13
Off-Shell Amplitudes
Abstract
While the previous chapter was purely geometrical, this one makes contact with
string theory through the worldsheet CFT. We continue the description of P g,n by
constructing p-forms. The reason why we need to consider the CFT is that ghosts
are necessary to build the p-forms: this can be understood from Chap. 2, where
we found that the ghosts must be interpreted as part of the measure on the moduli
space. Then, we build the off-shell amplitudes and discuss some properties.
13.1 Cotangent Spaces and Amplitudes
In this section, we construct the p-forms on P g,n that are needed for the amplitudes.
We first motivate the expressions from general ideas and check later that they have
the correct properties.
13.1.1 Construction of Forms
A p-form ω
(g,n)
p
∈
p T ∗ P g,n is a multilinear antisymmetric map from
p T P g,n
to a function of the moduli parameters. The superscript on the form is omitted when
there is no ambiguity about the space considered. The components ω i 1 ···i p of the
p-form are defined by inserting p basis vectors ∂ s 1 , . . . , ∂ s p
ω i 1 ···i p := ω p (∂ s 1 , . . . , ∂ s p ),
(13.1)
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_13
269
Off-Shell Amplitudes
Abstract
While the previous chapter was purely geometrical, this one makes contact with
string theory through the worldsheet CFT. We continue the description of P g,n by
constructing p-forms. The reason why we need to consider the CFT is that ghosts
are necessary to build the p-forms: this can be understood from Chap. 2, where
we found that the ghosts must be interpreted as part of the measure on the moduli
space. Then, we build the off-shell amplitudes and discuss some properties.
13.1 Cotangent Spaces and Amplitudes
In this section, we construct the p-forms on P g,n that are needed for the amplitudes.
We first motivate the expressions from general ideas and check later that they have
the correct properties.
13.1.1 Construction of Forms
A p-form ω
(g,n)
p
∈
p T ∗ P g,n is a multilinear antisymmetric map from
p T P g,n
to a function of the moduli parameters. The superscript on the form is omitted when
there is no ambiguity about the space considered. The components ω i 1 ···i p of the
p-form are defined by inserting p basis vectors ∂ s 1 , . . . , ∂ s p
ω i 1 ···i p := ω p (∂ s 1 , . . . , ∂ s p ),
(13.1)
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_13
269