Published for SISSA by Springer

Received: January 16, 2024

Accepted: January 26, 2024
Published: February 23, 2024

Bounds on field range for slowly varying positive

potentials

JHEP02(2024)175
Damian van de Heisteeg,a Cumrun Vafa,b Max Wiesner a,b and David H. Wu b

a
Center of Mathematical Sciences and Applications, Harvard University,
Cambridge, MA 02138, U.S.A.
b
Jefferson Physical Laboratory, Harvard University,
Cambridge, MA 02138, U.S.A.
E-mail: dvandeheisteeg@fas.harvard.edu, vafa@g.harvard.edu,
mwiesner@fas.harvard.edu, dwu@g.harvard.edu

Abstract: In the context of quantum gravitational systems, we place bounds on regions in

field space with slowly varying positive potentials. Using the fact that V < Λ2s , where Λs (ϕ)
is the species scale, and the emergent string conjecture, we show this places a bound on the
maximum diameter of such regions in field space: ∆ϕ ≤ a log(1/V ) + b in Planck units, where
p
a ≤ (d − 1)(d − 2), and b is an O(1) number and expected to be negative. The coefficient of
the logarithmic term has previously been derived using TCC, providing further confirmation.
For type II string flux compactifications on Calabi-Yau threefolds, using the recent results on
the moduli dependence of the species scale, we can check the above relation and determine
the constant b, which we verify is O(1) and negative in all the examples we studied.

Keywords: Cosmological models, Effective Field Theories, Superstring Vacua, Topological

Strings

ArXiv ePrint: 2305.07701

Open Access, © The Authors.

https://doi.org/10.1007/JHEP02(2024)175
Article funded by SCOAP3 .
Contents

1 Introduction 1

2 Species scale and scalar potentials 2

3 The main argument 5

4 Range of potentials in Calabi-Yau compactifications 6

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4.1 Species scale and fluxes in type IIB orientifolds 6
4.2 General considerations 8
4.2.1 Field range of constant potentials 9
4.2.2 Asymptotics of the field range 10
4.3 Examples 11
4.3.1 Example 1: (K3 × T 2 )/Z2 11
4.3.2 Example 2: mirror quintic X5 (15 ) 12
4.3.3 Example 3: mirror bicubic X3,3 (16 ) 13

5 Concluding remarks 14

1 Introduction

The species scale, Λs , introduced in [1–4] is an effective UV cutoff in theories of quantum

gravity capturing the number, N , of light degrees of freedom in a quantum theory of gravity
−(d−2)
via N ∼ Λs . In the presence of light modes, ϕ, the species scale can depend on these
fields. In particular in a recent work [5] we showed how one can use topological string
amplitudes at one loop to compute Λs (ϕ) for type II string theories compactified on Calabi-
Yau threefolds. In this paper we use this to place bounds on the field range for theories
which break supersymmetry mildly and lead to slowly varying V > 0. Here a mild breaking
of supersymmetry means that the breaking does not change the number of light degrees
of freedom dramatically. As we will show in this paper, an example of this is provided by
CY threefold flux compactifications. The basic idea is that in this case, we still can use the
Λs (ϕ) computed using topological strings. For slowly varying fields leading to a quasi-dS
space, the idea, that the effective theory does not break down, leads to the requirement that
V < Λ2s (ϕ). Using the fact that asymptotically the distance conjecture places exponential
bounds on the species scale [6], we find that
q
∆ϕ ≤ (d − 1)(d − 2) · log(1/V ) + b , (1.1)

where b ∼ O(1); moreover we find in all the examples we studied b < 0. The above bound,
except for the constraint on the linear shift term, was previously derived using the Trans-
Planckian Censorship Conjecture (TCC) in [7]. Our argument here is based on the emergent
string conjecture [8] and consistency of the effective theory, and provides further evidence

–1–
for the validity of TCC. Moreover for the first time in this paper we are able to determine
the value b in examples of type II compactifications on Calabi-Yau threefolds. A general
bound of the form ∆ϕ < a log 1/V for a ∼ O(1) was previously considered in [9] to put
bounds on inflationary models for which TCC fixes the coefficient a. Our results in this
paper reproduce the TCC bound from a completely different reasoning and moreover we
find the sub-leading correction to the field range encoded in b.
The organization of this paper is as follows. In section 2 we recall the behavior of the
species scale and scalar potentials in infinite distance limits, reviewing in particular the
bounds put by the TCC [7] on the latter. In section 3 we argue how the TCC bound can
be reproduced from species scale arguments and further use this to bound the field range of

JHEP02(2024)175
slowly varying scalar potentials. In section 4 we apply these principles to bound the field range
of scalar potentials arising in flux compactifications of type II string theory on Calabi-Yau
threefolds. We conclude in section 5. Two notebooks detailing the computations for the
quintic and bicubic examples have been attached to this paper as supplementary material.

