# Superfluid response of an atomically thin, gate-tuned van der Waals
superconductor

Alexander Jarjour1, G.M. Ferguson1, Brian T. Schaefer1, Menyoung Lee2,3,
Yen Lee Loh4, Nandini Trivedi5, Katja C. Nowack1,2,0

A growing number of two-dimensional superconductors are being discovered in
the family of layered van der Waals (vdW) materials. Due to small sample
volume, their characterization has been largely limited to electrical
transport measurements. As a consequence, characterization of the diamagnetic
response of the superfluid to an applied magnetic field, a defining property
of any superconductor, has been lacking. Here, we use a local magnetic probe
to directly measure the superfluid response of the tunable, gate-induced
superconducting state in MoS2. We find that the backgate changes the
superconducting transition temperature non-monotonically whereas the
superfluid stiffness at low temperature and the normal state conductivity
monotonically increase with backgate voltage. In some devices, we find direct
signatures in agreement with a Berezinskii-Kosterlitz-Thouless transition,
whereas in others we find a broadened, shallow onset of the superfluid
response. We show that the observed behavior is consistent with disorder
playing an important role in determining the superconducting properties in
superconducting MoS2. Our work demonstrates that magnetic property
measurements are within reach for vdW superconductors and reveals that the
superfluid response significantly deviates from simple BCS-like behavior.

11footnotetext: Laboratory of Atomic and Solid State Physics, Cornell
University, Ithaca, NY22footnotetext: Kavli Institute at Cornell for Nanoscale
Science, Ithaca, NY33footnotetext: Department of Electrical and Computer
Engineering, Cornell University, Ithaca, NY44footnotetext: Department of
Physics and Astrophysics, University of North Dakota, Grand Forks,
ND55footnotetext: Department of Physics, The Ohio State University, Columbus,
OH00footnotetext: Corresponding author, email<EMAIL_ADDRESS>
The two defining properties of a superconductor are a vanishing electrical
resistance and the expulsion of magnetic fields below a characteristic
critical temperature, $T_{c}$. Typically, superconductivity is first
identified in a material by observing a sharp drop in the resistance at
$T_{c}$. However, resistance measurements only give limited information about
the superconducting state forming below $T_{c}$, and other experimental probes
are needed to reveal its nature. Measurements of the strength with which the
superconductor screens a magnetic field directly probe the superfluid
stiffness, $\rho_{s}$, and have provided insight into the nature of
unconventional superconductors [1, 2, 3, 4]. From $\rho_{s}$, the superfluid
density can be extracted, which in a clean BCS superconductor at $T=0$ is
expected to be equal to the normal carrier density [5]. Comparing the
superfluid density to the normal carrier density, pair breaking by impurity
scattering and other mechanisms can be identified [6]. In two-dimensional
superconductors, the onset of $\rho_{s}$ may show fingerprints of the
Berezinskii-Kosterlitz-Thouless (BKT) transition [7, 8].

A growing family of atomically thin superconductors is realized by
mechanically exfoliated sheets of vdW materials. These include two-dimensional
(2D) superconductors based on bulk superconducting materials such as NbSe2 [9,
10], NbS2 [11], and TaS2 [12], as well as 2D superconductors that are induced
by electrostatic gating such as MoS2 [13], WS2 [14, 15], MoTe2 [16], WTe2 [17,
18], twisted bilayer graphene [19], and ABC stacked trilayer graphene [20]. A
variety of superconducting phenomena have been observed in atomically thin vdW
superconductors, such as robustness against large in-plane magnetic fields
[21, 22, 23], superconductivity in the vicinity of correlated electronic
states [19, 20], a dramatically enhanced $T_{c}$ in the monolayer limit [16,
12], and unusual symmetry breaking in the superconducting state [24]. A
detailed study of the transport properties of NbSe2 with varying thickness has
shown that dissipationless transport is highly fragile to temperature, applied
magnetic field and the employed bias current [25] further highlighting the
need for directly probing the phase coherence of the superconducting state in
vdW materials. However, due to the typically small sample size, only a few
measurements beyond electronic transport which directly probe the
superconducting state below $T_{c}$ are available [26, 27], and no
characterization of the magnetic response has been reported for any atomically
thin vdW superconductor.

Fig. 1:  Magnetic measurement of ionic gated MoS2. a, A flake of MoS2 (shown
in purple) on a SiO2/Si substrate is patterned into a disk and covered by a
spin coated ionic gel. The SQUID pickup loop (shown in silver), with
concentric field coil (shown in dark gray) is approached to the sample. A
current in the field coil produces a magnetic field, which results in an
opposing screening current in the superconductor. The strength of the
screening current is magnetically detected by the pickup loop. b, Optical
image of a 20 $\mu$m diameter circular MoS2 device (blue) with electrical
contacts. c, Image of the magnetic response of the device shown in b at 4 K.
The white dashed circle has a diameter of 20 $\mu$m and indicates the device
circumference.

Here, we report direct measurements of the magnetic response of the gate-
induced superconducting state in few-layer MoS2. Although MoS2 is a
semiconductor when undoped, ionic liquid gating can induce an electron
accumulation layer at the surface of a MoS2 flake which exhibits
superconductivity at carrier densities exceeding $0.5\times 10^{14}$ cm-2
[13]. $T_{c}$ changes non-monotonically with the carrier density with a
maximum of approximately 10 K. Superconductivity is retained in the monolayer
limit [28], but is always in the 2D limit regardless of the flake thickness
because the accumulation layer is approximately confined to the topmost layer
[29, 21]. Spin-valley locking in the electronic bandstructure of MoS2 gives
rise to an Ising protection of the superconducting state leading to an in-
plane critical field dramatically exceeding the Pauli limit [21, 29].
Recently, tunneling measurements have suggested that the order parameter is
not fully gapped [26], a possible signature of an unconventional
superconducting state.

We fabricate our devices from exfoliated MoS2 flakes with a thickness of 3-10
layers, and pattern them into disks with Ti/Au contacts to allow gating and
electrical transport measurements (Fig. 1b). To induce high carrier densities,
we use a spin coated, approximately 2-$\mu$m-thick ionic gel. In addition, we
apply a backgate voltage, $V_{\text{BG}}$, across 300 nm thick SiO2 to our
devices. Figure 1a schematically shows how we measure the magnetic response of
a device. A scanning superconducting quantum interference device (SQUID) [30,
31] with a pickup loop and a concentric field coil is centered above the
device. Here the pickup loop and field coil have an inner diameter of 1.5
$\mu$m and 8 $\mu$m, respectively. An AC current in the field coil produces a
small magnetic field, and we measure the resulting flux in the pickup loop
using a lock-in amplifier. Away from the sample, this signal corresponds to
the mutual inductance between the pickup loop and field coil. When the pickup
loop/field coil pair is brought close to the device, the magnetic response of
the device appears as a change in the mutual inductance. By measuring this
change, we directly probe the device’s magnetic response $\chi$.

Specifically, a superconductor generates currents to screen the applied
magnetic field. The strength of the screening currents can be related to the
superfluid stiffness $\rho_{s}$, the Pearl length $\Lambda$, and the
superfluid density $n_{s}$ which are connected through
$\rho_{s}=\hbar^{2}/(2\mu_{0}k_{B}e^{2}\Lambda)$ and
$n_{s}^{2D}/m^{*}=4k_{B}\rho_{s}/\hbar^{2}$, where $\hbar$ is the reduced
Planck’s constant, $\mu_{0}$ is the permeability of free space, $k_{B}$ is the
Boltzmann constant, $e$ is the elementary charge, and $m^{*}$ is the effective
mass. In the limit of weak screening, i.e., $\Lambda\gtrsim R$, where R is the
device size, the magnetic response $\chi$ is directly proportional to
$\rho_{s}$: $\chi=M_{geo}\rho_{s}$, which is why we refer to $\chi$ in the
following as the superfluid response. The proportionality factor $M_{geo}$
depends on the SQUID dimensions, its height above the sample, and the device
dimensions. We can model $M_{geo}$ to extract absolute values of $\rho_{s}$
and $\Lambda$ from our measurements (Supplementary Information). The estimate
of $M_{geo}$ has systematic uncertainties due to uncertainties in the SQUID
height, the exact device dimensions, and other geometrical factors; however,
relative changes of $\rho_{s}$ as a function of $V_{\text{BG}}$ and
temperature are captured with high accuracy. Figure 1c shows an image of
$\chi$ at a constant height. From this, we identify the center of the device
where we position the SQUID. We then measure $\chi$ at a fixed height as a
function of temperature and $V_{\text{BG}}$ and simultaneously record the
sheet resistance $R_{\square}$.

Fig. 2: Temperature dependence of the gate-tuned resistance and superfluid
response. a, Sheet resistance, $R_{\square}$, and b superfluid response,
$\chi$, of device A versus temperature. Different colors correspond to
different backgate voltage $V_{\text{BG}}$ as indicated by the color bar in b.
The black line corresponds to the universal BKT condition $\rho_{s}=2T/\pi$.
The gray shaded area indicates uncertainty in the universal condition arising
from uncertainty in the SQUID height. c,d, Same as in a,b but for device B.

Figure 2 shows $R_{\square}$ and $\chi$ as a function of temperature and
$V_{\text{BG}}$ for two devices, which we label A (20 $\mu$m diameter) and B
(15 $\mu$m diameter). We include data from an additional device in the
Supplementary Information (Fig. S3). A weakly temperature-dependent magnetic
response from the ionic liquid has been subtracted from $\chi$ (Supplementary
Information). For both devices, $R_{\square}$ drops sharply as we lower the
temperature, and the superfluid response appears when the drop in resistance
is completed. At $V_{\text{BG}}=0$, device A has a lower $T_{c}$ of
approximately 6 K, compared to 9 K in device B. For both devices, the critical
temperature changes with $V_{\text{BG}}$. In device A, the resistive
transition remains approximately 1.5 K wide across the backgate range, whereas
in device B the transition broadens with decreasing $V_{\text{BG}}$ and
changes shape. Likewise, we observe that the shape of the superfluid response
versus temperature for device A is qualitatively independent of
$V_{\text{BG}}$, but does vary for device B. However, in both devices, the
response measured at the lowest temperature increases monotonically with
$V_{\text{BG}}$. Finally, the resistance below the transition is finite for
both devices at low values of $V_{\text{BG}}$. Combined with the simultaneous
superfluid response of the sample, this raises the question if this behavior
is intrinsic to the superconducting state. A possible extrinsic explanation is
a non-superconducting region along the periphery of the device, which would
have a particularly pronounced effect in disk-shaped devices (Supplementary
Information). While determining the origin of the finite resistance is
interesting, it is outside the scope of this work. In the following, we first
focus on the superfluid response substantially below $T_{c}$. We then discuss
the onset of diamagnetism close to the superconducting transition.

Fig. 3: Correlation between device resistivity and superfluid stiffness. a-c,
Summary of properties of the superconducting state in device A versus
$V_{\text{BG}}$. a, Two measures of the critical temperature: $T_{c}^{R}$ at
which the resistance has decreased by 10% from the normal value and
$T_{c}^{\chi}$ at which the superfluid response starts to exceed our noise
floor. b $\chi$ extracted at $T=0.9T_{c}^{\chi}$. c $R_{\square}$ at 8 K. d-f,
Same as a-c but for device B. For e, $\chi$ is shown at $T=0.55T_{c}^{\chi}$,
for f $R_{\square}$ is shown at 12 K. g, Inverse of the superfluid stiffness
at zero temperature versus $R_{\square}$. See the main text for the method of
estimation of $\rho_{s}(T=0)$. The colors of the circles match the colors in
a-c. The uncertainty in the inverse superfluid stiffness from uncertainty in
the height of the SQUID is indicated by the gray band. Fit to Eq. (1) shown in
black. h, Same as g but for device B.

