Fiber bundles, Yang–Mills theory, and general relativity

Abstract

I articulate and discuss a geometrical interpretation of Yang–Mills theory. Analogies and disanalogies between Yang–Mills theory and general relativity are also considered.

Appendix 1: Primer on the geometry of Yang–Mills theory for philosophers of space and time

In this appendix, I provide background definitions and some further technical details regarding principal bundles and vector bundles. Unlike other presentations of this material, this appendix is targeted at philosophers of physics familiar with the formalism and conventions of the foundations of spacetime physics literature, as in (for instance) Friedman (1983), Wald (1984), Earman (1995), or Malament (2012). That said, this is not a complete pedagogical treatment of these topics. For further details, the books I find clearest are Palais (1981) and Bleecker (1981); Nakahara 1990) and Baez and Munian (1994) are also noteworthy. For additional background on the geometry, the classic source on the theory of connections on fiber bundles is Kobayashi and Nomizu (1963), which remains the most comprehensive text available, despite certain pre-modern tendencies; additional helpful sources are Kolář et al. (1993), Michor (2009), Lee (2009), and Taubes (2011). One important difference, however, between the presentation here and in the sources just cited is that, as noted in fn. 8, I use the “abstract index” notation developed by Penrose and Rindler (1984) and described in detail by Wald (1984) and Malament (2012), suitably modified to distinguish the range of vector spaces that one encounters in the theory of principal bundles.⁵⁰

1.1 Fiber bundles

A (smooth) ⁵¹ fiber bundle $B\xrightarrow {\pi } M$ is a smooth, surjective map $\pi$ from a manifold B to a manifold M satisfying the following condition: there exists a manifold F such that, given any point $p\in M$, there exists an open neighborhood $U\subseteq M$ containing p and a local trivialization of B over U, which is a diffeomorphism $\zeta : U\times F\rightarrow \pi ^{-1}[U]$ such that $\pi \circ \zeta :(q,f)\mapsto q$ for all $(q,f)\in U\times F$. The map $\pi$ is called the projection map; the manifold B is called the total space of the bundle; the manifold M is called the base space; and the manifold F is called the typical fiber. The local trivialization condition guarantees that the collection of points in B mapped by $\pi$ to a given point $p\in M$, denoted $\pi ^{-1}[p]$ and called the fiber at p, is an embedded submanifold of the total space diffeomorphic to the typical fiber. This fact supports the following picture: a fiber bundle may be thought of as an association of copies of F with each point of M in such a way that the result is “locally a product manifold” in much the same way that a manifold is “locally $\mathbb {R}^n$”. We will sometimes write $F\rightarrow B\xrightarrow {\pi } M$ when we want to emphasize the typical fiber of a given fiber bundle; under other circumstances, when no ambiguity can arise, we will use just the total space B to refer to the entire bundle.

A fiber bundle morphism $(\Psi ,\psi ):(B\xrightarrow {\pi } M)\rightarrow (B{^\prime }\xrightarrow {\pi {^\prime }} M)$ is a pair of smooth maps $\Psi :B \rightarrow B{^\prime }$ and $\psi :M\rightarrow M{^\prime }$ such that $\pi '\circ \Psi =\psi \circ \pi$. A fiber bundle morphism is said to be an isomorphism if both maps are diffeomorphisms. A fiber bundle isomorphism whose domain and codomain are the same (i.e., a fiber bundle automorphism) is said to be vertical if the associated map $\psi :M\rightarrow M$ is the identity—i.e., if the automorphism takes the fiber over each point back to itself. A (local) section of a fiber bundle $B\xrightarrow {\pi } M$ is a smooth map $\sigma : U\rightarrow B$ such that $\pi \circ \sigma = 1_U$, where $U\subseteq M$ is open and $1_U$ is the identity on M restricted to U. A local section of a fiber bundle may be thought of as a generalization of the ordinary notion of smooth (scalar, vector, tensor) “field” on a manifold: it is a smoothly-varying assignment of a fiber value to each point $p\in U$.

Examples of fiber bundles include product manifolds with the projection onto one of the factors as the projection map, such as $N\rightarrow M\times N\xrightarrow {pr_1} M$. A fiber bundle that can be written this way, i.e., any bundle admitting a global trivialization, is called a trivial bundle.⁵² An example of a non-trivial fiber bundle is the Möbius strip, $(-1,1)\rightarrow M\ddot{o}\xrightarrow {\pi } S^1$, which has the circle as base space and an open subset of the real line as fiber. Here the fibers are “twisted” in such a way that $M\ddot{o}$ is not isomorphic to the cylinder, $S^1\times (-1,1)$.

1.2 Vector bundles and principal bundles

We will be particularly interested in two special classes of fiber bundles: principal bundles and vector bundles. A vector bundle is a fiber bundle $V\rightarrow E \xrightarrow {\pi } M$ where the typical fiber V is a vector space and for each $p\in M$, there exists a neighborhood U of p and a local trivialization $\zeta :U\times V\rightarrow \pi ^{-1}[U]$ such that for any $q\in U$, the map $v\mapsto \zeta (q,v)$ is a vector space isomorphism. A smooth fiber bundle morphism $(\Psi ,\psi ):(E\xrightarrow {\pi } M)\rightarrow (E{^\prime }\xrightarrow {\pi {^\prime }} M{^\prime })$ is a vector bundle morphism if for each $p\in M$, the restricted map $\Psi _{|\pi ^{-1}[p]}:\pi ^{-1}[p]\rightarrow \pi {^\prime }^{-1}[\psi (p)]$ is linear; it is a vector bundle isomorphism if it is also a fiber bundle isomorphism.

