310
15 Closed String Field Theory
where we recall that the basis states satisfy
b 0 |φ ↓↓,r = ¯
b 0 |φ ↓↓,r = 0,
b 0 |φ ↓↑,r = ¯
c 0 |φ ↓↑,r = 0,
c 0 |φ ↑↓,r = ¯
b 0 |φ ↑↓,r = 0,
c 0 |φ ↑↑,r = ¯
c 0 |φ ↑↑,r = 0.
(15.3)
We recall the definition of the dual basis {φ c
r } through the BPZ inner product
φ
c
r |φ s = δ rs .
(15.4)
In terms of the ghost decomposition, the components of the dual states satisfy
φ
c
↓↓,r | c 0 = =φ
c
↓↓,r | ¯
c 0 = 0,
φ
c
↓↑,r | c 0 = =φ
c
↓↑,r | ¯
b 0 = 0,
φ
c
↑↓,r | b 0 = =φ
c
↑↓,r | ¯
c 0 = 0,
φ
c
↑↑,r | b 0 = =φ
c
↑↑,r | ¯
b 0 = 0,
φ
c
x,r |φ y,s = δ xy δ rs ,
(15.5)
where x, y =↓↓, ↑↓, ↓↑, ↑↑. The spacetime ghost number of the fields ψ r is
defined by
G(ψ r ) = 2 − n r .
(15.6)
Remember that the ghost number of the basis states is denoted by
n r = N gh (φ r ),
n
c
r = N gh (φ
c
r ) = 6 − n r .
(15.7)
15.2 Gauge Fixed Theory
Having built the kinetic term (Chap. 9), one needs to construct the interactions.
For the same reason—our ignorance of SFT first principles—that forced us to start
with the free equation of motion to derive the quadratic action (Chap. 10), we also
need to infer the interactions from the scattering amplitudes. Preparing the stage
for this analysis was the goal of Chap. 14, where we introduced the factorization of
amplitudes to derive the fundamental interactions.
Scattering amplitudes are expressed in terms of gauge fixed states since only
them are physical. This allows to give an alternative derivation of the kinetic term
by defining it as the inverse of the propagator, which is well-defined for gauge fixed
states. 1 The price to pay by constructing interactions in this way is that the SFT
1 This step is not necessary because the propagator corresponding to the plumbing fixture
(Sect. 14.2.2) matches the one found in Sect. 10.5 by considering the simplest gauge fixing.
However, this would have been necessary if the factorization had given another propagator, or
if the structure of the theory was more complicated, for example for the superstring.
15 Closed String Field Theory
where we recall that the basis states satisfy
b 0 |φ ↓↓,r = ¯
b 0 |φ ↓↓,r = 0,
b 0 |φ ↓↑,r = ¯
c 0 |φ ↓↑,r = 0,
c 0 |φ ↑↓,r = ¯
b 0 |φ ↑↓,r = 0,
c 0 |φ ↑↑,r = ¯
c 0 |φ ↑↑,r = 0.
(15.3)
We recall the definition of the dual basis {φ c
r } through the BPZ inner product
φ
c
r |φ s = δ rs .
(15.4)
In terms of the ghost decomposition, the components of the dual states satisfy
φ
c
↓↓,r | c 0 = =φ
c
↓↓,r | ¯
c 0 = 0,
φ
c
↓↑,r | c 0 = =φ
c
↓↑,r | ¯
b 0 = 0,
φ
c
↑↓,r | b 0 = =φ
c
↑↓,r | ¯
c 0 = 0,
φ
c
↑↑,r | b 0 = =φ
c
↑↑,r | ¯
b 0 = 0,
φ
c
x,r |φ y,s = δ xy δ rs ,
(15.5)
where x, y =↓↓, ↑↓, ↓↑, ↑↑. The spacetime ghost number of the fields ψ r is
defined by
G(ψ r ) = 2 − n r .
(15.6)
Remember that the ghost number of the basis states is denoted by
n r = N gh (φ r ),
n
c
r = N gh (φ
c
r ) = 6 − n r .
(15.7)
15.2 Gauge Fixed Theory
Having built the kinetic term (Chap. 9), one needs to construct the interactions.
For the same reason—our ignorance of SFT first principles—that forced us to start
with the free equation of motion to derive the quadratic action (Chap. 10), we also
need to infer the interactions from the scattering amplitudes. Preparing the stage
for this analysis was the goal of Chap. 14, where we introduced the factorization of
amplitudes to derive the fundamental interactions.
Scattering amplitudes are expressed in terms of gauge fixed states since only
them are physical. This allows to give an alternative derivation of the kinetic term
by defining it as the inverse of the propagator, which is well-defined for gauge fixed
states. 1 The price to pay by constructing interactions in this way is that the SFT
1 This step is not necessary because the propagator corresponding to the plumbing fixture
(Sect. 14.2.2) matches the one found in Sect. 10.5 by considering the simplest gauge fixing.
However, this would have been necessary if the factorization had given another propagator, or
if the structure of the theory was more complicated, for example for the superstring.