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# Active Region-based Flare Forecasting with Sliding Window Multivariate Time
Series Forest Classifiers

Anli Ji and Berkay Aydin  Department of Computer Science
Georgia State University
Atlanta, GA
{aji1<EMAIL_ADDRESS>
###### Abstract

Over the past few decades, many applications of physics-based simulations and
data-driven techniques (including machine learning and deep learning) have
emerged to analyze and predict solar flares. These approaches are pivotal in
understanding the dynamics of solar flares, primarily aiming to forecast these
events and minimize potential risks they may pose to Earth. Although current
methods have made significant progress, there are still limitations to these
data-driven approaches. One prominent drawback is the lack of consideration
for the temporal evolution characteristics in the active regions from which
these flares originate. This oversight hinders the ability of these methods to
grasp the relationships between high-dimensional active region features,
thereby limiting their usability in operations. This study centers on the
development of interpretable classifiers for multivariate time series and the
demonstration of a novel feature ranking method with sliding window-based sub-
interval ranking. The primary contribution of our work is to bridge the gap
between complex, less understandable black-box models used for high-
dimensional data and the exploration of relevant sub-intervals from
multivariate time series, specifically in the context of solar flare
forecasting. Our findings demonstrate that our sliding-window time series
forest classifier performs effectively in solar flare prediction (with a True
Skill Statistic of over 85%) while also pinpointing the most crucial features
and sub-intervals for a given learning task.

###### Index Terms:

Multivariate Time Series Classification, Solar Flare Prediction, Interval-
based Classification

##  I Introduction

Solar flares are intense localized bursts of electromagnetic energy
originating from the Sun’s atmosphere, varying in duration from minutes to
hours. These sudden energy releases travel nearly at the speed of light and
can coincide with other eruptive solar events such as coronal mass ejections
(CMEs). These events have the potential to trigger severe geomagnetic storms,
cause extensive radio disruptions on the sunlit side of Earth, and disturb
sensitive space equipment close to Earth. An illustrative example of an
X9.3-class solar flare (on September 6, 2017) accompanied by a CME is shown in
Figure 1, which led to an Earth-impacting solar energetic particle event the
same day. Since solar flares are central events, efforts to forecast them,
combining physics-based and data-driven methodologies, have been prominent in
the literature [28] [30] [20]. These methods often leverage data from solar
magnetograms gathered by diverse satellites and employ diverse learning
algorithms such as support vector machines (SVMs) [6] [7], logistic regression
[32], and decision trees [34] [16].

Figure 1: An X9.3-class flare occurred in 2017-09-06 captured by Solar
Dynamics Observatory (a) and its accompanying coronal mass ejection captured
by Large Angle and Spectrometric Coronagraph (LASCO) instrument (b)

Recently, many methodologies and corresponding models have been introduced to
tackle flare prediction as a classification task by using point-in-time
measurements. However, these approaches usually overlook the inherent time-
dependent (i.e., temporal evolution characteristics) nature of the data (see
[12] for a review), treating multiple physical observations as separate
entities and generating predictions solely based on instantaneous values [7].
This disregards the dynamic essence of flares, whose characteristics depend on
the evolutionary behavior of solar active regions [4] [11]. By exploring
temporal features spread across time series intervals, we can potentially
unveil implicit relationships and capture relevant patterns underneath. In our
earlier work [18], we employed interval-based time series classification
models on multivariate time series data using a set of interval features
extracted. However, the computational efficiency of this approach led to the
extraction of interval features from randomly sampled interval sets, lacking a
systematic evaluation of the internal process of determining features (or
intervals from these time series features) that are more important for
predictive tasks. Hence, our primary focus in this study is interpreting
statistical features obtained from multivariate time series and identifying
sequences of features (and related patterns) using multi-scale sliding windows
with varying interval windows and step sizes.

The rest of the paper is structured as follows: Section II provides background
information on existing time series classification models pertinent to flare
prediction. In Section III, we provide our problem formulation and introduce
our multivariate time series classification model and feature ranking method
used for extracting relevant feature intervals from provided time series data.
Our experimental setup and evaluation framework are presented in Section IV.
Finally, Section V provides conclusions drawn from our study and discusses
potential avenues for future research.

##  II Related Work

Within the proliferation of available time series data sets [31] and a wide
spectrum of machine learning-based techniques proposed for time series
classification, two notable categories utilized for these predictive tasks are
similarity-based and feature-based algorithms. The former group makes
predictions by measuring the similarity between training and testing
instances, while the latter generates predictions by utilizing temporal
features extracted from entire time series or subsequences within these time
series instances. Especially in the task of solar flare prediction or other
tasks (such as anomaly detection [14]), approaches proposed utilizing either
full-disk (e.g., [25][27] [26]) or active region-based (e.g., [18] [17] [8]
[15]) patches have shown great impact with respect to derived time series
features (i.e., features extracted from each subsequent time series). Among
the commonly employed similarity-based algorithms, some rely on nearest-
neighbor classifiers combined with metrics like Euclidean distance (NN-
Euclidean) or elastic measures like Dynamic Time Warping (NN-DTW) [3], [29],
[2],[21]. In contrast, feature-based algorithms employ derived time series
features to capture associations between target variables and time series
instances. For instance, in [24], basic statistical features (e.g., mean and
standard deviation) were extracted from global time series and used as input
for a multi-layer perceptron network, yet neglecting localized informative
characteristics and properties. Meanwhile, [13] considers local attributes
through piecewise constant modeling, coupled with extracted patterns to
enhance model interpretability. However, this approach also comes with the
issue of acquiring simplistic features during the feature selection process.
In [10], an extensive number of features derived from time series data was
incorporated (e.g., wavelets, Chebyshev coefficients, etc.), but the method
incurs high computational costs and lacks inherent interpretability in high-
dimensional data spaces.

Feature-based classification methods face challenges when dealing with
multivariate time series data, as they require additional intricate
information across features. Generating discriminating features in high-
dimensional spaces becomes difficult due to the unknown interrelations among
input parameters of the multivariate time series, therefore, adding complexity
to model construction. Various techniques have been attempted to address this
challenge, often utilizing ensembled univariate models across individual
feature spaces for predictions. These methods focus on extracting relevant
features in multivariate aspects and then applying traditional machine
learning algorithms for classification. Common features include statistical
measures (e.g., mean, variance), spectral features (e.g., Fast Fourier
Transform coefficients), or time-domain features (e.g., autocorrelation). For
example, Shapelet-based decision trees [33] combine the concept of shapelets
(i.e., discriminative sub-sequences that capture distinctive patterns in time
series data) with the ensemble approach. It extracts shapelets from the
training data and constructs an ensemble of decision trees (i.e., random
forest) where each estimator focuses on a different subset of shapelets,
usually measured by Euclidean distance (i.e., the distance to the finest
position in the time series). This method can effectively capture local
patterns in multivariate time series data but also can be computationally
expensive and hard to identify relevant shapelets (especially in high
dimensions) that are both informative and applicable across dimensions.
Another problem is that the shapelets extracted from one multivariate time
series dataset might not generalize well to other datasets with different
dimensionality, characteristics, and patterns. To mitigate these issues, the
Generalized random shapelet forest (gRSF) [19] enhances the original shapelet-
based method by measuring distances between randomly selected time series and
other time series falling within a threshold distance of the representative
shapelet. Similarly, the Time Series Forest (TSF) [9] also incorporates
subseries, but instead of relying on distance measurement from learned
subsequences, it derives summarized statistical features (such as mean,
standard deviation, and gradient) within randomly divided intervals of the
univariate time series. It treats each time step as a separate component and
constructs decision trees in each feature dimension to capture temporal
relationships and reduce the high-dimensional feature space. However,
important interrelationships between different components of the time series
might not be fully captured by treating them as separate features, which can
lead to a loss of inter-channel relationships and dependencies that are often
crucial in multivariate time series data. Furthermore, the Canonical Interval
Forest (CIF) [23] extends TSF by incorporating additional canonical
characteristics of the time series and _catch22_ [22] features extracted from
each interval. This approach is designed to capture both individual patterns
within each time series component and relationships between different
components but can only provide a certain level of interpretability as
interpreting an ensemble of decision trees can be more challenging.
Understanding the combined effects of multiple trees on multivariate time
series data might be less intuitive compared to single decision trees.

