∫DϕDψ e^-S_2(φ, ϕ) - S^(1)_ghost(φ, ψ) - S^(2)_ghost(φ, ϕ, ψ)   =   e^-S_nloc(φ) - S_1^(g)(φ) - S_nloc^(g)(φ), which would keep the form of $\hat{\bf S}_{\rm tot}(\varphi)$ from (<ref>) unchanged[It is clear that this is only possible if both ${\bf S}^{({\rm g})}_{1}(\varphi)$ and ${\bf S}_{\rm nloc}^{(g)}(\varphi)$ are expressed using convergent series. Failure to maintain this would result in changing ${\bf S}_{\rm NP}(\varphi)$.], but introduce two new ghost contributions: ${\bf S}_{1}^{({\rm g})}(\varphi)$ and ${\bf S}_{\rm nloc}^{({\rm g})}(\varphi)$, both defined in terms of on-shell field component $\varphi$. These changes (<ref>) to (<ref>). The EOM (<ref>), while much more manageable than (<ref>), still requires some adjustments to bring it in a form where we can identify it to a more realistic scenario. By realistic we mean a scenario close to the vacuum solution that we studied earlier in section <ref>. Looking at Table <ref>, we see that the terms in the energy-momentum tensors that would almost reproduce a de Sitter vacuum solution are the ones coming from the non-perturbative and the non-local parts of the action as elucidated in section <ref>. Unfortunately, and as discussed in section <ref>, the construction fails because of the absence of a well-defined Wilsonian effective action over a temporally varying de Sitter vacuum among other issues. On the other hand, if we demand that the equation in (<ref>) splits into the following two equations:

δŜ_tot(⟨φ⟩_σ)δ⟨φ⟩_σ   =   0

∑_{σ_i} σ⟨δŜ_tot(φ)δφ⟩_({σ_i}|σ) +
⟨δŜ_tot(φ)δφ⟩_(Q̂|σ) +  ⟨δS_1^(g)(φ)δφ⟩_σ+ ⟨δS_nloc^(g)(φ)δφ⟩_σ=   ⟨δδφlog(𝔻_o^†𝔻_o)⟩_σ

then we are very close to the same scenario that we studied in section <ref> and especially in section <ref>! However now there are a few key differences.

${\bf 1.}$ The first EOM in (<ref>) is now for the emergent on-shell degree of freedom $\langle\varphi\rangle_\sigma$ and not for the vacuum configuration. Resorting to actual metric, flux and fermionic degrees of freedom, an equivalent picture[For example if $\langle\Xi\rangle_\sigma = (\langle{\bf g}_{\rm AB}\rangle_\sigma, \langle{\bf C}_{\rm ABC}\rangle_\sigma, \langle\Psi_{\rm A}\rangle_\sigma, \langle \overline{\Psi}_{\rm A}\rangle_\sigma)$, then we are effectively looking at EOMs of the form ${\delta\hat{\bf S}_{\rm tot}(\langle\Xi\rangle_\sigma)\over \delta \langle\Xi\rangle_\sigma} ~ = ~ 0$. This would lead to EOMs as in Table <ref> but for the emergent on-shell degrees of freedom $\langle {\bf g}_{\rm AB}\rangle_\sigma, \langle {\bf C}_{\rm ABC}\rangle_\sigma$ and $\langle \Psi_{\rm A}\rangle_\sigma$.] would lead to the EOMs for the emergent on-shell degrees of freedom $\langle {\bf g}_{\rm AB}\rangle_\sigma, \langle {\bf C}_{\rm ABC}\rangle_\sigma$ and $\langle \Psi_{\rm A}\rangle_\sigma$, where $({\rm A, B, C}) \in {\bf R}^{2, 1} \times {\cal M}_4 \times {\cal M}_2\times {\mathbb{T}^2\over {\cal G}}$.

2. The second equation in (<ref>) is something that we did not encounter earlier. This equation weaves the three crucial elements in the construction together: the intermediate Glauber-Sudarshan states, the Faddeev-Popov ghosts (if any) and the linear term in the displacement operator. Note that all the three contributions cannot be visible at the “classical” level, i.e. at the level of the first equation in (<ref>), thus explaining the natural split of the EOM (<ref>) into a set of two EOMs.

${\bf 3.}$ The issues which prohibited us to declare the results from Table <ref> as evidence for the existence of a de Sitter vacuum solution now no longer plagues us. In fact the first equation in (<ref>) clearly suggests that de Sitter space-time should be an emergent solution appearing from the Glauber-Sudarshan state $\vert\sigma\rangle$. The emergent background is now backed by a well defined effective field theory description that is  consistent with the trans-Planckian bound and other constraints.

There is however a bit more to the above story. The second equation in (<ref>) is not the only non-classical equation that we encounter here. There is yet another set of equations that could appear from the EOM associated from the off-shell field $\phi$. This is (<ref>), and as before we have three possibilities.

${\bf 1.}$ Rewrite it using the non-perturbative completion that we did for the on-shell field $\varphi$ in (<ref>).

${\bf 2.}$ Add in the Faddeev-Popov ghosts and see how the EOM changes to compare with (<ref>).

${\bf 3.}$ Try to see if the equation could be split into two equations as in (<ref>) for the on-shell case.

All the three steps above are now more involved than their on-shell counterparts. To see this, let us start by answering the first two points above, namely the non-perturbative completion of the action and the insertion of the Faddeev-Popov ghosts. For that we can express the Schwinger-Dyson's equation for $\phi$ field in the following way:

∫DφDϕDψ e^-S_1(φ) - S_2(φ, ϕ) - S^(1)_ghost(φ, ψ) - S^(2)_ghost(φ, ϕ, ψ) 𝔻^†(σ, φ) 𝔻(σ, φ)
(δS_2(φ, ϕ)δϕ +
δS^(2)_ghost(φ, ϕ)δϕ) = 0,
where we have followed the same decomposition for ${\bf S}_{\rm tot}(\varphi, \phi)$ and ${\bf S}_{\rm ghost}(\varphi, \phi, \psi)$ as before. One may compare this equation with the one we got earlier, namely (<ref>), and it is clear that all the $\widetilde{f}_i$ functions appearing therein may be derived from ${\delta{\bf S}_2\over \delta\phi}$ above. A more elaborate comparison could be done, but we will leave this for our diligent readers to work them out. The above equation has the following non-perturbative completion:

∫Dφ   𝔻_o^†(σ, φ) 𝔻_o(σ, φ) e^-S_kin(φ) - S_NP(φ)- S_1^(g)(φ) ∫Dϕ δδϕ(e^- S_2(φ, ϕ)  - S_2^(g)(φ, ϕ)) = 0, where ${\bf S}_{1}^{({\rm g})}(\varphi)$ and ${\bf S}_{2}^{({\rm g})}(\varphi, \phi)$ appear from ${\bf S}^{(1)}_{\rm ghost}(\varphi, \psi)$ and ${\bf S}^{(2)}_{\rm ghost}(\varphi, \phi, \psi)$ respectively by integrating over the ghost field $\psi$, and $\mathbb{D}_o(\sigma,\varphi)$ is from (<ref>). ${\bf S}_1^{(g)}(\varphi)$ as before is given by a convergent series, so that it doesn't effect the non-perturbative action. The second term in (<ref>) can be integrated over to give us the following EOM:

∫Dφ e^-Ŝ_tot(φ) - S_1^(g)(φ) - S^(g)_nloc(φ)
(𝕋_nloc(φ) + 𝕋^(g)_nloc(φ))
𝔻_o^†(σ, φ) 𝔻_o(σ, φ) =
⟨𝕋_nloc(φ) + 𝕋^(g)_nloc(φ)⟩_σ= 0,
where both ${\bf S}_{\rm nloc}(\varphi)$ and $\mathbb{T}_{\rm nloc}(\varphi)$ are related and so are ${\bf S}^{(g)}_{\rm nloc}(\varphi)$ and $\mathbb{T}^{(g)}_{\rm nloc}(\varphi)$. Moreover, $\mathbb{T}_{\rm nloc}(\varphi) = \widetilde{f}_2(\varphi) + {\rm T}_{\rm nloc}(\varphi)$ from (<ref>), and $\hat{\bf S}_{\rm tot}(\varphi)$ is the same one that appeared in (<ref>). Using the resolution of identity now splits (<ref>) into the following two equations:

𝕋_nloc(⟨φ⟩_σ) = 0

⟨𝕋_nloc(φ)⟩_({σ_i}|σ) +
⟨𝕋_nloc(φ)⟩_(Q̂|σ) +
⟨𝕋^(g)_nloc(φ)⟩_σ= 0

where combining the first equations from (<ref>) and (<ref>) respectively would provide the necessary EOMs that would control the dynamics of the emergent de Sitter space-time from the Glauber-Sudarshan state. These are the equations that would replace the vacuum EOMs from Table <ref>. The remain two equations from (<ref>) and (<ref>) respectively would relate the dynamics of ghosts and the intermediate Glauber-Sudarshan states with the displacement operator $\mathbb{D}_o(\sigma, \varphi)$ from (<ref>).

The set of equations in (<ref>) signal the non-classical nature of the system and would not appear if we had taken coherent states. One may think of them as the first concrete sign of four-dimensional de Sitter space being an excited state coming from an expectation value of the graviton operator over a Glauber-Sudarshan state.
A specific consequence of these equations appear from the cross-term EOM of ${\bf g}_{0n}$ where $y^n \in {\cal M}_6$ despite the fact that ${\bf g}_{0n}$ is an off-shell state. A more detailed discussion of this will appear elsewhere [9].

Finally, in the third point after (<ref>), we argued that the Glauber-Sudarshan states are well within the temporal bound advocated by the trans-Planckian censorship, and they always satisfy the EFT criteria.  In fact satisfying the trans-Planckian bound is enough to justify the latter [41]. However we would like to dwell a bit on this topic with the aim to dispel any doubts that reader might have regarding the existence of these states without violating any EFT criteria. This will be the content of the following section.

§ GLAUBER-SUDARSHAN STATES AND EFFECTIVE FIELD THEORY CRITERIA

After having got all the EOMs for the emergent on-shell degrees of freedom, it is time to ask how the Glauber-Sudarshan states themselves satisfy the EFT criteria. Of course the way we constructed these states over supersymmetric time-independent background clearly follow the necessary EFT constraints $-$ coming from the ERG procedure and from the restriction we imposed earlier to remain withing the energy range $k_{\rm IR} < k < \mu << \hat\mu$ as depicted in figure <ref> $-$ so it is necessary to explain what we mean by “satisfying the EFT criteria”. Somewhat surprisingly, even after following the aforementioned procedure to construct the Glauber-Sudarshan states, and from there determine the dynamics of the emergent on-shell degrees of freedom, it is still necessary to satisfy the following two criteria [16]:

g_s   <   1,           ∂g_s∂t   ∝  g_s^+ive where $g_s$ is the one that we encountered in (<ref>), and is the dual type IIA coupling with $t$ being the conformal time. The first criterion leads to (<ref>) which restricts the Glauber-Sudarshan states to lie within the temporal domain (<ref>). The second criterion boils down to the statement that the Null Energy Condition (NEC) should not be violated: any violation of the NEC is a violation of the underlying EFT [16]. One might wonder how do (<ref>) appear from the two set of equations (<ref>) and (<ref>). In the following sections we will first give a warm-up example and then, in section <ref> onwards, we will take a more realistic example and argue that many of the EFT criteria alluded to earlier are in fact tied up to (<ref>). Readers wishing to delve directly to the application of (<ref>) on an actual de Sitter background, may skip the following section and go to section <ref>.

§.§ A warm-up example on the EFT criteria for the emergent states

The two sets of equations (<ref>) and (<ref>) contain all the necessary information to determine not only the evolution of the emergent on-shell degrees of freedom, but also to determine $\rho^\ast(k)$ in (<ref>) necessary to construct the Glauber-Sudarshan states. Moreover, it is only the first equations in each of the two sets, that are actually needed to fix the form of $\rho^\ast(k)$. The other two equations aren't directly involved but are essential for the consistency of the underlying picture. The two relevant equations from (<ref>) and (<ref>) are:

δŜ_tot(⟨φ⟩_σ)δ⟨φ⟩_σ   =   0   =   𝕋_nloc(⟨φ⟩_σ), with $\hat{\bf S}_{\rm tot}(\varphi)$ is given by (<ref>). Despite appearance, (<ref>) is not exactly related to the classical EOM. Three key differences are as follows.

${\bf 1.}$ The equations are for the emergent on-shell degree of freedom $\langle\varphi\rangle_\sigma$ and not for the on-shell field $\varphi$.

${\bf 2.}$ The action governing the emergent degree of freedom is not the classical action with field $\varphi$ replaced by $\langle\varphi\rangle_\sigma$, but differs from it by having non-local and non-perturbative contributions that are usually absent in a given classical action (or even with an action with perturbative corrections).

${\bf 3.}$ The second equation with $\mathbb{T}_{\rm nloc}$ is purely a consequence of having a Glauber-Sudarshan state, and therefore completely invisible from either the vacuum or the coherent state point of view. In a similar vein the other two equations, from the two sets (<ref>) and (<ref>) respectively, are also invisible from either the vacuum or the coherent state point of view.

To see the effect of (<ref>) on the EFT, it would be best to first start with a toy model and then go to the original de Sitter space-time from the Glauber-Sudarshan state. For that let us specify the perturbative action as:

S_pert(c; φ)     = M_p^11∫d^11 x (-φ(x)  □M_p^2  φ(x) + Q_pert(c; φ(x)))

= M_p^11∫d^11 x (-φ(x, y)  □M_p^2  φ(x, y) + 1M_p^2n∑_n, p c_np  ∂^2n ·φ^p(x)),

where the action of the derivative follows the procedure outlined in footnote <ref>, and $n \ge 0, p > 2$ to avoid generating mass term in the energy scale $k_{\rm IR} < k < \mu << \hat\mu$. The relevant action[This is an intermediate action because the actual action governing the evolution of the emergent state $\langle\varphi(x, y)\rangle_\sigma$ is $\hat{\bf S}_{\rm tot}(\langle\varphi\rangle_\sigma)$ which is clearly not a non-perturbative completion of ${\bf S}_{\rm pert}(c; \varphi)$ with $\varphi$ replaced by $\langle\varphi\rangle_\sigma$ from (<ref>) as cautioned earlier.] is $\hat{\bf S}_{\rm tot}$, which doesn't have any perturbative pieces, and from (<ref>) it may be expressed as:

Ŝ_tot(φ)     = M_p^11𝕍_2
∫d^3 x  d^6y {-φ(x, y)  □M_p^2  φ(x, y) +
∑_s = 0^∞d_s Q_pert(c̅; φ(x, y)) exp(-s M_p^6 ∫_0^y d^6y' |Q_pert(ĉ; φ(y', x))|)

+ ∑_p = 1^∞b_p [exp(-pM_p^6∫_M_6 d^6y' |𝔽(y -y') Q_pert(c̃; φ(y', x))|) - 1]},

where $x \in {\bf R}^{2, 1}, y \in {\cal M}_6$, $w \in {\mathbb{T}^2\over {\cal G}}$ and we have restricted the on-shell field to be independent of the coordinates of the toroidal direction (whose volume is $\mathbb{V}_2$).
In (<ref>), the first term is the kinetic term, the second  and the third are the non-perturbative and the non-local contributions respectively. Note that the latter two are expressed using ${\bf Q}_{\rm pert}$, but the coefficients differ. The non-perturbative term takes a trans-series form as one would have expected from the instanton saddles and the fluctuation determinants of (<ref>). This may also be justified from the growth of the non-perturbative and the non-local terms in the path integral. Fixing a particular value of $p$ in (<ref>), these two terms grow respectively as:

(pN)!N! exp(-N),
1N! exp(-N!), at ${\cal O}({\rm N})$. It is easy to see from figure <ref> that the first series is asymptotic[Since $p \ge 3$ in (<ref>), we have $\lim\limits_{{\rm N}\to \infty} ~{\left(p{\rm N}\right)!\over {\rm N}!}~{\rm exp}(-{\rm N}) = {\rm exp}\left[{\rm N}(p-1){\rm log}~{\rm N} + p{\rm N}({\rm log}~p - 1)\right]_{{\rm N} \to \infty} ~ \to ~ \infty$.] while the second one is convergent.
The asymptotic nature shows that the perturbative series from the fluctuation determinants knows about the next order non-perturbative effect. This is of course the precise requirement to apply the resurgence theory [43], implying that the non-perturbative trans-series from (<ref>) itself may be studied using the resurgent trans-series when inserted in the path integral. Additionally, the generic formulation[We have avoided using eight-dimensional integral $d^8y$ because of the assumption that all fields are independent of the toroidal direction ${\mathbb{T}^2\over {\cal G}}$ in M-theory. Thus generic here is up to this assumption. We will relax this condition when we study a more realistic scenario later.] of the two contributions is important as it would specify the structure of these effects which, with appropriate choices of the $(d, b, \bar{c}, \tilde{c}, \hat{c})$ coefficients, may be tuned to any specific construction. In fact a more appropriate way of representing the $c$-parameters would be to demand:

c̅ ≡c̅(s),      ĉ ≡ĉ(s),      c̃ ≡c̃(p), this way the non-perturbative series would spell out exactly what we had in (<ref>), namely, for every instanton saddle, the corresponding fluctuation determinant would be expectedly different. Similar story would resonate for the non-local terms too: at every order of the expansion, the corresponding quantum terms would be distributed differently. On the other hand, the absence of any fluctuation determinants in the non-local quantum series may strike as odd, although so far we have seen no inconsistencies with it. However there is an issue that remains invisible with scalar degrees of freedom till we study the EOMs with actual metric and flux components. We will come back to it soon. Meanwhile
the Schwinger-Dyson equation, from the first equation in (<ref>)
in the scalar field set-up, becomes:

The growths of the non-perturbative  series (in blue) and the non-local series (orange) at ${\cal O}({\rm N})$, with $c_{np} = 0$ except $c_{n3}$ in (<ref>). The first one is asymptotic while the second one is convergent. The trans-series form in (<ref>) leading to the asymptotic growth can be precisely tackled by the resurgence theory.

