de Sitter State in Heterotic String Theory

: Recent no-go theorems have ruled out four-dimensional classical de Sitter vacua in heterotic string theory. On the other hand, the absence of a well-deﬁned Wilsonian eﬀective action and other related phenomena also appear to rule out such time-dependent vacua with de Sitter isometries, even in the presence of quantum corrections. In this note, we argue that a four-dimensional de Sitter space can still exist in SO (32) heterotic string theory as a Glauber-Sudarshan state, i


Introduction and summary
Our modern understanding of quantum field theories is based on two recurring themes, one, on the existence of a Wilsonian effective action and two, on the asymptotic nature of the perturbation series.The latter, which was actually known for some time now [1], was surprisingly only appreciated more recently from some remarkable works [2] which showed clearly how non-perturbative effects manifest themselves naturally in correlation functions.
Extending both these themes to cosmological set-up wherein temporal dependences appear automatically is much more non-trivial.Even more challenging is the scenario where string theory is involved.In string theory, where the asymptotic nature of string perturbation theory is well documented, the existence of a Wilsonian effective action over a temporally varying cosmological background is not guaranteed.In fact there are strong evidences to suggest that a Wilsonian effective action may not exist because of the temporal dependences of the fluctuating frequencies, as well as of the massive stringy and the KK modes.For such a background, although we do expect some (as yet unknown) stringy description, it is a futile affair to search for a supergravity description where none exists.Equally futile then is the search for a vacuum solution for a cosmological background.These and other related arguments form the core of the so-called trans-Planckian problems in string-cosmology [3].
The situation, unfortunate as may seem, is not without hope.Solutions do exist, but not in a way envisioned earlier.Demanding the existence of a Wilsonian effective action then instructs us to realize the cosmological background − which is a de Sitter space in this case − as an excited state over a supersymmetric Minkowski background in string theory.We expect the excited state to break supersymmetry spontaneously, but the question is what kind of excited state are we looking for?Clearly, since the universe we live in is very close to a classical one, the only excited state that has any chance of reproducing the classical behavior is a coherent state which amounts to shifting the free vacua in field theory.Unfortunately, due to the non-existence of free vacua in string theory, such a state cannot be easily realized and the closest we can come to realizing a coherent state would be by shifting the interacting vacua.This actually turned out to be a reasonably viable option as amply demonstrated for the type IIB case in [4,5], and we called such a state as the Glauber-Sudarshan state to distinguish it from the coherent state 1 .
Our aim in this paper is to realize a de Sitter state in heterotic string theory.Due to various technical reasons, SO(32) heterotic theory appears to provide a more controlled laboratory than E 8 × E 8 theory to implement the computational technology.This computational technology involves performing a full-fledged path-integral along the lines of [6,7] over a Minkowski saddle which, expectedly, leads us to an asymptotic series of the Gevrey kind [8] thus requiring Borel resummation [9].The final answer we get matches somewhat with the type IIB case from [4][5][6], but the details are quite different.These differences are important and in section 2.3 we will spell them out.
The note is organized in the following way.In section 2.1 we point out the reason for uplifting the type IIB or the dual heterotic background to M-theory.In section 2.2 we provide the duality chain that relates a type IIB orientifold background to a heterotic SO(32) background, and in section 2.3 we present our main results of constructing the de Sitter Glauber-Sudarshan state using Borel resummation of a Gevrey series and point out the key differences from the type IIB case.We end with a discussion in section 3.
