# Phantom_Ghost

1. 上周
2019-02-02 05:13:01
Phantom_Ghost 更新于 K-theory and D-branes

Action of D-branes

Def. $E\to M$ rank-$n$ vector bundle, then there is polyforms
\begin{align*}
\hat{A}(E)&=\prod_i\frac{\frac{x_i}{2}}{\sinh(\frac{x_i}{2})}=1-\frac{1}{24}p_1(E)+\frac{1}{5760}(7p_1(E)^2-4p_2(E))+\cdots\\
p(E)&=\prod_i(1+x_i^2)
\end{align*}

The action: Let $f:W_{p+1}\to S$ be the embedding of $p+1$-dim. compact manifold $W_{p+1}$ into 10-dim. compact manifold $S$. $E\to W_{p+1}$ rank-$n$ complex vector bundle

Action over the brane for $\text{IIB}$-type:
$I_W=\int_{W_{p+1}}f^* C\wedge \text{ch}(E)\wedge\hat{A}(TW_{p+1})\wedge e^{\frac{c_1(NW_{p+1})}{2}}\wedge\frac{1}{f^*\left(\sqrt{\hat{A}(TS)}\right)}$
where $C=C_2+C_4+C_8+C_{10}$.

The total action:
$I=\int_S G\wedge *G+I_W$
$G$ is the curvature of $C$.

Def. The (push-forward) Gysin map acting on cohomology
\begin{align*}
f^C_!:H^k(W_{p+1})\to H^{k+10-(p+1)}(S)
\end{align*}
is given by $f_!^C:=\mathcal{D}_S^{-1}f_*\mathcal{D}_W$ with $\mathcal{D}_W:H^k(W_{p+1})\xrightarrow{\sim} H_{p+1-k}(W_p)$ Poincaré duality map.

Applying to the action $f_!^C:I_W\to I_S$
$I_S=\int_S f_!^C\left(f^*C\wedge\text{ch}(E)\wedge\hat{A}(TW)\wedge e^{\frac{c_1(NW)}{2}}\wedge\frac{1}{f^*\left(\sqrt{\hat{A}(TS)}\right)}\right)$
Fact: $\theta\in H^\bullet(S)\;,\;\eta\in H^\bullet(W)$, then $f_!^C(f^*\theta\wedge\eta)=\theta\wedge f_!^C(\eta)$,
\begin{align*}
I_S=\int_S C\wedge f_!^C(\text{ch}(E)\wedge\hat{A}(TW))\wedge e^{\frac{c_1(NW)}{2}}\wedge\frac{1}{f^*\left(\sqrt{\hat{A}(TS)}\right)}
\end{align*}

Thm. (Atiyah-Hirzebruch version of the Riemann-Roch theorem): If $f:X\to Y$ continuous map, $X,Y$ compact s.t. $\text{dim}\,Y\equiv\text{dim}\,X\,(\text{mod}\,2)$; $\nu_f=[f^*(TY)]-[TX]\in K(X)$, $\nu_f$ has a stable $\text{Spin}^c$-structure. Then it induces a homomorphism $f_!^K$ that $\text{ch}(f_!^K(E))=f_!^C(e^{\frac{c_1(NW)}{2}}\wedge\hat{A}^{-1}(\nu_f)\wedge\text{ch}(E))$ where $\hat{A}([E_1]-[E_2])=\hat{A}(E_1)\wedge\hat{A}^{-1}(E_2)$ and $\hat{A}^{-1}(\nu_f)=\hat{A}^{-1}(TS)\wedge\hat{A}(TW)$.

Let $Q=f_!^C(\text{ch}(E)\wedge\hat{A}(TW))\wedge e^{\frac{c_1(NW)}{2}}\wedge\frac{1}{\sqrt{\hat{A}(TS)}}$, apply the Thm. we then have $Q=\text{ch}(f_!^K(E))\wedge\sqrt{\hat{A}(TW)}$. D-branes are classified by $f_!^K(E)\in K(S)$.

2. 2周前
2019-02-01 02:34:36
Phantom_Ghost 更新于 K-theory and D-branes

Chern-Weil theory

Def. $G$ is a compact Lie group, $\pi:P\to M$ is a principle $G$-bundle $(P,\pi,M,G)$ when it satisfies the following properties:
(1) There is a free, fiberwise transitive right $G$-action on $P$.
(2) $(U_i,\varphi_i)$ charts, $U_i\subset M$ open, $\varphi_i\xrightarrow{\approx}\pi^{-1}(U_i)\to U_i\times G$ s.t. $\varphi_i(p\cdot g)=\varphi_i(p)\cdot g\;,\;\forall g\in G$ (equivariant).

