Gopakumar-Vafa Invariants and the
Emergent String Conjecture

Tom Rudelius1,2

1Department of Mathematical Sciences, Durham University, Durham DH1 3LE United
Kingdom
2Department of Physics, University of California, Berkeley, CA 94720 USA

The Emergent String Conjecture of Lee, Lerche, and Weigand holds that every
infinite-distance limit in the moduli space of a quantum gravity represents
either a decompactification limit or an emergent string limit in some duality
frame. Within the context of 5d supergravities coming from M-theory
compactifications on Calabi-Yau threefolds, we find evidence for this
conjecture by studying (a) the gauge couplings and (b) the BPS spectrum, which
is encoded in the Gopakumar-Vafa invariants of the threefold. In the process,
we derive a testable geometric consequence of the Emergent String Conjecture,
and we verify that it is satisfied in all complete intersection Calabi-Yau
threefolds in products of projective spaces (CICYs).

###### Contents

  1. 1 Introduction
  2. 2 Review of Supergravity, Calabi-Yau Geometry, and Dimensional Reduction
    1. 2.1 5d Supergravity and Calabi-Yau Threefolds
    2. 2.2 Dimensional Reduction
  3. 3 Decompactification and Emergent Strings
  4. 4 Gopakumar-Vafa Invariants and Infinite-Distance Limits
  5. 5 Numerical Sweep
  6. 6 Discussion

##  1 Introduction

The study of the Landscape and the Swampland of quantum gravity has seen a
surge of excitement in recent years. Among the plethora of new ideas, the
Emergent String Conjecture Lee:2019wij  has emerged as one of the most
exciting. This conjecture holds that any infinite-distance limit in the moduli
space of a quantum gravity theory is either a decompactification limit (in
which extra compact dimensions grow to infinite size) or else it is an
emergent string limit (in which a fundamental string becomes tensionless).

Compelling evidence for the Emergent String Conjecture has been given in
various contexts in string theory Lee:2019xtm ; Lee:2019wij ; Baume:2019sry ;
Xu:2020nlh ; Basile:2022zee ; Etheredge:2022opl ; Blumenhagen:2023yws .
However, to date there has been little progress towards a bottom-up argument
for the Emergent String Conjecture, as there is for other Swampland
conjectures like the absence of global symmetries. To this end, the first part
of this paper, §3, provides evidence for the Emergent String Conjecture in 5d
supergravity by studying the scaling of gauge couplings in infinite-distance
limits, expanding on previous work in Etheredge:2022opl . Notably, this
argument does not rely on any UV input from string/M-theory, and instead it
depends solely on the cubic structure of the prepotential in five dimensions.

In the second part of this paper, §4, we provide an additional top-down
argument for the Emergent String Conjecture by studying the spectrum of BPS
states that become light in infinite-distance limits in 5d supergravity. For
M-theory compactified on a Calabi-Yau threefold, this BPS spectrum is encoded
in the Gopakumar-Vafa invariants of the threefold, which can be computed in a
large class of examples using the methods of Hosono:1993qy ; Hosono:1994ax .
In a decompactification limit to six dimensions, we expect a light tower of
BPS Kaluza-Klein modes, each of which is labeled by a Kaluza-Klein momentum
$q\in\mathbb{Z}$, which does not grow indefinitely as $q$ increases. In
contrast, an emergent string limit features a tower of string oscillator
modes, which grows exponentially with increasing mass. By studying how the
Gopakumar-Vafa invariants $n_{q}$ scale with increasing $q$, therefore, we
find evidence as to whether the infinite-distance limit in question represents
a decompactification limit or an emergent string limit.

Comparing this with the analysis of §3 for several simple Calabi-Yau
threefolds, we find perfect agreement: the GV invariants scale like a Kaluza-
Klein tower precisely when the gauge couplings scale like a decompactification
limit, and the GV invariants scale like a tower of string oscillator modes
precisely when the gauge couplings scale like an emergent string limit. Taken
together, this gives strong evidence that these limits are indeed
decompactification limits and emergent string limits in appropriate duality
frames, consistent with the Emergent String Conjecture.

Our analysis also yields a novel and surprising geometric implication of the
Emergent String Conjecture for Calabi-Yau manifolds, which can be tested in
examples through the computation of Gopakumar-Vafa invariants and (classical)
triple intersection numbers. In §5, we carry out these tests in 7820 examples
of Calabi-Yau threefolds, extending the analysis of §4 to the entire set of
Calabi-Yau threefolds which arise as (favorable) complete intersections of
hypersurfaces in products of projective spaces (CICYs). For all such CICYs, we
again find perfect agreement between the scaling of the GV invariants and the
scaling of the gauge couplings, strongly suggesting that each infinite-
distance limit in the vector multiplet moduli spaces of these theories
represents either an emergent string limit or a decompactification limit. This
is remarkable because all of the limits we consider–even the
decompactification limits–arise from shrinking one of the $\mathbb{P}^{n}$
factors of the ambient space. It is a nontrivial prediction of the Emergent
String Conjecture that shrinking this cycle actually corresponds to
decompactifying a circle in a different duality frame, but our analysis
provides nontrivial evidence that this is indeed the case.

Before we get to the crux of the matter, we first review relevant aspects of
5d supergravity, Calabi-Yau geometry, and Kaluza-Klein compactifications from
six to five dimensions; this is covered in the following section.

##  2 Review of Supergravity, Calabi-Yau Geometry, and Dimensional Reduction

In this section, we first review supergravity theories in five dimensions and
their construction through compactification of M-theory on a Calabi-Yau
threefold. We then review the towers of light states that appear when such a
theory is decompactified to six dimensions by taking one of the extra
dimensions to be large.

###  2.1 5d Supergravity and Calabi-Yau Threefolds

Many features of a 5d supergravity theory are captured by its prepotential, a
cubic homogeneous polynomial:

$\mathcal{F}=\frac{1}{6}C_{IJK}Y^{I}Y^{J}Y^{K}.$ (1)

In an M-theory compactification to 5d on a Calabi-Yau threefold $X$, indices
$I,J,K$ run from $1$ to $h^{1,1}(X)$, the constants $C_{IJK}$ are the triple
intersection numbers of the manifold, and the moduli $Y^{I}$ are volumes of
certain two-cycles.

The prepotential is subject to the constraint $\mathcal{F}=1$, which follows
from the fact that the overall volume of the Calabi-Yau is not a vector
multiplet modulus in 5d, so the vector multiplet moduli space has dimension
$h^{1,1}(X)-1$. In this work, however, we will drop this constraint by instead
taking the $Y^{I}$ to be homogeneous coordinates, which are identified under
simultaneous rescaling, $Y^{I}\sim\lambda Y^{I}$, $\lambda>0$. Of course, this
is simply a choice of convention, which does not affect our results in any
way.

At a generic point in moduli space, the gauge group is $U(1)^{h^{1,1}(X)}$,
and the gauge kinetic matrix is given by

$a_{IJ}=\frac{\mathcal{F}_{I}\mathcal{F}_{J}}{\mathcal{F}^{4/3}}-\frac{\mathcal{F}_{IJ}}{\mathcal{F}^{1/3}},$
(2)

with

$\mathcal{F}_{I}=\partial_{I}\mathcal{F}=\frac{1}{2}C_{IJK}Y^{J}Y^{K},\qquad\mathcal{F}_{IJ}=\partial_{I}\partial_{J}\mathcal{F}=C_{IJK}Y^{K}.$
(3)

The eigenvalues of the gauge kinetic matrix correspond to the inverse-squares
of gauge couplings, $\lambda_{I}\sim 1/g_{I}^{2}$. This means that an
eigenvalue of $a_{IJ}$ diverges precisely when a gauge coupling vanishes.
Conversely, a gauge coupling diverges precisely when an eigenvalue of $a_{IJ}$
vanishes. By electromagnetic duality, this happens precisely when the gauge
coupling of a magnetic 2-form gauge field vanishes.

Meanwhile, the metric on moduli space in homogeneous coordinates is given by

$g_{IJ}=\frac{2}{3}\frac{\mathcal{F}_{I}\mathcal{F}_{J}}{\mathcal{F}^{2}}-\frac{\mathcal{F}_{IJ}}{\mathcal{F}}.$
(4)

This metric is positive-semidefinite: all eigenvalues are positive except for
one null eigenvalue, which corresponds to rescaling $Y^{I}\rightarrow\lambda
Y^{I}$.

The BPS bound is given by

$m(q_{I})\geq\left(\frac{\sqrt{2}\pi}{\kappa_{5}}\right)^{1/3}|Z|=\left(\frac{\sqrt{2}\pi}{\kappa_{5}}\right)^{1/3}\frac{|q_{I}Y^{I}|}{\mathcal{F}^{1/3}}\,,$
(5)

where $q_{I}\in\mathbb{Z}$ is the charge of the particle under the $I$th
$U(1)$, and $\kappa_{5}^{2}=1/(8\pi G)$. Particles that saturate the BPS bound
are called BPS particles, and the number of BPS particles of charge $q_{I}$ is
counted by the Gopakumar-Vafa invariant $n_{q_{I}}$ Gopakumar:1998jq ;
Gopakumar:1998ii .111More precisely, Gopakumar-Vafa invariants compute an
index of the number of BPS particles, so there can be a cancellation between
BPS particles of different spins. This will not be important for our purposes,
however.

This concludes our lightning review of 5d supergravity and Calabi-Yau
geometry. More details can be found in e.g. Bergshoeff:2004kh ; Alim:2021vhs .

