# Spatially Resolved Observations of Betelgeuse at $\lambda$7 mm and
$\lambda$1.3 cm Just Prior to the Great Dimming

L. D. Matthews11affiliation: Massachusetts Institute of Technology Haystack
Observatory, 99 Millstone Road, Westford, MA 01886 USA & A. K.
Dupree22affiliation: Center for Astrophysics $|$ Harvard & Smithsonian, 60
Garden Street, MS-15, Cambridge, MA 02138 USA

(Accepted to ApJ)

###### Abstract

We present spatially resolved observations of Betelgeuse ($\alpha$ Orionis)
obtained with the Karl G. Jansky Very Large Array (VLA) at $\lambda\sim$7 mm
(44 GHz) and $\lambda\sim$1.3 cm (22 GHz) on 2019 August 2, just prior to the
onset of the historical optical dimming that occurred between late 2019 and
early 2020. Our measurements suggest recent changes in the temperature and
density structure of the atmosphere between radii $r\sim 2$–$3R{\star}$. At
$\lambda$7 mm the star is $\sim$20% dimmer than in previously published
observing epochs between 1996–2004. We measure a mean gas temperature of
$T_{B}=2270\pm 260$ K at $r\sim 2.1R{\star}$, where $R_{\star}$ is the
canonical photospheric radius. This is $\sim 2\sigma$ lower than previously
reported temperatures at comparable radii and $>$1200 K lower than predicted
by previous semi-empirical models of the atmosphere. The measured brightness
temperature at $r\sim 2.6R_{\star}$ ($T_{B}=2580\pm 260$ K) is also cooler
than expected based on trends in past measurements. The stellar brightness
profile in our current measurements appears relatively smooth and symmetric,
with no obvious signatures of giant convective cells or other surface
features. However, the azimuthally averaged brightness profile is found to be
more complex than a uniform elliptical disk. Our observations were obtained
approximately six weeks before spectroscopic measurements in the ultraviolet
revealed evidence of increases in the chromospheric electron density in the
southern hemisphere of Betelgeuse, coupled with a large-scale outflow. We
discuss possible scenarios linking these events with the observed radio
properties of the star, including the passage of a strong shock wave.

###### Subject headings:

Stellar chromospheres (230); Stellar photospheres (1237); Red Giant stars
(1372); Stellar properties (1624); Stellar surfaces (1632)

##  1\. Introduction

The red supergiant Betelgeuse ($\alpha$ Orionis) has been the recent subject
of intense observational scrutiny owing to an unprecedented optical dimming of
the star observed in late 2019 and early 2020 (see Calderwood 2021). While the
visible light curve of Betelgeuse is known to undergo semi-regular variations
in optical brightness on timescales of $\sim$300-500 days and $\sim$2000 days,
respectively (e.g., Kiss et al. 2006), the “Great Dimming” of 2019/2020 (which
reached its brightness minimum from 2020 February 7–13; Guinan et al. 2020)
marked the faintest appearance of the star in nearly 200 years of recorded
photometry. Two hypotheses that emerged to explain the dramatic dimming are
recent dust formation or a reduction in photospheric temperature—or perhaps a
combination of these effects.

Based on optical spectrophotometric measurements of TiO from 2020 February,
Levesque & Massey (2020) argued that the dimming of the star could not be
explained by a temperature decrease alone. They found $T_{\rm eff}=3600\pm 25$
K, only marginally cooler than seen in measurements made in 2004 ($T_{\rm
eff}=3650\pm$25 K). These authors proposed instead episodic mass loss, coupled
with an increase in large-grain circumstellar dust along the line-of-sight.
However, using higher resolution spectra of atomic lines, Začs & Pukītis
(2021) found evidence for increased macroturbulence during the Great Dimming,
coupled with a statistically significant temperature decrease to $T_{\rm
eff}\approx 3500$ K. From the analysis of Wing TiO and near-infrared
photometry spanning five years, Harper et al. (2020) argued that no new dust
is required, and that the dimming can be explained by the presence of
photospheric inhomogeneities with a large covering fraction ($\geq$50%) and a
mean effective temperature$\geq$250 K cooler than expected for an M2 Iab
supergiant. A similar scenario was suggested by Dharmawardena et al. (2020),
who measured a flux decrease of $\sim$20% at submillimeter wavelengths during
the Great Dimming compared with measurements between 2007–2017 and concluded
that the observed flux change could not be explained by dust, but may result
either from changes in the temperature and/or radius of the star. Based on
aperture polarimetry, Cotton et al. (2020) found that polarization appeared
during the Great Dimming that could be explained by photospheric asymmetries
and/or obscuration by grains. Meanwhile George et al. (2020) have argued that
the Great Dimming may have been linked with a critical shift in the pulsation
dynamics of Betelgeuse.

Several key insights into the behavior of Betelgeuse near the time of the
Great Dimming were provided by spatially resolved ultraviolet (UV)
spectroscopy from the Hubble Space Telescope (HST), spanning several epochs in
2019 and 2020 (Dupree et al. 2020). Measurements of the Mg ii $h$ and $k$
lines revealed evidence for the passage of a shock or pressure wave through
the southwestern portion of the star’s atmosphere between 2019 September and
November, a time when the photosphere was at its maximum outflow velocity
(relative to the mean) as a result of the phase of the 400-day pulsation cycle
(see Figure 2b of Dupree et al.). Analysis of the C ii line revealed that the
passage of this wave was accompanied by increases in the temperature and
electron density in the southern hemisphere. A scenario proposed by Dupree et
al. is that an exceptionally strong convective upwelling from the photosphere
(enhanced in strength by the phase of the pulsation cycle) led to a major
ejection event that launched material through the chromosphere, beyond which
it may have cooled sufficiently to allow the formation of dust. High spatial
resolution optical images obtained by Montargès et al. (2021) appear to
reinforce this picture, showing that the southern hemisphere of the star was
ten times darker during the Great Dimming compared with images obtained a year
earlier. The authors attribute the observed darkening to obscuration by a
newly formed dust clump. Using a tomographic analysis, Kravchenko et al.
(2021) found evidence for successive shock waves that could explain the
outflow detected by Dupree et al. (2020), but argued that an increase in
molecular opacity rather than dust formation best explains the subsequent
dimming.

Past studies of red supergiants at radio (centimeter, millimeter, and
submillimeter) wavelengths have shown that they are sources of continuum
emission that is predominantly free-free in origin. At wavelengths longer than
a few millimeters, a significant fraction of their continuum emission appears
to arise from a component of chromospheric gas at cooler temperatures than the
material that emits in the UV, and with a larger filling factor (Lim et al.
1998; Harper, Brown, & Lim 2001; O’Gorman et al. 2020). At submillimeter
wavelengths, blackbody emission from the stellar disk becomes increasingly
dominant. Because the radio emission is thermal and optically thick, the
Rayleigh-Jeans limit applies to the radiative transfer equation, and radio
observations with sufficient angular resolution to spatially resolve the stars
permit direct measurements of the stellar diameter and the mean gas (electron)
temperature (e.g., Reid & Menten 1997; Lim et al. 1998; Harper et al. 2001;
O’Gorman et al. 2017; see also Section 5.2). For these reasons, radio
observations provide a valuable complement to UV measurements for the study of
red supergiant chromospheres and the adjacent layers of the atmosphere.

In conjunction with our ongoing multi-cycle HST program (Dupree 2018) we were
awarded a single epoch of observations of Betelgeuse with the Karl G. Jansky
Very Large Array (VLA)111The VLA of the National Radio Astronomy Observatory
(NRAO) is operated by Associated Universities, Inc. under cooperative
agreement with the National Science Foundation. at wavelengths of $\lambda$1.3
cm and $\lambda$7 mm. These two wavelengths are of particular interest with
respect to the UV observations of Dupree et al. (2020), since they probe gas
at radii comparable to those giving rise the UV continuum, and just interior
to that region, respectively (Lim et al. 1998). Here we present the results of
these new VLA observations, which were carried out on 2019 August 2, just
prior to the Great Dimming and the appearance of the outflow event reported by
Dupree et al. (2020).

##  2\. Previous Radio Observations of Betelgeuse

Betelgeuse is classified as an M2Iab supergiant of variability class SRc (Kiss
et al. 2006). Its proximity (222${}^{+40}_{-34}$ pc; Harper et al. 2017) makes
the angular size of its photosphere the largest of any star visible from the
northern hemisphere ($\theta_{\star}$=44.2$\pm$0.2 mas at $\lambda$2.2$\mu$m;
Dyck et al. 2002).

Betelgeuse has been the target of radio wavelength studies for more than half
a century (Kellermann & Pauliny-Toth 1966; Seaquist 1967). Early hints that
the radio emission has a significant chromospheric contribution came from the
multi-frequency study of Altenhoff, Oster, & Wendker (1979). The first
observations of Betelgeuse with the VLA were published by Newell & Hjellming
(1982) at $\nu$=1.46, 4.89, 15.0, and 22.5 GHz. Although these latter
measurements did not spatially resolve the star, the inferred spectral index
$\alpha$=1.32 (where flux density $S_{\nu}\propto\nu^{\alpha}$) led the
authors to conclude that the radio emission must be thermal in origin, and
they suggested that it arises predominantly from an optically thick
chromosphere with an extent of a few stellar radii.

The first spatially resolved radio observations of Betelgeuse were obtained by
Skinner et al. (1997) using the Multi-Element Radio Linked Interferometer
Network (MERLIN) and the VLA at $\lambda$6 cm, confirming that the radio
emission is extended to several times the photospheric diameter, consistent
with a chromospheric origin. Subsequently Lim et al. (1998) were able to
spatially resolve Betelgeuse at several wavelengths between 7 mm and 6 cm
using the VLA. Those observations revealed the temperature structure of the
atmosphere to be complex and established that shorter radio wavelengths sample
material at successively smaller radii. At $r\sim 2R_{\star}$ (which
corresponds to unity optical depth at $\lambda$7 mm) they measured a
brightness (electron) temperature of 3450$\pm$850 K, roughly consistent with
the photospheric temperature; however the temperatures were seen to steadily
decrease with larger $r$, reaching 1370$\pm$330 K at $r\sim 7R_{\star}$. These
temperatures are well below the values of $\geq$4000–8000 K inferred from
measurements of chromospheric tracers in the optical and UV across similar
radii (e.g., Gilliland & Dupree 1996; Dupree et al. 2020), leading Lim et al.
to conclude that the hot chromospheric material probed by the UV must have a
relatively small filling factor. The 7 mm image presented by Lim et al. also
implied that radio emission from Betelgeuse was highly asymmetric, which the
authors attributed to the possible presence of giant convective cells.

