# Using LoRa communication for Urban VANETs: Feasibility and Challenges

Thenuka Karunathilake
University of Bremen
Bremen, Germany
<EMAIL_ADDRESS>
Anna Förster
University of Bremen
Bremen, Germany
<EMAIL_ADDRESS>

###### Abstract

Vehicular Ad-Hoc Networks (VANETs) were introduced mainly to increase
vehicular safety by enabling communication between vehicles and infrastructure
to improve overall awareness. The vehicles in a VANET are expected to exchange
numerous messages generated by multiple applications, but mainly, these
applications can be subdivided into safety and non-safety. The main
communication technologies designed for VANETs, DSRC (Dedicated Short Range
Communication) and C-V2X (Cellular V2X), mainly focus on delay-sensitive
safety-related applications. However, sharing the same bandwidth for safety
and non-safety applications will increase the burden on the communication
channel and can cause an increase in the overall latencies. Therefore, this
work analyses the feasibility of using LoRa communication for non-safety-
related urban VANET applications. We conducted multiple real-world experiments
to analyse the performance of LoRa communication in various urban VANET
scenarios. Our results show that LoRa communication handles the Dopper shifts
caused by the urban VANET speeds with both Spreading Factor (SF) 7 and 12.
However, higher SF was more vulnerable to Doppler shifts than lower SF.
Furthermore, the results illustrate that the Line-of-Sight (LoS) condition
significantly affects the LoRa communication, especially in the case of lower
SF.

_K_ eywords LoRa $\cdot$ Vehicular Networks $\cdot$ Smart Cities

##  1 Introduction

In recent years, the applications of Vehicular Ad-Hoc Networks (VANETs) have
increased significantly, and VANETs have become the foundation for Intelligent
Transportation Systems (ITS) in future smart cities. There are many
applications of VANETs, mainly safety message exchange, platooning, road
traffic control, service advertisements, parking spot identification, and
infotainment. In many ways, VANETs are similar to conventional Wireless Sensor
Networks (WSNs), but some major significant differences exist due to the
mobility of the nodes. In WSNs, most nodes can be considered static or moving
at relatively low speeds, but in contrast, nodes associated with VANETs have
higher moving speeds. Therefore, traditional communication technologies do not
work reliably with VANETs due to frequency shifts caused by the Doppler
effect. Hence, several communication technologies are specifically designed to
work with VANETs and handle their special characteristics.

The main communication technologies associated with VANETs are Dedicated Short
Range communication (DSRC) [1] and Cellular V2X (C-V2X) [2]. DSRC was the
initial technology specially designed for VANET applications. It is based on
IEEE 802.11p (an improved version of 802.11 to handle mobility) and can handle
both Vehicle-to-Vehicle (V2V) and Vehicle-to-Infrastructure (V2I)
communications. After its introduction, DSRC was heavily tested in many VANET
applications, and now it has become the most mature VANET communication
technology. However, DSRC has several drawbacks: relatively expensive
equipment cost, shorter range, and carrier sense multiple access (CSMA)
mechanism. With CSMA, all network nodes must check the channel activity before
transmitting data. The CSMA mechanism can guarantee low latency in low-dense
networks due to fewer competing nodes. However, in a high-dense network, the
CSMA mechanism can increase latency.

Recently, C-V2X technology, which is based on cellular technologies like LTE
and 5G, was introduced to VANETs. C-V2X can handle longer distance
communications than DSRC with network-based communication through base
stations, additionally to direct V2X communications. Furthermore, C-V2X has
the ability to communicate with pedestrians through cellular-operated mobile
devices which is not possible with DSRC technology. However, C-V2X suffers
from longer latencies due to a semi-persistent scheduling mechanism where
nodes select random resource blocks for transmission.

The current market penetration of the number of connected vehicles (V2X
capable vehicles) is relatively small, but it is expected to increase
significantly in the near future. In a highly congested traffic scenario (ex,
intersections and crowded highway sections), the available bandwidth of DSRC
and C-V2X technologies may not be enough to cater to all the connected
vehicles simultaneously. In our previous work [3], we calculated how many
connected vehicles can be served by a single 5G base station, and the results
illustrated that in highly congested scenarios, the available bandwidth is not
sufficient, which can lead to increased latency and hinder the vehicular
safety. Therefore, it is beneficial to incorporate another supporting
communication technology for non-safety related applications to ease the
burden on the main communication technologies (DSRC and C-V2X) for such
congested scenarios, allowing primary technologies to focus on safety-related
applications solely.

Usually, most congested scenarios occur in urban areas due to the high density
of vehicles. Therefore, this work extensively investigates the feasibility and
challenges of using LoRa communication as a supporting communication
technology in urban VANETs. Since LoRa communication is designed for WSN
devices, the performance of LoRa has to be investigated in challenging urban
VANET characteristics like the presence of Non-LoS communication and Doppler
shifts. Therefore, we conducted multiple real-world experiments in different
locations with different speeds in V2V and V2I configurations. The performance
of LoRa is analysed in terms of packet delivery ratio and Received Signal
Strength Indicator (RSSI) with different LoRa SFs in urban speeds up to 58
km/h.

The rest of the paper is arranged as follows. In Section 2, we describe the
details of LoRa communication, and in Section 3, we discuss prior works done
on LoRa communication in V2X scenarios. The used experiment setup and
scenarios are explained in detail in Section 4, and in Section 5, a detailed
analysis of the results is provided. Finally, in Section 6, we conclude our
work.

##  2 LoRa Communication

Long Range (LoRa) communication has gained extensive attention after its
introduction in 2015, mainly in the domain of Wireless Sensor Networks (WSN)
and IoT. LoRa has several key benefits over other existing communication
technologies, mainly the extended communication range (10 kilometers in
optimal condition), extremely low power consumption, etc. These
characteristics have made LoRa an ideal solution for WSNs. However, LoRa also
has some disadvantages, mainly the relatively low data rate (0.3 kbit/s to
50kbit/s per channel) [4], which limits the use of LoRa communication in
applications where high-speed data links are essential.

LoRa is a mostly used LPWAN technology in WSN and IoT applications [4]. It is
based on the Chirp Spread Spectrum (CSS) modulation technique, which uses
chirp signals to transmit information. Over time, chirp signals can increase
in frequency (up-chirp) or decrease (down-chirp). LoRa uses sub-GHZ unlicensed
ISM frequency bands for communication, and frequency bands depend on the
country or region (in Europe, 433MHz and 863MHz). There is a common confusion
between LoRaWAN and LoRa in the literature, and they are often used
synonymously. However, there is a significant difference between these two
terms. LoRa handles the physical layer of the protocol stack with CSS
modulation for long-range communication. In contrast, LoRaWAN handles the MAC
layer, enabling communication between LoRa gateways and duty-cycled end
devices (class A,B and C). LoRaWAN specifications are developed and maintained
by the LoRa alliance, which consists of multiple companies and research
bodies.

In our real-world experiments, we only use the LoRa physical layer to allow
direct communication between LoRa nodes without LoRaWAN.

##  3 Related Works

In this section, we discuss some prior work done to evaluate LoRa
communication in the context of VANETs.

As discussed earlier, the Doppler shift is a significant concern when using
LoRa in highly mobile scenarios. Therefore, authors of [5] performed
simulation analysis to determine which LoRa parameters should be selected in
fast-fading scenarios caused by the Doppler effect. The results show that
higher bandwidth and lower spreading factors are more suitable for V2X
scenarios.

One of the major benefits of LoRa communication is its long communication
range. However, these longer communication ranges depend on many factors,
including LoS communication, mobility, and terrain distribution. Therefore,
the authors of [6] conducted an experiment in an urban scenario with one
moving transmitter and a static gateway. The results show that the obtained
delivery ratios depend significantly on speed. Furthermore, the collected RSSI
and SNR depend on terrain distribution and LoS communication rather than the
distance between the transmitter and the gateway. A similar work was carried
out by the authors of [7] and [8]. In [7], experiments were conducted in
multiple environments, including urban, suburban, and rural environments where
the average moving speed was different. The results illustrate that the LoRa
communication range is significantly longer in rural scenarios compared to
urban scenarios. Moreover, in mobility scenarios, LoRa performance depicted a
vulnerability related to the Doppler effect.

VANETs consist mainly of V2I and V2V communications, and in V2V scenarios, the
relative speed varies more than in V2I scenarios. The authors of [9] conducted
experiments in a University environment for both V2I and V2V scenarios
independently. Initially, 50 different locations were selected, and later, it
was reduced to 9 locations by simulation analysis. Two vehicles were used for
the experiments, and in the V2I scenario, one vehicle was stopped at these
selected locations while the other vehicle was moving around the university.
In the V2V scenario, both vehicles were moved randomly. The results show that
higher spreading factors have lower RSSI values in mobility scenarios, and
lower spreading factors depict lower communication ranges.

A LoRa-based architecture for V2X communication was proposed in [10]. The
authors used multiple RSUs and a vehicle equipped with an On-Board-Unit (OBU)
for the experiments. In the proposed architecture V2V and V2I communication
decisions are taken based on the RSSI values. In the case of V2I, when
multiple RSUs are in the communication range, the vehicle decides to
communicate with the RSU, which has the best channel quality (RSSI). Once the
RSSI value decreases below a preset threshold value the vehicle handover the
communication to another RSU which has better channel quality. The results
show that in the V2I configuration, the proposed architecture performed as
expected, up to the speed of 50 km/h.

All the above-mentioned prior work evaluates LoRa communication performance in
mobility scenarios. However, these evaluated scenarios combine LoS
communication, Non-LoS communication, and different Doppler shifts. Therefore,
each of these characteristics cannot be investigated independently. As a
solution, we conducted experiments in multiple locations to isolate these
characteristics. For example, in our LoS experiments, LoS communication is
guaranteed during the whole experiment, and we kept the moving speed of the
transmitter and receiver at a constant value during the experiments. Hence,
each of the characteristics can be isolated and can be analysed independently.

##  4 Experiment Setup

This section describes our hardware setup and used scenarios for the
experiments in detail.

We used multiple Pycom LoPy4 devices to conduct the experiments, as shown in
Figure 1. Pytrack expansion boards were used to extract GPS coordinates with
LoPy4 as the transmitters. For the receivers, a combination of expansion
boards, pysense, and pytracks was used with the LoPy4 devices. All the LoPy4
devices were configured with default LoRa parameters as shown in Table 1, two
different spreading factors were used (SF-7 and SF-12), and the packet
transmission interval was set to 1 second.

We mainly divide our experiments into two scenarios. The first set of
scenarios was conducted with LoS communication, therefore named as ’LoS
Scenario’ in the following. Later, we performed another set of experiments in
a scenario with both LoS and Non-LoS communication because, in real-world
urban VANET scenarios, both LoS and Non-LoS are combined. This scenario is
named as ’LoS + Non-LoS Scenario’ in the following.

