Counting $\mathcal{N}=8$ Black Holes as Algebraic Varieties

Sourav Maji, Abhishek Chowdhury

School of Basic Sciences, Indian Institute of Technology Bhubaneswar

Jatni, Khurda, Odisha, 752050, India

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Abstract

We calculate the helicity trace index $B_{14}$ for $\mathcal{N}=8$ pure
D–brane black holes using various techniques of computational algebraic
geometry and find perfect agreement with the existing results in the
literature. For these black holes, microstate counting is equivalent to
finding the number of supersymmetric vacua of a multi–variable supersymmetric
quantum mechanics which in turn is equivalent to solving a set of
multi–variable polynomial equations with syzygies. We explore four different
techniques to solve a set of polynomial equations, namely Newton Polytopes,
Homotopy continuation, Monodromy and Hilbert series. Unlike other schemes,
these methods are almost exact with very high success rate for finding all
solutions 111There are some caveats if the solutions take zero values.
Sometimes round–off errors can destroy a solution. These situations are rare,
at least in our systems.. A gauge invariant analysis is also possible using
the symbolic Hilbert series. Furthermore, exploiting various exchange
symmetries, we show the non–existence of a quartic and(or) higher order terms
in the potential which if present would have spoiled the counting.
Incorporating recent developments in mathematics, algorithms and
multi–threading, we have extended the scope of one of the authors previous
works [1, 2] to other charges and presented a new perspective for the
microstate counting problem. This further establishes the pure D–brane system
as a consistent model, bringing us a step closer to $\mathcal{N}=2$ black hole
counting.

###### Contents

  1. 1 Introduction
  2. 2 Review of the D–brane System
    1. 2.1 Action and Potentials
    2. 2.2 Symmetries
    3. 2.3 Supersymmetric Vacua
  3. 3 Counting Black Holes as Algebraic Varieties
    1. 3.1 Newton Polytope Method
      1. 3.1.1 Abelian case
      2. 3.1.2 Non–abelian cases
    2. 3.2 Homotopy Continuation Method
      1. 3.2.1 Abelian case
      2. 3.2.2 Non–abelian cases
    3. 3.3 Monodromy Method
    4. 3.4 Hilbert Series
  4. 4 Counting Black Holes as Hilbert series
    1. 4.1 Abelian case
    2. 4.2 Non–abelian cases
      1. 4.2.1 Charge $(1,1,1,2)$
      2. 4.2.2 Charge $(1,1,1,3)$
      3. 4.2.3 Charge $(1,1,1,4)$
  5. 5 Higher Order Terms in the Potential
  6. 6 Discussion
  7. A Hilbert series of charge $(1,1,1,1)$
  8. B The $14^{th}$ Helicity Trace Index
  9. C Gauge charges

##  1 Introduction

The interplay between mathematics and physics has been very rewarding both
from the perspective of mathematicians and physicists alike. Substantial
contribution can be attributed to String theory which represents a framework
to quantize gravity within the realm of theoretical physics and enjoy deep
connections with mathematics, especially in the areas of topology [3],
algebraic geometry [4] and number theory [5]. String theory in turn, has
benefited immensely from the extensive studies on black holes as theoretical
laboratories allowing for controlled theoretical predictions. One of the more
important successes of string theory is the validation of its internal
mathematical consistency spearheaded by the precise resolution of
Bekenstein–Hawking entropy [6, 7] as degeneracy of BPS states representing the
internal degrees of freedom of a black hole. Over the past two decades,
significant progress has been made towards counting BPS states of
supersymmetric extremal black holes in $\mathcal{N}=8$ [8, 9, 10, 1, 2],
$\mathcal{N}=4$ [11, 12, 13, 14, 15, 16, 17, 18] and $\mathcal{N}=2$ [19, 20]
theories.
For a typical 4–dimensional black hole, a weak coupling microscopic
description would have a higher dimensional type II or Heterotic string theory
appropriately compactified on a tori, $K3$ or Calabi–Yau spaces (preserving 8
or more supercharges) with a collection of intersecting D–branes and strings
wrapping around 1–cycles forming a bound state carrying charges (in the
process breaking most of the SUSY). The SUSY preserving BPS degeneracy ($d$)
of its low energy dynamics is captured by computing a supersymmetric/Witten
index of the low energy excitations. On the macroscopic side, where gravity is
relatively strong, the bound state is interpreted as a black hole solution of
an appropriate supergravity theory with entropy ($S_{BH}$) proportional to the
area of its event horizon. Barring the important caveat of degeneracy vs index
[18], the invariance of the index under deformations results in $d\approx
e^{S_{BH}}$.
Near horizon analysis of BPS black holes carrying four unbroken SUSY suggests
that the microstates of single center black holes carry zero angular momentum
leading to the positivity of the index [21, 22] 222The correct statement is
that the microstates are given by the tensor product of a BPS multiplet
obtained by quantizing goldstino fermion zero modes and a singlet
representation of the $SU(2)$ rotation group.. However, it is difficult to
test this conjecture directly as the regimes of microstate counting and black
hole description are different but positivity of the index has been checked in
many examples [23, 24, 25, 26] and so far no counterexample has been found. In
previous works of one of the authors [2] a ‘zero angular momentum conjecture’
was put forward which states that at a generic point in the moduli space in
both macroscopic and microscopic regimes, the spectrum of BPS states will
carry zero angular momentum 333In a non–generic or special point in the moduli
space where for example most moduli take zero values, the spectrum of BPS
states have different angular momenta and an unequal cancellation of the
positive and negative contribution to the index gives a net positive
contribution.. To test this conjecture it is best to start with
$\mathcal{N}=8$ theories where it is known that only single centers contribute
to the index [8] and we don’t have to the deal with the complications arising
from wall crossing and two or multi–center contributions to the index as are
the case for $\mathcal{N}=4$ [27, 28, 29, 30, 31] and $\mathcal{N}=2$ [32, 19]
theories.
In this paper we shall focus on a pure D–brane D2–D2–D2–D6 system in type IIA
theory compactified on $T^{6}$ that is dual to a D1–D5–P–KK monopole system in
type IIB on $T^{6}$ for which the microscopic index is known [10]. The
original motivation to study this system as a gateway to the counting of black
holes in $\mathcal{N}=2$ string theories compactified on Calabi–Yau manifolds
still stands [33, 34, 35]. As an opening move, we extend the analysis of [1,
2] where it was shown that for abelian charge $(1,1,1,1)$ and non–abelian
charges $(1,1,1,2)$ & $(1,1,1,3)$ the SUSY vacua manifold for the $0+1$
dimensional quantum mechanics with support on the intersection of the four
branes is a collection of 12, 56 and 208 points respectively. These results in
support of the proposed conjecture were found by solving the F–terms equations
which are multi–variable polynomials using Mathematica [36] which now we know
to be using an algorithm based on numerical Gr$\ddot{o}$bner basis. Extension
of these results to non–abelian charge $(1,1,1,4)$ which gives a SUSY vacua of
684 points again in support of the conjecture and the task of developing a
gauge invariant counting procedure lead us to the study of myriad
sophisticated techniques developed in computational algebraic geometry to deal
with algebraic or projective varieties [37, 38, 39]. For the known results,
some of these methods are more than 100 times faster than Mathematica, even on
a single core 444We have a Dell Workstation with two Intel Xeon Gold 6258R
processors (56 cores) and 128 GB of RAM. The main bottleneck in the
algorithmic implementation of most of these techniques seems to be the RAM..
Along the way, we provided an explicit check for the vacua count by computing
the Hessian around the critical points of the potential and argued the
consequences of introducing higher order terms in the potential.
A companion goal of this paper is to highlight the interplay between a robust,
sufficiently complex physical model with various symmetries and techniques of
computational algebraic geometry. The application of algebraic geometry to
black holes in string theory is not new, largely due to the study of
Calabi–Yau [40, 41], structure of various moduli spaces [42, 43] and mirror
symmetry [4] but to the best of our knowledge, application of numerical
algorithms developed in algebraic geometry has not featured in any previous
works of microstate counting in black holes. Among the many techniques we
studied [38, 44, 37], in this paper we present four of them, namely the Newton
polytope, Homotopy continuation, Monodromy and Hilbert series based on the
suitability to our problem, parallelizability and wider applicability. These
methods can also be applied to a large class of cost function/energy
minimization problems where the landscape of low lying critical points/states
is expected to grow exponentially with respect to some parameters, like the
string vacua landscape [45, 42] and the cost functions in machine learning
[46]. For mathematicians, these D–brane models presents an infinite class of
parametric weighted projective varieties whose number of critical points are
coefficients of a known generating function and for large values of the
parameters $N_{1}$, $N_{2}$, $N_{3}$ and $N_{4}$ (charges of the black hole),
the count is $\displaystyle d\approx
e^{2\pi\sqrt{N_{1}N_{2}N_{3}N_{4}}-2\ln(4N_{1}N_{2}N_{3}N_{4})}$.
The rest of the paper is organized as follows. In section 2 we review the
D–brane model focusing mainly on the potentials, the symmetries and the nature
of vacua manifold. In section 3 we briefly review the mathematical ideas
behind the computational algebraic geometry techniques of Newton polytope,
Homotopy continuation, Monodromy and Hilbert series and then discuss our
strategies to apply them in the context of solving the F–term equations
arising in our models, along with the results. In section 4 we continue with
Hilbert series, providing more details on gauge (un-)fixing and discuss the
results. In section 5 we show that addition of a quartic term to our potential
will potentially destroy the U–duality. We give symmetry arguments as to why
such terms are not allowed. We conclude in section 6 with some implications of
the zero angular momentum conjecture along with a broad overview of the
applicability and interplay of these methods with various fields of study.
There are three appendices, in appendix A we discuss in detail the contour
integrals involved in the Hilbert series of charge $(1,1,1,1)$, in appendix B
we summarize the extraction of the microstate counts from its generating
function, the weak Jacobi form and tabulate the relevent results, in appendix
C we tabulate the charge tables used in the Hilbert series computations for
various abelian and non–abelian charge configurations. The paper ends with the
references.

##  2 Review of the D–brane System

In this section we will review the D–brane model proposed in [1, 2] and give a
brief summary of its action, the potentials, various symmetries and finally
our expectation from the supersymmetric vacua manifold. The pure D–brane
D2–D2-D2–D6 system is a D–brane intersection model in type IIA string theory
compactified on $T^{6}$ with one D2–brane wrapping the $(x^{4},\,x^{5})$
directions, one D2–brane along $(x^{6},\,x^{7})$ directions, another D2–brane
along $(x^{8},\,x^{9})$ directions and finally the D6–brane along
$(x^{4},\cdots,x^{9})$ directions. As the system breaks 28 out of 32
supersymmetries, it can be described in terms of $\mathcal{N}=1$ supersymmetry
in $3+1$ dimensions and we can use the $\mathcal{N}=1$ superfield formalism to
organize the field content as $\mathcal{N}=1$ multiplets.
Associated with each D–brane is a low energy effective theory which in this
case is the dimensional reduction from $3+1$ to $0+1$ dimensions of a
$\mathcal{N}=4$ supersymmetric $U(N)$ Yang–Mills theory coming from string
fluctuations starting and ending on the same brane. It has one $\mathcal{N}=1$
vector multiplet $V^{(k)}$ and three $\mathcal{N}=1$ chiral multiplets
$\Phi_{1}^{(k)}$, $\Phi_{2}^{(k)}$ and $\Phi_{3}^{(k)}$, where $k$ labels the
brane index. On the other hand, the strings between a pair of branes, starting
at one and ending on another give rise to a $\mathcal{N}=2$ hyper–multiplet,
or equivalently two $\mathcal{N}=1$ chiral multiplets $Z^{(k\ell)}$ and
$Z^{(\ell k)}$.

###  2.1 Action and Potentials

Assuming the six circles of $T^{6}$ are orthonormal to each other and having
radius $\sqrt{\alpha^{\prime}}$, which we will set to $\alpha^{\prime}=1$, the
action of the pure D–brane system is given by,

$S_{\text{kinetic }}+\int dx^{0}\left[\int
d^{4}\theta\sum_{k=1}^{4}\sum_{\begin{subarray}{c}\ell=1\\\ \ell\neq
k\end{subarray}}^{4}\left\\{\bar{Z}^{(k\ell)}e^{2V^{(\ell)}-2V^{(k)}}Z^{(k\ell)}\right\\}+\int
d^{2}\theta\mathcal{W}+\int d^{2}\bar{\theta}\overline{\mathcal{W}}\right]\,,$
(2.1)

where $S_{\text{kinetic }}$ is the standard kinetic term involving the chiral
superfields and vector superfields. The interaction superpotential
$\mathcal{W}$ has three different terms, $\mathcal{W}_{1}$, $\mathcal{W}_{2}$
and $\mathcal{W}_{3}$. $\mathcal{W}_{1}$ describes the coupling between the
superfields $Z^{(k\ell)}$ and $\Phi^{(k)}_{m}$ and it is of the following
form,

$\displaystyle\mathcal{W}_{1}=\sqrt{2}$
$\displaystyle{\left[\sum_{k,\ell,m=1}^{3}\varepsilon^{k\ell
m}\operatorname{Tr}\left(\Phi_{m}^{(k)}Z^{(k\ell)}Z^{(\ell
k)}\right)+\sum_{k=1}^{3}\operatorname{Tr}\left(Z^{(4k)}\Phi_{k}^{(k)}Z^{(k4)}\right)\right.}$
(2.2)
$\displaystyle\left.-\sum_{k=1}^{3}\operatorname{Tr}\left(\Phi_{k}^{(4)}Z^{(4k)}Z^{(k4)}\right)\right]\,,$

where $\varepsilon^{k\ell m}$ is the totally antisymmetric symbol with
$\varepsilon^{123}=1$. The second component $\mathcal{W}_{2}$ describes the
cubic self–coupling between the $Z^{(k\ell)}$’s and takes the form,

$\mathcal{W}_{2}=\sqrt{2}\,C\left[\sum_{\begin{subarray}{c}k,l,m=1\\\
k<\ell,m;l\neq m\end{subarray}}^{4}(-1)^{{\delta_{k1}}{\delta_{\ell
3}}{\delta_{m4}}}\operatorname{Tr}\left(Z^{(k\ell)}Z^{(\ell
m)}Z^{(mk)}\right)\right]\,.$ (2.3)

After switching on small background values of the off–diagonal components of
the metric and 2–form $B$–fields, we get an extra superpotential term,

$\mathcal{W}_{3}=\sqrt{2}\left[\sum_{k,\ell,m=1}^{3}c^{(k\ell)}\varepsilon^{k\ell
m}N_{\ell}\operatorname{Tr}\left(\Phi_{m}^{(k)}\right)+\sum_{k=1}^{3}c^{(k4)}\left[N_{4}\operatorname{Tr}\left(\Phi_{k}^{(k)}\right)-N_{k}\operatorname{Tr}\left(\Phi_{k}^{(4)}\right)\right]\right]\,.$
(2.4)

There is also an additional superpotential

$\mathcal{W}_{4}=-\sqrt{2}\sum_{k=1}^{4}\operatorname{Tr}\left(\Phi_{1}^{(k)}\left[\Phi_{2}^{(k)},\Phi_{3}^{(k)}\right]\right)\,.$
(2.5)

where $C$ is a constant whose value has been computed [2] by analyzing the
coupling between open strings stretched between different branes and $c^{(k)}$
are Fayet–Iliopoulos (FI) parameters satisfying
$\displaystyle\sum_{k=1}^{4}c^{(k)}N_{k}=0$. By comparing the mass of open
strings stretched between pair of branes and the renormalized mass determined
from the quadratic term involving the fields $Z^{(k\ell)}$ and $Z^{(\ell k)}$,
we can write $c^{(k)}$ and $c^{(k\ell)}$ in terms of the off–diagonal
components of metric $g_{mn}$ and 2–form field $b_{mn}$. The explicit
expressions are given in the appendix A of [1].
Writing the action in terms of the component field, we get three different
potential terms,

$\displaystyle V_{\text{gauge }}=$
$\displaystyle\sum_{k=1}^{4}\sum_{\begin{subarray}{c}k=1\\\ l\neq
k\end{subarray}}^{4}\sum_{i=1}^{3}\operatorname{Tr}\left[\left(X_{i}^{(k)}Z^{(k\ell)}-Z^{(k\ell)}X_{i}^{(\ell)}\right)^{\dagger}\left(X_{i}^{(k)}Z^{(k\ell)}-Z^{(k\ell)}X_{i}^{(\ell)}\right)\right]$
(2.6)
$\displaystyle+\sum_{k=1}^{4}\sum_{i,j=1}^{3}\operatorname{Tr}\left(\left[X_{i}^{(k)},\Phi_{j}^{(k)}\right]^{\dagger}\left[X_{i}^{(k)},\Phi_{j}^{(k)}\right]\right)+\frac{1}{4}\sum_{k=1}^{4}\sum_{i,j=1}^{3}\operatorname{Tr}\left(\left[X_{i}^{(k)},X_{j}^{(k)}\right]^{\dagger}\left[X_{i}^{(k)},X_{j}^{(k)}\right]\right)\,.$

where ‘bar’ denotes complex conjugation. We abuse the notation here as the
scalar components are being denoted with the same name $Z^{(kl)}$,
$\Phi_{i}^{(k)}$ etc. as the superfields. $X_{i}^{(k)}$ are gauge fields
arising from the dimension reduction of the vector fields of the vector
multiplets and we choose the gauge $A_{0}=0$. The second term is the D–term
potential and it takes the form,

$V_{D}=\frac{1}{2}\sum_{k=1}^{4}\operatorname{Tr}\left[\left(\sum_{\begin{subarray}{c}l=1\\\
l\neq
k\end{subarray}}^{4}Z^{(k\ell)}Z^{(k\ell)\dagger}-\sum_{\begin{subarray}{c}i=1\\\
i\neq k\end{subarray}}^{4}Z^{(\ell k)\dagger}Z^{(\ell
k)}+\sum_{i=1}^{3}\left[\Phi_{i}^{(k)},\Phi_{i}^{(k)\dagger}\right]-c^{(k)}I_{N_{k}}\right)^{2}\right]\,.$
(2.7)

The third component is the F–term potential which takes the following form,

$V_{F}=\sum_{k=1}^{4}\sum_{i=1}^{3}\left|\frac{\partial
W}{\partial\Phi_{i}^{(k)}}\right|^{2}+\sum_{k=1}^{4}\sum_{\begin{subarray}{c}\ell=1\\\
\ell\neq k\end{subarray}}^{4}\left|\frac{\partial W}{\partial
Z^{(k\ell)}}\right|^{2}\,.$ (2.8)

###  2.2 Symmetries

The above Lagrangian has various continuous and discrete symmetries which we
list here. Throughout the rest of the paper we have used the Gauge symmetries
and the shift symmetries to constraint the system, ultimately solving for the
SUSY ground states. In section 5 we used the discrete symmetries to show that
quartic and higher order terms are not allowed in the potential.