###  2.2 Dimensional Reduction

Consider an Einstein-dilaton theory coupled to a 2-form gauge field in six
dimensions

$S=\int\textrm{d}^{6}x\sqrt{-g}\left(\frac{1}{2\kappa_{6}^{2}}{\cal
R}_{D}-\frac{1}{2}(\nabla\phi)^{2}-\frac{1}{2g_{0}^{2}}{\rm
e}^{2\phi}|H_{3}|^{2}\right)\,.$ (6)

An action of this form shows up, for instance, in minimal supergravity coupled
to a tensor multiplet. In the limit $\phi\rightarrow\infty$, the 2-form gauge
field becomes weakly coupled, with gauge coupling

$g_{B,6}^{2}=g_{0}^{2}{\rm e}^{-2\phi}\rightarrow 0\,.$ (7)

The Weak Gravity Conjecture for strings Arkanihamed:2006dz  implies that in
this limit, a tensionless string emerges, with tension $T\sim
g_{B,6}/\kappa_{6}$. This leads to an infinite tower of charged particles
beginning at the string scale, $M_{\rm string}=\sqrt{2\pi
T}\sim\sqrt{g_{B,6}/\kappa_{6}}\sim\exp(-\phi/2)$.

Next, suppose that we dimensionally reduce this theory on a circle, with
dimensional reduction ansatz

$ds_{6}^{2}={\rm e}^{\frac{-\rho(x)}{\sqrt{3}}}d{\hat{s}}_{5}^{2}(x)+{\rm
e}^{\sqrt{3}\rho(x)}dy^{2},$ (8)

This yields a canonically normalized radion field $\rho$ in five dimensions,
in addition to the canonically normalized dilaton $\phi$ that descends from
six dimensions. With this, the 5-dimensional gauge coupling for the string
picks up an exponential dependence on the radion:

$g_{B}^{2}=(2\pi R)g_{0}^{2}{\rm e}^{-2\phi-\frac{1}{\sqrt{3}}\rho}\,.$ (9)

This leads to a tower of string oscillator modes beginning at the string
scale, $M_{\rm string}=\sqrt{2\pi
T}\sim\sqrt{g_{B}/\kappa_{5}}\sim\exp(-\frac{1}{2}\phi-\frac{1}{2\sqrt{3}}\rho)$.

Circle reduction also leads to a Kaluza-Klein photon, with gauge coupling

$e_{\text{KK}}^{2}=\frac{2\kappa_{5}^{2}}{R^{2}}{\rm
e}^{-\frac{4}{\sqrt{3}}\rho}\,.$ (10)

The tower Weak Gravity Conjecture Arkanihamed:2006dz ; Heidenreich:2015nta ;
Andriolo:2018lvp  implies that in the decompactification limit
$e_{\text{KK}}\rightarrow 0$, there is a tower of particles beginning at the
scale $m_{\text{KK}}\sim e_{\text{KK}}/\kappa_{5}\sim\exp(-2\rho/\sqrt{3})$.
Of course, this is nothing other than the Kaluza-Klein scale, and the tower in
question is simply a tower of Kaluza-Klein modes.

Figure 1: The scalar charge-to-mass vectors, $\zeta_{i}=-\partial_{i}\log(m)$,
for Kaluza-Klein modes and string oscillator modes for a compactification from
six to five dimensions. In the string decompactification limit
$\rho\rightarrow\infty$, Kaluza-Klein modes and string oscillator modes become
light with coefficients $\lambda_{\text{KK}}=2/\sqrt{3}$ and
$\lambda_{\text{string}}=1/\sqrt{12}$. In the emergent string limit
$\rho,\phi\rightarrow\infty$ with $\textrm{d}\rho/\textrm{d}\phi=1/\sqrt{3}$,
Kaluza-Klein modes and string oscillator modes become light with coefficient
$\lambda_{\text{KK}}=\lambda_{\text{string}}=1/\sqrt{3}$. In the intermediate
regime (shaded blue), $\rho,\phi\rightarrow\infty$ with
$\textrm{d}\rho/\textrm{d}\phi=\tan\vartheta$, Kaluza-Klein modes and string
oscillator modes become light with coefficients
$\lambda_{\text{KK}}=\frac{2}{\sqrt{3}}\sin(\vartheta)$,
$\lambda_{\text{string}}=\frac{1}{2}\cos(\vartheta)+\frac{1}{2\sqrt{3}}\sin(\vartheta)$.

It is helpful to define the scalar charge-to-mass vector of a particle of mass
$m$ as Calderon-Infante:2020dhm ; Etheredge:2022opl

$\zeta_{i}\equiv-\frac{\partial}{\partial\phi^{i}}\log m\,,$ (11)

where differentiation is performed with the $d$-dimensional Planck mass held
fixed. With this, we have

$\vec{\zeta}_{\rm
string}=(\frac{1}{2},\frac{1}{2\sqrt{3}})\,,~{}~{}~{}~{}\vec{\zeta}_{\rm
string}=(0,\frac{2}{\sqrt{3}})\,.$ (12)

These vectors are plotted in Figure 1.

The Distance Conjecture Ooguri:2006in  holds that in any infinite-distance
limit in moduli space, a tower of particles must become light with a
characteristic mass scale that decays exponentially as
$m\sim\exp(-\lambda\kappa_{5}\phi)$. From the discussion above, we see that in
the limit $\rho\rightarrow\infty$, the Kaluza-Klein tower satisfies the
Distance Conjecture with a coefficient $\lambda=2/\sqrt{3}$. We refer to this
as the “strict” decompactification limit because the radion is taken to
infinity while the dilaton is held fixed. Meanwhile, in the limit
$\phi,\rho\rightarrow\infty$ with constant slope
$\textrm{d}\rho/\textrm{d}\phi=1/\sqrt{3}$, the tower of string oscillator
modes satisfies the Distance Conjecture with $\lambda=1/\sqrt{3}$. This is an
emergent string limit, in the terminology of Lee:2019wij . In a more general
decompactification limit $\rho,\phi\rightarrow\infty$ with constant slope
$\textrm{d}\rho/\textrm{d}\phi\equiv\mathfrak{m}$, let us define a normalized
tangent vector $\hat{t}=(\cos\vartheta,\sin\vartheta)$, so that
$\tan\vartheta=\mathfrak{m}$. With this, the Kaluza-Klein modes and the string
oscillator modes satisfy the Distance Conjecture in this limit with respective
coefficients

$\lambda_{\text{KK}}=\hat{t}\cdot\vec{\zeta}_{\rm
KK}=\frac{2}{\sqrt{3}}\sin(\vartheta)\,,~{}~{}~{}~{}\lambda_{\text{string}}=\hat{t}\cdot\vec{\zeta}_{\rm
string}=\frac{1}{2}\cos(\vartheta)+\frac{1}{2\sqrt{3}}\sin(\vartheta)\,.$ (13)

For $\mathfrak{m}\geq 1/\sqrt{3}$ (shaded blue in Figure 1), these
coefficients satisfy $1/\sqrt{12}\leq\lambda_{\text{string}}\leq
1/\sqrt{3}\leq\lambda_{\text{KK}}\leq 2/\sqrt{3}$.

##  3 Decompactification and Emergent Strings

Consider a path in vector multiplet moduli space of the form
$Y^{I}(s),s\in[0,1]$, and suppose that the limit $s\rightarrow 0$ lies at
infinite distance. As shown in Etheredge:2022opl , if we restrict the path to
take a “straight-line” form in homogeneous coordinates, i.e.,
$Y^{I}(s)=Y^{I}_{0}+sY^{I}_{1}$, then there are two possible behaviors for the
prepotential: either $\mathcal{F}$ vanishes linearly as $s\rightarrow 0$
($\mathcal{F}\sim s$) or else $\mathcal{F}$ vanishes quadratically as
$s\rightarrow 0$ ($\mathcal{F}\sim s^{2}$). In the former case, the smallest
and largest gauge couplings scale as

$g_{\min}\sim\exp(-\frac{2}{\sqrt{3}}\rho)\,,~{}~{}~{}\frac{1}{\sqrt{g_{\max}}}\sim\exp(-\frac{1}{\sqrt{12}}\rho)\,.$
(14)

where $\rho$ is a canonically normalized scalar field which diverges as
$\rho\rightarrow\infty$ as $s\rightarrow 0$. Comparing with §2.2, we see that
this precisely matches the scaling expected for a strict decompactification
limit to six dimensions, in which $g_{\min}$ represents the gauge coupling of
the Kaluza-Klein photon, $2\pi/g_{\max}$ represents the gauge coupling for the
2-form gauge field of a fundamental string in six dimensions, and $\rho$
represents the radion field.

The tower Weak Gravity Conjecture (WGC) implies that as a gauge coupling $g$
tends to zero, a tower of light charged particles will appear beginning at the
mass scale $m\sim g/\kappa_{5}$. In this case, setting $g=g_{\min}$, we find a
tower of particles whose mass decays in the limit $s\rightarrow 0$ as
$\exp(-\lambda\rho)$, with $\lambda=2/\sqrt{3}$. Meanwhile, the WGC for
strings implies that as a 2-form gauge coupling $g_{\text{string}}$ tends to
zero, a tower of string oscillator modes will appear at the string scale
$M_{\text{string}}\sim\sqrt{g_{\text{string}}}/\kappa_{5}^{1/2}$. Setting
$g_{\text{string}}\sim 1/g_{\max}$, we find a tower of string oscillator modes
whose mass decays in the limit $s\rightarrow 0$ as $\exp(-\lambda\rho)$, with
$\lambda=1/\sqrt{12}$. This precisely matches the expected scaling for a
circle compactification of string theory in six dimensions, and it strongly
suggests that the limit $\rho\rightarrow\infty$ represents a
decompactification limit in some duality frame.