Thm. $\text{Vect}_\mathbb{C}^n(M)\xrightarrow{1:1}\text{Prin}_{U(n)}(M)$
sketch of proof: $\bullet$ Associated vector bundle: $V$ is a $n$-dimensional complex vector space, $\pi: P\to M$ a $U(n)$-bundle, $E=(P\times V)/\sim$, $(p,v)\sim (p\cdot g,g^{-1}\cdot v)$, $\tilde{\pi}:E\to M\;,\;\tilde{\pi}([(p,v)])=\pi(p)$ ;
$\bullet$ $\pi:E\to M$, $P_x=\{s_x=(s_1,\dots,s_n)|s_x\;\text{is a basis in}\;E_x\}\;\;(x\in M)$.

Def. $A\in\Omega(P;\mathfrak{g})$, $\mathfrak{g}=\text{Lie}(G)$ in $(P,\pi,M,G)$ is a connection if following holds
i) $r^*_g A=\text{Ad}_{g^{-1}}(A)\;,\;\forall g\in G$
ii) $A(\tilde{X})=X\;,\;X\in\mathfrak{g}$, $\tilde{X}(u)=\frac{d}{dt}\Big|_{t=0} u e^{t}X\;,\;u\in P$

$F^A\in\Omega(P;\mathfrak{g})$ is a curvature if $F^A=dA+[A,A]$ and $F^A(v,w)=dA(v,w)$ for $v,w=\text{ker}(A)$.

Pick a basis in $\mathfrak{g}$, i.e. $\{b_1,\dots,b_r\}$.

Def. An $\text{Ad}$-invariant polynomial in $\mathfrak{g}$ is a map
\begin{align*}
\mathcal{P}:&\mathfrak{g}\to\mathbb{C}\\
X&\mapsto\mathcal{P}(X):=\sum_{i_1,\dots,i_k=1}^{r}a_{i_1\cdots i_k}x_{i_1}\cdots x_{i_k}
\end{align*}
for $X=\sum_{i=1}^{r}x_i b_i$, s.t. $\mathcal{P}(\text{Ad}_g(X))=\mathcal{P}(X)\;,\;\forall g\in G$.
Denote $\mathcal{P}^*_G(\mathfrak{g})$ as the algebra of $\text{Ad}$-invariant polynomial over $\mathfrak{g}$.

Thm. Expand $F^A=\sum_{i-1}^r F^{A_i}b_i$, then there is an homomorphism
\begin{align*}
\mathcal{P}^*_G(\mathfrak{g})&\to H^\bullet_\text{dR}(M)\\
\mathcal{P}&\mapsto\mathcal{P}(F^A):= \sum_{i_1,\dots,i_k=1}^{r}a_{i_1\cdots i_k}F^{A_{i_1}}\wedge\cdots\wedge F^{A_{i_k}}
\end{align*}

For connections $A^1,A^2$, then $\mathcal{P}(F^{A_1})=\mathcal{P}(F^{A_2})+d\phi$.

Chern class: let $f_1,\dots,f_n$ be polynomials over $\mathfrak{u}(n)$ defined via
$\text{det}(t\mathbb{I}+\frac{i}{2\pi}X)=\sum_{k=0}f_k(X)t^{n-k}\;,\;X\in\mathfrak{u}(n).$

Thm. $f_0,\dots,f_n$ are real valued $\text{Ad}_{U(n)}$-invariant and they generate $\mathcal{P}^*_{U(n)}(\mathfrak{u}(n))$.

Def. Let $E\to M$ be rank-$n$ vector bundle, then $c_k(E)=[f_k(F^A)]\in H^{2k}(M;\mathbb{Z})$
e.g. $c_1(E)=[\frac{i}{2\pi}\text{tr}(F^A)]$, $c_2(E)=[-\frac{1}{8\pi^2}\text{tr}(F^A\wedge F^2)+\text{tr}(F^A)\wedge\text{tr}(F^A)]$

Thm. $c(E):=\sum_i c_i(E)$, then
$\bullet$ $c(E_1\oplus E_2)=c(E_1)c(E_2)$
$\bullet$ $c(f^*E)=f^*c(E)$

Chern Character: $\text{ch}(E)=\sum_i e^{x_i}$ where $c(E)=\prod_i(1+x_i)$

Pontryagin class: For $R\to M$ real vector bundle, the $k$-th. Pontryagin class is given as
$p_k(R)=(-1)^k c_{2k}(R^\mathbb{C})$

