# BENCHMARKING HEADTAIL WITH ELECTRON CLOUD
INSTABILITIES OBSERVED IN THE LHC

H. Bartosik  W. Höfle  G. Iadarola  Y. Papaphilippou  G. Rumolo
CERN  Geneva  Switzerland

###### Abstract

After a successful scrubbing run in the beginning of 2011, the LHC can be
presently operated with high intensity proton beams with 50 ns bunch spacing.
However, strong electron cloud effects were observed during machine studies
with the nominal beam with 25 ns bunch spacing. In particular, fast transverse
instabilities were observed when attempting to inject trains of 48 bunches
into the LHC for the first time. An analysis of the turn-by-turn bunch-by-
bunch data from the transverse damper pick-ups during these injection studies
is presented, showing a clear signature of the electron cloud effect. These
experimental observations are reproduced using numerical simulations: the
electron distribution before each bunch passage is generated with PyECLOUD and
used as input for a set of HEADTAIL simulations. This paper describes the
simulation method as well as the sensitivity of the results to the initial
conditions for the electron build-up. The potential of this type of
simulations and their clear limitations on the other hand are discussed.

##  1 INTRODUCTION

At the early phase of the 2011 LHC run, seven days were devoted to scrubbing
using the 50 ns bunch spacing beam with gradually increasing number of bunches
circulating in the machine. This allowed to sufficiently condition the inner
surface of the LHC beam screens and vacuum chambers for running with the same
beam for physics production in routine operation throughout 2011. First
attempts to inject a beam with the nominal 25 ns bunch spacing were performed
at the end of June 2011 with bunch trains of 24 bunches. Pressure rise and
increased heat load in the arcs were observed. The studies continued with
injections of 48 bunches in August, where the beams became transversely
unstable after about 1000 turns with the transverse damper switched on and
after about 500 turns without transverse damper, for two injections
respectively. As will be discussed in more detail below, an analysis of the
turn-by-turn bunch-by-bunch data from the transverse damper pick-ups [1]
points to the observation of coherent electron cloud instabilities. During
these injection tests the chromaticity was set to about $Q^{\prime}\approx 2$
in both planes, as usually used on the LHC flat bottom in routine operation.
Only after increasing the chromaticity to about $Q^{\prime}\approx 15$ in both
planes during further studies in October, it was possible to perform nominal
injections of 288 bunches with 25 ns bunch spacing from the SPS. The high
chromaticity suppressed the fast instabilities observed before. However, the
beam suffered from slow losses and transverse emittance blow-up along the
bunch train and, as before, pressure rise and increased heat load were
measured in the cold arcs. The conditioning of the LHC beam screens due to the
beam based electron bombardment was demonstrated by estimating the secondary
electron yield (SEY) from a comparison of the measured heat load data in the
arcs with PyECLOUD simulations [2, 3]. Figure 1 shows the evolution of the
obtained maximum SEY ($\delta_{\text{max}}$) as a function of beam time in the
LHC together with the total intensity for both beams [4]. Since the heat load
data can be read only per half cell in the arcs, the SEY can be estimated only
for both beams at the same time (grey markers) unless there is only one beam
present in the machine (red markers for beam 2). A clear conditioning effect
from $\delta_{\text{max}}\\!=\\!2.1$ at the end of June when running with 50
ns bunch spacing to $\delta_{\text{max}}\\!=\\!1.52$ after approximately 50 h
beam time with 25 ns bunch spacing is observed. The analysis and simulation
studies presented in this paper will concentrate on the aforementioned
injection tests with 48 bunches in August, where $\delta_{max}$ was estimated
to be still around 2.1 (cf. Fig. 1).

Figure 1: Evolution of $\delta_{max}$ in the LHC arcs as obtained by
reproducing the measured heat load on the LHC beam screens using PyECLOUD [4].

##  2 Analysis of turn-by-turn data

Figure 2: Transverse oscillations measured with the pick-ups of the transverse
feedback for 73 turns close before the beam dump, for the first case with
transverse damper (left) and the second case without transverse damper (right)
in the horizontal (top) and vertical plane (bottom).

As mentioned before, a first attempt to inject bunch trains of 48 bunches
spaced by 25 ns into the LHC with the an intensity of about
$1.0\\!\times\\!10^{11}$ p/b was made for beam 2 during a machine development
session on August 26, 2011. Shorter bunch trains of 12 and 24 bunches with the
same bunch spacing had been successfully injected earlier in the session. It
was planned to establish the injection of longer bunch trains and complete the
transverse damper set-up for the nominal bunch spacing of 25 ns. A first
injection of 48 bunches with the transverse damper switched on was dumped
after 1000 turns due to a beam excursion interlock (referred to as dump 1).
Another injection of 48 bunches without transverse damper was aborted already
after around 500 turns due to a beam loss interlock (referred to as dump 2).
In both cases, chromaticity was set to the values usually used during
operation ($Q^{\prime}\approx 2$).

Data of the damper pick-ups for about 73 turns are stored for both beams in
the post mortem system at a beam abort. Each data set (channel) represents one
of the allowed 3564 bunch positions per ring. Figure 2 shows the turn-by-turn
oscillation of each of the 48 bunches around the closed orbit in both planes
for the 2 injections just before the respective beam dump. The oscillation
amplitude is very small for the first 25 bunches, especially when the damper
is switched on. In this case the last bunches of the bunch train reach peak
values of around 1 mm in the horizontal plane and up to 6 mm in the vertical
plane, thus the instability is mainly observed in the vertical plane. Without
damper, the last bunches of the train exhibit oscillations of up to 3 mm
amplitude in both planes. These observations are compatible with a coupled
bunch instability (slightly stronger in the horizontal plane than in the
vertical plane), which can be suppressed by the damper, and a high frequency
instability with a broad spectrum mainly in the vertical plane. The frequency
spectrum without damper is dominated by coupled bunch modes up to 1 MHz in the
horizontal plane and up to 2 MHz in the vertical plane. With damper on,
instabilities in both planes are damped up to a frequency of about 15 MHz [1].
This can be explained by a “single bunch” instability in the vertical plane
and a coupled bunch instability (mainly) in the horizontal plane, as has been
observed in the past already in the SPS [5]. The beam observations during the
injection tests in the LHC are used to benchmark numerical simulations as
described in the following sections.

##  3 Simulation procedure

Figure 3: Electron density per unit length along the bunch train as obtained
from a PyECLOUD simulation. A snapshot of the electron distribution before
each bunch entrance as indicated by the red dots is used as input for
individual HEADTAIL simulations.

The above described experimental observations were studied by combining
PyECLOUD [2] and HEADTAIL [6] simulations. As first step the electron cloud
build-up in the LHC dipoles is simulated with PyECLOUD for an r.m.s. bunch
length of 12 cm and equal bunch intensities of $N=1.0\times 10^{11}$ p/b along
the train111It was not possible to retrieve the actual bunch-by-bunch
intensity variation from the data logging during the injection tests due to
the short time of circulating beam before the dump.. Figure 3 shows the
obtained electron density per unit length along the bunch train for a
secondary electron yield of $\delta_{\text{max}}\\!=\\!2.1$. The simulation
was initialized with a uniform distribution using the same number of primary
electrons as in the simulations for determining the evolution of the SEY
discussed above [3]. The importance of the number of primary electrons and
their spatial distribution for the onset of the e-cloud instability will be
discussed in more detail in the next section. The red markers indicate the
time steps in the simulation just before the bunch passages. The electron
distributions sampled at these points are used as input for a set of 48
HEADTAIL simulations, one for each bunch.

Figure 4: Electron distribution before the passage of the last bunch of the
train as obtained from the PyECLOUD simulation. Note that the cross section of
the LHC beam screen is approximated as ellipse.

Figure 4 shows for example the electron distribution before the entrance of
the last bunch of the train. Note the accumulation of electrons in two
stripes, as typically obtained for the build-up with 25 ns bunch spacing in a
geometry as the LHC beam screen when the electrons move in a strong dipolar
magnetic field. In HEADTAIL the electron cloud is represented by thin slices
lumped at several accelerator sections and the electron motion is frozen in
the horizontal plane in order to account for the effect of the dipole magnetic
field. At each electron cloud section the electron proton interaction is
computed consecutively for longitudinal bunch slices. After a complete bunch
passage the electron cloud is reset to the initial distribution for the next
interaction section. Figure 5 shows the evolution of the transverse emittance
for a few selected bunches in the middle of the bunch train obtained with
HEADTAIL for the simulation of 500 turns at the LHC injection plateau using
the initial electron distribution as described above.

Figure 5: Evolution of the horizontal (top) and vertical (bottom) emittance
along 500 turns for a few selected bunches in the center of the bunch train.

No instability or coherent excitation of the horizontal bunch motion is
observed, as was expected since the electron cloud is located in dipole
regions. On the other hand the electron cloud drives a single bunch
instability in the vertical plane. This instability can be observed in the
form of an exponential emittance growth, as in the case studied here for all
bunches after bunch number 25. Figure 6 shows the bunch-by-bunch oscillations
in the vertical plane during the instability (from turn 50 to turn 120) as
obtained from HEADTAIL.

Figure 6: Bunch-by-bunch oscillations in the vertical plane as obtained with
HEADTAIL. For better comparison, only 70 turns during the instability (from
turn 50 to turn 120) are shown.

The simulation is in good agreement with the experimental observations during
the first injection of 48 bunches in August with the transverse damper on,
where the second half of the bunch train is unstable mainly in the vertical
plane (cf. Fig. 2), as the damper is suppressing horizontal coupled bunch
instabilities. Horizontal instabilities are not observed in the simulation,
since the coupling between bunches is not taken into account but each bunch is
treated by an independent HEADTAIL simulation. To include this coupling would
require a combined simulation of PyECLOUD and HEADTAIL, which is envisaged to
be implemented at a later stage. This will allow to reproduce better the
instability observed during the injection without transverse damper.

##  4 Sensitivity studies

It was shown in the previous section that the instability observed at LHC
injection can be reproduced with the presented simulation procedure. In the
following the sensitivity of the simulation result to the number and
distribution of the primary electrons will be studied. Furthermore, it is
observed that the electron cloud is formed mainly in two symmetric vertical
“stripes” around the beam center (cf. Fig. 4) and only a small number of
electrons is concentrated in the central part of the chamber. Thus it is
interesting to assess which part of the electron cloud distribution is mainly
responsible for driving the beam unstable.

###  4.1 Electron density

Figure 7 shows an example for a bunch in the middle of the train, which
encounters an electron density which is just above the instability threshold.
The top graph shows a histogram of the horizontal electron distribution before
the bunch arrival as obtained from the PyECLOUD simulation and used as input
for the HEADTAIL simulation. Note again

Figure 7: From top to bottom: Histogram of the horizontal electron
distribution; transverse electron distribution divided into colored regions;
evolution of the vertical emittance for the interaction with the electrons of
the respective colored area; evolution of the vertical emittance for different
horizontal cuts of the electron distribution.

that most of the electrons are concentrated in the two stripes far away from
the beam (the $3\,\sigma$ beam envelope covers roughly the two central bins).
The graph below shows for the same case the actual positions of the macro-
electrons in the transverse plane. The macro-particle distribution is divided
here into five regions as indicated by the color code (corresponding to the
bins in the histogram). In the following, the contribution from these five
regions to the vertical instability is studied. First, each of these regions
is considered independently and used as input for individual HEADTAIL
simulations. The evolution of the vertical emittance for these five cases is
shown for 500 turns in the third graph of Fig. 7. In the case studied here,
only the central part of the electron distribution is able to drive the beam
unstable. The bottom graph shows the vertical emittance evolution in case all
electrons enclosed by the respective colored areas have been removed, i.e.
cutting the electron distribution at the inner borders of the colored regions.
The instability appears only when the electrons in the central part of the
distribution are taken into account. It follows that for the typical electron
cloud distributions encountered in the simulations of the LHC dipole regions,
it is mainly the central electron density which determines the onset of the
instability. Therefore the instability threshold can be inferred roughly from
the central electron density. Only in cases where the central electron density
is _very_ small compared to the density in the stripes this approximation may
not hold.

In the following, the central density of the electron distribution is thus
used to determine the instability threshold. It should be noted that the
instability thresholds found in this manner are consistent with thresholds
previously found in HEADTAIL simulations assuming a uniform electron
distribution before the bunch passage, which is around $1\times
10^{12}\,e^{-}/$m3 [7].

###  4.2 Dependence on number of primary electrons

One of the main uncertainties of the electron cloud build-up simulations for
the LHC injection energy is the number of primary (seed) electrons. Since the
synchrotron radiation at 450 GeV is inefficient to generate photo electrons,
it is assumed that the primary electrons are created by rest gas ionization.
Therefore the number of primary electrons depends on the pressure. However the
static pressure without beam is much lower compared to the pressure levels
measured after beam injection. Figure 8 shows the central density along the
bunch train of 48 bunches for $\delta_{\text{max}}=2.1$ and two different
cases: for a small number of seed electrons equivalent to the static pressure
in the LHC cold sections (top) and for a large number of seed electrons
(bottom). The dashed red line shows the central electron density at the
instability threshold. As expected, it takes more bunch passages to build up
the electron cloud beyond the instability threshold for a smaller number of
primary electrons. The onset of the instability along the bunch train depends
thus on the pressure level assumed in the build-up simulation. For
completeness it should be emphasized that the saturation level of the central
density is very similar in both cases (as it depends mostly on
$\delta_{\text{max}}$).

Figure 8: Comparison of the central electron density along the bunch train for
a number of seeding electrons comparable to the static pressure in the LHC
dipole sections (top) and for a larger number of seeding electrons (bottom).

###  4.3 Dependence on primary electron distribution

Figure 9: Example for an electron cloud build-up simulation assuming a uniform
transverse distribution of the primary electrons all across the vacuum
chamber: central density along the bunch train (left), horizontal distribution
for bunch 33 (middle) and for the last bunch (right).

Figure 10: Example for an electron cloud build-up simulation assuming a
Gaussian distribution of the primary electrons on top of a 10% uniform
background all across the vacuum chamber: central density along the bunch
train (left), horizontal distribution for bunch 33 (middle) and for the last
bunch (right).

