13.1 Cotangent Spaces and Amplitudes
273
For simplicity, we will sometimes use the same notation for the section of P g,n and
its projection on the base M g,n . For this reason, the reader should assume that some
choice of local coordinates around the punctures is made except otherwise stated.
Given sections R g,n , the sum over all genus contribution is written formally as
R n :=
g≥0
R g,n ,
(13.19)
such that
R n (V 1 , . . . , V n ) :=
g≥0
R g,n (V 1 , . . . , V n ) =
g≥0
R g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(13.20)
A surface state is defined as a n-fold bra that reproduces the expression of a given
function when contracted with n states A i . The surface g,n |, formω g,n |, section
R g,n | and amplitudeA g,n | n-fold states are defined by the following expressions:
g,n | B s 1 · · · B s p | ⊗ i V i := ω s 1 ···s p (V 1 , . . . , V n ),
(13.21a)
ω
g,n
p | ⊗ i V i := ω p (V 1 , . . . , V n ),
(13.21b)
A g,n | ⊗ i V i := A g,n (V 1 , . . . , V n ).
(13.21c)
The last relation is generalized to any section R g,n
R g,n | ⊗ i V i := R g,n (V 1 , . . . , V n ).
(13.21d)
The reason for introducing these objects is that the form (13.12) is a linear map
from H ⊗n to a form on M g,n —see (13.4). Thus, there is always a state g,n | such
that its BPZ product with the states reproduces the form. In particular, the state
g,n | contains all the information about the local coordinates and the moduli (the
dependence is kept implicit). The definitions of the other states follow similarly.
These states are defined as bras, but they can be mapped to kets.
One finds the obvious relations
ω
g,n
p | == g,n | B s 1 dx
s 1 · · · B s p dx
s p ,
A g,n | =
M g,n
ω
g,n
p | .
(13.22)
The surface states do not contain information about the matter CFT: they collect
the universal data (like local coordinates) needed to describe amplitudes. Hence, it
is an important step in the description of off-shell string theory to characterize this
data. However, note that the relation between a surface state and the corresponding
form does depend on the CFT.
273
For simplicity, we will sometimes use the same notation for the section of P g,n and
its projection on the base M g,n . For this reason, the reader should assume that some
choice of local coordinates around the punctures is made except otherwise stated.
Given sections R g,n , the sum over all genus contribution is written formally as
R n :=
g≥0
R g,n ,
(13.19)
such that
R n (V 1 , . . . , V n ) :=
g≥0
R g,n (V 1 , . . . , V n ) =
g≥0
R g,n
ω
g,n
M g,n
(V 1 , . . . , V n ).
(13.20)
A surface state is defined as a n-fold bra that reproduces the expression of a given
function when contracted with n states A i . The surface g,n |, formω g,n |, section
R g,n | and amplitudeA g,n | n-fold states are defined by the following expressions:
g,n | B s 1 · · · B s p | ⊗ i V i := ω s 1 ···s p (V 1 , . . . , V n ),
(13.21a)
ω
g,n
p | ⊗ i V i := ω p (V 1 , . . . , V n ),
(13.21b)
A g,n | ⊗ i V i := A g,n (V 1 , . . . , V n ).
(13.21c)
The last relation is generalized to any section R g,n
R g,n | ⊗ i V i := R g,n (V 1 , . . . , V n ).
(13.21d)
The reason for introducing these objects is that the form (13.12) is a linear map
from H ⊗n to a form on M g,n —see (13.4). Thus, there is always a state g,n | such
that its BPZ product with the states reproduces the form. In particular, the state
g,n | contains all the information about the local coordinates and the moduli (the
dependence is kept implicit). The definitions of the other states follow similarly.
These states are defined as bras, but they can be mapped to kets.
One finds the obvious relations
ω
g,n
p | == g,n | B s 1 dx
s 1 · · · B s p dx
s p ,
A g,n | =
M g,n
ω
g,n
p | .
(13.22)
The surface states do not contain information about the matter CFT: they collect
the universal data (like local coordinates) needed to describe amplitudes. Hence, it
is an important step in the description of off-shell string theory to characterize this
data. However, note that the relation between a surface state and the corresponding
form does depend on the CFT.