11institutetext:  Institute for Astronomy, University of Hawai’i, Honolulu, HI
96822, USA
email<EMAIL_ADDRESS>22institutetext: Institute of Astronomy, KU Leuven,
Celestijnenlaan 200D, B-3001 Leuven, Belgium
email<EMAIL_ADDRESS>33institutetext: Department of Astrophysics,
IMAPP, Radboud University Nijmegen, PO Box 9010, 6500 GL, Nijmegen, The
Netherlands 44institutetext: Max Planck Institute for Astronomy, Koenigstuhl
17, 69117, Heidelberg, Germany

# Confronting sparse Gaia DR3 photometry with TESS for a sample of about
60,000 hot massive non-radial pulsators

Daniel Hey 1 1  Conny Aerts 22334 4

(Received February ??, 2024; Accepted ??, 2024)

###### Abstract

Context. The Gaia mission has delivered hundreds of thousands of variable star
light curves in multiple wavelengths. Recent work demonstrates that these
light curves can be used to identify (non-)radial pulsations in the OBAF-type
stars, despite the irregular cadence and low light curve precision of order a
few mmag. With the considerably more precise TESS photometry, we revisit these
candidate pulsators to conclusively ascertain the nature of their variability.

Aims. We seek to re-classify the Gaia light curves with the first two years of
TESS photometry for a sample of 58,970 p- and g-mode pulsators, encompassing
$\gamma$ Dor, $\delta$ Scuti, SPB, and $\beta$ Cep variables. From the TESS
data, we seek to assess the quality of Gaia’s classification of non-radial
pulsators which is based on sparse, years-long light curves of mmag precision.
We also supply four new catalogues containing the confirmed pulsators, along
with their dominant and secondary pulsation frequencies, the number of
independent mode frequencies, and a ranking according to their usefulness for
future asteroseismic ensemble analysis.

Methods. We first analyze the TESS light curves independent of their Gaia
classification by prewhitening all dominant pulsation modes down to a 1% false
alarm probability. Using this, in combination with a feature-based random
forest classifier, we identify different variability types across the sample.

Results. We find that the Gaia photometry is exceptionally accurate for
detecting the dominant and secondary frequencies, reaching approximately 80%
accuracy in frequency for p- and g-mode pulsators. The majority of Gaia
classifications are consistent with the classifications from the TESS data,
illustrating the power of the low-cadence Gaia photometry for pulsation
studies. We find that the sample of g-mode pulsators forms a continuous group
of variable stars along the main sequence across B, A, and F spectral types,
implying that the mode excitation mechanisms for all these pulsators need to
be updated with improved physics. Finally, we provide a rank-ordered table of
pulsators according to their asteroseismic potential for follow-up studies,
based on the number of sectors they have been observed in, their
classification probability, and the number of independent modes found in the
TESS light curves from the nominal mission.

Conclusions. Our catalogue offers a major increase in the number of confirmed
gravity-mode pulsators with an identified dominant mode suitable for follow-up
TESS ensemble asteroseismology of such stars.

###### Key Words.:

Techniques: photometric – Stars: Rotation – Stars: binaries: general – Stars:
oscillations (including pulsations) – Methods: statistical – Catalogs

##  1 Introduction

Over the course of the past century, variable stars have proven to be an
invaluable resource in the refinement of our understanding of stellar
structure and evolution. This has become notably apparent following the advent
of high-precision space photometry, a revolution which has provided
astronomers with a continuous stream of high-frequency micro-magnitude
precision light curves spanning the entirety of the Hertzsprung-Russell
diagram (HRD, Kurtz, 2022). The richness of this data has paved the way for
advanced asteroseismic investigations into the internal structure of stars
(Aerts, 2021).

Asteroseismic modelling has been utilized for thousands of red giants, with
stochastically excited identified modes exhibiting amplitudes around 0.1 mmag
and periodicities on the order of hours (e.g., Hekker & Christensen-Dalsgaard,
2017, for a review). Conversely, the sample of uniformly analysed white dwarf
successors to these red giants only includes a few tens of compact pulsators
with identified modes (e.g. Hermes et al., 2017). Despite the possibility of
their amplitudes reaching several mmag, the limited sample size is primarily
due to their characteristics as faint, fast gravity-mode pulsators (e.g.
Córsico et al., 2019, for a summary). Therefore, monitoring these pulsators
for asteroseismology necessitates a cadence under a minute, as opposed to the
half-hour cadences suitable for covering the hours-long periodicities of red
giant modes.

Asteroseismology of main sequence pulsators has so far also only been applied
to limited ensembles compared to red giants. For low-mass dwarfs like the Sun
this limitation is due to the low amplitudes (of order 10 $\mu$mag) and short
periods (of order minutes) of their solar-like pulsations (García & Ballot,
2019, for a review). For intermediate- and high-mass dwarfs the limitations
are mainly caused by the fast rotation (see Aerts & Tkachenko, 2024, for a
review), preventing identification of the asymmetrically split merged mode
multiplets for the majority of discovered class members. So far, homogeneous
asteroseismic modelling of intermediate-mass dwarfs, treating all pulsators in
the same way has only been done for a few modest ensembles:

  * •

60 young unevolved $\delta\,$Scuti stars very close to the zero-age main
sequence with high-frequency pressure modes (Bedding et al., 2020). As for
most of the slowly rotating red giant pulsators, their modelling was done
while ignoring the Coriolis force and relied on just the large frequency
separation and the frequency of maximum power (Panda et al., 2024), rather
than the fitting of individual mode frequencies;

  * •

26 slowly pulsating B stars (SPB stars, Pedersen et al., 2021) whose
individual identified mode frequencies were modelled based on the
methodological framework designed specifically for gravito-inertial modes in
fast rotators relying on the Traditional Approximation of Rotation (TAR), as
developed in Aerts et al. (2018, to which we refer for details);

  * •

490 $\gamma\,$Doradus ($\gamma\,$Dor) stars whose measured buoyancy travel
time and near-core rotation frequency deduced from identified gravito-inertial
modes adopting the TAR were modelled by Fritzewski et al. (2024a); 37 of these
$\gamma\,$Dor stars had their individual modes modelled by methodology based
on the TAR coupled to a neural network (Mombarg et al., 2021).

For the high-mass $\beta\,$Cep stars, homogeneous ensemble modelling of their
few (typically between 3 and 5) identified pressure modes has not yet been
done; Bowman (2020) summarised some of the results for seven individual class
members. Clearly, ensemble asteroseismology of intermediate- and high-mass
dwarfs is still in its early stages, despite the discovery of thousands of
class members (Handler et al., 2019; Burssens et al., 2023; Eze & Handler,
2024). Lack of mode identification prevents applications to large ensembles.
Novel approaches to filter thousands of main-sequence pulsators with proper
mode identification are therefore in order. The initial attempts for the
toughest case of gravito-inertial pulsators by Garcia et al. (2022a, b) based
on just the first year of TESS monitoring already illustrated the major
potential of this mission for ensemble asteroseismology of fast rotators
across the Milky Way, in line with the opportunities discussed in Aerts &
Tkachenko (2024).

The current study aims to increase the sample of main sequence pulsators of
spectral types O, B, A, and F (hereafter OBAF pulsators) with the potential
for ensemble asteroseismology with an order of magnitude. Our work is focused
on the availability of homogeneously assembled seismic and non-seismic
observables. Even if it was not designed for this type of research, the ESA
Gaia mission (Gaia Collaboration et al., 2016b, a) has a significant role to
play in this context. We tackle the challenging search for suitable large
ensembles of dwarf pulsators by screening the homogeneous database of stellar
properties and sparse time-series photometry offered by Gaia’s Data Release 3
(DR3, Gaia Collaboration et al., 2023b). Starting from the sample of 106,207
candidate OBAF pulsators classified by Gaia Collaboration et al. (2023a), we
analyse those targets among this large ensemble having high-cadence high-
precision light curves assembled by the NASA TESS space telescope (Ricker et
al., 2015).

In order to prepare the ‘industrialisation’ of asteroseismic ensemble
modelling of OBAF pulsators, we need to find targets with tens of identified
modes from the TESS photometry among the ‘zoo of variables’ along the main
sequence (Eyer et al., 2023). Thousands of variable intermediate- and high-
mass dwarfs have already been found from high-cadence uninterrupted space
photometry (e.g., Uytterhoeven et al., 2011; Balona et al., 2011a; Balona &
Dziembowski, 2011; Balona et al., 2011b; Balona, 2016; Balona et al., 2019;
Antoci et al., 2019; Pedersen et al., 2019; Burssens et al., 2019, 2020).
Their variability is caused by different physical phenomena, making
identification of the pulsation modes for large ensembles of OBAF pulsators a
major obstacle. In this era of high-cadence space photometry the variability
phenomena are better understood and include stellar flaring (e.g. Balona,
2015; Pedersen et al., 2017) along with other magnetic variability (e.g.
Shultz et al., 2019; David-Uraz et al., 2019; Sikora et al., 2019a),
rotational modulation (e.g., Bowman et al., 2018; Sikora et al., 2019b), low-
frequency variability interpreted in terms of internal gravity waves (e.g.,
Aerts & Rogers, 2015; Bowman et al., 2019, 2020) or subsurface convection
(e.g., Cantiello & Braithwaite, 2019; Cantiello et al., 2021), eclipses in
close binaries (e.g., Kirk et al., 2016; IJspeert et al., 2021) and so on.
This non-exhaustive list of variable phenomena often occurs in combination
with nonradial pulsations (as summarised in Kurtz, 2022).