In Fig. 3a,d, we summarize the backgate dependence of different temperature
scales characterizing the superconducting transition. We plot $T_{c}^{R}$ at
which $R_{\square}$ has decreased by 10% from the normal state value and
$T_{c}^{\chi}$ at which the superfluid response just rises above our noise
floor. In Fig. S8 we show these temperatures $T_{c}$ overlaid onto the data.
To compare changes in the overall strength of the diamagnetic response versus
$V_{\text{BG}}$, we plot values of $\chi$ at a fixed fraction of
$T_{c}^{\chi}$ for both devices. Due to the comparably low values of
$T_{c}^{\chi}$, we plot $\chi$(.9 $T_{c}^{\chi}$) for device A in Fig. 3b and
omit values corresponding to the two lowest values of $V_{\text{BG}}$. In
device B, $T_{c}^{\chi}$ is substantially higher across the backgate voltage
range. Therefore, we plot $\chi$(.55 $T_{c}^{\chi}$) in Fig. 3e. Lastly, we
plot the normal state $R_{\square}$ in Fig. 3c,f for devices A and B,
respectively.

$T_{c}^{R}$ and $T_{c}^{\chi}$ both have a non-monotonic dependence on
$V_{\text{BG}}$ for device A and B, which is also directly visible in the data
in Fig. 2. In device B, the superconducting transition broadens as
$V_{\text{BG}}$ decreases, which is reflected in the growing difference
between the two temperatures. In contrast to the non-monotonic dependencies of
$T_{c}^{R}$ and $T_{c}^{\chi}$, the superfluid response increases
monotonically with increasing $V_{\text{BG}}$ doping, whereas the normal-state
resistance decreases.

In principle, a monotonic increase in the superfluid density and therefore
$\chi$ can be expected as the normal carrier density, $n_{n}$, increases with
$V_{\text{BG}}$. However, the observed change in $\chi$ over the full backgate
range is larger than can be explained by a change of $n_{n}$ alone, especially
in device B. We estimate the carrier density induced by the backgate as
$7.0\times 10^{10}$ cm-2 per volt given the 300 nm thickness of the SiO2 with
an approximate dielectric constant of 3.8. Our setup is restricted to low
magnetic fields and therefore does not allow us to perform accurate
measurements of the Hall effect. To estimate the carrier density induced by
the ionic gate, we compare to Refs. [13] and [26], which establish a
relationship between $T_{c}$ and $n_{n}$ that is consistent across many
devices. Based on $T^{R}_{c}$ at $V_{\text{BG}}=0$, device B has a normal
carrier density of 1-2 $10^{14}$ cm-2 [26]. Device A has a lower critical
temperature, which could be due to under- or over-doping. At
$V_{\text{BG}}<0$, $T_{c}^{R}$ increases by approximately 0.4 K for a change
of 1012 cm-2 in the carrier density, which is in agreement with the increase
in $T_{c}$ observed in Refs. [32, 13, 26] on the underdoped side of the dome.
We, therefore, assume in the following that device A is underdoped with a
carrier density of 0.6–0.8 x $10^{14}$ cm-2. Compared to these carrier
densities, $n_{n}$ decreases by at most 17% and 11%, in device A and device B
respectively, across the accessible backgate range. However, we observe larger
decreases of 25% and 76%, respectively, in the magnitude of $\chi$. Similarly,
the normal-state resistance of both devices changes more significantly than
can be explained by the carrier density. This indicates that the electron
mobility is varying with $V_{\text{BG}}$. From $R_{\square}$, we estimate the
mobility at $V_{\text{BG}}=0$ to be 300 cm2/V s in device A and 40 cm2/V s in
device B assuming $0.7\times 10^{14}$ cm-2 and $1.5\times 10^{14}$ cm-2 for
the carrier densities respectively. These mobilities are within the range
observed in the literature.

Disorder in 2D superconductors can strongly reduce the superfluid density
compared to the normal-state carrier density even at $T=0$. In the dirty
limit, in which the elastic scattering rate $1/\tau$ exceeds the
superconducting gap, $\Delta$, the fraction of carriers forming the
superconducting condensate is expected to be $n_{s}(T=0)/n_{n}\approx
2\Delta/(\hbar/\tau)$ [6], where $n_{s}(T=0)$ is the superfluid density at
zero temperature. Using $R_{\square}=m^{*}/(n_{n}e^{2}\tau)$, we can relate
the superfluid stiffness and the sheet resistance through

$\rho_{s}(T=0)\approx\frac{\Delta\hbar}{2k_{B}e^{2}R_{\square}}.$ (1)

To compare this relationship to our data, we model $M_{geo}$ to convert $\chi$
into $\rho_{s}$. $M_{geo}$ depends on our measurement geometry, such as the
height and dimensions of the SQUID and the dimensions of the sample (see
Supplementary Information for details). We then estimate $\rho_{s}(T=0)$ by
fitting a phenomenological BCS model to the temperature dependence of
$\rho_{s}$ [1] as discussed in more detail below (see also Supplementary
Information for details). We constrain the fits to below a fixed fraction of
$T_{c}^{\chi}$. Due to the limited temperature range, we cannot perform this
analysis for traces corresponding to the most negative $V_{\text{BG}}$ values
from both devices. In Fig. 3g,h we plot $1/\rho_{s}(T=0)$ versus $R_{\square}$
for device A and B, respectively. The shaded gray areas reflect several
systematic uncertainties in estimating $M_{geo}$ which do not affect the
relative changes of $\rho_{s}$ with backgate or temperature. The fit to Eq.
(1) is shown in black and we extract $\Delta=0.4\pm 0.2$ meV and
$\Delta=2.6\pm 1.2$ meV for device A and B respectively. For device B, this is
in agreement with $\Delta=1.75$ meV extracted from tunneling spectroscopy
[26]. Tunneling has not been performed in underdoped devices; however,
$\Delta$ for device A is similar to the gap of overdoped devices of comparable
$T_{c}$. Combined, our measurements suggest that the backgate modifies the
superfluid stiffness via both disorder and carrier density, rather than
density alone. In some devices, such as device B, disorder tuning is the
predominant mechanism. Further, using the free electron mass and assuming
$n_{s}^{2D}=10^{14}$ cm-2, we would expect $1/\rho_{s}(T=0)=$ .005 K-1. The
much larger inverse stiffness we observe suggests that only a fraction of the
electrons pair.

Potential sources of electrostatic disorder in our devices are the SiO2
substrate and the ionic gate. As a function of backgate voltage, the carrier
density, screening properties, and the shape of the confinement potential may
change. A striking feature in device A is a pronounced asymmetry of $T_{c}$
with respect to $V_{\text{BG}}=0$. Refs. [33, 32] study few-layer MoS2 devices
dual-gated by an ionic gate and a backgate similar to ours. They report
signatures of a low-density metallic layer forming at the bottom of the MoS2
flake only at positive backgate voltage, whereas at negative backgate voltage,
the carrier density of the top layer decreases. Such a metallic layer could
modify the electron mean free path in the top layer by screening the disorder
from the SiO2 substrate, which is likely significant given the reported
enhancement in mobility of gated MoS2 devices that are placed on hBN [34].
However, we observe a change in mobility and $\rho_{s}$ at all backgate
voltages and no clear asymmetry in those properties with respect to
$V_{\text{BG}}=0$. The exact mechanism of the disorder tuning in these dual-
gated devices remains an open question.

Next, we discuss the magnetic response close to the superconducting
transition. Device A and B show significantly different behavior. In device A
(Fig. 2b), we observe a sharp rise in the superfluid response near
$T_{c}^{\chi}$, followed by a change to a lower slope giving rise to a kink in
the curves. In device B (Fig. 2d) we observe a smooth rise in the superfluid
response throughout the accessible temperature range. A jump in the superfluid
response is expected in a 2D superconductor close to $T_{c}$ due to the BKT
transition [7, 8]. Just below the transition, the superfluid stiffness is
predicted to satisfy the universal condition $\rho_{s}/T_{BKT}=2/\pi$. In Fig.
2b,d, we include a line corresponding to $2T/\pi$ converted to $\chi$ in black
and the systematic error from the uncertainty in $M_{geo}$ in gray. For device
A, the sharp rise in $\chi$ near $T_{c}$ ends at the lower edge of the
universal condition error band. Instead of a jump, there is a kink, which is
consistent with a BKT transition broadened and modified by a combination of
finite-size and disorder effects [35]. In device B, however, we do not observe
any features that match the expected form of a BKT transition.

Fig. 4: Normalized superfluid response curves. a, Gate-tuned superfluid
response $\chi$ from device A with the temperature axis scaled by
$T_{c}^{\chi}$ and the vertical axis by $\chi$ at 0.9 $T_{c}^{\chi}$ . Dashed
red curve shows a fit to $\chi$ below 0.9 $T_{c}^{\chi}$ to the
phenomenological model in Eq. (2) assuming a BCS dependence with
$\Delta/T_{c}^{BCS}=1.76$ and a shape parameter $\alpha=1$. b, Gate-tuned
superfluid response of device B with the axes scaled similarly as in a, but
using $\chi$ at .55 $T_{c}^{\chi}$. Red dashed curve shows same fit as in a to
data below 0.55 $T_{c}^{\chi}$. The green and blue dashed curves show fits of
Eq. (2) to the full temperature range at $V_{\text{BG}}=-70~{}\mathrm{V}$ and
$V_{\text{BG}}=70~{}\mathrm{V}$, respectively, with no constraints imposed on
$\Delta/T_{c}$ and $\alpha$. The inset shows the evolution of the shape
parameter, $\alpha$, obtained from fits at all values of $V_{\text{BG}}$.
Error bars are smaller than the markers.

To highlight how the temperature dependence of $\chi$ evolves as a function of
$V_{\text{BG}}$, we normalize the superfluid response versus temperature
curves in Fig. 4. The vertical axis is scaled by the superfluid response at
0.9 and 0.55 $T_{c}^{\chi}$ for device A and B, respectively. The horizontal
axis is scaled by $T_{c}^{\chi}$. For device A, the curves collapse.
Conversely, in device B, the curves differ in the range of 0.7 to 1.0
$T_{c}^{\chi}$.

