Stability of BPS States and Weak Coupling Limits

Eran Palti

_Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel_

###### Abstract

We study the stability and spectrum of BPS states in ${\cal N}=2$
supergravity. We find evidence, and prove for a large class of cases, that BPS
stability exhibits a certain filtration which is partially independent of the
value of the gauge couplings. Specifically, for any perturbative value of any
gauge coupling $g\ll 1$, a BPS state can only decay to some constituents if
those constituents do not become infinitely heavier than it in the vanishing
coupling limit $g\rightarrow 0$. This stability filtration can be
mathematically formulated in terms of the monodromy weight filtration of the
limiting mixed Hodge structure associated to the vanishing coupling limit. We
study various implications of the result for the Swampland program which aims
to understand such weak-coupling limits, specifically regarding the nature and
presence of an infinite tower of light charged BPS states.

###### Contents

  1. 1 Introduction
    1. 1.1 Motivation from the Swampland program
  2. 2 D3 branes as BPS states
    1. 2.1 ${\cal N}=2$ BPS states
    2. 2.2 Decay and binding of BPS states
    3. 2.3 The BPS index and wall crossing formula
    4. 2.4 BPS states in Calabi-Yau manifolds
    5. 2.5 Black hole attractors and BPS states
  3. 3 BPS states at infinite distance
    1. 3.1 Moduli space near monodromies
    2. 3.2 BPS states near monodromies
    3. 3.3 Walls of marginal stability at infinite distance
    4. 3.4 Single-centre attractor flows at infinite distance
  4. 4 Evidence for obstruction to split attractor flows
    1. 4.1 Obstruction for one-parameter two-centre electric flows
    2. 4.2 Obstruction for one-parameter $n$-centre electric flows
    3. 4.3 Generalizing to multi-parameter models
      1. 4.3.1 Obstruction in a two-parameter example
      2. 4.3.2 The attractor loci
      3. 4.3.3 The $n=d=1$ sector
      4. 4.3.4 The $n=d=3$ sector
  5. 5 Obstruction from moduli space singularities
    1. 5.1 Equivalence of stability and attractor loci
    2. 5.2 General proof of obstruction
    3. 5.3 A BPS Stability Filtration
  6. 6 Summary and discussion
  7. A Example one-parameter Calabi-Yau
    1. A.1 The period vector
    2. A.2 The BPS spectrum
    3. A.3 Conifold and Large Complex Structure loci
      1. A.3.1 The conifold locus
      2. A.3.2 The large complex structure locus

##  1 Introduction

In this paper we study the behaviour of the BPS spectrum of states in ${\cal
N}=2$ supergravity as we vary a gauge coupling to zero $g\rightarrow 0$. It is
known that there is a certain universal structure to any weak-coupling limit
(which reflects the nilpotent orbit theorem [1]), as was studied in detail in
[2, 3]. This universal structure allows for a general study of such limits.
The spectrum of BPS states in ${\cal N}=2$ theories has been the topic of much
work (see for example [4] for a review). It is interesting because the
spectrum is well controlled, specifically it can only change upon crossing
walls of marginal stability in the moduli space (or coupling space). How the
spectrum changes upon crossing these walls is described by the wall-crossing
formula [5].

We are motivated to study this for a number of reasons, which are all related
in some way to ideas that are part of the Swampland program [6, 7, 8]. The
central theme is the expectation that as $g\rightarrow 0$ one should find an
infinite tower of charged BPS states becoming massless. We would like to
understand various aspects of these states. We outline below the Swampland-
related motivations. However, the results we find are quite general and go
beyond the Swampland theme, and so we outline them first.

Our findings are then as follows: we find evidence that whether a state can
decay to some constituents depends not only on their masses at the point in
coupling space where the decay occurs, but on their masses all the way to the
weak-coupling limit. So find evidence for a certain filtration in the BPS
spectrum which is (partially) independent of the value of the couplings:

BPS Stability Filtration: In any weakly-coupled region $g\ll 1$, a BPS state
can only decay to constituents whose mass is not infinitely higher than it in
the $g\rightarrow 0$ limit.

We are able to prove this property for a large class (one of the three types)
of weakly-coupled regions.111By weakly-coupled we mean that instanton-type
effects, so exponential in the coupling, are suppressed. We also go some way
towards showing that it holds completely generally.

We find this result quite striking. As an example, it implies that an
electrically charged BPS black hole with a mass of our galaxy cannot emit a
dyonic sub-Planckian BPS particle (upon crossing a wall of marginal stability)
because in the $g\rightarrow 0$ limit the dyon will be infinitely heavier than
the black hole (which would transition to a particle).

The BPS filtration has a nice mathematical formulation: it is that the
stability of BPS states in any weakly-coupled region is determined by their
weight under the Monodromy Weight Filtration of the limiting mixed Hodge
structure associated to the $g\rightarrow 0$ limit.

###  1.1 Motivation from the Swampland program

In this part of the introduction we outline the motivation for studying the
spectrum and decay of BPS states approaching weak-coupling limits in the
context of the Swampland. See also [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] for studies of
BPS and charged states in weak-coupling limits in the Swampland context.

#### Black hole stability and the black hole to particle transition

The Swampland program aims to understand constrains on effective theories
coupled to gravity from their quantum gravity ultraviolet completion. One of
the approaches to this question is to try to utilize black hole physics. The
prototypical example is the Weak Gravity Conjecture (WGC), which postulates
the existence of a particle whose mass is less than its charge [36]

$m\leq gqM_{p}\;,$ (1.1)

where here $m$ is the mass of the particle, $q$ its integer quantized charge,
and $g$ the gauge coupling of the gauge field under which it is charged. It
was proposed in [36] that this constraint may be related to the discharge of
extremal black holes, a process which requires such a particle. This is an
attempt to connect physics of states whose mass is above the Planck scale,
such as black holes, to states whose mass is below the Planck scale, such as
the WGC particle. Crossing the Planck scale in this sense attempts to capture
the input of quantum gravity.222There are some arguments we may try to make
about how quantum gravity should behave to try and connect super-Planckian and
sub-Planckian physics. For example, starting with a sub-extremal black hole
and letting it shed mass through Hawking radiation, but no charge, until it is
a Planck-scale remnant. But it is difficult to understand the physics of any
such remnants, and whether they are problematic or not.

One can try to understand crossing the Planck scale in a more explicit way as
follows. Consider an extremal state with charge $Q$ such that for a certain
value of the gauge coupling $g=g_{i}$, we have

$g_{i}Q\gg 1\;.$ (1.2)

We may let this state be an extremal black hole with a super-Planckian mass,
so saturating the extremality bound

$M\geq g_{i}QM_{p}\;.$ (1.3)

Now vary the gauge coupling to $g=g_{f}$, say by moving along a moduli space,
into the regime of

$g_{f}Q\ll 1\;.$ (1.4)

Now the same state can no longer be considered as an extremal black hole, but
may be regarded as a particle. In practice, tracking the state (while
maintaining extremality) through this variation of the gauge coupling requires
knowing the microscopics of quantum gravity, or having sufficient
supersymmetry. We will return to this point, but let us continue to think
about that this means. In particular, a contemporary perspective on the
magnetic WGC [37, 38, 2, 39, 10, 22, 7, 8] is that there should be an infinite
tower of charged states whose mass scale is set by the gauge coupling

$m_{\infty}\sim gM_{p}\;,$ (1.5)

where $m_{\infty}$ is the mass scale of an infinite tower of states. Strictly
speaking we should not consider this tower extended beyond the Planck scale,
but it is infinite in the sense that as $g\rightarrow 0$ there are an infinite
number of states below the Planck scale in the tower. Now we see that the
state with charge $Q$ should be part of this tower at $g=g_{f}$, while it
should be a charged black hole at $g=g_{i}$. This leads to tension, because on
the one hand we were considering that charged black holes should be unstable
and decay, on the other hand, we want a tower of stable charged states.

We would like to understand what happens to a state as we vary the gauge
coupling

$gQ\gg 1\rightarrow\;gQ\ll 1\;.$ (1.6)

There is a way to understand this transition if there is sufficient
supersymmetry. For example, with ${\cal N}=4$ supersymmetry we can track such
a transition. Indeed, if we replace the gauge coupling by the string coupling
$g_{s}$, then the two regimes correspond to the different descriptions of
branes in string theory

$g_{s}Q\gg 1\mathrm{\;(Black\;Holes)}\rightarrow\;g_{s}Q\ll
1\mathrm{\;(Branes)}\;.$ (1.7)

Following this transition is precisely what allowed the extraction of the
microstates of black holes in string theory [40]. However, the case of ${\cal
N}=4$ supersymmetry is not so interesting for the questions we are after,
because there is no decay processes for BPS states upon variations of the
gauge couplings (paths in moduli space). Geometrically, the counting of the
D-brane states is topological.

The case of ${\cal N}=2$ supersymmetry is an interesting compromise between
the complete control (but stability) of ${\cal N}=4$, and the instability and
generality of no supersymmetry. With ${\cal N}=2$ supersymmetry BPS states are
not necessarily stable over paths in moduli space, so there is something to
understand about the transition (1.6). On the other hand, we have some control
over their stability and spectrum. In particular, it can be that a state at
one point in moduli space has no charged particles it can decay to, while
after moving to a different point it no longer is BPS and can, and does, decay
to some constituent particles. We will consider here specifically the case of
type IIB string theory compactified on a Calabi-Yau manifold. Geometrically,
in this setting the BPS states are no longer topologically counted, but
correspond to D3 branes wrapping special Lagrangian sub-manifolds.

It is worth emphasising that the sense of stability and decay of BPS states is
technically quite different to the picture of a charged black hole emitting a
charged particle, as in the Weak Gravity Conjecture. At a given point in
moduli space, BPS states are just stable and do not decay at all. Decays can
only be induced by varying the moduli, or equivalently, the couplings. Even
then, a state cannot decay to a constituent which is mutually local to it. So
an electric black hole will not decay by emitting an electric particle.
Nonetheless, the idea is that studying the interaction between stability of
black holes, the spectrum of particles, and the black hole to particle
transition, in a controlled setting, will shed light on the same type of
physics in less supersymmetric settings.

#### Population of the light tower of BPS states

Our analysis is based on a very general understanding of the behaviour of the
masses of charged BPS states near any weak-coupling limit, as developed in [2]
(see, [3, 13, 41, 42, 34, 43, 28] for follow-up work). One universal feature
is that we can split the states into two types. Those which become massless at
infinite distance in field space, or as $g\rightarrow 0$, which we term
electric states. And those which become infinitely massive in that limit,
which we term dyonic states. The light tower of charged states is therefore
composed of purely electric states. We are then concerned with the question of
the spectrum, and stability, of electric states along variations in moduli
space approaching weak coupling limits.333Of course, the tower of states also
relates to that of the distance conjecture [44, 45], but the focus on a
charged tower naturally places this in the context of the weak gravity
conjecture.

It was shown in [2] that approaching any weak-coupling limit $g\rightarrow 0$
there exists an infinite number of charges such that their associated BPS
states would become massless in the limit. However, it was not proven that
those BPS states are actually in the spectrum.444An argument was made for
certain limits which showed that if one state in the so-called monodromy orbit
is populated, they all are. This argument made an assumption that states in
the orbit do not undergo decay upon the monodromy path. Actually, this is
simple to prove: it follows from the fact that the decay can only occur with
decay products that are not mutually local. But all magnetically charged
states are heavier than the electric ones in the tower approaching infinite
distance. So the argument indeed holds. We would like to show that the states
are indeed populated.

How can we study the spectrum of BPS states in this setting? In regimes where
there is a geometric picture, such as type IIA mirrors in the large volume
regime, it is possible to study the spectrum directly. But these are only very
special loci in the type IIB moduli space, and more generally there are no
geometric tools that can be applied directly on the type IIB side. Another
method is to use duality with the Heterotic string [10, 46, 47, 18, 22, 29,
48]. In this work, we will utilise a tool which builds on the relation between
black holes and D-branes (1.7). In our setting, the string coupling $g_{s}$ is
part of the hypermultiplet moduli space, while the gauge couplings and BPS
state masses and stability are controlled by the vector-multiplet moduli
space, and the two moduli spaces decouple exactly.555At least they do so at
the two-derivative level which corresponds to infinite Calabi-Yau volume. But
since the volume is also a modulus, which also sits in the hypermultiplet
sector, this limit does not affect the vector-multiplet sector. We therefore
expect some controlled correspondence between the microscopic D-brane spectrum
and black hole physics.

Such a correspondence has been developed extensively. starting with the works
[49, 50, 51]. The idea is to associate a given charge with a black hole (even
if the charge is such that the mass of the state is smaller than the Planck
mass), and study how the moduli behave around this black hole. The moduli
fields follow so-called attractor flows from their values at infinity, into
the attractor loci on the black hole horizon, which is fixed uniquely by the
charges (it is in this sense that a charge can be associated with a black
hole, so more precisely is associated with an attractor locus). Then the claim
is: if the attractor locus is in a controlled region of moduli space, the
charge is populated by a BPS state. If the locus is in an uncontrolled region
of moduli space, or it diverges to infinity, then the result is inconclusive
regarding the presence of a BPS state with that charge.

A crucial aspect of the attractors/BPS states correspondence is that we are
considering the spectrum of a given charge at the value of the moduli at
spatial infinity of the associated black hole solution. We must then allow for
multi-centre black hole solutions, such that the multiple black holes have
charges which sum to the charge of interest. If those component black holes
have physical attractor loci, then the BPS state of the total charge is in the
spectrum. Such multi-centre black holes can be understood as attractor flows
which split on loci of marginal stability, and continue as separate flows to
the different centres.

If we try to probe the spectrum and stability of purely electrically charged
states, those relevant for the tower, we encounter an immediate problem: the
attractor flows of electric states diverge since it is possible to reduce
their energy indefinitely by going to arbitrary weak gauge coupling. However,
the existence of split attractor flows suggests a way around this issue: we
can consider starting with a total electric charge at infinity, but then
having a split attractor flow to a multi-centre black hole configuration where
the constituent black holes are dyonic and have physical attractor loci. This
is illustrated in figure 1. The existence of split flows corresponds to the
decay of the electric BPS states into two dyonic states. We therefore are
again motivated to understand BPS stability in the weakly-coupled regions of
moduli space.

Figure 1: Figure illustrating two attractor flows for a purely electric charge
${\bf q}_{e}$. $z_{\infty}$ denotes the values of the moduli at spatial
infinity, while $z_{\star}$ denotes their attractor values on the black hole
horizon. The single-centred flow on the left is divergent and so not well-
defined and cannot be used to deduce the presence of an associated BPS state.
The split attractor flow on the right is well-defined and ends on two dyonic
black holes with charges ${\bf q}_{d}^{1}$ and ${\bf q}_{d}^{2}$ at finite
distance in moduli space.

#### The emergence proposal

In [2] it was proposed that the weak coupling limit in ${\cal N}=2$ settings
arises purely from integrating out the tower of light charged BPS states in
that limit. This would be a controlled example of the more general emergence
proposal [52, 2, 7, 39, 53]. If this is the case, then we should expect that
the behaviour of BPS states, at least in weakly-coupled regions $g\ll 1$,
would be controlled by the tower of states and so by the limit $g\rightarrow
0$. This is related also to the notion of moduli space holography [54, 55],
where the whole moduli space is specified completely by its asymptotic limits
(see also [56, 57] for related ideas). So again we would like to understand
how the BPS spectrum behaves in the bulk $g\ll 1$ with respect to its
behaviour on the boundary $g\rightarrow 0$.

In fact, the discussion above about how the spectrum of states can be
understood from the existence of attractor flows already hints at some sort of
emergence/holography. This is because it proposes that the spectrum of BPS
states manifests non-locality in the moduli space. So the claim is that the
presence of a BPS state at the values of the moduli which are taken at spatial
infinity, is determined by the behaviour of the attractor loci on the black
hole horizon, which are at a completely unrelated locus in moduli space. Such
a notion of non-locality is naturally required for emergence/holography.

We will discuss the implications of the results for the motivation topics
above in section 6.

##  2 D3 branes as BPS states

In this work we are considering the setting of type IIB string theory on
Calabi-Yau manifolds which gives an ${\cal N}=2$ supergravity. Within this we
consider further only the vector-multiplet moduli space. The charged BPS
states whose mass depends on the vector-multiplet (complex-structure) moduli
are D3 branes wrapping 3-cycles in the Calabi-Yau. In this section, we discuss
some properties of these states.

