Connections, holonomy and curvature

The physical motivation for connections is that you can't do physics if you can't differentiate the fields! So a connection is a rule for differentiating sections of a line bundle. The important thing to remember is that there is no a priori way of doing this - a connection is a choice of how to differentiate. Making that choice is something extra, additional structure above and beyond the line bundle itself. The reason for this is that if $L \to M$ is a line bundle and $\gamma : (-\epsilon, \epsilon) \to M$ a path through $\gamma(0) = m$ say and $s$ a section of $L$ then the conventional definition of the rate of change of $s$ in the direction tangent to $\gamma$, that is:

$$\lim_{t \to 0} = \frac{s (\gamma(t)) - s (\gamma (0)) }{ t}$$

makes no sense as $s(\gamma(t) )$ is in the vector space $L_{\gamma(t)}$ and $s(\gamma(0))$ is in the different vector space $L_{\gamma(0)}$ so that we cannot perform the required subtraction.

So being people mathematicians we make a definition by abstracting the notion of derivative:

Definition 2.1   A connection $\nabla$ is a linear map

$$\nabla : \Gamma (M, L) \to \Gamma (M, T^{\ast} M \otimes L)$$

such that for all $s$ in $\Gamma (M,L)$ and $f \in C^{\infty} (M, L)$ we have the Liebniz rule:

$$\nabla (f s) = d f \otimes s + f \nabla s$$

If $X \in T_{x} M$ we often use the notation $\nabla_{X} s = (\nabla s) (X).$

Example 2.1 (The trivial bundle.)   $L = \mathbb{C}\times M.$ Then identifying sections with functions we see that (ordinary) differentiation $d$ of functions defines a connection. If $\nabla$ is a general connection then we will see in a moment that $\nabla s - ds$ is a 1-form. So all the connections on $L$ are of the form $\nabla = d + A$ for $A$ a 1-form on $M$ (any 1-form).

Example 2.2 (The tangent bundle to the sphere.)   $TS^2.$ If $s$ is a section then $s : S^{2} \to \mathbb{R}^3$ such that $s(u) \in T_{u} S^{2}$ that is $\langle s(u), u \rangle = 0$. As $s(u) \in \mathbb{R}^3$ we can differentiate it in $\mathbb{R}^3$ but then $ds$ may not take values in $T_uS^2$ necessarily. We remedy this by defining

$$\nabla(s) = \pi ( ds)$$

where $\pi$ is orthogonal projection from $\mathbb{R}^3$ onto the tangent space to $x$. That is $\pi(v) = v - \< x, v\> x$.

Example 2.3 (The tangent bundle to a surface.)   A surface $\Sigma$ in $\mathbb{R}^3$. We can do the same orthogonal projection trick as with the previous example.

Example 2.4 (The Hopf bundle.)   Because we have $H \subset \mathbb{C}^{2} \times \mathbb{C}P_{1}$ we can apply the same technique as in the previous sections. A section $s$ of $H$ can be identified with a function $s \colon \mathbb{C}P_{1} \to \mathbb{C}^{2}$ such that $s{[z]} = \lambda z$ for some $\lambda \in \mathbb{C}$. Hence we can differentiate it as a map into $\mathbb{C}^{2}$. We can then project the result orthogonally using the Hermitian connection on $\mathbb{C}^{2}$.

The name connection comes from the name infinitesimal connection which was meant to convey the idea that the connection gives an identification of the fibre at a point and the fibre at a nearby `infinitesimally close' point. Infinitesimally close points are not something we like very much but we shall see in the next section that we can make sense of the `integrated' version of this idea in as much as a connection, by parallel transport, defines an identification between fibres at endpoints of a path. However this identification is generally path dependent. Before discussing parallel transport we need to consider two technical points.

The first is the question of existence of connections. We have

Proposition 2.1   Every line bundle has a connection.