Furthermore, it has long been established that OBAF pulsators coexist with
various types of variable stars along the main sequence in the Hertzsprung-
Russell diagram (HRD) (e.g., Briquet et al., 2007). The recently released Gaia
Data Release 3 (Gaia DR3) data reaffirm this observation, demonstrating that
the classes of variability, including OBAF pulsators, encompass a substantial
proportion of stars distributed across nearly the entire main sequence within
the intermediate- and high-mass dwarf star population (Gaia Collaboration et
al., 2023a). This phenomenon is further substantiated by investigations of
young open clusters, which have been systematically monitored through joint
efforts involving the Gaia mission, as well as the refurbished Kepler (known
as K2) and Transiting Exoplanet Survey Satellite (TESS) missions (e.g., White
et al., 2017; Murphy et al., 2022; Bedding et al., 2023; Fritzewski et al.,
2024b; Li et al., 2024). Consequently, it is prudent to adopt an approach that
involves scrutinizing the time-series photometric data itself, rather than
solely focusing on the stellar positions within the HRD, to effectively
address our scientific objectives. Gaia DR3 provides us with the requisite
data to pursue this approach in our quest to identify the most appropriate
ensembles of OBAF pulsators.

In this paper, we revisit a large fraction of the 106207 OBAF pulsators
discovered in the sparse Gaia DR3 photometry by Gaia Collaboration et al.
(2023a) with the goal to derive and analyse their TESS light curves. We focus
on the poorly populated samples of nonradial pulsators having the highest
asteroseismic potential and thus revisit the candidate $\gamma\,$Dor stars,
Slowly pulsating B (SPB) stars, $\beta\,$Cep stars, and $\delta$ Scuti stars
identified by Gaia Collaboration et al. (2023a). The $\gamma\,$Dor and SPB
g-mode pulsators have been revisited already by Aerts et al. (2023) in terms
of their astrophysical properties, but the $\beta\,$Cep and $\delta$ Scuti
p-mode pulsators have not been evaluated as such. Here, we confront the Gaia
DR3 variability properties for the members assigned to these four classes with
their TESS data when available. Our work has the following two major aims:

  * •

to assess the quality of Gaia’s classification of nonradial pulsators, which
is only based on sparse years-long light curves of mmag precision, by
confronting its outcome with high-cadence $\mu$mag precision TESS space
photometry for all class members that have Gaia and TESS independent data sets
in the public domain;

  * •

to create four large new catalogues containing Gaia-discovered nonradial
pulsators confirmed by TESS, offering the community their TESS light curves
covering the first year of monitoring, their independent mode frequencies, and
identification of the dominant mode if possible. These catalogues are an
optimal starting point for future ensemble asteroseismic modelling.

##  2 Gaia and TESS Data

Figure 1: Left: Comparison of measured dominant TESS frequency against Gaia
frequency for the g-mode pulsators (top) and p-mode pulsators (bottom). Right:
The same for the secondary frequency. The dashed lines indicate the half,
unity, and twice relationships among the frequencies. We compare the sample of
g-mode candidate pulsators before removing combination frequencies, to match
the method of Aerts et al. (2023).

Figure 2:  Example light curves and amplitude spectra from the p-mode (top
three panels) and g-mode (bottom three panels) candidate sample. The green
vertical line marks the measured Gaia frequency. We show the following cases;
a) where the measured Gaia frequency is in good agreement with TESS, b) where
the measured Gaia frequency is not the true dominant one, and c) the measured
Gaia frequency is incorrect.  Figure 3: Distribution of amplitudes for the
dominant (blue) and secondary (orange) frequencies for each variable type in
our sample. Note that we use only stars where the dominant frequency lies on
the bisector within 0.1 d-1 tolerance for the TESS/Gaia overlap (cf. Fig. 1).

###  2.1 The four samples of candidate pulsators

Our samples consist of the p- and g-mode pulsators discussed in Gaia
Collaboration et al. (2023a). For the g-mode pulsators, we take the 15,062
candidates from Aerts et al. (2023). This sample contains 11,636 $\gamma$ Dor
and 3,462 SPB star candidates. In addition, we consider the 222 candidate
$\beta\,$Cep stars and 85,317 $\delta$ Scuti candidates classified as such by
Gaia Collaboration et al. (2023a). For the latter two classes, the extra
vetting based on the expected frequency range as done for the g-mode pulsators
by Aerts et al. (2023) is not meaningful, because their dominant p-mode
frequencies are expected to intervene with (multiples of) Gaia’s main
satellite frequency near 4 d-1 at mmag level. Moreover, large fractions among
the $\beta\,$Cep and $\delta\,$Scuti stars may have a dominant high-amplitude
radial mode, so a restriction on amplitude as extra vetting rule as adopted by
Aerts et al. (2023) for the $\gamma\,$Dor and SPB pulsators is less obvious to
define for the p-mode pulsators.

To construct the four samples for the current work, we cross-match their Gaia
Data Release 2 (DR2, Gaia Collaboration et al., 2018) identifications (IDs)
using the ‘nearest neighbours’ table. To obtain their TESS IDs, we then cross-
match the Gaia DR2 IDs against the TESS Input Catalog (TIC, Stassun et al.,
2018). The final cross-matched sample among DR3 (Gaia Collaboration et al.,
2023b), DR2, and TIC contains 85,313 $\delta$ Scuti stars, 11,636 $\gamma$ Dor
stars, 3,426 SPB stars, and 222 $\beta$ Cep stars. The loss of several stars
in the cross-matching process is a result of the DR2 to DR3 best neighbours
matching catalogue, which is not strictly one-to-one.

###  2.2 TESS photometry

The majority of the sample has not been targeted for dedicated observations by
TESS. With no official light curves delivered by the TESS team using the SPOC
pipeline, we instead use light curves from the TESS Gaia light curve (TGLC)
catalogue, produced from full-frame images by Han & Brandt (2023). TGLC light
curves were used as an alternative to the Quick-Look Pipeline (QLP, Huang et
al. (2020) because they reach fainter than 13.5 mag. The TGLC light curves
leverage Gaia photometry and astrometry to inform light curve extraction by
building a local point spread function forward model, making it a viable
source for fainter stars, where Gaia astrometry is useful for resolving
contaminated stars.

We use the TGLC light curves from the first 26 sectors of the TESS mission
calculated by the TGLC pipeline from the full-frame images. These light curves
are at a nominal 30-minute cadence. We use the calibrated aperture flux with
the “good” quality flags. For each light curve, we perform 3$\sigma$ clipping
of the flux and apply a Gaussian filter with a standard deviation of 100 to
remove significant long-term trends. A table containing the curl scripts used
to download the data is available in electronic format as supplementary
material. The TGLC light curves are available in the first 26 TESS sectors for
45,919 $\delta$ Scuti stars, 10,099 $\gamma$ Doradus stars, 2,777 SPB stars,
and 175 $\beta$ Cep stars, leading to a total analysis sample of 58,970
variables out of the original target list.

##  3 Confrontation between the two light curves per sample star

###  3.1 Prewhitening of dominant pulsation modes

To analyze the candidate pulsators in our sample we have developed an
automated prewhitening scheme based on the approach used for p-mode pulsators
in Hey et al. (2021), with several modifications. The algorithm functions as
follows: it begins by identifying the highest peak in the amplitude spectrum.
It then optimises the value of the peak frequency from a sinusoidal function
in the time domain characterized by a frequency, amplitude, and phase. This
fitted sinusoid is then subtracted from the light curve, repeating iteratively
until a predefined stopping condition. In contrast to Hey et al. (2021) where
the stopping condition was the signal-to-noise ratio (SNR) of the peak, our
method employs the false-alarm level for the highest amplitude peak, ensuring
it falls below a 1% probability threshold. If this condition is met, the peak
is deemed significant, removed, and the prewhitening process continues.

The prewhitening procedure is applied to each peak exceeding the 1%
significance level in the amplitude spectrum, or until a maximum of 100
iterations is reached, whichever occurs first. Additionally, for each peak, we
calculate its ‘prominence’, which quantifies the peak’s distinctiveness
relative to the surrounding amplitude spectrum. This metric serves as a useful
diagnostic tool for evaluating individual modes, especially in the scenario
where a single prewhitening step does not completely remove a peak (a common
occurrence in non-sinusoidal signals).

We perform the prewhitening for all stars in our sample. Stars with more than
one sector of TESS observations are stitched together prior to prewhitening,
with each sector having a simple 5$\sigma$ outlier removal and long-term trend
removal with a Gaussian filter. During the prewhitening routine, it is common
to encounter combination frequencies (Kurtz et al., 2015). These can be
identified and subsequently removed using the following equation (Li et al.,
2019):

$|f_{k}-(n_{i}f_{i}+n_{j}f_{j})|<\epsilon\,;$ (1)

where $i$, $j$, and $k$ are indices of the frequency peaks, $f_{i}$ and
$f_{j}$ are the parent frequencies, $f_{k}$ is the candidate for the
combination frequency, $n_{i}$ and $n_{j}$ are combination coefficients, and
$\epsilon$ is the threshold for identification. Following the approach of Li
et al. (2019), we limit our analysis to the 20 highest amplitude peaks,
considering them as potential parent frequencies. Our criterion for
combinations is restricted to cases where $|n_{i}|+|n_{j}|\leq 2$. In light of
the TESS data’s lower precision compared to the Kepler data, we opt for a
considerably larger $\epsilon$ value of 0.002 d-1, compared to that used by Li
et al. (2019). We also remove harmonics, defined as integer or half-integer
multiples of the parent frequency. Such harmonics are common, for example, in
eclipsing binary samples (e.g., IJspeert et al., 2024).

###  3.2 Comparison of the two dominant frequencies

Here we compare the results of our TESS light curves against the Gaia
photometry explored in Gaia Collaboration et al. (2023a) and Aerts et al.
(2023). Figure 1 shows the comparison between dominant and secondary modes
between the TESS and Gaia data for all the candidate pulsators in the four
samples, grouping the p-mode and g-mode pulsators in separate panels. The
agreement is particularly good considering the sparse sampling of the Gaia
data, with 69% of the sample lying along the bisector with a 0.1 d-1 tolerance
for the dominant frequencies of the p-modes, and 80% for the g-mode
frequencies. This result on comparisons between the dominant frequencies in
the Gaia and TESS light curves is superior to the 20% agreement in dominant
frequency between the Gaia and Kepler light curves for the few tens of
$\gamma\,$Dor and SPB stars found by Gaia Collaboration et al. (2023a, see
their Appendix A).

