How to get the anticommutator $\{b_m,c_n\}$?

Question

Suppose $b(z)$ and $c(z)$ are holomorphic fermionic ghost fields with conformal dimension 2 and -1, respectively, with mode expansions $$b(z)=\sum_n \frac{b_n}{z^{n+2}}$$ and $$c(z)=\sum_n \frac{c_n}{z^{n-1}}.$$ Also, they have OPE $$c(z)b(w)=\frac{1}{z-w}+\cdots \quad ,\quad b(z)c(w)=\frac{1}{z-w}+\cdots .$$ My question is that how to get the anticommutator $\{ b_m,c_n\}$?

I know that if we know the OPE of two conformal fields $b(z)$ and $c(z)$, then we can obtain $[b_m,c_n]$ using countor integral but I don't how to obtain $\{b_m ,c_n\}$.

Answer 1

That seems to be a misunderstanding. One can get the supercommutator (as opposed to the commutator) from the OPE, cf. e.g. this Phys.SE post. Since $b$ and $c$ are Grassmann-odd, their supercommutator is OP's sought-for anticommutator.