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# A Floer-theoretic interpretation of the polynomial representation of the
double affine Hecke algebra

Eilon Reisin-Tzur  University of California, Los Angeles, Los Angeles, CA
90095<EMAIL_ADDRESS>
###### Abstract.

We construct an isomorphism between the wrapped higher-dimensional Heegaard
Floer homology of $\kappa$-tuples of cotangent fibers and $\kappa$-tuples of
conormal bundles of homotopically nontrivial simple closed curves in
$T^{*}\Sigma$ with a certain braid skein group, where $\Sigma$ is a closed
oriented surface of genus $>0$ and $\kappa$ is a positive integer. Moreover,
we show this produces a (right) module over the surface Hecke algebra
associated to $\Sigma$. This module structure is shown to be equivalent to the
polynomial representation of DAHA in the case where $\Sigma=T^{2}$ and the
cotangent fibers and conormal bundles of curves are both parallel copies.

This work partially supported by NSF grant DMS-2003483.

###### Contents

  1. 1 Introduction
  2. 2 Review of HDHF, wrapped HDHF, and conormal boundary conditions
    1. 2.1 Review of HDHF
    2. 2.2 Wrapped Floer theory of conormal bundles
    3. 2.3 Wrapped HDHF
    4. 2.4 Wrapped HDHF example
  3. 3 Path space and wrapped HDHF
    1. 3.1 Dual Lagrangian formulation and perturbed geodesics
    2. 3.2 Path space and wrapped Floer homology
    3. 3.3 Unordered configuration space and wrapped HDHF
  4. 4 HDHF with conormal boundary conditions as a braid skein algebra module
    1. 4.1 The braid skein algebra of a surface
    2. 4.2 Hecke algebra realization in HDHF
    3. 4.3 The parameter $c$
    4. 4.4 The proof of Theorem 1.1, $\kappa=1$ case
    5. 4.5 The proof of Theorem 1.1, general case
  5. 5 Geometric realization of $CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$ as a module over the braid skein algebra
    1. 5.1 Algebraic action
    2. 5.2 Geometric action
    3. 5.3 Equivalence of DAHA modules
  6. 6 The enhanced polynomial representation
    1. 6.1 Double affine Hecke algebra and its polynomial representation
    2. 6.2 The enhanced polynomial representation

##  1\. Introduction

Higher-dimensional Heegaard Floer homology (HDHF) was developed by Colin,
Honda, and Tian in [CHT20] to analyze symplectic fillability questions in
higher-dimensional contact topology. As its name suggests, it is very closely
related to Heegaard Floer homology introduced by Ozsváth and Szabó in [OS04]
to study closed oriented 3-manifolds. HDHF models the Fukaya category of the
Hilbert scheme of points on a Liouville domain and has been used to produce an
invariant of links in $S^{3}$. In [HTY ̵a], Honda, Tian, and Yuan constructed
isomorphisms between the wrapped HDHF of cotangent fibers of cotangent bundles
of closed oriented surfaces $\Sigma$ with positive genus to Hecke algebras
($H_{\kappa}(\Sigma)$) associated with the $\Sigma$. In particular, they
showed that the wrapped HDHF of cotangent fibers of $T^{*}T^{2}$ is isomorphic
to the double affine Hecke algebra (DAHA) $\ddot{H}_{\kappa}$ introduced by
Cherednik in [Che95] for his proof of Macdonald’s conjectures. Using results
by Morton and Samuelson in [MS21], Honda, Tian, and Yuan were able to provide
a symplectic geometry (Floer-theoretic) interpretation of DAHA and various
other Hecke algebras.

The goal of this paper is to build on [HTY ̵a] to provide a symplectic
geometry interpretation of the polynomial representation of DAHA, closely
related to Cherednik’s basic representation.

Let $\kappa$ be a positive integer and fix $\kappa$ distinct points
$q_{1},\dots,q_{\kappa}\in\Sigma$. Given the isomorphism between the wrapped
HDHF $CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$ of cotangent fibers of
$T^{*}\Sigma$ with an additional parameter $c$ and the surface Hecke algebra
tensor product $H_{\kappa}(\Sigma)\otimes\mathbb{Z}[[\hbar]]$, there exists a
functor from the HDHF Fukaya category (with parameter $c$) to the category of
(right) $\hbar$-deformed $H_{\kappa}(\Sigma)$-modules which sends mutually
disjoint Lagrangians $L_{1},\dots,L_{\kappa}$ to
Hom$(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}L_{i})$.
This paper aims to shed more light on this functor by giving it a more
explicit topological interpretation.

Abbondandolo, Portaluri, and Schwarz proved in [APS08] that Floer homology
with conormal boundary conditions is isomorphic to the singular homology of
the natural path space associated to the boundary conditions. In the case of
cotangent fibers, Abouzaid improved this to an $A_{\infty}$-equivalence on the
chain level in [Abo12].

We discuss the generalization of these results to HDHF. More precisely, we
define the wrapped HDHF cochain complex between $\kappa$-tuples of mutually
disjoint conormal bundles. Restricting to the manifold $T^{*}\Sigma$ and
conormal bundles
$\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i}$,
where $\alpha_{1},\dots,\alpha_{\kappa}$ is a mutually disjoint collection of
homotopically nontrivial simple closed curves in $\Sigma$, we find that the
wrapped HDHF is concentrated in degree zero.

Our generalization of the path space in [APS08] is the path space of the
unordered configuration space $\text{UConf}_{\kappa}(\Sigma)$ of $\kappa$
points on $\Sigma$ satisfying boundary conditions to ensure the paths go
between our conormal bundles.

Following [Abo12] and [HTY ̵a], we define an evaluation map

$\mathcal{E}:CW(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma),\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\longrightarrow
C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))\otimes\mathbb{Z}[[\hbar]],$

where $\bm{q}=\\{q_{1},\dots,q_{\kappa}\\}\in\text{UConf}_{\kappa}(\Sigma)$,
$\bm{\alpha}=\alpha_{1}\times\cdots\times\alpha_{\kappa}$, and
$\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})$ is the space of
paths in $\text{UConf}_{\kappa}(\Sigma)$ starting at $\bm{q}$ and ending in
$\bm{\alpha}$. Here we are viewing an element
$(x_{1},\dots,x_{\kappa})\in\bm{\alpha}$ as an unordered tuple; this is
possible since the $\alpha_{1},\dots,\alpha_{\kappa}$ are mutually disjoint.

Taking homotopy classes of paths in
$C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))$ and
quotienting by the HOMFLY skein relation produces a map

$\mathcal{F}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha}),$

where $BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha})$ is a variant of the braid
skein algebra introduced by Morton and Samuelson in [MS21]. Informally,
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha})$ consists of homotopy classes of
braids which start at $\\{q_{1},\dots,q_{\kappa}\\}$ and end in
$\alpha_{1}\times\cdots\times\alpha_{\kappa}$, modulo the HOMFLY skein
relation. Morton and Samuelson also define the braid skein algebra on a
punctured surface, leading to our variant
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$, defined in Section 4.3. The
puncture $\ast\in\Sigma$ gives rise to a marked point relation which in turn
is interpreted as a $c$-deformed homotopy relation on our braids. Adding this
marked point into our formulation, we get a map

$\mathcal{F}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast).$

The first main result of this paper is Theorem 1.1, proved in Section 4.5 :

######  Theorem 1.1.

$\mathcal{F}$ is an isomorphism.

With this in mind, we show that the action of
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$ on
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
agrees with the action of the braid skein algebra
$BSk_{\kappa}(\Sigma,\bm{q},\ast)$ on
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$.

######  Lemma 1.2.

Let $\alpha_{1},\dots,\alpha_{\kappa}$ be parallel copies of the meridian on
$T^{2}$. Then

$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)\simeq(\mathbb{Z}[a_{1}^{\pm
1},\dots,a_{\kappa}^{\pm
1}]\otimes\mathbb{Z}[S_{\kappa}])\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]].$

For the configuration of points $q_{i}$ and curves $\alpha_{i}$ as in Lemma
1.2, we introduce an extra homological variable $d$ which keeps track of
sliding the ends of the braids past each other on the $\alpha_{i}$; see
Definition 6.6. The next main result is obtained after setting $d=s$ and
relating $(\mathbb{Z}[a_{1}^{\pm 1},\dots,a_{\kappa}^{\pm
1}]\otimes\mathbb{Z}[S_{\kappa}])\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]]\otimes\mathbb{Z}[d^{\pm 1}]$ to
$\mathbb{Z}[[s]][c^{\pm 1}][X_{1},\dots,X_{\kappa}]$ by averaging over the
permutation components:

######  Theorem 1.3.

The action of $CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}T^{2})_{c}$ on
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c,d}$
agrees with the polynomial representation of $\ddot{H}_{\kappa}$ on
$\mathbb{Z}[[s]][c^{\pm 1}][X_{1},\dots,X_{\kappa}]$ after setting $d=s$ and
averaging the permutation components of

$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c,d}$.

Organization: In Section 2, we give a brief review of HDHF, wrapped HDHF, and
conormal boundary conditions. In Section 3, we introduce the results of
Abbondandolo and Schwarz before generalizing to the case $\kappa\geq 1$. We
give a summary of [HTY ̵a] in Section 4, along with the proof of Theorem 1.1.
Section 5 discusses the action of
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$ on
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
in the general case, before specializing to $T^{2}$ and the curves
$\alpha_{i}$ in Section 6, where we prove Lemma 1.2 and Theorem 1.3.

Acknowledgements. I would like to thank Ko Honda for nearly countless
discussions and guidance through this project. I would also like to thank
Tianyu Yuan for helpful discussions and suggestions and Peter Samuelson for
discussions which motivated this study.