###  2.1 ${\cal N}=2$ BPS states

The moduli space of $\mathcal{N}=2$ supersymmetric theories splits into two
sectors, the vector multiplet moduli space and the hypermultiplet one. We will
be solely concerned with the vector multiplet sector and denote by $n_{V}$ the
number of vector multiplets. The moduli fields $z^{i}$, with $i=1,...,n_{V}$,
control the mass of (half-)BPS states in the theory. The mass is set by the
central charge $Z\left({\bf q}\right)$, which is a function of (integer
quantised) charge vectors ${\bf q}$. The charges are with respect to the gauge
fields of the vector multiplets plus the graviphoton in the graviton
multiplet. The moduli space and the central charge are controlled by the
period vector ${\bf\Pi}\left(z^{i}\right)$, which has $2\left(n_{V}+1\right)$
entries that are holomorphic in the $z^{i}$. Specifically,

$Z\left({{\bf q}}\right)=e^{\frac{K}{2}}{\bf q}^{T}\cdot\eta\cdot{\bf\Pi}\;.$
(2.1)

Here $\eta$ is a matrix defining a symplectic inner product, for example it
can be taken as

$\eta=\left(\begin{array}[]{cc}0&\mathbb{1}_{n_{V}+1}\\\
-\mathbb{1}_{n_{V}+1}&0\end{array}\right)\;.$ (2.2)

It is useful to denote the symplectic inner product as

$\left<{\bf q},{\bf p}\right>\equiv{\bf q}^{T}\cdot\eta\cdot{\bf p}\;.$ (2.3)

The Kahler potential, K, is itself determined by the period vector through

$K=-\log i\left<{\bf\Pi},\overline{{\bf\Pi}}\right>\;.$ (2.4)

Compactifications of type IIB string theory on a Calabi-Yau manifold have BPS
states corresponding to D3 branes wrapping three-cycles in the Calabi-Yau. The
charge vector ${\bf q}$ corresponds to the wrapping numbers of the $D3$ brane.
While this is determined topologically, the question of whether there is an
actual BPS state of a given charge is mapped to the existence of a special
Lagrangian sub-manifold in the given homology class, which is not a
topological invariant.

It is informative to look at the mass scale of the wrapped D3 branes in more
detail. The expression for BPS states (2.1) is given in units of the four-
dimensional (reduced) Planck mass $M_{p}$. This is related to the string scale
$M_{s}$ as

$M_{s}\sim\frac{g_{s}}{\sqrt{{\cal V}_{s}}}M_{p}\;.$ (2.5)

Here $g_{s}$ is the string coupling, which is a field appearing in the
hypermultiplet sector. ${\cal V}_{s}$ is the (dimensionless) volume of the
Calabi-Yau in units of the string length, it is also a field which appears in
the hypermultiplet sector. The mass of a $D3$ brane wrapped on a three-cycle
$\mathcal{C}$ takes the form

$M_{D3}\sim\frac{M_{s}}{g_{s}}{\cal
V}_{\mathcal{C}}\sim\frac{M_{s}}{g_{s}}\sqrt{{\cal V}_{s}}Z\left({\bf
q}\right)\sim Z\left({\bf q}\right)M_{p}\;,$ (2.6)

where here ${\cal V}_{\mathcal{C}}$ denotes the volume in string units of
$\mathcal{C}$.

We will be interested in this paper in limits in moduli space where
$Z\left({\bf q}\right)\rightarrow 0$ or $Z\left({\bf
q}\right)\rightarrow\infty$. We will refer to branes whose central charge goes
to zero as light (electric) states, while branes whose central charge diverges
will be referred to as heavy (dyonic) states. It is important to note that the
expression (2.6) shows that these two ways to refer to the states should be
interpreted with care. In particular, in the weakly-coupled supergravity
regime we must have ${\cal V}_{\mathcal{C}}\gg 1$ and $g_{s}\ll 1$ which means
that $M_{D3}\gg M_{s}$ irrespective of the value of $Z\left({\bf q}\right)$.
Therefore, in that regime even the light $D3$ states are always heavier than
the string states. On the other side, the heavy $D3$ states have $M_{D3}\gg
M_{p}$, and so should not really be interpreted as particle states.
Nonetheless, the presence of $\mathcal{N}=2$ supersymmetry in the theory will
allow us to probe certain aspects of the physics associated to both the light
and heavy $D3$ states. It is these ‘protected’ aspects of the $D3$ brane
physics which will be the central elements in this paper.

###  2.2 Decay and binding of BPS states

A given charge vector ${\bf q}$ does not necessarily correspond to a stable
BPS state in the theory. Further, the spectrum of stable BPS states depends on
the point in moduli space at which it is evaluated. The changes to the
spectrum of BPS states in the theory upon traversing paths in moduli space is
controlled by Walls of Marginal Stability (WMS) where BPS states may either
combine with other ones to form new stable bound states, or may become
unstable and split into stable constituents. In this paper we will be
primarily concerned with such process which involve only two constituents.

Consider three BPS states $A$, $B$ and $C$ with respective charges ${\bf
q}_{A}$, ${\bf q}_{B}$ and ${\bf q}_{C}$. We are interested in the process
where the state $A$ decays into $B$ and $C$, denoted as $A\rightarrow B+C$. We
therefore should impose charge conservation

${\bf q}_{A}={\bf q}_{B}+{\bf q}_{C}\;.$ (2.7)

The central charge is a linear function of the charge, and therefore (2.7)
implies

$Z\left({\bf q}_{A}\right)=Z\left({\bf q}_{B}\right)+Z\left({\bf
q}_{C}\right)\;.$ (2.8)

The mass of the BPS states is

$M\left({\bf q}\right)=\left|Z\left({\bf q}\right)\right|\;.$ (2.9)

We can then introduce the phase of the central charge $\alpha\left({\bf
q}\right)$ as

$Z\left({\bf q}\right)=M\left({\bf q}\right)e^{i\alpha\left({\bf
q}\right)}\;.$ (2.10)

From (2.8) we can write

$M\left({\bf q}_{A}\right)^{2}=\left(M\left({\bf q}_{B}\right)+M\left({\bf
q}_{C}\right)\right)^{2}-2M\left({\bf q}_{B}\right)M\left({\bf
q}_{C}\right)\big{[}1-\cos\left(\alpha\left({\bf
q}_{B}\right)-\alpha\left({\bf q}_{C}\right)\right)\big{]}\;.$ (2.11)

Since the last term in (2.11) is negative we have $M\left({\bf
q}_{A}\right)\leq M\left({\bf q}_{B}\right)+M\left({\bf q}_{C}\right)$ and
therefore the decay $A\rightarrow B+C$ cannot occur unless the phases of $B$
and $C$ align. This alignment defines a Wall of Marginal Stability (WMS) for
the decay $A\rightarrow B+C$,

$\mathrm{WMS}_{A\rightarrow B+C}\;:\;\alpha\left({\bf
q}_{B}\right)=\alpha\left({\bf q}_{C}\right)\;.$ (2.12)

It is useful to note that (2.12) implies also $\alpha\left({\bf
q}_{A}\right)=\alpha\left({\bf q}_{B}\right)$. However, the opposite direction
is not quite true. Defining a wall of marginal stability by through the phases
of $B$ and $A$ requires an additional condition on the masses

$\mathrm{WMS}_{A\rightarrow B+C}\;:\;\alpha\left({\bf
q}_{A}\right)=\alpha\left({\bf q}_{B}\right)\;\mathrm{and}\;M\left({\bf
q}_{A}\right)\geq M\left({\bf q}_{B}\right)\;.$ (2.13)

The condition on the wall of marginal stability (2.12) determines where such a
decay process may occur, but it does not guarantee that it does. This depends
on whether the constituent states are in the BPS spectrum themselves, and also
on which direction we cross the wall. If crossing the wall in one direction we
have the decay $A\rightarrow B+C$, then crossing he wall in the opposite
direction must correspond to the reverse binding process $B+C\rightarrow A$.
For small variations away from the WMS the condition on which process is
occurring, decay or binding, is determined as [50, 58, 51]

$\displaystyle\mathrm{Decay}\;\left(A\rightarrow B+C\right)\;:\left<{\bf
q}_{B},{\bf q}_{C}\right>\left(\alpha\left({\bf q}_{B}\right)-\alpha\left({\bf
q}_{C}\right)\right)<0\;,$
$\displaystyle\mathrm{Binding\;}\left(B+C\rightarrow A\right)\;:\;\left<{\bf
q}_{B},{\bf q}_{C}\right>\left(\alpha\left({\bf q}_{B}\right)-\alpha\left({\bf
q}_{C}\right)\right)>0\;.$ (2.14)

The microscopic physics associated to the decay or binding is the mass of open
strings stretching between the branes. When the strings are tachyonic the
branes form a bound state. Macroscopically, the condition corresponds to
having a split attractor flow with a positive value for the splitting radius.
The two pictures are related by varying the string coupling, as studied in
[51].

###  2.3 The BPS index and wall crossing formula

A crucial aspect of the spectrum of BPS states is the degeneracy of states of
a given charge. The deformation invariant quantity which provides a good
measure of the degeneracy of states is the BPS index. The index is denoted as
$\Omega\left({\bf q},z\right)$ for a charge ${\bf q}$ at a point in moduli
space $z$. It receives a $+1$ from a massive hypermultiplet and $-2$ from a
massive vector multiplet. The wrapped D3 branes give rise to hypermultiplets.

In section 2.2 we discussed walls of marginal stability. Upon crossing such
walls, BPS states can bind or decay and correspondingly the BPS index can
jump. Conversely, a change in the BPS index implies that the state has to
cross some wall of marginal stability. The change in the BPS index upon
crossing a wall is described in generality by the wall-crossing formula [5].
We will not utilise the full details of the formula, and just give a feeling
for it for sub-cases which were described in [59]. First we may consider the
primitive decay process ${\bf q}_{A}\rightarrow{\bf q}_{B}+{\bf q}_{C}$. By
primitive we mean that ${\bf q}_{B}$ and ${\bf q}_{C}$ cannot be written as an
integer multiple of some other quantized charge. In this case the change in
the BPS index $\Omega\left({\bf q},z\right)$ is given by [59]

$\Delta\Omega\left({\bf q}_{A},z\right)=\left(-1\right)^{\left<{\bf
q}_{B},{\bf q}_{C}\right>-1}\left|\left<{\bf q}_{B},{\bf
q}_{C}\right>\right|\Omega\left({\bf q}_{B},z_{ms}\right)\Omega\left({\bf
q}_{C},z_{ms}\right)\;,$ (2.15)

where $z_{ms}$ is the wall of marginal stability locus in moduli space.

If we allow one of the charges to not be primitive, since the marginal
stability locus corresponds to the alignment of the phases of $Z\left({\bf
q}_{B}\right)$ and $Z\left({\bf q}_{C}\right)$, it is also a locus of marginal
stability for the more general decay ${\bf q}_{A}\rightarrow{\bf
q}_{B}+M_{AC}\;{\bf q}_{C}$. The change in the BPS index for this decay is
given by the generating function [59]

$\Omega\left({\bf q}_{B}\right)+\sum_{X_{C\bar{A}}>0}\Delta\Omega\left({\bf
q}_{B}+M_{AC}\;{\bf q}_{C}\right)l^{M_{AC}}=\Omega\left({\bf
q}_{B}\right)\prod_{k>0}\left(1-(-1)^{k\left<{\bf q}_{B},{\bf
q}_{C}\right>}l^{k}\right)^{k\left|\left<{\bf q}_{B},{\bf
q}_{C}\right>\right|\Omega\left(k\;{\bf q}_{C}\right)}\;.$ (2.16)

Here $l$ is a dummy variable, and all the indices are evaluated on the locus
of marginal stability.

###  2.4 BPS states in Calabi-Yau manifolds

It is useful to gain some intuition about the spectrum of BPS states in
Calabi-Yau manifolds by looking at examples. We can consider the mirror IIA
setting in the geometric supergravity regime, and look at D2-branes wrapping
holomorphic curves. These can be calculated through mirror symmetry for
example, or by counting directly holomorphic curves. The spectrum of such
states for a two-parameter Calabi-Yau was calculated in [60]. The CY is
denoted $\mathbb{P}_{4}^{(1,1,2,2,2)}[8]$, which determines its construction
as a complete intersection in weighted projective space. It has two homology
classes for two-cycles, and associated Kahler moduli $v_{t}$ and $v_{s}$ which
are part of vector multiplets with scalar components $t$ and $s$ denoted as

$t=b_{t}+iv_{t}\;,\;\;s=b_{s}+iv_{s}\;.$ (2.17)

In the large volume geometric regime, the Kahler potential for the Kahler
moduli space is approximately given by

$e^{\frac{K}{2}}\sim\left(v_{t}^{3}-3v_{t}v_{s}^{2}-2v_{s}^{3}\right)^{-\frac{1}{2}}\;.$
(2.18)

The mass of the D2 BPS states is given by the general central charge formula
(2.1), except that now we are considering Kahler moduli of the CY. In this
case it reads

$Z\left(j,k\right)=e^{\frac{K}{2}}\left[\frac{j}{2}t+\left(\frac{j}{2}-k\right)s\right]+{\cal
O}\left(e^{2\pi it},e^{2\pi is}\right)\;.$ (2.19)

Here $j$ and $k$ are the two integers specifying the charges or wrapping
numbers of the D2 branes on the two homology classes. The number of BPS states
for $j$ and $k$ are shown in table 2.1.

$j$ / $k$ | $0$ | $1$ | $2$ | $3$
---|---|---|---|---
0 | 0 | 4 | 0 | 0
1 | 640 | 640 | 0 | 0
2 | 10032 | 72224 | 10032 | 0
3 | 288384 | 7539200 | 7539200 | 288384
4 | 10979984 | 757561520 | 2346819520 | 757561520
5 | 495269504 | 74132328704 | 520834042880 | 520834042880
6 | 24945542832 | 7117563990784 | 7117563990784 | 212132862927264
Table 2.1: Table taken from [60] showing the number of BPS states for
different wrapping numbers $j$ and $k$, in the Calabi-Yau manifold
$\mathbb{P}_{4}^{(1,1,2,2,2)}[8]$.

There are of course much more advanced and complete calculations of BPS states
in the geometric regime of type IIA, see for example [61] for the latest
cutting edge. However, we are only interested in certain features of the BPS
spectrum, which are universal to all examples. From table 2.1. we see observe:

  1. 1.

Not all charges are populated by BPS states.

  2. 2.

The populated charges have $k,j>0$ (or $k,j<0$ for the anti-branes).

  3. 3.

If a charge ${\bf q}$ supports a BPS state, this does not imply that $K{\bf
q}$ supports one, with $K$ some integer.

  4. 4.

If a charge $K{\bf q}$ supports a BPS state, with $K$ some integer, then so
does the charge ${\bf q}$.666In fact all charges $n{\bf q}$ with $n\leq K$
support BPS states.

  5. 5.

There is a $\mathbb{Z}_{2}$ symmetry acting as
$\left(j,k\right)\rightarrow\left(j,j-k\right)$

The number of BPS states in a given homology class is counted by the
holomorphic curves in the class. To gain some intuition for this it is
simplest to consider holomorphic divisors. These are given by some holomorphic
equations, say $f(z_{i})=0$. The degree of the polynomial is associated to the
homology class of the curve. The simplest such divisors to understand are
those where the Calabi-Yau is embedded in a weighted projective space (as
above) and the equations are in the ambient space coordinates which have
certain weights. The weights of the polynomial determine the homology class.
If we allow meromorphic functions, then we can span all the possible divisor
classes. But a given class may not have any holomorphic representation, and in
that case it will have a vanishing BPS count. This is the schematic
understanding of property 1.

Property 2 seems straightforward to understand: if we have opposite signs for
$k$ and $j$ then one wrapping direction will be anti-holomorphic with respect
to the other direction, so the overall state cannot be BPS. In fact, this type
of positivity property will play a crucial role in our analysis. In the wall-
crossing formula it is related to the restriction to stable states under a
certain quadratic form [5].

Now consider property 3. From the picture of holomorphic divisors, having
${\bf q}$ populated means we have some holomorphic $f(z_{i})$ in that class.
But then $f(z_{i})^{K}$ is in the class $K{\bf q}$ and so we may wonder how it
cannot be populated. The point is that $f(z_{i})^{K}$ is a factorised
polynomial, and the BPS states are counted by non-factorizable holomorphic
polynomials. So to have a representative we need some other monomial in the
class $K{\bf q}$, such that we can construct a non-factorized polynomial from
the monomials. This does not always exist, and so we obtain property 3.

Property 4 on the other hand says something like, if we have some holomorphic
polynomial in class $K{\bf q}$, then we must have one in ${\bf q}$. This would
be the case for example if the polynomial always contains a monomial
$f(z_{i})^{K}$. It is not clear why this should hold, but we can understand it
by thinking about the origin of the homology classes which are empty of BPS
states, which we denote as vanishing classes. Indeed, property 4 can also be
stated as: If ${\bf q}$ is a vanishing class, then so is $K{\bf q}$. We
observe in table 2.1 that there are infinite chains of vanishing classes. One
way to understand these infinite chains is through the property 5. First note
that holomorphicity is mapped to some positivity in the classes, $j,k\geq 0$,
since we can only include positive powers of the coordinates. On the other
hand, the $\mathbb{Z}_{2}$ symmetry maps any class with $k>j$ to a negative
class, which cannot have holomorphic curves. Hence all classes with $k>j$ must
be vanishing. So the infinite vanishing classes can be understood
schematically as: there is one charge direction for which increasing the
charge increases the negative powers of some coordinate. So that at some
charge, we run out of holomorphic monomials and only have meromorphic ones.

