AI

    artificial intelligence
AoA

    angle-of-arrival
AoD

    angle-of-departure
BS

    base station
D-MIMO

    distributed multiple-input multiple-output
DFTS

    discrete Fourier transform spread
EM

    electromagnetic
E2E

    end-to-end
GDOP

    geometric dilution of precision
GNSS

    global navigation satellite system
IQ

    in-phase and quadrature
JCS

    Joint Communication and Sensing
JRC

    joint radar and communication
JRC2LS

    joint radar communication, computation, localization, and sensing
ICI

    inter-carrier interference
IOO

    indoor open office
IoT

    Internet of Things
IN

    infrastructure node
KPI

    key performance indicator
LoS

    line-of-sight
MIMO

    multiple-input multiple-output
mmWave

    millimeter-wave
NLoS

    non-line-of-sight
NR

    new radio
OFDM

    orthogonal frequency-division multiplexing
OTFS

    orthogonal time-frequency-space
PRS

    positioning reference signal
QoS

    Quality of Service
RAN

    radio access network
RAT

    radio access technology
RedCap

    reduced capacity
RF

    radio frequency
RIS

    reconfigurable intelligent surface
RTK

    real-time kinematic
RTT

    round-trip-time
SLAM

    simultaneous localization and mapping
SNR

    signal-to-noise ratio
ToA

    time-of-arrival
TDoA

    time-difference-of-arrival
TR

    time-reversal
Tx/Rx

    transmitter/receiver
Tx

    transmitter
Rx

    receiver
UE

    user equipment

# 6G Radio Requirements to Support Integrated Communication, Localization, and
Sensing

Henk Wymeersch7, Aarno Pärssinen2, Traian E. Abrudan8, Andreas Wolfgang4,
Katsuyuki Haneda6,
Muris Sarajlic1, Marko E. Leinonen2, Musa Furkan Keskin7, Hui Chen7,
Simon Lindberg4, Pekka Kyösti2, Tommy Svensson7, Xinxin Yang4,
7Chalmers University of Technology, Sweden 1Ericsson Research, Sweden
4Qamcom Research and Technology, Sweden 2University of Oulu, Finland
6Aalto University, Finland 8Nokia Bell Labs, Finland

###### Abstract

6G will be characterized by extreme use cases, not only for communication, but
also for localization, and sensing. The use cases can be directly mapped to
requirements in terms of standard key performance indicators, such as data
rate, latency, or localization accuracy. The goal of this paper is to go one
step further and map these standard KPIs to requirements on signals, on
hardware architectures, and on deployments. Based on this, system solutions
can be identified that can support several use cases simultaneously. Since
there are several ways to meet the KPIs, there is no unique solution and
preferable configurations will be discussed.

##  I Introduction

6G will support extreme use cases, requiring gigabit-per-second peak data
rates with millisecond end-to-end (E2E) latency [1]. Secondly, 6G will rely on
a variety of new enablers at the physical layer [2], including the utilization
of frequency bands in the upper millimeter-wave (mmWave) range
($100\text{--}300\,\mathrm{GHz}$), novel radio frequency (RF) architectures,
exploration of hardware-friendly, energy-efficient waveforms and beamforming,
and the use of distributed large MIMO system and reconfigurable intelligent
surfaces. Finally, 6G will feature a tight integration among communication,
localization, and sensing [3, 4]. Integration will not only enable exciting,
but challenging, new use cases, but also improve communication functionality
as well as other services[5]. Use case requirements are generally specified in
terms of KPIs, which in turn impose requirements on the signals (e.g.,
bandwidth, waveform, and modulation) [6], the hardware architectures (e.g.,
carrier, channelization, array type, output power) [7], and deployments (e.g.,
placement of (distributed) base stations and RISs) [8].

In this paper, we investigate the requirements on signals, hardware
architectures, and deployments for 6G integrated communication, localization,
and sensing. Starting from several 6G use cases, we state expected KPIs, list
the possible options in terms of signals, hardware architectures, and
deployments, followed by discussing how they should be determined to support
the considered use cases. As there is no unique solution to this design
problem, several alternatives are discussed, with the aim of finding
commonalities and possibly a joint design that can serve several 6G use cases
simultaneously. The main contribution of this paper is to provide both a
methodology for determining such requirements, as well as listing initial
solutions. We also highlight several of the (sometimes subtle) synergies and
trade-offs that should be considered in overall 6G radio.

In terms of structure, this paper starts with a recap of 6G use cases and KPIs
(Section II), followed by a brief overview of the components of the radio
channel (antennas, RF technology, and wave propagation) at upper mmWave
frequencies (Section III). Section IV describes the identified degrees of
freedom in terms of signals, hardware architecture, and deployments. Then, in
Sections V–VII, the specific requirements are determined, needed to support
the use cases from Section II. We conclude the paper with Section VIII.

Figure 1: Communication, localization, and sensing each impose requirement on
signal, hardware architectures, and deployments, which should be met by 6G
radio.

##  II 6G Use Cases and KPIs

In this section, we recap several 6G use cases and list their KPIs, for all
three considered applications: communication (C1–C2), localization (L1–L3),
and sensing (S1–S2).

###  II-A Communication

Following [9], we propose two communication system design scenarios
corresponding to 6G use-cases with advanced design requirements. _Very short-
range wireless access (C1)_ $100\,\mathrm{Gbps}$ per-user rate,
$0.1\text{--}1\,\mathrm{ms}$ E2E latency, $10\,\mathrm{m}$ link range. _Short-
range wireless access (C2)_ $10\,\mathrm{Gbps}$ per-user rate,
${<}1\,\mathrm{ms}$ E2E latency, $100\,\mathrm{m}$ link range.

###  II-B Localization

Following [10], we divide localization scenarios into three categories, with
KPIs derived from [5]. _High-accuracy positioning (L1)_ for very fine
maneuvers or coordination. An example in this category is _augmented reality_
, with $1\,\mathrm{cm}$ location accuracy, $1\,\mathrm{{}^{\circ}}$
orientation accuracy, $100\,\mathrm{Hz}$ update rate, and $10\,\mathrm{m}$
link range. _Low-latency positioning (L2)_ , for example for _collaborative
robot localization_ , requiring $10\,\mathrm{cm}$ location accuracy,
$1\,\mathrm{{}^{\circ}}$ orientation accuracy, $1\,\mathrm{kHz}$ update rate,
and $30\,\mathrm{m}$ link range. _Low-complexity positioning (L3)_ for
Internet of Things (IoT) localization with reduced capacity (RedCap) devices.
An example is _remote sensing_ , with $1\text{--}10\,\mathrm{m}$ localization
accuracy, $1\,\mathrm{Hz}$ or lower update rate, and more than
$1\,\mathrm{km}$ link range.

