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16 Background Independence
computations—and in particular for the questions of dynamics and quantization.
Generally, these questions are subsumed into the problem of the emergence of time
in a generally covariant theory. In the previous language, QFTs without gravity
are (generically) manifestly background independent after minimal coupling. 2 For
example, a classical field theory is defined on a fixed Minkowski background, and
a well-defined time is necessary to perform its quantization and to obtain a QFT,
but it is not needed to choose a background for the other fields. For this reason, the
extension of a QFT on a curved background is generally possible if the spacetime
is hyperbolic, implying that there is a distinguished time direction. But the coupling
to gravity is difficult and restricted to a (semi-)classical description.
What is the status of background independence in string theory? The worldsheet
formulation requires to fix a background (usually Minkowski) to quantize the theory
and to compute scattering amplitudes. Thus, the quantum theory is at least not
manifestly background independent. On the other hand, the worldsheet action can be
modified to a generic CFT including a generic non-linear sigma model describing
an arbitrary target spacetime. Conformal invariance reproduces (at leading order)
Einstein equations coupled to various matters and gauge field equations of motion.
From this point of view, the classical theory can be written as a manifestly
background independent theory, and this provides hopes that the quantum theory
may also be background independent, even if non-manifestly. This idea is supported
by other definitions of string theory (e.g. through the AdS/CFT conjecture—and
other holographic realizations—or through matrix models) that provide, at least
partially, background independent formulations.
Ultimately, the greatest avenue to establish the background independence is
string field theory. Indeed, the form of the SFT action and of its properties (gauge
invariance, equation of motion. . . ) are identical irrespective of the background [13].
This provides a good starting point. The background dependence enters in the
precise definition of the string products (BRST operator and vertices). The origin of
this dependence lies in the derivation of the action (Chaps. 14 and 15): one begins
with a particular CFT describing a given background (spacetime compactifications,
fluxes, etc.) and defines the vertices from correlation functions of vertex operators,
and the Hilbert space from the CFT operators. As a consequence, even though
it is clear that no specific property of the background has been used in the
derivation—and that the final action describes SFT for any background—this is
not sufficient to establish background independence. Since the theory assumes
implicitly a background choice, one cannot guarantee that the physical quantities
have no residual dependence in the background, even if the action looks superficially
background independent. Background independence in SFT is thus the statement
that theories characterized by different CFTs can be related by a field redefinition.
In this chapter, we will sketch the proof of background independence for
backgrounds related by marginal deformations. 3 It is possible to prove it at the level
2 However, non-minimal coupling terms may be necessary to make the theory physical.
3 An alternative approach based on morphism of L ∞ algebra is followed in [4].
16 Background Independence
computations—and in particular for the questions of dynamics and quantization.
Generally, these questions are subsumed into the problem of the emergence of time
in a generally covariant theory. In the previous language, QFTs without gravity
are (generically) manifestly background independent after minimal coupling. 2 For
example, a classical field theory is defined on a fixed Minkowski background, and
a well-defined time is necessary to perform its quantization and to obtain a QFT,
but it is not needed to choose a background for the other fields. For this reason, the
extension of a QFT on a curved background is generally possible if the spacetime
is hyperbolic, implying that there is a distinguished time direction. But the coupling
to gravity is difficult and restricted to a (semi-)classical description.
What is the status of background independence in string theory? The worldsheet
formulation requires to fix a background (usually Minkowski) to quantize the theory
and to compute scattering amplitudes. Thus, the quantum theory is at least not
manifestly background independent. On the other hand, the worldsheet action can be
modified to a generic CFT including a generic non-linear sigma model describing
an arbitrary target spacetime. Conformal invariance reproduces (at leading order)
Einstein equations coupled to various matters and gauge field equations of motion.
From this point of view, the classical theory can be written as a manifestly
background independent theory, and this provides hopes that the quantum theory
may also be background independent, even if non-manifestly. This idea is supported
by other definitions of string theory (e.g. through the AdS/CFT conjecture—and
other holographic realizations—or through matrix models) that provide, at least
partially, background independent formulations.
Ultimately, the greatest avenue to establish the background independence is
string field theory. Indeed, the form of the SFT action and of its properties (gauge
invariance, equation of motion. . . ) are identical irrespective of the background [13].
This provides a good starting point. The background dependence enters in the
precise definition of the string products (BRST operator and vertices). The origin of
this dependence lies in the derivation of the action (Chaps. 14 and 15): one begins
with a particular CFT describing a given background (spacetime compactifications,
fluxes, etc.) and defines the vertices from correlation functions of vertex operators,
and the Hilbert space from the CFT operators. As a consequence, even though
it is clear that no specific property of the background has been used in the
derivation—and that the final action describes SFT for any background—this is
not sufficient to establish background independence. Since the theory assumes
implicitly a background choice, one cannot guarantee that the physical quantities
have no residual dependence in the background, even if the action looks superficially
background independent. Background independence in SFT is thus the statement
that theories characterized by different CFTs can be related by a field redefinition.
In this chapter, we will sketch the proof of background independence for
backgrounds related by marginal deformations. 3 It is possible to prove it at the level
2 However, non-minimal coupling terms may be necessary to make the theory physical.
3 An alternative approach based on morphism of L ∞ algebra is followed in [4].