the kinetic term is

$$⟨d_A\mathrm{\Phi },d_A\mathrm{\Phi }⟩=-{⟨{\nabla }^{\mathit{A\mu }}\mathrm{\Phi },{\nabla }^{\mathit{A\mu }}\mathrm{\Phi }⟩}_E.$$E59

In a local gauge $s$ for the principal bundle, the Klein-Gordon Lagrangian can be written as ${\left.\mathrm{\Phi }\right|}_U=\left[s,\varphi \right]$

$${\mathcal{L}}_{\mathit{KG}}\left(\mathrm{\Phi },A\right)={\left({\partial }^{\mu }\varphi \right)}^{†}\left({\partial }_{\mu }\varphi \right)-m^2{\varphi }^{†}\varphi +{\left({\partial }^{\mu }\varphi \right)}^{†}\left(A_{\mu }\varphi \right)-\left({\varphi }^{†}{A}_{\mu }\right)\left({\partial }^{\mu }\varphi \right)-{\varphi }^{†}{A}^{\mu }{A}_{\mu }\varphi .$$E60

As with the Yang-Mills Lagrangian, the Klein-Gordon Lagrangian of a multiplet of complex scalar fields coupled to a gauge field is gauge invariant.

To describe fermion fields classically using spinor fields on spacetime, a Lagrangian for fermions is defined. The setting for doing this is an $n$-dimensional oriented and time-oriented pseudo-Riemannian spin manifold $\left(M,g\right)$ of signature $\left(s,t\right)$, a spin structure ${\mathit{\text{Spin}}}^{†}\left(M\right)$ together with complex spin bundle $S\to M$, and a Dirac form $⟨,⟩$ on the Dirac spinor space, not necessarily positive definite, with Dirac bundle metric ${⟨,⟩}_D$. We abbreviate ${⟨\mathrm{\Psi },\mathrm{\Phi }⟩}_D$ as $\overline{\mathrm{\Psi }}\mathrm{\Phi }$.

Definition 3.2 The Dirac Lagrangian for a free spinor field $\psi \in \mathrm{\Gamma }\left(S\otimes E\right)$ mass $m$ is defined by

$${\mathcal{L}}_D\left(\psi \right)=Re{⟨\mathrm{\Psi },D_A\mathrm{\Psi }⟩}_{S\otimes E}-m{⟨\mathrm{\Psi },\mathrm{\Psi }⟩}_{S\otimes E}=Re\left(\overline{\mathrm{\Psi }}{D}_A\mathrm{\Psi }\right)-m\overline{\mathrm{\Psi }}\mathrm{\Psi },$$E61

where $D_A\mathrm{\Gamma }\left(S\otimes E\right)\to \mathrm{\Gamma }\left(S\otimes E\right)$ denotes the twisted Dirac operator, the first term the kinetic term and the second pertains to the mass of the particle. The associated action $S_D\left[\mathrm{\Psi },A\right]$ is the integral over the Dirac Lagrangian on a closed manifold $M$.

Based on the fundamental Lagrangians which couple the fields to the gauge field, the Lagrangian of the Standard Model can be built up as the sum of all the individual Lagrangians that are to be accounted for and required to describe all the observed fields. It could be referred to as the Yang-Mills-Dirac-Higgs-Yukawa Lagrangian

$$\mathcal{L}=Re\left(\overline{\mathrm{\Psi }}{D}_A\mathrm{\Psi }\right)+⟨d_A\mathrm{\Phi },d_A\mathrm{\Phi }⟩-V\left(\mathrm{\Phi }\right)-2g_{Y}Re\left({\overline{\mathrm{\Psi }}}_L{\mathrm{\Phi \Psi }}_R\right)-\frac{1}{2}{⟨F_M^A,F_M^A⟩}_{\mathit{Ad}\left(P\right)}.$$E62

Experiment informs us that a realistic theory of particle physics has to involve chiral fermions with a nonzero mass because the weak interaction is not invariant under parity inversion.

4. Yang-Mills on four-dimensional manifolds

The general overview of Yang-Mills theory is now restricted to four-dimensional compact Riemannian manifolds. This will emphasize how Yang-Mills relates to manifolds which are the natural context for Yang-Mills theory for more than one reason. First the four-dimensional action is bounded below by the characteristic number of the bundle so the field is constrained by the topology. By invariant theory, this is linked to the conformal invariance of the action occuring just in dimension four. The base manifold conformal structure leads to the relevant geometry. The curvature is given in terms of the connection form $\omega$, and the action is the sum of a gradient term and a non-linear self-interaction term. They are of comparable strength only in dimension four. Some of the symbols are adapted to the particular case studied here.

Riemannian geometry in dimension four is distinguished by the fact that the universal cover $\mathit{\text{Spin}}\left(4\right)$ of the rotation group $\mathit{SO}\left(4\right)$ is not a simple group, but factors $\mathit{\text{Spin}}\left(4\right)=\mathit{SU}\left(2\right)\times \mathit{SU}\left(2\right)$. One way to look at this is at the group level, ${\mathbb{R}}^4$ and ${\mathrm{ℂ}}^2$ can be identified with the quaternions $\mathrm{ℍ}$. Thus $\mathit{SO}\left(2\right)$ may be regarded as the unit quaternions. For unit quaternions, $g$ and $h$, the map $x\to g^{-1}\mathit{xh}$ is an orthogonal transformation of $\mathrm{ℍ}={\mathbb{R}}^4$ with determinant one, and hence yields a homeomorphism $\pi :\mathit{SU}\left(2\right)\times \mathit{SU}\left(2\right)\to \mathit{SO}\left(4\right)$. This map has kernel $\left\{-1,1\right\}$ and so indicates $\mathit{SU}\left(2\right)\times \mathit{SU}\left(2\right)$ as the two-fold universal covering group of $\mathit{SO}\left(4\right)$.

Suppose $M$ is a Riemannian manifold, so the metric determines the basic Levi-Civita connection on the cotangent space

$$\nabla :\mathrm{\Gamma }\left(T^{\ast }M\right)\to \mathrm{\Gamma }\left(T^{\ast }\otimes T^{\ast }M\right).$$E63

Choosing a local basis of sections $\left\{e^i\right\}$ of $T^{\ast }M$ we may write $\nabla e^i={\sum }_k{\omega }_k^i\otimes e^k$, where $\left\{{\omega }_k^i\right\}$ are the connection one-forms. The nature of these one-forms can be understood in the context of an arbitrary bundle. Let $G$ be a compact semi-simple Lie group with Lie algebra $g_L$ and let $\pi :P\to M$ be a principal $G$-bundle over $M$. A connection on $P$ is a choice of an equivariant horizontal subspace on $T_{\ast }P$ or a $g_L$-valued one-form on $P$ which has horizontal kernel and is equivariant $g^{\ast }\omega \left(X\right)=\left(\mathit{Ad}{g}^{-1}\right)\omega \left(X\right)$ for $x\in \mathrm{\Gamma }\left(T_{\ast }P\right)$ and $g\in G$.

