where $L_{0},\ldots,L_{k}$ are Lagrangians in $Y$.

Consider a map $y:{\rm D}\rightarrow Y$, from a $k+1$-pointed disk ${\rm D}$
to $Y$ interpolating from ${\cal P}_{i}\in CF^{*,*}({L}_{i-1},{L}_{i})$ for
$i$ running from $1$ to $k$, to ${\cal P}_{k+1}\in CF^{*,*}({L}_{0},{L}_{k})$.
The map can contribute to the coefficient of ${\cal P}_{k+1}$ in
$\mu^{k}({\cal P}_{1},\ldots,{\cal P}_{k})$, provided its
${\vec{J}}$-equivariant degree vanishes

${\vec{J}}(y)={\vec{J}}({\cal P}_{k+1})-\sum_{i=1}^{k}{\vec{J}}({\cal
P}_{i}),$

and the Maslov index

$ind(y)=M({\cal P}_{k+1})-\sum_{i=1}^{k}M({\cal P}_{i})$ (7.33)

equals to $2-k$, The former is required for the map to be in $Y$, the later
for the expected dimension of the reduced moduli space to vanish.

#### 7.4.4

Just like for $\mu^{1}=Q$, the key to computability is the reformulation of
the problem in terms of counting holomorphic curves in ${\rm D}\times{\cal
A}$. The ${J}_{i}$-equivariant degree of the map is simply the intersection of
$y$ with the divisor $F_{i}$, $J_{i}(y)=-\int_{\partial{\rm
D}}y^{*}c^{i}=-y\cdot F^{i}$, which we can compute from its projection to a
domain $A$ using the formulas given earlier in this section, in equations

To compute the Maslov index, we will again borrow the results from Heegard-
Floer theory, for which the corresponding index was computed in [155]:

$ind_{HF}(y)=i(y)+2e(A)-(k-1)d/2.$

The index $ind_{HF}(y)$ in Heegard-Floer theory differs from the index $i(y)$
of the theory here by the factor $i(y)$ in (7.32), so it follows that the
index of a $k+1$ pointed map to $Y$ is

$ind(y)=2e(A)-(k-1)d/2.$ (7.34)

We will use this result in the next section, to compute the algebra $A$.

###  7.5 The thimble algebras for ${\mathscr{D}_{Y}}$

From results earlier in this section, we can explicitly compute the tilting
thimble algebra $A$. The computation is simple, yet it gives a complete
description of the derived Fukaya-Seidel category of $Y$ – an answer to every
computation. While algebra makes computations easy, geometry is needed to tell
us which computation we should do to get homological link invariants, as we
will see in the later in this section.

The end result of this subsection is an algebraic formulation of
${\mathscr{D}_{Y}}$ as the derived category ${\mathscr{D}_{A}}$ of
$A$-modules,

${\mathscr{D}_{\cal Y}}\cong{\mathscr{D}_{A}}.$

We will find that the algebra $A$ is a quotient of the algebra ${\mathscr{A}}$
of ${\cal X}$, the equivariant mirror of $Y$, by an ideal that corresponds to
restricting to ${X}$. This proves the Conjecture 1 for
${}^{L}{\mathfrak{g}}=\mathfrak{su}_{2}$.

This generalizes the $d=1$ case of $Y$ which is the equivariant mirror to the
$A_{m-1}$ surface in sections 5.2 and 6.5. As elsewhere in this section,
$V=(V_{1\over 2})^{\otimes m}$, and $V_{\nu}$ is the weight $\nu$ subspace of
it $d$ levels below the top. The description of the algebra $A$ holds for any
integer $0\leq d\leq m$. (For $d>m/2$, the description of ${\cal X}$ as a
transversal slice in affine Grassmannian from section $2$ needs to be replaced
by the generalized slice from [39, 40]).

#### 7.5.1

Fix an ordering of punctures $0\leq{\rm Im}(\ln a_{1})<\ldots<{\rm Im}(\ln
a_{m})<2\pi$ on the $S^{1}$ in ${\cal A}$, partitioning the circle into $m$
intervals. A $[0,1]$ interval together with the $S^{1}$ make up a cylinder
${\rm C}=S^{1}\times[0,1]$ partitioned into $m$ bins, the boundaries of whom
are swept by the positions of the punctures.

For each critical point ${\cal C}$ we get a left thimble $T_{\cal C}$ which
is, in the chamber (5.5), a collection of $d$ real line Lagrangians:

$T_{\cal C}=T_{i_{1}}\times T_{i_{2}}\times\ldots\times T_{i_{d}}.$ (7.35)

Each $T_{i}$ is a one dimensional thimble of our $d=1$ example, running at
fixed ${\rm Im}\log y\in({\rm Im}\log a_{i},{\rm Im}\log a_{i+1})$. There are
$m$ such thimbles available, and the choice of ${\cal C}$ is which $d$
distinct one dimensional thimbles we pick to form $T_{\cal C}$. We will order
the thimbles in $T_{\cal C}$ so that $i_{1}<i_{2}<\ldots<i_{d}$ – this order,
by increasing value of ${\rm Im}W$ in the one dimensional case. To the
critical point ${\cal C}$ assign the bins numbered $i_{1},i_{2},\ldots,i_{d}$
on ${\rm C}$. (The interval in ${\rm C}$ is really the space direction in the
Landau-Ginsburg model.)

Figure 15: Two thimbles when $d=2$. Intersection points come in families of
$d!=2$. The orange point corresponds to $\sigma=1$, the green to
${\sigma}=(12)$.

#### 7.5.2

As in section 5.2, the elements of $A$ are the $Homs$

$A=Hom^{*}_{\mathscr{D}_{Y}}(T,T)$ (7.36)

where $T=\bigoplus_{\cal C}T_{\cal C}$ is the tilting generator. Explicitly,

$A=\bigoplus_{{\cal C},{\cal
C}^{\prime}}\bigoplus_{{\vec{J}}\in{\mathbb{Z}}^{2}}Hom_{\mathscr{D}_{Y}}(T_{\cal
C},T_{{\cal C}^{\prime}}\\{{\vec{J}}\\})$

where $Hom_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal C}^{\prime}}\\{{\vec{J}}\\})$
is the cohomology of the Floer complex.

The Floer differential is trivial, so as a vector space, $A$ is generated by
intersection points of the thimbles on ${\cal A}$:

$Hom^{*}_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal C}^{\prime}})=\bigoplus_{{\cal
P}\in T^{\zeta}_{\cal C}\cap T_{\cal C^{\prime}}}{\mathbb{C}}{\cal P},$ (7.37)

where $T^{\zeta}_{\cal C}$ is the thimble corresponding to ${\rm
Im}e^{-i\zeta}W$ being constant, for ${\zeta}>0$, as in section 5.2. Each
intersection point ${\cal P}$ is a $d$-tuple of intersections points of one
dimensional Lagrangians,

${\cal P}=({p}_{1},\ldots,p_{d}),\qquad p_{\alpha}\in
T^{\zeta}_{i_{\alpha}}\cap T_{\sigma{(i_{\alpha}^{\prime})}}.$ (7.38)

The intersections of one dimensional Lagrangians are those inherited from
section 5.2. For example, if all elements of ${\cal P}$ come from the region
near ${y\rightarrow\infty}$ on ${\cal A}$, then all the one dimensional
generators are of the form $a_{i,j}$ with $i>j$, and then

${\cal
P}=(a_{\sigma(i_{1}^{\prime}),i_{1}},\dots,a_{\sigma(i_{d}^{\prime}),i_{d}}).$
(7.39)

#### 7.5.3

The best way to capture elements of $A$ is graphically, in terms of ${\rm C}$.
Color the $m$ strings partitioning ${\rm C}$ into bins red. To

${\cal P}\in Hom_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal
C}^{\prime}}\\{\vec{J}\\})$

assign $d$ blue strings that start at the bottom of the cylinder, inside the
$d$ bins corresponding to ${\rm C}$ and end et the top in the bins
corresponding to ${\cal C}^{\prime}$. This is not a cartoon – recall from
section $4$ that to intersection point ${\cal P}$ there corresponds a one
dimensional flow $y=y(s)$, solving the equation (4.3). Here, $s\in[0,1]$
parametrizes the axis of ${\rm C}$ and the positions of the endpoints of the
string at $s=0,1$ are the locations of the actual thimbles on the $S^{1}$ in
${\cal A}$. Moreover, strings have tension and the equation (4.3) is for the
string to minimize energy, so the strings are taut - in particular, they have
no excess intersections with each other or with the red strings. Each of the
$d$ strings can wind arbitrarily many times around the cylinder, and the paths
they take determine their equivariant charge ${\vec{J}}$, as we will describe
below. Each of the $d$ individual strings is inherited from the $d=1$ case we
solved in section 5.2; and working with $d>1$ introduces one new ingredient –
the blue strings can cross each other. As we will see, these crossings will
correct the equivariant degree ${\vec{J}}$ away from the simple sum of its
$d=1$ contributions.

Figure 16: An algebra $A$ element is a configuration of strings on the
cylinder ${\rm C}$, with $d$ blue strings and $m$ red, satisfying the
relations in figures $18$-$21$.

#### 7.5.4

As in the one dimensional case, each of the $d$ strings is composed of string
”bits” which are one of four types:

Figure 17: The strings are composed of bits, one of the four types above. The
$x_{i}$ bit is a crossing of a pair of strings in the $i$’th bin.

from which we get all the other string configurations. The bits are useful
since the equivariant degree of any intersection point ${\cal P}$ is the sum
of the equivariant degrees of all of its bits. To spell out equivariant the
degrees of all elements ${\cal P}\in A$, decompose every one of its $d$
strings into products of string bits which are either constant configurations
– corresponding to simply $p_{i}$ for some $i$, or are of the form

$a_{i,j}=a_{i,i-1}\circ a_{i-1,i-2}\circ\cdots\circ a_{j+1,j},\qquad
b_{i,j}=b_{i,i+1}\circ b_{i+1,i+2}\circ\cdots\circ b_{j+1,j},$ (7.40)

for $i>j$. Here $a_{i+1,i}$ describes a string bit crossing from the $i$-th to
the $i+1$-th bin, $b_{i,i+1}$ follows the same has path but in reverse. The
product “$\circ$” is the ordinary product in the one dimensional problem. (The
decomposition into string is compatible with the product on $A$ away from
crossings. In presence of crossings, strings which are decomposable in the one
dimensional problem are not decomposable in $A$.) The equivariant degrees of
string bits can be read off from the torus action:

${\rm T}:\;\;\;\;\begin{aligned}
&\;\;\;\;\;\;\;\;p_{i},\;\;\;\;x_{i}\;\;\;\;\;\rightarrow\;\;\;\;\;\;\;\;\;\;p_{i},\;\;\;\;{\mathfrak{q}}^{-1}x_{i}\;\;\\\\[-7.0pt]
&\;a_{i+1,i},\;\;b_{i,i+1}\;\;\;\;\rightarrow\;\;\;\;\;\;a_{i+1,i},\;\;{\mathfrak{q}}\;b_{i,i+1},\qquad
i\neq m-1\\\\[-7.0pt]
&a_{0,m-1},\;b_{m-1,0}\;\;\rightarrow\;\;{\mathfrak{h}}\,a_{0,m-1},\;\;{\mathfrak{q}}{\mathfrak{h}}^{-1}\,b_{m-1,0},\\\\[7.0pt]
\end{aligned}$ (7.41)

borrowing the action on the $a$’s and $b$’s from table (5.39),

The equivariant degree ${\vec{J}}({\cal P})$ of any element ${\cal P}$ of the
algebra $A$, is a sum of the equivariant degrees of all of its bits. In other
words,

${\vec{J}}({\cal
P})=\sum_{\alpha=1}^{d}{\vec{J}}(p_{\alpha})+\sum_{\textup{blue
crossings}}{\vec{J}}(x),$ (7.42)

where $p_{\alpha}\in T^{\zeta}_{i_{\alpha}}\cap
T_{\sigma{(i_{\alpha}^{\prime})}}$ are the one dimensional intersection points
in (7.38), so ${\vec{J}}(p_{\alpha})$ is the equivariant degree of a single
blue string in the $d=1$ problem. The second term is the correction coming
from crossings of blue strings. A single non-degenerate crossing of two
strings in ${\cal P}$ corrects the equivariant degree by $\\{-1,0\\}$. For
degenerate crossing strings, one first makes all the intersection points non-
degenerate and then counts them. We will prove the expressions (7.41) and
(7.42) below, since for this we need to learn how to compute the algebra
products.

#### 7.5.5

Our algebra $A$ is a vector space with a non-commutative associative product

$A\times A\;\rightarrow\;A$ (7.43)

coming from the product on the Floer cohomology groups in (4.20). If

${\cal P}\in Hom_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal
C}^{\prime}}\\{\vec{J}\\})\;\;\;\textup{and}\;\;\;{\cal P}^{\prime}\in
Hom_{\mathscr{D}_{Y}}(T_{{\cal C}^{\prime}}\\{\vec{J}\\},T_{{\cal
C}^{\prime\prime}}\\{\vec{J}^{\prime}\\})$

are two elements of $A$, their product is a class

${\cal P}^{\prime}\cdot{\cal P}=\mu^{2}({\cal P}^{\prime},{\cal P})\in
Hom_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal
C}^{\prime\prime}}\\{\vec{J}^{\prime}\\}).$ (7.44)

Per definition, the coefficient of ${\cal P}^{\prime\prime}$ in the product is
the count of Maslov index zero maps $y:{\rm D}\rightarrow Y$, where the three
points marked points on the boundary of the disk ${\rm D}$ map to ${\cal P}$,
${\cal P}^{\prime}$ and ${\cal P}^{\prime\prime}$ in counter clockwise order,
and the boundary itself to the three Lagrangians in $Y$.

As we will see, the product in (7.44) is computable explicitly.

#### 7.5.6

First, we need to understand what Maslov index being zero requires. From our
index computation in (7.34), index being zero means that Euler measure of the
corresponding domain must equal

$e(A)=d/4.$ (7.45)

The only solution to (7.45) is a domain $A$ on ${\cal A}$ which is a union of
$d$ triangles each of which contributes $1-3/4=1/4$ to the Euler measure. To
see that, observe that we can arrange all the intersections of the three one
dimensional thimbles from $T_{\cal C}$, $T_{{\cal C}^{\prime}}$ and $T_{{\cal
C}^{\prime\prime}}$ to be at $\pi/3$ or $2\pi/3$ angles. Correspondingly, a
three-pointed disk projects to a domain $A$ which may be a union of hexagons
or triangles. Each corner of a hexagon contributes $-1/12$ to its Euler
measure, and each corner of a triangle $+1/12$. It follows the only way to
saturate (7.45) with $3d$ vertices is by $A$ which is a union of $d$
triangles. (This argument was used in [24] in a closely related setting).

#### 7.5.7

Since all Maslov index zero domains are unions of $d$ triangles, given ${\cal
P}$ and ${\cal P}^{\prime}$, at most one ${\cal P}^{\prime\prime}\in
Hom^{*}_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal
C}^{\prime\prime}}\\{\vec{J}^{\prime}\\})$ contributes to ${\cal
P}^{\prime}\cdot{\cal P}$ – the one determined by the third vertex of every
one of the $d$ triangles. It follows that, if ${\cal P}^{\prime}\cdot{\cal P}$
is not zero it equals to ${\cal P}^{\prime\prime}$ obtained by concatenating
each of $d$ strings in ${\cal P}$ with one string in ${\cal P}^{\prime}$,
since this is what the product in one dimension does. Next, we need to
determine the coefficient of ${\cal P}^{\prime\prime}$ in the product ${\cal
P}^{\prime}\cdot{\cal P}$.

Figure 18: Product of the algebra $A$ comes from holomorphic triangles,
reducing the problem of finding the relations in $A$ to one with $d=1$. The
diagram has $i^{\prime}(A)=0$ since all triangle intersections are ”head to.

#### 7.5.8

Like in the one dimensional problem, the product ${\cal P}^{\prime}\cdot{\cal
P}$ vanishes automatically if the end-points of one dimensional strings do not
match up. In this case, at least one of the $d$ one dimensional triangles on
${\cal A}$ does not exist.

Another elementary reason why the product may vanish is that the equivariant
degree of the map $y$ does not vanish. Then, the action of the map to $Y$ is
infinite, and the coefficient of ${\cal P}^{\prime\prime}$ in the product is
zero. An alternative way to see the vanishing is that the equivariant degrees
of ${\cal P}^{\prime\prime}$ and ${\cal P}^{\prime}\cdot{\cal P}$ are not
equal so the algebra cannot relate them. The $J_{1}$ equivariant degree
vanishes for trivially for any $A$ which is a compact domain on ${\cal A}$,
and as such does not intersect the punctures at $y=0$ and $y=\infty$.

The $J_{0}$ equivariant degree of the map is given by

$J_{0}(y)=i^{\prime}(A)-\sum_{i}n_{a_{i}}(A).$ (7.46)

The $i^{\prime}(A)$ term is the intersection of the map $y$ with with the
diagonal in $Y$. It vanishes automatically in the one dimensional problem, but
not for general $d$. (The prime is to differentiate the intersection with the
diagonal here, for maps $y:{\rm D}\rightarrow Y$ of a three-pointed disk to
$Y$, from $i(A)$ in (7.25) which is the intersection with the diagonal for
maps from a 2-pointed disk ${\rm D}$ to $Y$ instead.) The formula for
$i^{\prime}(A)$, derived in [154], is given by

$i^{\prime}(A)=n_{\cal P}(A)+n_{{\cal P}^{\prime}}(A)-e(A)+{b_{{\cal
P}^{\prime\prime}}}(A).$ (7.47)

where $n_{\cal P}(A)$ and $n_{{\cal P}^{\prime}}(A)$ are defined in the same
way as those in (7.25) – they are the average multiplicities at the vertices
of $A$ coming from ${\cal P}$ and ${\cal P}^{\prime}$. ${b_{{\cal
P}^{\prime\prime}}}(A)$, the precise definition of it may be found in [154],
turns out to be the average intersection of the boundary arcs at vertices
corresponding to ${\cal P}^{\prime\prime}$. The formula (7.47) is symmetric
under cyclic permutations of ${\cal P}$, ${\cal P}^{\prime}$ and ${\cal
P}^{\prime\prime}$, even if not manifestly so [154]. We can compute ${b_{{\cal
P}^{\prime\prime}}}(A)$ without defining it directly by noting that for a
triangle of unit multiplicity its intersection with the diagonal must vanish,
simply because there is no diagonal in the one dimensional problem. This is
reproduced by (7.47) if it assigns $b=-1/4$ for any of the $d$ points in
${\cal P}^{\prime\prime}$, which indeed is the result in [154].

#### 7.5.9

Enumeration of cases in [24] shows that $i^{\prime}(A)$, the intersection of
$A$ with the diagonal counts the excess of blue string intersections in ${\cal
P}^{\prime}\cdot{\cal P}$. This is the number of intersections between the
blue strings we get to remove by pulling them taut.

The second term in (7.46), coming from the intersection of $A$ with finite
punctures, also has a geometric meaning – it counts the excess intersection of
blue strings with the red.

