# Tilted Dark Halos are Common, Long-Lived, and can Warp Galactic Disks

Jiwon Jesse Han Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA 02138, USA  Vadim Semenov Center for Astrophysics $|$
Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Charlie
Conroy Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA  Lars Hernquist  Center for Astrophysics $|$ Harvard
& Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

###### Abstract

In the $\Lambda$-CDM paradigm, the dark halo governs the gravitational
potential within which a galaxy can form and evolve. In this Letter we show
that the present-day inner ($r<50\text{ kpc}$) dark halo can be significantly
misaligned with the stellar disk. To this end, we use the TNG50 run from the
cosmological magneto-hydrodynamic IllustrisTNG simulation suite. Such “tilted”
dark halos can arise from a variety of processes including major mergers,
massive fly-bys, or interactions with satellite companions. Furthermore, we
show that tilted dark halos: (1) are well traced by tilted stellar halos, (2)
can maintain their tilt for $>$ 5 Gyr in isolated evolution, and (3) can
generate warps in the outer disks that are stable over many Gyr. A tilted dark
halo holds clues to important events in the formation history of a galaxy, and
could help explain the abundance of warped disks in galaxy observations,
including the Milky Way.

##  1 Introduction

In a $\Lambda$-CDM universe, dark matter collapses into halos (White & Rees,
1978; Davis et al., 1985; Rubin et al., 1985; Frenk et al., 1985) that create
the gravitational wells within which galaxies form (Blumenthal et al., 1984).
While the spherically averaged profile of dark halos in $N$-body simulations
is roughly universal (Navarro et al., 1997), it was realized early on that
dark halos are likely aspherical in nature (Frenk et al., 1988; Dubinski &
Carlberg, 1991). On the other hand, luminous matter affects the shapes of dark
halos, preferentially causing the inner halo to be more spherical (Gnedin et
al., 2004; Kazantzidis et al., 2004; Gustafsson et al., 2006; Duffy et al.,
2010). Furthermore, hierarchically formed halos comprise a wealth of
substructure that ranges from intact to completely relaxed (Bullock &
Johnston, 2005), which also affects the overall shape of the “smooth” dark
halo (Moore et al., 2004; Cooper et al., 2010). An interesting regime is the
transition from the inner ($r<0.1\text{\>}r_{\text{virial}}$) to outer
($r>0.5\>r_{\text{virial}}$) halo, where the intrinsic asphericity of the dark
halo competes with the backreaction from luminous matter, potentially enhanced
or erased by major accretion events.

Connecting the properties of the dark halo to its directly observable
counterpart—the stellar halo—is a valuable tool that allows observations to
constrain the dark halo. Gaia (Gaia Collaboration et al., 2018) has revealed
that the stellar halo of the Milky Way is dominated by one major merger event
(Belokurov et al., 2018; Helmi et al., 2018), which may be connected to a
global asymmetry in its distribution (Iorio & Belokurov, 2019; Han et al.,
2022a). Furthermore, using the H3 Survey (Conroy et al., 2019), Han et al.
(2022a) find that the stellar halo from Galactocentric radius $r=5-50\text{
kpc}$ is $\sim 25^{\circ}$ tilted with respect to the plane of the disk.

An intriguing question is whether the dark halo can also be tilted in a
similar direction on these scales (Han et al., 2022b). Shao et al. (2021) and
Emami et al. (2021) find that the dark halos of Milky Way-like galaxies in
large-volume cosmological simulations EAGLE (Schaye et al., 2015) and
Illustris TNG50 (Pillepich et al., 2019) generally change orientations beyond
$0.1\>r_{\text{virial}}$, which is around $20-30\text{ kpc}$ for the Galaxy.
On the contrary, Orkney et al. (2023) have analyzed 10 Milky Way-like galaxies
in the zoom-in cosmological simulation AURIGA (Grand et al., 2017) and did not
find a present-day tilt of the halo with respect to the disk within 50 kpc.

Figure 1: Evolution of the tilt angle of the dark and stellar halo from 6 Gyr
ago to present day in TNG50 Milky Way analogs (Pillepich et al., 2023). Each
line represents a single halo. The first and second row panels show the
evolution of the tilt angle of the dark matter and stellar halo, and the third
row panels show the evolution of the dark halo triaxiality parameter $T\equiv
1-p^{2}/1-q^{2}$, where $p$ and $q$ are the major-to-intermediate and major-
to-minor axis ratios. We increase the line thickness with time, and color each
line according to its final dark halo tilt angle. On the right panel, we plot
the cumulative distribution function (CDF) of the tilt angles at the present
day, which reveals that around half of TNG50 Milky Way analogs have dark halo
tilt angles greater than $10^{\circ}$. We find a diversity of triaxiality
parameters for the tilted halos, ranging from prolate ($T>0.6$) to oblate
($T<0.3$). Figure 2: Relationship between the dark halo and the stellar halo
tilt angles. Each point shows the dark halo and stellar halo tilt angle at one
snapshot, spanning all of the halos and lookback times from Fig 1. The pink
shaded region marks the measured Galactic stellar halo tilt from Han et al.
(2022a), and the blue histogram on the right panel shows the marginal
distribution of the Milky Way’s dark halo. The blue horizontal line shows the
most likely value of the Galactic dark halo to be $20^{\circ}$.

In this Letter we address three questions. First, can a present-day tilt of
the inner 50 kpc dark halo with respect to the disk exist? Second, if those
features do exist, are they common and long-lived? Lastly, how do such
misalignments affect the stellar disk? To answer these questions, we use the
TNG50 run of the cosmological magneto-hydrodynamic simulation suite
IllustrisTNG (Nelson et al., 2019a, b; Pillepich et al., 2019). TNG50 is
uniquely suited for our study because it: (1) captures a wide range of galaxy
formation scenarios ($\sim\>$200 Milky Way-like halos), (2) incorporates
reasonably realistic baryonic physics, and (3) sufficiently resolves the
particle dynamics of the halo and the disk.

We organize the Letter as follows. In Section 2, we desribe the TNG50 sample
of galaxies that we use, then outline the method to analyze their tilt angles.
In Section 3, we identify an archetypal galaxy and closely follow the
evolution of its dark matter halo, stellar halo, and stellar disk. Finally, in
Section 4, we summarize the results and discuss their implications in a
broader context.

##  2 Milky Way Analogs in TNG50

In this Section, we describe the sample of galaxies in TNG50 that we use, and
how we define and measure the tilt angles of galaxies over their formation
histories. We then present the distribution of dark halo tilt angles in our
sample, and relate those measurements to the stellar halo.

###  2.1 Sample and Methods

TNG50 (Pillepich et al., 2019; Nelson et al., 2019a, b) is a cosmological
magneto-hydrodynamical simulation run with AREPO (Springel, 2010; Weinberger
et al., 2020) using the fiducial TNG galaxy formation model (Weinberger et
al., 2017; Pillepich et al., 2018a) and Planck Collaboration et al. (2016)
cosmological parameters. Like the other simulations in IllustrisTNG (Pillepich
et al., 2018b; Springel et al., 2018; Nelson et al., 2018; Marinacci et al.,
2018; Naiman et al., 2018), TNG50 follows the evolution of cold dark matter
(CDM), stars, gas, magnetic fields, and supermassive black holes from redshift
$z=127$ to $z=0$. In the case of TNG50, the volume modeled is ${51.7\text{
Mpc}}^{3}$, the mass resolution of the CDM particles is $4.5\times
10^{5}M_{\odot}$, and that of the stellar particles is within a factor of 2
from the target baryonic mass resolution of $8.5\times 10^{4}M_{\odot}$.
Combined with this resolution and the cosmological volume, TNG50 captures a
wide range of galaxy formation processes from massive galaxy clusters to
isolated dwarf galaxies. At $z=0$, there are $\sim 900,000$ halos and subhalos
with gravitationally bound mass greater than $10^{8}M_{\odot}$. Details of the
simulation can be found in Nelson et al. (2019a, b); Pillepich et al. (2019).

Using TNG50, Pillepich et al. (2023) identify Milky Way and M31-like galaxies
in the simulation. For this study, we use their “observable-based selection,”
which is the subset of three criteria: (1)
$10.5<\log_{10}(M_{*}/M_{\odot})<2\times 10^{12}$, (2) no massive galaxy
within $500\text{ kpc}$ and $M_{\text{host},200c}<10^{13}M_{\odot}$, and (3)
disky galaxies (based on Pillepich et al., 2019, and visual inspection). This
selection yields 198 galaxies and their host halos.