These infinite chains of vanishing classes are tied to the presence of a
conifold locus in the mirror Calabi-Yau moduli space. The most explicit way to
see this is that the $\mathbb{Z}_{2}$ symmetry is a sub-group of the monodromy
group. In particular it corresponds to a combination of the conifold monodromy
and another one [60]. Traversing the monodromy path, and acting with the
monodromy transformation on the charges is a gauge symmetry, see (3.13). In
general, this does not imply that the monodromy action on the charges is a
symmetry of the spectrum. This is because somewhere along this path the BPS
states can cross curves of marginal stability. If a state ${\bf q}$ does not
cross a wall of marginal stability upon a monodromy path, then the state
$T\cdot{\bf q}$ (with $T$ some matrix which is part of the monodromy group)
must have the same number of BPS states [2, 59]. In the case when the
monodromy action is a discrete finite one, say $\mathbb{Z}_{N}$, then the
condition on not crossing a wall of marginal stability is automatic. This is
because going around $N$ times with the same orientation must lead back to the
original spectrum, but any wall of marginal stabilities would be crossed in
the same direction upon traversing such a path, and so cannot switch between
decay and binding walls as would be required. This is the reason why the
$\mathbb{Z}_{2}$ monodromy subgroup is also a symmetry group of the spectrum.
Its origin is the conifold monodromy, even though it is applied at large
complex-structure (or more precisely at large volume in the IIA mirror).

It can be checked that property 4 above holds in many examples, possibly all
known ones. For example, we found them to hold in all cases we looked at
within the extensive database of [61]. We expect that it is general.

###  2.5 Black hole attractors and BPS states

Calculating the BPS index as a function of the moduli is too difficult. But
there is a very useful way to probe the index by using the relation (1.7).
Essentially, this is related to the fact that the entropy of a black hole is
associated with the number of its microstates. In our case, the black holes
are extremal supersymmetric black holes formed from D3 branes. The precise
relation between the macroscopic entropy and the microscopic BPS index has
been the subject of a large body of work (see [62] for a review). We will
primarily utilise the ideas first introduced in [50].

The entropy of black holes can be calculated through the horizon area, and
counts the degeneracy of microstates with the black hole charge. Therefore, we
can identify the macroscopic black hole entropy $S\left({\bf q}\right)$ with
the (log of the) BPS index $\Omega\left({\bf q}\right)$. But the BPS index
depends on the location in the moduli space, and therefore in this
identification we must associate some point in moduli space to the black hole
entropy. To understand what this means we need to introduce attractor flows.

The charged black holes have two natural points in moduli space associated to
them. The values of the moduli at the horizon of the black holes, these are
the attractor values $z^{i}_{\star}$. And the values of the moduli at spatial
infinity $z^{i}_{\infty}$. The central charge evaluated at these two values of
the moduli corresponds to the black hole ADM mass and its entropy

$M_{ADM}=\left|Z\left({\bf
q},z^{i}_{\infty}\right)\right|\;,\;\;\;S=\pi\left|Z\left({\bf
q},z^{i}_{\star}\right)\right|^{2}\;.$ (2.20)

The flow of the moduli between infinity and the horizon is described by the
attractor mechanism [63, 64] (see [62] for a review). For extremal black
holes, the attractor moduli values are fixed by the charges of the black
holes, and are independent of the moduli values at infinity. For a given
charge of the black hole, our freedom amounts to the choice of the moduli
values at infinity. The simplest cases, denoted double-extremal black holes in
[65], are when the moduli values at infinity are set equal to those on the
horizon $z^{i}_{\infty}=z^{i}_{\star}$. In such situations there is no
attractor flow at all. If instead we have $z^{i}_{\infty}\neq z^{i}_{\star}$
then we must have a non-trivial attractor flow. This flow is determined by the
attractor flow equations, and is uniquely fixed. The attractor flow is
monotonic in the central charge: throughout the flow the central charge is
decreasing. The flow ends when the central charge is minimized

$\partial_{i}\left|Z\left({\bf q}\right)\right|_{\star}=0\;.$ (2.21)

Let us now return to the question of how to identify the black hole entropy
with the BPS index. As explained in [50], we should match the values of the
moduli of the BPS index with the values of the moduli at spatial infinity, so
we are calculating $\Omega\left({\bf q},z^{i}_{\infty}\right)$. The index then
receives contributions from two types of black holes: single-centered and
multi-centered black holes. In the case of single-centered black holes we have
a direct single attractor flow of the moduli from infinity to the black hole
horizon. While in the case of multi-centered black holes the attractor flow
splits on loci of marginal stability and each branch then leads to a different
attractor point on different horizons of separate black holes. We can then
associate a total entropy $S\left({\bf
q},z^{i}_{\infty}\right)^{\mathrm{Tot}}$ to a given charge and moduli values
at infinity as the sum over the entropies of all the possible black hole
solutions which have that (total) charge and asymptotic moduli values. It is
this total entropy which can be identified with the (log of the) BPS
index777There are some subtleties to do with this identification associated to
potential black hole hair [66, 67]. We will ignore these subtleties as we do
not expect that they can modify the analysis in this work.

$S\left({\bf
q},z^{i}_{\infty}\right)^{\mathrm{Tot}}\equiv\sum_{n}^{\mathrm{Black\;Holes}}S_{n}\left({\bf
q},z^{i}_{\infty}\right)=\log\Omega\left({\bf q},z^{i}_{\infty}\right)\;.$
(2.22)

This is illustrated in figure 2.

Figure 2: Figure illustrating different black holes solutions which contribute
to the total entropy $S\left({\bf q},z^{i}_{\infty}\right)^{\mathrm{Tot}}$
associated with a fixed total charge and moduli values at infinity. Here
infinity is denoted by the large circle, and the the moduli flows are shown as
lines leading to attractor values on the black hole horizons.

While it is rather natural to identify the entropy of single-centered black
holes with the number of microscopic BPS states, the case of multi-centered
black holes appears a little strange. After all, the microscopic BPS states
are particles, while the multi-centered black holes have large spatial
separation. The crucial point, as discussed for example in [51], is the same
one which relates the macroscopic entropy with microstates in a quantitative
way: the string coupling $g_{s}$ does not correct the number of BPS states.
The black hole description is valid at $g_{s}\left|{\bf q}\right|\gg 1$, where
$\left|{\bf q}\right|$ is some appropriate measure of the amount of charge.
While the microscopic BPS D-brane picture is valid for $g_{s}\left|{\bf
q}\right|\ll 1$. The limit $g_{s}\rightarrow 0$ is such that the string length
becomes infinitely larger than the Planck length
$\frac{l_{s}}{l_{P}}\rightarrow\infty$. The separation of the black hole
centres is finite in Planck units, and therefore goes to zero in string units
in the $g_{s}\rightarrow 0$ limit. In that limit, from the perspective of the
microscopic string theory, they are a single object.

It is very important to emphasise that there are limitations to this
macroscopic calculation of the BPS index. The attractor flows give only
partial information about the microscopic BPS states. Certainly a well-
controlled attractor flow and locus will contribute BPS states for a given
charge, but an attractor flow which is not well-defined may or may not
correspond to a state. In particular, this is the case for purely electric (or
purely magnetic) charges where the attractor flow diverges due to the
possibility of decreasing the energy density in the electric field arbitrarily
by going to arbitrarily weak coupling.

To illustrate this problem explicitly we can consider the spectrum of BPS
states given in table 2.1. Since all the charges are purely electric, the
attractor flows for all the black holes with those charges (or charges
proportional to them) diverge to large volume. But the fact that some charges
are populated with BPS states, while some are not, shows that we cannot deduce
anything from these diverging flows.

There is also an important connection to the analysis of the example spectrum
in section 2.4 which we would like to utilise. The point is that if we accept
the proposal that the spectrum of states is such that the population of a
state of charge $K{\bf q}$ implies the population of a state of charge ${\bf
q}$, then we can utilise split attractor flows as probes for towers of stable
states. Specifically, we may construct a very massive electric state as a
split flow into two dyonic states. If we find such flows (and we will not)
then we deduce the population of a very massive super-Planckian electric
state, but which would imply also that light sub-Planckian electric states are
populated. Those, in particular, would be stable against any decay approaching
weak coupling in moduli space, since they are lighter than any dyon. This way
a split attractor flow can show the existence of a populated tower of states.

##  3 BPS states at infinite distance

The structure of the vector multiplet moduli space exhibits certain universal
features at infinite distance and weak coupling limits, which was developed in
detail in [2, 3]. In this section we apply these results to determine the
behaviour of the masses and phases of BPS states near those limits.

###  3.1 Moduli space near monodromies

The complex-structure, or vector multiplet, moduli space has certain singular
loci where the period vector undergoes monodromies. If we denote the local
coordinate about one of these singular loci as $z$, so that the locus is at
$z=0$, then the monodromy action $\gamma$ corresponds to circling this locus

$\gamma\;:\;z\rightarrow ze^{r2\pi i}\;.$ (3.1)

The constant $r$ can take the values $\pm 1$. The sign corresponds to going
around the monodromy locus clockwise or anti-clockwise. This is a free choice
at a global level, so we can choose to traverse all the monodromy loci
together either clockwise or anti-clockwise, but once this choice is made
there is a fixed global orientation which fixes the $\pm$ sign for each
monodromy locus. This global orientation ensures that the product of all the
monodromies gives the identity

$\prod_{I}\gamma_{I}=\mathbb{1}\;.$ (3.2)

While this is in general important, it will not play a role in our analysis
and so we henceforth set $r=+1$ for simplicity.

Note that for moduli spaces with multiple moduli there are monodromy loci with
complex codimension higher than one, and so have a number of monodromies
associated to them. These settings will not be considered in detail in this
paper, and we refer to, for example, [3] for an in-depth study of such
monodromy loci.

Under the monodromy the period vector transforms as

$\gamma\cdot{\bf\Pi}\left(z\right)={\bf\Pi}\left(ze^{2\pi
i}\right)=T\cdot{\bf\Pi}\left(z\right)\;.$ (3.3)

Here $T$ is the monodromy matrix which has integer entries in an appropriate
basis. Note that we will choose the coordinate $z$ in (3.3) such that it is
the holomorphic coordinate of minimal power leading to a period vector which
is holomorphic in it up to logarithmic factors (which then induce the
monodromy). One can physically motivate this as saying (3.3) is the minimal
rotation which leaves the physics invariant up to monodromy
transformations.888In general, the monodromy matrix $T$ can be factorised as
$T=T^{(o)}T^{(u)}$, where $T^{(o)}$ is a finite order matrix associated to a
possible orbifolding of the moduli space, while $T^{(u)}$ is a unipotent
matrix of infinite order. This will not be important for us because of this
definition of $z$ as the minimal power.

It is useful to introduce the matrix $N$ defined as

$T=e^{N}\;.$ (3.4)

The unipotency of $T$ then translates to nilpotency of $N$ to a certain order.
In particular, for Calabi-Yau threefolds $N^{4}=0$ for any monodromy locus.
Indeed, an important property of a monodromy locus is the nilpotency order of
its $N$. We define an integer $n$ such that

$n\;:\;N^{n+1}=0\;,\;\;N^{n}\neq 0\;.$ (3.5)

The matrix $N$ controls the behaviour of the period vector. This is the
Nilpotent Orbit Theorem [1, 2] which states that

${\bf\Pi}\left(z\right)=\mathrm{Exp}\left[N\frac{\log z}{2\pi
i}\right]\sum_{p=0}^{\infty}{\bf a}_{p}z^{p}\;,$ (3.6)

where the ${\bf a}_{p}$ are constant (in $z$) vectors. We see that as
$z\rightarrow 0$ the period vector is dominated by the $\log z$ parts with the
$z^{p}$ parts contributing exponentially small corrections. This is true at
least as long as $N{\bf a}_{0}\neq 0$. Indeed, an important quantity is the
integer $d$ defined such that

$d\;:\;N^{d+1}{\bf a}_{0}=0\;,\;\;N^{d}{\bf a}_{0}\neq 0\;.$ (3.7)

In particular, if $d=0$ then the monodromy locus is at finite distance in
moduli space, like the conifold. While if $d>0$ then the locus is at infinite
distance in moduli space. Conversely, any infinite distance locus in moduli
space, which is also the case for any vanishing gauge coupling limit, has a
monodromy with $d>0$.

###  3.2 BPS states near monodromies

Combining the expression for the central charge (2.1) with that of the period
vector (3.6) we can obtain an approximate expression for the central charge.
Consider the cases $d=1$ and $d=0$ where the relevant terms contributing to
the central charge take the form

$Z\left({\bf
q}\right)=\frac{1}{\left|c\log\left|z\right|\right|^{\frac{1}{2}}}\left[\left<{\bf
q},{\bf a}_{0}\right>+\left<{\bf q},N\cdot{\bf a}_{0}\right>\left(\frac{\log
z}{2\pi i}\right)+\;\left<{\bf q},{\bf a}_{1}\right>z+...\right]\;.$ (3.8)

Here $c$ is some unimportant constant. Let us consider the case $d=0$, which
corresponds to the conifold locus. In this case, since $N\cdot{\bf a}_{0}=0$,
we have two types of states. Those whose charges satisfy $\left<{\bf q},{\bf
a}_{0}\right>\neq 0$, which stay massive in the conifold limit $z\rightarrow
0$. And there are those where $\left<{\bf q},{\bf a}_{0}\right>=0$, which
become massless in the conifold limit.

Similarly, for the case $d=1$, there are states whose charges satisfy
$\left<{\bf q},N\cdot{\bf a}_{0}\right>\neq 0$ which become infinitely massive
in the degeneration limit $z\rightarrow 0$. And states which satisfy
$\left<{\bf q},N\cdot{\bf a}_{0}\right>=0$, that become massless in the
degeneration limit.

This type of decomposition into states that are massless and massive in the
degeneration limit $z\rightarrow 0$ is general. We discuss aspects of this in
section 5.3, and refer to [2, 3] for more details. The result is that charges
near the monodromy loci can be split into three types: dyonic charges labelled
as ${\bf q}_{d}$, magnetic charges labelled as ${\bf q}_{m}$ and electric
charges labelled as ${\bf q}_{e}$. In the degeneration limit dyonic and
magnetic states become infinitely massive, while electric states become
massless.

For $n=d=1$, they satisfy999It is worth noting that with these definitions of
electric and magnetic states, they are not related by an electric-magnetic
duality of the gauge coupling. In this sense, the magnetic states here should
be thought of as dyonic also.

$\displaystyle\mathrm{(Massive)\;dyonic\;}$ $\displaystyle:$
$\displaystyle\;\left<{\bf q}_{d},N\cdot{\bf a}_{0}\right>\neq
0\;,\;\;\left<{\bf q}_{d},{\bf a}_{0}\right>\neq 0\;,$
$\displaystyle\mathrm{(Massive)\;magnetic\;}$ $\displaystyle:$
$\displaystyle\;\left<{\bf q}_{m},N\cdot{\bf a}_{0}\right>\neq
0\;,\;\;\left<{\bf q}_{m},{\bf a}_{0}\right>\neq 0\;,$
$\displaystyle\mathrm{(Massless)\;electric\;}$ $\displaystyle:$
$\displaystyle\;\left<{\bf q}_{e},N\cdot{\bf a}_{0}\right>=0\;,\;\;\left<{\bf
q}_{e},{\bf a}_{0}\right>\neq 0\;.$ (3.9)

It is also the case that magnetic and electric states are mutually local

$\left<{\bf q}^{A}_{m},{\bf q}^{B}_{m}\right>=\left<{\bf q}^{A}_{e},{\bf
q}^{B}_{e}\right>=0\;.$ (3.10)

There is a natural action of the monodromy matrix $T$ on the charges ${\bf
q}$,

$\gamma\;:\;\;{\bf q}\rightarrow T^{-1}{\bf q}\;.$ (3.11)

To see the action (3.11) we can track the monodromy action on the central
charge

$\displaystyle Z\left({\bf q},z\right)$ $\displaystyle=$ $\displaystyle
e^{\frac{K}{2}}\left<{\bf q},{\bf\Pi}\left(z\right)\right>$ (3.12)
$\displaystyle\xrightarrow{\gamma}$ $\displaystyle e^{\frac{K}{2}}\left<{\bf
q},{\bf\Pi}\left(ze^{2\pi i}\right)\right>=e^{\frac{K}{2}}\left<{\bf
q},T\cdot{\bf\Pi}\left(z\right)\right>=e^{\frac{K}{2}}\left<T^{-1}\cdot{\bf
q},{\bf\Pi}\left(z\right)\right>$ $\displaystyle=$ $\displaystyle
Z\left(T^{-1}\cdot{\bf q},z\right)\;.$

The monodromy is a gauge transformation, which means are can always exchange
the monodromy path on the moduli space for the action on the charges

$\left\\{z\rightarrow ze^{2\pi i}\right\\}\leftrightarrow\left\\{{\bf
q}\rightarrow T^{-1}\cdot{\bf q}\right\\}\;.$ (3.13)

Note that BPS states satisfy an identity (see, for example, [68])

${\cal Q}\left({\bf q}\right)^{2}=\left|Z\left({\bf
q}\right)\right|^{2}+g^{ij}D_{i}Z\left({\bf
q}\right)\overline{D}_{j}\overline{Z}\left({\bf q}\right)\;.$ (3.14)

Here ${\cal Q}\left({\bf q}\right)^{2}$ is a measure of the physical charge of
the state, the precise form of which is not important but for a single $U(1)$
it is schematically like $g^{2}\left|{\bf q}\right|^{2}$ with $g$ the gauge
coupling. From this we see, as expected, that the degeneration locus
$\log\left|z\right|\rightarrow-\infty$ also corresponds to a vanishing gauge
coupling (at least for the linear combination of gauge fields associated to
the charge ${\bf q}$). Indeed, generally, electric charges are with respect to
electric gauge fields, all of which have a gauge coupling which behaves as [2]

$g\sim\left(\left|\log\left|z\right|\right|\right)^{-\frac{d}{2}}\;.$ (3.15)

###  3.3 Walls of marginal stability at infinite distance

In this section we study the structure of walls of marginal stability near
infinite distance loci. We consider for simplicity the $d=1$ case, but the
conclusions are general.

