# K-theory and D-branes

1. 2周前

### Phantom_Ghost

1楼 2月1日 数学版主, 物理版主, 优秀回答者
上周Phantom_Ghost 重新编辑

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.

2. ### Phantom_Ghost

2楼 2月1日 数学版主, 物理版主, 优秀回答者
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String theory

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

- string theory: strings

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

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

$U(N)$ 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.

3. ### Phantom_Ghost

3楼 2月1日 数学版主, 物理版主, 优秀回答者
上周Phantom_Ghost 重新编辑

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
$p_k(R)=(-1)^k c_{2k}(R^\mathbb{C})$

4. 上周

### Phantom_Ghost

4楼 2月2日 数学版主, 物理版主, 优秀回答者
3天前Phantom_Ghost 重新编辑

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 $f_!^K(E)\in K(S)$.