2 Quantum corrections, Glauber-Sudarshan state and M-theory In [4,5] we showed how a four-dimensional type IIB background with de Sitter isometries can be realized as a Glauber-Sudarshan state.As we also discussed therein, this background cannot appear as a vacuum configuration in IIB string theory due to numerous issues.The question that we want to ask here is that whether a generic background of the form: can also be realized as a Glauber-Sudarshan state.Here F i (t) captures the dominant temporal scalings, and in what sense they do will be elaborated when we lift this configuration to M-theory.Note that a 2 (t), with t being the dimensionless conformal time (measured with respect to M p = 1), is kept arbitrary with the only condition being that it becomes large at late time.This means the background (2.1) naturally expands at late time.For example when a 2 (t) = 1 Λt 2 , with Λ being the cosmological constant, we get an expanding de Sitter space in the flat slicing in type IIB as t → 0 at late time.The other factors, g ij (x), g 33 (x), g mn (y), g αβ (y) and H 2 (y) are the unwarped spatial metric components and the warp-factor respectively.The coordinate y ≡ (y m , y α ) ∈ M 4 × M 2 and x = (t, x) ∈ R 2,1 , so that nothing depends on the third spatial direction parametrized by x 3 here.We will soon make a further restriction by converting where Z 2 ≡ Ω(−1) F L I T 2 will be an orientifolding operation (I T 2 is the orbifold action.Details are in [14]).Such a choice will give us a way to reach the heterotic background by making a series of duality transformations.In that case y ≡ y m ∈ M 4 .For the time being, however, we will continue with the generic picture.
The reason for this genericity is simple.As alluded to above, for various choices of a2 (t), F i (t) and the internal sub-manifolds, we can study the possibilities of realizing de Sitter state in various string theories (including also in M-theory).Such realizations will involve duality transformations: for example appropriate T-dualities, with and without any orientifolding operations, can give rise to the possibiltites of de Sitter states in type I and type IIA theories respectively.With an additional S-duality, as mentioned earlier, we could study de Sitter state in heterotic SO(32) theory (appropriately broken to a suitable subgroup).We can even dualize to M-theory (see [5]) and from there dualize further to heterotic E 8 × E 8 theory.The questions that we want to investigate here is whether such possibilities could be explicitly realized.
Expectedly, there are also a few other changes from the construction in [4,5].We no longer impose any constraint on F i (t).This means the four-dimensional Newton constant can become time-dependent in the IIB side.What we do want, however, is that the Newton constant remains time-independent in the dual side (i.e. the dual side where we want to realize the de Sitter state).In a similar vein, the functional form for a 2 (t) will be determined by demanding a de Sitter space in the dual side.The precise conditions on a 2 (t) and F i (t) will be elucidated once we dualize to the corresponding theory.
The dualities to the various string and M-theory sides are more subtle now because the type IIB background (2.1) cannot be realized classically [11].Quantum mechanically we expect such a background to exist only in the presence of all possible perturbative, non-perturbative and non-local, including topological corrections.Additionally − on one hand − temporal dependences of the underlying degrees of freedom are absolutely essential for an Effective Field Theory (EFT) to exist.(The existence of EFT is in turn related to the existence of four-dimensional Null Energy Condition (NEC) [5].)On the other hand, existence of the temporal degrees of freedom, for example fluxes etc., are tightly constrained by the flux quantization and anomaly cancellation conditions.Thus the system is highly intertwined, and unless we demonstrate that a background like (2.1) can exist (at least as a Glauber-Sudarshan state), the duality chasing will be a meaningless exercise.
The last comment on the existence of the background (2.1) as a Glauber-Sudarshan state deserves some explanation.As we saw in [4,5], when a 2 (t) specifies a given de Sitter slicing, Wilsonian effective action can only be defined properly when the background becomes a Glauber-Sudarshan state 2 .Other issues like the existence of a Trans-Planckian Cosmic Censorship (TCC), moduli stabilization, Faddeev-Popov ghosts, Schwinger-Dyson equations etc., appear much more naturally in this framework.More so, the existence of the Glauber-Sudarshan state tells us how a de Sitter state may exist in the type IIB string landscape (and not in the so-called swampland).The question that we want to ask here is whether such a de Sitter Glauber-Sudarshan state can be found in the dual landscape.The answer, as we shall see, turns out to be more complex in a sense that shall be elaborated soon 3 .But first: does the background (2.1) exist in the IIB landscape as a Glauber-Sudarshan state?This is what we turn to next.