A principal bundle, meanwhile, is a fiber bundle $G\rightarrow P\xrightarrow {\wp } M$ where G is a Lie group—i.e., a smooth manifold endowed with a group structure in such a way that the group operations are smooth maps—and there is a smooth, free,⁵³ fiber-preserving right action of G on P such that given any point $p\in M$, there exists a neighborhood U of p and a local trivialization $\zeta :U\times G\rightarrow \wp ^{-1}[U]$ such that for any $q\in U$ and any $g,g'\in G$, $\zeta (q,g)g{^\prime }=\zeta (q,gg{^\prime })$. The group G is known as the structure group of the bundle. A principal bundle morphism consists of a fiber bundle morphism $(\Psi ,\psi ):(G\rightarrow P\xrightarrow {\wp } M)\rightarrow (G{^\prime }\rightarrow P{^\prime }\xrightarrow {\wp {^\prime }} M{^\prime })$ and a smooth homomorphism $h:G\rightarrow G{^\prime }$ such that for any $x\in P$ and $g\in G$, $\Psi (xg)=\Psi (x)h(g)$. A principal bundle morphism is a principal bundle isomorphism if $(\Psi ,\psi )$ and h are isomorphisms; it is a vertical principal bundle automorphism if $(\Psi ,\psi )$ is a vertical bundle automorphism and h is the identity map.

1.3 Notational conventions

Let $F\rightarrow B\xrightarrow {\pi }M$ be a fiber bundle. I will adopt the following notational conventions.⁵⁴ I will use lower-case Latin indices $a,b,c,\ldots$ for vectors and tensors that are tangent to the base space of a bundle, or generically when I am considering manifolds outside the context of a particular bundle. Lower-case Greek indices $\alpha ,\beta ,\gamma \ldots$ will label vectors and tensors that are tangent to the total space of a bundle. So given a point $x\in B$, a vector at x would be denoted (for instance) by $\xi ^{\alpha }$, while a vector at $\pi (x)$ would be denoted by $\eta ^a$. Capital Latin indices $A,B,C,\ldots$ will label vectors and tensors valued in other spaces, including the fibers of vector bundles. In cases where there are several such vector spaces under consideration, further decorations will be used to distinguish membership in the different spaces. Finally, if a given vector space has a Lie algebra structure, we will label vectors in that space using capital Fraktur indices $\mathfrak {A},\mathfrak {B},\mathfrak {C},\ldots$. In all cases, raised indices will indicate that an object is an element of a salient vector space; lowered indices will indicate that the object is a linear functional on the corresponding vector space.

This notation allows one to consider “mixed index” tensors and tensor fields on a manifold M, which represent multilinear maps between various vector spaces associated with a point of M. For instance, given a vector $\xi ^{\alpha }$ at a point x in the total space of a fiber bundle, one may think of the pushforward along $\pi$ at x, $(\pi _x)_*$, as a linear map from vectors at x to vectors at $\pi (x)$, which might then be written as $(\nabla \pi )^a{}_{\alpha }$. In fact, this notation also subsumes the pullback along $\pi$, since given a covector $\kappa _a$ at $\pi (x)$, $\kappa _a(\nabla \pi )^a{}_{\alpha }$ is precisely the covector at x whose action on a vector $\xi ^{\alpha }$ is the action of $\kappa _a$ on the pushforward of $\xi ^{\alpha }$, i.e., $(\kappa _a(\nabla \pi )^a{}_{\alpha })\xi ^{\alpha }=\kappa _a((\nabla \pi )^a{}_{\alpha }\xi ^{\alpha })$. In what follows, I will freely adopt both the mixed index and $*$ notations for the pushforward and pullback, depending on context. Other examples of mixed index tensors include principal connections (discussed in appendix section “Principal connections”), curvature tensors (discussed in appendix section “Curvature”), and solder forms (discussed in Sect. 5). Mixed index tensors on a manifold M are said to be smooth if their contraction with appropriate collections of smooth vectors (and covectors) is a smooth scalar field.⁵⁵

1.4 Tangent and frame bundles

Any manifold M is naturally associated with a vector bundle over M, known as the tangent bundle. Let $T_pM$ be the tangent space at $p\in M$ and let TM be the set of all of the tangent vectors at all point of M, $TM= \bigcup _{p\in M} T_pM$. A manifold structure on TM may be induced as follows. First note that any point $x\in TM$ may be written as $(p,\xi ^a)$, where $\xi ^a$ is a tangent vector at p. Then, given any chart $(U,\varphi )$ on M, one can associate a point $(p,\xi ^a)\in TM$ with an element of $\mathbb {R}^{2n}$ (where n is the dimension of M) by $(p,\xi ^a)\mapsto (\varphi ^1(p),\ldots ,\varphi ^n(p),\overset{1}{\xi },\ldots \overset{n}{\xi })$. Here $\varphi ^i(p)$ represents the ith coordinate of p relative to the chosen chart, and $\overset{i}{\xi }$ is the ith component of $\xi ^a$ in the basis determined by the chart. Requiring all such maps, for every chart on M, to be smooth and smoothly invertible induces a manifold structure on TM. The map $\pi _T:TM\rightarrow M$ that takes elements $x\in TM$ to the point $p\in M$ to which they correspond—that is, $\pi _T:(p,\xi ^a)\mapsto p$—is smooth relative to this manifold structure. Similarly, the maps used to induce the manifold structure on TM are local trivializations relative to which $\mathbb {R}^n\rightarrow TM\xrightarrow {\pi _T} M$ is a vector bundle over M. Sections of the tangent bundle are (ordinary, tangent) vector fields, i.e., smooth assignments of tangent vectors to points of (an open subset of) a manifold. Similar constructions may be used to define the cotangent bundle, $T^*M\xrightarrow {\pi _{T^*}}M$, and bundles of rank (r, s) tensors on M.