Table 1: LoRa Parameters for all our experiments Parameter | Values
---|---
Mode | Raw lora
Region | EU 868
Frequency | 868 MHz
TX power | 14 dBm
Bandwidth | 125 kHz
Spreading Factor (SF) | 7 and 12
Packet Interval | 1 s
Figure 1: Experiment Setup

###  4.1 LoS Scenario

The campus park (Figure 2) of the University of Bremen was used for LoS
experiments, and the route around the campus park is approximately 650 m long.
Three devices (one transmitter and two receivers) were used during the
experiments. The transmitter and one of the receivers were moved on this route
in opposite directions (to have the maximum relative speed), as shown in
Figure 2. Since this receiver is moving, it is considered to be the ’V2V
receiver’. The other receiver was placed in a nearby building, as shown in
Figure 2 with red color. This receiver is a stationary device; therefore, we
consider it to be the ’V2I receiver’. The V2I receiver was placed outside a
window of the building to improve LoS communication. In the initial set of
experiments, V2I and V2V configurations were conducted as two separate
experiments using only 2 devices (one transmitter and V2I receiver or V2V
receiver). However, we later conducted V2I and V2V experiments simultaneously
using all 3 devices (one transmitter and V2I receiver and V2V receiver).

We conducted multiple experiments by changing the moving speed of the
transmitter and the V2V receiver. Each experiment was conducted for 30
minutes, and we conducted experiments with different speeds of 4 km/h, 10
km/h, and 20 km/h. Both the transmitter and V2V receiver were moved at the
same speed during one experiment. For example, in the case of 20 km/h, both
the transmitter and V2V receiver were moved at 20 km/h for 30 minutes, and
during the entire 30 minutes, we tried to maintain the same constant speed.
However, for safety reasons, we had to reduce the speed slightly during
cornering maneuvers at higher speeds (10 km/h and 20 km/h). Furthermore, in
the V2V configuration, the maximum relative speed can reach up to twice the
moving speed of one node (20 km/h if the transmitter and receiver are moving
at 20 km/h). Different modes of transportation were used to achieve different
speeds. The experiments with 4 km/h were conducted by walking around the route
with two persons. One carried the transmitter, and the other person carried
the V2V Receiver. Similarly, 10 km/h was achieved by using bicycles, and 20
km/h was achieved by using electric scooters. We used speed-measuring
applications on mobile devices to measure the moving speed during the
experiments.

The first set of experiments was conducted in the Summer of 2022, and in the
following sections, we refer to these experiments as ’LoS-Summer’. Even though
no buildings affect the LoS communication, still, as shown in Figure 2, some
trees might slightly affect LoS conditions. Therefore, the same set of
experiments was conducted again in the Fall of 2022 because, during the fall
season, better LoS communication was expected because there were no leaves on
the trees. In the following, we refer to these experiments as ’LoS-Fall’.

Figure 2: LoS experiment scenario

The main goal of the work was to check the feasibility of using LoRa
communication for urban VANET applications. Since the driving speeds of
vehicles can be higher than 20 km/h in urban scenarios, more experiments had
to be conducted to investigate the LoRa communication behavior with higher
speeds. However, in reality, it is difficult to achieve higher speed while
maintaining LoS communication. For example, other road vehicles (cars, motor
bicycles and etc) can be used to achieve higher speeds, but in reality, it is
not possible to maintain the same constant speed due to many obstacles like
traffic lights, pedestrian crossings, etc., while maintaining LoS
communication.

To overcome these issues, we decided to use two drones to conduct experiments
with higher speeds. Since the drones fly above ground, the LOS communication
and constant speed are guaranteed. We decided to use two DJI Maverick M300 RTK
drones for the experiments, as shown in Figure 3. We selected this specific
drone model because it was able to reach the expected speed levels while
carrying the Pycom devices and the battery pack. These drones had a maximum
speed limit of 58 km/h; therefore, we conducted 40 km/h and 58 km/h
experiments. Predefined waypoints were used to define the path for the drones,
and the drone was able to maintain a constant speed while counteracting the
wind speed. The drone experiments were conducted in an open area without
obstacles in Colombo, Sri Lanka, as shown in Figure 4. One drone was used to
carry the transmitter, the other drone was used as the V2V Receiver, and the
V2I Receiver was placed on the ground level. The drone with the transmitter
was set to fly at a height of 70 m, and the V2V Receiver drone was flown at a
height of 50 m to avoid collision between the drones. Furthermore, each drone
experiment was limited to 15 minutes due to the battery life of the drones.

Figure 3: The used drones with attached Pycom devices Figure 4: LoS experiment
scenario with drones

###  4.2 LoS + Non-LoS Scenario

In real-world urban VANET scenarios, it is impossible to achieve LoS
communication continuously because of other objects (buildings, trees, etc.).
Therefore, most VANET scenarios are a combination of both LoS and Non-LoS
communication. Hence, we extended our experiments with another scenario by
introducing both LoS and Non-LoS communication. The 1.2 km long route around
the NW1 building of the University of Bremen, Germany, was selected to conduct
LoS + Non-LoS experiments. As shown in Figure 5, multiple buildings are
located in the surroundings; therefore, LoS communication is not guaranteed.
We conducted a similar set of experiments as explained in the section 4.1 with
LoS + Non-LoS scenario. However, LoS + Non-LoS scenario experiments were only
conducted during the Summer of 2022 because there was no significant
difference between the Fall and the Summer for this scenario. Furthermore, the
LoS + Non-Los Scenario was only investigated with speeds up to 20 km/h.

Figure 5: LoS + Non-LoS experiment scenario

##  5 Results Analysis

In this section, we analyse the obtained results from the experiments. As
explained in Section 4, we conducted experiments in the Summer and Fall
seasons. Therefore, our analysis in this section is categorised into two main
parts: Summer experiments and Fall experiments.

###  5.1 Summer experiments

Scenario | Speed |  Mode of Transport and Location | V2I Configuration | V2V Configuration
---|---|---|---|---
SF 7 | SF 12 | SF 7 | SF 12
|  |  | Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
LoS Scenario | 4 km/h |  | Walking
---
Bremen, Germany
749 | 690 | 92.12 | 447 | 447 | 100 | 747 | 743 | 99.46 | 465 | 465 | 100
10 km/h |  | Bicycle
---
Bremen, Germany
705 | 650 | 92.19 | 461 | 460 | 99.78 | 741 | 737 | 99.46 | 469 | 469 | 100
20 km/h |  | E-Scooter
---
Bremen, Germany
788 | 722 | 91.62 | 472 | 472 | 100 | 736 | 723 | 98.23 | 462 | 456 | 98.70
| 40 km/h |  | Drone
---
Colombo, Sri Lanka
589 | 579 | 98.30 | 424 | 414 | 97.64 | 567 | 555 | 97.88 | 265 | 256 | 96.60
| 58 km/h |  | Drone
---
Colombo, Sri Lanka
575 | 561 | 97.56 | 314 | 309 | 98.40 | 481 | 430 | 89.39 | 136 | 132 | 97.05
LoS + Non-LoS Scenario | 4 km/h |  | Walking
---
Bremen, Germany
757 | 429 | 56.57 | 452 | 392 | 86.72 | 762 | 649 | 85.17 | 471 | 465 | 98.72
10 km/h |  | Bicycle
---
Bremen, Germany
694 | 428 | 61.67 | 458 | 414 | 90.39 | 754 | 676 | 89.65 | 456 | 449 | 98.46
| 20 km/h |  | E-Scooter
---
Bremen, Germany
729 | 449 | 61.59 | 461 | 458 | 99.34 | 763 | 659 | 86.36 | 483 | 478 | 98.96

Table 2: Results of the Summer-Experiments

The Summer experiments contain both ’LoS Scenario’ and ’LoS + Non-LoS
Scenario’ as discussed in Section 4. The obtained results are depicted in
Table 2 and the same results are illustrated graphically in Figure 6. We
measured the delivery ratio as the ratio between the number of transmitted and
received packets. All of our experiments were limited to 30 minutes except the
experiments with the drones, which were limited to 15 minutes. However, the
number of transmitted packets varies in different experiments because it
depends on the hardware behavior, like the time required to obtain the GPS
coordinates. Furthermore, the number of packets transmitted with SF-12 is
significantly lower compared to SF-7 because SF-12 requires a longer
transmission time due to the longer symbol time.

(a) Summer experiments with V2I configuration

(b) Summer experiments with V2V configuration

Figure 6: Delivery ratios for the Summer experiments

The results with SF-12 have a better delivery ratio than SF-7 in all
experiments with both V2I and V2V configurations, as shown in Figure 6(a) and
Figure 6(b). Furthermore, the improvement with SF-12 compared to SF-7 is more
significant in the ’LoS + Non-LoS Scenario’. This improvement is due to the
longer symbol times of SF-12, which enable the detecting of weaker signals
more than SF-7. In addition, the wider bandwidth of SF-12 helps to reduce
interference from other signals and noise, especially in crowded environments.
However, SF-12 has the trade-off of a low data rate and, therefore, may not be
suitable for some VANET applications. During our experiments, we used the
frequency of 868 MHz for LoRa communication. However, in Europe, LoRa
communication is also allowed to work on the frequency of 433 MHz. Since the
lower frequencies are better at penetrating obstacles, the delivery ratios can
be expected to increase slightly with 433 MHz.

Furthermore, the delivery ratios for the ’LoS + Non-LoS scenario’ was improved
with V2V configuration (Figure 6(b)) than with V2I configuration (Figure
6(a)). This improvement is caused by our experiment scenario because, as
discussed in Section 4 in the V2V configuration, we moved the devices on the
same route in opposite directions. Therefore, transmitter and receiver meet
multiple times during one experiment depending on their relative speed,
resulting in better delivery ratios than the V2I configuration.

To better visualise the effect of LoS condition on LoRa with SF-7 and SF-12,
we mapped the distribution of received packets for the ’LoS + Non-LoS
scenario’ in the V2I configuration in Figure 7. The mapped data indicate that
with the SF-7, most of the packets were not received when the signal was
attenuated by surrounding objects (mainly the University building) between the
transmitter and the receiver. This behavior is shown in Figure 7 with very few
red dots in the area around the University building that obstruct the
communication. In contrast, in the same area, there are many more blue dots,
which implies that SF-12 manages to transmit packets successfully.
Furthermore, there are more blue dots around the University building with 20
km/h (Figure 7(c)) than 10 km/h (Figure 7(b)) and 4 km/h (Figure 7(a)) because
with higher speeds the transmitter was able to travel multiple rounds on the
route allowing more transmissions to be successful around the University
building.

(a) SF-7 and SF-12 with 4 km/h

(b) SF-7 and SF-12 with 10 km/h

(c) SF-7 and SF-12 with 20 km/h

Figure 7: The distribution of received packets in ’LoS + Non-LoS scenario’
with V2I configuration

In all the conducted experiments LoRa communication was tested with speeds up
to 58 km/h. In these considered speeds, the delivery ratios of LoRa
communication were not affected by the induced Doppler shift, as shown in
Table 2 and Figure 6. This implies that LoRa communication is robust against
the urban VANET speeds. However, our results show that LoS affects the
performance of LoRa significantly, and SF-12 performs better in the case of
Non-LoS communication than SF-7.

###  5.2 Fall experiments

As explained in Section 4, our experiments with the LoS scenario consist of
several trees along the route, which might slightly affect the LoS condition.
In order to evaluate the effect of these trees on LoRa communication, we
extended our ’LoS Scenario’ by repeating the same set of Summer experiments in
the following Fall season.

