# A GLSM realization of derived equivalence
in $U(1)\times U(2)$ models

Jirui<EMAIL_ADDRESS>School of Mathematical Sciences and Institute
for Advanced Study,
Tongji University, Shanghai 200092, China  Yau Mathematical Sciences Center,
Tsinghua University, Beijing 100084, China  Ban Lin
<EMAIL_ADDRESS>Yau Mathematical Sciences Center, Tsinghua
University, Beijing 100084, China  Department of Mathematical Sciences,
Tsinghua University, Beijing 100084, China  School of Mathematics, University
of Birmingham, Birmingham B15 2TT, UK  Hao<EMAIL_ADDRESS>Beijing Institute
of Mathematical Sciences and Applications, Beijing 101408, China  Yau
Mathematical Sciences Center, Tsinghua University, Beijing 100084, China

###### Abstract

The paper [1] has studied a derived equivalence between two Calabi-Yau
fivefolds. In this paper, we propose a new understanding from the viewpoint of
the GLSM, by realizing these two Calabi-Yau fivefolds in two different phases
of a $U(1)\times U(2)$ GLSM. We find that this method can be generalized to
construct other examples of derived equivalence. The functor for the
equivalence can be realized by the brane transport along a path across
different phases. To implement the brane transport, we compute the small
window categories for anomalous $U(2)$ theories, extending previous results on
window categories.

###### Contents

  1. 1 Introduction
  2. 2 Review on hemisphere partition functions and window categories
    1. 2.1 Hemisphere partition functions
    2. 2.2 Window and derived equivalence
    3. 2.3 Big window v.s. small window
  3. 3 Small window of $U(2)$ theories
    1. 3.1 One determinantal field
    2. 3.2 Two determinantal fields
  4. 4 Examples
    1. 4.1 Orbibundle over projective space
    2. 4.2 Product of a projective space and a Grassmannian
    3. 4.3 Two-step flag manifold
    4. 4.4 Derived equivalence between Calabi-Yau 5-folds
  5. 5 Conclusion
  6. A Count the Coulomb vacua
  7. B Phases of the GLSM for $Fl(1,2,N)$

##  1 Introduction

The gauged linear sigma model (GLSM) with $\mathcal{N}=(2,2)$ supersymmetry
has been a thriving research area that spawned many topics in both physics and
mathematics. The boundary conditions of GLSMs that preserve a particular
subset of the supersymmetries, known as B-branes, form a category, which is an
important tool in analyzing the GLSMs. The objects of this category are matrix
factorizations of the superpotential. The Kähler moduli space of a GLSM can be
divided into several phases. If the low energy behaviour of a phase can be
described by a nonlinear sigma model (NLSM) with target space $X$, then the
B-brane category projects onto $D^{b}(X)$ the derived category of $X$ [2, 3,
4]. On the other hand, if the low energy behaviour of a phase is described by
a Landau-Ginzburg (LG) model, then the low energy image of a GLSM brane in
that phase is a matrix factorization of the LG model. Different phases have
different sets of empty brane relations in general. Brane transport provides a
functor between the B-brane categories of different phases of a GLSM [2, 5,
6]. In certain cases, this functor gives rise to an equivalence between the
categories of B-branes. In particular, if two phases are described by two
NLSMs with different target space $X_{+}$ and $X_{-}$ and the functor induced
by the brane transport is an equivalence, then we get a derived equivalence
between $X_{+}$ and $X_{-}$, i.e. $D^{b}(X_{+})\cong D^{b}(X_{-})$. Therefore
GLSMs have been used to study a quite large class of manifolds and their
derived equivalence (see for example [7, 8, 9, 10, 11, 12]).

There are certain constraints on brane transport. The situation depends on
whether the Fayet-Iliopoulos (FI) parameter is renormalized under RG flow
along the direction of the brane transport. In the Calabi-Yau (CY) case (also
known as the non-anomalous case, meaning that the axial U(1) R-symmetry is
anomaly free), in which the FI parameter is marginal, there are a number of
sigular points on the Kähler moduli space (parameterized by the FI parameters)
between the two phases, so the path along which the brane is transported must
avoid these singularities. Given such a path, only the branes satisfying the
grade restriction rule (GRR) can be smoothly transported from one phase to the
other. The branes satisfying the grade restriction rule constitute a
subcategory of the category of matrix factorizations of the GLSM, which is
called the window category [2].

In the non-Calabi-Yau case (also known as the anomalous case, meaning that the
axial $U(1)$ R-symmetry has anomaly), there are no singularities in the
middle, but in order to determine the low energy image of a B-brane, one has
to find an IR equivalent brane that is in the big window category, namely the
category of branes satisfying the GRR. The low energy physics typically
include a Higgs branch and some Coulomb/mixed branches, therefore the low
energy limit of an anomalous GLSM matrix factorization has components on both
Higgs and Coulomb/mixed branches. The small window category is a subcategory
of the big window category, the objects in the small window category only have
components on the Higgs branch in the low energy limit [6]. The grade
restriction rules depend on the choice of the theta angle, and a shift of the
theta-angle by certain amount results in a different but equivalent window
category. In summary,

$\begin{gathered}\text{{Matrix factorization}: GLSM B-branes}\\\ \cup\\\
\text{{Big window}: Branes that flow to both Higgs and mixed branches}\\\
\cup\\\ \text{{Small window}: Branes that only flow to Higgs
branches}\end{gathered}$

The GRR in both CY and non-CY cases have been studied and determined for
abelian GLSMs [2, 5, 6]. However, the GRR for nonabelian GLSMs are not well
understood. At this stage, there exists no general results and one has to
resort to a case-by-case study. There are a few nonabelian examples where the
GRR can be deduced (see, for example [5, 13]). In this paper, we advance this
course by studying the GRR for nonabelian GLSMs with $U(2)$ gauge group and
chiral fields in fundamental and determinantal representations. To get the
result, we follow a categorical reasoning and an analysis of the hemisphere
partition function.

In addition, with the GRR in hand, we study the derived equivalence between
two manifolds, $X_{+}$ and $X_{-}$, by realizing them as the Higgs branches in
different phases of a two-parameter nonabelian GLSM (the GLSM has two
independent FI parameters) whose pure geometric phase realizes another
manifold $\hat{X}$444To be more accurate, when the GLSM is anomalous, we
should first take the gauge-decoupling limit and then consider the IR limit to
see this phase. Please see more discussions on these two limits in [6, section
1].. In the examples we are going to consider, $\hat{X}$ will be either Fano
or CY. Other than the pure geometric phase, there are also two other phases
consisting of the Higgs branches $X_{\pm}$ and mixed branches
$\mathcal{C}_{\pm}$. The derived equivalence between $X_{+}$ and $X_{-}$ is
implemented by the brane transport along a path across the phases of this
GLSM. This idea is sketched in Fig. 1. As a two-parameter model for the
variety $\hat{X}$, it has two paths to transport the branes, one is from
$\hat{X}$ to $(X_{+},\mathcal{C}_{+})$ and the other one is from $\hat{X}$ to
$(X_{-},\mathcal{C}_{-})$. The fact we will use extensively in this paper is
that the total Witten index in different phases are the same and this implies

$\chi(\hat{X})\>=\>\chi(\hat{X}_{+})+\chi(\mathcal{C}_{+})\>=\>\chi(\hat{X}_{-})+\chi(\mathcal{C}_{-})\,.$

With this picture in mind, we will show the derived equivalence between
$X_{+}$ and $X_{-}$ via brane transport: for any object in $D^{b}(X_{+})$, we
first lift it to a GLSM brane, use empty branes to grade restrict it if
necessary, then transport it to the phase of $X_{-}$, where it can be
projected to an object in $D^{b}(X_{-})$. One necessary condition for $X_{+}$
and $X_{-}$ to be derived equivalent is that their Euler characteristics
should be equal to each other, namely, $\chi(X_{+})=\chi(X_{-})$. This
condition implies nontrivial conditions on $X_{+}$ and $X_{-}$, and so on
$\hat{X}$. This point will be more clear in section 4. However, sometimes it
is difficult to compute their characteristics directly. For these cases, we
instead compute the Witten indices of the mixed branches $\mathcal{C}_{\pm}$
which are easier to calculate. Matching of the Witten indices of
$\mathcal{C}_{\pm}$ induces the same constraint due to the fact that the total
Witten index is always the same.

$\begindc{\commdiag}[300]\obj(1,3)[G]{GLSM}\obj(1,1)[xh]{\hat{X}}\obj(-1,0)[xp]{X_{+},\mathcal{C}_{+}}\obj(3,0)[xm]{X_{-},\mathcal{C}_{-}}\par\obj(-3,3)[uv]{UV}\obj(-3,1)[ir]{IR}\mor{uv}{ir}{}[\atright,\solidarrow]\par\mor{G}{xh}{\hat{\pi}}\mor{G}{xp}{\pi_{+}}\mor{G}{xm}{\pi_{-}}\par\mor{xp}{xh}{\mathbb{W}_{+}}[\atright,\dashArrow]\mor{xh}{xm}{\mathbb{W}_{-}}[\atright,\dashArrow]\enddc$
Figure 1: Schematic diagram for the derived equivalence.

In order to implement the brane transport, we need to know the window
categories of the local models at the phase boundaries [2, 12]. For the
examples explored in this paper, the corresponding local models could be
either abelian or non-abelian, and they could be non-anomalous or anomalous.
Some of these local models are abelian gauge theories which have been
previously studied [2, 6] and we will use their results directly. While the
rest of the local models we will meet are $U(2)$ gauge theories with
fundamental and determinantal representations. Although analysis of the window
categories for such theories in the non-anomalous case exists [11], there are
no known results in the anomalous cases. So before studying the B-branes in
the $U(1)\times U(2)$ theories, we first analyze the window categories of the
$U(2)$ theories in section 3.

Our original motivation of this work is to understand Segal’s proof of the
derived equivalence between two non-compact Calabi-Yau fivefolds in [1] from a
physics point of view. There, the two Calabi-Yau fivefolds, $X_{+}$ and
$X_{-}$, are defined as the following:

$X_{+}\>=\>\text{Tot}[(\wedge^{2}\mathcal{S})\otimes\mathcal{S}\to
SGr(2,4)]\,,\quad
X_{-}\>=\>\text{Tot}[(\mathcal{L}^{\perp}/\mathcal{L})\otimes\mathcal{L}^{2}\to\mathbb{P}^{3}]\,.$

We will see this derived equivalence can be realized by an anomalous two-
parameter GLSM with gauge group $U(1)\times U(2)$. The pure geometric phase
has the target space

$\hat{X}\>=\>{\rm Tot}[\mathcal{L}\otimes\wedge^{2}S\to SF(1,2,4)]\,,$

where $SF(1,2,4)$ is the symplectic flag manifold. The full analysis for this
example can be found in section 4.4. We also give a general construction and
go beyond this example.

The organization of this paper is the following: in section 2, we will give a
brief review of hemisphere partition functions and brane transport. Then, we
generalize the idea to $U(2)$ gauge theories and derive the window categories
in section 3. In section 4, we explicitly study several concrete examples, we
start with an abelian gauge theory with gauge group $U(1)\times U(1)$ to
illustrate the idea. Then we study the nonabelian theories with gauge group
$U(1)\times U(2)$ including the theory corresponding to the derived
equivalence between the Calabi-Yau fivefolds $X_{+}$ and $X_{-}$ mentioned
above. At the end, we will give some concluding remarks and potential future
directions in section 5. The appendix includes supplementary materials such as
counting of Coulomb vacua and the analysis for the GLSM for a class of flag
manifolds.

##  2 Review on hemisphere partition functions and window categories

In this section, we give a short review on the hemisphere partition functions
of GLSMs. In particular, we review how we can extract big/small window
categories from these hemisphere partitions, and this will be crucial for
later sections. For more details, please refer to [2, 5, 6] and references
therein.

###  2.1 Hemisphere partition functions

The data of a $\mathcal{N}=(2,2)$ GLSM model consists of gauge field content
$\sigma_{a}$ valued in the Cartan subalgebra of the gauge group $G$, and the
matter content:

$\begin{array}[]{c|cc}&G&U(1)_{R}\\\ \hline\cr
X_{I}^{a}&Q_{I}^{a}&R_{I}\end{array}$

for $I$ runs over all chiral fields and $a$ over weights under the Cartan
subalgebra of $G$. The gauge charge for non-abelian $G$ is defined with
respect to its Cartan subalgebra. $R_{I}$ is the charge under the $U(1)$
R-symmetry. A superpotential is a generic quasi-homogeneous polynomial $W(X)$
of matter fields, with its gauge charge $0$ and R-charge $2$. In the infrared,
the classical Higgs branch, if exists, of this theory is the field
configuration under gauging, known as geometric invariant theory quotient (GIT
quotient). This geometry is represented as

$X_{\xi}\>=\>\left\\{\frac{\mathbb{C}^{I}-\Delta_{\xi}}{G}\right\\}\cap\left\\{\partial_{I}W(X)=0\right\\}\,,$
(1)

where $X_{\xi}$ depends on the Kähler cone of FI parameter $\xi$, and
$\Delta_{\xi}$ is the discriminant locus in this phase. The dependence on
$\xi$ is what is called the phase structure in [14].

One can assign different types of boundary conditions when the GLSM is put on
a hemisphere [2, 5] and we are interested in the B-type boundary condition,
which corresponds to a bound state coming from boundary terms of GLSM action
of open string tachyon between Wilson line branes carrying gauge charges. The
boundary condition is represented by a pair $\mathcal{B}=(Q,M)$ as a matrix
factorization of the superpotential $W$ and a Lagrangian submanifold in the
complexified Cartan subalgebra of the gauge group [15]. The pair $\mathcal{B}$
consists of an odd endomorphism $Q$ on a graded vector space $M$ such that

$Q\circ Q=W\cdot\operatorname{Id}_{M}\,.$ (2)

This matrix $Q$ has entries in polynomial ring of matter fields $X_{I}^{a}$,
and it has gauge charge $0$ and R-charge $1$:

$\begin{gathered}\rho_{G}(g)^{-1}\circ Q(g\circ X)\circ\rho_{G}(g)=Q(X)\,,\\\
\rho_{R}(\lambda)\circ Q(\lambda\circ X)\circ\rho_{R}(\lambda)^{-1}=\lambda
Q(X)\,,\end{gathered}$

where $\rho_{R}$ and $\rho_{G}$ are the induced action on matrix
factorization. It is useful to realize such $Q$ and $M$ in Clifford algebra
and Clifford module. Given a factorization of polynomial $W$ as

$W(X)\>=\>\sum_{i}g_{i}(X)\cdot f_{i}(X)\,,$

if we introduce a pair of complex fermion operators $\psi_{i}$ and
$\overline{\psi}_{i}$ satisfying the following anti-commutator relation:

$\\{\psi_{i},\overline{\psi}_{j}\\}=\delta_{ij}\,.$ (3)

Then, after assigning the proper R charge and gauge charges to fermion
operators, one can write a matrix factorization as

$Q=\sum_{i}g_{i}(X)\psi_{i}+f_{i}(X)\overline{\psi}_{i}$ (4)

with graded module as the Hilbert space of free fermions

$M=\bigoplus_{n}\bigoplus_{i_{1}<\cdots<i_{n}}\overline{\psi}_{i_{1}}\cdots\overline{\psi}_{i_{n}}|0\rangle\,,$
(5)

where $\psi|0\rangle=0$. Using the anti-commutation relation (3), one can
immediately see that

$Q^{2}=\sum_{i}f_{i}(X)g_{i}(X)=W(X)\,.$ (6)

The support of this brane as the zero locus of boundary fermion energy

$\text{supp}(Q)=\big{\\{}X\in\mathbb{C}^{I},\\{Q(X),Q(X)^{\dagger}\\}=0\big{\\}}\,.$
(7)

Gauge invariance of matrix factorization permits it to flow to the complex of
coherent sheaves in the IR Higgs branch $X_{\xi}$, in which the support region
$\mathrm{supp}(Q)$ determines the tachyon decay mode in different phases
according to the discriminant locus $\Delta_{\xi}$.

The hemisphere partition function of a brane in $\mathcal{N}=(2,2)$ GLSM can
be computed via the supersymmetric localization technique and was given in [5]
as

$Z_{\mathcal{B}}(t)\>=\>\int_{\gamma}d^{|G|}\sigma\prod_{\alpha\in\Delta^{+}_{G}}\alpha(\sigma)\sinh\pi\alpha(\sigma)\
\prod_{I}\Gamma\left(\sum_{a}iQ_{I}^{a}\sigma_{a}+\frac{R_{I}}{2}\right)e^{it(\sigma)}f_{\mathcal{B}}(\sigma)\,,$
(8)

where $|G|$ denotes the dimension of Cartan subalgebra of $G$,
$t(\sigma):=\langle t,\sigma\rangle$ is the pairing between the complexified
FI-parameter $t=\xi-i\theta$ and the gauge field $\sigma$. Denote
${\hat{q}}_{a}$ and $\hat{R}$ as the charge operator for the induced action of
$\rho_{G}$ and $\rho_{R}$ respectively, $f_{\mathcal{B}}(\sigma)$ is the brane
factor evaluated on the gauge charge and R charges of graded module:

$f_{\mathcal{B}}(\sigma)={\rm
Tr}_{M}e^{i\pi\hat{R}}e^{2\pi\hat{q}_{a}\sigma_{a}}\,.$ (9)

The integral in Eq. (8) is evaluated along a contour $\gamma$, which is a
middle dimensional real class in the $\sigma$ space (a Lagrangian submanifold
of the complexified Cartan subalgebra). In particular, this contour should be
considered as a deformation from the real hyperplane in this space such that
it avoids singularities of the gamma functions. Then the partition function
can be computed as a sum of infinite many residues of gamma poles in a proper
region depending on the phase determined by the FI-parameter $\xi$. After a
specific choice of $\xi$, the poles are determined by the convergent region of
the contour $\gamma$. In the large volume limit where $e^{-t}\rightarrow 0$,
the leading term of $Z_{\mathcal{B}}$ evaluates the $\Gamma$-class of the
bundle corresponding to $\mathcal{B}$, which is the data computed by boundary
B-twisted theory, while the complete infinite summation comes from instanton
effect computed by bulk A-twisted theory [5].

