Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 306 / 2

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 $w$.

Applying Wick theorem to the stated holomorphic stress-energy tensor givesandThese are exactly the stress-tensor OPEs of primary operators. Hence $\gamma$ has holomorphic conformal weight $h$ and $\beta$ has weight $1-h$.

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

The $D$ free embedding scalars contribute $c_X=D$. The ordinary diffeomorphism ghosts contribute $c_{bc}=-26$. The fermionic $(m,n)$ pair has $h_n=0$ and contributes $-2$. Each of the $D/2$ fermionic $({\chi }_a,{\psi }^a)$ pairs has $h_{\psi }=1/2$ and contributes $+1$. Each bosonic $({\beta }_r,{\gamma }^r)$ pair has $h_{\gamma }=-1/2$ and contributes $+11$, and there are two such pairs. The total holomorphic central charge is thereforeCancellation of the worldsheet Weyl anomaly requires $c_{\mathrm{tot}}=0$, soThe antiholomorphic sector gives the same condition.