11institutetext: University of Tartu, Estonia
11email: {mahmoud.shoush<EMAIL_ADDRESS>
# When to intervene? Prescriptive Process Monitoring Under Uncertainty and
Resource Constraints††thanks: Supported by the European Research Council (PIX
Project).

Mahmoud Shoush  0000-0002-7423-9909 Marlon Dumas  0000-0002-9247-7476

###### Abstract

Prescriptive process monitoring approaches leverage historical data to
prescribe runtime interventions that will likely prevent negative case
outcomes or improve a process’s performance. A centerpiece of a prescriptive
process monitoring method is its intervention policy: a decision function
determining if and when to trigger an intervention on an ongoing case.
Previous proposals in this field rely on intervention policies that consider
only the current state of a given case. These approaches do not consider the
tradeoff between triggering an intervention in the current state, given the
level of uncertainty of the underlying predictive models, versus delaying the
intervention to a later state. Moreover, they assume that a resource is always
available to perform an intervention (infinite capacity). This paper addresses
these gaps by introducing a prescriptive process monitoring method that
filters and ranks ongoing cases based on prediction scores, prediction
uncertainty, and causal effect of the intervention, and triggers interventions
to maximize a gain function, considering the available resources. The proposal
is evaluated using a real-life event log. The results show that the proposed
method outperforms existing baselines regarding total gain.

###### Keywords:

Prescriptive Process Monitoring Causal Inference Uncertainty

##  1 Introduction

Prescriptive Process monitoring (PrPM) is a family of process mining methods
that trigger runtime actions to optimize a process’s performance [6, 14]. PrPM
methods use event logs describing past business process executions to train
machine learning (ML) algorithms for two goals. First, the trained ML models
predict how an instance of the process (a.k.a. case) will unfold. For example,
whether the case leads to a positive outcome (e.g., a customer is satisfied)
or a negative outcome (e.g., a customer launches a complaint) [20]. Second,
PrPM methods use ML to assess the effect of triggering an action (herein
called an intervention) on the probability of a negative outcome or a
performance measure.

Recently, various PrPM methods have been proposed [18, 6, 14, 2]. These
methods, however, implement intervention policies based on predictions of
negative outcomes without considering the uncertainty of these predictions.
Also, they trigger an intervention when the predicted probability of a
negative outcome is above a threshold, without considering potential increases
or decreases in this predicted probability that may occur as the case unfolds
further. Finally, these methods do not consider the fact that there are
limited resources available to perform the recommended interventions.

In this paper, we address the following problem: Given a set of cases, and
given a type of intervention that generally decreases the likelihood of a
negative case outcome. How do we select the cases for which applying the
intervention (now or later) maximizes a gain function, considering the
available resources to perform interventions? Here, the gain function
considers the tradeoff between the cost of applying the intervention to a case
and the cost of negative outcomes.

To address this problem, we first apply an ensemble-based predictive model to
estimate the negative outcome probability for each case, and we estimate the
associated uncertainty. Using a causal model, we then determine the causal
effect of applying an intervention on the negative outcome probability. We
then use the negative outcome probability, the uncertainty, and the estimated
causal effect and apply a filtering and ranking mechanism to identify cases
for which an intervention would be most profitable (highest gain). We also
consider the tradeoff (i.e., opportunity cost) between triggering an
intervention in the current state, given the level of uncertainty of the
predictive model, versus postponing the intervention to a later state. The
paper reports an empirical evaluation comparing the proposed approach against
state-of-the-art baselines.

The following section motivates the proposed method. Sect. 3 then presents
background concepts and related work. Sect. 4 explains the proposed method,
while Sect. 5 discusses the empirical evaluation. Finally, Sect. 6 concludes
and discusses future work directions.

##  2 Motivating Example

In a loan origination process, a case starts when a customer submits his
documents to obtain a loan. Then a process worker (or an employee) verifies
the submitted documents. When they are valid, the employee sends an offer to
the customer via different channels, such as phone calls or emails. This case
ends positively when the customer accepts the offer and receives the loan, or
negatively when the customer declines the offer or the employee rejects the
application.

The principal concern arises when cases end negatively, leading to less
payoff. One way to deal with this could be to predict negative cases based on
prediction scores. Then trigger an alarm to take a proactive action or an
intervention, e.g., making a follow-up call, when the prediction score exceeds
a certain threshold.

However, this strategy could be ineffective. Suppose an intervention policy
where interventions are triggered to cases that are likely to end negatively
based on low-quality prediction scores. Also, all employees are occupied and
cannot immediately perform the intervention for all cases. Additionally,
triggering interventions without considering their effect could be misleading
since they may provide low or negative impact when utilized.

A more proper method is to quantify the prediction uncertainty to estimate how
sure predictive models are with the prediction scores. Moreover, measuring the
causal effect of utilizing interventions and considering the availability of
resources. Another step that may enhance the overall payoff could be
considering the tradeoff between triggering interventions now versus
postponing them for a later state. In this paper, we discuss this method and
evaluate its performance.

##  3 Background and Related Work

###  3.1 Predictive process monitoring

Predictive process monitoring (PPM) [12] is a complementary set of process
mining methods to predict how ongoing cases will end. A PPM technique may, for
instance, predict the remaining time for an ongoing case to be executed
entirely [22], the following action or activity to be executed [15], or the
outcome w.r.t group of outcomes, e.g., positive or negative [20]. This paper
focuses on the latter technique, known as an outcome-oriented PPM.

Recent outcome-oriented PPM methods estimate the prediction scores, i.e.,
probability of negative outcomes, for ongoing cases and classify them
positively or negatively. If the prediction scores exceed a threshold, e.g.,
above $0.5$, the ongoing case is considered more likely to end negatively.

However, outcome-oriented PPM methods focus only on making predictions as
accurate as possible, regardless of the quality of the predictions. These
methods rely on several case bucketing techniques [4], e.g., a single bucket
where cases are made in the same bucket and train one ML algorithm instead of
several. Also, they rely on various feature encoding techniques [20] to map
each case into a feature vector to train the ML algorithm. For instance, an
aggregate encoding in which all events from the beginning of the case are
considered. Thus, several aggregate functions may be used to the values an
event has carried throughout the case. Also, a handful of possible inter-case
features are extracted [10] to enrich the training of ML or deep learning (DL)
algorithms [11]. Still, these techniques aim to improve the performance of the
prediction scores and ignore quantifying the prediction quality via measuring
prediction uncertainty.

To the best of our knowledge, only one work from the literature considers
estimating the model’s prediction uncertainty explicitly, tackling another PPM
task, i.e., remaining time [24]. They learn the prediction uncertainty with
artificial neural networks and a Monte Carlo (MC) [7] dropout technique that
is unreliable for out-of-distribution data (i.e., where there is an input case
from a region very far from the trained data) and computationally expensive
[1].

Metzger et al. [14] introduces an approach to measure the reliability of
prediction scores using an ensemble of DL classifiers at different process
states. This approach does not discuss the estimation of the prediction
uncertainty ignoring the situation of out-of-distribution input. Moreover,
where predictive models provide several prediction scores for the same input,
i.e., outcomes overlap.

###  3.2 Prescriptive process monitoring

Prescriptive process monitoring (PrPM) methods go beyond predictions to
prescribe runtime interventions to prevent or mitigate negative outcome
effects. These methods aim to improve the performance of business process
executions by determining if and when to trigger an intervention to maximize a
payoff.

Diverse PrPM methods have been proposed. Metzger et al. [14] suggest using
ongoing cases prediction scores and their reliability estimate with a
reinforcement learning technique to discover when to trigger runtime
interventions. Another work by Fahrenkrog et al. [6] proposes triggering one
or more alarms when cases are more likely to end negatively, followed by an
intervention.

Both the work of Metzger et al. and that of Fahrenkrog et al. identify cases
that need intervention. They assume that resources are unbounded and consider
only the current state of a given case to determine when to intervene.
Instead, we study the tradeoff between intervening now or later based on the
current and future prediction scores. Thus, we identify the most profitable
case and assign resources to it, considering that resources are limited.

Weinzerl et al. [23] suggest a PrPM method to recommend the next best activity
from a list of possible activities with a higher preference for a pre-defined
KPI. Khan et al. [11] introduce a memory-augmented neural net approach to
recommend the most suitable path (meaning a set of activities until the
completion of the process) based on pre-specified KPIs. Both the work of
Weinzerl et al. and that of Khan et al. do not discuss an exact idea of
interventions or when to trigger them to maximize payoff.

###  3.3 Causal Inference

Causal Inference (CI) [25] is a set of methods to predict what would occur if
we adjust the process during its execution time by finding a cause-effect
relationship between two variables, i.e., an intervention ($T$) and an outcome
(Y).

CI methods mainly unfold into two categories [8]. The first category is
structural causal models (SCMs), a multivariate statistical analysis method
exploring structural relationships between dependent and independent
variables. It depends mainly on discovering and building a causal graph by
domain experts.

The second category is potential outcome frameworks (a.k.a., the Neyman-Rubin
Causal Model). A statistical analysis method that does not require a pre-built
causal graph like the SCMs and relies on the concept of potential outcomes. We
use this category in this paper to automatically estimate the causal effect
(or conditional average treatment effect (CATE)) of intervention on negative
cases instead of manually building causal graphs.

For example, in a loan-origination process, a customer would have a loan if he
received an intervention (T), e.g., a follow-up call three days after
receiving the first offer; otherwise, he would have a different outcome (Y),
e.g., offer declined. Accordingly, to measure the $CATE$ of having a follow-up
call, we need to compare $Y$ for the same customer when receiving the follow-
up call, i.e., $T=1$, and not receiving the follow-up call, i.e., $T=0$.

Recent work uses the potential outcome method to estimate the $CATE$ of
utilizing interventions. Specifically, in [2], the Authors introduce a PrPM
technique to measure the effect of intervention at an individual case level to
reduce the cycle time of the process. It targets another PPM problem and
considers only one process execution state; it also assumes that resources are
unbounded.

In our previous work [18], we propose a PrPM method that utilizes the
potential outcome method and a resource allocator technique in the outcome-
oriented PPM to allocate resources to cases with max gain. That work considers
only a given case’s current state and triggers interventions when the
prediction score and the causal effect exceed a threshold. However, we did not
discuss the tradeoff between triggering an intervention now, given the level
of uncertainty of the underlying predictive models versus later.

##  4 Approach

Figure 1: An overview of the proposed approach.

The proposed PrPM method consists of two main phases, training and testing,
see Fig. 1. In the training phase, we train two ML models, i.e., predictive
and causal. While in the testing phase, we present filtering and ranking
techniques to determine the most profitable case. Then, decide when to trigger
an intervention for the selected case to maximize a gain function—considering
ongoing cases’ current and future state scores, uncertainty estimation, and
resource availability.

###  4.1 Training Phase

We first prepare the process execution data; then, we construct an ensemble-
based predictive model to estimate the prediction scores, i.e., the
probability of cases likely to end negatively and quantify the prediction
uncertainty. Further, we build a causal model to estimate the $CATE$.

