A *section* of a line bundle is like a vector field. It is a map $s: M\longrightarrow L$ such that $s(m)\in L_m$ or $\pi\circ s(m)=m$. Section of a line bundle is one-to-one.

*Example*. For the trivial bundle $L=M\times\mathbb{C}$, every section $s$ looks like $s(x)=(x,f(x))$ for some function $f$.

*Example*. For a tangent bundle $TM$, sections are vector fields.

\begin{align*}

s: M&\longrightarrow TM\\

x&\longmapsto v_x\in T_xM

\end{align*}

For the tangent bundle $TS^2$ (minitwistor space) over $S^2$, one can think of a section as a map $s: S^2\longrightarrow TS^2$ such that $\langle s(x),x\rangle=0$ for each $x\in S^2$.

*Proposition*. A line bundle $L$ is trivial if and only if it has a nowhere vanishing section.

*Proof*. Suppose that $L$ is trivial. Let $\varphi: L\longrightarrow M\times\mathbb{C}$ be the trivialization. Then $s: M\longrightarrow L$ defined by $s(m)=\varphi^{-1}(m,1)$ is a nowhere vanishing section. Conversely, if $s$ is a nowhere vanishing section, define a trivialization $M\times\mathbb{C}\longrightarrow L$ by $(m,\lambda)\longmapsto\lambda s(m)$. This is an isomorphism.

Physically sections are fields and if we cannot differentiate fields, we cannot do physics. Let $L\stackrel{\pi}{\longrightarrow}M$ be a line bundle and $s:M\longrightarrow L$ a section. Let $\gamma:(-\epsilon,\epsilon)\longrightarrow M$ be a path through $\gamma())=m$. The conventional definition of the derivative of $s$ would be

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

However, this definition makes no sense because $s(\gamma(t))\in L_{\gamma(t)}$ and $s(\gamma(0))\in L_{\gamma(0)}=L_m$ and we cannot perform the required subtraction

$$s(\gamma(t))-s(\gamma(0)).$$

So, we need to devise a way to differentiate sections of a line bundle. To get a clue, we need to review what we already know and maybe start from there since we cannot create something from nothing. At least we learned how to differentiate vector fields in Euclidean space, say $\mathbb{R}^3$. Let $X$ be a vector field in $\mathbb{R}^3$. The covariant derivative $\nabla_vX$ of $X$ in the direction of the tangent vector $v\in T_p\mathbb{R}^3$ is

\begin{align*}

\nabla_vX&=X'(p+tv)(0)\\

&=\lim_{t\to 0}\frac{X(p+tv)-X(p)}{t}.

\end{align*}

At this moment, one may say “Wait a minute! We have already examined the definition and know that it does not work for sections of a line bundle.” I know and please be patient. We haven’t got a clue yet and something useful may come out of this along the way. Since we cannot create the derivative of a section out of thin air, it is still important to review what we already know. We can naturally extend the above definition to the covariant derivative $\nabla_XY$ of a vector field $Y$ with respect to a vector field $X$. The covariant derivative $\nabla$ satisfies the following properties:

- $\nabla_{f X+gZ}Y=f\nabla_XY+g\nabla_ZY$;
- $\nabla_XfY=(Xf)Y+f\nabla_XY$ where $Xf$ denotes the directional derivative $Xf=\sum_{i=1}^n\alpha^i\frac{\partial f}{\partial x^i}$.

The first property is linearity and the second property is Leibniz rule. These are the most basic rules that you would expect from differentiation. Denote by $\mathfrak{X}(\mathbb{R}^3)$ the set of all tangent vector fields on $\mathbb{R}^3$. The covariant derivative may be regarded as a bilinear map $\nabla:\mathfrak{X}(\mathbb{R}^3)\times\mathfrak{X}(\mathbb{R}^3)\longrightarrow\mathfrak{X}(\mathbb{R}^3)$ satisfying the properties 1 and 2 and we write $\nabla(X,Y)$ as $\nabla_XY$. This gives us a clue on how to define the derivative of a section. It turns out that there isn’t a unique way to differentiate a section for there can be many different maps $\nabla$ satisfying the properties 1 and 2. In fact, one needs to make a choice. Such a choice of differentiation is called a connection.

*Definition*. Let $M$ be a differentiable manifold of dimension $n$ and $\mathfrak{X}(M)$ the set of all tangent vectors on $M$. A *connection* on $M$ is a bilinear map $\nabla:\mathfrak{X}(M)\times\mathfrak{X}(M)\longrightarrow\mathfrak{X}(M)$ such that

\begin{align*}

\nabla_{f X+gZ}Y&=f\nabla_XY+g\nabla_ZY\\

\nabla_XfY&=(Xf)Y+f\nabla_XY

\end{align*}

where $\nabla(X,Y)$ is written as $\nabla_XY$.

The one we defined here is a way of differentiating (connection) of vector fields on a differentiable manifold, but we still have not defined a way of differentiating sections of a line bundle. But we now have a much clearer picture about it. Before we continue, let us briefly discuss differentials because they are closely related to directional derivative. For each $i=1,\cdots,n$, the differential 1-form $dx^i$ is a 1-form on $T_\ast M$ such that

$$dx^i\left(\frac{\partial}{\partial x^j}\right)=\frac{\partial x^i}{\partial x^j}=\delta_{ij}$$

where $\delta_{ij}$ is the Kronecker’s delta. That is, $dx^i$ is is the dual vector of the tangent vector $\frac{\partial}{\partial x^i}$, $i=1,\cdots,n$ and that the $dx^i$, $i=1,\cdots,n$ form the standard basis for the cotangent space $T^\ast M$. For any tangent vector field $X=\sum_{j=1}^n\alpha^j\frac{\partial}{\partial x^j}$,

$$dx^i(X)=\alpha^i=Xx^i.$$

So if we define the differential of $f$ by

$$df:=\sum_{i=1}^n\frac{\partial f}{\partial x^i}dx^i,$$

then

$$df(X)=\sum_{i=1}^n\frac{\partial f}{\partial x^i}dx^i(X)=Xf.$$

Thus the Leibniz rule can be also written as

$$\nabla_XfY=df(X)Y+f\nabla_XY.$$

Let $\gamma: (-\epsilon,\epsilon)\longrightarrow M$ be a path with $\gamma(0)=p$. On a local coordinate neighborhood $(U(p),\varphi)$, $\gamma(t)$ is written as $\gamma(t)=(x^1(t),\cdots,x^n(t))$ and

$$\frac{d\gamma}{dt}(0)=\sum_{i=1}^n\frac{dx^i}{dt}(0)\left(\frac{\partial}{\partial x^i}\right)_p\in T_p(M).$$

Now,

\begin{align*}

df\left(\frac{d\gamma}{dt}(0)\right)&=\sum_{i=1}^n\left(\frac{\partial}{\partial x^i}\right)_pdx^i\left(\frac{d\gamma}{dt}(0)\right)\\

&=\sum_{i=1}^n\left(\frac{\partial}{\partial x^i}\right)_p\frac{dx^i}{dt}(0)\\

&=\frac{df}{dt}(0).

\end{align*}

For the tangent vector field $\dot{\gamma}(t)=\frac{d\gamma}{dt}$, we have

$$df(\dot{\gamma}(t))=\frac{df}{dt}.$$

We will discuss connection on a line bundle in the second part of this lecture. I would like to end this lecture with a physical motivation for considering bundles. The fields in physics are usually given by map $\phi:M\longrightarrow\mathbb{C}^n$ where $M$ is the spacetime. In quantum mechanics, particles are described by so-called complex-valued wave functions (also called state function) $\phi: M\longrightarrow\mathbb{C}$. Due to the Uncertainty Principle, one cannot pinpoint the exact location of a particle. The best thing one can do is to measure a probable location of the particle. The probability of a particle in the state $\phi$ to be discovered in the region $U\subset M$ is

$$\int_Udx\langle\phi(x)|\phi(x)\rangle.$$

To define probability, all we need to know is that $\phi(x)$ takes its value in $\mathbb{C}$ with Hermitian product, and there is no reason for this to be the same vector space for all values of $\phi(x)$. Functions like $\phi$ which are the generalization of complex-valued functions are called sections of vector bundles.

References:

[1] M. Murray, Notes on Line Bundles

[2] B. O’Neill, Elementary Differential Geometry, Academic Press, 1966

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