11institutetext: Ecole Polytechnique, Palaiseau, France 22institutetext:
Accenta, Paris, France 33institutetext: IRT SystemX, Palaiseau, France

# Conv-NILM-Net, a causal and multi-appliance model for energy source
separation††thanks: Supported by Accenta.ai

Mohamed Alami C 1122 0000-0002-9091-7095  Jérémie Decock 22 Rim kaddah 33
Jesse Read 11 0000-0002-1013-6724

###### Abstract

Non-Intrusive Load Monitoring (NILM) seeks to save energy by estimating
individual appliance power usage from a single aggregate measurement. Deep
neural networks have become increasingly popular in attempting to solve NILM
problems. However most used models are used for Load Identification rather
than online Source Separation. Among source separation models, most use a
single-task learning approach in which a neural network is trained exclusively
for each appliance. This strategy is computationally expensive and ignores the
fact that multiple appliances can be active simultaneously and dependencies
between them. The rest of models are not causal, which is important for real-
time application. Inspired by Convtas-Net, a model for speech separation, we
propose Conv-NILM-net, a fully convolutional framework for end-to-end NILM.
Conv-NILM-net is a causal model for multi appliance source separation. Our
model is tested on two real datasets REDD and UK-DALE and clearly outperforms
the state of the art while keeping a significantly smaller size than the
competing models.

###### Keywords:

NILM Single Channel Source Separation Deep Learning.

##  1 Introduction

In 2018, 26.1% of the total energy consumption in EU was attributed to
households. This consumption mainly serves a heating purpose (78.4%).
Moreover, most of the residential energy consumption is covered by natural gas
(32.1%) and electricity (24.7%), while renewables account for just 19.5% [6].
However, as solar and wind generation rely on weather conditions, challenges
due to intermittent generation have to be solved, and solutions for energy
management such as demand response and photovoltaic (PV) battery management
can play a key role in this regard. Machine Learning has proven to be a viable
solution for smart home energy management [32]. These methods autonomously
control heating and domestic hot water systems, which are the most relevant
loads in a dwelling, helping consumers to reduce energy consumption and also
improving their comfort.

An efficient energy management system has to take into account users habits in
order to anticipate their behaviour. However, comfort is hard to quantify as
it remains purely subjective. We argue that in an energy management context,
the users are the only ones that can offer a proper evaluation of their own
comfort. Hence, a satisfactory hypothesis is to consider that their current
behaviour and habits are the ones that optimise their comfort. Therefore, an
efficient energy management system is one that can anticipate users habits,
optimise consumption levels (for example by deciding which source to use,
temperature settings etc.) while offering solutions that alter users known
habits as little as possible.

Learning users’ habits in a household is a hard problem mainly regarding data
acquisition. The possible behaviours are diverse, if not unique, while
monitoring inhabitants is not acceptable as it is a privacy infringement. From
an energy provider perspective, the only available information is the
household’s total power consumption. A solution is therefore to decompose this
consumption into the consumptions induced by each appliance in the household.
The resulting disaggregated power time series can then be used as an input for
a machine learning algorithm in order to learn consumption habits.

Energy disaggregation (also called non-intrusive load monitoring or NILM) is a
computational technique for estimating the power demand of individual
appliances from a single meter which measures the combined demand of multiple
appliances. The NILM problem can be formulated as follows: Let $\bar{y}(t)$
the aggregated energy consumption measured at time $t$. With no loss of
generality, $\bar{y}(t)$ can represent the active power (The power which is
actually consumed or utilised in an AC Circuit in kW). Then $\bar{y}(t)$ can
be expressed as in:

$\bar{y}(t)=\sum_{i=1}^{C}y^{(i)}(t)+e(t)$ (1)

Where $C$ is the number of appliances, $y^{(i)}$ the consumption induced by
appliance $i$ and $e(t)$ some noise. The aim is to find $y^{(i)}$ given
$\bar{y}(t)$.

There exist two approaches for NILM, namely load identification and source
separation. In the first case, a first step called signature detection
corresponds to the activation of a given appliance then a classification
algorithm classifies the appliance category. The idea behind load
identification is to build appropriate features called load signatures that
allow to easily distinguish the referenced appliance from others within the
installation. In the latter case, separation is directly obtained while
retrieving the source signal.

In order to manage a building efficiently, for example using a reinforcement
learning agent, it is necessary to use a model that performs source separation
while being causal. In signal processing, a causal model is a model that
performs the needed task (here source separation) without looking beyond time
$t$ rather than having to look in the future as presented in figure 3. The
model can then be used as a backend for prediction, which is necessary for
energy management.

We propose Conv-NILM-net, a fully convolutional and causal neural network for
end-to-end energy source separation. Conv-NILM-net is inspired from Conv-
TasNet [20], a convolutional model for speech separation. The model does not
require more quantities than active power and disaggregates the signal for
multiple appliances at once. We evaluate it on REDD and UK-DALE datasets,
compared to recent models and achieves state of the art performance. Figure 1
presents an overview of the model and table 1 summarises the notations used
throughout this paper.

Figure 1: Overview of Conv-NILM-net. It is composed of 2 blocs, the encoder/decoder and the separator. The encoder first projects the signal into a latent space, the separator disaggregates it into $C$ corresponding to each appliance by learning $C$ masks applied to input signal, then the decoder projects the $C$ signals to the input space Table 1: Summary of notation. $\bar{y}(t)$ | Aggregated energy consumption at time $t$, $\bar{y}(t)\in\mathbb{R}^{T}$
---|---
$C$ | Number of appliances
$T$ | Length of a signal
$y^{(i)}(t)$ | True energy consumption of appliance $i$ at time $t$, $y^{(i)}(t)\in\mathbb{R}^{T}$
$\hat{y}^{(i)}(t)$ | Predicted energy consumption of appliance $i$ at time $t$, $y^{(i)}(t)\in\mathbb{R}^{T}$
$Z$ | latent space representation of the mixture signal, $Z\in\mathbb{R}^{N\times K}$
$m_{i}$ | learned mask for appliance $i$, $m_{i}\in\mathbb{R}^{N}$
$s_{i}$ | filtered signal from the encoder for appliance i, $s_{i}\in\mathbb{R}^{N}$
$N$ | Number of filters in encoder-decoder
$L$ | Length of the filters
$B$ | Number of channels in bottleneck and the residual paths’ $1\times 1$-conv blocks
$H$ | Number of channels in convolutional blocks
$P$ | Kernel size in convolutional blocks
$X$ | Number of convolutional blocks in each repeat
$R$ | Number of repeats

##  2 Related Work

Most approaches in the literature are load identification approaches that
predict the state of an appliance (on/off) and predict the average consumption
of the given appliance during a certain period of time. Four appliance models
are usually considered:

  * •

Type I On/off devices: most appliances in households, such as bulbs and
toasters;

  * •

Type II Finite-State-Machines (FSM): the appliances in this category present
states, typically in a periodical fashion. Examples are washer/dryers,
refrigerators, and so on;

  * •

Type III Continuously Varying Devices: the power of these appliances varies
over time, but not in a periodic fashion. Examples are dimmers and tools.

  * •

Type IV Permanent Consumer Devices: these are devices with constant power but
that operate 24 h, such as alarms and external power supplies.

Current NILM methods work well for two-state appliances, but it is still
difficult to identify some multi-state appliances, and even more challenging
with continuous-state appliances. One of the most noticeable approaches called
FHMM models each appliance as a hidden markov model (HMM) [13]. The HMM of
each appliance is modelled independently, each one contributing to the
aggregated power. AFAMAP [15] extends FHMM by predicting combinations of
appliances working states. In AFAMAP, the posterior is constrained into one
state change per time step. In [26], the authors propose a hierarchical FHMM
in order to stop imposing independence between between appliances. The
algorithm takes the active power as input and performs a clustering of the
correlated signals then trains an HMM on the identified clusters called super
devices. During the disaggregation step, the prediction is done using AFAMAP
on the super devices then the clustering is reversed to find the the original
appliance.

A critical step is the construction of load signatures or features that help
to uniquely identify all types of home appliances with different operation
modes. Event-based techniques have been employed to identify turn-on and turn-
off events using a variety of features like the active and reactive power [31,
2, 8], current and voltage harmonics [28, 27], transient behaviour
particularly during the activation and/or deactivation [17, 3], current
waveform characteristics [30]. Although the existing harmonic-based NILM
methods achieved high load identification accuracy, their applicability is
limited. The main drawback of this approach is that it requires harmonic
current signatures with respect to all possible combinations of devices.
Consequently, the complexity of this method increases exponentially with the
number of electrical devices.

Deep Learning approaches have consistently outperformed HMM-based methods.
Indeed, the number of features associated with the complexity induced by all
the possible devices combinations make deep learning a natural candidate for
NILM. In recent years, learning-based approaches were proposed to classify and
directly estimate the power consumption of type-1 and type-2 appliances from
an aggregated signal. Although FHMM-based NILM approaches are extensively used
for power disaggregation, their performance is limited to the accurate
approximation of appliance actual power consumption especially for type-2
(multi-state) and type-4 (always-on) appliances. Moreover, HMM-based methods
have been reported to suffer from scalability and generalisation, which limits
its real-world application. In contrast to classical event-based and state-
based approaches, deep neural networks are capable of dealing with time
complexity issues, scalability issues, and can learn very complex appliance
signatures if trained with sufficient data.

Most recent NILM works employing deep neural networks used 1/6-Hz or 1/3-Hz
sampled active power measurement as an input feature to train various deep
neural networks. such as long short-term memory (LSTM) networks [21, 24],
denoised autoencoder [11, 1] and Convolutional Neural Networks (CNN) [33, 29].
[11] proposed 3 different neural networks. A convolutional layer followed by
an LSTM to estimate the disaggregated signal from the global one. They also
used a denoising convolutional autoencoder to produce clean signals. The last
neural network estimates the beginning and the end time of each appliance
activation along with the mean consumption of each. [11] performs better than
FHMM however their model was unable to identify multi-state appliances. To
solve the multi-state appliance identification issue, [21] proposed a two-
layer bidirectional LSTM based DNN model. Similarly, [29] proposed a two-step
approach to identify multi-state appliances. They used a deep CNN model to
identify the type of appliances and then used a k-means clustering algorithm
to calculate the number of states of appliances.