2 Species scale and scalar potentials

In effective theories of gravity with scalar fields ϕ an important question concerns the
diameter of the scalar field space as this determines the maximal possible field variation
∆ϕ. In effective theories that allow for a consistent UV completion to quantum gravity,
the scalar field space M are generically expected to be non-compact [10] implying that the
diameter of scalar field spaces consistent with quantum gravity is infinite. On the other
hand, the distance conjecture [10] implies that close to the infinite distance limits a tower
of states becomes light as
m
∼ e−α∆ϕ , (2.1)
Mpl

with ∆ϕ being the distance in field space and Mpl the reduced Planck scale. The parameter
α can be further constrained using the emergent string conjecture [8] which states that the
light modes emerging at infinite distance are either KK modes or the excitations of light
fundamental strings. With this input, one can then show [11]
s
(D − 2) 1
≥α≥ √ ,
(D − d)(d − 2) d−2
where d is the spacetime dimension and D ≤ 11 is the higher-dimensional theory which one
may decompactify to (additional arguments for the lower bound have been given in [12]).
The presence of the light tower of states at large distances invalidates the EFT description
above some finite energy scale ΛEFT such that the part of the field space which allows for a
consistent EFT description is expected to have finite diameter.
While the distance conjecture clearly excludes infinite field variations ∆ϕ → ∞ within
a consistent EFT description, it does not tell us much about the diameter of the residual
field space once all infinite distance tails are removed. In fact even the refined distance
conjecture [13], stating that the exponential behavior in (2.1) should set in after one Planck
length is traversed, does not give much information about the diameter of the residual field

–2–
space since it does not constrain the overall normalization in (2.1). This bears the possibility
that for large distances the exponential behavior in (2.1) is realized but that the relevant
states still have masses well above the Planck scale. In this context it was recently conjectured
in [14] that in fact there is always a tower of states with mass below Mpl which would
effectively constrain the diameter of the effective field space.
Instead of considering the mass scales of individual towers of states, a more invariant
way to capture the moduli-dependence of the EFT cut-off is via the species scale Λs as this
quantity captures the effect of all light towers. The species scale is in general a function of
the scalar fields ϕ and in certain cases can be evaluated without detailed knowledge about the
exact spectrum of light states. In fact in [5] we showed that for the case of N = 2 theories

JHEP02(2024)175
in 4d arising from compactifications of type II string theory on Calabi-Yau threefolds, the
species scale can be related to the genus-one topological free energy, F1 , as F1 ∼ 1/Λ2s and
is readily computable. As noted in [6], the emergent string conjecture [8] also predicts the
species scale in asymptotic regions to fall off exponentially as
Λs
∼ e−λ∆ϕ , (2.2)
Mpl
where s
(D − d) 1
≤λ≤ √ , (2.3)
(D − 2)(d − 2) d−2
with the highest value corresponding to emergent string limits. Importantly, the lowest value
for λ is achieved for a decompactification of a single dimensions, i.e. D = d + 1, leading to
1
λ≥ p .
(d − 1)(d − 2)
Apart from the species scale, in theories that allow for a non-trivial scalar potential, there
exists a second fundamental scale set by V (ϕ). In case of relatively flat potentials (as those
required in models of inflation) the relevant scale is the Hubble parameter
√
H= V, (2.4)

where we have set Mpl = 1. In this paper we are mainly interested in studying situations where
we indeed have a relatively flat potential. Such scalar potentials have already been severely
constrained by the TCC [7]. The TCC asserts that in any consistent theory of quantum gravity
any sub-Planckian fluctuation should remain quantum during any cosmological expansion.
This leads to a bound on the integral of the Hubble parameter during the expansionary period
Z tf
Mpl
H ≤ log , (2.5)
ti Hf
with Hf the Hubble rate at the end of the expansion at time tf . Using the Friedmann equation

(d − 1)(d − 2)H 2 = ϕ̇2 + 2V , (2.6)

where ϕ̇ is the time derivative of the rolling scalar field, for V > 0 one finds
H 1
≥p . (2.7)
|ϕ̇| (d − 1)(d − 2)

–3–
 V (ϕ)

 
2∆ϕ
exp − √
(d−1)(d−2)

 
exp − √2∆ϕ
d−2

JHEP02(2024)175
ϕ

Figure 1. An illustration of a potential V (ϕ) (solid line) consistent with the constraint imposed
by the bound (2.8) (dotted line) and that asymptotically also satisfies the TCC bound (2.9) for
exponentially decaying potentials (dashed line).

