# Scaling Black Holes and Modularity

Aradhita Chattopadhyaya, Jan Manschot, Swapnamay Mondal
1 School of Mathematics, Trinity College, Dublin 2, Ireland
2 Hamilton Mathematical Institute, Trinity College, Dublin 2, Ireland

###### Abstract:

Scaling black holes are solutions of supergravity with multiple black hole
singularities, which can be adiabatically connected to a single center black
hole solution. We develop techniques to determine partition functions for such
scaling black holes, if each constituent carries a non-vanishing magnetic
charge corresponding to a D4-brane in string theory, or equivalently M5-brane
in M-theory. For three constituents, we demonstrate that the partition
function is a mock modular form of depth two, and we determine the appropriate
non-holomorphic completion using generalized error functions. From the four-
dimensional perspective, the modular parameter is the axion-dilaton, and our
results show that $S$-duality leaves this subset of the spectrum invariant.
From the five-dimensional perspective, the modular parameter is the complex
structure of a torus $T^{2}$, and the scaling black holes are dual to states
in the dimensional reduction of the M5-brane worldvolume theory to $T^{2}$. As
a case study, we specialize the compactification manifold to a K3 fibration,
and explicitly evaluate holomorphic parts of partition functions.

##  1 Introduction

Solutions of supergravity with multiple black hole singularities provide
interesting insights on the spectrum of quantum gravity and the dependence on
the compactification moduli [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Such multi-center
solutions are particularly intriguing if their solution space contains a
region, where the distances between the centers becomes arbitrarily small. In
other words, these solutions, known as “scaling” solutions [6, 11] in four
dimensions, are adiabatically connected to a supergravity solution with a
single black hole singularity.

We will study such scaling black holes formed from M5-branes in M-theory, or
equivalently D4-branes in Type IIA string theory [12]. Such black holes are
well studied using the AdS3/CFT2 correspondence [12, 13, 14, 15], as well as
the hypermultiplet geometry of the IIB string theory [16, 17, 18]. The
D4/M5-branes wrap a four-cycle (or divisor) $P\in H_{4}(X,\mathbb{Z})$ of the
Calabi-Yau threefold $X$, which can also be studied within algebraic geometry
[19, 20, 21, 22, 23].

Multi-center solutions exist with each center having a positive D4-brane
charge but vanishing D6-brane charge. Their walls of marginal stability extend
to the large volume regime of the Kähler moduli space [24, 25, 26, 27], which
corresponds to $\mathrm{Im}(t)=J=\lambda\,\underline{J}$, where
$\underline{J}$ is the normalized, dimensionless Kähler modulus,
$\underline{J}^{3}=1$, and $\lambda\gg 1$. While attractor points lie
typically in the interior of the Kähler moduli space, there is an analogue of
the attractor point at large volume $t^{\lambda}_{\gamma}$ defined in Eq. (58)
[18, 25, 27]. Similarly, single center black holes, whose internal degrees of
freedom are independent of the asymptotic moduli, have an analogue at large
volume, where families of black hole solutions appear effectively as single
center black holes. Much like nucleons can be considered as elementary
particles at sufficiently low energies.

To deal with these effectively single center solutions, we introduce the
notion of “$\lambda$-core”, or “core” for short. The defining property of a
“$\lambda$-core” is that the distances between constituents in a
$\lambda$-core are bounded by $C\lambda^{-3}$ for sufficiently large
$\lambda$, and for some fixed length $C$. Examples of such bound black holes
states are of course proper single center black holes, as well as bound states
of a D6-brane and anti-D6-brane, for which the wall of marginal stability lies
in the interior of the Kähler moduli space.

The $\lambda$-core solutions are also distinguished in the uplift to five
dimensions, and decoupling limit. Recall that the four dimensional Newton’s
constant equals $G_{4}=\ell_{5}^{3}/R$, with $\ell_{5}$ the five dimensional
Planck length and $R$ the radius of the M-theory circle. Then, $\lambda$ is
related to these variables by $\lambda=R/\ell_{5}$ [27]. When uplifted to five
dimensions and in the decoupling limit $\ell_{5}\to 0$, states with D4-D2-D0
charges and within a $\lambda$-core will develop an asymptotically AdS3
throat, whereas bound states with larger separation will decouple from the
spectrum since their energy diverges [27].

In a series of works [18, 28, 29, 30], the modular properties of partition
functions enumerating D4-D2-D0 black holes are studied from a complementary
perspective, namely by mapping this D-brane system to D3-D1-D(-1) instantons.
These D-instantons correct the hypermultiplet geometry [17, 31, 32, 33], which
is constrained by IIB $SL(2,\mathbb{Z})$ S-duality group. This duality group
acts on the axion-dilation field $\chi:=C^{(0)}+ie^{-\phi}$ by linear
fractional transformations. Here $C^{(0)}$ is the RR scalar and $\phi$ is the
dilaton. Alternatively, the duality can be identified with the modular
symmetry of the worldvolume reduction of M5-branes. These have proven to be
fruitful connections to determine the modular and analytic properties of the
partition functions for D4-D2-D0 black holes. In this way, non-holomorphic
contributions to the partition function are determined, which imply that the
partition functions involve mock modular forms [34, 35], and mock modular
forms of higher depth [36, 37, 38, 39]. This implies potentially interesting
arithmetic of the BPS indices, while the non-holomorphic contribution is also
interesting independently, and a generalization of similar non-holomorphic
terms in partition functions of $\mathcal{N}=4$ Yang-Mills on four-manifolds
[40, 41]. The origins and explicit expressions of these terms have been
understood better recently [30, 42, 43, 44, 45, 46, 47, 48].

We will study in this paper the partition function of scaling black holes with
three constituents. Each constituent is a $\lambda$-core carrying a positive
D4-brane charge, but vanishing D6-brane charge. Intriguingly, each core gives
rise to an associated AdS3/CFT2 after uplifting to M-theory, while the near
coincident region gives rise to an asymptotic AdS3 solution for the total
magnetic charge, and should thus be captured by the AdS3/CFT2 correspondence
for the total charge. In other words, they are examples of AdS fragmentation,
and attributed to the Coulomb branch of the CFT [1]. The existence of scaling
black holes with these charges puts constraints on the topology of the Calabi-
Yau threefold. In particular, these only exist if the second Betti number of
the Calabi-Yau manifold $b_{2}\geq 2$.

Our explicit analysis of scaling solutions gives rise to a decomposition for
the partition function of attractor black holes $\mathcal{Z}^{\lambda}_{P}$
with fixed magnetic charge $P$ reads (See also Eq. (128).) in terms of
partition functions of single core and $n$-core scaling black holes
$\mathcal{Z}^{nT}_{P}$. This reads explicitly,

$\mathcal{Z}^{\lambda}_{P}=\mathcal{Z}^{T}_{P}+\mathcal{Z}^{3T}_{P}+\dots,$
(1)

where $\mathcal{Z}^{T}_{P}$ is the partition function of single core black
holes, $\mathcal{Z}^{3T}_{P}$ the partition function of scaling black holes
consistenting of three cores, and the dots stand for scaling black holes with
$n>3$ cores. We determine the holomorphic part of the partition function using
formula’s for the degeneracies of black hole bound states such as studied in
[6, 11, 49, 50, 51, 52, 53], which gives rise to a holomorphic indefinite
theta series $\Psi_{\boldsymbol{\mu}}$ of signature $(2,2b_{2}-2)$. Since its
coefficients grow polynomially, this demonstrates that the entropy arising
from these solutions is exponentially smaller than the entropy of a single
center black hole. This is expected since we have not included pure Higgs
degeneracies [11, 49] or single center [51]. We have worked out two explicit
case studies, where we specialize the CY three-fold to a K3 fibration and
determine explicit $q$-series for the partition function. We do observe that
the exponent of the leading term in the $q$-expansions is rather large.

We have moreover demonstrated that the partition function
$\mathcal{Z}^{3T}_{P}$ admits a theta function decomposition as a consequence
of a spectral flow symmetry familiar from MSW CFT. This in contrast to generic
bound states with non-vanishing D4-brane charge for which this symmetry is not
present [25]. On the other hand for modular transformations,
$\mathcal{Z}^{3T}_{P}$ needs to be complemented with additional non-
holomorphic terms, a phenomena which is also familiar for the attractor
partition function $\mathcal{Z}^{\lambda}_{P}$ as mentioned above. We
distinguish the completed functions from the original functions by a hat, thus
$\widehat{\mathcal{Z}}^{\lambda}_{P}$ for $\mathcal{Z}^{\lambda}_{P}$ and
$\widehat{\mathcal{Z}}^{3T}_{P}$ for $\mathcal{Z}^{3T}_{P}$ and similarly for
other functions. We then establish that $\widehat{\mathcal{Z}}^{\lambda}_{P}$
and $\widehat{\mathcal{Z}}^{3T}_{P}$ transform identically. Alexandrov and
Pioline have derived in Ref. [29] the non-holomorphic terms for the rhs,
$\widehat{\mathcal{Z}}^{\lambda}_{P}$. It would be interesting to combine this
with the non-holomorphic terms derived in the present paper for
$\widehat{\mathcal{Z}}^{3T}_{P}$ to deduce the non-holomorphic terms of
$\widehat{\mathcal{Z}}^{T}_{P}$.

The identical transformation properties of
$\widehat{\mathcal{Z}}^{\lambda}_{P}$ and $\widehat{\mathcal{Z}}^{nT}_{P}$
raises the question, which term(s) correspond to the partition function of the
conformal field theory. Since upon taking the decoupling limit to vanishing 5d
Planck length $\ell_{5}\to 0$ [27], the supergravity solutions contributing to
$\widehat{\mathcal{Z}}^{3T}_{P}$ decouple from the AdS3 geometry. This term is
therefore not expected to correspond to states within the MSW conformal field
theory, and it seems therefore plausible to us to expect that
$\widehat{\mathcal{Z}}^{T}_{P}$ is to be identified with the MSW CFT partition
function. It would be interesting to understand whether the finite difference
between $\widehat{\mathcal{Z}}^{T}_{P}$ and
$\widehat{\mathcal{Z}}^{\lambda}_{P}$ can arise by turning on the irrelevant
perturbation in the CFT, which corresponds to moving away from the near
horizon geometry and up the attractor flow [54].

Determination of the completion $\widehat{\mathcal{Z}}^{3T}_{P}$ amounts to
determining the completion of the indefinite theta series
$\Psi_{\boldsymbol{\mu}}$. We determine furthermore the non-holomorphic
completion of a closely related function $\Phi_{\boldsymbol{\mu}}$ (123),
which enumerates the number of “scaling” charge configurations. Although still
involved, determination of the non-holomorphic completion of this function is
simpler than for generating function of the BPS indices. The analysis of
$\Phi_{\boldsymbol{\mu}}$ and $\Psi_{\boldsymbol{\mu}}$ demonstrates that
these are mock modular forms of depth 2. There are different representations
of the completion:

  1. 1.

As a non-holomorphic kernel of the theta series. This involves (generalized)
error functions [36]. The modular properties of the completion follow from
application of results by Vignéras [55]. This is applied to the partition
functions for scaling black holes in Section 4.4. For the partition function
$\widehat{\Phi}_{\boldsymbol{\mu}}$ see Equation (152), and for
$\widehat{\Psi}_{\boldsymbol{\mu}}$, see Equation (181).

  2. 2.

Another useful representation, well-known for mock modular forms [34, 35], is
as a holomorphic $q$-series plus an (iterated) integral of modular forms [36,
43, 56]. This form is determined for three-center scaling black holes in
equation (155) and (170). The modular transformations of the $q$-series follow
directly from those of the iterated integral. This representation is also
relevant physically, where the non-holomorphic part is contributed to
continuum of states [57], or the Coulomb branch [44, 48].

  3. 3.

The third representation we mention here is as an integral over a domain in a
symmetric space, studied by Funke, Kudla and Millson [38, 39, 58], also known
as Narain moduli space in the context of conformal field theory. In the case
of a 3-center scaling solution, this would be a union of triangles in
$SO(2,2b_{2}-2;\mathbb{Z})\backslash
SO(2,2b_{2}-2;\mathbb{R})/SO(2;\mathbb{R})\times SO(2b_{2}-2;\mathbb{R})$.
While we have not explored this representation in detail, it is interesting to
mention in light of recent discussion on averaging over Narian moduli space
and the AdS3/CFT2 correspondence [59, 60].

Though technically involved, we believe that our results can be extended to
scaling solutions with $n>3$ cores, and will give rise to mock modular forms
of depth $n-1$. Any Calabi-Yau manifold with $b_{2}\geq 2$ gives rise to such
mock modular forms, which thus provides a large resource of holomorphic higher
depth mock modular forms. Further, it will be interesting to include single
center degeneracies. Moreover, we hope that our results could be used for the
study of AdS3 fragmention, and the interpretation of these solutions in the
dual CFT. In this way, it may be possible to derive the non-holomorphic terms
within gravity or the worldvolume theory of intersecting D-branes.

The outline of this paper is as follows. Section 2 reviews aspects of multi-
center black holes, and in particular the index of scaling solutions. Section
3 reviews partition functions of D4-brane black holes. Section 4 discusses
charge lattices for D4-brane bound states, and defines the partition functions
of scaling black holes. We define here also the partition functions
$\Phi_{\boldsymbol{\mu}}$ and $\Psi_{\boldsymbol{\mu}}$, which enumerate
scaling configurations and their BPS indices, and determine its modular
completion. Section 5 discusses the relation to M-theory and the decoupling
limit AdS3. Section 6 considers two case studies for a specific Calabi-Yau
3-fold and charges, and presents the holomorphic $q$-series which are mock
modular forms of depth 2.

##  2 Black hole solutions in $\mathcal{N}=2$ Supergravity

We briefly review in this section supersymmetric black holes in
$\mathcal{N}=2$ supergravity and partition functions.

###  2.1 Black hole bound states

Let $X$ be a simply connected Calabi-Yau threefold, with triple intersection
product $d_{abc}$, $a,b,c=1,\dots,b_{2}$. The intersection product $d_{abc}$
is symmetric in its indices. The classical central charge of a BPS state is
given by

$Z(\gamma,t)=-\int_{X}e^{-t}\wedge\gamma,$ (2)

where $\gamma$ on the rhs is the Poincaré dual differential form of the
homology class of the cycle which supports the D-branes, and is in 1-to-1
correspondence with the electric-magnetic charge of the BPS state. Moreover,
$t$ is the Kähler modulus of the Calabi-Yau three-fold.

The scalar fields $X^{I}$, $I=0,\dots,b_{2}$ of the vector multiplets are
related to the Calabi-Yau moduli, $t^{a}=B^{a}+iJ^{a}$, as $t^{a}=X^{a}/X^{0}$
for $a=1,\dots,b_{2}$. Near the horizon, their values are determined by the
attractor equations at the horizon in terms of the electric-magnetic charges
of the black hole [61, 62, 63]. On the other hand, their asymptotic values for
$|\vec{r}|\to\infty$ are boundary conditions for the equations of motion.

Besides the single center black hole, the equations of motion of
$\mathcal{N}=2$ supergravity give rise to intricate multi-center black hole
solutions [3, 6, 64, 65]. Upon varying the asymptotic values of the scalar
fields, multi-center solutions can cease to exist as proper solutions to the
supergravity equations of motions, or reversely new solutions can appear. If
the asymptotic values are chosen equal to the attractor values only few multi-
center solutions exist. That is to say, only multi-center solutions which can
be continuously connected to a single center black hole exist. These are the
scaling solutions mentioned above, and are the main focus of this paper.

To understand this more explicitly, recall that an $n$-center solution, is
required to satisfy the following $n-1$ Denef equations,

$\sum_{i\neq
j}\frac{\left<\gamma_{i},\gamma_{j}\right>}{r_{ij}}=2\,\left.\mathrm{Im}\left(e^{-{\rm
i}\alpha}Z(\gamma_{i},t)\right)\right|_{r=\infty},$ (3)

where $\left<,\right>$ is the symplectic innerproduct between the charges,

$\left<\gamma_{1},\gamma_{2}\right>=-P_{1}^{0}Q_{0,2}+P_{1}\cdot
Q_{2}-P_{2}\cdot Q_{1}+P_{2}^{0}Q_{0,1}.$ (4)

Moreover, $r_{ij}=|\vec{r}_{i}-\vec{r}_{j}|$ is the distance between the
centers $i$ and $j$, and $\alpha$ is the phase of the central charge
$Z(\gamma,t)$ for the total charge $\gamma=\sum_{i=1}^{n}\gamma_{i}$. We set

$c_{j}=2\,\left.\mathrm{Im}\left(e^{-{\rm
i}\alpha}Z(\gamma_{j},t)\right)\right|_{r=\infty}.$ (5)

We fix $\vec{r}_{1}$ at the origin $\mathbb{R}^{3}$ and let $\mathcal{M}_{n}$
be the solution space for $\vec{r}_{j}\in\mathbb{R}^{3}$, $j=2,\dots,n$ to
(3). Then $\mathcal{M}_{n}$ has dimension $2n-2$. The low energy degrees of
freedom of the supersymmetric multi-center black hole give rise to
$\mathcal{N}=4$ quiver quantum mechanics [64]. The quiver for an $n$-center
bound state with charges $\\{\gamma_{j}\\}$ consists of $n$ nodes, and
$\gamma_{ij}>0$ arrows from node $i$ to node $j$.

Note that the equations (3) are necessary but not sufficient for the multi-
center solution to exist, to this end one needs to verify that the full
supergravity solution is regular away from the black hole singularities, and
without time-like curves [3]. Since we restrict to the large volume limit of
the Calabi-Yau moduli space, we assume that this is the case in the following,
and that we can determine the existence of bound states from (3).

The gravity perspective has led to the following form for the index of an
$n$-center bound state [6, 9]. To express this, we first introduce the
rational index $\bar{\Omega}(\gamma)$ associated to the integer index
$\Omega(\gamma)$,

$\bar{\Omega}(\gamma)=\sum_{m|\gamma}\frac{\Omega(\gamma/m)}{m^{2}}.$ (6)

The single center invariants $\Omega_{S}(\gamma)$ are the internal
degeneracies of a supersymmetric particle or black hole with charge $\gamma$.
It is expected to be a positive integer for a black hole. To analyze the
spectrum of bound states it is convenient to introduce a fugacity $y$ for
angular momentum. The rational variant of the refined index is defined as

$\bar{\Omega}(\gamma,y)=\sum_{m|\gamma}\frac{1}{m}\frac{y-y^{-1}}{y^{m}-y^{-m}}\,\Omega(\gamma/m,y^{m}).$
(7)

This reproduces (6) in the limit $y\to 1$. A few variants of BPS indices will
be important for us. We mention,

  * •

The BPS invariant $\Omega(\gamma;t)$, which enumerates BPS states for a given
value $t$ of the asymptotic moduli. This include single-center BPS states as
well as bound states.

  * •

The single-center invariant $\Omega_{S}(\gamma)$, which is the internal
degeneracy of a BPS particle or black hole center. This invariant is
independent of the moduli $t$.

  * •

The total invariant $\Omega_{T}(\gamma)$, which is a composite of
$\Omega_{S}(\gamma)$ and independent of the moduli $t$. We give the expression
below in (10).

The rational refined BPS index $\bar{\Omega}(\gamma,y;t)$ can be expressed as
a sum over partitions of $\gamma$, each representing a BPS bound state. It
takes the form [9]

$\bar{\Omega}(\gamma,y;t)=\sum_{\gamma=\sum_{i=1}^{n}\gamma_{i}}\frac{g_{C}(\\{\gamma_{j}\\},\\{c_{j}\\},y)}{|{\rm
Aut}(\\{\gamma_{j}\\})|}\,\prod_{j=1}^{n}\bar{\Omega}_{T}(\gamma_{j},y),$ (8)

with,

  * •

$|{\rm Aut}(\\{\gamma_{j}\\})|$ is the order of the subgroup of the
permutation group, which preserves the ordered set
$\\{\gamma_{1},\dots,\gamma_{n}\\}$.

  * •

The index $g_{C}$ can be determined using localization of the black hole
solution with respect to rotation around a fixed axis generated by $J_{3}$,
say the $z$-axis [66]. A fixed point $p\in\mathcal{M}_{n}$ corresponds to a
collinear solution with all centers placed on the $z$-axis. If the associated
bound state quiver has no oriented loop, $g_{C}$ is the refined index of the
$\mathcal{N}=4$ quiver quantum mechanics describing the bound state [9],

$g_{C}(\\{\gamma_{j}\\},\\{c_{j}\\},y)=\mathrm{Tr}^{\prime}_{\mathcal{H}_{\rm
qm}}(-y)^{2J_{3}},$ (9)

where the trace is over the BPS Hilbert space $\mathcal{H}_{\rm qm}$ of the
quiver quantum mechanics, and $J_{3}$ is one of the generators of $SU(2)$.

