15
Closed String Field Theory
Abstract
We bring together the elements from the previous chapters in order to write
the closed string field action. We first study the gauge fixed theory before
reintroducing the gauge invariance. We then prove that the action satisfies the
BV master equation meaning that closed SFT is completely consistent at the
quantum level. Finally, we describe the 1PI effective action.
15.1 Closed String Field Expansion
In Chaps. 11, 13 and 14, constraints on the external and internal states were found
to be necessary. But, to provide another perspective and decouple the properties of
the field from the ones of the state, we assume that the string field does not obey any
constraint. They will be derived later in order to reproduce the scattering amplitudes
from the action and to make the latter well-defined.
The string field is expanded on a basis {φ r } of the CFT Hilbert space H (see
Sect. 11.2 for more details)
| =
r
ψ r |φ r .
(15.1)
Using the decomposition (11.40) of the Hilbert space according to the ghost zeromodes, the string field can also be expanded as
| =
r
ψ ↓↓,r |φ ↓↓,r + ψ ↓↑,r |φ ↓↑,r + ψ ↑↓,r |φ ↑↓,r + ψ ↑↑,r |φ ↑↑,r
,
(15.2)
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_15
309
Closed String Field Theory
Abstract
We bring together the elements from the previous chapters in order to write
the closed string field action. We first study the gauge fixed theory before
reintroducing the gauge invariance. We then prove that the action satisfies the
BV master equation meaning that closed SFT is completely consistent at the
quantum level. Finally, we describe the 1PI effective action.
15.1 Closed String Field Expansion
In Chaps. 11, 13 and 14, constraints on the external and internal states were found
to be necessary. But, to provide another perspective and decouple the properties of
the field from the ones of the state, we assume that the string field does not obey any
constraint. They will be derived later in order to reproduce the scattering amplitudes
from the action and to make the latter well-defined.
The string field is expanded on a basis {φ r } of the CFT Hilbert space H (see
Sect. 11.2 for more details)
| =
r
ψ r |φ r .
(15.1)
Using the decomposition (11.40) of the Hilbert space according to the ghost zeromodes, the string field can also be expanded as
| =
r
ψ ↓↓,r |φ ↓↓,r + ψ ↓↑,r |φ ↓↑,r + ψ ↑↓,r |φ ↑↓,r + ψ ↑↑,r |φ ↑↑,r
,
(15.2)
© Springer Nature Switzerland AG 2021
H. Erbin, String Field Theory, Lecture Notes in Physics 980,
https://doi.org/10.1007/978-3-030-65321-7_15
309