When $i^{\prime}(A)$ vanishes the triangles in the domain $A$ either do not
intersect, or have “head-to-tail” intersections, where they intersect on a
triangle. When $i^{\prime}(A)$ fails to vanish, some triangles in the domain
$A$ either enclose each other, or intersect on a quadrangle. The second term
vanishes if and only if the triangles are empty.

#### 7.5.10

When strings have excess intersection, the product ${\cal
P}^{\prime}\cdot{\cal P}$ vanishes on equivariant degree grounds. Writing
${\cal P}^{\prime}\cdot{\cal P}$ as an algebra element ${\cal
P}^{\prime\prime}$ involves isotoping to remove excess intersection, since
elements of the algebra $A$ have none – they are all represented by taut
strings. Generically the equivariant degrees of ${\cal P}^{\prime}\cdot{\cal
P}$ and the algebra element ${\cal P}^{\prime\prime}$ obtained by pulling
strings taut do not coincide, because they have no reason to. As a result, the
equivariant $J^{0}(y)$ degree of the map that generates the product does not
vanish, so the coefficient it generates must. Alternatively, we can formally
shift the equivariant degree of ${\cal P}^{\prime\prime}$ so that the
instanton has zero equivariant degree, but then we get an element that is not
already in the algebra $A$ and the product again vanishes. There are a few
products that by accident end up having equivariant degree zero, but
associativity of the algebra products forces them to vanish.

Thus all non-vanishing products in $A$ come from products in the one
dimensional problem which minimize the intersections of blue with red strings,
also minimize the intersections of blue strings with themselves.

The rules of the algebra $A$ and its product $A\times A\rightarrow A$ can be
represented diagrammatically as follows.

Figure 19: Sliding over a crossing does not change the algebra element. Figure
20: Elements are defined up to isotopies. Figure 21: No two blue strings begin
or end in the same bin, or have excess intersection with the red. Figure 22:
Blue strings cannot have excess intersection with each other.

#### 7.5.11

Lagrangians on $Y$ are simply products of $d$ one-dimensional Lagrangians on
${\cal A}$, so there is no meaning to the ordering of the branes within

$T_{\cal C}=T_{i_{1}}\times T_{i_{2}}\times\ldots\times T_{i_{d}}.$

An element $p_{{\cal C},\sigma}$ of the algebra $A$ that corresponds to a
permutation $\sigma$ of the $d$ labels

$\sigma:\;\;T_{\cal C}=T_{i_{1}}\times T_{i_{2}}\times\ldots\times
T_{i_{d}}\;\;\rightarrow\;\;T_{{\cal C},\sigma}=T_{\sigma{(i_{1})}}\times
T_{\sigma{(i_{2})}}\times\ldots\times T_{\sigma{(i_{d})}},$

has equivariant degree $N_{\sigma}$ computed by summing up the equivariant
degrees of string bits in the corresponding string diagram. It follows that
viewing the thimbles as objects of the derived category, the element

$p_{{\cal C},\sigma}\in Hom_{\mathscr{D}_{Y}}(T_{\cal C},T_{{\cal C},\sigma})$

generates parallel transport that maps $T_{{\cal C}}$ to itself, but with a
degree shift:

$T_{{\cal C},\sigma}\cong T_{{\cal C}}\\{N_{\sigma}\\}.$ (7.48)

#### 7.5.12

It remains to prove that the assignment of equivariant degrees to algebra
elements from section is correct. We can do this by induction. Fix ${\cal C}$,
corresponding to choice of $d$ bins at the bottom of the cylinder. Given an
element ${\cal P}$ in (7.38), assume that the equations (7.41) and (7.42) hold
for all ${\cal P}^{\prime}<{\cal P}$, where the relation $``<"$ means that
${\cal P}$ can be obtained from ${\cal P}^{\prime}$ by adding one bit. If
${\cal P}_{\textup{bit}}$ is the algebra element that differs from the
identity by one of the bits in figure 17, than ${\cal P}^{\prime}$ satisfies
${\cal P}={\cal P}_{\textup{bit}}\cdot{\cal P}^{\prime}$. We will now prove
that (7.41) and (7.42) hold for ${\cal P}$ as well.

Since the algebra products preserve gradings, the grading of ${\cal P}$ and
${\cal P}^{\prime}$ differ by the grading of the bit algebra element:

${\vec{J}}({\cal P})={\vec{J}}({\cal P}^{\prime})+{\vec{J}}({\cal
P}_{\textup{bit}}).$ (7.49)

It follows we will prove ${\vec{J}}({\cal P})$ satisfies (7.41) and (7.42) if
we show that the gradings of the bit ${\cal P}_{\textup{bit}}$ satisfy (7.41).
These hold per definition for all the bits except for the ones involving a
crossing of $i$’th and $i+1$’st strand which we labeled $x_{i}$.

Suppose ${\cal P}$ differs from ${\cal P}^{\prime}$ by adding the bit $x_{i}$.
Then, ${\cal P}$ and ${\cal P}^{\prime}$ correspond to intersections of a pair
of thimbles $T_{{\cal C}}$ and $T_{{\cal C}^{\prime}}$ arranged so that the
disk that interpolates from ${\cal P}$ to ${\cal P}^{\prime}$ projects to a
domain $A$ which is a single empty rhombus. The equivariant degree of this
rhombus is difference between equivariant degrees of ${\cal P}$ and ${\cal
P}^{\prime}$, and

${J}^{0}({\cal P})=J^{0}({\cal P}^{\prime})+J^{0}(A)=J^{0}({\cal
P}^{\prime})-1.$ (7.50)

Namely, $A$ being a rhombus has Euler measure which rhombus since $A$ has $4$
acute angles: $e(A)=1-4\times{1\over 4}=0$. It also has $i(A)=4\times{1\over
4}-0=1$, and vanishing intersections with the punctures. Thus, (7.50) follows
from the expression for equivariant degree of a 2-pointed disk in (7.22). The
$J^{1}(A)$ vanishes trivially. Comparing (7.49) and (7.50), the equivariant
degrees of the rhombus $A$ is the equivariant degree of ${\cal
P}_{\textup{bit}}$ for the bit $x_{i}$.

Combining these, we have proven the equations (7.41) and (7.42) hold in $A$.

#### 7.5.13

$A$ is an ordinary associative algebra with trivial differential since all the
Massey products $\mu^{k}$ for $k\neq 2$ vanish. For $\mu^{k}$ not to vanish
the Maslov index $ind(y)$ of a map $y$ that generates if needs to equal $2-k$.
For us $ind(y)$ vanishes by (7.33) because Maslov degrees of all the algebra
$A$ elements are zero. Another way to understand why $ind(y)$ vanishes is
using its relation (7.34) to Euler measures of domains $A$. For a
$k+1$-pointed map the relation is $ind(y)=2e(A)-(k-1)d/2,$ but shown in [24],
the Euler measure of every domain that could contribute to $\mu^{k}$ equals to
$e(A)=d(k-1)/4$.

#### 7.5.14

The algebra $A$ we just computed is the endomorphism algebra of the tilting
generator $T=\bigoplus_{\cal C}T_{\cal C}$ and as a consequence,
${\mathscr{D}_{Y}}$,the derived Fukaya-Seidel category of $Y$, is equivalent
to ${\mathscr{D}_{A}}$, the derived category of $A$-modules

${\mathscr{D}_{Y}}\cong{\mathscr{D}_{A}}.$

This equivalence, which sends a brane $L\in\mathscr{D}_{Y}$ to an $A$-module
via (5.15) allows to translate any question about branes in $\mathscr{D}_{Y}$
to a question about algebra modules in ${\mathscr{D}_{A}}$, and vice versa, as
we described in section 5.5 and will elaborate on later in section 7.6.2.

#### 7.5.15

The algebra $A$ arizes as the quotient of a larger algebra ${\mathscr{A}}$,
which is a generalization of the algebra with the same name from section 6.5
associated to the resolution of the $A_{m-1}$ surface singularity. The
construction of such an algebra, with the property that

${\mathscr{D}_{\cal X}}={\mathscr{D}_{\mathscr{A}}},$ (7.51)

where ${\cal X}$ is the equivariant mirror of $Y$ is the recent result of
Webster [175, 176]. The algebra ${\mathscr{A}}$ is obtained as the
endomorphism algebra of the tilting line bundle ${\cal P}$ on ${\cal X}$
(sometimes also called the splitting bundle), obtained by methods of
Bezrukavninkov and Kaledin [29, 28, 100, 15], namely quantization of ${\cal
X}$ in large characteristic $p\gg 0$. The tilting line bundle has the property
that its line bundle components

${\cal P}=\bigoplus_{\cal C}{\cal P}_{\cal C}$

generate $\mathscr{D}_{\cal X}$, and ${\mathscr{A}}$ is their endomorphism
algebra

${\mathscr{A}}=Hom^{*}_{\mathscr{D}_{\cal X}}({\cal P},{\cal P}).$

The algebra ${\mathscr{A}}$ is given by similar terms as $A$, but it has one
extra ingredient - namely the blue strands can carry dots. In the algebra
${\mathscr{A}}$, two of the relations in figure 21 get replaced by

Figure 23: KLRW algebra relations

This comes from the fact that in the one dimensional version of
${\mathscr{A}}$ from section 6.5, instead of imposing the algebra $A$ relation

${\cal I}:\qquad a_{i,i-1}b_{i-1,i}=0=b_{i,i+1}a_{i+1,i}$ (7.52)

one merely imposes

$a_{i,i-1}b_{i-1,i}=b_{i,i+1}a_{i+1,i}.$ (7.53)

Correspondingly, the algebra $A$ is a quotient of ${\mathscr{A}}$ by the
relation in (7.52):

$A={\mathscr{A}}/{\cal I},$ (7.54)

which simply says to set the dot to zero.

The introduction of dots leads to a few other relations where ${\mathscr{A}}$
and $A$ differ. The third relation in figure 21 gets replaced by

Figure 24: A basic KLR algebra relation.

and another similar one. The other relations, corresponding to figures 20 and
22 are unchanged – they are the relations 2.2$c$ in [173], for example. There
is only one further local relation in [173], relation 2.2$d$, which we have
not listed. It is not difficult to show that the inhomogenous term present in
that relation gets set to zero when we set the right hand side of the relation
in figure 24 to zero.

#### 7.5.16

The algebra ${\mathscr{A}}$ is a cousin of the planar KLRW algebra written
down in [172], as the algebra that gives rise to homological
${}^{L}\mathfrak{g}=\mathfrak{su}_{2}$ link invariants on ${\mathbb{R}}^{3}$.
The algebra ${\mathscr{A}}$ does the same [176], except that since it
corresponds to working on the cylinder, it lets one capture homological
invariants of links on ${\mathbb{R}}^{2}\times S^{1}$, and not just
${\mathbb{R}}^{3}$.

Viewing ${\cal X}$ as the Coulomb branch of the 3d quiver gauge theory, the
algebra of dotted blue strands is the algebra of the pure gauge theory.
Opening up the cylinder to a plane, it becomes the KLR algebra of [50, 153,
110] of Khovanov and Lauda and Rouquier. Adding red strands is what one gets
when one adds matter.

#### 7.5.17

Taking the quotient of ${\mathscr{A}}$ with respect to the ideal ${\cal I}$ to
get $A={\mathscr{A}}/{\cal I}$ is the algebraic counterpart of restricting
${\cal X}$ to its core. Therefore, from the proof in [176] that the two
categories ${\mathscr{D}_{\mathscr{A}}}$ and $\mathscr{D}_{\cal X}$ are
equivalent it follows we also have the equivalence of ${\mathscr{D}_{A}}$, the
derived category of modules of our algebra $A$, and $\mathscr{D}_{X}$.
Therefore, we have the theorem

######  Theorem 5.

Homological mirror symmetry

${\mathscr{D}_{X}}\cong\mathscr{D}_{A}\cong\mathscr{D}_{Y},$ (7.55)

holds for ${}^{L}\mathfrak{g}=\mathfrak{su}_{2}$.

proving the Conjecture 1 in this case.

#### 7.5.18

We have used similarity of our thory to Heegard-Floer theory as a tool. Its
origin is a genuine relation between the $\mathfrak{su}_{2}$ and
$\mathfrak{gl}_{1|1}$ theories. One can formulate the Heegard-Floer link
homology $\widehat{HFL}$ by similar means as what we have done here [11], as a
Landau-Ginsburg model on the target $Y^{HF}$ which is $Y^{0}=Sym^{d}({\cal
A})$, the symmetric product of $d$ copies of ${\cal A}$ with some set $F_{HF}$
deleted, and together with a potential $W_{HF}$. The sets deleted in
$\mathfrak{su}_{2}$ and $\mathfrak{gl}_{1|1}$ theories are different, so
holomorphic maps which one counts in one theory one does not count in the
other, and vice versa. This is reflected in the action of the differential.

In replacing $\mathfrak{su}_{2}$ theory to $\mathfrak{gl}_{1|1}$ we can still
generate the entire derived Fukaya category of $Y_{HF}$ in terms of the same
set of thimbles $T_{\cal C}$ in equation (7.35), where the $m\choose d$
choices of ${\cal C}$ correspond to picking out $d$ out of $m$ bins on the
$S^{1}$. Correspondingly, as vector spaces irrespective of their Maslov and
equivariant grading, Floer complexes of the $\mathfrak{gl}_{1|1}$ and
$\mathfrak{su}_{2}$ theories are the same.

Geometry of thimbles says that the would-be maps that generate the action of
differential project in both cases, to rectangles on ${\cal A}$. If $A$ is one
such rectangle, it has Euler measure $e(A)=0$ and intersection with diagonal
$i(A)=1$. In the $\mathfrak{su}_{2}$ theory, the corresponding map $y$ has the
wrong index $ind(y)=2e(A)=0$ to contribute to the differential, so the thimble
algebra

$A=Hom_{\mathscr{D}_{Y}}^{*}(T,T),$

is ordinary associative algebra with zero differential. In Heegard-Floer
theory, the same rectangle generates the action of the differential $Q_{HF}$,
as it has Maslov index $1$ by (7.32)

$ind_{HF}(y)=i(A)+2e(A)=1,$

provided its equivariant degree is zero $J^{0}_{HF}(y)=0$. The equivariant
degree being zero, in Heegard-Floer theory, means the rectangle is empty. The
endomorphism algebra of the thimble $T$ in the $\mathfrak{gl}_{1|1}$ theory,
is a differential graded algebra

$A_{HF}=CF_{{\cal FS}(Y_{HF})}^{*,*}(T,T),$ (7.56)

based on Floer complexes in the Fukaya-Seidel theory of $Y_{HF}$ with
potential $W_{HF}$. The Floer differential, coming from empty rectangles, acts
by removing a crossing of $a$-strings, or of $b$-strings, leaving the $a-b$
crossings untouched. The striking fact is that while the gradings on $A_{HF}$
and $A$ are different (even for $d=1$ ), they have a compatible product
structure, which is in both cases generated by counts of empty triangles with
$i^{\prime}(A)=0$, and moreover the higher products vanish in both. Thus, up
to regrading, we have an equivalence of as $A_{\infty}$-algebras,

$A_{HF}=(A,Q_{HF}),$

equating the endomorphism algebra $A_{HF}$ of the $\mathfrak{gl}_{1|1}$
theory, with the endomorphism algebra $A$ of the $\mathfrak{su}_{2}$ theory
equipped with the differential $Q_{HF}$. This simple observation is no doubt
at the heart of the spectral sequences relating Khovanov and Heegard-Floer
knot homology, which go back to work of Rasmussen [150] and which was recently
proven by Dowlin [61].

#### 7.5.19

${\mathscr{D}_{Y}}$ has the two equivalent descriptions in section 5.4:

${\mathscr{D}_{A}}\cong{\mathscr{D}_{Y}}\cong{\mathscr{D}}_{A^{\vee}}.$

one based on the left thimbles $T_{\cal C}$, the other on the right $I_{\cal
C}$, where

$A^{\vee}=Hom^{*,*}_{\mathscr{D}_{Y}}(I,I).$

with $I=\bigoplus_{\cal C}I_{\cal C}$. The right thimbles

$I_{\cal C}=I_{i_{1}}\times\ldots\times I_{i_{d}},$ (7.57)

are products of $d$-touples of distinct the right thimbles $I_{i}$ from
section 5.6. Being dual to the left thimble $T_{\cal C}$, $I_{\cal C}$ passes
through the same critical point ${\cal C}$. The choice of ${\cal C}$ is the
choice of which $d$ out of $m$ one dimensional right thimbles from section 5.6
one picks. The places where $I_{i_{k}}$’s may intersect map to the divisors
$F_{a_{i}}$ (coming from the punctures on ${\cal A}$), and these are deleted
from $Y$. For this, the local model of $Y$ as any pair of points on ${\cal A}$
approach a puncture, from section 7.3.12, is useful. It follows $W^{0}$ is
well defined on $I_{\cal C}$, which in turn defines an object of
$\mathscr{D}_{Y}$.

###  7.6 Link homology from ${\mathscr{D}_{Y}}$

At the beginning of this section we described how ${\mathscr{D}_{\cal X}}$
categorifies $U_{\mathfrak{q}}(\mathfrak{su}_{2})$ invariants of links colored
by the fundamental representation of $\mathfrak{su}_{2}$. By equivariant
mirror symmetry, $\mathscr{D}_{Y}$ does the same. In the rest of this section
we will describe explicitly how this comes about.

#### 7.6.1

The brane ${\cal U}={\cal O}_{U}\in\mathscr{D}_{\cal X}$ which represent the
$d$ cups and caps in terms of ${\cal X}$ is the structure sheaf of a cycle
$U$, a product of $d$ non-intersecting vanishing ${\mathbb{P}}^{1}$’s in
(7.3):

${{U}}={S}_{1}\times{S}_{2}\times\ldots\times{S}_{d}.$ (7.58)

Equivariant homological symmetry Theorems 4 and 5 associates to it a pair of
branes $I_{{\cal U}},E_{{\cal U}}\in{\mathscr{D}_{Y}}$, where $I_{{\cal U}}$ a
product of $d$ specific interval branes, and $E_{U}$ is a product of $d$
corresponding figure eight Lagrangians:

$I_{{\cal U}}=I_{1}\times I_{2}\times\ldots\times
I_{d},\;\;\;\;\textup{and}\;\;\;\;E_{{\cal U}}=E_{1}\times
E_{2}\times\ldots\times E_{d}.$ (7.59)

The functor $h_{*}:{\mathscr{D}_{Y}}\rightarrow{\mathscr{D}_{\cal X}}$ sends
the $I_{{\cal U}}$-brane in $Y$ to ${\cal U}$, the structure sheaf of the
vanishing cycle ${U}$ in ${\cal X}$,

${\cal U}=h_{*}I_{{\cal U}},\qquad E_{{\cal U}}=h^{*}{\cal U},$

while the functor $h^{*}:{\mathscr{D}_{\cal X}}\rightarrow{\mathscr{D}_{Y}}$
sends the ${\cal U}$ brane back to the $E_{\cal U}=h^{*}h_{*}I_{\cal
U}$-brane.