In the second case, $\mathcal{F}\sim s^{2}$, the gauge couplings scale as

$g_{\min}\sim\exp(-\frac{1}{\sqrt{3}}\phi)\,,~{}~{}~{}\frac{1}{\sqrt{g_{\max}}}\sim\exp(-\frac{1}{\sqrt{3}}\phi)\,.$
(15)

where $\phi$ is a canonically normalized scalar field which diverges as
$\phi\rightarrow\infty$ as $s\rightarrow 0$. Comparing with §2.2, we see that
this precisely matches the expected scaling for an emergent string limit,
where.again $g_{\min}$ represents the gauge coupling of a Kaluza-Klein photon
and $2\pi/g_{\max}$ represents the gauge coupling for the 2-form gauge field
of a fundamental string. Invoking the tower WGC and the magnetic WGC for
strings, respectively, we find a tower of particles with $\lambda=1/\sqrt{3}$
and a tower of string oscillator modes with $\lambda=1/\sqrt{3}$; this
precisely matches the scaling expected for an emergent string limit.

This cannot be the full story, however. Dimensional reduction of a 6d string
theory gives a tower of Kaluza-Klein modes with
$m_{\text{KK}}\sim\exp(-2\rho/\sqrt{3})$ and a tower of string oscillator
modes with $M_{\text{string}}\sim\exp(-\phi/2-\rho/(2\sqrt{3}))$. The two
cases we considered above correspond respectively to the limit
$\rho\rightarrow\infty$, $\phi$ fixed and the limit
$\rho,\phi\rightarrow\infty$, $\textrm{d}\rho/\textrm{d}\phi=1/\sqrt{3}$
fixed. But this does not include the intermediate regime with
$\rho,\phi\rightarrow\infty$, $\textrm{d}\rho/\textrm{d}\phi>1/\sqrt{3}$
fixed. How are we to see these limits within the 5d vector multiplet moduli
space?

The answer is that these limits require curved paths in the space of
homogeneous coordinates. Consider once again a path of the form $Y^{I}(s)$,
and let the limit $s\rightarrow 0$ be at infinite distance. By a suitable
linear transformation, we may set $Y^{I}(s=0)=\delta_{1}^{I}$. Then, suppose
$Y^{I}(s)$ takes the form:

$Y^{1}=1\,,~{}~{}~{}Y^{2}=s\,,~{}~{}~{}Y^{3}=s^{\alpha}\,,~{}~{}~{}Y^{I}=0\,,~{}~{}~{}I>3\,.$
(16)

for $0<\alpha<1$. With this, the relevant terms in the prepotential
$\mathcal{F}=C_{IJK}Y^{I}Y^{J}Y^{K}/6$ are those with $I,J,K=1,2,3$. Since we
have assumed that $s\rightarrow 0$ lies at infinite distance, we must have
$C_{111}=0$. Let us further assume that $C_{333}=C_{133}=C_{113}=C_{112}=0$:
this assumption will prove necessary for our purposes, as we will see
momentarily. As a result, the nonzero terms in the prepotential are given by

$\displaystyle\mathcal{F}=\frac{1}{6}\Big{(}$ $\displaystyle
C_{222}(Y^{2})^{3}+3C_{122}Y^{1}(Y^{2})^{2}+6C_{123}Y^{1}Y^{2}Y^{3}+3C_{223}(Y^{2})^{2}Y^{3}+3C_{233}Y^{2}(Y^{3})^{2}\Big{)}\,.$
(17)

In the limit $s\rightarrow 0$, the proper distance along the path scales with
$s$ as

$\ell(s)=\frac{1}{\sqrt{2}}\int_{s}^{1}ds^{\prime}\sqrt{\mathfrak{g}_{IJ}\dot{Y}^{I}\dot{Y}^{J}}\,,$
(18)

where $\dot{~{}}$ indicates differentiation with respect to the parameter
$s^{\prime}$, and $g_{IJ}$ is given by (4). Plugging in the prepotential in
(17), we find

$\ell(s)=\frac{1}{\sqrt{3}}\sqrt{\alpha^{2}-\alpha+1}|\log(s)|+...$ (19)

where $...$ represents terms that are finite in the limit $s\rightarrow 0$.
Meanwhile, the smallest and largest gauge couplings scale with $s$ as

$\displaystyle g_{\min}$ $\displaystyle\sim\frac{1}{\sqrt{a_{11}}}\sim
s^{(2-\alpha)/3}\sim\exp(-\lambda_{\text{KK}}\ell(s))$ (20)
$\displaystyle\frac{1}{\sqrt{g_{\max}}}$ $\displaystyle\sim a_{00}^{1/4}\sim
s^{(\alpha+1)/6}\sim\exp(-\lambda_{\text{string}}\ell(s))\,,$ (21)

where

$\lambda_{\text{KK}}=\frac{1}{\sqrt{3}}\frac{2-\alpha}{\sqrt{\alpha^{2}-\alpha+1}}\,,~{}~{}~{}\lambda_{\text{string}}=\frac{1}{2\sqrt{3}}\frac{\alpha+1}{\sqrt{\alpha^{2}-\alpha+1}}\,.$
(22)

Recall that the tower WGC implies a tower of charged particles beginning at
the scale $g_{\text{min}}$, and the WGC for strings implies a tower of string
oscillator modes beginning at the scale $1/\sqrt{g_{\max}}$. The masses of
these towers therefore decay exponentially in the limit $s\rightarrow 0$ with
coefficients $\lambda_{\text{KK}}$ and $\lambda_{\text{string}}$,
respectively.

Next, we define an angle $\vartheta$ by

$\vartheta=\sin^{-1}\left(\frac{2-\alpha}{2\sqrt{\alpha^{2}-\alpha+1}}\right)\,.$
(23)

With this, we may rewrite the $\lambda$ coefficients as

$\lambda_{\text{KK}}=\frac{2}{\sqrt{3}}\sin(\vartheta)\,,~{}~{}~{}\lambda_{\text{string}}=\frac{1}{2}\cos(\vartheta)+\frac{1}{2\sqrt{3}}\sin(\vartheta)\,.$
(24)

This precisely matches the coefficients of (13)! We see that the tower implied
by the tower WGC matches the expected scaling for a Kaluza-Klein tower, and
the string scale implied by the WGC for strings matches the expected scaling
for a fundamental string in six dimensions. In other words, we have recovered
the “intermediate regime” between the strict decompactification limit
$\rho\rightarrow\infty$ and the emergent string limit
$\textrm{d}\rho/\textrm{d}\phi=1/\sqrt{3}$.

The crucial ingredient here was the introduction of the parameter $\alpha$.
For $\alpha=0$, we recover the expected scaling behavior for the strict
decompactification limit, whereas for $\alpha=1$ we recover the emergent
string limit. As $\alpha$ varies between 0 and 1, we interpolate between these
two limits, as shown in Figure 1. Notably, this result did not rely on any
top-down input from string theory: the observed scaling behavior of the gauge
couplings $g_{\min}$, $g_{\max}$ depends on a delicate interplay between the
gauge kinetic matrix $a_{IJ}$ and the metric $g_{IJ}$, but the low-energy 5d
supergravity relates these two through the prepotential $\mathcal{F}$,
irrespective of the details of the UV completion.

What happens if we drop the assumption $C_{333}=C_{133}=C_{113}=C_{112}=0$? If
any of these coefficients are nonzero, we find that the scaling above no
longer applies. Instead, the presence of any of these terms automatically
yields the limiting cases of a strict decompactification or an emergent string
limit and renders the intermediate regime
$\tan^{-1}(1/\sqrt{3})<\vartheta<\pi/2$ inaccessible.

In this example, the emergent string limit arises only when both $Y^{2}$ and
$Y^{3}$ are taken to zero with finite, nonzero $Y^{2}/Y^{3}$. In fact, for
$h^{1,1}>2$, emergent string limits always require at least two moduli to
vanish. To see this, suppose that only a single modulus vanishes in an
emergent string limit, $Y^{1}=s$ with all other moduli constant. Since
$\mathcal{F}$ must vanish quadratically in an emergent string limit, this
means that $\mathcal{F}$ must factorize as
$\mathcal{F}=(Y^{1})^{2}(a_{I}Y^{I})$, where each $a_{I}$ is c-number
coefficient. However, defining $X=a_{I}Y^{I}$, we find that that prepotential
is given simply by $\mathcal{F}=(Y^{1})^{2}X$: in particular, it depends on
only two homogeneous coordinates $X$, $Y^{1}$. This is possible only if
$h^{1,1}=2$.

For $h^{1,1}>2$, therefore, emergent string limits necessarily involve
multiple vanishing moduli. This strongly suggests that most emergent string
limits arise through the structure we have uncovered above, in which curved
paths interpolate smoothly between a strict decompactification limit and an
emergent string limit.

##  4 Gopakumar-Vafa Invariants and Infinite-Distance Limits

We have seen that infinite-distance limits in vector multiplet moduli space in
5d supergravity feature the correct scaling of the smallest and largest gauge
couplings $g_{\min}$, $g_{\max}$ to correspond to either emergent string
limits or decompactification limits to six dimensions. In the latter case, the
tower WGC implies a tower of light particles with precisely the correct
scaling behavior for a tower of Kaluza-Klein modes.

Towers of Kaluza-Klein modes are not only distinguished by the exponential
scaling of their masses with increasing field distance, however; they are also
distinguished by an approximately constant density of states. Said
differently, the number of particles in a Kaluza-Klein tower of mass $m\approx
nm_{\text{KK}}=n/R$ in 5d is approximately independent of $n$.