It is worth remarking that all methods above apply to the non-compact case
$W=0$, where matrix factorization of zero is simply a differential. In this
case, any R-charge of matter fields should be set to zero and separating poles
in gamma-function may fail. However, a compact brane factor can be evaluated
by regularizing gamma poles with small R-charge and taking the zero limit. A
more systematic approach can be done by using the transformation law of
residues in [16]. On the level of partition function, the existence of
superpotential is known by R-charge assignment. Later, in section 4, we will
see the example of Calabi-Yau 5-fold is a mixed case, i.e. it will have both a
non-trivial superpotential and the non-compact direction.

###  2.2 Window and derived equivalence

In practice, there is a subtlety in choosing the contour $\gamma$ for
$\mathcal{B}$, namely it may not have a convergent region for contour on a
phase boundary for an arbitrary brane factor. Using Stirling’s formula, the
partition function can be written asymptotically as

$\displaystyle Z_{\mathcal{B}}(t)\sim\int d\sigma e^{F(\sigma,q,t)}$ (10)
$\displaystyle\sim\int_{\gamma}d^{|G|}\sigma\exp\bigg{\\{}\sum_{I}iQ_{I}(\sigma)\log
Q_{I}(\sigma)-\sum_{I}iQ_{I}(\sigma)+\pi\sum_{\Delta^{+}_{G}}|\alpha(\sigma)|+it(\sigma)-\sum_{M}2\pi
q(\sigma)\bigg{\\}}\,.$

The absolutely convergent condition requires the negativity of the real part
of the exponential in the integrand, which is

$\displaystyle\mathrm{Re}[F(\sigma,q,t)]\>=$
$\displaystyle\sum_{\Delta^{+}}\pi|\alpha(\tau)|+\big{(}-\xi_{R}(\nu)+(\theta+2\pi
q)(\tau)\big{)}$ (11)
$\displaystyle+\sum_{I}Q_{I}(\nu)\big{[}\log|Q_{I}(\sigma)|-1\big{]}-|Q_{I}(\tau)|\left[\frac{\pi}{2}+\arctan\frac{Q_{I}(\nu)}{|Q_{I}(\tau)|}\right]\,,$

with

$\sigma=\tau+i\nu,\quad\operatorname{Arg}(iz)=\operatorname{sgn}(\mathrm{Re}[z])\left(\frac{\pi}{2}+\arctan\frac{\mathrm{Im}[z]}{\mathrm{Re}[z]}\right)\,,$

for sufficiently large $|\sigma|$ on the contour when we move the parameter
$t$ close to the radius of convergence. This condition picks out a specific
range of admissible gauge charges in the brane. This range of gauge charges is
called the window on a phase boundary. For example, a window of boundary
$\xi=0$ in the $U(1)$ model of [2, section 7.3.1] is obtained by setting
$\nu=0$ in the real exponential, $(\theta+2\pi
q)\tau-\frac{\pi}{2}\sum_{I}|Q_{I}|\tau<0$, resulting in the grade restriction
rule (GRR)

$-\frac{\sum_{I}|Q_{I}|}{4}<q+\frac{\theta}{2\pi}<\frac{\sum_{I}|Q_{I}|}{4}\,.$
(12)

One can see that the set of charges satisfying GRR depends on the choice of
the $\theta$-angle, and this set is subject to an overall shift upon a
$2\pi$-shift in the $\theta$ parameter. Different window categories defined by
different $\theta$, whose objects are the branes with gauge charges satisfying
the GRR, are categorically equivalent.

It was conjectured in [5] that branes with charges in the window correspond to
the exceptional collection of the derived category of the geometry in the IR
Higgs branch. The shift due to the $\theta$-angle corresponds to an
automorphism of this collection by the Serre twist. The windows of non-abelian
models are also systematically considered in [10, 11]. In the localization
formula for the hemisphere partition function of the non-abelian models [5],
the factors from positive roots of the gauge group will put non-trivial
constraints on non-abelian charges. Depending on the matter content, it turns
out that the window category of the GLSM for bundles over $Gr(k,n)$ can be
either Kapranov’s, Kuznetsov’s or other exceptional collection.

In the non-anomalous case, the window only occurs on a phase boundary where
parameters $e^{t}$ hit the radius of convergence. In the large volume limit,
there should be no window constraints. The singularity on a phase boundary of
$\xi$ is classical, however, this singularity can be lifted by quantum
corrections due to the $\theta$-angle. Therefore, $Z_{\mathcal{B}}(t)$ can be
analytically continued from one phase to another and this process is called
brane transport. The derived equivalence of branes (sheaves) in different
phases can be understood in this way.

In the case with superpotential, admissible branes are not only sheaves but
also complexes of sheaves corresponding to matrix factorizations, which may
not always lie in a window. As mentioned above, $\Delta_{\xi}$ corresponds to
the semi-stable locus which is deleted in $X_{\xi}$, and this means a brane
supported on this locus will be empty in this phase since it finds no where to
take in $X_{\xi}$ as boundary condition. Then this bound state of branes is
trivial as vacuum in $D^{b}(X_{\xi})$, or, equivalently, the corresponding
complex is exact. Exactness of this bound state implies the annihilation from
decayed open string tachyonic modes. This is known as the tachyon condensation
of brane-anti-brane system. These trivial branes are called empty branes. By
combining with empty branes, one can transform a brane complex exceeding the
window to create a new but homologically equivalent brane complex, such that
it is inside the window. This process is called grade restriction in [2].

In the IR limit, branes naturally decay via tachyon condensation. These IR
branes can be lifted back to the UV Wilson line branes up to empty brane
states. In conclusion, the derived equivalence in different phases of non-
anomalous GLSMs are realized by brane transport, which is understood as an
analytic continuation of hemisphere partition function of grade restricted
brane across the phase boundaries. Such transport connects boundary conditions
from one phase to another phase.

###  2.3 Big window v.s. small window

The discussion in the previous section only concerns the non-anomalous
(Calabi-Yau) case. For the anomalous case, the situation is different. As
mentioned in section 1, one has to distinguish between the big window and
small window categories. In this case, the FI-parameters are subject to RG-
flow. If the flow is from phase A to phase B, then in the IR we can only see
phase B. But phase A can be reached by taking the gauge-decoupling limit,
where the FI-parameter is fixed in the region corresponding to phase A at some
fixed energy scale while the gauge coupling constant is taken to infinity [6].
Therefore, we can still use the techniques of brane transport to construct a
functor between the categories of B-branes of phase A and phase B.

For any B-brane in phase A, we can lift it to a UV GLSM brane. Since there is
an RG-flow on the space of FI-parameters, in order to read off the IR image of
the GLSM brane, it has to be graded restricted so that the hemisphere
partition function can be computed by a saddle point analysis. These grade
restricted branes constitute the big window category. In general, phase B may
consist of a Higgs branch and several Coulomb/mixed branches, so the IR brane
will have components on Higgs and/or Coulomb/mixed branches. Only a subset of
branes in the big window category has images only on the Higgs branch, which
gives rise to the small window category. Therefore, the category of B-branes
of the Higgs branch in the IR limit is equivalent to the small window
category. Note that a $2\pi$-shift in $\theta$ angle will result in a
different but equivalent big/small window category. In the Calabi-Yau case,
the FI-parameters are marginal, and the big window and small window categories
coincide.

As shown in [6], for any brane in the big window category, the contour
integral can be computed by a saddle point analysis. The hemisphere partition
function can then be written as a sum of residues corresponding to the Higgs
branch and a contribution from the saddle points. The existence of the saddle
points is an indication that the brane has components on the Coulomb/mixed
branches in the IR limit. Therefore, for the image to be in the small window
category, it should be possible to choose the contour such that there are no
contribution to the hemisphere partition function from the saddle points. The
saddle points are given by solutions to

$\partial_{\sigma}F(\sigma,q,t)=0\,,$ (13)

where $F$ is defined by the first line of Eq. (10). We will test this
condition numerically and determine the possible region for small window. To
do that, we plot the hypersurfaces determined by the saddle point equations on
the contour, like in Fig.3, label the position of gauge charges admitting a
solution, and then derive the maximal set of possible charges within them. By
integrality we only need to test among the charges such that

$q+\frac{\theta}{2\pi}\in\frac{1}{2}{\mathbb{Z}}\,.$ (14)

We will demonstrate this analysis for $U(2)$ theories in the next section.

##  3 Small window of $U(2)$ theories

In this section, we will apply the above procedures in two concrete examples
with $U(2)$ gauge group and obtain their small window categories. These two
examples will later arise as the effective boundary theories (local model at
the phase boundary [12]) in section 4, either on $\xi_{1}=0$ or on
$\xi_{2}=0$. In principle, this method could be generalized to cases with
higher rank gauge groups, but we leave this to future work.

###  3.1 One determinantal field

Let us first consider an $U(2)$ gauge theory with $n$ fundamental fields,
$X_{i}$, $i=1,\dots,n$, and one field $P$ in the determinantal representation,
$\det^{-m}$, with $m\leq n$, as summarized below

| $X_{i}$ | $P$ |
---|---|---|---
$U(2)$ | ${\tiny{\yng(1)}}$ | $\det^{-m}$ | $\xi$
(15)

where $\xi$ is the associated FI parameter. This model has a vanishing
superpotential, $W=0$. For later purpose, we will only focus on the cases with
$n=3$ and $n=4$, and the same analysis in principle can be generalized to
arbitrary $n$.

According to the D-term equations, the region $\xi\gg 0$ is a phase described
by a total space of $m$-fold determinantal bundle over $Gr(2,n)$, i.e.

$\mathrm{Tot}\left({\det}^{-m}\mathcal{S}\to Gr(2,n)\right)\,,$

where $\mathcal{S}$ is the tautological bundle over the Grassmannian
$Gr(2,n)$. Therefore, the total Witten index in this phase should be the Euler
characteristic of this geometry, which is $\binom{n}{2}\,,$ This is also the
number of generators of the big window category as we will see later.

Another region where $\xi\ll 0$ consists of a Higgs branch and some Coulomb
vacua. The Coulomb branch behaviour at the large $\sigma$ region is determined
by the one-loop corrected twisted superpotential and the Coulomb vacua
equations are

$\sigma_{a}^{n}\>=\>-q(-m)^{m}(\sigma_{1}+\sigma_{2})^{m},\quad a=1,2\,,$

with excluded loci condition $\sigma_{1}\neq\pm\sigma_{2}$. One can count the
number of Coulomb vacua and obtain

$\left\lfloor\frac{n-1}{2}\right\rfloor(n-m)\,.$ (16)

(The detailed computation is presented in appendix A.) The Higgs branch in
this region is more complicated and, for our purpose, we need to extract the
small window category which has pure Higgs branch contributions. And we will
use the fact that the number of small window generators plus the number of
Coulomb vacua equals the number of big window generators, as a consistency
check.

To understand the Higgs branch of the phase with $\xi\ll 0$ and the small
window category, let us do a category-theoretical analysis first, in the cases
$n=3$ and $n=4$.

####  3.1.1 Higgs branch of the negative phase and its category of B-branes

When $m=n=3$, the negative phase can be understood via the IR duality of [17,
18]. The $U(2)$ theory (15) with $m=n=3$ is dual to the abelian GLSM, which
has gauge group $U(1)$, three chirals with charge $1$ and one chiral with
charge $-3$. The dual abelian GLSM has a phase corresponding to the limit
$\xi\gg 0$ with target
$\mathrm{Tot}(\mathcal{O}(-3)\rightarrow\mathbb{P}^{2})$, a noncompact Calabi-
Yau manifold. The target space of the phase with $\xi\ll 0$ is the orbifold
$\mathbb{C}^{3}/\mathbb{Z}_{3}$. The category of B-branes in this phase is the
derived category $D^{b}(\mathbb{C}^{3}/\mathbb{Z}_{3})$, which is equivalent
to the derived category of graded modules of
$\mathcal{A}=\mathbb{C}[x_{1},x_{2},x_{3}]$, i.e.
$D(\mathrm{gr}\text{-}\mathcal{A})$. This category is generated by
$\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2}$, where $\mathcal{A}_{l}$
denotes the graded $\mathcal{A}$-module $\mathcal{A}$ with grading $l$ (the
$\mathbb{Z}_{3}$-charge). $\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2}$
are mapped to
$\pi^{*}\mathcal{O}_{\mathbb{P}^{3}},\pi^{*}\mathcal{O}_{\mathbb{P}^{3}}(1),\pi^{*}\mathcal{O}_{\mathbb{P}^{3}}(2)$
respectively via brane transport to the phase with $\xi\gg 0$, where $\pi$ is
the projection onto the base $\mathbb{P}^{3}$. From the duality, we see the
category of B-branes in the negative phase of the original $U(2)$ theory with
$m=n=3$ is equivalent to
$D(\mathrm{gr}\text{-}\mathcal{A})=\langle\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2}\rangle$.
Because the orbifold group $\mathbb{Z}_{3}$ in this theory is a subgroup of
$\det U(2)$, we see that $\mathcal{A}_{i}$ can be lifted to the GLSM brane
$\mathcal{W}_{\det^{i+3l}}$. The window category is generated by
$\mathcal{W}_{\det^{3l}},\mathcal{W}_{\det^{3l+1}},\mathcal{W}_{\det^{3l+2}}$
($l$ depends on the choice of the theta-angle), which are mapped to
$\det(\mathcal{S}^{\vee})^{3l},\det(\mathcal{S}^{\vee})^{3l+1},\det(\mathcal{S}^{\vee})^{3l+2}$
in the positive phase.

When $0<m<n=3$, there are $3-m$ Coulomb vacua in the negative phase and the
category of B-branes of the Higgs branch, due to the $\mathbb{Z}_{m}$
orbifold, is generated by $m$ objects, namely
$\mathcal{A}_{0},\cdots,\mathcal{A}_{m-1}$. Accordingly, the small window
category is generated by
$\mathcal{W}_{\det^{ml}},\mathcal{W}_{\det^{ml+1}},\cdots,\allowbreak\mathcal{W}_{\det^{ml+(m-1)}}$.
The big window category remains the same as the window category of the $m=n=3$
theory.

The window category of the theory with $m=n=4$ is studied in [11]. The idea is
as follows. The target space of the positive phase is the canonical line
bundle $K_{Gr(2,4)}$. In the negative phase ($\xi\ll 0$), the $P$ field
acquires a nonzero vev, which breaks $\det U(2)$ to a $\mathbb{Z}_{4}$
subgroup. Upon integrating out the fluctuation of $P$, we are left with a
$SU(2)\times\mathbb{Z}_{4}$ gauge group, the $SU(2)$ fundamentals $X^{a}_{i}$
all have charge one under $\mathbb{Z}_{4}$. In the IR limit, the $SU(2)$ gauge
group is decoupled, and the low energy degrees of freedom are described by the
baryons

$B_{ij}=\epsilon_{ab}X^{a}_{i}X^{b}_{j},\quad 1\leq i<j\leq 4\,,$

which all have $\mathbb{Z}_{4}$ charge two.

Because $B_{ij}$ are the Plücker coordinates of the Grassmannian $Gr(2,4)$,
the negative phase is expected to be a NLSM with target space555This statement
was first conjectured in [19, appendix A].

$CGr(2,4)/\mathbb{Z}_{4}\,,$

where $CGr(2,4)$ is the affine cone of the Plücker embedding of $Gr(2,4)$ in
$\mathbb{P}^{5}$,

$CGr(2,4)=\\{(B_{12},B_{13},B_{14},B_{23},B_{24},B_{34})\in\mathbb{C}^{6}|f(B)=0\\}\,,$

where

$f(B)=B_{12}B_{34}-B_{13}B_{24}+B_{14}B_{23}\,.$

Indeed, there is a one-to-one correspondence between the generators of
$D^{b}(K_{Gr(2,4)})$ and the generators of $D^{b}(CGr(2,4)/\mathbb{Z}_{4})$.
Let $\mathcal{A}$ be the coordinate ring of $CGr(2,4)$, i.e.

$\mathcal{A}\>=\>\mathbb{C}[B_{12},B_{13},B_{14},B_{23},B_{24},B_{34}]/\langle
f(B)\rangle\,.$

Then $D^{b}(CGr(2,4))\cong D(\mathrm{gr}\text{-}\mathcal{A})$, the derived
category of graded $\mathcal{A}$-modules. Denote by $Perf(\mathcal{A})$ the
subcategory of $D(\mathrm{gr}\text{-}\mathcal{A})$ whose objects are
equivalent to finite-length complexes of free $\mathcal{A}$-modules of finite
rank, and

$D_{\mathrm{sg}}(\mathcal{A})\>=\>D(\mathrm{gr}\text{-}\mathcal{A})/Perf(\mathcal{A})\,.$

We know $Perf(\mathcal{A})=\langle\mathcal{A}\rangle$, and (see [20, 21])

$D_{\mathrm{sg}}(\mathcal{A})\>\cong\>\mathrm{MF}(f)\,,$

where $\mathrm{MF}(f)$ is the category of matrix factorizations of $f$.
Therefore there is a decomposition

$D^{b}(CGr(2,4))\>=\>\langle\mathcal{A},\mathrm{MF}(f)\rangle\,.$

Upon taking the $\mathbb{Z}_{4}$ orbifold action, we get

$D(CGr(2,4)/\mathbb{Z}_{4})\>=\>\langle\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3},\mathrm{MF}_{\mathbb{Z}_{2}}(f)\rangle\,,$
(17)

where $\mathcal{A}_{i}$ is the $\mathcal{A}$-module $\mathcal{A}$ with
$\mathbb{Z}_{4}$ charge $i$. Notice that the $B_{ij}$’s all have
$\mathbb{Z}_{4}$ charge 2 and $f$ is quadratic in $B_{ij}$, so we have
$\mathrm{MF}_{\mathbb{Z}_{2}}(f)$ in Eq. (17) instead of
$\mathrm{MF}_{\mathbb{Z}_{4}}(f)$.

