12.20主要内容

Poincare lemma

1. closed form:$d \omega = 0$, exact form: $\omega = d \eta$
   Cohomology group: $H^r(M) =$closed form/exact form $= \left\{[\omega]=\omega+d\eta | \eta \in \Omega^{r-1} \right\}$
2. Poincare lemma
   局域地, $d \omega = 0$ $\Rightarrow$ $\omega = d \eta$, 即局域上所有闭形式都是恰当形式.
   $\mathbb{R}^n$ space is trivial
3. Stokes定理
   $\int_{\Sigma} \omega = (\Sigma,\omega) = (\Sigma,\omega+d\eta)$ $\Rightarrow$ $(\Sigma,d\eta) = (\partial\Sigma,\eta)$
   注意: Failure of Stokes theorem!

Winding number in complex analysis

1. 问题引入
   $W\left\{\begin{array}{l} \text{直接积分}, \int_{S^1} \omega=1 \\ \text{Stokes定理}, \int_{S^{1}} d\eta = \int_{\partial S^{1}} \eta = 0 \end{array}\right.$ 矛盾出在$z=0$奇点导致Stokes定理失效.
2. 如何解决
   分成上下半区域, 通过Poincare lemma将每个区域上的微分形式都局域地变为恰当形式, 再分别利用Stokes定理

   

3. Homotopy group 与 topo term 的关系
   $\left\{\begin{array}{lll} {\pi_{k}(S^k)=\mathbb{Z}} & {\Leftrightarrow} & {\int d\Omega=\int \epsilon^{\mu\nu\rho\sigma \cdots} n_{\mu} \partial_x n_{\nu} \partial_y n_{\rho} \cdots dx dy dz \cdots} \\ {\pi_{k}(U(N))=0(k=2n),\ 1(k=2n+1)} & {\Leftrightarrow} & {\int Tr[(g^{-1}dg)^k]} \end{array}\right.$
4. 两个方向的推广: (1)考虑更高维度; (2)考虑non-Abelian, 即Matrix

   

Chern-Simons term in IQHE

1. 整数Hall效应(IQHE)
   $\sigma_{xy} = ne^2/h = c_1 e^2/h$
2. Field theory for the above phys
   $\vec{E}\cdot \vec{B}$这一项是关键, $j^{\mu}=\frac{C_{1}}{2 \pi} \epsilon^{\mu \nu \tau} \partial_{\nu} A_{\tau}$
   $S_{\mathrm{eff}}=\frac{C_{1}}{4 \pi} \int d^{2} x \int d t A_{\mu} \epsilon^{\mu \nu \tau} \partial_{\nu} A_{\tau}$
参考Xiao-Liang Qi, First Chern number and topological response function in (2+1)d
3. Chern character, n-th Chern number, Chern-Simons form, Winding number(详细内容将附上note)

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