# de Sitter State in Heterotic String Theory

Stephon Alexander1, Keshav Dasgupta2, Archana Maji3, P. Ramadevi3 and Radu
Tatar4
1 Brown Theoretical Physics Center and Department of Physics, Brown
University, Providence, RI 02912, USA
2 Department of Physics, McGill University, Montréal, Québec, H3A 2T8, Canada
3 Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India
4 Department of Mathematical Sciences, University of Liverpool, Liverpool, L69
7ZL, United Kingdom
<EMAIL_ADDRESS><EMAIL_ADDRESS>archana-
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

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.e. as a coherent state, over a
supersymmetric Minkowski background, albeit within a finite temporal domain.
Borel resummation and resurgence play a crucial role in constructing such a
state in the Hilbert space of heterotic theory governed entirely by the IR
degrees of freedom.

##  1 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 dyson , was surprisingly only appreciated more
recently from some remarkable works unsal  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 tcc .

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 coherbeta ;
coherbeta2 , and we called such a state as the Glauber-Sudarshan state to
distinguish it from the coherent state111Although we shall use both the
terminologies variably throughout, it is the former that is always meant..

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 $E_{8}\times E_{8}$ theory to
implement the computational technology. This computational technology involves
performing a full-fledged path-integral along the lines of borel2 ; borel3
over a Minkowski saddle which, expectedly, leads us to an asymptotic series of
the Gevrey kind gevrey  thus requiring Borel resummation borelborel . The
final answer we get matches somewhat with the type IIB case from coherbeta ;
coherbeta2 ; borel2 , 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 coherbeta ; coherbeta2  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:

$\displaystyle ds^{2}={a^{2}(t)\over{\rm
H}^{2}(y)}\left[-dt^{2}+g_{ij}dx^{i}dx^{j}+g_{33}(dx^{3})^{2}\right]+{\rm
H}^{2}(y)\left[{\rm F}_{2}(t)g_{mn}dy^{m}dy^{n}+{\rm
F}_{1}(t)g_{\alpha\beta}dy^{\alpha}dy^{\beta}\right],$ (1)

can also be realized as a Glauber-Sudarshan state. Here ${\rm F}_{i}(t)$
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
${\rm M}_{p}=1$), is kept arbitrary with the only condition being that it
becomes large at late time. This means the background (1) naturally expands at
late time. For example when $a^{2}(t)={1\over\Lambda t^{2}}$, with $\Lambda$
being the cosmological constant, we get an expanding de Sitter space in the
flat slicing in type IIB as $t\to 0$ at late time. The other factors,
$g_{ij}({\bf x}),g_{33}({\bf x}),g_{mn}(y),g_{\alpha\beta}(y)$ and ${\rm
H}^{2}(y)$ are the unwarped spatial metric components and the warp-factor
respectively. The coordinate $y\equiv(y^{m},y^{\alpha})\in{\cal
M}_{4}\times{\cal M}_{2}$ and $x=(t,{\bf x})\in{\bf R}^{2,1}$, so that nothing
depends on the third spatial direction parametrized by $x^{3}$ here. We will
soon make a further restriction by converting
$g_{\alpha\beta}=\delta_{\alpha\beta}$, so that ${\cal
M}_{2}={\mathbb{T}^{2}\over\mathbb{Z}_{2}}$ where
$\mathbb{Z}_{2}\equiv\Omega(-1)^{\rm F_{L}}{\bf I}_{\mathbb{T}^{2}}$ will be
an orientifolding operation (${\bf I}_{\mathbb{T}^{2}}$ is the orbifold
action. Details are in DRS ). Such a choice will give us a way to reach the
heterotic background by making a series of duality transformations. In that
case $y\equiv y^{m}\in{\cal M}_{4}$. 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 $\left(a^{2}(t),{\rm F}_{i}(t)\right)$ 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 coherbeta2 ) and from there dualize further
to heterotic $E_{8}\times E_{8}$ 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
coherbeta ; coherbeta2 . We no longer impose any constraint on ${\rm
F}_{i}(t)$. 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 ${\rm F}_{i}(t)$ 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 (1) cannot be realized classically GMN .
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) coherbeta2 .) 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 (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 (1) as a Glauber-Sudarshan
state deserves some explanation. As we saw in coherbeta ; coherbeta2 , 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
state222As alluded to in the introduction, the key differences between the
Glauber-Sudarshan state and the standard coherent state are described in
coherbeta ; coherbeta2 . We will not go into those details here and the
readers may pick up all the relevant informations from the references.. 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 soon333For example, one of the
question whose answer that we seek is as follows. Since heterotic and type IIB
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.. But first: does the background (1) exist in the IIB landscape as a
Glauber-Sudarshan state? This is what we turn to next.