□⟨φ(x, y)⟩_σ
-  ∑_s= 0^∞d_s δQ_pert(c̅(s); ⟨φ(x, y)⟩_σ)δ⟨φ(x, y)⟩_σ   exp (-sM_p^6 ∫_0^y d^6y' |Q_pert(ĉ(s);⟨φ(y', x)⟩_σ)|)

+  M_p^6 ∑_s = 1^∞s  d_s ∫_M_6 d^6 y'
Q_pert(c̅(s); ⟨φ(y', x)⟩_σ) δ|Q_pert(ĉ(s);⟨φ(y, x)⟩_σ)|δ⟨φ(y, x)⟩_σ  Θ(y' - y)  exp (-sM_p^6 ∫_0^y' d^6y” |Q_pert(ĉ(s);⟨φ(y”, x)⟩_σ)|)

+   M_p^6∑_p = 1^∞p  b_p ∫_M_6 d^6 y' δ|𝔽(y'-y)Q_pert(c̃(p); ⟨φ(x, y)⟩_σ)|δ⟨φ(x, y)⟩_σ
exp(-pM_p^6∫_M_6 d^6 y” |𝔽(y'-y”) Q_pert(c̃(p);⟨φ(x, y”)⟩_σ)|)   =   0,

where the perturbative series in (<ref>) is expressed using
${\textcolor{red}{\langle\varphi(x, y)\rangle_\sigma}}$. Note the conditional integral over $y'$ in the first line governed by the Heaviside step-function $\Theta(y' - y)$. The non-locality factor $\mathbb{F}(y'-y)$ is a bit of a concern now as one might worry that the EOM becomes non-local because of it's presence. This is actually not the case because the non-locality factor integrates out in the following way:

M_p^6∫d^6 y' 𝔽(y'-y)    exp(-pM_p^6∫d^6 y” |𝔽(y'- y”) Q_pert(c̃(p); ⟨φ(x, y”)⟩_σ)|)  ≡  F_p(x, y), thus leading to a local function! In doing the computation we have assumed that
$\mathbb{F}(y_1- y_2) > 0$ for simplicity. If $\mathbb{F}(y_1- y_2) < 0$, then we can simply take $\left\vert\mathbb{F}(y_1 - y_2)\right\vert$ or
$-\mathbb{F}(y_1 - y_2)$
in (<ref>). Combining (<ref>) with (<ref>), then provides the evolution dynamics for the emergent state $\langle\varphi(x, y)\rangle_\sigma$ in a potential given by:

V(⟨φ(x, y)⟩_σ) =      ⟨φ(x, y)⟩_σ  ^2M_p^2  ⟨φ(x, y)⟩_σ-∑_s = 0^∞d_s  Q_pert(c̅(s); ⟨φ(x, y)⟩_σ)  exp(-s M_p^6 ∫_0^y d^6y' |Q_pert(ĉ(s); ⟨φ(y', x)⟩_σ)|)

- ∑_p = 1^∞b_p [exp(-pM_p^6∫_M_6 d^6y' |𝔽(y -y') Q_pert(c̃(p); ⟨φ(y', x)⟩_σ)|) - 1],

where $\grad^2 = \eta^{ij}\partial_i\partial_j$ with mostly plus signature, and
the overall signs for second and the third terms are controlled by the signs of $d_s$ and $b_p$ $\forall ~(s, b)$. In fact for large values of $s$ and $p$, the terms in the two exponential series die off, which is of course the reason for their convergence properties (recall the trans-series structure discussed earlier). Taking a derivative with respect to $\langle\varphi(x, y)\rangle_\sigma$ gives us the following functional form:

δV(⟨φ(x, y)⟩_σ)δ⟨φ(x, y)⟩_σ  =
^2M_p^2  ⟨φ(x, y)⟩_σ-  ∑_s= 0^∞d_s δQ_pert(c̅(s); ⟨φ(x, y)⟩_σ)δ⟨φ(x, y)⟩_σ   exp (-sM_p^6 ∫_0^y d^6y' |Q_pert(ĉ(s);⟨φ(y', x)⟩_σ)|)

+   M_p^6∑_p = 1^∞p  b_p ∫_M_6 d^6 y'  δ|𝔽(y'-y) Q_pert(c̃(p); ⟨φ(x, y)⟩_σ)|δ⟨φ(x, y)⟩_σ
exp(-pM_p^6∫_M_6 d^6 y” |𝔽(y'-y”) Q_pert(c̃(p); ⟨φ(x, y”)⟩_σ)|),

+   M_p^6 ∑_s = 1^∞s  d_s ∫_M_6 d^6 y'
Q_pert(c̅(s); ⟨φ(y', x)⟩_σ) δ|Q_pert(ĉ(s); ⟨φ(y, x)⟩_σ)|δ⟨φ(y, x)⟩_σ  Θ(y' - y)  exp (-sM_p^6 ∫_0^y' d^6y” |Q_pert(ĉ(s); ⟨φ(y”, x)⟩_σ)|)

where the overall sign of (<ref>) now depends on the signs of
$b_p$ and $d_s$ as well as the signs of ${\delta \left\vert{\bf Q}_{\rm pert}(\hat{c}(s); \langle\varphi(y, x)\rangle_\sigma)\right\vert\over \delta\langle\varphi(y, x)\rangle_\sigma}, {\delta \big\vert \mathbb{F}(y'-y) {\bf Q}_{\rm pert}(\tilde{c}(p); \langle\varphi(x, y)\rangle_\sigma)\big\vert\over \delta \langle\varphi(x, y)\rangle_\sigma}$ and
${\delta {\bf Q}_{\rm pert}(\bar{c}(s); \langle\varphi(y, x)\rangle_\sigma)\over \delta\langle\varphi(y, x)\rangle_\sigma}$; and we have assumed that
$\mathbb{F}(y'-y) > 0$. In a more realistic case, which we are going to discuss soon, the signs of ${\delta \left\vert{\bf Q}_{\rm pert}(\hat{c}(s); \langle\varphi(y, x)\rangle_\sigma)\right\vert\over \delta\langle\varphi(y, x)\rangle_\sigma}, {\delta \big\vert \mathbb{F}(y'-y) {\bf Q}_{\rm pert}(\tilde{c}(p); \langle\varphi(x, y)\rangle_\sigma)\big\vert\over \delta \langle\varphi(x, y)\rangle_\sigma}$
and  ${\delta {\bf Q}_{\rm pert}(\bar{c}(s); \langle\varphi(y, x)\rangle_\sigma)\over \delta\langle\varphi(y, x)\rangle_\sigma}$
will depend on the specific configurations of local and global fluxes as well as the warp-factor and curvature components. Question is whether it is necessary for us to recover a condition like:

δV(⟨φ(x, y)⟩_σ)δ⟨φ(x, y)⟩_σ   ≥  c_1 V(⟨φ(x, y)⟩_σ), for the effective emergent potential ${\bf V}(\langle\varphi(x, y)\rangle_\sigma)$. To answer this, first note that the vacuum Minkowski space (or more appropriately a warped Minkowski spacetime with an internal compact non-Kähler space) does not allow the potential to be “uplifted” beyond the Minkowski minima. Secondly, the equation of motion from (<ref>), that fixes a form of $\langle\varphi(x, y)\rangle_\sigma$ as well as that of the Glauber-Sudarshan state $\vert\sigma\rangle$ will always be:

δV(⟨φ(x, y)⟩_σ)δ⟨φ(x, y)⟩_σ   =   ∂^2∂t^2 ⟨φ(x, y)⟩_σ, and never ${\delta{\bf V}(\langle\varphi\rangle_\sigma)\over \delta\langle\varphi\rangle_\sigma} = 0$,
implying that the emergent degrees of freedom $-$ from metric, fluxes and fermions $-$ should always have temporal dependence. This fits well with the EFT criteria analysed in [14, 15, 16], namely that if the emergent degrees of freedom become time-independent then there is a loss of both $g_s$ and ${\rm M}_p$ hierarchies implying a breakdown of EFT[One might question as to what happens in the static patch. First of course static patch doesn't mean that there are no temporal dynamics. As shown in figure <ref> the hidden temporal behavior actually requires us to keep the degrees of freedom with certain amount of temporal dependence. Moreover, the dual IIA string coupling $g_s$, albeit looks time independent in (<ref>), has other issues that we will discuss in section <ref>. These arguments confirm once again that (<ref>) is the correct EOM for the emergent states. We will discuss more on this as we go along. ]. This is captured by the second criterion in (<ref>).

The above discussion means that (<ref>) is unnecessary. The derivative of the emergent potential, i.e. ${\delta{\bf V}(\langle\varphi\rangle_\sigma)\over \delta\langle\varphi\rangle_\sigma}$ never hits zero, even at the so-called “classical” level because of
(<ref>). Thus there appears to be no strong reason why the EFT criteria of [6] should hold for the emergent potential (<ref>) although (<ref>) should continue to hold. The worry that de Sitter space may not exist due to vanishing minima of the potential is now unfounded. For historical reasons however we note that
the coefficient $c_1$ is positive, and it's value may be easily extracted from the integral form in (<ref>), but we will not do so here. In the presence of multiple fields, we can try to find the second derivative and again ask which of the following two conditions:

δV({⟨φ_l(x, y)⟩_σ})δ⟨φ_j(x, y)⟩_σ   ≥  c_1 V({⟨φ_l(x, y)⟩_σ})

min   (δ^2 V({⟨φ_l(x, y)⟩_σ})δ⟨φ_i(x, y)⟩_σδ⟨φ_j(x, y)⟩_σ)   ≤  -c_2 V({⟨φ_l(x, y)⟩_σ}),

are necessary to capture the EFT criteria alluded to earlier
for the set of emergent fields $\{\langle\varphi_l(x, y)\rangle_\sigma\}$. Following [6] one might expect the answer to be the first of the two in (<ref>), because the Glauber-Sudarshan states are defined in the temporal domain (<ref>) $-$ which appears from the first criterion in (<ref>) $-$ and so first condition in (<ref>) should be automatically satisfied. One might even take an emergent field
that satisfies $\grad^2 \langle\varphi(x, y)\rangle_\sigma = \partial_0^2 \langle\varphi(x, y)\rangle_\sigma = 0$ (a plot of such a scenario is presented in figure <ref>) and compare the first equation in (<ref>) with the potential in (<ref>).

Plot of two functions ${\bf V}_1 = e^{2\psi}$ (in blue) and
${\bf V}_2 = 10 - 10 e^{-\psi^6 - \psi^4 - \psi^2 - 1}$ (in orange), with $\psi \equiv \langle\varphi\rangle_\sigma$. ${\bf V}_1$ appears from the first equation in (<ref>) with $c_1 = 2$, and ${\bf V}_2$ appears from (<ref>) with $d_s = \bar{c}(s) = 0, b_1 = 10$,  $\grad^2\psi = \partial_0^2\psi = 0$ and $b_p = 0$ for $p > 1$. Such a configuration doesn't satisfy (<ref>), and therefore we expect the EFT criteria outlined in (<ref>) to not be satisfied either.

Needless to say, since the emergent potential has a minimum, the condition (<ref>), or the first criterion in (<ref>), is not satisfied. As mentioned earlier, this is of no concern because the emergent potential is not required to satisfy either of the two conditions in (<ref>). Additionally, the derivative of the
emergent potential in (<ref>) cannot vanish. We do however need to satisfy the EFT criteria outlined in (<ref>). For this  it is necessary to not only include the consequence from the temporal derivative of the emergent field, but to also keep $\bar{c}(s) \ne 0$ in (<ref>). The latter would imply the inclusion of the fluctuation determinants over each instanton saddles.
As an example, let us consider a toy example motivated from the scalar field theory that we discussed here. We will only consider the non-perturbative sector in the potential (<ref>) coming from the instanton saddles and their corresponding fluctuation determinants. To make this precise, we can choose $\hat{c}(s)$ and $\bar{c}(s)$ such that:

Q_pert(c̅(s); ⟨φ(x, y)⟩_σ) = ∑_q = 0^∞c_sq√(2^q q!)  ℍ_q(⟨φ(x, y)⟩_σ)

M_p^6∫_0^y d^6y'  |Q_pert(ĉ(s); ⟨φ(y', x)⟩_σ)|= 12s ∑_p = 0^∞|h_sp ℍ_p(⟨φ(x, y)⟩_σ)|√(2^p p!),

where $\mathbb{H}_q\left(\langle\varphi(x, y)\rangle_\sigma\right)$ are the dimensionless Hermite polynomials $\forall q \in \mathbb{Z}$, and $h_{sp}, c_{sq}$ are constants. The way we have expressed the above form of the perturbative corrections, one could make this fit with any given theory by appropriately taking a linear combination of the Hermite polynomials using $c_{sq}$ and $h_{sp}$ without compromising any generalities. Using these the non-perturbative potential may be expressed in the following way:

V_NP(⟨φ(x, y)⟩_σ) = -∑_s, q=0^∞d_s c_sq √(2^q q!)  ℍ_q(⟨φ(x, y)⟩_σ) exp(-12∑_p = 0^∞|h_sp ℍ_p(⟨φ(x, y)⟩_σ)|√(2^p p!)), with $d_s$ appearing from the second term in (<ref>). With appropriate choice of the coefficients $d_s$, the non-perturbative potential could either be a normalizable or a non-normalizable function (or a combination thereof) in the field space governed by $\langle\varphi(x, y)\rangle_\sigma$. (Demanding a potential whose derivative is always bigger than the potential itself in the field space can be easily constructed but, as mentioned earlier, is not necessary here.) The total package with the fluctuation determinants inserted as in (<ref>) contributes in the right way to precisely restore the EFT criteria outlined in (<ref>). This again justifies what we said earlier: the EFT criteria are simply the ones from (<ref>).

Therefore to summarize: The original motivation to impose (<ref>) or (<ref>) was to eliminate four-dimensional de Sitter spacetime which would appear, for example, in the static patch as a minimum of the potential. Unfortunately both these motivations are on the wrong footing: de Sitter spacetime is never a vacuum solution, nor is the static patch a good coordinate choice to study de Sitter. The correct picture is that the de Sitter spacetime is generated from a Glauber-Sudarshan state with an emergent potential (<ref>) satisfying (<ref>), implying that even in this language the de Sitter excited state is not a minimum of the emergent potential. This makes the two criteria in (<ref>) unnecessary but does lead to two other criteria (<ref>) as the necessary requirements for the validity of EFT.

§.§ Glauber-Sudarshan states and the first EFT criterion from (<ref>)

The lessons that we learnt from the warm-up example in the previous section with one on-shell field $\varphi(x, y)$ $-$ along-with the assumption that we have integrated out at least one off-shell field $\phi(x, y)$ $-$ are the following.

${\bf 1.}$ The action that controls the dynamics of the emergent field $\langle\varphi(x, y)\rangle_\sigma$ is neither ${\bf S}_{\rm pert}(c; \varphi)$ from (<ref>) nor $\hat{\bf S}_{\rm tot}(\varphi)$ from (<ref>). Rather it is given by
$\hat{\bf S}_{\rm tot}(\langle\varphi\rangle_\sigma)$.

${\bf 2.}$ The effective field theory criteria that control the dynamics of the emergent degree of freedom, in the M-theory uplift of the type IIB scenario, are completely given in terms of the type IIA dual string coupling $g_s$ as in (<ref>).

${\bf 3.}$ The effective field theory criteria proposed in [6] are designed to rule out four-dimensional de Sitter spacetime that is a minimum of a vacuum potential. Since the emergent de Sitter spacetime that we get (from an equally emergent potential) is neither a minimum of a potential nor controlled by a vacuum potential, the EFT criteria of [6] play no role here and may thus be ignored. The only useful aspect that survives from the conjectures in [6] is the trans-Planckian censorship conjecture which, here, appears from the first criterion of (<ref>).

The last point regarding (a) the trans-Planckian censorship conjecture, and (b) the de Sitter spacetime not being the minimum of the emergent potential, deserves some discussion in light of the EFT criteria proposed in (<ref>). We will start with the trans-Planckian censorship conjecture and then discuss the issues surrounding (<ref>).