Many of these methods focus solely on understanding how each feature behaves
on its own, without considering how different features might interact. There
are instances where a particular relationship within a single time series
parameter might be significant for a specific feature, but not necessarily for
others. The connections between distinct features are often not known upfront.
When trying to generate features that effectively differentiate classes, it is
more advantageous to select the most relevant time intervals to create a more
robust model. However, identifying these relevant intervals is not an easy
task as they typically cannot be directly determined and require an expensive
search across the entire series. Being able to extract the underlying mutual
information present in these relevant intervals can enhance our understanding
of the predictive process and accelerate the transition from research to
operations in flare forecasting models. Our objective in this work is to
establish a framework that can recognize these characteristics and offer
deeper insights into the behavior of classifiers during prediction tasks.

##  III Methodology

In this section, we will outline the core aspects of our approach. We will
start by defining the problem we are addressing and explaining how we generate
statistical features from intervals. Next, we will introduce the sliding-
window time series forest, a key component of our method. Lastly, we will
describe our feature ranking technique. A visual representation overview of
our methodology is provided in Figure 2.

Figure 2: The overview of the methodology presented in this paper. We first
generate subsequences (intervals) with a sliding window. Then, we create
vectorized features from these intervals where these features can be used as
input for the sliding window time series forest (a random forest built on
multivariate time series features) and features are ranked with aggregated
relevance scores.

###  III-A Problem Formulation

The sliding-window time series forest represents an intricate interval-based
classifier for time series, employing a random forest approach on interval
features extracted from individual univariate time series. In the following
parts, we outline the definitions and explanations concerning the manipulation
of multivariate time series data and the construction of a random forest using
these transformed features.

###### Definition 1

A time series $T$ is a sequence of real values observed in ascending time
order pair such that $<(t_{1},v_{1}),(t_{2},v_{2}),...,(t_{n},v_{n})>$ where
$t_{1}$ to $t_{n}$ represent the equally spaced timestamps $(\forall
i,t_{i+1}-t_{i}=\Delta t$ at which the values are being measured.

Such values (i.e., $v_{i}$) can be $P$ dimensional at each time point $t_{i}$
in terms of multivariate time series (i.e., $mvts$). We refer to each
distinctive value in the same dimension as a feature $f$ at the same time
point, meaning that each timestamp $t_{i}$ has $P$ features.

###### Definition 2

A multivariate time series dataset can be defined as
$D_{MVTS}=\\{mvts_{1},mvts_{2},...,mvts_{i}\\}$ where the length of each
series $|mvts_{i}|=k_{i}$ such that there are $k$ number of pairs (i.e.,
$<(t_{1},v_{1}),(t_{2},v_{2}),...,(t_{k},v_{k})>$) included and the value of
$k_{i}$ can be varied across the dataset.

To create an equal-length multivariate time series dataset, one approach
involves performing a slicing operation by sliding through each $mvts_{i}$
with a window size $l$ so that each slice would contain a set of
$<(t_{s},v_{s}),...,(t_{s+l},v_{s+l})>$ where $s$ refers to the starting time
of each slice.

###### Definition 3

An interval $T_{s,e}$ is defined as a subsequence of the original time series
$T$ such that $<(t_{s},v_{s}),(t_{s+l},v_{s+l}),...,(t_{e},v_{e})>$ with
starting point $s$ and ending point $e$ where $1\leq s<e\leq n$.

A dataset created using the sliding window technique and intervals can be
potentially different, despite both methods involving comparable operations.
The former (i.e., by incorporating the sliding window) augments the dataset
with more instances while the latter (i.e., by using intervals) encompasses
only a segment of the original dataset. It often refers to each instance
generated through a sliding window procedure as an interval, marked by its
initiation time $s$ and culmination time $s+l$.

For a comprehensive assessment of the impact of a time series feature on a
target variable, it is necessary to pinpoint the pertinent intervals and their
connections with other time series (and their intervals) within a dataset.
Essentially, understanding frequently occurring patterns in a time series
classification task allows us to identify strong correlations with others and
discover the hidden shared characteristics among them. Nevertheless, these
pertinent parameters might encompass only specific intervals rather than the
entire time series. In order to carry out classification tasks on a
multivariate time series entity $T$, the task of recognizing relevant
intervals involves determining a set of ranked intervals based on their
significance in contributing to the classification process. To achieve this,
it becomes essential to recognize and scrutinize the characteristics of
univariate features.

###### Definition 4

A univariate feature interval set (i.e., $UFI$, which we will denote as $I$
for brevity of notation) with a sliced starting point $s$ and ending point $e$
for each univariate feature $f$ is defined as
$I_{f}=\\{T_{s,e}:cv_{1},T_{s+1,e+1}:cv_{2},...,T_{s+m,e+m}:cv_{m-1}\\}$ where
$cv$ refers to the contribution value of such interval $T_{s,e}$. For every
feature $f_{i}$ in a multivariate time series dataset $D_{MVTS}$, we can
generate a set of $UFIs$ in the form of $I=\\{I_{f_{i}},...,I_{f_{j}}\\}$
where this set of $UFIs$ can be ordered based on the starting time of each
$UFI$ such that $\\{I_{f_{i}}:s_{f_{i}}\\}<\\{I_{f_{j}}:s_{f_{j}}\\}$.

In order to assess the relevance of an interval set, it is essential to
recognize how the feature impacts the target variable across various
parametric settings in the learning process. Several contribution functions
exist as alternatives for quantifying this influence, including supervised
feature ranking scores, direct rankings, or applying a Boolean function to the
top-k ranked list. These individual scores can subsequently be aggregated to
determine the overall relevance of each univariate feature interval set
($UFI$) in relation to the classification task.

###  III-B Interval Features

In order to generate well-ordered and pertinent intervals, we compute interval
statistical features (i.e., mean, standard deviation, and slope) along with
secondary transformed features (i.e., maximum, minimum, and mean pooled
features) for each subsequence. Initially, we obtain sets of potential
candidate intervals using pre-defined window and step sizes. Here, the window
size corresponds to the total number of equally-spaced timestamps that need to
be gathered from the original time series within each slice, while the step
size determines the incremental movement of the sliding window from one to the
next. Then, by employing a method similar to the one described in [9], we
compute statistical features (as indicated in Eq. 1, Eq. 2, and Eq. 3) from
each of these potential candidate interval sets
$\\{(t_{1},v_{1}),(t_{2},v_{2}),...,(t_{k},v_{k})\\}$, thereby creating a
collection of extracted interval features. Note here that these derived
statistical attributes serve as vectorized representations of a subsequence
generated through the application of a sliding window mechanism.

$f_{mean}(t_{1},t_{k})=\frac{\sum_{i=t_{1}}^{t_{k}}v_{i}}{t_{k}-t_{1}+1}$ (1)

$\begin{split}f_{std}(t_{1},t_{k})=\sqrt{\frac{\sum_{i=t_{1}}^{t_{k}}(v_{i}-f_{mean}(t_{1},t_{k}))^{2}}{t_{k}-t_{1}}}\text{,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}}\\\
\text{~{}~{}~{}where }t_{2}>t_{1}\end{split}$ (2)

$f_{slope}(t_{1},t_{k})=\widehat{\beta}$ (3)

where $t_{2}>t_{1}$ and $\widehat{\beta}$ is the slope of the least squares
regression line in candidate interval sets.

The secondary transformation involves an extra localized pooling process
applied to the interval features obtained from each individual time series.
This operation gathers all the potential interval sets derived from the same
time series and applies a statistical function (such as maximum, minimum, or
mean) for aggregation. In other words, during this phase, we calculate the
highest, lowest, and average values of statistical attributes (i.e., mean,
standard deviation, and slope) from the subseries obtained through sliding
window operations.

Figure 3: Results of our feature ranking experiments with only mean
($f_{mean}$ of slices used in the feature set and secondary transformation
applied. Results come from an aggregation of five different imbalance weight
settings and bars show how many times a feature has appeared in the top-5
ranking list. Orange bars show the features obtained after secondary
transformations while blue ones show the local features. Note that WS
represents the window size while SS represents the step size of candidate
interval settings.

