# Exploring the archives: A search for novae in UVIT snapshots of M31

Judhajeet Basu Indian Institute of Astrophysics, 2nd Block Koramangala,
560034, Bangalore, India  Pondicherry University, R.V. Nagar, Kalapet, 605014
Puducherry, India  Krishnendu S. Indian Institute of Astrophysics, 2nd Block
Koramangala, 560034, Bangalore, India  Amrita School of Physical Sciences,
Amrita School of Engineering, Amrita University, 641112, Ettimadai, India
Sudhanshu Barway Indian Institute of Astrophysics, 2nd Block Koramangala,
560034, Bangalore, India  Shatakshi Chamoli Indian Institute of Astrophysics,
2nd Block Koramangala, 560034, Bangalore, India  Pondicherry University, R.V.
Nagar, Kalapet, 605014 Puducherry, India  G.C. Anupama Indian Institute of
Astrophysics, 2nd Block Koramangala, 560034, Bangalore, India

###### Abstract

Extensive multi-wavelength studies of novae have been carried out in our
galaxy and in M31 for decades. However, UV studies of extragalactic novae are
limited, especially those in quiescence. For the first time, we present a UV
catalog of novae in M31 using the archival AstroSat UVIT imaging data. We used
two image subtraction techniques to retrieve objects located deep into the M31
central region. We have found 42 novae in total in the UVIT images, 15 of
which have been detected in multiple filters in FUV and NUV. The novae
detected at quiescence show signatures of accretion disk from their UV
spectral energy distributions, whereas those in the outburst phase show
signatures of pseudo-photosphere. A few novae were also detected in multiple
epochs. Some show a near-constant FUV magnitude at quiescence, while others
caught near the outburst reveal pre-eruption dips in their light curves. We
conclude with a discussion on the significance of UV surveys in illuminating
theoretical predictions for novae systems, including detecting the elusive
early UV flash.

galaxies: individual (M31) – novae, cataclysmic variables – stars: techniques:
image processing – techniques: photometric – transients: novae

††facilities:  AstroSat (UVIT) ††software: CCDLAB (Postma & Leahy, 2017), IRAF
v2.16.1 (Tody, 1993), Python v3.6.6 (Van Rossum & Drake, 2009), NumPy (Harris
et al., 2020), SciPy (Virtanen et al., 2020), Pandas (McKinney et al., 2010),
Matplotlib (Hunter, 2007), photutils (Bradley et al., 2023), SWarp (Bertin et
al., 2002), SAOImageDS9 (Joye & Mandel, 2003)

##  1 Introduction

Novae are cataclysmic events triggered by thermonuclear runaway reactions on
the surface of accreting White Dwarfs (WD) (Starrfield, 1999; Bode & Evans,
2008; Starrfield et al., 2016). These novae eruptions increase the brightness
of the binary system up to several orders of magnitude, triggering
nucleosynthesis and expelling material into the interstellar medium (ISM)
(Hernanz & Jose, 1998; Gehrz et al., 1998). Individual novae are excellent
laboratories for understanding accretion and mixing processes (Guo et al.,
2022), binary evolution (Kraft, 1964; Shen & Quataert, 2022), supernova type
Ia progenitors (Hillman et al., 2016; Wang, 2018), and shock mechanisms (Aydi
et al., 2020; Chomiuk et al., 2021). Nova studies are also important tracers
of galactic chemical evolution and ISM enrichment (Starrfield et al., 2020).

One of the galaxies in which novae are studied in detail is M31, owing to full
visibility for more than half a year, close distance, low inclination, and
high nova rate. High occurrences of recurrent novae in M31, particularly those
with short recurrence periods of fewer than 10 years such as M31N2008-12a
111All M31 novae denoted by M31NYYYY-MMx will be denoted by YYYY-MMx hereafter
(Basu et al., 2024b), 2017-01e, 2013-10c, 1990-10a, 1963-09c and others have
intrigued astronomers to explore the M31 galaxy. M31 has been surveyed
extensively for novae for several decades in optical wavebands (Arp, 1956;
Rosino, 1964, 1973; Ciardullo et al., 1987; Rosino et al., 1989; Sharov &
Alksnis, 1991; Tomaney & Shafter, 1992; Rector et al., 1999; Shafter & Irby,
2001; Darnley et al., 2004, 2006; Kasliwal et al., 2011; Shafter et al.,
2011b; Lee et al., 2012; Williams et al., 2014, 2016; Rector et al., 2022).
Over the last two decades, a few extensive studies have been performed in
X-rays as well (Pietsch et al., 2007; Pietsch, 2010; Henze et al., 2010, 2011,
2014). However, surveys are limited in IR (Shafter et al., 2011a) and UV (Cao
et al., 2012) bands. Quiescence studies of M31 novae were conducted by
Williams et al. (2014, 2016) to look for progenitors using HST optical data,
taking advantage of its high resolution.

Galactic novae have been studied in the UV since the launch of IUE (Starrfield
1986 and references therein). Selvelli & Gilmozzi (2013) studied a sample of
18 galactic novae in quiescence in UV, focusing on their spectral energy
distributions (SEDs), which were later combined with outburst phase data by
Selvelli & Gilmozzi (2019) for an overall understanding of the properties of
old novae. Page et al. (2020, 2022) summarized the UV and X-ray light curves
of galactic novae observed by Swift.

Rare UV studies of extra-galactic novae, such as Cao et al. (2012) for M31,
and Lessing et al. (2023) and Shara et al. (2023) for M87, concentrated
primarily on eruption characteristics, spatial distribution, nova rates, and
population studies. In the case of the Magellanic Clouds, their wide fields
and low nova rates (Mróz et al., 2016) have led to limited surveys in optical
and other bands. However, focusing on some individual novae, for instance,
SMCN 2016-10a (Aydi et al., 2018), Nova LMC 1971b (Bode et al. 2016), and Nova
LMC 1968 (Kuin et al. 2020) have revealed their UV quiescence features.
Nonetheless, no study has been reported in the literature honing on the
systematic study of the quiescence phase of extragalactic novae in UV. Though
nova systems in M31 and other galaxies are expected to be similar to those in
our Milky Way, the missing gap must be filled for completeness.

This work presents the search for novae in M31 in the UV bands using archival
AstroSat-UVIT data. The paper discusses our data reduction, analysis, and
detection methods and provides a catalog of the novae detected (including
those caught during outbursts) as well as non-detection upper limits. We also
briefly discuss the importance of UV missions and multi-wavelength facilities
as synoptic surveys for novae studies.

##  2 Data

###  2.1 Observations and data reduction

Table 1: UVIT observation log of all images, including co-added frames, in
which at least one nova was detected. The full table, which includes details
of all individual and co-added frames analyzed in this work, will be available
in the published version.

Observation ID | RA | DEC | Filter | Field | Obs Epoch | Exp time | Limiting Mag | Novae detected
---|---|---|---|---|---|---|---|---
| (deg) | (deg) |  |  | (BJD) | (s) | (AB) |
A02_028T01_9000000724 | 10.71683 | 41.12430 | F148W | M31 01 | 2457671.85662 | 7736.347 | 22.94 | 18
A02_028T01_9000000724 | 10.71683 | 41.12430 | F172M | M31 01 | 2457672.19496 | 3611.883 | 21.39 | 14
A02_028T01_9000000724 | 10.71683 | 41.12430 | N219M | M31 01 | 2457671.85657 | 7780.686 | 22.08 | 12
A02_028T01_9000000724 | 10.71683 | 41.12430 | N279N | M31 01 | 2457672.19491 | 3627.438 | 21.53 | 11
A07_007T04_9000003310 | 10.70412 | 41.27929 | F148W | M31 01 | 2458805.25645 | 17045.731 | 23.37 | 17
A07_007T04_9000003310 | 10.70412 | 41.27929 | F169M | M31 01 | 2458804.85075 | 10426.754 | 22.63 | 15
A07_007T04_9000003310 | 10.70412 | 41.27929 | F172M | M31 01 | 2458806.27133 | 16606.192 | 22.34 | 15
A10_002T04_9000004022 | 10.59843 | 41.21609 | F148W | M31 01 | 2459174.71396 | 11759.864 | 23.10 | 18
A02_028T03_9000000788 | 11.01776 | 41.54606 | F148W | M31 02 | 2457704.13548 | 7942.829 | 22.69 | 3
A02_028T03_9000000788 | 11.01776 | 41.54606 | F172M | M31 02 | 2457704.48574 | 5350.934 | 21.48 | 2
A02_028T03_9000000788 | 11.01776 | 41.54606 | N219M | M31 02 | 2457704.13543 | 7986.231 | 22.01 | 2
A02_028T03_9000000788 | 11.01776 | 41.54606 | N279N | M31 02 | 2457704.48567 | 5544.727 | 21.33 | 3
A10_002T05_9000004000 | 11.13728 | 41.48918 | F148W | M31 02 | 2459167.97966 | 12610.195 | 22.77 | 1
A04_022T02_9000002184 | 11.33933 | 41.88125 | F148W | M31 08 | 2458292.30948 | 3371.811 | 22.85 | 1
A04_022T02_9000002184 | 11.33933 | 41.88125 | F172M | M31 08 | 2458292.98608 | 9017.513 | 22.29 | 1
A05_004T03_9000002508 | 9.87699 | 40.36606 | F148W | M31 15 | 2458435.07551 | 4610.422 | 23.04 | 1
A05_004T03_9000002508 | 9.87699 | 40.36606 | F169M | M31 15 | 2458435.27774 | 9286.96 | 22.69 | 1
A05_004T03_9000002538 | 9.86651 | 40.37805 | F148W | M31 15 | 2458449.21674 | 2104.392 | 22.72 | 1
A05_004T03_9000002538 | 9.86651 | 40.37805 | F169M | M31 15 | 2458449.42980 | 9224.752 | 22.76 | 1
Co-added frames ∗ | 10.70959 † | 41.27957 † | F148W | M31 01 | 2458550.60901 ‡ | 36541.942 | 23.54 | 3 #
Co-added frames ∗ | 10.70959 † | 41.27957 † | F172M | M31 01 | 2458239.23315 ‡ | 20218.075 | 22.50 | 1 #

Notes
∗ Co-added frames include all the images of the same field in the same filter
combined together. For example, the co-added frame of the M31 01 field in the
F148W filter includes the sum of F148W images with Obs ID
A02_028T01_9000000724, A07_007T04_9000003310 and A10_002T04_9000004022.
† RA DEC of the coadded frames are the centers of the co-added images.
‡ Observation epochs for co-added frames are the mid-point of the first and
last observation.
# The number of novae detected in the co-added frames but not in the
individual images.

AstroSat is a space-based high-energy telescope in a low inclination Earth
orbit (Singh et al., 2014). The Ultraviolet Imaging Telescope (UVIT) is one of
the primary instruments onboard AstroSat, consisting of a twin telescope with
one FUV and one NUV/VIS channel, capable of observing simultaneously. Both
telescopes have a mirror of 38 cm diameter and a circular field of view of 28
arcmin. UVIT has one of the best spatial resolutions at 1.5 arcsec in UV,
making it suitable for studying extragalactic UV sources. UVIT’s FUV filters
offer a unique waveband (up to 1000 Å) to detect and study novae. Ground-based
and in-orbit calibrations of the UVIT instrument are reported in Subramaniam
et al. (2016) and Tandon et al. (2017a, b).

UVIT has observed multiple fields of M31 in different filters over multiple
epochs. The observations and datasets summarized by Leahy et al. (2020, 2021)
are used in this work. Additional fields observed later by the same PI and
other M31 fields observed by other PIs have also been looked into for this
work. All the data utilized in this work have been summarised in Table 1. The
Level 1 and Level 2 data have been made available online by the Indian Space
Science Data Center’s (ISSDC)
astrobrowse222https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp.
We obtained all the L1 data and processed it using CCDLAB (Postma & Leahy,
2017). The automated steps in CCDLAB include flat-fielding, drift correction,
and cosmic-ray correction, among others discussed in detail in Postma & Leahy
(2021). The orbit-wise images were registered and merged to get a single image
with a high signal-to-noise ratio. The final step in CCDLAB involved
performing astrometry on the merged and orbit-wise images.

The F148W images of all the 19 M31 fields were combined together to produce a
composite image of M31 in FUV. Individual images were normalized by their
exposure times to bring them to the same scale. The normalized images were
then tiled using SWarp (Bertin et al., 2002). The exposure array frames
generated by CCDLAB were used as weights to take care of the edges and
overlapping regions during tiling. The M31 FUV mosaic is shown in Figure 2.

Figure 1: UVIT F148W mosaic image of M31 generated using SWarp. The locations
of all the novae detected in the archival images are marked in the figure. The
inset shows a zoomed-in version of the M31 central region.