The category $\mathrm{MF}_{\mathbb{Z}_{2}}(f)$ is the category of B-branes of
the Higgs branch in the $\zeta\ll 0$ phase of the $U(1)$ GLSM with matter
content

$\begin{array}[]{c|ccccccc|c}&\Phi_{0}&\Phi_{1}&\Phi_{2}&\Phi_{3}&\Phi_{4}&\Phi_{5}&P&\\\
\hline\cr U(1)&1&1&1&1&1&1&-2&\zeta\end{array}$ (18)

and superpotential $W=Pf(\Phi)$. The phase with $\zeta\gg 0$ has pure Higgs
branch whose target space is the quadric $\\{f=0\\}$ in $\mathbb{P}^{5}$,
which is the image of $Gr(2,4)$ under the Plücker embedding. The category
$\mathrm{MF}_{\mathbb{Z}_{2}}(f)$ is generated by the matrix factorization

$\textstyle{{\mathcal{M}_{1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\begin{array}[]{cccc}\Phi_{3}&-\Phi_{4}&\Phi_{5}&0\\\
-\Phi_{1}&\Phi_{0}&0&\Phi_{5}\\\ -\Phi_{2}&0&\Phi_{0}&\Phi_{4}\\\
0&-\Phi_{2}&\Phi_{1}&\Phi_{3}\end{array}\right)}$$\textstyle{\mathcal{M}_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left(\begin{array}[]{cccc}\Phi_{0}&\Phi_{4}&-\Phi_{5}&0\\\
\Phi_{1}&\Phi_{3}&0&-\Phi_{5}\\\ \Phi_{2}&0&\Phi_{3}&-\Phi_{4}\\\
0&\Phi_{2}&-\Phi_{1}&\Phi_{0}\end{array}\right)}$

and its $\mathbb{Z}_{2}$-twisted version (where the $\mathbb{Z}_{2}$ charges
of the Chan-Paton spaces are flipped), where
$\mathcal{M}=\mathbb{C}[\Phi_{0},\cdots,\Phi_{5}]^{\oplus 4}$ and the
subscript indicates the $\mathbb{Z}_{2}$ charge. This fact can be seen from
the fact that the small window category of the GLSM specified by (18) can only
accommodate two consecutive $U(1)$ charges, and by counting the number of
Coulomb vacua in the negative phase, it is clear that
$\mathrm{MF}_{\mathbb{Z}_{2}}(f)$ is generated by two independent objects.
These two matrix factorizations are mapped to the two spinor bundles[22] of
the quadric via brane transport. The spinor bundles on $Gr(2,4)$ are the dual
of the tautological bundle $\mathcal{S}^{\vee}$ and the universal quotient
bundle $\mathcal{Q}$. Therefore, under the equivalence
$D^{b}(CGr(2,4)/\mathbb{Z}_{4})\cong D^{b}(K_{Gr(2,4)})$,
$\mathrm{MF}_{\mathbb{Z}_{2}}(f)$ is mapped to the subcategory of
$D^{b}(K_{Gr(2,4)})$ generated by $\pi^{*}\mathcal{S}^{\vee}$ and
$\pi^{*}\mathcal{Q}$, where $\pi$ is the projection
$\pi:K_{Gr(2,4)}\rightarrow Gr(2,4)$.

In the $U(2)$ GLSM (15) with $n=m=4$ describing $K_{Gr(2,4)}$, because
$\mathcal{A}_{i}$ has $\mathbb{Z}_{4}$ charge $i$, $\mathcal{A}_{i}$ can be
lifted to a GLSM brane in the $\det^{i}$ representation, and it is mapped to
$\pi^{*}(\det\mathcal{S}^{\vee})^{\otimes i}$ via brane transport.

Therefore, we have the following identification

$\mathcal{A}_{i}\leftrightarrow\pi^{*}(\det\mathcal{S}^{\vee})^{\otimes
i},\quad\mathrm{MF}_{\mathbb{Z}_{2}}(f)\leftrightarrow\langle\pi^{*}\mathcal{S}^{\vee},\pi^{*}\mathcal{Q}\rangle$
(19)

for the equivalence

$D^{b}(CGr(2,4)/\mathbb{Z}_{4})\>=\>\langle\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3},\mathrm{MF}_{\mathbb{Z}_{2}}(f)\rangle\cong
D^{b}(K_{Gr(2,4)})\,.$ (20)

Notice that

$\mathcal{Q}\>\cong\>\det\mathcal{Q}\otimes\mathcal{Q}^{\vee}\>\cong\>\det\mathcal{S}^{\vee}\otimes(\mathcal{O}^{\oplus
4}\rightarrow\mathcal{S}^{\vee})\,,$

the window category is generated by666Here and in the following, the Young
diagram with $a$ boxes in the first row and $b$ boxes in the second row are
used to denote the $U(2)$ representation with highest weight $(a,b)$. The
empty diagram $\cdot$ denotes the trivial representation.

${\mathcal{W}}_{\cdot},{\mathcal{W}}_{\tiny{\yng(1)}},{\mathcal{W}}_{\tiny{\yng(1,1)}},{\mathcal{W}}_{\tiny{\yng(2,1)}},{\mathcal{W}}_{\tiny{\yng(2,2)}},{\mathcal{W}}_{\tiny{\yng(3,3)}}$
(21)

up to a shift by $\mathcal{W}_{\det^{l}}$ induced by changing the theta-angle.
The set of branes in (21) reduces to the semi-orthogonal decomposition

$D^{b}(K_{Gr(2,4)})\>=\>\langle\mathcal{O},\mathcal{S}^{\vee},\det\mathcal{S}^{\vee},\det\mathcal{S}^{\vee}\otimes\mathcal{S}^{\vee},(\det\mathcal{S}^{\vee})^{\otimes
2},(\det\mathcal{S}^{\vee})^{\otimes 3}\rangle$

in the positive phase. The equivalence (20) is an evidence that the IR theory
of the negative phase is described by a NLSM with target space
$CGr(2,4)/\mathbb{Z}_{4}$.

When $0<m<n=4$, the nonzero vev of $P$ breaks $\det U(2)$ to $\mathbb{Z}_{m}$,
so we conjecture that the Higgs branch of the negative phase is described by a
NLSM with target space $CGr(2,4)/\mathbb{Z}_{m}$. The subcategory
$Perf(\mathcal{A})$ is thus generated by $m$ objects
$\mathcal{A}_{0},\cdots,\mathcal{A}_{m-1}$. In addition, there are $4-m$
Coulomb vacua in this phases, therefore the singular subcategory
$D_{\mathrm{sg}}(\mathcal{A})$ is still generated by ${4\choose 2}-m-(4-m)=2$
objects and we conjecture they are still mapped to $\pi^{*}\mathcal{S}^{\vee}$
and $\pi^{*}\mathcal{Q}$ in the positive phase. The small window category is
thus expected to be generated by some $\det^{l_{1}}$ shift of
$\langle\mathcal{W}_{\cdot},\mathcal{W}_{\tiny{\yng(1,1)}},\cdots,\mathcal{W}_{(m-1,m-1)}\rangle$
and some $\det^{l_{2}}$ shift of
$\langle\mathcal{W}_{\tiny{\yng(1)}},\mathcal{W}_{\tiny{\yng(2,1)}}\rangle$,
where $l_{1}-l_{2}$ can be determined by analyzing the hemisphere partition
function, and the overall shift is determined by the theta-angle.

####  3.1.2 Hemisphere partition function and the window categories

Now let us obtain the big and small windows from the hemisphere partition
functions, as discussed in section 2. For general $m\leq n$, the real
exponential factor is given by ($\sigma_{a}=\tau_{a}+i\nu_{a}$,
$t=\xi-i\theta$)

$\displaystyle A_{q}(t)\>=$
$\displaystyle\pi|\tau_{1}-\tau_{2}|-\xi(\nu_{1}+\nu_{2})+\theta(\tau_{1}+\nu_{1})+2\pi(q_{1}\tau_{1}+q_{2}\tau_{2})$
(22)
$\displaystyle+m(-\nu_{1}-\nu_{2})\left(\log|m(\sigma_{1}+\sigma_{2})|-1\right)-m|\tau_{1}+\tau_{2}|\left(\frac{\pi}{2}-\arctan\frac{\nu_{1}+\nu_{2}}{|\tau_{1}+\tau_{2}|}\right)$
$\displaystyle+n\left[\nu_{1}\left(\log|\sigma_{1}|-1\right)-|\tau_{1}|\left(\frac{\pi}{2}+\arctan\frac{\nu_{1}}{|\tau_{1}|}\right)\right]$
$\displaystyle+n\left[\nu_{2}\left(\log|\sigma_{2}|-1\right)-|\tau_{2}|\left(\frac{\pi}{2}+\arctan\frac{\nu_{2}}{|\tau_{2}|}\right)\right]\,.$

If we take the contour [11]

$\gamma\>=\>\left\\{\nu_{1}\>=\>-\nu_{2}\>=\>\tau_{1}^{2}-\tau_{2}^{2}\right\\}\,,$
(23)

then the following wedge conditions [5, 11]

$\displaystyle\tau_{1}\>=\>0\quad\Rightarrow$
$\displaystyle\quad\nu_{1}\>\leq\>0\,;$ (24)
$\displaystyle\tau_{2}\>=\>0\quad\Rightarrow$
$\displaystyle\quad\nu_{2}\>\leq\>0\,;$ (25)
$\displaystyle\tau_{1}+\tau_{2}\>=\>0\quad\Rightarrow$
$\displaystyle\quad\nu_{1}+\nu_{2}\>\geq\>0$ (26)

are satisfied so the contour will not hit the singularities. Along the contour
(23), we have

$\displaystyle A_{q}(t)\>=$
$\displaystyle\pi|\tau_{1}-\tau_{2}|+\left(\theta+2\pi
q_{1}\right)\tau_{1}+\left(\theta+2\pi
q_{2}\right)\tau_{2}-\pi\frac{m}{2}|\tau_{1}+\tau_{2}|-\pi\frac{n}{2}(|\tau_{1}|+|\tau_{2}|)$
(27)
$\displaystyle-n|\tau_{1}|\arctan\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{|\tau_{1}|}\right)-n|\tau_{2}|\arctan\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{|\tau_{2}|}\right)$
$\displaystyle+((\tau_{1})^{2}-(\tau_{2})^{2})\frac{n}{2}\log\frac{(\tau_{1})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}{(\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}\,.$

Then the convergence condition is given by [11, 12]

$\begin{gathered}\left|\frac{\theta}{2\pi}+q_{1}+\frac{\theta}{2\pi}+q_{2}\right|\>\leq\>\frac{n+m}{2}\,,\\\
\left|\frac{\theta}{2\pi}+q_{\alpha}\right|\>\leq\>\frac{2n-2+m}{4}\,,\\\
\left|q_{1}-q_{2}\right|\>\leq\>\frac{n-2}{2}\,.\end{gathered}$ (28)

However, this condition is not restrict enough and it gives more charges in
the window than expected. Fortunately, there are certain relationships from
the Eagon-Northcott complex associated to a matrix, among the branes in the
relaxed window [23], so the number of independent generators in the window
will be exactly the Euler characteristic of $Gr(2,n)$. For $m=n$, the
convergent region is shown in Fig. 2, in which the axis are parameterized by
$q_{a}+\theta/{2\pi}$, where $q_{1}$ and $q_{2}$ are the charges under the
Cartan of $U(2)$.

Figure 2: Admissible charges in $\frac{1}{2}\mathbb{Z}$ and an integral choice
of $\mathbb{W}_{\mathrm{Big}}$ in red for $n=3$ (left) and $n=4$ (right).

The admissible charges are marked as blue dots in Fig. 2, among which the big
window category consists of charges marked red, which correspond to
representations

$\mathbb{W}_{\mathrm{Big}}\>=\>\left\\{\begin{array}[]{cl}\left\langle\cdot,\
\tiny{\yng(1)},\ \tiny{\yng(1,1)}\right\rangle\,,&n=3\,,\\\ \\\
\left\langle\cdot,\ \tiny{\yng(1)},\ \tiny{\yng(1,1)},\ \tiny{\yng(2)},\
\tiny{\yng(2,2)},\ \tiny{\yng(3,3)}\right\rangle\,,&n=4\,.\end{array}\right.$
(29)

They correspond to the semi-orthogonal decomposition of the derived category
of ${\mathbb{P}}^{2}(\cong Gr(2,3))$ and $Gr(2,4)$ respectively. Note that in
the case of $n=4$, half of the off-diagonal charges is equivalent to the other
half by binding with the empty branes [11].

Figure 3: Numerical illustration of
$\partial_{\tau_{a}}A_{q}(\tau_{1},\tau_{2})$ and zero contours. The light
yellow sheet is the plot for $\partial_{\tau_{1}}A_{q}(\tau_{1},\tau_{2})$
while the red contour indicates the solutions to
$\partial_{\tau_{1}}A_{q}(\tau_{1},\tau_{2})=0$. The light blue sheet and the
blue contour are for $\partial_{\tau_{2}}A_{q}(\tau_{1},\tau_{2})$ and its
zero loci.

Next, in order to obtain the small window, one should rule out the solutions
to the saddle point equations

$\partial_{\tau_{\alpha}}A_{q}(\tau_{1},\tau_{2})\>=\>0,\quad\alpha=1,2\,,$
(30)

among the admissible charges on the contour $\gamma$, (23). More explicitly,
$\partial_{\tau_{\alpha}}A_{q}$ restricted on $\gamma$ are given as

$\displaystyle\partial_{\tau_{1}}A_{q}|_{\gamma}\>=$ $\displaystyle\left(2\pi
q_{1}+\theta\right)+n\tau_{1}\log\left(\frac{(\tau_{1})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}{(\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}\right)-n\tan^{-1}\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{\tau_{1}}\right)$
(31)
$\displaystyle-2n\frac{\tau_{1}(\tau_{1}-\tau_{2})(\tau_{1}+\tau_{2})(((\tau_{1})^{2}-(\tau_{2})^{2})^{2}-(\tau_{2})^{2})}{((\tau_{1})^{4}+(\tau_{2})^{4}+(\tau_{1})^{2}(1-2(\tau_{2})^{2}))((\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2})}$
$\displaystyle-\frac{\pi}{2}\left(n\operatorname{sign}(\tau_{1})+m\operatorname{sign}(\tau_{1}+\tau_{2})-2\operatorname{sign}(\tau_{1}-\tau_{2})\right)\,,$

and

$\displaystyle\partial_{\tau_{2}}A_{q}|_{\gamma}\>=$ $\displaystyle\left(2\pi
q_{2}+\theta\right)+n\tau_{2}\log\left(\frac{(\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}{(\tau_{1})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}\right)-n\tan^{-1}\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{\tau_{2}}\right)$
(32)
$\displaystyle-2n\frac{\tau_{2}(\tau_{2}-\tau_{1})(\tau_{1}+\tau_{2})(((\tau_{2})^{2}-(\tau_{1})^{2})^{2}-(\tau_{1})^{2})}{((\tau_{1})^{4}+(\tau_{2})^{4}+(\tau_{1})^{2}(1-2(\tau_{2})^{2}))((\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2})}$
$\displaystyle-\frac{\pi}{2}\left(n\operatorname{sign}(\tau_{2})+m\operatorname{sign}(\tau_{1}+\tau_{2})+2\operatorname{sign}(\tau_{1}-\tau_{2})\right)\,.$

To see when Eq. (30) has no solution, we apply a numerical method. For
example, for $n=3,m=1$ with $q_{\alpha}+\theta/2\pi=(-1/2,-1/2)$,
$\partial_{\tau_{1}}A_{q}(\tau_{1},\tau_{2})=0$ and
$\partial_{\tau_{2}}A_{q}(\tau_{1},\tau_{2})=0$ are plotted as the red and
blue contours in Fig. 3. Then, the transversal intersection between the red
and blue contours implies the solutions to the Coulomb vacuum equations, and
therefore the corresponding representation should be ruled out from the small
window.

The numerical results are summarized as in Fig. 4, where the gray background
indicates convergent region, black points are the charges such that Eq. (30)
has saddle point solutions, which should be ruled out, and the red points
correspond to the charges in the small window.

(a) n=3,m=1 (b) n=3,m=2

(c) n=4,m=1 (d) n=4,m=2

(e) n=4,m=3

Figure 4: Saddle point solutions (black) and the admissible integer charges in
small window (red).