Figure 4: Results of our feature ranking experiments with only standard
deviation ($f_{std}$ of slices used in the feature set and secondary
transformation applied (similar to Fig. 3). Results come from an aggregation
of five weight settings and bars show how many times a feature has appeared in
the top-5 ranking list. Orange bars show the features obtained after secondary
transformations and blue ones show the local features.

Figure 5: Results of our feature ranking experiments with only slope
($f_{slope}$ of slices used in the feature set and secondary transformation
applied (similar to Fig. 3). Results come from an aggregation of five weight
settings and bars show how many times a feature has appeared in the top-5
ranking list. Orange bars show the features obtained after secondary
transformations and no local feature was ranked in the top 5 across all the
experimental runs.

###  III-C Time Series Forest and Feature Ranking

After deriving interval features from subsequences and applying a secondary
transformation to these statistical features, we construct an input vector.
This vector is then utilized for constructing a random forest, denoted as the
_sliding window time series forest_. While various classifiers could be
employed for generating predictions, we opt for random forest classifiers for
two primary reasons: (1) their effectiveness and robustness with noisy high-
dimensional data, and (2) their inherent feature selection capabilities. It’s
worth noting that depending on the chosen parameter settings, such as
employing smaller window and step sizes, the data space for the interval
feature vectors can undergo significant expansion. Additionally, the process
of vectorization based on the sliding window approach can produce data points
that are relatively correlated and potentially noisy. Consequently, it becomes
crucial to effectively eliminate these features through information-theoretic
relevance metrics measures (e.g., entropy and Gini index, etc) to ensure the
effectiveness of our approach.

Next, the feature importance scores obtained through the proposed sliding
window time series forest are employed to establish a ranking for interval
features. Different alternatives for the importance scores have been
documented in the literature, such as the mean decrease in impurity or feature
permutation. These scores enable us to rank the transformed features vector
extracted from intervals within a framework of embedded feature selection
scenarios. The classification based on random forests provides an enhancement
of the robustness of classification, especially in tasks that need to be
highly generalized.

##  IV Experimental Evaluations

The experiments are structured with two objectives: (1) Showcasing the
effectiveness and conducting a performance comparison among time series
classifiers developed using distinct extracted interval features. (2)
Identifying the intervals that are most relevant to the initial time series.
Our goal is centered around offering interpretable insights into our model.
This entails pinpointing the segments of the time series that influence
predictions, along with comprehending the aggregation strategies that could
yield more advantageous outcomes.

###  IV-A Data Collection

For our solar flare prediction study, we used an open source multivariate time
series dataset named Space Weather ANalytics dataset for Solar Flare
prediction (SWAN-SF) by [1] as our benchmark dataset. This dataset provides
space weather-related physical parameters derived from solar magnetograms
onboarding the various observatory resources. We utilized six solar active
region parameters which are: the total unsigned magnetic flux (USFLUX), the
total unsigned vertical current (TOTUSJZ), the total unsigned current helicity
(TOTUSJH), and the total photospheric magnetic free-energy density (TOTPOT),
Sum of the Absolute Value of the Net Currents Per Polarity (SAVNCPP), Absolute
value of the net current helicity (ABSNJZH). These are considered to be the
most representative features of solar activity in [5]. Detailed descriptions
of these parameters can be found on [5] [1]. For time and space consideration,
we only use these six physical parameters, although the analysis can be
extended with many more parameters.

The SWAN-SF dataset consists of five time-segmented partitions meaning that
the data instances within each partition do not overlap with others in the
time domain. The active regions are sliced through a sliding observation
window (i.e., 12-hour intervals) over the multivariate time series. Each slice
has a corresponding active region number (i.e., National Oceanic and
Atmospheric Administration (NOAA) Active Region number matched with HMI Active
Region Patches (HARPs) identifiers), a class label (i.e., the maximum
intensity flare originating from that active region within the next 24 hours),
a start and end timestamp of each slice. Note here that the flare intensity is
determined by the logarithmic classification of peak X-ray flux. Within the
major flaring categories (X, M, C, B, or A) from such schema, we include
instances labeled with M- and X-class flares as the flaring (i.e., positive
class) while the relatively weak C- and B- class flare labels and flare-quiet
regions are considered as non-flaring (i.e., negative class). In this way, the
flare forecasting problem can be modeled as a binary multivariate time series
classification task.

Throughout the studies, we use Partitions 1,2,3, and 5 for training purposes
and Partition 4 for testing purposes as it has a relatively more balanced
flaring to non-flaring class ratio.

###  IV-B Experimental Settings

To evaluate the performance of our model, we employed a binary $2\times 2$
contingency matrix along with various evaluation metrics and crucial forecast
skill scores. Within these evaluation metrics, the positive class corresponds
to the occurrence of a large flare class (i.e., $\geq$M1.0 in the NOAA/GOES
flare classification) while the negative class refers to the relatively small
flare and flare-quite region (i.e., $<$M1.0 class flare). Therefore, true
positives (TPs) denote cases where the model accurately predicts a flare event
(positive class), and true negatives (TNs) refer to instances where the model
correctly predicts a small flare event (negative class). In contrast, false
positives (FPs) represent false alarms, occurring when the model predicts a
non-flaring event as flaring, and false negatives (FNs) indicate misses, which
happen when the model fails to predict an actual flare event. In our study, we
employ two well-known skill scores: the True Skill Statistic score (TSS) and
the Heidke Skill Score (HSS). The former can be calculated as the disparity
between the probability of detection (recall for the positive class) and the
probability of false detection (POFD). This measurement is determined using
the equation specified as Eq. 4.

$TSS=\frac{TP}{TP+FN}-\frac{FP}{FP+TN}$ (4)

The second measurement is the Heidke Skill Score and assesses how much the
forecast enhances compared to a random forecast. This score varies from
-$\infty$ to 1, where a score of 1 signifies flawless performance and 0
denotes a lack of skill. A score of 0 signifies that the forecast’s accuracy
is no better than a random binary forecast based on the provided class
distributions. This metric can be computed using Eq. 5, where $P$ represents
the sum of true positives ($TP$) and false negatives ($FN$), and $N$
corresponds to the sum of false positives ($FP$) and true negatives ($TN$) in
the observed data.

$HSS=\frac{2\cdot((TP\cdot TN)-(FN\cdot FP))}{P\cdot(FN+TN)+N\cdot(TP+FP)}$
(5)

###  IV-C Comparison between different window sizes

To identify the potential candidate intervals, we employ a fixed-size sliding
window approach with three configurations: window sizes of 8, 15, and 30, and
corresponding step sizes of 4, 8, and 15, respectively. This approach extracts
intervals from the original time series based on pre-defined window sizes (WS)
and traverses the time series using the designated step size (SS). These
identified intervals are then transformed with statistical functions (i.e.,
mean, standard deviation, slope) to generate informative interval features. To
assess the predictive capabilities of these extracted features, we train a
Random Forest (RF) classifier for each window setting. The outcomes are
presented in Table I, showcasing the forecast skill scores in terms of TSS,
HSS and recall.

TABLE I: The evaluation metrics for Random Forest with different window sizes and step sizes settings Model | TSS | HSS | Recall
---|---|---|---
WS=60 | 0.16 | 0.27 | 0.16
WS=8, SS=4 | 0.82 | 0.19 | 0.97
WS=15, SS=8 | 0.83 | 0.21 | 0.97
WS=30, SS=15 | 0.83 | 0.20 | 0.98

Results in Table I show that the random forest models constructed using
different configurations exhibit consistent performance across different
window sizes in terms of TSS and HSS. These models achieve an average TSS of
82.3% and an average HSS of 20%. Notably, these models are trained on a
substantially imbalanced class weight, with an imbalance ratio of
approximately 1:50 between our positive (flaring) and negative (non-flaring)
classes, based on the original SWAN-SF benchmark dataset. Given this
situation, the subsequent experiment involves adjusting the class weights to
achieve a balanced class distribution and to identify pertinent interval
features.

###  IV-D Class weights tuning

In this part of the study, we conduct an experiment involving the training of
five Random Forest (RF) models. These models vary in terms of the combinations
of class weights assigned to our flaring and non-flaring classes. The primary
aim is to reveal interval features that hold greater significance for a
predictive task, thereby facilitating the interpretation of the model’s
feature selection process. The models are all trained using consistent sliding
window settings, along with the separate utilization of each statistical
function. The outcomes obtained from all the combinations are aggregated to
generate a ranking list based on the occurrence frequency of the top 5
features. The visual representation of these results is provided in Figures 3
through 5.