Scenario | Speed | LoS-Summer | LoS-Fall
---|---|---|---
SF 7 | SF 12 | SF 7 | SF 12
|  |  | Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
| Transmitted
---
Packets
| Received
---
Packets
| Delivery
---
Ratio (%)
V2I Configuration | 4 km/h | 749 | 690 | 92.12 | 447 | 447 | 100 | 833 | 813 | 97.59 | 457 | 457 | 100
10 km/h | 705 | 650 | 92.19 | 461 | 460 | 99.78 | 764 | 752 | 98.42 | 454 | 454 | 100
| 20 km/h | 788 | 722 | 91.62 | 472 | 472 | 100 | 753 | 744 | 98.80 | 522 | 522 | 100
V2V Configuration | 4 km/h | 747 | 743 | 99.46 | 465 | 465 | 100 | 833 | 822 | 98.67 | 457 | 456 | 99.78
10 km/h | 741 | 737 | 99.46 | 469 | 469 | 100 | 764 | 751 | 98.29 | 454 | 453 | 99.77
| 20 km/h | 736 | 723 | 98.23 | 462 | 456 | 98.70 | 898 | 897 | 99.88 | 522 | 522 | 100

Table 3: Results of the Fall-Experiments

Table 3 summarises these results along with LoS-Summer experiments for
comparison, and Figure 8 illustrates the same results graphically. The
delivery ratios with SF-7 were noticeably increased with V2I configuration
compared to LoS-Summer, as shown in Figure 8(a). This improvement is caused by
the better LoS condition in the Fall, resulting in less attenuation of the
signals. There is no significant difference in results for SF-12 because SF-12
is more robust against attenuated signals, and as expected, SF-12 outperforms
SF-7 in all the experiments. Furthermore, as discussed in Section 5.1, V2V
configuration depicts slightly better results than V2I configuration because
of the mobility pattern of the devices.

Moreover, we decided to collect the RSSI values of the received packets for
the Fall experiments. The distribution of RSSI values obtained with V2V
configuration is illustrated in Figure 9 and summarised graphically in Figure
10. Overall, the RSSI values with SF-7 are better than SF-12. These better
RSSI values are caused by the shorter transmission time of SF-7, giving less
opportunity for the signal to be attenuated during the transmission, and once
attenuated, the signal is not received at the receiver. In contrast, SF-12 was
able to receive weaker signals than SF-7, as shown in Figure 10. In addition,
the RSSI with SF-12 was more affected by the higher moving speed, as shown in
10 with decreased RSSI values with increasing speed. This implies that the
longer transmission time of SF-12 increases the likelihood of frequency shifts
and makes it more vulnerable to Doppler shift.

(a) Fall and Summer results with V2I configuration

(b) Fall and Summer results with V2V configuration

Figure 8: Delivery ratios for the Fall and Summer Experiments Figure 9: The
RSSI distribution for LoS-Fall experiments with V2V configuration Figure 10:
Average RSSI for V2V configuration for Fall experiments

##  6 Conclusion

LoRa communication has gained a lot of attention in the domain of WSNs due to
its characteristics like long range and low power. In this study, we
investigate the possibility of using LoRa for VANETs in non-safety related
applications as a supporting communication technology to ease the burden on
the main communication technologies: DSRC and C-V2X. Our experimental analysis
was focused on testing LoRa in urban VANET scenarios where the most congested
and demanding communication channels are expected. We conducted an extensive
set of real-world experiments and tested the LoRa communication behavior in
terms of delivery ratio and RSSI.

Our results show that LoRa communication performs well with urban speeds up to
58 km/h in both V2I and V2V configurations. Therefore, LoRa communication was
robust against the Doppler shifts caused by these urban speeds. However, the
performance of LoRa depends significantly on LoS communication, and SF-12
depicted better performance than SF-7 in the presence of Non-LoS communication
due to its ability to detect weaker signals. Even though SF-12 depicts better
delivery ratios, it also showed more vulnerability toward Dopper shifts,
resulting in lower RSSI values at higher speeds.

Our results proved that LoRa can be used in urban VANET scenarios for non-
safety related applications. However, the most suitable LoRa parameters must
be selected depending on the application requirements, such as the required
data rate and surrounding environment.

##  7 Acknowledgement

We would like to thank Suleco (Pvt) Ltd and LHP Group for providing the
infrastructure for the drone experiments. Furthermore, we thank all the
following individuals for providing their support for conducting these
multiple experiments. Ajmal Azeez (drone pilot), Asanga Udugama, Jens Dede,
Saurabh Band, Cyrille Sepele, Purna Rathnayake, Jehan Jayawardana, and Dinithi
Amarawardana.

## References

  * [1] John B. Kenney.  Dedicated Short-Range Communications (DSRC) Standards in the United States.  Proceedings of the IEEE, 99(7):1162–1182, July 2011.  Conference Name: Proceedings of the IEEE.
  * [2] Muhammad Jalal Khan, Manzoor Ahmed Khan, Sumbal Malik, Parag Kulkarni, Najla Alkaabi, Obaid Ullah, Hesham El-Sayed, Amir Ahmed, and Sherzod Turaev.  Advancing C-V2X for Level 5 Autonomous Driving from the Perspective of 3GPP Standards.  Sensors, 23(4):2261, January 2023.  Number: 4 Publisher: Multidisciplinary Digital Publishing Institute.
  * [3] Thenuka Karunathilake and Anna Förster.  A Survey on Mobile Road Side Units in VANETs.  Vehicles, 4(2):482–500, June 2022.  Number: 2 Publisher: Multidisciplinary Digital Publishing Institute.
  * [4] Sadiq Idris, Thenuka Karunathilake, and Anna Förster.  Survey and Comparative Study of LoRa-Enabled Simulators for Internet of Things and Wireless Sensor Networks.  Sensors, 22(15):5546, January 2022.  Number: 15 Publisher: Multidisciplinary Digital Publishing Institute.
  * [5] Yuke Li, Shuangshuang Han, Linyao Yang, Fei-Yue Wang, and Hui Zhang.  LoRa on the Move: Performance Evaluation of LoRa in V2X Communications.  In 2018 IEEE Intelligent Vehicles Symposium (IV), pages 1107–1111, June 2018.  ISSN: 1931-0587.
  * [6] Rui Oliveira, Miguel Luís, and Susana Sargento.  LoRa Connectivity Analysis for Urban Coverage in Real Mobile Environments.  In 2021 Telecoms Conference (ConfE)), pages 1–6, February 2021.
  * [7] Ramon Sanchez-Iborra, Jesus Sanchez-Gomez, Juan Ballesta-Viñas, Maria-Dolores Cano, and Antonio F. Skarmeta.  Performance Evaluation of LoRa Considering Scenario Conditions.  Sensors, 18(3):772, March 2018.  Number: 3 Publisher: Multidisciplinary Digital Publishing Institute.
  * [8] Oscar Alvear, Jorge Herrera Tapia, Carlos T. Calafate, Enrique Hernandez Orallo, Juan Carlos Cano, and Pietro Manzoni.  Assessing the Impact of Mobility on LoRa Communications.  In Giancarlo Fortino, Carlos E. Palau, Antonio Guerrieri, Nora Cuppens, Frédéric Cuppens, Hakima Chaouchi, and Alban Gabillon, editors, Interoperability, Safety and Security in IoT, Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering, pages 75–81, Cham, 2018. Springer International Publishing.
  * [9] Ana Paula Alves Torres, Claudio Bastos Da Silva, and Horácio Tertuliano Filho.  An Experimental Study on the Use of LoRa Technology in Vehicle Communication.  IEEE Access, 9:26633–26640, 2021.  Conference Name: IEEE Access.
  * [10] Khandaker Foysal Haque, Ahmed Abdelgawad, Venkata Prasanth Yanambaka, and Kumar Yelamarthi.  LoRa Architecture for V2X Communication: An Experimental Evaluation with Vehicles on the Move.  Sensors, 20(23):6876, January 2020.  Number: 23 Publisher: Multidisciplinary Digital Publishing Institute.


$\mathcal{M}_{0}=\frac{\text{U}(1)_{\chi}/\mathbb{Z}_{2}\times\text{SU}(N_{f})}{\mathbb{Z}_{Q}}$
(97)

which we parametrize by the pair

$\left(e^{i\varphi},U\right)\in\text{U}(1)_{\chi}/\mathbb{Z}_{2}\times\text{SU}(N_{f})$
(98)

with the identification

$\left(e^{i\varphi},U\right)\sim\left(e^{2\pi i/Q}\text{e}^{i\varphi},e^{2\pi
iN/N_{f}}U\right)$ (99)

enforcing the $\mathbb{Z}_{Q}$ discrete quotient. Another way to see that we
need this $\mathbb{Z}_{Q}$ identification is by noting that such a
$\mathbb{Z}_{Q}$ transformation leaves the condensates

$\expectationvalue{\tr\lambda\lambda}\sim
e^{iQ\varphi},\quad\expectationvalue{\psi\tilde{\psi}}\sim e^{-iq\varphi}U$
(100)

invariant. Note also that our parametrization implies that $\varphi$ has
charge $2$ under the original chiral symmetry $\text{U}(1)_{\chi}$ like the
one-flavor case. The lowest derivative terms are given by

$\begin{split}\mathcal{L}_{0}&=\frac{\tilde{f}_{\varphi}^{2}}{2}\left(\partial\varphi\right)^{2}+\frac{f_{\pi}^{2}}{2}\tr\left(\partial_{\mu}(e^{-iq\varphi}U)\partial^{\mu}(e^{-iq\varphi}U)^{\dagger}\right)\\\
&=\frac{f_{\varphi}^{2}}{2}\left(\partial\varphi\right)^{2}+\frac{f_{\pi}^{2}}{2}\tr\left(\partial_{\mu}U\partial^{\mu}U^{{\dagger}}\right)+\frac{iqf_{\pi}^{2}}{2}\partial_{\mu}\varphi\tr\left(U^{{\dagger}}\partial^{\mu}U-U\partial^{\mu}U^{{\dagger}}\right)\end{split}$
(101)

where $f_{\pi}^{2}$ and
$f_{\varphi}^{2}:=\tilde{f}_{\varphi}^{2}+N_{f}q^{2}f_{\pi}^{2}$ are two
different ‘pion decay constants’, scaling with $N$ as $f_{\pi}^{2}\sim N^{2}$
and $f_{\varphi}^{2}\sim N^{3}$. Additional terms (including the WZW term) are
needed to match the ’t Hooft anomalies of the UV theory.

Turning on the positive masses $m_{adj}$ for the adjoint fermion $\lambda$ and
$m$ for the fundamental fermions $\psi$, $\tilde{\psi}$, as well as the
$\text{SU}(N)$ theta-angle $\theta$, induces a potential on $\mathcal{M}_{0}$:

$V=-\Lambda^{3}\left[(N^{2}-1)m_{adj}\cos\left(Q\varphi\right)+\frac{m}{2}\tr\left(\text{e}^{-i(q\varphi-\theta/N_{f})}U+\text{e}^{i(q\varphi-\theta/N_{f})}U^{{\dagger}}\right)\right]$
(102)

For the purpose of finding the vacua, we can focus on $U$ of the form
$U=\text{e}^{2\pi ik/N_{f}}\textbf{1}_{N_{f}}$ so that the symmetry
$\text{SU}(N_{f})_{V}$ is preserved. The potential now reads

$V(\varphi,k)=-\Lambda^{3}\left[(N^{2}-1)m_{adj}\cos(Q\varphi)+mNN_{f}\cos\left(q\varphi-\frac{2\pi
k+\theta}{N_{f}}\right)\right].$ (103)

To complete the chiral Lagrangian, additional terms (including the WZW term)
are needed to match the ’t Hooft anomalies of the UV theory, but these will
not be necessary for what we want to discuss next.