#### Gauge symmetries

The system originally has a $U(N_{1})\times U(N_{2})\times U(N_{3})\times
U(N_{4})$ gauge symmetry. As we have partially gauge fixed to $A_{0}=0$, we
can only focus on the global part of the gauge symmetry, with various fields
transforming as $Z^{(k\ell)}\rightarrow
U(N_{k})\,Z^{(k\ell)}\,U^{-1}(N_{\ell})$ and $\Phi^{(k)}_{i}\rightarrow
U(N_{k})\,\Phi^{(k)}_{i}\,U^{-1}(N_{k})$. In the later sections we will
discuss various ways to implement gauge fixing for various charge
configurations.

#### Shift symmetries

Type IIA string theory in ten space–time dimensions has 32 supercharges. The
Lagrangian described in the previous section is essentially that of a SUSY
quantum mechanics which has support on the intersection of four D–branes. It
breaks 28 out the 32 supercharges, resulting in 28 Goldstinos. The remaining
four supercharges ensure that the Goldstinos have their bosonic partners,
pairing up as $\mathcal{N}=1$ massless hyper–multiplets. These bosonic zero
modes are the consequence of spontaneous symmetry breaking due the insertion
of D–branes. We have called them shift symmetries and are given by,

$\displaystyle\Phi^{(k)}_{m}\to\Phi^{(k)}_{m}+\xi_{m}I_{N_{k}},\quad\hbox{for}\quad
1\leq k\leq 3,\quad k\neq m;\quad 1\leq m\leq 3,$
$\displaystyle\Phi^{(k)}_{k}\to\Phi^{(k)}_{k}+\zeta_{k}I_{N_{k}},\quad\Phi^{(4)}_{k}\to\Phi^{(4)}_{k}+\zeta_{k}I_{N_{4}},\quad\hbox{for}\quad
1\leq k\leq 3\,,$ $\displaystyle X^{(k)}_{i}\to
X^{(k)}_{i}+a_{i}\,I_{N_{k}}\,,\quad\hbox{for}\quad 1\leq i\leq 3\,.$ (2.9)

where $\xi_{m}$ and $\zeta_{k}$ are arbitrary complex numbers and $a_{i}$ are
arbitrary real numbers. The $a_{i}$’s represent overall translation of the
system along the non–compact directions. In the black hole microstate counting
programme, we mod out by these hyper–multiplets while computing the $B_{14}$
Helicity trace index. Therefore, we have to ‘gauge fix’ these flat directions
of the potential to compute the SUSY ground states 555As a consequence of
modding out by translations in the non–compact directions, the gauge
symmetries have to be modded out by an overall diagonal $U(1)$ i.e.
$(U(N_{1})\times U(N_{2})\times U(N_{3})\times U(N_{4}))/U(1).$ . Throughout
the paper while doing computations, we have chosen

$\Phi^{1}_{1}=0,\,\Phi^{1}_{2}=0,\,\Phi^{1}_{3}=0,\,\Phi^{2}_{1}=0,\,\Phi^{2}_{2}=0\,\text{and}\,\Phi^{3}_{3}=0\,.$
(2.10)

#### Discrete symmetries

Type IIA theory on $T^{6}$ exhibits various discrete exchange symmetries.
These symmetries constrain the form of the superpotentials $\mathcal{W}_{2}$
(2.3) and $\mathcal{W}_{4}$ (2.5) . Later, in section 5, we will use these
symmetries to show that the superpotential doesn’t admit additional terms of
quartic and beyond.

  * •

First Exchange Symmetry: We exchange the D2–brane stacks 1 and 2 i.e.
$x^{4}\leftrightarrow x^{6}$, $x^{5}\leftrightarrow x^{7}$ and
$N_{1}\leftrightarrow$ $N_{2}$. Moreover, $c^{(34)}$ changes sign, $c^{(12)}$
remains unchanged, and $c^{(1i)}$ and $c^{(2i)}$ get exchanged for $i=3,4$.
The superpotentials transform as $\mathcal{W}_{i}\rightarrow-\mathcal{W}_{i}$
for $1\leq i\leq 4$ and to satisfy this, various field should transform in the
following way,

$\displaystyle\left(\Phi_{3}^{(4)},\Phi_{3}^{(3)}\right)\rightarrow\left(\Phi_{3}^{(4)},\Phi_{3}^{(3)}\right),\quad\left(\Phi_{3}^{(2)},\Phi_{3}^{(1)}\right)\rightarrow\left(\Phi_{3}^{(1)},\Phi_{3}^{(2)}\right),$
(2.11)
$\displaystyle\left(\Phi_{1}^{(2)},\Phi_{1}^{(3)}\right)\leftrightarrow\left(\Phi_{2}^{(1)},\Phi_{2}^{(3)}\right),\quad\left(\Phi_{1}^{(4)},\Phi_{1}^{(1)}\right)\leftrightarrow-\left(\Phi_{2}^{(4)},\Phi_{2}^{(2)}\right),$
$\displaystyle Z^{(34)}\rightarrow Z^{(34)},\quad
Z^{(i1)}\leftrightarrow-Z^{(i2)},\quad
Z^{(1i)}\leftrightarrow-Z^{(2i)},\quad\text{ for }i=3,4,$ $\displaystyle
Z^{(12)}\leftrightarrow-Z^{(21)},\quad Z^{(43)}\rightarrow-Z^{(43)}\,.$

  * •

Second Exchange Symmetry: We exchange the D2–brane stacks 2 and 3 i.e.
$x^{6}\leftrightarrow x^{8}$, $x^{7}\leftrightarrow x^{9}$ and
$N_{2}\leftrightarrow N_{3}$. Moreover, we have
$c^{(14)}\rightarrow-c^{(14)},c^{(23)}\rightarrow
c^{(23)},c^{(12)}\leftrightarrow c^{(13)}$ and $c^{(24)}\leftrightarrow
c^{(34)}$. The superpotentials transform as
$\mathcal{W}_{i}\rightarrow-\mathcal{W}_{i}$ for $1\leq i\leq 4$ and to
satisfy this, various fields should transform in the following way,

$\displaystyle\left(\Phi_{1}^{(4)},\Phi_{1}^{(1)}\right)\rightarrow\left(\Phi_{1}^{(4)},\Phi_{1}^{(1)}\right),\quad\left(\Phi_{3}^{(1)},\Phi_{3}^{(2)}\right)\leftrightarrow\left(\Phi_{2}^{(1)},\Phi_{2}^{(3)}\right),$
(2.12)
$\displaystyle\left(\Phi_{1}^{(2)},\Phi_{1}^{(3)}\right)\rightarrow\left(\Phi_{1}^{(3)},\Phi_{1}^{(2)}\right),\quad\left(\Phi_{2}^{(4)},\Phi_{2}^{(2)}\right)\leftrightarrow-\left(\Phi_{3}^{(4)},\Phi_{3}^{(3)}\right),$
$\displaystyle Z^{(41)}\rightarrow Z^{(41)},\quad
Z^{(2i)}\leftrightarrow-Z^{(3i)},\quad
Z^{(i2)}\leftrightarrow-Z^{(i3)},\quad\text{ for }i=1,4,$ $\displaystyle
Z^{(32)}\leftrightarrow-Z^{(23)},\quad Z^{(14)}\rightarrow-Z^{(14)}\,.$

  * •

Third Exchange Symmetry: We exchange the D2–brane stacks 1 and 4. Here, it is
best to perform a T–duality transformation along 6–7–8–9 directions, exchange
$x^{6}\leftrightarrow x^{8}$, $x^{7}\leftrightarrow x^{9}$ and at the same
time exchange $N_{1}$ and $N_{4}$. Moreover, $c^{(14)}$ remains unchanged,
$c^{(23)}$ changes sign, $c^{(12)}\leftrightarrow ic^{(24)}$ and
$c^{(13)}\leftrightarrow ic^{(34)}$. The superpotentials transform as
$\mathcal{W}_{i}$ to $-\mathcal{W}_{i}$ for $1\leq i\leq 4$ and to satisfy
this, various fields transforms as follows:

$\displaystyle\left(\Phi_{1}^{(4)},\Phi_{1}^{(1)}\right)\rightarrow\left(\Phi_{1}^{(1)},\Phi_{1}^{(4)}\right),\quad\left(\Phi_{2}^{(1)},\Phi_{2}^{(3)}\right)\leftrightarrow
i\left(\Phi_{3}^{(4)},\Phi_{3}^{(3)}\right),$ (2.13)
$\displaystyle\left(\Phi_{1}^{(2)},\Phi_{1}^{(3)}\right)\rightarrow\left(\Phi_{1}^{(2)},\Phi_{1}^{(3)}\right),\quad\left(\Phi_{2}^{(4)},\Phi_{2}^{(2)}\right)\leftrightarrow
i\left(\Phi_{3}^{(1)},\Phi_{3}^{(2)}\right),$ $\displaystyle
Z^{(12)}\leftrightarrow-Z^{(42)},\quad Z^{(13)}\leftrightarrow-iZ^{(43)},\quad
Z^{(21)}\leftrightarrow-iZ^{(24)},\quad Z^{(31)}\leftrightarrow-Z^{(34)},$
$\displaystyle Z^{(14)}\leftrightarrow iZ^{(41)},\quad Z^{(32)}\rightarrow
Z^{(32)},\quad Z^{(23)}\rightarrow-Z^{(23)}\,.$

All other exchange symmetries are compositions of the above three symmetries.
There are other symmetries, for example, the worldsheet parity symmetry, under
which the NS–NS 2–form field changes sign. This is not a symmetry of type IIA
string theory, but if we accompany this by the parity transformation along the
non–compact directions and $(-1)^{F_{L}}$, this becomes a symmetry of the
theory. The details can be found in appendix A of [2].

###  2.3 Supersymmetric Vacua

The Lagrangian as described in section 2.1 has a potential,
$V=V_{\text{gauge}}+V_{D}+V_{F}$. The vacua manifold comprises of
supersymmetric solutions which are stationary solutions with energy $E=V=0$.
As the potential $V$ is a sum of square terms, vanishing of the potential
implies that each individual term equals to zero. Our first agenda is to
reduce as much as possible the number of fields over which we have to minimize
the potential. To that end, we pick the $V_{gauge}$ term, diagonalize the
gauge fields $X^{(k)}_{i}$ using the gauge symmetry, $U(N_{1})\times
U(N_{2})\times U(N_{3})\times U(N_{4})$ and following the arguments of
appendix A of [2], set all $X^{(k)}_{i}=0$ for $V_{gauge}=0$. We are now left
with the $V_{D}$ and $V_{F}$ terms in the potentials with active fields
$Z^{(k\ell)}$ and $\Phi^{(k)}_{i}$. Upon complexifying the original gauge
invariance to $(GL(N_{1},\mathcal{C}^{*})\times
GL(N_{2},\mathcal{C}^{*})\times GL(N_{3},\mathcal{C}^{*})\times
GL(N_{4},\mathcal{C}^{*}))/GL(1,\mathcal{C}^{*})$, we can effectively set
$V_{D}$ to zero 666For example, in the abelian case, the $Z^{(k\ell)}$ span a
projective variety but as no coordinates can go to zero as long as
$c^{(k\ell)}\neq 0\,$, it is a toric variety. In case coordinates are allowed
to go to zero, we have to perform the counting in different coordinate patches
and then take a union of the results.. We can now focus only on the vanishing
of the F–term potential, which results in the following F–term equations to be
solved to obtain the vacua manifold.

$\displaystyle Z^{(k\ell)}Z^{(\ell k)}$
$\displaystyle=-c^{(k\ell)}N_{\ell}I_{N_{k}}+\left[\Phi_{k}^{(k)},\Phi_{\ell}^{(k)}\right]\quad\text{
for }\quad 1\leq k,\ell\leq 3,$ (2.14) $\displaystyle Z^{(k4)}Z^{(4k)}$
$\displaystyle=-c^{(k4)}N_{4}I_{N_{k}}+\sum_{\ell,m=1}^{3}\varepsilon^{k\ell
m}\Phi_{\ell}^{(k)}\Phi_{m}^{(k)},\quad 1\leq k\leq 3,$ $\displaystyle
Z^{(4k)}Z^{(k4)}$
$\displaystyle=-c^{(k4)}N_{k}I_{N_{4}}-\sum_{\ell,m=1}^{3}\varepsilon^{k\ell
m}\Phi_{\ell}^{(4)}\Phi_{m}^{(4)},\quad 1\leq k\leq 3,$
$\displaystyle\sum_{m=1}^{3}\varepsilon^{k\ell m}\left(Z^{(\ell
k)}\Phi_{m}^{(k)}-\Phi_{m}^{(\ell)}Z^{(\ell
k)}\right)+C\sum_{\begin{subarray}{c}m=1\\\ m\neq
k,\ell\end{subarray}}^{4}Z^{(\ell m)}Z^{(mk)}(-1)^{\delta_{k1}\delta_{\ell
3}\delta_{m4}}=0\text{ for }1\leq k,\ell\leq 3,$ (2.15)
$\displaystyle\left(\Phi_{k}^{(k)}Z^{(k4)}-Z^{(k4)}\Phi_{k}^{(4)}\right)+C\sum_{\begin{subarray}{c}\ell=1\\\
\ell\neq k\end{subarray}}^{3}Z^{(k\ell)}Z^{(\ell
4)}(-1)^{\delta_{k1}\delta_{\ell 3}}=0\quad\text{ for }\quad 1\leq k\leq 3,$
$\displaystyle\left(Z^{(4k)}\Phi_{k}^{(k)}-\Phi_{k}^{(4)}Z^{(4k)}\right)+C\sum_{\begin{subarray}{c}m=1\\\
m\neq
k\end{subarray}}^{3}Z^{(4m)}Z^{(mk)}(-1)^{\delta_{m1}\delta_{k3}}=0\quad\text{
for }\quad 1\leq k\leq 3,$

To understand better the expectations from the vacua manifold lets look at a
standard Landau Ginsberg theory with $n$ chiral hyper–multiplets and four
supersymmetries [47, 48]. The Lagrangian is given by

$\begin{array}[]{r}\displaystyle\mathcal{L}=\sum_{i=1}^{n}\left(\left|\dot{z}_{i}\right|^{2}+i\tilde{\Psi}\partial_{t}\overline{\tilde{\Psi}}^{i}+i\bar{\Psi}^{\bar{i}}\partial_{t}\Psi^{i}-\frac{1}{4}\left|\partial_{i}\mathcal{W}\right|^{2}\right)\\\
\displaystyle-\frac{1}{2}\sum_{i,j}\left(\partial_{i}\partial_{j}\mathcal{W}\Psi^{i}\tilde{\Psi}^{j}+\bar{\partial}_{\bar{i}}\bar{\partial}_{\bar{j}}\bar{\mathcal{W}}\overline{\tilde{\Psi}}^{\bar{i}}\bar{\Psi}^{\bar{j}}\right),\end{array}$
(2.16)

where $\mathcal{W}(z)=-h(x,y)-if(x,y)$, the superpotential is a holomorphic
function of the complex fields, $z_{i}=x_{i}+iy_{i}\,$. Our lagrangian,
reduces to the above lagrangian after we remove the shift symmetries, put the
the gauge fields $X_{i}^{(k)}=0$ and complexify the gauge groups to set
$V_{D}=0$. Albeit, we still have to mod out by the complexified gauge orbits.
The critical points of the superpotential are associated with the
semi–classical ground states of the theory, i.e.