In addition to the number of primary electrons, also their distribution is not
well known in the case of the LHC at injection energy. In order to reproduce
the experimentally observed vertical stripes of the electron distribution in
saturation, primary electrons have to be assumed across the entire cross
section of the vacuum chamber (since the electrons are bound by the vertical
magnetic field lines). Therefore a uniform distribution of the primary
electrons can be optionally used in PyECLOUD simulations. Figure 9 shows an
example for a uniform distribution of the seed electrons together with the
histogram of the horizontal electron distribution for two selected bunches (a
bunch in the second half of the train where the saturation level is reached,
and the last bunch). The central density is very similar in the two cases, but
the stripes build-up further outside towards the end of the bunch train.

Considering that the generation of the primary electrons is caused by rest gas
ionization, it can be argued that the primary electrons should follow a
Gaussian distribution similar to the proton beam. In this case the formation
of the stripes can be achieved by adding about 10% of the total number of
electrons in the form of a uniform background all across the vacuum chamber.
Figure 10 shows the evolution of the central electron density along the bunch
train assuming such a distribution of seed electrons together with the
horizontal electron distribution for selected bunches as in Fig. 9. Here a
smaller number of seed electrons was chosen in order to obtain the instability
onset roughly at the same bunch along the bunch train as for the case of the
uniform distribution of seed electrons. However, the evolution of the electron
distribution along the bunch train is quite different for the two cases. In
comparison to the case of the uniform distribution, the central density
reaches much higher peak values in the saturated part of the train and it
takes longer to develop the stripes in the outer part of the vacuum chamber
assuming the Gaussian distribution. This can be understood intuitively, since
the center of the chamber is seeded with a larger number of electrons while
the outer regions of the vacuum chamber are seeded with a comparably smaller
number of electrons.

##  5 Compatible parameter space

In fact the number and distribution of the primary electrons are among the
biggest uncertainties in the present understanding of the electron cloud
build-up in the LHC dipoles at injection energy. On the other hand, the
estimation of $\delta_{\text{max}}$ from the reproduction of the measured heat
load in the LHC beam screens with PyECLOUD [4] is not so sensitive to the
assumptions on the seed electrons if the bunch train is long and the electron
saturation level is reached for many bunches. The estimation of
$\delta_{\text{max}}$ for the case studied here was not done for the same time
as the observations of the injections of the 48 bunches and therefore might be
slightly smaller than $\delta_{\text{max}}=2.1$. In the following, the
available parameter space is thus scanned for compatible solutions reproducing
the observed instability at LHC injection in August 2011. Figure 11 shows
compatible solutions, i.e. cases where the onset of the instability is close
to bunch 25 (between bunch 23 and bunch 27), for a range of possible values
for $\delta_{\text{max}}$ and for the number of primary electrons per bunch
passage. The two colored regions correspond to the cases of a purely uniform
distribution of the primary electrons and a Gaussian distribution with a 10%
uniform background, respectively. As expected, higher pressure levels (or
equivalently a larger number of primary electrons) and a larger secondary
electron yield is needed to reproduce the observations in the case of a
uniform electron distribution, due to the smaller central electron density
compared to the case with a Gaussian electron distribution. Note that the
range of the number of seed electrons per bunch passage explored in this study
corresponds to room-temperature equivalent pressures between 10 nTorr and 200
nTorr (the static pressure in the LHC cold sections is around 32 nTorr room-
temperature equivalent). Further studies will be needed in the future in order
to better understand the mechanism of the primary electron production and
their distribution at LHC injection energy.

Figure 11: Range of parameters reproducing the observed instability in the
LHC, as indicated by the colored areas assuming a uniform primary electron
distribution (blue) and the Gaussian distribution on top of a 10% background
(green).

##  6 CONCLUSIONS AND OUTLOOK

Fast instabilities were observed during the first two attempts of injecting
bunch trains of 48 bunches with the nominal 25 ns bunch spacing in August
2011. The onset of vertical single bunch instability can be reproduced in good
agreement by electron cloud simulations using a combination of PyECLOUD for
the build-up and HEADTAIL for the beam dynamics part. The simulations are
based on values of $\delta_{\text{max}}$ as estimated from the measured heat
load data in the LHC using PyECLOUD. The presented studies based on a
combination of PyECLOUD with the HEADTAIL code can therefore be considered as
consistency check of the current model of the electron-cloud effects on the
LHC flat bottom. The simulations show a vertical instability, fast emittance
growth and strong losses, similar to observations in the LHC. The instability
onset depends strongly on the central electron density seen by the beam, which
itself depends on the initial conditions assumed for the build-up, namely the
number of primary electrons and their distribution. Future studies should aim
at improving the understanding of the mechanism responsible for the generation
of the primary electrons. This will become important also for the estimation
of $\delta_{\text{max}}$ from the measured heat load close to the threshold of
the electron cloud build-up, despite the fact that the beam is not expected to
suffer from the instability any more. Apart from this, it would be interesting
to include the coupling between the bunches due to the electron cloud in the
simulations. This would require a self consistent model of the electron cloud
effects like a combination of PyECLOUD and HEADTAIL in one big simulation for
all bunches, which would then allow to study the observed horizontal coupled
bunch instability. Finally it might be interesting to include the effect of
the transverse feedback in HEADTAIL, as this would help to estimate better the
required settings of chromaticity and octupoles to stabilize the beam in a
strong electron-cloud regime.

##  7 ACKNOWLEDGEMENTS

The authors would like to thank D. Valuch for help extracting and analyzing
the turn-by-turn data from the damper pick-ups and feedback setup, G. Arduini
for valuable discussion, and K. Li for support for the HEADTAIL simulations.

## References

  * [1] H. Bartosik and W. Höfle, “Analysis of bunch by bunch oscillations with bunch trains at injection into LHC at 25 ns bunch spacing”, CERN-ATS-Note-2012-027 MD (LHC), Geneva 2012.
  * [2] G. Iadarola and G. Rumolo, “PyECLOUD and Build Up Simulations at CERN”, these proceedings.
  * [3] G. Rumolo et al., “Electron Cloud Effects in the LHC in 2011”, in Proceedings of the LHC Beam Operation Workshop - Evian 2011, Evian 2011.
  * [4] G. Rumolo et al., “LHC Experience with Different Bunch Spacings in 2011 (25, 50 and 75ns)”, in Proceedings of Chamonix 2012 workshop on LHC Performance, Chamonix 2012.
  * [5] A. Gianluigi et al., “Beam observations with electron cloud in the CERN PS and SPS complex”, 31st Advanced ICFA Beam Dynamics Workshop on Electron-Cloud Effects, Napa 2004.
  * [6] G. Rumolo and F. Zimmermann, “Electron cloud simulations: beam instabilities and wakefields”, Phys. Rev. STAccel. Beams 5, 121002, 2002.
  * [7] K. Li and G. Rumolo, “Review of Beam Instabilities in the Presence of Electron Clouds in the LHC”, in Proceedings of IPAC11, San Sebastian 2011.


Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence

Stringy Kähler Moduli for the Pfaffian–Grassmannian Correspondence††This paper
is a contribution to the Special Issue on Primitive Forms and Related Topics
in honor of Kyoji Saito for his 77th birthday. The full collection is
available at https://www.emis.de/journals/SIGMA/Saito.html

Will DONOVAN

W. Donovan

Yau Mathematical Sciences Center, Tsinghua University,
Haidian District, Beijing 100084, China<EMAIL_ADDRESS>
Received September 29, 2020, in final form March 10, 2021; Published online
March 24, 2021

The Pfaffian–Grassmannian correspondence relates certain pairs of derived
equivalent non-birational Calabi–Yau 3-folds. Given such a pair, I construct a
set of derived equivalences corresponding to mutations of an exceptional
collection on the relevant Grassmannian, and give a mirror symmetry
interpretation, following a physical analysis of Eager, Hori, Knapp, and Romo.

Calabi–Yau threefolds; stringy Kähler moduli; derived category; derived
equivalence; matrix factorizations; Landau–Ginzburg model; Pfaffian;
Grassmannian

14F08; 14J32; 14M15; 18G80; 81T30

##  1 Introduction

Birational Calabi–Yau $3$-folds are known to have equivalent derived
categories [4]. There also exist pairs of Calabi–Yau $3$-folds which are not
birational but may be proved to be derived equivalent. A much-studied class of
examples comes from the “Pfaffian–Grassmannian” correspondence, concerning
pairs of $3$-folds arising as linear sections of the Grassmannian
$\mathrm{G}(2,7)$ and its projective dual Pfaffian. Rødland conjectured that
such pairs share a mirror [15], leading to an expectation that they are
derived equivalent: this was proved by Borisov and Căldăraru [3], and
Kuznetsov [13].

Meanwhile, Hori and Tong [10] gave a physical explanation of how such pairs of
$3$-folds arise from the same gauged linear $\sigma$-model. In work of
Addington, Segal, and the author [1], a partial mathematical interpretation of
this was given by constructing a particular derived equivalence for each pair
using categories of matrix factorizations.

According to mirror symmetry, the derived symmetries of a variety may be
determined by monodromy on a stringy Kähler moduli space (SKMS). Hori and Tong
described this space for such $3$-folds: implicit in this was a prediction of
further equivalences, corresponding to “grade restriction windows”, see [8,
16]. Later physics work made these windows explicit, and argued that
differences between equivalences are given by spherical twists: see Eager,
Hori, Knapp, and Romo [6], and Hori [9, end of Section 5].

In this paper, I interpret this physics work by constructing, for each pair of
$3$-folds coming from the Pfaffian–Grassmannian correspondence, a set of
equivalences corresponding to the “windows” above by extending the methods of
[1]. I then show that these equivalences, along with appropriate spherical
twists, may be organized into an action of the fundamental group of the
relevant SKMS.

###  1.1 Calabi–Yau pairs

I recall the construction of the $3$-folds $Y_{\mathrm{G}}$ and
$Y_{\mathrm{P}}$. Start with a $7$-dimensional vector space $V$, and consider
the following.

  * •

The Grassmannian of $2$-planes in its Plücker embedding

$\mathrm{G}(2,V)\subset\mathbb{P}\big{(}{\wedge}^{2}V\big{)}.$

  * •

The Pfaffian of 2-forms on $V$ of rank at most $4$, denoted

$\mathrm{P}(4,V)\subset\mathbb{P}\big{(}{\wedge}^{2}V^{\vee}\big{)}.$

These varieties are projectively dual. The Pfaffian $\mathrm{P}(4,V)$ is
singular along the locus of forms of rank at most $2$, but taking sufficiently
generic hyperplane sections yields a smooth Calabi–Yau $3$-fold
$Y_{\mathrm{P}}$. Taking a dimension 7 subspace $\Pi\subset\wedge^{2}V^{\vee}$
and its annihilator $\Pi^{\circ}\subset\wedge^{2}V$, we then obtain smooth
Calabi–Yau $3$-folds as follows:111For details, see [3]. In fact, if $\Pi$ is
chosen such that $Y_{\mathrm{P}}$ is a smooth $3$-fold, then the same is true
of $Y_{\mathrm{G}}$ [3, Corollary 2.3].

$\displaystyle Y_{\mathrm{G}}=\mathrm{G}(2,V)\cap\mathbb{P}\Pi^{\circ},$
$\displaystyle Y_{\mathrm{P}}=\mathrm{P}(4,V)\cap\mathbb{P}\Pi.$

###  1.2 Equivalences

For each pair $Y_{\mathrm{G}}$ and $Y_{\mathrm{P}}$, I construct a set of
derived equivalences depending on a discrete parameter given as follows:

$\underline{m}=(m_{0},m_{1},m_{2})\in\mathbb{Z}^{3}\qquad\text{such that}\quad
m_{l}\leq m_{l+1}\leq m_{l}+1.$

Each choice of $\underline{m}$ gives an exceptional collection on
$\mathrm{G}(2,V)$ by successive mutations of a collection due to Kuznetsov
(Proposition 2.2). This collection is determined by a Lefschetz block

$\mathcal{O}_{\mathrm{G}}(m_{0}),\qquad
S(m_{1}),\qquad\operatorname{Sym}^{2}S(m_{2}),$

where $S$ is the rank $2$ tautological subspace bundle on $\mathrm{G}(2,V)$.

Now, by extending the construction of [1], I prove the following.

######  Theorem 1.1 (Theorem 5.8).

For each $\underline{m}\in\mathbb{Z}^{3}$ as above, an equivalence

$\Psi^{\underline{m}}\colon\
\mathrm{D^{b}}(Y_{\mathrm{G}})\overset{\sim}{\longrightarrow}\mathrm{D^{b}}(Y_{\mathrm{P}})$

may be constructed using the exceptional collection on $\mathrm{G}(2,V)$ given
by $\underline{m}$.

Section 3 outlines how $\Psi^{\underline{m}}$ is constructed: it is a “window
equivalence”, where generators are obtained from the collection given by
$\underline{m}$. The equivalence obtained in [1] corresponds to
$\underline{m}=(6,7,8)$ (Remark 4.3).

###  1.3 Physics

The SKMS in our case is described in the physics literature as a sphere with
five punctures: see [10, Fig. 1] and [6, Section 4]. Our $3$-folds correspond
to “large radius limits” near two of these punctures, shown below as the
poles. In this picture, homotopy classes of paths between large radius limits
are expected to correspond to derived equivalences, given by “grade
restriction windows”. Such equivalences are supplied by Theorem 1.1. Monodromy
around the other three punctures are then expected to correspond to spherical
twists [9, end of Section 5]. Theorems 1.2 and 1.4 below confirm these
expectations from the physics literature.

###  1.4 Groupoid action

I show that the fundamental groupoid of the SKMS acts on the derived
categories $\mathrm{D^{b}}(Y_{\mathrm{G}})$ and
$\mathrm{D^{b}}(Y_{\mathrm{P}})$.