###  II-C Sensing

A natural breakdown is into monostatic and bi-/multi-static sensing.
_Monostatic sensing (S1)_ involves a transmitter and receiver are co-located,
sharing common knowledge of the data and have a shared clock, and provides
radar-like mapping (e.g., for automotive applications), with $10\,\mathrm{cm}$
distance accuracy, $0.04\,\mathrm{m/s}$ velocity accuracy,
$3\,\mathrm{{}^{\circ}}$ angular resolution, $0.2\,\mathrm{{}^{\circ}}$
angular accuracy, maximum range of $50\,\mathrm{m}$, and $25\,\mathrm{Hz}$
update rate [11, Table C2]. This application is similar to the _sensor
infrastructure web_ use case from [5]. We only consider monostatic sensing
with communication signals, not dedicated sensing waveforms. _Bi-/multi-static
sensing (S2)_ involves physically separate transmitters and receivers, e.g.,
for _robotic object sensing_ , with $1\,\mathrm{cm}$ localization resolution,
as well as angular resolution below $1\,\mathrm{{}^{\circ}}$,
$0.1\,\mathrm{m/s}$ velocity resolution, and up to $1\,\mathrm{kHz}$ update
rate.

##  III Radio Channel and RF Technologies

The radio channel is composed of antennas/arrays connected to RF transceivers
at ends of the link, with the multipath wave propagation channel between them.
Combined, they determine the practical link performance within the constraints
imposed by the wave propagation and semiconductor/material physics. The
following summarizes some viewpoints about them, focusing on upper mmWave
frequencies.

###  III-A Antennas and Arrays

At both lower and upper mmWave, most likely hybrid architectures of analog RF
and digital baseband beamforming will be used to support a massive number of
antenna elements in the array. Due to challenges in integrated designs of an
antenna array and its feeding network for RF phase shifting, practical
implementation of an integrated phased antenna array may need a compromise in
a beam scanning range and some extents of grating lobes. Lens antennas and
radomes, despite bulkiness, allow improvement of the beam scan range of
antenna arrays, e.g., [12], and are applicable to upper mmWave RF. The
knowledge of radiation patterns of antennas and arrays is essential in
localization and sensing, while it is less important in communications but
would enhance the link performance in, e.g., beam searching. Antenna and array
calibration methods, in an anechoic chamber or over-the-air, would be an
important technical element [13].

###  III-B RF Technology

Array performance is dependent on the amplifier performance both in transmit
and receive side. The former defines the largest possible output power while
the latter minimum noise figure. Both are to certain extent bandwidth
dependent and even more carrier frequency dependent. Fundamental limits stem
from the properties of semiconductor technologies and wiring losses. Silicon
and compound semiconductors including different transistor types, sizes and
technology nodes determine maximum operating frequency as maximum unilateral
gain ($f_{\max}$) or transition frequency. As carrier frequency gets higher,
the gain, fundamental requirement of RF processing, approaches unity while
available output power decreases and noise increases rapidly. Recent example
of a state-of-the-art $290\,\mathrm{GHz}$ amplifier in SiGe BJT process
demonstrates more than $10\,\mathrm{dB}$ of gain at $2/3$ of $f_{\max}$ at the
cost of significantly lowered dynamic range compared to a low frequency
counterpart using the same technology [14]. RF transceivers for single link
have been demonstrated at frequencies above $200\,\mathrm{GHz}$ for short
range ($1\,\mathrm{m}$) communications trials achieving $100\,\mathrm{Gbps}$
[15].

###  III-C Wave Propagation

The most important characteristic of wave propagation is pathloss, with upper
mmWave RFs showing pathloss exponents of omni-directional channels around $2$
and $3$ in outdoor cellular line-of-sight (LoS) and non-line-of-sight (NLoS)
environments [16]. Even though diffraction coefficients are smaller as the RF
increases, reflections on concrete walls, metal lampposts and tinted glasses
can deliver power from one link end to another, making the link connectivity
in NLoS feasible through one or multiple reflections even for upper mmWave.
The finding of pathloss exponents does not differ from available insights for
lower mmWave RF according to the comparison of indoor hotspot channels [17].
The number of multipaths or clusters is an important degree of freedom. More
clusters indicate increased possibility of spatial multiplexing to send
different data sets over possibly orthogonal beams. Existence of multiple
clusters also implies the possibility of device localization and sensing
through them. Measurements show the number of spatial clusters to be $0.69$
and $1.82$ for LoS and NLoS urban microcellular links [18] while channels
support one to four beams in an indoor entrance hall [19].

