effectively one of the branes behaves as compact, we can use either the
definition of the Homs’s from section $4$, as $Hom_{\mathscr{D}_{\cal
Y}}({\cal L},{\cal L}^{\prime})=HF^{0,0}({\cal L}_{\cal W},{\cal
L}^{\prime}),$ where ${\cal L}_{\cal W}$ is obtained from ${\cal L}$ by the
Hamiltonian flow of $H_{\cal W}={\rm Re}{\cal W}$, or the $\zeta$ deformation
from above – the two give the same answer, as they should.

###  A.3 Central charges and analytic continuation

The equivariant central charge

${\cal Z}[{{\cal T}}]=\int_{{\cal T}}\Omega e^{-{\cal W}}.$

of the thimble brane ${\cal T}$, is just the $\Gamma$ function

${\cal Z}[{{\cal
T}}]=\int_{0}^{\infty}dy^{\prime}\,y^{\prime\lambda-1}\,e^{-y^{\prime}}=\Gamma(\lambda).$
(A.14)

Its ordinary central charge, defined naively as $\int_{\cal T}\Omega$ is
infinite. The definition from section 8.24, as the leading term in the
$\lambda$ to zero limit, regulates the divergence. For the thimble brane, it
evaluates to

${\cal Z}^{0}[{\cal T}]=\Bigl{(}\Gamma(\lambda)\Bigr{)}_{\lambda\rightarrow
0}={1\over\lambda}$ (A.15)

This is the equivariant volume of the 2-brane ${\cal P}$ on ${\mathbb{C}}$,
which is scaled by a ${\mathbb{C}}^{\times}$ action with parameter
${\mathfrak{h}}=e^{2\pi i\lambda}$.

The equivariant central charge of the ${\cal I}$ brane, computed from the
contour in figure 34, gives the integral representation of the inverse
$\Gamma$ function, or more precisely,

${\cal Z}[{\cal I}]=\int_{\cal I}\Omega e^{-{\cal W}}=-{2\pi
i\over\Gamma(1-\lambda)}e^{i\pi\lambda}$ (A.16)

The non-equivariant $\lambda\rightarrow 0$ limit of this is finite, so

${\cal Z}^{0}[{\cal I}]=\int_{\cal I}\Omega=2\pi i.$ (A.17)

This is the central charge of a $0$-brane on ${\cal X}$.

Another way to compute the equivariant central charge of the ${\cal I}$ brane
is from the complex that tells us how to obtain the brane from connected sums
of ${\cal T}$’s, $[{\cal I}]=[{\cal T}]-[{\cal T}\\{-1\\}]=(1-e^{2\pi
i\lambda})[{\cal T}]$ so that

${\cal Z}[{\cal I}]=(1-e^{2\pi i\lambda})\Gamma(\lambda).$ (A.18)

The fact that the two expressions for ${\cal Z}[{\cal I}]$ in (A.16) and
(A.18) are equal is the standard $\Gamma$-function identity.

#### A.3.1

So far we took $\lambda$ to be real, but now we will describe what happens as
we vary $\lambda$ in the complex plane. For all ${\rm Re}(\lambda)>0$ the
central charge of the brane ${\cal T}$ gives the integral representation of
the $\Gamma$-function ${\cal Z}[{\cal T}]=\Gamma(\lambda)$. Being an analytic
function of $\lambda$, the central charge of the brane ${\cal T}$ coincides
with the $\Gamma$-function for all $\lambda\in{\mathbb{C}}$. However, for
${\rm Re}(\lambda)<0$ the integral along the $Y=Y_{\star}+{\mathbb{R}}$ no
longer converges. Instead, one deforms the contour of integration so as to
obtain a convergent expression that defines the analytic continuation of the
$\Gamma$-function. As explained in [181], thimbles of downward gradient flow
of $H_{\cal W}={\rm Re}{\cal W}$ provide a tool for analytic continuation,
since if ${\cal L}$ is such a thimble, the integral in ${\cal Z}[{{\cal
L}}]=\int_{{\cal L}}\Omega e^{-{\cal W}}$ is guaranteed to converge. Analytic
continuation of the $\Gamma$-function was described in these terms in [89].

The actual thimble and the Lagrangian ${\cal T}$ define the same brane of
${\mathscr{D}_{Y}}$ for all ${\rm Re}(\lambda)>0$, since they are isotopic.
For ${\rm Re}(\lambda)<0$ this is no longer the case. Instead, for ${\rm
Re}(\lambda)<0$ the thimble becomes isotopic to the brane we called ${\cal I}$
instead, and the ${\cal I}$ brane and the ${\cal T}$ brane are different
objects of ${\mathscr{D}_{\cal Y}}$.

For ${\rm Re}(\lambda)>0$, because ${\cal Z}[{\cal T}]$ manifestly converges
as opposed through analytic continuation, in this regime, it is natural to
take the ${\cal T}$-brane as the generator of ${\mathscr{D}_{\cal Y}}$, and
describe the ${\cal I}$-brane as the complex in (A.13).

For ${\rm Re}({\lambda})<0$, it is natural to take the ${\cal I}$-brane as the
generator, and express the ${\cal T}$-brane in terms of it, for example we can
take the complex

${\cal T}\cong{\cal I}\xrightarrow{p_{1}}{\cal
I}[1]\\{1\\}\xrightarrow{p_{2}}\ldots$

of cones over the intersection points $p_{i}$ that generate
$Hom_{\mathscr{D}_{Y}}({\cal I}\\{i+1\\},{\cal I}\\{i\\}[1])$. This complex is
the natural description of the brane for ${\rm Re}\lambda<0$ and ${\rm
Im}\lambda<0$ for which one isotopes the right half of the ${\cal T}$ brane
into ${\rm Im}Y>0$. This Lagrangian describes ${\cal T}$ from taking a
connected sum of branes in $\oplus_{n=0}^{\infty}{\cal I}\\{n\\}$ over their
intersection points (after the $\zeta$-deformation). In K-theory, this relates
$[{\cal T}]=\oplus_{n=0}^{\infty}[{\cal I}\\{n\\}]=[{\cal I}]/(1-e^{2\pi
i\lambda}),$ inverting the formula we had earlier. The complex is not unique,
for example, the contour deformation natural for ${\rm Re}\lambda<0$, ${\rm
Im}\lambda>0$ will give a different one. The ${\cal T}$ brane, and either of
the two complexes describe the same object on ${\mathscr{D}_{\cal Y}}$.

#### A.3.2

While the ${\cal X}={\mathbb{C}}$ example is not part of the family of
theories that are the main interest in this paper, it is serves as a model.
For example, take ${\cal X}={\mathbb{C}}^{2}$, by setting $m=1$ in the example
of section 5. This is an instructive example, since it is just two copies of
the ${\mathbb{C}}$-theory. Its equivariant mirror $Y$ has just one thimble
brane $T$ which generates ${\mathscr{D}_{Y}}$, whose equivariant central
charge is the Beta function:

${\cal
Z}[{T}]=B(\lambda_{1},\lambda_{2})={\Gamma(\lambda_{1})\Gamma(\lambda_{2})\over\Gamma(\lambda_{1}+\lambda_{2})},$

where $\lambda_{1}$ and $\lambda_{2}=\lambda_{0}-\lambda_{1}$ are the pair of
equivariant weights of two ${\mathbb{C}}^{\times}$ actions on ${\cal X}$,
scaling its two complex coordinates $x_{1}$ and $x_{2}$ by
${\mathfrak{h}}_{1}=e^{2\pi i\lambda_{1}}$ and ${\mathfrak{h}}_{2}=e^{2\pi
i\lambda_{2}}$. The ordinary central charge, as defined in section 8.24 equals

${\cal Z}^{0}[{T}]={1\over\lambda_{1}}+{1\over\lambda_{2}}.$ (A.19)

The thimble brane ${T}$ is mirror to the line bundle $P$ on the core $X$,
which is just the $x_{1}x_{2}=0$ locus in ${\cal X}$, and whose equivariant
volume (A.19) computes, as the sum of equivariant volumes of the two copies of
${\mathbb{C}}$ in $X$. Equivalently, the ${T}$-brane is the equivariant mirror
of the holomorphic Lagrangian $f_{*}P={\mathscr{O}}_{X}$ in ${\cal X}$, which
is the structure sheaf of $X$.

###  A.4 Coherent sheaves and quivers

The fact that the derived category of equivariant coherent sheaves on ${\cal
X}$ is equivalent to the derived category of modules of an algebra
${\mathscr{A}}$, which is moreover a quiver path algebra was an important
simplification, both in this section, for ${\cal X}={\mathbb{C}}$ and in
section 5, for ${\cal X}$ which is the resolution of an $A_{m-1}$ surface
singularity. This was easy to understand what the corresponding quiver is the
case of ${\mathbb{C}}$, so here we will understand it in the case of the
$A_{m-1}$ surface, following [53, 35, 36, 19].

