String Theory and M-Theory

3.2 BRST quantization 79
in differential geometry d2 = 0. 9 In that case one considers various types
of differential forms ω. Ones that satisfy dω = 0 are called closed, and
ones that can be written in the form ω = dρ are called exact. Nilpotency
of d implies that every exact form is closed. If there are closed forms that
are not exact, this encodes topological information about the manifold M
on which the differential forms are defined. One defines equivalence classes
of closed forms by declaring two closed forms to be equivalent if and only
if their difference is exact. These equivalence classes then define elements
of the cohomology of M. More specifically, an equivalence class of closed
n-forms is an element of the nth cohomology group H n (M).
The idea is now clear. Physical string states are identified as BRST cohomology
classes. Thus, in the enlarged Fock space that includes the b and
c oscillators in addition to the α oscillators, one requires that a physical
on-shell string state is annihilated by the operator QB, that is, it is BRST
closed. Furthermore, if the difference of two BRST-closed states is BRST
exact, so that it is given as QB applied to some state, then the two BRSTclosed
states represent the same physical state. In the case of closed strings,
this applies to the holomorphic and antiholomorphic sectors separately.
Because of the ghost zero modes, b0 and c0, the ground state is doubly
degenerate. Denoting the two states by | ↑〉 and | ↓〉, c0| ↓〉 = | ↑〉 and
b0| ↑〉 = | ↓〉. Also, c0| ↑〉 = b0| ↓〉 = 0. The ghost number assigned to one
of these two states is a matter of convention. The other is then determined.
The most symmetrical choice is to assign the values ±1/2, which is what we
do. This resolves the ambiguity of a constant in the ghost-number operator
U = 1
2πi
: c(z)b(z) : dz = 1
2 (c0b0 − b0c0) +
∞
(c−nbn − b−ncn). (3.89)
Which one of the two degenerate ground states corresponds to the physical
ground state (the tachyon)? The fields b and c are not on a symmetrical footing,
so there is a definite answer, namely | ↓〉, as will become clear shortly.
The definition of physical states can now be made precise: they correspond
to BRST cohomology classes with ghost number equal to −1/2. In the case
of open strings, this is the whole story. In the case of closed strings, this
construction has to be carried out for the holomorphic (right-moving) and
antiholomorphic (left-moving) sectors separately. The two sectors are then
tensored with one another in the usual manner.
To make contact with the results of Chapter 2, let us construct a unique
9 This is the proper analogy for open strings. In the case of closed strings, the better analogy
relates QB and eQB to the holomorphic and antiholomorphic differential operators ∂ and ¯ ∂ of
complex differential geometry.
n=1