Use case |  C1 |  C2 |  L1 |  L2 |  L3 |  S1 |  S2
---|---|---|---|---|---|---|---
Signals |  |  |  |  |  |  |
Bandwidth |  (a) $4\,\mathrm{GHz}$
(b) $10\,\mathrm{GHz}$ |  (a) $0.4\,\mathrm{GHz}$
(b) $5\,\mathrm{GHz}$ |  (a) $2\text{--}4\,\mathrm{GHz}$
(b) $500\,\mathrm{MHz}$ |  $0.5\text{--}1\,\mathrm{GHz}$ |  ${<}500\,\mathrm{MHz}$ |  (a) $2\text{--}4\,\mathrm{GHz}$
(b) ${<}1\,\mathrm{GHz}$ |  $2\text{--}4\,\mathrm{GHz}$
Waveform |  any |  any |  (DFTS-)OFDM |  (DFTS-)OFDM |  any |  (DFTS-)OFDM |  (DFTS-)OFDM
Modulation |  coherent or
non-coherent |  coherent or
non-coherent |  coherent |  coherent |  coherent or non-coherent |  coherent |  coherent
Signal shaping |  space in
some scenarios |  space and/or freq.
in some scenarios |  freq., time, and space |  freq. and space |  no |  freq., time, and space |  trade-off with comm.
Hardware Arch. |  |  |  |  |  |  |
Carrier* |  $60\text{-}140\,\mathrm{GHz}$ |  (a) $\text{sub-}6\,\mathrm{GHz}$
(b) $60\text{-}140\,\mathrm{GHz}$ |  (a) $<30\,\mathrm{GHz}$
(b) $60\text{-}140\,\mathrm{GHz}$ |  $60\text{-}140\,\mathrm{GHz}$ |  $6\text{--}30\,\mathrm{GHz}$ |  $60\text{-}140\,\mathrm{GHz}$ |  $60\text{-}140\,\mathrm{GHz}$
Channelization |  optional |  optional |  no |  no |  no |  not preferred |  not preferred
IN array type |  analog or hybrid |  analog or hybrid |  analog or hybrid |  hybrid or digital |  analog |  analog or hybrid |  analog or hybrid
UE array type |  analog or hybrid |  analog or hybrid |  analog or hybrid |  analog or hybrid |  SNR boost |  analog or hybrid |  hybrid or digital
IN array size (per dim.) |  10-20 |  10-20 |  (a) 10–20
(b) 50–100 |  10–20 |  SNR boost |  (a) 10–20
(b) 40–50 |  (a) 10–20
(b) 40–50
UE array size (per dim.) |  4-8 |  4-8 |  10–20 |  10–20 |  SNR boost |  (a) 100
(b) 10–20 |  100
Transmit power |  medium |  medium |  low (see (1)) |  higher
(small $T$ in (1)) |  higher
(large $d$ in (1)) |  high (high path loss) |  high (high path loss)
Deployments |  |  |  |  |  |  |
Placement around each device |  (a) $\geq 1$ IN multi-stream or D-MIMO
(b) $\geq 1$ IN single stream |  (a) $\geq 1$ IN multi-stream or D-MIMO
(b) $\geq 1$ IN single stream |  $\geq 4$ INs in LoS (TDoA, 3D),
$\geq 2$ INs in LoS (AoA, 3D) |  same as L1 |  $\geq 4$ INs in LoS (TDoA, 3D) |  N/A |  (a) $\geq 1$ Rx INs (TDoA + AoA \+ AoD), Tx-Rx in LoS
(b) $\geq 1$ Rx INs (ToA \+ AoA \+ AoD), Tx-Rx in NLoS
Synchronization |  $1\text{--}10\,\mathrm{ns}$ (D-MIMO) |  $1\text{--}10\,\mathrm{ns}$ (D-MIMO) |  $100\,\mathrm{ps}$ (TDoA) |  $0.5\,\mathrm{ns}$ (TDoA) |  $1\text{--}10\,\mathrm{ns}$ (TDoA) |  N/A |  (a) N/A
(b) $100\,\mathrm{ps}$
IN knowledge |  position: area-level |  position: area-level |  position: mm-level
orientation: $\leq 0.1^{\circ}$ (AoA) |  position: cm-level,
orientation: $\leq 1^{\circ}$ (AoA) |  position: m-level,
orientation: $\leq 1^{\circ}$ (AoA) |  N/A |  position: mm-level,
orientation: $\leq 0.5^{\circ}$ (AoA)

TABLE I: Summarizing table on the requirements on signals, hardware
architectures, and deployments. Acronyms are defined in the text. *Caution:
numbers are in reality technology-dependent, with noise figure and maximum
transmit power being frequency-dependent.

##  IV Signals, Architectures, and Deployments

We describe the degrees of freedom in terms of signals, hardware
architectures, and deployments. These will then be elaborated in the
subsequent sections to support the use cases from Section II, and summarized
in Table I.

###  IV-A Signals

The radio signals are considered to have the following 4 degrees of freedom:
the _aggregate bandwidth_ 111The aggregate bandwidth does not have to be
contiguous in frequency and can be fragmented into multiple RF chains with
separate smaller-bandwidth ADCs. For full utilization of the entire bandwidth,
phase coherency should be maintained across the different chains. (in Hz), the
_waveform type_ (examples are orthogonal frequency-division multiplexing
(OFDM) [20], discrete Fourier transform spread (DFTS) OFDM [21, 22],
orthogonal time-frequency-space (OTFS) [23], single carrier [24]), the
_modulation type_ (e.g., constant-modulus or QAM, non-coherent or coherent),
the _signal shaping_ (in time-frequency-space domains, including precoding,
combining, pilot allocation, and power allocation).

###  IV-B Hardware Architectures

Hardware specifications originate from signal quality and range requirements.
The former sets bandwidth and signal-to-noise ratio (SNR). On the other hand,
underlying technologies determine minimum achievable noise and maximum
transmitted power that are physics, technology, bandwidth, and carrier
frequency dependent. In addition, also energy consumption and cost need to be
considered carefully if we need to parallelize the signal paths. Wider
bandwidth leads inevitably to higher noise and to the need to search for
available radio spectrum at higher carrier frequencies. Both are impacting
negatively on achievable performance. Taking these into account we have some
degrees of freedom in frequency to arrange signals in _carrier, and
channelization_ domains [25]. In the spatial domain, we can increase the range
by beamforming or the rate with MIMO, either in centralized or distributed
manner. Depending on the targets we have the choice of _array type_ (e.g.,
analog, hybrid, planar), _array size_ (number of elements). RF phased arrays
already exist in 5G NR operating at lower mmWave region, which are needed even
in mobile equipment for decent range [26]. Careful link analysis for line-of-
sight and other anticipated radio channel conditions is needed to balance all
the requirements with hardware properties including antennas to evaluate the
feasibility of the radio transceiver hardware for different deployments[27].
Hence, hardware architecture in digital systems needs to determine the number
of parallel digitized radio channels to achieve sufficient bandwidth and then
channelize it into different radio paths in frequency and spatial domain while
utilizing array gain for compensating radio path loss of every available
beam[19].

###  IV-C Deployments

The deployments are largely limited by cost and include the following 3
degrees of freedom: the _placement of infrastructure nodes_, including fixed
and mobile base stations and RISs (number of INs and their positions and
orientations), the _level of synchronization between INs_ (time
synchronization and phase synchronization), the _level of knowledge needed
regarding the IN configuration_ (e.g., location, orientation, phase center).
In distributed deployments, we can split the burden to several physically
separated radio units with additional cost.

##  V Requirements on Signals

$10^{0}$$10^{1}$$10^{2}$$10^{3}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$$10^{1}$distance
[m]distance estimation RMSE [m]AoA estimation RMSE [rad]positioning RMSE
[m]rate [Gb/s] Figure 2: Example showing how the ability to estimate distance
AoA is affected by the distance between user and base station, for an uplink
scenario according to [5, Annex C] using 120 OFDM symbols for localization
(lasting about 125 us).

To understand the involved trade-offs, Fig. 2 shows the performance of time-
based distance, angle-of-arrival (AoA) estimation, and positioning in an
uplink scenario under resolved LoS with respect to the distance, following the
parameters from [5, Annex C], with a $140\,\mathrm{GHz}$ carrier,
$2\,\mathrm{GHz}$ bandwidth, and $14\,\mathrm{dBm}$ transmit power. We see
that even though distance and AoA can be well estimated at far distances
thanks to the large integrated SNR, positioning (combining distance and angle)
quickly degrades. The rate, assuming perfect beam alignment at the BS,
degrades with distance.