There is a clear systematic in the TGLC light curves at 1 d-1 caused by what
we believe to be reflected light from Earth (‘earthshine’). We find no
correlation between the amplitude of this signal and whether the star falls
along a TESS bisector or not. For the p-mode pulsators, Fig. 1 reveals
additional criss-crossing structures aside from the 1 d-1 systematic in the
TESS data. This phenomenon is understood in terms of the 30-min sampling by
TESS, causing a mirroring effect around the Nyquist frequency that leads to an
aliased signal of the true frequency. For example, a $\delta$ Scuti variable
with a true pulsation at 44 d-1 will have an indistinguishable copy of the
signal appearing at around 4 d-1 if observed in 30-minute cadence. This effect
can be seen, for example, in the $\delta$ Scuti and rapidly oscillating Ap
stars observed by Kepler in long-cadence (Bell et al., 2017; Murphy et al.,
2019; Hey et al., 2019). Note that the true and aliased signals of coherent
pulsators can be distinguished in the Kepler data as a consequence of periodic
modulation of the light curve measurement times (Murphy et al., 2013).

We show a series of light curves in Fig. 2 to demonstrate pathological cases
of the Gaia and TESS data. For the p- and g-mode candidate sample, we
illustrate three occurring scenarios: a) the Gaia measured dominant frequency
is correct and confirmed by TESS, b) Gaia picks up a secondary peak of an
otherwise correctly classified pulsator, and c) the dominant Gaia frequency is
wrong and likely of instrumental origin. The latter situation calls for a re-
evaluation of the Gaia DR3 variability classification of the star, based on
its TESS light curves. We tackle this subject in Sect. 4. We also note that
the Gaia identification of dominant and secondary frequencies is dependent on
the scanning law (Steen et al., 2024).

Figure 3 shows the histograms of the primary and secondary amplitudes for the
TESS data of each variable type, where the dominant frequency is in good (that
is, to within 0.1 d-1) agreement with the measured Gaia frequency, and the
classification is accurate in both TESS and Gaia with a classification
probability $>0.5$ (Section 4). We perform a two-sided Kolmogorov-Smirnov test
to compare the amplitude distributions across each of the two variability
classes with the same type of modes. The null hypothesis for this test is that
the two distributions are identical. That is, they are drawn from the same
underlying distribution. We choose a confidence level of 95% for the test,
such that the null hypothesis is rejected if the p-value is less than 0.05.
For the $\delta$ Scuti and $\beta$ Cep sample, the test indicates that their
amplitude distributions are statistically different, with a p-value of around
$10^{-14}$. This result reflects that the two classes of p-mode pulsators are
subject to the same type of excitation mechanism, namely the opacity (or
$\kappa$) mechanism, but that it acts in a different layer in the outer
envelope of the star. For the $\beta\,$Cep stars it concerns the partial
ionisation zone of iron-like isotopes, while the heat engine in the case of
$\delta\,$Scuti stars is acting in the second ionization zone of helium
(Pamyatnykh, 1999; Guzik, 2021).

For the $\gamma$ Dor and SPB sample, however, we find that the null hypothesis
can not be rejected with a p-value of 0.055. This indicates that the
distributions of their amplitudes are statistically the same. From the
observational side, this is expected from prior work based on variability
classification – only colour information allows us to distinguish between the
SPB and $\gamma$ Dor classes (Audenaert et al., 2021), and even then, there is
still confusion for stars with an effective temperature between 8500-9500 K
(Aerts et al., 2023). The equal amplitude distributions among $\gamma$ Dor and
SPB stars are of interest to improve mode excitation computations, which
requires a non-adiabatic treatment of the pulsation equations. Indeed, while
the SPB stars have their g modes excited by the $\kappa$ mechanism acting in
the same partial ionisation zone of iron-like isotopes as the p modes in the
$\beta\,$Cep stars (Pamyatnykh, 1999), the $\gamma\,$Dor stars are subject to
at least two not mutually exclusive excitation mechanisms: the
$\kappa-$mechanism for the hotter class members and the mechanism based on
flux blocking at the base of the thin outer convective envelope for the cooler
class members (Dupret et al., 2004, 2005). Moreover, nonlinear mode coupling
was also found to occur in large-amplitude SPB stars (Van Beeck et al., 2021)
and may excite extra modes via energy exchange, aside from the self-driven
linear modes due to the $\kappa$ mechanism. While a similar study on nonlinear
mode coupling has not yet been done for $\gamma\,$Dor stars, their Kepler
light curves show similar cusp-like shapes near the maxima than the SPB stars
do (cf. compare the data in Van Reeth et al., 2015; Pedersen et al., 2021).

Finally, it has been shown that adding novel physical ingredients in mode
excitation computations may appreciably enlarge the instability strips, such
as the Coriolis force due to fast rotation (Szewczuk & Daszyńska-Daszkiewicz,
2017) and radiative levitation due to atomic diffusion (Deal et al., 2016;
Rehm et al., 2024). All of this makes comparing observational instability
strips from surveys of pulsators with predicted strips based on just one
choice of input physics of limited value. This was already highlighted from
the dominant frequencies found in the Gaia DR3 light curves of g-mode
pulsators in Paper I, stressed in the review on asteroseismology of fast
rotators by Aerts & Tkachenko (2024), and is reinforced here from our
indistinguishable TESS amplitude distributions for the $\gamma\,$Dor and SPB
classes shown in Fig. 3. It is for this reason that we merge the SPB and
$\gamma$ Dor classes in Sec. 4.

###  3.3 Gaia amplitude ratios

Figure 4:  Violin plot of the amplitude ratios for the four samples deduced
from the light curves in the three Gaia bandpasses with respect to the
amplitude in the TESS band (covering approximately 600 - 1,000 nm). The shaded
regions indicate the density distributions for each of the samples, with the
solid points showing the median. For clarity, we do not add the density
distribution centred around value 1.0 for the TESS bandpass itself.

So far we worked with the Gaia G passband. It covers the wavelengths between
330 nm to 1050 nm, with peak sensitivity at 640 nm. But DR3 also delivered the
light curves in the RP and BP colour bands. In practice, these BP and RP bands
are blue and red cuts of the broad G band, covering the ranges from 330 nm to
680 nm (BP) and from 630 nm to 1050 nm (RP), with maximum responses at 517 nm
and 783 nm, respectively (Weiler, 2018). On the other hand, the TESS detector
passband spans from 600 nm to 1000 nm with central wavelength at 787 nm
(Ricker et al., 2015).

Having time series data with colour information is advantageous for
asteroseismology in the case of ambiguous mode identification in terms of the
spherical harmonic wavenumbers $(l,m)$ characterising the geometry of the
mode. Indeed, the theoretical expression for the observed amplitudes of a mode
described by the spherical harmonic function $Y_{l}^{m}$ and viewed upon at an
inclination angle $i$ depends on the wavelength, via the perturbations of the
atmospheric flux and limb darkening caused by the mode (see Eq. (6.29) in
Aerts et al., 2010). The dependencies of this expression on the azimuthal
order $m$ and on the inclination angle $i$ of the star’s pulsation symmetry
axis drop out of the expression of amplitude ratios for different wavelengths.
This is why observed amplitude ratios deduced for light curves observed in
different passbands have been used to identify the mode degree $l$ of main-
sequence pulsators (e.g., Heynderickx et al., 1994; Breger, 2000; Dupret et
al., 2003; Aerts et al., 2004; De Cat, 2017; Brunsden et al., 2018). All these
applications of mode identification assumed one fixed set of input physics for
the theoretical predictions. We now know from space asteroseismology that this
is not appropriate for such pulsators (Aerts, 2021; Johnston, 2021; Pedersen,
2022).

Although the Coriolis and centrifugal forces complicate this capacity of mode
identification in fast rotators such as $\delta\,$Scuti stars (Daszyńska-
Daszkiewicz et al., 2002) and SPB stars (Townsend, 2003), we have a good
understanding of how they do so. Therefore, it was recently suggested by Aerts
& Tkachenko (2024) to exploit amplitude ratios for stars whose identification
of $(l,m)$ is already established. Indeed, in this case, any diversity in
observed amplitude ratios encapsulates differences in the internal,
atmospheric, and limb-darkening physics of the star. Figure 11 in Aerts &
Tkachenko (2024) illustrates this (future) potential opportunity from measured
amplitude ratios of prograde dipole gravito-inertial modes observed in both
Gaia and Kepler data of 23 $\gamma\,$Dor stars. Here, we illustrate the
potential of amplitude ratios from combined Gaia and TESS light curves for the
four samples of new Gaia DR3 pulsators.

For all the stars with consistent dominant frequency and consistent
classification (Section 4) in the Gaia G and TESS passbands, we computed the
ratios of the G, BP, and RP amplitudes of that frequency with respect to the
amplitude in the TESS passband. We show a violin plot of the results for the
four classes in Fig. 4. This figure is in line with expectations for low-
degree ($l\leq 2$) mode behaviour in stars with slow to moderate rotation,
whose ratios are predicted to decrease with increasing wavelength for the
three Gaia passbands (e.g., Heynderickx et al., 1994; De Ridder et al., 2004).
Comparison between Fig. 11 in Aerts & Tkachenko (2024) and the violin plot in
Fig. 4 illustrates the superiority of TESS over Kepler for this type of
exploratory research based on the dominant pulsation mode, given that the TESS
pulsators are generally much brighter than the Kepler $\gamma\,$Dor stars,
leading to more precise amplitude ratios. Given its potential for
asteroseismology, we provide the Gaia amplitude ratios alongside the database
of prewhitened modes as supplementary data.

