14
Amplitude Factorization and Feynman
Diagrams
Abstract
In the previous chapter, we built the off-shell amplitudes by integrating forms
on sections of P g,n . Studying their factorizations leads to rewrite them in terms
of Feynman diagrams, which allows to identify the fundamental interactions’
vertices. We will then be able to write the SFT action in the next chapter.
14.1 Amplitude Factorization
We have seen how to write off-shell amplitudes. The next step is to rewrite them as
a sum of Feynman diagrams through factorization of amplitudes.
Factorization consists in writing a g-loop n-point amplitude in terms of lowerorder amplitudes in both g and n connected by propagators. Since an amplitude
corresponds to a sum over all possible processes, which corresponds to integrating
over the moduli space, it is natural to associate Feynman diagrams to different
subspaces of the moduli space. One can expect that the plumbing fixture (Sect. 12.3)
is the appropriate translation of the factorization at the level of Riemann surfaces.
We will assume that it is the case and check that it is correct a posteriori.
To proceed, we consider the contribution to the amplitude A g,n of the family
of surfaces obtained by the plumbing fixture of two surfaces (separating case) or a
surface with itself (non-separating case).
14.1.1 Separating Case
In this section, we consider the separating plumbing fixture where part of the moduli
space M g,n is covered by M g 1 ,n 1 #M g 2 ,n 2 with g = g 1 + g 2 and n = n 1 + n 2 − 2
(Sect. 12.3.1). The local coordinates read w
(1)
i and w
(2)
j for i = 1, . . . , n 1 and j =
© 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_14
285
Amplitude Factorization and Feynman
Diagrams
Abstract
In the previous chapter, we built the off-shell amplitudes by integrating forms
on sections of P g,n . Studying their factorizations leads to rewrite them in terms
of Feynman diagrams, which allows to identify the fundamental interactions’
vertices. We will then be able to write the SFT action in the next chapter.
14.1 Amplitude Factorization
We have seen how to write off-shell amplitudes. The next step is to rewrite them as
a sum of Feynman diagrams through factorization of amplitudes.
Factorization consists in writing a g-loop n-point amplitude in terms of lowerorder amplitudes in both g and n connected by propagators. Since an amplitude
corresponds to a sum over all possible processes, which corresponds to integrating
over the moduli space, it is natural to associate Feynman diagrams to different
subspaces of the moduli space. One can expect that the plumbing fixture (Sect. 12.3)
is the appropriate translation of the factorization at the level of Riemann surfaces.
We will assume that it is the case and check that it is correct a posteriori.
To proceed, we consider the contribution to the amplitude A g,n of the family
of surfaces obtained by the plumbing fixture of two surfaces (separating case) or a
surface with itself (non-separating case).
14.1.1 Separating Case
In this section, we consider the separating plumbing fixture where part of the moduli
space M g,n is covered by M g 1 ,n 1 #M g 2 ,n 2 with g = g 1 + g 2 and n = n 1 + n 2 − 2
(Sect. 12.3.1). The local coordinates read w
(1)
i and w
(2)
j for i = 1, . . . , n 1 and j =
© 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_14
285