[a,b]Ralph Blumenhagen

# Reflections on an M-theoretic Emergence Proposal

Niccolò Cribiori  Aleksandar Gligovic  Antonia Paraskevopoulou

###### Abstract

In a pedagogical manner, we review recent developments in the investigation of
the Emergence Proposal. Although it is fair to say that this idea is still at
an exploratory level and a fully coherent picture has yet to be developed, we
put it into perspective to previous work on the swampland program and on
emergence in QG. In view of the emergent string conjecture, we argue and
provide evidence that it is not the emergent string but rather the
decompactification limit which is a natural candidate for the potential
realization of the Emergence Proposal. This resonates in a compelling way with
old ideas of emergence in M(atrix) theory and gives rise to a number of
further speculations.

##  1 Introduction

That the notion of emergence may be relevant in theories of Quantum Gravity
(QG) is not a new idea. While different interpretations of emergence might be
encountered across different areas of study, in the context of QG it has been
defined [1, 2] as the appearance of properties of a system that are novel with
respect to other (more fundamental) descriptions of the same system and robust
in the sense of characterizability and reproducibility. However, there remains
the more philosophical question of what is actually meant by emergence, if it
really describes a novel phenomenon or if it is just a relative, rather
epistemological notion reflecting the state of ignorance about all
implications of a given theory. Not entering this discussion, we take a more
pragmatic point of view and consider a standard string theory example to
elaborate on how emergence should be understood in the course of this article.

An open string stretched between two $D$-branes has a tower of excitations
whose massless modes are gauge fields confined to the branes. Computing the
one-loop annulus diagram and applying loop channel - tree channel equivalence,
the amplitude encodes the tree-level exchange of gravitons, i.e. of closed
string excitations. Here, we say that QG is emerging via quantum effects from
another theory not having the graviton as fundamental degree of freedom. On
the other side, in weakly coupled perturbative string theory one usually
concludes that, due to loop channel - tree channel equivalence again, a theory
of just open strings is not consistent and one always needs to include the
closed strings, as well. This is confirmed by all existing superstring
theories in 10D and their compactifications. Then, the leading order
gravitational interaction is just a tree-level effect in closed string theory.
Since the quantization of closed strings includes gravity, in these theories
we say that QG is not truly emerging.111One could say that gravity emerges
from the quantization of closed strings, but this is not how we want to
understand it.

A lesson we draw from this is that to realize emergence in the above sense we
need a theory featuring $D$-branes as fundamental light degrees of freedom,
with closed strings being heavy. As mentioned, this cannot happen in weakly
coupled string theory, $g_{s}=e^{\phi}\ll 1$, so that one is guided towards
other asymptotic limits of QG, that might be potentially realizing such a
hierarchical pattern of mass scales. Thanks to recent progress in the
swampland program (see e.g. [3, 4, 5] for reviews), we now have a more
systematic understanding of such infinite distance limits. There are a couple
of well established swampland conjectures particularly dealing with them,
known as the swampland distance conjecture [6] and one of its refinements, the
emergent string conjecture [7].

As will be reviewed in section 2, the lesson one can draw from them is that in
infinite distance limits, with a parameter $t\gg 1$, the degrees of freedom of
QG will show a hierarchical pattern such that one can distinguish between
light and heavy modes, with the former being the fundamental quantum degrees
of freedom while the latter can be thought of as classical. In terms of the
naturally small parameter $g=1/t\ll 1$, the mass scale of these modes behave
as

$m_{\rm pert}\simeq g^{\alpha}\Lambda\,,\qquad m_{\rm
class}\simeq\frac{\Lambda}{g^{\beta}}$ (1)

with $\alpha\geq 0$, $\beta>0$ and $\Lambda$ a characteristic mass scale of
the limit. This is the behavior also known from perturbative quantum field
theories (QFTs), where we distinguish between the classical non-perturbative
contributions to the path integral and the light quantum fluctuations around
them. Hence, it is in these limits that we can expect to find something like a
perturbative QG theory that shows certain resemblance to quantum field theory.

From this perspective, the usual perturbative string theory is just the
perturbative QG theory that arises in the small string coupling regime,
$g_{s}=e^{\phi}\ll 1$, i.e. the infinite distance limit $\phi\to-\infty$ for
the dilaton. The characteristic mass scale $\Lambda$ is given by the string
scale $\Lambda=M_{s}=(\alpha^{\prime})^{-1/2}$. The perturbative states are
the vibration modes of the string and upon compactification also its Kaluza-
Klein (KK) and winding modes. The non-perturbative states are the various
$p$-branes with tension $T_{p}\simeq M_{s}^{p+1}/g_{s}^{\beta}$, $\beta=1,2$.
For $D$-branes with $\beta=1$, the quantum fluctuations (of the closed string
modes) around them can be described by open string modes and in fact the
$D$-brane can be characterized as a coherent state (boundary state in CFT) of
closed strings. It is important to notice that the vibration modes of the
$p$-branes themselves are of the scale of the brane tension and hence are
heavy and freeze out in the limit $g_{s}\to 0$. While they seem similar at
first sight, this example reveals that QG is differing from QFT in one
important aspect, namely that it does not feature a finite number of
perturbative modes but rather infinite towers thereof already at first
quantization.

To connect to the notion of emergence described initially, the crucial
question is whether there exist asymptotic limits of QG where the light states
are not string excitations. What is the value of $\Lambda$ in these cases? If
such non-string limits exist, then the highly non-trivial question of how one
can mathematically describe these new perturbative QG theories arises. With no
string loop expansion available, as well as the techniques developed so far
for it evaluation available, how can one determine e.g. the low-energy
effective action for the massless modes? As we will describe, the answer to
these questions seems to be closely related to the two aforementioned
swampland conjectures, the notion of species scale [8, 9] and the so-called
Emergence Proposal [10, 11, 12]. The latter suggests that the kinetic terms in
the low-energy effective action of (all) QG theories are emerging quantum
mechanically from integrating out states below a certain ultraviolet (UV)
scale.

It is fair to say that the Emergence Proposal is currently on less firm ground
than other swampland conjectures. It is the purpose of this article to review
and conceptually reflect on some recent advances [13, 14, 15] in concretizing
this general idea and to connect it to the concept of emergence mentioned at
the very beginning of this introduction. At the end of section 3, we will also
comment on the recent work [16, 17] employing an approach slightly different
form ours. Admittedly, part of the results of [13, 14, 15] build upon previous
seminal work by e.g. Green-Gutperle-Vanhove [18] (and then [19, 20, 21, 22,
23, 24, 25]) and Gopakumar-Vafa [26, 27], which however at that time did not
explicitly emphasize the relation to emergence, let alone the swampland
program.

##  2 Preliminaries

In this section we first lay out some of the basic notions from the swampland
program predating the formulation of the Emergence Proposal. Then, we discuss
the Emergence Proposal in its initial form and argue that a refined
M-theoretic version of it has a real chance of being realized.

###  2.1 The swampland distance conjecture

The swampland distance conjecture [6] states that when approaching points at
infinite distance in the moduli space of an effective field theory (EFT)
arising from a viable QG theory, an infinite tower of states becomes
exponentially light. Denoting with $M_{n}$ the mass of the $n$-th level of the
tower, the statement is that for $\phi\to\infty$ we have

$M_{n}\simeq f(n)\,e^{-\alpha\phi}\,,$ (2)

where $\alpha$ is of order one in natural units where $M_{\rm pl}=1$. Since
the EFT comes with an UV cut-off above which all the states have been
integrated out, this implies a breakdown of the effective description for
field excursions $\phi\gtrsim\alpha^{-1}$. Hence, an EFT derived from QG only
has a finite range of validity.

In string theory two different types of such towers are usually encountered,
one is genuinely stringy in nature, while the other has to do with the fact
that string theory is a higher dimensional theory. Consider for instance
critical superstring theory, for which the relation between the 10D Planck
scale and the string scale is

$M_{\rm s}\simeq M_{\rm pl}\,g_{s}^{1/4}\simeq M_{\rm pl}\,e^{\phi/4}\,.$ (3)

Therefore, in the infinite distance limit of weak string coupling,
$\phi\to-\infty$, (for fixed Planck scale) the tower of string excitations of
mass $M_{n}=\sqrt{n}M_{s}$ becomes exponentially light. Asymptotically this
tower has an exponential degeneracy of states at mass level $n$ given by ${\rm
deg}_{n}\simeq\exp(\sqrt{n})$.

Compactifying string theory to lower dimensions, like e.g. to 9D on a circle,
the sizes of the compact dimensions will appear as extra scalars in the EFT.
Taking one of these scalars to infinite distance leads to a decompactification
limit, for which the KK modes become exponentially light. To see this,
consider the circle compactification on a radius of size $R$. Setting
$\phi=\gamma\log(M_{\rm pl}R)$, where $\gamma$ is fixed by canonically
normalizing the kinetic term, the mass of the KK tower is given by

$M_{n}\simeq\frac{n}{R}\simeq M_{\rm pl}\,n\,e^{-\phi/\gamma}\,.$ (4)

Hence, the KK tower shows an exponential scaling behavior with a dependence on
the level $n$ and a polynomial degeneracy ${\rm deg}_{n}$ (here ${\rm
deg}_{n}=1$ for a single $S^{1}$).

###  2.2 The Emergent String Conjecture

From these two simple string theory examples, it might seem as a big leap to
conjecture that this is essentially already the exhaustive list of different
behaviors in any theory of QG. Nevertheless, this is precisely what the
emergent string conjecture [7] states:

> Any infinite distance limit in QG is either an emerging string limit, where
> a fundamental string tower accompanied by particle-like towers becomes
> light, or a decompactification limit, where the lightest tower shows the
> behavior of a KK tower.

Evidence for this conjecture was initially collected for Calabi-Yau
compactifications of M-theory and type IIA superstring theory in infinite
distance limits in their vector-multiplet moduli space. To appreciate the
meaning of this conjecture, it is important to notice that these two limiting
behaviors can come in various disguises.

##### Emergent string limit

The most obvious such limit is the aforementioned coscaled weak coupling limit

$g_{s}\to\lambda g_{s}\,,\qquad M_{s}\to\lambda^{1/4}M_{s}\,,$ (5)

of type IIA/IIB string theory in 10D. Here we have $\lambda\ll 1$ while
keeping the 10D Planck scale constant. For the type IIB superstring, one can
also consider the analogous strong coupling regime, $\lambda\gg 1$, leading to
an emergent string limit where the string tower comes from $D1$-branes [14].
This can be generalized to compactifications on $k$-dimensional tori down to
$d=10-k$ dimensions. In this case, the coscaled emergent $D1$-string limit is

$g_{s}\to\lambda g_{s}\,,\qquad
M_{s}\to\lambda^{\frac{d-6}{2(d-2)}}M_{s}\,,\qquad\rho_{1}\to\lambda^{\frac{1}{2}}\rho_{1}\,,\qquad\rho_{i}\to\lambda^{\frac{1}{2}}\rho_{i}\,,$
(6)

where $\rho_{i}$ denote the internal radii in string units. After applying the
map

$M_{s}^{2}=M_{*}^{2}\,r_{11}\,,\qquad
g_{s}=\frac{r_{11}}{r_{1}}\,,\qquad\rho_{1}=\frac{1}{r_{1}\,r^{1/2}_{11}}\,,\qquad\rho_{i}=r_{i}\,r^{1/2}_{11}\,,$
(7)

between M-theory (on $T^{2}\times T^{k-1}$) and type IIB (on $T^{k}$), one can
transform this scaling to the M-theory quantities

$\begin{split}r_{11}\to\lambda^{\frac{1}{3}}r_{11}\,,\qquad
M_{*}\to\lambda^{\frac{(d-8)}{3(d-2)}}M_{*}\,,\qquad
r_{1}\to\lambda^{-{\frac{2}{3}}}r_{1}\,,\qquad
r_{i}\to\lambda^{\frac{1}{3}}r_{i}\,,\end{split}$ (8)

where $r_{11}$, $r_{1}$ and the $r_{i}$ denote the internal radii in 11D
Planck units and $M_{*}$ the 11D Planck scale. In this limit, the lightest
states are the $D1$-branes, whose mass scale is

$\begin{split}M_{D1}=T_{D1}^{1/2}\simeq{\frac{M_{s}}{g_{s}^{1/2}}}\simeq{\frac{M_{\rm
pl}^{(d)}}{\lambda^{\frac{2}{d-2}}}}\,.\end{split}$ (9)

However, there also exist more involved limits of this type. Consider e.g. the
compactification of the type IIA superstring on a Calabi-Yau which is
$K3$-fibered over a $\mathbb{P}^{1}$. Let us denote the size of the base as
$t_{b}$ so that the total volume of the Calabi-Yau is given by
$V=t_{b}\tau_{K3}+\ldots$ where $\tau_{K3}$ indicates the size of the $K3$
fiber. Now, we can consider the infinite distance limit where we scale
$t_{b}\to\lambda t_{b}$ with $\lambda\gg 1$ while keeping $\tau_{K3}$ finite.
To maintain also the 4D Planck scale finite, $M_{\rm
pl}^{2}=M_{s}^{2}V/g_{s}^{2}$, we have to coscale
$g_{s}\to\lambda^{1/2}g_{s}$. This means that we are taking a limit with a
large string coupling. Scanning through the list of states, we see that the
lightest modes scale like $M_{s}\simeq M_{\rm pl}/\lambda^{1/2}$ and are given
by the excitations of a string resulting from wrapping the $NS5$-brane on
$K3$, together with the particle-like states of $D0$\- and transverse (to the
large $\mathbb{P}^{1}$) $D2$\- and $D4$-branes. The excitations of the type
IIA fundamental string scale like $M\simeq\lambda^{0}$ and are parametrically
heavier. One can show that in this limit there exists a weakly coupled
heterotic dual model compactified on $K3\times T^{2}$ such that the
complexified Kähler modulus, $T_{B}=t_{B}+ib$, is mapped to the dilaton of the
heterotic string, $S=\exp(-\phi_{H})+iB$. The light towers of states are
mapped to the heterotic string excitations and to the KK and winding modes on
$K3\times T^{2}$. Hence this type IIA infinite distance limit is an emergent
string limit, where a (dual) fundamental string is among the lightest modes.