Let $\mathcal{C}$ denote the affine space of $C^{\infty }$ connections of $P$. Then $\mathcal{C}$ becomes a vector space when a base connection is fixed. The equivariance property shows that the difference $\eta =\omega -{\omega }_0$ pulls down to $M$ as a one-form with values in the adjoint bundle $P{\times }_{\mathit{Ad}}{g}_L$ also denoted $G_L$. As such it determines a covariant map $\nabla :\mathrm{\Gamma }\left(g_L\right)\to \mathrm{\Gamma }\left(g_L\right)\otimes T^{\ast }\left(M\right)$ by virtue of $\varphi \to {\nabla }_0+\left[\eta ,\varphi \right]$, where ${\nabla }_0$ is the covariant derivative corresponding to ${\omega }_0$. If $\rho :G\to \mathrm{Aut}\left(\overline{E}\right)$ is a representation and $E=P{\times }_{\rho }\overline{E}$ the associated vector bundle, then $\omega$ induces a covariant derivative

$${\nabla }^E:\mathrm{\Gamma }\left(E\right)\to \mathrm{\Gamma }\left(\otimes T^{\ast }M\right)$$E64

on $E$ by applying the Lie algebra representation $\rho :g\to \mathit{End}\left(\overline{E}\right)$ to $\nabla$ above. Suppose for example $P$ is the frame bundle of $T^{\ast }M$, the Riemannian connection can be described either in terms of the covariant derivative or in terms of the corresponding $\mathit{so}\left(n\right)$-valued connection form $\omega =\left\{{\omega }_k^i\right\}$.

Given a connection ${\nabla }^E$ on a vector bundle $E$, several related operations can be constructed from ${\nabla }^E$ a the symbol map. Extending ${\nabla }^E$ to the covariant derivative $\overline{\nabla }=\nabla \otimes 1+1\otimes \nabla$ on ${\mathrm{\Lambda }}^k\otimes E$, with $\nabla$ the Riemann connection on ${\mathrm{\Lambda }}^{\ast }$, and using exterior differentiation or its adjoint contraction as the symbol, an exterior differentiation $D:\mathrm{\Gamma }\left({\mathrm{\Lambda }}^{\ast }\otimes E\right)\to \mathrm{\Gamma }\left({\mathrm{\Lambda }}^{\ast +1}\otimes E\right)$ is obtained and its formal adjoint $D^{\ast }$. In a local orthonormal frame $\left(e^i\right)$, $\varphi \in \mathrm{\Gamma }\left({\mathrm{\Lambda }}^{\ast }\otimes E\right)$,

$$\mathit{D\varphi }=\sum _i{e}^i\wedge {\overline{\nabla }}_i\varphi ,D^{\ast }\varphi =-\sum _i{e}_i{\overline{\nabla }}^i\varphi .$$E65

There are also two second order operators. They are the trace Laplacian

$${\left({\nabla }^E\right)}^{\ast }{\overline{\nabla }}^E=-\sum _i{\overline{\nabla }}_i^E{\overline{\nabla }}_i^E-{\overline{\nabla }}_{{\nabla }_i{e}_i}^E$$E66

on $\mathrm{\Gamma }\left(E\right)$, and the bundle Laplace-Beltrami operator $\square =\left({\mathit{DD}}^{\ast }+D^{\ast }D\right)$ on $\mathrm{\Gamma }\left({\mathrm{\Lambda }}^{\ast }\otimes E\right)$. The covariant derivative of $\nabla :\mathrm{\Gamma }\left(g_L\right)\to \mathrm{\Gamma }\left(g_L\otimes T^{\ast }M\right)$ extends by virtue of (65) to an exterior differentiation $D$ on the space of sections ${\mathrm{\Lambda }}^{\ast }=\mathrm{\Gamma }\left({\mathrm{\Lambda }}^{\ast }\otimes g_L\right)$ by $D\varphi ={\nabla }_0\varphi +\left[\eta ,\varphi \right]$ where ${\nabla }_0$ is the covariant derivative corresponding to ${\omega }_0$.

The curvature of a connection $\omega$ on a principal bundle $P$ is the $g_L$-valued two-form $\mathrm{\Omega }\left(X,Y\right)=\mathit{d\omega }\left(\mathit{hX},\mathit{hY}\right)$ where $h$ is the projection onto the horizontal subspace of $\omega$. One can say $D=d\cdot h$ is a derivation on equivariant $g_L$-valued one-forms on $P$ given by $\mathit{D\varphi }=\mathit{d\varphi }+\left[\omega ,\varphi \right]$ for one-forms with vertical kernel and $\mathit{D\varphi }=d\varphi +\frac{1}{2}\left[\omega ,\varphi \right]$ for connection forms $\varphi$, in particular, $\mathrm{\Omega }=\mathit{d\omega }+\frac{1}{2}\left[\omega ,\omega \right]$ on $P$. Let us fix connection ${\omega }_0$, so for any other $\omega$, the difference $\eta =\omega -{\omega }_0$ descends to $M$ as an element of $A^1$ and the difference of the curvature is

$$\begin{array}{c}\mathrm{\Omega }-{\mathrm{\Omega }}_0=\mathit{d\eta }+\frac{1}{2}\left[\omega ,\omega \right]-\frac{1}{2}\left[{\omega }_0,{\omega }_0\right]=\mathit{d\eta }+\frac{1}{2}\left[\eta ,\eta \right]+\left[\omega ,\eta \right],\\ \mathrm{\Omega }={\mathrm{\Omega }}_0+D_0\eta +\frac{1}{2}\left[\eta ,\eta \right].\end{array}$$E67

Alternatively, $\varphi \in A^0$ lifts to an equivariant $g_L$-valued function on $P$ and $D\varphi$ has a vertical kernel, so

$$D\circ D\left(\varphi \right)=d\left(\mathit{D\varphi }\right)+\left[\omega ,D\varphi \right]=d\left(\mathit{d\varphi }+\left[\omega ,\varphi \right]\right)+\left[\omega ,\mathit{D\varphi }\right]=\left[\mathit{D\omega },\varphi \right]-\left[\omega ,\mathit{D\varphi }\right]+\left[\omega ,\mathit{D\varphi }\right]=\left[\mathit{D\omega },\varphi \right].$$E68

The formula descends to the base $D\circ \nabla \left(\varphi \right)=\left[\mathrm{\Omega },\varphi \right]$ for $\varphi \in A^0$. In terms of a local basis of vector fields $\left\{e_i\right\}$ and dual forms $\left\{e^i\right\}$

$$\mathrm{\Omega }\left(\varphi \right)=D\circ \nabla \varphi =D\left(\sum _i{\nabla }_i\varphi e^i\right)=\sum _{j,k}\left({\nabla }_j\varphi e^k\wedge {\nabla }_k{e}^j+{\nabla }_k{\nabla }_j\varphi e^k\wedge e^j\right).$$E69

For the Riemannian connection, ${\nabla }_i{e}^j-{\nabla }_j{e}^i=\left[e^i,e^j\right]$ so ${\mathrm{\Omega }}_{\mathit{ij}}={\nabla }_i{\nabla }_j-{\nabla }_j{\nabla }_i-{\nabla }_{\left[e_i,e_j\right]}$ and similarly for ${\mathrm{\Omega }}^E$.

