3
Worldsheet Path Integral: Scattering
Amplitudes
Abstract
In this chapter, we generalize the worldsheet path integral to compute scattering
amplitudes, which corresponds to insert vertex operators. The gauge fixing from
the previous chapter is generalized to this case. In particular, we discuss the 2point amplitude on the sphere. Finally, we introduce the BRST symmetry and
motivate some properties of the BRST quantization, which will be performed in
details later. The formulas in this chapter are all covariant: they will be rewritten
in complex coordinates in the next chapter.
3.1
Scattering Amplitudes on Moduli Space
In this section, we describe the scattering of n strings. The momentum representation is more natural for describing interactions, especially in string theory.
Therefore, each string is characterized by a state V α i (k i ) with momentum k i and
some additional quantum numbers α i (i = 1, . . . , n). We start from the worldsheet
path integral (2.28) before gauge fixing
Z g =
d g g ab
gauge [g]
Z m [g],
Z m [g] =
d g e
−S m [g,,] .
(3.1)
3.1.1 Vertex Operators and Path Integral
The external states are represented by infinite semi-tubes attached to the surfaces.
Under a conformal mapping, the tubes can be mapped to points called punctures
on the worldsheet. At g-loops, the resulting space is a Riemann surface g,n of
© 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_3
69
Worldsheet Path Integral: Scattering
Amplitudes
Abstract
In this chapter, we generalize the worldsheet path integral to compute scattering
amplitudes, which corresponds to insert vertex operators. The gauge fixing from
the previous chapter is generalized to this case. In particular, we discuss the 2point amplitude on the sphere. Finally, we introduce the BRST symmetry and
motivate some properties of the BRST quantization, which will be performed in
details later. The formulas in this chapter are all covariant: they will be rewritten
in complex coordinates in the next chapter.
3.1
Scattering Amplitudes on Moduli Space
In this section, we describe the scattering of n strings. The momentum representation is more natural for describing interactions, especially in string theory.
Therefore, each string is characterized by a state V α i (k i ) with momentum k i and
some additional quantum numbers α i (i = 1, . . . , n). We start from the worldsheet
path integral (2.28) before gauge fixing
Z g =
d g g ab
gauge [g]
Z m [g],
Z m [g] =
d g e
−S m [g,,] .
(3.1)
3.1.1 Vertex Operators and Path Integral
The external states are represented by infinite semi-tubes attached to the surfaces.
Under a conformal mapping, the tubes can be mapped to points called punctures
on the worldsheet. At g-loops, the resulting space is a Riemann surface g,n of
© 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_3
69