##### Decompactification limit

The type IIA Kähler moduli space of Calabi-Yau compactifications also admits a
coscaled decompactification limit, which is given by scaling all of its Kähler
moduli isotropically as $t_{I}\to\lambda^{2/3}t_{I}$ and coscaling the dilaton
as $g_{s}\to\lambda g_{s}$ to keep the 4D Planck scale finite. For later
purposes let us discuss this strong coupling limit more generally, i.e. upon
compactifying type IIA string theory on an internal manifold $X$ of dimension
$k$. Since this limit will be related to M-theory, we recall the dictionary
between the strong coupling limit of the type IIA superstring and M-theory.
The string scale $M_{s}$ and coupling $g_{s}$ are given in terms of the 11D
Planck scale $M_{*}$ and of the size $r_{11}$ of the eleventh direction as

$M_{s}^{2}=M_{*}^{2}\,r_{11}\,,\qquad g_{s}=r_{11}^{\frac{3}{2}}\,.$ (10)

Let us consider the strong coupling limit, $\lambda\to\infty$, such that the
$d=10-k$ dimensional Planck scale $M_{\rm pl}^{(d)}$ and the size of the
internal space remain finite in units of $M_{*}$. In terms of the type IIA
quantities it reads

$g_{s}\to\lambda g_{s}\,,\qquad
M_{s}\to\lambda^{\frac{d-4}{3(d-2)}}\,M_{s}\,,\qquad\rho_{I}\to\lambda^{\frac{1}{3}}\rho_{I}\,.$
(11)

Note that $d=4$ is special in the sense that the string scale does not scale
with $\lambda$. This can be translated to the M-theory quantities as

$r_{11}\to\lambda^{\frac{2}{3}}r_{11}\,,\qquad
M_{*}\to\frac{M_{*}}{\lambda^{\frac{2}{3(d-2)}}}\,,\qquad r_{I}\to r_{I}\,,$
(12)

where $r_{I}$ denote the radii of the internal space $X$ in units of $M_{*}$.
This means that all length scales of $X$ are scaled isotropically. One can
show that the $(d+1)$-dimensional Planck scale $M_{\rm pl}^{(d+1)}$ scales in
the same way as $M_{*}$.

From the M-theory perspective this particular type IIA strong coupling limit
corresponds to decompactification from $d$ to $d+1$ dimensions. The lightest
tower of states are particle-like $D0$-branes, or equivalently KK states of
the eleventh direction of mass

$\begin{split}M_{D0}\simeq{\frac{M_{s}}{g_{s}}}\simeq{\frac{M_{\rm
pl}^{(d)}}{\lambda^{\frac{2(d-1)}{3(d-2)}}}}\,.\end{split}$ (13)

The next lightest states are arising from wrapped $D2$\- and $NS5$-branes,
having a mass scale $M_{D2,NS5}\simeq M_{s}/g_{s}^{1/3}\simeq M_{\rm
pl}^{(d)}/\lambda^{\frac{2}{3(d-2)}}\simeq M_{*}$. KK modes along other
compact directions $I$ also have mass $M_{\rm KK}\simeq M_{*}/r_{I}$. All
other states, like wrapped $D4$-branes or the fundamental string, are
parametrically heavier. Hence, this is a typical example of a
decompactification limit where the lightest tower is (dual) to a KK tower of
particle-like states.

The evidence for the emergent string conjecture comes mostly from string
theory examples.222See [29] for a recent attempt to recover the emergent
string conjecture via bottom-up arguments based on black hole thermodynamics.
It is nevertheless remarkable that this string lamppost approach already led
to more than just emergent string limits, namely the existence of
decompactification limits. As we have seen, one of them is closely related to
the M-theory corner of the known string duality diagram. It would be
interesting to know how generic this limit is. The suspicion is that all
infinite distance decompactification limits are combinations of the M-theory
limit and its conventional further decompactification limits of additional
compact directions.

###  2.3 The species scale

Naively one would think that the mass scale where QG effects become important
is the Planck mass. However, already the weakly coupled string theory shows
the relevance of another mass scale, namely the string scale $M_{s}$, which in
10D is related to the Planck scale via (3). For fixed Planck scale and in the
infinite distance limit $g_{s}\to 0$, the string scale can become arbitrarily
small. This means that new, QG related physics will already occur at an energy
scale well below the Planck scale.

One can ask the question whether also the decompactification limit comes with
such an intermediate mass scale and how it can be determined. Quite
intriguingly, the general appearance of an effective QG cut-off in the case of
a large number of light states was pointed out independently of any string
theory reasoning in [8, 9] (see also [28] for earlier work). When considering
the quantum corrections to the 4D graviton propagator due to the coupling of a
large number $N_{\rm sp}$ of light species to gravity, the combination $N_{\rm
sp}p^{2}/M_{\rm pl}^{2}$ appears, with $p$ the momentum involved in the
process. If this combination becomes of order one perturbation theory
definitely breaks down revealing the mass scale $\tilde{\Lambda}\simeq M_{\rm
pl}/\sqrt{N_{\rm sp}}<M_{\rm pl}$ where quantum effects of gravity become
important. This is the so-called species scale, which in $d$ dimensions reads

$\tilde{\Lambda}\simeq\frac{M_{\rm pl}}{N_{\rm sp}^{\frac{1}{d-2}}}\,.$ (14)

For a given tower, the number of light species with mass below the species
scale can effectively be counted as

$N_{\rm sp}=\\#(m\leq\tilde{\Lambda})\,.$ (15)

Then, the latter two equations can be solved for the two unknowns $N_{\rm sp}$
and $\tilde{\Lambda}$. For KK towers this definition indeed gives the correct
species scale, whereas for string towers it gives an extra multiplicative
$\log$-factor that is not expected to be physical, for reasons that we will
review in the following.

It turns out that one could alternatively think of the species scale as the
scale where quantum corrections to the leading order Einstein-Hilbert term
become relevant. One way of seeing this is to define the species scale as the
radius $r_{0}=1/\tilde{\Lambda}$ of the minimal-sized black hole that can be
described within the EFT. The mass and Bekenstein-Hawking entropy of such a
black hole are

$M_{\rm BH}=\frac{M_{\rm pl}^{d-2}}{\tilde{\Lambda}^{d-3}}\,,\qquad\qquad
S_{\rm BH}=\frac{M_{\rm pl}^{d-2}}{\tilde{\Lambda}^{d-2}}\,.$ (16)

The number of species is defined via the statistical entropy as

$S_{\rm BH}=\log\Omega(M_{\rm BH})=:N_{\rm sp}\,,$ (17)

where $\Omega(M_{\rm BH})$ is the number of ways the macroscopic black hole of
mass $M_{\rm BH}$ can be realized by the microstates. Note that this
definition of the number of species satisfies the relation (14) and has also
been developed into a more complete thermodynamic picture in [30, 31].

Let us employ this definition for a string tower with mass levels
$M=M_{s}\sqrt{N}$ and degeneracy ${\rm deg}_{N}$. For sufficiently large mass
levels $N$, one can use the asymptotic expansion

${\rm deg}_{N}\sim e^{\beta\sqrt{N}}\,.$ (18)

Now, the excitation level required for the black hole mass is

$\sqrt{N_{\rm BH}}\simeq\frac{M_{\rm BH}}{M_{s}}\simeq\frac{M_{\rm
pl}^{d-2}}{M_{s}\,\tilde{\Lambda}^{d-3}}\,,$ (19)

which for small $g_{s}$ is expected to be a very large number. Then, up to a
$\beta$ factor, the black hole entropy is

$S_{\rm BH}\simeq\log\left({\rm deg}_{N_{\rm BH}}\right)\simeq\sqrt{N_{\rm
BH}}\,.$ (20)

Setting this equal to the Bekenstein-Hawking entropy (16) and using (19) gives
at leading order

$\tilde{\Lambda}\simeq M_{s}\,,\qquad N_{\rm sp}\simeq\left(\frac{M_{\rm
pl}}{M_{s}}\right)^{d-2}\,.$ (21)

Hence, the so defined species scale is equal to the string scale. Note that
here we have only exploited the presence of a string tower and the self-
consistency of the relations (16) and (17). This does not exclude that there
might be a lower scale where the black hole dynamically undergoes a phase
transition. This was discussed in the context of transitions from towers of
states to (minimal) black holes in [29] and was also employed for a conjecture
on the characteristic energy scales appearing in an EFT of QG more recently in
[32]. Let us mention that one could try to use the relations (14) and (15) to
determine the species scale [33, 34, 35], which leads to the result
$\tilde{\Lambda}\simeq M_{s}\log(M_{\rm pl}/M_{s})$.

The multiplicative $\log$-factor seems to be unphysical, as
$\tilde{\Lambda}\simeq M_{s}$ is consistent with the known string corrections
to the Einstein-Hilbert action, which include higher derivative terms
generically suppressed by the string scale. For the already mentioned Kähler
moduli of type IIA compactifications on Calabi-Yau manifolds, it was argued in
[36] that the one-loop topological free energy
$\mathbb{F}_{1}(T,\overline{T})$ provides a good measure for the number of
light species so that the species scale was proposed to be

$\tilde{\Lambda}\simeq\frac{M_{\rm pl}}{\sqrt{\mathbb{F}_{1}}}\,,$ (22)

which receives additive and not multiplicative corrections [37]. One can show
that for the aforementioned emerging string limit, $\mathbb{F}_{1}\simeq
t_{B}$ so that $\tilde{\Lambda}\simeq M_{s}$. Hence, we summarize that in an
emergent string limit the species scale coincides with the string scale (of
the emergent string) and that there are no towers of states with a
parametrically lighter mass.

Given the above generalization of the string scale as the scale where quantum
effects of gravity become important in the presence of light towers of states,
it is now straightforward to also apply it to the type IIA decompactification
limit discussed in the previous section. In this case, it is much shorter to
employ the definitions (14) and (15) for the computation of the species scale.
The corresponding black hole computation was presented in [35] and gives the
same result. Recall that the lightest states were BPS bound states of
$D0$-branes leading to a tower with masses

$M^{n}_{D0}\simeq\frac{M_{s}}{g_{s}}n\simeq\frac{M_{\rm
pl}^{(d)}}{\lambda^{\frac{2(d-1)}{3(d-2)}}}n\,.$ (23)

The number of light species is given by the maximal KK mode, i.e $N_{\rm
sp}=n_{\rm max}\simeq\lambda^{\frac{2(d-1)}{3(d-2)}}\,\tilde{\Lambda}/M_{\rm
pl}^{(d)}$ so that we can solve for

$\tilde{\Lambda}\simeq\frac{M_{\rm
pl}^{(d)}}{\lambda^{\frac{2}{3(d-2)}}}\simeq M_{\rm pl}^{(d+1)}\simeq
M_{*}\,.$ (24)

As expected for decompactification limits, the species scale is given by the
Planck scale of one dimension higher, which here scales in the same way as the
11D Planck scale. In the infinite distance limit this goes to zero, signalling
that the $d$-dimensional theory breaks down and one has to describe the theory
in $(d+1)$ dimensions. However, we are not yet done. Since $\tilde{\Lambda}$
is parametrically larger than the mass scale of the $D0$-brane tower, there
could still be other towers of states with a mass scale smaller than
$\tilde{\Lambda}$. These would further lower the species scale. However, the
next lightest states are the aforementioned bound states of wrapped $D2$\- and
$NS5$-branes as well as KK modes along internal directions, whose mass scales
precisely as $\tilde{\Lambda}$. Therefore, they do not further lower the
species scale which is indeed given by the higher dimensional Planck
scale.333An algorithm to calculate the species scale in the presence of
multiple towers can be found in [38].