Let $\mathcal{M}=S^{2}-\\{\text{$5$ points}\\}$ where two of the punctures are
the poles, and basepoints $m_{\mathrm{G}}$ and $m_{\mathrm{P}}$ are chosen
near them. Following the physics analysis [6, Section 4], I choose a finite
subset $\Psi^{k}$, $0\leq k\leq 3$, of the equivalences $\Psi^{\underline{m}}$
of Theorem 1.1 as follows:

$\Psi^{k}=\Psi^{\underline{m}},\qquad\text{where}\quad
m_{l}=\begin{cases}-1,&l<k,\\\ \hphantom{-}0,&l\geq k.\end{cases}$

The corresponding exceptional collections are illustrated in Section 2 as
Collections  ‣ 2–3. Combining the construction of Theorem 1.1 with standard
techniques for manipulating window equivalences, I obtain the following.

######  Theorem 1.2 (Theorem 6.3).

There is an action of the fundamental groupoid
$\pi_{1}(\mathcal{M},\\{m_{\mathrm{G}},m_{\mathrm{P}}\\})$ on
$\mathrm{D^{b}}(Y_{\mathrm{G}})$ and $\mathrm{D^{b}}(Y_{\mathrm{P}})$, given
by the following diagram:

$\mathcal{M}$$\scriptstyle m_{\mathrm{G}}$$\scriptstyle
m_{\mathrm{P}}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{G}}}\\!(1)$$\scriptstyle\Psi^{0}$$\scriptstyle\Psi^{1}$$\scriptstyle\Psi^{2}$$\scriptstyle\Psi^{3}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{P}}}\\!(1)$

###  1.5 Mutations and group action

The exceptional collections for the $\Psi^{k}$ are related by mutations of
exceptional objects as follows:

Collection  ‣
2123.$\mathcal{O}_{\mathrm{G}}$$S_{\phantom{\mathrm{G}}}$$\operatorname{Sym}^{2}S$

The restrictions of these three exceptional objects to $Y_{\mathrm{G}}$ are
the three spherical objects below (Proposition 7.6):

$\mathcal{O}_{Y_{\mathrm{G}}},\qquad
S_{Y_{\mathrm{G}}},\qquad\operatorname{Sym}^{2}S_{Y_{\mathrm{G}}}.$ (1.1)

######  Remark 1.3.

This phenomenon is analogous to exceptional objects restricting to spherical
objects on an anticanonical divisor [18, Example 3.14(c)], though here the
codimension of $Y_{\mathrm{G}}$ is $7$.

I then show (Proposition 7.7) that the “differences” between equivalences

$\big{(}\Psi^{j+1}\big{)}^{-1}\circ\Psi^{j}$

are spherical twists around the objects (1.1). Combining this with Theorem
1.2, I deduce the following.

######  Theorem 1.4 (Theorem 7.8).

There is an action of the fundamental group
$\pi_{1}(\mathcal{M},m_{\mathrm{G}})$ on $\mathrm{D^{b}}(Y_{\mathrm{G}})$
given by the following diagram:

$\mathcal{M}$$\scriptstyle\operatorname{Sym}^{2}\\!S$ $\scriptstyle S$
$\scriptstyle\mathcal{O}$$\scriptstyle
m_{\mathrm{G}}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{G}}}\\!(1)$$\scriptstyle\Psi^{-1}(\otimes\mathcal{O}_{Y_{\mathrm{P}}}\\!(1))\Psi$

Here we let $\Psi=\Psi^{3}$, and the loops around equatorial holes indicate
spherical twists around the objects (1.1).

It would be interesting to recover $\mathcal{M}$, and the above actions, from
an analysis of Bridgeland stability conditions.

###  1.6 Contents

Section 2 explains the exceptional collections in Theorem 1.1, Section 3
outlines the structure of the proof, and then Sections 4 and 5 give the
details. Section 6 proves Theorem 1.2, and Section 7 proves Theorem 1.4.

##  2 Exceptional collections

In this section, I explain the exceptional collections in Theorem 1.1, and
convenient ways to visualize them.

The construction in [1] used the following full exceptional collection on the
Grassmannian $\mathrm{G}(2,V)$ for a 7-dimensional vector space $V$ [14,
Theorem 4.1], where $S$ denotes the rank $2$ tautological subspace bundle.

$\big{\\{}\begin{minipage}[c][15pt][c]{164.81963pt}{$\operatorname{Sym}^{l}S^{\vee}\otimes(\det
S^{\vee})^{m}_{\phantom{,}{}}\colon
l\in\left[0,3\right),\,m\in\left[0,7\right)$}\end{minipage}\big{\\}}.$ (2.1)

I depict the collection (2.1) as follows, where as usual we write
$\mathcal{O}(1)$ for $\det S^{\vee}$, and for convenience in this section put
$S^{2}$ for $\operatorname{Sym}^{2}S$:

$\mathcal{O}\phantom{(}$$S^{\vee}\phantom{(}$${S}^{{2}\vee}\phantom{(}$$\cdots$$\cdots$$\mathcal{O}{({6})}$$S^{\vee}{({6})}$${S}^{{2}\vee}{({6})}$

######  Remark 2.1.

The diagonal display format matches with [9] and is used later to visualize
the direction of $\operatorname{Hom}$s in the collection: see Remark 4.10.

It will be helpful instead to take a dual exceptional collection as
follows.222Note that this is an exceptional collection because, for locally
free sheaves $\mathcal{E}$ and $\mathcal{F}$, we have
$\operatorname{Ext}^{\bullet}(\mathcal{E},\mathcal{F})=\operatorname{Ext}^{\bullet}\big{(}\mathcal{F}^{\vee},\mathcal{E}^{\vee}\big{)}$.

######  Collection 0.

$\mathcal{O}{({-6})}$$S{({-6})}$$S^{{2}}{({-6})}$$\cdots$$\cdots$$\mathcal{O}\phantom{(}$$S\phantom{(}$$S^{{2}}\phantom{(}$

From this collection, we deduce others by mutation. Namely, fixing an order
for the collection $\langle\mathcal{E}_{1},\dots,\mathcal{E}_{n}\rangle$, it
is standard that mutating $\mathcal{E}_{n}$ from right to left gives

$L_{1}\cdots L_{n-1}(\mathcal{E}_{n})=\mathbb{S}(\mathcal{E}_{n}),$

where the $L_{i}$ denote left mutation functors, and $\mathbb{S}$ is the Serre
functor

$\mathbb{S}=-\otimes\omega_{\mathrm{G}}[\dim\mathrm{G}]=-\otimes\mathcal{O}(-7)[\dim\mathrm{G}].$

Mutating in this way at $\mathcal{O}$, and noting that the property of being
an exceptional collection is invariant under the shift $[\dim\mathrm{G}]$, we
obtain the following:

######  Collection 1.

$\mathcal{O}{({-7})}$$S{({-6})}$$S^{{2}}{({-6})}$$\cdots$$\cdots$$\mathcal{O}{({-1})}$$S\phantom{(}$$S^{{2}}\phantom{(}$

Mutating at $S$ then gives:

######  Collection 2.

$\mathcal{O}{({-7})}$$S{({-7})}$$S^{{2}}{({-6})}$$\cdots$$\cdots$$\mathcal{O}{({-1})}$$S{({-1})}$$S^{{2}}\phantom{(}$

We now have two choices for how to mutate, at $\mathcal{O}(-1)$ or $S^{2}$.

Mutating at $S^{2}$ gives a twist by $\mathcal{O}(-1)$ of Collection  ‣ 2:

######  Collection 3.

$\mathcal{O}{({-7})}$$S{({-7})}$$S^{{2}}{({-7})}$$\cdots$$\cdots$$\mathcal{O}{({-1})}$$S{({-1})}$$S^{{2}}{({-1})}$

Mutating at $\mathcal{O}(-1)$ gives:

######  Collection 4.

$\mathcal{O}{({-8})}$$S{({-7})}$$S^{{2}}{({-6})}$$\cdots$$\cdots$$\mathcal{O}{({-2})}$$S{({-1})}$$S^{{2}}\phantom{(}$

Noting then that being a full exceptional collection is invariant under
twisting by $\mathcal{O}(k)$, we obtain further collections summarized in the
following.

######  Proposition 2.2.

Let $\underline{m}=(m_{0},m_{1},m_{2})$ be a non-decreasing sequence of
integers such that $m_{l+1}\leq m_{l}+1$. Then

$\big{\\{}\begin{minipage}[c][15pt][c]{152.29697pt}{$\operatorname{Sym}^{l}S(m)\colon
l\in\left[0,2\right],\,m\in\left(m_{l}-7,m_{l}\right]$}\end{minipage}\big{\\}}$
(2.2)

gives a full exceptional collection on $\mathrm{G}(2,V)$, for $\dim V=7$.

##  3 Structure of equivalence proof

I outline the proof of the derived equivalences $\Psi^{\underline{m}}$ between
$Y_{\mathrm{G}}$ and $Y_{\mathrm{P}}$ in Theorem 1.1, before giving the proof
in the following Sections 4 and 5.

These derived equivalences are obtained by showing that $Y_{\mathrm{G}}$ and
$Y_{\mathrm{P}}$ are derived equivalent, in an appropriate sense, to
Landau–Ginzburg models

$(X_{\mathrm{G}},f)\qquad\text{and}\qquad(X_{\mathrm{P}},f).$

Here the space $X_{\mathrm{G}}$ is a bundle over $\mathrm{G}(2,V)$, the space
$X_{\mathrm{P}}$ is an Artin stack which is described in the next section, and
$f$ is a function defined on both these spaces: $f$ arises by restriction of a
function from an Artin stack $\mathfrak{X}$, of which $X_{\mathrm{G}}$ and
$X_{\mathrm{P}}$ are open substacks. This follows a physics construction of
Hori–Tong [10], where this function is a “superpotential”.

Each equivalence $\Psi^{\underline{m}}$ is a composition of three equivalences
as below, where $\mathrm{D^{b}}(X_{\mathrm{G}},f)$ is a category of matrix
factorizations (see for instance [1, Section 2]), and
$\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$ is a closely related “B-brane
category” (Definition 4.5):

$\mathrm{D^{b}}(X_{\mathrm{G}},f)$$\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$$\mathrm{D^{b}}(Y_{\mathrm{G}})$$\mathrm{D^{b}}(Y_{\mathrm{P}})$$\scriptstyle\Psi_{\mathbb{W}}^{\underline{m}}$$\scriptstyle\sim$$\scriptstyle\Psi_{\mathrm{G}}$$\scriptstyle\sim$$\scriptstyle\Psi_{\mathrm{P}}^{\underline{m}}$$\scriptstyle\sim$$\scriptstyle\Psi^{\underline{m}}$

The vertical arrows are constructed as in [1], using that the spaces $Y$ may
be recovered from the spaces $X$. In particular, $Y_{\mathrm{G}}$ is the
critical locus of $f$ in $X_{\mathrm{G}}$, giving the left-hand equivalence
$\Psi_{\mathrm{G}}$ via Knörrer periodicity [19]. The right-hand equivalence
$\Psi_{\mathrm{P}}^{\underline{m}}$ is a generalized Knörrer periodicity,
established in [1, Section 5].

The horizontal equivalence $\Psi_{\mathbb{W}}^{\underline{m}}$ is a “window
equivalence”, which factors via a certain subcategory in
$\mathrm{D^{b}}(\mathfrak{X},f)$, determined by one of the exceptional
collections from Proposition 2.2. This is explained and proved in the
following Section 4. The rest of the proof is then given in Section 5.

######  Remark 3.1.

There is a general theory of window equivalences [2, 7], however it does not
yet apply in this case. In particular, it gives exceptional collections
related to that of Kapranov [11] rather than those in Proposition 2.2: this
seems to make it unsuitable for giving an equivalence with the Calabi–Yau
$Y_{\mathrm{P}}$. For more discussion, see [1, Remark 4.12].

######  Remark 3.2.

In the physics literature [6, Section 7] a grade restriction window for
$\underline{m}=(7,7,7)$ is associated with the derived equivalence of Borisov
and Căldăraru [3]. It would be interesting to compare the latter with the
$\Psi^{\underline{m}}$ obtained in this paper.

##  4 Equivalences for Landau–Ginzburg models

In this section, I construct the window equivalences
$\Psi_{\mathbb{W}}^{\underline{m}}$ used to prove Theorem 1.1, after reviewing
the construction of $X_{\mathrm{G}}$ and $X_{\mathrm{P}}$, the underlying
spaces of the Landau–Ginzburg models discussed above.

Let $S$ be a $2$-dimensional vector space, and consider the quotient stack

$\mathcal{G}=\big{[}\operatorname{Hom}(S,V)\big{/}\,\operatorname{GL}(S)\big{]}.$

The open substack of $\mathcal{G}$ of rank $2$ homomorphisms is equivalent to
the variety $\mathrm{G}(2,V)$. Furthermore, the vector bundle on $\mathcal{G}$
induced by the representation $S$ of $\operatorname{GL}(S)$ restricts to the
tautological subspace bundle on $\mathrm{G}(2,V)$. The representation $\det
S^{\vee}$ similarly corresponds to the bundle $\mathcal{O}(1)$ on
$\mathrm{G}(2,V)$.

Take the quotient stack

$\mathfrak{X}=\big{[}\operatorname{Hom}(S,V)\oplus\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}\big{/}\,\operatorname{GL}(S)\big{]}.$

Representations of $\operatorname{GL}(S)$ also give vector bundles on
$\mathfrak{X}$, and its substacks. In particular, the representation $S$ gives
a bundle on $\mathfrak{X}$, and the notation $S$ will also be used to denote
this bundle. For brevity, we write the line bundle corresponding to the
representation $\big{(}\det S^{\vee}\big{)}^{\otimes k}$ as $\mathcal{O}(k)$,
by analogy with the notation on $\mathrm{G}$.

