# Quiver Hecke algebras from Floer homology in Couloumb branches

Mina Aganagic1,2, Ivan Danilenko1, Yixuan Li1, Vivek Shende1,3, Peng Zhou1

###### Abstract

Homology theories categorifying quantum group link invariants are known to be
governed by the representation theory of quiver Hecke algebras, also called
KLRW algebras. Here we show that certain cylindrical KLRW algebras, relevant
in particular for cylindrical generalizations of link homology theories, can
be realized by Lagrangian Floer homology in multiplicative Coulomb branches.
This confirms a homological mirror symmetry prediction of the first author.

1 Departments of Mathematics, University of California, Berkeley, USA
2Department of Physics, University of California, Berkeley, USA
3 Center for Quantum Mathematics, Syddansk University, Odense, Denmark

###### Contents

  1. 1 Introduction
    1. 1.1 Geometry of Coulomb branches for quiver gauge theory
      1. 1.1.1 Coordinates and superpotential
      2. 1.1.2 Holomorphic curves and cylindrical model
    2. 1.2 Fukaya category calculations
    3. 1.3 Gradings
    4. 1.4 Homological mirror symmetry
    5. 1.5 Acknowledgements
  2. 2 Review of 3d N=4 Coulomb branches
    1. 2.1 Definition
    2. 2.2 Abelian case and “pure gauge theory” case
    3. 2.3 Localization, abelianization, and formulas for monopole operators
    4. 2.4 Root and Matter divisors
    5. 2.5 Deformation
  3. 3 Cylindrical model for disks in Coulomb branches
    1. 3.1 Quiver Gauge Theory
    2. 3.2 Examples
    3. 3.3 Deformed Coulomb branch and formulas for monopole operators
    4. 3.4 Some geometry of deformed Coulomb branch
    5. 3.5 Proof of Theorem 1.5
  4. 4 KLRW category
  5. 5 Floer homologies in $(\mathbf{T}_{O}\times\mathbf{T}^{\vee})/\mathbf{W}$
    1. 5.1 Fukaya category setup
    2. 5.2 Floer theory in $\mathbb{C}^{*},(\mathbb{C}^{*})^{2}$ , $(\mathbb{C}^{*})^{2}\backslash\Delta$ and $\operatorname{Sym}^{2}(\mathbb{C}^{*})\backslash\Delta$
    3. 5.3 Wrapping and Intersections
    4. 5.4 Composition
    5. 5.5 Comparison of Liouville structure
  6. 6 Gradings
  7. 7 Floer homologies with only root divisor removed
    1. 7.1 Hom spaces and taut diagrams
    2. 7.2 KLRW relations from disk counts, I: matter divisor
      1. 7.2.1 Bigon of the same node
      2. 7.2.2 Bigon between red and black
      3. 7.2.3 Bigon of black strands of adjacent nodes circle nodes
    3. 7.3 Proof of Theorem 7.1
  8. 8 Floer homologies in the multiplicative Coulomb branch
    1. 8.1 Crossing number filtration
    2. 8.2 Continuation maps and Robust diagrams
    3. 8.3 KLRW relations from disk counts II: root divisor
    4. 8.4 Proof of Main theorem 1.7

##  1 Introduction

Let $\Gamma$ be a directed graph with no multiple edges, or loops. Pick a
natural number $d_{i}\geq 0$ for each vertex $i$ of $\Gamma$, and a collection
$F$ of points on a line, labeled by vertices of $\Gamma$ and possibly empty.
In diagrams, we depict the points of $F$ as red. The KLRW category
$\mathcal{C}_{\Gamma,\vec{d},F}$ is defined as follows :

  * •

Objects are collections of points on a line $\mathbb{R}$, all distinct from
the points of $F$, with $d_{i}$ points labeled by the vertex $i$ of $\Gamma$.
In diagrams, we depict these points as black.

  * •

Morphism spaces are generated by strand diagrams in the plane
$\mathbb{R}\times[0,1]$, with no either horizontal or vertical tangencies, or
non-generic intersections. The strands may be decorated by dots.

  * •

Composition $D_{1}\circ D_{2}$ is given by stacking the diagram $D_{1}$ on top
of $D_{2}$.

  * •

Diagrams are considered up to isotopy and satisfy relations in Figure 1.

This category is relevant for categorification of
$U_{\mathfrak{q}}({\mathfrak{g}})$, where $\mathfrak{g}$ has Dynkin diagram
$\Gamma$ [27, 28, 36, 41, 26]. In particular, when $\Gamma$ is of ADE type, it
leads to corresponding invariants of links in ${\mathbb{R}}^{3}$ [42].

We will be interested in a cylindrical variant
$\mathcal{C}^{cyl}_{\Gamma,\vec{d},F}$, which is the analogous structure with
the line $\mathbb{R}$ replaced by a circle, and the plane correspondingly
replaced by a cylinder. $\mathcal{C}^{cyl}_{\Gamma,\vec{d},F}$ leads to
homological invariants of links in ${\mathbb{R}}^{2}\times S^{1}$ [44].

We will (re)discover the cylindrical KLRW categories as Fukaya-Seidel
categories of multiplicative Coulomb branches of quiver gauge theories whose
quiver is $\Gamma$. This identification, along with the specific form of the
superpotential, was proposed in [2, 3, 4].

$(i)$$(i)$$=0$ (a) bigon

$(j)$$(i)$$=\eta$$(j)$$(i)$$-\eta$$(j)$$(i)$ (b) bigon with neighbor
$(j)\to(i)$

$[i]$$(i)$$=\;\eta$$[i]$$(i)$ (c) bigon with red

$(i)$$(i)$$(j)$$-$$(i)$$(i)$$(j)$$=\eta\hbar$$(i)$$(i)$$(j)$ (d) braid with
neighbour $(j)\to(i)$.

$(i)$$(i)$$[i]$$-$$(i)$$(i)$$[i]$$=\eta\hbar$$(i)$$(i)$$[i]$ (e) braid with
red

$(i)$$(i)$$-$$(i)$$(i)$$(i)$$=\;\;\hbar$$(i)$$(i)$ (f) dot-pass-crossing

$(i)$$(i)$$-$$(i)$$(i)$$(i)$$=\;\;\hbar$$(i)$$(i)$ (g) another dot-pass-
crossing

Figure 1: Nontrivial KLRW relations. Exchanging $i$ and $j$ in diagrams (b)
and (d), i.e. if we have an arrow $(j)\leftarrow(i)$, the right-hand-side gets
an extra (-1) factor.

From the data $(\Gamma,{\vec{d}},F)$, we consider the following group and
vector spaces.

$\mathbf{G}:=\prod_{i\in\Gamma}GL(\mathbb{C}^{d_{i}}),\quad\mathbf{N}:=\bigoplus_{i\to
j}\operatorname{Hom}(\mathbb{C}^{d_{i}},\mathbb{C}^{d_{j}}),\quad\mathbf{N}_{F}:=\bigoplus_{i}\operatorname{Hom}(\mathbb{C}^{m_{i}},\mathbb{C}^{d_{i}}),$
(1)

Here, the integers $m_{i}$ are the number of points in $F$ labeled by the
$i$’th vertex of $\Gamma$.

The space $\mathbf{N}$ parameterizes representations of the quiver $\Gamma$
with dimension vector $\vec{d}$; automorphisms of such representations are
given by the action of $\mathbf{G}$. Meanwhile
$\mathbf{N}\oplus\mathbf{N}_{F}$ parameterizes representations of a framed
quiver $\overline{\Gamma}$, with dimension vector provided by $\vec{d}$ and
$F$.

In general, from any reductive Lie group $\mathbf{G}$ and representation
$\mathbf{N}$, the mathematical work [10] defines spaces
$\mathcal{M}^{+}(\mathbf{G},\mathbf{N})$ and
$\mathcal{M}^{\times}(\mathbf{G},\mathbf{N})$, termed the ‘additive’ and
‘multiplicative’ Coulomb branches, and expected to capture the appropriate
physical moduli of vacua of certain corresponding 3d $N=4$ gauge theories. The
additive Coulomb branches $\mathcal{M}^{+}(\mathbf{G},\mathbf{N})$ include
many well studied spaces in representation theory, such as cotangent bundles
of flag varieties, slices in affine Grassmanians, and Hilbert schemes of
points on $A_{n}$ surfaces. These affine algebraic varieties carry holomorphic
symplectic structures on their smooth loci and maps
$\mathcal{M}^{+}(\mathbf{G},\mathbf{N})\to\mathfrak{t}/W$, and
$\mathcal{M}^{\times}(\mathbf{G},\mathbf{N})\to\mathbf{T}/W$ and where
$\mathbf{T}\subset\mathbf{G}$ is a maximal torus, ${\bf t}$ is its Lie
algebra, and $W$ is the Weyl group of $\mathbf{G}$.

We will write
$\mathcal{M}^{\times}(\Gamma,\vec{d}):=\mathcal{M}^{\times}(\mathbf{G},\mathbf{N})$.
We will also use the Coulomb branch of the framed quiver
$\mathcal{M}^{\times}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})$ in order to
describe a certain function
${\mathcal{W}}_{F}:\mathcal{M}^{\times}(\Gamma,\vec{d})\rightarrow{\mathbb{C}}$,
along with some special coordinates on $\mathcal{M}^{\times}(\Gamma,\vec{d})$.

The purpose of the present article is to construct an embedding of
$\mathcal{C}^{cyl}_{\Gamma,\vec{d},F}$ in the Fukaya-Seidel category
$Fuk(\mathcal{M}^{\times}(\Gamma,\vec{d}),{\mathcal{W}}_{F})$, as conjectured
in [3].

###  1.1 Geometry of Coulomb branches for quiver gauge theory

####  1.1.1 Coordinates and superpotential

We fix a maximal torus ${\bf T}\subset\mathbf{G}$ along with an isomorphism
$\mathbf{T}\cong\prod_{i}\prod_{\alpha=1}^{d_{i}}\mathbb{C}^{*}$. For
$y\in\mathbf{T}$, we write its coordinates as $y_{i,\alpha}\in\mathbb{C}^{*}$.

######  Definition 1.1.

Fix $a\in\mathbf{T}_{F}=\prod_{i}\prod_{\alpha=1}^{m_{i}}\mathbb{C}^{*}$ so
that the coordinate entries $a_{i,\alpha}\in\mathbb{C}^{*}$ have distinct
arguments. We write: $\mathbf{T}_{O}\subset\mathbf{T}$ for the complement of
the following hyperplanes:

  1. 1.

$H_{R,i}$ for the locus where some of the $y_{i,\alpha}$ coincide, and
$H_{R}:=\bigcup H_{R,i}$.

  2. 2.

$H_{I,j\to i}$, if there is an arrow $j\to i$, for the locus where some
$y_{i,\alpha}$ coincides with some $y_{j,\beta}$

  3. 3.

$H_{F,i}$ for the locus where some $y_{i,\alpha}$ coincides with some
$a_{i,\beta}$.

The locus $\mathbf{T}_{O}$ is evidently $W$ invariant, and descends to the
quotient. We define by fiber products:

${\widetilde{\mathcal{M}}^{\times}(\Gamma,\vec{d})}$${\mathcal{M}^{\times}(\Gamma,\vec{d})}$
${\widetilde{\mathcal{M}}^{\times}(\Gamma,\vec{d})_{O}}$${\mathcal{M}^{\times}(\Gamma,\vec{d})_{O}}$${\mathbf{T}}$${\mathbf{T}/W}$
${\mathbf{T}_{O}}$${\mathbf{T}_{O}/W}$

We will explicitly construct coordinates trivializing the fibration, over the
torus $\mathbf{T}_{O}$. These coordinates are chosen so that our desired
superpotential function has a simple expression:

######  Theorem 1.2.

For $a\in\mathbf{T}_{F}$ as in Definition 1.1, there is a $W$-equivariant
isomorphism

$(u,y):\widetilde{\mathcal{M}}^{\times}(\Gamma,\vec{d})_{O}\to\mathbf{T}^{\vee}\times\mathbf{T}_{O}$

trivializing the fibration
$y:\widetilde{\mathcal{M}}^{\times}(\Gamma,\vec{d})_{O}\to\mathbf{T}_{O}$,
such that

$\mathcal{W}:=\sum_{i,\alpha}u_{i,\alpha}$

extends to a $W$-invariant regular function on
$\widetilde{\mathcal{M}}^{\times}(\Gamma,\vec{d})$.

###### Proof.

As we recall in Section 2.3, it follows immediately from the results of [11]
that there is such a trivialization $(x,y)$ for coordinates ‘$x$’ coming via a
comparison to the Coulomb branch for the Cartan. The $\mathcal{W}$ is a
certain ‘monopole operator’ (in particular a global function) whose expression
in the $x$ variables was calculated in [15] to be a sum of terms, each of
which is a product of one of the ‘$x$’ variables and a term invertible away
from the loci in Def. 1.1; see Prop. 3.7 below. The ‘$u$’ variables are
defined (Def. 3.8) to be the terms in this sum. The fact that $(u,y)$ gives an
isomorphism then follows immediately; it is stated in Proposition 3.6 and
Proposition 3.10 below. ∎

Note that $u$ and hence $\mathcal{W}$ depend on $a$, as does the subset
$\mathbf{T}_{O}\subset\mathbf{T}$, and thus also
$\mathcal{M}^{\times}(\Gamma,\vec{d})_{O}\subset\mathcal{M}^{\times}(\Gamma,\vec{d})$.
However, $\mathcal{M}^{\times}(\Gamma,\vec{d})$ itself does not depend on $a$.

The function $\mathcal{W}$ is the single valued part of the ‘upstairs
superpotential’ introduced in [3, Appendix B.2.3], as one can see by comparing
Def. 3.8 with that reference. The potential, and the space it is defined on,
are distinct relatives of that in [16, 9].

####  1.1.2 Holomorphic curves and cylindrical model

We will study holomorphic curves $C\to\mathcal{M}^{\times}(\Gamma,\vec{d})$ by
first considering their composition with the projection to
$\mathbf{T}/W=\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$.

We borrow the ‘cylindrical model’ from the Heegaard Floer literature. This is
the following assertion:

######  Lemma 1.3.

[29] For any target curve $X$, there is a bijection (a) between maps $C\to
Sym^{d}X$ transverse to the double point locus and (b) maps $S\to C\times X$,
where $S\to C$ is a $d:1$ cover with simple ramification.

We correspondingly identify: (a) maps
$C\to\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$ which are transverse to the ’big
diagonal divisor’ with (b) tuples of maps $S_{i}\to C\times\mathbb{C}^{*}_{y}$
such that $S_{i}\to C$ has simple ramification. We write $S=\coprod S_{i}$,
but always understand the components of $S$ to be labelled.

Now will classify lifts of a given
$C\to\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$ (always assumed transverse to the
double point locus) to $C\to\mathcal{M}^{\times}(\Gamma,\vec{d})$, in terms of
the corresponding maps $S\to C\times\mathbb{C}^{*}_{y}$.

Let us write $C_{O}=C\times_{\mathbf{T}/W}\mathbf{T}_{O}/W$, and
correspondingly $S_{O}=S\times_{C}C_{O}$. Since in $\mathbf{T}_{O}$ we have in
particular removed the double point locus, the map $S_{O}\to C_{O}$ is étale.
Since the $W$ action on $\mathbf{T}_{O}$ is free, the following is immediate:

######  Lemma 1.4.

Fix a map $C_{O}\to\mathbf{T}_{O}/W$ and consider the corresponding $S_{O}\to
C_{O}\times\mathbb{C}^{*}_{y}$. Then there is a canonical bijection between:
(a) lifts to
$C_{O}\to\mathcal{M}^{\times}(\Gamma,\vec{d})_{O}=(\mathbf{T}^{\vee}\times\mathbf{T}_{O})/W$
and (b) tuples of maps $S_{O}\to\mathbb{C}^{*}_{u}$, where the subscript $u$
in $\mathbf{T}_{u}^{\vee}$

Consider now the case of some
$C\to\mathbf{T}/W=\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$ transverse to
$(\mathbf{T}\setminus\mathbf{T}_{O})/W$. We write $C_{(i)}$, $C_{[i]\to(i)}$,
and $C_{(i)\to(j)}$ for preimages in $C$ of the corresponding hyperplanes. (By
assumption of transversality, these sets of points have multiplicity one and
are all distinct.)

Points in the first two classes each correspond to a single witness in $S$, we
write the sets of such points as $S_{(i)}$ and $S_{[i]\to(i)}$. Points in
$C_{(i)\to(j)}$ correspond to certain pairs of points $(s_{i},s_{j})$ with the
same image in $C$; we denote the set of such pairs $S_{(i)\to(j)}\subset
S\times_{C}S$.

We establish the following result.

######  Theorem 1.5.

(3.11) Fix a map $C\to\mathbf{T}/W=\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$
transverse to $(\mathbf{T}\setminus\mathbf{T}_{O})/W$, and consider the
corresponding $S\to C\times\mathbb{C}^{*}_{y}$. Lifts to
$C\to\mathcal{M}^{\times}(\Gamma,\vec{d})$ are in bijection with maps
$\Phi_{u}:S\to\mathbb{P}^{1}_{u}$ satisfying the following conditions:

  1. 0.

$\Phi_{u}(S_{O})\subset\mathbb{C}^{*}_{u}$.

  2. 1.

$\Phi_{u}$ has simple poles over points in $S_{(i)}$.

  3. 2.

$\Phi_{u}$ has simple zeros over $S_{[i]\to(i)}$.

  4. 3.

For $(s_{i},s_{j})\in S\times_{C}S$, the function
$\Phi_{u}(s_{i})\Phi_{u}(s_{j})$ has a simple zero at $(s_{i},s_{j})\in
S_{(i)\to(j)}$.

The basic point in the proof is that, according to [10, Section 6] the
geometry of the Coulomb branch over the various hyperplanes are given by a
trivial factor times a Coulomb branch with total dimension $2$. The ‘$u$’
variables transform conveniently along this reduction, and so the theorem
reduces to the case of such quivers (there are three), where it is elementary.

######  Remark 1.6.

Our approach is informed and inspired by some recent uses of cylindrical model
approaches to compute (or define) Fukaya categories of symmetric powers and
related spaces of higher dimensional symplectic varieties [14, 22, 30].

###  1.2 Fukaya category calculations

The trivialization 1.2 lets us explicitly describe certain special Lagrangians
${\mathcal{T}}_{\theta}$ in $\mathcal{M}^{\times}(\Gamma,\vec{d})$. We write
$\mathbf{T}=\mathbf{R}\times\mathbf{\Theta}$, for the coordinate-wise
splitting of the corresponding complex numbers into modulus and argument,
$y=e^{\rho}e^{i\theta}$, and similarly for the dual torus.

We define

${\bf\Theta}_{O}:=\\{\theta\in{\bf\Theta}\mid{\bf R}\times\theta\subset{\bf
T}_{O}\\}.$

Then ${\bf W}$ acts on ${\bf\Theta}_{O}$ freely.

For any $\theta\in{\bf\Theta}_{O}$, we define

$\widetilde{T}_{\theta}:=\mathbf{R}\times\theta\subset\mathbf{T}_{O}\qquad\qquad\widetilde{\mathcal{T}}_{\theta}:=(\mathbf{R}^{\vee}\times
0)\times(\mathbf{R}\times\theta)\subset\mathbf{T}^{\vee}\times\mathbf{T}_{O}$

Let $T_{\theta}\subset{\bf T}_{O}/W$ and $\mathcal{T}_{\theta}\subset({\bf
T}^{\vee}\times{\bf T}_{O})/W$ to be the images of these Lagrangians under the
free ${\bf W}$ quotient. We use the same name of the Lagrangians in the
(unpunctured) upstairs and downstairs spaces $T_{\theta}\in{\bf T}/{\bf W}$
and $\mathcal{T}_{\theta}\in\mathcal{M}^{\times}(\Gamma,\vec{d})$; these
remain closed submanifolds. Note that the $\mathcal{T}_{\theta}$ are
Lagrangians for the natural product Kähler form on
$\mathbf{T}^{\vee}\times\mathbf{T}_{O}$, not for the holomorphic symplectic
form.

We can record all the data relevant to ${\mathcal{T}}_{\theta}$ in terms of
configurations of red and black points on a circle. Write $[[a]]$ for a circle
decorated by red points, with one red point labelled $i$ at each
$\arg(a_{i,\alpha})$. Write $((\theta))$ for a circle which is decorated by
black points in addition to red, with one black point labeled $i$ at each
$\theta_{i,\alpha}$.

There is an evident correspondence of objects:

$\mathcal{T}_{\theta}\in
Fuk(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})\longleftrightarrow((\theta))\in\mathcal{C}.$

We will prove:

######  Theorem 1.7.

Assume $\mathcal{M}^{\times}(\Gamma,\vec{d})$ is smooth. (Smoothness is known
when $\Gamma$ is of ADE type [11].) There is an embedding

$\displaystyle KLRW$ $\displaystyle\hookrightarrow$
$\displaystyle\operatorname{Fuk}(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})$
$\displaystyle((\theta))$ $\displaystyle\mapsto$
$\displaystyle\mathcal{T}_{\theta}$

The proof proceeds by using the cylindrical model of Theorem 1.5 to reduce all
calculations to disk counts in a cylinder, which we then carry out explicitly.
The proof occupies the entirety of the final four sections of this article.