The idea behind the trans-Planckian censorship conjecture is simple. It states that a reverse time evolution of excitations with ${\cal O}(1)$ energies living on a late time slice leads to trans-Planckian energies for the same at a finite time in the past. In terms of future time evolution, trans-Planckian modes in the past become increasingly long wavelength and subsequently freeze upon exiting the Hubble horizon. Thus at late times, one can classically detect trans-Planckian physics at ${\cal O}(1)$ scales, for instance, in the CMB data.

If we are only considering a generic higher-derivative gravity theory then these trans-Planckian modes will be problematic because the short-distance behavior of such a theory itself is problematic. The non-normalizability issue, appearing at a scale ${\rm M}_p$ and discussed right at the start of section <ref>, is the root cause of the problem. Of course now we know that this isn't really an issue in string or M- theory because new degrees of freedom appear at that scale to render all UV amplitudes finite. In the original formulation [13], where the theory was restricted to obey Einstein gravity only,
the trans-Planckian censorship conjecture prohibits the existence of such a scenario by simply bounding the temporal domain for these UV modes to not exit the Hubble horizon. More precisely, the conjecture states that any solution where we can see a classical imprint of trans-Planckian features upon exiting the de Sitter horizon is not possible since the time scale is bounded by:

a_f/a_i   <   M_pl/H,
where the labels $i,f$ denote the initial and final times, $a$ is the expansion parameter, ${\rm M_{pl}}$ is the four-dimensional Planck mass, and $\cal{H}$ is the Hubble parameter (not to be confused with the warp-factor ${\rm H}(y)$ used earlier). If the Hubble parameter is constant over time, then the time scale where the de Sitter state can be valid is given by:

Δt_TCC  =  1/H  log(M_pl/H)  =
1/√(Λ)  log(M_pl/√(Λ)),
where $\Lambda$ is the four-dimensional cosmological constant. Assuming
$\Lambda = {10}^{-120} {\rm M}_p^2$, we see that $\Delta t_{\rm TCC} =
{138.55\over \sqrt{\Lambda}} ~\log\left({{\rm M_{pl}} \over {\rm M}_p}\right)$. In the flat-slicing, this temporal domain is precisely related to the temporal domain (<ref>) got by demanding the first EFT criterion from (<ref>), namely $g_s < 1$.

In light of what we said above, the similarity between the two temporal domain is a bit surprising. The original trans-Planckian censorship conjecture deals with Einstein gravity (or more appropriately, generic higher derivative gravity) only whereas our analysis is in full string theory with the UV completeness inherently built in. (In fact the UV finiteness manifests as finite Wilsonian coefficients of the interactions in the low energy EFT. These are exactly the finite coefficients $(d_s, b_p, \bar{c}(s), \hat{c}(s), \tilde{c}(p))$ that we encountered in the toy set-up of (<ref>).) There are no pathologies for the far UV modes in our set-up, so the natural question is why would the two temporal domain match-up so nicely. Moreover, nothing really bad happens if we go beyond the temporal domain (<ref>). We simply lose control of the non-perturbative and non-local interactions that go as ${\rm exp}\left(-{1\over g_s^p}\right)$, for $g_s > 1$ in the M-theory side, implying that the system no longer has a Glauber-Sudarshan state beyond the temporal domain (<ref>). Four-dimensional de Sitter space would then exist only in this limited temporal domain and beyond that the theory might simply go back to the warped-Minkowski description. This conclusion matches somewhat with the quantum breaking time of [32] although the specifics of the detail differ.

There is yet another matter of contention here.
It can be shown that imposing the trans-Planckian censorship conjecture for a scalar field with say, a monotonically increasing potential, leads to satisfying the first EFT condition of (<ref>) $-$ which we will call as the “de Sitter potential conjecture” [41, 6]. However in contrast to the de Sitter potential conjecture which does not allow for a de Sitter vacuum solution, the trans-Planckian censorship conjecture does allow for vacuum solution provided they satisfy the temporal bound (<ref>). Our analysis has ruled out any possibility of a de Sitter vacuum solution, so the trans-Planckian conjecture should be re-interpreted appropriately for the case with  Glauber-Sudarshan states. Putting things together, it appears that there are at least three different computations that lead to similar temporal domain of validity.

${\bf 1.}$ The weak coupling $g_s < 1$ limit for the Glauber-Sudarshan states.

${\bf 2.}$ The quantum breaking time within the standard formulation of the coherent states [32].

${\bf 3.}$ The trans-Planckian censorship conjecture with Einstein, or higher derivative, gravity [22].

All three computations are different, and in particular the first and the second computations differ because the Glauber-Sudarshan states by construction are different from the coherent states. Why would then the temporal domain of validity match up in the aforementioned three computations? First, of course this match-up is not precise. For example (<ref>) doesn't exactly equate to say
(<ref>): there is some difference of factors, although if we construct de Sitter Glauber-Sudarshan state in global coordinates we can come very close to matching with (<ref>). But this is besides the point, and it doesn't answer the pertinent question of why is there any equivalence at all?

The answer may lie on the fact that, a subset of these cases (especially the first and the third ones), rely on a specific ingredient of the computation: the temporal variation of the fluctuating frequencies. This temporal variation appears to source both the trans-Planckian issue with Einstein, or higher derivative, gravity and the breakdown of a Wilsonian integrating-out procedure within string theory. The latter of course led to the formulation of the Glauber-Sudarshan states as discussed in much detail in section <ref>. While the ingredient is similar in both cases, the consequences are quite different. In the third case, the UV finiteness becomes the main issue, whereas in the first case, while the UV finiteness is no longer an issue, the convergence property of the non-perturbative and the non-local quantum series now becomes problematic. (The second issue with quantum break time for the coherent states [32] should also come under similar scrutiny but we will not discuss it here.) Question is whether there is any quantity in string theory that can succinctly capture all the aforementioned issues. The answer turns out to be the dual type IIA coupling $g_s$. By definition it can be expressed in terms of the Hubble parameter $\cal{H}$ $-$ which appears from the T-duality rules $-$ and therefore keeping $g_s < 1$
takes care of both the convergence property of the instanton and non-local series, and the UV problems of Einstein gravity. Thus this is one of the main reason for choosing $g_s < 1$ as our first EFT criterion in (<ref>).

Another more subtle reason for choosing $g_s < 1$ appears when we try to
study a de Sitter state in a static patch (see also footnote <ref>). If $\vert\alpha\rangle$ denotes the Glauber-Sudarshan state in type IIB theory, then the emergent type IIB metric in a static patch takes the following form:

ds^2     = ⟨g_MN⟩_αd Y^M dY^N

= 1H^2(y)[-(1- Λr_s^2) dt_s^2 +
dr_s^21- Λr_s^2  + r_s^2(dθ_s^2 + sin^2θ_s dφ_s^2)] + H^2(y)  g_mn(y) dy^m dy^n

where ${\rm Y}^{\rm M} = \left(t_s, r_s, \theta_s, \varphi_s, y^m\right)$ and $y^m \in {\cal M}_6$, with ${\rm H}(y)$ and $\Lambda$ being the warp-factor and the four-dimensional cosmological constant. The radial coordinate $r_s$ is bounded by $r_s < {1\over \sqrt{\Lambda}}$ as shown as the middle figure in figure <ref>. One would expect some emergent static flux configuration to support a background like (<ref>). Unfortunately problem lies when we try to explore region $r_s > {1\over \sqrt{\Lambda}}$. The absence of a globally time-like Killing vector suggests that outside the static patch the Killing vector could become space-like or/and even change orientations. Interchanging the time-like and space-like coordinates would mean that outside the static patch a static flux configuration would develop temporal dependence leading to fluctuating frequencies shown on the extreme right of figure <ref>. The consequence of such a behavior has already been pointed out in section <ref>, so we will not elaborate the story further and instead discuss yet another problem  that may be tied up to the first EFT criterion of (<ref>). This appears when we uplift (<ref>) to M-theory by defining the type IIA dual coupling $g_s$ as:

g_s   =   1r_s(g_b  H(y)   cosec θ_s), where $g_b$ is the type IIB coupling that may be kept fixed at the constant coupling point of F-theory [17]. Since ${\rm H}(y)$ is a smooth function, and $r_s$ is bounded by $0 \le r_s < {1\over \sqrt{\Lambda}}$, we see that $g_s$ is typically bigger than 1 along specific paths, i.e. $g_s > 1$, because $1 \le {\rm cosec}~\theta_s \le \infty$; implying that the non-perturbative and the non-local series that go as ${\rm exp}\left(-{1\over g_s^p}\right)$ with $p > 0$ cannot be made convergent (see also figure <ref>)[More appropriately, at any point in $r_s$, there is at least one point in the angular direction $\theta_s$ for which $g_s$ blows up. What really matters is not that $g_s < 1$ at other points, but the fact that the instanton and non-local corrections are not convergent along certain directions in ${\bf R}^{2, 1}$ in M-theory. Since the action involves an integral over space-time, this naturally becomes a problem. Alternatively, if we absorb the spatial factors in the definition of ${\rm H}_o({\bf x})$, then in the static patch ${g_s\over {\rm H}(y) {\rm H}_o({\bf x})} = 1$.]. Thus the behavior in a static patch appears to be what we discussed earlier outside the trans-Planckian bound. Due to the UV completeness of string theory, one wouldn't expect things to go bad in this regime, but we do expect the Glauber-Sudarshan state to not exist in this regime.

Plot $g_s = {g_b~{\rm H}(y)\over r}~{\rm cosec}~\theta$ for $0 \le r \le 1$, and $-8 \le \theta \le 8$. We have also taken $g_b = {\rm H}(y) \equiv 1$. The former equality is to remain at the constant coupling point of F-theory, and the latter is to simplify the analysis.

One might however wonder why is this the case if we compare it to say the flat-slicing. Going from type IIB to M-theory involves one T-duality, so in both flat-slicing and the static patch should involve one T-duality in each case respectively. Why is then the physics different? The answer has to do with the duality directions. For the case with a flat-slicing, the duality direction is $x_3$, whereas in the case with the static patch, the duality direction is $\varphi_s$. Comparing the two set of coordinate systems, it is easy to convince oneself that $\varphi_s$ is at least a function of both $x_3$ and the conformal time $t$ of the flat-slicing coordinates. This means, while in the flat-slicing case we are making a T-duality along $x_3$ direction, in the static patch we are actually making a T-duality along a combination of $x_3$ and $t$ directions. Since the duality directions are different, we don't expect the physics to be the same in the two cases. This should also explain why we see the variation of the frequencies outside the static-patch in figure <ref>.

The above arguments should convince the readers that the static patch construction clashes with the first criterion of (<ref>), and therefore we expect only those patches of emergent de Sitter space to exist that allow explicit temporal dependence of the underlying on-shell degrees of freedom. This is consistent with the EOM (<ref>), justifying what we said earlier: a de Sitter emergent state is never a minimum of the emergent potential, rather it satisfies (<ref>). To summarize, it appears that:

${\bf 1.}$ The Minkowski minimum may not be uplifted further leading us to construct a four-dimensional de Sitter spacetime in type IIB theory as an emergent excited state over the Minkowski minimum.

${\bf 2.}$ The four-dimensional emergent de Sitter state may not be well defined using a static patch leading us to consider only those patches that allow some inherent temporal dependence implying (<ref>) as the correct EOM to study the dynamics of such a state.

Both the aforementioned results should also embody the second EFT criterion of (<ref>). How far this is realized in an actual set-up with metric and fluxes will be the topic of the following section.

§.§ Glauber-Sudarshan states and the second EFT criterion from (<ref>)

The second EFT criterion from (<ref>) deals with the derivative of $g_s$ with respect to the conformal time $t$. In [16] we have already connected this to the non-violation of the Null Energy Condition (NEC), so we will not discuss this connection any further here. Instead we will see how, in a more realistic case with metric and fluxes,
the second criterion from (<ref>) is embedded in the EOMs from (<ref>).

As mentioned earlier, the action governing the EOMs of the emergent space-time is neither the perturbative action (say (<ref>)), nor the one with non-perturbative and the non-local completion (say (<ref>)), rather it is the second one with the metric and the fluxes replaced by their expectation values over any arbitrary Glauber-Sudarshan state. Both the state and the dynamics of the emergent space-time are fixed by the Schwinger-Dyson equations:

δŜ_tot(⟨Ξ⟩_σ)δ⟨Ξ⟩_σ   =   0, where  ${\bf \Xi}$ is the set of on-shell fields
${\bf \Xi} = ({\bf g}_{\rm AB}, {\bf C}_{\rm ABD}, {\bf \Psi}_{\rm A}, \overline{\bf \Psi}_{\rm A})$ with $({\rm A, B, D}) \in {\bf R}^{2, 1} \times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$; and $\hat{\bf S}_{\rm tot}$ is from (<ref>). There are also other equations accompanying (<ref>), but we will not consider them here. A more complete analysis of the EOMs, including additional EOMs from the off-shell degrees of freedom (like the second constraint equation in (<ref>)), will be dealt in our upcoming work [9].

To get the form of $\hat{\bf S}_{\rm tot}$ we will follow the procedure as outlined earlier going from the perturbative action (<ref>) to the non-perturbative and the non-local completion as in (<ref>), and then finally replace the on-shell fields by their expectation value over any Glauber-Sudarshan state. The perturbative action takes the form:

S_pert(c; Ξ) = M_p^9∫d^11x √(-g_11)[R_11 + G_4 ∧∗_11G_4  + Q_pert(c; Ξ)] +
M_p^9∫C_3 ∧G_4 ∧G_4 + M_p^3 ∫C_3 ∧𝕏_8,
where we have included all the kinetic terms for the metric and the three-form field, the topological terms, and the perturbative interactions. These have already been discussed earlier after (<ref>). What we have ignored are the fermionic Rarita-Schwinger terms and the M2 and the M5-brane terms. (The latter should directly appear from the asymptotic nature of the perturbative series itself, so we will not worry too much about them now.) These contributions, and especially the fermionic and the world-volume perturbative terms, are pretty non-trivial and we will discuss them in our upcoming paper [9]. This also means that henceforth the set of on-shell fields will be
${\bf \Xi} = ({\bf g}_{\rm AB}, {\bf C}_{\rm ABD})$ without the fermionic fields. In addition to what we see from (<ref>), we have allowed an exhaustive list of perturbative terms denoted by ${\bf Q}_{\rm pert}(c; {\bf \Xi})$. This is similar to what we had in (<ref>), but without the non-perturbative contributions. For an easy access, we quote it again in the following way:

Q_pert(c; Ξ(x, y, w)) = ∑_{l_i}, {n_j} c_nl M_p^σ_nl [g^-1] ∏_j = 0^3 [∂]^n_j ∏_k = 1^60 (R_A_k B_k C_k D_k)^l_k ∏_p = 61^100 (G_A_p B_p C_p D_p)^l_p, where ${c}_{nl}$ are dimensionless constants for a given values in the set $(n, l) \equiv (\{n_i\}, \{l_j\})$; $(n_i, l_j) \in (+\mathbb{Z}, +\mathbb{Z})$;
$(n_0, n_1, n_2, n_3)$ are the number of derivatives along temporal, ${\bf R}^2, {\cal M}_6$ and ${\mathbb{T}^2\over {\cal G}}$ respectively; and
$\sigma_{nl} \equiv \sum\limits_{j = 0}^3 n_j + 2\sum\limits_{k = 1}^{60} l_k + \sum\limits_{p = 61}^{100} l_p$ fixes the ${\rm M}_p$ dimension of the operators for a given choice of $(n, l)$ [14, 15, 16]. All the curvature and the flux tensors, including the inverse metric components are expressed in terms of their warped form, and therefore once we take their expectation values over the appropriate Glauber-Sudarshan state they would include the $g_s$ factors. This in particular implies that (<ref>) may be re-expressed simply as:

ds^2     = ⟨g_AB⟩_σdx^A dx^B

= g_s^-8/3(-dt^2 + dx_1^2 + dx_2^2) + g_s^-2/3H^2(y) g_mn(y) dy^m dy^n + g_s^4/3 η_ab dw^a dw^b,

with $g_s \equiv {\rm H}(y) \sqrt{\Lambda} t$, $t$ being the conformal time, and ${\rm H}(y)$ being the smooth warp-factor. The coordinate choice is as follows. $(w^a, w^b) = (x_3, x_{11}) \in {\mathbb{T}^2\over {\cal G}}$ with ${\cal G}$ being an orbifold action without a fixed point; and $(y^m, y^n) \in {\cal M}_6$ such that the internal eight-manifold is locally ${\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$.
We have also generically taken the on-shell fields to depend on all the eleven-dimensional coordinates which are distributed as above, i.e. $({\bf x}, y, w) \in ({\bf R}^2, {\cal M}_6, {\mathbb{T}^2\over {\cal G}})$. One may check that ${\partial g_s\over \partial t} \propto g_s^0$, consistent with the second EFT criterion from (<ref>).

The quantum series ${\bf Q}_{\rm pert}(c; {\bf \Xi})$ from (<ref>) doesn't appear in the total effective action $\hat{\bf S}_{\rm tot}({\bf \Xi})$ directly as discussed earlier, but does appear in the construction of the non-perturbative and the non-local quantum terms.