Recall ${\cal X}$ is the holomorphic quotient of ${\mathbb{C}}^{m+1}$ with
coordinates $x^{0},\ldots,x^{m}$, by
$H_{\mathbb{C}}=({\mathbb{C}}^{\times})^{m-1}$, where the $i$’th factor in
$H_{\mathbb{C}}$ acts on $x_{i-1},x_{i},x_{i+1}$ with weights $+1,-2,+1$, and
with $0$ on the rest. More precisely, ${\cal X}$ is ${\cal
X}=({\mathbb{C}}^{m+1}-\Delta)/H_{\mathbb{C}},$ where the locus $\Delta$
consists of points where $x_{i}=0=x_{j}$ for any pair of $i,j$ such that
$i\neq j\pm 1$.

The ring of $H_{\mathbb{C}}$ invariant polynomials is generated by

$z=x_{0}x_{1}\ldots x_{m},\qquad u=x_{0}^{m}x_{1}^{m-1}x_{2}^{m-2}\ldots
x_{m-1},\qquad v=x_{1}x_{2}^{2}\ldots x_{m}^{m},$

which satisfy one relation, $uv=z^{m}$, which maps us back to the $A_{m-1}$
singularity.

Coherent sheaves on ${\cal X}$ have a description as $H$-graded modules for
the algebra ${\cal R}_{0}={\mathbb{C}}[x_{0},\ldots,x_{m}]$ of polynomials in
the $x$’s. Hom’s between sheaves on ${\cal X}$ correspond to $H$-invariant
maps between the corresponding modules for ${\cal R}_{0}$, i.e. sheaves on
${\mathbb{C}}^{m+1}$. We can make this ${\rm T}$-equivariant by grading
everything in addition by the ${\rm T}$-action, which we can take to be

${\rm
T}:(x_{0},x_{1},\ldots,x_{m-1},x_{m})\;\rightarrow\;(x_{0},x_{1},\ldots,\mathfrak{h}x_{m-1},\mathfrak{q}\mathfrak{h}^{-1}x_{m}).$

For example, the structure sheaf ${\cal P}_{0}={\cal O}_{{\cal X}}$ in (6.35),
the sheaf of all holomorphic functions on ${{\cal X}}$, corresponds to the
module for ${\cal R}_{0}$ equal to ${\cal R}_{0}$ itself. The Homs from ${\cal
P}_{0}$ to itself are the $H_{\mathbb{C}}$-invariant subset of elements of
${\cal R}_{0}$, which is just the ring ${\cal R}$ itself, so we recover
(6.39). The sheaf ${\cal P}_{i}={\cal O}_{{\cal X}}({\cal D}_{i})$ for $i$
ranging from $1$ to $m-1$ is a sheaf of holomorphic sections of a holomorphic
line bundle on ${{\cal X}}$ corresponding to a divisor ${\cal D}_{i}$,

${\cal D}_{i}:\qquad x^{{\cal D}_{i}}=x_{0}^{i}x_{1}^{i-1}\ldots x_{i-1}=0.$
(A.20)

One can obtain ${\cal P}_{i>0}$ from ${\cal P}_{0}$ by multiplying with
$x^{{\cal D}_{i}}$. Thus, if we associate the tilting sheaf ${\cal P}_{i}$ to
the $i$’th node of the quiver $Q_{\cal A}$, then we get the identification of
arrows with monomials as

$a_{i+1,i}=x_{0}\ldots x_{i},\;\;\;b_{i,i+1}=x_{i+1}\ldots x_{m},$ (A.21)

where $a_{i+1,i}\in hom_{{\cal R}_{0}}({\cal P}_{i},{\cal P}_{i+1})$, and so
on, and where the quiver relations in (6.37) follow from (A.21).

##  Appendix B The upstairs mirror symmetry

Here, we will describe some aspects of ordinary mirror symmetry, relating

${\cal X}\xlongrightarrow{mirror}{\cal Y}.$

Recall from section 2 that one of the interpretations of ${\cal X}$ is as the
Coulomb branch of a three dimensional quiver gauge theory, with quiver
${\mathcal{Q}}$ and eight supercharges.

The mirror of sigma model on ${\cal X}$ is the Landau-Ginsburg model with
target ${\cal Y}$ and potential ${\cal W}$, where ${\cal Y}$ is the Coulomb
branch of a four dimensional quiver gauge theory with the same quiver
$\mathcal{Q}$. Sometimes, ${\cal Y}$ is referred to as the ”multiplicative”
version of ${\cal X}$. The potential ${\cal W}$ gets additional terms if we
study ${\cal X}$ equivariantly, but it is non-vanishing even without. This
appendix will explain why this is the case.

The remainder of it describes Hori-Vafa mirror symmetry [92] for a hypertoric
${\cal X}$, and shows how to derive from it the equivariant mirror $Y$ and the
potential $W$.

###  B.1 Coulomb branches in three and four dimensions

While ${\cal X}$ is the Coulomb branch of a three dimensional quiver gauge
theory, to understand what its ordinary mirror is, it helps to consider the
Coulomb branch ${\cal Y}^{\vee}$ of a four dimensional gauge theory based on
the same quiver ${\mathcal{Q}}$, with eight supercharges, and compactified on
circle of radius $R^{\vee}$. While ${\cal X}$ is the moduli space of singular
monopoles on ${\mathbb{R}}^{3}$, ${\cal Y}^{\vee}$ is the moduli space of
singular monopoles on ${\mathbb{R}}^{2}\times S^{1}$, with the radius of the
$S^{1}$ inversely proportional to $R^{\vee}$ (see [6] for a string theory
derivation).

Like ${\cal X}$, ${\cal Y}^{\vee}$ is also a holomorphic symplectic manifold,
but unlike ${\cal X}$, ${\cal Y}^{\vee}$ turns out to be self mirror for
reasons we will review – this is why it is a useful starting point for
understanding the action of mirror symmetry on ${\cal X}$. (Homological mirror
symmetry for multiplicative hypertoric varieties was proven in [82].)

#### B.1.1

Classically ${\cal
Y}^{\vee}\sim({\mathbb{C}}_{\theta^{\vee}}^{\times}\times{\mathbb{C}}_{\varphi^{\vee}}^{\times})^{d}/\textup{Weyl}$,
where $d$ is the rank of the quiver gauge group, as before. The
${\mathbb{C}}_{\theta^{\vee}}^{\times}$ factor comes from the holonomy of the
four dimensional gauge field around the circle of radius $R^{\vee}$ that takes
us from four to three dimensions, and ${\mathbb{C}}_{\varphi^{\vee}}^{\times}$
from the dual of the three dimensional gauge field. In three dimensions, an
abelian gauge field $A$ is dual to a compact real scalar ${\varphi}^{\vee}$
known as the dual photon. The duality relates the two by
$dA=*d\varphi^{\vee}$, see [56] for more details. Compactifying the 3d sigma
model on a further circle of radius $R$, the long-distance theory is a sigma
model in two dimensions with target ${\cal Y}^{\vee}$ and no potential, ${\cal
W}^{\vee}\equiv 0$.

The sigma model on ${\cal Y}^{\vee}$ defined in this way is self mirror, in
that ${\cal Y}^{\vee}$ and its mirror ${\cal Y}$, are related by a hyper-
Kahler rotation. From four-dimensional perspective, two dimensional mirror
symmetry that exchanges ${\cal Y}$ and ${\cal Y}^{\vee}$ exchanges the roles
of the two circles the theory is compactified on. The easiest way to see this
is as follows. Rather than compactifying sequentially on circles of radii $R$
and $R^{\vee}$ (by taking one of the them to be much smaller than the other),
we can simply compactify the four dimensional theory on a square torus with
two small circles of radii $R$ and $R^{\vee}$. In this way, at long distance,
the theory is described by a two-dimensional sigma model on a target which is
neither ${\cal Y}$ nor ${\cal Y}^{\vee}$ – instead, it is a sigma model on a
manifold parameterized by holonomies of the four dimensional gauge fields
around the circles of radii $R$ and $R^{\vee}$. All three descriptions of two
dimensional physics we get are equivalent for the purposes of studying
topological, or homological mirror symmetry since neither are sensitive to
metric-type data – the values of $R$ and $R^{\vee}$ are irrelevant, as long as
they are not zero or infinite.

The description which treats $R$ and $R^{\vee}$ on the same footing is related
to the description based on ${\cal Y}$ or on ${\cal Y}^{\vee}$ by
$T$-dualities: taking $R^{\vee}\ll R$, one gets ${\cal Y}^{\vee}$ by dualizing
the holonomy $\theta$ around the circle of radius $R$ – trading it for for the
“dual photon” of the theory in three dimensions obtained by compactifying on a
small circle of radius $R^{\vee}$, and vice versa. The duality in two
dimensions reads $d\theta=*d{\varphi}^{\vee}$. If $\theta^{\vee}$’s stand for
holonomies around the circle of radius ${R}^{\vee}$, ${\cal Y}^{\vee}$ emerges
from the description based on $(\theta^{\vee},\varphi^{\vee})$ and ${\cal Y}$
from the one based on $(\varphi,\theta)$. The two dimensional dualities of
this kind are what underlies mirror symmetry – this perspective played the key
role in [92], for example. The fact that mirror symmetry is T-duality was
discovered by Strominger, Yau and Zaslow in [164].