##  2\. Review of HDHF, wrapped HDHF, and conormal boundary conditions

We give a brief summary of HDHF in Section 2.1 before defining the specific
wrapped version of interest in Section 2.2.

###  2.1. Review of HDHF

We refer the reader to [CHT20] for more details regarding HDHF.

######  Definition 2.1.

Let $(X,\alpha)$ be a $2n$-dimensional completed Liouville domain and let
$\omega=d\alpha$ be the exact symplectic form on $X$. The objects of the
$A_{\infty}$-category $\mathcal{F}_{\kappa}(X)$ are $\kappa$-tuples of
disjoint exact Lagrangians. The morphisms
Hom${}_{\mathcal{F}_{\kappa}(X)}(L_{0},L_{1})=CF(L_{0},L_{1})$ between two
such objects $L_{i}=L_{i1}\sqcup\dots\sqcup L_{i\kappa}$, $i=0,1$, with
mutually transverse components is the free abelian group generated by
$\kappa$-tuples of intersections where each component is used exactly once.
That is, the generators are $\bm{y}=\\{y_{1},\dots,y_{\kappa}\\}$ where
$y_{j}\in L_{0j}\cap L_{1\sigma(j)}$ for some permutation $\sigma$ of
$\\{1,\dots,\kappa\\}$. The coefficient ring is $\mathbb{Z}[[\hbar]]$ and the
$A_{\infty}$-operations $\mu^{m}$ will be defined by (2.2).

Similarly to the cylindrical reformulation of Heegaard Floer homology by
Lipschitz [Lip06], we introduce an extra direction to keep track of points in
the symmetric product of $X$. Let $D$ be the unit disk in $\mathbb{C}$ and
$D_{m}=D-\\{p_{0},\dots,p_{m}\\}$ be the disk with $m$ boundary punctures
arranged counterclockwise. Let $\partial_{i}D_{m}$ be the boundary component
from $p_{i}$ to $p_{i+1}$, with $\partial_{m}D_{m}$ going from $p_{m}$ to
$p_{0}$. We choose representatives of the moduli space of $D_{m}$ modulo
automorphisms and label these representatives $D_{m}$, for lack of a better
name. The $A_{\infty}$-base direction $D_{m}$ is shown in Figure 1.

Figure 1. The $A_{\infty}$ base

Consider the manifold $\tilde{X}=(D_{m}\times X,\omega_{m}+\omega)$, where
$\omega_{m}$ is an area form on $D_{m}$ which restricts to $ds_{i}\wedge
dt_{i}$ on each strip-like end $e_{i}$ near $p_{i}$. We take $s_{0}\to-\infty$
as we approach the negative end $p_{0}$ and $s_{i}\to+\infty$ for the other
punctures refered to as the positive ends. Given $m+1$ objects
$L_{0},\dots,L_{m}$, we let $\tilde{L}_{i}=\partial_{i}D_{m}\times L_{i}$. We
denote by $\pi_{X}:D_{m}\times X\to X$ the projection onto X and by
$\pi_{D_{m}}:D_{m}\times X\to D_{m}$ the symplectic fibration of the base.

There is a smooth assignment $D_{m}\mapsto J_{D_{m}}$ of almost complex
structures $J_{D_{m}}$ that are close to a split almost complex structure
$j_{m}\times J_{X}$ and which project holomorphically onto $D_{m}$. For more
details, the reader is referred to [CHT20] or [HTY ̵a].

######  Remark 2.2.

We call an assignment of almost complex structures sufficiently generic if all
of the moduli spaces under consideration are transversely cut out.

Let $\mathcal{M}(\bm{y}_{1},\dots,\bm{y}_{m},\bm{y}_{0})$ be the moduli space
of maps

$u:(\dot{F},j)\longrightarrow(D_{m}\times X,J_{D_{m}}),$

where $(F,j)$ is a compact Riemann surface with boundary, $\bm{p}_{0}$,
$\dots$, $\bm{p}_{m}$ are disjoint $\kappa$-tuples of boundary punctures of
$F$, and $\dot{F}=F\setminus\cup_{i}\bm{p}_{i}$, such that $u$ satisfies:

(2.1) $\begin{cases}$$du\circ j=J_{D_{m}}\circ du$$;\\\ $$\text{each component
of }\partial\dot{F}\text{ is mapped to a unique }\tilde{L}_{ij}$$;\\\
$$\pi_{X}\circ u\text{ approaches }\bm{y}_{i}\text{ as }s_{i}\to+\infty\text{
for }i=1,\dots,m$$;\\\ $$\pi_{x}\circ u\text{ tends to }\bm{y}_{0}\text{ as
}s_{0}\to-\infty$$;\\\ $$\pi_{D_{m}}\circ u\text{ is a }\kappa\text{-fold
branched cover of }D_{m}.$$\end{cases}$

Letting the boundary punctures $\bm{p}_{0}$, $\dots$, $\bm{p}_{m}$ vary, the
$A_{\infty}$ composition map

$\mu^{m}:CF(L_{m-1},L_{m})\otimes\cdots\otimes CF(L_{0},L_{1})\longrightarrow
CF(L_{0},L_{m})$

is then defined as

(2.2)
$\mu^{m}(\bm{y}_{1},\dots,\bm{y}_{m})=\sum_{\bm{y}_{0},\chi\leq\kappa}\\#\mathcal{M}^{ind=0,\chi}(\bm{y}_{1},\dots,\bm{y}_{m},\bm{y}_{0})\cdot\hbar^{\kappa-\chi}\cdot\bm{y}_{0},\
$

where $\chi$ is the Euler characteristic of $\dot{F}$ and $\\#$ is the signed
count of the moduli space.

######  Theorem 2.3.

The Fredholm index of
$\mathcal{M}^{\chi}(\bm{y}_{1},\dots,\bm{y}_{m},\bm{y}_{0})$ with a varying
complex structure on $D_{m}$ is

$ind(u)=(n-2)\chi+\mu+2\kappa-m\kappa n+m-2,$

where $\mu$ is the Maslov index of u. We refer the reader to [CHT20] for
details on a similar formula.

######  Remark 2.4.

In the case where $2c_{1}(TX)=0$ and the Maslov classes of the involved
Lagrangians vanish, we have that $|\hbar|=2-n$.

###  2.2. Wrapped Floer theory of conormal bundles

We give a quick summary of wrapped Floer homology of conormal bundles; more
details can be found in [APS08].

Let $M$ be a closed manifold and $T^{*}M$ be its cotangent bundle. Denoting
the elements of $T^{*}M$ as pairs $(q,p)\in M\times T_{q}^{*}M$, let
$\omega=dp\wedge dq$ be the the standard symplectic form on $T^{*}M$.
Furthermore, let $\eta$ be the Liouville vector field satisfying
$\mathcal{L}_{\eta}\omega=\omega$. Given a time-dependent Hamiltonian
$H:[0,1]\times T^{*}M\to\mathbb{R}$, let $X_{H}$ be the unique vector field
such that $-dH(Y)=\omega(X_{H},Y)$ for any vector field $Y$ on $T^{*}M$. We
look for solutions $x:[0,1]\to T^{*}M$ of the non-local boundary value
Hamiltonian equation

(2.3) $x^{\prime}(t)=X_{H}(t,x(t)),\ x(0)\in L_{0},\ x(1)\in L_{1},$

where $L_{0}$ and $L_{1}$ are Lagrangian submanifolds of $T^{*}M$, not of $M$.

With this in mind, we consider smooth Hamiltonians $H$ on $[0,1]\times T^{*}M$
such that:

  1. (H0)

every solution $x$ of the non-local boundary value Hamiltonian problem is
nondegenerate

  2. (H1)

there exist $h_{0}>0$ and $h_{1}\geq 0$ such that

$DH(t,q,p)[\eta]-H(t,q,p)\geq h_{0}|p|^{2}-h_{1},$

for every $(t,q,p)\in[0,1]\times T^{*}M$

  3. (H2)

there exists an $h_{2}\geq 0$ such that

$|\nabla_{q}H(t,q,p)|\leq h_{2}(1+|p|^{2}),\ |\nabla_{p}H(t,q,p)\leq
h_{2}(1+|p|)|,$

for every $(t,q,p)\in[0,1]\times T^{*}M$, where $\nabla_{q}$ and $\nabla_{p}$
denote the horizontal and vertical components of the gradient, respectively.

Condition (H0) holds for a generic choice of $H$ in the space that we are
considering. Conditions (H1) and (H2) ensure that $H$ grows quadratically on
the fibers of $T^{*}M$ and is radially convex for large $|p|$.

We reframe our non-local boundary value Hamiltonian problem (2.3) as is done
in [APS08].

Let $Q$ be a submanifold of $M\times M$ which is either compact or has
cylindrical ends, meaning it is the union of a compact submanifold and
submanifolds of the form $K\times[0,\infty)$, where $K$ is a Legendrian. Let
$N^{*}Q$ denote the conormal bundle of $Q$, i.e. the set of covectors $(q,p)$
in $T^{*}(M\times M)$ such that $p\in T^{*}_{q}(M\times M)$ vanishes
identically on $T_{q}Q$. The conormal bundle $N^{*}Q$ is a Lagrangian
submanifold of $T^{*}(M\times M)$ and so is a natural candidate for
investigation.