The $I_{{\cal U}}$-brane corresponds to a specific simple module of the
downstairs algebra $A$, the ${\cal U}$-brane to a simple module of the
upstairs algebra ${\mathscr{A}}$. The interval branes $I_{\beta}$ and their
figure eight counterparts $E_{\beta}$ are the branes that equivariant
homological mirror symmetry of the $A_{m-1}$ surface in section 6.5 relates to
the structure sheaf ${\cal S}_{\beta}={\mathscr{O}}_{S_{\beta}}$ of the
2-cycle $S_{\beta}$ and the simple module of the algebra ${\mathscr{A}}$.

#### 7.6.2

It is manifest from the proof of Theorem 5 that the action of braiding is
compatible with the functors $h_{*}$ and $h^{*}$, so we will use the same
letter ${\mathscr{B}}$ to denote the functors $B$ induces on
${\mathscr{D}_{\cal X}}$ and ${\mathscr{D}_{Y}}$. It follows from (6.31) that
the homology groups between branes in ${\mathscr{D}_{Y}}$ equal to those in
${\mathscr{D}_{\cal X}}$,

$Hom^{*,*}_{\mathscr{D}_{Y}}({\mathscr{B}}E_{\cal U},I_{\cal
U})=Hom^{*,*}_{\mathscr{D}_{\cal X}}({\mathscr{B}}\,{\cal U},{\cal U}).$
(7.60)

Theorem 5* of [5] and theorem F of [176] say that
$Hom^{*,*}_{\mathscr{D}_{\cal X}}({\mathscr{B}}\,{\cal U},{\cal U})$ are
homological invariants (of links in ${\mathbb{R}}^{2}\times S^{1}$ or by
specialization to braids that do not link the punctures at $0$ and $\infty$ of
${\cal A}$, of links in ${\mathbb{R}}^{3}$), so the same is true of
$Hom^{*,*}_{\mathscr{D}_{Y}}({\mathscr{B}}\,E_{\cal U},I_{\cal U})$.

#### 7.6.3

As we explained in section 5.5, there are two ways to go about computing the
link homology groups

$Hom^{*,*}_{\mathscr{D}_{Y}}({\mathscr{B}}\,E_{\cal U},I_{\cal
U})=\bigoplus_{M\in{\mathbb{Z}},{\vec{J}}\in{\mathbb{Z}}^{2}}Hom_{\mathscr{D}_{Y}}({\mathscr{B}}\,E_{\cal
U},I_{\cal U}[M]\\{{\vec{J}}\\}).$ (7.61)

The algebraic approach comes from describing ${\mathscr{B}}E$-branes and
$I$-branes as complexes of thimbles, and uses the equivalences
${\mathscr{D}_{A}}\cong{\mathscr{D}_{Y}}\cong{\mathscr{D}_{A^{\vee}}}$. In
fact, for the question at hand, it suffices to use only the first of these.
Describe the ${\mathscr{B}}E_{\cal U}$-brane as a complex of left thimble
$T_{\cal C}$-branes, bounded from the right,

${\mathscr{B}}E_{\cal
U}\cong\ldots\xrightarrow{e_{1}}{\mathscr{B}}E_{1}\xrightarrow{e_{0}}{\mathscr{B}}E_{0}.$
(7.62)

Each term ${\mathscr{B}}E_{k}={\mathscr{B}}E_{k}(T)$ is a direct sum of
$T_{\cal C}$-branes, or equivalently, of projective modules of algebra $A$ and
their equivariant (but not Maslov) degree shifts.

Projective resolution of brane ${\mathscr{B}}E$ in terms of the left thimble
branes is a prescription for how to start with a direct sum of sum of thimble
branes

$\bigoplus_{k}{\mathscr{B}}E_{k}(T)[k]$ (7.63)

and take a sequence of cones, or connected sums of these branes, at a specific
set of their intersections corresponding to $e_{k}\in
Hom_{\mathscr{D}_{Y}}({\mathscr{B}}E_{k+1},{\mathscr{B}}E_{k})$. This deforms
the differential $Q$ acting on (7.63) from trivial to

$Q=\sum_{k}e_{k}.$

Thus, in the algebraic approach, the action of the differential $Q$ is purely
classical. In particular, $Q$ squares to zero by definition of a brane as an
object of ${\mathscr{D}_{Y}}$.

The cohomology groups in (7.61), in fixed equivariant degree ${\vec{J}}$, are
the cohomology groups

$Hom^{*}_{\mathscr{D}_{Y}}({\mathscr{B}}E_{\cal U},I_{\cal
U}\,\\{{\vec{J}}\\})=H^{*}\bigl{(}hom_{A}({\mathscr{B}}E_{\cal U},I_{\cal
U}\\{{\vec{J}}\\})\bigr{)}$ (7.64)

are per definition the cohomologies of the complex

$0\rightarrow hom_{A}({\mathscr{B}}E_{0},I_{\cal
U}\\{{\vec{J}}\\})\xrightarrow{e_{0}}hom_{A}({\mathscr{B}}E_{1},I_{\cal
U}\\{{\vec{J}}\\})\xrightarrow{e_{1}}\ldots,$ (7.65)

which follows from (7.62). Thanks to the fact that the $I_{\cal U}$-brane is a
simple module of the algebra $A$, and each term of the complex is easily
computable by (5.30).

Thus, in the algebraic approach, link homologies have a simple geometric
meaning - they come directly from the geometry of the brane
${\mathscr{B}}E_{\cal U}$, as the cohomology of the complex in (7.64).

#### 7.6.4

The geometric approach to computing the cohomology groups starts with
intersection points ${\mathscr{B}}E_{\cal U}\cap I_{\cal U}$. The action of
differential $Q$ which turns the vector space

$CF^{*,*}({\mathscr{B}}E_{\cal U},I_{\cal U})=\bigoplus_{{\cal
P}\in{\mathscr{B}}E_{\cal U}\cap I_{\cal U}}{\mathbb{C}}\,{\cal P}.$ (7.66)

into a complex, is obtained by counting holomorphic maps from a disk ${\rm D}$
with two marked points to $Y$ with boundaries on the Lagrangians, with Maslov
index $1$ and equivariant degree zero. As we saw, the count of maps to $Y$ can
be simplified to a one dimensional problem, that involves instead studying
Riemann surfaces $S$ in ${\rm D}\times{\cal A}$ which are $d$-fold covers of
${\rm D}$ and project to a domain $A$ on ${\cal A}$ with boundaries on the one
dimensional Lagrangians. In this approach, it is straightforward to identify
domains which could not contribute to the coefficient of ${\cal P}$ in $Q{\cal
P}^{\prime}$, because their Maslov index is not one, or equivariant degree
does not vanish.

However, given a domain $A$ with Maslov index one and equivariant degree zero,
computing its contribution to the differential requires evaluating the signed
count $\\#{\cal M}({\cal P}^{\prime},{\cal P};y)$ of points in the reduced
moduli space, in (4.16). While this is a concrete problem, an exercise in
applications of Riemannian mapping theorem (in which, in addition to asking
that points $\pm 1$ on ${\rm D}$ get mapped to vertices of the domain $A$, and
its boundary to one dimensional Lagrangians on the boundary $\partial A$ or
the domain, we need every order one branch point of the projection
$S\rightarrow{\rm D}$ to map to a puncture in $A$ in such a way that
$f=e^{W^{0}}$ remains a regular function on ${\rm D}$), there is currently no
systematic approach to solving such problems.

An example of such a problem with known solution is if the domain $A$ is a
bigon with 2 acute vertices, whose contribution is always $\pm 1$. Namely, by
the Riemann mapping theorem there is a bi-holomorphic map of the unit disk
${\rm D}$ to any domain $A$ of disk topology (we used this once already, in
mapping the infinite strip which ${\rm D}$ starts out as, to the unit disk). A
holomorphic map from $A$ to ${\rm D}$ will take the two vertices of $A$ to a
pair of points on the boundary of ${\rm D}$; there is a one real parameter
family of Mobius transformations from ${\rm D}$ to itself that map the two
points on the boundary to $\pm 1$. This gives a reduced moduli space which is
single point, contributing $\pm 1$ to the count. The sign of its contribution
depends on the relative orientation of the boundary of the disk and the
Lagrangians. If in the conventions of figure 3 the orientation of the boundary
coincides with the orientation of $L_{1}$ the contribution is $+1$, and $-1$
otherwise. This enters holomorphic disk counts on a one dimensional Calabi-
Yau, see for example [21]. (Another example where such counts are known, which
we used earlier in this section, involve computing the action of $\mu^{2}$,
where empty triangles with three acute angles is always contribute $1$.) In
general, the problem is well defined, but hard.

In the geometric approach, the fact that the differential squares to zero
follows from the properties of the theory – provided one can prove
transversality and compactness of the unreduced moduli spaces ${\widehat{\cal
M}}(y)$. This requires showing that ${\widehat{\cal M}}(y)$ are smooth and
oriented manifolds of expected dimension $ind(y)$, see for example [23].
Proving transversality and compactness properties, as these are known [23], is
the “hard part” of symplectic geometry. In the current setting, these involve
some finite, but possibly non-trivial technical steps, since the “cylindrical”
approach to Floer theory we used is relatively novel. It is used in several
recent works [24, 25, 125].

#### 7.6.5

The action of the differential $Q$ in the algebraic approach is purely
classical. Computing the action of differential $Q$ in the geometric approach
involves counting instantons, a problem as far from classical as possible.

In both the geometric and the algebraic approaches, we end up computing the
cohomology of a complex of graded vector spaces, acted on by a differential
that squares to zero. Despite the superficially different appearances, the two
graded vector spaces are in fact the same

$hom_{A}({\mathscr{B}}E_{k},I_{\cal U}\\{\vec{J}\\})\cong
CF^{k}({\mathscr{B}}E_{\cal U},I_{\cal U}\\{\vec{J}\\}).$ (7.67)

On the left side is the $k$-the term in the complex of $hom_{A}$’s in (7.65)
and on the right the $k$-the term in the Floer complex (7.66), both in
equivariant degree ${\vec{J}}$.

The fact we are describing the same computation done in two different ways in
a single theory implies that, beyond the isomorphism of vector spaces, we have
an isomorphism of complexes (a chain homotopy equivalence, which is the map of
complexes that on their cohomology acts as identity).

#### 7.6.6

The isomorphism in (7.67) follows from the following simple fact. For any
brane $L\in\mathscr{D}_{Y}$ and any thimble $T_{\cal C}\\{\vec{J}\\}$, we get
a one-dimensional contribution to $hom_{A}(L_{k},I_{\cal C}\\{\vec{J}\\})$ for
every time $T_{\cal C}\\{\vec{J}\\}$ appears in $L_{k}$, the $k$-th term of
the complex that resolves $L$. This is the consequence of the duality between
the left to right thimbles in (5.30).

At the same time, for any brane $L\in\mathscr{D}_{Y}$, the number of times a
thimble $T_{\cal C}\\{\vec{J}\\}$ brane appears in the $k$-th term of its
projective resolution is the number of Maslov index $k$ and equivariant degree
$\\{\vec{J}\\}$ intersection points $L\cap I_{\cal C}$ – in other words, the
dimension of the vector space $CF^{k}(L,I_{\cal C}\\{\vec{J}\\})$.

Thus, as vector spaces, $hom_{A}(L_{k},I_{\cal C}\\{\vec{J}\\})$ and
$CF^{k}(L,I_{\cal C}\\{\vec{J}\\})$ are isomorphic, for any brane $L$ and any
right thimble $I_{\cal C}$. The statement in (7.67) is a special case with
$L={\mathscr{B}}E_{\cal U}$ and $I_{\cal C}=I_{\cal U}$.

#### 7.6.7

While computing the action of the differential $Q$ in the geometric
description is far harder than in the algebraic one, they are complementary in
that, in the geometric description it is easy to describe which problem we
need to solve to obtain link homologies, and in the algebraic one, to solve
it.

To translate a link $K$ in ${\mathbb{R}}^{3}$ to a pair of Lagrangians

$L={\mathscr{B}}E_{{\cal U}},\qquad L^{\prime}=I_{{\cal U}},$

in $Y$, choose a projection of the link to a plane, viewed as a small patch of
${\cal A}$. (To obtain invariants of a general link in ${\mathbb{R}}^{2}\times
S^{1}$ we would choose the projection to ${\cal A}={\mathbb{R}}\times S^{1}$
instead, everything else say stays the same.) Next, pick a 2-coloring of the
link projection, by $d$ red and blue segments. Require the coloring to be such
that no link component has a single color, that no crossing involves two
segments of the same color and that the red segments always underpass the
blue. Identify the $d$ red segments with the Lagrangians $I_{\beta}$ and get
the Lagrangian $L^{\prime}$ as their product, $I_{{\cal
U}}=I_{1}\times\ldots\times I_{d}$. The endpoints of the segments define the
positions of the $2d$ punctures, mapping to $a_{2b-1}$ and $a_{2b}$, for $b$
that runs from $1$ to $d$. To get $L$, replace each of the remaining $d$
segments with a figure eight Lagrangian. The result is ${\mathscr{B}}E_{{\cal
U}}={\mathscr{B}}E_{1}\times\ldots\times{\mathscr{B}}E_{d}$. The orientation
of the link components determines the (relative) grading of
${\mathscr{B}}E_{{\cal U}}$’s and $I_{{\cal U}}$’s.

###  7.7 ${\mathscr{D}_{Y}}$ and Khovanov homology

The discussion so far applied equally to links in ${\mathbb{R}}^{2}\times
S^{1}$ and in ${\mathbb{R}}^{3}$. Specialize now to the latter. In this case,
the homology groups $Hom_{\mathscr{D}_{Y}}^{*,*}({\mathscr{B}}E_{{\cal
U}},I_{{\cal U}})$ depend only a single equivariant grading
$\\{J\\}\equiv\\{J,0\\}$. This follows since the restriction of the algebra
$A$ from section 7.5 to strands that do not wind around the cylinder suffices,
and all such strands are composed of bits with vanishing $J_{1}$.

Combining the equivariant homological mirror symmetry, with theorems F and
7.12 of [176], it follows that the homology theory provided by
${\mathscr{D}_{Y}}$ is the one constructed by Khovanov in [108], so we have

######  Corrolary 1.

The homology group

$Hom_{\mathscr{D}_{Y}}({\mathscr{B}}E_{{\cal U}},I_{{\cal U}}[M]\\{J\\})$

coincides with the link homology $Kh^{i,j}(K)$ of the link $K$ defined in
[108], over complex numbers, with Khovanov’s $i$ and $j$-gradings related to
$M$ and $J$ by

$i=M+2J+i_{0},\qquad j=2J+j_{0}.$

The constants $i_{0}$ and $j_{0}$ depend on the link and its presentation. For
definiteness, take a link $K$ which is presented as a Markov trace of a braid
$\beta$ with $d$ strands all of which are oriented in the same direction, as
in figure 25. Then, $B$ is of the form $\beta\times 1^{d}$, where $1^{d}$ is a
trivial braid acting on the even punctures at $a_{2b}$ and $\beta$ on the odd
ones, at $a_{2b-1}$.

Figure 25: Markov closure of a braid $B=\beta\times 1^{4}$ with $n_{+}=5$ and
$n_{-}=2$.

The intersection points of of ${\mathscr{B}}E_{\beta}$’s with $I_{\beta}$’s
near the even punctures have equivariant and homological degree zero, which
fix the relative degrees of ${\mathscr{B}}E_{{\cal U}}$ and $I_{{\cal U}}$.
With this choice of link presentation we have

$i_{0}=0,\qquad j_{0}=d+w$ (7.68)

where $w$ is the writhe of the braid $B$, $w=n_{+}-n_{-}$ where $n_{+}$ is the
number of positive and $n_{-}$ the number of negative crossings.

#### 7.7.1

The Euler characteristic of the theory

$\chi({\mathscr{B}}E_{{\cal U}},I_{{\cal
U}})=\sum_{M,J\in{\mathbb{Z}}}(-1)^{M}{\mathfrak{q}}^{J}Hom_{{\mathscr{D}_{Y}}}({\mathscr{B}}E_{{\cal
U}},I_{{\cal U}}[M]\\{J\\}),$

per definition computes the weighted intersection number of Lagrangians $E$
and ${\mathscr{B}}I$

$\chi({\mathscr{B}}E_{{\cal U}},I_{{\cal U}})=\sum_{{{\cal
P}}\in{\mathscr{B}}E_{{\cal U}}\cap I_{{\cal U}}}(-1)^{M({\cal
P})}{\mathfrak{q}}^{J({\cal P})}.$ (7.69)

Its overall normalization depends on the relative grading of
${\mathscr{B}}E_{{\cal U}}$ and $I_{{\cal U}}$. The Euler characteristic is
related to the Jones polynomial of the link $K$ by:

$J_{K}({\mathfrak{q}})=(-1)^{i_{0}}(-{\mathfrak{q}}^{1\over
2})^{j_{0}}\chi({\mathscr{B}}E_{{\cal U}},I_{{\cal U}}).$ (7.70)

The fact that it computes the Jones polynomial follows on general grounds,
from the fact that the theory manifestly categorifies
$U_{\mathfrak{q}}(\mathfrak{su}_{2})$ quantum group invariants. This is also a
theorem of Bigelow, who proved in [31] that Jones polynomial $J_{K}$ computes
the weighted intersection number of a collection of curves in a plane,
consisting of figure eights and intervals, anticipating the geometric
interpretation in terms of our $Y$. The fact that the signs in this count come
from the Maslov grading $(-1)^{M({\cal P})}$ as we defined it, is the result
of Manolescu [123].

#### 7.7.2

Let us pause for a moment to explain the prefactor in (7.70). While it is just
an overall regrading, we can understand its actual value. The explanation is
somewhat involved however, as it gets contributioms from three different
places. Consider, for definiteness the braid of Markov trace form, so
$i_{0}=0$ and $j_{0}=d+w$ are as in (7.68).

The first source is shift of gradings of $Hom_{\mathscr{D}_{\cal X}}({\cal
F},{\cal G})$ by $[d]\\{-d/2\\}$ from [5], to center the degrees so that Serre
duality inverts them. This is responsible for the symmetry of knot invariants
which says that for a link $K$ and its mirror,
$J_{K}({\mathfrak{q}})=J_{K^{*}}({\mathfrak{q}}^{-1})$, see section 2.5.6.

The second source is that the conventional normalization of the
$U_{{\mathfrak{q}}}(\mathfrak{su}_{2})$ braiding matrix differs from the
normalization that naturally comes from geometry [5], see section 7. This
traces to the fact that natural normalization of conformal blocks of
$\widehat{{}^{L}\mathfrak{g}}$ differs from the natural normalization of the
equivariant central charge function of branes in $\mathscr{D}_{Y}$, or
${\mathscr{D}_{\cal X}}$, by a factor in (8.20) which the same for every
brane. To match the quantum group conventions, one has to multiply the
geometric Euler characteristic by ${\mathfrak{q}}^{-{1\over 4}}$, for each
positive crossing in $B=\beta\times 1^{d}$ and ${\mathfrak{q}}^{1/4}$ for each
negative crossing, or ${\mathfrak{q}}^{w\over 4}$ total, where $w$ is the
writhe of ${\beta}$.