####  4.1.1 Event Log Preprocessing

This step is vital for PPM or PrPM tasks. In PPM, it includes data cleaning,
prefix extraction, and prefix encoding see Fig. 2. These steps have been
discussed in Teinemaa et al. [20], and we follow their suggestions here. We
first pre-process the log to dismiss incomplete cases and then extract length
$k$ from every case that results in a so-called prefix log. This prefix
extraction ensures that our training data is equivalent to the testing data.
Finally, we encode each trace prefix into a feature vector ($X$) to train the
predictive ensemble model, see Fig. 2.

Figure 2: An overview of the ensemble model.

While in PrPM, one further step is needed to analyze and understand the data
and the business objective to identify an intervention $T$ that could
positively impact an outcome $Y$. Moreover, determining what other variables
($W$: a.k.a. confounders) affect the intervention and outcome.

####  4.1.2 Ensemble model

We construct an outcome-oriented predictive model (a classification problem
from an ML perspective) via ensemble learning [5], as shown in Fig. 2. The
principal assumption of ensemble learning is constructing one robust
predictive model from several weak ones. Accordingly, the overall prediction
scores performance would be superior where overfitting and the chance of
getting a local minimum are avoided, which has two advantages to our work.
First, it ensures that we will accurately predict the probability of negative
outcomes, i.e., the prediction score. Second, it allows estimating the
prediction uncertainty.

A single classifier in the ensemble is a probabilistic model
$cls_{i}:f(X,P_{uout_{i}})$. $X$ is the input feature vector, and the
$P_{uout_{i}}$ is the estimated probability of cases likely to end with
negative outcomes where $i\in\\{1,m\\}$, and $m$ is the number of classifiers
in the ensemble. We then define the prediction score as the average of
individual classifiers’ prediction scores $(avg_{pred})$, as shown in Eq. 1.
The following step is to estimate the prediction uncertainty.

$avg_{pred}=\frac{\sum_{i=1}^{m}P_{uout_{i}}}{m}$ (1)

There are two sources of uncertainties where predictive models become unsure
about predictions [9]. The first source is data (or aleatoric) uncertainty
$(\sigma)$. It is a property of the distribution that generates cases, and it
occurs when outcomes overlap or there is noise in the underlying data
distribution. The second source is knowledge (or epistemic) uncertainty
$(\rho)$. It is a property of the predictive model’s learning parameters and
arises due to a lack of model knowledge. It appears when there is out-of-
distribution input. Both sources form the prediction uncertainty, i.e., total
uncertainty ($total_{uncer}$), see Eq. 2.

$total_{uncer}=\sigma+\rho$ (2)

The proper level of $\sigma$ is defined as entropy of the actual underlying
data distribution [13]. Here, the entropy is the average level of surprise or
uncertainty inherent in the possible outcomes. It is calculated for a random
variable $S$ with $c$ in $C$ discrete states, see Eq. 3, where $P(c)$ is the
probability of $c$ to occur.

$Entropy(S)=-{\sum_{c\in C}P(c)\log(P(c)}$ (3)

However, we do not have access to the actual underlying data distribution, but
our model is probabilistic and is trained on this data. So measuring the
entropy of our probabilistic model, i.e., trained using negative log-
likelihood, estimates the level of $\sigma$. In particular, we obtain a
distribution over outcome labels from individual classifiers in the ensemble
for a given case, and $\sigma$ is the average entropy of the individual
prediction scores, as shown in Eq. 4.

$\sigma=\frac{\sum_{i=1}^{m}Entropy(P_{uout_{i}})}{m}$ (4)

Furthermore, we rely on the Bayes rule [19] to estimate the $\rho$ [13].
Assume we obtain a posterior over model parameters that give us the
distribution over likely models that have generated the data. Accordingly,
models sampled from the obtained distribution agree on data they have seen and
provide similar predictions, indicating low $\rho$. However, if the models do
not understand the input, they provide diverse predictions and strongly
disagree, indicating high $\rho$. Thus, $\rho$ is the mutual information
between the models’ parameters and the predictions.

Similarly, the total uncertainty is the entropy of the average prediction, see
Eq. 5. To exemplify the estimation of the prediction uncertainty
($total_{uncer}$), suppose we show the ensemble of classifiers several kinds
of input. (1) We give input that all classifiers understand and yield the
exact prediction scores. Consequently, classifiers are confident about their
prediction scores, and the $total_{uncer}$ is minimum. (2) We show input that
all classifiers understand and generate identical predictions but high entropy
distribution over outcomes; then, classifiers are uncertain with high
$\sigma$. (3) We show the ensemble something none of the classifiers
understand; hence, all classifiers yield different prediction scores because
the input comes from a very far region from the training data. Thus, the
ensemble is very diverse with high entropy because we average various
probability distributions together, then classifiers are uncertain with high
$\rho$.

$total_{uncer}=Entropy(avg_{pred})$ (5)

In addition to the ensemble model that estimates $avg_{pred}$ and
$total_{uncer}$ in the training phase, we train a causal model to measure the
$CATE$. In particular, we utilize an orthogonal random forest (ORF) algorithm
because it reasonably deals with high-dimensional variable spaces. Thus, it is
beneficial since the process execution logs have numerous event attributes
with categorical values.

Estimating the $CATE$ means we evaluate the difference between the probability
of a negative outcome if we intervene and if we do not intervene. The higher
the differences, the stronger the effect of the intervention. For explanation,
we recall the motivating example in Sect. 2, where the goal was to improve the
performance of a loan origination process by raising the number of successful
applications. Accordingly, we would give customers who are likely to decline
the first offer another offer in a way that affects the probability of
declining the first offer positively. Then we estimate what would happen for
the probability of negative outcome when we send customers a second offer and
when we do not.

###  4.2 Testing Phase

We first use the trained ensemble and causal models to obtain $avg_{pred}$,
$total_{uncer}$, and $CATE$ scores for ongoing cases. Then, we use these
scores to filter ongoing cases into candidate ones and rank them to choose the
most profitable case to maximize a gain function. We consider that resources
are bounded and compare ongoing cases’ current and future state scores.

####  4.2.1 Filtering

At run time, new events of ongoing cases keep coming continuously, and we
first collate them via a prefix collator to accumulate the sequence of events.
Thus, at any point in time, we could have one or multiple ongoing cases that
we choose from which one is the most profitable to trigger the intervention.
Hence, ongoing cases need to be filtered to minimize the search space.

We use $avg_{pred}$, $total_{uncer}$, and $CATE$ scores to filter ongoing
cases into candidate ones. The essence of these scores varies from one to
another. The $avg_{pred}$ gives information about whether ongoing cases are
likely to end negatively or not. At the same time, the $total_{uncer}$ shows
how sure the model is with its predictions. It ranges from $0$ to $1$, where
the predictive model is entirely certain or uncertain, respectively. Moreover,
the $CATE$ is crucial to any PrPM technique, representing the expected impact
of utilizing intervention on an ongoing case, e.g., when $CATE$ is above $0$,
it impacts positively.

All the abovementioned scores ($avg_{proba}$, $total_{uncer}$, $CATE$) are
vital to determining candidate cases from which we will choose the most
profitable. However, there are two other critical aspects to determine when to
intervene and to define the most profitable case; how the estimated scores
will change in the following state and whether resources are available or not.

####  4.2.2 Future state scores estimation

Considering only the current state scores of ongoing cases regardless of
investigating what would occur in the future could be misleading. Because if
we decide not to intervene, maybe it will be more effective to achieve higher
gain when we utilize the intervention later than now, or we will be more sure
about the prediction of the outcome and decide not to intervene at all. So,
discovering what will happen in the future of ongoing cases allows deciding
whether to intervene now or later.

To estimate what will happen in the future state, we predict the $avg_{pred}$,
$CATE$, and $total_{uncer}$ in the future. So for each score, we will have two
values: one representing the current state ($c\\_avg_{pred}$, $c\\_CATE$,
$c\\_total_{uncer}$) and the other representing the future ($f\\_avg_{pred}$,
$f\\_CATE$, $f\\_total_{uncer}$). We follow a technique inspired by the
k-nearest neighbors (KNN) [16] algorithm to get scores representing the future
state, given the degree of similarity and frequency.

We look at previous cases similar to the ongoing one at the next prefix. For
example, in a given case at prefix $4$, we want to know what will happen at
prefix $5$. Then, we capture similar prefixes and define an aggregate score,
i.e., the weighted average. We consider the degree of similarity using
Euclidean distance to the current prefix, their frequency because higher
frequency means more weight and the scores for all similar prefixes.
Accordingly, scores representing the future state ($f\\_avg_{pred}$,
$f\\_CATE$, $f\\_total_{uncer}$) of ongoing cases are the weighted average
from similar previous prefixes. The next step is to use current and future
estimates to rank candidate cases and select the most profitable.

####  4.2.3 Rank and retrieve the best case.

To define the most profitable case, we first distinguish between gain and
adjusted gain. The gain is the benefits we attain at one state only, either
current or future, and it means we estimate two gains, one for the current
state ($c\\_gain$) and the other for the future state ($f\\_gain$). In
contrast, the adjusted gain ($adj_{gain}$) is the benefits we attain
considering current and future states. We use the $adj_{gain}$ as a decision
function to determine whether to intervene now or later.

Triggering interventions may come with benefits and, at the same time, comes
at a cost. The costs can vary from one process to another. However, for a
given case $c_{id}$, there is generally a cost for applying the intervention
($cost(c_{id},T_{i=1})$)) when $T=1$ and not applying the intervention
($cost(c_{id},T_{i=0})$) when $T=0$.

The cost of not applying the intervention describes how much we lose if the
negative outcome occurs, and it relies on the $avg_{pred}$ and cost of
negative outcomes ($c_{uout}$), as shown in Eq. 6. In contrast, the cost of
utilizing the intervention refers to how much we decrease the probability of
negative outcomes considering the intervention cost ($c_{T_{1}}$), see Eq. 7,
which assumes $CATE$ is reliable. Costs are mainly identified via domain
knowledge; however, we assume that the $c_{T_{1}}$ is less than the $c_{uout}$
to obtain meaningful results.

$cost(c_{id},T_{i=0})=avg_{pred}*c_{uout}$ (6)
$cost(c_{id},T_{i=1})=(avg_{pred}-CATE_{1})*c_{uout}+c_{T_{1}}$ (7)

The corresponding gain ($gain(c_{id},T_{i=1})$) from utilizing the
intervention on $c_{id}$ is the benefits that allow the highest cost reduction
in Eq. 8

$gain(c_{id},T_{i=1})=cost(c_{id},T_{0})-cost(c_{id},T_{i=1})$ (8)

We estimate the current state gain ($c\\_gain(c_{id},T_{i=1})$) using scores
from the ensemble and causal models, see Eq. 9. In contrast, the gain for the
future state ($f\\_gain(c_{id},T_{i=1})$) is based on the weighted average
scores from previous similar prefixes, see Eq. 10.