Using that by (2.6) V is bounded by H 2 one concludes that if a distance ∆ϕ is traversed

in field space, in regions where the scalar potential is monotonic (say decreasing), V (in
Planck units) should satisfy the bound [7]

" #
−2∆ϕ
V ≤ A exp p . (2.8)
(d − 1)(d − 2)

This bound implies that any scalar potential can only stay approximately flat for a finite
range in the scalar field space since for sufficiently large distances the scalar potential needs
to decay at least exponentially in the canonically normalized scalar field. Its validity has been
confirmed in a large class of explicit type II compactifications in [15, 16]. Notice that the TCC
does not explain what forces the scalar potential into this exponential behavior. In this note,
by using the species scale, we provide an alternative explanation for the exponential bound
in (2.8). Moreover we find that the maximum range in field space before the exponential
behavior sets in occurs for decompactification of a single extra dimension. We show this by
p
connecting the factor of 2/ (d − 1)(d − 2) in (2.8), coming from the TCC, with the minimal
exponent possible for the asymptotic behaviour of the species bound (2.3). Notice that this
does not necessarily imply that the asymptotic shape of the potential is given by this exponent.
Indeed if we assume V has an exponential profile, as one would expect to be the case for
large enough field values of ϕ, V ∼ exp(−βϕ), it was shown in [7] that the TCC implies

2
β≥p . (2.9)
(d − 2)

This has also been argued for in [17]. In other words TCC would be compatible with a
potential which fits the profile in figure 1.

–4–
3 The main argument

To argue for the bound (2.8) in this paper we make use of the species scale. Recall that
the species scale gives a measure for the number, N , of light degrees of freedom in an
effective theory of gravity via
Mpl
Λs = 1 . (3.1)
N d−2
Let us consider a situation in which we have an approximately flat, positive potential leading
to a quasi-dS space. For this dS space the radius of the Hubble horizon is set by the

JHEP02(2024)175
scalar potential
(d−2)/2
1 Mpl
rH ∼ = √ , (3.2)
H V
leading to the Gibbons-Hawking dS entropy [18]

SdS ∼ (rH Mpl )d−2 . (3.3)

Even though in general it is not clear what are the microstates accounting for this entropy,
we know that the entropy should at least account for the light states in the theory which
are counted by N . Therefore we find the bound

N < SdS , (3.4)

which using (3.1) and (3.2) leads to

√
V
Λs ≥ (d−2)/2
. (3.5)
Mpl

Since the species scale can also be associated to the radius rmin of the horizon of the smallest
−1
black hole describable within the EFT via Λs = rmin , the above bound can equivalently be
obtained by requiring that this smallest black hole fits within the dS horizon.
Notice that the bound (3.5) is also valid in case we have exponentially decaying potentials
with V ∼ exp(−βϕ) because the Hubble parameter H and V are still related via the
Friedmann equation (2.6). Since H defines a horizon radius, we still have the bound (3.5) by
requiring that the horizon of the smallest black hole should be smaller than the cosmological
horizon set by H.
Let us mention that the bound (3.5) has been previously discussed in [19, 20] and used
in [9] to constrain the maximal field excursion for inflationary models, which was motivated
by the requirement that in a consistent EFT the Hubble scale should remain below the
quantum gravity cut-off. Identifying the quantum gravity cut-off with the species scale, our
previous discussion shows that this is consistent with dS entropy considerations. Using the
asymptotic behavior of the species scale as in (2.2) the bound (3.5) was used in [9] to put
an asymptotic bound on ∆ϕ given by
1 Mpl
∆ϕ ≤ log . (3.6)
λ H

–5–
The bound on the field range in this note is in spirit close to that argument, except that
in this note we can sharpen this result by finding the actual field range and not just the
parametric behavior since, as discussed in the next section, in certain cases we know the
explicit form of the species scale also in the interior of field space. In addition, this also
allows us to fix the constant coefficients in the expression for the field range ∆ϕ.
As reviewed in the previous section, asymptotically the species scale behaves as

Λs
= A e−λϕ , (3.7)
Mpl

with λ constrained by the emergent string conjecture as in (2.3). Notice that since λ ≥

JHEP02(2024)175
√ 1 the bound (3.5) leads to (2.8). The species scale constraint together with the
(d−1)(d−2)
emergent string conjecture hence reproduces the bound imposed by TCC [7]! We can further
bound the range, ∆ϕflat , in scalar field space over which the potential is approximately
flat with value of order V0 by
q V0
∆ϕflat ≤ − (d − 1)(d − 2) log d
+ log A + . . . , (3.8)
Mpl

where the . . . denote additional terms that are suppressed for V0 ≪ Mpl d . Let us stress that

for large A this would allow the effective field range to possibly be very large. This can
be understood by noticing that the species scale is bounded by Λs < Mpl : for large A this
implies that the exponential behavior can only set in for large values of ∆ϕ allowing for large
field ranges with approximately flat scalar potentials. Specifying to flux compactifications of
type II string theory on Calabi-Yau threefolds we show in the next section that generically
A ≲ O(1), thereby severely constraining the maximal field range before the exponential
behavior of the scalar potential sets in.