The third source is framing. Framing is an inherent ambiguity of the quantum
Chern-Simons theory [179, 180]. To write down the quantum invariant requires
specifying the self-linking number of each link. For this, one chooses a unit
normal vector field at each point of the link and defines the self linking
number as the linking number between the link, and its displacement by the
vector field. A change of framing that increases the self linking number by
$1$, results in multiplying the link invariant by $(-{\mathfrak{q}}^{{3\over
4}})$. The Jones polynomial turns out to be the invariant of the link in the
”standard” framing, in which the self-linking number of every link vanishes.
This choice of framing has an advantage that we get to forget about the
framing altogether, so the Jones polynomial is invariant under link isotopy.
By contrast, quantum group invariants are invariants of knots in “vertical”
framing. Given a projection of a link to a plane, the vertical framing is the
one where the framing vector field is everywhere normal to the plane of the
projection. The plane of the projection for us is the plane of the Riemann
surface ${\cal A}$ and the fact that vertical framing enters is due to the
special role ${\cal A}$ plays. The first Reidermeister move introduces a
$2\pi$ twist in the vertical vector field, so quantum group invariants only
satisfy a modified version of it. To get from a link $K$ with $n_{+}$ positive
and $n_{-}$ negative crossings in vertical framing to the same link in
standard framing, we have to increase framing by $w=n_{+}-n_{-}$ which
multiplies the link invariant a $(-{\mathfrak{q}}^{{3\over 4}})^{w}$ factor.

#### 7.7.3

The $\mathfrak{su}_{2}$ quantum group invariants are invariants of unoriented
framed links $K$, see [135] for example. From Chern-Simons theory perspective,
exchanging a link $K$ with its orientation reversal corresponds to replacing
the representation $R$ coloring the link by its conjugate $R^{*}$ [179]. For
the $\mathfrak{su}_{2}$ Lie algebra, the every representation $R$ is self
conjugate, so the theory apriori does not distinguish a link from its
orientation reversal. The Jones polynomial becomes an invariant of oriented
links in ${\mathbb{R}}^{3}$ only because there is an implicit choice of
orientation associated with defining the standard framing that corresponds to
it. (The standard framing is defined via the Seifert surface of the link, by
choosing the framing vector field to be tangent to Seifert surface and
pointing inward. Seifert surfaces are always oriented, and so they induce an
orientation on the link they bound.) The homological $\mathfrak{su}_{2}$ link
invariants are also essentially invariants of unoriented links – the
dependence on the orientation of the link comes only through the overall
grading, e.g. the constants $i_{0}$ and $j_{0}$ in the Corrolary 1.

###  7.8 Some examples

In this subsection we will work out some examples of link homology computation
from $\mathscr{D}_{Y}$. Not only do resulting cohomologies coincide with
Khovanov homology, illustrating Corollary 1, but for certain link projections
which lead to diagrams similar to grid diagrams of Heegard-Floer theory, the
complexes themselves coincide.

#### 7.8.1

The simplest example is the unknot, projected as in figure 26. The
corresponding $Y$ is the Riemann surface ${\cal A}$ with two points
$y=a_{1},a_{2}$ deleted. The red and the blue sections of the unknot get
replaced with Lagrangians $E_{{\cal U}}$ and $I_{{\cal U}}$.

Figure 26: Unknot from A-branes. The two dashed lines are identified,
compactifying the plane to a cylinder. We rotated its axis clockwise by $90$
degrees, relative to earlier such pictures.

With only a pair of punctures, ${\mathscr{D}_{Y}}$ is generated by the pair of
left thimbles $T_{0}$ and $T_{1}$, or by the dual right thimbles $I_{0}$ and
$I_{1}$. The cup brane is simply $I_{{\cal U}}=I_{1}$. In section 6.5.9 we saw
how to describe the figure eight brane $E_{{\cal U}}=E_{1}$-brane as a complex
of left thimble branes:

${E_{{\cal U}}}\;\cong\;{T}_{1}\\{-1\\}\xlongrightarrow{\text{
$\begin{pmatrix}\;a_{0,1}\\\
-b_{0,1}\end{pmatrix}$}}\;\;\begin{array}[]{c}{T}_{0}\\{-1\\}\\\ \oplus\\\
{T}_{0}\end{array}\xlongrightarrow{\text{
$\begin{pmatrix}b_{1,0}&\\!\\!\\!\\!\\!a_{1,0}\end{pmatrix}$}}\;\;{T}_{1}.$
(7.71)

From this we can read off the corresponding complex of $A$-module
homomorphisms, reversing the arrows in the complex in the process:

$0\;\rightarrow\;hom_{A}(T_{1},I_{1}\\{J\\})\;\rightarrow\;0\;\rightarrow\;hom_{A}(T_{1}\\{-1\\},I_{1}\\{J\\})\;\rightarrow\;0,$
(7.72)

The two non-zero homs in the complex (7.72) correspond to the two intersection
points ${\cal P}$ and ${\cal P}^{\prime}$ of the branes $E_{\cal U}$ and
$I_{\cal U}$ branes in figure 26, as we will see momentarily. Since all the
maps are zero, the non-trivial cohomology of this complex is

$Hom_{\mathscr{D}_{Y}}(E_{{\cal U}},I_{{\cal
U}})={\mathbb{C}}=Hom_{\mathscr{D}_{Y}}(E_{{\cal U}},I_{{\cal
U}}[2]\\{-1\\}).$ (7.73)

We could have discovered the same complex from geometry of the branes as
follows. The relative gradings of intersection points can be computed by
considering arbitrary domains interpolating between the intersection points.
Take the shaded disk in the figure, which we will call $A$. standard
orientation of the boundary is counter-clockwise, so For $A$ interpolating
from ${\cal P}^{\prime}$ to ${\cal P}$, one goes from ${\cal P}^{\prime}$ to
${\cal P}$ along the $I$-segments of $\partial A$ following the standard,
counterclockwise orientation of the boundary, and returns from ${\cal P}$ to
${\cal P}^{\prime}$ via the figure eights (this is per conventions in section
4, with ${\cal P}^{\prime}$ and ${\cal P}$ viewed as elements of
$CF^{*,*}(E_{{\cal U}},I_{{\cal U}})$).

From a domain such as $A$, one computes the relative equivariant $J$-degrees
by (7.22)

$J({\cal P})-J({\cal P}^{\prime})=i(A)-\sum_{i}n_{a_{i}}(A)$

The first term is twice the intersection with the diagonal $i(A)$, computed
from (7.25); second is twice the total intersection number of $A$ with the
punctures, or equivalently, the total winding number of $\partial A$ around
them, viewed as a loop going counterclockwise, from ${\cal P}^{\prime}$ to
${\cal P}$. The difference in Maslov degrees is

$M({\cal P})-M({\cal P}^{\prime})=2e(A)$

with $e(A)$ the Euler measure from equation (7.69).

Since $A$ in figure 26 is a disk with one acute angle at ${\cal P}$, and one
obtuse at ${\cal P}^{\prime}$, $e(A)=1-{1\over 4}+{1\over 4}=1$, so $M({\cal
P})-M({\cal P^{\prime}})=2e(A)=2$. It follows that ${\cal P}$ generates
$Hom_{\mathscr{D}_{Y}}(E_{{\cal U}},I_{{\cal U}}[2]\\{-1\\})$ and has $M({\cal
P})=2$ and $J({\cal P})=-1$. In addition, $J({\cal P})-J({\cal
P}^{\prime})=-1$, since $A$ intersects $a_{2}$ puncture once so that
$n_{a_{2}}(A)=1$, while the intersection with the diagonal vanishes, as it
must in one dimension, since $i(A)={1\over 4}+{3\over 4}-e(A)=0$. Therefore
${\cal P}^{\prime}$ generates $Hom_{\mathscr{D}_{Y}}(E_{{\cal U}},I_{{\cal
U}})$ and has $J({\cal P})=0$ and $M({\cal P})=0$. From the geometric
perspective, the differential is trivial $Q{\cal P}=0=Q{\cal P}^{\prime},$
because the Maslov degrees are all even.

The Euler characteristic, normalized as in (7.70) with $i_{0}=0$ and $j_{0}=1$
since $d$ is one and $w$ zero, gives the Jones polynomial of the unknot
$J_{\bigcirc}({\mathfrak{q}})=-({\mathfrak{q}}^{1\over
2}+{\mathfrak{q}}^{-{1\over 2}})$. The equivariant mirror of $Y$ is the
resolved $A_{1}$-surface from section 5. The cup and the cap $E$ and $I$ with
orientations are related to structure sheaf ${\cal S}$ of the vanishing
${\mathbb{P}}^{1}$ in ${\cal X}$ by equivariant mirror symmetry, ${\cal
S}=h_{*}I$, $E=h^{*}{\cal S}$. The corresponding cohomology groups
$Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal S}[m]\\{d,0\\})$ are given in
(6.45). From perspective of ${\cal X}$, because $w$ vanishes, the correct
normalization is already built into the definition of the Euler characteristic
in (2.23), as paragraph 7.7.2 explains.

#### 7.8.2

Consider now a more complicated way of presenting the unknot, as in figure 27,
where

${\mathscr{B}}E_{{\cal U}}={\mathscr{B}}E_{1}\times{\mathscr{B}}E_{2}\times
E_{4},\qquad I_{{\cal U}}=I_{1}\times I_{3}\times I_{5}.$

There are four intersection points

${\cal P}_{1}=ayp,\;\;{\cal P}_{2}=bxp,\;\;{\cal P}_{3}=awq,\;\;{\cal
P}_{4}=azp,$

with degrees:

$J$ $M$ | $0$ | $1$ | $2$
---|---|---|---
$-1$ |  | ${\cal P}_{1}$ |  ${\cal P}_{2}$, ${\cal P}_{3}$
$0$ | ${\cal P}_{4}$ |  |

These come about as two shaded domains $A_{1}$ and $A_{2}$ each have three
acute and one obtuse angle, so $e(A_{i})=1-3\times{1\over 4}+{1\over
4}={1\over 2}$, and Maslov index of both is $M(A_{i})=2e(A_{i})=1$. They have
equivariant degree zero, $J(A_{i})=0$, since $i(A_{i})=3\times{1\over
4}+{3\over 4}-e(A_{i})=1$, and intersect a puncture once. Hence they can each
contribute to differential.

Figure 27: Another projection of an unknot.

We will now show momentarily that they contribute to the differential by

$Q{\cal P}_{1}=e^{-A_{1}}{\cal P}_{2}+e^{-A_{2}}{\cal P}_{3},\qquad Q{\cal
P}_{2}=Q{\cal P}_{3}=0=Q{\cal P}_{4}.$ (7.74)

With this action of the differential, one linear combination of ${\cal P}_{2}$
and ${\cal P}_{3}$ is $Q$-exact; while ${\cal P}_{1}$ is not $Q$-closed. From
this, we recover the unknot homology in (7.73), with the non-vanishing
homology groups $Hom_{\mathscr{D}_{Y}}({\mathscr{B}}E_{{\cal U}},I_{{\cal
U}})$ generated by ${\cal P}_{4}$, and
$Hom_{\mathscr{D}_{Y}}({\mathscr{B}}E_{{\cal U}},I_{{\cal U}}[2]\\{-1\\})$
generated by one linear combination of ${\cal P}_{2}$ and ${\cal P}_{3}$. This
illustrates the fact homology depends only on the link, and not on its
projection.

In (7.74) $A_{1}$ and $A_{2}$ are the areas of shaded domains with the same
name. We can absorb them into the definition of the generators of the Floer
complex, but we included them here as book keeping devices. With the figure as
drawn $A_{1}$ and $A_{2}$ are just two of several domains that have Maslov
index one and equivariant degree $0$, so they can support holomorphic disks.
One can isotope the diagram, as in figure 28 until only of all the domains
visible in figure (7.74), only $A_{1}$ and $A_{2}$ survive as the domains that
can support holomorphic disks. All the all others involve domains with both
positive and negative coefficients, which cannot support holomorphic curves,
since holomorphic maps are orientation preserving. The deformation does
introduce new intersection points, which are easily seen not to contribute to
the cohomology of the Floer complex. The disk interpolating between newly
created intersections are the bigon in figure 28. Their contributions to the
differential are $\pm 1$, cancelling the newly created intersection points
pairwise. It follows, indirectly, that the action of the differential must be
as in (7.74), or else the cohomology of $Q$ would not have the correct Euler
characteristic.

Figure 28: An isotopy.

To show directly that (7.74) is indeed the action of the differential,
consider the resolutions of the three figure eight Lagrangians in terms of the
left thimbles:

${\mathscr{B}}{E}_{1}\;\cong\;{T}_{2}\\{-2\\}\xlongrightarrow{\text{
$\begin{pmatrix}\;a_{3,2}\\\
-b_{0,2}\end{pmatrix}$}}\;\;\begin{array}[]{c}{T}_{3}\\{-2\\}\\\ \oplus\\\
{T}_{0}\end{array}\xlongrightarrow{\text{
$\begin{pmatrix}b_{1,3}&\\!\\!\\!\\!\\!\;\;a_{1,0}\end{pmatrix}$}}\;\;{T}_{1}.$
(7.75) ${\mathscr{B}}{E}_{2}\;\cong\;\begin{array}[]{c}{T}_{5}\\{-3\\}\\\
\oplus\\\ {T}_{2}\\{-1\\}\end{array}\xlongrightarrow{\text{
$\begin{pmatrix}\;a_{6,5}&0\\\ b_{3,5}&a_{3,2}\\\
0&b_{1,2}\end{pmatrix}$}}\;\;\begin{array}[]{c}{T}_{6}\\{-3\\}\\\ \oplus\\\
{T}_{3}\\{-1\\}\\\ \oplus\\\ {T}_{1}\end{array}\xlongrightarrow{\text{
$\begin{pmatrix}b_{3,6}\;&0&a_{3,1}\end{pmatrix}$}}\;\;{T}_{3}.$ (7.76)

${E}_{4}\;\cong\;{T}_{4}\\{-2\\}\xlongrightarrow{\text{
$\begin{pmatrix}\;a_{5,4}\\\
-b_{3,4}\end{pmatrix}$}}\;\;\begin{array}[]{c}{T}_{5}\\{-1\\}\\\ \oplus\\\
{T}_{3}\end{array}\xlongrightarrow{\text{
$\begin{pmatrix}b_{4,5}&\\!\\!\\!\\!\\!\;\;a_{4,3}\end{pmatrix}$}}\;\;{T}_{4}.$
(7.77)

It is easy to check that the the maps indeed define a complex for each of the
brane. For the brane $E_{4}$, on which ${\mathscr{B}}$ acts as the identity,
the complex is just the specialization of (6.55). For ${\mathscr{B}}E_{1}$ and
${\mathscr{B}}E_{2}$, we need to translate the braided Lagrangians into a
sequence of cones over the intersection points of thimbles, just as we did in
section (6.5.9) for the basic figure eight. There is a systematic way to
compute this is from a functor that implements the action of braiding on all
branes in the theory, not just these two, but for the purpose at hand, where
understanding the action of ${\mathscr{B}}$ on just two Lagrangians is all we
need, this will do.

The actual Lagrangian ${\mathscr{B}}E_{\cal U}$ is a complex obtained by
taking a product of the three branes above, translating the maps between the
thimbles to algebra $A$ elements. In doing so, one recalls that the terms in
the product describe intersections of ${\mathscr{B}}E$-brane with the right
thimbles $I_{\cal C}$ of ${\mathscr{D}_{Y}}$. These are all of the form
$I_{\cal C}=I_{i_{1}}\times I_{i_{2}}\times I_{i_{3}}$, with the three one
dimensional branes $I_{i_{k}}$ all distinct. In particular, this means we will
be setting to zero any term where the same thimble $T_{i_{k}}$ appears
(regardless of its equivariant degree) appears more than once. From it, we can
read off the complex that computes
$Hom_{\mathscr{D}_{Y}}^{*,*}({\mathscr{B}}E,I)$, as follows.

To illustrate (7.67) and save space, note that we have the identifications,
writing both the vector space, and the intersection point on $Y$ that gives
rise to it:

$hom_{A}(T_{1}\times T_{3}\times T_{5}\\{-1\\},I_{1}\times I_{3}\times
I_{5})={\mathbb{C}}\,azp={\mathbb{C}}\,{\cal P}_{4}.$

This is placed in equivariant degree $-1$ and cohomological degree $1$ since
$T_{5}\\{-1\\}$ comes from the second from the top term in $E_{4}$, whereas
$T_{1}$ and $T_{3}$ come from the top degree term. Next:

$hom_{A}(T_{1}\times T_{3}\times T_{5}\\{-2\\},I_{1}\times I_{3}\times
I_{5})={\mathbb{C}}\,ayp={\mathbb{C}}\,{\cal P}_{1}.$

this gets assigned cohomological degree $2$, and equivariant degree $-2$.
Finally, we get

$hom_{A}(T_{3}\times T_{1}\times T_{5}\\{-3\\},I_{1}\times I_{3}\times
I_{5})={\mathbb{C}}\,bpx={\mathbb{C}}\,{\cal P}_{2},$

and

$hom_{A}(T_{1}\times T_{5}\times T_{3}\\{-3\\},I_{1}\times I_{3}\times
I_{5})={\mathbb{C}}\,awq={\mathbb{C}}\,{\cal P}_{3}.$

Both of these get placed in cohomological degree $3$ and equivariant degree
$-2$, using (7.48) to identify $T_{1}\times T_{5}\times T_{3}\cong T_{1}\times
T_{3}\times T_{5}\\{1\\}$, for example. (Alternatively, the identification
follows from existence of an algebra $A$ element that maps the brane to itself
permuting the terms. All algebra $A$ elements have equivariant degree zero,
per definition.)

From the algebra, we can also read off the maps. Since $d=3$, these are best
presented graphically, as follows:

which reproduces the complex in (7.74).

#### 7.8.3

Consider now the Hopf link, with projection as in figure 29. In the figure, we
picked a minimal choice of coloring, corresponding to $d=2$, with four
punctures on ${\cal A}$. Replacing two the red under-crossing sections with a
pair of figure eights, we get the pair of intersecting Lagrangians in the
figure.

Figure 29: Hopf link from A-branes

There are eight intersection points, corresponding to pairs ${\cal
P}_{{ij}}=p_{i}q_{j}$ and ${\cal
P}^{\prime}_{ij}={p_{i}^{\prime}}{q_{j}^{\prime}}$, for $i,j=1,2$. To
determine their degrees consider domains interpolating between them. For
example, from the domain labeled by $A_{2}$ in figure 30 which interpolates
from ${\cal P}^{\prime}={\cal P}_{11}$ to ${\cal P}={\cal P}_{22}^{\prime}$ we
read off that $M({\cal P}_{22}^{\prime})-M({\cal P}_{11})=1$, since $A_{2}$
has $3$ acute angles and one obtuse so $e(A_{2})=1-3\cdot{1\over 4}+{1\over
4}={1\over 2}$. We also read off that $J({\cal P}^{\prime}_{22})-J({\cal
P}_{11})=0$, since $i(A_{2})=3\cdot{1\over 4}+{3\over 4}-e(A_{2})=1$ and
$A_{2}$ intersects one puncture. While we used $A_{2}$ to compute the relative
degrees, any other domain interpolating from ${\cal P}_{11}$ to ${\cal
P}^{\prime}_{22}$ gives the same answer, because there are no anomalies in the
theory.