Now, with $x\in\operatorname{Hom}(S,V)$ and
$p\in\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}$, let

$X_{\mathrm{G}}=\\{\operatorname{rk}x=2\\}\qquad\text{and}\qquad
X_{\mathrm{P}}=\\{\operatorname{rk}p=1\\}$

be open substacks of $\mathfrak{X}$. These have the structure of a vector
bundle over $\mathrm{G}(2,V)$ and an Artin stack $\mathcal{P}$, respectively,
where

$\mathcal{P}=\big{[}\operatorname{Hom}^{\operatorname{rk}=1}\big{(}V,\wedge^{2}S\big{)}\big{/}\operatorname{GL}(S)\big{]}.$

We let $f$ denote a function on the stack $\mathfrak{X}$, and use the same
notation for its restriction to substacks. As the particular form of $f$ is
not used in this section, we defer its definition to the following Section 5.

The following categories correspond to the collections of Proposition 2.2.

######  Definition 4.1 (window subcategories $\mathbb{W}^{\underline{m}}$).

Let $\underline{m}=(m_{0},m_{1},m_{2})$ be a non-decreasing sequence of
integers such that $m_{l+1}\leq m_{l}+1$. Then

$\big{\\{}\begin{minipage}[c][15pt][c]{152.29697pt}{$\operatorname{Sym}^{l}S(m)\colon
l\in\left[0,2\right],\,m\in\left(m_{l}-7,m_{l}\right]$}\end{minipage}\big{\\}}$
(4.1)

gives a set of bundles on $\mathfrak{X}$, and we write
$\mathbb{W}^{\underline{m}}$ for the full subcategories of
$\mathrm{D^{b}}(\mathfrak{X})$ and $\mathrm{D^{b}}(\mathfrak{X},f)$ generated
by this set.

For brevity, we notate bundles as follows.

######  Notation 4.2.

Let $S_{l,m}$ denote the bundle $\operatorname{Sym}^{l}S(m)$ on
$\mathfrak{X}$, or its restriction to a substack.

######  Remark 4.3.

In [1] a set of generators $T_{l,m}=\operatorname{Sym}^{l}S^{\vee}(m)$ was
used. Note that $T_{l,m}\cong S_{l,m+l}$, using that $S^{\vee}\cong S(1)$
because $S^{\vee}\cong S\otimes\det S^{\vee}$. The window subcategory used in
[1], with generators given in (2.1) above, is therefore the window
$\mathbb{W}^{\underline{m}}$ with $\underline{m}=(6,7,8)$ in our notation
here.

We write embeddings of stacks as follows:

$X_{\mathrm{G}}\stackrel{{\scriptstyle
i_{\mathrm{G}}}}{{\longrightarrow}}\mathfrak{X}\stackrel{{\scriptstyle
i_{\mathrm{P}}}}{{\longleftarrow}}X_{\mathrm{P}}.$

######  Proposition 4.4.

For $\mathbb{W}^{\underline{m}}\subset\mathrm{D^{b}}(\mathfrak{X},f)$ the
derived functor

$i_{\mathrm{G}}^{*}\colon\
\mathbb{W}^{\underline{m}}\to\mathrm{D^{b}}(X_{\mathrm{G}},f)$

is an equivalence, and the derived functor

$i_{\mathrm{P}}^{*}\colon\
\mathbb{W}^{\underline{m}}\to\mathrm{D^{b}}(X_{\mathrm{P}},f)$

is fully faithful.

###### Proof.

We first prove the analogous statement with $\mathrm{D^{b}}(X_{\mathrm{G}},f)$
and $\mathrm{D^{b}}(X_{\mathrm{P}},f)$ replaced with
$\mathrm{D^{b}}(X_{\mathrm{G}})$ and $\mathrm{D^{b}}(X_{\mathrm{P}})$, before
explaining how to deduce the proposition.

To prove fully faithfulness, we generalize the argument given in [1, Lemma
4.3]. It suffices to check fully faithfulness on generators: take then two of
the generators $S_{l,m}$ and $S_{l^{\prime},m^{\prime}}$ of
$\mathbb{W}^{\underline{m}}$. The $\operatorname{Hom}$s between these are
equal to the $\operatorname{Hom}$s between their restrictions to
$X_{\mathrm{G}}$ and $X_{\mathrm{P}}$, respectively, because the complements
of the latter have codimensions at least $2$. Furthermore, there are no higher
$\operatorname{Ext}$s between them because they are vector bundles on an
affine stack, so it suffices to check that they do not acquire any higher
$\operatorname{Ext}$s after applying the functors $i^{*}$. Namely, we require
the following:

$\displaystyle\operatorname{Ext}^{>0}_{X_{\mathrm{G}}}\big{(}i_{\mathrm{G}}^{*}S_{l,m},\;i_{\mathrm{G}}^{*}S_{l^{\prime},m^{\prime}}\big{)}=0,$
(4.2)
$\displaystyle\operatorname{Ext}^{>0}_{X_{\mathrm{P}}}\big{(}i_{\mathrm{P}}^{*}S_{l,m},\;i_{\mathrm{P}}^{*}S_{l^{\prime},m^{\prime}}\big{)}=0.$
(4.3)

These vanishings are proved in Sections 4.1 and 4.2, respectively.

We then claim that $i_{\mathrm{G}}^{*}$ is essentially surjective. For this,
note that the generators $S_{l,m}$, when considered as sheaves on
$\mathrm{G}(2,V)$ give generators for the derived category by Proposition 2.2.
Essential surjectivity then follows by a general argument, exactly as in [1,
Lemma 4.6].

To complete the proof of the proposition, we repeat the arguments in [1,
Proposition 4.9, Lemma 4.10]. The important point here is that morphisms in
the categories $\mathrm{D^{b}}(X_{\mathrm{G}},f)$ and
$\mathrm{D^{b}}(X_{\mathrm{P}},f)$ are related to those in the categories
$\mathrm{D^{b}}(X_{\mathrm{G}})$ and $\mathrm{D^{b}}(X_{\mathrm{P}})$ by
certain spectral sequences. ∎

Using the above proposition we have the following.

######  Definition 4.5 (B-brane categories).

We define a subcategory

$\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)\subset\mathrm{D^{b}}(X_{\mathrm{P}},f)$

to be the image of
$\mathbb{W}^{\underline{m}}\subset\mathrm{D^{b}}(\mathfrak{X},f)$ under
$i_{\mathrm{P}}^{*}$.

######  Remark 4.6.

In other words, $\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$ is the full
subcategory of $\mathrm{D^{b}}(X_{\mathrm{P}},f)$ generated by the bundles
given by (4.1).

######  Definition 4.7 (window equivalences).

We write

$\Psi_{\mathbb{W}}^{\underline{m}}=i_{\mathrm{P}}^{*}\circ(i_{\mathrm{G}}^{*})^{-1}\colon\
\mathrm{D^{b}}(X_{\mathrm{G}},f)\to\mathbb{W}^{\underline{m}}\to\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f).$

###  4.1 Grassmannian side

I start with some vanishing results on $\mathrm{G}(2,V)$, before proving the
vanishing (4.2) on $X_{\mathrm{G}}$. Is a recall the following, again writing
$T_{l,m}=\operatorname{Sym}^{l}S^{\vee}(m)$.

######  Lemma 4.8 ([1, Lemma 4.5]).

Let $\mathrm{G}=\mathrm{G}(2,V)$, with $\dim V=n$ odd. If $0\leq
l,l^{\prime}\leq\frac{1}{2}n-1$ and $m^{\prime}\geq m$ then we have

$\operatorname{Ext}^{>0}_{\mathrm{G}}\big{(}T_{l,m},\,T_{l^{\prime},m^{\prime}}\big{)}=0.$

We quickly obtain the following, where again
$S_{l,m}=\operatorname{Sym}^{l}S(m)$.

######  Corollary 4.9.

In the setting of Lemma 4.8 above, so that in particular we have
$m^{\prime}\geq m$,

$\operatorname{Ext}^{>0}_{\mathrm{G}}\big{(}S_{l,m},\,S_{l^{\prime},m^{\prime}}\big{)}=0\qquad\text{and}\qquad\operatorname{Ext}^{>0}_{\mathrm{G}}\big{(}S_{l,m},\,S_{l^{\prime},m^{\prime}+l^{\prime}-l}\big{)}=0.$

###### Proof.

For the first, we exchange $l$ and $l^{\prime}$, and note that the result is
isomorphic to the $\operatorname{Ext}$-group in Lemma 4.8 above, using local
freeness of the $\operatorname{Sym}$s. For the second, we use that
$T_{l,m}\cong S_{l,m+l}$, and the local freeness of $\mathcal{O}(1)$. ∎

######  Remark 4.10.

In terms of the following picture, this corollary says that there are no
higher $\operatorname{Ext}$s on $\mathrm{G}(2,V)$ from an object on the lines
pictured, say $L$, to any object on a parallel line $L^{\prime}$ lying to the
right of $L$:

$\cdots$$\cdots$$S_{0,-1}$$S_{0,0}$$S_{1,0}$$S_{2,0}$$S_{0,1}$$S_{1,1}$$S_{2,1}$$S_{0,2}$$S_{1,2}$$S_{2,2}$$S_{2,3}$$\cdots$$\cdots$

I now show the vanishing (4.2). By a standard calculation using the structure
of $X_{\mathrm{G}}$ as a vector bundle over $\mathrm{G}$, we have

$\displaystyle\operatorname{RHom}_{X_{\mathrm{G}}}\big{(}i_{\mathrm{G}}^{*}S_{l,m},\,i_{\mathrm{G}}^{*}S_{l^{\prime},m^{\prime}}\big{)}\cong\bigoplus_{n\geq
0}\operatorname{RHom}_{\mathrm{G}}\big{(}S_{l,m},\,S_{l^{\prime},m^{\prime}}\otimes\operatorname{Sym}^{n}\mathcal{O}(1)^{\oplus
7}\big{)}.$

By expanding the $\operatorname{Sym}^{n}$ piece into irreducibles we see that
this splits into summands

$\operatorname{RHom}_{\mathrm{G}}\big{(}S_{l,m},\,S_{l^{\prime},m^{\prime}+n}\big{)},$

where $S_{l,m},S_{l^{\prime},m^{\prime}}\in\mathbb{W}^{\underline{k}}$ and
$n\geq 0$. It therefore suffices to know that there are no higher
$\operatorname{Ext}$s from the generators $S_{l,m}$ of
$\mathbb{W}^{\underline{m}}$ to other generators $S_{l^{\prime},m^{\prime}}$
of $\mathbb{W}^{\underline{m}}$ or to sheaves $S_{l^{\prime},m^{\prime}+n}$
“to their right”. But this is clear from the form of the exceptional
collections in Section 2, combined with Corollary 4.9 above.

###  4.2 Pfaffian side

We now show the vanishing (4.3). Similarly to the above, by a standard
calculation we have

$\displaystyle\operatorname{RHom}_{X_{\mathrm{P}}}\big{(}i_{\mathrm{P}}^{*}S_{l,m},\,i_{\mathrm{P}}^{*}S_{l^{\prime},m^{\prime}}\big{)}\cong\bigoplus_{n\geq
0}\operatorname{RHom}_{\mathcal{P}}\big{(}S_{l,m},\,S_{l^{\prime},m^{\prime}}\otimes\operatorname{Sym}^{n}S^{\oplus
7}\big{)}.$

We check the following very straightforward bound. This will give the
vanishing for the $\operatorname{Sym}^{0}$ piece: the vanishing for the
$\operatorname{Sym}^{n}$ piece with $n>0$ will then follow by induction.

######  Lemma 4.11.

For two generators $S_{l,m}$ and $S_{l^{\prime},m^{\prime}}$ of
$\mathbb{W}^{\underline{m}}$ we have

$m^{\prime}-m<\max(l^{\prime}-l,0)+7.$

###### Proof.

By assumption

$m\in(m_{l}-7,m_{l}]\qquad\text{and}\qquad
m^{\prime}\in(m_{l^{\prime}}-7,m_{l^{\prime}}].$

Notice then that

$\displaystyle
m^{\prime}-m<m_{l^{\prime}}-(m_{l}-7)=(m_{l^{\prime}}-m_{l})+7.$

Recall that the $m_{l}$ form a non-decreasing sequence with $m_{l+1}\leq
m_{l}+1$. If $l^{\prime}\geq l$ then

$m_{l^{\prime}}-m_{l}\leq l^{\prime}-l,$

and otherwise $m_{l^{\prime}}-m_{l}\leq 0$, hence the claim. ∎

######  Remark 4.12.

The collections $\mathbb{W}^{(0,0,0)}$ and $\mathbb{W}^{(-2,-1,0)}$ (given as
Collections  ‣ 2 and 4 in Section 2) show that, amongst such bounds which are
uniform for all the $\mathbb{W}^{\underline{m}}$, the bound of Lemma 4.11 is
the best possible.

For our induction, we now observe that

$\displaystyle S_{l,m}\otimes S$
$\displaystyle=\operatorname{Sym}^{l}S(m)\otimes S$
$\displaystyle\cong\big{(}\operatorname{Sym}^{l+1}S\,\oplus\,\operatorname{Sym}^{l}S\otimes\wedge^{2}S\,\big{)}(m)$
$\displaystyle\cong\operatorname{Sym}^{l+1}S(m)\,\oplus\,\operatorname{Sym}^{l}S(m-1)$
$\displaystyle=S_{l+1,m}\oplus S_{l,m-1}.$ (4.4)

Now suppose we use this repeatedly to expand the $\operatorname{Sym}^{n}$
piece, namely

$\operatorname{RHom}_{\mathcal{P}}\big{(}S_{l,m},S_{l^{\prime},m^{\prime}}\otimes\operatorname{Sym}^{n}S^{\oplus
7}\big{)}.$

I claim we get summands

$\operatorname{RHom}_{\mathcal{P}}\big{(}S_{l,m},S_{l^{\prime\prime},m^{\prime\prime}}\big{)}$
(4.5)

for which the inequality

$m^{\prime\prime}-m<\max(l^{\prime\prime}-l,0)+7.$ (4.6)

is satisfied. This follows by induction: the base case $n=0$ is shown in Lemma
4.11, and the inequality still holds when we increment $l^{\prime\prime}$ or
decrement $m^{\prime\prime}$ each time we apply (4.2).