Deep Learning also allowed source separation rather than load identification.
This approach is more difficult but offers precise estimation of the
consumption of each appliance in real time which includes continuous state
appliances. [33, 9] proposed sequence-to-point learning-based CNN architecture
with only active power as an input feature. In [4] gated linear unit
convolutional layers [5] are used to extract information from the sequences of
aggregate electricity consumption. In [23], the authors used a deep recurrent
neural network using multi-feature input space and post-processing. First, the
mutual information method was used to select electrical parameters that had
the most influence on the power consumption of each target appliance. Second,
selected parameters were used to train the LSTM for each target appliance.
Finally, a post-processing technique was used at the disaggregation stage to
eliminate irrelevant predicted sequences, enhancing the classification and
estimation accuracy of the algorithm.

In [8], the authors present WaveNILM which is a causal 1-D convolutional
neural network inspired by WaveNet [22] for NILM on low-frequency data. They
used various components of the complex power signal for NILM, current, active
power, reactive power, and apparent power. WaveNILM, requires no significant
alterations whether using one or many input signals. However, most of the
existing DNNs models for NILM use a single-task learning approach in which a
neural network is trained exclusively for each appliance. That is also the
case for WaveNILM. This strategy is computationally expensive and ignores the
fact that multiple appliances can be active simultaneously and dependencies
between them. In [7] the authors introduce UNet-NILM for multi-task
appliances’ state detection and power estimation, applying a multi-label
learning strategy and multi-target quantile regression. The UNet-NILM is a
one-dimensional CNN based on the U-Net architecture initially proposed for
image segmentation [25]. However, this model is not causal like WaveNILM.

Conv-NILM-net achieves the best of both worlds as it can handle source
separation for any type of appliance, for multiple appliances simultaneously,
it only needs the active power (although it is possible to other types of
current in the same time).

##  3 Conv-NILM-Net

The model aims at separating $C$ individual power sources
$y^{(i)}\in\mathbb{R}^{T}$, where $i\in\\{1,2,\dots,C\\}$ from a mixture of
signals representing the total consumption
$\bar{y}(t)=\sum_{i=1}^{C}y^{(i)}(t)+e(t)$ and $T$ is the length of the
waveform. Therefore it take as input a single channel time series
corresponding to the total consumption and outputs $C$ time series
corresponding to the consumption of each individual appliance. In this
section, we present and detail our proposed architecture. We will describe the
overall structure before focusing on the separation module.

###  3.1 Overall structure

Conv-NILM-Net is an adaptation of ConvTas-net [20]. Conv-Tasnet was originally
only designed for speech separation and limited to two speakers. We propose an
adaptation to energy load source separation with theoretically no limitation
to the number of appliances. Our fully convolutional model is trainable end-to
end and uses the aggregated active power as only input making the training
easily deployable (no additional costly features needed).

Conv-NILM-net architecture consists of two parts: an encoder/decoder, and a
separator. The encoder generates a multidimensional representation of the
mixture signal; the separator learns masks applied to this representation to
decompose the mixture signal, then the decoder translates the obtained signals
from the encoded representation to the classic active power. The masks are
found using a temporal convolutional network (TCN) consisting of stacked 1-D
dilated convolutional blocks, which allows the network to model the long-term
dependencies of the signal while maintaining a small model size.

Using encoder filters of legnth $L$, the model first segments the input total
consumption into $K$ overlapping frames $\bar{y}_{k}\in\mathbb{R}^{L}$,
$k=1,2,\dots,K$ each of length $L$ with stride $S$. $\bar{y}_{k}$ is
transformed into a $N$-dimensional representation, $Z\in\mathbb{R}^{N\times
K}$:

$Z=\mathcal{F}(w\cdot\bar{Y})$ (2)

where $Y\in\mathbb{R}^{L\times K}$ and $w\in\mathbb{R}^{N\times K}$ the $N$
learnable basis filters of length $L$ each. $Z$ represents the latent space
representation of the mixture series while $\mathcal{F}$ is a non-linear
function. To ensure that the representation is non-negative, Conv-tasnet [20]
uses the rectified linear unit (ReLU). However, this choice leads to a
vanishing gradient behaviour, driving the norm of the gradients towards 0 thus
making the model collapse as it eventually outputs null signals. Therefore, we
replace ReLU with Leaky ReLU and only use ReLU for the last layer of the
separation masks to enforce positive outputs.

The separator predicts a representation for each source by learning a mask in
this latent space. It is performed by estimating $C$ masks
$m_{i}\in\mathbb{R}^{N}$. The representation of each source
$s_{i}\in\mathbb{R}^{N}$, is then calculated by applying the corresponding
mask $m_{i}$ to the mixture representation, using element-wise multiplication:

$s_{i}=Z\odot m_{i}$ (3)

In Conv-tasnet as well as in [19], the masks were constrained such that
$\sum_{i=1}^{C}m_{i}=1$. This was applied based on the assumption that the
encoder-decoder architecture can perfectly reconstruct the input mixture.
indeed, in their model, $e(t)=0$, $\forall t$. This assumption cannot be made
in a NILM context, it is therefore relaxed.

The input signal of each source is then reconstructed by the decoder:

$\hat{y}^{(i)}=s_{i}\cdot V$ (4)

where $V\in\mathbb{R}^{N\times L}$ are the decoder filters, each of length
$L$.

###  3.2 Separation module

The separator is mainly based on a temporal convolutional network (TCN) [16]
and is detailed in figure 3. Temporal convolutions require the use of dilated
convolutions which aim to increase the receptive field. Indeed, pooling or
strided convolutions are usually implemented for this purpose, however they
reduce the resolution. Dilated convolutions allow exponential expansions of
the receptive field without loss of resolution, while achieving same
computation and memory costs. These are simply implemented by defining a
spacing between the values in a kernel as illustrated in figure 3.

Figure 2: Causal and dilated convolutions.

Figure 3: Detailed representation of the separator.

In [21], the authors used LSTM [18] for NILM. This architecture can handle
long sequences but suffers from the vanishing gradient issue while being
computationally costly. We argue that a more efficient approach is to make use
of 1D-convolutions. As illustrated in figure 3, convolutions for time series
require future values (compared to the point of reference). During inference,
these values are not accessible making the use of this model unpractical for
prediction. This can be solved by giving a causal formulation to convolutions
where the present value only depends of past ones. Moreover, the
implementation is easy as it only requires an asymetric padding.

However relevant values become sparse. As in [20], Conv-NILM-net uses dilated
layers with exponentially increasing dilation rates. The dilation factors
increase exponentially to ensure a sufficiently large temporal context window
to take advantage of the long-range dependencies in the signal. Therefore the
dilation factors increase exponentially to ensure a sufficiently large
temporal context window to take advantage of the long-range dependencies in
the signal. Therefore TCN consists of 1-D convolutional blocks with increasing
dilation factors. Given kernels of length $L$ and $l$ layers, the receptive
field of Conv-NILM-net is of size $RF=2^{l}(L-1)$.

The output of the TCN is passed to a $1\times 1$ conv block for mask
estimation. This block also serves for dimensionality reduction and together
with a nonlinear activation function estimates $C$ mask vectors for the $C$
target sources. The last layer of the last bloc uses a ReLU activation
function to ensure the non-negativity of the outputs.

Contrary to speech separation, where simultaneous speeches are independent
from one another, it is not the case in NILM context where appliance
activations can be highly dependent. An elegant solution proposed in [10, 4],
can be to use gated linear units (GLU) [5] to replace LeakyReLU activation
functions. GLU allow the model to decide itself the relative importance of the
kernels by using two parallel convolutions with the first followed by a
sigmoid which result is then multiplied with the second convolution. The
output of the sigmoid acts like a mask that activates depending on the input
of the second convolution.

MSE, $L_{1}$, or even SI-SNR [20] losses are often used for NILM or source
separation problems. The MSE takes the average squared error on all time steps
for all disaggregated signals (ie appliances). We found that taking the mean
on appliances is detrimental to the learning process as the error is
distributed over all appliances. Therefore, the signals get mixed and
artefacts of most consuming appliances appear on the remaining ones as
illustrated in figure 4. We therefore choose to sum the error over all
appliance rather than averaging it. The window mean squared error is
calculated as:

$\text{WMSE}=\frac{1}{T}\sum_{t=0}^{T}\sum_{i=1}^{C}\left(\hat{y}^{(i)}(t)-\bar{y}^{(i)}(t)\right)^{2}$
(5)

Figure 4: Outputs of Conv-NILM-net of top 5 appliances of REDD building 1 when
trained using classic MSE. We observe the presence of the artefacts although
the MSE is minimized.

##  4 Experimental Methodology, Results, and Discussion

###  4.1 Datasets and parameters

Our experiments are done on two real-world datasets, Reference Energy
Disaggregation Dataset (REDD) [14] and UK-DALE [12]. REDD records the power
for 6 houses with sampling frequency of 1Hz for mains meter and 1/3Hz for
appliance-channel meters. We choose to disaggregate the five top appliances
for each building. UK-DALE data set published in 2015, records the power
demand from five UK houses. In each house we record both the whole-house mains
power demand every six seconds as well as power demand from individual
appliances every six seconds.

For REDD, we converted the disaggregated data to 1Hz using linear
extrapolation and kept $1/6$ Hz frequency for UK-DALE. Usually the data are
normalised and the series of each appliance are scaled individually. We argue
that appliance scaling is not practical as it is not possible to apply the
scaling factors to the global signal available during inference. Therefore we
chose to only use a minmax scaler for all appliances combined directly on the
mixture power signal. In some contributions like seq2seq and seq2point [9,
33], a sliding window of the aggregate power is used as the input sequence,
and the midpoint of the window corresponding to the target device reading is
used as the output. This preprocessing smooths the power loads and makes the
target values to retrieve easier. All results presented in section 4.3 for
these implementations were obtained using smoothing. The results presented for
Conv-NILM-NET were obtained without smoothing, making the difference in
performance even more noticeable.