######  Conjecture 1.8.

The collection $\mathcal{T}_{\theta}$ generate
$\operatorname{Fuk}(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})$.

######  Remark 1.9.

The reader might wonder whether the problem of generation might be approached
by using the well known fact that, for a Lefschetz fibration in the sense of
Seidel, the thimbles are known to generate the Fukaya-Seidel category (see
e.g. [37, 8, 20]). We caution that this result does not apply here because
$\mathcal{W}$ is not a ‘fibration at infinity’ in the fibers. In fact, our
$\mathcal{W}$ has no critical points whatsoever, similar to the situation
$\mathcal{W}:\mathbb{C}^{*}\to\mathbb{C}$ given by the evident embedding. (The
analogue of $\mathcal{T}_{\theta}$ in that example is $\mathbb{R}e^{i\theta}$,
which does generate.)

There is a (multi-valued) ‘equivariant’ potential $\mathcal{W}_{eq}$ for
$\mathcal{M}^{\times}(\Gamma,\vec{d})$ (see (3) below), which does have
critical points, and whose thimbles include some, but not generally all, of
the $\mathcal{T}_{\theta}$. However, at present, there are no available
results guaranteeing generation by thimbles of such multi-valued
superpotentials.

###  1.3 Gradings

There is a long history of attempts to find a Floer theoretic realization of
categorified quantum group link invariants, pioneered by Seidel and Smith [38]
and pursued in [1, 31, 32, 30] and elsewhere. One virtue of such descriptions
was that the categorical braid group action had a simple geometric
description. However, the corresponding categories were invariably singly
graded: they had the homological grading, but lacked the grading associated to
the ‘$q$’ variable of the Jones polynomial.

In general, if $Y$ is affine and $H$ is a divisor, then the function cutting
out $H$ determines a map $Y\setminus H\to\mathbb{C}^{*}$. Such a map in
particular determines a $H^{1}$ class (by pulling back from $\mathbb{C}^{*}$).
We may use this $H^{1}$-class to define a $\mathbb{Z}$-gradings on the hom
space $End(L)$, where $L$ is any contractible Lagrangian in $Y\backslash H$.

In our setup, we can obtain the desired gradings in this manner. The relevant
divisor arises as follows. Our space
$\mathcal{M}^{\times}(\mathbf{G},\mathbf{N})$ arises from the multiplicative
Coulomb branch
$\mathcal{M}^{\times}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})$ by removing
the divisor $\\{r_{\vec{1},\cdots,\vec{1}}=0\\}$. The resulting
$\mathbb{Z}$-grading matches the KLRW $\mathbb{Z}$-grading, where crossing of
black strands of the same label contribute $-1$ and dot contribute $1$. See
Section 6.

Let us note that the target in [38] was shown in [31] to be a certain quiver
variety, now known to be isomorphic to
$\mathcal{M}^{+}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})$. Our space
differs both by being the multiplicative, instead of additive, version (this
is related to the fact that we find the cylindrical KLRW) and insofar as we
have deleted a divisor.

We refer to [2, 3, 4] and especially [5] for the realization in terms of
$Fuk(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})$ of the relation
between the KLRW category and the homological knot invariants. In [5] it is
explained how to explicitly compute the braid group action on in terms of
${\mathcal{T}}_{\theta}$ branes as well as link homology groups, as spaces of
morphisms between objects of the cylindrical KLRW category. We note in
particular that the cylindrical KLRW category gives rise to the (richer)
invariants of knots in $S^{1}\times\mathbb{R}^{2}$, of which knots in
$\mathbb{R}^{3}$ are a special case.

###  1.4 Homological mirror symmetry

It was conjectured in [3] that
$Fuk(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})$ is equvialent to the
category of coherent sheaves on the (resolved) additive Coulomb branch
$\mathcal{M}^{+}_{C}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})_{\chi}$ of a
3d $N=4$ theory based on the same quiver, where $\chi$ is a resolution
parameter.

For a general class of conical holomorphic symplectic varieties, including in
particular the additive Coulomb branches, Bezrukavnikov and Kaledin [7]
constructed a tilting vector bundle ${\mathcal{T}}$ [24]. For the case of
interest here, Webster computed the endomorphism algebra of this bundle in
[43, 44], and showed that it coincides with the cylindrical KLRW algebra.

Combining this result of Webster with Thm. 1.7, we have:

$Coh(\mathcal{M}^{+}_{C}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})_{\chi})=\text{KLRW-
Perf}\hookrightarrow Fuk(\mathcal{M}^{\times}(\Gamma,\vec{d}),\mathcal{W})$
(2)

Establishing surjectivity would amount to verifying Conjecture 1.8.

Let us turn to the question of how the gradings discussed in Section 1.3
interact with mirror symmetry. An ‘equivariant’ extension of $\mathcal{W}$ was
proposed in [3, Appendix B.2.3], it took the form:

$\mathcal{W}_{eq}=\mathcal{W}+\sum_{i\in\Gamma}\Lambda_{i}\log
f_{i}+\Lambda_{0}\log f_{0}.$ (3)

Let us recall the relation of equivariance and log terms in the
superpotential. It has long been known that considering torus-equivariant
Gromov-Witten invariants on one side of mirror symmetry corresponds to adding
log terms to the mirror superpotential on the other [21]. However, here we are
concerned instead with the other direction of mirror symmetry: how a torus
action in the B-model is reflected the A-model.

The prescription we follow is the following. First, delete the locus where the
argument of the logarithms in the equivariant superpotential
$\mathcal{W}_{eq}$ vanish. Then, form the pull-back square

${\widetilde{\mathcal{Y}}}$${\mathcal{Y}}$${\mathbb{C}^{r}}$${(\mathbb{C}^{*})^{r}}$$\scriptstyle{exp}$

where $r$ is the number of log terms. Now $\mathcal{W}_{eq}$ is a well-defined
function on $\widetilde{\mathcal{Y}}$, and the action by deck transformation
induces a $\mathbb{Z}^{r}$ action on the Fukaya-Seidel category of
$(\widetilde{\mathcal{Y}},\mathcal{W})$. The general expectation is that this
$\mathbb{Z}^{r}$ action should be mirror to the action by tensor product of
the characters of the mirror $(\mathbb{C}^{*})^{r}$ on the equivariant
coherent sheaves category of the mirror [40, Rem. 4.6]. (The case of toric
varieties is discussed in [39].)

In the present case, tracking the $\mathbb{Z}^{r}$ action across the mirror
symmetry (2) amounts to checking that the grading on KLRW is recovered on the
$Coh$ side via a torus action on
$\mathcal{M}^{+}_{C}(\mathbf{G},\mathbf{N}\oplus\mathbf{N}_{F})_{\chi}$. This
was established in [2, 44].

######  Remark 1.10.

In the abelian case (all $d_{i}=1$), a related mirror symmetry result (after
removing the superpotential on the A-model and replacing the B-model with the
multiplicative version) was established by Gammage, McBreen, and Webster [34,
17] by entirely different methods; namely, calculating the skeleton and
microsheaves in the sense of [35] and then appealing to [19].

######  Remark 1.11.

In the abelian setting, there is a Hamiltonian torus action on the target on
the $A$ side. Using the results of the aforementioned [17], it was shown in
[33] that ‘reduction commutes with Fukaya category’, which in turn amounted to
a mirror symmetry between the A-model on $\mathbf{T}_{O}$, and the B-model on
the core of a certain conical Lagrangian inside the original mirror dual
multiplicative Coulomb branch. We expect a similar phenomenon in general,
although on the A-side, rather than simply $\mathbf{T}_{O}$, we expect a more
general “Fukaya category with coefficients in a schober on $\mathbf{T}$”.

######  Remark 1.12.

Another mirror symmetry result in the context of Coulomb branches is Jin’s
study [23], of a Fukaya category of $\mathcal{M}^{+}(G,0)$, where the
symplectic form is taken to be the real part of the holomorphic symplectic
form $\Omega$; she showed an equivalence with
$\operatorname{Coh}(T^{\vee}/W)$. At present, we do not entirely understand
the relationship with the work in this article.

######  Remark 1.13.

Cautis and Kamnitzer [13, 12] previously studied coherent sheaves on moduli
spaces of Hecke modifications, to obtain invariants of links in
${\mathbb{R}}^{3}$. The spaces of Hecke modifications are (compact)
convolution products ${\rm Gr}^{\vec{\mu}}$ of orbits in the affine
Grassmannian ${\rm Gr}={\rm Gr}_{G_{\Gamma}}$ of the quiver group
$G_{\Gamma}$. In this setting, the desired $q$ grading also arises from a
torus action. Our space is a transversal slices inside ${\rm Gr}^{\vec{\mu}}$
to another orbit ${\rm Gr}_{\nu}$, whenever a ${\vec{\mu}}$ and $\nu$ satisfy
a certain dominance condition [10].

###  1.5 Acknowledgements

We would like to thank Mohammed Abouzaid, Ivan Smith, Ben Webster, Xin Jin for
many insightful discussion. We would like to especially thank Cheuk-Yu Mak,
Yin Tian and Tianyu Yuan for explaining the work in [30] and [22]. The main
disk counting are done with the generous help of Tianyu Yuan.

M.A. is supported, in part, by the NSF foundation grant PHY2112880 and by the
Simons Investigator Award. V.S. is supported by Novo Nordisk Foundation grant
NNF20OC0066298, Villum Fonden Villum Investigator grant 37814, and Danish
National Research Foundation grant DNRF157.

##  2 Review of 3d N=4 Coulomb branches

Let $G$ be a complex reductive group and $N$ a finite dimensional
representation. To this data, physicists associate a 3d N=4 gauge theory,
which has certain associated moduli spaces called the “Higgs branch” and the
“Coulomb branch”.

A direct mathematical definition of Coulomb branches was given in [10], who
describe them as algebraic varieties. There are two versions: additive
($\mathcal{M}^{+}(G,N)$) and multiplicative ($\mathcal{M}^{\times}(G,N)$).
Each carries a map:

$\mu^{+}:\mathcal{M}^{+}(G,N)\to\mathfrak{t}/W,\quad\mu^{\times}:\mathcal{M}^{\times}(G,N)\to
T/W.$

The main results of the present article concern Floer theory in multiplicative
Coulomb branches. The purpose of this section is to recall from the literature
various general results regarding the geometric properties of these spaces. We
focus on the multiplicative version.

###  2.1 Definition

Fix a connected complex reductive group $G$ and a finite dimensional
representation $N$ of $G$. We assume $\pi_{1}(G)$ has no torsion. Let
$T\subset B\subset G$, $B$ a Borel subgroup and $T$ a maximal torus. Let $W$
be the Weyl group. Let $X_{*}(T)=\operatorname{Hom}(\mathbb{C}^{*},T)$ be the
coweight lattice. Let
$\mathcal{K}=\mathbb{C}((t)),\mathcal{O}=\mathbb{C}[[t]]$. Let
$Gr_{G}=G(\mathcal{K})/G(\mathcal{O})$ be the affine Grassmannian. For any
$\lambda\in X_{*}(T)$, we have the $G(\mathcal{O})$ orbit
$Gr_{G}^{\lambda}=G(\mathcal{O})t^{\lambda}G(\mathcal{O})/G(\mathcal{O})\subset
Gr_{G}$.

Let $\mathbb{D}=\operatorname{Spec}\mathcal{O}$ denote the formal disk,
$\mathbb{D}^{o}=\operatorname{Spec}\mathcal{K}$ the punctured formal disk, and
$\mathbb{B}=\mathbb{D}\cup_{\mathbb{D}^{o}}\mathbb{D}$ the ‘bubble’ or the
‘raviolli’.

We have the following moduli stacks of $G$-bundles

$\operatorname{Bun}_{G}(\mathbb{D})=\text{pt}/G(\mathcal{O}),\quad\operatorname{Bun}_{G}(\mathbb{D}^{o})=\text{pt}/G(\mathcal{K}),\quad\operatorname{Bun}_{G}(\mathbb{B})=\operatorname{Bun}_{G}(\mathbb{D})\times_{\operatorname{Bun}_{G}(\mathbb{D}^{o})}\operatorname{Bun}_{G}(\mathbb{D}).$

where the map
$\operatorname{Bun}_{G}(\mathbb{D})\to\operatorname{Bun}_{G}(\mathbb{D}^{o})$
is the restriction of $G$-bundles. Let $\operatorname{Bun}_{G,N}$ denote the
moduli stack of $G$-bundles together with a section in the associated
$N$-bundle, then

$\operatorname{Bun}_{G,N}(\mathbb{D})=N(\mathcal{O})/G(\mathcal{O}),\quad\operatorname{Bun}_{G,N}(\mathbb{D}^{o})=N(\mathcal{K})/G(\mathcal{K}),\quad\operatorname{Bun}_{G,N}(\mathbb{B})=\operatorname{Bun}_{G,N}(\mathbb{D})\times_{\operatorname{Bun}_{G,N}(\mathbb{D}^{o})}\operatorname{Bun}_{G,N}(\mathbb{D}).$

Define
$\mathbb{B}_{123}:=\mathbb{D}_{1}\cup_{\mathbb{D}^{o}}\mathbb{D}_{2}\cup_{\mathbb{D}^{o}}\mathbb{D}_{3}$
and $\mathbb{B}_{ij}=\mathbb{D}_{i}\cup_{\mathbb{D}^{o}}\mathbb{D}_{j}$ for
$1=i<j=3$. We have restriction maps
$p_{ij}:\operatorname{Bun}_{G,N}(\mathbb{B}_{123})\to\operatorname{Bun}_{G,N}(\mathbb{B}_{ij})$.
In terms of these, one would expect a convolution product on

$\mathcal{A}^{\times}(G,N)\,\,\mbox{`:='}\,\,K(\operatorname{Bun}_{G,N}(\mathbb{B}))=K\left(\frac{N(\mathcal{O})}{G(\mathcal{O})}\bigtimes_{\frac{N(\mathcal{K})}{G(\mathcal{K})}}\frac{N(\mathcal{O})}{G(\mathcal{O})}\right)$

We put the quotes because the infinite nature of the spaces involved (and the
state of present technology) lead to some problems in defining the RHS
directly. Instead, [10] rewrite

$\frac{N(\mathcal{O})}{G(\mathcal{O})}\bigtimes_{\frac{N(\mathcal{K})}{G(\mathcal{K})}}\frac{N(\mathcal{O})}{G(\mathcal{O})}\cong\frac{\frac{G(\mathcal{K})\times
N(\mathcal{O})}{G(\mathcal{O})}\underset{N(\mathcal{K})}{\times}\frac{G(\mathcal{K})\times
N(\mathcal{O})}{G(\mathcal{O})}}{G(\mathcal{K})}\cong\frac{N(\mathcal{O})\underset{N(\mathcal{K})}{\times}\frac{G(\mathcal{K})\times
N(\mathcal{O})}{G(\mathcal{O})}}{G(\mathcal{O})}$

and introduce

$T_{G,N}:=\frac{G(\mathcal{K})\times N(\mathcal{O})}{G(\mathcal{O})},\quad
R_{G,N}=N(\mathcal{O})\underset{N(\mathcal{K})}{\times}T_{G,N}$ (4)

A main result of [10] is that $K^{G(\mathcal{O})}(R_{G,N})$ indeed admits a
natural structure of convolution algebra, which is moreover commutative. They
then define:

######  Definition 2.1 ([10]).

The affine multiplicative Coulomb branch algebra and spaces are defined as

$\mathcal{A}^{\times}(G,N):=K^{G(\mathcal{O})}(R_{G,N}),\quad\mathcal{M}^{\times}(G,N):=\operatorname{Spec}K^{G(\mathcal{O})}(R_{G,N})$

######  Remark 2.2.

For the discussion of equivariant K-theory and BM homology, we can replace
$G(\mathcal{O})$-equivariance with $G$-equivariance. See [6] Lemma 6.2 and the
comment above.

Using $R_{G,N}\to pt$, we get a pullback map of algebra
$K^{G(\mathcal{O})}(pt)\to K^{G(\mathcal{O})}(R_{G,N})$. Since
$\operatorname{Spec}K^{G(\mathcal{O})}(pt)=T/W$, we get a map of spaces

$\mu:\mathcal{M}^{\times}(G,N)\to T/W.$

Let $\pi_{R}:R_{G,N}\to Gr_{G}$ be the projection map. If $\lambda$ is a
minuscule coweight of $G$, then the $G(\mathcal{O})$ orbit $Gr_{G}^{\lambda}$
is closed, and the preimage
$R^{\lambda}_{G,N}:=\pi_{R}^{-1}(Gr_{G}^{\lambda})$ is closed.

######  Definition 2.3.

For any minuscule coweight $\lambda$ of $G$, then we write
$r_{\lambda}\in\mathcal{A}^{\times}(G,N)$ for the equivariant K-theory class
of the structure sheaf on $R^{\lambda}_{G,N}$. More generally, if
$\mathcal{V}$ is a $G(\mathcal{O})$-equivariant vector bundle over
$Gr^{\lambda}$, then we write
$r_{\lambda,\mathcal{V}}\in\mathcal{A}^{\times}(G,N)$ for the K-theory class
of the pullback bundle $\pi_{R}^{*}\mathcal{V}$ on $R^{\lambda}_{G,N}$.

###  2.2 Abelian case and “pure gauge theory” case

The simplest example is when $G=T=(\mathbb{C}^{*})^{k}$ and $N=0$. In this
case, we get

$\mathcal{M}^{\times}(T,0)=T^{\vee}\times T$

and the map $\mu$ is simply the projection $\mu:T^{\vee}\times T\to T$.

######  Proposition 2.4.

[10, Prop 4.1] Let $T$ be a torus, and $N$ be its representation with weights
$\xi_{1},\cdots,\xi_{n}$. For any two integers $k,l$, let

$d(k,l)=\begin{cases}0&\text{if $k$ and $l$ have the same sign}\\\
\min(|k|,|l|)&\text{otherwise}\end{cases}$

and for any $\xi\in X^{*}(T)$, and $\lambda,\mu\in X_{*}(T)$, let

$d_{\xi}(\lambda,\mu):=d(\langle\xi,\lambda\rangle,\langle\xi,\lambda\rangle).$

The multiplicative Coulomb branch algebra $\mathcal{A}^{\times}(T,N)$ is
generated as an algebra over $\mathcal{O}(T)$ with generators
$\\{r_{\lambda}\\}_{\lambda\in X_{*}(T)}$, satisfying relations

$r_{\lambda}r_{\mu}=r_{\lambda+\mu}\cdot\prod_{i=1}^{n}(1-y^{-\xi_{i}})^{d_{\xi_{i}}(\lambda,\mu)}$

where for $\xi\in X^{*}(T)$ we denote $y^{\xi}:T\to\mathbb{C}^{*}$ as the
character function on $T$.

In general, when $N=0$ (‘pure gauge theory’), there is an explicit description
of the Coulomb branch:

######  Proposition 2.5.

[6, Theorem 2.15] Let $G$ be a complex reductive group, $R\subset X^{*}(T)$
the roots of $G$, $W$ the Weyl group, then:

$K^{G(\mathcal{O})}_{*}(Gr_{G})\cong\mathbb{C}[T^{\vee}\times
T,\\{\frac{e^{\alpha^{\vee}}-1}{e^{\alpha}-1}\\}_{\alpha\in R}]^{W}.$

######  Example 2.6.

If $G=GL(d)$, we have

$K^{G(\mathcal{O})}(Gr_{G})\cong\mathbb{C}[x_{1}^{\pm},\cdots,x_{d}^{\pm},y_{1}^{\pm},\cdots,y_{d}^{\pm},\\{\frac{x_{i}/x_{j}-1}{y_{i}/y_{j}-1}\\}_{i\neq
j}]^{S_{d}}$

where $x_{i}^{\pm}$ means $x_{i}$ is a $\mathbb{C}^{*}$ variable.

###  2.3 Localization, abelianization, and formulas for monopole operators

We recall from [11] an embedding of the coordinate ring of
$\mathcal{M}^{\times}(G,N)$ into a localization of the coordinate ring of
$\mathcal{M}^{\times}(T,0)$.