According to the numerical results, one can conclude that small windows of
this non-abelian GLSM are given as the following:

  * •

$n=3$

${\begin{array}[]{ll}m=1,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\cdot\right\rangle\\\
m=2,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\cdot,\quad{\tiny\yng(1,1)}\right\rangle\\\
m=3,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\cdot,\quad{\tiny\yng(1,1)},\quad{\tiny\yng(2,2)}\right\rangle\end{array}}$
(33)

  * •

$n=4$

${\begin{array}[]{ll}m=1,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\cdot,\quad\tiny\yng(1),\quad\tiny\yng(1,1)\right\rangle\\\
m=2,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\tiny\yng(1),\quad\tiny\yng(1,1),\quad\tiny\yng(2,1),\quad\tiny\yng(2,2)\right\rangle\\\
m=3,&\mathbb{W}_{\mathrm{Small}}\>=\>\left\langle\tiny\yng(1),\quad\tiny\yng(1,1),\quad\tiny\yng(2,1),\quad\tiny\yng(2,2),\quad\tiny\yng(3,3)\right\rangle\\\
m=4,&\mathbb{W}_{\mathrm{Small}}=\left\langle\cdot,\quad\tiny\yng(1),\quad\tiny\yng(1,1),\quad\tiny\yng(2,1),\quad\tiny\yng(2,2),\quad\tiny\yng(3,3)\right\rangle\end{array}}$
(34)

If one counts the number of generators in the small windows, one can conclude
that, in the cases of $n=3$ and $n=4$,

$\chi_{-}\>=\>\text{the number of generators
of~{}}\mathbb{W}_{\mathrm{Small}}\>=\>\left\\{\begin{array}[]{ll}m\,,&n=3\,,\\\
m+2\,,&n=4\,,\end{array}\right.$

and together with the number of Coulomb vacua (16), one can see that they
match the total Witten index and the above analysis is consistent:

$\chi_{+}\>=\>\text{the number of generators
of~{}}\mathbb{W}_{\mathrm{Big}}\>=\>\chi_{-}+\left\lfloor\frac{n-1}{2}\right\rfloor(n-m)\>=\>\binom{n}{2}\,.$
(35)

###  3.2 Two determinantal fields

Let us consider a slightly generalized model with one more determinantal
fields, namely it is defined by the following GLSM data 777This field content
is a non-abelian generalization of the hybrid model studied in, e.g. [24, 25],
except in general we do not require $m_{1}=m_{2}$, however, here we only have
vanishing superpotentials.

| $X_{i}$ | $P_{1}$ | $P_{2}$
---|---|---|---
$U(2)$ | $\overline{\tiny{\yng(1)}}$ | $\det^{-m_{1}}$ | $\det^{-m_{2}}$

and the superpotential is again zero, $W=0$. One can see that this GLSM
realizes a similar pure Higgs branch in the positive FI region,

$\mathrm{Tot}\left[{\det}^{m_{1}}(\mathcal{S})\oplus{\det}^{m_{2}}(\mathcal{S})\rightarrow
Gr(2,n)\right]\,,$ (36)

which has the same Witten index as $Gr(2,n)$. On the other hand, the negative
FI region consists of a Higgs branch and some Coulomb vacua. The Coulomb vacua
are determined by the equations

$\sigma_{a}^{n}=-q(-m_{1})^{m_{1}}(-m_{2})^{m_{2}}(\sigma_{1}+\sigma_{2})^{m_{1}+m_{2}},\quad
a=1,2\,.$

Again, one can count the number of Coulomb vacua as in appendix A,

$N(n,m_{1},m_{2})\>=\>\left\lfloor\frac{n-1}{2}\right\rfloor(n-m_{1}-m_{2})\,.$

On the Higgs branch of the negative phase, the nonzero vev of $P_{1}$ and
$P_{2}$ furnishes the weighted projective space $W\mathbb{P}(m_{1},m_{2})$.
The baryons $B_{ij}=\varepsilon_{\alpha\beta}X^{\alpha}_{i}X^{\beta}_{j}$ are
coordinates along the fiber of the vector bundle $\mathcal{O}^{\oplus{n\choose
2}}(-1)$ over the weighted projective space. But there are relationships
between the baryon fields, so the target space in the negative phase is a
fiber bundle over $W\mathbb{P}(m_{1},m_{2})$ with each fiber being isomorphic
to the affine cone of $Gr(2,n)$.

In the case $n=3$, the affine cone of $Gr(2,3)$ is just $\mathbb{C}^{3}$,
therefore the target space is just $Y=\mathrm{Tot}(\mathcal{O}^{\oplus
3}(-1)\rightarrow W\mathbb{P}(m_{1},m_{2}))$, and $D^{b}(Y)$ is generated by
$\mathcal{O}(i)$, $i=0,1,\cdots,m_{1}+m_{2}-1$. So the number of generators of
$D^{b}(Y)$ plus the number of Coulomb vacua is 3, which is exactly the Euler
characteristic of $Gr(2,3)$ as expected.

In the case $n=4$, the target space $Y$ is a fiber bundle over
$W\mathbb{P}(m_{1},m_{2})$ with each fiber being the quadric in
$\mathbb{C}^{6}$ corresponding to the affine cone of the Grassmannian
$Gr(2,4)$. Even though the orbifold groups in the coordinate patches of
$W\mathbb{P}(m_{1},m_{2})$ are generally different from each other888For
example, in the patch where $P_{1}\neq 0$, the orbifold group is
$\mathbb{Z}_{m_{1}}$, while in the patch where $P_{2}\neq 0$, the orbifold
group is $\mathbb{Z}_{m_{2}}$., the number of generators corresponding to the
spinor bundles should be independent of the orbifold structure as argued in
section 3.1.1, and thus independent of $m_{1}$ and $m_{2}$. So the spinor
bundles constitute two generators of $D^{b}(Y)$. In addition, there are
$m_{1}+m_{2}$ generators given by $\mathcal{O}(i)$,
$i=0,1,\cdots,m_{1}+m_{2}-1$. Again, the the number of generators of
$D^{b}(Y)$, which is $m_{1}+m_{2}+2$, plus the number of Coulomb vacua, which
is $4-m_{1}-m_{2}$, is 6, which is equal to the Euler characteristic of
$Gr(2,4)$.

One can analyze the hemisphere partition function as before to find out the
small and big windows, and in this case the exponential factor is similarly
given by:

$\displaystyle A_{q}(t)\>=$
$\displaystyle\pi(\tau_{1}-\tau_{2})-\xi(\nu_{1}+\nu_{2})+\theta(\tau_{1}+\nu_{1})+2\pi(q_{1}\tau_{1}+q_{2}\tau_{2})$
(37)
$\displaystyle+(m_{1}+m_{2})(-\nu_{1}-\nu_{2})\left(\log|m_{1}(\sigma_{1}+\sigma_{2})|-1\right)$
$\displaystyle-(m_{1}+m_{2})|\tau_{1}+\tau_{2}|\left(\frac{\pi}{2}-\arctan\frac{\nu_{1}+\nu_{2}}{|\tau_{1}+\tau_{2}|}\right)$
$\displaystyle+n\left[\nu_{1}\left(\log|\sigma_{1}|-1\right)-|\tau_{1}|\left(\frac{\pi}{2}+\arctan\frac{\nu_{1}}{|\tau_{1}|}\right)\right]$
$\displaystyle+n\left[\nu_{2}\left(\log|\sigma_{2}|-1\right)-|\tau_{2}|\left(\frac{\pi}{2}+\arctan\frac{\nu_{2}}{|\tau_{2}|}\right)\right]\,,$

and on the contour $\gamma$, it is reduced to

$\displaystyle A_{q}(t)\>=$
$\displaystyle\pi|\tau_{1}-\tau_{2}|+\left(\theta+2\pi
q_{1}\right)\tau_{1}+\left(\theta+2\pi
q_{2}\right)\tau_{2}-\pi\frac{m_{1}+m_{2}}{2}|\tau_{1}+\tau_{2}|-\pi\frac{n}{2}(|\tau_{1}|+|\tau_{2}|)$
(38)
$\displaystyle-n|\tau_{1}|\arctan\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{|\tau_{1}|}\right)-n|\tau_{2}|\arctan\left(\frac{(\tau_{1})^{2}-(\tau_{2})^{2}}{|\tau_{2}|}\right)$
$\displaystyle+((\tau_{1})^{2}-(\tau_{2})^{2})\frac{n}{2}\log\frac{(\tau_{1})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}{(\tau_{2})^{2}+((\tau_{1})^{2}-(\tau_{2})^{2})^{2}}\,,$

which is the same as (27) when $m=m_{1}+m_{2}$. Therefore, the previous
discussions and computations can be directly applied here by changing $m$ to
$m_{1}+m_{2}$. So, the small and big windows are given by that of
$m=m_{1}+m_{2}$ in the examples given in the last subsection. For the cases we
are interested in, $n=3$ and $n=4$, the consistency relation still holds as in
Eq. (35).

##  4 Examples

In this section, we will implement the method broadcast in the introduction
and use results from the previous section to study the derived equivalence in
concrete examples. As a warmup, we first start with an abelian example in
section 4.1, which highlights many of the features of this analysis. In
sections 4.2, 4.3 and 4.4, we then study three nonabelian examples and analyze
the corresponding derived equivalences.

##### Remarks on the notation.

For all the examples considered in this section, we will always use $\hat{X}$
to denote the target space of the geometric phase of the GLSM, while $X_{\pm}$
as the target space of the Higgs branch, accompanied by a mixed branch
$\mathcal{C}_{\pm}$, in a different phase of the same GLSM.

###  4.1 Orbibundle over projective space

As the first example, we will consider the derived equivalence between two
orbibundles, $X_{+}$ and $X_{-}$, which are defined as the following999In our
notation, $\mathcal{O}(p/q)$, where $p,q\in\mathbb{Z}$, denotes an orbibundle
whose transition functions take value in $\mathbb{C}^{*}/\mathbb{Z}_{q}$. For
more information on orbibundles, we refer the readers to [26]. For any space
$X$ with a $G$-action, $[X/G]$ denotes the corresponding quotient stack. From
the point of view of B-branes,
$[\mathrm{Tot}(\mathcal{O}(p/q)\rightarrow\mathbb{P}^{n})/{\mathbb{Z}_{q}}]$
with trivial ${\mathbb{Z}}_{q}$-action on the base space can be viewed as a
noncommutative resolution of $\mathbb{P}^{n}$ [27]. The Witten index of a NLSM
with target space
$[\mathrm{Tot}(\mathcal{O}(p/q)\rightarrow\mathbb{P}^{n})/{\mathbb{Z}_{q}}]$
is $q(n+1)$, because as a line bundle which is contractible onto a
$\mathbb{Z}_{q}$-gerbe, its Euler characteristic is the same as this
$\mathbb{Z}_{q}$-gerbe.

$\displaystyle X_{+}\>=\>[{\rm Tot}(\mathcal{O}(-n/m)^{\oplus
M}\to\mathbb{P}^{N-1})/(\mathbb{Z}_{m^{\prime}}\times\mathbb{Z}_{g})]\,,$ (39)
$\displaystyle X_{-}\>=\>[{\rm Tot}(\mathcal{O}(-m/n)^{\oplus
N}\to\mathbb{P}^{M-1})/(\mathbb{Z}_{n^{\prime}}\times\mathbb{Z}_{g})]\,,$

where $g=\gcd(m,n)$, $n^{\prime}=n/g$, $m^{\prime}=m/g$. $X_{+}$ and $X_{-}$
can be realized in two different phases of a GLSM with $\hat{X}$ as its target
space in the pure geometric phase, where

$\hat{X}\>=\>{\rm
Tot}(\mathcal{O}(-m,-n)\to\mathbb{P}^{M-1}\times\mathbb{P}^{N-1})\,.$ (40)

In order to ensure the derived equivalence between $X_{+}$ and $X_{-}$, the
constraint $mN=nM$ must hold, which is also the Calabi-Yau condition for both
$X_{+}$ and $X_{-}$.

####  4.1.1 GLSM construction

To realize $\hat{X}$ , we consider the GLSM with $U(1)\times U(1)$ gauge group
and it is defined by the following matter content for the chiral fields

| $\Phi_{a}$ | $X_{i}$ | $P$ |
---|---|---|---|---
$U(1)_{1}$ | $1$ | $0$ | $-m$ | $\xi_{1}$
$U(1)_{2}$ | $0$ | $1$ | $-n$ | $\xi_{2}$

with $a=1,\dots,M$ and $i=1,\dots,N$. $\xi_{1}$ and $\xi_{2}$ are the FI
parameters. We assume $m<M,n<N$ so the axial $U(1)$ R-symmetry has anomaly.
There is no superpotential in this model and the classical vacua are
determined by the $D$-term equations, which are

$\sum_{a=1}^{M}|\Phi_{a}|^{2}-m|P|^{2}\>=\>\xi_{1}\,,\quad\sum_{i=1}^{N}|X_{i}|^{2}-n|P|^{2}\>=\>\xi_{2}\,.$
(41)

The effective twisted superpotential on the Coulomb branch is given as

$\displaystyle\widetilde{W}_{\rm eff}(\sigma_{1},\sigma_{2})\>=$
$\displaystyle-
t_{1}\sigma_{1}-t_{2}\sigma_{2}-M\sigma_{1}\left(\log\sigma_{1}-1\right)-N\sigma_{2}\left(\log\sigma_{2}-1\right)$
$\displaystyle-(-m\sigma_{1}-n\sigma_{2})\left(\log(-m\sigma_{1}-n\sigma_{2})-1\right)\,$

and therefore the Coulomb branch equations are

$\displaystyle\sigma_{1}^{M}$
$\displaystyle\>=\>q_{1}(-m\sigma_{1}-n\sigma_{2})^{m}\,,$ (42)
$\displaystyle\sigma_{2}^{N}$
$\displaystyle\>=\>q_{2}(-m\sigma_{1}-n\sigma_{2})^{n}\,,$

with $q_{a}=\exp(-t_{a})$ and the $t_{a}$’s are the FI-$\theta$ parameters.

####  4.1.2 Phases

As depicted in Fig. 5, this GLSM has three different phases: $\\{\xi_{1}\gg
0,\xi_{2}\gg 0\\}$, $\\{\xi_{1}\ll 0,n\xi_{1}\ll m\xi_{2}\\}$ and
$\\{\xi_{2}\ll 0,n\xi_{1}\gg m\xi_{2}\\}$. Let us go through these phases and
analyze their IR behaviour.

$I$$\xi_{1}$$\xi_{2}$$III$$II$$n\ \xi_{1}=m\ \xi_{2}$ Figure 5: Phases for the
abelian two-parameter model.

##### Phase I: $\xi_{1}\gg 0$ and $\xi_{2}\gg 0$.

This phase is a pure geometric phase, and the target space is

$\hat{X}\>=\>{\rm
Tot}(\mathcal{O}(-m,-n)\to\mathbb{P}^{M-1}\times\mathbb{P}^{N-1})\,,$

which can be read off from the D-term equations (41). One can also confirm
this by analysing the asymptotic behaviour of the Coulomb branch equations
(42) in the deep phase limit, $\xi_{1}\to+\infty$ and $\xi_{2}\to+\infty$.
101010This method has been used in [28, appendix B] to study the phases of
two-parameter anomalous models and we will also use this strategy in the
following examples. In this limit, the only solution to the Coulomb vacuum
equations is $\sigma_{1}=\sigma_{2}=0$, which indicates that this phase only
has a pure Higgs branch.

Then the Witten index in this phase only has contributions from the Euler
characteristic of the target space $\hat{X}$, which is $MN$.

##### Phase II: $\xi_{1}\ll 0$ and $n\xi_{1}\ll m\xi_{2}$.

This phase consists of a Higgs branch and some mixed Higgs-Coulomb branches.
One can see this by looking at the behaviour of solutions to the Coulomb
branch equations (42) in the limit $\xi_{1}\to-\infty$ and $\xi_{2}\to+\infty$
such that $n\xi_{1}<m\xi_{2}$. In this limit, $\sigma_{2}\to 0$ and then the
first equation becomes

$\sigma_{1}^{M}=q_{1}(-m)^{m}\sigma_{1}^{m}\,,$

which has total $M$ solutions and $m$ of them are zero solutions while the
other $(M-m)$ solutions are non-zero. Namely, now we have $m$ zero solutions
$(\sigma_{1},\sigma_{2})=(0,0)$ and $(M-m)$ solutions with
$(\sigma_{1},\sigma_{2})=(\ast,0)$ where $\ast$ is one of the solutions to
equation $\sigma_{1}^{M-m}=q_{1}(-m)^{m}$.

The solution $(\sigma_{1},\sigma_{2})=(0,0)$ suggests that there exists a pure
Higgs branch, which can be read from the D-term equations. To describe the
Higgs branch, one can change the basis for the Lie algebra of the gauge group
and obtain

| $\Phi_{a}$ | $X_{i}$ | P |
---|---|---|---|---
$n^{\prime}U(1)_{1}-m^{\prime}U(1)_{2}$ | $n^{\prime}$ | $-m^{\prime}$ | 0 | $n^{\prime}\xi_{1}-m^{\prime}\xi_{2}$
$\mu U(1)_{1}+\nu U(1)_{2}$ | $\mu$ | $\nu$ | $-g$ | $\mu\xi_{1}+\nu\xi_{2}$

where $g=\mathrm{gcd}(m,n)$, and $\mu$ and $\nu$ are a pair of integers such
that111111This is to ensure that the transformation given by the matrix
$\left(\begin{array}[]{cc}n^{\prime}&-m^{\prime}\\\ \mu&\nu\end{array}\right)$
is in $SL(2,{\mathbb{Z}})$. The choice of $(\mu,\nu)$ is not unique, but a
different choice, say $(\mu+ln^{\prime},\nu-lm^{\prime})$ for some integer
$l$, amounts to combining $\mu U(1)_{1}+\nu U(1)_{2}$ with $l$ multiples of
$n^{\prime}U(1)_{1}-m^{\prime}U(1)_{2}$, so we may choose one choice and fix
$(\mu,\nu)$. $\mu m^{\prime}+\nu n^{\prime}=1$. Since $\xi_{1}\ll 0$, the $P$
field should take the non-vanishing expectation value
$\left<P\right>=\sqrt{-\xi_{1}/m}$. As such, the gauge group breaks to a
subgroup $U(1)_{s}\times\mathbb{Z}_{g}$, where
$U(1)_{s}:=n^{\prime}U(1)_{1}-m^{\prime}U(1)_{2}$ and its associated FI
parameter is $\xi_{s}:=n^{\prime}\xi_{1}-m^{\prime}\xi_{2}$. Once the
fluctuation of the $P$ field is integrated out, the remaining massless degrees
of freedom are

| $\Phi_{a}$ | $X_{i}$ |
---|---|---|---
$U(1)_{s}$ | $n^{\prime}$ | $-m^{\prime}$ | $\xi_{s}$
$\mathbb{Z}_{g}$ | $\mu$ | $\nu$ |
(43)

and because $\xi_{s}\ll 0$, one can immediately find out that the pure Higgs
branch has target space

$X_{+}\>=\>[{\rm Tot}(\mathcal{O}(-n^{\prime}/m^{\prime})^{\oplus
M}\to\mathbb{P}^{N-1})/({\mathbb{Z}}_{m^{\prime}}\times{\mathbb{Z}}_{g})]\,,$

where the orbifold group acts trivally on the base space, and its action on
the fiber can be read from (43). The Euler characteristic of $X_{+}$ can be
computed as

$\chi(X_{+})\>=\>|{\mathbb{Z}}_{m^{\prime}}\times{\mathbb{Z}}_{g}|\cdot\chi(\mathbb{P}^{N-1})\>=\>mN\,.$

At the nonzero solutions $(\sigma_{1},\sigma_{2})=(\ast,0)$, the fields
$\Phi_{a}$ and $P$ will be massive and should be integrated out. What left is
an effective GLSM for $\mathbb{P}^{N-1}$ defined by $U(1)_{2}$ with $N$ chiral
fields $X_{i}$ of charge $1$. In summary, the whole structure of this phase
can be described as in Fig. 6.