#### B.1.2

In the limit where we take the radius ${R}^{\vee}$ to zero, ${\cal Y}^{\vee}$
becomes the Coulomb branch ${\cal X}$ of the three dimensional gauge theory
with quiver ${\mathcal{Q}}$. Classically, ${\cal
X}\sim({\mathbb{C}}_{\theta^{\vee}}\times{\mathbb{C}}_{\varphi^{\vee}}^{\times})^{d}/\textup{Weyl}$
because the ${\mathbb{C}}_{\theta^{\vee}}^{\times}$ factors previously
associated to the Wilson-lines of four dimensional gauge fields decompactify
in taking ${R}^{\vee}$ to zero – instead of being valued on a circle of radius
$1/R^{\vee}$, ${\theta}^{\vee}$’s become valued on ${\mathbb{R}}$.

The effect of this is that, as a complex manifold,

${\cal Y}^{\vee}={\cal X}\backslash D^{\vee},$

where

$D^{\vee}=\cup_{a}D^{\vee}_{a}$ (B.1)

is the union of divisors we have to remove to replace the $d$ factors of
${\mathbb{C}}_{\theta^{\vee}}$ by ${\mathbb{C}}_{\theta^{\vee}}^{\times}$. The
divisor $D^{\vee}_{a}$ is the locus in ${\cal X}$ where ${\rm
det}(\Phi_{a})=0$, where $\Phi_{a}$ is the complex scalar valued in the
adjoint representation of the $U(d_{a})$ gauge group.

Thus, as holomorphic symplectic manifolds, Coulomb branches ${\cal X}$ and
${\cal Y}^{\vee}$ of three and four dimensional gauge theories based on the
same quiver $\mathcal{Q}$ differ in that the locus corresponding to the union
of divisors in (B.1), which is deleted in the four dimensional case, gets
filled in by going down to three dimensions. In going from ${\cal Y}^{\vee}$
to ${\cal X}$, one also gains a symmetry corresponding to circle actions on
the newly acquired ${\mathbb{C}}_{\theta^{\vee}}$ factors. Among them is the
${\mathbb{C}}^{\times}$-action that scales the holomorphic symplectic form of
${\cal X}$.

###  B.2 Mirror symmetry, divisors and superpotentials

To understand what is the “upstairs” mirror of ${\cal X}$, one therefore needs
to understand what is the mirror to partially compactifying ${\cal Y}^{\vee}$,
by adding in the divisors $D^{\vee}_{a}$.

Mirror symmetry relates compactifying ${\cal Y}^{\vee}$ by adding divisors, to
turning on a superpotential on ${\cal Y}$. The terms in the superpotential
reflect the divisors we add. The argument for this originated in [94], based
on [92, 93]. It is developed, starting with [22], in many works, see for
example [86]. Consider the B-model on ${\cal Y}$ with a 0-brane probe, and its
mirror, which is the A-model on ${\cal Y}^{\vee}$ with an A-brane wrapping the
Lagrangian torus fiber $L_{T^{2d}}$. Compactifying ${\cal Y}^{\vee}$ by adding
divisors $D^{\vee}_{a}$ to get ${\cal X}$, some one cycles of the torus fiber
that were not contractible in ${\cal Y}^{\vee}$ become contractible in ${\cal
X}$. This introduces new holomorphic disks with boundary on $L_{T^{2d}}$,
passing through $D^{\vee}_{a}$, which generate a superpotential, as the
zero’th Massey product $\mu^{0}$ of the Lagrangian. It is obtained by counting
Maslov index $2$ disks with boundary on $L_{T^{2d}}$ and one boundary marked
point. The is reflected in the mirror ${\cal Y}$ as a term in the classical
superpotential felt by 0-branes. From perspective of ${\cal X}$, this gives a
way of calculating its SYZ mirror.

#### B.2.1

As an example, consider ${\cal X}$ which is the resolution of an $A_{m-1}$
surface singularity. In this case, both ${\cal Y}$ and ${\cal Y}^{\vee}$ are
“multiplicative” $A_{m-1}$ surfaces – more precisely, ${\cal Y}^{\vee}$ is a
resolution of an $A_{m-1}$ surface singularity

${\cal Y}^{\vee}:\qquad u^{\vee}v^{\vee}=(1-1/y^{\vee})^{m},$

which differs from ${\cal X}$ in (3.1) by having the set $y^{\vee}=0$ deleted
(and by a relabeling of coordinates), and ${\cal Y}$ a deformation of one,
which can be written as the hypersurface

${\cal Y}:\qquad uv=\prod_{i=1}^{m}(1-a_{i}/y).$ (B.2)

The $y$ and $y^{\vee}$ are ${\mathbb{C}}^{\times}$-valued, so they cannot
vanish, and $u$’s and $v$’s are ${\mathbb{C}}$-valued. In particular, ${\cal
Y}$ is a ${\mathbb{C}}^{\times}$ fibration over $Y={\mathbb{C}}^{\times}$
parameterized by $y\neq 0,\infty$. The parameters $a_{i}/a_{i+1}$ are mirror
to complexified Kahler moduli of ${\cal Y}^{\vee}.$ In terms of the discussion
above, ${\rm Im}(\log y^{\vee})$ gets identified with $\theta^{\vee}$, and
${\rm Im}(\log y)$ with $\theta$.

#### B.2.2

As proven in [2], the SYZ mirror to replacing ${\cal Y}^{\vee}$ by ${\cal X}$
is turning on a superpotential on ${\cal Y}$ equal to simply $v$. Turning on,
in addition, equivariant action on ${\cal X}$ is mirror to

${\cal W}=\lambda_{1}\log y-\lambda_{0}\log u+v.$

(For this superpotential to be well defined, requires $u\neq 0$ which removes
a collection of curves. This should have be part of the data of what ${\cal
Y}$ is, even before we turn on equivariance.) To make the structure of ${\cal
Y}$ as a ${\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}$ fibration obvious,
introduce a coordinate $z\neq 0$ by

$u=z,\;\;\;v=z^{-1}\prod_{i=1}^{m}(1-a_{i}/y),$

in terms of which the superpotential becomes

${\cal W}=\lambda_{1}\log y-\lambda_{0}\log
z+z^{-1}\prod_{i=1}^{m}(1-a_{i}/y).$

As we will see later in this section, this is the Hori-Vafa mirror of ${\cal
X}$, with equivariant action turned on. Projecting to $Y$, restrict ${\cal W}$
to its critical point $\partial_{z}{\cal W}=0$, we recover the superpotential
$W$ on $Y$ in (3.4).

From perspective of ${\cal Y}^{\vee}$, the coordinate $z$ has an
interpretation as the $e^{-\textup{Area}}$ of a holomorphic disk ending on the
Lagrangian fiber in ${\cal Y}^{\vee}$; it is also the variable whose ${\rm
Im}(\log z)$ part is identified with the dual photon $\varphi$. From either
perspective, unlike $u$, $v$ and $y$, $z$ is not globally defined. The
expression $v=z^{-1}\prod_{i=1}^{m}(1-a_{i}/y)$ holds only for $y\gg a_{i}$
for all $i$, since only then $v=v(z,y)$ is a term in the superpotential
generated by instantons; in other chambers on $Y$, one finds different
expressions. The advantage of the description based on $(Y,W)$, is that it is
global.

#### B.2.3

From gauge theory perspective, the good variables on the Coulomb branch are
not associated to dual photons, but rather to dressed monopole operators [55,
52]. This leads to a natural guess (which, should be ascribed to [76] as we
will elaborate on momentarily) for the potential ${\cal W}$ on ${\cal Y}$,
which is mirror to replacing ${\cal Y}^{\vee}$ by ${\cal X}$ in a general
case.

Start by parameterizing ${\cal Y}$ locally in terms of coordinates
$y_{a,\alpha}$ on $Y$, and local coordinates $z_{a,\alpha}$ valid for large
$y$. The $y$’s and the $z$’s correspond to ${\mathbb{C}}_{\theta}^{\times}$
and ${\mathbb{C}}_{\varphi}^{\times}$ directions of ${\cal Y}$, respectivelly.
The form of the superpotential ${\cal W}$ on ${\cal Y}$ is constrained by
asking that $i.)$ ”integrating out” $z$’s, corresponding to restricting to
$\partial_{z}{\cal W}=0$, we recover the superpotential $W$ from section 3,
that $ii.)$ the terms in ${\cal W}$ generated by replacing ${\cal Y}^{\vee}$
by ${\cal X}$ can be generated by instantons, and that $iii.)$ those terms can
be written in terms of globally defined coordinates on ${\cal Y}$. This leads
one to conjecture the following form of the potential:

${\cal
W}=\sum_{a}\sum_{\alpha}\bigl{(}\lambda_{a}\log(y_{a,\alpha})-\lambda_{0}\log(z_{a,\alpha})+z^{-1}_{a,\alpha}f_{a,\alpha}(y)\bigr{)},$
(B.3)

where $f_{a,\alpha}(y)$ are given in equation (3.10). It is easy to check that
this satisfies the constraints $i.)$ and $ii.)$. To see it satisfies $iii.)$
as well, recall that the dressed monopole operators take the form

$D^{+}_{a,k}=\sum_{\alpha}d^{+}_{a,\alpha}(y)y_{a,\alpha}^{k}z_{a,\alpha},\qquad
D^{-}_{a,k}=\sum_{\alpha}d^{-}_{a,\alpha}y_{a,\alpha}^{k}z^{-1}_{a,\alpha},$

where for large $y$, $d^{+}_{a,\alpha}(y)=f_{a,\alpha}(y)$ and
$d^{-}_{a,\alpha}={1/\prod_{\beta\neq\alpha}(1-y_{\beta,a}/y_{\alpha,a})}$.

