11institutetext:  Department of Applied Mathematics, FEECS, VŠB-TU Ostrava
22institutetext: Czech Academy of Sciences, Institute of Geonics 22email:
<EMAIL_ADDRESS>
# Balancing Predictive Relevance of Ligand Biochemical Activities

Marek Pecha

###### Abstract

In this paper, we present a technique for balancing predictive relevance
models related to supervised modelling ligand biochemical activities to
biological targets. We train uncalibrated models employing conventional
supervised machine learning technique, namely Support Vector Machines.
Unfortunately, SVMs have a serious drawback. They are sensitive to imbalanced
datasets, outliers and high multicollinearity among training samples, which
could be a cause of preferencing one group over another. Thus, an additional
calibration could be required for balancing a predictive relevance of models.
As a technique for this balancing, we propose the Platt’s scaling. The
achieved results were demonstrated on single-target models trained on datasets
exported from the ExCAPE database. Unlike traditional used machine techniques,
we focus on decreasing uncertainty employing deterministic solvers.

##  1  Ligand-based Supervised Biochemical Activity Modelling

In silico computational supervised modelling based on ligand profiles is
widely used as a complementary method in the pharmaceutical industry;
specifically in the early stage of drug design and development to obtain an
indication of off-target interactions. The aim is to train a model that
generalizes the biological activities of ligands, i.e. small biochemical
molecule, ion, or protein, by reusing information from former in vitro
laboratory experiments. A common approach to predicting activities of ligands
on biological targets, e.g. genes, is to exploit a panel of structure-analysis
models such as quantitative structure-activity relationship (QSAR) with one
model per a biological target. Then, using statistical learning, we train a
classification or regression model used to predict categorical (active or
inactive) or numerical values, e.g. probability of reaction, for a new unseen
ligand, respectively. The suggested strategies on how to train the models are
the single and multi-target schemes. In our text, we focus on a simplified
single-target modelling that is a baseline for more general approaches, e.g.
the multi-target ones mentioned above.

Several machine learning methods are typically employed for constructing such
models. Currently, the most popular ones are Deep Neural Network (DNNs) and
Support Vector Machines (SVMs). Despite the fact that Deep Learning (DL) is
getting popular in recent years, SVMs are still applicable. One of the most
advantages of the SVMs approach is that they find learning functions
maximizing geometric margins, unlike generally composed functions produced by
DNNs. A benefit of this additional implicit (in contrast with DNNS)
requirement is that reducing generalization error leading to preventing
overfitting of a model and improving its robustness in terms of a bias-
variance tradeoff. Respecting later, SVM seems to be appropriate as a
technique for modelling biochemical activity, which exploits ligand profiles,
for the reason that a multivariance among samples is typically small. This
fact is arising from the nature of in vitro experiments because laboratories
test biochemical activities of quite similar ligands to a particular target
commonly than testing ligands belonging to entirely different groups.

However, SVMs have a serious drawback; they are sensitive to imbalanced
datasets, outliers and multicollinearies among training samples, which could
be a cause of preferencing one group over another. Therefore, we propose to
use Platt scalling for an additional model calibration, which is based on
transforming the SVM classification model output into a posterior probability
distribution by fitting logistic regression model to SVM raw prediction
scores. This calibrating technique is practically used for reducing impact of
overfitting to predictor mainly caused by training data. After training
calibrated models, we demonstrate balanced predictive relevance of these
models by converting them to label prediction using an optimal threshold.

This text is organized as follows. In Section 2, the SVM formulations with a
relaxed-bias term are presented. A calibration technique based on the Platt
scaling is introduced in Section 3. Numerical experiments are presented in
Section 4.

##  2 Support Vector Machines for Relaxed-bias Classification

SVMs belong to conventional machine learning techniques, and they are
practically used for both classification and regression. Unlike the DL
underlying architecture, SVMs could be considered as the single perceptron
problems that find the learning functions maximizing the geometric margins.
Therefore, we can explain the qualities of a learning model and the underlying
solver behaviour straightforwardly. In this paper, we focus only on the linear
C-SVM for classification, in which a misclassifications error term is
penalized by a user-specified penalty $C$. We denote the C-SVM as SVM for
simplification in further text.

SVM was originally designed by CorVap-ML-1995  as a supervised binary
classifier, i.e. a classifier that decides whether a sample falls into either
Class A or Class B by means of a model determined from already categorised
samples in the training phase of the classifier. Let us denote the training
data as an ordered sample-label pairs such that

$T:=\\{\left(\mathbmit{x}_{1},y_{1}\right),\
\left(\mathbmit{x}_{2},y_{2}\right),\ \dots,\
\left(\mathbmit{x}_{m},y_{m}\right)\\},$ (1)

where $m$ is the number of samples, $\mathbmit{x}_{i}\in\mathbb{R}^{n},\
n\in\mathbb{N},$ is the $i$-th sample and $y_{i}\in\\{-1,1\\}$ denotes the
label of the $i$-th sample, $i\leq m$. The linear SVM solves the problem of
training the classification model in the form of a so-called maximal-margin
hyperplane:

$H=\langle\mathbmit{w},\mathbmit{x}\rangle+\hat{b},$ (2)

where $\mathbmit{w}$ is a normal vector of hyperplane $H$ and
$\hat{b}=\frac{b}{\|\mathbmit{w}\|}$ determines the offset of the hyperplane
$H$ from the origin along its normal vector $\mathbmit{w}\in\mathbb{R}^{n}$.
In the cases of the relaxed-bias classification, we do not consider $\hat{b}$
in a classification model, however we include it into the problem by means of
augmenting the vector $\mathbmit{w}$ and each sample $\mathbmit{x}_{i}$ with
an additional dimension so that
$\mathbmit{\widehat{w}}\leftarrow\begin{bmatrix}\mathbmit{w}\\\
B\end{bmatrix}$,
$\mathbmit{\widehat{x}_{i}}\leftarrow\begin{bmatrix}\mathbmit{x}_{i}\\\
\gamma\end{bmatrix}$, and $\gamma\in\mathbb{R}^{+}$ is a user defined
variable, typically set to $1$.

Let $p\in\\{1,2\\}$ for purposes related to our application, then the problem
of finding hyperplane
$\widehat{H}=\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}\rangle$ can
be formulated as a constrained optimization problem in the following primal
formulation:

$\operatorname*{arg\,min}_{\widehat{\mathbmit{w}},\ \widehat{\xi}_{i}}\
\frac{1}{2}\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{w}}\rangle+\frac{C}{p}\sum_{i=1}^{n}\widehat{\xi}_{i}^{p}\hskip
5.0pt\text{s.t.}\hskip 5.0pt\begin{cases}\
y_{i}\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}_{i}\rangle\geq
1-\widehat{\xi}_{i},\\\ \ \widehat{\xi}_{i}\geq 0\hskip 5.0pt\text{if $p=1$},\
i\in\\{1,2,\dots,n\\},\end{cases}$ (3)

where
$\widehat{\xi}_{i}=\max\\{0,1-y_{i}\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}_{i}\rangle\\}$
is the hinge loss function releated to augmented samples
$\widehat{\mathbmit{x}}_{i}$. Essentially, the hinge loss function quantifies
error between predicted and correct classification of sample
$\widehat{\mathbmit{x}}_{i}$. The variable $C\in\mathbb{R}^{+}$ is a penalty
parameter that penalizes misclassification error. Generally, a higher value of
$C$ increases the importance of minimising the hinge loss functions $\xi_{i}$
and, on the other hand, causes maximizing $\|\mathbmit{w}\|$, i.e. minimizing
the width of margin, leading to poor generalization capabilities of a
classification model. The goal is to find a reasonable value of $C$ such that
a resulting model balances the robustness and performance tradeoff.

Further, we can say in general, the minimizer associated with formulation (3)
corresponding to an optimal rotation of separating hyperplane $\widehat{H}$ in
one-dimension higher feature-space $\mathbb{R}^{n+1}$ than the original
feature space has.

To reduce a number of unknowns, we can dualize the primal formulation (3)
using the Lagrange duality so that, for both $p=1$ and $p=2$, they result in:

$\operatorname*{arg\,min}_{\boldsymbol{\alpha}}\
\frac{1}{2}\boldsymbol{\alpha}^{T}\mathbmit{Y}^{T}\mathbmit{K}\mathbmit{Y}\boldsymbol{\alpha}-\boldsymbol{\alpha}^{T}\mathbmit{e}\hskip
5.0pt\text{s.t.}\hskip 5.0pt\mathbmit{o}\leq\boldsymbol{\alpha}\leq
C\mathbmit{e},$ (4) $\operatorname*{arg\,min}_{\boldsymbol{\alpha}}\
\frac{1}{2}\boldsymbol{\alpha}^{T}\left(\mathbmit{Y}^{T}\mathbmit{K}\mathbmit{Y}+C^{-1}\mathbmit{I}\right)\boldsymbol{\alpha}-\boldsymbol{\alpha}^{T}\mathbmit{e}\hskip
5.0pt\text{s.t.}\hskip 5.0pt\mathbmit{o}\leq\boldsymbol{\alpha},$ (5)

respectively. $\mathbmit{K}\in\mathbb{R}^{m\times m}$ is matrix of inner
products called Gramian such that
$\mathbmit{K}:=\mathbmit{X}^{T}\mathbmit{X}$,
$\mathbmit{X}=\left[\mathbmit{x}_{1},\ \mathbmit{x}_{2},\
\dots,\mathbmit{x}_{m}\right]$ is data matrix of the training samples,
$\mathbmit{y}=\left[y_{1},\ y_{2},\ \dots,\ y_{m}\right]^{T}$ is vector of
corresponding labels, $\mathbmit{Y}=diag(\mathbmit{y})$, and
$\mathbmit{o}\in\mathbb{R}^{m}$, $\mathbmit{e}\in\mathbb{R}^{m}$ denote a
zero-vector and an all-ones vector, respectively. In general, the Gramian
$\mathbmit{K}$ is symmetric positive semi-definite (SPS) of a rank

$\operatorname{rank}\left\lparen\mathbmit{K}\right\rparen=\min\\{\min\\{m,n\\},N_{g}\\},$
(6)

where $N_{g}$ maximum number of linearly independent training samples, $m$ and
$n$ are a numbers of training samples and their features, respectively.

Comparing the dual SVM-QP formulations, specifically, $\mathcal{l}$1-loss (4)
and $\mathcal{l}$2-loss (5), we can see that they differ in the forms of the
related Hessians and constraints. While the Hessian is generally an SPS matrix
in a case of (4), the Hessian related to the formulation (5) is regularized by
means of the matrix $C^{-1}\mathbmit{I}$. This can provide a better
convergence rate for (5) and an associated optimization problem could be more
stable. On the other hand, $\mathcal{l}1$-loss SVM could produce a more robust
model in the sense of performance score, because using a linear sum of
$\xi_{i}$ leads to catching the outliers during a training phase of a
classifier.

Further, for obtaining a solution of the original primal problem, we introduce
dual to primal reconstruction formula as follows:

$\widehat{\mathbmit{w}}=\widehat{\mathbmit{X}}\mathbmit{Y}\boldsymbol{\alpha},$
(7)

Using the reconstructed normal vector $\widehat{\mathbmit{w}}$, we can set the
decision rule:

$\operatorname{sgn}\left(\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}_{i}\rangle\right)=\begin{cases}\
+1\dots\ \widehat{\mathbmit{x}}_{i}\in\text{Class A,}\\\ \ -1\dots\
\widehat{\mathbmit{x}}_{i}\in\text{Class B.}\end{cases}$ (8)

In sense of equivalence of solutions, we can easily show a connection between
the classification models associated with standard model (2) and relaxed-bias
formulations. Let us write the separating hyperplane equation in a component-
wise form such that:

$\widehat{H}:=\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}\rangle=\underbrace{w_{1}x_{1}+w_{2}x_{2}+\dots+w_{m}x_{m}}_{=\langle\mathbmit{w},\mathbmit{x}\rangle}+\underbrace{B\gamma}_{=:b},$
(9)

which is equivalent to (2). However, the bias term $b$ is incorporated into
the regularization term in sense of Tikhonov regularization, the resulting
model could slightly differ from this attained by standard (non-relaxed)
formulations. On the other side, we have not to deal with equality
constraints, which are appear in the standard dual formulations, and
sometimes, they are reason why solvers could diverge.

##  3 Model Calibration

Calibrating a classification model refers to a special type of statistical
inference that transforms a uncalibrated output (raw prediction),
particularly, decision function $f\\!\left(\mathbmit{x}\right)$, to a
probability of class membership $P(\textit{class}\ |\ \textit{input})$.
Commonly, the calibration is required when we need to adjust robustness of a
classification model, balances of class preference and provides a cost-
sensitive classification. In this section, we pay attention to an estimation
of the probability employing a well-known calibration technique called Platt’s
scaling – introduced by Platt-Advances-1999 . Commonly, this technique is
known as Platt’s calibration in the machine learning communities.

An idea beyond this technique is based on fitting a parametric form of
sigmoid-shaped function that maps the uncalibrated SVM output to the posterior
probability $P\left(y=1\ |\ \mathbmit{x}\right)\approx P_{A,B}\left(y=1\ |\
\mathbmit{x}\right)$, where

$P_{A,B}\left(y=1\ |\
\mathbmit{x}\right)=\frac{1}{1+\exp\left(Af(\mathbmit{x})+B\right)}.$ (10)

The parameters $A$, $B$ determine the slope of the sigmoidal curve and lateral
displacement, respectively, and they are practically fitted using maximum
likelihood estimation (MLE). In order to the relaxed-bias classification
mentioned in Section 2, we assume that the raw SVM output

$f(\mathbmit{x}):=\widehat{H}\\!\left(\widehat{\mathbmit{x}}\leftarrow\begin{bmatrix}\mathbmit{x}\\\
\gamma\end{bmatrix}\right)=\left\langle\widehat{\mathbmit{w}},\widehat{\mathbmit{x}}\right\rangle$
(11)

is proportional to the log odds of positive samples in the model (10).

