Beta-gamma system ($\beta \gamma$ system)

A beta-gamma system is a pair of commuting chiral fields with first-order action $\int \beta \bar{\partial }\gamma$. If $\gamma$ has weight $h$ and $\beta$ has weight $1-h$, its central charge is $2(6h^2-6h+1)$.

Using the radial-ordering sign in commuting beta-gamma contractions, single contractions give $T(z)\beta (w)\sim \lambda \beta (w)/(z-w{)}^2+\partial \beta (w)/(z-w)$ and the corresponding weight $1-\lambda$ for $\gamma$. Put $A=:\partial \beta \gamma :$ and $B=:\beta \partial \gamma :$. Their double-contraction coefficients in $AA,BB,AB,BA$ are respectively $1,1,2,2$. The fourth-order pole of $TT$ is therefore $(1-\lambda {)}^2+{\lambda }^2-4\lambda (1-\lambda )=6{\lambda }^2-6\lambda +1$, which equals $c/2$. There is no anticommuting Wick sign.

Commutation allows exchanging the factors at their existing arguments; it does not allow exchanging their arguments while retaining the same pole sign. With the displayed convention, differentiation produces $\partial \beta (z)\gamma (w)\sim (z-w{)}^{-2}$ and $\partial \gamma (z)\beta (w)\sim -(z-w{)}^{-2}$. These signs are essential for the conformal weights and central charge of a commuting beta-gamma system.