For each halo, we define the “tilt angle” to be a misalignment of the stellar
disk and the inner dark halo, and calculate this quantity using the following
method. We first select dark matter particles with $r\in(10\text{
kpc},50\text{ kpc})$. From the position and mass of each particle, we
calculate the moment of inertia tensor. By solving for the eigenvector-
eigenvalue pairs of the tensor, we find the three principal axes of rotation
and their respective moments of inertia. The major axis has the minimum moment
of inertia, and the minor axis has the maximum moment of inertia. The moments
of inertia $I_{i}$, $i\in\\{a,b,c\\}$, can be related to the “length” $r_{i}$
of the principal axes as the following:

$I_{i}\propto r_{j}^{2}+r_{k}^{2}\rightarrow
r_{i}^{2}\propto\frac{-I_{i}+I_{j}+I_{k}}{2}$ (1)

Using this relation, we can compute the major-to-intermediate and major-to-
minor axes ratios as follows:

$1:\frac{r_{b}}{r_{a}}:\frac{r_{c}}{r_{a}}=1:\sqrt{\frac{I_{a}-I_{b}+I_{c}}{-I_{a}+I_{b}+I_{c}}}:\sqrt{\frac{I_{a}+I_{b}-I_{c}}{-I_{a}+I_{b}+I_{c}}}$
(2)

Some studies calculate the “reduced” moment of inertia to down-weight the
outer halo particles (e.g., Allgood et al., 2006; Vera-Ciro et al., 2011;
Schneider et al., 2012; Emami et al., 2021). In this study, we are limiting
the analysis to particles with $10\text{ kpc}<r<50\text{ kpc}$, and do not
downweight the particles. The canonical moment of inertia is sufficient to
capture the misalignment of the halo and the disk, and also allows for an
easier interpretation of the result. Furthermore, the tilt angle is
insensitive to the actual values of the moments of inertia; it is only
sensitive to the direction of the principal axes.

Once we obtain the principal axes, we measure the tilt angle as the angle
between the minor axis and the $Z$-axis, which is set by the total angular
momentum of the stellar disk. There are two motivations to use the minor axis
as opposed to the major axis. First, many TNG50 galaxies show an oblate inner
halo (see last row of Fig. 1), meaning that the major and intermediate axes
are roughly degenerate with each other. This is consistent with previous
studies (e.g., Kazantzidis et al., 2004; Shao et al., 2021). In this scenario,
the degeneracy between the major/intermediate axes (and the corresponding
freedom of azimuthal rotation) causes their angle with respect to the $Z$-axis
to be an unstable measurement. On the other hand, the minor axis is almost
never degenerate, making its angle with the $Z$-axis a much more stable
measurement. Secondly, when the minor axis is aligned with the $Z$-axis, the
halo is stable against perturbations to the gravity of the disk, and when the
minor axis is $90^{\circ}$ to the $Z$-axis, the halo is unstable. This analogy
yields an intuitive interpretation of the tilt angle. We note that the stellar
halo tends to be more triaxial than the dark halo in TNG50, which means that
all three axes are nondegnerate and one can use either the major or minor axis
to measure the tilt angle.

###  2.2 Diversity of Tilt Angles

In Figure 1 we show the evolution of the tilt angle of all 198 Milky Way
analogs from 6 Gyr ago to the present day. Each colored line represents an
individual halo. The top panels show the tilt angle measured for the dark
halo, and the middle panels show that of the stellar halo. The bottom panels
show the dark halo triaxiality parameter $T\equiv 1-p^{2}/1-q^{2}$, where $p$
and $q$ are the major-to-intermediate and major-to-minor axis ratios. High
values of this parameter ($T>0.6$) indicate prolate halos, while low values
($T<0.3$) indicate oblate halos. For all rows, we show a cumulative
distribution function (CDF) on the right panel that encapsulates the
population trend. For example, 50% of dark halos have tilt angles greater than
$10^{\circ}$, 25% of dark halos have tilt angles greater than $20^{\circ}$,
and 15% of dark halos have tilt angles greater than $40^{\circ}$. This figure
demonstrates that the majority of halos, for the majority of their lifetimes,
have tilted dark halos. Furthermore, tilted dark halos show a broad range of
shapes ranging from prolate to oblate.

In Figure 2, we plot the dark halo tilt angle against the stellar halo tilt
angle at each snapshot, for all galaxies in Figure 1. The relationship is
strikingly linear. The pink region marks the posterior probability of the
Milky Way’s stellar halo tilt angle from Han et al. (2022a), and the
consequent marginal probability distribution of the dark halo is plotted in
the right panel. We overplot a fitted exponentially modified Gaussian
distribution in blue. From this Figure, we can infer that the tilt of the dark
halo is most likely greather than $20^{\circ}$ for the Milky Way at the
present day.

Figure 3: Evolution of Halo 533060 after a 10:1 merger at 7 Gyr. In the top
panel, we plot the tilt angle of the dark (stellar) halo in blue (red) dots
with $1\sigma$ error bars estimated from jack-knifing. X-marks indicate where
the tilt angle uncertainties are larger than $10^{\circ}$. The colored shaded
lines are cubic spline interpolations of the data, sampled from their
statistical errors. In the bottom panel, we show the major-to-intermediate and
the major-to-minor ratios of the dark/stellar halo, which we refer to as the
“flattening parameters.” We grey out the regions in which both flattening
parameters are close to 1 (i.e. roughly spherical), because the tilt angle is
not well defined in this region. Once the dark (stellar) halo settles into an
oblate (triaxial) shape at $\sim$5 Gyr, the tilt angle steadily decreases from
$50^{\circ}$ to $20^{\circ}$ at present day. Figure 4: Stellar disk of Halo
533060 plotted at four snapshots. The tilt of the dark halo settles at around
5 Gyr (see Fig 3). $Z$ denotes the vetical height of the star, and $R$ denotes
the cylindrical radius with positive (negative) sign where the angles are $\pm
90^{\circ}$ within the positive (negative) vertical extremum of the warp. We
overplot an analytic fit to the warp (power-law in radius and sinusoid in
azimuth) in blue, with dotted regions marking $1\sigma$ uncertainties of the
fit. A strong warp persists over $>4\text{ Gyr}$.

##  3 An Archetypal Galaxy

In this Section, we present a case-study of a galaxy with a tilted dark halo
and a warped disk. Halo 533060 has a present-day total mass of $8\times
10^{11}M_{\odot}$, and it experiences a 10:1 merger $\sim 7\text{ Gyr}$ ago
that induces a strong misalignment in the dark halo and the disk. This halo
does not undergo any other significant perturbations (e.g., mergers, fly-bys,
or massive companions) afterwards, allowing us to study the secular evolution
of the tilted dark halo. In the following, we explore the evolution of the
halo tilt angle and the response of the disk in Halo 533060.

###  3.1 A Tilted Halo

In Figure 3 we show the evolution of the tilt and shape of Halo 533060. In the
top panel, we plot the tilt angle of the dark (stellar) halo in blue (red)
circles, and indicate points with $>5^{\circ}$ statistical uncertainties with
x-marks. The colored shaded regions show a cubic spline interpolation of the
points and the $1\sigma$ variance in the interpolated curve. In the bottom
panel, we plot the major-to-intermediate and the major-to-minor axes ratios
for the dark/stellar halo, which refer to as the “flattening parameters.” We
grey out the regions in which both parameters are close to one, indicating
that the halo is roughly spherical and the tilt angle is not well defined. For
Halo 533060, this region corresponds to the immediate aftermath of the merger.
Once the merger is complete, the debris eventually settles and the dark
(stellar) halo takes on an oblate (triaxial) shape that is $\sim 50^{\circ}$
misaligned with the disk. Consistent with what we find in Figure 2, the
stellar halo tilt angle closely follows that of the dark halo. In subsequent
isolated evolution, the dark halo tilt angle declines to $20^{\circ}$ in 5
Gyr, likely due to a combination of processes such as dynamical friction,
phase mixing, and torque exerted by the disk. Furthermore, we see the dark
halo shape remains roughly constant, indicating that it rotates as a solid
body. This is consistent with previous work (e.g., Bailin & Steinmetz, 2004;
Perryman et al., 2014). At the present day, the tilt angle of the stellar halo
matches that of the dark halo.

Figure 5: Correspondence between the warp amplitude and the dark halo tilt. On
the top panel we plot the evolution of the dark halo tilt as in Fig 3. On the
bottom panel we plot the amplitude of the disk warp measured at twice the
scale radius. We grey out the same region as Fig 3 where the halo is roughly
spherical and the tilt angle is not well defined. When the halo is spherical,
the warp amplitude is zero. Once the halo settles to an oblate shape and a
tilt angle of $50^{\circ}$, the warp amplitude sharply increases over 1 Gyr.
There are no satellite galaxies or equivalent perturbers present at this
epoch. In subsequent evolution, the warp amplitude stays stable over 4 Gyr as
the tilt angle of the dark halo slowly declines.

###  3.2 A Warped Disk

In Figure 4 we show the stellar disk of Halo 533060 at four representative
snapshots. We plot stars in a cylindrical projection, where $R$ is defined to
be positive (negative) where the azimuthal angle is $\pm 90^{\circ}$ of the
maximum (minimum) vertical height of the disk. At $t_{\text{lookback}}=6\text{
Gyr}$, there is no discernible azimuthal asymmtery of the disk. However,
shortly after the tilted halo settles at $t_{\text{lookback}}=4\text{ Gyr}$,
we see a warp beginning to develop in the disk. In subsequent snapshots, a
strong warp persists in the disk. We fit the warp as a power-law in radius and
sinusoid in azimuth, plotted in blue lines.