…$X_{+}$$\sigma_{1}=0$$\mathbb{P}^{N-1}$$\mathbb{P}^{N-1}$$\mathbb{P}^{N-1}$$\sigma_{1}$
Figure 6: Mixed structure of Phase II.

Now, let us count the Witten index as a consistency check. There are two types
of contributions: one is from the pure Higgs branch $X_{+}$ and the other one
is from the $(M-m)$ mixed Higgs-Coulomb branches ($M-m$ copies of
$\mathbb{P}^{N-1}$). Therefore, the total Witten index is the sum of these two
contributions

$mN+(M-m)N=MN\,,$

matching the result in Phase I.

##### Phase III: $\xi_{2}\ll 0$ and $n\xi_{1}\gg m\xi_{2}$.

This phase has a similar mixed structure as Phase II, including: one Higgs
branch, which is $X_{-}=[{\rm Tot}(\mathcal{O}(-m^{\prime}/n^{\prime})^{\oplus
N}\to\mathbb{P}^{M-1})/{\mathbb{Z}}_{n^{\prime}}\times{\mathbb{Z}}_{g}]$, and
$(N-n)$ mixed Higgs-Coulomb branches, namely $\mathbb{P}^{M-1}$ located at
$(N-n)$ non-zero points on the $\sigma_{2}$-plane. We skipped the details
here, but one can also check the matching of the total Witten index

$nM+(N-n)M=MN.$

####  4.1.3 Effective theories on the phase boundaries

Now we want to implement the brane transport along the path illustrated in
Fig. 5 to find a functor realizing the equivalence between $D^{b}(X_{+})$ and
$D^{b}(X_{-})$. Apparently, this requires the understanding of the effective
local models at the phase boundaries, $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ and
$\\{\xi_{1}\gg 0,\xi_{2}=0\\}$, and the small window categories thereof.

The local model at the phase boundary $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ has the
following matter content

| $\Phi_{a}$ | $P$ |
---|---|---|---
$U(1)_{1}$ | $1$ | $-m$ | $\xi_{1}$
(44)

Therefore, the small window category consists of the branes with $U(1)_{1}$
charges $q_{1}$ satisfying

$\left|q_{1}+\frac{\theta_{1}}{2\pi}\right|<\frac{m}{2},$

which is termed band restriction rule in [2]. Thus, with a suitable choice of
the $\theta$-angle, the small window category at the phase boundary
$\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ is generated by branes of the form
$\mathcal{W}(i,q)$, where $0\leq i\leq m-1,q\in{\mathbb{Z}}$.

The local model at the phase boundary $\\{\xi_{1}\gg 0,\xi_{2}=0\\}$ has the
following matter content

| $X_{i}$ | $P$ |
---|---|---|---
$U(1)_{2}$ | $1$ | $-n$ | $\xi_{2}$
(45)

Therefore, the small window category consists of the branes with $U(1)_{2}$
charges $q_{2}$ satisfying the band restriction rule

$\left|q_{2}+\frac{\theta_{2}}{2\pi}\right|<\frac{n}{2}.$

Thus, with a suitable choice of the theta-angle, the small window category at
the phase boundary $\\{\xi_{1}\gg 0,\xi_{2}=0\\}$ is generated by branes of
the form $\mathcal{W}(q,j)$, where $-n+1\leq j\leq 0,q\in{\mathbb{Z}}$.

####  4.1.4 Brane transport and derived equivalence

We are interested in studying the derived equivalence between $X_{+}$ and
$X_{-}$. One necessary condition is that their Euler characteristics coincide,
namely $mN=nM$. As discussed before, we have realized them in different phases
of the GLSM for $\hat{X}={\rm
Tot}(\mathcal{O}(-m,-n)\to\mathbb{P}^{M-1}\times\mathbb{P}^{N-1})$. In the
GLSM construction, $X_{\pm}$ are accompanied with the mixed Higgs-Coulomb
branches $\mathcal{C}_{\pm}$ respectively. In this example,

$\displaystyle\mathcal{C}_{+}\>=\>\left\\{(M-m)\text{ copies of
}\mathbb{P}^{N-1}\ \text{located at $(M-m)$ nonzero points on
$\sigma_{1}$-plane}\right\\}\,,$
$\displaystyle\mathcal{C}_{-}\>=\>\left\\{(N-n)\text{ copies of
}\mathbb{P}^{M-1}\ \text{located at $(N-n)$ nonzero points on
$\sigma_{2}$-plane}\right\\}\,.$

The Witten index contributions from $\mathcal{C}_{\pm}$ should also match,
i.e. $(M-m)N=(N-n)M$, which is exactly the same as $mN=nM$ as one would
expect121212In later examples, direct computation of the Euler characteristics
of some pure Higgs branches might be difficult and so we might instead compute
the Witten indices of the mixed Higgs-Coulomb branches to derive the matching
conditions for the Euler characteristics of the pure Higgs branches.. One also
immediately recognize that the condition, $mN=nM$, is also the Calabi-Yau
condition for both $X_{+}$ and $X_{-}$.

The deleted sets on the phases are given by

$\displaystyle X_{+}:$ $\displaystyle\ \\{X=0\\}\cup\\{P=0\\}$ (46)
$\displaystyle\hat{X}:$ $\displaystyle\ \\{X=0\\}\cup\\{\Phi=0\\}$
$\displaystyle X_{-}:$ $\displaystyle\ \\{\Phi=0\\}\cup\\{P=0\\}$

and the corresponding Koszul complexes of empty branes on these deleted sets
are

$\begin{gathered}\mathcal{E}_{X}:\quad\mathcal{W}(q,-N)\xrightarrow{X}\mathcal{W}(q,-N+1)^{\oplus
N}\xrightarrow{X}\cdots\xrightarrow{X}\mathcal{W}(q,-1)^{\oplus
N}\xrightarrow{X}\mathcal{W}(q,0),\\\
\mathcal{E}_{\Phi}:\quad\mathcal{W}(-M,q)\xrightarrow{\Phi}\mathcal{W}(-M+1,q)^{\oplus
M}\xrightarrow{\Phi}\cdots\xrightarrow{\Phi}\mathcal{W}(-1,q)^{\oplus
M}\xrightarrow{\Phi}\mathcal{W}(0,q),\\\
\mathcal{E}_{P}:\quad\mathcal{W}(m,n)\xrightarrow{P}\mathcal{W}(0,0),\end{gathered}$
(47)

from which $\mathcal{E}_{P}$ will be used to grade restrict UV brane complexes
into the small windows. As we assume $m<M,n<N$, the FI parameters are subject
to renormalization. If the FI parameters $(\xi_{1},\xi_{2})$ are above the
line $n\xi_{1}=m\xi_{2}$, then they will end up in the phase of $X_{+}$ in the
IR limit under RG flow. The RG flow will take a UV GLSM brane
$\mathcal{W}(p_{1},p_{2})$ to an IR brane on $X_{+}$ as131313Here and in the
following, we will use ${\mathcal{O}}_{X}(q)$ to denote
$\pi^{*}{\mathcal{O}}_{{\mathbb{P}}^{r}}(q)$ for
$X=\mathrm{Tot}(\mathcal{F}\stackrel{{\scriptstyle\pi}}{{\rightarrow}}{\mathbb{P}}^{r})$,
where $\mathcal{F}$ is a vector bundle or an orbibundle.

$\mathcal{W}(p_{1},p_{2})\xrightarrow{\pi_{+}}\mathcal{O}_{X_{+}}\left(\frac{p_{2}m^{\prime}-p_{1}n^{\prime}}{m^{\prime}}\right)_{s},$
(48)

where $s\equiv\mu p_{1}+\nu p_{2}~{}(\mathrm{mod}~{}g)$ labels the twisted
sector of $\mathbb{Z}_{g}$. On the other hand, if the FI parameters
$(\xi_{1},\xi_{2})$ are below the line $n\xi_{1}=m\xi_{2}$, then they will end
up in the phase of $X_{-}$ in the IR limit. A UV GLSM brane
$\mathcal{W}(p_{1},p_{2})$ will descend to an IR brane on $X_{-}$ as

$\mathcal{W}(p_{1},p_{2})\xrightarrow{\pi_{-}}\mathcal{O}_{X_{-}}\left(\frac{p_{1}n^{\prime}-p_{2}m^{\prime}}{n^{\prime}}\right)_{s}.$
(49)

Notice that if we use $\mathcal{E}_{P}$ to replace $\mathcal{W}(p_{1},p_{2})$
with an IR equivalent brane $\mathcal{W}(p_{1}-lm,p_{2}-ln)$ for some integer
$l$, then both $i=q_{2}m^{\prime}-q_{1}n^{\prime}$ and
$s\equiv\mu(p_{1}-lm)+\nu(p_{2}-ln)\equiv\mu p_{1}+\nu
p_{2}~{}(\mathrm{mod}~{}g)$ are invariant, therefore the IR images under
$\pi_{+}$ and $\pi_{-}$ are indeed unchanged.

Consequently, the brane transport between $D^{b}(X_{+})$ and $D^{b}(X_{-})$ is
given by the following procedure:

  * •

For any $i\in\\{0,1,\cdots,m^{\prime}N-1\\},s\in{\mathbb{Z}}_{g}$, choose a
pair of integers $(p_{1},p_{2})$ such that $i=p_{2}m^{\prime}-p_{1}n^{\prime}$
and $s\equiv\mu p_{1}+\nu p_{2}~{}(\mathrm{mod}~{}g)$. For example,

$p_{1}=-\nu i+m^{\prime}s+lm,\ p_{2}=\mu i+n^{\prime}s+ln,\
\forall~{}l\in\mathbb{Z}.$

  * •

Then, from Eq. (48), The lift of a brane
$\mathcal{O}_{X_{+}}\left(\frac{i}{m^{\prime}}\right)_{s}$ in the phase of
$X_{+}$ (above the line $n\xi_{1}=m\xi_{2}$) is

$\mathcal{O}_{X_{+}}\left(\frac{i}{m^{\prime}}\right)_{s}\xrightarrow{\pi_{+}^{-1}}\mathcal{W}(p_{1},p_{2}),$
(50)

  * •

Use the IR empty brane $\mathcal{E}_{P}$ to grade restrict
$\mathcal{W}(p_{1},p_{2})$ so the resulting brane is in the small window
category at the phase boundary $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$, i.e.
$p_{1}=0,\cdots,m-1$, so it can pass through this phase boundary.

  * •

Once in the phase $\\{\xi_{1}\gg 0,\xi_{2}\gg 0\\}$, we can move it below the
line $n\xi_{1}=m\xi_{2}$ and use the IR empty brane $\mathcal{E}_{P}$ to grade
restrict it such that it is in the small window category at the phase boundary
$\\{\xi_{1}\gg 0,\xi_{2}=0\\}$, i.e. $p_{2}\in\\{-n+1,-n+2,\cdots,1,0\\}$.
Then the brane can be transported to the phase of $X_{-}$.

  * •

Due to Eq. (49), the brane can be projected to give the image:

$\mathcal{W}(\overline{p}_{1},\overline{p}_{2})\xrightarrow{\pi_{-}}\mathcal{O}_{X_{-}}\left(-\frac{i}{n^{\prime}}\right)_{s},$

where $\overline{p}_{1}$ and $\overline{p}_{2}$ are the resulting gauge
charges after the grade restriction has been applied.

In conclusion, with our choice of the small windows, the brane transport
yields the following map between the generators of $D^{b}(X_{+})$ and
$D^{b}(X_{-})$:

$\mathcal{O}_{X_{+}}\left(\frac{i}{m^{\prime}}\right)_{s}\Rightarrow\mathcal{O}_{X_{-}}\left(-\frac{i}{n^{\prime}}\right)_{s}$

for $i=0,1,\cdots,m^{\prime}N-1,~{}s\in{\mathbb{Z}}_{g}$.

##### Example:

$m=4,n=6$, $M=6$, $N=9$.
In this case, $g=\gcd(m,n)=2,m^{\prime}=2,n^{\prime}=3$ and we can take
$\mu=-1,\nu=1$. The geometries are

$\begin{gathered}X_{+}=[\mathrm{Tot}(\mathcal{O}(-3/2)^{\oplus
6}\rightarrow\mathbb{P}^{8})/(\mathbb{Z}_{2}\times\mathbb{Z}_{2})]\,,\\\
X_{-}=[\mathrm{Tot}(\mathcal{O}(-2/3)^{\oplus
9}\rightarrow\mathbb{P}^{5})/(\mathbb{Z}_{3}\times\mathbb{Z}_{2})]\,.\end{gathered}$

The small window at the phase boundary $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ is
generated by $\mathcal{W}(i,p_{2})$, where $i=0,1,2,3$ and $p_{2}$ is any
$U(1)_{2}$ charge. The small window at the phase boundary $\\{\xi_{1}\gg
0,\xi_{2}=0\\}$ is generated by $\mathcal{W}(p_{1},j)$, where
$j=-5,-4,\cdots,0$ and $p_{1}$ is any $U(1)_{1}$ charge.

The derived category $D^{b}(X_{+})$ is generated by
${\mathcal{O}}_{X_{+}}\left(\frac{i}{2}\right)_{s}$ with
$i=0,1,\cdots,17,~{}s=0,1$. The derived category $D^{b}(X_{-})$ is generated
by ${\mathcal{O}}_{X_{-}}\left(-\frac{j}{3}\right)_{t}$ with
$j=0,1,\cdots,17,~{}t=0,1$. For any $i$ and $s$ in the range indicated above,
take

$p_{1}=-i+2s,\quad p_{2}=-i+3s.$

The lift of ${\mathcal{O}}_{X_{+}}\left(\frac{i}{2}\right)_{s}$ is
$\mathcal{W}(p_{1}+4l,p_{2}+6l)$ for any integer $l$. We can first choose $l$
such that $p_{1}+4l\in\\{0,1,2,3\\}$. After the brane is transported to the
phase $\\{\xi_{1}\gg 0,\xi_{2}\gg 0\\}$, we can choose another integer
$l^{\prime}$ such that $p_{2}+6l+6l^{\prime}\in\\{-5,-4,\cdots,0\\}$ so the
grade restricted brane can be transported to the phase of $X_{-}$, where its
IR image is ${\mathcal{O}}_{X_{-}}\left(-\frac{i}{3}\right)_{s}$.

###  4.2 Product of a projective space and a Grassmannian

In the previous section, we have presented our strategy for showing the
derived equivalence in an abelian gauge theory. Now, in this section, we start
to consider a nonabelian example and show the derived equivalence between
$X_{+}$, a vector bundle over Grassmannian as defined by Eq. (56), and
$X_{-}$, a fibre bundle over projective space as defined by Eq. (58). We
follow a similar construction as discussed in the previous section and find
that $X_{+}$ and $X_{-}$ can be realized in two different phases of the GLSM
for $\hat{X}$, which is a fiber bundle over the product of a projective space
and a Grassmannian, Eq. (53).

####  4.2.1 GLSM construction

The GLSM has the gauge group $U(1)\times U(2)$ and the following matter
content

| $X_{a}$ | $\Phi^{\alpha}_{j}$ | $P$ |
---|---|---|---|---
$U(1)$ | 1 | 0 | $-m$ | $\xi_{1}$
$U(2)$ | $\cdot$ | $\tiny{\yng(1)}$ | $\det^{-n}$ | $\xi_{2}$
(51)

with $\alpha=1,2$ the $U(2)$ color index, $a=1,\cdots,M$ and $j=1,\dots,N$ are
flavor indices, and we assume $m\leq M,n\leq N$. In this model, the
superpotential $W=0$. The Coulomb vacuum equations are

$\displaystyle\sigma^{M}$
$\displaystyle=q_{1}(-m\sigma-n\sigma_{1}-n\sigma_{2})^{m}\,,$ (52)
$\displaystyle\sigma_{\alpha}^{N}$
$\displaystyle=-q_{2}(-m\sigma-n\sigma_{1}-n\sigma_{2})^{n}\,,\alpha=1,2\,,$

where $\sigma$ is the scalar component of the gauge field strength associated
with $U(1)$, while $\sigma_{\alpha},\alpha=1,2$ are those associated with the
Cartan subalgebra of $U(2)$. In the above, $q_{s}:=\exp(-t_{s})$ and $t_{s}$
is the FI-$\theta$ parameter corresponding to $\xi_{s}$.

####  4.2.2 Phases

The phase diagram for this GLSM is the same as the previous example, i.e. Fig.
5. Clearly, in the $\xi_{1}\gg 0,\xi_{2}\gg 0$ region, there is only a pure
Higgs branch with target space

$\hat{X}\>=\>\mathrm{Tot}\left(\mathcal{O}_{\mathbb{P}^{M-1}}(-m)\otimes\left(\det\mathcal{S}\right)^{\otimes
n}\rightarrow\mathbb{P}^{M-1}\times Gr(2,N)\right)\,,$ (53)

where $\mathcal{S}$ is the tautological bundle over $Gr(2,N)$. The total
Witten index in this phase equals the Euler characteristic of $\hat{X}$, which
is

$\chi(\hat{X})\>=\>M\binom{N}{2}\,.$ (54)

If either $\xi_{1}\ll 0$ or $\xi_{2}\ll 0$, the $P$ field gets a nonzero vev
due to the D-term equations and breaks the $U(1)\times\det U(2)$ gauge group
to a subgroup $U(1)_{s}\times{\mathbb{Z}}_{\gcd(m,n)}$ with the FI parameter
of $U(1)_{s}$ given by $\xi_{s}:=n^{\prime}\xi_{1}-m^{\prime}\xi_{2}$, where
$m^{\prime}=m/{\gcd(m,n)},n^{\prime}=n/{\gcd(m,n)}$. Under the $U(1)_{s}$
gauge group, the $SU(2)$ invariant fields $X_{a}$ and the baryon fields
$B_{ij}=\varepsilon_{\alpha\beta}\Phi^{\alpha}_{i}\Phi^{\beta}_{j}$ have
charges

| $X_{a}$ | $B_{ij}$
---|---|---
$U(1)_{s}$ | $n^{\prime}$ | $-m^{\prime}$

When $\xi_{s}\ll 0,\xi_{1}\ll 0$, $X_{a}$ and $B_{ij}$ become the fiber and
base coordinates of

$\mathrm{Tot}\left(\mathcal{O}(-n/m)^{\oplus
M}\rightarrow\mathbb{P}^{{N\choose 2}-1}\right)$ (55)

respectively at low energy scale. The relations satisfied by the $B_{ij}$’s
then tell us that in this phase, there is a Higgs branch whose target space
$X_{+}$ is an orbifold of the vector bundle (55) restricted to the image of
the Plücker embedding of $Gr(2,N)$ in $\mathbb{P}^{{N\choose 2}-1}$.
Therefore, $X_{+}$ is defined as 141414In fact, $\mathcal{O}(-n/m)$ should be
interpreted as the pullback of the orbibundle $\mathcal{O}(-n/m)$ on
$\mathbb{P}^{\binom{N}{2}-1}$ under the Plücker embedding and here we abuse
the notation to avoid clumsy symbols.

