MPP-2023-192

Demystifying the Emergence Proposal

Ralph Blumenhagen1,2, Niccolò Cribiori1, Aleksandar Gligovic1,2,

Antonia Paraskevopoulou1,3

_ 1 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
Föhringer Ring 6, 80805 München, Germany _

_2 Exzellenzcluster ORIGINS, Boltzmannstr. 2, D-85748 Garching, Germany_

_3 Ludwig-Maximilians-Universität München, Fakultät für Physik,
Theresienstr. 37, 80333 München, Germany_

###### Abstract

We revisit the Emergence Proposal in the vector multiplet moduli space of type
IIA $N=2$ supersymmetric string vacua in four dimensions, for which the string
tree-level prepotential and the string one-loop correction are exactly known
via mirror symmetry. We argue that there exists an exact version of the
proposal, according to which these four-dimensional couplings can be computed
exactly in any asymptotic limit in field space. In such limits, a perturbative
quantum gravity theory emerges, whose fundamental degrees of freedom include
all complete infinite towers of states with typical mass scale not larger than
the species scale. For a decompactification limit, this picture is closely
related to and in fact motivated by the computation of Gopakumar-Vafa
invariants. In addition, in the same limit our results suggest that the
emergent theory will also contain asymptotically tensionless wrapped
$NS5$-branes.

##  1 Introduction

Among the various swampland conjectures that have been proposed in the
endeavor of revealing the underlying principles of quantum gravity, see [1, 2,
3, 4] for reviews, the so-called Emergence Proposal [5, 6, 7, 8] is probably
the least established and understood one. It is closely tied to the swampland
distance conjecture [9] and suggests that terms in the low-energy effective
action emerge at one-loop from integrating out towers of states becoming
exponentially light in asymptotic regions of the moduli space. When following
an effective field theory (EFT) approach111We thank the authors of [6] for
reminding us that this EFT approach was only employed as an intermediate tool
until an ultimate quantum gravity approach will be available., one usually
integrates out those states whose masses are below the species scale [10, 11],
which is considered as the ultraviolet cut-off of quantum gravity.

For decompactification limits, or limits dual to them, it was shown in some
examples [12, 13] that the leading order singular behavior of field metrics
and gauge couplings comes out qualitatively correct by such a computation.
However, a couple of puzzling issues have been reported recently in [14], in a
set-up involving more than one modulus and especially in so-called emergent
string limits [15], where the lightest tower of states includes also a string
becoming tensionless asymptotically (we will refer to it as a string tower).
Here, one encounters first of all an issue with the very definition of the
species scale, since, for emergent string limits, quantum field theory
computations lead to results different from those obtained with black hole
arguments. In the first case, one gets a logarithmic enhancement factor, whose
role is unclear. Besides, even if qualitatively part of the structure of the
four-dimensional effective action does emerge, in the fine print there are a
couple of shortcomings, such as: incorrect (relative) numerical coefficients,
extra logarithmic factors for emergent string limits, and additional terms at
subleading order with no clear interpretation. We consider this as evidence
that the precise formulation of the Emergence Proposal in quantum gravity is
still unclear.

In this paper, we take a critical look at the Emergence Proposal and, in the
course of our analysis, we develop a general picture leading to exact
emergence of all terms in the effective action. From this perspective,
previous results using the field theory approach to the Emergence Proposal
constitute only an incomplete part of this more sophisticated computation
which leads to exact emergence, going beyond field theoretic reasoning and
involving the full quantum theory of gravity. This is also close in spirit to
the recent discussion in [16].

Crucially, this step is achieved by summing over all infinite states in the
towers and not only up to those states whose masses are below the species
scale. While this intuition is behind the proposal in [1, 6], it has not been
pursued quantitatively so far. The towers to be considered are those with a
_typical mass scale_ parametrically not larger than the species scale. In a
weakly coupled regime where also the quantization scheme of the theory is
known, such typical mass scale is simply the mass gap between consecutive
levels of the tower. At strong (string) coupling, where the quantization is
usually not known and the former weakly coupled states might migrate across
the mass spectrum or even become unstable, we still assume that the naive mass
scale, given for instance by the tension of a string, is a good measure for
the typical mass scale of the tower. This is certainly true for BPS states,
whose weakly coupled mass can be trusted also for strong coupling. The
situation becomes clearer in a weakly coupled dual description, whenever it
exists.

For concreteness, we work in a moduli space with eight preserved supercharges
whose asymptotic structure is well-understood, namely the vector multiplet
moduli space of type IIA string theory on a Calabi-Yau threefold. Infinite
distance limits in this setup have been studied and classified in [8, 15]. In
this paper, we will consider two kinds of such limits, namely
decompactification and emergent string limits, which are believed to be the
only two possibilities according to the emergent string conjecture [15].
Moreover, via mirror symmetry the exact form of a large portion of the $N=2$
low-energy effective action is known. Our focus here will be on the string
tree-level prepotential $\mathcal{F}_{0}$ and the string one-loop correction
$\mathcal{F}_{1}$, which are closely related to amplitudes of the associated
topological string.

Our strategy is the following. We consider an asymptotic limit in the Kähler
moduli space, in which a certain Kähler modulus $t$, or more concretely its
vacuum expectation value $t_{0}=\langle t\rangle$, is sent to infinity. This
limit is taken such that the four-dimensional dilaton residing in a hyper
multiplet stays fixed. This guarantees that the four-dimensional Planck-scale
does not diverge, so that we can meaningfully talk about (quantum) gravity in
four dimensions. We propose that in each of these limits a perturbative
quantum gravity theory emerges with a characteristic set of fundamental
degrees of freedom. They consist of the complete and infinite towers of states
whose typical mass scale is not larger than the species scale, $\Delta
m\leq\tilde{\Lambda}$. Note that it is not the mass of each state itself that
is required to be not larger than the species scale, but the typical mass
scale of each tower, which then contributes infinitely many individual states.
This prescription is in accordance with [1] (see footnote 46),[17] and with
the more recent work [18]. In addition, we propose that this emergent theory
admits a perturbative-like expansion in the naturally small parameter
$g_{E}\simeq t_{0}^{-1}\simeq N_{\rm sp}^{-1}$, where $N_{\rm sp}$ is the
number of species related to the species scale as $\tilde{\Lambda}=M_{\rm
pl}/\sqrt{N_{\rm sp}}$, in four dimensions. By this we mean that similar to
standard string theory with $g_{s}=\langle\exp(\phi)\rangle$, the expansion in
the coupling constant of the emergent theory $g_{E}$ will have perturbative
polynomial corrections as well as non-perturbative exponentially suppressed
corrections of the type $\exp(-1/g_{E})$. In general we do not know how to
quantize this emergent theory from first principles but, as we will see, in
certain limits one can understand it via an existing dual fundamental string
theory.

This work is organized as follows. In section 2, we review the main ideas
behind the Emergence Proposal and we recall some properties of the vector
multiplet moduli space of type IIA string theory compactified on Calabi-Yau
threefolds.

In section 3, we consider a decompactification limit with the four-dimensional
dilaton remaining constant. In this case, while $D0$-branes are certainly the
lightest degrees of freedom, one finds that towers of BPS bound states of
$D0$-branes, wrapped $D2$-branes and wrapped $NS5$-branes, carrying Kaluza-
Klein momentum along the Calabi-Yau threefold have a typical mass scale not
larger than the species scale. We do not know how to quantize this theory in
general, but luckily, thanks to the seminal work of Gopakumar-Vafa [19, 20],
certain topological quantities can be obtained via Schwinger one-loop
integrals.222Very recently, Gopakumar-Vafa invariants were also considered in
the context of the emergent string conjecture in [21]. Hence, we can setup an
emergence computation of the prepotential and the genus-one free energy. This
is performed via one-loop diagrams with the aforementioned BPS bound states
running in the loop. By summing over all the full infinite towers of such
states, we are able to recover the exact type IIA expressions, at both genus
zero and one, if we consistently treat the (divergent) infinite sums via
$\zeta$-function regularization. This constitutes the first fully fledged
realization of the Emergence Proposal in quantum gravity. We show how in an
EFT approach, cutting-off the sums at some finite level can only recover the
leading order behavior, but it crucially misses the correct normalization and
subleading corrections.

In section 4, we comment on an emergent string limit, which requires a $K3$
fibration over a base $\mathbb{P}^{1}$, whose volume is sent to infinity. The
light towers include an emergent four-dimensional string arising from an
$NS5$-brane wrapped on the $K3$ fiber and a couple of emergent particle-like
states arsing from wrapped type IIA $D$-branes. We do not know how to quantize
this four-dimensional string theory, but we can still understand it in terms
of its dual, namely the heterotic string on $K3\times T^{2}$. On the
quantitative level, since this emergent theory is just another perturbative
description of the same theory, the tree-level type IIA prepotential must also
be exactly realized in a perturbative expansion in $g\simeq t_{0}^{-1}\simeq
g_{H}$. All this is indeed the familiar story of a string-string duality,
where certainly no cut on the tower of states is imposed and the ultraviolet
cut-off is just the string scale. We realize that in this case the Schwinger
integral gives (only) the one-loop correction of the heterotic dual. This
means that the full prepotential is _not emerging_ (in the original sense of
[1, 6], which includes the tree level term) via integrating out towers of
states up to the species (i.e. string) scale. Whether an increase of the
latter ultraviolet cut-off cures this problem of the Emergence Proposal
remains to be seen.333We thank E. Palti for pointing out this idea to us.
Here, we are not following this approach but, more conservatively, we always
consider the species scale as the quantum gravity cut-off. Our notion of
_exact emergence_ simply means that all terms in the prepotential will arise
via the correct quantization of the perturbative quantum gravity theory that
emerges in the considered asymptotic field limit. In section 5, we draw our
conclusions and point out future directions of research.

In summary, as depicted in figure 1, the type IIA vector multiplet moduli
space in four dimensions features a pattern very similar to the familiar
duality star in ten/eleven dimensions.

Figure 1: Cartoon of type IIA vector multiplet moduli space and asymptotic
limits.

In the figure, we picture the difference between the mass spectrum in the
asymptotic limits and the mass spectrum in the bulk region, which has been
called the desert in [22]. For the latter, there are only a few light states,
while the majority of the towers and the species scale are all close to the
Planck scale. The decompactification limit sheds some new light on a
perturbative description of the still mysterious eleven dimensional M-theory,
which will be reported in [23].

##  2 Preliminaries

In this section, we provide a brief review of the original Emergence Proposal
and of some salient features of the $N=2$ vector multiplet moduli space of
type IIA string compactifications on Calabi-Yau threefolds.

###  2.1 The Emergence Proposal

In view of the swampland distance conjecture[9], which predicts infinite
towers of states becoming exponentially light at asymptotic distances in field
space, the Emergence Proposal [5, 6, 7, 8] offers a field-theoretic approach
to the effects that such towers have on certain physical quantities. In fact,
at present two possible ways to interpret the concept of emergence in the
context of quantum gravity have been put forward. According to [13], the
strong version of the Emergence Proposal [1] states the following.

  * •

Emergence Proposal (Strong): In a theory of quantum gravity all light
particles in a perturbative regime have no kinetic terms in the ultraviolet.
The required kinetic terms appear as an infrared effect due to loop
corrections involving the sum over a tower of massless states.