3. 2019-02-01 02:28:23
Phantom_Ghost 更新于 K-theory and D-branes

String theory

- Particle physics: particles are points "$\bullet$"

- string theory: strings

[attachment:5c5333aa06c70]

$X^\mu:\Sigma\to S$ embeds worldsheet into spacetime as submanifold

[attachment:5c53372b58df0]

action:
$S_\text{WS}\sim\int_\Sigma d\tau\,d\sigma\,\sqrt{-h}h^{\alpha\beta}\partial_\alpha X^\alpha\partial_\beta X_\mu+\text{fermion}$
mode expansion
$X^\mu(\tau-\sigma)\sim\sum_{n\in\mathbb{Z}}\alpha_n^\mu e^{i(t-\sigma)}\xrightarrow{\text{canonical quantization}}\hat{\alpha}_n^\mu\hat{\alpha}_m^\nu|0\rangle$

Massless fields:
$g^{\mu\nu}$ (spacetime metric tensor field), $B^{\mu\nu}$ (other rank-2 tensor), $\phi$ (scalar field)
$C_0,C_2,C_4,C_8,C_{10}$ are RR gauge field in $\text{IIB}$-type theory (2, 4, 8, 10-form field valued in vector spaces), in $\text{IIA}$-type $C_1,C_3$ are also present.

Spacetime dimension: 10 (vanishing of Weyl anomaly or center charge)

D-branes: in physics they are sources, "D" stands for Dirichlet boundary condition

[attachment:5c533cb7b07bc]

$U(N)$ gauge field lives on $\text{D}_p$-branes

$C_2$ coupled to $\text{D}_1$, $C_4$ coupled to $\text{D}_3$ ect.

Electromagnetic field $\int_\gamma A$ (1-form field), $\gamma$ is electron's worldline

General: embeds $\text{D}_p$-brane into spacetime $f:W_p\to S$ ($\text{dim}\,W_p=p+1$ the $p+1$-dimensional worldvolume of the $\text{D}_p$-brane.), $\int_{W_p}f^* C_{p+1}$

Interaction with $U(N)$-gauge field, gravity on $\text{D}_p$-branes.

4. 2019-02-01 01:38:13
Phantom_Ghost 发表了帖子 K-theory and D-branes

This is a short note recording the content of the talk from the seminar of ''topological K-theory'' during WS18/19.

Reference:
[1] Kasper Olsena,and Richard J. Szabo, CONSTRUCTING D-BRANESFROM K-THEORY.
[2] Jarah EVSLIN, What Does(n't) K-theory Classify?
[3] Edward Witten, D-BRANES AND K-THEORY.
[4] Petr Hořava, Type IIA D-Branes, K-Theory and Matrix Theory.