$X_{+}\>=\>[\mathrm{Tot}\left(\mathcal{O}(-n/m)^{\oplus M}\rightarrow
Gr(2,N)\right)/({\mathbb{Z}}_{m^{\prime}}\times{\mathbb{Z}}_{\gcd(m,n)})]\,,$
(56)

The Euler characteristic of $X_{+}$ can be computed directly,

$\chi(X_{+})\>=\>m\binom{N}{2}\,.$

In this region, there are also mixed Higgs-Coulomb branches, as can be read
from nonvanishing Coulomb vacua when $\xi_{s}\ll 0,\xi_{1}\ll 0$. When we take
$\xi_{s}\to-\infty$, one can see from (52) that

$\sigma_{\alpha}^{Nm}\>=\>-q_{s}\sigma^{Mn}\to 0\,,$

which means $\sigma_{\alpha}\to 0$. The remaining Coulomb vacua equation is

$\sigma^{M}\>=\>q_{1}(-m)^{m}\sigma^{m}\,,$

and this equation suggests that $\sigma$ has zero solutions, which is
consistent with the existence of the pure Higgs branch, namely $X_{+}$. In
addition, this equation also has $(M-m)$ nonzero solutions, which indicates
the existence of the mixed Higgs-Coulomb branches. One can find that the
effective theory corresponding to each mixed branch can be described by a NLSM
with target space the Grassmannian $Gr(2,N)$. Therefore, the mixed Higgs-
Coulomb branch is

$\mathcal{C}_{+}\>=\>\left\\{(M-m)\text{ copies of }Gr(2,N)\text{ located at
}(M-m)\text{ nonzero points on $\sigma$-plane}\right\\}\,.$

One can check that the total Witten index equals

$\chi(X_{+})+\chi(\mathcal{C}_{+})\>=\>m\binom{N}{2}+(M-m)\binom{N}{2}\>=\>M\binom{N}{2}\,,$

which is consistent with the total Witten index (54).

When $\xi_{s}\gg 0,\xi_{2}\ll 0$, $X_{a}$ and $B_{ij}$ become the base and
fiber coordinates of

$\mathrm{Tot}\left(\mathcal{O}(-m/n)^{\oplus{N\choose
2}}\rightarrow\mathbb{P}^{M-1}\right)$ (57)

respectively at low energy scale. The relations satisfied by the $B_{ij}$’s
then tell us that in this phase, there is a Higgs branch whose target space
$X_{-}$ is an orbifold of the fiber bundle over $\mathbb{P}^{M-1}$ with each
fiber being the affine cone of $Gr(2,N)$ (we denote the affine cone by
$CGr(2,N)$). Let us call the fiber bundle $\mathcal{X}(2,N)$. In short the
target space in this phase is

$X_{-}\>=\>\left[\left(\mathrm{Tot}\left(\mathcal{X}(2,N)\rightarrow\mathbb{P}^{M-1}\right)\right)/(\mathbb{Z}_{n^{\prime}}\times{\mathbb{Z}}_{\gcd(m,n)})\right]\,.$
(58)

Note that $X_{-}$ is not an ordinary smooth fiber bundle and it is not
straight forward to calculate its Euler characteristic, but we can easily
obtain the Euler characteristic of $X_{-}$ by subtracting the mixed Higgs-
Coulomb branch contributions from the total Witten index. One way to do this
computation is to take the standpoint that each fiber arises as the Higgs
branch of the negative phase of the $U(2)$ GLSM with $N$ fundamentals and one
field in the $\det^{-n}$ representation. The positive phase of this $U(2)$
theory has a pure Higgs branch which realizes a vector bundle over the
Grassmannian $Gr(2,N)$, so the total Witten index is ${N\choose 2}$. From the
computation in appendix A, we see there are $(N-n)\lfloor\frac{N-1}{2}\rfloor$
Coulomb vacua in the negative phase of the $U(2)$ theory. Therefore the Witten
index of the Higgs branch of the negative phase of this $U(2)$ theory is
${N\choose 2}-(N-n)\left\lfloor\frac{N-1}{2}\right\rfloor$. Consequently, the
Witten index of $X_{-}$ is

$\chi(X_{-})\>=\>M\binom{N}{2}-M(N-n)\left\lfloor\frac{N-1}{2}\right\rfloor\,,$

The mixed Higgs-Coulomb branch can be found following the same procedure as
before. In the limit $\xi_{s}\to+\infty$, one will find the mixed Higgs-
Coulomb branches located at $\sigma=0$ and $(N-n)\lfloor\frac{N-1}{2}\rfloor$
nonvanishing solutions for $(\sigma_{1},\sigma_{2})$ 151515This number can be
obtained following appendix A.. Further, on each such nonvainishing point in
the $\\{\sigma_{1},\sigma_{2}\\}$-plane, there is a projective space
$\mathbb{P}^{M-1}$. Namely, we have

$\mathcal{C}_{-}\>=\>\left\\{(N-n)\left\lfloor\frac{N-1}{2}\right\rfloor\text{
copies of }\mathbb{P}^{M-1}\text{ located at the nonzero points on
$\\{\sigma_{1},\sigma_{2}\\}$-plane}\right\\}\,.$

At the end, this leads to the matching of the Witten indices

$\chi(X_{-})\>=\>\chi(\hat{X})-\chi(\mathcal{C}_{-})\,.$

In particular,

$\chi(X_{-})\>=\>\begin{dcases}\frac{nM(N-1)}{2},&N\text{ is odd}\,,\\\
\frac{nM(N-2)}{2}+\frac{MN}{2},&N\text{ is even}\,.\end{dcases}$

Comparing $\chi(X_{+})$ and $\chi(X_{-})$, the derived equivalence between
$X_{+}$ and $X_{-}$ suggests

$\begin{dcases}mN\>=\>nM,&N\text{ is odd}\,,\\\
mN(N-1)-nM(N-2)\>=\>MN,&N\text{ is even}\,.\end{dcases}$ (59)

One shall note that the condition with $N$ odd is also the Calabi-Yau
condition for $X_{\pm}$, while $\hat{X}$ is not necessarily a Calabi-Yau. When
$N$ is even, if we further apply the Calabi-Yau condition for $X_{\pm}$, i.e.
$mN=nM$, then the second equation becomes

$(N-n)M\>=\>0\,,$

namely, we should have $M=m$ and $N=n$, which is also the Calabi-Yau condition
for $\hat{X}$.

In particular, when $N=3$, the first equation gives $nM=3m$. When $N=4$, the
second equation gives $(n+2)M=6m$ and in this case, if the Calabi-Yau
condition is further applied, we can only take $M=m$ and $n=N=4$. In the
following, we will consider the cases of $X_{\pm}$ being Calabi-Yau’s.

####  4.2.3 Effective theories on the phase boundaries

The local model at the phase boundary of $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ is a
$U(1)$ gauge theory with the following matter content

| $X_{a}$ | $P$
---|---|---
$U(1)$ | 1 | $-m$
(60)

The small window of this abelian GLSM is

$\left|q_{1}+\frac{\theta_{1}}{2\pi}\right|<\frac{m}{2}\,,$ (61)

where $q_{1}$ and $\theta_{1}$ are the gauge charge and theta angle of $U(1)$.
Eq. (61) suggests that the window category at the phase boundary
$\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ can be chosen to be generated by branes of the
form $\mathcal{W}(q_{1},\lambda)$, where $q_{1}\in\\{0,1,\cdots m-1\\}$ and
$\lambda$ is any $U(2)$ representation.

The local model at the phase boundary of $\\{\xi_{2}=0,\xi_{1}\gg 0\\}$ is the
$U(2)$ gauge theory with the following matter content

| $\Phi^{\alpha}_{j}$ | $P$
---|---|---
$U(2)$ | $\tiny{\yng(1)}$ | $\det^{-n}$
(62)

For the cases with $n\leq N$, $N=3$ and $4$, this model has been studied in
section 3 and the small window is given by Eq. (33) and Eq. (34). Therefore,
the small window category at the phase boundary $\\{\xi_{2}=0,\xi_{1}\gg 0\\}$
can be chosen to be generated by branes of the form
$\mathcal{W}(q_{1},\lambda)$, where $\lambda$ is given by Eq. (33) and Eq.
(34) in the cases of $N=3$ and $N=4$ respectively and $q_{1}$ is any $U(1)$
charge.

####  4.2.4 Brane transport and derived equivalence

The theory (51) with $N=3$ is dual in the IR limit to an abelian theory with
matter content

| $X_{a}$ | $\Phi_{j}$ | $P$ |
---|---|---|---|---
$U(1)$ | 1 | 0 | $-m$ | $\xi_{1}$
$U(1)$ | 0 | 1 | $-n$ | $\xi_{2}$

which reduces to one of the cases studied in section 4.1. In this section, we
focus on the behaviour of brane transport in the case of $N=4$.

As discussed before, the Calabi-Yau condition for $X_{\pm}$ requires that
$n=N=4$ and $m=M$. In this cae, $\hat{X}$ is also Calabi-Yau, and we have the
derived equivalence

$D^{b}(X_{+})\cong D^{b}(\hat{X})\cong D^{b}(X_{-}),$

where the geometries in the three phases are

$\displaystyle X_{+}\>=\>\left[\mathrm{Tot}\left(\mathcal{O}(-4/m)^{\oplus
m}\stackrel{{\scriptstyle\pi_{+}}}{{\rightarrow}}Gr(2,4)\right)/({\mathbb{Z}}_{m/\gcd(m,4)}\times{\mathbb{Z}}_{\gcd(m,4)})\right],$
$\displaystyle
X_{-}\>=\>\left[\left(\mathcal{X}(2,4)\stackrel{{\scriptstyle\pi_{-}}}{{\rightarrow}}\mathbb{P}^{m-1}\right)/({\mathbb{Z}}_{4/\gcd(m,4)}\times{\mathbb{Z}}_{\gcd(m,4)})\right],$
$\displaystyle\hat{X}\>=\>\mathrm{Tot}\left(\mathcal{O}_{{\mathbb{P}}_{m-1}}(-m)\otimes(\wedge^{2}\mathcal{S})^{\otimes
4})\stackrel{{\scriptstyle\pi}}{{\rightarrow}}\mathbb{P}^{m-1}\times
Gr(2,4)\right).$

In this case, the axial $U(1)$ R-symmetry is non-anomalous, the small window
category and the big window category coincide. Moreover, because the FI
parameters are not renormalized, the brane transport amounts to the following
steps161616Here we take the transport to be from the phase of $X_{+}$ to the
phase of $X_{-}$ for example, and the transport in the opposite direction
follows the same procedure in the reverse order.:

  * •

For any brane $B_{+}$ in $D^{b}(X_{+})$, lift it to a GLSM brane
$\mathcal{W}_{+}$ such that it is in the window category of the local model
(60), so it can be smoothly transported to the phase of $\hat{X}$.

  * •

Once in the phase of $\hat{X}$, we can read off its image in $D^{b}(\hat{X})$.

  * •

Use empty branes to grade restrict $\mathcal{W}_{+}$ and obtain an IR
equivalent brane $\mathcal{W}_{-}$, which is in the window category of the
local model (62).

  * •

$\mathcal{W}_{-}$ can be smoothly transported to the phase of $X_{-}$, where
its image in $D^{b}(X_{-})$ can be determined.

The affine cone of $Gr(2,4)$ is given by the pre-image of $\det U(2)$ GIT
quotient as

$CGr(2,4)/{\mathbb{Z}}_{4}=\mathbb{C}^{6}/(B_{12}B_{34}-B_{13}B_{24}+B_{14}B_{23}).$

As discussed in section 3.1.1, $CGr(2,4)/{\mathbb{Z}}_{4}$ can be viewed as
arising from the negative phase of the $U(2)$ GLSM with four fundamentals and
one field in the $\det^{-4}$ representation. There and also in [11], we see
$D^{b}(CGr(2,4)/{\mathbb{Z}}_{4})$ is generated by the perfect subcategory
$\langle\mathcal{A}_{0},\mathcal{A}_{1},\mathcal{A}_{2},\mathcal{A}_{3}\rangle$,
which correspond to the $\det^{l}$, $l=0,1,2,3$, representations of $U(2)$.
There is also a singular subcategory
$\langle\mathcal{M}_{0},\mathcal{M}_{1}\rangle$, where $\mathcal{M}_{0}$
corresponds to the $U(2)$ representation $\tiny\yng(1)$, and $\mathcal{M}_{1}$
corresponds to the $\tiny\yng(1)\otimes\det$ representation. Let us assume
that $\mathcal{M}_{0}$ can be glued together nicely to give a sheaf
$\tilde{\mathcal{S}}^{\vee}$ on $\mathcal{X}(2,4)$.

Let us investigate some examples to see how the above steps are implemented.
Assume that $M=m=3$ for simplicity. In this case, the empty branes for grade
restriction at the phase boundary $\\{\xi_{1}\gg 0,\xi_{2}=0\\}$ are derived
from exact sequences on $Gr(2,4)$ generated by Eagon-Northcott type complexes,
such as171717The Young diagram $\lambda$ is a shorthand notation for
$\Sigma_{\lambda}\mathcal{S}^{\vee}$, where $\Sigma_{\lambda}$ is the Schur
functor associated with $\lambda$.

${\tiny\yng(3,2)}\rightarrow{\tiny\yng(2,2)}^{\oplus
4}\rightarrow{\tiny\yng(1,1)}^{\oplus 4}\rightarrow{\tiny\yng(1)},$ (63)
${\tiny\yng(4,4)}\rightarrow{\tiny\yng(3,3)}^{\oplus
6}\rightarrow{\tiny\yng(3,2)}^{\oplus 4}\rightarrow{\tiny\yng(2,1)}^{\oplus
4}\rightarrow{\tiny\yng(1,1)}^{\oplus 6}\rightarrow{\mathcal{O}}.$ (64)

On the other hand, the empty branes for grade restriction at the phase
boundary $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ are of the form

$W(q_{1},\lambda)\rightarrow W(q_{1}+1,\lambda)^{\oplus 3}\rightarrow
W(q_{1}+2,\lambda)^{\oplus 3}\rightarrow W(q_{1}+3,\lambda),$

where the arrows are maps given by $X_{a},a=1,2,3$, and $\lambda$ is any
$U(2)$ representation. Notice that a GLSM brane of the form
$\mathcal{W}(z_{1},\det^{z_{2}})$ is projected to
${\mathcal{O}}_{X_{+}}((3z_{2}-4z_{1})/3)$ in the phase of $X_{+}$, and to
${\mathcal{O}}_{X_{-}}((4z_{1}-3z_{2})/4)$ in the phase of $X_{-}$, as
explained in section 4.1.4.

  * •

$\mathcal{O}_{X_{+}}(1)$ can be lifted to the GLSM brane
$\mathcal{W}(0,\det)$, which is in the window category of (60) so it can be
transported to the phase $\xi_{1}\gg 0,\xi_{2}\gg 0$. It can be projected to
$\pi^{*}({\mathcal{O}}_{{\mathbb{P}}^{2}}\otimes\wedge^{2}\mathcal{S}^{\vee})$
on $\hat{X}$. Because $\mathcal{W}(0,\det)$ is also in the window category of
(62), it can be transported to the phase of $X_{-}$, where it can be projected
to ${\mathcal{O}}_{X_{-}}(-3/4)$.