On a four-dimensional Riemannian manifold, the metric covariant derivative on the spin bundle $S_{+}$ is a map $\nabla :\mathrm{\Gamma }\left(S_{+}\right)\to \mathrm{\Gamma }\left(S_{+}{\otimes }^{\ast }M\right)$. Thus $\nabla$ on $S_{\pm }$ decomposes into two operators: first the Dirac operator $\mathcal{D}:\mathrm{\Gamma }\left(S_{\pm }\right)\to \mathrm{\Gamma }\left(S_{\mp }\right)$ where symbol is Clifford multiplication, and the twister operator $\overline{\mathcal{D}}:\mathrm{\Gamma }\left(V_{\pm }\right)\to \mathrm{\Gamma }\left({\mathrm{\Lambda }}_{\pm }^2\right)$, whose symbol is the orthogonal complement of Clifford multiplication. In a local orthonormal frame $\left\{e^i\right\}$ with $\varphi \in \mathrm{\Gamma }\left(V\right)$

$$\mathcal{D}\varphi =\sum _i{e}^i\cdot {\nabla }_i\varphi ,\overline{\mathcal{D}}\varphi =\nabla \varphi +\frac{1}{4}\sum _i{e}^i\mathcal{D}\varphi \otimes e^i.$$E70

The Dirac operator is elliptic and is formally self-adjoint on the total spin bundle $S$.

In four dimensions, the Riemannian curvature tensor $R\in {\mathrm{\Lambda }}^2\otimes {\mathrm{\Lambda }}^2$ decomposes under the splitting ${\mathrm{\Lambda }}^2={\mathrm{\Lambda }}_{+}^2\oplus {\mathrm{\Lambda }}_{-}^2$. Due to the symmetry $R_{\mathit{\text{ijkl}}}=R_{\mathit{\text{klij}}}$, this is an element of the symmetric tensor product ${\mathit{Sym}}^2\left({\mathrm{\Lambda }}_{+}^2\oplus {\mathrm{\Lambda }}_{-}^2\right)$, which is a Spin $\left(4\right)$ module, this breaks uo into five irreducible pieces.

The components of this tensor under this decomposition are $\left(W^{+},\frac{R_s}{12},2B,W^{-}\right)$_, where $R_s$ is the scalar curvature, $B$ the traceless Ricci tensor, and $W^{\pm }$ are the self-dual and anti-self-dual components of the conformally invariant Weyl tensor. This decomposition results in some important classes of four-manifolds: $M^4$ is Einstein if $B=0$, conformally flat if $W=0$, and self-dual (anti) if $W^{-}=0$ ($W^{+}=0$).

Suppose $M$ is a spin four-manifold with Riemannian connection $\nabla$ and $E$_, a vector bundle over $M$ with connection ${\nabla }^E$ and curvature ${\mathrm{\Omega }}^E$. Then the Dirac operator is $\mathcal{D}\mathrm{\Gamma }\left(V\otimes E\right)\to \mathrm{\Gamma }\left(V\otimes E\right)$ is defined for $E$-valued spinors by $\mathcal{D}={\sum }_i{e}^i\cdot {\overline{\nabla }}_i$ where $\overline{\nabla }$ is the total covariant derivative on $V\otimes E$. It is shown this operator has an algebraic decomposition into Laplacian and curvature terms. Such an expression is called a Weitzenböck operator. These encompass more than one kind of operator so it is worth showing how they can be developed. To get ${\mathcal{D}}^2$, choose an orthonormal basis $\left\{e^i\right\}$ around $x\in M$, vector fields $\left\{e_i\right\}$ dual to the $e^i$ such that ${\left({\nabla }_{e_i}{e}^j\right)}_x=0$ for all $i,j$. Squaring $\mathcal{D}$ and separating the symmetric and skew-symmetric parts

$${\mathcal{D}}^2=\left(\sum e^i\cdot {\overline{\nabla }}_i\right)\left(\sum e^j{\overline{\nabla }}_j\right)=\sum _{i,j}{e}^i\cdot e^j\cdot {\overline{\nabla }}_i{\overline{\nabla }}_j=-\sum _{i,j}{\overline{\nabla }}_j{\overline{\nabla }}_i+\sum _{i,j}{e}^i\cdot e^j\cdot \left({\overline{\nabla }}_i{\overline{\nabla }}_j-{\overline{\nabla }}_j{\overline{\nabla }}_i\right).$$E71

This can be summarized as

$${\mathcal{D}}^2={\overline{\nabla }}^{\ast }\overline{\nabla }+\frac{1}{2}\sum _{i,j}{e}^i\cdot e^j\cdot R_{\mathit{ij}}\otimes 1+\frac{1}{2}\sum _{i,j}{e}^i\cdot e^j\cdot \left(1\otimes {\mathrm{\Omega }}_{\mathit{ij}}^E\right).$$E72

Since $\nabla$ is torsionless, ${\left[e_i,e_j\right]}_x=0$, and the total curvature is ${\mathrm{\Omega }}_{\mathit{ij}}={\overline{\nabla }}_i{\overline{\nabla }}_j-{\overline{\nabla }}_j{\overline{\nabla }}_i$. The first term is (72) is the positive trace Laplacian of $\overline{\nabla }$, $R_{\mathit{ij}}$ can be written in terms of the irreducible components of $R$.

Compact four-dimensional manifolds $M^4$ possess two real characteristic classes. These can be expressed locally as polynomials in the curvature of $M$ and hence as polynomials in the irreducible components of the curvature $\left(R_s,B,W^{\pm }\right)$.

Topological invariants arise in the consideration of four-dimensional manifolds $M$. These have two real characteristic classes. They are the Pontryagin class $p_{1}M$ and the Euler class $\chi \left(M\right)$ given by

$$\begin{array}{l}{p}_{1}M=\frac{1}{4{\pi }^2}{\int }_M\left({\left|W^{+}\right|}^2-{\left|W^{-}\right|}^2\right){\mathit{dv}}_g,\\ \chi \left(M\right)=\frac{1}{8{\pi }^2}{\int }_M\left(\frac{R_s^2}{24}-2{\left|B\right|}^2+{\left|W^{+}\right|}^2+{\left|W^{-}\right|}^2\right){\mathit{dv}}_g.\end{array}$$E73