We compare the normalized curves to a phenomenological model for s-wave
superconductors [1]:

$\chi=\chi_{0}\left(1-\frac{1}{2T}\int_{0}^{\infty}\mathrm{cosh}^{-2}\left(\frac{\sqrt{\epsilon^{2}+\Delta^{2}}}{2T}\right)\mathrm{d}\epsilon\right).$
(2)

Here $\chi_{0}$ is the zero temperature response, and $\Delta$ is temperature
dependent gap given by:

$\displaystyle\Delta(T)=\Delta_{0}~{}\mathrm{tanh}\left(\frac{\pi
T_{c}^{\text{BCS}}}{\Delta_{0}}\sqrt{\alpha\left(\frac{T_{c}^{\text{BCS}}}{T}-1\right)}\right),$
(3)

where $\alpha$ is the shape parameter governing the opening of the gap,
$T_{c}^{\text{BCS}}$ is the critical temperature, and $\Delta_{0}$ is the size
of the gap at zero temperature. We first constrain
$\Delta_{0}/T_{c}^{\text{BCS}}=1.76$ and $\alpha=1$ as expected for a BCS
superconductor, leaving only two free parameters, $T_{c}^{\text{BCS}}$ and
$\chi_{0}$. We fit to data below .9 $T_{c}^{\chi}$ for device A and .55
$T_{c}^{\chi}$ for device B, because the superfluid response is in clear
disagreement with the phenomenological model above the cutoff temperature for
device A, and in device B, the shape of the onset is changing with
$V_{\text{BG}}$. The resulting fits for the highest backgate voltages are
shown as red dashed lines in Fig. 4a,b. In device A, $T_{c}^{\text{BCS}}$ is
slightly above the onset of susceptibility consistent with a small temperature
range above $T_{c}^{\chi}$ in which the superfluid response is suppressed due
to a BKT transition, which is not captured in the phenomenological model. The
fitted value of $T_{c}^{\text{BCS}}$ is sensitive to details of the fitting
such as an initial guess and the exact temperature range. However, the
extracted low-temperature value $\chi_{0}$ is comparably robust and was used
to estimate $\rho_{s}(T=0)$ for Fig. 2g,h (see Supplementary Information for
more detail). The onset of diamagnetism in both devices is not captured by the
simple model, which could be due to a number of reasons. In particular,
disorder can modify the onset of diamagnetism even in 3D superconductors.
Within the phenomenological model in Eq. 2, disorder causes an increase in the
shape parameter [36, 1]. To explore this further, we fit our data across the
entire temperature range without constraints on $\alpha$ and
$\Delta_{0}/T_{c}$. For device A, we cannot obtain a good fit of the model
near $T_{c}^{\chi}$, indicating that the kink in the curves is not due to a
disorder-modified BCS transition. In device B, good agreement between the
model and the data can be achieved for all backgate voltages. The fits for the
highest and lowest backgate voltages are shown as dashed blue and green lines,
respectively, in Fig. 4b (all fits are shown in the Supplementary
Information). The fitted $\Delta_{0}/T_{c}$ significantly exceeds the BCS
result, and $\alpha$ increases from less than 1 to almost 2 with increasing
$V_{\text{BG}}$ (see inset to Fig. 4b). This dependence is contrary to
expectation, as the model suggests that $\alpha$ should decrease with
decreasing disorder and hence with increasing $V_{\text{BG}}$, because the
change of $R_{\square}$ with $V_{\text{BG}}$ suggests lower disorder at more
positive $V_{\text{BG}}$.

For device A, the deviation of the superfluid response from Eq. (2) is likely
caused by phase fluctuations. That is, the kink in $\chi(T)$ results from a
BKT jump in $\rho_{s}(T)$ slightly broadened by the interplay of finite-size
and disorder effects. Although device B does not show a similarly clear
feature, it is likely that similar effects are at play. For a given system
size, within the disorder modified BKT paradigm we expect that stronger
disorder (i.e., decreasing $V_{\text{BG}}$) will cause the superfluid response
to become more shallow close to $T_{c}$ [37], which is what we observe. This
behavior is also consistent with the generally stronger disorder in device B
compared to device A as indicated by the normal state carrier mobility.
Therefore, our data suggest that MoS2 represents a crossover system, where the
superfluid stiffness near $T_{c}$ is governed by phase fluctuations, but a
clear signature of a BKT transition may or may not be present depending on
doping and other parameters.

In conclusion, we report the first measurements of the superfluid response of
an atomically thin van der Waals superconductor using a local probe that
provides sufficient sensitivity to the small sample volume typical in this
material family. We find that the superfluid stiffness monotonically increases
at low temperatures as the backgate is tuned, even when the critical
temperature decreases. Our analysis suggests that our devices are in the dirty
limit of superconductivity in which the superfluid stiffness responds to
changes in device resistivity. This demonstrates that disorder plays an
important role even in crystalline 2D superconductors. Further, we observe
direct signatures of a BKT transition in one device, whereas, in another the
universal jump is replaced by a broad region of suppressed superfluid response
close to $T_{c}$. This demonstrates that a clear BKT transition is not
ubiquitous in these systems, but can be substantially obscured by disorder. In
the present work, our 4 K base temperature prevented characterizing $\chi$ at
a small fraction of $T_{c}$. Future work extending to lower temperatures will
be sensitive to the presence of nodes in the superconducting gap, which would
be a sign of an unconventional order parameter [38, 39, 40, 26].

## Acknowledgment

We thank Eric Smith for advice on cryogenic instrumentation and Debdeep Jena,
Huili Xing, Guen Prawiroatmodjo, Eun-Ah Kim, Yi-Ting Hsu, and Debanjan
Chowdhury for fruitful discussions. This work was primarily supported by the
National Science Foundation under Grant No. DMR-2004864, and partially
supported by the Cornell Center for Materials Research with funding from the
NSF MRSEC program (DMR-1719875). This work was performed in part at the
Cornell NanoScale Facility, an NNCI member supported by NSF Grant
NNCI-2025233.

## Author Contributions

A.J. and K.C.N. designed the experiments. A.J., B.T.S., and G.M.F. built the
instrument. A.J. fabricated the devices, performed the measurements, processed
the data, and performed modeling to estimate the SQUID height and relate
stiffness and magnetic response. M.L. advised on implementing liquid gating
and cryogenic measurements. A.J., Y.L.L., N.T., and K.C.N. analyzed the data.
A.J. and K.C.N. wrote the manuscript with input from all authors.

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$\operatorname{\textup{(lco)}}$ and $\operatorname{\textup{(lsco)}}$. Let $c$
be a chamber of $\Delta$ and let $\mathcal{F}:=\mathcal{F}(\Delta,c)$. As in
the previous corollary, each rank $3$-residue of $\mathcal{F}$ is integrable.
Theorem (5.2) implies that $\mathcal{D}_{\mathcal{F}}$ is an RGD-system and
$\mathcal{F}\cong\mathcal{F}(\Delta(\mathcal{D}_{\mathcal{F}}),B_{+})$. Using
Proposition (3.4), we deduce $\Delta\cong\Delta(\mathcal{D}_{\mathcal{F}})$.

Now we assume that there exists a panel containing at most $5$ chambers. Then
[24, $(34.5)$] implies that every panel contains only finitely many chambers.
But then $\Delta$ is known by the Main result of [11]. We note that the Main
result of [11] uses the fact that no rank $2$ residue is associated with
$B_{2}(2)$ in order the use the extension theorem of [15]. But as it is shown
in [4, Corollary $6.4$], the extension theorem holds for arbitrary thick
$3$-spherical twin buildings. Thus every irreducible locally finite
$3$-spherical twin building is known by [4] and [11]. ∎

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$\mathds{1}+\varphi^{\prime}_{\mu}a\frac{dx^{\mu}}{dt}+\frac{\partial
a}{\partial
x^{\nu}}\frac{dx^{\nu}}{dt}=\mathds{1}+a\varphi_{\nu}\frac{dx^{\nu}}{dt}-\underline{\alpha_{a}}(A^{\prime}_{\mu})\frac{dx^{\mu}}{dt}.$
(3.114)

Note that if $h\in H,$ the function $\alpha_{h}:G\to H$ is defined by $G\ni
g\mapsto\alpha_{g}(h)$ so that
$\underline{\alpha_{h}}:\mathfrak{g}\to\mathfrak{h}$ is the derivative. This
result gives the condition (after multiplying by $a^{-1}$ on the right)

$\varphi^{\prime}_{\mu}=a\varphi_{\mu}a^{-1}-\underline{\alpha_{a}}(A^{\prime}_{\mu})a^{-1}-(\partial_{\mu}a)a^{-1}$
(3.115)

on components and

$\varphi^{\prime}=a\varphi
a^{-1}-daa^{-1}-\underline{\alpha_{a}}(A^{\prime})a^{-1}$ (3.116)

as $\mathfrak{h}$-valued differential forms. This and (3.111) exactly agree
with Proposition 2.11 of [SW2]. Physically, a second order gauge
transformation is a gauge transformation for the 1-form $\varphi,$ which
itself is a type of field even though it appears as a (fat) gauge
transformation for the 1-form and 2-form gauge potentials.

###  3.4 Orientations and inverses

It is well-known that given a path $y\xleftarrow{\gamma}x,$ the parallel
transport along the reversed oriented path $\gamma^{-1}$ is the inverse

${{\mathrm{triv}}}(\gamma^{-1})={{\mathrm{triv}}}(\gamma)^{-1},$ (3.117)

where ${{\mathrm{triv}}}:\mathcal{P}^{1}M\to{{\mathbb{B}}}G$ is the (local)
parallel transport functor. This can be viewed as a consequence of thin
homotopy invariance and functoriality of parallel transport. Namely, although
the paths $\gamma\gamma^{-1}$ and $\gamma^{-1}\gamma$ are _not_ the constant
paths (the notation $\gamma^{-1}$ is therefore a bit abusive), they are
_thinly homotopic_ to constant paths and hence give the same value on
${{\mathrm{triv}}}.$ Thus,

$\textstyle{\displaystyle{{\mathrm{triv}}}(\gamma^{-1}\gamma)}$$\textstyle{\displaystyle{{\mathrm{triv}}}(\mathrm{id}_{x})}$$\textstyle{\quad\displaystyle{{\mathrm{triv}}}(\gamma^{-1}){{\mathrm{triv}}}(\gamma)}$$\textstyle{\displaystyle
1}$ (3.118)

verifying (3.117). In this section, we will explore analogous results for
reversing different kinds of orientations on bigons. We therefore include
arrows for clarity.

$y$$\gamma$$<$$<$$\delta$$\textstyle{\displaystyle\Sigma}$$x$

$y$$\gamma$$<$$<$$\delta$$\textstyle{\displaystyle\overline{\Sigma}}$$x$

$y$$\gamma^{-1}$$>$$>$$\delta^{-1}$$\textstyle{\displaystyle\Sigma^{-1}}$$x$

Technically, there is one more possibility given by
$\underline{\Sigma}:\delta^{-1}\Rightarrow\gamma^{-1}.$ However, this
possibility is a combination of the above two, namely
$\underline{\Sigma}=\overline{\Sigma^{-1}}.$ The meaning of these different
possibilities is given physically as follows. A given string may be given the
additional datum of an orientation. Furthermore, as it moves in time, it has
an additional directionality. These two directional orientations are precisely
encoded in the definition of a bigon/2-morphism. These different orientation
reversals are given by time reversal for $\overline{\Sigma}$ and spatial
orientation reversal for $\Sigma^{-1}.$ The case $\underline{\Sigma}$
corresponds to both reversals (the order of reversal does not matter since the
operations commute). Note that the different orientations on a bigon can be
expressed as an orientation of edges on the boundary and an orientation of the
surface. The above bigons correspond to the following surfaces with associated
orientations

$y$$\gamma$$<$$<$$\delta$$\circlearrowleft$$\Sigma$$x$

$y$$\gamma$$<$$<$$\delta$$\circlearrowright$$\overline{\Sigma}$$x$

$y$$\gamma^{-1}$$>$$>$$\delta^{-1}$$\circlearrowright$$\Sigma^{-1}$$x$

respectively.