Ŝ_tot(Ξ) = S_kin(Ξ) + S_NP(Ξ) + S_nloc(Ξ)

= M_p^9
∫d^3 x  d^6y d^2w √(-g_11(X))
[R_11(X) + G_4(X) ∧∗_11G_4(X)] + M_p^9∫C_3(X) ∧G_4(X) ∧G_4(X) + M_p^3∫C_3(X) ∧𝕏_8(X)

+ M_p^11∫d^3x d^6y d^2w √(-g_11(X))
∑_s = 0^∞d_s Q_pert(c̅(s); Ξ(X)) exp(-s M_p^8 ∫_0^y ∫_0^w d^6y' d^2w' √(g_8(Y', x)) |Q_pert(ĉ(s); Ξ(Y', x))|)

+M_p^11∫d^3x d^6y d^2w √(-g_11(X)) ∑_p = 1^∞b_p [exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(Y', x))|𝔽(y -y'; w -w') Q_pert(c̃(p); Ξ(Y', x))|) - 1],

where ${\rm X} \equiv (x, y, w)$, ${\rm Y}' \equiv (y', w')$, ${\cal M}_8 \equiv {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$, and we have generically taken all the fields to be functions of $(x, y, w)$. The second line is the set of kinetic and the topological terms, the third line is the non-perturbative series expressed using sum over all the instanton saddles and their corresponding fluctuation determinants;
and the last line is
the non-local quantum series expressed using the non-locality function $\mathbb{F}(y-y', w-w')$. Finally, all the quantum series are expressed using
(<ref>), but with different values of the coefficients $c_{nl}$.

The action (<ref>) is a more elaborate version of (<ref>) because the former includes all the real on-shell degrees of freedom like metric and fluxes. The non-perturbative term has the standard form of instanton saddles accompanied by the corresponding fluctuation determinants, but the non-local terms (as derived in (<ref>)) appear with no such pre-factors. Are these pre-factors necessary, or we can simply suffice with what we have now? Clearly it appears that such pre-factors are not necessary at least in (<ref>): the EOM in (<ref>) shows no untoward pathologies. We will however show that, in a more realistic case, the $g_s$ scaling forces us to introduce additional fluctuation determinants around each of the non-local saddles. To show this we will have to develop the story a bit more, and in the following we elaborate on this.

The action controlling the dynamics of the evolution of the Glauber-Sudarshan states is not (<ref>), rather it is $\hat{\bf S}_{\rm tot}(\langle {\bf \Xi}\rangle_\sigma)$. The typical Schwinger-Dyson equations governing the evolutions of the emergent on-shell states are now:

δŜ_tot(⟨Ξ⟩_σ)δ⟨g_AB⟩_σ   =    δŜ_tot(⟨Ξ⟩_σ)δ⟨C_ABD⟩_σ   =   δŜ_tot(⟨Ξ⟩_σ)δ⟨Ψ_A⟩_σ   =   0, where to derive the fermionic EOMs one will have to introduce the fermionic perturbative series in (<ref>). This will be discussed in [9]. There are also other equations coming from integrating out the off-shell degrees of freedom. Those will also be discussed in [9].

With this we are ready to tackle the Schwinger-Dyson equations for the emergent on-shell degrees of freedom. Here however we will only discuss the dynamics of the metric components $\langle {\bf g}_{\rm AB}\rangle_\sigma$ and leave the discussion on other components for a future work (see also [9]). Using the first equation in (<ref>), the EOM takes the following form:

R_AB(⟨Ξ(X)⟩_σ) -
12 ⟨g_AB(X)⟩_σR(⟨Ξ(X)⟩_σ)

2 √(g_11(⟨Ξ(X)⟩_σ)) δδ⟨g^AB(X)⟩_σ (√(g_11(⟨Ξ(X)⟩_σ)) G_4( ⟨Ξ(X)⟩_σ) ∧∗_11G_4( ⟨Ξ(X)⟩_σ))

-  2M_p^2√(g_11(⟨Ξ(X)⟩_σ))
∑_s= 0^∞d_s δδ⟨g^AB(X)⟩_σ
[√(g_11(⟨Ξ(X)⟩_σ))Q_pert(c̅(s); ⟨Ξ(X)⟩_σ)]

× exp(-s M_p^8 ∫_0^y ∫_0^w d^6y' d^2w' √(g_8(⟨Ξ(Y', x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y', x)⟩_σ)|)

+  M_p^10 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) ∑_s = 0^∞ s d_s
Q_pert(c̅(s); ⟨Ξ(Y', x)⟩_σ)

×  δδ⟨g^AB(X)⟩_σ
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(ĉ(s); ⟨Ξ(X)⟩_σ)]
Θ(y'-y) Θ(w'-z)

× exp(-s M_p^8 ∫_0^y' ∫_0^w' d^6y” d^2w” √(g_8(⟨Ξ(Y”, x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y”, x)⟩_σ)|)

- M_p^2⟨g_AB(X)⟩_σ∑_p = 1^∞b_p [exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(⟨Ξ(Y', x)⟩_σ))|𝔽(Y - Y') Q_pert(c̃(p); ⟨Ξ(Y', x)⟩_σ)|) - 1]

+   M_p^10 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) 𝔽(Y'-Y)∑_p = 1^∞pb_p
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(c̃(p); ⟨Ξ(X)⟩_σ)]

×  exp(-pM_p^8∫_M_8 d^6y” d^2w”
√(g_8(⟨Ξ(Y”, x)⟩_σ))|𝔽(y' -y”; w' -w”) Q_pert(c̃(p); Ξ(Y”, x))|)

where ${\rm Y} = (y, w), {\rm X} = (x, {\rm Y}) = (x, y, w)$, ${\rm Y}' = (y', w')$ and $\mathbb{F}({\rm Y' - Y}) \equiv \mathbb{F}(y' - y; w' - w) > 0$. We will consider the EOM along the space-time direction so that
$({\rm A, B}) \equiv (\mu, \nu) \in {\bf R}^{2, 1}$ in M-theory. As a warm-up let us consider the following metric ansätze:

⟨g_μν⟩_σ= g_s^-83η_μν,
⟨g_mn⟩_σ= g_s^-23 g_mn(y),
⟨g_ab⟩_σ= g_s^43 δ_ab, which is similar to the scalings that we discussed in sections <ref> and <ref>, with ${\rm H}(y) = {\rm H}_o({\bf x}) \equiv 1$ taken to simplify the ensuing analysis. A detailed analysis will be presented in our upcoming paper [9].

Terms in the EOM (<ref>)     $g_s$ scalings

$ \langle {\bf g}_{\mu\nu}\rangle_\sigma$     $g_s^{-{8\over 3}}$
${\bf R}_{\mu\nu}(\langle {\bf \Xi}\rangle_\sigma) -
{1\over 2} \langle {\bf g}_{\mu\nu}\rangle_\sigma {\bf R}(\langle {\bf \Xi}\rangle_\sigma)$     ${1\over g_s^2}$

${1 \over \sqrt{{\bf g}_{11}(\langle{\bf \Xi}\rangle_\sigma)}} {\delta\over \delta \langle{\bf g}^{\mu\nu}\rangle_\sigma} \left(\sqrt{{\bf g}_{11}(\langle{\bf \Xi}\rangle_\sigma)} {\bf G}_4( \langle{\bf \Xi}\rangle_\sigma) \wedge \ast_{11}{\bf G}_4( \langle{\bf \Xi}\rangle_\sigma)\right)$     $g_s^{\theta_{\rm class} - {8\over 3}}$

${1 \over \sqrt{{\bf g}_{11}(\langle{\bf \Xi}\rangle_\sigma)}}
~{\delta\over \delta \langle{\bf g}^{\mu\nu}\rangle_\sigma}
\left[\sqrt{{\bf g}_{11}(\langle{\bf \Xi}\rangle_\sigma)}~{\bf Q}_{\rm pert}(\bar{c}; \langle{\bf \Xi}\rangle_\sigma)\right]$
$g_s^{\theta_{\bar{c}} - {8\over 3}}$

${\rm exp}\left(-{\rm M}_p^6 \int_0^y d^6y'\sqrt{{\bf g}_6(\langle {\bf \Xi}(y')\rangle_\sigma)} \big\vert {\bf Q}_{\rm pert}(\hat{c}; \langle{\bf \Xi}(y')\rangle_\sigma\big\vert \right)$     ${\rm exp}\left(-{g_s^{\theta_{\hat{c}}}\over g_s^2}\right)$

${\bf Q}_{\rm pert}(\bar{c}; \langle{\bf \Xi}\rangle_\sigma)~
{\delta\over \delta \langle{\bf g}^{\mu\nu}\rangle_\sigma}
\left[\sqrt{{\bf g}_{6}(\langle{\bf \Xi}\rangle_\sigma)}{\bf Q}_{\rm pert}(\hat{c}; \langle{\bf \Xi}\rangle_\sigma)\right]$     ${g_s^{\theta_{\bar{c}} + \theta_{\hat{c}} -{8\over 3}}\over g_s^2}$

${\rm exp}\left({-{\rm M}_p^6\int_{{\cal M}_6} d^6y'
\sqrt{{\bf g}_6(\langle{\bf \Xi}(y')\rangle_\sigma)}\big\vert \mathbb{F}({y - y'}) {\bf Q}_{\rm pert}(\tilde{c}; \langle{\bf \Xi}({ y}')\rangle_\sigma)\big\vert}\right)$     ${\rm exp}\left(-{g_s^{\theta_{\tilde{c}}}\over g_s^2}\right)$

${\delta\over \delta \langle{\bf g}^{\mu\nu}\rangle_\sigma}
\left[\sqrt{{\bf g}_{6}(\langle{\bf \Xi}\rangle_\sigma)}{\bf Q}_{\rm pert}(\tilde{c}; \langle{\bf \Xi}\rangle_\sigma)\right]$     ${g_s^{\theta_{\tilde{c}} -{8\over 3}}\over g_s^2}$

[]The dominant $g_s$ scalings of various terms appearing the EOMs (<ref>) in the presence of BBS like M5-brane instantons. Details about how these $g_s$ scalings help us to solve EOM are described in the text.

§.§.§ Five-brane instantons wrapped on six-manifold ${\cal M}_6$

To proceed with (<ref>), we need to quantify the non-perturbative and the non-local interactions carefully right at the level of the action (<ref>). We will start by dealing with BBS like instanton effects which are related to M5-brane instantons wrapped on
${\cal M}_6$ [11]. For this, let us define the perturbative series in the following way:

Q_pert(c; Ξ(Y', x)) ≡Q_pert^BBS(c; Ξ(y', x))  δ^2(w'-w_o)M_p^2 √(g_2(w', x)), where $c = (\hat{c}(s), \bar{c}(s), \tilde{c}(p))$ and ${\bf g}_8({\rm Y}', x) = {\bf g}_6(y', x) {\bf g}_2(w', x)$. The delta function localizes these M5-brane instantons to be at a point $w_o$ on the toroidal manifold ${\mathbb{T}^2\over {\cal G}}$ parametrized by $w'$ and wrapped on ${\cal M}_6$. In the language of [15], these are the localized BBS instantons. (One could also allow delocalized instantons, but we will not deal with them here to avoid complicating the story.) Plugging (<ref>) in (<ref>), would convert all the terms in the EOM (<ref>) in a way shown on the LHS of
Table <ref>.

There is still a couple more adjustments need to be done before we can use the EOM (<ref>). One is to make all the emergent degrees of freedom to be independent of the toroidal direction $w \in {\mathbb{T}^2\over {\cal G}}$, and two, is to convert the metric ansätze (<ref>) to a slightly more refined form as in (<ref>) but with $\alpha(t) = \beta(t) = f_k(t) = \widetilde{f}_k(t) = 0, \alpha_k(t) = \beta_k(t) = g_k(t) = 1$. This
is a simplified version that doesn't take into account the temporal variation of the cosmological constant. In section <ref> we discussed how one might solve the EOMs order by order in $\check{\bf g}_s$. It is quite straightforward to extend the present set-up to a more involved one from section <ref> but we will not study this any further here. Instead we will take the following emergent metric and flux components:

⟨g_μν⟩_σ= g_s^-83η_μν,
⟨g_mn⟩_σ= ∑_k = 0^∞g^(k)_mn(y)  g_s^-23 + 2k3,
⟨g_ab⟩_σ= g_s^43 δ_ab,
G_ABCD(⟨Ξ⟩_σ) = ∑_k = 0^∞G^(k)_ABCD(y)  g_s^2k3 + l^CD_AB,

where $g^{(k)}_{mn}(y)$ are chosen such that the volume of the internal six-manifold ${\cal M}_6$ is time-independent keeping, in turn, the four-dimensional Newton's constant time independent[Although this is not necessary if we want to realize the configuration with temporally varying cosmological constant in section <ref>. For genericity, we will not impose this constraint in solving the Schwinger-Dyson equations. ]; and $({\rm A, B}) \in {\bf R}^{2, 1} \times {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$.
$l^{\rm AB}_{\rm CD} > 0$ is the dominant scaling for the emergent G-flux degrees of freedom, and is non-zero to keep in tune with the conditions
studied in [14, 15, 16]. Thus the emergent flux components are never time-independent.
Note that we have kept the definition of the emergent four-form flux, compared to (<ref>), generic to include fermionic terms. This is not important here but will become important once we study the full EOMs in [9].

With these at hand, we are ready to tackle the EOM (<ref>) in the presence of M5-brane instantons using (<ref>). Because of the complexity of the form of (<ref>), we will follow the strategy of comparing the EOM order-by-order in $g_s$. The dominant $g_s$ scalings of the various terms of (<ref>) is shown in Table <ref>. The first two set of terms are easy to quantify, and their
full $g_s$ scaling take the following form:

R_AB(⟨Ξ(X)⟩_σ) -
12 ⟨g_AB(X)⟩_σR(⟨Ξ(X)⟩_σ) =   1g_s^2 ∑_k = 0^∞G^(k)_μν(y) g_s^2k3

1 √(g_11(⟨Ξ(X)⟩_σ)) δδ⟨g^AB(X)⟩_σ (√(g_11(⟨Ξ(X)⟩_σ)) G_4( ⟨Ξ(X)⟩_σ) ∧∗_11G_4( ⟨Ξ(X)⟩_σ)) =   g_s^θ_class - 83 ∑_k = 0^∞H^(k)_μν(y) g_s^2k3

where ${\bf G}_{\mu\nu}^{(k)}(y)$ appears from the higher order series in $g_s$ for the metric components $\langle {\bf g}_{mn}\rangle_\sigma$, that enter via the emergent Ricci scalar ${\bf R}(\langle{\bf \Xi}\rangle_\sigma)$. In a similar vein we define the Hodge star using the emergent metric components from (<ref>) and use the flux definition from there to construct the emergent energy-momentum tensor from the G-fluxes. All these information appear in the tensor
${\bf H}^{(k)}_{\mu\nu}(y)$ with the $g_s$ scaling shown above. The dominant $g_s$ scaling from the fluxes, coming from a combination of $l^{\rm AB}_{\rm CD}$ and the metric scalings, is given by $\theta_{\rm class}$. For $k = 0$, we can compare the $g_s$ scaling of the two set of terms above to get:

θ_class   =   23, which, while allowing the fluxes to contribute to the EOM, does not help in solving it. (The contributing flux components at this order are only ${\bf G}_{mnab}(\langle{\bf \Xi}\rangle_\sigma)$ with $(m, n) \in {\cal M}_6, (a, b) \in {\mathbb{T}^2\over {\cal G}}$ and
$l^{ab}_{mn} = 1$.)
The underlying argument is similar to what we discussed in section <ref>, so we will not repeat it here[Recall that we need a positive cosmological constant solution. The fluxes can only solve the EOM with negative cosmological constant.].

