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BRST Transformations
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Next we look at BRST quantization by considering a set of BRST transformations which are derived using a path integral approach. This is a bit beyond the level of the discussion used in the book, so we simply state the results. Working in lightcone coordinates, we de ne a ghost eld c and an antighost eld b where c has
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components c + , c , and b has components b++ and b . We also introduce an energygh momentum tensor for the ghost elds T with components given by
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gh T++ = i (2b++ + c + + + b++ c + ) gh T = i (2b c + b c )
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The BRST transformations, using a small anticommuting operator are
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X = i ( c + + + c ) X c = i ( c + + + c )c
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gh b = i (T + T )
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The action for the ghost elds is Sgh = d 2 (b++ c + + b + c ) From which the following equations of motion follow: b++ = + b = c + = + c = 0 We can write down modal expansions of the ghost elds. These are given by c + = cn e in ( + )
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c = cn e in ( )
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b++ = bn e
in ( + )
b = bn e in ( )
The modes satisfy the following anticommutation relation: {bm , cn } = m+n ,0 with {bm , bn } = {cm , cn } = 0. Virasoro operators are de ned for the ghost elds using the modes. Using normal-ordered expansions, these are Lgh = (m n) : bm+n c n : m
Lgh = (m n) : bm+n c n : m
CHAPTER 6 BRST Quantization
To write down the total Virasoro operator for the real elds + ghost elds, we form a sum of the respective operators. That is Ltot = Lm + Lgh a m ,0 m m where the last term on the right is the normal-ordering constant for m = 0. It can be shown that the commutation relation for the total Virasoro operator is of the form: Ltot , Ltot = (m n) Ltot+n + A(m) m+n ,0 m m n Notice that the presence of the term A(m) on the right keeps us from obtaining a relation that preserves the classical Virasoro algebra. As such, this term is called an anomaly. The anomaly is determined in terms of two unknown constants which you might guess by now are D and a. It has the form A(m) = D 1 m(m 2 1) + (m 13m 3 ) + 2am 6 12
To make the anomaly vanish, we take D = 26, a = 1, which is consistent with the other results obtained so far in the book for bosonic string theory. The BRST current is given by j = cT + 1 3 : cT gh : + 2c 2 2
The BRST charge is given by the mode expansion: Q = cn L n +
1 (m n) : cmcn b m n : c0 2 m ,n
Using tedious algebra one can show that Q2 = 1 1 Ltot , Ltot (m n) Ltot+n c mc n 12 ( D 26) m m n 2
Hence the requirement that Q 2 = 0 forces us to take D = 26. Going back to the original BRST approach outlined in Eqs. (6.1) to (6.5), using the classical algebra for the Virasoro operators: [ Lm , Ln ] = ( m n ) Lm + n
We can identify the structure constants as
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k fmn = (m n) m+n ,k
To see how the physical spectrum can be constructed in string theory, we consider the open string case. The states are built up from the ghost vacuum state. Let s call the ghost vacuum state . This state is annihilated by all positive ghost modes. Let n > 0, then bn = cn = 0 The zero modes of the ghost elds are a special case. They can be used to build the physical states of the theory. Using the anticommutation relations [Eq. (6.3)], the zero modes satisfy {b0 , c0 } = 1
2 2 Using Eq. (6.3) it should also be obvious that b0 = c0 = 0. We also require that b0 = 0 for physical states . Now we can construct a two-state system from the zero modes of the ghost states. The basis states are denoted by , . The ghost states act as
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