In the original paper, Platt suggested to use an additional training set, i.e.
a calibration set, for training calibration curve on output of general
instance-based SVM to avoid incorporating bias failures, i.e. cases when
$1-y_{i}f_{i}>0$, on functional-margin $\gamma=|1|$. Let us denote such
dataset as an ordered set:

$T_{CA}:=\\{\left(f_{1},y_{1}\right),\left(f_{2},y_{2}\right),\dots\left(f_{l},y_{l}\right)\\},$
(12)

where $l$ is a number of the calibration samples, $f_{j}$ is estimate of
$f\left(\mathbmit{x}_{j}\right)$ for $j\in\\{1,2,\dots,l\\}$. On the other
hand, when an optimal model performance is attained in a reasonably small
value of the penalty $C$, e.g., in real-world applications employing linear
SVMs, see Platt-Advances-1999 , and data is well-behaved, bias on margin
failures usually become small. Therefore, it often possible to simply fit the
sigmoid on the training dataset.

To prevent model overfitting, Platt proposed additional transformation of
binary labels $y_{j}$ to target probabilities $t_{j}$ such that
$t_{j}=\frac{N_{p}+1}{N_{p}+2}\ \text{iff}\ y=+1$, or
$t_{j}=\frac{1}{N_{n}+2}\ \text{iff}\ y=-1$, where $N_{p}$ and $N_{n}$ are
numbers of positive and negative calibration samples, respectively.

The best parameter setting $\left(A^{*},B^{*}\right)$ is determined by
minimizing negative log likehood (cross-entropy error function) on calibration
data so that:

$\left(A^{*},B^{*}\right)=\operatorname*{arg\,min}_{A,B}-\sum_{j=1}^{l}\left[\
t_{j}\log\left(p_{j}\right)+\left(1-t_{j}\right)\log\left(1-p_{j}\right)\
\right],$ (13)

where $p_{j}=\frac{1}{1+\exp\left(Af_{j}+B\right)}$. To solve (13), an author
in Platt-Advances-1999  proposed to use Levenberg–Marquardt (LM) algorithm
Levenberg-QAM-1944 . Unfortunately, a technique for updating a damping factor
for LM introduced by Platt causes that solver could not converge to a minimum
of (12). It is discussed in Lin-ML-2007 . To avoid issue arising from a
damping factor associated with the LM method, the authors suggested the Newton
method with backtracking line-search.

Though the proposed approach is favourable due to its simplicity, the trust-
region methods are more robust. Since we focus on training robust predictors
in this paper, we exploit the Newton method with trust region in all numerical
experiments presented in Section 4.

##  4 PermonSVM

The PermonSVM package is a part of the PERMON toolbox designed for usage in a
massively parallel distributed environment containing hundreds or thousands
computational cores. Technically, it is an extension of the core PERMON
package called PermonQP, from which it inherits basic data structures,
initialization routines, build system, and utilizes computational and QP
transformation routines, e.g. normalization of an objective function,
dualization, etc. Programmatically, core functionality of PERMON toolbox is
written on the top of the PETSc framework, follows the same design and coding
style, making it easy-to-use for anyone familiar with PETSc. It is usable on
all main operating systems and architectures consisting of smartphones through
laptops to high-end supercomputers.

PermonSVM supports distributed parallel (through MPI) reading from formats
like SVMLight, HDF5, PETSc binary file formats, more than 4 problem
formulations of classification problem, two types of parallel cross-
validation, namely k-fold and stratified k-fold cross-validation. The
resulting QP-SVM problem with implicitly represented Hessian, in which Gram
matrix $\mathbmit{X}^{T}\mathbmit{X}$ is not assembled, is proceeded by
solvers provided by the PermonQP package or the PETSc framework. Unlike
standard machine learning libraries, PERMON toolbox provides interface
functions to change underlying QP-SVM solver, monitoring and tweaking the
algorithms. In Code 1, we present an example of a usage PermonSVM API. Our
libraries are developed as an open-source project under the BSD 2-Clause
Licence.

⬇

MPI_comm comm = PETSC_COMM_WORLD;

SVM svm;

PetscViewer v;

char f_train[PETSC_MAX_PATH_LEN] = "cnr1.h5";

char f_test[PETSC_MAX_PATH_LEN] = "cnr1.t.h5";

TRY( SVMCreate(comm,&svm) );

TRY( SVMSetType(svm,SVMPC) ); /* Platt calibration type */

TRY( SVMSetFromOptions(svm) );

TRY( PetscViewerHDF5Open(comm,f_train,FILE_MODE_READ,&v) );

TRY( SVMLoadTrainingDataset(svm,viewer) );

TRY( PetscViewerDestroy(&v) );

TRY( PetscViewerHDF5Open(comm,f_train,FILE_MODE_READ,&v) );

TRY( SVMLoadCalibrationDataset(svm,viewer) );

TRY( PetscViewerDestroy(&v) );

TRY( PetscViewerHDF5Open(comm,f_test,FILE_MODE_READ,&v) );

TRY( SVMLoadTestDataset(svm,v) );

TRY( PetscViewerDestroy(&v) );

TRY( SVMSetHyperOpt(svm,PETSC_TRUE) );

TRY( SVMSetNfolds(svm,3) );

TRY( SVMTrain(svm) );

TRY( SVMTest(svm) );

TRY( SVMDestroy(&svm) );

Code 1: Example of calling PermonSVM API.

##  5 Numerical Experiments

In this section, we analyze numerical experiments related to balancing
predictive relevance of the single-target relaxed-bias classification model
using the Platt’s Calibration technique, which we introduced in Section 2 and
Section 3, respectively. We benchmark this approach on datasets associated
with biochemical activities of ligands on $4$ biological targets, namely
abl$1$ (Abelson murine leukemia viral oncogene homolog $1$ protein), adora$2$a
(Adenosine A$2$A receptor), cnr$1$ (cannabinoid receptor type $1$), and cnr$2$
(cannabinoid receptor type $2$). These datasets were exported from the ExCAPE
database, which was developed by Sun-JOC-2017 . While it is possible to
calibrate a model on the same dataset, on which the model was trained, see
Section 3, it could be problematic to decide if a bias of an uncalibrated
model is small enough. Therefore, we split training samples into the training
and calibration datasets. After training models, we evaluate their performance
on the test dataset using precision, sensitivity, and area under the curve
receiver operating characteristic (AUC) performance scores. The input datasets
were divided into training, calibration and test datasets such that they
consist of $640$, $200$, $160$ ligands, respectively, and a ratio of active
and inactive ones is sufficiently preserved. Characteristics associated with
these datasets are summarized in Table 1.

Table 1: The characteristics of training, calibration and test dataset related to abl$1$, adora$2$a, cnr$1$, cnr$2$ biological targets. Target (dataset) | #ligands
---|---
#active | #incative | $\sum$
abl$1$ (training) | $312\ (48.75\%)$ | $328\ (51.25\%)$ | $640$
abl$1$ (calibration) | $92\ (46.00\%)$ | $108\ (54.00\%)$ | $200$
abl$1$ (test) | $81\ (50.62\%)$ | $79\ (49.38\%)$ | $160$
adora$2$a (training) | $343\ (53.59\%)$ | $297\ (46.40\%)$ | $640$
adora$2$a (calibration) | $105\ (52.50\%)$ | $95\ (47.50\%)$ | $200$
adora$2$a (test) | $95\ (59.38\%)$ | $65\ (40.62\%)$ | $160$
cnr$1$ (training) | $392\ (61.25\%)$ | $248\ (38.75\%)$ | $640$
cnr$1$ (calibration) | $123\ (61.50\%)$ | $77\ (38.50\%)$ | $200$
cnr$1$ (test) | $110\ (68.75\%)$ | $50\ (31.25\%)$ | $160$
cnr$2$ (training) | $405\ (63.28\%)$ | $235\ (36.72\%)$ | $640$
cnr$2$ (cablibration) | $127\ (63.50\%)$ | $73\ (36.50\%)$ | $200$
cnr$2$ (test) | $112\ (70.00\%)$ | $48\ (36.50\%)$ | $160$

For training uncalibrated classification models, we choose the best penalty
$C_{BE}$ from the set $\widehat{C}=\\{2^{p},p\in\\{-7,-6,\dots,6,7\\}\\}$
algorithmically employing the hyperparameter optimization (HyperOpt) by means
of grid-search combined with stratified $3$-fold cross validation (CV). The
value of the best penalty $C_{BE}$ is selected so that accumulated related
precision and sensitivity during CV are maximized. All components of the
initial guess $\mathbmit{x}_{0}$ are set to $0.99*C$, proposed in Pecha-
LNEE-2019 . The relative norm of projected gradient being smaller than $1e-1$,
discussed in Pecha-SVM-AIP-2018 , is used as stopping criterion for the MPRGP
(Modified Proportioning and Reduced Gradient Projection) algorithm, see Dos-
book-09 , in all presented experiments. The expansion step-length $\alpha$ is
fixed and determined such as $\alpha=1.95/\|\mathbmit{A}\|_{2}$, where
$\|\mathbmit{A}\|_{2}=\sqrt{\lambda_{max}\left(\mathbmit{A}^{T}\mathbmit{A}\right)}$,
where $\mathbmit{A}$ denotes the Hessian matrix associated with (4) and (5).

Using PETSc implementation of the Newton method without preconditioning with
default setting, the sigmoid-shaped calibration function is computed by
minimizing cross-entropy (13) on calibration data. Since the Newton method
converges quickly to optimal solution $\mathbmit{x}^{*}$ when vector
$\mathbmit{x}$ is close enough to $\mathbmit{x}^{*}$, Platt-Advances-1999
proposed initial guesses for parameters of sigmoid such that $A_{0}=0$ and
$B_{0}=\log{\frac{l^{+}+1}{l^{-}+1}}$, where $l^{+}$ and $l^{-}$ denote
numbers of active and inactive samples associated with the calibration dataset
$T_{CA}$. To avoid numerical difficulties or catastrophic cancellations that
could arise from evaluation $1-p_{i}$, where $p_{i}$ is close to $1$, we
evaluate (10) by using
$\frac{\exp\left(-Af(\mathbmit{x})-B\right)}{1+\exp\left(-Af(\mathbmit{x})-B\right)}$
when $Af(\mathbmit{x})+B\geq 0$ else we use (10). This numerical improvements
were proposed by Lin-ML-2007 . Other numerical obstacles could arise from
evaluating Hessian

$H=\begin{bmatrix}\sum_{i=1}^{l}f_{i}^{2}p_{i}\left(1-p_{i}\right)&\sum_{i=1}^{l}f_{i}p_{i}\left(1-p_{i}\right)\\\
\sum_{i=1}^{l}f_{i}p_{i}\left(1-p_{i}\right)&\sum_{i=1}^{l}p_{i}\left(1-p_{i}\right)\end{bmatrix}$
(14)

associated with cross-entropy function (13). Thus, we replace the term
$\left(1-p_{i}\right)$ by means of
$\frac{1}{1+\exp\left(-Af(\mathbmit{x})-B\right)}$ when
$Af(\mathbmit{x})+B\geq 0$, else
$\frac{\exp\left(Af(\mathbmit{x})+B\right)}{1+\exp\left(Af(\mathbmit{x})+B\right)}$,
see Lin-ML-2007 . Since the Hessian (14) is SPS in general Lin-ML-2007 , we
regularize it by the matrix $\sigma\mathbmit{I}$, where
$\sigma=1\mathrm{e}{-12}$ in our experiments.

Instead of stochastic optimization, which commonly used in the machine
learning community, our used solvers are deterministic in these experiments.
They pass all training samples in one iteration. Therefore we consider terms
epoch and iteration as identical in this text. In other words, we do not take
into account a batch of a training dataset during the training phase of the
classifier. By this, we obtain strictly settled and reproducible training
pipelines, unlike employing DNNs or other traditionally used techniques. On
the other hand, deterministic solvers suffer on their cost in the sense of
computational resources. Thus, training predictors can take longer than in the
case of stochastic optimization. At this expense, we reduce uncertainty during
the training process, which could be crucial for some scientific application,
e.g. ones related to the pharmaceutical industry.

After calibrating a classification model, we convert probabilities to label
prediction using optimal threshold (thr.), i.e. $y=1\ \text{if
$p>\text{thr.}$}$ to demonstrate the balanced class predictive relevance of
calibrated models. The optimal threshold is determined using grid-search so
that absolute value of the difference of precision score (Pre.) and
sensitivity (Sen.) on the test dataset is minimized, and F$1$ score must be
greater than $0.50$, i.e. predictive ability of model must be better than
random.

Because of all datasets are too small to utilize more than one processor core,
all experiments were run on $1$ MPI process pinned to a processor core. In all
presented experiments, we utilized the same node of the ANSELM supercomputer
at IT4Innovations. Evaluations of performance scores are summarized in Table 2
and Table 3 for uncalibrated models and models after calibration,
respectively.

Table 2: abl$1$, adora$2$a, cnr$1$, cnr$2$ uncalibrated single-target models: evaluation of performance scores of models trained with parameter $C_{BE}$ determined by means of HyperOpt, particularly, grid-search combined with $3$-fold cross-validation. We report precision (Pre.), sensitivity (Sen.), F1 score, and AUC as qualitative metrics of a model performance for the both $\mathcal{l}1$-loss and $\mathcal{l}2$-loss SVM. Target |  Loss | Uncalibrated model
---|---|---
$C_{BE}$ | Pre. [%] | Sen. [%] | F$1$ | AUC
abl$1$ | $\mathcal{l}1$ | $2^{-6}$ | $71.60$ | $65.17$ | $0.68$ | $0.66$
$\mathcal{l}2$ | $2^{-5}$ | $69.14$ | $60.22$ | $0.64$ | $0.64$
adora$2$a | $\mathcal{l}1$ | $2^{-6}$ | $70.53$ | $82.72$ | $0.76$ | $0.74$
$\mathcal{l}2$ | $2^{-7}$ | $70.53$ | $83.75$ | $0.77$ | $0.74$
cnr$1$ | $\mathcal{l}1$ | $2^{-6}$ | $90.00$ | $82.50$ | $0.86$ | $0.77$
$\mathcal{l}2$ | $2^{-6}$ | $87.27$ | $81.36$ | $0.84$ | $0.74$
cnr$2$ | $\mathcal{l}1$ | $2^{-6}$ | $83.93$ | $82.46$ | $0.83$ | $0.72$
$\mathcal{l}2$ | $2^{-5}$ | $86.61$ | $82.91$ | $0.85$ | $0.74$

Looking at performance scores presented in Table 2, we can see that
$\mathcal{l}1$-loss SVM outperforms $\mathcal{l}2$-loss in overall performance
scores (F$1$ and AUC) in order to abl$1$ and cnr$1$ datasets, while
$\mathcal{l}2$-loss SVM provides slightly better models for adora$2$a and
cnr$2$ datasets. As we mentioned in Section $2$, $\mathcal{l}$1-loss SVM
commonly produces better quality models. However, we have to take into an
account that we relax bias $b$ in our approaches. Therefore the models are
relaxed as well, which could be a cause of these unexpected results in the
sense of performance score of models.