5. 2019-01-27 00:20:55

投影Hilbert空间可以被赋予一种量子度量——Fubini-Study度量，从而形成一种复流形的几何。这种几何化的描述最先源自于数学上的相干态量子化方案：令$\mathcal{H}$为一$N$维Hilbert空间（包括无穷维情形），考虑其投影Hilbert空间$\mathbb{P}(\mathcal{H})$等同于所有正交投影子$\Psi_\psi=\frac{|\psi\rangle\langle\psi|}{\|\psi\|}$(投影到$\mathcal{H}$的一维子空间)集合，赋予其一个解析图册$\{V_\Phi,\mathbf{h}_\Phi,\mathcal{H}_\Phi|\Phi\in\mathbb{P}(\mathcal{H})\}$，其中$V_\Phi=\{\Psi\in\mathbb{P}(\mathcal{H})|\langle\phi|\psi\rangle\neq 0\}$, $\mathcal{H}_\Phi=(\text{Id}-\Phi)\mathcal{H}$, $\mathbf{h}_\Phi:V_\Phi\to\mathcal{H}_\Phi\,,\,\mathbf{h}_\Phi(\Psi)=\frac{(\text{Id}-\Phi)}{\langle\hat{\phi}|\psi\rangle}\psi\,\,(\|\hat{\phi}\|=1)$为微分同胚；通过正交基表示$z_i=\langle e_i|\psi\rangle$, $Z_i=\frac{z_i}{z_0}=\langle e_i|\mathbf{h}_\Phi(\Psi\rangle)$可以把$\mathbb{P}(\mathcal{H})$等同为$\mathbb{C}P^N$。$\pi:\mathcal{H}\backslash\{0\}\to\mathbb{P}(\mathcal{H})$视为$\mathbb{P}(\mathcal{H})$上的典范线丛(一个主$\mathbb{C}^\times$-丛)记为$\mathbb{L}(\mathcal{H})$。$\mathcal{H}_\Psi$上的复结构赋予切空间$T_\Psi\mathbb{P}(\mathcal{H})$一个可积复结构$J_\Psi$，在整个$\mathbb{P}(\mathcal{H})$上为可积近复结构从而使得$\mathbb{P}(\mathcal{H})$成为Kähler流形，它上面的典范2-形式(兼容的辛形式)就叫Fubini-Study 2-形式$\omega_\text{FS}$，于是相配的Riemann度量$g_\text{FS}(\cdot,\cdot)_\Psi=\omega_\text{FS}(\cdot,J_\Psi\cdot)$，并诱导出Hermite内积$h_\text{FS}=g_\text{FS}+i\omega_\text{FS}$。Hermite线丛$(\mathbb{L}(\mathcal{H}),h_\text{FS},\nabla_\text{FS})$视为$(\mathbb{P}(\mathcal{H}),\omega_\text{FS})$的几何量子化中的预量子化线丛。接下来对于作为描述经典力学的任何一个辛流形$(M,\omega)$，相干态量子化就是找到一个辛同胚$\text{Coh}:M\to\mathbb{P}(\mathcal{H})$, $\omega=\text{Coh}^*\omega_\text{FS}$, $h_K=\text{Coh}^*h_\text{FS}$, $\nabla_K=\text{Coh}^*\nabla_\text{FS}$，$\mathbb{L}=\text{Coh}^*\mathbb{L}(\mathcal{H})$，这使得$(\mathbb{L},h_K,\nabla_K)$为$(M,\omega)$上的预量子化线丛。线丛上的光滑截面即为相干态$\varepsilon:U\subset M\to\mathcal{H}$，也就有$\text{Coh}(x)=\frac{|\varepsilon(x)\rangle\langle\varepsilon(x)|}{\|\varepsilon(x)\|}$, $\int_M |\varepsilon(x)\rangle\langle\varepsilon(x)| d\mu_L(x)=\text{Id}_\mathcal{H}$。

这种几何化的表述数学上有着很丰富的内容，物理上也可以提供人们一些新的工具或语言来作为理论描述，如这些工作：
1.Extracting the quantum metric tensor through periodic driving
2.Revealing Tensor Monopoles through Quantum-Metric Measurements
3.Quantum mechanics in an evolving Hilbert space

6. 3周前
2019-01-24 01:55:13
Phantom_Ghost 更新于 Kane-Mele Model

我觉得学习物理方面最为重要的是看到这些理论模型的物理来源和图像，固然一些技术性的细节如怎么去计算这个模型的能谱以及如何弄个程序能呈现出来也挺不错，但更重要的是这里面蕴含的物理内容。我在以下段落给出我的点评：

要了解清楚这个模型最开始是怎么出来：以前人们在研究Haldane模型（基于蜂窝点阵上的紧束缚模型跃迁引入次近邻的时间反演对称破坏项和空间反演对称破坏项）具有能隙相和手征边缘态（所以也需要进一步追问为什么研究Haldane模型，这源于最初的IQHE方面的内容，人们从想仿照IQHE那样但不想加磁场这种动机构造并研究Haldane模型，这后来也成为了发现QAHE的基础），事实上这是第一类Chern绝缘体。于是后来人们开始考虑能不能不破坏时间反演对称，然后产生有能隙相，这就是Kane-Mele模型被提出的动机；而它也就可以理解为双份Haldane模型堆叠而成因此而有了时间反演对称（我用“堆叠”stack这个词在后来的拓扑相里面是有更深的含义的，例如两份手性相反的p-波超导的堆叠形成新的螺旋性拓扑超导，这就是一个由可逆拓扑序堆叠形成SPT的例子，放在这里也是一样的含义）。

所以这模型的体能带不是人们最关心的，其螺旋性边缘态的出现才是这个模型最重要的特点。进而要知道边缘态是受时间反演对称保护的，这一点在他们的文章里面也有描述，这也是非常重要的一点，是催生SPT这个领域的关键性的现象。整体理论方面人们有TKNN不变量刻画IQHE（Chern数），然而在这里两个自旋相反的能带贡献的Chern数符号相反进而抵消，所以人们不得不寻找新的拓扑不变量来刻画这种相，如所谓自旋Chern数以及$Z_2$不变量，在物理上对应着边缘态的自旋流（这里所谓体-边缘对应在数学上有个广为人知的Atiyah-Singer指标定理作为其背景）。