As depicted in Figure 3, the blue bars represent the identified relevant
intervals whose features contribute the most to the prediction, while the
orange bars signify the relevant intervals and features from the secondary
transformation. The observation from Figure 3 highlights that the segments
starting at the index of 40 within the TOTUSJH (the total unsigned current
helicity) parameter are frequently repeated across all window settings, which
suggests a potential starting point for the most relevant intervals.
Considering we use a 12-min cadence and length-60 time series, this data point
corresponds to approximately 4 hours before the end time of the observation
window.

Upon employing secondary transformations, we observe that distinct patterns
emerge. For instance, we note a strong repetitive pattern of occurrences of
the minimum/mean values of TOTUSJH within a size-8 window and size-4 steps (in
terms of timestamps), the minimum value of SAVNCPP (Sum of the Absolute Value
of the Net Currents Per Polarity) within a size-15 window and size-8 step, as
well as the maximum value of TOTUSJH within a size-30 window and size-15 step.
This occurrence frequency implies that these physical parameters could serve
as promising candidates for temporal dynamics leading to a relatively large
flare event.

We also observe that only a minimal number of local slice features from
standard deviation and slope features are considered significant for solar
flare prediction. This suggests that our classifiers’ predictions do not
exclusively rely on the interval features from individual slices but also
consider features we obtain after secondary transformation.

##  V Conclusions and Future Work

In this study, we have undertaken the training of classifiers for multivariate
time series using intervals and introduced an innovative approach for ranking
features obtained from sliding window intervals. Our work concentrates on
including interpretability in interval-based multivariate time series
classifiers. Our work’s primary achievement is in connecting the black-box
models used for high-dimensional data and investigating the key sub-intervals
within multivariate time series, particularly their relevance in the context
of solar flare prediction. Our findings demonstrate a skill score of
considerable effectiveness across various window size configurations.
Furthermore, we present an outline of our ranking framework, highlighting the
significance of identified statistical features in predicting solar flares.

There are a few potential directions available for future research. These
include the extension of the sliding time window dimensions and step sizes,
the incorporation of alternative ranking metrics and physical parameters, as
well as the exploration of diverse time series models.

## Acknowledgment

This work is supported in part under two grants from NSF (Award #2104004) and
NASA (SWR2O2R Grant #80NSSC22K0272).

## References

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$A=c^{\prime}\sum_{\mu={0}}^{25-d}k_{\mu}k^{\mu}+c^{\prime}\sum_{I,J,L=1}^{d}\Big{(}k_{I}-\mathcal{B}_{IJ}\omega^{J}\Big{)}G^{IL}\Big{(}k_{L}-\mathcal{B}_{LK}\omega^{K}\Big{)}.$
(181)

Following the same line of argument provided in Bagchi:2020fpr  it can be
shown that $A=0,\hskip 2.84526pt\text{and}\hskip
2.84526pta_{\mu}=b_{\mu},\hskip 2.84526pt\mu=\\{0,1,...,25-d\\}$. $A=0$
implies that the mass of the state has to be same as (177). The norm of the
state is then given by $\braket{1}{1}=a_{\mu}a^{\mu}-b_{\mu}b^{\mu}=0$,
implying that level 1 physical states are null states. The remaining physical
state conditions give us the following additional constraints on the
coefficients $a_{\mu}$ themselves:

$\displaystyle\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}a_{I}+\sum_{\mu=0}^{25-d}k^{\mu}a_{\mu}=0.$
(182)

### Level 2 states

We start from a generic state of level 2, which is given below in terms of all
possible polarisations

$\displaystyle\ket{2,k_{I},\omega^{I}}_{A}$
$\displaystyle=a_{\mu}C^{\mu}_{-2}\ket{0}_{A}+e_{\mu\nu}C^{\mu}_{-1}C^{\nu}_{-1}\ket{0}_{A}+h_{\mu\nu}C^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}$
$\displaystyle+b_{\mu}\tilde{\mathcal{C}}^{\mu}_{-2}\ket{0}_{A}+f_{\mu\nu}\tilde{\mathcal{C}}^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}+j_{\mu\nu}C^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}.$
(183)

In the above expression for level 2, $\ket{0}_{A}$ is written as short form of
$\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{A}$, $e_{\mu\nu}$ and $f_{\mu\nu}$ are
essentially symmetric, $h_{\mu\nu}$ and $j_{\mu\nu}$ are by definition
symmetric and antisymmetric respectively. Applying the $L_{0}$ condition on
this state we inevitably get

$\displaystyle\sum_{i=1}^{d}k_{I}\omega^{I}=0.$ (184)

For level 2, apart from $L_{0}$ condition there will be five more non-trivial
physical state conditions. The $M_{0}$ condition is given by

$\begin{split}M_{0}\ket{2,k_{I},\omega^{I}}&=\left[(A+2)a_{\mu}-2b_{\mu}\right]C^{\mu}_{-2}\ket{0}_{A}+\left[(A-2)b_{\mu}-2a_{\mu}\right]\tilde{\mathcal{C}}^{\mu}_{-2}\ket{0}_{A}\\\
&+\left[(A+2)e_{\mu\nu}-h_{\mu\nu}\right]C^{\mu}_{-1}C^{\nu}_{-1}\ket{0}_{A}+\left[(A-2)f_{\mu\nu}+h_{\mu\nu}\right]\tilde{\mathcal{C}}^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}\\\
&+\left[A(h_{\mu\nu}+j_{\mu\nu})+2(e_{\mu\nu}-f_{\mu\nu})\right]C^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A},\end{split}$
(185)

where expression of $A$ is given in (6). The remaining physical state
conditions are given below.

$\begin{split}L_{1}\ket{2,k_{I},\omega^{I}}&=\Bigg{[}2a_{\nu}+c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(2e_{I\nu}+h_{I\nu}-j_{I\nu})\\\
&+\sum_{I=1}^{d}\omega^{I}(2e_{I\nu}-h_{I\nu}+j_{I\nu})+c^{\prime}\sum_{\mu=0}^{25-d}(2e_{\mu\nu}+h_{\mu\nu}-j_{\mu\nu})k^{\mu}\Bigg{]}C^{\nu}_{-1}\ket{0}_{A}\\\
&+\Bigg{[}2b_{\nu}+c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(h_{I\nu}+j_{I\nu}+2f_{I\nu})\\\
&+\sum_{I=1}^{d}\omega^{I}(h_{I\nu}+j_{I\nu}-2f_{I\nu})+c^{\prime}\sum_{\mu=0}^{25-d}(h_{\mu\nu}+j_{\mu\nu}+2f_{\mu\nu})k^{\mu}\Bigg{]}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}=0\end{split}$
(186)
$\begin{split}M_{1}\ket{2,k_{I},\omega^{I}}&=2\Bigg{[}(a_{\nu}-b_{\nu})+c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(2e_{I\nu}+j_{I\nu}-h_{I\nu})\\\
&+c^{\prime}\sum_{\mu=0}^{25-d}(2e_{\mu\nu}+j_{\mu\nu}-h_{\mu\nu})k^{\mu}\Bigg{]}C^{\nu}_{-1}\ket{0}_{A}+2\Bigg{[}(a_{\nu}-b_{\nu})\\\
&+c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(j_{I\nu}+h_{I\nu}-2f_{I\nu})\\\
&+c^{\prime}\sum_{\mu=0}^{25-d}(h_{\mu\nu}+j_{\mu\nu}-2f_{\mu\nu})k^{\mu}\Bigg{]}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}=0\end{split}$
$\begin{split}L_{2}\ket{2,k_{I},\omega^{I}}=\Bigg{[}c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(a_{I}+b_{I})+\sum_{I=1}^{d}\omega^{I}(a_{I}&-b_{I})+c^{\prime}\sum_{\mu=0}^{25-d}(a_{\mu}+b_{\mu})k^{\mu}\\\
&+\frac{1}{2}(e^{\mu}_{\mu}-f^{\mu}_{\mu})\Bigg{]}\ket{0}_{A}=0\end{split}$
(187)
$\begin{split}M_{2}\ket{2,k_{I},\omega^{I}}=\Bigg{[}c^{\prime}\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}(a_{I}-b_{I})&+c^{\prime}\sum_{\mu=0}^{25-d}(a_{\mu}-b_{\mu})k^{\mu}\\\
&+\frac{1}{4}(e^{\mu}_{\mu}+f^{\mu}_{\mu}-h^{\mu}_{\mu})\Bigg{]}\ket{0}_{A}=0\end{split}$
(188)