The formula (B.3) says that mirror to compactifying ${\cal Y}^{\vee}$ by
adding a divisor ${D}^{\vee}_{a}$ is changing the potential by adding a term
${\cal W}\rightarrow{\cal W}+D^{+}_{a,0}$, which is a special $k=0$ case of
monopole operator. The $\lambda_{a>0}$-dependent term in ${\cal W}$ is mirror
to the ${\Lambda\subset{\rm T}}$ action that preserves the holomorphic
symplectic form, and $\lambda_{0}$ term to the
${\mathbb{C}}^{\times}_{\mathfrak{q}}\subset{\rm T}$ action that scales it.
One can show the latter can easily be rewritten in terms of the other set of
dressed monopole operators, $D^{-}_{a,k}$ and functions on $Y$.

#### B.2.4

The potential in (B.3) is, up to details, the effective potential on the
monopole moduli space of Gaiotto and Witten [76]. The potential, and the two-
dimensional Landau-Ginsburg model that comes with it, was considered by [76]
to be an effective two dimensional description of Witten’s five-dimensional
gauge theory [182, 183, 184, 185], the theory that should allow one to
categorifying Chern-Simons knot invariants for all Lie algebras
${}^{L}{\mathfrak{g}}$ in a way complementary to the two approaches described
here.444The superpotential in [76] differs from ${\cal W}$ in (B.3) in that
the term is mirror of the equivariant $\Lambda\subset{\rm T}$ action on ${\cal
X}$, and comes from the pair of Verma module states inserted at $y=0,\infty$
in (2.11). This term is important to get invariants of knots in
${\mathbb{R}}^{2}\times S^{1}$, but it is not important for knots in
${\mathbb{R}}^{3}$. See [77, 78] for an attempt to use [76] and the
description of the A-brane category in [73, 74] to obtain homological link
invariants.

As I will elaborate on in [6], the five dimensional gauge theory of [182] and
two dimensional theories studied in this paper and in [5] both describe the
same physics of the six dimensional $(0,2)$ theory with two dimensional
defects which lead to knots. They just approach the problem from two different
perspectives – the perspective of [182] is from the bulk (or rather, its
reduction to five dimensions), the perspective here is of the theory on the
defects themselves. The fact that the six dimensional $(0,2)$ theory contains
in it information to solve the categorification problem is implicit in the
work of Ooguri and Vafa [140], but the connection was first made in [87].

The explanation of [76] for how the two dimensional theory arises from five
dimensions should not be regarded as the derivation of the $(Y,W)$ Landau-
Ginsburg model. The Landau-Ginsburg model $(Y,W)$ follows from a more
fundamental theory on defects in little string theory [8]. That theory is the
quiver gauge theory from section 2, compactified on a circle, which itself
leads to ${\hbar}$-deformations of KZ equation and of
$\widehat{{}^{L}\mathfrak{g}}$, namely the qKZ equation and the quantum affine
algebra $U_{\hbar}(\widehat{{}^{L}\mathfrak{g}})$ studied in [7, 14]. It also
explains why three dimensional gauge theories should enter the problem at all.
Rather, [76] are explaining why the two approaches, the five dimensional
approach of [182] and the two dimensional ones of [5] and here, should be
equivalent.

###  B.3 Hori-Vafa mirror of hyper-toric ${\cal X}$

When ${\cal X}$ is hypertoric, we can derive $Y$ and the potential $W$ on it,
from Hori-Vafa mirror symmetry [92]. A hypertoric ${\cal X}$ is the Coulomb
branch of a three dimensional abelian gauge theory, with ${\cal N}=4$
supersymmetry. This is a special case of our 3d quiver gauge theory from
section 2 in which the gauge group is ${G}_{\mathcal{Q}}=U(1)^{d}$, and one
has $n$ hypermultiplets. It is economical to forget the origin of the
hypermultiplets in the quiver, and simply encode their ${G}_{\mathcal{Q}}$
charges in the charge matrix $Q_{i}^{a}$ which is charge of the $i$’th
hypermultiplet under $a$’th of $d$ $U(1)$ gauge groups. If the theory comes
from the quiver, as we will assume, all the matter fields are fundamentals and
bifundamentals, and then the charges take one of three values, $Q_{i}^{a}$ are
either $0$ or $\pm 1$. The hypermultiplets parameterize $T^{*}V$ where
$V={\mathbb{C}}^{n}$.

#### B.3.1

Since ${\cal X}$ is hypertoric, it has a quotient construction - which
corresponds to viewing it as the Higgs branch of the 3d mirror theory. The
mirror theory has gauge group ${G}_{\mathcal{Q}^{\vee}}=U(1)^{n-d}$ and $n$
hypermultiplets whose charges ${Q^{\vee}}$ satisfy orthogonality condition
[98, 103, 9]

$\sum_{i}Q_{i}^{a}\,{Q}^{\vee}_{i,\alpha}=0,$ (B.4)

for $\alpha=1,\ldots,n-d$. We have

${\cal
X}=\Bigl{(}\mu_{R}^{-1}(m),\mu_{\mathbb{C}}^{-1}(0)\Bigr{)}/\,U(1)^{n-d}\,,$

where

$\mu^{\alpha}_{R}(m)=\sum_{i=1}^{n}Q^{\vee}_{\alpha,i}(|{u}_{i}|^{2}-|u_{-i}|^{2})-m^{\alpha},\qquad\mu^{\alpha}_{\mathbb{C}}(0)=\sum_{i=1}^{n}Q^{\vee}_{\alpha,i}\,{u}_{i}\,u_{-i},$
(B.5)

are the real and complex moment map constraints.

#### B.3.2

Hori and Vafa [92] derived the Landau-Ginzburg mirror of ${\cal X}$, by a
version of $T$-duality. Mirror to $u_{i}$, $u_{-i}$ on ${\cal X}$ are dual
variables $Y_{i},$ $Y_{-i}$ on the mirror ${\cal Y}$, which are periodic,
$Y_{\pm i}\sim Y_{\pm i}+2\pi i$, related by ${\rm Re}(Y_{\pm i})=|u_{\pm
i}|^{2}$. This maps the real moment map constraints in (B.5) to holomorphic
constraints in the mirror

$\sum_{i=1}^{n}Q^{\vee}_{\alpha,i}(Y_{i}-Y_{-i})-m^{\alpha}=0.$ (B.6)

They also derived the Landau-Ginsburg superpotential, given by

${W}_{HV}=\sum_{i}(e^{-Y_{i}}+e^{-Y_{-i}})+\sum_{i}\lambda_{i}(Y_{-i}-Y_{i})+\lambda_{0}\sum_{i}Y_{i},$

where the $\lambda$ dependent terms are mirror to the equivariant action on
${\cal X}$. More precisely, here we are viewing ${\cal X}$ not as a hypertoric
manifold, but as a hypersurface obtained by setting
$\mu^{\alpha}_{\mathbb{C}}$ to zero in a toric manifold. The choice of
hypersurface of degree zero does not affect $W_{LG}$ in any way.

To make contact with formulas in the rest of the paper, let

$y_{i}=-e^{Y_{-i}-Y_{i}},$

so that we can solve for $Y_{-i}$ in terms of $Y_{i}$ and $y_{i}$. The
effective potential on ${\cal Y}$ can be written as

$W_{HV}=\sum_{i}e^{-Y_{i}}(1-y_{i}^{-1})+\sum_{i}\lambda_{i}\ln(y_{i})+\lambda_{0}\sum_{i}Y_{i},$

where the constraint (B.6) becomes

$a_{\alpha}\cdot\prod_{i=1}^{n}y_{i}^{Q^{\vee}_{\alpha,i}}=1,$ (B.7)

for some complex $a_{\alpha}=a_{\alpha}(m)$. Finally, by arguments similar to
those used in [88], we can integrate over $Y_{i}$’s in all the periods, which
is equivalent to restricting $W_{HV}$ to critical points of

$\partial_{Y_{i}}W^{HV}=0.$

The value of $W^{HV}$ on the critical locus is, up to an additive constant,
equal to

${W}=\sum_{i}\lambda_{i}\ln(y_{i})+\lambda_{0}\sum_{i}\log(1-y_{i}^{-1})$
(B.8)

where

$y_{i}=a_{i}^{-1}\cdot\prod_{a}y_{a}^{Q_{i}^{a}}$ (B.9)

solve the $n-d$ constraints in (B.6) in terms of $d$ ”variables” $y_{a}$.
Above, $a_{i}$ are chosen to satisfy
$a_{\alpha}=\prod_{i=1}^{n}a_{i}^{Q^{\vee}_{\alpha,i}}$ – the fact that
$a_{\alpha}$’s do not uniquely determine $a_{i}$’s corresponds to the freedom
of shifting $y_{a}$’s by arbitrary constants. Since
$\sum_{i}\lambda_{i}\ln(y_{i})$ equals $\sum_{a}\lambda_{a}\ln(y_{a})$ up to a
constant (because $\lambda_{a}=\sum_{i}{Q_{a,i}}\lambda_{i}$), we have
recovered the superpotential $W$ on the mirror of a hypertoric ${\cal X}$.