In Figure 5, we show the time evolution of the amplitude of the warp alongside
the time evolution of the tilted dark halo. We define the warp amplitude to be
that of the sinusoid measured at twice the scale radius of the disk, which
grows with time (see, e.g., Frankel et al., 2019, for evidence of such
“inside-out” growth in the Galaxy). The warp amplitude is zero when the halo
is roughly spherical, as shown in the grey shaded region. Once the dark halo
settles into an oblate shape with a $50^{\circ}$ tilt, the warp amplitude
sharply increases up to $8\text{ kpc}$ at an approximate rate of $4\text{
kpc}/\text{Gyr}$, then slowly declines to $4\text{ kpc}$ as the tilt angle
decreases.

In a companion study, Han et al. (in prep) use idealized simulations to show
that a tilted dark halo can induce a warp in the galactic disk within a Gyr.
Furthermore, there are no massive satellites around Halo 533060 at the time of
the onset of the warped disk. All of these pieces of evidence indicate that
the warped disk of Halo 533060 is driven by its tilted dark halo.

##  4 Summary and Discussion

In this Letter, we have analyzed the “tilt angle”—a misalignment of the inner
($r<50\text{ kpc}$) dark halo and the stellar disk—in the Illustris TNG50
simulation. We find an abundance of Milky Way-like halos that are
significantly tilted at the present day, and identify an isolated halo that
allows us to study the secular evolution of a merger-induced tilted dark halo.
We find that the tilt angle decreases over time, likely due to dynamical
friction and phase mixing of the merger debris. The timescale of this decay is
long: the tilt angle declines from $50^{\circ}$ to $20^{\circ}$ over 5 Gyr.
Furthermore, the stellar halo is a good tracer of the underlying dark halo
tilt angle over all epochs. Lastly, we find a compelling relationship between
the tilted dark halo and a persistent warp in the galactic disk. The warp
amplitude increases sharply after the tilt angle reaches its maximum, and
subsequently follows the gradual decline of the tilt angle over $\sim 4\text{
Gyr}$.

The shape and configuration of the dark halo is primarily set by gravity, and
is affected by baryonic feedback physics (e.g., Debattista et al., 2008; Duffy
et al., 2010). TNG50 implements a wide range of gravitational and baryonic
physical processes including stellar feedback, and reproduces several aspects
of observed galaxies (Pillepich et al., 2019). Thus, following our
demonstration of the diversity of tilted dark halos in TNG50, we believe that
real galactic halos must also exhibit such diversity.

The origin of the warped disks in galaxies including the Milky Way is a
longstanding mystery (Kerr, 1957; Burke, 1957). Bosma (1991) find that at
least half of spiral galaxies are warped, which suggests that the warping
mechanism must be universal and long-lasting. Semczuk et al. (2020) have shown
in TNG100 that 35% of the warped disks are caused by satellite encounters
(half of which lead to a merger). These satellite-driven warps are transient,
surviving on average about 1 Gyr. In this study, we have shown that a tilted
dark halo can sustain a long-lasting warp ($>$ 4 Gyr), and that half of the
Milky Way analogs have tilt angles greater than $10^{\circ}$. For the Milky
Way, Han et al. (in prep.) demonstrate the same process in an idealized
simulation. Using a tilted, triaxial dark halo component based on the observed
stellar halo—i.e., adopting a “minimal-tuning” model for the dark halo—they
calculate the orbits of disk particles. Remarkably, the disk develops a stable
warp within 1 Gyr at precisely the observed amplitude, direction, and shape of
the Galactic disk warp and flare (Grabelsky et al., 1987; Bosma, 1991; Levine
et al., 2006). Thus, the results from TNG50 and the isolated simulation make a
compelling case for tilted dark halos being the primary driver of warped
galaxies in the observed universe.

## References

  * Allgood et al. (2006) Allgood, B., Flores, R. A., Primack, J. R., et al. 2006, MNRAS, 367, 1781, doi: 10.1111/j.1365-2966.2006.10094.x
  * Bailin & Steinmetz (2004) Bailin, J., & Steinmetz, M. 2004, ApJ, 616, 27, doi: 10.1086/424912
  * Belokurov et al. (2018) Belokurov, V., Erkal, D., Evans, N. W., Koposov, S. E., & Deason, A. J. 2018, MNRAS, 478, 611, doi: 10.1093/mnras/sty982
  * Blumenthal et al. (1984) Blumenthal, G. R., Faber, S. M., Primack, J. R., & Rees, M. J. 1984, Nature, 311, 517, doi: 10.1038/311517a0
  * Bosma (1991) Bosma, A. 1991, Warped and Flaring HI Disks (Cambridge University Press), 181
  * Bullock & Johnston (2005) Bullock, J. S., & Johnston, K. V. 2005, ApJ, 635, 931, doi: 10.1086/497422
  * Burke (1957) Burke, B. F. 1957, AJ, 62, 90, doi: 10.1086/107463
  * Conroy et al. (2019) Conroy, C., Bonaca, A., Cargile, P., et al. 2019, ApJ, 883, 107, doi: 10.3847/1538-4357/ab38b8
  * Cooper et al. (2010) Cooper, A. P., Cole, S., Frenk, C. S., et al. 2010, MNRAS, 406, 744, doi: 10.1111/j.1365-2966.2010.16740.x
  * Davis et al. (1985) Davis, M., Efstathiou, G., Frenk, C. S., & White, S. D. M. 1985, ApJ, 292, 371, doi: 10.1086/163168
  * Debattista et al. (2008) Debattista, V. P., Moore, B., Quinn, T., et al. 2008, ApJ, 681, 1076, doi: 10.1086/587977
  * Dubinski & Carlberg (1991) Dubinski, J., & Carlberg, R. G. 1991, ApJ, 378, 496, doi: 10.1086/170451
  * Duffy et al. (2010) Duffy, A. R., Schaye, J., Kay, S. T., et al. 2010, MNRAS, 405, 2161, doi: 10.1111/j.1365-2966.2010.16613.x
  * Emami et al. (2021) Emami, R., Genel, S., Hernquist, L., et al. 2021, ApJ, 913, 36, doi: 10.3847/1538-4357/abf147
  * Frankel et al. (2019) Frankel, N., Sanders, J., Rix, H.-W., Ting, Y.-S., & Ness, M. 2019, ApJ, 884, 99, doi: 10.3847/1538-4357/ab4254
  * Frenk et al. (1988) Frenk, C. S., White, S. D. M., Davis, M., & Efstathiou, G. 1988, ApJ, 327, 507, doi: 10.1086/166213
  * Frenk et al. (1985) Frenk, C. S., White, S. D. M., Efstathiou, G., & Davis, M. 1985, Nature, 317, 595, doi: 10.1038/317595a0
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Gnedin et al. (2004) Gnedin, O. Y., Kravtsov, A. V., Klypin, A. A., & Nagai, D. 2004, ApJ, 616, 16, doi: 10.1086/424914
  * Grabelsky et al. (1987) Grabelsky, D. A., Cohen, R. S., Bronfman, L., Thaddeus, P., & May, J. 1987, ApJ, 315, 122, doi: 10.1086/165118
  * Grand et al. (2017) Grand, R. J. J., Gómez, F. A., Marinacci, F., et al. 2017, MNRAS, 467, 179, doi: 10.1093/mnras/stx071
  * Gustafsson et al. (2006) Gustafsson, M., Fairbairn, M., & Sommer-Larsen, J. 2006, Phys. Rev. D, 74, 123522, doi: 10.1103/PhysRevD.74.123522
  * Han et al. (2022a) Han, J. J., Conroy, C., Johnson, B. D., et al. 2022a, AJ, 164, 249, doi: 10.3847/1538-3881/ac97e9
  * Han et al. (2022b) Han, J. J., Naidu, R. P., Conroy, C., et al. 2022b, ApJ, 934, 14, doi: 10.3847/1538-4357/ac795f
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85, doi: 10.1038/s41586-018-0625-x
  * Iorio & Belokurov (2019) Iorio, G., & Belokurov, V. 2019, MNRAS, 482, 3868, doi: 10.1093/mnras/sty2806
  * Kazantzidis et al. (2004) Kazantzidis, S., Kravtsov, A. V., Zentner, A. R., et al. 2004, ApJ, 611, L73, doi: 10.1086/423992
  * Kerr (1957) Kerr, F. J. 1957, AJ, 62, 93, doi: 10.1086/107466
  * Levine et al. (2006) Levine, E. S., Blitz, L., & Heiles, C. 2006, ApJ, 643, 881, doi: 10.1086/503091
  * Marinacci et al. (2018) Marinacci, F., Vogelsberger, M., Pakmor, R., et al. 2018, MNRAS, 480, 5113, doi: 10.1093/mnras/sty2206
  * Moore et al. (2004) Moore, B., Kazantzidis, S., Diemand, J., & Stadel, J. 2004, MNRAS, 354, 522, doi: 10.1111/j.1365-2966.2004.08211.x
  * Naiman et al. (2018) Naiman, J. P., Pillepich, A., Springel, V., et al. 2018, MNRAS, 477, 1206, doi: 10.1093/mnras/sty618
  * Navarro et al. (1997) Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493, doi: 10.1086/304888
  * Nelson et al. (2018) Nelson, D., Pillepich, A., Springel, V., et al. 2018, MNRAS, 475, 624, doi: 10.1093/mnras/stx3040
  * Nelson et al. (2019a) Nelson, D., Springel, V., Pillepich, A., et al. 2019a, Computational Astrophysics and Cosmology, 6, 2, doi: 10.1186/s40668-019-0028-x
  * Nelson et al. (2019b) Nelson, D., Pillepich, A., Springel, V., et al. 2019b, MNRAS, 490, 3234, doi: 10.1093/mnras/stz2306
  * Orkney et al. (2023) Orkney, M. D. A., Laporte, C. F. P., Grand, R. J. J., et al. 2023, arXiv e-prints, arXiv:2303.02147, doi: 10.48550/arXiv.2303.02147
  * Perryman et al. (2014) Perryman, M., Spergel, D. N., & Lindegren, L. 2014, ApJ, 789, 166, doi: 10.1088/0004-637X/789/2/166
  * Pillepich et al. (2018a) Pillepich, A., Springel, V., Nelson, D., et al. 2018a, MNRAS, 473, 4077, doi: 10.1093/mnras/stx2656
  * Pillepich et al. (2018b) Pillepich, A., Nelson, D., Hernquist, L., et al. 2018b, MNRAS, 475, 648, doi: 10.1093/mnras/stx3112
  * Pillepich et al. (2019) Pillepich, A., Nelson, D., Springel, V., et al. 2019, MNRAS, 490, 3196, doi: 10.1093/mnras/stz2338
  * Pillepich et al. (2023) Pillepich, A., Sotillo-Ramos, D., Ramesh, R., et al. 2023, arXiv e-prints, arXiv:2303.16217, doi: 10.48550/arXiv.2303.16217
  * Planck Collaboration et al. (2016) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13, doi: 10.1051/0004-6361/201525830
  * Rubin et al. (1985) Rubin, V. C., Burstein, D., Ford, W. K., J., & Thonnard, N. 1985, ApJ, 289, 81, doi: 10.1086/162866
  * Schaye et al. (2015) Schaye, J., Crain, R. A., Bower, R. G., et al. 2015, MNRAS, 446, 521, doi: 10.1093/mnras/stu2058
  * Schneider et al. (2012) Schneider, M. D., Frenk, C. S., & Cole, S. 2012, J. Cosmology Astropart. Phys, 2012, 030, doi: 10.1088/1475-7516/2012/05/030
  * Semczuk et al. (2020) Semczuk, M., Łokas, E. L., D’Onghia, E., et al. 2020, MNRAS, 498, 3535, doi: 10.1093/mnras/staa2609
  * Shao et al. (2021) Shao, S., Cautun, M., Deason, A., & Frenk, C. S. 2021, MNRAS, 504, 6033, doi: 10.1093/mnras/staa3883
  * Springel (2010) Springel, V. 2010, MNRAS, 401, 791, doi: 10.1111/j.1365-2966.2009.15715.x
  * Springel et al. (2018) Springel, V., Pakmor, R., Pillepich, A., et al. 2018, MNRAS, 475, 676, doi: 10.1093/mnras/stx3304
  * Vera-Ciro et al. (2011) Vera-Ciro, C. A., Sales, L. V., Helmi, A., et al. 2011, MNRAS, 416, 1377, doi: 10.1111/j.1365-2966.2011.19134.x
  * Weinberger et al. (2020) Weinberger, R., Springel, V., & Pakmor, R. 2020, ApJS, 248, 32, doi: 10.3847/1538-4365/ab908c
  * Weinberger et al. (2017) Weinberger, R., Springel, V., Hernquist, L., et al. 2017, MNRAS, 465, 3291, doi: 10.1093/mnras/stw2944
  * White & Rees (1978) White, S. D. M., & Rees, M. J. 1978, MNRAS, 183, 341, doi: 10.1093/mnras/183.3.341


