# Wess-Zumino-Witten模型

## 作用

G緊緻單連通李羣，設g為其李代數。設γ為黎曼球面${\displaystyle S^{2}}$ 複平面之一點緊緻化）上一G-值場

Wess-Zumino-Witten 模型是γ所定義之非線性 sigma 模型，其作用

${\displaystyle S_{k}(\gamma )=-\,{\frac {k}{8\pi }}\int _{S^{2}}d^{2}x\,{\mathcal {K}}(\gamma ^{-1}\partial ^{\mu }\gamma \,,\,\gamma ^{-1}\partial _{\mu }\gamma )+2\pi k\,S^{\mathrm {W} Z}(\gamma )}$

SWZ 項人稱 Wess-Zumino 項，其定義為

${\displaystyle S^{\mathrm {W} Z}(\gamma )=-\,{\frac {1}{48\pi ^{2}}}\int _{B^{3}}d^{3}y\,\epsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{i}}}\,,\,\left[\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{j}}}\,,\,\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{k}}}\right]\right)}$

### 拉回

${\displaystyle S^{\mathrm {W} Z}(\gamma )=\int _{B^{3}}\gamma ^{*}c}$

### 拓撲障礙

γ 有多種延拓至球${\displaystyle B^{3}K}$ 之內部；若要求物理現象不依賴於特定之延拓，則常數k需符合以下「量子條件」：

• 取γ 到球內部之任何兩種延拓。是為平三維區域至李羣G之兩支影射。在其邊界 ${\displaystyle S^{2}}$ 黏起此兩個三維球，則成一三維球面${\displaystyle S^{3}}$ ；其中每一三維半球面來自一${\displaystyle B^{3}}$ 。 γ 之兩種延拓則成為一影射： ${\displaystyle S^{3}\rightarrow G}$ 。然而，任何緊緻單連通李羣G之同倫羣

${\displaystyle \pi _{3}(G)=\mathbb {Z} }$  。故

${\displaystyle S^{\mathrm {W} Z}(\gamma )=S^{\mathrm {W} Z}(\gamma ')+n}$

${\displaystyle \exp \left(i2\pi kS^{\mathrm {W} Z}(\gamma )\right)=\exp \left(i2\pi kS^{\mathrm {W} Z}(\gamma ')\right)}$

## 參攷

• J. Wess, B. Zumino, "Consequences of anomalous Ward identities", Physics Letters B, 37 (1971) pp. 95-97.
• E. Witten, "Global aspects of current algebra", Nuclear Physics B 223 (1983) pp. 422-432.
• V. Kac, Infinite dimensional Lie algebras

## 註

1. ^ Kac, Victor, Infinite dimensional Lie algebras, ISBN 0-521-46693-8 第十章，
2. ^ 反德西特空間，為SL(2,R)之通用覆蓋