4 Range of potentials in Calabi-Yau compactifications

In this section we bound the field ranges for slowly varying scalar potentials arising from
Calabi-Yau compactifications of string theory. First we review the species scale definition
of [5], and argue how it extends to N = 1 supergravity theories with scalar potentials. We
then consider a set of examples where we explicitly compute the range of the field space
for a fixed scalar potential V = V0 .

4.1 Species scale and fluxes in type IIB orientifolds

In [5] it was argued that the species scale in Type II Calabi-Yau compactifications can be
related to the one-loop topological free energy F1 . This free energy can be defined from the
perspective of the underlying 2d N = 2 CFT as the index [21, 22]

1
Z h i
F1 = Tr (−1)F FL FR q H0 q̄ H̄0 , (4.1)
2 F

where FL(R) denotes the left-moving (right-moving) fermion number, F the fundamental
domain of SL(2, Z) and H0 the Hamiltonian of the CFT. In [5] the species scale was related

–6–
to F1 as (in the following we set Mpl = 1)

1
Λs = √ . (4.2)
F1

In [23] this relation was discussed from the perspective of the smallest black holes described
by the effective theory.
The most direct argument given in [5] for this relation (4.2) between Λs and F1 follows
from the higher-derivative corrections in the 4d N = 2 supergravity theory: F1 appears as
coefficient of the higher-derivative term [24, 25]

JHEP02(2024)175
Z
SN =2 ⊃ d4 x d4 θ F 1 W 2 , (4.3)

where θ denotes the fermionic superspace coordinates, and Wµν the Weyl superfield. By
expanding Wµν one obtains the (R− )2 term in the effective action, with R− the anti-self-dual
part of the curvature. The identification of F1 with the species scale Λs then follows by
noting that the coefficient of this term should be given by Mpl 2 /Λ2 [6].
s
Importantly, when breaking the N = 2 theory to N = 1 by means of fluxes, the topological
string amplitudes do not get affected [26]. Hence the genus-g topological free energies still
determine the coefficients of higher-derivative terms in the effective action. In particular, in
the N = 1 effective action there are two sorts of higher-derivative corrections [27, 28]

∂Fg
Z
SN =1 ⊃ d4 x d2 θ F 2g Ni , (4.4)
∂Si
and Z
SN =1 ⊃ g d4 x d2 θ W 2 F 2(g−1) Fg , (4.5)

where W now denotes the reduction to an N = 1 multiplet, Ni the flux quanta, Si chiral
superfields, and F the anti-self-dual part of the graviphoton field strength. Note in particular
that the genus-0 answer from (4.4) has been used to compute exact corrections to N = 1
superpotential terms using the breaking of N = 2 to N = 1 by turning on fluxes [29]. For
us the crucial point is that the fluxes Ni do not appear in equation (4.5) and in particular
the Weyl-squared term, W 2 , is unaffected. This allows us to again relate the species scale
Λs to the genus-one free energy F1 . In this sense flux compactifications of Type II string
theory on CY3-folds provide an example for which the supersymmetry breaking is mild in
the sense that the species scale is not affected considerably by the breaking.
In order to obtain a closed form for F1 , its holomorphic anomaly equation can be
integrated [24]. For Type IIA compactifications this results in an expression that depends
only on the Kähler moduli. In the N = 1 theory this is still the case where additionally we
have to project out all orientifold-even coordinates. The genus-one free energy F1 then reads

1 χ 1
 
F1 = 3 + h1,1 + K N =2 − log det GN
ij̄
=2
+ log |f |2 , (4.6)
2 12 2

where h1,1 and χ denote the dimension of the vector multiplet moduli space and Euler
characteristic of the Calabi-Yau threefold respectively, K N =2 and GN
ij̄
=2 the Kähler potential

–7–
 Λ2s

JHEP02(2024)175
V0

ϕ
∆ϕL ∆ϕbulk ∆ϕR
Figure 2. Sketch of the field range ∆ϕ for a constant scalar potential V (ϕ) = V0 (dotted, red) cut off
by the species scale Λ2s going to zero asymptotically (blue, solid). It has been decomposed according
to (4.8) into a fixed interior segment ∆ϕbulk and two tails ∆ϕL,R probing the infinite distance limits.
On the left the species scale goes to zero at a slower exponential rate compared to the right such that
the field range extends further into deeper into the left asymptotic regime.

and metric of the N = 2 parent theory, and the holomorphic ambiguity f can be fixed by
matching the asymptotics of F1 at the boundaries of the moduli space.
Finally, let us recall that F1 given by (4.6) is only defined up to an additive constant,
as the zero-mode contribution in (4.1) diverges. From the perspective of the relation (4.2)
between Λs and F1 this requires us to introduce an additional coefficient that parametrizes
this ambiguity. If we define F1 such that it vanishes at the desert point, this parameter
reduces to the number of species Ndes at the desert point

1
Λs = √ . (4.7)
F1 + Ndes

In the following we use this form of the species scale to bound the field range of constant
scalar potentials resulting from Calabi-Yau compactifications of Type IIA by using (3.5).