Clearly, this is a very non-trivial statement which goes beyond our usual
understanding of string theory. As formulated above, it is not clear if only
the kinetic terms of the moduli approaching asymptotic regions are supposed to
emerge, or if the full field metric should be recovered, as claimed in [1].

There exists a milder formulation, called the weak version of the Emergence
Proposal [13].

  * •

Emergence Proposal (Weak): In a consistent effective theory of quantum
gravity, for any singularity at infinite distance in the moduli space, there
is an associated infinite tower of states becoming light and inducing quantum
corrections to the metrics which match the ‘tree level’ singular behavior.

This statement is more specific and only refers to the components of the field
metric which diverge in the asymptotic limit. In [14], the possibility to
extract information for the non-singular components was also explored.

The towers mentioned in both versions of the Emergence Proposal are typically
infinite, as predicted by the swampland distance conjecture. However, the
states that are usually integrated out in an EFT setup are only those lying
below the ultraviolet quantum gravity cut-off. Naively, one would expect the
latter to be the four-dimensional Planck mass $M_{\rm pl}$, but it has been
proposed [10, 11] (see also [24] for earlier work) that the gravitational cut-
off should be lowered down to the species scale $\tilde{\Lambda}$. In four
dimensions, this is given by

$\tilde{\Lambda}=\frac{M_{\rm pl}}{\sqrt{N_{\rm sp}}}\,.$ (2.1)

The interpretation of $N_{\rm sp}$ may change if one derives (2.1) by using a
quantum field theory calculation or rather black hole arguments. In the first
case, the species scale is the energy at which the one-loop correction to the
graviton propagator, due to the coupling to $N_{\rm sp}$ light states, becomes
of the same order as the tree level term. Hence, $N_{\rm sp}$ is the number of
light states with masses lying below the cut-off $\tilde{\Lambda}$. In the
second case, $\tilde{\Lambda}$ is the inverse of the radius of the smallest
possible black hole which can be described by the effective theory. Hence,
$N_{\rm sp}$ is the entropy of such a black hole.

A particular feature, which will be relevant to us, is that in the setup we
are interested in $N_{\rm sp}$ has been related to the genus-one free energy
of the topological string. This has been proposed in [25] and investigated
further in [26, 27, 28, 29]. Besides, the interpretation of $N_{\rm sp}$ as
entropy has been developed into a more complete thermodynamic picture in [30].

While in principle we have two ways at our disposal to estimate the species
scale, the Emergence Proposal in its standard formulation can be compatible
only with a quantum field theory setup. Recently, the application of the
Emergence Proposal to moduli space metrics and gauge kinetic functions has
been studied in detail in [12, 13, 31]. From these investigations, one learns
that e.g. in four dimensions a gauge coupling $g$ emerges at one loop from
integrating out a tower of charged states with mass $m_{n}=n\Delta m$ and
charge $q_{n}=nq$, namely

$\frac{1}{g^{2}}\sim\sum_{n=1}^{N_{\rm
sp}}n^{2}q^{2}\log\left({\frac{n\,\Delta m}{\tilde{\Lambda}}}\right)\,.$ (2.2)

Notice that in an EFT approach the sum stops at $n=N_{\rm sp}$, when indeed
the mass of the state is $N_{\rm sp}\Delta m=\tilde{\Lambda}$. States in the
tower with mass larger than $\tilde{\Lambda}$ are thus not included. These
loop corrections have been evaluated for a couple of asymptotic limits,
confirming the weak version of the Emergence Proposal.

A step further has been performed in the recent work [14], where the Emergence
Proposal has been tested in an emergent string limit within a toroidal
orbifold model with multiple moduli. This allowed to compute also the one-loop
corrections for the non-asymptotic field directions and to compare them to
their known tree-level values. As a result, it has been pointed out that the
quantum field theory approach to the species scale seems to be always plagued
by multiplicative logarithmic corrections when string towers are included.
This is inconsistent with the fact that the genus-one topological string free
energy receives additive logarithmic corrections. Moreover, while the one-loop
diagrams reproduce the leading order functional form of the gauge couplings
and of the moduli metric correctly, the prefactors do not match and also the
subleading terms are not related to known string amplitudes. All of these
issues suggest that a complete and precise understanding of the Emergence
Proposal is still missing in the literature. This is the main motivation
behind our investigation.

###  2.2 Kähler moduli space in type IIA and limits

The setup we work in is the Kähler moduli space of type IIA string theory
compactified on a Calabi-Yau threefold. We review some of its properties
following mainly [32, 33].

The moduli space of type II string theory compactified on a Calabi-Yau
threefold is locally a product of two factors,

$\mathcal{M}_{V}\times\mathcal{M}_{H}\,,$ (2.3)

which do not mix due to supersymmetry. The space $\mathcal{M}_{V}$ is special
Kähler and parametrized locally by scalar fields in vector multiplets. The
space $\mathcal{M}_{H}$ is quaternionic and parametrized locally by scalar
fields in hypermultiplets. We concentrate only on the manifold
$\mathcal{M}_{V}$ associated to type IIA compactifications, for which scalar
fields in vector multiplets are complexified Kähler moduli. We denote them as
$T^{i}=t^{i}+ib^{i}$, with $i=1,\dots,h_{11}(\text{CY}_{3})$, 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 $h_{11}(\text{CY}_{3})$ homologically two-cycles.
There is one more vector field, the graviphoton, residing in the $N=2$
supergravity multiplet. This is simply given by the type IIA RR $C_{1}$-form.

The kinetic terms for the Kähler moduli and the gauge couplings are determined
by a holomorphic, homogeneous of degree two prepotential $F=F(X)$. It is
customary to introduce symplectic coordinates $X^{\Lambda}=(X^{0},X^{i})$,
such that local coordinates on $\mathcal{M}_{V}$ are recovered as
$T^{i}=-iX^{i}/X^{0}$. In our conventions, the Kähler potential is then given
by
$K=-\log\left[i\left(\overline{X}^{\Lambda}F_{\Lambda}-X^{\Lambda}\overline{F}_{\Lambda}\right)\right]$.
Apart from being constrained by supersymmetry, the form of $F(X)$ in type II
Calabi-Yau compactifications is restricted by discrete Peccei-Quinn
symmetries, which act on $T^{i}$ and are almost exact for $t^{i}\gg 1$, and
also by non-renormalization theorems. Since in these vacua the four-
dimensional dilaton resides in a hypermultiplet, the prepotential $F(X)$ is
exact at perturbative string tree level and loop corrections can only affect
$\mathcal{M}_{H}$. Nevertheless, there are non-perturbative corrections from
worldsheet instantons, such that one can write schematically

$F(X)=F_{\rm pert}(X)+F_{\rm np}(X)\,.$ (2.4)

At large volume, one has [34, 35]

$\displaystyle F_{\rm pert}(X)$
$\displaystyle=i(X^{0})^{2}\left(\frac{1}{3!}C_{ijk}T^{i}T^{j}T^{k}+\frac{\zeta(3)}{2}\chi(\text{CY}_{3})\right),$
(2.5) $\displaystyle F_{\rm np}(X)$
$\displaystyle=-i(X^{0})^{2}\sum_{\vec{\beta}\in
H_{2}(\rm{CY_{3}},\mathbb{Z})}\alpha^{\vec{\beta}}_{0}\,\,\,{\rm
Li}_{3}\left(e^{-\vec{\beta}\cdot\vec{T}}\right),$ (2.6)

where $\chi(\text{CY}_{3})$ is the Euler characteristic of the Calabi-Yau
threefold, while $\alpha^{\vec{\beta}}_{0}$ are genus-zero Gopakumar-Vafa
invariants [19, 20]. The latter count the number of BPS configurations from
the $D2$-branes wrapping a (genus-zero) curve in the class $\vec{\beta}\in
H_{2}(\rm{CY_{3}})$. That the terms above arise at string tree-level, i.e. at
genus zero, can be seen after the gauge fixing $|X^{0}|\sim M_{\rm pl}\sim
1/g_{s}$ in string units, where $g_{s}$ is the string coupling. Notice that in
$F_{pert}(X)$ we are omitting those linear and quadratic terms in $T^{i}$ that
can be absorbed by a symplectic transformation [36, 37].

So far, we have discussed (non-)perturbative corrections in the string
coupling constant to a two-derivative four dimensional $N=2$ supergravity
action. It is known that the Wilsonian effective action of type IIA Calabi-Yau
compacifications contains also a whole series of higher derivative
corrections. They can be encoded into a holomorphic function

$\mathcal{F}=\sum_{g=0}^{\infty}\mathcal{F}_{g}(X)\,\mathcal{W}^{2g}\,,$ (2.7)

where $\mathcal{W}^{2}$ is the square of the lowest component of the Weyl
multiplet introducing higher derivative terms into the supergravity action
[38].444Notice that in four dimensions a derivative expansion is not an
expansion in $\alpha^{\prime}$, since the moduli also contain
$\alpha^{\prime}$. Due to the decoupling of the dilaton from the vector
multiplet moduli space, there cannot be (non-)perturbative corrections to a
given $\mathcal{F}_{g}$ appearing at order $g^{\prime}>g$. Hence,
$\mathcal{F}$ is for us the complete holomorphic prepotential, while the
object $\mathcal{F}_{0}\equiv F$ corresponds to the two-derivative
(non-)perturbative couplings discussed before. The corrections
$\mathcal{F}_{g}$ are intimately related to topological string amplitudes at
genus $g$. The precise relation is nevertheless subtle and we refer to [39]
for more details. What will be important for us in the following is that
$\mathcal{F}_{g}$ can be calculated by counting BPS states as originally
proposed in the seminal work of Gopakumar and Vafa [19, 40, 20].

The holomorphic terms in (2.7) contribute to higher derivative interactions of
the type $\mathbb{F}_{g}(T,\overline{T})R_{+}^{2}F_{+}^{2g-2}$, where $R_{+}$
and $F_{+}$ are respectively 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

$\mathbb{F}_{g}(T,\overline{T})={\rm
Re}(\mathcal{F}_{g}(T))+f_{g}^{anom}(T,\overline{T})\,,$ (2.8)

with $\partial_{i}\partial_{\bar{\jmath}}f_{g}^{anom}(T,\overline{T})\neq 0$.
Of particular interest for us is the term with $g=1$, which governs the
coefficient of an $R^{2}$-correction [41], arising from an $R^{4}$-term in
eleven dimensions [42, 43]. Indeed, such term a gives an estimate for the
number of species

$\mathbb{F}_{1}\simeq N_{\rm sp}\,,$ (2.9)

as recently proposed in [25] and subsequently developed in [26, 27, 28, 30,
29].

The moduli-dependent expression of the number of species thus obtained is
valid over the entire vector multiplet moduli space. We propose and we will
make concrete in examples that the quantity

$g_{E}\simeq\frac{1}{\mathbb{F}_{1}}\simeq\frac{1}{N_{\rm sp}}$ (2.10)

naturally provides a small parameter in which to organize pertubatively the
various quantum gravity theories emerging in the asymptotic regions of the
moduli space. In other words, the coupling of the emergent theory, $g_{E}$,
will serve as a compass in our analysis.