##### $m_{adj}=0$

When we tune the adjoint mass to zero, the theta angle becomes unphysical as
it can be rotated away by an anomalous chiral rotation. The potential for
$\varphi$ thus reduces to

$V(\varphi,k)=-mNN_{f}\Lambda^{3}\cos\left(q\varphi-\frac{2\pi
k}{N_{f}}\right),$ (104)

with obvious minima at $\varphi=2\pi k/QN$. However, the number of distinct
vacua are smaller than $QN$ due to the $\mathbb{Z}_{Q}$ quotient. Denote the
$k$th vacuum by

$\ket{k}:=\left(\text{e}^{\frac{2\pi ik}{QN}},\text{e}^{\frac{2\pi
ik}{N_{f}}}\textbf{1}_{N_{f}}\right).$ (105)

Then the equivalence (99) implies that

$\ket{N}=\left(\text{e}^{2\pi i/Q},\text{e}^{2\pi
iN/N_{f}}\textbf{1}_{N_{f}}\right)\sim\left(1,\textbf{1}_{N_{f}}\right)=\ket{0}.$
(106)

Thus, there are only $N$ distinct vacua. See Figure 12 for the visualization
of a specific case.

Figure 12: $V(\varphi,k)$ for the case $N=4$, $N_{f}=6$. The $\mathbb{Z}_{Q}$
quotient means we should identify $\pi/Q$ and $-\pi/Q$. It is clear there are
$N=4$ vacua here.

There are domain walls connecting neighbouring vacua $\ket{k}$ and
$\ket{k+1}$. The domain wall configurations cannot preserve the full global
symmetry

$\frac{\text{SU}(N_{f})_{V}\times\text{U}(1)_{q}\times\mathbb{Z}_{2}}{\mathbb{Z}_{N_{f}}\times\mathbb{Z}_{N}}$

of each vacuum. Following Gaiotto:2017tne , one can show that the non-Abelian
symmetry $\text{SU}(N_{f})_{V}$ is necessarily broken down to
$S[\text{U}(1)\times\text{U}(N_{f}-1)]$ by domain wall configurations,
resulting in a non-linear $\sigma$-model on the domain wall with the target
space

$\frac{\text{SU}(N_{f})_{V}}{S[\text{U}(1)\times\text{U}(N_{f}-1)]}\cong\mathbb{CP}^{N_{f}-1}$
(107)

coupled to a topological term induced by the WZW term in the bulk. We can also
conclude that there must be a phase transition on the domain wall as we crank
up the fundamental mass $m$, just like in the one-flavor case, because in the
large mass limit, the domain-wall theory is a TQFT with no massless degrees of
freedom.

##### $m=0$

When we set $m=0$ instead of $m_{adj}$, the potential becomes

$V(\varphi)=-\Lambda^{3}m_{adj}(N^{2}-1)\cos\left(Q\varphi\right),$ (108)

independent of $U$. The potential has its minima at $\varphi=2\pi k/Q$ for any
integer $k$, and any $U\in\text{SU}(N_{f})$. The vacuum manifold
$\mathcal{M}_{0}=(\frac{\text{U}(1)_{\chi}}{\mathbb{Z}_{2}}\times\text{SU}(N_{f}))/\mathbb{Z}_{Q}$
is reduced to to

$\frac{\mathbb{Z}_{Q}\times\text{SU}(N_{f})}{\mathbb{Z}_{Q}}\cong\text{SU}(N_{f}).$

Therefore, when $m=0$ and $0<m_{adj}\ll\Lambda$, only the non-Abelian NGBs
remain massless. The Abelian NGB $\varphi$ becomes massive.

##### $m,m_{adj}>0$

When both masses are non-zero, we need to look at the full potential (103):

$V(\varphi,k)=-\Lambda^{3}\left[(N^{2}-1)m_{adj}\cos(Q\varphi)+mNN_{f}\cos\left(q\varphi-\frac{2\pi
k+\theta}{N_{f}}\right)\right].$

There is a $\mathbb{Z}_{2}$ time-reversal symmetry at $\theta=0,\pi$, which is
broken explicitly at other values of $\theta$. The symmetry transformation is
given by

$\varphi\mapsto-\varphi,\quad k\mapsto-k,$

at $\theta=0$, and by

$\varphi\mapsto-\varphi,\quad k\mapsto-k-1,$

at $\theta=\pi$. Most of the vacuum degeneracy is lifted by the non-zero $M$
term in the potential, due to the fact that $Q$ and $q$ are coprime. The new
vacua are those closest to $\varphi=0$. At $\theta=0$, there is a single
vacuum at $\varphi=0,k=0$, invariant under the $\mathbb{Z}_{2}$ time-reversal
symmetry. At $\theta=\pi$, however, there are two degenerate vacua related to
each other by the time-reversal symmetry (see Figure 13). It is clear that
there can be no second order phase transition to the trivially gapped phase,
unlike what we saw earlier in the one-flavor cases for both the Dirac fermion
and the scalar fields. There is a $\mathbb{CP}^{N_{f}-1}$ non-linear sigma-
model on the domain wall connecting the two vacua just like in the case with
$m_{adj}=0$ consistent with the proposal Gaiotto:2017tne  for QCD.

(a)

(b)

Figure 13: The potential $V(\varphi,k)$ for $N=4$, $N_{f}=6$ when
$m,m_{adj}\neq 0$ at (a) $\theta=0$ and (b) $\theta=\pi$. Figure 14: Phase
diagram for the $\text{SU}(N)$ QCD (f/adj) with fundamental Dirac fermions. It
is color-coded as follows. Purple means it is conformal; red means it is in a
Goldstone phase; green means there is a broken discrete symmetry with domain
walls joining different vacua, called the domain wall (DW) phase. The theory
loses asymptotic freedom and is not well-defined in the UV when $N_{f}/N>4.5$

To summarise, when the adjoint fermion is massless, we can combine our results
from §§ 2, 4 together to obtain a phase diagram in terms of the fundamental
fermions’ mass $m$ and , the ratio of the number of flavors to the number of
colors $\mathcal{R}=N_{f}/N$. The phase diagram is shown in Figure 14.

On the other hand, when $N_{f}$ is fixed such that there is chiral symmetry
breaking, we can vary the adjoint mass $m_{adj}$ and obtain the phase diagram
shown in as shown in Figure 15 by piecing together various limits explored
earlier. When all masses vanish, there are both a massless Abelian
Nambu–Goldstone boson as well as non-Abelian Nambu–Goldstone bosons; only the
non-Abelian ones remain when we turn on the adjoint mass. Contrast this with
the single fundamental fermion case in Figure 2. There, the existence of an
Abelian Nambu–Goldstone boson persists for all value of the adjoint fermion
mass, whereas in the multi-flavor case, it only appears when all fermions are
massless. There are no phase transition in the bulk as we dial the mass $m$
down to zero, but there are phase transitions on the domain walls, just like
in the one-flavor case, from a TQFT to a $\mathbb{CP}^{N_{f}-1}$ non-linear
sigma-model (NLSM) as shown in Figure 16.

Figure 15: The $T$-invariant slice of the phase diagram for QCD (f/adj) with
multiple fundamental fermion flavors. Note that when the fundamental fermions’
mass vanishes, the theta-angle is not physical. We choose to represent the
adjoint mass along the negative mass axis to emphasize that it is the end
point of the $\mathbb{Z}_{2}$ broken phase.  Figure 16: The same as Figure 15,
but showing the phases on the domain walls instead of the bulk phase.

##  5 Theory with multiple fundamental scalars

To cap off our analysis, let us turn to multiple scalar flavors scenario in
this section. Even though the anomaly story goes much the same way as before,
we will see that the IR behaves qualitatively differently as we increase the
number of the scalar fields.

The action for the matter fields reads

$S_{\text{matter}}=\int\text{d}^{4}x\,\left(i\bar{\lambda}\left(\not{\partial}-i\not{a}_{adj}\right)\lambda+\sum_{i=1}^{N_{b}}\absolutevalue{(\partial-
ia)\phi_{i}}^{2}-V(\phi_{i})\right)\;,$ (109)

with the potential

$V(\phi_{i})=\sum_{i=1}^{N_{b}}m^{2}\absolutevalue{\phi_{i}}^{2}+\mathcal{O}(\absolutevalue{\phi}^{4})\;.$
(110)

We give the same mass to all $N_{b}$ scalar fields, just like what we did in §
4, to preserve as much global symmetry as possible. Terms with larger power in
$\phi$ are also assumed to preserve the maximal symmetry.

###  5.1 Symmetry and anomalies

The global symmetry group is

$\begin{split}G^{\text{Global}}&=\frac{\text{SU}(N_{b})\times\text{U}(1)_{B}}{\mathbb{Z}_{N}\times\mathbb{Z}_{N_{b}}}\times\mathbb{Z}^{\chi}_{2N}\\\
&=\frac{\text{U}(N_{b})}{\mathbb{Z}_{N}}\times\mathbb{Z}^{\chi}_{2N}\;,\end{split}$
(111)

under which the matter content transforms in the representations given by
Table 9. There is a mixed anomaly between $\text{U}(N_{f})/\mathbb{Z}_{N}$ and
$\mathbb{Z}^{\chi}_{2N}$, exact as explained in §4.1.

| $\text{SU}(N)$ | $\text{SU}(N_{b})$ | $\text{U}(1)_{B}$ | $\mathbb{Z}^{\chi}_{2N}$
---|---|---|---|---
$\phi$ | ${\tiny\yng(1)}$ | ${\tiny\yng(1)}$ | $+1$ | $0$
$\lambda$ | adj | 1 | $0$ | $+1$
Table 9: Representations of the matter content under the gauge and the global
symmetry groups in the multiple scalar flavors case

###  5.2 The IR phases

When the mass parameter $m^{2}>0$, the scalar fields do not condense. At the
scale below the mass scale, we can integrate them out and again obtain the
same phase as the $\mathcal{N}=1$ $\text{SU}(N)$ SYM: there are $N$ distinct
vacua and there are domain walls connecting them.

When $m^{2}<0$, we need to include the quartic terms in the potential for
stability. Let us combine the scalar fields into an $N\times N_{b}$ matrix
$\Phi$, which transform under the $\text{SU}(N)$ gauge group and the
$\text{SU}(N_{b})$ global symmetry group as

$\Phi\mapsto U\Phi V,\quad U\in\text{SU}(N),\;V\in\text{SU}(N_{b})\;.$

Then the most general potential invariant under the gauge and the global
symmetry group is

$V(\Phi)=-\absolutevalue{m}^{2}\text{~{}Tr~{}}\Phi^{{\dagger}}\Phi+\kappa_{1}\text{~{}Tr~{}}\left(\Phi^{{\dagger}}\Phi\Phi^{{\dagger}}\Phi\right)+\kappa_{2}\left(\text{~{}Tr~{}}\Phi^{{\dagger}}\Phi\right)^{2}+\ldots$
(112)

and we require $\kappa_{1}+N_{b}\kappa_{2}>0$ for stability. By adding
appropriate constant terms to this potential, we can complete the square and
write the potential as

$V(\Phi)=\kappa_{1}\text{~{}Tr~{}}\left(\Phi^{{\dagger}}\Phi-v^{2}\textbf{1}_{N_{b}}\right)^{2}+\kappa_{2}\left[\text{~{}Tr~{}}\left(\Phi^{{\dagger}}\Phi-v^{2}\textbf{1}_{N_{b}}\right)\right]^{2}$
(113)

with $v^{2}=\absolutevalue{m}^{2}/(\kappa_{1}+N_{b}\kappa_{2})$. $\Phi$ must
now acquire vacuum expectation value to minimize the potential, which is
achieved by the configuration $\Phi^{{\dagger}}\Phi=v^{2}\textbf{1}_{N_{b}}$.
In this Higgs regime, the IR phases are sensitive to the number of scalar
flavors $N_{b}$. We will now consider each different scenario in turn.