The third set of terms is more non-trivial, because this is where the non-perturbative corrections first enter. The non-perturbative corrections in
(<ref>) are given in terms of the instanton saddles and their corresponding fluctuation determinants. Including them, the contribution to the EOM may be quantified in the following way:

1 √(g_11(⟨Ξ(X)⟩_σ))
∑_s= 0^∞d_s δδ⟨g^AB(X)⟩_σ
[√(g_11(⟨Ξ(X)⟩_σ))Q_pert(c̅(s); ⟨Ξ(X)⟩_σ)]

× exp(-s M_p^8 ∫_0^y ∫_0^w d^6y' d^2w' √(g_8(⟨Ξ(Y', x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y', x)⟩_σ)|)

=   ∑_r = 0^∞d_r g_s^θ_c̅(r) -83 ∑_k = 0^∞I^(k)_μν(c̅(r); y) g_s^2k3  exp(-r M_p^6 g_s^θ_ĉ(r) - 2 ∫_0^y d^6y'∑_l = 0^∞|I^(l)_1(ĉ(r); y) |g_s^2l3 Θ(w_1-w_2)),

where ${\bf Q}_{\rm pert}(c; \langle {\bf \Xi}\rangle_\sigma)$ is from (<ref>) and
$c = \bar{c}(s)$ and $c = \hat{c}(s)$ with appropriate choices for these parameters. The LHS is over the full eight dimensional internal
manifold ${\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$, and the RHS is over the six-dimensional base ${\cal M}_6$ where we used (<ref>). The Heaviside step-function $\Theta(w_1-w_2)$ with
$w_i \in {\mathbb{T}^2\over {\cal G}}$ is used here to localize the integrals over the six-dimensional base. The $g_s$ scalings of the two aforementioned perturbative series appear as $\theta_{\bar{c}(r)}$ and
$\theta_{\hat{c}(r)}$ for $r \in \mathbb{Z}$ alongwith the corresponding functions ${\bf I}_{\mu\nu}^{(k)}(\bar{c}(r); y)$ and ${\bf I}_1^{(l)}(\hat{c}(r); y)$ respectively. As before, the dominant scaling will appear when $k = l = 0$. Ignoring the functional dependence, there is now a puzzle when we try to compare the $g_s$ scalings on the RHS of (<ref>) with the dominant $g_s$ scaling from the emergent Einstein tensor of (<ref>). A naive comparison gives us the following:

θ_c̅(r) - 23 = g_s^θ_ĉ(r) - 2 log g_s, $\forall r \in \mathbb{Z}$. There is already an issue here as seen from
figure <ref>: there is no solution to the system for $g_s < 1$. (Only the orange curve intersects with the blue curve in figure <ref>.) If we try to solve (<ref>), we find the following solution for
$\theta_{\bar{c}(r)} =\theta_{\hat{c}(r)} \equiv\theta_{{c}(r)}$:

θ_c(r) = 23 - 𝕎(-g_s^-4/3)log g_s, where $\mathbb{W}(z)$ is the Lambert W-function (also known as the Product Log function). Generically $\mathbb{W}(z)$ has both real and complex parts, and therefore $\theta_{c(r)}$ will have both real and complex pieces as shown in figure <ref>. Solution would exist in the regime where the imaginary part vanishes, and this happens
for $g_s > 1$ as shown in figure <ref>. Unfortunately, even in the regime where solutions exist (ignoring the fact that $g_s > 1$ there), this is unacceptable because $\theta_{c(r)}$ is not a constant but is a function of $g_s$. More so, it will also be a function of $y \in {\cal M}_6$ because of the functions ${\bf I}_{\mu\nu}^{(k)}(y)$ and
${\bf I}^{(l)}(y)$ appearing in (<ref>).

Plot of the functions $y = \theta -{2\over 3}$ in blue and
$y = {g_s^{\theta-2}\over {\rm log}~g_s}$ for $g_s = 5$ in orange and $g_s = 0.2$ in green. We see that there are points of intersection between the
blue curve and the orange
curve only when $g_s > 1$. When $g_s < 1$,
there are no points of intersection between the blue curve and the green curve.

All in all it appears that equating the $g_s$ scalings in the aforementioned way is not the correct way to go. How should we then proceed to solve the EOM (<ref>)?
The hint appears from the $g_s$ scaling itself: Since the $g_s$ dependence in the exponential factor is the main source of contention,
we cannot presume that $\theta_{\bar{c}(r)} =\theta_{\hat{c}(r)}$ for
$k \ge 0, l\ge 0$ in (<ref>). Instead we can equate ${1\over g_s^2}$ from the emergent Einstein tensor in (<ref>) with the dominant scaling $g_s^{\theta_{\bar{c}(r)} - {8\over 3}}$ to demand the following scaling of the quantum terms from the fluctuation determinants:

θ_c̅(r)   =   23, and take $\theta_{\hat{c}(r)} \ge {2\over 3}$. Interestingly, for
${2\over 3} \le \theta_{\hat{c}(r)} < 2$, and $l = 0$, the instanton contributions are heavily suppressed (recall that ${\rm M}_p >> 1$ from the point of view of the low energy scale that we have been using). On the other hand, if $g_s$ and ${\rm M}_p$ go to zero and infinity respectively as $g_s = \epsilon$ and ${\rm M}_p = \epsilon^{-b}$ for
$b > 0$, then the exponential part can be perturbatively expanded if the quantum terms from the instanton saddles scale as:

θ_ĉ(r)   >   2 + 6b, implying that even in the range $2 \le \theta_{\hat{c}(r)} < 2 + 6b$, we expect suppressions of the exponential parts. Unfortunately, for
$\theta_{\bar{c}(r)} ~ = ~ {2\over 3}$ in (<ref>), there are no quantum terms that could contribute! In fact this is the same issue that we encountered with the perturbative quantum series, implying that to this order there are no non-trivial contributions from (<ref>).

Plot of the real and imaginary parts of the function $y = {2\over 3} - {\mathbb{W}(-g^{-4/3})\over {\rm log}~g}$ in blue and
orange respectively. Note that solution exists only for $g > 2$ here because ${\bf Im}~y = 0$. For $g < 1$, both
${\bf Im}~y$ and ${\bf Re}~y$ are non-zero.

What would then contribute? Contributions can come from the two set of quantum terms with $\theta_{\bar{c}(r)} > {2\over 3}$ and $\theta_{\hat{c}(r)} > {2\over 3}$ which are not suppressed by the exponential terms. This is possible for contributions to the EOM (<ref>) that take the following form:

∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) ∑_s = 0^∞ s d_s
Q_pert(c̅(s); ⟨Ξ(Y', x)⟩_σ)

×  δδ⟨g^AB(X)⟩_σ
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(ĉ(s); ⟨Ξ(X)⟩_σ)]
Θ(y'-y) Θ(w'-z)

× exp(-s M_p^8 ∫_0^y' ∫_0^w' d^6y” d^2w” √(g_8(⟨Ξ(Y”, x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y”, x)⟩_σ)|)

=  ∑_r = 0^∞rd_r ∫d^6y'∑_k = 0^∞∑_l = 0^∞I_2^(k)(c̅(r); y') J_μν^(l)(ĉ(r); y) g_s^θ_c̅(r) + θ_ĉ(r) - 143 + 23(k + l)  Θ(y'-y) Θ(w_1-w_2)

×  exp(-r M_p^6 g_s^θ_ĉ(r) - 2 ∫_0^y' d^6y”∑_n = 0^∞|I^(n)_1(ĉ(r); y”) |g_s^2n3 Θ(w_1-w_2))

where ${\bf I}_2^{(k)}(\bar{c}(r); y')$ is similar to the quantum series
${\bf I}_1^{(l)}(\hat{c}(r); y'')$ but defined with different coefficients $\bar{c}(r)$ and $\hat{c}(r)$ respectively. (The latter is also similar to one in (<ref>).) ${\bf J}_{\mu\nu}^{(l)}(\hat{c}(r); y)$ appears from the instanton contributions, and as before the dominant contribution comes from $k = l = n = 0$ in (<ref>). Since now the $g_s$ scalings have changed, we can try to equate them with the dominant ${1\over g_s^2}$ scaling of the emergent Einstein tensor from (<ref>). As before we will assume that
$\theta_{\bar{c}(r)} = \theta_{\hat{c}(r)} \equiv \theta_{c(r)}$, and ignore the terms that depend on $y \in {\cal M}_6$ in the exponential part. Doing this leads to:

θ_c(r) = 43 - 𝕎(-12g_s^2/3)log g_s, showing that this cannot be the right answer because of the $g_s$ dependence. Worse, solutions exist only for $g_s > 1$; and $\mathbb{W}(z)$ is the Lambert W-function. The plots are very similar to figure <ref> and figure <ref>. Of course now we know how to proceed. We start by not identifying $\theta_{\bar{c}(r)}$ to $\theta_{\hat{c}(r)}$, and then by comparing the $g_s$ scaling with the emergent Einstein tensor from (<ref>) we get the following relation:

θ_c̅(r) + θ_ĉ(r)   =   83, which is a significant improvement over (<ref>) because solutions would now exist where none existed before. For example consider
$\theta_{\bar{c}(r)} = 0$, and $\theta_{\hat{c}(r)} = {8\over 3}$. Since the emergent Riemann curvature tensors generically scale as $g_s^{2/3}$ (see for example [14]), it would imply a quartic order curvature contribution among other stuff. They are all coming from the instanton sectors and interestingly it was predicted earlier [4] that such terms would be necessary to allow for a de Sitter solution to exist. We now see that for a de Sitter excited state to occur, at least quartic order curvature corrections are necessary. A similar conclusion can be made for the fluctuation determinants if we take $\theta_{\bar{c}(r)} = {8\over 3}$, and $\theta_{\hat{c}(r)} = 0$. However now two questions arise.

${\bf 1.}$ What happens to the exponential pieces with the choice (<ref>)?

${\bf 2.}$ How and where does the second EFT criterion from (<ref>) fits inside (<ref>)?

$\theta_{\bar{c}(r)}$       $\theta_{\hat{c}(r)}$      $\theta_{\hat{c}(r)}-2$

${8\over 3}$     0     $-2$

${5\over 3}$     1     $-1$

${4\over 3}$     ${4\over 3}$     $-{2\over 3}$

${1}$     ${5\over 3}$     $-{1\over 3}$

${2\over 3}$     ${2}$     $~~0$

${0}$     ${8\over 3}$     $~~{2\over 3}$

[]The $g_s$ scalings $\theta_{\bar{c}(r)}$ and $\theta_{\hat{c}(r)}$ satisfying (<ref>).

We will start with the first question and to answer this we refer the readers to Table <ref> where the various distributions of $\theta_{\bar{c}(r)}$ and $\theta_{\hat{c}(r)}$ satisfying (<ref>) are laid out. Let ${\rm M}_p = \epsilon^{-b}$ and $g_s = \epsilon^a$, for $(a, b) > (0, 0)$ in the limit $\epsilon \to 0$.  The exponential piece in (<ref>), behaves as:

lim_ϵ→0 exp[-r ϵ^aθ_ĉ(r) - 2a -6b], for $r \in \mathbb{Z}$ and $n = k = l = 0$. The choice of $g_s = \epsilon^a$ with $a > 0$ is the late time, but we can keep $a \to 0$ to cover the whole range in the temporal domain
$-{1\over \sqrt{\Lambda}} < t \le 0$. Even with this, the first four entries in Table <ref> suggests that the exponential piece dies-off pretty fast, implying that the contributions from the instanton sector may be made arbitrarily small. Note that the choice
$\theta_{\bar{c}(r)} = {2\over 3}$ makes the exponential piece to be independent of $g_s$. This is not a classical contribution because $\theta_{\hat{c}(r)} = 2$ adds in the necessary quantum corrections both to the fluctuation determinants and to the exponential factors.  From
(<ref>), the exponential piece would still appear to die-off, but if the value of the integral coming from ${\bf I}_1^{(0)}(\hat{c}(r), y'')$ is kept small then we can have finite answers for small $r$.
${\bf I}_1^{(0)}(\hat{c}(r), y'')$ can have contributions from cubic order curvature terms, but also from the emergent Rarita-Schwinger fermionic condensates.
Although we don't discuss the fermionic story here (see [9] for details on this), it is known for sometime now [2] that type IIB gaugino condensates on seven-branes play a crucial role in generating a positive cosmological constant solution.
The series constructed out of them is clearly convergent as one would have expected from it's trans-series form, thus quantifying the precise contributions from the instanton sectors for this scenario. For the choice $\theta_{\bar{c}(r)} = 0$, we have $\theta_{\hat{c}(r)} = {8\over 3}$ from (<ref>). On the other hand from Table <ref> and (<ref>),
we see that as long as $a > 9b$ $-$ which is late time $-$ the exponential piece may be perturbatively expanded, otherwise it will die-off pretty fast. Finally the choice of $\theta_{\hat{c}(r)} = {2\over 3}$ doesn't produce anything quantum so we will ignore it. Putting everything together would tell us how much do the instantons contribute to the EOM (<ref>).

This brings us to the final question: where does the second EFT criterion
from (<ref>) fits in the aforementioned discussion? The answer lies in the form of the two scalings $\theta_{\bar{c}(r)}$ and $\theta_{\hat{c}(r)}$. We can rewrite them in the following way:

θ_c̅(r) = θ^(1)_c̅(r) + |θ^(2)_c̅(r)| sign  θ^(2)_c̅(r),
θ_ĉ(r) = θ^(1)_ĉ(r) + |θ^(2)_ĉ(r)| sign  θ^(2)_ĉ(r), where $\theta^{(i)}_{c(r)}$ for $i = 1, 2$ are used to split the scaling $\theta_{c(r)}$ into two pieces with the second one being sensitive to the second criterion of (<ref>). This is captured by ${\rm sign}~\theta_{c(r)}$ defined in the following way:

sign θ_c(r) ≡   +1
for   ∂g_s∂t ∝g_s^+ive

-1         for   ∂g_s∂t ∝g_s^-ive

Plugging this in either (<ref>) or (<ref>) would create relative minus signs implying that there are now infinite possible solutions for $\theta^{(i)}_{c(r)}$. This would clearly be a breakdown of the $g_s$ hierarchy. The ${\rm M}_p$ hierarchy is also broken because of the presence of the emergent localized G-fluxes in a similar vein as in
[14, 16]. Thus if the second criterion of (<ref>) is not satisfied then EFT would break down because of the presence of an infinite number of operators at order ${g_s^a\over {\rm M}_p^b}$, $\forall (a, b)$ with $(a, b) \in (\mathbb{Z}, \mathbb{Z})$.

Therefore to summarize: the second criterion in (<ref>) is absolutely essential for the consistency of the computations presented in this section done by including the instanton corrections. What about the non-local interactions? It turns out that the story is bit more subtle than what we discussed earlier. In the following we elaborate on this.

§.§.§ Revisiting the non-local interactions from (<ref>) and (<ref>)

In the three points mentioned just below (<ref>), we cautioned the readers that (<ref>) may not be the full answer, as there could exist the possibility of additional corrections to (<ref>). However the analysis that we performed to reach to
(<ref>) did not suggest any such possibilities, so the question is how and why do we need additional corrections to (<ref>). Note that a similar form for the instanton terms did require additional contributions coming from the fluctuation determinants around the instantons, so therein the corrected action did have a solid reasoning for back up. On the other hand, the computation that we performed to convert (<ref>) to (<ref>) was aimed to get a convergent form for the non-local interactions without losing any information contained in (<ref>). This resulted in a specific choice for the coefficients in (<ref>) related to the ones from
(<ref>) via the matrix $\mathbb{M}$ in (<ref>). The only way to relax this, yet still keeping the convergence property intact, is to convert (<ref>) to a trans-series much like what we had for the instanton terms. However this begs the question as to why should we do this. Is there anything in particular missing from
(<ref>) that requires us to change the format of (<ref>)?

To see the short-comings of the non-local interactions (<ref>) appearing in the action (<ref>), let us go back to the terms  on the eighth line of (<ref>). Our aim would be to check the $g_s$ scalings of the various terms therein using the metric scalings (<ref>). For simplicity, we can restrict ourselves to the space-time directions only. The $g_s$ scaling then becomes:

⟨g_AB(X)⟩_σ∑_p = 1^∞b_p [exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(⟨Ξ(Y', x)⟩_σ))|𝔽(Y - Y') Q_pert(c̃(p); ⟨Ξ(Y', x)⟩_σ)|) - 1]

=   g_s^-83 ∑_p = 1^∞b_p [exp(-pM_p^6 g_s^θ_c̃(p) - 2 ∫_M_6 d^6y' ∑_n = 0^∞|𝔽(y - y') I_3^(n)(c̃(p); y')|g_s^2n3) - 1]

where ${\bf I}_3^{(n)}(\tilde{c}(p); y')$ is similar to ${\bf I}_1^{(n)}(\hat{c}(r); y'')$ appearing in say (<ref>) but expressed using different coefficients $\tilde{c}(p)$ in (<ref>). Comparing the $g_s$ scalings of this term with the ones from (<ref>), we already see an insurmountable issue: the $g_s$ scaling of the last term above can never match with the $g_s$ scalings of (<ref>)! However in the limit $g_s \to 0$ this terms is cancelled by the exponential piece whenever $\theta_{\tilde{c}(p)} > 2$, so doesn't give enough reason to discard it yet. On the other hand, comparing the $g_s$ scaling of the exponential piece with the ones from (<ref>), we get the following value for $\theta_{\tilde{c}(p)}$:

θ_c̃(p)   =   2 + log(2r log g_s) - log 3log g_s ,

Plot of the real and imaginary parts of the function $\theta_{\tilde{c}(p)} = 2 + {{\rm log}\left(-{2\over 3}~{\rm log}~g\right)\over {\rm log}~g}$ in blue and
orange respectively. Note that although solutions exist for $g < 1$, because ${\bf Im}~\theta_{\tilde{c}(p)} = 0$, they are functions of $g$. For $g > 1$, clearly both
${\bf Im}~\theta_{\tilde{c}(p)}$ and ${\bf Re}~\theta_{\tilde{c}(p)}$ are non-zero, so no solutions exist.

with $r = -1$ and $r = 3$. For generic value of $g_s < 1$ this is problematic because it will make $\theta_{\tilde{c}(p)}$ to be $g_s$ dependent, which is unacceptable. In fact for $r = -1$ in (<ref>), solutions for  $\theta_{\tilde{c}(p)}$  appear to exist for $g_s < 1$ but they are still slowly varying functions of $g_s$ as shown in figure <ref>. For $r = 3$ in (<ref>), there are no solutions for $g_s < 1$ as shown in figure <ref>. From this figure one may see that solutions exist, albeit as functions of $g_s$, only for $g_s \ge \left[\mathbb{W}(1)\right]^{-{1\over 2}} \approx 1.328$. ($\mathbb{W}(z)$ is the Lambert W-function.) This implies that $\theta_{\tilde{c}(p)}$ we got in (<ref>) for either values of $r$ is unacceptable.