#### The resolved conifold

A setup that will be highly illustrative for our purposes is the resolved
conifold, which has been analysed in [40] by developing a duality to Chern-
Simons theory already pointed out in [19]. In this case, we have only a single
2-cycle of topology $S^{2}$ so that the holomorphic prepotential takes a very
simple form

$\displaystyle\mathcal{F}_{0}$
$\displaystyle=-\frac{1}{g_{s}^{2}}\left(\zeta(3)-\frac{\pi^{2}}{6}T-i\left(m+\frac{1}{4}\right)\pi
T^{2}+\frac{T^{3}}{12}-{\rm Li}_{3}(e^{-T})\right)$ (2.11)
$\displaystyle\mathcal{F}_{1}$ $\displaystyle=\frac{T}{24}-\frac{1}{12}{\rm
Li}_{1}(e^{-T})=\frac{T}{24}-\frac{1}{12}\sum_{m\geq 1}\frac{1}{m}e^{-mT}\,,$
(2.12)

where $T$ is the complexified Kähler parameter of the $S^{2}$. For $g>1$, one
gets

$\mathcal{F}_{g}=g_{s}^{2g-2}\left((-1)^{g}\chi_{g}\frac{2\zeta(2g-2)}{(2\pi)^{2g-2}}+\frac{\chi_{g}}{(2g-3)!}{\rm
Li}_{3-2g}(e^{-T})\right)\,,$ (2.13)

where $\chi_{g}$ is the Euler characteristic of the moduli space of all
Riemann surfaces with genus $g$ and $h$ punctures. While the integer $m$ in
(2.12) is naively related to $h$ as $m=\frac{h}{2}-1$, it is actually not
uniquely fixed.

##  3 Decompactification limit

We begin by examining decompactification limits, the analysis of which will
prove to be richer than expected. First, we identify the towers of states
becoming light in this limit. Then, we employ a Schwinger computation to show
how the prepotential emerges by integrating these states out. Crucially, to
recover the known expression, we will need to include all those entire towers
with a typical mass scale not larger than the species scale. Finally, we
compare and contrast our methodology with the simplified EFT approach to the
Emergence Proposal, where one usually integrates out only those states with
masses up to the species scale.

### Towers of light states

For the sake of our analysis, it is sufficient to consider the standard
example used to motivate the Emergence Proposal, namely a Calabi-Yau manifold,
$Y$, with a single complexified Kähler modulus, $T=t+ib$. A similar analysis
can be performed for the more generic case with an arbitrary number of Kähler
moduli. The infinite distance limit we are interested in is $t\to\lambda t$
and $\lambda\to\infty$.555One can naturally think of $\lambda$ as the ratio
between the final and the initial value of the vacuum expectation value
$t_{0}$ along the trajectory over the moduli space. Besides, here and in the
following, when writing $t$ we actually refer to $t_{0}$ and omit the
subscript for convenience. This is accompanied by a rescaling of the string
coupling, $g_{s}\to\lambda^{3/2}g_{s}$, in such a way that the four-
dimensional dilaton, $\sigma\simeq M_{\rm
pl}/M_{s}\simeq\mathcal{V}^{1/2}/g_{s}$, remains constant. Hence, we are
dealing with strong (string) coupling limit. Here and in the following,
$M_{s}$ is the string mass while $\mathcal{V}$ is the volume of the Calabi-Yau
threefold in units of the string length, which in this example is
$\mathcal{V}\simeq t^{3}$. Note that from the dual perspective of M-theory
compactified on $Y\times S^{1}$ this limit is precisely the decompactification
limit of the $S^{1}$, i.e. $R_{11}\to\lambda R_{11}$. The question is which
states have a typical mass scale not larger than the quantum gravity cut-off
in this decompactification limit.

The lightest degrees of freedom in the spectrum are particle-like and given by
the tower of BPS bound states of $D0$-branes with mass

$m^{D0}_{n}=n(\Delta m)_{D0}\,,\qquad(\Delta m)_{D0}=\frac{M_{\rm pl}}{{\cal
V}^{\frac{1}{2}}}\,.$ (3.1)

Since these are BPS states, we can trust their mass also for large $g_{s}$, so
that to this tower of states we associate the typical mass scale

${\mathfrak{m}}_{D0}=\frac{M_{\rm pl}}{{\cal
V}^{\frac{1}{2}}}\simeq\frac{M_{\rm pl}}{\lambda^{\frac{3}{2}}}\,.$ (3.2)

By looking at the $R^{2}$-correction in the spacetime effective action, we can
read off the species scale. In the limit of interest, we have
$\mathbb{F}_{1}\simeq N_{\rm sp}\simeq t\simeq{\lambda}$ and thus [25]

$\tilde{\Lambda}=\frac{M_{\rm pl}}{{\cal V}^{\frac{1}{6}}}\simeq\frac{M_{\rm
pl}}{\lambda^{\frac{1}{2}}}\,.$ (3.3)

Alternatively, one can obtain the same result by recalling that for a
decompactification from four to five dimensions the species scale is given by
$\tilde{\Lambda}=M_{\rm KK}^{1/3}M_{\rm pl}^{2/3}$, and then by substituting
$M_{\rm KK}$ with $(\Delta m)_{D0}$. Due to the scaling of $g_{s}$, we can see
that $\lambda\to\infty$ is a strong coupling limit in the type IIA frame.
However, for the emergent theory we have in mind a perturbative expansion in
the small parameter $g_{E}\simeq 1/N_{\rm sp}\simeq\lambda^{-1}$.

Since $\tilde{\Lambda}$ is parametrically larger than $\mathfrak{m}_{D0}$,
there is room for additional degrees of freedom to become light. In this
respect, consider a $D2$-brane wrapped around a 2-cycle in
$H_{2}(Y,\mathbb{Z})$. We get towers of particle-like states with typical mass
scale

${\mathfrak{m}}_{D2}=\frac{M_{\rm pl}}{{\cal
V}^{\frac{1}{2}}}t\simeq\frac{M_{\rm pl}}{\lambda^{\frac{1}{2}}}\,,$ (3.4)

which is of the same order as the species scale. Since there will be only a
few such states below the species scale, the latter will not change
parametrically by their inclusion. We note that $D0$\- and $D2$-branes are the
objects which are electrically charged under the graviphoton and the gauge
fields residing in the $N=2$ vector multiplets, respectively.

However, this is not the end of the story, as there are more towers of states
whose mass spacing is parametrically not larger than the species scale. First,
we expect to have towers of Kaluza-Klein modes along the Calabi-Yau threefold,
whose mass spacing and typical mass scale is

${\mathfrak{m}}_{\rm KK}\simeq\frac{M_{\rm pl}}{{\cal
V}^{\frac{1}{6}}\sigma}\simeq\frac{M_{\rm
pl}}{\lambda^{\frac{1}{2}}\sigma}\,.$ (3.5)

For constant $\sigma$, this is parametrically of the same order as the species
scale.

These are all the light particle-like towers that we found. However, we can
also have BPS $NS5$-branes wrapped on a 4-cycle $\Sigma$ in
$H_{4}(Y,\mathbb{Z})$. They give rise to $h_{11}(Y)$ emergent strings in four-
dimensions [44] with tension

$T_{\rm str}\sim\frac{M_{s}^{2}}{g_{s}^{2}}{\rm vol}(\Sigma)\sim M_{\rm
pl}^{2}\frac{{\rm vol}(\Sigma)}{{\cal V}}\sim\frac{M_{\rm
pl}^{2}}{\lambda}\,,$ (3.6)

where ${\rm vol}(\Sigma)$ denotes the volume of the four-cycle $\Sigma$ in
units of the string length. We do not know what the quantization of this
effective string in four dimensions and at strong string coupling $g_{s}$ is,
but the natural scale for its excitations is certainly

${\mathfrak{m}}_{\rm str}\simeq T_{\rm str}^{\frac{1}{2}}\simeq\frac{M_{\rm
pl}}{\lambda^{\frac{1}{2}}}\,.$ (3.7)

In case the four-cycle is the $K3$ fiber of a $K3$ fibered Calabi-Yau
threefold, the wrapped $NS5$-brane is dual to a heterotic string at weak
string coupling (see section 4). Via duality, its quantization leads indeed to
this mass spacing for its string excitations. In the generic case, we expect
that the spectrum will likely not be equidistant, but the mass levels will
fluctuate around this typical value. We note that we arrive at the same scale
by just taking the sixth route of the $NS5$-brane tension

${\mathfrak{m}}_{NS5}\simeq T_{NS5}^{\frac{1}{6}}\simeq\frac{M_{\rm
pl}\,g_{s}^{\frac{2}{3}}}{{\cal V}^{\frac{1}{2}}}\simeq\frac{M_{\rm
pl}}{\lambda^{\frac{1}{2}}}\,.$ (3.8)

Hence, in contrast to what we will find in section 4 for an emergent string
limit, in the decompactification limit one should better not speak of an
emergent string but of an emergent $NS5$-brane.666In [23] we will argue that
the four-dimensional theory discussed here can be understood as a
compactification of an emergent perturbative M-theory on a Calabi-Yau
threefold. If this reasoning is correct, then the wrapped $NS5$-branes can
also contribute BPS excitations with mass at the species scale.

One can then check that all other BPS modes, like wrapped $D4$-branes or the
type IIA perturbative string states, are heavier than the species scale.
Indeed, in terms of $g_{E}\simeq 1/N_{sp}\simeq 1/\lambda$ the typical mass
scale in their towers is

${\mathfrak{m}}_{D4}\simeq\frac{\tilde{\Lambda}}{g_{E}}\,,\qquad{\mathfrak{m}}_{F1}\simeq\frac{\tilde{\Lambda}}{g^{1/2}_{E}}\,,$
(3.9)

and they are non-perturbative in the regime $g_{E}\to 0$.

To summarize, in this decompactification limit we propose to find a new
perturbative quantum gravity theory, whose fundamental degrees of freedom are
given by (towers of) bound states of $D0$-branes, wrapped $D2$-branes and
wrapped $NS5$-branes, possibly carrying quantized Kaluza-Klein momentum along
the Calabi-Yau threefold. Since the wrapped $NS5$-branes give rise to
$h_{11}(Y)$ strings in four-dimensions, this is not just a theory of
particles. From the dual perspective of M-theory on $Y\times S^{1}$, the
corresponding light degrees of freedom are bound states of transversal (to
$S^{1}$) $M2$-branes and $M5$-branes carrying Kaluza-Klein momentum along the
$S^{1}$ and possibly along $Y$. This will be further elaborated upon in [23].

We do not know what this new perturbative quantum gravity theory really is and
how to quantize it. In the following, due to its relation to M-theory, we will
call it $M_{\rm CY}$. Following the strategy outlined in the introduction, for
our proposal of a new quantum gravity theory emerging at each infinite
distance limit to make sense, we should be able to obtain the prepotential
terms $\mathcal{F}_{g}$ directly within $M_{\rm CY}$. Furthermore, the result
should be organized in a perturbative expansion in the small parameter
$g_{E}\simeq\lambda^{-1}$. If this turns out to be possible, then it would
represent a non-trivial check for our claim and it will shed light on the role
that the Emergence Proposal plays in this context.