In [14] a similar analysis was also done for the coscaled strong coupling
limit of the 10D type IIB superstring. As expected, in this case the lightest
towers are the string towers of the $D1$-branes, so that the species scale is
nothing else than the $D1$-string mass scale $\tilde{\Lambda}\simeq
T_{D1}^{1/2}\simeq M_{\rm pl}/g_{s}^{1/4}$ with all other mass scales being
parametrically larger than $\tilde{\Lambda}$. The analogous result holds for
the coscaled type IIB limit in lower dimensions with the resulting species
scale $\tilde{\Lambda}\simeq T_{D1}^{1/2}\simeq M_{\rm
pl}^{(d)}/\lambda^{\frac{2}{d-2}}$.

###  2.4 The Emergence Proposal

We have seen that a genuine feature of QG is the existence of infinite
distance limits which come in two different types, namely emergent string and
decompactification limits. The QG cut-off is the species scale, which is the
string scale for emergent string limits and the higher dimensional Planck
scale for decompactification limits. In these limits towers of states become
asymptotically massless and one has a naturally small parameter in which one
can hope to formulate perturbation theory. The prime example is (fundamental)
string theory itself, where this parameter is just the string coupling,
$g_{s}=\exp(\phi)$.

In this context, in [10, 11, 12] an interesting observation was made, namely
that the metric on moduli space can be recovered by integrating out the tower
of asymptotically massless states at one-loop. As a simple toy model, consider
a light modulus $\phi$ and a tower of massive KK states $h_{n}$ with mass
$M_{n}=n\,\Delta m(\phi)$ governed by a $d$-dimensional effective action

$S=M_{\rm pl}^{d-2}\int
d^{d}x\left(\frac{1}{2}G_{\phi\phi}\,\partial_{\mu}\phi\partial^{\mu}\phi+\sum_{n}\frac{1}{2}\partial_{\mu}h_{n}\partial^{\mu}h_{n}+\frac{1}{2}m^{2}_{n}(\phi)h_{n}^{2}\right)\,,$
(25)

where $G_{\phi\phi}$ denotes the metric on field space. The moduli-dependent
mass terms for $h_{n}$ leads to three-point couplings $y=[m_{n}(\phi)\partial
m_{n}(\phi)]\,h_{n}^{2}\,\phi$, inducing a one-loop correction to the kinetic
term for $\phi$ with the KK modes $h_{n}$ running in the loop. Integrating out
these modes up to the UV cut-off, which is taken to be the species scale,
leads to the leading order one-loop correction (see e.g. [34] for more
details)

$G_{\phi\phi}^{\rm 1-loop}\simeq\frac{\tilde{\Lambda}^{d-1}}{M_{\rm
pl}^{d-2}}\frac{\left(\partial_{\phi}\Delta m(\phi)\right)^{2}}{\left(\Delta
m(\phi)\right)^{3}}+\ldots\,.$ (26)

For a KK tower with $\Delta m=M_{\rm pl}/r$, the species scale is the
$(d+1)$-dimensional Planck scale, i.e. $\tilde{\Lambda}^{d-1}\simeq M_{\rm
pl}^{d-1}/r$. Then we find $G_{rr}^{\rm 1-loop}\simeq 1/r^{2}$, which has the
same functional dependence on the modulus $r$ as the tree level metric,
$G_{rr}^{0}$, resulting from the dimensional reduction of the Einstein-Hilbert
action. Even though this is just a simple toy model, the above was considered
quite a remarkable correlation leading to the formulation of the so-called
Emergence Proposal:

> Emergence Proposal (Strong): The dynamics (kinetic terms) for all fields are
> emergent in the infrared by integrating out towers of states down from an
> ultraviolet scale $\Lambda_{s}$, which is below the Planck scale.

There have been slightly different formulations and also a weak version [34],
but for the purpose of this presentation let us stick to this version
formulated in the review [3].

As it stands, this proposal is very generic and it is not clear what its realm
of validity could be. Of course, it is certainly meant in the context of QG
and the swampland program, where one usually identifies the UV cut-off scale
$\Lambda_{s}$ with the species scale, i.e. $\Lambda_{s}\simeq\tilde{\Lambda}$.
Moreover, the towers of states to be integrated out are those described in the
previous sections and which become light in infinite distance limits of the
moduli space.

In the previous toy example, a hard UV cut-off for the one-loop integral was
introduced, i.e both internal momenta and the mass of the states running in
the loop were cut-off at the species scale. However, when extending this
proposal beyond the pure EFT setting to theories of QG, the example of the
fundamental string tells us that one should not cut-off the loop-integrals at
a finite energy scale and keep only the string modes with masses below that
scale. In fact, such string loop amplitudes have nice UV properties precisely
by including all infinitely many states from the tower, as only then we have
modular invariance and we can restrict the integration over the fundamental
domain of $SL(2,\mathbb{Z})/\mathbb{Z}_{2}$. Hence, despite the treatment of
the simple toy example, in the Emergence Proposal it is implicitly meant that
one really integrates out the full infinite tower of states. This is also
compatible with the calculations of [3] (see footnote 46 therein).444We thank
E. Palti for confirming this point.

Then the question of which towers one has to integrate out arises: are they
only the lightest one or even all conceivable towers? The latter option can be
excluded, as in weakly coupled string theory, one only integrates out those
towers with a mass scale $M_{s}$. As mentioned at the beginning of this
section, $p$-brane towers are considered as classical non-perturbative objects
and are not running in the loop. In the next section, we will provide evidence
that for decompactification limits one also has to integrate out more than
just the lightest tower. Hence, we think that the answer is likely in the
middle and one has to integrate out all full infinite towers of states with
mass scale not larger than the species scale. In analogy to string theory,
these will be considered as the perturbative states in the effective
description of QG, while all the heavier towers of states will be classical
and non-perturbative. This means that we identify the scale $\Lambda$ from (1)
with the species scale, i.e $\Lambda=\tilde{\Lambda}$. In addition, while the
Emergence Proposal explicitly mentions kinetic terms, one could conceive that
in a fully emerging effective theory also all higher derivative terms are
generated by quantum effects.

The emergent string conjecture tells us that there are only two different
kinds of infinite distance limits: the emerging string limit and the
decompactification limit. Can the Emergence Proposal be true for an emerging
string limit? We think that the answer is negative for the following reasons.
First, we notice that even the naive computation around (25)-(26) for a string
tower does not give correct leading order results, as the previously mentioned
multiplicative $\log$-terms in the species scale are transferred to the one-
loop corrections [35]. Second and at a more fundamental level, we know how to
quantize the weakly coupled fundamental string and in fact, similarly to QFT,
one obtains a loop expansion in terms of higher genus Riemannian surfaces.
Here, e.g. the one-loop Schwinger integral over the tower of string states
gives (tautologically) only the one-loop correction to certain terms in the
low energy effective action. Hence, none of the tree-level terms in the
$g_{s}$ expansion are generated from quantum effects. This will also be true
for other emerging string limits, like the one discussed in the type IIA
Kähler moduli space for a $K3$-fibered Calabi-Yau, as these are conjectured to
be dual to a fundamental string. We refer to [13] for more details. The
exclusion of weakly coupled string limits resonates with the pragmatic concept
of emergence described at the beginning of this article.

With the emergent string limit excluded, it is only the decompactification
limit that remains, of which the M-theory limit is the typical example and
perhaps the only non-trivial one. In this respect, we note that in standard
(non-coscaled) decompactification limits $R\to\infty$ with $g_{s}\ll 1$ and
the volume ${\cal V}$ (in string units) of the orthogonal $9-d$ dimensional
compact space held fixed, the perturbative quantum gravity theory still
contains light strings. Certainly, the lightest tower of states is the KK
tower with $m_{\rm KK}\sim 1/R$, whose induced species scale is the finite
$(d+1)$-dimensional Planck scale. However, this is related to the string scale
via

$\begin{split}\tilde{\Lambda}\simeq
M_{s}\left(\frac{\mathcal{V}}{g_{s}^{2}}\right)^{\frac{1}{d-1}}\,,\end{split}$
(27)

which for $g_{s}\ll 1$ and ${\cal V}>1$ is larger than the string scale
$M_{s}$. Therefore, as expected, such a limit is just a higher dimensional
perturbative string theory. Even though the lightest states are given by KK
towers, the QG theory is described by quantized strings and as for the
aforementioned emergent string limit, the Emergence Proposal is not
realized.555In fact, there have been examples of the emergence proposal not
being straightforwardly realized in such decompactification limits, like the
partial emergence of certain quartic gauge couplings analyzed in [39].

As we have seen, the M-theoretic decompactification limit is of a different
type as string towers are heavier than the species scale. The QG theory of
M-theory is arguably one of the deepest mysteries and only partial results are
available at present, like a formulation in terms of $D0$-branes, the BFSS
matrix model [40] (see [41, 42, 43] for reviews). While later on we will
present a more detailed discussion, we can already state that in the BFSS
matrix model the interaction between gravitons was indeed found to be absent
classically and only generated via quantum (loop) effects. This can indicate
that there is a good chance for the M-theory limit to be the natural home of
the Emergence Proposal. In this spirit, from our discussion we extrapolate a
lesson in the form of an M-theoretic refinement of the Emergence Proposal:

> Emergence Proposal (M-theory): In the infinite distance M-theory limit
> $M_{*}R_{11}\gg 1$ with the Planck scale kept fixed, a perturbative QG
> theory arises whose low energy effective description emerges via quantum
> effects by integrating out the full infinite towers of states with a mass
> scale parametrically not larger than the 11D Planck scale. These are
> transverse $M2$-, $M5$-branes carrying momentum along the eleventh direction
> ($D0$-branes) and along any potentially present compact direction.

Note that in this limit the longitudinally wrapped $M2$-brane, i.e. the type
IIA fundamental string, and the longitudinally wrapped $M5$-brane, i.e. the
type IIA $D4$-brane, have masses

$M_{F1}=\frac{M_{*}}{g_{M}^{1/2}}\,,\qquad\qquad
M_{D4}=\frac{M_{*}}{g_{M}^{1/5}}\,,$ (28)

with the formal coupling constant $g_{M}=1/(M_{*}R_{11})\ll 1$. Hence, they
are not among the light modes and are considered as non-perturbative classical
objects. We note that the BFSS matrix model is indeed containing these two
longitudinal branes as bound states of $D0$-branes, which one might speculate
to be the analogue of the description of non-perturbative $D$-branes as
coherent (boundary) states of weakly coupled closed string modes. However, the
transverse $M5$-brane was lacking in the original BFSS matrix model which in
view of the Emergence Proposal might indicate that it is not yet the complete
description of quantum M-theory.666Transverse $M5$-branes appear in the BMN
version of the BFSS matrix model [44] and, more recently, they have been
investigated in cohomotopy and in connection to “Hypothesis H” [45, 46].
Before delving too deeply into such speculations, we provide more evidence for
our M-theoretic Emergence Proposal.

##  3 Evidence for the M-theoretic Emergence Proposal

The central challenge is to provide evidence despite the obvious shortcoming
that we do not understand the full quantization of M-theory, yet. In addition,
as M(matrix) theory teaches us, namely that the leading order supergravity
action at the second derivative level is emerging via loops, also space-time
itself should be somehow emergent.

The loophole bypassing these difficulties is that there are certain couplings
in the effective action that are protected by supersymmetry and do only
receive contributions from 1/2 BPS states. These states are under good control
and, up to a certain extend, can already be reliably described by their weak
string coupling counterparts, i.e. in the weakly coupled type IIA theory.
Hence, this sector of M-theory is special and admits the usual geometric
interpretation we are used to from string theory. We will see that indeed the
string one-loop evaluation for 1/2 BPS states can be extended to M-theory,
providing very reasonable results.

This is reminiscent of the working extension from Double Field Theory (DFT) to
Exceptional Field Theory (ExFT) (see [47, 48, 49] for reviews). The section
conditions are in fact the 1/2 BPS conditions for the corresponding $M2$-,
$M5$-branes and KK modes, written as differential operators on the extended
space made from usual and (brane) winding coordinates [50]. In DFT and ExFT
one also truncates the complete spectrum to just KK and brane-wrapping modes
leaving out the string, respectively M-theory, excitations.

In theories with 32 supercharges, the higher derivative $R^{4}$-term is 1/2
BPS saturated. Longer supermultiplets, preserving less supersymmetry, do not
contribute to it (see e.g. [51]). This term has received attention lately in
the context of species scale calculations [52, 53] and of the emergence of
species scale black hole horizons [54]. In the former, the coefficient of the
$R^{4}$-term has been shown to give the expected cut-off for emergent string
and decompactification limits, i.e. the string scale and the higher
dimensional Planck mass respectively. In theories with 16 supercharges, like
type IIA on $K3$ or the dual heterotic string on $T^{4}$, the $F^{4}$-coupling
is 1/2 BPS saturated. In theories with 8 supercharges, like $N=2$ supergravity
in 4D, the topological string couplings ${\cal F}_{g}$ at arbitrary genus $g$
are 1/2 BPS saturated. Note that ${\cal F}_{0}$ contains information about the
second order supergravity action, namely about the gauge couplings and kinetic
terms for the Kähler moduli. In this sense, such a coupling is special and
might be sensitive to issues related to the emergence of space-time itself.

