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String Theory Demysti ed
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This implies that for a state with zero ghost number Ki = 0 (6.9)
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So, we can say that a state which is BRST invariant with zero ghost number is also invariant under the symmetry described by the generators K i. Furthermore, if a state has ghost number zero, this tells us that the state is not a ghost state, hence we avoid negative probabilities.
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We will take a look at the BRST formalism in string theory by brie y considering two approaches. The derivation of this approach is based on the use of path integrals, which we are purposely avoiding due to the level of this text. So some results will simply be stated, the reader who is interested in their derivation is encouraged to check the references at the back of the book. The application of BRST quantization to string theory can be done easily using conformal eld theory. The advantage of this approach is that the critical dimension D = 26 arises in a straightforward manner. We work in the conformal gauge where we take h = . In this case the energy-momentum tensor has a holomorphic component Tzz ( z ) and an antiholomorphic component Tzz ( z ) where Tzz ( z ) was given in Eq. (5.33) as Tzz ( z ) =
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In Example 5.5 we worked out the OPE of Tzz ( z )Tww ( w) and found Tzz ( z )Tww ( w) = 2T ( w) T ( w) D/2 ww 2 w ww 4 ( z w) ( z w) z w
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Where for simplicity of notation we have omitted the multiplying 2 factor. The ghost s elds are introduced as functions of a complex variable z as follows. We de ne b( z )c( w) = 1 z w b ( z )c ( w) = 1 z w
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Next we write down an energy-momentum tensor Tgh ( z ) for the ghost elds. This is given by Tgh ( z ) = 2b( z ) z c( z ) z b( z )c( z )
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CHAPTER 6 BRST Quantization
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The conformal dimension [Eq. (6.4)] of the ghost elds follows from this de nition. Using the ghost energy-momentum tensor together with the energymomentum tensor for the string we can arrive at the BRST current: 1 j ( z ) = c( z ) Tzz ( z ) + Tgh ( z ) = c( z )Tzz ( z ) + c( z ) z c( z )b( z ) 2 The BRST charge is given by Q= dz j( z ) 2 i
Now, the central charge (i.e., the critical space-time dimension) comes from the leading term in the OPE of the energy-momentum tensor, which is D/2 ( z w) 4 The presence of this extra term is called the conformal anomaly since it prevents the algebra from closing. So we would like to get rid of it. This is done by considering a total energy-momentum tensor, which is the sum of the string energy-momentum tensor and the ghost energy-momentum tensor, that is, T = Tzz ( z ) + Tgh ( z ). It can be shown that the OPE of the ghost energy-momentum tensor is Tgh ( z )Tgh ( w) = 2Tgh ( w) w Tgh ( w) 13 4 z w ( z w) ( z w) 2
Taking the leading term in this expression to be of the form ( D/2)/( z w)4, we see that the ghost elds contribute a central charge of 26 which precisely cancels the conformal anomaly that arises from the matter energy-momentum tensor. This result actually follows from the nilpotency requirement (i.e., Q 2 = 0 ) of the BRST charge.