######  Remark 2.5.

Given two Lagrangians $L_{0}$ and $L_{1}$ in $T^{*}M$, it is not necessarily
true that $L_{0}\times L_{1}$ is the conormal bundle of some submanifold
$Q\subset M\times M$. On the other hand, not every $N^{*}Q$ can be written as
$L_{0}\times L_{1}$ for Lagrangians $L_{0},L_{1}\in T^{*}M$. Therefore, the
problem we are tackling neither contains nor is contained in the original
formulation above. For the purposes of this paper, we will be interested in
Lagrangians $L_{i}$ of $T^{*}M$ which are conormal bundles of submanifolds
$Q_{i}$ in $M$. Under these conditions, this new formulation is equivalent to
the first one since $N^{*}(Q_{1}\times Q_{2})=L_{1}\times L_{2}$.
Specifically, we will be interested in the case where $M$ is a closed oriented
surface of positive genus and $Q=\\{\text{pt}\\}\times\\{\text{pt}\\}$ or
$Q=\\{\text{pt}\\}\times\alpha$ for a homotopically nontrivial simple closed
curve $\alpha$.

Let $H$ be a time-dependent Hamiltonian on $T^{*}M$ satisfying (H0), (H1), and
(H2) and consider the set of solutions, $\mathcal{P}^{Q}(H)$, to the
Hamiltonian equation with conormal boundary conditions.

That is, $\mathcal{P}^{Q}(H)$ is the set of $x:[0,1]\to T^{*}M$ satisfying

$x^{\prime}(t)=X_{H}(t,x(t)),$

subject to the boundary conditions

$(x(0),\mathcal{C}x(1))\in N^{*}Q,$

where $\mathcal{C}$ is the anti-symplectic involution $T^{*}M\to T^{*}M$,
$(q,p)\mapsto(q,-p)$. Note that if $Q=Q_{1}\times Q_{2}\in M\times M$, then
$\mathcal{P}^{Q}(H)$ is the set of trajectories from $Q_{1}$ to $Q_{2}$ along
our Hamiltonian vector field $X_{H}$.

Let $\theta$ be the Liouville one-form on $T^{*}M$ and consider the
Hamiltonian action functional given by

$\mathbb{A}_{H}(x):=\int x^{*}(\theta-Hdt).$

The first variation of $\mathbb{A}_{H}(x)$ on the space of free paths is

$d\mathbb{A}_{H}(x)[\xi]=\int_{0}^{1}\omega(\xi,x^{\prime}(t)-X_{H}(t,x))dt+\theta(x(1))[\xi(1)]-\theta(x(0))[\xi(0)],$

where $\xi$ is a section of $x^{*}(TT^{*}M)$. Since $\theta$ vanishes on the
conormal bundle of every submanifold of $M$, the extremal curves of
$\mathbb{A}_{H}(x)$ are precisely the elements of $\mathcal{P}^{Q}(H)$. This
is the reason we set this up with conormal bundles.

The conditions (H0), (H1), and (H2) that we imposed on our Hamiltonian $H$
imply that the set of solutions $x\in\mathcal{P}^{Q}(H)$ such that
$\mathbb{A}_{H}(x)\leq A$ is finite.

Given a smoothly time-dependent $\omega$-compatible almost complex structure
$J$ on $T^{*}M$, we consider the Floer equation

(2.4) $\partial_{s}u+J(t,u)(\partial_{t}u-X_{H}(t,u))=0,$

where $u:\mathbb{R}\times[0,1]\to T^{*}M$.

Let $x^{-},x^{+}\in\mathcal{P}^{Q}(H)$ and denote by
$\mathcal{M}(x^{-},x^{+})$ the set of all solutions of the Floer equation
(2.4) with the non-local boundary condition such that

$\lim_{s\to\pm\infty}u(s,t)=x^{\pm}(t),\ \forall t\in[0,1].$

Then one can show that we have an energy identity

(2.5) $E(u):=\mathbb{A}_{H}(x^{-})-\mathbb{A}_{H}(x^{+}).$

Furthermore, $\mathcal{M}(x^{-},x^{+})$ is empty whenever
$\mathbb{A}_{H}(x^{-})\leq\mathbb{A}_{H}(x^{+})$ and $x^{-}\neq x^{+}$ and it
consists of only the element $u(s,t)=x(t)$ when $x^{-}=x^{+}=x$.

By perturbing the almost complex structure $J$, we can give
$\mathcal{M}(x^{-},x^{+})$ a smooth structure and its dimension is the
difference of Maslov indices,

$\text{dim}\mathcal{M}(x^{-},x^{+})=\mu^{Q}(x^{-})-\mu^{Q}(x^{+}).$

It follows that when $\mu^{Q}(x^{-})-\mu^{Q}(x^{+})=1$, we get an oriented
one-dimensional manifold. Moreover, we have a free $\mathbb{R}$-action given
by translation of the $s$ variable and so we arrive at a compact zero-
dimensional manifold $\mathcal{M}(x^{-},x^{+})/\mathbb{R}$. Let
$\epsilon([u])\in\\{-1,1\\}$ be $+1$ if the $\mathbb{R}$-action is
orientation-preserving on the component of $\mathcal{M}(x^{-},x^{+})$
containing $u$, and $-1$ otherwise. Define

$n_{F}(x^{-},x^{+}):=\sum_{[u]\in\mathcal{M}(x^{-},x^{+})/\mathbb{R}}\epsilon([u]),$

and denote by $F_{k}^{Q}(H)$ the free Abelian group generated by the elements
$x\in\mathcal{P}^{Q}(H)$ with Maslov index $k$. The boundary morphism

$\partial_{k}:F_{k}^{Q}(H)\longrightarrow F_{k-1}^{Q}(H)$

is defined by

$\partial_{k}x^{-}:=\sum_{x^{+}\in\mathcal{P}^{Q}(H),\mu^{Q}(x^{+})=k-1}n_{F}(x^{-},x^{+})x^{+}.$

Since the set of elements with an upper bound on the action is finite, the
above sum is finite. It can be shown that $\partial_{k-1}\circ\partial_{k}=0$,
and so $\\{F_{*}^{Q}(H),\partial_{*}\\}$ is a complex of free Abelian groups,
called the Floer complex of $(T^{*}M,Q,H,J)$. The Floer homology is then
defined as usual from the complex.

As usual, different choices of the Hamiltonian $H$ produce chain homotopy
equivalent complexes (provided the Hamiltonians are close enough).

###  2.3. Wrapped HDHF

We offer a different but equivalent formulation of the wrapped Floer homology
defined in the previous subsection and extend it to wrapped HDHF.

Let $(M,g)$ be a compact Riemmanian manifold of dimension $n$ with the induced
norm $|\cdot|$ on $T^{*}M$. Choose a time-dependent Hamiltonian
$H_{V}:[0,1]\times T^{*}M\to\mathbb{R}$:

$H_{V}(t,q,p)=\frac{1}{2}|p|^{2}+V(t,q),$

where $q\in M,p\in T^{*}_{q}M$ and $V$ is a perturbation term with small
$W^{1,2}$-norm. This Hamiltonian satisfies the conditions mentioned in the
previous subsection.
Taking the standard symplectic form $\omega=dq\wedge dp$ on $T^{*}M$, let
$X_{H_{V}}$ be the Hamiltonian vector field and $\phi^{t}_{H_{V}}$ be the
time-$t$ flow of $X_{H_{V}}$.

Let $L_{0},L_{1}$ be Lagrangian submanifolds of $T^{*}M$ with cylindrical
ends. The time-1 flow, $\phi^{1}_{H_{V}}(L_{0})$, is again a Lagrangian
submanifold of $T^{*}M$ with cylindrical ends. We define the wrapped Floer
chain complex $CW(L_{0},L_{1})$ of $L_{0}$ and $L_{1}$ to be the Floer chain
complex $CF(\phi^{1}_{H_{V}}(L_{0}),L_{1})$ of $\phi^{1}_{H_{V}}(L_{0})$ and
$L_{1}$.

######  Proposition 2.6.

Let $Q=Q_{1}\times Q_{2}\in M\times M$ be a product of submanifolds whose
conormal bundles have cylindrical ends. Then given a Hamiltonian $H_{V}$
satisfying conditions as before,

$CW(N^{*}Q_{1},N^{*}Q_{2})=F^{Q}(H_{V}).$

###### Proof.

In both cases, the generators are time-1 Hamiltonian chords from $N^{*}Q_{1}$
to $N^{*}Q_{2}$. Moreover, the differentials for both count the same
0-dimensional moduli space of pseudoholomorphic curves between generators. ∎

We now generalize the definition of wrapped Floer theory to wrapped HDHF.

Consider disjoint $\kappa$-tuples of Lagrangians
$L_{i}=\sqcup_{i=j}^{\kappa}L_{ij}$ for $i=1,2$. We can ensure that the
Hamiltonian chords between all of the Lagrangians involved are non-degenerate
by choosing $g$ and $V$ generically.

######  Definition 2.7.

The wrapped higher dimensional Heegaard Floer chain complex is given by

$CW(L_{1},L_{2}):=CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(L_{1i}),\sqcup_{i=1}^{\kappa}L_{2i}).$

In the case of $L_{1}=L_{2}$, we let
$CW(L_{1}):=CW(L_{1},L_{1})=CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(L_{1i}),\sqcup_{i=1}^{\kappa}L_{1i})$.
The $A_{\infty}$-operation

$\mu^{m}:CW(L_{1})\otimes\cdots\otimes CW(L_{1})\longrightarrow CW(L_{1})$

does not immediately follow from the $A_{\infty}$-operations in the non-
wrapped HDHF. Writing the Lagrangians out carefully, we see that the map is
actually

$\mu^{m}:CF(\sqcup_{i}\phi^{1}_{H_{V}}(L_{1i}),\sqcup_{i}L_{1i})\otimes\cdots\otimes
CF(\sqcup_{i}\phi^{m}_{H_{V}}(L_{1i}),\sqcup_{i}\phi^{m-1}_{H_{V}}(L_{1i}))\to
CF(\sqcup_{i}\phi^{m}_{H_{V}}(L_{1i}),\sqcup_{i}L_{1i}),$

where $i=1,\cdots,\kappa$. The subtlety is that while
$CF(\sqcup_{i=1}^{\kappa}\phi^{l}_{H_{V}}(L_{1i}),\sqcup_{i=1}^{\kappa}\phi^{l-1}(L_{1i}))$
is naturally isomorphic to
$CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(L_{1i}),\sqcup_{i=1}^{\kappa}L_{1i})$
for all $l\in\mathbb{Z}$, it is not the case for the chain complex
$CF(\sqcup_{i=1}^{\kappa}\phi^{m}_{H_{V}}(L_{1i}),\sqcup_{i=1}^{\kappa}L_{1i})$.
Luckily, this is resolved by a rescaling argument outlined in [HTY ̵a],
following [Abo10].