A necessary and sufficient condition for such orientations on surfaces and
edges to give rise to a bigon is the following. Given a map of a polygon
$\Sigma$ into $M,$ the boundary consists of the edges of the polygon. The
union of the oriented edges consistent with the orientation of the polygon
must be connected. Similarly, the union of the orientated edges with negative
orientation with respect to the induced one from the polygon must also be
connected. Then, the source of the bigon is the union of the consistent edges
and the target is the union of the oppositely oriented edges. An example
together with a non-example are

$>$

$\gg$

$\gg$

$>$

$<$

$\circlearrowleft$

$<$

$>$

$\gg$

$<$

$>$

$<$

$\circlearrowleft$

$\gg$

& ,

respectively (blue, with arrows written using $>,$ corresponds to an
orientation agreeing with the induced one from the surface while yellow, with
arrows written using $\gg,$ disagrees with that orientation).

Going back to the three bigons and their orientations at the beginning of this
section, we notice that several of these bigons can be composed with one
another. For instance,

$y$$<$$\gamma$$<$$\overline{\Sigma}$$\delta$$<$$\gamma$$\Sigma$$x$$=$$y$$\gamma$$<$$x$

and

$x$$<$$\delta$$<$$\gamma$$\Sigma$$y$$<$$\delta^{-1}$$<$$\gamma^{-1}$$\Sigma^{-1}$$=$$x$$x$

after applying thin homotopies. Therefore, these bigons provide inverses in
series and in parallel, respectively, of $\Sigma.$ This implies, together with
functoriality of ${{\mathrm{triv}}}$ and the inverses discussed in Example
2.45,

${{\mathrm{triv}}}\big{(}\overline{\Sigma}\big{)}={{\mathrm{triv}}}(\Sigma)^{-1}\qquad\&\qquad{{\mathrm{triv}}}\big{(}\Sigma^{-1}\big{)}=\alpha_{{{\mathrm{triv}}}(\gamma)^{-1}}\big{(}{{\mathrm{triv}}}(\Sigma)^{-1}\big{)}$
(3.119)

and therefore describes how parallel transport along surfaces changes under
reversals in surface orientations and boundary orientations, respectively. For
completeness, for the last possible orientation
$\underline{\Sigma}:\delta^{-1}\Rightarrow\gamma^{-1},$ we have

${{\mathrm{triv}}}\left(\underline{\Sigma}\right)={{\mathrm{triv}}}\left(\overline{\Sigma^{-1}}\right)=\left(\alpha_{{{\mathrm{triv}}}(\gamma)^{-1}}\big{(}{{\mathrm{triv}}}(\Sigma)\big{)}^{-1}\right)^{-1}=\alpha_{{{\mathrm{triv}}}(\gamma)^{-1}}\big{(}{{\mathrm{triv}}}(\Sigma)\big{)}.$
(3.120)

###  3.5 The 3-curvature

In the following, we make some further calculations. Just as the curvature $F$
of a 1-form connection $A$ can be obtained by calculating the parallel
transport along an infinitesimal loop, the 2-curvature of a 2-form connection
$(A,B)$ can be obtained by calculating the surface transport along an
infinitesimal sphere, which on a lattice corresponds to a cube. We will
perform this calculation explicitly and study some properties of the resulting
3-form curvature. Similar analysis was done on a tetrahedron in [GP].

Let $(r,s,t)\mapsto x(r,s,t)$ be an infinitesimal cube and consider the
following domain for that cube along with the infinitesimal path that goes
first along the $r$ direction, then in the $s$ direction, and finally in the
$t$ direct. Our convention is that $(r,s,t)$ is a right-handed coordinate
frame, i.e. $dr\wedge ds\wedge dt$ is the volume form.

$r$$s$$t$$(0,0,0)$$(\Delta r,\Delta s,\Delta t)$

$<$

$>$

$>$

Such a cube can be expressed as a bigon by the following composition of
plaquette bigons that begin and end at the same path starting at the top left
and moving clockwise.

$r$$s$$t$

$\Rightarrow$

$r$$s$$t$

$\Rightarrow$

$r$$s$$t$

$\Rightarrow$

$\rightsquigarrow$ $\rightsquigarrow$

$r$$s$$t$

$\Rightarrow$

$\rightsquigarrow$

$r$$s$$t$

$\Rightarrow$

$\rightsquigarrow$

$\rightsquigarrow$

$r$$s$$t$

$\Rightarrow$

The corresponding 2-group elements are given as follows. We begin with the
first surface

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}$$e^{-A_{\nu}\partial_{s}x^{\nu}|_{(\epsilon,0,0)}}$$e^{-A_{\lambda}\partial_{r}x^{\lambda}|_{(0,0,0)}}$$e^{B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}|_{(0,0,0)}}$$e^{-A_{\rho}\partial_{r}x^{\rho}|_{(0,\epsilon,0)}}$$e^{-A_{\sigma}\partial_{s}x^{\sigma}|_{(0,0,0)}}$

$\leftrightarrow$

where we use the shorthand notation

$\partial_{r}x:=\frac{\partial x}{\partial
r},\qquad\partial_{s}x:=\frac{\partial x}{\partial
s},\qquad\&\qquad\partial_{t}x:=\frac{\partial x}{\partial t}$ (3.121)

as well as

$e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}:=\exp\left\\{-A_{\mu}\big{(}x(\epsilon,\epsilon,0)\big{)}\frac{\partial
x^{\mu}}{\partial t}\Big{|}_{(\epsilon,\epsilon,0)}\right\\}$ (3.122)

and similarly for the other terms. We also write $\epsilon$ instead of $\Delta
r,\Delta s,$ or $\Delta t$ and use the derivatives to remind ourselves of the
direction. We have also assumed for simplicity that our coordinates are
centered at the origin and the lattice spacing is $\epsilon$ in each
direction. Working out this diagram infinitesimally on the 0-d defect gives

$\alpha_{e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}}\left(e^{B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}|_{(0,0,0)}}\right)=\mathds{1}+B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}-\underline{\alpha}_{A_{\mu}}(B_{\lambda\nu})\partial_{t}x^{\mu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}$
(3.123)

to lowest order. As usual, rather than writing out the $\Delta r,\Delta
s,\Delta t,$ we use the number and type of derivatives appearing to keep track
of the order. The other terms are given by the following

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}$$e^{-A_{\rho}\partial_{r}x^{\rho}|_{(0,\epsilon,0)}}$$e^{B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}|_{(0,\epsilon,0)}}$$e^{-A_{\sigma}\partial_{s}x^{\sigma}|_{(0,0,0)}}$$e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,\epsilon,\epsilon)}}$$e^{-A_{\tau}\partial_{t}x^{\tau}|_{(0,\epsilon,0)}}$

$\leftrightarrow$

$\begin{split}e^{B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}|_{(0,\epsilon,0)}}&=\mathds{1}+B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}+\partial_{c}B_{\rho\pi}\partial_{s}x^{c}\partial_{r}x^{\rho}\partial_{t}x^{\pi}\\\
&\quad+B_{\rho\pi}\partial_{s}\partial_{r}x^{\rho}\partial_{t}x^{\pi}+B_{\rho\pi}\partial_{r}x^{\rho}\partial_{s}\partial_{t}x^{\pi}\end{split}$
(3.124)

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,\epsilon,\epsilon)}}$$e^{-A_{\tau}\partial_{t}x^{\tau}|_{(0,\epsilon,0)}}$$e^{-A_{\sigma}\partial_{s}x^{\sigma}|_{(0,0,0)}}$$e^{B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}|_{(0,0,0)}}$$e^{-A_{\beta}\partial_{s}x^{\beta}|_{(0,0,\epsilon)}}$$e^{-A_{\gamma}\partial_{t}x^{\gamma}|_{(0,0,0)}}$

$\leftrightarrow$

$\alpha_{e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,0,\epsilon)}}}\left(e^{B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}|_{(0,0,0)}}\right)=\mathds{1}+B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}-\underline{\alpha}_{A_{\alpha}}(B_{\sigma\tau})\partial_{r}x^{\alpha}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}$
(3.125)

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,\epsilon,\epsilon)}}$$e^{-A_{\beta}\partial_{s}x^{\beta}|_{(0,0,\epsilon)}}$$e^{B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}|_{(0,0,\epsilon)}}$$e^{-A_{\gamma}\partial_{t}x^{\gamma}|_{(0,0,0)}}$$e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}$$e^{-A_{\eta}\partial_{r}x^{\eta}|_{(0,0,\epsilon)}}$

$\leftrightarrow$

$\begin{split}e^{B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}|_{(0,0,\epsilon)}}&=\mathds{1}+B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}+\partial_{b}B_{\beta\kappa}\partial_{t}x^{b}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}\\\
&\quad+B_{\beta\kappa}\partial_{t}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}+B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{t}\partial_{r}x^{\kappa}\end{split}$
(3.126)

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}$$e^{-A_{\eta}\partial_{r}x^{\eta}|_{(0,0,\epsilon)}}$$e^{-A_{\gamma}\partial_{t}x^{\gamma}|_{(0,0,0)}}$$e^{B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}|_{(0,0,0)}}$$e^{-A_{\omega}\partial_{t}x^{\omega}|_{(\epsilon,0,0)}}$$e^{-A_{\lambda}\partial_{r}x^{\lambda}|_{(0,0,0)}}$

$\leftrightarrow$

$\alpha_{e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}}\left(e^{B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}|_{(0,0,0)}}\right)=\mathds{1}+B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}-\underline{\alpha}_{A_{\theta}}(B_{\gamma\eta})\partial_{s}x^{\theta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}$
(3.127)

$r$$s$$t$

$\Rightarrow$

$e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}$$e^{-A_{\omega}\partial_{t}x^{\omega}|_{(\epsilon,0,0)}}$$e^{B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}|_{(\epsilon,0,0)}}$$e^{-A_{\lambda}\partial_{r}x^{\lambda}|_{(0,0,0)}}$$e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}$$e^{-A_{\nu}\partial_{s}x^{\nu}|_{(\epsilon,0,0)}}$

$\leftrightarrow$

$\begin{split}e^{B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}|_{(\epsilon,0,0)}}&=\mathds{1}+B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}+\partial_{a}B_{\omega\psi}\partial_{r}x^{a}\partial_{t}x^{\omega}\partial_{s}x^{\psi}\\\
&\quad+B_{\omega\psi}\partial_{r}\partial_{t}x^{\omega}\partial_{s}x^{\psi}+B_{\omega\psi}\partial_{t}x^{\omega}\partial_{r}\partial_{s}x^{\psi}\end{split}$
(3.128)

The composition of all of these elements is given by the following diagram
(with the light shaded blue squares depicting the faces on the cube).

