34
3 String Theory
Fig. 3.4 Contact vertex for
three strings
Fig. 3.5 A tetrahedron
describing the scattering of
four strings
rotation of these curves around the puncture. Due to the restriction b
−
0 ψ = L
−
0 ψ =
0, ψ ∈ V , this prescription is well defined.
Let us first consider a contact interaction for 3 strings. Without loss of generality,
we can assume that each of the three external strings has circumference 2π. The 3string vertex, denoted by ν 3 , can be represented by a 2-sphere with three semicircles
from the north to the south pole at the relative angle of 2π/3 as in Fig. 3.4.
This defines a symmetric product on V upon gluing of the three semicircles with
the boundaries of the disks used for representing the respective states as in (3.6).
For the point particle, this vertex already provides a consistent action as we saw in
Chap. 2. However, for the string that cannot be so as can be seen by examining the
four-point vertex represented by a spherical tetrahedron in Fig. 3.5 subject to the
condition that the boundary of each face has length 2π (four equations).
Since the tetrahedron has six edges, we are left with two variables which in
turn parametrize the moduli space ˆ
P 4 of 4-punctured spheres together with the
corresponding coordinate curves around each puncture. The solution to the four
constraint equations is best represented by a tetrahedron with opposite edges of
equal length a, b, and c, respectively, see Fig. 3.5, subject to the constraint
a + b + c = 2π .
(3.10)
Solutions to this constraint give a parameterization of the relevant moduli space
for four punctures. It is easy to see that not all of the moduli space can be covered by
joining two cubic vertices with a closed string propagator represented by an internal
cylinder of circumference 2π. Indeed, the tetrahedron obtained in this way, in the
limit of a collapsing propagator, has two edges with length π, see Fig. 3.6.
The result is that by joining two cubic vertices in all possible, nonequivalent
ways with a propagator of positive or vanishing length, one covers the part of the
moduli space of the 4-vertex satisfying (3.10), which is the complement to the subset
parametrized by (3.10) subject to the condition a, b, c < π. Gluing two cubic
vertices in all possible, nonequivalent ways gives the boundary of that region. In
order to cover the whole moduli space one needs to add an elementary four-vertex
3 String Theory
Fig. 3.4 Contact vertex for
three strings
Fig. 3.5 A tetrahedron
describing the scattering of
four strings
rotation of these curves around the puncture. Due to the restriction b
−
0 ψ = L
−
0 ψ =
0, ψ ∈ V , this prescription is well defined.
Let us first consider a contact interaction for 3 strings. Without loss of generality,
we can assume that each of the three external strings has circumference 2π. The 3string vertex, denoted by ν 3 , can be represented by a 2-sphere with three semicircles
from the north to the south pole at the relative angle of 2π/3 as in Fig. 3.4.
This defines a symmetric product on V upon gluing of the three semicircles with
the boundaries of the disks used for representing the respective states as in (3.6).
For the point particle, this vertex already provides a consistent action as we saw in
Chap. 2. However, for the string that cannot be so as can be seen by examining the
four-point vertex represented by a spherical tetrahedron in Fig. 3.5 subject to the
condition that the boundary of each face has length 2π (four equations).
Since the tetrahedron has six edges, we are left with two variables which in
turn parametrize the moduli space ˆ
P 4 of 4-punctured spheres together with the
corresponding coordinate curves around each puncture. The solution to the four
constraint equations is best represented by a tetrahedron with opposite edges of
equal length a, b, and c, respectively, see Fig. 3.5, subject to the constraint
a + b + c = 2π .
(3.10)
Solutions to this constraint give a parameterization of the relevant moduli space
for four punctures. It is easy to see that not all of the moduli space can be covered by
joining two cubic vertices with a closed string propagator represented by an internal
cylinder of circumference 2π. Indeed, the tetrahedron obtained in this way, in the
limit of a collapsing propagator, has two edges with length π, see Fig. 3.6.
The result is that by joining two cubic vertices in all possible, nonequivalent
ways with a propagator of positive or vanishing length, one covers the part of the
moduli space of the 4-vertex satisfying (3.10), which is the complement to the subset
parametrized by (3.10) subject to the condition a, b, c < π. Gluing two cubic
vertices in all possible, nonequivalent ways gives the boundary of that region. In
order to cover the whole moduli space one needs to add an elementary four-vertex