Kiefer C. Quantum gravity

286 STRING THEORY
The full action is invariant under BRST transformations, which were already
briefly mentioned in Section 2.2.3. This is an important concept, since it encodes
the information about gauge invariance at the gauge-fixed level. For this reason,
we shall give a brief introduction here (see e.g. Weinberg 1996 for more details).
BRST transformations mix commuting and anticommuting fields (ghosts) and
are generated by the ‘BRST charge’ Q B .Letφ a be a general set of first-class
constraints (see Section 3.1.2),
The BRST charge then reads
{φ a ,φ b } = f c abφ c . (9.28)
Q B = η a φ a − 1 2 P cf c abη b η a , (9.29)
where η a denotes the Faddeev–Popov ghosts and P a their canonically conjugate
momenta (‘anti-ghosts’) obeying [η a ,P b ] + = δb a . We have assumed here that the
physical fields are bosonic; for a fermion there would be a plus sign in (9.29).
One can show that Q B is nilpotent,
Q 2 B =0. (9.30)
This follows from (9.28) and the Jacobi identities for the structure constants.
BRST invariance of the path integral leads in the quantum theory to the
demand that physical states should be BRST-invariant, that is,
ˆQ B |Ψ〉 =0. (9.31)
This condition is less stringent than the Dirac condition, which states that physical
states be annihilated by all constraints. Equation (9.31) can be fulfilled for
the quantized bosonic string, which is not the case for the Dirac conditions. The
quantum version of (9.30) reads
[
ˆQB , ˆQ
]
B =0. (9.32)
For this to be fulfilled, the total central charge of the X µ -fields and the Faddeev–
Popov ghosts must vanish,
+
c tot = c + c ghost = D − 26 = 0 , (9.33)
since it turns out that the ghosts have central charge −26. The string must therefore
move in 26 dimensions. In the case of the superstring (see Section 9.2.4),
the corresponding condition leads to D = 10. The condition (9.32) thus carries
information about quantum anomalies (here, the Weyl anomaly) and their
possible cancellation by ghosts. 1 One can prove the ‘no-ghost theorem’ (see e.g.
Polchinski 1998a): the Hilbert space arising from BRST quantization has a positive
inner product and is isomorphic to the Hilbert space of transverse string
excitations.
1 One can also discuss non-critical strings living in D ≠ 26 dimensions. They have a Weyl
anomaly, which means that different gauge choices are inequivalent.