15.2 Gauge Fixed Theory
315
the interactions (15.22) are compatible with the definition of the kinetic term and
the restriction of the string field.
15.2.3 Action
The interacting gauge fixed action is built from the kinetic term V 0,2 (15.17) and
from the interactions V g,n (15.22) with χ g,n < 0. However, this is not sufficient: we
have seen in Sect. 14.3 that it makes sense to consider the vertices with χ g,n ≥ 0.
First, we should consider the 1-loop cosmological constant V 1,0 . Then, we can also
add the classical source V 0,1 and the tree-level cosmological constant V 0,0 . With all
the terms together, the action reads
S =
g,n≥0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n )
:=
1
2
| c
−
0 c
+
0 L
+
0 δ L
−
0 ,0 | +
g,n≥0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n ),
(15.24)
where V n was defined in (14.58). A prime on the sum indicates that the term g =
0, n = 2 is removed, such that one can single out the kinetic term. We will often
drop the delta function imposing L
−
0 = 0 because the field is taken to satisfy this
constraint.
Rewriting the vertices in terms of the products g,n defined in (14.65)
V g,n ((
n ) := =| c
−
0
g,n−1((
n−1 )
(15.25)
leads to the alternative form
S =
g,n≥0
¯
h
g g
2g−2+n
s
n!
| c
−
0
g,n−1((
n−1 )
.
(15.26)
The definition (14.56) leads to the following explicit expression for 0,1 :
0,1 (( cl ) = c
+
0 L
+
0 | cl .
(15.27)
In most cases, the terms g = 0, n = 0, 1 vanish such that the action reads
S =
g,n≥0
χ g,n ≤0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n ).
(15.28)
315
the interactions (15.22) are compatible with the definition of the kinetic term and
the restriction of the string field.
15.2.3 Action
The interacting gauge fixed action is built from the kinetic term V 0,2 (15.17) and
from the interactions V g,n (15.22) with χ g,n < 0. However, this is not sufficient: we
have seen in Sect. 14.3 that it makes sense to consider the vertices with χ g,n ≥ 0.
First, we should consider the 1-loop cosmological constant V 1,0 . Then, we can also
add the classical source V 0,1 and the tree-level cosmological constant V 0,0 . With all
the terms together, the action reads
S =
g,n≥0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n )
:=
1
2
| c
−
0 c
+
0 L
+
0 δ L
−
0 ,0 | +
g,n≥0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n ),
(15.24)
where V n was defined in (14.58). A prime on the sum indicates that the term g =
0, n = 2 is removed, such that one can single out the kinetic term. We will often
drop the delta function imposing L
−
0 = 0 because the field is taken to satisfy this
constraint.
Rewriting the vertices in terms of the products g,n defined in (14.65)
V g,n ((
n ) := =| c
−
0
g,n−1((
n−1 )
(15.25)
leads to the alternative form
S =
g,n≥0
¯
h
g g
2g−2+n
s
n!
| c
−
0
g,n−1((
n−1 )
.
(15.26)
The definition (14.56) leads to the following explicit expression for 0,1 :
0,1 (( cl ) = c
+
0 L
+
0 | cl .
(15.27)
In most cases, the terms g = 0, n = 0, 1 vanish such that the action reads
S =
g,n≥0
χ g,n ≤0
¯
h
g g
2g−2+n
s
n!
V g,n ((
n ).
(15.28)