###  V-A Communication

Assuming single-stream spectrum efficiencies of $2\text{--}6\,\mathrm{bps/Hz}$
and $1\text{--}4$ parallel spatial streams (depending on the array and spatial
richness of the channel), the _aggregate bandwidth_ requirement for supporting
C1 is $4\text{--}10\,\mathrm{GHz}$, whereas it is
$0.4\text{--}5\,\mathrm{GHz}$ for C2. The choice of _waveform type_ and
_modulation_ is largely dependent on the choice of hardware architecture, but
in general waveforms with low envelope variations and inherent robustness to
RF impairments are preferred. Waveforms based on OFDM have benefits in terms
of backward compatibility and adaptability. Finally, if multi-stream
transmission is to be employed, beamspace _signal shaping_ is required for
both C1 and C2; furthermore, flexible frequency-domain multi-user allocation
may be of importance for C2, whereas it can be deemed of lesser importance for
C1.

###  V-B Localization

The _aggregate bandwidth_ plays a role in terms of delay/distance resolution
and accuracy. For L1, a cluttered environment and high accuracy demand a
bandwidth on the order of $2\text{--}4\,\mathrm{GHz}$ if the resolution is to
be achieved in the delay domain, while it can go down to $500\,\mathrm{MHz}$
when resolution is achieved in the angular domain (see later in Section VI-B).
For L2, a bandwidth of $0.4\text{--}1\,\mathrm{GHz}$ is sufficient. To meet
the L2 update rate requirement, the E2E latency should be below
$1\,\mathrm{ms}$, which can be met with the considered bandwidth, though
$1\,\mathrm{GHz}$ is preferred.222Considering OFDM-like waveforms with 4096
subcarriers and 7% cyclic prefix overhead, OFDM symbol duration are
approximately $1\text{--}10\,\mathrm{us}$ for bandwidths in
$0.4\text{--}4\,\mathrm{GHz}$. Considering slots of 14 symbols, physical layer
latency with bi-directional transmission is on the order of
$28\text{--}280\,\mathrm{us}$. Under L3, a bandwidth of less than
$400\,\mathrm{MHz}$ is sufficient. The _waveform type_ is largely irrelevant
for localization, provided it can be flexible in duration and has a suitable
ambiguity function with narrow main lobe and suppressed sidelobes. In terms of
_modulation type_ , since localization is based on pilots, low-order constant-
modulus signals are preferred for L1–L3. Communication should be coherent for
L1 and L2 to achieve accuracy beyond the resolution limit, while L3 can rely
on more simple non-coherent measurements. To optimize tracking performance,
_signal shaping_ should be employed in L1 and L2, both for delay estimation
(shaping in frequency, leading to a preference for OFDM-like signals) [28] and
angle estimation (shaping in beamspace) [29].

###  V-C Sensing

To support cm-level ranging accuracy and resolution in S1 and S2, the
_aggregate bandwidth_ should be on the order of $2\text{--}4\,\mathrm{GHz}$.
Similar to localization, the _waveform type_ affects the sensing performance
only through the main-lobe and side-lobe characteristics of the corresponding
ambiguity function. Multi-carrier waveforms, such as OFDM and OTFS, have the
advantages of lower side-lobe levels [30] and greater flexibility in signal
shaping [31] and removal of the transmit data, over single-carrier ones, at
the cost of being less hardware-friendly. With regard to the _modulation type_
, sensing requires constant-modulus signals to have favorable side-lobe
behavior. S1 relies on both random data and pilots (constant-modulus), while
S2 uses only pilots. Hence, S1 can be more sensitive to the modulation type
than S2. To achieve high range and angular accuracy in S1 and S2, both use
cases should employ coherent measurements. Finally, for the _signal shaping_ ,
pilot allocation could be crucial for S2, while for S1, there are trade-offs
between communication rate and sensing performance [28]

##  VI Requirements on Hardware Architectures

The larger bandwidths identified in Section V are available at the upper
mmWave and terahertz frequency range. High propagation loss per patch type
antenna element at these frequencies drastically limits the signal
transmission distance and communication coverage range. In this case, Txs with
high output power using antenna arrays is of particular importance, because
the degradation of transmitted signal power due to high-frequency hardware
technologies highly limits performance. The key hardware impairments include
nonlinear distortion (due to power amplifiers, leading also to intermodulation
products outside the channel bandwidth), phase noise (induced by RF
oscillators at the Tx and Rx chains), quantization noise and mutual coupling
between antenna ports.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$5$$10$$15$100 m distance, 10 Gbps10
m distance, 100 Gbpstransmit power per antenna [dBm]bandwidth
[GHz]uplinkdownlink Figure 3: Example of required bandwidth vs required Tx
power per antenna element for achieving a link distance of 10 m and 100 m. A
point-to-point communication link with free-space propagation at 140 GHz was
considered. Other parameters: 256 BS antennas, 64 UE antennas, BS noise figure
5 dB, UE noise figure is 10 dB, pathloss exponent 2. The sum of power
amplifier output back off and Rx sensitivity degradation due to implementation
imperfections is 20 dB. For details of the calculation, see [9, Section
3.4.2.2.1].

###  VI-A Communication

In Fig. 3 the tradeoff between bandwidth and output power for the use-cases C1
and C2 is illustrated. For the simulation, a point-to-point communication link
at a carrier frequency of 140 GHz was assumed. For C2, a bandwidth of around 2
GHz is sufficient, and that for both the uplink and downlink a saturated
output power per antenna of around 5 dBm is needed. Increasing the power will
not significantly reduce the bandwidth required. For realizing C1 at the same
power levels, a bandwidth of around 13 GHz is needed. Note that for the uplink
case the required power per element is larger since a smaller number of
transmit elements is used compared to the downlink. To mitigate the need for a
higher output power per element on the UE side, it is possible to use more
than one IN in the uplink, distributed multiple-input multiple-output (D-MIMO)
and perform multi-stream transmission. Achieving C1 with a bandwidth of 7 GHz
and a single stream is very challenging due to the high output power that
would be required, which is another argument for introducing multi-
stream/multi-IN/D-MIMO transmission.

