16.2 Problem Setup
329
of the action [11, 12], or at the level of the equations of motion [10]. The advantage
of the second approach is that one can use the 1PI theory, which simplifies vastly
the analysis. It also generalizes directly to the super-SFT.
Remark 16.1 (Field Theory on the CFT Space) As mentioned earlier, the string
field is defined as a functional on the state space of a given CFT and not as a
functional on the field theory space (off-shell states would correspond to general
QFTs, and only on-shell states are CFTs). In this case, background independence
would amount to reparametrization invariance of the action in the theory space and
would thus almost automatically hold. A complete formulation of SFT following
this line is currently out of reach, but some ideas can be found in [15].
16.2 Problem Setup
Given a SFT on a background, there are two ways to describe it on another
background:
• deform the worldsheet CFT and express the SFT on the new background;
• expand the original action around the infinitesimal classical solution (to the
linearized equations of motion) corresponding to the deformation.
Background independence amounts to the equivalence of both theories up to a field
redefinition. The derivation can be performed at the level of the action or of the
equations of motion. To prove the background independence at the quantum level,
one needs to take into account the changes in the path integral measure or to work
with the 1PI action.
The simplest case is when the two CFTs are related by an infinitesimal marginal
deformation
δS cft =
λ
2π
d
2 z ϕ(z, ¯
z),
(16.1)
with ϕ a (1, 1) primary operator and λ infinitesimal. The two CFTs are denoted
by CFT 1 and CFT 2 , and quantities associated to each CFTs are indexed with the
appropriate number.
Establishing background independence in this case also implies it for finite
marginal deformation since they can be built from a series of successive deformations. In the latter case, the field redefinition may be singular, which reflects that the
parametrization of one CFT is not adapted for the other (equivalently, the coordinate
systems for the string field break down), which is expected if both CFTs are far in
the field theory space.
329
of the action [11, 12], or at the level of the equations of motion [10]. The advantage
of the second approach is that one can use the 1PI theory, which simplifies vastly
the analysis. It also generalizes directly to the super-SFT.
Remark 16.1 (Field Theory on the CFT Space) As mentioned earlier, the string
field is defined as a functional on the state space of a given CFT and not as a
functional on the field theory space (off-shell states would correspond to general
QFTs, and only on-shell states are CFTs). In this case, background independence
would amount to reparametrization invariance of the action in the theory space and
would thus almost automatically hold. A complete formulation of SFT following
this line is currently out of reach, but some ideas can be found in [15].
16.2 Problem Setup
Given a SFT on a background, there are two ways to describe it on another
background:
• deform the worldsheet CFT and express the SFT on the new background;
• expand the original action around the infinitesimal classical solution (to the
linearized equations of motion) corresponding to the deformation.
Background independence amounts to the equivalence of both theories up to a field
redefinition. The derivation can be performed at the level of the action or of the
equations of motion. To prove the background independence at the quantum level,
one needs to take into account the changes in the path integral measure or to work
with the 1PI action.
The simplest case is when the two CFTs are related by an infinitesimal marginal
deformation
δS cft =
λ
2π
d
2 z ϕ(z, ¯
z),
(16.1)
with ϕ a (1, 1) primary operator and λ infinitesimal. The two CFTs are denoted
by CFT 1 and CFT 2 , and quantities associated to each CFTs are indexed with the
appropriate number.
Establishing background independence in this case also implies it for finite
marginal deformation since they can be built from a series of successive deformations. In the latter case, the field redefinition may be singular, which reflects that the
parametrization of one CFT is not adapted for the other (equivalently, the coordinate
systems for the string field break down), which is expected if both CFTs are far in
the field theory space.