Walls of Marginal Stability (WMS) are defined by the phase equalities (2.12)
and (2.13). This is a real condition and therefore they trace out real
dimension one lines in the moduli space. We will often work with a change of
coordinates

$t=\frac{\log z}{2\pi i}\;.$ (3.16)

So that $\mathrm{Im\;}t\rightarrow+\infty$ is the degeneration limit, and the
monodromy path $\gamma$ is $\mathrm{Re\;}t\rightarrow\mathrm{Re\;}t+1$.

Let us consider electric states near infinite distances. From the central
charge expression (3.8), and the definitions (3.9), we see that electric
states have central charges of the form

$Z\left({\bf
q}\right)=\frac{1}{\left|c\;\mathrm{Im\;}t\right|^{\frac{1}{2}}}\left[\left<{\bf
q}_{e},{\bf a}_{0}\right>+\left<{\bf q}_{e},{\bf a}_{1}\right>e^{2\pi
it}+...\right]\;.$ (3.17)

We see that variations in moduli space do not change the phase of the central
charge (up to exponentially small corrections). So approaching infinite
distance, electric states have a static phase. Since the charge space of
electric states is at least two (real) dimensional, the phases of electric
states are static points distributed throughout the phase circle.

There are no walls of marginal stability (in the $t$ plane) for purely
electric states to decay to other purely electric states when approaching
infinite distance. So if they are present in the spectrum at some large
$\mathrm{Im\;}t$, increasing $\mathrm{Im\;}t$ further (and varying
$\mathrm{Re\;}t$ as we wish) will not lead to a decay to other purely electric
states. Even without the phase, since the inner product of electric states
with each other vanishes (3.10), we see that they cannot decay to other
electric states.

Magnetically charged states (which could be purely magnetic or dyonic) have a
central charge such that all the terms in (3.8) are generically non-vanishing.
We can write this as

$Z\left({\bf
q}\right)=\frac{\mathrm{Im\;}t}{\left|c\;\mathrm{Im\;}t\right|^{\frac{1}{2}}}\left[i\left<{\bf
q}_{m},N\cdot{\bf a}_{0}\right>+\frac{\left<{\bf q},{\bf
a}_{0}\right>+\left<{\bf q}_{m},N\cdot{\bf
a}_{0}\right>\mathrm{Re\;}t}{\mathrm{Im\;}t}+...\right]\;,$ (3.18)

where ${\bf q}_{m}$ is the magnetic component of the dyonic charge. The
variation of the phase under paths in moduli space is suppressed again but now
only by $\mathrm{Im\;}t$ and not exponentially. This means that dyonic states
can cross walls of marginal stability with respect to other dyonic states and
also electric states.

With respect to the stability of purely electric states, we see that purely
electric states which satisfy

$\frac{\left<{\bf q}_{e},{\bf
a}_{0}\right>}{\left|\;\mathrm{Im\;}t\right|^{\frac{1}{2}}}\ll 1\;,$ (3.19)

are stable because they cannot decay to dyonic states as they are too light.
So at infinite distance we have an infinite number of states that, if
populated, would be stable [2]. But importantly, such states are sub-Planckian
in mass, so they are not electric black holes. Black holes, so states which
satisfy instead

$\frac{\left<{\bf q}_{e},{\bf
a}_{0}\right>}{\left|\;\mathrm{Im\;}t\right|^{\frac{1}{2}}}\gg 1\;,$ (3.20)

can decay to dyonic states if those are present in the spectrum.

It is simple to state the wall of marginal stability between two general
dyonic states. Let us consider three charges ${\bf q}^{A}={\bf q}^{B}+{\bf
q}^{C}$, and we are interested in a wall of marginal stability for the process

${\bf q}^{A}\rightarrow{\bf q}^{B}+{\bf q}^{C}\;.$ (3.21)

There are two ways to formulate the condition for the wall of marginal
stability, corresponding to (2.12) and (2.13).

In terms of the charges ${\bf q}^{B}$ and ${\bf q}^{C}$, as in (2.12), we can
write the condition for the wall of marginal stability as

$\mathrm{Re\;}Z\left({\bf q}^{B},t_{WMS}\right)\mathrm{Im\;}Z\left({\bf
q}^{C},t_{WMS}\right)=\mathrm{Re\;}Z\left({\bf
q}^{C},t_{WMS}\right)\mathrm{Im\;}Z\left({\bf q}^{B},t_{WMS}\right)\;,$ (3.22)

where additionally we must impose

$\frac{\mathrm{Im\;}Z\left({\bf
q}^{B},t_{WMS}\right)}{\mathrm{Im\;}Z\left({\bf q}^{C},t_{WMS}\right)}>0\;.$
(3.23)

The second condition ensures that (3.22) corresponds to alignment rather than
anti-alignment.

Here we have introduced the notation $t_{WMS}$, which refers to the value of
$t$ on the wall of marginal stability. Note that this is a wall, so a one real
parameter line in complex $t$ space, and so $t_{WMS}$ is a parameter.

In terms of the charges ${\bf q}^{A}$ and ${\bf q}^{B}$, as in (2.13), we can
write the condition for the wall of marginal stability as

$\mathrm{Re\;}Z\left({\bf q}^{A},t_{WMS}\right)\mathrm{Im\;}Z\left({\bf
q}^{B},t_{WMS}\right)=\mathrm{Re\;}Z\left({\bf
q}^{B},t_{WMS}\right)\mathrm{Im\;}Z\left({\bf q}^{A},t_{WMS}\right)\;,$ (3.24)

where additionally we must impose

$\frac{Z\left({\bf q}^{A},t_{WMS}\right)}{Z\left({\bf
q}^{B},t_{WMS}\right)}\geq 1\;.$ (3.25)

The condition (3.25) ensures that it is alignment, rather than anti-alignment,
which is solving (3.24), as well as ensuring the second constraint in (2.13).

###  3.4 Single-centre attractor flows at infinite distance

In this section we study single-centre attractor flows near infinite distance
loci corresponding to $n=d=1$ in the classification of (3.7). We aim to
determine which charges support attractor loci which are in a controlled
region of moduli space. For the modulus $t$ associated to the infinite
distance, this requirement on the attractor locus is

$\mathrm{Im\;}t_{\star}\gg 1\;.$ (3.26)

For the other moduli, which we will refer to as $z_{i}$, to clearly
distinguish them from $t$, the condition can be stated generally that they
should have finite attractor values within the bulk of the moduli space. This
need not be near some infinite distance locus, or where any Nilpotent orbit
expansion is manifest. Let us denote the space of such physical values of the
moduli as ${\cal M}_{\mathrm{Phys}}$. We therefore require for a viable black
hole to exist that (2.21) has solutions such that

$z^{i}_{\star}\in{\cal M}_{\mathrm{Phys}}\;.$ (3.27)

The central charge is a restriction of (3.8) and takes the form

$Z\left({\bf
q},t,z^{i}\right)=\frac{1}{\left|c\left(z^{i}\right)\mathrm{Im\;}t\right|^{\frac{1}{2}}}\left[\left<{\bf
q},{\bf a}_{0}\left(z^{i}\right)\right>+\left<{\bf q},N\cdot{\bf
a}_{0}\left(z^{i}\right)\right>t+...\right]\;.$ (3.28)

Here we dropped the terms exponentially suppressed in $\mathrm{Im\;}t$, and
have manifested the dependence on the other moduli $z_{i}$. The attractor
locus for the modulus $t$ is then

$\bar{t}_{\star}=-\frac{\left<{\bf q},\left({\bf
a}_{0}\right)_{\star}\right>}{\left<{\bf q},N\cdot\left({\bf
a}_{0}\right)_{\star}\right>}\;,$ (3.29)

where $\left({\bf a}_{0}\right)_{\star}$ means evaluated on the attractor
solution for the other fields $z^{i}=z^{i}_{\star}$. This gives

$\mathrm{Im\;}t_{\star}=\frac{\mathrm{Re\;}\left<{\bf q},N\cdot\left({\bf
a}_{0}\right)_{\star}\right>\mathrm{Im\;}\left<{\bf q},\left({\bf
a}_{0}\right)_{\star}\right>-\mathrm{Re\;}\left<{\bf q},\left({\bf
a}_{0}\right)_{\star}\right>\mathrm{Im\;}\left<{\bf q},N\cdot\left({\bf
a}_{0}\right)_{\star}\right>}{\left|\left<{\bf q},N\cdot\left({\bf
a}_{0}\right)_{\star}\right>\right|^{2}}\;.$ (3.30)

The central charge evaluated on the attractor locus is

$\left|Z\left({\bf
q}\right)\right|^{2}_{\star}=\frac{2\;\mathrm{Im}\left(\left<{\bf
q},\left({\bf a}_{0}\right)_{\star}\right>\left<{\bf
q},N\cdot\left(\overline{{\bf
a}}_{0}\right)_{\star}\right>\right)}{\left<\left({\bf
a}_{0}\right)_{\star},N\cdot\left(\overline{{\bf
a}}_{0}\right)_{\star}\right>}\;.$ (3.31)

If the conditions (3.26) and (3.27) are satisfied, so the attractor locus is
physical, then this gives the entropy of the single-centered black hole
solution and so the associated contribution to the spectrum of BPS states of
that charge. It is worth noting that the central charge on the attractor locus
(3.31) is invariant under the monodromy transformation (3.11). This is a
general result, holding not only in this specific case.

The existence of a nilpotent orbit expansion implies that there is an
associated splitting of the charges into electric and magnetic as in (3.9). A
charge which admits a controlled and physical attractor locus, so satisfying
(3.26) and (3.27), must be magnetic. This is clear from the expression of the
central charge (3.28): purely electric charges only depend on the moduli
through an overall factor which cannot lead to a solution to the attractor
equations.

In general, it is not difficult to find infinite classes of charges which have
controlled attractor loci, so satisfy (3.26) and (3.27). Let us consider such
a charge and denote it ${\bf q}^{\mathrm{Phys}}$. It can have a magnetic and
an electric component, so we can write

${\bf q}^{\mathrm{Phys}}={\bf q}_{m}+Q{\bf q}_{e}\;.$ (3.32)

Here ${\bf q}_{m}$ is the magnetic component of the charge, ${\bf q}_{e}$ is
the electric component, and $Q$ is some integer. We can then consider,
schematically,

${\bf q}_{m}\sim{\cal O}(1)\;,\;\;{\bf q}_{e}\sim{\cal
O}(1)\;,\;\;\left|Q\right|\gg 1\;.$ (3.33)

Substituting the form (3.32) into (3.29), and using the properties (3.9) and
(3.33) gives

$\mathrm{Im\;}t_{\star}\simeq Q\;\mathrm{Im}\left(\frac{\left<{\bf
q}_{e},\left({\bf a}_{0}\right)_{\star}\right>}{\left<{\bf
q}_{m},N\cdot\left({\bf a}_{0}\right)_{\star}\right>}\right)\;.$ (3.34)

We see that by choosing $Q$ large and of appropriate sign we can always
satisfy (3.26).

What about the condition (3.27)? Let us ‘integrate out’ the field $t$ by using
its attractor locus (3.29), but keeping ${\bf a}_{0}$ as a function of the
$z^{i}$. We reach

$Z\left({\bf
q},t_{\star},z^{i}\right)\simeq\frac{Q}{\left|c\left(z^{i}\right)\mathrm{Im\;}t_{\star}\right|^{\frac{1}{2}}}\left[\left<{\bf
q}_{e},{\bf a}_{0}\left(z^{i}\right)\right>-\left<{\bf q}_{m},N\cdot{\bf
a}_{0}\left(z^{i}\right)\right>\frac{\left<{\bf q}_{e},\overline{{\bf
a}_{0}\left(z^{i}\right)}\right>}{\left<{\bf q}_{m},N\cdot\overline{{\bf
a}_{0}\left(z^{i}\right)}\right>}+...\right]\;.$ (3.35)

This expression gives us the (approximate) form of the potential for the
$z^{i}$, and the condition (3.27) is the assumption that there is a minimum to
this potential in the bulk of the moduli space. An important point is that $Q$
factors out in (3.35) which means that the attractor locus for the $z^{i}$ is
independent of $Q$. Since it is determined only by the choices of ${\bf
q}_{m}$ and ${\bf q}_{e}$, which are completely general, it is in general a
rather weak assumption that for some choice of charges there is a physical
attractor locus satisfying (3.27).

In conclusion, we have for charges of the form (3.32), we expect to be able to
find a physical attractor locus given the freedom to choose $Q$.

##  4 Evidence for obstruction to split attractor flows

In this section we study multi-centre, so split, attractor flows. These are
solutions where the attractor flow crosses a wall of marginal stability and
splits on it. Such flows combine the analysis of single-centred attractor
flows in section 3.4, and the analysis of walls of marginal stability in
section 3.3. Our analysis will be restricted first to single-parameter moduli
spaces, and to infinite distance loci of type $d=1$. In such cases, ${\bf
a}_{0}$ is a true constant, which makes things much simpler to analyse. We
will then go on to discuss increasingly general cases building on this.

While we analyse things completely generally here, it is always useful to be
guided by an example. In appendix A we present such an example.

We begin with the fact that if we are free to choose the charges ${\bf q}$,
then this translates into freedom in choosing $\left<{\bf q},{\bf
a}_{0}\right>$ and $\left<{\bf q},N\cdot{\bf a}_{0}\right>$. This follows from
the fact that minimally there are two real charge components of ${\bf q}$
which appear in $\left<{\bf q},{\bf a}_{0}\right>$, that allow us to choose
its real and imaginary parts arbitrarily. This is true up to quantization of
the charges, but of course if we there is an obstruction to split flows
allowing for continuous charges, then it will hold for quantized
charges.101010In any case, it turns out that quantization of the charges does
not place strong restrictions as all the relevant physical quantities have
rational coefficients. More precisely, $\left<{\bf q},{\bf a}_{0}\right>$
contains in general four real parameters (exactly four for one-parameter
models), corresponding to two electric and two magnetic charges. It is
possible to go to a basis, which we will not utilise, where it contains only
two electric charges. The expression $\left<{\bf q},N\cdot{\bf a}_{0}\right>$
contains at least two real magnetic charges. Choosing these charges is mapped
to choosing the real and imaginary parts of $\left<{\bf q},N\cdot{\bf
a}_{0}\right>$.

We will sometimes encounter rather lengthy expressions, and so it is useful to
introduce some condensed notation:

$\displaystyle\mathrm{Re}\left<{\bf q}^{A},{\bf a}_{0}\right>$
$\displaystyle\equiv$ $\displaystyle q^{A,+}\;,\;\;\mathrm{Im}\left<{\bf
q}^{A},{\bf a}_{0}\right>\equiv q^{A,-}\;,$
$\displaystyle\mathrm{Re}\left<{\bf q}^{A},N\cdot{\bf a}_{0}\right>$
$\displaystyle\equiv$ $\displaystyle q_{N}^{A,+}\;,\;\;\mathrm{Im}\left<{\bf
q}^{A},N\cdot{\bf a}_{0}\right>\equiv q_{N}^{A,-}\;.$ (4.1)

Since, as explained above, we are free to choose $q^{\pm}$ and $q_{N}^{\pm}$
arbitrarily as real parameters, the notation should hopefully not lead to any
loss of clarity. Also, we recall, that since we are in one-parameter models,
the $q$’s are all true constants.

###  4.1 Obstruction for one-parameter two-centre electric flows

Let us consider a setup where a purely electric charge flow splits into two
dyonic flows. We would like to know if the dyonic flows can end on physical
attractor loci. We take the split as ${\bf q}^{A}={\bf q}^{B}+{\bf q}^{C}$,
and take the magnetic component of ${\bf q}^{A}$ to vanish, ${\bf
q}^{A}_{m}=0$.

