10
1 Introduction
2 particle (with low-energy action being the Einstein–Hilbert action) is a key result
and originally raised interest for string theory.
Remark 1.1 (Reparametrization Constraints) Reparametrization invariance leads
to other constraints than H = 0. They imply in particular that the massless fields
have the correct gauge invariance and hence the correct degrees of freedom.
Note that, after taking into account these constraints, the remaining modes
correspond to excitations of the string in the directions transverse to it.
Hence, each vibrational mode of the string corresponds to a spacetime field for
a point-particle (and linear superpositions of modes can describe several fields).
This is how string theory achieves unification since a single type of string (of each
topology) is sufficient for describing all the possible types of fields encountered in
the standard model and in gravity. They correspond to the lowest excitation modes,
the higher massive modes being too heavy to be observed at low energy.
Bosonic string theory includes tachyons and is thus unstable. While the instability of the open string tachyon is well understood and indicates that open strings
are unstable and condense to closed strings, the status of the closed string tachyon
is more worrisome (literally interpreted, it indicates a decay of spacetime itself).
In order to solve this problem, one can introduce supersymmetry: in this case,
the spectrum does not include the tachyon because it cannot be paired with a
supersymmetric partner.
Moreover, as its name indicates, the bosonic string possesses only bosons in
its spectrum (perturbatively), which is an important obstacle to reproduce the
standard model. By introducing spacetime fermions, supersymmetry also solves this
problem. The last direct advantage of the superstring is that it reduces the number
of dimensions from 26 to 10, which makes the compactification easier.
1.2.2 Classification of Superstring Theories
In this section, we describe the different superstring theories (Chap. 17). In order to
proceed, we need to introduce some new elements.
The worldsheet field theory of the closed string is made of two sectors, called
the left- and right-moving sectors (the α n and ¯
α n modes). While they are treated
symmetrically in the simplest models, they are in fact independent (up to the zeromode) and the corresponding CFT can be chosen to be distinct.
The second ingredient already evoked earlier is supersymmetry. This symmetry
associates a fermion to each boson (and conversely) through the action of a
supercharge Q
|boson = Q |fermion .
(1.20)
More generally, one can consider N supercharges which build up a family of several
bosonic and fermionic partners. Since each supercharge increases the spin by 1/2 (in
D = 4), there is an upper limit for the number of supersymmetries—for interacting
1 Introduction
2 particle (with low-energy action being the Einstein–Hilbert action) is a key result
and originally raised interest for string theory.
Remark 1.1 (Reparametrization Constraints) Reparametrization invariance leads
to other constraints than H = 0. They imply in particular that the massless fields
have the correct gauge invariance and hence the correct degrees of freedom.
Note that, after taking into account these constraints, the remaining modes
correspond to excitations of the string in the directions transverse to it.
Hence, each vibrational mode of the string corresponds to a spacetime field for
a point-particle (and linear superpositions of modes can describe several fields).
This is how string theory achieves unification since a single type of string (of each
topology) is sufficient for describing all the possible types of fields encountered in
the standard model and in gravity. They correspond to the lowest excitation modes,
the higher massive modes being too heavy to be observed at low energy.
Bosonic string theory includes tachyons and is thus unstable. While the instability of the open string tachyon is well understood and indicates that open strings
are unstable and condense to closed strings, the status of the closed string tachyon
is more worrisome (literally interpreted, it indicates a decay of spacetime itself).
In order to solve this problem, one can introduce supersymmetry: in this case,
the spectrum does not include the tachyon because it cannot be paired with a
supersymmetric partner.
Moreover, as its name indicates, the bosonic string possesses only bosons in
its spectrum (perturbatively), which is an important obstacle to reproduce the
standard model. By introducing spacetime fermions, supersymmetry also solves this
problem. The last direct advantage of the superstring is that it reduces the number
of dimensions from 26 to 10, which makes the compactification easier.
1.2.2 Classification of Superstring Theories
In this section, we describe the different superstring theories (Chap. 17). In order to
proceed, we need to introduce some new elements.
The worldsheet field theory of the closed string is made of two sectors, called
the left- and right-moving sectors (the α n and ¯
α n modes). While they are treated
symmetrically in the simplest models, they are in fact independent (up to the zeromode) and the corresponding CFT can be chosen to be distinct.
The second ingredient already evoked earlier is supersymmetry. This symmetry
associates a fermion to each boson (and conversely) through the action of a
supercharge Q
|boson = Q |fermion .
(1.20)
More generally, one can consider N supercharges which build up a family of several
bosonic and fermionic partners. Since each supercharge increases the spin by 1/2 (in
D = 4), there is an upper limit for the number of supersymmetries—for interacting