4
1 Introduction
in two dimensions. Hence, strings should display nice properties and are thus of
special interest.
Worldvolume Theory The point-particle (0-brane) and the string (1-brane) are
also remarkable in another aspect: it is possible to construct a simple worldvolume
field theory (and the associated functional integral) in terms of a worldvolume
metric. All components of the latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorphisms and Weyl invariance for the string). This
ensures the reparametrization invariance of the worldvolume without having to use
a complicated action. Oppositely, the worldvolume metric cannot be completely
gauge fixed for p > 1.
Summary As a conclusion, strings achieve an optimal balance between spacetime and worldsheet divergences, as well as having a simple description with
reparametrization invariance.
Since the construction of a field theory is difficult, it is natural to start with a
worldsheet theory and to study it in the first-quantization formalism, which will
provide a guideline for writing the field theory. In particular, this allows to access
the physical states in a simple way and to find other general properties of the
theory. When it comes to the interactions and scattering amplitudes, this approach
may be hopeless in general since the topology of the worldvolume needs to be
specified by hand (describing the interaction process). In this respect, the case
of the string is again exceptional: because Riemann surfaces have been classified
and are well-understood, the arbitrariness is minimal. Combined with the tools
of conformal field theory, many computations can be performed. Moreover, since
the modes of vibrations of the strings provide all the necessary ingredients to
describe the Standard model, it is sufficient to consider only one string field
(for one type of strings), instead of the plethora found in point-particle field
theory (one field for each particle). Similarly, non-perturbative information (such
as branes and dualities) could be found only due to the specific properties of
strings.
Coming back to the question which opened this section, higher-dimensional
branes of all the allowed dimensions naturally appear in string theory as bound
states. Hence, even if the worldvolume formulation of branes with p > 1
looks pathological, 2 string theory hints towards another definition of these
objects.
2 Entering in the details would take us too far away from the main topic of this book. Some of
the problems found when dealing with (p > 2)-branes are: how to define a Wick rotation for 3manifolds, the presence of Lorentz anomalies in target spacetime, problems with the spectrum, lack
of renormalizability, impossibility to gauge-fix the worldvolume metric [1–7,13,16–18,44,58,61–
63, 71, 78].
1 Introduction
in two dimensions. Hence, strings should display nice properties and are thus of
special interest.
Worldvolume Theory The point-particle (0-brane) and the string (1-brane) are
also remarkable in another aspect: it is possible to construct a simple worldvolume
field theory (and the associated functional integral) in terms of a worldvolume
metric. All components of the latter are fixed by gauge symmetries (diffeomorphisms for the particle, diffeomorphisms and Weyl invariance for the string). This
ensures the reparametrization invariance of the worldvolume without having to use
a complicated action. Oppositely, the worldvolume metric cannot be completely
gauge fixed for p > 1.
Summary As a conclusion, strings achieve an optimal balance between spacetime and worldsheet divergences, as well as having a simple description with
reparametrization invariance.
Since the construction of a field theory is difficult, it is natural to start with a
worldsheet theory and to study it in the first-quantization formalism, which will
provide a guideline for writing the field theory. In particular, this allows to access
the physical states in a simple way and to find other general properties of the
theory. When it comes to the interactions and scattering amplitudes, this approach
may be hopeless in general since the topology of the worldvolume needs to be
specified by hand (describing the interaction process). In this respect, the case
of the string is again exceptional: because Riemann surfaces have been classified
and are well-understood, the arbitrariness is minimal. Combined with the tools
of conformal field theory, many computations can be performed. Moreover, since
the modes of vibrations of the strings provide all the necessary ingredients to
describe the Standard model, it is sufficient to consider only one string field
(for one type of strings), instead of the plethora found in point-particle field
theory (one field for each particle). Similarly, non-perturbative information (such
as branes and dualities) could be found only due to the specific properties of
strings.
Coming back to the question which opened this section, higher-dimensional
branes of all the allowed dimensions naturally appear in string theory as bound
states. Hence, even if the worldvolume formulation of branes with p > 1
looks pathological, 2 string theory hints towards another definition of these
objects.
2 Entering in the details would take us too far away from the main topic of this book. Some of
the problems found when dealing with (p > 2)-branes are: how to define a Wick rotation for 3manifolds, the presence of Lorentz anomalies in target spacetime, problems with the spectrum, lack
of renormalizability, impossibility to gauge-fix the worldvolume metric [1–7,13,16–18,44,58,61–
63, 71, 78].