$\partial_{i}\mathcal{W}=0\text{ for all }i=1,\ldots,n$ (2.17)

which are critical points of $h(x,y)$ too. Assuming that $h(x,y)$ is suitably
generic so that there are no degenerate critical points, the Witten Index is
given by

$\operatorname{Tr}(-1)^{F}e^{-\beta H}=\sum_{(X,Y)}(-1)^{\mu(X,Y)}\,,$ (2.18)

where the sum is over all critical points $(X,Y)$ of $h(x,y)$. Here,
$\mu(X,Y)$, is the Morse Index,

$\mu(X,Y)=\text{ The number of negative eigenvalues of the {Hessian}
}-\partial_{i}\partial_{j}h(X,Y)\,.$ (2.19)

The ground state around the critical point $(X,Y)$ sits in the sector with
$\mu(X,Y)$ excited fermions, i.e. the ground state wavefunction is a $p$–form,
where $p=\mu(X,Y)$ is the Morse index 777 Generically, not all critical points
of $h(X,Y)$ correspond to true $E=0$ ground states of the theory. It may well
be that some get lifted to non–zero energy, but that is not relevant for
computing the Witten index since any such states must get lifted in pairs and
so cancel out..
Fortunately, for the SUSY Landu Ginzberg models $h(x,y)$ is not completely
generic. For example, if there is a critical point at say $(X,Y)=(0,0)$, then
after suitable diagonalization

$\mathcal{W}(z)\approx\sum_{i}\alpha_{i}\left(z^{i}\right)^{2}+\ldots$ (2.20)

while our original function $h(x,y)$ is given by

$h(x,y)=-\operatorname{Re}\mathcal{W}\approx-\sum_{i}\alpha_{i}\left(\left(x^{i}\right)^{2}-\left(y^{i}\right)^{2}\right)+\ldots$
(2.21)

Holomorphy of $\mathcal{W}$ ensures that for every positive eigenvalue of the
Hessian of $h$, there is a corresponding negative eigenvalue, which in turn
ensures that every critical point has Morse index given by $n$ and each
contributes exactly the same to the Witten index which becomes

$\operatorname{Tr}(-1)^{F}e^{-\beta H}=(-1)^{n}\times\text{ Number of critical
points of }\mathcal{W}$ (2.22)

i.e. every critical point of $\mathcal{W}$ is a true $E=0$ ground state of the
quantum theory. No semi–classical ground state gets lifted by quantum
tunneling/instanton effects, hence in our case degeneracy of critical points
equals Witten index up to a sign 888For an alternate perspective involving the
Lefschetz $SU(2)$ and middle cohomology of the vacua manifold, see section 2
of [2]..
In the next section we will solve the set of F–term equations by treating them
as polynomial varieties and discuss various numerical and exact methods to
solve a set of equations. We have also computed the Hessian for various
D–brane charges and explicitly verified the above claim.

##  3 Counting Black Holes as Algebraic Varieties

In this section we discuss various ways to solve the F–term and D–term
equations to get information about the vacuum moduli space of the the pure
D–brane system. The vacuum moduli space is best described as an algebraic
variety. The geometries we are interested in are affine varieties, which are
points, curves, surfaces and higher dimensional objects defined by polynomial
equations over complex field $\mathbb{C}$ 999For computer algorithms, we
mostly used rational number field $\mathbb{Q}$. Sometimes, using a Galois
field of integers mod $p$ where $p$ is a prime number speeds up the
computations.. The set of all polynomials over the field $\mathbb{C}$ form a
commutative ring $\mathbb{C}[x_{1},\ldots x_{n}]$, referred to as a polynomial
ring 101010$\mathbb{C}[x_{1},\ldots x_{n}]$ satisfies all field axioms except
for the existence of multiplicative inverses as $1/x$ is not a polynomial..
Given the field $\mathbb{C}$ and $n\in\mathbb{Z}_{\geq 0}$ , we can define the
$n$–dimensional affine space over $\mathbb{C}$ to be the set

$\mathbb{C}^{n}=\left\\{(a_{1},\ldots,a_{n})\,|\,a_{1},\cdots,a_{n}\in\mathbb{C}\right\\}\,.$
(3.1)

The link between algebra and geometry comes from the fact that a polynomial
$f(x_{1},\ldots,x_{n})$ can be regarded as a function over the affine space.
The set of all simultaneous solutions
$\left(a_{1},\ldots,a_{n}\right)\in\mathbb{C}^{n}$ of a system of polynomial
equations

$\displaystyle f_{1}\left(x_{1},\ldots,x_{n}\right)$ $\displaystyle=0$ (3.2)
$\displaystyle f_{2}\left(x_{1},\ldots,x_{n}\right)$ $\displaystyle=0$
$\displaystyle\vdots$ $\displaystyle f_{s}\left(x_{1},\ldots,x_{n}\right)$
$\displaystyle=0$

is known as an affine variety, $V=\mathcal{V}\left(f_{1},\ldots,f_{s}\right)$.
To make the connection to our system, we consider the fields $Z^{(k\ell)}$ and
$\Phi^{(k)}_{i}$ as the coordinates $x_{1},\ldots,x_{n}$ and the F–term
equations (2.14), (2.15) as the set of polynomials 111111Our D–brane system
have gauge symmetries which have been complexified, therefore we have a
projective space and the homogeneous F–term equations defines a projective
variety..
To make progress with Hilbert series to be discussed in the last subsection,
we recall the definition of an algebraic object, the ideal (similar to the
idea of a sub–space) such that
$\mathcal{I}\subseteq\mathbb{C}\left[x_{1},\ldots,x_{n}\right]$ satisfies the
following properties:

(i) $\displaystyle 0\in\mathcal{I}.$ (3.3a) (ii)
$\displaystyle\,\text{If}\,f,g\in\mathcal{I},\,\text{then}\,f+g\in\mathcal{I}.$
(3.3b) (iii)
$\displaystyle\,\text{If}\,f\in\mathcal{I}\,\text{and}\,h\in\mathbb{C}\left[x_{1},\ldots,x_{n}\right],\,\text{then}\,hf\in\mathcal{I}.$
(3.3c) (iv) $\displaystyle\,\text{The set}\,\left\langle
f_{1},\ldots,f_{s}\right\rangle=\left\\{\sum_{i=1}^{s}h_{i}f_{i}\mid
h_{1},\ldots,h_{s}\in\mathbb{C}\left[x_{1},\ldots,x_{n}\right]\right\\}\,\text{is
an ideal }\,.$ (3.3d)

By applying the Hilbert Basis theorem [49] which in turn relies upon the
existence of the Gr$\ddot{o}$bner basis [49, 50] for the ideal $\mathcal{I}$,
it can be shown that affine varieties are determined by ideals i.e.
$\mathcal{V}(\mathcal{I})=\mathcal{V}(f_{1},\ldots,f_{s})$ 131313In reverse we
have $\left\langle
f_{1},\ldots,f_{s}\right\rangle\subseteq\mathcal{I}(\mathcal{V}(f_{1},\ldots,f_{s}))$.
But, for algebraically closed fields like $\mathbb{C}$, we can complete the
ideal–variety correspondence by referring to the much celebrated Hilbert’s
Nullstellensatz [51] which states that
$\mathcal{I}(\mathcal{V}(\mathcal{I}))=\sqrt{\mathcal{I}}$ and
$\mathcal{V}(\sqrt{\mathcal{I}})=\mathcal{V}(\mathcal{I})$ where
$\sqrt{\mathcal{I}}$ is the radical ideal..
We will be mostly interested in the quotients of the polynomial rings, written
as a commutative ring $\mathbb{C}[x_{1},\ldots,x_{n}]/\mathcal{I}$, which is
the set of equivalence classes for congruence modulo $\mathcal{I}$, i.e.

$\mathbb{C}\left[x_{1},\ldots,x_{n}\right]/\mathcal{I}=\left\\{[g]\mid
g\in\mathbb{C}\left[x_{1},\ldots,x_{n}\right]\right\\}$ (3.4)

where the equivalence class $[g]$ is

$[g]=\left\\{f\in\mathbb{C}\left[x_{1},\ldots,x_{n}\right]\mid
f-g\in\mathcal{I}\right\\}\,.$ (3.5)

Polynomials with support on the variety $V$ forms a ring $\mathbb{C}[V]$
namely the coordinate ring which is isomorphic to
$R_{V}=\mathbb{C}[x_{1},\ldots,x_{n}]/\mathcal{I}(V)$, therefore the
“algebra–geometry” dictionary works for any variety $V$ and $\mathbb{C}[V]$
[49].
We will add more to this minimal introduction to computational algebraic
geometry as we move along to discuss various methods to efficiently count or
in some cases explicitly find solutions to a set of polynomial equations. We
shall now discuss four methods in their order of significance, namely the
Newton polytope, Homotopy continuation, Monodromy and Hilbert series. To put
things in perspective, for the first three methods, it is best to think of the
fields in our D–brane system as coordinates and the F–term equations as
generating a variety. A priori due to gauge symmetries we expect infinite
number of solutions as we move along the gauge orbits but upon gauge fixing we
expect and get finite number of solutions i.e. zero dimensional varieties with
degree equal to the number of solutions [52]. If this has not been the case,
the first three methods would have been practically useless 141414As mentioned
in the next section 3.1, for simpler cases like the case of abelian charges,
it is possible to enumerate the independent gauge invariant operators and
write the F–term equations in a gauge invariant way. It is increasingly
difficult to do so for the non–abelian cases due to the presence of unwanted
syzygies.. On the contrary, Hilbert series is a more robust quantity which
characterizes various important aspects of an algebraic variety. Here, it is
best to think of the F–term equations as an ideal modding out the polynomial
ring generated by the fields of the D–brane system. We will discuss both gauge
fixed and gauge invariant ways of computing the Hilbert series and extract the
information about the dimension and degree of the corresponding varieties.

###  3.1 Newton Polytope Method

The study of geometric and combinatorial properties of convex polytopes is an
old subject, yet applications of these techniques to questions of mathematical
programming is only few decades old [53, 54]. Here, we are mainly interested
in Newton polytopes which continue to play a pivotal role in analyzing
polynomial systems of equations. We will use two central results for a system
of polynomial equations:

  * •

Bezout’s Theorem: It states that the number of isolated roots of the system is
bounded by the product of their degrees [55, 56].

  * •

Bernstein’s Theorem: It states that the number of solutions of a generic
system almost equals 151515The theorem is for sparse–square systems i.e. for
equal number of variables and polynomials. Our systems are sparse–square
systems. Also, the solutions are assumed to be in the dense tori
$(\mathbb{C}^{*})^{s}$. This is true for abelian charges with $c^{(k\ell)}\neq
0$. For non–abelian case some components of the fields might take zero values
at the solutions. To get around it one may use Cox homotopy [57] by thinking
of the sparse system as a toric variety. the mixed volume of Newton polytopes
corresponding to the polynomial system [58, 54].

Even for the Homotopy continuation method to be discussed in the next section
3.2, the construction of the start system requires computing mixed volumes of
newton polytopes (polyhedral homotopy) [59]. We can potentially solve our
F–term polynomial equations using this method which we have done for the
abelian charges.
A polytope is a subset of $\mathbb{R}^{s}$ that is the convex hull of a finite
set of points,
$\mathcal{A}=\left\\{m_{1},\ldots,m_{\ell}\right\\}\subset\mathbb{R}^{s}$ and
expressed as,

$\operatorname{Conv}(\mathcal{A})=\left\\{\lambda_{1}m_{1}+\cdots+\lambda_{\ell}m_{\ell}:\lambda_{i}\geq
0,\,\sum_{i=1}^{\ell}\lambda_{i}=1\right\\}$ (3.6)

where, $\operatorname{Conv}(\mathcal{A})$ is the convex hull of
$\mathcal{A}\subset\mathbb{R}^{s}$. A $s$–dimensional polytope has many faces,
which are again polytopes of various dimensions between 0 and $(s-1)$.
Vertices, edges and facets are the zero, one and $(s-1)$–dimensional faces
respectively 161616For example the convex hull of
$\\{(0,0,0),\,(0,1,0),(0,0,1),\,(0,1,1),\,(1,0,0)$,
$(1,1,0),\,(1,0,1),\,(1,1,1)\\}$ in $\mathbb{R}^{3}$ is the cube. It has has 8
vertices, 12 edges, and 6 facets.. Consider the following polynomial,

$f(x,y)=a_{1}x^{u_{1}}y^{v_{1}}+a_{2}x^{u_{1}}y^{v_{1}}+\cdots+a_{m}x^{u_{m}}y^{v_{m}}\,.$
(3.7)

Each monomial $x^{u_{i}}y^{v_{i}}$ appearing in $f(x,y)$ corresponds to a
lattice point $\left(u_{i},v_{i}\right)$ in the plane $\mathbb{R}^{2}$. The
convex hull of all these points (known as ‘exponent vectors’) is the Newton
polytope of $f(x,y)$ 171717We always look for the minimal convex hull, some
vertices can be on facets or inside the hull. Therefore, the hull has at most
$m$ vertices.,

$\operatorname{New}(f)=\operatorname{Conv}\left\\{\left(u_{1},v_{1}\right),\left(u_{2},v_{2}\right),\ldots,\left(u_{m},v_{m}\right)\right\\}\,.$
(3.8)

Another important notion is that of a Minkowski sum. If $f_{1}$ and $f_{2}$
are any two polytopes in $\mathbb{R}^{2}$, then their Minkowski sum is the
polytope,

$f_{1}+f_{2}:=\\{p_{1}+p_{2}\,|\,p_{1}\in f_{1},\,p_{2}\in f_{2}\\}.$ (3.9)

where each edge of $f_{1}+f_{2}$ is parallel to an edge of either $f_{1}$ or
$f_{2}$. Geometrically taking the Minkowski sum is mirrored as the algebraic
operation of multiplying the polynomials

$\operatorname{New}(f_{1}\,f_{2})=\operatorname{New}(f_{1})+\operatorname{New}(f_{2})\,.$
(3.10)

The number of solutions in a square system of $s$ polynomials in $s$ unknowns
equals the mixed volume $\mathcal{M}$ of the $s$ Newton polytopes. If
$f_{1},f_{2},\ldots,f_{s}$ are polytopes in $\mathbb{R}^{s}$ then their mixed
volume can be defined by the inclusion–exclusion formula

$\mathcal{M}\left(f_{1},f_{2},\ldots,f_{s}\right)=\sum_{T\subseteq\\{1,2,\ldots,,s\\}}(-1)^{s-\\#(T)}\cdot\operatorname{Vol}\left(\sum_{s\in
T}f_{T}\right).$ (3.11)

Equivalently, $\mathcal{M}\left(f_{1},f_{2},\ldots,f_{s}\right)$ is the
coefficient of the monomial $\lambda_{1}\lambda_{2}\cdots\lambda_{s}$ in the
expansion of

$V\left(\lambda_{1},\ldots,\lambda_{s}\right)=\operatorname{Vol}\left(\lambda_{1}\,f_{1}+\lambda_{2}\,f_{2}+\cdots+\lambda_{s}\,f_{s}\right)\,,$
(3.12)

which is a homogeneous polynomial of degree $s$. Here ‘$\operatorname{Vol}$’
denotes the usual Euclidean volume in $\mathbb{R}^{s}$ and
$\lambda_{1},\ldots,\lambda_{s}\geq 0$. From Bernstein’s Theorem, the number
of solutions to the polynomial equations (3.2) is
$\mathcal{M}\left(f_{1},f_{2},\ldots,f_{s}\right)$ 181818It also works for
Laurent polynomials with negative or mixed exponents..

