# Orbital and stellar parameters for 2M06464003+0109157: a double-lined
eclipsing binary of spotted, sub-solar twins

Annaliese Miller  Marina Kounkel  Chase Boggio  Kevin Covey  Western
Washington University, 516 High St., Bellingham, WA 98225, USA  Adrian M.
Price-Whelan  Center for Computational Astrophysics, Flatiron Institute, New
York, NY, 10010, USA

###### Abstract

We calculated physical and orbital properties for 2M06464003+0109157, a 1.06
day eclipsing double lined spectroscopic binary. We modelled the system’s
ASAS-SN and TESS light curves, measuring the system’s inclination and radii of
each component. Extracting radial velocities for each component from 6
SDSS/APOGEE spectra, we measured the system’s mass ratio and performed a full
orbital fit. Our analysis indicates that 2M06464003+0109157 has components
with nearly equal masses ($m_{1}/m_{2}=0.99\pm 0.01$;
$M_{1}{{}_{,}}{{}_{2}}=~{}0.57\pm 0.015$ M☉) and comparable radii
($R_{1}=~{}0.66\pm 0.05$ R☉, $R_{2}=~{}0.57\pm 0.06$ R☉). The solution
required two star spots to incorporate the out of eclipse variation that is
seen in the light curve. We report our full characterization of this system,
and prospects for similar analyses using survey data to measure precise
physical and orbital properties for EBs.

Eclipsing binary stars, Light curves, Radial velocity, Starspots,
Spectroscopic binary stars

††software: ellc (Maxted, 2016), eleanor (Feinstein et al., 2019), The Joker
(Price-Whelan et al., 2017, 2018),emcee (Foreman-Mackey et al., 2013)

##  1 Introduction

Eclipsing binaries (EBs) are extremely valuable objects, as they provide an
opportunity to test stellar evolutionary models. Through characterizing a
light curve of an EB, it is possible to constrain not only the inclination and
eccentricity of an orbit, but also measure the radius of both stars (as a
fraction of their semi-major axis). Furthermore, if complementary high
resolution multi-epoch spectra are available, and both stars have a
sufficiently high luminosity ratio and large enough separation in their
respective radial velocities (RVs), the system may be detectable as a double-
lined spectroscopic binary (SB2). In this case, from analyzing the RVs, it is
possible to solve the full orbit and determine masses and true radii of the
individual stars. Thus, EBs that are also SB2s can help to better constrain
the mass-radius relation. There have been previous studies of the mass-radius
relation on some stars in the past (e.g., Demircan & Kahraman, 1991; Feiden &
Chaboyer, 2012; Han et al., 2019; Southworth, 2015), however the number of
such systems is still relatively low, particularly on the low mass end, as
identification and analysis of EBs and SB2s require numerous photometric and
spectroscopic observations spread over a large baseline.

In recent years, large surveys have made it possible to increase the sample of
EBs with known masses and radii. In this study, we analyze a specific EB
system to validate a workflow for deriving stellar & orbital parameters using
only serendipitous survey measurements, without any dedicated follow-up
observations. By analyzing All-Sky Automated Survey for Supernovae (ASAS-SN)
photometric monitoring of sources that had also been observed by Apache Point
Observatory Galactic Evolution Experiment (APOGEE), Pawlak et al. (2019)
identified 430 EBs and reported their period estimates. We examined these
systems using routines previously developed to identify and analyze SB2s in
the APOGEE dataset (Kounkel et al., 2019). In total we identified 53 that had
a clear and well-resolved presence of both the primary and the secondary in
the cross-correlation function of the individual spectral visits.
Unfortunately, for most of these EBs, only a few epochs of APOGEE spectra are
available, making it difficult to independently reconstruct a full orbit.

In this pilot study we characterize the properties of one system in
particular, 2M06464003+0109157, hereafter 2M0646. This system is an EA type
eclipsing binary with a 1.06493 day period (Pawlak et al., 2019). At the time
of the analysis, this particular system was a detached binary with the largest
number of RV epochs to construct a full orbit with clear phasing of RVs to the
eclipsing period.

2M0646 has kinematics and metallicity that are consistent with the stellar
population typically identified as the Galaxy’s Thick Disk component. Located
just over 100 pc away, it has very high proper motion, in excess of 250 mas
yr-1, or over 120 km s-1. When combined with the system’s center of mass
velocity ($\sim$67 km/s; see section 4.2) and distance, these proper motions
imply UVW velocities (U${}_{lsr}=-19.2$ km s-1; V${}_{lsr}=-130.3$ km s-1;
W${}_{lsr}=-20.1$ km s-1) that are consistent with thick disk membership (see
Toomre diagrams in, e.g. Bensby et al., 2005; Reddy et al., 2006; Nissen &
Schuster, 2009). Furthermore, 2M0646 has been reported to have low metallicity
(Fe/H $\sim$ -0.6 dex, Ahumada et al., 2020). Both of these point to the
system likely being quite old, with the age of $\sim$10 Gyr (e.g., Kilic et
al., 2017).

Although originally identified as an EB using the ASAS-SN data, 2M0646 has now
been observed by TESS, allowing for a more pristine light curve analysis. The
system was also observed in 6 epochs of the APOGEE spectra, with individual
epochs well separated in the phase space for a full orbital fitting, making it
a suitable target for mass and radius determination of both stars in the
system.

In Section 2 we describe the data available for 2M0646, both spectra and light
curves. In Section 3 we perform fitting of all the available data, including
light curves and the measured radial velocities to determine masses and radii
of each star. Finally, we conclude in Section 4 by comparing the properties of
this system to current mass-radius relations.

##  2 Identification and Observations

###  2.1 APOGEE spectra and RVs

The APOGEE spectrograph (Wilson et al., 2010, 2012; Majewski et al., 2017)
covers the wavelength range of the H band (1.51-1.7 $\micron$), with a
spectral resolution of R$\sim$22,500. 2M0646 has been observed by APOGEE over
the course of 6 epochs, from HJD of 2456312 to 2456673, with distinct spectral
reductions in the 14th and 16th data releases (DR14 and DR16) of the Sloan
Digital Sky Survey (Blanton et al., 2017; Abolfathi et al., 2018; Ahumada et
al., 2020). Olney et al. (2020) report parameters for 2M0646 of Teff
=3870$\pm$26 K and $\log g$ = 4.416$\pm$0.027 dex, consistent with a low mass
main sequence dwarf. Using the DR14 reduction of the data, 2M0646 is reported
to have a metallicity of [Fe/H] = -0.302$\pm$0.016 dex; however, it drops to
-0.6 using the DR16 reduction. We should note, that all of these pipelines are
assuming a single star. It is difficult to estimate how an SB2 should affect
the stellar parameters, though Olney et al. (2020) note that they do not
appear to systematically affect $T_{eff}$ or $\log g$ distribution compared to
single stars in a single age populations. Thus, most likely, they represent
the parameters of the flux weighted mean of the two stars, which should
usually gravitate to that of the primary.

In examining the cross-correlation function (CCF) of the system’s APOGEE
spectra (Nidever et al., 2015), we found that there was a signature of two
strong components, indicating that this system is a double lined spectroscopic
binary. In the DR14 data, components can be resolved in 4 out of 6 epochs,
with the other two epochs unresolved due to the closeness of the velocities of
the two stars. In the DR16 reduction, both components are resolved in all 6
epochs. The velocities of these components were extracted by fitting a
Gaussian profile to the CCF using GaussPy (Lindner et al., 2015). These
velocities extracted from each reduction are presented in Table 1; when
resolved in both data releases, individual component velocities are typically
consistent to within 1 sigma, though the velocities measured at MJD=56316.7579
do differ by $\sim$2 sigma in each component. At the two epochs where the two
components are not resolved in the DR14 CCFs, we report the velocity measured
from the system’s unresolved CCF in the $v_{1}$ column, but note this quantity
may better represent the system’s bulk heliocentric velocity than the velocity
of any individual component at this epoch.

We report velocities measured from both reductions, and we use DR16 reduction
for the orbital fit (discussed in Section 3.2).

Table 1: Radial velocities for 2M0646, measured from APOGEE spectra Date |  $v_{1}$ (DR14) |  $v_{2}$ (DR14) |  $v_{1}$ (DR16) |  $v_{2}$ (DR16)
---|---|---|---|---
MJD | km s-1 | km s-1 | km s-1 | km s-1
56312.7647605 | 17.6$\pm$4.57 | 114.5$\pm$6.4 | 17.4$\pm$2.9 | 110.8$\pm$4.4
56313.7623219 | -15.2$\pm$3.0 | 145.4 $\pm$ 4.3 | -17.6$\pm$3.4 | 143.6$\pm$6.0
56316.7578667 | -27.1$\pm$2.7 | 162.3 $\pm$ 4.2 | -30.5$\pm$1.5 | 168.4$\pm$3.3
56349.676885 | 16.2$\pm$6.01 | 114.9$\pm$9.4 | 18.2$\pm$ 3.7 | 114.6$\pm$5.7
56350.6830319 | 65.2$\pm$2.6 | N/A | 45.7$\pm$18.2 | 82.9$\pm$27.1
56672.8065246 | 64.2$\pm$ 2.5 | N/A | 90.9$\pm$14.7 | 52.5$\pm$17.1

###  2.2 Light curves

Figure 1: TESS light curve for 2M0646 (black dots) overlaid by multiple ellc
models computed using maximum likelihood stellar & orbital parameters with
differing amounts of star spots. Figure 2: Final jointly fit LC models for the
TESS (top), ASAS-SN V-band (middle) and ASAS-SN g-band (bottom) light curves.
Spot modulation is best fit in the TESS light curve, as out-of-eclipse
datapoints in ASAS-SN light curves were intentionally down-weighted. The spot
modulation signature that is fit to the TESS data appears consistent with the
ASAS-SN g-band data, which is consistent with the relatively small temporal
lag between the two datasets; the ASAS-SN V-band data was obtained somewhat
earlier, and thus agrees less well with the spot modulation seen in the TESS
data.

2M0646 was identified as a $P=1.06493$ d eclipsing binary (EB) by Pawlak et
al. (2019) using light curves from the ASAS-SN survey (Kochanek et al., 2017).
ASAS-SN conducts nightly observations from multiple observing stations around
the world. Each station contains four 14 cm aperture telephoto lenses, each
equipped with a CCD camera that provides a field of view of 4.5 deg2. ASAS-SN
has observed 2M06464 over 1878 epochs from December 2014 to December 2020 at
the time of analysis. Between 2014 and 2018, all epochs were observed in
V-band. In 2018, ASAS-SN brought online their first g-band cameras and thus
the observations were able to continue in both g-band and V-band. Even with
the two different bands, the light curve was still sparsely populated.

These data have been since superseded by the light curves from Transiting
Exoplanet Survey Satellite (TESS, Ricker et al., 2015), which offer better
cadencing and precision in the flux. TESS began its survey in July 2018. It is
equipped with four CCD cameras that provide a simultaneous coverage of a 96x24
deg2 area on the sky. Each sector of the sky is observed for a period of
$\sim$27.4 days with a 30 minute cadence. 2M0646 is located in sector 6,
producing a TESS light curve that spans the period from December 15th 2018 to
January 7th 2019.