We consider the specification of the wall of marginal stability given by
(3.24) and (3.25). We can solve (3.24) for $\mathrm{Re\;}t_{WMS}$ as

$\mathrm{Re\;}t_{WMS}=\frac{-q^{A,-}q^{B,+}+q^{B,-}q^{A,+}+\left(q^{A,-}q_{N}^{B,-}+q^{A,+}q_{N}^{B,+}\right)\mathrm{Im\;}t_{WMS}}{-q_{N}^{B,-}q^{A,+}+q^{A,-}q_{N}^{B,+}}\;.$
(4.2)

Since $\mathrm{Re\;}t_{WMS}$ has no physical constraints on its sign or
magnitude, there is always such a solution. Note that we are here allowing
$\mathrm{Re\;}t$ to be arbitrary, while it is often considered to lie in the
range $0\leq\mathrm{Re\;}t\leq 1$. It can always be brought into this range by
acting with the monodromy (gauge) transformation
$\mathrm{Re\;}t\rightarrow\mathrm{Re\;}t+n$ and ${\bf q}\rightarrow
T^{n}\cdot{\bf q}$. In other words, there is a choice of charge representative
within each monodromy orbit for which $\left|\mathrm{Re\;}t\right|\leq 1$, but
since we do not want to restrict the charge to that representative, we should
equally keep $\mathrm{Re\;}t$ arbitrary.

We still have the condition (3.25) to impose. First though, let us solve for
the attractor loci $\mathrm{Im\;}t_{\star}^{B}$ and
$\mathrm{Im\;}t_{\star}^{C}$ of the two dyonic states ${\bf q}^{B}$ and ${\bf
q}^{C}$, as given by (3.30). We do this by keeping
$\mathrm{Im\;}t_{\star}^{B}$ and $\mathrm{Im\;}t_{\star}^{C}$ arbitrary, and
solving for some of the flux parameters, or equivalently the $q$’s, in terms
of them as

$\displaystyle q^{B,-}$ $\displaystyle=$
$\displaystyle\frac{q_{N}^{B,-}q^{B,+}+\left|q_{N}^{B}\right|^{2}\mathrm{Im\;}t_{\star}^{B}}{q_{N}^{B,+}}\;,$
$\displaystyle q^{A,-}$ $\displaystyle=$
$\displaystyle\frac{q_{N}^{B,-}q^{A,+}+\left|q_{N}^{B}\right|^{2}\left(\mathrm{Im\;}t_{\star}^{B}-\mathrm{Im\;}t_{\star}^{C}\right)}{q_{N}^{B,+}}\;,$
(4.3)

where
$\left|q_{N}^{B}\right|^{2}=\left(q_{N}^{B,+}\right)^{2}+\left(q_{N}^{B,-}\right)^{2}$.

Now we can return to the second constraint for the wall of marginal stability
(3.25). Substituting (4.2) and (4.3) into (3.25) gives

$\frac{\mathrm{Im\;}t^{B}_{\star}-\mathrm{Im\;}t^{C}_{\star}}{\mathrm{Im\;}t^{B}_{\star}+\mathrm{Im\;}t_{WMS}}\geq
1\;.$ (4.4)

It is manifest now that there is no way to satisfy this inequality while
maintaining the physical constraint on both the attractor loci as well as the
location of the wall of marginal stability

$\mathrm{Im\;}t^{B}_{\star}>0\;,\;\;\mathrm{Im\;}t^{C}_{\star}>0\;,\;\;\mathrm{Im\;}t_{WMS}>0\;.$
(4.5)

We have therefore proven that, in the one-parameter moduli space case, that
there are no two-centre physical split attractor flows with total electric
charge, at least not in the weakly-coupled region of moduli space
$\mathrm{Im\;}t\gg 1$.

There is no obstruction to a split flow which has a negative attractor locus
for one of the dyons. We can take, for example,
$-\mathrm{Im\;}t^{C}_{\star}>\mathrm{Im\;}t_{WMS}>0$ and satisfy the condition
(4.4). What does it mean that we have an attractor locus which has
$\mathrm{Im\;}t_{\star}<0$? This just means that the attractor flow will drive
$\mathrm{Im\;}t\rightarrow 1$ approaching the black hole horizon. What happens
after that is not clear because by then we are in the strongly-coupled regime
of the moduli space, where the exponentially suppressed (instanton-type)
corrections in $e^{-\mathrm{Im\;}t}$ become important and the Nilpotent orbit
approximation breaks down.

It is worthwhile looking at why the obstruction comes about in a less sharp
but more direct way. The requirement that the total charge is purely electric
imposes that the magnetic charges of the two dyons are opposite ${\bf
q}_{m}^{B}=-{\bf q}_{m}^{C}$. From (3.30) we therefore see that if
$\mathrm{Im\;}\left<{\bf q}_{e}^{B},\left({\bf a}_{0}\right)_{\star}\right>$
and $\mathrm{Im\;}\left<{\bf q}_{e}^{C},\left({\bf
a}_{0}\right)_{\star}\right>$ have the same sign (and also the real parts),
then we would have that $\mathrm{Im\;}t^{B}_{\star}$ and
$\mathrm{Im\;}t^{C}_{\star}$ must have opposite signs and so they cannot both
be positive.111111The magnetic parts of the charges in $\left<{\bf
q}^{B},\left({\bf a}_{0}\right)_{\star}\right>$ cancel out in (3.30). We also
have that the sum of the electric charges sum to the total electric charge
${\bf q}^{A}_{e}={\bf q}^{B}_{e}+{\bf q}^{C}_{e}$. But we need that the
electric state is more massive than the two dyonic states, and it obtains all
its mass from the electric charge. So we expect, schematically, $\left|{\bf
q}^{A}_{e}\right|>\left|{\bf q}^{B}_{e}\right|$ and $\left|{\bf
q}^{A}_{e}\right|>\left|{\bf q}^{C}_{e}\right|$. So the electric charges ${\bf
q}^{B}_{e}$ and ${\bf q}^{C}_{e}$ must ‘add up’ rather than ‘subtract’, and
this results in $\mathrm{Im\;}\left<{\bf q}_{e}^{B},\left({\bf
a}_{0}\right)_{\star}\right>$ and $\mathrm{Im\;}\left<{\bf
q}_{e}^{C},\left({\bf a}_{0}\right)_{\star}\right>$ having the same sign. This
picture is of course more schematic than the sharp result (4.4), but it makes
manifest the underlying cause.

What can we make of this no-go in terms of the stability of the electric
states? If the attractor flows ended in well-controlled loci, we would be able
to deduce that the dyonic constituents were in the spectrum of BPS states.
Therefore, crossing the wall of marginal stability in the opposite direction,
also the electric state would be in the spectrum. Since it would be heavier
than the dyonic constituents, which are themselves heavier than the Planck
scale, it would be a black hole. This black hole would then be unstable and
decay to the dyons upon the path to weak coupling which would make it sub-
Planckian in mass eventually. Also, following the discussion in section 2.4,
we would expect that by choosing the electric charge to be of the form ${\bf
q}^{A}_{e}=K{\bf q}_{e}$, for some large integer $K$, we can deduce the
existence of an electric state of charge ${\bf q}_{e}$ in the spectrum. But
this state would be sub-Planckian and cannot decay to dyonic states upon the
path to weak coupling. It would therefore remain stable, and this way we could
populate the BPS states of the light tower.

If the no-go would imply that there are just no physical attractor flows for
both the dyons, then we would deduce that ${\bf q}^{B}$ and ${\bf q}^{C}$
cannot both be populated by BPS states at the same time. This means that if
the black hole ${\bf q}^{A}$ was in the spectrum, so is physical, then it will
remain stable upon the path to weak coupling and become part of the sub-
Planckian light tower.

The fact that the attractor flows must go into the strongly-coupled region,
means at this general level, the results remain ambiguous.

###  4.2 Obstruction for one-parameter $n$-centre electric flows

After the first split of the attractors, one of the dyonic flows is towards
small $\mathrm{Im\;t}$. But we can consider the flow splitting again, this
time into two dyons with well controlled attractor loci. This can occur an
arbitrary number of times, and so we should consider in generality an
$n$-centre black hole configuration, where each centre is a dyonic black hole
with a controlled attractor locus, and the total sum of the charges is purely
electric.

To show that this is not possible, we will show the following result: a dyonic
flow towards a negative attractor value for $\mathrm{Im\;}t_{\star}$ cannot
split into two dyonic flows into positive attractor values. So this means that
there is no way to get rid of the negative attractor value we found from the
two-centre solution, so the one induced after the first split, no matter how
many subsequent splits there are.

This is a little more involved to show than the result of section 4.1, but
follows similar logic. We will eliminate fluxes for the attractor and wall of
marginal stability values of the modulus, and show an inconsistency. So we
consider ${\bf q}^{A}={\bf q}^{B}+{\bf q}^{C}$, but now all the charges are
dyonic. The only constraint we will impose is that ${\bf q}^{A}$ has a
negative value for its attractor locus

$\mathrm{Im\;}t^{A}_{\star}<0\;.$ (4.6)

We can eliminate some fluxes for the attractor loci of B and C by solving
(3.30) to give

$q^{B,-}=\frac{q_{N}^{B,-}q^{B,+}+\left|q_{N}^{B}\right|^{2}\mathrm{Im\;}t_{\star}^{B}}{q_{N}^{B,+}}\;,\;\;q^{C,-}=\frac{q_{N}^{C,-}q^{C,+}+\left|q_{N}^{C}\right|^{2}\mathrm{Im\;}t_{\star}^{C}}{q_{N}^{C,+}}\;.$
(4.7)

We can then substitute (4.7) into the attractor locus for state A, and solve
that as

$\displaystyle q^{B,+}=$ $\displaystyle-$
$\displaystyle\frac{1}{q_{N}^{C,+}\left(-q_{N}^{C,-}q_{N}^{B,+}+q_{N}^{B,-}q_{N}^{C,+}\right)}$
$\displaystyle\Bigg{[}q_{N}^{C,-}q_{N}^{B,+}\left(q^{C,+}q_{N}^{B,+}-2q_{N}^{B,-}q_{N}^{C,+}\mathrm{Im\;}t_{\star}^{A}\right)$
$\displaystyle+\left(q_{N}^{C,-}\right)^{2}q_{N}^{B,+}\left(q_{N}^{B,+}\mathrm{Im\;}t_{\star}^{C}+q_{N}^{C,+}\left(\mathrm{Im\;}t_{\star}^{C}-\mathrm{Im\;}t_{\star}^{A}\right)\right)$
$\displaystyle+q_{N}^{C,+}\left(-q_{N}^{B,-}q^{C,+}q_{N}^{B,+}+\left(q_{N}^{B,-}\right)^{2}\left(q_{N}^{C,+}\mathrm{Im\;}t_{\star}^{B}+q_{N}^{B,+}\left(\mathrm{Im\;}t_{\star}^{B}-\mathrm{Im\;}t_{\star}^{A}\right)\right)\right)$
$\displaystyle+q_{N}^{B,+}\left(q_{N}^{B,+}+q_{N}^{C,+}\right)\left(q_{N}^{B,+}\left(\mathrm{Im\;}t_{\star}^{B}-\mathrm{Im\;}t_{\star}^{A}\right)+q_{N}^{C,+}\left(\mathrm{Im\;}t_{\star}^{C}-\mathrm{Im\;}t_{\star}^{A}\right)\right)\Bigg{]}$

For the wall of marginal stability we will take the conditions (3.22) and
(3.23). We again would like to solve the constraint (3.22) for
$\mathrm{Re\;}t_{WMS}$. However, unlike the purely electric case, the
constraint (3.22) is now quadratic in $\mathrm{Re\;}t_{WMS}$ which means that
a real solution is not guaranteed. We should require some positivity
constraint on the coefficients. More precisely, we can solve (3.22) as

$\displaystyle\mathrm{Re\;}t_{WMS}=\frac{-q_{N}^{C,-}q^{B,+}+q_{N}^{B,-}q^{C,+}-q^{C,-}q^{B,+}+q_{N}^{C,+}q^{B,-}-\sqrt{P}}{2\left(q_{N}^{C,-}q_{N}^{B,+}-q_{N}^{B,-}q_{N}^{C,+}\right)}\;,$
(4.9)

where $P$ is a quantity which we require to be positive to obtain a real
solution

$\displaystyle P$ $\displaystyle=$
$\displaystyle\left(q_{N}^{C,-}q^{B,+}-q_{N}^{B,-}q^{C,+}+q^{C,-}q_{N}^{B,+}-q^{B,-}q_{N}^{C,+}\right)^{2}$
(4.10)
$\displaystyle-4\left(q_{N}^{C,-}q_{N}^{B,+}-q_{N}^{B,-}q_{N}^{C,+}\right)\Bigg{[}-q^{B,-}q^{C,+}+q^{C,-}\left(q^{B,+}-q_{N}^{B,-}\;\mathrm{Im\;}t_{WMS}\right)$
$\displaystyle+\mathrm{Im\;}t_{WMS}\left(q^{B,-}q_{N}^{C,-}-q^{C,+}q_{N}^{B,+}+q^{B,+}q_{N}^{C,+}\right.$
$\displaystyle\left.+q_{N}^{C,-}q_{N}^{B,+}\;\mathrm{Im\;}t_{WMS}-q_{N}^{B,-}q_{N}^{C,+}\;\mathrm{Im\;}t_{WMS}\right)\Bigg{]}>0\;.$

The constraint for a solution (4.10) must then be imposed along with the
second constraint on the wall of marginal stability (3.23). After substituting
the attractor values (4.7) and (4.2), the two constraints can be written as

$P=X^{2}-Y>0\;,\;\;X+\sqrt{X^{2}-Y}>0\;,$ (4.11)

where

$\displaystyle X$ $\displaystyle=$ $\displaystyle
2\left(q^{B,-}_{N}q^{C,-}_{N}+q^{B,+}_{N}q^{C,+}_{N}\right)\mathrm{Im\;}t_{WMS}$
$\displaystyle+\left[\left(q^{B,-}_{N}+q^{C,-}_{N}\right)^{2}+\left(q^{B,+}_{N}+q^{C,+}_{N}\right)^{2}\right]\mathrm{Im\;}t_{\star}^{A}-\left|q^{B}_{N}\right|^{2}\mathrm{Im\;}t_{\star}^{B}-\left|q^{C}_{N}\right|^{2}\mathrm{Im\;}t_{\star}^{C}\;,$
$\displaystyle Y$ $\displaystyle=$ $\displaystyle
4\left|q^{B}_{N}\right|^{2}\left|q^{C}_{N}\right|^{2}(\mathrm{Im\;}t_{WMS}+\mathrm{Im\;}t_{\star}^{B})(\mathrm{Im\;}t_{WMS}+\mathrm{Im\;}t_{\star}^{C})$
(4.12)

A solution to the constraints (4.11) requires $X>0$. Now since we are
interested in attractor loci of the form

$\mathrm{Im\;}t_{\star}^{A}<0\;,\;\;\mathrm{Im\;}t_{\star}^{B}>0\;,\;\;\mathrm{Im\;}t_{\star}^{C}>0\;,$
(4.13)

we see that

$2\left(q^{B,-}_{N}q^{C,-}_{N}+q^{B,+}_{N}q^{C,+}_{N}\right)\mathrm{Im\;}t_{WMS}>X>0\;.$
(4.14)

But (4.11) also requires $X^{2}>Y$, so we require

$\left(2\left(q^{B,-}_{N}q^{C,-}_{N}+q^{B,+}_{N}q^{C,+}_{N}\right)\mathrm{Im\;}t_{WMS}\right)^{2}>X^{2}>Y>\Big{(}2\left|q^{B}_{N}\right|\left|q^{C}_{N}\right|\mathrm{Im\;}t_{WMS}\Big{)}^{2}\;.$
(4.15)

It is a simple identity that

$\left(q^{B,-}_{N}q^{C,-}_{N}+q^{B,+}_{N}q^{C,+}_{N}\right)^{2}<\left|q^{B}_{N}\right|^{2}\left|q^{C}_{N}\right|^{2}\;,$
(4.16)

and therefore (4.15) cannot be satisfied. Hence, it is not possible to satisfy
the constraints for the wall of marginal stability (4.11) and the attractor
loci (4.13) simultaneously. This proves that any attractor flow towards a
negative attractor locus, cannot split into two attractor flows which are both
towards positive attractor loci.

###  4.3 Generalizing to multi-parameter models

We have shown that for one-parameter moduli spaces, there are no split
attractor flows near $n=d=1$ infinite distance loci which have a total
electric charge and terminate on controlled attractor loci. In this section we
generalise this result.

Generalising to the case of an arbitrary number of moduli still allows the
natural restriction of one-parameter approaches to infinite distances, and so
there is a singled-out coordinate $t$ which controls the infinite distance
limit. The crucial difference from the analysis of sections 4.1 and 4.2 is
that now the limiting period vector is not a constant but a function of the
other moduli ${\bf a}_{0}={\bf a}_{0}\left(z_{i}\right)$. This first of all
places additional constraints on the attractor loci, that they should be at
controlled and physical values of the $z_{i}$. However, this is expected to be
rather mild, and in any case can only make the constraints on the existence of
split flows stronger.