然后我觉得他们仿照探测IQHE那样设计了一个四通道的设备来测纯自旋流也是非常有意思的东西，详细研究一下这里的输运的内容（弹性弹道输运、Landauer-Büttiker公式计算电导，以及定义自旋流导等等）都是不简单的。

然后就是这个模型的具体物理实现问题，在文中提到可以根据自旋-轨道耦合来引入这种打开能隙而又保持时间反演对称的机制。但这个东西你们想必是没有仔细研究过的，怎么从最开始的格点模型中的原子自旋-轨道耦合来导出他们的低能有效能带理论里面那些项？这涉及固体物理里面很基本的计算方法（例如$k\cdot p$-微扰），在这里完全可以作为最简单的练习，既可以练理论推导能力，然后还可以练数值计算能力（把那些耦合常数从格点模型的参数出发计算出有效能带模型里的参数）；显然做些这个比单纯解个能带模型要多花许多时间精力。最后还要讨论一下这里面为什么拿石墨烯实现这个模型是不现实的。然后Kane和Mele最后还想通过在石墨烯里面引入Coulomb相互作用然后用一点单圈自能修正的RG流来论证在低能下这个自旋-轨道耦合常数是会增强的，由它打开的能隙是可以变大的（大到可以勉强观察得到），以此来鼓吹人们关注这个工作。事实上哪怕这点小的论证，你们要是去做也是得花上些许功夫的。

后面这个模型更多的意义就是指导了BHZ模型的提出从而能描述HgTe量子阱中出现的拓扑绝缘体相，可以参考
正方晶格模型与HgTe量子井的关系（因此在这里实际上人们会说QSH就是TI，然而根据文小刚的定义QSH是不需要时间反演对称保护的，而是由电荷也即$U(1)$对称和$s_z$对称保护的，这个$s_z$就看做自旋荷，所以他会说Kane-Mele模型在描述2+1维TI，自然BHZ模型也是）。张首晟等人研究3+1维TI并提出时间反演对称的Chern-Simons拓扑场论描述，开启了SPT的拓扑场论时代。

7. 2019-01-21 01:18:09
Phantom_Ghost 更新于 Chern characteristic class

Chern character
\begin{align*}
\text{ch}(E_1\oplus E_2)&=\text{ch}(E_2\oplus E_1)=\text{ch}(E_1)+\text{ch}(E_2)\\
\text{ch}(E_1\otimes E_2)&=\text{ch}(E_1)\smile \text{ch}(E_2)\\
\text{ch}(L)&=e^{c_1(L)}=1+c_1(L)+\frac{1}{2!}(c_1(L))^2+\cdots\\
\text{ch}(L_1\otimes L_2)&=e^{c_1(L_1\otimes L_2)}=e^{c_1(L_1)+c_1(L_2)}=e^{c_1(L_1)}e^{c_1(L_1)}
\end{align*}

Proposition 4: $c_1:\text{Vect}_\mathbb{C}^1(B)\to H^2(B;\mathbb{Z})$ is a homomorphism of rings.
(Here the vector bundles equipped with operation $\oplus,\otimes$ become a ring, while the integral cohomology equipped with $+,\smile$ become a graded ring)