####  3.1.1 Abelian case

We now apply the above method to solve the system of F–term equations coming
from (2.15) after eliminating the $\Phi$ fields. The system of equations are

$\displaystyle
Z^{(23)}Z^{(31)}Z^{(12)}+Z^{(23)}Z^{(34)}Z^{(42)}=Z^{(32)}Z^{(21)}Z^{(13)}+Z^{(32)}Z^{(24)}Z^{(43)}$
(3.13) $\displaystyle
Z^{(24)}Z^{(41)}Z^{(12)}+Z^{(24)}Z^{(43)}Z^{(32)}=Z^{(42)}Z^{(21)}Z^{(14)}+Z^{(42)}Z^{(23)}Z^{(34)}$
$\displaystyle
Z^{(34)}Z^{(41)}Z^{(13)}+Z^{(34)}Z^{(42)}Z^{(23)}=Z^{(43)}Z^{(31)}Z^{(14)}+Z^{(43)}Z^{(32)}Z^{(24)}$

and

$Z^{(k\ell)}Z^{(\ell K)}=-c^{(k\ell)}\quad\text{ for }\quad 1\leq k<\ell\leq
4\,.$ (3.14)

We now define gauge invariant variables $u$’s, invariant under the $U(1)\times
U(1)\times U(1)\times U(1)$ gauge transformations as

$\displaystyle u_{1}\equiv Z^{(12)}Z^{(21)},\quad u_{2}\equiv
Z^{(23)}Z^{(32)},\quad u_{3}\equiv Z^{(31)}Z^{(13)}$ (3.15) $\displaystyle
u_{4}\equiv Z^{(14)}Z^{(41)},\quad u_{5}\equiv Z^{(24)}Z^{(42)},\quad
u_{6}\equiv Z^{(34)}Z^{(43)}\text{, }$ $\displaystyle u_{7}\equiv
Z^{(12)}Z^{(24)}Z^{(41)},\quad u_{8}\equiv Z^{(13)}Z^{(34)}Z^{(41)},\quad
u_{9}\equiv Z^{(23)}Z^{(34)}Z^{(42)}\,.$

The variables $(u_{1},\ldots,u_{6})$ simply take the values $-c^{(k\ell)}$ and
the remaining equations (3.13) can be written as

$\displaystyle
a_{1}u_{7}^{2}u_{9}^{2}-a_{2}u_{7}u_{8}-a_{3}u_{8}^{2}-u_{7}u_{8}u_{9}^{2}$
$\displaystyle=0$ (3.16) $\displaystyle
u_{7}^{2}u_{9}-u_{7}u_{9}^{2}-a_{4}u_{9}+a_{5}u_{7}$ $\displaystyle=0$
$\displaystyle u_{8}^{2}u_{9}-a_{6}u_{8}-a_{7}u_{9}+u_{8}u_{9}^{2}$
$\displaystyle=0\,,$

where $a_{i}$ are generic functions of the tunable constants $c^{(k\ell)}$. To
apply the Newton polytope method, it is important that $a_{i}$’s are generic
and $a_{i}\neq 0$.
The exponent vectors of the above three polynomials $(f_{1},\,f_{2},\,f_{3})$
are $\\{(2,0,2),\,(1,1,0),\,(0,2,0),\\\ (1,1,2)\\}$,
$\\{(2,0,1),\,(1,0,2),\,(0,0,1),\,(1,0,0)\\}$ and
$\\{(0,2,1),\,(0,1,0),\,(0,0,1),\,(0,1,2)\\}$ respectively, see figure 1. The
number of solution to these three polynomial equations will be the mixed
volume of $f_{1}$, $f_{2}$ and $f_{3}$, i.e.

$\displaystyle\mathcal{M}\left(f_{1},\,f_{2},\,f_{3}\right)=$
$\displaystyle\operatorname{Vol}(f_{1}+f_{2}+f_{3})-\left\\{\operatorname{Vol}(f_{1}+f_{2})+\operatorname{Vol}(f_{1}+f_{3})+\operatorname{Vol}(f_{2}+f_{3})\right\\}$
(3.17)
$\displaystyle+\left\\{\operatorname{Vol}(f_{1})+\operatorname{Vol}(f_{2})+\operatorname{Vol}(f_{3})\right\\}\,.$

It turns out that $f_{1}$, $f_{2}$ and $f_{3}$ are 2–dimensional polytopes in
$\mathbb{R}^{3}$, hence they have zero volume. Remaining volumes are non zero
and has been computed by writing a code in Mathematica [36] and by using
software package SAGE [60]. The values of the volumes are as below,

$\operatorname{Vol}(f_{1}+f_{2})=\operatorname{Vol}(f_{1}+f_{3})=\operatorname{Vol}(f_{1}+f_{3})=8,\,\operatorname{Vol}(f_{1}+f_{2}+f_{3})=36\,$
(3.18)

and $\mathcal{M}\left(f_{1},\,f_{2},\,f_{3}\right)=12$, which matches with
earlier results [1, 2]. We can start with the gauge fixed version of the
abelian charges and repeat the calculations with more sophisticated algorithms
using mixed subdivision of newton polytopes [61, 59], the result remains the
same.

(a) Newton Polytope for $f_{1}+f_{2}$
(14 vertices)

(b) Newton Polytope for $f_{1}+f_{3}$
(14 vertices)

(c) Newton Polytope for $f_{2}+f_{3}$
(14 vertices)

(d) Newton Polytope for $f_{1}+f_{2}+f_{3}$
(26 vertices)

Figure 1: Newton Polytopes for charge $(1,1,1,1)\,$.

####  3.1.2 Non–abelian cases

In principle, by applying the above method we should be able to obtain the
correct number of solutions to the gauge fixed F–term equations, but there are
caveats.

  * •

As we increase the charges $(N)$, the number of variables grows as
$\mathcal{O}(N^{2})$. Newton polytope method is a $\mathbb{P}$–hard problem
[62], so computations will take forever 191919The number of
$\operatorname{Vol}$ computations scales as $2^{s}=e^{s\ln 2}$. So, complexity
scales as $2^{s}\times$ $\mathbb{P}$–hardness of Convex hull volume
computation..

  * •

The solutions are supposed to be on a dense tori $(\mathbb{C}^{*})^{s}$ but
for non–abelian cases some field components could take values in $\mathbb{C}$
i.e. be zero.

  * •

Most importantly, the method is valid for generic coefficients of the
polynomials and for non–abelian cases the coefficients of our F–term equations
are not generic.

Let us expand on the last point. In [2], it was shown that if we choose the
superpotential term $\mathcal{W}_{4}$ as

$\displaystyle\mathcal{W}_{4}=$
$\displaystyle-\sqrt{2}\,C\left[\operatorname{Tr}\left(\Phi_{1}^{(1)}\Phi_{2}^{(1)}\Phi_{3}^{(1)}-\Phi_{1}^{(1)}\Phi_{3}^{(1)}\Phi_{2}^{(1)}\right)-\operatorname{Tr}\left(\Phi_{1}^{(2)}\Phi_{2}^{(2)}\Phi_{3}^{(2)}-\Phi_{1}^{(2)}\Phi_{3}^{(2)}\Phi_{2}^{(2)}\right)\right.$
(3.19)
$\displaystyle\left.+\operatorname{Tr}\left(\Phi_{1}^{(3)}\Phi_{2}^{(3)}\Phi_{3}^{(3)}-\Phi_{1}^{(3)}\Phi_{3}^{(3)}\Phi_{2}^{(3)}\right)+\operatorname{Tr}\left(\Phi_{1}^{(4)}\Phi_{2}^{(4)}\Phi_{3}^{(4)}-\Phi_{1}^{(4)}\Phi_{3}^{(4)}\Phi_{2}^{(4)}\right)\right]$

with $C=1$, we get the correct counting 56 for charges $(1,1,1,2)$ and 208 for
charges $(1,1,1,3)$. If we choose $C\neq 1$, we get wrong counts or if we
choose $\mathcal{W}_{4}$ to be

$\displaystyle\mathcal{W}_{4}=$
$\displaystyle-\sqrt{2}\left[\operatorname{Tr}\left(\Phi_{1}^{(1)}\Phi_{2}^{(1)}\Phi_{3}^{(1)}-\Phi_{1}^{(1)}\Phi_{3}^{(1)}\Phi_{2}^{(1)}\right)+\operatorname{Tr}\left(\Phi_{1}^{(2)}\Phi_{2}^{(2)}\Phi_{3}^{(2)}-\Phi_{1}^{(2)}\Phi_{3}^{(2)}\Phi_{2}^{(2)}\right)\right.$
(3.20)
$\displaystyle\left.+\operatorname{Tr}\left(\Phi_{1}^{(3)}\Phi_{2}^{(3)}\Phi_{3}^{(3)}-\Phi_{1}^{(3)}\Phi_{3}^{(3)}\Phi_{2}^{(3)}\right)+\operatorname{Tr}\left(\Phi_{1}^{(4)}\Phi_{2}^{(4)}\Phi_{3}^{(4)}-\Phi_{1}^{(4)}\Phi_{3}^{(4)}\Phi_{2}^{(4)}\right)\right]\,,$

we get 60 for charges $(1,1,1,2)$ and 232 for charges $(1,1,1,3)$, again wrong
counts. Therefore, the polynomials of our system are not generic and the
answer depends on the specific coefficients. Though, we will not pursue it any
further, we have explained the method in some detail as it forms the basis for
Homotopy continuation method discussed in section 3.2.

###  3.2 Homotopy Continuation Method

A fundamental tool in computational algebraic geometry is homotopy
continuation, which computes the isolated solutions to a polynomial system
$F=\\{f_{1},\ldots,f_{s}\\}$ by numerically tracking paths that interpolate
between the solutions of $F$ and an already known solutions of a similar
system $G=\\{g_{1},\ldots,g_{s}\\}$ [63, 64, 38]. For the purposes of this
paper we assume both $F$ and $G$ are square sparse systems that define zero
dimensional varieties 202020Both over–determined and under–determined systems,
as well as projective varieties can be solved.. We have used a modified
version of HomotopyContinuation.jl [65], a Julia package designed for this
purpose to solve the F–term equations for charges $(1,1,1,1)$, $(1,1,1,2)$,
$(1,1,1,3)$ and the previously unreported $(1,1,1,4)$. In all these cases, the
supersymmetric vacua count matches with the numbers coming from the U–dual
system, appendix B.
We begin by defining $H(x;\tau)\subset$
$\mathbb{C}\left[x_{1},\ldots,x_{s}\right][\tau]$ as a one–parameter family of
equations with parameter $\tau\in\mathbb{C}_{\tau}$ such that both $H(x;0)=G$
and $H(x;1)=F$ belong to the family. Then
$\mathcal{V}(H)\subset\mathbb{C}^{s}\times\mathbb{C}_{\tau}$ is a complex
family of varieties realized as a fibre bundle by the projection
$\pi:\mathcal{V}(H)\rightarrow\mathbb{C}_{\tau}$ i.e. $\mathcal{V}(G)$ in the
fiber over $\tau=0$ and $\mathcal{V}(F)$ over $\tau=1\,$. Let $C$ denote the
union of all the one–dimensional components of $\mathcal{V}(H)$ such that its
restriction, $\left.C\right|_{\gamma}\,$, to an arc
$\gamma\subset\mathbb{C}_{\tau}$ between 0 and 1 is smooth and allows for
tracking of every point in $\mathcal{V}(G)$ at $\tau=0$ to a point in
$\mathcal{V}(F)$ at $\tau=1$ 212121This is possible if a point
$(x;\tau)\in\mathcal{V}(H)$ is non–degenerate i.e. the Jacobian of $H$ with
respect to the $x$–variables, $D_{x}H$, is invertible at $(x;\tau)\,$.. An
example of a simple homotopy is the straight–line homotopy, which is defined
by $H(x;\tau)=G(1-\tau)+F\tau$.

#### Path tracking

Figure 2: Schematic of path tracking, showing prediction and correction steps.

Path tracking amounts to numerically solving an initial value problem. For
every solution, by the Implicit Function Theorem, there is a parametrization
$x(\tau)$ for each arc such that $H(x(\tau);\tau)=0$ and $x(0)=x_{0}$ is
known. Differentiating $H(x(\tau);\tau)=0$ with respect to $\tau$ gives a
Davidenko differential equation [66] as a initial value problem,

$\left\\{\begin{array}[]{l}\displaystyle\frac{dx}{d\tau}=-\left(D_{x}H\right)^{-1}\frac{\partial
H}{\partial\tau}\vspace{0.25cm}\\\ x(0)=x_{0}.\end{array}\right.$ (3.21)

which can be solved by applying any numerical differential equation solver. To
maintain numerical stability, it is best to use the predictor–corrector
algorithm 222222Starting from a point close to the solution curve Euler
Forward method (predictor) generates a solution one step further, then
Newton’s method of root approximation (corrector) is used to refine the
solution at that step and the iterations continues till $x(\tau)$ is found at
$\tau=1$. It is reasonably straightforward to parallelize the path tracking.
It can be further improved by the use of GPU [67]. [44], see Figure 2. Path
tracking rely on inverting $D_{x}H$, and will fail close to $\tau=1$ if
$\mathcal{V}(F)$ contains singular solutions. In such cases tracking is
stopped very close to $\tau=1$ and an endgame algorithm like Cauchy endgame
[68] takes over.

#### Start system

The choice of start system $G$ and homotopy $H$ is important to maintain the
efficacy and efficiency of homotopy algorithms. The start system $G$ should
have at least the same number of solutions as $F$ and if it has more solutions
then excess paths diverge to infinity 232323This happens for example in Bézout
or total–degree homotopy, where
$G=\\{x_{1}^{d_{1}}-1=0,\,x_{2}^{d_{2}}-1=0,\cdots,x_{s}^{d_{s}}-1=0\\}$ and
$d_{i}=\operatorname{deg}(f_{i})$. The Bézout bound $d_{1}d_{2}\ldots d_{s}$
is usually much larger than the $\operatorname{deg}(\mathcal{V}(F))$.. A
homotopy $H(x;\tau)$ is optimal if $F$ and $G$ have the same number of
solutions when counted with multiplicity, and if the paths connecting
$\mathcal{V}(G)$ and $\mathcal{V}(F)$ are smooth everywhere except possibly at
$\tau=1$. The algorithms we have used relied on polyhedral homotopy [59] which
used toric degenerations [69] such that the start system, $\mathcal{V}(G)$, is
a toric variety and used tropical geometry to enumerate the mixed cells of a
mixed subdivisions [70].

#### Our strategy

The F–term equations (2.14), (2.15) for both the abelian and non–abelian cases
are not a square sparse system; not all of them are independent. We can
convert it to a square system by appropriately fixing the gauge and using the
phase symmetries (2.10), along with the removal of the corresponding F–term
equations from the polynomial equations list. In all cases, some of the
remaining F–term equations now become linear, we can use this opportunity to
eliminate them and reduce the system a bit further. With these reduced set of
variables and F–term equations forming a square system, we discuss below the
cases of our interest.

####  3.2.1 Abelian case

$\begin{array}[]{|c|c|}\hline\cr Z^{(12)}&1\\\ \hline\cr Z^{(34)}&1\\\
\hline\cr Z^{(13)}&1\\\ \hline\cr Z^{(21)}&-1.11\\\ \hline\cr
Z^{(31)}&-1.13\\\ \hline\cr Z^{(14)}&0.589628+0.0708712i\\\ \hline\cr
Z^{(41)}&-1.90589+0.229081i\\\ \hline\cr Z^{(23)}&-0.424973-0.792931i\\\
\hline\cr Z^{(32)}&0.588095-1.09729i\\\ \hline\cr
Z^{(24)}&0.41384-0.601617i\\\ \hline\cr Z^{(42)}&-0.892554-1.29754i\\\
\hline\cr Z^{(43)}&-1.16\\\ \hline\cr\Phi^{(2)}_{3}&0.153779-1.92561i\\\
\hline\cr\Phi^{(3)}_{1}&0.151189-1.92426i\\\
\hline\cr\Phi^{(3)}_{2}&1.10894+0.875142i\\\
\hline\cr\Phi^{(4)}_{1}&-1.10085-0.888016i\\\
\hline\cr\Phi^{(4)}_{2}&0.145618-1.89443i\\\
\hline\cr\Phi^{(4)}_{3}&-1.08305-0.887995i\\\ \hline\cr\end{array}$

Table 1: One of the 12 SUSY vacuum for charges $(1,1,1,1)$. First three
entries denote the gauge fixing.

