Lectures on String Theory

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The expression in the bracket vanishes because this is the Jacobi identity written for
structure constants of the Lie algebra
f k ijf p km + f k mif p kj + f k jmf p ki = 0 .
Thus, we found that the BRST operator is nilpotent, i.e. it its square is zero
Q 2 = 1 {Q, Q} = 0 .
2
This is the fundamental property of the BRST operator. We also assume that Q is
hermitian, i.e. Q † = Q.
Let H k will be a Hilbert space of states with fixed ghost number U = k. An element
|χ〉 ∈ H k is called BRST-invariant if
Qχ = 0 (4.19)
Clearly, any state of the form Q|λ〉, where |λ〉 is any state with the ghost number 14
k − 1, is BRST-invariant because
The state Q|λ〉 has zero norm because
Q(Q|λ〉) = Q 2 |λ〉 = 0 .
〈λ|Q † Q|λ〉 = 〈λ|Q 2 |λ〉 = 0 .
The most important BRST-invariant states are those which can not be written in
the form |χ〉 = Q|λ〉. We will regard two solutions of equation (4.19) equivalent if
for some λ.
|χ ′ 〉 − |χ〉 = Q|λ〉
In fact, we recognize that the BRST-operator mimics all the properties of the de-Rahm operator d which acts on the space of external
(differential) forms on a manifold M. Indeed, it has a property that d 2 = 0. A differential form ω is called closed if dω = 0 and it is
called exact if there is another form θ such that ω = dθ. The factor-space of all closed forms over all exact forms of a given degree n
H n closed forms
(M) =
exact forms
is called n-th cohomology group of the manifold M. In our present case the operator Q takes values in the Lie algebra and it defines
cohomologies with values in an given representation of the Lie algebra.
Furthermore, the states with zero ghost charge are of special importance. Such
a state must be annihilated by all b k . For such states the BRST operator reduces to
Q|χ〉 = c i K i |χ〉 = 0 .
14 As we found the BRST-operator increases the ghost number by one.