In simple cases, the number of Kaluza-Klein modes of a given KK momentum $n$
is exactly independent of $n$, as each particle in 6d gives rise to one
Kaluza-Klein mode of Kaluza-Klein momentum $n$ for each integer $n$. However,
in the presence of discrete symmetries or orbifold actions, certain Kaluza-
Klein may be projected out of the spectrum, resulting in a density of states
that is periodic rather than constant. One example of this is the “simple
orbifold” in §2.2 of Heidenreich:2016aqi : in this example, many Kaluza-Klein
modes of odd charge are projected out of the spectrum by the orbifold action,
and the number of Kaluza-Klein modes of KK momentum $n$ depends only on
whether $n$ is odd or even, but it is order-one in either case.

This is in stark contrast to the exponentially large Hagedorn density of
string oscillator modes which become light in an emergent string limit; here,
the mass of the oscillator mode at level $k$ is given by
$m_{k}\sim\sqrt{k}M_{\text{string}}$, and the number of oscillator modes at
level $k$ scales exponentially with $\sqrt{k}$.222For the heterotic string in
ten dimensions, for example, the degeneracy of oscillator modes at level $n$
scales as $d(k)\sim\exp(2\pi(2+\sqrt{2})\sqrt{k})$ Heidenreich:2017sim .
Sometimes, as in the case of heterotic string theory, string oscillator modes
may carry gauge charge. In the heterotic case, the charge under a $U(1)$ in
the Cartan of the gauge group is quantized, the mass of the lightest state of
a given charge $n$ grows linearly with $n$, while its level $k$ scales as
$k\sim n^{2}$. As a result, the number of states with this mass and charge
grows exponentially with $\sqrt{k}\sim n$.333For the heterotic string, this
exponential growth comes from the exponentially-large number of choices for
the right-moving oscillators needed to level-match the left-moving charge. We
expect that this exponential growth with charge should apply more generally,
and we will see that this expectation is borne out in the examples below.

Within the vector multiplet moduli space of a 5d supergravity arising from
M-theory on a Calabi-Yau threefold, the spectrum of charged BPS states that
become light in an infinite-distance limit are encoded in the Gopakumar-Vafa
(GV) invariants of the threefold. By studying how these GV invariants depend
on the charge $n$, we can put the Emergent String Conjecture to the test.

$\begin{array}[]{c|ccccc}\hbox{\diagbox[width=28.45274pt,height=21.33955pt,innerleftsep=2.84544pt,innerrightsep=0.0pt]{{\shortstack[l]{$q_{2}$}}}{{\shortstack[r]{$q_{1}$}}}{}}&0&1&2&3&4\\\
\hline\cr 0&-&40&4&0&0\\\
1&\pagecolor{orange!25}144&496&496&23953120&2388434784\\\
2&\pagecolor{orange!25}164&5616&23100&34528&23100\\\
3&\pagecolor{orange!25}144&44384&602016&2471824&4709216\\\
4&\pagecolor{orange!25}88&279976&10439512&97922024&398659384\\\
5&\pagecolor{orange!25}144&1482384&136431424&2616030416&20133562480\\\
6&\pagecolor{orange!25}164&6751472&1439003864&52447406096&707697743208\\\
7&\pagecolor{orange!25}144&27208608&12779098368&841622542048&18899196173440\\\
8&\pagecolor{orange!25}88&99569856&98370714948&11277044593704&405560003481888\\\
\end{array}$

Table 1: Genus 0 GV invariants of degree $(q_{1},q_{2})$ for the geometry
$X_{7643}$. The BPS particles that become light in the candidate
decompactification limit $Y\rightarrow 0$ are shaded orange; these GV
invariants are order-one and periodic (of order 4) as a function of $q_{2}$,
consistent with the fact that they represent Kaluza-Klein modes for a circle
in another duality frame, which grows to infinite size as $Y\rightarrow 0$.

As a first example, consider the case of the complete intersection Calabi-Yau
(CICY) $X_{7643}$ with configuration matrix444For a pedagogical introduction
to CICYs and their configuration matrices, see Hubsch:1992nu .

$\left(\begin{array}[]{c || cccc}\mathbb{P}^{2}{}&~{}0&0&2&1\\\
\mathbb{P}^{5}{}&~{}2&2&1&1\\\ \end{array}\right)\,.$ (25)

In other words, this Calabi-Yau consists of the intersection of four
polynomials of bidegree $(0,2)$, $(0,2)$, $(2,1)$, and $(1,1)$, respectively,
inside $\mathbb{P}^{2}\times\mathbb{P}^{5}$. The GV invariants for this
geometry can be computed using the Mathematica notebook of Klemm and Kreuzer
Klemm:2001aaa , which was based on the earlier work Hosono:1994ax ;
Hosono:1993qy , and the result is shown in Table 1.

The mass of a BPS particle of charge $(q_{1},q_{2})$ is given by

$m_{q_{1},q_{2}}=\left(\frac{\sqrt{2}\pi}{\kappa_{5}}\right)^{1/3}\frac{|q_{1}X+q_{2}Y|}{\mathcal{F}^{1/3}}\,,$
(26)

where $X\equiv Y^{1}$ measures the volume of the $\mathbb{P}^{2}$ factor and
$Y\equiv Y^{2}$ measures the volume of the $\mathbb{P}^{5}$ factor. In the
limit $Y\rightarrow 0$ the tower of particles with $q_{1}=0$, $q_{2}=1,2,...$
become massless. Furthermore, from the form of the prepotential,

$\mathcal{F}=2X^{2}Y+6XY^{2}+\frac{4}{3}Y^{3}\,,$ (27)

we see that the limit $Y\rightarrow 0$ represents a candidate
decompactification limit, since the prepotential vanishes linearly in this
limit, $\mathcal{F}\sim Y$. This agrees with the fact that the GV invariants
$n_{0,q_{2}}$ (shaded orange in Table 1) are order-one and periodic as a
function of $q_{2}$, as expected for a Kaluza-Klein tower. This is especially
remarkable when these GV invariants are compared with the GV invariants for
$q_{1}\neq 0$, which grow exponentially with increasing $q_{1}$ or $q_{2}$.

$\begin{array}[]{c|ccccc}\hbox{\diagbox[width=28.45274pt,height=21.33955pt,innerleftsep=2.84544pt,innerrightsep=0.0pt]{{\shortstack[l]{$q_{2}$}}}{{\shortstack[r]{$q_{1}$}}}{}}&0&1&2&3&4\\\
\hline\cr 0&-&24&0&0&0\\\ 1&\pagecolor{blue!25}396&1152&396&0&0\\\
2&\pagecolor{blue!25}2610&53136&112068&53136&2610\\\
3&\pagecolor{blue!25}35640&2377728&15951564&28024704&15951564\\\
4&\pagecolor{blue!25}605844&103323672&1602730872&6746381496&10576809936\\\
5&\pagecolor{blue!25}12212172&4400303616&132192153792&1084701369600&3472953972948\\\
6&\pagecolor{blue!25}273644244&184590071136&9593083752300&135592015659408&762494479579314\\\
\end{array}$

Table 2: Genus 0 GV invariants of degree $(q_{1},q_{2})$ for the geometry
$X_{7806}$. The BPS particles that become light in the emergent string limit
$Y\rightarrow 0$ are shaded blue; the exponential growth of these GV
invariants with $q_{2}$ is consistent with the exponential, Hagedorn growth of
string states with increasing mass.

As a second example, we consider the case of the CICY $X_{7806}$ with
configuration matrix

$\left(\begin{array}[]{c || cccc}\mathbb{P}^{1}{}&~{}0&2\\\
\mathbb{P}^{4}{}&~{}3&2\\\ \end{array}\right)\,.$ (28)

In other words, this Calabi-Yau consists of the intersection of two
polynomials of bidegree $(0,3)$ and $(2,2)$, respectively, inside
$\mathbb{P}^{1}\times\mathbb{P}^{4}$. The GV invariants for this geometry are
shown in Table 2. Here, the mass of a BPS particle of charge $(q_{1},q_{2})$
is again given by (26), where now $X\equiv Y^{1}$ measures the size of
$\mathbb{P}^{1}$ and $Y\equiv Y^{2}$ measures the size of $\mathbb{P}^{4}$. In
the limit $Y\rightarrow 0$, the tower of particles with $q_{1}=0$,
$q_{2}=1,2,...$ become massless. Furthermore, from the form of the
prepotential,

$\mathcal{F}=3XY^{2}+Y^{3}\,,$ (29)

we see that the limit $Y\rightarrow 0$ represents a candidate emergent string
limit, since the prepotential vanishes quadratically in this limit,
$\mathcal{F}\sim Y^{2}$. This agrees with the fact that the GV invariants
$n_{0,q_{2}}$ (shaded blue in Table 2) seem to grow exponentially with
increasing $q_{2}$, as expected for a tower in an emergent string
limit.555Throughout this paper, we used a simple eye test to assess
exponential growth of the GV invariants, checking that the number of digits in
each respective GV invariant grows at a roughly constant rate as charge
increases. It would be interesting to measure the exponential rate of growth
more precisely and see how it varies across different examples of emergent
string limits, but we leave this for future work.