The manifold M also naturally determines a principal bundle over M, known as the frame bundle. The construction is similar to the tangent bundle. Suppose n is the dimension of M. Then a frame at a point p is an ordered collection of n linearly independent vectors at p. Let LM be the collection of all frames at all points of M. Analogously to the tangent bundle, any point $x\in LM$ may be written (p, u), where $u=(\overset{1}{u}{}^a,\ldots ,\overset{n}{u}{}^a)$ is a frame at p. Now given a chart $(U,\varphi )$ on M, we may associate any point $(p,u)\in LM$ with an element of $\mathbb {R}^{n^2+n}$ by $(p,u)\mapsto (\varphi ^1(p),\ldots ,\varphi ^n(p),\overset{11}{u},\ldots ,\overset{ij}{u},\ldots ,\overset{nn}{u})$, where now $\overset{ij}{u}$ is the jth component in the basis determined by the chart of $\overset{i}{u}{}^a$, the ith element of the frame. The image of this map is $\varphi [U]\times F$, where $F\subset \mathbb {R}^{n^2}$ is the collection of all $n^2$-tuples corresponding to invertible $n\times n$ matrices. Thus F is an open subset of $\mathbb {R}^{n^2}$ (because the determinant map $det:\mathbb {R}^{n^2}\rightarrow \mathbb {R}$ is continuous and therefore the set of matrices with determinant 0, $det^{-1}[0]$, is closed), diffeomorphic to the Lie group $GL(n,\mathbb {R})$ of invertible real valued matrices. (The Lie algebra associated with $GL(n,\mathbb {R})$, $\mathfrak {gl}(n,\mathbb {R})$, is the algebra of all $n\times n$ matrices, not necessarily invertible.) Requiring all such maps, for all charts on M, to be smooth and smoothly invertible induces a manifold structure on LM. The map $\wp _L:LM\rightarrow M$ where $\wp _L:(p,u)\mapsto p$ is smooth with respect to this structure, and once again the chart-relative maps defined above are local trivializations relative to which $GL(n,\mathbb {R})\rightarrow LM\xrightarrow {\wp _L} M$ is a principal bundle, where the associated right $GL(n,\mathbb {R})$ action corresponds to a smooth change of frame at each point. Sections of the frame bundle are (local) frame fields, which are smoothly varying bases of the tangent space assigned to each point of (an open subset of) a manifold.

More generally, given any vector bundle $V\rightarrow E\rightarrow M$, one may readily construct an associated principal bundle $GL(V)\rightarrow LE\rightarrow M$, called the frame bundle for E, whose fibers correspond to the frames for the fibers of E.

1.5 Associated bundles

The frame bundle construction provides a sense in which a given vector bundle may be used to build a principal bundle. But one can also move in the other direction, from a given principal bundle to a vector bundle. Let $G\rightarrow P\xrightarrow {\wp } M$ be a principal bundle, and let V be some vector space with a (fixed) representation $\rho :G\rightarrow GL(V)$ of G. Now for each point $p\in M$, consider maps $v^A:\wp ^{-1}[p]\rightarrow V$ that are equivariant in the sense that for any $x\in \wp ^{-1}[p]$, $v^A(xg)=(\rho (g^{-1}))^A{}_B v^B$ and smooth in the sense that, given any fixed linear functional $u_A$ on V, $u_Av^A$ is a smooth scalar field on $\pi ^{-1}[p]$. Let $P\times _{G} V$ denote the set of all such maps, for all points $p\in M$. Then there is a unique manifold structure on $P\times _{G} V$ such that $V\rightarrow P\times _G V\xrightarrow {\pi _{\rho }} M$, where $\pi _{\rho }:(v^A:\pi ^{-1}[p]\rightarrow V)\mapsto p)$, is a vector bundle over M. This bundle is called an associated vector bundle. Under this construction, if V is an n dimensional vector space, then any vector bundle $V\rightarrow E\xrightarrow {\pi } M$ is isomorphic to $LE\times _{GL(V)} V\xrightarrow {\pi _{\rho }}M$, for any faithful representation of GL(V) on V, where $LE\xrightarrow {\wp _L} M$ is the frame bundle for E.

1.6 Vector valued forms

Let M be a manifold and let V be a vector space. A vector valued n form (or a V valued n form) on an open set U is a mixed index tensor field $\kappa ^A{}_{a_1\ldots a_n}$ that is totally antisymmetric in its covariant (tangent) indices, and where the A index indicates membership in the (fixed) vector space V.⁵⁶ These fields are required to satisfy the following (degenerate) smoothness condition: given any (fixed) linear functional $\beta _A$ on V, we require that $\beta _A\alpha ^A{}_{a_1\ldots a_n}$ is a smooth (ordinary) n form on U. Note that from this perspective, ordinary n forms are also vector valued forms, where the vector space in which they are valued is $\mathbb {R}$; in such cases, one simply drops the index corresponding to membership in $\mathbb {R}$.