Figure 2: Difference images with inset showing detection of novae (a) M31N
2016-03e, (b) M31N 2015-05a, (c) M31N 2016-05b and (d) M31N 2016-03c. Left:
Median-combined template subtraction method. Right: Isophote model subtraction
method.

###  2.2 Removal of background contribution from the bulge

The bulge of M31 is shrouded by galactic light. This light contamination must
be removed to search for any point source in this region. Since traditional
image subtraction techniques do not entirely remove the light excess in
crowded fields, we have used two methods to create templates for detecting
novae in the bulge region.

The first method involves a median combination of multi-epoch images of the
same field in the same filter. Tandon et al. (2020) showed that the
sensitivity of the UVIT detectors has been constant over the years. Flux-
scaling of the multi-epoch images thus only involved normalizing by their
respective exposure times. Since UVIT is a space-based telescope, we expect
the point spread function (PSF) to be constant across all epochs; hence, PSF
matching was not performed before combining. The flux-scaled images were
registered and then median-combined to make the template. This template was
subtracted from the normalized image of each epoch to look for novae.

In the second method, we generated an isophote model of the bulge. We modeled
the bulge of M31 by performing an isophote fit using the isophote module of
the photutils python package (Bradley et al., 2023). The isophote model
generated for a given range of semi-major axis (sma) ($\sim 400$ pixels = 2.78
arcmin) for each epoch was used as a template. Subtracting this template from
its original image reveals all sources above the nuclear brightness level
predicted by the model.

Both techniques were useful in detecting four novae, otherwise hidden in the
bright M31 bulge. The difference images and the results from both methods are
shown in Figure 2.

###  2.3 Source detection and photometry

A list of all M31 novae, as of 19 May 2022, is available at a website hosted
by the Max Planck Institute for Extraterrestrial Physics
333https://www.mpe.mpg.de/$\sim$m31novae/opt/m31/M31_table.html (Pietsch,
2010). We performed forced photometry at the locations of the novae listed to
check for their detection in the UVIT data. We also looked into novae that
erupted after the latest UVIT observations (Nov 2020), for which the data are
publicly available, to check for pre-eruption UV magnitudes or upper limits.
All the detected novae were carefully inspected visually to confirm the
detections. Sources in crowded fields, where source confusion could occur
within a 5 arcsec region of the nova, were rejected. We used a detection
threshold of $3\sigma$ to eliminate faint sources. The limiting magnitude in
the F148W filter for deep exposures ($>$ 10 ks) was found to be around 23 mags
for a $3\sigma$ detection near the central region of M31. The detection limit
for regions away from the center was around 0.5 mag deeper. The average
limiting magnitude for each image is given in Table 1. However, it should be
noted that the limiting magnitude varies spatially in an image. All the
detected sources are shown in the UVIT F148W mosaic image in Figure 2, and
their details are mentioned in Table LABEL:catalog.

Availability of multiple images in the same field and the same filter
encouraged us to sum them and look for fainter novae in the deeper exposures.
Summing the images increased the limiting magnitude of the images by 0.2–0.4
mags (see Table 1). We found three old novae, namely 1985-09c, 1998-09e, and
2010-03a, in the co-added frame of the central region of M31. All three novae
were fainter than 23 mag (see Table LABEL:catalog). The FUV and NUV magnitudes
of detected novae and upper limits of all undetected novae in the deepest
available images in each filter were estimated, and have been provided in the
appendix A Table 4.

Leahy et al. (2020) and Devaraj et al. (2023) found that the PSF of UVIT
images is better represented by Moffat functions rather than elliptical or
circular Gaussian. However, neither Moffat nor Gaussian PSF could accurately
estimate the flux from stars, which was underestimated due to the presence of
extended PSF wings in the UVIT images (Tandon et al., 2017b; Leahy et al.,
2020). In such cases, aperture photometry gives the best results, but due to
crowded fields of M31, aperture photometry is not a feasible solution. PSF
photometry was performed using an elliptical Moffat function to fit the
sources. Standard routines in IRAF (Tody, 1993) were used for photometric
measurements. To accurately measure the flux, we performed aperture photometry
with an aperture of 15 pixels ($\sim 4\times$FWHM) and PSF photometry with a
PSF radius of 11 pixels ($\sim 3\times$FWHM) of isolated bright sources to
determine the aperture correction term. This correction took care of the
emission from the extended PSF wings. It was then applied to the magnitudes of
the novae obtained by PSF photometry. The zero-point calibrations in the AB
system were adopted from Tandon et al. (2020). We have not corrected the
magnitudes for extinction.

For the novae in the central fields, we tried aperture and PSF photometry on
both sets of difference images. In aperture photometry, the background
subtraction led to inconsistencies, whereas in PSF photometry, the PSF model
fitting diverged. We thus performed PSF photometry with aperture correction of
the bulge novae on the original science images using the same technique we
used for the other novae. It ensured the measurements were consistent for all
novae reported in this work.

The photometry of all the detected sources in various filters at different
epochs is given in Table LABEL:catalog.

Notes
${\dagger}$ Also reported by Leahy et al. (2021)
o Also detected around outburst (within $\sim$ 100 days after outburst and
$\sim$ 10 days before outburst)
q Also detected at quiescence (more than 100 days after outburst or 30 days
before outburst)
∗ $t_{2}$ determined from linear interpolation of the optical data around
$(\rm m_{max}-2)$
${\ddagger}$ Observation date for co-added images is the midpoint of the first
and last observation

References \- (1) Ovcharov et al. (2015a), (2) Fabrika et al. (2015), (3)
Delvaux et al. (2015), (4) Williams & Darnley (2015a), (5) Williams & Darnley
(2015b), (6) Darnley & Williams (2016a), (7) Kasliwal & Cao (2018a), (8)
Hornoch & Kucakova (2016a), (9) Hornoch & Kucakova (2016b), (10) Fabrika et
al. (2016c), (11) Hornoch et al. (2016c), (12) Fabrika et al. (2016a), (13)
Hornoch & Kucakova (2016c), (14) Valcheva et al. (2016), (15) Fabrika et al.
(2016b), (16) Conseil (2018), (17) Campaner (2019), (18) Soraisam et al.
(2019b), (19) Zhang & Gao (2020a), (20) Williams et al. (2020a), (21) Hornoch
& Kucakova (2020a), (22) Williams et al. (2020c), (23) Zhang & Gao (2020b),
(24) Valcheva et al. (2020), (25) Soraisam et al. (2020), (26) Zhang & Gao
(2020c), (27) Srivastav et al. (2020), (28) Belligoli (2020), (29) Fabrika et
al. (2020), (30) Balcon (2020), (31) Hornoch & Kucakova (2020b), (32) Tomaney
& Crotts (1996), (33) Pietsch et al. (2010a), (34) Cao et al. (2012), (35)
Ovcharov et al. (2013), (36) Hornoch et al. (2013), (37) Kasliwal & Cao
(2018b), (38) Darnley et al. (2018), (39) Zhang & Gao (2018a), (40) Valcheva
et al. (2018), (41) Hornoch & Kucakova (2018), (42) Williams et al. (2019a),
(43) Zhang & Gao (2018b), (44) Hornoch & Kusnirak (2019), (45) Hornoch &
Kucakova (2019a), (46) Conseil (2019), (47) Williams et al. (2019b), (48)
Hornoch & Kucakova (2019b), (49) Zhang & Gao (2019a), (50) Lee et al. (2019),
(51) Carey (2019), (52) Dahiwale & Fremling (2020), (53) Soraisam et al.
(2019a), (54) Nordin et al. (2019), (55) Hornoch & Kucakova (2019c), (56)
Madrigal-Aguado et al. (2019), (57) Williams et al. (2019c), (58) Fabrika et
al. (2019), (59) Zhang & Gao (2019b), (60) Jiang et al. (2019), (61) Hornoch
et al. (2019), (62) Williams et al. (2020b), (63) Zhang & Gao (2020d), (64)
Hornoch & Kucakova (2020c), (65) Zhang & Gao (2020e), (66) Fremling (2020),
(67) Hornoch & Kucakova (2016d), (68) Martin et al. (2018), (69) Hornoch &
Honkova (2015), (70) Ovcharov et al. (2015b), (71) Hornoch et al. (2016a).
(72) Hornoch et al. (2016b), (73) Hornoch & Kucakova (2016e), (74) Darnley &
Williams (2016b), (75) Clark et al. (2024), (76) Shumkov et al. (2017), (77)
Ciardullo et al. (1987), (78) Sharov et al. (2000), (79) Pietsch et al.
(2010b), (80) Hornoch et al. (2010)

For more references check https://www.mpe.mpg.de/
m31novae/opt/m31/M31_table.html