The above analysis suggests that the non-local interactions appearing in the fourth line of (<ref>) cannot be the complete answer. Note that this mismatch is only visible once we used the Schwinger-Dyson equations. In fact the reason why we couldn't match the $g_s$ scalings was because of the absence of any operators accompanying the exponential piece. These operators would be similar to the fluctuation determinants that appear in a similar setting with the instanton terms, but now there is a difference: we can use both local and non-local operators as the fluctuation determinants now. This means we can quantify the “fluctuation determinants$"$ by the following set of operators:

Q_pert(č(p); Ξ(X)) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(Y”, x))  Q_pert(c̀(p); Ξ(Y”, x)) 𝔽(y-y”; w-w”),
where ${\bf Q}_{\rm pert}$ is defined as in (<ref>) but now with coefficients $\check{c}(p) \equiv \check{c}_{nl}(p)$ and $\grave{c}(p) \equiv \grave{c}_{nl}(p)$. The other notations are: ${\rm X} = (x, y, w)$,
${\rm Y}'' = (y'', w'')$ and ${\bf \Xi}$ is the set of on-shell fields as before. In the Schwinger-Dyson equations the latter has to be replaced by its expectation value $\langle {\bf \Xi}\rangle_\sigma$, but there is still one more subtlety remaining related to the piece $-b_p \sqrt{-{\bf g}_{11}({\rm X})}$ that appears on the fourth line of (<ref>) without an exponential factor. This is also the piece that led to an earlier mismatch with the $g_s$ scalings. In the presence of the operators (<ref>), this particular piece will lead to two set of operator series: a perturbative quantum series and a non-local quantum series. The former is what we had as ${\bf S}_{\rm pert}$ in (<ref>) and the latter as ${\bf S}_{\rm nloc}$ in (<ref>) before we converted them to ${\bf S}_{\rm NP}$ and ${\bf S}_{\rm nloc}$  in (<ref>) and (<ref>) respectively. The former simply renormalizes the coefficient $d_s$ in the third line of (<ref>), while the latter $-$ taking the form  (<ref>) $-$ will again have the same issue as before because of the $-1$ factor. The correct way out of this is to rewrite the fourth line of (<ref>) in the following way:

S_nloc(Ξ)     =
M_p^11∫d^3x d^6y d^2w √(-g_11(X))

×∑_p = 0^∞b_p [Q_pert(č(p); Ξ(X)) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(Y”, x))  Q_pert(c̀(p); Ξ(Y”, x)) 𝔽(y-y”; w-w”)]

×exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(Y', x))|𝔽(y -y'; w -w') Q_pert(c̃(p); Ξ(Y', x))|)

where the second line above may be interpreted as the fluctuation determinants and the third line as the saddles from the non-local effects. This will effect the EOM in (<ref>), and consequently the eighth, ninth and the tenth lines will get modified, including some of the entries in Table <ref>. These changes are easy to quantify and
in the following we will elaborate on them.

Plot of the real and imaginary parts of the function $\theta_{\tilde{c}(p)}= {2} + {{\rm log}\left({2}~{\rm log}~g\right)\over {\rm log}~g}$ in blue and
orange respectively. Note that no solutions exist for $g < 1$ here because  both
${\bf Im}~\theta_{\tilde{c}(p)}$ and ${\bf Re}~\theta_{\tilde{c}(p)}$ are non-zero. For $g \ge 1.33$ solutions exist, because ${\bf Im}~\theta_{\tilde{c}(p)} = 0$, but they are functions of $g$.

Let us start with the simpler piece in (<ref>) where the fluctuation determinant does not depend on the non-locality factor. The Schwinger-Dyson equation corresponding to this term may be expressed in the following way:

-1√(g_11(⟨Ξ(X)⟩_σ)) ∑_p = 0^∞b_pδδ⟨g^AB(X)⟩_σ
(√(g_11(⟨Ξ(X)⟩_σ))Q_pert(č(p); ⟨Ξ(X)⟩_σ))

× exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(⟨Ξ(Y', x)⟩_σ))|𝔽(y -y'; w -w') Q_pert(c̃(p); ⟨Ξ(Y', x)⟩_σ)|)

=    ∑_p = 0^∞b_p g_s^θ_č(p)-83  ∑_r = 0^∞K^(r)_μν(č(p); y) g_s^2r3 exp(-pM_p^6 g_s^θ_c̃(p) - 2 ∫_M_6 d^6y' ∑_n = 0^∞|𝔽(y - y') I_3^(n)(c̃(p); y')⟩_σ)|g_s^2n3)

where ${\bf I}_3^{(n)}(\tilde{c}(p); y')$ is the same operator that we encountered in (<ref>), and ${\bf K}^{(r)}_{\mu\nu}(\check{c}(p); y)$ is constructed from the first line above. The dominant $g_s$ scaling, for $r = n = 0$ and $p =1$, when compared to (<ref>) satisfies:

θ_č(p) - 23 = g_s^θ_c̃(p) - 2 log g_s, which is similar to what we had in (<ref>), and motivated from there, as well as from figures <ref> and <ref>, we cannot make $\theta_{\check{c}(p)} =\theta_{\tilde{c}(p)}$. One solution is that $\theta_{\check{c}(p)} = {2\over 3}$, and for
${2\over 3} \le \theta_{\tilde{c}(p)} < 2$, the non-local contributions are heavily suppressed. Additionally the absence of quantum terms for the scaling $\theta_{\check{c}(p)} = {2\over 3}$, along with similar arguments following (<ref>), shows that (<ref>) does not contribute significantly to the Schwinger-Dyson equations.

The next contribution would appear from the non-local fluctuation determinants from (<ref>). It contains more complicated set of terms, so one might wonder if they can contribute to the Schwinger-Dyson equations. To see this let us express the EOMs in the following way:

- M_p^8∑_p = 0^∞⟨g_AB(X)⟩_σ∫_M_8 d^6y' d^2w'√(g_8(⟨Ξ(Y', x)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(Y', x)⟩_σ) 𝔽(y-y'; w-w')

× exp(-pM_p^8∫_M_8 d^6y” d^2w”
√(g_8(⟨Ξ(Y”, x)⟩_σ))|𝔽(y -y”; w -w”) Q_pert(c̃(p); ⟨Ξ(Y”, x)⟩_σ)|)

=    M_p^6∑_p = 0^∞b_p g_s^θ_c̀(p)-143  ∑_r = 0^∞∫_M_6 d^6y'L^(r)_μν(c̀(p); y) 𝔽(y - y')g_s^2r3
exp(-pM_p^6 g_s^θ_c̃(p) - 2 ∫_M_6 d^6y' ∑_n = 0^∞|𝔽(y - y') I_3^(n)(c̃(p); y')|g_s^2n3)

where ${\bf I}_3^{(n)}(\tilde{c}(p); y')$ is the same operator that we encountered earlier, but ${\bf L}^{(r)}_{\mu\nu}(\grave{c}(p); y)$ is new and is constructed from the first line above. A naive $g_s$ scaling of the dominant term for $r = n = 0$ and $p = 1$ with the $g_s$ scalings from (<ref>) gives us:

θ_c̀(p) - 83 = g_s^θ_c̃(p) - 2 log g_s, which while doesn't have the same problem that we faced with (<ref>), doesn't also provide a $g_s$ independent solution
for $\theta_{\grave{c}(p)} =\theta_{\tilde{c}(p)}$. This is of course expected, and therefore we can keep $\theta_{\grave{c}(p)} \ne \theta_{\tilde{c}(p)}$ and assume the following solution for $\theta_{\grave{c}(p)}$:

θ_c̀(p) = 83, implying that for ${2\over 3} \le \theta_{\tilde{c}(p)} < 2$ the exponential piece will continue to be suppressed for $p \ge 1$ at late time. For $\theta_{\tilde{c}(p)} \ge 2$, one may perturbatively expand the exponential piece in powers of $g_s$. They will then contribute to the higher order Schwinger-Dyson equations. Again, we see that the contributing terms come typically from quartic order curvature terms satisfying (<ref>). In fact we expect similar contributions from another related term from the non-local action (<ref>), namely:

-M_p^8 ∫d^6y' d^2w' √(g_11(⟨Ξ(Y', x)⟩_σg_11(⟨Ξ(X)⟩_σ)∑_p = 0^∞δδ⟨g_AB(X)⟩_σ (√(g_8(⟨Ξ(X)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(X)⟩_σ)) 𝔽(y'-y; w'-w)

× exp(-pM_p^8∫_M_8 d^6y” d^2w”
√(g_8(⟨Ξ(Y”, x)⟩_σ))|𝔽(y' -y”; w' -w”) Q_pert(c̃(p); ⟨Ξ(Y”, x)⟩_σ)|)

=    M_p^6∑_p = 0^∞b_p g_s^θ_c̀(p)-143  ∑_r = 0^∞∫_M_6 d^6y'M^(r)_μν(c̀(p); y) 𝔽(y - y')g_s^2r3
exp(-pM_p^6 g_s^θ_c̃(p) - 2 ∫_M_6 d^6y” ∑_n = 0^∞|𝔽(y' - y”) I_3^(n)(c̃(p); y”)|g_s^2n3)

which has similar behavior as (<ref>) except with a different function ${\bf M}^{(r)}_{\mu\nu}(\grave{c}(p); y)$ and a different distribution of integrals in the exponential term. This also means that (<ref>) continues to be the quantum scaling here too, and the term could contribute non-trivially as long as $\theta_{\tilde{c}(p)} \ge 2$. The dominant contribution would come from say a quartic order curvature piece from the fluctuation determinants and a $g_s$ independent $\theta_{\tilde{c}(p)} = 2$ piece from the exponential factor.

There is still one more contribution to the Schwinger-Dyson equation coming from a mixture of the terms from the fluctuation determinants and the exponential factors. This may be presented in the following way:

M_p^8 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) 𝔽(Y'-Y)∑_p = 1^∞pb_p
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(c̃(p); ⟨Ξ(X)⟩_σ)]

×[Q_pert(č(p); ⟨Ξ(Y', x)⟩_σ) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(⟨Ξ(Y”, x)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(Y”, x)⟩_σ) 𝔽(y'-y”; w'-w”)]

×  exp(-pM_p^8∫_M_8 d^6y”' d^2w”'
√(g_8(⟨Ξ(Y”', x)⟩_σ))|𝔽(y' -y”'; w' -w”') Q_pert(c̃(p); ⟨Ξ(Y”', x)⟩_σ)|)

= M_p^6 ∑_p = 1^∞pb_p ∫_M_6 d^6y' g_s^θ_c̃(p) - 143
exp(-pM_p^6 g_s^θ_c̃(p) - 2 ∫_M_6 d^6y”' ∑_n = 0^∞|𝔽(y' - y”') I_3^(n)(c̃(p); y”')|g_s^2n3)

×[∑_k = 0^∞P_μν^(k)(c̃(p), č(p); y')  g_s^θ_č(p) + 2k3 + M_p^6 ∫_M_6 d^6y”
∑_k = 0^∞Q_μν^(k)(c̃(p), c̀(p); y”)  𝔽(y' - y”)   g_s^θ_c̀(p) + 2k3] 𝔽(y' - y)

where ${\bf P}_{\mu\nu}^{(k)}(\tilde{c}(p), \check{c}(p); y')$ and
${\bf Q}_{\mu\nu}^{(k)}(\tilde{c}(p), \grave{c}(p); y'')$ are two new functions appearing from the local and non-local pieces from the fluctuation determinants; and ${\bf I}_3^{(n)}(\tilde{c}(p); y''')$ is the same function that appeared earlier. One may also easily see that the quantum scalings now satisfy the following set of relations:

θ_c̃(p) +  θ_č(p) = 83,
θ_c̃(p) +  θ_c̀(p) = 83, where each piece is similar to what we encountered in (<ref>), and therefore would be classified individually as in Table <ref>. To summarize: including the fluctuation determinants to modify the non-local interactions as in (<ref>), all ambiguities seem to go away and consistent matching of the $g_s$ scalings may be easily performed.

§.§.§ Five-brane instantons wrapped on six-manifold ${\cal M}_4 \times {\mathbb{T}^2\over {\cal G}}$

A detailed study done in the previous section instructed us how to deal with five-brane instantons wrapped on the six-dimensional base ${\cal M}_6$ of the eight-manifold ${\cal M}_8 = {\cal M}_6 \times {\mathbb{T}^2\over {\cal G}}$. We saw the contributions from both the non-perturbative as well as the non-local terms in the action (<ref>), although a more careful study revealed that (<ref>) needs to be modified because the non-local interactions given in (<ref>) are incomplete. The action that actually takes care of all the aforementioned subtleties and from where the correct Schwinger-Dyson equations can be derived is given by the following:

Ŝ_tot(Ξ) = S_kin(Ξ) + S_NP(Ξ) + S_nloc(Ξ)

= M_p^9
∫d^3 x  d^6y d^2w √(-g_11(X))
[R_11(X) + G_4(X) ∧∗_11G_4(X)] + M_p^9∫C_3(X) ∧G_4(X) ∧G_4(X) + M_p^3∫C_3(X) ∧𝕏_8(X)

+ M_p^11∫d^3x d^6y d^2w √(-g_11(X))
∑_s = 0^∞d_s Q_pert(c̅(s); Ξ(X)) exp(-s M_p^8 ∫_0^y ∫_0^w d^6y' d^2w' √(g_8(Y', x)) |Q_pert(ĉ(s); Ξ(Y', x))|)

M_p^11∫d^3x d^6y d^2w √(-g_11(X))
∑_p = 0^∞b_p  exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(Y', x))|𝔽(y -y'; w -w') Q_pert(c̃(p); Ξ(Y', x))|)

×[Q_pert(č(p); Ξ(X)) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(Y”, x))  Q_pert(c̀(p); Ξ(Y”, x)) 𝔽(y-y”; w-w”)],

where ${\bf Q}_{\rm pert}(c; {\bf \Xi}(x, y, w))$ is defined in terms of the on-shell degrees of freedom ${\bf \Xi}(x, y, w)$ in (<ref>) with $c \equiv \{c_{nl}\}$ forming the set of coefficients for the perturbative series in (<ref>). These set of coefficients may be used to differentiate between the various contributions from the instanton and the non-local saddles and their corresponding fluctuation determinants as shown in (<ref>). For the five-brane instantons wrapped on the six-manifold base, the perturbative series may be expressed as (<ref>). On the other hand, to study the five-brane instantons wrapped on the toroidal manifold and a four-cycle in the six-dimensional base one can take ${\cal M}_6$ locally as ${\cal M}_4 \times {\cal M}_2$, and study the instantons wrapped on ${\cal M}_4 \times {\mathbb{T}^2\over {\cal G}}$. These instantons are related to the three-branes instantons in the type IIB side, which we can call as the KKLT instantons [2]. For such a case the perturbative series may be defined as:

Q_pert(c; Ξ(y^m, y^α, w, x)) ≡Q_pert^KKLT(c; Ξ(y^m, w, x))  δ^2(y^α-y^α_o)M_p^2 √(g̅_2(y^α, x)), where $y^m \in {\cal M}_4, y^\alpha \in {\cal M}_2, w \in {\mathbb{T}^2\over {\cal G}}$ and $x \in {\bf R}^{2, 1}$. We have also assume that ${\bf g}_8(y^m, y^\alpha, w, x) = {\bf g}_4(y^m, x) \bar{\bf g}_2(y^\alpha, x) {\bf g}_2(w, x)$, where the coordinate dependence specifies the corresponding sub-manifold for which we take the metric determinant.