We used the eleanor pipeline (Feinstein et al., 2019) to download and extract
the light curve of 2M0646 from the TESS full frame images archived in STScI’s
MAST database. This pipeline provides an option for several different
calibrations to detrend the light curves and reduce the systematics of the
instrument. We adopt the light curve produced by the point spread function
modelling, which performs well for brighter targets such as 2M0646.

The light curve is extracted with the photometry in the units of flux.
However, the light curve fitting routine (Section 3.1) expects light curves to
be magnitudes. We convert the flux to the T magnitude, ensuring that the
average flux of the system is consistent with the magnitude reported in the
TESS Input Catalog (T=11.442 mag, Stassun et al., 2019). The resulting light
curve, phase-folded on a 1.0649d period, is shown in Figure 1, which shows a
significant improvement in precision and coverage compared to the ASAS-SN data
in Figure 2. In addition to better sampling of the eclipse themselves, the
TESS light curve also reveals the presence of out-of-eclipse variations likely
due to starspots.

Table 2: Spot Parameter Study – System Parameters  |  | One Spot | Two Spots |
---|---|---|---|---
Parameter | units | per component | on Primary | No Spots
Orbital Parameters
Period | days | 1.065$\pm$0.019 | 1.065$\pm$0.008 | 1.065$\pm$0.009
Inclination | ∘ | 83.4$\pm$1.6 | 83.3$\pm$8.5 | 83.7$\pm$8.1
Stellar Parameters
Primary Radius | (R/a) | 0.136$\pm$0.025 | 0.127$\pm$0.052 | 0.141 $\pm$ 0.059
Secondary Radius | (R/a) | 0.136$\pm$0.024 | 0.143$\pm$0.067 | 0.136 $\pm$ 0.061
Surface Brightness Ratio | N/A | 1.00$\pm$0.11 | 0.96$\pm$0.16 | 0.90$\pm$ 0.22
Table 3: Spot Parameter Study – Starspot Parameters  | Spot |  |  |
---|---|---|---|---
| Size | Longitude | Latitude | Spot
Model | degrees (∘) | degrees (∘) | degrees (∘) | Contrast
One Spot Per Component
Primary Spot | 34$\pm$16 | 214$\pm$53 | 52$\pm$25 | 0.68$\pm$0.25
Secondary Spot | 31$\pm$15 | 210$\pm$140 | 35$\pm$31 | 0.70$\pm$0.24
Two Spots on Primary
Sub-equatorial Spot | 37$\pm$15 | 210$\pm$70 | -35$\pm$31 | 0.72 $\pm$ 0.28
Equatorial Spot | 36$\pm$17 | 205$\pm$92 | -1$\pm$38 | 0.84 $\pm$ 0.31

##  3 Measuring Stellar and Orbital Parameters from LC & RV fits

###  3.1 Light curve solutions: Degeneracies due to starspots

ellc (Maxted, 2016) is a detached binary star modeler that is designed to
allow a Markov Chain Monte Carlo (MCMC) sampling mode. We used ellc, as driven
by the emcee MCMC package Foreman-Mackey et al. (2013) to fit 2M0646’s light
curve. Before performing a joint fit of the ASAS-SN, TESS, and APOGEE
observations of this system (see next section), we conducted a parameter study
to identify the sensitivity of the model results to the number and location of
the starspots adopted in the fitting procedure. As the spot modulation is more
precisely measured, and less temporally blurred, in the TESS light curve, we
focused this parameter study on fitting that light curve in isolation.

For each trial in this parameter study, we initialized 200 MCMC walkers and
computed chains for each walker with 20,000 steps. Starting with the broad
distribution of parameters that were largely unconstrained, we allowed the
model to converge and examined the outputs. Based on these solutions, we put
increasingly restrictive priors on the model parameters to exclude unphysical
solutions (e.g., negative periods, face-on inclination, unrealistic ratio of
the two radii, etc.), without cutting into the physically plausible parameter
space to which most of the chains had converged.

This process identified a core set of well motivated priors for the system’s
stellar and orbital parameters, which we used as consistent initial conditions
while varying the number and location of spots in the model. Specifically, we
set a Gaussian prior on the period, centered at 1.06 days with a width of
0.025 days. The stellar radii are expressed by ellc as $R/a$, where $R$ is the
physical radius of a given star, and $a$ is the semi-major axis of the system,
thus the radius can be inferred from the width of the system’s eclipses in
phase space. We set a Gaussian prior on $R/a$ centered on $R/a$ $\sim$ 0.13
with a width of $\pm$0.06. The inclination of the system was set at 83$\pm$15
degrees, and the surface brightness ratio as 0.95$\pm$0.25. Eccentricity was
set as zero. The time of passage of the eclipse $T_{0}$ was restricted with a
uniform prior from 0 to 1.1 (to prevent the solution wrapping around the
phase), and was set centered at 0.75 days.

Using these priors, we then attempted to fit the TESS light curve with models
assuming no star spots, one spot (on the system’s primary), one spot per star,
and two spots on the primary component. The best fitting model from each of
these spot configurations is shown in Figure 1, and system and spot parameters
are given in Tables 2 and 3, respectively. As Figure 1 shows, the best fit
from the no-spot model is unable to reproduce the system’s out-of-eclipse
variations. Adding a single spot to the model significantly improves the fit,
but still leaves systematic residuals in phase.

Ultimately, we find two spots are necessary to fully fit the out-of-eclipse
light curve. In the two spots, one-per-component model, the spots were
initialized with a size of $40\pm 20$ degrees, a longitude ($l$) of $180\pm
180$ degrees, a latitude ($b$) of $30\pm 30$ degrees, and a brightness factor
of $0.7\pm 0.3$. At the final initialization of the system, only one uniform
prior was placed on $l$ from 0 to 360 degrees to keep it physically plausible.
However, the model is not sensitive to the distribution of spots - the fit is
comparable if both spots are located on the same stars versus on one star
each. Similarly, the model is not sensitive to discriminating between dark
spots and bright faculae. The resulting parameters across different models are
shown in Table 2. The preferred spot sizes ($\sim 35$ degrees) and contrasts
($\sim 70$%) are consistent across both dual-spot configurations.

Although the average $l$ and $b$ show a wide distribution in Table 3, this
average is a superposition of various discrete locations that are equally
favored. If faculae are used, they tend to be positioned facing towards the
other star in the binary, whereas darker spots tend to be located on the
opposite side. We interpret this as two stars that are tidally locked
irradiating each other’s photosphere, and consider the model where both spots
are equally distributed across the stars as most likely.

The addition of the star spots to the solution does shrink the radii of the
individual stars and increase the surface brightness ratio compared to the
spotless model. This is due to ellc suppressing the average brightness of the
spotted primary. Both single and double spot models produce a comparable
effect. The spotted models are more consistent with the near-equal mass ratio
system (see Section 3.2). Although both solutions that have two star spots fit
the LC well, whether both of the spots are on the primary, or there is one on
each of the stars, there is an 6% radii difference between the two which
represents the systematic uncertainty in the solution. We chose, in our final
fit, to adopt the ’one-spot-per-component’ model, as this seemed most
consistent with the underlying symmetry in the system’s parameters/mass ratio.

###  3.2 Full orbital fits: joint light curve and radial velocity fits

Table 4: Priors & Initialization of stellar + orbital parameters |  | Initial | Initial
---|---|---|---
Parameter | Priors | Median | $\sigma$
$P$ (days) | NA | 1.06494 | 1e-6
$T_{0}$ | NA | 2458472.537 | 0.001
$\gamma$ (km s-1) | -200 – 200 | 67 | 2
$K_{1}$ (km s-1) | NA | 107 | 2.5
$q$ | 0 – 1 | 0.98 | 0.01
$e$ | 0 – 1 | 0.01 | 0.02
$i$ (deg) | 0 – 90 | 83.4 | 2.0
$\omega$ (radians) | 0 – 2$\pi$ | 4.97 | 0.52
Surface brightness ratio |  $>$0 | 0.94 | 0.02
R1/a | 0 – 1 | 0.145 | 0.02
R2/a | 0 – 1 | 0.13 | 0.02
Primary spot size (deg) | 0 – 90 | 31 | 10
Spot longitude (deg) | 0 – 360 | 207 | 30
Spot latitude (deg) | -90 – 90 | 40 | 30
Spot contrast ratio | 0 – 1 | 0.65 | 0.2
Secondary spot size (deg) | 0 – 90 | 18 | 10
Spot longitude (deg) | 0 – 360 | 260 | 30
Spot latitude (deg) | -90 – 90 | 22 | 30
Spot contrast ratio | 0 – 1 | 0.65 | 0.2

Initially, we explored extracting stellar and orbital parameters by using ellc
to fit the system’s light curve and The Joker (Price-Whelan et al., 2017,
2018) to fit the system’s radial velocity measurements. Given the disjoint
epochs of the TESS light curve and the APOGEE radial velocities, and the
longer but sparser ASAS-SN monitoring, however, fitting the light curves and
radial velocities separately led to non-negligible uncertainties in the
system’s period and T0 ($\sim$0.02 days).

To best constrain the system’s orbital period and ephemeris, we utilized
ellc’s light curve and radial velocity modelling capabilities to jointly fit
the system’s TESS & ASAS-SN (V- and g-band) light curves, as well as the
APOGEE radial velocities. In this joint fit, we utilize the ASAS-SN light
curves mainly for the constraints their in-eclipse datapoints provide for the
system’s period and ephemeris; as 2M0646’s starspots may evolve substantially
over the baseline of the ASAS-SN monitoring, we artificially down-weight the
ASAS-SN out-of-eclipse data points (which we define as any datapoint with
phase between 0.05 – 0.45, and 0.55 – 0.95) by inflating their errors by a
factor of 100. We then use emcee to identify 2M0646’s maximum likelihood
stellar and orbital parameters by using ellc to compute self-consistent light
curve and radial velocity solutions for a given set of model parameters, and
then compute a joint chi square value from the residuals of the fits to the
TESS, ASAS-SN and APOGEE data. Given the relatively small number of radial
velocity datapoints, and their orthogonal information content relative to the
light curve, we apply a 20x multiple to the APOGEE residuals in computing the
joint chi square value. In computing the light curve models, we adopt limb
darkening coefficients computed by Claret & Bloemen (2011) for the V- and
g-bands (for the ASAS-SN light curves), and for a white-light/Kepler bandpass
(for the TESS light curve). In this joint fit, we applied priors sparingly,
primarily to limit the solution to physically plausible solutions (i.e,
restricting eccentricity to 0 $<e<$ 1, the mass ratio to 0 $<q<$ 1, etc.). We
initialized the chain with Gaussian distributions around parameters which were
found to well fit the light curve in the parameter study described earlier.
Specifically, we initialized the walkers using the mean values and standard
deviations listed in Table 4.

Figure 3: RV vs. time plot, showing individual sample orbits fit by ellc (blue
& red lines) to the velocities measured for 2M0646’s primary (red dots) and
secondary (blue dots) respectively. Two different models are shown, the top
one has included the two RV points close to the eclipses, the bottom one has
excluded them from the fit.

The remaining parameters that are customizable by ellc (e.g., third light,
irradiation, apsidal motion rate, asynchronous rotation factor, surface
gravity, $[M/H]$, reflection coefficient, external noise, $T_{\rm{exp}}$) were
not modified from their default values, as appropriate for a detached system
of solar-metallicity main sequence dwarfs, or set to zero in the initial run.