To fix the absolute degrees of intersection points, and relative degrees of
${\mathscr{B}}E_{\cal U}$ and $I_{{\cal U}}$, choose the orientation of
strands of the link $K$. (This choice has nothing to do with the choice of
orientation of the Lagrangians themselves). With choice as in figure 29, $K$
is the Markov trace closure of a braid $B={\beta}\times 1^{2}$ where $\beta$
has two negative crossings, $n_{+}=0$ and $n_{-}=2$. This identifies the point
${\cal P}_{12}=p_{1}q_{2}$ as the intersection point closest to ”all even”
punctures in figure 25, and we assign to it equivariant and Maslov degree
zero. Combining this with the relative degrees of intersection points,
$J({\cal P})$ and $M({\cal P})$ for all of the points in ${\mathscr{B}}E_{\cal
U}\cap I_{{\cal U}}$ are fixed to be those in the following table:

$M$ $J$ | $-3$ | $-2$ | $-1$ | $0$
---|---|---|---|---
$0$ |  |  |  | ${\cal P}_{{12}}$
$1$ |  |  |  |
$2$ |  | ${\cal P}_{{11}},{\cal P}_{{22}}$ | ${\cal P}_{{12}}^{\prime}$ |
$3$ |  | ${\cal P}^{\prime}_{{11}},{\cal P}^{\prime}_{{22}}$ |  |
$4$ | ${\cal P}_{{21}}^{\prime}$ | ${\cal P}_{{21}}$ |  |

The Maslov degrees in the table match those induced from the Lagrangians
${\mathscr{B}}E_{\cal U}=E_{1}\times E_{2}$ and $I_{\cal U}=I_{2}\times I_{4}$
oriented as in figure 29, after assigning to $p_{1}$ and $q_{2}$ in ${\cal
P}_{12}$ degrees $M(p_{1})=0=M(q_{2})$.

Consider now the action of the differential $Q$. To compute it, we proceed as
before, writing ${\mathscr{B}}E_{\cal U}$ in terms of its resolution by the
left thimbles, and finally, compute the cohomology of the complex
$0\rightarrow hom_{A}({\mathscr{B}}E_{\cal U},I_{{\cal U}})\rightarrow 0$,
which each term of which can be written in terms of the intersection points
above. The computation is purely algorithmic. This leads to the following
complex:

The differential squares to zero since the algebra element that enters is
nilpotent, due to the relation in figure 22.

The complex is a prediction for disk instanton counts in this problem. From
geometric perspective, the four non-trivial terms in the complex come from
holomorphic disks that project to domains labeled $A_{2},A_{3},A_{5}$ and
$A_{6}$ in figure 30. The domains $A_{5}$ and $A_{3}$ look at first sight to
be able to pair up into a single domain, which is their union, and which has
Maslov index $2$ and equivariant degree zero.

Figure 30: Four Maslov index one domains, corresponding to $A_{2}$ and
$A_{6}$, and to $A_{3}$ and $A_{5}$ which generate the non-trivial
differential.

It is easy to see that there is no other way to break the domain $A_{3}\cup
A_{5}$ into two, each of which could contribute to the differential (the only
contributions which one could write down do not correspond to holomorphic maps
$y:{\rm D}\rightarrow Y$). Consequently it follows that the moduli space of
honest Maslov index 2 disks is empty - it is generated entirely by a pair of
broken maps, which square to zero on their own. The same considerations follow
for the other pair, $A_{2}$ and $A_{6}$.

The complex has cohomology $Hom_{{\mathscr{D}_{Y}}}({\mathscr{B}}E_{\cal
U},I_{{\cal U}}[M]\\{J\\}),$ encoded by the table below:

$M$ $J$ | $-3$ | $-2$ | $-1$ | $0$
---|---|---|---|---
$0$ |  |  |  | ${\mathbb{C}}$
$1$ |  |  |  |
$2$ |  | ${\mathbb{C}}$ | ${\mathbb{C}}$ |
$3$ |  |  |  |
$4$ | ${\mathbb{C}}$ |  |  |

To relate it to Khovanov homology recall that, with orientations as in figure
29, $K$ is a left handed Hopf link which may be obtained as Markov closure of
a braid $B=\beta\times 1^{2}$ where $\beta$ with $n_{+}=0$, and $n_{-}=2$. It
follows $i_{0}=0=j_{0}$ so the homology groups in the table get identified
with the non-vanishing Khovanov cohomology groups $Kh^{i,j}$ of the left-
handed Hopf link by $i=M+2J$ and $j=2J$. It follows the non-vanishing homology
groups are those of left-handed Hopf link: $Kh^{-2,-6}$, $Kh^{-2,-4}$,
$Kh^{0,-2}$ and $Kh^{0,0}$ are the only non zero ones, and they are all one
dimensional.

The orientation of link components enters only through overall grading. We
could flip the orientation of one of the strands, to get a right-handed Hopf
link instead. The corresponding braid $B^{\prime}={\beta}^{\prime}\times
1^{2}$ has $n_{+}=2$ and $n_{-}=0$, so we would get that $i^{\prime}_{0}=0$
and $j^{\prime}_{0}=4$, and where we assign the point ${\cal
P}_{22}=p_{2}q_{2}$ vanishing Maslov and equivariant degrees. With regrading,
$J$ gradings shift by to $J^{\prime}({\cal P})=J({\cal P})+2$ and
$M^{\prime}({\cal P})=M({\cal P})-2$. We would find the four cohomology groups
in the table correspond to $Kh^{0,0}$, $Kh^{0,2}$, $Kh^{2,4}$ and $Kh^{2,6}$,
again in agreement with Khovanov homology.

##  8 Fusion and filtration from mirror symmetry

In this section I will show that, near a wall in complex structure moduli of
$Y$, where $|a_{i}|\rightarrow|a_{j}|$ for some pair of punctures on ${\cal
A}$, the category of its equivariant A-branes $\mathscr{D}_{Y}$ develops a
filtration with very special properties. In [51] this kind of filtration was
named “perverse filtration”. The key property of the filtration is that
braiding $a_{i}$ and $a_{j}$ preserves it, and acts only by degree shifts on
quotient subcategories.

This filtration is the geometric counterpart of fusion in conformal field
theory, which describes what happens as we bring a pair of vertex operators
together. In conformal field theory, fusion diagonalizes the action of
braiding [131, 132], so it provides the “right”, or at least simplest, setting
to understand it. Similarly, perverse filtrations were envisioned by [51] to
provide the right language to understand the action of braiding on derived
categories.

In [5], I described an analogous filtrations on ${\mathscr{D}_{\cal X}}$. The
filtration allowed to identify branes which correspond to cups or caps, and it
was a key ingredient that made it simple to prove theorem $5^{\star}$, which
says that the link homology groups one obtains from $\mathscr{D}_{\cal X}$ are
link invariants. The $\star$ in the theorem is to indicate the assumption that
braiding acts on the filtration as a perverse equivalence, with stated degree
shifts. One can infer that it does, from the theorem of [30, 136] (or its
physical counterpart which is theorem $4^{\star}$ of [5]). A more direct way
of understanding this is homological mirror symmetry, Conjecture 1 which
identifies $\mathscr{D}_{X}$ and $\mathscr{D}_{Y}$ directly, via the branes
that generate them. It implies same filtration exists on ${\mathscr{D}_{\cal
X}}$, because ${\mathscr{D}_{\cal X}}$ can be generated by holomorphic
Lagrangians all of which come from branes in $\mathscr{D}_{X}$, via the
functor $f_{\star}:{\mathscr{D}_{X}}\rightarrow{\mathscr{D}_{\cal X}}$.

The action of braiding $a_{i}$ and $a_{j}$ on the corresponding categories
comes from variations of stability condition defined with respect to the
central charge ${\cal Z}^{0}$. We get a richer category by allowing non-
compact branes. I will explain how to define ${\cal Z}^{0}$ for such branes in
section 8.3. By conjecture 2, this lets one extend Bridgeland stability
conditions to all holomorphic Lagrangians in ${\cal X}$, compact or not. These
branes, working equivariantly, generate the entire category $\mathscr{D}_{\cal
X}$.

###  8.1 Filtration, braid group actions and link invariants

Just as fusion provides the right language to understand the action of
braiding in confromal field theory, perverse filtrations provide the right
language to describe the action of braiding on derived categories.

Traditionally, describing braid group actions on derived categories of
coherent sheaves, or B-branes, is difficult or at least technical, see for
example [46, 47]. Braid group actions in the A-model are much easier to
describe via Picard-Lefshetz theory and its categorical uplifts [158], see
e.g. [103, 23, 168, 169] for examples. The theory of variations of Bridgeland
stability conditions is invented, by Douglas and Bridgeland, for the purposes
of bridging the two [21, 95].

The vision of Chuang and Rouquier is that perverse equivalences are the right
language to describe how derived equivalences, which come from variations of
the space of stability conditions, act on triangulated categories. In the
current setting, of our ${\mathscr{D}_{Y}}$ and ${\mathscr{D}_{\cal X}}$, this
comes to life.

#### 8.1.1

As an illustration, to show that the move (equivalent to the pitchfork move)
in the figure below is satisfied in ${\mathscr{D}_{\cal X}}$:

Figure 31: A move equivalent to the pitchfork move.

we need to show we get a derived equivalence

${\mathscr{B}}\circ{\mathscr{C}}_{i}\cong{\mathscr{C}}_{i}^{\prime\prime},$
(8.1)

where ${\mathscr{C}}_{i}$ and ${\mathscr{C}}_{i}^{\prime\prime}$ are cup
functors on the right and the left in figure 31, respectively. They increase
the number of strands by two and map

${\mathscr{C}}_{i}:{\mathscr{D}}_{{\cal
X}_{n-2}}\longrightarrow{\mathscr{D}}_{{\cal
X}_{n}},\;\;\textup{and}\;\;{\mathscr{C}}^{\prime\prime}_{i}:{\mathscr{D}}_{{\cal
X}_{n-2}}\longrightarrow{\mathscr{D}}_{{\cal X}^{\prime\prime}_{n}},$

where the subscript serves to indicate the number of strands. In conformal
field theory, they correspond to pair creating
$\Phi_{V_{i}}(a_{i})\otimes\Phi_{{V}_{i}^{*}}(a_{j})$ from the identity $1$,
the inverse of fusing. The functor ${\mathscr{B}}$ is an equivalence of
categories

${\mathscr{B}}:{\mathscr{D}}_{{\cal X}_{n}}\cong{\mathscr{D}}_{{\cal
X}^{\prime\prime}_{n}},$

corresponding to braiding $\Phi_{{V}_{k}}(a_{k})$ with
$(\Phi_{V_{i}}(a_{i})\otimes\Phi_{{V}_{i}^{*}}(a_{j}))$ where
$V_{i},V_{i}^{*}$ color the red and $V_{k}$ the black strand in figure 31.

The identity (8.1) was proven in [5] by noting that

${\mathscr{C}}_{i}{\mathscr{D}}_{{\cal X}_{n-2}}\subset{\mathscr{D}}_{{\cal
X}_{n}}\;\;\textup{and}\;\;{\mathscr{C}}^{\prime\prime}_{i}{\mathscr{D}}_{{\cal
X}_{n-2}}\subset{\mathscr{D}}_{{\cal X}_{n}^{\prime\prime}},$ (8.2)

are the derived sub-categories that are the bottom-most terms of double
filtrations which ${\mathscr{D}}_{{\cal X}_{n}}$ and ${\mathscr{D}}_{{\cal
X}_{n}^{\prime\prime}}$ develop near the intersection of a pair of walls, as
three vertex operators $\Phi_{{V}_{k}}(a_{k})$, $\Phi_{V_{i}}(a_{i})$ and
$\Phi_{{V}_{i}^{*}}(a_{j})$ come together. The functor ${\mathscr{B}}$ acts at
a bottom part of a double filtration at most by degree shifts – this is a
basic property of perverse filtrations. The degree shifts are trivial too,
since if they were not, the relation we are trying to prove would not hold
even in conformal field theory, and we know it does.

A perverse equivalence that acts by degree shifts that are trivial is an
equivalence of categories [51], which proves (8.1). One similarly proves the
other moves needed for homology groups to be link invariants. One should
compare this to proofs of the same relations in [46, 47], which are more
complicated and less general.

###  8.2 Fusion and vanishing cycles

To understand the behavior of conformal blocks as a pair of vertex operators
$\Phi_{V_{i}}(a_{i})$ and $\Phi_{V_{j}}(a_{j})$ come together in (2.11), it
suffices to study conformal blocks which contain only these two operators:

$\langle\lambda|\Phi_{V_{i}}(a_{i})\Phi_{V_{j}}(a_{j})|\lambda^{\prime}\rangle,$
(8.3)

as they contain all the information needed. As in section 2, the
${}^{L}{\mathfrak{g}}$ representations $V_{i}$ and $V_{j}$ are minuscule. We
also choose a weight $\mu_{k}$ in the representation $V_{i}\otimes V_{j}$, and
fix $\lambda^{\prime}=\lambda+\mu_{k}$. In the notation of section 2,
$\nu=\mu_{k}$.

#### 8.2.1

Analogously, to understand filtrations on any ${\mathscr{D}_{\cal X}}$ from
section 2, it suffices to understand them for ${\cal X}$ that corresponds to
conformal blocks in (8.3). Specializing from section 2, ${\cal X}$ is simply

${\cal X}={\rm Gr}_{\mu_{k}}^{(\mu_{i},\mu_{j})}=T^{*}F_{k},$

where $\mu_{i}$, $\mu_{j}$ are the highest weights of minuscule
representations $V_{i}$ and $V_{j}$ and $\mu_{k}$ is the highest weight of a
representation $V_{k}$, in the tensor product of $V_{i}$ and $V_{j}$,

${V}_{i}\otimes{V}_{j}=\bigotimes_{m=0}^{m_{max}}{V}_{k_{m}}.$ (8.4)

In particular

$\mu_{k}=\mu_{i}+\mu_{j}-\sum_{\alpha=1}^{D_{k}}{{}^{L}e}_{\alpha},$ (8.5)

where ${}^{L}e$’s are the simple positive roots, not necessarily distinct.
${\cal X}$ is the moduli space of smooth monopoles, the total number of which
is $D_{k}$, of charges ${}^{L}e_{1},\ldots,^{L}e_{D_{k}}$, in presence of a
pair of singular monopoles of charges $\mu_{i}$ and $\mu_{j}$.

We order the vertex operators in (8.3) so that $|a_{i}|<|a_{j}|$ – this is a
choice of chamber in the Kahler moduli of ${\cal X}$. While this is a special
simple case, it contains all the essential features of the general case.

#### 8.2.2

Since ${\cal X}=T^{*}F_{k}$, the zero section of the cotangent bundle $F_{k}$
is holomorphic Lagrangian in ${\cal X}$ which vanishes at $|a_{i}|=|a_{j}|$
which is the wall in Kahler moduli. This vanishing cycle gives rize to a very
special B-brane

${\cal F}_{k}={\cal O}_{F_{k}}\in\mathscr{D}_{\cal X},$

which is its structure sheaf. From (2.17), the central charge of this brane is

${\cal Z}^{0}[{\cal
F}_{k}]=\int_{F_{k}}\bigl{(}\omega^{1,1}+iB\bigr{)}^{d}/d!\;=\bigl{(}\log(a_{j}/a_{i})\bigr{)}^{D_{k}}/s_{k}.$
(8.6)

where $s_{k}$ is a constant we will not need and $D_{k}$ is the dimension of
the vanishing cycle $F_{k}$. It equals the number of simple roots in (8.5),
which can be written as (2.28)

$D_{k}=d_{i}+d_{j}-d_{k},$ (8.7)

where $d_{i}=\langle\mu_{i},\rho\rangle.$ Since ${\cal X}$ is hyper-Kahler,
there are no quantum corrections to its classical geometry, and the formula
(8.6) is exact.

The equivariant central charge ${\cal Z}[{\cal F}_{k}]$ and its generalization
${\cal V}[{\cal F}_{k}]$, receive instanton corrections computed by
equivariant Gromov-Witten theory of ${\cal X}$, since the deformation that
scales the holomorphic symplectic form of ${\cal X}$ with weight
${\mathfrak{q}}\neq 1$ breaks the hyper-Kahler symmetry, so simple exact
formulas as in (8.6) do not exist.

#### 8.2.3

The equivariant mirror $Y$ of ${\cal X}$ is an open subset of the
appropriately symmetrized product of $d=D_{k}$ copies of ${\cal A}$,
parameterized by $y_{1},\ldots,y_{D_{k}}$, with holomorphic form

$\Omega=\prod_{\alpha=1}^{D_{k}}\,dy_{\alpha}/y_{\alpha}$

and potential

$\displaystyle e^{W}=$
$\displaystyle\prod_{\alpha=1}^{D_{k}}y_{\alpha}^{\lambda_{\alpha}}\;(1-a_{i}/y_{\alpha})^{\langle\mu_{i},^{L}e_{\alpha}\rangle/\kappa}\;(1-a_{j}/y_{\alpha})^{\langle\mu_{j},^{L}e_{\alpha}\rangle/\kappa}$
(8.8) $\displaystyle\;\;\;\;\;\;\;\;\;\times\prod_{1\leq\alpha\neq\gamma\leq
D_{k}}(1-y_{\gamma}/y_{\alpha})^{-\langle^{L}e_{\alpha},^{L}e_{\gamma}\rangle/2\kappa}.$

The equivariant parameters $\lambda_{\alpha}$ depends not on $y_{\alpha}$ but
only on the simple root ${}^{L}e_{\alpha}$ corresponding to it. In $Y$, there
is a single complex structure modulus, which we can identify as
$\log(a_{i}/a_{j})$ and which mirrors the single Kahler modulus in ${\cal X}$.
The parameter $\kappa$ is related to $\lambda_{0}$ we had elsewhere by
$\lambda^{0}=1/\kappa$.

#### 8.2.4

There is a special A-brane in $L_{k}\in\mathscr{D}_{Y}$ of the form

$L_{k}:\qquad a_{i}\;\leq\;y_{1},\ldots,y_{D_{k}}\;\leq\;a_{j},$ (8.9)

where $y$’s run along a straight line from $y=a_{i}$ to $y=a_{j}$ on the
${\cal A}$-cylinder,

$\log(y_{\alpha}/a_{i})=\log(a_{j}/a_{i})t_{\alpha},\qquad
t_{\alpha}\in[0,1],$

and avoid the deleted set ${\rm F}$ in (3.7) by restricting $t$’s coming from
the same root ${}^{L}e_{\alpha}$ to not coincide. The brane is a special
Lagrangian in $Y$, since the phase of $\Omega$ restricted to the brane is
constant:

$\Omega|_{L_{k}}=\bigl{(}\log(a_{j}/a_{i})\bigr{)}^{D_{k}}dt_{1}\wedge\ldots\wedge
dt_{D_{k}}.$

The central charge of $L_{k}$ is

${\cal
Z}^{0}[L_{k}]=\int_{L_{k}}\Omega\;=\;\bigl{(}\log(a_{j}/a_{i})\bigr{)}^{D_{k}}/s_{k},$
(8.10)

which equals the central charge of ${\cal F}_{k}\in\mathscr{D}_{\cal X}$.