$e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}$$e^{B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}|_{(0,0,0)}}$$e^{B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}|_{(0,\epsilon,0)}}$$e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,0,\epsilon)}}$$e^{B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}|_{(0,0,0)}}$$e^{B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}|_{(0,0,\epsilon)}}$$e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}$$e^{B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}|_{(0,0,0)}}$$e^{B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}|_{(\epsilon,0,0)}}$

And the result of multiplying these out gives

$\begin{split}e^{B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}|_{(\epsilon,0,0)}}\alpha_{e^{-A_{\theta}\partial_{s}x^{\theta}|_{(\epsilon,0,\epsilon)}}}\left(e^{B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}|_{(0,0,0)}}\right)e^{B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}|_{(0,0,\epsilon)}}\\\
\times\alpha_{e^{-A_{\alpha}\partial_{r}x^{\alpha}|_{(0,0,\epsilon)}}}\left(e^{B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}|_{(0,0,0)}}\right)e^{B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}|_{(0,\epsilon,0)}}\alpha_{e^{-A_{\mu}\partial_{t}x^{\mu}|_{(\epsilon,\epsilon,0)}}}\left(e^{B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}|_{(0,0,0)}}\right).\end{split}$
(3.129)

This is yet another manifestation of two-dimensional algebra. The result of
multiplying all these terms is given as follows, order by order. The zeroth
order term is $\mathds{1}.$ There are no first order terms. The second order
terms are given by

$\begin{split}B_{\omega\psi}\partial_{t}x^{\omega}\partial_{s}x^{\psi}&+B_{\gamma\eta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}+B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}+B_{\sigma\tau}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}+B_{\rho\pi}\partial_{r}x^{\rho}\partial_{t}x^{\pi}+B_{\lambda\nu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}\\\
&=(B_{\sigma\tau}+B_{\tau\sigma})\partial_{s}x^{\sigma}\partial_{t}x^{\tau}+(B_{\lambda\nu}+B_{\nu\lambda})\partial_{r}x^{\lambda}\partial_{s}x^{\nu}+(B_{\rho\pi}+B_{\pi\rho})\partial_{r}x^{\rho}\partial_{t}x^{\pi}\\\
&=0\end{split}$ (3.130)

by anti-symmetry of $B_{\mu\nu}$ in the $\mu$ and $\nu$ indices. Thus, the
only non-zero terms are the zeroth and third order terms (up to third order).
One type of the third order terms are given by

$\begin{split}\Big{(}&B_{\omega\psi}\partial_{r}\partial_{t}x^{\omega}\partial_{s}x^{\psi}+B_{\omega\psi}\partial_{t}x^{\omega}\partial_{r}\partial_{s}x^{\psi}+B_{\beta\kappa}\partial_{t}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}\Big{)}\\\
+\Big{(}&B_{\beta\kappa}\partial_{s}x^{\beta}\partial_{t}\partial_{r}x^{\kappa}+B_{\rho\pi}\partial_{s}\partial_{r}x^{\rho}\partial_{t}x^{\pi}+B_{\rho\pi}\partial_{r}x^{\rho}\partial_{s}\partial_{t}x^{\pi}\Big{)}\\\
&=(B_{\beta\kappa}+B_{\kappa\beta})\partial_{s}x^{\beta}\partial_{t}\partial_{r}x^{\kappa}+(B_{\omega\psi}+B_{\psi\omega})\partial_{t}x^{\omega}\partial_{r}\partial_{s}x^{\psi}+(B_{\rho\pi}+B_{\pi\rho})\partial_{r}x^{\rho}\partial_{s}\partial_{t}x^{\pi}\end{split}$
(3.131)

and vanish again by anti-symmetry of $B_{\mu\nu}$ and commutativity of partial
derivatives. The final result is therefore

$\begin{split}\mathds{1}&+\partial_{a}B_{\omega\psi}\partial_{r}x^{a}\partial_{t}x^{\omega}\partial_{s}x^{\psi}-\underline{\alpha}_{A_{\theta}}(B_{\gamma\eta})\partial_{s}x^{\theta}\partial_{t}x^{\gamma}\partial_{r}x^{\eta}+\partial_{b}B_{\beta\kappa}\partial_{t}x^{b}\partial_{s}x^{\beta}\partial_{r}x^{\kappa}\\\
&-\underline{\alpha}_{A_{\alpha}}(B_{\sigma\tau})\partial_{r}x^{\alpha}\partial_{s}x^{\sigma}\partial_{t}x^{\tau}+\partial_{c}B_{\rho\pi}\partial_{s}x^{c}\partial_{r}x^{\rho}\partial_{t}x^{\pi}\
-\underline{\alpha}_{A_{\mu}}(B_{\lambda\nu})\partial_{t}x^{\mu}\partial_{r}x^{\lambda}\partial_{s}x^{\nu}\\\
&=\mathds{1}-\Big{(}\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}B_{\mu\nu}+\partial_{\nu}B_{\lambda\mu}+\underline{\alpha}_{A_{\mu}}(B_{\nu\lambda})+\underline{\alpha}_{A_{\nu}}(B_{\lambda\mu})+\underline{\alpha}_{A_{\lambda}}(B_{\mu\nu})\Big{)}\partial_{r}x^{\mu}\partial_{s}x^{\nu}\partial_{t}x^{\lambda}.\end{split}$
(3.132)

In analogy to the curvature 2-form associated to a 1-form potential $A$
obtained by calculating the holonomy along an infinitesimal square, we define
this third order term to be the _3-form curvature_ associated to the pair
$(A,B)$ and denote it by $H.$ In terms of components, it is given by

$H_{\mu\nu\lambda}:=\partial_{\mu}B_{\nu\lambda}+\partial_{\lambda}B_{\mu\nu}+\partial_{\nu}B_{\lambda\mu}+\underline{\alpha}_{A_{\mu}}(B_{\nu\lambda})+\underline{\alpha}_{A_{\nu}}(B_{\lambda\mu})+\underline{\alpha}_{A_{\lambda}}(B_{\mu\nu})$
(3.133)

and using differential form notation

$H:=dB+\underline{\alpha}_{A}(B).$ (3.134)

This definition and result agrees with (3.28) of [GP] and Lemma A.11 in [SW2].
As was also pointed out in [GP],

$\underline{\tau}(H)=\underline{\tau}(dB)+\underline{\tau}\big{(}\underline{\alpha}_{A}(B)\big{)}=d\underline{\tau}(B)+[A,\underline{\tau}(B)]=dF+[A,F]=0$
(3.135)

by the Bianchi identity. Since $\ker\underline{\tau}$ is a central Lie
subalgebra of $\mathfrak{h},$ this means $H$ is a 3-form with values in an
abelian Lie algebra (see Remark 2.44). Under a first order gauge
transformation $(A,B)\xrightarrow{(g,\varphi)}(A^{\prime},B^{\prime})$ as in
(3.109) and using (3.110), the 3-form curvature changes to

$\begin{split}H^{\prime}&=dB^{\prime}+\underline{\alpha}_{A^{\prime}}(B^{\prime})\\\
&=d\Big{(}\underline{\alpha_{g}}(B)+\varphi\wedge\varphi-d\varphi-\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\Big{)}\\\
&\quad+\underline{\alpha}_{gAg^{-1}-dgg^{-1}-\underline{\tau}(\varphi)}\Big{(}\underline{\alpha_{g}}(B)+\varphi\wedge\varphi-d\varphi-\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\Big{)}\\\
&=\underbrace{d\big{(}\underline{\alpha_{g}}(B)\big{)}}_{(\ref{eq:3curvdagB})}+\uwave{d\varphi\wedge\varphi-\varphi\wedge
d\varphi}-\underbrace{d\big{(}\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\big{)}}_{(\ref{eq:3curvdaAprimephi})}\\\
&\quad+\underbrace{\underline{\alpha}_{gAg^{-1}-dgg^{-1}}\big{(}\underline{\alpha_{g}}(B)\big{)}}_{(\ref{eq:3curvagaAB})}-\underline{\alpha}_{dgg^{-1}}\big{(}\underline{\alpha_{g}}(B)\big{)}+\underbrace{\underline{\alpha}_{gAg^{-1}-dgg^{-1}}\big{(}\varphi\wedge\varphi\big{)}}_{(\ref{eq:3curveaAprimephiphi})}\\\
&\quad-\underbrace{\underline{\alpha}_{gAg^{-1}-dgg^{-1}}\big{(}\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\big{)}}_{(\ref{eq:3curvaAprimeaAprimephi})}-\underbrace{\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(d\varphi)}_{(\ref{eq:3curvdaAprimephi})}\\\
&\quad-[\varphi,\underline{\alpha_{g}}(B)]-\underbrace{[\varphi,\varphi\wedge\varphi]}_{0}+\uwave{[\varphi,d\varphi]}+[\varphi,\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)],\end{split}$
(3.136)

where the underlined terms cancel and the other terms with underbraces will be
calculated and compared momentarily. At this point, it is useful to simplify
some of these terms by applying $\underline{\tau}$ and calculating the results
in terms of commutators and such. For example,

$\begin{split}\underline{\tau}\Big{(}d\underline{\alpha_{g}}(B)\big{)}\Big{)}&=d\Big{(}\underline{\tau}\big{(}\underline{\alpha_{g}}(B)\big{)}\Big{)}\\\
&=d\big{(}g\underline{\tau}(B)g^{-1}\big{)}\\\
&=dg\underline{\tau}(B)g^{-1}+g\underline{\tau}(dB)g^{-1}+g\underline{\tau}(B)dg^{-1}\\\
&=dgg^{-1}g\underline{\tau}(B)g^{-1}+\underline{\tau}\big{(}\underline{\alpha_{g}}(dB)\big{)}-g\underline{\tau}(B)g^{-1}dgg^{-1}\\\
&=\underline{\tau}\Big{(}\underline{\alpha_{g}}(dB)+\underline{\alpha}_{dgg^{-1}}\big{(}\underline{\alpha_{g}}(B)\big{)}\Big{)}.\end{split}$
(3.137)

We can safely equate the terms inside the $\underline{\tau}.$ This is a
helpful trick and we will use it to calculate all other terms. For instance,
one can show using this trick that

$\underline{\alpha}_{gAg^{-1}}\big{(}\underline{\alpha_{g}}(B)\big{)}=\underline{\alpha_{g}}\big{(}\underline{\alpha}_{A}(B)\big{)}.$
(3.138)

Although this is a trick and not completely rigorous, it works for all of the
calculations we will do. These formulas can all be found in Appendix A of
[Wal1] and can be derived more rigorously. Since $\underline{\alpha}$ is a
derivation,

$\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi\wedge\varphi)=\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\wedge\varphi-\varphi\wedge\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi),$
(3.139)

which cancels with the term
$[\varphi,\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)]$ in (3.136).
Furthermore, note that

$\begin{split}\underline{\tau}\Big{(}\underline{\alpha}_{X}\big{(}\underline{\alpha}_{X}(Y)\big{)}\Big{)}&=\Big{[}X,\underline{\tau}\big{(}\underline{\alpha}_{X}(Y)\big{)}\Big{]}\\\
&=\Big{[}X,[X,\underline{\tau}(Y)\big{]}\Big{]}\\\
&=\big{[}X,X\wedge\underline{\tau}(Y)+\underline{\tau}(Y)\wedge X\big{]}\\\
&=X\wedge X\wedge\underline{\tau}(Y)-X\wedge\underline{\tau}(Y)\wedge
X+X\underline{\tau}(Y)\wedge X-\underline{\tau}(Y)\wedge X\wedge X\\\
&=\underline{\tau}\big{(}\underline{\alpha}_{X\wedge X}(Y)\big{)}\end{split}$
(3.140)

for any $\mathfrak{g}$-valued 1-form $X$ and for any $\mathfrak{h}$-valued
1-form $Y.$ Although $\underline{\tau}$ has been applied to derive these
equalities, the expressions inside $\underline{\tau}$ are still equal. This
equality implies