For UK-DALE dataset, we compare our results to UNET-NILM [7] and seq2point
[9]. The constructed artificial aggregate consumption is obtained by taking
the summation of selected appliances plus additional one appliance (Television
in this setting). For UNET-NILM, the authors used a quantile filter to smooth
the signal. This is not required for Conv-NILM-Net.

In our implementation for Conv-NILM-Net, we used for, each dataset, one day as
input. This means that with 1Hz frequency, the input to Conv-NILM-net was
86400 points for REDD and 14400 for UK-DALE ($1/6$ Hz). The used parameters
for Conv-NILM net are: $N=32$; $L=48$; $B=2$; $H=P=X=3$; $R=2$. The meaning of
each notation is made available in table 1 where we kept the same notation as
in [20]. The model was trained for 2000 epochs using 10-fold cross-validation
and a batch size of $5$. We used Adam optimiser with an initial learning rate
$lr=0.01$, $\text{betas}=(0.9,0.999)$, $\text{eps}=0.01$.

###  4.2 Metrics

We evaluate the performance of the framework using the mean absolute error
(MAE). Estimated accuracy is also a common metric for evaluating disaggregated
power.

$\text{Est.Acc.}=1-\frac{\sum_{t=1}^{T}\sum_{i=1}^{C}|\hat{y}^{(i)}(t)-\bar{y}^{(i)}(t)|}{2\sum_{t=1}^{T}\sum_{i=1}^{C}\bar{y}^{(i)}(t)}$
(6)

Where $\hat{y}^{(i)}(t)$ is the predicted power level of appliance $i$ at time
$t$, and $\bar{y}^{(i)}(t)$ is the ground truth. The above expression yields
total estimated accuracy; if needed, the summation over $i$ can be removed
creating an appliance-specific estimation accuracy. We also report the Signal
Aggregate Error (SAE):

$\text{SAE}=\frac{|\hat{r}-r|}{r}$ (7)

where $\hat{r}$ and r represent the predicted total energy consumption of an
appliance and the ground truth one. SAE measures the total error in the energy
within a period, which is accurate for reporting daily power consumption even
if its consumption is inaccurate in every time point.

###  4.3 Results on REDD

Table 2 presents the results obtained on REDD dataset for five building. For
each building with disaggregated the top five appliances and reported the MAE,
estimated accuracy ans SAE. We tested 3 versions on Conv-NILM-net. We observe
that the causal+GLU tend to perform better on average but its results are very
close to the causal implementation while increasing the number of parameters
dramatically. We therefore tend to prefer the causal version of our model.

|  | Model
---|---|---
|  | Conv-NILM-NET | Causal | Causal + GLU
Building | Appliance | MAE | Est.Acc | SAE | MAE | Est.Acc | SAE | MAE | Est.Acc | SAE
Building 1 |  | Fridge
---
Washer dryer
Light
Sockets
Dishwasher
Total
| 0.049
---
0.005
0.063
0.015
0.006
0.027
| 0.900
---
0.937
0.970
0.874
0.916
0.919
| 0.058
---
0.074
0.030
0.092
0.102
0.071
| 0.006
---
0.002
0.007
0.005
0.002
0.005
| 0.981
---
0.993
0.980
0.984
0.993
0.986
| 0.053
---
0.088
0.021
0.131
0.095
0.078
| 0.004
---
0.002
0.008
0.006
0.003
0.004
| 0.987
---
0.990
0.984
0.992
0.997
0.989
| 0.051
---
0.103
0.033
0.191
0.070
0.054
Building 2 |  | Fridge
---
Washer dryer
Light
Sockets
Dishwasher
Total
| 0.038
---
0.018
0.012
0.002
0.0006
0.014
| 0.912
---
0.914
0.986
0.993
0.997
0.960
| 0.052
---
0.109
0.024
0.059
0.115
0.0718
| 0.032
---
0.031
0.002
0.001
0.001
0.014
| 0.939
---
0.940
0.981
0.991
0.993
0.969
| 0.068
---
0.098
0.104
0.032
0.029
0.066
| 0.041
---
0.034
0.008
0.006
0.003
0.018
| 0.956
---
0.967
0.987
0.990
0.992
0.978
| 0.054
---
0.087
0.099
0.058
0.031
0.066
Building 3 |  | Fridge
---
Washer dryer
Light
Sockets
Dishwasher
Total
| 0.006
---
0.007
0.009
0.009
0.007
0.006
| 0.820
---
0.997
0.863
0.961
0.960
0.920
| 0.089
---
0.061
0.123
0.091
0.085
0.90
| 0.072
---
0.003
0.037
0.040
0.032
0.037
| 0.822
---
0.993
0.854
0.941
0.940
0.910
| 0.078
---
0.065
0.098
0.080
0.096
0.083
| 0.009
---
0.004
0.006
0.007
0.005
0.006
| 0.856
---
0.993
0.900
0.942
0.989
0.936
| 0.068
---
0.087
0.098
0.126
0.078
0.091
Building 4 |  | Fridge
---
Washer dryer
Light
Sockets
Dishwasher
Total
| 0.003
---
0.002
0.002
0.001
0.0006
0.002
| 0.961
---
0.947
0.936
0.981
0.994
0.964
| 0.054
---
0.105
0.076
0.098
0.132
0.093
| 0.007
---
0.016
0.019
0.008
0.002
0.011
| 0.982
---
0.933
0.940
0.980
0.995
0.966
| 0.021
---
0.087
0.055
0.016
0.098
0.055
| 0.050
---
0.031
0.002
0.006
0.020
0.022
| 0.930
---
0.954
0.901
0.937
0.892
0.923
| 0.043
---
0.056
0.080
0.024
0.069
0.054
Building 5 |  | Fridge
---
Washer dryer
Light
Sockets
Dishwasher
Total
| 0.003
---
0.003
0.0006
0.0006
0.0008
0.002
| 0.883
---
0.992
0.999
0.966
0.913
0.950
| 0.005
---
0.001
0.0001
0.002
0.004
0.002
| 0.005
---
0.0009
0.0001
0.002
0.004
0.002
| 0.880
---
0.993
0.999
0.97
0.91
0.95
| 0.004
---
0.001
0.002
0.005
0.002
0.002
| 0.005
---
0.003
0.002
0.007
0.010
0.005
| 0.991
---
0.983
0.977
0.931
0.990
0.974
| 0.093
---
0.012
0.10
0.037
0.078
0.064
Table 2: Conv-NILM-Net scaled results on top five appliances REDD dataset.
Best average results are highlighted in bold.

Table 3 compares the performance of Conv-NILM-net with state of the art models
on 3 appliances that appear on REDD dataset. These appliances were selected as
they are the only one presented in [34]. We therefore were limited to these
appliance to compare our framework. We observe that our models outperform the
state of the art by a margin. It decreases the MAE by $45\%$ for the fridge,
$51\%$ for the microwave and even by $80\%$ for the dishwaser. The best
performing model is the causal model. In the appendix we present some outputs
of the model for buildings 1 to 4 from REDD. These were obtained when
disaggregating the top 5 appliances detailed in the same order as in table 2.

Model | Fridge | Microwave | Dishwasher
---|---|---|---
| seq2point
---
seq2seq
GLU-Res [4]
CNN-DI[34]
Conv-NILM-NET
Conv-NILM-NET (causal)
Conv-NILM-NET (GLU, causal)
| 28.104
---
30.63
21.97
26.801
14.67
14.21
15.02
| 28.199
---
33.272
25.202
19.455
9.67
8.51
9.76
| 20.048
---
19.449
33.37
17.665
3.56
3.29
3.31
Table 3: MAE results for Buiding 1 of REDD dataset.

###  4.4 Results on UK-DALE

Table 4 compares the MAE of our model on UK-DALE dataset with UNET-NILM and
seq2point. Our model outperforms the state of the art on the selected
appliances. The causal model performs the best again while the total average
is decreased by $33\%$ compared to seq2pont.

Appliance | Model
---|---
| 1D-CNN | UNET-NILM | Seq2point | Ours | Ours (causal) | Ours (causal + GLU)
Kettle | $20.390$ | $16.003$ | $2.16$ | $1.85$ | $1.9$ | $2.5$
Freezer | $18.583$ | $15.124$ | $8.136$ | $5.32$ | $5.01$ | $6.1$
Dish washer | $9.884$ | $6.764$ | $3.49$ | $2.42$ | $2.01$ | $2.55$
Washing machine | $15.758$ | $11.506$ | $4.063$ | $2.3$ | $2.15$ | $2.39$
Microwave | $9.690$ | $6.475$ | $1.305$ | $0.902$ | $0.91$ | $1.05$
Total | $14.86$ | $11.174$ | $3.831$ | $2.56$ | $2.40$ | $2.918$
Table 4: Experimental results (MAE) in the UK-DALE dataset.

Table 5 compares the size of Conv-NILM-net with state of the art models in
terms of number of parameters. We observe that the fully convolution
architecture of our model along with its particular architecture
(encoder/decoder + separator) allow to obtain state of the art results with a
model of approximately 40K parameters. This also possible because, contrary to
other models, we use a unique loss for only one task. For instance UNET-NILM
uses two separate loss functions, one to detect activation and an other to
regress the average consumption while Seq2point [9] uses bidirectional
residual networks which are very deep. It is also valuable to notice that
models like UNET-NILM are specialized on individual appliances, meaning that
in one needs to disaggregate 5 appliances, it requires 5 models, multiplying
the number of parameters.

Models | # parameters
---|---
seq2point | $29.2M$
seq2seq | $29.8M$
GLU-Res | $1.2M$
CNN-DI | $738K$
Conv-NILM-net | $41088$
Table 5: number of parameters

Finally, figure 5 presents some results on top five appliances of first 4
building of REDD dataset. For each building the appliances are presented in
the same order as in table 2. The left panels corresponds to the disaggregated
target signals and the right panels presents the predicted output from Conv-
NILM-net.

(a) building 1

(b) Building 2

(c) Building 3

(d) Building 4

Figure 5: Selected results on top five appliances of first 4 building of REDD
dataset. For each building the appliances are presented in the same order as
in table 2. The left panels correspond to the disaggregated target signals and
the right panels presents the predicted output from Conv-NILM-net.