Harper et al. (2001) used the observations of Lim et al. (1998) to develop a
semi-empirical model for the extended atmosphere of Betelgeuse with a mean
chromospheric temperature of 3800 K, significantly lower than other models
(cf. Hartmann & Avrett 1984). An updated version of this model was recently
discussed by O’Gorman et al. (2020), but across the range of radii of
relevance for comparsion with our VLA measurements, the difference is
negligible (G. Harper, private communication). The Harper et al. model
includes a region between $r\sim$1.6–1.9$R_{\star}$ where the temperature dips
below the mean effective temperature.222The values quoted here have been
radially scaled by a factor of (56/44) to account for the difference in
angular diameter of Betelgeuse’s photosphere adopted in the present paper. The
presence of this temperature minimum was first indicated by infrared
observations (Tsuji 2000, 2006) and was further confirmed by O’Gorman et al.
(2017) using spatially resolved submillimeter (338 GHz) observations from the
Atacama Large Millimeter/submillimeter Array (ALMA). The ALMA data of O’Gorman
et al. (2017) also showed evidence of inhomogeneities in the atmosphere which
the authors suggested may arise from localized heating.

The temporal behavior of Betelgeuse’s radio emission has been examined by
O’Gorman et al. (2015) using resolved imaging from the VLA and the Pie Town
antenna of the Very Long Baseline Array at wavelengths ranging from 7 mm to 6
cm. These authors found evidence for modest flux density variations with time
($~{}\hbox to0.0pt{$<$\hss}{\lower 4.30554pt\hbox{$\sim$}}\pm$25%; see also
Drake et al. 1992) that appear to mirror the star’s cyclic $V$ magnitude
variations—except at 7 mm where no evidence of variations was seen to within
measurement uncertainties. In addition to the cyclic variations, O’Gorman et
al. found that the flux density between $\lambda$1.3–6 cm decreased overall by
$\sim$20% between the 1970s and 1980s and the early 2000s, despite maintaining
a roughly constant spectral index.

Figure 1.—  CLEAN contour maps of Betelgeuse obtained on 2019 August 2 at 7 mm
(44 GHz; left) and 1.3 cm (22 GHz; right). Contour levels are ($-$10[absent],
10, 14.1, 20, 28.3, 40, 56.6, 80, 113.1, 160, 200)$\times 29.1\mu$Jy beam-1
(44 GHz) and ($-$10[absent], 10, 14.1, 20, 28.3, 40, 56.6, 80, 113.1, 160,
226, 319)$\times 15.7\mu$Jy beam-1 (22 GHz) where the lowest contours are
10$\sigma$. The adopted restoring beam is shown in the lower left of each
panel (see Table 2). The red ellipses indicate the best-fitting uniform
elliptical disk models (Table 3).

##  3\. Observations

On 2019 August 2 we carried out new continuum observations of Betelgeuse at 44
GHz (Q-band; $\lambda\sim 7$ mm) and 22 GHz (K-band; $\lambda\sim 1.3$ cm)
using the VLA in its most extended (A) configuration. Antenna separations
ranged from 0.68 km to 36.4 km. At $\lambda$7 mm, this configuration provided
angular resolution of $\sim$42 mas, comparable to the size of Betelgeuse’s
disk in the near infrared (see Section 2). At $\lambda$1.3 cm the angular
resolution was $\sim$80 mas.

The Q-band observations were obtained using the 3-bit observing mode and dual
circular polarizations. The WIDAR correlator was configured with four baseband
pairs tuned to contiguously cover a frequency window of $\sim$7.9 GHz,
centered near 44 GHz. Each baseband pair contained either 15 or 16 subbands,
each of which had a bandwidth of 128 MHz and 128 spectral channels. An
analogous setup was used for the K-band observations, but with a center
frequency of $\sim$22 GHz. To improve $u$-$v$ coverage, observations in the
two bands were interleaved during a 4.0 hour session, spanning approximately
14:45–18:45 UTC. Total integration time on Betelgeuse was $\sim$62 min in Q
band and $\sim$42 min in K band and the observed elevation of the star ranged
from 43 deg to 63 deg. Data were recorded with 2-second time resolution.

The observations were carried out during the late morning and early afternoon
local time. Weather conditions ranged from clear to partly cloudy with wind
speeds $\leq$4.2 m s-1. The RMS atmospheric phase fluctuations reported at the
VLA site were 6–7 deg during the first half of the observations and 12–17 deg
during the second half. Antenna pointing corrections were evaluated hourly
using observations of a strong point source at X-band (8 GHz). 3C48 was
observed in both K and Q band to allow calibration of the absolute flux
density scale (see Section 4). To calibrate the atmospheric phases, fast
switching was used between Betelgeuse and two neighboring gain calibrators:
J0532+0732 (“cal 1”) and J0552+0313 (“cal 2”), which lie at projected
separations of $5^{\circ}\,.58$ and $4^{\circ}\,.22$, respectively, from the
star. For Q-band, the adopted observing sequence was: [$t_{\rm cal1},(t_{\rm
star},t_{\rm cal2})\times 4,t_{\rm star},t_{\rm cal1}...$], where $t_{\rm
cal1}=$ 44 sec, $t_{\rm star}=$46 sec, and $t_{\rm cal2}=$38 sec. For K-band
the sequence was: [$t_{\rm cal1},(t_{\rm star},t_{\rm cal2})\times 2,t_{\rm
star},t_{\rm cal1}...$], where $t_{\rm cal1}=$ 42 sec, $t_{\rm star}=$ 132
sec, and $t_{\rm cal2}=$ 36 sec.

##  4\. Data Reduction

Data reduction and calibration were performed using the Astronomical Image
Processing System (AIPS; Greisen 2003). The original archival science data
model files were loaded into AIPS using the Obit software package (Cotton
2008). However, the default calibration (‘CL’) table was regenerated to update
the gain and opacity information, and antenna positions were updated to the
best available values.

After flagging visibly corrupted data, a requantizer gain correction was
applied using the AIPS task TYAPL. Instrumental delays were corrected via
fringe fitting to a 1-minute segment of 3C48 data, with separate delay
solutions determined for each of the four independent basebands. Bandpass
calibration was performed in the standard manner using J0532+0732 as the
calibrator; a spectral index of $\alpha\approx$0 was assumed (Healey et al.
2007).

The flux density calibrator 3C48 was known to be undergoing a flare during
2019333https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fdscale;
therefore special care was taken with the absolute flux density calibration of
our data. Clean component models of 3C48 appropriate for the two observing
bands were obtained from NRAO via anonymous ftp.444ftp.aoc.nrao.edu, directory
pub/staff/rperley/MODELMAPS2019. The models were derived by R. Perley from
observations of 3C48 in the VLA A configuration in 2019 October and were used
to set the absolute flux density scale, assuming the standard coefficients of
Perley & Butler (2017). We found that the differences in the derived flux
density scales based on the 2019 October Perley models compared with the
default (non-flaring) models for 3C48 were $\sim$2% in Q band and $\sim$3% in
K band, respectively. Thus uncertainty in the absolute flux calibration scale
resulting from the flare is expected to be quite small.

As a check that the 2019 October Perley model is suitable for calibrating our
measurements, we imaged 3C48 using our new data and found the source structure
to be virtually indistinguishable from that seen in the Perley model images,
consistent with minimal change in the source at K and Q band frequencies
between the date of our observations (2019 August 2) and the date of Perley’s
observations (2019 October 24). As a final check on the robustness of our flux
density scale, we compared our VLA measurements of the bandpass/complex gain
calibrator J052+0732 with a 91.5 GHz measurement from the ALMA obtained on
2019 July 26 with an independent absolute flux density calibrator. Because of
the flat spectral index of the source, we expect similar flux densities at
91.5 GHz, 44 GHz, and 22 GHz, and consistent with this, the reported ALMA flux
density was 1.22$\pm$0.06 Jy555Value taken from the ALMA Calibrator Source
Catalogue; https://almascience.nrao.edu/sc/., in good agreement with our VLA
measurements (Table 1).

Table 1Calibration Sources

Source |  $\alpha$(J2000.0) |  $\delta$(J2000.0) | Flux Density (Jy) |  $\nu$ (GHz)
---|---|---|---|---
3C48a | 01 37 41.2994 | 33 09 35.133 | 0.6134∗ | 44.0
… | … | … | 1.2194∗ | 22.0
J0532+0732b | 05 32 38.9985 | 07 32 43.346 | 1.131$\pm$0.034 | 44.0
… | … | … | 1.172$\pm$0.009 | 22.0
J0552+0313c | 05 52 50.1015 | 03 13 27.243 | 0.569$\pm$0.020 | 44.0
… | … | … | 0.734$\pm$0.006 | 22.0

**footnotetext: Adopted value, calculated at the frequency in column 5 using
the 2019 October clean component models from R. Perley and the coefficients
from Perley & Butler (2017; see Section 4).

aafootnotetext: Flux density calibrator.

bbfootnotetext: Complex gain and bandpass calibrator.

ccfootnotetext: Complex gain calibrator.

Note. — Units of right ascension are hours, minutes, and seconds, and units of
declination are degrees, arcminutes, and arcseconds. Explanation of columns:
(1) source name; (2) & (3) right ascension and declination (J2000.0); (4)
derived flux density in Jy at the frequency indicated in column 5; (5)
frequency at which the flux density in the fourth column was computed.

Calibration of the frequency-independent portion of the complex gains was
performed in the standard manner using the measurements of the gain
calibrators J0532+0732 and J0552+0313. Initially, phase-only corrections were
solved for using these sources and applied to Betelgeuse using linear
interpolation in time. This was followed by similarly computed amplitude and
phase corrections. The rms scatter in the phase solutions for the two
calibrators was $<$$\sim$5 deg in Q band and $<$$\sim$2 deg in K band,
respectively, for all antennas.

