8
BRST Quantization
Abstract
The BRST quantization can be introduced either by following the standard QFT
treatment (outlined in Sect. 3.2) or by translating it into the CFT language. One
can then use all the CFT techniques to extract information on the spectrum, which
makes this approach more powerful. Moreover, this also provides an elegant
description of states and string fields. In this chapter, we set the stage of the BRST
quantization using the CFT language, and we apply it to string theory. The main
results of this chapter are a proof of the no-ghost theorem and a characterization
of the BRST cohomology (physical states).
8.1
BRST for Reparametrization Invariance
The BRST symmetry we are interested in results from gauge fixing the
reparametrization invariance. In this chapter, we focus on the holomorphic
sector: since both sectors are independent, most results follow directly, except
those concerning the zero-modes. We consider a generic matter CFT coupled to
reparametrization ghosts
1. matter: central charge c m , energy–momentum tensor T m and Hilbert space H m ;
2. reparametrization ghosts: bc ghost system (Sects. 2.3 and 7.2) with = +1 and
λ = 2, c gh = −26, energy–momentum tensor T gh and Hilbert space H gh .
The formulas for the reparametrization ghosts are summarized in Appendix B.3.5.
For modes, the system (m, gh, b or c) is indicated as a superscript to not confuse it
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_8
181
BRST Quantization
Abstract
The BRST quantization can be introduced either by following the standard QFT
treatment (outlined in Sect. 3.2) or by translating it into the CFT language. One
can then use all the CFT techniques to extract information on the spectrum, which
makes this approach more powerful. Moreover, this also provides an elegant
description of states and string fields. In this chapter, we set the stage of the BRST
quantization using the CFT language, and we apply it to string theory. The main
results of this chapter are a proof of the no-ghost theorem and a characterization
of the BRST cohomology (physical states).
8.1
BRST for Reparametrization Invariance
The BRST symmetry we are interested in results from gauge fixing the
reparametrization invariance. In this chapter, we focus on the holomorphic
sector: since both sectors are independent, most results follow directly, except
those concerning the zero-modes. We consider a generic matter CFT coupled to
reparametrization ghosts
1. matter: central charge c m , energy–momentum tensor T m and Hilbert space H m ;
2. reparametrization ghosts: bc ghost system (Sects. 2.3 and 7.2) with = +1 and
λ = 2, c gh = −26, energy–momentum tensor T gh and Hilbert space H gh .
The formulas for the reparametrization ghosts are summarized in Appendix B.3.5.
For modes, the system (m, gh, b or c) is indicated as a superscript to not confuse it
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_8
181