Table 3: abl$1$, adora$2$a, cnr$1$, cnr$2$ calibrated single-target models: quality of models in probabilistic sense (Brier score) and performance scores in sense of binary classification on test datasets related to models, which are converted from target probabilities to labels, using the optimal threshold (Thr.). Results are presented for the both $\mathcal{l}1$-loss and $\mathcal{l}2$-loss SVM.  Target | Loss | Calibrated model
---|---|---
Brier score | Binary classification
Thr. | Pre. [%] | Sen. [%] | AUC
abl$1$ | $\mathcal{l}1$ | $0.1105$ | $0.52$ | $65.43$ | $64.63$ | $0.64$
$\mathcal{l}2$ | $0.0947$ | $0.54$ | $60.49$ | $60.49$ | $0.60$
adora$2$a | $\mathcal{l}1$ | $0.1280$ | $0.41$ | $78.95$ | $78.95$ | $0.74$
$\mathcal{l}2$ | $0.1222$ | $0.44$ | $78.95$ | $78.95$ | $0.74$
cnr$1$ | $\mathcal{l}1$ | $0.0905$ | $0.63$ | $83.64$ | $83.64$ | $0.74$
$\mathcal{l}2$ | $0.0710$ | $0.58$ | $83.64$ | $83.64$ | $0.74$
cnr$2$ | $\mathcal{l}1$ | $0.0889$ | $0.53$ | $83.04$ | $83.04$ | $0.72$
$\mathcal{l}2$ | $0.0611$ | $0.53$ | $83.93$ | $83.93$ | $0.73$

Analysing predictive relevance of uncalibrated models, we can see that active
ligands are preferred in cases of models related to abl$1$, cnr$1$, and
cnr$2$, on the other hand, inactive ligands are prefered for adora$2$a
dataset. To balance predictive relevance, we perform model calibration. After
this, we can see in Table 3, the models trained using $\mathcal{l}$2-loss seem
to be better calibrated by comparing Brier scores for all cases than ones
related to the $\mathcal{l}$1-loss SVM. This could simple consequence of
underlying model robustness. Even we use relaxed approach, the
$\mathcal{l}$1-loss SVM still tries to produce a more robust model than the
$\mathcal{l}$2-loss SVM, since using a linear sum of hinge loss functions
instead of a sum of squared hinge loss functions leads to slightly better
catching the outliers.

Table 4: abl$1$, adora$2$a, cnr$1$, cnr$2$ biological targets: elapsed time related to training of models including HyperOpt and calibration. Loss |  Elapsed time [s] (HyperOpt + Training + Calibration)
---|---
|  abl$1$ |  adora$2$a |  cnr$1$ |  cnr$2$
$\mathcal{l}1$ | $2.15$ | $2.61$ | $1.95$ | $2.57$
$\mathcal{l}2$ | $1.38$ | $1.86$ | $1.57$ | $1.58$

Thus, calibrating models related to the $\mathcal{l}$2-loss SVM has
significant impact than in a case of the $\mathcal{l}$1-loss SVM; as we can
see, predictive relevances of classes are well balanced. Moreover, from the
Table 4, we can observe speedups $1.56$ (abl$1$), $1.40$ (adora$2$a), $1.24$
(cnr$2$), and $1.62$ (cnr$1$) in order to using the $\mathcal{l}2$-loss SVM
against the $\mathcal{l}1$-loss SVM.

However, calibrating models could cause a deterioration of overall model
performance determined by means of AUC as we can see in Table 4. Specifically,
the over performance scores of models decrease by $1\%$ to $4\%$ in the cases
of the abl$1$ (both loss-type models), cnr$1$ ($\mathcal{l}1$-loss model) and
cnr$2$ ($\mathcal{l}2$-loss model). In order to models related to adora$2$a
target, $\mathcal{l}1$-loss and $\mathcal{l}2$-loss models associated with
cnr$1$ and cnr$2$, respectively, AUC scores are same. Since models were
trained, calibrated and tested on different datasets, we can consider the
calibrated models have the same overall performance score in the sense of AUC
as their related uncalibrated models. Comparing the quality of the remaining
calibrated and uncalibrated models is application-specific. In some
application, they could be considered as models of same quality. For more
strict quality merits, the models can be considered that differ significantly.

From achieved results, it seems that it is better to train models using the
$\mathcal{l}2$-loss SVM that are not such robust as in the case of the
$\mathcal{l}1$-loss SVM and, then, perform their calibration. Moreover, we can
obtain a better convergence rate by employing this approach. We observe
speedup up to $1.62$ in case of training model on the cnr$2$ dataset.

##  6 Conclusion

In this paper, we focused on a problem dealing with balancing predictive
relevance of single-target models trained using SVMs. This calibration could
be required since SVMs is sensitive to imbalanced datasets, outliers and high
multicorrelation among training samples.

Regarding calibration improvements of models, we observe that an additional
calibration works significantly better for models trained using the
$\mathcal{l}2$-loss SVM with relaxed-bias from achieved results. It seems this
could be a consequence of that models are not such robust as in the case of
the $\mathcal{l}1$-loss SVM. Moreover, we achieve speedup up to $1.62$ by
means of the approach based on $\mathcal{l}2$-loss SVM. On the other hand,
calibrating models could cause a deterioration of overall model performance as
we saw in the presented numerical experiments. Therefore, it makes sense
calibrating models just for critical applications, e.g. biochemical modelling
presented in this paper, where balanced predictive relevance is required.

Since we achieved some unexpected results in the sense of model performance
scores, which are probably caused relaxing the bias term of the hyperplane
$H$, we are going to focus on calibrating model trained to employ training
based on full-formulation dual formulation of SVM, i.e. with equality
constraint. Further, we are going to test another calibration technique, e.g.
the isotonic regression.

## Acknowledgments

The author acknowledge the support of The Ministry of Education, Youth and
Sports from the National Programme of Sustainability (NPU II) project
“IT4Innovations excellence in science - LQ1602”; the grant programme “Support
for Science and Research in the Moravia–Silesia Region 2017” (RRC/10/2017),
financed from the budget of the Moravian–Silesian region; and the Grant of SGS
No. SP2020/84, VSB - Technical University of Ostrava. The author would like to
thank a reviewer for the constructive feedback as well.

## References

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Fermion zero modes of Denef Bates black holes

Palash Dubey

National Institute of Science Education and Research Bhubaneshwar,

P.O. Jatni, Khurdha, 752050, Odisha, INDIA

Homi Bhabha National Institute, Training School Complex,

Anushakti Nagar, Mumbai 400094, INDIA

E-mail<EMAIL_ADDRESS>
Abstract

We study the fermion zero modes of Denef Bates black holes in $\mathcal{N}=2$
Supergravity. The goldstino zero modes are constructed explicitly and their
boundary conditions, charge, and normalizability are studied. We then study if
there can exist fermion zero modes that are normalizable and not gauge
equivalent to the goldstino modes. We find that such modes are not possible if
we want to preserve the continuity of these solutions. The non-existence of
these modes shows that Denef Bates black holes must contribute to the index.

##  1 Introduction and summary

Black holes are classical objects in general relativity that are regions of
space-time enclosed by an event horizon that even massless particles cannot
escape. But, quantum-mechanically these objects are not at zero temperature.
Their temperature and Bekenstein-Hawking entropy is:

$T=\frac{\kappa}{2\pi},\;\;\;S=\frac{A}{4},$ (1.1)

where $\kappa$ is the surface gravity and $A$ is the area of the event
horizon.

Being a quantum theory of gravity string theory can be used to count the
microscopic degrees of freedom of a black hole and match it with the entropy
of the black hole with the relation

$S=\ln\Omega.$ (1.2)

This has been checked successfully in string theory for supersymmetric black
holes [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In this program, one identifies a
supersymmetric black hole carrying some charges and then using the charges we
compute the entropy of the black hole in Supergravity which can have
corrections to 1.1. Then we identify quantum states with the same charges in
string theory. This allows us to match the logarithm of the number of states
with the entropy of the black hole in Supergravity. The computation of the
number of states can be done in a region of moduli space where gravity is
weak. We compute an index which receives contributions from BPS states alone.
This makes the index protected by supersymmetry, that is to say that it does
not change with coupling.

If a BPS state breaks 4n supersymmetries then there are 4n goldstino fermion
zero modes. Quantization of these modes produces an equal number of fermionic
and bosonic states; hence the Witten index $Tr(-1)^{F}$ receives no
contribution from these states. To have a non-zero contribution from these
states the helicity trace index is used. This has been reviewed in [12]. It is
given by

$B_{2n}=\frac{1}{2n!}Tr\left[{(-1)^{F}(2h)^{2n}}\right]$ (1.3)

where $h$ is the third component of angular momentum in the rest frame. The
trace is computed over states with a fixed set of charges. If a state has more
than 4n fermion zero modes then those states will not contribute to the index.
This property makes only BPS states contribute to it and also can be used to
check which states contribute to the index and which do not. In this paper, we
are asking if Denef Bates black holes [13] contribute to the index. More
specifically we ask the question: are there any fermion zero modes apart from
the broken zero modes for Denef Bates black holes in $\mathcal{N}=2$
Supergravity; which would make the index associated with these black holes
vanish.

In [14] the authors have shown that a charge $\Lambda\Gamma$ supporting single
cenered Denef-Bates black holes of entropy $\sim\Lambda^{2}S(\Gamma)$ also
admits two-centered BPS black hole realizations whose entropy grows like
$\Lambda^{3}$ when $\Lambda\rightarrow\infty$. This shows that the entropy
associated to two centered black holes is extensive and macroscopic and the
two centered black holes contribute to the index. Hence, we expect to find no
extra fermion zero modes apart from the goldstino zero modes.

Fermion zero modes were first computed in [15, 16, 17, 18] for Majumdar
Papapetrou solutions. The norm on these fermion zero modes, which is used as a
central tool in this paper, was proposed in [19]. Extra fermion zero modes for
the Majumdar Papapetrou black holes were studied in [20].

In this paper, we compute the goldstino fermion zero modes for multicentered
Denef Bates black holes in $\mathcal{N}=2$ Supergravity and calculate the
supercharge of these modes. Then we study the normalizability and boundary
conditions of these modes. We find that the boundary condition for these zero
charge modes is $r^{2}\psi\rightarrow 0$ asymptotically, also they are
continuous all over $\mathbb{R}^{3}$. Which shows that the norm of these zero
charge modes recieves no contribution in the asymptotic limit. Then we show
that to have a non-zero contribution in the near horizon limit we must have a
discontinuous gravitini. Thus, we do not have a contribution to the norm from
the near horizon limit also.

The nonexistence of a zero charge and non-zero norm mode implies that there
cannot be any gauge inequivalent extra fermion zero modes in the spectrum.
Hence, the Denef Bates black holes must contribute to the index.

The paper is organized as follows. In section 2 we gather the needed
ingredients of $\mathcal{N}=2$ supergravity. In section 3 we introduce the
Denef Bates background and also compute the Killing spinors for the
background. In section 4 we find the goldstino fermion zero modes, show that
they satisfy the equations of motion, find the supercharge and the boundary
conditions of these solutions. In section 5 we show that you need to have non-
trivial zero charge solutions to have gauge inequivalent solutions; then we
argue that such zero charge modes satisfying the gauge condition and equations
of motion must have zero norm, and hence will not be proper states. In
appendix A we calculate spin connection and other quantities needed for
calculating the fermion zero modes. In appendix B we compute the equations of
motion of the theory.

##  2 $\mathcal{N}=2$ Supergravity

In this paper, we are concerned with Denef Bates black holes in the context of
string theory. If we reduce 10 dimensional Type IIA Supergravity (the low
energy limit of type IIA string theory) on a six dimensional Calabi Yau
manifold $X$, which preserves only a quarter of the supersymmetry, we get 4
dimensional $\mathcal{N}=2$ Supergravity coupled to
$h^{1,1}=\text{dim}\,H^{1,1}(X)$ vector multiplets and $h^{2,1}+1$ hyper
multiplets. The hyper multiplets can be completely decoupled, and we can set
them to arbitrary constant values. Hence, we are interested in Denef Bates
black holes as solutions to $\mathcal{N}=2$ Supergravity coupled to
$n_{V}=h^{1,1}$ vector multiplets.

The field content of this theory contains a graviton $g_{\mu\nu}$, two
gravitini which are Rarita Schwinger fields $\psi_{\mu}^{(i)}$, $n_{V}+1$
gauge fields (containing the graviphoton) $A_{\mu}^{I}$, $2\;n_{V}$ gaugini
$\chi_{(i)}^{\alpha}$ and $n_{V}$ complex scalar fields $z^{\alpha}$. $I$
takes values $0\cdots n_{V}$ and $\alpha$ takes values $1\cdots n_{V}$.

The metric we use is mostly positive. The clifford algebra in this paper is
defined by

$\displaystyle\gamma^{a}\gamma^{b}+\gamma^{b}\gamma^{a}=2\eta^{ab},\;\;\;\gamma_{\mu\nu\cdots}=\gamma_{[\mu}\gamma_{\nu}\cdots_{]},$
$\displaystyle\gamma_{*}=i\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}.$ (2.1)

The Chiral projection operators are

$P_{L}=\frac{1}{2}\left(1+\gamma_{*}\right),\;\;\;P_{R}=\frac{1}{2}\left(1-\gamma_{*}\right).$
(2.2)

The $(i)$ indices are the rigid R-symmetry $SU(2)$ indices, they are raised
and lowered by charge(or complex) conjugation.

$\psi^{(i)}=P_{L}\psi^{(i)},\;\;\;\psi_{(i)}=P_{R}\psi_{(i)},\;\;\psi^{(i)}=\psi_{(i)}^{C}.$
(2.3)

We also use the Levi-Civita $\varepsilon^{(ij)}$ for which the important
property is that

$\varepsilon_{(ij)}\varepsilon^{(jk)}=-\delta_{i}^{k}.$ (2.4)

These are the conventions and definitions of [21].