$\begin{array}[]{c|ccccc}\hbox{\diagbox[width=28.45274pt,height=21.33955pt,innerleftsep=2.84544pt,innerrightsep=0.0pt]{{\shortstack[l]{$q_{3}$}}}{{\shortstack[r]{$q_{2}$}}}{}}&0&1&2&3&4\\\
\hline\cr
0&-&\pagecolor{blue!25}16&\pagecolor{blue!25}0&\pagecolor{blue!25}0&\pagecolor{blue!25}0\\\
1&\pagecolor{orange!25}128&\pagecolor{blue!25}128&\pagecolor{blue!25}0&\pagecolor{blue!25}0&\pagecolor{blue!25}0\\\
2&\pagecolor{orange!25}144&\pagecolor{blue!25}960&\pagecolor{blue!25}144&\pagecolor{blue!25}0&\pagecolor{blue!25}0\\\
3&\pagecolor{orange!25}128&\pagecolor{blue!25}5120&\pagecolor{blue!25}5120&\pagecolor{blue!25}128&\pagecolor{blue!25}0\\\
4&\pagecolor{orange!25}80&\pagecolor{blue!25}20640&\pagecolor{blue!25}70272&\pagecolor{blue!25}20640&\pagecolor{blue!25}80\\\
5&\pagecolor{orange!25}128&\pagecolor{blue!25}70656&\pagecolor{blue!25}626688&\pagecolor{blue!25}626688&\pagecolor{blue!25}70656\\\
6&\pagecolor{orange!25}144&\pagecolor{blue!25}218752&\pagecolor{blue!25}4265600&\pagecolor{blue!25}10349760&\pagecolor{blue!25}4265600\\\
\end{array}$

Table 3: Genus 0 GV invariants of degree $(0,q_{2},q_{3})$ for the geometry
$X_{7465}$. The BPS particles that become light in the decompactification
string limit $Z\rightarrow 0$ are shaded orange, while the additional BPS
particles that become light in the emergent string limit $Y,Z\rightarrow 0$
are shaded blue. The GV invariants grow exponentially with increasing charge
in the emergent string limit, whereas they are order-one and periodic in the
decompactification limit, as expected.

As a third and final example, consider the $h^{1,1}=3$ CICY $X_{7465}$ with
configuration matrix

$\left(\begin{array}[]{c || ccccc}\mathbb{P}^{1}{}&~{}1&1&0&0&0\\\
\mathbb{P}^{2}{}&~{}1&0&2&0&0\\\ \mathbb{P}^{5}{}&~{}0&1&1&2&2\\\
\end{array}\right)\,.$ (30)

In other words, this Calabi-Yau consists of the intersection of five
polynomials of tridegree $(1,1,1)$, $(1,0,1)$, $(0,2,1)$, $(0,0,2)$, and
$(0,0,2)$, respectively, inside
$\mathbb{P}^{1}\times\mathbb{P}^{2}\times\mathbb{P}^{5}$. The GV invariants
with $q_{1}=0$ for this geometry are shown in Table 3. Here, the mass of a BPS
particle of charge $(q_{1},q_{2},q_{3})$ is given by

$m_{q_{1},q_{2},q_{3}}=\left(\frac{\sqrt{2}\pi}{\kappa_{5}}\right)^{1/3}\frac{|q_{1}X+q_{2}Y+q_{3}Z|}{\mathcal{F}^{1/3}}\,,$
(31)

where $X\equiv Y^{1}$ measures the volume of the $\mathbb{P}^{1}$ factor,
$Y\equiv Y^{2}$ measures the volume of the $\mathbb{P}^{2}$ factor, and
$Z\equiv Y^{3}$ measures the size of the $\mathbb{P}^{5}$ factor. Meanwhile,
the prepotential is given by

$\mathcal{F}=4XYZ+2Y^{2}Z+4XZ^{2}+6YZ^{2}+\frac{4}{3}Z^{3}\,,$ (32)

In the limit $Z\rightarrow 0$, we see that the prepotential vanishes linearly,
$\mathcal{F}\sim Z$. Thus, this corresponds to a candidate decompactification
limit. On the other hand, in the limit $Y,Z\rightarrow 0$ with fixed
$\textrm{d}Y/\textrm{d}Z$, the prepotential vanishes quadratically; this
represents a candidate emergent string limit. Indeed, this prepotential takes
precisely the form (17), and thus we interpolate between a decompactification
limit and an emergent string limit when $X$ is constant, $Z\rightarrow 0$, and
$Y\sim Z^{\alpha}$ for $\alpha\in\\{0,1\\}$.

The Emergent String Conjecture suggests that the behavior seen in these
particular examples is in fact rather generic. Given any asymptotic boundary
in which the prepotential vanishes linearly with some modulus $Y^{I}$, we
expect a periodic tower of order-one GV invariants, which correspond to
Kaluza-Klein modes in some duality frame. Meanwhile, any asymptotic boundary
in which the prepotential vanishes quadratically with the moduli $Y^{I}$
corresponds to an emergent string limit, and correspondingly we expect an
exponentially-growing tower of charged string modes. If the prepotential
vanishes linearly as $Y^{I}\rightarrow 0$ and quadratically as
$Y^{I},Y^{J}\rightarrow 0$, then we expect a periodic tower of order-one GV
invariants in the former limit and an exponentially-growing tower of charged
string modes in the latter limit. All this may be viewed as a geometric
consequence of the Emergent String Conjecture for Calabi-Yau manifolds.

##  5 Numerical Sweep

Complete intersection Calabi-Yau manifolds in products of projective spaces
(CICYs) offer a simple but sizeable arena to test this geometric consequence
of the Emergent String Conjecture. In the remainder of this paper, we consider
all asymptotic limits of the Kähler moduli spaces of CICYs that arise as a
single $\mathbb{P}^{n}$ factor shrinks to zero size. In the language of 5d
supergravity, this means that a single homogenous coordinate on vector
multiplet moduli space is taken to zero. We further consider all examples of
the asymptotic limits discussed in §3, in which the prepotential takes the
form (17). If the Emergent String Conjecture is true, these limits should
interpolate between decompactification limits and emergent string limits
depending on the rate at which two of the moduli are taken to zero size.

The GV invariants for all of these geometries can be found in the database of
Carta, Mininno, Righi, and Westphal Carta:2021sms  or, alternatively, computed
using the Mathematica code of Klemm and Kreuzer Klemm:2001aaa . We examine the
GV invariants up to degree 10 to confirm that they exhibit the periodic
behavior expected for a decompactification limit and the exponential growth
expected for an emergent string limit.

We have seen that decompactification limits are characterized by a
prepotential that vanishes linearly with some modulus, $\mathcal{F}\sim
Y^{I}$, and without loss of generality we take $I=1$. In terms of the triple
intersection numbers, this means that $C_{IJK}=0$ if $I,J,K\neq 1$, but
$C_{IJ1}\neq 0$ for some choice of $I,J\neq 1$.

Given a CICY with configuration matrix

$\left(\begin{array}[]{c ||
ccccc}\mathbb{P}^{n_{1}}{}&~{}d_{1}^{(1)}&d_{2}^{(1)}&\cdots&d_{k}^{(1)}\\\
\mathbb{P}^{n_{2}}{}&~{}d_{1}^{(2)}&d_{2}^{(2)}&\cdots&d_{k}^{(2)}\\\
\vdots{}&~{}\vdots&&&\vdots\\\
\mathbb{P}^{n_{h^{1,1}}}{}&~{}d_{1}^{(h^{1,1})}&d_{2}^{(h^{1,1})}&\cdots&d_{k}^{(h^{1,1})}\\\
\end{array}\right)\,,$ (33)

the condition $C_{IJK}=0$ if $I,J,K\neq 1$ is satisfied precisely when the
first row $\\{d_{j}^{(1)}\\}$ has fewer than $n_{1}$ nonzero entries.
Meanwhile, the condition $C_{IJ1}\neq 0$ for some choice of $I,J\neq 1$
requires $\\{d_{j}^{(1)}\\}$ to have at least $n_{1}-1$ nonzero entries. The
Calabi-Yau condition implies $\sum_{j}d_{j}^{(i)}=n_{i}$, which means that
there are only two ways $\\{d_{j}^{(1)}\\}$ can have exactly $n_{1}-1$ nonzero
entries:

$\\{d_{j}^{(1)}\\}=\\{2,2,\underset{n_{1}-3}{\underbrace{1,...,1}},0,...,0\\}\text{~{}~{}~{}or~{}~{}~{}}\\{d_{j}^{(1)}\\}=\\{3,\underset{n_{1}-2}{\underbrace{1,...,1}},0,...,0\\}\,,$
(34)

or else $\\{d_{j}^{(1)}\\}$ is given by some permutation of the above entries.

We performed a sweep of the favorable CICY database of Anderson:2017aux
(based on the previous work Candelas:1987kf ), finding that just 140 of the
7820 favorable CICYs have rows $\\{d_{j}^{(I)}\\}$ of the above form (and only
one of these 140, number 7884 in the database, has two rows of this form666 It
is worth noting that CICY number 7884 has $h^{1,1}=2$ and thus a one-
dimensional vector multiplet moduli space. The two decompactification limits
lie at opposite ends of this moduli space, so they do not intersect, and there
is no way to decompactify both circles at once. ). Computing the GV
invariants, we confirm that every row $\\{d_{j}^{(I)}\\}$ of the form
corresponds to a periodic tower of order-one GV particles with
$q_{I}=1,2,...$, $q_{K}=0$ for $K\neq I$. One such example–number 7643–was
discussed in the previous section, and its GV invariants are shown in Table 1.

Physically speaking, this means that every candidate decompactification limit,
for which the prepotential vanishes linearly with the modulus, indeed features
a tower of exponentially light particles with an approximately constant
density of states, precisely as we would expect for a tower of Kaluza-Klein
modes. This gives strong evidence that these limits are indeed
decompactification limits in some duality frame, as required by the Emergent
String Conjecture.

Another interesting result from our sweep of candidate decompactification
limits in CICYs is that the order of periodicity $p$ of the GV invariants
Kaluza-Klein tower is always less than or equal to 4. Furthermore, the GV
invariants $n_{q}$ of the tower satisfy not only the periodicity condition
$n_{q}=n_{q+p}$, but also the condition $n_{q}=n_{p-q}$ for $0<q<p$. This can
be understood as a consequence of charge conjugation, which implies
$n_{q}=n_{-q}$, in conjunction with the periodicity condition.777We thank Ben
Heidenreich for discussions on this point.