Table 2: List of M31 novae detected in UVIT images and their F148W mags. Additional photometric data of these novae at different epochs, and in other filters will be available in the published version. Name | RA | DEC | Discovery | Observation | Mag (F148W) | Class | $t_{2}$ | Ref.
---|---|---|---|---|---|---|---|---
M31N | (HH:MM:SS) | (DD:MM:SS) | Date (JD) | Date (JD) | AB | (Phase) | (days) |
Detected at post outburst quiescence
2015-02a † | 00:42:33.06 | +41:13:08.90 | 2457047.810 | 2457671.8566 | 23.27 $\pm$ 0.24 | Fe II | – | 1-3
2015-10b † | 00:43:15.34 | +41:20:16.60 | 2457321.460 | 2457671.8566 | 22.54 $\pm$ 0.13 | Fe II | 100.3 $\pm$ 75.8 (R) | 4,5,75
2016-02b | 00:44:37.03 | +41:42:26.40 | 2457430.370 | 2457430.3700 | 22.01 $\pm$ 0.12 | Fe II | – | 6
2016-03b | 00:42:19.51 | +41:11:13.70 | 2457446.280 | 2457671.8567 | 22.16 $\pm$ 0.14 | Nova | 41.5 $\pm$ 7.1 (R) | 67,68,75
1994-09b | 00:42:58.52 | +41:15:49.80 | 2449622.800 | 2457671.8566 | 21.88 $\pm$ 0.13 | – | – | 32
2010-12b | 00:42:31.08 | +41:27:20.30 | 2455540.620 | 2457671.8566 | 21.64 $\pm$ 0.09 | – | 3$(R)^{*}$ | 33,34
2013-10c | 00:43:09.32 | +41:15:41.60 | 2456574.800 | 2457671.8566 | 24.32 $\pm$ 0.30 | Nova | 5.5 $\pm$ 1.7 (R) | 35,36,75
2015-12d | 00:43:07.79 | +41:22:19.30 | 2457387.220 | 2457671.8566 | 21.72 $\pm$ 0.10 | – | – | –
2018-12a | 00:42:40.65 | +41:11:08.00 | 2458455.320 | 2458788.0143 | 23.31 $\pm$ 0.14 | Fe II | – | 39-42
2018-12d | 00:43:24.62 | +41:20:22.30 | 2458474.090 | 2458788.0143 | 23.33 $\pm$ 0.12 | Nova | – | 42,43
2019-06b | 00:42:41.73 | +41:17:53.70 | 2458661.560 | 2458788.0143 | 22.75 $\pm$ 0.14 | – | – | 44
2019-07c † | 00:43:00.90 | +41:19:19.40 | 2458672.510 | 2458788.0143 | 21.53 $\pm$ 0.06 | – | – | 45
2019-08b | 00:44:18.25 | +41:32:47.20 | 2458725.440 | 2459167.9797 | 22.00 $\pm$ 0.08 | Fe II |  $>$57($r^{\prime}$), $>$53($g^{\prime}$) ∗ | 46,47
2015-05a | 00:42:46.60 | +41:17:55.3 | 2457153.56 | 2457671.8566 | 23.66 $\pm$ 0.31 | Nova | – | 69,70
2016-03c | 00:42:49.14 | +41:16:40.1 | 2457430.24 | 2457671.8566 | 21.85 $\pm$ 0.13 | Nova | – | 71,68
2016-03e | 00:42:48.05 | +41:17:13.4 | 2457464.27 | 2457671.8566 | 22.78 $\pm$ 0.18 | – | – | 72
2016-05b | 00:42:42.88 | +41:15:27.4 | 2457536.55 | 2457671.8566 | 22.89 $\pm$ 0.29 | Fe II | – | 68,73,74
2017-06g | 00:46:17.94 | +41:56:26.1 | 2457922.69 | 2458061.9118 | 20.87 $\pm$ 0.12 | – | – | 76
Detected at post outburst quiescence in coadded frames ${\ddagger}$
1985-09c | 00:42:44.50 | +41:17:35.0 | 2446320.74 | 2458550.6090 | 23.24 $\pm$ 0.10 | – | – | 77
1998-09e | 00:42:49.53 | +41:16:47.9 | 2451083.28 | 2458550.6090 | 23.02 $\pm$ 0.09 | – | – | 78
2010-03a | 00:42:47.74 | +41:17:01.4 | 2455257.25 | 2458550.6090 | 23.18 $\pm$ 0.10 | Nova | – | 79,80
Detected at pre outburst quiescence
2019-12b | 00:42:20.47 | +41:15:00.00 | 2458833.090 | 2457671.8566 | 23.42 $\pm$ 0.25 | Fe II | 50($r^{\prime}$), 27($g^{\prime}$) ∗ | 60-62
2021-07d | 00:42:00.58 | +41:08:48.30 | 2459420.380 | 2458788.0143 | 22.99 $\pm$ 0.13 | – | – | –
2019-11f o | 00:42:51.76 | +41:11:26.60 | 2458814.160 | 2457671.8566 | 22.95 $\pm$ 0.21 | – | – | 59
2020-10b o | 00:42:30.67 | +41:21:28.50 | 2459127.230 | 2457671.8566 | 22.15 $\pm$ 0.13 | – | – | 63,64
2020-10d o | 00:42:36.20 | +41:20:35.50 | 2459101.180 | 2457671.8566 | 23.12 $\pm$ 0.20 | – | – | 65
Detected during outburst
2016-08b † | 00:41:56.82 | +41:11:27.60 | 2457611.050 | 2457671.8566 | 20.71 $\pm$ 0.12 | – | 42.6 $\pm$ 4.3 (R) | 7,8,75
2016-08e | 00:43:53.29 | +41:26:21.60 | 2457630.420 | 2457704.1355 | 20.83 $\pm$ 0.06 | Fe II | 60.6 $\pm$ 16.7 (R) | 9,10,75
2016-09a | 00:42:55.66 | +41:19:14.50 | 2457654.300 | 2457671.8566 | 23.56 $\pm$ 0.22 | Fe II | 57.0 $\pm$ 35.1 (R) | 11,12,75
2016-09b | 00:42:17.29 | +41:09:44.40 | 2457657.660 | 2457671.8566 | 21.18 $\pm$ 0.09 | Fe II | 43.9 $\pm$ 17.3 (R) | 13-15,75
2018-05a | 00:45:34.90 | +41:57:40.40 | 2458246.700 | 2458292.3095 | 20.84 $\pm$ 0.12 | – | – | 16
2019-11a | 00:42:30.37 | +41:03:29.50 | 2458789.760 | 2458805.2565 | 22.92 $\pm$ 0.15 | Nova | 49($r^{\prime}$) ∗ | 17,18
2020-09b | 00:42:46.30 | +41:13:35.50 | 2459117.200 | 2459174.7140 | 23.36 $\pm$ 0.18 | Fe IIb | – | 19,20
2020-09c | 00:43:08.38 | +41:08:50.00 | 2459121.790 | 2459174.7140 | 21.64 $\pm$ 0.08 | Fe II | 17($r^{\prime}$), 15($g^{\prime}$) ∗ | 21,22
2020-10f | 00:42:20.17 | +41:14:28.00 | 2459154.530 | 2459174.7140 | 20.74 $\pm$ 0.05 | Nova | – | 23-25
2020-11a | 00:42:46.79 | +41:07:03.10 | 2459155.080 | 2459174.7140 | 21.82 $\pm$ 0.09 | Nova | – | 26,27
2020-11c | 00:43:13.15 | +41:24:56.00 | 2459159.240 | 2459174.7140 | 19.05 $\pm$ 0.03 | Fe IIb | 20($r^{\prime}$) ∗ | 28-31
2019-09b | 00:42:42.32 | +41:17:45.80 | 2458740.280 | 2458788.0143 | 21.72 $\pm$ 0.07 | He/N | – | 48-50
2019-10c † | 00:43:01.56 | +41:17:59.60 | 2458776.280 | 2458788.0143 | 21.09 $\pm$ 0.05 | Fe II | 18($r^{\prime}$), $>$22($g^{\prime}$) ∗ | 54-58
2021-03a | 00:42:34.49 | +41:17:29.80 | 2459188.680 | 2459174.7140 | 22.89 $\pm$ 0.15 | – | – | 66
2018-08b | 00:39:21.90 | +40:15:48.80 | 2458344.080 | 2458435.0755 | 20.92 $\pm$ 0.10 | Fe II | – | 37,38
2019-10a q | 00:42:23.53 | +41:11:52.20 | 2458759.300 | 2458788.0143 | 21.78 $\pm$ 0.08 | Nova |  $>$18($r^{\prime}$), $>$29($g^{\prime}$) ∗ | 51-53
|  |  |  |  |  |  |  |

##  3 Results and Discussion

All the novae detected in the archival UVIT images are shown in Figure 2 and
listed in Table LABEL:catalog. Nova outbursts can last for a wide range of
duration, with a varied rate of decline ($t_{2}$ or $t_{3}$) of the light
curve (Warner, 2008). These parameters, in turn, depend on the physical nature
of the system, i.e., the primary WD and the binary system. Theoretical models
of Hachisu & Kato (2006) suggest the duration of the outburst depends on the
WD mass and composition. Strope et al. (2010) studied the optical light curves
of around a hundred novae to look for various processes after the outburst
capable of altering the light curve shapes. UV light curves have also been
seen to vary in shape and duration. This is usually related to the super soft
X-ray phase (Page et al., 2022), which, in turn, is dependent on the WD mass
(Hachisu & Kato, 2006; Henze et al., 2011). Due to a wide spectrum of speed
class of novae, we decided to take 100 days as the cut-off for novae in
outburst, corresponding to 7 mag decline from maxima for a 0.9 $M_{\odot}$ CO
WD (Hachisu & Kato, 2006). Additionally, considering pre-eruption dips and
early UV flash, we consider novae detected within 20 days before the eruption
to be in the outburst phase. For any other case, the detection is considered
to be at quiescence.

Overall, we have detected 42 novae in various fields of M31 observed by UVIT.
Almost 90% of the detected novae are concentrated within a 90 $\times$ 90
arcmin region around the M31 center. This is expected because the nova rates
follow the galactic light in optical/IR bands tracing the older stellar
population of M31 (Rector et al., 2022). The following sections discuss the
novae detected at different phases and their SEDs.

Figure 3: UVIT light curves of some interesting candidates

###  3.1 Novae detected at quiescence

23 novae were detected during quiescence, 21 of which were detected post-
outburst. 2021-07d was detected pre-outburst, and 2019-12b was detected both
before and after the outburst. The quiescent F148W AB magnitudes of novae vary
from 21.5 to 24 mags. Depending on the mass of the WD and accretion rate, the
contribution in this filter could be due to both the WD and the disk. The
quiescent magnitude also strongly depends on the system’s orbital inclination
(Warner, 2008).

The light curve of nova 2019-12b (middle panel in Figure 3) shows a
variability in the F148W filter, notably a dip of $\sim$ 0.5 mag around 28
days before the eruption. Zamanov et al. (2023) pointed out that dips in light
curves are caused by the accumulation of an optically thick, dense shell
surrounding the WD prior to the nova eruption. Pre-eruption dips of around 1
mag have been noticed in other novae, such as T CrB in optical wavelengths
(Schaefer, 2023; Maslennikova et al., 2024). If caused by extinction due to an
accumulated shell, these dips will be enhanced in the bluer bands. With the
available data set, we cannot confirm the maximum drop in magnitude before the
eruption. Nonetheless, a drop in FUV magnitude weeks before the eruption hints
at the presence of a pre-eruption dip in the light curve.

Most other novae detected more than once at post-outburst quiescence in the
same filter do not show much variability between epochs. It strengthens the
claim of nearly constant flux at quiescence by Selvelli & Gilmozzi (2013). The
disks of such systems have steady accretion rates and radiate at constant
luminosity.

Among the old novae, only 2010-12b and 1994-09b have been detected at each
epoch in multiple filters. Further, three more old novae, 2010-03a, 1998-09e,
and 1985-09c, were detected in the co-added images. Detection of these old
novae in multiple filters indicates a bright accretion disk. A high accretion
rate during quiescence can make the disk brighter. The inclination of the
system can also play a major factor in determining if such systems should be
so bright long after the eruption. Another factor that could be responsible is
the intrinsic brightness of the WD, which illuminates the accretion disk. The
FUV flux in the F148W filter could even arise from novae systems with long SSS
phases.

###  3.2 Novae detected during outburst

15 novae were detected within 100 days after the outburst, and 5 of them were
detected within 20 days from the outburst. The only nova detected just before
the outburst is 2021-03a, 14 days before its eruption. Most of the novae
detected only during outbursts are bright ($\sim$ 19-20 mag) and fast. 6 novae
were detected only once in the F148W filter. 4 novae were detected in multiple
filters, and their SEDs are discussed in §3.4.

In the early outburst phase, the photospheric emission is the major source of
UV photons. As the photosphere moves inward, the super-soft source dominates
the FUV photon contribution. In the case of partially disrupted disks after
eruption, contributions from the inner regions of the accretion disk to the
FUV luminosity cannot be ruled out.

###  3.3 Novae detected during quiescence and at outburst

4 novae were detected both during quiescence and at outburst. 2019-10a was
observed in the F148W filter at 46 and 415 days after the outburst, and it
indicates a declining trend in its light curve as shown in the top panel of
Figure 3.

2019-11f shows a pre-eruption dip similar to 2019-12b (see §3.1). In the F148W
filter, it was observed 1140 and 8 days before the eruption and 360 days after
the eruption. Additionally, the pre-eruption dip of 2019-11f is confirmed in
the F172M filter observed 1140 and 8 days before the eruption. As shown in the
bottom panel of Figure 3, the pre-eruption dip is $\sim$ 8 days before the
eruption.

The other two novae, 2020-10b and 2020-10d, also show 0.4 and 0.7 mag
variability, respectively, but the causes are unclear. However, we would like
to point out that these variations are close to the limits of their respective
errors.

###  3.4 The spectral energy distributions (SEDs)

Figure 4: SED of novae detected after outburst are shown in the top left panel
with AB magnitude in the y-axis and wavelength in the x-axis. The UVIT filter
bands are also marked. The extinction-corrected flux of novae detected at
post-outburst quiescence (top right), detected close to outburst (bottom left)
and detected in the central region (bottom right) are also shown.

13 novae out of the 35 detected have near-simultaneous (within two days)
observations in multiple filters in NUV and FUV. These data sets encouraged us
to look into their SEDs. The observed SEDs are shown in the top-left subplot
of Figure 4. To correct for the extinction towards M31, we derived the $A_{V}$
values at each nova location from the M160 resolution dust mass maps of Draine
et al. (2014), owing to its complete M31 coverage. However, compared to the
PHAT survey of Dalcanton et al. (2015), the extinction values were
overestimated by a factor of $\sim$ 2 (B.T. Draine 2024, private
communication). The $A_{V}$ values derived from Draine et al. (2014) were
corrected before applying to the SED of each nova. The UVIT AB magnitudes were
converted to flux using the relations given in Tandon et al. (2017b). We used
the extinction functions from Cardelli et al. (1989) to deredden the UV flux
using the extinction (Barbary, 2021) package in python. The extinction-
corrected SEDs in terms of $F_{\lambda}$ against $\lambda$ are shown in Figure
4.

The most interesting candidates are those detected at post-outburst
quiescence, shown in the upper-right panel of Figure 4. The power law
dependence of $F_{\lambda}\propto\lambda^{-\alpha}$ was fitted to obtain the
best-fit power-laws given in Table 3. The radiation from accretion disks
follows a power-law with a slope of $\alpha=7/3$. The best-fit $\alpha$ values
for the novae are within the error bars of this, indicating the presence of
accretion disks (Frank et al., 1992). This confirms that most of the UV
radiation originates from the accretion disk at quiescence (Selvelli &
Gilmozzi, 2013). Another point to note is that the flux in the F148W filter is
comparatively less than in F172M. It could indicate that bluewards of 1600 Å,
the flux from the disk starts to diminish and could have a contribution from
the WD’s blackbody tail.

Table 3: $\alpha$ values of novae with accretion disk signatures. $\alpha$ corresponds to the power-law between $F_{\lambda}$ vs $\lambda$ when errors in $A_{V}$ are not taken into account, whereas $\alpha_{2}$ corresponds for the same power-law when errors in $A_{V}$ are taken into account. M31N | $\alpha$ | $\alpha_{2}$
---|---|---
1994-09b | 2.71 $\pm$ 0.81 | 3.28 $\pm$ 0.79
2010-12b | 2.29 $\pm$ 0.002 | 2.29 $\pm$ 0.94
2015-10b | 2.01 $\pm$ 0.96 | 1.34 $\pm$ 1.32
2015-12d | 2.15 $\pm$ 0.33 | 1.92 $\pm$ 0.82

4 novae were caught during their eruption in multiple filters, and their SEDs
are shown in the lower-left panel of Figure 4. Two novae, 2016-09a and
2016-09b, were observed within 18 days of eruption. Their speed class suggests
that they belong to a moderately fast category, and both were also confirmed
to be in their Fe II phase (see Table LABEL:catalog). Their SEDs indicate an
increasing flux from FUV to NUV, hinting the peak to be in the optical region.
The peak brightness of the nova eruption corresponds to the maximum extent of
the photosphere. During the maxima, the peak of the SED moves to longer
wavelengths, which could be in UV for very fast novae but also in optical or
even IR for slow novae. As the photosphere recedes, hotter regions closer to
the WD are revealed, and the SED peak moves blue wards. A SED with its peak in
optical bands could indicate that these two novae were observed at a stage
when their pseudo-photosphere was shrinking.