###  3.1 Emergence of $R^{4}$-terms

Let us start with a theory featuring maximal supersymmetry, i.e. the 10D type
IIA superstring compactified to $d$ dimensions on a $k$-dimensional torus. The
higher derivative $R^{4}$-term arises in the low-energy effective action as

$S_{R^{4}}\simeq M^{d-8}_{s}\,V_{k}\int
d^{d}x\sqrt{-g}\,a_{d}\,t_{8}t_{8}\,R^{4}\,,$ (29)

where $g_{\mu\nu}$ denotes the string frame metric and $V_{k}$ the volume of
the internal torus in string units. For simplicity, we restrict ourselves to
rectangular tori and set $B_{2}=0$. In the emergent string limit, i.e. for
small $g_{s}$, the coefficient $a_{d}$ receives at most one-loop perturbative
corrections and space-time instanton corrections. The general form can be
schematically written as

$a_{d}=\frac{c_{0}}{g_{s}^{2}}+\underbrace{\left(c_{1}+{\cal
O}\left(e^{-S_{\rm ws}}\right)\right)}_{\rm one-loop}+{\cal O}\left(e^{-S_{\rm
st}}\right)\,,$ (30)

where $S_{\rm ws}$ denotes the action of world-sheet instantons and $S_{\rm
st}$ that of space-time instantons. The tree-level and one-loop coefficients
are known to be $c_{0}=2\zeta(3)$ and $c_{1}=2\pi^{2}/3$ where
$\zeta(s)=\sum_{n=1}^{\infty}n^{-s}$ is the Riemann zeta function. Our task is
to compute $a_{d}$ in the M-theory limit, where in particular $g_{s}\gg 1$.
Since the coupling is 1/2 BPS, in both limits one should get the same result
and the Emergence Proposal claims that, in the M-theory limit, $a_{d}$ should
stem entirely from quantum effects without any classical contribution.

Before discussing the M-theory limit, it is worthwhile to recall a few aspects
of the one-loop computation for $a_{d}$ in the weakly coupled type IIA string.
This will help us to sharpen our technical tools and to understand certain
analogies between the string and the M-theory computation.

####  3.1.1 Emergent string limit

An essential step of computing the one-loop diagram is a proper regularization
method for the real Schwinger integrals that are naively UV divergent, such as

$\log(p^{2}+m^{2})\sim\int_{0}^{\infty}\frac{dt}{t}e^{-\pi t(p^{2}+m^{2})}\,.$
(31)

In string theory it is well established that the real Schwinger parameter $t$
is complexified to $\tau=\theta+it$ by implementing the string level-matching
condition,

$L_{0}-\overline{L}_{0}=m_{i}\,n^{i}+N-\overline{N}=0\,,$ (32)

via a Lagrange multiplier $\theta$ and then using modular invariance to
restrict the complex integration to the fundamental domain $\mathcal{F}$ of
$SL(2,\mathbb{Z})$, thus avoiding the UV singularity. Notice that for
vanishing string excitations, $N=\overline{N}=0$, the level matching condition
becomes the 1/2 BPS condition $m_{i}n^{i}=0$ for KK momenta $m_{i}$ and string
winding modes $n^{i}$. Proceeding in this manner, the one-loop contribution to
the coefficient of the $R^{4}$-term in $d$ dimensions can be expressed as

$a_{d,{\rm
string}}^{(1)}\simeq\frac{2\pi}{{V}_{k}}\sum_{m_{i},n^{i}\in\mathbb{Z}}\int_{\cal
F}\frac{d^{2}\tau}{\tau_{2}^{\frac{d-6}{2}}}\,e^{-\pi\tau_{2}M^{2}-2\pi
i\tau_{1}m_{i}n^{i}}\,,$ (33)

with

$M^{2}=m_{i}G^{ij}m_{j}+n^{i}G_{ij}n^{j}\,,$ (34)

and $G_{ij}$ the metric on the torus. In this expression the Gaussian integral
over the continuous momenta along the $d$ non-compact directions has already
been carried out. Going one step back, this is the stringy regularization of
the initially UV divergent expression

$a_{d}^{(1)}\simeq\frac{2\pi}{V_{k}}\sum^{\\_\\_\\_}_{m_{i},n^{i}\in\mathbb{Z}}\int_{0}^{\infty}\frac{dt}{t^{\frac{d-6}{2}}}\,\delta({\rm
BPS})\,e^{-\pi tM^{2}}\,,$ (35)

which is sort of not taking into account the extended nature of the string,
and hence is affected by UV divergence close to $t=0$. In (35), the symbol
$\overline{\sum}$ is denoting the sum with the term with all $m_{i}$ and
$n^{i}$ vanishing excluded, so that the expression is related to the
definition of constrained Eisenstein series in [24], while $\delta({\rm BPS})$
arises from carrying out the unfolded integral over $\theta$ using
$\delta(x)=\int_{-\infty}^{\infty}d\theta e^{-2\pi ix\theta}$.

Let us consider the simplest case, $d=10$. Evaluating the string integral (33)
gives the finite result

$a_{10,{\rm string}}^{(1)}\simeq
2\pi\int_{\mathcal{F}}\frac{d^{2}\tau}{\tau_{2}^{2}}=\frac{2\pi^{2}}{3}\,.$
(36)

On the other hand, introducing a UV cut-off $\epsilon>0$ in the divergent
integral (35), one can write

$a_{10}^{(1)}\simeq
2\pi\int_{\epsilon}^{\infty}\frac{dt}{t^{2}}=\frac{2\pi}{\epsilon}\,.$ (37)

One could just minimally subtract this term, but then in 10D one would arrive
at $a_{10}^{(1)}=0$. Baring this limitation of (35) in mind, let us have a
look at the $d=9$ case, where the wrapped strings are particle-like. Here we
have KK and winding modes with mass $M^{2}=m^{2}/\rho^{2}+n^{2}\rho^{2}$, with
$\rho$ the radius of the circle in string units. The evaluation of (33) yields

$a_{9,{\rm
string}}^{(1)}\simeq\frac{2\pi^{2}}{3}\left(1+\frac{1}{\rho^{2}}\right)\,.$
(38)

Now let us try again to regularize the divergent integral (35). From the BPS
condition $m\cdot n=0$ we see that two different sectors contribute, namely
one with only winding, $n\neq 0$, and the other with only KK momentum, $m\neq
0$. We proceed as in the previous 10D case and introduce a UV regulator
$\epsilon>0$ to get

$\int_{\epsilon}^{\infty}\frac{dt}{t^{3/2}}e^{-\pi
tA}=\frac{2}{\sqrt{\epsilon}}-2\pi\sqrt{A}+\mathcal{O}(\sqrt{\epsilon})\,,$
(39)

where we expanded around $\epsilon\simeq 0$. We regularize this expression via
minimal subtraction of the divergent term, $2/\sqrt{\epsilon}$, and by sending
$\epsilon\to 0$ afterwards. Thus, the winding sector contribution becomes

$a_{9,m=0}^{(1)}\simeq\frac{2\pi}{\rho}\sum_{n\neq
0}\int_{0}^{\infty}\frac{dt}{t^{3/2}}\,e^{-\pi
t\rho^{2}n^{2}}=-4\pi^{2}\sum_{n\neq 0}|n|=\frac{2\pi^{2}}{3}\,,$ (40)

where we regularized the sum over $n$ via the zeta-function,
$\zeta(-1)=-1/12$. Following the same procedure for the KK contribution one
obtains

$a_{9,n=0}^{(1)}\simeq\frac{2\pi}{\rho}\sum_{m\neq
0}\int_{0}^{\infty}\frac{dt}{t^{3/2}}\,e^{-\pi
t\frac{m^{2}}{\rho^{2}}}=\frac{2\pi^{2}}{3}\frac{1}{\rho^{2}}\,,$ (41)

so that by combining (40) and (41) the full string one-loop result (38) is
recovered. In [15] this regularization method was applied also for lower non-
compact spacetime dimensions and was shown to give consistent results. For
later purposes, we provide here the result in 8D

$a^{(1)}_{8}\simeq-\frac{2\pi}{T}\log\left(T\,\left|\eta(iT)\right|^{4}\right)-\frac{2\pi}{T}\log\left(U\,\left|\eta(iU)\right|^{4}\right)\,,$
(42)

with $T=\rho_{1}\rho_{2}$ and $U=\rho_{2}/\rho_{1}$. Notice that the first
term shows the presence of world-sheet instantons.

To summarize, via minimal subtraction of the UV divergence and zeta-function
regularization of the infinite sums, we have found an alternative way to
regularize the divergent real Schwinger integral giving the same result as the
known regularization performed in string theory. Only in 10D the method does
not apply directly, and we believe this to be related to the fact that the
generic winding string is not particle-like. Nevertheless, one can recover the
correct 10D result from the 9D one via decompactification, $\rho\to\infty$, so
that in principle the full information about the one-loop correction to the
$R^{4}$-term can be computed via the regularization of the divergent
expression (35) just mentioned. In the same vein, one could calculate the one-
loop contributions in other emergent string limits, like e.g. the coscaled
strong coupling limit of type IIB. Here one would expect that a Schwinger
integral gives the one-loop corrections in $g_{E}=1/g_{s}\ll 1$ to the
$R^{4}$-coupling.

####  3.1.2 Decompactification limit

The question is if and how one can compute the $R^{4}$-coefficient $a_{d}$
also in the M-theoretic decompactification limit, $r_{11}=M_{*}R_{11}\gg 1$.
As we will argue, the method for the evaluation of the real Schwinger integral
introduced previously will be very useful in this regard.

Let us recall that, in this limit of strong type IIA coupling, swampland
arguments suggest that we need to integrate out the towers of states with mass
scale below the species scale, which is the 11D Planck scale. These are
transverse $M2$\- and $M5$-branes carrying KK momentum along all compact
directions, including the very large eleventh direction. For $d\leq 3$ ($k\geq
7$), also the transverse KK-monopole should be taken into account. For
simplicity, here we do not consider it and restrict ourselves to $d\geq 4$.
Note that the light modes do not include the type IIA fundamental string, i.e.
the longitudinal $M2$-brane, so that it is a priori non-trivial that the
former perturbative (in $g_{s}$) one-loop correction $a_{d}^{(1)}$ can be
recovered.

In the pioneering work of Green-Gutperle-Vanhove (GGV) [18], such a
computation was performed for the first time. The coefficients $a_{d}$ in 10D
and 9D were given by a natural generalization of the weakly coupled one-loop
string formula (35), where one was summing over the KK spectrum along the
eleventh direction, i.e. bound states of $D0$-branes. This was generalized in
[24] (see also the closely related work [19, 20, 21, 22, 23]) to include the
(full) 1/2 BPS particle-like states of M-theory in $d$ dimensions. The final
expression for the one-loop Schwinger integral in perturbative M-theory can be
compactly written as

$a_{d,{\rm M}}^{(1)}\simeq\frac{2\pi}{r_{11}{\cal
V}_{k}}\sum^{\\_\\_\\_}_{N^{I},m\in\mathbb{Z}}\int_{0}^{\infty}\frac{dt}{t^{\frac{d-6}{2}}}\;\delta({\rm
BPS})\;e^{-\pi t\,N^{I}{\cal M}_{IJ}N^{J}-\pi t\,\frac{m^{2}}{r_{11}^{2}}}\,,$
(43)

where all volumes and masses are measured in M-theory units. Apparently, one
now also integrates out the KK momentum $m$ along the eleventh direction,
which has been isolated from the rest. We have collectively denoted the
transverse KK momenta $m_{i}$, $i\in\\{1,\ldots,k\\}$, and the various
$M$-brane wrapping numbers as

$N^{I}=\left(m_{i},n^{ij},n^{ijklm}\right)\,.$ (44)

The mass matrix for 1/2 BPS states is given by

${\cal M}={\rm
diag}\left(\frac{1}{r_{i}^{2}},t_{ij}^{2},t_{ijklm}^{2}\right)\,,$ (45)

where, as throughout the paper, we made the simplifying assumption of a
rectangular torus with vanishing axionic fields. Turning on the latter induces
off-diagonal terms in the mass matrix. The quantity $t_{ij}=r_{i}r_{j}$ (and
similarly for $t_{ijklm}$) denotes the volume wrapped by the corresponding
transverse $M2$-brane (and $M5$-brane). Finally, the 1/2 BPS conditions
involving KK modes and the transverse $M2$\- and $M5$\- wrapping numbers read
[23]

$\displaystyle n^{ij}m_{j}=0\,,\hskip 92.47145pt\\#=k\,,$ (46) $\displaystyle
n^{[ij}\,n^{kl]}+m_{p}\,n^{pijkl}=0\,,\hskip 28.45274pt\\#={\binom{k}{4}}\,,$
(47) $\displaystyle n^{i[j}\,n^{klmnp]}=0\,,\hskip
64.87224pt\\#=k\binom{k}{6}\,,$ (48)

where we also indicated their number. The first condition says that the
momentum has to be orthogonal to the world-volume of the $M2$-brane. For
vanishing $M5$-brane wrapping number, the second condition means that the
matrix $n^{ij}$ has rank two.