The equations (185)-(188) yield the following constraints on the state (6)

$a_{\mu}=b_{\mu}=0,\hskip 14.22636pth_{\mu\nu}=e_{\mu\nu}=2h_{\mu\nu}$ (189a)
$A=0$ (189b)
$\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}e_{I\nu}+\sum_{\nu=0}^{25-d}e_{\mu\nu}k^{\mu}=\sum_{I,J,K=1}^{d}(k_{J}-\mathcal{B}_{JK}\omega^{k})G^{IJ}j_{I\nu}+\sum_{\nu=0}^{25-d}j_{\mu\nu}k^{\mu}=0.$
(189c) $\sum_{I=1}^{d}j_{I\nu}W^{I}=0.$ (189d)

The constraint (189b) leads us to the mass spectrum identical to (177). The
only difference is that {$k_{I},\omega^{I}$} for level 2 has to satisfy the
constraint (184). The remaining constraints are on the coefficients and fixing
those lead us to the following final form of the level 2 states, which
decouple into symmetric and antisymmetric parts:

$\displaystyle\ket{2,k_{I},\omega^{I}}=e_{\mu\nu}\Big{[}C^{\mu}_{-1}C^{\nu}_{-1}\ket{0}_{A}+2C^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}+\tilde{\mathcal{C}}^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}\Big{]}+j_{\mu\nu}C^{\mu}_{-1}\tilde{\mathcal{C}}^{\nu}_{-1}\ket{0}_{A}.$
(190)

### Higher level states

In this part we generalise our analysis presented above to an arbitrary level
$l=r+s$. A generic state of this level is given by

$\displaystyle\ket{r,s,k^{\mu},k_{I},\omega^{I}}=\sum_{j}\rho_{j}\Bigg{(}\prod_{i=1}^{p}C^{a_{i}}_{-m_{i}}\prod_{j=1}^{q}\tilde{\mathcal{C}}^{b_{j}}_{-n_{j}}\Bigg{)}_{j}\ket{0,0,k^{\mu},k_{I},\omega^{I}}_{c},$
(191)

$a_{j}$ and $b_{j}$ being the powers of $C$ and $\tilde{\mathcal{C}}$
respectively, and the level is given by

$\displaystyle l=r+s=\sum_{i=1}^{p}a_{i}m_{i}+\sum_{i=1}^{q}b_{i}n_{i}.$ (192)

The $L_{0}$ condition for this level will be

$\displaystyle\sum_{I}k_{I}\omega^{I}=2-l.$ (193)

After applying this condition, we shall be left with $(2l+1)$ number of
physical state conditions. The $M_{0}$ condition will lead us to the same mass
spectrum we have derived before for level 0,1 and 2 and the remaining
conditions will restrict the coefficients $\rho_{j}$. We do not present all
those details here for sake of brevity.

Another thing which is important to highlight here is that the mass spectrum
obtained in this theory happens to be identical to the mass spectrum we
previously obtained for oscillator vacuum and induced vacuum. Hence the
symmetry structure of the mass spectrum too is identical to the other two
theories discussed before.

###  6.3 Limit from tensile flipped string

The level matching condition and the mass spectrum of the tensile counterpart
of the flipped string theory has already been calculated. Let us rewrite the
results

$\begin{split}&\hskip 5.69054ptr+s+\sum_{I=1}^{d}k_{I}\omega^{I}=2\\\
m^{2}=k^{T}G^{-1}k+\frac{1}{\alpha^{\prime
2}}&\omega^{T}(G-bG^{-1}b)\omega-\frac{2}{\alpha^{\prime}}k^{T}bG^{-1}\omega+\frac{2}{\alpha^{\prime}}(r-s).\end{split}$

As we have already highlighted, the level matching conditions for both tensile
and tensionless flipped string theory are identical. From this we get an
indication that the physical state with level $l$ of tensile flipped theory
will reduce to the physical state of level $l$ in the tensionless theory. As
we have seen in Banerjee:2023ekd , that is indeed the case when we work in a
background with only one dimension compactified. Working in a target space
with $d$-dimensions compactified does not change anything: here too, the
tensile physical states will reduce to tensionless physical states of the very
same level.

We should also note here, taking direct tensionless limit of the mass spectrum
in equation (166b) in the spirit of earlier computations, we can directly
reproduce the mass spectrum in (177). Hence the limiting analysis of
tensionless twisted string theory also turns out to be consistent with the
intrinsic flipped construction of the same.

##  7 Conclusions and Discussions

###  7.1 Summary

In this paper, we have extensively investigated the impact of a constant $B$
field on the three distinct quantum theories of tensionless strings, namely,
those defined on Oscillator, Induced, and Flipped vacua. Our analysis,
complemented with earlier works, revealed that while the $B$ field does not
affect the level matching constraint of the states, it does modify the mass-
shell condition when the target space is compactified (on a $T^{d}$ in this
case). In our previous work, we have observed that the mass spectrum of all
the three quantum theories seemingly violated T-duality as the string length
became infinitely large in the tensionless limit. However, tensile string with
constant $B$ field has additional symmetries which turns out to be preserved
even at tensionless limit, restoring a sense of T-duality to the spectrum.
This has been a key point in our manuscript.

We also explicitly worked out the fate of $O(d,d)$ symmetries associated to
tensile generalised metric in this case, and it turned out to be a degenerate
one as well, leading to a broken version of the tensile self. Notably, for all
the vacua, introduction of a constant $B$ field leads to the appearance of
winding number dependence in the mass spectrum, distinguishing it from the
case without a $B$ field. This creates an important difference in the spectrum
of states. We verified the consistency of our results by comparing them with
the direct tensionless limit of the corresponding parent tensile theories for
the Induced and Flipped cases. Furthermore, we discovered that in multi-
dimensional compactification, as opposed to circle compactification, an
infinite number of physical states emerge at each level for the flipped
spectrum. Overall, our findings shed light on the intricate interplay between
constant $B$ fields, compactification, and the tensionless limit, deepening
our understanding of these fundamental aspects in string theory.

###  7.2 Future directions

As we mentioned at the very beginning, the study of null strings from a
formalism viewpoint is still very nascent and requires close investigation. In
this work, which is hopefully another one in a series of many, we are just
starting to delve deeper into the theory of null string compactifications.
With the inclusion of the antisymmetric field, we are one step closer to
understanding tensionless sigma models and role of interactions therein. The
road ahead is intriguing, and one can look at various related problems in
close connection with the current one. We discuss some immediate avenues
below.

Null duality and connection to other “corners”: As already discussed in the
text, the manifestations of sectorised T-duality for our case takes the theory
out of the null form of $\mathcal{G}$. Although one could argue these two
theories are related via the very same spectrum, more is required to show the
equivalence of these theories. Fuelled by the recent observations
Gomis:2023eav , we also theorized that our formalism indicates sectorised
T-duality transformations on Euclidean compact spaces for null strings may
lead to tensile strings in target space equipped with a degenerate metric. By
construction, our compact directions are spacelike, unlike the ones in
Gomis:2023eav  giving rise to pure Carrollian target space structures. However
this observation keeps open the possibility that something similar could also
be done for our case, involving a timelike compact circle, so that a
sectorised duality transforms our null string to some opposite unexplored
corner with tensile strings on Carroll backgrounds.

Bose-Einstein condensation and $B$ fields: As we mentioned in the Induced
spectrum discussion, the Induced vacuum state in the non-compact case simply
boils down to a space-filling Neumann boundary state Bagchi:2019cay . There
has been hints that this appearance of what essentially are D-brane boundary
states, are nothing but a Bose-Einstein like condensation of all closed
perturbative degrees of freedom on open ones. For compact cases these new
degrees of freedom are even more interesting, however they lack an inherent
T-dual picture, which is explicitly remedied by the presence of a constant $B$
field. It is imperative to understand whether the physics of this phase
transition can be directly linked to the more widely known phenomenon of open-
closed duality Seiberg:1999vs  in presence of a $B$ field.