For the most part, we will only be interested in the vanishing behavior of the
central charges as $a_{j}$ approaches $a_{i}$. To only capture only the
vanishing behavior of ${\cal Z}^{0}[L_{k}]$, for instance, we will write it as

${\cal Z}^{0}[L_{k}]=(a_{j}-a_{i})^{{\rm D}_{k}}\times\rm{finite}$ (8.11)

where “finite” stands for terms that remain non-vanishing as $a_{i}\rightarrow
a_{j}$.

#### 8.2.5

The equivariant central charge of $L_{k}$ has the vanishing behavior:

${\cal
Z}[L_{k}]=\int_{L_{k}}\Omega\,e^{-W}=(a_{i}-a_{j})^{D_{k}-C_{k}^{g}/\kappa}\times\textup{finite}.$
(8.12)

where $D_{k}$ is as in (8.7) and where

$C_{k}^{g}=-(c_{k}-c_{i}-c_{j})+c_{ij},$ (8.13)

is written in terms of

$c_{i}={1\over 2}\langle\mu_{i},\mu_{i}+2^{L}\rho\rangle,\qquad
c_{ij}=\langle\mu_{i},\mu_{j}\rangle.$ (8.14)

Showing this is elementary. To capture its vanishing behavior of the integral
in (8.12) as $a_{i}\rightarrow a_{j}$, change variables as
$y_{\alpha}=a_{i}+(a_{j}-a_{i})u_{\alpha}.$ This leads to

$\Omega=(a_{i}-a_{j})^{n(\Omega)}\prod_{\alpha=1}^{D_{k}}du_{\alpha}\times\textup{finite},$
(8.15) $\displaystyle e^{W}=(a_{i}-a_{j})^{n(W)/\kappa}$
$\displaystyle\prod_{\alpha=1}^{D_{k}}\;(u_{\alpha})^{-\langle\mu_{i},^{L}e_{\alpha}\rangle/\kappa}\;(u_{\alpha}-1)^{-\langle\mu_{j},^{L}e_{\alpha}\rangle/\kappa}$
(8.16) $\displaystyle\times$ $\displaystyle\prod_{1\leq\alpha<\gamma\leq
n}(u_{\alpha}-u_{\gamma})^{\langle^{L}e_{\alpha},^{L}e_{\gamma}\rangle/\kappa}\times\textup{finite},$

where $n(\Omega)=D_{k}$ and where,

$n(W)=\sum_{\alpha<\beta}\langle^{L}e_{\alpha},^{L}e_{\beta}\rangle-\sum_{\alpha}\langle^{L}e_{\alpha},\mu_{i}+\mu_{j}\rangle=-C_{k}^{g}.$
(8.17)

It follows that a braid $B$ that exchanges $a_{j}$ and $a_{i}$
counterclockwise, takes

$L_{k}\rightarrow{\mathscr{B}}L_{k}=L_{k}[D_{k}]\\{C_{k}^{g}\\},$ (8.18)

where the shift in the homological degree of $L_{k}$ comes from change of the
phase of $\Omega$ restricted to the brane, and the shift in equivariant degree
from the change of $e^{-W}$.

#### 8.2.6

The vector conformal blocks differ from the scalar ones by inserting operators
$\Phi_{\sigma}$ as in (3.17),

${\cal V}_{\sigma}[L_{k}]=\int_{L_{k}}\Phi_{\sigma}\,\Omega\,e^{-W},$

which gives us an integral formulation of the conformal blocks of
$\widehat{{}^{L}\mathfrak{g}}_{\kappa}$ affine Lie algebra corresponding to
$L_{k}$. The explicit expressions for the insertions $\Phi_{\sigma}$ due to
[156, 157] (they can also be obtained, as explained in [7], as limits of
elliptic stable envelopes [13] of the Nakajima quiver variety which is the
Higgs branch of the 3d quiver gauge theory in section 2) imply that as
$a_{i}\rightarrow a_{j}$,

$\Phi_{\sigma}\,\Omega\;=\;\prod_{\alpha=1}^{D_{k}}\;du_{\alpha}\;\Phi_{\sigma}(u),$

remains finite, as the vanishing of $\Phi_{\sigma}$ exactly cancels that of
$\Omega$ in (8.15), so that

${\cal V}[L_{k}]=(a_{i}-a_{j})^{-C_{k}^{g}/\kappa}\times\textup{finite}.$
(8.19)

The formula (8.19) takes the more familiar form if we use the standard
normalization of conformal blocks, obtained by multiplying ours by an overall
factor

$\Omega\,e^{-W}\rightarrow(a_{i}-a_{j})^{\langle\mu_{i},\mu_{j}\rangle/\kappa}\;\Omega\,e^{-W}.$
(8.20)

The choice of normalization does not affect the eigenvectors of the braiding
matrix. It affects the eigenvalues of the braiding matrix only by framing
factors. Geometrically, it leads to an overall shift of equivariant degrees of
the Homs, by replacing

$C_{k}^{g}\rightarrow C_{k}=C_{k}^{g}-c_{ij}=c_{i}+c_{j}-c_{k}$ (8.21)

(see eqns. (5.35) and (5.50) of [5]). While (8.22) is more familiar and
natural from representation theory perspective, $C_{k}$ is in general not
integral, while $C_{k}^{g}$ always is. In the more conventional normalization,
(8.19) turns into

${\cal
V}[L_{k}]=(a_{i}-a_{j})^{(c_{k}-c_{i}-c_{j})/\kappa}\times\textup{finite}.$
(8.22)

The combination $h_{k}=c_{k}/\kappa$ is the
$\widehat{{}^{L}\mathfrak{g}}_{\kappa}$ conformal weight of the vertex
operator ${\Phi}_{V_{k}}$, so (8.22) is exactly the vanishing behavior (2.26)
of the very special conformal block, the one associated to fusing vertex
operators $\Phi_{V_{i}}(a_{i})$ and $\Phi_{V_{j}}(a_{j})$ to
$\Phi_{V_{k}}(a_{j})$

$\Phi_{V_{i}}(a_{i})\otimes\Phi_{V_{j}}(a_{j})\;\;=\;\;(a_{i}-a_{j})^{h_{k}-h_{i}-h_{j}}\Phi_{V_{k}}(a_{j})+\ldots.$
(8.23)

#### 8.2.7

The fact that both ${\cal V}[L_{k}]$ obtained from geometry as the generalized
central charge of the brane $L_{k}\in\mathscr{D}_{Y}$ and ${\cal V}_{k}$ the
conformal block coming from fusing the vertex operators in (8.23) are both
eigenvectors of braiding $a_{i}$ and $a_{j}$ with the same eigenvalue uniquely
identifies them

${\cal V}[L_{k}]={\cal V}_{k},$ (8.24)

where we are using the fact, since $V_{i}$ and $V_{j}$ are minuscule, all
representations in (8.4) have unit multiplicity. Similarly, the generalized
central charge and its scalar conformal block counterpart

${\cal Z}[L_{k}]={\cal Z}_{k},$ (8.25)

are also both eigenvectors of braiding, with the same eigenvalue as ${\cal
V}_{k}$, but different vanishing behavior, written in (8.12). As we will see,
the simple exact relations in (8.24) and (8.25) between brane central charges
and products of fusion in conformal field theory are very special, for the
reason the branes ${\cal F}_{k}$ and $L_{k}$ are special.

#### 8.2.8

There are other Lagrangians in $Y$ whose central charge also vanishes as
$a_{i}\rightarrow a_{j}$, except that it vanishes slower. They are associated
with representations

$V_{k_{m}}\;\subset\;V_{i}\otimes V_{j},$

in the tensor product which are “above” $V_{k}$, in the sense that weight
spaces of $V_{k_{m}}$ contain $\mu_{k}$ as a weight – we will indicate this by
writing

$\mu_{k}<\mu_{k_{m}}.$

It is also useful to identify $k$ with $k_{0}$, so this is what we shall do
going forward. In particular, $V_{k}=V_{k_{0}}$. Associated to representation
$V_{k_{m}}$ for $m\neq 0$ is a non-compact special Lagrangian
$L_{k_{m}}\subset{\mathscr{D}_{Y}}$, of the form

$\displaystyle L_{k_{m}}:\;\;\;\;a_{i}$ $\displaystyle\leq
y_{\beta(1)},\ldots,y_{\beta(D_{k_{m}})}\leq a_{j},\;$ (8.26) $\displaystyle
y_{\beta({D_{k_{m}}}+1)},\ldots,y_{\beta(D_{k_{m}}+n_{-})}$ $\displaystyle\leq
a_{i},\qquad a_{j}\leq
y_{\beta({D_{k_{m}}}+n_{-}+1)},\ldots,y_{\beta(D_{k_{m}}+n_{-}+n_{+})},$

Here, $y_{\beta(1)},\ldots,y_{\beta(D_{k_{m}})}$ correspond to the $D_{k_{m}}$
simple roots in

$\mu_{k_{m}}={\mu}_{i}+{\mu}_{j}-\sum_{k=1}^{D_{k_{m}}}{{}^{L}e}_{\beta(k)}.$

There are $n_{-}$ $y$’s which run over a non-compact cycle that starts at
$y=0$ and runs to $y\rightarrow a_{i}$, in a straight line at fixed ${\rm
Im}\,y={\rm Im}\,a_{i}$. Finally, the remaining $n_{+}$ $y$’s run over the
cycle that starts at $y=a_{j}$ and runs off to $y=\infty$, with ${\rm
Im}\,y={\rm Im}\,a_{j}$ fixed. (This is best pictured by viewing ${\cal A}$ as
an infinite cylinder, in terms of a coordinate $Y\sim Y+2\pi i$ related to $y$
by $y=e^{-Y}$.) Together

$n_{-}+n_{+}=D_{k}-D_{k_{m}}.$

The very definition of the Lagrangian $L_{k_{m}}$ depends on our choice of the
chamber $|a_{i}|<|a_{j}|$. Unlike $L_{k}$, which is unique, there is more than
one special Lagrangian brane we can associate to $V_{k_{m}}\neq V_{k}$
depending on how we choose the non-compact directions of the branes. What is
common to all such branes is their vanishing behavior, which comes from the
compact part of the cycle in (8.26). In what follows, we will refer to the
class of special Lagrangians associated to $V_{k_{m}}$ branes of “type
$L_{k_{m}}$”.

#### 8.2.9

Since the branes of type $L_{k_{m}}$ for $m\neq 0$ are not compact, their
ordinary central charge ${\cal Z}^{0}[L_{k_{m}}]$ apriori diverges.
Equivariant central charge ${\cal Z}[L_{k_{m}}]$ and its generalization ${\cal
V}[L_{k_{m}}]$ converge, provided we choose the appropriate chamber of
equivariant parameters, which we will explain how to do that in sec.8.3.

We can now repeat the derivation above, to show that any brane of type
$L_{k_{m}}$ leads to a solution ${\cal V}[L_{k_{m}}]$ of the KZ equation in
(2.9) which has the same behavior as $a_{i}\rightarrow a_{j}$ as the conformal
block ${\cal V}_{k_{m}}$, corresponding to fusing

$\Phi_{V_{i}}(a_{i})\otimes\Phi_{V_{j}}(a_{j})\;\;=\;\;(a_{i}-a_{j})^{h_{k_{m}}-h_{i}-h_{j}}\Phi_{V_{k_{m}}}(a_{j})+\ldots.$

However, in general unless $V_{k_{m}}=V_{k}$, or equivalently, unless $m=0$,
the relation between ${\cal V}[L_{k_{m}}]$ and ${\cal V}_{k_{m}}$ is only an
asymptotic equality – which holds for large $\kappa$,

${\cal V}[L_{k_{m}}]\sim{\cal V}_{k_{m}},\qquad{\cal Z}[L_{k_{m}}]\sim{\cal
Z}_{k_{m}}$ (8.27)

where the $"\sim"$ means equality to all orders in $1/\kappa$, but not
exactly. We will now explain why this is the case, and what consequences this
has.

#### 8.2.10

Conformal blocks ${\cal V}_{k_{m}}$ obtained by fusing vertex operators are
always eigenvectors of braiding $a_{i}$ and $a_{j}$, by the same analysis that
leads to (2.26). The generalized central charges of branes ${\cal
V}[L_{k_{m}}]$, by contrast, do not diagonalize braiding – braiding acts on
them by generalized Picard-Lefshetz monodromies. The exact formula relating
them is of the form

${\cal V}[L_{k_{m}}]={\cal
V}_{k_{m}}+\sum_{\mu_{k_{\ell}}<\mu_{k_{m}}}n_{\ell}\cdot{\cal V}_{k_{\ell}},$
(8.28)

with coefficients $n_{\ell}$ which are rational functions of ${\mathfrak{q}}$,
which depend on the specific brane $L_{k_{m}}$ chosen. We get the same formula
with ${\cal Z}$ in place of ${\cal V}$ since the formulas depend only on the
K-theory class of the brane, and not on the integrands in (3.17) or (3.14).
Thus, conformal blocks which diagonalize braiding, obtained by fusing
$\Phi_{V_{i}}(a_{i})$ and $\Phi_{V_{j}}(a_{j})$ to $\Phi_{V_{k_{m}}}(a_{j})$,
do not come from branes – instead, they are formal linear combination of
central charges of branes.

#### 8.2.11

It may be helpful to give a simple example. Take ${\cal X}$ which is a
resolution of the $A_{1}$ surface singularity, corresponding to
${}^{L}{\mathfrak{g}}=\mathfrak{su}_{2}$ with two vertex operators colored by
fundamental, spin ${1\over 2}$ representation $V_{1\over 2}$ of
${}^{L}{\mathfrak{g}}$. There are two terms in the fusion product,

$V_{1\over 2}\otimes V_{1\over 2}=V_{0}\oplus V_{1}$

where $V_{0}$ is the trivial, and $V_{1}$ the spin one representation of
$\mathfrak{su}_{2}$.

The monodromy that exchanges $a_{i}$ and $a_{j}$ has eigenvectors ${\cal
Z}_{0}$ and ${\cal Z}_{1}$, which, in their geometric normalization, scale as

${\cal Z}_{0}\sim(a_{i}-a_{j})^{1-2/\kappa}\times\textup{finite},\qquad{\cal
Z}_{1}\sim 1\times\textup{finite},$

so under half monodromy that exchanges $a_{i}$ and $a_{j}$ counterclockwise,
${\mathfrak{B}}$ takes ${\cal Z}_{1}$ to itself and ${\cal Z}_{0}$ to
${\mathfrak{B}}{\cal Z}_{0}=e^{i\pi}{\mathfrak{q}}^{-1}{\cal Z}_{0}$. The
central charges of branes, by contrast equal

${\cal Z}[L_{0}]={\cal Z}_{0},\qquad{\cal Z}[L_{1}]={\cal
Z}_{1}+{{\mathfrak{q}}^{1/2}\over{\mathfrak{q}}^{1/2}+{\mathfrak{q}}^{-1/2}}{\cal
Z}_{0},$ (8.29)

which one shows by from manipulating the Gauss hypergeometric function
identities.

This originates from the action of braiding on ${\mathscr{D}_{Y}}$, which
takes

${\mathscr{B}}L_{0}=L_{0}[1]\\{1\\},\qquad{\mathscr{B}}L_{2}=L_{1}\\#L_{0}[1],$

where ${\mathscr{B}}L_{2}$ is same as the $Cone(L_{0}\xrightarrow{{\cal
P}}L_{1})$. Consequently, only $L_{0}$, corresponding to the trivial
representation, is the eigenbrane of ${\mathscr{B}}$, but not $L_{1}$. One
checks from (8.29) that indeed ${\mathfrak{B}}{\cal Z}[L_{1}]={\cal
Z}[L_{1}]-{\cal Z}[L_{0}]$. The would-be eigenbrane corresponding to $V_{1}$,
with central charge ${\cal Z}_{1}$, is an infinite complex of $L_{1}$ and
$L_{0}$ branes that does not come from any actual Lagrangian in the geometry.
Note that there is nothing a-priori wrong with infinite compexes of branes –
some of them do converge to an actual Lagrangian, as we have seen in the
example of section 5.

Figure 32: The brane ${\mathscr{B}}L_{1}$ obtained from $L_{1}$ by braiding
$a_{i}$ and $a_{j}$ counterclockwise. Refer to section 5 for explanation how
to compute the $Hom_{\mathscr{D}_{Y}}(L_{0},L_{1})$ corresponding to point
${\cal P}$.

###  8.3 Equivariant Bridgeland stability

Apriori, there are no stability conditions for categories of branes on non-
compact manifolds, if one wants to include all branes, as opposed to only
those with compact support, because central charges of non-compact branes
diverge.

In the case at hand, there is a natural way to extend the definition of ${\cal
Z}^{0}$ to all branes in ${\mathscr{D}_{Y}}$ and ${\mathscr{D}_{X}}$ which
uses only the ingredients at hand in the problem anyhow, and is compatible
with both mirror symmetry and braid group actions on these categories. By
Conjecture 2, this central charge should give rise to a Bridgeland stability
condition on all branes in these categories, as well as to a subcategory of
${\mathscr{D}_{\cal X}}$ generated by holomorphic Lagrangians, since they come
as the image of functor $f_{*}:{\mathscr{D}_{X}}\rightarrow{\mathscr{D}_{\cal
X}}$.

To keep the discussion concrete, we will illustrate this in the context of the
current example, which is ${\cal X}={\rm Gr}^{\mu_{i},\mu_{j}}_{\mu_{k}}$ and
its equivariant mirror $Y$, however the construction is general and applies to
any ${\cal X}$ from section 2.

#### 8.3.1

Fix the chamber in the space of equivariant parameters
${\vec{\lambda}}\in{\mathbb{R}}^{{\rm rk}+1}$ in such a way that the
Lagrangians on $Y$ which are mirror to the tilting set of line bundles on
${X}$ coincide with the actual thimbles of downward gradient flows of
$H_{W}={\rm Re}W$. Each of such brane is a product of real-line Lagrangians on
${\cal A}$ which run along the axis of the cylinder, from $y=0$ to $y=\infty$.
In this chamber, the integral that defines the equivariant central charge of
the Lagrangian brane and its mirror, the line bundle on $X$ is finite without
any need to deform the contour and hence the brane itself. This follows per
definition of a thimble. (An instructive toy example, with most of the
essential features, is ${\cal X}={\mathbb{C}}$ and its ordinary mirror ${\cal
Y}$, in appendix A.) A short calculation shows that this corresponds to the
window of equivariant parameters in (5.5) which for ${\cal X}={\rm
Gr}^{\mu_{i},\mu_{j}}_{\mu_{k}}$ specializes to

${1\over 2}\,\langle
e_{\alpha},\mu_{k}+e_{\alpha}\rangle\,\lambda_{0}\;>\;\lambda_{\alpha}^{\prime}\;>\;-{1\over
2}\,\langle e_{\alpha},\mu_{k}+e_{\alpha}\rangle\,\lambda_{0}$ (8.30)

where $\lambda_{0}=1/\kappa$, and where we defined
$\lambda_{\alpha}^{\prime}=\lambda_{\alpha}-{1\over 2}\langle
e_{\alpha},\mu_{i}+\mu_{j}\rangle$.

