String Theory Demystified

CHAPTER 6 BRST Quantization 123
To write down the total Virasoro operator for the “real” fi elds + ghost fi elds, we
form a sum of the respective operators. That is
tot
gh
L L L a
m
= m + m − δ m,0
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:
tot tot
tot
⎡⎣ Lm, Ln⎤⎦ = ( m− n) Lm+ n+
A( m)
m+ n,
δ 0
Notice that the presence of the term Am ( ) 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
D 2 1
3
Am ( ) = mm ( − 1)
+ ( m− 13m ) + 2am
12
6
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
1 gh 3 2
j= cT + : cT : + ∂ c
2 2
The BRST charge is given by the mode expansion:
1
Q= ∑ cnL−n + ∑ ( m−n): cmcnb−m−n : −c
2
n
mn ,
Using tedious algebra one can show that
2 1 tot tot
tot 1
Q = ∑ ( ⎡LmLnm n Lm n c mc n
2
⎣ , ⎤⎦ −( − ) + ) − − ≈ ( D − 26)
12
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
0