1.2 String Theory
5
1.2
String Theory
1.2.1 Properties
The goal of this section is to give a general idea of string theory by introducing
some concepts and terminology. The reader not familiar with the points described
in this section is advised to follow in parallel some standard worldsheet string theory
textbooks.
Worldsheet CFT
A string is characterized by its worldsheet field theory (Chap. 2). 3 The worldsheet
is parametrized by coordinates σ a = (τ, σ ). The simplest description is obtained
by endowing the worldsheet with a metric g ab (σ a ) (a = 0, 1) and by adding
a set of D scalar fields X μ (σ a ) living on the worldsheet (μ = 0, . . . , D − 1).
The latter represents the position of the string in the D-dimensional spacetime.
From the classical equations of motion, the metric g ab is proportional to the metric
induced on the worldsheet from its embedding in spacetime. More generally, one
ensures that the worldsheet metric is non-dynamical by imposing that the action
is invariant under (worldsheet) diffeomorphisms and under Weyl transformations
(local rescalings of the metric). The consistency of these conditions at the quantum
level imposes that D = 26, and this number is called the critical dimension. Gauge
fixing the symmetries, and thus the metric, leads to the conformal invariance of
the resulting worldsheet field theory: a conformal field theory (CFT) is a field
theory (possibly on a curved background) in which only angles and not distances
can be measured (Chaps. 5–7). This simplifies greatly the analysis since the twodimensional conformal algebra (called the Virasoro algebra) is infinite-dimensional.
CFTs more general than D free scalar fields can be considered: fields taking
non-compact values are interpreted as non-compact dimensions while compact or
Grassmann-odd fields are interpreted as compact dimensions or internal structure,
like the spin.
While the light-cone quantization allows to find quickly the states of the theory,
the simplest covariant method is the BRST quantization (Chap. 8). It introduces
ghosts (and super-ghosts) associated to the gauge fixing of diffeomorphisms
(and local supersymmetry). These (super)ghosts form a CFT which is universal
(independent of the matter CFT).
The trajectory of the string is denoted by x c (τ, σ ). It begins and ends respectively
at the geometric shapes parametrized by x c (τ i , σ ) = x i (σ ) and by x c (τ f , σ ) =
x f (σ ). Note that the coordinate system on the worldsheet itself is arbitrary. The
spatial section of a string can be topologically closed (circle) or open (line)
(Fig. 1.3), leading to cylindrical or rectangular worldsheets as illustrated in Figs. 1.4
3 We focus mainly on the bosonic string theory, leaving aside the superstring, except when
differences are important.
5
1.2
String Theory
1.2.1 Properties
The goal of this section is to give a general idea of string theory by introducing
some concepts and terminology. The reader not familiar with the points described
in this section is advised to follow in parallel some standard worldsheet string theory
textbooks.
Worldsheet CFT
A string is characterized by its worldsheet field theory (Chap. 2). 3 The worldsheet
is parametrized by coordinates σ a = (τ, σ ). The simplest description is obtained
by endowing the worldsheet with a metric g ab (σ a ) (a = 0, 1) and by adding
a set of D scalar fields X μ (σ a ) living on the worldsheet (μ = 0, . . . , D − 1).
The latter represents the position of the string in the D-dimensional spacetime.
From the classical equations of motion, the metric g ab is proportional to the metric
induced on the worldsheet from its embedding in spacetime. More generally, one
ensures that the worldsheet metric is non-dynamical by imposing that the action
is invariant under (worldsheet) diffeomorphisms and under Weyl transformations
(local rescalings of the metric). The consistency of these conditions at the quantum
level imposes that D = 26, and this number is called the critical dimension. Gauge
fixing the symmetries, and thus the metric, leads to the conformal invariance of
the resulting worldsheet field theory: a conformal field theory (CFT) is a field
theory (possibly on a curved background) in which only angles and not distances
can be measured (Chaps. 5–7). This simplifies greatly the analysis since the twodimensional conformal algebra (called the Virasoro algebra) is infinite-dimensional.
CFTs more general than D free scalar fields can be considered: fields taking
non-compact values are interpreted as non-compact dimensions while compact or
Grassmann-odd fields are interpreted as compact dimensions or internal structure,
like the spin.
While the light-cone quantization allows to find quickly the states of the theory,
the simplest covariant method is the BRST quantization (Chap. 8). It introduces
ghosts (and super-ghosts) associated to the gauge fixing of diffeomorphisms
(and local supersymmetry). These (super)ghosts form a CFT which is universal
(independent of the matter CFT).
The trajectory of the string is denoted by x c (τ, σ ). It begins and ends respectively
at the geometric shapes parametrized by x c (τ i , σ ) = x i (σ ) and by x c (τ f , σ ) =
x f (σ ). Note that the coordinate system on the worldsheet itself is arbitrary. The
spatial section of a string can be topologically closed (circle) or open (line)
(Fig. 1.3), leading to cylindrical or rectangular worldsheets as illustrated in Figs. 1.4
3 We focus mainly on the bosonic string theory, leaving aside the superstring, except when
differences are important.