More generally, if $G$ is a compact simple Lie group, $H^i\left(\mathit{BG};\mathbb{R}\right)$ vanishes for $i=\mathrm{1,2,3}$ and is $\mathbb{R}$ for $i=4$. Thus, there is a single real characteristic class for principal $G$-bundles over $M^4$ and it resides in dimension $4$. In Yang-Mills theory, the corresponding characteristic number is called the Pontryagin index $\kappa$ of the bundle. It is obtained by substituting the curvature $\mathrm{\Omega }$ of $P$ into the Killing form. In terms of the anti-self-dual components ${\mathrm{\Omega }}^{\pm }$ of $\mathrm{\Omega }$, $\kappa$ is

$$\kappa =\frac{1}{8{\pi }^2}{\int }_M\left({\left|{\mathrm{\Omega }}^{+}\right|}^2-{\left|{\mathrm{\Omega }}^{-}\right|}^2\right){\mathit{dv}}_g.$$E74

For functions on a bounded domain in ${\mathbb{R}}^n$ the Sobolov space $L_{k,p}\left(D\right)$ is the completion of the space of $C^{\infty }$ functions in the norm

$${‖f‖}_{k,p}={\left({\int }_D\sum _{\mid \alpha \mid =1}^k{\left|{\partial }_{\alpha }f\right|}^p\right)}^{1/p}.$$E75

These spaces are related by the Sobolev embedding theorems: for $p,q\ge 1$, the inclusion $K_{k,p}\left(D\right)\to L_{l.q}\left(D\right)$ is continuous for $k-n/p\ge 1-n/q$, compact for $k-n/p>1-n/q$. This extends to vector bundles $E$ with metric over $M$.

5. Deriving coupled Yang-Mills equations

It is the case that once the geometrical setting for a gauge theory has been set out, the requirement of naturality then determines the theory. Let $\pi :P\to M$ be a principal bundle over a four-dimensional $M$ with compact simple structure group $G$, $\rho :G\to \mathrm{Aut}\left(\overline{E}\right)$ a unitary representation of $G$, $E=p{\times }_{\rho }\overline{E}$ the associated vector bundle, and $W$ the bundle associated to the frame bundle of $M$. To get a mathematically rigorous development of the field equations, assume that they are variational equations and arise as the stationary points of an action integral

$$A\left(g,\nabla ,\varphi \right)={\int }_{M}L\left(g,\omega ,\varphi \right){\mathit{dv}}_g,$$E76

where the Lagrangian is a $4$-form constructed from $g$, $\nabla$_, and $\varphi$.

Then $P$ is a manifold with a certain geometric structure. There is a free right action of $G$, so an automorphism of $P$ is a map $f:P\to P$_, which preserves this structure $f\left({\mathit{xg}}^{-1}\right)=f\left(x\right)g^{-1}$ for all $x\in P$ and $g\in G$. Let $\mathrm{Aut}\left(P\right)$ denote the group of all bundle automorphisms $f$ such that the induced map $\pi \cdot f:M\to M$ preserves orientation, ${\mathrm{Aut}}_0\left(P\right)$ the subgroup which induces the identity on $M$. In the language of physics, a section $s:M∍U\to P$ is called a local choice of gauge, an automorphism $f\in {\mathrm{Aut}}_0\left(P\right)$ is a gauge transformation, and the group $\mathcal{G}={\mathrm{Aut}}_0\left(P\right)$ is the gauge group of the bundle.

These properties of the Lagrangian are required $\left(i\right)$ in a local coordinate system and choice of gauge, $L$ should be a universal polynomial in $g,h,\mathrm{\Gamma },\varphi ,{\left(detg\right)}^{-1/2},{\left(deth\right)}^{-1/2}$_, and their derivatives, $\mathrm{\Gamma }$ the Christoffel symbols. $\left(\mathit{ii}\right)$ the map $L$ should be a natural transformation with respect to the bundle automorphism $f$, $L\left({\left(\pi \cdot f\right)}^{\ast }g,f^{\ast }\nabla ,\rho \left(f^{\ast }\right)\varphi \right)=f^{\ast }L\left(g,\nabla ,\varphi \right)$$\left(\mathit{iii}\right)$ it should have conformal invariance, for any $\sigma$ on $M$, $L\left(e^{\mathit{i\sigma }}g,\nabla ,\varphi \right)=L\left(g,\nabla ,\varphi \right)$. Naturality with respect to ${\mathrm{Aut}}_0\left(P\right)$ means that $L\left(g,f^{\ast }\nabla ,\rho \left(f^{\ast }\right)\varphi \right)=L\left(g,\nabla ,\varphi \right)$. This is Weyl’s principle of gauge invariance. For the case in which $L=L\left(g,\nabla \right)$ requiring naturality under orientation preserving diffeomorphisms of $P$, $\mathit{SO}\left(4\right)$ invariant theory implies

$$L=c_1{\left|R_s\right|}^2+c_2{\left|B\right|}^2+c_3{\left|W^{+}\right|}^2+c_4{\left|W^{-}\right|}^2+c_5\mathrm{\Omega }\wedge \mathrm{\Omega }+c_6\mathrm{\Omega }\wedge \ast \mathrm{\Omega },$$E77

where $\left(R_s,B,W^{\pm }\right)$ are the components of the Riemann curvature of $g$, $\mathrm{\Omega }$ the curvature of $\omega$ and the $c_i$ are real numbers. The actions of the various values of the $c_i$ include topological invariants $p_1\left(M\right)$, $\chi \left(M\right)$ for example.

Let us be concerned with the action which depends on the bundle curvature which is called the Yang-Mills action

$$A\left(g,\nabla \right)={\int }_M\mathrm{\Omega }\wedge \ast \mathrm{\Omega }{\mathit{dv}}_g={\int }_M{\left|\mathrm{\Omega }\right|}^2\sqrt{det\left(g\right)}{\mathit{dx}}^1\wedge \cdots \wedge {\mathit{dx}}^4.$$E78

The action is evidently regular and $\text{Diff}\left(M\right)$ covariant. It is conformally invariant because the ${}^{\ast }$ operator on two-forms is

$$A\left(e^{2\sigma }g,V\right)={\int }_M{e}^{-4\sigma }{g}^{\mathit{ij}}{g}^{\mathit{kl}}⟨{\mathrm{\Omega }}_{\mathit{ik}},{\mathrm{\Omega }}_{\mathit{jl}}⟩(det{\left(e^{2\mathit{\sigma g}}\right)}^{1/2}{\mathit{dx}}^1\wedge \cdots \wedge {\mathit{dx}}^4=A\left(g,\omega \right).$$E79

A gauge transformation $g\in \mathcal{G}$ takes $\nabla$ to $g\nabla g^{-1}$ and $\mathrm{\Omega }=D\circ \nabla$ to $g\mathrm{\Omega }{g}^{-1}$. The Lagrangian ${\left|\mathrm{\Omega }\right|}^2$ is then unchanged because the Killing form is invariant. Since ${\left|\mathrm{\Omega }\right|}^2={\left|{\mathrm{\Omega }}^{-}\right|}^2+{\left|{\mathrm{\Omega }}^{+}\right|}^2$, (71) shows that $A\left(g,\omega \right)\ge 8{\pi }^{2}k$ with equality if and only if ${\mathrm{\Omega }}^{-}=0$. Consequently, self-dual connections are absolute minima of the Yang-Mills action. There are two action integrals considered by physicists. They are the fermionic and the bosonic types.