Proof: First show for line bundles: $c_1(L_1\otimes L_2)=c_1(L_1)+c_1(L_2)$ where $L_1\otimes L_2\to\mathbb{C}P^\infty\times\mathbb{C}P^\infty$ , $L_{1,2}\hookrightarrow\mathbb{C}P^\infty$ are the pullbacks of canonical line bundle. Since $c_1(L_i)$ is a generator of $H^2(\mathbb{C}P^\infty)$, we get $H^\bullet(\mathbb{C}P^\infty\times\mathbb{C}P^\infty)\cong\mathbb{Z}[\alpha_1,\alpha_2]$ by Künneth formula where $\alpha_i=\text{pr}_i^*(c_1(L_i))$, $\text{pr}_{1,2}:\mathbb{C}P^\infty_1\times\mathbb{C}P^\infty_2\to \mathbb{C}P^\infty_{1,2}$.
\begin{align*}
H^2(\mathbb{C}P^\infty\times\mathbb{C}P^\infty)&\cong[H^2(\mathbb{C}P^\infty)\otimes H^0(\mathbb{C}P^\infty)]\oplus[H^0(\mathbb{C}P^\infty)\otimes H^2(\mathbb{C}P^\infty)]\\
&\bigoplus_{i+j=2+1}\underset{=0\;\text{(torsion free)}}{\underbrace{\text{Tor}(H^i(\mathbb{C}P^\infty),H^j(\mathbb{C}P^\infty))}}
\end{align*}
$\Rightarrow$ $\mathbb{C}P^\infty\vee\mathbb{C}P^\infty\hookrightarrow\mathbb{C}P^\infty\times\mathbb{C}P^\infty$ induces isomorphism on cohomology $H^2(\mathbb{C}P^\infty\vee\mathbb{C}P^\infty)\cong H^2(\mathbb{C}P^\infty\times\mathbb{C}P^\infty)$.
Therefore to compute $c_1(L_1\otimes L_2)$ it is sufficient to compute $c_1(L_1\otimes L_2|_{\mathbb{C}P^\infty\vee\mathbb{C}P^\infty})$:
$L_2$ is trivial on the first factor of $\mathbb{C}P^\infty\vee\mathbb{C}P^\infty$ while $L_1$ is trivial on the second factor: $c_1(L_1\otimes L_2|_{\mathbb{C}P^\infty\times\{\text{pt.}\}})=c_1(L_1)=\alpha_1$ , $c_1(L_1\otimes L_2|_{\{\text{pt.}\}\times\mathbb{C}P^\infty})=c_1(L_2)=\alpha_2$ $\Rightarrow$ $c_1(L_1\otimes L_2)=c_1(L_1)+c_1(L_2)$.

For the general case $E\cong L_1\oplus\cdots\oplus L_n$, $t_i:=c_1(L_i)$, then $\text{ch}(E)=\sum_i e^{t_i}=n+(t_1+\cdots+t_n)+\cdots+\frac{1}{k!}(t_1^k+\cdots+t_n^k)+\cdots$
$t_1^k+\cdots+t_n^k=s_k(\sigma_1,\dots,\sigma_n)$ with $\sigma_i=c_i(E)$ where $s_k(\cdot)$ is Newton polynomial. Then $\text{ch}(E)=\text{dim}\,E+\sum_{k>0}s_k(c_1(E),\dots,c_n(E))$.

Proposition 5 $\text{ch}(E_1\oplus E_2)=\text{ch}(E_1)+\text{ch}(E_2)$ ; $\text{ch}(E_1\otimes E_2)=\text{ch}(E_1)\text{ch}(E_2)$.

Proof: Splitting principle works also for $\mathbb{Q}$. This time we can pull back $E_1$ to a sum of $L_1\oplus\cdots L_n$ over $F(E_1)$. Then pull back $E_2$ over $F(E_1)$ so as a result $F(E_1,E_2)\to B$, where we pull back both $E_1$ and $E_2$ to the line budle.
$\text{ch}(E_1\oplus E_2)=\text{ch}(\bigoplus_{i=1,2,j>0}L_{i,j})e^{c_1(L_{i,j})}=\text{ch}(E_1)+\text{ch}(E_2)$
$\text{ch}(E_1\otimes E_2)=\text{ch}(\bigoplus_{i,j>0}L_{1,i}\otimes L_{2,j})=\sum_{i,j}\text{ch}(L_{1,i}\otimes L_{2,j})=\text{ch}(E_1)\text{ch}(E_2)$

Fact: $K(X)\xrightarrow{\text{ch}}H^\bullet(X;\mathbb{Q})$ is homomorphism
\begin{array}[c]{ccc}
K(X)&\xrightarrow{\text{ch}}&H^\bullet(X;\mathbb{Q})\\
\downarrow&&\downarrow\\
K(\{\text{pt.}\})&\xrightarrow{\text{ch}}&H^\bullet(\{\text{pt.}\};\mathbb{Q})
\end{array}
This yields the functoriality and there is a natural transormation between the functors $K(X)\xrightarrow{\text{ch}}H^\bullet(X;\mathbb{Q})$ and $\widetilde{K}(X)\xrightarrow{\text{ch}}\widetilde{H}^\bullet(X;\mathbb{Q})$.

Proposition 6: $\widetilde{K}(S^{2n})\xrightarrow{\text{ch}}\widetilde{H}^{2n}(S^{2n};\mathbb{Q})$ injective, $\text{im}(\text{ch})\subset\widetilde{H}^{2n}(S^{2n};\mathbb{Z})\subset\widetilde{H}^{2n}(S^{2n};\mathbb{Q})$.