4.2 General considerations

In this (and the following) subsection we consider constant potentials V (ϕ) = V0 in Calabi-Yau
compactifications, and ask ourselves how the bound V ≤ Λ2s precisely restricts their field
range. This range often extends across multiple phases of the moduli space, so we will have
to carefully examine both the size of the interior of the compact moduli space, as well as the
precise asymptotics near its infinite distance boundaries. In this first subsection we approach
this problem from a general viewpoint, decomposing the field range into multiple pieces and
estimating the contributions coming from each of them, as depicted in figure 2.

–8–
4.2.1 Field range of constant potentials
We denote the field range for a given energy scale V0 by ∆ϕ(V0 ). This field range is computed
by considering the longest minimal geodesics in the moduli space along which V0 ≤ Λ2s
is satisfied. Concretely, this geodesic has two endpoints that we may move further into
infinite distance regimes as we decrease V0 . We may therefore break down the range ∆ϕ(V0 )
into three contributions1

∆ϕ(V0 ) = ∆ϕL (V0 ) + ∆ϕbulk + ∆ϕR (V0 ) . (4.8)

The middle contribution ∆ϕbulk comes from the interior of the moduli space; throughout

JHEP02(2024)175
most of this paper we assume V0 to be sufficiently small, so we traverse this phase in its
entirety, and hence ∆ϕbulk represents a constant contribution to the field range. Estimating
the size of ∆ϕbulk is a crucial piece in order to determine when the exponential behavior
of the distance conjecture [10] sets in.2
The other two contributions ∆ϕL (V0 ) and ∆ϕR (V0 ) come from the two endpoints along
infinite distance limits where Λ2s goes to zero. These distances can be computed by considering
just the asymptotic behaviors associated to these phases. Taking ∆ϕR for definiteness, it
behaves as
1 1
∆ϕR (V0 ) = − log[V0 ] + log[AR ] , (4.9)
2λR 2λR

where we are using that the species scale behaves asymptotically as Λ2s = AR e−2λR ∆ϕR . In
total the field range may thus be estimated as
1 1 1 1
 
∆ϕ(V0 ) = − + log[V0 ] + log[AL ] + log[AR ] + ∆ϕbulk . (4.10)
2λL 2λR 2λL 2λR
| {z }
=:b

Thus we can maximize the field range for small V0 by selecting limits with the smallest
coefficient λL/R . From the discussion in the previous section we know that this corresponds to
the case that both asymptotic limits to the left and the right correspond to decompactification
limits to one dimension higher. The last three terms on the r.h.s. in (4.10) are independent of
V0 and can be identified with the constant shift b in (1.1). The sign of b is hence determined
by the value of log AL/R relative to ∆ϕbulk . From the analysis of [30] we expect that the
contribution of non-geometric phases to ∆ϕbulk is of O(1) in Planck units.3 On the other
1
Notice that for certain moduli spaces there is only one infinite distance limit in the fundamental domain,
as happens for instance for the quintic. In that case we fix one of the endpoints at the desert point, meaning
that one tail has zero length ∆ϕL = 0, and we only get contributions from the interior phase ∆ϕbulk and the
other endpoint ∆ϕR .
2
A closely related problem concerns the computation of the diameter of the non-geometric phase of the
moduli space: in [30, 31] this problem has been addressed in examples by calculating the length of geodesics
between, e.g., the LG and conifold points without any reference to scalar potentials. Instead our definition of
∆ϕbulk in terms of the species scale applies to more general situations, as it allows to compute the bulk field
range in any direction of the field space even in the absence of LG points.
3
Reference [31] identifies possible geometries for which ∆ϕbulk can be made larger than unity in Planck
units. However, in the examples studied a hierarchy ∆ϕbulk ≫ 1 cannot be achieved. It would be interesting
to study the relation between ∆ϕbulk and the scaling of AL/R in these examples in more detail.

–9–
hand, by studying the asymptotics of the species scale we show in the following that typically
AL/R ≪ 1, such that their respective contributions to b overcompensate the positive ∆ϕbulk
leading to a negative sign for b. We further confirm this expectation in explicit examples
in section 4.3.