Definition 5.1 The fermion action is defined on $E$-valued spinors $\psi \in \mathrm{\Gamma }\left(V\otimes E\right)$ as

$$A\left(g,\nabla ,\psi \right)={\int }_M\left({\left|\mathrm{\Omega }\right|}^2+⟨\psi ,\mathcal{D}\psi ⟩\right){\mathit{dv}}_g,$$E80

where $\mathcal{D}$ is the Dirac operator and $⟨,⟩$ is the inner product on $V\otimes E$ and ${\mathit{dv}}_g$ the volume form.

Definition 5.2 The boson action is defined on $E$-valued scalars $\varphi \in \mathrm{\Gamma }\left(E\right)$ by

$$A\left(g,\nabla ,\varphi \right)={\int }_M\left({\left|\mathrm{\Omega }\right|}^2+{\left|\nabla \varphi \right|}^2+\frac{5}{6}{\left|\varphi \right|}^2-V\left(\varphi \right)\right){\mathit{dv}}_g.$$E81

where $V:E\to \mathbb{R}$ is a gauge invariant polynomial on the fiber such that $deg\left(V\right)\le 4$.

Both Lagrangians are regular $\text{Diff}\left(M\right)$ invariant and gauge invariant. The degree requirement comes about as we wish to vary the action over a Sobolev space and by the Sobolev inequality, any polynomial in $\varphi$ whose degree does not exceed four is then integrable. Note the second term in the fermion Lagrangian is not positive definite, for suppose $\psi ={\psi }_{+}-{\psi }_{-}\in \mathrm{\Gamma }\left(Q_{+}\oplus Q_{-}\right)$ satisfies $\mathcal{D}\psi =\mathit{\lambda \psi }$ for some eigenvalue $\lambda$, $\overline{\varphi }={\varphi }_{+}-{\varphi }_{-}$ satisfies $\mathcal{D}\overline{\varphi }=-\lambda \overline{\varphi }$. This gives that the spectrum of $\mathcal{D}$ is symmetric about zero.

6. Theorems in four dimensions for the Yang-Mills system

Let us calculate the first variation of the action for a spinor field. Introduce two real parameters $\left(u,v\right)$ and pick a one-parameter family of connections ${\nabla }_u={\nabla }_0+\mathit{u\eta }+\cdots$, $\eta \in \mathrm{\Gamma }\left({\mathrm{\Lambda }}^1\otimes g_L\right)$ and a one-parameter family of spinors ${\psi }_v={\psi }_0+\mathit{v\psi }+\cdots$, $\psi \in \mathrm{\Gamma }\left(V\otimes E\right)$. The curvature and total covariant derivative on $V\otimes E$ are

$${\mathrm{\Omega }}_u={\mathrm{\Omega }}_0+{\mathit{uD}}_0\eta +\frac{R_s}{2}\left[\eta ,\eta \right],{\overline{\nabla }}_u={\overline{\nabla }}_0+\mathit{u\rho }\left(\eta \right).$$E82

Expanding the action, it is given by

$$A\left({\nabla }_u,{\psi }_v\right)=\begin{array}{l}{\int }_M({\left|{\mathrm{\Omega }}_0\right|}^2+2u⟨{\mathrm{\Omega }}_0,D_0\eta ⟩+v⟨\psi ,\mathcal{D}\varphi ⟩+v⟨\varphi ,\mathcal{D}\psi ⟩\\ +u⟨\psi ,\sum _i{e}^i\rho \left({\eta }_i\right)\psi ⟩+\cdots ){\mathit{dv}}_g.\end{array}$$E83

In (83), $\left\{e^i\right\}$ is a local orthonormal basis. This implies the equations, which result from the first variation are for $\eta \in {\mathrm{\Gamma }}^{\infty }\left({\mathrm{\Lambda }}^1\otimes g_L\right)$ and $\psi \in {\mathrm{\Gamma }}^{\infty }\left(V\otimes E\right)$

$${\int }_M\left(2⟨D^{\ast }\mathrm{\Omega },\eta ⟩+⟨\psi ,\sum _i{e}^i\rho \left({\eta }_i\right)\psi )⟩\right){\mathit{dv}}_g=0,{\int }_M\left(⟨\psi ,\mathcal{D}\psi ⟩+⟨\varphi ,\mathcal{D}\psi ⟩\right){\mathit{dv}}_g=0,$$E84

Recall that $\mathcal{D}$ is self adjoint, so (84) gives the pair of equations

$$D^{\ast }\mathrm{\Omega }=J\left(\varphi \right)=-\frac{1}{2}\sum _i⟨\psi ,e^i\rho \left({\sigma }^a\right)\psi ⟩{\sigma }_a\otimes e_i,\mathcal{D}\psi =0.$$E85

In (85), $\left\{{\sigma }_a\right\}$ is a local orthogonal basis of sections of $g_L$, $\left\{{\sigma }^a\right\}$ the dual basis in $\mathrm{\Gamma }\left(g_L^{\ast }\right)$. The current due to $\psi$ is $J\left(\psi \right)$ and it is real-valued since $⟨\psi ,e^i\rho \left({\eta }_i\right)\psi ⟩=⟨e^i\rho \left({\eta }_i\right)\psi ,\psi ⟩$. It is interpreted as a one-form on the space of connections.

The boson action is defined on $E$-valued scalars $\varphi \in \mathrm{\Gamma }\left(E\right)$ by

$$A_b\left(g,\nabla ,\varphi \right)={\int }_M\left({\left|\mathrm{\Omega }\right|}^2+{\left|\overline{\nabla }\varphi \right|}^2+\frac{m}{6}{\left|\varphi \right|}^2-V\left(\varphi \right)\right){\mathit{dv}}_g.$$E86

The first variation of this action is computed as follows. Choose a one-parameter family of connections ${\nabla }_u={\nabla }_0+u\eta +\cdots$, $\eta \in \mathrm{\Gamma }\left(\mathrm{\Lambda }\otimes g_L\right)$ and a one-parameter family ${\varphi }_v={\varphi }_0+\mathit{v\tau }+\cdots$, $\tau \in \mathrm{\Gamma }\left(V\otimes E\right)$

$${\mathrm{\Omega }}_u={\mathrm{\Omega }}_0+u{D}_0\eta +\frac{u^2}{2}\left[\eta ,\eta \right],{\overline{\nabla }}_u={\overline{\nabla }}_0+u\rho \left(\eta \right),$$E87