##  5 Conclusion

In this work, we presented Conv-NILM-net, an adaptation of Convtas-net to non
intrusive load monitoring. We tested our model on two real world dataset and
showed that Conv-NILM-net outperforms the state of the art by a margin. We
presented 2 alternate models, one being causal and other using Gated Linear
Units (GLU). These models allowed accurate disaggregation of several
appliances at once while being much more smaller than their existing
counterparts. Finally, the causal model allows consumption prediction and is
ideal as input to an energy management system or a reinforcement learning
model. In future work, we will use causal conv-NILM-net as a prediction model
and test it in a reinforcement learning context. We will also test the GLU
augmented model to verify if this implementation effectively takes into
account appliances inter-dependencies and helps achieve better consumption
predictions.

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# Spherical objects and stability conditions on 2-Calabi–Yau quiver categories

Asilata Bapat and Anand Deopurkar and Anthony M. Licata

###### Abstract.

Consider a 2-Calabi–Yau triangulated category with a Bridgeland stability
condition. We devise an effective procedure to reduce the phase spread of an
object by applying spherical twists. Using this, we give new proofs of the
following theorems for 2-Calabi–Yau categories associated to ADE quivers:  (1)
all spherical objects lie in a single orbit of the braid group, and  (2) the
space of Bridgeland stability conditions is connected.

##  1\. Introduction

Suppose we have a category $\mathcal{C}$ with a measure of complexity of its
objects. To what extent can we simplify an object by applying suitable auto-
equivalences of $\mathcal{C}$? In this paper, we explore this question for
$2$-Calabi–Yau triangulated categories with respect to a measure of complexity
provided by a Bridgeland stability condition. We focus on the question of
simplifying spherical objects by applying a sequence of auto-equivalences
known as spherical twists.

Spherical objects play an important role in Calabi–Yau categories. Recall that
an object of an $n$-Calabi–Yau category is spherical if its endomorphism ring
is isomorphic to the cohomology ring of the $n$-sphere—this is the simplest
possible endomorphism ring, given the $n$-Calabi–Yau condition. Every
spherical object gives an auto-equivalence of the category, called a spherical
twist [7].

Spherical objects interact wonderfully with Bridgeland stability conditions.
For example, the Harder–Narasimhan factors of any spherical object with
respect to a Bridgeland stability condition are themselves spherical (see,
e.g. [3, Corollary 2.3]). In some cases (for example, for derived categories
of K3 surfaces), a stability condition is determined by its behaviour on
spherical objects (see [3]).

Our first set of results (§ 3) discusses how to simplify arbitrary objects of
a 2-Calabi–Yau triangulated category $\mathcal{C}$ by applying twists by
spherical objects that are stable with respect to a given stability condition.
We show that under suitable hypotheses, we can strictly decrease the phase
spread of an object by applying appropriate positive and negative twists. If
the set of possible spreads is a discrete subset of $\mathbb{R}$, then this
procedure must terminate. As a result, we get a particularly simple
representative of the orbit of the object under the spherical twist group.

The phase improvement results raise the question of classifying the stable
spherical objects with respect to a given stability condition. Our next set of
results (§ 4) achieves this for the 2-Calabi–Yau category
$\mathcal{C}_{\Gamma}$ associated to a quiver $\Gamma$. We briefly describe
the classification. The Grothendieck group
$K\left(\mathcal{C}_{\Gamma}\right)$, together with the $\hom$ pairing is
naturally isomorphic to the root lattice of $\Gamma$. The spherical objects of
$C_{\Gamma}$ map to real roots of $K\left(C_{\Gamma}\right)$. The action of
the Coxeter group $W_{\Gamma}$ on the root lattice lifts to an action of the
Artin–Tits braid group $B_{\Gamma}$ on $\mathcal{C}_{\Gamma}$. The standard
generators of $B_{\Gamma}$ act on $\mathcal{C}_{\Gamma}$ by spherical twists.

Let $\tau$ be a generic stability condition on $\mathcal{C}_{\Gamma}$ whose
heart is the standard heart. Given a root $w\in K\left(C_{\Gamma}\right)$,
there are multiple (in fact, infinitely many, even modulo triangulated shifts)
spherical objects with class $w$. However, only one of them (up to
triangulated shift) is $\tau$-stable. We give an explicit combinatorial
algorithm to construct this object. More precisely, suppose we have a minimal
expression for $w$ in terms of simple reflections applied to a simple root
$v$, say

$w=s_{v_{n}}\cdots s_{v_{1}}v.$

Let $P_{v}$ be the simple object of the standard heart of class $v$. We prove
that the unique $\tau$-stable object $P_{w}$ of class $w$ can be expressed as
a lift of the above expression:

$P_{w}=\sigma_{v_{n}}^{\pm 1}\cdots\sigma_{v_{1}}^{\pm 1}P_{v},$

and moreover we explicitly identify the exponents. The choice of signs in the
exponents is governed by the interaction of the root sequence for the
expression for $w$ with the central charge.

In the last section (§ 5), we give two applications of the simplification
procedure and of the classification of spherical stable objects mentioned
above. The first application is a new proof of the following theorem.

######  Theorem 1.1.

Let $\Gamma$ be a quiver of type $A_{n}$, $D_{n}$, or $E_{6}$, $E_{7}$,
$E_{8}$. The spherical objects of $\mathcal{C}_{\Gamma}$ lie in the
$B_{\Gamma}$ orbit of the simple objects of the standard heart.

In the main text, Theorem 1.1 is 5.2. A proof of this theorem in type $A$
appears in [6, 5] and may also follow for all $ADE$ types from the ideas of
[1].

The second application is a new proof of the following theorem.

######  Theorem 1.2.

Let $\Gamma$ be a quiver of type $A_{n}$, $D_{n}$, or $E_{6}$, $E_{7}$,
$E_{8}$. Any stability condition
$\tau\in\operatorname{Stab}(\mathcal{C}_{\Gamma})$ is in the $B_{\Gamma}$
orbit of a standard stability condition. Furthermore,
$\operatorname{Stab}(\mathcal{C}_{\Gamma})$ is connected.

In the main text, Theorem 1.2 is 5.5. Theorem 1.2 has been proved in [6, 5]
for type $A$, and in [1] for types $A$, $D$, and $E$.

We highlight that the classification of spherical stable objects and the phase
reduction procedure holds for any quiver $\Gamma$. We require the finite type
assumption to guarantee the termination of the procedure. It is likely that
termination works more generally (we have not found any counter-examples). We
hope to address this question in the future.

### Acknowledgements

We are grateful to Fabian Haiden, Ailsa Keating, Alexander Polishchuk, and
Catharina Stroppel for discussions that prompted us to write this paper. We
are indebted to Tom Bridgeland for his encouragement and for discussions and
suggestions about phase improvement.

##  2\. Background on stability conditions and spherical objects

Let $\mathcal{C}$ be an arbitrary 2-Calabi–Yau triangulated category. We begin
by recalling the notion of spherical objects, spherical twists, and bottom/top
phases with respect to a stability condition. Recall that an object $X$ of
$\mathcal{C}$ is spherical if we have an isomorphism of $\mathbf{k}$-algebras

$\operatorname{Hom}^{*}(X,X)\cong H^{*}(S^{2},\mathbf{k}).$

In what follows, we write $\operatorname{Hom}(X,Y)$ for the graded vector
space $\operatorname{Hom}^{*}(X,Y)$, namely the direct sum
$\bigoplus_{n}\operatorname{Hom}^{0}(X,Y[n])$. We have a pairing on the
Grothendieck group $K(\mathcal{C})$, defined as

$\langle X,Y\rangle=\sum_{i=-\infty}^{\infty}\dim\operatorname{Hom}^{i}(X,Y).$

We call this the _$\operatorname{Hom}$ pairing_.

A spherical object $X$ defines a triangulated auto-equivalence of
$\mathcal{C}$, called the spherical twist in $X$, denoted by $\sigma_{X}$. An
object $Y$ and its spherical twist $\sigma_{X}(Y)$ are related by the exact
triangle

(1) $\operatorname{Hom}(X,Y)\otimes X\to Y\to\sigma_{X}(Y)\xrightarrow{+1},$

where the map $\operatorname{Hom}(X,Y)\otimes X\to Y$ is the evaluation map.
Likewise, $Y$ and the inverse twist $\sigma^{-1}_{X}(Y)$ are related by the
exact triangle

(2) $\sigma^{-1}_{X}(Y)\to Y\to
X\otimes\operatorname{Hom}(Y,X)^{\vee}\xrightarrow{+1},$

where the map $Y\to X\otimes\operatorname{Hom}(Y,X)^{\vee}$ is obtained from
the evaluation adjoint to the evaluation map.

Let $\tau$ be a stability condition on $\mathcal{C}$. Denote by
$\phi_{\tau}(X)$ or simply $\phi(X)$ the phase of a semi-stable object $X$. An
arbitrary object $X$ has a unique Harder–Narasimhan (HN) filtration

$0=X_{0}\to X_{1}\to\dots\to X_{n}=X,$

where the sub-quotients $Z_{i}$, defined by triangles $X_{i-1}\to X_{i}\to
Z_{i}\xrightarrow{+1}$, are $\tau$-semistable and satisfy

$\phi(Z_{1})>\cdots>\phi(Z_{n}).$

With the notation above, set

$\displaystyle\phi^{+}(X)=\phi(Z_{1}),$
$\displaystyle\quad\phi^{-}(X)=\phi(Z_{n}),$ $\displaystyle
X^{\operatorname{top}}=Z_{1},$ $\displaystyle\text{ and
}X^{\operatorname{bot}}=Z_{n}.$

We call $\phi^{+}(X)$ (resp. $\phi^{-}(X)$) the _top_ (resp. _bottom_) phase
of $X$.

Recall that the definition of a stability condition implies that
$\operatorname{Hom}(X,Y)=0$ if $X$ an $Y$ are semistable with
$\phi(X)>\phi(Y)$. More generally, an easy induction shows that
$\operatorname{Hom}(X,Y)=0$ when $\phi^{-}(X)>\phi^{+}(Y)$.

##  3\. Phase impromevent using spherical twists

The goal of this section is to prove that by applying suitable spherical
twists, we can predictably increase/decrease the bottom/top phase of an
object. Throughout, fix an arbitrary $\mathbf{k}$-linear 2-Calabi–Yau
triangulated category $\mathcal{C}$ and a stability condition $\tau$ on
$\mathcal{C}$.