  * •

$\mathcal{S}^{\vee}\otimes{\mathcal{O}}_{X_{+}}(1/3)$ can be lifted to the
GLSM brane $\mathcal{W}(3,{\tiny\yng(1)}\otimes\det^{2})$, which is in the
window category of (60) so it can be transported to the phase $\xi_{1}\gg
0,\xi_{2}\gg 0$, where it can be projected to
$\pi^{*}({\mathcal{O}}_{{\mathbb{P}}^{2}}(3)\otimes(\wedge^{2}\mathcal{S}^{\vee})^{\otimes
2}\otimes\mathcal{S}^{\vee})$ on $\hat{X}$. As
$\mathcal{W}(3,{\tiny\yng(1)}\otimes\det^{2})$ is not in the window category
of (62), we need to grade restrict it. This can be done by using the exact
sequence (63). The grade restricted brane is then

$\mathcal{W}(3,{\det}^{2})^{\oplus 4}\rightarrow\mathcal{W}(3,\det)^{\oplus
4}\rightarrow\mathcal{W}(3,{\tiny\yng(1)}),$

which is now in the window category so it can be transported to give the image

${\mathcal{O}}_{X_{-}}\left(\frac{3}{2}\right)^{\oplus
4}\rightarrow{\mathcal{O}}_{X_{-}}\left(\frac{9}{4}\right)^{\oplus
4}\rightarrow{\mathcal{O}}_{X_{-}}\left(3\right)\otimes\tilde{\mathcal{S}}^{\vee}.$

  * •

${\mathcal{O}}_{X_{+}}(4/3)$ can be lifted to the GLSM brane
$\mathcal{W}(2,\det^{4})$, which is in the window category of (60) so it can
be transported to the phase $\xi_{1}\gg 0,\xi_{2}\gg 0$, where it can be
projected to
$\pi^{*}({\mathcal{O}}_{{\mathbb{P}}^{2}}(2)\otimes(\wedge^{2}\mathcal{S}^{\vee})^{\otimes
4})$ on $\hat{X}$. $\mathcal{W}(3,{\tiny\yng(1)}\otimes\det^{2})$ is not in
the window category of (62), but it can be grade restricted with the help of
the cone of the exact sequences (64) and four copies of (63), in which the
representation ${\tiny\yng(3,2)}$ is cancelled. The grade restricted brane
reads

$\mathcal{W}(2,{\det}^{3})^{\oplus
6}\rightarrow\mathcal{W}(2,{\det}^{2})^{\oplus
16}\rightarrow\mathcal{W}(2,\det)^{\oplus
16}\rightarrow\mathcal{W}(2,{\tiny\yng(1)})^{\oplus 4},$

which is now in the window category so it can be transported to give the image

${\mathcal{O}}_{X_{-}}\left(-\frac{1}{4}\right)^{\oplus
6}\rightarrow{\mathcal{O}}_{X_{-}}\left(\frac{1}{2}\right)^{\oplus
16}\rightarrow{\mathcal{O}}_{X_{-}}\left(\frac{5}{4}\right)^{\oplus
16}\rightarrow\left({\mathcal{O}}_{X_{-}}\left(2\right)\otimes\tilde{\mathcal{S}}^{\vee}\right)^{\oplus
4}.$

  * •

Let us look at an example of brane transport in the reverse direction.
Starting with the brane ${\mathcal{O}}_{X_{-}}(3/4)$ in the phase of $X_{-}$,
we get a GLSM brane $\mathcal{W}(3,\det^{3})$, which is in the window category
of (62), so it can be transported to the phase of $\hat{X}$ and projected to
$\pi^{*}({\mathcal{O}}_{{\mathbb{P}}^{2}}(3)\otimes(\wedge^{2}\mathcal{S}^{\vee})^{\otimes
3})$. $\mathcal{W}(3,\det^{3})$ is not in the window category of (60), but can
be grade restricted using the exact sequence

$W(0,{\det}^{3})\rightarrow W(1,{\det}^{3})^{\oplus 3}\rightarrow
W(2,{\det}^{3})^{\oplus 3}\rightarrow W(3,{\det}^{3}).$

Therefore, the image of the brane transport in the phase of $X_{+}$ is

${\mathcal{O}}_{X_{+}}(3)\rightarrow{\mathcal{O}}_{X_{+}}\left(\frac{5}{3}\right)^{\oplus
3}\rightarrow{\mathcal{O}}_{X_{+}}\left(\frac{1}{3}\right)^{\oplus 3}.$

Other cases can be analyzed in the same way.

###  4.3 Two-step flag manifold

In this subsection, we discuss the derived equivalence between $X_{+}$ and
$X_{-}$ defined as

$X_{+}\>=\>{\rm Tot}[\mathcal{S}\otimes\det\mathcal{S}\to Gr(2,N)],\quad\quad
X_{-}\>=\>{\rm Tot}[\mathcal{L}^{2}\otimes\left(\mathcal{O}^{\oplus
N}/\mathcal{L}\right)\to\mathbb{P}^{N-1}]\,,$

where $\mathcal{S}$ and $\mathcal{L}$ are the tautological bundles over
$Gr(2,N)$ and $\mathbb{P}^{N-1}$ respectively. It turns out that $X_{+}$ and
$X_{-}$ can be constructed by a $U(1)\times U(2)$ GLSM whose target space in
the pure geometric phase is

$\hat{X}\>=\>{\rm Tot}[\mathcal{L}\otimes\det\mathcal{S}\to Fl(1,2,N)]\,,$

where $\mathcal{L}$ and $\mathcal{S}$ are the first and second tautological
bundles of the two-step flag manifold $Fl(1,2,N)$.

Note that some analysis in this section can also be applied to the two-step
flag manifolds without fiber directions. Therefore, we have included some
discussions on $Fl(1,2,N)$ in appendix B for comparison.

####  4.3.1 GLSM construction for $\hat{X}$

The GLSM data for $\hat{X}$ is given as the following

| $\Phi_{\alpha}$ | $X^{\alpha}_{i}$ | $P$ |
---|---|---|---|---
$U(1)$ | 1 | 0 | $-1$ | $\xi_{1}$
$U(2)$ | $\overline{\tiny{\yng(1)}}$ | $\tiny{\yng(1)}$ | $\det^{-1}$ | $\xi_{2}$

with the color index $\alpha=1,2$ and the flavor index $i=1,\dots,N$. The
D-terms associated with $U(1)$ and $U(2)$ are

$\displaystyle D$
$\displaystyle\sim\sum_{\alpha=1}^{2}|\Phi_{\alpha}|^{2}-|P|^{2}-\xi_{1}\,,$
(65) $\displaystyle D^{\alpha}_{\beta}$
$\displaystyle\sim-\overline{\Phi}^{\alpha}\Phi_{\beta}+\sum_{i}\overline{X}_{\beta,i}X^{\alpha}_{i}-|P|^{2}\delta^{\alpha}_{\beta}-\xi_{2}\delta^{\alpha}_{\beta}\,.$

The equations for Coulomb vacua derived from the effective twisted
superpotential are

$\displaystyle\prod_{\alpha=1}^{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)=-q_{1}\left(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}\right)\,,$
(66)
$\displaystyle{\sigma^{(2)}_{\alpha}}^{N}=q_{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)\left(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}\right)\,,$

for $\alpha=1,2$ and $q_{a}=\exp(-t_{a})$ with $t_{1}$ and $t_{2}$ the complex
FI parameter associated with $U(1)$ and $U(2)$ respectively. In the above
equations, $\sigma^{(1)}$ is the scalar component of the $U(1)$ gauge field
strength, while $\sigma_{\alpha}^{(2)}$ are the scalar components of the
Cartan part of $U(2)$ gauge field strength.

####  4.3.2 Phase structure

The phase structure is shown in Fig. 7. For the purpose of this paper, we are
only interested in Phase I, II and III below.

$I$$\xi_{1}$$\xi_{2}$$III$$II$$\xi_{1}+2\xi_{2}=0$ Figure 7: Phase diagram.

##### Phase I: $\xi_{1}\gg 0$ and $\xi_{2}\gg 0$.

In this phase, the gauge groups are totally broken. Following the analysis of
the asymptotic behaviour of solutions to Coulomb branch equations in the limit
of $\xi_{1}\to+\infty$ and $\xi_{2}\to+\infty$, one can see that the only
solution for $\sigma^{(1)}$ and $\sigma^{(2)}_{\alpha}$ is the zero solution.
Therefore, this phase only has a Higgs branch, whose target space is
determined by the D-term equations $\\{D=0,D^{\alpha}_{\beta}=0\\}$, which is
the space $\hat{X}$ as expected,

$\hat{X}\>=\>{\rm Tot}[\mathcal{L}\otimes\det\mathcal{S}\to Fl(1,2,N)]\,.$

The base $Fl(1,2,N)$ is determined by $\Phi_{\alpha}$ and $X^{\alpha}_{i}$
(see e.g. [29] for more details) and the fiber direction is described by the
field $P$.

##### Phase II: $\xi_{1}\ll 0$ and $\xi_{2}\gg 0$.

Let us first analyze the asymptotic behaviour of the Coulomb vacua in the
large $\xi$ region, namely, the limit $\xi_{1}\to-\infty$ and
$\xi_{2}\to+\infty$. In this limit, $\sigma^{(2)}_{\alpha}\to 0$ and the first
equation of motion becomes

$\left(\sigma^{(1)}\right)^{2}=-q_{1}\sigma^{(1)}\,,$

which has two solutions, $\sigma^{(1)}=0$ and $\sigma^{(1)}=-q_{1}$.
Therefore, there is a Higgs branch suggested by the solution
$\sigma^{(1)}=\sigma^{(2)}_{\alpha}=0$ and a mixed Higgs-Coulomb branch at
$\sigma^{(1)}=-q_{1}$ and $\sigma^{(2)}_{\alpha}=0$.

The pure Higgs branch can be described by an effective $U(2)$ GLSM defined by
fundamentals $X^{\alpha}_{i}$ and composite fields
$Y_{\alpha}:=P\Phi_{\alpha}$, namely, $Y_{\alpha}$ is in the representation
$\det^{-1}\otimes\mathbb{\bar{2}}$ 181818Similar analysis for pure flag
manifolds has been used in [29, section 2.1], in which they have defined the
(forgetful) projection map. Here, we slightly generalize this idea to discuss
bundles on flag manifolds.. One can read off the geometry from the
corresponding D-term equation, which is

$X_{+}\>=\>{\rm Tot}[\mathcal{S}\otimes\det\mathcal{S}\to Gr(2,N)]\,.$

For the mixed Higgs-Coulomb branch, since $\Phi_{\alpha}$ and $P$ are charged
under $U(1)$, they are massive on this branch and should be integrated out.
What is left is an effective $U(2)$ GLSM for the Grassmannian $Gr(2,N)$,
defined by $N$ chiral fields $X^{\alpha}_{i}$ in the fundamental
representation of $U(2)$ gauge group. In other words, the mixed Higgs-Coulomb
in this phase is

$\mathcal{C}_{+}\>=\>\left\\{\text{one }Gr(2,N)\text{ located at one non-zero
point on the }\sigma\text{-plane}\right\\}\,.$

The contributions to the total Witten index from the mixed Higgs-Coulomb
branch $\mathcal{C}^{N}_{+}$ is easy to count, which is just the Euler
characteristic of $Gr(2,N)$, i.e. $\binom{N}{2}\,.$

##### Phase III: $\xi_{1}\gg 0$ and $\xi_{1}+2\xi_{2}\gg 0$.

Again, let us get some hints from the asymptotic behaviour of the Coulomb
vacua equations in this phase. In the limit, $\xi_{1}\to+\infty$,
$\prod_{\alpha=1}^{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)=0$ and so
there are two solutions for $\sigma^{(1)}$, namely
$\sigma^{(1)}=\sigma^{(2)}_{1}$ or $\sigma^{(1)}=\sigma^{(2)}_{2}$. Also, note
that

$\left(\sigma^{(2)}_{1}\sigma^{(2)}_{2}\right)^{N}=q_{2}^{2}\left(\prod_{\alpha=1}^{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)\right)\left(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}\right)=-q_{1}q_{2}^{2}\left(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}\right)^{2}\,,$

and, due to $\xi_{1}+2\xi_{2}>0$, $q_{1}q_{2}^{2}\to 0$ in the large $\xi$
region of this phase, we shall have the solution $\sigma^{(2)}_{1}=0$ or
$\sigma^{(2)}_{2}=0$. We choose $\sigma^{(2)}_{1}=0$ for convenience in the
following discussion. Due to the Weyl symmetry, the case of
$\sigma^{(2)}_{2}=0$ follows the same analysis but will not contribute more to
the Witten index. If take $\sigma^{(1)}=\sigma^{(2)}_{1}=0$, Eq. (66) reduces
to an equation for $\sigma^{(2)}_{2}$ taking the following form,

$\left(\sigma^{(2)}_{2}\right)^{N}=-q_{2}\,\left(\sigma^{(2)}_{2}\right)^{2}\,.$

This equation has $N-2$ non-vanishing solutions and the rest is
$\sigma^{(2)}_{2}=0$. While if take $\sigma^{(1)}=\sigma^{(2)}_{2}$, then the
remaining equation for $\sigma^{(2)}_{2}$ only has zero solutions. In summary,
the asymptotic behaviour of the Coulomb vacua suggests that this phase
consists of a Higgs branch together with $(N-2)$ mixed Higgs-Coulomb branches.

Now let us give more details on the Higgs branch and the mixed Higgs-Coulomb
branches in phase III. To describe the effective theory, we follow the
strategy described in [29, section 2.2] by Plücker embedding into a product of
projective space. See also some similar discussions in appendix B. We need to
introduce two sets of $SU(2)$ invariant coordinates:
$Z_{i}:=\Phi_{\alpha}X^{\alpha}_{i}$ and
$B_{ij}:=\epsilon_{\alpha\beta}X^{\alpha}_{i}X^{\beta}_{j}$. The effective
theory is a gauged nonlinear sigma model defined by

| $Z_{i}$ | $B_{ij}$ | $P$
---|---|---|---
$U(1)$ | 1 | 0 | $-1$
$\det{U(2})$ | $0$ | $1$ | $-1$

with nontrivial nonlinear relations for $N\geq 4$ among coordinates $B_{ij}$’s
determined by their definitions. Moreover, the definition of $Z_{i}$ suggests
that $Z_{i}$ is a vector in the vector space spanned by $X^{1}$ and $X^{2}$.
However, different from the example discussed in appendix B, in this example
both $D$-term equations will have solutions.

Because $\xi_{2}\ll 0$, $P$ gets a nonzero vev and break the gauge group to
$U(1)-\det U(2)$ whose FI-parameter is large and positive. Moreover, due to
the D-term equations from (65),

$\displaystyle\sum_{i=1}^{N}|Z_{i}|^{2}$
$\displaystyle\>=\>\sum_{i=1}^{N}\sum_{\alpha,\beta}\overline{\Phi}^{\beta}\Phi_{\alpha}\overline{X}_{\beta,i}X^{\alpha}_{i}=\sum_{\alpha,\beta}\overline{\Phi}^{\beta}\Phi_{\alpha}(\overline{\Phi}^{\alpha}\Phi_{\beta}+|P|^{2}\delta^{\alpha}_{\beta}+\xi_{2}\delta^{\alpha}_{\beta}),$
$\displaystyle\>=\>\sum_{\alpha=1}^{2}|\Phi_{\alpha}|^{2}(\sum_{\beta=1}^{2}|\Phi_{\beta}|^{2}+|P|^{2}+\xi_{2})=(2|P|^{2}+\xi_{1}+\xi_{2})\sum_{\alpha=1}^{2}|\Phi_{\alpha}|^{2}>0\,,$

thus the $Z_{i}$’s cannot vanish simultaneously in this phase. Therefore the
$Z_{i}$’s become the homogeneous coordinates of $\mathbb{P}^{N-1}$, and the
$B_{ij}$’s represent the vector subspace in $\mathbb{C}^{N}$ spanned by
$X^{1}$ and $X^{2}$ which contains $Z_{i}$. Since such a subspace can be
determined by a vector in $\mathbb{C}^{N}$ which is either a nonzero vector in
a direction different from that of $Z_{i}$ or a zero vector (In the latter
case, it simply means $X^{1}$ and $X^{2}$ are linearly dependent so their span
is one dimensional), we see the $B_{ij}$’s furnish the vector bundle
$\mathcal{L}^{2}\otimes\left(\mathcal{O}^{\oplus N}/\mathcal{L}\right)$ over
$\mathbb{P}^{N-1}$. The extra $\mathcal{L}$ factor is due to the fact that
$B_{ij}$ has $U(1)-\det U(2)$ charge $-1$. Therefore, we have

$X_{-}\>=\>\mathrm{Tot}\left(\mathcal{L}^{2}\otimes\left(\mathcal{O}^{\oplus
N}/\mathcal{L}\right)\rightarrow\mathbb{P}^{N-1}\right)\,,$

and

$\mathcal{C}_{-}\>=\>\left\\{(N-2)\text{ copies of }\mathbb{P}^{N-1}\text{
located at }(N-2)\text{~{}non-zero points in the
}\sigma\text{-plane}\right\\}\,.$

Since we will discuss the derived equivalence between $X_{+}$ and $X_{-}$, the
first requirement is the matching of their Euler characteristics. Instead of
direct computation, we compare the Witten indices of the mixed Higgs-Coulomb
branches. Since the total Witten index should be invariant across different
phases, matching of contributions from mixed Higgs-Coulomb branches will
indicate the Euler characteristics of the Higgs branches are the same. In the
current example, we need

$\chi(Gr(2,N))\>=\>(N-2)\chi(\mathbb{P}^{N-1})\,,$

namely, $N=3$ is the only choice.

####  4.3.3 Effective theories on the phase boundaries

The local model at the phase boundary of $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ is the
$U(1)$ gauge theory with the following matter content

| $\Phi_{\alpha}$ | $P$
---|---|---
$U(1)$ | 1 | $-1$
(67)

The small window of this model consists of only one $U(1)$ charge. By a
suitable choice of theta angle, this charge can be taken to be zero.

The local model at the phase boundary of $\\{\xi_{2}=0,\xi_{1}\gg 0\\}$ is the
$U(2)$ gauge theory with the following matter content

| $X_{i}^{\alpha}$ | $P$
---|---|---
$U(2)$ | $\tiny{\yng(1)}$ | $\det^{-1}$
(68)

The small window is given by Eq. (33), from the $m=1$ case, we see the small
window consists of a single $U(2)$ representation with dimensional one. We can
choose the theta angle such that the representation in the small window is the
trivial representation.

####  4.3.4 Brane transport and derived equivalence

Now we can compute the functor for the equivalence of $D^{b}(X_{+})$ and
$D^{b}(X_{-})$. Let $\pi_{\pm}$ be the projection of the vector bundle
$X_{\pm}$ onto the corresponding base space. We choose the generators of
$D^{b}(X_{+})$ to be $\pi_{+}^{*}{\mathcal{O}}_{Gr(2,3)}$,
$\pi_{+}^{*}\mathcal{S}_{Gr(2,3)}^{\vee}$ and
$\pi_{+}^{*}\wedge^{2}\mathcal{S}_{Gr(2,3)}^{\vee}$.

$\pi_{+}^{*}{\mathcal{O}}_{Gr(2,3)}$ can be lifted to the GLSM brane
$W(0,\cdot)$, which fits in the small windows of both (67) and (68), so it can
be directly transported and projected to give the image
$\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}$.

$\pi_{+}^{*}\wedge^{2}\mathcal{S}_{Gr(2,3)}^{\vee}$ can be lifted to the GLSM
brane $W(0,\wedge^{2}\tiny{\yng(1)})$, which fits in the small window of the
local model (67) at the first boundary. Due to the empty brane

${W(0,\wedge^{2}\tiny{\yng(1)})}\stackrel{{\scriptstyle
P}}{{\longrightarrow}}W(-1,\cdot),$

it is IR equivalent to $W(-1,\cdot)$, which is then in the small window of the
local model (68) at the second boundary. Consequently, it can be projected to
give the image $\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(-1)$.