#### $N_{b}<N$

When $N_{b}<N$, $\expectationvalue{\Phi}\neq 0$ partially higgses the gauge
group down to $\text{SU}(N-N_{b})$ without spontaneous breaking of the global
symmetry, similar to the one flavor case. To see this, let us first rotate
$\expectationvalue{\Phi}$ by $\text{SU}(N)$ and $\text{SU}(N_{b})$
transformations to be of the diagonal form 343434This is none other than the
singular value decomposition (SVD). We can always write any $N\times N_{b}$
complex matrix $\Phi$ as $\Phi=U\Phi_{D}V^{\dagger}$ where $\Phi_{D}$ is
diagonal, and $U,V$ are $\text{SU}(N)$ and $\text{SU}(N_{b})$ matrices.

$\expectationvalue{\Phi}=\left(\begin{array}[]{ccccc}v_{1}&0&\cdots&0&0\\\
0&v_{2}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&0&v_{N_{b}}\\\ \hline\cr 0&0&\cdots&0&0\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&0&0\end{array}\right)\,.$
(114)

This configuration minimizes the potential if and only if $v_{i}=v$ for all
$i$. To see what symmetry is left unbroken by such a VEV, let’s act on
$\expectationvalue{\Phi}$ by a combined $\text{SU}(N)\times\text{U}(N_{b})$
transformation of the form

$\expectationvalue{\Phi}\to U\expectationvalue{\Phi}V,\quad
U=\begin{pmatrix}U_{1}&\\\ &U_{2}\end{pmatrix}\in\text{SU}(N),\quad
V\in\text{U}(N_{b}),$ (115)

where $U_{1}$ and $U_{2}$ are $N_{b}\times N_{b}$ and
$(N-N_{b})\times(N-N_{b})$ unitary matrices, respectively. The transformed VEV
is

$U\expectationvalue{\Phi}V=\begin{pmatrix}vU_{1}V\\\ \textbf{0}^{T}\\\
\vdots\\\ \textbf{0}^{T}\end{pmatrix}\,.$ (116)

We see that the transformation leaves the VEV invariant if and only if we take
$U_{1}=V^{{\dagger}}$, and $U_{2}=e^{i\theta}\tilde{U}$, where the phase
$\theta$ is determined by $V$ (because we need $\det U_{1}\det U_{2}=1$), but
are free to choose $\tilde{U}$ to be any $\text{SU}(N-N_{b})$ matrix. We can
clearly see that the gauge group $\text{SU}(N)$ is higgsed down to
$\text{SU}(N-N_{b})$, while the rest of the gauge group is locked with the
flavor symmetry $\text{U}(N_{b})$, resulting in the color-flavor locked
$\text{U}(N_{b})_{cf}$ global symmetry that survives in the Higgs regime. The
discrete chiral symmetry $\mathbb{Z}^{\chi}_{2N}$ does not act on the scalar
fields, so it remains intact in this phase.

The $\text{SU}(N)$ adjoint fermion $\lambda$ decomposes into an
$\text{SU}(N-N_{b})$ adjoint fermion $\tilde{\lambda}$, $N_{b}$ Weyl fermions
in the fundamental representation of $\text{SU}(N-N_{b})$, $\psi$, $N_{b}$
Weyl fermions in the anti-fundamental representation, $\tilde{\psi}$, a
neutral fermions in the adjoint representation of $\text{U}(N_{b})$, $\eta$,
and one neutral Weyl fermion $\nu$. The matter fields transform under the IR
gauge group $\text{SU}(N-N_{b})$ and the IR global symmetry

$G^{\text{Global}}_{\text{IR}}=\frac{\text{SU}(N_{b})\times\text{U}(1)^{\prime}_{B}}{\mathbb{Z}_{N_{b}}\times\mathbb{Z}_{N-N_{b}}}\times\mathbb{Z}^{\chi}_{2N}$
(117)

in the representations given in Table 10. Again, even though the free massless
fermion matter content has an enhanced global symmetry, as discussed in §3.1,
there are irrelevant terms in the Lagrangian that reduce the symmetry down to
the one we have in the UV. The enhancement that lifts $\mathbb{Z}^{\chi}_{2N}$
to a continuous chiral symmetry happens only at $v=\infty$. The difference in
the discrete quotient between $G^{\text{Global}}_{\text{IR}}$ in Eq. (117) and
$G^{\text{Global}}$ in the UV is not a cause for concern. It simply reflects
the fact that we assign the $\text{U}(1)^{\prime}_{B}$ charge $\pm 1$ to the
fundamental and anti-fundamental fermions $\psi$, $\tilde{\psi}$. Note that
the flavor symmetry that acts on the scalar fields in the UV now acts on the
fermions in the IR through color-flavor locking, as we have already seen in
the one-flavor scalar case.

| $\text{SU}(N-N_{b})$ | $\text{SU}(N_{b})$ | $\text{U}(1)^{\prime}_{B}$ | $\mathbb{Z}^{\chi}_{2N}$
---|---|---|---|---
$\tilde{\lambda}$ | adj | 1 | $0$ | $1$
$\psi$ | ${\tiny\yng(1)}$ | ${\tiny\yng(1)}$ | $1$ | $1$
$\tilde{\psi}$ | $\overline{{\tiny\yng(1)}}$ | $\overline{{\tiny\yng(1)}}$ | $-1$ | $1$
$\eta$ | 1 | adj | $0$ | $1$
$\nu$ | 1 | 1 | $0$ | $1$
Table 10: Representations of the fermions in the Higgs regime when $N_{b}<N$

When $N_{b}/N>9/11$, the resulting gauge theory is IR-free, leading to the
free fermion CFT phase. When $N_{b}/N<9/11$, this $\text{SU}(N-N_{b})$ gauge
theory is asymptotically free. We learn about its IR dynamics by looking back
at the dynamics of the fermionic theory that we studied in § 4, because it is
the theory that emerges in the intermediate region, barring a few extra Weyl
fermions neutral under the gauge group that only couple to the rest via
higher-order terms. Thus, for any finite $v$, we expect to have domain walls
connecting $N$ vacua in the IR. The fermions that transform non-trivially
under the $\text{SU}(N-N_{b})$ gauge group are gapped out by the gauge
dynamics, while the neutral fermions are gapped out by the interaction with
the would-be Goldstone boson for the $\text{U}(1)$ chiral symmetry.

In the deep Higgs regime where $v\gg\Lambda$, still assuming asymptotic
freedom, there are two options for the IR phases, depending on the ratio
between the number of flavors and the number of colors. There exists a
critical point $N_{b}/N=\mathcal{R}_{b*}$, below which we have chiral symmetry
breaking, and above which the phase enters a conformal window. The value of
$R_{b*}$ cannot be ascertained in the generic case due to the strong dynamics
involved. In the Veneziano limit, however, the Banks-Zaks computation can be
trusted, and we can estimate $\mathcal{R}_{b*}$ to be

$\mathcal{R}_{b*}=\frac{18}{31}\approx 0.58\;,$ (118)

up to corrections in $1/N$.

#### $N_{b}=N-1$

When $N_{b}=N-1$, the gauge group is completely higgsed. The global symmetry
is still not spontaneously broken. The IR dynamics is that of free fermions in
multiplets of the global symmetry $G^{\text{global}}$. The massless composite
fermions $\eta$, $\psi$, and $\tilde{\psi}$, now transform under the global
symmetry group

$G^{\text{Global}}_{\text{IR}}=\frac{\text{U}(1)^{\prime}_{B}\times\text{SU}(N_{b})}{\mathbb{Z}_{N_{b}}}\times\mathbb{Z}^{\chi}_{2N}\cong\text{U}(N_{b})\times\mathbb{Z}^{\chi}_{2N}$
(119)

in the adjoint representation, the fundamental representation, and the anti-
fundamental representation of $\text{U}(N_{b})$, respectively. All of them
have unit charge under the discrete chiral symmetry $\mathbb{Z}^{\chi}_{2N}$.
We can dress these fermions with scalars to form gauge-invariant composites,
as we did in (1.1). The special case $N=2$ was considered as a warmup exercise
in Section 1.1. Here, the global symmetry $\text{U}(1)_{B}$ is enhanced to
$SO(3)$ custodial symmetry; see Footnote 4.

#### $N_{b}=N$

When the number of flavors $N_{b}$ is equal to the number of colors $N$, the
gauge group is completely higgsed just like in the case when $N_{b}$ is one
less than $N$. However, there is not enough room in the gauge group
$\text{SU}(N)$ to fully preserve $\text{U}(N_{b})=\text{U}(N)$ global symmetry
through color-flavor locking when $\expectationvalue{\Phi}\neq 0$. The
$\text{U}(1)$ baryon symmetry must now be broken spontaneously by the VEV
scalar fields, while the non-abelian flavor symmetry $\text{SU}(N_{b})$
remains unbroken. The IR phase consists of one $\text{U}(1)$ Goldstone boson,
as well as a massless composite Weyl fermion in the adjoint representation of
the global symmetry $\text{SU}(N_{b})$.

#### $N_{b}>N$

Things get more complicated when $N_{b}>N$. We see from the SVD that the
equation $\Phi^{{\dagger}}\Phi=v^{2}\textbf{1}_{N_{b}}$ has no solutions.
Nonetheless, the diagonal form

$\expectationvalue{\Phi}=\begin{pmatrix}v\textbf{1}_{N}&\textbf{0}&\ldots&\textbf{0}\end{pmatrix}$
(120)

still minimizes the potential. The gauge group is still fully higssed.
Moreover, the global symmetry now breaks spontaneously to
$\text{SU}(N)_{cf}\times\text{U}(N_{b}-N)$, which can be shown by the same
argument around Eq. (116) but in the opposite direction. In the IR, there are
Goldstone bosons described by the coset space

$\mathcal{M}=\frac{\text{U}(N_{b})}{\text{SU}(N)\times\text{U}(N_{b}-N)}$
(121)

apart from the composite fermions that we have previously. The ’t Hooft
anomalies in the UV can be matched by the composite fermions and the WZW term
in the effective Lagrangian.