The following is standard.

######  Lemma 3.1 (Sandwich lemma).

Let $X\to Y\to Z\xrightarrow{+1}$ be an exact triangle. Then

$\displaystyle\phi^{-}(Y)$
$\displaystyle\geq\min\\{\phi^{-}(X),\phi^{-}(Z)\\},\text{ and }$
$\displaystyle\phi^{+}(Y)$
$\displaystyle\leq\max\\{\phi^{+}(X),\phi^{+}(Z)\\}.$

###### Proof.

We prove the first inequality; the second is analogous. We have a nonzero map
$Y\to Y^{\operatorname{bot}}$. As a result, there is either a nonzero map
$X\to Y^{\operatorname{bot}}$ or a nonzero map $Z\to Y^{\operatorname{bot}}$.
This shows that $\phi(Y^{\operatorname{bot}})\geq\phi^{-}(X)$ or
$\phi(Y^{\operatorname{bot}})\geq\phi^{-}(Z)$. Equivalently,
$\phi^{-}(Y)\geq\min\\{\phi^{-}(X),\phi^{-}(Z)\\}$. ∎

We now investigate the effect of applying suitable spherical twists on the
bottom and the top phase.

######  Proposition 3.2.

Let $X$ be a spherical stable object of $\mathcal{C}$ such that $X$ is the
unique stable object of its phase. Let $Y$ be any object of $\mathcal{C}$. We
have the following.

  1. (1)

If $\phi(X)\leq\phi^{-}(Y)$, then $\phi(X)<\phi^{-}(\sigma_{X}^{-1}(Y))$.

  2. (2)

If $\phi^{+}(Y)\leq\phi(X)$, then $\phi^{+}(\sigma_{X}(Y))<\phi(X)$.

###### Proof.

We prove the first inequality; the second is analogous. The key is the exact
triangle

(3) $X\otimes\operatorname{Hom}(Y,X)^{\vee}[-1]\to\sigma_{X}^{-1}(Y)\to
Y\xrightarrow{+1},$

obtained by rotating the triangle in (2) in § 2.

We first prove the proposition assuming the strict inequality

(4) $\phi(X)<\phi^{-}(Y).$

Let $V=\operatorname{Hom}(Y,X)^{\vee}[-1]$. Since $V$ is a complex of vector
spaces, it is quasi-isomorphic to a direct sum of shifts of copies of
$\mathbf{k}$. The inequality (4) implies that $\operatorname{Hom}^{i}(Y,X)=0$
for $i\leq 0$. Therefore, $V$ is a direct sum of copies of $\mathbf{k}[j]$ for
$j\geq 0$. As a result, we have

(5) $\phi(X)\leq\phi^{-}(X\otimes V).$

Thanks to (4) and (5), we can apply the sandwich lemma (3.1) to the key
triangle (3) to get

$\phi(X)\leq\phi^{-}(\sigma_{X}^{-1}(Y)).$

To show that the inequality is strict, it suffices to show that
$\sigma_{X}^{-1}Y$ does not have a nonzero map to any stable object of phase
$\phi(X)$. By our assumption, the only such stable object is $X$ itself.
Consider a map $f\colon\sigma_{X}^{-1}(Y)\to X$. Applying $\sigma_{X}$ gives a
map $\sigma_{X}(f)\colon Y\to X[-1]$. Since $\phi(X[-1])<\phi(X)<\phi^{-}(Y)$,
the map $\sigma_{X}(f)$ must be zero. Therefore $f$ is zero. The proof is thus
complete, assuming $\phi(X)<\phi^{-}(Y)$.

We now treat the case $\phi(X)=\phi^{-}(Y)$. By our assumption on $X$, in this
case, $Y^{\operatorname{bot}}$ must be a direct sum of copies of $X$. This
means that we have an exact triangle

(6) $Z\to Y\to X^{\oplus n}\xrightarrow{+1},$

where $\phi(X)<\phi^{-}(Z)$. The previous argument now applies to $Z$, and we
get $\phi(X)<\phi^{-1}\left(\sigma_{X}^{-1}Z\right)$. Applying
$\sigma_{X}^{-1}$ to the triangle in (6) gives the triangle

(7) $\sigma_{X}^{-1}(Z)\to\sigma_{X}^{-1}(Y)\to X[1]^{\oplus
n}\xrightarrow{+1}.$

By applying the sandwich lemma (3.1) to (7), we conclude that
$\phi(X)<\phi^{-}(\sigma_{X}^{-1}(Y))$. ∎

The following shows that the improvement on one end achieved by 3.2 does not
cause a deterioration at the other end, assuming that the object $Y$ has a
sufficiently large phase spread.

######  Proposition 3.3.

Let $X$ be a spherical stable object of $\mathcal{C}$. Assume that $X$ is the
unique stable object of its phase. Let $Y$ be any object of $\mathcal{C}$ such
that $\operatorname{Hom}^{i}(Y,Y)=0$ for any $i<0$ and
$\phi^{+}(Y)-\phi^{-}(Y)\geq 1$. The following hold.

  1. (1)

If $\phi^{-}(Y)=\phi(X)$, then $\phi^{+}(Y)\geq\phi^{+}(\sigma_{X}^{-1}(Y))$.

  2. (2)

If $\phi^{+}(Y)=\phi(X)$, then $\phi^{-}(Y)\leq\phi^{-}(\sigma_{X}(Y))$.

###### Proof.

We prove the first statement; the second is analogous. Again, the key is the
exact triangle

(8) $X\otimes\operatorname{Hom}(Y,X)^{\vee}[-1]\to\sigma_{X}^{-1}(Y)\to
Y\xrightarrow{+1},$

obtained by rotating the triangle (2) in § 2. Let
$V=\operatorname{Hom}(Y,X)^{\vee}[-1]$. By the sandwich lemma (3.1), it
suffices to show that

(9) $\phi^{+}(Y)\geq\phi^{+}(X\otimes V).$

Let us compute $\phi^{+}(X\otimes V)$. Let $\ell$ be the largest integer such
that $\operatorname{Hom}^{\ell}(Y,X)\neq 0$. Then $V$ is a direct sum of
copies of $\mathbf{k}[j]$ with $j\leq\ell-1$, and including at least one copy
of $\mathbf{k}[\ell-1]$. Therefore, we get

(10) $\phi^{+}(X\otimes V)=\phi(X[\ell-1]).$

Thus, showing (9) is equivalent to showing

$\phi^{+}(Y)\geq\phi(X[\ell-1]).$

First suppose $\ell\leq 2$. Then we have

$\phi^{-}(Y)+1=\phi(X[1])\geq\phi(X[\ell-1]).$

Since $\phi^{+}(Y)-\phi^{-}(Y)\geq 1$, we conclude that

$\phi^{+}(Y)\geq\phi(X[\ell-1]),$

as desired.

Next suppose $\ell>2$. Then the 2-Calabi–Yau property implies that

$\operatorname{Hom}^{2-\ell}(X,Y)\neq 0.$

By our assumptions on $X$ and $Y$, the object $Y^{\operatorname{bot}}$ is a
direct sum of copies of $X$. Therefore, we also have

$\operatorname{Hom}^{2-\ell}(Y^{\operatorname{bot}},Y)\neq 0.$

Define the object $K$ by the following exact triangle:

$K\to Y\to Y^{\operatorname{bot}}\xrightarrow{+1}.$

Let $f\in\operatorname{Hom}^{2-\ell}(Y^{\operatorname{bot}},Y)$ be a non-zero
element. The composition of $Y\to Y^{\operatorname{bot}}$ with $f$ gives a map
$Y\to Y[2-\ell]$. Since this is a map of negative degree from $Y$ to itself,
it must be zero. Therefore $f$ factors as the composite of
$Y^{\operatorname{bot}}\to K[1]$ with a (non-zero) map $g\colon K[1]\to
Y[2-\ell]$

${Y}$${Y^{\operatorname{bot}}}$${K[1]}$${Y[2-\ell].}$$\scriptstyle{0}$$\scriptstyle{f}$$\scriptstyle{+1}$$\scriptstyle{g}$

Since $g$ is non-zero, we get

$\phi^{+}(Y[2-\ell])\geq\phi^{-}(K[1]).$

By construction, we have

$\phi^{-}(K)>\phi^{-}(Y)=\phi(X).$

By combining the last two inequalities, we see that

$\phi^{+}(Y[2-\ell])>\phi(X[1]).$

Therefore, we get

$\phi^{+}(Y)>\phi(X[\ell-1]),$

as desired. ∎

The following is an analogue of 3.3 for $Y$ of small spread.

######  Proposition 3.4.

Let $X$ be a spherical stable object of $\mathcal{C}$. Assume that $X$ is the
unique stable object of its phase. Let $Y$ be any object of $\mathcal{C}$ such
that $\phi^{+}(Y)-\phi^{-}(Y)<1$. Assume, furthermore, that
$\operatorname{Hom}^{0}(Y,Y)=\mathbf{k}$ and that $X$ is not a direct summand
of $Y$. The following hold.

  1. (1)

If $\phi^{-}(Y)=\phi(X)$, then $\phi^{+}(Y)\geq\phi^{+}(\sigma_{X}^{-1}(Y))$.

  2. (2)

If $\phi^{+}(Y)=\phi(X)$, then $\phi^{-}(Y)\leq\phi^{-}(\sigma_{X}(Y))$.

###### Proof.

We prove the first statement; the second is analogous. We begin as in the
proof of 3.3. Consider the triangle

(11) $X\otimes\operatorname{Hom}(Y,X)^{\vee}[-1]\to\sigma_{X}^{-1}(Y)\to
Y\xrightarrow{+1}.$

Let $V=\operatorname{Hom}(Y,X)^{\vee}[-1]$. It suffices to show that

$\phi^{+}(Y)\geq\phi^{+}(X\otimes V).$

Let $\ell$ be the largest integer such that
$\operatorname{Hom}^{\ell}(Y,X)\neq 0$. Then we must show that

$\phi^{+}(Y)\geq\phi(X[\ell-1]).$

By the 2-Calabi–Yau property, we have
$\operatorname{Hom}^{\ell}(Y,X)\cong\operatorname{Hom}^{2-\ell}(X,Y)^{\vee}$.
Since

$\phi^{+}(Y)<\phi^{-}(Y)+1=\phi(X)+1,$

there cannot be a non-zero map from $X$ to $Y[k]$ for $k<0$. As a result, we
must have $\ell\leq 2$.