The SEDs of the novae 2016-02b, 2016-08e, and 2016-08b deviate from the power
law expected from accretion disks. One possible cause for this could be the
presence of spectral features in the UV bands. The quiescent novae sample of
Selvelli & Gilmozzi (2013) consisted majorly of C iv 1550 Å, along with some
fainter features like N v 1240 Å, Si iv 1400 Å, He ii 1640 Å, O iii] 1666 Å,
and N iii] 1750 Å in emission. FUV absorption features arising from metals
such as Si ii 1190 Å, Si ii 1260 Å, O i+Si ii 1303 Å, C ii 1335 Å, Si iv 1400
Å, C iv 1550 Å, and N v 1240 Å can also distort the SEDs. During outburst,
however, the UV spectral features are mostly emission until the Fe curtains
stage sets in. The list of emission lines and how they evolve have been
described in Shore (2008).

Another possibility for the deviation of the SED could be the improper
determination of the extinction parameter, which would amplify any errors in
the FUV bands as well as in the N219M band (since it covers the UV extinction
bump at 2175 Å). The SED of the quiescent nova 2016-02b could be under-
corrected for extinction. The SED of 2016-08e might be over-corrected, whereas
that of 2016-08b might have been under-corrected, both of which were captured
during outburst. The over-correction in some of the SEDs of novae could imply
that the sources are away from the galaxy’s disk towards us, i.e., these
sources suffer less extinction than predicted by the dust maps. Similarly,
sources with under-correction may be located on the far side of M31, thus
suffering more extinction than obtained from the dust maps.

The bulge of M31 in FUV is complicated. Leahy et al. (2023) could fit the
bulge with a single component but showed that an 8-component model fits best.
Such a complicated bulge region will lead to contamination, dependent on the
spatial location of the novae and on the different UV wavebands. Furthermore,
the location of these novae inside the nuclear region is unknown, leading to
uncertainties in determining their column densities and extinction. It thus
becomes difficult to extract much information from the SEDs of novae in the
central region of M31.

##  4 Summary and Conclusion

Astronomers have extensively studied novae in M31, in the optical for more
than a century, and in X-rays over the last couple of decades to understand
their distribution, rates, and evolution at different phases. However, UV
surveys remain limited. This study utilizes multi-band AstroSat UV data to
explore cataclysmic variables, novae in particular, during outbursts and
quiescence. Below, we present the key findings from our investigation

  * •

We have examined 91 sets of UVIT images of M31 from 2016 onwards, uncovering
at least one nova in 19 of those images. Over 80% of the identified novae were
situated within or close to the central bulge.

  * •

We employed two image subtraction methods to eliminate the light from the
galactic bulge, leading to the detection of four novae previously obscured by
the brightness of the central region of M31.

  * •

A total of 42 novae were identified, with 15 observed across multiple filters
in both FUV and NUV. Additionally, several novae were detected in multiple
epochs. Magnitude upper limits were estimated for more than 1000 novae
undetected in individual and co-added frames.

  * •

During quiescence, the spectral energy distributions (SEDs) of novae exhibit
indications of the accretion disk’s influence, confirming its dominance in UV
radiation during this phase. Analysis of the SEDs of two novae during outburst
indicates they were observed while their photospheres were receding.

  * •

The FUV light curves of two novae exhibit a pre-eruption dip that can possibly
be attributed to the accumulation of accreted material in a shell before the
nova eruption. Multi-epoch photometry of the novae at quiescence indicates a
near-constant magnitude, a sign of a steady accretion rate.

Upcoming UV missions like INSIST (Subramaniam, 2022), CASTOR (Côte et al.,
2012), UVEX (Kulkarni et al., 2021), and ULTRASAT (Shvartzvald et al., 2024)
offer the potential for detecting novae at different stages of their UV
evolution. These observations can provide insights into disk activity,
accretion processes, and the characteristics of WD during dormancy phases.
Additionally, by employing high-frequency observations, we anticipate
elucidating the UV flash preceding eruptions, as predicted by the theoretical
models of Kato et al. (2022), along with the rapid UV evolution post-outburst.
We have initiated a UV survey of M31’s central region using the FUV F148W
filter on UVIT/AstroSat. Initial observations have revealed the detection of
the newly identified RN M31N 2013-10c during our monitoring activities (Basu
et al., 2024a). The FUV survey was designed to complement the simultaneous
monitoring of M31 by GIT (Kumar et al., 2022) in $g^{\prime},r^{\prime}$
bands, which has detected numerous novae and even discovered the nova AT
2023tkw (GIT230919aa, Kumar 2023). The integration of UV monitoring with
multi-wavelength surveys such as ZTF (Bellm et al., 2019) and GIT in optical,
XRISM in X-rays (XRISM Science Team, 2020), and the forthcoming Roman Space
Telescope in IR (Akeson et al., 2019), alongside Daksha in X-rays (Bhalerao et
al., 2024), promises to contribute significantly to resolving outstanding
questions concerning nova phenomena.

##  5 Acknowledgement

We appreciate the anonymous reviewer’s insightful comments and suggestions,
which have enhanced the quality of this article. This work uses the archival
UVIT data from the AstroSat mission of the Indian Space Research Organisation
(ISRO). We thank the PIs of the proposals that were observed. The proposal IDs
are the first seven characters of the observation IDs given in Table 1. We
thank the UVIT payload operation centers for verifying and releasing the raw
and processed data via the ISSDC data archive and providing the necessary
software tools. GCA thanks the Indian National Science Academy for support
under the INSA Senior Scientist Programme. KS thanks the Indian Institute of
Astrophysics for supporting her work while at IIA.

##  Appendix A Catalog Table

Table 4: UVIT catalog of all M31 novae detection and non-detection in each
filter, along with available archival data. A sample table with ten rows is
shown below. The full version of the table will be made available in machine-
readable format in the published paper.

Name | RA | DEC | UVIT field | Individual Detection | Discovery Date | Max mag | Filter | $t_{2}$ | Alternative names | F148W mag | F172M mag | F169M mag | N219M mag | N279N mag
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
M31N | hh:mm:ss | dd:mm:ss |  | * | JD |  |  | days |  | AB | AB | AB | AB | AB
2022-04d | 00:38:48.05 | +40:15:26.7 | 15 | N | 2459695.52 | 16.5 | w | – | AT2022iro |  $>$ 23.34 |  |  $>$ 24.30 |  |
2022-04c | 00:42:57.33 | +41:16:18.4 | 1 | N | 2459692.41 | 17.3 | w | – | AT2022icc |  $>$ 23.31 |  $>$ 22.29 |  $>$ 22.67 |  $>$ 21.86 |  $>$ 21.05
2021-07d | 00:42:00.58 | +41:08:48.3 | 1 | Y | 2459420.38 | 19.3 | w | – | XM57MZ | 23.21 $\pm$ 0.09 | 23.76 $\pm$ 0.32 | 22.40 $\pm$ 0.17 |  $>$ 22.08 |  $>$ 21.53
2021-07c | 00:42:17.92 | +41:00:18.5 | 3 | N | 2459399.89 | 18.3 | r | – | AT2021scd, ZTF21abjiotr |  $>$ 22.83 |  $>$ 22.07 |  |  $>$ 22.36 |  $>$ 21.87
2020-10f | 00:42:20.17 | +41:14:28.0 | 1 | Y | 2459154.53 | 16.3 | R | – | AT2020yky, XM42MZ | 21.99 $\pm$ 0.042 |  $>$ 23.12 |  $>$ 22.63 |  $>$ 22.08 |  $>$ 21.48
2020-10e | 00:42:37.35 | +41:20:52.2 | 1 | N | 2459144.24 | 16.3 | R | 4 | AT2020xyv, XM41MZ |  $>$ 24.02 |  $>$ 23.18 |  $>$ 22.63 |  $>$ 22.08 |  $>$ 21.53
2020-01b | 00:42:52.34 | +41:16:13.1 | 1 | N | 2458877.04 | 17.40 $\pm$ 0.15 | R | 11.4 $\pm$ 2.5 | AT2020ber, XM24MZ |  $>$ 22.96 |  $>$ 22.37 |  $>$ 22.60 |  $>$ 21.61 |  $>$ 20.68
2017-06g | 00:46:17.94 | +41:56:26.1 | 8 | Y | 2457922.69 | 18.1 | w | – | MASTER OT J004617.94+415626.1 | 21.89 $\pm$ 0.10 |  $>$ 23.28 |  |  $>$ 22.06 | 20.70 $\pm$ 0.18
2016-11b | 00:42:42.81 | +41:12:33.4 | 1 | N | 2457720.26 | 16.03 $\pm$ 0.05 | R | 12.6 $\pm$ 1.8 | PNV J00424281+4112334 |  $>$ 23.53 |  $>$ 22.91 |  $>$ 22.59 |  $>$ 22.08 |  $>$ 21.53
2016-03e | 00:42:48.05 | +41:17:13.4 | 1 | Y | 2457464.27 | 16.9 | w | – | PNV J00424805+4117134 | 23.98 $\pm$ 0.21 | 23.59 $\pm$ 0.52 |  $>$ 22.15 | 21.28 $\pm$ 0.18 | 20.16 $\pm$ 0.18
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮

Notes:
* Y indicates detection in at least one individual frame. N indicates non-detection in individual frames.
The UVIT magnitudes were estimated from the co-added images where more than
one epoch of data was available in the same field and filter. In other cases,
magnitudes were estimated from the individual frames.
Blank fields in the UVIT magnitude columns indicate the absence of data of
that field in that filter.

