de Sitter Vacua in the String Landscape: La Petite Version
449
description in four-dimensions with de Sitter isometries [13, 14, 16]. Thus these
backgrounds are truly in the swampland [7]. On the other hand, once the internal
degrees of freedom become time-dependent, miraculously the effective field theory
description reappears [13, 14].
Why is this the case? The answer can be ascertained from many angles, but
we shall adopt the strategy of uplifting the IIB background to M-theory and then
studying the dynamics from there. The reason for choosing M-theory as opposed
to IIB is not just for the sheer compactness of the representation of the degrees of
freedom (the number of degrees of freedom of course does not change from IIB
to M-theory), but also for the fact that M-theory allows a Lagrangian formulation
(even with non-local counter-terms, as we shall discuss a bit later) as opposed
to IIB, where a Lagrangian formulation is harder to come by. Existence of a
Lagrangian formulation then allows us to express the most generic form of the
quantum corrections as a series in polynomial powers of four-form flux components
and Riemann curvature tensors as well as with spatial and temporal derivatives as
shown in eq. (2.6) of [14]. The spatial derivatives are with respect to the internal
eight-manifold which is an orbifolded torus fibered over the six-dimensional base
that we have in the IIB side. The existence of an effective field theory in lower
dimensions then depends on whether the quantum series allows a hierarchy or not.
This hierarchy should of course be with respect to M p , but also with respect to any
other relevant expansion parameter. The only other allowed expansion parameter is
the type IIA string coupling g s , so the question of hierarchy boils down to finding a
finite number of operators at any order in g
|a|
s /M b
p , where (a, b) ∈
Z
3 , Z
, and the
1/3 moding has been explained in [13, 14]. Note that while b can have any sign, we
take only positive powers of g s . This is because all the negative powers of g s can be
resummed into powers of exp
−
1
g
1/3
s
, taking the expected non-perturbative form
and expressing the full set of corrections as a resurgent trans-series.
All of these would make sense when g s 1. As a happy coincidence, the type
IIA coupling g s turns out to be related to the dimensionless temporal coordinate
in the IIB side (the dimension being determined by the cosmological constant Λ).
This means at late time, where we expect Λ|t| 2 to vanish (recall that we are in the
flat slicing), the type IIA string coupling g s also vanishes, implying that the weak
coupling limit is also the late time limit in our case. This is good, but the fact that
g s now becomes time-dependent implies that one needs to deal with the quantum
series (eq (2.6) in [14]) much more carefully. In fact one would also have to assign
certain temporal dependences of the background G-flux components in M-theory. A
simple ansatz for the G-flux is given in eq. (2.5) of [14] where, note that, we have
traded the temporal dependences with g s dependences. This means generically all
degrees of freedom may be expanded into a series of perturbative as well as nonperturbative g s -dependent terms. At late time, the non-perturbative terms decouple,
so we are left with a perturbative series in g s .
A careful study of the quantum series then reveals the following interesting fact.
One can easily isolate the dominant g s coefficient of the quantum series, which we
449
description in four-dimensions with de Sitter isometries [13, 14, 16]. Thus these
backgrounds are truly in the swampland [7]. On the other hand, once the internal
degrees of freedom become time-dependent, miraculously the effective field theory
description reappears [13, 14].
Why is this the case? The answer can be ascertained from many angles, but
we shall adopt the strategy of uplifting the IIB background to M-theory and then
studying the dynamics from there. The reason for choosing M-theory as opposed
to IIB is not just for the sheer compactness of the representation of the degrees of
freedom (the number of degrees of freedom of course does not change from IIB
to M-theory), but also for the fact that M-theory allows a Lagrangian formulation
(even with non-local counter-terms, as we shall discuss a bit later) as opposed
to IIB, where a Lagrangian formulation is harder to come by. Existence of a
Lagrangian formulation then allows us to express the most generic form of the
quantum corrections as a series in polynomial powers of four-form flux components
and Riemann curvature tensors as well as with spatial and temporal derivatives as
shown in eq. (2.6) of [14]. The spatial derivatives are with respect to the internal
eight-manifold which is an orbifolded torus fibered over the six-dimensional base
that we have in the IIB side. The existence of an effective field theory in lower
dimensions then depends on whether the quantum series allows a hierarchy or not.
This hierarchy should of course be with respect to M p , but also with respect to any
other relevant expansion parameter. The only other allowed expansion parameter is
the type IIA string coupling g s , so the question of hierarchy boils down to finding a
finite number of operators at any order in g
|a|
s /M b
p , where (a, b) ∈
Z
3 , Z
, and the
1/3 moding has been explained in [13, 14]. Note that while b can have any sign, we
take only positive powers of g s . This is because all the negative powers of g s can be
resummed into powers of exp
−
1
g
1/3
s
, taking the expected non-perturbative form
and expressing the full set of corrections as a resurgent trans-series.
All of these would make sense when g s 1. As a happy coincidence, the type
IIA coupling g s turns out to be related to the dimensionless temporal coordinate
in the IIB side (the dimension being determined by the cosmological constant Λ).
This means at late time, where we expect Λ|t| 2 to vanish (recall that we are in the
flat slicing), the type IIA string coupling g s also vanishes, implying that the weak
coupling limit is also the late time limit in our case. This is good, but the fact that
g s now becomes time-dependent implies that one needs to deal with the quantum
series (eq (2.6) in [14]) much more carefully. In fact one would also have to assign
certain temporal dependences of the background G-flux components in M-theory. A
simple ansatz for the G-flux is given in eq. (2.5) of [14] where, note that, we have
traded the temporal dependences with g s dependences. This means generically all
degrees of freedom may be expanded into a series of perturbative as well as nonperturbative g s -dependent terms. At late time, the non-perturbative terms decouple,
so we are left with a perturbative series in g s .
A careful study of the quantum series then reveals the following interesting fact.
One can easily isolate the dominant g s coefficient of the quantum series, which we