#### 8.3.2

In the chamber (8.30), not only the thimble branes, but all branes of type
$L_{k_{m}}$, for any $m$ also have finite equivariant central charge. This
let’s us define the ordinary central charge ${\cal Z}^{0}[L_{k_{m}}]$ of any
brane in ${\mathscr{D}_{Y}}$ as the leading term of the equivariant central
charge of the brane in the non-equivariant $\lambda_{0}\rightarrow 0$ limit,
which has the form

${\cal
Z}^{0}[L_{k_{m}}]\equiv\log(a_{j}/a_{i})^{D_{k_{m}}}\times\bigl{(}1/\lambda\bigr{)}^{D_{k}-D_{k_{m}}}/s_{L_{k_{m}}},$
(8.31)

Above $\log(a_{j}/a_{i})^{D_{k_{m}}}$ is the holomorphic volume of the compact
part of $L_{k_{m}}$, since $D_{k_{m}}$ is the dimension of the vanishing
cycle,

$\bigl{(}1/\lambda\bigr{)}^{D_{k}-D_{k_{m}}}=\prod_{i=1}^{D_{k}-D_{k_{m}}}(1/\lambda_{i}),$
(8.32)

is the product over the non-compact directions in $L_{k_{m}}$, and
$s_{L_{k_{m}}}$ is a symmetry factor that depends on the brane but not on any
parameters.

For the compact Lagrangian $L_{k}$, from (8.31) we immediately recover the
original definition of the central charge. For non-compact Lagrangians of type
$L_{k_{m}}$ the central charge is made finite by turning on equivariance, but
the dependence on equivariant parameters is canonically fixed.

To be explicit, equivariant weights $\lambda_{i}$ in (8.32) are all real and
positive and go to zero as $\lambda_{0}$ goes to zero by the virtue of being
in chamber (8.30). Moreover, $\lambda_{i}$ equals ${\lambda}_{\alpha}^{-}$ for
the direction in $L_{k_{m}}$ associated to a root $e_{\alpha}$ that runs off
to $y=0$ and equals ${\lambda}_{\alpha}^{+}$ for a root that runs off to
$y=\infty$ instead, where

$\lambda^{+}_{\alpha}=\lambda_{\alpha}^{\prime}-{1\over 2}\,\langle
e_{\alpha},\mu_{k}+e_{\alpha}\rangle\,\lambda_{0},\qquad\lambda^{-}_{\alpha}={1\over
2}\,\langle
e_{\alpha},\mu_{k}+e_{\alpha}\rangle\,\lambda_{0}-\lambda_{\alpha}^{\prime}.$

The key property of this definition of the central charge is that it is
compatible with both mirror symmetry and with the action of braiding on
${\mathscr{D}_{Y}}$ and ${\mathscr{D}_{\cal X}}$. This is the case simply
because the equivariant central charge, which is used as the starting point,
has these properties.

#### 8.3.3

The equivariant mirror of $L_{k}\in\mathscr{D}_{Y}$ is the brane ${\cal
F}_{k}\in\mathscr{D}_{\cal X}$ we started the section with, supported on the
holomorphic Lagrangian $F_{k}$ which is the zero section of ${\cal
X}=T^{*}F_{k}$. Both branes are the stable objects of the respective
categories, with respect to the central charge ${\cal Z}^{0}$. $L_{k}$ is
stable because as we have seen, it is a special Lagrangian in $Y$; ${\cal
F}_{k}$ is stable as the structure sheaf of a holomorphic Lagrangian, since
there are no quantum corrections on ${\cal X}.$ $L_{k}$ is the ordinary
homological mirror of the brane $F_{k}\in\mathscr{D}_{X}$, the structure sheaf
of the vanishing cycle in $X$ with the same name, and ${\cal
F}_{k}={\mathscr{O}}_{F_{k}}$ comes from it via ${\cal F}_{k}=f_{*}F_{k}$.

The equivariant mirror of the special Lagrangian of type $L_{k_{m}}$ is a
sheaf of type ${\cal F}_{k_{m}}\in\mathscr{D}_{\cal X}$ described explicitly
in section 5 of [5], which also comes via the functor $f_{*}$ from a brane on
$X$, which is the ordinary homological mirror or $L_{k_{m}}$. ${\cal
F}_{k_{m}}$ is the structure sheaf of a holomorphic Lagrangian in ${\cal X}$
which restricts to a compact cycle $F_{k_{m}}\subset F_{k}$ of dimension
$D_{k_{m}}$. $F_{k_{m}}$ arizes as the zero section of ${\rm
Gr}^{\mu_{i},\mu_{j}}_{{\mu_{k_{m}}}}=T^{*}F_{k_{m}}\subset{\cal X}$, just as
$F_{k}$ does in ${\cal X}={\rm Gr}^{\mu_{i},\mu_{j}}_{{\mu_{k_{m}}}}$. The
first factor in (8.31)

$\log(a_{j}/a_{i})^{D_{k_{m}}}$

is, up to a numerical coefficient, equal the holomorphic volume of
$F_{k_{m}}$. The choice of a specific Lagrangian $L_{k_{m}}$ determines what
the brane ${\cal F}_{k_{m}}$ looks like in the $T^{*}$ fiber over $F_{k}$ in
${\cal X}$. The second factor of ${\cal Z}^{0}$, given in (8.32), coincides
with the equivariant volume of the non-compact directions of the mirror brane
${\cal F}_{k_{m}}$.

It follows that central charges of these branes are equal,

${\cal Z}^{0}[L_{k_{m}}]={\cal Z}^{0}[{\cal F}_{k_{m}}].$

On both sides, ${\cal Z}^{0}$ makes use of equivariance to effectively
compactify $Y$ and ${X}$, in a way compatible with mirror symmetry.

#### 8.3.4

With ${\cal Z}^{0}$ in hand, we can define ${\mathscr{D}_{\cal X}}$,
$\mathscr{D}_{X}$ and ${\mathscr{D}_{Y}}$ as categories of branes whose
central charge ${\cal Z}^{0}$ is finite,

$|{\cal Z}^{0}|<\infty.$

Per construction, this includes the tilting set of line bundles on $X$. Every
such line bundle $F\in\mathscr{D}_{X}$ gives rise to a holomorphic Lagrangian
${\cal F}=f^{*}F$ in $\mathscr{D}_{\cal X}$, so correspondingly,
${\mathscr{D}_{\cal X}}$ should be thought of as generated by holomorphic
Lagrangians on ${\cal X}$. Since we are working equivariantly, holomorphic
Lagrangians generate all of ${\mathscr{D}_{\cal X}}$ – the entire derived
category of ${\rm T}$-equivariant coherent sheaves on ${\cal X}$ including,
for example, line bundles on ${\cal X}$. Furthermore, holomorphic Lagrangians
on ${\cal X}$ and special Lagrangians on $Y$ are both stable objects with
respect to ${\cal Z}^{0}$.

From physics perspective, ${\cal Z}^{0}$ defined in this way shares all the
properties of the $\Pi$-stability central charge of Douglas [59, 60], which is
what motivated Bridgeland’s work [41, 42, 43]. It follows:

######  Conjecture 2.

The central charge ${\cal Z}^{0}$ defined by the leading $\lambda_{0}$ to zero
limit of equivariant central charge

${\cal Z}={\cal Z}^{0}+{\cal O}(\lambda^{0}),$ (8.33)

in the chamber of equivariant parameters (5.5) for which the tilting line
bundles on ${X}$ have finite equivariant central charge ${\cal Z}$, defines
the Bridgeland stability condition on ${\mathscr{D}_{Y}}$, on
${\mathscr{D}_{X}}$ and on ${\mathscr{D}_{\cal X}}$.

Next, we will learn how to describe derived equivalences that come from
variations of this stability condition. We will keep $\lambda$’s fixed, and
vary complexified Kahler moduli of ${\cal X}$, and complex structure moduli on
$Y$.

###  8.4 Perverse filtration from mirror symmetry

The reason ${\cal V}[L_{k_{m}}]$ fails to diagonalize braiding is that, unless
$m=0$, the braiding functor ${\mathscr{B}}$ which lifts the half-monodromy
that exchanges $a_{i}$ and $a_{j}$ counterclockwise to $\mathscr{D}_{Y}$, does
not bring the brane $L_{k_{m}}$ back to itself up to degree shifts

${\mathscr{B}}L_{k_{m}}\neq L_{k_{m}}[D_{k_{m}}]\\{C^{g}_{k_{m}}\\},$

like it did for $L_{k}=L_{k_{0}}$. Instead, it mixes up the brane $L_{k_{m}}$,
with those that have a higher degree of support on the vanishing cycle.

#### 8.4.1

The special Lagrangian $L_{k}=L_{k_{0}}$ and the special Lagrangians of the
type $L_{k_{1}},\ldots,L_{k_{max}}$ generate the heart

${\mathscr{A}}\subset{\mathscr{D}_{Y}}$

of the derived category ${\mathscr{D}_{Y}}$ in the chamber where
$|a_{j}|>|a_{i}|$. The heart ${\mathscr{A}}$ is a subcategory of
$\mathscr{D}_{Y}$ obtained from the collection of special Lagrangians we
started with by taking Lagrangian connected sums $\\#$ (these mirror taking
extensions of coherent sheaves in ${\mathscr{D}_{\cal X}}$, see e.g. [169,
162]). ${\mathscr{A}}$ consists of Lagrangians whose central charge is the
upper half of the complex ${\cal Z}^{0}$ plane, since the central charge of a
connected-sum brane is the sum of central charges of branes it consists of.
From the heart ${\mathscr{A}}$ we recover the entire the derived category
${\mathscr{D}_{Y}}$ by further degree shifts and taking direct sums and cones.

#### 8.4.2

The heart ${\mathscr{A}}$ is filtered by subcategories
${\mathscr{A}}_{k_{m}}\subset{\mathscr{A}}$, where ${\mathscr{A}}_{k_{m}}$ is
generated from $L_{k_{0}}$ and Lagrangians of type
$L_{k_{1}},\ldots,L_{k_{m}}$, by taking connected sums, etc. This way, we have
a filtration of ${\mathscr{A}}$

${{\mathscr{A}}}_{k_{0}}\subset{{\mathscr{A}}}_{k_{1}}\ldots\subset{{\mathscr{A}}}_{k_{max}}={{\mathscr{A}}},$
(8.34)

whose terms are labeled by representations $V_{k_{m}}$ in the tensor product,

${V}_{i}\otimes{V}_{j}=\bigoplus_{m=0}^{\rm{max}}{V}_{k_{m}},$

ordered so that

$\mu_{k_{m}}<\mu_{k_{m+1}},$ (8.35)

where $\mu_{k_{m}}$ are the highest weights of ${V}_{k_{m}}$, for all $m$.
This is the same as ordering by the dimension of the vanishing cycle, since
(8.35) implies

$D_{k_{m}}>D_{k_{m+1}}.$

This filtration induces a corresponding filtration of $\mathscr{D}_{Y}$

${{\mathscr{D}}}_{k_{0}}\subset{{\mathscr{D}}}_{k_{1}}\ldots\subset{{\mathscr{D}}}_{k_{max}}={{\mathscr{D}}}_{Y},$
(8.36)

where the derived category ${{\mathscr{D}}}_{k_{m}}$ is generated by taking
direct sums, cones and shifts of the branes in ${\mathscr{A}}_{k_{m}}$.

On the other side of the wall, for $|a_{i}|>|a_{j}|$, we get the same kind of
filtration

${{\mathscr{D}}}_{k_{0}}^{\prime}\subset{{\mathscr{D}}}_{k_{1}}^{\prime}\ldots\subset{{\mathscr{D}}}_{k_{max}}^{\prime}={{\mathscr{D}}}_{Y},$
(8.37)

obtained from the filtration of the heart ${{\mathscr{A}}^{\prime}}$ of
${{\mathscr{D}}}_{Y}$,

${{\mathscr{A}}^{\prime}}_{k_{0}}\subset{{\mathscr{A}}^{\prime}}_{k_{1}}\ldots\subset{{\mathscr{A}}^{\prime}}_{k_{max}}={{\mathscr{A}}^{\prime}},$
(8.38)

except that the heart ${\mathscr{A}}^{\prime}$ is different, because the set
of special Lagrangians which have central charge ${\cal Z}^{0}$ in the upper
half of the complex plane is different, as is clear from their construction
earlier in this section.

#### 8.4.3

The key aspect of the filtrations is that the derived equivalence
${\mathscr{B}}$, that corresponds to braiding $a_{i}$ and $a_{j}$, preserves
them. Pick a clockwise path $B$ across the wall, avoiding the singularity at
$a_{i}=a_{j}$. Along such a path, branes whose central charge vanishes to any
given order $m$ in the filtration can get mixed up with those at lower order,
whose central charge vanishes faster near the wall, but not the other way
around. This is a simple consequence of Picard-Lefshetz monodromies of
vanishing cycles. In fact, we can describe very precisely how derived
equivalence ${\mathscr{B}}$ acts.

Consider the quotient category,

${\rm gr}_{m}({\mathscr{A}})={\mathscr{A}}_{k_{m}}/{\mathscr{A}}_{k_{m-1}},$
(8.39)

obtained from ${\mathscr{A}}_{k_{m}}$ by treating all objects that come from
${\mathscr{A}}_{k_{m-1}}$ as zero. Even though ${\mathscr{B}}$ does not
preserve ${\mathscr{A}}_{k_{m}}$ or ${\mathscr{A}}_{k_{m}}^{\prime}$
themselves, it preserves the filtration and acts on ${\rm
gr}_{m}({\mathscr{A}})$ at most by degree shifts. This is because any brane
whose central charge vanishes faster is an object that is treated as zero in
the quotient categories ${\mathscr{A}}_{k_{m}}/{\mathscr{A}}_{k_{m-1}}$ and
${\mathscr{A}}^{\prime}_{k_{m}}/{\mathscr{A}}^{\prime}_{k_{m-1}}$, and there
are no contributions from branes whose central charge vanishes slower to
either ${\mathscr{A}}_{k_{m}}$ or to ${\mathscr{A}}_{k_{m}}^{\prime}$, per
their definitions.

To read off the degree shifts note that the equivairant central charge any
object of either ${\rm gr}_{m}({\mathscr{A}})$ or of ${\rm
gr}_{m}({\mathscr{A}}^{\prime})$ is

${\cal Z}_{k_{m}}=(a_{i}-a_{j})^{D_{{m}}-C^{g}_{m}/\kappa}\times{\rm finite},$
(8.40)

per (8.28), and ${\cal Z}_{k_{m}}$ is an eigenvector of the action of
braiding. It follows that, along the path $B$, the central charge ${\cal
Z}[L]$ of an arbitrary object $L\in{\rm gr}_{m}({\mathscr{A}})$ changes by the
following phase:

${\cal Z}[L]\;\longrightarrow e^{-\pi iD_{m}}\,{\mathfrak{q}}^{{1\over
2}C^{g}_{m}}\,{\cal Z}[L],$ (8.41)

which reflects the cohomological and equivariant degree shifts. The phase does
not depend on the Lagrangian $L$ at all, but only on the order $m$ in the
filtration it belongs to, and the path $B$ we chose. Thus the functor
${\mathscr{B}}$ composed with a degree shift $[D_{m}]\\{C_{m}^{g}\\}$ is an
equivalence of abelian categories

${\mathscr{B}}[D_{m}]\\{C_{m}^{g}\\}:\;{\rm
gr}_{m}({\mathscr{A}})\;\cong\;{\rm gr}_{m}({\mathscr{A}}^{\prime}).$ (8.42)

Derived equivalences with these properties are called a “perverse
equivalences” by Chuang and Rouquier [51]. This equivalence of abelian
categories induces the corresponding equivalence of their derived categories,

${\mathscr{B}}:\;{\mathscr{D}}_{k_{m}}/{\mathscr{D}}_{k_{m-1}}\cong{\mathscr{D}}_{k_{m}}^{\prime}/{\mathscr{D}}_{k_{m-1}}^{\prime}.$

#### 8.4.4

By construction of ${\cal Z}^{0}$, ${\mathscr{D}_{\cal X}}$ also has a heart
${\mathscr{A}}\subset{\mathscr{D}_{\cal X}}$, filtered by subcategories
${\mathscr{A}}_{k_{m}}$ which are generated by structure sheaves of
holomorphic Lagrangian branes, ${\cal F}_{k}$ and together with branes of type
${\cal F}_{k_{1}},\ldots,{\cal F}_{k_{m}}$. They are all stable with respect
to ${\cal Z}^{0}$ and mirror to special Lagrangian branes ${L}_{k}$ and
together with branes of type ${L}_{k_{1}},\ldots,{L}_{k_{m}}$, that generate
the filtration on $\mathscr{D}_{Y}$, see section 8.3.3. One clearly gets such
a filtration on both sides of the wall, for $|a_{j}|>|a_{i}|$ on a manifold we
called ${\cal X}$, and for $|a_{i}|>|a_{j}|$ on its generalized flop which was
called ${\cal X}^{\prime}$ in [5].

What remains to show is that the derived equivalence functor ${\mathscr{B}}$,
that corresponds to braiding $a_{i}$ and $a_{j}$ clockwise preserves the
filtrations, and acts by degree shifts in (8.42). The fact that it preserves
the filtrations can be seen by a hyper-Kahler rotation of ${\cal X}$, which
maps the holomorphic to special Lagrangians on ${\cal X}$. The complete
existence proof would be provided by equivariant mirror symmetry. More
abstractly, it also follows from the theorem of Bezrukavnikov and Okounkov
[30, 137], and its physics reinterpretation, theorem $4^{\star}$ of [5], which
says that monodromies of the KZ equation, viewed as the quantum differential
equation of ${\cal X}$, lift to derived equivalences of $\mathscr{D}_{\cal
X}$.

##  Appendix A Homological mirror symmetry for ${\mathbb{C}}$

In this section will work out a classic example of homological mirror symmetry
for ${\cal X}={\mathbb{C}}$. The only new ingredient is that we will be
working equivariantly on both sides.

###  A.1 Equivariant derived category of coherent sheaves on ${\mathbb{C}}$

Take ${\cal X}={\mathbb{C}}$ with complex coordinate $x$. Coherent sheaves on
${\cal X}$ can be identified with modules for the ring

${\mathscr{R}}={\mathbb{C}}[x],$

of holomorphic functions on ${\cal X}$. Elements of ${\mathscr{R}}$ are
polynomials in $x$. From this perspective, the trivial line bundle on ${\cal
X}$

${\cal P}={\cal O}_{\cal X},$

is the structure sheaf of ${\cal X}$ and a module for ${\mathscr{R}}$ equal to
${\mathscr{R}}$ itself. Another example is ${\cal O}_{a}$, the structure sheaf
of a point $x=a$, on which elements of ${\mathscr{R}}$ act by evaluation.