Taking this one step further, given $\kappa$-tuples of Lagrangians $L_{1}$ and
$L_{2}$ with cylindrical ends, we can give $CW(L_{1},L_{2})$ the structure of
a (right) $A_{\infty}$-module over $CW(L_{1})$. Using a similar rescaling
argument for $d\geq 2$ we can define the $A_{\infty}$-maps

$\mu^{d}:CW(L_{1},L_{2})\otimes CW(L_{1})\otimes\cdots\otimes
CW(L_{1})\longrightarrow CW(L_{1},L_{2}),$

giving us a (right) $A_{\infty}$-module structure.

###  2.4. Wrapped HDHF example

We perform a model calculation with $\kappa=2$. Identify $M=T^{2}$ with
$S^{1}\times S^{1}=\mathbb{R}/\mathbb{Z}\times\mathbb{R}/\mathbb{Z}$. Fix
points $q_{1}=(\frac{1}{6},\frac{1}{6})$, $q_{2}=(\frac{2}{6},\frac{2}{6})$
and curves $\alpha_{1}=\\{\frac{4}{6}\\}\times S^{1}$ and
$\alpha_{2}=\\{\frac{5}{6}\\}\times S^{1}$. Let
$L_{1}=\sqcup_{i=1}^{2}T^{*}_{q_{i}}T^{2}$ and
$L_{2}=\sqcup_{i=1}^{2}N^{*}\alpha_{i}$. Note that
$N^{*}\alpha_{i}=\alpha_{i}\times(\mathbb{R}\times\\{0\\})\subset T^{*}T^{2}$.

The perturbation term $V(t,q)$ can be chosen arbitrarily small and we will
disregard it for the sake of this model computation. Then taking
$H_{V}(t,q,p)=\frac{1}{2}|p|^{2}$, we have that
$X_{H}=-p_{1}\partial_{q_{1}}-p_{2}\partial_{q_{2}}$ and the flow is given by
$\phi^{t}_{H}(q_{1},q_{2},p_{1},p_{2})=(q_{1}-p_{1}t,q_{2}-p_{2}t,p_{1},p_{2})$,
where $(q_{i},p_{i})$ are viewed as coordinates on $T^{*}S^{1}$. Since the
cotangent fibers are based at $(\frac{i}{6},\frac{i}{6})$, the time-1 flow is
given by
$\phi^{1}_{H}(\frac{i}{6},\frac{i}{6},p_{1},p_{2})=(\frac{i}{6}-p_{1},\frac{i}{6}-p_{2},p_{1},p_{2})$.

We are interested in intersections of $\phi^{1}_{H}(T^{*}_{q_{i}}T^{2})$ with
$N^{*}\alpha_{j}$, so we want solutions to
$(\frac{i}{6}-p_{1},\frac{i}{6}-p_{2},p_{1},p_{2})$ =
$(\frac{1}{2}+\frac{j}{6},a,p,0)$ for some $a\in\mathbb{R}/\mathbb{Z},\
p\in\mathbb{R}$. Then $p_{2}$ must be 0. (Note that with a perturbation term,
this is no longer necessarily true since there will be some flow in the fiber
direction, but $p_{2}$ would be very small.) Continuing with this example, we
see that $\frac{i}{6}-p_{1}=\frac{1}{2}+\frac{j}{6}$, and so
$p_{1}=-\frac{1}{2}+\frac{i-j}{6}$. Given one such $p_{1}$, we see that adding
or subtracting 1 from $p_{1}$ gives another intersection. Let
$\pi:T^{*}T^{2}\to T^{2}$, $(q,p)\mapsto q$, be the projection onto the zero
section. Then adding (subtracting) 1 along the fiber direction $p_{1}$ simply
wraps clockwise (counterclockwise) once more around the
$S^{1}\times\\{\frac{i}{6}\\}$ direction on the torus before intersecting
$\alpha_{j}$. The wrapping, along with some Hamiltonian chords, is shown in
Figure 2 for $\kappa=1$.

Figure 2. The case where $\kappa$=1. Here we view $T^{*}T^{2}$ as two copies
of $T^{*}S^{1}$. The curve in red is $T^{*}_{q}T^{2}$,
$\phi^{1}_{H_{V}}(T_{q}^{*}T^{2})$ is in blue, and $N^{*}\alpha$ is in green.
The arrows in orange represent Hamiltonian chords going from $T^{*}_{q}T^{2}$
to $N^{*}\alpha$.

######  Claim 2.8.

The generators of
$CF(\sqcup_{i=1}^{2}\phi^{1}_{H_{V}}(T^{*}_{q_{i}}T^{2}),\sqcup_{i=1}^{2}N^{*}\alpha_{i})$
are elements of the form $(a_{1}^{n_{1}}a_{2}^{n_{2}},\sigma)$ where
$\sigma\in S_{2}$ indicates intersections of
$\phi^{1}_{H_{V}}(T^{*}_{q_{i}}T^{2})$ and $N^{*}\alpha_{\sigma(i)}$ for
$i=1,2$ and the exponent of $a_{i}$ indicates the number of times the
intersection point, viewed as a Hamiltonian chord, wraps around the
$(-1,0)$-direction when projected onto the zero section.

More generally we have the following:

For $n_{i}\in\mathbb{Z}$, let $(a_{1}^{n_{1}}\cdots
a_{\kappa}^{n_{\kappa}},\sigma)$ denote intersections between
$\phi^{1}_{H_{V}}(T^{*}_{q_{i}}T^{2})$ and $N^{*}\alpha_{\sigma(i)}$ for
$i=1,\dots,\kappa$ whose corresponding Hamiltonian chords wrap $n_{i}$ times
around the torus in the $(-1,0)$-direction when projected onto the zero
section.

######  Lemma 2.9.

The generators of
$CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(T^{*}_{q_{i}}T^{2}),\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})$
for points $q_{i}$ and curves $\alpha_{i}$ as in the model case are given by

$\mathbb{Z}[(\textstyle\prod_{i=1}^{\kappa}a_{i}^{n_{i}},\sigma)\ |\ \sigma\in
S_{\kappa},\ n_{i}\in\mathbb{Z}].$

###### Proof.

We choose coordinates $(q_{i1},q_{i2},p_{i1},p_{i2})$ on $T^{*}_{q_{i}}T^{2}$.
Ignoring the perturbation term $V(t,q)$ for now, let $p_{i1}>0$ be the
smallest such number such that
$\phi^{1}_{H_{V}}(q_{i1},q_{i2},p_{i1},p_{i2})\cap
N^{*}\alpha_{j}\neq\emptyset$. Following the calculation done in our model
example, we see that every such intersection occurs at
$\phi^{1}(q_{i1},q_{i2},p_{i1}+n,0)$, for some $n\in\mathbb{Z}$. Then we can
describe the set of intersections by the form in the lemma.

Adding the perturbation term has little effect on the Hamiltonian chords, and
it can be chosen small enough and such that $\frac{\partial V}{\partial q}$ is
small. It follows that each intersection will occur at
$\phi^{1}(q_{i1},q_{i2},p_{i1}+n,p_{i2}(n))$ where $p_{i2}(n)$ will be small
for all $n$. While $n$ may not be an integer, it will be very close to one.
Alternatively, one could order all such $n$’s, giving a bijection with
$\mathbb{Z}$. ∎

######  Remark 2.10.

One way to visualize the wrapping is to view the image of the cotangent fibers
projected onto $T^{2}$. If the $\alpha$ curves lie in only one component of
$S^{1}\times S^{1}$, the Hamiltonian chords look like geodesics from $q_{i}$
to $\alpha_{j}$ in the other $S^{1}$ component; see Figures 2 and 3. This will
be made more explicit in the next section.

##  3\. Path space and wrapped HDHF

We restrict to the case $\kappa=1$ and make use of an isomorphism of the
wrapped Floer homology with the singular homology of a certain path space. We
will review the Morse complex in this context in general before specializing
to the surface $\Sigma$ and Lagrangian submanifolds of interest. For more
details, the reader is encouraged to look at [AS10, Sections 2, 3, and 4] .

###  3.1. Dual Lagrangian formulation and perturbed geodesics

We recall some facts about Legendre transforms and perturbed geodesics that
will help establish a dual Lagrangian formulation from which to do Morse
theory.

Let $H\in C^{\infty}([0,1]\times T^{*}M)$ be a Hamiltonian satisfying the
classical Tonelli assumptions. The Fenchel transform defines a smooth, time-
dependent Lagrangian on $TM$,

$L(t,q,v):=\max_{p\in T_{q}^{*}M}(\langle p,v\rangle)-H(t,q,p),\
(t,q,v)\in[0,1]\times TM.$

We call this Lagrangian the Fenchel dual of our Hamiltonian $H$. Similarly,
given a Lagrangian satisfying equivalent assumptions, we can dualize to get
the Hamiltonian

$H(t,q,p)=\max_{p\in T_{q}M}(\langle p,v\rangle)-L(t,q,v),\
(t,q,p)\in[0,1]\times T^{*}M.$

Moreover, we have a diffeomorphism known as the Legendre transform

$\mathcal{L}:[0,1]\times TM\longrightarrow[0,1]\times T^{*}M,\
(t,q,v)\mapsto(t,q,D_{v}L(t,q,v)),$

such that $\mathcal{L}(t,q,v)=(t,q,p)$ if and only if $L(t,q,v)=\langle
p,v\rangle-H(t,q,p)$.