Consistency of M-theory uplift and Glauber-Sudarshan state
As in [4,5], we will lift the background (2.1) to M-theory by T-dualizing along x 3 and then uplifting the configuration to eleven-dimensions.There are various reasons why such an uplift becomes necessary.
• The type IIB background (2.1) is supported at a constant coupling point in F-theory [12].This means axio-dilaton vanishes and the IIB coupling g b = 1.This is a strong coupling point where S-duality doesn't help.More so, the vanishing axio-dilaton for example is necessary to get a gauge group of D 4  4 = [SO( 8)] 4 in the dual heterotic side.The E 8 heterotic theory appears in a more non-trivial way as shown in [13].
• The internal space with topology M 4 × M 2 is not only a non-Kähler manifold but is also non-complex.Additionally, various parts of the internal space evolve differently with time, as shown by the temporal factors F i (t).The space-time has positive cosmological constant, and is therefore highly non-supersymmetric 4 .Putting everything together we see that all conventional techniques that we have learnt so far would fail to quantify the dynamics of the system.
• The unit coupling in the IIB side means that no controlled computation can be performed there.The only leverage we can get is by dualizing to the IIA side where the IIA coupling g s becomes gs HHo = 1 a(t) , where H(y) is the warp-factor and H o (x) is related to g 33 (x).For an expanding cosmology, the system then is naturally weakly coupled at late times.
• Uplifting this to M-theory converts most of the IIB fluxes to four-form G-fluxes.Moreover, the IIB seven-branes become geometric spaces in M-theory.The D3-branes, which are instantons on the seven-branes, naturally also become geometric.The temporal dependences of the G-flux components will make the D3-branes dynamical.
• In the small-instanton limit, these D3-branes dualize to either D2 or D4-branes in IIA, that are uplifted to M2 and M5-branes respectively.Since both D2 and D4-branes dissolve as instantons or as first Chern classes respectively on the IIA D6-branes, they naturally become dynamical in M-theory 5 .These dynamical M5-branes are responsible for the flux quantization procedure as shown in [5,10].
are related by a set of duality transformations, doesn't that naturally guarantee a de Sitter state in the heterotic side?The answer is no.In fact, as we shall see, duality chasing only works if the seed background in the IIB side exists.Our analysis will hopefully reveal that this is not guaranteed a-priori. 4If we can realize the background as a Glauber-Sudarshan state over a supersymmetric Minkowski space (with a non-Kähler non-complex internal manifold), much like in [4,5,10], then the supersymmetry is broken spontaneously.The supersymmetric vacuum appears from the self-dual G-fluxes.Once we take the expectation values of the G-fluxes over the Glauber-Sudarshan state, they no longer remain self-dual and therefore break supersymmetry spontaneously.See [4,5] for details. 5 Recall that the gauge fluxes on the D6-branes appear from localized G-fluxes of the form G MNab and G 0Mab that are generically time-dependent [4,5,10].This would make both the world-volume gauge fluxes as well as the branes dynamical.
• The IIB D-string dualize to either D0 or D2-branes in IIA.The D0-branes uplift to the massive gravity multiplet (which makes sense because the magnetic dual of the D0-branes, namely the D6-branes, dualize to Taub-NUT spaces (or KK-monopoles) in M-theory).The type IIB fundamental string dualize to a wrapped M2-brane when uplifted.
• The Glauber-Sudarshan states are most succinctly presented from M-theory perspective as they could easily reproduce any metric and flux configurations.Since most of the IIB branes dualize to either geometry or flux configurations when uplifted to M-theory, the Glauber-Sudarshan states could in principle reproduce these configurations too.From the IIB side we could probably view them from a string field theory set-up, where the shifting of the interacting IIB vacuum could be naturally realized6 .
• Non-existence of a well-defined action in the IIB side is also another reason for the uplift to M-theory.As we saw in [4,5,10] it is absolutely essential to spell out the precise set of perturbative, non-perturbative, non-local and topological quantum corrections.It is only in M-theory that such a procedure may be explicitly performed.Existence of a Wilsonian effective action − as guaranteed from the existence of a Glauber-Sudarshan state 7 − means that all the short-distance degrees of freedom can be integrated out to express the quantum corrections in the form given in [4,5,10].