$c\\_gain(c_{id},T_{i=1})=c\\_cost(c_{id},T_{0})-c\\_cost(c_{id},T_{i=1})$ (9)
$f\\_gain(c_{id},T_{i=1})=f\\_cost(c_{id},T_{0})-f\\_cost(c_{id},T_{i=1})$
(10)

Determining the gain for candidate cases’ current and future states is vital
to define the adjusted gain. To explain the adjusted gain, we first define an
opportunity cost that measures what we lose when choosing between two or more
alternatives—for example, utilizing the intervention now or later. The
opportunity cost ($opp_{cost}$) is the difference between the gain we could
achieve in the future state of a given case and the gain in the current state,
as shown in Eq. 11.

$opp_{cost}=f\\_gain(c_{id},T_{i=1})-c\\_gain(c_{id},T_{i=1})$ (11)

Given the opportunity cost, we define the adjusted gain as the payoff (or
gain) we acquire from utilizing the intervention on candidate cases
considering current and future states. It is the difference between the
$c\\_gain(c_{id},T_{i=1})$ and the $opp_{cost}$, as shown in Eq. 12. Thus, we
define the most profitable case as the one with the highest $adj_{gain}$,
which means the lowest $opp_{cost}$.

$adj_{gain}=c\\_gain(c_{id},T_{i=1})-opp_{cost}$ (12)

Table 1: An example of defining gain. $c_{id}$ | $c\\_gain(c_{id},T_{i=1})$ | $f\\_gain(c_{id},T_{i=1})$ | $opp_{cost}$ | adju | decision
---|---|---|---|---|---
A | 7 | 12 | 5 | 3 | Wait
B | 5 | 1 | -4 | 9 | Treat
C | 3 | 3 | 0 | 3 | Neutral

For example, suppose we filtered ongoing cases into three candidates (see Tab.
1) eligible for the intervention. Also, there is an available resource to do
the intervention; we need to choose which one is more suitable to intervene
now or later. If we consider only the gain from the current state, we assign
resources to $c_{id}=A$ and treat it now. However, If we think about the gain
from the future state, we observe that later we can achieve more gain if we do
not intervene and previously assigned resources to $c_{id}=A$ inaccurately.
Hence, it is more beneficial to allocate resources to $c_{id}=B$ and apply the
intervention now since we might lose current gain later. Thus, using the
adjusted gain to decide when to intervene could enhance the performance of
PrPM.

Selecting the best or most profitable case can be judged efficiently based on
the adjusted gain to maximize the total gain. However, triggering
interventions as often as we want and immediately is impossible since
resources are bounded in practice, which is the last aspect we need to
consider.

####  4.2.4 Resource Allocator.

Monitoring resources and assigning them to cases that need intervention is
critical. The resource allocator checks the availability of resources. Once
the most profitable case is selected and a free resource is available, we
assign that resource to the selected case and block it for a certain time,
i.e., treatment duration ($T_{dur}$). The number of available resources and
the time required to perform the intervention could be identified via domain
knowledge [18].

##  5 Evaluation

To verify the effectiveness and relevance of the proposed method, we
experimentally investigate whether we can learn when to trigger an
intervention to maximize the total gain at the run time, considering the
tradeoff between intervening now or later. We compare our results to baselines
that consider either predictive models without quantifying the prediction
uncertainty [6, 14] or only the current case’s execution state scores [18, 2]
as state-of-the-art baselines by addressing the following research questions:

  1. RQ1.

To what extent does taking into account the current uncertainty prediction
enhance the total gain?

  2. RQ2.

To what extent does taking into account the derivative of the uncertainty
prediction and the adjusted gain enhance the total gain?

###  5.1 Dataset

We use a real-life event log, named _BPIC2017_
111https://doi.org/10.4121/uuid:5f3067df-f10b-45da-b98b-86ae4c7a310b, publicly
available from the 4TU.ResearchData. The log describes the execution of a loan
origination process, and we choose this log for several reasons. First, it
contains a clear notion for outcome definition and interventions utilized to
show our method’s efficiency. Second, this log contains $31,413$ applications
and $1,202,267$ events, which is large enough and frequently used for
predictive and prescriptive methods.

The BPIC2017 log is characterized by various case and event attributes, and we
include all of them in our experiments. Additionally, we extracted other
essential critical features in our work, such as the number of sent offers,
event number, and additional temporal features.

We used all original and extracted attributes as input for ensemble and causal
models in the preprocessing step. Then we defined case outcomes based on the
end state of cases, i.e., “A_Pending” state means positive outcome and
“A_canceled or A_Denied” means negative outcome. After that, we defined the
intervention that positively impacts the negative outcome as sending a second
offer (or ”Creat_Offer” activity) to all customers who received only one
offer. Accordingly, we denoted cases with $T=1$ based on the number of offers
sent to each, i.e., cases that receive only one offer. Then, we extracted
length prefixes no more than the $90^{th}$ percentile for each case to avoid
bias from lengthy cases. Finally, we used an aggregate encoding to encode the
extracted prefixes.

###  5.2 Experimental Setup

The experiments show our method’s effectiveness during operation time with two
main objectives: deciding when to intervene, either now or later, and
selecting the most profitable case among all candidates to maximize the total
gain.

We adapted an ensemble model based on a Gradient Boosting Decision Tree (GBDT)
method to estimate the $avg_{pred}$ and $total_{uncer}$. In particular, we
used Catboost [17], an open-source GBDT library with several tools to quantify
the prediction uncertainty and automatically handle categorical features.

Catboost is trained with a negative log-likelihood loss and Langevin
optimization [21] to ensure global conversion instead of a local optimum and
generate an ensemble of several independent GBDT. Additionally, we used the
following parameters during training: ensemble size of $50$, a learning rate
of $0.05$, a subsample of $0.82$, and a max tree depth of $12$.

Catboost returns a probability distribution over the case outcomes. This
distribution is based on a given model version, i.e., on the seed used to
initialize the model parameters before training. So we train the same model
using different seed initialization and evaluating these models on the same
input to obtain the $avg_{pred}$ and its $total_{uncer}$. We use an ORF
algorithm implemented in the _EconMl_ 222https://github.com/microsoft/EconML
to train a causal model to estimate the CATE.

We follow the machine learning workflow to train both ensemble and causal
models. We temporally split the data into training ($60\%$), validation
($20\%$ ), and testing ($20\%$) sets. Training and validation sets are used to
train and tune model parameters, while the testing set is used to evaluate the
model’s performance.

Table 2: Parameter settings of the introduced method

$avg_{proba}$ | $CATE$ |  $total_{uncer}$ | $\Delta total_{uncer}\hskip 14.22636pt$ |  #resources |  $c_{uout}$ | $c_{T_{1}}$ |  $T_{dur}$ (sec)
---|---|---|---|---|---|---|---
>$0.5$ | >$0$ | $<0.25,0.5,0.75$ | $<0,-0.5,-0.1,-0.15,-0.2,-0.25,-0.3$ | $1,2,...10$ | $20$ | $1$ |  | Fixed = $60$
---
Normal $\in$ {$1,60$}
Exponential $\in$ {$1,60$}

During the testing time, we follow the configurations shown in Tab. 2. First,
ongoing cases are filtered into candidates. We filter cases based on the
estimated probability of $avg_{pred}$ >$0.5$ to ensure that cases are highly
probable to end negatively and $CATE$ >$0$ to guarantee that intervention has
a positive impact.

Additionally, we filter cases using the $total_{uncer}$ to see how sure the
predictive model is with the predictions. The estimated $total_{uncer}$ ranges
from $0$, where the model is certain, to $1$, which is entirely uncertain. We
experiment with three thresholds (see Tab. 2) when considering only the
current state of cases.

On the other hand, we use the $opp_{coast}$ or $adj_{gain}$ when we consider
both current and future states of ongoing cases. At the same time, with and
without a derivative of the total uncertainty ($\Delta total_{uncer}$),
representing the difference between $c\\_total_{uncer}$ and
$f\\_total_{uncer}$. Accordingly, we filter cases based on the
$c\\_avg_{pred}$ and $c\\_CATE$ and then select the case with the highest
$adj_{gain}$ with and without $\Delta total_{uncer}$ when it is below $0$ or
negative values (see Table 2), which means the predictive model becomes more
uncertain in the future state.

###  5.3 Results

We show results here based on one $T_{dur}$ distribution, i.e., fixed. Since
the normal distribution achieves similar results to the fixed in terms of the
total gain and is higher than the exponential distribution as the variability
is higher in the latter. Also, we observed that the behavior for different
thresholds for $total_{uncer}$ and $\Delta total_{uncer}$ is the same, i.e.
when we lower the threshold, fewer cases are treated with no substantial
effect on the total gain per case. Hence, we present results using one
threshold, i.e., the higher value for $total_{uncer}<0.75$ and $\Delta
total_{uncer}<0$. However, The full results of experimenting with all
thresholds and $T_{dur}$ distributions are available in supplementary material
333https://zenodo.org/record/6381445#.YjwaFfexWuA.

To discuss RQ1, we consider only the current state scores of ongoing cases.
Then, examine how the total gain evolves when adding the current prediction
uncertainty ($c\\_total_{uncer}$) to the filtering step. Besides the estimated
prediction score ($c\\_avg_{proba}$ ) and the causal effect ($c\\_CATE$). We
compare this to baselines shown in [18, 3], where only the $avg_{proba}$ and
$CATE$ are set, see Fig. 3(a).

(a) RQ1: Where $avgProba\\_CATE$ means $c\\_avg_{proba}>0.5$ & $c\\_CATE>0$.
Also, $avgProba\\_CATE\\_tUncer$ means $c\\_avg_{proba}>0.5$ & $c\\_CATE>0$ &
$c\\_total_{uncer}<0.75$.

(b) RQ2: Where $avgProba\\_CATE\\_oppCost$ refers to the $adj_{gain}$ is
maximized. Also, $avgProba\\_CATE\\_oppCost\\_dUncer$ refers to the
$adj_{gain}$ is maximized & $\Delta total_{uncer}<0$.

Figure 3: The total gain progress under the same number of available
resources.

The results in Fig. 3(a) explain that adding the current prediction
uncertainty improves the total gain when the available resources exceed 80%.
Also, the baseline treats more cases with less total gain (as shown in the
supplementary 3). In contrast, adding the current prediction uncertainty
allows efficient allocation of resources since fewer cases are treated but
with a higher total gain than the baseline. Because considering the current
prediction uncertainty does not trigger the intervention until the predictive
model becomes more sure about its predictions and achieves higher total gain.

Turning to discuss RQ2, we consider two states’ scores of ongoing cases, i.e.,
current and future, to estimate the $opp_{cost}$ and the $adj_{gain}$ and
their impacts on the total gain. Hence, we decide when to intervene and
allocate the available resources to ongoing cases based on two things: first,
cases with the maximum $adj_{gain}$ only, and second, cases with the maximum
$adj_{gain}$ and the $\Delta total_{uncer}<0$. We compare this method to the
baseline in RQ1 and consider the current prediction uncertainty only, see Fig.
3(b).