## References

  * Akeson et al. (2019) Akeson, R., Armus, L., Bachelet, E., et al. 2019, arXiv e-prints, arXiv:1902.05569, doi: 10.48550/arXiv.1902.05569
  * Arp (1956) Arp, H. C. 1956, AJ, 61, 15, doi: 10.1086/107284
  * Aydi et al. (2018) Aydi, E., Page, K. L., Kuin, N. P. M., et al. 2018, MNRAS, 474, 2679, doi: 10.1093/mnras/stx2678
  * Aydi et al. (2020) Aydi, E., Sokolovsky, K. V., Chomiuk, L., et al. 2020, Nature Astronomy, 4, 776, doi: 10.1038/s41550-020-1070-y
  * Balcon (2020) Balcon, C. 2020, Transient Name Server Classification Report, 2020-3397, 1
  * Barbary (2021) Barbary, K. 2021, extinction: Dust extinction laws, Astrophysics Source Code Library, record ascl:2102.026
  * Basu et al. (2024a) Basu, J., Chamoli, S., Barway, S., & Anupama, G. 2024a, The Astronomer’s Telegram, 16526, 1
  * Basu et al. (2024b) Basu, J., Pavana, M., Anupama, G. C., et al. 2024b, ApJ, 966, 44, doi: 10.3847/1538-4357/ad2c8e
  * Belligoli (2020) Belligoli, R. 2020, Transient Name Server Discovery Report, 2020-3354, 1
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbe
  * Bertin et al. (2002) Bertin, E., Mellier, Y., Radovich, M., et al. 2002, in Astronomical Society of the Pacific Conference Series, Vol. 281, Astronomical Data Analysis Software and Systems XI, ed. D. A. Bohlender, D. Durand, & T. H. Handley, 228
  * Bhalerao et al. (2024) Bhalerao, V., Sawant, D., Pai, A., et al. 2024, Experimental Astronomy, 57, 23, doi: 10.1007/s10686-024-09923-1
  * Bode & Evans (2008) Bode, M. F., & Evans, A. 2008, Classical Novae, Vol. 43
  * Bode et al. (2016) Bode, M. F., Darnley, M. J., Beardmore, A. P., et al. 2016, ApJ, 818, 145, doi: 10.3847/0004-637X/818/2/145
  * Bradley et al. (2023) Bradley, L., Sipőcz, B., Robitaille, T., et al. 2023, astropy/photutils: 1.8.0, 1.8.0, Zenodo, doi: 10.5281/zenodo.7946442
  * Campaner (2019) Campaner, P. 2019, Transient Name Server Discovery Report, 2019-2301, 1
  * Cao et al. (2012) Cao, Y., Kasliwal, M. M., Neill, J. D., et al. 2012, ApJ, 752, 133, doi: 10.1088/0004-637X/752/2/133
  * Cardelli et al. (1989) Cardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245, doi: 10.1086/167900
  * Carey (2019) Carey, G. 2019, Transient Name Server Discovery Report, 2019-1970, 1
  * Chomiuk et al. (2021) Chomiuk, L., Metzger, B. D., & Shen, K. J. 2021, ARA&A, 59, 391, doi: 10.1146/annurev-astro-112420-114502
  * Ciardullo et al. (1987) Ciardullo, R., Ford, H. C., Neill, J. D., Jacoby, G. H., & Shafter, A. W. 1987, ApJ, 318, 520, doi: 10.1086/165388
  * Clark et al. (2024) Clark, J. G., Hornoch, K., Shafter, A. W., et al. 2024, arXiv e-prints, arXiv:2404.04733, doi: 10.48550/arXiv.2404.04733
  * Conseil (2018) Conseil, E. 2018, Transient Name Server Discovery Report, 2018-597, 1
  * Conseil (2019) —. 2019, Transient Name Server Discovery Report, 2019-1718, 1
  * Côte et al. (2012) Côte, P., Scott, A., Balogh, M., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8442, Space Telescopes and Instrumentation 2012: Optical, Infrared, and Millimeter Wave, ed. M. C. Clampin, G. G. Fazio, H. A. MacEwen, & J. Oschmann, Jacobus M., 844215, doi: 10.1117/12.926198
  * Dahiwale & Fremling (2020) Dahiwale, A., & Fremling, C. 2020, Transient Name Server Classification Report, 2020-2139, 1
  * Dalcanton et al. (2015) Dalcanton, J. J., Fouesneau, M., Hogg, D. W., et al. 2015, ApJ, 814, 3, doi: 10.1088/0004-637X/814/1/3
  * Darnley et al. (2018) Darnley, M. J., Healy, M. W., & Williams, S. C. 2018, The Astronomer’s Telegram, 11983, 1
  * Darnley & Williams (2016a) Darnley, M. J., & Williams, S. C. 2016a, The Astronomer’s Telegram, 8710, 1
  * Darnley & Williams (2016b) —. 2016b, The Astronomer’s Telegram, 9143, 1
  * Darnley et al. (2004) Darnley, M. J., Bode, M. F., Kerins, E., et al. 2004, MNRAS, 353, 571, doi: 10.1111/j.1365-2966.2004.08087.x
  * Darnley et al. (2006) —. 2006, MNRAS, 369, 257, doi: 10.1111/j.1365-2966.2006.10297.x
  * Delvaux et al. (2015) Delvaux, C., Greiner, J., & Pietsch, W. 2015, The Astronomer’s Telegram, 7104, 1
  * Devaraj et al. (2023) Devaraj, A., Joseph, P., Stalin, C. S., Tandon, S. N., & Ghosh, S. K. 2023, ApJ, 946, 65, doi: 10.3847/1538-4357/acba9c
  * Draine et al. (2014) Draine, B. T., Aniano, G., Krause, O., et al. 2014, ApJ, 780, 172, doi: 10.1088/0004-637X/780/2/172
  * Fabrika et al. (2015) Fabrika, S., Barsukova, E. A., Valeev, A. F., et al. 2015, The Astronomer’s Telegram, 7158, 1
  * Fabrika et al. (2020) Fabrika, S., Sholukhova, O., Sarkisyan, A., et al. 2020, The Astronomer’s Telegram, 14217, 1
  * Fabrika et al. (2019) Fabrika, S., Sholukhova, O., Solovyeva, Y., et al. 2019, The Astronomer’s Telegram, 13230, 1
  * Fabrika et al. (2016a) Fabrika, S., Sholukhova, O., Valeev, A. F., et al. 2016a, The Astronomer’s Telegram, 9557, 1
  * Fabrika et al. (2016b) —. 2016b, The Astronomer’s Telegram, 9563, 1
  * Fabrika et al. (2016c) —. 2016c, The Astronomer’s Telegram, 9443, 1
  * Frank et al. (1992) Frank, J., King, A., & Raine, D. 1992, Accretion power in astrophysics., Vol. 21
  * Fremling (2020) Fremling, C. 2020, Transient Name Server Discovery Report, 2020-3674, 1
  * Gehrz et al. (1998) Gehrz, R. D., Truran, J. W., Williams, R. E., & Starrfield, S. 1998, PASP, 110, 3, doi: 10.1086/316107
  * Guo et al. (2022) Guo, Y., Wu, C., & Wang, B. 2022, A&A, 660, A53, doi: 10.1051/0004-6361/202142163
  * Hachisu & Kato (2006) Hachisu, I., & Kato, M. 2006, ApJS, 167, 59, doi: 10.1086/508063
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357–362, doi: 10.1038/s41586-020-2649-2
  * Henze et al. (2010) Henze, M., Pietsch, W., Haberl, F., et al. 2010, A&A, 523, A89, doi: 10.1051/0004-6361/201014710
  * Henze et al. (2011) —. 2011, A&A, 533, A52, doi: 10.1051/0004-6361/201015887
  * Henze et al. (2014) —. 2014, A&A, 563, A2, doi: 10.1051/0004-6361/201322426
  * Hernanz & Jose (1998) Hernanz, M., & Jose, J. 1998, in Astronomical Society of the Pacific Conference Series, Vol. 137, Wild Stars in the Old West, ed. S. Howell, E. Kuulkers, & C. Woodward, 368
  * Hillman et al. (2016) Hillman, Y., Prialnik, D., Kovetz, A., & Shara, M. M. 2016, ApJ, 819, 168, doi: 10.3847/0004-637X/819/2/168
  * Hornoch & Honkova (2015) Hornoch, K., & Honkova, K. 2015, The Astronomer’s Telegram, 7492, 1
  * Hornoch & Kucakova (2016a) Hornoch, K., & Kucakova, H. 2016a, The Astronomer’s Telegram, 9373, 1
  * Hornoch & Kucakova (2016b) —. 2016b, The Astronomer’s Telegram, 9431, 1
  * Hornoch & Kucakova (2016c) —. 2016c, The Astronomer’s Telegram, 9552, 1
  * Hornoch & Kucakova (2016d) —. 2016d, The Astronomer’s Telegram, 8785, 1
  * Hornoch & Kucakova (2016e) —. 2016e, The Astronomer’s Telegram, 9098, 1
  * Hornoch & Kucakova (2018) —. 2018, The Astronomer’s Telegram, 12262, 1
  * Hornoch & Kucakova (2019a) —. 2019a, The Astronomer’s Telegram, 12925, 1
  * Hornoch & Kucakova (2019b) —. 2019b, The Astronomer’s Telegram, 13109, 1
  * Hornoch & Kucakova (2019c) —. 2019c, The Astronomer’s Telegram, 13207, 1
  * Hornoch & Kucakova (2020a) —. 2020a, The Astronomer’s Telegram, 14055, 1
  * Hornoch & Kucakova (2020b) —. 2020b, The Astronomer’s Telegram, 14155, 1
  * Hornoch & Kucakova (2020c) —. 2020c, The Astronomer’s Telegram, 14068, 1
  * Hornoch et al. (2016a) Hornoch, K., Kucakova, H., & Wolf, M. 2016a, The Astronomer’s Telegram, 8787, 1
  * Hornoch et al. (2019) —. 2019, The Astronomer’s Telegram, 13355, 1
  * Hornoch & Kusnirak (2019) Hornoch, K., & Kusnirak, P. 2019, The Astronomer’s Telegram, 12895, 1
  * Hornoch et al. (2010) Hornoch, K., Kusnirak, P., Wolf, M., & Zasche, P. 2010, Central Bureau Electronic Telegrams, 2212, 3
  * Hornoch et al. (2013) Hornoch, K., Manilla-Robles, A., Tudor, V., Vaduvescu, O., & Ramsay, G. 2013, The Astronomer’s Telegram, 5503, 1
  * Hornoch et al. (2016b) Hornoch, K., Vrastil, J., Honkova, K., & Kucakova, H. 2016b, The Astronomer’s Telegram, 8877, 1
  * Hornoch et al. (2016c) Hornoch, K., Wolf, M., & Kucakova, H. 2016c, The Astronomer’s Telegram, 9545, 1
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Jiang et al. (2019) Jiang, X., Sun, G., Zhang, M., & Gao, X. 2019, Transient Name Server Discovery Report, 2019-2620, 1
  * Joye & Mandel (2003) Joye, W. A., & Mandel, E. 2003, in Astronomical Society of the Pacific Conference Series, Vol. 295, Astronomical Data Analysis Software and Systems XII, ed. H. E. Payne, R. I. Jedrzejewski, & R. N. Hook, 489
  * Kasliwal & Cao (2018a) Kasliwal, M., & Cao, Y. 2018a, Transient Name Server Discovery Report, 2018-1344, 1
  * Kasliwal & Cao (2018b) —. 2018b, Transient Name Server Discovery Report, 2018-1069, 1
  * Kasliwal et al. (2011) Kasliwal, M. M., Cenko, S. B., Kulkarni, S. R., et al. 2011, ApJ, 735, 94, doi: 10.1088/0004-637X/735/2/94
  * Kato et al. (2022) Kato, M., Saio, H., & Hachisu, I. 2022, PASJ, 74, 1005, doi: 10.1093/pasj/psac051
  * Kraft (1964) Kraft, R. P. 1964, ApJ, 139, 457, doi: 10.1086/147776
  * Kuin et al. (2020) Kuin, N. P. M., Page, K. L., Mróz, P., et al. 2020, MNRAS, 491, 655, doi: 10.1093/mnras/stz2960
  * Kulkarni et al. (2021) Kulkarni, S. R., Harrison, F. A., Grefenstette, B. W., et al. 2021, arXiv e-prints, arXiv:2111.15608, doi: 10.48550/arXiv.2111.15608
  * Kumar et al. (2022) Kumar, H., Bhalerao, V., Anupama, G. C., et al. 2022, AJ, 164, 90, doi: 10.3847/1538-3881/ac7bea
  * Kumar (2023) Kumar, R. 2023, Transient Name Server Discovery Report, 2023-2354, 1
  * Leahy et al. (2021) Leahy, D., Buick, M., Postma, J., & Morgan, C. 2021, AJ, 161, 215, doi: 10.3847/1538-3881/abe9b3
  * Leahy et al. (2023) Leahy, D., Craiciu, T., & Postma, J. 2023, ApJS, 265, 6, doi: 10.3847/1538-4365/acae90
  * Leahy et al. (2020) Leahy, D. A., Postma, J., Chen, Y., & Buick, M. 2020, ApJS, 247, 47, doi: 10.3847/1538-4365/ab77a9
  * Lee et al. (2019) Lee, C.-H., Soraisam, M., Narayan, G., Matheson, T., & Saha, A. 2019, The Astronomer’s Telegram, 13141, 1
  * Lee et al. (2012) Lee, C. H., Riffeser, A., Seitz, S., et al. 2012, A&A, 537, A43, doi: 10.1051/0004-6361/201117068
  * Lessing et al. (2023) Lessing, A. M., Shara, M. M., Hounsell, R., et al. 2023, arXiv e-prints, arXiv:2309.16856, doi: 10.48550/arXiv.2309.16856
  * Madrigal-Aguado et al. (2019) Madrigal-Aguado, A., Alvarez-Hernandez, L., Bayer, J., et al. 2019, The Astronomer’s Telegram, 13272, 1
  * Martin et al. (2018) Martin, T. B., Drissen, L., & Melchior, A.-L. 2018, MNRAS, 473, 4130, doi: 10.1093/mnras/stx2513
  * Maslennikova et al. (2024) Maslennikova, N. A., Tatarnikov, A. M., Tatarnikova, A. A., et al. 2024, Astronomy Letters, 49, 501, doi: 10.1134/S1063773723090037
  * McKinney et al. (2010) McKinney, W., et al. 2010, in Proceedings of the 9th Python in Science Conference, Vol. 445, Austin, TX, 51–56
  * Mróz et al. (2016) Mróz, P., Udalski, A., Poleski, R., et al. 2016, ApJS, 222, 9, doi: 10.3847/0067-0049/222/1/9
  * Nordin et al. (2019) Nordin, J., Brinnel, V., Giomi, M., et al. 2019, Transient Name Server Discovery Report, 2019-2130, 1
  * Ovcharov et al. (2015a) Ovcharov, E., Kostov, A., Kurtenkov, A., Valcheva, A., & Nedialkov, P. 2015a, The Astronomer’s Telegram, 7065, 1
  * Ovcharov et al. (2013) Ovcharov, E., Kurtenkov, A., Nikolov, G., et al. 2013, The Astronomer’s Telegram, 5475, 1
  * Ovcharov et al. (2015b) Ovcharov, E., Minev, M., Bozhilov, V., et al. 2015b, The Astronomer’s Telegram, 7789, 1
  * Page et al. (2020) Page, K. L., Beardmore, A. P., & Osborne, J. P. 2020, Advances in Space Research, 66, 1169, doi: 10.1016/j.asr.2019.08.003
  * Page et al. (2022) Page, K. L., Kuin, N. P. M., & Osborne, J. P. 2022, Universe, 8, 643, doi: 10.3390/universe8120643
  * Pietsch (2010) Pietsch, W. 2010, Astronomische Nachrichten, 331, 187, doi: 10.1002/asna.200911324
  * Pietsch et al. (2010a) Pietsch, W., Henze, M., Burwitz, V., et al. 2010a, The Astronomer’s Telegram, 3076, 1
  * Pietsch et al. (2010b) Pietsch, W., Lloyd, J., Henze, M., et al. 2010b, The Astronomer’s Telegram, 2964, 1
  * Pietsch et al. (2007) Pietsch, W., Haberl, F., Sala, G., et al. 2007, A&A, 465, 375, doi: 10.1051/0004-6361:20066812
  * Postma & Leahy (2017) Postma, J. E., & Leahy, D. 2017, PASP, 129, 115002, doi: 10.1088/1538-3873/aa8800
  * Postma & Leahy (2021) —. 2021, Journal of Astrophysics and Astronomy, 42, 30, doi: 10.1007/s12036-020-09689-w
  * Rector et al. (1999) Rector, T. A., Jacoby, G. H., Corbett, D. L., Denham, M., & RBSE Nova Search Team. 1999, in American Astronomical Society Meeting Abstracts, Vol. 195, American Astronomical Society Meeting Abstracts, 36.08
  * Rector et al. (2022) Rector, T. A., Shafter, A. W., Burris, W. A., et al. 2022, ApJ, 936, 117, doi: 10.3847/1538-4357/ac87ad
  * Rosino (1964) Rosino, L. 1964, Annales d’Astrophysique, 27, 498
  * Rosino (1973) —. 1973, A&AS, 9, 347
  * Rosino et al. (1989) Rosino, L., Capaccioli, M., D’Onofrio, M., & della Valle, M. 1989, AJ, 97, 83, doi: 10.1086/114959
  * Schaefer (2023) Schaefer, B. E. 2023, MNRAS, 524, 3146, doi: 10.1093/mnras/stad735
  * Selvelli & Gilmozzi (2013) Selvelli, P., & Gilmozzi, R. 2013, A&A, 560, A49, doi: 10.1051/0004-6361/201220627
  * Selvelli & Gilmozzi (2019) —. 2019, A&A, 622, A186, doi: 10.1051/0004-6361/201834238
  * Shafter et al. (2011a) Shafter, A. W., Bode, M. F., Darnley, M. J., et al. 2011a, ApJ, 727, 50, doi: 10.1088/0004-637X/727/1/50
  * Shafter & Irby (2001) Shafter, A. W., & Irby, B. K. 2001, ApJ, 563, 749, doi: 10.1086/324044
  * Shafter et al. (2011b) Shafter, A. W., Darnley, M. J., Hornoch, K., et al. 2011b, ApJ, 734, 12, doi: 10.1088/0004-637X/734/1/12
  * Shara et al. (2023) Shara, M. M., Lessing, A. M., Hounsell, R., et al. 2023, ApJS, 269, 42, doi: 10.3847/1538-4365/ad02fd
  * Sharov & Alksnis (1991) Sharov, A. S., & Alksnis, A. 1991, Ap&SS, 180, 273, doi: 10.1007/BF00648182
  * Sharov et al. (2000) Sharov, A. S., Alksnis, A., Zharova, A. V., & Shokin, Y. A. 2000, Astronomy Letters, 26, 433, doi: 10.1134/1.20416
  * Shen & Quataert (2022) Shen, K. J., & Quataert, E. 2022, ApJ, 938, 31, doi: 10.3847/1538-4357/ac9136
  * Shore (2008) Shore, S. N. 2008, in Classical Novae, ed. M. F. Bode & A. Evans, Vol. 43, 194–231, doi: 10.1017/CBO9780511536168.011
  * Shumkov et al. (2017) Shumkov, V., Lipunov, V., Rebolo, R., et al. 2017, The Astronomer’s Telegram, 10506, 1
  * Shvartzvald et al. (2024) Shvartzvald, Y., Waxman, E., Gal-Yam, A., et al. 2024, ApJ, 964, 74, doi: 10.3847/1538-4357/ad2704
  * Singh et al. (2014) Singh, K. P., Tandon, S. N., Agrawal, P. C., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray, ed. T. Takahashi, J.-W. A. den Herder, & M. Bautz, 91441S, doi: 10.1117/12.2062667
  * Soraisam et al. (2019a) Soraisam, M., Lee, C.-H., Narayan, G., Matheson, T., & Saha, A. 2019a, The Astronomer’s Telegram, 13210, 1
  * Soraisam et al. (2019b) Soraisam, M., Lee, C.-H., Nunez, R., et al. 2019b, The Astronomer’s Telegram, 13317, 1
  * Soraisam et al. (2020) Soraisam, M., McKinnon, K., Nunez, R., et al. 2020, The Astronomer’s Telegram, 14150, 1
  * Srivastav et al. (2020) Srivastav, S., Smartt, S. J., McBrien, O., et al. 2020, Transient Name Server Classification Report, 2020-3471, 1
  * Starrfield (1986) Starrfield, S. 1986, in ESA Special Publication, Vol. 263, New Insights in Astrophysics. Eight Years of UV Astronomy with IUE, ed. E. J. Rolfe & R. Wilson, 239
  * Starrfield (1999) Starrfield, S. 1999, Phys. Rep., 311, 371, doi: 10.1016/S0370-1573(98)00116-1
  * Starrfield et al. (2020) Starrfield, S., Bose, M., Iliadis, C., et al. 2020, ApJ, 895, 70, doi: 10.3847/1538-4357/ab8d23
  * Starrfield et al. (2016) Starrfield, S., Iliadis, C., & Hix, W. R. 2016, PASP, 128, 051001, doi: 10.1088/1538-3873/128/963/051001
  * Strope et al. (2010) Strope, R. J., Schaefer, B. E., & Henden, A. A. 2010, AJ, 140, 34, doi: 10.1088/0004-6256/140/1/34
  * Subramaniam (2022) Subramaniam, A. 2022, Journal of Astrophysics and Astronomy, 43, 80, doi: 10.1007/s12036-022-09870-3
  * Subramaniam et al. (2016) Subramaniam, A., Tandon, S. N., Hutchings, J., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051F, doi: 10.1117/12.2235271
  * Tandon et al. (2017a) Tandon, S. N., Hutchings, J. B., Ghosh, S. K., et al. 2017a, Journal of Astrophysics and Astronomy, 38, 28, doi: 10.1007/s12036-017-9445-x
  * Tandon et al. (2017b) Tandon, S. N., Subramaniam, A., Girish, V., et al. 2017b, AJ, 154, 128, doi: 10.3847/1538-3881/aa8451
  * Tandon et al. (2020) Tandon, S. N., Postma, J., Joseph, P., et al. 2020, AJ, 159, 158, doi: 10.3847/1538-3881/ab72a3
  * Tody (1993) Tody, D. 1993, in Astronomical Society of the Pacific Conference Series, Vol. 52, Astronomical Data Analysis Software and Systems II, ed. R. J. Hanisch, R. J. V. Brissenden, & J. Barnes, 173
  * Tomaney & Crotts (1996) Tomaney, A. B., & Crotts, A. P. S. 1996, AJ, 112, 2872, doi: 10.1086/118228
  * Tomaney & Shafter (1992) Tomaney, A. B., & Shafter, A. W. 1992, ApJS, 81, 683, doi: 10.1086/191702
  * Valcheva et al. (2018) Valcheva, A., Kostov, A., Minev, M., Ovcharov, E., & Nedialkov, P. 2018, The Astronomer’s Telegram, 12261, 1
  * Valcheva et al. (2016) Valcheva, A., Kurtenkov, A., Ovcharov, E., et al. 2016, The Astronomer’s Telegram, 9559, 1
  * Valcheva et al. (2020) Valcheva, A., Minev, M., Ovcharov, E., & Nedialkov, P. 2020, The Astronomer’s Telegram, 14212, 1
  * Van Rossum & Drake (2009) Van Rossum, G., & Drake, F. L. 2009, Python 3 Reference Manual (Scotts Valley, CA: CreateSpace)
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Wang (2018) Wang, B. 2018, Research in Astronomy and Astrophysics, 18, 049, doi: 10.1088/1674-4527/18/5/49
  * Warner (2008) Warner, B. 2008, in Classical Novae, ed. M. F. Bode & A. Evans, Vol. 43, 16–33, doi: 10.1017/CBO9780511536168.004
  * Williams & Darnley (2015a) Williams, S. C., & Darnley, M. J. 2015a, The Astronomer’s Telegram, 8218, 1
  * Williams & Darnley (2015b) —. 2015b, The Astronomer’s Telegram, 8281, 1
  * Williams et al. (2014) Williams, S. C., Darnley, M. J., Bode, M. F., Keen, A., & Shafter, A. W. 2014, ApJS, 213, 10, doi: 10.1088/0067-0049/213/1/10
  * Williams et al. (2016) Williams, S. C., Darnley, M. J., Bode, M. F., & Shafter, A. W. 2016, ApJ, 817, 143, doi: 10.3847/0004-637X/817/2/143
  * Williams et al. (2019a) Williams, S. C., Darnley, M. J., & Healy, M. W. 2019a, The Astronomer’s Telegram, 12349, 1
  * Williams et al. (2019b) Williams, S. C., Darnley, M. J., Healy, M. W., Murphy-Glaysher, F. J., & Ransome, C. L. 2019b, The Astronomer’s Telegram, 13078, 1
  * Williams et al. (2019c) —. 2019c, The Astronomer’s Telegram, 13228, 1
  * Williams et al. (2020a) —. 2020a, The Astronomer’s Telegram, 14053, 1
  * Williams et al. (2020b) —. 2020b, The Astronomer’s Telegram, 13384, 1
  * Williams et al. (2020c) Williams, S. C., Darnley, M. J., Healy, M. W., et al. 2020c, The Astronomer’s Telegram, 14121, 1
  * XRISM Science Team (2020) XRISM Science Team. 2020, arXiv e-prints, arXiv:2003.04962, doi: 10.48550/arXiv.2003.04962
  * Zamanov et al. (2023) Zamanov, R. K., Marchev, V., Marti, J., & Latev, G. Y. 2023, arXiv e-prints, arXiv:2308.04104, doi: 10.48550/arXiv.2308.04104
  * Zhang & Gao (2018a) Zhang, M., & Gao, X. 2018a, Transient Name Server Discovery Report, 2018-2050, 1
  * Zhang & Gao (2018b) —. 2018b, Transient Name Server Discovery Report, 2018-2069, 1
  * Zhang & Gao (2019a) —. 2019a, Transient Name Server Discovery Report, 2019-1827, 1
  * Zhang & Gao (2019b) —. 2019b, Transient Name Server Discovery Report, 2019-2512, 1
  * Zhang & Gao (2020a) —. 2020a, Transient Name Server Discovery Report, 2020-2924, 1
  * Zhang & Gao (2020b) —. 2020b, Transient Name Server Discovery Report, 2020-3315, 1
  * Zhang & Gao (2020c) —. 2020c, Transient Name Server Discovery Report, 2020-3322, 1
  * Zhang & Gao (2020d) —. 2020d, Transient Name Server Discovery Report, 2020-3028, 1
  * Zhang & Gao (2020e) —. 2020e, Transient Name Server Discovery Report, 2020-3130, 1