Hence, the action is

$$\begin{array}{rcl}A\left(g,{\nabla }_u,{\varphi }_v\right)& =& {\int }_M{\left|{\mathrm{\Omega }}_0+{\mathit{uD}}_0\eta \right|}^2+{\left|{\overline{\nabla }}_0{\varphi }_0\right|}^2+u\rho \left(\eta \right){\varphi }_0+v{\left|{\nabla }_0\tau \right|}^2+\frac{m}{6}{\left|{\varphi }_0+\mathit{v\tau }\right|}^2\\ & & +V\left({\varphi }_0+\mathit{v\tau }\right){\mathit{dv}}_g\\ & =& {\int }_M{\left|{\mathrm{\Omega }}_0\right|}^2+2u⟨{\mathrm{\Omega }}_0,D_0\eta ⟩+{\left|{\overline{\nabla }}_0{\varphi }_0\right|}^2+2u⟨{\overline{\nabla }}_0{\varphi }_0,\rho \left(\eta \right){\varphi }_0⟩\\ & & +2v⟨{\overline{\nabla }}_0{\varphi }_0,{\overline{\nabla }}_0\tau ⟩\end{array}$$E88

$$+\frac{m}{6}{\left|{\varphi }_0\right|}^2+\frac{m}{6}v⟨\varphi ,\tau ⟩+\frac{m}{6}v⟨\tau ,{\varphi }_0⟩-V\left({\varphi }_0+\mathit{v\tau }\right.⟩{\mathit{dv}}_g.$$E89

Differentiating with respect to $u$ and $v$ then setting $u=v=0$,

$$\begin{array}{c}{\left.\frac{\partial A}{\partial u}\right|}_0=2{\int }_M\left(⟨D_0^{\ast }{\mathrm{\Omega }}_0,\eta ⟩+⟨{\overline{\nabla }}_0\varphi ,\rho \left(\eta \right){\varphi }_0⟩\right)d{\mu }_g,\end{array}$$

$$\begin{array}{c}{\left.\frac{\partial A}{\partial v}\right|}_0={\int }_M\left(2⟨{\overline{\nabla }}_0{\varphi }_0,{\overline{\nabla }}_0\tau ⟩+\frac{m}{6}⟨{\varphi }_0,\tau ⟩+\frac{m}{6}⟨\tau ,{\varphi }_0⟩+⟨V^{\prime }\left({\varphi }_0\right),\tau ⟩\right)d{\mu }_g.\end{array}$$E90

Equating the results, (90) to zero yields the coupled fermion and boson equations of motion taking $V^{\prime }=a{\left|\varphi \right|}^2\varphi +m_b^2\varphi$

$$\begin{array}{cc}{D}^{\ast }\mathrm{\Omega }=J=-\frac{1}{2}\sum ⟨\varphi ,e^i\cdot \rho \left({\sigma }^a\right)\varphi ⟩{\sigma }_a\otimes e_i,& \mathcal{D}\varphi =\mathit{m\varphi },\\ D^{\ast }\mathrm{\Omega }=J=-Re\sum _i⟨{\overline{\nabla }}_i\varphi ,\rho \left({\sigma }^a\right)\varphi ⟩{\sigma }_a\otimes e_i,& {\overline{\nabla }}^{\ast }\overline{\nabla }\varphi =\frac{m}{6}\varphi +a{\left|\varphi \right|}^2\varphi +m_b^2\varphi .\end{array}$$E91

In physics, one says $\mathrm{\Omega }$ is a gauge field, $\omega$ its gauge potential, and $\psi ,\varphi$ the field of a massive particle interacting with $\mathrm{\Omega }$. When the fields are set equal to zero, the fermion and boson actions reduce to the Yang-Mills field equations. Self-dual connections satisfy this equation because they are absolute minima of the action. In fact, the first field equation can be used to get

$$D^{\ast }J={\mathit{DD}}^{\ast }\mathrm{\Omega }=\left[\mathrm{\Omega },{}^{\ast }\mathrm{\Omega }\right]=\ast \sum _{\alpha ,\beta }⟨{\mathrm{\Omega }}_{\alpha },{\mathrm{\Omega }}_{\beta }⟩\left[{\sigma }^{\alpha },{\sigma }^{\beta }\right]=0.$$E92

When the structure group is abelian, the equation $D^{\ast }J=0$. This expresses the fact that electric charge is conserved in electromagnetism.

The field Eqs. (91) simplify considerably when we take $a=m=0$. Then either $\mathcal{D}\psi =0$ on E-valued spinors or ${\overline{\nabla }}^{\ast }\overline{\nabla }\varphi =\left(m/6\right)\varphi$ with $\varphi$ an E-valued scalar.

Theorem 6.1 Let $E$ be a vector bundle over a manifold $M$ and $\left(\varphi ,\mathrm{\Omega }\right)$ a solution of the coupled boson equations $D^{\ast }\mathrm{\Omega }=J$, ${\overline{\nabla }}^{\ast }\overline{\nabla }\varphi =\left(R_s/6\right)\varphi$. If $M$ is a compact manifold with positive scalar curvature, or if $M={\mathbb{R}}^4$ and $\varphi$ vanishes at infinity, then $\varphi =0$ and $\mathrm{\Omega }$ is Yang-Mills.

Proof: If $M$ is compact and $s>0$, integration by parts yields

$${\int }_M\left({\left|\overline{\nabla }\varphi \right|}^2+\frac{m}{6}{\left|\varphi \right|}^2\right){\mathit{dv}}_g=0.$$E93

Thus $\varphi =0$ and $J=0$. The equation ${\overline{\nabla }}^{\ast }\overline{\nabla }\varphi =0$ can be converted to a differential inequality for $\mid \varphi \mid$

$$\begin{array}{c}{d}^{\ast }d{\left|\varphi \right|}^2=2d^{\ast }⟨\varphi ,\overline{\nabla }\varphi ⟩=-2{\left|\overline{\nabla }\varphi \right|}^2+2⟨\varphi ,{\overline{\nabla }}^{\ast }\overline{\nabla }\varphi ⟩=-2{\left|\overline{\nabla }\varphi \right|}^2,\\ {d^{\ast }d{\left|\varphi \right|}^2=2d^{\ast }\left(|d|\varphi \right)=-2|d|\varphi ‖}^2+2\mid \varphi \mid d^{\ast }d\mid \varphi \mid .\end{array}$$E94

Thus upon solving the second in (94) for $\mid \varphi \mid d^{\ast }d\mid \varphi \mid$ and using the first, we get

$${|\varphi |d^{\ast }d|\varphi |=|d|\varphi ‖}^2-{\left|\overline{\nabla }\varphi \right|}^2\le 0.$$E95

Consequently, $\mathrm{\Delta }\mid \varphi \mid \ge 0$ If $\mid \varphi \mid$ vanishes at infinity, the maximum principle implies that $\varphi =0$, hence the current $J$ vanishes and $\mathrm{\Omega }$ is a Yang-Mills field. $\square$.

Theorem 6.2 Let $M$ be a compact Riemannian four-manifold with $R_s/3-\mid W^{-}\mid \ge \epsilon >0$. There is a constant $\alpha$ such that $\left(i\right)$ Any Yang-Mills $\mathrm{\Omega }$ such that ${‖{\mathrm{\Omega }}^{-}‖}_{0,2}<\alpha$ is self-dual (ii) Any solution $\left(\mathrm{\Omega },\varphi \right)$ to the massless coupled fermion Eqs. (90) with ${‖{\mathrm{\Omega }}^{-}‖}_{0,2}<\alpha$ satisfies ${\mathrm{\Omega }}^{-}=J={\varphi }^{-}=0$.