The results in Fig. 3(b) show that the total gain is highly affected by the
trade off of current and future state scores under the same quantity of
consumed resources. The proposed method achieves a significant improvement in
terms of the total gain and outperforms the baselines (cf. Fig 3(a)). We found
that considering the $adj_{gain}$ as a decision function to determine when to
intervene is more efficient than considering it with the $\Delta
total_{uncer}$. Still, considering the derivative of the prediction
uncertainty could achieve a higher gain when the available resources are below
10%. Because when the number of available resources increases, more cases are
treated without ensuring that the $total_{uncer}$ is sufficient.

The proposed PrPM method results in Fig. 3 demonstrate a higher total gain
than the baselines. Accordingly, estimating the prediction uncertainty of the
underlying predictive model and analyzing the tradeoff between triggering an
intervention now versus later based on the adjusted gain and the opportunity
cost can optimize the performance of PrPM methods, hence, business processes.

###  5.4 Threats to Validity

Our method’s evaluation has an external validity threat (lack of
generalizability) because of its dependence on only one dataset. Accordingly,
the evaluation is vague and needs more experiments using other logs to be
followed up.

We assume that $CATE$ is accurate and will reduce the probability of negative
outcomes. Also, the intervention will be triggered only once for each ongoing
case. Accordingly, there is a threat to ecological reality where cases may be
treated more than once via multiple interventions. Moreover, the $CATE$ may
not represent the natural causal effect because of unobserved confounders.

The experiments’ setup used one feature encoding technique and did not discuss
the selection bias of the causal model due to variables that affect both the
outcome and the intervention. Utilizing other encoding and selection bias
techniques is a hint for future work to improve the performance of PrPM
methods.

##  6 Conclusion and Future Work

We presented a prescriptive monitoring method to determine if and when an
intervention should be triggered on ongoing cases to maximize a gain function.
The method leverages an ensemble model to estimate the probability of negative
case outcomes and the associated prediction uncertainty. They are combined
with a causal model to assess the effect of an intervention on the case
outcome. These estimates and a filtering and ranking method are embedded in a
resource allocator. It assigns resources to perform interventions to maximize
total gain, considering the tradeoff between intervening now or later. An
initial evaluation shows that taking into account the possible future states
of ongoing cases and the level of prediction uncertainty leads to a higher
gain than baselines that rely only on the current state of each case.

The proposed prescriptive method does not include constraints on when an
intervention may be triggered. In practice, it is often not possible to
trigger an intervention at any point in a case. For example, sending a second
loan offer to a customer who has already accepted an offer at another bank is
counter-productive. A direction for future work is to extend the approach with
temporal rules on the interventions and consider these rules in the
intervention policy.

The proposed method is limited to handling one type of intervention. Also, an
intervention is applied at most once to a case. Lifting these restrictions is
another direction for future work.

Reproducibility. The implementation and source code of the method, together
with instructions to reproduce the evaluation, can be found at:
https://github.com/mshoush/prescriptive-monitoring-uncertainty.

## References

  * [1] Abdar, M., Pourpanah, F., Hussain, S., Rezazadegan, D., Liu, L., Ghavamzadeh, M., Fieguth, P., Cao, X., Khosravi, A., Acharya, U.R., et al.: A review of uncertainty quantification in deep learning: Techniques, applications and challenges. Information Fusion 76, 243–297 (2021)
  * [2] Bozorgi, Z.D., Teinemaa, I., Dumas, M., La Rosa, M.: Prescriptive process monitoring for cost-aware cycle time reduction. ICPM (2021)
  * [3] Bozorgi, Z.D., Teinemaa, I., Dumas, M., La Rosa, M., Polyvyanyy, A.: Process mining meets causal machine learning: Discovering causal rules from event logs. In: ICPM. pp. 129–136. IEEE (2020)
  * [4] Di Francescomarino, C., Dumas, M., Maggi, F.M., Teinemaa, I.: Clustering-based predictive process monitoring. IEEE Trans. Serv. Comput. 12(6), 896–909 (2016)
  * [5] Dietterich, T.G., et al.: Ensemble learning. The handbook of brain theory and neural networks 2(1), 110–125 (2002)
  * [6] Fahrenkrog-Petersen, S.A., Tax, N., Teinemaa, I., Dumas, M., Leoni, M.d., Maggi, F.M., Weidlich, M.: Fire now, fire later: alarm-based systems for prescriptive process monitoring. Knowledge and Information Systems pp. 1–29 (2021)
  * [7] Gal, Y., Ghahramani, Z.: Dropout as a bayesian approximation: Representing model uncertainty in deep learning. In: ICML. pp. 1050–1059. PMLR (2016)
  * [8] Guo, R., Cheng, L., Li, J., Hahn, P.R., Liu, H.: A survey of learning causality with data: Problems and methods. ACM Comput. Surv. 53(4), 1–37 (2020)
  * [9] Hüllermeier, E., Waegeman, W.: Aleatoric and epistemic uncertainty in machine learning: An introduction to concepts and methods. Machine Learning (2021)
  * [10] Kim, J., Comuzzi, M., Dumas, M., Maggi, F.M., Teinemaa, I.: Encoding resource experience for predictive process monitoring. Decis. Support Syst. 153 (2022)
  * [11] Kratsch, W., Manderscheid, J., Röglinger, M., Seyfried, J.: Machine learning in business process monitoring: a comparison of deep learning and classical approaches used for outcome prediction. Bus. Inf. Syst. Eng. 63(3), 261–276 (2021)
  * [12] Maggi, F.M., Francescomarino, C.D., Dumas, M., Ghidini, C.: Predictive monitoring of business processes. In: Int. Conf. Adv. Comput. Appl. Springer (2014)
  * [13] Malinin, A., Prokhorenkova, L., Ustimenko, A.: Uncertainty in gradient boosting via ensembles. arXiv preprint arXiv:2006.10562 (2020)
  * [14] Metzger, A., Kley, T., Palm, A.: Triggering proactive business process adaptations via online reinforcement learning. In: BPM. pp. 273–290. Springer (2020)
  * [15] Pauwels, S., Calders, T.: Incremental predictive process monitoring: The next activity case. In: BPM conference. pp. 123–140. Springer (2021)
  * [16] Peterson, L.E.: K-nearest neighbor. Scholarpedia 4(2), 1883 (2009)
  * [17] Prokhorenkova, L., Gusev, G., Vorobev, A., Dorogush, A.V., Gulin, A.: Catboost: unbiased boosting with categorical features. NeurIPS 31 (2018)
  * [18] Shoush, M., Dumas, M.: Prescriptive process monitoring under resource constraints: A causal inference approach. 2021 3rd ICPM (2021)
  * [19] Stone, J.V.: Bayes’ rule: a tutorial introduction to bayesian analysis (2013)
  * [20] Teinemaa, I., Dumas, M., Rosa, M.L., Maggi, F.M.: Outcome-oriented predictive process monitoring: Review and benchmark. ACM TKDD 13(2), 1–57 (2019)
  * [21] Ustimenko, A., Prokhorenkova, L.: Sglb: Stochastic gradient langevin boosting. In: International Conference on Machine Learning. pp. 10487–10496. PMLR (2021)
  * [22] Verenich, I., Dumas, M., Rosa, M.L., Maggi, F.M., Teinemaa, I.: Survey and cross-benchmark comparison of remaining time prediction methods in business process monitoring. ACM TIST 10(4), 1–34 (2019)
  * [23] Weinzierl, S., Dunzer, S., Zilker, S., Matzner, M.: Prescriptive business process monitoring for recommending next best actions. In: ICPM. Springer (2020)
  * [24] Weytjens, H., Weerdt, J.D.: Learning uncertainty with artificial neural networks for improved remaining time prediction of business processes. In: International Conference on Business Process Management. pp. 141–157. Springer (2021)
  * [25] Xu, G., Duong, T.D., Li, Q., Liu, S., Wang, X.: Causality learning: A new perspective for interpretable machine learning. arXiv preprint arXiv:2006.16789 (2020)


Flow Equations In Arbitrary Signature

W. A. Sabra

Physics Department, American University of Beirut
Lebanon

Abstract

> We discuss general bosonic configurations of four-dimensional $N=2$
> supergravity coupled to vector multiplets in $(t,s)$ space-time. The
> supergravity theories with Euclidean and neutral signature are described by
> the so-called para-special Kähler geometry. For extremal solutions, we
> derive in a unified fashion, using the equations of motion, the flow
> equations for all space-time signatures. Demanding that the solutions with
> neutral and Euclidean signatures admit unbroken supersymmetry, we derive the
> constraints, known as the stabilisation equations, on the para-covariantly
> holomorphic sections expressed in terms of the adapted coordinates. The
> stabilisation equations expressed in terms of the para-complex sections
> imply generalised flow equations in terms of para-complex central charge.
> For Euclidean and neutral signature, it is demonstrated that solutions for
> either signs of gauge kinetic terms are mapped into each other via field
> redefinitions.

##  1 Introduction

The study of supersymmetric and BPS gravitational backgrounds of supergravity
theories in various space-time dimensions and signatures has been an active
area of research in recent years. Such backgrounds are of importance for a
better understanding of the non-perturbative structure of M theory, the
stringy duality symmetries as well as quantum geometry. The first step in the
systematic classification of solutions admitting supersymmetry in Einstein-
Maxwell theory was taken in [1] building on the earlier work of [2]. A result
obtained is that supersymmetric solutions with a time-like Killing vector are
the well known IWP solutions [3]. Later, a great deal of progress has been
made in the study of solutions of the more general Lorentzian $N=2$ four-
dimensional supergravity theories coupled with vector multiplets. Extensive
details on $N=2$ supergravity with vector multiplets and their associated
special Kähler geometry can be found in [4].

Supersymmetric solutions of the general $N=2$ supergravity theories were first
explored in [5]. A main result of [5] is that the scalar fields of the
supersymmetric solutions follow attractor equations with fixed points on the
horizon which are independent of their values at infinity. The solutions at
the near horizon are fully determined in terms of algebraic conditions, the
so-called stabilisation equations,

$i\left(\bar{Z}L^{I}-Z\bar{L}^{I}\right)=p^{I},\text{ \ \ \ \ \
}i\left(\bar{Z}M_{I}-\bar{Z}\bar{M}_{I}\right)=q_{I},$ (1.1)

where $Z$ is the central charge and $p^{I}$ and $q_{I}$ being the magnetic and
electric charges.