The aspect which may seem to allow to evade the no-go results, at least on
first thought, is that the $z_{i}$ moduli can vary between the locus of
marginal stability and the attractor loci. So $\left.{\bf
a}_{0}\left(z_{i}\right)\right|_{WMS}$ and ${\bf
a}_{0}\left(z_{i}\right)_{\star}$ need not be the same which would imply that
the $q$’s in (4.1) also need not be the same. Recall that it was the
interaction between these two loci in moduli space which led to the
obstruction of the split attractor flows. The wall of marginal stability and
the attractor loci could individually be made positive (physical), but not all
simultaneously. In this section we will study in examples if this additional
freedom allows to avoid the obstruction.

####  4.3.1 Obstruction in a two-parameter example

In this section we study the case of a two-parameter model near the large
complex-structure point. We will consider the mirror to the Calabi-Yau
$\mathbb{P}_{4}^{(1,1,2,2,2)}[8]$ as studied in [60]. There are two
monodromies associated to the two complex moduli fields $t_{1}$ and $t_{2}$.
These take the form

$N_{1}=\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\\ 1&0&0&0&0&0\\\
0&0&0&0&0&0\\\ -2&-4&0&0&0&0\\\ 0&-8&-4&0&0&0\\\
-\frac{22}{3}&0&-2&0&-1&0\end{array}\right)\;,\;\;N_{2}=\left(\begin{array}[]{cccccc}0&0&0&0&0&0\\\
0&0&0&0&0&0\\\ 1&0&0&0&0&0\\\ 0&0&0&0&0&0\\\ -2&-4&0&0&0&0\\\
-2&-2&0&-1&0&0\end{array}\right)\;.$ (4.17)

We note that the nilpotency indices are $n_{1}=3$ and $n_{2}=1$, so
$N_{1}^{4}=0$ and $N_{2}^{2}=0$. We can write the Nilpotent orbit of the
period vector as

$\Pi\left(t_{1},t_{2}\right)=e^{N_{1}t_{1}}{\bf
a}^{(1)}_{0}\left(t_{2}\right)=e^{N_{2}t_{2}}{\bf
a}^{(2)}_{0}\left(t_{1}\right)=e^{N_{2}t_{2}+N_{1}t_{1}}{\bf a}_{0}\;,$ (4.18)

where the limiting vectors are

${\bf a}^{(1)}_{0}\left(t_{2}\right)=\left(\begin{array}[]{c}1\\\ 0\\\
t_{2}\\\ -1\\\ -\frac{11}{3}-2t_{2}\\\
-t_{2}+2i\xi\end{array}\right)\;,\;\;{\bf
a}^{(2)}_{0}\left(t_{1}\right)=\left(\begin{array}[]{c}1\\\ t_{1}\\\ 0\\\
-1-2t_{1}-2t_{1}^{2}\\\ -\frac{11}{3}-4t_{1}^{2}\\\
\frac{1}{3}\left(-11t_{1}+4t_{1}^{2}+6i\xi\right)\end{array}\right)\;,\;\;{\bf
a}_{0}=\left(\begin{array}[]{c}1\\\ 0\\\ 0\\\ -1\\\ -\frac{11}{3}\\\
2i\xi\end{array}\right)\;,$ (4.19)

where $\xi$ is an order-one real constant whose value is not important for our
purposes. Note that in this example we have taken the symplectic inner-product
matrix as

$\eta=\left(\begin{array}[]{cccccc}0&0&0&0&0&-1\\\ 0&0&0&0&-1&0\\\
0&0&0&-1&0&0\\\ 0&0&1&0&0&0\\\ 0&1&0&0&0&0\\\
1&0&0&0&0&0\end{array}\right)\;,$ (4.20)

which is slightly different to (2.2).

Denoting the real and imaginary parts of the $t_{i}$ as

$t_{i}=b_{i}+iv_{i}\;,$ (4.21)

there are two types of infinite distance limits we can consider which match
onto the formulation in this paper:

$\displaystyle n=d=1\;$ $\displaystyle:$
$\displaystyle\;v_{1}\mathrm{\;finite}\;,\;\;v_{2}\rightarrow\infty\;,$
$\displaystyle n=d=3\;$ $\displaystyle:$
$\displaystyle\;v_{1}\rightarrow\infty\;,\;\;v_{2}\mathrm{\;finite}\;.$ (4.22)

Note that to keep control of the nilpotent orbit expressions we need to keep
always $v_{1},v_{2}\gg 1$.

#### Simplifying assumption

Let us denote the general charge vector as

${\bf q}=\left(q_{0},q_{1},q_{2},q_{3},q_{4},q_{5}\right)^{T}\;.$ (4.23)

It is difficult to analyse the attractor loci for charges in full generality.
We will therefore simplify the analysis by restricting to the case where all
charges have

$q_{0}=0\;.$ (4.24)

It is important to keep in mind that the obstruction we find may be related to
this restriction, and there exists the possibility that relaxing it will allow
a realisation of the split attractor flows. But in any case this is an example
analysis, not a general one, and it aim is to test for an obstruction once we
have multi-parameters, as relevant for the discussion in section 4.3. At least
this aspect is not affected by the restriction (4.24).

####  4.3.2 The attractor loci

We would like to solve for the attractor loci, specifically in order to test
if they are at physical well-controlled points in moduli space. Keeping in
mind this aim of the analysis, we can make a further simplification

$\xi\rightarrow 0\;.$ (4.25)

The justification for this is that because of the restriction (4.24), the only
place where $\xi$ appears is in the Kahler potential for the moduli. As long
as $v_{1},v_{2}\gg 1$, it gives a small correction which cannot change the
validity of an attractor locus.

With the states simplifications, the attractor loci are

$\displaystyle\left(v_{1}\right)_{\star}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sqrt{Q_{\star}}\;,\;\;\left(v_{2}\right)_{\star}=\frac{q_{2}}{q_{1}}\left(v_{1}\right)_{\star}\;,$
$\displaystyle\left(b_{1}\right)_{\star}$ $\displaystyle=$
$\displaystyle-\frac{2q_{1}+q_{3}}{4q_{1}}\;,\;\;\left(b_{2}\right)_{\star}=\frac{4q_{1}^{2}+2q_{1}q_{3}+q_{2}q_{3}-q_{1}q_{4}}{4q_{1}^{2}}\;,$
(4.26)

where

$Q_{\star}=\frac{-112q_{1}^{3}-3q_{2}q_{3}^{2}-6q_{1}q_{3}\left(q_{3}-q_{4}\right)-12q_{1}^{2}\left(q_{2}+2q_{3}-q_{4}+2q_{5}\right)}{q_{1}^{2}\left(2q_{1}+3q_{2}\right)}\;.$
(4.27)

Here we have assumed $q_{1}\neq 0$ which, as we will see, guarantees that the
state is dyonic with respect to the electromagnetic splitting induced by both
$N_{1}$ and $N_{2}$. Recall that being dyonic is a necessary requirement for a
non-divergent attractor locus.

####  4.3.3 The $n=d=1$ sector

Let us consider first the case of $v_{2}\rightarrow\infty$ as the degeneration
limit, so it is an $n=d=1$ degeneration. First we can split the charges into
electric and magnetic as in (3.9). We have

${\bf q}_{e}=\left(\begin{array}[]{c}0\\\ 0\\\ q_{2}\\\ 0\\\ q_{4}\\\
q_{5}\end{array}\right)\;,\;\;{\bf q}_{m}=\left(\begin{array}[]{c}q_{0}\\\
q_{1}\\\ 0\\\ q_{3}\\\ 0\\\ 0\end{array}\right)\;,\;\;$ (4.28)

We are interested in split attractor flows of the type ${\bf
q}^{A}_{e}\rightarrow{\bf q}^{B}_{d}+{\bf q}^{C}_{d}$, and define the wall of
marginal stability by the constraints (3.22) and (3.23). We can solve (3.22)
for $b_{2}$ as

$\displaystyle b_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{3v_{1}\left(8b_{1}^{2}q^{A}_{2}q^{B}_{1}-4q^{A}_{5}q^{B}_{1}+2q^{A}_{2}q^{B}_{3}+4b_{1}q^{A}_{2}\left(2q^{B}_{1}+q^{B}_{3}\right)+q^{A}_{4}(2q^{B}_{1}+q^{B}_{3})+8q^{A}_{2}q^{B}_{1}v_{1}^{2}\right)}\times$
(4.29)
$\displaystyle\Bigg{[}3q^{A}_{5}\left(2q^{B}_{2}+4b_{1}\left(2q^{B}_{1}+q^{B}_{2}\right)+q^{B}_{4}\right)v_{1}+q^{A}_{4}v_{1}\left[-11q^{B}_{1}+12b_{1}^{2}q^{B}_{1}+6b_{1}^{2}q^{B}_{2}-3\left(q^{B}_{2}+q^{B}_{5}\right)\right.$
$\displaystyle\left.+6\left(2q^{B}_{1}+q^{B}_{2}\right)v_{1}^{2}\right]+q^{A}_{2}v_{1}\left[q^{B}_{1}\left(-22+4b_{1}\left(-5+6b_{1}\right)+24v_{1}^{2}\right)\right.$
$\displaystyle\left.-3\left(2q^{B}_{5}+4b_{1}q^{B}_{5}+q^{B}_{4}\left(-1+2b_{1}^{2}+2v_{1}^{2}\right)\right)\right]$
$\displaystyle+3\left(q^{A}_{2}+2b_{1}\left(1+b_{1}\right)q^{A}_{2}+b_{1}q^{A}_{4}+q^{A}_{5}\right)\left(\left(2+4b_{1}\right)q^{B}_{1}+q^{B}_{3}\right)v_{2}$
$\displaystyle+6\left(2\left(q^{A}_{2}+2b_{1}q^{A}_{2}+q^{A}_{4}\right)q^{B}_{1}-q^{A}_{2}q^{B}_{3}\right)v_{1}^{2}v_{2}\Bigg{]}\;.$

Substituting this solution into (3.23), gives the constraint we need to
satisfy

$\displaystyle
R_{AB}\equiv\Bigg{(}3v_{1}\left(8b_{1}^{2}q^{A}_{2}q^{B}_{1}-4q^{A}_{5}q^{B}_{1}+2q^{A}_{2}q^{B}_{3}+4b_{1}q^{A}_{2}\left(2q^{B}_{1}+q^{B}_{3}\right)+q^{A}_{4}\left(2q^{B}_{1}+q^{B}_{3}\right)+8q^{A}_{2}q^{B}_{1}v_{1}^{2}\right)\Bigg{)}$
$\displaystyle\Bigg{[}v_{1}\left(3q^{B}_{3}\left(\left(2+4b_{1}\right)q^{B}_{2}+q^{B}_{4}\right)+\left(q^{B}_{1}\right)^{2}\left(-44+48b_{1}\left(1+b_{1}\right)+48v_{1}^{2}\right)\right.$
$\displaystyle\left.+6q^{B}_{1}\left(4b_{1}\left(q^{B}_{2}+b_{1}q^{B}_{2}+q^{B}_{3}\right)+q^{B}_{4}-2q^{B}_{5}+4q^{B}_{2}v_{1}^{2}\right)\right)$
$\displaystyle+3\left(\left(\left(2+4b_{1}\right)q^{B}_{1}+q^{B}_{3}\right)^{2}+16\left(q^{B}_{1}\right)^{2}v_{1}^{2}\right)v_{2}\Bigg{]}^{-1}\geq
1\;.$ (4.30)

At this point we need to impose that the attractor values for states $B$ and
$C$ are physical. The attractor solutions are given by (4.26-4.27). We can
eliminate generally $q^{B}_{5}$ and $q^{A}_{5}$ for $Q_{\star}^{B}$ and
$Q_{\star}^{C}$. Once we do that, the physical attractor loci conditions are

$Q_{\star}^{B}>0\;,\;\;Q_{\star}^{C}>0\;,\;\;\frac{q_{2}^{B}}{q^{B}_{1}}>0\;,\;\;\frac{q_{2}^{B}-q^{A}_{2}}{q^{B}_{1}}>0\;.$
(4.31)

Now let us return to the constraint (4.30). We note that that we have a free
real unconstrained parameter $b_{1}$ in the constraint. We can consider
eliminating $b_{1}$ for $R_{AB}$ (as defined in (4.30)). However, because
$b_{1}$ appears quadratically, there is a resulting condition on the
positivity of the square root to obtain a real solution (much like the
constraint (4.10)). After substituting the elimination of $q^{B}_{5}$ and
$q^{A}_{5}$ for $Q_{\star}^{B}$ and $Q_{\star}^{C}$ into this expression, we
recover the constraint

$\displaystyle-\Bigg{[}\left(q^{B}_{2}-q^{A}_{2}\right)v_{1}+q^{B}_{2}v_{1}\left(-1+R_{AB}\right)+2q^{B}_{1}\left(v_{1}+v_{2}\right)R_{AB}\Bigg{]}\times$
$\displaystyle\Bigg{[}Q_{\star}^{C}\left(3\left(q^{B}_{2}-q^{A}_{2}\right)+2q^{B}_{1}\right)+Q_{\star}^{B}\left(2q^{B}_{1}+3q^{B}_{2}\right)\left(-1+R_{AB}\right)$
$\displaystyle+3v_{1}\left(\left(q^{B}_{2}-q^{A}_{2}\right)v_{1}+q^{B}_{2}v_{1}\left(-1+R_{AB}\right)+2q^{B}_{1}\left(v_{1}+v_{2}\right)R_{AB}\right)\Bigg{]}>0\;.$
(4.32)

It is manifest that this constraint is incompatible with (4.30), which implies
$R_{AB}\geq 1$, and with (4.31). To see this note that (4.31) implies that
$q^{B}_{1}$, $q^{B}_{2}$ and $q^{B}_{2}-q^{A}_{2}$ must all have the same
sign, and therefore all the terms in the two square brackets have the same
signs.

We have therefore shown that, up to the restriction (4.24), there are no
electric split attractor flows in this example. So again we find an
obstruction.

####  4.3.4 The $n=d=3$ sector

We can repeat the same analysis but now with the electric-magnetic splitting
dictated by $N_{1}$. This corresponds to the limit $v_{1}\rightarrow\infty$
with $v_{2}$ finite. In this case the electric-magnetic splitting is

${\bf q}_{e}=\left(\begin{array}[]{c}0\\\ 0\\\ 0\\\ q_{3}\\\ q_{4}\\\
q_{5}\end{array}\right)\;,\;\;{\bf q}_{m}=\left(\begin{array}[]{c}q_{0}\\\
q_{1}\\\ q_{2}\\\ 0\\\ 0\\\ 0\end{array}\right)\;.$ (4.33)

The attractor loci solutions in (4.26) are still valid, but with the further
restriction that, since state $A$ is electric, $q^{A}_{2}=0$. This therefore
gives the constraints

$Q_{\star}^{B}>0\;,\;\;Q_{\star}^{C}>0\;,\;\;\frac{q_{2}^{B}}{q^{B}_{1}}>0\;.$
(4.34)

To determine the wall of marginal stability we can again solve for $b_{2}$.
However, to keep things tractable in this case we make the further simplifying
assumption

$q^{A}_{3}=0\;.$ (4.35)

We can then go through the same steps as in section 4.3.3 and we arrive at the
final positivity constraint (the analogue of (4.3.4))

$\displaystyle-
R_{AB}\Bigg{[}q^{B}_{2}v_{1}+2q^{B}_{1}\left(v_{1}+v_{2}\right)\Bigg{]}\times$
(4.36)
$\displaystyle\Bigg{[}Q_{\star}^{C}\left(3q^{B}_{2}+2q^{B}_{1}\right)+Q_{\star}^{B}\left(2q^{B}_{1}+3q^{B}_{2}\right)\left(-1+R_{AB}\right)+3v_{1}R_{AB}\left(q^{B}_{2}v_{1}+2q^{B}_{1}\left(v_{1}+v_{2}\right)\right)\Bigg{]}>0\;.$

Again there is an obstruction to (4.3.4) for $R_{AB}\geq 1$ and the
constraints (4.34).

##  5 Obstruction from moduli space singularities

In section 4 we presented evidence that there are no split attractor flows
where a total electric charge splits into a number of dyonic ones with well-
controlled attractor loci. This suggests that electric states do not decay to
dyonic states upon crossing walls of marginal stability.

In this section we will present further strong evidence for this, and also
generalise it significantly. The key idea is that in any weakly-coupled region
of moduli space, where the nilpotent orbit approximation holds, we know that
there are no singularities in the moduli space away from the degeneration
limit $g\rightarrow 0$.121212By this we mean we consider the complex plane of
the modulus associated to the nilpotent orbit expansion. The other moduli can
lead to singularities for non-generic values, but this is irrelevant for the
flow to $g\rightarrow 0$ which can be achieved by varying the nilpotent orbit
modulus. This places constraints on the BPS spectrum, because we cannot have a
populated state become massless in that region. So any charge ${\bf q}$ for
which the central charge $Z\left({\bf q}\right)$ vanishes in a weakly-coupled
asymptotic region must be absent from the spectrum. We will show that the
electric to dyonic decays are obstructed by this, so one of the dyonic
constituents would always have a vanishing point along the wall of marginal
stability of the decay and should therefore not be in the spectrum.

