BRST quantization of string question

[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..

 

[simic4]

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.

 

[Entropia]

Hello,

I understand your confusion regarding the use of ghost field equations of motion inside actions. This is allowed in the context of BRST quantization because the ghost fields are auxiliary fields that do not have physical degrees of freedom. Therefore, they can be treated as mathematical tools to simplify the calculations and obtain the correct quantum theory.

Regarding the arbitrariness of deriving the BRST current from Noether's theorem, it is important to note that Noether's theorem is a powerful tool for deriving conserved quantities from symmetries of the action. However, in the case of BRST symmetry, the symmetry is not a physical symmetry but rather a mathematical one. This is why the process of deriving the BRST current can seem arbitrary, but it is necessary to ensure the correct quantization of the theory.

As for the issue with the holomorphicity of the ghost field c in the "other" action, this is a technical requirement that ensures the invariance of the bosonic part of the action. It may seem arbitrary, but it is a necessary condition for the consistency of the theory.

I hope this helps clarify some of your concerns. If you have any further questions, please do not hesitate to ask.