Consider the closed embeddings (recall the definition of $R_{G,N},T_{G,N}$
from Eq. (4)),

$\iota:R_{T,N}\hookrightarrow R_{G,N}$ $h:R_{T,N}\hookrightarrow T_{T,N},$

and the inclusion of the zero-section

$z:Gr_{T}\hookrightarrow T_{T,N}.$

Consider the pushforward in K-theory

$h_{*}:K^{T}(R_{T,N})\to K^{T}(T_{T,N}),\quad\iota_{*}:K^{T}(R_{T,N})\to
K^{T}(R_{G,N}).$

An explicit calculation in the abelian gauge theory case (see section 4(vi) of
[10]) shows $h_{*}$ is injective. Since $T_{T,N}$ is a vector bundle over
$Gr_{T}$, we have an isomorphism

$z^{*}:K^{T}(T_{T,N})\xrightarrow{\sim}K^{T}(Gr_{T})$

Note $K^{T}(R_{T,N})$ and $K^{T}(R_{G,N})$ are modules over $K^{T}(pt)=O(T)$,
and for any root $\alpha$ of $G$ we have root hyperplane
$H_{\alpha}=\\{e^{\alpha}=1\\}$ in $T$. Let
$O(T)_{loc}=O(T\backslash\cup_{\alpha}H_{\alpha})$ denote the coordinate ring
of $T$ removing all the root hyperplanes, and let $K^{T}(R_{T,N})_{loc}$ and
$K^{T}(R_{G,N})_{loc}$ denote the corresponding localization as modules. After
such localization $\iota_{*}$ is an isomorphism:

$(\iota_{*})_{loc}:K^{T}(R_{T,N})_{loc}\xrightarrow{\sim}K^{T}(R_{G,N})_{loc}.$

Combined, this determines the desired morphism

$z^{*}h_{*}(\iota_{*})_{loc}^{-1}:K^{T}(R_{G,N})_{loc}\hookrightarrow
K^{T}(Gr_{T})_{loc}$ (5)

which restricts to the subring $\mathcal{O}(\mathcal{M}^{\times}(G,N))\cong
K^{T}(R_{G,N})^{W}\hookrightarrow K^{T}(R_{G,N})_{loc}$ to get the
‘abelianization’ map

$ab:=z^{*}h_{*}(\iota_{*})_{loc}^{-1}\quad:\mathcal{O}(\mathcal{M}^{\times}(G,N))\hookrightarrow\mathcal{O}(\mathcal{M}^{\times}(T,0))_{loc}.$

We introduce the following notation for writing in abelianized coordinates:

######  Definition 2.7.

For any $\lambda\in X_{*}(T)$, let $x^{\lambda}$ denote the corresponding
element in $\mathbb{C}[X_{*}(T)]=\mathcal{O}(T^{\vee})$. For any $\xi\in
X^{*}(T)$, let $y^{\xi}$ denote the element in
$\mathbb{C}[X^{*}(T)]=\mathcal{O}(T)$.

Then we have

$\mathcal{O}(\mathcal{M}^{\times}(T,0))=\mathcal{O}(T\times
T^{\vee}),\quad\mathcal{O}(\mathcal{M}^{\times}(T,0))_{loc}=\mathcal{O}(T\times
T^{\vee})[\\{(y^{\alpha}-1)^{-1}\mid\alpha\in R\\}]$

where $R\subset X^{*}(T)$ is the set of roots in $G$.

We turn to recall some formulas for the monopole operators (Def. 2.3).

######  Proposition 2.8.

[11, Proposition A.2] If $\lambda$ is a minuscule coweight of $G$, and
$r_{\lambda}\in\mathcal{O}(\mathcal{M}^{\times}(G,N))$ is the K-theory class
of the structure sheaf on $R^{\lambda}_{G,N}$, then

$ab(r_{\lambda})=\sum_{\lambda^{\prime}\in
W\lambda}x^{\lambda^{\prime}}\cdot\frac{\chi_{T}(t^{\lambda^{\prime}}N(\mathcal{O})/t^{\lambda^{\prime}}N(\mathcal{O})\cap
N(\mathcal{O}))}{\chi_{T}(T_{\lambda^{\prime}}Gr^{\lambda})}$ (6)

where $W\lambda$ is the $W$-orbit of $\lambda$,
$T_{\lambda^{\prime}}Gr^{\lambda}$ is the tangent space at the $T$-fixed point
$t^{\lambda^{\prime}}$ in $Gr^{\lambda}$, and $\chi_{T}$ is the map

$\chi_{T}:Rep(T)\to K(Rep(T)),\quad V\mapsto[\wedge^{*}(V^{\vee}[1])],\text{
i.e.}\quad\bigoplus_{i=1}^{m}\mathbb{C}_{\xi_{i}}\mapsto\prod_{i=1}^{m}(1-y^{-\xi_{i}}).$

More generally, if $\mathcal{V}$ is a $G(\mathcal{O})$-equivariant vector
bundle over $Gr^{\lambda}$, and $r_{\lambda,\mathcal{V}}$ is the K-theory
class of the pullback of $\mathcal{V}$ to $R^{\lambda}_{G,N}$, then

$ab(r_{\lambda,\mathcal{V}})=\sum_{\lambda^{\prime}\in
W\lambda}x^{\lambda^{\prime}}\cdot[\mathcal{V}|_{t^{\lambda^{\prime}}}]\cdot\frac{\chi_{T}(t^{\lambda^{\prime}}N(\mathcal{O})/t^{\lambda^{\prime}}N(\mathcal{O})\cap
N(\mathcal{O}))}{\chi_{T}(T_{\lambda^{\prime}}Gr^{\lambda})}.$ (7)

Consider the pure gauge theory of $\mathcal{M}^{\times}(GL_{d},0)$. [15] gives
the abelianized coordinate for some monopole operators as following.

For $r=1,\cdots,d-1$, let $\varpi_{r}$ be the $r$-th fundamental coweight,
i.e. $\varpi_{r}=(1,\cdots,1,0,\cdots,0)$ where $1$ appears $r$ times. Let
$w_{0}\in S_{d}$ be the total order reversing element, i.e., the longest word
in the Weyl group $W$. Then $-w_{0}\varpi_{r}=(0,\cdots,0,-1,\cdots,-1)$ where
$-1$ appears $r$ times. Let $L_{0}=(\mathbb{C}[[t]])^{d}$ be the standard
lattice, then we have

$Gr^{\varpi_{r}}=\\{tL_{0}\subset L\subset L_{0}\mid\dim(L_{0}/L)=r.\\}$

and

$Gr^{-w_{0}\varpi_{r}}=\\{L_{0}\subset L\subset
t^{-1}L_{0}\mid\dim(L/L_{0})=r.\\}$

Then $Gr^{\varpi_{r}}$ parametrizes a codimension $r$ subspace in an
$d$-dimensional space $L_{0}/tL_{0}$, and $Gr^{-w_{0}\varpi_{r}}$ parametrizes
a dimension $r$ subspace in an $d$-dimensional space $t^{-1}L_{0}/L_{0}$.

######  Proposition 2.9.

[15] Let $Q_{r}$ denote the tautological rank $r$ quotient vector bundle
$L_{0}/L$ on $Gr^{\varpi_{r}}$, and $S_{r}$ denote the tautological rank $r$
vector sub-bundle $L/L_{0}$ on $Gr^{-w_{0}\varpi_{r}}$. For $p=0,\cdots,r$,
let $\wedge^{p}(Q_{r})$ be the $p$-th exterior power of $Q_{r}$ (rank $r$
vector bundle), and let $e_{p}(x_{1},\cdots,x_{r})$ be the $p$-th elementary
symmmetric function in $r$ variables. Then, we have abelianization of K-theory
classes:

$ab([\wedge^{p}(Q_{r})\otimes\mathcal{O}_{Gr^{\varpi_{r}}}])=\sum_{J\subset\\{1,\cdots,d\\},\\#J=r}\frac{e_{p}(\\{y_{j}\\}_{j\in
J})}{\prod_{\alpha\in J,\beta\notin
J}(1-y_{\beta}/y_{\alpha})}\cdot\prod_{\alpha\in J}x_{\alpha}$ (8)
$ab([\wedge^{p}(S_{r})\otimes\mathcal{O}_{Gr^{-w_{0}\varpi_{r}}}])=\sum_{J\subset\\{1,\cdots,d\\},\\#J=r}\frac{e_{p}(\\{y_{j}\\}_{j\in
J})}{\prod_{\alpha\in J,\beta\notin
J}(1-y_{\alpha}/y_{\beta})}\cdot\prod_{\alpha\in J}x^{-1}_{\alpha}.$ (9)

We also recall a result about changing the representation $N$ to a
subrepresentation.

######  Proposition 2.10.

[10, Section 4(vi)] If $G=T$ is a torus and $N^{\prime}\subset N$ is a sub-
representation where $N/N^{\prime}=\oplus_{i}\mathbb{C}_{\xi_{i}}$, then we
have canonical morphism (not necessarily an embedding)

$q:\mathcal{M}^{\times}(T,N^{\prime})\to\mathcal{M}^{\times}(T,N).$

If we use $r_{\lambda}$ for the monopole operator on
$\mathcal{M}^{\times}(T,N)$ and $r_{\lambda}^{\prime}$ for
$\mathcal{M}^{\times}(T,N^{\prime})$, then

$q^{*}(r_{\lambda})=r^{\prime}_{\lambda}\prod_{i=1}^{m}(1-y^{-\xi_{i}}))^{\max(0,-\langle\xi_{i},\lambda\rangle)}.$
(10)

######  Proposition 2.11.

[25, Theorem 2.9] Let $\mathcal{A}(G,N)$ denote the (additive or
multiplicative) Coulomb branch algebra. Let $\xi:\mathbb{C}^{*}\to T$ be a
central cocharacter $G$. Assume that the representation $N$ restricted to this
$\mathbb{C}^{*}$ has only non-negative weights, and let $N^{\xi}$ denote the
fixed subspace of the $\mathbb{C}^{*}$-action. Then the natural inclusion
$\mathcal{A}(G,N)\hookrightarrow\mathcal{A}(G,N^{\xi})$ induces an isomorphism

$\mathcal{A}(G,N)[r_{\xi}^{-1}]\xrightarrow{\sim}\mathcal{A}(G,N^{\xi}).$

###  2.4 Root and Matter divisors

To study the geometry of $\mathcal{M}^{\times}(G,N)$, we can study the fibers
of $\mu:\mathcal{M}^{\times}(G,N)\to T/W$. First, we shall unfold the base
$T/W$ back to $T$ and base change $\mathcal{M}^{\times}(G,N)$.

######  Lemma 2.12.

[10, Lemma 5.3]. Let
$\widetilde{\mathcal{M}}^{\times}(G,N):=\operatorname{Spec}K^{T}(R_{G,N}).$
Then there is a pullback square:

${\widetilde{\mathcal{M}}^{\times}(G,N)}$${\mathcal{M}^{\times}(G,N)}$${T}$${T/W}$$\scriptstyle{\widetilde{\mu}}$$\scriptstyle{\mu}$
(11)

######  Definition 2.13.

There are two collections of hyperplanes (i.e. codimension-1 subtori) in $T$,

  1. 1.

If $\alpha\in X^{*}(T)$ is a root of $G$, we define

$\widetilde{H}_{\alpha}=\\{y^{\alpha}=1\\}\subset T$

as the downstairs root hyperplane for root $\alpha$.

  2. 2.

If $w\in X^{*}(T)$ is a weight of $N$ under action of $T$, then we define

$\widetilde{H}_{w}=\\{y^{w}=1\\}\subset T$

as the downstairs matter hyperplane for weight $w$.

For $\bullet=\alpha$ or $w$, let
$\widetilde{\mu}^{-1}(\widetilde{H}_{\bullet})$ be the unfolded upstairs (root
or matter) hyperplane. Let the downstairs root or matter divisor
$H_{\bullet}\subset T/W$ be the $W$ quotient image of
$\widetilde{H}_{\bullet}$, and let the upstairs root or matter divisors be
$\mu^{-1}(H_{\bullet})$.

Now we discuss the geometry near certain nice part of the divisor. We first
introduce the following notation.

######  Definition 2.14.

Let $X$ be a smooth variety and let $D_{1},\cdots,D_{n}$ be a collection of
divisor that pairwise intersects transversely in $X$. Let $D_{i}^{sing}$ be
the singularity loci of $D_{i}$, and $D_{i}^{sm}=D_{i}\backslash
D_{i}^{sing}$. We define the ’punctured’ divisor $D_{i}^{o}$ and its open
neighborhood $U(D_{i}^{o})$

$D_{i}^{o}=D_{i}\backslash(D_{i}^{sing}\cup\cup_{j\neq i}D_{j}),\quad
U(D_{i}^{o})=X\backslash(D_{i}^{sing}\cup\cup_{j\neq i}D_{j}).$

We also define two open subsets in $X$, $X_{O}\subset X_{o}\subset X$,

$X_{o}:=\cup_{i=1}^{n}U(D_{i}^{o}),\quad X_{O}:=\cap_{i=1}^{n}U(D_{i}^{o}).$

Note that $X_{O}=X\backslash\cup_{i}D_{i}$.

######  Proposition 2.15.

[10, Section 6] Consider the collection of all root hyperplanes
$\\{\widetilde{H}_{\alpha}\\}$ and matter hyperplanes
$\\{\widetilde{H}_{w}\\}$ in $T$, assuming that no root hyperplane coincides
with any weight hyperplane. Let $t\in T_{o}$, then we have the following three
cases:

  1. 1.

If $t\in T_{O}$, i.e. $t$ is not on any matter or root hyperplanes, then the
fiber $\widetilde{\mathcal{M}}^{\times}(G,N)|_{t}=T^{\vee}$. More precisely,
we have

$\widetilde{\mathcal{M}}^{\times}(G,N)|_{T_{O}}\cong\widetilde{\mathcal{M}}^{\times}(T,0)|_{T_{O}}=T^{\vee}\times
T_{O}.$

  2. 2.

If $t\in\widetilde{H}_{\alpha}^{o}$ for a root hyperplane, then over the
neighborhood $U(\widetilde{H}_{\alpha}^{o})$ of $t$, we have

$\widetilde{\mathcal{M}}^{\times}(G,N)|_{U(\widetilde{H}_{\alpha}^{o})}\cong\widetilde{\mathcal{M}}^{\times}(G_{\alpha},0)|_{U(\widetilde{H}_{\alpha}^{o})}$

where $T\subset G_{\alpha}\subset G$ is closed reductive subgroup that only
contains root $\pm\alpha$. Note that explicit coordinates on the RHS are given
by Prop. 2.5.

  3. 3.

If $t\in\widetilde{H}_{w}^{o}$ for a matter hyperplane, then over the
neighborhood $U(\widetilde{H}_{w}^{o})$ of $t$, we have

$\widetilde{\mathcal{M}}^{\times}(G,N)|_{U(\widetilde{H}_{w}^{o})}\cong\widetilde{\mathcal{M}}^{\times}(T,N_{w})|_{U(\widetilde{H}_{w}^{o})}$

where $N_{w}$ is the $w$-weight subspace of $N$ under the action of $T$. Note
that explicit coordinates on the RHS are given by Prop. 2.4.

###  2.5 Deformation

######  Definition 2.16.

[10, Section 3(viii)] If we have a torus $T_{F}$ that acts on $N$, and the
action commutes with the $G$ action. Then, we have the $T_{F}$-family Coulomb
branch

$\mathcal{M}^{\times}(G,N)_{T_{F}}:=\operatorname{Spec}K^{G(\mathcal{O})\times
T_{F}}(R_{G,N}).$

with a map

$\mu_{F}:\mathcal{M}^{\times}(G,N)_{T_{F}}\to T_{F}.$

For $\beta\in T_{F}$, we define the deformed multiplicative Coulomb branch as

$\mathcal{M}^{\times}(G,N)_{\beta}:=\mu_{F}^{-1}(\beta).$

In the abelian gauge theory with matter, we have a deformed version of Prop
2.4.

######  Proposition 2.17.

Let $T$ be a torus, $N=\oplus_{i=1}^{n}\mathbb{C}_{\xi_{i}}$ be its
representation with weights $\xi_{1},\cdots,\xi_{n}$. Let $T_{F}$ acts on
$\mathbb{C}_{\xi_{i}}$ with weight $\eta_{i}\in X^{*}(T_{F})$. The, the
$T_{F}$-deformed multiplicative Coulomb branch algebra
$\mathcal{A}_{C}^{\times}(T,N)_{T_{F}}$ is generated as an algebra over
$\mathcal{O}(T)\otimes\mathcal{O}(T_{F})$ with generators
$\\{r_{\lambda}\\}_{\lambda\in X_{*}(T)}$, satisfying relations

$r_{\lambda}r_{\mu}=r_{\lambda+\mu}\cdot\prod_{i=1}^{n}(1-a^{-\eta_{i}}y^{-\xi_{i}})^{d(\xi_{i}(\lambda),\xi_{i}(\mu))}$

where we use $y^{\xi}$ and $a^{\eta}$ as character functions on $T$ and
$T_{F}$, respectively.

###### Proof.

Comparing with Prop 2.4, we only enlarged the equivariant group from $T$ to
$T\times T_{F}$. A one dimensional representation of $T\times T_{F}$ of weight
$(\xi,\eta)\in X^{*}(T)\times X^{*}(T_{F})$ have Euler characteristic
$1-a^{-\eta}y^{-\xi}$. Hence we only need to replace $1-y^{-\xi_{i}}$ by
$1-y^{-\xi_{i}}a^{-\eta_{i}}$ in the formulas in Prop 2.4. See see BFN section
4(iii) for more details. ∎

######  Example 2.18.

Let $G=\mathbb{C}^{*}$, $V=\mathbb{C}^{2}$ with $\mathbb{C}^{*}$ acting
diagonally. The undeformed Coulomb branch is given by

$\mathcal{M}^{\times}(G,N)=\\{uv=(1-1/y)^{2}\\}\subset\mathbb{C}_{u}\times\mathbb{C}_{v}\times\mathbb{C}^{*}_{y}.$

Let $T_{F}=(\mathbb{C}^{*})^{2}$ acts on $\mathbb{C}^{2}$ canonically. Then,

$\mathcal{M}^{\times}(G,N)_{T_{F}}=\\{uv=(1-a_{1}/y)(1-a_{2}/y)\\}\subset\mathbb{C}_{u}\times\mathbb{C}_{v}\times\mathbb{C}^{*}_{y}\times(\mathbb{C}^{*})^{2}_{a_{1},a_{2}}.$

##  3 Cylindrical model for disks in Coulomb branches

###  3.1 Quiver Gauge Theory

Let $\Gamma$ be a quiver (i.e. directed graph), with (abusing notation)
$\Gamma=\\{(1),\cdots,(r)\\}$ the set of nodes. Let $\bar{\Gamma}$ be the
framed quiver associated with $\Gamma$, that has an additional ‘framing node’
$[i]$ and additional arrow $[i]\to(i)$ for each node $(i)\in\Gamma$. Here we
use ‘box’ $[\,]$ and ‘circle’ $(\,)$ to distinguish the added framing nodes
and the original nodes in $\Gamma$.

A representation of the framed quiver $\bar{Q}$ is given by a collection of
vector spaces for each node and a linear map for each arrow. Concretely, for a
‘circle’ node $(i)$ we attach vector space $V_{(i)}\cong\mathbb{C}^{d_{i}}$,
for a framing node $[i]$ we attach $W_{[i]}\cong\mathbb{C}^{m_{i}}$. The tuple
$\vec{d}=(d_{i}),\vec{m}=(m_{i})$ are called dimension vectors. The moduli
stack of representations is $N/G$ where

$N=N_{B}\oplus N_{F},\quad
N_{B}=\bigoplus_{e:(i)\to(j)}\operatorname{Hom}(\mathbb{C}^{d_{i}},\mathbb{C}^{d_{j}}),\quad
N_{F}=\bigoplus_{[i]\to(i)}\operatorname{Hom}(\mathbb{C}^{m_{i}},\mathbb{C}^{d_{i}})$

and

$G=\prod_{(i)}GL(d_{i}),\quad T=\prod_{(i)}(\mathbb{C}^{*})^{d_{i}},\quad
T_{F}=\prod_{[i]}(\mathbb{C}^{*})^{m_{i}}.$

An element $y\in T$ will have coordinates $(y_{i,\alpha})$ where
$i=1,\cdots,r$, and $1\leq\alpha\leq d_{i}$. An element $\beta\in T_{F}$ will
have coordinates $(a_{i,\alpha})$ where $i=1,\cdots,r$, and $1\leq\alpha\leq
m_{i}$.

###  3.2 Examples

Here we give some simple examples to illustrate the abelianization procedure.
We draw gauge nodes as circles, framing nodes as boxes, and the number inside
the quiver are the dimensions of the vector spaces.

######  Example 3.1.

1

Let $G=GL_{1}$ and $N=0$. In this case,
$T_{G,N}=R_{G,N}=Gr_{T}\cong\mathbb{Z}$, and the $G(\mathcal{O})$ action on
$R_{G,N}$ is trivial. Hence we have the multiplicative Coulomb branch algebra

$\mathcal{A}^{\times}(T,0)=K_{*}^{T(\mathcal{O})}(Gr_{T})=K_{T}^{*}(pt)\otimes\mathbb{C}[X_{*}(T)]=\mathcal{O}(T\times
T^{\vee}).$

The same is true for $G=T=(\mathbb{C}^{*})^{n}$.

######  Example 3.2.

11

$G=GL_{1}\times GL_{1},N=C$ with weight $1$ under the first $GL_{1}$ and $-1$
under the second $GL_{1}$. We have monopole generators $r_{i,j}$, for
$i,j\in\mathbb{Z}$, they satisfy relation as in Proposition 2.4. In
particular, we have

$r_{1,1}r_{-1,-1}=1,\quad r_{1,0}r_{-1,0}=1-y_{2}/y_{1},\quad
r_{0,1}r_{0,-1}=1-y_{2}/y_{1}.\quad r_{1,0}r_{0,1}=r_{1,1}(1-y_{2}/y_{1}).$

One can see that $r_{1,0},r_{0,1},r_{1,1},r_{-1,-1}$ are generators of the
coordinate ring over the base ring $\mathbb{C}[y_{1}^{\pm},y_{2}^{\pm}]$. If
we write $u=r_{1,0},v=r_{0,1},z=r_{-1,-1}$, then we have

$\mathcal{M}^{\times}(G,N)=\\{uvz=1-y_{2}/y_{1}\\}\subset\mathbb{C}_{u}\times\mathbb{C}_{v}\times\mathbb{C}^{*}_{z}\times(\mathbb{C}^{*})^{2}_{y_{1},y_{2}}$

The map $\mu:\mathcal{M}^{\times}(G,N)\to T$ is taking the $(y_{1},y_{2})$
coordinate.