  * •

$\Omega_{T}$ the total invariant defined by

$\Omega_{T}(\gamma,y)=\Omega_{S}(\gamma,y)+\sum_{\sum_{j=1}^{n}m_{j}\gamma_{j}=\gamma}H(\\{\gamma_{i},m_{i}\\},y)\,\prod_{i=1}^{n}\Omega_{S}(\gamma_{i},y^{m_{i}}),$
(10)

where $m_{j}\in\mathbb{N}$ are multiplicities of the charges in the partition
of $\gamma$. For bound states whose associated quiver has no closed loops, the
$H(\\{\gamma_{i},m_{i}\\},y)$ vanish. Otherwise they are determined by the
“minimal modification hypothesis”. This has the effect that if we express (8)
as

$\bar{\Omega}(\gamma,y;t)=\sum_{\gamma=\sum_{i=1}^{n}\gamma_{i}}\frac{\bar{g}_{C}(\\{\gamma_{j}\\},\\{c_{j}\\},y)}{|{\rm
Aut}(\\{\gamma_{j}\\})|}\,\prod_{j=1}^{n}\bar{\Omega}_{S}(\gamma_{j},y),$ (11)

then the $\bar{g}_{C}$ are $SU(2)$ characters.

To determine $g_{C}$ using localization, it is convenient to introduce the
refined index

$g_{C}(\\{\gamma_{j}\\},\\{c_{j}\\},y)=\mathrm{Tr}^{\prime}_{\mathcal{H}_{\rm
qm}}(-y)^{2J_{3}}.$

Let $z_{j}$ be the position of the center with charge $\gamma_{j}$. The
localization technique then gives the following sum over collinear fixed
points with respect to rotation around this axis,

$g_{C}(\\{\gamma_{j}\\},\\{c_{j}\\};y)=(-1)^{n-1}(y-y^{-1})^{-n+1}\sum_{p\in\mathcal{M}_{n}}s(p)\,(-y)^{\sum_{i<j}\gamma_{ij}\mathop{\mathrm{sgn}}(z_{j}-z_{i})},$
(12)

where $s(p)\in\pm 1$ is a sign depending on the details of the fixed point,
and $z_{j}$ is the $z$-coordinate of center $j$. If the associated quiver does
not contain loops, this is the complete index and the $y\to 1$ limit is well-
defined. However, if the quiver contains loops, the distances between the
black hole centers may be arbitrarily small [6]. Such solutions are known as
scaling solutions, and additional fixed points need to be included in (12). An
($n-$center) scaling black hole is a multi-center solution of $n$ black holes,
whose phase space $\mathcal{M}_{n}$ contains a region where the centers can
approach each other arbitrarily close. Thus, while the centers are spatially
separated for generic points of $\mathcal{M}_{n}$, they are adiabatically
connected to the black hole solution with a single center.

While many BPS bound states decay if we tune the moduli to their attractor
values, scaling solutions remain part of the BPS spectrum. Since the index is
evaluated at the attractor point $c_{j}^{*}$, each term on the rhs of (8) with
$n\geq 3$ corresponds to a scaling solution.

Various quantities may diverge in the limit $y\to 1$, such as $\Omega_{T}$ and
$g_{C}$. In order to arrive at numerical counterparts for these quantities, we
propose to regularize a rational function of the form

$\frac{f(y)}{(y-y^{-1})^{\ell}},\qquad\text{with}\qquad\lim_{y\to 1}f(y)\neq
0,$ (13)

as follows

$\frac{f(y)}{(y-y^{-1})^{\ell}}\longrightarrow\frac{1}{2^{\ell}\,\ell!}\,\left.\left(y\frac{d}{dy}\right)^{\ell}f(y)\right|_{y=1}.$
(14)

###  2.2 Bound state indices

Let us consider the equations (3) for small values of $n$. For $n=2$, there is
a single equation,

$\frac{\gamma_{12}}{r_{12}}=c_{1}.$ (15)

We deduce that the two-center solution only exists as a physical solution if
$\gamma_{12}\,c_{1}>0$. This depends on the moduli $t$. If $t$ approaches a
value where $c_{1}$ vanishes, $r_{12}$ diverges and the solution disappears as
a solution to the supergravity equations of motion. In particular for the
attractor point, the two-center solution never exists.

The quantum states of the two-center solution correspond to the product of the
internal degeneracies of the centers times the states of a spin
$(|\gamma_{12}|-1)/2$ multiplet, which arises due to the electric-magnetic
fields sourced by the charges of the two-centers [64]. We express it here as
the product

$\begin{split}\Omega_{2}(\gamma_{1}+\gamma_{2};t)&=g_{C}(\\{\gamma_{1},\gamma_{2}\\},\\{c_{1},c_{2}\\})\,\Omega(\gamma_{1})\,\Omega(\gamma_{2}),\end{split}$
(16)

with

$\begin{split}&g_{C}(\\{\gamma_{1},\gamma_{2}\\},\\{c_{1},c_{2}\\})=\tfrac{1}{2}\left(\mathop{\mathrm{sgn}}(\gamma_{12})+\mathop{\mathrm{sgn}}(c_{1})\right)\,(-1)^{\gamma_{12}-1}\,\gamma_{12},\end{split}$
(17)

and $\Omega(\gamma_{j})$ are degeneracies of the individual centers. For
$\mathop{\mathrm{sgn}}$, we use the definition

$\mathop{\mathrm{sgn}}(x)=\left\\{\begin{array}[]{rl}1,&x>0,\\\ 0,&x=0,\\\
-1,&x<0.\end{array}\ \right.$ (18)

The function
$\frac{1}{2}\left(\mathop{\mathrm{sgn}}(\gamma_{12})+\mathop{\mathrm{sgn}}(c_{1})\right)$
equals 1 if the solution to (15) is physical, i.e. $r_{12}>0$, and it vanishes
if the sign of $r_{12}$ is unphysical, $r_{12}<0$. The factor
$(-1)^{\gamma_{12}-1}\gamma_{12}$ is the number of states of the bound state,
assuming that it exists. The case that $c_{1}=0$ is a special case, we aim to
avoid. At the attractor point for the total charge, the $c_{j}$ (5) are equal
to $c^{*}_{j}$, given by [3, 4]

$c^{*}_{j}=|Z(\gamma,t^{*}_{\gamma})|\,\left<\gamma,\gamma_{j}\right>.$ (19)

As a result, two-center solutions do not exist at the attractor point, because
substituting $c^{*}_{1}$ in (15) gives a negative value for $r_{12}$ which is
unphysical.

For $n=3$ distinct charges, (3) gives two independent equations,

$\begin{split}\frac{\gamma_{12}}{r_{12}}+\frac{\gamma_{13}}{r_{13}}=c_{1},\\\
\frac{\gamma_{21}}{r_{12}}+\frac{\gamma_{23}}{r_{23}}=c_{2}.\\\ \end{split}$
(20)

An intriguing aspect of these equations is that for appropriate values of
$\gamma_{ij}$, they can be satisfied with positive $r_{ij}$ for all pairs
$i\neq j$, for $t$ at the attractor point. Then, there is a one-parameter
family of solutions [6], with

$r_{ij}=\varepsilon\,\gamma_{ij}+O(\varepsilon^{2}),\qquad\varepsilon\to 0,$
(21)

for $ij$ equal to $12$, $23$ and $31$.

We can in fact do better, and give an all order solution in $\varepsilon$. The
parameter $\varepsilon$ together with three angular variables form the
4-dimensional solution space to (20). We set $\gamma_{12}=a$, $\gamma_{23}=b$
and $\gamma_{31}=c$ in the following, and assume $a,b,c>0$.111We apologize for
the multiple use of $a$, $b$ and $c$. , we can then verify that the following
1-parameter family of distances $r_{ij}$ satisfy Denef’s equations (3),

$\displaystyle\frac{1}{r_{12}}$
$\displaystyle=\frac{1}{a\varepsilon}-\rho_{a}\,,\quad\,\frac{1}{r_{23}}=\frac{1}{b\varepsilon}-\rho_{b}\,,\quad\,\frac{1}{r_{31}}=\frac{1}{c\varepsilon}-\rho_{c}\,,$
(22)

where $\rho_{a},\rho_{b},\rho_{c}$ satisfy
$\left(\rho_{c}c-\rho_{a}a\right)=c_{1},\,\left(\rho_{a}a-\rho_{b}b\right)=c_{2},\,\left(\rho_{b}b-\rho_{c}c\right)=c_{3}$.
The range of $\varepsilon$ is determined by triangle inequalities and
positivity of $r_{12},r_{23},r_{31}$. The equations for $\rho_{a,b,c}$ allow
for a shift, which modifies the range of $\varepsilon$ accordingly.

We discuss this now in detail for the attractor poin, where we can use (19)
such that $|Z(\gamma,t^{*}_{\gamma})|=M$ and $\rho_{a}=\rho_{b}=\rho_{c}=M$.
We then have,

$\frac{1}{r_{12}}=\frac{1}{a\varepsilon}-M\,,\quad\frac{1}{r_{23}}=\frac{1}{b\varepsilon}-M\,,\quad\frac{1}{r_{31}}=\frac{1}{c\varepsilon}-M\,.$
(23)

The free parameter $\varepsilon$ is bounded from below by 0. In
$\varepsilon\ll 1/M$ regime, this solution reduces to Eq. (21); thus existence
of the scaling solution requires $a,b,c$ to obey triangle inequalities [6]. As
$\varepsilon$ is increased, the shape of the triangle changes, and we need to
determine the correct maximum of $\varepsilon$ domain. First, positivity of
$r_{12},r_{23},r_{31}$ imposes the upper bound
$\varepsilon\leq\frac{1}{M\max{}(a,b,c)}$. However, we also need to impose
that $r_{ij}$ satisfy the triangle inequalities, which imposes an even
stronger upper bound. E.g. the condition $r_{12}+r_{23}\geq r_{31}$ requires

$(a+b-c)-2ab\varepsilon M+abc\varepsilon^{2}M^{2}\geq 0.$ (24)

If $(c-a)(c-b)<0$, this condition is always satisfied since $a+b-c>0$ and the
lhs does not have real roots in this case. Moreover if $(c-a)(c-b)\geq 0$, the
condition is saturated for
$\varepsilon_{c}^{\pm}=\frac{1}{Mc}\left[1\pm\sqrt{\frac{(c-a)(c-b)}{ab}}\right]$
and violated for $\varepsilon_{c}^{-}<\varepsilon<\varepsilon_{c}^{+}$. Both
roots are non-negative provided $(a,b,c)$ obeys the triangle inequality
$a+b\geq c$. Noting that $\varepsilon_{c}^{+}>\frac{1}{Mc}$, we see
$\varepsilon\geq\varepsilon_{c}^{+}$ makes $r_{31}$ negative. Thus we must
have $\varepsilon\leq\varepsilon_{c}^{-}$. Using two other triangle
inequalities, we have
$\varepsilon\in(0,\min{}(\varepsilon_{a}^{-},\varepsilon_{b}^{-},\varepsilon_{c}^{-})_{\mathbb{R}}]$
where
$\varepsilon_{a}^{-}=\frac{1}{Ma}\left[1-\sqrt{\frac{(a-b)(a-c)}{bc}}\right],\,\varepsilon_{b}^{-}=\frac{1}{Mb}\left[1-\sqrt{\frac{(b-a)(b-c)}{ac}}\right]$,
and
$\min{}(\varepsilon_{a}^{-},\varepsilon_{b}^{-},\varepsilon_{c}^{-})_{\mathbb{R}}$
means the minimum among the $\varepsilon^{-}_{a,b,c}\in\mathbb{R}$. One of
$\varepsilon^{-}_{a,b,c}$ may be complex. At the maximal value of
$\varepsilon$ the configuration is collinear. With the three additional
angular variables, one can aligned the solution along the $z$-axis, thus
giving a fixed point for rotations around this axis.

For three centers, the sum over fixed points reads

$\displaystyle
g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\};y\right)=\frac{(-1)^{a+b+c}}{(y-y^{-1})^{2}}\Bigg{[}F(123)y^{a+b-c}+F(321)y^{-a-b+c}+F(213)y^{-a+b-c}$
$\displaystyle+F(312)y^{a-b+c}+F(132)y^{a-b-c}+F(231)y^{-a+b+c}\Bigg{]},$ (25)

where

$F(ijk):=F(ijk;\\{c_{j}\\})=\left\\{\begin{array}[]{rl}s(p),&\qquad\exists\,{\rm
a\,\,fixed\,\,point\,\,}p\in\mathcal{M}_{n}\,\,{\rm
with\,\,}z_{i}<z_{j}<z_{k},\\\ 0,&\qquad\nexists\,{\rm
a\,\,fixed\,\,point\,\,}p\in\mathcal{M}_{n}{\rm\,\,with\,\,}z_{i}<z_{j}<z_{k}.\end{array}\right.$
(26)

Since $a,b,c\in\mathbb{Z}$, the signs $(-1)^{a\pm b\pm c}$ equal
$(-1)^{a+b+c}$, and we can factor this out from the sum over permutations. The
dependence on the rhs on $\\{c_{j}\\}$ is through $\mathcal{M}_{n}$.

Since $F(123;\\{c_{j}\\})=F(321;\\{c_{j}\\})$, we can shorten $g_{C}$ to

$\begin{split}&g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\};y\right)=\frac{(-1)^{a+b+c}}{(y-y^{-1})^{2}}\Bigg{[}F(123;\\{c_{j}\\})\left(y^{a+b-c}+y^{-a-b+c}\right)\\\
&+F(213;\\{c_{j}\\})\left(y^{-a+b-c}+y^{a-b+c}\right)+F(132;\\{c_{j}\\})\left(y^{a-b-c}+y^{-a+b+c}\right)\Bigg{]}.\end{split}$
(27)

This expression does not have a smooth $y\rightarrow 1$ limit if
$\mathcal{M}_{n}$ is non-compact and contains a scaling region. In that case,
only one of $F(ijk;\\{c_{j}\\})$’s is non-vanishing. Turning on a fugacity is
indeed known to be subtle for non-compact phase spaces [67].

In the present context, the minimal modification hypothesis is put forward in
[66] to correct this. It adds a term with minimal angular momentum
corresponding to the fixed point with coincident centers. The effect of the
minimal modification hypothesis for three distinct charges is that the refined
index $g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\}\right)$ is completed to

$\bar{g}_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\},y\right)=g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\},y\right)+H(\\{\gamma_{j}\\},y),$
(28)

with

$H(\\{\gamma_{j}\\},y)=\left\\{\begin{array}[]{ll}-\frac{2}{(y-y^{-1})^{2}},&\quad{\rm
if}~{}a+b+c\in 2\mathbb{Z},\\\ \frac{y+y^{-1}}{(y-y^{-1})^{2}},&\quad{\rm
if}~{}a+b+c\in 2\mathbb{Z}+1,\end{array}\right.$ (29)

$\bar{g}_{C}$ has a well-defined $y\to 1$ limit, which reads,

$\begin{split}\bar{g}_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\}\right)&=\lim_{y\to
1}\bar{g}_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\};y\right)\\\
&=(-1)^{a+b-c}\left[F(123)\,s(a,b,c)+F(213)\,s(a,c,b)+F(132)\,s(c,b,a)\right],\end{split}$
(30)

with

$s(a,b,c)=\left\\{\begin{array}[]{rl}\frac{(a+b-c)^{2}}{4},&\qquad\text{if}~{}a+b+c\in
2\mathbb{Z},\\\ \frac{(a+b-c)^{2}-1}{4},&\qquad\text{if}~{}a+b+c\in
2\mathbb{Z}+1.\end{array}\right.\,$ (31)

We note that for degenerate cases, where one or more among $a,b$ and $c$
vanish, $g_{C}$ can be a multiple of $1/2$ or $1/4$ rather than in
$\\{-1,0,1\\}$.

Using the regularization (14), we obtaine for the numerical version (29) of
$H$,

$H(\\{\gamma_{j}\\})=\left\\{\begin{array}[]{rl}0,&\quad{\rm if}~{}a+b+c\in
2\mathbb{Z},\\\ \frac{1}{4},&\quad{\rm if}~{}a+b+c\in
2\mathbb{Z}+1,\end{array}\right.$ (32)

which in turn gives the numerical counterpart for $\Omega_{T}$,

$\Omega_{T}(\gamma)=\Omega_{S}(\gamma)+\left\\{\begin{array}[]{rl}0,&\quad{\rm
if}~{}a+b+c\in 2\mathbb{Z},\\\
\frac{1}{4}\prod_{j=1}^{3}\Omega_{S}(\gamma_{j}),&\quad{\rm if}~{}a+b+c\in
2\mathbb{Z}+1.\end{array}\right.$ (33)

We obtain similarly using (14) for the numerical version of $g_{C}$,

$\begin{split}&g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}\\}\right)=\frac{(-1)^{a+b+c}}{4}\times\left[F(123;\\{c_{j}\\})\,(a+b-c)^{2}\right.\\\
&\quad\left.+F(213,\\{c_{j}\\})\,(a-b+c)^{2}+F(132,\\{c_{j}\\})\,(a-b-c)^{2}\right].\end{split}$
(34)

Thus the numerical $g_{C}$ essentially corresponds to the $y\to 1$ limit of
the equivariant volume of the solution space $\mathcal{M}_{3}$ [66].

The term $F(123,\\{c_{j}\\})$ is determined in [51, Equation (2.57)]. For our
purposes we rewrite this in terms of $\mathop{\mathrm{sgn}}$ rather than the
step function. This reads,

$F(123,\\{c_{j}\\})=F_{1}(a,b,c,\\{c_{j}\\})+F_{2}(a,b,c),$ (35)

with

$\begin{split}F_{1}(a,b,c,\\{c_{j}\\})&=\frac{1}{4}(\mathop{\mathrm{sgn}}(a)+\mathop{\mathrm{sgn}}(c_{1}))\,(\mathop{\mathrm{sgn}}(b)-\mathop{\mathrm{sgn}}(c_{3})),\\\
F_{2}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(a)+\mathop{\mathrm{sgn}}(b)\right)\left(\mathop{\mathrm{sgn}}(a+b-c)-\mathop{\mathrm{sgn}}(a+b)\right).\end{split}$
(36)

At special charge configurations, where one or more arguments of the signs
vanish, this may differ from [51, Equation (2.57)]. In such cases, the $F_{j}$
can be a fraction rather than an integer. To deal properly with such cases, we
include additional terms below in Eqs (42) and (44).

Our interest in this paper is the index at the attractor point, thus

$g_{C}\left(\\{\gamma_{j}\\};\\{c_{j}^{*}\\}\right),$ (37)

where $c_{j}^{*}$ (19) is $c_{j}$ evaluated at the attractor point
$t^{*}_{\gamma}$. Then $F_{1}$ reads

$\begin{split}F_{1}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(a)+\mathop{\mathrm{sgn}}(c-a)\right)\left(\mathop{\mathrm{sgn}}(b)+\mathop{\mathrm{sgn}}(c-b)\right).\end{split}$
(38)

Assuming non-vanishing arguments of the sign functions, we can simplify the
sum $F^{*}(123)=F_{1}+F_{2}$ using the identity [68, Eq. (A.1)]222We thank
Sergey Alexandrov for stressing the simplifying power of this identity.

$(\mathop{\mathrm{sgn}}(x_{1})+\mathop{\mathrm{sgn}}(x_{2}))\,\mathop{\mathrm{sgn}}(x_{1}+x_{2})-\mathop{\mathrm{sgn}}(x_{1})\,\mathop{\mathrm{sgn}}(x_{2})=1,\qquad(x_{1},x_{2})\neq(0,0),$
(39)

to $F^{*}(123)=F_{1}(a,b,c)+F_{2}(a,b,c)$ [68, Eq. (4.10)],

$F^{*}(123)=\frac{1}{4}(1+\mathop{\mathrm{sgn}}(a-c)\,\mathop{\mathrm{sgn}}(b-c)+\mathop{\mathrm{sgn}}(b-c)\,\mathop{\mathrm{sgn}}(c-a-b)+\mathop{\mathrm{sgn}}(c-a-b)\,\mathop{\mathrm{sgn}}(a-c)\,).$
(40)

We stress that this expression may differ from (35) if the arguments of some
products of $\mathop{\mathrm{sgn}}$’s vanish. For example for $a=0,b=c=1$,
$F(123)=0$, while $F^{*}(123)$ equals $\frac{1}{4}$.

