# BRST quantization of string question

• simic4
In summary, the conversation discusses the BRST invariance of the bosonic string action, specifically equations 4.3.1a-c. It is noted that the ghost field equations of motion must be used, which may seem unusual. However, it is allowed and the process of deriving the BRST current from Noether's theorem is not entirely arbitrary. It is suggested to check BRST invariance of the "other" action first and then use the equation of motion of the ghost in the free bosonic part of the action. This can be seen as a delta function enforcing the ghost equation of motion.

#### simic4

Hi,

I am confused about the following, I was hoping someone could help:

The context: Polchinski Ch. 4.2, specifically equations 4.3.1a-c

I am verifying the BRST invariance of the bosonic string action (after one has integrated out B, and the weyl ghost),, I notice that one must use the ghost field equations of motion! d_bar c = 0 = d_bar b = 0.

i am not accustomed to being able to use equations of motion inside actions! why is this allowed..? Furthermore,, doesn't this make the whole process of deriving the brst current (eq 4.3.3) from Noether's theorem rather arbitrary?

This problem cannot be totally removed by checking BRST invariance from the "other" action (the action in 4.2.3,, the one before one integrates out B) because for the bosonic part to be invariant, one must assume c holomorphic,, ie d_bar c = 0.

perhaps i am missing a rather subtle point..

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Ah indeed,, I think I have it ( all these statements are in conformal gauge):

First off, one should check BRST invariance of the "other" action (the action in 4.2.3,, the one before one integrates out B), this is obviously the more correct way to go: then the only place one must use the equation of motion of c, the ghost, is in the free bosonic part of the action (S_1 in polchinski 4.2.3). However, this is perfectly fine, because one notices that the ghost part, consisting of B, c, and b, is in fact nothing but a delta function (in disguise) enforcing the c equation of motion! this can be seen by integrating out B, then b.. what is left, although perhaps in an uncommon representation,, is a delta function of d_bar c.

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