After fixing the gauge, the F–term equations become a square system of 15
polynomials in 15 variables. With parallel path tracking, 16 paths were
tracked very quickly (seconds) to give 12 non–singular solutions which of
course matches with the expectation. Here, we would like to discuss further
two points related to the Hessian and the D–terms which are also valid for the
non–abelian cases.
The real Hessian $h$ around all explicit solutions for both abelian and
non–abelian cases can be computed. It is also possible to diagonalize the
Hessian and look at it eigenvalues and eigenvectors 242424For non–abelian
cases, even with large number of variables, standard routines in python will
do the job.. For concreteness, we look at eigenvalues of the real Hessian for
a particular solution of the abelian charges, Table 1 with $C=1$,
$c^{(12)}=1.11$, $c^{(23)}=1.12$, $c^{(13)}=1.13$, $c^{(14)}=1.14$,
$c^{(24)}=1.15$ and $c^{(34)}=1.16\,$. The list of eigenvalues is
$\\{113.174,\,-113.174,\,113.171,\,-113.171,\,112.223,\\\
\,-112.223,\,-112.197,\,112.197,\,-112.052,\,112.052,\,112.03,\,-112.03,\,1.90029,\,-1.90029,\,\\\
1.89812,\,-1.89812,\,-1.15207,\,1.15207,\,-1.15027,\,1.15027,\,1.02881,\,-1.02881,\,-1.01489,\,\\\
1.01489,\,-0.852902,\,0.852902,\,-0.841017,\,0.841017,\,-0.00169278,\,0.00169278,\,0,\,0,\,0,\,0,\,0,\,0\\}$.
As discussed in section 2.3, there are equal numbers of positive and negative
eigenvalues. The six zero eigenvalues are the massless Goldstone modes coming
from gauge fixing the global complexified gauge $U(1)\times U(1)\times U(1)$.
The same can be verified for the other 11 solutions and also for all solutions
of the non–abelian cases.
Starting with a particular gauge fixed solution of the F–term equations, we
can move along the complex gauge orbits to generate an infinite set of
equivalent F–term solutions. Picking one such solution from the orbit will not
automatically solve the D–term equations which are invariant under the
physical gauge transformations. Therefore, for both abelian and non–abelian
cases, our F–terms solutions don’t satisfy the D–term equations and the
potential $V_{D}$ (2.7) evaluates to a non zero number. Then for each
solution, we retain the ‘physical gauge fixing’ and set up a minimization
problem of $V_{D}=0$ with the additional ‘complex’ part of the gauge group as
parameters. Upto physical gauge transformations, in all cases we find unique
solutions which satisfy both the F–term and D–term equations simultaneously
252525As an illustration, take the abelian case where the superpotential is
invariant under the complexified $U(1)$ gauge transformation
$Z^{(k\ell)}\rightarrow a_{k}\left(a_{\ell}\right)^{-1}Z^{(k\ell)}$ for $1\leq
k,\,\ell\leq 4,\,k\neq\ell$ and all $\Phi$ fields are gauge invariant. We can
set $a_{4}=1$ to mod out the overall diagonal $U(1)$. Moving along the gauge
orbits, suppose we find on set of $a_{1}$, $a_{2}$ and $a_{3}$ such that the
D–terms are satisfied. The physical gauge transformation $a_{k}\rightarrow
e^{i\phi_{k}}a_{k}$ also satisfies the D–term equations. Therefore, using the
transformations generated by the $\phi_{k}$’s we can fix the ‘gauge’ for
$a_{1},a_{2}$ and $a_{3}$ by setting $a_{k}$’s to be real and positive and
then minimize $V_{D}$ over these parameters..

####  3.2.2 Non–abelian cases

We report here the results of applying the homotopy continuation method to
F–term equations of non–abelian charges reduced to square sparse systems.

  * •

Charge $(1,1,1,2)$: This sparse system of 24 variable and 24 equations
resulted in the tracking of about 297 paths resulting in 56 non–singular
solutions as expected.

  * •

Charge $(1,1,1,3)$: This sparse system of 32 variable and 32 equations
resulted in the tracking of about 3736 paths resulting in 208 non–singular
solutions as expected 262626It should be noted that it took only a minute in
our PC while Mathematica latest version took more than 5 hours. For charges
$(1,1,1,4)$ Mathematica failed but homotopy took a little more than two
hours..

  * •

Charge $(1,1,1,4)$: This sparse system of 47 variables resulted in the
tracking of about 2.5 lakh paths resulting in 684 non–singular solutions as
expected.

It is clear from these numbers that complexity is increasing at a fast pace.
Again it is the mixed cell computations that are $\mathbb{P}$–hard but a few
more higher charges can be tried if one is willing to wait for hours. In the
next section we will describe an arguably better way to construct the start
systems.

###  3.3 Monodromy Method

In the previous subsection on Homotopy continuation, the main bottleneck has
been the construction of a generic start system $G$ such that
$\operatorname{deg}(\mathcal{V}(G))\geq\operatorname{deg}(\mathcal{V}(F))$. As
an alternative and often faster route, we could exploit the action of a
monodromy group to explore the variety $\mathcal{V}(F)$ [71, 37]. We begin by
deforming the square sparse system $F$ to a $k$–parameter family of
polynomials,
$F(x;p):\mathbb{C}^{s}\times\mathbb{C}^{k}\rightarrow\mathbb{C}^{s}$, such
that

$F(x,p)=\left[\begin{array}[]{c}f_{1}\left(x_{1},\ldots,x_{s};p_{1},\ldots,p_{k}\right)\\\
\vdots\\\
f_{s}\left(x_{1},\ldots,x_{s};p_{1},\ldots,p_{k}\right)\end{array}\right]=0\,,$
(3.22)

For generic choices of parameters $p\in\mathbb{C}^{k}$ it is expected that
isolated solutions of $F(x;p)$ will remain constant. Hence, by using parameter
homotopy $\displaystyle H(x,\tau)=F(x,\gamma(\tau))=0$ [38], specific
solutions for parameters $p$ can be tracked to other parameter values
$p^{\prime}$ by following continuous paths
$\gamma(\tau):[0,1]\rightarrow\mathbb{C}^{k}$ such that $\gamma(0)=p$ and
$\gamma(1)=p^{\prime}$. To see what happens if $\gamma$ is chosen to be closed
path, we notice that the projection
$\pi:\mathcal{V}(F(x;p))\rightarrow\mathbb{C}^{k}$ defined by $(x,p)\mapsto
p$, has an open set $U\subset\mathbb{C}^{k}$ where the restriction of $\pi$
onto $U$ is generically a finite morphism of degree
$d=\operatorname{deg}(\mathcal{V}(F))$. A loop in $U$ based at $p$ has $d$
lifts to $\pi^{-1}(U)$, one for each point in the fiber $\pi^{-1}(p)$. The
monodromy group of the map $\pi$ is the image of the usual permutation action
of the fundamental group of $U$ in the space of solutions $S_{d}$. Therefore,
if one solution for some $p^{*}\in\mathbb{C}^{k}$ is known, then we track the
solution $x^{*}\in\mathbb{C}^{s}$ to a new solution
$x^{\prime}\in\mathbb{C}^{s}$ of $F(x;p)$ by following a closed path (loop) in
parameter space such that $\gamma(0)=p^{*}=\gamma(1)$. If $S_{r}\subseteq
S_{d}$ is a set of solutions obtained by repeating this process $r$ times, we
obtain a new solution set by taking a new random loop to track $S_{r}$ to
$S_{r}^{\prime}$ and letting $S_{r+1}=S_{r}\cup S_{r}^{\prime}$. Notice that
$S_{r}^{\prime}$ may coincide with $S_{r}$, but we always have the inclusions
$S_{r}\subseteq S_{r+1}\subseteq S_{d}$. We continue constructing random
monodromy paths until $S_{r+1}=S_{d}$.

#### Our Strategy

We make the following trivial parameter deformation on our F–term equations

$F(x,p)=\left[\begin{array}[]{c}f_{1}(x_{1},\ldots,x_{s})-p_{1}\\\ \vdots\\\
f_{s}(x_{1},\ldots,x_{s})-p_{s}\end{array}\right]=0\,.$ (3.23)

such that it is trivial to find a solution for $p^{*}$ for any random values
for $x^{*}$. We take one such pair $(x^{*},p^{*})$ as the seed solution of
$F(x;p)$ and then carry out the monodromy anchored at $p^{*}$. Once we have
the set of solutions for $F(x;p^{*})=0$, we run a parameter homotopy on this
set tracking solutions from $\gamma(0)=p^{*}$ to $\gamma(1)=0$, thereby
generating the final set of solutions for $F(x;0)$ 272727Alternatively, we
could have performed the parameter homotopy at the beginning to go from the
solution $F(x^{*},p^{*})$ to $F(x^{\prime},0)$ and then performed the
monodromy around $p=0$ to generate all solution of $F(x;0)$.. The
computational advantage of the monodromy method is the parallelizability of
both the monodromy computations and the path tracking of parameter homotopy.

#### Results

The results for the abelian charge and non–abelian charges up to $(1,1,1,3)$
are as expected. For charges $(1,1,1,4)$ it missed two solutions. On an
average the monodromy method is about ten times faster than the polyhedral
homotopy discussed in the previous subsection. We believe this method requires
further attention primarily because of two reasons

  * •

Many other ways are available to $k$–parameterize the F–term equations. We
took the most straightforward one.

  * •

The solution sets have discrete symmetries, for example a discrete
$\mathbb{Z}_{2}$ coming from complex conjugation of all fields. It should be
possible to incorporate them automatically while populating the fibres
$\pi^{-1}(p^{*})$ or $\pi^{-1}(0)$.

Its original implementation in terms of minimizing a cost function also
deserves further attention [72]. Recently, the efficacy of Stochastic Gradient
Descent [73] in minimizing the square potential $V_{F}$ for $\mathcal{N}=8$
black holes is discussed in [74]. The results are not encouraging beyond
charges $(1,1,1,2)$. We believe that even in applications to machine learning,
where the loss function in many neural network architectures, have
exponentially many low lying local minima [75], more sophisticated techniques
like the current method will do better.
We will now move from numerical solutions to exact (symbolic) techniques in
the next sub–section, where similar to the Newton polytope method discussed at
the beginning of this section we will just count the number of solutions to
the F–term equations without ever having to know the solutions explicitly.

###  3.4 Hilbert Series

So far we have discussed three methods to deal with solving polynomial
equations but they have their limitations, the biggest of all being limited to
mostly zero dimensional varieties. We shall now discuss a very important
quantity characterizing an algebraic variety, namely the Hilbert series. We
have used it in a limited capacity to count the number of gauge invariant
operators which tells us about the dimension and degree of the variety – the
vacua manifold. In our cases the structure of the vacua manifold are not
complicated, it is essentially the ‘Higgs branch’ that too zero dimensional.
We believe that moving away from the generic background metric and $B$–field
moduli space will result in more complicated structures for the vacua
manifolds but hopefully the Witten index will remain the same. In such cases
the machinery developed by Hanany and friends [76, 77, 78, 79, 80, 52]
applying Hilbert series techniques to SQCD’s, Quiver gauge theories, Conifolds
etc. to probe the structure of the vacua manifolds will be useful. We will
venture along these lines in future works.
One of the key insights of Hilbert in [81] was that the dimension of a variety
associated to an ideal could be characterized by the growth of the number of
polynomials not in the ideal as the total degree increases. This later formed
the basis for the Hilbert functions, Hilbert polynomials and subsequently the
Hilbert series. The Hilbert series of a variety is not a topological quantity
as it depends on the embedding $\mathbb{C}^{n}$ for the variety. A specific
embedding then induces a grading on the coordinate ring, $\displaystyle
R_{V}=\bigoplus_{i\geq 0}R_{i}$, where $R_{i}$ is the collection of
independent degree $i$ polynomials on the variety $V$. The Hilbert series is
the generating function for the dimensions of the $\mathbb{C}$–vector space
spanned by these graded pieces of $R_{V}$,

$H(t;V)=\sum_{i=0}^{\infty}\left(\operatorname{dim}_{\mathbb{C}}R_{i}\right)t^{i}\,.$
(3.24)

Here, $t$ is a dummy variable and it is called ‘fugacity’ much like the
‘chemical potential’ in statistical mechanics. Hilbert series is a rational
function in $t$ which can be written in two different ways [82]:

$H(t;V)=\begin{cases}\frac{Q(t)}{(1-t)^{n}}&\text{ Hilbert series of First
kind }\\\ \frac{P(t)}{(1-t)^{\operatorname{dim}(V)}}&\text{ Hilbert series of
Second kind }\end{cases}$ (3.25)

where $P(t)$ and $Q(t)$ are polynomials with integer coefficients. The powers
of the denominators are such that the leading pole captures the dimension of
the embedding space and the variety $V$, respectively 282828As an example,
take the Ring of polynomials on $R=\mathbb{C}[z_{1},z_{2}]$. At level $n$, the
vector space is spanned by
$\left(z_{1}^{n},z_{1}^{n-1}z_{2},\ldots,z_{2}^{n}\right)$ and
$\operatorname{dim}_{\mathbb{C}}R_{i}=(n+1)$. The corresponding Hilbert series
is  $H(t,R)=\sum_{i=0}^{\infty}(i+1)t^{i}=\frac{1}{(1-t)^{2}}$ (3.26) i.e. $R$
as a variety is a two dimensional space. Now, if we create a new variety
$R_{V}=R/(z_{1}z_{2}-1)$, then the new Hilbert series is
$H(t,R_{V})=\frac{1-t^{2}}{(1-t)^{2}}=\frac{1+t}{1-t}\,.$ (3.27) The $(-ve)$
sign in front of the $t^{2}$ term in the numerator signifies that we have a
degree 2 constraint coming from $z_{1}z_{2}-1=0$.. In particular, $P(1)$
always equals the degree of the variety which for the case of zero dimensional
varieties is the number of solutions to the polynomial equations [52].
For a dimension $n$ variety $V$, a Laurent series expansion for the Hilbert
series of second kind in (3.25) can be developed as a partial fraction
expansion:

$H(t,V)=\frac{V_{n}}{(1-t)^{n}}+\cdots\frac{V_{3}}{(1-t)^{3}}+\frac{V_{2}}{(1-t)^{2}}+\frac{V_{1}}{1-t}+V_{0}+\mathcal{O}(1-t)\,,$
(3.28)

where we can see explicitly that the Hilbert series is a rational function and
the degree of its most singular pole is the dimension of $V$.
It is possible to generalize the grading to a $m$–multiple grading of the
coordinate ring $R_{V}$ [83], such that $\displaystyle
R_{V}=\bigoplus_{i_{1},i_{2},\ldots,i_{m}\geq 0}R_{i_{1},i_{2},\ldots,i_{m}}$,
and the corresponding Hilbert series is

$H\left(t_{1},t_{2},\ldots,t_{m};V\right)=\sum_{i_{1},i_{2},\ldots,i_{m}=0}^{\infty}\left(\operatorname{dim}_{\mathbb{C}}R_{i_{1},i_{2},\ldots,i_{m}}\right)t_{1}^{i_{1}}t_{2}^{i_{2}}\ldots
t_{m}^{i_{m}}\,.$ (3.29)

We will use the multi–graded Hilbert series in extracting the count of gauge
invariant operators.

#### Our Strategy

We follow the procedure as discussed in [39] to construct the Hilbert series
from the F–term constraints 292929We could have started with a more satisfying
approach of writing everything in terms of gauge invariant coordinates. The
gauge invariant F–term equations are generators of an ideal of the ring of
invariants. For the abelian charges we can do it in terms of the $u$ variables
discussed in 3.1. However for the non–abelian cases, finding generators of the
ring of invariants is a difficult task due to presence of hidden constraints
called syzygies. Take for example, the case of a three node abelian quiver
with bi–fundamental fields
$z_{12},\,z_{21},\,z_{13},\,z_{31},\,z_{23},\,z_{32}$. Naively, the gauge
invariant coordinates would be
$x_{1}=z_{12}z_{21},\,x_{2}=z_{13}z_{31},\,x_{3}=z_{23}z_{32},\,x_{4}=z_{12}z_{23}z_{31}$
and $x_{5}=z_{13}z_{32}z_{21}$, but they are not all independent. There is a
relation, the syzygy $x_{1}x_{2}x_{3}=x_{4}x_{5}\,$. Hence even without any
F–terms the coordinate ring is
$\displaystyle\mathbb{C}\left[x_{1},x_{2},x_{3},x_{4},x_{5}\right]/\left(x_{1}x_{2}x_{3}-x_{4}x_{5}\right)$.
To the best of our knowledge, there is no known algorithm to lay out the set
of genuinely independent gauge invariant coordinates..