The vanishing of the central charge along walls of marginal stability is
something that played an important role in the formulation of the Kontsevich-
Soibelman (KS) wall-crossing formula [5]. It was assumed that this does not
occur, or more precisely the formula was restricted to states which do not
have a vanishing central charge at the point where the wall is crossed. The
formula applies to charges which satisfy, on the wall of marginal stability, a
certain positivity constraint. Namely, there is a quadratic form on the
lattice of charges, so an inner product which maps a charge to a real number
${\cal Q}\;:\;{\bf q}\rightarrow\mathbb{R}$. Then the formula applies to
stable charges, so which satisfy

${\cal Q}\left({\bf q}\right)\geq 0\;.$ (5.1)

The form further has the property that

$\left.{\cal Q}\right|_{\mathrm{Ker}Z\left({\bf q}\right)}<0\;.$ (5.2)

So, charges which lead to a vanishing central charge $Z\left({\bf q}\right)=0$
are not stable charges. It was discussed in [5] that there is a natural
candidate for such a quadratic form on the Calabi-Yau.

The stability constraint is essential because it fixes the finiteness of the
number of combinations of charges which align on a wall of marginal stability.
Specifically, on the wall we may consider alignment

${\bf q}^{A}\left(r_{B},r_{C}\right)=r_{B}\;{\bf q}^{B}+r_{C}\;{\bf q}^{C}\;,$
(5.3)

with $r_{B}$ and $r_{C}$ some integers. So we let ${\bf q}^{A}$ be a charge
which depends on $r_{B}$ and $r_{C}$. Then the stability condition ensures
that there is a basis of the alignment charges such that $r_{B},r_{C}\geq 0$
(or $r_{B},r_{C}\leq 0$) for all populated BPS states. This positivity
constraint is crucial to guarantee finiteness in the wall crossing formula [5,
69, 70].

Returning to the singularities on the moduli space. We see that it is the same
as requiring that all the populated states on the wall of marginal stability
are stable (5.1). It is in this sense the statement that the wall-crossing
formula should apply to all BPS states at any point along the wall, at least
in any weakly-coupled region.

What we are able to show, below, is that an electric to dyonic decay implies
one of the dyonic states is not stable at some point on the wall. This
suggests that it cannot be part of the BPS spectrum, matching the results from
the attractor loci analysis. The final part of the proof is to show that it
cannot be that it participated in the decay, and then itself decayed away from
the spectrum before it became unstable. We can prove this for the case
$n=d=1$, but have not done so completely generally for any $n$ and $d$.

A key aspect of the analysis is that we find that the absence of a
singularity, or the stability constraint, leads to the same expressions as the
positivity of the attractor loci. This is expected if the correspondence
between attractors and the spectrum of BPS states holds, and can be said to
provide strong support for it. Indeed, since the attractor locus minimizes the
absolute value of the central charge, we could not have a physical attractor
locus with a non-vanishing central charge, and also a physical locus where the
central charge vanishes.

###  5.1 Equivalence of stability and attractor loci

Let us consider the one-parameter $n=d=1$ case, as studied in section 4.1. The
wall of marginal stability is given by (4.2). We can then substitute this into
the central charge formula and solve for vanishing central charge constraint
on the wall

$\left.Z\left({\bf q},\mathrm{Im\;}t_{0}\right)\right|_{WMS}=0\;.$ (5.4)

So here we are restricting the central charge to the wall, by solving for
$\mathrm{Re\;}t$ as in (4.2), and then defining $\mathrm{Im\;}t_{0}$ as the
value of $\mathrm{Im\;}t$ where the central charge vanishes. Doing this for
the two charges ${\bf q}^{B}$ and ${\bf q}^{C}$ gives

$\displaystyle\mathrm{Im\;}t_{0}^{B}$ $\displaystyle=$
$\displaystyle-\left[\frac{-q^{B,-}_{N}q^{B,+}+q^{B,-}q^{B,+}_{N}}{\left|q_{N}^{B}\right|^{2}}\right]=-\mathrm{Im\;}t_{\star}^{B}\;,$
$\displaystyle\mathrm{Im\;}t_{0}^{C}$ $\displaystyle=$
$\displaystyle-\left[\frac{-q^{B,-}_{N}q^{B,+}+q^{B,-}q^{B,+}_{N}-q^{A,-}q_{N}^{B,+}+q^{A,+}q^{B,-}_{N}}{\left|q_{N}^{B}\right|^{2}}\right]=-\mathrm{Im\;}t_{\star}^{C}\;.$
(5.5)

We see that the modulus value on the wall where the central charge vanishes
for the states $B$ and $C$ is exactly the negative of the attractor locus for
those states. Since the wall is defined by (4.2) for any $\mathrm{Im\;}t>0$,
having no point on the wall of marginal stability where the central charge
vanishes is the same as the statement that the two attractor loci are
positive. This is precisely the problem we have analysed in section 4.1, and
have shown that it leads to an obstruction.

Actually, the whole analysis is only valid for $\mathrm{Im\;}t\gg 1$, since
that is required for the use of the nilpotent orbit approximation. But the
power of the stability constraint is that if we found an attractor locus which
is very negative $\mathrm{Im\;}t_{\star}\ll-1$, then we can translate that
into a zero of the central charge in the controlled region. The only remaining
refuge for states could be in the strong-coupling regime
$\left|\mathrm{Im\;}t\right|<1$. So the only escape from the obstruction is if
we pick charges such that both of $\mathrm{Im\;}t_{0}^{B}$ and
$\mathrm{Im\;}t_{0}^{C}$ are either small or negative. However, looking at the
constraint (4.4) we see that it requires at least one of the attractor loci to
be large and negative, which means at least one large and positive
$\mathrm{Im\;}t_{0}$.

The second powerful aspect of the stability constraint it that it applies on
the wall of marginal stability. That means that the multi-parameter case can
be analysed in the same way as the one-parameter case, as long as the
degeneration itself is by variation of only one parameter. So the value of the
$q$’s for the positivity constraints (5.5), is the same as the one for the
wall of marginal stability. There is no need to track them to the attractor
loci. Their dependence on the other moduli $z^{i}$ is inconsequential for the
constraints.

There is an important way to attempt to avoid the problem of the vanishing
central charge point $\mathrm{Im\;}t_{0}$. We can postulate that between the
point on the wall of marginal stability which was crossed for the decay, and
the vanishing central charge locus, the state decayed and left the spectrum.
So it participated in the decay, and then after moving along the marginal
stability locus, it decayed away so that it did not lead to a singularity in
the moduli space at $\mathrm{Im\;}t_{0}$. It is difficult to believe that this
could occur, because recall that the states have an associated BPS index which
is typically very large, and this would have to completely vanish through
decays upon the variation to $\mathrm{Im\;}t_{0}$. But we can be even more
precise and show that in fact one can never avoid the singularity. Recall that
we faced a similar possibility in section 4.2 when studying attractor loci. We
considered whether it is possible to avoid a negative attractor locus by
having the state decay, so have a split attractor flow, to two states with
positive attractor loci. We showed that this is not possible because one of
the decay products would always have a negative attractor locus. We can import
this result into this setting to show that if the state which has a vanishing
point decays somewhere along the wall of marginal stability, then moving along
the marginal stability wall of that decay will always lead to a zero locus for
one of the products in a physical regime for $\mathrm{Im\;}t$. The only thing
we need to show is that the correspondence between the attractor locus and the
zero locus, as in (5.5), is general. So the negative attractor locus which we
cannot remove by decays turns into a positive vanishing point.131313One might
try to hide the vanishing of this second decay product in the strong-coupling
region, but this is not possible. The simplest way to see this is that the
analysis in section 4.2 shows that having both
$\left|\mathrm{Im\;}t_{\star}^{B}\right|\ll\left|\mathrm{Im\;}t_{WMS}\right|$
and
$\left|\mathrm{Im\;}t_{\star}^{C}\right|\ll\left|\mathrm{Im\;}t_{WMS}\right|$
is not possible for $\mathrm{Im\;}t_{\star}^{A}<0$. Therefore, at least one of
$\mathrm{Im\;}t_{\star}^{B}$ or $\mathrm{Im\;}t_{\star}^{C}$ must be negative
and large in magnitude.

It is simple to show that indeed the correspondence between the attractor loci
and the vanishing loci is general. The general central charge, dropping the
overall irrelevant Kahler-potential normalization, takes the form

$\mathrm{Re\;}Z\left({\bf q}\right)\sim
q^{+}+q_{N}^{+}\;\mathrm{Re\;}t-q_{N}^{-}\;\mathrm{Im\;}t\;,\;\;\mathrm{Im\;}Z\left({\bf
q}\right)\sim q^{-}+q_{N}^{+}\;\mathrm{Im\;}t+q_{N}^{-}\;\mathrm{Re\;}t\;.$
(5.6)

For a charge participating in a decay, the wall of marginal stability can be
defined by

$\mathrm{Re\;}Z\left({\bf
q}\right)A\left(t\right)=B\left(t\right)\;\mathrm{Im\;}Z\left({\bf
q}\right)\;,$ (5.7)

where $A\left(t\right)$ and $B\left(t\right)$ are in general real functions of
the moduli (in the case where one of the charges is purely electric it is a
constant). Then if such a wall exists in a physical regime, we can combine
(5.6) and (5.7) to write that everywhere on the wall we have

$\mathrm{Re\;t}_{WMS}=\frac{-Bq^{-}+Aq^{+}-\left(Aq_{N}^{-}+Bq_{N}^{+}\right)\mathrm{Im\;t}}{-Aq_{N}^{+}+Bq_{N}^{-}}\;.$
(5.8)

Substituting this into (5.6) we can solve for the vanishing locus

$\mathrm{Re\;}Z\left({\bf q},t_{0}\right)=0\;,$ (5.9)

which gives

$\mathrm{Im\;}t_{0}=-\left[\frac{-q^{-}_{N}q^{+}+q^{-}q^{+}_{N}}{\left|q_{N}\right|^{2}}\right]=-\mathrm{Im\;}t_{\star}\;.$
(5.10)

The only subtlety here is if the vanishing locus is such that also
$A\left(t_{0}\right)=B\left(t_{0}\right)=0$, but then we would have a
vanishing locus (5.4) for one of the other states.

###  5.2 General proof of obstruction

It is possible to prove very generally, and relatively simply, the first part
of the obstruction: that for electric to dyonic decays, one of the decay
products must have a vanishing central charge point on the wall of marginal
stability. In fact, one can prove a more general statement which we will
outline below.

Let us begin by noting the reason why the solutions (5.5) even exist. This is
highly non-trivial because if we take a generic central charge, and look for
its vanishing loci, they are two constraints on one coordinate and so pick out
a point in moduli space. On the other hand, the wall of marginal stability is
a line in the moduli space. And generic lines and points do not intersect.

The point is that we are considering electric states for which the phase is a
constant, so independent of $t$. The constraint for the wall of marginal
stability (3.24) then ensures that the ratio of the real and imaginary parts
of the central charges for ${\bf q}^{B}$ (and ${\bf q}^{C}$) are also a
constant. They therefore become proportional to each other. This means that
the requirement for the wall of marginal stability to have a vanishing point
on it becomes the intersection of two real lines: the marginal stability line
and the line of vanishing real part of the central charge. Once they
intersect, the vanishing of the imaginary part of the central charge is
guaranteed. In a real two-dimensional space, two generic real lines intersect.

Since on the wall of marginal stability the real and imaginary parts of the
central charge are proportional to each other, the second constraint for the
decay (3.25) can be written as

$\frac{\mathrm{Re\;}Z\left({\bf q}^{A}\right)}{\mathrm{Re\;}Z\left({\bf
q}^{B},t_{WMS}\right)}\geq 1\;.$ (5.11)

It is important to note that in (5.11) we have manifested the point that the
numerator is independent of $t_{WMS}$, it is a constant. The denominator is a
function of $t_{WMS}$, but we only need (5.11) to hold for some point on the
wall of marginal stability, so for some value of $t_{WMS}$.

Now let us consider the vanishing point for the central charge of ${\bf
q}^{C}$, so $\mathrm{Im\;}t_{0}^{C}$. Since the marginal stability line spans
all of $\mathrm{Im\;}t$, for this point not to lie one it we must have that
there is no positive value of $\mathrm{Im\;}t$ for which
$\mathrm{Re\;}Z\left({\bf q}^{A}\right)=\left.\mathrm{Re\;}Z\left({\bf
q}^{B},\mathrm{Im\;}t\right)\right|_{WMS}$. Since they must have the same sign
to satisfy (5.11), we see that we must therefore have

$\left.\mathrm{Re\;}Z\left({\bf
q}^{B},\mathrm{Im\;}t\right)\right|_{WMS}>\mathrm{Re\;}Z\left({\bf
q}^{A}\right)\mathrm{\;or\;\;}\left.\mathrm{Re\;}Z\left({\bf
q}^{B},\mathrm{Im\;}t\right)\right|_{WMS}<\mathrm{Re\;}Z\left({\bf
q}^{A}\right)\;\;\;\forall\;\mathrm{Im\;}t\;.$ (5.12)

But $\left.\mathrm{Re\;}Z\left({\bf q}^{B},\mathrm{Im\;}t\right)\right|_{WMS}$
is unbounded in magnitude as a function of $\mathrm{Im\;}t$, and therefore
only the first option in (5.12) is possible. Therefore, (5.11) can never be
satisfied.

The analysis presented generalises simply since it does not rely on a
particular choice of $n$ or $d$. The only assumption was that electric charges
are such that $\mathrm{Re\;}Z\left({\bf q}^{A}\right)$ is a constant (up to
the overall factor from the Kahler potential). We can also see that this
restriction is simple to generalise. The absence of a zero locus along the
wall of marginal stability is the same as the statement that (5.11) must hold
everywhere on the wall of marginal stability.

It is worth showing this explicitly. Let us consider in general a decay
$A\rightarrow B+C$. Then we require the generalisation of (5.11), so

$\left.\frac{\mathrm{Re\;}Z\left({\bf
q}^{A},t_{c}\right)}{\mathrm{Re\;}Z\left({\bf
q}^{B},t_{c}\right)}\right|_{WMS}\geq 1\;.$ (5.13)

Here we made things more explicit than (5.11) by labelling explicitly the
point on the wall of marginal stability where the decay is occurring as
$t_{c}$. Now let us violate (5.13) at some point by moving along the wall of
marginal stability, so that eventually we reach some point $t_{0}$ such that

$\left.\frac{\mathrm{Re\;}Z\left({\bf
q}^{A},t_{0}\right)}{\mathrm{Re\;}Z\left({\bf
q}^{B},t_{0}\right)}\right|_{WMS}=1\;.$ (5.14)

At this point we have

$\mathrm{Re\;}Z\left({\bf q}^{C},t_{0}\right)=\mathrm{Re\;}Z\left({\bf
q}^{A},t_{0}\right)-\mathrm{Re\;}Z\left({\bf q}^{B},t_{0}\right)=0\;.$ (5.15)

But we also know that everywhere on the wall of marginal stability we have

$\displaystyle\mathrm{Re\;}Z\left({\bf q}^{C},t\right)\mathrm{Im\;}Z\left({\bf
q}^{A},t\right)$ $\displaystyle=$ $\displaystyle\mathrm{Re\;}Z\left({\bf
q}^{A},t\right)\mathrm{Im\;}Z\left({\bf q}^{C},t\right)\;,$
$\displaystyle\mathrm{Re\;}Z\left({\bf q}^{C},t\right)\mathrm{Im\;}Z\left({\bf
q}^{B},t\right)$ $\displaystyle=$ $\displaystyle\mathrm{Re\;}Z\left({\bf
q}^{B},t\right)\mathrm{Im\;}Z\left({\bf q}^{C},t\right)\;.$ (5.16)

So at $t_{0}$ we have only two possibilities

$\mathrm{Re\;}Z\left({\bf q}^{C},t_{0}\right)=\mathrm{Im\;}Z\left({\bf
q}^{C},t_{0}\right)=0\;,$ (5.17)

or

$\mathrm{Re\;}Z\left({\bf q}^{A},t_{0}\right)=\mathrm{Re\;}Z\left({\bf
q}^{B},t_{0}\right)=\mathrm{Re\;}Z\left({\bf q}^{C},t_{0}\right)=0\;.$ (5.18)

The second option (5.18) is not viable because $t_{0}$ is defined by (5.14).
It is possible to consider the possibility that in (5.14) the numerator and
denominator share a zero at $t_{0}$ while leaving a finite ratio, but this
seems highly implausible since they are different functions in one real
variable. In other words, the wall of marginal stability guarantees that
$\mathrm{Re\;}Z\left({\bf q}^{C},t_{0}\right)$ is proportional to
$\mathrm{Im\;}Z\left({\bf q}^{C},t_{0}\right)$, but not that
$\mathrm{Re\;}Z\left({\bf q}^{A},t_{0}\right)$ is proportional to
$\mathrm{Re\;}Z\left({\bf q}^{B},t_{0}\right)$. We are therefore left with the
first option (5.17), and so $Z\left({\bf q}^{C}\right)$ vanishes somewhere on
the wall of marginal stability.