Before presenting the evaluation of this compact expression for $a_{d,{\rm
M}}^{(1)}$ in a few examples, let us note that there is a group-theoretic
structure behind the number of particle-like states and their 1/2 BPS
conditions, which is closely related to a similar structure in exceptional
field theory. Following [24], let us collect all light transverse particle
states that we integrate out in the Schwinger integral. These states form
bound states with the unrestricted $D0$-branes, i.e. the KK modes along the
eleventh direction, and fit nicely into representations of
$E_{k(k)}(\mathbb{Z})$, which we denote as $E_{k(k)}$ to simplify our
notation. We define as usual $E_{2(2)}=SL(2)$, $E_{3(3)}=SL(3)\times SL(2)$,
$E_{4(4)}=SL(5)$ and $E_{5(5)}=SO(5,5)$. In table 1, we list all particle
states and how they fit into representations $\Lambda_{E_{k}}$ of $E_{k(k)}$
as well as the representations $\lambda_{E_{k}}$ of the 1/2 BPS conditions
(the latter coinciding with the representation of the string multiplet [55]).

d | k | Particles $SL(k)$ reps. | $E_{k(k)}(\mathbb{Z})$ | $\Lambda_{E_{k}}$ | 1/2 BPS: $\lambda_{E_{k}}$
---|---|---|---|---|---
9 | 1 | $[1]_{p}$ | $1$ | 1 | 0
8 | 2 | $[2]_{p}+[1]_{M2}$ | $SL(2)$ | 3 | 2
7 | 3 | $[3]_{p}+[3]_{M2}$ | $SL(3)\times SL(2)$ | (3,2) | (3,1)
6 | 4 | $[4]_{p}+[6]_{M2}$ | $SL(5)$ | 10 | 5
5 | 5 | $[5]_{p}+[10]_{M2}+[1]_{M5}$ | $SO(5,5)$ | 16 | 10
4 | 6 | $[6]_{p}+[15]_{M2}+[6]_{M5}$ | $E_{6}$ | 27 | 27
Table 1: Particle states, 1/2 BPS conditions and their $E_{k}$ representations
for $k\leq 6$. $[k]_{p}$ denotes KK momenta along the $k$ transverse
directions.

Due to this structure, the Schwinger integral (43) may be viewed as a
constrained Eisenstein series

$a_{d,{\rm M}}^{(1)}={\cal E}^{E_{k(k)}}_{\Lambda_{E_{k}}\oplus
1,s={\frac{k}{2}}-1}\,.$ (49)

Notice that there is a shift by one dimension relatively to the full U-duality
group $E_{k+1(k+1)}$ of toroidal compactifications of M-theory on $T^{k+1}$
down to $d$ dimensions. The latter was the guiding principle in [24], where
all 1/2 BPS states were considered, including also the longitudinal
$M$-branes, which are instead excluded in our counting. In particular, in [24]
the resulting total Schwinger integral is identified with a constrained
Eisenstein series of the type

$a_{d,{\rm bulk}}^{(1)}={\cal
E}^{E_{k+1(k+1)}}_{\Lambda_{E_{k+1}},s={\frac{k}{2}}-1}\,,$ (50)

with $d+k=10$ for $k>2$. In other words, since we are consistently working in
the large $r_{11}\gg 1$ region the full U-duality group $E_{k+1(k+1)}$ is
broken to a subgroup $E_{k(k)}\times 1$, which distinguishes the eleventh
direction. As depicted in figure 1,

Figure 1: Schematic view of the coscaled dilaton moduli space.

physically the difference between our approach and [24] is that we are
consistently working in the perturbative decompactification limit where we
only integrate out the perturbative states, i.e. transverse $M$-branes,
whereas in [24] the Schwinger integral involves all states, meaning that they
work in the interior (or the desert) of the $r_{11}$ moduli space. Due to
supersymmetric protection, in the end both results for the $R^{4}$-term should
be equal, as we have shown explicitly for $d\geq 8$ in [15], while a more
general proof is available for $d\geq 4$ in the previous work [50, 56].

We notice that in the bulk region, $g_{s}=\mathcal{O}(1)$, of the type IIB
superstring, the complete $R^{4}$-coupling can also be derived from the
M-theory expression (50) for $d\leq 9$ by applying the map (7) between
M-theory and type IIB quantities. As usual, the 10D decompactification limit
of type IIB is given by the $t=r_{1}\,r_{11}\to 0$ limit in M-theory. Hence,
in the bulk one always deals with M-theory and the $R^{4}$-couplings are
emerging by integrating out all 1/2-BPS states, including transverse and
longitudinal ones. This is certainly the least understood regime of QG, where
the species scale is of the same order as the Planck scale. The lesson one
might draw from this is that this genuine QG theory is expected to feature
emergence of space-time and all its interactions. From this perspective, the
all familiar emergent string limits are special in that they admit a weakly
coupled perturbation theory, still fairly analogous to the description of
perturbative QFTs.

#### Emergence of $R^{4}$-term in 10D

Let us evaluate the expression (43) in 10D, i.e. for $d=10$ and $k=0$. As
already mentioned, the coefficient of the $R^{4}$-term contains only a string
tree-level and a one-loop term,

$a_{10}\simeq\frac{2\zeta(3)}{g_{s}^{2}}+\frac{2\pi^{2}}{3}\,.$ (51)

The M-theoretic Schwinger integral involves only a sum over KK modes and hence
simplifies considerably to

$a^{(1)}_{10,{\rm M}}\simeq\frac{2\pi}{r_{11}}\sum_{m\neq
0}\int_{0}^{\infty}\frac{dt}{t^{2}}\;e^{-\pi t\frac{m^{2}}{r_{11}^{2}}}\,.$
(52)

As in the string case, this real integral is divergent for $t\to 0$, so that
we proceed with its regularization along the same line as for the divergent
real string-theoretic Schwinger integrals (35). First, we introduce a
regulator $\epsilon>0$ and perform a minimal subtraction of the divergence to
arrive at

$a^{(1)}_{10,{\rm M}}\simeq\frac{2\pi^{2}}{r_{11}^{3}}\sum_{m\neq
0}m^{2}\log\left(\frac{m^{2}}{\mu^{2}}\right)\,,$ (53)

with a constant $\mu^{2}$. Next, the sum over $m$ is carried out employing
zeta-function regularization. Using $\zeta(-2)=0$, the $\mu$-dependent term
drops out and what remains can be expressed as

$a^{(1)}_{10,{\rm
M}}\simeq\frac{2\zeta(3)}{r_{11}^{3}}=\frac{2\zeta(3)}{g_{s}^{2}}\,,$ (54)

where we employed

$\sum_{m\geq 1}m^{2}\log(m)=-\zeta^{\prime}(-2)=\frac{\zeta(3)}{4\pi^{2}}\,.$
(55)

Thus, the Schwinger integral for the tower of $D0$-branes reproduces precisely
the tree-level term in the expansion of (51) at small $g_{s}\,$. This result
was already obtained in the original work of GGV [18], but can now be
interpreted as first evidence for the M-theoretic Emergence Proposal.

However, the string one-loop term in (51) is missing in our computation
above.777In this respect, we recall that GGV added an (infinite) constant $C$
that was fixed in a subsequent step to the correct value $2\pi^{2}/3$ by
invoking T-duality in 9D. As we will see, this is completely analogous to the
missing $2\pi^{2}/3$ term in the 10D real Schwinger integral (35). More
specifically, the object that generates it is not particle-like in 10D. Let us
mention that in the $g_{s}\gg 1$ limit, the one-loop term in (51) is leading
over the tree-level term.

#### Emergence of $R^{4}$-term in 9D

For $d=9$ and $k=1$, the Schwinger integral (43) with still only KK
contributions is given by

$a^{(1)}_{9,{\rm
M}}\simeq\frac{2\pi}{r_{11}\,r_{1}}\sum_{(m,n)\neq(0,0)}\int_{0}^{\infty}\frac{dt}{t^{3/2}}\;e^{-\pi
t\left(\frac{m^{2}}{r_{11}^{2}}+\frac{n^{2}}{r_{1}^{2}}\right)}\,.$ (56)

In order to compare the result to weakly coupled type IIA, one has to apply a
couple of steps to eventually obtain an expression that admits an
interpretation in terms of string loop and instanton corrections. For that
purpose, one first splits the sum according to
$\sum_{(m,n)\neq(0,0)}=\sum_{m=0,n\neq 0}+\sum_{m\neq 0,n\in\mathbb{Z}}\,$.
The first piece ($m=0$) can be treated straightforwardly to obtain

$a^{(1)}_{9,{\rm
M};(m=0)}\simeq\frac{2\pi^{2}}{3}\frac{1}{r_{1}^{2}\,r_{11}}\,.$ (57)

For the second piece ($m\neq 0$), one performs a Poisson resummation of the
sum over $n\in\mathbb{Z}$ to obtain

$a^{(1)}_{9,{\rm M};(m\neq 0)}\simeq\frac{2\pi}{r_{11}}\sum_{m\neq
0}\sum_{n\in\mathbb{Z}}\int_{0}^{\infty}\frac{dt}{t^{2}}\;e^{-\pi
t\frac{m^{2}}{r_{11}^{2}}-\frac{\pi}{t}n^{2}r_{1}^{2}}\,.$ (58)

The $n=0$ term gives precisely the sum (52) from the previous section, that is
equal to $2\zeta(3)/r_{11}^{3}$, while for the remaining sum over $n\neq 0$
one employs the relation

$\int_{0}^{\infty}\frac{dx}{x^{1-\nu}}\,e^{-{\frac{b}{x}}-cx}=2\left|{\frac{b}{c}}\right|^{\frac{\nu}{2}}K_{\nu}\left(2\sqrt{|b\,c|}\right)\,,$
(59)

where $K_{\nu}(x)$ denotes the modified Bessel-function of order $\nu$. In
this manner, one obtains

$a^{(1)}_{9,{\rm M};(m\neq
0)}\simeq\frac{2\zeta(3)}{r_{11}^{3}}+\frac{8\pi}{r^{2}_{11}\,r_{1}}\sum_{m\neq
0}\sum_{n\geq
1}\left|{\frac{m}{n}}\right|K_{1}\left(2\pi|m|n\,{\frac{r_{1}}{r_{11}}}\right)\,.$
(60)

Expressing the two terms (57) and (60) in string units, one recovers almost
all contributions from the expected result

$a_{9}\simeq\frac{2\zeta(3)}{g_{s}^{2}}+\frac{2\pi^{2}}{3}\left(1+\frac{1}{\rho^{2}_{1}}\right)+\frac{8\pi}{g_{s}}\sum_{m\neq
0}\sum_{n\geq
1}\left|\frac{m}{n}\right|K_{1}\left(2\pi|m|n\,\frac{\rho_{1}}{g_{s}}\right)\,,$
(61)

except again for the one-loop term $2\pi^{2}/3$. The second term in (60) has
the correct dependence on $g_{s}$ to be interpreted as the contribution from
$E\\!D0$-brane instantons wrapping the circle of radius $\rho_{1}$. This
result can be considered as evidence for emergence, arguably even more
striking than the previous example, as not only the correct tree-level but
also the space-time instantons are recovered from a single Schwinger integral.
The full Schwinger integral including also the longitudinal $M2$-brane, i.e.
the type IIA fundamental string, was evaluated in [51], which also reproduced
the constant $2\pi^{2}/3$ term.

#### Emergence of $R^{4}$-term in 8D

In 8D, i.e. $d=8$ and $k=2$, an interesting novelty arises, namely the fact
that one has to include also contributions from wrapped transverse $M2$-branes
in the Schwinger integral. As we will see, these are the particle-like states
generating the missing $2\pi^{2}/3$ term. The full coefficient of the
$R^{4}$-term in perturbative type IIA theory is [18]

$\displaystyle a_{8}\simeq\frac{2\zeta(3)}{g_{s}^{2}}$
$\displaystyle-\frac{2\pi}{T}\log\left(\rho_{2}^{2}\,\left|\eta(iU)\,\eta(iT)\right|^{4}\right)$
(62)
$\displaystyle+\frac{8\pi}{\rho_{1}g_{s}}\\!\\!\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}m\geq
1\\\
(n_{1},n_{2})\neq(0,0)\end{subarray}}\\!\\!\\!\\!\\!\\!\\!\\!\frac{m}{|n_{1}+in_{2}U|}K_{1}\left(2\pi\frac{\rho_{1}}{g_{s}}m\,|n_{1}+in_{2}U|\right)\,,$

with $T=\rho_{1}\rho_{2}$ and $U=\rho_{2}/\rho_{1}$. Similarly to the 9D case,
the M-theoretic Schwinger integral with only $D0$-brane contributions gives
the tree-level term, the $E\\!D0$ instanton corrections, the $U$-dependent
part of the one-loop contribution and in principle also the logarithmic
contribution $-2\pi\log(\rho_{2}^{2})/T$. More details can be found in [15].

