2
Worldsheet Path Integral: Vacuum Amplitudes
Abstract
In this chapter, we develop the path integral quantization for a generic closed
string theory in worldsheet Euclidean signature. We focus on the vacuum
amplitudes, leaving scattering amplitudes for the next chapter. This allows to
focus on the definition and gauge fixing of the path integral measure.
The exposition differs from most traditional textbooks in three ways: (1) we
consider a general matter CFT, (2) we consider the most general treatment (for
any genus) and (3) we do not use complex coordinates but always a covariant
parametrization.
The derivation is technical and the reader is encouraged to not stop at this
chapter in case of difficulties and to proceed forward: most concepts will be
reintroduced from a different point of view later in other chapters of the book.
2.1
Worldsheet Action and Symmetries
The string worldsheet is a Riemann surface W = g of genus g: the genus counts
the number of holes or handles. Coordinates on the worldsheet are denoted by σ a =
(τ, σ ). When there is no risk of confusion, σ denotes collectively both coordinates.
Since closed strings are considered, the Riemann surface has locally the topology of
a cylinder, with the spatial section being circles S 1 with radius taken to be 1, such
that
σ ∈ [0, 2π),
σ ∼ σ + 2π.
(2.1)
The string is embedded in the D-dimensional spacetime M with metric G μν
through maps X μ (σ a ) : W → M with μ = 0, . . . , D − 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_2
29
Worldsheet Path Integral: Vacuum Amplitudes
Abstract
In this chapter, we develop the path integral quantization for a generic closed
string theory in worldsheet Euclidean signature. We focus on the vacuum
amplitudes, leaving scattering amplitudes for the next chapter. This allows to
focus on the definition and gauge fixing of the path integral measure.
The exposition differs from most traditional textbooks in three ways: (1) we
consider a general matter CFT, (2) we consider the most general treatment (for
any genus) and (3) we do not use complex coordinates but always a covariant
parametrization.
The derivation is technical and the reader is encouraged to not stop at this
chapter in case of difficulties and to proceed forward: most concepts will be
reintroduced from a different point of view later in other chapters of the book.
2.1
Worldsheet Action and Symmetries
The string worldsheet is a Riemann surface W = g of genus g: the genus counts
the number of holes or handles. Coordinates on the worldsheet are denoted by σ a =
(τ, σ ). When there is no risk of confusion, σ denotes collectively both coordinates.
Since closed strings are considered, the Riemann surface has locally the topology of
a cylinder, with the spatial section being circles S 1 with radius taken to be 1, such
that
σ ∈ [0, 2π),
σ ∼ σ + 2π.
(2.1)
The string is embedded in the D-dimensional spacetime M with metric G μν
through maps X μ (σ a ) : W → M with μ = 0, . . . , D − 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_2
29