16.6 Idea of the Proof
335
Next, the BRST identity (13.46) and the equation of motion F 1 (( ) = 0 allow to
rewrite the first term as
B n+2 (( 0 , ,
n , Q B A) = ∂B n+2 (( 0 , ,
n , A) + n B n+2 (( 0 , ,
n−1 , Q B
, A)
(16.30a)
= ∂B n+2 (( 0 , ,
n , A)
−
m
n
m!
B n+2 (( 0 , ,
n−1 , , m ((
m ), A).
(16.30b)
In the second term, the sum over n is shifted. Combining everything together gives
=
n
1
n!
∂B n+2 (( 0 , ,
n , A) −
m,n
1
m!n!
B n+3 (( 0 , ,
n , , m ((
m ), 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 ).
(16.31)
Solving for = 0 requires that each term with a different power of vanishes
independently:
∂B n+2 (( 0 , ,
n , A) = − V
n+2 (A, , 0 , ,
n ) + V n+2 (A, , 0 , ,
n )
+
m 1 ,m 2
m 1 +m 2 =n
n!
m 1 !m 2 !
B m 1 +3 (( 0 , ,
m 1 , , m 2 ((
m 2 ), A)
−
m 1 ,m 2
m 1 +m 2 =n
n!
m 1 !m 2 !
B m 1 +2
0 , ,
m 1 , , m 2 +1 (A, ,
m 2 )
.
(16.32)
In order to proceed, one needs to perform a genus expansion of the various
spaces: this allows to solve recursively for all B g,n starting from B 0,3 . One can
then build |δδ recursively, which provides the field redefinition. Indeed, the RHS
of this equation contains only B g ,n for g < g or n < n, and the equation for
B 0,3 contains no B g,n in the RHS. It should be noted that the field redefinition
is not unique, but there is the freedom of performing (infinite-dimensional) gauge
transformations. Finding an obstruction to solve these equations means that the field
redefinition does not exist, and thus that the theory is not background independent.