Let us now calculate the cohomology of the summands (4.5). First note

$\displaystyle\operatorname{R\mathcal{H}om}_{\mathcal{P}}\big{(}S_{l,m},S_{l^{\prime\prime},m^{\prime\prime}}\big{)}$
$\displaystyle\cong\operatorname{Sym}^{l}S^{\vee}(-m)\otimes\operatorname{Sym}^{l^{\prime\prime}}S(m^{\prime\prime})$
$\displaystyle\cong\operatorname{Sym}^{l}S(l-m)\otimes\operatorname{Sym}^{l^{\prime\prime}}S(m^{\prime\prime})$
$\displaystyle\cong\big{(}\operatorname{Sym}^{l}S\otimes\operatorname{Sym}^{l^{\prime\prime}}S\big{)}(l-m+m^{\prime\prime}).$

Applying $\mathrm{R}\Gamma_{\mathcal{P}}$ to this, we will obtain
$\operatorname{RHom}_{\mathcal{P}}\cong\mathrm{R}\Gamma_{\mathcal{P}}\circ\operatorname{R\mathcal{H}om}_{\mathcal{P}}$.
Observe then that there is a morphism of stacks
$\delta\colon\mathcal{P}\to\mathbb{P}^{6}$ induced by
$\det\colon\operatorname{GL}(S)\to\mathbb{C}^{*}$, and we have
$\mathrm{R}\Gamma_{\mathcal{P}}\cong\mathrm{R}\Gamma_{\mathbb{P}^{6}}\circ\delta_{*}.$
The functor $\delta_{*}$ takes $\ker(\det)$-invariants where
$\ker(\det)=\operatorname{SL}(S)$. Expanding
$\operatorname{Sym}^{l}S\otimes\operatorname{Sym}^{l^{\prime\prime}}S$ into
irreducibles, the only sheaf that survives this is $(\wedge^{2}S)^{\otimes
l}\cong\mathcal{O}(-l)$ in the case that $l=l^{\prime\prime}$.

We thence deduce that the only contribution to the cohomology of (4.5) comes
from a$\mathrm{R}\Gamma_{\mathcal{P}}\mathcal{O}(-m+m^{\prime\prime})$ s

$\mathrm{R}\Gamma_{\mathcal{P}}\mathcal{O}(-m+m^{\prime\prime})$

in the case $l=l^{\prime\prime}$ and that, setting $n=m^{\prime\prime}-m$,
this contribution is

$\mathrm{R}\Gamma_{\mathbb{P}^{6}}\delta_{*}\mathcal{O}(n)\cong\mathrm{R}\Gamma_{\mathbb{P}^{6}}\mathcal{O}_{\mathbb{P}^{6}}(-n).$
(4.7)

But (4.6) implies that if $l=l^{\prime\prime}$ then $n<7$, so the required
vanishing of higher $\operatorname{Ext}$s in (4.3) follows from standard
cohomology vanishing on $\mathbb{P}^{6}$.

######  Remark 4.13.

The minus sign in (4.7) appears because the bundle $\mathcal{O}(1)$ on

$\mathcal{P}=\big{[}\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}-\\{0\\}\big{/}\,\operatorname{GL}(S)\big{]}$

corresponds to the representation $\wedge^{2}S^{\vee}$ by the definitions at
the beginning of this section. The bundle $\mathcal{O}_{\mathbb{P}^{6}}(1)$,
on the other hand, corresponds to $\wedge^{2}S$.

######  Remark 4.14.

The last part of the argument above, the calculation of the cohomology of the
summands (4.5), parallels the proof of [1, Lemma 4.3]. We include it here to
explain some details, and for convenience.

##  5 Equivalences for Calabi–Yau $\boldsymbol{3}$-folds

We now show that the categories $\mathrm{D^{b}}(X_{\mathrm{G}},f)$ and
$\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$ are equivalent, respectively,
to the derived categories of our Calabi–Yau $3$-folds $Y_{\mathrm{G}}$ and
$Y_{\mathrm{P}}$, completing the proof of Theorem 1.1.

In this section we require the definition of the function $f$ on
$\mathfrak{X}$ from [1], as follows. Take a surjective map
$A\colon\wedge^{2}V\to V$ such that the dual subspaces $\Pi$ and $\Pi^{\circ}$
used to construct the $3$-folds in Section 1.1 are given by

$\Pi=\ker A^{\vee}\qquad\text{and}\qquad\Pi^{\circ}=\operatorname{Im}A.$

Recalling that

$\mathfrak{X}=\big{[}\operatorname{Hom}(S,V)\oplus\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}\big{/}\,\operatorname{GL}(S)\big{]}$

with $x\in\operatorname{Hom}(S,V)$ and
$p\in\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}$ take a function on
$\mathfrak{X}$ defined by

$f(x,p)=p\circ A\circ\wedge^{2}x$

via the canonical isomorphism
$\operatorname{Hom}\big{(}{\wedge}^{2}S,\wedge^{2}S\big{)}\cong\mathbb{C}.$

In addition to the data $(\mathfrak{X},f)$, the Landau–Ginzburg model includes
the data of a $\mathbb{C}^{*}$-action on $\mathfrak{X}$ for which $f$ has
weight $2$. For this, we let $\mathbb{C}^{*}$ act with weight $0$ on $x$, and
weight $2$ on $p$. Note that this action has been suppressed in our notation
so far. Following physics terminology, it is sometimes know as the “R-charge”.
For more details, see [1, Section 2.1].

###  5.1 Grassmannian side

Recall that $X_{\mathrm{G}}$ is by definition the Artin stack

$X_{\mathrm{G}}=\big{[}\big{\\{}(x,p)\in\operatorname{Hom}(S,V)\oplus\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}\colon\operatorname{rk}(x)=2\big{\\}}\big{/}\operatorname{GL}(S)\big{]},$

which is isomorphic to the total space of the vector bundle
$\mathcal{O}(-1)^{\oplus 7}$ over the Grassmannian $\mathrm{G}(2,V)$. Denote
the projection morphism for this bundle by $\pi$. The Calabi–Yau 3-fold
$Y_{\mathrm{G}}$ defined in Section 1.1 is the zero locus of a transverse
section of $\mathcal{O}(1)^{\oplus 7}$ given by

$s=A\circ\wedge^{2}x.$

Then we have $f=sp$, where $p$ is the tautological section of
$\pi^{*}\mathcal{O}(-1)^{\oplus 7}$. The $\mathbb{C}^{*}$-action preserves
each fibre of the bundle, and acts with weight $2$ on those fibres.

Given this geometric situation, an equivalence from
$\mathrm{D^{b}}(Y_{\mathrm{G}})$ to $\mathrm{D^{b}}(X_{\mathrm{G}},f)$ follows
from a version of Knörrer periodicity, as follows. Let
$X_{\mathrm{G}}|_{Y_{\mathrm{G}}}$ denote the base change of the bundle
$X_{G}$ over $G$ to the base $Y_{G}$. Then we have

$Y_{\mathrm{G}}\stackrel{{\scriptstyle\pi}}{{\longleftarrow}}X_{\mathrm{G}}|_{Y_{\mathrm{G}}}\stackrel{{\scriptstyle
k}}{{\longrightarrow}}X_{\mathrm{G}},$

where $k$ is the inclusion.

######  Definition 5.1.

We define an equivalence $\Psi_{\mathrm{G}}$ by the composition

$\Psi_{\mathrm{G}}\colon\
\mathrm{D^{b}}(Y_{\mathrm{G}})\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}\mathrm{D^{b}}(X_{\mathrm{G}}|_{Y_{\mathrm{G}}})\stackrel{{\scriptstyle
k_{*}}}{{\longrightarrow}}\mathrm{D^{b}}(X_{\mathrm{G}},f).$

This is an equivalence by a result of Shipman [19, Theorem 3.4].

###  5.2 Calculations

Later, in Section 7, I study certain bundles on $Y_{\mathrm{G}}$ to prove
Theorem 1.4: for use there, in this subsection I calculate the images of these
objects under $\Psi_{\mathrm{G}}$. I first give the method in a simpler case,
before applying it to $Y_{\mathrm{G}}$.

######  Example 5.2.

Consider a line bundle $L$ over a variety $B$, and a subvariety $Y\subset B$
cut out by a transverse section $s$ of $L$. Let $X$ be the total space of
$L^{\vee}$, with projection $\pi$, and $p$ the tautological section of
$\pi^{*}L^{\vee}$. We make the setup $\mathbb{C}^{*}$-equivariant so that $s$
and $p$ have weights $0$ and $2$, respectively. Take notation as follows,
where $X|_{Y}$ is the total space of the base change of the bundle $L$ over
$B$ to $Y$, and $j$ is the inclusion of the zero section $B$ in $X$:

$X$$X|_{Y}$$B$$Y$$\scriptstyle k$$\scriptstyle
k$$\scriptstyle\pi$$\scriptstyle j$$\scriptstyle\pi$

Then in $\mathrm{D^{b}}(X,f)$, where $f=sp$, we have an object $\mathcal{K}$
given by

$\pi^{*}L^{\vee}[-1]$$\mathcal{O}_{X}$,$\scriptstyle s$$\scriptstyle p$

where the shift $[-1]$ denotes a change in $\mathbb{C}^{*}$-weight. Forgetting
the right- and left-moving morphisms respectively in $\mathcal{K}$ gives
Koszul resolutions for $\mathbb{C}^{*}$-equivariant sheaves

$\pi^{*}k_{*}\mathcal{O}_{Y}\qquad\text{and}\qquad
j_{*}\mathcal{O}_{B}\otimes\pi^{*}L^{\vee}[-1].$ (5.1)

Note that these determine objects of $\mathrm{D^{b}}(X,f)$ because they are
supported on the zero locus of $f$, namely $X|_{Y}\cup B$. Indeed, they may be
written as pushforwards from this locus using isomorphisms

$\pi^{*}k_{*}\mathcal{O}_{Y}\cong
k_{*}\pi^{*}\mathcal{O}_{Y}\qquad\text{and}\qquad
j_{*}\mathcal{O}_{B}\otimes\pi^{*}L^{\vee}\cong
j_{*}\big{(}\mathcal{O}_{B}\otimes j^{*}\pi^{*}L^{\vee}\big{)}\cong
j_{*}\big{(}\mathcal{O}_{B}\otimes L^{\vee}\big{)}$

which follow by flat base change, and the projection formula for $j$,
respectively. Now, by a standard argument with matrix factorizations, both the
objects of (5.1) are isomorphic to $\mathcal{K}$ in $\mathrm{D^{b}}(X,f)$: see
for instance [19, proof of Lemma 3.2]. Putting all this together, we obtain an
isomorphism in $\mathrm{D^{b}}(X,f)$ as follows.

$k_{*}\pi^{*}\mathcal{O}_{Y}\cong j_{*}(\mathcal{O}_{B}\otimes L^{\vee})[-1].$

We generalize the method of this example to deduce the following.

######  Proposition 5.3.

We have that

$\displaystyle\Psi_{\mathrm{G}}\colon\ \mathcal{O}_{Y_{\mathrm{G}}}$
$\displaystyle\mapsto j_{*}\mathcal{O}_{\mathrm{G}}(-7)[-7]\cong
j_{!}\mathcal{O}_{\mathrm{G}},$

where $j$ is the inclusion of the zero section of $X_{\mathrm{G}}$, and
similarly with occurences of $\mathcal{O}$ replaced with $S$ or
$\operatorname{Sym}^{2}S$.

###### Proof.

Take $F=\mathcal{O}(1)^{\oplus 7}$, a bundle on $\mathrm{G}(2,V)$ of rank
$r=\dim V=7$, and set notation for morphisms as follows:

$X_{\mathrm{G}}$$X_{\mathrm{G}}|_{Y_{\mathrm{G}}}$$\mathrm{G}$$Y_{\mathrm{G}}$$\scriptstyle
k$$\scriptstyle\pi$$\scriptstyle j$$\scriptstyle\pi$

Following the argument of the example above, we take $\mathcal{K}$ in
$\mathrm{D^{b}}(X_{\mathrm{G}},f)$ as follows:

$\wedge^{r}\pi^{*}F^{\vee}[-r]$$\dots$$\pi^{*}F^{\vee}[-1]$$\mathcal{O}_{X_{\mathrm{G}}}$,$\scriptstyle
s$$\scriptstyle p$$\scriptstyle s$$\scriptstyle p$$\scriptstyle
s$$\scriptstyle p$

which gives an isomorphism of objects

$k_{*}\pi^{*}\mathcal{O}_{Y_{\mathrm{G}}}\cong
j_{*}\big{(}\mathcal{O}_{\mathrm{G}}\otimes\det F^{\vee}\big{)}[-r].$

Recalling that $j_{!}=j_{*}(\omega_{j}[\dim j]\otimes-)$, and noting that
$\omega_{j}=\det F^{\vee}$ and $\dim j=-r$, the first claim follows. Twisting
$\mathcal{K}$ by $S$ or $\operatorname{Sym}^{2}S$ then gives the further
claim.∎

###  5.3 Pfaffian side

Recall that $X_{\mathrm{P}}$ is the Artin stack

$X_{\mathrm{P}}=\big{[}\big{\\{}(x,p)\in\operatorname{Hom}(S,V)\oplus\operatorname{Hom}\big{(}V,\wedge^{2}S\big{)}\colon
p\neq 0\big{\\}}\big{/}\operatorname{GL}(S)\big{]}.$

Given $A\colon\wedge^{2}V\to V$, each value of $p$ determines a
$\wedge^{2}S$-valued 2-form on $V$, namely $\omega_{p}=p\circ A$. We then have
that $f(x,p)=\omega_{p}\circ\wedge^{2}x$ by the definition of $f$.

In this subsection, we construct an equivalence

$\mathrm{D^{b}}(Y_{\mathrm{P}})\to\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$

for each choice of $\underline{m}$. In [1], such an equivalence was
constructed for $\underline{m}={(6,7,8)}$ by considering isotropic subspaces
for the $\omega_{p}$ as $p$ varies. An outline is given in [1, Section 5.1]
with details in the subsections following. The argument is lengthy and subtle,
but to obtain an equivalence for general $\underline{m}$ I only need to modify
the very last step of it: this subsection explains the modification.