We now specialize to the Lagrangian function $L_{V}:[0,1]\times
TM\to\mathbb{R}$ given by

$L_{V}(t,q,v)=\frac{1}{2}|v|^{2}-V(t,q).$

Its Fenchel dual Hamiltonian is then given by
$H(t,q,p)=\frac{1}{2}|p|^{2}+V(t,q)$, the same Hamiltonian we considered
earlier when defining the Floer complex.

With these functions in mind, there is an equivalence between Hamiltonian
orbits $x:[0,1]\to T^{*}M$ solving $x^{\prime}(t)=X_{H}(t,x(t))$ and curves
$\gamma:[0,1]\to M$ which are extremals of the Lagrangian action functional
$\mathcal{A}_{V}(\gamma)=\int_{0}^{1}L_{V}(t,\gamma,\dot{\gamma})dt$. More
precisely, $x$ is a Hamiltonian orbit if and only if $\gamma:=\pi_{M}\circ x$
is an absolutely continuous extremal of $\mathcal{A}_{V}$.

We conclude this subsection with a further equivalence, recalling the
definition of a perturbed geodesic.

######  Definition 3.1.

A $V$-perturbed geodesic $\gamma$ is a map $[0,1]\to M$ such that

$\nabla_{\dot{\gamma}}\dot{\gamma}=-\nabla V,$

where $\nabla V$ denotes the gradient of $V$ with respect to the metric $g$.

The critical points of $\mathcal{A}_{V}$ are precisely the $V$-perturbed
geodesics. Putting the two equivalences together, we have that the Hamiltonian
orbits are in bijection with the $V$-perturbed geodesics. We can impose
similar boundary conditions to the perturbed geodesics as we do the
Hamiltonian chords in the case of our wrapped Floer theory, giving us a map
$\mathcal{L}:\mathcal{P}^{Q}(H)\to\mathcal{P}^{Q}(L)$, where
$\mathcal{P}^{Q}(H)$ are the Hamiltonian chords with the boundary condition
and $\mathcal{P}^{Q}(L)$ are the $V$-perturbed geodesics satisfying the same
boundary condition. Moreover, condition (H0) placed on the Hamiltonian $H$
will ensure that all critical points will be non-degenerate.

######  Definition 3.2.

We call $\mathcal{L}$ the Legendre transform. If we let
$x\in\mathcal{P}^{Q}(H)$, then

$\mathcal{L}(x)(t):=\pi_{M}\circ x(t).\\\ $

Figure 3 below shows the perturbed geodesics corresponding to the generator
$(a_{2},(12))$ of our model calculation in Section 2.4.

Figure 3. Perturbed geodesics (in blue) representing the generator
$(a_{2},(12))$.

###  3.2. Path space and wrapped Floer homology

Let $Q\subset M\times M$ be a closed submanifold as before. Consider the path
space

$\Omega_{Q}(M)=\\{\gamma\in C^{0}([0,1],M)~{}|~{}(\gamma(0),\gamma(1))\in
Q\\}.$

We restrict to a subset consisting of paths in the class $W^{1,2}$, and wish
to do Morse theory on this new path space, denoted $\Omega_{Q}^{1,2}(M)$.

Given the same Hamiltonian $H_{V}$, Hamiltonian vector field $X_{H_{V}}$, and
flow $\phi^{t}_{H_{V}}$ as in the previous section, we define the function
$L:[0,1]\times TM\to\mathbb{R}$ satisfying second derivative conditions:

  1. (L1)

There exists $l_{1}>0$ such that $\nabla_{vv}L(t,q,v)\geq l_{1}I$

  2. (L2)

There exists $l_{2}>0$ such that

$|\nabla_{qq}L(t,q,v)|\leq l_{2}(1+|v|^{2}),\ |\nabla_{qv}L(t,q,v)|\leq
l_{2}(1+|v|),\ |\nabla_{vv}L(t,q,v)|\leq l_{2},$

ensuring that $L$ grows quadratically in the tangent direction. These
conditions are equivalent to conditions (H1), (H2) imposed on our Hamiltonian,
and will guarantee that $H_{V}$, the Fenchel dual of $L_{V}$, satisfies those.
We can take

$L_{V}(t,q,v)=\frac{1}{2}|v|^{2}-V(t,q),$

where $t\in[0,1],q\in M,\text{and }v\in T_{q}M$.

Considering the Morse function

$\mathcal{A}_{V}(\gamma)=\int_{0}^{1}L_{V}(t,\gamma,\dot{\gamma})dt,$

defined for $\gamma\in\Omega_{Q}^{1,2}(M)$, we define the Morse complex
generated by its critical points. This is well defined and the Morse homology
$HM_{*}(\mathcal{A}_{V})$ is isomorphic to the singular homology of
$\Omega_{Q}^{1,2}(M)$. Since the inclusion
$\Omega_{Q}^{1,2}(M)\hookrightarrow\Omega_{Q}(M)$ is a homotopy equivalence,
the isomorphism extends to the singular homology of $\Omega_{Q}(M)$.

######  Theorem 3.3 (Abbondandolo-Schwarz).

There is an isomorphism between the wrapped Floer homology
$HF_{Q}^{*}(T^{*}M)$ and the singular homology of the path space
$\Omega_{Q}(M)$.

For the rest of the paper, we specialize $M$ to a closed oriented surface
$\Sigma$ of genus $>0$.

Let $Q=\\{q\\}\times\alpha\in\Sigma\times\Sigma$, where $q\in\Sigma$ is a
point and $\alpha$ is a homotopically nontrivial simple closed curve in
$\Sigma$.

######  Lemma 3.4.

$H_{*}(\Omega_{Q}(\Sigma))$ is supported in degree 0.

###### Proof.

If $\Sigma$ is a torus, we can assume that $g$ is the flat metric, where all
$V$-perturbed geodesics with sufficiently small perturbation $V$ are minimal
and isolated. If the genus of $\Sigma$ is greater than 1, then we can assume
that $g$ is the hyperbolic metric with constant curvature $-1$. It is well
known that on a hyperbolic surface, there is a unique $V$-perturbed geodesic
in each homotopy class of paths for $V$ sufficiently small. Hence the Morse
indices of all critical points are 0. ∎

Next let $q_{1},\dots,q_{\kappa}$ be $\kappa$ distinct points on $\Sigma$ and
$\alpha_{1},\dots,\alpha_{\kappa}$ be $\kappa$ mutually disjoint homotopically
nontrivial simple closed curves on $\Sigma$. We use $T^{*}_{\bm{q}}\Sigma$ to
denote $\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma$ and $N^{*}\bm{\alpha}$ to
denote $\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i}$.

######  Corollary 3.5.

$HF_{Q}^{*}(T^{*}\Sigma)$ is supported in degree 0. In particular, the grading
$|\bm{y}|=0$ for every generator $\bm{y}\in
CW(T^{*}_{\bm{q}}\Sigma,N^{*}\bm{\alpha})$.

###### Proof.

Each $y_{i}\in\phi^{1}_{H_{V}}(T_{q_{i}}^{*}\Sigma)\cap N^{*}\alpha_{j}$
corresponds to a time-1 Hamiltonian chord from $T_{q_{i}}^{*}\Sigma$ to
$N^{*}\alpha_{j}$. Its Legendre transform gives a $V$-perturbed geodesic
$\gamma$ on $\Sigma$. The Conley-Zehnder index of $y_{i}$ is equal to the
Morse index of $\gamma$ with respect to its Lagrangian action. Lemma 3.4 above
implies that $|y_{i}|=0$ for all $i$, and so $|\bm{y}|=0$. ∎

###  3.3. Unordered configuration space and wrapped HDHF

Let

$\text{UConf}_{\kappa}(\Sigma)=\\{\\{q_{1},\dots,q_{\kappa}\\}~{}|~{}q_{i}\in\Sigma,q_{i}\neq
q_{j}\ \text{for}\ i\neq j\\}$

be the configuration space of $\kappa$ unordered (distinct) points on
$\Sigma$.

We wish to generalize Theorem 3.3 to the case where $\kappa>1$. Let
$\bm{q}=\\{q_{1},\dots,q_{\kappa}\\}\in\text{UConf}_{\kappa}(\Sigma)$ and
$\bm{\alpha}=\alpha_{1}\times\cdots\times\alpha_{\kappa}$, where we view
$\bm{\alpha}$ as a subset of $\text{UConf}_{\kappa}(\Sigma)$ by viewing
$(x_{1},\dots,x_{\kappa})$ as an unordered tuple. The natural analog to
consider on the path space side is the path space on
$\text{UConf}_{\kappa}(\Sigma)$, which we denote by

$\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})=\\{\gamma\in
C^{0}([0,1],\text{UConf}_{\kappa}(\Sigma))~{}|~{}\gamma(0)=\bm{q},\
\gamma(1)\in\bm{\alpha}\\}.$

We can identify each generator of
$CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma),\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})$
with a $\kappa$-tuple of $V$-perturbed geodesics from the $q_{i}$ to the
$\alpha_{\sigma(i)}$ using the duality established in Section 3.2. This is an
element of $\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})$.