• Lifting the metric configuration (2.1) to M-theory, one may easily see that the toroidal direction, parametrized by (x 3 , x 11 ), scales as which becomes arbitrarily small in the limit g s → 0. Since we are always in the limit8 g s < 1, the M-theory degrees of freedom do capture the type IIB behavior exactly.This also means that, for example if we are allowed to keep the M 2 cycle small compared to α ′ we can, in the orientifold limit, capture the SO(32) heterotic dynamics using the M-theory degrees of freedom.Thus the M-theory uplift has a dual advantage: provide both the IIB and the heterotic dynamics under appropriate conditions.In fact, as it turns out, the M-theory configuration is a master theory from where the de Sitter dynamics can be determined for all the string theories (including de Sitter state directly in M-theory).The proof of the statement is beyond the scope of this paper and will be demonstrated elsewhere.
Interestingly, the form of the M-theory metric always remains the same for any type IIB cosmology expressed using conformal coordinates as in (2.1).The only change is the value of the dual IIA string coupling: gs HHo = 1 a(t) which is sensitive to the functional form of a(t).Clearly, and as mentioned earlier, for expanding cosmologies the IIA coupling can be made small.In fact demanding gs HHo < 1, provides the temporal domain in which controlled quantum computations may be performed in M-theory.For the usual de Sitter case, irrespective of the choice of the de Sitter slicings, this temporal domain remains perfectly consistent with the so-called Trans-Planckian Cosmic Censorship (TCC) [3], as shown in [4,5,10].We now proceed to find the functional form of a(t) that provides a de Sitter metric in the dual heterotic side.In what follows we elaborate on this solution.
2.2 M-theory uplift of a heterotic SO(32) background and dualities The duality from type IIB to heterotic SO(32) theory, in the presence of orientifolds and fluxes, has been explicitly shown in [14].We will basically follow similar duality chasing here too, but not before we elucidate the consistency of the IIB background (2.1) from M-theory.In M-theory, the uplifted metric takes the following standard form: where H 1 (x, y) ≡ H(y)H o (x), which means F i (g s /H 1 ) depends on the temporal factor a(t), and we shall discuss their functional form soon.The other metric components may be related to the metric components in (2.1) in the following way: where we have taken M, N ∈ M 4 × M 2 and (w a , w b ) ≡ (x 3 , x 11 ).Note that we have taken the un-warped metric components along the toroidal direction to depend on both (x i , y M ).
In fact, for the computations of the curvature scalings, as shown in [5,10], one may take both the un-warped and the warped metric components to depend on all the coordinates (except of course the toroidal direction).Once we go to the heterotic side, we will see that the dependence on the coordinates of M 2 have to be removed.
Let us now come to the functional form for the temporal factors F i (g s /H 1 ).In our earlier works [4,5,10], these factors did not change the dominant scalings of the metric components as they were constrained by F i (g s /H 1 ) → 1, g s → 0 and F 1 F 2 2 = 1 to preserve the Newton's constant and to avoid late-time singularities.Both of these conditions are not essential now if we want to dualize to any of the other string and M-theories because only in the dual landscape we want a time-independent Newton's constant with no late time singularities.This means the dominant scalings of the internal metric could in-principle change, implying changes to the curvature scalings from what we had in [4,5,10].We can then propose the following scalings: where (A k , B k , C k ) are all integers, positive or negative with (α o , β o , γ o ) being the dominant scalings and k ∈ Z + .Note that, as we demonstrated rigorously in [5], when γ o < 0 EFT breaks down along-with a violation of the four-dimensional NEC.Here, in the generic setting, we will see whether this continues to hold or not.On the other hand (α o , β o ) are not a-priori required to be positive definite.An interesting question would be to find whether there is a connection between the three dominant scalings.If there is one then it would lead to an even deeper connection between the three disparate facts: existence of EFT from M-theory, preserving four-dimensional NEC from IIB, and temporal dependence of the internal six-dimensional manifold.Before going into this, let us clarify couple more things about the temporal dependence of the internal manifold.One, taking (2.4) at face-value might