Recall that there is a natural notion of differentiation on (ordinary, $\mathbb {R}$ valued) n forms on M, given by the exterior derivative $d_a$, which takes n forms $\kappa _{a_1\ldots a_n}$ to $(n+1)$ forms $d_a\kappa _{a_1\ldots a_n}$. The exterior derivative extends to vector valued n forms as follows. Given a (smooth) V valued n form $\kappa ^A{}_{a_1\ldots a_n}$, we take the exterior derivative of $\kappa ^A{}_{a_1\ldots a_n}$, written $d_a\kappa ^A{}_{a_1\ldots a_n}$, to be the unique V valued $(n+1)$ form whose action on any (fixed) linear functional $\beta _A$ on V is given by $\beta _A(d_n\alpha ^A{}_{a_1\ldots a_n})=d_n(\beta _A\alpha ^A{}_{a_1\ldots a_n})$, where the expression on the right should be interpreted as the (ordinary) exterior derivative of the (ordinary, smooth) n form $\beta _A\alpha ^A{}_{a_1\ldots a_n}$.⁵⁷ Similarly, given a smooth map $\varphi :M\rightarrow N$ and a V valued n form $\alpha ^A{}_{a_1\ldots a_n}$ on N, one can define the pullback of $\alpha ^A{}_{a_1\ldots a_n}$ along $\varphi$, $\varphi ^*(\alpha ^A{}_{a_1\ldots a_n})$, as the unique V valued n form on M such that, given any (fixed) linear functional $\beta _A$ acting on V, $\beta _A\varphi ^*(\alpha ^A{}_{a_1\ldots a_n})=\varphi ^*(\beta _A\alpha ^A{}_{a_1\ldots a_n})$.

Finally, consider the special case of vector valued n forms on the total space P of a principal bundle $G\rightarrow P\xrightarrow {\wp } M$ (or, more generally, forms defined on $\wp ^{-1}[U]$, for some open $U\subset M$). In this context, it is often natural to fix a representation $\rho$ of G, the structure group of the bundle, on the vector spaces in which forms of interest are valued. One can then consider forms that are equivariant with respect to the G action on P, in the sense that, given a V valued n form $\kappa ^A{}_{\alpha _1\ldots \alpha _n}$ on P, $\kappa ^A{}_{\alpha _1\ldots \alpha _n}$ is such that given any element $g\in G$, any point $x\in P$, and any vectors $\overset{1}{\eta }{}^{\alpha },\ldots ,\overset{n}{\eta }{}^{\alpha }$ at x, the following condition holds:

$$\begin{aligned} (\kappa ^{A}{}_{\alpha _1\ldots \alpha _n})_{|xg}(R_g)_*(\overset{1}{\eta }{}^{\alpha _1}\ldots \overset{n}{\eta }{}^{\alpha _n})=((\rho (g^{-1}))^{A}{}_{B}\kappa ^{B}{}_{\alpha _1\ldots \alpha _n})_{|x}\overset{1}{\eta }{}^{\alpha _1}\ldots \overset{n}{\eta }{}^{\alpha _n}. \end{aligned}$$

(6)

Here $(\rho (g^{-1}))^{A}{}_{B}$ is the tensor acting on V corresponding to $g^{-1}$ in the representation $\rho$, and $R_g$ is the smooth right action of g on P. Note that this equation makes sense, since both sides are elements of the fixed vector space V. Note, too, that, following the construction of the previous section, we now see that sections $\kappa ^A:U\rightarrow P\times _G V$ of the associated bundle $V\rightarrow P\times _G V\xrightarrow {\pi _{\rho }}M$ may be identified with equivariant V valued 0 forms on $\wp ^{-1}[U]$.

1.7 Connections and parallel transport

We now turn to connections. Some preliminary definitions are in order. Fix an arbitrary fiber bundle $F\rightarrow B\xrightarrow {\pi } M$ and let $\xi ^{\alpha }$ be a vector at a point $x\in B$. We will say $\xi ^{\alpha }$ is vertical if $(\nabla \pi )^a{}_{\alpha }\xi ^{\alpha }=\mathbf {0}$. Since $(\nabla \pi )^a{}_{\alpha }$ is a linear map, its kernel forms a linear subspace of $T_x B$, written $V_x$ and called the vertical subspace at x; in general, the dimension of $V_x$ will be the dimension of F. Indeed, one can think of the vertical vectors at x as “tangent to the fiber” in the precise sense that they are tangents to curves through x that remain in the fiber over $\pi (x)$. If a vector $\xi ^{\alpha }$ at a point x is not vertical, then it is horizontal. A subspace $H_x$ of $T_xB$ will be called a horizontal subspace if any vector $\xi ^{\alpha }$ at x may be uniquely written as the sum of one vector in $V_x$ and one vector in $H_x$. It immediately follows that, given any horizontal subspace $H_x$ at x, $(\nabla \pi )^a{}_{\alpha }:H_x\rightarrow T_{\pi (x)}$ is a vector space isomorphism, and the dimension of any horizontal subspace is the same as the dimension of M.