Following application of the computed gain solutions to Betelgeuse, an
additional iteration of phase-only self-calibration was performed for each of
the bands using the emission from Betelgeuse itself, with 30 s solution
intervals. This step resulted in improvements in the rms noise of of images of
18% and 37%, respectively, for Q and K bands, and the correlated amplitude of
the star increased by 52% and 19%, in the two bands, respectively, following
the self-calibration.

At this stage, optimized weights for the visibility data were calculated using
the AIPS task REWAY. Finally, the data in each band were averaged in frequency
to produce 8 spectral channels in each of the 62 subbands.

Images of Betelgeuse were produced from the fully calibrated data using CLEAN
deconvolution as implemented in AIPS. Robust weighting with R=0 was adopted,
along with a circular restoring beam (see Table 2). Images of the star at the
two observing wavelengths are shown in Figure 1. The images appear relatively
smooth and symmetric at both wavelengths (but see also Section 5). Note that
the apparent six-pointed shape seen in the outer contours of the 1.3 cm image
in Figure 1 is an artifact caused by imperfect deconvolution of the VLA’s
dirty beam, which has a six-pointed pattern whose first sidelobes overlap with
the outskirts of the stellar emission distribution in this band.

Table 2CLEAN Image Properties

Band | $\nu_{\rm GHz}$ | $\theta_{\rm a}$ | $\theta_{\rm b}$ | PA | $\theta_{\rm circ}$ | RMS noise
---|---|---|---|---|---|---
| (GHz) | (mas) | (mas) | (degrees) | (mas) | ($\mu$Jy beam-1)
Q | 44 | 43.8 | 40.7 | $+13.7$ | 42.0 | 29.0
K | 22 | 87.0 | 73.8 | $+10.2$ | 80.0 | 15.9

Note. — CLEAN images were produced using robust (${\cal R}$=0) weighting.
$\nu_{\rm GHz}$ is the center frequency in GHz. $\theta_{\rm a}$ and
$\theta_{\rm b}$ are the FWHM major and minor axes, respectively, of the dirty
beam. The position angle (PA) of the dirty beam was measured east from north.
The images presented in Figure 1 were produced using circular restoring beams
with the FWHM diameter $\theta_{\rm circ}$ (column 6); these values represent
the geometric mean of the dirty beam dimensions in columns 3 & 4.

Table 3Measured Stellar Parameters

Band | $\nu$ | $\theta_{\rm maj}$ | $\theta_{\rm min}$ | PA | $e$ | $S_{\nu}$ | $D$ | $T_{b}$
---|---|---|---|---|---|---|---|---
| (GHz) | (mas) | (mas) | (degrees) |  | (mJy) | (AU) | (K)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
Uniform Elliptical Disk Fits to Visibility Data
Q | 43.9 | 96.9$\pm$2.9 (0.2) | 92.6$\pm$2.8 (0.2) |  $111.0\pm$5.1 (1.2) | 0.04$\pm$0.04 | 22.34$\pm$2.41 (0.03) | 21.1 | 2270$\pm$260
K | 22.0 | 118.8$\pm$5.9 (0.3) | 114.5$\pm$5.7 (0.3) | 54.7$\pm$5.2 (1.5) | 0.04$\pm$0.07 | 9.64$\pm$0.97 (0.01) | 25.9 | 2580$\pm$260
Elliptical Gaussian Fits to Image Data
Q | 43.9 | 66.8$\pm$2.1 (0.5) | 63.6$\pm$2.0 (0.5) | 111.0$\pm$7.8 (6.0) | 0.05$\pm$0.04 | 23.0$\pm$2.5 (0.01) | … | …
K | 22.0 | 79.4$\pm$4.0 (0.6) | 73.4$\pm$3.7 (0.6) | 67.5$\pm$6.6 (4.3) | 0.07$\pm$0.07 | 9.63$\pm$1.04 (0.01) | … | …

Note. — Quoted uncertainties include formal, systematic, and calibration
errors (see Text for details) but not uncertainties in the stellar distance.
For measured quantities the value in parentheses indicates the error budget
contribution from formal fitting uncertainties. For a uniform elliptical disk,
$\theta_{\rm maj}$ and $\theta_{\min}$ are the major and minor axis
dimensions, respectively; for a Gaussian fit they represent the the FWHM
dimensions of the elliptical Gaussian after deconvolution from the dirty beam.
Explanation of columns: (1) observing band; (2) mean observing frequency; (3)
FWHM diameter of the major axis in mas; (4) FWHM diameter of the minor axis in
mas; (5) PA of the major axis in degrees, measured east from north; (6)
ellipticity, defined as $e=(\theta_{\rm maj}-\theta_{\rm min})/(\theta_{\rm
maj})$; (7) flux density in mJy; (8) mean size in AU, derived using the
geometric mean angular diameter and assuming a distance of 222 pc; (9)
brightness temperature in Kelvin, as derived assuming a uniform elliptical
disk morphology (see Equation 1).

##  5\. Results

###  5.1. Measured Stellar Parameters

To characterize the size and flux density of Betelgeuse in our two VLA
observing bands we performed fits of a two-dimensional (2D) uniform elliptical
disk to the visibility data using the AIPS OMFIT task. As a consistency check,
we also fitted elliptical Gaussian models to CLEAN deconvolved images (see
Table 2 and Figure 1) using the AIPS JMFIT task. Results are summarized in
Table 3. The quoted uncertainties include contributions from the formal
fitting uncertainties (Condon 1997), as well as from calibration and
systematic errors (see Matthews, Reid, & Menten 2015 for details). The
dominant source of uncertainty in the derived flux densities is the absolute
calibration uncertainty, assumed to be 10% in both observing
bands.666https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/fdscale.

At both observed wavebands the angular diameter measurements are consistent to
within uncertainties with previous spatially resolved observations of
Betelgeuse at comparable wavelengths (Lim et al. 1998; O’Gorman et al. 2015).
As expected for optically thick emission, we measure a larger angular size for
the star at the longer of our two observing wavelengths. The mean shape of the
radio emitting surface was nearly circular during the epoch of our
observations, with an ellipticity of at most a few per cent. The mean angular
diameters (taken as the geometric mean of the major and minor axes measured
from the visibility data) are 94.7 mas at $\lambda$7 mm and 116.6 mas at
$\lambda$1.3 cm, respectively, corresponding to $\sim 2.1$ and $\sim 2.6$
times the near-infrared diameter of the star (see Section 2). For comparison,
the angular extent of chromospheric emission traced by the UV “continuum”
(centered near $\lambda$2500 Å) is approximately 125$\pm$5 mas, or $\sim
2.8R_{\star}$ in radial units (Gilliland & Dupree 1996)777Near this wavelength
the UV “continuum” from Betelgeuse is dominated by a blend of numerous
emission lines (see e.g., Figure 10 of Brandt et al. 1995) and its angular
extent will be dependent on the exact filter passband used., while
chromospheric Mg ii emission lines are detected out to projected radii of
$\sim$48–135 mas ($\sim$1.1–6$R_{\star}$; Uitenbroek et al. 1998; Dupree et
al. 2020).

While the measured angular size of Betelgeuse at both $\lambda$7 mm and
$\lambda$1.3 cm is consistent with past observations, the total flux densities
in both observing bands are notably lower than previously published values.
Among six published $\lambda$1.3 cm measurements between 1996–2004 (Lim et al.
1998; O’Gorman et al. 2015), only one (8.96$\pm$0.24 mJy; obtained in 2002
April) was comparably faint as our current measurement. O’Gorman et al. (2015)
found that the 1.3 cm flux density appears to be correlated with the optical
($V$) magnitude, and the 2002 measurement corresponded to a phase when the
star was near its minimal optical brightness during its 400-day pulsation
cycle ($\phi\approx$0.5). However, at the time of our latest measurements,
Betelgeuse was at an intermediate phase ($\phi\approx$0.09). There is no
optical photometry available during a window from approximately 100 days
preceding our VLA measurements on 2019 August 2 until 10 days after that date,
since the star was a daytime object for ground-based observers. Based on an
extrapolation of the light curve presented by Dupree et al. (2020), the
estimated $V$ magnitude is $\sim$0.65, which is within the typical range for
this phase. To our knowledge, no published 1.3 cm measurements are available
between 2005–2019, so the more recent behavior of the star at this wavelength
is unknown.

At $\lambda$7 mm, the discrepancy between our new measurement and previous
results is even more pronounced. Based on four measurements taken between 1996
and 2004, the 7 mm flux density of Betelgeuse remained nearly constant
(28.4$\pm 0.5$ mJy, where the error bar indicates the measurement dispersion;
Lim et al. 1998; O’Gorman et al. 2015). However, the 7 mm flux density we
measure in 2019 August is $\sim$20% (2$\sigma$) lower, even after accounting
for calibration uncertainties.

O’Gorman et al. (2015) noted that despite evidence for a systematic dimming of
Betelgeuse at radio wavelengths, its spectral index has remained effectively
constant over several decades: $\alpha=1.33\pm 0.02$. Based on our current
measurements we find $\alpha=1.22\pm 0.22$. Thus to within uncertainties the
spectral index remains consistent with previous measurements, although it is
poorly constrained since we have only two observing bands.

###  5.2. Brightness Temperature

Because the radio emission from Betelgeuse is optically thick and our VLA
observations are spatially resolved, it is possible to use the angular sizes
and flux density measurements from Table 3 to derive a brightness temperature
that provides a direct measurement of the mean gas (electron) temperature via
the Rayleigh-Jeans expression:

$T_{B}=2S_{\nu}c^{2}(4.25\times 10^{-28})/(k\nu^{2}\pi\theta_{\rm
maj}\theta_{\rm min})$ (1)

where $S_{\nu}$ is in mJy, $\theta_{\rm maj}$ and $\theta_{\rm min}$ are the
dimensions of the best-fitting disk model in units of mas (see Table 3), $c$
is the speed of light in cm s-1, $k$ is the Boltzmann constant in cgs units,
and $\nu$ is the mean observing frequency in GHz. At $\lambda$7 mm we find
$T_{B}=2270\pm 260$ K (corresponding to $r\sim 2.1R_{\star}$) and at
$\lambda$1.3 cm $T_{B}=2580\pm$260 K (corresponding to $r\sim 2.6R_{\star}$).
The derived brightness temperature at $\lambda$1.3 cm is at the low end of
previously observed values at this wavelength (O’Gorman et al. 2015) and is
markedly cooler than the value of $\sim$3300 K reported at this wavelength by
Lim et al. (1998). At $\lambda$7 mm, the $T_{B}$ value measured in 2019 August
is the lowest ever reported at this wavelength. In comparison, Lim et al.
(1998), measured a brightness temperature at $\lambda$7 mm of 3450$\pm$850 K
(based on data from 1996), comparable to the mean photospheric temperature of
Betelgeuse, while O’Gorman et al. (2015) reported values of 3040$\pm$80 K,
2760$\pm$200 K, 2940$\pm$140 K for data obtained in 2000, 2003, and 2004,
respectively. The semi-empirical model of Harper et al. (2001) (which was
originally based on the Lim et al. measurements) predicts a characteristic
temperature at $r\sim 2.1R_{\star}$ of $\sim$1500 K hotter than seen in our
current data (see Figure 4, discussed below). Indeed, the temperature that we
measure based on our new 7 mm data is only slightly warmer than that within
the molecular layer known as the “MOLsphere”, which extends from
$r\sim$1.3–1.5$R_{\star}$ and has an estimated temperature of $\sim$1500-2000
K (Tsuji 2000, 2006). We discuss the implications of these temperature
measurements further in Section 5.4.