  1. 1.

We start with the polynomial ring $\mathbb{C}\left[x_{1},..x_{n}\right]$ of
the basic variables $Z^{(k\ell)}$ and $\Phi^{(k)}_{i}$ (which are not gauge
invariant).

  2. 2.

Impose $F$–term equations as an ideal $\mathcal{I}$ on this ring and compute
the quotient ring $\mathbb{C}\left[x_{1},..x_{n}\right]/\mathcal{I}$ 303030In
practice, in several situations we have replaced the ideal $\mathcal{I}$ by
its standard or reduced Gröbner basis ideal. These Gröbner bases has been
enumerated using a parallel implementation of the F4 algorithm as discussed in
[84]. .

  3. 3.

Compute the Hilbert series and extract the gauge invariant part of
$\mathbb{C}\left[x_{1},..x_{n}\right]/\mathcal{I}$, which would then be
identified with the coordinate ring of the vacua manifold as we expect it to
be spanned by gauge invariant operators.

We will now expand on the grading and extract part of the procedure.

#### Grading and Extraction

We endow appropriate gauge and global/bookkeeping charges to the fields which
are the basic variables which generate the polynomial ring. In most cases, it
is trivial to assign the charges for the Cartan generators of Lie groups by
looking at the transformation of the fields 313131We will discuss in the next
section a case of partial gauge fixing which has residual gauge symmetry and
the charge assignment will be different.. These charges are then used as the
grading exponents in the multi–graded Hilbert series. More details can be
found in the next section 4 where we discuss various abelian and non–abelian
cases. We set fugacities $\left\\{t_{\text{global}}\right\\}$ and
$\left\\{s_{\text{gauge}}\right\\}$ for a set of commuting abelian charges
under the bookkeeping $U(1)_{B}$ and the cartans of $U(N)$ gauge group(s)
respectively. We use modified versions of the algorithms available in computer
packages such as Macaulay 2[85] and Singular[86] to compute the multi–graded
Hilbert series of the quotient ring $R_{V}$,

$H(t;s_{1},\ldots
s_{m})=\sum_{p,i_{1},\ldots,i_{m}}h(p;i_{1},\ldots,i_{m})\,t^{p}s_{1}^{i_{1}}\ldots
s_{m}^{i_{m}}=\sum_{p,i_{1},\ldots,i_{m}}\chi_{i_{1},\ldots,i_{m}}(s_{1},\ldots
s_{m})\,t^{p}\,,$ (3.30)

where $\chi_{i_{1},\ldots,i_{m}}(s_{1},\ldots s_{m})$ are the characters of
some representation (not necessarily irreducible) of the $U(N)$ gauge group(s)
.
For the purposes of this paper we are not interested in the properties of
these characters. We are only interested in the extracting the zero net charge
sector from these characters i.e. the part independent of $(s_{1},\ldots
s_{m})$. This will be the ring of gauge invariant operators. We use the
Molien–Weyl formula [76, 87] to extract the gauge invariant part of the
Hilbert series

$\int\ldots\int d\mu_{G}(s_{1},\ldots s_{m})\,H(t;s_{1},\ldots
s_{m})=H_{GI}(t)\,,$ (3.31)

where $d\mu_{G}(s_{1},\ldots s_{m})$ is the Haar measure on the gauge
group(s), such that

$\int
d\mu_{G}=\begin{cases}\displaystyle\int_{U(1)}d\mu_{U(1)}(s)\rightarrow\frac{1}{2\pi
i}\oint_{|s|=1}\frac{ds}{s}&\text{$:U(1)$ group.}\vspace{0.25cm}\\\
\displaystyle\int_{SU(2)}d\mu_{SU(2)}(s)\rightarrow\frac{1}{2}\frac{1}{2\pi
i}\oint_{|s|=1}\frac{ds}{s}\left(1-s^{2}\right)\left(1-s^{-2}\right)&\text{$:SU(2)$
group .}\end{cases}$ (3.32)

For zero dimensional varieties, $H_{GI}(1)=\text{Number of points}=\text{No.
of BH microstates}$.

##  4 Counting Black Holes as Hilbert series

In this section we will construct the Hilbert series for the abelian and a few
non–abelian charge configurations. The computational complexity of Hilbert
series is similar to that of finding the Gröbner basis. The number of
variables and their corresponding F–term equations scales as $3N^{2}+6N+15$
for charge configurations of the type $(1,1,1,N)$. So, even after decent
speedup due to modifications in existing algorithms and parallelizing the
processes whenever possible, we were limited by our modest computational
resources 333333We are using a 56 core 128 GB machine but the algorithms tend
to store history of a large number of matrix manipulations which quickly
exhaust the RAM..
In the Hilbert series computations, all the $\Phi$ and $Z$ fields carry unit
charge under a bookkeeping abelian group $U(1)_{B}$. In the language of
computational algebraic geometry, the bookkeeping charges form the ‘heft
vector’ of the polynomial ring. We calculate the Hilbert series with different
degrees of gauge fixing:

  * •

Full gauge fixing.

  * •

Partial gauge fixing 343434Charges under the residual gauge symmetries need to
be assigned properly. See subsections 4.1 and 4.2..

  * •

No gauge fixing.

For cases we could do all three of them with the current computational
capacity, they all agree.

###  4.1 Abelian case

The abelian case has charge configuration $N_{1}=1$, $N_{2}=1$, $N_{3}=1$ and
$N_{4}=1$. The relative gauge symmetry is the complexified version of
$U(1)_{12}\times U(1)_{23}\times U(1)_{14}$ with fugacities $T_{0}$, $T_{1}$
and $T_{2}$ respectively. $T_{3}$ is the fugacity for the bookkeeping
$U(1)_{B}$.

#### Full gauge fixing

We can gauge fix the system by choosing $Z^{(12)}=1$, $Z^{(23)}=1$ and
$Z^{(14)}=1$ keeping an overall diagonal $U(1)$ charge arbitrary. This implies
that the all relative $U(1)$ charges are now set to zero. We now construct the
multigraded polynomial ring with $Z$ and $\Phi$ fields as variables, each
carrying charges/weights ${(0,0,0,1)}$. The corresponding Hilbert series
points to a zero dimensional variety,

$H(T_{3})=1+7\,T_{3}+4\,T_{3}^{2}\,,$ (4.1)

where substituting $T_{3}=1$ gives $H(1)=12$. Thus we have a vacua manifold
made up of 12 isolated points which are the 12 supersymmetric ground states.
Of course, the result matches with that reported in section 3 and the result
from the U–dual system of appendix B.

#### Partial gauge fixing

In situations where one would like the Hilbert series to keep track of gauge
invariant operators of subgroups of the full group, we might resort to partial
gauge fixing. In our case we can

  * •

gauge fix $Z^{(34)}=1$. The corresponding charges are in Table 3 of Appendix
C. The Hilbert series after performing the contour integration for the
residual gauge group $U(1)_{12}\times U(1)_{23}$ is

$H(T_{3})=T_{3}^{3}+6T_{3}^{2}+4T_{3}+1\,,$ (4.2)

where note that the coefficients of various powers of $T_{3}$ are different
from (4.1); yet $H(1)=12\,$.

  * •

gauge fix $Z^{(12)}=1$ and $Z^{(23)}=1$. The corresponding charges are in
Table 4 of Appendix C. Again, the Hilbert series gives $H(1)=12\,$.

#### No gauge fixing

We now compute the Hilbert series without any gauge fixing. As discussed
earlier, for zero dimensional varieties, the total number of GIOs is the
degree of the variety and count the number of supersymmetric ground states.
The gauge charges of various fields are in Table 5 of Appendix C which can be
used to construct the multi–graded Hilbert series
$H(T_{0},T_{1},T_{2},T_{3})$. The details of the contour integrations are in
Appendix A. The final gauge invariant Hilbert series is

$H(T_{3})=\left(T_{3}+1\right){}^{2}\left(2T_{3}+1\right)$ (4.3)

and yet again $H(1)=12$.

###  4.2 Non–abelian cases

For non–abelian cases, the complex scalars $\Phi_{i}^{(k)}$ are $N_{k}\times
N_{k}$ hermitian matrices transforming in the adjoint representation of
$U(N_{k})$. On the other hand, the complex scalar $Z^{(k\ell)}$ becomes a
$N_{k}\times N_{\ell}$ complex matrix transforming in the
$\left(N_{k},\bar{N}_{\ell}\right)$ representation of
$U\left(N_{k}\right)\times U\left(N_{\ell}\right)$. With this information, we
can assign gauge charges to our fields and construct the multi–graded
polynomial ring and compute the corresponding Hilbert series. Here, we discuss
two cases $N_{1}=1$, $N_{2}=1$, $N_{3}=1$, $N_{4}=2$ and $N_{1}=1$, $N_{2}=1$,
$N_{3}=1$, $N_{4}=3$ and the previously unreported case $N_{1}=1$, $N_{2}=1$,
$N_{3}=1$, $N_{4}=4$.
For the charges $(1,1,1,2)$ we computed the Hilbert series with three
different degrees of complexified gauge fixing. Due to limitations of
computational facilities we could compute the Hilbert series of charges
$(1,1,1,3)$ and $(1,1,1,4)$ by doing full complexified gauge fixing only.

####  4.2.1 Charge $(1,1,1,2)$

Here we get 56 as the degree of the zero dimensional variety which matches
with the results calculated in 3 and the result from the U–dual system [10].

#### Full gauge fixing

The relative gauge symmetry is the complexified version of $U(1)\times
U(1)\times U(2)$. To completely fix the gauge, we set $Z^{(12)}=1$,
$Z^{(23)}=1$, $Z^{(14)}=\left(\begin{array}[]{ll}1&0\end{array}\right)$ and
$Z^{(24)}=\left(\begin{array}[]{ll}0&1\end{array}\right)$. All the fields
carry charge $(0,0,0,0,1)$, last one being the bookkeeping charge. The Hilbert
series is

$H(T_{4})=1+16\,T_{4}+39\,T_{4}^{2}\,,$ (4.4)

where $T_{4}$ is the fugacity corresponding to the bookkeeping charge and
$H(1)=56\,$.

#### Partial gauge fixing

Various combinations of partial gauge fixing can be performed but here we
highlight the case of fixing one of the $U(1)$ and the $U(2)$ gauge symmetry
such that the residual gauge symmetry induce charges to the fixed gauge
groups. The fields under the complexified gauge group transform as

$\begin{array}[]{r}Z^{(k\ell)}\rightarrow
a_{k}\left(a_{\ell}\right)^{-1}Z^{(k\ell)},\quad
Z^{(4k)}\rightarrow\left(a_{k}\right)^{-1}MZ^{(4k)},\quad Z^{(k4)}\rightarrow
a_{k}Z^{(k4)}M^{-1}\\\ \text{ for }\quad 1\leq k\leq 3,\quad k\neq\ell,\\\
\Phi_{i}^{(k)}\rightarrow\Phi_{i}^{(k)},\quad\Phi_{i}^{(4)}\rightarrow
M\Phi_{i}^{(4)}M^{-1},\quad\text{ for }\quad 1\leq i\leq 3,\quad 1\leq k\leq
3,\end{array}$ (4.5)

where $a_{k}$ for $1\leq k\leq 3$ are complex numbers and $M$ is a $2\times 2$
complex matrix. We gauge fix $Z^{(23)}=1$,
$Z^{(14)}=\left(\begin{array}[]{ll}1&0\end{array}\right)$ and
$Z^{(24)}=\left(\begin{array}[]{ll}0&1\end{array}\right)$ leaving behind a
residual gauge symmetry 353535For the purpose of assigning charges let
$U(1)_{1}=e^{ia},\,U(1)_{2}=e^{ib},\,U(1)_{3}=e^{ic}$ and $\displaystyle
U(2)=M=\left(\begin{array}[]{cc}p&q\\\ r&s\end{array}\right)\,$. To get the
residual gauge symmetry, we have  • from invariance of $Z^{(14)}$,
$e^{ia}\left(\begin{array}[]{ll}1&0\end{array}\right)M^{-1}=\left(\begin{array}[]{ll}1&0\end{array}\right)\,$.
• from invariance of $Z^{(24)}$,
$e^{ib}\left(\begin{array}[]{ll}0&1\end{array}\right)M^{-1}=\left(\begin{array}[]{ll}0&1\end{array}\right)\,$.
• from invariance of $Z^{(23)}$, $1=e^{ib}\,1\,e^{-ic}\,$. Therefore, the
residual gauge group is $\displaystyle M=\left(\begin{array}[]{cc}e^{ia}&0\\\
0&e^{ib}\end{array}\right)$ and $e^{ib}=e^{ic}$ and we will use two diagonal
$U(1)_{a}$ and $U(1)_{b}$ of $M$ to assign the charges.. The charge assignment
is given in Table 6 and Table 7 of Appendix C with fugacities $T_{0}$,
$T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$ and $T_{5}$. After computing the the
multi–graded Hilbert series we have to set $T_{3}=T_{0}$, $T_{4}=T_{1}$ and
$T_{2}=T_{1}$ as per our gauge choice. The overall $U(1)$ drops out at the end
and finally we get $H(1)=56\,$.

#### No gauge fixing

The relative gauge symmetry is the complexified version of $U(1)\times
U(1)\times U(2)\equiv U(1)\times U(1)\times U(1)\times SU(2)$. The charges are
mentioned in Table 8 and Table 9 of Appendix C where 2 and $-2$ are the
adjoint $SU(2)$ gauge charges. For doing the contour integrals we used the
$SU(2)$ Haar measure mentioned in (3.32). The answer again is $H(1)=56\,$.

####  4.2.2 Charge $(1,1,1,3)$

Here we get 208 as the degree of the zero dimensional variety which matches
with the results discussed in 3 and the result from the U–dual system [10].

#### Full gauge fixing

In this case, the relative gauge symmetry is the complexified version of
$U(1)\times U(1)\times U(3)$. We fix the gauge completely by setting
$Z^{(12)}=1$, $Z^{(23)}=1$,
$Z^{(14)}=\left(\begin{array}[]{lll}1&0&0\end{array}\right)$,
$Z^{(24)}=\left(\begin{array}[]{lll}0&1&0\end{array}\right)$ and
$Z^{(34)}=\left(\begin{array}[]{lll}0&0&1\end{array}\right)$ 363636An
alternate gauge fixing would be to set $Z^{(12)}=1$, $Z^{(23)}=1$,
$Z^{(14)}=\left(\begin{array}[]{lll}1&0&0\end{array}\right)$,
$Z^{(24)}=\left(\begin{array}[]{lll}0&1&0\end{array}\right)$ and to choose the
first row of any of the $\Phi^{(4)}_{i}$ field to be
$\left(\begin{array}[]{lll}a_{1}&a_{2}&a_{3}\end{array}\right)$ where $a_{1}$,
$a_{2}$ and $a_{3}\neq 0$ are random complex numbers. For arbitrary choices
discrete residual gauge symmetries may remain.. For both of these gauge
fixing, we have the Hilbert series with gauge charges set to $(0,0,0,0,1)$ for
all fields. The final Hilbert series is gives $H(1)=208\,$.

####  4.2.3 Charge $(1,1,1,4)$

Here we get 684 as the degree of the zero dimensional variety which matches
with the result from the U–dual system[10].

#### Full gauge fixing

In this case, the relative gauge symmetry is the complexified version of
$U(1)\times U(1)\times U(4)$. We can gauge fix it by setting, $Z^{(12)}=1$,
$Z^{(23)}=1$, $Z^{(14)}=\left(\begin{array}[]{llll}1&0&0&0\end{array}\right)$,
$Z^{(24)}=\left(\begin{array}[]{llll}0&1&0&0\end{array}\right)$ and
$Z^{(34)}=\left(\begin{array}[]{llll}0&0&1&0\end{array}\right)$. As we don’t
have enough $Z$’s to fix the gauge completely, we also set
$\left(\begin{array}[]{llll}(\Phi^{(4)}_{3})_{11}&(\Phi^{(4)}_{3})_{12}&(\Phi^{(4)}_{3})_{13}&(\Phi^{(4)}_{3})_{14}\end{array}\right)=\left(\begin{array}[]{llll}0&0&0&1\end{array}\right)\,$
373737In the gauge choices we have been making for the non–abelian cases,
there are a few caveats. For example, we are throwing away the possibility
that $Z^{(14)}$ and $Z^{(24)}$ could be parallel to each other or say
$\left(\begin{array}[]{llll}(\Phi^{(4)}_{3})_{11}&(\Phi^{(4)}_{3})_{12}&(\Phi^{(4)}_{3})_{13}&(\Phi^{(4)}_{3})_{14}\end{array}\right)=\left(\begin{array}[]{llll}0&0&0&0\end{array}\right)$.
We have accounted for these cases.. Finally, from the Hilbert series we get
$H(1)=684\,$.