imply that all components of the internal metric should scale in a certain way temporally.This is actually not the case as long as we maintain the dominant scalings.For example we can start by generalizing the internal metric components as: αβ (x, y), where the repeated indices are not summed over, and the g MN (x, y) are the various possible metric components.It is easy to see that, unless we impose g (k) MN (x, y) ≡ gMN (x, y) from (2.3), it will in general be hard to keep the Newton's constant time-independent even in the type IIB side (although we are not required to do so now).Thus as long as the dominant scalings are divided into two sets: (α o , β o ), the system can be generalized without violating the EFT constraints.Note that this implies that further splitting into three or more dominant scalings are not necessary as higher order splittings can always be brought back to the two-splitting case by choosing appropriate values for A 3 , it would mean that the size of the two-cycle M 2 shrinks at late time when g s → 0. Beyond certain short-distance scale, T-duality would become necessary, and if M 2 ≡ T 2 Z 2 − with Z 2 being the orientifold operation [14] − we see that the late time physics may be succinctly captured by either the Type I or the heterotic theory.On the other hand if α o > + 2  3 , the late time physics may still be given by the IIB theory, assuming no orientifolding operation, although problems like late-time singularities could arise in the absence of local isometries and in the presence of orientifolds.Note that we did not encounter any of these subtleties in [4,5] because both the space-time and the internal six-manifold had dominant scalings of − 8 3 , − 2 3 and therefore the late-time physics was well within the supergravity limits 9 .(The subtleties with the M-theory torus are explained earlier.)With this we are almost ready to make the duality transformations to the heterotic side.We will assume that g αβ = δ αβ specifies a square torus capturing the local metric of M 2 = T 2 Z 2 , and y ≡ y m ∈ M 4 with M 4 being a generic non-Kähler manifold, as mentioned earlier.There are also NS and RR two-forms with components B αm (y, g s ) and C αm (y, g s ) respectively with one of their legs along the toroidal directions.The heterotic metric then takes the following form10 : where note that the toroidal directions no longer allow a warp-factor, but do allow a nontrivial fibration coming from the NS two-forms.Additionally, since the NS two-forms are time-dependent, the fibration naturally becomes time-dependent too.On the other hand, the temporal dependence of the RR two-forms provides the necessary torsion to support the non-Kähler, time-dependent metric (2.6) in the heterotic side.
We now want the four-dimensional part of the metric (2.6) to be a de Sitter metric in some specific slicing.For simplicity we will choose a flat slicing.(We avoid static patch or any other slicings related to the static patch because of the issues mentioned in [5].)We also want the four-dimensional Newton's constant to be time-independent.Putting these two together implies that11 : in the original type IIB metric (2.1), where Λ is the cosmological constant, t is the conformal time in flat-slicing and (g ij , g 33 ) = (δ ij , 1).(We can keep everything dimensionless by measuring with respect to M p ≡ 1 as shown in [6,7].)The choice (2.7) tells us that the original type IIB metric (2.1) is now no longer required to have a time-independent fourdimensional Newton's constant, or have a metric with four-dimensional de Sitter isometries.
The M-theory uplift however takes the same form as in (2.2) but with the following choices for g s , H o (x), α o and β o : showing that there is a dynamical possibility of the type IIB metric (2.1) to go to the heterotic side at late time.This also fixes the form for F 2 (t).One might worry that this could in-principle over-constrain the system, but a similar computation in the type IIB side done in [10] shows that this may not be the case.Of course we will have to fix the functional form for F 1 (t) using the Schwinger-Dyson type equations (in the presence of all non-perturbative and non-local corrections), but there is an additional condition on F 1 (t) borne out of the non-violating-NEC criterion from [5], namely: where γ o is defined in (2.4).Once we determine the form of F 1 (t), the above equation will then fix the values of the constant coefficients C k and γ o .If γ o < 0, then unfortunately such a system cannot be embedded in a UV complete theory, i.e. the solution (as a Glauber-Sudarshan state) cannot exist in heterotic string theory.