It is still worthwhile to consider the abelianization process. We have the
abelianization map

$\mathcal{O}(\mathcal{M}^{\times}(G,N))\to\mathcal{O}(\mathcal{M}^{\times}(T,0))_{loc},$

where we use $x_{i}^{\pm}$ as coordinates on
$T^{\vee}\cong(\mathbb{C}^{*})^{2}$ and $y_{i}^{\pm}$ as coordinates on
$T\cong(\mathbb{C}^{*})^{2}$. The abelianization map does

$r_{1,0}\mapsto x_{1},\quad r_{-1,0}\mapsto x_{1}^{-1}(1-y_{2}/y_{1}),$
$r_{0,1}=x_{2}(1-y_{2}/y_{1})\quad r_{0,-1}\mapsto x_{2}^{-1},$
$r_{1,1}\mapsto x_{1}x_{2},\quad r_{-1,-1}\mapsto x_{1}^{-1}x_{2}^{-1}.$

######  Example 3.3.

11

Let $G=GL_{1}$ and $N=\mathbb{C}$. We have

$\mathcal{A}^{\times}(G,N)=\oplus_{\lambda\in\mathbb{Z}}\mathbb{C}[y,y^{-1}]r_{\lambda},$

such that

$r_{\lambda}r_{\mu}=r_{\lambda+\mu}\cdot\begin{cases}1&\text{if $\lambda,\mu$
have the same sign}\\\
(1-1/y)^{\min(|\lambda|,|\mu|)}&\text{otherwise}.\end{cases}.$

We write $r_{0}$ as $1$, and $r_{1}=u,r_{-1}=v$, then we have

$\mathcal{A}^{\times}(G,N)=\mathbb{C}[y,y^{-1},u,v]/(uv=1-1/y).$

We have the map

$\mu:\mathcal{M}^{\times}(GL_{1},\mathbb{C})\to\mathbb{C}^{*},\quad(u,v,y)\mapsto
y.$

then the fiber over $y\neq 1$ is isomorphic to $\mathbb{C}^{*}_{u}$, and the
fiber over $y=1$ is a cone $uv=0$.

######  Example 3.4.

2

$G=GL_{2},N=0$. See [6, Section 3.8]. The coordinate ring of
$\mathcal{M}^{\times}(G,N)$ is

$\mathbb{C}\left[x_{1}^{\pm},x_{2}^{\pm},y_{1}^{\pm},y_{2}^{\pm},\frac{x_{1}-x_{2}}{y_{1}-y_{2}}\right]^{S_{2}}.$

where the symmetric group $S_{2}$ permute the indices of $x_{i}$ and $y_{i}$.

Using Proposition 2.8, we write miniscule monopole operators in abelianized
coordinates. Let
$\lambda=[n_{1},n_{2}]\in\mathbb{Z}^{2}/S_{2}=X_{*}(T)/S_{2}$. We have:

$r_{[1,1]}\mapsto(x_{1}x_{2}),\quad
r_{[1,0]}\mapsto\left(\frac{x_{1}}{1-y_{2}/y_{1}}+\frac{x_{2}}{1-y_{1}/y_{2}}\right).$
$r_{[-1,-1]}\mapsto(x_{1}x_{2})^{-1},\quad
r_{[-1,0]}\mapsto\left(\frac{x_{1}^{-1}}{1-y_{1}/y_{2}}+\frac{x_{2}^{-1}}{1-y_{2}/y_{1}}\right).$

######  Example 3.5.

21

$G=GL_{2}\times GL_{1},N=\mathbb{C}^{2}$, where $N$ is the standard
representation under the first factor $GL_{2}$, and of weight $(-1,-1)$ under
$GL_{1}$.

Using Proposition 2.8 and Eqs. (8), (9), we can compute the abelianization of
some minuscule operators,

$r_{[1,0],0}\mapsto\frac{x_{1,1}}{1-y_{1,2}/y_{1,1}}+\frac{x_{1,2}}{1-y_{1,1}/y_{1,2}}$
$r_{[1,1],0}\mapsto x_{1,1}x_{1,2},\quad
r_{[-1,-1],0}\mapsto(x_{1,1}x_{1,2})^{-1}(1-y_{2}/y_{1,1})(1-y_{2}/y_{1,2}).$
$r_{[-1,0],0}\mapsto\frac{x_{1,1}(1-y_{2}/y_{1,1})}{1-y_{1,1}/y_{1,2}}+\frac{x_{1,2}(1-y_{2}/y_{1,2})}{1-y_{1,2}/y_{1,1}}$
$r_{[0,0],1}\mapsto x_{2}(1-y_{2}/y_{1,1})(1-y_{2}/y_{1,2}),\quad
r_{[0,0],-1}\mapsto x_{2}^{-1}.$

###  3.3 Deformed Coulomb branch and formulas for monopole operators

Let $Q,\vec{d},\vec{m}$ be the fixed quiver data; we consider the
corresponding group $G$ and representation $N$. We write $\beta\in T_{F}$ for
a deformation parameter. Let us observe:

######  Proposition 3.6.

There is a canonical isomorphism

$\mathcal{M}^{\times}(G,N)_{\beta}\backslash\\{r_{(\vec{1},\cdots,\vec{1})}=0\\}=\mathcal{M}^{\times}(G,N_{B}).$

In particular, the left hand side is independent of the parameter $\beta$.

###### Proof.

We can apply [25, Theorem 2.9](recalled here as Proposition 2.11) to the
central cocharacter $\xi=(\vec{1},\cdots,\vec{1})$, and observe that if
$N=N_{B}\oplus N_{F}$, then $N_{F}$ is a direct sum of representations with
weight $1$, and $N_{B}$ is a direct sum representations with weight $0$. Note
that $T_{F}$ acts on $N_{B}$ trivially, hence the space is independent of the
deformation parameters. ∎

In particular, from Proposition 3.6 we obtain from (a $T_{F}$ family of)
restrictions of the monopole operators on $\mathcal{M}^{\times}(G,N)$ to
functions on $\mathcal{M}^{\times}(G,N_{B})$.

For any $\lambda$ minuscule coweight of $G$, we have monopole operator
$r_{\lambda}$ as a global function on $\mathcal{M}^{\times}(G,N)_{\beta}$.
Using Proposition 2.8 and 2.9, with the adaptation to the deformed Coulomb
branch as in Proposition 2.17, we can express its restriction to
$(T^{\vee}\times T_{O})/W$ using abelianized coordinates
$(y_{i,\alpha},x_{i,\alpha})$.

For $GL(d,\mathbb{C})$, $l=1,\cdots,d-1$, let

$\omega_{l}^{\vee}=(\underbrace{1,\cdots,1}_{\text{$l$ many $1$s}},0,\cdots
0)\subset\mathbb{Z}^{d}$

be $l$-th fundamental coweight, where $1$ appear $l$ times. Let
$\vec{1}=(1,\cdots,1)\in\mathbb{Z}^{d}$. We also note that

$-w_{0}\omega_{l}^{\vee}=(0,\cdots,0,\underbrace{-1,\cdots,-1}_{\text{$l$ many
$-1$s}}).$

For $G=\prod_{i=1}^{r}GL(d_{i})$, let $\omega_{i,j}^{\vee}$ be the coweight of
$G$ that equals $\omega_{j}^{\vee}$ on the $i$-th factor.

$\omega_{i,j}^{\vee}=(\vec{0},\cdots,\vec{0},\underbrace{(1,\cdots,1,0,\cdots
0)}_{\text{the $i$-th factor, with $j$ many
$1$s.}},\vec{0},\cdots,\vec{0})\in\mathbb{Z}^{d_{1}}\times\cdots\times\mathbb{Z}^{d_{n}}.$

In general, we will use
$(\vec{a}_{1},\cdots,\vec{a}_{r})\in\mathbb{Z}^{d_{1}}\times\cdots\mathbb{Z}^{d_{r}}$
for a cocharacter of $G$.

######  Proposition 3.7.

For $i\in 1,\cdots,r$, the monopole operator $r_{-w_{0}\omega_{i,1}^{\vee}}$
restricted to $(T^{\vee}\times T_{O})/W$ can be written in the abelianized
coordinates as

$ab(r_{-w_{0}\omega_{i,1}^{\vee}})=\sum_{(i,\alpha_{i})}x_{i,\alpha_{i}}^{-1}\frac{\prod_{(j,\alpha_{j})\to(i,\alpha_{i})}\left(1-\frac{y_{j,\alpha_{j}}}{y_{i,\alpha_{i}}}\right)\prod_{[i,\beta_{i}]\to(i,\alpha_{i})}\left(1-\frac{a_{i,\beta_{i}}}{y_{i,\alpha}}\right)}{\prod_{(i,\beta_{i})\neq(i,\alpha_{i})}(1-\frac{y_{i,\alpha_{i}}}{y_{i,\beta_{i}}})}$

where the product is over all possible indices with fixed $i$.

###### Proof.

This follows from Propositions 2.8, 2.9 (Eq (9)), and 2.10. The effect of
deformation is to associate the framing edge $[i,\beta_{i}]\to(i,\alpha_{i})$
the Euler characteristic of $T\times T_{F}$ instead of $T$ only, hence we get
the factor $1-\frac{a_{i,\beta_{i}}}{y_{i,\alpha}}$ instead of
$1-\frac{1}{y_{i,\alpha}}$ (see Prop 2.17). ∎

######  Definition 3.8.

For $y\in T_{O}$, we define fiberwise coordinates for $T^{\vee}$, where
$i=1,\cdots,r$, and $\alpha_{i}=1,\cdots,d_{i}$.

$u_{i,\alpha_{i}}:=x_{i,\alpha_{i}}^{-1}\frac{\prod_{(j,\alpha_{j})\to(i,\alpha_{i})}\left(1-\frac{y_{j,\alpha_{j}}}{y_{i,\alpha_{i}}}\right)\prod_{[i,\beta_{i}]\to(i,\alpha_{i})}\left(1-\frac{a_{i,\beta_{i}}}{y_{i,\alpha}}\right)}{\prod_{(i,\beta_{i})\neq(i,\alpha_{i})}(1-\frac{y_{i,\alpha_{i}}}{y_{i,\beta_{i}}})}$
(12)

The definition is arranged to ensure

$ab(r_{-w_{0}\omega_{i,1}^{\vee}})=\sum_{(i,\alpha_{i})}u_{i,\alpha_{i}}$ (13)

######  Proposition 3.9.

The monopole operator $r_{(\vec{1},\cdots,\vec{1})}$ restricted to
$(T^{\vee}\times T_{O})/W$ can be written in the abelianized coordinates as

$ab(r_{(\vec{1},\cdots,\vec{1})})=\prod_{i,\alpha}x_{i,\alpha}$

###### Proof.

We use Proposition 2.8. The $G(\mathcal{O})$ orbit $Gr_{G}^{\lambda}$ for the
coweight $\lambda=(\vec{1},\cdots,\vec{1})$ is a point. ∎

###  3.4 Some geometry of deformed Coulomb branch

We now study the geometry of $\mathcal{M}^{\times}(G,N)_{\beta}$. As in Eq
(11), we consider the $W$-fold cover, where $W$ is the Weyl group of $G$.

${\widetilde{\mathcal{M}}^{\times}(G,N)_{\beta}}$${\mathcal{M}^{\times}(G,N)_{\beta}}$${T}$${T/W}$$\scriptstyle{\widetilde{\mu}}$$\scriptstyle{\mu}$
(14)

Over $T_{O}$, we have fiber $T^{\vee}$ with fiber coordinates
$x_{i,\alpha_{i}}$, where $i=1,\cdots,r$ and $1\leq\alpha_{i}\leq d_{i}$.

There are three types of hyperplanes in $T$, over which the fiber of
$\widetilde{\mu}$ degenerate. For each such hyperplane $H$, we will describe
the fiber behavior over the neighborhood $U(\widetilde{H}^{o})$.

  1. 1.

(Root Hyperplane) For each circle node $(i)$, we have downstairs root
hyperplanes $H_{i,\alpha_{i},\alpha_{i}^{\prime}}$ labelled by
$1\leq\alpha_{i}<\alpha_{i}^{\prime}\leq d_{i}$:

$\widetilde{H}_{i,\alpha_{i},\alpha_{i}^{\prime}}:=\\{y_{i,\alpha_{i}}=y_{i,\alpha^{\prime}_{i}}\\}.$

We write the function cutting out the union of all root hyperplanes as

$\Delta_{R}:=\prod_{(i,\alpha)\neq(i,\beta)}(y_{i,\alpha}-y_{i,\beta})$ (15)

  2. 2.

(Internal matter hyperplane) For each internal edge $(i)\to(j)$, we have
downstairs weight hyperplanes $H_{(i,\alpha_{i})\to(j,\alpha_{j})}$ labelled
by $1\leq\alpha_{i}\leq d_{i}$, $1\leq\alpha_{j}\leq d_{j}$

$\widetilde{H}_{(i,\alpha_{i})\to(j,\alpha_{j})}:=\\{y_{i,\alpha_{i}}=y_{j,\alpha_{j}}\\}.$

We write the function cutting out the union of all the internal matter
hyperplanes as

$\Delta_{I}:=\prod_{(i,\alpha_{i})\to(j,\alpha_{j})}(y_{i,\alpha_{i}}-y_{j,\alpha_{j}})$
(16)

  3. 3.

(Framing matter hyperplane) For each framing edge $(i)\leftarrow[i]$, we have
downstairs weight hyperplanes $H_{(i,\alpha_{i})\leftarrow[i,\beta_{i}]}$
(resp. $\widetilde{\mathcal{H}}_{(i,\alpha_{i})\leftarrow[i,\beta_{i}]}$)
labelled by $1\leq\alpha_{i}\leq d_{i}$, $1\leq\beta_{i}\leq m_{i}$

$\widetilde{H}_{(i,\alpha_{i})\leftarrow[i,\beta_{i}]}=\\{y_{i,\alpha_{i}}=a_{i,\beta_{i}}\\}.$

We write the function cutting out the union of all the framing matter
hyperplanes as

$\Delta_{F}:=\prod_{(i,\alpha_{i})\leftarrow[i,\beta_{i}]}(y_{i,\alpha_{i}}-a_{i,\beta_{i}})$
(17)

The geometry with the three kinds of hyperplane divisors removed is given by
the following Proposition.

######  Proposition 3.10.

The affine variety
$\widetilde{\mathcal{M}}^{\times}(G,N)_{\beta}^{O}\setminus\\{r_{(\vec{1},\cdots,\vec{1})}=0\\}$
has ring of global functions given by

$\widetilde{\mathcal{A}}(G,N)_{\beta}^{O}:=\mathbb{C}[\\{x_{i,\alpha}^{\pm
1},y_{i,\alpha}^{\pm
1}\\}_{i,\alpha},\Delta_{R}^{-1},\Delta_{F}^{-1},\Delta_{I}^{-1}]=\mathbb{C}[\\{u_{i,\alpha}^{\pm
1},y_{i,\alpha}^{\pm
1}\\}_{i,\alpha},\Delta_{R}^{-1},\Delta_{F}^{-1},\Delta_{I}^{-1}].$

###### Proof.

The variety is affine, having been obtained from the definitionally affine
multiplicative Coulomb branches by removing divisors. The equality of the two
purported expressions for the ring of global functions is apparent from the
definition of the $u$ variables (Definition 3.8).

To compute the ring of global functions, we begin by noting that the global
functions on $\mathcal{M}^{\times}(G,N_{B})^{O}$ are, by Proposition 2.15,
item 1, given by

$\widetilde{\mathcal{A}}(G,N_{B})^{O}:=\mathbb{C}[\\{x_{i,\alpha}^{\pm
1},y_{i,\alpha}^{\pm 1}\\}_{i,\alpha},\Delta_{R}^{-1},\Delta_{I}^{-1}]$

Under the isomorphism of Proposition 3.6, we obtain
$\widetilde{\mathcal{M}}^{\times}(G,N)_{\beta}^{O}$ by further removing the
preimages of the framing matter hyperplanes (i.e. inverting $\Delta_{F}$). ∎

###  3.5 Proof of Theorem 1.5

We restate the result:

######  Theorem 3.11.

Fix a map $C\to\mathbf{T}/W=\prod_{i}Sym^{d_{i}}\mathbb{C}^{*}_{y}$ transverse
to $(\mathbf{T}\setminus\mathbf{T}_{O})/W$, and consider the corresponding
$S\to C\times\mathbb{C}^{*}_{y}$.

Lifts to $C\to\mathcal{M}^{\times}(\Gamma,\vec{d})$ are in bijection with maps
$\Phi_{u}:S\to\mathbb{P}^{1}_{u}$ satisfying the following conditions:

  1. 0.

$\Phi_{u}(S_{O})\subset\mathbb{C}^{*}_{u}$.

  2. 1.

$\Phi_{u}$ has simple poles over points in $S_{(i)}$.

  3. 2.

$\Phi_{u}$ has simple zeros over $S_{[i]\to(i)}$.

  4. 3.

For $(s_{i},s_{j})\in S_{(i)\to(j)}$, we require that either $\Phi_{u}$ has a
simple zero at $s_{i}$ and $\Phi_{u}(s_{j})\in\mathbb{C}^{*}$, or $\Phi_{u}$
has a simple zero at $s_{j}$ and $\Phi_{u}(s_{i})\in\mathbb{C}^{*}$.

###### Proof.

Such sections form a sheaf of sets over $C$. By hypothesis, $C\to\mathbf{T}/W$
meets all hyperplanes distinctly and transversely; we are (by the sheaf
property of sections) therefore reduced to the case where $C$ is a small disk
meeting at most one of the hyperplanes. In the complement of the hyperplanes,
the result is immediate from the identification of Proposition 3.6 and the
trivialization of Proposition 3.10.

Thus, we may compute in one of the charts described in Proposition 2.15.
Observe that, in the quiver gauge theory setting, the chart itself is again
the Coulomb branch of a quiver gauge theory, which moreover is either abelian
(all entries of dimension vector are $1$) and has a single arrow, or has no
arrows and only a single non-abelian node, which has dimension $2$.

Under the isomorphism of Proposition 2.15, we may compare the $u$ coordinates
of the more complicated original quiver and of the simplified quiver. From the
explicit defining formula of Definition 3.8, we see that the ratio is a
regular invertible function on $T_{O}$, and extends to a regular invertible
function on $U(H^{o})$ when restricted to the smaller locus.

More precisely, for $p\in C\cap(T\backslash T_{O})$, one can choose a lift
$\widetilde{p}\in T$, and find an open neighborhood $\widetilde{U}$ of
$\widetilde{p}$, and a simplified quiver $Q^{\prime}$, such that the
abelianized Coulomb branch $\widetilde{\mathcal{M}}_{Q}$ with rational
functions $u_{i,a}$, and the simplified one
$\widetilde{\mathcal{M}}_{Q^{\prime}}$ with rational function
$u^{\prime}_{i,a}$ satisfies the condition that
$\widetilde{\mathcal{M}}_{Q}\cong\widetilde{\mathcal{M}}_{Q^{\prime}}$ over
$\widetilde{U}$, and there is an invertible function $R_{i,a}$ on $U_{p}$,
such that $u_{i,a}=R_{i,a}u^{\prime}_{i,a}$ on $\widetilde{U}$. That is,
$u_{i,a}$ and $u^{\prime}_{i,a}$ have the same pole and zero structure when
restricted to $C$.

In particular, if we consider the assertion of the theorem on the smaller
locus, it does not matter whether we use the $u$ coordinates of the original
ambient quiver, or of the simplied quiver. We are thus reduced to proving the
theorem for quivers which have either no arrows and a single nonabelian node
which moreover has dimension $2$, or abelian quivers with one arrow. As the
$u$ coordinates are compatible with disjoint union of quivers, we are reduced
to checking for the situations of Examples 3.1, 3.2, 3.3, and 3.4. These can
be checked explicitly by hand. ∎

##  4 KLRW category

Here we fix some terminology for discussing the KLRW categories, and establish
some recognition principles.

For any $n\in\mathbb{N}=\mathbb{Z}_{\geq 0}$, let $[n]=\\{1,\cdots,n\\}$.

Let $\Gamma$ be a framed quiver with nodes labelled by $[n]$. We will also use
$\Gamma$ to denote the set of non-framing nodes. The non-framing node is
written as $(i)$ where $i\in[n]$, and the framing node is $[i]$. The arrows
are written as $(i)\to(j)$ between non-framing nodes, and $[i]\to(i)$ from the
framing node to the non-framing node. For each $i\in[n]$, we have dimension
$d_{i}\in\mathbb{N}$ for the non-framing node and $m_{i}\in\mathbb{N}$ for the
framing node. We use $\vec{d},\vec{m}$ for the tuples of numbers. We use
$(i,j)$ for $i\in[n],j\in[d_{i}]$, and use $[i,j]$ for $i\in[n],j\in[m_{i}]$.