Though the vertical subspace is uniquely determined by the map $\pi$, there is considerable freedom in the choice of horizontal subspace. A connection on B, roughly speaking, is a smoothly varying choice of horizontal subspace at each point $x\in B$. This idea of “smoothly varying” may be made precise as follows. Given any point $x\in B$ and a horizontal subspace $H_x$, one can always find a tensor $\omega ^{\alpha |}{}_{\beta }$ that acts on any vector $\xi ^{\alpha }$ at x by projecting $\xi ^{\alpha }$ onto its (unique, relative to $H_x$) vertical component, $\omega ^{\alpha |}{}_{\beta }\xi ^{\beta }$. (Here we have used a | following a Greek index to emphasize that the index is vertical.) Conversely, given any tensor $\omega ^{\alpha |}{}_{\beta }$ at x such that (a) $\omega ^{\alpha |}{}_{\beta }\omega ^{\beta |}{}_{\kappa }=\omega ^{\alpha |}{}_{\kappa }$ and (b) given any vertical vector $\xi ^{\alpha |}$ at x, $\omega ^{\alpha |}{}_{\beta }\xi ^{\beta |}=\xi ^{\alpha |}$, one can always define a horizontal subspace $H_x$ at x as the kernel of $\omega ^{\alpha |}{}_{\beta }$. This permits us to adopt the following official definition of a connection: a connection on a fiber bundle $B\xrightarrow {\pi } M$ is a smooth tensor field $\omega ^{\alpha |}{}_{\beta }$ on B satisfying conditions (a) and (b) above. Every fiber bundle admits connections.

A connection provides a notion of parallel transport of fiber values along curves in the base space M. This works as follows. Consider a smooth curve $\gamma :I\rightarrow M$. One can always lift such a curve $\gamma$ into the total space B, by defining a new curve $\hat{\gamma }:I\rightarrow B$ with the property that $\pi \circ \hat{\gamma }=\gamma$. This curve $\hat{\gamma }$ is generally not unique. One gets a unique lift by specifying some additional data: fix a connection $\omega ^{\alpha |}{}_{\beta }$ and choose some $t_0\in I$ and some $x\in \pi ^{-1}[\gamma (t_0)]$—that is, choose some point in the fiber above $\gamma (t_0)$. Then there is a unique horizontal lift of $\gamma$ through x, that is, a unique lift $\hat{\gamma }:I\rightarrow E$ such that (1) $\hat{\gamma }(t_0)=x$ and (2) the vector tangent to $\hat{\gamma }$ at each point in its image is horizontal relative to $\omega ^{\alpha |}{}_{\beta }$. Then, given any $t\in I$, we say the parallel transport of x to $\gamma (t)$ along $\gamma$ is $\hat{\gamma }(t)$, which is a point in the fiber above $\gamma (t)$.

1.8 Principal connections

Now suppose one has a principal bundle $G\rightarrow P\xrightarrow {\wp } M$. Here, one is interested in connections that are compatible with the principal bundle structure in the sense that they are equivariant under the right action of G on P. A bit of work is required to make this precise. First, recall that given any Lie group G, $T_eG$, the tangent space at the identity element, is endowed with a natural Lie algebra structure induced from the Lie group structure as follows. Take any vector $\xi ^a\in T_eG$. One can define a (smooth) vector field on G, called a left-invariant vector field, by assigning to each point $g\in G$ the vector $({}^g\ell _e)_*(\xi ^a)$, i.e., the pushforward of $\xi ^a$ along the left action on G determined by g. The Lie bracket of two vectors at e, then, is just the ordinary commutator of the corresponding vector fields induced by the left action.⁵⁸ The Lie algebra associated in this way with a Lie group G is often denoted $\mathfrak {g}$. The Lie algebra $\mathfrak {g}$, understood as a vector space, comes with a privileged representation of G, called the adjoint representation, $ad:G\rightarrow GL(\mathfrak {g})$, defined by $(ad(g))^{\mathfrak {A}}{}_{\mathfrak {B}}=(\nabla \Upsilon ^g_{|e})^{\mathfrak {A}}{}_{\mathfrak {B}}$, where (1) $\Upsilon ^g:G\rightarrow G$ acts as $\Upsilon ^g:h\mapsto ghg^{-1}$, and (2) $(\nabla \Upsilon ^g_{|e})^{\mathfrak {A}}{}_{\mathfrak {B}}$ should be understood as the pushforward along $\Upsilon ^g$ at the identity, which maps $T_eG$ to itself because $\Upsilon ^g(e)=geg^{-1}=e$ for every $g\in G$.

The right action of the structure group G on the principal bundle P allows us to define a canonical isomorphism between the vertical space $V_x$ at each point $x\in P$ and the Lie algebra $\mathfrak {g}$ associated with G. Given any vector $\xi ^{\mathfrak {A}}\in T_eG$, let $\gamma _{\xi }:I\rightarrow G$ be the (sufficiently unique) integral curve of the left-invariant vector field associated with $\xi ^{\mathfrak {A}}$. We assume $\gamma (0)=e$. Then, given any point $x\in P$, one can define a curve $\tilde{\gamma }_{\xi }:I\rightarrow P$ through x by setting $\tilde{\gamma }_{\xi }(t)=x\gamma _{\xi }(t)$. We will take the tangent to this curve, $\overrightarrow{(\tilde{\gamma }_{\xi })}^{\alpha |}$, which is necessarily vertical because the right action of G on P is fiber-preserving, to be the vertical vector at x associated with $\xi ^{\mathfrak {A}}$. This construction defines a linear bijection that can be represented by a mixed index tensor field on P, $\mathfrak {g}^{\mathfrak {A}}{}_{\alpha |}$, with inverse $\mathfrak {g}^{\alpha |}{}_{\mathfrak {A}}$. (Here the | next to a covariant index means that $\mathfrak {g}^{\mathfrak {A}}{}_{\alpha |}$ is only defined for vertical vectors.) Thus, we can always think of vertical vectors at a point x of a principal bundle as elements of a fixed Lie algebra, independent of x.