There are however reasons to believe that γ o could simply be zero if not a positive integer, but never negative.This is because the solution with β o = α o = 0 has been studied earlier in [10], where no apparent inconsistencies were detected.In the following section, we will demonstrate, under some mild approximations, that γ o can indeed be made positive definite here as long as β o ≥ 0.

The resurgence of de Sitter space as a Glauber-Sudarshan state
Our aforementioned discussion should convince the readers that an M-theory uplift of the heterotic background (2.6) − via type IIB − is possible and is given by (2.2) with the choice of parameters from (2.8).Unfortunately however, such a background cannot be realised as a vacuum solution either classically or quantum mechanically.The former is ruled out in [15] from the constraints coming from the Kac-Moody algebras with SO(4, 1) global symmetries 12 .The latter is ruled out in [4][5][6][7] from the absence of a well-defined Wilsonian effective action for an accelerating background like (2.1) or (2.6) 13 .This means the only way to realize the background (2.2) would be as a Glauber-Sudarshan state over a supersymmetric Minkowski background 14 .How should we go about constructing such a state now?
Fortunately, since the form of the metric (2.2) in M-theory remains unchanged from what we had in [4,5], the procedure to explicitly construct such a state, will follow [6], namely, doing a path-integral over the Minkowski saddle.In other words: .10)where (M, N, is the Glauber-Sudarshan state associated with 12 See also the last reference in [11] for a different take on the no-go conditions stemming directly from the energy conditions. 13One might argue that the situation could be dealt using open quantum field theories [16] which is suited to tackle scenarios where energy is not conserved.However in string or M-theories it is not a priori clear how to separate degrees of freedom to implement the energy loss.Plus other issues pointed out in [6] suggest that this may not be a viable option. 14 Being supersymmetric Minkowski (or more appropriately, warped supersymmetric Minkowski with compact non-Kähler internal eight-manifold), there is no longer any Kac-Moody constraints from [15], or any energy constraints from [11].
({g MN }, {C MNP }, {Ψ M }) degrees of freedom (see also [4,5]); g µν is the graviton operator related to g µν ; the measure Dg MN ≡ Dg µν Dg AB with (A, B) ∈ M 4 × M 2 × T 2 Z 2 ; D(σ) is a non-unitary displacement operator, i.e.D † (σ)D(σ) = D(σ)D † (σ) = 1 (see [4] for details); and the total action: where the perturbative part of S int comes from an interaction term like eq. ( 4.81) in [5] and S gf is the gauge-fixing term.The ghosts and the gauge-fixing terms are necessary to bring the propagators in the standard forms and remove the redundant degrees of freedom.In writing the expectation value, we have suppressed the measure associated with the ghosts (while this is not important for the present work, it will be elaborated in [7]).