# Double Inozemtsev Limits of the Quantum DELL System

Alexander Gorsky
Institute for Information Transmission Problems,
Russian Academy of Sciences,
Moscow, Russia
Email<EMAIL_ADDRESS>
http://iitp.ru/en/users/3804.htm ,  Peter Koroteev  Department of Mathematics,
University of California, Berkeley, CA 94720, USA and Rutgers University,
Piscataway, NJ, 08854, USA
Email<EMAIL_ADDRESS>
https://math.berkeley.edu/$\sim$pkoroteev ,  Olesya Koroteeva
School of Physics and Astronomy,
University of Minnesota
Minneapolis MN 55455
United States of America
Email<EMAIL_ADDRESS>
https://www.physics.umn.edu/people/koroteeva.html and  Shamil Shakirov
University of Geneva,
Department of Mathematics,
1211 Geneva 4, Switzerland
Email<EMAIL_ADDRESS>
###### Abstract.

In this letter we study various Inozemtsev-type limits of the quantum double
elliptic (DELL) system when both elliptic parameters are sent to zero at
different rates, while the coupling constant is sent to infinity, such that a
certain combination of the three parameters is kept fixed. We find a regime in
which such double Inozemtsev limit of DELL produces the elliptic Ruijsenaars-
Schneider (eRS) Hamiltonians albeit in an unconventional normalization. We
discuss other double scaling limits and anisotropic scaling of coordinates and
momenta. In addition we provide a formal expression for the eigenvalues of the
eRS Hamiltonians solely in terms of their eigenfunctions.

##  1\. The DELL System

The double elliptic model (DELL) is a remarkably sophisticated integrable
system which has been a point of interest of many researchers during the last
couple of decades. Since the advent of the Seiberg-Witten solution
[Seiberg:1994rs, Seiberg:1994aj] of the low energy sector of $\mathcal{N}=2$
supersymmetric gauge theories in four dimensions and its connection to
classical integrable systems [Gorsky:1993dq, Gorsky:1995zq, Martinec:1995by,
Donagi:1995cf] it became clear that there must exist integrable systems which
are dual to five and six dimensional gauge theories compactified on a circle
and on a torus respectively. These generalization have been later studied in
the literature (see [Koroteev:2019tz] for review).

The classical DELL model was introduced in [Hollowood:2003cv, Braden:2001yc,
Braden:2003gv] using six dimensional gauge theories with eight supercharges
compactified on a torus. It was shown how to construct the spectral curve, or,
equivalently, the Seiberg-Witten curve of the dual 6d theory, from M-theory.
The two elliptic parameters of DELL, $p$ and $w$ are related to the gauge
coupling of the 6d theory and to the elliptic modulus of the compactification
torus respectively.

The quantization is an art as well as a science and thus it requires novel
ideas. Nekrasov and Shatashvili [Nekrasov:2009rc] suggested a physics approach
which utilizes the duality between integrable systems and gauge theories with
eight supercharges. The key idea is to deform the gauge theory by the Omega-
background along one of the complex directions of the Euclidian
$\mathbb{R}^{4}$ worldvolume of the theory. The corresponding equivariant
parameter becomes the Planck’s constant of the dual quantum integrable system.

In [Bullimore:2015fr] it was demonstrated, and later proved by Nekrasov (in
the case of the four-dimensional $U(2)$ theory), see [Nekrasov:2017ab,
Nekrasov:2017aa], how to find a formal spectrum of quantum elliptic integrable
systems of Calogero or Ruijsenaars type. The wave functions of the quantum
Hamiltonians are the supersymmetric partition functions in the presence of
codimension-two defects of the dual 4d and 5d theories with adjoint matter
respectively. The corresponding eigenfunctions are therefore represented by
vacuum expectation values of local chiral observables (in 4d) and Wilson lines
(in 5d). Later in [Koroteev:2019tz] this approach was extended to 6d theories.

###  1.1. Overview of Quantum DELL System

First we review the basics of the quantum double elliptic system which was
discovered by two of the authors [Koroteev:2019tz] and further developed in
[Grekov:2021zqq, Grekov:2020jpk]. (see also [Braden:1999aj, Mironov:1999vi,
Aminov:2013asa, Aminov:2014wra, Aminov:2016ruk, Aminov:2017dud, Fock:1999ae,
Braden:2001yc] for different approach to double elliptic models).

The DELL Hamiltonians for $N$ particles read

(1.1) $\mathcal{H}_{a}=\mathcal{O}_{0}^{-1}\mathcal{O}_{a}\,,\qquad
a=1,\ldots,N-1\,,$

where operators $\mathcal{O}_{0},\mathcal{O}_{1},\dots,\mathcal{O}_{N-1}$ are
Fourier modes of the following current

(1.2) $\mathcal{O}(z)\ =\ \sum\limits_{n\in{\mathbb{Z}}}\ \mathcal{O}_{n}\
z^{n}\ =\ \sum\limits_{n_{1},\ldots,n_{N}=-\infty}^{\infty}\ (-z)^{\sum
n_{i}}\ w^{\sum\frac{n_{i}(n_{i}-1)}{2}}\
\prod\limits_{i<j}\theta\left(t^{n_{i}-n_{j}}\frac{x_{i}}{x_{j}}\Big{|}p\right)\
p_{1}^{n_{1}}\ldots p_{N}^{n_{N}}\,.$

In the above formula $\theta(x|p)$ is the odd theta function111The use of the
odd theta function is more preferable to study the Inosemtsev limit; we
changed our conventions from [Koroteev:2019tz].