Static solutions with non-constant scalar fields were later constructed in
[6]. The analogue of the stationary IWP solutions in the general Lorentzian
four-dimensional $N=2$ supergravity were first analysed in [7]. The special
Kähler geometry of the scalar fields proved to be an essential ingredient in
the analysis and the study of solutions admitting supersymmetry. A
rederivation of these solutions for either sign of the gauge fields kinetic
terms was performed in [8] employing spinorial geometry method as outlined in
[9]. The solutions with the standard sign of gauge kinetic terms are given by

$ds^{2}=-\frac{1}{|\beta|^{2}}\left(d\tau+\sigma\right)^{2}+|\beta|^{2}\left(dx^{2}+dy^{2}+dz^{2}\right),$
(1.2)

with the conditions

$\displaystyle i\left(\bar{\beta}L^{I}-\beta\bar{L}^{I}\right)$
$\displaystyle=$ $\displaystyle\tilde{H}^{I},\text{ \ \ \ \ \
}i\left(\bar{\beta}M_{I}-\beta\bar{M}_{I}\right)=H_{I},$ (1.3) $\displaystyle
d\sigma$ $\displaystyle=$
$\displaystyle\ast\left(H_{I}d\tilde{H}^{I}-\tilde{H}^{I}dH_{I}\right)$ (1.4)

where $\tilde{H}^{I}$ and $H_{I}$ are harmonic functions on the flat
coordinate space $\mathbb{R}^{3}$ and $\beta$ is a complex function. Here
$\ast$ is the Hodge star operator on flat $\mathbb{R}^{3}.$ Note that
solutions with non-canonical sign of gauge fields kinetic term have a space-
like Killing vector with the harmonic functions and the $\ast$ are on
$\mathbb{R}^{1,2}$[8]. Composite BPS configurations were considered in [10].
There, the correspondence between BPS states in type II string theory
compactified on Calabi-Yau manifolds and BPS solutions of the four-dimensional
$N=2$ supergravity were investigated.

A thorough and detailed study of four-dimensional Euclidean supersymmetric
theories has recently been conducted in [12, 13, 14, 15]. In particular,
Euclidean $N=2$ supergravity theories were obtained as dimensional reductions
of five-dimensional $N=2$ supergravity theories [16] on a time-like circle.
The couplings of the Euclidean theory were found to be described in terms of a
para-special Kähler geometry. More on the construction and the study of $N=2$
Euclidean supergravity theories can be found in [17, 18].

A large class of Lorentzian four-dimensional $N=2$ supergravities are
obtainable from five-dimensional $N=2$ supergravity theory with
$\left(1,4\right)$ signature via a reduction on a space-like circle [16]. In
turn, the five-dimensional theories are reductions of $\left(1,10\right)$
supergravity on a Calabi-Yau threefold $CY_{3}$ [19]111We use the notation
$(t,s)$ for space-time signature where $t$ is the number of time dimensions
and $s$ is the number of spatial dimensions.. Four-dimensional supergravity
theories with various space-time signatures were obtained in [20] by reducing
the eleven-dimensional supergravity theories [21, 22] on a $CY_{3}$ and a
circle. A rigorous study of the four-dimensional theories in various space-
time signatures and the classification of their four-dimensional $N=2$
supersymmetry algebras have been performed in [23].

In the present work we are mainly interested in the extension of the results
obtained for the solutions of the Lorentzian supergravity theories to all
four-dimensional supergravities with vector multiplets in various space-time
signatures. Our work is organized as follows. In section two we discuss
general solutions for all signatures following on the analysis of [24] . We
derive flow equations for extremal solutions through the analysis of the
equations of motion. In section three, using spinorial geometry, we analyse
solutions for a specific choice of spinor orbit and construct the
corresponding IWP-like Euclidean and neutral signature solutions. We derive
the stabilisation equations in terms of the so-called adapted coordinates and
also in terms para-complex variables and derive generalised flow equations. We
summarize and conclude in section four.

##  2 General Solutions and Flow equations

Ignoring the hypermultiplets, the $N=2$ supergravity theories have the bosonic
part of their Lagrangian given by

$\mathbf{e}^{-1}\mathcal{L}=R-2g_{A\bar{B}}\partial_{\mu}z^{A}\partial^{\mu}\bar{z}^{B}-\frac{\kappa^{2}}{2}F^{I}\cdot\left(\mathop{\mathrm{I}m}\mathcal{N}_{IJ}F^{J}+\mathop{\mathrm{R}e}\mathcal{N}_{IJ}\tilde{F}^{J}\right)\;.$
(2.1)

For theories with Lorentzian signature, the fields $z^{A}$ are $n$ complex
scalar fields, $F^{I}$ and $\tilde{F}^{I}$ $(I=0,...,n)$ are two-forms
representing the gauge field-strengths and their duals and $\kappa^{2}=\pm 1.$
The Lorentzian theories are described in terms of special Kähler geometry
which can be formulated in terms of the covariantly holomorphic sections

$V=\left(\begin{array}[]{c}L^{I}\\\ M_{I}\end{array}\right),\text{ \ \ \
}I=0,...,n,\text{ \ \ }$ (2.2)

satisfying

$\displaystyle i\langle V,\bar{V}\rangle$ $\displaystyle=$ $\displaystyle
i\left(\bar{L}^{I}M_{I}-L^{I}\bar{M}_{I}\right)=1,$ (2.3)
$\displaystyle\text{\ }D_{\bar{A}}V$ $\displaystyle=$
$\displaystyle\left(\partial_{\bar{A}}-\frac{1}{2}\partial_{\bar{A}}K\right)V=0.$
(2.4)

where $K$ is the Kähler potential. Defining

$U_{A}=D_{A}V=\left(\partial_{A}+\frac{1}{2}\partial_{A}K\right)V,$ (2.5)

we have the useful relations

$M_{I}=\mathcal{N}_{IJ}L^{J},\ \ \ \
D_{A}M_{I}=\mathcal{\bar{N}}_{IJ}D_{A}L^{J}$ (2.6)

and

$\displaystyle\langle D_{A}V,\bar{V}\rangle$ $\displaystyle=$
$\displaystyle\langle D_{A}V,V\rangle=\langle D_{A}V,D_{B}V\rangle=0,$
$\displaystyle g_{A\bar{B}}$ $\displaystyle=$ $\displaystyle-i\langle
D_{A}V,D_{\bar{B}}\bar{V}\rangle.$ (2.7)

The theories with Euclidean and neutral signature are formulated in terms of
para-special Kähler geometry. Roughly speaking, the equations of para special
Kähler geometry can be obtained from those of special Kähler geometry by
replacing $i$ with $\iota,$ with $\bar{\iota}=-\iota$ and $\ \iota^{2}=1$.

Following the analysis of [24] for static solutions, we start by considering
the following metrics

$ds^{2}=\epsilon_{1}e^{2U}d\tau^{2}+e^{-2U}\left(\epsilon_{0}{l}^{4}d\rho^{2}+l^{2}ds^{2}(\mathcal{M}_{2})\right),$
(2.8)

where $U$ and ${l}$ are functions of $\rho$ only and $\mathcal{M}_{2}$ is a
two-dimensional Einstein manifold with Ricci tensor

$\mathrm{R}_{ab}=kh_{ab}.$ (2.9)

The two-dimensional space $\mathcal{M}_{2}$ can be represented by the metric

$ds^{2}(\mathcal{M}_{2})=h_{ab}dx^{a}dx^{b}=\epsilon_{2}d\theta^{2}+\epsilon_{3}f^{2}d\phi^{2}.$
(2.10)

Here $\epsilon_{i}=\pm 1,$ $i=0,1,2,3.$ Euclidean and neutral signature
solutions have $\epsilon_{0}\epsilon_{1}\epsilon_{2}\epsilon_{3}=\epsilon=1,$
while Lorentzian solutions have
$\epsilon_{0}\epsilon_{1}\epsilon_{2}\epsilon_{3}=\epsilon=-1.$ Moreover, the
function $f$ can take the values $\sin\theta$, $\sinh\theta$ or $1$. The
manifold $\mathcal{M}_{2}$ can be of spherical $\left(k=1\right)$, hyperbolic
$\left(k=-1\right)$ and flat topology $\left(k=0\right)$ depending on the
choice of values of $\epsilon_{2},\epsilon_{3}$ and $f$.

The Einstein equations of motion derived from the action (2.1) are given by

$R_{\mu\nu}=2g_{A\bar{B}}\partial_{\mu}z^{A}\partial_{\nu}\bar{z}^{B}+2\kappa^{2}\mathop{\mathrm{I}m}\mathcal{N}_{IJ}\left(F{}_{\mu\lambda}^{I}F^{J}{}_{\nu}{}^{\lambda}-{\frac{1}{4}}g_{\mu\nu}F^{I}{}_{\alpha\beta}F^{J}{}^{\alpha\beta}\right).$
(2.11)

The non-vanishing components of the Ricci-tensor of the metric (2.8) are

$\displaystyle R_{\rho\rho}$ $\displaystyle=2\left(\partial_{\rho}\log
l\right)^{2}-2\partial_{\rho}^{2}\log
l+\partial_{\rho}^{2}U-2\left(\partial_{\rho}U\right)^{2},$ $\displaystyle
R_{\tau\tau}$
$\displaystyle=-\frac{\epsilon_{0}\epsilon_{1}}{l^{4}}e^{4U}\partial_{\rho}^{2}U,$
$\displaystyle R_{ab}$
$\displaystyle=\left(k-\frac{\epsilon_{0}}{l^{2}}\left(\partial_{\rho}^{2}\log
l-\partial_{\rho}^{2}U\right)\right)h_{ab}\ .$ (2.12)

Thus one obtains from (2.11)

$\displaystyle\left(\partial_{\rho}U\right)^{2}+g_{A\bar{B}}\partial_{\rho}z^{A}\partial_{\rho}\bar{z}^{B}-e^{2U}V$
$\displaystyle=\left(\partial_{\rho}\log{l}\right)^{2}-\partial_{\rho}^{2}\log{l,}$
$\displaystyle\partial_{\rho}^{2}U$ $\displaystyle=e^{2U}V,$
$\displaystyle\partial_{\rho}^{2}\log l$ $\displaystyle=k\epsilon_{0}l^{2},$
(2.13)

where $V$ is given by

$V=-\frac{\epsilon_{1}\kappa^{2}}{2}\mathop{\mathrm{I}m}\mathcal{N}_{IJ}\left(e^{-4U}F^{I}{}_{\tau\rho}F^{J}{}_{\tau\rho}{}-\frac{\epsilon}{f^{2}}F^{I}{}_{\theta\phi}F_{\theta\phi}^{J}\right).$
(2.14)

The equations (2.13) simplify further if we consider solutions with

$\left(\partial_{\rho}\log l\right)^{2}-\partial_{\rho}^{2}\log l=c^{2},$
(2.15)

where the constant $c$ can be regarded as a non-extremality parameter. Then (
2.15) together with (2.13) imply the three possibilities $l=\frac{c}{\sinh
c\rho},$ $l=\frac{c}{\cosh c\rho}$ and $l=e^{c\rho}$ for ${\
\epsilon_{0}}k=1,$ ${\epsilon_{0}}k=-1$ and $k=0,$ respectively. The Einstein
equations of motion, for various topologies of $\mathcal{M}_{2}$, reduce to

$\displaystyle\left(\partial_{\rho}U\right)^{2}+g_{A\bar{B}}\partial_{\rho}z^{A}\partial_{\rho}\bar{z}^{B}-{e}^{2U}{V}$
$\displaystyle={c}^{2}{,}$ $\displaystyle\partial_{\rho}^{2}U$
$\displaystyle={e^{2U}V.}$ (2.16)