For the other permutations, we also define

$\begin{split}F_{3}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(a)+\mathop{\mathrm{sgn}}(b-a)\right)\left(\mathop{\mathrm{sgn}}(c)+\mathop{\mathrm{sgn}}(b-c)\right),\\\
F_{4}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(a)+\mathop{\mathrm{sgn}}(c)\right)\left(\mathop{\mathrm{sgn}}(a+c-b)-\mathop{\mathrm{sgn}}(a+c)\right),\\\
F_{5}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(c)+\mathop{\mathrm{sgn}}(a-c)\right)\left(\mathop{\mathrm{sgn}}(b)+\mathop{\mathrm{sgn}}(a-b)\right),\\\
F_{6}(a,b,c)&=\frac{1}{4}\left(\mathop{\mathrm{sgn}}(c)+\mathop{\mathrm{sgn}}(b)\right)\left(\mathop{\mathrm{sgn}}(c+b-a)-\mathop{\mathrm{sgn}}(c+b)\right),\end{split}$
(41)

and

${}F^{*}(213)=\frac{1}{4}(1+\mathop{\mathrm{sgn}}(a-b)\,\mathop{\mathrm{sgn}}(c-b)+\mathop{\mathrm{sgn}}(c-b)\,\mathop{\mathrm{sgn}}(b-a-c)+\mathop{\mathrm{sgn}}(b-a-c)\,\mathop{\mathrm{sgn}}(a-b)\,),$
${}F^{*}(132)=\frac{1}{4}(1+\mathop{\mathrm{sgn}}(c-a)\,\mathop{\mathrm{sgn}}(b-a)+\mathop{\mathrm{sgn}}(b-a)\,\mathop{\mathrm{sgn}}(a-b-c)+\mathop{\mathrm{sgn}}(a-b-c)\,\mathop{\mathrm{sgn}}(c-a)\,).$

Having defined the $F^{*}(ijk)$, we turn to
$g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})$ and take care of the special cases
where both arguments of a products of $\mathop{\mathrm{sgn}}$’s vanish. Let us
first determine for which values of $a$, $b$ and $c$,
$g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})$ is affected. When the last two
products of $\mathop{\mathrm{sgn}}$’s of $F^{*}(123)$ (40) vanish, the angular
momentum factor $(a+b-c)^{2}/4$ also vanishes. Thus replacing these products
of $\mathop{\mathrm{sgn}}$’s with a non-vanishing value will not affect
$g_{C}$. For the remaining product,
$\mathop{\mathrm{sgn}}(a-c)\,\mathop{\mathrm{sgn}}(b-c)$’s, the arguments
vanish in the equilateral case $a=b=c$ for which the angular momentum factor
is $a^{2}/4$. This is the same as for the other permutations, $F^{*}(213)$ and
$F^{*}(132)$, such that we can take all three into account by adding a single
additional term with yet undetermined coefficient $A$. We obtain thus for
$g_{C}$ at the attractor point,

$\begin{split}g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})&=\frac{(-1)^{a+b+c}}{4}\left[F^{*}(123)\,(a+b-c)^{2}\,+F^{*}(213)\,(a-b+c)^{2}\,\right.\\\
&\quad\left.+F^{*}(132)\,(-a+b+c)^{2}\,+\frac{1}{4}A\,\delta_{a,c}\,\delta_{b,c}\,a^{2}\right].\end{split}$
(42)

At this point, we can make a “guess” of the preferred “physical” value for
$A$. From the gravity perspective, the equilateral configurations with $a=b=c$
are perfectly fine multi-center solutions, such that it is most natural that
these contribute with $1\times(-1)^{a}\,a^{2}/4$ to $g_{C}$. For (42), we have
instead $(3+A)\times(-1)^{a}\,a^{2}/16$, thus suggesting $A=1$. We will
demonstrate in Section 4.5 that modularity of the $q$-series leads to exactly
the same value.

Besides the bound state index $g_{C}$, we are also interested to enumerate the
number of charge configurations giving rise to scaling black holes. Up to
vanishing arguments, the sum $F_{\rm total}(a,b,c)=\sum_{j=1}^{6}F_{j}(a,b,c)$
can be further simplified to

$\begin{split}F_{\rm
total}(a,b,c)&=\frac{1}{4}\Big{[}1+\mathop{\mathrm{sgn}}{}(a+b-c)\mathop{\mathrm{sgn}}{}(a+c-b)\\\
&+\mathop{\mathrm{sgn}}{}(a+c-b)\mathop{\mathrm{sgn}}{}(b+c-a)+\mathop{\mathrm{sgn}}{}(b+c-a)\mathop{\mathrm{sgn}}{}(a+b-c)\Big{]}\,.\end{split}$
(43)

This expression has the advantage that for $a+b+c$ odd, the arguments of the
$\mathop{\mathrm{sgn}}$’s never vanish. With inclusion of additional terms to
deal with vanishing arguments, we define number $f_{C}$,

$\begin{split}f_{C}(\\{\gamma_{j}\\},\\{c_{j}^{*}\\})&=F_{\rm
total}(a,b,c)\,(-1)^{a+b+c}\\\
&+\tfrac{1}{4}A_{1}\,\delta_{a,0}\,\delta_{b,c}+\tfrac{1}{4}A_{2}\,\delta_{c,0}\,\delta_{a,b}+\tfrac{1}{4}A_{3}\,\delta_{b,0}\,\delta_{a,c},\end{split}$
(44)

where the constants $A_{j}$ are yet to be determined. We will determine these
from the modular completion. To our surprise, these numbers are typically
irrational for $a+b+c\in 2\mathbb{Z}$, while they do not contribute for
$a+b+c\in 2\mathbb{Z}+1$. We find the irrationality quite peculiar. On the
other hand, $f_{C}$ is not a BPS index, such that it is not really violating
any physical principles.

##  3 Review of D4-brane black holes

We review in this section aspects of partition functions of D4-brane black
holes. We start in subsection 5 by reviewing the uplift of D4-brane black
holes to M-theory. In Subsection 3.1, we discuss the “supergravity” partition
function, which enumerates D4-brane BPS indices $\Omega(\gamma;t)$ for fixed
Kähler modulus $t$. In Subsection 3.2, we discuss the “attractor” partition
function, which is a generating function of BPS indices
$\Omega(\gamma;t_{\gamma}^{*})$ evaluated at the attractor point
$t_{\gamma}^{*}$ of the corresponding charge $\gamma$.

###  3.1 Supergravity partition function

From the perspective of asymptotically flat $\mathbb{R}^{4}$, the most natural
BPS partition function is enumerates the BPS indices for a fixed, asymptotic
value of the (Kähler) moduli $t$, and in the mixed ensemble where the magnetic
charge $P$ is fixed [69, 13, 14, 6] and the electric charge $Q$ is varied. The
electric charge takes value in the lattice $\Lambda\simeq\mathbb{Z}^{b_{2}}$
with bi-linear form $D_{ab}=d_{abc}P^{c}$. For a positive magnetic charge
$P^{a}$, $d_{abc}P^{c}$ provides a non-degenerate quadratic form with
signature $(1,b_{2}-1)$ for the lattice $\Lambda$ of magnetic charges. The
electric charges $Q_{a}$ take values in dual lattice $\Lambda^{*}$, with
quadratic form $D^{ab}=(d_{abc}P^{c})^{-1}$. We abbreviate the pairing between
an element $Q\in\Lambda^{*}$ and $k\in\Lambda$ as

$\sum_{a=1}^{b_{2}}Q_{a}P^{a}=Q.P$ (45)

and extend this by linearity in each argument to
$\Lambda^{*}\otimes\mathbb{R}$ and $\Lambda\otimes\mathbb{R}$. For later use,
we also introduce the notation $\mu\in\Lambda^{*}/\Lambda$,

$\Lambda_{\mu}^{*}=\left\\{Q\in\Lambda+\mu+P/2\right\\}.$ (46)

We stress that the elements of $\Lambda_{\mu}^{*}$ do not necessarily lie in
$\Lambda^{*}$ due to the shift by $P/2$. Using this notation, the partition
function for D4-D2-D0 black holes reads schematically

$\displaystyle\mathcal{Z}_{SG}(\tau,C,t)$
$\displaystyle=\sum_{Q_{0},Q_{a}}\bar{\Omega}(\gamma,t)\,(-1)^{P.Q}\,e^{-2\pi\tau_{2}|Z(\gamma,t)|+2\pi
i\tau_{1}\left(Q_{0}-Q.B+B^{2}/2\right)+2\pi iC.(Q-B/2)},$ (47)

where $\bar{\Omega}(\gamma,t)$ is the rational index (6),
$\tau_{1}\in\mathbb{R}$ is the RR 1-form $C_{1}$, and
$\tau_{2}=e^{-\phi}\in\mathbb{R}_{+}$ with $\phi$ being the dilaton, $C$ the
RR 3-form and $B$ the $B$-field.

Here $Z(\gamma,t)$ is the central charge of the $\mathcal{N}=2$ algebra.
Ignoring the non-perturbative terms in the strict large volume limit, $Z$
reads

$\displaystyle Z(\gamma,t)$
$\displaystyle=\frac{1}{2}P.(J^{2}-B^{2})+Q.B-Q_{0}+i(Q-BP).J\,.$ (48)

The BPS mass becomes in this limit,

$\displaystyle|Z(\gamma,t)|$
$\displaystyle=\frac{1}{2}P.(J^{2}-B^{2})+Q.B-Q_{0}+\frac{\left((Q-BP).J\right)^{2}}{P.J^{2}}\,,$
(49)

up to inverse powers of $J$. By assumption $J^{2}>0$, hence $J^{a}$ lies in
the positive cone of $\Lambda$. Thus $\frac{k.J\,J}{J^{2}}$ is the component
of the vector $k$ along $J$.

In the large volume limit, supergravity has an $Sp(2b_{2}+2,\mathbb{Z})$
symmetry, generated by matrices

$\displaystyle\mathbb{K}(k)$ $\displaystyle=\begin{pmatrix}1&0&0&0\\\
k^{a}&\mathbb{I}_{b_{2}}&0&0\\\
\frac{1}{2}d_{abc}k^{b}k^{c}&d_{abc}k^{c}&\mathbb{I}_{b_{2}}&0\\\
\frac{1}{6}d_{abc}k^{a}k^{b}k^{c}&\frac{1}{2}d_{abc}k^{b}k^{c}&k^{a}&1\end{pmatrix},\,\,k\in\mathbb{Z}^{b_{2}}\,,$
(50)

and acts linearly on $\gamma=(P^{0},P^{a},Q_{a},Q_{0})$. For $P^{0}=0$, these
transformations preserve the magnetic charge, and act on remaining charges and
moduli as follows:

$\displaystyle{}Q_{a}$ $\displaystyle\rightarrow Q_{a}+d_{abc}k^{b}P^{c}\,,$
$\displaystyle Q_{0}$ $\displaystyle\rightarrow
Q_{0}+k^{a}Q_{a}+\frac{1}{2}d_{abc}k^{a}k^{b}P^{c}\,,$ (51)
$\displaystyle{}t^{a}$ $\displaystyle\rightarrow t^{a}+k^{a}\,.$

We introduce the abbreviations:

$\begin{split}\hat{Q}_{\bar{0}}&:=-Q_{0}+\frac{1}{2}Q^{2}\,,\\\
\,\hat{Q}&:=Q-B\,,\\\ \,\hat{Q}^{2}_{-}&:=\hat{Q}^{2}-\hat{Q}^{2}_{+}\,,\\\
\,(Q-B)^{2}_{+}&:=\frac{\left((Q-BJ).P\right)^{2}}{PJ^{2}}\,.\end{split}$ (52)

Note that the combinations $\hat{Q},\hat{Q}_{\bar{0}}$ and the conjugacy class
$\mu$ of electric charge vector are invariant under the transformations (51).
Invariance of the conjugacy class is seen by noting that spectral flow shifts
$Q$ by integer lattice vectors, when mapped to magnetic lattice $\Lambda$ and
such shifts do not change the conjugacy class.

The holomorphic and anti-holomorphic dependence on $\tau$ can be made more
manifest, if we rewrite $\mathcal{Z}_{SG}$ as

$\displaystyle\mathcal{Z}_{SG}(\tau,C,t)$
$\displaystyle=e^{-\pi\tau_{2}J^{2}}\sum_{Q_{0},Q}\bar{\Omega}(P,Q,Q_{0};t)\,(-1)^{P.Q}\bar{q}^{\hat{Q}_{\bar{0}}-\hat{Q}^{2}_{-}/2}q^{\hat{Q}^{2}_{+}/2}e^{2i\pi
C.(Q-B/2)},$ (53)

where $q=e^{2\pi i\tau}$. The transformations (51) act on $\mathcal{Z}_{SG}$
as

$\displaystyle\mathcal{Z}_{SG}(\tau,C,t)$
$\displaystyle\rightarrow(-1)^{P.k}e^{\pi iC.k}\mathcal{Z}_{SG}(\tau,C,t)\,,$
(54)

under the action of $\mathbb{K}(k)$.

We can map the system of D4-D2-D0 branes to IIB string theory, by a T-duality
along time circle. The RR 1-form $C_{1}$ is mapped to the RR 0-form $C_{0}$,
while the D4, D2 and D0-branes respectively become D3, D1 branes and
D(-1)-instantons. Moreover, the action of the IIB strong-weak duality on the
instantonic branes carries over to the spectrum of D4-D2-D0 branes in IIA
string theory. The duality acts as follows. The modular parameter is
$\tau=C_{1}+ie^{-\phi}$, with $\phi$ being the dilaton and $C_{1}$ the Ramond-
Ramond 1-form flux along $S^{1}_{t}$. For an element $\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2,\mathbb{Z})$. This duality acts as

$\begin{split}&\tau\rightarrow\frac{a\tau+b}{c\tau+d},\\\ &\begin{pmatrix}C\\\
B\end{pmatrix}\rightarrow\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\begin{pmatrix}C\\\ B\end{pmatrix},\\\
&J\rightarrow|c\tau+d|J\,,\end{split}$ (55)

In a series of papers [6, 13, 14, 18, 25, 26, 29], the supergravity partition
function (47) has been analyzed. We summarize the main properties:

  * •

Quasi-periodicity in the two-form fields $B$ and $C$:
For $k\in\Lambda$, we have

$\begin{split}&\mathcal{Z}_{SG}(\tau,C,t+k)=(-1)^{P.k}\,e^{\pi
iC.k}\,\mathcal{Z}_{SG}(\tau,C,t),\\\
&\mathcal{Z}_{SG}(\tau,C+k,t)=(-1)^{P.k}\,e^{-\pi
iB.k}\,\mathcal{Z}_{SG}(\tau,C,t).\end{split}$ (56)

These can be seen as large gauge transformations.

  * •

$SL(2,\mathbb{Z})$ S-duality:

$\begin{split}S&:\qquad\,\mathcal{Z}_{SG}(-1/\tau,-B,C+|\tau|J)=\tau^{1/2}\bar{\tau}^{-3/2}\varepsilon(S)\,\mathcal{Z}_{SG}(\tau,C,t)\,,\\\
T&:\qquad\,\mathcal{Z}_{SG}(\tau+1,C+B,t)=\varepsilon(T)\,\mathcal{Z}_{SG}(\tau,C,t)\,,\end{split}$
(57)

where
$\varepsilon(S)=\varepsilon(T)^{-3},\varepsilon(T)=e^{-i\pi\frac{c_{2}(X).P}{12}}$
are phases. The factor $\bar{\tau}^{-3/2}$ can be understood from the non-
compact bosons in the CFT due to the center of mass in $\mathbb{R}^{3}$.

  * •

The partition function involves an intricate dependence on the moduli $t$
through wall-crossing. The proper modular invariant partition function differs
from (53) by additional non-holomorphic, subleading terms which are non-
holomorphic in $\tau$.

###  3.2 Attractor partition function

The supergravity partition function is a rather complicated function. It has
become clear that an alternative partition function is a more amenable object
to study [18]. This is the attractor partititon function
$\mathcal{Z}_{P}^{\lambda}$, which is the generating function of indices
$\bar{\Omega}(\gamma;t^{*}_{\gamma})$, where each index is evaluated at its
(large volume) attractor point $t^{\lambda}_{\gamma}$. The indices
$\bar{\Omega}(\gamma;t^{\lambda}_{\gamma})$ are also referred to as MSW
invariants [18]. For irreducible magnetic charge $P$ and magnetic charges
which can be written as a sum of at most 2 magnetic charges, these indices are
conjectured to coincide with the CFT indices. However as discussed in Section
5, our findings in this paper suggest that there is a difference for generic
magnetic charges due to scaling black holes.

This partition is obtained by replacing $\Omega(\gamma,t)$ in the definition
of $\mathcal{Z}_{SG}$ by its attractor value
$\Omega(\gamma,t^{\lambda}_{\gamma})$. For a D4-D2-D0 black hole with charge
$\gamma=(0,P^{a},Q_{a},Q_{0})$, we have for the “large volume” attractor value
$(t_{\gamma}^{*})^{a}$,

$(t^{\lambda}_{\gamma})^{a}=D^{ab}Q_{b}+i\lambda\,P^{a},$ (58)

with $\lambda$ sufficiently large, such that subleading terms in a $\lambda$
expansion can be ignored. Eq. (58) is equivalent to

$B^{a}_{\gamma}=D^{ab}Q_{b},\qquad(J^{\lambda}_{\gamma})^{a}=\lambda\,P^{a}.$
(59)

The precise proportionality constant $\lambda$ between $J^{a}$ and $P^{a}$
will not be important for us, since we will restrict to the large volume
limit, $|J|\gg 1$. On the other hand, we do not take the limit
$\lambda\to\infty$, since physical quantities such as the BPS mass diverge in
this limit.

The attractor or MSW partition function then reads,

$\displaystyle\mathcal{Z}^{\lambda}_{P}(\tau,C,t)$
$\displaystyle=\sum_{Q_{0},Q_{a}}\bar{\Omega}(\gamma,t_{\gamma}^{\lambda})\,(-1)^{P.Q}\,e^{-2\pi\tau_{2}|Z(\gamma,t)|+2\pi
i\tau_{1}\left(Q_{0}-Q.B+B^{2}/2\right)+2\pi iC.(Q-B/2)}.$ (60)

Note that although the degeneracy is evaluated at attractor point
$t^{\lambda}_{\gamma}$ , the mass in the exponent is the ADM mass evaluated
for moduli at infinity being $t$. The other moduli dependence in the exponent
is also similar to that of $\mathcal{Z}_{SG}(\tau,C,t)$.

Since black holes states contributing to the attractor index exist everywhere
in moduli space, their quantum degeneracies should be moduli independent.
Moreover, one can show that $\Omega(\gamma;t^{\lambda}_{\gamma})$ are
invariant under spectral flow transformations (51). This entails that
$\Omega(\gamma;t^{\lambda}_{\gamma})$ depends on the charges only through the
invariant combination $\hat{Q}_{\bar{0}}$ and the conjugacy class $\mu$. These
imply the sum

$\displaystyle h_{P,\mu}(\tau)$
$\displaystyle:=\sum_{Q_{0}}\overline{\Omega}(\gamma;t^{\lambda}_{\gamma})\,q^{\hat{Q}_{\bar{0}}}\,,$
(61)

for fixed $Q$ is also invariant under spectral flow transformations and apart
from the magnetic charge, depends solely on the conjugacy class $\mu$. It has
been understood from wall-crossing and the perspective of D-instantons, that
$h_{P,\mu}$ receives additional non-holomorphic contributions $h_{P,\mu}$ [18,
29]. The resultant is the non-holomorphic function
$\widehat{h}_{P,\mu}(\tau,\bar{\tau})$.

Hence one is led to the following theta function decomposition of the
attractor partition function,

$\displaystyle\mathcal{Z}^{\lambda}_{P}(\tau,C,t)$
$\displaystyle=e^{-\pi\tau_{2}J^{2}}\sum_{\mu\in\Lambda^{*}/\Lambda}\overline{h_{P,\mu}(\tau)}\,\Theta_{\mu}(\tau,\bar{\tau},C,B)\,,$
(62) $\displaystyle\text{where}\,\,\Theta_{\mu}(\tau,\bar{\tau},C,B)$
$\displaystyle=\sum_{Q\in\Lambda^{*}_{\mu}}(-1)^{P.Q}q^{\hat{Q}^{2}_{+}/2}\bar{q}^{-\hat{Q}^{2}_{-}/2}e^{2\pi
iC.(Q-B/2)}\,,$ (63)

with the set $\Lambda^{*}_{\mu}$ is given in (46). We define analogously the
completion $\widehat{\mathcal{Z}}^{\lambda}_{P}$ by replacing $h_{P,\mu}$ by
$\widehat{h}_{P,\mu}$ in (62).

Various factors in the right hand side of (62) also have definite modular
transformation properties. The prefactor $e^{-\pi\tau_{2}J^{2}}$ is modular
invariant. Moreover, the Siegel-Narain theta function $\Theta_{\mu}(\tau,C,B)$
transforms as

$\begin{split}S&:\,\Theta_{\mu}(-1/\tau,-1/\bar{\tau},-B,C)=\frac{1}{\sqrt{|\Lambda^{*}/\Lambda|}}(-i\tau)^{b_{2}^{+}/2}(i\bar{\tau})^{b_{2}^{-}/2}e^{-i\pi
P^{2}/2}\\\
&\qquad\qquad\qquad\qquad\times\sum_{\nu\in\Lambda^{*}/\Lambda}e^{-2\pi
i\mu.\nu}\Theta_{\nu}(\tau,\bar{\tau},C,B)\,,\\\
T&:\,\Theta_{\mu}(\tau+1,\bar{\tau}+1,C+B,B)=e^{i\pi(\mu+P/2)^{2}}\Theta_{\mu}(\tau,\bar{\tau},B,C)\,.\end{split}$
(64)

This implies that the appropriate completion $\widehat{h}_{P,\mu}$ transforms
as a vector valued modular form,

$\begin{split}S&:\,\widehat{h}_{P,\mu}(-1/\tau,-1/\bar{\tau})=-\frac{1}{\sqrt{|\Lambda^{*}/\Lambda|}}(-i\tau)^{-b_{2}/2-1}\varepsilon(S)^{*}e^{-i\pi
P^{2}/2}\sum_{\delta\in\Lambda^{*}/\Lambda}e^{-2\pi
i\delta.\mu}\widehat{h}_{P,\delta}(\tau,\bar{\tau})\,,\\\
T&:\,\widehat{h}_{P,\mu}(\tau+1,\bar{\tau}+1)=\varepsilon(T)^{*}e^{i\pi(\mu+P/2)^{2}}\,\widehat{h}_{P,\mu}(\tau,\bar{\tau}),\end{split}$
(65)

where $\varepsilon(S)$ and $\varepsilon(T)$ are as below (57).