##  4 Re-classification of the pulsators from TESS

Figure 5:  Confusion matrix for the Random Forest classifier normalized against the true values. Here, hybrid refers to the simultaneous presence of $\delta$ Scuti p-mode pulsations and $\gamma$ Dor or SPB g-mode pulsations. Table 1: Classifications of the p- and g-mode candidate sample. The full table in electronic format, with probabilities for each class, is available online. DR3 Source ID | Sector | Class | Probability
---|---|---|---
2070667440659268352 | 14 |  $\delta$ Scuti | 0.26
2247307763228746240 | 14 | Hybrid | 0.67
5311721917688649856 | 10 |  $\delta$ Scuti | 0.35
5617799085534387968 | 7 |  $\delta$ Scuti | 0.67
5657691905702501888 | 9 |  $\gamma$ Dor/SPB | 0.93
| ⋮ |  |
2925330438953873024 | 7 | Hybrid | 0.95
2164506978535115776 | 16 | Hybrid | 0.41
5516571310575369472 | 8 |  $\delta$ Scuti | 0.74
5964132569560169472 | 12 |  $\delta$ Scuti | 0.57
5661189937524776320 | 9 |  $\gamma$ Dor/SPB | 0.90

We now re-evaluate the Gaia DR3 variability classification from Gaia
Collaboration et al. (2023a) by relying on the highly sampled and more precise
TESS light curves.

###  4.1 Training sample

To distinguish between various classes of variability, in particular,
intrinsic and extrinsic variability, we have implemented a simple feature-
based Random Forest classifier, similar to Audenaert et al. (2021) and Barbara
et al. (2022). This classifier seeks to identify different types of
variability based on extracted singular value features of the light curve.
These features, calculated with pycatch22 (Lubba et al., 2019), are spread
across different categories concerning linear autocorrelation and periodicity
of the flux, extreme events, distributions that ignore time ordering, and
more. These features have been chosen such that they are minimally redundant
and capture the largest possible variance of the input time series.

Our training sample for the classifier is sourced from both Kepler and TESS.
For Kepler, we use the sample compiled by Barbara et al. (2022), which
consists of high-level variability classifications pertaining to A/F-type
stars, including contact and detached binaries, $\delta$ Scuti stars, $\gamma$
Dor stars, RR Lyrae variables, along with rotational and non-variables. Given
that this sample relies heavily on data derived from the Kepler mission, it
poses certain challenges when applied to classify TESS data; the majority of
the stars within the sample are of such faint magnitude that their variability
signal cannot be observed within the TESS data, hence directly comparing the
TESS light curves with the Kepler labels is not feasible. As an alternative,
we have devised a strategy where we modify the Kepler light curves to mirror
the single-sector observations of the TESS photometry. The modifications we
have implemented on the Kepler light curves are as follows; limiting the time
span to a duration of less than 27 days, adding noise proportional to the
magnitude of the star, and introducing a data gap at 13.7 days to simulate the
TESS downlink.

We further construct a TESS training sample compiled against a series of A/F
variability papers (Skarka et al., 2022; Sikora et al., 2019a; Garcia et al.,
2022a, b; Shi et al., 2023), focusing on either the classification of A/F
stars or targeting a specific variable type in TESS. The Skarka sample
consists of variable A/F stars in the Northern continuous viewing zone, the
Sikora sample contains rotationally variable A-type stars, the Garcia sample
contains 60 $\gamma$ Dor stars with a long observational baseline, and the Shi
sample contains 286 SPB stars.

Each sample contains a slightly different type of classification, which we
homogenize into new categories. In particular, we merge all the ellipsoidal
and semi-detached binaries into the “contact” class, leaving the eclipsing
binary (EB) class for purely detached cases. We also merge the two hybrid
classes ($\gamma$ Dor + $\delta$ Scuti vs. $\delta$ Scuti + $\gamma$ Dor) into
a single hybrid class regardless of which variability type is more prominent.
We also merge the SPB and $\gamma$ Dor pulsators into a single class, as
already motivated in the previous section (additional reasons are discussed
below). The remaining classes are the pure $\delta$ Scuti pulsators and pure
rotational variables (‘Rot’). We discard the Skarka sample containing ”VAR”
sources – stars deemed to be variable with an indeterminate classification.

The majority of the stars in this additional training sample are located in
the TESS continuous viewing zone (CVZ), such that each star has multiple
sectors of observations up to almost a year in length. On the other hand, our
classification sample – the candidate p- and g-mode pulsators – are typically
observed in only one or two sectors as a consequence of being distributed
randomly across the sky. As a result, we do not stitch the light curves of any
of the targets in the training sample. Instead, we compute their features on a
per-sector basis and consider each sector of observations as a separate input.
That is, a single target in the training sample can contribute to the final
sample multiple times. We note that this will lead to larger ambiguity in the
classification sample. For example, observations of true $\gamma$ Dor
pulsations might be unresolved in a single sector, such that they are confused
with a rotational signal. Likewise, variables such as eclipsing or ellipsoidal
variables may have variability periods which exceed the single-sector
observations.

###  4.2 Feature extraction and classification

Table 2: Results of classification for each sample, showing the breakdown of
individual classifications as a fraction of the total sample. The number in
brackets represents the fraction of the sample for the class. $\gamma$ Dor
(N=10,047) $\displaystyle\left\\{\begin{array}[]{rl}\textrm{{$\gamma$ Dor /
SPB}}&6,489\leavevmode\nobreak\ (0.65)\\\
\textrm{Rotation}&2,416\leavevmode\nobreak\ (0.24)\\\
\textrm{{Hybrid}}&618\leavevmode\nobreak\ (0.06)\\\ \textrm{Eclipsing
binary}&251\leavevmode\nobreak\ (0.02)\\\ \textrm{Contact
binary}&205\leavevmode\nobreak\ (0.02)\\\ \textrm{$\delta$
Scuti}&58\leavevmode\nobreak\ (\sim 0)\\\ \end{array}\right.$ $\delta$ Scuti
(N=45,648) $\displaystyle\left\\{\begin{array}[]{rl}\textrm{{$\delta$
Scuti}}&19,226\leavevmode\nobreak\ (0.42)\\\
\textrm{{Hybrid}}&15,395\leavevmode\nobreak\ (0.34)\\\
\textrm{Rotation}&4,371\leavevmode\nobreak\ (0.10)\\\ \textrm{Eclipsing
binary}&3,656\leavevmode\nobreak\ (0.08)\\\ \textrm{$\gamma$ Dor /
SPB}&2,962\leavevmode\nobreak\ (0.06)\\\ \textrm{Contact
binary}&38\leavevmode\nobreak\ (\sim 0)\\\ \end{array}\right.$ SPB (N=2,795)
$\displaystyle\left\\{\begin{array}[]{rl}\textrm{{$\gamma$ Dor /
SPB}}&1,481\leavevmode\nobreak\ (0.53)\\\
\textrm{Rotation}&948\leavevmode\nobreak\ (0.34)\\\
\textrm{Hybrid}&209\leavevmode\nobreak\ (0.07)\\\ \textrm{Eclipsing
binary}&88\leavevmode\nobreak\ (0.03)\\\ \textrm{Contact
binary}&59\leavevmode\nobreak\ (0.02)\\\ \textrm{$\delta$
Scuti}&10\leavevmode\nobreak\ (\sim 0)\\\ \end{array}\right.$

We apply a uniform processing of each light curve prior to feature extraction.
This processing includes applying a Gaussian high-pass filter to remove long-
term trends and dividing the light curve by the standard deviation of its flux
(Z-scoring) to ensure normality across the light curve sample. Similar to the
training sample, each target is classified on a sector-by-sector basis, so
that a single target can have multiple classifications across different
sectors.

We use a greedy feature-selection algorithm to pick out a sample of 7 highly
orthogonal features from the original 22 features calculated. These are the
features that best discriminate amongst the variability classes. We also
include several additional features we consider important to the
classification which are not calculated in pycatch22 but are known to help
distinguish variability (see, e.g., Murphy et al. 2019). These features are
the skewness of the flux for discriminating eclipsing binaries, as well as the
skewness of the amplitude spectrum at frequencies less than 1 d-1, less than 5
d-1, and greater than 5 d-1. The frequency domain skewness indicators measure
the effective power contained in different regions of the amplitude spectrum:
$\delta$ Scuti variables will typically have significantly higher skewness at
higher frequencies, and hybrid pulsators will have strong skewness in both
regions. Finally, we include the dominant frequency and amplitude of pulsation
(in Z-scored units), the Gaia BP-RP colour index, and the Gaia reduced unit
weight error.

Figure 5 shows the confusion matrix for our training and test data. For
single-sector observations, the classifier appears accurate, especially for
hybrid pulsators and rotational variables. Unsurprisingly, the $\delta$ Scuti
class is strongly confused with the hybrid class since the training sample was
mostly based on TESS data exceeding multiple sectors. As a result, not all
modes in the hybrid pulsators are resolved in only a single sector. Curiously,
the eclipsing binary class has some overlap with the rotational variable
class. This is likely because the EB class consists of semi-detached (EA) and
W Ursae Majoris (EW) binaries. Only the contact class contains ellipsoidal
binaries. Finally, the $\gamma$ Dor class is weakly confused with the
rotational variables. Again, this is expected; a single sector of observations
limits the ability to resolve modes, thus the single rotation peak can be
mistaken for a $\gamma$ Dor pulsation and vice-versa.

We run the classifier on our sample of candidate $\delta$ Scuti and $\gamma$
Dor pulsators. Note that we do not classify the $\beta$ Cep sample which we
instead manually inspect. The results of the classifier for the sample along
with their class probability are presented in Table 1, with the breakdown of
each sample and resulting classification in Table 2. The number of classified
stars is slightly less than the number of available light curves discussed in
Sec. 2.2 because some light curves are completely dominated by poor data and
were discarded from the sample.

It is important to note that a single star observed in multiple sectors will
have a classification for each sector in the table. For example, TIC 38458616
has been observed for 13 sectors in the first two years of TESS and has an
independent classification per sector. Nine of the sectors predict it to be
$\gamma$ Dor variable, and four predict it to be a contact binary, with the
majority of sectors having a low probability for their respective class. A
closer inspection of the stitched light curve reveals the target to indeed be
a binary system. While stitching individual light curves may give better
results for poorly resolved modes, we strive to instead maintain a uniform
input sample for the classifier. We show examples of high probability
classifications in Fig. 8.