###  2.1 Special Kähler manifolds

The $n_{V}$ complex scalar fields of the gauge multiplet define a special
Kähler manifold. Now we collect the needed facts about these manifolds.

Special Kähler manifolds are Kähler manifolds that possess a symplectic
structure $Sp(2(n_{V}+1))$. The basic ingredients are symplectic vectors

$V=\begin{pmatrix}X^{I}\\\
F_{I}\end{pmatrix}=yv,\;\;v=\begin{pmatrix}Z^{I}(z)\\\
F_{I}(z)\end{pmatrix}_{.}$ (2.5)

The vector $V$ obeys the following

$\langle
V,\nabla_{\alpha}V\rangle=0,\;\;\;\nabla_{\alpha}V=y\nabla_{\alpha}v,\;\;\nabla_{\alpha}v=\partial_{\alpha}v+(\partial_{\alpha}\mathcal{K})v.$
(2.6)

Here,

$\langle V,\bar{V}\rangle\equiv X^{I}\bar{F_{I}}-F_{I}\bar{X^{I}}.$ (2.7)

The Kahler potential $\mathcal{K}$ is determined by

$\langle V,\bar{V}\rangle=i,(y\bar{y})^{-1}=e^{-\mathcal{K}}=-i\langle
v,\bar{v}\rangle,$ (2.8)

which in turn defines the metric on the Kähler manifold as

$g_{\alpha\bar{\beta}}=\partial_{\alpha}\partial_{\bar{\beta}}\mathcal{K}=i\langle\nabla_{\alpha}V,\bar{\nabla}_{\bar{\beta}}\bar{V}\rangle.$
(2.9)

The matrix ${\mathcal{N}}_{IJ}$, which couples the scalar fields to gauge
fields, is defined as

$\bar{\mathcal{N}}_{IJ}=(\bar{F}_{I}\;\;\;\nabla_{\alpha}F_{I})(\bar{X}^{J}\;\;\;\nabla_{\alpha}X^{J})^{-1}$
(2.10)

###  2.2 Action

The action for this theory upto quadratic terms in fermions 111we are studying
fermion zero modes so the quartic fermionic terms, which would give cubic
terms in fermions in the equation of motion, are irrelevant here is [21]

$\displaystyle
e^{-1}\mathcal{L}=\left(\frac{R}{2}-\bar{\psi}_{(i)\mu}\gamma^{\mu\nu\rho}D_{\nu}\psi_{\rho}^{(i)}\right)-g_{\alpha\bar{\beta}}\partial_{\mu}z^{\alpha}\partial^{\mu}\bar{z}^{\bar{\beta}}$
$\displaystyle+\Big{(}-\frac{1}{4}i\mathcal{N}_{IJ}F^{+I}_{\mu\nu}F^{+\mu\nu
J}+F^{+I}_{\mu\nu}Im{\mathcal{N}}_{IJ}Q^{\mu\nu+J}$
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{(i)}^{\alpha}\not{D}\chi^{(i)\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\bar{\psi}_{(i)a}\not{\partial}z^{\alpha}\gamma^{a}\chi^{(i)\bar{\beta}}+h.c.\Big{)}$
(2.11)

where

$Q^{ab-J}\equiv\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}\bar{\chi}^{\bar{\alpha}(i)}\gamma^{a}\psi^{b(j)}\varepsilon_{(ij)}+X^{J}\bar{\psi}^{a}_{(i)}\psi^{b}_{(j)}\varepsilon^{(ij)}+\frac{1}{8}\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}g^{\beta\bar{\alpha}}C_{\beta\gamma\delta}\bar{\chi}_{(i)}^{\gamma}\gamma^{ab}\chi_{(j)}^{\delta}\varepsilon^{(ij)}.$
(2.12)

The covariant derivatives are given by

$\displaystyle D_{\mu}\psi_{\nu(i)}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}+\frac{1}{2}i\mathcal{A}_{\mu}\right)\psi_{\nu(i)}$
$\displaystyle D_{\mu}\chi_{(i)}^{\alpha}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}+\frac{1}{2}i\mathcal{A}_{\mu}\right)\chi_{(i)}^{\alpha}+\Gamma^{\alpha}_{\beta\gamma}\chi_{(i)}^{\gamma}\partial_{\mu}z^{\beta}.$
(2.13)

They contain the Kähler connection

$\displaystyle\mathcal{A}_{\mu}=\frac{1}{2}i\partial_{\alpha}\mathcal{K}\partial_{\mu}z^{\alpha}-\frac{1}{2}i\partial_{\bar{\alpha}}\mathcal{K}\partial_{\mu}\bar{z}^{\bar{\alpha}}.$
(2.14)

###  2.3 Fermion variation equations

The supersymmetry variation equations for the fermions are [21]

$\displaystyle\delta\psi^{(i)}_{\mu}$ $\displaystyle=$
$\displaystyle\left(\partial_{\mu}+\frac{1}{4}\omega^{ab}_{\mu}\gamma_{ab}-\frac{1}{2}i\mathcal{A}_{\mu}\right)\epsilon^{(i)}-\frac{1}{16}\gamma^{ab}T^{-}_{ab}\varepsilon^{(ij)}\gamma_{\mu}\epsilon_{(j)}$
$\displaystyle\delta\chi_{(i)}^{\alpha}$ $\displaystyle=$
$\displaystyle\not{\partial}z^{\alpha}\epsilon_{(i)}-\frac{1}{2}G^{-\alpha}_{ab}\gamma^{ab}\varepsilon_{(ij)}\epsilon^{(j)}$
(2.15)

where,

$\displaystyle T^{+}_{ab}=2i(\bar{X}^{I}G^{+}_{Iab}-\bar{F}_{I}F^{+I}_{ab})$
$\displaystyle
G^{-\alpha}_{ab}=g^{\alpha\bar{\beta}}\bar{\nabla}_{\bar{\beta}}\bar{X}^{I}Im\,\mathcal{N}_{IJ}F^{-J}_{ab}.$
(2.16)

##  3 Denef Bates black holes

A multicentered black hole solution for $\mathcal{N}=2$ Supergravity,
described in [13], is given by a BPS metric,

$ds^{2}=-e^{2U}(dt+\omega)^{2}+e^{-2U}dx^{i}dx^{i}$ (3.1)

where $U$ and $\omega$ together with the moduli fields $z^{\alpha}$, are time-
independent solutions of the following equations

$\displaystyle 2e^{-U}Im(e^{-i\alpha}V)=-H$ $\displaystyle d\omega=\langle
dH,H\rangle$ (3.2)

where $\alpha$ is an unknown real function, $H(x)$ is a given harmonic
function, the Hodge star operates on flat $\mathbb{R}^{3}$.

For $N$ charges located at $x_{s}$, $p=1,\cdots,N$; in asymptotically flat
space, one has:

$H=\sum_{s=1}^{N}\frac{\Gamma_{s}}{|x-x_{s}|}-2Im(e^{-i\alpha}V)_{r=\infty}$
(3.3)

The boundary condition on $\alpha$ at $r=\infty$ is that it equals the phase
of the total central charge, $\alpha=arg\,Z$.

$\Gamma=\begin{pmatrix}p^{I}\\\ q_{I}\end{pmatrix}\;\;\;Z=2\langle
V,\Gamma\rangle$ (3.4)

3.2 gives us an integrability condition (by applying the operator $d*$ on both
sides of the equation), which in turns gives us a restriction on the distances
between the centers of the black holes

$\langle\Delta H,H\rangle=0$ (3.5)

For a generic $H$, $\Delta H=\sum_{s=1}^{N}\Gamma_{s}\delta^{3}(x-x_{s})$.
Then the integrability conditon becomes
$\langle\Gamma_{s},H\rangle|_{x=x_{s}}=0$ or,

$\sum_{r\neq
s}\frac{\langle\Gamma_{r},\Gamma_{s}\rangle}{|x_{r}-x_{s}|}=-2\langle\text{Im}(e^{-i\alpha}V)_{r=\infty},\Gamma_{s}\rangle$
(3.6)

this shows that the distances between the centres are constrained and the
black holes form bound states.

This is an important ingredient, if suppose we are to study the effective
supersymmetric quantum mechanics of the centres of the black holes then we
would find that these constraints are constraints on the bosonic degrees of
freedom, which would certainly reduce the fermionic degrees of freedom also.
The fermion zero modes we are interested in here are these fermionic degrees
of freedom in this effective theory.

With no loss of generality we can assume $\alpha=0$, or $(Im\langle
V,\Gamma\rangle)_{r=\infty}=0$. Then, 3.2 becomes

$ie^{-U}e^{\mathcal{K}/2}\begin{pmatrix}Z^{I}-\bar{Z}^{I}\\\
F_{I}-\bar{F}_{I}\end{pmatrix}=H\equiv\begin{pmatrix}\tilde{H}^{I}\\\
H_{I}\end{pmatrix}$ (3.7)

or,

$\displaystyle U=\frac{\mathcal{K}}{2}$ $\displaystyle i(v-\bar{v})=H.$ (3.8)

Then by definition,

$\displaystyle e^{-2U}=e^{-\mathcal{K}}=-i(Z^{I}\bar{F}_{I}-F_{I}\bar{Z}^{I})$
$\displaystyle\partial_{\alpha}\mathcal{K}=ie^{\mathcal{K}}(\partial_{\alpha}Z^{I}\bar{F}_{I}+Z^{I}\partial_{\alpha}\bar{F}_{I}-\partial_{\alpha}F_{I}\bar{Z}^{I}-F_{I}\partial_{\alpha}\bar{Z}^{I})$
(3.9)

since $F_{I}$ and $Z^{I}$ are holomorphic $\partial_{\alpha}\bar{F}_{I}=0$ and
$\partial_{\alpha}\bar{Z}^{I}=0$.

$\displaystyle\partial_{{\alpha}}\mathcal{K}=ie^{\mathcal{K}}(\partial_{\alpha}Z^{I}\bar{F}_{I}-\partial_{\alpha}F_{I}\bar{Z}^{I})$
$\displaystyle\partial_{\bar{\alpha}}\mathcal{K}=ie^{\mathcal{K}}(Z^{I}\partial_{\bar{\alpha}}\bar{F}_{I}-F_{I}\partial_{\bar{\alpha}}\bar{Z}^{I})$
$\displaystyle\mathcal{A}_{m}=i\frac{1}{2}(\partial_{\alpha}\mathcal{K}\partial_{m}z^{\alpha}-\partial_{\bar{\alpha}}\mathcal{K}\partial_{m}\bar{z}^{\bar{\alpha}})$
$\displaystyle\mathcal{A}_{m}=-\frac{1}{2}e^{\mathcal{K}}(\partial_{m}Z^{I}\bar{F}_{I}-\partial_{m}F_{I}\bar{Z}^{I}+F_{I}\partial_{m}\bar{Z}^{I}-Z^{I}\partial_{m}\bar{F}_{I})$
(3.10)

also from $\langle
v,\nabla_{\alpha}v\rangle=Z^{I}\partial_{\alpha}F_{I}-F_{I}\partial_{\alpha}Z^{I}=0.$
we have

$\displaystyle\mathcal{A}_{m}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{\mathcal{K}}\left[(Z^{I}-\bar{Z}^{I})\partial_{m}(F_{I}-\bar{F}_{I})-(F_{I}-\bar{F}_{I})\partial_{m}(Z^{I}-\bar{Z}^{I})\right]$
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{2U}\langle{d(v-\bar{v})},(v-\bar{v})\rangle$
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{2U}\langle{H,dH}\rangle$
$\displaystyle\mathcal{A}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}e^{2U}*d\omega$ (3.11)

3.8 and 3.11 are the same equations which define black holes in [22].

The gauge field of the background is given by [13]

$\displaystyle
F^{I}_{ij}=e^{2U}\epsilon_{ijk}\delta^{k}_{m}\partial_{m}\tilde{H}^{I}$
$\displaystyle G_{Iij}=e^{2U}\epsilon_{ijk}\delta^{k}_{m}\partial_{m}H_{I}.$
(3.12)

where,

$\displaystyle G^{+}_{\mu\nu I}=\mathcal{N}_{IJ}F^{+J}_{\mu\nu}.$ (3.13)

###  3.1 Killing Spinors

These black holes are supersymmetric by construction. We can also check their
supersymmetry by solving the fermion variation equations 2.3 to find the
Killing spinors. The Killing spinors are needed to find the broken zero modes.