The isomorphism just described allows one to think of a connection $\omega ^{\alpha |}{}_{\beta }$ on a principal bundle as a vector valued one form (sometimes called a Lie algebra valued one form), given by $\omega ^{\mathfrak {A}}{}_{\beta }=\mathfrak {g}^{\mathfrak {A}}{}_{\alpha |}\omega ^{\alpha |}{}_{\beta }$. Since this is a vector valued form on P, we also fix a representation of G on $\mathfrak {g}$, the vector space in which the form is valued (recall appendix section “Vector valued forms”); here, as with all Lie algebra valued forms we will consider, we take G to act on $\mathfrak {g}$ in the adjoint representation. Finally, we may define a principal connection as a connection $\omega ^{\mathfrak {A}}{}_{\alpha }$ on P that is equivariant in the sense of Eq. (6). This condition guarantees that for any point $x\in P$ and any $g\in G$, the horizontal subspace determined by $\omega ^{\mathfrak {A}}{}_{\beta }$ at xg equals the pushforward of the horizontal space at x along the right action of g, i.e., $({}^gR)_*[H_x]=H_{xg}$. Every principal bundle admits principal connections.

1.9 Covariant derivatives

Consider a vector bundle $V\rightarrow E\xrightarrow {\pi } M$. A covariant derivative operator $\nabla$ on E is a map from sections $\sigma ^A:U\rightarrow E$ of E to mixed index tensor fields $\sigma ^A\mapsto \nabla _a\sigma ^A$ on U satisfying the following conditions: (1) given two smooth sections $\sigma ^A,\nu ^A:U\rightarrow M$, $\nabla _a(\sigma ^A + \nu ^A)=\nabla _a\sigma ^A + \nabla _a\nu ^A$ and (2) given any smooth scalar field $\lambda :M\rightarrow \mathbb {R}$, $\nabla _a (\lambda v^A)=v^A d_a\lambda + \lambda \nabla _a v^A$, where $d_a$ is the exterior derivative. The covariant derivative of a section $\sigma ^A:U\rightarrow E$ has the following interpretation. Given a vector $\xi ^a$ at a point $p\in U$, $\xi ^a\nabla _a\sigma ^A$ is the derivative of $\sigma ^A$ in the direction of $\xi ^a$, relative to a standard of fiber-to-fiber constancy given by $\nabla$. The covariant derivatives one encounters in general relativity are special cases of this more general definition, where the vector bundle in question is the tangent bundle (and, by extension, various bundles of tensors constructed out of tangent vectors). Note that a covariant derivative operator on an arbitrary vector bundle also provides a notion of parallel transport, by a construction directly analogous to that for covariant derivatives on the tangent bundle.⁵⁹

1.10 Exterior and induced covariant derivatives

Now consider a principal bundle $G\rightarrow P\xrightarrow {\wp } M$ endowed with a principal connection $\omega ^{\mathfrak {A}}{}_{\alpha }$ and a vector space V with a fixed representation $\rho :G\rightarrow GL(V)$ of the structure group of P on V. In this case, we can define a second notion of differentiation for V valued n forms, called the exterior covariant derivative relative to $\omega ^{\mathfrak {A}}{}_{\alpha }$. It is denoted by $\overset{\omega }{D}$. The action of $\overset{\omega }{D}$ on a V valued n form $\kappa ^A{}_{\alpha _1\ldots \alpha _n}$ on $U\subseteq P$ is given by $\overset{\omega }{D}_{\alpha }\kappa ^A{}_{\alpha _1\ldots \alpha _n}=(d_{\beta }\kappa ^A{}_{\beta _1\ldots \beta _n})\bar{\omega }{}^{\beta }{}_{\alpha }\bar{\omega }{}^{\beta _1}{}_{\alpha _1}\ldots \bar{\omega }{}^{\beta _n}{}_{\alpha _n}$, where d is the ordinary exterior derivative and where $\bar{\omega }^{\alpha }{}_{\beta }=\delta ^{\alpha }{}_{\beta }-\mathfrak {g}^{\alpha |}{}_{\mathfrak {A}}\omega ^{\mathfrak {A}}{}_{\beta }$ is the horizontal projection relative to $\omega ^{\mathfrak {A}}{}_{\alpha }$. (Recall appendix section “Principal connections”.)

A V valued n form $\kappa ^A{}_{\alpha _1\ldots \alpha _n}$ on $\wp ^{-1}[U]$, for some open $U\subseteq M$, is said to be horizontal and equivariant if (1) it is horizontal in the sense that given any vertical vector $\xi ^{\alpha }$ at a point $p\in U$, $\kappa ^A{}_{\alpha _1\ldots \alpha _i\ldots \alpha _n}\xi ^{\alpha _i}=\mathbf {0}$ for all $i=1,\ldots , n$ and (2) it is equivariant in the sense of Eq. (6) . The important feature of the exterior covariant derivative is that if a V valued n form $\kappa ^A{}_{\alpha _1\ldots \alpha _n}$ is horizontal and equivariant, so is its exterior covariant derivative, $\overset{\omega }{D}_{\alpha }\kappa ^A{}_{\alpha _1\ldots \alpha _n}$. In the special case where V is the Lie algebra associated with the principal bundle and $\kappa ^{\mathfrak {A}}{}_{\alpha _1\ldots \alpha _n}$ is a horizontal and equivariant Lie algebra valued n form, we have the relation $\overset{\omega }{D}_{\alpha }\kappa ^{\mathfrak {A}}{}_{\alpha _1\ldots \alpha _n}=d_{\alpha }\kappa ^{\mathfrak {A}}{}_{\alpha _1\ldots \alpha _n}+[\omega ^{\mathfrak {A}}{}_{\alpha },\kappa ^{\mathfrak {A}}{}_{\alpha _1\ldots \alpha _n}]$, where the bracket is the Lie bracket.