For $\pi_{+}^{*}\mathcal{S}_{Gr(2,3)}^{\vee}$, we need another empty brane
because the $P$ field cannot turn a fundamental representation into a trivial
representation. Instead, we use the IR empty brane given by the short exact
sequence191919Note that $\Phi_{1}$ and $\Phi_{2}$ cannot both be zero,
otherwise the $Z_{i}$’s would vanish simultaneously.

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(69)

Together with the empty brane

$W(-1,\det)\stackrel{{\scriptstyle P}}{{\longrightarrow}}W(-2,\cdot)\,,$

we get the IR equivalence

$W(0,{\tiny\yng(1)})\cong\text{Cone}\left(W(1,\cdot)[-1]\rightarrow
W(-2,\cdot)\right)\,.$

Therefore the IR image in the phase $X_{-}$ is the complex
$\text{Cone}(\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)[-1]\rightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(-2))$.

In summary, the brane transport with the choice of small windows mentioned
above gives rise to the following map between the generators of $D^{b}(X_{+})$
and $D^{b}(X_{-})$:

$\displaystyle\pi_{+}^{*}{\mathcal{O}}_{Gr(2,3)}$
$\displaystyle\Longrightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}\,,$
$\displaystyle\pi_{+}^{*}\mathcal{S}_{Gr(2,3)}^{\vee}$
$\displaystyle\Longrightarrow\text{Cone}\left(\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(1)[-1]\rightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(-2)\right)\,,$
$\displaystyle\pi_{+}^{*}\wedge^{2}\mathcal{S}_{Gr(2,3)}^{\vee}$
$\displaystyle\Longrightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{2}}(-1)\,.$

###  4.4 Derived equivalence between Calabi-Yau 5-folds

In this section, we generalize the example considered in the previous section
by introducing a symplectic structure, which is realized by a superpotential.
We will consider $\hat{X}$ as the total space of a fiber bundle over the two-
step symplectic flag manifold $SF(1,2,N)$,

$SF(1,2,N)\>=\>\\{\mathbb{C}\subset\mathbb{C}^{2}\subset V~{}|~{}V\text{ is an
$N$-dimensional symplectic vector space },\mathbb{C}^{2}\text{ is isotropic
}\\}.$

When $N=4$, this becomes Segal’s example in [1], which has

$\hat{X}\>=\>{\rm Tot}[\mathcal{L}\otimes\wedge^{2}\mathcal{S}\to
SF(1,2,4)]\,,$

where $\mathcal{L}$ and $\mathcal{S}$ are the first and second tautological
bundles over $SF(1,2,4)$. The Calabi-Yau fivefolds, $X_{+}$ and $X_{-}$, are
defined as

$X_{+}\>=\>{\rm Tot}[S\otimes\wedge^{2}S\to SGr(2,4)]\ ,$

and

$X_{-}\>=\>{\rm
Tot}[(\mathcal{L}^{\perp}/\mathcal{L})\otimes\mathcal{L}^{2}\to{\mathbb{P}}^{3}]\
,$

where $\mathcal{L}^{\perp}$ is the symplectic orthogonal to $\mathcal{L}$ with
respect to the symplectic structure on ${\mathcal{O}}^{\oplus 4}$. This
example was studied in detail in [1] from the mathematical perspective, in
which it was showed that there are birational maps from $\hat{X}$ to $X_{+}$
and $X_{-}$ which makes $X_{+}$ and $X_{-}$ birational, and further arguments
showed that there is a derived equivalence between $X_{+}$ and $X_{-}$. Here
we use a GLSM construction to realize this equivalence and derive the functor
between $D^{b}(X_{+})$ and $D^{b}(X_{-})$ induced by brane transport.

####  4.4.1 GLSM for $\hat{X}$

The GLSM for $\hat{X}$ is a $U(1)\times U(2)$ gauge theory with chiral matters
$\Phi_{\alpha}$, $X^{\alpha}_{i}$, $Q_{\alpha\beta}$ and $P_{\alpha\beta}$
with $\alpha=1,2$ and $i=1,2,3,4$. These fields transform under the gauge
group action as follows

| $\Phi_{\alpha}$ | $X^{\alpha}_{i}$ | $Q_{\alpha\beta}$ | $P_{\alpha\beta}$ |
---|---|---|---|---|---
$U(1)$ | 1 | 0 | 0 | $-1$ | $\xi_{1}$
$U(2)$ | $\overline{\tiny{\yng(1)}}$ | $\tiny{\yng(1)}$ | $\wedge^{2}\overline{\tiny{\yng(1)}}$ | $\wedge^{2}\overline{\tiny{\yng(1)}}$ | $\xi_{2}$

and there is a superpotential

$W\>=\>\sum_{i,j}\sum_{\alpha,\beta}Q_{\alpha\beta}X_{i}^{\alpha}X_{j}^{\beta}\Omega^{ij}\>\equiv\>\sum_{i,j}\sum_{\alpha,\beta}Q_{\alpha\beta}X_{i}^{\alpha}\wedge
X_{j}^{\beta}\,,$ (70)

where $\Omega^{ij}$ is a symplectic form in ${\mathbb{C}}^{4}$. The $D$-term
equations read

$\displaystyle\sum_{\alpha}|\Phi_{\alpha}|^{2}-\sum_{\alpha<\beta}|P_{\alpha\beta}|^{2}\>=\>\sum_{\alpha}|\Phi_{\alpha}|^{2}-|P|^{2}$
$\displaystyle\>=\>\xi_{1}\ ,$ (71)
$\displaystyle\sum_{i}\overline{X}_{i,\alpha}X_{i}^{\beta}-\Phi_{\alpha}\overline{\Phi}^{\beta}-\left(|P|^{2}+|Q|^{2}\right)\delta_{\alpha}^{\beta}$
$\displaystyle\>=\>\xi_{2}\delta_{\alpha}^{\beta}\ .$

In the above, we have denoted $P_{\alpha\beta}=P\epsilon_{\alpha\beta}$ and
$Q_{\alpha\beta}=Q\epsilon_{\alpha\beta}$, $\alpha,\beta=1,2$, where
$\epsilon_{\alpha\beta}$ is the Levi-Civita symbol. To see this GLSM indeed
realizes the space $\hat{X}$, one can do a similar analysis following [30, 31]
and find that, in the phase $\xi_{1},\xi_{2}\gg 0$, the solutions to the
$D$-term and $F$-term equations tell us that the vevs of $\Phi_{\alpha}$ and
$X_{i}^{\alpha}$ constitute the symplectic flag manifold $SF(1,2,4)$ (the
F-term equations derived from (70) imposes the symplectic condition and force
the $Q$ field to vanish), which is the base for $\hat{X}$, while
$P_{\alpha\beta}$ contributes the fibers of $\hat{X}$.

The Coulomb branch equations are

$\displaystyle\prod_{\alpha=1}^{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)=q_{1}(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}),$
$\displaystyle\left(\sigma^{(2)}_{\alpha}\right)^{N}=q_{2}\left(\sigma^{(1)}-\sigma^{(2)}_{\alpha}\right)(\sigma^{(2)}_{1}+\sigma^{(2)}_{2})(\sigma^{(1)}+\sigma^{(2)}_{1}+\sigma^{(2)}_{2}).$

One can check that $\hat{X}$ is a Fano variety by investigating at the charges
of any $U(1)\subset U(1)\times U(2)$.

####  4.4.2 The phases for $X_{+}$ and $X_{-}$

The phase diagram of the GLSM for $\hat{X}$ can also be illustrated by Fig. 7.
The previous subsection has shown that phase I, namely, $\xi_{1},\xi_{2}\gg
0$, consists of a pure Higgs branch, realizing the manifold $\hat{X}$. In this
subsection, we explore other phases and we will see that the Calabi-Yau
fivefolds $X_{\pm}$ arise as Higgs branches in these phases.

##### Phase II: $\xi_{1}\ll 0$ and $\xi_{2}\gg 0$.

The $U(2)$ $D$-term equation tells that $X_{i}^{\alpha}$ is non-degenerate, so
they become homogeneous coordinates of $Gr(2,4)$ at low energy scale. The
F-term equations impose the symplectic condition and force $Q$ to vanish. The
$U(1)$ invariant field $P\Phi_{\alpha}$ is in the $U(2)$ representation
$\overline{\tiny{\yng(1)}}\otimes\wedge^{2}\overline{\tiny{\yng(1)}}$, which
become the coordinates along the fiber. Therefore the vacuum manifold in this
phase is

$X_{+}\>=\>{\rm Tot}\left(\mathcal{S}\oplus\wedge^{2}\mathcal{S}\to
SGr(2,4)\right)\ .$

##### Phase III: $\xi_{1}\gg 0$, $\xi_{2}\ll 0$ and $\xi_{1}+2\xi_{2}\gg 0$.

If we take the trace of the second equation of (71), we get

$\sum_{\alpha,i}|X^{\alpha}_{i}|^{2}-\sum_{\alpha}|\Phi_{\alpha}|^{2}-2\left(|P|^{2}+|Q|^{2}\right)=2\xi_{2}\,.$

Adding the first equation of (71) to the equation above, we get

$\sum_{\alpha,i}|X^{\alpha}_{i}|^{2}-3|P|^{2}-2|Q|^{2}=\xi_{1}+2\xi_{2}\,,$

therefore, $X^{\alpha}_{i}$ cannot vanish simultaneously in this phase.
Consequently, the F-term equations force the $Q$ field to vanish. Define the
$SU(2)$ invariant fields $Z_{i}=\Phi_{\alpha}X^{\alpha}_{i}$ and
$B_{ij}=\epsilon_{\alpha\beta}X^{\alpha}_{i}X^{\beta}_{j}$, then the remaining
degrees of freedom can be summarized as

| $Z_{i}$ | $B_{ij}$ | $P$
---|---|---|---
$U(1)$ | 1 | 0 | $-1$
$\det{U(2})$ | $0$ | $1$ | $-1$

Moreover, from the D-term equations,

$\displaystyle\sum_{i}|Z_{i}|^{2}$
$\displaystyle\>=\>\sum_{i=1}\sum_{\alpha,\beta}\overline{\Phi}^{\beta}\Phi_{\alpha}\overline{X}_{\beta,i}X^{\alpha}_{i}=\sum_{\alpha,\beta}\overline{\Phi}^{\beta}\Phi_{\alpha}(\overline{\Phi}^{\alpha}\Phi_{\beta}+(|P|^{2}+|Q|^{2}+\xi_{2})\delta^{\alpha}_{\beta})\,,$
$\displaystyle\>=\>\sum_{\alpha}|\Phi_{\alpha}|^{2}(\sum_{\beta}|\Phi_{\beta}|^{2}+|P|^{2}+|Q|^{2}+\xi_{2})=(\xi_{1}+\xi_{2}+2|P|^{2}+|Q|^{2})(\xi_{1}+|P|^{2})>0\,,$

so the $Z_{i}$’s cannot vanish simultaneously. Then following the same
analysis as that for phase III of section 4.3.2, we see $Z_{i}$ become the
homogeneous coordinates of $\mathbb{P}^{3}$ and $B_{ij}$’s encode the subspace
spanned by $X^{1}$ and $X^{2}$ in $\mathbb{C}^{4}$ containing $Z_{i}$. In
addition, $X^{1}$ and $X^{2}$ must be orthogonal to each other with respect to
the symplectic form on $\mathbb{C}^{4}$ due to the F-term equations, so the
$B_{ij}$’s give us the vector bundle
$(\mathcal{L}^{\perp}/\mathcal{L})\otimes\mathcal{L}^{2}$. As such, the vacuum
manifold in this phase is

$X_{-}\>=\>{\rm
Tot}\left((\mathcal{L}^{\perp}/\mathcal{L})\otimes\mathcal{L}^{2}\to{\mathbb{P}}^{3}\right)\,.$

####  4.4.3 Effective theories on the phase boundaries

The local model at the phase boundary of $\\{\xi_{1}=0,\xi_{2}\gg 0\\}$ is the
$U(1)$ gauge theory with the following matter content

| $\Phi_{\alpha}$ | $P$
---|---|---
$U(1)$ | 1 | $-1$
(72)

The small window of this model consists of only one $U(1)$ charge. By a
suitable choice of theta angle, this charge can be taken to be zero.

The local model at the phase boundary of $\\{\xi_{2}=0,\xi_{1}\gg 0\\}$ is the
$U(2)$ gauge theory with the following matter content

| $X_{i}^{\alpha}$ | Q | $P$
---|---|---|---
$U(2)$ | $\tiny{\yng(1)}$ | $\det^{-1}$ | $\det^{-1}$
(73)

Based on the discussion of section 3.2, the small window is given by Eq. (34)
with $m=2$. With a suitable choice of the theta angle, we can take the small
window to be

$\left\langle{\tiny\yng(1)}\otimes{\det}^{-2},~{}{\det}^{-1},~{}{\tiny\yng(1)}\otimes{\det}^{-1},~{}\cdot\right\rangle\,.$

####  4.4.4 Derived equivalence induced by brane transport

Now we can compute the functor for the equivalence of $D^{b}(X_{+})$ and
$D^{b}(X_{-})$. Let $\pi_{\pm}$ be the projection of the vector bundle
$X_{\pm}$ onto the corresponding base space. We choose the generators of
$D^{b}(X_{+})$ to be $\pi_{+}^{*}{\mathcal{O}}_{SGr(2,4)}$,
$\pi_{+}^{*}\mathcal{S}_{SGr(2,4)}^{\vee}$,
$\pi_{+}^{*}\wedge^{2}\mathcal{S}_{SGr(2,4)}^{\vee}$ and
$\pi_{+}^{*}(\wedge^{2}\mathcal{S}_{SGr(2,4)}^{\vee})^{\otimes 2}$. Since now
we have a superpotential $W=Q\Omega(X^{1},X^{2})$, the lift of an IR brane is
a matrix factorization of $W$.

$\pi_{+}^{*}{\mathcal{O}}_{SGr(2,4)}$ can be lifted to the matrix
factorization

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which fits in the small windows of both (72) and (73), so it can be directly
transported and projected to give the image
$\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}$.

$\pi_{+}^{*}\wedge^{2}\mathcal{S}_{SGr(2,4)}^{\vee}$ can be lifted to the
matrix factorization

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which fits in the small window of the local model (72) at the first boundary.
Because the vev of the $P$ field, which is in the $(-1,{\det}^{-1})$
representation, is nonzero in the IR limit, the matrix factorization above is
IR equivalent to

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0.0pt\hbox{$\textstyle{{W(-1,\cdot)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.31639pt\raise-8.22498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
29.08333pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\,,$

which is in the small window of the local model (73) at the second boundary.
Consequently, it can be projected to give the image
$\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}(-1)$.

Similarly, $\pi_{+}^{*}(\wedge^{2}\mathcal{S}_{SGr(2,4)}^{\vee})^{\otimes 2}$
can be transported to give the image
$\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}(-2)$.

As in the previous example, for $\pi_{+}^{*}\mathcal{S}_{SGr(2,4)}^{\vee}$, we
need another empty brane. We can use the IR empty brane given by the exact
sequence

$\textstyle{W(-1,\cdot)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left({\tiny\begin{array}[]{c}\Phi_{1}\\\
\Phi_{2}\end{array}}\right)}$$\scriptstyle{\Omega(X^{1},X^{2})}$$\textstyle{{W(0,{\tiny\overline{\yng(1)}})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\Phi_{2},-\Phi_{1})}$$\scriptstyle{\Omega(X^{1},X^{2})}$$\textstyle{W(1,{\det}^{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Omega(X^{1},X^{2})}$$\textstyle{W(-1,\det)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\left({\tiny\begin{array}[]{c}\Phi_{1}\\\
\Phi_{2}\end{array}}\right)}$$\scriptstyle{Q}$$\textstyle{{W(0,{\tiny\yng(1)})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\Phi_{2},-\Phi_{1})}$$\scriptstyle{Q}$$\textstyle{W(1,\cdot)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{Q}$
(74)

to construct an empty brane. The lift of
$\pi_{+}^{*}\mathcal{S}_{SGr(2,4)}^{\vee}$ is

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
29.48955pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W(0,\tiny{\yng(1)}\otimes{\det}^{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.91872pt\raise 9.06387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1611pt\hbox{$\scriptstyle{\Omega(X^{1},X^{2})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.48955pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.48955pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 83.48955pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{W(0,\tiny{\yng(1)})}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.72261pt\raise-8.22498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
29.48955pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\,.$
(75)

From the empty brane (74) and the equivalence

$\left(\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
20.36111pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W(-1,\cdot)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
29.79028pt\raise 9.06387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1611pt\hbox{$\scriptstyle{\Omega(X^{1},X^{2})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 74.36111pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
44.36111pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 74.36111pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{W(-1,\det)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.59418pt\raise-8.22498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
20.36111pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\right)\cong\left(\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
29.08331pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{W(-2,\det^{-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
38.51248pt\raise 9.06387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.1611pt\hbox{$\scriptstyle{\Omega(X^{1},X^{2})}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 83.08331pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
53.08331pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 83.08331pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{W(-2,\cdot)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
50.31639pt\raise-8.22498pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.71112pt\hbox{$\scriptstyle{Q}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
29.08333pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces}}}}\ignorespaces\right)$

induced by the action of the $P$ field, we see the IR image of (75) in the
phase of $X_{-}$ is the complex
$\text{Cone}(\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}(1)[-1]\rightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}(-2))$.

In summary, the brane transport with the choice of small windows mentioned
above gives rise to the following map between the generators of $D^{b}(X_{+})$
and $D^{b}(X_{-})$:

$\displaystyle\pi_{+}^{*}{\mathcal{O}}_{SGr(2,4)}$
$\displaystyle\Longrightarrow\pi_{-}^{*}{\mathcal{O}}_{\mathbb{P}^{3}}\,,$
$\displaystyle\pi_{+}^{*}\mathcal{S}_{SGr(2,4)}^{\vee}$