56
4 Open and Closed Strings
Fig. 4.1 Open-closed
world-sheet with open string
punctures on the boundary
and closed string punctures in
the bulk
where the conformal weights are h ∂ z φ = 1, h c = −1, h b = 2, and the modes satisfy
the commutation relations
[α
μ
m , α
ν
n ] = mg
μν δ m+n,0 , {c m , b n } = δ m+n,0 .
Similarly,
L n =
dz
2π
z
n+1 T (z) ,
T (z)| Im(z<0) = ¯
T (¯ z)| Im(z>0) .
In the operator formalism, the space of states V o is generated by acting with the
creation operators on the vacuum |0, k. The grading on V o is again induced by
assigning ghost number one to c, minus one to b and zero to φ, i.e. every c mode
increases the ghost number by one, whereas the b modes decrease the ghost number
by one. The operator-state correspondence for open string states works much the
same as for the closed case by mapping τ = −∞ to z = 0. We then define
the corresponding Fock space as the set of polynomials in the fields and their
derivatives Φ(φ, b, c, ∂ z φ, · · · ) at the origin. The vector Φ[ ¯
φ, ¯
b, ¯
c] ∈ V o is then
defined by evaluating the path integral measure in (3.3) on the half disk with
Φ(φ, b, c, ∂ z φ, · · · ) inserted at the origin and subject to boundary conditions at
|z| = 1
Φ[ ¯
φ, ¯
b, ¯
c] =
{φ,b,c}| ∂À ={ ¯
φ, ¯
b, ¯
c}
D[φ, b, c] e
−I (φ.b.c.h) Φ(φ, ∂ z φ, · · · )(0) .
A physical open string state is then a ghost number one state that has a representative
as a conformal primary with conformal weight h = 0 of the form
Φ(0) = Φ(φ(0), ∂ z φ(0), · · · )c(0) ,
where the c(0)-insertion represents the globally defined vector field on H that is nonvanishing at z = 0 and which generates the translation of the origin. Nonphysical
states correspond to non-primary insertions with arbitrary dependence on c and its
derivatives. Utilizing the operator-state correspondence, we can identify every state
φ ∈ ˜
V o with a local operator O φ and define the BPZ inner product by
1 , Φ 2 := lim
z→0
(I
∗
O Φ 1 )(z)O Φ 2 (z)
H
,
4 Open and Closed Strings
Fig. 4.1 Open-closed
world-sheet with open string
punctures on the boundary
and closed string punctures in
the bulk
where the conformal weights are h ∂ z φ = 1, h c = −1, h b = 2, and the modes satisfy
the commutation relations
[α
μ
m , α
ν
n ] = mg
μν δ m+n,0 , {c m , b n } = δ m+n,0 .
Similarly,
L n =
dz
2π
z
n+1 T (z) ,
T (z)| Im(z<0) = ¯
T (¯ z)| Im(z>0) .
In the operator formalism, the space of states V o is generated by acting with the
creation operators on the vacuum |0, k. The grading on V o is again induced by
assigning ghost number one to c, minus one to b and zero to φ, i.e. every c mode
increases the ghost number by one, whereas the b modes decrease the ghost number
by one. The operator-state correspondence for open string states works much the
same as for the closed case by mapping τ = −∞ to z = 0. We then define
the corresponding Fock space as the set of polynomials in the fields and their
derivatives Φ(φ, b, c, ∂ z φ, · · · ) at the origin. The vector Φ[ ¯
φ, ¯
b, ¯
c] ∈ V o is then
defined by evaluating the path integral measure in (3.3) on the half disk with
Φ(φ, b, c, ∂ z φ, · · · ) inserted at the origin and subject to boundary conditions at
|z| = 1
Φ[ ¯
φ, ¯
b, ¯
c] =
{φ,b,c}| ∂À ={ ¯
φ, ¯
b, ¯
c}
D[φ, b, c] e
−I (φ.b.c.h) Φ(φ, ∂ z φ, · · · )(0) .
A physical open string state is then a ghost number one state that has a representative
as a conformal primary with conformal weight h = 0 of the form
Φ(0) = Φ(φ(0), ∂ z φ(0), · · · )c(0) ,
where the c(0)-insertion represents the globally defined vector field on H that is nonvanishing at z = 0 and which generates the translation of the origin. Nonphysical
states correspond to non-primary insertions with arbitrary dependence on c and its
derivatives. Utilizing the operator-state correspondence, we can identify every state
φ ∈ ˜
V o with a local operator O φ and define the BPZ inner product by
1 , Φ 2 := lim
z→0
(I
∗
O Φ 1 )(z)O Φ 2 (z)
H
,