We initialize a model with 15 walkers, each of which takes 5000 steps. We used
the MCMC chain to estimate parameters for $r_{1}$, $r_{2}$, surface brightness
ratio, inclination, $T_{0}$, and period. Before measuring parameters from the
chain, we reject the first 10% of the chain to ensure that it has fully burned
in (i.e., become relatively independent of where it was initialized). We then
calculate the median and standard deviation of each parameter in the remainder
of the chain, and adopt these values to characterize the system.

To investigate the effect of the two RV pairs of points near the eclipses
(using DR16 data) on the derived orbit, we repeat the joint ellc fit and
exclude them. The two RV fits are shown in Figure 3, and have negligible
difference, either visually in examining the fitted orbit, or in the derived
parameters.

Best fitting parameters for the model containing all the data are given in
Table 5. The derived masses and radii are show in Figure 4. The full corner
plots of star+orbit and spot parameters are shown in Figures 5 and 6
respectively. To convert the radii from units of semi-major axis to solar
radii, we used a variation of Kepler’s 3rd Law to calculate the semi-major
axis of the system. We then multiply the resultant semi-major axis by the
relative radii ($r/a$) produced by ellc. Performing this conversion and
propagating the errors in $r/a$, $P$, $i$, and $v_{1}+v_{2}$, we find
$R_{1}=0.66\pm 0.05$ R☉ and $R_{2}=0.57\pm 0.06$ R☉.

Figure 4: Mass and radius relation of the two components of 2M0646 compared against the PARSEC isochrones of different ages and metallicities. Figure 5: Corner plot showing distribution of stellar and orbital parameters after final iteration of two spot model (with one spot per component). The parameters are labelled as, orbital period ($P$, in days), time of passage of periastron ($T_{0}$, julian date), radius of the primary and the secondary ($r_{1}/r_{2}$ in units of the system’s semi-major axis), orbital inclination ($i$, in degrees), eccentricity of the system ($e$), argument of periapsis ($\omega$, in radians), surface brightness ratio (sb_rat), velocity semi-amplitude of the primary ($K_{1}$, in km s-1), the center of mass velocity ($\gamma$, in km s-1), and mass ratio ($q$). Figure 6: Corner plot showing distribution of starspot properties in final two-spot ellc model (w/ one spot per component). The labelled parameters are the size of the spots on both primary and the secondary ($s_{1}/s_{2}$), in degrees, the surface longitude of the spots on each star ($l_{1}/l_{2}$, in degrees), surface latitude ($b_{1}/b_{2}$, in degrees), and the respective ratio of the contrast of the spot relative to the photosphere ($f_{1}/f_{2}$). Table 5: Final physical parameters for two spot solution Parameter | Value
---|---
$P$ (days) | 1.06493559$\pm$7.5e-07
$\gamma$ (km s-1) | 67.13$\pm$0.52
$K_{1}$ (km s-1) | 107.6$\pm$0.7
$K_{2}$ (km s-1) | 108.6$\pm$1.4
$q$ | 0.99$\pm$0.01
$e$ | 0.016 $\pm$0.007
$i$ (deg) | 83.6 $\pm$ 0.3
$\omega$ (deg) | 276$\pm$6
Surface brightness ratio | 0.94 $\pm$ 0.02
$T_{0}$ | 2458472.5371 $\pm$ 0.0003
$M_{1}$ (M☉) | 0.571$\pm$0.017
$M_{2}$ (M☉) | 0.565$\pm$0.013
$R_{1}$ (R☉) | 0.659$\pm$0.053
$R_{2}$ (R☉) | 0.570$\pm$0.060
Primary spot size (deg) | 27.7 $\pm$ 7.2
Spot longitude (deg) | 205.7 $\pm$3.2
Spot latitude (deg) | 40 $\pm$18
Spot contrast | 0.54 $\pm$0.20
Secondary spot size (deg) | 22.1 $\pm$ 8.2
Spot longitude (deg) | 258.5 $\pm$9.6
Spot latitude (deg) | 31 $\pm$29
Spot contrast | 0.58 $\pm$0.24

##  4 Discussion and conclusions

We have identified an eclipsing SB, 2M0646, and have performed a orbital and
light curve fitting to measure masses and radii of the individual stars of the
binary. We measure the properties of each component to be 0.57$\pm$0.02 M☉ and
0.66$\pm$0.05 R☉ for the primary, and 0.56$\pm$0.01 M☉ and 0.57$\pm$0.06 R☉
for the secondary. The secondary’s derived properties are consistent within
the $\sim$10% errors with the mass-radius relations suggested by the PARSEC
isochrones (Marigo et al., 2017), but the primary appears over-inflated at the
1.5$\sigma$ level with respect to those isochrones’ predictions. We note that
there various degeneracies in radius determination with respect to e.g.,
inclination (Figure 5), as well as spots coverage (Table 2). Given the
similarity of the masses, it is unlikely that there is a significant
difference in the radii, and, indeed the individual models of various subsets
of the data do suggest it. Thus, most likely, the radii for both stars are
likely to be $\sim$0.6 $R_{\odot}$, i.e., consistent with the overlap in the
1$\sigma$ between the two measurements reported in Table 5 and Figure 4, with
both stars moderately inflated in comparison to the isochrones.

Lower metallicity isochrones with older ages (which are more appropriate for
thick-disk sources such as 2M0646) tend to have marginally larger radii at a
given mass than the solar metallicity isochrones. This is likely insufficient
to fully compensate for the over-inflation, furthermore, the differences
between various isochrones are smaller than the uncertainties in our radius
measurements. Additional monitoring, and more detailed abundance measurements,
would provide a better basis for a precision test of the isochrones’ accuracy
for stars of non-solar age and metallicity.

For nearly a decade and a half precision measurements have identified that
theoretical models often underpredict the empirically measured radii of low-
mass stars. The presence and magnitude of these inflated radii have been
reported to correlate with metallicity (e.g., Berger et al., 2006), direct
signatures of stellar activity (e.g., López-Morales, 2007), and/or
orbital/rotational velocities indicative of likely induced activity (Kraus et
al., 2011; Somers & Stassun, 2017). These observational results have spurred
the development of theoretical models that seek to resolve this discrepancy by
incorporating new physics into the calculation of isochrones and mass radius
relations for low-mass stars, such as direct magnetic inhibition of convection
(Feiden & Chaboyer, 2014; MacDonald & Mullan, 2017) and the reduced efficiency
of radiative losses by spotted photospheres (Somers et al., 2020). The picture
is muddled, however, by other recent observational studies which find either
no evidence for inflated radii (e.g., Han et al., 2017) , or no clear
correlation with metallicity/rotation/activity (e.g., Kesseli et al., 2018;
Parsons et al., 2018; Morrell & Naylor, 2019).

To the extent that our measurements do suggest that the system’s primary may
be inflated at the 1.5 sigma level relative to the predictions of theoretical
isochrones, it is difficult to draw broader conclusions about the likely cause
of that inflation. Effects related to magnetic activity and/or sunspots are a
plausible explanation: with clear spot modulation on a period that matches the
system’s short orbital period, the system appears tidally locked and a good
candidate for long-lived magnetic activity with a rotationally synchronized
dynamo. Similarly, the two stars’ close proximity and near-equal mass ratio
suggests that irradiation from the companion will allow each component to
affect it’s neighbor’s surface temperature and thermal structure, potentially
inhibiting convection and producing an inflated radius.

In the future, additional photometric monitoring of 2M0646 to obtain color
information may help to further diagnose the temperature contrast and location
of spots.

Spectrophotometric monitoring of the system would provide even more
information: a detailed map not only of the location and magnetic properties
of the spots, but also an opportunity to perform detailed spectral fitting to
test for compositional differences. More broadly, more observations by large
spectroscopic surveys such as SDSS that allow serendipitous detections of
SB2s, combined with the new EB detections in the light curves that are now
available across much of the sky with TESS, will provide additional systems
with reliable masses and radii to conduct statistical tests of potential
inflation mechanisms.

##  5 Acknowledgements

We thank the anonymous referee whose useful and constructive feedback improved
the substance and presentation of our analysis. M.K. and K.C. acknowledge
support provided by the NSF through grant AST-1449476; A. M. acknowledges
additional support from the National Aeronautics and Space Administration,
provided by the Washington NASA Space Grant Consortium and Chandra Award
Number GO9-20006X issued by the Chandra X-ray Center, which is operated by the
Smithsonian Astrophysical Observatory for and on behalf of the National
Aeronautics Space Administration under contract NAS8-03060.

Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.
Sloan Foundation, the U.S. Department of Energy Office of Science, and the
Participating Institutions.

SDSS-IV acknowledges support and resources from the Center for High
Performance Computing at the University of Utah. The SDSS website is
www.sdss.org.

SDSS-IV is managed by the Astrophysical Research Consortium for the
Participating Institutions of the SDSS Collaboration including the Brazilian
Participation Group, the Carnegie Institution for Science, Carnegie Mellon
University, Center for Astrophysics — Harvard & Smithsonian, the Chilean
Participation Group, the French Participation Group, Instituto de Astrofísica
de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and
Mathematics of the Universe (IPMU) / University of Tokyo, the Korean
Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut
für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA
Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-
Institut für Extraterrestrische Physik (MPE), National Astronomical
Observatories of China, New Mexico State University, New York University,
University of Notre Dame, Observatário Nacional / MCTI, The Ohio State
University, Pennsylvania State University, Shanghai Astronomical Observatory,
United Kingdom Participation Group, Universidad Nacional Autónoma de México,
University of Arizona, University of Colorado Boulder, University of Oxford,
University of Portsmouth, University of Utah, University of Virginia,
University of Washington, University of Wisconsin, Vanderbilt University, and
Yale University.

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# Commutativity of quantization with conic reduction for torus actions on
compact CR manifolds

Andrea Galasso111Address: Room 407, Chee-Chun Leung Cosmology Hall, National
Taiwan University; ORCID iD: 0000-0002-5792-1674; e-mail:
<EMAIL_ADDRESS>
###### Abstract

We define conic reduction $X^{\mathrm{red}}_{\nu}$ for torus actions on the
boundary $X$ of a strictly pseudo-convex domain and for a given weight $\nu$
labelling a unitary irreducible representation. There is a natural residual
circle action on $X^{\mathrm{red}}_{\nu}$. We have two natural decompositions
of the corresponding Hardy spaces $H(X)$ and $H(X^{\mathrm{red}}_{\nu})$. The
first one is given by the ladder of isotypes $H(X)_{k\nu}$, $k\in\mathbb{Z}$,
the second one is given by the $k$-th Fourier components
$H(X^{\mathrm{red}}_{\nu})_{k}$ induced by the residual circle action. The aim
of this paper is to prove that they are isomorphic. Furthermore, we sketch the
proof of a similar result for spaces of $(0,q)$-forms with $L^{2}$-coefficient
when $X$ is a CR manifold with non-degenerate Levi-curvature for $k$
sufficiently large.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Geometric setting
    2. 2.2 Hardy spaces
  3. 3 Proof of Theorem 1.1
    1. 3.1 Ohsawa–Takegoshi extension theorem
    2. 3.2 The proof
  4. 4 Proof of Theorem 1.2

Keywords: Hardy space, torus action, CR conic reduction

Mathematics Subject Classification: 32A2

##  1 Introduction

Let $X$ be the boundary of a strictly pseudo-convex domain $D$ in
$\mathbb{C}^{n+1}$. Then $(X,\,T^{1,0}X)$ is a contact manifold of dimension
$2n+1,\,n\geq 1$, where $T^{1,0}X$ is the sub-bundle of $TX\otimes\mathbb{C}$
defining the CR structure. We denote by
$\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ the contact $1$-form whose
kernel is the horizontal bundle $HX\subset TX$, we refer to Section 2.1 for
definitions. Associated with this data we can define the Hardy space $H(X)$,
it is the space of boundary values of holomorphic functions in $D$ which lie
in $L^{2}(X)$, the Hilbert space of square integrable functions on $X$.
Suppose a contact and CR action of a $t$-dimensional torus $\mathbb{T}^{t}$ is
given; we denote by $\mu:X\rightarrow\mathfrak{t}^{*}$ the associated CR
moment map, we identify $\mathfrak{t}\cong\imath\mathbb{R}^{t}$ with its dual
$\mathfrak{t}$. Fix a weight $\imath\,\nu$ in the lattice
$\imath\,\mathbb{Z}^{t}\subset\mathfrak{t}^{*}$, if $0\in\mathfrak{t}^{*}$
does not lie in the image of the moment map the isotypes

$H(X)_{k\nu}=\\{f\in H(X)\,:\,(e^{\imath\theta}\cdot
f)(x)=e^{\imath\,k\langle\nu,\theta\rangle}f(x),\,\theta\in\mathbb{R}^{t}\\},\quad
k\in\mathbb{Z},$

are finite dimensional.