Proof: Since
\begin{array}[c]{ccc}
\widetilde{K}(X)&\xrightarrow{\sim}&\widetilde{K}(\Sigma^2X)\\
\scriptstyle{\text{ch}}\downarrow&&\downarrow\scriptstyle{\text{ch}}\\
\widetilde{H}^{2n}(X;\mathbb{Q})&\xrightarrow{\sim}&\widetilde{H}^{2n}(\Sigma^2X;\mathbb{Q})
\end{array}
where $\Sigma^2X$ is the double suspension of $X$. The upper row in the diagram is due to Bott periodicity while the lower row is due to Künneth formula.
$\text{ch}(H-1)=\text{ch}(H)-\text{ch}(1)=1+c_1(H)-1=c_1(H)$ ($H$ denotes the canonical line bundle over $S^2\approx \mathbb{C}P^1$).
Perform induction on $n$:
$X=S^0$, it holds.
$n\to n+1$ follows by the commutative diagram
\begin{array}[c]{ccc}
\widetilde{K}(X)&\xrightarrow{\sim}&\widetilde{K}(\Sigma^2X)&\to& \widetilde{K}(\Sigma^4X)&\to& \cdots\\
\scriptstyle{\text{ch}}\downarrow&&\downarrow\scriptstyle{\text{ch}}&&\downarrow\\
\widetilde{H}^{2n}(X;\mathbb{Q})&\xrightarrow{\sim}&\widetilde{H}^{2n}(\Sigma^2X;\mathbb{Q})&\to&\widetilde{H}^{2n+4}(\Sigma^4X;\mathbb{Q})&\to& \cdots
\end{array}

Corollary 7: A calss in $H^{2n}(S^{2n};\mathbb{Z})$ is a Chern characteristic $c_n(E)$ $\Rightarrow$ It is divisble by $(n-1)!$

Proof: $E\to S^{2n}$ we get $c_i(E)=0$ for $n>i>0$.
$\text{ch}(E)=\text{dim}\,E+\frac{1}{n!}s_n(\sigma_1,\dots,\sigma_n)=\text{dim}\,E+\frac{n}{(n+1)!}\sigma_n$ since $s_n(\sigma_1,\dots,\sigma_n)=\sigma_1 s_{n-1}-\sigma_2 s_{n-2}+\cdots+(-1)^{n-2}\sigma_{n-1}s_1+(-1)^{n-1}n\sigma_n$.

8. 2019-01-20 18:51:41
Phantom_Ghost 更新于 Chern characteristic class

Theorem 1: There is a unique sequence of maps $c_i$ assigning to vector bundle $E\to B$ a class $c_i(E)\in H^{2i}(B;\mathbb{Z})$, which depends on isomorphism class of $E$ and the following properties:

$i)$ $c_{i}(f^*E)=f^*c_i(E)$ ($f^*E$ is a pullback bundle)

$ii)$ $c(E_1\oplus E_2)=c_1(E_1)\smile c(E_2)$ where $c=1+\sum_{i>0}c_i$ or equivalently $c_k(E_1\oplus E_2)=\sum_{i+j=k}c_i(E_1)\smile c_j(E_2)$ where $c_0=1$.

$iii)$ $c_i(E)=0$ $i>\text{rank}\,E$

$iv)$ For the canonical line bundle $E\to\mathbb{C}P^\infty$, $c_1(E)$ is a fixed generator pf $H^2(\mathbb{C}P^\infty;\mathbb{Z})$

The proof of the theorem is based on Leray-Hirsch theorem:
Theorem 2 (Leray-Hirsch): Let $\pi:E\to B$ be a fiber bundle, $H^\bullet(E;R)$ is a module over $H^\bullet(B;R)$ ($R$ is commutative ring); $\alpha\in H^\bullet(B;R)$ and $\beta\in H^\bullet(E;R)$, $\alpha\cdot\beta=\pi^*\alpha\smile\beta$; $H^\bullet(E;R)$ is free over $H^\bullet(B;R)$. Evidently, $\forall x\in B$, $F_x\overset{i_x}{\hookrightarrow}E$ induces a surjection on $H^\bullet(E;R)\to H^\bullet(B;R)$ and $H^\bullet(F;R)$ is a free $R$-modlue of finite rank. Moreover we can obtain a basis by choosing elements that map to a basis in $H^\bullet(F;R)$ under $i^*$.