$\begin{split}\underline{\alpha}_{gAg^{-1}-dgg^{-1}}\big{(}\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\big{)}&=\underline{\alpha}_{gAAg^{-1}}(\varphi)-\underline{\alpha}_{dgAg^{-1}}(\varphi)\\\
&\quad-\underline{\alpha}_{gAg^{-1}dgg^{-1}}(\varphi)+\underline{\alpha}_{dgg^{-1}dgg^{-1}}(\varphi).\end{split}$
(3.141)

One of the more cumbersome set of terms is

$\begin{split}\underline{\tau}\Big{(}&d\big{(}\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(\varphi)\big{)}+\underline{\alpha}_{gAg^{-1}-dgg^{-1}}(d\varphi)\Big{)}=d[gAg^{-1},\underline{\tau}(\varphi)]-d[dgg^{-1},\underline{\tau}(\varphi)]\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+[gAg^{-1},\underline{\tau}(d\varphi)]-[dgg^{-1},\underline{\tau}(d\varphi)]\\\
&=dgAg^{-1}\underline{\tau}(\varphi)+gdAg^{-1}\underline{\tau}(\varphi)-gAdg^{-1}\underline{\tau}(\varphi)-\uuline{gAg^{-1}\underline{\tau}(d\varphi)}\\\
&+\dashuline{\underline{\tau}(d\varphi)gAg^{-1}}-\underline{\tau}(\varphi)dgAg^{-1}-\underline{\tau}(\varphi)gdAg^{-1}+\underline{\tau}(\varphi)gAdg^{-1}\\\
&+dgdg^{-1}\underline{\tau}(\varphi)+\dotuline{dgg^{-1}\underline{\tau}(d\varphi)}-\uwave{\underline{\tau}(d\varphi)dgg^{-1}}-\underline{\tau}(\varphi)dgdg^{-1}\\\
&+\uuline{gAg^{-1}\underline{\tau}(d\varphi)}-\dashuline{\underline{\tau}(d\varphi)gAg^{-1}}-\dotuline{dgg^{-1}\underline{\tau}(d\varphi)}+\uwave{\underline{\tau}(d\varphi)dgg^{-1}}\\\
&=\underline{\tau}\Big{(}\underline{\alpha}_{dgAg^{-1}}(\varphi)+\underline{\alpha}_{gdAg^{-1}}(\varphi)-\underline{\alpha}_{gAdg^{-1}}(\varphi)+\underline{\alpha}_{dgdg^{-1}}(\varphi)\Big{)}.\end{split}$
(3.142)

Combining this with the result preceding it gives just a single term
$\underline{\alpha}_{gFg^{-1}}(\varphi).$ Putting all of this together, we
obtain

$\begin{split}H^{\prime}&=\underline{\alpha_{g}}\big{(}dB+\underline{\alpha}_{A}(B)\big{)}-[\varphi,\underline{\alpha_{g}}(B)]-\underline{\alpha}_{gFg^{-1}}(\varphi)\\\
&=\underline{\alpha_{g}}(H)-[\varphi,\underline{\alpha_{g}}(B)]-\underline{\alpha}_{gFg^{-1}}(\varphi).\end{split}$
(3.143)

Finally, by the properties of crossed modules and by the vanishing of the fake
curvature,

$\underline{\alpha}_{gFg^{-1}}(\varphi)=\underline{\alpha}_{g\underline{\tau}(B)g^{-1}}(\varphi)=\underline{\alpha}_{\underline{\tau}(\underline{\alpha_{g}}(B))}(\varphi)=\big{[}\underline{\alpha_{g}}(B),\varphi\big{]}$
(3.144)

so that the above formula reduces further to just simply

$H^{\prime}=\underline{\alpha_{g}}(H).$ (3.145)

In particular, $H$ is invariant under fat gauge transformations. This result
agrees with what was discovered in [GP].

##  4 Conclusion

We have illustrated that 2-category theory can be implemented and used in such
a way as to calculate parallel transport along two-dimensional surfaces, such
as worldsheets of strings, explicitly for gauge groups that are not
necessarily abelian via an approximation technique that can be implemented
numerically. We have done this using string diagram techniques to facilitate
2-categorical techniques and bring higher category theory to a wider audience.
Although Girelli and Pfeiffer have calculated infinitesimal gauge
transformations and curvature forms via similar techniques [GP] and Schreiber
and Waldorf provided a formula for the parallel transport along a surface
[SW2], our infinitesimal methods give a much more explicit and direct
construction of the iterated surface integral from elementary building blocks
filling in some of the arguments sketched by Baez and Schreiber in [BS],
particularly in Section 2.3.2 (Section 5.1 of a draft of this paper even
contains a nice picture that unfortunately did not make it to the final
version of their paper but is present in Section 11.4.1 of Schreiber’s thesis
[Scthesis]). Schreiber and Waldorf’s integral in [SW2] was obtained from
consistency conditions and then they proved that it satisfies the necessary
functorial properties expected of surface holonomy. In relation to other work,
such surface-ordered integrals have been used recently in constructing a
Hochschild complex for surface transport [Mi15]. The novelty of our result is
that we derived the formula for surface parallel transport from scratch using
a discretization of our surface. To our knowledge, this is the first
appearance of such an explicit construction together with analytical results
on convergence (Proposition 3.57) and a simplification providing a manageable
surface-ordered integral by reducing the surface ordering to a single
direction as opposed to two (this result is embodied in Theorem 3.78). By
implementing string diagrams, we have also provided a more friendly
visualization. Furthermore, we have avoided using path spaces explicitly and
have simplified many arguments.

We hope that we have illustrated how two-dimensional algebra can be used for
explicit calculations. If developed further, these ideas might be used to
explain physical phenomena that utilize algebraic manipulations in more than
one dimension more naturally. Such higher-dimensional algebra appears in many
situations. For example, elements and molecules combine in a variety of ways
forming complicated compounds, amino acids, and proteins. These are objects
that use three dimensions to configure themselves. Therefore a natural and
faithful representation of them would involve a sort of 3-dimensional algebra.
Another example occurs in painting. Given a painting, it is much simpler for
us to “read” a 2-dimensional painting than to view all the pixels making it up
in a straight 1-dimensional list. Both perspectives contain the same
information-theoretic data, but the 2-d form is naturally and immediately
recognized. As another even more speculative example, it is known that the
entropy of a black hole is proportional to the surface area of its horizon.
This may lead one to believe that the microstates of the theory can be
expressed as living on a lower-dimensional world. This in turn then suggests
the possibility that a lower-dimensional algebra might be useful in describing
some of the properties of these microstates. Although these ideas are entirely
speculative, our point is that one can imagine that the one-dimensional
algebra we have forced upon ourselves is only the tip of an iceberg of
algebraic structures. Higher category theory opens us to these other
possibilities.

There are still many open questions in this relatively young field. One is how
to construct useful Actions in physics that model phenomena with non-abelian
higher form gauge fields and also the interactions with matter fields. Some
recent progress in this direction has been made by Sämann and others—see
[SS17] and the references therein. Work on the pure gauge field side was
initiated in the work of Pfeiffer [Pf] using a 2-categorical approach. To
proceed, it seems that a better suited representation theory for 2-categories
will be useful [BBFW]. Furthermore, characters for 2-groups [GK], [GU] and
traces [PoSh], [HPT] may need to be studied further to better understand what
gauge invariant combinations are possible. Although the number of higher gauge
theory examples are increasing [BW], [CLS], [FSS], [MM], [PaSa], [Pa], [SS17],
some work is still required to solidify the role of higher gauge theory in
lattice gauge theory and other areas of physics. Other lattice gauge theory
approaches existed earlier [Or80], [Or83], [Or84] with a renewed interest in
[LR] but it is not clear to us how these approaches to higher lattice gauge
theory are related to the rest of the literature.

In the realm of string theory and M-theory, beginning with early work of
Witten, Myers, and others [Wit96], [My], a more precise construction of the
non-abelian gauge theories on a stack of $D$-branes [Zw] and its low energy
effective Action is still lacking. These effective Actions are swarmed with
higher form non-abelian gauge fields, but the precise mathematical formulation
is still lacking though it is likely that non-abelian differential cohomology
[Sc] is relevant suggested by recent work on M5-branes in which it plays an
essential role [FSS]. Some arguments used to describe such effective Actions
are not always entirely straightforward and involve consistency conditions
(such as T-duality [My] and scattering amplitude calculations [DST]) rather
than direct derivations. It is therefore possible that a more thorough
investigation may involve understanding nonperturbative effects, one of which
is dictated by transport. On the other hand, due to the non-commutative nature
of the normal coordinates to these branes [Mo], this may involve a
modification of such transport to the setting of non-commutative geometry.
These and many other ideas have also been briefly discussed in [Scthesis], and
several such open questions can be found there.

##  Appendix A Differential Lie crossed modules

Here we briefly review the infinitesimal version of a Lie crossed module
$(H,G,\tau,\alpha),$ which we write as
$(\mathfrak{h},\mathfrak{g},\underline{\tau},\underline{\alpha}),$ including
the many relations that these maps satisfy that we use throughout our
calculations. We also make some comments on how this is used for differential
forms with values in $\mathfrak{g}$ and $\mathfrak{h}.$ This information can
also be found in many articles on the subject of higher gauge theory such as
[BS], [GP], and especially Waldorf’s concise one page formula sheet in
Appendix A of [Wal1]. Martins and Miković also have an exceptionally clear and
thorough exposition in Section 2.1 of [MM].

$\underline{\tau}:\mathfrak{h}\to\mathfrak{g}$ is the derivative of $\tau:H\to
G$ at the identity and is a Lie algebra homomorphism since $\tau$ is a Lie
group homomorphism. Notice that $\alpha$ can be equivalently described as a
function $\alpha:G\times H\to H$ that is a group homomorphism in each
component separately. As a result, for any fixed $g\in G,$ $\alpha_{g}:H\to H$
is a Lie group homomorphism and hence has a derivative at the identity denoted
by $\underline{\alpha_{g}}:\mathfrak{h}\to\mathfrak{h}.$ This map, besides
being a Lie algebra homomorphism, satisfies the additional property that

$\underline{\tau}\big{(}\underline{\alpha_{g}}(Y)\big{)}=g\underline{\tau}(Y)g^{-1}$
(A.1)

for all $Y\in\mathfrak{h}$ and $g\in G.$ Similarly, although $\alpha:G\times
H\to H$ is not a group homomorphism, it is smooth and its derivative
$\underline{\alpha}:\mathfrak{g}\times\mathfrak{h}\to\mathfrak{h}$ is a well-
defined linear map. It is a derivation once the $\mathfrak{g}$ coordinate is
fixed, i.e.