Suppose that the ray $\imath\,\mathbb{R}_{+}\cdot\nu\in\mathfrak{t}^{*}$ is
transversal to $\mu$, then $X_{\nu}:=\mu^{-1}(\imath\,\mathbb{R}_{+}\cdot\nu)$
is a sub-manifold of $X$ of codimension $t-1$. There is a well-defined locally
free action of
$\mathbb{T}^{t-1}_{\nu}:=\exp_{\mathbb{T}^{t}}(\imath\,\ker\nu)$ on $X_{\nu}$,
the resulting orbifold $X^{\mathrm{red}}_{\nu}$ is called conic reduction of
$X$ with respect to the weight $\nu$. The locus $X_{\nu}$ is
$\mathbb{T}$-invariant, we will always assume that the action of $\mathbb{T}$
on $X_{\nu}$ is locally free. After replacing $\mathbb{T}$ with its quotient
by a finite subgroup, we may and will assume without loss of generality that
the action is generically free. In Section 2.1, we show that
$X^{\mathrm{red}}_{\nu}$ is a strictly pseudo-convex domain of dimension
$2n-2t+3$. There is a natural locally free circle action
$\mathbb{T}^{1}_{\nu}:=\exp_{\mathbb{T}^{t}}(\imath\,\nu)$ on
$X^{\mathrm{red}}_{\nu}$ which induces an action on the Hardy space
$H(X^{\mathrm{red}}_{\nu})$. We denote by $H(X^{\mathrm{red}}_{\nu})_{k}$ the
corresponding $k$-th Fourier component and we call the action of
$\mathbb{T}^{1}_{\nu}$ on $X^{\mathrm{red}}_{\nu}$ residual circle action. The
aim of this paper is to prove the following theorem.

######  Theorem 1.1.

Suppose $X_{\nu}\neq\emptyset$ and assume $\mathbb{T}^{1}_{\nu}$ defines a
transversal locally free CR circle action on $X$. There is a natural
isomorphism of vector spaces $\delta_{k}\,:\,H(X)_{k\nu}\rightarrow
H(X^{\mathrm{red}}_{\nu})_{k}$ for each $k$.

From the point of view of symplectic geometry given a Hamiltonian and
holomorphic action with moment map $\Phi$ of a compact Lie Group $G$ on a
Hodge manifold $(M,\omega)$ with quantizing circle bundle $\pi:X\rightarrow
M$, one can always define an infinitesimal action of the Lie algebra
$\mathfrak{g}$ on $X$. If it can be integrated to an action of the whole group
$G$, then one has a representation of $G$ on $H(X)$. In [GS2] it was observed
that associated to group actions one can define reductions
$M_{\mathrm{red}}^{\Theta}$ by pulling back a $G$-invariant and proper sub-
manifold $\Theta$ of $\mathfrak{g}^{*}$ via the moment map $\Phi$. When
$\Theta$ is chosen to be a cone through a co-adjoint orbit
$C(\mathcal{O}_{\nu})$ one has associated reduction whose Hardy space of its
quantization is

$H(X_{\mathrm{red}}^{C(\mathcal{O}_{\nu})})=\bigoplus_{k\in\mathbb{Z}}H(X_{\mathrm{red}}^{C(\mathcal{O}_{\nu})})_{k}\,,$

where $k$ labels an irreducible representation of the residual circle action
described above. Now, it is natural to ask if this spaces are related with the
decomposition induced by $G$ on $H(X)$. When $G=\mathbb{T}^{r}$ Theorem 1.1
states that they are isomorphic to $H_{k\nu}(X)$; the canonical decomposition
of the Hardy space $H(X)$ of the quantitation by the built-in circle action
does not play any role. The semi-classical parameter $k$ is the one induced by
the ladder $k\,\nu$, $k=0,1,2,\dots$, labelling unitary irreducible
representations.

Further geometrical motivations for this theorem are explained in paper [P2],
where it is proved that $\delta_{k}$ is an isomorphism for $k$ large enough in
the setting when $X$ is the circle bundle of a polarized Hodge manifold whose
Grauert tube is $D$; we also refer to [P1] for examples and the explicit
expression of the leading term of the asymptotic expansion of $\dim
H(X)_{k\nu}$ as $k$ goes to infinity. The proof of Theorem 1.1 is an
application of the Ohsawa–Takegoshi extension theorem.

We prove Theorem 1.1 in the more general setting for spaces of $(0,q)$-forms
when $k$ is large. In fact, suppose more generally that the Levi form is non-
degenerate and the number of either positive or negative eigenvalues of the
Levi form is not equal to $n$, then the Szegő projector for functions
$S:L^{2}(X)\rightarrow H(X)$ is a smoothing operator, see [Hs]. Hence, in the
presence of a transversal locally free circle action, the $k$-th Fourier
component $S_{k}$ has a rapidly decreasing asymptotic expansion as $k$ goes to
infinity. From the standpoint of semi-classical analysis it is natural to look
for a replacement of these spaces. We then consider $(0,q)$ forms with $L^{2}$
coefficients and the corresponding projector $S^{(q)}$ onto the kernel of the
Köhn Laplacian $H^{q}(X)$.

Let $X^{\mathrm{red}}_{\nu}:=X_{\nu}/\mathbb{T}^{t-1}_{\nu}$, defined in the
same way as before. We will show that $X^{\mathrm{red}}_{\nu}$ is a CR
manifold with natural CR structure induced by $T^{1,0}X$ of dimension
$2n-2(t-1)+1$. Let $L_{X^{\mathrm{red}}_{\nu}}$ be the Levi form on
$X^{\mathrm{red}}_{\nu}$ induced naturally from the Levi form $L$ on $X$. Note
that the bilinear form $b$ defined in Section 2.1 is non-degenerate on
$X_{\nu}$. For a given subspace $\mathfrak{s}$ of $\mathfrak{t}$, we denote
$\mathfrak{s}_{X}$ the subspace of infinitesimal vector fields on $X$. Hence,
on

$\ker{\nu}_{X}\times\ker{\nu}_{X}\subseteq TX_{\nu}\times TX_{\nu}\,,$

$b$ has constant signature, suppose $b$ has $r$ negative and $t-1-r$ positive
eigenvalues on $X_{\nu}$. Hence $L_{X^{\mathrm{red}}_{\nu}}$ has $q-r$
negative and $n-t+1-q+r$ positive eigenvalues at each point of
${X^{\mathrm{red}}_{\nu}}$. We refer to Section 2.1 for definitions, we have:

######  Theorem 1.2.

Assume $\mathbb{T}^{1}_{\nu}$ defines a transversal locally free CR circle
action on a compact orientable CR manifold $X$ with non-degenerate Levi form.
Then $X^{\mathrm{red}}_{\nu}$ is a compact CR manifold with non-degenerate
Levi form having $q-r$-negative eigenvalues. There is a natural isomorphism of
vector spaces $\sigma_{k}\,:\,H^{q}(X)_{k\nu}\rightarrow
H^{q-r}(X^{\mathrm{red}}_{\nu})_{k}$ for $k$ sufficiently large.

We only give a sketch of the proof in Section 4, which can be deduced as in
[HH] and it is a consequence of the microlocal properties of the projector
$S^{(q)}_{k\nu}$ described in Section 2.2 and calculus of Fourier integral
operators of complex type, see [MS].

##  2 Preliminaries

###  2.1 Geometric setting

We recall some notations concerning CR and contact geometry. Let
$(X,T^{1,0}X)$ be a compact, connected and orientable CR manifold of dimension
$2n+1$, $n\geq 1$, where $T^{1,0}X$ is a CR structure of $X$. There is a
unique sub-bundle $HX$ of $TX$ such that $HX\otimes\mathbb{C}=T^{1,0}X\oplus
T^{0,1}X$. Let $J:HX\rightarrow HX$ be the complex structure map given by
$J(u+\overline{u})=\imath u-\imath\overline{u}$, for every $u\in T^{1,0}X$. By
complex linear extension of $J$ to $TX\otimes\mathbb{C}$, the
$\imath$-eigenspace of $J$ is $T^{1,0}X$. We shall also write $(X,HX,J)$ to
denote a CR manifold.

Since $X$ is orientable, there always exists a real non-vanishing $1$-form
$\omega_{0}\in\mathcal{C}^{\infty}(X,T^{*}X)$ so that
$\langle\,\omega_{0}(x)\,,\,u\,\rangle=0$, for every $u\in H_{x}X$, for every
$x\in X$; $\omega_{0}$ is called contact form and it naturally defines a
volume form on $X$. For each $x\in X$, we define a quadratic form on $HX$ by

${L}_{x}(U,V)=\frac{1}{2}\mathrm{d}\omega_{0}(JU,V),\qquad\forall\ U,V\in
H_{x}X.$

Then, we extend ${L}$ to $HX\otimes\mathbb{C}$ by complex linear extension;
for $U,V\in T^{1,0}_{x}X$,

${L}_{x}(U,\overline{V})=\frac{1}{2}\,\mathrm{d}\omega_{0}(JU,\overline{V})=-\frac{1}{2i}\,\mathrm{d}\omega_{0}(U,\overline{V}).$

The Hermitian quadratic form ${L}_{x}$ on $T^{1,0}_{x}X$ is called Levi form
at $x$. In the case when $X$ is the circle bundle of an Hodge manifold
$(M,\omega)$, the positivity of $\omega$ implies that the number of negative
eigenvalues of the Levi form is equal to $n$. The Reeb vector field
$R\in\mathcal{C}^{\infty}(X,TX)$ is defined to be the non-vanishing vector
field determined by

$\omega_{0}(R)=-1,\quad\mathrm{d}\omega_{0}(R,\cdot)\equiv 0\ \ \mbox{on
$TX$}.$

Fix a smooth Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$\mathbb{C}TX$ so that $T^{1,0}X$ is orthogonal to $T^{0,1}X$,
$\langle\,u\,|\,v\,\rangle$ is real if $u,v$ are real tangent vectors,
$\langle\,R\,|\,R\,\rangle=1$ and $R$ is orthogonal to $T^{1,0}X\oplus
T^{0,1}X$. For $u\in\mathbb{C}TX$, we write
$|u|^{2}:=\langle\,u\,|\,u\,\rangle$. Denote by $T^{*1,0}X$ and $T^{*0,1}X$
the dual bundles of $T^{1,0}X$ and $T^{0,1}X$, respectively. They can be
identified with sub-bundles of the complexified cotangent bundle
$\mathbb{C}T^{*}X$.