Double field theory with degenerate metric: A nice observation coming out of
our investigation was the presence of a degenerate DFT metric in the
description of such tensionless strings. As we mentioned earlier, embedding of
degenerate (non-Riemannian) target spaces in the DFT metric has been explored
in recent literature. Although in these cases, the DFT metric itself is non-
degenerate and $O(d,d)$ covariance in unbroken, it turns out novel connections
to Carroll and Newton-Cartan geometries can be established Morand:2017fnv . It
would be important to explore how our setup fits into this picture, and how
degenerate and non-degenrate DFT metrics encode non-Riemannian geometric data
in different ways. It will also be interesting to find whether this degenerate
theory has an explicit worldsheet embedding, in line with the $O(d,d)$
covariant doubled string sigma model formulation Aldazabal:2013sca .

Target space complex structures: Most intriguingly, a lot still remains to be
understood about the target space of such tensionless strings. The absence of
part of $O(2,2;\mathbb{Z})$ symmetry in the $2d$ compact case can be
visualised in another way. As is well known, the $O(2,2;\mathbb{Z})$ has
$SL(2,\mathbb{Z})\otimes SL(2,\mathbb{Z})$ as subgroup. In case of usual
tensile string theory with a $B$ field $B\epsilon_{IJ}$ this symmetry can be
observed by defining complexified Kähler modulus ($T$) and complex structure
modulus ($U$) on the torus as below Blumenhagen:2013fgp .

$\begin{split}T=\frac{1}{\alpha^{\prime}}(BR_{1}R_{2}+i\sqrt{\det
G})=(T_{1}+iT_{2}),\leavevmode\nobreak\ \leavevmode\nobreak\
U=\frac{1}{G_{11}}(G_{12}+i\sqrt{\det G})=U_{1}+iU_{2}\end{split}$ (194)

The squares of the left and right moving momentum can be expressed in terms of
these structure constants as

$p^{2}_{L}=\frac{\alpha^{\prime}}{4T_{2}U_{2}}|k_{2}-Uk_{1}+\overline{T}(\omega_{1}+U\omega_{2})|^{2},\leavevmode\nobreak\
\leavevmode\nobreak\
p^{2}_{R}=\frac{\alpha^{\prime}}{4T_{2}U_{2}}|k_{2}-Uk_{1}+T(\omega_{1}+U\omega_{2})|^{2}$
(195)

In the above $\overline{T}$ denotes the complex conjugate of $T$. Since the
mass spectrum of the tensile theory is the sum of $p^{2}_{L}$ and $p^{2}_{R}$,
the compactified part of mass spectrum was entirely expressed in terms of
these two complex structures. The mass spectrum for tensile string had an
$SL(2,\mathbb{Z})_{T}$ symmetry, with the sectorised duality transformation
leading to $T\leftrightarrow U$ Blumenhagen:2013fgp . The implication is that
the mass spectrum will also have an $SL(2,\mathbb{Z})_{U}$ symmetry.

In case of tensionless strings too, the momenta $K^{2}_{L}$ and $K^{2}_{R}$
can be expressed in terms of appropriately defined complex
structures161616$\alpha^{\prime}$ in (194) and (195) in this case has to be
replaced by $c^{\prime}$.. However, the mass spectrum of the tensionless
string, as we can see in (92) also has a $2K^{I}_{L}K_{I\hskip 1.42262ptR}$
part, which cannot be expressed as a function of the complex structure
constants. As we have already discussed, the $g_{D_{i}}$ transformation does
not lead us to a definite $G$ and $B$, which means we won’t get any
transformed values of $T$ and $U$ either, so $T\leftrightarrow U$ cannot
happen here. As a result, we cannot observe the $SL(2,\mathbb{Z})_{U}$
symmetry in the mass spectrum. However, the $SL(2,\mathbb{Z})_{T}$ symmetry
can still be observed. The direct tensionless limit of the complex Kähler
modulus ($T$) evolves to a real number:

$\displaystyle\lim_{\alpha^{\prime}\to\infty}T_{1}=bR_{1}R_{2}\hskip
14.22636pt\lim_{\alpha^{\prime}\to\infty}T_{2}=0.$ (196)

We know for tensile strings, the shift symmetry implies symmetry under the
transformation $T\to T+1$ i.e. $T_{1}\to T_{1}+1$, $T_{2}\to T_{2}$. Since
$T\to T_{1}$ at tensionless limit, here the mass spectrum is invariant under
the $T_{1}\to T_{1}+1$ transformation (also called the modular
T-transformation). Also the inversion symmetry leads to the transformation
$T_{1}\to-1/T_{1}$ (also called the modular S-transformation). It is well
known that the modular T-transformation and modular S-transformation together
generate the entire $SL(2,\mathbb{Z})$ group, which means
$SL(2,\mathbb{Z})_{T}$ still survives in the tensionless limit. In earlier
sections we found indication that for $T^{2}$ compactification, individual
$g_{D_{i}}$ transformations connect tensionless theory to other limits of
string theory involving a degenerate torus. Our expectation is that the lost
$SL(2,\mathbb{Z})_{U}$ symmetry probably resides on those yet to be explored
corners. We plan to come back to these pressing questions in future
communications.

## Acknowledgements

It is a pleasure to thank Arjun Bagchi for numerous illuminating discussions
and comments on the manuscript. The authors would also like to thank Stephane
Detournay, Sabyasachi Dhar, Daniel Grumiller, Jelle Hartong, Johannes
Lahnsteiner, Shailesh Lal, Hisayoshi Muraki, Niels Obers, Shahin Sheikh-
Jabbari, Stefan Vandoren, and Ziqi Yan for many useful discussions. Initial
version of this work was presented at OIST Okinawa and BITS Goa. ArB is
supported in part by an OPERA grant and a seed grant NFSG/PIL/2023/P3816 from
BITS-Pilani. RC would like to acknowledge the partial support of CSIR grant
File No: 09/092(0991)/2018-EMR-I and PHY/SERB/2019525. RC would also like to
thank TU Wien, University of Utrecht, University of Amsterdam, NORDITA, DESY
Hamburg, ULB Brussels, I.I.T. Bombay and BITS Goa for hospitality during
various stages of this work. PP is partially supported by SPO/SERB/PHY/2019525
and by Institute Post Doctoral Fellowship (IPDF) from IIT kanpur. PP would
also like to thank Kyoto University, OIST Okinawa and BITS Goa for kind
hospitality during various stages of this work.

##  Appendix A Symmetries of tensile string spectrum over two compact
directions

Let us consider Polyakov action for the tensile string theory compactified on
$T^{2}$. The mass spectrum for this theory is given in (138). Considering the
$B$ field along the torus to be $B_{IJ}=b\epsilon_{IJ}$ the mass spectrum of
tensile string is as given in (150). For finite tension one can always take
$\alpha^{\prime}=1$. In that case the compactfied part of the mass spectrum
will be

$\displaystyle
m^{2}=\frac{k_{1}^{2}}{R_{1}^{2}}+\frac{k_{2}^{2}}{R_{2}^{2}}+2b\Bigg{(}\frac{R_{1}}{R_{2}}k_{2}\omega_{1}-\frac{R_{2}}{R_{1}}k_{1}\omega_{2}\Bigg{)}+(1+b^{2})(R_{1}^{2}\omega_{1}^{2}+R_{2}^{2}\omega_{2}^{2})$
(197)

This mass spectrum can be rewritten in the following form.