###  5.3. Deviations from a Uniform Disk

####  5.3.1 Evidence for a Multi-component Brightness Profile

The uniform elliptical disk fits we have employed above are useful for
providing a characterization of the mean properties of the star (angular size,
temperature). However, the high signal-to-noise ratio of our current
data—roughly an order of magnitude higher than previous radio images of
Betelgeuse at comparable wavelengths—allows us to see clear evidence of
deviation from this simple model.

In Figure 2 we plot the azimuthally averaged visibility data (real and
imaginary parts) as a function of $u$-$v$ distance for our two observing
bands, with the best-fitting uniform elliptical disk models overplotted. In
each band we see that the uniform elliptical disk model under-represents the
observed flux on the smallest observed angular scales (longest baselines)
while systematically slightly over-predicting the flux density on scales
sampled by intermediate baselines ($\sim$0.5–1.5M$\lambda$).

If we subtract the best-fitting uniform elliptical disk model from the
visibility data and Fourier transform the residuals to make an image, an outer
“ring” of positive residuals is visible in both the 7 mm and 1.3 cm data,
surrounding a negative trough and a positive central peak coincident with the
star’s center (Figure 3, top row). The peak residuals are $~{}\hbox
to0.0pt{$<$\hss}{\lower 4.30554pt\hbox{$\sim$}}12\sigma$ in the 7 mm image and
$~{}\hbox to0.0pt{$<$\hss}{\lower 4.30554pt\hbox{$\sim$}}6.5\sigma$ in the 1.3
cm image.

The fits to both the 7 mm and 1.3 cm data are improved by the introduction of
a second component to the model. Results of a two-component elliptical disk
model (Model 1) and an elliptical disk+ring model (Model 2) are shown on the
upper panels of Figure 2 as dashed lines and dotted lines, respectively. The
model parameters are summarized in Table 4. Both of these models result in a
significant reduction of the residuals in the image domain (Figure 3, middle
and lower panels), although for the 7 mm data both models still underpredict
the visibility amplitude at large $u$-$v$ distances ($<5$M$\lambda$; Figure
2), with the discrepancy being larger for the disk+ring model. At $\lambda$1.3
cm the fit quality for Models 1 and 2 is similar. We stress that neither of
these two-component models is unique, nor do they have an obvious underlying
physical underpinning. However, it is noteworthy that because the two
components are concentric, the presence of a large convective cell (cf. Lim et
al. 1998) or other transient surface feature cannot readily account for
presence of the second component.

It is unclear whether this multi-component brightness profile is unique to our
2019 observations, as previous spatially resolved observations of Betelgeuse
at comparable wavelengths were obtained prior to the VLA’s sensitivity upgrade
(Perley et al. 2011), making the unambiguous identification of similar
features more difficult. However, hints of deviation from a simple uniform
elliptical disk model are seen in fits to earlier data (cf. Figure 2 of
O’Gorman et al. 2015). In addition, O’Gorman et al. (2017) also found a multi-
component model necessary to fit their 2015 ALMA 338 GHz observations of
Betelgeuse, and a similar behavior was seen in 7 mm observations of the nearby
carbon star IRC+10216 (Matthews et al. 2018). In general the identification of
such trends is important for constraining models of the electron temperature
as a function of radius in red supergiant atmospheres (see, e.g., Figure 3 of
Harper et al. 2001) and underscores the unique power of high-sensitivity,
spatially resolved radio observations.

####  5.3.2 No Evidence for Giant Convective Cells

Based on their 7 mm VLA observations, Lim et al. (1998) reported a significant
brightness asymmetry that they interpreted as possible evidence of a large
convective cell in the atmosphere of Betelgeuse. They noted that such features
may be linked with upwellings of cool photospheric gas and may play a role in
the mass-loss process of red supergiants. Indeed, it has been suggested that
the recent Great Dimming may have been caused by a mass-loss event tied to the
surfacing of a giant convective cell (Montargès et al. 2021). However, our VLA
measurements from 2019 August do not reveal any direct evidence for the
presence of such features; we see neither morphological evidence of giant
“cells” nor signatures of enhanced radio emission. As discussed in Sections
5.1 and 5.3.1, the star appears largely symmetric and uniform, although the
imaginary parts of the visibility data (Figure 2, right panels) do show some
low-level fluctuations that may reflect small brightness variations across the
star. One possible explanation is that any observable effects of a
photospheric convective cell within the regions sampled by our observations
had already disappeared after “surfacing” between 2019 January–April.

###  5.4. Discussion

As shown in Figure 4, the brightness temperature values that we derive from
our 2019 August 2 observations of Betelgeuse at $\lambda$7 mm and $\lambda$1.3
cm lie significantly below the predictions of the semi-empirical model of
Harper et al. (2001). The compendium of Betelgeuse radio measurements
presented by O’Gorman et al. (2015) shows that within the period from
1996–2004, the 1.3 cm flux density appears to have systematically decreased by
$\sim$20% compared with the earlier measurements of Newell & Hjellming (1982)
(which spanned 1972–1981) and Drake et al. (1992) (which spanned 1986–1990).
The physical mechanism that might explain such systematic dimming is unclear,
although it may be linked with long-term changes in the underlying dynamics of
the star (e.g., George et al. 2020). To our knowledge, no radio light curve
data for Betelgeuse are available between 2005 and mid-2019. Therefore we are
unable to determine conclusively whether our present measurements point to a
continued, systematic dimming trend of Betelgeuse at cm wavelengths, reflect
normal temporal variations—or alternatively, may be linked with atmospheric
changes leading up to the UV outburst detected by Dupree et al. (2020) and/or
the subsequent Great Dimming. Here we briefly discuss this latter possibility.

As discussed in the previous section, our VLA observations of Betelgeuse on
2019 August 2 do not appear to show any unambiguous morphological signatures
of an ongoing atmospheric disturbance in early 2019 August, roughly one month
prior to the onset of the outflow event documented by Dupree et al. (2020).
Despite this, our radio measurements are consistent with the possibility that
the atmosphere of Betelgeuse may have undergone marked physical changes near
the time of the outburst. Indeed, the anomalously low brightness temperatures
that we measure between $r\sim 2$–$3R{\star}$, as well as the apparent
temperature inversion within this radial range, point to significant changes
density and/or ionization fraction within this region compared with previous
epochs.

As briefly described in Section 1, the tomographic analysis of Kravchenko et
al. (2021) revealed evidence for the the occurrence of successive shock waves
through the photosphere of Betelgeuse in 2018 February and 2019 January, with
the initial shock serving to amplify the second one. Subsequently, spatially
resolved spectroscopic measurements in the UV by Dupree et al. (2020) revealed
that a shock or pressure wave passed through the southwestern portion of
Betelgeuse’s chromosphere between 2019 September and November. (The exact time
of the onset of this latter event is not precisely known, but evidence for a
disturbance was not seen during the previous epoch of UV observations in 2019
March). Concurrent measurements of the C ii lines at 2325 and 2328Å also point
to increases in the temperature and electron density in the southern
hemisphere of Betelgeuse. Importantly, the UV measurements of Betelgeuse’s
chromosphere from Dupree et al. (including the Mg ii lines and continuum)
probe approximately the same range of projected radii as our current VLA
observations ($\sim$2–3$R_{\star}$). These events are therefore also expected
to impact the radio continuum emission near the same time period.

As discussed by Reid & Menten (1997), the propagation of pulsationally-induced
shocks through the extended atmosphere of a red giant ($r~{}\hbox
to0.0pt{$>$\hss}{\lower 4.30554pt\hbox{$\sim$}}2R_{\star}$) may produce
temperature and density variations as a function of radius that may in turn
lead to variations in the observed radio emission. In a simple version of this
model, the temperature as a function of radius, $T(r_{0})$, predicted by a
hydrostatic model of the extended atmosphere becomes instead:

$T(r_{0})=\left<T(r_{0})\right>+\Delta T_{s}{\rm
cos}\left(\phi_{p}+\phi_{s}\right)$ (2)

where $\Delta T_{s}$ is the amplitude of the temperature contrast between the
pre- and post-shock gas, $\phi_{p}$ is the stellar phase, and $\phi_{s}$ is
the phase of the propagating disturbance. Following Reid & Menten, if one
assumes a shock propagation speed $V_{s}$ and a pulsation period $P$, then
$\phi_{s}$ can be defined as

$\phi_{s}=2\pi\frac{r-R_{\star}}{V_{s}P}$ (3)

Figure 4 (dotted line) shows an example of the effect of this toy model on the
radial temperature profile. We based the shape of the input temperature
profile on the semi-empirical model atmosphere of Harper et al. (2001), but
have applied a temperature offset of $-1000~{}K$ (thin solid line). Here we
have adopted $P$=400 days (comparable to Betelgeuse’s “short” pulsation
period; see Section 1), $V_{s}$=7.0 km s-1, and $\Delta T_{s}$=250 K. We show
the results for a single value of the shock excitation phase, $\phi_{p}$=0.09.
The exact combination of parameters is somewhat arbitrary. Nonetheless, this
toy model qualitatively illustrates that a disturbance of this kind may be
able to account for: (i) a (temporary) distortion of the temperature profile,
including a possible temperature reversal between the radial regions sampled
by the 7 mm and 1.3 cm radio emission, respectively; (ii) corresponding
modulations in the radio flux densities; and (iii) temperature and electron
density fluctuations across the chromospheric regions probed by the HST UV
measurements of Dupree et al. 2020). One important test of this scenario will
be follow-up observations of Betelgeuse at radio wavelengths to assess whether
the star has since returned to its historical flux density levels.