Finally, recall that in appendix section “Vector valued forms”, we observed that sections $\kappa ^A:U\rightarrow P\times _G V$ of the associated vector bundle $V\rightarrow P\times _G V\xrightarrow {\pi _{\rho }} M$ determined by P, V, and $\rho$ are naturally understood as V valued 0 forms on (subsets of) P. We now see that in fact they are horizontal and equivariant 0 forms. We may thus define an induced covariant derivative operator $\overset{\omega }{\nabla }$ on $P\times _G V$ as follows: given any section $\kappa ^A:U\rightarrow P\times _G V$ of $P\times _G V$, we take $\overset{\omega }{\nabla }_a\kappa ^A$ to be the unique mixed index tensor on U with the property that, given any point $p\in U$ and any vector $\xi ^a$ at p, $\xi ^a\overset{\omega }{\nabla }_a\kappa ^A$ is the vector in the fiber of $P\times _G V$ over p corresponding to the equivariant V valued 0 form $\xi ^{\alpha }\overset{\omega }{D}_{\alpha }\kappa ^A$ defined on the fiber of P over p, where $\xi ^{\alpha }$ is any vector field on the fiber with the property that at every point $x\in \wp ^{-1}[p]$, $(\nabla \wp )^a{}_{\alpha }\xi ^{\alpha }=\xi ^a$.⁶⁰ Note that this fully determines the action of $\overset{\omega }{\nabla }$ on sections of $P\times _G V$, and that, with this definition, $\overset{\omega }{\nabla }$ is both additive and satisfies the Leibniz rule (recall appendix section “Covariant derivatives”). Conversely, given a principal connection on a principal bundle, this construction yields a unique covariant derivative operator on every associated vector bundle, and given a covariant derivative operator on a vector bundle, there is always a unique principal connection on the frame bundle of that vector bundle that induces the covariant derivative in this way.

1.11 Curvature

In general, the standards of parallel transport given by a principal connection or a covariant derivative operator are path-dependent. The degree of path-dependence is measured by the curvature of a connection or derivative operator. Given a principal bundle $G\rightarrow P\xrightarrow {\wp } M$ and a principal connection $\omega ^{\mathfrak {A}}{}_{\alpha }$ on P, the curvature of $\omega ^{\mathfrak {A}}{}_{\alpha }$ is a horizontal and equivariant Lie algebra valued two form $\Omega ^{\mathfrak {A}}{}_{\alpha \beta }$ on P, defined by

$$\begin{aligned} \Omega ^{\mathfrak {A}}{}_{\alpha \beta } = \overset{\omega }{D}_{\alpha }\omega ^{\mathfrak {A}}{}_{\beta }. \end{aligned}$$

(7)

Its interpretation is as follows. Given a point $p\in M$, an infinitesimal closed curve through p may be represented by a pair of vectors, $\xi ^a$ and $\eta ^a$, at p, corresponding to the “incoming” and “outgoing” directions of the curve at p. Given an arbitrary point $x\in \wp ^{-1}[p]$, the (infinitesimal, or limiting) parallel transport of x along this infinitesimal curve in M is encoded by the vertical vector $\mathfrak {g}^{\kappa |}{}_{\mathfrak {A}}\Omega ^{\mathfrak {A}}{}_{\alpha \beta }\xi ^{\alpha }\eta ^{\beta }$ at x, where $\xi ^{\alpha }$ and $\eta ^{\beta }$ are arbitrary vectors at x with the properties that $(\nabla \wp )^a{}_{\alpha }\xi ^{\alpha }=\xi ^a$ and $(\nabla \wp )^a{}_{\alpha }\eta ^{\alpha }=\eta ^a$, respectively. This vertical vector represents the direction and magnitude of displacement of x under the infinitesimal parallel transport.

It is often convenient to express this curvature in a slightly different form, using the so-called “structure equation” (Bleecker 1981, p. 37):

$$\begin{aligned} \Omega ^{\mathfrak {A}}{}_{\alpha \beta } = d_{\alpha }\omega ^{\mathfrak {A}}{}_{\beta } + \frac{1}{2} \left[ \omega ^{\mathfrak {A}}{}_{\alpha },\omega ^{\mathfrak {A}}{}_{\beta }\right] . \end{aligned}$$

(8)

Here the bracket $[\cdot ,\cdot ]$ is the Lie bracket on the Lie algebra $\mathfrak {g}$. It is in this form that the curvature appears in Sect. 2. (Observe that this relation is a special case of the general fact concerning exterior covariant derivatives of Lie algebra valued forms stated in appendix section “Exterior and induced covariant derivatives”.)