Finally, we make an additional cut on the resulting classifications. To ensure
that the rotational and eclipsing binary variables are reasonably well-
separated from the g-mode pulsators, we apply a data cut such that the second-
highest amplitude frequency can not be half of the dominant frequency within a
tolerance of 0.01 d-1 based on the prewhitening performed in Sec. 3.1. This is
done because rotational variables and eclipsing binaries typically show a
subharmonic frequency at half the dominant due to their strongly non-
sinusoidal light curves. Indeed, for eclipsing binaries, this subharmonic is
usually the true frequency. Making this additional cut removes 498 candidates
from the $\gamma$ Dor sample.

Figure 6:  Stacked amplitude spectra of the g-mode candidate sample (left, in
period space) and $\delta$ Scuti candidate sample (right, in frequency space)
for which the prediction probability is greater than 0.5. Each star forms a
single row, sorted by the dominant pulsation, with color corresponding to
amplitude. For the g-mode sample, which combines both $\gamma$ Dor and SPB
stars, we see a distinct secondary ridge associated with either a harmonic of
the dominant frequency or the expected $\ell=2$ dipole modes seen in Li et al.
(2020). For the $\delta$ Scuti sample, we see ridges associated with the first
and second overtones, as well as a harmonic line.  Figure 7:  Stacked
amplitude spectra of the $\beta$ Cep sample sorted by dominant pulsation
frequency.

Using these classifications, we construct the stacked mplitude spectra. That
is, we calculate the amplitude spectrum for each star and stack it according
to the dominant pulsation frequency for the classified g-mode sample ($\gamma$
Dor / SPB), the $\delta$ Scuti stars, and the $\beta$ Cep stars, whose
prediction probability is above 0.5. We jointly visualize the $\gamma$ Dor and
SPB sample on the same figure. We show the stacked amplitude spectra in Fig.
6. Each row displays the amplitude spectrum of one star sorted vertically by
the dominant pulsation period. For the g-mode sample, we observe two distinct
ridges, along with a third very faint ridge. The primary ridge is likely
dominated by $\ell=1$ $m=1$ g-modes, with the secondary by either lower
amplitude $\ell=2$ modes similar to what was seen by Li et al. (2020) for the
Kepler data, or caused by the harmonic of the dominant mode. We note that the
highest amplitude ridge is likely not all pure dipole modes as the ridge is
formed from sorting by the dominant period. Note also the presence of a purely
vertical ridge and sidelobes at 1 d-1 caused by the known TGLC systematic.

The stacked $\delta$ Scuti figure shows four obvious ridges, corresponding to
the primary, first and second overtones, and harmonic frequency. These lines
are known properties of $\delta$ Scuti stars, visualized more commonly in
Peterson diagrams (Netzel et al., 2022; Pietrukowicz et al., 2020). Finally,
the $\beta$ Cep sample shows no obvious ridges as expected for these low-order
p- and g-mode pulsators (Stankov & Handler, 2005).

###  4.3 Prioritised catalogue of new pulsators ranked by asteroseismic
potential

Table 3: Rank ordered tables for the $\gamma$ Dor, $\delta$ Scuti, and SPB, and $\beta$ Cep classified pulsators. Note that the tables are separated according to their candidate and classification status – g-mode pulsators are the g-mode candidates from Aerts et al. (2023), and the p-mode pulsators are the candidates from Gaia Collaboration et al. (2023a). In the table, P and Nf refer to the probability of the classification and the number of independent modes respectively. Sectors is the number of non-contiguous sectors in which the target falls on TESS cameras calculated up to Cycle 6. The full version in electronic format is available online. |  |  |  |  $\gamma$ Dor sample |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---
TIC ID | DR3 ID | RA | Dec | Prediction | P | Nf | $f_{1}$ | $f_{2}$ | $A_{1}$ | $A_{2}$ | Sectors | Score
|  | deg | deg |  |  | d-1 | d-1 | ppt | ppt |  |  |
326258494 | 2249180128450868992 | 298.84 | 67.87 | GDOR/SPB | 0.98 | 17 | 2.62 | 2.24 | 12.21 | 0.72 | 37 | 0.79
259127682 | 2260889652408354944 | 291.59 | 67.19 | GDOR/SPB | 0.84 | 24 | 1.08 | 1.18 | 9.07 | 4.61 | 36 | 0.79
364588332 | 4648571270586910976 | 85.13 | -75.69 | GDOR/SPB | 0.91 | 22 | 1.31 | 1.30 | 9.71 | 5.38 | 35 | 0.79
|  |  |  | ⋮ |  |  |  |  |  |  |  |
|  |  |  |  $\delta$ Scuti sample |  |  |  |  |  |  |  |
TIC ID | DR3 ID | RA | Dec | Prediction | P | Nf | $f_{1}$ | $f_{2}$ | $A_{1}$ | $A_{2}$ | Sectors | Score
233617727 | 2253706710446026496 | 284.02 | 64.80 | DSCT | 0.75 | 26 | 13.02 | 13.09 | 15.41 | 3.60 | 36 | 0.77
176960346 | 5266903384176544640 | 101.21 | -70.16 | DSCT | 0.60 | 37 | 6.54 | 10.97 | 14.79 | 2.83 | 33 | 0.77
38461275 | 4670142206953240448 | 61.21 | -64.17 | DSCT | 0.62 | 36 | 9.82 | 10.53 | 8.04 | 2.46 | 32 | 0.76
|  |  |  | ⋮ |  |  |  |  |  |  |  |
|  |  |  | SPB sample |  |  |  |  |  |  |  |
TIC ID | DR3 ID | RA | Dec | Prediction | P | Nf | $f_{1}$ | $f_{2}$ | $A_{1}$ | $A_{2}$ | Sectors | Score
267543987 | 2264463198340787968 | 289.43 | 72.95 | GDOR/SPB | 0.62 | 16 | 1.28 | 1.00 | 20.90 | 0.57 | 36 | 0.66
349784439 | 5288831253807922560 | 113.20 | -62.78 | GDOR/SPB | 0.79 | 7 | 2.98 | 1.49 | 7.93 | 5.09 | 34 | 0.63
300382254 | 5269074232447890048 | 111.28 | -67.76 | GDOR/SPB | 0.64 | 14 | 1.24 | 0.62 | 16.69 | 0.80 | 34 | 0.63
|  |  |  | ⋮ |  |  |  |  |  |  |  |
|  |  |  |  $\beta$ Cep sample |  |  |  |  |  |  |  |
TIC ID | DR3 ID | RA | Dec | Prediction | P | Nf | $f_{1}$ | $f_{2}$ | $A_{1}$ | $A_{2}$ | Sectors | Score
145594454 | 5328039318762759552 | 132.55 | -49.00 | — | —- | 40 | 5.71 | 5.80 | 15.24 | 8.36 | 6 | 0.49
276171115 | 2055651749653738112 | 303.31 | 34.02 | — | — | 34 | 5.21 | 5.12 | 15.52 | 4.23 | 8 | 0.46
90964721 | 2055288395420325760 | 302.00 | 33.67 | — | — | 34 | 4.06 | 3.69 | 12.94 | 1.54 | 8 | 0.46
|  |  |  | ⋮ |  |  |  |  |  |  |  |

Finally, we present a catalogue of all the new pulsators ranked by their
asteroseismic potential (Table 3). As an example for the $\gamma$ Dor
variables, we quantify how likely they are to be true $\gamma$ Dor stars, and
how viable we believe their pulsation modes are for typical g-mode analyses
(i.e., Li et al. 2020). This table is a combination of the class prediction
probability, the number of currently available sectors (calculated using tess-
point; Burke et al. 2020), and cuts made based on prewhitening and combination
frequency removal. We weigh the contribution to the ‘score’ equally with the
following criteria; 1. The number of sectors that the target falls on a TESS
camera up to Cycle 6 of TESS observations divided by the maximum number of
sectors possible for a CVZ target. 2. The predicted class probability,
calculated as the mean probability of each sector. For targets with multiple
sectors of observation, we find the mean probability of each predicted class
and choose the class with the highest probability. Since the $\beta$ Cep
sample has no predictions, we do not include this in their scoring. 3. the
number of observed independent modes found during the prewhitening process
after removal of harmonics and combination frequencies.

We include all columns used to calculate this score for users wishing to
prioritize follow-up studies or work with alternative features, and show a few
high-probability classifications in Fig. 8.

##  5 Conclusions

In this paper we have examined the pulsators of spectral type O, B, A, or F
classified from Gaia DR3 photometry by Gaia Collaboration et al. (2023a) and
having TESS high-cadence light curves. A comparison of dominant frequencies
present in these independent light curves indicates that Gaia is extremely
good at measuring g-mode frequencies (approximately 80% precision when
compared against TESS), and reasonably effective at higher frequencies (69%).
We note that for the higher frequency p-modes, it is unclear whether the Gaia
or TESS data is more accurate for measuring the ‘true’ dominant frequency. The
30-minute cadence of the TESS data precludes accurate measurement of signals
above 24 d-1, whereas the Gaia photometry suffers from instrumental effects at
mmag level and its unequal sampling means there is no clearly defined Nyquist
limit. As such, we consider the 69% precision for the p-modes to be a worst-
case scenario.