Proof. Let $L \to M$ be the line bundle. Choose an open covering of $M$ by open sets $U_\alpha$ over which there exist nowhere vanishing sections $s_\alpha$. If $\xi$ is a section of $L$ write it locally as $\xi_{\vert U_{\alpha }} = \xi_{\alpha }s_{\alpha }$. Choose a partition of unity $\rho_{\alpha }$ for subordinate to the cover and note that $\rho_{\alpha }s_{\alpha }$ extends to a smooth function on all of $M$. Then define

$$\nabla(\xi) = \sum d\xi_{\alpha }\rho_{\alpha } s_{\alpha }.$$

We leave it as an exercise to check that this defines a connection. $\qedsymbol$

The second point is that we need to be able to restrict a connection to a open set so that we can work with local trivialisations. We have

Proposition 2.2   If $\nabla$ is a connection on a line bundle $L \to M$ and $U \subset M$ is an open set then there is a unique connection $\nabla_{U}$ on $L_{\vert U} \to U$ satisfying

$$\nabla(s)_{IU} = \nabla_{U}(s_{\vert U}).$$

Proof. We first need to show that if $s$ is a section which is zero in a neighbourhood of a point $x$ then $\nabla(s)(x) = 0$. To show this notice that if $s$ is zero on a neighbourhood $U$ of $x$ then we can find a function $\rho$ on $M$ which is $1$ outside $U$ and zero in a neighbourhood of $x$ such that $\rho s = s$. Then we have

$$\nabla(s)(x) = \nabla(\rho s)(x) = d\rho(x) s(x) + \rho(x) \nabla(s)(x) = 0.$$

It follows from linearity that if $s$ and $t$ are equal in a neighbourhood of $x$ then $\nabla(s)(x) = \nabla(t)(x)$. If $s$ is a section of $L$ over $U$ and $x \in U$ then we can multiply it by a bump function which is $1$ in a neighbourhood of $x$ so that it extends to a section $\hat s$ of $L$ over all of $M$. Then define $\nabla_{U}(s)(x) = \nabla(\hat s)(x)$. If we choose a different bump function to extend $s$ to a different section $\tilde s$ then $\tilde s$ and $\hat s$ agree in a neighbourhood of $x$ so that the definition of $\nabla_{U}(s)(x)$ does not change. $\qedsymbol$

From now on I will drop the notation $\nabla_{\vert U}$ and just denote it by $\nabla$.

Let $L \to M$ be a line bundle and $s_{\alpha} : U_{\alpha} \to L$ be local nowhere vanishing sections. Define a one-form $A_\alpha$ on $U_\alpha$ by $\nabla s_{\alpha} = A_{\alpha}\otimes s_{\alpha}$. If $\xi \in \Gamma(M, L)$ then $\xi\vert _{U \alpha} = \xi_{\alpha} s_{\alpha}$ where $\xi_{\alpha} : U_{\alpha} \to {\bf C}$ and

$$\begin{array}{rcl} \nabla \xi\vert _{U_\alpha} & = & d \xi_{\... ..._{\alpha} + A_{\alpha} \xi_{\alpha}) s_{\alpha}. \\ \end{array}$$ (2.1)

Recall that $s_{\alpha} = g_{\alpha \beta} s_{\beta}$ so $\nabla s_{\alpha} = dg_{\alpha \beta} s_{\beta} + g_{\alpha \beta} \nabla s_{\beta}$ and hence $A_{\alpha} s_{\alpha} = g_{\alpha \b }^{-1} dg_{\alpha \beta} g_{\alpha \beta} s_{\alpha} + s_{\alpha} A_{\beta}$. Hence

$$A_{\alpha} = A_{\beta} + g_{\alpha \beta}^{-1}dg_{\alpha \beta}$$ (2.2)

The converse is also true. If $\{A_{\alpha}\}$ is a collection of 1-forms satisfying the equation (2.2) on $U_{\alpha} \cap U_{\beta}$ then there is a connection $\nabla$ such that $\nabla s_{\alpha} = A_{\alpha} s_{\alpha}$. The proof is an exercise using equation (2.1) to define the connection.