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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