###  3.1 The Schwinger Computation

In the spirit of the Emergence Proposal, we will now make an attempt to
recover the prepotential terms $\mathcal{F}_{g}$ from a one-loop computation
within $M_{\rm CY}$. The states to be integrated out have been identified in
the previous section: they are bound states of $D0$-branes, wrapped
$D2$-branes, excitations of wrapped $NS5$-branes, carrying Kaluza-Klein
momentum along the Calabi-Yau threefold. The problem is well-posed, but the
tools to address it in general are lacking at present. However, for the
holomorphic quantities of interest there is a simplification, for we only have
to consider BPS states running in the loop. Indeed, in the context of the
topological string, the derivation of amplitudes at genus $g$ from $D2$-$D0$
BPS bound states has been carried out (from the dual M-theory perspective) in
the seminal work of Gopakumar and Vafa [19, 20]. In this sense, the Gopakumar-
Vafa idea is in fact an example of emergence and it is closely related to what
we are aiming at.

From [19, 20] we recall that the full genus expansion of the topological
amplitudes can be resummed by Schwinger-type integrals as

$\sum_{g}\mathcal{F}_{g}{W}^{2g}=\sum_{\vec{\beta},r,n}\alpha_{r}^{\vec{\beta}}\int_{0}^{\infty}\frac{ds}{s}\frac{W^{2}}{\left(2i\sinh\left(\frac{sW}{2}\right)\right)^{2-2r}}e^{-sZ_{\vec{\beta},n}}.$
(3.10)

Here, $\alpha_{r}^{\vec{\beta}}$ are genus $r\geq 0$ Gopakumar-Vafa
invariants, $s$ the Schwinger parameter,
$W=\langle\sqrt{\mathcal{W}^{2}}\rangle$ the vacuum expectation value of the
graviphoton field strength and

$Z_{\vec{\beta},n}={\frac{M_{s}}{g_{s}}}\left(\vec{\beta}\cdot\vec{T}+in\right)$
(3.11)

is the mass (equal to the central charge) of the BPS state with contributions
from the wrapping class $\vec{\beta}\in H_{2}(Y,\mathbb{Z})$ and from $n$
units of Kaluza-Klein momentum along the M-theory circle. In type IIA
language, $Z_{\vec{\beta},n}$ is the mass of $D2$-branes wrapping
$\vec{\beta}$ and bound to $n$ $D0$-branes; they arise from dimensional
reduction of wrapped $M2$-branes on the circle. A few comments on formula
(3.10) are in order. First, the factor $\sinh(sW/2)$ encodes the effect of a
non-vanishing background field $W$, namely the graviphoton background. Second,
to each $\mathcal{F}_{g}$ on the left hand side contribute only terms on the
right hand side such that $g\geq r$. The index $r$ can be understood from the
five-dimensional perspective as labelling a massive BPS state with
SO$(4)$-spin content $I_{1}\otimes\sum_{r}\alpha_{r}^{\vec{\beta}}I_{r}$,
where $I_{1}=2[0]\oplus[1/2]$ and $I_{r}=(I_{1})^{r}$. The simplest
configuration of bound states of $D2$-$D0$-branes correspond to $r=0$, in
which case the $D2$-brane is wrapping an $S^{2}$ [19].

If we want to employ these ideas, we face the problem that in the original
work [19, 20] there is no explicit mentioning of BPS bound states involving
also wrapped $NS5$-branes and Kaluza-Klein modes along the Calabi-Yau.
However, our previous swampland-based analysis suggests that their
contribution is of the same mass scale as the pure BPS $D2$-$D0$ bound states,
and thus we would expect them to contribute as well. We propose that this
issue can be resolved by noting that the wrapped $NS5$-branes and the Kaluza-
Klein modes are uncharged under the vector multiplets, which in type IIA arise
from the RR sector. As a consequence, these states do not contribute to the
central charge, which will still be given by the expression (3.11). Besides,
since the bound states remain BPS also when the $NS5$-branes and the Kaluza-
Klein modes are included, their mass will coincide again with
$Z_{\vec{\beta},n}$. In short, considering more general bound states with also
$NS5$-branes and Kaluza-Klein modes does not change the functional dependence
of ${\cal F}_{g}$, but only contributes to the number of BPS states, i.e. the
Gopakumar-Vafa invariants $\alpha_{r}^{\vec{\beta}}$. Formally these are
infinitely many states, so that a regularization has to be performed. However,
like for string theory, a proper (so far unknown) quantization of the
$NS5$-brane will presumably take care of it. Since, to our knowledge, the
Gopakumar-Vafa invariants have not yet been computed from first principles in
type II string theory by literally counting BPS states on a Calabi-Yau, our
swampland-based arguments suggest that these integers will also involve
contributions from BPS states involving wrapped $NS5$-branes and potentially
Kaluza-Klein modes along the Calabi-Yau.777Recently, impressive progress has
been accomplished in computing Gopakumar-Vafa invariants at genus zero [45,
46], extending on [47, 35]. Since the employed methodology exploits mirror
symmetry to extract the invariants from the periods of the mirror Calabi-Yau,
we would call this an indirect computation, in contrast to actually counting
$D2$-$D0$ BPS bound states. In view of this, we believe it to be possible that
the Gopakumar-Vafa invariants determined in this indirect way include
contributions from $NS5$-branes and Kaluza-Klein states as well. Additionally,
a formula to extract directly the degeneracy of states (whose weighted sum
enters the Gopakumar-Vafa invariants) on K3 fibered Calabi-Yau threefolds has
been conjectured in [48] (building on [49]) and tested recently in [50].

The above story is expected to be similar to the pattern of perturbative BPS
states of the heterotic string on $K3\times T^{2}$. In this case, the central
charge of the supersymmetry algebra is given by the right moving Kaluza-Klein
momentum $p_{R}$ along $T^{2}$ and the mass spectrum can be expressed as [37]

$\alpha^{\prime}M^{2}={\frac{1}{2}}p_{R}^{2}+N_{R}-\frac{1}{2}\,.$ (3.12)

Therefore, BPS states arise from right-moving ground states,
$N_{R}=\frac{1}{2}$, but can carry unrestricted left-moving excitations
$N_{L}$. This heterotic picture is precisely dual to what happens in the
emergent string limit to be discussed in section 4. For completeness, let us
mention that the heterotic/IIA duality has been exploited in the seminal work
[51] to compute topological amplitudes and to match the result with curve-
counting formulae.

###  3.2 Exact emergence on the resolved conifold

For the purpose of our analysis, it is sufficient to evaluate the Schwinger
integrals for a simple example, namely the (non-compact) resolved conifold,
which has a single 2-cycle of genus zero, i.e. a rigid $S^{2}$. In this
example, there is no compact 4-cycle so that there is no $NS5$-brane
contribution. Notice that a 1/2-BPS $D2$-$D0$ bound state wrapping the $S^{2}$
cannot have Kaluza-Klein momentum along it. This is the curved version of the
familiar result that 1/2-BPS $D$-brane boundary states carry only momentum
along transverse directions while winding in the longitudinal
directions.888For the perturbative type II theories, this is also the case for
Kaluza-Klein and winding modes of the fundamental string. Denoting with $p$
and $w$ the momentum and winding numbers respectively, only the configurations
$(p,w)=(p,0)$ or $(p,w)=(0,w)$ are 1/2-BPS. Therefore, only bound states of
$D0$-branes admit discrete Kaluza-Klein modes.

Hence, let us consider a bound state of a $D2$-brane wrapping the rigid
2-cycle and $n$ $D0$-branes, as in [19]. Then, the mass spectrum is
$Z_{1,n}\equiv Z_{n}$ with

$Z_{n}=\frac{M_{s}}{\lambda^{3/2}g_{s}}\Big{(}\lambda
t+i(b+n)\Big{)}=\frac{M_{\rm
pl}}{\lambda^{3/2}\mathcal{V}^{1/2}}\Big{(}\lambda t+i(b+n)\Big{)}\,,$ (3.13)

and a pure $D0$-tower is obtained by setting $t=b=0$ in the above expression.
Since the rigid 2-cycle is topologically an $S^{2}$, the sum over $r$ in the
Schwinger integral (3.10) collapses to the value $r=0$ (genus zero) and, after
expanding the denominator, we get999One may use
$\frac{1}{\sinh^{2}x}=-\frac{d}{dx}\coth
x=\sum_{n=0}^{\infty}\frac{2^{2n}(2n-1)}{(2n)!}B_{2n}x^{2n-2}\,,\,0<|x|<\pi\,.$

$\mathcal{F}_{g}=-\frac{(2g-1)B_{2g}}{(2g)!}\sum_{n}\int_{0}^{\infty}ds\,s^{2g-3}e^{-sZ_{n}}\,,$
(3.14)

where the $B_{2g}$ denote Bernoulli numbers. As shown in [19], for $g>1$ the
integral and the sum over all $n\in\mathbb{Z}$ can be carried out explicitly
and turn out to be finite. For $g=0,1$, the objects we are interested in are
at first sight divergent, so that one has to introduce an ultraviolet
regulator, $\epsilon>0$, in the integral. Therefore, we will be dealing with
the quantities

$\mathcal{F}_{1}=-\frac{1}{12}\sum_{n\in\mathbb{Z}}\int_{\epsilon}^{\infty}\frac{ds}{s}\,e^{-sZ_{n}}\,,\qquad\mathcal{F}_{0}=\sum_{n\in\mathbb{Z}}\int_{\epsilon}^{\infty}\frac{ds}{s^{3}}\,e^{-sZ_{n}},$
(3.15)

where we used $B_{0}=1$ and $B_{2}=1/6$.

To proceed, we first perform the integral over $s$, then the sum over $n$. As
for $\mathcal{F}_{1}$, we get

$\mathcal{F}_{1}=\frac{1}{12}\sum_{n\in\mathbb{Z}}\Big{(}\gamma_{E}+\log(\epsilon\,Z_{n})+\mathcal{O}(\epsilon)\Big{)}\,,$
(3.16)

where $\gamma_{E}$ is the Euler-Mascheroni constant. As for $\mathcal{F}_{0}$,
we get

$\mathcal{F}_{0}=\frac{1}{g_{s}^{2}}\sum_{n\in\mathbb{Z}}\left(\frac{1}{2(\epsilon^{\prime})^{2}}-\frac{g_{s}\,Z_{n}}{\epsilon^{\prime}}-\frac{(g_{s}Z_{n})^{2}}{4}\Big{(}2\log\left(\frac{\epsilon^{\prime}Z_{n}}{g_{s}}\right)+2\gamma_{E}-3\Big{)}+\mathcal{O}(\epsilon^{\prime})\right)\,,$
(3.17)

where we defined $\epsilon^{\prime}=g_{s}\,\epsilon$ to display an overall
multiplicative factor $g_{s}^{-2}$, matching with the resolved conifold
(2.11). Next, we carry out the sum over the infinite number of $D0$-branes.
Since it constitutes the main new quantitative result of the paper and it is
central to our whole argument, we show the computation in some detail.