Besides, one receives a contribution from transverse $M2$-branes carrying KK
momentum only along the eleventh direction. (Recall that additional KK
momentum along $T^{2}$ would spoil the 1/2 BPS property.) The masses of the
particle-like states arising from these wrapped transverse $M2$-branes are

$M^{2}=n^{2}t_{12}^{2}+\frac{m^{2}}{r_{11}^{2}}\,,$ (63)

where $t_{12}=r_{1}r_{2}$ denotes the area of $T^{2}$ in M-theory units. Thus,
the total Schwinger integral also includes a contribution

$a^{(1)}_{8,{\rm M};M2}\simeq\frac{2\pi}{r_{11}t_{12}}\sum_{n\neq
0}\sum_{m\in\mathbb{Z}}\int_{0}^{\infty}\frac{dt}{t}\;e^{-\pi
t\left(n^{2}t_{12}^{2}+\frac{m^{2}}{r_{11}^{2}}\right)}\,.$ (64)

This integral can be regularized following our usual procedure. After
performing Poisson resummation with respect to the KK momentum $m$ and
applying (59), one gets

$a^{(1)}_{8,{\rm
M};M2}\simeq\frac{2\pi}{r_{11}t_{12}}\left(\frac{\pi}{3}r_{11}\,t_{12}+4\\!\\!\sum_{n_{1},n_{2}\geq
1}\frac{1}{n_{2}}e^{-2\pi
n_{1}n_{2}r_{11}t_{12}}\right)=-\frac{2\pi}{T}\log\left(\left|\eta(iT)\right|^{4}\right)\,,$
(65)

where we used $K_{1/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}$. Thus, this $M2$-brane
contribution is indeed the missing part so that the complete M-theoretic
Schwinger integral, obtained by combining the pure $D0$-brane contribution
with (65), reproduces (62). Let us notice that in the original work of GGV
[18], the term (65) had to be somehow added by hand. In contrast, here it
stems from considering wrapped transverse $M2$-branes as required by the
physical prescription of integrating out towers of states with typical mass
not larger than the species scale.

As explained in [18], the exponential terms in (65) describe type IIA
(fundamental) string instantons wrapped on the $T^{2}$. However, let us
emphasize once more that in our approach the $M2$-branes are transversely
wrapped so that they are not type IIA fundamental string winding modes. In
addition, $a^{(1)}_{8,{\rm M};M2}$ also contains the constant term
$2\pi^{2}/3$, which was so far missing in 9D and 10D. This is completely
analogous to the string story, where our method in 10D was also missing this
constant term, while it could be obtained in 9D from integrating out the
string winding modes along the compact direction. Here, the relevant object is
the transverse $M2$-brane which can be particle-like only for $d\leq 8$.
Similar to string theory, upon decompactification of the 8D result, we can
deduce the presence of the same constant term, $2\pi^{2}/3$, also in 9D and
10D.

#### Comments on emergence in $d\leq 7$

One could now move forward and consider compactifications on higher
dimensional tori. The complete evaluation of the Schwinger integral (43)
becomes increasingly complicated, as there will be additional sectors
contributing and the 1/2 BPS conditions become more involved and harder to
solve explicitly. The $d=7$ case is discussed in detail in [15] and confirms
the expectation that the Schwinger integral (43) produces the full
$R^{4}$-term, including also all instanton corrections, which will now also
comprise type IIA Euclidean $E\\!D2$-brane instantons. It would be very
interesting to go to $d=5$, where also particle-like $M5$-branes would
contribute for the first time. In [15] we only provide partial results, as the
full amplitude turns out to be highly complicated.

On the technical level, we have seen that instantons arise via Poisson
resummation of certain wrapping numbers and by applying then the relation
(59). In particular, from the argument of the appearing Bessel function we
could read off the action of the corresponding instanton. In this way, we have
been recovering the $E\\!D0$ and $E\\!F1$ instantons from combinations of
$(D0,{\rm KK})$ and $(D2,D0)$ particles upon Poisson resummation of the
wrapping number of the second entry. Going through all possible such
combinations one arrives at table 2.

Particle states | Instantons
---|---
$(D0,{\rm KK}_{(k)})$ | $E\\!D0_{(k)}$
$(D2_{(ij)},{\rm KK}_{(k)})$ | $E\\!D2_{(ijk)}$
$(N\\!S5_{(ijklm)},{\rm KK}_{(n)})$ | $E\\!N\\!S5_{(ijklmn)}$
$(D2_{(ij)},D0)$ | $E\\!F1_{(ij)}$
$(N\\!S5_{(ijklm)},D0)$ | $E\\!D4_{(ijklm)}$
$(N\\!S5_{(ijklm)},D2_{(lm)})$ | $E\\!D2_{(ijk)}$
Table 2: Particle (in loop) - instanton correspondence for elementary states.

This constitutes another non-trivial check of the M-theoretic Emergence
Proposal. Even though in the Schwinger integral one only integrates out the
perturbative towers of states, all expected instanton actions eventually
appear, i.e. also $E\\!F1$, $E\\!D4$ instantons which correspond to
longitudinal Euclidean $M2$\- and $M5$-branes.

###  3.2 Emergence of 4D $N=2$ topological couplings

So far we discussed only a single term in the derivative expansion of the
spacetime low energy effective theory. The leading order two-derivative terms,
like the Einstein-Hilbert term and the kinetic terms for the type IIA $C_{1}$
and $C_{3}$ gauge fields, are not 1/2 BPS saturated and currently out of reach
with our method. However, upon compactification on a space that breaks part of
the supersymmetry these kinetic terms can become 1/2 BPS saturated as well.
This happens for compactifications of type IIA string theory on a Calabi-Yau
threefold $X$ yielding $N=2$ supergravity in 4D.

Since this is a standard class of models, let us recall some relevant facts
just very briefly. The resulting moduli space is locally a product of
decoupled hyper- and vector-multiplet moduli spaces. The scalar fields in the
vector-multiplets are complexified Kähler moduli denoted as
$T^{i}=t^{i}+ib^{i}$, with $i=1,\dots,h_{11}(X)$, where $b^{i}$ are Kalb-
Ramond axions, while $t^{i}$ are real Kähler moduli, defining the Kähler cone
$t^{i}>0$. The vector fields are given by the RR $C_{3}$-form dimensionally
reduced on $h_{11}(X)$ homologically two-cycles. There is one more vector
field, the graviphoton, residing in the $N=2$ gravity multiplet and given by
the type IIA RR $C_{1}$-form. The kinetic terms for the Kähler moduli and the
gauge couplings are determined by the holomorphic prepotential ${\cal
F}_{0}(T)$, which is known to only receive perturbative corrections up to one-
loop. However, for type IIA the prepotential is even completely classical, as
the four-dimensional dilaton lies in a hypermultiplet.888For $K3$ fibrations,
the heterotic dual models indeed feature non-vanishing one-loop corrections.
In addition, there are non-perturbative corrections from world-sheet
instantons, so that the full prepotential enjoys an expansion999Note that
mirror symmetry also fixes the a priori ambiguous quadratic and linear terms
in the prepotential [57].

${\cal
F}_{0}(T)=-\frac{1}{g_{s}^{2}}\bigg{[}\frac{1}{3!}C_{ijk}T^{i}T^{j}T^{k}+\frac{\zeta(3)}{2}\chi(X)-\\!\\!\\!\sum_{\beta\in
H_{2}(X,\mathbb{Z})}\alpha^{\beta}_{0}\,\,\,{\rm Li}_{3}\left(e^{-\beta\cdot
T}\right)\bigg{]}\,,$ (66)

where $C_{ijk}$ denote the triple intersection numbers and $\chi(X)$ the Euler
characteristic of the Calabi-Yau threefold. Moreover, the integers
$\alpha^{\beta}_{0}$ are the genus zero Gopakumar-Vafa invariants [26, 27],
which are topological invariants of the Calabi-Yau. For a recent work relating
the Gopakumar-Vafa invariants to the emergent string conjecture see [58].

The prepotential is only the first in an infinite series of topological higher
derivative couplings of the form
$\mathbb{F}_{g}(T,\overline{T})R_{+}^{2}F_{+}^{2g-2}$, where $R_{+}$ and
$F_{+}$ are the self-dual parts of the Riemann tensor and of the graviphoton
field strength. Up to an additive term independent of the Kähler moduli, the
coupling splits into a harmonic piece and the so-called holomorphic anomaly

$\mathbb{F}_{g}(T,\overline{T})={\rm
Re}(\mathcal{F}_{g}(T))+f_{g}^{anom}(T,\overline{T})\,,$ (67)

with $\partial_{i}\partial_{\bar{\jmath}}f_{g}^{anom}(T,\overline{T})\neq 0$.
The genus $g$ holomorphic topological string amplitudes $\mathcal{F}_{g}(T)$
are not corrected in type IIA beyond the string $g$-loop level.

The one-loop topological free energy $\mathbb{F}_{1}(T,\overline{T})$ is the
quantity that was proposed to be a measure for the number of light species, as
recalled in section 2.3. Its holomorphic part has an expansion in terms of a
linear term and an infinite sum of instanton corrections

${\cal F}_{1}(T)=-\frac{1}{24}c_{2,i}\,T^{i}-\sum_{\beta\in
H_{2}(X,\mathbb{Z})}\left(\frac{\alpha_{0}^{\beta}}{12}+\alpha_{1}^{\beta}\right){\rm
Li}_{1}(e^{-\beta\cdot T})\,,$ (68)

where the integers $\alpha^{\beta}_{1}$ are the genus one Gopakumar-Vafa
invariants and $c_{2,i}=\int_{X}c_{2}(T_{X})\wedge\omega_{i}$ denotes the
coefficient of the second Chern class of the tangent bundle of $X$, with the
Kähler form expanded in a basis of cohomological 2-forms as
$J=\sum_{i=1}^{h_{11}}t_{i}\,\omega_{i}$.

#### Emergence of the topological amplitudes

In the weakly coupled emergent string limit, $g_{s}\ll 1$, mirror symmetry to
type IIB is a very powerful tool to derive the exact expansion (66). For a
concretely chosen Calabi-Yau, this allows to read off the genus zero
Gopakumar-Vafa invariants from the periods upon employing the mirror map. Like
for the $R^{4}$-term, let us now consider the M-theory limit. Since the
$\mathcal{F}_{g}(T)$ are known to be 1/2 BPS saturated, in this limit one
might expect to find a Schwinger-like integral where one integrates out the
light towers of states, which are 1/2 BPS bound states of KK momentum along
the M-theory circle ($D0$-branes) and transverse $M2$-branes. Indeed, these
are the objects that carry charge under the central extensions of the $N=2$
supersymmetry algebra,101010To compare to (66) and (68), $T$ has to be
rescaled by a factor $2\pi$. We have also set $M_{s}=2\pi$ throughout this
calculation.

$Z_{n}(\beta)=\frac{2\pi}{g_{s}}\left(\beta\cdot T+in\right).$ (69)

Here $n$ is the number of $D0$-branes and $\beta\in H_{2}(X,\mathbb{Z})$
denotes the homology class wrapped by the $M2$-brane, whose complexified size
is $\beta\cdot T$. Remarkably, this is precisely the proposal of Gopakumar-
Vafa [26, 27], who provided such a Schwinger integral for all of the
holomorphic $\mathcal{F}_{g}(T)$. For our purposes it is sufficient to
concentrate on $\mathcal{F}_{0}(T)$ and $\mathcal{F}_{1}(T)$, for which
Gopakumar-Vafa provided the expressions

$\displaystyle\mathcal{F}_{0}$
$\displaystyle=\sum_{\beta}\alpha^{\beta}_{0}\sum_{n\in\mathbb{Z}}\int_{0}^{\infty}\frac{ds}{s^{3}}\,e^{-sZ_{n}(\beta)}\,,$
(70) $\displaystyle\mathcal{F}_{1}$
$\displaystyle=-\sum_{\beta}\alpha^{\beta}_{1}\sum_{n\in\mathbb{Z}}\int_{0}^{\infty}\frac{ds}{s}\,e^{-sZ_{n}(\beta)}-\frac{1}{12}\sum_{\beta}\alpha^{\beta}_{0}\sum_{n\in\mathbb{Z}}\int_{0}^{\infty}\frac{ds}{s}\,e^{-sZ_{n}(\beta)}\,.$

Note that $\mathcal{F}_{1}$ receives contributions both from genus one and
genus zero curves. In general, the Gopakumar-Vafa invariants
$\alpha^{\beta}_{g}$ count the number of BPS configurations from the
transverse $M2$-branes wrapping genus $g$ curves in the class $\beta\in
H_{2}(X)$.

It is important to keep in mind that Gopakumar-Vafa invariants are topological
index-like quantities hard to determine from first principles. Our previous
general analysis revealed that there are more light modes in the
decompactification limit, which do not carry any central charge appearing in
the $N=2$ supersymmetry algebra. These are discrete KK momenta and transverse
$M5$-branes wrapping 4-cycles of the Calabi-Yau. We observe that the latter
would give strings in 4D and upon quantization could result in contributions
to the Gopakumar-Vafa invariants that grow exponentially. Whether this is the
right picture remains to be seen.

Evidently, the Schwinger integrals (70) are of the same type as those
encountered in the previous section on $R^{4}$-terms. In particular, the
integrals for both ${\cal F}_{0}$ and ${\cal F}_{1}$ are UV divergent close to
$s\simeq 0$, so that we can proceed by regularizing them in the same way as
the $R^{4}$-terms. To be concrete, let us focus on two simple examples.