######  Definition 4.1.

A red points configuration on $S^{1}$ of type $\vec{m}$ is a finite subset
$R\subset S^{1}$, where $R=\sqcup_{[i]}R_{i}$ and $|R_{i}|=m_{i}$. Given a red
points configuration, a black points configuration on $S^{1}$ of type
$\vec{d}$ is finite subset $B\subset S^{1}\backslash R$, where
$B=\sqcup_{(i)}B_{i}$ and $|B_{i}|=d_{i}$.

A red points configuration is denoted as $\\{\theta_{[i,j]}\\}$ where
$\theta_{[i,j]}\in S^{1}$ and the ordering of the $\theta_{[i,j]}$ for a fixed
$i$ is irrelevant. Similarly, and a black points configuration is denoted as
$\\{\theta_{(i,j)}\\}$.

Typically we will have fixed a red point configuration. We use $\mathcal{C}$
to denote the set of black points configurations.

Recall that the KLRW category (for a fixed red points configuration) is the
following. The set of objects is the set of black points configurations. The
morphism space $Hom_{KLRW}(c_{1},c_{2})$ is the vector space generated by KLRW
strand diagrams with $c_{1}$ at the bottom and $c_{2}$ at the top modulo
relations, and composition is vertical concatenations.

Recall we have introduced parameters $\hbar$ and $\eta$, and defined the KLRW
category over $\mathbb{C}[\hbar,\eta]$. These parameters appear in the KLRW
relations.

We will be interested in certain degenerate cases, first $\hbar=0=\eta$, and
also just $\hbar=0$. In these cases, it is convenient to describe the
morphisms in terms of the following basis.

######  Definition 4.2.

A taut strands diagram on the cylinder from black points configuration $c_{1}$
to $c_{2}$, is a collection of strands that ends on $c_{1}$ at bottom and
$c_{2}$ at top, such that the strands have no horizontal tangency, no triple
intersection points, and no bigon between any two strands (red or black). Two
taut strands diagrams are equivalent, if they are isotopic modulo braid
relation.

A weighted taut strands diagram is a taut strands diagram where each strand is
decorated by a non-negative number (called weight).

We write $\mathcal{P}(c_{1},c_{2})$ as the equivalence class of weighted taut
strands diagram from black points configuration $c_{1}$ to $c_{2}$.

If $D_{12}\in\mathcal{P}(c_{1},c_{2})$ is a weighted taut strand diagram, a
taut decomposition of $D_{12}$ are two diagrams
$D_{1a}\in\mathcal{P}(c_{1},c_{a}),D_{a2}\in\mathcal{P}(c_{a},c_{2})$ such
that the concatenation $concat(D_{a2},D_{1a})$ is taut and equal to $D_{12}$
and the sum of weights of each strands matches with $D_{12}$. We write this as

$D_{12}=concat(D_{a2},D_{1a}).$

and say $D_{12}$ is the taut concatenation of $D_{a2}$ and $D_{1a}$.

######  Proposition 4.3.

For any black points configurations $c_{1},c_{2}\in\mathcal{C}$, we have
canonical isomorphism of vector spaces

$Hom_{KLRW}(c_{1},c_{2})_{\hbar=\eta=0}\cong\mathbb{C}\mathcal{P}(c_{1},c_{2}).$

###### Proof.

We may use KLRW relations to straighten the strands in any KLRW diagrams and
move dots to the top of the diagram, then by viewing the number of dots on a
strand as the weight on that strand we have the bijection. ∎

######  Lemma 4.4.

Let $D_{12}\in\mathcal{P}(c_{1},c_{2})$ and
$D_{23}\in\mathcal{P}(c_{2},c_{3})$ be weighted taut strands diagrams, and let
$D_{13}$ be the concatenation and straightening of $D_{23}\circ D_{12}$. If we
view $D_{ij}$ as morphisms in the KLRW category with $\hbar=\eta=0$, then
$D_{23}\circ D_{12}$ either equals zero or equals to $D_{13}$, and equals
$D_{13}$ if and only if one of the following happens

  1. (1)

There is no bigon cancellation between black strands of same label.

  2. (2)

There is no bigon cancellation between black strands with labels $(i)$ and
$(i^{\prime})$ if $(i)\to(i^{\prime})$.

  3. (3)

There is no bigon cancellation between black strands and red strands of the
same label (corresponding to a framing edge $[i]\to(i)$).

###### Proof.

This is immediate following the KLRW relations. ∎

######  Proposition 4.5.

When $\hbar=0$, we have a canonical isomorphism of vector spaces

$Hom_{KLRW}(c_{1},c_{2})_{\hbar=0}\cong\mathbb{C}[\eta]\cdot\mathcal{P}(c_{1},c_{2}).$

###### Proof.

First, any KLRW diagram can be expressed as a linear combination of taut KRLW
diagrams with dots at the bottom. Next, since $\hbar=0$, for any 3 strands in
a KLRW diagram we have braid relation, hence we have a bijection between taut
KLRW diagram (modulo relations) and taut weighted strands diagram. ∎

######  Definition 4.6.

A weighted taut diagram (resp. KLRW diagram) is called elementary if it is one
of the two types

  1. 1.

all strands vertical, and exactly one black strand has weight $1$ (resp.
carries a dot).

  2. 2.

there is at most one crossing of any type, and all strands have zero weights
(resp. no dots).

There is an obvious bijection between elementary KLRW diagram and elementary
weighted taut diagram, by exchanging weight $1$ with one dot. For this reason,
we do not distinguish elementary KLRW or weighted elementary diagram. Let
$Elem(c_{1},c_{2})\subset\mathcal{P}(c_{1},c_{2})$ be the subset of elementary
diagrams from $c_{1}$ to $c_{2}$.

The morphisms in KLRW category can be generated by elementary diagrams modulo
relations, which we formalize below.

######  Definition 4.7.

Let $FreeElem$ be the category with objects $\mathcal{C}$ and hom space be

$Hom_{FreeElem}(c,c^{\prime})=\oplus_{k\geq
0}\oplus_{c=c_{0},c_{1},\cdots,c_{k}=c^{\prime}}\mathbb{C}[\eta,\hbar]Elem(c_{0}\to
c_{1}\cdots\to c_{k})$

where $Elem(c_{0}\to\cdots\to c_{k})=\prod_{i=1}^{k}Elem(c_{i-1},c_{i})$ is a
chain of elementary weighted taut diagrams of length $k$, and composition is
concatenation of chains.

There is an obvious functor from $FreeElem$ to $KLRW$ by composing the chain
of elementary diagrams, and the map on hom space is surjective.

######  Proposition 4.8.

Let $KLRW_{\hbar=0}$ be the KLRW category with $\hbar=0$, where we identify
the morphism space by
$Hom_{KLRW}(c_{1},c_{2})_{\hbar=0}\cong\mathbb{C}[\eta]\cdot\mathcal{P}(c_{1},c_{2})$
and we denote the composition of
$c_{1}\xrightarrow{D_{12}}c_{2}\xrightarrow{D_{23}}c_{3}$ as
$D_{23}\circ_{K}D_{12}$.

Let $\mathcal{A}$ be a category with the same set of objects and morphism
space as $KLRW_{\hbar=0}$ and we denote the composition in $\mathcal{A}$ of
$c_{1}\xrightarrow{D_{12}}c_{2}\xrightarrow{D_{23}}c_{3}$ as
$D_{23}\circ_{\mathcal{A}}D_{12}$.

If the two compositions matches

$D_{23}\circ_{K}D_{12}=D_{23}\circ_{\mathcal{A}}D_{12}$ (18)

for the following subset of diagrams

  1. 1.

$D_{12},D_{23}$ are elementary strands diagram.

  2. 2.

$D_{12},D_{23}$ are such that $concat(D_{12},D_{23})$ is taut.

Then the two composition matches(18) for all diagrams $D_{23},D_{12}$. In
particular, the category $\mathcal{A}$ and $KLRW_{\hbar=0}$ are equivalent.

###### Proof.

We first prove that, for $D_{23}$ elementary (resp. $D_{12}$ elementary), Eq
(18) holds. We only prove the case for $D_{23}$.

If $concat(D_{23},D_{12})$ is taut, then there is nothing to prove by the
hypothesis. Hence we only need to consider the case $D_{23}$ has a crossing
and $concat(D_{23},D_{12})$ has a bigon. By braid isotopy, we may move the
bigon to the top and write $D_{12}$ as

$D_{12}=concat(D_{42},D_{14})=D_{42}\circ_{K}D_{14}=D_{42}\circ_{\mathcal{A}}D_{14},$

where $D_{42}$ is elementary and the concatenation is taut. Since we know

$D_{23}\circ_{K}D_{42}=D_{23}\circ_{\mathcal{A}}D_{42}$

by hypothesis, and the result is $\sum_{i}c_{i}(D_{43})_{i}$ where
$(D_{43})_{i}$ are diagrams without crossing, hence the concatenation of
$(D_{43})_{i}$ with $D_{14}$ is taut, and we get

$(D_{43})_{i}\circ_{K}D_{14}=(D_{43})_{i}\circ_{\mathcal{A}}D_{14}.$

Putting these together, we have

$D_{23}\circ_{K}D_{12}=D_{23}\circ_{K}(D_{42}\circ_{K}D_{14})=(D_{23}\circ_{K}D_{42})\circ_{K}D_{14}=[\sum_{i}c_{i}(D_{43})_{i}]\circ_{K}D_{14}=\sum_{i}c_{i}[(D_{43})_{i}\circ_{K}D_{14}]$
$=\sum_{i}c_{i}[(D_{43})_{i}\circ_{\mathcal{A}}D_{14}]=[\sum_{i}c_{i}(D_{43})_{i}]\circ_{\mathcal{A}}D_{14}=(D_{23}\circ_{\mathcal{A}}D_{42})\circ_{\mathcal{A}}D_{14}=D_{23}\circ_{\mathcal{A}}(D_{42}\circ_{\mathcal{A}}D_{14})=D_{23}\circ_{\mathcal{A}}D_{12}.$

Finally, we prove for general $D_{23}$, by induction on length of $D_{23}$
where length of a diagram is the number of total number of crossings (of any
type). Assume the composition matches for all $D_{23}$ of length $\leq N-1$
and we have $length(D_{23})=N$. We do a taut decomposition

$D_{23}=concat(D_{43},D_{24})=D_{43}\circ_{K}D_{24}=D_{43}\circ_{\mathcal{A}}D_{24}$

where $D_{24}$ is elementary, and $length(D_{43})=N-1$. Then

$D_{23}\circ_{K}D_{12}=D_{43}\circ_{K}D_{24}\circ_{K}D_{12}=D_{43}\circ_{K}(D_{24}\circ_{K}D_{12})$
$=D_{43}\circ_{\mathcal{A}}(D_{24}\circ_{\mathcal{A}}D_{12})=(D_{43}\circ_{\mathcal{A}}D_{24})\circ_{\mathcal{A}}D_{12}=D_{23}\circ_{\mathcal{A}}D_{12}.$

∎

Finally we consider KLRW category for $\hbar\neq 0$.

Let $D$ be a KLRW diagram, let $cross_{i}(D)$ be the number of crossings among
black strands with label $(i)$, and let
$cross(D)=\sum_{i\in\Gamma}cross_{i}(D)$. Note that the function $cross$ is
defined on KLRW diagrams, not modulo equivalence relations.

For any non-negative integer $k$, let
$\mathcal{F}_{k}Hom_{KLRW}(c_{1},c_{2})\subset Hom_{KLRW}(c_{1},c_{2})$ be the
subspace spanned by the equivalence classes of KRLW diagrams $D$ from $c_{1}$
to $c_{2}$ with $cross(D)\leq k$. Clearly $\mathcal{F}_{k}$ defines an
increasing filtration on $Hom_{KLRW}(c_{1},c_{2})$, and we have

######  Lemma 4.9.

The composition in KLRW category preserves filtration, namely

$-\circ_{KLRW}-:\mathcal{F}_{l}Hom_{KLRW}(c_{2},c_{3})\otimes\mathcal{F}_{k}Hom_{KLRW}(c_{1},c_{2})\to\mathcal{F}_{k+l}Hom_{KLRW}(c_{1},c_{3}).$

###### Proof.

Indeed, the composition in KLRW category is by concatenation, and crossing
number is additive on concatenation of diagrams. ∎

For weighted taut strand diagram $D\in\mathcal{P}(c_{1},c_{2})$, we may also
define $cross_{i}(D)$ for and $cross(D)$ as above. Let
$\mathcal{P}_{k}(c_{1},c_{2})$ be the subspace spanned by $D$ with
$cross(D)=k$. Then we have direct sum decomposition

$\mathcal{P}(c_{1},c_{2})=\oplus_{k\geq 0}\mathcal{P}_{k}(c_{1},c_{2}).$

######  Lemma 4.10.

For any $c_{1},c_{2}\in\mathcal{C}$, we have canonical isomorphism

$\Phi_{k}:Gr_{k}^{\mathcal{F}}Hom_{KLRW}(c_{1},c_{2})\cong\mathbb{C}[\eta,\hbar]\cdot\mathcal{P}_{k}(c_{1},c_{2}).$

###### Proof.

Let $D$ be a KLRW diagram with $cross(D)\leq k$, we may use KRLW relations to
express $D$ as a linear combination of taut KRLW diagrams (i.e. with out any
bigon) $D=\sum_{i}c_{i}D_{i}.$ We define
$\Phi_{k}(D)=\sum_{i}c_{i}\Phi_{k}(D_{i})$, where $\Phi_{k}(D_{i})=0$ if
$cross(D_{i})<k$ and otherwise $\Phi_{k}(D_{i})$ is defined by viewing $D_{i}$
as taut weighted diagram where weight is the number of dots. One can check
that different decomposition of $D$ as taut KLRW diagrams will give the same
$\Phi_{k}$ image and $\Phi_{k}$ naturally descend to $Gr_{k}^{\mathcal{F}}$. ∎

Let $Gr(KLRW)$ denote the associated graded category, where the set of objects
is still $\mathcal{C}$, and the hom space is
$Hom_{Gr(KLRW)}(c_{1},c_{2})=\oplus_{k}Gr_{k}^{\mathcal{F}}(Hom_{KLRW}(c_{1},c_{2}))$,
and composition is naturally induced.

######  Lemma 4.11.

Let $\mathcal{A},\mathcal{B}$ be two $R$-linear categories. Assume given an
exhausting increasing filtration indexed by $\mathbb{Z}_{\geq 0}$ on the hom
spaces and with compositions compatible with filtrations. If
$\psi:\mathcal{A}\to\mathcal{B}$ is a functor that respects the filtration,
and $Gr\psi:Gr\mathcal{A}\to Gr\mathcal{B}$ is an isomorphism, then $\psi$ is
an equivalence of category

###### Proof.

If $F:V\to W$ are morphisms of filtered $R$-module with isomorphism on
associated graded, then $V\xrightarrow{\sim}W$. ∎

######  Lemma 4.12.

Let $\mathcal{A}$ be a category defined over $\mathbb{C}[\hbar,\eta]$ with
objects $\mathcal{C}$, and let there be a functor

$\psi:FreeElem\to\mathcal{A}$

such that $\psi$ satisfies the following condition

  1. 1.

annihilate all KRLW relations except possibly the braid relations (Figure
1(d), 1(e))

  2. 2.

Let $D$ be the elementary diagram of a crossing of type $(i)-[i]$, i.e. a
black strand of label $(i)$ and a red strand with label $[i]$. Then $\psi(D)$
is a morphism in $\mathcal{A}$ that is not a zero-divisor, i.e., for any pre-
composable morphism $E$,

$\psi(D)\circ E=0\Leftrightarrow E=0.$

and for any post-composable morphism $E$ for $\psi(D)$,

$E\circ\psi(D)=0\Leftrightarrow E=0.$

  3. 3.

Let $D$ be the elementary diagram of a crossing of type $(i)-(j)$, where
$(i),(j)$ are adjacent nodes. Then $\psi(D)$ is not a zero-divisor (in the
above sense).

then $\psi$ in fact also annihilates the braid relations, and hence descends
to a functor $\psi:KRLW\to\mathcal{A}$.

###### Proof.

The two braid relations that we want to show are as in Figure 1(d) and 1(e).
We pre-compose the equation with a non-zero divisor and show the new equation
holds.

The braid with red-strand relation is equivalent to Figure 2, which holds by
dot pass crossing relation.

Figure 2: Proof for braid relations between two black strands and a red
strands of the same label.

The braid relation between black strands of adjacent nodes is equivalent to
the following equations of diagrams (Figure 3), which does hold.

Figure 3: Proof for braid relations between black strands labelled by adjacent
circle nodes.

∎

######  Proposition 4.13.

Let $\mathcal{A}$ be a category with objects $\mathcal{C}$ and with hom spaces
$Hom_{\mathcal{A}}(c_{1},c_{2})$ as a $\mathbb{C}[\eta,\hbar]$-free module. We
assume that

  1. 1.

The hom spaces $Hom_{\mathcal{A}}(c_{1},c_{2})$ has an increasing filtration
$\mathcal{F}_{k}(Hom_{\mathcal{A}}(c_{1},c_{2}))$ compatible with composition.

  2. 2.

The associated graded has a canonical isomorphism

$\Psi_{k}:Gr_{k}^{\mathcal{F}}(Hom_{\mathcal{A}}(c_{1},c_{2}))\cong\mathbb{C}[\eta,\hbar]\cdot\mathcal{P}_{k}(c_{1},c_{2}).$

  3. 3.

$\Psi_{k}$ defines an isomorphism of graded category

$\Psi:Gr(\mathcal{A})\cong Gr(KLRW).$

  4. 4.

There is a functor

$\psi:KLRW\to\mathcal{A}$

that respects the filtration, and $Gr\psi=\Psi$.

Then we have $\psi$ is an equivalence of category.

###### Proof.

We may apply Lemma 4.11 to $\psi:KLRW\to\mathcal{A}$. ∎

##  5 Floer homologies in $(\mathbf{T}_{O}\times\mathbf{T}^{\vee})/\mathbf{W}$

Our eventual goal is to compute Floer homology in the Coulomb branch for
certain distinguished Lagrangians. It is possible to describe the Lagrangians,
their positive wrapping and their intersections in an explicit way, and the
only difficulty is to count disks that defines the $A_{\infty}$ structure. We
will introduce two formal parameters $\eta$ and $\hbar$, where a factor of
$\eta$ indicate a disk that intersects the matter divisor, and a factor of
$\hbar$ indicate a disk intersects the root divisor. In this section, we set
$\eta,\hbar=0$, which means the computation are in the complement of these
divisors, which we call the punctured space.

This section will describe the punctured space, certain distinguished
Lagrangians in it and their hom spaces. In the next two sections, we will
gradually add back the divisors. This section can be read without any
knowledge of the Coulomb branches. We use the same notation as section 4.

###  5.1 Fukaya category setup

We define

${\bf T}_{y}=\prod_{(i,j)}\mathbb{C}^{*}_{y_{(i,j)}},\quad{\bf
T}^{\vee}_{u}=\prod_{(i,j)}\mathbb{C}^{*}_{u_{(i,j)}},\quad{\bf
W}=\prod_{(i)}S_{d_{i}}.$

with coordinates as subscript. We also define a punctured version of ${\bf
T}_{y}$

${\bf T}_{O}=\left\\{(y_{(i,j)})\in{\bf T}\mid\begin{cases}y_{(i,j)}\neq
y_{(i,j^{\prime})}&\forall(i,j)\neq(i,j^{\prime})\\\ y_{(i,j)}\neq
a_{[i,k]}&\forall(i,j),[i,k]\\\ y_{(i,j)}\neq
y_{(i^{\prime},j^{\prime})}&\forall(i)\to(i^{\prime}).\end{cases}\right\\}.$

We choose a product holomorphic volume form (used only to fix
grading/orientation data for Floer homologies) on ${\bf T}^{\vee}\times{\bf
T}$

$\Omega=\prod_{{(i,j)}}d\log u_{{(i,j)}}\wedge d\log y_{{(i,j)}}.$ (19)

Note that $\Omega$ is invariant under $\bf W$ hence descends to
$(\mathbf{T}_{O}\times\mathbf{T}^{\vee})/W$.

A product Lagrangian is

${\bf L}=\prod_{(i,\alpha)}L_{u_{{(i,j)}}}\times L_{y_{{(i,j)}}}\subset{\bf
T}^{\vee}\times{\bf T}_{O},$

where $L_{u_{(i,j)}}\subset\mathbb{C}^{*}_{u_{(i,j)}}$ and
$L_{y_{(i,j)}}\subset\mathbb{C}^{*}_{y_{(i,j)}}$ are Lagrangians (i.e. closed
1-dimensional submanifold without boundary).