The vierbeins for the Denef Bates black hole are

$e^{0}=e^{U}(dt+\omega),\;\;e^{i}=e^{-U}\delta^{i}_{m}\;dx^{m}.$ (3.14)

In appendix A we show that the spin connections for this background are,

$\displaystyle\omega^{-0i}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}e^{2U}\delta^{i}_{m}(\partial_{m}U+i\mathcal{A}_{m})=-\frac{1}{4}e^{U}T^{-0i}$
$\displaystyle\omega^{-ij}_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}ie^{2U}\epsilon_{ijk}\delta^{k}_{m}(\partial_{m}U+i\mathcal{A}_{m})=\frac{1}{4}ie^{U}\epsilon_{ijk}T^{-}_{0k}=-\frac{1}{4}e^{U}T^{-ij}$
$\displaystyle\omega_{m}^{-0i}$ $\displaystyle=$
$\displaystyle-\frac{1}{4}e^{U}\omega_{m}T^{-0i}+\frac{i}{4}e^{-U}\delta^{i}_{n}\epsilon_{nmk}T^{-0k}.$
(3.15)

Computing the time component of the gravitini variation equation 2.3

$\displaystyle\delta\psi^{(i)}_{t}=\left(\partial_{t}+\frac{1}{4}\omega^{ab}_{t}\gamma_{ab}-\frac{1}{2}i\mathcal{A}_{t}\right)\epsilon^{(i)}-\frac{1}{16}\gamma^{ab}T^{-}_{ab}\varepsilon^{(ij)}\gamma_{t}\epsilon_{(j)}$
$\partial_{t}\epsilon^{(i)}=0\;\;\;\mathcal{A}_{t}=0\;\;\;\gamma_{t}=e^{U}\gamma_{0}$
(3.16)
$\displaystyle\delta\psi^{(i)}_{t}=-\frac{1}{16}e^{U}T^{-}_{ab}\gamma^{ab}\epsilon^{(i)}-\frac{1}{16}e^{U}\gamma^{ab}T^{-}_{ab}\varepsilon^{(ij)}\gamma_{0}\epsilon_{(j)}=-\frac{1}{16}e^{U}T^{-}_{ab}\gamma^{ab}(\epsilon^{(i)}+\gamma_{0}\varepsilon^{(ij)}\epsilon_{(j)})$

This shows that the Killing spinors obey,

$\displaystyle\epsilon^{(i)}=-\gamma_{0}\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle\gamma_{0}\epsilon^{(i)}=\varepsilon^{(ij)}\epsilon_{(j)}$
(3.17)

Now for the spatial components

$\displaystyle\delta\psi^{(i)}_{m}=\left(\partial_{m}+\frac{1}{4}\omega^{ab}_{m}\gamma_{ab}-\frac{1}{2}i\mathcal{A}_{m}\right)\epsilon^{(i)}-\frac{1}{16}\gamma^{ab}T^{-}_{ab}\varepsilon^{(ij)}\gamma_{m}\epsilon_{(j)}$
$\gamma_{m}=e^{U}\omega_{m}\gamma_{0}+e^{-U}\delta^{k}_{m}\gamma_{k}$
$\displaystyle\delta\psi^{(i)}_{m}=\left(\partial_{m}-\frac{1}{2}i\mathcal{A}_{m}\right)\epsilon^{(i)}-\left(\frac{1}{4}e^{U}\omega_{m}T^{-0i}\gamma_{0i}P_{L}-\frac{i}{4}e^{-U}\epsilon_{ijk}\delta^{j}_{m}T^{-0k}\gamma_{0i}P_{L}\right)\epsilon^{(i)}$
$\displaystyle-\frac{1}{4}T^{-0i}\gamma_{0i}P_{L}(e^{U}\omega_{m}\gamma_{0}+e^{-U}\delta^{k}_{m}\gamma_{k})\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle=\left(\partial_{m}-\frac{1}{2}i\mathcal{A}_{m}\right)\epsilon^{(i)}-\frac{1}{4}e^{U}\omega_{m}T^{-0i}\gamma_{0i}(\epsilon^{(i)}+\gamma_{0}\varepsilon^{(ij)}\epsilon_{(j)})$
$\displaystyle-\frac{1}{4}e^{-U}\left(-i\epsilon_{ijk}\delta^{j}_{m}T^{-0k}\gamma_{0i}\epsilon^{(i)}+T^{-0i}\gamma_{0i}\delta^{k}_{m}\gamma_{k}\varepsilon^{(ij)}\epsilon_{(j)}\right)$
$i\epsilon_{ijk}\gamma_{0i}\epsilon^{(i)}=\gamma_{jk}\epsilon^{(i)}$ (3.18)

for the Killing spinor,
$\epsilon^{i}=-\gamma_{0}\varepsilon^{ij}\epsilon_{j}$,

$\displaystyle\delta\psi^{(i)}_{m}=\left(\partial_{m}-\frac{1}{2}i\mathcal{A}_{m}\right)\epsilon^{(i)}+\frac{1}{4}e^{-U}\delta^{j}_{m}T^{-0j}\epsilon^{(i)}$
$\displaystyle\delta\psi^{(i)}_{m}=\left(\partial_{m}-\frac{1}{2}i\mathcal{A}_{m}\right)\epsilon^{(i)}-\frac{1}{2}(\partial_{m}U+i\mathcal{A}_{m})\epsilon^{(i)}$
$\displaystyle\delta\psi^{(i)}_{m}=\left(\partial_{m}-\frac{1}{2}\partial_{m}U-i\mathcal{A}_{m}\right)\epsilon^{(i)}$
(3.19)

Hence the Killing spinor is a solution of the equation

$\displaystyle\left(\partial_{m}-\frac{1}{2}\partial_{m}U-i\mathcal{A}_{m}\right)\epsilon^{(i)}=0.$
(3.20)

The integrability condition for this equation is

$d\mathcal{A}=0.$ (3.21)

The Gaugino variation equation is given by 2.3

$\displaystyle\delta\chi_{i}^{\alpha}=\not{\partial}z^{\alpha}\epsilon_{(i)}-\frac{1}{2}G^{-\alpha}_{ab}\gamma^{ab}\varepsilon_{(ij)}\epsilon^{(j)}$
$G^{-\alpha}_{ab}=g^{\alpha\bar{\beta}}\bar{\nabla}_{\bar{\beta}}\bar{X}^{I}Im\;\mathcal{N}_{IJ}F^{-J}_{ab}$

from appendix A

$G^{-\beta}_{0i}=\frac{1}{2}e^{U}\delta^{i}_{m}\partial_{m}z^{\beta}$ (3.22)

substituting in the gaugini variation equation

$\displaystyle\delta\chi_{(i)}^{\alpha}=\not{\partial}z^{\alpha}\epsilon_{(i)}-2G^{-\alpha}_{0i}\gamma^{0i}\varepsilon_{(ij)}\epsilon^{(j)}$
$\displaystyle=\not{\partial}z^{\alpha}\epsilon_{(i)}-e^{U}\partial_{m}z^{\beta}\gamma^{i}\delta^{i}_{m}\epsilon_{(i)}=0$
(3.23)

The gaugini variation equations do not impose any more constraints. Hence, we
have shown that the Denef Bates solutions are supersymmetric and explicitly
found the Killing spinors.

##  4 Fermion zero modes

The Killing spinors are defined by

$\displaystyle\gamma_{0}\epsilon^{(i)}=\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle\left(\partial_{m}-\frac{1}{2}\partial_{m}U-i\mathcal{A}_{m}\right)\epsilon^{(i)}=0.$
(4.1)

The orthogonal space of spinors which breaks supersymmetry for the Denef Bates
background is defined by

$\displaystyle\gamma_{0}\epsilon^{(i)}=-\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle\left(\partial_{m}-\frac{1}{2}\partial_{m}U-i\mathcal{A}_{m}\right)\epsilon^{(i)}=0.$
(4.2)

This space is called the anti Killing spinor space, and since these spinors
break supersymmetry we can use these spinors to generate extra solutions to
the equations of motion. Here we are interested only in counting fermion zero
modes hence we are not concerned with the back reaction of these modes.

The fermion zero modes can be found by using our anti Killing spinors defined
by 4 in the supersymmetry variation equations 2.3. Doing so we find,

$\displaystyle\psi_{0}^{(i)}=-\frac{1}{2}T^{-0i}\gamma_{0i}\epsilon^{(i)}$
$\displaystyle\psi_{j}^{(i)}=-\frac{1}{2}T^{-0i}\gamma_{i}\gamma_{j}\epsilon^{(i)}$
$\displaystyle\chi_{(i)}^{\alpha}=2\not{\partial}z^{\alpha}\epsilon_{(i)}.$
(4.3)

If these are fermion zero modes they will satisfy the equations of motion for
the fermions. We will now check this.

###  4.1 Equations of motion for the fermions

The equations of motion for the fermions are shown in appendix B to be

$\displaystyle-\gamma^{abc}D_{b}\psi_{c}^{(i)}-g_{\alpha\bar{\beta}}G^{+\bar{\beta}ab}\gamma_{b}\chi^{\alpha}_{j}\varepsilon^{(ij)}-\frac{1}{2}T^{-ab}(\psi_{(j)})_{b}\varepsilon^{(ij)}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{a}\chi^{(i)\bar{\beta}}=0$
$\displaystyle\frac{1}{4}G^{+\bar{\gamma}ab}C_{\bar{\gamma}\bar{\beta}\bar{\delta}}\gamma_{ab}\chi^{(j)\bar{\delta}}\varepsilon_{(ij)}+g_{\alpha\bar{\beta}}G^{-\alpha
ab}\gamma_{a}\psi^{(j)}_{b}\varepsilon_{(ij)}-\frac{1}{2}g_{\alpha\bar{\beta}}\not{D}\chi_{(i)}^{\alpha}=0.$
(4.4)

For $a=0$ the gravitini equation of motion is,

$\displaystyle-\gamma^{0ij}D_{i}\psi_{j}^{(i)}-g_{\alpha\bar{\beta}}G^{+\bar{\beta}0i}\gamma_{i}\chi_{(j)}^{\alpha}\varepsilon^{(ij)}-\frac{1}{2}T^{-0i}\gamma_{i(j)}\varepsilon^{(ij)}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{0}\chi^{(i)\bar{\beta}}=0$
(4.5)

We can show that,

$\displaystyle\gamma_{0}\gamma_{ij}D_{i}\psi_{j}^{(i)}$ $\displaystyle=$
$\displaystyle-2e^{2U}\left(\partial_{m}(\partial_{m}U+i\mathcal{A}_{m})\right)\gamma_{0}\epsilon^{(i)}$
(4.6)
$\displaystyle-e^{2U}\left(\partial_{m}U-i\mathcal{A}_{m}\right)\left(\partial_{m}U+i\mathcal{A}_{m}\right)\gamma_{0}\epsilon^{(i)}$
$\displaystyle+2e^{2U}\left(\partial_{m}U+i\mathcal{A}_{m}\right)\left(\partial_{m}U+i\mathcal{A}_{m}\right)\gamma_{0}\epsilon^{(i)}.$
$\displaystyle-\frac{1}{2}T^{-0j}\psi_{(j)j}\varepsilon^{(ij)}=e^{2U}(\partial_{m}U+i\mathcal{A}_{m})\delta^{j}_{m}(\partial_{n}U-i\mathcal{A}_{n})\delta^{k}_{n}\gamma_{k}\gamma_{j}\epsilon_{(j)}\varepsilon^{(ij)}$
$\displaystyle=e^{2U}(\partial_{m}U\partial_{m}U+\mathcal{A}_{m}\mathcal{A}_{m})\epsilon_{(j)}\varepsilon^{(ij)}+ie^{2U}(\partial_{n}U\mathcal{A}_{m}-\partial_{m}U\mathcal{A}_{n})\delta^{j}_{m}\delta^{k}_{n}\gamma_{kj}\epsilon_{(j)}\varepsilon^{(ij)}$

3.21 implies $\partial_{n}U\mathcal{A}_{m}-\partial_{m}U\mathcal{A}_{n}=0$

$\displaystyle\gamma_{0}\gamma_{ij}D_{i}\psi_{j}^{(i)}-\frac{1}{2}T^{-0j}\psi_{(j)j}\varepsilon^{(ij)}=-2e^{2U}\left(\partial_{m}(\partial_{m}U+i\mathcal{A}_{m})\right)\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle+2e^{2U}\left(\partial_{m}U+i\mathcal{A}_{m}\right)\left(\partial_{m}U+i\mathcal{A}_{m}\right)\varepsilon^{(ij)}\epsilon_{(j)}$
(4.8) $\displaystyle-
g_{\alpha\bar{\beta}}G^{+\bar{\beta}0i}\gamma_{i}\chi^{\alpha}_{j}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{0}\chi^{i\bar{\beta}}=2e^{2U}g_{\alpha\bar{\beta}}\partial_{m}z^{\alpha}\partial_{m}\bar{z}^{\bar{\beta}}\varepsilon^{(ij)}\epsilon_{(j)}$
$\displaystyle\partial_{m}(\partial_{m}U+i\mathcal{A}_{m})=g_{\alpha\bar{\beta}}\partial_{m}z^{\alpha}\partial_{m}z^{\bar{\beta}}+\partial_{\bar{\alpha}}\mathcal{K}\partial_{\bar{\beta}}\mathcal{K}\partial_{m}\bar{z}^{\bar{\alpha}}\partial_{m}\bar{z}^{\bar{\beta}}$
(4.9)

Hence, we can see that the fermion zero modes satisfy the $a=0$ equation of
motion,

$\displaystyle\gamma_{0}\gamma_{ij}D_{i}\psi_{j}^{(i)}-\frac{1}{2}T^{-0j}\psi_{(j)j}\varepsilon^{(ij)}-g_{\alpha\bar{\beta}}G^{+\bar{\beta}0i}\gamma_{i}\chi^{\alpha}_{j}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{0}\chi^{i\bar{\beta}}=0.$

By similar substitutions, we can show that the equation of motion for $a=i$
and also the gaugini equations of motion 4.4 are satisfied.

###  4.2 Supercharge, boundary conditions and norm

The supercharge for this theory has been shown to be [23, 24]

$\displaystyle
Q^{(i)}=-i\int_{S^{2}_{\infty}}d\Sigma_{i}\gamma^{0ij}\psi_{j}^{(i)}.$ (4.10)

Using 4 and 4 we get

$\displaystyle
Q^{(i)}=-2i\int_{S^{2}_{\infty}}d\Sigma_{i}T^{-0i}\varepsilon^{(ij)}\epsilon_{(j)}$

using 2.3 and 3.4

$\displaystyle Q^{(i)}=2Z|_{r=\infty}\varepsilon^{(ij)}\epsilon_{(j)}.$ (4.11)

To have a sensical supercharge the fermion zero modes must behave
asymptotically as

$\displaystyle r^{2}\psi_{\mu}\xrightarrow{r=\infty}c_{\mu},$ (4.12)

where $c_{\mu}$ is a constant spinor.

The norm for fermion zero modes was introduced in [19], it was used by the
authors to show that extremal Reissner-Nordström and extremal Dilation black
holes have normailizable fermion zero modes. We borrow the same norm here,

$\displaystyle||\psi||^{2}=\int_{\Sigma}d^{3}x\sqrt{-g_{(3)}}\psi_{\mu{(i)}}^{\dagger}\psi_{\nu}^{(i)}g^{\mu\nu}=\int_{\Sigma}d\Sigma_{\mu}\bar{\psi}_{\nu(i)}\gamma^{\mu\nu\rho}\psi_{\rho}^{(i)}$
(4.13)

where $\Sigma$ is a space-like hypersurface and $g_{(3)}$ is the determinant
of the metric induced on it. We take $\Sigma$ to be $\mathbb{R}^{3}$ and
substitute 4

$\displaystyle||\psi||^{2}=\frac{1}{4}||\epsilon||^{2}\int_{\mathbb{R}^{3}}d^{3}x\;((\partial_{m}U)^{2}+(\mathcal{A}_{m})^{2})$
(4.14)

Hence, generally we have $||\psi||^{2}>0$. The goldstino modes are
normalizable. Now, we can search for extra fermion zero modes which are
normalizable and have the same boundary conditions as the goldstino modes.