To make this more precise, we construct an evaluation map

$\mathcal{E}:CF(\sqcup_{i=1}^{\kappa}\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma),\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\longrightarrow
C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))\otimes\mathbb{Z}[[\hbar]],$

where $C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))$ is the
space of 0-chains of the path space
$\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})$.

The map $\mathcal{E}$ counts pseudo-holomorphic curves between the conormal
Lagrangians and the zero section. Parametrizing the boundary of the curves
along the zero section produces a path in the unordered configuration space.
We keep the parameter $\hbar$ around to track the Euler characteristic of the
map, which will later relate to the HOMFLY skein relation on braids.

Let $T_{1}:=D_{2}$ be our $A_{\infty}$-base where
$\partial_{i}T_{1}=\partial_{i}D_{2}$; shown in Figure 4. Let
$\mathcal{T}_{1}$ be the moduli space of $T_{1}$ modulo automorphisms, and
choose representatives $T_{1}$ of equivalence classes in a smooth manner. Let
$\pi_{T^{*}\Sigma}$ be the projection $T_{1}\times T^{*}\Sigma\to T^{*}\Sigma$
and choose a sufficiently generic consistent collection of compatible almost
complex structures such that they are close to a split almost complex
structure projecting holomorphically to $T_{1}$, as in Section 2.1. We denote
by $\mathcal{H}(\bm{q}^{\prime},\bm{y},\ \bm{x})$ the moduli space of maps

$u:(\dot{F},j)\longrightarrow(T_{1}\times T^{*}\Sigma,J_{T_{1}}),$

where $(F,j)$ is a compact Riemann surface with boundary, $\bm{p}_{0}$,
$\bm{p}_{1}$, $\bm{p}_{2}$ are disjoint tuples of boundary punctures of $F$
and $\dot{F}=F\setminus\cup_{i}$$p_{i}$, satisfying:

(3.1) $\begin{cases}du\circ j=J_{T_{1}}\circ du;\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\phi^{1}_{H_{V}}(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)\text{ if
}\pi_{T_{1}}\circ u(z)\subset\partial_{0}T_{1};\\\ \text{each component of
}\partial\dot{F}\text{ that projects to }\partial_{0}T_{1}\text{ maps to a
distinct }\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma);\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i}\text{ if }\pi_{T_{1}}\circ
u(z)\subset\partial_{1}T_{1};\\\ \text{each component of
}\partial\dot{F}\text{ that projects to }\partial_{1}T_{1}\text{ maps to a
distinct }N^{*}\alpha_{i};\\\ \pi_{T^{*}\Sigma}\circ u(z)\in\Sigma\text{ if
}\pi_{T_{1}}\circ u(z)\subset\partial_{2}T_{1};\\\ \pi_{T^{*}\Sigma}\circ
u\text{ tends to }\bm{q}^{\prime},\bm{y},\ \bm{x}\text{ as
}s_{0},s_{1},s_{2}\to+\infty;\\\ \pi_{T_{1}}\circ u\text{ is a
}\kappa\text{-fold branched cover of a fixed }T_{1}\in\mathcal{T}_{1}.\\\
\end{cases}$

Figure 4. The $A_{\infty}$-base $T_{1}$

In simpler terms, we look at the moduli space of holomorphic curves between
the Lagrangians involved and the zero section of $T^{*}\Sigma$ in the
framework of HDHF with only positive punctures.

######  Remark 3.6.

While the intersections $\bm{y}$ and $\bm{q}^{\prime}$ are discrete, we have
an $S^{1}$-worth of choices for each $x_{i}\in\alpha_{i}\cap\Sigma$. To
resolve this issue, we can either use the Morse-Bott (clean intersection)
formalism or perturb the zero section near the $N^{*}\alpha_{i}$. This results
in two intersection points $x_{1},x_{2}\in
N^{*}\alpha_{i}\cap(\Sigma\times\\{0\\})$. If we are interested in index 0
curves in our moduli space then only one of these intersections will have the
right grading. This is shown in Figure 5 for our model $\Sigma=T^{2},\kappa=1$
case by counting pseudo-holomorphic triangles bounded by the conormal
Lagrangians and the perturbed zero section.

Figure 5. Perturbed zero section in one of the $T^{*}S^{1}$ directions. Only
$x_{2}\in N^{*}\alpha\cap\phi_{H^{\prime}}(T^{2})$ has the right Maslov index.

Let $\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{x})$ be the subset of
$\mathcal{H}(\bm{q}^{\prime},\bm{y},\bm{x})$ such that $\chi(\dot{F})=\chi$.
Moreover, let

$\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{\alpha})=\sqcup_{\bm{x}\in\bm{\alpha}}\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{x}).$

######  Lemma 3.7.

For fixed generic $J_{T_{1}}$,
$\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{x})$ is of dimension 0 and
consists of discrete regular curves for all $\bm{q}^{\prime},\ \bm{y},$ and
$\bm{x}$ such that $\bm{x}$ is a tuple of bottom generators for the Morse-Bott
intersections $N^{*}\alpha_{i}\cap(\Sigma\times\\{0\\})$.

###### Proof.

By Corollary 3.5, we have $|\bm{y}|=0$. Computing the grading for
intersections of cotangent fibers with the zero section gives that
$|\bm{q}^{\prime}|=0$; see [HTY ̵a]. Since the $\bm{x}$ are bottom generators
for our Morse-Bott intersection, it follows that $|\bm{x}|=0$ and so the
virtual dimension of $\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{x})$ is 0.
The rest follows from standard transversality arguments. ∎

######  Lemma 3.8.

Given $\bm{q}^{\prime},\bm{y},\bm{\alpha}$, the moduli space
$\mathcal{H}^{\chi}(\bm{q}^{\prime},\bm{y},\bm{\alpha})$ consists of finitely
many curves for each Euler characteristic $\chi$.

###### Proof.

Each $\bm{y}$ determines a unique $\bm{q}^{\prime}$ and
$\bm{x}\in\bm{\alpha}$. The energy bound along with Gromov compactness gives
the result. ∎

Fix a parametrization of the arc $\partial_{2}T_{1}$ from $p_{0}$ to $p_{2}$
by $\tau:[0,1]\to\partial_{2}T_{1}$. There exists a sufficiently generic
consistent collection of almost complex structures such that for all
$u\in\mathcal{H}(\bm{q}^{\prime},\bm{y},\bm{\alpha})$, $(\pi_{\Sigma}\circ
u)\circ(\pi_{T_{1}\circ u})^{-1}\circ\tau(t)$ consists of $\kappa$ distinct
points on $\Sigma$ for each $t\in[0,1]$ and hence gives a path in
$\text{UConf}_{\kappa}(\Sigma)$:

$\gamma(u):[0,1]\longrightarrow\text{UConf}_{\kappa}(\Sigma),$
$t\mapsto(\pi_{\Sigma}\circ u)\circ(\pi_{T_{1}\circ u})^{-1}\circ\tau(t).$

Since $\gamma(0)=\bm{q}^{\prime}$ and $\gamma(1)\in\bm{\alpha}$, it follows
that
$\gamma(u)\in\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q}^{\prime},\bm{\alpha})$.

Define the evaluation map

$\mathcal{E}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\longrightarrow
C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q}^{\prime},\bm{\alpha}))\otimes\mathbb{Z}[[\hbar]]$
$\bm{y}\mapsto\sum_{u\in\mathcal{H}(\bm{q}^{\prime},\bm{y},\bm{\alpha})}(-1)^{\natural(u)}\cdot\hbar^{\kappa-\chi(u)}\cdot\gamma(u),$

where $(-1)^{\natural(u)}$ is the sign assigned to $u$.

Since the perturbation term $V$ has small $W^{1,2}$-norm, the Hamiltonian
vector field $X_{H_{V}}$ has small norm near the zero section $\Sigma$ and
hence $\bm{q}^{\prime}$ is close to $\bm{q}$. We choose non-intersecting short
paths $\gamma_{i}$ on $\Sigma$ from $q_{i}$ to $q^{\prime}_{i}$ for
$i=1,\dots,\kappa$. Pre-concatenating with $\\{\gamma_{i}\\}$ allows us to
identify $\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q}^{\prime},\bm{\alpha})$
with $\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})$. We make this
identification whenever possible.

Next, we have a projection

$\mathcal{P}:C_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))\otimes\mathbb{Z}[[\hbar]]\to(H_{0}(\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha}))\otimes\mathbb{Z}[[\hbar]])/\text{HOMFLY
skein}$

given by first taking the homotopy class of the path $\gamma(u)$ and then,
viewing $\gamma(u)$ as a braid, quotienting by the HOMFLY skein relation
(given in Definition 4.1).

Composing the evaluation map and the projection, we arrive at the map
$\mathcal{F}=\mathcal{P}\circ\mathcal{E}$.

Let $BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha})$ denote the free
$\mathbb{Z}[[\hbar]]$-module generated by homotopy classes of braids
$\gamma\in\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})$ modulo the
HOMFLY skein relation. Then

$\mathcal{F}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha})$

is given by

$\bm{y}\mapsto\sum_{u\in\mathcal{H}(\bm{q}^{\prime},\bm{y},\bm{\alpha})}(-1)^{\natural(u)}\cdot\hbar^{\kappa-\chi(u)}\cdot[\gamma(u)],$

where $[\gamma(u)]$ is viewed as an equivalence class of braids modulo the
HOMFLY skein relation.

##  4\. HDHF with conormal boundary conditions as a braid skein algebra module

In this section we recall the equivalence between wrapped HDHF for cotangent
fibers and the braid skein algebra of a surface. The reader is encouraged to
look at [HTY ̵a] and [MS21] for a more detailed exposition of this section’s
content. Let $\Sigma$ be a closed oriented surface of genus $>0$ and
$q_{1},\dots,q_{\kappa}\in\Sigma$ be distinct points.