Proof of thm1(existence): $P(\pi):P(E)\to B$ is universal bundle, remove zero section from $\mathbb{C}P^{n-1}$ out of $\mathbb{C}^\times$. Wee want to find $x_i\in H^{2i}(P(E);\mathbb{Z})$ s.t. they restrict to generators of $H^{2i}(\mathbb{C}P^{n-1};\mathbb{Z})$. From universal bundle, there is $g:E\to\mathbb{C}P^\infty$ with linear injections as fibers. Projectivizing $g$ by deleting zero section and then factoring out scalar multiplication induces $P(g):P(E)\to\mathbb{C}P^\infty$
\begin{array}[c]{cc}
P(E) & \\
\scriptstyle{i}\uparrow&\searrow{}^{P(g)}\\
\;\;\;\;\;\;\mathbb{C}P^{n-1}\hookrightarrow& \mathbb{C}P^\infty\\
\end{array}

\begin{array}[c]{cc}
& H^{2i}(P(E);\mathbb{Z})\\
{}^{{}^{P(g)^*}\nearrow}&\downarrow\scriptstyle{i^*}\\
\;\;\;\;\;\;H^{2i}(\mathbb{C}P^\infty;\mathbb{Z})\rightarrow& H^{2i}(\mathbb{C}P^{n-1};\mathbb{Z})
\end{array}
Let $\alpha$ be a generator of $H^2(\mathbb{C}P^\infty;R)$, then $x^i=P(g)^*\alpha^i$, $1,x,x^2,\dots, x^{n-1}$ form a basis of $H^\bullet(P(E);R)$ as a $H^\bullet(E;R)$-module. Their images under $i^*$ map form a basis for the cohomology of fiber $H^\bullet(\mathbb{C}P^{n-1})$. Therefore $x^n+c_1(E)x^{n-1}+\cdots+c_n(E)\cdot 1=0$.
This is how we obtain the Chern class.

Proof of uniqueness follows from the proposition below.

Proposition 3 (Splitting principle): For each vector bundle $E\to B$, there is a space $F(E)$ with a map $F(E)\xrightarrow{p}B$ s.t. the pullback bundle $p^*E\to F(E)$ splits as a direct sum of line bundles and $p^*:H^\bullet(B;\mathbb{Z})\to H^\bullet(P(E);\mathbb{Z})$ is injective.
(This is a similar phenomenon compare with Bott periodicity for K-theory)

Proof of prop.: We have the map $P(E)\xrightarrow{P(\pi)}B$ and pull back $E$ along this map. There is a subbundle in bundle on $P(E)$: $L:=\{(\ell,v)\in P(E)\times|v\in\ell\}$ , $P(\pi)^*E=L\oplus L^\perp$. $H^\bullet(P(E);\mathbb{Z})$ is a free module with basis $1,x,x^2,\dots,x^{n-1}$. Repeat the same construction for $L^\perp$ and we are done.

Remark
1) $E=B\times\mathbb{C}$, it is a pullback of a bundle over one point, since $H^{2i}(\{\text{pt.}\};\mathbb{Z})=0$ for $i>0$, hence $c_i(E)=0$ for $i>0$.

2) $E=E_1\oplus\mathbb{n}$, $c(E)=c(E_1\oplus\mathbb{n})=c(E_1)\smile \underset{=1}{\underbrace{c(\mathbb{n})}}=c(E_1)$.

3) $E\cong L_1\oplus L_2\oplus\cdots\oplus L_n$, uniqueness comes from splitting principle and 4)

4) There is no bundle over $E\to\mathbb{C}P^\infty$ whose sum with the canonical line bundle over $\mathbb{C}P^\infty$ is trivial. i.e assume $E\oplus L\cong\mathbb{n}$, then $1=c(\mathbb{n})=c(E\oplus L)=c(E)\smile c(L)=(1+c_1(E)+\cdots)(1+c_1(L))$ by 1); hence $c(E)=(1+c_1(L))^{-1}=1-c_1(L)+c_1(L)^2+\cdots$ contradicts to 3).

9. 2019-01-20 18:45:33
Phantom_Ghost 发表了帖子 Chern characteristic class

A short note recording the content of the talk from the seminar of ''topological K-theory'' during WS18/19.

Reference: Allen Hatcher, Vector Bundles and K-Theory.

10. 4周前
2019-01-14 22:16:06
Phantom_Ghost 更新于 非欧几何是向量空间不

这也要封？我认为只要是没有逾越理性讨论这点原则都不可以实施封禁处罚，别人能不能理解那是他自己的事，你解释得好不好也是你自己的事情，自由理性讨论是每个用户在学术论坛上应享有的基本权利。如果觉得对方不能理解而懒得进一步交流你完全可以不回复不再参与讨论。