Assume that $X$ admits an action of $t$-dimensional torus $\mathbb{T}^{t}$. In
this work, we assume that the $\mathbb{T}^{t}$-action preserves $\omega_{0}$
and $J$; that is, $t^{\ast}\omega_{0}=\omega_{0}$ on $X$ and
$t_{\ast}J=Jt_{\ast}$ on $HX$. Let $\mathfrak{t}$ denote the Lie algebra of
$T$, we identify $\mathfrak{t}$ with its dual $\mathfrak{t}^{*}$ by means of
the scalar product $\langle\cdot,\cdot\rangle$. For any $\xi\in\mathfrak{t}$,
we write $\xi_{X}$ to denote the vector field on $X$ induced by $\xi$. The
moment map associated to the form $\omega_{0}$ is the map
$\mu:X\to\mathfrak{t}^{*}$ such that, for all $x\in X$ and
$\xi\in\mathfrak{t}$, we have

$\langle\mu(x),\xi\rangle=\omega_{0}(\xi_{X}(x)).$ (1)

We can also define a non-degenerate bilinear form $b$ on $HX$ such that

$b(\cdot,\cdot)=\mathrm{d}\omega_{0}(\cdot,J\cdot).$ (2)

Fix a maximal weight $\nu\neq 0$ in the lattice inside $\mathfrak{t}^{*}$.
Suppose that $\imath\mathbb{R}_{+}\cdot\nu$ is transversal to $\mu$, so we
have

$\mathfrak{t}=\imath\mathbb{R}_{+}\cdot\nu\oplus\mathrm{d}_{p}\mu(T_{p}X)$ (3)

and $X_{\nu}$ is a sub-manifold of $X$ of codimension $2n+2-t$. We claim that
the action of $\mathbb{T}^{t-1}_{\nu}$ on $X_{\nu}$ is locally free. In fact,
by the contrary suppose that there exists $\xi\in\mathfrak{t}$ such that
$\xi_{X}(x)=0$ on $T_{x}X_{\nu}$, $x\in X_{\nu}$. For each $v\in T_{x}X$, we
have

$(\mathrm{d}_{x}\mu(v))(\xi)=\mathrm{d}_{x}\omega_{0}(\xi_{X},v)=0$

which contradicts (3). For simplicity, in the next Lemma we assume the action
of $\mathbb{T}^{t-1}_{\nu}$ on $X_{\nu}$ to be free.

######  Lemma 2.1.

The manifold $X_{\nu}/\mathbb{T}^{t-1}_{\nu}$ is a CR manifold. Let
$\pi:X_{\nu}\rightarrow X^{\mathrm{red}}_{\nu}$ and
$\iota:X_{\nu}\hookrightarrow X$ be the natural quotient and inclusion,
respectively, then there is a unique induced contact form
$\omega_{0}^{\mathrm{red}}$ on $X_{\nu}/\mathbb{T}^{t-1}_{\nu}$ such that

$\pi^{*}\omega_{0}^{\mathrm{red}}=\iota^{*}\omega_{0}\,.$

In particular, set $HX_{\nu}=TX_{\nu}\cap HX$, we have

$HX=HX_{\nu}\oplus J\,\imath\nu^{\perp}_{X}\quad\text{ and }\quad
HX_{\nu}=\imath\nu^{\perp}_{X}\oplus\mathrm{d}\pi^{*}HX^{\mathrm{red}}_{\nu}\,.$

###### Proof.

It is an easy consequence of

$v\in T_{x}X_{\nu}\Leftrightarrow\mathrm{d}_{x}\mu(v)(\xi)=0\quad\text{for all
}\xi\in\nu^{\perp},\,v\in T_{x}X$

which proves that

$\mathrm{val}_{x}(\nu^{\perp}_{X}(x))=(T_{x}X_{\nu})^{\mathrm{d}_{x}\omega_{0}}\,.$

∎

We will also assume that $\mathbb{T}^{1}_{\nu}$-action is transversal CR, that
is, the infinitesimal vector field

$(\nu_{X}u)(x)=\frac{\partial}{\partial t}\left(u(\exp(\imath t\,\nu)\circ
x)\right)|_{t=0}\,,\quad\text{for any }u\in C^{\infty}(X)\,,$

preserves the CR structure $T^{1,0}X$, so that $\nu_{X}$ and $T^{1,0}X\oplus
T^{0,1}X$ generate the complex tangent bundle to $X$,

$\mathbb{C}T_{x}X=\mathbb{C}\nu_{X}(x)\oplus\mathbb{C}T_{x}^{1,0}X\oplus\mathbb{C}T_{x}^{0,1}X\qquad(x\in
X)\,.$

We define local coordinates that will be useful later. Recall that $X$ admits
a CR and transversal $\mathbb{T}^{1}_{\nu}$-action which is locally free on
$X_{\nu}$, $T\in\mathcal{C}^{\infty}(X,\,TX)$ denotes the global real vector
field given by this infinitesimal circle action. We will take $T$ to be our
Reeb vector field $R$. In a similar way as in Theorem $3.6$ in [HH], there
exist local coordinates $v=(v_{1},\dots,v_{t-1})$ of $\mathbb{T}^{t-1}_{\nu}$
in a small neighborhood $V_{0}$ of the identity $e$ with
$v(e)=(0,\,\dots,\,0)$, local coordinates $x=(x_{1}\,\dots,x_{2n+1})$ defined
in a neighborhood $U_{1}\times U_{2}$ of $p\in X_{\nu}$, where
$U_{1}\subseteq\mathbb{R}^{t-1}$ (resp. $U_{2}\subseteq\mathbb{R}^{2n+2-t}$)
is an open set of $0\in\mathbb{R}^{t-1}$ (resp. $0\in\mathbb{R}^{2n+2-t}$) and
$p\equiv 0\in\mathbb{R}^{2n+1}$, and a smooth function
$\gamma=(\gamma_{1},\dots,\,\gamma_{t-1})\in\mathcal{C}^{\infty}(U_{2},U_{1})$
with $\gamma(0)=0$ such that

$\displaystyle(v_{1},\dots,v_{t-1})\circ(\gamma(x_{t},\dots,x_{2n+1}),x_{t},\dots,x_{2n+1})$
$\displaystyle=(v_{1}+\gamma_{1}(x_{t},\dots,x_{2n+1}),\dots,\,v_{t-1}+\gamma_{d}(x_{t},\dots,x_{2n+1}),\,x_{t},\dots,x_{2n+1})$

for each $(v_{1},\dots,\,v_{t-1})\in V_{0}$ and $(x_{t},\dots,x_{2n+1})\in
U_{2}$. Furthermore, we have

$\mathfrak{t}=\mathrm{span}\left\\{\partial_{{x}_{j}}\right\\}_{j=1,\dots,t-1}\,,\quad\mu^{-1}(\imath\,{\mathbb{R}}_{\nu}\cdot\nu)\cap
U=\\{x_{2d-t+1}=\dots=x_{2d}=0\\}\,,$

on $\mu^{-1}(\imath\,{\mathbb{R}}_{\nu}\cdot\nu)\cap U$ there exist smooth
functions $a_{j}$’s with $a_{j}(0)=0$ for every $0\leq j\leq t-1$ and
independent on $x_{1},\dots,x_{2(t-1)},\,x_{2n+1}$ such that

$J\left(\partial_{{x}_{j}}\right)=\partial_{{x}_{t-1+j}}+a_{j}(x)\partial_{{x}_{2n+1}}\qquad
j=1,\dots,t-1\,,$

the Levi form ${L}_{p}$, the Hermitian metric
$\langle\,\cdot\,|\,\cdot\,\rangle$ and the $1$-form $\omega_{0}$ can be
written

${L}_{p}(Z_{j},\,\overline{Z}_{k})=\mu_{j}\,\delta_{j,k},\qquad\langle
Z_{j}|\,\overline{Z}_{k}\rangle=\delta_{j,k}\qquad(1\leq j,k\leq n)\,,$

and

$\displaystyle\omega_{0}(x)=$ $\displaystyle(1+O(\lvert
x\rvert))\mathrm{d}x_{2n+1}+\sum_{j=1}^{t-1}4\mu_{j}x_{t-1+j}\mathrm{d}x_{j}+\sum_{j=t}^{n}2\mu_{j}x_{2j}\mathrm{d}x_{2j-1}$
$\displaystyle-\sum_{j=t}^{n}2\mu_{j}x_{2j-1}\mathrm{d}x_{2j}+\sum_{j=r}^{2n}b_{j}x_{2n+1}\mathrm{d}x_{j}+O(\lvert
x\rvert^{2})$

where $b_{r},\dots,b_{2n}\in\mathbb{R}$,

$T_{p}^{1,0}X=\mathrm{span}\\{Z_{1},\dots,Z_{n}\\}$

and

$\displaystyle Z_{j}$
$\displaystyle=\frac{1}{2}\,(\partial_{{x}_{j}}-\imath\,\partial_{{x}_{t-1+j}})(p)\qquad(j=1,\dots,t-1)\,,$
$\displaystyle Z_{j}$
$\displaystyle=\frac{1}{2}\,(\partial_{{x}_{2j-1}}-\imath\,\partial_{{x}_{2j}})(p)\qquad(j=t,\dots,n)\,.$

We need to define in local coordinates we just introduced the phase function
of the $\mathbb{T}^{t-1}_{\nu}$-invariant Szegő kernel
$\Phi_{-}(x,y)\in\mathcal{C}^{\infty}(U\times U)$ which is independent of
$(x_{1},\ldots,x_{t-1})$ and $(y_{1},\ldots,y_{t-1})$. Hence, we write
$\Phi_{-}(x,y)=\Phi_{-}((0,x^{\prime\prime}),(0,y^{\prime\prime})):=\Phi_{-}(x^{\prime\prime},y^{\prime\prime})$.
Moreover, there is a constant $c>0$ such that

${\rm Im\,}\Phi_{-}(x^{\prime\prime},y^{\prime\prime})\geq
c\left(\lvert{\hat{x}^{\prime\prime}}\rvert^{2}+\lvert{\hat{y}^{\prime\prime}}\rvert^{2}+\lvert{\mathring{x}^{\prime\prime}-\mathring{y}^{\prime\prime}}\rvert^{2}\right),\
\ \text{for all }((0,x^{\prime\prime}),(0,y^{\prime\prime}))\in U\times U.$
(4)

Furthermore,

$\displaystyle\Phi_{-}$
$\displaystyle(x^{\prime\prime},y^{\prime\prime})=-x_{2n+1}+y_{2n+1}+2i\sum^{t-1}_{j=1}\left|\mu_{j}\right|y^{2}_{t-1+j}+2i\sum^{t-1}_{j=1}\left|\mu_{j}\right|x^{2}_{t-1+j}+i\sum^{n}_{j=t}\left|\mu_{j}\right|\left|z_{j}-w_{j}\right|^{2}$
(5)
$\displaystyle+\sum^{n}_{j=t}i\mu_{j}(\overline{z}_{j}w_{j}-z_{j}\overline{w}_{j})+\sum^{d}_{j=1}(-b_{t-1+j}x_{d+j}x_{2n+1}+b_{t-1+j}y_{t-1+j}y_{2n+1})$
$\displaystyle+\sum^{n}_{j=t}\frac{1}{2}(b_{2j-1}-ib_{2j})(-z_{j}x_{2n+1}+w_{j}y_{2n+1})+\sum^{n}_{j=t}\frac{1}{2}(b_{2j-1}+ib_{2j})(-\overline{z}_{j}x_{2n+1}+\overline{w}_{j}y_{2n+1})$
$\displaystyle+(x_{2n+1}-y_{2n+1})f(x,y)+O(\left|(x,y)\right|^{3}),$

where $z_{j}=x_{2j-1}+ix_{2j}$, $w_{j}=y_{2j-1}+iy_{2j}$, $j=t,\ldots,n$,
$\mu_{j}$, $j=1,\ldots,n$, $f$ is smooth and satisfies $f(0,0)=0$,
$f(x,y)=\overline{f}(y,x)$.