###  VI-B Localization

L1 and L2 require large bandwidths, which is only available at a high _carrier
frequency_. _Channelization_ can be used, but phase coherence must be
maintained across the entire band for coherent angle and delay estimation (and
thus localization). For L3, lower bands (e.g., at $30\,\mathrm{GHz}$ or even
$6\,\mathrm{GHz}$) are preferred with lower path loss. Localization relies on
pilot signals, so that single-stream transmission is sufficient for broadcast
signals. When dedicated pilots are used for individual devices or groups of
devices, multi-stream pilots are preferred. Hence, in terms of _array types_ ,
planar analog and hybrid structures are sufficient for L1. If stringent
latency is required as in L2, digital array structure can help with multi-
streams and highly flexible pilot signal design to perform localization within
fewer snapshots. Regarding the use case L3, single-antenna UEs are
appropriate, due to lower hardware and computation cost. In addition to the
beamforming gain, the array size at the BS determines the positioning
accuracy, whereas the array size at the UE affects the orientation estimation.
The requirements on _array size_ are thus variable: for L3, a single omni-
directional antenna at the UE and an array at the INs can be sufficient. For
L1 and L2, INs array sizes depend on the SNR, position resolution, and
accuracy requirements, as well as the considered bandwidth. If a large
bandwidth is employed (L1 and L2), this relaxes requirements on the array
sizes down to $10$–$20$ elements per dimension are sufficient, with an angular
resolution of $5\text{--}10\,\mathrm{{}^{\circ}}$. With bandwidths below
$500\,\mathrm{MHz}$, resolution in the angular domain requires on the order of
$50\text{--}100$ elements per dimension in L1
($1\text{--}2\,\mathrm{{}^{\circ}}$ resolution). High orientation accuracy
demands large array sizes on the user-side, where $10\text{--}20$ elements per
dimension should be considered. To understand the impact of the _transmission
power_ , it is instructive to consider a simplified 2D uplink scenario (as in
Fig. 2), where the localization accuracy in uplink can be expressed as [32]:

$\displaystyle\text{SPEB}[\text{m}^{2}]=\frac{N_{0}Bd^{2}}{TP_{\text{tx}}\lambda^{2}}\left(\frac{c^{2}\alpha_{\text{range}}}{B^{2}N_{\text{rx}}}+\frac{d^{2}\alpha_{\text{angle}}}{N_{\text{rx}}^{3}}\right),$
(1)

where $T$ is the integration time (in number of transmissions), $B$ is the
aggregate bandwidth, $N_{0}$ the noise power spectral density, $d$ is the
distance between UE and IN, $\lambda$ the wavelength, $c$ the speed of light,
$N_{\text{rx}}$ the number of IN antennas, and $\alpha_{\text{range}}$,
$\alpha_{\text{angle}}$ are constants. Finally, $P_{\text{tx}}$ is the
transmit power. The first term in (1) captures the distance estimation error,
while the second the angle estimation error. Hence, solving (1) for
$P_{\text{tx}}$ yields a minimum transmission power for a certain target
accuracy. We see that $P_{\text{tx}}$ can be reduced by increased number of
antennas $N_{\text{rx}}$ or longer integration times $T$. The latter approach
places demands on the stability of oscillators. For this reason, L1 (operating
at short ranges with relaxed integration times) does not require higher
transmission power, while L2 and L3 do (the former due to the limited
integration time, the latter due to the long link range).

###  VI-C Sensing

Both S1 and S2 require operation at high _carrier frequency_ due to
utilization of large bandwidths for high accuracy and resolution in range.
From the perspective of frequency _channelization_ [25], S1 and S2 should
perform coherent processing across the entire bandwidth. In terms of _array
types_ , monostatic sensing (S1) can exploit (re-use) both data and pilot
symbols generated by the communication system (i.e., opportunistic sensing
[30]); hence, the Tx array structure will mainly be determined by the
communication requirements (see Sec. VI-A), while analog and hybrid arrays are
sufficient for sensing purposes at the Rx side. On the other hand, bi-static
sensing (S2) favor pilot symbols due to physically separate
transmitter/receiver (Tx/Rx) hardware, which might degrade the performance of
range estimation (availability of a smaller portion of time-frequency
resources) and angle estimation (decrease in SNR) compared to S1. Hence, S2 Rx
can be equipped with a hybrid or digital array to compensate for this
performance loss. Moreover, to support $1^{\circ}$ degrees of angular
resolution for S2, the Rx _array size_ should be at least 100 elements per
dimension (35 for S1). Contrary to the Rx side, the array size at the Tx has
no impact on the angular resolution. The array size can thus vary depending on
requirements on the range and angle accuracy, which, in turn, are functions of
SNR, bandwidth and Rx array size. In the case of large bandwidths
($2\text{--}4\,\mathrm{GHz}$) and/or large Rx arrays (100 elements per
dimension), small Tx arrays with $10$–$20$ elements per dimension can be
sufficient to support both S1 and S2 accuracy requirements, while for smaller
bandwidths (below $1\,\mathrm{GHz}$) and smaller Rx arrays ($10$–$20$ elements
per dimension), S1 and S2 requires $40$–$50$ elements per dimension at the Tx
array to boost SNR towards the desired object locations. An important
distinction between localization and sensing pertains to the _transmission
power_. Since sensing (especially, S1) needs to combat higher path loss
($d^{4}$, where $d$ is the distance) than localization ($d^{2}$), the transmit
power should be larger for sensing. However, such high power requirements for
sensing may bring additional hardware complexities. For monostatic sensing
(S1), the Tx and receiver are co-located and work in full-duplex mode, which
necessitates perfect decoupling of Tx/Rx antenna arrays to prevent self-
interference [33, 24, 34]. On the other hand, for bi-static sensing (S2), the
Tx/Rx arrays are far from each other, meaning that perfect isolation is not an
issue.

##  VII Requirements on Deployments

Seen from a joint communication, localization, and sensing performance
perspective, the network planning problem is complex. The main reason is that
multiple metrics need to be optimized simultaneously: coverage, capacity,
positioning error, sensing error.

###  VII-A Communication

Traditionally, the deployment of communication networks targeted wide coverage
and high capacity [35] subject to cost constraints. Except for the compulsory
emergency location services provided by the cellular network operators under
regulatory mandates, localization and sensing services were disregarded in the
network planning phase. By contrast, the future 6G networks shall provide
integrated communication and sensing services to support emerging use cases
such as industrial applications, augmented reality, etc. Moreover, fulfilling
challenging joint coverage, reliability and throughput requirements can be
made possible by distributed deployments. The promising D-MIMO communication
technology inherently implies (sub-)nanosecond synchronization among densely
deployed network nodes, as well as multiple antennas both at Tx and Rx, which
benefit from reduced path loss and macro-diversity mitigating
shadowing/blocking.