$\begin{split}m^{2}=\big{[}\omega^{T}\hskip 8.53581ptk^{T}\big{]}\hskip
5.69054pt\mathcal{G}^{-1}_{\alpha^{\prime}}\begin{bmatrix}\omega\\\
k\end{bmatrix},\hskip
14.22636pt\mathcal{G}^{-1}_{\alpha^{\prime}}=\begin{bmatrix}(1+b^{2})R^{2}_{1}&0&0&\frac{bR_{1}}{R_{2}}\\\
0&(1+b^{2})R^{2}_{2}&-\frac{bR_{2}}{R_{1}}&0\\\
0&-\frac{bR_{2}}{R_{1}}&\frac{1}{R^{2}_{1}}&0\\\
\frac{bR_{1}}{R_{2}}&0&0&\frac{1}{R^{2}_{2}}\end{bmatrix}\end{split}$ (198)

This mass spectrum shows complete $O(2,2;\mathbb{Z})$ symmetry. For $T^{2}$,
there can be only two sectorised duality transformation

$\displaystyle g_{D_{1}}=\begin{bmatrix}0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\
0&0&0&1\end{bmatrix},\hskip 14.22636ptg_{D_{2}=}\begin{bmatrix}1&0&0&0\\\
0&0&0&1\\\ 0&0&1&0\\\ 0&1&0&0\end{bmatrix}$ (199)

This transformation leads to the following transformations on
$\mathcal{G}^{-1}_{\alpha^{\prime}}$

$\begin{split}\mathcal{G}^{-1}_{\alpha^{\prime}}\xrightarrow{g_{D_{1}}}{}g_{D_{1}}\mathcal{G}^{-1}_{\alpha^{\prime}}g^{T}_{D_{1}}&=\begin{bmatrix}\frac{1}{R^{2}_{1}}&-\frac{bR_{2}}{R_{1}}&0&0\\\
-\frac{bR_{2}}{R_{1}}&(1+b^{2})R^{2}_{2}&0&0\\\
0&0&(1+b^{2})R^{2}_{1}&\frac{bR_{1}}{R_{2}}\\\
0&0&\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}\end{bmatrix}\\\
\mathcal{G}^{-1}_{\alpha^{\prime}}\xrightarrow{g_{D_{2}}}{}g_{D_{2}}\mathcal{G}^{-1}_{\alpha^{\prime}}g^{T}_{D_{2}}&=\begin{bmatrix}(1+b^{2})R^{2}_{1}&\frac{bR_{1}}{R_{2}}&0&0\\\
\frac{bR_{1}}{R_{2}}&\frac{1}{R^{2}_{2}}&0&0\\\
0&0&\frac{1}{R^{2}_{1}}&-\frac{bR_{2}}{R_{1}}\\\
0&0&-\frac{bR_{2}}{R_{1}}&(1+b^{2})R^{2}_{2}\end{bmatrix}.\end{split}$ (200)

By definition $g_{D_{1}}$ and $g_{D_{2}}$ flips
$k_{1}\leftrightarrow\omega_{1}$ and $k_{2}\leftrightarrow\omega_{2}$ and the
transformed inverse generalised metric along with the transformed
$\\{\omega,k\\}$ vector gives us the same mass spectrum. As highlighted
earlier in (5.3), the transformed inverse generalised metric correspond to a
new $G_{IJ}$ with off diagonal terms and $B_{IJ}=0$. This spectrum also obeys
the shift symmetry. As one can see, the mass spectrum is invariant under the
transformation (107), i.e.

$\displaystyle b\to b+\frac{n}{R_{1}R_{2}}\hskip 14.22636ptk_{I}\to
k_{I}+n\epsilon_{IJ}\omega^{J}\hskip 14.22636ptn\in\mathbb{Z}.$

We recall that (107) belongs to the category defined in (94). This is also
invariant under the basis change transformation defined in (85). It can be
shown that under the basis change transformation the metric of the torus
$G_{IJ}$ and the $B$ field $B_{IJ}$ will transform as below

$\displaystyle\mathbb{B}^{\prime}=C\mathbb{B}(C^{T})\hskip
14.22636ptG^{\prime}=CG(C^{T}),$ (201)

which is the same as (87).

##  Appendix B Double sectorised duality calculation for $d=3$

Let us recall $\mathbb{B}_{IJ}^{1}$ from (105)

$\displaystyle\mathbb{B}_{IJ}^{1}=\left(\begin{array}[]{ccc}0&a{R_{1}}{R_{2}}&0\\\
-a{R_{1}}{R_{2}}&0&0\\\ 0&0&0\\\ \end{array}\right)$ (205)

Using $\eqref{a1}$ along with $G_{IJ}=R_{I}^{2}\delta_{IJ}$, we compute
$\mathcal{G}$ as defined in (78) and obtain the following

$\displaystyle\mathcal{G}=\left(\begin{array}[]{cccccc}a^{2}R_{1}^{2}&0&0&0&-\frac{aR_{1}}{R_{2}}&0\\\
0&a^{2}R_{2}^{2}&0&\frac{aR_{2}}{R_{1}}&0&0\\\ 0&0&0&0&0&0\\\
0&-\frac{aR_{2}}{R_{1}}&0&\frac{1}{R_{1}^{2}}&0&0\\\
\frac{aR_{1}}{R_{2}}&0&0&0&\frac{1}{R_{2}^{2}}&0\\\
0&0&0&0&0&\frac{1}{R_{3}^{2}}\\\ \end{array}\right)$ (212)

From the definition of $g_{D_{i}}$ given in (90), it can be easily seen that

$\displaystyle g_{D_{1}}g_{D_{2}}=\left(\begin{array}[]{cccccc}0&0&0&1&0&0\\\
0&0&0&0&1&0\\\ 0&0&1&0&0&0\\\ 1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\ 0&0&0&0&0&1\\\
\end{array}\right)$ (219)

Now let us recall that, under any transformation $A\in O(d,d;\mathbb{Z})$,
$\mathcal{G}$ will transform like

$\mathcal{G}\to\mathcal{G}^{\prime}=A\mathcal{G}A^{T}$

For $A=g_{D_{1}}g_{D_{2}}$, $\mathcal{G}^{\prime}$ will be given by

$\displaystyle\mathcal{G}^{\prime}=\left(\begin{array}[]{cccccc}\frac{1}{R_{1}^{2}}&0&0&0&-\frac{aR_{2}}{R_{1}}&0\\\
0&\frac{1}{R_{2}^{2}}&0&\frac{aR_{1}}{R_{2}}&0&0\\\ 0&0&0&0&0&0\\\
0&-\frac{aR_{1}}{R_{2}}&0&a^{2}R_{1}^{2}&0&0\\\
\frac{aR_{2}}{R_{1}}&0&0&0&a^{2}R_{2}^{2}&0\\\
0&0&0&0&0&\frac{1}{R_{3}^{2}}\\\ \end{array}\right)$ (226)

This $\mathcal{G}^{\prime}$ can be computed from $B$ field
$\mathbb{B}_{IJ}^{\prime}$ and metric $G_{IJ}^{\prime}$ using the same
expression given in (78), where $\mathbb{B}_{IJ}^{\prime}$ and
$G_{IJ}^{\prime}$ are given by

$\displaystyle\mathbb{B}_{IJ}^{\prime}=$
$\displaystyle\begin{bmatrix}0&&-\frac{1}{aR_{1}R_{2}}&&0\\\
\frac{1}{aR_{1}R_{2}}&&0&&0\\\ 0&&0&&0\\\ \end{bmatrix},\hskip
14.22636ptG_{IJ}^{\prime}=\begin{bmatrix}\frac{1}{a^{2}R_{1}^{2}}&&0&&0\\\
0&&\frac{1}{a^{2}R_{2}^{2}}&&0\\\ 0&&0&&R_{3}^{2}\\\ \end{bmatrix}.$

##  Appendix C Tensionless closed strings from limit

The generic mode expansion for the tensile bosonic string is given by

$X^{\mu}(\tau,\sigma)=x^{\mu}+\sqrt{\frac{\alpha^{\prime}}{2}}\big{[}(\alpha_{0}^{\mu}+\tilde{\alpha}_{0}^{\mu})\tau+(\alpha_{0}^{\mu}-\tilde{\alpha}_{0}^{\mu})\sigma\big{]}+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\left[\alpha^{\mu}_{n}e^{-in(\tau+\sigma)}+\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\sigma)}\right].$
(227)

Here $\alpha_{n}^{\mu}$ and $\tilde{\alpha}_{n}^{\mu}$ are left moving and
right moving oscillators. For closed bosonic string, periodicity condition
$(X^{\mu}(\tau,\sigma+2\pi)=X^{\mu}(\tau,\sigma))$ demands,
$\alpha_{0}^{\mu}=\tilde{\alpha}_{0}^{\mu}$, which reduces the above mode
expansion to,