To summarise, there are many IR phases of the theory as the number $N_{b}$ of
the fundamental scalars and the mass squared $m^{2}$ of the scalars varies.
The ultraviolet (UV) theory has $\mathbb{Z}_{2N}^{\chi}$ symmetry that acts on
the massless adjoint. In addition, baryon-number $\text{U}(1)_{B}$ and flavor
$\text{SU}(N_{b})$ symmetries353535The global symmetry is actually
$\text{SU}(N_{b})/\mathbb{Z}_{\text{GCD}(N_{b},N)}$, because the would-be
global $\text{SU}(N_{b})$ and the $\text{SU}(N)$ may have a common center. act
on the scalars. The theory admits mixed anomalies between
$\mathbb{Z}_{2N}^{\chi}$ and $\text{U}(1)_{B}$ as well as between
$\mathbb{Z}_{2N}^{\chi}$ and gravity. For $m^{2}>0$, we can integrate out the
scalars ending up with ${\cal N}=1$ super Yang-Mills theory. When $m^{2}<0$,
we need to distinguish between different scenarios depending on $N$ and
$N_{b}$. (A) $N_{b}<9N/11$, the gauge group is higgsed down to
$\text{SU}(N-N_{b})$ that is still strongly-coupled. In the IR, the theory
enjoys emergent continuous symmetries. Assuming that the Higgs vev is not much
larger than $\Lambda$, the continuous symmetries are explicitly broken down to
$\mathbb{Z}_{2N}^{\chi}$ by dangerously irrelevant operators. The IR theory
breaks the $\mathbb{Z}_{2N}^{\chi}$ chiral symmetry spontaneously, leading to
$N$ vacua and domain walls. (B) $9N/11<N_{b}<N-1$, the gauge group is again
higgsed down to $\text{SU}(N-N_{b})$. However, the new gauge coupling becomes
weaker under the RG flow. The composite fermions, i.e., the UV fermions
dressed by scalars that transform covariantly under $\text{SU}(N-N_{b})$,
decouples from the gauge fields in the IR. The theory then thus flows to the
composite free fermion phase. (C) $N_{b}=N-1$, the gauge group is fully
higgsed, in which case composite free fermions match the anomalies. These are
the UV fermions dressed by scalars in a fashion that preserves gauge
invariance. (D) $N_{b}=N$, the gauge group is fully higgsed, the flavor
$\text{SU}(N_{b})$ is intact, and $\text{U}(1)_{B}$ is spontaneously broken.
The IR phase contains one massless adjoint fermion in the global flavor group
and one Goldstone boson. (E) $N_{b}>N$, the gauge group is fully higgsed, and
the continuous symmetry is broken, leading to many Goldstones. These various
cases are neatly summarised in a phase diagram, shown in Fig. 17 below.

Figure 17: Phase diagram for the $\text{SU}(N)$ QCD (f/adj) with fundamental
scalars. “Composite fermions” here means the UV adjoint fermion dressed with
the scalar fields in a gauge-covariant way under the gauge group
$\text{SU}(N-N_{b})$ after higgsing. The theory loses asymptotic freedom and
is not well-defined in the UV when $N_{b}/N>18$.

##  6 Conclusions and future prospects

In this work, we analyzed the $\text{SU}(N)$ gauge theory with one massless
adjoint fermion and massive matter, either bosonic or fermionic. The special
case of one fundamental flavor of varying mass was analyzed in detail. The
abundance of ’t Hooft anomalies involving discrete chiral symmetry
$\mathbb{Z}_{2N}^{\chi}$, combined with Vafa-Witten-Weingarten theorems,
restricts the bulk phase quite strongly, which results in spontaneously broken
discrete chiral symmetry for any mass of the fundamental fermion or boson,
leaving $N$ degenerate vacua in the bulk. Further, the bosonic theory is
related to fermionic theory in the sense that when the boson condenses, the
theory is higgsed down to $\text{SU}(N-1)$ with one fundamental fermion. So
most conclusions about the theory with fundamental bosons can be drawn by
studying the fermionic theory.

The massless fundamental fermion limit is particularly interesting, where the
discrete chiral symmetry enhances to $\text{U}(1)_{\chi}$, and the domain
walls melt into Goldstone bosons. Studying the small fundamental mass regime
can be done systematically by perturbing the Goldstone theory. In particular,
we studied the domain walls between the $N$ vacua and, with the exception of
the one $T$-preserving domain wall, found them all to be trivial. On the other
hand, in the opposite limit of infinite fundamental fermion mass, the theory
becomes Super Yang-Mills, of which many things are known. The domain walls of
Super Yang-Mills are conjectured to hold a TQFT. As the fundamental fermion
mass is dialed from small to large, no bulk phase transition occurs, but our
analysis shows that a transition must occur on the domain wall. Further, we
conjectured that the domain wall theory is IR dual to the corresponding
$\text{SU}(N)$ gauge theory with both fundamental and adjoint matter in 3d. Of
particular interest is the $T$-preserving domain wall, which exists if $N$ is
even. This domain wall cannot be made trivial as it carries a
$\text{U}(1)_{B}-T$ mixed anomaly. The corresponding 3d theories were studied
in Lohitsiri:2022jyz , where it was proposed that anomalies can be saturated
either by composite fermions or spontaneous $T$-breaking, with large $N$
arguments favoring the latter. The 4d domain wall analysis indicates that $T$
breaking is preferred, at least for a small enough mass of the fundamentals.
We, however, speculate that there may be a composite fermion phase on the
domain wall for some intermediate mass of the fundamentals for $N=2$. We also
discuss the decoupling limit of adjoints which results in the usual QCD.

We generalized the one fundamental flavor case to multi-flavors and discussed
the bounds on the conformal window from the a-theorem and the Banks-Zacks
2-loop fixed point. For a sufficiently low number of flavors, we analyze the
chiral Lagrangian and map out the phase diagram in the bulk and on the domain
wall. As with one flavor, we discuss the decoupling limit of adjoints.

We end this section by discussing some future prospects. The conjectures about
the bulk phase analyticity can be tested on the lattice. More interesting
would be to study domain walls on the lattice, or the corresponding 3d gauge
theories. The lattice studies of domain walls would require using twisted
boundary conditions or spatially varying $\theta$-term, which generically
introduce a complex action problem that hinders numerical simulations. Studies
of corresponding 3d theories directly also generically require bare Chern-
Simons terms and complex fermionic measures, which again hinders lattice
studies. Another interesting approach is to study soft SUSY deformations of
super QCD setup, and see carefully what happens on the domain walls as
supersymmetry is broken.

###### Acknowledgements.

We would like to thank Joe Davighi, Thomas Dumitrescu, Avner Karasik, Zohar
Komargodski, Erich Poppitz and David Tong. TS would like to thank the
organizers of the Confinement Workshop 2023 in Minneapolis, where interesting
discussions took place. This work is supported by the STFC consolidated grant
in Particles, Strings and Cosmology number ST/T000708/1. TS and NL are also
supported by the Royal Society of London.

##  Appendix A Majorana and Weyl Fermions in the Real Gauge Group
Representation

###  A.1 Converting to the Majorana in 4d

Consider a Weyl fermion in 4d Lorentz space given by the action

$\bar{\lambda}iD\lambda\;,$ (122)

where $D=\sigma^{\mu}D_{\mu}$ with

$\sigma^{\mu}=(\mathbb{I},\tau^{1},\tau^{2},\tau^{3})\;,$ (123)

and $\tau^{i}$ are the Pauli matrices. We will take $D_{\mu}$ to be the
covariant derivative in a real representation of some gauge group, so that
$D_{\mu}^{*}=D_{\mu}$.

We can write the complex Weyl fermion as a real fermion by writing
$\lambda=\lambda_{1}+i\lambda_{2}$. Then we have that

$\bar{\lambda}iD\lambda=\lambda_{1}^{T}i\sigma^{\mu}D_{\mu}\lambda_{1}+i\lambda_{2}^{T}\sigma^{\mu}D_{\mu}\lambda_{2}-\lambda_{1}^{T}\sigma^{\mu}D_{\mu}\lambda_{2}+\lambda_{2}^{T}\sigma^{\mu}D_{\mu}\lambda_{1}\;.$
(124)

Then we have that $\lambda_{1}i\sigma^{\mu}D_{\mu}\lambda_{1}$ and
$\lambda_{2}i\sigma^{\mu}D_{\mu}\lambda_{2}$ do not get a contribution from
the anti-symmetric $\sigma^{2}$ matrix, while
$-\lambda_{1}\sigma^{\mu}D_{\mu}\lambda_{2}$ and
$\lambda_{2}\sigma^{\mu}D_{\mu}\lambda_{1}$ do not get a contribution from the
symmetric matrices $\mathbb{I},\sigma^{1},\sigma^{3}$. Now organize
$\lambda_{1}$ and $\lambda_{2}$ into a column vector
$\Psi=\begin{pmatrix}\lambda_{1}\\\ \lambda_{2}\end{pmatrix}$. We can write
the above action as

$\Psi^{T}i\Gamma^{\mu}D_{\mu}\Psi\;,$ (125)

where

$\displaystyle\Gamma^{0}=\begin{pmatrix}\mathbb{I}&0\\\
0&\mathbb{I}\end{pmatrix}$
$\displaystyle\Gamma^{1}=\begin{pmatrix}\sigma^{1}&0\\\
0&\sigma^{1}\end{pmatrix}$ (126)
$\displaystyle\Gamma^{2}=\begin{pmatrix}0&i\sigma^{2}\\\
-i\sigma^{2}&0\end{pmatrix}$
$\displaystyle\Gamma^{3}=\begin{pmatrix}\sigma^{3}&0\\\
0&\sigma^{3}\end{pmatrix}$ (127)

Now let us set

$C=\begin{pmatrix}0&i\sigma^{2}\\\ i\sigma^{2}&0\end{pmatrix}\;.$ (128)

Notice that the matrix is real and anti-symmetric. It is also unitary because
$C^{\dagger}=C^{-1}$. We want to set $\Gamma^{\mu}=C\gamma^{\mu}$, so that
$\gamma^{\mu}=C^{-1}\Gamma^{\mu}$. We then have that

$\bar{\lambda}iD\lambda=i\Psi^{T}C\gamma^{\mu}D_{\mu}\Psi\;,$ (129)

with

$\displaystyle\gamma^{0}=\begin{pmatrix}0&-i\sigma^{2}\\\
-i\sigma^{2}&0\end{pmatrix}$
$\displaystyle\gamma^{1}=\begin{pmatrix}0&-\sigma^{3}\\\
-\sigma^{3}&0\end{pmatrix}$ (130)
$\displaystyle\gamma^{2}=\begin{pmatrix}-\mathbb{I}&0\\\
0&\mathbb{I}\end{pmatrix}$
$\displaystyle\gamma^{3}=\begin{pmatrix}0&\sigma^{1}\\\
\sigma^{1}&0\end{pmatrix}\;,$ (131)

Note that all the Gamma-matrices are purely real. One can also check that
$\gamma^{\mu}$ as defined above satisfy the Clifford algebra

$\\{\gamma^{\mu},\gamma^{\nu}\\}=2\;\text{diag}(-1,1,1,1)_{\mu\nu}$ (132)

###  A.2 The Weight of the Dirac and Majorana Fermion in 4d, with real gauge
representation

If we have a Dirac fermion in $4d$ spacetime, we can write its Euclidean
action as

$S=\int d^{4}x\;\bar{\Psi}i\not{D}\Psi\;,$ (133)

where $\not{D}=\gamma^{\mu}D_{\mu}$ is the Dirac operator, and $D_{\mu}$ is a
covariant derivative $D_{\mu}=\partial_{\mu}+A_{\mu}$, where $A_{\mu}$ is the
gauge field in the real representation of some group G, i.e.
$A_{\mu}^{*}=A_{\mu}$.