###  VII-B Localization

Similar to communications, the _placement_ of localization infrastructure also
targets wide coverage, but its primary goal is to minimize the localization
error rather than rate. Also different from communication, localization of a
device requires a plurality of INs and the corresponding performance depends
not only on the SNR, but also on the relative geometry of the INs, through the
so-called geometric dilution of precision (GDOP): with delay-only measurements
each device needs LoS to at least 3 BSs in LoS under round-trip-time (RTT) and
4 under TDoA for UEs in the convex hull of these BSs. With angle-only
measurements, at least 2 INs (2 BSs or one BS and one RIS, which is a minimal
configuration). In addition to adopting multiple INs, geometry diversity can
be achieved by exploiting the multipath components [36]. In this case, the
surrounding environment can serve as passive INs (e.g., virtual anchors) for a
better localization performance. While BS placement will be limited to
existing sites or other constraints, RISs can be placed in an optimized manner
to meet GDOP requirements. Based on these considerations, L1 and L2 would need
at least 2 INs visible at any time, while L3 should rely on at least 3–4 INs,
as long-range links rely on time-based measurements (cf. Fig. 2). The
_synchronization_ requirements are a function of the employed time-based
measurements.333AoA measurements only need rough synchronization to know which
BS precoder or RIS configuration is used at which time. TDoA requires fine
timing synchronization among the INs, on the order of
$1\text{--}10\,\mathrm{ns}$ for L3, around $0.5\,\mathrm{ns}$ for L2, and
around $100\,\mathrm{ps}$ for L1. For coherent combining across different INs
(e.g., using D-MIMO), synchronization requirements are even more demanding, as
they relate to the signal phase. To relieve these requirements, RTT
measurements can be used, which only impose local clock stability
requirements, but scale worse than time-difference-of-arrival (TDoA) under
high user density. Estimating the synchronization error (clock offset) during
the localization phase is another option to mitigate the effect of clock
drift. Finally, accurate localization requires accurate _knowledge of the
locations_ of the INs: both the 3D location, the 3D orientation (for AoA), and
the phase center must be determined. L3 can cope with errors on the order of
tens of centimeters. For L2 and L1, extreme demands are placed on calibration,
which will likely require novel procedures (either offline or online). Since
small orientation errors lead to large location errors (see Fig. 2), extremely
precise orientation calibration is needed for INs, when UEs are farther away
from the INs.

###  VII-C Sensing

In monostatic sensing (S1), the _placement_ should be such that it provides
sufficient coverage in the deployment region. Multiple INs need to coordinate
transmissions to avoid interference. For bi- and multistatic sensing (S2), the
placement problem is similar to localization (see Section VII-B with related
GDOP notions), though with larger path loss. The combination of angle and TDoA
(with respect to the LoS path) measurements provides a 3D picture of the
environment. Statistical error bounds may be used to optimize the deployment
jointly for localization and sensing in both S1 and S2. In terms of
_synchronization_ , S1 does not impose any requirement, while under S2, tight
inter-IN synchronization is needed when no LoS link is present between the Tx
and Rxs. Finally, in terms of _IN knowledge_, S1 imposes no requirements, as
sensing is performed in the frame of reference of the monostatic sensor. For
S2, requirements are similar to localization.

##  VIII Conclusions

6G use cases feature extreme requirements in terms of communication,
localization, and sensing KPIs. The purpose of this paper was to
systematically investigate the corresponding requirements in signals, hardware
architectures, and deployments. From the summarizing Table I, it is apparent
that to support a combination of use cases, the following combination is
preferred: aggregate bandwidths around $5\text{--}10$ GHz with channels of
around $2$ GHz at carriers above 100 GHz, flexible multi-carrier waveforms and
coherent communication, with possibility to shape signals in space, relying on
analog or hybrid arrays with tens of elements per dimension at both
transmitter and receiver side. Transmit power requirements are moderate to
high, but each user must be able to have LoS visibility to several INs, which
may include RISs. Very high demands are placed on calibration in terms of
location and orientation knowledge of the INs, as well the synchronization.

## Acknowledgment

This work was supported, in part, by the European Commission through the H2020
project Hexa-X (Grant Agreement no. 101015956) and the MSCA-IF grant 888913
(OTFS-RADCOM).

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  *[KPIs]: key performance indicator
  *[E2E]: end-to-end
  *[mmWave]: millimeter-wave
  *[RF]: radio frequency
  *[RISs]: reconfigurable intelligent surface
  *[IoT]: Internet of Things
  *[RedCap]: reduced capacity
  *[LoS]: line-of-sight
  *[NLoS]: non-line-of-sight
  *[IN]: infrastructure node
  *[UE]: user equipment
  *[INs]: infrastructure node
  *[TDoA]: time-difference-of-arrival
  *[AoA]: angle-of-arrival
  *[Rx]: receiver
  *[ToA]: time-of-arrival
  *[D-MIMO]: distributed multiple-input multiple-output
  *[OFDM]: orthogonal frequency-division multiplexing
  *[DFTS]: discrete Fourier transform spread
  *[OTFS]: orthogonal time-frequency-space
  *[SNR]: signal-to-noise ratio
  *[ INs]: infrastructure node
  *[ IN]: infrastructure node
  *[Txs]: transmitter
  *[Tx]: transmitter
  *[BS]: base station
  *[UEs]: user equipment
  *[Tx/Rx]: transmitter/receiver
  *[GDOP]: geometric dilution of precision
  *[BSs]: base station
  *[RTT]: round-trip-time
  *[RIS]: reconfigurable intelligent surface
  *[Rxs]: receiver


#  RUNHETC-2024-15
Dumbbell Fermions and Fermi-Pauli Duality

Tom Banks
NHETC and Department of Physics
Rutgers University, Piscataway, NJ 08854-8019
E-mail<EMAIL_ADDRESS>

###### Abstract

We use the Kantor-Susskind[1] model of fermions as ”dumbbells” connecting
points on a cubic lattice to points on its dual, to define a duality between
local fermionic models invariant under a $Z_{2}$ gauge symmetry and models of
bosonic variables (generalizations of Pauli matrices) defined on the lattice.

###### Contents

  1. 1 Introduction

##  1 Introduction

There have been many attempts to generalize bosonization beyond the $1+1$
dimensional context in which it was first discovered. In continuum field
theory, these usually involve the idea of gauging the $Z_{2}$ symmetry
$(-1)^{F}$ and this program has succeeded for a variety of models with Chern-
Simons gauge fields in $2+1$ dimensions[2]. We also know how to construct
fermions from bosons using the Dirac quantization condition[3] in $3+1$
dimensions.