A symmetric product Lagrangian is the quotient image of a product Lagrangian
by ${\bf W}$

${\bf L/W}\subset({\bf T}^{\vee}\times{\bf T}_{O})/{\bf W}.$

Given two product Lagrangians $\bf L_{1},L_{2}$, their intersections are
product of intersections

${\bf L}_{1}\cap{\bf L}_{2}=\prod_{(i,j)}(L_{1,u_{(i,j)}}\cap
L_{2,u_{(i,j)}})\times(L_{1,y_{(i,j)}}\cap L_{2,y_{(i,j)}}).$

Similarly, given two symmetric product Lagrangians, their intersections are

${\bf L}_{1}/{\bf W}\cap{\bf L}_{2}/{\bf W}=\coprod_{\sigma\in{\bf
W}}\prod_{(i,j)}(L_{1,u_{(i,j)}}\cap
L_{2,u_{\sigma}{(i,j)}})\times(L_{1,y_{(i,j)}}\cap L_{2,y_{\sigma{(i,j)}}}).$

######  Definition 5.1.

Let $c=\\{\theta_{(i,j)}\\}$ be a black points configuration. We define the
following symmetric product Lagrangian

$\mathcal{T}_{c}=\\{((u_{(i,j)},y_{(i,j)}))\in{\bf T}^{\vee}\times{\bf
T}_{O}\mid\arg(y_{{(i,j)}})=\theta_{(i,j)},\arg(u_{(i,j)})=0\\}/{\bf W}.$

###  5.2 Floer theory in $\mathbb{C}^{*},(\mathbb{C}^{*})^{2}$ ,
$(\mathbb{C}^{*})^{2}\backslash\Delta$ and
$\operatorname{Sym}^{2}(\mathbb{C}^{*})\backslash\Delta$

We study some model cases in this subsection.

######  Definition 5.2.

Let $(\mathbb{C}^{*}_{z},\omega=i|z|^{-2}dz\wedge d\bar{z},\Omega=dz/z)$ be
the symplectic cylinder. Let $T_{1},T_{2}$ be two graded Lagrangians with
conical ends that are Lagrangian section for the torus fibration
$\log|z|:\mathbb{C}^{*}\to\mathbb{R}$. We say $T_{1}$ intersects $T_{2}$
positively and write $T_{1}>T_{2}$, if $T_{1},T_{2}$ intersects transversely
and the chain complex $CF(T_{1},T_{2})$ is concentrated in degree $0$. In
general, we write $T_{1}>T_{2}>\cdots>T_{n}$ if for any $i<j$ we have
$T_{i}>T_{j}$.

######  Lemma 5.3.

Given $T_{1}>T_{2}>T_{3}$ in $\mathbb{C}^{*}_{z}$, if $p_{12}\in T_{1}\cap
T_{2}$ and $p_{23}\in T_{2}\cap T_{3}$, then there exists a unique $p_{13}\in
T_{1}\cap T_{3}$ as the composition output of $p_{23}\circ p_{12}$.

###### Proof.

We consider the universal cover $\mathbb{C}_{u}\to\mathbb{C}^{*}_{z}$,
$z=e^{u}$. Choose a lift $\widetilde{T}_{i}$ of $T_{i}$ for $i=1,2,3$, such
that the intersection points $p_{12}$ and $p_{23}$ lifts to
$\widetilde{p}_{12}\in\widetilde{T}_{1}\cap\widetilde{T}_{2}$ and
$\widetilde{p}_{23}\in\widetilde{T}_{2}\cap\widetilde{T}_{3}$, respectively.
For $i<j$, we also have $CF(\widetilde{T}_{i},\widetilde{T}_{j})$ in degree
$0$ (where $\Omega_{\mathbb{C}_{u}}=du$), hence
$\widetilde{T}_{i}\cap\widetilde{T}_{j}$ at most at one point. We have seen
that $\widetilde{T}_{1}\cap\widetilde{T}_{2}=\\{\widetilde{p}_{12}\\}$ and
$\widetilde{T}_{2}\cap\widetilde{T}_{3}=\\{\widetilde{p}_{23}\\}$, it remains
to show that $\widetilde{T}_{1}\cap\widetilde{T}_{3}\neq\emptyset$. Since
$T_{i}$ are eventually conical, hence $Im(u)$ on $\widetilde{T}_{i}$ are
eventually constant when $|Re(u)|$ are large enough. Let $R_{i},L_{i}$ be the
eventual value of $Im(u)|_{\widetilde{T}_{i}}$ on the far right and left ends,
then we have

$R_{1}>R_{2}>R_{3},\quad L_{1}<L_{2}<L_{3}.$

Since $R_{1}>R_{3},L_{1}<L_{3}$, we see
$\widetilde{T}_{1}\cap\widetilde{T}_{3}\neq\emptyset$, and denote the unique
intersection point as $\widetilde{p}_{13}$. Then, there is an obvious disk in
$\mathbb{C}_{u}$ that realizes
$\widetilde{p}_{13}=\widetilde{p}_{23}\circ\widetilde{p}_{12}$. Let
$p_{13}\in\mathbb{C}^{*}_{z}$ be the image of $\widetilde{p}_{13}$, then the
image of the disk in $\mathbb{C}_{u}$ witnesses the composition
$p_{13}=p_{23}\circ p_{12}$. ∎

As a warm-up for using strand diagrams to represent morphisms in Fukaya
category, we define the set of states to be $\mathcal{C}_{1}=S^{1}$, and for
$c_{1},c_{2}\in\mathcal{C}_{1}$, the set of diagrams
$\mathcal{P}_{1}(c_{1},c_{2})$ to be the taut strands in $S^{1}\times[0,1]$
that connects $c_{1}$ at the bottom $S^{1}\times 0$ to $c_{2}$ at the top
$S^{1}\times 1$ modulo isotopy. The composition of diagrams $D_{23}\circ
D_{12}$ is concatenation, and is denoted as $concat(D_{23},D_{12})$.

For any $c\in\mathcal{C}_{1}$, let $T_{c}=c\times\mathbb{R}\subset
S^{1}\times\mathbb{R}\cong\mathbb{C}^{*}$ and let $T^{w}_{c}$ denote a wrapped
version of $T_{c}$.

######  Lemma 5.4.

(1) For any $c_{1},c_{2}\in\mathcal{C}_{1}$, if $T^{w}_{c_{1}}>T^{w}_{c_{2}}$,
then there is a natural injection

$\Phi:T^{w}_{c_{1}}\cap
T^{w}_{c_{2}}\hookrightarrow\mathcal{P}_{1}(c_{1},c_{2}).$

(2) For any $c_{1},c_{2},c_{3}\in\mathcal{C}_{1}$, if
$T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$, and $p_{12},p_{23}$ are
intersection points corresponding to diagrams $D_{12},D_{23}$ where $p_{ij}\in
T^{w}_{c_{i}}\cap T^{w}_{c_{j}}$, then

$\Phi:p_{23}\circ p_{12}\mapsto concat(D_{23},D_{12}).$

###### Proof.

(1) Given a point $p\in T^{w}_{c_{1}}\cap T^{w}_{c_{2}}$, let
$c_{i}^{w}=T^{w}_{c_{i}}\cap U(1)$, where $U(1)\subset\mathbb{C}^{*}$ is the
unit circle. Then we have a natural sequence of paths,

$c_{1}\xrightarrow{U(1)}c_{1}^{w}\xrightarrow{T^{w}_{c_{1}}}p\xrightarrow{T^{w}_{c_{2}}}c^{w}_{2}\xrightarrow{U(1)}c_{2},$

where the label on the arrow indicate where the path is along, the first and
last arrow are given by the isotopy from $T_{c_{i}}$ to $T^{w}_{c_{i}}$, and
the middle two arrow are the unique path along $T^{w}_{c_{i}}\cong\mathbb{R}$
between two points. This path from $c_{1}$ to $c_{2}$ in $\mathbb{C}^{*}$
defines a path on $U(1)$ by taking argument, hence defines a (taut) strand
diagram in $\mathcal{P}(c_{1},c_{2})$. Clearly, different point $p$
corresponds to differnt diagram $D(p)$ (we may isotope $T^{w}_{c_{2}}$ to be
the unwrapped versino $T_{c_{2}}$, then different point $p$ corresponds to
different winding number, hence different $D(p)$).

(2) By Lemma 5.3, we see $p_{23}\circ p_{12}$ is non-zero and is a unique
intersection point $p_{13}\in T^{w}_{c_{1}}\cap T^{w}_{c_{3}}$. It is also
easy to check that composition are compatible, since the holomorphic disk
provides the homotopy of two paths from $c_{1}$ to $c_{3}$. ∎

Next, we consider Fukaya category of $(\mathbb{C}^{*})^{2}$. Let
$\mathcal{C}_{1}^{2}=(S^{1})^{2}$. Then for
$c_{1},c_{2}\in\mathcal{C}_{1}^{2}$, we write $c_{i}=(c_{i,1},c_{i,2})$. If
$c_{1},c_{2}\in\mathcal{C}_{1}^{2}$, then let

$\mathcal{P}_{1}^{2}(c_{1},c_{2}):=\mathcal{P}_{1}(c_{1,1},c_{2,1})\times\mathcal{P}_{1}(c_{1,2},c_{2,2}).$

The composition of $\mathbb{C}\mathcal{P}_{1}^{2}(c_{1},c_{2})$ is given by
concatenation of each strand, denoted as $concat(D_{23},D_{12})$.

######  Lemma 5.5.

(1) For any $c_{1},c_{2}\in\mathcal{C}_{1}^{2}$, if
$T^{w}_{c_{1}}>T^{w}_{c_{2}}$, then there is a natural injection

$\Phi:T^{w}_{c_{1}}\cap
T^{w}_{c_{2}}\hookrightarrow\mathcal{P}_{1}^{2}(c_{1},c_{2}).$

(2) For any $c_{1},c_{2},c_{3}\in\mathcal{C}_{1}^{2}$, if
$T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$, and $p_{12},p_{23}$ are
intersection points corresponding to diagrams $D_{12},D_{23}$ where $p_{ij}\in
T^{w}_{c_{i}}\cap T^{w}_{c_{j}}$, then

$\Phi:p_{23}\circ p_{12}\mapsto concat(D_{23},D_{12}).$

###### Proof.

This follows by taking product of Lemma 5.4. ∎

Finally, we consider Fukaya category of
$(\mathbb{C}^{*})^{2}\backslash\Delta$. Let

$\mathcal{C}_{2}=(S^{1})^{2}\backslash\Delta$

and for all $c_{1},c_{2}\in\mathcal{C}_{2}$, let
$\mathcal{P}_{2}(c_{1},c_{2})$ be the set of taut strands diagrams (modulo
isotopy) from $c_{1}$ to $c_{2}$, and we write $c_{1}\xrightarrow{D}c_{2}$ for
a diagram $D\in\mathcal{P}_{2}(c_{1},c_{2})$. Let

$c_{1}\xrightarrow{D_{12}}c_{2}\xrightarrow{D_{23}}c_{3}$

we define

$D_{23}\circ D_{12}=\begin{cases}concat(D_{23},D_{12})&\text{if the
concatenation is taut}\\\ 0&\text{if the concatenation has a
bigon}\end{cases}$ (20)

For $c\in\mathcal{C}_{2}$, let

$T_{c}=c\times\mathbb{R}^{2}\subset(\mathbb{C}^{*})^{2}\backslash\Delta.$

######  Lemma 5.6.

(1) For $c_{1},c_{2}\in\mathcal{C}_{2}$, if $T^{w}_{c_{1}}>T^{w}_{c_{2}}$,
then there is a natural injection

$\Phi:T^{w}_{c_{1}}\cap
T^{w}_{c_{2}}\hookrightarrow\mathcal{P}_{2}(c_{1},c_{2}).$

(2) For $c_{1},c_{2},c_{3}\in\mathcal{C}_{2}$, and
$T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$, if $p_{12},p_{23}$ are
intersection points ($p_{ij}\in T^{w}_{c_{i}}\cap T^{w}_{c_{j}}$),
corresponding to diagrams $D_{12},D_{23}$, then composition in Fukaya category
is compatible with composition in $\mathcal{C}_{2}$,

$\Phi:p_{23}\circ p_{12}\mapsto D_{23}\circ D_{12}.$

###### Proof.

(1) We note that $\mathcal{C}_{2}\hookrightarrow\mathcal{C}_{1}^{2}$, and if
$c_{1},c_{2}\in\mathcal{C}_{2}$, then
$\mathcal{P}_{2}(c_{1},c_{2})\cong\mathcal{P}_{1}^{2}(c_{1},c_{2})$ where we
choose representative of taut strands by straight strands.

If $T^{w}_{c_{1}}>T^{w}_{c_{2}}$, we may view them as in
$(\mathbb{C}^{*})^{2}$ then apply Lemma 5.5 to get

$T^{w}_{c_{1}}\cap
T^{w}_{c_{2}}\hookrightarrow\mathcal{P}_{1}^{2}(c_{1},c_{2})\cong\mathcal{P}_{2}(c_{1},c_{2})$

(2) Let
$T^{w}_{c_{1}}\xrightarrow{p_{12}}T^{w}_{c_{2}}\xrightarrow{p_{23}}T^{w}_{c_{3}}$.
By Lemma 5.5(2), there exists an intersection point
$T^{w}_{c_{1}}\xrightarrow{p_{13}}T^{w}_{c_{3}}$ and a disk
$\mathcal{D}\hookrightarrow(\mathbb{C}^{*})^{2}$ that realizes the Floer
composition $p_{13}=p_{23}\circ p_{12}$ in $(\mathbb{C}^{*})^{2}$. If the disk
does not intersects diagonal $\Delta$, then in
$(\mathbb{C}^{*})^{2}\backslash\Delta$, we still have $p_{13}=p_{23}\circ
p_{12}$ and if the disk intersects $\Delta$, then $p_{23}\circ p_{12}=0$. The
the desired result follow from the following Lemma. ∎

(a) Bigon cancellation implies disk intersects with diagonal.

(b) No bigon in diagram concatenation corresponds to disk avoiding diagonal.

Figure 4: Caption

######  Lemma 5.7.

Let $T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$ in
$(\mathbb{C}^{*})^{2}\backslash\Delta$. Let $p_{12},p_{23},p_{13}$ be
intersection points corresponding to diagrams $D_{12},D_{23},D_{13}$, and
$\mathcal{D}\to(\mathbb{C}^{*})^{2}$ realizes the composition
$p_{13}=p_{23}\circ p_{12}$ in $(\mathbb{C}^{*})^{2}$.

${T^{w}_{c_{1}}}$${T^{w}_{c_{2}}}$${T^{w}_{c_{3}}}$$\scriptstyle{p_{12}}$$\scriptstyle{p_{13}}$$\scriptstyle{p_{23}}$

Then disk $\mathcal{D}\to(\mathbb{C}^{*})^{2}$ intersects diagonal $\Delta$ if
and only if the concatenation $concat(D_{23},D_{12})$ has a bigon.

###### Proof.

We consider the universal cover $\mathbb{C}_{Z}$ of $\mathbb{C}^{*}_{z}$ and
$(\mathbb{C}_{Z})^{2}$ for $(\mathbb{C}^{*}_{z})^{2}$. The Lagrangians
$T^{w}_{c_{i}}$, intersection points $p_{ij}$ and disk $\mathcal{D}$ in
$(\mathbb{C}^{*}_{z})^{2}$ each admits $\mathbb{Z}^{2}$-many choices of lifts.
We consider all compatible lifts of Lagrangians , intersections and disks

${\widetilde{T}^{w}_{c_{1}}}$${\widetilde{T}^{w}_{c_{2}}}$${\widetilde{T}^{w}_{c_{3}}}$$\scriptstyle{\widetilde{p}_{12}}$$\scriptstyle{\widetilde{p}_{13}}$$\scriptstyle{\widetilde{p}_{23}}$

with a disk $\widetilde{D}\to(\mathbb{C}_{Z})^{2}$ realizing the composition.
We may also consider lifting of diagrams to $\mathbb{R}\times[0,1]$.

Let $\widetilde{\Delta}$ be the diagonal in $(\mathbb{C}_{Z})^{2}$. We claim
the following are equivalent.

  1. (1)

$\mathcal{D}\cap\Delta=\emptyset$.

  2. (2)

For all such lift $\widetilde{D}$ we have
$\widetilde{D}\cap\widetilde{\Delta}=\emptyset$

  3. (3)

For all lift of $concat(D_{23},D_{12})$ to $\mathbb{R}\times[0,1]$, there is
no bigon.

  4. (4)

$concat(D_{23},D_{12})$ has no bigon.

It is clear that $(1)\Leftrightarrow(2)$ and $(3)\Leftrightarrow(4)$, we only
need to show $(2)\Leftrightarrow(3)$.

$(2)\Rightarrow(3)$. We only need to show that, if there is a lift of diagram
with bigon, then the corresponding lifts of Lagrangians, intersection and disk
satisfies $\widetilde{D}\cap\widetilde{\Delta}\neq\emptyset$. Consider Figure
4(a). We have map
$f=(f_{1},f_{2}):\widetilde{D}\hookrightarrow(\mathbb{C}_{Z})^{2}$. We need to
show that there exists $p\in\widetilde{D}$, such that $f_{1}(p)=f_{2}(p)$.
Such $p$ exists by contraction mapping theorem.

$(3)\Rightarrow(2)$. We need to show that, if a lift of diagram with no bigon,
then the corresponding lifts of Lagrangians, intersection and disk satisfies
$\widetilde{D}\cap\widetilde{\Delta}=\emptyset$. Consider Figure 4(b). We have
map $f=(f_{1},f_{2}):\widetilde{D}\hookrightarrow(\mathbb{C}_{Z})^{2}$. We
need to show that there exists no $p\in\widetilde{D}$, such that
$f_{1}(p)=f_{2}(p)$. Indeed, as $p\in\partial\mathcal{D}$, we have the winding
number of $f_{1}(p)-f_{2}(p)$ being zero, hence $f_{1}(p)-f_{2}(p)\neq 0$ for
all $p\in\mathcal{D}$ by argument principle. This finishes the proof of the
Lemma. ∎

Here is another useful Lemma with a similar proof.

######  Lemma 5.8.

Let $a\in\mathbb{C}^{*}$ with $\theta=\arg(a)$. Let
$c_{1},c_{2},c_{3}\in\mathcal{C}_{1}\backslash\theta$, and
$T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$ be wrapping in
$\mathbb{C}^{*}\backslash a$. Let $p_{12},p_{23},p_{13}$ be intersection
points of $T^{w}_{c_{i}}$ with diagram $D_{12},D_{23},D_{13}$, and assume
$\mathcal{D}\to\mathbb{C}^{*}$ realizes the composition of $p_{13}=p_{23}\circ
p_{12}$ in $\mathbb{C}^{*}$. Then the image of $\mathcal{D}$ covers $a$ if and
only if the concatenated diagram $concat(D_{23},D_{12})$ forms a bigon with
the ’red’ line $\theta\times[0,1]\subset S^{1}\times[0,1]$.

###### Proof.

We may use the same argument as in Lemma 5.7 to reduce the discussion to
planar diagram in $\mathbb{R}\times[0,1]$ and Lagrangians in $\mathbb{C}$
(instead of $\mathbb{C}^{*}$). If the composition of diagram has a bigon with
the red line, then the corresponding Lagrangians will be as in Figure 5, and
the disk clearly covers the point $a$. Conversely, if there is no bigon with
the red line, then the disk will not cover point $a$.

Figure 5: Bigon cancellation between a black and a red strand would involve a
disk that pass through the red puncture, hence forbidden in the $\hbar=\eta=0$
setting.

∎

Finally, we consider Fukaya category of
$\operatorname{Sym}^{2}(\mathbb{C}^{*})/\Delta$. Let

$\mathcal{C}_{[2]}=\operatorname{Sym}^{2}(S^{1})\backslash\Delta.$

For $c_{1},c_{2}\in\mathcal{C}_{[2]}$, let $\mathcal{P}_{[2]}(c_{1},c_{2})$ be
the set of taut diagrams. We define composition with the same formula as in
Eq20. For $c\in\mathcal{C}_{[2]}$, we may pick a lift
$\widetilde{c}\in\mathcal{C}_{2}$, then we define

$T_{c}=T_{\widetilde{c}}/S_{2}.$

######  Lemma 5.9.

(1) For $c_{1},c_{2}\in\mathcal{C}_{[2]}$, if $T^{w}_{c_{1}}>T^{w}_{c_{2}}$,
then there is a natural injection

$\Phi:T^{w}_{c_{1}}\cap
T^{w}_{c_{2}}\hookrightarrow\mathcal{P}_{[2]}(c_{1},c_{2}).$

(2) For $c_{1},c_{2},c_{3}\in\mathcal{C}_{2}$, and
$T^{w}_{c_{1}}>T^{w}_{c_{2}}>T^{w}_{c_{3}}$, if $p_{12},p_{23}$ are
intersection points ($p_{ij}\in T^{w}_{c_{i}}\cap T^{w}_{c_{j}}$),
corresponding to diagrams $D_{12},D_{23}$, then composition in Fukaya category
is compatible with composition in $\mathcal{C}_{[2]}$,

$\Phi:p_{23}\circ p_{12}\mapsto D_{23}\circ D_{12}.$

###### Proof.

(1) Choose a lifting of $c_{i}$ to $\widetilde{c}_{i}\in\mathcal{C}_{2}$. Let
$S_{2}=\\{e,\sigma\\}$. Then we have natural isomorphism

$\mathcal{P}_{[2]}(c_{1},c_{2})=\sqcup_{g\in
S_{2}}\mathcal{P}_{2}(\widetilde{c}_{1},g\widetilde{c}_{2})$

and

$T^{w}_{c_{1}}\cap T^{w}_{c_{2}}=\sqcup_{g\in
S_{2}}T^{w}_{\widetilde{c}_{1}}\cap T^{w}_{g\widetilde{c}_{2}}$

Hence the result follows from Lemma 5.6.