#### The resolved conifold

First, we look at the non-compact resolved conifold that has only a single
$S^{2}$ of size $T$ on which an $M2$-brane can wrap. This can be considered
the prototypical example for learning how to evaluate the integrals before
eventually taking the infinite sum over all 2-cycles in the homology lattice
for a compact Calabi-Yau $X$. Here, we only sketch the computation and refer
to [13] for more details.

Starting with the simpler integral, namely that of $\mathcal{F}_{1}$, we
introduce a UV cut-off and minimally subtract the divergent terms to arrive at

$\mathcal{F}_{1}^{M2}=\frac{1}{12}\sum_{n\in\mathbb{Z}}\log\left(\frac{T+in}{\mu}\right)\,,$
(71)

with a constant $\mu$ depending on the cut-off and on some numerical factors.
Applying zeta-function regularization for the infinite sum over $n$ and using
the identity

$\frac{\sinh(\pi T)}{\pi
T}=\prod_{n=1}^{\infty}\left(1+\frac{T^{2}}{n^{2}}\right)$ (72)

we can bring this to the form

${\cal F}_{1}^{M2}=\frac{2\pi T}{24}-\frac{1}{12}{\rm Li}_{1}(e^{-2\pi T})\,.$
(73)

We observe that this is identical to the topological free energy computed for
the resolved conifold in [59], including the linear term. Formally, the
resolved conifold has $c_{2}=-1$ and the single non-vanishing Gopakumar-Vafa
invariant $\alpha^{1}_{0}=1$. The sum over only the $D0$-branes gives an
ambiguous logarithmic factor ${\cal F}_{1}^{D0}\simeq\log(2\pi\mu)$, possibly
reflecting the existence of the holomorphic anomaly.

For the holomorphic prepotential, $\mathcal{F}_{0}$, one can proceed
analogously. After introducing a UV cut-off and minimally subtracting the
divergent terms one gets the infinite sum

$\mathcal{F}^{M2}_{0}=-\frac{2\pi^{2}}{g_{s}^{2}}\sum_{n\in\mathbb{Z}}(T+in)^{2}\,\log\left(\frac{T+in}{\mu}\right)\,,$
(74)

with another constant $\mu$, related to the cut-off. Again, applying zeta-
function regularization and a descendant of the relation (72) obtained after
performing two integrations, one finally arrives at

$\mathcal{F}_{0}^{M2}=\frac{1}{g_{s}^{2}}\left[-\frac{(2\pi T)^{3}}{12}+{\rm
Li}_{3}(e^{-2\pi T})\right]\,.$ (75)

In addition, there is a also a non-trivial contribution from bound states of
only $D0$-branes, given by

$\mathcal{F}_{0}^{D0}=-\frac{2\pi^{2}}{g_{s}^{2}}\sum_{n\neq
0}n^{2}\log\left(\frac{n}{\mu}\right)=-\frac{1}{g_{s}^{2}}\zeta(3)\,.$ (76)

The total, unambiguous part of the prepotential also agrees with [59], where
the triple intersection number is formally $C=1/2$ and the Euler
characteristic is $\chi=2(h_{11}-h_{21})=2$.

One might be worried that this is a too trivial example and that a large
portion of the complications for compact Calabi-Yau threefolds is actually
absent. These complications involve the sum over the full infinite homology
lattice and in particular the a priori unknown index-like Gopakumar-Vafa
invariants. However, for each individual genus zero curve, the evaluation of
the Schwinger integrals will proceed as for the resolved conifold and for both
${\cal F}_{0}$ and ${\cal F}_{1}$ we will get a cubic and a linear
contribution, respectively. How these contributions do add up to finally give
the triple intersection numbers and the second Chern class is far from being
obvious and deserves further investigation.

#### The Enriques Calabi-Yau

To better illustrate what the challenge of taking the sum over the infinite
homology lattice is, let us consider a second simple example, this time on a
compact Calabi-Yau. The Enriques Calabi-Yau manifold is defined as the free
quotient $X=(K3\times T^{2})/\mathbb{Z}_{2}$, where the $\mathbb{Z}_{2}$ acts
via a free action on $K3$ and an inversion $z\to-z$ on the $T^{2}$. The free
quotient of $K3$ is the Enriques surface ${\cal E}$ with Euler characteristic
$\chi({\cal E})=c_{2}(T_{\cal E})=12$, leading to a Calabi-Yau with Hodge
numbers $(h_{11}(X),h_{21}(X))=(11,11)$. This Calabi-Yau is a $K3$ fibration
over a base $\mathbb{P}^{1}$, with the fibration reducing to ${\cal E}$ over
the four $\mathbb{Z}_{2}$ fixed points in the base. In [60, 61] the one-loop
topological free energy has been computed using the duality to the heterotic
string on $K3\times T^{2}$ giving

${\cal F}_{1}=-\pi T-12\sum_{n\geq 1}{\rm Li}_{1}(e^{-2\pi nT})+\ldots\,,$
(77)

where we omitted the known contribution depending on the 10 Kähler moduli
related to the 2-cycles of the Enriques surface.

The question is whether we can reproduce this result by adding up appropriate
$D2/D0$ bound state contributions (73); we are especially interested in the
linear term, $-\pi T$, which is usually obtained upon dimensional reduction
but not really from a Schwinger computation. Since the action of the
$\mathbb{Z}_{2}$ orbifold is free on the $K3$, the relevant 2-cycle is indeed
a genus one curve. This curve is the $T^{2}$ sitting at any point of the
Enriques surface and which can be multiply wrapped. Apparently, its
contribution to the Gopakumar-Vafa invariant $\alpha_{1}^{n}$ is just the
Euler characteristic of the Enriques surface $\alpha_{1}^{n}=\chi({\cal
E})=12$, which can be read off from the instanton series in (77). Taking into
account that for the contribution of a genus one curve the prefactor $1/12$ in
(73) is replaced by one, the linear term becomes

$\sum_{n\geq 1}\chi({\cal E})n\,\pi T=-\pi\,T\,,$ (78)

where for the sum over $n$ we employed zeta-function regularization, i.e.
$\zeta(-1)=-1/12$. Since this expression agrees precisely with the general
expansion (68) for $c_{2}({\cal E})=12$, by taking the sum over the homology
of all individual contributions we have reproduced the known result including
in particular the linear term. We conclude that the full holomorphic one-loop
topological free energy (for the $T$ modulus) emerges just from the Schwinger
integral111111The logarithmic ambiguity can be used to include the non-
holomorphic term needed for modular invariance of
$F_{1}(T,\overline{T})=-6\log\big{(}(T+\overline{T})|\eta(iT)|^{4}\big{)}$..

This example reveals also a puzzle. In [60] it was shown that the prepotential
is just

${\cal F}_{0}\simeq-\frac{1}{g_{s}^{2}}C_{IJ}T^{I}T^{J}T^{B},$ (79)

with $C_{IJ}$ denoting the Cartan matrix of $E_{8}\times\Gamma^{1,1}$ and
$T^{B}$ the Kähler modulus of the base $\mathbb{P}^{1}$. This means that all
genus zero Gopakumar-Vafa invariants are vanishing.

Hence, we face the following question: how can a sum over the individual
contributions (75) ever lead to a finite cubic term? While we have no clear
answer yet, considering all the evidence we have collected for emergence, we
believe that this puzzle will likely not falsify it but reveal some important
subtleties on how emergence is realized for ${\cal F}_{0}$. Recall that the
prepotential contains information on kinetic terms. Since space-time is
expected to emerge together with these terms121212We thank Ivano Basile and
Eran Palti for comments on this point., it is conceivable that one has to go
beyond this quasi-geometric approach, with (BPS) branes wrapping cycles, to
reliably compute ${\cal F}_{0}$.

Perhaps this is also related to the more radical proposal made in [16], where
the initial real Schwinger integral for ${\cal F}_{0}$ was deformed in two
ways. First, the integral was taken over a contour in the complex plane and
the integrand was changed from the simple exponential form,
$\exp(-sZ_{n}(\beta))$, so that in the end one arrives at a contour
representation of ${\cal F}_{0}$ familiar from period computations. It was
shown that this yields the full expansion of the prepotential as derived from
the period computation in the weakly coupled emergent string limit, $g_{s}\ll
1$. This calculation was supported and refined by relating the UV degrees of
freedom that are being integrated out to Fermi gas degrees of freedom [17].
The relation between this and our approach remains to be investigated. We have
now reached our present level of understanding of the subject and hence refer
the reader to future research.

##  4 Final comments on emergence

In the previous sections, we reviewed our current indications for the
M-theoretic Emergence Proposal. These are based mainly on the evaluation of
certain 1/2 BPS saturated couplings that were also subject to perturbative
non-renormalization theorems beyond the one-loop level. In these cases it was
possible to recover the full couplings in a small $g_{s}$ expansion from just
a single one-loop Schwinger integral in M-theory, where one integrates out
solely the light, perturbative towers of particle-like states with masses not
larger than the species scale, i.e. the 11D Planck scale. From where we stand,
let us reflect on two natural next questions.

First let us recall that, in string theory, by implementing the 1/2 BPS
condition via a Lagrange multiplier and using modular invariance, one could
define a complex version of the Schwinger integral that also gives the correct
constant term, $2\pi^{2}/3$, in 10D. The non-trivial question is whether one
can generalize this to M-theory and as such provide a higher-dimensional
definition of the Schwinger integrals that upon evaluation also yields the
correct result in 10D and 9D. The essential difference to the string-theoretic
case is that, in M-theory, the number of 1/2 BPS conditions increases with the
number $k$ of compactified directions. As we have seen, the latter transform
in the representation $\lambda_{E_{k}}$ of the group $E_{k(k)}$ so that after
imposing them via Lagrange multipliers, we arrive at an integral over ${\rm
dim}(\lambda_{E_{k}})+1$ dimensions. The question is whether this can be
interpreted as the moduli space ${\cal M}_{k}$ of something like an $M2$-$M5$
world-sheet. At the moment, this is not obvious (to us) at all and it might
not even be the right way to proceed. Indeed, the BFSS matrix model has taught
us that the quantization of the $M2$-brane should rather be thought of as a
second quantized theory and not as a first quantized world-volume theory, like
for the string.

Second, the Schwinger integral approach has been performed in this still sort
of geometric manner thanks to the 1/2 BPS nature of the involved states. The
ultimate question is whether emergence is just a special aspect of such a
simplified set-up or whether it is a general property of M-theory. In the
latter case, all terms in the low energy effective action, including the 10D
tree-level kinetic terms, have to arise from quantum effects. However, the
high degree of supersymmetry forbids such couplings to be generated via loops
so that we cannot expect to find a simple Schwinger integral that gives these
second derivative couplings right away.

Nevertheless, to conclude that this already spoils the M-theoretic Emergence
Proposal might be premature for the following reasons. First, for non-BPS
saturated couplings, like the kinetic terms in 10D with maximal supersymmetry,
all the excitations of M-theory will matter and contribute. The only candidate
we have at present to deal with this situation is the BFSS matrix model.
Second, recall that also in string theory one is computing the low-energy
effective action only indirectly. Indeed, one has to compute the appropriate
on-shell scattering amplitudes that allow to gain information about the
underlying EFT. This is done by comparing part of the string amplitude with
the corresponding amplitude computed in the expected or proposed EFT.
Therefore, to see the emergence of the 2-derivative kinetic terms one has to
compute the appropriate object. In the context of the BFSS matrix model this
problem has already been approached. Here, we are not reviewing all these
efforts, but we just recall that in such matrix model two gravitons (i.e. two
$D0$-branes) do not interact classically. Instead, the leading order
interaction is generated at the one-loop level (in matrix theory) leading to
an effective potential

$V=-\frac{15}{16}\frac{v^{4}}{r^{7}}+\ldots\,,$ (80)

where $v$ denotes the relative velocity of the two gravitons. Note that a non-
vanishing $v$ breaks supersymmetry. This potential is precisely the leading
order classical interaction in the supergravity theory. Due to supersymmetry,
the latter is vanishing for $v=0$, i.e. the forces due to graviton and
$p$-form exchange do cancel. Only upon breaking supersymmetry one gets a non-
trivial potential but this is generated at one-loop in the M(atrix) theory. We
think that this is very much in accordance with the Emergence Proposal and
also features the expected correlated emergence of the kinetic terms and space
itself, here from non-commutative matrix degrees of freedom. As a word of
caution, it is not a priori clear that the BFSS matrix model is the theory
that emerges in the coscaled infinite distance limit we are taking. This
deserves further study.

Connecting to the concept of emergence from the very beginning of this
article, we note that the behavior (80) can already be derived from the
familiar annulus diagram [62] for an open string connecting two $D0$-branes
with relative velocity $v$ and distance $r$. Hence, we are tempted to
speculate that the M-theoretic Emergence Proposal might be closely related to
the celebrated loop-channel tree-channel equivalence for the $D0$-branes (and
$D2$-branes), which happen to be the lightest states in the $M_{*}R_{11}\gg 1$
region of M-theory. In this respect, the transverse $M5$-brane, i.e. the type
IIA $NS5$-brane, is of a different type, pointing to the main open issue of
the BFSS matrix model.