Table 4Multi-Component Model Fit Parameters

Comp. | E offset (mas) | N offset (mas) |  $\theta_{a}$ (mas) |  $\theta_{b}$ (mas) | PA (deg) |  $S_{\nu}$ (mJy)
---|---|---|---|---|---|---
Model 1 ($\lambda$=7 mm)
Uniform Elliptical Disk | 0.25$\pm$0.22 | 0.09$\pm$0.21 | 125.6$\pm$1.6 | 116.5$\pm$1.3 | $-33.6\pm 1.7$ | 12.5$\pm$0.5
Uniform Elliptical Disk |  $-$0.24$\pm$0.16 | 0.05$\pm$0.13 | 78.3$\pm$0.1 | 69.1$\pm$0.1 | $-93.7\pm 1.2$ | 10.6$\pm 0.6$
Model 2 ($\lambda$=7 mm)
Uniform Elliptical Disk |  $-$0.07$\pm$0.08 | 0.01$\pm$0.07 | 81.2$\pm$0.8 | 76.1$\pm$0.8 | $95.6\pm 1.1$ | 17.2$\pm 0.6$
Elliptical Ring | 0.29$\pm$0.26 |  $-$0.01$\pm$0.03 | 110.5$\pm$1.4 | 101.6$\pm$1.1 | 147.6$\pm 1.7$ | 5.9$\pm$0.2
Model 1 ($\lambda$=1.3 cm)
Uniform Elliptical Disk | 0.26$\pm$0.12 | 0.70$\pm$0.15 | 98.8$\pm$1.9 | 91.6$\pm$2.4 | 174.7$\pm$2.0 | 6.5$\pm$0.3
Uniform Elliptical Disk | $-1.20\pm 0.49$ | $-3.6\pm 0.63$ | 183.3$\pm$5.3 | 166.8$\pm$4.3 | 120.0$\pm$2.4 | 3.3$\pm$0.3
Model 2 ($\lambda$=1.3 cm)
Uniform Elliptical Disk | 0.03$\pm$0.08 | 0.06$\pm$0.09 | 98.9$\pm$2.0 | 92.9$\pm$2.2 | 77.5$\pm$1.7 | 7.7$\pm$0.2
Elliptical Ring | $-1.19\pm 0.48$ | $-3.7\pm 0.65$ | 151.1$\pm$4.1 | 137.3$\pm$3.3 | 26.9$\pm$2.2 | 2.1$\pm$0.2

Note. — Quoted error bars include only formal fitting uncertainties. The
models are shown overplotted on the visibility data in Figure 2.

##  6\. Summary

We have presented spatially resolved VLA imaging observations of the radio
emission from Betelgeuse at wavelengths of 7 mm and 1.3 cm. The data were
obtained on 2019 August 2, just prior to the onset of the Great Dimming of the
star in late 2019 and early 2020. At both observed wavelengths the star
appears spherically symmetric with no evidence of large-scale brightness non-
uniformities. However, the brightness profile in both bands is found to be
more complex than a simple uniform elliptical disk.

We find the 7 mm flux density of Betelgeuse to be $\sim$20% lower than
previously published measurements during the past 25 years. The mean
brightness temperature derived from the 7 mm observations, which probes gas at
a mean projected radius of $r\sim 2.1R_{\star}$, is also significantly lower
compared with previous measurements at comparable radii and lies $>$$\sim$1200
K below the prediction of published semi-empirical models of Betelgeuse’s
extended atmosphere. The brightness temperature derived from the 1.3 cm data
(which probes material at $r\sim 2.6R_{\star}$) is also lower than expected
based on trends in historical measurements. Our new data therefore suggest
recent changes in the density and/or temperature structure of the atmosphere
between $\sim 2$–$3R_{\star}$. One possible explanation is the recent passage
of a large-amplitude shock or pressure wave through the outer atmosphere. This
picture can also account for atmospheric disturbances revealed by UV imaging
spectroscopy by Dupree et al. (2020) during the two months following the VLA
observations. Such an event may be linked to a large-scale mass ejection from
the star that has been postulated as an explanation for the steep decline in
optical magnitude associated with the Great Dimming.

The observations presented here were part of NRAO program SQ0474. This work
was supported by Space Telescope Science Institute Grant HST-GO-15641.014-A.
We thank G. Harper for helpful discussions.

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Figure 2.— Real (top) and imaginary (bottom) parts of the visibility amplitude
vs. baseline length for observations of Betelgeuse at $\lambda$7 mm (left) and
$\lambda$1.3 cm (right). On the upper panels the solid red lines indicate the
best-fitting uniform elliptical disk models from Table 3. The two-component
models from Table 4 are overplotted as black dashed lines (Model 1) and dotted
lines (Model 2).

Figure 3.— Residual maps obtained by imaging the $\lambda$7 mm data (left) and
$\lambda$1.3 cm data (right) from 2019 August 2, after subtraction of model
fits from the visibilities. Top row: residuals from best-fitting uniform
elliptical disk model (see Table 3); Middle row: residuals from a model
comprising two uniform elliptical disk components (Model 1, Table 4); Bottom
row: residuals from a model comprising a uniform elliptal disk and elliptical
ring (Model 2, Table 4). Intensity units are $\mu$Jy beam-1. Contours are
[$-12,-8.5,-6,-4.2,-3,3,...,12]\times\sigma_{\nu}$ where $\sigma_{\nu}$ is the
1$\sigma$ RMS noise in the respective band (Table 2). Negative contours are
shown as dashed lines.

Figure 4.— Schematic illustration of the effects of a propagating shock wave
on the radial temperature profile of the extended atmosphere of Betelgeuse.
The assumed shape of the underlying temperature profile is based on the semi-
empirical model of Harper et al. (2001). The thick solid line shows the
temperature predicted by the original model; the thin solid line shows the
same model with a temperature offset of $-$1000 K. The dotted line shows the
temperature perturbations to the latter model resulting from a periodic
disturbance with $P$=400 days, amplitude $\Delta T_{s}=\pm 250$ K, and
propagation speed $V_{s}=$7.0 km s-1 (see text for details). The triangle
symbols indicate the brightness temperatures derived from the 2019 August 2
VLA measurements.


$\sigma_{1}=[v_{0},...,v_{n}]$ and $\sigma_{2}=[w_{0},...,w_{n}]$. Then

$\displaystyle a(2\sigma_{1}-\sigma_{2})=2a(\sigma_{1})-a(\sigma_{2})$ (A.1)

Given any $n$-cochain $a$, we can create an $n+1$ cochain $\text{d}a$ via:

$\displaystyle\text{d}a([v_{0},...,v_{n+1}])=\sum_{i=0}^{n+1}(-1)^{i}a([v_{0},...,\hat{v_{i}},...,v_{n+1}])$
(A.2)

where $\hat{v_{i}}$ means that we omit that index. One can then see that
$\text{d}(\text{d}a)=0$. This allows us to construct cohomology groups as the
cohomology of the complex of $n$-chains:

$\displaystyle\leftarrow C^{n}\leftarrow C^{n-1}...~{}C^{1}\leftarrow
C^{0}\leftarrow 0$ (A.3)

here the maps are just given by $d_{n}$, eg $d$ acting on elements of $C^{n}$,
and the cohomology groups are just $\Im~{}d_{n}/\ker d_{n}$.

In the case that the target space is a ring, we have an additional structure
called the cup product. The cup product takes an $m$-form $a_{m}$ and an
$n$-form $b_{n}$ and returns an $n+m$ form $a\smile b$, defined by:

$\displaystyle\ \ \ \ a_{m}\smile b_{n}([v_{0},...,v_{n+m}])$
$\displaystyle=a_{m}([v_{0},...,v_{m}])b_{n}([v_{m},...,v_{m+n}])$ (A.4)

Furthermore, when considered on cohomology classes, this cup product is graded
anticommutative. This means that there is a $n+m-1$ cochain $c_{m+n-1}$ such
that:

$\displaystyle a_{m}\smile b_{n}=(-1)^{nm}b_{n}\smile a_{m}+\text{d}c_{m+n-1}$
(A.5)

To get an explicit expression for $c_{m+n-1}$, we need to introduce higher cup
product $a_{m}\underset{{\scriptscriptstyle k}}{\smile}b_{n}$ which gives rise
to a $(m+n-k)$-cochain[89]:

$\displaystyle\ \ \ \ a_{m}\underset{{\scriptscriptstyle
k}}{\smile}b_{n}([0,1,\cdots,m+n-k])$
$\displaystyle=\hskip-10.00002pt\sum_{0\leq i_{0}<\cdots<i_{k}\leq
n+m-k}\hskip-30.00005pt(-)^{p}a_{m}([0\to i_{0},i_{1}\to i_{2},\cdots])\times$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ b_{n}([i_{0}\to
i_{1},i_{2}\to i_{3},\cdots]),$ (A.6)

and $a_{m}\underset{{\scriptscriptstyle k}}{\smile}b_{n}=0$ for $k<0$ or for
$k>m\text{ or }n$. Here $i\to j$ is the sequence $i,i+1,\cdots,j-1,j$, and $p$
is the number of permutations to bring the sequence

$\displaystyle 0\to i_{0},i_{1}\to i_{2},\cdots;i_{0}+1\to i_{1}-1,i_{2}+1\to
i_{3}-1,\cdots$ (A.7)

to the sequence

$\displaystyle 0\to m+n-k.$ (A.8)

For example

$\displaystyle a_{m}\underset{{\scriptscriptstyle 1}}{\smile}b_{n}([0\to
m+n-1])=\sum_{i=0}^{m-1}(-)^{(m-i)(n+1)}\times$ $\displaystyle a_{m}([0\to
i,i+n\to m+n-1])b_{n}([i\to i+n]).$ (A.9)

We can see that $\underset{{\scriptscriptstyle 0}}{\smile}=\smile$. Unlike cup
product at $k=0$, the higher cup product of two cocycles may not be a cocycle.
For cochains $a_{m},b_{n}$, we have[89]

$\displaystyle\text{d}(a_{m}\underset{{\scriptscriptstyle
k}}{\smile}b_{n})=\text{d}a_{m}\underset{{\scriptscriptstyle
k}}{\smile}b_{n}+(-)^{m}a_{m}\underset{{\scriptscriptstyle
k}}{\smile}\text{d}b_{n}+$ (A.10) $\displaystyle\ \ \
(-)^{m+n-k}a_{m}\underset{{\scriptscriptstyle
k-1}}{\smile}b_{n}+(-)^{mn+m+n}b_{n}\underset{{\scriptscriptstyle
k-1}}{\smile}a_{m}$

The above result also allows us to see that the cup product interacts with d
in the familiar way:

$\displaystyle\text{d}(a_{m}\smile b_{n})=(\text{d}a_{m})\smile
b_{n}+(-1)^{m}a_{m}\smile\text{d}b_{n}$ (A.11)

which we can interpret as the Leibniz rule. In the case that there is no
boundary, we can interpret this as yielding a form of integration by parts, so
that $\text{d}a_{m}\smile b_{n}=-(-1)^{m}a_{m}\smile\text{d}b_{n}$. We will
abbreviate the cup product with using ‘$\smile$,’ so that $ab\equiv a\smile
b$.