(2) Let $c_{1}\xrightarrow{D_{12}}c_{2}\xrightarrow{D_{23}}c_{3}$ be diagrams.
Choose a lifting of $c_{1}$ to $\widetilde{c}_{1}\in\mathcal{C}_{2}$. Then the
diagrams determine a lifting of $c_{2}\leadsto\widetilde{c}_{2}$ and
$c_{3}\leadsto\widetilde{c}_{3}$. We have corresponding lift of
$T^{w}_{c_{i}}\subset\operatorname{Sym}^{2}(\mathbb{C}^{*})/\Delta$ to
$T^{w}_{\widetilde{c}_{i}}\subset(\mathbb{C}^{*})^{2}/\Delta$, and the disk
$\mathcal{D}$ bounded by $T^{w}_{c_{i}}$, if exists, would lift to
$\widetilde{D}$ bounded by $\widetilde{T}^{w}_{c_{i}}$. Hence the result of
Lemma 5.6 implies the desired result here. ∎

###  5.3 Wrapping and Intersections

We will consider the partial wrapping associated to the potential

$\mathcal{W}=\sum_{{(i,j)}}u_{{(i,j)}}:({\bf T}^{\vee}\times{\bf T}_{O})/{\bf
W}\to\mathbb{C}.$

A cofinal sequence of wrappings of $\mathcal{T}_{c}$ may be realized by a
family which remain symmetric products (we will explain briefly why at the end
of the section; alternatively, the reader could choose simply to view this as
our definition of wrapping in this space).

Because we have a superpotential $\sum_{{(i,j)}}u_{{(i,j)}}$, the wrapping in
the $\mathbb{C}^{*}_{u}$ factors is partially stopped as in the model case
$(\mathbb{C}^{*}_{u},W=u)$. 111We will actually choose the superpotential to
be $W=e^{-i\epsilon}u$, for $0<\epsilon\ll 1$, so that the stop will be at
$u=\infty e^{i\epsilon}\in\partial_{\infty}\mathbb{C}^{*}$. We make such a
perturbation so that the Lagrangian $\mathbb{R}_{+}\subset\mathbb{C}^{*}_{u}$
will have its ideal boundary just before the stop with respect to the Reeb
flow. The wrapping in the $\mathbb{C}^{*}_{y}$ factor is unstopped. See Figure
6.

Fix black points configurations $c_{1},c_{2}$, and correspondingly consider
$\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}}$. Let $\mathcal{T}_{c_{1}}^{w}$,
$\mathcal{T}_{c_{2}}^{w}$ be certain positived wrapped versions of them.

  1. 1.

We write $\mathcal{T}_{c_{1}}^{w}>\mathcal{T}_{c_{2}}^{w}$ if for each node
$(i)\in\Gamma$, and for each $j_{1},j_{2}\in\\{1,\cdots,d_{i}\\}$, we have
$\mathcal{T}_{c_{1},y_{(i,j_{1})}}^{w}>\mathcal{T}_{c_{2},y_{(i,j_{2})}}^{w}$
and
$\mathcal{T}_{c_{1},u_{(i,j_{1})}}^{w}>\mathcal{T}_{c_{2},u_{(i,j_{2})}}^{w}$.
See Figure 6 for an illustration.

  2. 2.

We write
$\mathcal{T}_{c_{1}}^{w}>\mathcal{T}_{c_{2}}^{w}>\cdots>\mathcal{T}_{c_{n}}^{w}$
if for any $i<j$ we have $\mathcal{T}^{w}_{c_{i}}>\mathcal{T}^{w}_{c_{j}}$.

(a) Base factor $\mathbb{C}^{*}_{y}$ wrapping

(b) Fiberwise $\mathbb{C}^{*}_{u}$ wrapping

Figure 6: Wrapping of $\mathcal{T}$ in the base and in the fiber

Next, we describe how to use weighted taut strands diagram to describe
intersections of wrapped $\mathcal{T}_{c}$. This follows from the model cases
in the previous subsection, we still give a complete description. Let
$c_{1},c_{2}$ be two black strands configurations, and consider the
corresponding Lagrangians $\mathcal{T}_{c_{i}}$. Let $\mathcal{T}^{w}_{c_{i}}$
be their wrapped versions, such that
$\mathcal{T}^{w}_{c_{1}}>\mathcal{T}^{w}_{c_{2}}$. Let
$p\in\mathcal{T}^{w}_{c_{1}}\cap\mathcal{T}^{w}_{c_{2}}$ be an intersection
point. We now describe how to assign a weighted taut strand diagram
$D(p)\in\mathcal{P}(c_{1},c_{2})$ for the intersection point $p$.

We make a choice and label the points in $c_{1}$ and $c_{2}$ as
$(c_{1})_{{(i,j)}}$ and $(c_{2})_{(i,j)}$, then we may write factors of the
symmetric product Lagrangian as $\mathcal{T}_{c_{k},y_{(i,j)}}$ etc. The
intersection point is

$p=\prod_{(i,j)}(p_{y,{(i,j)}},p_{u,{(i,j)}}),\quad\text{where
}p_{y,{(i,j)}}\in\widetilde{\mathcal{T}}_{c_{1},y_{{(i,j)}}}^{w}\cap\widetilde{\mathcal{T}}_{c_{2},y_{\sigma{(i,j)}}}^{w},\text{
and
}p_{u,{(i,j)}}\in\widetilde{\mathcal{T}}_{c_{1},u_{{(i,j)}}}^{w}\cap\widetilde{\mathcal{T}}_{c_{2},u_{\sigma{(i,j)}}}^{w},$

for some $\sigma\in{\bf W}$. The taut strands diagram lives in
$U(1)_{y}\times[0,1]$, where the bottom $U(1)_{y}\times\\{0\\}$ is decorated
by $c_{1}$ and the top is decorated by $c_{2}$. Take a black point
$(c_{1})_{{(i,j)}}$ in $c_{1}$, find the corresponding factor
$\mathcal{T}_{c_{1},y_{{(i,j)}}}$. Write
$(c_{1})_{(i,j)}=\mathcal{T}_{c_{1},y_{{(i,j)}}}\cap U(1)$,
$(c_{1})^{w}_{(i,j)}=\mathcal{T}^{w}_{c_{1},y_{{(i,j)}}}\cap U(1)$, and
similar for $c_{2}$. Then we get a path in $\mathbb{C}^{*}_{y}$ as following

$(c_{1})_{(i,j)}\leadsto(c_{1})^{w}_{(i,j)}\leadsto
p_{y,{(i,j)}}\leadsto(c_{2})^{w}_{{(i,j)}}\to(c_{2})_{{(i,j)}},$

where the first and last segments of path are along $U(1)_{y}$ and are induced
by Lagrangian isotopy, and the second and third are along
$\mathcal{T}^{w}_{c_{1},y_{(i,j)}}$ and
$\mathcal{T}^{w}_{c_{2},y_{\sigma{(i,j)}}}$ respectively. Take the projection
$\mathbb{C}^{*}_{y}\to U(1)_{y}$, we then get a path from $(c_{1})_{{(i,j)}}$
to $(c_{2})_{\sigma{(i,j)}}$ in $U(1)_{y}$ which determines a straight-line
strand in $S^{1}\times[0,1]$ from $(c_{1})_{(i,j)}$ to
$(c_{2})_{\sigma{(i,j)}}$. Similar consideration in the $u$-factors gives a
winding number (winding counter-clockwise around $u=0$ contributes to $+1$)
which we label as weight on the strand. Repeat this construction for all
$(i,j$ determines the weighted taut strands diagram $D(p)$.

Thus, we have shown that

######  Proposition 5.10.

If $\mathcal{T}_{c_{1}}^{w}>\mathcal{T}_{c_{2}}^{w}$, then we have a natural
injection

$\mathcal{T}_{c_{1}}^{w}\cap\mathcal{T}_{c_{2}}^{w}\hookrightarrow\mathcal{P}(c_{1},c_{2});$

Every element in $\mathcal{P}(c_{1},c_{2})$ is realized by some wrapping.

(a) An example of one strands winding around in the base and in the fiber.

(b) An example of two strands crossing

Figure 7: Two examples of intersection points as weighted taut strands diagram

###  5.4 Composition

Recall that a black points configuration on $S^{1}$, $c=\\{\theta_{(i,j)}\\}$,
defines an element of the appropriate KLRW category, with the individual
$\theta_{(i,j)}$ giving the positions of the black dots. Recall also the ‘taut
strands’ basis of Def. 4.2, and the correspondence with intersection points
from Prop 5.10.

Consider the pull-back diagram

${\widetilde{\mathcal{C}}}$${\prod_{i}(S^{1})^{d_{i}}\backslash\Delta}$${\mathcal{C}}$${(\prod_{i}(S^{1})^{d_{i}}\backslash\Delta)/{\bf
W}}$

For $c\in\mathcal{C}$, if $\widetilde{c}\in\widetilde{\mathcal{C}}$ is a lift
of $c$, then we have a lift of $\mathcal{T}_{c}$ to
$\widetilde{\mathcal{T}}_{\widetilde{c}}\subset{\bf T}^{\vee}\times{\bf
T}_{O}.$

We claim:

######  Proposition 5.11.

Let $c_{1},c_{2},c_{3}\in\mathcal{C}$, and let $\mathcal{T}_{c_{i}}$ be the
corresponding Lagrangians. Choose any positive wrappings of
$\mathcal{T}_{c_{i}}$ such that
$\mathcal{T}^{w}_{c_{1}}>\mathcal{T}^{w}_{c_{2}}>\mathcal{T}^{w}_{c_{3}}$.
Choose any intersection points
$p_{ij}\in\mathcal{T}^{w}_{c_{i}}\cap\mathcal{T}^{w}_{c_{j}}$ for
$ij=12,23,13$, and let $D_{ij}$ denote the corresponding diagram in
$\mathcal{P}(c_{i},c_{j})$. Then $p_{13}$ appears in $p_{23}\circ p_{12}$ if
and only if the following holds

  1. (1)

$D_{13}$ is the straightening out of $concat(D_{23},D_{12})$ with weights
added on each strands.

  2. (2)

There is no bigon cancellation between black strands of same label.

  3. (3)

There is no bigon cancellation between black strands of adjacent labels $(i)$
and $(i^{\prime})$, i.e., $(i)\to(i^{\prime})$ has an arrow.

  4. (4)

There is no bigon cancellation between black strands and red strands of the
same label (corresponding to a framing edge $[i]\to(i)$).

(That is, bigon cancellations are allowed only for strands labelled by quiver
nodes which are not connected by an edge.)

###### Proof.

To determine compositions, we are counting disks $\mathcal{D}$ with three
boundary marked points mapping to the target space with prescribed Lagrangian
boundary condition. We use the cylindrical model of Theorem 3.11. Note that
because the Lagrangians are of symmetric product type, the boundary conditions
may also be expressed in the cylindrical model.

Since we deleted all root and matter divisors, the conditions from Theorem
3.11 on the map
$\mathcal{S}\to\mathcal{D}\times\mathbb{C}^{*}_{y}\times\mathbb{C}^{*}_{u}$
simplify to the following:

  1. (i)

Each map $\pi_{i,z}:\mathcal{S}_{i}\to\mathcal{D}_{z}$ is a unramified cover
of degree $d_{i}$, namely $\mathcal{S}_{i}$ is a disjoint union of $d_{i}$
disks, which we denote as
$\mathcal{S}_{i}=\sqcup_{j=1}^{d_{i}}\mathcal{S}_{i,j}$.

  2. (ii)

Each map
$(\pi_{i,z},\pi_{i,y}):\mathcal{S}_{i}\to\mathcal{D}_{z}\times\mathbb{C}^{*}_{y}$
is an embedding.

  3. (iii)

For each arrow $(i)\to(i^{\prime})$, the embedding images of $\mathcal{S}_{i}$
and $S_{j}$ in $\mathcal{D}_{z}\times\mathbb{C}^{*}_{y}$ are disjoint. Since
otherwise the map $\mathcal{D}\to{\bf T}/{\bf W}$ would intersect with the
loci where $y_{{(i,j)}}=y_{(i^{\prime},j^{\prime})}$.

  4. (iv)

For each arrow $[i]\to(i)$, the projection image
$\mathcal{S}_{i}\to\mathbb{C}^{*}_{y}$ is disjoint from any $a_{{[i,j]}}$.

We only need to show that the four conditions (i) - (iv) implies condition (1)
- (4) in the proposition.

$(i)\Leftrightarrow(1)$. We first show that there exists a unique such disk
$\mathcal{S}\to\mathcal{D}\times\mathbb{C}^{*}_{y}\times\mathbb{C}^{*}_{u}$
satisfing condition (i) above. Indeed, we may choose a lift of
$\mathcal{T}^{w}_{c_{k}}$ to
$\widetilde{\mathcal{T}}^{w}_{\widetilde{c}_{k}}\subset{\bf
T}^{\vee}\times{\bf T}$, for $k=1,2,3$, such that $p_{12}$ and $p_{23}$ lift
as intersections of the lifted $\mathcal{T}^{w}_{c_{k}}$. For each index
$(i,j)$, we have
$\mathcal{T}^{w}_{c_{1},y_{(i,j)}}>\mathcal{T}^{w}_{c_{2},y_{(i,j)}}>\mathcal{T}^{w}_{c_{3},y_{(i,j)}}$
in $\mathbb{C}^{*}_{y}$, and similarly for $\mathbb{C}^{*}_{u}$ factor. By
Lemma 5.3, there exists a unique disk $\mathcal{S}_{i,j}\to\mathbb{C}^{*}_{y}$
realizing the composition of $p_{12,y_{(i,j)}}$ and $p_{23,y_{(i,j)}}$,
similarly for $\mathbb{C}^{*}_{u}$ direction. Thus $D_{13}$ is necessarily is
the concatenation of $D_{23}$ and $D_{12}$ with weights added on the
corresponding strands.

$(ii)\Leftrightarrow(2)$ and $(iii)\Leftrightarrow(3)$ follows from Lemma 5.7.

$(iv)\Leftrightarrow(4)$ follows from Lemma 5.8. ∎

######  Theorem 5.12.

There is a fully faithful functor

$\displaystyle KLRW_{Q}|_{\hbar=\eta=0}$ $\displaystyle\to$
$\displaystyle\operatorname{Fuk}((\mathbf{T}_{O}\times\mathbf{T}^{\vee})/\mathbf{W},\mathcal{W})$
$\displaystyle c$ $\displaystyle\mapsto$ $\displaystyle\mathcal{T}_{c}$

###### Proof.

Fix any $c_{1},c_{2}\in\mathcal{C}.$ Recall we can choose wrappings where all
the intersections have zero cohomological degree. Using also Proposition 5.10,
we have:

$Hom_{Fuk}(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})=HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})=\text{colim}_{\mathcal{T}^{w}_{c_{1}}>\mathcal{T}_{c_{2}}}CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}_{c_{2}})=\text{colim}_{\mathcal{T}^{w}_{c_{1}}>\mathcal{T}_{c_{2}}}\mathbb{C}\mathcal{P}(\mathcal{T}^{w}_{c_{1}},\mathcal{T}_{c_{2}})$

Note that the above considerations alone does not imply that the maps in the
colimit on the RHS are compatible with the natural inclusions into
$\mathbb{C}\mathcal{P}(c_{1},c_{2})$. However, said compatibility is
established by Proposition 5.11. Combining this with Proposition 4.3, we get a
canonical vector space isomorphism:

$Hom_{KLRW}(c_{1},c_{2})_{\hbar=\eta=0}\cong\mathbb{C}\mathcal{P}(c_{1},c_{2})\cong
Hom_{Fuk}(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})$ (21)

Also by Proposition 5.11 and Lemma 4.4, this carries composition in the KLRW
category to the composition in the Fukaya category. ∎

###  5.5 Comparison of Liouville structure

Let us return briefly to clarify some issues about how our wrapping
prescription above compares with the general setting of [18, 20]. The space
$(\mathbf{T}_{O}\times\mathbf{T}^{\vee})/W$ acquires the structure of
Weinstein symplectic manifold via a Kähler form arising from the structure of
this space as an affine algebraic variety. We will want to compare this
symplectic structure to both the symplectic structure descended from the
product $\mathbf{T}\times\mathbf{T}^{\vee}$, and later to compare to
$\mathcal{M}^{\times}(\Gamma,\vec{d})$ (of which
$(\mathbf{T}_{O}\times\mathbf{T}^{\vee})/W$ is an open subset by Theorem 1.2).
The basic point is the following fact:

######  Lemma 5.13.

Suppose $Z$ is a smooth affine algebraic variety, and $f:Z\to\mathbb{C}$ a
function. Let $\lambda_{Z}$ and $\lambda_{Z\setminus f^{-1}(0)}$ be Kähler
Liouville structures on $Z$ and $Z\setminus f^{-1}(0)$.

Fix $\epsilon>0$ sufficiently small. Then there is a Liouville domain
$\overline{Z}\subset Z$ completing to $Z$ and an isotopy of Liouville
structures $\lambda_{Z\setminus f^{-1}(0)}\sim\widetilde{\lambda}_{Z\setminus
f^{-1}(0)}$ such that $\lambda_{Z}=\widetilde{\lambda}_{Z\setminus f^{-1}(0)}$
on $\overline{Z}\cap\\{|f|\geq\epsilon\\}$.

######  Corollary 5.14.

Let $L_{t}\subset\bar{Z}\cap\\{|f|\geq\epsilon\\}$ be a family of Lagrangians
with Legendrian boundary $\partial L_{t}\subset\bar{Z}\setminus Z$. Then
$L_{t}$ satisfies the cofinality criterion of [20] with respect to
$\lambda_{Z}$ iff it does with respect to $\widetilde{\lambda}_{Z\setminus
f^{-1}(0)}$.

##  6 Gradings

In general, if $L$ is a contractible Lagrangian in a Weinstein manifold $M$,
and $\lambda\in H^{1}(M,\mathbb{Z})\cong H^{1}(M,L;\mathbb{Z})$ is an integral
class, then given a path $\gamma$ in $M$ from $L$ to $L$, we can get a integer
$\langle\lambda,\gamma\rangle$ using the pairing

$H_{1}(M,L;\mathbb{Z})\times H^{1}(M,L;\mathbb{Z})\to\mathbb{Z}.$

Then $\operatorname{End}(L)$ is a $\mathbb{Z}$-graded algebra.

If $f:M\to\mathbb{C}^{*}$ is a continuous function, then $(2\pi i)^{-1}d\log
f\in H^{1}(M,\mathbb{Z})$ give an integral class. We have two kinds of such
functions

  1. 1.

For each circle node $(i)$, we have

$f_{i}(u,y)=\prod_{{(i,j)}}y_{{(i,j)}}$

This is called the equivariant $q_{i}$-grading.

  2. 2.

By Proposition 3.9, we have a monopole operator

$f_{0}=r_{(\vec{1},\cdots,\vec{1})}$

corresponding to the monopole operator for the diagonal embedding cocharacter
$\mathbb{C}^{*}\to G=\prod_{i}GL(d_{i})$. On $({\bf T}_{O}\times{\bf
T}^{\vee})/{\bf W}$, we have an explicit expression

$f_{0}(u,y)=\prod_{{(i,j)}}u_{{(i,j)}}^{-1}\cdot\prod_{(i,j)\to(i^{\prime},j^{\prime})}(1-y_{(i,j)}/y_{(i^{\prime},j^{\prime})})\cdot\prod_{[i,k]\to(i,j)}(1-a_{[i,k]}/y_{(i,j)})\cdot\prod_{(i,j)\neq(i,j^{\prime})}(1-y_{(i,j)}/y_{(i,j^{\prime})})^{-1}$
(22)

This is called the $q$-grading.

For each $c\in\mathcal{C}$, $\mathcal{T}_{c}\cong\mathbb{R}^{2d}$ is simply
connected, hence $\operatorname{End}(\mathcal{T}_{c})$ has well-defined $q$
and $q_{i}$ grading.

For $c\neq c^{\prime}\in\mathcal{C}$, the space
$\operatorname{Hom}(\mathcal{T}_{c},\mathcal{T}_{c^{\prime}})$ is not
naturally graded but instead a torsor for the grading space for either $c$ or
$c^{\prime}$. One choice of trivialization of the torsor is:

  1. 1.

a weight $w$ on any strand has $q$-grading $+w$.

  2. 2.

a crossing between strands of the same type has $q$-grading $-1$.

  3. 3.

if there is an arrow $[i]\to(i)$, then a strand of type $(i)$ crossing the red
strand $[i]$ has $q$-grading $+1/2$.

  4. 4.

if there is an arrow $(i)\to(j)$, then a strand of type $(i)$ crossing strand
$(j)$ has $q$-grading $+1/2$.

  5. 5.

all other crossings has $q$-grading $0$.

  6. 6.