##  5 Extra fermion zero modes

If extra fermion zero modes exist then they must be necessarily gauge
inequivalent to the goldstino modes we have constructed in 4. Now, we ask the
question when is it that two configurations $\psi$ and $\xi$ have the same
supercharge but are not gauge equivalent

$\displaystyle
Q^{(i)}=-i\int_{S^{2}_{\infty}}d\Sigma_{i}\gamma^{0ij}\psi_{j}^{(i)}=-i\int_{S^{2}_{\infty}}d\Sigma_{i}\gamma^{0ij}\xi_{j}^{(i)}$
(5.1)

This happens if and only if there exists a non trivial zero charge gravitini,
which is not gauge equivalent to zero. A zero charge gravitini has the
boundary condition

$\displaystyle r^{2}\psi\xrightarrow{r=\infty}0.$ (5.2)

We should note that the goldstino gravitini 4 are continious all over
$\mathbb{R}^{3}$, which can be seen from the continuity of $T^{-0i}$ at the
centers

$\displaystyle
T^{-0i}=2e^{3U}\delta^{i}_{m}(Z^{I}\partial_{m}H_{I}-F_{I}\partial_{m}\tilde{H}^{I})$
$\displaystyle e^{-U}\sim
r^{-1},\;\;\partial_{m}H_{I}\sim\frac{x_{m}}{r^{3}}e^{3U},\;\;T^{-0i}\sim\delta^{i}_{m}\frac{x_{m}}{r}.$
(5.3)

Also, since in a normed vector space every non zero state is normalizable, we
demand that these zero charge modes are normalizable.

###  5.1 Single center

We first consider the single centered Denef Bates black hole. For such
backgrounds $\omega$ in 3.1 is zero,

$\displaystyle ds^{2}=-e^{2U}dt^{2}+e^{-2U}(dr^{2}+r^{2}d\Omega^{2})$ (5.4)

here we are looking for fermion zero modes which are solutions to the equation
of motion, satisfy the boundary conditions
$r^{2}\psi_{\mu}\xrightarrow{r=\infty}0$, continuity of the zero modes, and
are normalizable.

Normalizability implies that the integral

$\int_{\Sigma}d^{3}x\sqrt{-g_{(3)}}\psi_{\mu{(i)}}^{\dagger}\psi_{\nu}^{(i)}g^{\mu\nu}$
(5.5)

is finite. The asymptotic boundary condition implies that the integral will
only recieve contributions from the near horizon limit.

$\int_{\Sigma}dr\,d\theta\,d\phi
e^{-3U}r^{2}\sin\theta\left(e^{-2U}\psi_{t(i)}^{\dagger}\psi_{t}^{(i)}+e^{2U}\psi_{r(i)}^{\dagger}\psi_{r}^{(i)}+r^{-2}e^{2U}\psi_{\theta(i)}^{\dagger}\psi_{\theta}^{(i)}+r^{-2}e^{2U}sin^{-2}\theta\psi_{\phi(i)}\psi_{\phi}^{(i)}\right)$
(5.6)

$e^{U}$ approches 1 asymptotically. This integral recieves no contribution in
the asymptotic limit.

Hence, we need to study the solutions of the equation of motion in the near
horizon limit with the fermion zero mode continuous at $r=0$.

The most general background for a single centre black hole in the near horizon
limit is

$\displaystyle
ds^{2}=a^{2}\left(-r^{2}\,dt^{2}+\frac{dr^{2}}{r^{2}}\right)+a^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ $\displaystyle z^{\alpha}=u^{\alpha},\;\;F^{I}_{tr}=q^{I}$
(5.7)

where $a$, $u^{\alpha}$ and $q^{I}$ are constants which can be determined by
Sen’s entropy function formalism. This is a maximally supersymmetric
background.

The norm in this limit is

$\displaystyle||\psi||^{2}=\int_{\Sigma}dr\,d\theta\,d\phi\frac{a^{3}\sin\theta}{r}\left(\frac{1}{a^{2}r^{2}}\psi^{\dagger}_{(i)t}\psi^{(i)}_{t}+\frac{r^{2}}{a^{2}}\psi_{r(i)}^{\dagger}\psi_{r}^{(i)}+\frac{1}{a^{2}}\psi_{\theta(i)}^{\dagger}\psi_{\theta}^{(i)}+\frac{1}{a^{2}\sin^{2}\theta}\psi^{\dagger}_{\phi(i)}\psi_{\phi}^{(i)}\right)$
(5.8)

This shows that to have a finite non zero contribution in the near horizon
limit to the norm we must have

$\displaystyle\psi_{t(i)}$ $\displaystyle\sim$ $\displaystyle
r^{n}\;\;n\in[0,3/2]$ $\displaystyle\psi_{r(i)}$ $\displaystyle\sim$
$\displaystyle r^{0}$ $\displaystyle\psi_{\theta(i)}$ $\displaystyle\sim$
$\displaystyle r^{n}\;\;n\in[0,1/2]$ $\displaystyle\psi_{\phi(i)}$
$\displaystyle\sim$ $\displaystyle r^{n}\;\;n\in[0,1/2]$ (5.9)

the lower limit on $n$ is fixed by requiring the gravitini to be continious
all over $\mathbb{R}^{3}$.

###  5.2 Near horizon equation of motion

We transform the metric to

$\displaystyle
ds^{2}=\left(-\frac{r^{2}}{a^{2}}{dt^{2}}+a^{2}\frac{dr^{2}}{r^{2}}\right)+a^{2}(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})$
(5.10)

This shows

$\displaystyle
e^{U}=\frac{r}{a},\;\;\;\partial_{r}U=\frac{1}{r},\;\;\partial_{\theta}U=0,\;\;\;\partial_{\phi}U=0$

Since, the scalars are constants all over space in the near horizon limit
$\partial_{\mu}z^{\alpha}=0$ ($G^{-\alpha}_{ab}=0$). This reduces the terms in
the equation of motion for the gravitini 4.4

$\displaystyle\gamma^{abc}D_{b}\psi_{c}^{(i)}+\frac{1}{2}T^{-ab}\psi_{b(j)}\varepsilon^{(ij)}=0$
(5.12)

The spin connections are

$\displaystyle\omega^{01}_{t}=\frac{r}{a^{2}},\;\;\omega^{23}_{\phi}=-\cos\theta,$
(5.13)

and the tensor T is given by

$\displaystyle
T^{-\theta\phi}=-\frac{2i}{a^{3}\sin\theta},\;\;T^{-tr}=-\frac{2}{a}.$ (5.14)

The equations of motion then are

$\displaystyle\gamma^{012}\frac{1}{a}\left(\partial_{r}\psi_{\theta}^{(i)}-\partial_{\theta}\psi_{r}^{(i)}\right)+\gamma^{013}\frac{1}{a\sin\theta}\left(\partial_{r}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{r}^{(i)}+\frac{1}{2}\cos\theta\;\gamma_{23}\;\psi_{r}^{(i)}\right)$
$\displaystyle+\gamma^{023}\frac{1}{ar\sin\theta}\left(\partial_{\theta}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{\theta}^{(i)}+\frac{1}{2}\cos\theta\,\gamma_{23}\,\psi_{\theta}^{(i)}\right)-\frac{1}{a}\psi_{r(j)}\varepsilon^{(ij)}=0$
$\displaystyle\gamma^{102}\frac{1}{a}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{\theta}^{(i)}-\partial_{\theta}\psi_{t}^{(i)}\right)+\frac{1}{a\sin\theta}\gamma^{103}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{t}^{(i)}+\frac{1}{2}\cos\theta\,\gamma_{23}\,\psi_{t}^{(i)}\right)$
$\displaystyle+\gamma^{123}\frac{r}{a^{3}\sin\theta}\left(\partial_{\theta}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{\theta}^{(i)}+\frac{1}{2}\cos\theta\,\gamma_{23}\,\psi_{\theta}^{(i)}\right)+\frac{1}{a}\psi_{t(j)}\varepsilon^{(ij)}=0$
$\displaystyle\gamma^{201}\frac{1}{a}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{r}^{(i)}-\partial_{r}\psi_{t}^{(i)}\right)+\gamma^{203}\frac{1}{ar\sin\theta}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{t}^{(i)}+\frac{1}{2}\cos\theta\,\gamma_{23}\,\psi_{t}^{(i)}\right)$
$\displaystyle+\gamma^{213}\frac{r}{a^{3}\sin\theta}\left(\partial_{r}\psi_{\phi}^{(i)}-\partial_{\phi}\psi_{r}^{(i)}+\frac{1}{2}\cos\theta\;\gamma_{23}\;\psi_{r}^{(i)}\right)-\frac{i}{a^{3}\sin\theta}\psi_{\phi(j)}\varepsilon^{(ij)}=0$
$\displaystyle\gamma^{301}\frac{1}{a\sin\theta}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{r}^{(i)}-\partial_{r}\psi_{t}^{(i)}\right)+\gamma^{302}\frac{1}{ar\sin\theta}\left(\frac{1}{2}\frac{r}{a^{2}}\gamma_{01}\psi_{\theta}^{(i)}-\partial_{\theta}\psi_{t}^{(i)}\right)$
$\displaystyle+\gamma^{312}\frac{r}{a^{3}\sin\theta}\left(\partial_{r}\psi_{\theta}^{(i)}-\partial_{\theta}\psi_{r}^{(i)}\right)+\frac{i}{a^{3}\sin\theta}\psi_{\theta(j)}\varepsilon^{(ij)}=0$
(5.15)

If we assume that $\psi_{t}^{(i)}\sim r^{a}$, $\psi_{r}^{(i)}\sim r^{b}$,
$\psi_{\theta}^{(i)}\sim r^{c}$, $\psi_{\phi}\sim r^{d}$. Then from the above
equations we get the relations

$\displaystyle b=c-1=d-1$ $\displaystyle a=c+1=d+1$ $\displaystyle b+1=a-1=d$
$\displaystyle b+1=a-1=c$

which shows that

$\displaystyle b=a-2=c-1=d-1.$ (5.16)

The lowest value of $b$ that can occur is $0$ (due to continuity). Which shows
that the lowest values of these constants are $a=2,b=0,c=1,d=1$. For these
values from 5.1 we get no contribution to the norm.

The gauge condition $\gamma^{a}\psi^{(i)}_{a}=0$ shows

$\displaystyle\frac{a}{r}\gamma^{0}\psi^{(i)}_{t}+\frac{r}{a}\gamma^{1}\psi^{(i)}_{r}+\frac{1}{a}\gamma^{2}\psi^{(i)}_{\theta}+\frac{1}{a\sin\theta}\gamma^{3}\psi^{(i)}_{\phi}=0.$
(5.17)

This also agrees with 5.16 .

Thus the dependence of $r$ in the equation of motion and the gauge condition
shows that to have a contribution to the norm we must have a fermion zero mode
which is discontinious at $r=0$; which would disagree with the boundary
conditions of the broken modes.

###  5.3 Multicentred black holes

For multicentred Denef Bates black holes the near horizon metric for any of
the horizons is 5.7 and the same arguments will show that near every horizon
there is no contribution to the norm. And, for zero charge black holes we must
also have no contribution to the norm in the asymptotic limit. Thus the norm
would vanish, and consecutively the black holes will contribute to the index.

##  6 Conclusion

We set out to show that there are no extra fermion zero modes for multicentred
Denef Bates black holes apart from the goldstino zero modes. We constructed
the goldstino modes and computed their norm and supercharge. To find solutions
to the fermion equations of motion, we argued that if such solutions exist
they must be exist as normalizable zero charge solutions. From the boundary
condition of the zero charge solutions we showed that the norm will recieve
zero contribution in the asymptotic limit. Then the only contribution to the
norm can come from the near horizon limit. We wrote our norm in the near
horizon limit and we saw that for small $r$ if there is a contribution to then
the gravitini will be discontinious. But, our fermion zero modes are required
to be continious all over $\mathbb{R}^{3}$. This shows that the norm of such
zero charge modes must be zero. Hence, as argued in the introduction we have
proved that multicentered Denef Bates black holes contribute to the index.

The goldstino fermion zero modes are hair degrees of freedom of the black hole
since they have support at infinity. The charge and the norm of these modes
depend directly on the value of the fermion zero mode at infinity. In trying
to solve this problem we showed that to have gauge inequivalent extra modes we
must construct fermion zero modes which have zero charge which are not gauge
equivalent to zero. Since they have zero charge they must be zero at infinity.
Hence these modes we set out to find are not hair modes. If these modes were
present they would have some corrections at the horizon of the black hole.

In [25] the authors have argued that in $\mathcal{N}=4$ Supergravity there
exist dyons which can decay into multicentered bound states. There they have
assumed the result that such decays do not contribute to the index due to
presence of additional fermion zero modes. This work has been a prelude to
checking this more complicated setting. Also, constructing extra zero modes
can be used to determine if the fuzzball configurations contribute to the
index. We hope to report progress on these fronts in a future paper.