In Euclidean space $i\not{D}$ is Hermitian, and so its eigenvalues are real.
Let $\psi_{n}$ be the eigenfunctions of $i\not{D}$ with eigenvalues
$\lambda_{n}$. Then we can decompose

$\displaystyle\Psi=\sum_{n}\alpha_{n}\psi_{n}\;,$ (134)
$\displaystyle\bar{\Psi}=\sum_{n}\bar{\alpha}_{n}\psi_{n}^{\dagger}\;,$ (135)

where $\alpha_{n}$ and $\bar{\alpha}_{n}$ are independent Grassmann numbers,
and where363636If there are degeneracies $\lambda_{n}=\lambda_{m}$ for $n\neq
m$ we can still chose that degenerate eigenstates are orthogonal. $\int
d^{4}x\;\psi_{n}^{\dagger}\psi_{m}=\delta_{nm}$. We have that the action is
given by

$S=\sum_{n}\lambda_{n}\bar{\alpha}_{n}\alpha_{n}\;,$ (136)

so if we define the measure of the Grassmann integral as373737the factor of
$i$ is there by convention, and is just an overall normalization. In this
convention the weight is always positive.
$\prod_{n}(id\bar{\alpha}_{n}d\alpha_{n})$ we have that the path-integral
weight is

$\prod_{n}(i\lambda_{n})=\det\not{D}\;.$ (137)

Notice that because $\\{\not{D},\gamma^{5}\\}=0$ we have that for every
eigenstate $\psi_{n}$ with eigenvalue $\lambda_{n}$ there exists an eigenstate
$\gamma^{5}\psi_{n}$ with the eigenvalue $-\lambda_{n}$. So we can rewrite
(assuming no $\lambda_{n}=0$)

$\prod_{n}(i\lambda_{n})=\prod_{n|\lambda_{n}>0}\lambda_{n}^{2}\;.$ (138)

so that the weight is positive definite. In addition we will see that each
eigenvalue $\lambda_{n}$ is twice degenerate because it forms a Kramers
doublet.

Now let us move to Majorana fermions. In this case $\bar{\Psi}=\psi^{T}C$
where $C$ is an unitary, anti-symmetric matrix with the property

$C\gamma^{\mu}C^{-1}=-(\gamma^{\mu})^{T}\;.$ (139)

Now notice that the Dirac operator $i\not{D}$ has a degeneracy, because if
$\psi_{n}$ has an eigenvalue $\lambda_{n}$, then $C^{-1}\psi_{n}^{*}$ has the
same eigenvalue. Indeed, since $(\gamma^{\mu})^{\dagger}=\gamma^{\mu}$, we
have

$i\not{D}C^{-1}\psi_{n}^{*}=-C^{-1}i\not{D}^{T}\psi_{n}^{*}=C^{-1}(i\not{D}^{\dagger}\psi_{n})^{*}=C^{-1}(i\not{D}^{\dagger}\psi_{n})^{*}=\lambda_{n}C^{-1}\psi_{n}^{*}\;.$
(140)

Moreover $C^{-1}\psi_{n}^{*}$ is orthogonal to $\psi_{n}$ by the anti-symmetry
of $C$, i.e.

$(C^{-1}\psi_{n}^{*})^{\dagger}\psi_{n}=\psi_{n}^{T}C\psi_{n}=0\;.$ (141)

So $i\not{D}$ has at least a double degeneracy of the spectrum.

One can also see this as a Kramers degeneracy Witten:2015aba . Indeed if $K$
is a complex conjugation operator, we define $\mathcal{T}=C^{-1}K$ an operator
which commutes with $i\not{D}$. Now
$\mathcal{T}^{2}=C^{-1}KC^{-1}K=-C^{-1}C=-1$, where we used the unitarity and
anti-symmetry of $C$.

Let us hence label $\psi_{n}^{i}$ with $i=1,2$ labels the Kramers doublet. Now
we expand the Majorana fermion fields as

$\displaystyle\Psi=\sum_{n,i}\alpha_{n}^{i}\psi_{n}^{i}\;,$ (142)
$\displaystyle\bar{\Psi}=\sum_{n,i}\alpha_{n}^{i}(\psi_{n}^{i})^{T}C\;.$ (143)

Now notice that

$\int
d^{4}x\;(\psi_{n}^{i})^{T}Ci\not{D}\psi_{m}^{j}=\lambda_{m}\int(\psi_{n}^{i})^{T}C\psi_{m}$
(144)

On the other hand we have that, by partially integrating,

$\int d^{4}x\;(\psi_{n}^{i})^{T}Ci\not{D}\psi_{m}^{j}=\int
d^{4}x(\not{D}\gamma^{\mu}\psi_{n}^{i})^{T}C\psi_{m}^{j}=\lambda_{n}\int
d^{4}x(\psi_{n}^{i})^{T}C\psi_{m}^{j}\;.$ (145)

Combining the two expressions we have that383838We will assume that the only
degeneracy in the spectrum is the Kramers degeneracy so that
$\lambda_{n}=\lambda_{m}$ implies $n=m$.

$\int d^{4}x(\psi_{n}^{i})^{T}C\psi_{m}^{j}=0\;\quad\text{ if $n\neq m$}\;.$
(146)

On the other hand if $n=m$ then we see that the expression $\int
d^{4}x(\psi_{n}^{i})^{T}C\psi_{n}^{j}$ is anti-symmetric in $i$ and $j$. We
use a natural normalization

$\int d^{4}x(\psi_{n}^{i})^{T}C\psi_{n}^{j}=\epsilon^{ij}\;.$ (147)

Then the action becomes

$S=\sum_{n}\lambda_{n}\alpha_{n}^{i}\alpha_{n}^{j}\epsilon_{ij}\;.$ (148)

Now we define the measure to be

$\prod_{n}(id\alpha_{n}^{1}d\alpha_{n}^{2})\;,$ (149)

so that the weight is

$\prod_{n}(i\lambda_{n})=\prod_{n|\lambda_{n}>0}\lambda_{n}^{2}\;.$ (150)

where the product over $\lambda_{n}$ is only over one of the Kramers doublet
eigenvalue. The above is manifestly positive.

##  Appendix B Spectral flow

Consider a first-order differential operator

$D=\mathbb{I}\partial_{\tau}+A(\tau)\;.$ (151)

where $A(\tau)$ and $\mathbb{I}$ are an $N\times N$ Hermitian and identity
matrices respectively. We want to look for the zero modes of the above
operator. We solve the differential equation

$D\psi=0\;.$ (152)

Now let us decompose $\psi=\sum_{n=1}^{N}c_{n}(\tau)\chi_{n}(\tau)$ into the
instantaneous eigenvectors $\chi_{n}(\tau)$ of $A(\tau)$, i.e.

$A(\tau)\chi_{n}(\tau)=\lambda_{n}(\tau)\chi_{n}(\tau)\;.$ (153)

We have that the zero-mode equation becomes equivalent to

$\partial_{\tau}c_{n}+\lambda_{n}c_{n}+\sum_{m=1}^{N}M_{nm}c_{m}=0$ (154)

where the matrix $M_{nm}$ is an anti-Hermitian matrix given by

$M_{nm}=\chi^{\dagger}_{m}\partial_{\tau}\chi_{n}\;.$ (155)

Let us put the coefficients $c_{n}$ into a complex vector $\bm{c}$, and
$\lambda_{n}$ into a diagonal matrix $A_{D}$. Then we have that

$\bm{c}(\tau)=Pe^{-\int_{0}^{\tau}d\tilde{\tau}(A_{D}(\tilde{\tau})+M(\tilde{\tau}))}\bm{c}(0)\;.$
(156)

Now let us discuss this solution in the adiabatic approximation limit. Namely
if we differentiate the equation (153), ignoring the derivative of $A$ and the
derivative of $\lambda_{n}$ as small, we have that (sum over $m$ implied)

$\lambda_{m}M_{mn}-M_{mn}\lambda_{n}=0\Leftrightarrow\Lambda M-MA_{D}=0\;,$
(157)

so $M_{mn}$, commuting with $A_{D}$ must be diagonal unless $A_{D}$ has exact
degeneracies. Let us assume that this is the case. Then the equation (156)
implies that if we start with $c_{n}(0)$ not equal to zero for only some $n$
and zero for others, it will stay that way. The diagonal matrix $M$ is just
the Barry phase of individual eigenstates. Now notice that only $c_{n}(\tau)$
for which $\lambda_{n}$ is positive for $\tau\rightarrow\infty$ and negative
for $\tau\rightarrow-\infty$ can be kept if we want normalizable $\psi(\tau)$.
Hence we conclude that the operator $D$ has as many zero modes as the net
spectral flow. If operator $A(\tau)$ still has some degeneracies, the story is
similar because we can always diagonalize $M$ in the subspace of the
degeneracies without affecting the discussion.

##  Appendix C $\beta$-function

The 3-loop $\beta$ function for $n_{R}$ Weyl fermions in representation $R$ of
$\text{SU}(N)$ Yang-Mills theory is given by (see Dietrich:2006cm ;
Zoller:2016sgq )

$\displaystyle\begin{split}\beta(g)=&-\beta_{0}\frac{g^{3}}{(4\pi)^{2}}-\beta_{1}\frac{g^{5}}{(4\pi)^{4}}-\beta_{2}\frac{g^{7}}{(4\pi)^{6}}~{},\\\\[3.0pt]
\beta_{0}=&\frac{11}{6}C_{2}(G)-\sum_{R}\frac{1}{3}T_{R}n_{R}~{},\\\\[3.0pt]
\beta_{1}=&\frac{34}{12}C_{2}^{2}(G)-\sum_{{R}}n_{R}\left\\{\frac{5}{6}C_{2}(G)T_{R}+\frac{1}{2}C_{2}({R})T_{R}\right\\}~{},\\\\[3.0pt]
\beta_{2}=&\frac{2857}{432}C_{2}^{3}(G)-\sum_{{R}}\frac{n_{R}T_{R}}{4}\left[-\frac{C_{2}^{2}({R})}{2}+\frac{205C_{2}(G)C_{2}({R})}{36}+\frac{1415C_{2}^{2}(G)}{108}\right]\\\\[3.0pt]
&+\quad\sum_{{R},{R}^{\prime}}\frac{n_{R}n_{R^{\prime}}T_{R}T_{R^{\prime}}}{16}\left[\frac{44C_{2}({R})}{18}+\frac{158C_{2}(G)}{54}\right]~{},\end{split}$
(158)

where $G$ denotes the adjoint representation. The quadratic Casimir of
representation $R$, $C_{2}({R})$, is

$\displaystyle t^{a}_{R}t^{a}_{R}=C_{2}({R})\mathbf{1}_{R}~{},$ (159)

and $C_{2}(G)$ is the quadratic Casimir of the adjoint representation. $T_{R}$
is the Dynkin index, which is defined via

$\displaystyle\mbox{tr}\left[t^{a}_{R}t^{b}_{R}\right]=T_{R}\delta^{ab}~{}.$
(160)

From Eqs. (159) and (160), we obtain the relation

$\displaystyle T_{R}\mbox{dim}_{G}=C_{2}({R})\mbox{dim}_{R}~{},$ (161)

where $\mbox{dim}_{R}$ is the dimension of $R$. In particular, using the
convention $T_{R}=1$ for the fundamental representation $R={\tiny\yng(1)}$, we
have $C_{2}(G)=2N$, $\mbox{dim}_{G}=N^{2}-1$.