Candidate emergent string limits, on the other hand, have a prepotential which
vanishes quadratically with the moduli. To begin, we focus on limits in which
a single $\mathbb{P}^{n_{I}}$ factor shrinks to zero size. Without loss of
generality, suppose $I=1$, so that the prepotential vanishes as
$\mathcal{F}\sim(Y^{1})^{2}$. This means that $C_{IJK}=0$ if $I,J,K\neq 1$,
$C_{IJ1}=0$ for $I,J\neq 1$, but $C_{11K}\neq 0$ for some $K\neq 1$.

For a Calabi-Yau with configuration matrix (33), these conditions on the
$C_{IJK}$ imply that the first row $\\{d_{j}^{(1)}\\}$ must have precisely
$n_{1}-2$ nonzero entries. There are, once again, only two ways this can
happen:

$\\{d_{j}^{(1)}\\}=\\{3,2,\underset{n_{1}-4}{\underbrace{1,...,1}},0,...,0\\}\text{~{}~{}~{}or~{}~{}~{}}\\{d_{j}^{(1)}\\}=\\{4,\underset{n_{1}-3}{\underbrace{1,...,1}},0,...,0\\}\,,$
(35)

or else $\\{d_{j}^{(1)}\\}$ is given by some permutation of these entries.

A sweep of the CICY database reveals that only 11 of the 7820 favorable CICYs
have rows of this form. All of these have $h^{1,1}=2$, as required for an
emergent string limit that involves only one vanishing modulus, as shown
above. Computing the GV invariants, we find that every row $\\{d_{j}^{(I)}\\}$
of this form corresponds to a tower of exponentially-growing GV invariants
with $q_{I}=1,2,...$, $q_{K}=0$ for $K\neq I$. One example of this type–number
7806–was discussed in the previous subsection, and the GV invariants are shown
in Table 2.

Finally, we consider candidate emergent string limits which occur in
conjunction with candidate decompactification limits, as in (30) above, so
that a decompactification limit appears when one modulus $Y^{1}$ is taken to
zero while an emergent string limit appears when another modulus $Y^{2}$ is
taken to zero simultaneously. This means, in particular, that the prepotential
vanishes linearly as $Y^{1}\rightarrow 0$, so $\mathcal{F}\sim Y^{1}$, whereas
it vanishes quadratically as $Y^{1},Y^{2}\rightarrow 0$. Such a prepotential
necessarily takes the form in (17).

Since these emergent string limits necessarily feature decompactification
limits and moduli spaces of dimension larger than one, the only geometries
that need to be considered in this sweep are the 127 CICYs with candidate
decompactification limits and $h^{1,1}>2$, a subset of the 140 total CICYs
with candidate decompactification limits. Given such a small set, it is
straightforward to simply compute the triple intersection numbers (and hence
the prepotential for every Calabi-Yau) to see if it takes the form in (17).

We find that of the 127 CICYs with candidate decompactification limits and
$h^{1,1}>2$, 63 also have candidate emergent string limits in which two
$\mathbb{P}^{n_{I}}$ factors shrink to zero size, and all of these have
$h^{1,1}=3$. Furthermore, 40 of the 63 CICYs have two candidate emergent
string limits; this means that the prepotential vanishes linearly as
$Y^{1}\rightarrow 0$, quadratically as $Y^{1},Y^{2}\rightarrow 0$, and
quadratically as $Y^{1},Y^{3}\rightarrow 0$. In terms of the prepotential in
(17), this happens precisely when $C_{233}=0$.

Studying the GV invariants of these 63 CICYs, we find once again that every
candidate emergent string limit features an exponentially growing tower of BPS
particles. One example of this type–number 7465–was discussed in the previous
subsection, and the GV invariants are shown in Table 3.

Thus, the candidate emergent string limits feature exponential growth of GV
invariants, as expected. In summary, our numerical sweep of CICYs provides
strong evidence for the Emergent String Conjecture in vector multiplet moduli
spaces of 5d supergravity theories

##  6 Discussion

In this paper, we have seen perfect agreement between the rate of growth of GV
invariants along certain rays in the charge lattice and the values of the
triple intersection numbers for a large class of Calabi-Yau manifolds. This
agreement conforms exactly to the expectations of the Emergent String
Conjecture, and thus this paper provides strong evidence in favor of this
conjecture based on topological invariants of Calabi-Yau manifolds.

To turn this around, our paper has also uncovered strong constraints on
certain GV invariants from the Emergent String Conjecture, just as previous
work has led to strong constraints on GV invariants from the tower/sublattice
WGC Alim:2021vhs ; Gendler:2022ztv . Hopefully, the Swampland program will
someday yield rigorous predictions that can be tested experimentally, but it
is encouraging that it has already given us rigorous, precise, and testable
predictions for Calabi-Yau geometry. So far, the Emergent String Conjecture
(and the sublattice WGC) have passed these tests with flying colors.

We also found compelling bottom-up evidence for the Emergent String Conjecture
by studying curved paths in the vector multiplet moduli spaces of 5d
supergravities. Such paths lead to precisely the scaling behavior of gauge
couplings for particles and strings that we would expect after circle
compactification of a six-dimensional string theory.

A few of our results warrant further study. We saw above many examples of
candidate decompactification limits in which the GV invariants are periodic
rather than constant with increasing charge. However, in all of the examples
we saw, the order of periodicity was less than or equal to 4. This is
reminiscent of the counterexamples to the lattice Weak Gravity Conjecture
found in Heidenreich:2016aqi ; such counterexamples have BPS particles of
charge 0 mod $k$ but no BPS particles of charge $q$, $q\neq 0$ mod $k$, and in
all examples considered, the order of periodicity satisfies $k\leq 3$.

A key difference between those examples and the examples we have seen in this
work is that our examples do not violate the lattice WGC: in Table 1, for
instance, we see that the GV invariants $n_{0,q}$ are all nonzero, indicating
that there are BPS particles of all charges $(0,q),q\neq 0$. This is true more
generally for all 141 examples of candidate decompactification limits that we
have studied, and we find no evidence for violations of the lattice WGC in our
examples. However, one intriguing similarity between our results and these is
that the periodicity in either case never grows parametrically larger than 1.
This is likely also related to the observation that discrete gauge groups in
string theory seem to have order-one cardinality (see e.g. Lee:2022swr ). It
would be nice to have a bottom-up argument as to why quantum gravity seems to
eschew large periodicities and discrete symmetries.

The CICY database studied here–while sizable–nonetheless represents only a
small fraction of all Calabi-Yau manifolds. A logical next step would be to
extend the numerical sweep considered here to the full Kreuzer-Skarke database
of Calabi-Yau hypersurfaces in ambient toric varieties Kreuzer:2000xy .

The bottom-up argument for the Emergent String Conjecture we have given above
is suggestive, but it is nonetheless restricted to the case of vector
multiplet moduli spaces in 5d supergravity. It would be nice to find a bottom-
up argument that doesn’t rely on the scaling of gauge couplings, since
infinite-distance limits in 5d hypermultiplet moduli spaces do not feature
weakly coupled gauge fields, and it would be especially exciting to find an
argument for the Emergent String Conjecture that does not rely on
supersymmetry at all. We leave these as worthwhile targets for future study.

## Acknowledgements

It is a pleasure to thank Jonathan Heckman, Ben Heidenreich, and Andrew Turner
for useful discussions and Matthew Reece for comments on a draft of this
paper. The author is also indebted to Lara B. Anderson, Xin Gao, James Gray,
and Seung-Joo Lee; Albrecht Klemm and Maximilian Kreuzer; Federico Carta,
Alessandro Mininno, Nicole Righi, and Alexander Westphal; Philip Candelas,
A.M. Dale, Carsten Andrew Lütken, and Rolf Schimmrigk for their publicly
available CICY databases and Mathematica notebooks, which played the starring
role in the numerical sweep performed in this work. This work was supported in
part by STFC through grant ST/T000708/1, the Berkeley Center for Theoretical
Physics; by the Department of Energy, Office of Science, Office of High Energy
Physics under QuantISED Award DE-SC0019380 and under contract DE-
AC02-05CH11231; and by the National Science Foundation under Award Number
2112880.