Attempts to find an analog of the Jordan-Wigner transformation in lattice
models have been somewhat less successful[4]. In this note we want to report
on a simple general procedure, which seems to do the trick. Our basic starting
point is the observation[5] that Hamiltonian lattice gauge theory for the
discrete groups $Z_{N}$ is most elegantly formulated by assigning the
canonical conjugate $V_{S(l)}$ of $U(l)$ to the $d-1$ dimensional face $S(l)$
of the hypercube on the dual lattice, that is pierced by the lattice link $l$.
We have

$U(l)V_{S(l)}=V_{S(l)}U(l)e^{2\pi i/N}.$ (1.1)

We will be interested only in the case $N=2$. We take our lattice to be
toroidal, as is the dual lattice. The requisite identifications are shown in
two dimensions in Figure 1.

Figure 1: Periodic boundary conditions: the Clifford variables at the two ends
of each pair of a line and a dual line are identical.

We can also denote a link variable $U(l)$ by the pair of lattice points
$U(m,n)$ that it connects. Then the $Z_{2}$ gauge transformation is

$U(m,n)\rightarrow W(m)U(m,n)W(n)$ (1.2)

where $W(n)$ belongs to an independent $Z_{2}$ group at each point. There is
also a higher form gauge symmetry associated with the $d-1$ dimensional
hypercube variables $V(S(l))$. This is only a symmetry if we restrict
attention to the topological $Z_{2}$ gauge theory[6]. In a Euclidean
formulation, the topological model is defined by forcing all plaquettes to be
equal to $1$, leaving only Wilson lines tied to an open boundary as physical.
In Hamiltonian formulation, we forbid the usual electric field term in the
Hamiltonian and allow only functions of Wilson loops.

The boundary of $S(l)$ is a union of $d-2$ dimensional hypercubes and we can
define a $Z_{2}$ group associated with each of these. $V(S(l))$ transforms
under the higher form symmetry by being multiplied with all of the $Z_{2}$
transformations on its boundary. Products of $V(S(l))$ variables over closed
$d-1$ dimensional surfaces are invariant under these gauge transformations.
The higher form gauge transformation on a particular $d-2$ dimensional face of
the $d-1$ dimensional open hypercube $S(l)$ is implemented by the Wilson loop
operator $U(\Gamma)$ that includes the link $l$ and encircles that face (see
Figure 2 for the $3$ dimensional case).

Figure 2: The red Wilson loop operator generates the gauge transformation on
the encircled link of the indicated electric hypercubic face.

The usual electric flux term in the $Z_{2}$ lattice gauge theory is just the
sum of the $V(S(l))$ variables over all $l$ (or what is the same, over all
”beginning” lattice sites $m$ in the $(m,n)$ parametrization of links)111Note
that for higher $Z_{N}$ groups this would be replaced by a sum over
$V(S(l))+V^{\dagger}(S(l))$ . . It is not invariant under the higher form
gauge symmetry, and therefore forbidden in our construction. The only gauge
invariant operators are $Z_{2}$ Wilson loops constructed from the $U$
variables, and the Gauss law operators at each lattice site $m$, which are the
products of the $V(S(l))$ over all dual hypercubes adjacent to that site. In a
moment we will introduce anti-commuting variables and a Gauss law constraint,
which enables us to write all of these gauge theory operators as monomials in
fermions.

Now let us introduce dumbbell fields, $\gamma(m,M)$ connecting a point $m$ on
the lattice to each of the points on the dual hypercube surrounding it. These
are independent Clifford variables

$[\gamma(m,M),\gamma(n,N)]_{+}=2\delta_{mn}\delta_{MN}.$ (1.3)

We impose periodic boundary conditions on the toroidal lattice. For each point
$M$ on the dual lattice and each lattice direction $\hat{e}$, we have gauge
invariant bilinears

$S(m,m+\hat{e})=\gamma(m,M)\gamma(m+\hat{e},M)U(m,m+\hat{e}).$ (1.4)

Since these commute with all Wilson loops, they are also invariant under the
higher form gauge symmetry. We also impose the gauge constraint on the Hilbert
space that the product of all the fermionic generators in a dual hypercube is
equal to the product of the $V_{S(l)}$ operators on the faces of that
hypercube. Note that we have to choose an order for the $\gamma$ operators in
imposing this constraint.

In order to do this, we first choose an orientation on the dual lattice. This
labels the points at the vertices of a hypercube as $1\ldots d$ . We first
multiply together $\gamma(m,1)\ldots\gamma(m,d)$. Now we are at the vertex
diagonally opposite $1$ on the hypercube. As shown in Figure 3 for $d=2,3$,
there is now a unique path through the other vertices, passing through each of
them once.

Figure 3: The path orderings that define the Gauss law constraint in 2 and 3
dimensions.

The resulting product of $2^{d}$ Clifford elements is called $\Gamma$.
$(1\pm\Gamma)/2$ defines chiral projectors on the $SO(2^{d})$ spinor
representation, decomposing it into representations of $SU(2^{d-1})\times
U(1)/Z_{2^{d-1}}$. The gauge generator is $(-1)^{N_{F}}$, with $N_{F}$ the
$U(1)$ generator, so if we introduce complex linear combinations of
$\gamma(j,J)$ variables to create and annihilate fermionic particles, the
topological gauge group is gauging $(-1)^{N_{F}}$.

The huge symmetry group generated by the Clifford elements at a point has an
$SO(d)$ subgroup corresponding to rotations around the point $m$ on the
lattice surrounded by the dual hypercube to which the operators $\gamma(m,M)$
belong. This is not of course a symmetry of the lattice itself, and only a
cubic subgroup of it is realized, but we can think of the operators as
transforming under it. They transform as a direct sum of irreducible
representations, corresponding to $p$ forms of all ranks between $0$ and $d$.
We have one Clifford variable for each independent component of each $p$ form.
The product of all the Clifford elements is invariant under $SO(d)$ rotations,
but can be either even or odd under reflection through the lattice point $m$.
We choose the ordering that is reflection invariant.