4.2.2 Asymptotics of the field range

While we cannot fix the radius of the interior ∆ϕbulk on general grounds, we can bound
the coefficients appearing in the infinite distance limits ∆ϕL,R in (4.9). The coefficients
λL,R of the species scale are already constrained by (2.3). As we will show now, the O(1)
factors AL,R can be bounded as well when considering the asymptotics of F1 in various

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limits in the large volume regime.
We parametrize this phase by coordinates ti = bi + iv i , with the large volume regime
corresponding to v i ≫ 1. By using the asymptotic behavior of [22] for F1 in this limit, we
can give the behavior of the species scale as
12
Λ2s = + O(v −2 log[v]) , (4.11)
2πc2,i v i

where c2,i are the second Chern class numbers of the Calabi-Yau threefold. We now want to
consider sublimits in the large volume regime, where we take a subset of volumes to infinity.
To this end we split the coordinates as ti = (ba + iv a , bα + iv α ), and scale the volumes as

v a = v̂ a s , v α = v̂ α , (4.12)

where we send s → ∞, with the coefficients v̂ i = (v̂ a , v̂ α ) fixed. Setting the point where
the large volume phase starts to be s = 1 (which may be achieved by rescaling the v̂ i
appropriately), we can compute the distance in terms of s as
r
n
∆ϕ = log[s] , (4.13)
2
where the integer n = 1, 2, 3 depends on the choice of limiting coordinates v a and the fibration
structure of the Calabi-Yau manifold. This allows us to express the species scale (4.11)
in terms of the distance as
12 p2
− n ∆ϕ
Λ2s = e . (4.14)
2πc2,a v̂ a

From this expression we can identify the order one coefficient A in (3.7) as
12
A= , (4.15)
2πc2,a v̂ a

where we note that the sum over second Chern class coefficients c2,a runs only over the
volumes v a which are taken to infinity in our limit. For the case of an emergent string limit
(n = 1) — which corresponds to a K3-fibration where we send only the volume v 1 of the
P1 base to infinity — we have c2,1 = 24 leading to
1
AES = . (4.16)
4πv̂ a

– 10 –
Identifying v̂ 1 with the string coupling in the dual heterotic picture, we are led to bounding
v̂ 1 ≳ 1 and thus AES ≲ 1/4π. In one-dimensional (n = 3) and two-dimensional (n = 2)
decompactification limits demanding the volumes to be bounded from below yields similar
upper bounds v̂ a ≳ 1 on A. Together with the condition that the second Chern class
coefficients are integer quantized, c2,i ∈ N, this leads to A ≤ O(1). As a consequence of
the non-negativity of each of the c2,i , in multi-moduli limits the parameter is generically
much smaller than one. In the following we show that even for one-parameter models A is
sufficiently small such that the constant shift b in (4.10) is negative.

4.3 Examples

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In this section we use the bound V ≤ Λ2s to determine the range of constant potentials
V (ϕ) = V0 in examples of Calabi-Yau compactifications, taking the same models as considered
in [5, 6]. For this field range we consider geodesic paths whose endpoints move further towards
infinite distance points as we decrease V0 .4 By using the explicit form (4.7) of the species
scale we are able to give a numerical relation between the distance ∆ϕ and the value of
the potential V0 in all examples.

4.3.1 Example 1: (K3 × T 2 )/Z2

We begin with a compactification of Type IIA on Enriques Calabi-Yau (K3 × T 2 )/Z2 . We
suppress all dependence on the K3-moduli, considering only the Kähler parameter of the
two-torus T 2 . In that case the topological free energy is given by [22]
"
3Γ( 13 )
#
h i
2 2
F1 = −6 log i(t̄ − t)|η (t)| + 6 log , (4.17)
16π 4

which we normalized such that F1 (e2πi/3 ) = 0. In order to gain intuition for the behavior of
the species scale we have provided a plot of Λs against the field space distance ϕ in figure 3.
Using the explicit expression (4.17) for F1 — and thus via (4.7) for the species scale
Λs — we can compute the range of the potential as a function of V0 by requiring V0 ≤ Λ2s .
We compute this diameter by considering a path5
1 √
t(s) = − + is , 3/2 ≤ s ≤ sES , (4.18)
2
with s the affine parameter. The starting point is located at the desert point, while the
endpoint sES lies along the emergent string limit s ≫ 1, defined by where the inequality
V0 ≤ Λ2s is saturated. The distance between these two endpoints is computed with the
standard hyperbolic metric Kss = 1/2s2 on the moduli space, which yields
1  √ 
∆ϕ = √ log[sES ] − log[ 3/2] . (4.19)
2
4
When the fundamental patch of the moduli space has only one infinite distance limit, such as e.g. the
quintic, we fix one of the two endpoints at the desert point.
5
Note that we have made the simplifying assumption here that the axion stays fixed for the geodesic of
maximal length in our reduced field space. In principle all axion values − 12 ≤ a ≤ 12 should be considered for
the endpoint where Λ2s = V0 . However, as we go further along the infinite distance limit, the contribution
from the axionic field displacement becomes exponentially small in the distance, as it is bounded from above
by a triangle inequality argument by the length of the segment − 12 ≤ a ≤ 12 at the endpoint s = sES .