##  Appendix B Physical Interpretation of Level Shift Symmetry

In the main text, we saw that the fact that the cocycle was a surface term for
our model implied that the bulk correlation functions of local operators are
identical to the trivial case, and that this implied that the critical
exponents for the SF-tMI transition were the same as for the SF-MI transition.
Now we discuss why we might expect that on general grounds.

We wish to consider the SF-tMI transition, and we begin in the tMI phase.
Rather than directly breaking the symmetry in the tMI, suppose that we stack
the tMI with a trivial MI. We can then break the $U(1)$ symmetry in the
trivial state, which involves a phase transition with the usual critical
exponents. After the symmetry is broken in the trivial phase, we couple in the
tMI. With the provision of symmetry breaking, the tMI can be trivialized, and
also reduced to a trivial state. The critical exponents for this transition
are thus the same as the usual SF-MI transition.

This argument shows that there must always be a SF-tMI critical point with the
same exponents as the SF-MI critical point. What we have shown in this paper
is that the XY SF-tMI transition, in the absence of a background gauge field,
is precisely this transition.

A similar argument may be applied to any SPT, thus showing that there exists a
symmetry-breaking transition out of the SPT that has the same exponents as the
symmetry breaking transition of a symmetric trivial state. On the mathematical
side, this reflects the fact that all group cocycles $\nu$ have the form
$\nu=d\mu$, where $\mu$ may not be $G$-symmetric, i.e. that all SPTs are
trivial in the absence of symmetry.

##  Appendix C Cech-Deligne-Beilinson Approach to Chern-Simons Theory

###  C.1 Introduction

It is quite rare in theoretical physics for a system to be both exactly
soluble and well defined. When we find a system that is, like the harmonic
oscillator, we tend to make extensive use of it. Topological Field Theories
(TFTs) are indeed exactly soluble, but their realization as a field integral
is difficult to define, while the mathematical approaches to TQFTs tend to
abandon the field integral entirely. In this Appendix, we propose a method to
define TQFTs on a spacetime lattice that remains exactly soluble. We argue
that we can reduce the field integral to a few real integrals and sums that
can be evaluated using standard methods of calculus, and we calculate the
partition function exactly in several example cases. While we largely focus
here on Chern-Simons theories and $\mathbb{Z}_{N}$ gauge theories in three
spacetime dimensions, the method we have developed can be extended to
topological field theories in any dimension.

The model we present is a lattice gauge theory. Gauge invariance is central to
the calculations we present, and will allow us to calculate exact partition
functions on various manifolds. In that sense, the integral measure—and hence
any Hilbert space of a related Hamiltonian formulation—is not a product of
local integrals. This stands in contrast to the models presented in the main
body of this thesis.

Throughout this Appendix, as in Chapters 5-7, we use a normalization so that
the cycles of $U(1)$ variables are quantized to unity. This means that for a
scalar field $\phi$, $\phi\sim\phi+1$, and that the naïve level-$k$ Chern-
Simons Lagrangian is $\pi kada$. One can always return to the usual
normalization by replacing $\phi\to\phi/2\pi,a\to a/2\pi$.

Furthermore, we freely use notation from Differential Geometry and Algrebraic
Topology [82, 51]. When working in the continuum, we represent differential
forms as, e.g., $a\equiv a_{\mu}dx^{\mu}$. We will make extensive use of the
wedge product without writing it, so that $ab\equiv
a_{\mu}b_{\nu}dx^{\mu}dx^{\nu}$. When working on the lattice, we will denote
lattice indices by latin letters $i,j,k,...$ and dual lattice indices greek
letters $\alpha,\beta,\gamma,...$ Like in the continuum case, we will
extensive use of the cup product without writing it, though we symmetrize it
as we describe in Section LABEL:sec:Sum.

The best way to demonstrate our formalism is by example. In this section, we
demonstrate the calculation of the partition function of Chern-Simons Theory
in three spacetime dimensions on $T^{2}\times[0,1]$. We assume that the
lattice is cubical. The generalization to other lattices and spacetimes of
general topology will be treated in the subsequent sections.

Figure C.1: _(color online)_ a) and b) A gauge field $A$ may only be expressed
as a one-form $a^{\alpha}$ in open sets $U_{\alpha}$, represented by the
dashed lines. To define physical quantities, such as an action, we must divide
the manifold into closed sets with overlaps of zero measure, within which we
use a particular one-form $a^{1},a^{2},...$. Here we denote such a choice by
the solid black lines. We refer to this choice as a _division_. On the
lattice, the division is exactly the lattice cells. c) and d) Schematic
depiction of two choices of division, with open sets not drawn. In order to
ensure that the physical action is independent of choice of division, we have
to add extra terms to the action along the black 1-cell and 0-cell boundaries.
It is this amended action that we are able to put on the lattice.

To define gauge theories on a lattice, we rely on a simple general idea from
continuum field theory. The basic principle that we review in Section
LABEL:sec:Classical is that $U(1)$ gauge fields can only be defined as one-
forms on contractible patches, while on overlaps they differ by a gauge
transformation. To define physical quantities, such as an action or a
Lagrangian, we cut the open patches into closed sets with overlaps of zero
measure, as shown in Figure C.1. At these lower-dimensional overlaps, we must
add extra terms to the Lagrangian to ensure that the Lagrangian is independent
of how we cut the open patches into closed sets. Here we will simply quote the
resulting Lagrangian, with a derivation to follow.

We focus only on square and cubic lattices. We denote zero-cells of the dual
lattice by $V_{\alpha}^{0}$, one-cells of the dual lattice by
$V^{1}_{\alpha\beta}\equiv V_{\alpha}^{0}\cap V_{\beta}^{0}$, two-cells of the
dual lattice as $V^{2}_{\alpha\beta\gamma\sigma}\equiv V_{\alpha}^{0}\cap
V_{\beta}^{0}\cap V_{\gamma}^{2}\cap V_{\sigma}^{2}$, and so forth. We assume
that $V^{1}_{\alpha\beta}$ inherits its orientation from $V_{\alpha}^{0}$,
$V^{2}_{\alpha\beta\gamma\sigma}$ inherits its orientation from
$V^{1}_{\alpha\beta}$, etc.

The crux of our lattice formalism is each lattice cell $V_{\alpha}^{0}$
carries its own gauge field $a^{\alpha}_{ij}$ that lives on the links of the
cell. When a link is shared by two lattice cells
$V_{\alpha}^{0},V_{\beta}^{0}$, we do not require the gauge fields in the
lattice cells to be equal, but allow them to differ by a gauge transformation:

$\displaystyle a^{\alpha}_{ij}-a^{\beta}_{ij}=(d\phi^{\alpha\beta})_{ij}\neq
0$ (C.1)

A gauge field in our formulation is the collection $a^{\alpha}$, together with
the $\phi^{\alpha\beta}$ that glue cells together. One can do the same thing
for a scalar field—on overlaps we require that a scalar field differ by an
integer $\theta^{\alpha}_{i}-\theta^{\beta}_{i}=m^{\alpha\beta}\in\mathbb{Z}$.
Below we formalize these ideas for both a scalar field and a gauge field.

####  C.1.1 Definition of a Scalar Field on the Lattice

Let us examine first how this works for a $U(1)$ scalar field $\Theta$. Recall
that we enumerate lattice sites by $i,j,k...$ and dual lattice sites by
$\alpha,\beta,\gamma...$.

In our formalism, a scalar field $\Theta$ consists of:

  * •

On each lattice cell (i.e. zero-cell of the dual lattice), a zero cochain
$\theta^{\alpha}_{i}$

  * •

One each overlap of lattice cells (i.e. one-cell of the dual lattice), a
constant111To unify notation with the situation of a gauge field $A$ described
below, one may consider a integer constant to be a “$-1$” form, as
$\mathbb{Z}\to U(1)\to 0$ is exact. $m^{\alpha\beta}\in\mathbb{Z}$

such that:

  * •

On each one-cell of the dual lattice,

$\displaystyle\theta^{\alpha}_{i}-\theta^{\beta}_{i}=m^{\alpha\beta}$ (C.2)

  * •

On each two-cell of the dual lattice

$\displaystyle
m^{\alpha\beta}+m^{\beta\sigma}-m^{\alpha\gamma}-m^{\gamma\sigma}=0$ (C.3)

where $\alpha<\beta<\gamma<\sigma$. One can in fact derive (C.3) from (C.2).
Finally, a scalar field comes with the following redundancy:

$\displaystyle\theta^{\alpha}_{i}\to\theta^{\alpha}_{i}+\ell^{\alpha}$ (C.4)
$\displaystyle m^{\alpha\beta}\to m^{\alpha\beta}+\ell^{\alpha}-\ell^{\beta}$

In our formalism, a scalar field $\Theta$ consists of both the collection of
zero-cochains $\theta^{\alpha}_{i}$ and constants $m^{\alpha\beta}$. We will
often write this as $\Theta=(\theta^{\alpha},m^{\alpha\beta})$.

