# User:hillgentleman/bc CFT

bc CFT

b-c 共形場論(bc CFT[1] (或 b-c system) 係種簡單嘅共形場論模型，由兩隻反交換嘅自由 bc 組成。渠會出現喺 量子化Polyakov 弦gauge fixing 度出現，做Faddev-Popov 鬼( en:Faddeev-Popov ghost)[2]

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

Action
${\displaystyle S={1 \over {2\pi \alpha '}}\int d^{2}b{\bar {\partial }}c}$
conformal weight
${\displaystyle h_{b}=\lambda }$
${\displaystyle h_{c}=1-\lambda }$, where ${\displaystyle \lambda }$ is the parameter
central charge
${\displaystyle c=-3(2\lambda -1)^{2}+1}$ ; ${\displaystyle {\tilde {c}}=0}$
ghost number current
j = -:bc:
energy-momentum tensor
${\displaystyle T(z)=:(\partial b)c:-\lambda \partial (:bc:)}$ ; ${\displaystyle {\tilde {T}}({\bar {z}})=0}$
equations of motion
${\displaystyle {\bar {\partial }}c(z)=\partial b(z)=0}$
${\displaystyle \partial b(z)c(0)=2\pi \delta ^{2}(z,{\bar {z}})}$
OPE
${\displaystyle b(z_{1})c(z_{2})=:b(z_{1})c(z_{2}):+{\frac {1}{z_{12}}}}$
${\displaystyle b(z_{1})b(z_{2})=O(z_{12})}$
${\displaystyle c(z_{1})c(z_{2})=O(z_{12})}$ ${\displaystyle T(z)T(0)\thicksim {\frac {c}{2z^{4}}}+{\frac {2}{z^{2}}}T(0)+{\frac {1}{z}}\partial T(0)}$
${\displaystyle T(z)j(0)\thicksim {\frac {1-2\lambda }{z^{3}}}+{\frac {1}{z^{2}}}j(0)+{\frac {1}{z}}\partial _{j}(0)}$
• When ${\displaystyle \lambda =1/2}$, ${\displaystyle h_{b}=h_{c}=1/2}$ and the theory can be split conformally invariantly into ${\displaystyle \psi _{1}}$ and ${\displaystyle \psi _{2}}$ , where
${\displaystyle \psi ={\frac {1}{2^{1/2}}}(\psi _{1}+i\psi _{2})}$ ; ${\displaystyle {\bar {\psi }}={\frac {1}{2^{1/2}}}(\psi _{1}-i\psi _{2})}$......
• ${\displaystyle \lambda =2}$${\displaystyle h_{b}=2}$ 同埋 ${\displaystyle h_{c}=-1}$，咁渠就描述Faddeev-Popov 鬼 [3]

## 註

1. Polchinski, p.50, etc
2. Polchinski, p.51
3. Polchinski, p.51