###  2.1 Consistency of M-theory uplift and Glauber-Sudarshan state

As in coherbeta ; coherbeta2 , we will lift the background (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.

$\bullet$ The type IIB background (1) is supported at a constant coupling
point in F-theory senkd . 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 ${\bf D}_{4}^{4}=\left[{\rm SO}(8)\right]^{4}$ in the dual
heterotic side. The $E_{8}$ heterotic theory appears in a more non-trivial way
as shown in horava .

$\bullet$ The internal space with topology ${\cal M}_{4}\times{\cal M}_{2}$ 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 ${\rm F}_{i}(t)$. The space-time has positive cosmological
constant, and is therefore highly non-supersymmetric444If 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 desitter2 ;
coherbeta ; coherbeta2 , 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
coherbeta ; coherbeta2  for details.. Putting everything together we see that
all conventional techniques that we have learnt so far would fail to quantify
the dynamics of the system.

$\bullet$ 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
${g_{s}\over{\rm HH}_{o}}={1\over a(t)}$, where ${\rm H}(y)$ is the warp-
factor and ${\rm H}_{o}({\bf x})$ is related to $g_{33}({\bf x})$. For an
expanding cosmology, the system then is naturally weakly coupled at late
times.

$\bullet$ 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.

$\bullet$ 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-theory555Recall that the gauge fluxes on the D6-branes appear from localized
G-fluxes of the form ${\bf G}_{{\rm MN}ab}$ and ${\bf G}_{{\rm 0M}ab}$ that
are generically time-dependent desitter2 ; coherbeta ; coherbeta2 . This would
make both the world-volume gauge fluxes as well as the branes dynamical..
These dynamical M5-branes are responsible for the flux quantization procedure
as shown in coherbeta2 ; desitter2 .

$\bullet$ 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.

$\bullet$ 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 realized666This shifting is
of course an essential ingredient in constructing the Glauber-Sudarshan
states. In M-theory this is easily realized at low energies and is directly
related to the wave-function renormalization of the usual coherent states.
There are quite a few subtleties that we have kept under the rug here which
the readers could get from section 6.1 of the first reference in coherbeta2 ..

$\bullet$ 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 desitter2 ; coherbeta
; coherbeta2  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 state777As demonstated carefully in coherbeta , these
two facts go hand-in-hand. One implies the other. This is also a stronger
reason for viewing de Sitter space as a Glauber-Sudarshan state and not as a
vacuum configuration. $-$ means that all the short-distance degrees of freedom
can be integrated out to express the quantum corrections in the form given in
desitter2 ; coherbeta ; coherbeta2 .

$\bullet$ Lifting the metric configuration (1) to M-theory, one may easily see
that the toroidal direction, parametrized by $(x^{3},x^{11})$, scales as
$\left({g_{s}\over{\rm HH}_{o}}\right)^{4/3}$ which becomes arbitrarily small
in the limit $g_{s}\to 0$. Since we are always in the limit888The reason for
this is to control the non-perturbative corrections of the form ${\rm
exp}\left[-\left({g_{s}\over{\rm HH}_{o}}\right)^{-2k/3}\right]$, where
$k\in\mathbb{Z}_{+}$. So while M-theory is usually defined for $g_{s}>>1$, we
want to be in the opposite limit. $g_{s}<1$, 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 ${\cal M}_{2}$ cycle small compared to
$\alpha^{\prime}$ 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 (1). The only
change is the value of the dual IIA string coupling: ${g_{s}\over{\rm
HH}_{o}}={1\over a(t)}$ 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 ${g_{s}\over{\rm HH}_{o}}<1$, 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) tcc , as shown in desitter2
; coherbeta ; coherbeta2 . 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 DRS . We will basically
follow similar duality chasing here too, but not before we elucidate the
consistency of the IIB background (1) from M-theory. In M-theory, the uplifted
metric takes the following standard form:

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle\left({g_{s}\over{\rm
H}{\rm
H}_{o}}\right)^{-8/3}\left(-\tilde{g}_{00}dt^{2}+\tilde{g}_{ij}dx^{i}dx^{j}\right)+\left({g_{s}\over{\rm
H}{\rm H}_{o}}\right)^{-2/3}\bigg{[}{\rm F}_{1}\left({g_{s}\over{\rm
H}_{1}}\right)\tilde{g}_{\alpha\beta}dy^{\alpha}dy^{\beta}+{\rm
F}_{2}\left({g_{s}\over{\rm H}_{1}}\right)\tilde{g}_{mn}dy^{m}dy^{n}\bigg{]}$
(2) $\displaystyle~{}~{}~{}~{}~{}~{}~{}+\left({g_{s}\over{\rm H}{\rm
H}_{o}}\right)^{4/3}\tilde{g}_{ab}dw^{a}dw^{b},$

where ${\rm H}_{1}({\bf x},y)\equiv{\rm H}(y){\rm H}_{o}({\bf x})$, which
means ${\rm F}_{i}(g_{s}/{\rm H}_{1})$ 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 (1) in the following way:

$\displaystyle\tilde{g}_{ab}({\bf x},y)\equiv\Big{[}{\rm H}(y){\rm H}_{o}({\bf
x})\Big{]}^{4/3}g_{ab}({\bf x},y)$ $\displaystyle\tilde{g}_{\mu\nu}({\bf
x},y)\equiv{g_{\mu\nu}({\bf x})\over\Big{[}{\rm H}^{4}(y){\rm H}_{o}({\bf
x})\Big{]}^{2/3}},~{}~{}~{}~{}\tilde{g}_{\rm MN}({\bf x},y)\equiv\left[{{\rm
H}^{2}(y)\over{\rm H}_{o}({\bf x})}\right]^{2/3}g_{\rm MN}(y)$ (3)

where we have taken ${\rm M,N}\in{\cal M}_{4}\times{\cal M}_{2}$ and
$(w^{a},w^{b})\equiv(x^{3},x^{11})$. Note that we have taken the un-warped
metric components along the toroidal direction to depend on both
$(x^{i},y^{\rm M})$. In fact, for the computations of the curvature scalings,
as shown in desitter2 ; coherbeta2 , 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 ${\cal M}_{2}$ have to be removed.

Let us now come to the functional form for the temporal factors ${\rm
F}_{i}(g_{s}/{\rm H}_{1})$. In our earlier works desitter2 ; coherbeta ;
coherbeta2 , these factors did not change the dominant scalings of the metric
components as they were constrained by ${\rm F}_{i}(g_{s}/{\rm H}_{1})\to
1,g_{s}\to 0$ and ${\rm F}_{1}{\rm F}_{2}^{2}=1$ 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 desitter2 ; coherbeta ; coherbeta2
. We can then propose the following scalings:

$\displaystyle{\rm F}_{1}\equiv\sum_{k=0}^{\infty}{\rm
A}_{k}\left({g_{s}\over{\rm HH}_{o}}\right)^{\beta_{o}+2k/3},~{}~{}{\rm
F}_{2}\equiv\sum_{k=0}^{\infty}{\rm B}_{k}\left({g_{s}\over{\rm
HH}_{o}}\right)^{\alpha_{o}+2k/3},~{}~{}{\partial\over\partial
t}\left({g_{s}\over{\rm HH}_{o}}\right)\equiv\sum_{k=0}^{\infty}{\rm
C}_{k}\left({g_{s}\over{\rm HH}_{o}}\right)^{\gamma_{o}+2k/3},$ (4)

where $({\rm A}_{k},{\rm B}_{k},{\rm C}_{k})$ are all integers, positive or
negative with $(\alpha_{o},\beta_{o},\gamma_{o})$ being the dominant scalings
and $k\in\mathbb{Z}_{+}$. Note that, as we demonstrated rigorously in
coherbeta2 , when $\gamma_{o}<0$ 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 $(\alpha_{o},\beta_{o})$ 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 (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:

$\displaystyle{\bf g}_{\alpha\beta}({\bf x},y;g_{s})=\sum_{k=0}^{\infty}{\rm
B}_{k}^{(\alpha,\beta)}\left({g_{s}\over{\rm HH}_{o}}\right)^{-{2\over
3}+\beta_{o}+{2k\over 3}}{g}^{(k)}_{\alpha\beta}({\bf x},y),$
$\displaystyle{\bf g}_{mn}({\bf x},y;g_{s})=\sum_{k=0}^{\infty}{\rm
A}_{k}^{(m,n)}\left({g_{s}\over{\rm HH}_{o}}\right)^{-{2\over
3}+\alpha_{o}+{2k\over 3}}{g}^{(k)}_{mn}({\bf x},y),$ (5)

where the repeated indices are not summed over, and the $g_{\rm MN}^{(k)}({\bf
x},y)$ are the various possible metric components. It is easy to see that,
unless we impose $g^{(k)}_{\rm MN}({\bf x},y)\equiv\tilde{g}_{\rm MN}({\bf
x},y)$ from (2.2), 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:
$(\alpha_{o},\beta_{o})$, 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
${\rm A}_{k}^{(m,n)}$ and ${\rm B}_{k}^{(\alpha,\beta)}$.

Two, the signs of $(\alpha_{o},\beta_{o})$ are important. If
$\beta_{o}>+{2\over 3}$, it would mean that the size of the two-cycle ${\cal
M}_{2}$ shrinks at late time when $g_{s}\to 0$. Beyond certain short-distance
scale, T-duality would become necessary, and if ${\cal
M}_{2}\equiv{\mathbb{T}^{2}\over\mathbb{Z}_{2}}$ $-$ with $\mathbb{Z}_{2}$
being the orientifold operation DRS  $-$ 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 $\alpha_{o}>+{2\over 3}$, 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 coherbeta ; coherbeta2  because both the space-time
and the internal six-manifold had dominant scalings of $\left(-{8\over
3},-{2\over 3}\right)$ and therefore the late-time physics was well within the
supergravity limits999This doesn’t mean that the dynamics are captured by
classical supergravity. The late time physics is captured by weakly curved
manifolds, but all perturbative, non-perturbative, non-local and topological
corrections are necessary to solve the EOMs as shown in desitter2 ; coherbeta
; coherbeta2 .. (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 $g_{\alpha\beta}=\delta_{\alpha\beta}$
specifies a square torus capturing the local metric of ${\cal
M}_{2}={\mathbb{T}^{2}\over\mathbb{Z}_{2}}$, and $y\equiv y^{m}\in{\cal
M}_{4}$ with ${\cal M}_{4}$ being a generic non-Kähler manifold, as mentioned
earlier. There are also NS and RR two-forms with components ${\bf B}_{\alpha
m}(y,g_{s})$ and ${\bf C}_{\alpha m}(y,g_{s})$ respectively with one of their
legs along the toroidal directions. The heterotic metric then takes the
following form101010This metric was first derived by Evan McDonough in 2015
(and later by Bohdan Kulinich in 2022) by following the duality chasing
argument mentioned above. We thank them for many discussions related to
heterotic de Sitter solutions.:

$\displaystyle ds^{2}_{\rm het}$ $\displaystyle=$ $\displaystyle{\rm
F}_{1}(t)a^{2}(t)\left[-dt^{2}+g_{ij}dx^{i}dx^{j}+g_{33}(dx^{3})^{2}\right]$
$\displaystyle+$ $\displaystyle{\rm H}^{4}(y){\rm F}_{1}(t){\rm
F}_{2}(t)~{}g_{mn}dy^{m}dy^{n}+\delta_{\alpha\beta}\big{(}dy^{\alpha}+{\bf
B}^{\alpha}_{m}dy^{m}\big{)}\big{(}dy^{\beta}+{\bf
B}^{\beta}_{n}dy^{n}\big{)},$

where note that the toroidal directions no longer allow a warp-factor, but do
allow a non-trivial 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 (2.2) in the heterotic side.

We now want the four-dimensional part of the metric (2.2) 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 coherbeta2 .) We also want the four-
dimensional Newton’s constant to be time-independent. Putting these two
together implies that111111Note that the volume ${\rm V}_{6}$ of the internal
non-Kähler six-manifold in (2.2) is in general time-dependent but independent
of the fibration structure and is given by ${\rm V}_{6}={\rm H}^{8}(y){\rm
F}^{2}_{1}(t){\rm F}^{2}_{2}(t)\sqrt{{\rm det}(g_{mn})}$. Once the second
condition of (7) is taken into account, the volume becomes time-independent.:

$\displaystyle a^{2}(t)={1\over\Lambda t^{2}{\rm
F}_{1}(t)},~{}~{}~{}~{}~{}{\rm F}_{1}(t){\rm F}_{2}(t)=1,$ (7)

in the original type IIB metric (1), where $\Lambda$ is the cosmological
constant, $t$ is the conformal time in flat-slicing and
$(g_{ij},g_{33})=(\delta_{ij},1)$. (We can keep everything dimensionless by
measuring with respect to ${\rm M}_{p}\equiv 1$ as shown in borel2 ; borel3 .)
The choice (7) tells us that the original type IIB metric (1) is now no longer
required to have a time-independent four-dimensional 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) but with the following choices for
$g_{s}$, ${\rm H}_{o}({\bf x}),\alpha_{o}$ and $\beta_{o}$:

$\displaystyle{g_{s}\over{\rm HH}_{o}}=t\sqrt{\Lambda{\rm
F}_{1}(t)},~{}~{}~{}~{}~{}{\rm H}_{o}({\bf
x})=1,~{}~{}~{}~{}~{}\beta_{o}=-\alpha_{o},~{}~{}~{}~{}~{}\beta_{o}\geq 0,$
(8)

showing that there is a dynamical possibility of the type IIB metric (1) to go
to the heterotic side at late time. This also fixes the form for ${\rm
F}_{2}(t)$. One might worry that this could in-principle over-constrain the
system, but a similar computation in the type IIB side done in desitter2
shows that this may not be the case. Of course we will have to fix the
functional form for ${\rm F}_{1}(t)$ using the Schwinger-Dyson type equations
(in the presence of all non-perturbative and non-local corrections), but there
is an additional condition on ${\rm F}_{1}(t)$ borne out of the non-violating-
NEC criterion from coherbeta2 , namely:

$\displaystyle{d{\rm F}_{1}\over dt}={2\sqrt{{\rm F}_{1}}\over
t\sqrt{\Lambda}}\left[\sum_{k=0}^{\infty}{\rm C}_{k}\left(t\sqrt{\Lambda{\rm
F}_{1}}\right)^{\gamma_{o}+2k/3}-\sqrt{\Lambda{\rm F}_{1}}\right],$ (9)

where $\gamma_{o}$ is defined in (4). Once we determine the form of ${\rm
F}_{1}(t)$, the above equation will then fix the values of the constant
coefficients ${\rm C}_{k}$ and $\gamma_{o}$. If $\gamma_{o}<0$, 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 $\gamma_{o}$ could simply be zero if
not a positive integer, but never negative. This is because the solution with
$\beta_{o}=\alpha_{o}=0$ has been studied earlier in desitter2 , where no
apparent inconsistencies were detected. In the following section, we will
demonstrate, under some mild approximations, that $\gamma_{o}$ can indeed be
made positive definite here as long as $\beta_{o}\geq 0$.

###  2.3 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 (2.2) $-$ via type IIB $-$ is possible and
is given by (2) with the choice of parameters from (8). Unfortunately however,
such a background cannot be realised as a vacuum solution either classically
or quantum mechanically. The former is ruled out in sethiK  from the
constraints coming from the Kac-Moody algebras with $SO(4,1)$ global
symmetries121212See also the last reference in GMN  for a different take on
the no-go conditions stemming directly from the energy conditions.. The latter
is ruled out in borel3 ; borel2 ; coherbeta ; coherbeta2  from the absence of
a well-defined Wilsonian effective action for an accelerating background like
(1) or (2.2)131313One might argue that the situation could be dealt using open
quantum field theories vernon  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 borel2  suggest that this may not be a viable
option. . This means the only way to realize the background (2) would be as a
Glauber-Sudarshan state over a supersymmetric Minkowski background141414Being
supersymmetric Minkowski (or more appropriately, warped supersymmetric
Minkowski with compact non-Kähler internal eight-manifold), there is no longer
any Kac-Moody constraints from sethiK , or any energy constraints from GMN ..
How should we go about constructing such a state now?

