References
201
Since the mass is negative, this state is a tachyon. The vertex operator associated
to the state reads
V (k, z, ¯
z) = c(z) ¯
c(¯ z)e
ik·X(z,¯ z) .
(8.98)
8.4
Summary
In this chapter, we have described the BRST quantization from the CFT point of
view. We have first considered only the holomorphic sector (equivalently, the open
string). We proved that the cohomology does not contain negative-norm states, and
we provided an explicit way to construct the states. Finally, we glued together both
sectors and characterized the BRST cohomology of the closed string.
What is the next step? We could move to computations of on-shell string
amplitudes, but this falls outside the scope of this book. We can also start to consider
string field theory. Indeed, the BRST equation Q B |ψ = 0 and the equivalence
|ψ ∼ |ψ + Q B | completely characterize the states. In QFT, states are solutions
of the linearized equations of motion; hence, the BRST equation can provide a
starting point for building the action. This is the topic of Chap. 10.
8.5
Suggested Readings
• The general method to construct the absolute cohomology follows [5, 12]. Other
works and reviews include [2, 4, 7–11].
• String states are discussed in [3, sec. 3.3, , 12, sec. 4.1].
References
1. M. Asano, M. Natsuume, The no-ghost theorem for string theory in curved backgrounds with
a flat timelike direction. Nuclear Phys. B 588(1–2), 453–470 (2000). https://doi.org/10.1016/
S0550-3213(00)00495-8. arXiv: hep-th/0005002
2. A. Bilal, Remarks on the BRST-cohomology for c M > 1 matter coupled to Liouville gravity. Phys. Lett. B 282(3–4), 309–313 (1992). https://doi.org/10.1016/0370-2693(92)90644-J.
arXiv: hep-th/9202035
3. R. Blumenhagen, D. Lüst, S. Theisen, Basic Concepts of String Theory. English. 2013 edition
(Springer, Berlin, 2014)
4. P. Bouwknegt, J. McCarthy, K. Pilch, BRST analysis of physical states for 2D (super) gravity
coupled to (super) conformal matter, in New Symmetry Principles in Quantum Field Theory
(Springer, Berlin, 1992), pp. 413–422. https://doi.org/10.1007/978-1-4615-3472-3_17. arXiv:
hep-th/9110031
201
Since the mass is negative, this state is a tachyon. The vertex operator associated
to the state reads
V (k, z, ¯
z) = c(z) ¯
c(¯ z)e
ik·X(z,¯ z) .
(8.98)
8.4
Summary
In this chapter, we have described the BRST quantization from the CFT point of
view. We have first considered only the holomorphic sector (equivalently, the open
string). We proved that the cohomology does not contain negative-norm states, and
we provided an explicit way to construct the states. Finally, we glued together both
sectors and characterized the BRST cohomology of the closed string.
What is the next step? We could move to computations of on-shell string
amplitudes, but this falls outside the scope of this book. We can also start to consider
string field theory. Indeed, the BRST equation Q B |ψ = 0 and the equivalence
|ψ ∼ |ψ + Q B | completely characterize the states. In QFT, states are solutions
of the linearized equations of motion; hence, the BRST equation can provide a
starting point for building the action. This is the topic of Chap. 10.
8.5
Suggested Readings
• The general method to construct the absolute cohomology follows [5, 12]. Other
works and reviews include [2, 4, 7–11].
• String states are discussed in [3, sec. 3.3, , 12, sec. 4.1].
References
1. M. Asano, M. Natsuume, The no-ghost theorem for string theory in curved backgrounds with
a flat timelike direction. Nuclear Phys. B 588(1–2), 453–470 (2000). https://doi.org/10.1016/
S0550-3213(00)00495-8. arXiv: hep-th/0005002
2. A. Bilal, Remarks on the BRST-cohomology for c M > 1 matter coupled to Liouville gravity. Phys. Lett. B 282(3–4), 309–313 (1992). https://doi.org/10.1016/0370-2693(92)90644-J.
arXiv: hep-th/9202035
3. R. Blumenhagen, D. Lüst, S. Theisen, Basic Concepts of String Theory. English. 2013 edition
(Springer, Berlin, 2014)
4. P. Bouwknegt, J. McCarthy, K. Pilch, BRST analysis of physical states for 2D (super) gravity
coupled to (super) conformal matter, in New Symmetry Principles in Quantum Field Theory
(Springer, Berlin, 1992), pp. 413–422. https://doi.org/10.1007/978-1-4615-3472-3_17. arXiv:
hep-th/9110031