$\underline{\alpha}_{X}\big{(}[Y,Z])=\big{[}\underline{\alpha}_{X}(Y),Z\big{]}+\big{[}Y,\underline{\alpha}_{X}(Z)\big{]}$
(A.2)

for all $X\in\mathfrak{g}$ and $Y,Z\in\mathfrak{h}.$ $\underline{\alpha}$ also
satisfies

$\underline{\alpha}_{[X,X^{\prime}]}(Y)=\underline{\alpha}_{X}\big{(}\underline{\alpha}_{X^{\prime}}(Y)\big{)}-\underline{\alpha}_{X^{\prime}}\big{(}\underline{\alpha}_{X}(Y)\big{)}$
(A.3)

for all $X,X^{\prime}\in\mathfrak{g}$ and $Y\in\mathfrak{h}.$ Finally,

$\underline{\tau}\big{(}\underline{\alpha}_{X}(Y)\big{)}=\big{[}X,\underline{\tau}(Y)\big{]}$
(A.4)

and

$\underline{\alpha}_{\tau(Y)}(Z)=[Y,Z]$ (A.5)

for all $X\in\mathfrak{g}$ and $Y,Z\in\mathfrak{h}.$

Once combined with differential forms, the maps $\underline{\alpha}$ and
$\underline{\tau}$ are extended in the appropriate way (see Part II Chapter 3
in the section on the Bianchi Identity in [BaMu] for details on differential
forms with values in Lie algebras). For instance, $\underline{\alpha}$ is a
graded derivation in its second coordinate. To clarify the notation used
throughout, consider differential forms $A\in\Omega^{1}(M;\mathfrak{g}),$
$F\in\Omega^{2}(M;\mathfrak{g}),$ $\varphi\in\Omega^{1}(M;\mathfrak{h}),$
$B\in\Omega^{2}(M;\mathfrak{h}).$ When we write expressions such as
$\underline{\alpha}_{A}(\varphi)$ or $\underline{\alpha}_{A}(B)$ we mean the
following. First, let $\\{t^{a}\\}_{a\in\\{1,\dots,\dim\mathfrak{g}\\}}$ be a
basis for $\mathfrak{g}$ and
$\\{s^{b}\\}_{b\in\\{1,\dots,\dim\mathfrak{h}\\}}$ be a basis for
$\mathfrak{h}.$ Then

$A=A_{a}t^{a},\qquad
F=F_{a}t^{a},\qquad\varphi=\varphi_{b}s^{b},\qquad\&\qquad B=B_{b}s^{b},$
(A.6)

where a summation over repeated indices is assumed and where
$A_{a},\varphi_{b}\in\Omega^{1}(M)$ and $F_{a},B_{b}\in\Omega^{2}(M)$ for all
indices. Then by definition,

$\underline{\alpha}_{A}(\varphi)\equiv\underline{\alpha}_{A_{a}t^{a}}\big{(}\varphi_{b}s^{b}\big{)}:=(A_{a}\wedge\varphi_{b})\underline{\alpha}_{t^{a}}\big{(}s^{b}\big{)}$
(A.7)

and similarly for any other forms. Because we use Lie algebra valued forms,
the bracket is graded, so for instance

$[\varphi,\varphi]:=(\varphi_{b}\wedge\varphi_{b^{\prime}})\big{[}s^{b},s^{b^{\prime}}\big{]}=\varphi\wedge\varphi+\varphi\wedge\varphi$
(A.8)

but

$[\varphi,B]:=(\varphi_{b}\wedge
B_{b^{\prime}})\big{[}s^{b},s^{b^{\prime}}\big{]}=\varphi\wedge
B-B\wedge\varphi$ (A.9)

since $\varphi$ is a 1-form and $B$ is a 2-form. The last two equalities
follow if we think of our Lie algebras as coming from matrix Lie algebras,
which we often do. The general formula is

$[\omega,\eta]=\omega\wedge\eta-(-1)^{|\omega||\eta|}\eta\wedge\omega,$ (A.10)

where $|\omega|$ and $|\eta|$ are the degrees of the forms
$\omega\in\Omega^{|\omega|}(M;\mathfrak{h})$ and
$\eta\in\Omega^{|\eta|}(M;\mathfrak{h}).$ Other properties are derived as
needed in calculations in the body of the article.

##  Appendix B Surface product convergence

This appendix serves to prove the convergence of the surface-ordered product
(3.53) as $n\to\infty$ and to also prove upper bounds on the $k$-th order
terms when expanded out. For this, we will first relax our conditions and work
with arbitrary partitions of the unit square. We will follow the conventions
of [Mu91] and use the results there without further reference. The surface-
ordered product is well-defined for each partition $P$ and will be denoted by
${{\mathrm{triv}}}_{P}.$ We will also use the notation $\gamma_{s,t}$ to
denote the path defined after Theorem 3.78. It will be helpful to define the
function

$[0,1]\times[0,1]\ni(s,t)\mapsto\mathcal{B}(s,t):=\left\lVert\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s,t})}}\left(B_{\mu\nu}\big{(}x(s,t)\big{)}\frac{\partial
x^{\mu}}{\partial t}\frac{\partial x^{\nu}}{\partial s}\right)\right\rVert.$
(B.1)

###### Proof of Proposition 3.57.

By smoothness of $A,B,$ and parallel transport along paths, $\mathcal{B}$ is a
smooth function. First note that for any point $(s_{i},t_{j})\in[0,1]$ on the
intersection points of the grid formed by some partition $P$

$\lim_{Q\geq
P}\underline{\alpha_{\gamma_{s_{i},t_{j}}^{Q}}}\left(B_{\mu_{i}\nu_{j}}\big{(}x(s_{i},t_{j})\big{)}\frac{\partial
x^{\mu_{i}}}{\partial t}\frac{\partial x^{\nu_{j}}}{\partial
s}\Big{|}_{(s_{i},t_{j})}\right)=\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s_{i},t_{j}})}}\left(B_{\mu\nu}\big{(}x(s_{i},t_{j})\big{)}\frac{\partial
x^{\mu}}{\partial t}\frac{\partial x^{\nu}}{\partial
s}\Big{|}_{(s_{i},t_{j})}\right)$ (B.2)

where $\gamma_{s_{i},t_{j}}^{Q}$ is the path obtained from expanding out the
2-group multiplication with respect to the partition $Q.$ The ordering on
partitions is given by refinement and the above limit is taken over all
refinements of $P.$ This limit is valid due to the smoothness of all
expressions. Visually, this limit is also reasonable. For example, consider
the following examples of refinements

$\textstyle{\small t}$$\textstyle{\small s}$

$\textstyle{\small t}$$\textstyle{\small s}$

$\textstyle{\small t}$$\textstyle{\small s}$

$\qquad\quad\to$ $\qquad\quad\to$

Taking the limit over partitions

$\textstyle{\small t}$$\textstyle{\small s}$

shows that the paths converge to $\gamma_{s,t}$ and the value of the alpha
action along the path at $B$ depends only on the point $(s,t).$ Hence, the
partial _products_ of terms of the form

$\mathds{1}+\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s_{i},t_{j}})}}\left(B_{\mu_{i}\nu_{j}}\big{(}x(s_{i},t_{j})\big{)}\frac{\partial
x^{\mu_{i}}}{\partial t}\frac{\partial x^{\nu_{j}}}{\partial
s}\Big{|}_{(s_{i},t_{j})}\right)\Delta s_{i}\Delta t_{j},$ (B.3)

which define ${{\mathrm{triv}}}_{P}$ to lowest order, converge if and only if
the _sum_ of terms of the form

$\left\lVert\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s_{i},t_{j}})}}\left(B_{\mu_{i}\nu_{j}}\big{(}x(s_{i},t_{j})\big{)}\frac{\partial
x^{\mu_{i}}}{\partial t}\frac{\partial x^{\nu_{j}}}{\partial
s}\Big{|}_{(s_{i},t_{j})}\right)\right\rVert\Delta s_{i}\Delta t_{j}$ (B.4)

converges (cf Section 8.10 in [We64]). However, the limit of the sum of these
terms over all partitions is exactly the definition of the Riemann integral of
the (smooth) function $\mathcal{B}$ over the unit cube

$\lim_{P}\sum_{(i,j)\in
P}\left\lVert\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s_{i},t_{j}})}}\left(B_{\mu_{i}\nu_{j}}\big{(}x(s_{i},t_{j})\big{)}\frac{\partial
x^{\mu_{i}}}{\partial t}\frac{\partial x^{\nu_{j}}}{\partial
s}\Big{|}_{(s_{i},t_{j})}\right)\right\rVert\Delta s_{i}\Delta
t_{j}=\iint\displaylimits_{[0,1]\times[0,1]}\mathcal{B}(s,t)\;dsdt.$ (B.5)

Since the integral exists, the sum converges, and hence the product converges.
∎

###### Proof of Proposition 3.61.

For the first claim, set $M_{n}$ to be the maximum value of the norm of the
expressions of the form
$\alpha_{a_{56}^{s}a_{46}^{s}a_{36}^{s}a_{35}^{t}a_{25}^{s}}(B_{14}).$ Then

$\left\lVert{{\mathrm{triv}}}_{n,k}\right\rVert\leq\left(\frac{M_{n}}{n^{2}}\right)^{k}\binom{n^{2}}{k}=M_{n}^{k}\cfrac{\binom{n^{2}}{k}}{n^{2k}}\leq\frac{M_{n}^{k}}{k!}$
(B.6)

for all $k\in\\{0,1,\dots,n^{2}\\}.$ For the second claim, we note that
$\displaystyle\lim_{n\to\infty}M_{n}$ converges to a finite value due to the
arguments preceding the proof of Proposition 3.57 in this appendix since the
sequence of such terms converges, upon refinement, to a term of the form
$\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s,t})}}\big{(}B(s,t)\big{)}.$
Set

$\tilde{M}:=\max_{(s,t)}\left\lVert\underline{\alpha_{{{\mathrm{triv}}}(\gamma_{s,t})}}\big{(}B(s,t)\big{)}\right\rVert,$
(B.7)

which is finite and well-defined by compactness and smoothness of all inputs.
For each $\epsilon>0,$ let $\tilde{N}_{\epsilon}\in{{\mathbb{N}}}$ be large
enough so that $|M_{n}-\tilde{M}|<\epsilon$ for all
$n\geq\tilde{N}_{\epsilon}.$ Choose any $\epsilon>0$ and set
$M:=\tilde{M}+\epsilon$ and
$N:=\max\left\\{\tilde{N}_{\epsilon},\left\lfloor\sqrt{2k}\right\rfloor+1\right\\}.$
From this it follows that

$\left\lVert{{\mathrm{triv}}}_{n,k}\right\rVert\leq\left(\frac{M}{n^{2}}\right)^{k}\binom{n^{2}}{k}=M^{k}\cfrac{\binom{n^{2}}{k}}{n^{2k}}\leq\frac{M^{k}}{k!}$
(B.8)

for all $k\in\\{0,1,\dots,n^{2}\\}$ for all $n\geq N.$ The final claim follows
from the triangle inequality and the fact that $\epsilon$ can be taken to be
arbitrarily small. ∎

##  Appendix C Proof of configurations Lemma

This appendix serves to give a rigorous proof of Lemma 3.72. For the proof of
this Lemma, it is useful to rewrite $S_{n,k}$ as

$\begin{split}S_{n,k}&=\frac{1}{k!}\sum_{2n-1\geq i_{k}\neq
i_{k-1}\neq\cdots\neq i_{1}\geq 1}l_{n}(i_{1})\cdots l_{n}(i_{k})\\\
&=\frac{1}{k!}\sum_{i_{k}=1}^{2n-1}l_{n}(i_{k})\sum_{\begin{subarray}{c}i_{k-1}=1\\\
i_{k-1}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{k-1})\cdots\sum_{\begin{subarray}{c}i_{2}=1\\\
i_{2}\neq i_{3}\\\ \vdots\\\ i_{2}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{2})\sum_{\begin{subarray}{c}i_{1}=1\\\
i_{1}\neq i_{2}\\\ \vdots\\\ i_{1}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{1}),\end{split}$ (C.1)

where it is understood that any sum operation on the left acts on everything
to the right. Before working out this summation to obtain a more explicit
formula, for each $n\in{{\mathbb{Z}}}^{+},$ define the function

$\begin{split}\\{1,2,\dots,2n-1\\}&\xrightarrow{\phi_{n}}{{\mathbb{Z}}}\\\
p&\mapsto\phi_{n}(p):=\sum_{i=1}^{2n-1}l_{n}(i)^{p}.\end{split}$ (C.2)

Explicitly, this can be calculated as follows [We].

