de Sitter Vacua in the String Landscape: La Petite Version
451
The answer lies in the careful mapping between the degrees of freedom of IIB in
M-theory. For example not all G-flux components can be allowed without breaking
the four-dimensional de Sitter isometries in the IIB side. In particular, if we allow the
internal G-flux components with two components along the toroidal directions, they
cannot be global fluxes. In fact they can only be localized fluxes so that they appear
as gauge fluxes on seven branes in the IIB side (the seven branes being related to
the orbifold points in M-theory). In the simplest case, the localized pieces of the Gflux components can be viewed as a Gaussian over the toroidal space. The spread of
such Gaussian pieces is measured in terms of M p , at least for the time-independent
cases, implying that the localized fluxes could in principle have hidden M p scalings
implicit in their definitions themselves. Such M p scalings are in general harmless
because they do not interfere or change the M p suppressions of the quantum terms.
However, the quantum terms, i.e. eq. (2.6) of [14], do also have derivatives along the
toroidal directions which we called n 3 earlier. If we switch on these n 3 derivatives,
we see that now they are able to influence the M p scalings of the quantum terms!
As shown in [14], the n 3 derivatives change the M p scalings of the quantum terms
in an interesting way: they introduce a relative minus sign there. The g s scaling θ 0
already had their share of minus signs, and now putting everything together we can
ask, for the time-independent case, how many operators are allowed at zeroth order
in g s and M p (we can even ask at zeroth order in g s but second order in M p ). The
answer is as shown in eq. (2.23), defined in terms of certain variables that may be
expressed in terms of (l i , n i , n 0 ) in eq. (2.16), both equations being from [14]. The
worrisome thing is the relative minus signs in eq. (2.23) whose RHS is 8, related
to the eighth order polynomials in curvature and G-flux components. There are an
infinite number of integer solutions to this equation, all to zeroth order in g s and M p .
Now we have clearly lost both the g s and M p hierarchies, implying that there cannot
be an effective field theory description in the IIB side with de Sitter isometries and
with time-independent internal degrees of freedom.
The above conclusion is definitely intriguing, but couldn’t we say the same thing
for the time-dependent cases too when we switch on the n 3 derivatives? Why should
the time-dependent cases be any different? Answering this will take us to the second
level of subtleties in our construction, that we hitherto kept under the rug.
In the time-dependent case, most of the arguments related to the localized fluxes
go through in a similar way to the time-independent case. However there are now
a few subtle differences. The first difference lies in the scale of the Gaussian
spread of the localized function. As shown in eq. (2.25) in [14], the Gaussian
spread now involves both g s and M p , simply because the distances are measured
in terms of warped metric components (that involve g s dependences). This means
the n 3 derivatives in the quantum terms of eq. (2.6) will contribute to both g s and
M p scalings. We already discussed how the M p scalings change, so the question
now is whether the change in the g s scalings can alter our earlier conclusion. The
answer turns out to be a bit more subtle. If the G-flux ansatz for the localized
flux, as given by eq. (2.15) and eq (2.25) of [14], has g s independent coefficients
(i.e. A n0 in eq. (2.25)), then even the extra g s contributions from n 3 derivative
cannot change the conclusion. The reason is simple to state: with time-independent
451
The answer lies in the careful mapping between the degrees of freedom of IIB in
M-theory. For example not all G-flux components can be allowed without breaking
the four-dimensional de Sitter isometries in the IIB side. In particular, if we allow the
internal G-flux components with two components along the toroidal directions, they
cannot be global fluxes. In fact they can only be localized fluxes so that they appear
as gauge fluxes on seven branes in the IIB side (the seven branes being related to
the orbifold points in M-theory). In the simplest case, the localized pieces of the Gflux components can be viewed as a Gaussian over the toroidal space. The spread of
such Gaussian pieces is measured in terms of M p , at least for the time-independent
cases, implying that the localized fluxes could in principle have hidden M p scalings
implicit in their definitions themselves. Such M p scalings are in general harmless
because they do not interfere or change the M p suppressions of the quantum terms.
However, the quantum terms, i.e. eq. (2.6) of [14], do also have derivatives along the
toroidal directions which we called n 3 earlier. If we switch on these n 3 derivatives,
we see that now they are able to influence the M p scalings of the quantum terms!
As shown in [14], the n 3 derivatives change the M p scalings of the quantum terms
in an interesting way: they introduce a relative minus sign there. The g s scaling θ 0
already had their share of minus signs, and now putting everything together we can
ask, for the time-independent case, how many operators are allowed at zeroth order
in g s and M p (we can even ask at zeroth order in g s but second order in M p ). The
answer is as shown in eq. (2.23), defined in terms of certain variables that may be
expressed in terms of (l i , n i , n 0 ) in eq. (2.16), both equations being from [14]. The
worrisome thing is the relative minus signs in eq. (2.23) whose RHS is 8, related
to the eighth order polynomials in curvature and G-flux components. There are an
infinite number of integer solutions to this equation, all to zeroth order in g s and M p .
Now we have clearly lost both the g s and M p hierarchies, implying that there cannot
be an effective field theory description in the IIB side with de Sitter isometries and
with time-independent internal degrees of freedom.
The above conclusion is definitely intriguing, but couldn’t we say the same thing
for the time-dependent cases too when we switch on the n 3 derivatives? Why should
the time-dependent cases be any different? Answering this will take us to the second
level of subtleties in our construction, that we hitherto kept under the rug.
In the time-dependent case, most of the arguments related to the localized fluxes
go through in a similar way to the time-independent case. However there are now
a few subtle differences. The first difference lies in the scale of the Gaussian
spread of the localized function. As shown in eq. (2.25) in [14], the Gaussian
spread now involves both g s and M p , simply because the distances are measured
in terms of warped metric components (that involve g s dependences). This means
the n 3 derivatives in the quantum terms of eq. (2.6) will contribute to both g s and
M p scalings. We already discussed how the M p scalings change, so the question
now is whether the change in the g s scalings can alter our earlier conclusion. The
answer turns out to be a bit more subtle. If the G-flux ansatz for the localized
flux, as given by eq. (2.15) and eq (2.25) of [14], has g s independent coefficients
(i.e. A n0 in eq. (2.25)), then even the extra g s contributions from n 3 derivative
cannot change the conclusion. The reason is simple to state: with time-independent