##  4 Partition functions for scaling black holes

We consider multi-core black holes, where each core carries a positive
D4-brane charge and vanishing D6-brane charge. The notion of “core” is as
introduced in the Introduction; it is a bound state with spatial magnitude of
the order $\ell_{5}^{3}/R^{2}$, which decreases as $\lambda^{-3}$ in the large
$\lambda$ limit. We will determine the partition function at the large volume
attractor point (59) for such black holes. For this value of the moduli and
assuming that $\gamma$ and $\gamma_{j}$ both carry D4-brane charge, the
$c_{j}$ (5) are to leading order equal to $c^{\lambda}_{j}$,

$c^{\lambda}_{j}=2\lambda\left<\gamma,\gamma_{j}\right>.$ (66)

Since the factor $2\lambda$ on the rhs is positive, the large volume attractor
point is similar in nature to the $c_{j}^{*}$ (19) for the proper attractor
point. Restricting to three center solutions with D4-brane charge, we have for
$a$, $b$ and $c$,

$\begin{split}a&=P_{1}Q_{2}-P_{2}Q_{1},\\\ b&=P_{2}Q_{3}-P_{3}Q_{2},\\\
c&=P_{3}Q_{1}-P_{1}Q_{3}.\end{split}$ (67)

This gives for the $c_{j}^{\lambda}$,
$c_{1}^{\lambda}=2\lambda\,(c-a),c_{2}^{\lambda}=2\lambda\,(a-b)$. Since the
$c^{*}_{j}$ (19) and $c^{\lambda}_{j}$ (66) are simply related by the
substitution $|Z(\gamma;t_{\gamma}^{*})|\to 2\lambda$, the discussion on the
solution to Denef’s equations below (23) is applicable, and in particular
gives the separation of collinear solutions.

At the attractor point, many multi-center configurations do not exist as
physical solutions. However we have seen that some multi-center solutions do
exist. These solutions are distinguished from generic solutions, since the
centers can approach each other arbitrarily close. The scaling solutions do
respect the spectral flow symmetry, since the symplectic innerproducts are
invariant under (51).

We note that the existence scaling solutions poses constraints on the magnetic
charges $P_{j}$. For example since for a scaling solution $a,b,c$ must have
the same sign, we deduce that there are no scaling solutions for
$P_{1}=P_{2}=P_{3}=P/3$, since this gives rise to
$\gamma_{12}+\gamma_{23}+\gamma_{31}=a+b+c=0$. Therefore, there must be an
asymmetry in the magnetic charge of the centers for such scaling solutions to
exist. To see that $F$ vanishes in the case of three equal magnetic charges,
note that the second factors in $F_{2},F_{4},F_{6}$ vanish immediately as
${\rm sgn}(2x)-{\rm sgn}(x)=0$. Moreover, in $F_{1}$ we can replace $c=-a-b$
and when $a>0$ the first factor imposes the constraint $-2a-b\geq 0$ so
$b<-2a$ and $b$ must be negative. Hence the second factor in $F_{1}$ vanishes.
The argument goes similarly for $F_{3},F_{5}$. Similarly one may show that
such scaling black holes only exist for $b_{2}>1$.

###  4.1 Bound states and lattices

We discuss in this subsection various aspects of charge lattices of bound
states, and the projection of the charge lattice to a sublattice with fixed
total charge. The discussion in this subsection does not rely on the existence
of scaling black holes for these charges and lattices.

Lattice decomposition
We apply some techniques of decompositions and gluings of integral lattices to
charge lattices of bound states. See for example [70, 71]. We will consider a
black hole bound state of $n$ centers, with non-vanishing, positive magnetic
charge $P_{j}\in\Lambda_{j},$ $j=1,2,\dots,n$, where $\Lambda_{j}$ is the
$b_{2}$-dimensional lattice associated to the $j$’th center with innerproduct
$D_{j}$,

$D_{jab}=d_{abc}P^{c}_{j},\qquad a,b,c=1,\dots,b_{2}.$ (68)

The total magnetic charge is $P=\sum_{j=1}^{n}P_{j}$ with quadratic form $D$
and $b_{2}$-dimensional lattice $\Lambda$. The signature $\Lambda_{j}$ and
$\Lambda$ is $(1,b_{2}-1)$.

We use boldface notation for the lattices for boundstates. For $n$ centers, we
introduce $(n\,b_{2})$-dimensional lattices and vectors,

$\begin{split}\vec{k}&=(k_{1},k_{2},\dots,k_{n})\in{\boldsymbol{\Lambda}}:=\Lambda_{1}\oplus\Lambda_{2}\oplus\dots\oplus\Lambda_{n},\\\
\vec{x}&=(x_{1},x_{2},\dots,x_{n})\in{\boldsymbol{\Lambda}^{*}}:=\Lambda_{1}^{*}\oplus\Lambda_{2}^{*}\oplus\dots\oplus\Lambda_{n}^{*},\\\
\vec{Q}&=(Q_{1},Q_{2},\dots,Q_{n})\in{\boldsymbol{\Lambda}}^{*}+\vec{P}/2=\Lambda_{1}^{*}\oplus\Lambda_{2}^{*}\oplus\dots\oplus\Lambda_{n}^{*}+(P_{1},P_{2},\dots,P_{n})/2.\end{split}$
(69)

We denote the quadratic form for ${\boldsymbol{\Lambda}}$ by $\vec{D}={\rm
diag}(D_{1},D_{2},\dots,D_{n})$.

Since we typically sum over bound states with fixed total electric charge, we
aim to decompose the lattice ${\boldsymbol{\Lambda}}$ in a $b_{2}$-dimensional
sublattice $\overline{\boldsymbol{\Lambda}}\subset{\boldsymbol{\Lambda}}$
representing the total charge, and its orthogonal complement
$\underline{{\boldsymbol{\Lambda}}}$ with dimension $(n-1)b_{2}$ representing
the relative charge distribution over the constituents. To introduce this
properly, let $\overline{\boldsymbol{\Lambda}}$ be the sublattice
$\overline{\boldsymbol{\Lambda}}\subset{\boldsymbol{\Lambda}}$,

$\overline{\boldsymbol{\Lambda}}=\\{\vec{k}=(k,k,\dots,k)\in{\boldsymbol{\Lambda}}\,|\,k\in\mathbb{Z}^{b_{2}}\\}.$
(70)

The lattice ${\boldsymbol{\Lambda}}$ induces a quadratic form on
$\overline{\boldsymbol{\Lambda}}$. Namely for
$\vec{k}\in{\boldsymbol{\Lambda}}$,
$\vec{D}(\vec{k})=\sum_{j=1}^{n}D_{j}(k)=D(k)$. For $k\in\mathbb{Z}^{b_{2}}$,
this is the quadratic form for the total charge $P$ as desired. As a result,
we have the group isomorphism
$\Lambda^{*}/\Lambda=\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}$.

Let $\overline{\pi}$ be the orthogonal projection,

$\overline{\pi}:{\boldsymbol{\Lambda}}\to\overline{\boldsymbol{\Lambda}}\otimes\mathbb{Q}.$
(71)

For $\vec{k}\in{\boldsymbol{\Lambda}}$ and
$\vec{m}\in\overline{\boldsymbol{\Lambda}}$, we have

$\vec{D}(\overline{\pi}(\vec{k}),\vec{m})=\vec{D}(\vec{k},\overline{\pi}(\vec{m}))=\vec{D}(\vec{k},\vec{m})\in\mathbb{Z}.$
(72)

Therefore, $\overline{\pi}$ gives an injection of ${\boldsymbol{\Lambda}}$ to
the dual lattice $\overline{\boldsymbol{\Lambda}}^{*}$,

$\overline{\pi}:{\boldsymbol{\Lambda}}\to\overline{\boldsymbol{\Lambda}}^{*}.$
(73)

Moreover, if we extend $\overline{\pi}$ by linearity to
${\boldsymbol{\Lambda}}^{*}$, a similar argument to (72) shows that
$\overline{\pi}:{\boldsymbol{\Lambda}}^{*}\to\overline{\boldsymbol{\Lambda}}^{*}$.
The projection $\overline{\pi}$ is explicitly given for
$\vec{k}=(k_{1},k_{2},\dots,k_{n})\in{\boldsymbol{\Lambda}}$ by,

$\overline{\pi}(\vec{k})=(k,k,\dots,k)\in(\overline{\boldsymbol{\Lambda}})^{*}\quad{\rm
with}\quad k=D^{-1}\sum_{j=1}^{n}D_{j}k_{j}.$ (74)

We define furthermore the kernel $\underline{{\boldsymbol{\Lambda}}}={\rm
Ker}(\overline{\pi})$,

$\begin{split}&\underline{{\boldsymbol{\Lambda}}}\coloneqq\left\\{\vec{k}\in{\boldsymbol{\Lambda}}\,\left|\,\sum_{j=1}^{n}D_{j}k_{j}=0\right.\right\\}.\end{split}$
(75)

Elements in $\underline{{\boldsymbol{\Lambda}}}$ have vanishing innerproduct
with $\overline{\boldsymbol{\Lambda}}$, such that they are indeed each others
orthogonal complement in ${\boldsymbol{\Lambda}}$.

Since $\overline{\pi}(\vec{k})\in\overline{\boldsymbol{\Lambda}}$ for
$\vec{k}\in{\boldsymbol{\Lambda}}$, we have that
$\vec{k}-\overline{\pi}(\vec{k})\in\underline{{\boldsymbol{\Lambda}}}$. Thus
$\overline{\boldsymbol{\Lambda}}\oplus\underline{{\boldsymbol{\Lambda}}}$ is
in the kernel of the homomorphism
$\overline{h}:{\boldsymbol{\Lambda}}\to\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}$.
We call

$G={\boldsymbol{\Lambda}}/(\overline{\boldsymbol{\Lambda}}\oplus\underline{{\boldsymbol{\Lambda}}}),$
(76)

the glue group for the decomposition of ${\boldsymbol{\Lambda}}$, and the
image of $G$ under $\overline{h}$,
$\overline{h}(G)\subset\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}$,
the glue group for $\overline{\boldsymbol{\Lambda}}$ [70]. The homomorphism
$\overline{h}$ gives an injection of $G$ to the subgroup
$\overline{h}(G)\subset\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}$.
Therefore, the number of glue vectors,
$N_{g}=|{\boldsymbol{\Lambda}}/(\overline{\boldsymbol{\Lambda}}\oplus\underline{{\boldsymbol{\Lambda}}})|$
is a factor in
$|\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}|=\det(\overline{D})$.
In the special case that $\det(\overline{D})$ is prime, or more generally co-
prime with $\det(D_{j})$ for all $j$, $N_{g}=\det(\overline{D})$. By the same
arguments, there is a projection
$\underline{\pi}:{\boldsymbol{\Lambda}}\to\underline{{\boldsymbol{\Lambda}}}^{*}$,
and a homomorphism
$\underline{h}:G\to\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}$.
Therefore, $N_{g}$ is also a factor in $\det(\underline{D})$. This gives for
the number of glue vectors $N_{g}$ in general

$N_{g}=\sqrt{\frac{\det(\overline{D})\,\det(\underline{D})}{\prod_{j=1}^{n}\det(D_{j})}}.$
(77)

The order of the quotient group
$(\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}})/\underline{h}(G)$
is

$N_{q}=\frac{\det(\underline{D})}{N_{g}}.$ (78)

This will be useful for us in the following way. If we consider the class of
vectors $\vec{k}\in{\boldsymbol{\Lambda}}$ with fixed projection to
$\overline{\boldsymbol{\Lambda}}^{*}$, for example
$\overline{\pi}(\vec{k})=0$, this fixes an element of the glue group $G$, and
thus also of the images of $G$ in
$\overline{\boldsymbol{\Lambda}}^{*}/\overline{\boldsymbol{\Lambda}}$ and
$\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}$.
As a result, the number of possible conjugacy classes of
$\underline{\pi}(\vec{k})\in\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}$
of $\underline{\pi}(\vec{k})$ is $N_{q}$ (78). This is also the case for a
vector $\vec{x}\in{\boldsymbol{\Lambda}}^{*}$ with fixed projection
$\overline{\pi}(\vec{x})\in\overline{\boldsymbol{\Lambda}}^{*}$.

Similarly to $\Lambda^{*}_{\mu}$ (46), we introduce the notation
$\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}$,

$\begin{split}&\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}\coloneqq\left\\{\vec{Q}=(Q_{1},Q_{2},\dots,Q_{n})\in{\boldsymbol{\Lambda}}+(\mu_{1},\mu_{2},\dots,\mu_{n})+\vec{P}/2\,\left|\,\sum_{j=1}^{n}Q_{j}=\mu+P/2\right.\right\\},\end{split}$
(79)

where the subscript
${\boldsymbol{\mu}}\in\underline{\boldsymbol{\Lambda}}^{*}/\underline{\boldsymbol{\Lambda}}$,
$\mu=\sum_{j=1}^{n}\mu_{j}\in\Lambda^{*}$ and
$P=\sum_{j=1}^{n}P_{j}\in\Lambda$. We define the quadratic form
${\boldsymbol{Q}}^{2}$ for
$\underline{{\boldsymbol{\Lambda}}}^{*}_{\boldsymbol{\mu}}$,

$\begin{split}{\boldsymbol{Q}}^{2}&=-Q^{2}+\sum_{j=1}^{3}(Q_{j})_{j}^{2}\\\
&=-(\mu+P/2)^{2}+\sum_{j=1}^{3}(Q_{j})_{j}^{2}.\end{split}$ (80)

This form of the quadratic form appears naturally, when we determine the
partition function $h^{s}_{\mu}$ (118) of scaling solutions in the next
subsection. On the other hand, it should match with the inverse of the
quadratic form for $\underline{{\boldsymbol{\Lambda}}}$. In the cases
considered below, we find that this is indeed the case.

Note that in components, $P$ in (79) has a lower index and equals
$P_{a}=d_{abc}P^{b}P^{c}=d_{abc}(\sum_{j}P_{j})^{b}(\sum_{j}P_{j})^{c}$. To
understand constraint $\sum_{j}Q_{j}=\mu+P/2$ better, we change variables from
$Q_{j}\in\Lambda^{*}_{j}$ to $k_{j}\in\Lambda_{j}$ using

$\begin{split}&Q_{1,a}=\mu_{1,a}+D_{1ab}(k^{b}_{1}+P_{2}^{b}+P_{1}^{b}/2),\\\
&Q_{2,a}=\mu_{2,a}+D_{2ab}(k^{b}_{2}+P_{3}^{b}+P_{2}^{b}/2),\\\
&\qquad\vdots\\\
&Q_{n-1,a}=\mu_{n-1,a}+D_{(n-1)ab}(k^{b}_{n}+P_{n}^{b}+P_{n-1}^{b}/2),\\\
&Q_{n,a}=\mu_{n,a}+D_{nab}(k^{b}_{n}+P_{1}^{b}+P_{n}^{b}/2),\end{split}$ (81)

with $k_{j}^{b}\in\mathbb{Z}^{b_{2}}$ for $j=1,\dots,n$. The shifts of
$k_{j}^{b}$ by $P_{j}^{b}$ are included such that the required identity for
the $k_{j}$ takes a compact form,

$\sum_{j=1}^{n}D_{jab}\,k_{j}^{b}=0,\qquad k_{j}\in\mathbb{Z}^{b_{2}},$ (82)

which is indeed identical to the defining condition for the lattice
$\underline{{\boldsymbol{\Lambda}}}$ in (75). Solving this relation over the
integers, $k_{j}\in\mathbb{Z}^{b_{2}}$, is in general a complicated problem
depending on the $D_{j}$. Solving say for $k_{n}$, we have

$k_{n}=-D_{n}^{-1}\sum_{j=1}^{n-1}D_{j}k_{j}.$ (83)

Thus we find that if $D_{n}^{-1}D_{j}$, $j=1,\dots n-1$ are not integral
matrices, not all $k_{j}\in\mathbb{Z}^{b_{2}}$ can correspond to bound state
charges for these conjugacy classes. In some cases, this problem can be avoid
by solving for another $k_{j}$ instead of $k_{n}$, but not in general. We
restrict to special cases in the following.
2- and 3-center bound states for $b_{2}=1$
For $b_{2}=1$, the $D_{j}$ are simply positive numbers. For $n=2$, the
relation (82) becomes

$D_{1}k_{1}+D_{2}k_{2}=0.$ (84)

The solutions with $k_{1,2}\in\mathbb{Z}$ are

$k_{1}=\frac{D_{2}}{\gcd(D_{1},D_{2})}\,m,\qquad
k_{2}=-\frac{D_{1}}{\gcd(D_{1},D_{2})}\,m,\qquad m\in\mathbb{Z},$ (85)

where $\gcd$ stands for the greatest common divisor. Substituting (85) in the
quadratic form $\sum_{j=1,2}D_{j}k_{j}^{2}$, gives for the quadratic form
$\underline{D}$ on $\underline{{\boldsymbol{\Lambda}}}$,

$\underline{D}=\frac{D_{1}D_{2}(D_{1}+D_{2})}{\gcd(D_{1},D_{2})^{2}}.$ (86)

Thus the number of glue vectors (77) and order $N_{q}$ (78) are in this case,

$n=2:\qquad N_{g}=\frac{D_{1}+D_{2}}{\gcd(D_{1},D_{2})},\qquad
N_{q}=\frac{D_{1}D_{2}}{\gcd(D_{1},D_{2})}.$ (87)

Moving on to $n=3$, we need to find the integral solutions to

$D_{1}k_{1}+D_{2}k_{2}+D_{3}k_{3}=0.$ (88)

To this end, we first consider $k_{3}=0$. Then, the solutions are obviously
given by (85). Next for a fixed non-vanishing $k_{3}$, there are only
solutions if $\gcd(D_{1},D_{2})$ divides $D_{3}k_{3}$ by Bézout’s identity. As
a result, if we choose $\gcd(D_{1},D_{2})/\gcd(D_{1},D_{2},D_{3})$ for
$k_{3}$, Bézout’s identity asserts that there is an integral solution
$(\ell_{1},\ell_{2})$ for $(k_{1},k_{2})$,

$D_{1}\ell_{1}+D_{2}\ell_{2}+D_{3}\frac{\gcd(D_{1},D_{2})}{\gcd(D_{1},D_{2},D_{3})}=0.$
(89)

Since this choice for $k_{3}$ has the smallest non-vanishing magnitude with
integral solutions, the other choices for $k_{3}$ follow by multiplication by
an integer $m_{1}$. Including also the solutions (85), we find for the general
solution,

$\begin{split}k_{1}&=\ell_{1}m_{1}+\frac{D_{2}}{\gcd(D_{1},D_{2})}\,m_{2},\\\
k_{2}&=\ell_{2}m_{1}-\frac{D_{1}}{\gcd(D_{1},D_{2})}\,m_{2},\\\
k_{3}&=\frac{\gcd(D_{1},D_{2})}{\gcd(D_{1},D_{2},D_{3})}m_{1}.\end{split}$
(90)

Substitution of (90) in $\sum_{j=1}^{3}D_{j}k_{j}^{2}$, one finds for the
2-dimensional quadratic form $\underline{D}$ for $(m_{1},m_{2})$,

$\underline{D}=\left(\begin{array}[]{cc}D_{1}\ell_{1}^{2}+D_{2}\ell_{2}^{2}+D_{3}\frac{\gcd(D_{1},D_{2})^{2}}{\gcd(D_{1},D_{2},D_{3})^{2}}&\quad\frac{D_{1}D_{2}(\ell_{1}-\ell_{2})}{\gcd(D_{1},D_{2})}\\\
\frac{D_{1}D_{2}(\ell_{1}-\ell_{2})}{\gcd(D_{1},D_{2})}&\frac{D_{1}D_{2}(D_{1}+D_{2})}{\gcd(D_{1},D_{2})^{2}}\end{array}\right).$
(91)

Using (89), we find for its determinant,

$\det(\underline{D})=\frac{D_{1}D_{2}D_{3}(D_{1}+D_{2}+D_{3})}{\gcd(D_{1},D_{2},D_{3})^{2}},$
(92)

which is symmetric in the $D_{j}$. For the number of glue vectors $N_{g}$ and
order $N_{q}$, we now obtain,

$n=3:\qquad N_{g}=\frac{D_{1}+D_{2}+D_{3}}{\gcd(D_{1},D_{2},D_{3})},\qquad
N_{q}=\frac{D_{1}D_{2}D_{3}}{\gcd(D_{1},D_{2},D_{3})}.$ (93)

Together with the discussion below, this is very suggestive that the
generalization to $n>3$ is

$\det(\underline{D})=\frac{(\sum_{j=1}^{n}D_{j})\,\prod_{j=1}^{n}D_{j}}{\gcd(D_{1},D_{2},\dots,D_{n})^{2}}.$
(94)

2- and 3-center bound states for $b_{2}\geq 1$ with simplifications
We continue to discuss the general case $b_{2}\geq 1$. Let us first consider
$n=2$. We make the assumption that $D_{2}^{-1}D_{1}$ is an integral matrix.
The equation (82) can be solved by setting $k_{2}=-D_{2}^{-1}D_{1}k_{1}$.
Substituting this in the quadratic form $\vec{D}$, gives us for the
$b_{2}$-dimensional quadratic form on $\underline{{\boldsymbol{\Lambda}}}$

$\underline{D}=D_{1}+D_{1}D_{2}^{-1}D_{1}.$ (95)

This agrees with (86) upon specialization to $b_{2}=1$. We find then for
$N_{g}$ and $N_{q}$,

$n=2:\qquad N_{g}={\rm det}(D_{2}^{-1}D),\qquad N_{q}={\rm det}(D_{1}).$ (96)

To deal with our main case, $n=3$, we make two technical simplifications:

  1. 1.

