334
16 Background Independence
To simplify the computations, it is simpler to consider the inner product of this
quantity with an arbitrary state A (assumed to be even):
:= λ A| c
−
0 Q B |δδ
+ λ
n
1
n!
V n+2 (A, δδ
, ,
n ) ++A| c
−
0 δQ B |
+
n
1
n!
δV n+1 (A, ,
n ) − λ
n
1
n!
V n+2 (A, , 0 , ,
n ).
(16.25)
The goal is to prove the existence of δδ such that = 0 up to the zero-order
equation of motion F 1 (( ) = 0.
16.6 Idea of the Proof
In this section, we give an idea of how the proof ends, referring to [10] for the
details.
The first step is to introduce new vertices V
0,3 and V
n parametrizing the variations
of the string vertices:
A| c
−
0 δQ B |B = λ V
0,3 (( 0 , B, A),
δV n ((
n ) = λ V
n+1 (( 0 , ,
n ),
(16.26)
where the notation (13.19) has been used. Each subspace V
g,n is defined such
that the LHS is recovered upon integrating the appropriate ω g,n over this section
segment. Next, the field redefinition δδ is parametrized as
A| c
−
0 |δδ
=
n
1
n!
B n+2 (( 0 , ,
n , A).
(16.27)
The objective is to prove the existence (and if possible the form) of the subspaces
B n+2 . Both the vertices V
n and B n admit a genus expansion:
V
n =
g≥0
V
g,n ,
B n =
g≥0
B g,n .
(16.28)
Plugging the new expressions in (16.25) gives
= −
n
1
n!
B n+2 (( 0 , ,
n , Q B A) +
m,n
1
m!n!
B n+2 (( 0 , ,
m , , n+1 (A, ,
n ))
+
n
1
n!
V
n+2 (A, , 0 , ,
n ) −
n
1
n!
V n+2 (A, , 0 , ,
n ).
(16.29)
16 Background Independence
To simplify the computations, it is simpler to consider the inner product of this
quantity with an arbitrary state A (assumed to be even):
:= λ A| c
−
0 Q B |δδ
+ λ
n
1
n!
V n+2 (A, δδ
, ,
n ) ++A| c
−
0 δQ B |
+
n
1
n!
δV n+1 (A, ,
n ) − λ
n
1
n!
V n+2 (A, , 0 , ,
n ).
(16.25)
The goal is to prove the existence of δδ such that = 0 up to the zero-order
equation of motion F 1 (( ) = 0.
16.6 Idea of the Proof
In this section, we give an idea of how the proof ends, referring to [10] for the
details.
The first step is to introduce new vertices V
0,3 and V
n parametrizing the variations
of the string vertices:
A| c
−
0 δQ B |B = λ V
0,3 (( 0 , B, A),
δV n ((
n ) = λ V
n+1 (( 0 , ,
n ),
(16.26)
where the notation (13.19) has been used. Each subspace V
g,n is defined such
that the LHS is recovered upon integrating the appropriate ω g,n over this section
segment. Next, the field redefinition δδ is parametrized as
A| c
−
0 |δδ
=
n
1
n!
B n+2 (( 0 , ,
n , A).
(16.27)
The objective is to prove the existence (and if possible the form) of the subspaces
B n+2 . Both the vertices V
n and B n admit a genus expansion:
V
n =
g≥0
V
g,n ,
B n =
g≥0
B g,n .
(16.28)
Plugging the new expressions in (16.25) gives
= −
n
1
n!
B n+2 (( 0 , ,
n , Q B A) +
m,n
1
m!n!
B n+2 (( 0 , ,
m , , n+1 (A, ,
n ))
+
n
1
n!
V
n+2 (A, , 0 , ,
n ) −
n
1
n!
V n+2 (A, , 0 , ,
n ).
(16.29)