3.2 BRST Quantization
85
3.2.2 BRST Cohomology and Physical States
Physical state |ψ is the element of the absolute cohomology of the BRST operator
|ψ ∈ H(Q B ) :=
ker Q B
Im Q B
,
(3.60)
or, more explicitly, closed but non-exact states
Q B |ψ = 0,
|χ : |ψ = Q B |χ .
(3.61)
The adjective “absolute” is used to distinguish it from two other cohomologies
(relative and semi-relative) defined below. Two states of the cohomology differing
by an exact state represent identical physical states
|ψ ∼ |ψ + Q B | .
(3.62)
This equivalence relation, translated in terms of spacetime fields, corresponds
to spacetime gauge transformations. In particular, it contains the (linearized)
reparametrization invariance of the spacetime metric in the closed string sector, and
for the open string sector, it contains Yang–Mills symmetries. We will find that it
corresponds to the gauge invariance of free string field theory (Chap. 10).
However, physical states satisfy two additional constraints (remember that b ab is
traceless symmetric)
dσ b ab |ψ = 0.
(3.63)
These conditions are central to string (field) theory, so they will appear regularly
in this book. For this reason, it is useful to provide first some general motivations
and to refine the analysis later since the CFT language will be more appropriate.
Moreover, these two conditions will naturally emerge in string field theory.
In order to introduce some additional terminology, let us define the following
quantities: 5
b
+
:=
dσ b 00 ,
b
−
:=
dσ b 01 .
(3.64)
5 The objects b ± are zero-modes of the b ghost fields. They correspond (up to a possible irrelevant
factor) to the modes b
±
0 in the CFT formulation of the ghost system (7.132)
85
3.2.2 BRST Cohomology and Physical States
Physical state |ψ is the element of the absolute cohomology of the BRST operator
|ψ ∈ H(Q B ) :=
ker Q B
Im Q B
,
(3.60)
or, more explicitly, closed but non-exact states
Q B |ψ = 0,
|χ : |ψ = Q B |χ .
(3.61)
The adjective “absolute” is used to distinguish it from two other cohomologies
(relative and semi-relative) defined below. Two states of the cohomology differing
by an exact state represent identical physical states
|ψ ∼ |ψ + Q B | .
(3.62)
This equivalence relation, translated in terms of spacetime fields, corresponds
to spacetime gauge transformations. In particular, it contains the (linearized)
reparametrization invariance of the spacetime metric in the closed string sector, and
for the open string sector, it contains Yang–Mills symmetries. We will find that it
corresponds to the gauge invariance of free string field theory (Chap. 10).
However, physical states satisfy two additional constraints (remember that b ab is
traceless symmetric)
dσ b ab |ψ = 0.
(3.63)
These conditions are central to string (field) theory, so they will appear regularly
in this book. For this reason, it is useful to provide first some general motivations
and to refine the analysis later since the CFT language will be more appropriate.
Moreover, these two conditions will naturally emerge in string field theory.
In order to introduce some additional terminology, let us define the following
quantities: 5
b
+
:=
dσ b 00 ,
b
−
:=
dσ b 01 .
(3.64)
5 The objects b ± are zero-modes of the b ghost fields. They correspond (up to a possible irrelevant
factor) to the modes b
±
0 in the CFT formulation of the ghost system (7.132)