We assume that $D_{3}^{-1}D_{1}$ and $D_{3}^{-1}D_{2}$ are integral matrices,
such that $k_{3}\in\mathbb{Z}^{b_{2}}$ in (83) if $k_{1}$ and
$k_{2}\in\mathbb{Z}^{2}$.

  2. 2.

We assume that $d_{abc}\in 2\mathbb{Z}$ for all $a$, $b$, $c$. Then
$\Lambda^{*}+P/2=\Lambda^{*}$ for any $P$ such that shifts by $P$ (and
$P_{j}$) in (79), (80) and (81) are unnecessary.

These assumptions are satisfied in the examples in Section 6.

If we substitute now (83) in ${\boldsymbol{Q}}^{2}$, we arrive at

$\begin{split}{\boldsymbol{Q}}^{2}&=-\mu^{2}+(\mu_{1})^{2}_{1}+(\mu_{2})^{2}_{2}+(\mu_{3})^{2}_{3}+2(\mu_{1}-D_{3}^{-1}D_{1}\mu_{3}).k_{1}+2(\mu_{2}-D_{3}^{-1}D_{2}\mu_{3}).k_{2}\\\
&\quad+(k_{1},k_{2})\,\underline{D}\,(k_{1},k_{2})^{T},\end{split}$ (97)

where $\underline{D}$ is the quadratic form of the lattice
$\underline{{\boldsymbol{\Lambda}}}$,

$\underline{D}=\left(\begin{array}[]{cc}D_{1}+D_{1}D_{3}^{-1}D_{1}&\quad
D_{1}D_{3}^{-1}D_{2}\\\ D_{2}D_{3}^{-1}D_{1}&\quad
D_{2}+D_{2}D_{3}^{-1}D_{2}\end{array}\right).$ (98)

The determinant of $\underline{D}$ is given by333We use that for an invertible
$n\times n$ matrix $A$ and $n\times m$ matrices $U$ and $V$, we have
$\det(A+UV^{T})=\det(A)\,\det(1_{m}+V^{T}A^{-1}U)$,
en.wikipedia.org/wiki/Matrix$\\_$determinant$\\_$lemma

$\begin{split}\det(\underline{D})&=\det(D_{1})\,\det(D_{2})\,\det(D_{3}^{-1})\,\det(D_{1}+D_{2}+D_{3})\\\
&=\det(D_{1})\,\det(D_{2})\,\det(D_{3}^{-1}D).\end{split}$ (99)

We have for $N_{g}$ and $N_{q}$ in this case,

$N_{g}=\det(D_{3}^{-1}D),\qquad N_{q}=\det(D_{1})\det(D_{2}).$ (100)

These formulas are in agreement with (92) for $b_{2}=1$, and
$\gcd(D_{1},D_{2},D_{3})=D_{3}$. Moreover for generic $b_{2}$, we can consider
cases where we can solve (82) in terms of $k_{2}$ as well as $k_{3}$, and we
expect a symmetry in $D_{2}\leftrightarrow D_{3}$. Indeed, then
$D_{2}^{-1}D_{1}$ and $D_{2}^{-1}D_{3}$ are integral matrices too. The only
way for $D_{2}^{-1}D_{3}$ and $D_{3}^{-1}D_{2}$ both to be integral is to
satisfy $|\det(D_{2})|=|\det(D_{3})|$, in which case (99) reduces to
$\det\underline{D}=\pm\det(D_{1})\,\det(D_{1}+D_{2}+D_{3})$, which is
symmetric under the exchange of 2 and 3. Similar comments hold of course as
well for $k_{1}$.

Using general formula’s for inverses of block matrices [72], we derive that
the inverse of $\underline{D}$ reads

$\begin{split}\underline{D}^{-1}&=\left(\begin{array}[]{cc}D^{-1}(D_{2}+D_{3})D_{1}^{-1}&\quad-D^{-1}\\\
-D^{-1}&D^{-1}(D_{1}+D_{3})D_{2}^{-1}\end{array}\right)\\\
&=\left(\begin{array}[]{cc}D_{1}^{-1}-D^{-1}&\quad-D^{-1}\\\
-D^{-1}&D_{2}^{-1}-D^{-1}\end{array}\right),\end{split}$ (101)

One can verify that the determinant of $\underline{D}^{-1}$ is indeed the
inverse of (99). Indeed, if we introduce the two components
${\boldsymbol{\mu}}_{1}$ and ${\boldsymbol{\mu}}_{2}$,

$\begin{split}{\boldsymbol{\mu}}_{1}&=\mu_{1}-D_{1}D_{3}^{-1}\mu_{3},\\\
{\boldsymbol{\mu}}_{2}&=\mu_{2}-D_{2}D_{3}^{-1}\mu_{3},\end{split}$ (102)

the quadratic form ${\boldsymbol{Q}}^{2}$ (97) for $k_{1}=k_{2}=0$,
${\boldsymbol{Q}}^{2}={\boldsymbol{\mu}}^{2}$, can be written as

${\boldsymbol{\mu}}^{2}=-({\boldsymbol{\mu}}_{1}+{\boldsymbol{\mu}}_{2})^{2}+({\boldsymbol{\mu}}_{1})_{1}^{2}+({\boldsymbol{\mu}}_{2})_{2}^{2}=({\boldsymbol{\mu}}_{1},{\boldsymbol{\mu}}_{2})\underline{D}^{-1}({\boldsymbol{\mu}}_{1},{\boldsymbol{\mu}}_{2})^{T}.$
(103)

For $k_{1},k_{2}$ non-zero, we have
${\boldsymbol{Q}}=({\boldsymbol{Q}}_{1},{\boldsymbol{Q}}_{2})$ with

$\displaystyle{\boldsymbol{Q}}_{1}$
$\displaystyle=\mu_{1}-D_{1}D_{3}^{-1}\mu_{3}+(D_{1}+D_{1}D_{3}^{-1}D_{1})k_{1}+D_{1}D_{3}^{-1}D_{2}k_{2},$
(104) $\displaystyle{}{\boldsymbol{Q}}_{2}$
$\displaystyle=\mu_{2}-D_{2}D_{3}^{-1}\mu_{3}+D_{2}D_{3}^{-1}D_{1}k_{1}+(D_{2}+D_{2}D_{3}^{-1}D_{2})k_{2},$

or equivalently

$\begin{split}{\boldsymbol{Q}}_{1}&=Q_{1}-D_{1}D_{3}^{-1}Q_{3},\\\
{\boldsymbol{Q}}_{2}&=Q_{2}-D_{2}D_{3}^{-1}Q_{3},\end{split}$ (105)

which we can write more compactly as
${\boldsymbol{Q}}={\boldsymbol{\mu}}+\underline{D}\,{\boldsymbol{k}}$ with
${\boldsymbol{k}}=({\boldsymbol{k}}_{1},{\boldsymbol{k}}_{2})^{T}$. Thus an
element
${\boldsymbol{\mu}}\in\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}$
is completely determined by

${\boldsymbol{\mu}}=\\{(\mu_{1},\mu_{2},\mu_{3},\mu)|\,\,\mu_{j}\in\Lambda_{j}^{*},\quad\mu_{1}+\mu_{2}+\mu_{3}=\mu\in\Lambda^{*}\\}.$
(106)

Generic number of constituents (with simplifications)
The analysis for a generic number $n$ of constituents (including $n=2$)
follows analogously. The quadratic form reads as in (80), but with 3 replaced
by $n$ in the summation, and with constraint $\sum_{j=1}^{n}Q_{j}=\mu+P/2$.
Similarly to (82), the constraint can be expressed as

$\sum_{j=1}^{n}D_{jab}k_{j}^{b}=0.$ (107)

With the assumption that $D_{n}^{-1}D_{j}$ is an integer matrix for all
$j=1,\dots,n$, this can be solved and the quadratic form becomes

$\underline{D}={\rm
diag}(D_{1},\dots,D_{n-1})+\left(\begin{array}[]{c}D_{1}\\\ \dots\\\
D_{n-1}\end{array}\right)\,(D_{n}^{-1}D_{1},\dots,D_{n}^{-1}D_{n-1}),$ (108)

with inverse

$\underline{D}^{-1}={\rm
diag}(D_{1}^{-1},\dots,D_{n-1}^{-1})-\left(\begin{array}[]{c}D^{-1}\\\
\dots\\\ D^{-1}\end{array}\right)\,(1,\dots,1),$ (109)

where $D=\sum_{j=1}^{n}D_{j}$. Moreover, the determinant of $\underline{D}$ is

$\begin{split}{\rm
det}(\underline{D})&=\det(D_{n}^{-1})\,\det(D)\,\prod_{j=1}^{n-1}\det(D_{j})\\\
&=\det(D_{n}^{-1}D)\,\prod_{j=1}^{n-1}\det(D_{j}),\end{split}$ (110)

from which $N_{g}$ and $N_{q}$ are easily determined.
Characteristic vectors
We briefly discuss here a characteristic vector for the lattice
$\underline{{\boldsymbol{\Lambda}}}$ with $n=3$, which is important for the
theta series of the scaling solutions. A sign which frequently occured in
Section 2 is $(-1)^{a+b+c}$. Such signs in a theta series are typically
written in terms of a characteristic vector. We therefore express $a+b+c$ as

$a+b+c={\boldsymbol{K}}.{\boldsymbol{Q}},$ (111)

with

${\boldsymbol{K}}=(P_{3}-P_{2},P_{1}-P_{3}).$ (112)

This is a characteristic vector of $\underline{{\boldsymbol{\Lambda}}}$.
Indeed, we have with $\vec{P}=(P_{1},P_{2},P_{3})$,

$\vec{k}\cdot\vec{P}+\vec{k}^{2}\in 2\mathbb{Z},$ (113)

since $P_{j}$ is a characteristic vector for $\Lambda_{j}$, $j=1,2,3$. We can
decompose with respect to the lattice decomposition
$\underline{{\boldsymbol{\Lambda}}}\oplus\overline{\boldsymbol{\Lambda}}$,

${\boldsymbol{k}}\cdot{\boldsymbol{K}}+{\boldsymbol{k}}^{2}+k\cdot P+k^{2}.$
(114)

Since $P$ is a characteristic vector for $\Lambda$, this shows that
${\boldsymbol{K}}$ is a characteristic vector for ${\boldsymbol{\Lambda}}$.

However, if we express $a+b+c+P.Q$ in terms of vectors in
${\boldsymbol{\Lambda}}=\sum_{j}\Lambda_{j}$, such that
$a+b+c=\vec{K}.\vec{Q}\mod 2$, then
$\vec{K}=(P_{3}-P_{2},P_{1}-P_{3},P_{2}-P_{1})$. Then
$(\vec{K}+\vec{P})\cdot\vec{Q}={\boldsymbol{K}}\cdot{\boldsymbol{Q}}+\vec{P}\cdot\vec{Q}=P\cdot
Q\mod 2$. Moreover, $\vec{P}$ is a characteristic vector of
${\boldsymbol{\Lambda}}$, and

$\vec{P}^{2}=P^{3}+{\boldsymbol{K}}^{3}\mod 4.$ (115)

###  4.2 Partition functions

We consider black hole bound states with three cores. The $j^{\rm th}$ core
carries electric and magnetic charges $Q_{j}$ and $P_{j}$ respectively. It is
natural to work with a mixed ensemble with total magnetic charge $P$ held
fixed. For the present purpose, we shall fix the total electric D2-brane
charge $Q=\sum_{j}Q_{j}\in\Lambda^{*}+P/2$ with $\mu\in\Lambda^{*}/\Lambda$ as
well. We work at the attractor value of the moduli, corresponding to total
charge vector $(P,Q)$. At this point apart from single core black holes, the
only other black holes to survive are the scaling black holes. A natural
question is - for a fixed total charge, how many scaling black holes are there
and what is their contribution to the index?

To this end, we define the generating function $h^{T}_{\mu}(\tau)$ of
numerical total core invariants $\Omega_{T}$ in analogy to the attractor
indices,

$h^{T}_{P,\mu}(\tau)=\sum_{Q_{0}}\bar{\Omega}_{T}(\gamma)\,q^{\hat{Q}_{\bar{0}}}.$
(116)

We can similarly define the partition function of single core indices
$h^{S}_{P,\mu}(\tau)$, with $\Omega_{T}$ replaced by $\Omega_{S}$. The
$\Omega_{T}(\gamma)$ are determined from the refined ones (10) using the
regularization (14). For the 3-core case, this gives (33). If $P$ is
irreducible, i.e. it can not be written as as sum of more than 1 positive
magnetic charge, and the three partition functions agree.

Based on (8), we can express the attractor partition function $h_{P,\mu}$ in
terms of the partition function $h^{T}_{P,\mu}$. We have schematically

$h_{P,\mu}(\tau)=h^{T}_{P,\mu}(\tau)+\sum_{n>1}\sum_{\sum_{j=1}^{n}P_{j}=P\atop\sum_{j=1}^{n}Q_{j}=Q}\frac{g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})}{|{\rm
Aut(\\{\gamma_{j}\\})}|}q^{\mu^{2}/2-\sum_{j}(Q_{j})_{j}^{2}/2}\prod_{j=1}^{n}h^{T}_{P_{j},\mu_{j}}(\tau).$
(117)

Recall that there are no 2-center/core scaling black holes, such that there is
no contribution from $n=2$ on the rhs.

We will proceed by considering the term in (117) with $n=3$. Using the
notation introduced in Section 4.1, we can enumerate the number of three-core
scaling black holes as

$h^{3T}_{\\{P_{j}\\},\mu}(\tau)=\sum_{\mu_{j}\in\Lambda^{*}_{j}/\Lambda_{j},\,\,j=1,2,3,\atop{\mu_{1}+\mu_{2}+\mu_{3}=\mu}}h^{T}_{P_{1},\mu_{1}}(\tau)\,h^{T}_{P_{2},\mu_{2}}(\tau)\,h^{T}_{P_{3},\mu_{3}}(\tau)\,\Psi_{{\boldsymbol{\mu}}}(\tau)\,,$
(118)

with ${\boldsymbol{\mu}}$ as in (106), and where $\Psi_{\boldsymbol{\mu}}$ is
the indefinite theta series

$\Psi_{{\boldsymbol{\mu}}}(\tau)=\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}}g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})\,q^{-{\boldsymbol{Q}}^{2}/2}.$
(119)

With $y=e^{2\pi iz}$, we define the refined series as

$\Psi_{{\boldsymbol{\mu}}}(\tau,z)=(y-y^{-1})^{2}\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}}g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\};y)\,q^{-{\boldsymbol{Q}}^{2}/2},$
(120)

with $g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})$ as in (42) and
$g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\};y)$ as in (27). Note that
$\Psi_{\boldsymbol{\mu}}$ is symmetric as function of $z$,

$\Psi_{{\boldsymbol{\mu}}}(\tau,-z)=\Psi_{{\boldsymbol{\mu}}}(\tau,z).$ (121)

The two functions are related by (14),

$\Psi_{{\boldsymbol{\mu}}}(\tau)=\left(\frac{1}{4\pi
i}\frac{\partial}{\partial
z}\right)^{2}\left.\Psi_{{\boldsymbol{\mu}}}(\tau,z)\right|_{z=0}.$ (122)

The kernel $g_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})$ and therefore
$\Psi_{{\boldsymbol{\mu}}}$ is unchanged under a symplectic transformation
(51), such that $h^{T}_{\\{P_{j}\\},\mu}$ is invariant under spectral flow as
required. The number of terms in the sum over $\mu_{j}$ in (118) is given by
$N_{q}$ (78).

To determine the modular properties of $\Psi_{\boldsymbol{\mu}}$, we consider
first the generating function of $f_{C}$ (44), which enumerates the number of
scaling charge configurations for a given total charge. We define this
function $\Phi_{{\boldsymbol{\mu}}}$ as the following theta series,

$\begin{split}\Phi_{{\boldsymbol{\mu}}}(\tau)&=\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}}f_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})\,q^{-{\boldsymbol{Q}}^{2}/2},\end{split}$
(123)

with $f_{C}(\\{\gamma_{j}\\},\\{c_{j}^{\lambda}\\})$ as in (44).

The following subsections will demonstrate that
$\Phi_{{\boldsymbol{\mu}}}(\tau)$ is a convergent $q$-series, which can be
completed to a function $\widehat{\Phi}_{\boldsymbol{\mu}}$ which transform as
a vector-valued modular form. The transformation properties under the $S$ and
$T$ transformations are

$\begin{split}\widehat{\Phi}_{{\boldsymbol{\mu}}}(-1/\tau,-1/\bar{\tau})&=-\frac{(-i\tau)^{b_{2}}}{\sqrt{|\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}|}}\,e^{\pi
i{\boldsymbol{K}}^{2}/2}\sum_{{\boldsymbol{\nu}}\in\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}}e^{2\pi
i{\boldsymbol{\mu}}.{\boldsymbol{\nu}}}\,\widehat{\Phi}_{{\boldsymbol{\nu}}}(\tau,\bar{\tau}),\\\
\widehat{\Phi}_{{\boldsymbol{\mu}}}(\tau+1,\bar{\tau}+1)&=e^{\pi
i({\boldsymbol{\mu}}+{\boldsymbol{K}}/2)^{2}}\,\widehat{\Phi}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau}).\end{split}$
(124)

The partition function $\widehat{\Psi}_{{\boldsymbol{\mu}}}(\tau)$ can be
obtained by introducing a suitable elliptic variable in
$\widehat{\Phi}_{{\boldsymbol{\mu}}}$ and subsequently differentiating twice
to this variable. As a result, the modular transformations of the completed
function $\widehat{\Phi}_{{\boldsymbol{\mu}}}$ equal those of the completed
$\widehat{\Psi}_{{\boldsymbol{\mu}}}$ except that the weight of
$\widehat{\Psi}_{{\boldsymbol{\mu}}}$ is increased by two compared to
$\widehat{\Phi}_{{\boldsymbol{\mu}}}$. The weight of
$\widehat{\Psi}_{{\boldsymbol{\mu}}}$ is thus $b_{2}+2$. The non-holomorphic
terms are determined in this way in Section 4.5, specifically Eq. (181). The
end result is that $\widehat{\Psi}_{{\boldsymbol{\mu}}}$ transforms as

$\begin{split}\widehat{\Psi}_{{\boldsymbol{\mu}}}(-1/\tau,-1/\bar{\tau})&=\frac{(-i\tau)^{b_{2}+2}}{\sqrt{|\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}|}}\,e^{\pi
i{\boldsymbol{K}}^{2}/2}\sum_{{\boldsymbol{\nu}}\in\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}}e^{2\pi
i{\boldsymbol{\mu}}.{\boldsymbol{\nu}}}\,\widehat{\Psi}_{{\boldsymbol{\nu}}}(\tau,\bar{\tau}),\\\
\widehat{\Psi}_{{\boldsymbol{\mu}}}(\tau+1,\bar{\tau}+1)&=e^{\pi
i({\boldsymbol{\mu}}+{\boldsymbol{K}}/2)^{2}}\,\widehat{\Psi}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau}).\end{split}$
(125)

Therefore, the completion of $h^{3T}_{\\{P_{j}\\},\mu}$ (118),

$\widehat{h}^{3T}_{\\{P_{j}\\},\mu}(\tau,\bar{\tau})=\sum_{\mu_{j}\in\Lambda^{*}_{j}/\Lambda_{j},\,\,j=1,2,3,\atop{\mu_{1}+\mu_{2}+\mu_{3}=\mu}}\widehat{h}^{T}_{P_{1},\mu_{1}}(\tau)\,\widehat{h}^{T}_{P_{2},\mu_{2}}(\tau)\,\widehat{h}^{T}_{P_{3},\mu_{3}}(\tau)\,\widehat{\Psi}_{{\boldsymbol{\mu}}}(\tau)\,,$
(126)

transforms as $\widehat{h}_{P,\mu}$ (65) as we aimed to show. We can
furthermore combine $\widehat{h}_{P,\mu}$ with the theta series
$\Theta_{\mu}$,

$\widehat{\mathcal{Z}}^{3T}_{P}(\tau,C,t)=\sum_{\sum_{j=1}^{3}P_{j}=P}\sum_{\mu\in\Lambda^{*}/\Lambda}\widehat{h}^{3T}_{\\{P_{j}\\},\mu}(\tau,\bar{\tau})\,\Theta_{\mu}(\tau,\bar{\tau},C,B).$
(127)

We can then decompose the attractor partition
$\widehat{\mathcal{Z}}^{\lambda}_{P}$ in terms of the multi-core partition
functions $\widehat{\mathcal{Z}}^{nT}_{P}$,

$\widehat{\mathcal{Z}}^{\lambda}_{P}(\tau,C,t)=\widehat{\mathcal{Z}}^{T}_{P}(\tau,C,t)+\widehat{\mathcal{Z}}^{3T}_{P}(\tau,C,t)+\dots.$
(128)

Since the partition functions transform the same way, this raises the question
which terms are captured by the MSW conformal field theory. As mentioned in
the introduction, it will also be interesting to deduce the non-holomorphic
terms of $\widehat{\mathcal{Z}}^{T}_{P}$ using those determined for
$\widehat{\mathcal{Z}}^{3T}_{P}$ in this paper, and those for
$\widehat{\mathcal{Z}}^{\lambda}_{P}$ in [29].