Yet another, the most simple, way to see this is to note that (5.14) is the
statement that state $A$ and $B$ have equal masses. But the wall of marginal
stability is precisely the locus where the masses of the constituents, $B$ and
$C$, sum up to the mass of $A$. So state $C$ must be massless there.

There is an important subtlety with connecting this to the $g\rightarrow 0$
limit, specifically, it is not always the case that the wall of marginal
stability stretches to the degeneration limit
$\mathrm{Im\;t}\rightarrow\infty$. In the $n=d=1$ case, for an electric to
dyonic decay, it did because the condition defining the wall of marginal
stability (3.24) was a linear equation in $\mathrm{Im\;t}$, and so has
solutions for all its values. On the other hand, the wall for a dyonic state
to decay to a dyonic state led to a quadratic equation which for fixed charges
had no solution as $\mathrm{Im\;t}\rightarrow\infty$.

In order to have a solution for arbitrarily large $\mathrm{Im\;t}$ we need to
look for a solution scaling of type

$\mathrm{Re\;t}=x\;\mathrm{Im\;t}\;,\;\;x\in\mathbb{R}\;.$ (5.19)

Whether such a solution exists for a real $x$ depends on the highest power of
$t$ which appears in the central charge for the states $A$ and $B$, and is
shown in table 5.1. We see that if the power of $t$ is larger in state $B$
than $A$, the wall of marginal stability stretches out to
$\mathrm{Im\;t}\rightarrow\infty$. So let us assume this for now, and we will
explain how this highest power is classified mathematically in section 5.3.
With this assumption, taking the degeneration limit
$\mathrm{Im\;t}\rightarrow\infty$, which corresponds to $g\rightarrow 0$,
gives the BPS Stability Filtration of the introduction.

A / B | 0 | 1 | 2 | 3
---|---|---|---|---
0 | - | ✓ | ✓ | ✓
1 | ✓ | $\times$ | ✓ | ✓
2 | ✓ | ✓ | $\times$ | ✓
3 | ✓ | ✓ | ✓ | $\times$
Table 5.1: Table showing if a solution of type (5.19) exists, and therefore
the wall of marginal stability stretches to the limit
$\mathrm{Im\;t}\rightarrow\infty$ (and so $g\rightarrow 0$), for the highest
powers of $t$ in the (holomorphic part of the) central charge of states $A$
and $B$. When a solution exists, there are no constraints on the charges.

The reason that the BPS Stability Filtration is a proposal and not proven is
that there remains the issue of showing that it is not possible for the state
with the vanishing locus to participate in the decay, and then decay away from
the spectrum by the time it reaches the vanishing point. For the case $n=d=1$
we have proven that indeed this cannot happen, or more precisely, that it will
lead to some other decay product having a zero locus. We have not done so for
general $n$ and $d$, which is the missing step for a general proof.
Unfortunately, showing this requires some further work, which we leave for the
future. We expect that this is only technical and not conceptual, the
accumulated evidence so far seems sufficiently strong to propose a BPS
Stability Filtration.

We can make some simple statement about the general case by restricting to
certain charge sectors. In general, the central charge takes the form

$\displaystyle Z\left({\bf q},t,z^{i}\right)$ $\displaystyle=$
$\displaystyle\frac{1}{\left|c\left(z^{i}\right)\mathrm{Im\;}t\right|^{\frac{d}{2}}}\left[\left<{\bf
q},{\bf a}_{0}\left(z^{i}\right)\right>+\left<{\bf q},N\cdot{\bf
a}_{0}\left(z^{i}\right)\right>t+\right.$ (5.20)
$\displaystyle\frac{1}{2}\left.\left<{\bf q},N^{2}\cdot{\bf
a}_{0}\left(z^{i}\right)\right>t^{2}+\frac{1}{6}\left<{\bf q},N^{3}\cdot{\bf
a}_{0}\left(z^{i}\right)\right>t^{3}\right]\;.$

We can choose our black hole charges such that

$\left<{\bf q},{\bf a}_{0}\left(z^{i}\right)\right>=\left<{\bf
q},N^{3}\cdot{\bf a}_{0}\left(z^{i}\right)\right>=0\;.$ (5.21)

We then define

$\tilde{{\bf q}}=N\cdot{\bf q}\;.$ (5.22)

so that the central charge takes the form (for $d=3$)

$Z\left({\bf
q},t,z^{i}\right)=\frac{t}{\left|c\left(z^{i}\right)\mathrm{Im\;}t\right|^{\frac{3}{2}}}\left[\left<\tilde{{\bf
q}},{\bf a}_{0}\left(z^{i}\right)\right>+\frac{1}{2}\left<\tilde{{\bf
q}},N\cdot{\bf a}_{0}\left(z^{i}\right)\right>t\right]\;.$ (5.23)

The analysis of the obstruction to decays from the electric to dyonic charges
is insensitive to overall factors in the central charge. Therefore, this is
essentially the same form as (3.28) and we can carry our results over.

It is worth discussing the relation of this analysis to that of [71]. There,
paths in moduli space which connect marginal stability walls to anti-marginal
stability walls (where the central charges anti-align) were studied. In our
setting, the locus where the central charge of one of the constituents
vanishes turns the marginal stability wall into an anti-marginal stability
wall. So one can consider circling that locus, thereby traversing a path as
studied in [71]. There it was argued that either there is a singularity in the
moduli space (which we are forbidding) that induces a conjugation wall, or the
constituents themselves were never fundamental in the first place, and their
constituents rearrange along the path (this was termed crossing a
recombination wall). This is consistent with our analysis. Of course, the
assumption of [71] is that BPS constituents are populated in the first place,
which need not be the case here.

###  5.3 A BPS Stability Filtration

The analysis in section 5.2 culminated in the point that if we consider a
decay of some state $A$ to some state $B$ (plus something else), then the
decay requires that the mass of $A$ is larger than the mass of $B$ over the
full locus of marginal stability. We also showed that if the highest power of
$\mathrm{Im\;}t$ in the central charge of $B$ is higher than the highest power
of $\mathrm{Im\;}t$ in the central charge of $A$, then the wall of marginal
stability stretches to the limit $\mathrm{Im\;t}\rightarrow\infty$ (and so
$g\rightarrow 0$). This also automatically implies that for some sufficiently
large value of $\mathrm{Im\;t}$ the mass of $B$ will become higher than the
mass of $A$, and so is a sufficient condition to forbid such a decay.

There is a mathematical formulation which captures precisely the highest power
of $\mathrm{Im\;}t$ in the central charge, termed the monodromy weight
filtration. Here we outline some parts of it, and refer to [2, 3] for much
more detail about this structure.

Consider the rational vector space of charges ${\bf q}\in
V\left(\mathbb{Q}\right)$. Then the monodromy matrix $N$ induces a monodromy
weight filtration $W_{i}\left(N\right)$ on $V$.

$W_{-1}\equiv 0\ \subset\ W_{0}\ \subset\ W_{1}\ \subset\ ...\ W_{6}=V\ .$
(5.24)

This filtration is uniquely specified through the following defining
properties

$\displaystyle\bullet\quad NW_{i}\subset W_{i-2}$ (5.25)
$\displaystyle\bullet\quad N^{j}:Gr_{3+j}\rightarrow Gr_{3-j}\ \ \text{is an
isomorphism,}\;\;\;\;Gr_{j}\equiv W_{j}/W_{j-1}\ .$ (5.26)

Note that there is a simple representation of the $W_{i}$ in terms of the
kernels and images of $N^{j}$ as

$W_{0}=\text{im}\,N^{3}\ ,\quad W_{1}=\text{im}\,N^{2}\cap\text{ker}\,N\ ,\
\ldots\ ,\quad W_{5}=\text{ker}N^{3}\ .$ (5.27)

This implies immediately that if the unipotency index is smaller than the
complex dimension of the manifold, $n<3$, some of the previous subspaces will
be empty. In particular, for all $j>n$ we have $W_{3+j}=W_{3+n}$ and
$W_{3-j}=0$.

Note that using the uniqueness of the filtration on can show that if we have
${\bf q}_{A}\in W_{i}$ and ${\bf q}_{B}\in W_{6-i-j}$ then

$\left<{\bf q}_{A},{\bf q}_{B}\right>=0\;,\mathrm{if\;}j>0\ .$ (5.28)

The splitting into electric and magnetic states can be done as follows. We
define electric states as those lying in $W_{i}$ with $i<3$. Dyonic states are
charges lying in $W_{i}$ with $i>3$. Magnetic states are dyonic states where
the electric components, so those that are in $W_{i}$ with $i<3$, are removed.
The simplest way to identify those is by acting with $N$ and using (5.25).

The $W_{i}$ are related to the highest power of $t$ which appears in the
(holomorphic part of the) central charge. This is because (from (5.25))

$\left<{\bf q},N^{w}\cdot{\bf
a}_{0}\right>=\left(-1\right)^{w}\left<N^{w}\cdot{\bf q},{\bf
a}_{0}\right>\neq 0\implies{\bf q}\notin W_{2\left(w+1\right)-n}\;.$ (5.29)

When using this we should keep in mind that $W_{4+n}$ does not exist, and
$W_{2-n}$ is empty. We can therefore assign an s-weight to a charge as

$\mathrm{s-weight}\left[{\bf
q}\right]=\mathrm{Largest\;integer\;}w\;\mathrm{such\;that\;}{\bf q}\notin
W_{2\left(w+1\right)-n}\;.$ (5.30)

Then the s-weight of a BPS state corresponds to the highest allowed power of
$t$ in its central charge.

We therefore can identify the BPS stability filtration with the monodromy
weight filtration as follows: a BPS state cannot decay to a BPS state of a
higher s-weight. The remarkable thing is that this is true for any
perturbative $g\ll 1$, and not only in the $g\rightarrow 0$ limit.

There is a subtlety with such an identification of the BPS stability
filtration with the monodromy weight filtration which is that it assumed the
the maximum allowed power of $t$ is realised by the charges. This assumes that
the charge is a sufficiently generic element of the (maximal) $W_{i}$ that it
belongs to. A better way to phrase this genericity condition is that the
moduli other than $t$, so the $z^{i}$, are at generic values. In that case a
vanishing $\left<{\bf q},N^{w}\cdot{\bf a}_{0}\right>$ implies the full
constraint on the charges in terms of the monodromy weight filtration.

If we would like to drop the assumption of generic values for the other
moduli, then there is still a mathematical formulation which relates the
highest power of $t$ which appears in the central charge to the charge ${\bf
q}$, but it is slightly more involved. It corresponds to placing the charge
${\bf q}$ into the primitive spaces of the limiting mixed Hodge structure (the
$P_{i}$ in [2, 3]). On those spaces there exists a polarized inner product,
which would guarantee the non-vanishing of inner products of type $\left<{\bf
q},N^{w}\cdot{\bf a}_{0}\right>$. Indeed, the most precise way to formulate
the stability constraints is by using the Deligne splittings in a basis where
the $Sl_{2}$-algebras associated to the different $N_{i}$ are fully commuting.
Such a Deligne splitting is so-called $\mathbb{R}$-split and so any real
charge can be placed into the space of a Deligne sub-space plus its conjugate.
The process of reaching such a basis, and the properties of the splitting, is
explained in detail in [3].141414Similarly, the identification with the
monodromy weight filtration assumes $n=d$, which to our knowledge is correct
for any infinite distance locus, but can again be removed as an assumption by
going to an $\mathbb{R}$-split Deligne basis.

##  6 Summary and discussion

In this paper we studied the decay and spectrum of BPS states in ${\cal N}=2$
theories, in particular considering weak-coupling limits in moduli space
$g\rightarrow 0$. We found evidence for a certain filtration structure in BPS
states, which is determined by their mass in the $g\rightarrow 0$ limit.
Specifically, that a BPS state at any (perturbative) value of $g$ cannot decay
to a BPS state whose mass at $g\rightarrow 0$ is infinitely larger. This
appears somewhat surprising, for example, an electrically charged black hole
cannot emit a dyonic particle upon crossing a wall of marginal stability in
any perturbative regime, because in the $g\rightarrow 0$ limit the black hole
would be infinitely lighter than the dyon.

Although the analysis was sometimes involved, at least some of the key points
are extremely simple and so we can summarise them here using quite general
notation. Consider a decay of states $A\rightarrow B+C$. Let this decay occur
at vanishing kinetic energy, like in ${\cal N}=2$. Let the masses of the
states depend on some couplings, say $g$ for simplicity. Then at the decay
point in coupling space, say $g=g_{c}$, we have an equality in the masses
$M_{A}\left(g_{c}\right)=M_{B}\left(g_{c}\right)+M_{C}\left(g_{c}\right)$. Now
consider this equality of the masses as a constraint on the value of the
couplings. It picks out a real co-dimension one subspace of the coupling
space. In ${\cal N}=2$ this is a wall of marginal stability. Now let this
subspace be non-compact, so that it stretches out to the limit $g\rightarrow
0$. This is sometimes, not always, true in ${\cal N}=2$. Then we see that if
somewhere between $g=g_{c}$ and $g\rightarrow 0$ the ordering of the masses
changes, so say $M_{B}$ becomes larger than $M_{A}$, we must have that one of
the constituents of the decay becomes massless, in this case $M_{C}=0$. A
massless charged state induces a singularity in the low-energy effective
theory, so in ${\cal N}=2$ would manifest as a singularity in the moduli
space. If we insist on the absence of such a massless state, or singularity,
then we forbid a change in the ordering of the masses. This induces a certain
filtration on the decay of states which is independent of $g$, so in
particular can be deduced at $g\rightarrow 0$.

The absence of a singularity in perturbative regimes in ${\cal N}=2$ is
guaranteed by the nilpotent orbit theorem, which gives the universal behaviour
around any infinite distance (or weakly-coupled) locus in moduli
space.151515This gives also a rather precise statement of what we mean by the
perturbative regime: it is one where instantons are suppressed. Note that we
do not consider cases where $g\rightarrow 0$ at finite distances in moduli
space, such as the conifold locus, as perturbative regimes. The vanishing
coupling there is only due to integrating out a massless state, and would be
finite at any finite mass scale. The only remaining subtlety is that it could
be that state $C$ participated in the decay process, and then as we varied the
coupling, itself decayed away so that it was not present in the spectrum at
$M_{C}=0$. This seems a contrived situation to occur, but we need to rule it
out to be sure. We can prove that this is not possible in certain (one of the
three possible) weak coupling limits. More precisely, we showed that if it
does decay, then at least one of the constituents would become massless in the
perturbative regime still leading to a singularity in the moduli space. So the
singularity is robust against decays in this sense. We were not able to prove
this for all possible weak-coupling limits, simply because the algebraic
equations are of order $2d$, with $d=1,2,3$ being the different possible
limits (we solved $d=1$, so the quadratic case). But we believe it is a
technical task to show it generally. This belief led to the proposal of a BPS
Stability Filtration, as stated in the introduction.

We gave further evidence, and developed the understanding around the
filtration, by studying also attractor flows for multi-centred black holes.
These should capture the spectrum of BPS states, and we find agreement with
the filtration proposal. So a decay which would violate the filtration,
manifested as a split attractor flow, leads to an unphysical attractor locus
for one of the constituents. Also, the filtration can be understood as the
statement that the wall crossing formula [5] should apply to all states in the
theory, everywhere on the wall (so imposing the absence of non-stable states
anywhere on the wall).

The filtration on the BPS states can be mathematically formulated in terms of
the Monodromy Weight Filtration associated to the limiting mixed Hodge
structure of the $g\rightarrow 0$ limit. The statement is that a state cannot
decay to one which has a higher s-weight (where s-weight is defined in
(5.30)). This condition also automatically captures the requirement that the
wall of marginal stability stretches to the $g\rightarrow 0$ point: it is true
whenever the s-weights of the decaying state and one of its products differ.
Of course, since BPS states are mapped to the existence of certain geometric
sub-manifolds such as special-Lagrangians, the filtration implies a
corresponding geometric statement which is mathematically interesting.

It is natural to propose that our analysis can be extended to all $g$, so away
from the perturbative regime, by assuming the absence of moduli space
singularities on the walls of marginal stability. So in this sense, it is the
statement that the stability of BPS states depends only on the singularities
in the moduli space, whether they are at finite or infinite distance.

If, as we propose, a filtration exists on BPS states in ${\cal N}=2$ theories,
then what does this imply for the Swampland motivations discussed in the
introduction? Regarding the first topic, the black hole to particle transition
as $g\rightarrow 0$. Electrically charged black holes, if we can make sense of
their divergent attractor flows (see below), cannot emit any dyons, and
therefore all their constituents are completely stable throughout the
transition. The tower of light states as $g\rightarrow 0$ therefore start