#### Evaluating $\mathcal{F}_{1}$

Since it is simpler, we start from $\mathcal{F}_{1}$. We substitute the
central charge (3.13) into (3.16) and keep only the terms which are non-
vanishing in the limit $\epsilon\to 0$, giving

$\mathcal{F}_{1}^{D0,D2}=\frac{1}{12}\sum_{n\in\mathbb{Z}}\log\left(\frac{T+in}{\mu}\right)\,$
(3.18)

with $\mu=e^{-\gamma_{E}}{\cal V}^{\frac{1}{2}}/(\epsilon M_{\rm pl})$. We
omit the factors $\lambda$ in the calculation and re-introduce them in the
final result. Then, the sum can be decomposed as

$\mathcal{F}_{1}^{D0,D2}=\frac{1}{12}\bigg{(}\log\left(T\right)+\sum_{n\geq
1}\log\left(1+\frac{T^{2}}{n^{2}}\right)+2\sum_{n\geq
1}\log(n)-\log(\mu)\sum_{n\in\mathbb{Z}}1\bigg{)}\,.$ (3.19)

The second term on the right hand side can be evaluated with (A.9), while the
third and the fourth terms are divergent and need to be regularized. Next
comes the crucial step, where instead of cutting these sums off at $n=N_{\rm
sp}$ we keep all of the terms and exploit $\zeta$-function regularization. As
for the third term, we employ

$\sum_{n\geq 1}\log(n)\equiv-\zeta^{\prime}(0)=\frac{1}{2}\log(2\pi)\,,$
(3.20)

while for the fourth and last term we invoke

$\sum_{n\geq 1}1\equiv\zeta(0)=-\frac{1}{2}\,,$ (3.21)

implying that

$\sum_{n\in\mathbb{Z}}1\equiv 1+2\zeta(0)=0\ .$ (3.22)

Hence, thanks to this regularization scheme, the dependence of (3.19) on the
scale $\mu$ drops out, since the last term vanishes. Putting everything
together, we obtain eventually

$\displaystyle\mathcal{F}_{1}^{D0,D2}$ $\displaystyle=\frac{2\pi(\lambda
t+ib)}{24}+\frac{1}{12}\log\Big{(}1-e^{-2\pi(\lambda t+ib)}\Big{)}$ (3.23)
$\displaystyle=\frac{2\pi}{24}\Big{(}\frac{t}{g_{E}}+ib\Big{)}-\frac{1}{12}\sum_{m\geq
1}\frac{1}{m}e^{-2\pi m\left(\frac{t}{g_{E}}+ib\right)}\,,$

where in the last step we recalled the series expansion of the logarithm and
restored the scaling parameter $\lambda\simeq g_{E}^{-1}$.

We have just shown that by summing over the entire infinite tower of BPS
states and treating the divergences via $\zeta$-function regularization one
obtains the exact one-loop topological free energy of the resolved conifold
(2.12) (upon rescaling $T$ by $2\pi$ appropriately to match conventions). In
particular, this includes the term linear in $T$, which is usually obtained
upon dimensional reduction [43, 42, 41, 52]. Instead, we have been able to
reconstruct it from the Schwinger integral using the light degrees of freedom
of the emergent theory $M_{\rm CY}$. To our knowledge, such a term has not
been obtained in this manner before.

To generalize the analysis to a compact Calabi-Yau threefold $Y$, one
considers the limit $t_{i}\to\lambda t_{i}$ with $\lambda\to\infty$ and
$i=1,\ldots,h_{11}(Y)$. With the four-dimensional dilaton kept constant, this
limit is dual to the decompactification limit to five-dimensions of M-theory
on $Y\times S^{1}$. The light degrees of freedom now include $D2$-$D0$ bound
states wrapping all 2-cycles in the homology lattice $H_{2}(Y,\mathbb{Z})$.
Then, one must sum over them in the Schwinger integral so that the result will
be sensitive to all Kähler moduli $t_{i}$. This means that the full
topological amplitude ${\cal F}_{1}$, including the linear terms, will emerge
from a Schwinger one-loop computation. Notice that the concrete evaluation
might require yet another regularization for the sum over the linear terms in
the Kähler moduli. As previously discussed, we expect also contributions from
BPS states involving $NS5$-branes and Kaluza-Klein modes along $Y$.
Determining their finite contributions requires a proper quantization of the
wrapped $NS5$-branes, which has been so far out of reach. Moreover, from the
general Schwinger integral (3.10) one infers that there will be also similar
contributions from $D2$-branes wrapping genus $g=r=1$ one curves in $Y$.

While this will certainly be an interesting future research direction, in
order to proceed with the conifold example we need just to consider the
contribution of a tower of pure $D0$-brane bound states. In this case, the
central charge does not depend on $T$ so that we obtain101010Note that
compared to eq.(3.18) we have introduced by hand an overall relative minus
sign in order to be in agreement with the extra $-\chi(Y)/2$ factor appearing
for pure $D0$ bound states in [19, 40]. We expect this factor to arise from
including BPS states with Kaluza-Klein modes along the $S^{2}$ followed by a
proper regularization.

$\mathcal{F}_{1}^{D0}=-\frac{1}{12}\sum_{n\neq
0}\log\left(\frac{in}{\mu}\right)\,.$ (3.24)

Notice that we excluded $n=0$, as some particle should really run in the loop.
Proceeding with $\zeta$-function regularization and neglecting an irrelevant
imaginary constant depending on the branch of the complex logarithm, we get

$\mathcal{F}_{1}^{D0}=-\frac{1}{12}\log\left(2\pi\mu\right)\,.$ (3.25)

Since $\mu$ contains a factor of ${\cal V}^{\frac{1}{2}}$, it is tempting to
speculate that this non-holomorphic logarithmic term is related to the
holomorphic anomaly of $\mathcal{F}_{1}$. Furthermore, this time the result is
not independent of the regulator $\epsilon$, arising due to the $\log(\mu)$
factor in $\mathcal{F}_{1}^{D0}$, but we can avoid this by subtracting this
term from the final result.

#### Evaluating $\mathcal{F}_{0}$

Next, we look at the prepotential $\mathcal{F}_{0}$. Using (3.22), one can see
that at fixed (small) $\epsilon^{\prime}$ the sum over the first two terms in
the expansion (3.17) vanishes. The remaining terms can then be written
explicitly as

$\mathcal{F}_{0}=-\frac{M_{s}^{2}}{2g_{s}^{2}}\sum_{n\in\mathbb{Z}}(T+in)^{2}\,\log\left(\frac{T+in}{\nu}\right)\,,$
(3.26)

where $\nu=e^{-\gamma_{E}+{\frac{3}{2}}}{\cal V}^{\frac{1}{2}}/(\epsilon
M_{\rm pl})$. We note that an expression of this kind arises also in the EFT
approach to the Emergence Proposal, see e.g. formula (2.2), where for instance
the gauge couplings are computed via the one-loop corrections with the light
tower of states running in the loop. However, in that approach the tower is
cut-off when the mass of the states exceeds the species scale.

To calculate $\mathcal{F}_{0}$, we first notice that the dependence on $\nu$
vanishes, due to (3.22) and $\zeta(-2)=0$. Then, for a $D2$-$D0$ bound state
we decompose the sum as

$\displaystyle\mathcal{F}_{0}^{D0,D2}=-{\frac{M_{s}^{2}}{2g_{s}^{2}}}\bigg{[}$
$\displaystyle T^{2}\log\left(T\right)+\sum_{n\geq
1}\left(T^{2}-n^{2}\right)\log\left(1+\frac{T^{2}}{n^{2}}\right)$ (3.27)
$\displaystyle+2\sum_{n\geq 1}(T^{2}-n^{2})\log(n)-4\,T\sum_{n\geq 1}\,n\;{\rm
arccot}\left(\frac{T}{n}\right)\bigg{]}\,,$

where for the inverse trigonometric functions we use the principal domain. To
proceed, we first employ the identity (A.11) and then regularize all infinite
sums over $n$ via $\zeta$-function regularization. Using in particular

$2\sum_{n\geq 1}\left(T^{2}-n^{2}\right)\log(n)\equiv
T^{2}\log(2\pi)-\frac{\zeta(3)}{2\pi^{2}}\,,$ (3.28)

we finally arrive at the simple expression

$\mathcal{F}_{0}^{D0,D2}=\left({\frac{M_{s}}{2\pi g_{s}}}\right)^{2}\left[{\rm
Li}_{3}(e^{-2\pi T})-\frac{(2\pi T)^{3}}{12}\right]\,,$ (3.29)

with $T=t/g_{E}+ib$. Note that after rescaling $T$ by $2\pi$ and setting
$M_{s}=2\pi$ this is equal to the result (2.11) for the resolved conifold up
the non-physical linear and quadratic terms in $T$. In appendix B we provide
an alternative derivation employing the Hurwitz $\zeta$-function.

Finally, we consider a bound state of only $D0$-branes, which is given by

$\mathcal{F}_{0}^{D0}=-\frac{M_{s}^{2}}{2g_{s}^{2}}\sum_{n\neq
0}n^{2}\log\left(\frac{n}{\nu}\right)=-\left(\frac{M_{s}}{2\pi
g_{s}}\right)^{2}\zeta(3)\ .$ (3.30)

In contrast to the analogous contribution in $\mathcal{F}_{1}$, this term does
not have any dependence on $\nu$ upon using once more $\zeta(-2)=0$. Hence,
$\mathcal{F}_{0}^{D0}$ comes out holomorphic.

We have just shown that, similarly to $\mathcal{F}_{1}$, also the full
prepotential $\mathcal{F}_{0}$ of the resolved conifold, including the
polynomial terms, can be obtained by a loop computation in the emergent theory
$M_{\rm CY}$. Once more, it has been essential to sum over the full towers of
BPS states and appropriately regularize via $\zeta$-function the infinite
sums. In addition, despite its simplicity this example has served as a proof
of principle that exact emergence is possible when including all towers whose
mass spacing is smaller or equal to the species scale. Had we neglected, say,
the tower of $D2$-$D0$ bound states, which are heavier than the pure
$D0$-brane tower but nevertheless below the species scale, we would not have
reached exact emergence. Then, if these BPS states need to be included, it
seems inconsistent to not include as well bound states involving $NS5$-branes.

For completeness, let us recall how these couplings usually arise from the
perspective of the dual M-theory. The polynomial terms arise via dimensional
reduction of the eleven-dimensional supergravity action, whereas the
exponential terms come from $M2$-brane instantons wrapping the $S^{1}$ and a
2-cycle on $Y$. In the decompactification limit of an $S^{1}$ of infinite
size, i.e. in the five-dimensional theory, the contribution of these latter
terms vanishes.

