Ehresmann Connection

Prerequisites: Connections. Full path (7 pages)

• Intuitive
• Concrete
• Abstract
• Why is it interesting?

Intuitive

Many objects in physics live "above" each point of space or spacetime: a vector attached to each point of a curved surface, the phase of a wave function at each point, an internal orientation of a field in color space. Mathematically this is a fiber bundle: a base space (where we are) and, above every point, a copy of some space of possible values (the fiber).

Nothing in this setup tells us which value at one point corresponds to which value at a neighbouring point. There is no built-in meaning of "the same phase over there" or "the same direction over there". An Ehresmann connection is the extra piece of information that supplies this. At every point of the bundle it splits the possible directions of motion into two kinds. Vertical directions change the value while staying at the same base point. Horizontal directions move along the base while keeping the value "constant" in the sense defined by the connection.

Once we have this split, we can take any path in the base and a starting value, and follow the path by always moving horizontally. This is parallel transport. If we go around a closed loop, we may come back with a different value than the one we started with. This mismatch is called the holonomy. For small loops it measures the curvature of the connection. In electromagnetism the connection is the vector potential, its curvature is the electromagnetic field, and the holonomy around a loop is the Aharonov-Bohm phase, fixed by the magnetic flux through the loop. That phase can be nonzero even when the field vanishes everywhere along the loop, because the loop can enclose a region where it does not vanish.

Concrete

Let $\pi : E \to M$ be a smooth fiber bundle. At each point $e \in E$ the tangent space $T_e E$ contains a canonical subspace, the vertical space

$$V_e = \ker\big(d\pi_e : T_e E \to T_{\pi(e)} M\big) ,$$

the directions tangent to the fiber. There is no canonical complement. An Ehresmann connection is a smooth choice of complement $H_e$ at every point,

$$T_e E = H_e \oplus V_e .$$

Equivalently it is a vertical-valued 1-form $\Phi$ on $E$ (the connection form) with $\Phi(v) = v$ for vertical $v$, and $H = \ker \Phi$. Since $d\pi_e$ restricted to $H_e$ is an isomorphism onto $T_{\pi(e)}M$, every tangent vector $X$ on $M$ has a unique horizontal lift $X^h$.

Local coordinates. Use coordinates $x^\mu$ on $M$ and $y^a$ on the fiber. The vertical space is spanned by $\partial/\partial y^a$, and the horizontal space can be written as the span of

$$h_\mu = \frac{\partial}{\partial x^\mu} - \Gamma^a_\mu(x, y) \frac{\partial}{\partial y^a}$$

for some functions $\Gamma^a_\mu$, which specify the connection. A curve $(x^\mu(t), y^a(t))$ is horizontal when its tangent vector is a combination of the $h_\mu$, which gives the parallel transport equation

$$\frac{dy^a}{dt} + \Gamma^a_\mu(x, y)\, \frac{dx^\mu}{dt} = 0 .$$

Given a curve $x^\mu(t)$ in the base and an initial value $y^a(0)$, this ODE defines the horizontal lift of the curve.

Linear connections. If $E$ is a vector bundle and parallel transport is required to be linear on the fibers, then $\Gamma^a_\mu(x, y) = \Gamma^a_{\mu b}(x)\, y^b$, and the transport equation becomes $\dot y^a + \Gamma^a_{\mu b} \dot x^\mu y^b = 0$. This is the condition $\dot x^\mu \nabla_\mu y = 0$ for the covariant derivative

$$\nabla_\mu s^a = \partial_\mu s^a + \Gamma^a_{\mu b}\, s^b .$$

For the tangent bundle the $\Gamma^a_{\mu b}$ are the Christoffel symbols, and the metric-compatible torsion-free choice is the Levi-Civita connection.

Curvature. The horizontal subspaces fit together into submanifolds (the distribution $H$ is integrable) exactly when the Lie brackets $[h_\mu, h_\nu]$ are again horizontal. A direct computation gives

$$[h_\mu, h_\nu] = -R^a_{\mu\nu} \frac{\partial}{\partial y^a}, \qquad R^a_{\mu\nu} = \partial_\mu \Gamma^a_\nu - \partial_\nu \Gamma^a_\mu + \Gamma^b_\nu \frac{\partial \Gamma^a_\mu}{\partial y^b} - \Gamma^b_\mu \frac{\partial \Gamma^a_\nu}{\partial y^b} ,$$

which is vertical. So $H$ is integrable if and only if $R = 0$, and then parallel transport around small contractible loops is trivial. For a linear connection, $R^a_{\mu\nu} = R^a{}_{b\mu\nu}\, y^b$ with

$$R^a{}_{b\mu\nu} = \partial_\mu \Gamma^a_{\nu b} - \partial_\nu \Gamma^a_{\mu b} + \Gamma^a_{\mu c} \Gamma^c_{\nu b} - \Gamma^a_{\nu c} \Gamma^c_{\mu b} ,$$

the usual Riemann tensor when $E = TM$.