The analysis with the KKLT instantons is very similar to the ones we did with the BBS instantons, except that the determinant of the metric for ${\cal M}_4 \times {\mathbb{T}^2\over {\cal G}}$ has zero scaling with $g_s$, so the quantum terms do not contribute much to the EOMs. Therefore
we will not go in details here and leave the analysis for the diligent readers. The procedure follows the strategy we laid out in section <ref>, namely substitute (<ref>) in the Schwinger-Dyson equations and then compute the contributions from all the terms. The Schwinger-Dyson equation coming from the action (<ref>) may be succinctly presented as:

R_AB(⟨Ξ(X)⟩_σ) -
12 ⟨g_AB(X)⟩_σR(⟨Ξ(X)⟩_σ)

2 √(g_11(⟨Ξ(X)⟩_σ)) δδ⟨g^AB(X)⟩_σ (√(g_11(⟨Ξ(X)⟩_σ)) G_4( ⟨Ξ(X)⟩_σ) ∧∗_11G_4( ⟨Ξ(X)⟩_σ))

-  2M_p^2√(g_11(⟨Ξ(X)⟩_σ))
∑_s= 0^∞d_s δδ⟨g^AB(X)⟩_σ
[√(g_11(⟨Ξ(X)⟩_σ))Q_pert(c̅(s); ⟨Ξ(X)⟩_σ)]

× exp(-s M_p^8 ∫_0^y ∫_0^w d^6y' d^2w' √(g_8(⟨Ξ(Y', x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y', x)⟩_σ)|)

+  M_p^10 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) ∑_s = 0^∞ s d_s
Q_pert(c̅(s); ⟨Ξ(Y', x)⟩_σ)

×  δδ⟨g^AB(X)⟩_σ
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(ĉ(s); ⟨Ξ(X)⟩_σ)]
Θ(y'-y) Θ(w'-z)

× exp(-s M_p^8 ∫_0^y' ∫_0^w' d^6y” d^2w” √(g_8(⟨Ξ(Y”, x)⟩_σ)) |Q_pert(ĉ(s); ⟨Ξ(Y”, x)⟩_σ)|)

-1√(g_11(⟨Ξ(X)⟩_σ)) ∑_p = 0^∞b_pδδ⟨g^AB(X)⟩_σ
(√(g_11(⟨Ξ(X)⟩_σ))Q_pert(č(p); ⟨Ξ(X)⟩_σ))

× exp(-pM_p^8∫_M_8 d^6y' d^2w'
√(g_8(⟨Ξ(Y', x)⟩_σ))|𝔽(y -y'; w -w') Q_pert(c̃(p); ⟨Ξ(Y', x)⟩_σ)|)

- M_p^8∑_p = 0^∞⟨g_AB(X)⟩_σ∫_M_8 d^6y' d^2w'√(g_8(⟨Ξ(Y', x)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(Y', x)⟩_σ) 𝔽(y-y'; w-w')

× exp(-pM_p^8∫_M_8 d^6y” d^2w”
√(g_8(⟨Ξ(Y”, x)⟩_σ))|𝔽(y -y”; w -w”) Q_pert(c̃(p); ⟨Ξ(Y”, x)⟩_σ)|)

-M_p^8 ∫d^6y' d^2w' √(g_11(⟨Ξ(Y', x)⟩_σg_11(⟨Ξ(X)⟩_σ)∑_p = 0^∞δδ⟨g_AB(X)⟩_σ (√(g_8(⟨Ξ(X)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(X)⟩_σ)) 𝔽(y'-y; w'-w)

× exp(-pM_p^8∫_M_8 d^6y” d^2w”
√(g_8(⟨Ξ(Y”, x)⟩_σ))|𝔽(y' -y”; w' -w”) Q_pert(c̃(p); ⟨Ξ(Y”, x)⟩_σ)|)

+   M_p^8 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) 𝔽(Y'-Y)∑_p = 1^∞pb_p
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(c̃(p); ⟨Ξ(X)⟩_σ)]

×[Q_pert(č(p); ⟨Ξ(Y', x)⟩_σ) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(⟨Ξ(Y”, x)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(Y”, x)⟩_σ) 𝔽(y'-y”; w'-w”)]

×  exp(-pM_p^8∫_M_8 d^6y”' d^2w”'
√(g_8(⟨Ξ(Y”', x)⟩_σ))|𝔽(y' -y”'; w' -w”') Q_pert(c̃(p); ⟨Ξ(Y”', x)⟩_σ)|)

which modifies (<ref>) appropriately. Therefore instead of taking the configuration (<ref>), we can take the configuration from (<ref>) with the simplifying conditions from (<ref>), except now $\alpha(t) \ne 0$ and $\beta(t) \ne 0$. Allowing a non-zero $\alpha(t)$ and $\beta(t)$ could in principle lead to late time singularities, which we should avoid. Since we have chosen $f_1(t)$ as the defining function for all the flux and the metric components, it will be useful to also define $\alpha(t)$ and $\beta(t)$ in terms of $f_1(t)$ in the following way:

α(t) = ∑_n = 1^∞l_n(f_1(t) - 3a2)^n,     β(t) = ∑_n = 1^∞p_n(f_1(t) - 3a2)^n, with appropriate choice for the coefficients $(l_n, p_n)$ such that
$(\alpha(t), \beta(t)) < \left({2\over 3}, {2\over 3}\right)$. The latter requirement is to ensure the correct limit of type IIB theory from M-theory. Note that as $t \to 0$, both $\alpha(t)$ and $\beta(t)$ vanishes as can be inferred from section <ref>. The Einstein term in (<ref>) scales in the following way:

R_AB(⟨Ξ(X)⟩_σ) -
12 ⟨g_AB(X)⟩_σR(⟨Ξ(X)⟩_σ) =   1g_s^2 ∑_k = 0^∞G^(k)_μν(y, w) ǧ_s^2k3 L_1(f_1(t)) + log corrections, which is significantly different from the scaling in (<ref>) because of the presence of $\hat{g}_s \equiv g_s^{f_1(t)}$, ${\rm L}_1(f_1(t))$ and log corrections. They provide sub-dominant contributions in
(<ref>) because at late time $f_1(t)$ approaches a constant value of ${3a\over 2}$, and log corrections are typically very small. The explicit form for ${\rm L}_1(f_1(t))$ can be derived but the analysis is quite technical and the readers may get the derivation from [9]. One may now try to compare the $g_s$ scalings of the Einstein term with all the other terms from (<ref>). We will however not do the detailed analysis here, but only discuss one such term and leave the others for future work. The term that we will like to study is the following:

M_p^8 ∫d^6y'  d^2w' √( g_11(⟨Ξ(Y', x)⟩_σ) g_11(⟨Ξ(X)⟩_σ)) 𝔽(Y'-Y)∑_p = 1^∞pb_p
[√(g_8(⟨Ξ(X)⟩_σ))Q_pert(c̃(p); ⟨Ξ(X)⟩_σ)]

×[Q_pert(č(p); ⟨Ξ(Y', x)⟩_σ) +
M_p^8∫_M_8 d^6y” d^2w”√(g_8(⟨Ξ(Y”, x)⟩_σ))  Q_pert(c̀(p); ⟨Ξ(Y”, x)⟩_σ) 𝔽(y'-y”; w'-w”)]

×  exp(-pM_p^8∫_M_8 d^6y”' d^2w”'
√(g_8(⟨Ξ(Y”', x)⟩_σ))|𝔽(y' -y”'; w' -w”') Q_pert(c̃(p); ⟨Ξ(Y”', x)⟩_σ)|)

= M_p^6 ∑_p = 1^∞pb_p ∫_M_6 d^4y' d^2w' g_s^θ_c̃(p)(g_s) - 143 ǧ_s^log corrections

×exp(-pM_p^6 g_s^θ_c̃(p)(g_s) - 2ǧ_s^log corrections ∫_M_6 d^4y”' d^2w”' ∑_n = 0^∞|𝔽(Y' - Y”') I_3^(n)(c̃(p); Y”')|ǧ_s^2n3 L_2(f_1(t)))

×[∑_k = 0^∞P_μν^(k)(c̃(p), č(p); y')  g_s^θ_č(p)(g_s) ǧ_s^2k3L_3(f_1(t)) + log corrections

+ M_p^6 ∫_M_6 d^4y”d^2w”
∑_k = 0^∞Q_μν^(k)(c̃(p), c̀(p); Y”)  𝔽(Y' - Y”)   g_s^θ_c̀(p)(g_s) ǧ_s^2k 3L_4(f_1(t)) + log corrections] 𝔽(Y' - Y)

where ${\rm Y} = (y, w)$ and $\widetilde{\cal M}_6 = {\cal M}_4 \times {\mathbb{T}^2\over {\cal G}}$. The matching of the $g_s$ scalings with the ones from (<ref>) is now more complicated because the quantum scaling $\theta_c$ of (<ref>) is now a function of $g_s$ itself because of its temporal dependence. In addition to that we now have three new functions of $f_1(t)$, namely ${\rm L}_i(f_1(t))$ for $i = 2, 3, 4$, plus the expected log corrections. The dominant scaling appears from $n = k = 0$ and, with the log corrections being very small, we have a similar relation like (<ref>):

θ_c̃(p)(g_s) +  θ_č(p)(g_s) = 83,
θ_c̃(p)(g_s) +  θ_c̀(p)(g_s) = 83, except that the LHS are functions of $g_s$. As it stands, (<ref>) doesn't make any sense because the RHS are $g_s$ independent factors of ${8\over 3}$ whereas the LHS are functions of $g_s$, and therefore functions of the conformal time $t$. On the other hand, if $\theta_{{c}}(g_s)$ are very slowly varying functions of time
then (<ref>) could still make sense and using it we can construct the required quantum terms that would support a configuration like (<ref>) with the simplifying conditions (<ref>). Recall that, one of the requirement on the temporally varying cosmological constant in section <ref> is to take a very slow variation. We now see that this is a self-consistent assumption from various angles. What this means with respect to the DESI BAO result [10] remains to be seen, although issues pointed out in [21] may no longer persist. We will discuss more on this in future publication.

§ CONCLUSION, SUMMARY AND OPEN QUESTIONS

We conclude with a non-technical summary of our work on the construction and the consistency of the de Sitter Glauber-Sudarshan (GS) states in a question-and-answer format. Afterwards, we list some future directions stating some important open questions.

§.§ A non-technical summary on the construction of de Sitter GS states

As unearthed[Also by G. Galilei and S. Coleman before.] by G. Moore and J. Harvey in [44] towards the end of last century, the discussion between Filippo Salviati, a scholar and a close friend of Galileo Galilei, and  Giovanni Francesco Sagredo, an intelligent layman, helped us to understand some basic principles behind the membrane instanton computations. Interestingly, unbeknownst to the general public, Salviati and Sagredo also had a detailed discussion on the two chief stringy systems related to positive cosmological constant solution in four spacetime dimensions, namely, the vacuum solution versus the excited “Glauber-Sudarshan$"$ states. In the following we reproduce an unabridged version of the dialogue[Needless to say, any errors in translation leading to any subsequent misinterpretations of the text are the faults of the present authors of this paper.].

§.§.§ The prologue: On the basics of the construction

Sagredo: I heard from Simplicio that you reproduced our four-dimensional de Sitter spacetime as a coherent state from M-theory...

Salviati: Indeed but you got the details wrong: We reproduced four-dimensional de Sitter spacetime in string theory as a Glauber-Sudarshan state, and not as a coherent state. The difference between them is very important, but we are getting ahead of the story. And sorry, I interrupted you midway in your question. Please proceed.

Sagredo: Yes, and of course that is one of my concern, but before I go into this I have some even simpler set of questions related to your construction. I understand that you realized de Sitter spacetime in type IIB theory, but I don't see why you have to do this from M-theory. Why not directly in type IIB theory?

Salviati: That's a good question. You can of course do everything from type IIB, but the construction that we're looking at has vanishing axio-dilaton which is the constant coupling point of F-theory [17]. This makes the type IIB string coupling to be exactly identity where even S-duality doesn't help. Plus the non-existence of a well-defined action in type IIB is yet another concern, although in recent times this has been surmounted [42].

Sagredo: OK, but why do you need to be at a constant coupling point of F-theory? You can surely always go away from that point to be somewhere in the moduli space of F-theory where the coupling is weak..

Salviati: Sure you can be at a generic point in the moduli space of F-theory, but this will make the axio-dilaton to become time-dependent which, in turn, will make the seven-branes dynamical. The fluxes are already time-dependent,  so the system will be highly non-trivial and we're not sure if it is easy to tackle the dynamics with the limited knowledge we have.

Sagredo: I'm still a bit confused. We could always be in the so-called static-patch of de Sitter where there is no time-dependence. Couldn't we just study the dynamics from this point of view?

Salviati: You could in principle be any patch of a de Sitter space, but that doesn't mean that the system will be time-independent. Static patch in particular has a very deceptive dynamics. Since you asked me the question, let me clarify by first pointing you to
figure <ref> in section <ref>. Please look at the middle and the right-most figures. In the middle figure the frequencies are not changing, however this doesn't mean that there is no time-dependence. Outside the static patch, the frequencies do change (the space-like vectors become time-like), and from Fourier decomposition we know that the frequencies are also changing with respect to time despite the metric appearing to be time-independent. (View this as though you are using global or Poincare patch temporal coordinate.) The difference is that we don't see these temporal variations from inside the static patch. Additionally, the dual IIA coupling (<ref>) as discussed in section <ref> shows that it becomes strongly coupled along certain paths (see figure <ref>).

Sagredo: Thank you, but your answer has raised more questions for me. First, you said that the dual type IIA coupling becomes strong. But, aren't you supposed to allow strong IIA coupling since you are going to M-theory?

Salviati: Yes we are in M-theory, but we are in the opposite limit where the dual IIA coupling $g_s < 1$. Now you might ask: why is that? The reason is that we are not restricted to perturbative computations only. In fact, as will become clear soon, we really do not have any perturbative interactions at all. Due to an underlying resurgence structure (which I shall also clarify soon), the action that actually controls the dynamics of the de Sitter space only has non-perturbative and non-local interactions. Both these effects go as ${\rm exp}\left(-{1\over g_s^a}\right)$, and to allow for a convergent series of such terms, we need $g_s < 1$.

Sagredo: OK, this looks even more complicated than what I thought, as I do not understand many of the points that you referred to related to resurgence, convergence and the system not having any perturbative interactions at all. I also do not understand how and where are you getting the non-local interactions from. But that's for later. Meanwhile, even in field theory, don't we have perturbative interactions? For example $\lambda\varphi^4$ theory or even QED. They all have perturbative interactions as far as I know.

Salviati: I see that you are going for even more basic questions. But I'm sorry to break it to you: perturbative interactions which were long thought to be useful techniques to study quantum field theory, are not very reliable as we go to higher orders in perturbation theory. This is in the light of more recent developments in the subject [43]. One needs to tread carefully here. The problem lies in the asymptotic nature of a perturbative series itself. The factorial growth of the Feynman diagrams calls for Borel resumming a perturbative series. Such a summation actually counts the number of instanton contributions to the system coming from both the real and the complex instantons. The net effect of this is to convert a perturbative series to a trans-series, or more appropriately to a resurgence trans-series. The meaning of such a trans-series should now be clear: in a path-integral representation this corresponds to taking the instanton saddles and the corresponding fluctuation determinants into account. This implies that your $\lambda\varphi^4$ theory and QED should be expressed in terms of such a trans-series, and not as perturbative theories, which is what Dyson [45] and other mathematical physicists [43] have been emphasizing for some time now. Of course perturbation theory does give correct answer up to some order, for example QED up to say three-hundredth loop, beyond which the series stops being convergent.

Sagredo: OK, I think I understand what you are saying, but what it has to do with your construction of a de Sitter spacetime? However before you answer that, maybe I want to take a few steps back and ask you an even more basic question. You are claiming to study de Sitter space from M-theory. But we know nothing of M-theory, except maybe the low energy dynamics. How are you going to tackle that?

Salviati: Yes we know nothing of the UV behavior of either string theory or M-theory, but we do know one thing for sure. No matter how complicated the short distance behavior of these theories are, or what degrees of freedom reside there, the far UV dynamics are well-defined without any pathologies. This is important for the consistency of these theories. The point is that, no matter what degrees of freedom exist in the far UV, an Exact Renormalization Group (ERG) procedure [46] will tell us that once we integrate from $\Lambda_{\rm UV}$ till $\mu$ (see figure <ref>) all the information of the UV can be encoded in the coefficients of the infinite towers of marginal, relevant and irrelevant interactions of the massless states! Of course this means that we have to keep track of all possible operators constructed from the massless degrees of freedom. This way we lose no information of the UV, but the subtlety is that some of these massless states could be off-shell. Question then is how to deal with the off-shell degrees of freedom? I'll answer that question after I have motivated the construction of the Glauber-Sudarshan states. Meanwhile I'll let you ask other basic questions you might have in mind.

Sagredo: Thank you, that discussion is useful, and I'll ask about the off-shell states soon. As I understood, the infinite interactions of the massless states are important and truncating the series to only relevant and marginal operators is definitely not the right way to go. I suspected something along that direction, but I don't see why we cannot allow for de Sitter spacetime to be a vacuum solution to the aforementioned system. Of course the actual analysis could become technical, but is there a fundamental reason why such a solution cannot appear in string theory?

Salviati: Well, I'm guessing that you may have already heard of the no-go theorem [1] that forbids a de Sitter spacetime with only classical sources. If not, the reasoning therein is rather simple and has to do with the strange behavior of gravity itself. Once we put matter or energy, the spacetime gets curved due to the back-reactions from these sources, and the resulting curvature creates a negative potential. This negative potential nullifies any positive energy that we put in the system. This is why a negative or a zero cosmological constant solution is ubiquitous in string theory (or in any theory of gravity), but a positive cosmological constant solution is rare. In string theory the no-go theorem states that any kind of matter, including fluxes, branes, anti-branes and orientifold planes, cannot give rise to a positive energy solution. We need something that does not back-react so strongly $-$ so that it is not nullified by the negative gravitational potential $-$ yet creates enough excess positive energy to give a positive cosmological constant solution. This is where the quantum terms become immensely useful. One of the surprise however is that the perturbative corrections fail to do this, so all responsibilities now lie on the shoulders of the non-perturbative and the non-local quantum terms...

Sagredo: OK, so the non-perturbative and the non-local quantum terms can give rise to a de Sitter vacuum solution. What prohibits us to use these corrections to get a de Sitter vacuum solution here?

Salviati:  Mi dispiace, questa $\grave{\rm e}$ una conclusione affrettata. You are forgetting one thing that I emphasized earlier, namely the existence of a low energy effective action coming from implementing an Exact Renormalization Group procedure. What does such a procedure entail?