###  3.3 Comparison with the Emergence Proposal

In order to appreciate the difference between the calculation above and the
simplified EFT approach usually taken in the Emergence Proposal, let us sketch
how the latter would work for the case of $\mathcal{F}_{0}$. The logic would
be to sum only over those states whose masses lie below the quantum gravity
cut-off, i.e. the species scale, $\tilde{\Lambda}$. In fact, the application
of this method to the sums that we are working with has been already proposed
in [13]: one is instructed to choose the scale $\nu=\tilde{\Lambda}$ and to
explicitly cut-off the sum in (3.27) at the level $n=N_{\rm sp}={\cal
V}^{1/3}$. This leads to the following expression for the prepotential

$\displaystyle\mathcal{F}_{0}=-\frac{M_{s}^{2}}{2g_{s}^{2}}\sum_{n=0}^{{\cal
V}^{\frac{1}{3}}}\Big{[}(T^{2}-n^{2})\log\left({\frac{T^{2}+n^{2}}{2\,{\cal
V}^{\frac{2}{3}}}}\right)-4nT{\rm arccot}\left({\frac{T}{n}}\right)\Big{]}\,,$
(3.31)

where a factor of $1/2$ has been introduced in the $\log$ for convenience (in
any case, we have not been keeping track of several constant factors and
normalizations). Proceeding then along with the logic same as in standard
emergence computations, one considers the scaling $t\to\lambda t$, ${\cal
V}^{\frac{1}{3}}\to\lambda{\cal V}^{\frac{1}{3}}$, with $\lambda\to\infty$,
and thus one is allowed to approximate the sum by an integral. Hence, one gets
the leading order result

$\displaystyle\mathcal{F}_{0}\approx-\frac{M_{s}^{2}\lambda^{3}}{18g_{s}^{2}}\bigg{[}$
$\displaystyle 3\pi(t^{3}-3t{\cal V}^{\frac{2}{3}})+\Big{(}-6t^{3}+18t{\cal
V}^{\frac{2}{3}}\Big{)}\arctan\left({\frac{t}{{\cal
V}^{\frac{1}{3}}}}\right)+$ (3.32) $\displaystyle\Big{(}-3t^{2}{\cal
V}^{\frac{1}{3}}+{\cal V}\Big{)}\left(2+3\log
2-3\log\left(1+{\frac{t^{2}}{{\cal
V}^{\frac{2}{3}}}}\right)\right)\bigg{]}\,.$

From this expression, it can be seen that only when ${\cal V}\simeq t^{3}$,
i.e. for the one modulus case, one gets the simple leading order expression

$\mathcal{F}_{0}\approx\frac{M_{s}^{2}}{18g_{s}^{2}}(3\pi+4)(\lambda
t)^{3}\,.$ (3.33)

This result agrees with the weak version of the Emergence Proposal, but
compared to the exact emergence that we have been able to recover it features
certain obvious shortcomings:

  * •

The full $\mathcal{F}_{0}$ is supposed to be holomorphic, but the explicit
cut-off for the sum introduces a non-holomorphic dependence on the moduli.

  * •

Even if a cubic polynomial part in $t$ arises, the prefactor is not correct
and the higher-order corrections are certainly not only those known for the
prepotential of type IIA Calabi-Yau compactification. Indeed, the computation
of the exact discrete sum instead of the continuous integral would produce
these exponential terms, but also several non-holomorphic corrections.

  * •

The higher-order corrections arising when insisting on this approach would
also depend on whether one takes the sum up to $N_{\rm sp}$ or to, say,
$N_{\rm sp}+1$. In this sense, such a cut-off procedure is not well defined if
one wants to extract more information than the qualitative leading cubic
behavior.

In view of the very simple and yet successful $\zeta$-function regularization
that we performed, we interpret all these issues as shortcomings of the too
naive EFT approach to the Emergence Proposal. Instead, our computation shows
that exact emergence is realized in this asymptotic decompactifcation limit if
all of the states in the towers with a typical mass scale $\mathfrak{m}$ up to
the species scale $\tilde{\Lambda}$ are running in the one-loop diagram. This
confirms quantitatively the intuition behind the Emergence Proposal of [1, 6],
for this limit. Similarly to asymptotic string limits, such as the fundamental
string at weak string coupling, the emergent perturbative quantum gravity
theory that we have been probing does not need an explicit ultraviolet cut-
off, but it provides finite answers for loop diagrams by its own. Recall that
for the weakly coupled fundamental string, one also sums over all infinite
string excitations and finiteness stems from modular invariance inducing a
restriction of the integral over $\tau$ to the fundamental domain.

##  4 Emergent String Limit

Given the concerns raised in [14] regarding the inclusion of string towers in
the standard emergence calculations, one might expect that an emergent string
limit shall be more complicated than the previous one. Instead, this limit is
actually more familiar from the point of view of string dualities. Given that
much of the following discussion is rather well-known, we can be brief and
just touch upon the main points. There is an important difference between
decompactification and emergent string limits. In the first case, as we have
just seen, one can encounter multiple string-like objects which are actually
emergent (wrapped) branes becoming asymptotic tensionless. In the second case,
there is really only a single and fundamental string emerging. Mathematically,
this is possible if the Calabi-Yau admits a certain $K3$ fibration.

In order to have a $K3$ fibration over a base $\mathbb{P}^{1}$, one needs to
have at least two Kähler moduli, see e.g. [53]. A well-studied example is the
Calabi-Yau $\mathbb{P}_{1,1,2,2,6}[12]$, which has Hodge numbers
$(h_{21},h_{11})=(128,2)$ and for which the prepotential has an expansion [54,
47]

$\mathcal{F}_{0}^{IIA}=T_{2}\,T_{1}^{2}+\underbrace{(T_{1}^{3}+\mathcal{O}(e^{-2\pi
T_{1}}))}_{g^{(1)}(T_{1})}+\mathcal{O}(e^{-2\pi T_{2}})\,.$ (4.1)

Here, $t_{2}={\rm Re}(T_{2})$ is measuring the size of the $\mathbb{P}^{1}$
which in the emergent string limit it is sent to infinity, i.e.
$t_{2}\rightarrow\lambda t_{2}$, with $\lambda\rightarrow\infty$. Again, we
rescale $g_{s}\rightarrow\lambda^{1/2}g_{s}$ accordingly to keep the four-
dimensional dilaton fixed and whenever referring to moduli we are implicitly
looking at their vacuum expectation values.111111In this limit, a natural way
to define $\lambda$ is as the ratio between the final and initial value of
$t_{2}$ along the trajectory over the moduli space that we are following.

#### Towers of light states

We start by identifying the species scale and the light towers of states in
the strongly coupled region $\lambda\to\infty$. Wrapping an $NS5$-brane on the
$K3$ fiber yields a four-dimensional string of tension

$T_{\rm str}\sim M_{\rm pl}^{2}\frac{{\cal V}_{K3}}{{\cal V}}\sim\frac{M_{\rm
pl}^{2}}{\lambda}\,,$ (4.2)

and whose excitations have a typical mass scale

$\mathfrak{m}_{\rm str}={\frac{M_{\rm pl}}{\sqrt{t_{2}}}}\simeq\frac{M_{\rm
pl}}{\sqrt{\lambda}}\,.$ (4.3)

Here, we denoted with $\mathcal{V}$ the volume of the Calabi-Yau, while
$\mathcal{V}_{K3}$ is the volume of the $K3$ fiber. The latter is indendent of
the scaling of $t_{2}$. In the presence of a string tower, the species scale
is equal to the string mass scale itself, hence we identify

$\tilde{\Lambda}\simeq\mathfrak{m}_{\rm str}=\frac{M_{\rm
pl}}{\sqrt{\lambda}}\,.$ (4.4)

As opposed to the decompactification limit, in this case the sixth route of
the $NS5$-brane tension is such that

$\mathfrak{m}_{NS5}\sim T^{\frac{1}{6}}_{NS5}\sim\frac{M_{\rm
pl}g_{s}^{\frac{2}{3}}}{{\cal V}^{\frac{1}{2}}}\sim\frac{M_{\rm
pl}}{\lambda^{\frac{1}{6}}}$ (4.5)

and hence gives a scale larger than $\mathfrak{m}_{\rm
str}\simeq\tilde{\Lambda}$, providing another justification for talking about
an emergent string limit in this case. While there are no other states which
are parametrically lighter than the species scale, there are a couple of
particle-like towers with a mass equal to it. First, due to the large size of
$\mathbb{P}^{1}$, we have light Kaluza-Klein modes with a typical mass scale

$\mathfrak{m}_{\rm KK}=\frac{M_{s}}{\sqrt{t_{2}}}\simeq\frac{M_{\rm
pl}}{\sqrt{\lambda}}\,.$ (4.6)

The remaining light states are once again given by $D$-branes, namely
$D0$-branes as well as $D4$-branes wrapped around $K3$ and $D2$-branes wrapped
around the second homological 2-cycle, sitting inside the $K3$ fiber. Their
typical masses read

$\mathfrak{m}_{D0}=\frac{M_{\rm
pl}}{\sqrt{t_{2}\mathcal{V}_{K3}}}\,,\qquad\mathfrak{m}_{D2}=\frac{M_{\rm
pl}}{\sqrt{t_{2}}}\,,\qquad\mathfrak{m}_{D4}=M_{\rm
pl}\sqrt{\frac{\mathcal{V}_{K3}}{t_{2}}}\,$ (4.7)

and all of them scale as $M_{\rm pl}/\sqrt{\lambda}$. Again, the wrapped
$NS5$-branes and the Kaluza-Klein modes are uncharged under the three RR
vector fields. Furthermore, note that a $D2$-brane wrapping the large base
${\mathbb{P}}^{1}$ is now too heavy to be included in the perturbative states.

#### Exact emergence of the prepotential

Our claim of exact emergence turns out to hold also in this new perturbative
theory, where the full prepotential can be completely recovered again, even if
not from a Schwinger integral exclusively. Indeed, we are presently in the
convenient situation in which a known dual string description exists, namely
the weakly coupled heterotic string, and we will exploit it to our advantage
in order to recover the full prepotential. This is a special case of the $N=2$
heterotic/IIA duality in four dimensions [55].

The dual heterotic model is a compactifation on $K3\times T^{2}$. Denoting the
toroidal Kähler and complex structure moduli with $T$ and $U$ respectively,
the dual model sits at the special locus $T=U$, where the Abelian gauge group
$U(1)_{U}\times U(1)_{T}$ is enhanced to $SU(2)\times U(1)$. The former type
IIA abelian gauge fields can be identified with the heterotic gauge fields
coming from the $T^{2}$ factor. Moreover, the particle-like states arising
from wrapped $D0$-, $D2$\- and $D4$-branes correspond to heterotic Kaluza-
Klein and winding modes along the $T^{2}$. Kaluza-Klein modes on the
$\mathbb{P}^{1}$ correspond to Kaluza-Klein modes along the $K3$ directions in
the heterotic dual picture, hence they are uncharged under the gauge fields.
The discussion can also be generalized to the case $T\neq U$, which requires a
Calabi-Yau with $h_{11}=3$ and whose prepotential has been originally derived
in [56].

With respect to the type IIA frame, a crucial difference is that the heterotic
dilaton $S$ sits in a vector multiplet so that the prepotential can receive
corrections at string one-loop and at the string non-perturbative level.
Therefore, it takes the schematic form [56, 57]

$\mathcal{F}^{\rm het}_{0}=\frac{1}{2}ST^{2}+h^{(1)}(T)+h^{(\rm
np)}(e^{-8\pi^{2}S})\,.$ (4.8)

Here, the first term arises at string tree-level, while the second is the
dilaton-independent one-loop correction. The function $h^{(1)}(T)$ is
determined by its transformation properties under the exact quantum symmetry
$SL(2,\mathbb{Z})_{T}$, acting on $T$ as in (A.13), and by the singular
structure at special points in moduli space. Modular invariance imposes that
its second derivative reads

$\partial_{T}^{2}h^{(1)}(T)=\frac{1}{4\pi^{2}}\log(j(iT)-j(i))+\text{finite
terms}\,,$ (4.9)

where $j(iT)$ is the modular invariant $j$-function (see appendix A for more
details). Microscopically, $h^{(1)}(T)$ receives contributions from heterotic
1/2-BPS states which, as reviewed in the discussion around (3.12), also
contain contributions from left-moving string excitations. Via mapping the
moduli according to

$T_{1}\rightarrow T\quad\text{and}\quad T_{2}\rightarrow 4\pi S\,,$ (4.10)

it has been shown in [55, 56, 57] that the part $g^{(1)}(T_{1})$ of the type
IIA prepotential nicely matches the heterotic 1-loop correction $h^{(1)}(T)$.
The other terms in (4.8) are then trivially mapped to those in (4.1).