After solving for the gauge fields, the potential $V$ takes the form

$V=\frac{\epsilon_{1}\kappa^{2}}{2}\left[\mathop{\mathrm{I}m}\mathcal{N}^{MI}\left(\mathop{\mathrm{R}e}\mathcal{N}_{MN}q_{I}p^{N}+\mathop{\mathrm{R}e}\mathcal{N}_{IL}p^{L}\left(q_{M}-\mathop{\mathrm{R}e}\mathcal{N}_{MN}p^{N}\right)-q_{I}q_{M}\right)+\epsilon\mathop{\mathrm{I}m}\mathcal{N}_{IJ}p^{I}p^{J}\right]$
(2.17)

where $p$ and $q$ represent the magnetic and electric charges. The potential
$V$ can be expressed as

$V=-\epsilon\epsilon_{1}\kappa^{2}\left(\left|Z\right|^{2}+g^{A\bar{B}}D_{A}ZD_{\bar{B}}\bar{Z}\right)$
(2.18)

with $Z$ being the central charge given by

$Z=L^{I}q_{I}-M_{I}p^{I},$ (2.19)

where $L^{I}$ and $M_{I}$ are para-complex for theories with neutral and
Euclidean signature and complex for Lorentzian theories. For extremal
spherically symmetric Lorentzian theories with ${\kappa^{2}=-1,}$
$\epsilon=\epsilon_{1}=-1,$ one reproduces the results of [24] where two
gradient flow equations were obtained

$\displaystyle\partial_{\rho}U$ $\displaystyle=$ $\displaystyle\pm
e^{U}\left|Z\right|,$ (2.20) $\displaystyle\partial_{\rho}z^{A}$
$\displaystyle=$ $\displaystyle\pm
2e^{U}g^{A\bar{B}}\partial_{\bar{B}}\left|Z\right|.$ (2.21)

These first order differential equations were also obtained in [5] using
supersymmetry. It can be shown that the flow equations (2.20) and ( 2.21)
imply all the equations of motion. The plus sign possibility on the right hand
side of (2.20) and (2.21) leads to physically unacceptable solutions [10]. The
same analysis holds for extremal Lorentzian solutions with non-canonical sign
of gauge kinetic terms where we have ${\kappa^{2}=1,}$
$\epsilon=\epsilon_{0}=-1.$ For extremal Euclidean and neutral solutions,
taking ${\epsilon_{i}=-\kappa}^{2}=\epsilon=1,$ one also gets the relation

$V=\left|Z\right|^{2}+g^{A\bar{B}}D_{A}ZD_{\bar{B}}\bar{Z}$ (2.22)

and as such we obtain the same flow equations expressed in terms of the para-
complex central charge.

For $c^{2}\neq 0,$ one obtains non-extremal solutions. A general analysis for
the study of non-supersymmetric solutions in arbitrary dimensions and metric
signatures was recently given in [25]. We must note that the non-extremal
metric ansatz (2.8) is related by a coordinate transformation to that used in
[25] and [26].

##  3 Supersymmetric Euclidean and Neutral Solutions

In this section, we shall consider Euclidean as well as neutral signature
solutions with either sign of the gauge kinetic term in a unified setting. A
systematic analysis of Euclidean and neutral solutions for Einstein-Maxwell
theory were recently considered in [27]. The Killing spinor equations for
these theories are given by [20]:

$\displaystyle\left(\nabla_{\mu}-{\frac{1}{2}}A_{\mu}\gamma_{5}+\frac{\kappa}{4}\gamma\cdot
F^{I}\left(\mathrm{Im}L^{J}+\gamma_{5}\mathrm{Re}L^{J}\right)(\mathrm{Im}\mathcal{N})_{IJ}\gamma_{\mu}\right)\varepsilon$
$\displaystyle=0,$ (3.1)
$\displaystyle\frac{\kappa}{2}(\mathrm{Im}\mathcal{N})_{IJ}\gamma\cdot
F^{J}\left[\mathrm{Im}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})+\gamma_{5}\mathrm{Re}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})\right]\varepsilon+\gamma^{\mu}\partial_{\mu}\left(\mathrm{Re}z^{A}-\gamma_{5}\mathrm{Im}z^{A}\right)\varepsilon\,$
$\displaystyle=0,\ $ (3.2)

with $\kappa=i$ or $\kappa=-1.$ Here $A$ is the $U(1)$ Kähler connection given
by

$A=-\frac{\iota}{2}(\partial_{A}Kdz^{A}-\partial_{\bar{A}}Kd\bar{z}^{A}).$

We start our analysis by considering metrics of the form

$ds^{2}=2\left(\mathbf{e}^{1}\mathbf{e}^{\bar{1}}+\eta^{2}\mathbf{e}^{2}\mathbf{e}^{\bar{2}}\right),$
(3.3)

where $\eta^{2}=-1$ for solutions with $(2,2)$ space-time signature and
$\eta^{2}=1$ for Euclidean solutions. Using spinorial geometry, we shall
analyze equations (3.1) and (3.2) for the spinor orbit $\varepsilon=\lambda
1+\sigma e_{1}$, with real functions $\lambda$ and $\sigma$. The action of the
Dirac matrices on the Dirac spinors is given by

$\gamma_{2}=\sqrt{2}\eta i_{e^{2}},\ \ \ \gamma_{\bar{2}}=\sqrt{2}\eta
e^{2}\wedge,\ \ \ \gamma_{1}=\sqrt{2}i_{e^{1}},\ \
\gamma_{\bar{1}}=\sqrt{2}e^{1}\wedge$ (3.4)

where $\\{1,e_{1},e_{2},e_{12}=e_{1}\wedge e_{2}\\}$ is the basis space of
forms on $\mathbb{R}^{2}.$ Plugging $\varepsilon=\lambda 1+\sigma e_{1}$ in
(3.1) and using (3.4), we obtain the following geometric conditions on the
spin connection

$\displaystyle\omega_{1\bar{1}}$ $\displaystyle=$
$\displaystyle\left(\partial_{2}\log\frac{\lambda}{\sigma}-A_{2}\right)\mathbf{e}^{2}-\partial_{1}\log\lambda\sigma\mathbf{e}^{1}-\left(\partial_{\bar{2}}\log\frac{\lambda}{\sigma}-A_{\bar{2}}\right)\mathbf{e}^{\bar{2}}+\partial_{\bar{1}}\log\lambda\sigma\mathbf{\
e}^{\bar{1}},$ $\displaystyle\omega_{2\bar{2}}$ $\displaystyle=$
$\displaystyle\eta^{2}\left(\left(\partial_{\bar{1}}\log\frac{\lambda}{\sigma}-A_{\bar{1}}\right)\mathbf{e}^{\bar{1}}-\left(\partial_{1}\log\frac{\lambda}{\sigma}-A_{1}\right)\mathbf{e}^{1}+\partial_{2}\log\lambda\sigma\mathbf{e}^{2}-\partial_{\bar{2}}\log\lambda\sigma\mathbf{e}^{\bar{2}}\right),$
$\displaystyle\omega_{12}$ $\displaystyle=$
$\displaystyle\kappa^{2}\left(\partial_{2}\log\lambda^{2}-A_{2}\right)\mathbf{e}^{1}+\eta^{2}\left(\partial_{1}\log\lambda^{2}-A_{1}\right)\mathbf{e}^{\bar{2}},\text{
\ \ \ }$ $\displaystyle\text{\ }\omega_{\bar{1}2}$ $\displaystyle=$
$\displaystyle\kappa^{2}\left(\partial_{2}\log\sigma^{2}+A_{2}\right)\mathbf{e}^{\bar{1}}+\eta^{2}\left(\partial_{\bar{1}}\log\sigma^{2}+A_{\bar{1}}\right)\mathbf{e}^{\bar{2}},$
(3.5)

together with

$\displaystyle\mathrm{Im}\mathcal{N}_{IJ}L_{-}^{I}\left(F_{1\bar{1}}^{J}-\eta^{2}F_{2\bar{2}}^{J}\right)$
$\displaystyle=$
$\displaystyle-\frac{\kappa\sqrt{2}}{\sigma\lambda}\left(\partial_{1}+A_{1}\right)\sigma^{2},$
$\displaystyle\mathrm{Im}\mathcal{N}_{IJ}L_{+}^{I}\left(F_{1\bar{1}}^{J}+\eta^{2}F_{2\bar{2}}^{J}\right)$
$\displaystyle=$
$\displaystyle\frac{\sqrt{2}\bar{\kappa}}{\sigma\lambda}\left(\partial_{1}-A_{1}\right)\lambda^{2},$
$\displaystyle\mathrm{Im}\mathcal{N}_{IJ}L_{-}^{I}F_{\bar{1}2}^{J}$
$\displaystyle=$
$\displaystyle\frac{\kappa}{\sqrt{2}\lambda\sigma}\left(\partial_{2}+A_{2}\right)\sigma^{2},$
$\displaystyle\mathrm{Im}\mathcal{N}_{IJ}L_{+}^{I}F_{12}^{J}$ $\displaystyle=$
$\displaystyle-\frac{\kappa}{\sqrt{2}\lambda\sigma}\left(\partial_{2}-A_{2}\right)\lambda^{2}.$
(3.6)

where we have expressed our relations in terms of the so-called adapted
coordinates (real light-cone coordinates) [15] defined as
$X_{\pm}=\mathop{\mathrm{R}e}X\pm\mathop{\mathrm{I}m}X.$ The analysis of (3.2)
gives

$\displaystyle\partial_{1}z_{-}^{A}$ $\displaystyle=$
$\displaystyle\frac{\kappa\sigma}{\sqrt{2}\lambda}\mathrm{Im}\mathcal{N}_{IJ}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})_{-}\left(F_{1\bar{1}}^{J}-\eta^{2}F_{2\bar{2}}^{J}\right),$
$\displaystyle\partial_{\bar{1}}z_{+}^{A}$ $\displaystyle=$
$\displaystyle\frac{\kappa\lambda}{\sqrt{2}\sigma}\mathrm{Im}\mathcal{N}_{IJ}\text{\
}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})_{+}\left(F_{1\bar{1}}^{J}+\eta^{2}F_{2\bar{2}}^{J}\right),$
$\displaystyle\partial_{2}z_{+}^{A}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}\kappa\lambda}{\sigma}\mathrm{Im}\mathcal{N}_{IJ}\text{\
}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})_{+}F_{12}^{J},$
$\displaystyle\partial_{2}z_{-}^{A}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}\kappa\sigma}{\lambda}\mathrm{Im}\mathcal{N}_{IJ}(D_{\bar{B}}{\bar{L}}^{I}g^{A\bar{B}})_{-}F_{\bar{1}2}^{J}.$
(3.7)

One also obtains

$\left(\partial_{\bar{1}}+\kappa^{2}\partial_{1}\right)\lambda=\left(\partial_{\bar{1}}+\kappa^{2}\partial_{1}\right)\sigma=\left(\partial_{1}+\kappa^{2}\partial_{\bar{1}}\right)z_{\pm}^{A}=\left(A_{\bar{1}}+\kappa^{2}A_{1}\right)=0.$
(3.8)

