8.3 BRST Cohomology: Two Flat Directions
191
The advantage of the subspace b 0 = 0 is to precisely pick the representative
of H abs , which lies in ker L 0 . In particular, the operator L 0 is simple and has a
direct physical interpretation as the worldsheet Hamiltonian. This condition is also
meaningful in string theory because these states are also mass eigenstates, which
have a nice spacetime interpretation, and it will later be interpreted in SFT as
fixing the Siegel gauge. Moreover, it is implied by the choice of in (8.39) as
the contracting homotopy operator, which is particularly convenient to work with
to derive the cohomology. However, there are other possible choices, which are
interpreted as different gauge fixings.
After having built this cohomology, we can look for the full cohomology by
relaxing the condition b 0 = 0. In view of the structure of the ghost Hilbert
space (7.169), one can expect that H abs (Q B ) = H rel (Q B ) ⊕ c 0 H rel (Q B ), which
is indeed the correct answer. But, we will see (building on Sect. 3.2.2) that, in fact,
it is this cohomology that contains the physical states in string theory, instead of the
absolute cohomology.
As a summary, we are looking for Q B -closed non-exact states annihilated by b 0
and L 0
Q B |ψ = 0,
L 0 |ψ = 0,
b 0 |ψ = 0.
(8.50)
8.3.2 Relative Cohomology
In (8.14a), the BRST operator was decomposed as
Q B = c 0 L 0 − b 0 M +
Q B ,
Q
2
B = L 0 M.
(8.51)
This shows that, on the subspace L 0 = b 0 = 0,
Q B is nilpotent and equivalent to
Q B
|ψ ∈ H 0 ∩ ker L 0 ⇒ Q B |ψ =
Q B |ψ ,
Q
2
B |ψ = 0.
(8.52)
Hence, this implies that
Q B is a proper BRST operator, and the relative cohomology
of Q B is isomorphic to the cohomology of
Q B
H 0 (Q B ) = H 0 (
Q B ).
(8.53)
Next, we introduce light-cone coordinates in the target spacetime. While it
does not allow to write Lorentz covariant expressions, it is helpful mathematically
because it introduces a grading of the Hilbert space, for which powerful theorems
exist (even if we will need only basic facts for our purpose).
191
The advantage of the subspace b 0 = 0 is to precisely pick the representative
of H abs , which lies in ker L 0 . In particular, the operator L 0 is simple and has a
direct physical interpretation as the worldsheet Hamiltonian. This condition is also
meaningful in string theory because these states are also mass eigenstates, which
have a nice spacetime interpretation, and it will later be interpreted in SFT as
fixing the Siegel gauge. Moreover, it is implied by the choice of in (8.39) as
the contracting homotopy operator, which is particularly convenient to work with
to derive the cohomology. However, there are other possible choices, which are
interpreted as different gauge fixings.
After having built this cohomology, we can look for the full cohomology by
relaxing the condition b 0 = 0. In view of the structure of the ghost Hilbert
space (7.169), one can expect that H abs (Q B ) = H rel (Q B ) ⊕ c 0 H rel (Q B ), which
is indeed the correct answer. But, we will see (building on Sect. 3.2.2) that, in fact,
it is this cohomology that contains the physical states in string theory, instead of the
absolute cohomology.
As a summary, we are looking for Q B -closed non-exact states annihilated by b 0
and L 0
Q B |ψ = 0,
L 0 |ψ = 0,
b 0 |ψ = 0.
(8.50)
8.3.2 Relative Cohomology
In (8.14a), the BRST operator was decomposed as
Q B = c 0 L 0 − b 0 M +
Q B ,
Q
2
B = L 0 M.
(8.51)
This shows that, on the subspace L 0 = b 0 = 0,
Q B is nilpotent and equivalent to
Q B
|ψ ∈ H 0 ∩ ker L 0 ⇒ Q B |ψ =
Q B |ψ ,
Q
2
B |ψ = 0.
(8.52)
Hence, this implies that
Q B is a proper BRST operator, and the relative cohomology
of Q B is isomorphic to the cohomology of
Q B
H 0 (Q B ) = H 0 (
Q B ).
(8.53)
Next, we introduce light-cone coordinates in the target spacetime. While it
does not allow to write Lorentz covariant expressions, it is helpful mathematically
because it introduces a grading of the Hilbert space, for which powerful theorems
exist (even if we will need only basic facts for our purpose).