I have a question with conformal field theory in Polchinski's string theory vol 1 p. 51.
For bc conformal field theory S=12π∫d2zbˉ∂c
T(z)=:(∂b)c:−λ∂(:bc:)
with central charge c=−3(2λ−1)2+1=1. Introducing ψ and ˉψ to replace the anticommuting fields b and c as following b→ψ=2−1/2(ψ1+iψ2)
and c→ˉψ=2−1/2(ψ1−iψ2)
It is claimed that S=14π∫d2zψ1ˉ∂ψ1+ψ2ˉ∂ψ2(2.5.18b)
T=−12ψ1∂ψ1−12ψ2∂ψ2(2.5.18c)
I cannot obtain the above expressions of S and T. Here is my derivations. First I try to recover the anti-commuting characters of fields b and c by ψ1 and ψ2. For bc+cb=0
I have ψ1ψ1+ψ2ψ2=0
Then for the action S=14π∫d2z(ψ1ˉ∂ψ1+iψ2ˉ∂ψ1−iψ1ˉ∂ψ2+ψ2ˉ∂ψ2)
(1) [Solved] Why the term iψ2ˉ∂ψ1−iψ1ˉ∂ψ2 does not contribute to the action?
(2) How to derive (2.5.18c)?
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