3
String Theory
A natural 1-dimensional generalization of the point particle is its blow-up into an
open or closed string, cf. Fig. 3.1. It turns out that both possibilities are meaningful
and, in fact, at quantum level the former implies the latter. The action for the open
string turns out to be structurally very similar to that of the point particle described
in the previous chapter. Indeed, one can show that the action (2.16) with suitably
defined (ω, Q, ∗) corresponds to a decomposition of the moduli space of bordered
Riemann surfaces with punctures on the boundary. For the closed string, however,
this is not the case and an infinite number of higher order vertices has to be added
to (2.16). In this chapter we will describe some features of string theory relevant for
the rest of this book. For more details we refer to the original literature listed at the
end of the chapter.
3.1
Closed Strings
Let us start with a one-dimensional generalization of the world-line action (2.1)
I [φ, h] =
1
4πα
Σ
g(dφ, ∧
∗ dφ) ,
(3.1)
where now φ : Σ → M, while Σ is topologically a cylinder equipped with
a pseudo-Riemannian world-sheet metric. However, in order to have a welldefined measure on the space of world-sheets we consider the Wick-rotated, or
Riemannian world-sheet metric h. This is in analogy with the discussion after (2.24).
Furthermore, 1/α is the string tension which sets the unit for the masses of the
excitations of the string. In what follows, we work in units where α = 1 since we
are not interested in the particle limit (α = 0), nor the tensionless limit (α = ∞)
where all excitations are massless.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_3
27
String Theory
A natural 1-dimensional generalization of the point particle is its blow-up into an
open or closed string, cf. Fig. 3.1. It turns out that both possibilities are meaningful
and, in fact, at quantum level the former implies the latter. The action for the open
string turns out to be structurally very similar to that of the point particle described
in the previous chapter. Indeed, one can show that the action (2.16) with suitably
defined (ω, Q, ∗) corresponds to a decomposition of the moduli space of bordered
Riemann surfaces with punctures on the boundary. For the closed string, however,
this is not the case and an infinite number of higher order vertices has to be added
to (2.16). In this chapter we will describe some features of string theory relevant for
the rest of this book. For more details we refer to the original literature listed at the
end of the chapter.
3.1
Closed Strings
Let us start with a one-dimensional generalization of the world-line action (2.1)
I [φ, h] =
1
4πα
Σ
g(dφ, ∧
∗ dφ) ,
(3.1)
where now φ : Σ → M, while Σ is topologically a cylinder equipped with
a pseudo-Riemannian world-sheet metric. However, in order to have a welldefined measure on the space of world-sheets we consider the Wick-rotated, or
Riemannian world-sheet metric h. This is in analogy with the discussion after (2.24).
Furthermore, 1/α is the string tension which sets the unit for the masses of the
excitations of the string. In what follows, we work in units where α = 1 since we
are not interested in the particle limit (α = 0), nor the tensionless limit (α = ∞)
where all excitations are massless.
© Springer Nature Switzerland AG 2020
M. Doubek et al., Algebraic Structure of String Field Theory, Lecture Notes
in Physics 973, https://doi.org/10.1007/978-3-030-53056-3_3
27