##  Appendix A Spin connection, T and G

The vierbeins are

$e^{0}=e^{U}(dt+\omega),\;\;e^{i}=e^{-U}\delta^{i}_{m}\;dx^{m}$ (A.1)

finding spin connections

$\displaystyle de^{0}=e^{U}\,dU\wedge(dt+\omega)+e^{U}d\omega\;$
$\displaystyle de^{i}=-e^{-U}\,dU\wedge dx^{m}\delta^{i}_{m}$ (A.2)

substituting in the structure equation $de^{a}+\omega^{a}_{\;b}\wedge e^{b}=0$

$de^{0}+\omega^{0i}\wedge e^{i}=0,\;\;de^{i}+\omega^{ij}\wedge
e^{j}+\omega^{0i}\wedge e^{0}=0$ (A.3)

Expanding the first equation we get

$\displaystyle
de^{0}+e^{-U}(\omega^{0i})_{t}dt\wedge\delta^{i}_{m}dx^{m}+e^{-U}(\omega^{0i})_{n}dx^{n}\wedge\delta^{i}_{m}dx^{m}=0$
$\displaystyle e^{U}\,dU\wedge
dt+e^{-U}(\omega^{0i})_{t}dt\wedge\delta^{i}_{m}dx^{m}=0$ $\displaystyle
e^{U}\,dU\wedge\omega+e^{U}d\omega+e^{-U}(\omega^{0i})_{n}dx^{n}\wedge\delta^{i}_{m}dx^{m}=0$
$\displaystyle e^{U}\,\partial_{m}U=e^{-U}(\omega^{0i})_{t}\delta^{i}_{m}$
$\displaystyle
e^{U}dU\wedge\omega+e^{U}\,d\omega+e^{-U}(\omega^{0i})_{n}dx^{n}\wedge\delta^{i}_{m}dx^{m}=0$
(A.4)

define
$\omega_{nm}=\frac{1}{2}(\partial_{n}\omega_{m}-\partial_{m}\omega_{n})$

$\displaystyle
e^{2U}\partial_{n}U\omega_{m}+e^{2U}\omega_{nm}+(\omega^{0i})_{n}\delta^{i}_{m}=0$
(A.5)

Expanding the second equation we get

$\displaystyle de^{i}+\omega^{ij}\wedge e^{j}+\omega^{0i}\wedge e^{0}=0$
$\displaystyle de^{i}+e^{-U}\delta^{j}_{m}(\omega^{ij})_{t}dt\wedge
dx^{m}+e^{-U}\delta^{j}_{m}(\omega^{ij})_{n}dx^{n}\wedge dx^{m}$
$\displaystyle+e^{U}(\omega^{0i})_{m}\,dx^{m}\wedge(dt+\omega)+e^{U}(\omega^{0i})_{t}dt\wedge\omega=0$
$\displaystyle e^{-U}\delta^{j}_{m}(\omega^{ij})_{t}dt\wedge
dx^{m}+e^{U}(\omega^{0i})_{m}dx^{m}\wedge
dt+e^{U}(\omega^{0i})_{t}\omega_{m}dt\wedge dx^{m}=0$
$\displaystyle\delta^{j}_{m}(\omega^{ij})_{t}=e^{2U}(\omega^{0i})_{m}-e^{2U}(\omega^{0i})_{t}\omega_{m}$
$\displaystyle\delta^{j}_{m}(\omega^{ij})_{t}=e^{4U}\delta^{n}_{i}(\partial_{n}U\omega_{m}+\omega_{nm})-e^{4U}\delta^{n}_{i}\partial_{n}U\omega_{m}$
$\displaystyle(\omega^{ij})_{t}=e^{4U}\delta^{i}_{n}\delta^{j}_{m}\omega_{nm}$
(A.6) $\displaystyle de^{i}+e^{-U}\delta^{j}_{m}(\omega^{ij})_{n}dx^{n}\wedge
dx^{m}+e^{U}(\omega^{0i})_{m}\omega_{n}dx^{m}\wedge dx^{n}=0$
$\displaystyle\delta^{j}_{m}(\omega^{ij})_{n}dx^{n}\wedge
dx^{m}=\partial_{n}Udx^{n}\wedge
dx^{m}\delta^{i}_{m}+e^{4U}\delta^{i}_{p}(\partial_{p}U\omega_{m}+\omega_{pm})\omega_{n}\,dx^{m}\wedge
dx^{n}$
$\displaystyle\delta^{j}_{m}(\omega^{ij})_{n}=-\partial_{[m}U\delta^{i}_{n]}-e^{4U}\delta^{i}_{p}\omega_{p[m}\omega_{n]}$

We also have from 3.11,

$\omega_{nm}=-e^{-2U}\epsilon_{nmp}\mathcal{A}_{p}$ (A.8)
$\displaystyle\omega^{0i}_{t}=e^{2U}\partial_{m}U\delta^{i}_{m}$
$\displaystyle\omega^{ij}_{t}=-e^{2U}\delta^{i}_{n}\delta^{j}_{m}\epsilon_{nmp}\mathcal{A}_{p}$
$\displaystyle\omega^{0i}_{n}=-e^{2U}\partial_{[n}U\omega_{m]}\delta^{i}_{m}+\delta^{i}_{m}\epsilon_{nmp}\mathcal{A}_{p}$
$\displaystyle\omega^{ij}_{n}=-\partial_{[m}U\delta^{i}_{n]}\delta^{j}_{m}-e^{2U}\delta^{i}_{p}\delta^{j}_{m}\epsilon_{pq[m}\omega_{n]}\mathcal{A}_{q}$
(A.9)
$\displaystyle\omega^{-0i}_{t}=\frac{1}{2}(\omega^{0i}_{t}-\tilde{\omega}^{0i}_{t})$
$\displaystyle\tilde{\omega}^{0i}_{t}=-\frac{i}{2}\epsilon^{0ijk}(\omega_{jk})_{t}=\frac{i}{2}\epsilon_{ijk}\omega^{jk}_{t}=\frac{i}{2}e^{2U}\epsilon_{ijk}\delta^{j}_{n}\delta^{k}_{m}\epsilon_{nmp}\mathcal{A}_{p}=-ie^{2U}\delta^{i}_{p}\mathcal{A}_{p}$
$\omega^{-0i}_{t}=\frac{1}{2}e^{2U}\delta^{i}_{m}(\partial_{m}U+i\mathcal{A}_{m})$
(A.10)
$\displaystyle\omega^{-ij}_{t}=\frac{1}{2}(\omega^{ij}_{t}-\tilde{\omega}^{ij}_{t})$
$\displaystyle\tilde{\omega}^{ij}_{t}=-i\epsilon_{ijk}\omega^{0k}_{t}=-ie^{2U}\epsilon_{ijk}\partial_{m}U\delta^{k}_{m}$
$\displaystyle\omega^{-ij}_{t}=\frac{1}{2}ie^{2U}\epsilon_{ijk}\delta^{k}_{m}(\partial_{m}U+i\mathcal{A}_{m})$
(A.11)

Solving for the spatial components of the gauge fields one obtains

$\displaystyle
F^{I}_{ij}=e^{2U}\epsilon_{ijk}\delta^{k}_{m}\partial_{m}\tilde{H}^{I}$
$\displaystyle G_{Iij}=e^{2U}\epsilon_{ijk}\delta^{k}_{m}\partial_{m}H_{I}$
(A.12)

Here, we can use the defining relations for $G_{I}$ to find the other
components, and they are

$\displaystyle F^{I}_{0i}=e^{2U}\delta^{i}_{m}\partial_{m}(Z^{I}+\bar{Z}^{I})$
$\displaystyle G_{I0i}=e^{2U}\delta^{i}_{m}\partial_{m}(F_{I}+\bar{F}_{I})$
(A.13) $\displaystyle
T^{-}_{ij}=2ie^{\mathcal{K}/2}\left(F_{I}F^{-I}_{ij}-Z^{I}G^{-}_{Iij}\right)$
$\displaystyle
T^{-}_{ij}=2ie^{3U}\epsilon_{ijk}\delta^{k}_{m}\left(F_{I}\partial_{m}\tilde{H}^{I}-Z^{I}\partial_{m}H_{I}\right)$
(A.14)

Also,

$\displaystyle\tilde{T}^{-}_{ab}=-T^{-}_{ab}$ $\displaystyle
T^{-}_{ab}=\frac{i}{2}\epsilon_{abcd}T^{-cd}$ $\displaystyle
T^{-}_{0i}=\frac{i}{2}\epsilon_{ijk}T^{-jk}$ $\displaystyle
T^{-}_{0i}=-2e^{3U}\delta^{i}_{m}\left(F_{I}\partial_{m}\tilde{H}^{I}-Z^{I}\partial_{m}H_{I}\right)$
(A.15)

We know that,

$\displaystyle e^{-2U}=e^{-\mathcal{K}}=-i(Z^{I}\bar{F}_{I}-F_{I}\bar{Z}^{I})$
$\displaystyle\partial_{m}U=\frac{1}{2}ie^{2U}(\partial_{m}Z^{I}\bar{F}_{I}+Z^{I}\partial_{m}\bar{F}_{I}-\partial_{m}F_{I}\bar{Z}^{I}-F_{I}\partial_{m}\bar{Z}^{I})$
$\displaystyle
i\mathcal{A}_{m}=-\frac{1}{2}ie^{2U}\left[(Z^{I}-\bar{Z}^{I})\partial_{m}(F_{I}-\bar{F}_{I})-(F_{I}-\bar{F}_{I})\partial_{m}(Z^{I}-\bar{Z}^{I})\right]$
$\displaystyle\partial_{m}U+i\mathcal{A}_{m}=e^{2U}(Z^{I}\partial_{m}H_{I}-F_{I}\partial_{m}\tilde{H}^{I})=\frac{1}{2}e^{-U}\delta^{i}_{m}T^{-}_{0i}$
$\displaystyle\omega^{-0i}_{t}=\frac{1}{2}e^{2U}\delta^{i}_{m}(\partial_{m}U+i\mathcal{A}_{m})=-\frac{1}{4}e^{U}T^{-0i}$
$\displaystyle\omega^{-ij}_{t}=\frac{1}{2}ie^{2U}\epsilon_{ijk}\delta^{k}_{m}(\partial_{m}U+i\mathcal{A}_{m})=\frac{1}{4}ie^{U}\epsilon_{ijk}T^{-}_{0k}=-\frac{1}{4}e^{U}T^{-ij}$
$\displaystyle\omega_{m}^{-0i}=-\frac{1}{4}e^{U}\omega_{m}T^{-0i}+\frac{i}{4}e^{-U}\delta^{i}_{n}\epsilon_{nmk}T^{-0k}$
(A.17)

finding $G_{ab}^{-\alpha}$,

$\displaystyle\nabla_{\alpha}X^{I}G^{+}_{Iab}-\nabla_{\alpha}F_{I}F^{+I}_{ab}=2ig_{\alpha\bar{\beta}}G^{+\bar{\beta}}_{ab}$
$\displaystyle\nabla_{\alpha}X^{I}G_{Iab}-\nabla_{\alpha}F_{I}F^{I}_{ab}=2ig_{\alpha\bar{\beta}}G^{+\bar{\beta}}_{ab}$
$\displaystyle\epsilon_{ijk}\delta^{k}_{m}e^{3U}(\nabla_{\alpha}Z^{I}\partial_{m}H_{I}-\nabla_{\alpha}F_{I}\partial_{m}\tilde{H}^{I})=2ig_{\alpha\bar{\beta}}G^{+\bar{\beta}}_{ij}$
$\displaystyle
g_{\alpha\bar{\beta}}G^{+\bar{\beta}}_{ij}=\frac{1}{2i}e^{3U}\epsilon_{ijk}\delta^{k}_{m}i(\nabla_{\alpha}F_{I}\partial_{\bar{\beta}}\bar{Z}^{I}-\nabla_{\alpha}Z^{I}\partial_{\bar{\beta}}\bar{F}_{I})\partial_{m}\bar{z}^{\bar{\beta}}$
$\displaystyle=-\frac{1}{2}e^{3U}\epsilon_{ijk}\delta^{k}_{m}\langle\nabla_{\alpha}v,\bar{\nabla}_{\bar{\beta}}\bar{v}\rangle\partial_{m}\bar{z}^{\bar{\beta}}$
$\displaystyle=\frac{i}{2}e^{U}\epsilon_{ijk}\delta^{k}_{m}g_{\alpha\bar{\beta}}\partial_{m}\bar{z}^{\bar{\beta}}$
$\displaystyle
G^{-\beta}_{ij}=-\frac{i}{2}\epsilon_{ijk}\delta^{k}_{m}\partial_{m}z^{\beta}$
$\displaystyle G^{-\beta}_{0i}=\frac{i}{2}\epsilon_{ijk}G^{-\beta jk}$
$\displaystyle
G^{-\beta}_{0i}=\frac{1}{2}e^{U}\delta^{i}_{m}\partial_{m}z^{\beta}$ (A.18)

##  Appendix B Equations of motion for the fermions

expanding 2.2 we get

$\displaystyle-\bar{\psi}_{ia}\gamma^{abc}D_{b}\psi_{c}^{i}+\Big{(}F^{+I}_{ab}Im{\mathcal{N}}_{IJ}{\nabla}_{{\alpha}}{X}^{J}\Big{(}\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\chi^{j\bar{\delta}}\varepsilon_{ij}+\bar{\chi}^{{\alpha}}_{i}\gamma^{a}\psi^{b}_{j}\varepsilon^{ij}\Big{)}$
$\displaystyle+F^{+I}_{ab}Im{\mathcal{N}}_{IJ}\bar{X}^{J}\bar{\psi}^{ai}\psi^{bj}\varepsilon_{ij}-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\not{D}\chi^{i\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}$
$\displaystyle+F^{-}_{ab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}\Big{(}\frac{1}{8}g^{\beta\bar{\alpha}}C_{\beta\gamma\delta}\bar{\chi}_{i}^{\gamma}\gamma_{ab}\chi_{j}^{\delta}\varepsilon^{ij}+\bar{\chi}^{\bar{\alpha}i}\gamma^{a}\psi^{bj}\varepsilon_{ij}\Big{)}$
$\displaystyle+F^{-}_{ab}Im\mathcal{N}_{IJ}X^{J}\bar{\psi}^{a}_{i}\psi^{b}_{j}\varepsilon^{ij}-\frac{1}{4}g_{\bar{\alpha}{\beta}}\bar{\chi}^{i\bar{\alpha}}\not{D}\chi^{{\beta}}_{i}+\frac{1}{2}g_{\bar{\alpha}{\beta}}\bar{\psi}^{i}_{a}\not{\partial}\bar{z}^{\bar{\alpha}}\gamma^{a}\chi^{{\beta}}_{i}\Big{)}$
(B.1)

Now varying the action w.r.t. fermions to find the equations of motion of the
fermions we get