We now consider $\mathbb{T}^{1}_{\nu}$ circle action on $X$. Let
$p\in\mu^{-1}(\imath\mathbb{R}_{+}\cdot\nu)$, there exist local coordinates
$v=(v_{1},\dots,v_{t-1})$ of $\mathbb{T}^{t-1}$ in a small neighborhood
$V_{0}$ of $e$ with $v(e)=(0,\,\dots,\,0)$, local coordinates
$x=(x_{1}\,\dots,x_{2n+1})$ defined in a neighborhood $U_{1}\times U_{2}$ of
$p$, where $U_{1}\subseteq\mathbb{R}^{t-1}$ (resp.
$U_{t-1}\subseteq\mathbb{R}^{2n+t}$) is an open set of $0\in\mathbb{R}^{t-1}$
(resp. $0\in\mathbb{R}^{2n+t}$) and $p\equiv 0\in\mathbb{R}^{2n+1}$, and a
smooth function
$\gamma=(\gamma_{1},\dots,\,\gamma_{t})\in\mathcal{C}^{\infty}(U_{2},U_{1})$
with $\gamma(0)=0$ such that $T=-\frac{\partial}{\partial x_{2n+1}}$ and all
the properties for the local coordinates defined before hold. The phase
function $\Psi$ satisfies
$\Psi(x,y)=-x_{2n+1}+y_{2n+1}+\hat{\Psi}(\mathring{x}^{\prime\prime},\mathring{y}^{\prime\prime})$,
where
$\hat{\Psi}(\mathring{x}^{\prime\prime},\mathring{y}^{\prime\prime})\in\mathcal{C}^{\infty}(U\times
U)$, $\mathring{x}^{\prime\prime}=(x_{t},\ldots,x_{2n})$,
$\mathring{y}^{\prime\prime}=(y_{t},\ldots,y_{2n})$, and $\Psi$ satisfies (5).

###  2.2 Hardy spaces

We denote by $L^{2}_{(0,q)}(X)$ the completion of $\Omega^{0,q}(X)$ with
respect to $(\,\cdot\,|\,\cdot\,)$. We extend $(\,\cdot\,|\,\cdot\,)$ to
$L^{2}_{(0,q)}(X)$ in the standard way. We extend $\overline{\partial}_{b}$ to
$L^{2}_{(0,r)}(X)$, $r=0,1,\ldots,n$, by

$\overline{\partial}_{b}:{\rm Dom\,}\overline{\partial}_{b}\subset
L^{2}_{(0,r)}(X)\rightarrow L^{2}_{(0,r+1)}(X)\,,$

where ${\rm Dom\,}\overline{\partial}_{b}:=\\{u\in
L^{2}_{(0,r)}(X);\,\overline{\partial}_{b}u\in L^{2}_{(0,r+1)}(X)\\}$ and, for
any $u\in L^{2}_{(0,r)}(X)$, $\overline{\partial}_{b}u$ is defined in the
sense of distributions. We also write

$\overline{\partial}^{*}_{b}:{\rm Dom\,}\overline{\partial}^{*}_{b}\subset
L^{2}_{(0,r+1)}(X)\rightarrow L^{2}_{(0,r)}(X)$

to denote the Hilbert space adjoint of $\overline{\partial}_{b}$ in the
$L^{2}$ space with respect to $(\,\cdot\,|\,\cdot\,)$. There is a well-defined
orthogonal projection

$S^{(q)}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{q}_{b}$ (6)

with respect to the $L^{2}$ inner product $(\,\cdot\,|\,\cdot\,)$ and let

$S^{(q)}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$

denote the distribution kernel of $S^{(q)}$. We will write $H^{q}(X)$ for
$\mathrm{Ker}\Box^{q}_{b}$ and for functions we simply write $H(X)$ to mean
$H^{0}(X)$. The distributional kernel $S^{(q)}(x,y)$ was studied in [Hs],
before recalling an explicit description describing the oscillatory integral
defining the distribution $S^{(q)}(x,y)$ we need to fix some notation.
Firstly, we recall Hörmander symbol space. Let $D\subset X$ be a local
coordinate patch with local coordinates $x=(x_{1},\ldots,x_{2n+1})$.

######  Definition 2.1.

For every $m\in\mathbb{R}$, we denote with

$S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\subseteq\mathcal{C}^{\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ (7)

the space of all $a$ such that, for all compact $K\Subset D\times D$ and all
$\alpha,\beta\in\mathbb{N}^{2n+1}_{0}$, $\gamma\in\mathbb{N}_{0}$, there is a
constant $C_{\alpha,\beta,\gamma}>0$ such that

$\left|\partial^{\alpha}_{x}\partial^{\beta}_{y}\partial^{\gamma}_{t}a(x,y,t)\right|\leq
C_{\alpha,\beta,\gamma}(1+\left|t\right|)^{m-\gamma},\ \ \mbox{for every
$(x,y,t)\in K\times\mathbb{R}_{+},t\geq 1$}.$

For simplicity we denote with $S^{m}_{1,0}$ the spaces defined in (7);
furthermore, we write

$S^{-\infty}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}):=\bigcap_{m\in\mathbb{R}}S^{m}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}).$

Let $a_{j}\in S^{m_{j}}_{1,0}$, $j=0,1,2,\ldots$ with
$m_{j}\rightarrow-\infty$, as $j\rightarrow\infty$. Then there exists $a\in
S^{m_{0}}_{1,0}$ unique modulo $S^{-\infty}$, such that

$a-\sum^{k-1}_{j=0}a_{j}\in S^{m_{k}}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*}\big{)}$

for $k=0,1,2,\ldots$. If $a$ and $a_{j}$ have the properties above, we write
$a\sim\sum^{\infty}_{j=0}a_{j}$ in $S^{m_{0}}_{1,0}$.

It is known that the characteristic set of $\Box^{q}_{b}$ is given by

$\Sigma=\Sigma^{-}\cup\Sigma^{+},\quad\Sigma^{-}=\left\\{(x,\lambda\omega_{0}(x))\in
T^{*}X;\,\lambda<0\right\\},$

and $\Sigma^{+}$ is defined similarly for $\lambda>0$. We recall the following
theorem (see [Hs, Theorem 1.2]).

######  Theorem 2.1.

Since the Levi form is non-degenerate, $\Box^{q}_{b}$ has $L^{2}$ closed
range. Then, there exist continuous operators
$S_{-},S_{+}:L^{2}_{(0,q)}(X)\rightarrow{\rm Ker\,}\Box^{q}_{b}$ such that

$S^{(q)}=S_{-}+S_{+},\quad S_{+}\equiv 0\ \ \mbox{if $q\neq n_{+}$}$

and

${\rm WF^{\prime}\,}(S_{-})={\rm diag\,}(\Sigma^{-}\times\Sigma^{-}),\quad{\rm
WF^{\prime}\,}(S_{+})={\rm diag\,}(\Sigma^{+}\times\Sigma^{+})\ \ \mbox{if
$q=n_{-}=n_{+}$},$

where ${\rm WF^{\prime}\,}(S_{-})=\left\\{(x,\xi,y,\eta)\in T^{*}X\times
T^{*}X;\,(x,\xi,y,-\eta)\in{\rm WF\,}(S_{-})\right\\}$, ${\rm WF\,}(S_{-})$ is
the wave front set of $S_{-}$ in the sense of Hörmander.

Moreover, consider any small local coordinate patch $D\subset X$ with local
coordinates $x=(x_{1},\ldots,x_{2n+1})$, then $S_{-}(x,y)$, $S_{+}(x,y)$
satisfy

$S_{\mp}(x,y)\equiv\int^{\infty}_{0}e^{i\varphi_{\mp}(x,y)t}s_{\mp}(x,y,t)dt\
\ \mbox{on $D$},$

with

$s_{\mp}(x,y,t)\sim\sum^{\infty}_{j=0}s^{j}_{\mp}(x,y)t^{n-j}\text{ in
}S^{n}_{1,0}(D\times
D\times\mathbb{R}_{+}\,,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\,,$

$s_{+}(x,y,t)=0$ if $q\neq n_{+}$ and $s^{0}_{-}(x,x)\neq 0$ for all $x\in D$.
The phase functions $\varphi_{-}$, $\varphi_{+}$ satisfy

$\varphi_{+},\varphi_{-}\in\mathcal{C}^{\infty}(D\times D),\ \ {\rm
Im\,}\varphi_{\mp}(x,y)\geq 0,\,\varphi_{-}(x,x)=0,\ \ \varphi_{-}(x,y)\neq 0\
\ \mbox{if}\ \ x\neq y,$

and

$d_{x}\varphi_{-}(x,y)\big{|}_{x=y}=-\omega_{0}(x),\ \
d_{y}\varphi_{-}(x,y)\big{|}_{x=y}=\omega_{0}(x),\,-\overline{\varphi}_{+}(x,y)=\varphi_{-}(x,y)\,.$

######  Remark 2.1.

Kohn [Kohn] proved that if $q=n_{-}=n_{+}$ or $\left|n_{-}-n_{+}\right|>1$
then $\Box^{q}_{b}$ has $L^{2}$ closed range. For a description of the phase
function in local coordinates see chapter 8 of part I in [Hs].