A comparison of amplitudes for the dominant and secondary frequencies for each
variable class reveals that $\gamma$ Dor and SPB variables have
indistinguishable amplitude distributions. Prior work on variability
classification supports this result. While colour information has been used to
distinguish between the two classes of pulsators on the basis of instability
computations, Gaia Collaboration et al. (2023a) and Aerts et al. (2023) found
there to be a ‘continuum’ of g-mode pulsators bridging the predicted strips.
Said differently, both Gaia DR3 and TESS reveal that g-mode pulsators occur
along the main sequence covering an effective temperature ranging from roughly
6500 K to 18 000 K. According to instability computations available in the
literature, a fraction of these g-mode pulsators are too cold to be SPB stars
and too hot to be $\gamma\,$Dor stars. The large ranges in effective
temperature and luminosity for the Gaia DR3 $\gamma\,$Dor and SPB stars
discussed in Aerts et al. (2023) and now confirmed with TESS point to a lack
of physical ingredients in excitation predictions, such as rotation, radiative
levitation, nonlinear mode coupling (and tidal mode excitation not discussed
here). The combined Gaia DR3 and TESS light curves make us conclude that there
is one large global region of g-mode pulsations excited along the main
sequence, which is likely caused by a multitude of non-exclusive physical
phenomena. This suggestion from Aerts et al. (2023) is now solidified here
from our confirmation of the nature of these g-mode pulsators in our
catalogue, thanks to their TESS light curves. We also note that although
g-modes appear to be found across that entire temperature range, not all stars
pulsate in g-modes and not all pulsators are hybrids.

The classification of the TESS light curves reveals that the original Gaia
variability classification done by Coordination Unit 7 of the mission (see
Holl et al., 2018; Gaia Collaboration et al., 2019; Eyer et al., 2023;
Rimoldini et al., 2023; Gaia Collaboration et al., 2023a) is remarkably
accurate. For each candidate variable from Gaia, we find that the majority of
their TESS light curve classifications are in good agreement with their Gaia
classification. These results point to around 6,000 new $\gamma$ Dor, 20,000
new $\delta$ Scuti, and 1,481 new SPB pulsators confirmed by TESS. While the
TESS light curve classification is expected to be more accurate than Gaia, we
note that it is not perfect. Notably, the low-frequency g-mode pulsators are
easily confused with rotational variables.

We have made available several tables and datasets from the results of this
paper, including; prewhitened frequencies, amplitudes (in Gaia and TESS),
phases, and false alarm levels to 1% significance level for every target,
classifications and their respective probabilities for each sector of
observation, and a rank-ordered table of useful candidate pulsators. It is our
hope that the results presented here will enable future ensemble asteroseismic
studies of hot non-radial pulsators, especially with the release of Gaia DR4
and DR5, as well as with the upcoming PLATO mission (Rauer et al., 2024).

###### Acknowledgements.

The research leading to these results has received funding from the KU Leuven
Research Council (grant C16/18/005: PARADISE) and from the European Research
Council (ERC) under the Horizon Europe programme (Synergy Grant agreement
N∘101071505: 4D-STAR). While partially funded by the European Union, views and
opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Research
Council. Neither the European Union nor the granting authority can be held
responsible for them. This research was supported by the Munich Institute for
Astro-, Particle and BioPhysics (MIAPbP) which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany´s
Excellence Strategy – EXC-2094 – 390783311. The MIAPbP research program
facilitated the interactions between the two authors, which ultimately led to
the current paper.

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##  Appendix A Example classified light curves

Figure 8:  A series of prototypical classified light curves from Sec. 4.2,
where the classification probability is greater than 0.8. Note that we have
stitched multiple TESS sectors to increase the frequency resolution.


# An analogue of Stone duality via support

Henning Krause  Fakultät für Mathematik
Universität Bielefeld
D-33501 Bielefeld
Germany<EMAIL_ADDRESS>
(Date: July 23, 2023)

###### Abstract.

The notion of support provides an analogue of Stone duality, relating lattices
to topological spaces. This note aims to explain in lattice theoretic terms
what has been developed in the context of triangulated categories. In
particular, the parallel between support via closed and open sets is addressed
in terms of Hochster duality. As an application we indicate some consequences
for tensor exact categories.

###### Key words and phrases:

Exact category, Hochster duality, lattice, Stone duality, support, thick
subcategory, triangulated category

###### 2020 Mathematics Subject Classification:

18F70 (primary), 18G30 (secondary)

The traditional way of defining support assigns to an object a closed subset
of an appropriate topological space. On the other hand, there is Stone duality
which provides a correspondence between spaces and their lattices of open
subsets. This note aims to explain the connection between both concepts which
is based on Hochster duality.

Interest in this topic started with the seminal work of Balmer on support for
tensor triangulated categories [1]. Soon after the first paper appeared,
several authors noticed the connection with Stone duality via Hochster duality
[6, 11]. More recent work focusses on support for triangulated categories,
without assuming a tensor product, or at least without assuming it to be
symmetric [3, 9, 12, 13]. This note is following these ideas, without claiming
any originality. The purpose is to treat the subject purely in terms of
lattices, to make some beautiful ideas accessible for a wider audience. In
particular, one may apply the notion of support in other algebraic contexts,
beyond triangulated categories. As an illustration we indicate the rudiments
of ‘tensor exact geometry’. A further bonus is a version of Stone duality that
does not require lattices to be distributive.

##  1\. Spaces

We write $\mathbf{2}=\\{0,1\\}$ and view it either as a lattice via the usual
partial ordering or as topological space, where $\\{1\\}$ is open but not
closed. For a lattice $L$ let $L^{\mathrm{op}}$ denote its dual lattice. For a
topological space $X$ we write
$\Omega(X)=\operatorname{Hom}_{\mathbf{Top}}(X,\mathbf{2})$ for the lattice of
open sets and
$\operatorname{Cl}(X)=\operatorname{Hom}_{\mathbf{Top}}(X,\mathbf{2})^{\mathrm{op}}$
for the lattice of closed sets.

##  2\. Join-semilattices

Let $L$ be a _join-semilattice_. Thus $L$ is a poset in which all finite joins
exist, including the join over the empty set which is the unique minimal
element. A morphism of join-semilattices is a map that preserves all finite
joins. In particular, the partial order is preserved since

$a\leq b\;\iff\;a\vee b=b\quad\text{for}\quad a,b\in L.$

An _ideal_ of $L$ is a subset $I\subseteq L$ that is closed under finite joins
and such that $a\leq b$ in $L$ with $b\in I$ implies $a\in I$. Equivalently,
$I$ is of the form $\phi^{-1}(0)$ for a morphism $\phi\colon L\to\mathbf{2}$.
We write $\operatorname{Id}(L)$ for the lattice of ideals in $L$, with partial
order given by inclusion. Let us turn $\operatorname{Id}(L)$ into a
topological space which we denote $\operatorname{Sp}(L)$. Set

$\operatorname{supp}(a)\colonequals\\{I\in\operatorname{Sp}(L)\mid a\not\in
I\\}\quad\text{for}\quad a\in L.$

Note that $\operatorname{supp}(a\vee
b)=\operatorname{supp}(a)\cup\operatorname{supp}(b)$ for $a,b\in L$, and
$\bigcap_{a\in L}\operatorname{supp}(a)=\varnothing$. Thus sets of the form
$\operatorname{supp}(a)$ yield a basis of closed sets for a topology on
$\operatorname{Sp}(L)$.

######  Definition 1.

A _support datum_ on $L$ is a pair $(X,\sigma)$ consisting of a topological
space $X$ and a morphism $\sigma\colon L\to\operatorname{Cl}(X)$. A morphism
of support data $(X,\sigma)\to(Y,\tau)$ is a continuous map $f\colon X\to Y$
such that $\sigma=\operatorname{Cl}(f)\circ\tau$.

A support datum is nothing but a map that assigns to each $a\in L$ a closed
subset $\sigma(a)\subseteq X$ such that

  1. ($\varnothing$)

$\sigma(0)=\varnothing$,

  2. ($\vee$)

$\sigma(a\vee b)=\sigma(a)\cup\sigma(b)$ for all $a,b\in L$.

Predecessors of the following result are [3, Theorem 2] and [9, Proposition
5.4.2] in the context of triangulated categories.

######  Theorem 2.

The functor $X\mapsto\operatorname{Cl}(X)$ from topological spaces to join-
semilattices admits $L\mapsto\operatorname{Sp}(L)$ as a right adjoint. Thus
there is a natural bijection

(I)
$\Sigma\colon\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Sp}(L))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{JSLat}}(L,\operatorname{Cl}(X))$

which takes a continuous map $f\colon X\to\operatorname{Sp}(L)$ to the support
datum

$a\longmapsto f^{-1}(\operatorname{supp}(a))\quad\text{for}\quad a\in L.$

In particular, the pair $(\operatorname{Sp}(L),\operatorname{supp})$ is the
final support datum on $L$.

###### Proof.

It is easily checked that the map $\Sigma$ is well defined; it takes the
identity $\operatorname{Sp}(L)\to\operatorname{Sp}(L)$ to the support datum
$(\operatorname{Sp}(L),\operatorname{supp})$. Given any support datum
$(X,\sigma)$, we define $f\colon X\to\operatorname{Sp}(L)$ by setting

$f(x)\colonequals\\{a\in L\mid x\not\in\sigma(a)\\}\quad\text{for}\quad x\in
X.$

Then we have

$f^{-1}(\operatorname{supp}(a))=\\{x\in X\mid a\not\in
f(x)\\}=\sigma(a)\quad\text{for}\quad a\in L.$

Thus $f$ is continuous and we see that $\Sigma$ is surjective. For the
injectivity of $\Sigma$, observe that for any map $f\colon
X\to\operatorname{Sp}(L)$ and $x\in X$ we have

$f(x)=\\{a\in L\mid a\in f(x)\\}=\\{a\in L\mid x\not\in
f^{-1}(\operatorname{supp}(a))\\}.$

Thus $f$ is determined by $\Sigma(f)$. ∎

######  Remark 3.

(1) The assignment $\phi\mapsto\phi^{-1}(1)$ identifies
$\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2}^{\mathrm{op}})$ with the
ideal lattice $\operatorname{Id}(L)$. Taking $a\in L$ to the principal ideal
$\\{b\in L\mid b\leq a\\}$ identifies $L$ with the poset of _compact elements_
in $\operatorname{Id}(L)$.

