User:hillgentleman/bc CFT

bc CFT

b-c 共形場論(bc CFT (或 b-c system) 係種簡單嘅共形場論模型，由兩隻反交換嘅自由場 b、c 組成。渠會出現喺 量子化Polyakov 弦 時 gauge fixing 度出現，做Faddev-Popov 鬼( en:Faddeev-Popov ghost)

Let $\lambda \in \mathbb {C}$.

Action

$S={1 \over {2\pi \alpha '}}\int d^{2}b{\bar {\partial }}c$

conformal weight

$h_{b}=\lambda$
$h_{c}=1-\lambda$, where $\lambda$ is the parameter

central charge

$c=-3(2\lambda -1)^{2}+1$ ; ${\tilde {c}}=0$

ghost number current

j = -:bc:

energy-momentum tensor

$T(z)=:(\partial b)c:-\lambda \partial (:bc:)$ ; ${\tilde {T}}({\bar {z}})=0$

equations of motion

${\bar {\partial }}c(z)=\partial b(z)=0$
$\partial b(z)c(0)=2\pi \delta ^{2}(z,{\bar {z}})$

OPE

$b(z_{1})c(z_{2})=:b(z_{1})c(z_{2}):+{\frac {1}{z_{12}}}$
$b(z_{1})b(z_{2})=O(z_{12})$
$c(z_{1})c(z_{2})=O(z_{12})$ $T(z)T(0)\thicksim {\frac {c}{2z^{4}}}+{\frac {2}{z^{2}}}T(0)+{\frac {1}{z}}\partial T(0)$
$T(z)j(0)\thicksim {\frac {1-2\lambda }{z^{3}}}+{\frac {1}{z^{2}}}j(0)+{\frac {1}{z}}\partial _{j}(0)$

• When $\lambda =1/2$, $h_{b}=h_{c}=1/2$ and the theory can be split conformally invariantly into $\psi _{1}$ and $\psi _{2}$ , where
  $\psi ={\frac {1}{2^{1/2}}}(\psi _{1}+i\psi _{2})$ ; ${\bar {\psi }}={\frac {1}{2^{1/2}}}(\psi _{1}-i\psi _{2})$......
• 當 $\lambda =2$、 $h_{b}=2$ 同埋 $h_{c}=-1$，咁渠就描述Faddeev-Popov 鬼 。

參攷

• Joseph Polchinski, "String Theory, Volume I", ISBN 0521-633036