(1.3)
$\theta(x|p)=(x^{\frac{1}{2}}-x^{-\frac{1}{2}})\prod_{i=1}^{\infty}(1-xp^{i})(1-x^{-1}p^{i})\,,$

$t$ is the exponentiated coupling constant, $z$ is an auxiliary counting
parameter, and the canonically conjugate position and momentum operators
obeying canonical q-commutation relation
$x_{i}p_{j}=q^{\delta_{ij}}p_{j}x_{j}$ which act on functions of positions as
follows

(1.4) $x_{i}f(x_{1},\ldots,x_{N})=x_{i}f(x_{1},\ldots,x_{N}),\qquad
p_{i}f(x_{1},\ldots,x_{N})=f(x_{1},\ldots,qx_{i},\ldots,x_{N})\,.$

The eigenvalue problem, which as of this writing, is a mathematical
conjecture, states that a properly normalized equivariant elliptic genus of
the affine Laumon space in the Nekrasov-Shatashvili limit [Nekrasov:2009rc] is
the eigenfunction of the quantum DELL Hamiltonians

(1.5)
$\widehat{\mathcal{H}}_{n}{Z}^{6d/4d}_{\text{inst}}(w,p,\textbf{x})=\lambda_{n}(\textbf{a},w,p){Z}^{6d/4d}_{\text{inst}}(w,p,\textbf{x})\,.$

We refer the reader to [Koroteev:2019tz] for more details and references. In
this paper we discuss double scaling limits of DELL of Inozemtsev type
[inozemtsev1989], in which the coupling $t\to\infty$ while the elliptic
parameters $p$ and $w$ go to zero in the presence of additional scaling of the
coordinates and momenta.

###  1.2. DELL-RS-Calogero Hierarchy

The DELL system lives on top of the hierarchy of integrable many-body systems
and all other known models (without spin degrees of freedom, we shall comment
on spin-DELL later) – Calogero-Moser-Sutherland (CMS) and Ruijsenaars-
Schneider (RS) systems can be obtained by decoupling certain parameters in
DELL Hamiltonians, see Fig. 1222In math literature (part of) this diagram is
sometimes referred to as Etingof diamond. .

Figure 1. The ITEP table [Mironov:2000if] of integrable many body systems
according to their periodicity properties in coordinates $q$ (columns) and
momenta $p$ (rows) together with their geometric interpretations.

The Calogero and Ruijsenaars families (first and second rows of the table
respectively) have a well-established geometric interpretation.

###  1.3. Spectrum of the Elliptic RS Model

In this subsection we would like to prove one of the conjectures of
[Koroteev:2019tz].

######  Theorem 1.1.

Let $\textbf{x}=(x_{1},\dots x_{N})$ be the position vector of the eRS model
and $\mathcal{Z}^{{\text{RS}}}(\textbf{a},\textbf{x})=\lim\limits_{w\to
0}\mathcal{Z}^{6d/4d}_{\text{inst}}(w,p,\textbf{x})$ is its wavefunction. Then
the following equality holds

(1.6)
$\mathcal{H}_{k}\mathcal{Z}^{{\text{RS}}}(\textbf{a},\textbf{x})=\lambda_{k}(\textbf{a})\mathcal{Z}^{{\text{RS}}}(\textbf{a},\textbf{x})\,,\qquad
k=1,\dots,N-1\,.$

where the eigenvalues read

(1.7)
$\lambda_{k}(\textbf{a})=\prod\limits_{n=0}^{k-1}\frac{\theta(t^{N-n})}{\theta(t^{n+1})}\cdot\frac{\mathcal{Z}^{{\text{RS}}}(\textbf{a},t^{\vec{\rho}}q^{\vec{\omega_{k}}})}{\mathcal{Z}^{{\text{RS}}}(\textbf{a},t^{\vec{\rho}})}\,,\qquad
k=1,\dots,N-1$

where $\vec{\omega_{k}}$ is the $k$-th fundamental weight of representation of
$SU(N)$ and $\vec{\rho}=\big{(}(N-1)/2,(N-3)/2,\dots,(3-N)/2,(1-N)/2\big{)}$
is the $SU(N)$ Weyl vector.

###### Proof.

In the above system of difference equations, the Hamiltonians have the form

(1.8)
$\mathcal{H}_{k}=\sum\limits_{\begin{subarray}{c}\mathcal{I}\subset\\{1\ldots
N\\}\\\ |\mathcal{I}|=k\end{subarray}}\
\prod\limits_{\begin{subarray}{c}j\notin\mathcal{I}\\\
i\in\mathcal{I}\end{subarray}}\dfrac{\theta(tx_{i}/x_{j})}{\theta(x_{i}/x_{j})}\
\prod\limits_{i\in I}p_{i}$

where $p_{i}$ are the shift operators as defined earlier. While for generic
$x_{i}$ this is a non-trivial system relating values of the eigenfunctions at
multiple different points, there exists a specific value of $x_{i}$ at which
all but one terms in the left hand side of each equation in the above system
vanish. This is the point $x_{i}=t^{\rho_{i}}=t^{(N+1)/2-i}$. Indeed it is
easy to see that, in the $k$-th equation of (1.6), the product of theta
functions at $x_{i}=t^{\rho_{i}}$ necessarily contains a factor of $\theta(1)$
because of the linear dependence of $i$ in $\rho_{i}$, unless
$I=\\{1,2,\ldots,k\\}$. Recall from (1.3) that $\theta(1)=0$, therefore the
$k$-th equation of the system at $x_{i}=t^{\rho_{i}}$ specializes to

(1.9)
$\prod\limits_{i=1}^{k}\prod\limits_{j=k+1}^{N}\dfrac{\theta(t^{j-i+1})}{\theta(t^{j-i})}\
\mathcal{Z}^{{\text{RS}}}(\textbf{a},t^{\vec{\rho}}q^{\vec{\omega_{k}}})=\lambda_{k}(\textbf{a})\mathcal{Z}^{{\text{RS}}}(\textbf{a},t^{\vec{\rho}})\,,\qquad
k=1,\dots,N-1\,.$

where
$\vec{\omega_{k}}=\big{(}\underbrace{1,\ldots,1}_{k},\underbrace{0,\ldots,0}_{N-k}\big{)}$.
Simplifying the product in the l.h.s. and dividing by the value of the
eigenfunction in the r.h.s., we obtain the desired expression for the
eigenvalues.

∎

##  2\. Double Inozemtsev Limits of DELL

The usual Inozemtsev limit [inozemtsev1989] describes a transition from
elliptic integrable systems – Calogero or Ruijsenaars (respectively the first
and the second rows of the ITEP table) – into Toda and q-Toda models
correspondingly. More precisely, trigonometric models (second column of the
table) are sent to open (q)Toda chains, while elliptic models (third column)
become affine (q)Toda chains after applying Inozemtsev (see [Gorsky:2019yqp]
for review).

Upon the Inozemtsev limit on first scales the coordinates of Calogero or
Ruijsenaars particles by a factor which is proportional to the order number of
each particle and proportional to the coupling constant. The the coupling
constant of the integrable system in question is sent to infinity while the
elliptic modulus is sent to zero provided that a certain combination thereof
remains finite (see below).

Since we are dealing with the double elliptic model which two elliptic
parameters $p$ and $w$ one may try to build a more complex limit which
involves scaling of the form $t\to\infty,\,p,w\to 0$ such that certain
combination of the above three parameters $F(t,p,w)$ remains constant. We
refer to such limits as double Inozemtsev limits.

More presicely the double Inozemtsev limit consists of the following rescaling

(2.1) $x_{a}\mapsto t^{-a}x_{a},\qquad p_{a}\mapsto t^{-a-1/2}p_{a},\qquad
p=t^{\alpha}\Lambda,\qquad w=t^{\beta}M\,,$

and taking the limit $t\to 0$. Here $\Lambda$ and $M$ are nonzero constants.
We study this limit for different values of $\alpha$ and $\beta$.

###  2.1. $\alpha=N$, $\beta>1$

In this case, due to the quadratic dependence of the Hamiltonians on $w$,
higher elliptic corrections vanish and the model becomes equivalent to the
elliptic Ruijsenaars-Schneider model. Thus its Inozemtsev limit is the quantum
affine q-Toda system [Gorsky:2019yqp]. Its first Hamiltonian reads

(2.2) $H^{\text{aff
q-Toda}}_{1}=\sum_{i=2}^{N}\left(1-\frac{x_{i-1}}{x_{i}}\right)p_{i}+\left(1-\Lambda\frac{x_{N}}{x_{1}}\right)p_{1}\,,$

Upon this limit $\mathcal{O}_{0}=1$.