(2) The join-semilattice $L$ can be recovered from the space
$\operatorname{Sp}(L)$ as follows. The _specialisation order_

$x\leq
y\;:\iff\;x\in\operatorname{cl}\\{y\\}\qquad(x,y\in\operatorname{Sp}(L))$

recovers the partial order on $\operatorname{Id}(L)=\operatorname{Sp}(L)$ that
is given by the inclusion of ideals. Taking the lattice $\operatorname{Id}(L)$
to its poset of compact elements yields $L$.

(3) The assignment $\phi\mapsto\phi^{-1}(0)$ identifies
$\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2})$ with
$\operatorname{Sp}(L)$ as sets. This yields the identification

$\operatorname{supp}(a)=\\{\phi\in\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2})\mid\phi(a)=1\\}\quad\text{for}\quad
a\in L.$

The choice of the topology (by declaring $\operatorname{supp}(a)$ to be
closed) suggests to write
$\operatorname{Sp}(L)=\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2})^{\vee}$.
Then one may rewrite the bijection (I) as

$\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2})^{\vee})\simeq\operatorname{Hom}_{\mathbf{JSLat}}(L,\operatorname{Hom}_{\mathbf{Top}}(X,\mathbf{2})^{\mathrm{op}}).$

##  3\. Bounded lattices

Let $L$ be a _bounded lattice_. Thus $L$ is a poset in which all finite joins
and all finite meets exist. A morphism of bounded lattices is a map that
preserves all finite joins and all finite meets. An ideal $I\subseteq L$ is
_prime_ if $I\neq L$ and $a\wedge b\in I$ implies $a\in I$ or $b\in I$.
Equivalently, $I$ is of the form $\phi^{-1}(0)$ for a morphism $\phi\colon
L\to\mathbf{2}$. We write $\operatorname{Spc}(L)$ for the set of prime ideals
of $L$ and turn this into a topological space. Set

$\operatorname{supp}(a)\colonequals\\{I\in\operatorname{Spc}(L)\mid a\not\in
I\\}\quad\text{for}\quad a\in L.$

Note that $\operatorname{supp}(a\vee
b)=\operatorname{supp}(a)\cup\operatorname{supp}(b)$ for $a,b\in L$, and
$\bigcap_{a\in L}\operatorname{supp}(a)=\varnothing$. Thus sets of the form
$\operatorname{supp}(a)$ yield a basis of closed sets for a topology on
$\operatorname{Spc}(L)$.

######  Definition 4.

A _closed support datum_ on $L$ is a pair $(X,\sigma)$ consisting of a
topological space $X$ and a morphism $\sigma\colon L\to\operatorname{Cl}(X)$.
A morphism of closed support data $(X,\sigma)\to(Y,\tau)$ is a continuous map
$f\colon X\to Y$ such that $\sigma=\operatorname{Cl}(f)\circ\tau$.

A closed support datum is nothing but a map that assigns to each $a\in L$ a
closed subset $\sigma(a)\subseteq X$ such that

  1. ($\varnothing$)

$\sigma(0)=\varnothing$ and $\sigma(1)=X$,

  2. ($\vee$)

$\sigma(a\vee b)=\sigma(a)\cup\sigma(b)$ for all $a,b\in L$,

  3. ($\wedge$)

$\sigma(a\wedge b)=\sigma(a)\cap\sigma(b)$ for all $a,b\in L$.

The predecessor of the following result is [1, Theorem 3.2] in the context of
tensor triangulated categories.

######  Theorem 5.

The functor $X\mapsto\operatorname{Cl}(X)$ from topological spaces to bounded
lattices admits $L\mapsto\operatorname{Spc}(L)$ as a right adjoint. Thus there
is a natural bijection

$\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Spc}(L))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{BLat}}(L,\operatorname{Cl}(X))$

which takes a continuous map $f\colon X\to\operatorname{Spc}(L)$ to the closed
support datum

$a\longmapsto f^{-1}(\operatorname{supp}(a))\quad\text{for}\quad a\in L.$

In particular, the pair $(\operatorname{Spc}(L),\operatorname{supp})$ is the
final support datum on $L$.

###### Proof.

The proof of Theorem 2 carries over without any change. The additional
structure on $L$ (given by the join $\wedge$) corresponds to the additional
condition on the ideals in $\operatorname{Spc}(L)$ (to be prime). ∎

##  4\. Hochster duality

Let $L$ be a bounded lattice. We consider the space $\operatorname{Spc}(L)$ of
prime ideals and provide a dual topology on this set. Observe that
$\operatorname{supp}(a\wedge
b)=\operatorname{supp}(a)\cap\operatorname{supp}(b)$ for all $a,b\in L$, and
$\bigcup_{a\in L}\operatorname{supp}(a)=\operatorname{Spc}(L)$. Thus sets of
the form $\operatorname{supp}(a)$ yield a basis of open sets for a topology on
$\operatorname{Spc}(L)$. We denote this space $\operatorname{Spc}(L)^{\vee}$
and call it the _Hochster dual_ of $\operatorname{Spc}(L)$. Note that

$\operatorname{Spc}(L)^{\vee}\cong\operatorname{Spc}(L^{\mathrm{op}})$

since

$\operatorname{Hom}_{\mathbf{BLat}}(L,\mathbf{2}^{\mathrm{op}})=\operatorname{Hom}_{\mathbf{BLat}}(L^{\mathrm{op}},\mathbf{2}).$

######  Definition 6.

An _open support datum_ on $L$ is a pair $(X,\sigma)$ consisting of a
topological space $X$ and a morphism $\sigma\colon L\to\Omega(X)$. A morphism
of open support data $(X,\sigma)\to(Y,\tau)$ is a continuous map $f\colon X\to
Y$ such that $\sigma=\Omega(f)\circ\tau$.

There is a natural bijection

$\displaystyle\operatorname{Hom}_{\mathbf{BLat}}(L,\Omega(X))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{BLat}}(L^{\mathrm{op}},\operatorname{Cl}(X))$

that takes an open support datum $(X,\sigma)$ to the closed support datum
$(X,\tau)$ with

$\tau(a)\colonequals X\setminus\sigma(a)\quad\text{for}\quad a\in
L^{\mathrm{op}}.$

This yields the following reformulation of Theorem 5.

######  Theorem 7.

The functor $X\mapsto\Omega(X)$ from topological spaces to bounded lattices
admits $L\mapsto\operatorname{Spc}(L)^{\vee}$ as a right adjoint. Thus there
is a natural bijection

(II)
$\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Spc}(L)^{\vee})\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{BLat}}(L,\Omega(X)).\qed$

######  Remark 8.

(1) The usual setting for Hochster duality are _spectral spaces_ , so spaces
of the form $\operatorname{Spc}(L)$ for some bounded distributive lattice $L$
[10, II.3.4].

(2) The adjunction formula (II) may be rewritten as

$\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Hom}_{\mathbf{BLat}}(L,\mathbf{2}))\simeq\operatorname{Hom}_{\mathbf{BLat}}(L,\operatorname{Hom}_{\mathbf{Top}}(X,\mathbf{2})).$

##  5\. Frames

We recall some well known facts from Stone duality. A _frame_ is a complete
lattice in which finite meets distribute over arbitrary joins, so

$a\wedge\big{(}\bigvee_{i}b_{i}\big{)}=\bigvee_{i}(a\wedge
b_{i})\quad\text{for all}\quad a,b_{i}\in F.$

A morphism of frames is a map that preserves all joins and finite meeets. The
functor sending a topological space $X$ to its frame $\Omega(X)$ of open sets
admits a right adjoint, which sends a frame $F$ to its space
$\operatorname{Pt}(F)$ of points [10, II.1.4]. A _point_ of $F$ is by
definition a frame morphism $F\to\mathbf{2}$, and an open set is one of the
form

$U(a)=\\{\phi\in\operatorname{Pt}(F)\mid\phi(a)=1\\}\quad\text{for some}\quad
a\in F.$

This adjunction amounts to a natural bijection

(III) $\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Pt}(F))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{Frm}}(F,\Omega(X)).$

A frame $F$ is _spatial_ if there are enough points, which means that the unit
of the adjunction (given by evaluation) yields an isomorphism

(IV) $F\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\Omega(\operatorname{Pt}(F)).$

Let $L$ be a bounded lattice that is distributive. Then its ideal lattice
$\operatorname{Id}(L)$ is a spatial frame [10, II.3.4]. A frame isomorphic to
one of the form $\operatorname{Id}(L)$ is called _coherent_. Let us consider
the embedding $L\to\operatorname{Id}(L)$ which takes $a\in L$ to the principal
ideal

$\left\downarrow a\right.\colonequals\\{b\in L\mid b\leq a\\}.$

######  Lemma 9.

Restriction along $L\to\operatorname{Id}(L)$ induces for any frame $F$ a
natural bijection

(V) $\operatorname{Hom}_{\mathbf{Frm}}(\operatorname{Id}(L),F)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{BLat}}(L,F).$

###### Proof.

The inverse map takes a morphism $\phi\colon L\to F$ to
$\operatorname{Id}(L)\to F$ given by

$I\longmapsto\bigvee_{a\in I}\phi(a)\quad\text{for}\quad
I\in\operatorname{Id}(L).\qed$

Take $F=\mathbf{2}$. Then the above bijection identifies each point
$\phi\in\operatorname{Pt}(\operatorname{Id}(L))$ with the prime ideal
$\phi^{-1}(0)\cap L$ of $L$, and this yields an isomorphism

(VI) $\operatorname{Pt}(\operatorname{Id}(L))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spc}(L)^{\vee}$

between the space of points of $\operatorname{Id}(L)$ and the Hochster dual of
$\operatorname{Spc}(L)$. More precisely, for any ideal
$I\in\operatorname{Id}(L)$ we have

$U(I)=\bigcup_{a\in I}U(\left\downarrow a\right.)$

and (VI) identifies $U(\left\downarrow a\right.)$ with
$\operatorname{supp}(a)$ for each $a\in L$.

