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K. Dasgupta et al.
coefficients we are as though allowing g s independent pieces in our original G-flux
ansatz eq. (2.5) of [14]. This would be like having k 2 = 0 in eq (2.6) of [14],
leading to the above-mentioned problems. Thus the coefficients of the localized
G-flux components should also have g s dependent factors. Such factors easily
arise by simply normalizing the localized functions! Putting everything together
immediately reproduces eq. (2.28) of [14], implying finite number of operators at
zeroth orders in g s and M p with time-dependent internal degrees of freedom.
The story does not end here because M-theory could also have non-local quantum
terms, these terms are sometime christened as non-local counter-terms. A simple
way to infer their presence is to go to the infinite derivative limits of eq. (2.6) in
[14]. More generic form of non-localities are possible in the limit where (n 0 , n i )
themselves become negative. Such negative-derivative actions may be rewritten as
nested integrals as shown in [13] and [14], thus allowing a Lagrangian formulation
of the scenario. One however needs to be careful in introducing such terms because
we do not expect the non-localities to show up in the low energy limit of M-theory.
The validity of the non-local quantum terms has been discussed in great details in
[13], and we urge the readers to look up the discussions therein. What we want to
question here is what happens to the two cases once we switch on these non-local
quantum terms.
The answer as meticulously shown in section 3 of [14], to any given order in
g
|a|
s /M b
p , there are only finite number of operators possible when all the internal
degrees of freedom become time-dependent, whereas the number of such operators
tend to become infinite once time-dependences are switched off. These have been
demonstrated in figure 1–4 in [14], where moving vertically down in any figure
counts the number of operators. The vertical axis in each figure represents the level
of non-locality (i.e. how many nested integrals are taken into account), and the
horizontal axis represents the M p scalings. Thus it appears that non-localities do
not seem to change the generic idea that four-dimensional effective field theory
description with de Sitter isometries remains valid whenever the internal degrees of
freedom become time-dependent, but fails when the time-dependences are switched
off. Interestingly all of these happen while still keeping the four-dimensional
Newton constant time-independent.
There are many other questions that may be asked at this stage, for example: How
are the moduli stabilized? How are the equations of motion satisfied? How does the
flux quantization, and anomaly cancellation work out in the time-dependent cases?
All of these have been answered, and to preserve the brevity of the note we will refer
the interested readers to our longer works [13] and [14]. We will however make two
comments, one, related to the quantum potential as it appears in eq. (2.7) of [14] and
two, related to the possibility of realizing inflation in our construction.
The quantum potential, with contributions from both local and non-local quantum terms, gives rise to an exact expression for the renormalized cosmological
constant Λ as given in eq. (4.1) of [14]. This contribution appears from the zeroth
order in g s and second order in M p (i.e. to order g 0
s /M 2
p once we assume the nonlocal contributions are suppressed by the toroidal volume in M-theory). Why don’t
K. Dasgupta et al.
coefficients we are as though allowing g s independent pieces in our original G-flux
ansatz eq. (2.5) of [14]. This would be like having k 2 = 0 in eq (2.6) of [14],
leading to the above-mentioned problems. Thus the coefficients of the localized
G-flux components should also have g s dependent factors. Such factors easily
arise by simply normalizing the localized functions! Putting everything together
immediately reproduces eq. (2.28) of [14], implying finite number of operators at
zeroth orders in g s and M p with time-dependent internal degrees of freedom.
The story does not end here because M-theory could also have non-local quantum
terms, these terms are sometime christened as non-local counter-terms. A simple
way to infer their presence is to go to the infinite derivative limits of eq. (2.6) in
[14]. More generic form of non-localities are possible in the limit where (n 0 , n i )
themselves become negative. Such negative-derivative actions may be rewritten as
nested integrals as shown in [13] and [14], thus allowing a Lagrangian formulation
of the scenario. One however needs to be careful in introducing such terms because
we do not expect the non-localities to show up in the low energy limit of M-theory.
The validity of the non-local quantum terms has been discussed in great details in
[13], and we urge the readers to look up the discussions therein. What we want to
question here is what happens to the two cases once we switch on these non-local
quantum terms.
The answer as meticulously shown in section 3 of [14], to any given order in
g
|a|
s /M b
p , there are only finite number of operators possible when all the internal
degrees of freedom become time-dependent, whereas the number of such operators
tend to become infinite once time-dependences are switched off. These have been
demonstrated in figure 1–4 in [14], where moving vertically down in any figure
counts the number of operators. The vertical axis in each figure represents the level
of non-locality (i.e. how many nested integrals are taken into account), and the
horizontal axis represents the M p scalings. Thus it appears that non-localities do
not seem to change the generic idea that four-dimensional effective field theory
description with de Sitter isometries remains valid whenever the internal degrees of
freedom become time-dependent, but fails when the time-dependences are switched
off. Interestingly all of these happen while still keeping the four-dimensional
Newton constant time-independent.
There are many other questions that may be asked at this stage, for example: How
are the moduli stabilized? How are the equations of motion satisfied? How does the
flux quantization, and anomaly cancellation work out in the time-dependent cases?
All of these have been answered, and to preserve the brevity of the note we will refer
the interested readers to our longer works [13] and [14]. We will however make two
comments, one, related to the quantum potential as it appears in eq. (2.7) of [14] and
two, related to the possibility of realizing inflation in our construction.
The quantum potential, with contributions from both local and non-local quantum terms, gives rise to an exact expression for the renormalized cosmological
constant Λ as given in eq. (4.1) of [14]. This contribution appears from the zeroth
order in g s and second order in M p (i.e. to order g 0
s /M 2
p once we assume the nonlocal contributions are suppressed by the toroidal volume in M-theory). Why don’t