## References

  * (1) S.-J. Lee, W. Lerche, and T. Weigand, “Emergent Strings from Infinite Distance Limits,” arXiv:1910.01135 [hep-th].
  * (2) S.-J. Lee, W. Lerche, and T. Weigand, “Emergent Strings, Duality and Weak Coupling Limits for Two-Form Fields,” arXiv:1904.06344 [hep-th].
  * (3) F. Baume, F. Marchesano, and M. Wiesner, “Instanton Corrections and Emergent Strings,” JHEP 04 (2020) 174, arXiv:1912.02218 [hep-th].
  * (4) F. Xu, “On TCS G2 manifolds and 4D emergent strings,” JHEP 10 (2020) 045, arXiv:2006.02350 [hep-th].
  * (5) I. Basile, “Emergent strings at infinite distance with broken supersymmetry,” arXiv:2201.08851 [hep-th].
  * (6) M. Etheredge, B. Heidenreich, S. Kaya, Y. Qiu, and T. Rudelius, “Sharpening the Distance Conjecture in Diverse Dimensions,” arXiv:2206.04063 [hep-th].
  * (7) R. Blumenhagen, A. Gligovic, and A. Paraskevopoulou, “The Emergence Proposal and the Emergent String,” arXiv:2305.10490 [hep-th].
  * (8) S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror symmetry, mirror map and applications to Calabi-Yau hypersurfaces,” Commun. Math. Phys. 167 (1995) 301–350, arXiv:hep-th/9308122 [hep-th].
  * (9) S. Hosono, A. Klemm, S. Theisen, and S.-T. Yau, “Mirror symmetry, mirror map and applications to complete intersection Calabi-Yau spaces,” Nucl. Phys. B433 (1995) 501–554, arXiv:hep-th/9406055 [hep-th]. [AMS/IP Stud. Adv. Math.1,545(1996)].
  * (10) R. Gopakumar and C. Vafa, “M theory and topological strings. 2.,” arXiv:hep-th/9812127 [hep-th].
  * (11) R. Gopakumar and C. Vafa, “M theory and topological strings. 1.,” arXiv:hep-th/9809187 [hep-th].
  * (12) E. Bergshoeff, S. Cucu, T. de Wit, J. Gheerardyn, S. Vandoren, and A. Van Proeyen, “N = 2 supergravity in five-dimensions revisited,” Class. Quant. Grav. 21 (2004) 3015–3042, arXiv:hep-th/0403045.
  * (13) M. Alim, B. Heidenreich, and T. Rudelius, “The Weak Gravity Conjecture and BPS Particles,” arXiv:2108.08309 [hep-th].
  * (14) N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, “The String landscape, black holes and gravity as the weakest force,” JHEP 0706 (2007) 060, arXiv:hep-th/0601001 [hep-th].
  * (15) B. Heidenreich, M. Reece, and T. Rudelius, “Sharpening the Weak Gravity Conjecture with Dimensional Reduction,” JHEP 02 (2016) 140, arXiv:1509.06374 [hep-th].
  * (16) S. Andriolo, D. Junghans, T. Noumi, and G. Shiu, “A Tower Weak Gravity Conjecture from Infrared Consistency,” Fortsch. Phys. 66 no. 5, (2018) 1800020, arXiv:1802.04287 [hep-th].
  * (17) J. Calderón-Infante, A. M. Uranga, and I. Valenzuela, “The Convex Hull Swampland Distance Conjecture and Bounds on Non-geodesics,” JHEP 03 (2021) 299, arXiv:2012.00034 [hep-th].
  * (18) H. Ooguri and C. Vafa, “On the Geometry of the String Landscape and the Swampland,” Nucl.Phys. B766 (2007) 21–33, arXiv:hep-th/0605264 [hep-th].
  * (19) B. Heidenreich, M. Reece, and T. Rudelius, “Evidence for a Lattice Weak Gravity Conjecture,” JHEP 08 (2017) 025, arXiv:1606.08437 [hep-th].
  * (20) B. Heidenreich, M. Reece, and T. Rudelius, “The Weak Gravity Conjecture and Emergence from an Ultraviolet Cutoff,” Eur. Phys. J. C78 no. 4, (2018) 337, arXiv:1712.01868 [hep-th].
  * (21) T. Hubsch, Calabi-Yau manifolds: A Bestiary for physicists.  World Scientific, Singapore, 1994.
  * (22) A. Klemm and M. Kreuzer, “Instanton (1.0).” http://hep.itp.tuwien.ac.at/~kreuzer/pub/prog/inst.m, 2001.  [Online].
  * (23) F. Carta, A. Mininno, N. Righi, and A. Westphal, “Gopakumar-Vafa hierarchies in winding inflation and uplifts,” JHEP 05 (2021) 271, arXiv:2101.07272 [hep-th].
  * (24) L. B. Anderson, X. Gao, J. Gray, and S.-J. Lee, “Fibrations in CICY Threefolds,” JHEP 10 (2017) 077, arXiv:1708.07907 [hep-th].
  * (25) P. Candelas, A. M. Dale, C. A. Lutken, and R. Schimmrigk, “Complete Intersection Calabi-Yau Manifolds,” Nucl. Phys. B 298 (1988) 493.
  * (26) N. Gendler, B. Heidenreich, L. McAllister, J. Moritz, and T. Rudelius, “Moduli Space Reconstruction and Weak Gravity,” arXiv:2212.10573 [hep-th].
  * (27) S.-J. Lee and P.-K. Oehlmann, “Geometric Bounds on the 1-Form Gauge Sector,” arXiv:2212.11915 [hep-th].
  * (28) M. Kreuzer and H. Skarke, “Complete classification of reflexive polyhedra in four-dimensions,” Adv. Theor. Math. Phys. 4 (2000) 1209–1230, arXiv:hep-th/0002240.


Orbital varieties in types $B$ and $C$

WILLIAM M. MCGOVERN
Department of Mathematics, Box 354350, University of Washington, Seattle, WA,
98195

## Introduction

The purpose of this paper is to correct the proof of the main result of [5],
which parametrizes orbital varieties in the classical cases in terms of domino
tableaux and gives an algorithm for computing the tableau of the orbital
variety corresponding to a classical Weyl group element. We treat types $B$
and $C$ in this paper; a later one will deal with type $D$. As in [5], we will
rely heavily on three papers of Garfinkle [1, 2, 3] and one of the author and
Pietraho [7], which use standard domino tableaux to classify Kazhdan-Lusztig
cells in types $B,C$, and $D$ .

## Main result

We recall the main result of [5], using the notation and definitions of that
paper.

######  Theorem.

Assume that the simple Lie algebra $\mathfrak{g}$ is of type $X=B$ or $C$.
Orbital varieties contained in the orbit $\mathcal{O}_{\mathbf{p}}$ with
partition $\mathbf{p}$ are parametrized by standard domino tableaux of shape
$\mathbf{p}$. For any $w$ in the Weyl group $W$, the tableau parametrizing the
corresponding variety $V(w)$ may be obtained from the left tableau $T_{L}(w)$
attached by Garfinkle to $w$ by moving through certain open cycles of type $X$
or $D$ whose holes and corners lie in rows of odd length in type $C$, or even
length in type $B$, lowering the partition corresponding to the shape of the
resulting tableau in the standard dominance order throughout, until this shape
becomes an $X$-partition $\mathbf{q}$. The orbit $\mathcal{O}_{w}$ then
coincides with $\mathcal{O}_{\mathbf{q}}$.

###### Proof.

As in [5], we write $w_{1}\sim w_{2}$ whenever $w_{,}w_{2}\in W$ lie in the
same geometric left cell (so that $V(w_{1})=V(w_{2}))$. Let $S$ be a subset of
the set $\Delta$ of simple roots. Writing any $w\in W$ uniquely as
$w_{S}w^{\prime}$ for some $w_{S}\in W_{S}$, the parabolic subgroup of $W$
generated by the reflections in $S$, and $w^{\prime}$ a minimal length coset
representative of $W_{S}$ in $W$, we argue as in [5] to show that $v\sim w$
relative to $W$ whenever $v^{\prime}=w^{\prime}$ and $v_{S}\sim w_{S}$
relative to $W_{S}$.

In particular whenever $\alpha,\beta$ are adjacent simple roots of the same
length, the operator $T_{\alpha\beta}^{R}$ defined in [2] preserves geometric
left cells wherever it is defined. If $\alpha,\beta$ are adjacent but have
different lengths, then we need a single-valued truncation
$V_{\alpha\beta}^{R}$ of this operator. Write
$(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$ for the image of
$V_{\alpha\beta}^{R}$ applied to a pair $(\mathbf{T}_{1},\mathbf{T}_{2})$ on
which it is defined. In type $C$, $V_{\alpha\beta}^{R}$ is defined on
$(\mathbf{T}_{1},\mathbf{T}_{2})$ if the 1- and 2-dominos of $\mathbf{T}_{2}$
form a subtableau of shape either $(3,1)$ or $(2,2)$; in the former case we
obtain $(\mathbf{T}_{1}^{\prime},\mathbf{T}_{2}^{\prime})$ from
$(\mathbf{T}_{1},\mathbf{T}_{2})$ by moving both tableaux through the extended
cycle of the 2-domino in $\mathbf{T}_{2}$ relative to $\mathbf{T}_{1}$ and
then transposing the 1- and 2-dominos in the $2\times 2$ box they now occupy
in the right tableau. In the latter case we set
$\mathbf{T}_{1}^{\prime}=\mathbf{T}_{1}$ and obtain $\mathbf{T}_{2}^{\prime}$
from $\mathbf{T}_{2}$ by transposing the 1- and 2-dominos in the $2\times 2$
box they occupy (while leaving all other dominos unchanged). In type
$B,V_{\alpha\beta}^{R}$ is defined if the 0 square and 1- and 2-dominos in
$\mathbf{T}_{2}$ form a subtableau of shape $(3,2)$ or $(3,1,1)$; in the
former case we move $(\mathbf{T}_{1},\mathbf{T}_{2})$ through the extended
cycle of the 2-domino in $\mathbf{T}_{2}$ relative to $\mathbf{T}_{1}$ and
then interchange the positions of the 1- and 2-dominos in the right tableau,
while in the latter we just interchange the positions of the 1- and 2-dominos
in the right tableau. Using [2, 2.1.4,2.3.4] and the above paragraph, we see
that $V_{\alpha\beta}^{R}$ preserves the geometric left cell whenever it is
defined.

We also need an analogous operator $V_{D}^{R}$, which is again a single-valued
truncation of a larger operator, this time the operator
$T_{\alpha_{1}^{\prime},C}^{R}$ defined in [7, 4.4.10]. In type $C$, this
operator is defined on pairs $(\mathbf{T}_{1},\mathbf{T}_{2})$ for which the
first four dominos in $\mathbf{T}_{2}$ form a subtableau with shape $(4,3,1)$
and the 2-domino is vertical (thus in the first column). Its image is then
obtained by moving $(\mathbf{T}_{1},\mathbf{T}_{2})$ through the extended
cycle in type $D$ of the 4-domino in $\mathbf{T}_{2}$ relative to
$\mathbf{T}_{1}$ and then transposing the positions of the 2- and 4-dominos in
the $2\times 2$ box they occupy in $\mathbf{T}_{2}$. In type $B$, this
operator is defined on pairs $(\mathbf{T}_{1},\mathbf{T}_{2})$ for which the 0
square and first three dominos in $\mathbf{T}_{2}$ form a subtableau of shape
$(4,2,1)$; its image is obtained by moving $(\mathbf{T}_{1},\mathbf{T}_{2})$
through the extended cycle in type $D$ of the 3-domino in $\mathbf{T}_{2}$
relative to $\mathbf{T}_{1}$ and then transposing the positions of the 2- and
3-dominos in the right tableau in the $2\times 2$ box they occupy in
$\mathbf{T}_{2}$. As before, we check that this operator preserves the
geometric left cell whenever it is defined.