First suppose $\ell\leq 1$. Then we have

$\phi^{+}(Y)\geq\phi^{-}(Y)=\phi(X)\geq\phi(X[\ell-1]),$

as desired.

We rule out $\ell=2$. Let us show that if $\ell=2$, then $X$ must be a direct
summand of $Y$.

Let $\mathcal{P}$ be the slicing defined by the stability condition $\tau$.
Since $\phi^{+}(Y)-\phi^{-}(Y)<1$ and $\phi(X)=\phi^{-}(Y)$, both $X$ and $Y$
lie in the abelian category $\mathcal{P}[\alpha,\alpha+1)$ for
$\alpha=\phi^{-}(Y)$. By our assumptions, $Y^{\operatorname{bot}}$ is a direct
sum of copies of $X$, say $Y^{\operatorname{bot}}=X^{\oplus n}$.

If $\ell=2$, we have a non-zero map $i\colon X\to Y$. Consider the composite

(12) $X\xrightarrow{i}Y\to Y^{\operatorname{bot}}.$

We show that the composite is non-zero. Equivalently, we must show that $i$
does not factor through the kernel $K$ of $Y\to Y^{\operatorname{bot}}$. In
fact, let us prove that there are no non-zero maps from $X$ to $K$.

Since $\operatorname{Hom}^{0}(Y,Y)=\mathbf{k}$, every non-zero map from $Y$ to
itself is an isomorphism. A non-zero map $X\to K$ gives a non-zero map $Y\to
Y$ that is not an isomorphism, namely the composite

$Y\twoheadrightarrow Y^{\operatorname{bot}}=X^{\oplus n}\twoheadrightarrow
X\to K\hookrightarrow Y.$

Therefore, there are no non-zero maps $X\to K$.

Since we have $Y^{\operatorname{bot}}=X^{\oplus n}$ and the composite in (12)
is non-zero, there is a map $\pi\colon Y\to X$ such that $\pi\circ i\colon
X\to X$ is non-zero. But $X$ is spherical, so $\pi\circ i$ must be an
isomorphism. That is, $X$ is a direct summand of $Y$, as desired. ∎

######  Remark 3.5.

In 3.2, 3.3, 3.2, suppose we know _a priori_ that the stable factors of $Y$,
and any spherical twist applied to $Y$, are spherical. This is true, for
example, if $Y$ is itself a spherical object, or a direct sum of spherical
objects (see [3, Corollary 2.3]). Then we may weaken the uniqueness assumption
on $X$ to the following: $X$ is the unique _spherical_ stable object of its
phase.

##  4\. Spherical stable objects in quiver categories

In this section we will consider the 2-Calabi–Yau category associated to an
arbitrary quiver (not necessarily of finite type), as defined, e.g., in [2,
Section 2].

In the remainder of this section, take $\Gamma$ to be an arbitrary quiver, not
necessarily of finite type. Let $\mathcal{C}$ be the 2-Calabi–Yau category
associated to $\Gamma$ (see, e.g. [2, Section 2]).

The Grothendieck group $K(\mathcal{C})$ with the $\operatorname{Hom}$ pairing
is naturally identified with the root lattice of $\Gamma$. Since
$\dim(\operatorname{Hom}^{*}(X,X))=2$ for any spherical object $X$, its class
$[X]$ in $K(\mathcal{C})$ is a real root.

Let $\tau$ be a stability condition on $\mathcal{C}$ with central charge
$Z\colon K(\mathcal{C})\to\mathbb{C}$ and slicing $\mathcal{P}$.

######  Proposition 4.1.

Assume that $\tau$ is generic in the following sense: $Z$ maps distinct real
roots to complex numbers of distinct phase. Suppose $X$ is a $\tau$-semistable
spherical object. Then $X$ is $\tau$-stable, and it is the unique
$\tau$-stable spherical object of phase $\phi=\phi(X)$.

###### Proof.

To show that $X$ is stable, we must show that $X$ is simple in the abelian
category $\mathcal{P}(\phi)$. Let $S\subset X$ be a non-zero simple sub-
object. By [3, Corollary 2.3], $S$ must be spherical. Then the class of $S$ in
$K(\mathcal{C})$ is a real root. Since $Z(S)$ has the same argument as $Z(X)$,
the genericity assumption on $\tau$ means that $X=S$ in $K(\mathcal{C})$. But
then $X/S=0$ in $K(\mathcal{C})$, and since $X/S$ is in a heart of $\tau$,
this implies that $X=S$.

Next, suppose $Y$ is another $\tau$-stable spherical object of the same phase
as $X$. Again by the genericity of $\tau$, we have $X=Y$ in $K(\mathcal{C})$.
But then we get

$\displaystyle\langle X,Y\rangle$
$\displaystyle=\dim\operatorname{Hom}^{0}(X,Y)-\dim\operatorname{Hom}^{1}(X,Y)+\dim\operatorname{Hom}^{2}(X,Y)$
$\displaystyle=\langle X,X\rangle=2,$

which implies $\operatorname{Hom}^{0}(X,Y)\neq 0$ by the Calabi–Yau property.
Since $X$ and $Y$ are simple objects of $\mathcal{P}(\phi)$, this forces
$X\cong Y$. ∎

For every simple root $v$, we have a spherical object $P_{v}\in\mathcal{C}$.
Recall that the extension closure of these objects is the heart of a bounded
$t$-structure on $\mathcal{C}$. We call this the _standard heart_ and denote
it $\heartsuit$. The objects $P_{v}$ corresponding to simple roots are simple
in $\heartsuit$. We say that a stability condition is _standard_ if its
$[0,1)$ heart is $\heartsuit$. We now give an effective construction of a
$\tau$-stable spherical object of every possible class, for a generic standard
stability condition $\tau$.

Let $w$ be an arbitrary positive root. Write

(13) $w=s_{v_{n}}\cdots s_{v_{1}}v,$

where $v$ is a simple root and $s_{v_{1}},\dots,s_{v_{n}}$ are reflections in
the simple roots $v_{1},\dots,v_{n}$. Set $v_{0}=v$. Associate to (13) a
sequence of roots $R_{0},\dots,R_{n}$ defined by

$R_{i}=s_{v_{n}}\cdots s_{v_{i+1}}(v_{i}).$

Note that $R_{0}=w$ and if (13) is a minimal expression for $w$, then all the
roots in the root sequence are positive.

Let $\epsilon_{1},\dots,\epsilon_{n}$ be $\pm 1$. Consider an object $X$ of
$\mathcal{C}$ defined by

(14)
$X=\sigma^{\epsilon_{n}}_{v_{n}}\circ\dots\circ\sigma_{v_{1}}^{\epsilon_{1}}(P_{v}),$

where $\sigma_{v_{i}}$ is the spherical twist in $P_{v_{i}}$. The $\epsilon$’s
allow us to divide the root sequence into _positive_ and _negative_ sub-
sequences, defined by

$\displaystyle R_{+}$ $\displaystyle=\left(R_{i}\mid\epsilon_{i}=1\right),$
$\displaystyle R_{-}$ $\displaystyle=\left(R_{i}\mid\epsilon_{i}=-1\right).$

We call $R_{0}$ the _neutral root_.

Let $\tau$ be a stability condition on $\mathcal{C}$ such that the $[0,1)$
heart of $\tau$ is the standard heart. Let $Z$ be the central charge of
$\tau$. Let $\mathbb{H}\subset\mathcal{C}$ be the half-open upper half plane:

$\mathbb{H}=\\{z\mid\Im(z)>0\\}\cup\mathbb{R}_{>0}.$

Let $\alpha=Z(R_{0})$. Since $R_{0}$ is a positive root, $Z(R_{0})$ lies in
$\mathbb{H}$. It divides $\mathbb{H}$ into two pieces

$\displaystyle\mathbb{H}_{+}$ $\displaystyle=\\{z\mid\arg z>\arg\alpha\\},$
$\displaystyle\mathbb{H}_{-}$ $\displaystyle=\\{z\mid\arg z<\arg\alpha\\},$

where $\arg$ is taken in $[0,\pi)$.

Figure 1 shows an example of the construction above for a stability condition
on the $A_{3}$-category.

$\mathbb{H}_{-}$$\mathbb{H}_{+}$$R_{3}$$R_{2}$$R_{1}$$R_{0}$ Figure 1.
Consider the $A_{3}$ quiver with simple roots $\alpha_{i}$ and simple
reflections $s_{i}$. Consider the root sequence for
$w=s_{2}s_{3}s_{1}(\alpha_{2})$. The central charge chosen for the diagram
above maps $R_{1}$ to $\mathbb{H}_{+}$ and $R_{2},R_{3}$ to $\mathbb{H}_{-}$.
By 4.2, the stable object of class $w$ is
$\sigma^{-1}_{2}\sigma^{-1}_{3}\sigma_{1}(P_{2})$.

Since all the roots in the root sequence $R$ of $X$ are positive, they are
mapped to the upper half plane $\mathbb{H}$ by the central charge. The semi-
stability of $X$ depends on their position with respect to $R_{0}$.

######  Proposition 4.2.

With the notation above, the object $X$ defined by (14) is $\tau$-semistable
if and only if $Z(R_{+})\subset\mathbb{H}_{+}$ and
$Z(R_{-})\subset\mathbb{H}_{-}$.

Let $\tau$ be generic. 4.2 gives an effective construction of the unique
$\tau$-stable object of class $w=R_{0}$. Indeed, we compute the root sequence
$R$, look at its image in $Z(R)\subset\mathbb{H}$, and take $\epsilon_{i}=\pm
1$, depending on whether $Z(R_{i})$ lies in $\mathbb{H}_{\pm}$.

We need some preparation to prove 4.2, including the definitions and basic
properties of spherical twists from the beginning of § 3. Let
$\heartsuit\subset\mathcal{C}$ be the standard heart. Set

$K=K(\heartsuit)=K(\mathcal{C}).$

Denote by $[X]$, the class in $K$ of an object $X$.