Now suppose one has a vector bundle $V\rightarrow E\xrightarrow {\pi } M$ with a covariant derivative $\nabla$ on E. In this case, we may define the curvature tensor $R^A{}_{Bcd}$ as the unique mixed index tensor on M such that, given any section $\kappa ^A:U\rightarrow E$, the action of $R^A{}_{Bcd}$ on $\kappa ^A$ at any point $p\in U$ is:⁶¹

$$\begin{aligned} R^A{}_{Bcd}\kappa ^B=-2\nabla _{[c}\nabla _{d]}\kappa ^A. \end{aligned}$$

(9)

Note that in the special case where the vector bundle is the tangent bundle, this curvature tensor corresponds exactly to the Riemann tensor. To see the relationship between the curvature tensors defined in Eqs. (7) and (9) more clearly, note that $R^A{}_{Bcd}$ may be thought of as a mixed index tensor $\Omega ^A{}_{\alpha \beta }$ on E, whose action at a point $v^A$ of E on vectors $\xi ^{\alpha },\eta ^{\alpha }$ at $v^A$ is given by $(\Omega ^A{}_{\alpha \beta }\xi ^{\alpha }\eta ^{\beta })_{|v^A}=R^A{}_{Bcd}v^B(\nabla \pi )^c{}_{\alpha }\xi ^{\alpha }(\nabla \pi )^d{}_{\beta }\eta ^{\beta }$. In this form, the interpretation given above for $\Omega ^{\mathfrak {A}}{}_{\alpha \beta }$ carries over essentially unchanged.⁶²

1.12 Hodge star operation on horizontal and equivariant vector valued forms

Now suppose we have a principal bundle $G\rightarrow P\xrightarrow {\wp } M$, a principal connection $\omega ^{\mathfrak {A}}{}_{\alpha }$ on P, a metric $g_{ab}$ on M, and a volume element $\epsilon _{a_1\ldots a_n}$ on M.⁶³ (Here we assume M is n dimensional.) Then, given any vector space V, we may define a Hodge star operation, $\star$, on horizontal and equivariant V valued forms on P. First, note that the volume form $\epsilon _{a_1\ldots a_n}$ on M determines an n form $\hat{\epsilon }_{\alpha _1\ldots \alpha _n}=\wp ^*(\epsilon _{a_1\ldots a_n})=\epsilon _{a_1\ldots a_n}(\nabla \wp )^{a_1}{}_{\alpha _1}\ldots (\nabla \wp )^{a_n}{}_{\alpha _n}$ on P. (Observe that although $\hat{\epsilon }_{\alpha _1\ldots \alpha _n}$ is not a volume form on P, it is of maximal rank in the sense that any horizontal and equivariant form on P with rank greater than n vanishes.) Next, we define a smooth mixed index tensor field on P, $\bar{\omega }^{\alpha }{}_{a}$, which is uniquely characterized by the properties that (1) $(\nabla \wp )^a{}_{\alpha }\bar{\omega }^{\alpha }{}_{b}=\delta ^a{}_b$ and (2) $\omega ^{\mathfrak {A}}{}_{\alpha }\bar{\omega }^{\alpha }{}_a=\mathbf {0}$. At any point $x\in P$, this tensor maps vectors $\xi ^a$ at $\wp (x)$ to the unique horizontal vector $\xi ^{\alpha }=\bar{\omega }{}^{\alpha }{}_a\xi ^a$ at x satisfying $(\nabla \wp )^a{}_{\alpha }\xi ^{\alpha }=\xi ^a$. Then, for any $k\le n$, we may define a tensor field $\hat{\epsilon }_{\alpha _1\ldots \alpha _{n-k}}{}^{\beta _1\ldots \beta _k}=\epsilon _{a_1\ldots a_{n-k}}{}^{b_1\ldots b_k}(\nabla \wp )^{a_1}{}_{\alpha _1}\ldots (\nabla \wp )^{a_{n-k}}{}_{\alpha _{n-k}}\bar{\omega }^{\beta _1}{}_{b_1}\ldots \bar{\omega }^{\beta _k}{}_{b_k}$. (Here the indices on $\epsilon _{a_1\ldots a_n}$ are raised with $g^{ab}$.) Finally, given any V valued horizontal and equivariant k form $\kappa ^A{}_{\alpha _1\ldots \alpha _k}$, we may define a horizontal and equivariant V valued $(n-k)$ form, $\star \kappa ^A{}_{\alpha _1\ldots \alpha _{n-k}}$, by $\star \kappa ^A{}_{\alpha _1\ldots \alpha _{n-k}}=\hat{\epsilon }_{\alpha _1\ldots \alpha _{n-k}}{}^{\beta _1\ldots \beta _k}\kappa ^A{}_{\beta _1\ldots \beta _k}$.

Notice that this Hodge star operator has the property that, given any V valued k form $\kappa ^A{}_{a_1\ldots a_k}$ on M and any section $\sigma :U\rightarrow P$ of P, $\sigma ^*(\star \kappa ^A{}_{\alpha _1\ldots \alpha _k})=\star \sigma ^*(\kappa ^A{}_{\alpha _1\ldots \alpha _k})$, where the first $\star$ (acting on $\kappa ^A{}_{a_1\ldots a_k}$) is the ordinary Hodge star operation, defined by $\star \kappa ^A{}_{a_1\ldots a_k}=\epsilon _{a_1\ldots a_{n-k}}{}^{b_1\ldots b_k}\kappa ^A{}_{b_1\ldots b_{n-k}}$. Since acting on a horizontal and equivariant k form on P with the covariant exterior derivative $\overset{\omega }{D}$ yields a horizontal and equivariant $(k+1)$ form, one may always take the Hodge dual of the exterior covariant derivative of a horizontal and equivariant k form $\kappa ^A{}_{\alpha _1\ldots \alpha _k}$ to yield a horizontal and equivariant $(n-k-1)$ form, $\star \overset{\omega }{D}_{\alpha }\kappa ^A{}_{\alpha _1\ldots \alpha _k}$.