Fortunately, since the form of the metric (2) in M-theory remains unchanged
from what we had in coherbeta ; coherbeta2 , the procedure to explicitly
construct such a state, will follow borel2 , namely, doing a path-integral
over the Minkowski saddle. In other words:

$\displaystyle\langle{\bf g}_{\mu\nu}\rangle_{\sigma}\equiv{\int[{\cal
D}g_{\rm MN}][{\cal D}{\rm C}_{\rm MNP}][{\cal D}\overline{\Psi}_{\rm
M}][{\cal D}\Psi_{\rm N}]~{}e^{i{\bf S}_{\rm
tot}}~{}\mathbb{D}^{\dagger}({\alpha},{\beta},{\gamma})g_{\mu\nu}(x,y,z)\mathbb{D}({\alpha},{\beta},{\gamma})\over\int[{\cal
D}g_{\rm MN}][{\cal D}{\rm C}_{\rm MNP}][{\cal D}\overline{\Psi}_{\rm
M}][{\cal D}\Psi_{\rm N}]~{}e^{i{\bf S}_{\rm
tot}}~{}\mathbb{D}^{\dagger}({\alpha},{\beta},{\gamma})\mathbb{D}({\alpha},{\beta},{\gamma})},$
(10)

where $({\rm M,N,P})\in{\bf R}^{2,1}\times{\cal M}_{4}\times{\cal
M}_{2}\times{\mathbb{T}^{2}\over{\bf Z}_{2}}$ with ${\bf Z}_{2}$ being the
orbifold action; $x\equiv({\bf x},t)\in{\bf R}^{2,1},y\in{\cal
M}_{4}\times{\cal M}_{2},z\in{\mathbb{T}^{2}\over{\bf Z}_{2}}$;
${\sigma}=({\alpha},{\beta},{\gamma})$ is the Glauber-Sudarshan state
associated with $(\\{{g}_{\rm MN}\\},\\{{\rm C}_{\rm MNP}\\},\\{{\Psi}_{\rm
M}\\})$ degrees of freedom (see also coherbeta ; coherbeta2 ); ${\bf
g}_{\mu\nu}$ is the graviton operator related to $g_{\mu\nu}$; the measure
${\cal D}g_{\rm MN}\equiv{\cal D}g_{\mu\nu}{\cal D}g_{\rm AB}$ with $({\rm
A,B})\in{\cal M}_{4}\times{\cal M}_{2}\times{\mathbb{T}^{2}\over{\bf Z}_{2}}$;
$\mathbb{D}(\sigma)$ is a non-unitary displacement operator, i.e.
$\mathbb{D}^{\dagger}(\sigma)\mathbb{D}(\sigma)\neq\mathbb{D}(\sigma)\mathbb{D}^{\dagger}(\sigma)\neq
1$ (see coherbeta  for details); and the total action:

$\displaystyle{\bf S}_{\rm tot}\equiv{\bf S}_{\rm kin}+{\bf S}_{\rm int}+{\bf
S}_{\rm ghost}+{\bf S}_{\rm gf},$ (11)

where the perturbative part of ${\bf S}_{\rm int}$ comes from an interaction
term like eq. (4.81) in coherbeta2  and ${\bf S}_{\rm gf}$ 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 borel3 ).

Unfortunately dealing with the path-integral structure in (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
borel2 , 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 (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 borel2 , it is shown that the path-integral
(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 gevrey  implying that Borel
resummation borelborel  of the Gevrey series is now necessary for the path-
integral (10) to make any sense! Putting everything together, and using the
computational procedure outlined in borel2  gives us the following:

$\displaystyle\langle{\bf g}_{\mu\nu}\rangle_{\sigma}=\sum_{\\{{\bf
s}\\}}\Bigg{[}{1\over g_{({\bf s})}^{1/l}}\int_{0}^{\infty}d{\rm S}~{}{\rm
exp}\left(-{{\rm S}\over g_{({\bf s})}^{1/l}}\right){1\over 1-{\cal A}_{({\bf
s})}{\rm
S}^{l}}\Bigg{]}_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rm
P.V}}\int_{k_{\rm IR}}^{\mu}d^{11}k~{}{\overline{\alpha}_{\mu\nu}(k)\over
a(k)}~{}{\bf Re}\left(\psi_{k}({\rm X})~{}e^{-i(k_{0}-\overline{\kappa}_{\rm
IR})t}\right),$ (12)