Now we focus on the decomposition induced by the group action on $H^{q}(X)$.
Fix $t\in\mathbb{T}^{r}$ and let
$t^{*}:\Lambda^{r}_{x}(\mathbb{C}T^{*}X)\rightarrow\Lambda^{r}_{t^{-1}\circ
x}(\mathbb{C}T^{*}X)$ be the pull-back map. Since $\mathbb{T}^{r}$ preserves
$J$, we have $t^{*}:T^{*0,q}_{x}X\rightarrow T^{*0,q}_{g^{-1}\circ x}X$, for
all $x\in X.$ Thus, for $u\in\Omega^{0,q}(X)$, we have
$t^{*}u\in\Omega^{0,q}(X)$. Put

$\Omega^{0,q}(X)_{k\nu}:=\left\\{u\in\Omega^{0,q}(X);\,(e^{\imath\theta})^{*}u=e^{\imath\,k\langle\nu,\,\theta\rangle}u,\
\ \forall\theta\in\mathbb{R}^{r}\right\\}.$

Since the Hermitian metric $\langle\,\cdot\,|\,\cdot\,\rangle$ on
$\mathbb{C}TX$ is $\mathbb{T}^{r}$-invariant, the $L^{2}$ inner product
$(\,\cdot\,|\,\cdot\,)$ on $\Omega^{0,q}(X)$ induced by
$\langle\,\cdot\,|\,\cdot\,\rangle$ is $\mathbb{T}^{r}$-invariant. Let $u\in
L^{2}_{(0,q)}(X)$ and $t\in\mathbb{T}^{r}$, we can also define $t^{*}u$ in the
standard way. We introduce the following notation

$L^{2}_{(0,q)}(X)_{k\nu}:=\left\\{u\in
L^{2}_{(0,q)}(X);\,(e^{\imath\theta})^{*}u=e^{\imath\,k\langle\nu,\,\theta\rangle}u,\
\ \forall\theta\in\mathbb{R}^{r}\right\\},$

and put

$({\rm Ker\,}\Box^{q}_{b})_{k\nu}:={\rm Ker\,}\Box^{q}_{b}\cap
L^{2}_{(0,q)}(X)_{k\nu}\,.$

The equivariant Szegő projection is the orthogonal projection

$S^{(q)}_{k\nu}:L^{2}_{(0,q)}(X)\rightarrow({\rm Ker\,}\Box^{q}_{b})_{k\nu}$

with respect to $(\,\cdot\,|\,\cdot\,)$. Let
$S^{(q)}_{k\nu}(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$ be the distribution kernel of
$S^{(q)}_{k\nu}$. The asymptotic expansion for the distributional kernel of
the projector $S^{(0)}_{k\nu}$ was studied in [P1] when $X$ is the quantizing
circle bundle of a given Hodge manifold in Heisenberg local coordinates. Using
similar ideas as in [HH] one can generalize the results for $(0,q)$-forms in
the following theorem, we give a sketch of the proof.

######  Theorem 2.2.

Suppose that $\Box^{q}_{b}$ has $L^{2}$ closed range. Then, there exist
continuous operators
$S^{-}_{k\nu},S^{+}_{k\nu}:L^{2}_{(0,q)}(X)\rightarrow({\rm
Ker\,}\Box^{q}_{b})_{k\nu}$ such that

$S^{(q)}_{k\nu}=S^{-}_{k\nu}+S^{+}_{k\nu},\quad S_{+}\equiv 0\ \ \mbox{if
$q\neq n_{+}$}$

Suppose for simplicity that $q\neq n_{+}$. If $q\neq n_{-}$, then
$S^{(q)}_{k\nu}\equiv O(k^{-\infty})$ on $X$.

Suppose $q=n_{-}$ and let $D$ be an open set in $X$ such that the intersection
$\mu^{-1}(\imath\,\mathbb{R}_{+}\cdot\nu)\cap D=\emptyset$. Then
$S^{(q)}_{k\nu}\equiv O(k^{-\infty})$ on $D$.

Let $p\in\mu^{-1}(\imath\,\mathbb{R}_{+}\cdot\nu)$ and let $U$ a local
neighborhood of $p$ with local coordinates $(x_{1},\dots\,x_{2n+1})$. Then, if
$q=n_{-}$, for every fix $y\in U$, we consider $S^{(q)}_{k\nu}(x,y)$ as a
$k$-dependent smooth function in $x$, then

$S^{(q)}_{k\nu}(x,\,y)=e^{\imath
k\lVert\nu\rVert\,\Psi(x,\,y)}\,b(x,\,y,\,k\lVert\nu\rVert)+O(k^{-\infty})\,.$

for every $x\in U$. The phase function $\Psi(x,\,y)$ is defined in local
coordinates in the end of Section 2.1, the symbol satisfies

$b(x,\,y,\,k\lVert\nu\rVert)\in S^{n+(1-t)/2}_{\mathrm{loc}}(1,\,U\times
U,\,T^{*\,(0,q)}X\boxtimes(T^{*\,(0,q)})^{*})\,$

and the leading term of $b(x,x,\lVert\nu\rVert)$ is non-zero.

###### Proof.

Suppose $q=n_{-}$, on small local neighbourhood $D$ of a point $p\in X_{\nu}$
we have

$S_{k\nu}^{(q)}(x,\,y)=\frac{1}{(2\,\pi)^{t}}\int_{-\pi}^{\pi}\dots\int_{-\pi}^{\pi}e^{\imath
k\,\langle\nu,\,\theta\rangle}\,S^{(q)}(x,\,e^{\imath\,\theta}\cdot
y)\,\mathrm{d}\theta\,$

where $\theta=(\theta_{1},\dots,\theta_{t})$. We set $\theta_{t}$ the variable
for circle action of $\mathbb{T}^{1}_{\nu}$ and
$(\theta_{1},\dots,\theta_{t-1})$ the variables for $\mathbb{T}^{t-1}_{\nu}$.
We can then use local coordinates defined in 2.1, we have

$S_{k\nu}^{(q)}(x,\,y)=\frac{1}{2\,\pi}\int_{-\pi}^{\pi}e^{\imath
k\,\lVert\nu\rVert\theta_{t}-\imath\,kx_{2n+1}+\imath\,ky_{2n+1}}\,S_{\mathbb{T}^{t-1}}^{(q)}(\mathring{x},\,e^{\imath\,\theta_{t}}\cdot\mathring{y})\,\mathrm{d}\theta_{t}$

where

$S_{\mathbb{T}^{t-1}}^{(q)}({x},\,{y})=\frac{1}{(2\,\pi)^{t-1}}\int_{-\pi}^{\pi}\dots\int_{-\pi}^{\pi}\,S^{(q)}(x,\,e^{\imath\,\mathrm{diag}(\theta_{1},\dots,\theta_{r-1})}\cdot
y)\,\mathrm{d}\theta_{1}\,\dots\mathrm{d}\theta_{t-1}\,,$ (8)

and $\mathring{x}=(x_{1},\dots,x_{n},0)$ and
$\mathring{y}=(y_{1},\dots,y_{n},0)$. The action of $\mathbb{T}^{t}$ restricts
to an action of $\mathbb{T}^{t-1}$ whose moment map is given by

$\mu_{|\mathbb{T}^{r-1}}=p_{\nu}\circ\mu\,,\quad\text{ where
}p_{\nu}\,:\,\mathfrak{t}^{*}\rightarrow\mathfrak{t}_{\nu}^{*}$

is the canonical projection onto the $t-1$-dimensional subspace $\nu^{\perp}$
in $\mathfrak{t}^{*}$. Since we are assuming $X_{\nu}\neq\emptyset$ we have
that $0\in\mathfrak{t}_{t-1}^{*}$ does lie in the image of the moment map
$\mu_{|\mathbb{T}^{r-1}}$. We can apply Theorem $1.5$ in [HH] for the action
of $\mathbb{T}^{r-1}$ with moment map $\mu_{\mathbb{T}^{r-1}}$ to the
$\mathbb{T}^{r-1}$-invariant Szegő kernel in (8). The $k\nu$ Fourier
components are the one induced by the transversal
$\mathbb{T}^{1}_{\nu}$-action and can be computed in a similar way as in
Theorem $1.8$ of [HH]. Thus the proof can be deduced by applying Theorem $1.5$
and $1.8$ in [HH]. ∎

For ease of notation we introduce a definition to say that an operator behaves
micro-locally as equivariant Szegő projector we just studied. For simplicity
we assume $q=n_{-}$.

######  Definition 2.2 (Equivariant Szegő type operator).

Suppose that $q=n_{-}$ and consider
$H:\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ be a continuous operator with
distribution kernel

$H(x,y)\in\mathcal{D}^{\prime}(X\times
X,T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})\,.$

We say that $H$ is a complex Fourier integral operator of equivariant Szegő
type of order $n+(1-t)/2\in\mathbb{Z}$ if $H$ is smoothing away
$\mu^{-1}(\imath\,\mathbb{R}_{+}\cdot\nu)$ and for given
$p\in\mu^{-1}(\imath\,\mathbb{R}_{+}\cdot\nu)$ let $D$ a local neighborhood of
$p$ with local coordinates $(x_{1},\dots\,x_{2n+1})$. Then, the distributional
kernel of $H$ satisfies

$H_{k}(x,y)=e^{\imath k\,\Psi(x,\,y)}\,a(x,\,y,\,k)+O(k^{-\infty})\quad\text{
on }D$

where $a\in S^{n+(1-t)/2}_{1,0}(D\times
D\times\mathbb{R}_{+},T^{*0,q}X\boxtimes(T^{*0,q}X)^{*})$, and $\Psi$ is as in
the end of Section 2.1.

For $k=0$, $H_{0}^{(q)}(X)$ is the space of $\mathbb{T}$-fixed vectors, by
Theorem $1.5$ in [HH] we have $\dim H_{0}^{(q)}(X)<+\infty$ since when
$0\notin\mu(X)$ the projector $S^{(q)}_{\mathbb{T}^{t}}$ is smoothing. Now,
given $k_{1},k_{2}\in\mathbb{Z}$ with $k_{1},k_{2}\neq 0$, consider $f_{1}\in
H_{k_{1}\nu}^{(q)}(X)$ and $f_{2}\in H_{k_{2}\nu}^{(q)}(X)$, we have
$f_{1}\cdot f_{2}\in H_{(k_{1}+k_{2})\nu}^{(q)}(X)$; we can study $\dim
H_{k\nu}(X)$ for $k$ large, we have

$\dim H_{k\nu}^{(q)}(X)=\int_{X}S_{k\nu}^{(q)}(x,\,x)\,\mathrm{dV}_{X}(x)\,.$

These dimensions can be studied as $k\rightarrow+\infty$ by using the
microlocal properties of the Szegő kernel and Stationary Phase Lemma in a
similar way as in Corollary $1.3$ in [P1] we have $\dim
H_{k\nu}^{(q)}(X)=O(k^{d+1-t})$. So, we have the following generalization of
Theorem $A.3$ in [GS1], where it is proved for spaces of functions
$H(X)_{k\nu}$.

######  Lemma 2.2.

If $0\notin\mu(X)$, then $H_{k\nu}^{q}(X)$ are finite dimensional.

##  3 Proof of Theorem 1.1

###  3.1 Ohsawa–Takegoshi extension theorem

We recall the Ohsawa–Takegoshi extension theorem from [GZ] in the form we need
in this paper. Before stating the theorem we need some notation. Since the
domain $D$ is strictly pseudo-convex and the action of $\mathbb{T}^{1}_{\nu}$
is transversal for $(X,T^{1,0}X)$ we get a complex manifold
$N:=X/\mathbb{T}^{1}_{\nu}$ and a ample line bundle $(B,h)$ over it such that
curvature of the unique compatible and holomorphic connection is
$-2\imath\,\eta_{\nu}$, where $\mathrm{d}\omega_{0}=2\pi^{*}\eta_{\nu}$ and
$\pi:X\rightarrow N$, $X\subset B^{*}$. In the situation we are considering
here the transversal circle action is the one induced by restriction of the
action of $\mathbb{T}^{r}$ to $\mathbb{T}^{1}_{\nu}$. Notice that when $X$ is
the boundary of the Grauert tube of a given Hodge manifold $M$, one also has a
built-in circle action; we will not discuss this situation and we instead
refer to [P1] and [P2].