######  Lemma 4.3.

Let $v$ be a simple root and $X\in\heartsuit$ be any object. The twist
$\sigma^{-1}_{P_{v}}X$ lies in $\heartsuit$ if and only if $P_{v}$ is not a
sub-object of $X$. Similarly, the twist $\sigma_{P_{v}}X$ lies in $\heartsuit$
if and only if $P_{v}$ is not a quotient of $X$.

###### Proof.

We prove the first statement; the second is analogous. Set $P=P_{v}$. We have
the exact triangle

(15) $P\otimes\operatorname{Hom}(X,P)^{\vee}[-1]\to\sigma^{-1}_{P}(X)\to
X\xrightarrow{+1}.$

Since both $P$ and $X$ lie in $\heartsuit$, the 2-Calabi–Yau property implies
that $\operatorname{Hom}^{i}(X,P)$ is zero for $i<0$ and $i>2$.

Assume that $P$ is not a sub of $X$ in $\heartsuit$. Since $P$ is simple in
$\mathcal{A}$, we must have $\operatorname{Hom}^{0}(P,X)=0$. By the
2-Calabi–Yau property, this implies $\operatorname{Hom}^{2}(X,P)=0$.

Let $V=\operatorname{Hom}(X,P)^{\vee}[-1]$. Since $V$ is a complex of vector
spaces, it is quasi-isomorphic to a direct sum of shifted copies of
$\mathbf{k}$. Since $\operatorname{Hom}^{i}(X,P)=0$ if $i\notin\\{0,1\\}$, the
complex $V$ must be a direct sum of copies of $\mathbf{k}[j]$ for $j=-1,0$.

In the exact triangle (15), the two extreme terms lie in
$\mathcal{C}_{\geq-1}\cap\mathcal{C}_{<1}$. As a result,
$\sigma^{-1}_{P}(X)_{<0}$ also lies in
$\mathcal{C}_{\geq-1}\cap\mathcal{C}_{<1}$. We must show that it lies in
$\heartsuit=\mathcal{C}_{\geq 0}\cap\mathcal{C}_{<1}$. That is, we must show
that its truncation to $\mathcal{C}_{<0}$ is zero.

Note that the truncation $\sigma^{-1}_{P}(X)_{<0}$ lies in $\heartsuit[-1]$
and coincides with $H^{1}(\sigma^{-1}_{P}(X))[-1]$. Since $H^{1}(X)=0$, the
cohomology long exact sequence applied to the triangle (15) shows that
$\sigma^{-1}_{P}(X)_{<0}$ is a quotient of a direct sum of copies of $P[-1]$
in $\heartsuit[-1]$. Since $P$ is simple in $\heartsuit$, the object
$\sigma^{-1}_{P}(X)_{<0}$ must itself be a direct sum of copies of $P[-1]$.

If $\sigma^{-1}_{P}(X)_{<0}$ were non-zero, then we would have a non-zero map
$\sigma^{-1}_{P}(X)_{<0}\to P[-1]$, and hence a non-zero map
$\sigma^{-1}_{P}(X)\to P[-1]$. By applying $\sigma_{P}$, we would then obtain
a non-zero map $X\to P[-2]$, which is a contradiction. We conclude that
$\sigma_{P}^{-1}X$ lies in $\heartsuit$.

Conversely, if $P$ is a sub of $X$, then we have a non-zero map $P\to X$ and
hence a non-zero map $\sigma^{-1}_{P}P=P[1]\to\sigma^{-1}_{P}X$. It follows
that $\sigma^{-1}_{P}X$ is not in $\heartsuit$. ∎

Consider an object $X\in\heartsuit$. We say that a subset $S\subset K$
_envelops_ the subs (resp. quotients) of $X$ if for every sub (resp. quotient)
object $Y$ of $X$, the class $[Y]$ can be expressed as non-negative linear
combination of the elements of $S$ and $\pm[X]$. Observe that if $S$ envelops
the subs of $X$ then $-S$ envelops the quotients of $X$, and vice-versa.

######  Lemma 4.4.

Let $X$ be an object of $\heartsuit$ and let $v\in K$ be a simple root. Let
$S$ be a subset of $K$ and set $S^{\prime}=s_{v}(S)\cup\\{v\\}$.

  1. (1)

If $S$ envelops the subs of $X$ and $\sigma^{-1}_{P_{v}}X$ lies in
$\heartsuit$, then $S^{\prime}$ envelops the subs of $\sigma^{-1}_{P_{v}}X$.

  2. (2)

If $S\subset K$ envelops the quotients of $X$ and $\sigma_{P_{v}}X$ lies in
$\heartsuit$, then $S^{\prime}$ envelops the quotients of $\sigma_{P_{v}}X$.

###### Proof.

We prove the first assertion; the second is similar.

Set $P=P_{v}$. Let $Y$ be any sub of $\sigma^{-1}_{P}X$. We must prove that
$[Y]$ is a non-negative linear combination of the elements of $S^{\prime}$ and
$\pm[\sigma_{P}^{-1}(X)]$.

First suppose $P$ is not a quotient of $Y$. Set

$Q=\operatorname{coker}(Y\to\sigma^{-1}_{P}X).$

Since $\sigma_{P}\sigma^{-1}_{P}X$ lies in $\heartsuit$, by 4.3, $P$ is not a
quotient of $\sigma^{-1}_{P}X$. Therefore $P$ is not a quotient of $Q$. By
applying $\sigma_{P}$ to the exact sequence

$0\to Y\to\sigma^{-1}_{P}X\to Q\to 0,$

we get an exact triangle

$\sigma_{P}Y\to X\to\sigma_{P}Q\xrightarrow{+1},$

whose terms are in $\heartsuit$ by 4.3. Therefore, it is an exact sequence in
$\heartsuit$. Since $S$ envelops the subs of $X$, the class
$[\sigma_{P}Y]=s_{v}[Y]$ is a non-negative linear combination of elements of
$S$ and $\pm[X]$. Equivalently, the class $[Y]$ is a non-negative linear
combination of the elements of $s_{v}(S)$ and $\pm[\sigma_{P}^{-1}X]$.

It remains to treat the case when $P$ is a quotient of $Y$. Define
$Y^{\prime}\subset Y$ be such that we have an exact sequence

$0\to Y^{\prime}\to Y\to P^{\oplus n}\to 0$

for some $n$ and $P$ is not a quotient of $Y$ (such a $Y^{\prime}$ exists
because $\heartsuit$ is a finite-length category). By the argument above,
$[Y^{\prime}]$ is a non-negative linear combination of the elements of
$s_{v}(S)$ and $\pm[\sigma_{P}^{-1}X]$. But then $[Y]$ is a non-negative
linear combination of the elements of $s_{v}(S)\cup\\{v\\}$ and
$\pm[\sigma_{P}^{-1}X]$, as desired. ∎

Consider an object $X$ of $\mathcal{C}$ defined as in (14):

$X=\sigma^{\epsilon_{n}}_{v_{n}}\circ\dots\circ\sigma_{v_{1}}^{\epsilon_{1}}(P_{v}),$

and the root sequence $R_{i}$, divided into a positive sub-sequence $R_{+}$, a
negative sub-sequence $R-$, and the neutral root $R_{0}$.

######  Lemma 4.5.

In the above setup, suppose there exists a linear functional $\lambda\colon
K(\mathcal{C})\to{\mathbb{R}}$ such that $\lambda(R_{0})=0$, and
$\lambda(R_{+})\subset\mathbb{R}_{>0}$, and
$\lambda(R_{-})\subset\mathbb{R}_{<0}$. Then $X$ lies in the heart
$\heartsuit$. Furthermore, the set $R_{-}$ (resp. $R_{+}$) envelops the subs
(resp. quotients) of $X$.

###### Proof.

We induct on $n$. If $n=0$, then $X=P_{v}$ is simple, both $R_{+}$ and $R_{-}$
are empty, and the statement holds.

Assume the statement for $(n-1)$. Let

$X^{\prime}=\sigma^{\epsilon_{n-1}}_{v_{n-1}}\circ\cdots\circ\sigma_{v_{1}}^{\epsilon_{1}}(P_{v}),$

and let $R^{\prime}$ denote the root sequence for $X^{\prime}$. Then we have
$R^{\prime}_{i}=s_{v_{n}}R_{i}$ for $i=0,\dots,n-1$. Note that

$\lambda^{\prime}=\lambda\circ s_{v_{n}}\colon K(\mathcal{C})\to\mathbb{R}$

is a linear functional that vanishes on the neutral root $R^{\prime}_{0}$ for
$X^{\prime}$ and takes opposite signs on the positive and the negative sub-
sequences $R^{\prime}_{+}$ and $R^{\prime}_{-}$. By the induction hypothesis,
$X^{\prime}$ lies in the heart $\heartsuit$ and its subs (resp. quotients) are
enveloped by $R^{\prime}_{-}$ (resp. $R^{\prime}_{+}$).

Observe that $R_{n}=v_{n}$. Suppose $\epsilon_{n}=-1$. Then $R_{n}\in R_{-}$.
Since $\lambda$ is negative on $R_{-}$, it is positive on
$-R_{n}=s_{v_{n}}(R_{n})$. By construction, $\lambda^{\prime}$ is negative on
$R^{\prime}_{-}$, positive on $R_{n}$, and zero on $[X^{\prime}]$. So
$R_{n}=[P_{v_{n}}]$ cannot be a positive linear combination of elements of
$R^{\prime}_{-}$ together with $\pm[X^{\prime}]$.

Since $R^{\prime}_{-}$ envelops the subs of $X^{\prime}$, we conclude that
$P_{v_{n}}$ is not a sub of $X^{\prime}$. Hence, by 4.3, $X$ is in the heart
and its subs are enveloped by $s_{1}(R^{\prime}_{s})\cup\\{v_{n}\\}=R_{-}$.
The proof when $\epsilon_{n}=+1$ is similar. ∎

We now have the tools to finish the proof of 4.2.