$\phi_{n}(p)=2\sum_{q=1}^{n}q^{p}-n^{p}=\frac{2}{p+1}\left(\sum_{q=1}^{p+1}(-1)^{\delta_{qp}}\binom{p+1}{q}B_{p+1-q}n^{q}\right)-n^{p},$
(C.3)

where $\delta_{qp}$ is the Kronecker delta function and $B_{r}$ is the
Bernoulli number defined, for instance, by the power series expansion (thought
of as a formal power series in the variable $x$)

$\frac{x}{e^{x}-1}=\sum_{r=0}^{\infty}\frac{B_{r}x^{r}}{r!}.$ (C.4)

The first few of these Bernoulli numbers are

$\begin{split}B_{0}&=1\\\ B_{1}&=-\frac{1}{2}\\\ B_{2}&=\frac{1}{6}\\\
B_{3}&=0\\\ B_{4}&=-\frac{1}{30}\end{split}$ (C.5)

while the first few $\phi_{n}$ are

$\begin{split}\phi_{n}(1)&=n^{2}\\\ \phi_{n}(2)&=\frac{n(n^{2}+1)}{3}\\\
\phi_{n}(3)&=\frac{n^{2}(n^{2}+1)}{2}\\\
\phi_{n}(4)&=\frac{n(6n^{4}+10n^{2}-1)}{15}.\end{split}$ (C.6)

Examining $\phi_{n}(p)$ a little more, one immediately notices the crucial
result

$\lim_{n\to\infty}\frac{\phi_{n}(p)}{n^{2p}}=0\qquad\mbox{ for }p\geq 2.$
(C.7)

Now, $S_{n,k}$ can be written as a polynomial in the $\phi_{n}$’s

$\begin{split}S_{n,k}&=\frac{1}{k!}\sum_{i_{k}=1}^{2n-1}l_{n}(i_{k})\sum_{\begin{subarray}{c}i_{k-1}=1\\\
i_{k-1}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{k-1})\cdots\sum_{\begin{subarray}{c}i_{2}=1\\\
i_{2}\neq i_{3}\\\ \vdots\\\ i_{2}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{2})\left[\phi_{n}(1)-\sum_{j_{1}=2}^{k}l_{n}(i_{j_{1}})\right]\\\
&=\frac{1}{k!}\sum_{i_{k}=1}^{2n-1}l_{n}(i_{k})\cdots\sum_{\begin{subarray}{c}i_{3}=1\\\
i_{3}\neq i_{4}\\\ \vdots\\\ i_{3}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{3})\left[\phi_{n}(1)^{2}-\phi_{n}(2)-2\phi_{n}(1)\sum_{j_{1}=3}^{k}l_{n}(i_{j_{1}})\right.\\\
&\qquad\qquad\qquad\qquad\qquad\qquad\left.+\sum_{j_{2}=3}^{k}l_{n}(i_{j_{2}})^{2}+\sum_{j_{2}=3}^{k}\sum_{j_{1}=3}^{k}l_{n}(i_{j_{2}})l_{n}(i_{j_{1}})\right]\\\
&=\frac{1}{k!}\sum_{i_{k}=1}^{2n-1}l_{n}(i_{k})\cdots\sum_{\begin{subarray}{c}i_{4}=1\\\
i_{4}\neq i_{5}\\\ \vdots\\\ i_{4}\neq
i_{k}\end{subarray}}^{2n-1}l_{n}(i_{4})\left[*_{n,4}\right],\end{split}$ (C.8)

where

$\begin{split}*_{n,4}&=\phi_{n}(1)^{3}-3\phi_{n}(1)\phi_{n}(2)+2\phi_{n}(3)+3\Big{(}\phi_{n}(2)-\phi_{n}(1)^{2}\Big{)}\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})\\\
&+3\phi_{n}(1)\left(\
\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})^{2}+\sum_{j_{2}=4}^{k}\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})l_{n}(i_{j_{2}})\right)\\\
&-2\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})^{3}-3\sum_{j_{2}=4}^{k}\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})^{2}l_{n}(i_{j_{2}})-\sum_{j_{3}=4}^{k}\sum_{j_{2}=4}^{k}\sum_{j_{1}=4}^{k}l_{n}(i_{j_{1}})l_{n}(i_{j_{2}})l_{n}(i_{j_{3}}),\end{split}$
(C.9)

and so on (a more explicit formula will be given momentarily). For example,
one obtains the following expressions for small values of $k$:

$\begin{split}S_{n,1}&=\phi_{n}(1)\\\
S_{n,2}&=\frac{1}{2!}\Big{(}\phi_{n}(1)^{2}-\phi_{n}(2)\Big{)}\\\
S_{n,3}&=\frac{1}{3!}\Big{(}\phi_{n}(1)^{3}-3\phi_{n}(1)\phi_{n}(2)+2\phi_{n}(3)\Big{)}\\\
S_{n,4}&=\frac{1}{4!}\Big{(}\phi_{n}(1)^{4}-6\phi_{n}(1)^{2}\phi_{n}(2)+8\phi_{n}(1)\phi_{n}(3)+3\phi_{n}(2)^{2}-6\phi_{n}(4)\Big{)}\\\
S_{n,5}&=\frac{1}{5!}\Big{(}\phi_{n}(1)^{5}+10\phi_{n}(1)^{3}\phi_{n}(2)+20\phi_{n}(1)^{2}\phi_{n}(3)+15\phi_{n}(1)\phi_{n}(2)^{2}\\\
&\qquad\qquad-30\phi_{n}(1)\phi_{n}(4)-20\phi_{n}(2)\phi_{n}(3)+24\phi_{n}(5)\Big{)}\end{split}$
(C.10)

Looking back at the expressions for $S_{n,k},$ one sees that there is a
recursion relation for $S_{n,k}.$ Setting $S_{n,0}:=1,$ this recursion
relation reads

$S_{n,k}=\frac{1}{k}\sum_{j=1}^{k}(-1)^{j+1}S_{n,k-j}\phi_{n}(j).$ (C.11)

This recursion relation can be used to express $S_{n,k}$ purely in terms of
the $\phi_{n}$’s and is given by

$S_{n,k}=\sum_{j_{1}=1}^{k}\sum_{j_{2}=1}^{k=j_{1}}\sum_{j_{3}=1}^{k-j_{1}-j_{2}}\cdots\sum_{j_{k}=1}^{k-j_{1}-j_{2}-\cdots-
j_{k-1}}\frac{(-1)^{k+j_{1}+j_{2}+\cdots+j_{k}}\phi_{n}(j_{1})\phi_{n}(j_{2})\cdots\phi_{n}(j_{k})}{k(k-j_{1})(k-j_{1}-j_{2})\cdots(k-j_{1}-j_{2}-\cdots-
j_{k-1})},$ (C.12)

where it is understood that the sum terminates earlier if any of the $j$’s are
larger than $1.$ For example, if there are $s$ of them, then

$\sum_{r=1}^{s}j_{r}=k.$ (C.13)

Therefore, fixing $k,$ one obtains

$\lim_{n\to\infty}\cfrac{\prod_{r=1}^{s}\phi_{n}(j_{r})}{\phi_{n}(1)^{k}}=\begin{cases}0&\mbox{
if }s<k\\\ 1&\mbox{ if }s=k\end{cases}.$ (C.14)

To see this, first notice that $j_{r}=1$ for all $r\in\\{1,\dots,s\\}$ when
$s=k$ in which case the denominator and numerator in (C.14) are equal and the
limit is $1.$ However, when $s<k,$ by the formula for $\phi_{n}(p)$ in (C.3)
and the asymptotics of this given in (C.7),

$\begin{split}\lim_{n\to\infty}\cfrac{\prod_{r=1}^{s}\phi_{n}(j_{r})}{\phi_{n}(1)^{k}}&=\lim_{n\to\infty}\cfrac{\prod_{r=1}^{s}\phi_{n}(j_{r})}{n^{2k}}\\\
&=\lim_{n\to\infty}\cfrac{\prod_{r=1}^{s}\phi_{n}(j_{r})}{n^{2(j_{1}+\cdots+j_{s})}}\\\
&=\lim_{n\to\infty}\prod_{r=1}^{s}\frac{\phi_{n}(j_{r})}{n^{2j_{r}}}\\\
&=0.\end{split}$ (C.15)

Hence,

$\lim_{n\to\infty}\frac{k!S_{n,k}}{n^{2k}}=1.$ (C.16)

Finally going back to $R_{n,k}$ and using this fact gives

$\begin{split}\lim_{n\to\infty}R_{n,k}&=\lim_{n\to\infty}\frac{k!S_{n,k}}{n^{2}(n^{2}-1)\cdots(n^{2}-k+1)}\\\
&=\lim_{n\to\infty}\frac{k!S_{n,k}}{n^{2k}\left(1-\frac{1}{n^{2}}\right)\cdots\left(1-\frac{k-1}{n^{2}}\right)}\\\
&=1.\end{split}$ (C.17)

## Index of notation

Notation | Name/description | Location | Page
---|---|---|---
$\mathcal{C}$ | category/2-category |  Def 2.1/2.11 |  2.1/2.11
$G$ | group |  Def 2.4 | 2.4
${{\mathbb{B}}}G$ | one-object groupoid |  Ex 2.4 | 2.4
$\sigma:F\Rightarrow G$ |  | natural transformation
---
from functor $F$ to $G$
Def 2.7/2.25 |  2.7/2.25
$\mathcal{G}\equiv(H,G,\tau,\alpha)$ | crossed module |  Def 2.36 | 2.36
${{\mathbb{B}}}\mathcal{G}$ | one-object 2-groupoid |  Ex 2.45 | 2.45
$A$ | 1-form potential |  Eq (3.5) | 3.5
$M$ | smooth manifold |  Sec 3.1 | 3.1
$\mathds{1}$ | identity matrix |  Eq (3.7) | 3.7
${{\mathrm{triv}}}(\gamma)$ |  local transport along a path $\gamma$ |  Eq (3.8) | 3.8
$\mathcal{P}^{1}M$ |  path groupoid of $M$ |  After Eq 3.14 | 3.14
$g$ | (thin) gauge transformation |  Eq (3.17) | 3.17
$\mathcal{P}^{2}M$ |  path 2-groupoid of $M$ |  Before Def 3.26 | 3.26
${{\mathrm{triv}}}(\Sigma)$ |  local transport along a bigon $\Sigma$ |  Eq (3.29)&(3.84) |  3.29&3.84
$B$ | 2-form potential |  Eq (3.35) | 3.35
$F$ |  2-form curvature of $A$ |  Eq (3.40) | 3.40
$\varphi$ | fat gauge transformation |  Eq (3.100) | 3.100
$H$ |  3-form curvature of $(A,B)$ |  Eq (3.134) | 3.134
$(\mathfrak{h},\mathfrak{g},\underline{\tau},\underline{\alpha})$ | differential crossed module |  Appendix A | A.1

## References