The sub-manifold $X_{\nu}\subset X$ defines a circle bundle over a real sub-
manifold $\iota_{N}:N_{\nu}\hookrightarrow N$, which is co-isotropic with
respect to $\eta_{\nu}$. Then the space of leaves $N^{\mathrm{red}}_{\nu}$ of
the characteristic foliation of $\iota_{N}^{*}\eta_{\nu}$ inherits a unique
symplectic form $\eta_{\nu}^{\mathrm{red}}$ such that

$\pi^{*}_{N}\omega_{0}^{\mathrm{red}}=\iota_{N}^{*}\omega_{0}\,,$

where $\pi_{N}:N_{\nu}\rightarrow N^{\mathrm{red}}_{\nu}$ is the natural
projection.

######  Theorem 3.1 ([GZ]).

Let $N$ be a complex manifold of dimension $n$ and let
$N^{\mathrm{red}}_{\nu}$ be a complex sub-manifold of codimension $r-1$. Let
$h$ be a smooth metric on the holomorphic ample line bundle $B$. Then, there
exists a uniform constant $C$, such that, for any holomorphic section $s$ of
$B_{|N^{\mathrm{red}}_{\nu}}$ on $N^{\mathrm{red}}_{\nu}$, there exists a
holomorphic section $S$ of $B$ on $N$ satisfying $S=s$ on
$N^{\mathrm{red}}_{\nu}$ and

$\int_{N}\lvert S\rvert^{2}_{h}\,\mathrm{dV}_{N}\leq
C\,\int_{N^{\mathrm{red}}_{\nu}}\lvert
f\rvert^{2}_{h}\,\mathrm{dV}_{N^{\mathrm{red}}_{\nu}}\,.$

###  3.2 The proof

Consider the restriction to $X_{\nu}$ of a function $f\in H(X)_{k\nu}$. Then,
it is $\mathbb{T}^{r-1}_{\nu}$-invariant on $X_{\nu}$; in fact for each
$\theta\in\nu^{\perp}$ we have

$e^{\imath\theta}\cdot f=t^{\imath k\langle\nu,\theta\rangle}f=f\,;$

thus, there is a natural injection

$\delta_{k}\,:\,H(X)_{k\nu}\hookrightarrow H(X^{\mathrm{red}}_{\nu})\,.$

Furthermore, notice that the map $\delta_{k}$ induces a map onto
$H_{k}(X^{\mathrm{red}}_{\nu})$ by composing with

$F_{k}\,:\,L^{2}(X^{\mathrm{red}}_{\nu})\rightarrow
L^{2}_{k}(X^{\mathrm{red}}_{\nu})$

which takes the $k$-th component of $L^{2}(X^{\mathrm{red}}_{\nu})$ for the
$\mathbb{T}^{1}_{\nu}$-action, since it is CR transversal.

Now, given $f\in H_{k}(X^{\mathrm{red}}_{\nu})$, we recall that
$X^{\mathrm{red}}_{\nu}$ is the total space of the circle bundle over
$N^{\mathrm{red}}_{\nu}$, thus $f$ is a holomorphic section of the $k$-th
tensor power of a line bundle over $N^{\mathrm{red}}_{\nu}$ by the usual
identification. The complex manifold $N^{\mathrm{red}}_{\nu}$ lies naturally
in $N:=X/\mathbb{T}^{1}_{\nu}$, it is a complex sub-orbifold of complex
codimension $r-1$. By the Ohsawa–Takegoshi extension theorem there is a global
holomorphic section $F$ in $H_{k\nu}(X)$ whose restriction to
$X^{\mathrm{red}}_{\nu}$ equals $f$. The Theorem is thus concluded.

##  4 Proof of Theorem 1.2

In this section we shall explain how to prove asymptotic commutativity for
quantization and reduction for spaces of $(0,q)$-forms. We recall that
$X^{\mathrm{red}}_{\nu}$ is a CR manifold whose Levi form is non-degenerate,
with $r$ negative eigenvalues; let us denote with $S^{(q)}$ the corresponding
Szegő kernel $S^{(q)}_{\mathrm{red}}$ for $(0,q)$-forms whose $k$-the Fourier
components are the one induced by the $\mathbb{T}^{1}_{\nu}$-action on
$X^{\mathrm{red}}_{\nu}$. Since $\nu^{\perp}_{X}:=\mathrm{val}(\nu^{\perp})$
is orthogonal to $H_{x}X_{\nu}\cap J_{x}H_{x}X_{\nu}$ and $H_{x}X_{\nu}\cap
J_{x}H_{x}X_{\nu}\subset(\nu^{\perp}_{X})^{\perp_{b}}$ for every $x\in
X_{\nu}$, we can find a $\mathbb{T}$-invariant orthonormal basis
$\\{Z_{1},\,\dots,\,Z_{n}\\}$ of $T^{*\,0,1}X$ on $X_{\nu}$ such that for each
$j,\,k=1,\dots n$,

$L_{x}(Z_{j}(x),\,\overline{Z}_{j}(x))=\delta_{jk}\,\lambda_{j}(x)\,,$

where

$Z_{j}(x)\in(\nu^{\perp}_{X,x}+\imath J\nu^{\perp}_{X,x})\text{ for each
$j=1,\,\dots,\,r-1$ }\,,$ $Z_{j}(x)\in H_{x}X_{\nu}\cap
J_{x}(H_{x}X_{\nu})\text{ for each $j=r,\,\dots,\,n$ }\,.$

Let $\\{Z_{1}^{*},\,\dots,\,Z_{n}^{*}\\}$ denote the orthonormal basis of
$T^{*\,0,1}X$ on $X_{\nu}$, dual to $Z_{1},\,\dots,\,Z_{n}$. Fix
$s=0,\,1,\,2,\dots,\,n-r+1$. For $x\in X_{\nu}$, put

$B_{x}^{*\,0,s}X=\left\\{\sum_{r\leq j_{1}<\dots<j_{s}\leq
n}a_{j_{1},\dots,j_{s}}Z_{j_{1}}^{*},\,\dots,Z_{j_{s}}^{*};\,a_{j_{1},\dots,j_{s}}\in\mathbb{C},\,\right\\}$

and let $B^{*\,0,s}X$ be the vector bundle of $X_{\nu}$ with fibre
$B_{x}^{*\,0,s}$, $x\in X_{\nu}$. Let
$C^{\infty}(X_{\nu},B^{*\,0,s}X)^{\mathbb{T}}$ denote the set of all
$\mathbb{T}$-invariant sections of $X_{\nu}$ with values in $B_{x}^{*\,0,s}X$;
let

$\iota_{\mathrm{red}}\,:\,C^{\infty}(X_{\nu},B^{*\,0,s}X)^{\mathbb{T}}\rightarrow\Omega^{0,s}(X^{\mathrm{red}}_{\nu})$

be the natural identification. Before defining the map between
$H^{q}(X)_{k\nu}$ and $H^{q-r}(X^{\mathrm{red}}_{\nu})_{k}$ we need one more
piece of notation. We can assume that $\lambda_{1}<0,\cdots,\lambda_{r}<0$ and
also $\lambda_{t}<0,\dots,\,\lambda_{n_{-}-r+t-1}<0$. For $x\in X_{\nu}$, set

$\hat{\mathcal{N}}(x,n_{-})=\\{c\,Z_{t}^{*}\wedge\,\cdots\,\wedge
Z_{n_{-}-r+t-1}^{*}\,,c\in\mathbb{C}\\}$

and let
$\hat{p}\,:\,\mathcal{N}(x,n_{-})\rightarrow\hat{\mathcal{N}}(x,n_{-})$ to be

$p(c\,Z_{1}^{*}\wedge\,\cdots\,\wedge Z_{r}^{*}\wedge
Z_{t}^{*}\wedge\,\cdots\,\wedge
Z_{n_{-}-r+t-1}^{*}):=c\,Z_{t}^{*}\wedge\,\cdots\,\wedge Z_{n_{-}-r+t-1}^{*}$

where $c$ is a complex number. Put $\iota_{\nu}\,:\,X_{\nu}\rightarrow X$ be
the natural inclusion, let
$\iota_{\nu}^{*}\,:\,\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X_{\nu})$ be the
pull-back of $\iota_{\nu}$.

Let $q=n_{-}$; now, inspired by [HH], we define the map

$\sigma_{k\nu}\,:\,H^{(q)}_{k\nu}(X)\rightarrow H(X^{\mathrm{red}}_{\nu})_{k}$

given by

$\sigma_{k\nu}(u):=k^{(t-1)/4}\,S^{(q-r)}_{\mathrm{red},k}\circ\iota_{\mathrm{red}}\circ\hat{p}\circ\tau_{x,n_{-}}\circ
e\circ\iota^{*}_{\nu}\circ S^{(q)}_{k\nu}(u)\,,$ (9)

here, $e(x)$ is a $\mathbb{T}$-invariant smooth function on $X_{\nu}$ that can
be found explicitly. We recall the idea of the proof which then follows from
calculation of complex Fourier integral operators. Notice that operators
$S^{(q-r)}_{\mathrm{red},k}$ and $S^{(q)}_{k\nu}$ appearing in (9) are known
explicitly. Micro-locally $S^{(q-r)}_{\mathrm{red},k}$ is the $k$-th Fourier
component of the standard Szegő kernel for the CR manifold
$X^{\mathrm{red}}_{\nu}$, on the other hand $S^{(q)}_{k\nu}$ is described in
Section 2.2. In fact, $\sigma_{k\nu}$ is an isomorphism for $k$ large if we
can prove that

$\sigma_{k\nu}\,:\,H^{q}(X)_{k\nu}\rightarrow H^{q-r}(X_{\nu})_{k}\quad\text{
and }\quad\sigma_{k\nu}^{*}\,:\,H^{q-r}(X^{\mathrm{red}}_{\nu})_{k}\rightarrow
H^{q}(X)_{k\nu}$

are injective for $k$ large.

The injectivity of the map $\sigma_{k\nu}$ is a consequence of the following
equality of operators

$\sigma_{k\nu}^{*}\sigma_{k\nu}\equiv c_{0}\,(1+R_{k})\,S^{(q)}_{k\nu}$

where $c_{0}$ is a positive constant and
$1+R_{k}\,:\,\Omega^{0,q}(X)\rightarrow\Omega^{0,q}(X)$ is an injective
integral operator of equivariant Szegő type, see Definition 2.2. The
injectivity of operator $1+R_{k}$ follows by direct computations of the
leading symbol of $R_{k}$ which vanishes at $\mathrm{diag}(X_{\nu}\times
X_{\nu})$. In fact, as a consequence of this, by Lemma $6.8$ in [HH] we have
that $\lVert R_{k}u\lVert\leq\epsilon_{k}\lVert u\rVert$ for all
$u\in\Omega^{0,q}(X)$, for all $k\in\mathbb{N}$ where $\epsilon_{k}$ is a
sequence with $\lim_{k\rightarrow+\infty}\epsilon_{k}=0$; this in turn
implies, if $k$ is large enough, that the map $1+R_{k}$ is injective.

The injectivity of the map $\sigma_{k\nu}^{*}$ follows by studying
$\sigma_{k\nu}\sigma_{k\nu}^{*}$. We don’t repeat the proof here since it is
an application of the stationary phase formula for Fourier integral operators
of complex type, see [MS], and it follows in a similar way as in [HH].

Acknowledgements: This project was started during the author’s postdoctoral
fellowship at the National Center for Theoretical Sciences in Taiwan; the
author thanks the Center for the support.

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