– 11 –
 Λ2s

0.01

10-6

10-10

JHEP02(2024)175
ϕ
0 5 10 15 20

Figure 3. Logarithmic plot of the square of the species scale for Enriques Calabi-Yau (K3 × T 2 )/Z2
against the distance ϕ in scalar field space. As path in field space we take the desert point t = 12 (−1 +
√
i 3) as starting point, and move towards t = i∞ while keeping the real part of t — the axion —
fixed. We see that the asymptotic exponential behavior of Λ2s , corresponding to linear behavior in this
logarithmic plot, sets in within roughly one Planck length ϕ ≳ 1.

In order to give a precise characterization of the relation between the range ∆ϕ and the gap
V0 we expand Λs for large imaginary t = − 12 + is. In this limit the exact expression (4.17)
can be approximated by
1
Λ2s = + O(s−2 log[s]) . (4.20)
2πs
This approximation suffices to determine the relation between ∆ϕ and V0 for small V0 as
1 1 √ 1
∆ϕ = − √ log[V0 ] − √ log[ 3π] ≃ − √ log[V0 ] − 1.198 . (4.21)
2 2 2
√
The first term gives the expected exponential relation between V0 and ∆ϕ, with 1/ 2 the
coefficient of the emergent string limit. In the light of (4.10) the negative second term can
be attributed to the sum of the λ1 log A = √12 log 2π 1
≈ −1.3 and ∆ϕbulk ≈ 0.1 measuring
the distance between the desert point and t = − 12 + i where the exponential behavior of
the species scale sets in.

4.3.2 Example 2: mirror quintic X5 (15 )

Next we consider scalar potentials over the one-dimensional Kähler moduli space of the mirror
quintic. We parametrize this moduli space by a coordinate x; it has an infinite distance
limit corresponding to the large volume point(s) at x = ∞, conifold point(s) at x = 1, and
a Landau-Ginzburg point at x = 0. As studied in [5], the desert point is located at the
LG-point. We take this desert point as the starting point for computing the range of the
potentials, and move the other endpoint out towards the large volume point. To be precise,
we consider the geodesics parameterized by

x(s) = eπi/5 s , 0 ≤ s < sLCS , (4.22)

– 12 –
 Λ2s
Λ2s
1.0

0.100
0.9

0.010
0.8

0.001

0.7

10-4

0.6

10-5

0.5

JHEP02(2024)175
10-6

0.4
10-7
0.1 0.2 0.3 0.4 0.5 0.6
ϕ 0 5 10 15
ϕ

(a) LG phase. (b) Large volume phase.

Figure 4. Plot of the square of the species scale for the quintic against the distance ϕ in scalar field
space along (4.22). In 4(a) we considered the LG phase, with the right-end corresponding to its end
at |x| = 1; in 4(b) the large volume phase starting from this edge |x| = 1.

avoiding the conifold singularities. The endpoint sLCS is defined as the point where the
species scale Λ2s crosses the scalar potential V0 . We have provided two plots for the species
scale as a function of distance along this path for both the non-geometric phase close to the
LG point and as we move towards the large volume point in figure 4.
We next want to provide a numerical relation between the field range ∆ϕ and the gap of
the scalar potential V0 in the large volume phase. To this end we can use the asymptotics
of the species scale as
3
Λ2s = + O(s−2 log[s]) , (4.23)
25πs
where s is the Kähler parameter. For the field range ∆ϕ we can compute the distance in
both the LG phase and the large volume phase along the geodesic (4.22), where we take the
endpoint to be given by where Λ2s in (4.23) crosses V0 . Altogether this yields
r
3
∆ϕ ≃ − log[V0 ] − 3.798 , (4.24)
2
p
where the constant −3.79 indeed matches with the plot in figure 4(b). The coefficient 3/2
matches with the expectation for a decompactification limit from four to five dimensions,
in which case the species scale bound scales as
− √2 d
V0 ≲ Λ2s ∼ e 6 . (4.25)

4.3.3 Example 3: mirror bicubic X3,3 (16 )

For our next example we consider the bicubic: its moduli space contains a K-point, conifold
point and a large volume point, located at x = 0, x = xc = 3−9 and x = ∞, respectively. Note
that there is no orbifold point in this moduli space, so this allows us to test our bound (3.5)

– 13 –
in an example with a desert point at a generic point in moduli space; in [5] it was found
to be located on the real line between the conifold point and K-point.
For our geodesic we consider a path along the real line x > 0 that crosses through
the desert and conifold point, with its endpoints in the asymptotic regimes of the large
volume and K-point, given by

x(s) = s , sK ≤ s ≤ sLV , (4.26)

where the endpoints sK and sLV correspond to where the species scale Λ2s crosses the gap
of the scalar potential V = V0 .