We now argue as in [5, p. 2987], with a small modification. In type $C$, given
$w\in W$ whose left tableau $T_{L}(w)$ is not already an $X$-partition, look
at the largest odd part $p$ occurring with odd multiplicity in the partition
corresponding to the shape of $T_{L}(w)$. If the lowest row of length $p$ in
this shape is an odd-numbered row, then it is easy to construct a tableau
$\mathbf{T}$ having the same shape as $T_{R}(w)$ with the $m(m+1)$\- and
$(m+1)^{2}$-dominos having the same orientation as in $T_{R}(w)$, for all
$m\geq 2$ such that dominos with these labels appear in $T_{R}(w)$, and such
that the first two dominos in $\mathbf{T}$ form a subtableau of shape $(3,1)$;
moreover, the proof of the main result of [6] shows that we may apply a
sequence $\Sigma$ of operators $T_{\alpha\beta}^{R}$ and $V_{\alpha\beta}^{R}$
to $T_{R}(w)$ in such a way that tableau shapes are preserved throughout and
the resulting tableau is $\mathbf{T}$. Applying the operator
$V_{\alpha\beta}^{R}$ one more time, this time lowering the shape in the
dominance order, we arrive at $w^{\prime}\in W$ in the same geometric left
cell as $w$ whose left tableau $T_{L}(w^{\prime})$ has lower shape than that
of $T_{L}(w)$. If instead the lowest row of length $p$ in the shape as
$T_{R}(w)$ is an even row, we argue similarly, producing a tableau
$\mathbf{T}$ in the domain of $V_{D}^{R}$ whose first four dominos form a
subtableau of shape $(4,3,1)$, such that some sequence $\Sigma$ of operators
$T_{\alpha\beta}^{R}$ and $V_{\alpha\beta}^{R}$ applied to
$(T_{L}(w),T_{R}(w))$ preserves tableau shapes throughout and transforms the
right tableau to $\mathbf{T}$; then applying $V_{D}^{R}$ we again get
$w^{\prime}\in W$ in the same geometric left cell as $w$ whose left tableau
$T_{L}(w^{\prime})$ has a lower shape than that of $T_{L}(w)$. Iterating this
process, we eventually replace $w$ by an element $v$ in the same geometric
left cell the shape of whose left tableau $T_{L}(v)$ is a $C$-partition.
Further applications of $V_{\alpha\beta}^{R}$ and $V_{D}^{R}$ cannot lower the
shape of this partition in the dominance order, since for example if the first
two dominos of $T_{L}(v)$ form a subtableau of shape $(3,1)$, then the cycle
of the 2-domino is unboxed in the sense of [1]; if it were open, then it would
be a down cycle and moving through this cycle would produce a tableau whose
row lengths do not weakly decrease as one moves down, a contradiction. In type
$B$, we argue similarly, interchanging the roles of even and odd parts, and
using the operators $T_{\alpha\beta}^{R}$ and $V_{D}^{R}$ as defined in that
case.

We are therefore reduced to the case where $T_{L}(w)$ is an $X$-partition. Now
we recall that orbital varieties, like primitive ideals of a fixed regular
infinitesimal character, have $\tau$-invariants. Given a pair $\alpha\beta$ of
adjacent simple roots there is a wall-crossing operator $T_{\alpha\beta}^{V}$
defined on orbital varieties having one of $\alpha,\beta$ in their
$\tau$-invariants but not the other [4, 9.9,9.11]. If $\alpha$ and $\beta$
have the same length then the image of this operator, say on a variety $V$
with $\alpha$ in its $\tau$-invariant but not $\beta$, is a variety with
$\beta$ in its $\tau$-invariant but not $\alpha$, obtained by taking the image
of $V$ under the minimal nonsolvable parabolic subgroup $P_{\beta}$
corresponding to $\beta$ and then intersecting with the nilradical
$\mathfrak{n}_{\beta}$ of its Lie algebra. In types $B$ and $C$ we have
$T_{\alpha\beta}^{V}V(w)=V(T_{\alpha\beta}^{R}(w))$ for $\alpha,\beta$ of the
same length [4, 9.11], defining $T_{\alpha\beta}^{R}(w)$ as in [2]. If instead
$\alpha$ and $\beta$ have different lengths, then the operator
$T_{\alpha\beta}^{V}$ is defined by the same recipe, but behaves in a slightly
more complicated way. It is defined on a variety $V$ having $\alpha$ in its
$\tau$-invariant but not $\beta$ if some component of
$P_{\beta}(V)\cap\mathfrak{n}_{\beta}$ has the same dimension as $V$ and can
take one or two values in that case (that is, there can be one or two
components in the intersection of the same dimension as $V$); these values are
then again those of $V(T_{\alpha\beta}^{R}(w))$. Specifically, in type $C$, if
the first two dominos of $T_{L}(w)$ form a subtableau of shape $(2,1,1)$, so
that the simple root $\beta=e_{2}-e_{1}$ lies in the $\tau$-invariant
$\tau(w)$ but the root $\alpha=2e_{1}$ does not, and if the cycle of the
2-domino in $T_{L}(w)$ is open, then moving through this cycle raises the
shape of $T_{L}(w)$ in the dominance order and $T_{\alpha\beta}^{V}$ is not
defined on $V(w)$. Similarly, if the first two dominos of $T_{L}(w)$ are
vertical and form a $2\times 2$ box, then one of the images of
$T_{\beta\alpha}^{R}(w)$, namely the one with left tableau obtained from
$T_{L}(w)$ by transposing the positions of its first dominos within their
$2\times 2$ box and then moving the 2-domino through its cycle, does not
appear as an image of $T_{\beta\alpha}(w)$ if this 2-cycle is open. Otherwise,
this map coincides with $T_{\alpha\beta}^{R}$. A similar situation holds in
type $B$. We use the maps $T_{\alpha\beta}^{V}$ to define the notion of
generalized $\tau$-invariant of a variety as in [3] and argue as in that paper
to show that the generalized $\tau$-invariant of a variety is a complete
invariant, so that two varieties in the same Lie algebra with the same
generalized $\tau$-invariant coincide.

Now it is easy to complete the proof. Given two Weyl group elements
$w,w^{\prime}$ with the same left tableau $T_{L}(w)=T_{L}(w^{\prime})$ having
the shape of an $X$-partition, any operator $T_{\alpha\beta}^{V}$ defined on
$V(w)$ is also defined on $V(w^{\prime})$ and preserves the shapes of
$T_{L}(w)$ and $T_{L}(w^{\prime})$, whence in fact it fixes these tableaux, so
that $V(w)$ and $V(w^{\prime})$ have the same generalized $\tau$-invariant and
must coincide. Hence there are at most as many distinct orbital varieties as
there are standard domino tableaux with shape an $X$-partition; but the number
of such tableaux is the sum of the dimensions of the Springer representations
attached to the corresponding nilpotent orbits in the ambient Lie algebra,
which is well known to be the number of orbital varieties, so we get a natural
bijection between orbital varieties and standard domino tableaux with shape
that of an $X$-partition. Since we know by [5, pp. 2985,6] that given an
$X$-partition $\mathbf{p}$ at least one $w\in W$ has left tableau $T_{L}(w)$
of shape $\mathbf{p}$ and orbital variety $V(w)$ lying in the orbit
$\mathcal{O}_{\mathbf{p}}$, the bijection between standard domino tableaux and
orbital varieties must send tableaux with shape $\mathbf{p}$ to varieties
lying in $\mathcal{O}_{\mathbf{p}}$, as claimed. ∎

The above argument also validates the proof of [5, Theorem 2], which asserts
that one component of the variety $\mathcal{V}(w)$ of the simple highest
weight module $L(w\gamma)$ of highest weight $w\gamma-\rho$, where $\gamma$ is
an antidominant regular integral weight and $\rho$ is half the sum of the
positive roots, is the one corresponding to the tableau $S_{R}(w)$, the unique
tableau of special shape obtained from the right tableau $T_{R}(w)$ by moving
through open $X$-cycles (keeping $X$ fixed this time and allowing either up or
down cycles).

## References

  * [1] D. Garfinkle,  On the classification of primitive ideals for complex classical Lie algebras (I),  Compositio Math., 75(2):135–169, 1990.
  * [2] D. Garfinkle,  On the classification of primitive ideals for complex classical Lie algebras (II),  Compositio Math., 81(3):307–336, 1992.
  * [3] D. Garfinkle,  On the classification of primitive ideals for complex classical Lie algebras (III),  Compositio Math., 88:187–234, 1993.
  * [4] A. Joseph,  On the variety of a highest weight module,  J. Alg., 88:238–278, 1984.
  * [5] W. M. McGovern,  On the Spaltenstein-Steinberg map for classical Lie algebras,  Comm. in Alg., 27:2979–2993, 1999.
  * [6] W. M. McGovern,  A family of operators generating domino tableaux of a fixed shape and a decomposition of left cells into isotypic components,  preprint, 2020.
  * [7] W. M. McGovern and T. Pietraho,  On the classification of primitive ideals for complex classical Lie algebras (IV),  preprint, 2016.