where we restricted the metric function $g_{\mu\nu}=g_{\mu\nu}(x,y)$ with
$y\in{\cal M}_{4}$ only in (10); P. V is the principal value of the enclosed
integral; $g_{({\bf s})}$ is a set of coupling constants defined by inverse
powers of ${\rm M}_{p}$ in a coupling-constant space parametrized by the set
${\\{{\bf s}\\}}$ 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); ${\cal A}_{({\bf s})}$ is the result of computing
all the nodal diagrams, including the NLO diagrams for all interactions in the
set $\\{{\bf s}\\}$, that are combinatorially suppressed but not volume ${\rm
V}$ suppressed where the volume ${\rm V}$ refers to the IR volume appearing
from the IR/UV mixing iruv ;
$\overline{\alpha}_{\mu\nu}(k)={\alpha_{\mu\nu}(k)\over{\rm
V}}\equiv{\alpha(k)\eta_{\mu\nu}(k)\over{\rm V}},a(k)={k^{2}\over{\rm V}}$ if
we choose the scale to be ${\rm M}_{p}$ and $\alpha(k)$ is related to the
Glauber-Sudarshan state $|\sigma\rangle$ discussed earlier; and $\psi_{k}({\rm
X})$, with ${\rm X}\equiv({\bf x},y)\in({\bf R}^{2},{\cal M}_{4})$, is the
spatial wave-function over the solitonic background with
$\overline{\kappa}_{\rm IR}(>k_{\rm IR})$ being an IR scale. (See borel2  for
details on the aforementioned computations.)

The summing over the set $\\{{\bf s}\\}$ 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 $\Lambda$ small
borel3 , where the closed form expression of the dimensionless cosmological
constant151515Recall that both the cosmological constant $\Lambda$ and the
conformal time $t$ are made dimensionless here using the scale ${\rm
M}_{p}\equiv 1$. 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 borel2 . appears from the first
integral over $d{\rm S}$ in (12), namely:

$\displaystyle{1\over\Lambda^{\kappa}}\equiv\sum_{\\{{\bf s}\\}}\left[{1\over
g_{({\bf s})}^{1/l}}\int_{0}^{\infty}d{\rm S}~{}{\rm exp}\Bigg{(}-{{\rm
S}\over g_{({\bf s})}^{1/l}}\Bigg{)}{1\over 1-{\cal A}_{({\bf s})}{\rm
S}^{l}}\right]_{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\rm
P.V}},$ (13)

with all the parameters appearing above being defined earlier, and $\kappa$
will be determined later (see (17)). This integral form of the cosmological
constant has already been shown to be positive definite in borel2 ,
irrespective of the signs of ${\cal A}_{({\bf s})}$ and in borel3  we argue
that this can be made small.

All in all, although it may appear that the analysis follows closely the ones
from borel2 , 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
$\alpha_{\mu\nu}(k)$. Can they be precisely determined here?

The answer turns out to be yes provided we know the functional form for ${\rm
F}_{1}(t)$. This is represented by a series in ${g_{s}\over{\rm HH}_{o}}$ in
(4). Compared to the type IIB case studied in desitter2  the situation is
different because ${g_{s}\over{\rm HH}_{o}}$ itself depends on ${\rm
F}_{1}(t)$ as shown in (8). This means, even before we invoke the consequence
from the Schwinger-Dyson’s equations, ${\rm F}_{1}(t)$ has to satisfy:

$\displaystyle{\rm F}_{1}(t)=\left(t\sqrt{\Lambda{\rm
F}_{1}(t)}\right)^{\beta_{o}}\sum_{k=0}^{\infty}{\rm
A}_{k}\left(t\sqrt{\Lambda{\rm F}_{1}(t)}\right)^{2k/3},$ (14)

where ${\rm A}_{k}$ are time independent constants; and in the limit ${\rm
M}_{p}\equiv 1$ we take both $\Lambda$ 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 coherbeta2  as in (9). The
coefficients ${\rm A}_{k}$ can only be fixed using the Schwinger-Dyson’s
equations (see equivalent example for the type IIB case in desitter2 ;
coherbeta ; coherbeta2 ). 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 ${\rm A}_{0}>>{\rm A}_{k}$ for
$k\geq 1$. For such a case, defining ${\rm A}_{0}\equiv 1$, we find:

$\displaystyle{\rm F}_{1}(t)=\left(\Lambda t^{2}\right)^{\beta_{o}\over
2-\beta_{o}},~{}~{}~{}~{}~{}-{1\over\sqrt{\Lambda}}~{}<t~{}\leq 0,$ (15)