The fact that

$\int_{L}\Omega\,e^{-W}=\int_{L}\prod_{a=1}^{k}{dy_{a}\over
y_{a}}y_{a}^{-\lambda_{a}}\prod_{i}(1-a_{i}/\prod_{a}y_{a}^{Q_{i}^{a}})^{-\lambda_{0}}$

solves the quantum differential equation of ${\cal X}$ for any integration
cycle $L$ is a theorem of [131].

###  B.4 Lagrangian correspondence and the figure eight

We will see how the figure eight Lagrangian from section 5.3 follows from
Lagrangian correspondences, the pair of functors that relate

$k_{*}:\mathscr{D}_{Y}\rightarrow\mathscr{D}_{\cal Y},\qquad
k^{*}:\mathscr{D}_{\cal Y}\rightarrow\mathscr{D}_{Y}.$

Recall from section 3 and appendix $B$ that mirror to ${\cal X}$ which is a
resolution of an $A_{m-1}$ surface singularity is ${\cal Y}$ which is the
multiplicative $A_{m-1}$ surface from equation (B.2), which we will rewrite as

$x^{2}+z^{2}=f(y),$ (B.10)

where $x$ and $z$ are related to $u$ and $v$ in (B.2) by $u=x+iz$ and
$v=x-iz$, and where $f(y)=\prod_{i}(1-a_{i}/y)$. ${\cal Y}$ is a
${\mathbb{C}}^{\times}$ fibration over $Y$, obtained by forgetting $x$ and
$z$.

#### B.4.1

As explained in section 5, the symplectic form $\omega_{Y}$ on $Y$ is obtained
from a symplectic form $\omega_{\cal Y}$ on ${\cal Y}$ by restricting to the
vanishing $S^{1}$ fiber at each point in the base. Take the symplectic form
$\omega_{\cal Y}$ on ${\cal Y}$ to be the one obtained from the symplectic
form on ${\mathbb{C}}^{2}\times{\mathbb{C}}^{\times}$ by restriction,

$\omega_{\cal Y}={i\over 2}(dx\wedge d{\overline{x}}+dz\wedge
d{\overline{z}}+dy\wedge d{\overline{y}}/|y|^{2})$

The vanishing cycle is parameterized by $s$ and $t$ real and satisfying
$s^{2}+t^{2}=1$, and embedded into ${\cal Y}$ by $x=s\sqrt{f(y)}$,
$z=t\sqrt{f(y)}$, so

$\omega_{Y}={i\over 2}dy\wedge
d{\overline{y}}\;\bigl{(}{1\over|y|^{2}}+|(\sqrt{f(y)})^{\prime}|^{2}\bigr{)},$

or alternatively, $\omega_{Y}={i\over 2}d{\overline{d}}(|\log
y|^{2}+\sqrt{|f(y)|^{2}})$. With this Kahler potential, the naive Kahler
metric on $Y$ becomes incomplete at $y=a_{i}$, where $f(y)$ vanishes, so one
should picture the geometry as developing long tubes there.

The Lagrangian ${\mathscr{K}}$ which leads to the correspondences $k_{*}$ and
$k^{*}$ lives on the product

${\mathscr{K}}\in Y_{-}\times{\cal Y},$

of ${\cal Y}$, and $Y$ with the sign of its symplectic form reversed.
${\mathscr{K}}$ is a fibration over $Y$ with vanishing $S^{1}$ fibers.
Concretely, ${\mathscr{K}}$ is the set of all points of the form
$y^{\prime}\in Y$, and $(x,y,z)\in{\cal Y}$, with $y^{\prime}=y$ and
$x=s\sqrt{P(y)}$, $z=t\sqrt{P(y)}$ where $s,t$ are real and parameterize the
vanishing $S^{1}$, $s^{2}+t^{2}=1$. It is immediate that the symplectic form
$\omega_{Y^{-}\times{\cal Y}}=\omega_{\cal Y}-\omega_{Y}$ vanishes on
${\mathscr{K}}$, so it is indeed a Lagrangian.

#### B.4.2

For simplicity, zoom in next one of the vanishing $S^{2}$’s, so that, after
changing variables

$f(y)\sim a^{2}-y^{2},$

with $a$ which is real, and where $y$ is now a local coordinate. ${\cal Y}$
has a vanishing cycle ${\cal I}$ which is an $S^{2}$ of radius $a$ given by

${\cal I}:\qquad q_{x}^{2}+q_{y}^{2}+q_{z}^{2}=a^{2},\qquad
p_{x}=p_{y}=p_{z}=0,$ (B.11)

where we put $x=q_{x}+ip_{x}$, $y=q_{y}+ip_{y}$ and $z=q_{z}+ip_{z}$, with
$q$’s and $p$’s real.

First, note that

${\cal I}=k_{*}I,$

where $I$ is a straight line in $Y$ between $y=\pm a$, the set of points with
$p_{y}=0$ and $q_{y}\in[-a,a]$. Per definition, $k_{*}I$ is obtained by
intersecting $I$ with ${\mathscr{K}}$ in $Y_{-}\times{\cal Y}$, and projecting
to ${\cal Y}$. The result is the family of vanishing $S^{1}$’s over the
interval in $Y$, viewed as the base of ${\cal Y}$, which is ${\cal I}$ itself.
Thus, the ${\cal I}$ brane on ${\cal Y}$ comes from the $I$-brane on $Y$, via
the correspondence ${\mathscr{K}}$.

#### B.4.3

To compute

$E=k^{*}{\cal I}=k^{*}k_{*}I,$

we need to intersect ${\cal I}$ with ${\mathscr{K}}$ in $Y_{-}\times{\cal Y}$
and project to $Y$. As is, ${\cal I}$ does not intersect ${\mathscr{K}}$
transversely. To get a Lagrangian with transverse intersections with
${\mathscr{K}}$, we deform ${\cal I}$ by a flow of some Hamiltonian $H$ which
is Morse function on ${\cal I}$ in infinitesimal time $\epsilon$. Taking
$H={q_{x}/(q_{x}^{2}+q_{y}^{2})^{1/2}}$, to the first order in $\epsilon$, the
Lagrangian ${\cal I}$ deforms to ${\cal I}_{\epsilon}$ given by (B.11), with
the $p_{x}$ and $p_{y}$ deformed to
$p_{x}=\epsilon\,q_{y}^{2}/(q_{x}^{2}+q_{y}^{2})^{3/2}$, and
$p_{y}=-\epsilon\,q_{x}q_{y}/(q_{x}^{2}+q_{y}^{2})^{3/2}$.

The intersection points of ${\cal I}_{\epsilon}$ with ${\mathscr{K}}$ have
$t=0$ (for $\theta\neq 0,\pi$). Thus they all have $z=0$ and lie on
$q_{x}^{2}+q_{y}^{2}-p_{x}^{2}-p_{y}^{2}=a^{2}$. From this it follows that
$q_{x}^{2}\sim a^{2}-(1-\epsilon^{2}/a^{4})q_{y}^{2},$ and therefore $q_{y}$
and $p_{y}$ satisfy

$E:\qquad a^{6}p_{y}^{2}\approx\epsilon^{2}\,q_{y}^{2}(a^{2}-q_{y}^{2}).$

This describes a figure eight curve in the $y$-plane, except that the two
points where the $S^{1}$ vanishes, $y=\pm a$ are on the curve itself. Keeping
the next order in $\epsilon$, points $y=\pm a$ move inside two leafs of a
figure eight, so we recover the Lagrangian $E$ in figure 12.