To see these transformation properties, start with $d=2$. The sum over all
points is the $0$ form and the sum over all points with negative signs for
those shifted by one unit in (say) the $\hat{1}$ direction is the two form.
The two differences between points in the $\hat{1}$ and $\hat{2}$ directions
are the two components of the $1$ form. Now take $d=3$. We can define the $0$
and $3$ forms in a manner completely analogous to the two dimensional case, in
terms of their transformation under rotations and reflections. The one form
components are again differences of points in the three lattice directions.
The two form components are then defined for each two plane by the rules for
$d=2$. It’s now clear how to proceed to general $d$ by induction.

Note that under the $SO(d)$ subgroup of $SU(2^{d-1})$, which we might wish to
interpret as performing internal spin rotations on particles localized at
points on the lattice, all fields transform in tensor representations. The
model does not have any spinor excitations. This is related to the fact that
in the continuum limit, the dual lattice rotation symmetry is really a mixture
of a continuum rotation and an emanant internal symmetry[7][8]. Our dumbbell
fermions are related to Susskind’s staggered fermions and, in the continuum to
the Kahler-Dirac equation.

If we multiply the gauge invariant link bilinears around a plaquette, then the
Clifford variables all disappear, and we just get the plaquette variables of
the $Z_{2}$ gauge theory. Thus, the gauge invariant subspace of pure gauge
theory is entirely spanned by $S$ operators acting on the function $1$ in the
Hilbert space of $L^{2}$ functions of the Wilson loops of the pure gauge
theory.

We can write other gauge invariant operators in the fermion Hilbert space by
constructing bilinears at a single lattice site. These are the $SO(2^{d})$
generators referred to previously. The Hilbert space at a point is $2^{2^{d}}$
dimensional and can be viewed as constructed from $2^{d}$ independent q-bits
in the usual tensor product construction of Clifford algebras from Pauli
matrices. Link local bilinears in the $S(i,i+\hat{e})$ operators can be
rewritten in terms of these Pauli variables.

The $S(i,i+\hat{e})$ variables do not have a closed algebra. Rather, they
close on an algebra of $Z_{2}$ strings. These include Wilson loops of the
gauge theory and open strings with fermion fields on their ends. As noted,
fermion fields have a $2^{d}$ valued index on them, labeling the dual point to
which the dumbbell is attached. It transforms as a vector under the dual
$SO(2^{d})$ group generated by the operators at a single lattice point. These
fields do not create particle like excitations because they disappear, leaving
behind a closed string if they act twice at the same point. Complex linear
combinations behave like $d$ pairs of fermion creation and annihilation
operators and can support a $U(2^{d-1})$ symmetry including a particle number
$N_{F}$. $(-1)^{N_{F}}$ is the $Z_{2}$electric flux through the dual
hypercube.

In fact, all operators in the model can be generated by starting with the link
variable $\gamma(m,M)U(m,m+\hat{e})\gamma(m,M+\hat{e})$ on a single link, and
multiplying it by local operators $\gamma(n,M)\gamma(n,K)$ from all possible
points. The entire physical operator algebra of the model differs by a single
operator from that of an $SO(2^{d})$ spin model with a local Hilbert space in
the spinor representation.

So far we have not made a commitment to any particular form of the
Hamiltonian. The system we’ve defined has an additional $Z_{2}$ $d-1$ form
gauge symmetry, whose gauge potentials are the $V_{S(l)}$ variables defined on
faces of dual $d-1$ cubes. Gauge invariant functions of these variables are
products of $V$’s over closed $d-1$ surfaces. These all detect open string
operators, but only topologically, if the two ends of the string are on
opposite sides of the surface. Thus, if we impose $d-1$ form gauge invariance
on the Hamiltonian the fermionic strings are purely topological in nature. As
noted above, the usual electric field strength term in $Z_{2}$ lattice gauge
theory would violate the $d-1$ form gauge invariance.

It is important to add a term that depends on the $Z_{2}$ Wilson loops to the
Hamiltonian. Of course, since Clifford variables square to one, a Wilson loop
is equivalent to a fermion bilinear $\gamma(m,M)U_{W}(m,m)\gamma(m,M)$. A
general quadratic fermion Hamiltonian involving terms that are maximally local
on the lattice takes the form.

$\sum_{M,m,\hat{e}}\gamma(M,m)[\gamma(M,m+\hat{e})U(m,m+\hat{e})+c\gamma(M+\hat{e},m)]+h\sum_{plaquettes\
P}U(P).$ (1.5)

We’ve also imposed invariance under cubic rotations and reflections in both
the lattice and the dual lattice. The ground state energy of the fermions has
an expansion in Wilson loops of the gauge theory, which are all products of
single plaquette loops. It can be solved exactly when the plaquettes are
uniform, so a large $h$ expansion leads to a systematic perturbation theory in
terms of free fermions.

An extremely important identity is

$\gamma(m,M)\gamma(m,K)\gamma(m+\hat{e},M)\gamma(m+\hat{e},K)=\gamma(m,M)\gamma(m+\hat{e},K)\gamma(m,K)\gamma(m+\hat{e},M).$
(1.6)

This means that a local interaction between purely bosonic spins can be viewed
as separating two pairs of fermion fields. As noted above, by iterating the
spin interaction we can create arbitrarily long topological fermionic strings.
To put this in other terms, we could have formulated models without the
explicit fermion kinetic term linear in the $S$ operators, as models of $d$
Pauli spins interacting locally on the lattice $m$ with no mention of fermions
or fluxes, but their underlying algebra has a hidden fermionic structure with
a topological $Z_{2}$ gauge invariance.

It is clear that the dumbbell fermions we have constructed are, from the point
of view of the dual lattice, a real version of the staggered fermions of
Susskind[7] which are in turn related to the Kahler-Dirac
equation[kahler]222I’ve known about the connection between Susskind fermions
and Kahler fermions since[9], as well as the connection to the topological
twist of SUSY gauge theories[12] that makes them into topological field
theories. S. Catterall[10] has written extensively on this subject and has
recently made the fascinating point that a gravitational anomaly of Kahler
fermions can be seen on the lattice[11]. There’s also likely to be a
connection to the work of[13].. This implies that when we take the continuum
limit of appropriate Hamiltonians, the geometrical lattice symmetries are
revealed to be discrete combinations of continuum symmetries under which the
fermion fields transform as spinors, and a continuous internal symmetry acting
on a multiplet of spinor fields. In most applications, Susskind fermions are
endowed with extra indices and the resulting symmetries coupled to gauge
fields. One may anticipate an interesting interplay between the bosonization
proposed here and those applications.

Acknowledgments

The work of T.B. was supported by the Department of Energy under grant DE-
SC0010008. Rutgers Project 833012. Conversations with N. Seiberg about $Z_{2}$
gauge theory and Susskind fermions are gratefully acknowledged.

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