17.3 Superstring Field Theory
355
17.3.2 Constraint Approach
Two new PCO operators must be introduced
X = G 0 δ(β 0 ) + b 0 δ
(β 0 ),
Y = −c 0 δ
(γ 0 ).
(17.63)
The first operator commutes with the BRST operator
[Q B , X] = 0.
(17.64)
The product of these operators is a projector
XY X = X.
(17.65)
Then, the R string field is constrained to satisfy
XY | −1/2 = | −1/2 .
(17.66)
A state satisfying this condition is said to be in the restricted Hilbert space. It can be
shown that it reproduces the cohomology of Q B on-shell.
Remark 17.6 Since G 0 contains derivatives, the restriction is not purely algebraic
as in the bosonic case. It prevents the degeneration due to γ n
0 .
Remark 17.7 (Comparison with Level-Matching) The conditions b
−
0 = L
−
0 = 0
can be rephrased as the statement that the string field is invariant under the action
of the projector Bc
−
0
Bc
−
0 | = | ,
(17.67)
where
B = b
−
0
2π
0
dθ
2π
e
iθL
−
0 = δ(b
−
0 )δ(L
−
0 ).
(17.68)
The kinetic term (after unfixing the gauge) reads
S 0,2 = −
1
2
−1 | c
−
0 Q B | −1 −
1
2
−1/2 | c
−
0 Y Q B | −1/2 .
(17.69)
The action is invariant under the gauge transformation
δ | = Q B | ,
(17.70)