###### Proof of 4.2.

Once again, set

$X=\sigma^{\epsilon_{n}}_{v_{n}}\circ\dots\circ\sigma_{v_{1}}^{\epsilon_{1}}(P_{v}),$

with $R_{+}$ and $R_{-}$ defined as above. Suppose $Z$ maps $R_{+}$ and
$R_{-}$ to $\mathbb{H}_{+}$ and $\mathbb{H}_{-}$, respectively. Choose a
linear functional $\ell\colon\mathbb{C}\to\mathbb{R}$ that vanishes on
$\alpha=Z(R_{0})$ and takes positive (resp. negative) values on
$\mathbb{H}_{+}$ (resp. $\mathbb{H}_{-}$). Set $\lambda=\ell\circ Z$. Then
$\lambda$ satisfies the hypotheses of 4.5. As a result, $X$ is in the heart
$\heartsuit$.

To show that $X$ is semi-stable, consider a sub $Y\subset X$. But $R_{-}$
envelops the subs of $X$, that is, $[Y]$ is a non-negative linear combination
of elements of $R_{-}$ and $\pm[X]$. By applying $Z$, we obtain that $Z(Y)$ is
a non-negative linear combination of elements of $Z(R_{-})$ and $\pm\alpha$.
Note that $Z(R_{-})\subset\mathbb{H}_{-}$. Since we know that
$Z(Y)\subset\mathbb{H}$, we conclude that $Z(Y)$ lies in
$\mathbb{H}_{-}\cup\mathbb{R}_{>0}\cdot\alpha$. As a result, we have $\arg
Y\leq\arg X$. Since this is true for any sub $Y\subset X$, we conclude that
$X$ is semi-stable. ∎

##  5\. Applications

In this section, we reap the benefits of the results proved in the previous
sections.

Fix the following notation:

  * $\mathcal{C}$

a $\mathbf{k}$-linear 2-Calabi–Yau triangulated category,

  * $\tau$

a stability condition on $\mathcal{C}$,

  * $\Phi$

the subset of $\mathbb{R}$ consisting of the phases of $\tau$-stable spherical
objects

  * $G$

the group of auto-equivalences of $\mathcal{C}$ generated by the twists in
$\tau$-stable spherical objects.

######  Proposition 5.1.

With the notation introduced at the beginning of § 5, assume that $\tau$
admits at most one spherical stable object of every phase and
$\Phi\subset\mathbb{R}$ is discrete. Then every spherical object in
$\mathcal{C}$ is in the $G$-orbit of a $\tau$-stable spherical object.

###### Proof.

Let $Y$ be any spherical object of $\mathcal{C}$. We denote by $|Y|$ the
_spread_ of $Y$, which is the quantity

$|Y|=\phi^{+}(Y)-\phi^{-}(Y).$

Note that since the stable factors of $Y$ must be spherical [3, Corollary
2.3], both $\phi^{+}(Y)$ and $\phi^{-}(Y)$ lie in $\Phi$, and hence their
difference lies in the set $\\{a-b\mid a,b\in\Phi,a\geq b\\}$, which is a
discrete subset of $\mathbb{R}$.

We induct on $|Y|$. If $|Y|=0$, then $Y$ is $\tau$-stable, and we are done.

Otherwise, let $X$ be the unique $\tau$-stable spherical object of phase
$\phi^{-}(Y)$. Since $Y$ is spherical and not stable, $X$ is not a direct
summand of $Y$. Consider $Y^{\prime}=\sigma^{-1}_{X}Y$. By 3.2 along with
either 3.3 or 3.4, we have $|Y^{\prime}|<|Y|$. By the induction hypothesis,
$Y^{\prime}$ lies in the $G$ orbit of $P$, and hence so does $Y$.

Note that the same argument works with $Y^{\prime}=\sigma_{Z}Y$ where $Z$ is
the unique $\tau$-stable spherical object of phase $\phi^{+}(Y)$. ∎

Now suppose that $\Gamma$ is a finite quiver. Recall that the braid group of
$\Gamma$ acts on the associated 2-Calabi–Yau category via spherical twists.

######  Corollary 5.2 (Theorem 1.1).

Let $\mathcal{C}$ be the 2-Calabi–Yau category associated to a quiver of
finite (ADE) type. Then every spherical object of $\mathcal{C}$ is in the
braid group orbit of a simple object of the standard heart.

###### Proof.

Choose a stability condition $\tau$ on $\mathcal{C}$ with the standard heart
and generic central charge $Z\colon K(\mathcal{C})\to\mathbb{C}$. In
particular, assume that $Z$ maps distinct roots to complex numbers of distinct
arguments. Then, by 4.1, there is at most one spherical stable object of every
phase.

Let $\Phi\subset\mathbb{R}$ be the set of phases of spherical stable objects.
Since the class of a spherical object in $K(\mathcal{C})$ is a root, of which
there are only finitely many, the set $\Phi$ consists of integer translates of
a finite set. In particular, $\Phi$ is discrete.

The simple objects of the standard heart are $\tau$-stable and spherical. So
the image of the braid group lies in $G$. It is not hard to see that the image
is, in fact, $G$. To see this, let $X$ be a $\tau$-stable spherical object. We
must show that $\sigma_{X}$ lies in the image of the braid group. From 4.2, we
know that $X=\beta Y$, where $Y$ is a simple object in the standard heart, and
$\beta$ is in the image of the braid group. Then

$\sigma_{X}=\beta\sigma_{Y}\beta^{-1}$

is also in the image of the braid group.

We now apply 5.1 and conclude that every spherical object is in the braid
group orbit of a $\tau$-stable spherical object. But we already know that
every $\tau$-stable spherical object is in the braid group orbit of a simple
object of the standard heart. The proof is now complete. ∎

######  Remark 5.3 (Choice of writing).

Note that in 5.1, we have a choice of applying a positive or a negative twist,
leading to different expressions for a given spherical object as a braid image
of a simple object. It is an interesting problem to understand these different
expressions.

######  Proposition 5.4.

With the notation introduced at the beginning of § 5, assume that $\tau$
admits at most one spherical stable object of every phase and
$\Phi\subset\mathbb{R}$ is discrete. Let $Y$ be a direct sum of spherical
objects of $\mathcal{C}$ such that $\operatorname{Hom}^{i}(Y,Y)=0$ for $i<0$.
Then there exists a stability condition $\omega$ in the $G$-orbit of $\tau$
such that $Y$ lies in the $[\alpha,\alpha+1)$-heart of $\omega$ for some
$\alpha$.

###### Proof.

We induct on the spread $|Y|=|Y|_{\tau}$. Note that this quantity lies in the
discrete set $\\{a-b\mid a,b\in\Phi\\}$.

If $|Y|_{\tau}<1$, then we are done. Simply take $\omega=\tau$ and
$\alpha=\phi^{-}(Y)$.

Suppose $|Y|_{\tau}\geq 1$. Let $X$ be the unique spherical $\tau$-stable
object of phase $\phi^{-}(Y)$. Let $\tau^{\prime}=\sigma_{X}\tau$. Note that
the group $G$ and the set $\Phi$ are unchanged if we replace $\tau$ by
$\tau^{\prime}$. By 3.2 and 3.3, we have

$|Y|_{\tau^{\prime}}=|\sigma_{X}^{-1}Y|_{\tau}<|Y|_{\tau}.$

We conclude the result by the induction hypothesis. ∎

######  Corollary 5.5 (Theorem 1.2).

Let $\mathcal{C}$ be the 2-Calabi–Yau category associated to a quiver of
finite (ADE) type. Then the stability manifold of $\mathcal{C}$ is connected.
Furthermore, up to rotation, every stability condition is in the braid group
orbit of a standard stability condition.

###### Proof.

Let $\tau$ be an arbitrary stability condition on $\mathcal{C}$. Let $Z\colon
K(\mathcal{C})\to\mathbb{C}$ be its central charge. Perturb $\tau$ so that $Z$
maps distinct roots to complex numbers of distinct arguments. Note that the
perturbed $\tau$ lies in the same connected component of the stability
manifold as the original $\tau$. There is now a unique spherical $\tau$-stable
object of every phase.

We have shown in 5.1 that every spherical object is in the braid group orbit
of the simple objects of the standard heart. Therefore the subgroup $G$ of
$\operatorname{Aut}(\mathcal{C})$ generated by twists in $\tau$-stable
spherical objects is a subgroup of the image of the braid group in
$\operatorname{Aut}(\mathcal{C})$.

Let $\Phi\subset\mathbb{R}$ be the set of phases of spherical $\tau$-stable
objects. Since there are finitely many roots, $\Phi$ consists of integer
translates of a finite set, and hence is discrete.

Let $Y$ be the direct sum of the simple objects in the standard heart of
$\mathcal{C}$. Note that $Y$ satisfies the hypotheses of 5.4. By 5.4, there is
a stability condition $\omega$ in the braid group orbit of $\tau$ such that
$Y$ is in the $[\alpha,\alpha+1)$ heart of $\omega$. Let $\omega^{\prime}$ be
the rotation of $\omega$ by $\alpha$, so that $Y$ lies in the $[0,1)$ heart of
$\omega^{\prime}$. Note that $\omega^{\prime}$ is in the same connected
component as $\omega$.

The direct summands of $Y$ generate the standard heart of $\mathcal{C}$ (under
taking extensions). Therefore, the $[0,1)$ heart of $\omega^{\prime}$ contains
the standard heart. Since both hearts are hearts of a $t$-structure, they must
in fact be equal. In other words, $\omega^{\prime}$ is a standard stability
condition.

Let $\operatorname{Stab}_{0}\mathcal{C}$ be the connected component of
$\operatorname{Stab}\mathcal{C}$ that contains the standard stability
conditions. We have shown that an arbitrary
$\tau\in\operatorname{Stab}\mathcal{C}$ is in the braid group orbit of a
stability condition $\omega$ in $\operatorname{Stab}_{0}\mathcal{C}$. But we
know that the braid group preserves the connected component
$\operatorname{Stab}_{0}\mathcal{C}$ and in fact every
$\tau\in\operatorname{Stab}_{0}$ is, up to rotation, in the braid group orbit
of a standard stability condition [4, § 4]. Hence, we conclude that
$\operatorname{Stab}\mathcal{C}=\operatorname{Stab}_{0}\mathcal{C}$ and that
every stability condition is, up to rotation, in the braid group orbit of a
standard one. ∎

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