## References

  * (1) A. Kapustin and N. Seiberg, _Coupling a QFT to a TQFT and Duality_ , _JHEP_ 04 (2014) 001, [1401.0740].
  * (2) D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 (2015) 172, [1412.5148].
  * (3) D. Gaiotto, A. Kapustin, Z. Komargodski and N. Seiberg, _Theta, Time Reversal, and Temperature_ , _JHEP_ 05 (2017) 091, [1703.00501].
  * (4) D. Gaiotto, Z. Komargodski and N. Seiberg, _Time-reversal breaking in QCD 4, walls, and dualities in 2 + 1 dimensions_, _JHEP_ 01 (2018) 110, [1708.06806].
  * (5) Z. Komargodski, A. Sharon, R. Thorngren and X. Zhou, _Comments on Abelian Higgs Models and Persistent Order_ , _SciPost Phys._ 6 (2019) 003, [1705.04786].
  * (6) Z. Komargodski, T. Sulejmanpasic and M. Ünsal, _Walls, anomalies, and deconfinement in quantum antiferromagnets_ , _Phys. Rev. B_ 97 (2018) 054418, [1706.05731].
  * (7) Z. Komargodski and N. Seiberg, _A symmetry breaking scenario for QCD 3_, _JHEP_ 01 (2018) 109, [1706.08755].
  * (8) T. Sulejmanpasic and Y. Tanizaki, _C-P-T anomaly matching in bosonic quantum field theory and spin chains_ , _Phys. Rev. B_ 97 (2018) 144201, [1802.02153].
  * (9) Y. Tanizaki and T. Sulejmanpasic, _Anomaly and global inconsistency matching: $\theta$-angles, $SU(3)/U(1)^{2}$ nonlinear sigma model, $SU(3)$ chains and its generalizations_, _Phys. Rev. B_ 98 (2018) 115126, [1805.11423].
  * (10) Y. Tanizaki, _Anomaly constraint on massless QCD and the role of Skyrmions in chiral symmetry breaking_ , _JHEP_ 08 (2018) 171, [1807.07666].
  * (11) A. Karasik and Z. Komargodski, _The Bi-Fundamental Gauge Theory in 3+1 Dimensions: The Vacuum Structure and a Cascade_ , _JHEP_ 05 (2019) 144, [1904.09551].
  * (12) M. M. Anber and E. Poppitz, _Two-flavor adjoint QCD_ , _Phys. Rev. D_ 98 (2018) 034026, [1805.12290].
  * (13) M. M. Anber and E. Poppitz, _Anomaly matching, (axial) Schwinger models, and high-T super Yang-Mills domain walls_ , _JHEP_ 09 (2018) 076, [1807.00093].
  * (14) M. M. Anber, _Self-conjugate QCD_ , _JHEP_ 10 (2019) 042, [1906.10315].
  * (15) M. M. Anber, _Condensates and anomaly cascade in vector-like theories_ , _JHEP_ 03 (2021) 191, [2101.04132].
  * (16) T. Sulejmanpasic, Y. Tanizaki and M. Ünsal, _Universality between vector-like and chiral quiver gauge theories: Anomalies and domain walls_ , _JHEP_ 06 (2020) 173, [2004.10328].
  * (17) P. B. Smith, A. Karasik, N. Lohitsiri and D. Tong, _On discrete anomalies in chiral gauge theories_ , _JHEP_ 01 (2022) 112, [2106.06402].
  * (18) M. M. Anber and E. Poppitz, _On the baryon-color-flavor (BCF) anomaly in vector-like theories_ , _JHEP_ 11 (2019) 063, [1909.09027].
  * (19) C. Vafa and E. Witten, _Restrictions on Symmetry Breaking in Vector-Like Gauge Theories_ , _Nucl. Phys. B_ 234 (1984) 173–188.
  * (20) V. Bashmakov, F. Benini, S. Benvenuti and M. Bertolini, _Living on the walls of super-QCD_ , _SciPost Phys._ 6 (2019) 044, [1812.04645].
  * (21) E. Poppitz and M. Unsal, _Conformality or confinement (II): One-flavor CFTs and mixed-representation QCD_ , _JHEP_ 12 (2009) 011, [0910.1245].
  * (22) M. M. Anber and L. Vincent-Genod, _Classification of compactified $su(N_{c})$ gauge theories with fermions in all representations_, _JHEP_ 12 (2017) 028, [1704.08277].
  * (23) T. DeGrand, M. Golterman, E. T. Neil and Y. Shamir, _One-loop Chiral Perturbation Theory with two fermion representations_ , _Phys. Rev. D_ 94 (2016) 025020, [1605.07738].
  * (24) V. Ayyar, T. DeGrand, M. Golterman, D. C. Hackett, W. I. Jay, E. T. Neil et al., _Spectroscopy of SU(4) composite Higgs theory with two distinct fermion representations_ , _Phys. Rev. D_ 97 (2018) 074505, [1710.00806].
  * (25) G. Cossu, L. Del Debbio, M. Panero and D. Preti, _Strong dynamics with matter in multiple representations: $\mathrm{SU}(4)$ gauge theory with fundamental and sextet fermions_, _Eur. Phys. J. C_ 79 (2019) 638, [1904.08885].
  * (26) G. Bergner and S. Piemonte, _Lattice simulations of a gauge theory with mixed adjoint-fundamental matter_ , _Phys. Rev. D_ 103 (2021) 014503, [2008.02855].
  * (27) H. Shimizu and K. Yonekura, _Anomaly constraints on deconfinement and chiral phase transition_ , _Phys. Rev. D_ 97 (2018) 105011, [1706.06104].
  * (28) M. M. Anber, S. Hong and M. Son, _New anomalies, TQFTs, and confinement in bosonic chiral gauge theories_ , _JHEP_ 02 (2022) 062, [2109.03245].
  * (29) C.-T. Hsieh, _Discrete gauge anomalies revisited_ , 1808.02881.
  * (30) D. Delmastro, J. Gomis, P.-S. Hsin and Z. Komargodski, _Anomalies and Symmetry Fractionalization_ , 2206.15118.
  * (31) J. Davighi and N. Lohitsiri, _The algebra of anomaly interplay_ , _SciPost Phys._ 10 (2021) 074, [2011.10102].
  * (32) C. Córdova and K. Ohmori, _Anomaly Obstructions to Symmetry Preserving Gapped Phases_ , 1910.04962.
  * (33) C. Córdova and K. Ohmori, _Anomaly Constraints on Gapped Phases with Discrete Chiral Symmetry_ , _Phys. Rev. D_ 102 (2020) 025011, [1912.13069].
  * (34) D. Weingarten, _Mass inequalities for quantum chromodynamics_ , _Phys. Rev. Lett._ 51 (Nov, 1983) 1830–1833.
  * (35) A. Cherman, S. Sen and L. G. Yaffe, _Anyonic particle-vortex statistics and the nature of dense quark matter_ , _Phys. Rev. D_ 100 (2019) 034015, [1808.04827].
  * (36) N. Lohitsiri and T. Sulejmanpasic, _Comments on QCD 3 and anomalies with fundamental and adjoint matter_, _JHEP_ 10 (2022) 081, [2205.07825].
  * (37) Y. Tachikawa and K. Yonekura, _On time-reversal anomaly of 2+1d topological phases_ , _PTEP_ 2017 (2017) 033B04, [1610.07010].
  * (38) C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski and N. Seiberg, _Comments on Chern-Simons Contact Terms in Three Dimensions_ , _JHEP_ 09 (2012) 091, [1206.5218].
  * (39) B. S. Acharya and C. Vafa, _On domain walls of N=1 supersymmetric Yang-Mills in four-dimensions_ , hep-th/0103011.
  * (40) J. Gomis, Z. Komargodski and N. Seiberg, _Phases Of Adjoint QCD 3 And Dualities_, _SciPost Phys._ 5 (2018) 007, [1710.03258].
  * (41) D. Delmastro and J. Gomis, _Domain walls in 4d $\mathcal{N}$ = 1 SYM_, _JHEP_ 03 (2021) 259, [2004.11395].
  * (42) C. Cordova, P.-S. Hsin and N. Seiberg, _Global Symmetries, Counterterms, and Duality in Chern-Simons Matter Theories with Orthogonal Gauge Groups_ , _SciPost Phys._ 4 (2018) 021, [1711.10008].
  * (43) P.-S. Hsin and N. Seiberg, _Level/rank Duality and Chern-Simons-Matter Theories_ , _JHEP_ 09 (2016) 095, [1607.07457].
  * (44) E. Witten, _Fermion Path Integrals And Topological Phases_ , _Rev. Mod. Phys._ 88 (2016) 035001, [1508.04715].
  * (45) C.-T. Hsieh, G. Y. Cho and S. Ryu, _Global anomalies on the surface of fermionic symmetry-protected topological phases in (3+1) dimensions_ , _Phys. Rev. B_ 93 (2016) 075135, [1503.01411].
  * (46) I. Hason, Z. Komargodski and R. Thorngren, _Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism_ , _SciPost Phys._ 8 (2020) 062, [1910.14039].
  * (47) Y. Tachikawa and K. Yonekura, _More on time-reversal anomaly of 2+1d topological phases_ , _Phys. Rev. Lett._ 119 (2017) 111603, [1611.01601].
  * (48) P. Di Vecchia, G. Rossi, G. Veneziano and S. Yankielowicz, _Spontaneous $CP$ breaking in QCD and the axion potential: an effective Lagrangian approach_, _JHEP_ 12 (2017) 104, [1709.00731].
  * (49) R. Kitano, N. Yamada and M. Yamazaki, _Is $N=2$ Large?_, _JHEP_ 02 (2021) 073, [2010.08810].
  * (50) E. Witten, _Supersymmetric index in four-dimensional gauge theories_ , _Adv. Theor. Math. Phys._ 5 (2002) 841–907, [hep-th/0006010].
  * (51) W. BROWDER and E. THOMAS, _AXIOMS FOR THE GENERALIZED PONTRYAGIN COHOMOLOGY OPERATIONS_ , _The Quarterly Journal of Mathematics_ 13 (01, 1962) 55–60, [https://academic.oup.com/qjmath/article-pdf/13/1/55/7288473/13-1-55.pdf].
  * (52) O. Aharony, N. Seiberg and Y. Tachikawa, _Reading between the lines of four-dimensional gauge theories_ , _JHEP_ 08 (2013) 115, [1305.0318].
  * (53) M. M. Anber and E. Poppitz, _Generalized ’t Hooft anomalies on non-spin manifolds_ , _JHEP_ 04 (2020) 097, [2002.02037].
  * (54) A. Kapustin, R. Thorngren, A. Turzillo and Z. Wang, _Fermionic Symmetry Protected Topological Phases and Cobordisms_ , _JHEP_ 12 (2015) 052, [1406.7329].
  * (55) T. Appelquist, A. G. Cohen and M. Schmaltz, _A New constraint on strongly coupled gauge theories_ , _Phys. Rev. D_ 60 (1999) 045003, [hep-th/9901109].
  * (56) D. D. Dietrich and F. Sannino, _Conformal window of SU(N) gauge theories with fermions in higher dimensional representations_ , _Phys. Rev. D_ 75 (2007) 085018, [hep-ph/0611341].
  * (57) M. F. Zoller, _Four-loop QCD $\beta$-function with different fermion representations of the gauge group_, _JHEP_ 10 (2016) 118, [1608.08982].