Let us unpack the physical picture behind these definitions. Most of the
lattices we will refer to in this paper—such as square lattices, cubic
lattices, and so forth—carry a one-to-one correspondence of points to lattice
cells. Denote the lattice cell (i.e. zero-cell of the dual lattice)
corresponding to the $ith$ lattice point as $\alpha(i)$. Then one can fix the
redundancy (C.4) by requiring that $\theta^{\alpha(i)}_{i}\in[0,1)$. Thus at
each lattice site, we have an angular variable $\theta^{\alpha(i)}_{i}$, but
between lattice sites we also have an integer $m^{\alpha(i)\alpha(j)}$. The
‘continuum’ picture to have in mind is that of of scalar field $\theta$ that
has the value $\theta^{\alpha(i)}_{i}$ at $i$ and $\theta^{\alpha(j)}_{j}$ at
$j$, but winds $m^{\alpha\beta}$ times between $i$ and $j$. For instance, the
winding number about a nontrivial cycle is just the sum of the
$m^{\alpha\beta}$ along that cycle.

In the conventional lattice formulation of a scalar field
$\tilde{\theta}_{i}$, we lack the integers $m^{\alpha\beta}$, and in lieu of
(C.4) we impose the redundancy
$\tilde{\theta}_{i}\to\tilde{\theta}_{i}+\ell_{i}$, $\ell_{i}\in\mathbb{Z}$.
As such, a typical ‘kinetic’ term is
$-\cos(\tilde{\theta}_{i}-\tilde{\theta}_{j})$ and a typical ‘potential’ term
would be $-\cos(\tilde{\theta}_{i})$. Instead, in our regularization, the
difference of any two field values in the same patch $\alpha$ is well defined
as a value in $\mathbb{R}$, and so a typical ‘kinetic term’ might be

$\displaystyle-\frac{1}{2}(\theta_{i}^{\alpha(i)}-\theta_{j}^{\alpha(i)})^{2}=-\frac{1}{2}(\theta_{i}^{\alpha(i)}-\theta_{j}^{\alpha(j)}+m^{\alpha(i)\alpha(j)})^{2}$
(C.5)

On the other hand, a ‘potential’ term with just a single $\theta^{\alpha}_{i}$
must still be periodic:

$\displaystyle-\cos(\theta^{\alpha(i)}_{i})$ (C.6)

The inclusion of the integers $m^{\alpha\beta}$ are why this regularization
differs radically from the usual, but they are also what will render our model
exactly soluble.

####  C.1.2 Definition of a Gauge Field on the Lattice

A gauge field $A$ in our approach consists of:

  * •

On each lattice cell (i.e. zero cell of the dual lattice), a one cochain
$a^{\alpha}_{ij}$

  * •

On each overlap (i.e. one-cell of the dual lattice), a scalar field
$\Phi^{\alpha\beta}_{i}=(\phi^{\alpha\beta}_{i},m^{\alpha\beta\gamma\sigma})$.

such that:

  * •

on each one-cell of the dual lattice:

$\displaystyle a^{\alpha}_{ij}-a^{\beta}_{ij}=(d\phi^{\alpha\beta})_{ij}$
(C.7)

  * •

on each two-cell of the dual lattice:

$\displaystyle\phi^{\alpha\beta}_{i}+\phi^{\beta\sigma}_{i}-\phi^{\alpha\gamma}_{i}-\phi^{\gamma\sigma}\equiv
n^{\alpha\beta\gamma\sigma}\in\mathbb{Z}$ (C.8)

Every lattice gauge field comes equipped with the gauge redundancy:

$\displaystyle a^{\alpha}_{ij}\to a^{\alpha}_{ij}+(d\theta^{a})_{ij}$ (C.9)
$\displaystyle\phi^{\alpha\beta}_{i}\to\phi^{\alpha\beta}_{i}+\theta^{\alpha}_{i}-\theta^{\beta}_{i}$

where $\theta^{\alpha}_{i}$ is a scalar field independently defined on each
lattice cell.

As in the case of a scalar field, our treatment of a gauge field is radically
different from the usual. For one, the flux through a plaquette is unbounded,
whereas typical treatments of lattice field theory have flux that is only
defined modulo $2\pi$. Again, this property is essential in making our
approach well defined.

Furthermore, one can show [4] that on any closed, oriented two-dimensional
surface $S$, the sum of $da$ over the generator of $H_{2}(S)$ is given by the
sum of the $n^{\alpha\beta\gamma\sigma}$ over $S$. In other words, flux is
still quantized on closed surfaces. This will be a useful tool in the
partition functions below.

###  C.2 Lattice Lagrangians

We use the Alvarez approach [4] to derive lattice Lagrangians that can
evaluate the action for gauge fields $A$ that are only defined by one-forms
$a^{\alpha}$ locally. In our lattice formalism, we translate these Lagrangians
to the lattice in the most naïve possible way: substitute the continuum
differential “$d$” for the lattice differential “$d$”, replace the wedge
product “$\wedge$” with a symmetrized cup product “$\cup$,” and exchange
integration for evaluation against restrictions of the generator of the top
cohomology of the manifold.

Figure C.2: _(color online)_ a) Example of a cubic lattice on two cells. For
more larger cubic lattices, we number lattice points first in the $x$
direction, then in the $y$ direction, and lastly in the $z$ direction, as we
have done for this two cell lattice. b) Visual representation of the
symmetrized cup product of a one-cochain and a two cochain. c) The interwoven
branching structures of the lattice (black) and dual lattice (pink), where we
use the self-duality of the cubic lattice to assign a branching structure to
the dual lattice.

We focus on the case of a cubic lattice on $S^{1}\times S^{1}\times I$ where
$I=[0,1]$. Labeling the three directions $x,y,$ and $z$, we assume that
lattice sites have been numbered by proceeding first in the $x$ direction,
then in the $y$ direction, and then in the $z$ direction. To each lattice
cell, we assign the number of the site in the $(-\hat{x},-\hat{y},-\hat{z})$
corner. See Figure C.2. In later sections, we can remove this awkward
requirement by phrasing it as a certain branching structure condition, which
will be essential to generalize approach to more complicated manifolds.

For boundary conditions, we take the gauge field to have zero flux on
$T^{2}\times\\{0\\}$ and $T^{2}\times\\{1\\}$, as well as identical holonomy
at both times. The zero flux condition is typical and represents a truncation
of the Hilbert space to ‘low-energy states’, while the requirement of
identical holonomy reflects the fact that the partition function is a trace.

We we also need a cyclic symmetrization of the usual cup product on this cubic
lattice. Let $\omega_{ij}$ be a one-cochain and $\nu_{ijkl}$ a two cochain on
our cubic lattice. Then we define the cup product as:

$\displaystyle(\omega\cup\nu)_{ijklmnop}=\omega_{ij}\nu_{ikmo}+\omega_{ik}\nu_{imjn}+\omega_{im}\nu_{ijkm}$
(C.10)

This may be understood as a symmetrization of the usual cup product over
cyclic permutations of indices; a visual demonstration is given in Figure
C.2b. As before, we will always drop the cup “$\cup$.”

Figure C.3: a) Schematic depiction of the Wilson line when $A$ is only
represented by one-forms $a^{\alpha}$ locally. The corresponding picture on
the lattice, where $A$ is represented by one-cochains $a^{\alpha}_{ij}$ in
each lattice cell.

We will also need the expression for the holonomy. Consider the situation in
Figure C.3. In each closed interval, we use a particular one form $aa$,
however where two closed sections intersect between regions $\alpha$ and
$\beta$ we must add a term $-\phi^{\alpha\beta}$, where
$a^{\alpha}-a^{\beta}=\phi^{\alpha\beta}$. So the continuum expression for a
Wilson line is

The action we obtain from the Alvarez procedure is:

$\displaystyle S=\pi
k\left\\{\sum_{\alpha}a^{\alpha}da^{\alpha}-\sum_{V_{\alpha\beta}}\phi^{\alpha\beta}da^{\beta}+\sum_{V_{\alpha\beta\gamma\sigma}}n^{\alpha\beta\gamma\sigma}a^{\sigma}\right.$
(C.11)
$\displaystyle\left.-\sum_{V_{\alpha\beta\gamma\sigma\delta\lambda\rho\zeta}}(n^{\alpha\beta\delta\lambda}\phi^{\lambda\zeta}-n^{\alpha\beta\gamma\sigma}\phi^{\sigma\zeta}-n^{\alpha\gamma\lambda\rho}\phi^{\rho\zeta})\right\\}$

where we sum over all lattice cubes $V_{\alpha}$, faces $V_{\alpha\beta}$,
edges $V_{\alpha\beta\gamma\sigma}$ and points
$V_{\alpha\beta\gamma\sigma\delta\lambda\rho\zeta}$. Each term is evaluated on
the appropriate restriction of the top cohomology of $T^{2}\times I$. For
example, consider the cube in Figure C.2c above. The terms in the action for
this cube are:

$\displaystyle
a^{1}_{12}(da^{1})_{1,4,7,10}+a^{1}_{14}(da^{1})_{1728}+a^{1}_{17}(da^{1})_{1245}-\phi^{12}_{1}(da^{\beta})_{1357}-\phi^{13}_{2}(da^{3})_{2637}-\phi^{15}(da^{5})_{4567}$
(C.12)
$\displaystyle+n^{1256}a^{6}_{57}+n^{1357}a^{7}_{67}+n^{1234}a^{4}_{37}-(n^{1256}\phi^{68}_{7}+n^{1234}\phi^{48}_{7}+n^{1367}\phi^{78}_{7})$

where
$da^{\alpha}_{ijkl}=a^{\alpha}_{ij}+a^{\alpha}_{jl}-a^{\alpha}_{ik}-a^{\alpha}_{kl}$.
One can then proceed over all cubes of an arbitrarily large cubic lattice,
assigning terms as above.

At this point one can see already because $\phi^{\alpha\beta}$ carries an
integer redundancy $\phi\to\phi+\ell_{i}$, the action is only well defined if
$k\in 2\mathbb{Z}$, reflecting the fact that our theory describes bosonic
Chern Simons theory222Indeed, the Alvarez approach [4] was developed
specifically to explain level quantization without resorting to the homology
of the underlying manifold.. For convenience, we relax this restriction to
$k\in Z$below.

####  C.2.1 Evaluation of Partition Functions

We believe that our theory describes a renormalization group fixed point, in
that one can evaluate the integrals over certain lattice links and end up with
the same action on a reduced lattice. With that in mind, here we evaluate the
partition function on the simplest possible lattice: a single lattice cell
covering $M=S^{1}\times S^{1}\times I$, ‘glued’ to itself around the
nontrivial loops of $M$.