## References

  * [1] M. Abouzaid, D. Auroux, A. I. Efimov, L. Katzarkov, D. Orlov, “Homological mirror symmetry for punctured spheres”, J. Amer. Math. Soc. 26 (2013), 1051-1083; arXiv:1103.4322.
  * [2] M. Abouzaid, D. Auroux, L. Katzarkov, “Lagrangian fibrations on blowups of toric varieties and mirror symmetry for hypersurfaces” arXiv:1205.0053
  * [3] M. Abouzaid, I. Smith, “The symplectic arc algebra is formal”, arXiv:1311.5535.
  * [4] M. Abouzaid, I. Smith, “Khovanov homology from Floer cohomology”, arXiv:1504.01230.
  * [5] M. Aganagic, ”Knot Categorification from Mirror Symmetry, Part I: Coherent Sheaves”, arXiv:2004.14518.
  * [6] M. Aganagic, ”Knot Categorification from Mirror Symmetry, Part III: String Theory Origin”, to appear.
  * [7] M. Aganagic, E. Frenkel and A. Okounkov, “Quantum q-Langlands Correspondence,” arXiv:1701.03146.
  * [8] M. Aganagic and N. Haouzi, “ADE Little String Theory on a Riemann Surface (and Triality),” arXiv:1506.04183.
  * [9] M. Aganagic, K. Hori, A. Karch and D. Tong, “Mirror symmetry in (2+1)-dimensions and (1+1)-dimensions,” JHEP 0107, 022 (2001) [hep-th/0105075].
  * [10] M. Aganagic, A. Klemm and C. Vafa, “Disk instantons, mirror symmetry and the duality web,” Z. Naturforsch. A 57 (2002), 1-28 arXiv:hep-th/0105045.
  * [11] M. Aganagic, E. LePage and M. Rapcak, to appear.
  * [12] M. Aganagic, M. McBreen and V. Shende, to appear.
  * [13] M. Aganagic and A. Okounkov, “Elliptic stable envelope,” arXiv:1604.00423.
  * [14] M. Aganagic and A. Okounkov, “Quasimap counts and Bethe eigenfunctions,” Moscow Math. J. 17, no. 4, 565 (2017) arXiv:1704.08746.
  * [15] R. Anno, R. Bezrukavnikov, I. Mirkovic, “Stability conditions for Slodowy slices and real variations of stability”, arXiv:1108.1563.
  * [16] P. S. Aspinwall and M. R. Douglas, “D-brane stability and monodromy,” JHEP 0205, 031 (2002) [hep-th/0110071].
  * [17] P. S. Aspinwall, “D-branes on Calabi-Yau manifolds,” [arXiv:hep-th/0403166].
  * [18] P. S. Aspinwall and I. V. Melnikov, “D-branes on vanishing del Pezzo surfaces,” JHEP 12 (2004), 042 [arXiv:hep-th/0405134].
  * [19] P. S. Aspinwall, “D-Branes on Toric Calabi-Yau Varieties,” [arXiv:0806.2612].
  * [20] P. S. Aspinwall and A. E. Lawrence, “Derived categories and zero-brane stability,” JHEP 0108, 004 (2001) [hep-th/0104147].
  * [21] P. S. Aspinwall et al., “Dirichlet branes and mirror symmetry,” 681 pages, Clay Mathematics Monographs, 4 Providence, RI: AMS (2009)
  * [22] D. Auroux, “Special Lagrangian fibrations, wall-crossing, and mirror symmetry”, arXiv:0902.1595
  * [23] D. Auroux, “A beginner’s introduction to Fukaya categories”, arXiv:1301.7056.
  * [24] D. Auroux, “Fukaya categories of symmetric products and bordered Heegaard-Floer homology.” arXiv: 1001.4323.
  * [25] D. Auroux,“Fukaya categories and bordered Heegaard-Floer homology”, arXiv:1003.2962.
  * [26] A. Beilinson ,V. Ginzburg, and W. Soergel, “Koszul duality patterns in representation theory”, J. Amer. Math. Soc., 9(2):473–527, 1996.
  * [27] M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, “Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes,” Commun. Math. Phys. 165 (1994), 311-428 [arXiv:hep-th/9309140 [hep-th]].
  * [28] R.Bezrukavnikov, “Noncommutative Counterparts of the Springer Resolution,” math/0604445.
  * [29] R. Bezrukavnikov, D. Kaledin, ”Fedosov quantization in positive characteristic,” 2005, arXiv:math/0501247.
  * [30] R. Bezrukavnikov, A. Okounkov, to appear.
  * [31] S. Bigelow, “A homological definition of the Jones polynomial”, Geom. & Top. Monogr. 4, 29-41, 2002. math.GT/0201221.
  * [32] J. Birman, “Braids, links, and mapping class groups”, Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1974.
  * [33] J. Birman, “On the stable equivalence of plat representations of knots and links”, Canad. J. Math. 28 (1976), no. 2.
  * [34] A. I. Bondal, “Representations of Associative Algebras and Coherent Sheaves,” Math. USSR Izvestiya 34 (1990) 23–42.
  * [35] A. Bondal, “Helixes, Representations of Quivers and Koszul Algebras”. In A. Rudakov (Ed.), Helices and Vector Bundles: Seminaire Rudakov (London Mathematical Society Lecture Note Series, pp. 75-96). Cambridge: Cambridge University Press.
  * [36] A. Bondal, “Derived categories of toric varieties.” In Convex and Algebraic Geometry, Oberwolfach Conference Reports, volume 3, pages 284–286. 2006.
  * [37] G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, “Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants,” Commun. Math. Phys. 333 (2015) no.2, 717-760 [arXiv:1307.5997 [hep-th]].
  * [38] T. Braden, A. Licata, N. Proudfoot, and B. Webster, “Gale duality and koszul duality”, Adv. Math. 225 (2010) 2002–2049, arXiv:0806.3256v2.
  * [39] A. Braverman, M. Finkelberg and H. Nakajima, “Towards a mathematical definition of Coulomb branches of $3$-dimensional ${\cal N}=4$ gauge theories, II,” arXiv:1601.03586 [math.RT].
  * [40] A. Braverman, M. Finkelberg and H. Nakajima, “Coulomb branches of $3d$ ${\cal N}=4$ quiver gauge theories and slices in the affine Grassmannian” (with appendices by A. Braverman, M. Finkelberg, J. Kamnitzer, R. Kodera, H. Nakajima, B. Webster, and A. Weekes), arXiv:1604.03625 [math.RT].
  * [41] T. Bridgeland, “Stability conditions on triangulated categories”, math/0212237, Annals of Mathematics, 166 (2007), 317-345.
  * [42] T. Bridgeland, “Stability conditions and Kleinian singularities”, math/0508257.
  * [43] T. Bridgeland, “Spaces of stability conditions”, math/0611510.
  * [44] M. Bullimore, T. Dimofte and D. Gaiotto, “The Coulomb Branch of 3d ${\mathcal{N}=4}$ Theories,” Commun. Math. Phys. 354 (2017) no.2, 671 [arXiv:1503.04817 [hep-th]].
  * [45] A. Caldararu, “Derived categories of sheaves: a skimming,” arXiv:math/0501094.
  * [46] S. Cautis, J. Kamnitzer, “Knot homology via derived categories of coherent sheaves I, $sl(2)$ case”, 2007, arXiv:math/0701194
  * [47] S. Cautis, J. Kamnitzer, “Knot homology via derived categories of coherent sheaves II, ${sl}_{m}$ case”, J. 2008, Inventiones Mathematicae, 174, 165.
  * [48] S. Cecotti, D. Gaiotto and C. Vafa, “$tt^{*}$ geometry in 3 and 4 dimensions,” JHEP 1405, 055 (2014) doi:10.1007/JHEP05(2014)055 [arXiv:1312.1008 [hep-th]].
  * [49] S. Cecotti and C. Vafa, “Topological antitopological fusion,” Nucl. Phys. B367 (1991) 359.
  * [50] J. Chuang and R. Rouquier, “Derived equivalences for symmetric groups and $sl_{2}$-categorification,” math/0407205
  * [51] J. Chuang and R. Rouquier, “Perverse Equivalence”, preprint, available at https://www.math.ucla.edu/ rouquier/papers/perverse.pdf
  * [52] T. Coates, A. Givental, “Quantum Riemann-Roch, Lefschetz and Serre,” math/0110142.
  * [53] D. A. Cox, Erratum to “The Homogeneous Coordinate Ring of a Toric Variety”, along with the original paper; alg-geom/9210008.
  * [54] A. Craw, ”Explicit Methods for Derived Categories of Sheaves”, available at https://www.math.utah.edu/dc/tilting.pdf
  * [55] S. Cremonesi, A. Hanany, and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories, JHEP 1401 (2014) 005, [arXiv:1309.2657].
  * [56] P. Deligne, P. Etingof, D. S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. R. Morrison and E. Witten, eds., “Quantum Fields and Strings: A Course For Mathematicians”, 2 vols., American Mathematical Society, Providence, 1999.
  * [57] I. Danilenko, _Slices of the Affine Grassmannian and Quantum Cohomology_ , PhD thesis in preparation, Columbia University.
  * [58] R. Dijkgraaf, H. L. Verlinde and E. P. Verlinde, “Topological strings in d $<$ 1,” Nucl. Phys. B 352 (1991), 59-86
  * [59] M. R. Douglas, B. Fiol and C. Romelsberger, “Stability and BPS branes,” JHEP 0509, 006 (2005) [hep-th/0002037].
  * [60] M. R. Douglas, “D-branes, categories and N=1 supersymmetry,” J. Math. Phys. 42, 2818 (2001) doi:10.1063/1.1374448 [hep-th/0011017].
