14.1 Amplitude Factorization
287
Computation: Equation (14.3)
Starting from (14.1), vary q → q + δq while keeping w
(2)
n 2 fixed:
w
(1)
n 1
w
(2)
n 2
= q + δq
w
(1)
n 1
=
w
(1)
n 1
q
(q + δq) = w
(1)
n 1
+
q
w
(1)
n 1
δq.
The second line follows by replacing w
(2)
n 2 using (14.1).
The M g,n -form for the moduli described by the plumbing fixture can be expressed
as
ω M g,n
V
(1)
1 , . . . , V
(1)
M g 1 ,n 1
, ∂ q , ∂ ¯
q , V
(2)
1 , . . . , V
(2)
M g 2 ,n 2
= (2π i)
−M c
g,n
M g 1 ,n 1
λ=1
B
V
(1)
λ
B(∂ q )B(∂ ¯
q )
M g 2 ,n 2
κ=1
B
V
(2)
κ
n 1 −1
i=1
V
(1)
i
n 2 −1
j =1
V
(2)
j
g,n
.
(14.5)
We introduce the surface states n 1 and n 2 such that the BPZ inner product
with the new states V
(1)
n 1 and V
(2)
n 2 reproduces the M g 1 ,n 1 - and M g 2 ,n 2 -forms:
n 1 |V
(1)
n 1
:= ω M g 1 ,n 1 (V
(1)
1 , . . . , V
(1)
n 1
)
= (2π i)
−M c
g 1 ,n 1
M g 1 ,n 1
λ=1
B
V
(1)
λ
n 1 −1
i=1
V
(1)
i
g 1 ,n 1
,
(14.6a)
n 2 |V
(2)
n 2
:= ω M g 2 ,n 2 (V
(2)
1 , . . . , V
(2)
n 2
)
= (2π i)
−M c
g 2 ,n 2
M g 1 ,n 1
λ=1
B
V
(2)
λ
n 2 −1
j =1
V
(2)
j
g 2 ,n 2
.
(14.6b)
As described in Sect. 13.1.2, these states exist since the p-form is linear in each of
the external state and the BPZ inner product is non-degenerate. Each of the surface
states corresponds to an operator
n 1 | ==0| I ◦ n 1 (0),
n 2 | ==0| I ◦ n 2 (0),
(14.7)