Fix a representative in each connected component of $\mathcal{C}$ (there are
finitely many). Fix a generic $\theta\in S^{1}$, away from any red points and
any black points in the representatives. If a black strand of type $(i)$
crosses this line from right to left ($\arg$ is increasing), then $q_{i}$
grading is $+1$; if the crossing is from left to right, then $q_{i}$ grading
is $-1$.

From Eq (22), rule 1 comes from the factor $\prod_{{(i,j)}}u_{(i,j)}^{-1}$;
rule 2 comes from
$\prod_{(i,j)\neq(i,j^{\prime})}(1-y_{(i,j)}/y_{(i,j^{\prime})})^{-1}$; rule 3
comes from factor $\prod_{[i,k]\to(i,j)}(1-a_{[i,k]}/y_{(i,j)})$; rule 4 comes
from
$\prod_{(i,j)\to(i^{\prime},j^{\prime})}(1-y_{(i,j)}/y_{(i^{\prime},j^{\prime})})$.

##  7 Floer homologies with only root divisor removed

Consider the following (less) punctured torus

${\bf T}_{o}=\\{(y_{(i,j)})\in{\bf T}\mid y_{(i,j)}\neq
y_{(i,j^{\prime})}\;\;\forall(i,j)\text{ and }(i,j^{\prime})\\},$

and the corresponding base change:

${\mathcal{Y}_{o}}$${\mathcal{M}^{\times}(\Gamma,\vec{d})}$${{\bf T}_{o}/{\bf
W}}$${{\bf T}/{\bf W}}$

Let $\overline{H}_{matter}=({\bf T}_{o}-{\bf T}_{O})/{\bf W}$, and
$H_{matter}$ be the preimage of $\overline{H}_{matter}$ in $\mathcal{Y}_{o}$.
Let
$\operatorname{Fuk}_{\eta}(\mathcal{Y}_{o},\mathcal{W}|_{\mathcal{Y}_{o}};H_{matter})$
be the Fukaya category defined over $\mathbb{C}[\eta]$, where we weight the
count of a disk $\mathcal{D}$ by $\eta^{\mathcal{D}\cdot H_{matter}}$.

Our main result in this section is the following embedding.

######  Theorem 7.1.

There is a fully faithful functor

$\displaystyle KLRW_{Q}|_{\hbar=0}$ $\displaystyle\to$
$\displaystyle\operatorname{Fuk}_{\eta}(\mathcal{Y}_{o},\mathcal{W}|_{\mathcal{Y}_{o}};H_{matter})$
$\displaystyle c$ $\displaystyle\mapsto$ $\displaystyle\mathcal{T}_{c}$

###  7.1 Hom spaces and taut diagrams

######  Proposition 7.2.

Let $c_{1},c_{2},c_{3}\in\mathcal{C}$, and consider wrappings
$\mathcal{T}^{w}_{c_{1}}>\mathcal{T}^{w}_{c_{2}}>\mathcal{T}^{w}_{c_{3}}$. Let
$D_{12},D_{23}$ be weighted taut strands diagrams that represent intersection
points (hence morphisms)

$\mathcal{T}^{w}_{c_{1}}\xrightarrow{D_{12}}\mathcal{T}^{w}_{c_{2}}\xrightarrow{D_{23}}\mathcal{T}^{w}_{c_{3}}.$

If the concatenation $concat(D_{23},D_{12})$ is taut, denoted as $D_{13}$ then
$D_{13}$ represent an intersection point in
$CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}^{w}_{c_{3}})$ and the output of the
Fukaya category composition is

$\mu_{2}(D_{23},D_{12})=D_{13}.$

###### Proof.

From Prop 5.11, we see there is a disk $\mathcal{D}\to({\bf
T}^{\vee}\times{\bf T}_{O})/{\bf W}$ that realizes the composition
$\mu_{2}(D_{23},D_{12})=D_{13}$. The argument in 5.3 shows that such disk
$\mathcal{S}\to\mathcal{D}\times\mathbb{C}^{*}_{y}$ is unique, hence this is
the only disk contributing to $\mu_{2}(D_{23},D_{12})$. ∎

######  Corollary 7.3.

Given
$\mathcal{T}^{w}_{c_{1}}>\mathcal{T}^{w^{\prime}}_{c_{1}}>\mathcal{T}^{w}_{c_{2}}$,
and
$e_{11^{\prime}}\in\mathcal{T}^{w}_{c_{1}}\cap\mathcal{T}^{w^{\prime}}_{c_{1}}$
corresponds to the identity diagram, and
$p_{1^{\prime}2}\in\mathcal{T}^{w^{\prime}}_{c_{1}}\cap\mathcal{T}^{w}_{c_{2}}$
corresponds to some diagram $D_{12}$, then there is an intersection
$p_{12}\in\mathcal{T}^{w}_{c_{1}}\cap\mathcal{T}^{w}_{c_{2}}$ corresponding to
$D_{12}$, and

$\mu_{2}(p_{1^{\prime}2},e_{11^{\prime}})=p_{12}.$

###### Proof.

The identity diagram composed with any taut diagram is still taut. ∎

######  Corollary 7.4.

For any $c_{1},c_{2}\in\mathcal{C}$, we have canonical isomorphism

$HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})\cong\mathbb{C}\mathcal{P}(c_{1},c_{2}).$

###### Proof.

Since for each wrapped versions $\mathcal{T}^{w}_{c_{1}}>\mathcal{T}_{c_{2}}$
we have natural basis (intersection points) in
$CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}_{c_{2}})$ identified with diagrams in
$\mathcal{P}(c_{1},c_{2})$ (Proposition 5.10), and the continuation map
identifies these basis (Corollary 7.3), hence
$HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})$ has a natural basis given by
diagrams in $\mathbb{C}\mathcal{P}(c_{1},c_{2})$. Further more, any diagrams
in $\mathbb{C}\mathcal{P}(c_{1},c_{2})$ can be realized in wrapped Floer
cohomology by sufficient wrapping, hence we have the desired isomorphism. ∎

###  7.2 KLRW relations from disk counts, I: matter divisor

Here we will count certain disks, which will ultimately give rise to the
bigon-dot relations in the KLRW category.

####  7.2.1 Bigon of the same node

$(i)$$(i)$$=0$

This follows from Lemma 5.7 and we removed root divisor, ie, diagonal divisor.

####  7.2.2 Bigon between red and black

Figure 8: KLRW relation for bigon between a black strand and a red strand

Consider three Lagrangans $\mathcal{T}_{1}>\mathcal{T}_{2}>\mathcal{T}_{3}$,
drawn with color brown, green and blue in Figure 8. On the base
$\mathbb{C}^{*}_{y}$ direction, we have $\mathcal{T}_{1},\mathcal{T}_{3}$ are
on the same side of the marked point, and $\mathcal{T}_{2}$ on the other side.
On the fiber $\mathbb{C}^{*}_{u}$ direction, we pick the intersection points
in $\mathcal{T}_{1}\cap\mathcal{T}_{2}$ and
$\mathcal{T}_{2}\cap\mathcal{T}_{3}$ with winding number zero (marked by $1$
and $2$), and the intersection point in $\mathcal{T}_{1}\cap\mathcal{T}_{3}$
with winding number $1$ (marked as $3$).

Now we are ready to count disk. Consider a disk with three boundary marked
points and one interior marked point, denoted as $\mathcal{D}_{3,1}$. Such a
disk has a real 2 dimensional moduli space $\mathcal{M}_{\mathcal{D}_{3,1}}$.

Recall from Theorem 1.5, for the base direction map
$\mathcal{D}\to\mathbb{C}^{*}_{y}$, the interior marked point in $\mathcal{D}$
maps to the marked point $a\in\mathbb{C}^{*}_{y}$; for the fiber direction
$\mathcal{D}\to\mathbb{P}^{1}_{u}$, the interior marked point maps to $u=0$.

${\mathcal{M}_{\mathcal{D}_{3,1}\to\mathbb{C}^{*}_{y}\times\mathbb{P}^{1}_{u}}(\dim_{\mathbb{R}}=0)}$${\mathcal{M}_{\mathcal{D}_{3,1}\to\mathbb{C}^{*}_{y}}(\dim_{\mathbb{R}}=0)}$${\mathcal{M}_{\mathcal{D}_{3,1}\to\mathbb{P}^{1}_{u}}(\dim_{\mathbb{R}}=2)}$${\mathcal{M}_{\mathcal{D}_{3,1}}(\dim_{\mathbb{R}}=2)}$

The base direction map has no moduli and is consist of one point. The fiber
direction map has two degrees of freedom from the ’obtuse angle’ at points $2$
and $3$. A more careful analysis shows that the map
$\mathcal{M}_{\mathcal{D}_{3,1}\to\mathbb{P}^{1}_{u}}\to\mathcal{M}_{\mathcal{D}_{3,1}}$
is a bijection. Hence the fiber product
$\mathcal{M}_{\mathcal{D}_{3,1}\to\mathbb{C}^{*}_{y}\times\mathbb{P}^{1}_{u}}$
consist of a single point.

####  7.2.3 Bigon of black strands of adjacent nodes circle nodes

Figure 9: KLRW relation for bigon between a black strand and a red strand.
Figure (c) shows the first composition output term in (a).

Consider two adjacent nodes in the quiver $(j)\to(i)$. We draw the Lagrangians
base and fiber picture as in Figure 9.

We show that the first output (dot on the (i) strand) can be realized, the
second output can be similarly analysed which we omit. We are looking for two
disks $\mathcal{S}_{i},\mathcal{S}_{j}$ with three marked points (i.e.
triangle), with maps
$\mathcal{S}_{i}\sqcup\mathcal{S}_{j}\to\mathcal{D}\times\mathbb{C}^{*}_{y}\times\mathbb{C}_{u}$,
such that

  1. 1.

In $\mathbb{C}^{*}_{y}$, $\mathcal{S}_{j}$ (resp $\mathcal{S}_{i}$) maps to
the triangle with vertex $1,2,3$ (resp $4,5,6$).

  2. 2.

In $\mathbb{C}_{u}$, $\mathcal{S}_{j}$ maps to the triangle with vertex
$1,2,3$, and $S_{i}$ maps into the region with corner $4,5,6$.

  3. 3.

There is a $s_{i}\in\mathcal{S}_{i}$ and $s_{j}\in\mathcal{S}_{j}$ that maps
to the same point $(z_{0},y_{0})$ in $\mathcal{D}\times\mathbb{C}^{*}_{y}$,
and such that under $S_{j}\to\mathbb{C}_{u}$, we have $s_{j}\mapsto 0$.

We now show that there is a unique such disk.

  1. 1.

Let $\mathcal{M}_{\mathcal{S}}$ be the moduli space of
$\\{(\mathcal{S}_{i},s_{i}),(\mathcal{S}_{j},s_{j})\\}$ of two disks, each
with 3 boundary marked points and one interior marked points,
$\mathcal{M}_{\mathcal{S}}\cong\mathcal{M}_{\mathcal{D}_{3,1}}^{2}$.

  2. 2.

Let $\mathcal{M}_{\mathcal{S}\to\mathcal{D}}$ be the moduli space of disk
$\mathcal{D}$ with 3 boundary marked points and 1 interior marked point, and
isomorphism $\mathcal{S}_{i}\to\mathcal{D},\mathcal{S}_{j}\to\mathcal{D}$,
hence $\mathcal{M}_{\mathcal{S}\to\mathcal{D}}$ is the diagonal in
$\mathcal{M}_{\mathcal{S}}$.

  3. 3.

Let $\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{y}^{*}}$ be the moduli space of
$\mathcal{S}_{j}$ (resp $\mathcal{S}_{i}$) maps to the triangle with vertex
$1,2,3$ (resp $4,5,6$), and such that the marked points
$s_{i}\in\mathcal{S}_{i},s_{j}\in\mathcal{S}_{j}$ maps to the same point. Then
the moduli space is determined by the image $y_{0}$ of $s_{i}$, hence
$\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{y}^{*}}$ is isomorphic to choice of
$y_{0}$, namely the triangle with vertex $123$ in $\mathbb{C}^{*}_{y}$.

  4. 4.

Let $\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{u}}$ be the moduli space where
$\mathcal{S}_{j}$ is mapped to the triangle $123$ in $\mathbb{C}_{u}$, and
$\mathcal{S}_{i}$ is mapped to a degenerated ’triangle’ where the three
boundary marked points are mapped to $4,5,6$ and the interior point is mapped
to $0\in\mathbb{C}_{u}$ (see the green shaded region in Figure 9 (c)). The
obtuse corner $5,6$ gives two real degrees of freedom for $\mathcal{S}_{j}$
with marked points, and a more careful analysis shows that the forgetful map
$\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{u}}\to\mathcal{M}_{\mathcal{S}}$ is an
isomorphism.

By contraction mapping argument, the following fiber product
$\mathcal{M}_{\mathcal{S}\to\mathcal{D}\times\mathbb{C}_{y}^{*}\times\mathbb{C}_{u}}$
is one point.

${\mathcal{M}_{\mathcal{S}\to\mathcal{D}\times\mathbb{C}_{y}^{*}\times\mathbb{C}_{u}}(0\dim_{\mathbb{R}})}$${\mathcal{M}_{\mathcal{S}\to\mathcal{D}}(2\dim_{\mathbb{R}})}$${\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{y}^{*}}(2\dim_{\mathbb{R}})}$${\mathcal{M}_{\mathcal{S}\to\mathbb{C}_{u}}(4\dim_{\mathbb{R}})}$${\mathcal{M}_{\mathcal{S}}(4\dim_{\mathbb{R}})}$

###  7.3 Proof of Theorem 7.1

From Proposition 4.5 and Corollary 7.4, we see for any
$c_{1},c_{2}\in\mathcal{C}$, there is a vector space isomorphism

$\operatorname{Hom}_{KLRW,\hbar=0}(c_{1},c_{2})\cong\operatorname{Hom}_{Fuk}(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})\cong\mathbb{C}\mathcal{P}(c_{1},c_{2}).$

Let $\circ_{KLRW}$ and $\circ_{Fuk}$ denote composition in the two categories,
and let $\circ$ denote concatenation of diagram. Then, for any weighted taut
diagrams $c_{1}\xrightarrow{D_{12}}c_{2}\xrightarrow{D_{23}}c_{3}$, we need to
verify

$\Phi(D_{23}\circ_{KLRW}D_{12})\overset{?}{=}D_{23}\circ_{Fuk}D_{12}.$ (23)

Proposition 7.2 asserts that (23) holds when $concat(D_{23},D_{12})$ is itself
taut (Prop 7.2).

Let us show that if $D_{12}$ and $D_{23}$ are elementary weighted strands
diagram, which can also be viewed as KLRW diagrams, then

$\Phi(D_{23}\circ_{KLRW}D_{12})=D_{23}\circ_{Fuk}D_{12}.$

That is, that (23) holds for compositions of elementary diagrams.

Indeed, if $D_{12}$ or $D_{23}$ has all vertical strands, then the
concatenation $concat(D_{23},D_{12})$ is taut thus Prop 7.2 implies the result
holds.

If $concat(D_{23},D_{12})$ is not taut, then we have exactly one bigon. If the
bigon is of type $(i)-(i)$, $(i)-[i]$ or $(i)-(j)$ where $(i)$ and $(j)$ are
adjacent in $\Gamma$, then section 7.2 shows that the KLRW composition
coincide with the Fukaya composition. If the bigon is of type $(i),(j)$ where
$(i)$ $(j)$ are not adjacent, then KLRW and Fukaya composition (Lemma 5.5)
will cancel the bigon.

Finally, by applying Prop 4.8 to the full subcategory
$\\{\mathcal{T}_{c},c\in\mathcal{C}\\}$, we finishes the proof of Theorem 7.1.
∎

##  8 Floer homologies in the multiplicative Coulomb branch

In this section we consider the Fukaya category of the original space
$\mathcal{M}^{\times}(\Gamma,\vec{d})$ with no divisor removed.

Two new difficulties are that (1) when $\hbar\neq 0$ there is no longer an
isomorphism
$\operatorname{Hom}_{KLRW}(c_{1},c_{2})\cong\mathbb{C}\mathcal{P}(c_{1},c_{2})$,
and, in fact correspondingly, (2) the continuation maps determining wrapped
Floer homology no longer preserve the diagram basis of Proposition 5.10. That
is, both isomorphisms of Equation (21) fail.

Instead we proceed by implementing the recognition principle of Proposition
4.13.

Recall we consider a Fukaya category where we count disks
$\mathcal{D}\to\mathcal{M}^{\times}(\Gamma,\vec{d})$ with weight
$\eta^{\mathcal{D}\cdot H_{matter}}\hbar^{\mathcal{D}\cdot H_{root}}$. Let
$\mathcal{A}$ be a full subcategory of the Fukaya category with objects
$\\{\mathcal{T}_{c}\mid c\in\mathcal{C}\\}$. Our result is that
$KLRW\cong\mathcal{A}$, as categories defined over $\mathbb{C}[\eta,\hbar]$.

###  8.1 Crossing number filtration

Let $c_{1},c_{2}\in\mathcal{C}$ be two black points configuration, for any
diagram $D\in\mathcal{P}(c_{1},c_{2})$, define ”crossing number of type (i)”
as

$cross_{i}(D)=\text{number of crossings of black strands of type $(i)$},$

and define the ”total crossing number” as

$cross(D)=\sum_{i\in\Gamma}cross_{i}(D).$

We may define a (multi-index) increasing filtration on
$\mathbb{C}\mathcal{P}(c_{1},c_{2})$.

######  Definition 8.1.

For any $(i)\in\Gamma$, let $k_{i}\geq 0$ be an integer and let
$\vec{k}=(k_{i})$. Define an increasing filtration on
$\mathbb{C}\mathcal{P}(c_{1},c_{2})$ called ‘crossing number filtration’,

$\mathcal{F}_{\vec{k}}\mathbb{C}\mathcal{P}(c_{1},c_{2})=\mathbb{C}\cdot\\{D\in\mathcal{P}(c_{1},c_{2})\mid
cross_{i}(D)\leq k_{i},\forall i\\}.$

For any $k\in\mathbb{Z}_{\geq 0}$, we may define

$\mathcal{F}_{k}\mathbb{C}\mathcal{P}(c_{1},c_{2})=\mathbb{C}\cdot\\{D\in\mathcal{P}(c_{1},c_{2})\mid
cross(D)\leq
k\\}=\sum_{\vec{k},|\vec{k}|=k}\mathcal{F}_{\vec{k}}\mathbb{C}\mathcal{P}(c_{1},c_{2}).$

Let $Gr^{\mathcal{F}}_{k}$ denote the associated graded for the total crossing
number filtration, and let $Gr^{\mathcal{F}}$ be the direct sum over $k$.

######  Proposition 8.2.

For any $c_{1},c_{2},c_{3}\in\mathcal{C}$, and Lagrangians
$\mathcal{T}^{w}_{c_{1}}>\mathcal{T}^{w}_{c_{2}}>\mathcal{T}^{w}_{c_{3}}$, the
composition

$\mu_{2}:CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}^{w}_{c_{2}})\otimes
CF(\mathcal{T}^{w}_{c_{2}},\mathcal{T}^{w}_{c_{3}})\to
CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}^{w}_{c_{3}})$

preserves the crossing number filtration. Namely, if diagram $D_{13}\in
CF(\mathcal{T}_{1},\mathcal{T}_{3})$ appears in $\mu_{2}(D_{12},D_{23})$, then
for any $i\in\Gamma$,

$cross_{i}(D_{13})\leq cross_{i}(D_{12})+cross_{i}(D_{23}).$

###### Proof.

If $\varphi:\mathcal{D}\to\mathcal{Y}$ is a holomorphic triangle interpolating
$D_{12},D_{23},D_{13}$, then under the projection
$\mathcal{Y}\to\operatorname{Sym}^{d_{i}}(\mathbb{C}^{*})$, we have
$\mathcal{D}\to\operatorname{Sym}^{d_{i}}(\mathbb{C}^{*})$. Let $\Delta$ be
the big diagonal divisor in $\operatorname{Sym}^{d_{i}}(\mathbb{C}^{*})$, then
we have

$0\leq\\#(\mathcal{D}\cap\Delta)=cross_{i}(D_{12})+cross_{i}(D_{23})-cross_{i}(D_{13}),$

hence the desired inequality. ∎

######  Corollary 8.3.

There is a natural total crossing number filtration on
$HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})$, and

$Gr_{k}(HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}}))\cong\mathbb{C}[\eta,\hbar]\cdot\mathcal{P}_{k}(c_{1},c_{2}),\forall
k\in\mathbb{Z}_{\geq 0}.$

###### Proof.

Recall that

$HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})=co\lim_{\mathcal{T}_{c_{1}}^{w}}CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}_{c_{2}})$

where various $CF(\mathcal{T}^{w}_{c_{1}},\mathcal{T}_{c_{2}})$ are related by
continuation maps. The continuation map is realized by composition with an
intersection that is labelled by the ’identity diagram’ in
$\mathcal{P}(c_{1},c_{2})$, hence is in $\mathcal{F}_{0}$. By Prop 8.2,
continuation map does not increase the filtration level, i.e., respects
filtration. Hence the colimit $HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}})$ is
filtered. On the associated graded level, we have continuation map is identity
on the basis of weighted taut strands diagrams, hence
$Gr(HW(\mathcal{T}_{c_{1}},\mathcal{T}_{c_{2}}))$ has a basis which are in