Sagredo: I'm guessing that this would entail integrating out the high frequency $-$ and consequently the high energy $-$ modes as well as integrating out the massive states above a certain energy scale.

Salviati: Indeed! But what if we couldn't do such an integrating-out procedure? You may ask when and how could such a situation arise. Such a situation would appear if there is no well-defined UV completion or if the frequencies themselves are changing with respect to time. We can clearly rule out the former because, as I described a moment ago, no matter how complicated the short-distance behavior of string or M-theory is, the far UV is by definition a well-defined theory. However the latter is now an issue: one may easily check that the fluctuating frequencies over a four-dimensional de Sitter spacetime do become time-dependent. Such time-dependence would rule out the Wilsonian integrating-out procedure because  $-$ as the frequencies are constantly red-shifted  $-$ there is no meaning of the integrating out process now! You may immediately ask why we couldn't do this over a static patch. I already answered that: the dynamics over a static patch is highly deceptive. This should also be clear from figure <ref>: looking at the right-most diagram, the frequencies are actually changing even inside the static patch although they may not look like to the static patch observer. It would clearly be wrong to implement the ERG procedure and write an effective action at low energies. Moreover quantization of strings now becomes rather complicated because of this time-dependence. It is not at all clear that the same massless modes $-$ that we got from quantizing a string over a flat background $-$ continues to provide the dynamics over the static patch. Plus of course having no supersymmetry at the vacuum level simply adds more issues to an already problematic edifice.

Sagredo: But if we are inside the static patch, we are confined to be within the boundary of the patch. Does it matter what happens outside the boundary as we have no access to it?

Salviati: This is a wrong conception of the dynamics in the static patch. You have chosen a bad coordinate system that appears to disallow you to go outside, but that doesn't mean you have no access beyond the confines of the static patch. Outside the static patch what really happens is that the relevant time-like vector becomes space-like and vice versa. Thus all degrees of freedom that are not changing with respect to time inside the static patch will start having temporal dependence outside the static patch. This is of course the content of figure <ref> that I referred to you earlier.

Sagredo: I'm still a bit puzzled by this. For the Schwarzschild black hole the static patch argument still works, isn't it? Since de Sitter spacetime is very similar to a Schwarzschild black hole, shouldn't the argument go through here too?

Salviati: Thank you for asking this, but it is a misconception that the physics for the two cases are similar. I don't want to go through the details of the black-hole physics here, but let me just point out the following. A drawback of analysing the static patch alone is that the cosmological horizon, and consequently the static patch, are both observer-dependent concepts. What is typically assumed when one draws the Penrose diagram of a de Sitter space is that there is a static observer sitting at the North Pole, and the static patch is the causal region of the de Sitter space accessible to the observer since this is the region to which he can both send and receive information from. However, for any two physical observers, their respective static patches would be different. This is in contrast to the
case of the Schwarzschild black hole, whose exterior Schwarzschild patch is the same for all asymptotic observers.

Sagredo: Thank you for the explanation. However I'll need some time to digest what you said as they are rather non-trivial concepts. Shall we break for lunch and we can resume after?

Salviati:  Fare una pausa per il pranzo sembra un'ottima idea. Incontriamoci dopo.

§.§.§ The intermezzo: On the subtleties of the construction

Sagredo: I had some time to ponder over the lunch. I think I'm beginning to understand the problem, but surely such a startling thing must have been noticed earlier? How did the physicists deal with such an outcome?

Salviati: For a long time static patch was considered to be the way out by somehow restricting the dynamics within the finite radius of the patch. However the study of de Sitter in string theory is relatively new and with it new surprises are coming forth. For example, there are various constraints that come with such a study that involve many things and not just reproducing the metric. Moreover in string theory we simply cannot restrict the fields to lie within a finite region of space and forget that there is something beyond the confines of the static patch! Additionally, ignoring the fact that we have no spacetime supersymmetry anymore,
a consistent compactification would imply quantized fluxes over the internal compact non-Kähler manifold. Outside the static patch, such a manifold itself will become time-dependent, implying subtleties with flux quantization, anomaly cancellation, dynamical motion of the branes, e cos$\grave{i}$ via. All these issues have to be solved simultaneously
before we can claim that we have a consistent solution. Putting it differently, in string theory there is no meaning of one or the other, rather all have to be solved together especially when the background has time-dependence. Going to
other patches related to the static patch, like the Eddington-Finkelstein or the Kruskal slicings or the so-called “nice-slices$"$, will unfortunately not alleviate the problem. They may hide it, but cannot remove the problem.

Sagredo: Sorry, but I need to interrupt as I don't understand how the branes enter the picture. I thought you said that you have integrated out all massive degrees of freedom beyond some energy scale. Aren't branes included in those massive degrees of freedom?

Salviati: First let me explain where the branes enter the story. Remember, I mentioned the asymptotic nature of the perturbative series, and how this is replaced by a resurgent trans-series? A part of the contribution comes from the brane-instantons as was first discussed in [20]. However in the low energy limit that we study here, we don't see the branes, although we do see the instantons. The effects of the branes come from their classical back-reactions and from their massless world-volume interactions. The latter is again asymptotic and may be rewritten in terms of a resurgent trans-series. This is what I emphasised earlier: there are no perturbative interactions.

Sagredo: Thank you for the brane discussion, and I understand the seriousness of the problem regarding the Wilsonian integrating out procedure. But you haven't answered my previous question: How did the physicists deal with such an outcome?

Salviati: Yes, sorry I didn't answer that. There are at least two ways I know by which this problem was handled: one, using open quantum field theory, and two, using a certain censorship criterion called the trans-Planckian censorship. The first one, namely the
open QFT [24], is traditionally described by isolating relevant degrees of freedom from an “environment$"$ which allows them to gain or lose energies to the environment. Since neither energy is conserved, nor an EFT description exists, the dynamics of the theory typically follows some Markovian process which may be quantified in certain settings [25]. The problem with this picture appears when we try to use it in string theory: in a UV complete theory, like string theory or M-theory, there appears to be no clear demarcation between the relevant degrees of freedom and the environment, and therefore a concrete realization over a temporally varying background is equally hard. So far there has not been much progress implementing this idea in string theory, although some recent works in field theories have shed some interesting light on the de Sitter problem [26].

Sagredo: This means,
if I understood correctly, since we are dealing with only the low-energy degrees of freedom in string or M-theory, there is no simple way to isolate the relevant degrees of freedom from an “environment$"$. In fact ambiguities like whether we should view the massive $-$ or what fractions of the massless on-shell and off-shell $-$ degrees of freedom to be the “environment$"$ now appear.

Salviati: Indeed that is the point. Because of these ambiguities there is no simple way to implement the idea of an open QFT in string theory.

Sagredo: Thank you, although I still haven't understood how you are dealing with the massless off-shell degrees of freedom, and I'm guessing you'll explain that soon. But before we go into that, what about the trans-Planckian censorship criterion?

Salviati: Yes, first let me explain what this entails.
The idea behind the trans-Planckian censorship conjecture is simple. It states that a reverse time evolution of excitations with ${\cal O}(1)$ energies living on a late time slice leads to trans-Planckian energies for the same at a finite time in the past. In generic theories of higher derivative gravity, including Einstein gravity, this is problematic because the short-distance behaviors are not well-defined. In terms of future time evolution, trans-Planckian modes in the past become increasingly long wavelength and subsequently freeze upon exiting the Hubble horizon. Thus at late times, one can classically detect trans-Planckian physics at ${\cal O}(1)$ scales, for instance, in the CMB data.  In the original formulation [13], where the theory was restricted to obey Einstein gravity only, the trans-Planckian censorship conjecture prohibits the existence of such a scenario by simply bounding the temporal domain for these UV modes to not exit the Hubble horizon. More precisely, the conjecture states that any solution where we can see a classical imprint of trans-Planckian features upon exiting the de Sitter horizon is not possible since the time scale is bounded by (<ref>). Saying differently, as long as we are bounded by the temporal interval (<ref>) we will not see the trans-Planckian problems.

Sagredo: Thank you, but I'm puzzled by your explanation. I thought you said earlier that the short distance behaviors of string and M-theory have no pathologies. The UV dynamics therein are all well-defined, so why should we worry about the trans-Planckian modes? The new degrees of freedom that enter at scale ${\rm M}_p$ would render all UV amplitudes finite, so any modes frozen at late time should show no trans-Planckian issues.

Salviati: I'm glad you asked this question. Indeed in string or M theories there are no such trans-Planckian issues because of the UV finiteness, although one might be concerned that the UV degrees of freedom are somehow getting classicalized due to the Hubble expansion. This is not a worrisome point but more like a technical challenge, so imposing the trans-Planckian bound here would simply amount to hiding it. However something else gets manifested here due to the temporal variation of the fluctuating frequencies: the convergence of the non-perturbative series. How do we see this? Remember I mentioned the fact that the dual IIA coupling has to satisfy $g_s < 1$? In a flat-slicing this actually leads to (<ref>). Now compare (<ref>) to
(<ref>): there appears to be an amazing match despite the fact that the two ways of reaching the temporal domain are quite different. This means, while for the generic higher derivative or Einstein gravities going beyond the temporal domain (<ref>) classicalizes the trans-Planckian issues, for string or M theories it is the convergence of the non-perturbative series going as ${\rm exp}\left(-{1\over g_s^a}\right)$, $a > 0$,
that becomes an issue beyond the temporal domain (<ref>).

Sagredo: I thought the instanton saddles go as ${\rm exp}\left(-{n\over g_s^a}\right)$ where $n \in \mathbb{Z}$ is the instanton number. So even if $g_s = {\cal O}(1)$ number, shouldn't the series be always convergent for large $n$?

Salviati: Well, yes and no. If the instanton series were as simple as what you mentioned, then yes, the convergence is guaranteed for large instanton numbers. However the actual computation, with say M5-brane instantons wrapped on the six-dimensional non-Kähler base, is far more complicated. For example if you look at the third line in (<ref>), with ${\bf Q}_{\rm pert}$ therein given by the series (<ref>), it is not guaranteed that the series would converge even for large $s \in \mathbb{Z}$. However for $g_s << 1$, convergence can actually happen as discussed in the analysis presented in section <ref>.

Sagredo: I see that the quantum series ${\bf Q}_{\rm pert}$ appears in both the fluctuation determinants as well as the instanton saddles, with different choices of the coefficients. This already looks pretty complicated, and I was wondering how do we even go about using it in the EOMs? But I think I'm getting ahead of myself. I need to first understand how and why you are constructing de Sitter spacetime as a Glauber-Sudarshan state. Could you please start by clarifying that?

Salviati: Yes, it is high time we get into the main issue of de Sitter spacetime as a Glauber-Sudarshan state. Let me ask you something before we start. Do you agree that the kind of potentials we see in string or M theories only have Minkowski and AdS minima, but no minima with positive energies? In other words, in string/M theory we can see potentials like the ones in figure <ref> $-$ with possible additional AdS minima $-$ but not a potential like the one on the right of figure <ref> with unequal minima?

Sagredo: Not really. Why do you say that?

Salviati: Well, think of it in the following way. If the potential has a minimum like the one on the right of figure <ref>, then we will have a positive energy solution in the static patch. (Or generically in any other patch.) From our discussion above, such a solution cannot allow a Wilsonian integrating out procedure resulting in an ill-defined EFT at low energies. Since string or M theories cannot allow this, the only way this can happen is by modifying the profile of the potential using the non-pertubative and the non-local quantum corrections so that the minimum is removed or smoothened out. In other words, the low energy EFT that I told you earlier must have come from potentials like the ones in figure <ref> with additional AdS minima, but not the one on the right of figure <ref> with unequal minima. You see that demanding Wilsonian integration procedure or an Exact Renormalization Group technique essentially rules out a four-dimensional de Sitter vacuum solution in string theory.

Sagredo: This reminds be of the conjectures I read about in [3] and [6]. Your conclusion seems to coincide with what those papers said, but as far as I see, your methods are quite different. Is this correct?

Salviati: Yes, we do seem to agree with the final conclusion that a four-dimensional de Sitter space cannot exist as a vacuum solution in string theory. And you are also correct to say that the techniques we used to get to our conclusion are quite different from the ones used in the aforementioned papers. To be honest, I do not understand how the EFT criteria mentioned in [6] were derived. I understand that they were proposed to rule out accelerating solutions in string theory, but I didn't see any derivations there. Our analysis, which has its roots in the ERG techniques, does not rule out de Sitter space in string theory. Instead, four-dimensional de Sitter space can exist in string theory $-$ and even within the temporal bound advocated by the so-called trans-Planckian censorship conjecture $-$ except not as a vacuum state but as an excited state in the low energy effective action of string theory.

Sagredo: Actually I heard that the modified version of the criteria in [6], motivated from the trans-Planckian censorship conjecture [13], does allow a four-dimensional de Sitter space to exist.

Salviati: Yes, but there is a difference: nothing really bad happens in our set-up beyond the temporal domain (<ref>). The convergence issue of the instanton series does come under scrutiny, but this simply means that the de Sitter excited state may no longer be sustained by the non-perturbative effects and may just go back to the vacuum Minkowski configuration. On the other hand, in the set-up of [6], the theory is strictly restricted to be within the temporal domain (<ref>) because beyond which there are short distance pathologies.

Sagredo: OK that explains things quite well. Now could you kindly explain how the de Sitter excited state is constructed? Plus you said at the begining that it is different from a coherent state. Could you elaborate on it?

Salviati: Yes, after a long preparation I can finally explain to you our construction. However it is getting late, and we have been talking for quite a while. How about we resume our discussion tomorrow?

Sagredo: S$\grave{\rm i}$ riprendiamo la discussione domani.

§.§.§ The dialogue: On the construction of the GS states

Sagredo: Yesterday before breaking off, you said that you'll motivate the construction of four-dimensional de Sitter space as an excited state in the low energy effective field theory of type IIB string theory.

Salviati: Yes indeed. The fact that string or M theory appears to  allow potentials that only have Minkowski and AdS minima, like the ones in figure <ref>, but no potentials with unequal minima, like the one in right of figure <ref>, convinced us to look for four-dimensional de Sitter space as an excited state over a supersymmetric Minkowski minimum in string theory. Now if you look around yourself, the surrounding spacetime is pretty classical in the sense that you can measure distances and do other experiments classically. So whatever excited state we get from the low energy effective action must be very close to the classical picture. Now what quantum state is closest to classical physics? The answer is clearly a coherent state so naively one would expect de Sitter space to be a coherent state.

Sagredo: Yes, that would also be my expectation. However you are going to say that it is incorrect. Why is that?

Salviati: The reasoning is simple if you look at the definition of the coherent state in quantum mechanics. Coherent state is constructed by shifting the free vacuum by a unitary displacement operator. One immediate problem is that this definition is in quantum mechanics, but can be extended to field theory because quantization of modes does lead to an infinite collection of simple harmonic oscillators.
However the issue with the low energy effective field theory is that it is not a free theory. Recall our earlier discussion of the constant coupling limit of F-theory where I emphasised that the type IIB coupling is fixed to 1 and going away from that point entails complicated and possibly uncontrolled dynamics involving the seven-branes and whatnot.

Sagredo: Which means, if I understood correctly, construction of a coherent state doesn't make any sense here and probably doesn't even exist in a simple form that we usually see in quantum mechanics. Is that the only reason to discard this idea?

Salviati: No, there are many other issues. Even if I grant you a weak coupling scenario, the construction of coherent state cannot be simple because of the numerous tunnelling effects from the nearby vacua in potentials  like the ones from figure <ref>. These tunnelling effects do not just come from the real instantons, but they have sizable contributions from the complex instantons also. (These complex instantons come from the complex solutions of the turning points in an inverted potential.) Secondly, a coherent state over the lower minimum in (say) a potential like the one on the right in figure <ref> is simply a Bogoliubov transformation of the state over the positive energy minimum. The second point is crucial. It simply says that a coherent state construction is nothing but a fancy way of saying that a vacuum state exists in a nearby minimum with positive energy. Plus the fact that the construction uses a unitary displacement operator means that the resulting analysis is just the tree-level answer in a free field theory. de Sitter space cannot appear from any tree-level computations, so coherent state cannot give rise to the kind of quantum state we want which overcomes the no-go conditions of [1].

Sagredo: I see the issues with a coherent state.
But what stops us to make simple modifications of the coherent state by adding in quantum corrections by hand?
Are there problems with these kind of scenarios?

Salviati: Yes because adding in pertubative quantum corrections cannot help simply because of the asymptotic nature of such quantum series. You may now say that we can allow only the non-perturbative corrections. However an unambiguous set of non-perturbative corrections
could only appear once we fix the form of the perturbative correction unambiguously. The latter is only possible if we know what kind of quantum state we are looking for..

Sagredo: I'm guessing that the state should be in the full interacting theory that produces a potential like the ones in figure <ref> with possible additional AdS minima.

Salviati: Yes, and in the full interacting theory we can choose the interacting vacuum $\vert\Omega\rangle$ and shift this by a displacement operator $\mathbb{D}(\sigma)$ to create the so-called Glauber-Sudarshan state $\vert\sigma\rangle$ associated to every on-shell degrees of freedom over a supersymmetric Minkowski minimum. Such a state is far from the Gaussian approximation we use in perturbative field theories.