1.2 String Theory
13
Fig. 1.6 General Riemann surfaces with boundaries and punctures
Due to the topological equivalence between surfaces, a conformal map can
be used in order to work with simpler surfaces. In particular, the external tubes
and strips are collapsed to points called punctures (or marked points) on the
corresponding surfaces or boundaries. A general amplitude then looks like a sphere
from which holes and disks have been removed and to which marked points have
been pierced (Fig. 1.6).
Amplitudes
In order to compute an amplitude for the scattering of n strings (Chaps. 3 and 4),
one must sum over all the inequivalent worldsheets through a path integral weighted
by the CFT action chosen to describe the theory. 8 At fixed n, the sum runs over the
genus g, such that each term is described by a Riemann surface g,n of genus g
with n punctures.
The interactions between strings follow from the graph topologies: since the
latter are not encoded into the action, the dependence in the coupling constant must
be added by hand. For closed strings, there is a unique cubic vertex with coupling
g s . A direct inspection shows that the correct factor is g
n−2+2g
s
:
• for n = 3 there is one factor g s , and every additional external string leads to the
addition of one vertex with factor g s , since this process can be obtained from the
n − 1 process by splitting one of the external string in two by inserting a vertex;
• each loop comes with two vertices, so g-loops provide a factor g
2g
s .
Remark 1.2 (Status of g s as a Parameter) It was stated earlier that string theory has
no dimensionless parameter, but g s looks to be one. In reality it is determined by
the expectation value of the dilaton g s = e . Hence the coupling constant is not a
parameter defining the theory but is rather determined by the dynamics of the theory.
8 For simplicity we focus on closed string amplitudes in this section.
13
Fig. 1.6 General Riemann surfaces with boundaries and punctures
Due to the topological equivalence between surfaces, a conformal map can
be used in order to work with simpler surfaces. In particular, the external tubes
and strips are collapsed to points called punctures (or marked points) on the
corresponding surfaces or boundaries. A general amplitude then looks like a sphere
from which holes and disks have been removed and to which marked points have
been pierced (Fig. 1.6).
Amplitudes
In order to compute an amplitude for the scattering of n strings (Chaps. 3 and 4),
one must sum over all the inequivalent worldsheets through a path integral weighted
by the CFT action chosen to describe the theory. 8 At fixed n, the sum runs over the
genus g, such that each term is described by a Riemann surface g,n of genus g
with n punctures.
The interactions between strings follow from the graph topologies: since the
latter are not encoded into the action, the dependence in the coupling constant must
be added by hand. For closed strings, there is a unique cubic vertex with coupling
g s . A direct inspection shows that the correct factor is g
n−2+2g
s
:
• for n = 3 there is one factor g s , and every additional external string leads to the
addition of one vertex with factor g s , since this process can be obtained from the
n − 1 process by splitting one of the external string in two by inserting a vertex;
• each loop comes with two vertices, so g-loops provide a factor g
2g
s .
Remark 1.2 (Status of g s as a Parameter) It was stated earlier that string theory has
no dimensionless parameter, but g s looks to be one. In reality it is determined by
the expectation value of the dilaton g s = e . Hence the coupling constant is not a
parameter defining the theory but is rather determined by the dynamics of the theory.
8 For simplicity we focus on closed string amplitudes in this section.