2
1 Introduction
the latter (in its supersymmetric version) contains all the necessary ingredients for a
fully consistent high-energy model: 1
• quantum gravity (quantization of general relativity plus higher-derivative corrections);
• grand unification (of matter, interactions and gravity);
• no divergences, UV finiteness (finite and renormalizable theory);
• fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for
the supersymmetric version);
• existence of all possible branes;
• no dimensionless parameters and one dimensionful parameter (the string length
s ).
It can be expected that a theory of fundamental strings (1-branes) occupies a
distinguished place among fundamental p-branes for the following reasons.
Interaction Non-locality In a QFT of point-particles, UV divergences arise
because interactions (defined as the place where the number and/or nature of
the objects change) are arbitrarily localized at a spacetime point. In Feynman
graphs, such divergences can be seen when the momentum of a loop becomes
infinite (two vertices collide): this happens when trying to concentrate an infinite
amount of energy at a single point. However, these divergences are expected to
be reduced or absent in a field theory of extended objects: whereas the interaction
between particles is perfectly local in spacetime and agreed upon by all observers
(Fig. 1.1), the spatial extension of branes makes the interactions non-local. This
means that two different observers will neither agree on the place of the interactions (Fig. 1.2), nor on the part of the diagram which describes one or two
branes.
The string lies at the boundary between too much local and too much non-local:
in any given frame, the interaction is local in space, but not in spacetime. The reason
is that a string is one-dimensional and splits or joins along a point. For p > 1,
the brane needs to break/join along an extended spatial section, which looks nonlocal.
Another consequence of the non-locality is a drastic reduction of the possible
interactions. If an interaction is Lorentz invariant, Lorentz covariant objects can be
attached at the vertex (such as momentum or gamma matrices): this gives Lorentz
invariants after contracting with indices carried by the field. But, this is impossible
if the interaction itself is non-local (and thus not invariant): inserting a covariant
object would break Lorentz invariance.
1 There are also indications that a theory of membranes (2-branes) in 10 + 1 dimensions, called
M-theory, should exist. No direct and satisfactory description of the latter has been found and we
will thus focus on string theory in this book.
1 Introduction
the latter (in its supersymmetric version) contains all the necessary ingredients for a
fully consistent high-energy model: 1
• quantum gravity (quantization of general relativity plus higher-derivative corrections);
• grand unification (of matter, interactions and gravity);
• no divergences, UV finiteness (finite and renormalizable theory);
• fixed number of dimensions (26 = 25 + 1 for the bosonic string, 10 = 9 + 1 for
the supersymmetric version);
• existence of all possible branes;
• no dimensionless parameters and one dimensionful parameter (the string length
s ).
It can be expected that a theory of fundamental strings (1-branes) occupies a
distinguished place among fundamental p-branes for the following reasons.
Interaction Non-locality In a QFT of point-particles, UV divergences arise
because interactions (defined as the place where the number and/or nature of
the objects change) are arbitrarily localized at a spacetime point. In Feynman
graphs, such divergences can be seen when the momentum of a loop becomes
infinite (two vertices collide): this happens when trying to concentrate an infinite
amount of energy at a single point. However, these divergences are expected to
be reduced or absent in a field theory of extended objects: whereas the interaction
between particles is perfectly local in spacetime and agreed upon by all observers
(Fig. 1.1), the spatial extension of branes makes the interactions non-local. This
means that two different observers will neither agree on the place of the interactions (Fig. 1.2), nor on the part of the diagram which describes one or two
branes.
The string lies at the boundary between too much local and too much non-local:
in any given frame, the interaction is local in space, but not in spacetime. The reason
is that a string is one-dimensional and splits or joins along a point. For p > 1,
the brane needs to break/join along an extended spatial section, which looks nonlocal.
Another consequence of the non-locality is a drastic reduction of the possible
interactions. If an interaction is Lorentz invariant, Lorentz covariant objects can be
attached at the vertex (such as momentum or gamma matrices): this gives Lorentz
invariants after contracting with indices carried by the field. But, this is impossible
if the interaction itself is non-local (and thus not invariant): inserting a covariant
object would break Lorentz invariance.
1 There are also indications that a theory of membranes (2-branes) in 10 + 1 dimensions, called
M-theory, should exist. No direct and satisfactory description of the latter has been found and we
will thus focus on string theory in this book.