Unfortunately dealing with the path-integral structure in (2.10) with 44 metric degrees of freedom, 84 three-form degrees of freedom and 128 Rarita-Schwinger degrees of freedom (plus the ghost terms) is clearly beyond the scope of the present treatment.We will then follow the simplifying procedure adopted in [6], namely, consider three scalar degrees of freedom which are the representative samples from the set of 44 gravitons, 84 fluxes and fermionic condensates from the 128 Rarita-Schwinger fermions.The latter is chosen to avoid Grassmanian integrals in (2.10), and the representative sample of gravitons include the graviton degrees of freedom along the non-compact directions.Even for such simplifying choice, the analysis of the path-integral is still complicated.In [6], it is shown that the path-integral (2.10) may be analyzed using the so-called nodal diagrams instead of the usual Feynman diagrams because of the shifted vacuum structure.In fact the nodal diagrams show growth of the Gevrey kind [8] implying that Borel resummation [9] of the Gevrey series is now necessary for the path-integral (2.10) to make any sense!Putting everything together, and using the computational procedure outlined in [6] gives us the following: where we restricted the metric function g µν = g µν (x, y) with y ∈ M 4 only in (2.10); P. V is the principal value of the enclosed integral; g (s) is a set of coupling constants defined by inverse powers of M p in a coupling-constant space parametrized by the set {s} of interactions; the first integral is the result of Borel resumming the Gevrey-l growth where l is one less than the total number of fields (i.e.l = 2 here); A (s) is the result of computing all the nodal diagrams, including the NLO diagrams for all interactions in the set {s}, that are combinatorially suppressed but not volume V suppressed where the volume V refers to the IR volume appearing from the IR/UV mixing [17]; if we choose the scale to be M p and α(k) is related to the Glauber-Sudarshan state |σ discussed earlier; and ψ k (X), with X ≡ (x, y) ∈ (R 2 , M 4 ), is the spatial wave-function over the solitonic background with κ IR (> k IR ) being an IR scale.(See [6] for details on the aforementioned computations.) The summing over the set {s} of interactions in the coupling constant space means that we are summing over Borel-resummed series.This double-summing is necessary to make the cosmological constant Λ small [7], where the closed form expression of the dimensionless cosmological constant 15 appears from the first integral over dS in (2.12), namely: .13)with all the parameters appearing above being defined earlier, and κ will be determined later (see (2.17)).This integral form of the cosmological constant has already been shown to be positive definite in [6], irrespective of the signs of A (s) and in [7] we argue that this can be made small.All in all, although it may appear that the analysis follows closely the ones from [6], there are few key differences.First is the temporal domain of the validity of the Glauber-Sudarshan state, i.e. the temporal domain where the state remains approximately coherent, and second is the functional form of α µν (k).Can they be precisely determined here?
The answer turns out to be yes provided we know the functional form for F 1 (t).This is represented by a series in gs HHo in (2.4).Compared to the type IIB case studied in [10] the situation is different because gs HHo itself depends on F 1 (t) as shown in (2.8).This means, even before we invoke the consequence from the Schwinger-Dyson's equations, F 1 (t) has to satisfy: where A k are time independent constants; and in the limit M p ≡ 1 we take both Λ and t to be dimensionless as explained earlier.In addition to this, there is also a derivative constraint coming from the NEC non-violating criterion [5] as in (2.9).The coefficients A k can only be fixed using the Schwinger-Dyson's equations (see equivalent example for the type IIB case in [4,5,10]).This is in general a non-trivial exercise because of the mixing of the various degrees of freedom (including ghosts), but we can try a toy example.A simple case would be when A 0 >> A k for k ≥ 1.For such a case, defining A 0 ≡ 1, we find: with 0 ≤ β o < 2 for the system to remain consistent 16 , and not violate the NEC criterion.
Recall that as long as β o > + 2 3 , the type IIB system can dynamically go to SO(32) heterotic (broken to a suitable subgroup that we will discuss soon).The NEC non-violating criterion allows 0 ≤ β o < 2, so is compatible with the aforementioned condition.More so, the size of 15 Recall that both the cosmological constant Λ and the conformal time t are made dimensionless here using the scale Mp ≡ 1.One could use a different scale, namely the size of the internal eight-manifold from the supersymmetric Minkowski background, but the final answer remains unaffected by this choice [6]. 16Note that, with the choice of F1(t) from (2.15), both the type IIA coupling gs HHo = Λt 2 1 2−βo and the heterotic coupling g het H 2 = Λt 2 βo 2−βo are small at late time, i.e. when t → 0. Thus both the system are naturally weakly coupled at late time as long as 0 ≤ βo < 2. Beyond this regime of βo, there are multiple issues with the Glauber-Sudarshan states in both heterotic and the dual type IIB theories.
M 4 grows at late time keeping the four-dimensional Newton's constant time-independent in the heterotic side.