###  4.3 Convergence

A crucial aspect of $\Phi_{\boldsymbol{\mu}}$ (and $\Psi_{\boldsymbol{\mu}}$)
is whether the sum on the rhs of (123) (and (119)) is convergent. If
${\boldsymbol{Q}}^{2}$ would be negative definite, convergence of these series
would be guaranteed. However this is not the case since the electric charge
lattice has signature $(2,2b_{2}-2)$, i.e. has 2 positive directions.

To prove the convergence, we first introduce a theta series
$\Theta_{\mu}[\mathcal{K}](\tau)$ with kernel $\mathcal{K}$ for a generic
indefinite theta lattice $L$ and $\mu\in L^{*}$,

$\displaystyle\Theta_{\mu}[\mathcal{K}](\tau;L)$ $\displaystyle=\sum_{x\in
L+\mu}\mathcal{K}(x)\,q^{-B(x)/2}\,,$ (129)

with integral quadratic form $B$. If $L$ is negative definite, we also use

$\displaystyle\theta_{\mu}(\tau;L)$ $\displaystyle=\Theta_{\mu}[1](\tau;L).$
(130)

For an indefinite lattice the kernel
$\mathcal{K}(x)=\mathcal{K}(x,\mathcal{V})$ depends on a collection

$\mathcal{V}=\\{V_{1},V_{2},\dots,V_{N}\\},$

of positive vectors. For signature $(2,2b_{2}-2)$,
$\mathcal{K}(x,\mathcal{V})$ can be expressed as [36, 28, 73]

$\mathcal{K}(x,\mathcal{V})=\frac{1}{4}\left(w(\mathcal{V})+\sum_{j=1}^{N}\mathop{\mathrm{sgn}}(B(x,V_{j}))\mathop{\mathrm{sgn}}(B(x,V_{j+1}))\right)\,,$
(131)

where for any strictly positive vector $v\in L$, $v^{2}>0$,

$w(\mathcal{V})=-\sum_{j=1}^{N}\mathop{\mathrm{sgn}}(B(v,V_{j}))\mathop{\mathrm{sgn}}(B(v,V_{j+1})),$
(132)

which is independent of the choice of positive vector $v$ [73]. There are
various sufficient conditions for convergence put forward in the literature
[28, 36, 38, 39, 73]. We will consider here the following $N$-gon conditions
put forward in [28, 36, 73], which read

$\begin{split}&B(V_{j},V_{j})>0,\\\
&B(V_{j},V_{j})\,B(V_{j+1},V_{j+1})-B(V_{j},V_{j+1})^{2}>0,\\\
&B(V_{j},V_{j})\,B(V_{j-1},V_{j+1})-B(V_{j},V_{j-1})\,B(V_{j},V_{j+1})<0.\end{split}$
(133)

Now let us return to the sum (123) at hand. It comprises of six individual
sums, which are each of the form

$\displaystyle\begin{split}&\sum_{Q_{i}\in\mu_{i}+\Lambda_{i}+P_{i}/2\atop{Q_{1}+Q_{2}+Q_{3}=\mu+P/2}}F_{\ell}(a,b,c)\,(-1)^{a+b+c}\,q^{Q^{2}/2-\sum_{i}(Q_{i})_{i}^{2}/2}=\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{{\boldsymbol{\mu}}}}F_{\ell}(a,b,c)\,(-1)^{a+b+c}\,q^{-{\boldsymbol{Q}}^{2}/2},\end{split}$
(134)

with $\ell=1,\dots,6$.

The simplification $F_{\rm total}(a,b,c)$ for $\sum_{j=1}^{6}F_{j}(a,b,c)$ put
forward in (43) is precisely of the form (131) with $N=3$. To present the
vectors, we define

$\begin{split}&C_{a}=(-P_{2},P_{1},0),\\\ &C_{b}=(0,-P_{3},P_{2}),\\\
&C_{c}=(P_{3},0,-P_{1}),\end{split}$ (135)

such that $C_{a}.{\boldsymbol{Q}}=a$, $C_{b}.{\boldsymbol{Q}}=b$ and
$C_{c}.{\boldsymbol{Q}}=c$, with $a,b$ and $c$ as in (67). The $V_{j}$ are
then identified with $C_{j}\in\underline{{\boldsymbol{\Lambda}}}$ with the
$C_{j}$ given by,

$\begin{split}&C_{1}=C_{a}+C_{b}-C_{c}=(-P_{2}-P_{3},\,P_{1}-P_{3},P_{1}+P_{2}),\\\
&C_{2}=C_{a}-C_{b}+C_{c}=(-P_{2}+P_{3},\,P_{1}+P_{3},\,-P_{1}-P_{2}),\\\
&C_{3}=-C_{a}+C_{b}+C_{c}=(P_{2}+P_{3},\,-P_{1}-P_{3},\,-P_{1}+P_{2})\,.\end{split}$
(136)

If we assume that $P_{j}$ is an ample divisor for each $j\in 1,2,3$, triple
intersections $P_{i}P_{j}P_{k}>0$ for all $i,j,k\in\\{1,2,3\\}$. The
conditions for convergence (133) are then satisfied.

It is also useful to consider the convergence for the kernel due to a single
permutation separately, since the different permuations are weighted by a
different factor in $g_{C}$. The vectors $V_{j}$ for the kernel $F^{*}(123)$
(40) can be chosen as $C_{j}^{(123)}$,

$\begin{split}C_{1}^{(123)}&=C_{a}-C_{c}=(C_{1}-C_{2})/2,\\\
C_{2}^{(123)}&=C_{b}-C_{c}=(C_{1}-C_{3})/2,\\\
C_{3}^{(123)}&=C_{c}-C_{a}-C_{b}=-C_{1}.\end{split}$ (137)

Again one may verify that with the assumption $P_{i}P_{j}P_{k}>0$ for all
$i,j,k\in\\{1,2,3\\}$, these vectors satisfy the conditions for convergence
(133).

###  4.4 Modular completion of $\Phi_{\boldsymbol{\mu}}$

Having discussed the convergence of $\Phi_{\boldsymbol{\mu}}$, we proceed in
this Section to discuss its modularity. Since $\Phi_{\boldsymbol{\mu}}$ is a
sum over a subset of an indefinite lattice, the function is not modular in the
classical sense. Our task is to determine a modular completion
$\widehat{\Phi}_{\boldsymbol{\mu}}$, which differes from
$\Phi_{\boldsymbol{\mu}}$ by subleading non-holomorphic terms, and which does
transform as a modular form. Essentially, products of sgn-functions are
replaced by a generalized error function [36]. Following this approach, we
will demonstrate that the difference between
$\widehat{\Phi}_{\boldsymbol{\mu}}$ and $\Phi_{\boldsymbol{\mu}}$, is given by
iterated integrals of modular forms.

Such non-holomorphic contributions have appeared in similar contexts. In
specific cases, the non-holomorphic contributions are derived from different
physical points of view, for example the continuum of multi-particle states in
$\mathbb{R}^{4}$ [57], or the quantum field theory on the world volume of the
D-branes [44, 47, 48], or the perspective of D3-instantons in the
hypermultiplet moduli space [18].

To this end let us consider $F_{\rm total}(a,b,c)$ in (43). Under modular
completion, one adds certain extra terms to
$\mathop{\mathrm{sgn}}(V_{1},x)\mathop{\mathrm{sgn}}(V_{2},x)$, therefore
replacing $\mathop{\mathrm{sgn}}(V_{1},x)\mathop{\mathrm{sgn}}(V_{2},x)$ with
the double error function $E_{2}(\alpha,u_{1},u_{2})$:

$\mathop{\mathrm{sgn}}(V_{1},x)\mathop{\mathrm{sgn}}(V_{2},x)\to
E_{2}(\alpha,\sqrt{2\tau_{2}}\,{\boldsymbol{u}}),$ (138)

with [36]

$E_{2}(\alpha;{\boldsymbol{u}})=\int_{\mathbb{R}^{2}}e^{-\pi(u_{1}-u_{1}^{\prime})^{2}-\pi(u_{2}-u_{2}^{\prime})^{2}}\,\mathop{\mathrm{sgn}}(u_{2}^{\prime})\,\mathop{\mathrm{sgn}}(u_{1}^{\prime}+\alpha\,u_{2}^{\prime})\,du_{1}^{\prime}\,du_{2}^{\prime},$
(139)

whose arguments are given in terms of $V_{1}$, $V_{2}$ and $x$ by

$\displaystyle\alpha=\alpha(V_{1},V_{2})=\frac{(V_{1},V_{2})}{\sqrt{V_{1}^{2}\,V_{2}^{2}-(V_{1},V_{2})^{2}}},$
(140)
$\displaystyle{\boldsymbol{u}}={\boldsymbol{u}}(V_{1},V_{2};x)=(u_{1}(V_{1},V_{2};x),u_{2}(V_{1},V_{2};x)),$

with

$\displaystyle u_{1}(V_{1},V_{2};x)=\frac{(V_{1\perp 2},x)}{\sqrt{(V_{1\perp
2},V_{1\perp 2})}}\,,$ (141) $\displaystyle
u_{2}(V_{1},V_{2};x)=\frac{(V_{2},x)}{\sqrt{(V_{2},V_{2})}}\,$ (142)

and $V_{1\perp 2}$ the component of $V_{1}$ orthogonal to $V_{2}$,

$V_{1\perp 2}=V_{1}-\frac{(V_{1},V_{2})}{(V_{2},V_{2})}\,V_{2}.$ (143)

To stress the dependance of $E_{2}$ on the vectors $V_{1}$, $V_{2}$ and $x$,
we will also use $E_{2}$ with alternative arguments,

$E_{2}(\alpha,{\boldsymbol{u}})\equiv E_{2}(V_{1},V_{2};x),$ (144)

with the identifications as in (140).

In addition to (139), another (equivalent) expression for $E_{2}$ is in terms
of Eichler integrals. To this end, we first define the Eichler integrals
$M_{1}$,

$M_{1}(u)=\left\\{\begin{array}[]{cc}\frac{iu}{\sqrt{2\tau_{2}}}q^{\frac{u^{2}}{4\tau_{2}}}\int_{-\bar{\tau}}^{i\infty}\frac{e^{\frac{i\pi
u^{2}w}{2\tau_{2}}}}{\sqrt{-i(w+\tau)}}dw,&\qquad u\neq 0,\\\ 0,&\qquad
u=0.\end{array}\right.\\\ $ (145)

and the (iterated) Eichler integral $M_{2}$ and $m_{2}$, for $u_{1}\neq 0$,
and $u_{2}-\alpha u_{1}\neq 0$,

$\begin{split}m_{2}(u_{1},u_{2})&=\left\\{\begin{array}[]{cc}\frac{u_{1}u_{2}}{2\tau_{2}}q^{\frac{u_{1}^{2}}{4\tau_{2}}+\frac{u_{2}^{2}}{4\tau_{2}}}\int_{-\bar{\tau}}^{i\infty}dw_{2}\int_{w_{2}}^{i\infty}dw_{1}~{}\frac{e^{\frac{\pi
iu_{1}^{2}w_{1}}{2\tau_{2}}+\frac{\pi
iu_{2}^{2}w_{2}}{2\tau_{2}}}}{\sqrt{-(w_{1}+\tau)(w_{2}+\tau)}},&\qquad
u_{1}\neq 0\\\ 0,&\qquad u_{1}=0.\end{array}\right.\,\\\
M_{2}(\alpha;u_{1},u_{2})&=\left\\{\begin{array}[]{rl}&-m_{2}(u_{1},u_{2})-m_{2}\left(\frac{u_{2}-\alpha
u_{1}}{\sqrt{1+\alpha^{2}}},\frac{u_{1}+\alpha
u_{2}}{\sqrt{1+\alpha^{2}}}\right)\,\quad u_{1}\neq 0,u_{2}-\alpha u_{1}\neq
0\,,\\\ &-m_{2}\left(\frac{u_{2}-\alpha
u_{1}}{\sqrt{1+\alpha^{2}}},\frac{u_{1}+\alpha
u_{2}}{\sqrt{1+\alpha^{2}}}\right)\,\hskip 76.82243ptu_{1}=0,u_{2}\neq 0\,,\\\
&-m_{2}(u_{1},u_{2})\,\hskip 128.0374ptu_{1}\neq 0,u_{2}-\alpha u_{1}=0\,,\\\
&\frac{2}{\pi}\arctan\alpha\,\hskip 139.4185ptu_{1}=u_{2}=0.\end{array}\
\right.\\\ \end{split}$ (146)

With ${\boldsymbol{u}}=(u_{1},u_{2})$ as before, the double error function
$E_{2}$ is then defined as a linear combination of $M_{1}$ and $M_{2}$ [36,
56]

$\begin{split}E_{2}(\alpha;{\boldsymbol{u}})&=\mathop{\mathrm{sgn}}(u_{2})\mathop{\mathrm{sgn}}(u_{1}+\alpha
u_{2})+\mathop{\mathrm{sgn}}(u_{1})M_{1}(u_{2})\\\
&\quad+\mathop{\mathrm{sgn}}(u_{2}-\alpha
u_{1})\,M_{1}\\!\left(\frac{u_{1}+\alpha
u_{2}}{\sqrt{1+\alpha^{2}}}\right)+M_{2}(\alpha;u_{1},u_{2})\,.\end{split}$
(147)

See [36] for other representations of $E_{2}$.

Thus $E_{2}$ consists of the original
$\mathop{\mathrm{sgn}}(V_{1},x)\mathop{\mathrm{sgn}}(V_{2},x)$ plus four more
terms. Noting that

$\displaystyle{}\frac{u_{2}(V_{2},x)-\alpha
u_{1}(V_{1},V_{2},x)}{\sqrt{1+\alpha^{2}}}$
$\displaystyle=u_{1}(V_{2},V_{1},x)\,,$
$\displaystyle\frac{u_{1}(V_{1},V_{2},x)+\alpha
u_{2}(V_{2},x)}{\sqrt{1+\alpha^{2}}}$ $\displaystyle=u_{2}(V_{1},x)\,,$ (148)

we can write

$\displaystyle{}E_{2}(\alpha;{\boldsymbol{u}})$
$\displaystyle=\mathop{\mathrm{sgn}}{}(V_{1}.x)\,\mathop{\mathrm{sgn}}{}(V_{2}.x)+\mathop{\mathrm{sgn}}{}(u_{1}(V_{1},V_{2},x))\,M_{1}(u_{2}(V_{2},x))$
$\displaystyle+\mathop{\mathrm{sgn}}{}(u_{1}(V_{2},V_{1},x))\,M_{1}(u_{2}(V_{1},x))$
(149)
$\displaystyle{}-m_{2}(u_{1}(V_{1},V_{2},x),\,u_{2}(V_{2},x))-m_{2}(u_{1}(V_{2},V_{1},x),\,u_{2}(V_{1},x))\,.$

$E_{2}$ satisfies an identity similar to (39). This reads

$E_{2}(V_{1},V_{1}+V_{2};x)+E_{2}(V_{2},V_{1}+V_{2};x)-E_{2}(V_{1},V_{2};x)=1$
(150)

and is valid for any choice of the arguments such that the corresponding
$\alpha$, $u_{1}$ and $u_{2}$’s are in $\mathbb{R}$.

The sum at hand (43) has the form

$\displaystyle\mathop{\mathrm{sgn}}{}(C_{1}.x)\,\mathop{\mathrm{sgn}}{}(C_{2}.x)\,+\,\mathop{\mathrm{sgn}}{}(C_{2}.x)\,\mathop{\mathrm{sgn}}{}(C_{3}.x)\,+\,\mathop{\mathrm{sgn}}{}(C_{3}.x)\,\mathop{\mathrm{sgn}}{}(C_{1}.x)\,,$
(151)

with $C_{1},C_{2},C_{3}$ given in (136). The modular completion
$\widehat{\Phi}_{\boldsymbol{\mu}}$ of $\Phi_{\boldsymbol{\mu}}$ (123) follows
by the adding to the coefficient $f_{C}$ (44) the following terms

$\frac{1}{4}\sum_{\ell=1,2,3}\left[E_{2}(C_{\ell},C_{\ell+1};\sqrt{2\tau_{2}}x)-\mathop{\mathrm{sgn}}{}(C_{\ell}.x)\,\mathop{\mathrm{sgn}}{}(C_{\ell+1}.x)-A_{\ell}\,\delta_{(C_{\ell}.x)}\,\delta_{(C_{\ell+1}.x)}\right].$
(152)

This essentially amounts to replacing $f_{C}$ by a linear combination of
$E_{2}$’s. Since the latter satisfies Vignéras equation, modular
transformation properties are ensured [55].

Our first aim is to determine the value of $A_{\ell}$ such that the completion
is subleading, i.e. it vanishes in the limit that ${\rm
Im}(\tau)=\tau_{2}\to\infty$. This follows from realizing that the difference
$E_{2}(C_{\ell},C_{\ell+1};\sqrt{2\tau_{2}}x)-\mathop{\mathrm{sgn}}{}(C_{\ell}.x)\,\mathop{\mathrm{sgn}}{}(C_{\ell+1}.x)$
vanishes in this limit except if $(C_{\ell}.x)=(C_{\ell+1}.x)=0$, when it
equals $\frac{2}{\pi}\arctan(\alpha_{\ell})$,
$\alpha_{\ell}=\alpha(C_{\ell},C_{\ell+1})$ (140). Requiring that (152)
vanish, we thus arrive at

$A_{\ell}=\frac{2}{\pi}\arctan(\alpha_{\ell}).$ (153)

Surprisingly this implies that $A_{\ell}$ can be irrational, as we will see in
the explicit case studies in Section 6. This of course obstructs an
interpretation as a “counting” function for the coefficients to which
$A_{\ell}$ contribute. On the other hand, since the function
$\Phi_{\boldsymbol{\mu}}$ is not a proper physical partition function summing
over a Hilbert space, we are not very concerned about this.

To proceed with determining the completion, we rearrange the terms and write
(152) as

$\begin{split}&\frac{1}{4}\sum_{\ell=1,2,3}\left[\mathop{\mathrm{sgn}}{}(u_{1}(C_{\ell+1},C_{\ell},x))+\mathop{\mathrm{sgn}}{}(u_{1}(C_{\ell-1},C_{\ell},x))\right]M_{1}(\sqrt{2\tau_{2}}\,u_{2}(C_{\ell},x))\\\
&-m_{2}(\sqrt{2\tau_{2}}u_{1}(C_{\ell+1},C_{\ell},x),\sqrt{2\tau_{2}}u_{2}(C_{\ell},x))-m_{2}(\sqrt{2\tau_{2}}u_{1}(C_{\ell-1},C_{\ell},x),\sqrt{2\tau_{2}}u_{2}(C_{\ell},x)).\end{split}$
(154)

We determine now the $A_{\ell}$ by requiring that this expression vanishes in
the limit $y\to\infty$.