###  2.2. $\alpha=N$, $\beta=1$

This is a new limit which at first glance yields a new elliptic model. The
normal ordered DELL operators $\mathcal{O}_{k}$ in this limit read as follows

(2.3)
$\widetilde{\mathcal{O}}_{k}=c_{N}:\sum_{i_{1}<\dots<i_{k}}\prod\limits_{a<b}\theta\left(M^{b-a+m_{b}-m_{a}}\frac{x_{a}}{x_{b}}\frac{p_{b}}{p_{a}}\Big{|}\widetilde{p}\right):\
p_{i_{1}}\cdots p_{i_{k}}\,,$

where $\widetilde{p}=\Lambda M^{N}$ and $m_{a}=\delta_{a\in I}$, and $c_{N}$
is a certain constant. However, the above Hamiltonians can undergo the
following chain of transformations. First, let $x_{a}\mapsto M^{-a}x_{a}$,
then we perform the $SL(2;\mathbb{Z})$ $T$-transformation which in the
multiplicative form reads

(2.4) $x_{a}\mapsto x_{a}p_{a},\,\qquad\qquad p_{a}\mapsto p_{a}\,.$

This transformation is none other than conjugation of the operators by the
Gaussian term $\exp(\partial^{2}/2)$ (recall that the shift operator is
$p=q^{\partial}$). After that (2.3) becomes

(2.5)
$\widetilde{\mathcal{O}}_{k}=c_{N}\sum_{i_{1}<\dots<i_{k}}\prod\limits_{a<b}\theta\left(M^{m_{b}-m_{a}}\frac{x_{a}}{x_{b}}\Big{|}\widetilde{p}\right)p_{i_{1}}\cdots
p_{i_{k}}\,.$

According to [Koroteev:2019tz] the above Hamiltonians lead to the elliptic
Ruijsenaars-Schneider model
$\mathcal{H}^{\text{eRS}}_{k}=\widetilde{\mathcal{O}}_{0}^{-1}\widetilde{\mathcal{O}}_{k}$
with coupling constant $M$ and elliptic parameter $\widetilde{p}$.

The above limit can be illustrated using the following brane description (see
Fig. 2). Here vertically we have NS5 branes and horizontally we find D5
branes. The starting point is the little string theory with periodic NS5
direction which can be then scaled to the field theory limit by sending the
vacuum expectation of the dilaton to infinity.

Figure 2. Brane description of the 6d theory whose infrared regime is
described by the DELL model with coupling $t$ (left) and the 5d theory whose
Seiberg-Witten solution gives rise to the eRS model with coupling $M$ (right).

The left figure describes 6d $\mathcal{N}=(1,0)^{*}$ theory on
$\mathbb{R}^{4}\times S^{1}_{R}\times S^{1}_{R^{\prime}}$ which can be viewed
as a 5d $\mathcal{N}=1^{*}$ theory on $\mathbb{R}^{4}\times
S^{1}_{R^{\prime}}$ whose Yang-Mills coupling depends on the compactification
radius $g_{\text{YM}}^{-2}=1/R$ along the sixth direction. The complexified 5d
gauge coupling is equal to $p\sim e^{-4\pi R^{\prime}/R}$ while the elliptic
modulus of the compactification torus is $w$. The $\mathcal{N}=2^{*}$
supersymmetry is softly broken to $\mathcal{N}=1^{*}$ by introducing the mass
parameter to the adjoint hypermultiplet $t=e^{R^{\prime}m}$.

In the Inozemtsev limit $m\to\infty$ and $R\to 0$ such that $pt^{N}=\Lambda$
or, equivalently, $Nm\sim 1/R$. This results in collapsing of one of the
periodic directions in the 6d description. Then we apply a T-duality
transformation along the compact NS5 brane direction which results in the NS5
brane with infinite radius as is shown on the right side of Fig. 2.
Interestingly, during these chain of limits and transformations the former
elliptic modulus parameter $w$ scalers to the new mass parameter $M$ while the
5d Yang-Mills gauge coupling of the limiting theory becomes
$\widetilde{p}=\Lambda M^{N}$.

###  2.3. Fractional Limits

Let us consider a different scaling

(2.6) $x_{a}\mapsto t^{-\frac{a}{\gamma}}x_{a},\qquad p_{a}\mapsto
t^{-\frac{a}{\gamma}-\frac{1}{2\gamma}}p_{a},\qquad p=t^{\alpha}\Lambda,\qquad
w=t^{\beta}M\,.$

Note that this scaling is not equivalent to the previous family via
redefinition $u=t^{1/2}$ since the dependence on $w$ of DELL Hamiltonians is
nonlinear. This scaling, as we shall see below, leads to q-Toda-like systems,
albeit with certain restrictions.

Consider an element in the product inside theta-function

(2.7) $\theta(z|p)\supset\prod_{k=0}^{\infty}(1-p^{k}z)$

where in our case $z=t^{n_{i}-n_{j}}\frac{x_{i}}{x_{j}}$. After the rescaling
the power of $t$ in the $k$-th term in the product will become

(2.8) $P_{ij}=\alpha k+n_{i}-n_{j}+\frac{j-i}{\gamma}$

Our goal is to find the most singular terms in each theta function ratio thus
we need to make sure that the above exponent is negative (or zero). This
singular contributions will be compensated by $w$ in the expression for the
DELL Hamiltonian. Consider the $w$-terms in more detail.

  * •

$w^{0}$: This corresponds to vector $n_{i}=0$ or $n_{i}=1$ since the function
$n(n-1)/2$ vanishes at $0$ and $1$ and otherwise is positively defined at
integer $n$.

  * •

$w^{1}$: Notice that $n(n-1)/2=1$ for $n=-1$ and $n=2$. However, we need to
satisfy the constraint $\sum n_{i}=k$ for $\mathcal{O}_{k}$. Consider $k=1$
for now. Thus we need $\sum_{i}n_{i}(n_{i}-1)/2=1$ as well. One can see that
this can be achieved by having vector $\textbf{n}=(1,-1,1,0,\dots,0)$ and its
permutations. Therefore the values of $n_{i}-n_{j}$ in $P_{ij}$ range over
$-2,-1,0,1,2$. In order to obtain singular terms we need the whole expression
to be negative so $\gamma$ can be chosen in such a way that
$n_{i}-n_{j}+\frac{j-i}{\gamma}\leq 0$ or

(2.9) $\gamma\geq\frac{j-i}{2}$

In other words $\gamma=\frac{j-i}{2}$ defines the range of the interaction
between the particles (the interaction becomes infinitely weak beyond this
range). Once $\gamma$ is set we can adjust $\alpha$. For instance,
$\gamma=2,\alpha=3/2,\beta=3/2$ will provide one term proportional to $M$ in
the expression for $\mathcal{O}_{1}$.

The above analysis leads to the following operators for two and three
particles.

##### Two Particles

Consider $\alpha=\beta=\frac{3}{2}$ in (2.6), then

(2.10) $\displaystyle\mathcal{O}_{0}$ $\displaystyle=1-\Lambda
M\frac{x_{2}}{x_{1}}\frac{p_{1}}{p_{2}}\,,$ (2.11)
$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\left(1-\Lambda\frac{x_{2}}{x_{1}}+\Lambda^{3}M^{2}\frac{x_{2}^{2}}{x_{1}^{2}}\frac{p_{1}}{p_{2}}\right)p_{1}-\frac{x_{1}}{x_{2}}p_{2}\,,$

##### Three Particles

Consider $\alpha=\beta=2$, then

(2.12) $\displaystyle\mathcal{O}_{0}$ $\displaystyle=1-\Lambda
M\frac{x_{3}}{x_{1}}\frac{p_{1}}{p_{3}}\,,$ (2.13)
$\displaystyle\mathcal{O}_{1}$
$\displaystyle=\left(1-\Lambda\frac{x_{3}}{x_{1}}\right)p_{1}-\frac{x_{1}}{x_{2}}p_{2}+\Lambda^{2}M\frac{x_{3}^{2}}{x_{1}x_{2}}\frac{p_{1}p_{2}}{p_{3}}\,,$
(2.14) $\displaystyle\mathcal{O}_{2}$
$\displaystyle=\left(p_{2}-\Lambda\frac{x_{3}}{x_{1}}p_{2}-\frac{x_{2}}{x_{3}}p_{3}\right)p_{1}+\Lambda^{2}M\frac{x_{2}x_{3}}{x_{1}^{2}}p_{1}^{2}$

However, for higher number of particles this limit produces Hamiltonians which
only involve interaction with a fewer number of particles and completely
ignore interactions with the others. Notice also that the above Hamiltonians
are deformation of finite q-Toda chains (not affine).

Therefore at this moment we cannot find a novel limit of DELL with $N>3$
particles which would present a novel integrable system which lies ‘in
between’ eRS and DELL in the hierarchy. Most likely such model does not exist.

One may still be able to find a different scaling other than (2.6) of the form

(2.15) $x_{a}\mapsto t^{f(a)}x_{a},\qquad p_{a}\mapsto t^{g(a)}p_{a},\qquad
p=t^{\alpha}\Lambda,\qquad w=t^{\beta}M\,,$

where $f(a)$ and $g(a)$ are certain functions, in order to capture more terms
from the DELL Hamiltonians in the limit. We encourage the reader to try.