The torsion free condition implies that

$\displaystyle d\left(\mathbf{e}^{1}-\kappa^{2}\mathbf{e}^{\bar{1}}\right)$
$\displaystyle=$
$\displaystyle-d\log\lambda\sigma\wedge\left(\mathbf{e}^{1}-\kappa^{2}\mathbf{e}^{\bar{1}}\right)$
$\displaystyle d\mathbf{e}^{2}$ $\displaystyle=$
$\displaystyle-d\log\lambda\sigma\wedge\mathbf{e}^{2}.$ (3.9)

All the above conditions indicates that we can introduce the coordinates
$\tau,$ $x,$ $y$, and $z,$ and write

$\mathbf{e}^{1}=-\frac{\kappa}{\sqrt{2}}\left(\lambda\sigma(d\tau+\phi)+\frac{1}{\lambda\sigma}idx\right),\text{
\ \ \ }\mathbf{e}^{2}=\frac{1}{\sqrt{2}\lambda\sigma}\left(dy+idz\right)\ .$
(3.10)

The metric solution obtained from (3.3) is independent of the coordinate
$\tau$, and is given by

$ds^{2}=\left(\lambda\sigma\right)^{2}(d\tau+\phi)^{2}+\frac{1}{\left(\lambda\sigma\right)^{2}}\left(dx^{2}+\eta^{2}\left(dy^{2}+dz^{2}\right)\right)\
$ (3.11)

with

$d\phi=\frac{2}{\left(\lambda\sigma\right)^{2}}\ast_{3}\left(d\log\frac{\lambda}{\sigma}-A\right).$
(3.12)

Here $\ast_{3}$ is the Hodge dual defined with respect to the metric
$\left(dx^{2}+\eta^{2}\left(dy^{2}+dz^{2}\right)\right).$ Our orientation is
such that $\epsilon_{1\bar{1}2\bar{2}}=\eta^{2}.$ Using ( 3.6), (3.7), (3.10),
(3.12) and the relations of the para-special Kähler geometry

$\displaystyle g^{A\bar{B}}D_{A}L^{I}D_{\bar{B}}\bar{L}^{J}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\mathop{\mathrm{I}m}\mathcal{N}\right)^{IJ}-\bar{L}^{I}L^{J},$
(3.13) $\displaystyle dM_{I}$ $\displaystyle=$
$\displaystyle\mathcal{\bar{N}}_{IJ}dL^{J}+2A\mathop{\mathrm{I}m}\mathcal{N}_{IJ}L^{J},$
(3.14)

we obtain after some calculation

$\displaystyle F^{I}$ $\displaystyle=$ $\displaystyle
d\left[\left(\sigma^{2}L_{+}^{I}-\kappa^{2}\lambda^{2}L_{-}^{I}\right)(d\tau+\phi)\right]+\ast
d\left(\frac{\kappa^{2}L_{-}^{I}}{\sigma^{2}}+\frac{L_{+}^{I}}{\lambda^{2}}\right),$
(3.15) $\displaystyle G_{I}$ $\displaystyle=$ $\displaystyle
d\left[\left(\sigma^{2}M_{I+}-\kappa^{2}\lambda^{2}M_{I-}\right)(d\tau+\phi)\right]+\ast
d\left(\kappa^{2}\frac{M_{I-}}{\sigma^{2}}+\frac{M_{I+}}{\lambda^{2}}\right),$
(3.16)

where
$G_{I}=\mathop{\mathrm{R}e}\mathcal{N}_{IJ}F^{I}+\mathop{\mathrm{I}m}\mathcal{N}_{IJ}\tilde{F}^{I}$.
The Bianchi identities together with Maxwell equations

$dF^{I}=dG_{I}=0,$ (3.17)

give

$\nabla^{2}\left(\frac{L_{+}^{I}}{\lambda^{2}}+\frac{\kappa^{2}L_{-}^{I}}{\sigma^{2}}\right)=0\text{,
\ \ \ \ \
}\nabla^{2}\left(\frac{M_{+I}}{\lambda^{2}}+\frac{\kappa^{2}M_{-I}}{\sigma^{2}}\right)=0,$
(3.18)

where
$\nabla^{2}=\partial_{x}^{2}+\eta^{2}\left(\partial_{y}^{2}+\partial_{z}^{2}\right),$
thus one obtains the stabilisation equations

$\frac{L_{+}^{I}}{\lambda^{2}}+\frac{\kappa^{2}L_{-}^{I}}{\sigma^{2}}=\tilde{H}^{I}\text{,
\ \ \ \ \
}\frac{M_{+I}}{\lambda^{2}}+\frac{\kappa^{2}M_{-I}}{\sigma^{2}}=H_{I}.$ (3.19)

Here $\tilde{H}^{I}$ and $H_{I}$ are harmonic functions in the three-
dimensional space with metric $dx^{2}+\eta^{2}\left(dy^{2}+dz^{2}\right)$.
Furthermore, (3.19) imply the relations

$\displaystyle A$ $\displaystyle=$
$\displaystyle-\frac{\kappa^{2}}{2}\left(\lambda\sigma\right)^{2}\left(H_{I}d\tilde{H}^{I}-\tilde{H}^{I}dH_{I}\right)+d\log\frac{\lambda}{\sigma},$
(3.20) $\displaystyle d\phi$ $\displaystyle=$
$\displaystyle\kappa^{2}\ast_{3}\left(H_{I}d\tilde{H}^{I}-\tilde{H}^{I}dH_{I}\right)\
.$ (3.21)

In what follows, we recast the solutions in terms of para-complex functions.
For definiteness, we consider the case with $\kappa^{2}=-1$ and introduce the
para-complex functions

$\mathbb{Y}^{I}=\bar{\beta}L^{I},\ \ \ \ \ \mathbb{F}_{I}=\bar{\beta}M_{I},$
(3.22)

where $L^{I},$ $M_{I}$ and $\beta$ are all para-complex function. The metric
(3.11) then takes the form

$ds^{2}=\frac{1}{|\beta|^{2}}(d\tau+\omega)^{2}+|\beta|^{2}\left(dx^{2}+\eta^{2}\left(dy^{2}+dz^{2}\right)\right).$

The function $\beta$ can be related to $\lambda$ and $\sigma$ as

$\beta=\frac{1}{2\lambda^{2}\sigma^{2}}\left[\left(\lambda^{2}+\sigma^{2}\right)+\iota\left(\lambda^{2}-\sigma^{2}\right)\right].$
(3.23)

The symplectic constraint (2.3) implies

$e^{-2U}=|\beta|^{2}=\iota(\mathbb{\bar{Y}}^{I}\mathbb{F}_{I}-\mathbb{Y}^{I}\mathbb{\bar{F}}_{I})$
(3.24)

and the stabilisation equations (3.19) take the familiar form

$\iota\left(\mathbb{Y}^{I}-\mathbb{\bar{Y}}^{I}\right)=\tilde{H}^{I}\text{ , \
\ \ \ \ \ \ \ }\iota\left(\mathbb{F}_{I}-\mathbb{\bar{F}}_{I}\right)=H_{I}.$
(3.25)

which can be written in a more compact form

$2\mathop{\mathrm{I}m}\left(\bar{\beta}V\right)=H\text{, \ \ \ \ \ \ \ \
}H=\left(\begin{array}[]{c}\tilde{H}^{I}\\\ H_{I}\end{array}\right).$ (3.26)

Using the relations (2.7) (for para-complex variables) together with

$dV=D_{A}Vdz^{A}+\iota AV,$

we obtain from (3.26) the relations

$\displaystyle Zd\rho$ $\displaystyle=$ $\displaystyle\left(d-\iota
A\right)\beta,$ (3.27) $\displaystyle\partial_{\rho}z^{A}\ \ \ $
$\displaystyle=$
$\displaystyle-e^{U}g^{A\bar{B}}e^{\iota\gamma}D_{\bar{B}}\bar{Z}$ (3.28)

where

$\beta=e^{-U}e^{\iota\gamma}=e^{-U}\left(\cosh\gamma+\iota\sinh\gamma\right).$
(3.29)

Note that the relation (3.27) implies

$\displaystyle\partial_{\rho}U$ $\displaystyle=$
$\displaystyle-\mathop{\mathrm{R}e}\left(\frac{Z}{\beta}\right),$ (3.30)
$\displaystyle\left(d\gamma-A\right)$ $\displaystyle=$
$\displaystyle\mathop{\mathrm{I}m}\left(\frac{Z}{\beta}\right)d\rho.$ (3.31)

The flow equations discussed in the previous section are obtained by setting

$A=d\gamma.$ (3.32)

Inspecting the stabilisation equations as expressed in terms of the adapted
coordinates in (3.19) and following on the arguments presented in [18], it can
be seen that the solution corresponding to $\kappa^{2}=-1$ can be mapped into
the solution with $\kappa^{2}=1,$ via the symmetry

$\displaystyle M_{-I}^{\prime}$ $\displaystyle=$ $\displaystyle-
L_{-}^{I},\text{ \ \ \ }M_{+I}^{\prime}=L_{+}^{I},$ (3.33) $\displaystyle
L_{-}^{\prime I}$ $\displaystyle=$ $\displaystyle-M_{-I},\text{ \ \ \
}L_{+}^{\prime I}=M_{+I},$ (3.34)

together with interchanging magnetic with electric charges. Here the prime
coordinates represent the fields in $\kappa^{2}=1$ theory. Note that in terms
of para-complex sections, this transformation reads

$\left(\begin{array}[]{c}L^{\prime I}\\\
M_{I}^{\prime}\end{array}\right)=\iota\left(\begin{array}[]{cc}\mathrm{0}&\mathrm{1}\\\
\mathrm{1}&\mathrm{0}\end{array}\right)\left(\begin{array}[]{c}L^{I}\\\
M_{I}\end{array}\right)$ (3.35)

which together with electric-magnetic duality imply

$Z^{\prime}=-\iota Z.$ (3.36)

Note also that

$\iota\left(\bar{L}^{\prime I}M_{I}^{\prime}-L^{\prime
I}\bar{M}_{I}^{\prime}\right)=\iota\left(\bar{L}^{I}M_{I}-L^{I}\bar{M}_{I}\right)=1.$
(3.37)

It is also evident from (3.19) that solutions for $\kappa^{2}=1$ (with primed
variables) can be obtained form the solutions corresponding to
$\kappa^{2}=-1,$ via the field redefinitions

$L_{+}^{\prime I}=L_{+}^{I},\text{\ \ \ \ \ }L_{-}^{\prime
I}=-L_{-}^{I},\text{ \ \ \ \ }M_{+I}^{\prime}=M_{+I},\text{ \ \
}M_{-I}^{\prime}=-M_{-I}.$ (3.38)

This is the transformation discussed in [23]. In terms of the para-complex
variables this is given by

$\left(\begin{array}[]{c}L^{\prime I}\\\
M_{I}^{\prime}\end{array}\right)=\iota\left(\begin{array}[]{cc}\mathrm{1}&\mathrm{0}\\\
\mathrm{0}&\mathrm{1}\end{array}\right)\left(\begin{array}[]{c}L^{I}\\\
M_{I}\end{array}\right)$ (3.39)

and it implies that

$Z^{\prime}=\iota Z.$ (3.40)

In this case we have

$\iota\left(\bar{L}^{\prime I}M_{I}^{\prime}-L^{\prime
I}\bar{M}_{I}^{\prime}\right)=-\iota\left(\bar{L}^{I}M-L^{I}\bar{M}_{I}\right)=-1.$
(3.41)

The transformation (3.39) is local and connects the Lagrangians of theories
with either sign of gauge kinetic terms [23]. As explained in [23], the
transformation is induced by an isomorphism between the two $N=2$
superalgebras arising from the dimensional reduction of the five-dimensional
supersymmetry algebras.