#### A.1.1

The derived category $\mathscr{D}_{\cal X}$ of coherent sheaves on ${\cal X}$
has as its objects complexes of coherent sheaves, where one identifies
complexes whose cohomologies are the same. Given any coherent sheaf ${\cal M}$
on ${\cal X}$, we get an object of the derived category ${\cal M}[n]$ which is
a 1-term complex,

$\ldots\rightarrow 0\rightarrow{\cal M}\rightarrow 0\rightarrow\ldots.$

in a fixed cohomological degree $-n$, with maps that are all zero. These
identifications simplify the theory, since many distinct complexes describe
the same object of $\mathscr{D}_{\cal X}$. For example, in the derived
category, the structure sheaf ${\cal O}_{a}[n]$ is identified in
${\mathscr{D}_{\cal X}}$ with the cohomology of the complex

$\cdots\rightarrow 0\rightarrow{\cal P}\xrightarrow{x-a}{\cal P}\rightarrow
0\rightarrow\cdots,$

where all but one map are zero, and the rightmost copy of ${\cal P}$ is placed
in degree cohomological $-n$. Often, we will omit the zeros.

In fact, $\mathscr{D}_{\cal X}$ is in this case generated by ${\cal P}$, as
every one of its objects has a free resolution in terms of ${\cal P}$, for
example. A free resolution of ${\cal M}\in\mathscr{D}_{\cal X}$ is a complex
all of whose terms are direct sums of copies of ${\cal P},$

$\ldots\rightarrow{\cal M}^{-2}\rightarrow{\cal M}^{-1}\rightarrow{\cal
M}^{0}\rightarrow 0\rightarrow\ldots,$ (A.1)

where ${\cal M}^{-i}\equiv{\cal P}^{\oplus r_{i}}$ is a direct sum of $r_{i}$
copies of ${\cal P}$, placed in cohomological degree $-i$. The complex that
resolves ${\cal M}$ has cohomology in degree zero equal to ${\cal M}$ and zero
otherwise, so in ${\mathscr{D}_{\cal X}}$, ${\cal M}$ and its resolutions are
isomorphic.

#### A.1.2

As in the examples of section 5, we get a further equivalence of the derived
category $\mathscr{D}_{\cal X}$ of coherent sheaves on ${\cal X}$ with derived
category of modules of an algebra ${\mathscr{A}}$,

$\mathscr{D}_{\cal X}\cong\mathscr{D}_{\mathscr{A}},$

see for example, where ${\mathscr{A}}$ is the algebra of endomorphisms of the
line bundle ${\cal P}$, which here coincides with the ring ${\mathscr{R}}$:

${\mathscr{A}}=Hom_{\mathscr{D}_{\cal X}}({\cal P},{\cal P})={\mathscr{R}}.$

Thinking about coherent sheaves as modules for ${\mathscr{A}}$, simplifies
computations. As in section 5, given the resolution of ${\cal M}$ in (A.1),
and any other object ${\cal N}$ of ${\mathscr{D}_{\cal X}}$, the Homs of the
derived category are given by [18, 99, 91]

$Hom_{\mathscr{D}_{\cal X}}({\cal M},{\cal
N}[k])=H^{k}(hom_{\mathscr{A}}({\cal M},{\cal N})),$ (A.2)

where $hom_{\mathscr{A}}({\cal M},{\cal N})$ are ${\mathscr{A}}$-module
homomorphisms from ${\cal M}$ to ${\cal N}$. It follows

$Hom_{\mathscr{D}_{\cal X}}({\cal M},{\cal N}[k])=Hom_{\mathscr{D}_{\cal
X}}({\cal M}[j],{\cal N}[j+k]),$

so we can work only with homology in degree zero, at the expense of working
with branes of all degrees.

#### A.1.3

The algebra ${\mathscr{A}}$ is the path algebra of a quiver $Q_{\mathscr{A}}$,
which has a single node, one arrow which represents multiplication by $x$, and
no relations. The quiver $Q_{\mathscr{A}}$ is familiar to physicists – it
describes the 0-branes on ${\cal X}={\mathbb{C}}$. More precisely, the 0-brane
at $x=0$ corresponds to the quiver representation of rank one, on which the
arrow $x$ acts by zero. It is also the simple module ${\cal S}$ of the algebra
${\mathscr{A}}$, and the structure sheaf of the origin in ${\cal X}$

${\cal S}={\cal O}_{0}.$

The brane ${\cal P}={\cal O}_{\cal X}$, is a 2-brane on ${\cal X}$ with
trivial bundle on it, and a the projective module of the path algebra. The
fact that ${\cal P}$ generates $\mathscr{D}_{\cal X}$ reflects the fact that
any B-brane on ${\cal X}$ can be obtained as a bound state of 2-branes and
their anti-branes on ${\cal X}$. For example, the fact that, as objects of
${\mathscr{D}_{\cal X}}$, the sheaf ${\cal O}_{a}$ is indistinguishable from
the complex ${\cal P}\xrightarrow{x-a}{\cal P}$ corresponds to the fact that,
as B-branes, a 0-brane at $x=a$ is indistinguishable from a bound state of a
2-brane and its anti-brane, obtained by giving an expectation value $x-a$ to
the open string tachyon between them. One gets the 0-brane where the tachyon
field vanishes. More generally, the complex such as (A.1) describes a brane
${\cal M}\in\mathscr{D}_{\cal X}$, obtained by successive gluings $r_{i}$
2-branes with $r_{i+1}$ anti-2-branes by giving specific expectation values to
tachyon fields between them. Many additional details may be found in [90, 19,
18, 95, 21, 17, 167].

#### A.1.4

Working equivariantly with respect to a torus ${\rm T}={\mathbb{C}}^{\times}$
which acts by $x\rightarrow\mathfrak{h}x$ for $\mathfrak{h}$ a non-zero
complex number, the algebra ${\mathscr{A}}$ becomes graded

${\mathscr{A}}=\oplus_{n=0}^{\infty}{\mathscr{A}}_{n},$

where ${\mathscr{A}}_{n}={\mathbb{C}}x^{n}$ is generated by a single element
$x^{n}$ for $n\geq 0$, and the grading is in this case just the path length.
Equivariant coherent sheaves on ${\cal X}$ correspond to graded
${\mathscr{A}}$-modules. A graded module ${\cal M}$ is simply ${\cal M}$
itself, where we keep track of equivariant degree: ${\cal M}=\oplus_{n}{\cal
M}_{n}$ where ${\cal M}_{n}$ has degree $n$. For example, working
equivariantly, the structure sheaf ${\cal P}$ splits ${\cal P}=\oplus_{n}{\cal
P}_{n}$ where ${\cal P}_{n}$ is generated by $x^{n}$, for $n\geq 0$, and $0$
otherwise. A simple operation on graded modules is an equivariant degree shift
that maps a module ${\cal M}$ to ${\cal M}\\{d\\}$, defined by ${\cal
M}\\{d\\}_{n}={\cal M}_{n+d}$, which decreases the degree of all elements of
${\cal M}$ by $d$.

The ${\mathscr{A}}$-module homomorphisms also become graded, where
homomorphisms in degree $n$ map ${\cal M}_{m}\rightarrow{\cal N}_{m+n}$ for
all $m$. The theory is invariant under overall degree shifts, so the
homomorphism in degree $n$ from ${\cal M}$ to ${\cal N}$ is the same as
homomorphism in degree zero from ${\cal M}$, to ${\cal N}\\{n\\}$. For this
reason, we loose nothing by declaring that

$Hom_{\mathscr{D}_{\cal X}}({\cal M},{\cal N}[k]\\{n\\}),$

contain only the Hom’s in degree zero, both the homological and equivariant.
Taking a direct sum,

$Hom_{\mathscr{D}_{\cal X}}^{*,*}({\cal M},{\cal
N})=\bigoplus_{k,n\in{\mathbb{Z}}}Hom_{\mathscr{D}_{\cal X}}({\cal M},{\cal
N}[k]\\{n\\}),$

contains all the Hom’s of the ungraded theory.

#### A.1.5

For example, ${\cal P}\\{d\\}_{n}$ is generated by $x^{n+d}$, for $n\geq-d$.
For $d\geq 0$ we get an ${\mathscr{A}}$-module homomorphism from ${\cal P}$ to
${\cal P}\\{d\\}$, which is just multiplication by $x^{d}$. This takes
monomials $1,x,\ldots$ in degrees $0,1,\ldots$ viewed as elements of ${\cal
P}$ and maps them to $x^{d},x^{d+1},\ldots$ in ${\cal P}\\{d\\}$, in the same
degrees, and acts by zero otherwise. This, together (A.2) leads to

${\mathscr{A}}_{d}=Hom_{\mathscr{D}_{\cal X}}({\cal P},{\cal
P}\\{d\\})={\mathbb{C}}x^{d},\textup{ for }d\geq 0.$

Hom’s in the other direction, or in other homological degrees, are all zero.

As another example, 0-brane at the origin has a projective resolution as the
complex

${\cal S}\;\;\cong\;\;\ {\cal P}\\{-1\\}\xrightarrow{x}{\cal P}$ (A.3)

of graded modules where all the omitted terms and maps are $0$. Here, one
views ${\cal P}\\{-1\\}$ as placed in cohomological degree $-1$, ${\cal P}$ in
degree $0$, and $x$ deforms the otherwise trivial differential on ${\cal
P}[1]\\{-1\\}\oplus{\cal P}$.

As additional practice, by using the resolution in (A.3), and computing
cohomologies in (A.2) for ${\cal M}={\cal S}$ and various ${\cal N}$’s, one
finds

$Hom_{\mathscr{D}_{\cal X}}({\cal P},{\cal
S})={\mathbb{C}}=Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal P}[1]\\{-1\\})$
(A.4)

Homs in all other degrees are zero. Similarly, one finds

$Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal
S})={\mathbb{C}}=Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal S}[1]\\{-1\\}).$
(A.5)

This is as expected since $Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal
S})={\mathbb{C}}$ reflects the gauge group of the single 0-brane being $U(1)$,
and $Hom_{\mathscr{D}_{\cal X}}({\cal S},{\cal S}[1]\\{-1\\})={\mathbb{C}}$
corresponds to deformations of the 0-brane by changing its position on ${\cal
X}={\mathbb{C}}$ [21]. Both are consistent with Serre duality for a one
dimensional ${\cal X}$, with trivial canonical bundle whose section $dx$
scales with degree $1$.

###  A.2 Homological mirror symmetry, equivariantly

Now we will see how the above structure emerges from mirror symmetry. The
mirror to ${\cal X}={\mathbb{C}}$ working equivariantly with respect to the
${\rm T}={\mathbb{C}}^{\times}$-action is the Landau-Ginsburg model on target
${\cal Y}={\mathbb{C}}^{\times}$, with potential

${\cal W}(y)=\lambda\log y+y^{-1},$

and points $y=0,\infty$ deleted, and holomorphic form

$\Omega=dY={dy\over y}.$

This Landau-Ginsburg model is the Hori-Vafa mirror to ${\mathbb{C}}$ [92]. The
parameter ${\mathfrak{h}}$ of the equivariant $\Lambda$-action on ${\cal X}$
maps to the parameter $\lambda$ in the superpotential by

$\mathfrak{h}=e^{2\pi i\lambda}.$

With $\lambda\neq 0$, the superpotential is not single valued. For now, we
will fix $\lambda\in{\mathbb{R}}_{>0}$.

#### A.2.1

An equivariant A-brane on ${\cal Y}$ is a Maslov graded Lagrangian ${\cal L}$,
together with a lift ${\cal W}$ to a single valued function on ${\cal L}$, as
explained in section 4. In particular, given an A-brane ${\cal L}$, we denoted
by ${\cal L}\\{d\\}$ an A-brane with a different lift of ${\cal W}$,

${\cal W}|_{{\cal L}\\{d\\}}={\cal W}|_{{\cal L}}+2\pi id\,\lambda.$

which follows from (4.13). The equivariant central charge of the brane is
given by

${\cal Z}[{{\cal L}}]=\int_{{\cal L}}\Omega e^{-{\cal W}}.$ (A.6)

The definition of the integral requires a choice of the lift of ${\cal W}$,
which is why we needed to distinguish the A-branes.

#### A.2.2

Mirror to a line bundle brane ${\cal P}$ on ${\cal X}$, is the brane

${{\cal T}},$

given by the set of points of the form $Y=Y_{\star}+{\mathbb{R}}$. As an
object of ${\mathscr{D}_{Y}}$ this Lagrangian, and any other in the same
isotopy class (for all branes in this class, the two infinite ends of the
brane approach the two infinities on ${\cal Y}={\mathbb{C}}^{\times}$), give
an equivalent brane.

If we choose $Y_{\star}$ to be the critical point of the potential, then
${\cal T}$ is the thimble of the potential $W$, the set of all initial
conditions for downward gradient flows of the Hamiltonian $H_{\cal W}={\rm
Re}({\cal W})$, which get attracted to the critical point of the potential.
Along the thimble, ${\rm Im}\,{\cal W}$ is conserved, equal to its value at
the critical point. We will call ${\cal T}$ the thimble brane.

#### A.2.3

The thimble brane ${{\cal T}}$ generates the derived category
${\mathscr{D}_{\cal Y}}$ of A-branes on ${\cal Y}$, just like the line bundle
brane ${\cal P}$ generates ${\mathscr{D}_{\cal X}}$ [158]. To show that
$\mathscr{D}_{\cal X}$ and $\mathscr{D}_{\cal Y}$ are mirror

$\mathscr{D}_{\cal X}\cong\mathscr{D}_{\cal Y}$

it suffices to show that the branes ${\cal T}$ and ${\cal P}$ which generate
the two categories are mirrors of each other

${{\cal T}}\xleftrightarrow{mirror}{\cal P}.$

For this, we want to show that their bi-graded Homs are equal. In other words,
we want to show that

$Hom^{*,*}_{\mathscr{D}_{Y}}({\cal T},{\cal
T})=\bigoplus_{n,d\in{\mathbb{Z}}}Hom_{\mathscr{D}_{Y}}({\cal T},{\cal
T}[n]\\{d\\})$ (A.7)

coincide with the algebra ${\mathscr{A}}$. The calculation follows
straightforwardly applying section 4, and its specialization to non-compact
branes described in section 5. Namely, as in section 5, since ${\cal T}$ is
non-compact, we define

$Hom_{\mathscr{D}_{\cal Y}}({\cal T},{\cal T}[n]\\{d\\})=HF^{0,0}({\cal
T}_{\zeta},{\cal T}[n]\\{d\\}),$ (A.8)

where ${\cal T}_{\zeta}$ is the thimble corresponding to downward gradient
flow of $H_{R}={\rm Re}(e^{-i\zeta}{\cal W})$, where $\zeta>0$, and $HF^{n,d}$
is Floer cohomology in Maslov degree $n$ and equivariant degree $d$.

#### A.2.4

Since ${\cal W}\sim e^{-Y}$, for ${\rm Re}Y\rightarrow-\infty$, and ${\cal
W}\sim\mu Y$ for ${\rm Re}Y\rightarrow\infty$, we get the configuration of
branes as in the figure 33. In particular ${\cal T}_{\zeta}\cap{\cal T}$
intersect over points $x_{0},x_{1},\ldots$. The intersection point
$x_{d}\in{\cal T}_{\zeta}\cap{\cal T}$ has

$J(x_{d})=-({\cal W}(x_{d})|_{\cal T}-{\cal W}(x_{d})|_{{\cal
T}_{\zeta}})/2\pi=d,$ (A.9)

since, because ${\cal T}$ is a thimble, ${\rm Im}W$ is constant on it, while
it increases along ${\cal T}_{\zeta}$ as $y\rightarrow\infty$. In addition,
all intersection points $x_{d}$ have all Maslov degree zero, since $Y$ so the
differential is necessarily trivial.

Since $x_{d}$ has equivariant degree $d$ as intersection of ${\cal T}$ with
itself, by (A.9), it is the generator of

$Hom_{\mathscr{D}_{\cal Y}}({\cal T},{\cal T}\\{d\\})={\mathbb{C}}x_{d},\qquad
d\geq 0.$ (A.10)

per definition if section 4, and these only non-vanishing Hom’s. It follows
that on a thimble ${\cal T}\\{d\\}$,

${\cal W}(x_{d})|_{{\cal T}\\{d\\}}={\cal W}(x_{d})|_{{\cal T}}+2\pi
id\lambda.$ (A.11)

Figure 33: Brane ${\cal T}_{\zeta}$ winds infinitely many times around
$y=\infty$, but not around $y=0$. It intersects ${\cal T}$ at
$x_{0},x_{1},\ldots$

The theory has more structure yet, since the generators in (A.10) are not
independent. The product on Floer cohomology groups, defined as in (5.10),
with $L_{0}={\cal T}$, $L_{1}={\cal T}\\{d\\}$, and $L_{2}={\cal
T}\\{d\\!+d^{\prime}\\},$ leads to

$x_{d}\cdot x_{d^{\prime}}=x_{d+d^{\prime}}$

as the three branes intersect analogously to those figure 6. It follows that
they have a single independent generator

$x_{d}=x^{d},\qquad d\geq 0.$

All the higher $A_{\infty}$ products vanish, since for them not to vanish we
would need intersection points with no-zero Maslov degrees which we don’t
have. Consequently,

$Hom^{*,*}_{\mathscr{D}_{Y}}({\cal T},{\cal T})=\mathscr{A},$ (A.12)

from which the equivalence of ${\mathscr{D}_{\cal X}}$ and ${\mathscr{D}_{\cal
Y}}$ follows.

#### A.2.5

As an exercise, consider the geometric representation of

${\cal I}\;\;\cong\;\;{{\cal T}}\\{-1\\}\xrightarrow{x}{{\cal T}}.$ (A.13)

The ${\cal I}$ brane is the mirror of the brane ${\cal S}$ on ${\cal X}$,
given by the complex in (A.3). The brane ${\cal S}$ is the 0-brane.

The complex (A.13) is a prescription get ${\cal I}$ by gluing ${\cal T}$’s:
start with the pair of branes ${{\cal T}}\\{-1\\}$ and ${{\cal T}}$, which
intersect at a single point corresponding to $x$. Reverse the orientation of
${{\cal T}}\\{-1\\}$ to get ${{\cal T}}[1]\\{-1\\}$. The string at the
intersection of ${{\cal T}}[1]\\{-1\\}$ and ${{\cal T}}$ is a tachyon [21],
and giving it an expectation value deforms ${{\cal T}}[1]\\{-1\\}\oplus{{\cal
T}}$ to the new brane, as in the figure 34.

Figure 34: Brane ${\cal I}$ is mirror to a D0 brane, drawn here on the cover
of ${\cal Y}$ where we open up the $S^{1}$.

Provided we set $\lambda$ to zero, we can deform the ${\cal I}$ brane further
to an $S^{1}$ that wraps around the cylinder in figure 31, to get the naive
mirror of a 0-brane on ${\cal X}$. For $\lambda$ that does not vanish, the
superpotential does not come back to itself along the brane, and only the
brane in the figure makes sense.

It is not difficult to show that the only non-vanishing Hom’s between ${\cal
I}$’s and ${\cal T}$’s mirror those in (A.4) and (A.5). Because now