##### Acknowledgments.

We thank Carlo Angelantonj, Ivano Basile, Álvaro Herráez, Elias Kiritsis,
Wolfgang Lerche, Dieter Lüst, Eran Palti, Andreas Schachner, Timo Weigand and
Max Wiesner for useful discussions. The work of R.B. and A.G. is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy – EXC-2094 – 390783311. R.B. and N.C. thank the
hospitality of the Corfu Summer Institute 2023, where part of the research
reviewed here has been performed.

## References

  * [1] J. Butterfield, “Emergence, Reduction and Supervenience: A Varied Landscape,” Foundations of Physics 41, no. 6, 920-959 (2011)
  * [2] D. Oriti, “Levels of spacetime emergence in QG,” [arXiv:1807.04875 [physics.hist-ph]].
  * [3] E. Palti, “The Swampland: Introduction and Review,” Fortsch. Phys. 67, no.6, 1900037 (2019) [arXiv:1903.06239 [hep-th]].
  * [4] M. van Beest, J. Calderón-Infante, D. Mirfendereski and I. Valenzuela, “Lectures on the Swampland Program in String Compactifications,” Phys. Rept. 989, 1-50 (2022) [arXiv:2102.01111 [hep-th]].
  * [5] N. B. Agmon, A. Bedroya, M. J. Kang and C. Vafa, “Lectures on the string landscape and the Swampland,” [arXiv:2212.06187 [hep-th]].
  * [6] H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland,” Nucl. Phys. B 766, 21-33 (2007) [arXiv:hep-th/0605264 [hep-th]].
  * [7] S. J. Lee, W. Lerche and T. Weigand, “Emergent strings from infinite distance limits,” JHEP 02, 190 (2022) [arXiv:1910.01135 [hep-th]].
  * [8] G. Dvali, “Black Holes and Large N Species Solution to the Hierarchy Problem,” Fortsch. Phys. 58, 528-536 (2010) [arXiv:0706.2050 [hep-th]].
  * [9] G. Dvali and M. Redi, “Black Hole Bound on the Number of Species and Quantum Gravity at LHC,” Phys. Rev. D 77, 045027 (2008) [arXiv:0710.4344 [hep-th]].
  * [10] B. Heidenreich, M. Reece and T. Rudelius, “The Weak Gravity Conjecture and Emergence from an Ultraviolet cut-off,” Eur. Phys. J. C 78, no.4, 337 (2018) [arXiv:1712.01868 [hep-th]].
  * [11] T. W. Grimm, E. Palti and I. Valenzuela, “Infinite Distances in Field Space and Massless Towers of States,” JHEP 08, 143 (2018) [arXiv:1802.08264 [hep-th]].
  * [12] B. Heidenreich, M. Reece and T. Rudelius, “Emergence of Weak Coupling at Large Distance in Quantum Gravity,” Phys. Rev. Lett. 121, no.5, 051601 (2018) [arXiv:1802.08698 [hep-th]].
  * [13] R. Blumenhagen, N. Cribiori, A. Gligovic and A. Paraskevopoulou, “Demystifying the Emergence Proposal,” JHEP 04 (2024), 053 [arXiv:2309.11551 [hep-th]].
  * [14] R. Blumenhagen, N. Cribiori, A. Gligovic and A. Paraskevopoulou, “Emergent M-theory limit,” Phys. Rev. D 109, no.2, L021901 (2024) [arXiv:2309.11554 [hep-th]].
  * [15] R. Blumenhagen, N. Cribiori, A. Gligovic and A. Paraskevopoulou, “Emergence of $R^{4}$-terms in M-theory,” [arXiv:2404.01371 [hep-th]].
  * [16] J. Hattab and E. Palti, “On the particle picture of Emergence,” JHEP 03, 065 (2024) [arXiv:2312.15440 [hep-th]].
  * [17] J. Hattab and E. Palti, “Emergence in String Theory and Fermi Gases,” [arXiv:2404.05176 [hep-th]].
  * [18] M. B. Green, M. Gutperle and P. Vanhove, “One loop in eleven-dimensions,” Phys. Lett. B 409, 177-184 (1997) [arXiv:hep-th/9706175 [hep-th]].
  * [19] M. B. Green and P. Vanhove, “D instantons, strings and M theory,” Phys. Lett. B 408, 122-134 (1997) [arXiv:hep-th/9704145 [hep-th]].
  * [20] E. Kiritsis and B. Pioline, “On R**4 threshold corrections in IIb string theory and (p, q) string instantons,” Nucl. Phys. B 508, 509-534 (1997) [arXiv:hep-th/9707018 [hep-th]].
  * [21] J. G. Russo and A. A. Tseytlin, “One loop four graviton amplitude in eleven-dimensional supergravity,” Nucl. Phys. B 508, 245-259 (1997) [arXiv:hep-th/9707134 [hep-th]].
  * [22] B. Pioline and E. Kiritsis, “U duality and D-brane combinatorics,” Phys. Lett. B 418, 61-69 (1998) [arXiv:hep-th/9710078 [hep-th]].
  * [23] N. A. Obers and B. Pioline, “U duality and M theory,” Phys. Rept. 318, 113-225 (1999) [arXiv:hep-th/9809039 [hep-th]].
  * [24] N. A. Obers and B. Pioline, “Eisenstein series and string thresholds,” Commun. Math. Phys. 209, 275-324 (2000) [arXiv:hep-th/9903113 [hep-th]].
  * [25] C. Angelantonj, I. Florakis and B. Pioline, “A new look at one-loop integrals in string theory,” Commun. Num. Theor. Phys. 6, 159-201 (2012) [arXiv:1110.5318 [hep-th]].
  * [26] R. Gopakumar and C. Vafa, “M theory and topological strings. 1.,” [arXiv:hep-th/9809187 [hep-th]].
  * [27] R. Gopakumar and C. Vafa, “M theory and topological strings. 2.,” [arXiv:hep-th/9812127 [hep-th]].
  * [28] G. Veneziano, “Large N bounds on, and compositeness limit of, gauge and gravitational interactions,” JHEP 06 (2002), 051 [arXiv:hep-th/0110129 [hep-th]].
  * [29] I. Basile, D. Lüst and C. Montella, “Shedding black hole light on the emergent string conjecture,” [arXiv:2311.12113 [hep-th]].
  * [30] N. Cribiori, D. Lüst and C. Montella, “Species entropy and thermodynamics,” JHEP 10 (2023), 059 [arXiv:2305.10489 [hep-th]].
  * [31] I. Basile, N. Cribiori, D. Lüst and C. Montella, “Minimal Black Holes and Species Thermodynamics,” [arXiv:2401.06851 [hep-th]].
  * [32] A. Bedroya, C. Vafa and D. H. Wu, “The Tale of Three Scales: the Planck, the Species, and the Black Hole Scales,” [arXiv:2403.18005 [hep-th]].
  * [33] F. Marchesano and L. Melotti, “EFT strings and emergence,” JHEP 02, 112 (2023) [arXiv:2211.01409 [hep-th]].
  * [34] A. Castellano, A. Herráez and L. E. Ibáñez, “The emergence proposal in QG and the species scale,” JHEP 06, 047 (2023) [arXiv:2212.03908 [hep-th]].
  * [35] R. Blumenhagen, A. Gligovic and A. Paraskevopoulou, “The emergence proposal and the emergent string,” JHEP 10, 145 (2023) [arXiv:2305.10490 [hep-th]].
  * [36] D. van de Heisteeg, C. Vafa, M. Wiesner and D. H. Wu, “Moduli-dependent Species Scale,” [arXiv:2212.06841 [hep-th]].
  * [37] N. Cribiori and D. Lüst, “A Note on Modular Invariant Species Scale and Potentials,” Fortsch. Phys. 71 (2023) no.10-11, 2300150 [arXiv:2306.08673 [hep-th]].
  * [38] A. Castellano, A. Herráez and L. E. Ibáñez, “IR/UV mixing, towers of species and swampland conjectures,” JHEP 08 (2022), 217 [arXiv:2112.10796 [hep-th]].
  * [39] S. J. Lee, W. Lerche and T. Weigand, “Physics of infinite complex structure limits in eight dimensions,” JHEP 06 (2022), 042 [arXiv:2112.08385 [hep-th]].
  * [40] T. Banks, W. Fischler, S. H. Shenker and L. Susskind, “M theory as a matrix model: A Conjecture,” Phys. Rev. D 55, 5112-5128 (1997) [arXiv:hep-th/9610043 [hep-th]].
  * [41] A. Bilal, “M(atrix) theory : A Pedagogical introduction,” Fortsch. Phys. 47, 5-28 (1999) [arXiv:hep-th/9710136 [hep-th]].
  * [42] D. Bigatti and L. Susskind, “Review of matrix theory,” NATO Sci. Ser. C 520, 277-318 (1999) [arXiv:hep-th/9712072 [hep-th]].
  * [43] W. Taylor, “M(atrix) Theory: Matrix Quantum Mechanics as a Fundamental Theory,” Rev. Mod. Phys. 73, 419-462 (2001) [arXiv:hep-th/0101126 [hep-th]].
  * [44] J. M. Maldacena, M. M. Sheikh-Jabbari and M. Van Raamsdonk, “Transverse five-branes in matrix theory,” JHEP 01 (2003), 038 [arXiv:hep-th/0211139 [hep-th]].
  * [45] H. Sati and U. Schreiber, “Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams,” Adv. Theor. Math. Phys. 26 (2022) no.4, 957-1051 [arXiv:1912.10425 [hep-th]].
  * [46] D. Corfield, H. Sati and U. Schreiber, “Fundamental weight systems are quantum states,” Lett. Math. Phys. 113 (2023) no.6, 112 [arXiv:2105.02871 [math.GT]].
  * [47] G. Aldazabal, D. Marques and C. Nunez, “Double Field Theory: A Pedagogical Review,” Class. Quant. Grav. 30, 163001 (2013) [arXiv:1305.1907 [hep-th]].
  * [48] O. Hohm, D. Lüst and B. Zwiebach, “The Spacetime of Double Field Theory: Review, Remarks, and Outlook,” Fortsch. Phys. 61, 926-966 (2013) [arXiv:1309.2977 [hep-th]].
  * [49] D. S. Berman and C. D. A. Blair, “The Geometry, Branes and Applications of Exceptional Field Theory,” Int. J. Mod. Phys. A 35, no.30, 2030014 (2020) [arXiv:2006.09777 [hep-th]].
  * [50] G. Bossard and A. Kleinschmidt, “Loops in exceptional field theory,” JHEP 01, 164 (2016) [arXiv:1510.07859 [hep-th]].
  * [51] B. de Wit and D. Lüst, “BPS amplitudes, helicity supertraces and membranes in M theory,” Phys. Lett. B 477, 299-308 (2000) [arXiv:hep-th/9912225 [hep-th]].
  * [52] D. van de Heisteeg, C. Vafa, M. Wiesner and D. H. Wu, “Species Scale in Diverse Dimensions,” [arXiv:2310.07213 [hep-th]].
  * [53] A. Castellano, A. Herráez and L. E. Ibáñez, “On the Species Scale, Modular Invariance and the Gravitational EFT expansion,” [arXiv:2310.07708 [hep-th]].
  * [54] J. Calderón-Infante, M. Delgado and A. M. Uranga, “Emergence of species scale black hole horizons,” JHEP 01, 003 (2024) [arXiv:2310.04488 [hep-th]].
  * [55] B. Pioline, “R**4 couplings and automorphic unipotent representations,” JHEP 03 (2010), 116 [arXiv:1001.3647 [hep-th]].
  * [56] G. Bossard and B. Pioline, “Exact $\nabla^{4}R^{4}$ couplings and helicity supertraces,” JHEP 01, 050 (2017) [arXiv:1610.06693 [hep-th]].
  * [57] S. Hosono, A. Klemm, S. Theisen and S. T. Yau, “Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces,” Nucl. Phys. B 433, 501-554 (1995) [arXiv:hep-th/9406055 [hep-th]].
  * [58] T. Rudelius, “Gopakumar-Vafa invariants and the Emergent String Conjecture,” JHEP 03, 061 (2024) [arXiv:2309.10024 [hep-th]].
  * [59] R. Gopakumar and C. Vafa, “On the gauge theory / geometry correspondence,” Adv. Theor. Math. Phys. 3, 1415-1443 (1999) [arXiv:hep-th/9811131 [hep-th]].
  * [60] A. Klemm and M. Marino, “Counting BPS states on the enriques Calabi-Yau,” Commun. Math. Phys. 280, 27-76 (2008) [arXiv:hep-th/0512227 [hep-th]].
  * [61] T. W. Grimm, A. Klemm, M. Marino and M. Weiss, “Direct Integration of the Topological String,” JHEP 08, 058 (2007) [arXiv:hep-th/0702187 [hep-th]].
  * [62] M. R. Douglas, D. N. Kabat, P. Pouliot and S. H. Shenker, “D-branes and short distances in string theory,” Nucl. Phys. B 485, 85-127 (1997) [arXiv:hep-th/9608024 [hep-th]].