Figure C.4:  Pruning of the lattice for Chern-Simons Theory. a) We begin with
a large cubic lattice on $T^{2}\times I$. b) In Section (), we show that by
integrating out lattice variables, we can obtain the same theory on a reduced
lattice. We do so gradually until it is a single “stack” of non-simply-
connected cubes along $I$. In this frame, we denote the flat-connection
boundary conditions by shading. c) After integrating out yet more lattice
points, we can reduce the lattice until it is a single cubic cell, non-simply
connected to itself. In this diagram, $1\sim 2\sim 3\sim 4$ are a single
actual point, though we denote them differently to make it easier to notate
links. Similarly, $5\sim 6\sim 7\sim 8$ are a single point. We have dropped
the shading that was present in (b).

The single lattice is shown in Figure . Since there is only one cell, we drop
the cell index.

Instead of periodic boundary conditions, we allow the links to differ by a
gauge transformation. Specifically, we assume that there is a field $\phi$
defined on faces $5678,2468,$ and $3478$ (see Figure C.5) such that:

$\displaystyle a_{34}-a_{12}=(d\phi)_{34}\hskip
28.45274pta_{78}-a_{56}=(d\phi)_{56}$ (C.13) $\displaystyle
a_{37}-a_{15}=(d\phi)_{48}=a_{15}-a_{26}$ (C.14) $\displaystyle
a_{24}-a_{13}=(d\phi)_{24}\hskip 28.45274pta_{68}-a_{57}=(d\phi)_{57}$ (C.15)

Note that, because $1\sim 2\sim 3\sim 4$ represent the same point as do $5\sim
6\sim 7\sim 8$, $(d\phi)_{12},(d\phi)_{13},(d\phi)_{56},(d\phi)_{57}\in Z$. On
the other hand, $(d\phi)_{15}\in\mathbb{R}$, and because the link $15$ is
contractible we can take the temporal gauge with $a_{14}=0$.

Now we can write down the Lagrangian on our lattice. Directly translating eq.
(C.2), we obtain:

$\displaystyle S=\pi
k\Big{[}a_{13}(da)_{1256}+a_{12}(da)_{1357}-\phi_{1}(da)_{1256}-\phi_{1}(da)_{1357}$
(C.16) $\displaystyle
n_{1256}a_{13}+n_{1357}a_{12}-n_{1256}\phi_{1}-n_{1357}\phi_{1}\Big{]}$

Figure C.5: Detailed view of the single-cell lattice. a) We do not require
that the lattice variables are periodic, but instead allow them to differ by a
gauge transformation $d\phi$, where $\phi$ is defined on the orange faces. b)
Taking the temporal gauge with all links in the vertical direction zero, we
can rewrite the Lagrangian in terms of the holonomy in the $x$ and $y$
directions, $H_{x}$ and $H_{y}$ respectively, as well as the flux on the $xz$
and $yz$ faces.

Let us rename these quantities in a more transparent way. We rename the
holonomies as:

$\displaystyle H_{x}\equiv\text{Hol}(A,12)\hskip
56.9055ptH_{y}=\text{Hol}(A,13)$ (C.17)

and the fluxes as:

$\displaystyle F_{zx}=(da)_{1256}\hskip 56.9055ptF_{yz}=(da)_{1357}$ (C.18)

The action becomes:

$\displaystyle S=2\pi k\left[H_{y}F_{zx}+H_{x}F_{yz}\right]$ (C.19)

which has the virtue of being written in completely gauge-invariant, non-
redundant quantities.

Now we can evaluate the partition function. When doing so, we can sum over the
‘$x$’ variables first or the ‘$y$’ variables first. Whichever variable we sum
over first plays the role of the canonical momentum from canonical
quantization, while the second play the role the canonical position. Let us
sum over the ‘$x$’ variables first, $H_{x},F_{zx}$. The partition function is:

$\displaystyle
Z=\sum_{F_{yz}\in\mathbb{Z}}\int\limits_{0}^{1}dH_{y}\sum_{F_{zx}\in\mathbb{Z}}\int\limits_{0}^{1}dH_{x}\exp\left\\{2\pi
ik\left[H_{y}F_{zx}+H_{x}F_{yz}\right]\right\\}$ (C.20)

We have reduced the quantum mechanical problem to two real integrals over
$[0,1)$, and two sums over $\mathbb{Z}$. Evaluating from right to left, we
have:

  1. 1.

Integrate $H_{x}$ from $0$ to $1$. Recalling that $F_{yz}\in\mathbb{Z}$ we
have,

$\displaystyle\int\limits_{0}^{1}dHxe^{2\pi
ikH_{x}F_{yz}}=\begin{cases}1&F_{yz}=0\\\ 0&F_{yz}\neq 0\end{cases}$ (C.21)

Hence the integral over $H_{x}$ sets up a Kronecker $\delta$-function
$\delta_{F_{yz},0}$.

  2. 2.

Sum $F_{zx}$ over $\mathbb{Z}$. This sets up a Dirac comb:

$\displaystyle\sum_{F_{zx}\in\mathbb{Z}}e^{2\pi
ikH_{y}F_{zx}}=\sum_{\ell\in\mathbb{Z}}\delta(\ell-
kH_{y})=\frac{1}{k}\sum_{\ell\in\mathbb{Z}}\delta\left(H_{y}-\frac{\ell}{k}\right)$
(C.22)

These Dirac $\delta$-functions force the $y$-holonomy $H_{y}$ to be in
$\mathbb{Z}_{k}$.

Let us unpack the physical picture here. The Kronecker $\delta$-function sets
the flux $F_{yz}$ to be zero. This is analogous to setting $H_{y}$ to be time-
independent. In the Hamiltonian picture, the holonomy around one nontrivial
loop is the canonical position and uniquely labels ground states. Similiarly,
the Dirac $\delta$-function constrains that same holonomy to be in
$\mathbb{Z}_{k}$, reflecting the $k$ ground states on a torus. One may
exchange the roles of $H_{x}$ and $H_{y}$ by exchanging the order of
integrating over variables, namely by integrating over $H_{y}$ and summing
over $F_{yz}$ and then integrating over $H_{x}$ and summing over $F_{xz}$.

Performing the remaining integrations, we have

$\displaystyle
Z=\sum_{F_{yz}\in\mathbb{Z}}\int\limits_{0}^{1}dH_{y}~{}\left(\delta_{F_{yz},0}\right)\sum_{\ell=0}^{k-1}\frac{1}{k}\delta\left(H_{y}-\frac{\ell}{k}\right)=1$
(C.23)

The $k$ contributions to the partition function from the $k$ ground states on
a torus are multiplied by the $\frac{1}{k}$, reflecting the fact that our
formalism calculates a normalized partition function; future work will
generalize this calculation to the ground state degeneracy and braiding of
Wilson loops.

##  Appendix D Commentary

I have not asked him, but I believe that Xiao-Gang and I began to work with
each other because we both believe that the universe must fundamentally be
computable. To that end, I took the least computable ideas I could imagine:
chiral quantum field theories, and sought to render them well defined on the
lattice so that we can simulate them on a computer. Of course, the great joy
of reality becomes apparent when, instead of somehow taming the wildness of
Quantum Field Theory, defining them on the lattice only reveals an even more
extraordinary nature.

Not all of my drive comes from abstract considerations. I hope deeply that
this work proves useful to other scientists over the long term. Most of my
work in the latter days of my PhD has been seeking to communicate these ideas
to those from the Lattice Gauge Theory and Quantum Information fields, in the
hopes that it solve several related problems for them not only in principle,
but in practice. In my experience, progress in abstract considerations is best
measured by its contribution to hard, down-to-earth problems.

Many physicists would ask why, given my obsession with computability, I did
not work in quantum information. On the one hand, I will spend my coming post-
doctoral fellowship, and likely the rest of my career, building quantum
computers, and I could not be more excited for that. But I do not regret
working in condensed matter quantum field theory at all. There is something
extraordinary about quantum mechanics in spacetime, where dimensionality plays
a central, even defining role. I did not want to study states in an abstract
Hilbert space, but rather correlation functions stretched out across space and
time. And, most of all, I wanted to work with Xiao-Gang.

Feynman first said that we do not understand a theory unless we can explain it
to freshmen. About halfway through my PhD, I first heard the “Wen Principle:”
that we do not understand a theory unless we can program it into a
computer111This is very similar to Wilson’s principle, named for the founder
of modern renormalization group theory Kenneth Wilson. From his Nobel lecture
[110]:  “In thinking and trying out ideas about ‘what is a field theory’ I
found it very helpful to demand that a correctly formulated field theory
should be soluble by computer, the same way an ordinary differential equation
can be solved on a computer, namely with arbitrary accuracy in return for
sufficient computing power.” Yet there is a key difference between the
viewpoints of Wen and Wilson. Wilson saw the lattice as a useful way to define
a quantum field theory, while field theory itself was closer to the
fundamental theory of the universe. Wen sees the lattice as a candidate for a
fundamental theory of our universe, and quantum field theories as useful ways
to reason about many-body systems. I am clearly sympathetic to Wen’s view. .
(Surely this is more flexible than Feynman’s rule.) The Wen principle has
drastic implication for the nature of our world: If you believe that the
universe can be understood, or even, as I do, that it is our destiny to
understand it, then the universe must be computable. All of the quantum field
theories that control the behavior of the physical world must be computable.
That is what my PhD work, and this thesis, have been about.

The connection between understanding and computation goes much farther. In the
“it-from-bit” and “it-from-qubit” formulations, we often speak of the universe
_as a_ computation. However, it is not that reality is modeled after
computation, but the opposite: all of our computation, and even all of our
cognition, is modeled after the nature of the universe. Equations encode
words, words communicate ideas, and ideas emerge from the world around us. I
think often of C.S. Lewis’ belief that there is an order to things, and the
thing in our experience which that order most resembles is a mind. The Wen
principle then follows naturally: computers model minds, and minds model
reality.

Again from Feynman: “what I cannot create I do not understand.” Perhaps to
understand _is_ to create [100]. Or, from Wen’s principle, to compute. Because
I suspect that, at the center of things, at the beginning of it all, lays the
idea, the order. The _logos_.