Using these expressions, we thus naturally separate the holomorphic part
$\Phi_{\boldsymbol{\mu}}$ from the completion
$\widehat{\Phi}_{\boldsymbol{\mu}}$,

$\widehat{\Phi}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau})=\Phi_{\boldsymbol{\mu}}(\tau)+R^{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau}),$
(155)

with the non-holomorphic completion $R^{\Phi}_{\boldsymbol{\mu}}$ defined by

$\begin{split}R^{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})=&\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{\mu}}\,\,\sum_{\ell=1,2,3}\\\
&\left[\left[\mathop{\mathrm{sgn}}{}(u_{1}(C_{\ell+1},C_{\ell},x))+\mathop{\mathrm{sgn}}{}(u_{1}(C_{\ell-1},C_{\ell},x))\right]M_{1}(\sqrt{2\tau_{2}}u_{2}(C_{\ell},x))\right.\\\
&\left.-m_{2}(\sqrt{2\tau_{2}}u_{1}(C_{\ell+1},C_{\ell},x),\sqrt{2\tau_{2}}u_{2}(C_{\ell},x))\right.\\\
&\left.-m_{2}(\sqrt{2\tau_{2}}u_{1}(C_{\ell-1},C_{\ell},x),\sqrt{2\tau_{2}}u_{2}(C_{\ell},x))\right]\,\\\
&\times(-1)^{{\boldsymbol{K}}.{\boldsymbol{Q}}}q^{-{\boldsymbol{Q}}^{2}/2}.\end{split}$
(156)

Our next aim is to write the non-holomorphic part as an (iterated) integral
over modular forms. This makes the modular properties of the holomorphic
$q$-series manifest, since the modular properties of integrals of modular
forms are readily determined. Moreover, it is straightforward to determine the
non-holomorphic anomaly.

To determine this form of the non-holomorphic part, we write
$R^{\Phi}_{\boldsymbol{\mu}}$ as

$\displaystyle R^{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})$
$\displaystyle=\sum_{\ell=1,2,3}\left[R_{{\boldsymbol{\mu}},1,\ell}(\tau,\bar{\tau})+R_{{\boldsymbol{\mu}},2,(\ell-1,\ell)}(\tau,\bar{\tau})+R_{{\boldsymbol{\mu}},2,(\ell+1,\ell)}(\tau,\bar{\tau})\right]$
(157)

and $R_{{\boldsymbol{\mu}},1,\ell},R_{{\boldsymbol{\mu}},2,(k,\ell)}$ are
defined as

$\displaystyle{}R_{{\boldsymbol{\mu}},1,\ell}(\tau,\bar{\tau})$
$\displaystyle=\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{\mu}}\,\,\left[\mathop{\mathrm{sgn}}{}(u_{1,(\ell+1,\ell)}+\mathop{\mathrm{sgn}}{}(u_{1,(\ell-1,\ell)})\right]M_{1}(\sqrt{2y}u_{2,\ell})(-1)^{{\boldsymbol{K}}.{\boldsymbol{Q}}}q^{-{\boldsymbol{Q}}^{2}/2}\,,$
$\displaystyle R_{{\boldsymbol{\mu}},2,(k,\ell)}(\tau,\bar{\tau})$
$\displaystyle=-\sum_{{\boldsymbol{Q}}\in\underline{\boldsymbol{\Lambda}}^{*}_{\mu}}\,\,m_{2}(\sqrt{2\tau_{2}}u_{1,(k,l)},\,\sqrt{2\tau_{2}}u_{2,\ell})\,(-1)^{{\boldsymbol{K}}.{\boldsymbol{Q}}}q^{-{\boldsymbol{Q}}^{2}/2}\,,$
(158)

where we have used the abbreviations
$u_{1,(k,\ell)}=u_{1}(C_{k},C_{\ell},x),\,u_{2,\ell}=u_{2}(C_{\ell},x)$.

To evaluate the sums (158), it is useful to split ${\boldsymbol{Q}}^{2}$ as
follows

$\displaystyle{\boldsymbol{Q}}^{2}$
$\displaystyle=u_{1,(k,\ell)}^{2}+u_{2,\ell}^{2}+({\boldsymbol{Q}}^{2}-u_{1,(k,\ell)}^{2}-u_{2,\ell}^{2})\,,$
(159)

$u_{1,(k,\ell)}^{2},\,u_{2,\ell}^{2}$ and
$({\boldsymbol{Q}}^{2}-u_{1,(k,\ell)}^{2}-u_{2,\ell}^{2})$ are naturally
associated with quadratic forms and lattices as described in Table 1.

term | form | dual lattice | signature
---|---|---|---
${\boldsymbol{Q}}^{2}$ | $\underline{D}$ | $\underline{{\boldsymbol{\Lambda}}}^{*}_{\boldsymbol{\mu}}$ | $(2,2b_{2}-2)$
$u_{2,\ell}^{2}$ | $\underline{D}_{\ell}=|C_{\ell}|^{2}$ | $(L_{\ell})^{*}_{\boldsymbol{\mu}}\subset\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$ | (1,0)
${\boldsymbol{Q}}^{2}-u_{2,\ell}^{2}$ | $\underline{D}_{\perp\ell}$ | $(L_{\ell}^{\perp})^{*}_{\boldsymbol{\mu}}\subset\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$ | $(1,2b_{2}-2)$
$u_{1,(k,l)}^{2}$ | $\underline{D}_{(k,l)}=|C_{k\perp\ell}|^{2}$ | $(L_{k\ell})^{*}_{\boldsymbol{\mu}}\subset(L_{\ell}^{\perp})^{*}_{\boldsymbol{\mu}}$ | (1,0)
${\boldsymbol{Q}}^{2}-u_{1,(k,l)}^{2}-u_{2,\ell}^{2}$ | $\underline{D}^{\perp}_{(k\ell)}$ | $(L_{k\ell}^{\perp})^{*}_{\boldsymbol{\mu}}\subset(L_{\ell}^{\perp})^{*}_{\boldsymbol{\mu}}$ | $(0,2b_{2}-2)$
Table 1: Lattices and quadratic forms associated to the splitting in Equation
(159).

We let $L_{\ell}$ be the 1-dimensional lattice spanned by $C_{\ell}$. The
quadratic form is a number in this case, $D_{\ell}=|C_{\ell}|^{2}$. We denote
the dual lattice with quadratic form $|C_{\ell}|^{-2}$ by $(L_{\ell})^{*}$.
The projection of ${\boldsymbol{\mu}}$ to $(L_{\ell})^{*}$ is
$\mu_{\ell}=({\boldsymbol{\mu}}.C_{\ell})\,C_{\ell}\in\mathbb{Z}^{2b_{2}}$. To
express a generic vector in
$\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$ as an element of
$(L_{\ell})^{*}\oplus(L_{\ell}^{\perp})^{*}$, we introduce glue vectors
$\rho$. We denote by $(L_{\ell})^{*}_{\boldsymbol{\mu}}$ the set of vectors
$\mu_{\ell}\mod L_{\ell}\in(L_{\ell})^{*}$, and by
$(L_{\ell})^{*}_{{\boldsymbol{\mu}}+\rho}$ the set of vectors
$\mu_{\ell}+\rho\mod L_{\ell}\in(L_{\ell})^{*}$. We introduce similar notation
for $L_{\ell}^{\perp}$. Using this notation, the direct sum
$(L_{\ell})^{*}_{\boldsymbol{\mu}}\oplus(L_{\ell}^{\perp})^{*}_{\boldsymbol{\mu}}$
is a subset of the lattice
$\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$,
$(L_{\ell})^{*}_{\boldsymbol{\mu}}\oplus(L_{\ell}^{\perp})^{*}_{\boldsymbol{\mu}}\subset\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$.

A generic element ${\boldsymbol{k}}$ of
$\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}$ can be written as
a sum,
${\boldsymbol{k}}={\boldsymbol{l}}_{\ell}+{\boldsymbol{l}}_{\ell}^{\perp}\in(L_{\ell})^{*}_{{\boldsymbol{\mu}},\rho}\oplus(L_{\ell}^{\perp})^{*}_{{\boldsymbol{\mu}},\rho}$
for some $\rho$, where the projection ${\boldsymbol{l}}_{\ell}$ to
$(L_{\ell}^{\perp})^{*}$ vanishes, and similarly for the projection of
${\boldsymbol{l}}_{\ell}^{\perp}$ to $(L_{\ell})^{*}$. We have a similar
decomposition of vectors in $(L_{\ell})^{\perp}$ with respect to the
decomposition $(L_{k\ell})^{*}\oplus(L_{k\ell}^{\perp})^{*}$. The
representatives of minimal length in
$(L_{\ell})^{*}\oplus(L_{\ell}^{\perp})^{*}$ appearing in such splits are
called “glue vectors”. We use glue vectors $\rho$ for the splitting
$(L_{\ell})^{*}_{{\boldsymbol{\mu}},\rho}\oplus(L_{\ell}^{\perp})^{*}_{{\boldsymbol{\mu}},\rho}$
and $\nu$ for the splitting $(L_{k\ell})^{*}\oplus(L_{k\ell}^{\perp})^{*}$. We
then have,

$\begin{split}&\underline{{\boldsymbol{\Lambda}}}_{{\boldsymbol{\mu}}}^{*}=\sum_{\rho}(L_{\ell})^{*}_{{\boldsymbol{\mu}}+\rho}\oplus(L_{\ell}^{\perp})^{*}_{{\boldsymbol{\mu}}+\rho},\\\
&(L^{\perp}_{\ell})^{*}_{\rho}=\sum_{\nu}\subset(L_{k})^{*}_{\rho+\nu}\oplus(L_{k\ell}^{\perp})^{*}_{\rho+\nu}.\end{split}$
(160)

Now let us evaluate the sum $R_{{\boldsymbol{\mu}},1}$ and
$R_{{\boldsymbol{\mu}},2}$ (158). The embedding of $L_{\ell}$ in
$\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}\otimes\mathbb{Q}$ is
spanned by $k\,C_{\ell}$ with $k\in\mathbb{Z}+\rho$ where $\rho\in\mathbb{Q}$
is (the projection of) the glue vector. Then, $u_{2,\ell}=|C_{\ell}|k$, such
that $R_{{\boldsymbol{\mu}},1,\ell}$ reads

$\displaystyle R_{{\boldsymbol{\mu}},1,\ell}(\tau,\bar{\tau})$
$\displaystyle=\sum_{\rho}\sum_{{\boldsymbol{l}}_{\ell}^{\perp}\in(L_{\ell}^{\perp})^{*}_{{\boldsymbol{\mu}}+\rho}\atop
k\in\mathbb{Z}+{\boldsymbol{\mu}}+\rho}(\mathop{\mathrm{sgn}}(u_{1,(\ell-1,\ell)})+\mathop{\mathrm{sgn}}(u_{1,(\ell+1,\ell)}))$
(161)
$\displaystyle\qquad\times(-1)^{{\boldsymbol{K}}.{\boldsymbol{l}}_{\ell}^{\perp}+{\boldsymbol{K}}.C_{\ell}\,k}\,M_{1}\\!\left(\sqrt{2\tau_{2}}\,|C_{\ell}|\,k\right)q^{-({\boldsymbol{l}}_{\ell}^{\perp})^{2}/2-C_{\ell}^{2}\,k^{2}/2}.$

To write this more compactly, we define for a generic lattice $L$ of signature
$(1,\dim(L)-1)$ and characteristic vector $K$,

$\displaystyle\Theta_{\alpha}(\tau;L,\\{V,V^{\prime}\\})$
$\displaystyle:=\sum_{x\in
L+\alpha}\left(\mathop{\mathrm{sgn}}(V,x)-\mathop{\mathrm{sgn}}(V^{\prime},x)\right)(-1)^{K.x}q^{-x^{2}/2}\,.$
(162)

Convergence of $\Theta_{\alpha}$ requires [34]

$\displaystyle V^{2}$
$\displaystyle>0\,,\,(V^{\prime})^{2}>0\,,\,(V,V^{\prime})>0\,,$ (163)

Moreover, we define the unary theta series $\Upsilon_{\alpha}$,

$\Upsilon_{\alpha}(\tau,M,N)=\sum_{x\in\mathbb{Z}+\alpha}x\,(-1)^{Nx}\,e^{\pi
i\tau Mx^{2}}.$ (164)

For $\sigma\in\bar{\mathbb{H}}$, we introduce the period integral

$\begin{split}R_{\alpha}(\tau,\sigma;M,N)&:=i\sum_{x\in\mathbb{Z}+\alpha}(-1)^{Nx}x\int_{-\sigma}^{i\infty}dw\,\frac{e^{\pi
iwMx^{2}}}{\sqrt{-i(w+\tau)}}\,\\\
&=i\int_{-\sigma}^{i\infty}dw\,\frac{\Upsilon_{\alpha}(w,M,N)}{\sqrt{-i(w+\tau)}}\,.\end{split}$
(165)

The non-holomorphic modular completion $\widehat{\Theta}_{\alpha}$ of
$\Theta_{\alpha}$ is expressed in terms of $R_{\alpha}$ as

$\begin{split}\widehat{\Theta}_{\alpha}(\tau,\bar{\tau};L,\\{V,V^{\prime}\\})&=\Theta_{\alpha}(\tau;L,V,V^{\prime}\\})\\\
&\quad+\sum_{\nu}\theta_{\alpha+\nu}(\tau;L^{\perp}_{V})\,R_{\alpha+\nu}(\tau,\bar{\tau};V^{2},K_{L}.V)\\\
&\quad-\sum_{\nu^{\prime}}\theta_{\alpha+\nu^{\prime}}(\tau;L^{\perp}_{V^{\prime}})\,R_{\alpha+\nu^{\prime}}(\tau,\bar{\tau};(V^{\prime})^{2},K_{L}.V^{\prime}),\end{split}$
(166)

with $\theta_{\alpha}$ as in (130).

Returning to $R_{{\boldsymbol{\mu}},1,\ell}$, we can now write this as

$R_{{\boldsymbol{\mu}},1,\ell}(\tau,\bar{\tau})=\sum_{\rho}\Theta_{{\boldsymbol{\mu}}+\rho}(\tau,L_{\ell}^{\perp},\\{C_{\ell-1},\,C_{\ell+1}\\})\,R_{{\boldsymbol{\mu}}+\rho}(\tau,\bar{\tau};C_{\ell}^{2},{\boldsymbol{K}}.C_{\ell}).$
(167)

The relations for convergence of $\Theta_{{\boldsymbol{\mu}}+\rho}$ are indeed
satisfied for the vectors given in (136). We will continue to demonstrate that
these are captured by the term $R_{{\boldsymbol{\mu}},2}$ in (157). We have

$\begin{split}R_{{\boldsymbol{\mu}},2,(k,\ell)}(\tau,\bar{\tau})&=-\sum_{{\boldsymbol{Q}}\in\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}}m_{2}(\sqrt{2\tau_{2}}\,u_{1}(C_{k},C_{\ell},{\boldsymbol{Q}}),\sqrt{2\tau_{2}}\,u_{2}(C_{\ell},{\boldsymbol{Q}}))\,(-1)^{{\boldsymbol{K}}\cdot{\boldsymbol{Q}}}\,q^{-{\boldsymbol{Q}}^{2}/2}\\\
&=-\sum_{{\boldsymbol{Q}}\in\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}}u_{1}(C_{k},C_{\ell},{\boldsymbol{Q}})u_{2}(C_{\ell},{\boldsymbol{Q}})\,(-1)^{{\boldsymbol{K}}\cdot{\boldsymbol{Q}}}\,q^{(u_{1}(C_{k},C_{\ell},{\boldsymbol{Q}})^{2}+u_{2}(C_{\ell},{\boldsymbol{Q}})^{2}-{\boldsymbol{Q}}^{2})/2}\\\
&\quad\times\int_{-\bar{\tau}}^{i\infty}dw_{2}\int_{w_{2}}^{i\infty}dw_{1}\frac{e^{\pi
i(w_{1}u_{1}(C_{k},C_{\ell},{\boldsymbol{Q}})^{2}+w_{2}u_{2}(C_{\ell},{\boldsymbol{Q}})^{2})}}{\sqrt{-(w_{1}+\tau)(w_{2}+\tau)}}.\end{split}$
(168)

Using the splits of the lattices and $\Upsilon_{\alpha}$ (164), we can express
this as

$\begin{split}&R_{{\boldsymbol{\mu}},2,(k,\ell)}(\tau,\bar{\tau})=-\sum_{\rho}\theta_{{\boldsymbol{\mu}}+\rho+\nu}(\tau;L_{k\ell}^{\perp})\\\
&\quad\times\,\int_{-\bar{\tau}}^{i\infty}dw_{2}\int_{w_{2}}^{i\infty}dw_{1}\frac{\Upsilon_{{\boldsymbol{\mu}}+\rho}(w_{2};C_{\ell}^{2},{\boldsymbol{K}}\cdot
C_{\ell})\,\Upsilon_{{\boldsymbol{\mu}}+\rho+\nu}(w_{1};C_{k\perp\ell}^{2},{\boldsymbol{K}}\cdot
C_{k\perp\ell})}{\sqrt{-(w_{1}+\tau)(w_{2}+\tau)}}.\end{split}$ (169)

Next, we aim to combine the sums in (156). We recognize the two $\Upsilon$’s
in the integrand in (169). The first, $\Upsilon_{{\boldsymbol{\mu}}+\rho}$,
appears in the integrand of $R_{{\boldsymbol{\mu}}+\rho}$ on the rhs of (167),
whereas the second $\Upsilon_{{\boldsymbol{\mu}}+\rho+\nu}$ matches with one
of the non-holomorphic terms on the rhs of (166). More precisely and
concisely, we arrive at

$R^{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})=\sum_{\ell=1,2,3}i\int_{-\bar{\tau}}^{i\infty}dw\frac{\sum_{\rho}\widehat{\Theta}_{{\boldsymbol{\mu}}+\rho}(\tau,-w;L^{\perp}_{\ell},\\{C_{\ell-1}C_{\ell+1}\\})\,\Upsilon_{{\boldsymbol{\mu}}+\rho}(w;C_{\ell}^{2},{\boldsymbol{K}}\cdot
C_{\ell})}{\sqrt{-i(w+\tau)}}.$ (170)

We thus have succeeded to express $R^{\Phi}_{\boldsymbol{\mu}}$ in (155) as an
iterated integral of modular forms.

Note that for $\ell=1$, there is a symmetry exchanging $P_{1}\leftrightarrow
P_{2}$, and there are similar symmetries for $\ell=2$ and $\ell=3$. The form
(170) makes the determination of the anti-holomorphic derivative of
$\widehat{\Phi}_{\boldsymbol{\mu}}$ immediate. We have,

$\frac{\partial\widehat{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})}{\partial\bar{\tau}}=\sum_{\ell=1,2,3}\sum_{\rho}\frac{\widehat{\Theta}_{{\boldsymbol{\mu}}+\rho}(\tau,\bar{\tau};L^{\perp}_{\ell},\\{C_{\ell-1},C_{\ell+1}\\})\,\Upsilon_{{\boldsymbol{\mu}}+\rho}(-\bar{\tau};C_{\ell}^{2},{\boldsymbol{K}}\cdot
C_{\ell})}{\sqrt{2\tau_{2}}}.$ (171)

###  4.5 Modular completion of $\Psi_{\boldsymbol{\mu}}$

We will treat in this subsection the completion of $\Psi_{\boldsymbol{\mu}}$.
Eq. (122) related the holomorphic $q$-series $\Psi_{\boldsymbol{\mu}}(\tau)$
to that the second derivative of the function
$\Psi_{\boldsymbol{\mu}}(\tau,z)$. To determine the non-holomorphic completion
$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})$, we consider the non-
holomorphic differential operator

$-\frac{1}{4\pi^{2}}\left(\frac{\partial^{2}}{\partial z^{2}}+\frac{2\pi
m}{\tau_{2}}\right),$ (172)

on the completion
$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau},z,\bar{z})$ of
$\Psi_{\boldsymbol{\mu}}(\tau,z)$ (120). The differential operator maps a
Jacobi form of weight $k$ and index $m$ to a Jacobi form of weight $k+2$ and
index $m$. The second non-holomorphic term in (172) is required for modularity
but does is not relevant for the holomorphic part.

