## 数学代写|拓扑学代写Topology代考|MATH6204

2022年7月28日

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## 数学代写|拓扑学代写Topology代考|Chern numbers

Consider a compact two-dimensional manifold $M$ and a map to the two-sphere, $\phi: M \rightarrow S^{2}$. In other words, for $\boldsymbol{r} \in M, \phi(\boldsymbol{r})=\left(\phi_{x}(\boldsymbol{r}), \phi_{y}(\boldsymbol{r})\right)$, with $\phi_{x}^{2}+\phi_{y}^{2}=1$, where $\boldsymbol{r}=(x, y)$ in some coordinate system on $M$. Something that looks suspiciously like a curvature tensor can be defined, $F_{i j}=\partial_{i} \phi_{j}-\partial_{j} \phi_{i}$, and from this curvature we may construct a topological invariant:
$$c_{1}=\frac{1}{2 \pi} \int_{M} d x d y F_{x y} .$$
This is the first Chern number, and it takes values in the integers. Notice that equation (5.14) looks very reminiscent of the expression for the Euler characteristic as an integral over Gaussian curvature (the Gauss-Bonnet theorem); the Chern number is in fact one of many generalizations of the Euler characteristic.

Even for non-compact spaces, the integral defining the Chern number may converge if the curvature vanishes at large distances. It is often convenient to replace non-compact two-dimensional surfaces by equivalent compact surfaces. For example, the plane $M=\mathbb{R}^{2}$ can be compactified into a two-sphere by, collapsing the circle at infinity to a point, effectively sewing together the ‘edges’ at infinity. Topologically, this means identifying $M$ with a two-sphere, $M \sim S^{2}$, so that the mapping $\phi$ takes spheres to spheres, $\phi: S^{2} \rightarrow S^{2}$. In the notation of chapter 3, the compactification can be written $S^{2} \sim M / \partial M . c_{1}$ essentially counts the number of times the first sphere wraps around the second, and its integer values will characterize the second homotopy class, $\pi_{2}\left(S^{2}\right)$.

In physical applications, $\phi(\boldsymbol{r})$ will often represent a point on the Bloch or Poincaré sphere, $x, y$ will be momentum components, and the two dimensional manifold $M$ will be a torus $T^{2}$ representing a Brillouin zone. In this context, $c_{1}$ is often referred to as the TKNN invariant (after Thouless, Kohomoto, Nightingale, and den Nijs [1]); it turns out to be important in the quantum Hall effect, and it will make an appearance in chapter $10 .$

## 数学代写|拓扑学代写Topology代考|Pontrjagin index

The second and third homotopy classes of the two-spheres are both labeled by integers:
$$\pi_{2}\left(S^{2}\right)=\pi_{3}\left(S^{2}\right)=\mathbb{Z} .$$
These labels are known, respectively, as the Pontrjagin and Hopf indices, and are discussed, respectively, in this section and the next.

Consider a map $\boldsymbol{\phi}: I \times I \rightarrow S^{2}$. This could, for example, represent a scalar field with internal degrees of freedom forming a three-dimensional vector lying on the unit sphere in the internal space. Such internal variables could be the three ‘color’ degrees of freedom in the theory of strong nuclear interactions or the components of a polarization vector in optics. The condition that the field be a unit vector in the internal space is written as $\phi \cdot \phi=1$ or $\phi_{a} \phi^{a}=1$, where the dot product and the latin indices are in the internal target space of the mapping. Components in the original square $I \times I$ will be given Greek indices. Let $x^{\mu}$ with $\mu=1,2$ be coordinates on the unit square $I \times I$. We can compactify the square by collapsing all the points on the boundary to a single point, thereby identifying the square with the two-sphere. Therefore, $\boldsymbol{\phi}$ can be viewed as a map between spheres: $\boldsymbol{\phi}: S^{2} \rightarrow S^{2}$, thereby defining a homotopy class in $\pi_{2}\left(S^{2}\right)$.
Define a quantity
$$Q=\frac{1}{8 \pi} \int_{I \times I} \epsilon^{\mu \nu} \boldsymbol{\phi} \cdot\left(\partial_{\mu} \boldsymbol{\phi} \times \partial_{\nu} \boldsymbol{\phi}\right) d x^{1} d x^{2}=\frac{1}{4 \pi} \int_{I \times I} A,$$
where
$$A=\phi \cdot\left(\frac{\partial \phi}{\partial x^{1}} \times \frac{\partial \phi}{\partial x^{2}}\right) d x^{1} \wedge d x^{2}=\frac{1}{2} \epsilon_{u b \iota} \phi^{a} \partial_{\mu} \phi^{b} \partial_{\nu} \phi^{c} d^{\mu} \wedge d x^{\nu}$$

# 拓扑学代考

## 数学代写|拓扑学代写Topology代考|Chern numbers

$$c_{1}=\frac{1}{2 \pi} \int_{M} d x d y F_{x y} .$$

## 数学代写|拓扑学代写Topology代考|Pontrjagin index

$$\pi_{2}\left(S^{2}\right)=\pi_{3}\left(S^{2}\right)=\mathbb{Z} .$$

$$Q=\frac{1}{8 \pi} \int_{I \times I} \epsilon^{\mu \nu} \phi \cdot\left(\partial_{\mu} \phi \times \partial_{\nu} \phi\right) d x^{1} d x^{2}=\frac{1}{4 \pi} \int_{I \times I} A$$

$$A=\phi \cdot\left(\frac{\partial \phi}{\partial x^{1}} \times \frac{\partial \phi}{\partial x^{2}}\right) d x^{1} \wedge d x^{2}=\frac{1}{2} \epsilon_{u b l} \phi^{a} \partial_{\mu} \phi^{b} \partial_{\nu} \phi^{c} d^{\mu} \wedge d x^{\nu}$$

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