  * [61] N. Dowlin, “A spectral sequence from Khovanov homology to knot Floer homology”. arXiv:1811:07848.
  * [62] V. G. Drinfeld,“Quasi-Hopf Algebras and Knizhnik-Zamolodchikov Equations”. In: Belavin A.A., Klimyk A.U., Zamolodchikov A.B. (eds) Problems of Modern Quantum Field Theory. Research Reports in Physics. Springer, Berlin, Heidelberg, (1989)
  * [63] B. Dubrovin, “Geometry of 2d topological field theories.” hep-th/9407018.
  * [64] P. Etingof, I. Frenkel, and A. A. Kirillov, “Lectures on representation theory and Knizhnik-Zamolodchikov equations”,Jr., Math. Surveys and Monographs, vol. 58, Amer. Math. Soc., Providence, RI, 1998, 198 pp.
  * [65] P. Etingof, N. Geer, “Monodromy of trigonometric KZ equations”, arXiv:math/0611003.
  * [66] B. Feigin, E. Frenkel, “A family of representations of affine Lie algebras”, Uspekhi Matem. Nauk, 43:5 (1988), 227-228 [English translation: Russ. Math. Surv., 43:5 (1988), 221-222].
  * [67] B. Feigin, E. Frenkel and N. Reshetikhin, “Gaudin model, Bethe ansatz and correlation functions at the critical level,” Commun. Math. Phys. 166 (1994), 27-62 [arXiv:hep-th/9402022 [hep-th]].
  * [68] B. Feigin, E. Frenkel, L. Rybnikov, “Opers with irregular singularity and spectra of the shift of argument subalgebra”, arXiv:0712.1183.
  * [69] B. Feigin, E. Frenkel and V. Toledano Laredo, “Gaudin models with irregular singularities,” Adv. Math. 223 (2010), 873-948 [arXiv:math/0612798 [math.QA]].
  * [70] A. Floer, “Morse Theory for Lagrangian Intersections”,J . Differential Geometry, 28. (1988) 513-547
  * [71] E. Frenkel, “Free Field Realizations in Representation Theory and Conformal Field Theory”, ICM lecture, 1994, hep-th/9408109
  * [72] I. B. Frenkel, N.Yu. Reshetikhin, “Quantum affine algebras and holonomic difference equations,” Comm. Math. Phys. 146 (1992) 1-60.
  * [73] D. Gaiotto, G. W. Moore and E. Witten, “An Introduction To The Web-Based Formalism,” arXiv:1506.04086 [hep-th].
  * [74] D. Gaiotto, G. W. Moore and E. Witten, “Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions,” arXiv:1506.04087 [hep-th].
  * [75] D. Gaiotto and E. Witten, “S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory,” Adv. Theor. Math. Phys. 13 (2009) no.3, 721-896 [arXiv:0807.3720 [hep-th]].
  * [76] D. Gaiotto and E. Witten, “Knot Invariants from Four-Dimensional Gauge Theory,” Adv. Theor. Math. Phys. 16 (2012) 935 [ar Xiv:1106.4789 [hep-th]].
  * [77] D. Galakhov and G. W. Moore, “Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology,” [arXiv:1607.04222 [hep-th]].
  * [78] D. Galakhov, “Why Is Landau-Ginzburg Link Cohomology Equivalent To Khovanov Homology?,” JHEP 05 (2019), 085, arXiv:1702.07086.
  * [79] S. Ganatra, J. Pardon, V. Shende, “Covariantly functorial wrapped Floer theory on Liouville sectors”, arXiv:1706.03152.
  * [80] S. Ganatra, J. Pardon, V. Shende, “Sectorial descent for wrapped Fukaya categories”, arXiv:1809.03427.
  * [81] S. Ganatra, J. Pardon, V. Shende, “Microlocal Morse theory of wrapped Fukaya categories”, arXiv:1809.08807.
  * [82] B. Gammage, M. McBreen, B. Webster, “Homological mirror symmetry for hypertoric varieties II”. arXiv e-prints.arXiv:1903.07928.
  * [83] V. Ginzburg, “Perverse sheaves on a loop group and Langlands duality”, alg-geom/9511007.
  * [84] A. Givental, “Equivariant Gromov-Witten invariants”, Int. Math. Res. Notices 13 (1996), 613-663.
  * [85] A. Givental, “Semisimple Frobenius structures at higher genus”, arXiv:math/0008067.
  * [86] M. Gross, P. Hacking, S. Keel, M. Kontsevich, “Canonical bases for cluster algebras”. arXiv:1411.1394.
  * [87] S. Gukov, A. S. Schwarz and C. Vafa, “Khovanov-Rozansky homology and topological strings,” Lett. Math. Phys. 74 (2005), 53-74 [arXiv:hep-th/0412243 [hep-th]].
  * [88] A. Hanany and E. Witten, “Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics,” Nucl. Phys. B 492, 152 (1997) [hep-th/9611230].
  * [89] D. Harlow, J. Maltz and E. Witten, “Analytic Continuation of Liouville Theory,” JHEP 12 (2011), 071 [arXiv:1108.4417 [hep-th]].
  * [90] M. Herbst, K. Hori and D. Page, “Phases Of N=2 Theories In 1+1 Dimensions With Boundary,” arXiv:0803.2045 [hep-th].
  * [91] T. Holm, P. Jorgensen, R. Rouquier, “Triangulated Categories”, Cambridge University Press, 2010
  * [92] K. Hori and C. Vafa, “Mirror symmetry,” hep-th/0002222.
  * [93] K. Hori, A. Iqbal and C. Vafa, “D-branes and mirror symmetry,” hep-th/0005247.
  * [94] K. Hori, “Mirror symmetry and quantum geometry,” [arXiv:hep-th/0207068 [hep-th]].
  * [95] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil and E. Zaslow, “Mirror symmetry,” 2006. 929 pp.
  * [96] J. Hu, A. Mathas, “Graded cellular bases for the cyclotomic Khovanov-Lauda-Rouquier algebras of type A,” arXiv:0907.2985.
  * [97] D. Huybrechts, R. P.Thomas, “$\mathbb{P}$-objects and autoequivalences of derived categories”, math/0507040.
  * [98] K. A. Intriligator and N. Seiberg, “Mirror symmetry in three-dimensional gauge theories,” Phys. Lett. B 387 (1996) 513 [hep-th/9607207].
  * [99] P. Jorgensen, “Homological algebra”, Lecture slides available at https://sites.google.com/view/peterjorgensen/
  * [100] D. Kaledin, ”Derived equivalences by quantization”, arXiv:math/0504584
  * [101] D. Kaledin, ”Geometry and topology of symplectic resolutions”, math/0608143
  * [102] M. Kapranov, M. Kontsevich and Y. Soibelman, “Algebra of the infrared and secondary polytopes,” Adv. Math. 300 (2016), 616-671 [arXiv:1408.2673 [math.SG]].
  * [103] A. Kapustin and M. J. Strassler, “On mirror symmetry in three-dimensional Abelian gauge theories,” JHEP 9904, 021 (1999) [hep-th/9902033].
  * [104] A. Kapustin and E. Witten, “Electric-Magnetic Duality And The Geometric Langlands Program,” Commun. Num. Theor. Phys. 1 (2007) 1 arXiv:hep-th/0604151.
  * [105] D. Kazhdan and G. Lusztig, ”Tensor structures arising from affine Lie algebras. I-IV”, J. Amer. Math. Soc. 6 (1993), 905-947,949-1011; 7 (1994), 355-381, 383-453.
  * [106] Louis H. Kauffman, “An introduction to Khovanov Homology”, Contemporary Mathematics Volume 670, 2016.
  * [107] B. Keller, “A-Infinity Algebras in Representation Theory”, 2001.
  * [108] M. Khovanov, “A categorification of the Jones polynomial”, Duke Math. J. 101 (2000), no. 3, 359–426.
  * [109] M. Khovanov, “A functor-valued invariant of tangles”, math/0103190.
  * [110] M. Khovanov, A. D. Lauda, “A diagrammatic approach to categorification of quantum groups” I. arXiv:0803.4121
  * [111] V. G. Knizhnik and A. B. Zamolodchikov, “Current Algebra and Wess-Zumino Model in Two-Dimensions,” Nucl. Phys. B247 (1984) 83.
  * [112] T. Kohno, “Monodromy representations of braid groups and Yang Baxter equations”. Ann. Inst. Fourier 37, 139 160 (1987)
  * [113] M. Kontsevich, Y. Manin, “Gromov-Witten classes, quantum cohomology, and enumerative geometry”, Comm. Math. Phys. 164(3): 525-562 (1994).
  * [114] S.-C. Lau, X. Zheng, “SYZ Mirror Symmetry for Hypertoric Varieties.” Communications in Mathematical Physics 373.
  * [115] R. J. Lawrence, “Homology representations of braid groups”, D.Phil. Thesis, Univerity of Oxford (June 1989)
  * [116] R. J. Lawrence, “A functorial approach to the one-variable Jones Polynomial”, J. Diff. Geom. 37 (1993) 689-710.
  * [117] R. J. Lawrence, “The homological approach applied to higher representations,” Harvard preprint (1990), available at http://www.ma.huji.ac.il/ ruthel/
  * [118] Y. Lekili, A. Polishchuk, “Homological mirror symmetry for higher-dimensional pairs of pants”, Compositio Mathematica, vol. 7, pp 1310–1347, 2020, arXiv:1811.04264v6.
  * [119] R. Lipshitz, “A cylindrical reformulation of Heegaard Floer homology,” Geom. Topol., 10:955–1097, 2006.
  * [120] R. Lipshitz, P. Ozsvath, D. Thurston, “Bordered Heegaard Floer homology: invariance and pairing,” arXiv:0810.0687.