The argument uses the following locus in $X_{\mathrm{P}}$. Note that
$X_{\mathrm{P}}$ is a bundle over $\mathcal{P}$ and also over $\mathbb{P}^{6}$
by composition with the morphism $\delta\colon\mathcal{P}\to\mathbb{P}^{6}$
induced by $\det\colon\operatorname{GL}(S)\to\mathbb{C}^{*}$.

######  Definition 5.4 ([1, Definition 5.5]).

Let $\Gamma\subset X_{\mathrm{P}}$ be the closed substack with points $(x,p)$,
where $p$ corresponds to a point of $Y_{\mathrm{P}}\subset\mathbb{P}^{6}$, and
the composition of $x$ with a quotient morphism

$S\overset{x}{\longrightarrow}V\to V/\ker\omega_{p}$

has rank at most 1.

######  Remark 5.5.

Note that, when $\operatorname{rk}(x)=2$, the above condition matches the
condition defining the correspondence between $Y_{\mathrm{G}}$ and
$Y_{\mathrm{P}}$ which appears in Borisov and Căldăraru’s work [3, Section
0.7]. They show that the ideal sheaf of this correspondence gives a derived
equivalence.

Note that $\Gamma$ is a flat family of stacks over $Y_{\mathrm{P}}$. The
following proposition gives the construction of
$\Psi_{\mathrm{P}}^{\underline{m}}$. Take notation as below, where $j$ and $k$
denote inclusions:

$X_{\mathrm{P}}$$X_{\mathrm{P}}|_{Y_{\mathrm{P}}}$$\mathbb{P}^{6}$$Y_{\mathrm{P}}$$\Gamma$$\scriptstyle
k$$\scriptstyle k$$\scriptstyle\pi$$\scriptstyle\pi$$\scriptstyle j$

######  Proposition 5.6.

There exists an equivalence

$\Psi_{\mathrm{P}}^{\underline{m}}=k_{*}\circ j_{*}\circ
j^{*}\circ\pi^{*}\colon\
\mathrm{D^{b}}(Y_{\mathrm{P}})\overset{\sim}{\longrightarrow}\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)\subset\mathrm{D^{b}}(X_{\mathrm{P}},f).$

###### Proof.

I explain how choices can be made so that the given composition has essential
image in $\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$. This refines the
argument of [1, proof of Proposition 5.9].

I first explain why, for $\mathcal{E}$ a sheaf on $Y_{P}$, we may make choices
so that
$\Psi_{\mathrm{P}}^{\underline{m}}(\mathcal{E})\in\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$.
The result then follows by generation. Using that $\pi j$ is flat,
$\Psi_{\mathrm{P}}^{\underline{m}}(\mathcal{E})$ is a sheaf on
$X_{\mathrm{P}}$. Furthermore, by the projection formula,

$\Psi_{\mathrm{P}}^{\underline{m}}(\mathcal{E})=k_{*}j_{*}j^{*}{\pi}^{*}(\mathcal{E})\cong
k_{*}(j_{*}\mathcal{O}_{\Gamma}\otimes\pi^{*}\mathcal{E}).$

Now $j_{*}\mathcal{O}_{\Gamma}$ has an Eagon–Northcott resolution

$\wedge^{4}Q^{\vee}\otimes\operatorname{Sym}^{2}S(1)\to\wedge^{3}Q^{\vee}\otimes
S(1)\to\wedge^{2}Q^{\vee}(1)\to\mathcal{O},$

where $Q=V/\ker\omega_{p}$, by for instance [20, Section 6.1.6]. This may be
made $\mathbb{C}^{*}$-equivariant by adding appropriate shifts of
$\mathbb{C}^{*}$-weight, and it follows that
$\Psi_{\mathrm{P}}^{\underline{m}}(\mathcal{E})$ has a
$\mathbb{C}^{*}$-equivariant resolution

$\pi^{*}\mathcal{F}_{2}\otimes\operatorname{Sym}^{2}S\to\pi^{*}\mathcal{F}_{1}\otimes
S\to\pi^{*}\mathcal{F}_{0}\to\pi^{*}\mathcal{F},$

where $\mathcal{F}$ and $\mathcal{F}_{0},\ldots,\mathcal{F}_{2}$ are sheaves
in the image of
$j_{*}\colon\operatorname{Coh}(Y_{\mathrm{P}})\to\operatorname{Coh}(\mathbb{P}^{6})$.
Using a Beilinson collection on $\mathbb{P}^{6}$, each of these $\mathcal{F}$s
can be replaced by a quasi-isomorphic complex whose terms are direct sums of
line bundles $\mathcal{O}_{\mathbb{P}^{6}}(-m)$ with $m\in(n-7,n]$, for any
choice of integer $n$.

Now $\pi^{*}\mathcal{O}_{\mathbb{P}^{6}}(-m)\cong\mathcal{O}(m)$, with the
sign appearing as in Remark 4.13. Therefore, we may obtain a resolution of
$\Psi_{\mathrm{P}}^{\underline{m}}(\mathcal{E})$ by vector bundles in the
window subcategory $\mathbb{W}^{\underline{m}}$, by letting $n=m_{l}$ for
$\mathcal{F}_{l}$, and $n=m_{0}$ for $\mathcal{F}$. Doing likewise for
$\mathcal{E}$ ranging over a set of generators for
$\mathrm{D^{b}}(Y_{\mathrm{P}})$, it follows that
$\Psi_{\mathrm{P}}^{\underline{m}}$ may be constructed to have essential image
in $\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f)$. The proof of the
equivalence property is exactly as in [1, Theorem 5.12]. ∎

######  Remark 5.7.

A different construction of such a functor is given in [17, Section 4.2],
avoiding much of the difficult analysis in [1]. However, the discussion in the
latter suffices for our purposes.

###  5.4 Equivalences

We now complete the proof of Theorem 1.1.

As before, let $\underline{m}=(m_{0},m_{1},m_{2})$ be a non-decreasing
sequence of integers such that $m_{l+1}\leq m_{l}+1$.

######  Theorem 5.8.

For each $\underline{m}$ as above, there is an equivalence

$\Psi^{\underline{m}}\colon\
\mathrm{D^{b}}(Y_{\mathrm{G}})\overset{\sim}{\longrightarrow}\mathrm{D^{b}}(Y_{\mathrm{P}}),$

which factors through the subcategory $\mathbb{W}^{\underline{m}}$ of
$\mathrm{D^{b}}(\mathfrak{X},f)$.

###### Proof.

Recall that in the previous section we obtained (Definition 4.7) an
equivalence as follows.

$\Psi_{\mathbb{W}}^{\underline{m}}=i_{\mathrm{P}}^{*}\circ(i_{\mathrm{G}}^{*})^{-1}\colon\
\mathrm{D^{b}}(X_{\mathrm{G}},f)\to\mathbb{W}^{\underline{m}}\to\mathrm{Br}^{\underline{m}}(X_{\mathrm{P}},f).$

Composing this with the equivalences of this section, from Definition 5.1 and
Proposition 5.6, we get

$\Psi^{\underline{m}}=(\Psi_{\mathrm{P}}^{\underline{m}})^{-1}\circ\Psi_{\mathbb{W}}^{\underline{m}}\circ\Psi_{\mathrm{G}}\colon\
\mathrm{D^{b}}(Y_{\mathrm{G}})\to\mathrm{D^{b}}(Y_{\mathrm{P}})$

and the theorem is proved. ∎

##  6 Groupoid action

In this section I construct the groupoid action of Theorem 1.2. Let us first
recall the notation which was used to state this theorem in Section 1.4.

######  Notation 6.1.

For brevity, we write windows, and equivalences,

$\begin{split}\mathbb{W}^{0}&=\mathbb{W}^{(0,0,0)},\\\
\mathbb{W}^{1}&=\mathbb{W}^{(-1,0,0)},\\\
\mathbb{W}^{2}&=\mathbb{W}^{(-1,-1,0)},\\\
\mathbb{W}^{3}&=\mathbb{W}^{(-1,-1,-1)},\end{split}\qquad\qquad\begin{split}\Psi^{0}&=\Psi^{(0,0,0)},\\\
\Psi^{1}&=\Psi^{(-1,0,0)},\\\ \Psi^{2}&=\Psi^{(-1,-1,0)},\\\
\Psi^{3}&=\Psi^{(-1,-1,-1)},\end{split}$

and similarly write $\Psi^{l}_{\mathbb{W}}$ for $l=0,\dots,3.$

Theorem 1.2 will follow from the fact that
$\mathbb{W}^{3}=\mathbb{W}^{0}\otimes\mathcal{O}(-1)$, and the following
proposition.

######  Proposition 6.2.

There are natural isomorphisms as follows

$\displaystyle(-\otimes\mathcal{O}_{X_{\mathrm{G}}}(1))\circ\Psi_{\mathrm{G}}\cong\Psi_{\mathrm{G}}\circ(-\otimes\mathcal{O}_{Y_{\mathrm{G}}}(1)),$
$\displaystyle(-\otimes\mathcal{O}_{X_{\mathrm{P}}}(1))\circ\Psi_{\mathrm{P}}\cong\Psi_{\mathrm{P}}\circ(-\otimes\mathcal{O}_{Y_{\mathrm{P}}}(-1)).$

###### Proof.

Recall from Definition 5.1 that $\Psi_{\mathrm{G}}$ is the composition

$\mathrm{D^{b}}(Y_{\mathrm{G}})\stackrel{{\scriptstyle\pi^{*}}}{{\longrightarrow}}\mathrm{D^{b}}(X_{\mathrm{G}}|_{Y_{\mathrm{G}}})\stackrel{{\scriptstyle
k_{*}}}{{\longrightarrow}}\mathrm{D^{b}}(X_{\mathrm{G}},f),$

where morphisms are as follows:

$X_{\mathrm{G}}$$X_{\mathrm{G}}|_{Y_{\mathrm{G}}}$$\mathrm{G}$$Y_{\mathrm{G}}$$\scriptstyle
k$$\scriptstyle\pi$

Now assume given sheaves $\mathcal{L}$ on $X_{\mathrm{G}}$ and $\mathcal{M}$
on $Y_{\mathrm{G}}$ such that $k^{*}\mathcal{L}\cong\pi^{*}\mathcal{M}$. Then,
using the projection formula, we have isomorphisms as follows:

$\displaystyle(-\otimes\mathcal{L})\circ\Psi_{\mathrm{G}}$
$\displaystyle=(-\otimes\mathcal{L})\circ(k_{*}\pi^{*})\cong
k_{*}\circ(-\otimes k^{*}\mathcal{L})\circ\pi^{*}$ $\displaystyle\cong
k_{*}\circ(-\otimes\pi^{*}\mathcal{M})\circ\pi^{*}\cong(k_{*}\pi^{*})\circ(-\otimes\mathcal{M})$
$\displaystyle\cong\Psi_{\mathrm{G}}\circ(-\otimes\mathcal{M}).$

We immediately deduce the first statement, by taking
$\mathcal{L}=\mathcal{O}_{X_{\mathrm{G}}}(1)$ and
$\mathcal{M}=\mathcal{O}_{Y_{\mathrm{G}}}(1)$.

From Proposition 5.6 we have that $\Psi_{\mathrm{P}}^{\underline{m}}$ is
isomorphic to the composition

$\mathrm{D^{b}}(Y_{\mathrm{P}})\xrightarrow{(\pi
j)^{*}}\mathrm{D^{b}}(\Gamma)\xrightarrow{(kj)_{*}}\mathrm{D^{b}}(X_{\mathrm{P}},f),$

where morphisms are as shown below. Here we write $\mathrm{D^{b}}(\Gamma)$
rather than $\mathrm{D^{b}}(\Gamma,f)$ because $f$ restricts to zero on
$\Gamma\subset X_{\mathrm{P}}$ by Definition 5.4.

$X_{\mathrm{P}}$$X_{\mathrm{P}}|_{Y_{\mathrm{P}}}$$\mathbb{P}^{6}$$Y_{\mathrm{P}}$$\Gamma$$\scriptstyle
k$$\scriptstyle\pi$$\scriptstyle j$

Applying a similar argument to $\Psi_{\mathrm{P}}$ using sheaves
$\mathcal{L}=\mathcal{O}_{X_{\mathrm{P}}}(1)$ and
$\mathcal{M}=\mathcal{O}_{Y_{\mathrm{P}}}(-1)$, we deduce the second
statement. In this case it suffices that
$k^{*}\mathcal{L}\cong\pi^{*}\mathcal{M}$, as this gives
$(kj)^{*}\mathcal{L}\cong(\pi j)^{*}\mathcal{M}$ after applying $j^{*}$. The
minus sign here arises as in Remark 4.13. ∎

######  Theorem 6.3.

There is an action of the groupoid
$\pi_{1}(\mathcal{M},\\{m_{\mathrm{G}},m_{\mathrm{P}}\\})$ for
$\mathcal{M}=S^{2}-\\{\text{$5$ points}\\}$ on
$\mathrm{D^{b}}(Y_{\mathrm{G}})$ and $\mathrm{D^{b}}(Y_{\mathrm{P}})$, given
by the following diagram.

