11
Introduction to Off-Shell String Theory
Abstract
In this chapter, we introduce a framework to describe off-shell amplitudes in
string theory. We first start by motivating various concepts—in particular, local
coordinates and factorization—by focusing on the 3- and 4-point amplitudes. We
then prepare the stage for a general description of off-shell amplitudes. We focus
again on the closed bosonic string only.
11.1 Motivations
11.1.1 3-Point Function
The tree-level 3-point amplitude of 3 weight h i vertex operators 1 V i is given by
A 0,3 =
3
i=1
V i (z i )
S 2
∝ (z 1 − z 2 )
h 3 −h 1 −h 2 × perms × c.c.
(11.1)
There is no integration since dim M 0,3 = 0.
The amplitude is independent of the z i only if the matter state is on-shell, h i = 0,
for example, if V i = c ¯
cV i with h(V i ) = 1. Indeed, if h i = 0, then A 0,3 is not
1 The quantum number (k, j ) of the vertex operator is mostly irrelevant for the discussion of
the current and next chapters, and they are omitted. We will distinguish them by a number and
reintroduce the momentum k when necessary. We also omit the overall normalization of the
amplitudes.
© 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_11
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