  * [121] G. Lusztig, “Singularities, character formulas and a q-analog of weight multiplicities”, Asterisque 101-102 (1983), 208-229.
  * [122] C. Y. Mak, I. Smith, “Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$”, arXiv:1907.07624.
  * [123] C. Manolescu, “Nilpotent slices, Hilbert schemes, and the Jones polynomial,” arXiv:math/0411015.
  * [124] C. Manolescu, “An introduction to knot Floer homology,” arXiv:1401.7107.
  * [125] C.Y. Mak, I. Smith, “Fukaya-Seidel categories of Hilbert schemes and parabolic category $\mathcal{O}$”. arXiv:1907.0762
  * [126] D. Maulik and A. Okounkov, “Quantum Groups and Quantum Cohomology,” arXiv:1211.1287 [math.AG].
  * [127] V. Mazorchuk, S. Ovsienko and C. Stroppel, ”Quadratic duals, Koszul dual functors, and applications”, math/0603475.
  * [128] M. B. McBreen, D. K. Shenfeld, “Quantum Cohomology of Hypertoric Varieties”, 2013, Letters in Mathematical Physics, 103, 1273 .
  * [129] B. Fang, C.-C. M. Liu, D. Treumann, E. Zaslow, “T-Duality and Homological Mirror Symmetry of Toric Varieties”. arXiv:0811.1228.
  * [130] I. Mirkovic and K. Vilonen, “Geometric Langlands duality and representations of algebraic groups over commutative rings”, math.RT/0401222.
  * [131] G. W. Moore and N. Seiberg, “Classical and Quantum Conformal Field Theory,” Commun. Math. Phys. 123 (1989) 177.
  * [132] G. W. Moore and N. Seiberg, “Lectures On RCFT,” RU-89-32, YCTP-P13-89.
  * [133] C. Nayak, S. H. Simon, A. Stern, M. Freedman, S. Das Sarma, “Non-Abelian anyons and topological quantum computation.” Reviews of Modern Physics 80, 1083–1159.
  * [134] H. Nakajima, “Towards a mathematical definition of Coulomb branches of $3$-dimensional ${\cal N}=4$ gauge theories, I,” arXiv:1503.03676 [math-ph].
  * [135] T. Ohtsuki, “Quantum Invariants: A Study of Knots, Three Manifolds and Their Sets”, World Scientific, 2001.
  * [136] A. Okounkov, “Lectures on K-theoretic computations in enumerative geometry,” arXiv:1512.07363.
  * [137] A. Okounkov,“Enumerative geometry and geometric representation theory”, arxiv:1701.00713.
  * [138] A. Okounkov, “On the crossroads of enumerative geometry and geometric representation theory,” arXiv:1801.09818 [math.RT].
  * [139] H. Ooguri, Y. Oz and Z. Yin, “D-branes on Calabi-Yau spaces and their mirrors,” Nucl. Phys. B 477, 407 (1996) [hep-th/9606112].
  * [140] H. Ooguri and C. Vafa, “Knot invariants and topological strings,” Nucl. Phys. B577 (2000) 419 [arXiv:hep-th/9912123].
  * [141] P. S. Ozsvath, A. I. Stipsicz and Z. Szabo, “Grid Homology for Knots and Links”, Mathematical Surveys and Monographs Volume: 208; 2015; 410 pp.
  * [142] P. S. Ozsvath and Z. Szabo, “Holomorphic disks and knot invariants,” Adv. Math., 186(1):58–116, 2004.
  * [143] P. S. Ozsvath and Z. Szabo, “An overview of knot Floer homology,” arXiv:1706.07729.
  * [144] R. Pandharipande, “Rational curves on hypersurfaces [after A. Givental]”, arXiv:math/9806133
  * [145] R. Pandharipande, H. H. Tseng, “Higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and CohFTs associated to local curves”. arXiv:1707.01406
  * [146] R. Pandharipande, “Cohomological field theory calculations”, Proceedings of the ICM 2018, arXiv:1712.02528
  * [147] Y. Lekili, A. Polishchuk, “Homological mirror symmetry for higher dimensional pairs of pants”, arXiv:181104264.
  * [148] Y. Qi, J. Sussan, “A Categorification of the Burau representation at prime roots of unity”, arXiv:1312.7692.
  * [149] J. A. Rasmussen, “Floer homology and knot complements,” PhD thesis, Harvard University, 2003\.
  * [150] J. A. Rasmussen, “Knot polynomials and knot homologies”. Geometry and Topology of Manifolds (Fields Institute Communications), 47:261–280, 2005.
  * [151] N. Reshetikhin, A. Varchenko, “Quasiclassical asymptotics of solutions to the KZ equations”, hep-th/9402126.
  * [152] R. Rouquier, P. Shan, M. Varagnolo, and E. Vasserot, “Categorifications and cyclotomic rational double affine hecke algebras”, Inventiones mathematicae (2015) 1–116.
  * [153] R. Rouquier, “2-Kac-Moody algebras,” arXiv:0812.5023.
  * [154] S. Sarkar, “Maslov index of holomorphic triangles,” math/0609673 (v1).
  * [155] S. Sarkar, “Maslov index formulas for Whitney n-gons,” math/0609673 (v2).
  * [156] V. Schechtman and A. Varchenko, “Quantum groups and homology of local systems,” in Algebraic Geometry and Analytic Geometry, ICM-90 Satellite Conference Proceedings, pp. 182-197, Springer, 1991.
  * [157] V. Schechtman and A. Varchenko, “Arrangements of hyperplanes and Lie algebra homology,” in Inventiones mathematicae 106, 139-194 (1991)
  * [158] P. Seidel, “ Fukaya categories and Picard-Lefschetz theory”, 2007, book, 292 pp., Vol. 10 of Zurich lectures in advanced mathematics.
  * [159] P. Seidel, “Homological mirror symmetry for the genus two curve”, arXiv:0812.1171
  * [160] P. Seidel, and I. Smith, “A link invariant from the symplectic geometry of nilpotent slices”, Duke Math. J. 134, no. 3 (2006), 453-514; math.SG/0405089.
  * [161] P. Seidel, J. P. Solomon, “Symplectic cohomology and q-intersection numbers”, arXiv:1005.5156
  * [162] P. Seidel and R. P. Thomas, “Braid group actions on derived categories of coherent sheaves,” arXiv:math/0001043.
  * [163] P. Shan, M. Varagnolo, and E. Vasserot, Koszul duality of affine Kac-Moody algebras and cyclotomic rational double affine Hecke algebras, Adv. Math. 262 (2014) 370–435
  * [164] A. Strominger, S. T. Yau and E. Zaslow, “Mirror symmetry is T duality,” Nucl. Phys. B 479, 243 (1996) [hep-th/9606040].
  * [165] C. Stroppel, B. Webster, “Quiver Schur algebras and q-Fock space.” arXiv:1110.1115
  * [166] C. Teleman,“The structure of semi-simple field theory at higher genus”, arXiv:0712.0160.
  * [167] R. P. Thomas, “Derived categories for the working mathematician,” AMS/IP Stud. Adv. Math. 23 (2001), 349-361 [arXiv:math/0001045 [math.AG]].
  * [168] R. P. Thomas, “An exercise in mirror symmetry”, Proceedings of the International Congress of Mathematicians, Hyderabad, India, 2010.
  * [169] R. P. Thomas, “Stability conditions and the braid group”, math/0212214.
  * [170] A. Tsuchia and Y. Kanie, ”Vertex Operators in Conformal Field Theory on ${\mathbb{P}}^{1}$ and Monodromy Representations of Braid Group,” Conformal Field Theory and Solvable Lattice Models, 297-372, Mathematical Society of Japan, Tokyo, 1988.
  * [171] B. Webster, “Weighted Khovanov-Lauda-Rouquier algebras”, arXiv:1209.2463
  * [172] B. Webster, “Knot invariants and higher representation theory”, arXiv:1309.3796.
  * [173] B. Webster, “Tensor product algebras, Grassmannians and Khovanov homology,” arXiv:1312.7357.
  * [174] B. Webster, “On generalized category O for a quiver variety”, arXiv:1409.4461
  * [175] B. Webster, “Coherent sheaves and quantum Coulomb branches I: tilting bundles from integrable systems,” arXiv:1905.04623.
  * [176] B. Webster, “Coherent sheaves and quantum Coulomb branches II: quiver gauge theories and knot homology,” to appear.
  * [177] Charles A. Weibel, “An introduction to homological algebra”, Cambridge University Press, 1994
  * [178] E. Witten, “Supersymmetry and Morse Theory”, J. Differential Geometry, 17 (1982).
  * [179] E. Witten, “Quantum Field Theory and the Jones Polynomial,” Commun. Math. Phys. 121 (1989) 351.
  * [180] E. Witten, “Gauge Theories and Integrable Lattice Models,” Nucl. Phys. B 322, 629 (1989).
  * [181] E. Witten, “Analytic Continuation Of Chern-Simons Theory,” AMS/IP Stud. Adv. Math. 50 (2011), 347-446 [arXiv:1001.2933 [hep-th]].
  * [182] E. Witten, “Fivebranes and Knots,” arXiv:1101.3216 [hep-th].
  * [183] E. Witten, “Khovanov Homology And Gauge Theory,” [arXiv:1108.3103 [math.GT]].
  * [184] E. Witten, “Two Lectures On The Jones Polynomial And Khovanov Homology,” [arXiv:1401.6996 [math.GT]].
  * [185] E. Witten, “Two Lectures on Gauge Theory and Khovanov Homology,” [arXiv:1603.03854 [math.GT]].