##  5 Higher Order Terms in the Potential

In the Lagrangian for the pure D–brane system we presented in section 2.1 we
have accounted for cubic interactions involving a closed string connecting
three branes (2.3). It is natural to wonder if we could have quartic
interaction term involving fours branes, like

$\mathcal{W}_{5}=\sqrt{2}C_{4}\sum_{\begin{subarray}{c}k,\ell,m,n\\\
k<\ell,m,n\\\ \ell\neq m\neq
n\end{subarray}}^{4}\text{(sign)}\,\operatorname{Tr}\left[Z^{(k\ell)}Z^{(\ell
m)}Z^{(mn)}Z^{(nk)}\right]\,.$ (5.1)

In general, the expectation with Witten index is that it will remain invariant
as we change the parameters of the superpotential. For Landau–Ginsberg like
models, there are two possibilities as we change the parameters:

  * •

Separate critical points come together to form one or more degenerate critical
points.

  * •

One or more critical points slide towards the far end i.e. infinity.

As an illustration, we take a toy version of the Landau–Ginsberg model
discussed in (2.16) with one hyper–multiplet and a holomorphic superpotential
$\mathcal{W}(z)$ such that

$\frac{\partial\mathcal{W}(z)}{\partial z}=\lambda z^{2}+2z+1\,,$ (5.2)

where $\lambda$ is a real tunable parameter. For a moderate range of values
for $\lambda$ we get two critical points

$z_{1}=\frac{\sqrt{1-\lambda}+1}{\lambda}\quad\text{and}\quad
z_{2}=\frac{\sqrt{1-\lambda}-1}{\lambda}$ (5.3)

and the Witten index is $2$. Now, lets take $\lambda$ to the extremes,

  * •

For $\lambda\rightarrow\infty$, we have both $z_{1}\approx 0$ and
$z_{2}\approx 0$ .

  * •

For $\lambda\rightarrow 0$, we have
$z_{1}\approx-\frac{2}{\lambda}+\frac{1}{2}+\frac{\lambda}{8}+\frac{\lambda^{2}}{16}+O\left(\lambda^{3}\right)$
and
$z_{2}\approx-\frac{1}{2}-\frac{\lambda}{8}-\frac{\lambda^{2}}{16}+O\left(\lambda^{3}\right)$
.

In the first case, the two separate critical points becomes doubly degenerate
and the contribution of the critical point $z=0$ to the Witten index should be
counted with its multiplicity, here it is $2$. On the other hand, in the
second case the critical point $z_{1}$ moves to infinity, and the
corresponding quantum SUSY ground state is now outside the Hilbert space. The
point $z_{2}$ remains a valid critical point and the Witten index jumps down
from $2$ to $1$.
We witness the similar behaviour in our system of D–branes if we add the extra
quartic term (5.1). For concreteness, lets focus on the abelian case where the
system has D–brane charges set as $(1,1,1,1)$. In the previous work of one of
the current authors [1, 2], looking at the F–term equations (2.14), (2.15), it
had been argued that if we parametrically scale down the the coefficients
$c^{(k\ell)}$’s by multiplying $\lambda\ll 1$, then the values of the
$Z^{(k\ell)}$’s and $\Phi^{(k)}_{i}$ scale down as $\sqrt{\lambda}$ and hence
the cubic interaction term $\mathcal{W}_{2}$ (2.3) dominates over the quartic
term $\mathcal{W}_{5}$ (5.1). We now think that this argument is only
partially correct. If we include the quartic term, the number of critical
points jumps from 12 to 24. Interestingly, out of these 24 critical points, 12
are the critical points of the potential with the original cubic interaction
but corrected by $\mathcal{O}(\lambda)$ terms. The other 12 solutions have
either some $Z^{(k\ell)}$ or $\Phi^{(k)}_{i}$ parametrically larger
$\displaystyle\mathcal{O}\left(\frac{1}{\lambda^{\\#}}\right)$ than the rest
of the numbers. So, it seems Witten index jumps from 12 to 24, spoiling the
microstate counting. Same behaviour can also be seen for the non–abelian cases
with the addition of many more critical points if quartic superpotential is
included in the potential.
We will now argue that in both the abelian and non–abelian cases, no such
extra superpotential terms survive the constraints coming from various
exchange symmetries of type IIA string theory on $T^{6}$ as mentioned in
section 2.2. To begin with, lets take the cubic $\mathcal{W}_{2}$ term with
arbitrary coefficients

$\displaystyle\mathcal{W}_{2}=$
$\displaystyle\alpha_{1}\operatorname{Tr}\left(Z^{(31)}Z^{(12)}Z^{(23)}\right)+\alpha_{2}\operatorname{Tr}\left(Z^{(13)}Z^{(32)}Z^{(21)}\right)+\alpha_{3}\operatorname{Tr}\left(Z^{(12)}Z^{(24)}Z^{(41)}\right)$
(5.4)
$\displaystyle+\alpha_{4}\operatorname{Tr}\left(Z^{(42)}Z^{(21)}Z^{(14)}\right)+\alpha_{5}\operatorname{Tr}\left(Z^{(13)}Z^{(34)}Z^{(41)}\right)+\alpha_{6}\operatorname{Tr}\left(Z^{(31)}Z^{(14)}Z^{(43)}\right)$
$\displaystyle+\alpha_{7}\operatorname{Tr}\left(Z^{(34)}Z^{(42)}Z^{(23)}\right)+\alpha_{8}\operatorname{Tr}\left(Z^{(43)}Z^{(32)}Z^{(24)}\right)\,.$

The invariance of the Lagrangian under the three exchange symmetries
constrains the $\alpha_{i}$’s to specific $\pm\alpha$ and the value of
$\alpha$ can be determined by performing a first principle three point vertex
amplitude calculation on the open–disk [2]. Similarly, $\mathcal{W}_{3}$ can
also be constrained. Coming back to $\mathcal{W}_{5}$, the gauge invariant
quartic term can be expanded as

$\displaystyle\mathcal{W}_{5}=$
$\displaystyle\beta_{1}\operatorname{Tr}\left(Z^{(12)}Z^{(23)}Z^{(34)}Z^{(41)}\right)+\beta_{2}\operatorname{Tr}\left(Z^{(12)}Z^{(24)}Z^{(43)}Z^{(31)}\right)$
(5.5)
$\displaystyle+\beta_{3}\operatorname{Tr}\left(Z^{(13)}Z^{(34)}Z^{(42)}Z^{(21)}\right)+\beta_{4}\operatorname{Tr}\left(Z^{(13)}Z^{(32)}Z^{(24)}Z^{(41)}\right)$
$\displaystyle+\beta_{5}\operatorname{Tr}\left(Z^{(14)}Z^{(42)}Z^{(23)}Z^{(31)}\right)+\beta_{6}\operatorname{Tr}\left(Z^{(14)}Z^{(43)}Z^{(32)}Z^{(21)}\right).$

Applying the exchange symmetries one by one, we get

  * •

First exchange:
$\beta_{1}=\beta_{3},\,\beta_{2}=-\beta_{6},\,\beta_{4}=-\beta_{5}$ .

  * •

Second exchange:
$\beta_{1}=\beta_{4},\,\beta_{2}=-\beta_{3},\,\beta_{5}=-\beta_{6}$ .

  * •

Third exchange:
$\beta_{1}=i\beta_{5},\,\beta_{2}=\beta_{3},\,\beta_{4}=i\beta_{6}$ .

As is evident there is no common solutions to the above sets of equations,
except for all $\beta_{i}$ put to zero. Other higher order terms which
schematically can be written as $\Phi\Phi ZZ$, $\Phi ZZZ$, $(ZZ)(ZZ)$,
$(ZZZ)(ZZ)$ etc. are also absent for not being compatible with the three
exchange symmetries. All these cases and more have been checked both for
abelian and non–abelian cases. This ensures that the Witten index is now
protected which in turn protects the microstate counting.

##  6 Discussion

In this paper we have presented a bouquet of techniques coming from
computational algebraic geometry to probe the SUSY vacua manifold of black
holes in $\mathcal{N}=8$ string theory compactified on $T^{6}$. These
$\frac{1}{8}$–BPS black holes have a pure D–brane, D2–D2–D2–D6 configuration
wrapping various cycles of $T^{6}$ and admits an U–dual description in terms
of a D1–D5–P–KK monopole system [10]. In previous works of one of the authors
[1, 2], it has been shown that for very low lying charges the U–duality
remains intact and for generic choices of the metric and $B$–field moduli the
SUSY vacua manifold presents itself as a collection of isolated points
carrying zero angular momentum. This is consistent with the near horizon
$AdS_{2}\times S^{2}$ geometry of extremal black holes and puts strong
constraints on the fuzzball program [88, 89]. It is therefore important that
we extend the validity of this zero angular momentum conjecture to more charge
configurations.
Our primary motivation to take up this work is to eventually count black holes
in $\mathcal{N}=2$ theories which are compactified on Calabi-Yau spaces [90,
20, 19]. As these manifolds don’t have non–trivial 1–cycles, the corresponding
black holes can’t admit electric or winding charges associated with the
$S^{1}$. Therefore, black holes in Calabi–Yau spaces only carry pure D–brane
charges. Currently, we are pushing the counting for $\mathcal{N}=8$ theories
with the hope that once enough data (count) is collected, we would be able to
write a generating function for different charges and compare with the the
U–dual side. Next steps would be to do twining and twisting ($\mathcal{N}=4$
theory on K3 [13, 16, 14]) of the theory and perform the counting. We are
actively pursuing these directions and hopefully will report new results soon.
The results presented in [1, 2] is still at the limit of what the latest
version of Mathematica can do in terms of solving a system of polynomial
equations – the F–term equations enumerating the isolated SUSY ground states.
In our quest to break past the glass ceiling of computational complexity, we
ended up discovering a number of very interesting and efficient ways to probe
algebraic varieties, especially the zero–dimensional varieties [42, 54]. The
techniques discussed in this paper have a far wider scope than black hole
physics. Polynomial systems describe a wide variety of systems in physics [91,
92], chemistry [93, 94], mathematics [49], biology [95, 96], engineering [97,
98], machine learning [99, 100, 101], economics [102] among other fields of
study. We hope that other researchers in our community would take up these
techniques and with suitable additions and(or) modifications apply to other
interesting settings.
For mathematicians, one can take a slightly different view of our non–abelian
system for general charges $(N_{1},N_{2},N_{3},N_{4})$. The F–terms can now be
thought of as a 4–parameter family of sparse polynomial systems where unlike
most cases in literature where the number of equations and monomials don’t
change with parameters, here the number of equations scale like
$\mathcal{O}(N_{1}^{2}N_{2}^{2}N_{3}^{2}N_{4}^{2})$ and for each equation the
number of monomials at max scale as $\mathcal{O}(N_{1}+N_{2}+N_{3}+N_{4})$.
This provides an infinite class of sufficiently complex polynomial families
with both discrete and continuous symmetries which can be used to benchmark
current and future techniques & algorithms arising in computational algebraic
geometry. They have a nice advantage over other such large polynomial systems
as the numbers of solutions for all charges is known and admits a generating
function, a weak Jacobi form of weight $-2$ and index 1 coming from the U–dual
description of the system, see Appendix B. We would like to explore if
specialized techniques could be developed for such scaling behaviour as many
quiver gauge theories share similar scaling [103]. The problem can also be
recast in the language of optimization theory, where a cost function, the
potential $V_{F}$ needs minimization. Our current endeavours are at the moment
a bit handicapped due to lack of access to a state of the art computing
facility but hopefully, it will get resolved in future.

### Observations

During the course of analysing the pure D–brane system in various limits, we
gathered some interesting observations which were not resolved to our
satisfaction. We list them below:

  * •

Moduli $c^{(k\ell)}=0$ : The $c^{(k\ell)}$’s depend on the background metric
and $B$–field moduli and have been assigned non–zero generic values in current
and previous works [1, 2]. For abelian charges, lets set, $c^{(13)}=0$ with
gauge choice $Z^{(12)}=1$, $Z^{(23)}=1$ and $Z^{(14)}=1$. We get 4 solutions
which is obviously wrong !
Tracking the 12 solutions for $c^{(13)}\rightarrow 0$ limit, we noticed that
in 8 solutions the distance between brane 1 and brane 3 in the 6, 7 directions
i.e. $(\Phi^{(3)}_{2}-\Phi^{(1)}_{2})\rightarrow\infty$ such that the term
$c^{(13)}(\Phi^{(3)}_{2}-\Phi^{(1)}_{2})$ in the superpotential (2.4) remains
finite. If we do $\Phi\rightarrow\frac{1}{\Phi^{\prime}}$ and solve with
$\Phi^{\prime}$, we get 12 solutions grouped into multiplicities of
$(2,2,2,2,1,1,1,1)$ and in 8 of them $\Phi^{\prime}\rightarrow 0$. Either way,
it looks like a degeneration limit whose resolution is $c^{(13)}\neq 0$.
Situation worsens if we set four $c^{(k\ell)}$’s to zero. Then a gauge
invariant Hilbert series computation points to a 4–dimensional vacua manifold.

  * •

Charge $(1,1,2,2)$ : This configuration is now accessible with the modern
techniques after gauge fixing, $Z^{(12)}=1$,
$Z^{(13)}=\left(\begin{array}[]{ll}1&1\end{array}\right)$ ,
$Z^{(23)}=\left(\begin{array}[]{ll}-1&1\end{array}\right)$,
$Z^{(14)}=\left(\begin{array}[]{ll}1&0\end{array}\right)$ and
$Z^{(24)}=\left(\begin{array}[]{ll}0&1\end{array}\right)$. But the Hilbert
series points to a 1–dimensional variety ! Again, a degeneration is at play,
we are looking into it more closely.

  * •

Three charges : We can set the charge $N_{1}=0$ and analyse the three charge
systems. The D2–D2–D6 system is equivalent to a D1–D5–P system via U–duality
whose $\frac{1}{8}$–BPS microstate counting can be found in [104, 10]. The
abelian case of charge $(1,1,1)$ results in 2 solutions which matches the
U–dual counting only if it carries a spin $J^{3}_{L}=1$. For non–abelian
charges $(1,1,2)$ we get a 1-dimensional SUSY vacua manifold, for charges
$(1,1,3)$ we are getting no solutions. Clearly, $N_{1}\rightarrow 0$ limit of
the 4–charge system is not kosher !

Acknowledgements: We wish to thank Ashoke Sen for encouraging us to look
beyond Mathematica and suggesting the twined case. A.C. would like to thank
Swapnamay Mondal for introducing him to the works of Hanany & friends. The
work of A.C. is supported by IIT Bhubaneswar Seed Grant SP–103. The work of
S.M. is supported by fellowship from CSIR, Govt. of India.