###  4.1. The braid skein algebra of a surface

Consider the braid group $B_{\kappa}(\Sigma\setminus\\{\ast\\},\bm{q})$ of
$\kappa$-braids in the punctured surface $\Sigma\setminus\\{\ast\\}$ based at
$\bm{q}=\\{q_{1},\dots,q_{\kappa}\\}$. One way to view this is to take the
thickened surface $\Sigma\times I$ with a fixed base string $\\{\ast\\}\times
I$. In this case, the elements are made up of $\kappa$ strings oriented
monotonically from $\Sigma\times\\{0\\}$ to $\Sigma\times\\{1\\}$ which do not
intersect each other or the base string. Two braids are equivalent if they are
isotopic to each other, with the isotopy avoiding the base string.

######  Definition 4.1.

The braid skein algebra $BSk_{\kappa}(\Sigma,\bm{q},\ast)$ (or the surface
Hecke algebra) of the surface $\Sigma$ is the free $\mathbb{Z}[s^{\pm
1},c^{\pm 1}]$-module generated by $\kappa$-braids in the punctured surface
$\Sigma\setminus\\{\ast\\}$ based at $\bm{q}$, up to isotopy which does not
intersect $\\{\ast\\}\times[0,1]$, subject to the local relations:

  1. (1)

the HOMFLY skein relation

  2. (2)

the marked point relation

where the string in blue corresponds to the marked point strand
$\\{\ast\\}\times[0,1]$.

Using the marked point relation, we define a slightly more useful $c$-deformed
homotopy relation.

######  Definition 4.2.

The $c$-deformed braid group $B_{\kappa}(\Sigma,\bm{q})_{c}$ of $\Sigma$ based
at $\bm{q}$ is generated by $B_{\kappa}(\Sigma\setminus\\{\ast\\},\bm{q})$ and
a central element $c$, subject to the following $c$-deformed homotopy
relation:

(4.1) $[\gamma_{2}]=c^{2\langle H,\ast\rangle}[\gamma_{1}],$

where
$\gamma_{1},\gamma_{2}\in\Omega(\text{UConf}_{\kappa}(\Sigma\setminus\\{\ast\\}),\bm{q})$,
$H$ is a homotopy between $\gamma_{1}$ and $\gamma_{2}$, $\langle
H,\ast\rangle$ is the algebraic intersection number for the homotopy $H$
defined in [HTY ̵a], and $[\gamma_{i}]$ is the homology class of the braid
$\gamma_{i}$.

The quotient of $\mathbb{Z}[s^{\pm 1}][B_{\kappa}(\Sigma,\bm{q})_{c}]$ by the
HOMFLY skein relation gives the braid skein algebra
$BSk_{\kappa}(\Sigma,\bm{q},\ast)$.

###  4.2. Hecke algebra realization in HDHF

We briefly review the connection between the wrapped HDHF of contangent fibers
on surfaces and the surface Hecke algebras defined in [HTY ̵a].

######  Definition 4.3.

The wrapped HDHF chain complex of disjoint cotangent fibers with parameter
$c$, $CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$, is given by
$CF(\phi^{1}_{H_{V}}(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma),\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)\otimes\mathbb{Z}[c^{\pm
1}]$ as a $\mathbb{Z}$-module and has enhanced $A_{\infty}$-operations to
include $c$-coefficients. Specifically,

$\mu^{m}(\bm{y}_{1},\dots,\bm{y}_{m})=\sum_{u\in\mathcal{M}^{\text{ind}=0}(\bm{y}_{1},\dots,\bm{y}_{m},\bm{y}_{0})}(-1)^{\natural(u)}\cdot
c^{2\langle u,\ast\rangle}\cdot\hbar^{\kappa-\chi(u)}\cdot\bm{y}_{0}.$

Given a map $u\in\mathcal{M}(\bm{y}_{1},\dots,\bm{y}_{m},\bm{y}_{0})$,
$\langle u,\ast\rangle$ is the intersection number of $\pi_{\Sigma}(u)$ and
$\ast$, after some modifications to $\pi_{\Sigma}(u)$ to ensure this is well
defined.

######  Proposition 4.4 (Honda-Tian-Yuan).

The $A_{\infty}$-algebra $CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$ is
supported in degree zero, and hence is an ordinary algebra.

With this on hand, HTY go on to prove the following theorem connecting the
wrapped HDHF to the surface Hecke algebra $H_{\kappa}(\Sigma,\bm{q})$.

######  Theorem 4.5 (Honda-Tian-Yuan).

There is an isomorphism of algebras

$\mathcal{F}:CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}\longrightarrow
H_{\kappa}(\Sigma,\bm{q})\otimes\mathbb{Z}[[\hbar]],$

where the surface Hecke algebra $H_{\kappa}(\Sigma,\bm{q})$ is naturally
isomorphic to the braid skein algebra $BSk_{\kappa}(\Sigma,\bm{q},\ast)$.

We run through an (informal) overview of the proof as the main ideas will be
modified and applied in the proof of Theorem 1.1 in Section 4.5.

The map $\mathcal{F}$ is an evaluation map similar to that in Section 3.3. It
is constructed by considering a moduli space of pseudo-holomorphic curves
bounded by $\phi^{1}_{H_{V}}(T^{*}_{\bm{q}}\Sigma),\ T^{*}_{\bm{q}}\Sigma,$
and $\Sigma$. The authors then make use of the construction of the braid skein
algebra coming from the unordered configuration space to relate this to the
surface Hecke algebra.

The next step is to show that $\mathcal{F}$ is indeed a homomorphism of
algebras. This is done by taking the moduli space of index 1 curves projecting
to a 4-punctured disk with boundary conditions given by
$\phi^{2}_{H_{V}}(T^{*}_{\bm{q}}\Sigma),$
$\phi^{1}_{H_{V}}(T^{*}_{\bm{q}}\Sigma),$ $T^{*}_{\bm{q}}\Sigma,$ and
$\Sigma$. An inspection of the boundary of the compactification of this space
concludes that $\mathcal{F}$ respects the algebra structure. More
specifically, the only breakings that occur are ones that correspond to
$\mathcal{F}(\mu^{2}(\bm{y,y^{\prime}}))$ or
$\mathcal{F}(\bm{y})\mathcal{F}(\bm{y}^{\prime})$.

The final step is to show that $\mathcal{F}|_{\hbar=0}$ is an isomorphism and
then use the algebra homomorphism properties to prove it is a bijection when
$\hbar$ is reintroduced.

We will repeat and modify much of the argument in our proof of Theorem 1.1.

###  4.3. The parameter $c$

We enhance
$CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})$
to include the parameter $c$. Let

$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}:=CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})\otimes\mathbb{Z}[c^{\pm
1}].$

Consider the updated evaluation map:

$\mathcal{E}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\longrightarrow
C_{0}(\Omega(\text{UConf}_{\kappa}((\Sigma),\bm{q},\bm{\alpha}))\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]]$

given by

$\bm{y}\mapsto\sum_{u\in\mathcal{H}(\bm{q}^{\prime},\bm{y},\bm{\alpha})}(-1)^{\natural(u)}\cdot
c^{2\langle u,\ast\rangle}\cdot\hbar^{\kappa-\chi(u)}\cdot\gamma(u),$

where $\langle u,\ast\rangle:=\langle[\pi_{\Sigma}(u)]^{\prime},\ast\rangle$
is defined in [HTY ̵a, Section 5] .

######  Definition 4.6.

The braid skein group $BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$ from
$\bm{q}$ to $\bm{\alpha}$ on $\Sigma\setminus\\{\ast\\}$ is the free
$\mathbb{Z}[[\hbar]]$-module generated by $c$-deformed homotopy classes of
paths in

$\Omega(\text{UConf}_{\kappa}(\Sigma),\bm{q},\bm{\alpha})=\\{\gamma\in
C^{0}([0,1],\text{UConf}_{\kappa}(\Sigma))~{}|~{}\gamma(0)=\bm{q},\
\gamma(1)\in\bm{\alpha}=\alpha_{1}\times\dots\times\alpha_{\kappa}\\}$

modulo the HOMFLY skein relation.

######  Remark 4.7.

Given a braid $\gamma$, its $c$-deformed homotopy class $[\gamma]_{c}$ is the
set of braids equivalent to it under the local relations given in Definition
4.1. Using these relations, every braid is equivalent to a
$\mathbb{Z}[\hbar,c^{\pm 1}]$-combination of $\kappa$-tuples of perturbed
geodesics. In the spirit of [AS10], we can flow the braid $\gamma$ by the
Morse function defined in Section 5.2. Every time a crossing switches from
positive to negative, or vice versa, the flow will bifurcate according to the
HOMFLY skein relation. Similarly, crossing the strand over the puncture $\ast$
will pick up factors of $c$. Continuing this flow, we arrive at a
$\mathbb{Z}[\hbar,c^{\pm 1}]$-combination of $\kappa$-tuples of perturbed
geodesics. Since the relations of our $c$-deformed homotopy classes are the
same as those during the Morse flow, the two equivalence classes will be the
same.

Composing $\mathcal{E}$ with the projection which takes the $c$-deformed
homotopy class and quotients out by the HOMFLY skein relation gives the map

(4.2)
$\mathcal{F}:CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast).$

###  4.4. The proof of Theorem 1.1, $\kappa=1$ case

In this subsection we prove Theorem 1.1 for $\kappa=1$. Let $\mathcal{F}_{0}$
denote the specialization $\mathcal{F}_{\hbar=0}$.