Example: a $U(1)$ phase. Take the fiber coordinate to be a phase $\theta$ and $\Gamma_\mu = A_\mu(x)$, independent of $\theta$. Horizontal lifts satisfy $\dot\theta = -A_\mu \dot x^\mu$, so going around a closed loop $C$ shifts the phase by $-\oint_C A_\mu dx^\mu$, which by Stokes' theorem is minus the flux of $\partial_\mu A_\nu - \partial_\nu A_\mu$ through the loop. Here $R_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$, the field strength.

Abstract

For a general smooth fiber bundle $\pi : E \to M$, an Ehresmann connection is a smooth subbundle $H \subset TE$ complementary to $V = \ker d\pi$. Horizontal lifts of curves always exist for short times. They need not exist for all times when the fiber is non-compact, so a general Ehresmann connection need not define parallel transport along every curve. A connection is called complete when it does. Connections on principal bundles, and linear connections on vector bundles, are always complete.

Principal bundles. Let $P \to M$ be a principal $G$-bundle with right action $R_g$, and let $\mathfrak{g}$ be the Lie algebra of $G$. Each $X \in \mathfrak{g}$ defines a vertical vector field $X^\#$ on $P$ (the fundamental vector field), and $X \mapsto X^\#_p$ identifies $V_p$ with $\mathfrak{g}$. A principal connection is an Ehresmann connection that is $G$-invariant, $(R_g)_* H_p = H_{pg}$. Equivalently it is a $\mathfrak{g}$-valued 1-form $\omega \in \Omega^1(P, \mathfrak{g})$ with

$$\omega(X^\#) = X, \qquad R_g^* \omega = \mathrm{Ad}_{g^{-1}} \omega ,$$

and $H = \ker \omega$. The curvature is

$$\Omega = d\omega + \tfrac{1}{2}[\omega \wedge \omega] ,$$

which equals $d\omega$ evaluated on the horizontal parts of its arguments, and satisfies the Bianchi identity $D\Omega = 0$ with $D$ the exterior covariant derivative. By the Frobenius theorem $\Omega = 0$ exactly when $H$ is integrable.

Gauge fields. A local section $\sigma : U \to P$ pulls $\omega$ back to a $\mathfrak{g}$-valued 1-form $A = \sigma^* \omega$ on $U$, the gauge potential, and $\sigma^* \Omega = F = dA + A \wedge A$ (for matrix groups). For another section $\sigma' = \sigma g$ with $g : U \to G$,

$$A' = g^{-1} A g + g^{-1} dg, \qquad F' = g^{-1} F g ,$$

which is the transformation law of a Yang-Mills field. In this convention $\mathfrak{g}$ for $U(1)$ is $i\mathbb{R}$ and physicists' real potential appears as $A = -i a$ or $A = i a$ depending on the sign convention for the covariant derivative.

Associated bundles and holonomy. A representation $\rho : G \to GL(W)$ gives an associated vector bundle $P \times_G W$, and a principal connection induces a linear connection on it, with covariant derivative $\nabla s = ds + \rho_*(A) s$ in a local trivialization. Parallel transport around loops based at $p$ defines the holonomy group $\mathrm{Hol}_p(\omega) \subset G$. The Ambrose-Singer theorem states that its Lie algebra is spanned by the values $\Omega_q(X, Y)$ for horizontal $X, Y$ at all points $q$ that can be joined to $p$ by a horizontal curve. Linear connections on a vector bundle with fiber $\mathbb{R}^n$ are the same as principal connections on its frame bundle, with $G = GL(n, \mathbb{R})$.

Why is it interesting?

Ehresmann developed the notion of connections on principal fiber bundles, which turns out to be the natural mathematical framework for Yang-Mills field theories.Gauge field theories: various mathematical approaches by Jordan François, Serge Lazzarini and Thierry Masson

Path to this page

1. Calculus
2. Linear Algebra
3. Group Theory
4. Manifold
5. Fiber Bundles
6. Parallel Transport
7. Connections