# Topology of Singularities on Algebraic Loop Spaces

EMILE BOUAZIZ

###### Abstract.

We study the topology of some simple infinite dimensional singularities
arising from spaces of _algebraic formal loops_. We prove that in some simple
cases the natural analogue of nearby cycles cohomology for a function on the
loop space vanishes, and show further that a suitably renormalized version of
the above cohomology produces a simple (non-zero) result.

##  1\. introduction

The goal of this note is to present some rather simple computations of the
cohomology of some infinite dimensional spaces. If $X$ is an affine variety
(over $\mathbb{C}$) with a regular function $F:X\rightarrow\mathbb{A}^{1}$
which is topologically a fibration over $\mathbb{G}_{m}$, then a measure of
the singularity of $F$ over $\\{0\\}$ is given by the functor of vanishing
cycles and its close relative, nearby cycles. At a crude approximation (which
will be sufficient for our purposes) these functors measure the difference in
central and nearby fibre cohomology.

If one restricts to the simple case where the central fibre is contractible,
for example if $F$ is homogeneous, then to compute vanishing cycles one must
simply compute the cohomology of the nearby fibre. According to a result of
Milnor, and under the additional assumption that the singularity at the origin
be isolated, vanishing cohomology of such a function is concentrated in degree
$\dim X-1$, and of dimension the _Milnor number_ , $\mu(X)$.

