Lectures on String Theory

– 73 –
Introduce the ghost number operator U:
U = ∑ i
c i b i .
The eigenvalues of this operator are integers ranging from 0 up to dimK.
The BRST operator is defined as
Q = c i K i − 1f k 2 ijc i c j b k . (4.18)
First we compute a commutator
[Q, U] = [c i K i − 1f k 2 ijc i c j b k , c m b m ]
= −c m {c i , b m }K i − c m 1f k 2 ij{c i c j , b m }b k − 1f k 2 ijc i c j {b k , c m }b m
= −c i K i + fijc k i c j b k − 1f k 2 ijc i c j b k = −Q .
Thus, the BRST operator has the following commutator with U:
[U, Q] = Q
and as the result it increases the ghost number by one:
UQ|χ〉 = (QU + Q)|χ〉 = (N gh + 1)Q|χ〉 .
Second, compute the anticommutator
{Q, Q} = {c i K i − 1 2 f k ijc i c j b k , c s K s − 1 2 f p mnc m c n b p }
= c i c s K i K s − c s c i K s K i − 1 2 f k ijc i c j {b k , c s }K s − 1 2 f p mnc m c n {c i , b p }K i
+ 1 4 f k ijf p mn{c i c j b k , c m c n b p } .
It is easy to find
f k ijf p mn{c i c j b k , c m c n b p } = 4f k ijf p km ci c j c m b p .
Therefore, the expression we are interested in reduces to
{Q, Q} = c i c j [K i , K j ] − 1 2 f k ijc i c j K k − 1 2 f k ijc i c j K k + f k ijf p km ci c j c m b p .
Due to the algebra relation [K i , K j ] = f k ijK k the first three terms in the last expression
cancel out and we are left with
{Q, Q} = f k ijf p km ci c j c m b p .
Due to the anti-commuting property of the ghosts the last expression can be rewritten
as
{Q, Q} = 1 3 (f k ijf p km + f k mif p kj + f k jmf p ki )ci c j c m b p .