with $0\leq\beta_{o}<2$ for the system to remain consistent161616Note that,
with the choice of ${\rm F}_{1}(t)$ from (15), both the type IIA coupling
${g_{s}\over{\rm HH}_{o}}=\left(\Lambda t^{2}\right)^{1\over 2-\beta_{o}}$ and
the heterotic coupling ${g_{\rm het}\over{\rm H}^{2}}=\left(\Lambda
t^{2}\right)^{\beta_{o}\over 2-\beta_{o}}$ are small at late time, i.e. when
$t\to 0$. Thus both the system are naturally weakly coupled at late time as
long as $0\leq\beta_{o}<2$. Beyond this regime of $\beta_{o}$, there are
multiple issues with the Glauber-Sudarshan states in both heterotic and the
dual type IIB theories., and not violate the NEC criterion. Recall that as
long as $\beta_{o}>+{2\over 3}$, 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 $0\leq\beta_{o}<2$, so is compatible
with the aforementioned condition. More so, the size of ${\cal M}_{4}$ 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 tcc , at least for this simple toy
example. Additionally, plugging (15) in (9) $-$ to see whether the non-
violating NEC criterion from coherbeta2  is satisfied or not $-$ gives us the
following solution:

$\displaystyle\gamma_{o}={\beta_{o}\over 2},~{}~{}~{}~{}~{}{\rm
C}_{0}={2\sqrt{\Lambda}\over 2-\beta_{o}},~{}~{}~{}~{}~{}{\rm
C}_{k}=0,~{}~{}~{}~{}~{}\forall k>0.$ (16)

This is clearly consistent as long as $\beta_{o}\geq 0$. Thus combining the
compatibility criteria from both TCC tcc  and non-violating NEC coherbeta2
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 $\alpha_{\mu\nu}(k)$
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 (2.2) 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 (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 ${\bf D}_{4}^{4}$ 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 elsewhere171717A generic study has to deal with two
aspects of the internal space: one, the temporal dependence of the fibration
structure and two, the non-Kähler (and subsequently, non-complex) nature of
the six-manifold. The latter has been addressed in the past, for example in
DRS , but the former is new. Other issues like anomaly cancellations and flux
quantizations follow the route laid out in desitter2 ..

Finally let us figure out the functional form for $\alpha_{\mu\nu}(k)$. It is
now related to the Fourier transform of the temporal part of the M-theory
metric (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 $\alpha_{\mu\nu}(k)$ and the value of the parameter
$\kappa$ in the expression for the cosmological constant $\Lambda$ in (13).
They are:

$\displaystyle\alpha_{\mu\nu}(k)=\sqrt{2\over\pi}\Gamma\Big{[}1-{16\over
3(2-\beta_{o})}\Big{]}~{}{\rm sin}\Big{[}{8\pi\over
3(2-\beta_{o})}\Big{]}\left[k^{1+{16\over
3(2-\beta_{o})}}+\chi(k)\right]\eta_{\mu\nu},~{}~{}~{}\kappa={8\over
3(2-\beta_{o})},$ (17)

where $\chi(k)$ may be determined by restricting $\alpha_{\mu\nu}(k)$ to its
temporal part. For $\beta_{o}=0$ we recover exactly the results of borel2
showing that the system is consistent. Thus together with (17), (16), and (15)
we have a $SO(32)$ heterotic de Sitter background (2.2) realized as a Glauber-
Sudarshan state with a positive cosmological constant given by (13).

##  3 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 susha , 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
vernon , there are numerous issues with this181818See however recent attempts
to realize de Sitter vacuum solution using loop-holes in the no-go theorems
westphal , or using AdS spaces mav . It will be interesting to find some
connections to our work.. 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 (15), which is consistent
with the trans-Planckian bound tcc . Moreover (16) reveals that the solution
is consistent with the non-violating NEC condition of coherbeta2 . With some
mild approximations, one can even achieve a surprisingly precise determination
of such a state in (17) with a closed-form expression for the positive
cosmological constant in (13).

Few detail still remains to be investigated. For example, we haven’t been able
to analyze the case for the $E_{8}\times E_{8}$ heterotic theory. The duality
chain connecting this to M-theory from horava  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 desitter2  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.

## Acknowledgements:

We would like to thank Heliudson Bernardo, Suddhasattwa Brahma, Mir-Mehedi
Faruk, Bohdan Kulinich, Evan McDonough, Brent Pym, Mark Van Raamsdonk, Savdeep
Sethi and Alexander Westphal for many discussions related to de Sitter space
in heterotic string theory. The work of SA is supported in part by the Simons
Foundation award number 896696. The work of KD is supported in part by a
Discovery Grant from the Natural Sciences and Engineering Research Council of
Canada (NSERC). The work of AM is supported in part by the Prime Minister’s
Research Fellowship provided by the Ministry of 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.

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