Line Bundle, Topological Quantization and Berry Phase

A Generalization of Gauss-Bonnet

Let $E\to M$ be a complex line bundle, we suppose the structure group is $U(1)$,

$$\begin{array}{}\text{(1)}& {\psi }_V=e^{i\alpha }{\psi }_U.\end{array}$$

Let $\omega$ be a $U(1)$ connection, which is a $\mathfrak{g}$-valued 1-form on $M$. It is also pure imaginary.

If

$$\begin{array}{}\text{(2)}& \mathrm{\Psi }={\mathbf{e}}_U{\psi }_U,\mathrm{\Phi }={\mathbf{e}}_U{\varphi }_U\end{array}$$

be two sections. Then we can define a coordinate independent scalar product between these two sections,

$$\begin{array}{}\text{(3)}& {\bar{\psi }}_V{\bar{\varphi }}_V={\bar{\psi }}_U{\bar{\varphi }}_U,\end{array}$$

where $\bar{\psi }$ means the complex conjugate of $\psi$.

We write

$$\begin{array}{}\text{(4)}& ⟨\mathrm{\Psi },\mathrm{\Phi }⟩:=\bar{\psi }\varphi .\end{array}$$

We then say that $E$ is a hermitian line bundle.

If we put

$$\begin{array}{}\text{(5)}& \mathrm{\nabla }\psi =(d+\omega )\psi \end{array}$$

then

$$\begin{array}{}\text{(6)}& ⟨\mathrm{\nabla }\mathrm{\Psi },\mathrm{\Phi }⟩+⟨\mathrm{\Psi },\mathrm{\nabla }\mathrm{\Phi }⟩=d(\psi \varphi )\end{array}$$

where the term involving $\omega$ cancels out due to the fact the $\bar{\omega }=-\omega$.

Let ${\mathbf{e}}_U$ be a frame over $U$, that is a norm one complex line section. Then the coordinates of $\mathrm{\Psi }$ takes value in $U(1)$ group,

$$\begin{array}{}\text{(7)}& \mathrm{\Psi }={\mathbf{e}}_U{\psi }_U={\mathbf{e}}_U{e}^{i\alpha }.\end{array}$$

This is the most general section of $E$ over $U$.

Let $V$ be an oriented closed surface (dimension 2) embedded in $M$. The points of $FM$ over $V$ defines a new frame bundle over $V$, again we call it $FV\to V$ or $E_V$, which means $E$ restricted to $V$.

We wish to consider a smooth section of $E$ over $V$, but it may not exist, for example consider the hairy ball theorem, a tangent vector on ${\mathbb{S}}^2$ must have at least one singularity. Then we can take a step back and be satisfied with a section with some singularities, which is always possible. To see it, first consider a section $s:V\to E$ of the complex line bundle, then putting $\mathrm{\Psi }=s/\Vert s\Vert$ at those $p\in V$ where $s\ne 0$. Now consider the zero section $0$. The questions is, what could the intersection between $\mathrm{\Psi }$ and $0$ be? Such intersections must be singularities because there is no smooth way for a norm one complex line to go to zero. Now, both $\mathrm{\Psi }$ and $0$ are submanifolds of $E_V$ which is dimension $4$. All sections have the same dimension as the base space so both $\mathrm{\Psi }$ and $0$ are $2$ dimensional. Then the section in general is or dimension $4-2-2=0$. Thus the intersection comprises of a set of points. In other words, we expect to find a nonvanishing section of $E$, and a resulting frame bundle $FM$, over all $V$ except for perhaps some finite points $p_1,\dots ,p_N$.

let then $\mathrm{\Psi }$ be such a section. We construct the connection form

$$\begin{array}{}\text{(8)}& {\omega }^{\ast }={\pi }^{\ast }\omega +id\alpha .\end{array}$$

We define the index of $\mathrm{\Psi }=\mathbf{e}\psi =\mathbf{e}{e}^{i\alpha }$ at the zero $p_k$ to be

$$\begin{array}{}\text{(9)}& j_{\mathrm{\Psi }}(p_k):=\frac{1}{2\pi }{\oint }_{\partial D}d\alpha \end{array}$$

where $D$ is some infinitesimal disk surrounding $p_k$.

Then, there a theorem telling us that

$$\begin{array}{}\text{(10)}& \frac{i}{2\pi }{\iint }_V\theta =\sum _k{j}_{\psi }(p_k)\in \text{integer}.\end{array}$$

Theorem. Let $E$ be a hermitian line bundle with pure imaginary connections $\omega$ and curvature $\theta$, over a manifold $M$. Let $V$ be any oriented closed surface embedded in $M$. Then

$$\begin{array}{}\text{(11)}& \frac{i}{2\pi }{\iint }_V\theta \end{array}$$

is an integer and represents the sum of the indices of any sections

$$\begin{array}{}\text{(12)}& s:V\to E.\end{array}$$

It is assumed that $S$ has finitely many zeros on $V$. $i\theta /2\pi$ is called the Chern form of $E$.

Intersection number

The intersection number tells us in quality how a section $s$ intersects the $0$-section $0$, counted with multiplicity. By this we mean the following. Let $E$ be a rank-$n$ vector bundle over an $n$-dimensional manifold $M$. We assume that

• $M$ is oriented,
• The vector fiber $F$ is orientable in a continuous way. This is the case when the structure group of $F$ is a connected group. Let $x^{}1,\dots ,x^n$ be positively-oriented coordinates of $M$ and let $u^1,\dots ,u^n$ be the coordinates of the fiber. Let the intersection point be $x=0,u=0$. The section $s$ can be described by the $n$ functions $n^i(x),i=1,\dots ,n$.

We say that the section $s$ intersects $0$ transversely if the Jacobian

$$\begin{array}{}\text{(13)}& \left|\frac{\partial u}{\partial x}\right|\ne 0\end{array}$$

at $x=0$. It is also said that $s$ has a nondegenerate zero at $x=0$. From

$$\begin{array}{}\text{(14)}& \frac{d{u}^j}{dt}=\frac{\partial u^j}{\partial x^i}\frac{d{x}^i}{dt}\end{array}$$

we see that transversality simply means that $s$ and $o$ don’t have any nontrivial tangent vectors in common. So, when $s$ intersects with $0$ transversally at $x=0$, we define the local intersection number to be $+1$ (-1), provided the Jacobian is positive (negative). The total intersection number is the sum of all the local intersection numbers at all the intersections.

Consider a complex line bundle $E\to {\mathbb{S}}^2$ over the Riemann sphere. We may use $z=x+iy$ as the local coordinates of the Riemann sphere near $z=0$. Use $\zeta =u+iv$ for fiber coordinates. A section $s$ is given by

$$\begin{array}{}\text{(15)}& u=u(x,y),v=v(x,y)\text{ or }\zeta =\zeta (z,\bar{z}).\end{array}$$

We do not assume $\zeta$ is holomorphic, however if it is then we have a holomorphic bundle. In this case, by Cauchy-Riemann relation we have

$$\begin{array}{}\text{(16)}& {\left|\frac{d\zeta }{dz}\right|}^2\ge 0,\end{array}$$

thus if a holomorphic section intersects with zero then the local intersection number is $+1$.