One might wonder what the role of the former type IIA Schwinger integral is in
this setting. In the present case, when expressed in terms of $M_{s}$, all
light towers listed in (4.7) are insensitive to the size of the large base, so
that the sum over the Schwinger integrals will depend only on the type IIA
modulus $T_{1}$. The Kaluza-Klein modes on the large $\mathbb{P}_{1}$ do
depend on $t_{2}$ but, just like the $NS5$-branes, they do not contribute to
the Schwinger-integral, as we have already explained in section 3. Hence, the
(regularized) Schwinger integral just gives the part $g^{(1)}(T_{1})$ in the
prepotential, which from the dual heterotic point of view is the string one-
loop correction.

All this is supporting our claim that in the asymptotic limit, we can consider
the type IIA prepotential (4.1) as a perturbative expansion in the small
parameter $(\lambda\,t_{2})^{-1}$, which is the inverse of the number of light
species. Moreover, via duality we are also led to the conclusion that
$g^{(1)}(T_{1})$ does indeed receive contributions not only from $D2$-$D0$
bound states but also from excitations of $NS5$-$D2$-$D0$ bound states. The
latter are dual to heterotic 1/2-BPS left-moving string excitations with
Kaluza-Klein momentum and winding.

##  5 Conclusion

The Emergence Proposal is arguably among the less established swampland
conjectures, but it may potentially have important consequences on our
understanding of gravitational effective theories. When looking at it from a
field theory perspective, one would usually let terms emerge in the low-energy
effective action by integrating out (all) states whose masses are below the
cut-off, which is upper bounded by the species scale. While this strategy
seems to give answers which are qualitatively correct, it nevertheless leads
to several puzzles when quantitative tests are performed in concrete string
setups [14].

Motivated by this, in the present work we performed a quantitative analysis of
the Emergence Proposal, aimed at understanding its precise place and
formulation within string theory. Our results indicate that there exists an
exact version of the Emergence Proposal, which goes beyond the naive EFT
approach usually employed in qualitative tests and extends emergence
considerations to the full prepotential. With the tools available at present,
this exact version does not always lead to the emergence of tree level terms,
as one might expect from the original strong Emergence Proposal of [1, 6].
Concretely, we focussed on the Kähler moduli space of type IIA string theory
compactified on a Calabi-Yau threefold. In this framework, we looked at both
decompactification and emergent string limits.

As for decompactification limits, we found that exact emergence is realized if
one integrates out all states in towers with a typical mass scale smaller than
or equal to the species scale (here taken as the actual ultraviolet cut-off).
This led us to the conclusion that considering only the lightest states, which
would be just $D0$-branes in our setup, is certainly not sufficient, as it
misses great part (and in fact almost all) of the moduli dependence. Hence,
one definitely needs to include wrapped $D2$-branes. In addition, our
swampland arguments suggest that appart from them also wrapped $NS5$-branes
and Kaluza-Klein modes should be included, as they have the same typical mass
scale. This implies that the emergent (perturbative) quantum gravity theory in
the decompactification limit is not just a theory of particles, for it
includes also multiple four-dimensional strings arising from wrapped
$NS5$-branes.

At the quantitative level, we showed that when summing over towers of states
running in the loop diagrams, cutting off the sum at a certain level of the
tower fails in reproducing the desired terms in the effective action: an
integration over the full tower is needed. It would be clearly important to
collect more evidence to substantiate our proposal concerning the implicit
inclusion of wrapped $NS5$-branes and Kaluza-Klein modes in the Gopakumar-Vafa
invariants; we hope to come back to this problem in the future. However, even
if this proposal will turn out to be incorrect, our computation of ${\cal
F}_{0}$ and ${\cal F}_{1}$ for the conifold will still be valid and likewise
our general conclusion on the Emergence Proposal.

As for emergent string limits, the situation appears to be simpler, since in
the setup under investigation exact emergence is just another name for the
well-known string-string duality, where once more one needs to consider
infinite towers of states and no sharp cut-off is needed.

More in general, we propose the following picture. In each of the asymptotic
corners of the moduli space, assuming a fixed four-dimensional dilaton, a new
perturbation theory emerges which is organized by the inverse species number
associated to that limit (recall that the species scale is a function of the
moduli). Then, one can either quantize this theory, a step which is typically
out of reach at present, or hopefully understand it in terms of its dual,
which is the strategy we followed in the present work. The nice aspect of the
$N=2$ moduli space of type IIA string theory that we considered is that the
exact solution for at least the holomorphic prepotential is known via non-
renormalization theorems and mirror symmetry. This solution holds even in the
desert of the moduli space, where no towers of states are becoming light and
all mass scales are close to the Planck scale. Then, the various perturbative
expansions with different light towers of states are doomed to reproduce that
result in an expansion of the respective small parameter.

One can consider this moduli space in four dimensions as a well controlled
raw-model for the quantum gravity theories in higher dimensions. In
particular, it will be shown in [23] that the decompactification limit to five
dimensions studied in the present paper can be considered as a
compactification on a Calabi-Yau of a perturbative description of M-theory in
terms of transversal $M2$ and $M5$-branes carrying Kaluza-Klein momentum along
the compact (large) eleventh directions.

Our analysis has implications on central ideas in the swampland program: it
supports the existence of a unified portrait involving the distance
conjecture, the (exact) Emergence Proposal and the species scale, suggesting
that any investigation focusing only on a subset of these ideas is necessarily
incomplete. Furthermore, it provides also a novel perspective on the distance
conjecture itself, which should be understood not really as a statement within
effective field theory but rather about emergent (perturbative) theories of
quantum gravity. To understand this point better, let us recall that in
asymptotic limits, $\phi\to\infty$, there can really be three kind of towers
of states with typical mass $\mathfrak{m}_{\rm tower}$:

  * •

$\lim_{\phi\to\infty}\mathfrak{m}_{\rm tower}/\tilde{\Lambda}=\infty$, these
states are decoupled in the emergent perturbative quantum gravity theory;

  * •

$\lim_{\phi\to\infty}\mathfrak{m}_{\rm tower}/\tilde{\Lambda}=0$, these states
are part of the emergent perturbative quantum gravity theory;

  * •

$\lim_{\phi\to\infty}\mathfrak{m}_{\rm tower}/\tilde{\Lambda}=1$, these states
are at the threshold of the ultraviolet cutoff and must be included in
perturbative quantum gravity theory in order to have exact emergence. In
particular, they do not decouple.

Hence, one can argue that the swampland distance conjecture reflects a
fundamental property of quantum gravity, namely that in asymptotic regions in
field space there exists a perturbative description with a new emerging mass
scale $\tilde{\Lambda}<M_{\rm pl}$ and whose consistent description requires
to keep the full towers of states with typical mass scale not larger than
$\tilde{\Lambda}$.

Our results can be extended in various ways. One interesting extension would
be to perform an orientifold projection on the setup and then study a moduli
space with only four preserved supercharges, which is closer to
phenomenological applications. Alternatively, one can improve the quantitative
aspects of our analysis. For example, one can trying to reconstruct precisely
the triple intersection numbers (in $\mathcal{F}_{0}$) and the second Chern
class (in $\mathcal{F}_{1}$) by summing over all BPS bound states on the
Calabi-Yau. Otherwise, one can also apply our methodology systematically to
all possible asymptotic limits of the moduli space that we considered, which
have been classified in [15]. Our analysis can also be repeated in
compactifications letting a higher number of spacetime dimensions be non-
compact; recent work in this respect has been performed in [58]. We hope to
come back to these questions in the future.

##### Acknowledgments.

We are grateful to C. Fierro Cota, T. Grimm, E. Palti, I. Valenzuela and T.
Weigand for 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. The work of N.C. is supported by
the Alexander-von-Humboldt foundation.

##  Appendix A Special functions

In this appendix, we collect some useful functions and relations (see e.g.
[59]) that we employed in the main part of the work.

###  $\zeta$-functions

We recall the definition of the Riemann $\zeta$-function and its first
derivative which, for ${\rm Re}(s)>1$, can be written as

$\zeta(s)=\sum_{n\geq 1}{n^{-s}}\,,\qquad\zeta^{\prime}(s)=-\sum_{n\geq
1}\frac{\log(n)}{n^{s}}\,.$ (A.1)

Outside ${\rm Re}(s)>1$ an analytic continuation is understood. The trivial
zeros of the $\zeta$-function are located at the negative integers, while for
its derivative evaluated at these points we have

$\zeta^{\prime}(-2n)=(-1)^{n}\frac{(2n)!}{2(2\pi)^{2n}}\zeta(2n+1),\qquad
0\neq n\in\mathbb{N}\,.$ (A.2)

Some particular values that are of importance in our calculations are

$\zeta(0)=-\frac{1}{2}\,,\quad\zeta(2)=\frac{\pi^{2}}{6},\quad\zeta(-2)=0,\quad\zeta^{\prime}(0)=-\frac{1}{2}\log(2\pi)\,,\quad\zeta^{\prime}(-2)=-\frac{\zeta(3)}{4\pi^{2}}\,,$
(A.3)

where $\zeta(3)=1,2020569\ldots$ is Apéry’s constant. The Riemann
$\zeta$-function is generalized by the Hurwitz $\zeta$-function, $\zeta(s,z)$.
For ${\rm Re}(s)>1$ and $z\neq 0,-1,-2,\ldots$, such function and its
derivative with respect to $s$ are represented by the series

$\zeta(s,z)=\sum_{n\geq 0}{(n+z)^{-s}},\qquad\zeta^{\prime}(s,z)=-\sum_{n\geq
0}(n+z)^{-s}\log(n+z)\,,$ (A.4)

otherwise one can perform an analytic continuation for $s\neq 1$.