######  Corollary 10.

Let $L$ be a bounded lattice that is distributive. Then the assignment

$I\longmapsto\bigcup_{a\in I}\operatorname{supp}(a)$

yields an isomorphism

(VII) $\operatorname{Id}(L)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\Omega(\operatorname{Spc}(L)^{\vee}).$

###### Proof.

The isomorphism is a consequence of (IV) since $\operatorname{Id}(L)$ is a
spatial frame; its inverse sends an open set $U$ to $\\{a\in
L\mid\operatorname{supp}(a)\subseteq U\\}$. ∎

######  Remark 11.

(1) Stone duality yields an alternative proof of Theorem 7 when the lattice
$L$ is distributive, by combining (III), (V), and (VI).

(2) The adjunction formula (III) may be rewritten as

$\operatorname{Hom}_{\mathbf{Top}}(X,\operatorname{Hom}_{\mathbf{Frm}}(F,\mathbf{2}))\simeq\operatorname{Hom}_{\mathbf{Frm}}(F,\operatorname{Hom}_{\mathbf{Top}}(X,\mathbf{2})).$

(3) With the canonical identification
$\operatorname{Id}(L)=\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2}^{\mathrm{op}})$
for a join-semilattice $L$, the isomorphism (VI) may be rewritten as

$\operatorname{Hom}_{\mathbf{Frm}}(\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2}^{\mathrm{op}}),\mathbf{2})\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{BLat}}(L,\mathbf{2})$

whereas the isomorphism (VII) becomes

$\operatorname{Hom}_{\mathbf{JSLat}}(L,\mathbf{2}^{\mathrm{op}})\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Hom}_{\mathbf{Top}}(\operatorname{Hom}_{\mathbf{BLat}}(L,\mathbf{2}),\mathbf{2}).$

##  6\. Triangulated categories and beyond

Let $\mathcal{T}$ be an essentially small triangulated category and let
$\operatorname{Ob}\mathcal{T}$ denote the class of objects. For objects $X,Y$
in $\mathcal{T}$ we write

$X\sim Y\;:\iff\;[X]=[Y]$

where $[X]$ denotes the thick subcategory generated by $X$. This provides an
equivalence relation and the set of equivalence classes

$L(\mathcal{T})\colonequals\operatorname{Ob}\mathcal{T}/{\sim}$

is partially ordered via inclusion; it is a join-semilattice with

$[X]\vee[Y]=[X\oplus Y].$

Let $\operatorname{Thick}(\mathcal{T})$ denote the lattice of thick
subcategories of $\mathcal{T}$.

######  Lemma 12.

The assignment

$\mathcal{T}\supseteq\mathcal{S}\longmapsto\\{[X]\in L(\mathcal{T})\mid
X\in\mathcal{S}\\}\subseteq L(\mathcal{T})$

yields a lattice isomorphism
$\operatorname{Thick}(\mathcal{T})\xrightarrow{\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}}\operatorname{Id}(L(\mathcal{T}))$.

###### Proof.

The inverse map sends an ideal $I\subseteq L(\mathcal{T})$ to the full
subcategory of $\mathcal{T}$ that is given by the objects $X$ with $[X]\in I$.
∎

Now let $(\mathcal{T},\otimes)$ be a tensor triangulated category, i.e. a
triangulated category equipped with a monoidal structure (not necessarily
symmetric) which is exact in each variable. A thick subcategory
$\mathcal{S}\subseteq\mathcal{T}$ is a _tensor ideal_ if $X\otimes Y$ is in
$\mathcal{S}$ provided that one of $X,Y$ is in $\mathcal{S}$, and an ideal is
_radical_ if $X$ is in $\mathcal{S}$ provided that $X\otimes X$ is in
$\mathcal{S}$.

For objects $X,Y$ in $\mathcal{T}$ we write

$X\approx Y\;:\iff\;\langle X\rangle=\langle Y\rangle$

where $\langle X\rangle$ denotes the radical thick tensor ideal generated by
$X$. We obtain an equivalence relation and the set of equivalence classes

$L(\mathcal{T},\otimes)\colonequals\operatorname{Ob}\mathcal{T}/{\approx}$

is partially ordered via inclusion.

######  Lemma 13.

For objects $X,Y$ in $\mathcal{T}$ we have $\langle X\rangle\cap\langle
Y\rangle=\langle X\otimes Y\rangle$.

###### Proof.

One inclusion is clear. Let $Z\in\langle X\rangle\cap\langle Y\rangle$. Then
we compute

$\langle Z\rangle\subseteq\langle Z\otimes Z\rangle\subseteq\langle Z\otimes
Y\rangle\subseteq\langle X\otimes Y\rangle.$

Thus $Z\in\langle X\otimes Y\rangle$. ∎

We record the basic properties of $L(\mathcal{T},\otimes)$.

######  Proposition 14.

Let $(\mathcal{T},\otimes)$ be a tensor triangulated category. Then the poset
$L(\mathcal{T},\otimes)$ is a distributive lattice and its ideal lattice
$\operatorname{Id}(L(\mathcal{T},\otimes))$ identifies with the lattice of
radical thick tensor ideals of $\mathcal{T}$.

###### Proof.

For objects $X,Y$ in $\mathcal{T}$ we have

$\langle X\rangle\vee\langle Y\rangle=\langle X\oplus
Y\rangle\qquad\text{and}\qquad\langle X\rangle\wedge\langle Y\rangle=\langle
X\otimes Y\rangle$

thanks to Lemma 13. The tensor product distributes over sums and this implies
the distributivity of $L(\mathcal{T},\otimes)$. The second assertion is the
analogue of Lemma 12 and its proof carries over without change. ∎

Let us mention a couple of consequences. For example, Corollary 10 appplies
and yields a description of the lattice of radical thick tensor ideals of
$\mathcal{T}$ in terms of the space
$\operatorname{Spc}(L(\mathcal{T},\otimes))$ [1, Theorem 10.4]. Also, we see
that the lattice of radical thick tensor ideals of $\mathcal{T}$ is a coherent
frame [11, Theorem 3.1.9].

We conclude that the material developed in the previous sections can be
applied towards the description of thick subcategories and thick tensor
ideals. For some recent applications of this lattice theoretic approach, see
for example [4].

The thoughtful reader will notice that other settings (different from
triangulated categories) are perfectly feasible. Classical examples are
categories of modules or sheaves and their Serre subcategories; this is in
line with Gabriel’s reconstruction of a noetherian scheme from its category of
coherent sheaves [8].

##  7\. Tensor exact geometry

For exact categories we indicate the rudiments of ‘tensor exact geometry’
which may be viewed as the analogue of tensor triangular geometry [2]. Let
$(\mathcal{C},\otimes)$ be a _tensor exact category_ ; thus $\mathcal{C}$ is
an exact category in the sense of Quillen (so equipped with a distinguished
class of exact sequences) and there is a monoidal structure (not necessarily
symmetric) which is exact in each variable. A _thick subcategory_ is a full
additive subcategory $\mathcal{B}\subseteq\mathcal{C}$ satisfying the _two-
out-of-three property_ , so any exact sequence from $\mathcal{C}$ lies in
$\mathcal{B}$ if two of its three terms are in $\mathcal{B}$. We make the same
definitions as before for a tensor triangulated category and obtain with same
proof the analogue of Proposition 14.

######  Proposition 15.

Let $(\mathcal{C},\otimes)$ be a tensor exact category. Then the poset
$L(\mathcal{C},\otimes)$ is a distributive lattice and its ideal lattice
$\operatorname{Id}(L(\mathcal{C},\otimes))$ identifies with the lattice of
radical thick tensor ideals of $\mathcal{C}$.∎

A cornerstone of tensor triangular geometry is the classification of radical
thick tensor ideals [1, Theorem 10.4]; its analogue for tensor exact
categories is now a consequence of Corollary 10.

######  Corollary 16.

Taking an object to its support yields an isomorphism between the lattice of
radical thick tensor ideals of $\mathcal{C}$ and
$\Omega(\operatorname{Spc}(L(\mathcal{C},\otimes))^{\vee})$.∎

######  Example 17.

Let $G$ be a finite group scheme over a field $k$. We consider the category
$(\mathcal{C},\otimes)=(\operatorname{mod}kG,\otimes)$ of finite dimensional
$k$-linear representations of $G$ with the usual tensor exact structure. We
write $H^{*}(G,k)$ for the cohomology ring of $G$ with coefficients in $k$ and
$\operatorname{Spec}(H^{*}(G,k))$ for its spectrum of homogeneous prime ideals
(with the Zariski topology). Then results from [5, 7] yield a homeomorphism

$\operatorname{Spec}(H^{*}(G,k))\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}\operatorname{Spc}(L(\operatorname{mod}kG,\otimes))$

given by

$H^{*}(G,k)\supseteq\mathfrak{p}\longmapsto\\{M\in\operatorname{mod}kG\mid\operatorname{Ext}_{kG}^{*}(M,M)_{\mathfrak{p}}=0\\}\subseteq
L(\operatorname{mod}kG,\otimes).$

Let $\mathbf{D}^{b}(\operatorname{mod}kG)$ denote the bounded derived category
of $\operatorname{mod}kG$ with the induced tensor structure. It is interesting
to note that the inclusion
$\operatorname{mod}kG\to\mathbf{D}^{b}(\operatorname{mod}kG)$ induces an
isomorphism

$L(\mathbf{D}^{b}(\operatorname{mod}kG),\otimes)\xrightarrow{\
\raisebox{-1.72218pt}[0.0pt][0.0pt]{$\scriptstyle{\sim}$}\
}L(\operatorname{mod}kG,\otimes),$

reconciling triangulated and exact structure.

### Acknowledgements

It is a pleasure to thank Greg Stevenson for several helpful discussions.
Also, I am grateful to Paul Balmer for sharing some private notes [3]. This
work was supported by the Deutsche Forschungsgemeinschaft (SFB-TRR 358/1 2023
- 491392403).

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