16
1 Introduction
the gauge fixing condition. 10 Since these spurious singularities of the amplitudes
are not physical, one needs to ensure that they can be removed, which is indeed
possible to achieve.
Hence, only IR divergences present a real challenge to string theory. Dealing with
these divergences requires renormalizing the amplitudes, but this is not possible in
the standard formulation of worldsheet string theory since the states are on-shell. 11
1.3
String Field Theory
1.3.1 From the Worldsheet to Field Theory
The first step is to solve the IR divergences problem is to go off-shell (Chaps. 11
and 13). This is made possible by introducing local coordinates around the
punctures of the Riemann surface (Chap. 12).
The IR divergences originate from Riemann surfaces close to degeneration, that
is, surfaces with long tubes. The latter can be of separating and non-separating
types, depending on whether the Riemann surface splits in two pieces if the tube is
cut (Fig. 1.8). By exploring the form of the amplitudes in this limit (Chap. 14),
the expression naturally separates into several pieces, to be interpreted as two
amplitudes (of lower n and g) connected by a propagator. The latter can be
reinterpreted as a standard (k 2 + m 2 ) −1 term, hence solving the divergence problem
for k 2 + m 2 < 0. Taking this decomposition seriously leads to identify each
contribution with a Feynman graph.
Decomposing the amplitude recursively, the next step consists in finding the
elementary graphs, i.e. the interaction vertices from which all other graphs (and
amplitudes) can be built. These graphs are the building blocks of the field theory
(Chap. 15), with the kinetic term given by the inverse of the propagator. Having
Feynman diagrams and a field theory allows to use all the standard tools from
QFT.
However, this field theory is gauge fixed because on-shell amplitudes are gauge
invariant and include only physical states. For this reason, one needs to find how to
re-establish the gauge invariance. Due to the complicated structure of string theory,
the full-fledged Batalin–Vilkovisky (BV) formalism must be used (Chap. 15): it
basically amounts to introduce ghosts before the gauge fixing. The final stage is to
obtain the 1PI effective action from which the physics is more easily extracted. But,
it is useful to study first the free theory (Chaps. 9 and 10) to gain some insights.
The book ends with a discussion of the momentum-space representation and of
background independence (Chaps. 16 and 18).
10 Such spurious singularities are also found in supergravity.
11 The on-shell condition is a consequence of the BRST and conformal invariance. While the first
will be given up, the second will be maintained to facilitate the computations.
1 Introduction
the gauge fixing condition. 10 Since these spurious singularities of the amplitudes
are not physical, one needs to ensure that they can be removed, which is indeed
possible to achieve.
Hence, only IR divergences present a real challenge to string theory. Dealing with
these divergences requires renormalizing the amplitudes, but this is not possible in
the standard formulation of worldsheet string theory since the states are on-shell. 11
1.3
String Field Theory
1.3.1 From the Worldsheet to Field Theory
The first step is to solve the IR divergences problem is to go off-shell (Chaps. 11
and 13). This is made possible by introducing local coordinates around the
punctures of the Riemann surface (Chap. 12).
The IR divergences originate from Riemann surfaces close to degeneration, that
is, surfaces with long tubes. The latter can be of separating and non-separating
types, depending on whether the Riemann surface splits in two pieces if the tube is
cut (Fig. 1.8). By exploring the form of the amplitudes in this limit (Chap. 14),
the expression naturally separates into several pieces, to be interpreted as two
amplitudes (of lower n and g) connected by a propagator. The latter can be
reinterpreted as a standard (k 2 + m 2 ) −1 term, hence solving the divergence problem
for k 2 + m 2 < 0. Taking this decomposition seriously leads to identify each
contribution with a Feynman graph.
Decomposing the amplitude recursively, the next step consists in finding the
elementary graphs, i.e. the interaction vertices from which all other graphs (and
amplitudes) can be built. These graphs are the building blocks of the field theory
(Chap. 15), with the kinetic term given by the inverse of the propagator. Having
Feynman diagrams and a field theory allows to use all the standard tools from
QFT.
However, this field theory is gauge fixed because on-shell amplitudes are gauge
invariant and include only physical states. For this reason, one needs to find how to
re-establish the gauge invariance. Due to the complicated structure of string theory,
the full-fledged Batalin–Vilkovisky (BV) formalism must be used (Chap. 15): it
basically amounts to introduce ghosts before the gauge fixing. The final stage is to
obtain the 1PI effective action from which the physics is more easily extracted. But,
it is useful to study first the free theory (Chaps. 9 and 10) to gain some insights.
The book ends with a discussion of the momentum-space representation and of
background independence (Chaps. 16 and 18).
10 Such spurious singularities are also found in supergravity.
11 The on-shell condition is a consequence of the BRST and conformal invariance. While the first
will be given up, the second will be maintained to facilitate the computations.