$\displaystyle
e^{-1}\delta\mathcal{L}_{fermi}=-\delta\bar{\psi}_{ia}\gamma^{abc}D_{b}\psi_{c}^{i}-\bar{\psi}_{ia}\gamma^{abc}D_{b}\delta\psi_{c}^{i}$
$\displaystyle+F^{+Iab}Im{\mathcal{N}}_{IJ}{\nabla}_{{\alpha}}{X}^{J}\Big{(}\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\chi^{j\bar{\delta}}\varepsilon_{ij}+\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\delta\chi^{j\bar{\delta}}\varepsilon_{ij}$
$\displaystyle+\delta\bar{\chi}^{{\alpha}}_{i}\gamma_{a}\psi_{bj}\varepsilon^{ij}+\bar{\chi}^{{\alpha}}_{i}\gamma_{a}\delta\psi_{bj}\varepsilon^{ij}\Big{)}+F^{+I}_{ab}Im{\mathcal{N}}_{IJ}\bar{X}^{J}\delta\bar{\psi}^{ai}\psi^{bj}\varepsilon_{ij}+F^{+I}_{ab}Im{\mathcal{N}}_{IJ}\bar{X}^{J}\bar{\psi}^{ai}\delta\psi^{bj}\varepsilon_{ij}$
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}_{i}^{\alpha}\not{D}\chi^{i\bar{\beta}}-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\not{D}\delta\chi^{i\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\delta\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\delta\chi^{i\bar{\beta}}$
$\displaystyle+F^{-ab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}\Big{(}\frac{1}{8}g^{\beta\bar{\alpha}}C_{\beta\gamma\delta}\delta\bar{\chi}_{i}^{\gamma}\gamma_{ab}\chi_{j}^{\delta}\varepsilon^{ij}+\frac{1}{8}g^{\beta\bar{\alpha}}C_{\beta\gamma\delta}\bar{\chi}_{i}^{\gamma}\gamma_{ab}\delta\chi_{j}^{\delta}\varepsilon^{ij}$
$\displaystyle+\delta\bar{\chi}^{\bar{\alpha}i}\gamma_{a}\psi^{j}_{b}\varepsilon_{ij}+\bar{\chi}^{\bar{\alpha}i}\gamma_{a}\delta\psi^{j}_{b}\varepsilon_{ij}\Big{)}+F^{-I}_{ab}Im\mathcal{N}_{IJ}X^{J}\delta\bar{\psi}^{a}_{i}\psi^{b}_{j}\varepsilon^{ij}+F^{-I}_{ab}Im\mathcal{N}_{IJ}X^{J}\bar{\psi}^{a}_{i}\delta\psi^{b}_{j}\varepsilon^{ij}$
$\displaystyle-\frac{1}{4}g_{\bar{\alpha}{\beta}}\delta\bar{\chi}^{i\bar{\alpha}}\not{D}\chi^{{\beta}}_{i}-\frac{1}{4}g_{\bar{\alpha}{\beta}}\bar{\chi}^{i\bar{\alpha}}\not{D}\delta\chi^{{\beta}}_{i}+\frac{1}{2}g_{\bar{\alpha}{\beta}}\delta\bar{\psi}^{i}_{a}\not{\partial}\bar{z}^{\bar{\alpha}}\gamma^{a}\chi^{{\beta}}_{i}+\frac{1}{2}g_{\bar{\alpha}{\beta}}\bar{\psi}^{i}_{a}\not{\partial}\bar{z}^{\bar{\alpha}}\gamma^{a}\delta\chi^{{\beta}}_{i}$
(B.2)

collecting terms with $\delta\bar{\psi}_{ia}$

$\displaystyle-\delta{\bar{\psi}}_{ia}\gamma^{abc}D_{b}\psi_{c}^{i}+F^{+Iab}Im\mathcal{N}_{IJ}\nabla_{\alpha}X^{J}\bar{\chi}_{i}^{\alpha}\gamma_{a}\delta\psi_{jb}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\delta\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}$
$\displaystyle+F^{-Iab}Im\mathcal{N}_{IJ}X^{J}\delta\bar{\psi}_{ia}\psi_{jb}\varepsilon^{ij}+F^{-Iab}Im\mathcal{N}_{IJ}X^{J}\bar{\psi}_{ia}\delta\psi_{jb}\varepsilon^{ij}$
(B.3)

using spinor flip properties
$\bar{\lambda}\gamma_{\mu_{1}\cdots\mu_{r}}\chi=t_{r}\bar{\chi}\gamma_{\mu_{1}\cdots\mu_{r}}\lambda$

$\displaystyle-\delta{\bar{\psi}}_{ia}\gamma^{abc}D_{b}\psi_{c}^{i}-F^{+Iab}Im\mathcal{N}_{IJ}\nabla_{\alpha}X^{J}\delta\bar{\psi}_{ia}\gamma_{b}\chi_{j}^{\alpha}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\delta\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}$
$\displaystyle+2F^{-Iab}Im\mathcal{N}_{IJ}X^{J}\delta\bar{\psi}_{ia}\psi_{jb}\varepsilon^{ij}$
(B.4)

Hence one of the equations of motion is

$\displaystyle-\gamma^{abc}D_{b}\psi_{c}^{i}-F^{+Iab}Im\mathcal{N}_{IJ}\nabla_{\alpha}X^{J}\gamma_{b}\chi_{j}^{\alpha}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}+2F^{-Iab}Im\mathcal{N}_{IJ}X^{J}\psi_{jb}\varepsilon^{ij}=0$

from the definitions

$\displaystyle G^{-\alpha
ab}=g^{\alpha\bar{\beta}}F^{-Iab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\beta}}\bar{X}^{J}$
$\displaystyle T^{+}_{ab}=-4F^{+I}_{ab}Im\mathcal{N}_{IJ}\bar{X}^{J}$ (B.6)

The equations of motion for the gravitini can be written as

$\displaystyle\frac{\delta{\mathcal{L}}}{\delta\bar{\psi}_{ia}}=-\gamma^{abc}D_{b}\psi_{c}^{i}-F^{+Iab}Im\;\mathcal{N}_{IJ}\nabla_{\alpha}X^{J}\gamma_{b}\chi_{j}^{\alpha}\varepsilon^{ij}+2F^{-Iab}Im\;\mathcal{N}_{IJ}X^{J}\psi_{jb}\varepsilon^{ij}$
$\displaystyle+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}=0$
$\displaystyle-\gamma^{abc}D_{b}\psi_{c}^{i}-g_{\alpha\bar{\beta}}G^{+\bar{\beta}ab}\gamma_{b}\chi^{\alpha}_{j}\varepsilon^{ij}-\frac{1}{2}T^{-ab}(\psi_{j})_{b}\varepsilon^{ij}+\frac{1}{2}g_{\alpha\bar{\beta}}\not{\partial}z^{\alpha}\gamma^{a}\chi^{i\bar{\beta}}=0$

now we collect the terms containing $\delta\chi^{i\bar{\alpha}}$

$\displaystyle
F^{+Iab}Im{\mathcal{N}}_{IJ}{\nabla}_{{\alpha}}{X}^{J}\Big{(}\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\chi^{j\bar{\delta}}\varepsilon_{ij}+\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\delta\chi^{j\bar{\delta}}\varepsilon_{ij}\Big{)}$
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\not{D}\delta\chi^{i\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\delta\chi^{i\bar{\beta}}+F^{-ab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}\delta\bar{\chi}^{\bar{\alpha}i}\gamma_{a}\psi^{j}_{b}\varepsilon_{ij}$
$\displaystyle-\frac{1}{4}g_{\bar{\alpha}{\beta}}\delta\bar{\chi}^{i\bar{\alpha}}\not{D}\chi^{{\beta}}_{i}$
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\not{D}\delta\chi^{i\bar{\beta}}=-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\left(\gamma^{\mu}\partial_{\mu}+\frac{1}{4}\gamma^{\mu}\omega_{\mu}^{ab}\gamma_{ab}-\frac{i}{2}\gamma^{\mu}\mathcal{A}_{\mu}\right)\delta\chi^{i\bar{\beta}}-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\delta\chi^{i\bar{\gamma}}$
$\displaystyle=-\frac{1}{4}g_{\alpha\bar{\beta}}\left(-\partial_{\mu}\bar{\delta{\chi}}^{i\bar{\beta}}\gamma^{\mu}\chi_{i}^{\alpha}+\frac{1}{4}\omega_{\mu}^{ab}\delta{\chi}^{i\bar{\beta}}\gamma_{ab}\gamma^{\mu}\chi_{i}^{\alpha}+\frac{i}{2}\delta\chi^{i\bar{\beta}}\gamma^{\mu}\mathcal{A}_{\mu}\chi_{i}^{\alpha}\right)\delta\chi^{i\bar{\beta}}+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\chi_{i}^{\alpha}$
$\displaystyle=-\frac{1}{4}g_{\alpha\bar{\beta}}\left({\delta\bar{\chi}}^{i\bar{\beta}}\gamma^{\mu}\partial_{\mu}\chi_{i}^{\alpha}+\frac{1}{4}\omega_{\mu}^{ab}\delta\bar{\chi}^{i\bar{\beta}}\gamma^{\mu}\gamma_{ab}\chi_{i}^{\alpha}+\frac{i}{2}\delta\bar{\chi}^{i\bar{\beta}}\gamma^{\mu}\mathcal{A}_{\mu}\chi_{i}^{\alpha}\right)+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\chi_{i}^{\alpha}$
$\left(\partial_{\mu}e^{a}_{\nu}+\left(\omega^{a}_{b}\right)_{\mu}e^{b}_{\nu}\right)dx^{\mu}\wedge
dx^{\nu}=0$ (B.9)
$\displaystyle=-\frac{1}{4}g_{\alpha\bar{\beta}}\left({\delta\bar{\chi}}^{i\bar{\beta}}\gamma^{\mu}\partial_{\mu}\chi_{i}^{\alpha}+\frac{1}{4}\omega_{\mu}^{ab}\delta\bar{\chi}^{i\bar{\beta}}\gamma^{\mu}\gamma_{ab}\chi_{i}^{\alpha}+\frac{i}{2}\delta\bar{\chi}^{i\bar{\beta}}\gamma^{\mu}\mathcal{A}_{\mu}\chi_{i}^{\alpha}\right)$
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\Gamma^{\alpha}_{\delta\gamma}\gamma^{\mu}\chi^{\gamma}_{i}\partial_{\mu}z^{\delta}+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\chi_{i}^{\alpha}$
$\displaystyle+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\Gamma^{\alpha}_{\delta\gamma}\gamma^{\mu}\chi^{\gamma}_{i}\partial_{\mu}z^{\delta}$
$\displaystyle=-\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\not{D}\chi_{i}^{\alpha}+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\chi_{i}^{\alpha}+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\Gamma^{\alpha}_{\delta\gamma}\gamma^{\mu}\chi^{\gamma}_{i}\partial_{\mu}z^{\delta}$
(B.10)
$\displaystyle\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma^{\mu}\Gamma^{\bar{\beta}}_{\bar{\delta}\bar{\gamma}}\partial_{\mu}\bar{z}^{\bar{\delta}}\chi_{i}^{\alpha}+\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\Gamma^{\alpha}_{\delta\gamma}\gamma^{\mu}\chi^{\gamma}_{i}\partial_{\mu}z^{\delta}$
$\displaystyle=\frac{1}{4}\delta\chi^{i\bar{\beta}}\gamma^{\mu}\chi_{i}^{\alpha}(\partial_{\delta}g_{\alpha\bar{\beta}}\partial_{\mu}z^{\delta}+\partial_{\bar{\delta}}g_{\alpha\bar{\beta}}\partial_{\mu}\bar{z}^{\bar{\delta}})=\frac{1}{4}\delta\chi^{i\bar{\beta}}\gamma^{\mu}\chi_{i}^{\alpha}\partial_{\mu}g_{\alpha\bar{\beta}}$
(B.11)
$\displaystyle-\frac{1}{4}g_{\alpha\bar{\beta}}\bar{\chi}_{i}^{\alpha}\not{D}\delta\chi^{i\bar{\beta}}=-\frac{1}{4}g_{\alpha\bar{\beta}}\delta\bar{\chi}^{i\bar{\beta}}\not{D}\chi_{i}^{\alpha}+\frac{1}{4}\delta\chi^{i\bar{\beta}}\gamma^{\mu}\chi_{i}^{\alpha}\partial_{\mu}g_{\alpha\bar{\beta}}$
(B.12)

Now, the terms of the action containing $\delta\chi^{i\bar{\alpha}}$ can be
written as

$\displaystyle
F^{+Iab}Im{\mathcal{N}}_{IJ}{\nabla}_{{\alpha}}{X}^{J}\Big{(}\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\delta\bar{\chi}^{i\bar{\gamma}}\gamma_{ab}\chi^{j\bar{\delta}}\varepsilon_{ij}+\frac{1}{8}g^{{\alpha}\bar{\beta}}C_{\bar{\beta}\bar{\gamma}\bar{\delta}}\delta\bar{\chi}^{i\bar{\delta}}\gamma_{ab}\chi^{j\bar{\gamma}}\varepsilon_{ij}\Big{)}$
$\displaystyle+\frac{1}{4}\delta\chi^{i\bar{\beta}}\gamma^{\mu}\chi_{i}^{\alpha}\partial_{\mu}g_{\alpha\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\bar{\psi}_{ia}\not{\partial}z^{\alpha}\gamma^{a}\delta\chi^{i\bar{\beta}}+F^{-ab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\alpha}}\bar{X}^{J}\delta\bar{\chi}^{\bar{\alpha}i}\gamma_{a}\psi^{j}_{b}\varepsilon_{ij}$
$\displaystyle-\frac{1}{2}g_{\bar{\alpha}{\beta}}\delta\bar{\chi}^{i\bar{\alpha}}\not{D}\chi^{{\beta}}_{i}$

The equation of motion for the gaugino then is

$\displaystyle\frac{1}{4}F^{+Iab}Im{\mathcal{N}}_{IJ}{\nabla}_{{\alpha}}{X}^{J}g^{{\alpha}\bar{\gamma}}C_{\bar{\gamma}\bar{\beta}\bar{\delta}}\gamma_{ab}\chi^{j\bar{\delta}}\varepsilon_{ij}+\frac{1}{4}\gamma^{\mu}\chi_{i}^{\alpha}\partial_{\mu}g_{\alpha\bar{\beta}}+\frac{1}{2}g_{\alpha\bar{\beta}}\gamma^{a}\not{\partial}z^{\alpha}{\psi}_{ia}$
$\displaystyle+F^{-Iab}Im\mathcal{N}_{IJ}\bar{\nabla}_{\bar{\beta}}\bar{X}^{J}\gamma_{a}\psi^{j}_{b}\varepsilon_{ij}-\frac{1}{2}g_{\alpha\bar{\beta}}\not{D}\chi_{i}^{\alpha}=0.$
(B.13)

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