##  Appendix A Hilbert series of charge $(1,1,1,1)$

In this appendix we shall discuss in detail the procedure leading up to the
gauge–invariant Hilbert series for the abelian charges. For the (1,1,1,1)
charge system, the relative gauge symmetry is $U(1)\times U(1)\times U(1)$.
Consider, $T_{0}$, $T_{1}$ and $T_{2}$ as the fugacities corresponding to the
relative $U(1)$’s, i.e. $U(1)_{12}$, $U(1)_{23}$ and $U(1)_{23}$ respectively.
The fugacity, $T_{3}$ corresponds to the bookkeeping $U(1)_{B}$. The charges
are given in Table 5 and we choose ‘degree–reverse–lexicographic’ monomial
ordering for fast computation of the multi–graded Hilbert series. The Hilbert
series is computed using Macaulay2 [85], which has the structure

$H(T_{0},T_{1},T_{2},T_{3})=\frac{N(T_{0},T_{1},T_{2},T_{3})}{D(T_{0},T_{1},T_{2},T_{3})}$
(A.1)

where $N(T_{0},T_{1},T_{2},T_{3})$ is a very long numerator which we will not
present here and $D(T_{0},T_{1},T_{2},T_{3})$ is the denominator

$\displaystyle D(T_{0},T_{1},T_{2},T_{3})=$
$\displaystyle\left(1-T_{3}\right){}^{6}\left(1-\frac{T_{3}}{T_{0}}\right)\left(1-T_{0}T_{3}\right)\left(1-\frac{T_{3}}{T_{1}}\right)\left(1-\frac{T_{3}}{T_{0}T_{1}}\right)\left(1-T_{1}T_{3}\right)$
(A.2)
$\displaystyle\left(1-T_{0}T_{1}T_{3}\right)\left(1-\frac{T_{3}}{T_{2}}\right)\left(1-\frac{T_{3}}{T_{1}T_{2}}\right)\left(1-\frac{T_{3}}{T_{0}T_{1}T_{2}}\right)\left(1-T_{2}T_{3}\right)$
$\displaystyle\left(1-T_{1}T_{2}T_{3}\right)\left(1-T_{0}T_{1}T_{2}T_{3}\right).$

After simplification and appropriate scaling the final denominator is

$\displaystyle D(T_{0},T_{1},T_{2},T_{3})=$ $\displaystyle
T_{0}^{4}T_{1}^{8}T_{2}^{4}\left(T_{0}-T_{3}\right)\left(T_{1}-T_{3}\right)\left(T_{0}T_{1}-T_{3}\right)\left(T_{2}-T_{3}\right)\left(T_{1}T_{2}-T_{3}\right)$
(A.3)
$\displaystyle\left(T_{0}T_{1}T_{2}-T_{3}\right)\left(T_{0}T_{3}-1\right)\left(T_{1}T_{3}-1\right)\left(T_{0}T_{1}T_{3}-1\right)\left(T_{2}T_{3}-1\right)$
$\displaystyle\left(T_{1}T_{2}T_{3}-1\right)\left(T_{0}T_{1}T_{2}T_{3}-1\right)\,.$

Now, we shall extract the count of independent gauge–invariant operators from
the Hilbert series i.e. the part independent of $T_{0},T_{1},T_{2}$. To that
end, we have to perform contour integrals over the poles of these the
variables i.e.

$H(T_{3})=\int\int\int
d\mu_{{U(1)}_{12}}d\mu_{{U(1)}_{23}}d\mu_{{U(1)}_{13}}H(T_{0},T_{1},T_{2},T_{3})\,.$
(A.4)

Here, the Haar measure for the $U(1)$ is given by,

$\int_{\mathrm{U}(1)}\mathrm{d}\mu_{\mathrm{U}(1)}(T)=\frac{1}{2\pi
i}\oint_{|T|=1}\frac{\mathrm{d}T}{T}\,.$ (A.5)

Therefore, the integral we have to perform is

$H(T_{3})=\frac{1}{(2\pi
i)^{3}}\oint\frac{dT_{2}}{T_{2}}\oint\frac{dT_{1}}{T_{1}}\oint\frac{dT_{0}}{T_{0}}\frac{N(T_{0},T_{1},T_{2},T_{3})}{D(T_{0},T_{1},T_{2},T_{3})}\,.$
(A.6)

To perform the integrals we choose circles as contours with an ordering of
variable radii as, $T_{3}<T_{2}<T_{1}<T_{0}$ 383838It can be shown that any
other ordering gives the same result.. To do the $T_{0}$ integral we choose
the contour $|T_{0}|=1$ and look at its poles at,

$T_{0}=0,\,T_{3},\frac{T_{3}}{T_{1}},\,\frac{T_{3}}{T_{1}T_{2}},\,\frac{1}{T_{3}},\,\frac{1}{T_{1}T_{3}}\quad\text{and}\quad\frac{1}{T_{1}T_{2}T_{3}}\,.$
(A.7)

The Hilbert series should be expandable as power series in terms of the
fugacities which puts a convergence condition on the terms in the denominator,

$\displaystyle\left|\frac{T_{3}}{T_{0}}\right|<1,\,\left|\frac{T_{3}}{T_{0}T_{1}}\right|<1,\,\left|\frac{T_{3}}{T_{0}T_{1}T_{2}}\right|<1,\,\left|T_{3}\right|<1,\left|\frac{T_{3}}{T_{1}}\right|<1,\,\left|\frac{T_{3}}{T_{2}}\right|<1,\,\left|\frac{T_{3}}{T_{1}T_{2}}\right|<1,\,\left|T_{0}T_{3}\right|<1,\,$
$\displaystyle\left|T_{1}T_{3}\right|<1,\left|T_{0}T_{1}T_{3}\right|<1,\left|T_{2}T_{3}\right|<1,\left|T_{1}T_{2}T_{3}\right|<1\quad\text{and}\quad\left|T_{0}T_{1}T_{2}T_{3}\right|<1\,.$
(A.8)

From the above condition we see that the poles at $\displaystyle
T_{0}=0,\,T_{3},\,\frac{T_{3}}{T_{1}},\,\frac{T_{3}}{T_{1}T_{2}}$ are inside
the contour $|T_{0}|=1$ and the rest are to be neglected. After performing the
integration, we get the Hilbert series $H(T_{1},T_{2},T_{3})$. We now perform
the $|T_{1}|=1$ contour integration with poles at $\displaystyle
T_{1}=0,\,T_{3},\,\frac{T_{3}}{T_{2}},\,\frac{1}{T_{3}},\,\frac{1}{T_{2}T_{3}}$.
The above convergence condition dictates that we keep the poles at
$\displaystyle T_{1}=0,\,T_{3}$ and $\displaystyle\frac{T_{3}}{T_{2}}$. After
performing the contour integration, we get a relatively simple looking Hilbert
series

$H(T_{2},T_{3})=\frac{\left(T_{3}^{2}-1\right)\left(T_{2}^{2}T_{3}^{2}+T_{3}^{2}+T_{2}\left(5T_{3}^{2}+4T_{3}+1\right)\right)}{\left(T_{2}-T_{3}\right)\left(T_{2}T_{3}-1\right)}\,.$
(A.9)

The poles for the $|T_{2}|=1$ integration are at $\displaystyle
T_{2}=0,\,T_{3},\,\frac{T_{2}}{T_{3}}$. The convergence condition reduces to
$\displaystyle\left|\frac{T_{3}}{T_{2}}\right|<1$ and
$\left|T_{2}T_{3}\right|<1$. This implies that we should take contribution
only from the poles $T_{2}=0$ and $T_{3}$. Finally, integrating over $T_{2}$,
we get the gauge–invariant Hilbert series as,

$H(T_{3})=\left(T_{3}+1\right){}^{2}\left(2T_{3}+1\right)\,.$ (A.10)

We can see that $H(1)=12$ which is the correct count for microstates for the
abelian case. Similar calculations follow for the non–abelian cases with
appropriate Haar measures.

##  Appendix B The $14^{th}$ Helicity Trace Index

In this appendix, we will discuss the $14^{th}$ helicity trace index in the
context of $\frac{1}{8}$–BPS, 4–charge black holes. Following the notation of
[17], and by appropriate duality transformation as described in [1], we can
write the electric $Q$ and magnetic charge vector $P$ as,

$Q=\left(\begin{array}[]{c}0\\\ -N_{2}\\\ 0\\\ -N_{1}\end{array}\right),\quad
P=\left(\begin{array}[]{c}N_{3}\\\ 0\\\ N_{4}\\\ 0\end{array}\right)\,.$ (B.1)

The T–duality invariant combinations of charges are

$Q^{2}=2N_{1}N_{2},\quad P^{2}=2N_{3}N_{4},\quad Q\cdot P=0\,.$ (B.2)

With the following definition of ‘gcd’

$\displaystyle\ell_{1}=\operatorname{gcd}\left\\{Q_{i}P_{j}-Q_{j}P_{i}\right\\}=\operatorname{gcd}\left\\{N_{1}N_{3},N_{1}N_{4},N_{2}N_{3},N_{2}N_{4}\right\\},$
(B.3) $\displaystyle\ell_{2}=\operatorname{gcd}\left\\{Q^{2}/2,P^{2}/2,Q\cdot
P\right\\}=\operatorname{gcd}\left\\{N_{1}N_{2},N_{3}N_{4}\right\\}\,,$

the $14^{th}$ helicity super–trace $B_{14}$, can be written as [9]

$B_{14}=-\sum_{s\mid\ell_{1}\ell_{2}}s\,\hat{c}\left(\Delta/s^{2}\right),\quad\Delta\equiv
Q^{2}P^{2}=4N_{1}N_{2}N_{3}N_{4}\,,$ (B.4)

where $\hat{c}(u)$ is defined through the relation [10, 104]

$-\vartheta_{1}(z\mid\tau)^{2}\,\eta(\tau)^{-6}\equiv\sum_{k,l}\widehat{c}\left(4k-l^{2}\right)e^{2\pi
i(k\tau+lz)}\,.$ (B.5)

$\vartheta_{1}(z\mid\tau)$ and $\eta(\tau)$ are respectively the odd Jacobi
theta function and the Dedekind eta function.
Table 2 shows helicity trace index for different 4–charges relevant for this
paper. Charge ${(1,1,2,2)}$ presents a system for which
$\operatorname{gcd}\left\\{\ell_{1},\ell_{2}\right\\}=2\,$.

$\begin{array}[]{|c|c|c|c|c|}\hline\cr\text{Charges}&\operatorname{gcd}\left\\{\ell_{1},\ell_{2}\right\\}&s&\Delta&B_{14}\\\
\hline\cr(1,1,1,1)&1&1&4&-\widehat{c}(4)=12\\\
\hline\cr(1,1,1,2)&1&1&8&-\widehat{c}(8)=56\\\
\hline\cr(1,1,1,3)&1&1&12&-\widehat{c}(12)=208\\\
\hline\cr(1,1,1,4)&1&1&16&-\widehat{c}(16)=684\\\
\hline\cr(1,1,2,2)&2&1,2&16&-\widehat{c}(4)-\widehat{c}(16)=696\\\
\hline\cr(1,1,2,3)&1&1&24&-\widehat{c}(24)=5616\\\ \hline\cr\end{array}$

Table 2: $14^{th}$ helicity trace index for different 4–charges.

##  Appendix C Gauge charges

In this appendix we collect various gauge charge assignment tables arising in
section 4 discussing Hilbert series. In the tables we have removed the
$\Phi$’s which are set to zero to ‘gauge fix’ the phase symmetries.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{12}&1&-1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{23}&0&0&1&-1&1&-1&1&-1&1&-1&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{34}&0&0&0&0&1&-1&0&0&1&-1&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 3: Gauge charges for fields in the order
$\\{Z^{(12)},Z^{(21)},Z^{(13)},Z^{(31)},Z^{(14)},Z^{(41)},Z^{(23)},Z^{(32)},\\\
Z^{(24)},Z^{(42)},Z^{(34)},Z^{(43)},\Phi^{(2)}_{3},\Phi^{(3)}_{1},\Phi^{(3)}_{2},\Phi^{(4)}_{1},\Phi^{(4)}_{2},\Phi^{(4)}_{3}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{12}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{23}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{34}&0&0&0&0&1&-1&0&0&1&-1&1&-1&0&0&0&0&0&0\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 4: Gauge charges for fields in the order
$\\{Z^{(12)},Z^{(21)},Z^{(13)},Z^{(31)},Z^{(14)},Z^{(41)},Z^{(23)},Z^{(32)},\\\
Z^{(24)},Z^{(42)},Z^{(34)},Z^{(43)},\Phi^{(2)}_{3},\Phi^{(3)}_{1},\Phi^{(3)}_{2},\Phi^{(4)}_{1},\Phi^{(4)}_{2},\Phi^{(4)}_{3}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{12}&1&-1&1&-1&1&-1&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{23}&0&0&1&-1&1&-1&1&-1&1&-1&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{34}&0&0&0&0&1&-1&0&0&1&-1&1&-1&0&0&0&0&0&0\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 5: Gauge charges for fields in the order
$\\{Z^{(12)},Z^{(21)},Z^{(13)},Z^{(31)},Z^{(14)},Z^{(41)},Z^{(23)},Z^{(32)},\\\
Z^{(24)},Z^{(42)},Z^{(34)},Z^{(43)},\Phi^{(2)}_{3},\Phi^{(3)}_{1},\Phi^{(3)}_{2},\Phi^{(4)}_{1},\Phi^{(4)}_{2},\Phi^{(4)}_{3}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{1}&1&-1&1&-1&0&0&1&1&-1&-1&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{2}&-1&1&0&0&1&-1&0&0&0&0&1&1&-1&-1&0&0&0&0\\\ \hline\cr
U(1)_{3}&0&0&-1&1&-1&1&0&0&0&0&0&0&0&0&1&1&-1&-1\\\ \hline\cr
U(1)_{a}&0&0&0&0&0&0&-1&0&1&0&-1&0&1&0&-1&0&1&0\\\ \hline\cr
U(1)_{b}&0&0&0&0&0&0&0&-1&0&1&0&-1&0&1&0&-1&0&1\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 6: Gauge charges for $Z^{(k\ell)}$ fields in the order
$\\{Z^{(12)},Z^{(21)},Z^{(13)},Z^{(31)},Z^{(23)},Z^{(32)},Z^{(14)}_{1},\\\
Z^{(14)}_{2},Z^{(41)}_{1},Z^{(41)}_{2},Z^{(24)}_{1},Z^{(24)}_{2},Z^{(42)}_{1},Z^{(42)}_{2},Z^{(34)}_{1},Z^{(34)}_{2},Z^{(43)}_{1},Z^{(43)}_{2}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{1}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{2}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{3}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{a}&0&0&0&0&1&-1&0&0&1&-1&0&0&1&-1&0\\\ \hline\cr
U(1)_{b}&0&0&0&0&-1&1&0&0&-1&1&0&0&-1&1&0\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 7: Gauge charges for $\Phi^{(k)}_{\ell}$ fields in the order
$\\{\Phi^{(2)}_{3},\Phi^{(3)}_{1},\Phi^{(3)}_{2},\Phi^{(4)}_{1,11},\Phi^{(4)}_{1,12},\Phi^{(4)}_{1,21},\Phi^{(4)}_{1,22},\Phi^{(4)}_{2,11},\\\
\Phi^{(4)}_{2,12},\Phi^{(4)}_{2,21},\Phi^{(4)}_{2,22},\Phi^{(4)}_{3,11},\Phi^{(4)}_{3,12},\Phi^{(4)}_{3,21},\Phi^{(4)}_{3,22}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{12}&1&-1&1&-1&0&0&1&1&-1&-1&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{23}&0&0&1&-1&1&-1&1&1&-1&-1&1&1&-1&-1&0&0&0&0\\\ \hline\cr
U(1)_{34}&0&0&0&0&0&0&1&1&-1&-1&1&1&-1&-1&1&1&-1&-1\\\ \hline\cr
SU(2)&0&0&0&0&0&0&1&-1&-1&1&1&-1&-1&1&1&-1&-1&1\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 8: Gauge charges for $Z^{(k\ell)}$ fields in the order
$\\{Z^{(12)},Z^{(21)},Z^{(13)},Z^{(31)},Z^{(23)},Z^{(32)},Z^{(14)}_{1},\\\
Z^{(14)}_{2},Z^{(41)}_{1},Z^{(41)}_{2},Z^{(24)}_{1},Z^{(24)}_{2},Z^{(42)}_{1},Z^{(42)}_{2},Z^{(34)}_{1},Z^{(34)}_{2},Z^{(43)}_{1},Z^{(43)}_{2}\\}$.

$\begin{array}[]{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline\cr
U(1)_{12}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{23}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
U(1)_{34}&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ \hline\cr
SU(2)&0&0&0&0&-2&2&0&0&-2&2&0&0&-2&2&0\\\ \hline\cr
U(1)_{B}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\\\ \hline\cr\end{array}$

Table 9: Gauge charges for $\Phi^{(k)}_{\ell}$ fields in the order
$\\{\Phi^{(2)}_{3},\Phi^{(3)}_{1},\Phi^{(3)}_{2},\Phi^{(4)}_{1,11},\Phi^{(4)}_{1,12},\Phi^{(4)}_{1,21},\Phi^{(4)}_{1,22},\Phi^{(4)}_{2,11},\\\
\Phi^{(4)}_{2,12},\Phi^{(4)}_{2,21},\Phi^{(4)}_{2,22},\Phi^{(4)}_{3,11},\Phi^{(4)}_{3,12},\Phi^{(4)}_{3,21},\Phi^{(4)}_{3,22}\\}$.

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