Proof: (i) Start with the equations $D^{\ast }\mathrm{\Omega }=D\mathrm{\Omega }=0$ to obtain $D{\mathrm{\Omega }}^{+}\pm D{\mathrm{\Omega }}^{-}=0$ so $D{\mathrm{\Omega }}^{-}=D^{+}{\mathrm{\Omega }}^{-}=0$ and hence $\square {\mathrm{\Omega }}^{-}=0$. Here, $\square ={\mathit{DD}}^{\ast }+D^{\ast }D$ is the Laplace Beltrami operator. Integrate $⟨{\mathrm{\Omega }}^{-},\square {\mathrm{\Omega }}^{-}⟩$ by parts over $M$ using the Weizenbock formula,

$$\square ={\overline{\nabla }}^{\ast }\overline{\nabla }+\frac{R_s}{3}+W^{+}\left(\right)-\left[{\left({\mathrm{\Omega }}^E\right)}^{+},\cdot \right]$$E96

and apply Kato’s inequality gives

$$\begin{array}{c}0=⟨{\mathrm{\Omega }}^{-},\square {\mathrm{\Omega }}^{-}⟩={\int }_M\left({\overline{\mathrm{\Omega }}}^{-}|{\nabla }^{\ast }|\nabla {\mathrm{\Omega }}^{-}+\frac{m}{3}{\left|{\mathrm{\Omega }}^{-}\right|}^2+W^{-}{\left|{\mathrm{\Omega }}^{-}\right|}^2-{\mathrm{\Omega }}^{-}\left[{\left({\mathrm{\Omega }}^E\right)}^{-},{\mathrm{\Omega }}^{-}\right]\right){\mathit{dv}}_g\\ ={\int }_M\left({\left|\overline{\nabla }{\mathrm{\Omega }}^{-}\right|}^2+\left(\frac{m}{3}+W^{-}\right){\left|{\mathrm{\Omega }}^{-}\right|}^2+{\overline{\mathrm{\Omega }}}^{-}\left[{\left({\mathrm{\Omega }}^E\right)}^{-},{\mathrm{\Omega }}^{-}\right]\right){\mathit{dv}}_g\\ \ge {\int }_M\left({|d|{\mathrm{\Omega }}^{-}‖}^2+\epsilon {\left|{\mathrm{\Omega }}^{-}\right|}^2-{\left|{\mathrm{\Omega }}^{-}\right|}^3\right){\mathit{dv}}_g.\end{array}$$E97

By Hölder’s inequality, followed by Sobolev’s inequality, the last term is bounded by

$${\int }_M{\left|{\mathrm{\Omega }}^{-}\right|}^3{\mathit{dv}}_g\le \alpha {‖{\mathrm{\Omega }}^{-}‖}_{0,2}\cdot \left({‖d{\mathrm{\Omega }}^{-}‖}_{0,2}^2+{‖{\mathrm{\Omega }}^{-}‖}_{0,2}^2\right).$$E98

This is dominated by the first two terms whenever ${‖{\mathrm{\Omega }}^{-}‖}_{0,2}$ is sufficiently small. But this means the right-hand side is positive and a contradiction. The only way this can be is that ${\mathrm{\Omega }}^{-}=0$.

(ii) If $\varphi =\left({\varphi }^{+},{\varphi }^{-}\right)\in V_{+}\oplus V_{-}$ satisfies $\mathcal{D}\psi =0$, then $⟨{\varphi }^{-},{\mathcal{D}}^2{\psi }^{-}⟩=0$. Using the Weitzenböck formula for the squared Dirac operator ${\mathcal{D}}^2$, we obtain

$$\begin{array}{l}0={\int }_M\left({\left|\overline{\nabla }{\varphi }^{=}\right|}^2+\frac{R_s}{2}{\left|{\varphi }^{-}\right|}^2+\frac{1}{2}\sum _{i,j}{e}^i{e}^j\cdot {\overline{\varphi }}^{-}{\mathrm{\Omega }}_{\mathit{ij}}^E{\varphi }^{-}\right){\mathit{dv}}_g\\ {\ge {\int }_M|d|{\varphi }^{-}‖}^2+\frac{R_s}{2}{\left|{\varphi }^{-}\right|}^2+\frac{1}{2}\sum _{i,j}{e}^i{e}^j{\overline{\varphi }}^{-}{\mathrm{\Omega }}_{\mathit{ij}}{\varphi }^{-}{\mathit{dv}}_g.\end{array}$$E99

Whenever ${‖{\mathrm{\Omega }}^{-}‖}_{0,2}$ is sufficiently small this inequality can apply, provided that ${\mathit{\varphi \varphi }}^{-}=0$ so then

$$-2D^{\ast }\mathrm{\Omega }=-2J=⟨{\varphi }^{-},e^i\rho \left({\sigma }^{\alpha }\right)\varphi ⟩{\sigma }_{\alpha }\otimes e_i+⟨{\varphi }^{+},e^i\rho \left({\sigma }^{\alpha }\right){\varphi }^{-}⟩{\sigma }_{\alpha }\otimes e_i=0.$$E100

Therefore, ${\mathrm{\Omega }}^{-}=0$ by $\left(i\right)$.

Solutions of the coupled field equations have the properties expected of elliptic equations, specifically, for $p>2$ an $L_{2p}$ weak solution is $C^{\infty }$_. This is basically elliptic regularity. There is a subtle point in that the coupled equations are elliptic only after a choice of gauge. Rather than using a connection to identify $\mathcal{C}=A^1$, we shall choose a point $x\in M$ and ball $B=B\left(x;r\right)$ around $x$ and fix a gauge, considered as a section of the frame bundle of $E$, to pull down connections. This identifies the space of connections over $B$ with ${\left.A^1\right|}_B$. Let $V_0$ be the connection corresponding to ${\left.0\in A^1\right|}_B$ under this identification. Then in terms of covariant derivatives, the original connection is $\nabla =d+\omega$, and $V_0$ is simply exterior differentiation $d$.

The tangent space to the orbit of the gauge group through $\left(\nabla ,\varphi \right)\in \mathcal{C}\in \mathcal{E}$ is the image of $K:A^0\to A^1+\mathcal{E}$ by $X\to \left(\nabla X,\rho \left(X\right),\varphi \right)$. The $L^2$ orthogonal complement of the image, which is the kernel of the adjoint operator $K^{\ast }$, provides a natural slice for the gauge orbit. This adjoint is $K^{\ast }:\left(\eta ,\psi \right)\to {\nabla }^{\ast }\eta +⟨\psi ,\rho \left(\right)\varphi ⟩$, where this last term selects an element of $A^0={\left(A^0\right)}^{\ast }$ via the Killing metric. There is a theorem which applies at the regular points of $\mathcal{C}\times \mathcal{E}$, where the action of the gauge algebra is free which is just stated: Suppose $M$ is a compact Riemannian $4$-manifold possibly with boundary. If a regular field $\left(V,\varphi \right)\in {\left(\mathcal{C}\times \mathcal{E}\right)}_{k+1,p}$ with $k\ge 0$, $2<p<4$. Then there is a constant $c$ such that for every field $\left(\eta ,\psi \right)$ with ${‖\left(\eta ,\psi \right)‖}_{k+1,p}<c$ there is a gauge transformation $g\in {\mathcal{C}}_{k+2,p}$ unique is a neighborhood of the identity, with $K_{\lambda }^{\ast }(g\cdot \left(V+\eta \right)-V$, $g\cdot \left(\varphi +\psi \right)-\varphi )=0$ weakly. If $\nabla ,\varphi ,\eta$ and $\psi$ are $C^{\infty }$, then $g$ is $C^{\infty }$.