As examples, we construct some explicit solutions corresponding to the minimal
models defined by

$M_{I}=Q_{IJ}L^{J}$ (3.42)

with $Q_{IJ}$ being the elements of a constant (para)-complex symmetric
matrix. The stabilisation equations (3.25) lead to the solutions

$\mathbb{Y}^{K}=\bar{\beta}L^{K}=-\frac{1}{2}\mathop{\mathrm{I}m}Q^{IK}\left(\bar{Q}_{IJ}\tilde{H}^{J}-H_{I}\right),\
$

We then obtain from (3.24)

$e^{-2U}=\frac{1}{2}H^{t}XH$ (3.43)

where

$X=\left(\begin{array}[]{cc}\mathop{\mathrm{R}e}Q\left(\mathop{\mathrm{I}m}Q\right)^{-1}\mathop{\mathrm{R}e}Q-\mathop{\mathrm{I}m}Q&-\mathop{\mathrm{R}e}Q\left(\mathop{\mathrm{I}m}Q\right)^{-1}\\\
-\left(\mathop{\mathrm{I}m}Q\right)^{-1}\mathop{\mathrm{R}e}Q&\left(\mathop{\mathrm{I}m}Q\right)^{-1}\end{array}\right).$
(3.44)

Next we can construct supersymmetric solutions for theories with cubic
prepotential which can be obtained from eleven dimensions via a
compactification on a $CY_{3}$ and a circle. These theories can be defined by
the relations

$M_{0}=-C_{IJK}\frac{L^{I}L^{J}L^{K}}{\left(L^{0}\right)^{2}},\text{ \ \ \ \
}M_{I}=3C_{IJK}\frac{L^{J}L^{K}}{L^{0}}.$ (3.45)

The stabilization equations, for the specific choice $\tilde{H}^{0}=H_{I}=0,$
can be solved by

$\mathbb{Y}^{I}=\frac{1}{2}\iota\tilde{H}^{I},\text{ \ \ \ \ \ \ \
}\mathbb{Y}^{0}=\frac{1}{2}\text{\ \
}\sqrt{-\frac{C_{IJK}\tilde{H}^{I}\tilde{H}^{J}\tilde{H}^{K}}{H_{0}}}$ (3.46)

and we obtain

$e^{-2U}=2\text{\ }\sqrt{-C_{IJK}H_{0}H^{I}H^{J}H^{K}}.$ (3.47)

##  4 Final Remarks

In this paper we have considered solutions to four-dimensional supergravity
theories in arbitrary space-time signature. The first order differential flow
equations, in terms of the (para)-complex central charge, for the metric and
scalar fields were derived via the analysis of the equations of motion. The
spinorial geometry methods were employed in the analysis of the Killing spinor
equations and a class of supersymmetric Euclidean and neutral IWP-like
solutions were derived. The solutions and in particular the stabilisation
equations characterizing the evolution of the scalar fields were expressed in
terms of adapted coordinates (real light-cone coordinates). The stabilisation
conditions expressed in terms of para-complex variables were used to find the
generalised flow equations characterizing our supersymmetric solutions. For
different signs of the gauge kinetic terms, the solutions were related to each
other via a field redefinition in line with the discussions in [18, 23].

BPS states in type II theory compactified on $CY_{3}$ can be described in
terms of branes wrapped on various supersymmetric cycles. To match the two
spectrums, multi-centered composite solutions, corresponding to multi-centered
harmonic functions, must be taken into consideration [10]. Some explicit
solutions of such composite solutions were also given in [11]. It would be of
interest to perform similar analysis for Euclidean and neutral solutions in
relation to the various $CY_{3}$ compactifications of type II string theories
obtained as circle reductions of M-theory with space-time signatures
$(1,10)$,($2,9)$ and $(5,6)$ and their mirrors [21]. It is also of interest to
generalise the results of [27] to include vector multiplets and thus have a
complete classification of supersymmetric solutions. Another important
direction is the construction of supergravity theories with higher derivative
terms in Euclidean and neutral signatures and the study of their BPS
solutions. We hope to report on some of these research directions in future
publications.

Acknowledgements: The work is supported in part by the National Science
Foundation under grant number PHY-1620505.

## References

  * [1] K. P. Tod All Metrics Admitting Supercovariantly Constant Spinors, Phys. Lett. B121 (1983) 241.
  * [2] G.W. Gibbons and C. M. Hull, A Bogomolny Bound for General Relativity and Solitons in N=2 Supergravity, Phys. Lett.  B109 (1982) 190.
  * [3] Z. Perjés, Solutions of the coupled Einstein-Maxwell equations representing the fields of spinning sources, Phys. Rev. Lett. 27 (1971) 1668; W. Israel and G. A. Wilson, A Class of stationary electromagnetic vacuum fields, J. Math. Phys. 13 (1972) 865.
  * [4] A. Van Proeyen, N = 2 Supergravity in D = 4, 5, 6 Dimensions, arXiv:2004.11433 [hep-th].
  * [5] S. Ferrara, R. Kallosh and A. Strominger, N=2 Extremal Black Holes, Phys. Rev. D52 (1995) 5412; S. Ferrara and R. Kallosh, Supersymmetry and Attractors, Phys. Rev. D54 (1996) 1514.
  * [6] K. Behrndt, Quantum corrections for D=4 black holes and D=5 strings, Phys. Lett. B396 (1997) 77; K. Behrndt and W. A. Sabra, Static N = 2 black holes for quadratic prepotentials, Phys. Lett. B401 (1997) 258; W. A. Sabra, General static N = 2 black holes, Mod. Phys. Lett. A12 (1997) 2585; W. A. Sabra,  Black holes in N = 2 supergravity and harmonic functions, Nucl. Phys. B510 (1998) 247.
  * [7] K. Behrndt, D. Lüst and W. A. Sabra,  Stationary solutions of N=2 supergravity, Nucl. Phys. B510 (1998) 264\.
  * [8] W. A. Sabra, Phantom Metrics With Killing Spinors , Phys. Lett. B750 (2015) 237; M. Bu Taam and W. A. Sabra,  Phantom space–times in fake supergravity, Phys. Lett. B751 (2015) 297.
  * [9] J. Grover, J. B. Gutowski and W. A. Sabra,  Maximally Minimal Preons in Four Dimensions, Class. Quant. Grav. 24 (2007) 32.
  * [10] F. Denef, Supergravity flows and D-brane stability, JHEP 08 (2000) 050.
  * [11] B. Bates and F. Denef, Exact solutions for supersymmetric stationary black hole composites, JHEP 11 (2011) 127\.
  * [12] V. Cortés, C. Mayer and T. Mohaupt and F. Saueressig, Special Geometry of Euclidean Supersymmetry I: Vector Multiplets, JHEP 03 (2004) 028.
  * [13] V. Cortés, C. Mayer and T. Mohaupt and F. Saueressig, Special geometry of Euclidean supersymmetry II: hypermultiplets and the c-map, JHEP 06 (2005) 024\.
  * [14] V. Cortés, P. Dempster, T. Mohaupt and O. Vaughan, “Special Geometry of Euclidean Supersymmetry IV: the local c-map, JHEP 10 (2015) 066.
  * [15] V. Cortés and T. Mohaupt, Special Geometry of Euclidean Supersymmetry III: the local r-map, instantons and black holes, JHEP 07 (2009) 066.
  * [16] M. Gunaydin, G. Sierra and P. K. Townsend, The Geometry of N=2 Maxwell-Einstein Supergravity And Jordan Algebras, Nucl. Phys. B242 (1984) 244.
  * [17] J. B. Gutowski, and W. A. Sabra, Euclidean N=2 supergravity, Phy. Lett. $\mathbf{B718}$ (2012) 610; J. B. Gutowski and W. A. Sabra, Para-complex geometry and gravitational instantons, Class. Quantum Grav. 30 (2013) 195001; W. A. Sabra and O. Vaughan, 10D to 4D Euclidean Supergravity over a Calabi-Yau three-fold, Class. Quant. Grav. 33 (2015) 1033.
  * [18] W. A. Sabra and O. Vaughan, Euclidean Supergravity in Five Dimensions, Phy. Lett. $\mathbf{B760}$ (2016) 14.
  * [19] A. Ceresole, R. D’Auria, and S. Ferrara,  11-Dimensional Supergravity Compactified on Calabi-Yau Threefolds, Phys. Lett. B357 (1995) 76.
  * [20] W. A. Sabra, Special geometry and space–time signature , Phys. Lett. B773 (2017) 191.
  * [21] C. M. Hull, Duality and the signature of space-time, JHEP 11 (1998) 017.
  * [22] E. Cremmer, B. Julia and J. Scherk, Supergravity Theory in Eleven-Dimensions, Phys. Lett. B76 (1978) 409.
  * [23] V. Cortés, L. Gall and T. Mohaupt,  Four-dimensional vector multiplets in arbitrary signature (I),  International Journal of Geometric Methods in Modern Physics, Vol. 17, No. 10, (2020) 2050150; V. Cortés, L. Gall and T. Mohaupt,  Four-dimensional vector multiplets in arbitrary signature (II),  International Journal of Geometric Methods in Modern Physics, Vol. 17, No. 10, (2020) 2050151.
  * [24] S. Ferrara, G. W. Gibbons and R. Kallosh, Black holes and critical points in moduli space, Nucl. Phys. B500 (1997) 75.
  * [25] J. B. Gutowski and W. A. Sabra, On Non-extremal Instantons and Black Holes, arXiv:1505.05963 [hep-th].
  * [26] P. Galli, T. Ortín, J. Perz and C.S. Shahbazi,  Non-Extremal Black Holes of N=2, d=4 Supergravity, JHEP 07 (2011) 041\.
  * [27] J. B. Gutowski and W. A. Sabra, Gravitational Instantons and Euclidean Supersymmetry, Phys. Lett. B693 (2010) 498; D. Klemm and M. Nozawa, Geometry of Killing spinors in neutral signature, Class. Quant. Grav. 32 (2015) 185012; J. B. Gutowski and W. A. Sabra, Real Killing Spinors in Neutral Signature, JHEP 11 (2019) 173, J. B. Gutowski and W. A. Sabra, Neutral signature gauged supergravity solutions, JHEP 02 (2021) 030.