Attached to $X$ is a certain infinite dimensional space (in general not
representable by a scheme, but still fairly manageable) of loops into $X$,
denoted $\mathcal{L}X$. Further, the data of $F$ endows this space of loops
with a natural global function, denoted $\Lambda(F)$ which can roughly be
defined as the constant (in $t$) term obtained upon replacing local coordinate
functions $z^{i}$, in $F(z^{1},...,z^{d})$, with Laurent series
$z^{i}(t)=\sum_{j}z^{i}_{j}t^{j}$, so that the variables $z^{i}_{j}$ are
functions on the loop space of $X$. Everything in this context is highly
infinite dimensional and so one certainly should not expect a simple analogue
of the entire vanishing cycles package, but it is of course still possible to
compare central and nearby fibre cohomology, which will be the subject of this
note.

######  Example.

A useful example to bear in mind is the pair $(\mathbb{A}^{1},z^{2})$. We have

$\Lambda(z^{2})=z_{0}^{2}+2\sum_{j>0}z_{j}z_{-j}\in\mathcal{O}(\mathcal{L}\mathbb{A}^{1}).$

We think of this as an (ind-) infinite dimensional affine quadric. Truncating
the Laurent tails by requiring no poles of order greater than $n$, we obtain
the function $z_{0}^{2}+2\sum_{j=1}^{n}z_{j}z_{-j}$. The nearby fibre cuts out
the product of a quadric in $\mathbb{A}^{2n+1}$ with an infinite dimensional
affine space, and is thus homeomorphic to
$T^{*}S^{2n}\times\mathbb{R}^{\infty}$. The nearby fibres are therefore
homotopic to $S^{2n}$, so that in the limit we obtain a weakly contractible
space. If we instead computed a colimit of maps along Gysin maps on cohomology
we would obtain a one dimensional space in degree $0$, which we note agrees
with vanishing cohomology of the pair $(\mathbb{A}^{1},z^{2})$.

We compute the relevant spaces explicitly in the case of homogeneous $F$ with
an isolated singularity, that is to say the cone singularity for the affine
cone on a smooth projective hypersurface. Further, we observe that, in the
spirit of [1], there is a natural method by which to renormalize the
cohomology spaces, so as to avoid classes running off to infinity and thus
vanishing in the limit. It is in fact this renormalized/ stabilized cohomology
with which we deal primarily in this note. We remark that the methods
presented here are quite elementary, we only make use of basic properties of
Gysin maps and residues, the reader familiar with loop spaces and cohomology
of ind-schemes more broadly can safely skip straight to Section 3, and indeed
is encouraged to do so.

We state here our main theorem, undefined terms will be defined in the
following sections.

######  Theorem 1.1.

Let $F:\mathbb{A}^{d}\rightarrow\mathbb{A}^{1}$ be a homogeneous function with
an isolated singularity at $0$. Then the renormalized nearby cohomology of the
pair $(\mathcal{L}\mathbb{A}^{d},\Lambda(F))$ is concentrated in degree $d-1$
and of dimension $\mu(F)$.

We remark that the above computation agrees with the corresponding (reduced)
nearby cohomology of the pair $(\mathbb{A}^{d},F)$, which maps to
$(\mathcal{L}\mathbb{A}^{d},\Lambda(F))$ via the inclusion of constant loops
into loops. As such, the above could be phrased as a cohomology vanishing
theorem for the open complement.

Further, over the course of the proof of the above we will also prove the
following,

######  Theorem 1.2.

The reduced (co)homology of the nearby fibre to $\Lambda(F)$ vanishes.

###  1.1. Acknowledgements

Some of the results presented here were obtained when the author was a PhD
student at the University of Cambridge under the supervision of Ian
Grojnowski, whom the author wishes to warmly thank.

##  2\. basics

We will recall here some basic notions in the theory of loop spaces. We will
work throughout over the field $\mathbb{C}$ of complex numbers. By a _space_
we will mean a pre-sheaf on the category of affine $\mathbb{C}$-schemes, so
that to give a space is simply to give its value on all test algebras $A$. We
will deal in particular with _ind-schemes_ , which are filtered colimits
(taken inside the pre-sheaf category) of schemes along closed embeddings.
Given a set $S$, throughout we will write $\mathbb{A}\\{S\\}$ for the affine
space with ring of functions, $\mathbb{C}[S]$, the free commutative
$\mathbb{C}$-algebra generated by $S$.

###  2.1. Loop Spaces

We now define some of our basic objects of study, mostly to fix notation. $X$
will denote an arbitrary space below.

######  Definition 2.1.

  1. (1)

The _arc space_ of $X$, $\mathcal{L}^{+}X$, is defined as the space with
$A$-points $\mathcal{L}^{+}X(A)=X(A[[t]])$

  2. (2)

The _loop space_ , denoted $\mathcal{L}X$, is defined as the space with
$A$-points $\mathcal{L}X(A)=X(A((t)))$.

  3. (3)

There is an evident $\mathbb{G}_{m}$ action on $\mathcal{L}X$. Functions on
$\mathcal{L}X$ of weight $w$ for this action will be said to be of _conformal
weight_ $w$.

We record some simple and well-known facts below. For a more detailed
introduction to these objects we recommend [2] and its sequels.

######  Lemma 2.1.

  1. (1)

If $X$ is a scheme then the functor $\mathcal{L}^{+}X$ is representable by a
scheme.

  2. (2)

If $X$ is moreover finite type and affine, then the functor $\mathcal{L}X$ is
representable by an ind-scheme, and in fact by an ind-affine scheme, that is
to say a filtered colimit of affine schemes along closed embeddings.

  3. (3)

If $X$ is smooth then both the loop and the arc space are formally smooth.

######  Remark.

Giving an ind-affine scheme is equivalent to giving a complete and _linearly
topologized_ $\mathbb{C}$-algebra, that is a complete topological algebra with
a neighbourhood basis at $0$ consisting of ideals.

######  Example.

The case of $\mathcal{L}\mathbb{A}^{1}$ corresponds to the linearly
topologized algebra $\lim_{n}\,\mathbb{C}[\mathbb{Z}_{\geq-n}]$. The function
$z_{i}$ is of conformal weight $i$. The topology is such that for example the
infinite sum $\sum_{j}z_{j}z_{-j}$ is convergent.

We now introduce the functions on the loop space that we wish to consider. We
will assume $X$ is an affine scheme in what follows. If $X=spec(A)$, then we
will write

$X\widehat{\times}D^{*}=spec(A((t))).$

We extend the definition of $(-)\widehat{\times}D^{*}$ to ind-affine schemes
by imposing compatibility with filtered colimits. Note that if an ind-affine
scheme $\mathcal{X}$ corresponds to a linearly topologized algebra $A$, then
the ind-affine scheme $\mathcal{X}\widehat{\times}D^{*}$ corresponds to the
algebra, $A\\{t\\}$, of _topological Laurent series_ over $A$, i.e. the
algebra of formal series in $t$ and $t^{-1}$ with negative tails tending to
$0$; $\\{\sum_{i}a_{i}t^{i}\mid a_{i}\rightarrow 0,i\rightarrow-\infty\\}$,
this is of course the natural completion of the naive algebra of $A$-valued
Laurent series.

######  Definition 2.2.

The functor of points definition of $\mathcal{L}X$ produces a universal map

$\mathcal{L}X\widehat{\times}D^{*}\longrightarrow X.$

We call this the _evaluation map_ and denote it $\eta$. If $f$ is a function
on $X$, we denote by $\Lambda(f)$ the constant term of $\eta^{*}(f)$.

######  Example.

In the case of $\mathbb{A}^{1}$, $\eta$ can be understood very easily. Indeed,
by definition it is a global function on
$\mathcal{L}\mathbb{A}^{1}\widehat{\times}\,D^{*}$, the algebra of such is the
algebra of topological power series over thecomplete linearly topologized
algebra $\lim_{n}\,\mathbb{C}[\mathbb{Z}_{\geq-n}]$. It is then readily seen
that $\eta$ corresponds to the (universal) topological Laurent series

$z(t):=\sum_{i}z_{i}t^{i}.$

As such we see for example that

$\Lambda(z^{2})=\sum_{i}z_{i}z_{-i}.$

In general, working with coordinates $\\{z^{j}\\}_{j}$, one substitutes the
sums $z^{j}(t)=\sum_{i}z^{j}_{i}t^{i}$ into $f$ and formally expands, then
takes the constant term.

###  2.2. Homology and Cohomology of Ind-Schemes

If $\mathcal{Y}$ is an ind-scheme space, then the set of $\mathbb{C}$ points
of $\mathcal{Y}$ is topologized as the colimit of spaces $Y(\mathbb{C})$, with
$Y\rightarrow\mathcal{Y}$ a closed embedding of a scheme $Y$ into
$\mathcal{Y}$.

######  Example.

Consider the ind-scheme $\mathbb{A}^{\infty}=\lim_{n}\,\mathbb{A}^{n}$, and
let $\mathcal{Y}=\mathbb{A}^{\infty}\setminus\\{0\\}$. Then
$\mathcal{Y}(\mathbb{C})$ is the weakly contractible space of non-zero
finitely supported sequences of elements of $\mathbb{C}$.

We remark that the above example illustrates a fairly typical phenomenon, the
support of the reduced homology of the spaces in a colimit presentation of the
ind-scheme $\mathcal{Y}$ tends to infinity, and so nothing survives to the
limit. Note that the same thing happens for cohomology, as the relevant
$\mathbb{R}\lim$ terms vanish in such cases, as the resulting pro-systems are
manifestly Mittag-Leffler. One might then hope that a suitable renormalization
procedure could remedy this. In [1] the authors note that such is possible for
sufficiently well behaved ind-schemes.

######  Definition 2.3.

If $X$ is a scheme then we call it _smooth_ if it can be represented as the
filtered limit of smooth finite type schemes.

######  Remark.

This implies formal smoothness of $X$.

We extend this to ind-schemes in an evident manner:

######  Definition 2.4.

If $\mathcal{X}$ is an ind-scheme, we call it _smooth_ if it can be
represented as the filtered colimit of smooth schemes in the sense of the
definition above.

######  Remark.

This is a strong condition, and indeed is not fulfilled by the loop space to
an arbitrary smooth scheme, for example $\mathcal{L}\mathbb{G}_{m}$ is non-
reduced, and thus certainly not smooth in this sense. Note that on the other
hand $\mathcal{L}\mathbb{A}^{1}$ certainly is smooth.

Henceforth, for a scheme $X$, when we write $H^{*}(X)$ it is to be understood
as $H^{*}(X(\mathbb{C}),\mathbb{C})$. As explained in [1], a codimension $d$
embedding, $X\rightarrow Y$, of smooth and possibly infinite dimensional
schemes induces a _Gysin map_ on cohomologies, $H^{*}(X)\rightarrow
H^{*+2d}(Y)$. We require one more definition:

######  Definition 2.5.

Let $\mathcal{Y}=\underrightarrow{\lim}_{j}\mathcal{Y}^{j}$ be an ind-scheme.
Then a _dimension theory_ , $\Delta$, for $\mathcal{Y}$ is a rule assigning an
integer $\Delta(j)$ to each scheme $\mathcal{Y}^{j}$ in such a way that for
$i<j$, the codimension of $\mathcal{Y}^{i}$ inside $\mathcal{Y}^{j}$ is equal
to $\Delta(j)-\Delta(i)$.

######  Remark.

  * •

It is clear that a dimension theory always exists for $\mathcal{Y}$ and the
set of such is a $\mathbb{Z}$-torsor for connected $\mathcal{Y}$.

  * •

Given two presentations of $\mathcal{Y}$ as a colimit of schemes along finite
codimensional closed embeddings, a dimension theory with respect to one
presentation canonically produces a dimension theory with respect to the
other.

We require one final piece of data, here we assume that we are given a smooth
ind-scheme $\mathcal{Y}$ equipped with a global function
$F:\mathcal{Y}\rightarrow\mathbb{A}^{1}$.

######  Definition 2.6.

We say that $F$ is smooth if there is a presentation of $\mathcal{Y}$ as an
ind-limit of smooth closed subschemes on which $F$ is smooth in the ususal
sense.

######  Remark.

If $F$ is smooth over $\mathbb{G}_{m}$ then the nearby fibres are endowed with
a smooth structure as ind-schemes.

We are finally in the position to define the renormalized cohomology of a
smooth ind-scheme, $\mathcal{Y}$, equipped with a dimension theory $\Delta$.

