# Why can the commutator of a general expression be replaced by the anti-commutator in the $bc$ CFT theory?

Polchinski states in his equation 2.6.14 (in his book String Theory Vol. 1, Introduction to the Bosonic String) that for charges $$Q_1$$ and $$Q_2$$ the following equation holds, where $$j_i$$ is the corresponding current:

$$[Q_1,Q_2]=\oint\!\frac{\mathrm{d}w}{2\pi i}\text{Res}_{z\rightarrow w}j_1(z)j_2(w).\tag{2.6.14}$$

Now, for the $$bc$$ conformal field theory, one find that $$\{b_m,c_n\}=\delta_{m,-n}$$, which can be shown if above equations holds for the anticommutator.

The equation 2.6.14 is stated with the commutator. Why can it also be used with the anticommutator of $$b$$ and $$c$$?

• Where is it said that it holds for the commutator $[b_m,c_n]$? Commented Feb 11, 2023 at 20:36
• @Qmechanic: There's a catch: It is for example stated in the solutions of UChicago to the course PHYS 483, where question 2.12 from Polchinski was part of a problem set. The question is to show that $\{b_m,c_n\}=\delta_{m,-n}$ and the solution just states that the relation 2.6.14 holds in this case also for the commutator instead of the anti-commutator. Maybe the solution is wrong, but the desired result actually follows. This is why I am asking. Commented Feb 11, 2023 at 20:43
• homes.psd.uchicago.edu/~sethi/Teaching/P483-W2018/p483-sol3.pdf, Page 4, eq. 29. Commented Feb 11, 2023 at 22:22

To generalize the bosonic eq. (2.6.14) to operators with arbitrary definite Grassmann parity, the commutator on the LHS of eq. (2.6.14) should be replaced with a supercommutator. Similar superization should be done with the implicitly written radial operator ordering $${\cal R}$$ on the RHS of eq. (2.6.14). For details, see e.g. my related Phys.SE answer here.