##  3\. DELL System in the Dual Frame

In this section we would like to comment on the recent results of
[Mironov:2021tx]. In particular, the authors conjecture the existence of a pq-
self-dual DELL system, which, according to the reasons listed below, does not
exist without invoking spin degrees of freedom (see Fig 3 for the toric
diagram).

Figure 3. Toric diagram for affine $\widehat{A}_{1}$ 6d quiver with two $U(3)$
nodes whose Seiberg-Witten geometry is described by spin-$\frac{1}{2}$ DELL
system with four particles.

The pq-duality is manifested differently in physical setting in various
dimensions. In 5d it is realized as the S-duality of the pq-brane web, while
in 3d it is manifested by the 3d mirror symmetry. Since the description of the
DELL model uses periodic pq-webs the pq-duality should also be manifested via
the S-duality (90-degree rotation of the brane web). For instance, for the
theory in Fig. 3 the pq-dual quiver is the following

which yields spin-$\frac{3}{2}$ DELL model with two particles.

Every Hamiltonian of the Calogero-Ruijsennars-DELL family has two
geometric/physics interpretations – the so-called ‘electric frame’ (or
$a$-frame) and the ‘magnetic frame’ (or $x$-frame). The two frames are related
via the pq-duality which interchanges $a$ and $x$ variables together with the
involution on the coupling constant parameter $\hbar\mapsto\hbar^{-1}$.

In addition to the pq-duality described above, in both electric and magnetic
frames (see Fig. 4) there are Fourier transforms which interchange momenta and
coordinates in the Hamiltonians.

Figure 4. Network of degeneration limits from the DELL models (up to the tRS
models) whose Hamiltonians act on $x$-variables (DELLx) and on $a$-variables
(DELLa). The two families (two dual ITEP tables, see Fig. 1) are related via
3d mirror symmetry which interchanges parameters $x$ and $a$.

Therefore the only self-dual family of double elliptic integrable models are
spin-DELL systems whit spin-$\frac{N-1}{2}$ and $N$ particles, since their
toric diagrams are pq-invariant. These models describe Seiberg-Witten geometry
of affine $\widehat{A}_{N}$ quiver gauge theory with $U(N)$ gauge groups by
means of the following generating function

$\displaystyle\widehat{\mathcal{O^{S}}}(z)\ $ $\displaystyle=\
\sum\limits_{n\in{\mathbb{Z}}}\ \widehat{\mathcal{O^{S}}}_{n}\ z^{n}$ (3.1)
$\displaystyle=\sum\limits_{a=1}^{N}\sum\limits_{\textbf{n}_{a}\in\mathbb{Z}^{N}}\
(-z)^{\sum n_{a,i}}\ w^{\sum\frac{n_{a,i}(n_{a,i}-1)}{2}}\
\prod\limits_{a=1}^{N}\prod\limits_{i<j}\theta\left(t^{n_{a,i}-n_{a,j}}\frac{x_{a,i}}{x_{a,j}}\Big{|}p\right)\
p_{a,1}^{n_{a,1}}\ldots p_{a,N}^{n_{a,N}}\,.$

We expect to study the properties of spin-DELL system in the near future.

Below we shall describe electric and magnetic frames for Ruijsenaars models in
more details.

###  3.1. 3d Mirror Symmetry and pq-Duality in tRS models

First, we remind the reader about the solutions of quantum trigonometric
Ruijsenaars-Schneider model in electric and magnetic frames using quantum
geometry of quiver varieties. At generic values of the equivariant parameters
${\bf a}$ the eigenfunctions are series. As we shall review below at certain
resonance values (3.6) these series truncate to Macdonald polynomials which
are most commonly discussed in the representation theory literature. The Pieri
rule and its elliptic generalization, which were discussed in
[Mironov:2021tx], will therefore follow from the tRS eigenvalue problem in the
electric frame (3.13), after the resonance conditions have been imposed.

####  3.1.1. Magnetic Frame

In the magnetic frame the tRS operators act on the Kähler parameters
$\zeta_{i}$ of the quiver variety (in physics language on the Fayet-Iliopoulos
couplings, see [Bullimore:2015fr] for gauge theory interpretation of electric
and magnetic frames).

######  Definition 3.1.

The difference operators of trigonometric Ruijsenaars-Schneider model are
given by

(3.2)
$T_{r}(\bm{\zeta})=\sum_{\begin{subarray}{c}\mathcal{I}\subset\\{1,\dots,n\\}\\\
|\mathcal{I}|=r\end{subarray}}\prod_{\begin{subarray}{c}i\in\mathcal{I}\\\
j\notin\mathcal{I}\end{subarray}}\frac{\hbar\,\zeta_{i}-\zeta_{j}}{\zeta_{i}-\zeta_{j}}\prod\limits_{i\in\mathcal{I}}p_{k}\,,$

where $\bm{\zeta}=\\{\zeta_{1},\dots,\zeta_{n}\\}$, the shift operator
$p_{k}f(\zeta_{k})=f(q\zeta_{k})$.

The spectrum of the tRS operators is described geometrically by K-theory
vertex function for the cotangent bundle to the complete flag variety
$T^{*}\mathbb{F}l_{n}$ [Koroteev:2018a, Koroteev:2018cmp] and physically as
vortex partition functions of the $T[U(n)]$ gauge theory on
$\mathbb{R}^{2}\times S^{1}$ [Bullimore:2015fr]. We remind the reader the
definitions and theorems.

######  Theorem 3.2 ([Koroteev:2018cmp] Thm. 2.10).

Let $V^{(1)}_{\textbf{p}}$ be the coefficient for the vertex function for the
cotangent bundle to the complete flag variety $T^{*}\mathbb{F}l_{n}$. Define

(3.3)
$\mathsf{V}^{(1)}_{\textbf{p}}=\prod\limits_{i=1}^{n}\frac{\theta(\hbar^{i-n}\zeta_{i},q)}{\theta(a_{i}\zeta_{i},q)}\cdot
V^{(1)}_{\textbf{p}}\,,$

where $\theta(x,q)=(x,q)_{\infty}(qx^{-1},q)_{\infty}$ is basic theta-function
and

(3.4) $V^{(1)}_{\textbf{p}}(z)=\sum\limits_{d_{i,j}\in
C}\prod_{i=1}^{n-1}\left(t\frac{\zeta_{i}}{\zeta_{i+1}}\right)^{d_{i}}\prod\limits_{j,k=1}^{i}\frac{\left(q\frac{x_{i,j}}{x_{i,k}},q\right)_{d_{i,j}-d_{i,k}}}{\left(\hbar\frac{x_{i,j}}{x_{i,k}},q\right)_{d_{i,j}-d_{i,k}}}\cdot\prod_{j=1}^{i}\prod_{k=1}^{i+1}\frac{\left(\hbar\frac{x_{i+1,k}}{x_{i,j}},q\right)_{d_{i,j}-d_{i+1,k}}}{\left(q\frac{x_{i+1,k}}{x_{i,j}},q\right)_{d_{i,j}-d_{i+1,k}}}\,,$

are the coefficient functions for the vertex function for fixed point ${\bf
p}$. Then $\mathsf{V}_{\textbf{p}}$ are eigenfunctions for tRS difference
operators (3.2) for all fixed points p

(3.5)
$T_{r}(\bm{\zeta})\mathsf{V}^{(1)}_{\textbf{p}}=e_{r}(\mathbf{a})\mathsf{V}^{(1)}_{\textbf{p}}\,,\qquad
r=1,\dots,n\,,$

where $e_{r}$ is elementary symmetric polynomial of degree $r$ of
$a_{1},\dots,a_{n}$ .

The following statement describes truncation of the eigenfunctions
$\mathsf{V}^{(1)}_{\textbf{p}}$ from the above theorem. In the presence of
certain resonance conditions these series become Macdonald polynomials.

######  Proposition 3.3 ([Koroteev:2018cmp] Prop. 2.11).

Consider coefficient functions for K-theory of QM to $X_{n}$ (3.3) for all
fixed points of the maximal torus. Let $\lambda$ be a partition of $k$
elements of length $n$ and $\lambda_{1}\geq\dots\geq\lambda_{n}$. Let

(3.6)
$\frac{a_{i+1}}{a_{i}}=q^{\ell_{i}}\hbar\,,\quad\ell_{i}=\lambda_{i+1}-\lambda_{i}\,,\quad
i=1,\dots,n-1\,.$

Then there exists a fixed point q for which

(3.7) $\mathsf{V}_{\textbf{q}}=P_{\lambda}(\bm{\zeta};q,\hbar)\,.$

For instance, for $T^{*}\mathbb{P}^{1}$ the condition (3.6) reads
$a_{1}=aq^{\lambda_{1}}\hbar^{-1}\,,a_{2}=aq^{\lambda_{2}}$. Then we have the
following Macdonald polynomials for some simple tableaux.