Interestingly the temporal domain of the validity of our analysis matches exactly with the TCC bound advocated in [3], at least for this simple toy example.Additionally, plugging (2.15) in (2.9) − to see whether the non-violating NEC criterion from [5] is satisfied or not − gives us the following solution: .16)This is clearly consistent as long as β o ≥ 0. Thus combining the compatibility criteria from both TCC [3] and non-violating NEC [5] gives us a stronger reason to justify the existence of a de Sitter space as a Glauber-Sudarshan state in SO(32) heterotic string theory.Two points still remain.One, the functional form for α µν (k) which would specify the Glauber-Sudarshan state in the supergravity configuration space, and two, the form of the vector bundle over the internal six-manifold.The latter is a bit subtle because, while the four-dimensional base of the internal six-manifold in (2.6) is time-independent, the fibration structure is time-dependent because of its dependence on time-dependent NS two-form fluxes from the dual type IIB side.Nevertheless the vector bundle can be studied from localized G-flux components whose expectation values may be extracted from the corresponding Glauber-Sudarshan state.The computation should be similar to the path-integral analysis we did for the metric in (2.10), which in turn means Gevrey growth of the corresponding nodal diagrams and the subsequent Borel resummation to extract finite answer.For the simpler case studied here, we can restrict the D 4 4 bundle, alluded to earlier, on the non-Kähler four-dimensional base of the internal six-manifold.Unfortunately a generic study of the vector bundle is beyond the scope of this paper and will be dealt elsewhere 17 .
Finally let us figure out the functional form for α µν (k).It is now related to the Fourier transform of the temporal part of the M-theory metric (2.2).Since we are only dealing with real fields in the path-integral, it suffices to take the cosine Fourier transform.Doing this gives us both the functional form for α µν (k) and the value of the parameter κ in the expression for the cosmological constant Λ in (2.13).They are: where χ(k) may be determined by restricting α µν (k) to its temporal part.For β o = 0 we recover exactly the results of [6] showing that the system is consistent.Thus together with (2.17), (2.16), and (2.15) we have a SO(32) heterotic de Sitter background (2.6) realized as a Glauber-Sudarshan state with a positive cosmological constant given by (2.13).

Discussions and conclusions
The quest for the existence of a four-dimensional de Sitter space in type II string theories is a non-trivial problem [18], partly because of the existence of various no-go theorems, ruling out classical de Sitter vacuum, and partly because of the absence of a well-defined Wilsonian action over an accelerating background, ruling out quantum de Sitter vacuum.
(Although arguments can be made in favor of a quantum de Sitter vacuum using open QFTs [16], there are numerous issues with this 18 .See footnote 13.) Similar fate befalls de Sitter vacua in heterotic theories.In this paper we have steered clear of both classical and quantum vacuum solutions, and instead argued for the existence of de Sitter space as a Glauber-Sudarshan state.Such a state is the closest we can come to a classical solution, yet the analysis is fully quantum.In fact our analysis reveals that the quantum computations are more subtle because of the asymptotic natures of the perturbation series.The Gevrey growths of the perturbation series then allow us to use the powerful machinery of resurgence and Borel resummations to argue for the existence of such a state.
Our studies have shown that heterotic SO(32) theory does allow such a state to exist within a finite temporal domain given in (2.15), which is consistent with the trans-Planckian bound [3].Moreover (2.16) reveals that the solution is consistent with the non-violating NEC condition of [5].With some mild approximations, one can even achieve a surprisingly precise determination of such a state in (2.17) with a closed-form expression for the positive cosmological constant in (2.13).
Few detail still remains to be investigated.For example, we haven't been able to analyze the case for the E 8 ×E 8 heterotic theory.The duality chain connecting this to M-theory from [13] certainly looks promising, but the advantage we had from the eight-manifold in M-theory for the present analysis cannot be replicated so easily using seven-manifold for the E 8 case.We also haven't said much on moduli stabilization, flux-quantizations, anomaly cancellations or vector bundles either.All of these should follow the path laid out for the type IIB case in [10] because our heterotic background is dual to an orientifold background in type IIB.Nevertheless a direct analysis from the heterotic side is needed.All these and other related issues will be discussed elsewhere.
Education, Government of India.PR would like to acknowledge the ICTP's Associate programme where progress on the ongoing work continued during her visit as senior associate.For the purpose of open access, the authors have applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission.