314
15 Closed String Field Theory
possibilities. The above prescription also yields directly the insertion of the
momentum necessary if the interaction contains derivatives.
In Sect. 14.2, the Feynman rule for a g-loop n-point fundamental vertex of states
(V 1 , . . . , V n ) was found to be given by (14.33):
V g,n ( 1 , . . . , n ) =
R g,n
ω
g,n
M g,n
( 1 , . . . , n
n
) = 1
2
,
(15.21)
where R g,n is a section over the fundamental region V g,n ⊂ M g,n (12.42b) that
cannot be covered from the plumbing fixture of lower-dimensional surfaces.
From Example 15.1, it should be clear that the g-loop n-point contribution to the
action can be obtained simply by replacing every state with a string field in V g,n :
S g,n = ¯
h
g g
2g−2+n
s
n!
V g,n ((
n ),
(15.22)
where n := ⊗n . The power of the coupling constant has been reinstated: it can
be motivated by the fact that it should have the same power as the corresponding
amplitude (Sect. 3.1.1). Note that the interactions are defined only when the power
of g s is positive: χ g,n = 2 − 2g − n < 0. We have also written explicitly the power
of ¯
h, which counts the number of loops.
Before closing this section, we need to comment on the effect of the constraints (15.14) on the interactions. Building a Feynman graph by gluing two mand n-point interactions with a propagator, one finds that the states proportional to
φ x,r for x =↓↓ do not propagate inside internal legs
V g,m (V 1 , . . . , V m−1 , φ r )φ
c
r | b
+
0 b
−
0
1
L
+
0
|φ
c
s V g ,n (W 1 , . . . , W n−1 , φ s )
= V g,m (V 1 , . . . , V m−1 , φ ↓↓,r )φ
c
↓↓,r | b
+
0 b
−
0
1
L
+
0
|φ
c
↓↓,s
× V g ,n (W 1 , . . . , W n−1 , φ ↓↓,s ).
(15.23)
Thus, they do not contribute to the final result even if the interactions contain them.
While the conditions L
−
0 = b
−
0 = 0 were found to be necessary for defining off-shell
amplitudes, the condition b
+
0 = 0 does not arise from any consistency requirement.
But, it is also consistent with the interactions, since only fundamental vertices have
a chance to give a non-vanishing result for states that do not satisfy (15.14). Hence,
15 Closed String Field Theory
possibilities. The above prescription also yields directly the insertion of the
momentum necessary if the interaction contains derivatives.
In Sect. 14.2, the Feynman rule for a g-loop n-point fundamental vertex of states
(V 1 , . . . , V n ) was found to be given by (14.33):
V g,n ( 1 , . . . , n ) =
R g,n
ω
g,n
M g,n
( 1 , . . . , n
n
) = 1
2
,
(15.21)
where R g,n is a section over the fundamental region V g,n ⊂ M g,n (12.42b) that
cannot be covered from the plumbing fixture of lower-dimensional surfaces.
From Example 15.1, it should be clear that the g-loop n-point contribution to the
action can be obtained simply by replacing every state with a string field in V g,n :
S g,n = ¯
h
g g
2g−2+n
s
n!
V g,n ((
n ),
(15.22)
where n := ⊗n . The power of the coupling constant has been reinstated: it can
be motivated by the fact that it should have the same power as the corresponding
amplitude (Sect. 3.1.1). Note that the interactions are defined only when the power
of g s is positive: χ g,n = 2 − 2g − n < 0. We have also written explicitly the power
of ¯
h, which counts the number of loops.
Before closing this section, we need to comment on the effect of the constraints (15.14) on the interactions. Building a Feynman graph by gluing two mand n-point interactions with a propagator, one finds that the states proportional to
φ x,r for x =↓↓ do not propagate inside internal legs
V g,m (V 1 , . . . , V m−1 , φ r )φ
c
r | b
+
0 b
−
0
1
L
+
0
|φ
c
s V g ,n (W 1 , . . . , W n−1 , φ s )
= V g,m (V 1 , . . . , V m−1 , φ ↓↓,r )φ
c
↓↓,r | b
+
0 b
−
0
1
L
+
0
|φ
c
↓↓,s
× V g ,n (W 1 , . . . , W n−1 , φ ↓↓,s ).
(15.23)
Thus, they do not contribute to the final result even if the interactions contain them.
While the conditions L
−
0 = b
−
0 = 0 were found to be necessary for defining off-shell
amplitudes, the condition b
+
0 = 0 does not arise from any consistency requirement.
But, it is also consistent with the interactions, since only fundamental vertices have
a chance to give a non-vanishing result for states that do not satisfy (15.14). Hence,