$\mathcal{M}$$\scriptstyle m_{\mathrm{G}}$$\scriptstyle
m_{\mathrm{P}}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{G}}}\\!(1)$$\scriptstyle\Psi^{0}$$\scriptstyle\Psi^{1}$$\scriptstyle\Psi^{2}$$\scriptstyle\Psi^{3}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{P}}}\\!(1)$

###### Proof.

Removing a point from the far side of the sphere, the claim is that the
groupoid acts by the following diagram, subject to a relation: that monodromy
starting at either $\mathrm{D^{b}}(Y_{\mathrm{G}})$ or
$\mathrm{D^{b}}(Y_{\mathrm{P}})$ and then going around a large circle gives
the identity up to isomorphism:

$\mathrm{D^{b}}(Y_{\mathrm{G}})$$\mathrm{D^{b}}(Y_{\mathrm{P}})$$\Psi^{3}$$\Psi^{2}$$\Psi^{1}$$\Psi^{0}$$\otimes\mathcal{O}_{Y_{\mathrm{P}}}(1)$$\otimes\mathcal{O}_{Y_{\mathrm{G}}}(1)$

Now we have that
$\mathbb{W}^{3}=\mathbb{W}^{0}\otimes\mathcal{O}_{\mathfrak{X}}(-1)$, and note
that

$i_{\mathrm{G}}^{*}\mathcal{O}_{\mathfrak{X}}(m)\cong\mathcal{O}_{X_{\mathrm{G}}}(m)\qquad\text{and}\qquad
i_{\mathrm{P}}^{*}\mathcal{O}_{\mathfrak{X}}(m)\cong\mathcal{O}_{X_{\mathrm{P}}}(m).$

It then follows easily from Definition 4.7 of the window equivalences
$\Psi_{\mathbb{W}}^{\underline{m}}$, of which the $\Psi_{\mathbb{W}}^{l}$ are
examples, that

$\Psi_{\mathbb{W}}^{3}\cong(-\otimes\mathcal{O}_{X_{\mathrm{P}}}(-1))\circ\Psi_{\mathbb{W}}^{0}\circ(-\otimes\mathcal{O}_{X_{\mathrm{G}}}(1)),$

noting the opposite signs in the line bundle twists. Combining with the above
Proposition 6.2 then gives an isomorphism

$\Psi^{3}\cong(-\otimes\mathcal{O}_{Y_{\mathrm{P}}}(1))\circ\Psi^{0}\circ(-\otimes\mathcal{O}_{Y_{\mathrm{G}}}(1)),$

which yields the required relation, and completes the proof.∎

##  7 Monodromy action

In this section I describe the differences between the equivalences $\Psi^{l}$
in Theorem 1.2 (Theorem 6.3 above) as spherical twists on the Calabi–Yau
$Y_{\mathrm{G}}$, and thereby prove Theorem 1.4.

To describe the difference $\Psi^{\prime-1}\circ\Psi$ between such
equivalences $\Psi$ and $\Psi^{\prime}$, we will use a functor
$\operatorname{Tr}$ that “transfers” between the two corresponding windows
$\mathbb{W}$ and $\mathbb{W}^{\prime}$. The difference is then described as a
twist functor, denoted $\operatorname{Tw}$. The following proposition gives
the formal part of this procedure.

For convenience, take two functors from
$\mathbb{W}=\mathbb{W}^{\underline{m}}$ as follows:

$\Phi_{\mathrm{G}}=(\Psi_{\mathrm{G}})^{-1}\circ
i_{\mathrm{G}}^{*},\qquad\Phi_{\mathrm{P}}=(\Psi^{\underline{m}}_{\mathrm{P}})^{-1}\circ
i_{\mathrm{P}}^{*}$

so that the factorization of $\Psi=\Psi^{\underline{m}}$ via $\mathbb{W}$
takes the form

$\Psi\cong\Phi_{\mathrm{P}}\circ(\Phi_{\mathrm{G}})^{-1}\colon\
\mathrm{D^{b}}(Y_{\mathrm{G}})\to\mathbb{W}\to\mathrm{D^{b}}(Y_{\mathrm{P}}).$

######  Proposition 7.1 ([5, Proposition 2.2]).

For windows $\mathbb{W}$ and $\mathbb{W}^{\prime}$, assume given a functor
$\operatorname{Tr}\colon\mathbb{W}\to\mathbb{W}^{\prime}$ that intertwines
with an autoequivalence $\operatorname{Tw}$ of
$\mathrm{D^{b}}(Y_{\mathrm{G}})$ and with the identity on
$\mathrm{D^{b}}(Y_{\mathrm{P}})$, namely

$\Phi_{\mathrm{G}}\circ\operatorname{Tr}\cong\operatorname{Tw}\circ\,\Phi_{\mathrm{G}}\qquad\text{and}\qquad\Phi^{\prime}_{\mathrm{P}}\circ\operatorname{Tr}\cong\Phi_{\mathrm{P}}.$
(7.1)

Then there is an isomorphism

$\Psi^{\prime-1}\circ\Psi\cong\operatorname{Tw},$

where $\Psi$ and $\Psi^{\prime}$ are the window equivalences associated to
$\mathbb{W}$ and $\mathbb{W}^{\prime}$.

###### Proof.

This is a diagram chase of functors. In the reference $\operatorname{Tr}$ was
assumed to be a restriction of an endofunctor of the category containing
$\mathbb{W}$, in our case $\mathrm{D^{b}}(\mathfrak{X},f)$, but the proof
proceeds without this assumption. ∎

To construct functors $\operatorname{Tr}$ we define endofunctors of
$\mathrm{D^{b}}(\mathfrak{X},f)$, and then show that they restrict to functors
between windows. Recall that $\mathfrak{X}$ is a vector bundle over a stack

$\mathcal{G}=\big{[}\operatorname{Hom}(S,V)\big{/}\,\operatorname{GL}(S)\big{]}$

and write $j\colon\mathcal{G}\to\mathfrak{X}$ for the inclusion of the zero
section. Recall also that

$j_{!}=j_{*}(\omega_{j}[\dim j]\otimes-)\cong
j_{*}(\mathcal{O}_{\mathcal{G}}(-7)[-7]\otimes-),$

where we use that $\omega_{j}$ is the determinant of the normal bundle of
$\mathcal{G}$.

######  Definition 7.2.

Take endofunctors of $\mathrm{D^{b}}(\mathfrak{X},f)$

$\operatorname{Tr}^{0}=\operatorname{Tr}(j_{!}\mathcal{O}_{\mathcal{G}}),\qquad\operatorname{Tr}^{1}=\operatorname{Tr}(j_{!}S),\qquad\operatorname{Tr}^{2}=\operatorname{Tr}\big{(}j_{!}S^{2}\big{)},$

where

$\operatorname{Tr}(\mathcal{E})=\mathrm{Cone}\big{(}{\mathcal{E}}\otimes\operatorname{Hom}_{\mathfrak{X}}({\mathcal{E}},-)\to\operatorname{id}\big{)}.$

Here, and elsewhere in this section, we again put $S^{2}$ for
$\operatorname{Sym}^{2}S$.

######  Remark 7.3.

The functorial cone above is shorthand for the usual Fourier–Mukai
constructions.

I then claim the following.

######  Proposition 7.4.

The $\operatorname{Tr}^{l}$ restrict to functors
$\operatorname{Tr}^{l}\colon\mathbb{W}^{l}\to\mathbb{W}^{l+1}.$

###### Proof.

I first show this for $\operatorname{Tr}^{0}$, as the others are similar. I
claim that the functor acts as the identity on all generators of the window
$\mathbb{W}^{0}$ except $\mathcal{O}$. First observe that

$\displaystyle\operatorname{Hom}_{\mathfrak{X}}(j_{!}\mathcal{O}_{\mathcal{G}},-)$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{G}}(\mathcal{O}_{\mathcal{G}},j^{*}-).$
(7.2)

This is zero except on $\mathcal{O}$ by inspection of Collection  ‣ 2, noting
the vanishing in Corollary 4.9. It also implies, using that objects in the
collection are exceptional, that

$\operatorname{Hom}_{\mathfrak{X}}(j_{!}\mathcal{O}_{\mathcal{G}},\mathcal{O})=\mathbb{C}.$

A generator of this $\operatorname{Hom}$ may be seen explicitly by writing
down the Koszul resolution of $j_{!}\mathcal{O}_{\mathcal{G}}$, namely

$\mathcal{O}\to\mathcal{O}(-1)^{\oplus 7}\to\dots\to\mathcal{O}(-6)^{\oplus
7}\to\mathcal{O}(-7).$ (7.3)

We thus see that
$\operatorname{Tr}^{0}(\mathcal{O})=\mathrm{Cone}(j_{!}\mathcal{O}_{\mathcal{G}}\to\mathcal{O})$
is quasi-isomorphic to a complex given by

$\mathcal{O}(-1)^{\oplus 7}\to\dots\to\mathcal{O}(-6)^{\oplus
7}\to\mathcal{O}(-7)$

which, in particular, lies in the window $\mathbb{W}^{1}$ given as Collection
1.

For the other two functors $\operatorname{Tr}^{l}$ for $l=1,2$, we replace
(7.2) with

$\displaystyle\operatorname{Hom}_{\mathfrak{X}}\\!\big{(}j_{!}S^{l},-\big{)}$
$\displaystyle\cong\operatorname{Hom}_{\mathcal{G}}\\!\big{(}S^{l},j^{*}\\!-\\!\big{)}$

and repeat the same argument, where we tensor (7.3) by $S^{l}$. ∎

I now complete the proof of the assumptions of Proposition 7.1, in particular
property (7.1), after constructing autoequivalences of
$\mathrm{D^{b}}(Y_{\mathrm{G}})$ as follows.

######  Definition 7.5.

Take endofunctors of $\mathrm{D^{b}}(Y_{\mathrm{G}})$

$\operatorname{Tw}^{0}=\operatorname{Tw}(\mathcal{O}_{Y_{\mathrm{G}}}),\qquad\operatorname{Tw}^{1}=\operatorname{Tw}(S_{Y_{\mathrm{G}}}),\qquad\operatorname{Tw}^{2}=\operatorname{Tw}\big{(}S^{2}_{Y_{\mathrm{G}}}\big{)},$

where

$\operatorname{Tw}(\mathcal{F})=\mathrm{Cone}\big{(}\mathcal{F}\otimes\operatorname{Hom}_{Y_{\mathrm{G}}}(\mathcal{F},-)\to\operatorname{id}\big{)}.$

######  Proposition 7.6.

Each of the $\operatorname{Tw}^{l}$ is a twist by a spherical object, and
therefore is an autoequivalence.

###### Proof.

To see this for $\operatorname{Tw}^{0}$, write $k\colon
Y_{\mathrm{G}}\hookrightarrow\mathrm{G}$ and note that

$\operatorname{Ext}^{\bullet}_{Y_{\mathrm{G}}}(\mathcal{O}_{Y_{\mathrm{G}}},\mathcal{O}_{Y_{\mathrm{G}}})=\operatorname{Ext}^{\bullet}_{Y_{\mathrm{G}}}(k^{*}\mathcal{O}_{\mathrm{G}},k^{*}\mathcal{O}_{\mathrm{G}})\cong\operatorname{Ext}^{\bullet}_{{\mathrm{G}}}(\mathcal{O}_{\mathrm{G}},k_{*}k^{*}\mathcal{O}_{\mathrm{G}}).$

After taking a Koszul resolution of $k_{*}k^{*}\mathcal{O}_{\mathrm{G}}$, this
may be calculated by a spectral sequence from
$\operatorname{Ext}^{\bullet}_{\mathrm{G}}(\mathcal{O}_{\mathrm{G}},\wedge^{\bullet}\mathcal{O}_{\mathrm{G}}(-1)^{\oplus
7})$. By examination of the exceptional collectional given as Collection  ‣ 2,
the only non-trivial terms occuring are
$\operatorname{Ext}^{0}(\mathcal{O}_{\mathrm{G}},\mathcal{O}_{\mathrm{G}})\cong\mathbb{C}$
and

$\operatorname{Ext}^{\bullet}_{\mathrm{G}}(\mathcal{O}_{\mathrm{G}},\mathcal{O}_{\mathrm{G}}(-7))\cong
H^{\bullet}_{\mathrm{G}}(\mathcal{O}_{\mathrm{G}}(-7))\cong\mathbb{C}[-\dim\mathrm{G}]$

by duality. Noting that $Y_{\mathrm{G}}$ is Calabi–Yau, the result follows. A
similar argument applies to the other $\operatorname{Tw}^{l}$. ∎

###### Proof of intertwinement (7.1).

The isomorphism
$\Phi^{\prime}_{\mathrm{P}}\circ\operatorname{Tr}^{0}\cong\Phi_{\mathrm{P}}$
follows immediately from the cone construction of $\operatorname{Tr}^{0}$:
recall that $\Phi_{\mathrm{P}}=(\Psi^{\underline{m}}_{\mathrm{P}})^{-1}\circ
i_{\mathrm{P}}^{*}$ and note that
$i_{\mathrm{P}}^{*}\,j_{!}\,\mathcal{O}_{\mathcal{G}}=0$ for support reasons.
The same argument applies to the other $\operatorname{Tr}^{l}$.

The other intertwinements

$\Phi_{\mathrm{G}}\circ\operatorname{Tr}^{l}\cong\operatorname{Tw}^{l}\circ\,\Phi_{\mathrm{G}}$
(7.4)

are more involved, but essentially follow from the exceptional collection
property. We give the argument for $\operatorname{Tr}^{0}$ before explaining
how it adapts to the other $\operatorname{Tr}^{l}$.

As preparation, recall that $\Phi_{\mathrm{G}}=\Psi_{\mathrm{G}}^{-1}\circ
i_{\mathrm{G}}^{*}$, and write $i$ in place of $i_{\mathrm{G}}$ for clarity.
It will be more convenient to prove the following, which is equivalent to
(7.4).