(3.8) $\displaystyle\mathrm{P}_{{\tiny\yng(1)}}$
$\displaystyle=\zeta_{1}+\zeta_{2},$ (3.9)
$\displaystyle\mathrm{P}_{{\tiny\yng(1,1)}}$
$\displaystyle=\zeta_{1}^{2}+\zeta_{2}^{2}+\frac{(q+1)(\hbar-1)}{q\hbar-1}\zeta_{1}\zeta_{2},$
(3.10) $\displaystyle\mathrm{P}_{{\tiny\yng(1,1,1)}}$
$\displaystyle=\zeta_{1}^{3}+\zeta_{2}^{3}+\frac{\left(q^{2}+q+1\right)(\hbar-1)}{q^{2}\hbar-1}\zeta_{2}\zeta_{1}^{2}+\frac{\left(q^{2}+q+1\right)(\hbar-1)}{q^{2}\hbar-1}\zeta_{2}^{2}\zeta_{1}\,.$

####  3.1.2. Electric Frame

In the $a$-frame the tRS Hamiltonians read

(3.11)
$T_{r}(\textbf{a})=\sum_{\begin{subarray}{c}\mathcal{I}\subset\\{1,\dots,n\\}\\\
|\mathcal{I}|=r\end{subarray}}\prod_{\begin{subarray}{c}i\in\mathcal{I}\\\
j\notin\mathcal{I}\end{subarray}}\frac{t\,a_{i}-a_{j}}{a_{i}-a_{j}}\prod\limits_{i\in\mathcal{I}}p_{i}\,,$

where $\textbf{a}=\\{a_{1},\dots,a_{n}\\}$, the shift operator
$p_{i}f(a_{i})=f(qa_{i})$ and we denoted $t=\frac{q}{\hbar}$.

######  Theorem 3.4.

The following function constructed for the cotangent bundle to the complete
flag variety

(3.12)
$\mathrm{V}(\textbf{a},{\bm{\zeta}})=\frac{e^{\frac{\log\zeta_{n}\sum_{i=1}^{n-1}\log
a_{i}}{\log q}}}{2\pi
i}\int\limits_{C}\prod_{m=1}^{n-1}\prod_{i=1}^{m}\frac{ds_{m,i}}{s_{m,i}}E(s_{m,i})\,\,e^{-\frac{\log\zeta_{m}/\zeta_{m+1}\cdot\log
s_{m,i}}{\log
q}}\cdot\prod_{j=1}^{\textbf{v}_{m+1}}H_{m,m+1}\left(s_{m,i},s_{m+1,j}\right)\,,$

where $E(s_{m,i})$ and $H_{m,m+1}$ are certain rational functions and contour
$C$ is chosen in such a way that shifts of the contour $\textbf{s}\to q^{\pm
1}\textbf{s}$ do not encounter any poles, satisfies tRS difference relations

(3.13)
$T_{r}(\textbf{a})\mathrm{V}(\textbf{a},{\bm{\zeta}})=e_{r}({\bm{\zeta}})\mathrm{V}(\textbf{a},{\bm{\zeta}})\,,\qquad
r=1,\dots,n$

where function $e_{r}$ is $r$th elementary symmetric polynomial
$\zeta_{1},\dots,\zeta_{n}$.

The 3d mirror symmetry/pq-duality states that for the cotangent bundle to the
complete flag variety the eigenfunctions and eigenvalues are related to each
other via the interchange of the Kähler and equivariant parameters. Namely,
for a certain fixed point of the maximal torus p we have

(3.14) $V_{\textbf{p}}({\bf
a},{\bm{\zeta}},\hbar)=V_{\textbf{p}}({\bm{\zeta}},{\bf a},q/\hbar)\,,$

and the tRS operators (3.2) and (3.11) are related in the same way.

After imposing the resonance conditions (3.6) the tRS eigenvalue relations in
the electric frame (3.13) turns into the Pieri rules (upon applying a
conjugation to the tRS Hamiltonians) if we look at it from right to left.
Consider the relation for $r=1$

(3.15)
$\mathrm{P}_{{\tiny\yng(1)}}(q^{\lambda_{i}}\hbar^{n-i})\cdot\mathrm{P}_{\lambda}({\bm{\zeta}})=\sum_{i=1}^{n}\prod_{j\neq
i}\frac{q^{\lambda_{i}-\lambda_{j}+1}\hbar^{j-i-1}-1}{q^{\lambda_{i}-\lambda_{j}}\hbar^{j-i}-1}\mathrm{P}_{\lambda+{\tiny\yng(1)\,}_{i}}({\bm{\zeta}})\,,$

where $\lambda+{\tiny\yng(1)\,}_{i}$ denotes a diagram obtained from the
diagram $\lambda$ by adding one box in column $i$.

###  3.2. PQ-duality in eRS models

Analogously one can explain the pq-duality between RSa and
$\widehat{\text{eRS}}_{x}$ from Fig. 4. In [Mironov:2021tx] the former
eigenvalue problem was presented as an elliptic Pieri rule which can be
understood as truncation of the eigenvector of $H^{a}_{eRS}$ which
geometrically is the equivariant Euler characteristic of the affine Laumon
space. The elliptic Pieri rule reads

(3.16)
$\mathrm{E}_{{\tiny\yng(1)}}(p_{k})\cdot\mathrm{E}_{\lambda}(p_{k})=\sum_{i=1}^{l(\lambda)+1}\prod_{j\neq
i}\psi\left(\frac{q^{\lambda_{i}}\hbar^{i-1}}{q^{\lambda_{i}}\hbar^{j-1}}\right)\mathrm{E}_{\lambda+{\tiny\yng(1)\,}_{i}}(p_{k})\,,$

where $p_{k}=\sum_{i}z_{i}^{k}$ is the $k$-th power symmetric polynomial and
function

$\psi(x)=\frac{\theta(q\hbar
x|w)\theta(\hbar^{-1}x|w)}{\theta(x|w)\theta(qx|w)}$

in the limit $w\to 0$ reduces to the fraction in the right hand side of (3.15)
with an extra factor which is and artifact of different normalization of
Ruijsenaars Hamiltonians in [Mironov:2021tx].

Meanwhile the $\widehat{\text{eRS}}_{x}$ in [Mironov:2021tx] was referred to
as the eigenfunction of the DELL Hamiltonians in the elliptic-trigonometric
limit ($w\to 0$).

##  4\. DELL and Separated Variables

One more approach to the Inozemtsev limit for DELL is based on the separation
of variables – the procedure effective for the interacting many-body systems.
The system of $N$ interacting degrees of freedom with a simple phase-space
manifold gets mapped via canonical transformation into the system of $N$ non-
interacting degrees of freedom with complicated phase-space manifold. The
derivation of the separated variables and the corresponding canonical
transformation is not a simple issue and it was identified in the brane terms
[separation] as the representation of the integrable model associated with the
geometry involving several types of branes into a system of D0 branes on some
manifold via Fourier-Mukai transforms. More formally we map the initial phase
space of the $N$-body holomorphic system $M$ into the phase space identified
with Hilbert scheme of $N$ points on some two-dimensional complex manifold
$X$: Hilb${}^{N}(X)$. This manifold can be considered as the mirror to some
two dimensional Calabi-Yau manifold and the prepotential attributed to the
integrable system is identified with the partition function of the topological
string on $X$.

Such approach has been applied for the two-body DELL system in
[Braden:2001yc]. It was argued in [Braden:2001yc] that the two-body case the
manifold $X$ is the two-torus bundled over another two-torus, where one torus
modulus corresponds to the gauge coupling while the second corresponds to the
elliptic modulus of the compactification torus. The Inozemtsev limit in terms
of the separation of variables corresponds to the peculiar degeneration of the
manifold $X$. The clear-cut example of resulting geometry for two-body
holomorphic periodic q-Toda model can be read off from [Hatsuda:2016mdw]. It
was shown there that the energy level for the two-body system coincides with
the mirror curve for the $\mathbb{P}_{1}\times\mathbb{P}_{1}$ manifold which
in this case plays the role of manifold $X$. In our paper we have discussed at
the algebraic level the possible intermediate degenerations of two-torus
bundled over two-torus manifold into $\mathbb{P}_{1}\times\mathbb{P}_{1}$.

Let us remark that it was argued in [Hatsuda:2016mdw] that the periodic q-Toda
Hamiltonian describes the particle on the lattice in the magnetic field where
the lattice is modeled by the cosine potential. The spectrum of this model
enjoys the Hofstadter butterfly pattern. However the generalization of such
magnetic interpretation for interacting N-body system in magnetic field is
more complicated. For instance the Lauphlin wave function in the continuum
case is related with the wave function of rational Calogero model in the
external oscillatory potential which can be mapped into the trigonometric
Calogero model [Nekrasov:1997jf] describing the particles on the cylinder when
the magnetic field gets mapped into the radius of the cylinder. It is this
interacting system to which the Inozemtsev limit can be applied and upon the
proper rescaling the trigonometric Calogero model reduces into the open Toda
chain. However it is not clear how this simple model of interacting particles
in magnetic field can be extended to the elliptic case.

## Acknowledgments

The work of AG was supported by Basis Foundation grant 20-1-1-23-1 and by
grant RFBR-19-02-00214. PK is partially supported by AMS Simons Travel Grant.

## References