Theorem 6.3 Let $\nabla$ be an $L_{k+1,p}$, $k\ge 0$, $2<p<4$ connection on a bundle $E$ over a four-manifold $M$ and let $\sigma :M\to \text{Frame}\left(E\right)$ be a $C^{\infty }$ gauge for $E$. Then there exists a constant $c>0$ depending only on $M$ such that if $\nabla =d+\omega$ and ${‖\omega ‖}_{k+1,p}<c$ in the gauge $\sigma$, then there is a gauge transformation $g\in {\mathcal{G}}_{k+2,p}$ such that $d^{\ast }\omega =0$ in the gauge $g\cdot \sigma$. If $\nabla$ is $C^{\infty }$, then $g$ is $C^{\infty }$.

We can choose a $C^{\infty }$ gauge around a given point $x_0$ and modify this to a gauge in which $d^{\ast }\omega =0$ using Theorem 6.2. To achieve this, it is necessary to make the $L_{k,p}$ norm of the fields small.

Theorem 6.4 Let $\nabla$ be an $L_{1,p}$, $2<p<4$ connection on a domain $D\subset M^4$. Then there is a $C^{\infty }$ gauge $\sigma$ and a gauge transformation $g\in {\mathcal{G}}_{2,p}$ such that, after a constant conformal change of metric, $d^{\ast }\omega =0$ in a neighborhood of $0\in D$ in the gauge $g\cdot \sigma$ and the new metric.

Proof: Given $\epsilon >0$ choose a $C^{\infty }$ gauge in a neighborhood of $0\in D$ and a small ball $B\left(1,\tau \right)$, $\tau >1$ around $0$ with ${‖\omega ‖}_{0,p}<\epsilon$ is the required scale. Take $B\left(1,{\tau }^2\right)$ to the unit disk $B\left(\tau ,1\right)$ by a conformal change of metric. Since ${\left|\omega \right|}^{2p}$ and ${\left|\nabla \omega \right|}^p={\left|\sum e^k\otimes {\nabla }_k\omega \right|}^p$ have conformal weight $2p$, rescaling gives

$${‖\omega ‖}_{0,2p,B\left(\tau ,1\right)}^{2p}={\tau }^{2p-4}{‖\omega ‖}_{0,2p,B\left(1,{\tau }^2\right)}^p,{‖\nabla \omega ‖}_{0,p,B\left(\tau ,1\right)}^p={\tau }^{2p-4}{‖\nabla \omega ‖}_{0,p,B\left(1,{\tau }^2\right)}^p.$$E101

In the new metric, Hölder’s inequality gives,

$${‖\omega ‖}_{1,p,B\left(\tau ,1\right)}\le {‖\nabla \omega ‖}_{0,p,B\left(\tau ,1\right)}+c{‖\omega ‖}_{0,2p,B\left(\tau ,1\right)}\le {‖\nabla \omega ‖}_{0,p,B\left(1,{\tau }^2\right)}+c{‖\omega ‖}_{0,2p,B\left(1,{\tau }^2\right)}\le \left(1+c\right)\epsilon ,$$E102

where $c^{2p}$ is the volume of the unit ball in the rescaled metric, which is uniformly bounded in $\tau$ for $\tau <1$. When $\epsilon$ is sufficiently small, Theorem 6.3 applies.

Uhlenbeck has proved the much more difficult fact that the rescaling used here depends only on ${‖\mathrm{\Omega }‖}_{0,p}$.

Theorem 6.5 Let $\left(\nabla \varphi \right)\in {\left(\mathcal{C}\times \mathcal{E}\right)}_{1,p}$, $p>2$ be a weak solution to the coupled Yang-Mills Eq. (90). Then there is an $L_{2,p}$ gauge in which $\left(\nabla ,\varphi \right)$ is $C^{\infty }$.

Proof: Fix an $x\in M$, By Theorem 6.2, there is an $L_{2,p}$ gauge defined in a neighborhood of $x$ such that $\nabla =d+\omega$ with $d^{\ast }\omega =0$ in this gauge. Expanding the field equations in this gauge, we have $J=D^{\ast }\mathrm{\Omega }=d^{\ast }d\omega +\omega \mathit{d\omega }+\left(1/2\right)\omega \left[\omega ,\omega \right]$. Hence ${\mathit{dd}}^{\ast }\omega =0$, so $d^{\ast }d\omega =\square \omega ={\nabla }^{\ast }\nabla \omega +\mathrm{Ric}\left(\omega \right)$ by the Weitzenböck formula, $\square ={\nabla }^{\ast }\nabla +\mathrm{Ric}+\left(1/2\right)\sum e^i\cdot e^j\cdot {\mathrm{\Omega }}_{\mathit{ij}}^E$. A boson field then weakly satisfies

$$\begin{array}{c}\mathrm{\Delta }\omega -\mathrm{Ric}\left(\omega \right)-\omega \mathit{d\omega }-\frac{1}{2}\omega \left[\omega ,\omega \right]-Re⟨\left(\nabla +\omega \right)\varphi ,\rho \left(\right)\varphi ⟩=0,\\ \mathrm{\Delta }\varphi +2\omega \varphi +\omega \omega \left(\varphi \right)+\frac{R_s}{6}\varphi +a{\left|\varphi \right|}^2\varphi +m^2\varphi =0,\end{array}$$E103

where $\mathrm{\Delta }$ is the metric Laplacian on functions. Applying $\mathcal{D}$ to $\mathcal{D}\varphi =\mathit{m\varphi }$ and using the Weizenböck formula (72) for the square of the Dirac operator on $E$-valued spinors, gives equations for the fermion fields. These are uniformly elliptic systems. Regularity follows by usual elliptic theory.

7. Conclusions

An extensive theory of Yang-Mills fields coupled to scalar and spinor fields on finite dimensional manifolds has been established. As well as differential geometric ideas, the appearence and systematic use of non-abelian Lie groups is also crucial and as such play a deep role in the study of elementary particles. The Yang-Mills fields represent forces or more accurately, they can be thought of as carriers of those fundamental forces. The presentation has been innovative and proofs have been given for all of the theorems that were introduced. It can also be looked at as a starting point for the study of other topics such as the existence of singularities or isolated singularities.