### Polylogarithms

We recall the definition of a polylogarithm

${\rm Li}_{s}(z)=\sum_{n\geq 1}\frac{z^{n}}{n^{s}}\,,$ (A.5)

which is valid for any complex number $s$ and for $|z|<1$. The natural
logarithm is recovered for the specific value $s=1$,

${\rm Li}_{1}(z)=-\log(1-z)\,,$ (A.6)

while for ${\rm Re}(s)>1$ the $\zeta$-function corresponds to $z=1$

${\rm Li}_{s}(1)=\zeta(s)\,.$ (A.7)

### Useful identities

Recalling that

$\displaystyle\log\left(\frac{\sinh(\pi z)}{\pi z}\right)$
$\displaystyle=\log\left[\prod_{n=1}^{\infty}\left(1+\frac{z^{2}}{n^{2}}\right)\right]\,,$
(A.8)

one can prove the identity

$\sum_{n\geq 1}\log\left(1+\frac{z^{2}}{n^{2}}\right)=\log\left(\frac{e^{\pi
z}-e^{-\pi z}}{2\pi z}\right)=\pi z-\log{2\pi z}+\log(1-e^{-2\pi z})\,,$ (A.9)

which is used to obtain (3.23). Furthermore, integrating (A.9) twice and using

$\int_{0}^{A}dz\,{\rm Li}_{n}(e^{2\pi z})=\frac{1}{2\pi}\left(-\zeta(n+1)+{\rm
Li}_{n+1}(e^{2\pi A})\right)\,,$ (A.10)

one can derive the curious identity

$\displaystyle\sum_{n\geq 1}\Big{[}\left(\frac{z^{2}-n^{2}}{2}\right)$
$\displaystyle\log\left(1+\frac{z^{2}}{n^{2}}\right)+2zn\arctan\left(\frac{z}{n}\right)-{\frac{3}{2}}z^{2}\Big{]}=$
(A.11) $\displaystyle-$ $\displaystyle\frac{1}{4\pi^{2}}{\rm Li}_{3}(e^{-2\pi
z})-\frac{\pi z}{12}+\frac{3}{4}z^{2}+\frac{\pi
z^{3}}{6}-\frac{1}{2}z^{2}\log(2\pi z)+\frac{\zeta(3)}{4\pi^{2}}\,,$

which is used to obtain (3.29). Another useful identity relates derivatives of
the Hurwitz $\zeta$-function, polylogarithms and Bernoulli polynomials
$B_{n}(x)$ as [60, 61, 62]

$\zeta^{\prime}(-s,z)+(-1)^{s}\zeta^{\prime}(-s,1-z)=\pi
i\frac{B_{s+1}(z)}{s+1}+e^{-\pi is/2}\frac{s!}{(2\pi)^{s}}{\rm
Li}_{s+1}(e^{2\pi iz})\,.$ (A.12)

This is employed in appendix B to provide an alternative derivation of the
objects $\mathcal{F}_{0}$ and $\mathcal{F}_{1}$ for the resolved conifold.

### Modular forms

The modular group maps a complex modulus $T$ according to

$iT\rightarrow\frac{iaT+b}{icT+d}\quad\text{with}\quad ad-bc=1\,,\quad
a,b,c,d\in\mathbb{Z}\,.$ (A.13)

A modular form $F_{r}(T)$ of weight $r$ is a holomorphic function from the
upper half plane transforming as

$F_{r}(T)\rightarrow(icT+d)^{r}F_{r}(T)$ (A.14)

under modular transformations.

One example is the modular invariant $j$-function, defined for ${\rm Re}(T)>0$
as

$j(iT)=\frac{E_{4}^{3}(iT)}{\eta^{24}(iT)}=\frac{E_{6}^{2}(iT)}{\eta^{24}(iT)}+1728\,.$
(A.15)

It is a combination of modular forms of weight 4 and 6, called Eisenstein
series, which, for $q=e^{-2\pi T}$, are given by

$E_{4}(iT)=1+240\sum_{n=1}^{\infty}\frac{n^{3}q^{n}}{1-q^{n}}\,,\quad
E_{6}(iT)=1-504\sum_{n=1}^{\infty}\frac{n^{5}q^{n}}{1-q^{n}}$ (A.16)

and the Dedekind $\eta$-function

$\eta(iT)=q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})\,,$ (A.17)

which is of weight 1/2, so that the $j$-function is of weight 0 and hence
called the $j$-invariant. One of its properties that is particularly useful in
our context is given by Borcherds’ product formula [63]:

$j(iT)-j(iU)=p^{-1}\prod_{m>0,n\in\mathbb{Z}}(1-p^{m}q^{n})^{c(mn)}\,,$ (A.18)

where $p=e^{-2\pi T}$ and $q=e^{-2\pi U}$ and the coefficients $c$ are given
by the expansion

$j(iT)-744=\sum_{n=-1}^{\infty}c(n)q^{n}=q^{-1}+196884q+21493760q^{2}\ldots\,.$
(A.19)

The infinite product converges only for $|p|,|q|<e^{-2\pi}$ and $p\neq q$,
while the infinite series converges for $|q|<1$, but the relations may be
extended on the entire complex plane via analytic continuation. Taking the
logarithm of (A.18) we obtain

$\log(j(iT)-j(iU))=2\pi
T+\sum_{m>0,n\in\mathbb{Z}}c(mn)\log(1-e^{-2\pi(mT+nU)})\,.$ (A.20)

While $T=U$ is a branch singularity of the above expression, one can already
recognize the functional behaviors justifying the equivalence of (4.1) and
(4.8).

##  Appendix B Alternative derivation of $\mathcal{F}_{0}$ and
$\mathcal{F}_{1}$

In this appendix, we provide an alternative derivation of $\mathcal{F}_{0}$
and $\mathcal{F}_{1}$ for the resolved conifold by means of Riemann and
Hurwitz $\zeta$-function regularization. We start from $\mathcal{F}_{0}$,
whose computation is more involved.

#### Evaluating $\mathcal{F}_{0}$

We look at the contribution from $D2$-$D0$ bound states, which reads

$-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}=\sum_{n\in\mathbb{Z}}(T+in)^{2}\log\left(\frac{T+in}{\mu}\right)\,,$
(B.1)

and it can be decomposed as

$-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}=-\sum_{n\in\mathbb{Z}}(T+in)^{2}\log{\mu}+\sum_{n\in\mathbb{Z}}(T+in)^{2}\log(T+in)\,.$
(B.2)

One can check that the first term vanishes thanks to (3.22) and $\zeta(-2)=0$,
so that we have

$\displaystyle-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}$
$\displaystyle=-\log(i)\sum_{n\in\mathbb{Z}}(n-iT)^{2}-\sum_{n\in\mathbb{Z}}(n-iT)^{2}\log(n-iT).$
(B.3)

Now, we proceed by regularizing the first sum with the Riemann
$\zeta$-function, while the second with the Hurwitz $\zeta$-function. In this
way, the coefficient of $\log(i)$ vanishes (thanks to (3.22) and $\zeta(-2)=0$
again) and we arrive at

$\displaystyle-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}$
$\displaystyle=-\sum_{n\geq
0}(n-iT)^{2}\log(n-iT)-\sum_{n>0}(n+iT)^{2}\log(n+iT)$ (B.4)
$\displaystyle\quad-\log(-1)\sum_{n>0}(n+iT)^{2}.$

We regularize the first and the second sum with the Hurwitz $\zeta$, while the
third one with the Riemann $\zeta$. Using $\zeta(-2)=0$,
$\zeta(-1)=-\frac{1}{12}$, $\zeta(0)=-\frac{1}{2}$ and
$\zeta^{\prime}(s,z)\equiv-\sum_{n\geq 0}(z+n)^{-s}\log(z+n)$, we have

$\displaystyle-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}$
$\displaystyle=\zeta^{\prime}(-2,-iT)+\zeta^{\prime}(-2,iT)+(iT)^{2}\log(iT)$
(B.5)
$\displaystyle\quad-\frac{1}{6}T(2k_{1}+1)\pi-\frac{T^{2}}{2}(2k_{2}+1)i\pi,$

where we introduced $k_{1},k_{2}\in\mathbb{Z}$ to keep track of the
monodromies of the complex logarithm. Specializing (A.12) for $s=-2$ and
$z=iT$, one finds

$\zeta^{\prime}(-2,+iT)+\zeta^{\prime}(-2,-iT)=i\frac{\pi}{3}B_{3}(iT)-\frac{1}{2\pi^{2}}{\rm
Li}_{3}(e^{-2\pi T})-(-iT)^{2}\log(-iT),$ (B.6)

with the Bernoulli polynomial
$B_{3}(iT)=\frac{i}{2}T+\frac{3}{2}T^{2}-iT^{3}$. Inserting this into (B.5),
we get

$\displaystyle-\frac{2g_{s}^{2}}{M_{s}^{2}}\mathcal{F}_{0}^{D0,D2}$
$\displaystyle=-\frac{\pi}{6}\left(2k_{1}+2\right)T-\frac{i}{2}\pi(2k_{2}+4k_{3}+2)T^{2}$
(B.7) $\displaystyle\quad+\frac{\pi}{3}T^{3}-\frac{1}{2\pi^{2}}{\rm
Li}_{3}(e^{-2\pi T}),$

where $k_{3}\in\mathbb{Z}$ has been introduced to keep track of the branch of
another complex logarithm, $\log(-1)=(2k_{3}+1)i\pi$. Observe that the
integers $k_{i}$ may be tuned so that the non-physical terms of quadratic and
linear order in $T$ vanish.

#### Evaluating $\mathcal{F}_{1}$

The same logic may be applied to $\mathcal{F}_{1}^{D0,D2}$, namely

$\mathcal{F}_{1}^{D0,D2}=\frac{1}{12}\sum_{n\in\mathbb{Z}}\log\left(\frac{T+in}{\mu}\right)=+\frac{1}{12}\log\left(\frac{i}{\mu}\right)\sum_{n\in\mathbb{Z}}1+\frac{1}{12}\sum_{n\in\mathbb{Z}}\log(n-iT)\,.$
(B.8)

The first term vanishes due to (3.22), so we get

$\displaystyle 12\mathcal{F}_{1}^{D0,D2}$
$\displaystyle=\sum_{n=1}^{\infty}\log(n+iT)+\sum_{n=0}^{\infty}\log(n-iT)+\log(-1)\sum_{n=1}^{\infty}1$
(B.9)
$\displaystyle=-\zeta^{\prime}(0,iT)-\zeta^{\prime}(0,-iT)-\log(+iT)+\frac{1}{2}(2k_{1}+1)\pi
i\,,\quad k_{1}\in\mathbb{Z}\,.$

Setting this time $s=0$ and $z=iT$ in (A.12), we get

$\zeta^{\prime}(0,iT)+\zeta^{\prime}(0,-iT)=\pi iB_{1}(iT)+{\rm
Li}_{1}(e^{-2\pi T})-\log(-iT)\,,$ (B.10)

where the relevant Bernoulli polynomial is now $B_{1}(iT)=iT-\frac{1}{2}$ .
Thus,

$\displaystyle\mathcal{F}_{1}^{D0,D2}$ $\displaystyle=\frac{1}{12}\left(-{\rm
Li}_{1}(e^{-2\pi T})+\pi T-\frac{\pi i}{2}+(2k_{2}+1)\pi
i+\frac{1}{2}(2k_{1}+1)\pi i\right)$ (B.11)
$\displaystyle=\frac{1}{12}\left(-{\rm Li}_{1}(e^{-2\pi T})+\frac{(2\pi
T)}{2}+\frac{1}{2}(4k_{2}+2k_{1}+3)\pi i\right)\,,\quad
k_{1},k_{2}\in\mathbb{Z}\,,$

which, after rescaling $T\rightarrow T/2\pi$, is in agreement with (2.12), up
to a modulus-independent additive constant. Once again the integers $k_{i}$
were added to keep track of the complex logarithm branch ambiguities and are
only affecting non-physical terms.

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