To determine the completion
$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau},z,\bar{z})$, we set
$z=\beta\tau+\delta$ with $\beta,\delta\in\mathbb{R}$, so that $\beta={\rm
Im}(z)/\tau_{2}$. We first note that completing the square gives for a generic
vector $V\in\underline{{\boldsymbol{\Lambda}}}$,

$y^{V.{\boldsymbol{Q}}}\,q^{-\frac{1}{2}{\boldsymbol{Q}}^{2}}=q^{\frac{\beta^{2}}{2}V^{2}-\frac{1}{2}({\boldsymbol{Q}}-\beta
V)^{2}}e^{2\pi i(V.{\boldsymbol{Q}})\delta},$ (173)

for an arbitrary vector $V$. To write the modular completion, we can treat the
three permutations separately. With $C_{1}$ as in (136), and
$C^{(123)}_{\ell}$ as in (137), we find that the kernel
$(-1)^{a+b-c}F^{*}(123)\,(y^{a+b-c}+y^{-a-b+c})$ is to be completed to

$\widehat{F}^{*}(123,y)=\frac{(-1)^{a+b-c}}{4}\sum_{\pm}\left[1+\sum_{\ell=1}^{3}E_{2}(C^{(123)}_{\ell},C^{(123)}_{\ell+1};\sqrt{2\tau_{2}}\,({\boldsymbol{Q}}\mp\beta
C_{1}))\right]y^{\pm(a+b-c)}.$ (174)

Then the completion
$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau},z,\bar{z})$ reads,

$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau},z,\bar{z})=\sum_{{\boldsymbol{Q}}\in\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}}[\widehat{F}^{*}(123,y)+\widehat{F}^{*}(213,y)+\widehat{F}^{*}(132,y)]\,q^{-{\boldsymbol{Q}}^{2}/2}.$
(175)

The completion $\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau},z,\bar{z})$
transforms as a Jacobi form of weight $b_{2}/2$ and index $m_{P}$ given by,

$m_{P}=-\frac{1}{2}C_{1}^{2}=-P_{1}P_{2}P_{3}-\frac{1}{2}\sum_{\ell=1}^{3}P_{\ell}\,(P_{\ell+1}^{2}+P_{\ell+2}^{2}).$
(176)

This is symmetric under permutations, and thus also equal to
$-C_{2}^{2}/2=-C_{3}^{2}/2$.

To write the modular completion, we define the function

$\begin{split}G(V_{1},V_{2},V_{3};{\boldsymbol{Q}},\tau_{2})&=-\frac{1}{4\pi^{2}}\partial_{z}^{2}\left(E_{2}(V_{1},V_{2};\sqrt{2\tau_{2}}\,({\boldsymbol{Q}}-\beta
V_{3}))\,y^{V_{3}.{\boldsymbol{Q}}}\right)|_{z=0}\\\
&=(V_{3}.{\boldsymbol{Q}})^{2}E_{2}(V_{1},V_{2};\sqrt{2\tau_{2}}\,{\boldsymbol{Q}})\\\
&\quad-\frac{1}{4\pi^{2}}\partial_{z}^{2}E_{2}(V_{1},V_{2};\sqrt{2\tau_{2}}\,({\boldsymbol{Q}}-\beta
V_{3}))|_{z=0}\\\ &\quad+\frac{1}{2\pi
i}(V_{3}.{\boldsymbol{Q}})\,\partial_{z}E_{2}(V_{1},V_{2};\sqrt{2\tau_{2}}\,({\boldsymbol{Q}}-\beta
V_{3}))|_{z=0}.\end{split}$ (177)

The limit $\tau_{2}\to\infty$ is determined by the first term,

$\lim_{\tau_{2}\to\infty}G(V_{1},V_{2},V_{3};{\boldsymbol{Q}},\tau_{2})=\left\\{\begin{array}[]{rl}(V_{3}.{\boldsymbol{Q}})^{2}\,\arctan(\alpha),&\quad{\rm
if}\,\,V_{1}.{\boldsymbol{Q}}=V_{2}.{\boldsymbol{Q}}=0,\\\
(V_{3}.{\boldsymbol{Q}})^{2}\,\mathop{\mathrm{sgn}}(V_{1}.{\boldsymbol{Q}})\,\mathop{\mathrm{sgn}}(V_{2}.{\boldsymbol{Q}}),&\quad{\rm
otherwise},\end{array}\right.$ (178)

with $\alpha$ as in (140).

To write the modular completion, we can treat the three permutations
separately. We set

$\widehat{G}^{*}(123)=\frac{(-1)^{a+b-c}}{4}\left((a+b-c)^{2}+\sum_{\ell=1}^{3}\,G(C^{(123)}_{\ell},C^{(123)}_{\ell+1},C_{1};{\boldsymbol{Q}},\tau_{2})\right),$
(179)

with $C_{1}$ as in (136), and $C^{(123)}_{\ell}$ as in (137) The other
permutations $(213)$ and $(132)$ give similarly rise to $\widehat{F}^{*}(213)$
and $\widehat{F}^{*}(132)$. We define,

$\begin{split}\widehat{g}_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})&=\frac{1}{4}\left[\widehat{G}^{*}(123)+\widehat{G}^{*}(213)+\widehat{G}^{*}(132)\right].\end{split}$
(180)

The modular completion $\widehat{\Psi}_{\boldsymbol{\mu}}$ now reads

$\widehat{\Psi}_{\boldsymbol{\mu}}(\tau,\bar{\tau})=\sum_{{\boldsymbol{Q}}\in\underline{{\boldsymbol{\Lambda}}}_{\boldsymbol{\mu}}^{*}}\widehat{g}_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})\,q^{-{\boldsymbol{Q}}^{2}/2}-\frac{m_{P}}{2\pi\tau_{2}}\,\widehat{\Phi}_{\boldsymbol{\mu}}(\tau,\bar{\tau}).$
(181)

where the last term is due to the $m$-dependent term in (172). We let this
expression be our final form for
$\widehat{\Psi}_{{\boldsymbol{\mu}}}(\tau,\bar{\tau})$.

Our last task is to determine the constant $A$ in
$g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})$ (42). To this end, we consider the
$\tau_{2}\to\infty$ limit of
$\widehat{g}_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})$ and require that it
reduces to $g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})$. If we consider the term
$F^{*}(123)\,(a+b-c)^{2}$, we see that only the completion of the term
$\mathop{\mathrm{sgn}}(a-c)\,\mathop{\mathrm{sgn}}(b-c)$ in $g_{C}$ can
contribute a non-vanishing difference. Indeed, if we include the other
permutations, the only remaining terms in the difference are the equilateral
cases with $a=b=c$,444We stress that the equilateral condition $a=b=c$, is the
condition on ${\boldsymbol{Q}}$ to satisfy
$C_{a}.{\boldsymbol{Q}}=C_{b}.{\boldsymbol{Q}}=C_{c}.{\boldsymbol{Q}}$, and
does not imply equalities among $C_{a}$, $C_{b}$ and $C_{c}$.

$\begin{split}&\lim_{\tau_{2}\to\infty}\widehat{g}_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})-g_{C}(\\{\gamma_{j}\\};\\{c_{j}^{*}\\})=\frac{(-1)^{a}}{4}\,a^{2}\,\delta_{a,b}\,\delta_{b,c}\\\
&\qquad\times\lim_{\tau_{2}\to\infty}\left(E_{2}(C_{a}-C_{c},C_{b}-C_{c};\sqrt{2\tau_{2}}\,{\boldsymbol{Q}})+E_{2}(C_{a}-C_{b},C_{c}-C_{b};\sqrt{2\tau_{2}}\,{\boldsymbol{Q}})\right.\\\
&\qquad\left.+E_{2}(C_{c}-C_{a},C_{b}-C_{a};\sqrt{2\tau_{2}}\,{\boldsymbol{Q}})-A)\right).\end{split}$
(182)

Now the sum of $E_{2}$’s is precisely of the form (150), and thus equals 1\.
As a result, we find that with $A=1$ the limit vanishes for any choice of
Calabi-Yau and charge configurations. This matches perfectly with the physical
expectation. Note that the individual values for $E_{2}$ are given by an
arctan, and are generically irrational, but that this combination adds up to
1.

It is quite striking that the values of $E_{2}$’s precisely confirm the
physical expectation. Also in other cases [43, 30], the values of the
generalized error functions for vanishing arguments has matched with the
expectations based on BPS invariants.

##  5 Relation to M5-branes and AdS3/CFT2

In this brief section, we discuss our findings from the point of view of
M-theory and the MSW CFT. We discuss how the partition functions
$\widehat{Z}_{P}^{nT}$, $n>2$, may arise from the 2-dimensional perspective.

The uplift of the D4-branes to M5-branes in M-theory is useful to understand
the modular properties of the partition functions [6, 12, 13, 14]. The spatial
dimensions of the M5-brane are $P\times S^{1}_{M}$, where $P\in
H_{4}(X,\mathbb{Z})$ is a four-cycle in the Calabi-Yau threefold and
$S^{1}_{M}$ is the M-theory circle. The D2 branes of IIA string theory are
realized as excitations of the self-dual 2-form field on M5 brane world
volume, while the D0-branes are realized as momenta of the brane system along
$S^{1}_{M}$ with radius $R=g_{s}\,\ell_{s}=\ell_{11}^{3}/\ell_{s}^{2}$ in
terms of the string coupling $g_{s}$, string length $\ell_{s}$, and eleven
dimensional Planck length $\ell_{11}$. The world volume theory of the M5 brane
gives a low energy description of the system, provided gravitational effects
can be ignored. This can be ensured by taking the volume of the CY3 to be
large, namely $V_{X}/\ell_{11}^{6}$ large but fixed. Supersymmetry of the M5
world volume theory implies that the MSW CFT has (0,4) supersymmetry, and is
dual to the near horizon AdS3 geometry [12, 74]. Bosons of this CFT include,
moduli of the divisor $P$ inside $X$, translations along flat $\mathbb{R}^{3}$
and (anti)-chiral scalars coming from reduction of self-dual two-form field in
the M5 brane world volume. The number of fields can be determined using
geometric data of $X$, when $P$ is a very ample divisor. Consequently the
central charges can be determined. The central charge of the left-moving, non-
supersymmetric sector is

$c_{L}=P^{3}+c_{2}(X)\cdot P,$ (183)

where $c_{2}(X)$ is the second Chern class of the Calabi-Yau three-fold $X$.
Using Cardy’s formula, Reference [12] demonstrated that microscopic entropy is
in agreement with one loop corrected Bekenstein-Hawking entropy.

As for any CFT, a key feature of the MSW CFT is the modular symmetry. When
time direction is compactified, the CFT lives on a 2-torus
$T^{2}=S^{1}_{M}\times S^{1}_{t}$. The linear fractional transformation of the
complex structure modulus $\tau$ of $T^{2}$ by an element of
$SL(2,\mathbb{Z})$ corresponds to the same torus $T^{2}$. Thus the
$SL(2,\mathbb{Z})$ symmetry, or modular symmetry, appears as a symmetry of the
CFT. This is a strong constraint on the degeneracies. Another property of the
CFT is the spectral flow of the $U(1)^{b_{2}}$ current algebra. This leads to
a symmetry of degeneracies as function of the charges, such that the partition
function can be decomposed as a finite sum of theta series $\times$ modular
functions, much as is the case for the attractor partition function in Section
3.2.

We proceed by considering the AdS3 dual to the MSW CFT following [27]. The
relation of $\lambda$ in (58) to the five-dimensional quantities can be
understood from four-dimensional Newton’s constant. From the IIA perspective,
we have

$G_{4}=\frac{g_{s}^{2}\,\ell_{s}^{8}}{V_{X}}=\frac{g_{s}^{2}\,\ell_{s}^{2}}{\lambda^{3}},$
(184)

while in terms of the five-dimensional quantities,

$G_{4}=\frac{\ell_{5}^{3}}{R}=g_{s}^{2}\,\ell_{s}^{2}\frac{\ell_{5}^{3}}{R^{3}},$
(185)

such that one has,

$\lambda=\frac{R}{\ell_{5}}.$ (186)

Starting from five dimensional asymptotically flat geometries with appropriate
charges, one takes the decoupling limit as follows: size of M-theory circle
$R$ is kept fixed (in absolute units), Calabi-Yau volume is kept finite in
units of five dimensional Planck length $\ell_{5}$. In the decoupling limit to
AdS3, $\ell_{5}\rightarrow 0$, hence $\lambda\to\infty$ by (186). Multi-
centered geometries where centers have mutual distance of the order of
$\ell_{5}^{3}\sim\lambda^{-3}$ or less, survive this limit and go over to
asymptotically AdS${}_{3}\times S^{2}$ geometries, with asymptotic moduli
fixed to their attractor values. These are the $\lambda$-core geometries
mentioned in the introduction, and include centers with non-vanishing
individual D6 brane charges, which add up to zero. On the other hand, if the
distances between the centers is larger than $\sim\ell_{5}^{3}/R^{2}$ for
$\ell_{5}\to 0$, the bound states decouple from the spectrum. As we can see
from the Denef equations (3), this is for example the case if the centers
carry a non-vanishing D4-brane charge with vanishing D6-brane charge. Then the
distances between the centers scale as $\ell_{5}$ for non-scaling bound
states, and the centers give rise to disconnected AdS3 throats in the
decoupling limit. On the other hand for scaling solutions, the distance
between the centers contains a regime where the centers can come arbitrarily
close.

As explained in Section 2, the BPS index can be determined using localization
with respect to rotation around the $z$-axis, and we can therefore concentrate
on collinear solutions to Denef’s equations [66]. These collinear solutions
admit two branches, one corresponding to centers at finite distances, and the
other one is when the centers are nearly coincident. The second branch,
sometimes referred to as “deep scaling regime” reproduces pure Higgs states
and goes to a single throat in $\ell_{5}\rightarrow 0$ limit [53]. This is in
accordance with the expectation that MSW CFT captures the near coincident
regime of scaling solutions since these are smoothly connected to the single
center black hole. The separation between the centers at the collinear fixed
point is of order $\lambda^{-1}\sim\ell_{5}$, and these therefore decouple
from the AdS3 geometry. As a result, these do not appear to be captured by the
CFT. This leads us to speculate that the first term on the rhs of (1),
$\widehat{\mathcal{Z}}_{P}^{T}$, corresponds to the AdS3/CFT2 partition
function while the other terms on the rhs do not.

It is intriguing that the terms due to scaling solutions
$\widehat{\mathcal{Z}}_{P}^{nT}$ with $n\geq 3$ do satisfy the restrictive
modular transformations as well as the spectral flow symmetry, and it is
desirable to understand the origin. We think that these terms can appear after
turning on an irrelevant deformation in the $(0,4)$ CFT, away from the
conformal fixed point and reversing the attractor flow and $\ell_{5}\to 0$
limit. While such a deformation does not lead to a finite change in the
partition function for $(4,4)$ CFT [54], it seems plausible to us that this
can happen with reduced supersymmetry. This deformation is distinguished among
other deformations spanning the space of attractor moduli, since this
deformation does preserve the spectral flow symmetry which is in general not
the case for variations of the Kähler moduli orthogonal to $P$ [25]. We leave
a further exploration of these interesting aspects for future work.

##  6 Case studies

To make a suitable choice of a Calabi-Yau threefold, we note that the lattices
involved have dimensions linear in $b_{2}$, the second Betti number. Thus the
computations are less involved for smaller $b_{2}$. However for the simplest
case $b_{2}=1$, there are no cyclic quivers and hence no scaling
solutions.555Consider triangular quiver to start with. For $b_{2}=1$, electric
and magnetic charges are numbers and satisfy the identity
$aP_{3}+bP_{1}+cP_{2}=0$. Since $P_{i}>0$ this implies $a,b,c$ can not all
have the same sign hence the quiver can not be cyclic. This is easily
generalized to any cyclic quiver. Also, the lattices concerned are positive
definite, and therefore do not give rise to mock modular forms. So we settle
for the next simplest case $b_{2}=2$.

In order to define quadratic forms on the lattices, one needs the intersection
numbers. For a Calabi Yau threefold with $b_{2}=2$, there are $2^{3}=8$
intersection numbers of which only 4 are independent. For the sake of
simplicity we shall choose a Calabi Yau threefold $X$, for which all
intersection numbers are $\in 2\mathbb{Z}$ such that Simplification 2 of
Section 4.1 holds, and for which most of these numbers vanish. To be specific,
we choose $X$ to be a $K3$ fibration over $\mathbb{P}^{1}$. This Calabi-Yau
manifold corresponds to Polytope ID # =14 in the online database [75]. $X$ can
be realized as a divisor in an ambient toric variety
$\mathbb{P}^{3}\times\mathbb{P}^{1}$, specified by the weight matrix

$\displaystyle W$ $\displaystyle=\begin{pmatrix}0&0&0&1&1&0\\\
1&1&1&0&0&1\end{pmatrix}\,.$ (187)

$W$ defines the toric variety through the identification on
$\mathbb{C}^{6}/\\{0\\}$

$\displaystyle(z_{1},\dots,z_{6})$
$\displaystyle\sim(\lambda^{W_{i1}}z_{1},\dots,\lambda^{W_{i6}}z_{6}),\,\,\lambda\in\mathbb{C}^{\star},\,\,i=1,2\,,$
(188)

where $z_{1},\dots,z_{6}$ are coordinates of $\mathbb{C}^{6}$. Anti-canonical
divisor of this toric variety defines the Calabi Yau threefold under
consideration. It has the following Hodge numbers

$\displaystyle h^{1,1}$ $\displaystyle=2,\quad\,h^{2,1}=86\,$ (189)

and consequently the Euler number is $\chi=2(h^{1,1}-h^{2,1})=-168$.
Intersection polynomial and second Chern class of $X$ are given respectively
by

$\begin{split}\text{intersection polynomial}&=4J_{1}J_{2}^{2}+2J_{2}^{3},\,\\\
\text{second Chern class}&=8J_{1}J_{2}+6J_{2}^{2}\,,\end{split}$ (190)

where $J_{i}$’s are a basis of 2-forms on $X$ and generators of the Kähler
cone. $X$ is favorable, meaning that Kähler forms on $X$ descend from the
ambient toric variety. In our notation, the intersection numbers $d_{abc}$ and
second Chern class $c_{2,a}$ read

$\begin{split}&d_{111}=d_{112}=0,\qquad d_{122}=4,\qquad d_{222}=2,\\\
&c_{2,1}=24,\qquad c_{2,2}=44\end{split}$ (191)

and the $d_{abc}$ for other indices follow by permutation.

###  6.1 Charge configuration 1: $P_{1}=P_{2}=(0,1),\;P_{3}=(1,1)$

In this example we choose the total magnetic charge $P=(1,3)$ magnetic charges
split into 3 centers as

$\displaystyle P_{1}=P_{2}=(0,1),\quad P_{3}=(1,1).$ (192)

We note that $P_{1,2}$ are irreducible while $P_{3}$ is reducible. Therefore
$h_{P_{1,2},\mu}$ is a weakly holomorphic modular form, while $h_{P_{3},\mu}$
is a mock modular form. The corresponding central charges of the left-moving
sector of the CFT (183) are,

$c_{L}(P_{1,2})=46,\qquad c_{L}(P_{3})=92,\qquad c_{L}(P)=318.$ (193)

For the $j$-th center, we have for the innerproduct $D_{j}=d_{abc}P_{j}^{c}$,

$\displaystyle D_{1}=D_{2}=\begin{pmatrix}0&4\\\ 4&2\end{pmatrix},\quad
D_{3}=\begin{pmatrix}0&4\\\ 4&6\end{pmatrix}.$ (194)

For this choice, $D_{3}^{-1}D_{1}=D_{3}^{-1}D_{2}$ is an integral matrix, such
that Simplication 1. of Section 4.1 is satisfied, and we can use the results
of that section. We have, $D_{1}D_{3}^{-1}D_{1}=\begin{pmatrix}0&4\\\
4&-2\end{pmatrix}$ which produces the quadratic form of
$\underline{{\boldsymbol{\Lambda}}}$ (98),

$\displaystyle\underline{D}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}[c]0&8&0&4\\\ 8&0&4&-2\\\ 0&4&0&8\\\
4&-2&8&0\end{pmatrix},\quad\det{\underline{D}}=2304.$ (195)

The quadratic form on the lattice of electric charges $\underline{D}^{-1}$ is
given by,

$\displaystyle\underline{D}^{-1}=\left(\begin{array}[]{cccc}-\frac{1}{18}&\frac{1}{6}&\frac{5}{72}&-\frac{1}{12}\\\
\frac{1}{6}&0&-\frac{1}{12}&0\\\
\frac{5}{72}&-\frac{1}{12}&-\frac{1}{18}&\frac{1}{6}\\\
-\frac{1}{12}&0&\frac{1}{6}&0\\\ \end{array}\right).$ (200)

Electric charge vectors $Q_{j}\in\Lambda_{j}^{*}$ have the following form

$Q_{j}=\begin{pmatrix}q_{j1}\\\ q_{j2}\end{pmatrix}.$

There are $\det(D_{1})\,\det(D_{2})=256$ conjugacy classes of the form

$\displaystyle q_{11}$ $\displaystyle=$ $\displaystyle 4k_{12}+\mu_{11},\quad
q_{12}=4k_{11}+2k_{12}+\mu_{12},$ (201) $\displaystyle{}q_{21}$
$\displaystyle=$ $\displaystyle 4k_{22}+\mu_{21},\quad
q_{22}=4k_{21}+2k_{22}+\mu_{22},$

where $\mu_{ij}\in\\{0,1,2,3\\}$, $k_{ij}\in\mathbb{Z}$. These conjugacy
classes have the exchange symmetry between $\mu_{1},\mu_{2}$. Since $D_{3}$
divides $D_{1}$ and $D_{2}$, $N_{q}=\det(D_{1})\,\det(D_{2})$ (100) is the
number of conjugacy classes of
$\underline{{\boldsymbol{\Lambda}}}^{*}/\underline{{\boldsymbol{\Lambda}}}$
for fixed $\mu_{1}+\mu_{2}+\mu_{3}=\mu\in\Lambda^{*}/\Lambda$, since the class