332
16 Background Independence
The terms with p = q vanish because the contour integrals are performed around
circles of unit radius centred at the origin. Moreover, the terms p = q are identical
and cancel with each other, showing that δL
−
0 = 0 when acting on states satisfying
L
−
0 = 0.
The SFT action S 2 [ 1 ] in the new background reads
S 2 [ 1 ] = S 1 [ 1 ] + δS 1 [ 1 ],
(16.12)
where the change δS 1 in the action is induced by the changes in the string vertices:
δS 1 [ 1 ] =
1
g 2
s
⎛
⎝ 1
2
1 | c
−
0 δQ B | 1 +
n≥0
1
n!
δV n ((
n
1 )
⎞
⎠ .
(16.13)
The equation of motion is
F 2 (( 1 ) = F 1 (( 1 ) + λ δF 1 (( 1 ) = 0,
(16.14)
where F 1 is given in (16.3) and
λ δF 1 (( 1 ) = δQ B | 1 +
n
1
n!
δδ n ((
n
1 ).
(16.15)
16.4 Expansion of the Action
Given a (1, 1) primary ϕ, a BRST invariant operator is c ¯
cϕ. Hence, the field
| 1 = λ | 0 ,
| 0 = c 1 ¯
c 1 (0) |ϕ
(16.16)
is a classical solution to first order in λ since the interactions on the sphere are at
least cubic.
Separating the string field as the contribution from the (fixed) background and a
fluctuation
| 1 = λ | 0 + |
,
(16.17)
the action expanded to first order in λ reads
S 1 [ 1 ] = S 1 [ 0 ] + S
[
],
(16.18)