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A bc system is the anticommuting counterpart of a beta-gamma system. For weights $h$ and $1-h$, its central charge is $-2(6h^2-6h+1)$.

The quadratic kinetic operator of the beta-gamma system couples $\beta$ only to $\gamma$. Its Green-function equation isin the stated normalization. Since $\bar{\partial }(1/(z-w))=2\pi {\delta }^{(2)}(z-w)$, the singular part isThe inverse kinetic matrix has no $\beta \beta$ or $\gamma \gamma$ entry, so those two operator product expansions are nonsingular. For commuting fields, reversing the order gives $\gamma (z)\beta (w)\sim 1/(z-w)$ after expanding about $w$.

In any two-dimensional conformal field theory,Double contractions in the bosonic beta-gamma system giveThe polynomial is unchanged by $h\mapsto 1-h$, as required when the roles of the two fields are interchanged. If the fields anticommute, a closed fermionic contraction contributes an additional minus sign, giving the fermionic bc-system result