Quantum Gravity

286 STRING THEORYThe full action is invariant under BRST transformations, which were alreadybriefly mentioned in Section 2.2.3. This is an important concept, since it encodesthe 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) andare generated by the ‘BRST charge’ Q B .Letφ a be a general set of first-classconstraints (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 conjugatemomenta (‘anti-ghosts’) obeying [η a ,P b ] + = δb a . We have assumed here that thephysical 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 thedemand 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 physicalstates be annihilated by all constraints. Equation (9.31) can be fulfilled forthe quantized bosonic string, which is not the case for the Dirac conditions. Thequantum 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 thereforemove 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 carriesinformation about quantum anomalies (here, the Weyl anomaly) and theirpossible 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 positiveinner product and is isomorphic to the Hilbert space of transverse stringexcitations.1 One can also discuss non-critical strings living in D ≠ 26 dimensions. They have a Weylanomaly, which means that different gauge choices are inequivalent.