######  Definition 2.7.

We define
$H^{*}_{\Delta}(\mathcal{Y})=\underrightarrow{\lim}_{j}H^{*+2\Delta(j)}(\mathcal{Y}^{j})$,
where $\\{\mathcal{Y}^{j}\\}_{j}$ is a given smooth presentation of
$\mathcal{Y}$ and the limit is taken with respect to the Gysin maps. In the
presence of a global function $F$ which is smooth over $\mathbb{G}_{m}$ in the
above sense, the natural renormalized cohomology of the nearby fibre will be
referred to as _renormalized nearby cohomology_ , and denoted
$H^{*}_{\Delta}(\mathcal{Y},F)$.

######  Remark.

This doesn’t depend on the presentation. The dependence on the choice of
dimension theory is only up to a shift in degree. We will often drop the
$\Delta$ subscript in what follows, and instead write $H^{*}_{ren}$, which
we’ll refer to as _renormalised cohomology_.

######  Remark.

We remark that the definition given above makes perfect sense for $*<0$, if we
agree that for a topological space $Y$, $H^{<0}(Y):=0$. Further, the
definition above may be non-zero for negative $*$. Indeed let us fix the
unique dimension theory on $\mathbb{A}^{\infty}\setminus\\{0\\}$ such that
$\Delta(\mathbb{A}^{1})=0$. Then there is a stable class in $H^{-1}_{\Delta}$
corresponding to the natural classes in
$H^{2d-1}(\mathbb{A}^{d}\setminus\\{0\\})$. Now, in this case the choice of
another dimension theory allows us to fix this. Note however that since
dimension theories form a $\mathbb{Z}$ torsor, if there are non-zero classes
in arbitrarily negative degrees with respect to one dimension theory, then
this is true with respect to any other dimension theory. Such can certainly be
the case, as the example of $\mathbb{P}^{\infty}$ shows. A hyperplane class in
$H^{2}(\mathbb{P}^{n})$ can easily be checked to survive in the colimit, and
as $n$ tends to infinity, these live in more and more negative degrees.

##  3\. The Computation

###  3.1. The Set-Up

For ease of reference we recall the set-up. Let
$F\in\mathcal{O}(\mathbb{A}^{d})$ be a homogenous polynomial with isolated
singularity at $0$. The degree of homogeneity of $F$ will be denoted $\delta$.
The coordinates on the affine space $\mathbb{A}^{d}$ will be denoted with
upper indices, $\\{z^{1},...,z^{d}\\}$. The loop space
$\mathcal{L}\mathbb{A}^{d}$ is then
$\underrightarrow{\lim}_{n}\,\mathbb{A}\\{z^{i}_{j}\mid
i\in\big{[}1,d\big{]},j\in\mathbb{Z}_{\geq-n}\\}$ and it carries the function
$\Lambda(F)=CT(F(z^{1}(t),...,z^{d}(t)))$, where $CT$ denotes the constant
term and $z(t)=\sum_{i}{z_{i}}t^{i}$. The closed subscheme
$\\{\Lambda(F)=1\\}$ is denoted $\mathcal{X}$.

Let us say something about the smooth structure: we write
$\mathcal{X}^{n}_{m}=\mathcal{X}\cap\mathcal{L}^{n}_{m}\mathbb{A}^{d}$, where
$\mathcal{L}^{n}_{m}\mathbb{A}^{d}$ is defined to be
$\mathbb{A}\\{z^{i}_{j}\mid i\in\big{[}1,d\big{]},j\in\big{[}-n,m\big{]}\\}$.
Evidently we have that

$\mathcal{X}=\underrightarrow{\lim}_{n}\,\underleftarrow{\lim}_{m}\,\mathcal{X}^{n}_{m}.$

We now record a trivial lemma, guaranteeing that we may define the
renormalised cohomology of $\mathcal{X}$.

######  Lemma 3.1.

The family $\\{\mathcal{X}^{n}_{m}\\}_{n,m}$ gives a smooth presentation for
$\mathcal{X}$, i.e. every space $\mathcal{X}^{n}_{m}$ is smooth.

###### Proof.

Recall that $F$ is homogeneous of weight $\delta>0$ and hence the
$\mathbb{G}_{m}$ action on $\mathbb{A}^{d}$ induces one on
$\mathcal{L}\mathbb{A}^{d}$ for which $\Lambda(F)$ has weight $\delta$. Each
$\mathcal{X}^{n}_{m}$ is then the nearby fibre of a homogeneous function on an
affine space. However Euler’s formula guarantees that the singularities of
such a function are in the central fibre, so we are done. ∎

We will write $\mathcal{X}^{n}=\underleftarrow{\lim}_{m}\mathcal{X}^{n}_{m}$
in what follows. We have seen above that $\mathcal{X}^{n}$ is a smooth
(infinite type) scheme and of course the colimit of the $\mathcal{X}^{n}$ is
$\mathcal{X}$.

###  3.2. Proof of Main Theorem

###### Proof.

We’ll break the proof up into a few steps.

(_Step 1_) Observe that if we bound the conformal degree of our variables
below by $-n$, then homogeneity of $F$ implies that no variables of conformal
degree strictly greater than $n(\delta-1)$ can show up in $\Lambda(F)$ (more
properly $\Lambda(F)$ restricted to $\mathcal{X}^{n}$). This is simply because
the conformal weight would become strictly positive otherwise. This
immediately implies that we have an isomorphism:

$\mathcal{X}^{n}\xrightarrow{\sim}\mathcal{X}^{n}_{n(\delta-1)}\times\mathbb{A}\\{z^{i}_{j}\mid
i\in\big{[}1,d\big{]},j>n(\delta-1)\\}.$

(_Step 2_). We wish to calculate the Gysin maps in cohomology for the
inclusions $\mathcal{X}^{n}\hookrightarrow\mathcal{X}^{n+1}$. Now, assume
given two smooth finite type spaces $X$ and $Y$ and a closed inclusion
(relative to $\mathbb{A}\\{\mathbb{Z}_{\geq 0}\\}$),
$X\times\mathbb{A}\\{\mathbb{Z}_{\geq 0}\\}\hookrightarrow
Y\times\mathbb{A}\\{\mathbb{Z}_{\geq 0}\\}$. Then to compute the Gysin map in
cohomology for this inclusion we need only do it for $X\hookrightarrow Y$.

Using (_step 1_) we see that we need to compute the Gysin map for the closed
inclusion
$\mathcal{X}^{n}_{(n+1)(\delta-1)}\hookrightarrow\mathcal{X}^{n+1}_{(n+1)(\delta-1)}$.
We denote by $\mathcal{U}^{n}$ the open complement to this inclusion. So as to
avoid having to write $(n+1)(\delta-1)$ repeatedly we hereby denote this
number $N$. Recall the Gysin exact sequence:

$\dotsm\rightarrow H^{*}(\mathcal{X}^{n}_{N})\rightarrow
H^{*+2d}(\mathcal{X}^{n+1}_{N})\rightarrow
H^{*+2d}(\mathcal{U}^{n})\rightarrow
H^{*+1}(\mathcal{X}^{n}_{N})\rightarrow\dotsm;$

we’d like to use this to compute the Gysin maps by computing the cohomology of
$\mathcal{U}^{n}$.

(_Step 3_) Here we compute the cohomology of $\mathcal{U}^{n}$ explicitly.
This relies crucially on the hypothesis that $F$ be a homogenous isolated
singularity. We claim that $\mathcal{U}^{n}$ has the same cohomology as
$\mathbb{A}^{d}\backslash\\{0\\}$, i.e. the cohomology of $S^{2d-1}$. In order
to prove this we firstly define $\mathcal{U}^{n}(j)$ to be the locus of non-
vanishing of $\partial_{z^{j}_{N}}\Lambda(F)$. Monomials showing up in
$\Lambda(F)$ can contain at most one variable of conformal degree $N$ for
obvious degree reasons. This implies that $\Lambda(F)$ looks like

$\sum_{j}z^{j}_{N}(\partial_{z^{j}_{N}}\Lambda(F))+\mathcal{O}(<N),$

where the big $\mathcal{O}$ notation denotes a sum of monomials with no
conformal degree $N$ terms. Further, let us note that the chain rule for
differentiation implies that we have $\partial_{z^{j}_{N}}\Lambda(F)$ is the
$t^{-N}$ coefficient of $\partial_{j}F(z^{1}(t),...,z^{d}(t))$. Now
homogeneity of $F$ is used to note that we thus have

$\partial_{z^{j}_{N}}\Lambda(F)=\partial_{j}F(z_{-n}^{1},...,z^{d}_{-n}).$

We claim further that $\mathcal{U}^{n}=\bigcup_{j}\mathcal{U}^{n}(j)$. To see
this, recall that $\mathcal{U}^{n}(j)$ is the locus of non-vanishing of
$\partial_{j}F(z^{1}_{-(n+1)},...,z^{d}_{-(n+1)})$ and the simultaneous
vanishing locus of these partials is precisely the locus of vanishing of all
the functions $z^{i}_{-(n+1)}$, since $F$ was assumed to have an isolated
singularity at the origin. That is to say the locus of vanishing is precisely
$\mathcal{X}^{n}_{N}$, which is the complement to $\mathcal{U}^{n}$, and so
the claim holds. Now consider the map

$\mathcal{U}^{n}\longrightarrow\mathbb{A}^{d}\setminus\\{0\\},$

defined as $(z^{1}_{-n},...,z^{d}_{-n})$. We claim that this map is a Zariski
locally trivial affine space bundle. Indeed, let us cover
$\mathbb{A}^{d}\setminus\\{0\\}$ by opens $\mathcal{V}(j)$ defined by the non-
vanishing of $\partial_{j}F$. Recalling that we have

$\Lambda(F)=\sum_{j}z^{j}_{N}(\partial_{z^{j}_{N}}\Lambda(F))+\mathcal{O}(<N),$

we easily see that

$\mathcal{U}^{n}(j)\longrightarrow\mathcal{V}(j)$

is a globally trivial affine space bundle with fibre
$\mathbb{A}\\{z^{i}_{j}\mid(i,j)\in[1,d]\times[-n,N-1]\\}$. This suffices to
prove that $\mathcal{U}^{n}$ has the cohomology of a $2d-1$-sphere,
independent of $n$.

_Step 4_ We can now conclude the proof quite easily. We begin with $*>0$.
Examining the Gysin sequence

$\dotsm\rightarrow H^{*}(\mathcal{X}^{n}_{N})\rightarrow
H^{*+2d}(\mathcal{X}^{n+1}_{N})\rightarrow
H^{*+2d}(\mathcal{U}^{n})\rightarrow
H^{*+1}(\mathcal{X}^{n}_{N})\rightarrow\dotsm$

we see stabilisation in the limit defining $H^{*}_{ren}$ at $n=0$ for any
$*>0$, because $H^{>2d-1}(\mathcal{U}^{n})=0$ by the result above. By Milnor’s
results (cf. [3]) this will produce Milnor fibre cohomology concentrated in
$*=d-1$.

The same argument implies stabilisation at $n=1$ for $*=0$. It then suffices
to prove that this is $0$, i.e. that $H^{2d}(\mathcal{X}^{1})=0$. Looking at
the Gysin sequence we must show that the residue map

$H^{2d-1}(\mathcal{U}^{n})\longrightarrow H^{0}(\mathcal{X}^{1})$

is non-zero, as it is then an isomorphism. A representative for the non-zero
class in $H^{2d-1}(\mathcal{U}^{n})$ is given with respect to the Cech cover

$\mathcal{U}^{n}=\bigcup_{j}\mathcal{U}^{n}(j),$

by $d\log z^{1}_{-n}...d\log z^{d}_{-n}$ restricted to
$\cap_{j}\mathcal{U}^{n}(j)$, which manifestly has non-vanishing residue,
whence we are done. Finally, the case of $*<0$ splits into two subcases,
according to whether $2d$ divides $*$. One takes the minimum $n>0$ so that
$*+2nd$ is non-negative and applies arguments exactly as above to conclude
stabilization at said $n$ if $*$ is not divisible by $2d$ and at $n+1$
otherwise, the vanishing is proven similarily. ∎

We can now prove the vanishing of nearby cycles cohomology for the function
$\widehat{F}$. We note that the central fibre is contractible so that we need
only prove that the reduced cohomology of the nearby fibre vanishes.

######  Theorem 3.2.

The reduced (co)homology of the ind-scheme $\mathcal{X}$ vanishes.

###### Proof.

One sees in the course of the above proof that we have that $\mathcal{X}^{n}$
has no reduced cohomology below degree $2(n+1)d-1$ and so the result is
proven. ∎

## References

  * [1] S. Arkhipov, M. Kapranov; Toric arc schemes and quantum cohomology of toric varieties, Mathematische Annalen volume 335, pages 953 -964 (2006),
  * [2] M. Kapranov, E. Vasserot; Vertex algebras and the formal loop space, Publications mathématiques de l’IHÉS volume 100, pages 209-269 (2004)
  * [3] J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, no. 61.