$i^{*}\circ\operatorname{Tr}^{l}\cong\Psi_{\mathrm{G}}\circ\operatorname{Tw}^{l}\circ\Psi_{\mathrm{G}}^{-1}\circ\,i^{*}.$
(7.5)

By standard facts about spherical twists
$\Psi_{\mathrm{G}}\circ\operatorname{Tw}^{0}\circ\Psi_{\mathrm{G}}^{-1}$ is
given by the spherical twist around
$\Psi_{\mathrm{G}}(\mathcal{O}_{Y_{\mathrm{G}}})$. By Proposition 5.3 this is
isomorphic to $j_{!}\mathcal{O}_{\mathrm{G}}$, and therefore

$\Psi_{\mathrm{G}}\circ\operatorname{Tw}^{0}\circ\Psi_{\mathrm{G}}^{-1}\cong\mathrm{Cone}(j_{!}\mathcal{O}_{\mathrm{G}}\otimes\operatorname{Hom}_{X_{\mathrm{G}}}(j_{!}\mathcal{O}_{\mathrm{G}},-)\to\operatorname{id}).$

Now we have a fibre product diagram

$\mathfrak{X}$$X_{\mathrm{G}}$$\mathcal{G}$$\mathrm{G}$$\scriptstyle
i$$\scriptstyle i$$\scriptstyle j$$\scriptstyle j$

and noting that the $i$ are open immersions and therefore flat, we find by
base change that

$j_{!}\mathcal{O}_{\mathrm{G}}=j_{!}i^{*}\mathcal{O}_{\mathcal{G}}\cong
i^{*}j_{!}\mathcal{O}_{\mathcal{G}}.$

We thence have isomorphisms of functors
$\mathrm{D^{b}}(\mathfrak{X},f)\to\mathrm{D^{b}}(X_{\mathrm{G}},f)$ as
follows:

$\displaystyle
i^{*}\circ\operatorname{Tr}^{0}\cong\mathrm{Cone}(i^{*}j_{!}\mathcal{O}_{\mathcal{G}}\otimes\operatorname{Hom}_{\mathfrak{X}}(j_{!}\mathcal{O}_{\mathcal{G}},-)\to
i^{*}),$
$\displaystyle\Psi_{\mathrm{G}}\circ\operatorname{Tw}^{0}\circ\Psi_{\mathrm{G}}^{-1}\circ\,i^{*}\cong\mathrm{Cone}(i^{*}j_{!}\mathcal{O}_{\mathcal{G}}\otimes\operatorname{Hom}_{X_{\mathrm{G}}}(i^{*}j_{!}\mathcal{O}_{\mathcal{G}},i^{*}-)\to
i^{*}).$

Comparing the two cones above suggests that the intertwinement may follow from
an isomorphism

$\operatorname{Hom}_{\mathfrak{X}}(j_{!}\mathcal{O}_{\mathcal{G}},-)\cong\operatorname{Hom}_{X_{\mathrm{G}}}(i^{*}j_{!}\mathcal{O}_{\mathcal{G}},i^{*}-)\text{
on }\mathbb{W}^{0}.$ (7.6)

This isomorphism does indeed hold: we follow an argument for this, which
furthermore gives (7.5), in [5, proof of Lemma 3.17, end of Section 3.2.2].
(Note that the setting there is more general, involving a spherical functor,
not just a spherical object.) According to the argument, it suffices if the
natural morphism of functors from $\mathrm{D^{b}}(\mathcal{G})$

$\tau\colon\ R\Gamma_{\mathcal{G}}\to R\Gamma_{\mathrm{G}}\circ i^{*}$

is an isomorphism on the subcategory $j^{*}\mathbb{W}^{0}$ of
$\mathrm{D^{b}}(\mathcal{G})$. To see why we take this subcategory, note that
$j^{*}\colon\mathrm{D^{b}}(\mathfrak{X},f)\to\mathrm{D^{b}}(\mathcal{G})$
because $f|_{\mathcal{G}}=0$, and the left-hand side of (7.6) is given by

$\displaystyle\operatorname{Hom}_{\mathfrak{X}}(j_{!}\mathcal{O}_{\mathcal{G}},-)\cong\operatorname{Hom}_{\mathcal{G}}(\mathcal{O}_{\mathcal{G}},j^{*}-)\cong
R\Gamma_{\mathcal{G}}(j^{*}-).$ (7.7)

Now we have that $j^{*}-\cong\mathcal{H}{\rm
om}_{\mathcal{G}}(\mathcal{O},j^{*}-).$ It therefore suffices to check if
$\tau$ is an isomorphism on objects

$\mathcal{A}=\mathcal{H}{\rm om}_{\mathcal{G}}(\mathcal{O},j^{*}\mathcal{B}),$

where $\mathcal{B}$ is a generator of $\mathbb{W}^{0}$.

We determine $\tau_{\mathcal{A}}$. Firstly,
$R^{>0}\Gamma_{\mathcal{G}}(\mathcal{A})=0$ because $\mathcal{A}$ is a vector
bundle on an affine stack. Furthermore,
$R^{>0}\Gamma_{\mathrm{G}}(i^{*}\mathcal{A})\cong\operatorname{Ext}^{>0}_{\mathrm{G}}(\mathcal{O},i^{*}j^{*}\mathcal{B})$.
But then this vanishes by the exceptional collection property, as
$\mathcal{O}$ is on the right-hand side of the collection associated to
$\mathbb{W}^{0}$, namely Collection  ‣ 2. Finally, we have that
$H^{0}\tau_{\mathcal{A}}$ is an isomorphism by normality of $\mathcal{G}$, as
the codimension of $\mathcal{G}-\mathrm{G}$ is greater than two. We deduce
that $\tau_{\mathcal{A}}$ is an isomorphism, and thence $\tau$ is an
isomorphism, thereby proving the intertwinement (7.4) for
$\operatorname{Tr}^{0}$.

For the other transfer functors $\operatorname{Tr}^{1}$ and
$\operatorname{Tr}^{2}$, it now suffices that $\tau$ is an isomorphism on
$S^{\vee}\otimes j^{*}\mathbb{W}^{1}$ and $S^{\vee 2}\otimes
j^{*}\mathbb{W}^{2}$, respectively. These categories appear because the role
of (7.7) is replaced by

$\displaystyle\operatorname{Hom}_{\mathfrak{X}}(j_{!}S,-)\cong
R\Gamma_{\mathcal{G}}\big{(}S^{\vee}\otimes j^{*}(-)\big{)},$
$\displaystyle\operatorname{Hom}_{\mathfrak{X}}\big{(}j_{!}S^{2},{-}\big{)}\cong
R\Gamma_{\mathcal{G}}\big{(}S^{\vee 2}\otimes j^{*}(-)\big{)}.$

But then the above argument suffices, using that $S$ is on the right-hand side
for Collection 1, and $S^{2}$ is on the right-hand side for Collection 2, and
that Proposition 5.3 continues to hold with occurences of $\mathcal{O}$
replaced with $S$ or $S^{2}$.∎

Combining the above, and applying Proposition 7.1, we can describe differences
between equivalences as follows.

######  Proposition 7.7.

The twists $\operatorname{Tw}$ correspond to “window shifts” as follows:

$\displaystyle\big{(}\Psi^{1}\big{)}^{-1}\circ\Psi^{0}\cong\operatorname{Tw}^{0}=\operatorname{Tw}(\mathcal{O}_{Y_{\mathrm{G}}}),$
$\displaystyle\big{(}\Psi^{2}\big{)}^{-1}\circ\Psi^{1}\cong\operatorname{Tw}^{1}=\operatorname{Tw}(S_{Y_{\mathrm{G}}}),$
$\displaystyle\big{(}\Psi^{3}\big{)}^{-1}\circ\Psi^{2}\cong\operatorname{Tw}^{2}=\operatorname{Tw}(S^{2}_{Y_{\mathrm{G}}}).$

We may then complete the proof of Theorem 1.4, as follows.

######  Theorem 7.8.

There is an action of the fundamental group
$\pi_{1}(\mathcal{M},m_{\mathrm{G}})$ on $\mathrm{D^{b}}(Y_{\mathrm{G}})$
given by the following diagram, with $\Psi=\Psi^{3}{:}$

$\mathcal{M}$$\scriptstyle\operatorname{Tw}^{2}$
$\scriptstyle\operatorname{Tw}^{1}$ $\scriptstyle\operatorname{Tw}^{0}$
$\scriptstyle
m_{\mathrm{G}}$$\scriptstyle\otimes\mathcal{O}_{Y_{\mathrm{G}}}\\!(1)$$\scriptstyle\Psi^{-1}(\otimes\mathcal{O}_{Y_{\mathrm{P}}}\\!(1))\Psi$

###### Proof.

We take the groupoid action of Theorem 1.2 (Theorem 6.3) and forget one of the
basepoints $m_{\mathrm{P}}$, as follows. Restricting that action to a chart
containing both basepoints, we obtain the left-hand picture below. Proposition
7.7 then says that monodromies at $\mathrm{D^{b}}(Y_{\mathrm{G}})$ around
equatorial holes are given by the right-hand picture:

$\mathrm{D^{b}}(Y_{\mathrm{G}})$$\operatorname{Tw}^{2}$$\operatorname{Tw}^{1}$$\operatorname{Tw}^{0}$$\mathrm{D^{b}}(Y_{\mathrm{G}})$$\mathrm{D^{b}}(Y_{\mathrm{P}})$$\Psi^{3}$$\Psi^{2}$$\Psi^{1}$$\Psi^{0}$

The remaining two monodromies are immediate, giving the result. ∎

######  Remark 7.9.

It is natural to ask for a Pfaffian analogue of Theorem 7.8 with a basepoint
$m_{\mathrm{P}}$ in place of $m_{\mathrm{G}}$. This seems an interesting and
approachable problem, but it appears that the method here would need
modification to solve it, as follows.

A first step could be to establish spherical objects on the $3$-fold
$Y_{\mathrm{P}}$ and prove an analogue of Proposition 5.3, giving their images
under appropriate equivalences $\Psi_{\mathrm{P}}^{\underline{m}}$. To
continue the strategy above, we would then need manageable resolutions for the
images, as in the Koszul resolution of $j_{!}\mathcal{O}_{\mathcal{G}}$ ina s
(7.3). These should be obtainable, but can be expected to be more complicated
than Koszul resolutions, because of the Eagon–Northcott resolutions in the
construction of $\Psi_{\mathrm{P}}^{\underline{m}}$ from the proof of
Proposition 5.6.

######  Remark 7.10.

I do not discuss whether the action of Theorem 7.8 is faithful. If it is, then
the spherical twists by the objects in Proposition 7.7 act as a free group: it
could be interesting to try to prove this by some B-side analogue of results
of Keating on free group actions by symplectic Dehn twists [12].

### Acknowledgements

In celebration of his 77th birthday, I am pleased to express my gratitude to
Kyoji Saito for his kindness and interest over the years. I also want to thank
N. Addington and E. Segal for the great experience of working on our paper
[1], of which this work is a continuation.

I am supported by the Yau MSC, Tsinghua University, and the Thousand Talents
Plan. I also acknowledge the support of WPI Initiative, MEXT, Japan, and of
EPSRC Programme Grant EP/R034826/1.

I am grateful for conversations with K. Hori, who made me aware of the windows
used here, for discussions with R. Eager, A. Kuznetsov, and M. Romo, and for
helpful suggestions from anonymous referees. In the last stage of this
project, I was away from my home institution due to the coronavirus pandemic,
so I am especially appreciative of hospitality and support from Kavli IPMU, in
particular Y. Ito and M. Kapranov, and from B. Kim at KIAS, and M. Wemyss at
Glasgow University.

## References

  * [1] Addington N., Donovan W., Segal E., The Pfaffian–Grassmannian equivalence revisited, Algebr. Geom. 2 (2015), 332–364, arXiv:1401.3661.
  * [2] Ballard M., Favero D., Katzarkov L., Variation of geometric invariant theory quotients and derived categories, J. Reine Angew. Math. 746 (2019), 235–303, arXiv:1203.6643.
  * [3] Borisov L., Căldăraru A., The Pfaffian–Grassmannian derived equivalence, J. Algebraic Geom. 18 (2009), 201–222, arXiv:math.AG/0608404.
  * [4] Bridgeland T., Flops and derived categories, Invent. Math. 147 (2002), 613–632, arXiv:math.AG/0009053.
  * [5] Donovan W., Segal E., Window shifts, flop equivalences and Grassmannian twists, Compos. Math. 150 (2014), 942–978, arXiv:1206.0219.
  * [6] Eager R., Hori K., Knapp J., Romo M., Beijing lectures on the grade restriction rule, Chin. Ann. Math. Ser. B 38 (2017), 901–912.
  * [7] Halpern-Leistner D., The derived category of a GIT quotient, J. Amer. Math. Soc. 28 (2015), 871–912, arXiv:1203.0276.
  * [8] Herbst M., Hori K., Page D., Phases of $\mathcal{N}=2$ theories in $1+1$ dimensions with boundary, arXiv:0803.2045.
  * [9] Hori K., Grade restriction rule and equivalences of categories, in Kinosaki Symposium in Algebraic Geometry, Kyoto University, 2016, 75–88.
  * [10] Hori K., Tong D., Aspects of non-abelian gauge dynamics in two-dimensional $\mathcal{N}=(2,2)$ theories, J. High Energy Phys. 2007 (2007), no. 5, 079, 41 pages, arXiv:hep-th/0609032.
  * [11] Kapranov M.M., On the derived categories of coherent sheaves on some homogeneous spaces, Invent. Math. 92 (1988), 479–508.
  * [12] Keating A.M., Dehn twists and free subgroups of symplectic mapping class groups, J. Topol. 7 (2014), 436–474, arXiv:1204.2851.
  * [13] Kuznetsov A., Homological projective duality for Grassmannians of lines, arXiv:math.AG/0610957.
  * [14] Kuznetsov A., Exceptional collections for Grassmannians of isotropic lines, Proc. Lond. Math. Soc. 97 (2008), 155–182, arXiv:math.AG/0512013.
  * [15] Rødland E.A., The Pfaffian Calabi–Yau, its mirror, and their link to the Grassmannian $G(2,7)$, Compositio Math. 122 (2000), 135–149, arXiv:math.AG/9801092.
  * [16] Segal E., Equivalence between GIT quotients of Landau–Ginzburg B-models, Comm. Math. Phys. 304 (2011), 411–432, arXiv:0910.5534.
  * [17] Segal E., Thomas R., Quintic threefolds and Fano elevenfolds, J. Reine Angew. Math. 743 (2018), 245–259, arXiv:1410.6829.
  * [18] Seidel P., Thomas R., Braid group actions on derived categories of coherent sheaves, Duke Math. J. 108 (2001), 37–108, arXiv:math.AG/0001043.
  * [19] Shipman I., A geometric approach to Orlov’s theorem, Compos. Math. 148 (2012), 1365–1389, arXiv:1012.5282.
  * [20] Weyman J., Cohomology of vector bundles and syzygies, Cambridge Tracts in Mathematics, Vol. 149, Cambridge University Press, Cambridge, 2003.