$X^{\mu}(\tau,\sigma)=x^{\mu}+\sqrt{2\alpha^{\prime}}\alpha_{0}^{\mu}\tau+i\sqrt{\frac{\alpha^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\left[\alpha^{\mu}_{n}e^{-in(\tau+\sigma)}+\tilde{\alpha}^{\mu}_{n}e^{-in(\tau-\sigma)}\right].$
(228)

Now in order to obtain closed bosonic tensionless string mode expansion, we
consider the following limit on the worldsheet coordinates,

$\tau\to\epsilon\tau,\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sigma\to\sigma\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\alpha^{\prime}\to c^{\prime}/\epsilon,\leavevmode\nobreak\
\leavevmode\nobreak\ \epsilon\to 0,$ (229)

where $c^{\prime}$ is a finite parameter. The mode expansion (228) in this
limit takes the following form

$X^{\mu}(\tau,\sigma)=x^{\mu}+\sqrt{2\epsilon
c^{\prime}}\alpha_{0}^{\mu}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}\left[\frac{\alpha_{n}^{\mu}-\tilde{\alpha}_{-n}^{\mu}}{\sqrt{\epsilon}}-in\tau\sqrt{\epsilon}(\alpha_{n}^{\mu}+\tilde{\alpha}_{-n}^{\mu})\right]e^{-in\sigma}.$
(230)

We now compare (230) with the tensionless mode expansion (31) given by

$X^{\mu}(\tau,\sigma)=x^{\mu}+\sqrt{\frac{c^{\prime}}{2}}B^{\mu}_{0}\tau+i\sqrt{\frac{c^{\prime}}{2}}\sum_{n\neq
0}\frac{1}{n}(A^{\mu}_{n}-in\tau B^{\mu}_{n})e^{-in\sigma}$ (231)

and obtain the relation between ($\alpha,\tilde{\alpha}$) with $A$’s and $B$’s
as:

$A_{n}^{\mu}=\frac{1}{\sqrt{\epsilon}}(\alpha_{n}^{\mu}-\tilde{\alpha}_{-n}^{\mu}),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
B_{n}^{\mu}=\sqrt{\epsilon}(\alpha_{n}^{\mu}+\tilde{\alpha}_{-n}^{\mu}).$
(232)

##  Appendix D More oscillator states from $T^{2}$ compactification

Here we will consider the states with {$r=1,s=0$} and $\\{r=0,s=1\\}$ for the
oscillator spectrum for $T^{2}$ compactification as given in (112). We begin
with the constraint satisfied by {$k_{I},\omega^{I}$} for the state
{$r=1,s=0$} which is given by

$\displaystyle k_{1}\omega_{1}+k_{2}\omega_{2}=-1$ (233)

$k_{1},\omega_{1}$ | $k_{2},\omega_{2}$ | $m^{2}$ |  Condition for $m^{2}<0$
---|---|---|---
$k_{1}=\omega_{1}=0$ | $k_{2}=1,\omega_{2}=-1$ | $m^{2}=\frac{1}{R^{2}}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\frac{1}{R^{2}}+b^{2}R^{2}<\frac{1}{c^{\prime}}$
$k_{2}=-1,\omega_{2}=1$ |  |
$k_{1}=0,\leavevmode\nobreak\ \omega_{1}\neq 0$ | $k_{2}=1,\omega_{2}=-1$ | $m^{2}=\left(\frac{1}{R}-bR\omega_{1}\right)^{2}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\left(\frac{1}{R}-bR\omega_{1}\right)^{2}+b^{2}R^{2}<-\frac{1}{c^{\prime}}$
$k_{2}=-1,\leavevmode\nobreak\ \omega_{2}=1$ | $m^{2}=\left(\frac{1}{R}+bR\omega_{1}\right)^{2}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\left(\frac{1}{R}+bR\omega_{1}\right)^{2}+b^{2}R^{2}<-\frac{1}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}=0$ | $k_{2}=1,\omega_{2}=-1$ | $m^{2}=\left(\frac{k_{1}}{R}-bR\right)^{2}+\frac{1}{R^{2}}-\frac{1}{c^{\prime}}$ | $\left(\frac{k_{1}}{R}-bR\right)^{2}+\frac{1}{R^{2}}<-\frac{1}{c^{\prime}}$
$k_{2}=-1,\leavevmode\nobreak\ \omega_{2}=1$ | $m^{2}=\left(\frac{k_{1}}{R}+bR\right)^{2}+\frac{1}{R^{2}}-\frac{1}{c^{\prime}}$ | $\left(\frac{k_{1}}{R}+bR\right)^{2}+\frac{1}{R^{2}}<-\frac{1}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}\neq 0$ | $k_{2}\neq 0,\leavevmode\nobreak\ \omega_{2}\neq 0$ | $m^{2}=(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}$ | $k_{2}=\omega_{1}bR^{2}$
| $-\frac{1}{c^{\prime}}$ | $(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}<\frac{1}{c^{\prime}}$
Table 4: Mass square of states belonging to level $r=1,s=0$ for different
$k_{I},\omega^{I}$.

The Table (4) provides us with the expression of $m^{2}$ for the state
{$r=1,s=0$} with the relevant constraint and the condition for getting
tachyons for given values of $k_{1},k_{2},\omega_{1}$ and $\omega_{2}$.
Similarly, we write the constraint satisfied by {$k_{I},\omega^{I}$} for the
state {$r=0,s=1$} which is given by

$\displaystyle k_{1}\omega_{1}+k_{2}\omega_{2}=1.$ (234)

The corresponding Table (5) with expression for $m^{2}$ for different values
of $k_{1},k_{2},\omega_{1}$ and $\omega_{2}$ is given below.

$k_{1},\omega_{1}$ | $k_{2},\omega_{2}$ | $m^{2}$ |  Condition for $m^{2}<0$
---|---|---|---
$k_{1}=\omega_{1}=0$ | $k_{2}=1,\omega_{2}=1$ | $m^{2}=\frac{1}{R^{2}}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\frac{1}{R^{2}}+b^{2}R^{2}<\frac{1}{c^{\prime}}$
$k_{2}=-1,\omega_{2}=-1$ |  |
$k_{1}=0,\leavevmode\nobreak\ \omega_{1}\neq 0$ | $k_{2}=1,\omega_{2}=1$ | $m^{2}=\left(\frac{1}{R}-bR\omega_{1}\right)^{2}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\left(\frac{1}{R}-bR\omega_{1}\right)^{2}+b^{2}R^{2}<-\frac{1}{c^{\prime}}$
$k_{2}=-1,\leavevmode\nobreak\ \omega_{2}=-1$ | $m^{2}=\left(\frac{1}{R}+bR\omega_{1}\right)^{2}+b^{2}R^{2}-\frac{1}{c^{\prime}}$ | $\left(\frac{1}{R}+bR\omega_{1}\right)^{2}+b^{2}R^{2}<-\frac{1}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}=0$ | $k_{2}=1,\omega_{2}=1$ | $m^{2}=\left(\frac{k_{1}}{R}+bR\right)^{2}+\frac{1}{R^{2}}-\frac{1}{c^{\prime}}$ | $\left(\frac{k_{1}}{R}+bR\right)^{2}+\frac{1}{R^{2}}<-\frac{1}{c^{\prime}}$
$k_{2}=-1,\leavevmode\nobreak\ \omega_{2}=-1$ | $m^{2}=\left(\frac{k_{1}}{R}-bR\right)^{2}+\frac{1}{R^{2}}-\frac{1}{c^{\prime}}$ | $\left(\frac{k_{1}}{R}-bR\right)^{2}+\frac{1}{R^{2}}<-\frac{1}{c^{\prime}}$
$k_{1}\neq 0,\leavevmode\nobreak\ \omega_{1}\neq 0$ | $k_{2}\neq 0,\leavevmode\nobreak\ \omega_{2}\neq 0$ | $m^{2}=(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}$ | $k_{2}=\omega_{1}bR^{2}$
| $-\frac{1}{c^{\prime}}$ | $(\omega_{1}^{2}+\omega^{2}_{2})\left(\frac{k_{2}}{R\omega_{1}}-bR\right)^{2}<\frac{1}{c^{\prime}}$

Table 5: Mass square of states belonging to level $r=0,s=1$ for different
$k_{I},\omega^{I}$.

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