Thursday, August 7, 2014

string theory - Expressions of action and energy momentum tensor in bc conformal field with central charge equals one


I have a question with conformal field theory in Polchinski's string theory vol 1 p. 51.


For bc conformal field theory S=12πd2zbˉc

T(z)=:(b)c:λ(:bc:)
with central charge c=3(2λ1)2+1=1. Introducing ψ and ˉψ to replace the anticommuting fields b and c as following bψ=21/2(ψ1+iψ2)
and cˉψ=21/2(ψ1iψ2)
It is claimed that S=14πd2zψ1ˉψ1+ψ2ˉψ2(2.5.18b)
T=12ψ1ψ112ψ2ψ2(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=14πd2z(ψ1ˉψ1+iψ2ˉψ1iψ1ˉψ2+ψ2ˉψ2)
(1) [Solved] Why the term iψ2ˉψ1iψ1ˉψ2 does not contribute to the action?


(2) How to derive (2.5.18c)?




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