JHEP02(2024)175
Let us now compute the field range of this constant scalar potential by breaking up the
distance as described by (4.10) into two asymptotic segments and a bulk piece. Assuming V0
is sufficiently small we traverse through the entire conifold phase; we compute the diameter
of this interior phase between the two half-points x = xc /2 and x = 3xc /2 to be

∆ϕbulk ≃ 0.210 . (4.27)

In turn, we consider the large volume and K-point phase separately. For both we compute
the distance from the respective half-points xc /2 and 3xc /2 up to the points sLV and sK
where V0 = Λ2s . This yields functional expressions for the distances as a function of the
gap V0 , where for the LCS phase we find
r
3
∆ϕLV ≃ − log[V0 ] − 4.334 , (4.28)
2
while for the K-point we find
1
∆ϕK ≃ − √ log[V0 ] − 0.692 . (4.29)
2
Putting all these three distances together we find
√
1+ 3
∆ϕ = ∆ϕbulk + ∆ϕLV + ∆ϕK ≃− √ log[V0 ] − 4.815 . (4.30)
2
Note that this yields a larger field range for the potential compared to the quintic in (4.24),
√
as we have an additional log[V0 ]/ 2 coming from moving the other endpoint to the K-point
(instead of fixing it at the desert point).

5 Concluding remarks

In this paper we obtained a bound on the range of V for slowly varying fields using the
fact that in a consistent EFT, V < Λ2s . Moreover, using bounds on the asymptotic form
of Λs coming from the distance conjecture and the emergent string conjecture, we found
that our result is consistent with an asymptotic bound that was obtained from a completely
different reasoning namely by using the TCC [7]. Our result in this paper refines this result
in that it leads to a computable bound everywhere, not just asymptotically, and allows
in the asymptotic regions to find the subleading O(1) shifts in the bound. Given the fact

– 14 –
that the nature of our argument is completely distinct from reasoning behind the TCC, it
lends further support to the TCC itself.
It is natural to wonder what implications our results have for inflationary models, as
having a large field range is a desirable feature in many inflation models. In these models
the field range is given in terms of the Lyth bound (setting, again, the reduced Planck
scale Mpl = 1) [32]
√
∆ϕ = 2ϵ N ,

where N is the number of e-folds and ϵ is the slow roll parameter ϵ = 12 (V ′ /V )2 . The

JHEP02(2024)175
condition for inflation to address the horizon problem the number, N , of e-fold expansion
during inflation should satisfy
1/4
V

N
e ≥ ,
Λ

where V is the inflation potential and Λ ∼ 0.7 × 10−120 is the dark energy in reduced Planck
units (where here we are using the coincidence of matter-radiation equality temperature with
√
the onset of dark energy dominance). Using the bound in this paper (∆ϕ ≲ 6 log(1/V )),
and requiring ϵ to be a small number we find (in reduced Planck units)
√
V ≲ 5 × 10−18 ϵ
,

which is not a strong bound. This bound is very sensitive to the largest possible value for
the coefficient of the log(1/V )-term in ∆ϕ, which we fixed here using the emergent string
conjecture. Notice that inflation in the context of TCC [33] puts much stronger restrictions
on V and ϵ, since TCC leads to an upper bound on the scale of inflation:
1/4
1 V 1

N
e < → < √ → V 1/4 < Λ1/12 .
Hinf Λ V

Interestingly enough this scale Λ1/12 = M̂pl ∼ 1010 GeV is the Planck scale in the dark
dimension scenario, which predicts one extra dimension in our universe [34] in the micron
range! In that context the smallness of V requires no fine tuning, namely it states:

V 1/4 < Λ1/12 → Λinf < M̂pl .

Moreover to get the observed power spectrum for fluctuations one would need

V
 
∼ 10−9 ,
ϵ

which leads to ϵ ∼ 109 V ∼ 109 Λ1/3 ∼ 10−30 for the upper range of V . Thus the bound we
get for the V is well satisfied with such a small ϵ. Equivalently, the Lyth bound implies that
√
the field range for the inflaton field is related to N by an extra factor of ϵ leading to

∆ϕ ∼ 10−15 Mpl ∼ 10−5 M̂pl .

– 15 –
which is far smaller than the upper bound derived in this work. This small value for the
potential and field range is a strong fine tuning from the 4d perspective. However from the
view point of the dark dimension scenario [34] all we are led to is that the inflation scale is
no higher than the 5d Planck scale. The inflationary period then starts near the top of the
potential where it is flat (as ϵ ≪ 1) and the required field range is still a bit fine tuned to
10−5 , but not as much as before. So the dark dimension scenario somewhat reduces, but does
not eliminate the tension, between inflation and the Swampland conditions. 6

Acknowledgments

JHEP02(2024)175
We would like to thank L. Anchordoqui, I. Antoniadis, A. Bedroya, D. Lüst, and I. Valenzuela
for interesting discussions and correspondence. The work of CV, MW and DW is supported
by a grant from the Simons Foundation (602883,CV), the DellaPietra Foundation, and by
the NSF grant PHY-2013858.

Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.

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