We start by briefly reviewing the chain map

$\Theta:CM_{\ast}(\Omega^{1,2}(\Sigma,q,\alpha),\mathcal{A}_{V})\longrightarrow
CW(T_{q}^{*}\Sigma,N^{*}\alpha)$

constructed by Abbondandolo and Schwarz in [AS10, Theorem 3.3], where the
domain $CM_{\ast}(\Omega^{1,2}(\Sigma,q,\alpha),\mathcal{A}_{V})$ is the Morse
complex of the function $\mathcal{A}_{V}$ defined by:

$\mathcal{A}_{V}(\gamma)=\int_{0}^{1}L_{V}(t,\gamma,\dot{\gamma})dt,$

for $\gamma\in\Omega^{1,2}(\Sigma,q,\alpha)$. Both complexes are concentrated
at grading 0, so $\Theta$ is an isomorphism from the group generated by index
0 critical points of $\mathcal{A}_{V}$ to the wrapped Floer group
$CW(T_{q}^{*}\Sigma,N^{*}\alpha)$.

In what follows, we identify
$CM_{0}(\Omega^{1,2}(\Sigma,q,\alpha),\mathcal{A}_{V})$ with
$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})$.

Given $y\in CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V}),x\in
CW(T_{q}^{*}\Sigma,N^{*}\alpha)$, AS construct the space $\mathcal{M}(y,x)$ of
maps

$u:(-\infty,0]\times[0,1]\longrightarrow T^{*}\Sigma$

solving the Floer equation (2.4), which converge to the Hamiltonian chord
representing $x$ at $-\infty$, satisfy boundary conditions $T^{*}_{q}\Sigma$
along $(-\infty,0]\times\\{0\\}$, $N^{*}\alpha$ along
$(-\infty,0]\times\\{1\\}$, and such that the image of $u(\\{0\\}\times[0,1])$
under the projection from $T^{*}\Sigma$ to $\Sigma$ is a path on $\Sigma$
lying in the descending manifold of $y$ with respect to the negative gradient
flow of $\mathcal{A}_{V}$.

Letting $\\#\mathcal{M}(y,x)$ be the count of such maps, we have

$\Theta(y)=\sum\\#\mathcal{M}(y,x)x.$

Let

$\Theta_{c}:CM_{\ast}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm
1}]\longrightarrow CW(T_{q}^{*}\Sigma,N^{*}\alpha)_{c}$

be the $\mathbb{Z}[c^{\pm 1}]$-linear extension of $\Theta$.

We make an additional identification which allows us to apply results of
[AS10].

######  Proposition 4.8.

$BSk_{\kappa=1}(\Sigma,q,\alpha,\ast)\simeq
CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm 1}]$

###### Proof.

Elements in $BSk_{\kappa=1}(\Sigma,\bm{q},\bm{\alpha},\ast)$ are $c$-deformed
homotopy classes of paths starting at $q$ and ending on $\alpha$. Every such
path is homotopic to a unique perturbed geodesic on $\Sigma$ with the same
boundary conditions. Let $\gamma_{1}$ be a path homotopic to a perturbed
geodesic $\gamma_{2}$ and let $n$ be the signed intersection number of this
homotopy with the marked point $\ast$. The map sending
$[\gamma_{1}]_{c}\mapsto c^{2n}\gamma_{2}$ is the desired isomorphism. ∎

We prove the following proposition, recreating a proof similar to Abouzaid’s
[Abo12] in the case of cotangent fibers. Let $\tilde{\mathcal{F}}_{0}$ be the
composition of $\mathcal{F}_{0}$ with the isomorphism described in Proposition
4.8.

######  Proposition 4.9.

Let $\kappa=1$. The map

$\tilde{\mathcal{F}}_{0}:CW(T_{q}^{*}\Sigma,N^{*}\alpha)_{c}\longrightarrow
CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm 1}]$

is an isomorphism of chain complexes. Moreover, $\tilde{\mathcal{F}}_{0}$ is
an inverse to $\Theta_{c}$.

###### Proof.

Since $\Theta$ is an isomorphism of chain complexes, it suffices to show that
the composition

$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm
1}]\overset{\Theta_{c}}{\longrightarrow}CW(T_{q}^{*}\Sigma,N^{*}\alpha)_{c}\overset{\tilde{\mathcal{F}}_{0}}{\longrightarrow}CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm
1}]$

is homotopic to the identity map on
$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm 1}]$.
We show that it is not just homotopic to the identity, but is in fact the
identity map on the nose. It follows then that
$\tilde{\mathcal{F}}_{0}=\Theta^{-1}_{c}$. Let $y$ and $z$ be critical points
of the Lagrangian actional $\mathcal{A}_{V}$ which is Fenchel dual to $H_{V}$
and let $a$ be a positive real number.

Define $\mathcal{C}(y,z;a)$ to be the moduli space of maps

$u:[0,a]\times[0,1]\longrightarrow T^{*}\Sigma$

which solve the Floer equation (2.4) with boundary conditions
$T^{*}_{q}\Sigma$ along $[0,a]\times\\{0\\}$, $N^{*}\alpha$ along
$[0,a]\times\\{1\\}$, and such that $u(\\{0\\}\times[0,1])$ is contained in
the zero section and, considered as a path on $\Sigma$, lies on the ascending
manifold of $z$, while the image of $u(\\{a\\}\times[0,1])$ under the
projection from $T^{*}\Sigma$ to $\Sigma$ is a path on $\Sigma$ lying on the
descending manifold of $y$ with respect to the the negative gradient flow of
$\mathcal{A}_{V}$. This is shown in the central part of Figure 6.

Write $\mathcal{C}(y,z):=\sqcup_{a\in[0,\infty)}\mathcal{C}(y,z;a)$ and let
$\overline{\mathcal{C}}(y,z)$ be its Gromov compactification.

When $a=0$, any solution in $\mathcal{C}(y,z;a)$ is necessarily constant.
Thus, the count of rigid elements of $\mathcal{C}(y,z;0)$ gives the identity
map on
$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm 1}]$.

Letting $a$ go to $+\infty$, a family of maps $u_{a}$ defined on finite strips
$[0,a]\times[0,1]$ breaks into two maps $u_{-},u_{+}$ defined on semi-infinite
strips, as shown in the bottom right part of the figure below. This boundary
component is precisely the composition $\tilde{\mathcal{F}}_{0}\circ\Theta$.

The remaining boundary strata occur at finite $a$ when the projection of
$u(\\{a\\}\times[0,1])$ to $\Sigma$ escapes to the ascending manifold of a
critical point $y^{\prime}$ which differs from $y$. Similarly, the image of
$u(\\{0\\}\times[0,1])$ may converge to the descending manifold of a critical
point $z^{\prime}\neq z$. Such boundary strata are in bijective correspondence
with

$\mathcal{T}(y,y^{\prime})\times\overline{\mathcal{C}}(y^{\prime},z)\cup\overline{\mathcal{C}}(y,z^{\prime})\times\mathcal{T}(z^{\prime},z),$

where $\mathcal{T}(y,y^{\prime})$ is the moduli space of gradient trajectories
from $y$ to $y^{\prime}$.

For a general manifold this would give a chain homotopy. However, all critical
points of $\mathcal{A}_{V}$ have grading 0 and so there are no gradient
trajectories between critical points. Thus there are no boundary strata of
this form.

It follows then that, up to signs,
$\mbox{id}=\tilde{\mathcal{F}}_{0}\circ\Theta$, where id is the identity map
on

$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm 1}].$

Moreover, identifying $x\in CW(T_{q}^{*}\Sigma,N^{*}\alpha)_{c}$ with a time-1
Hamiltonian chord, $\tilde{\mathcal{F}}_{0}$ maps $x$ to the homotopy class of
its Legendre transform $[\mathcal{L}(x)]$. ∎

(a) $a=0$ boundary

(b) $a\to\infty$ boundary

Figure 6. $\overline{\mathcal{C}}(y,z;a)$ with its boundary. The arrows
represent downward gradient flows.

Composing $\tilde{\mathcal{F}}_{0}$ with the isomorphism

$CM_{0}(\Omega(\Sigma,q,\alpha),\mathcal{A}_{V})\otimes\mathbb{Z}[c^{\pm
1}]\hookrightarrow BSk_{1}(\Sigma,q,\alpha,\ast)$

given by viewing a geodesic as a path with the same boundary conditions proves
that $\mathcal{F}_{0,\kappa=1}$ is an isomorphism.

###  4.5. The proof of Theorem 1.1, general case

We first extend Proposition 4.9 to $\kappa\geq 1$.

######  Lemma 4.10.

Let $\kappa\geq 1$. Then $\mathcal{F}_{0}$ is an isomorphism.

###### Proof.

Let $\kappa\geq 1$ and $\hbar=0$.
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)_{c}|_{\hbar=0}$ no longer keeps
track of braid crossings and thus is generated over $\mathbb{Z}[c^{\pm 1}]$ by
elements of the form
$\otimes_{i=1}^{\kappa}H_{0}(\Omega(\Sigma,q_{i},\alpha_{\rho(i)}))_{c}=\otimes_{i=1}^{\kappa}BSk_{1}(\Sigma,q_{i},\alpha_{\rho(i)},\ast)_{c}$,
where $\rho\in S_{\kappa}$. Each homotopy class of paths in
$H_{0}(\Omega(\Sigma,q_{i},\alpha_{\rho(i)}))_{c}$ contains a unique
$V$-perturbed geodesic from $q_{i}$ to $\alpha_{\rho(i)}$. Applying the
Fenchel duality in Section 3.1 gives us a bijection between $V$-perturbed
geodesics and their dual Hamiltonian chords. By Proposition 4.9, this
bijection is given by $\mathcal{F}_{0,\kappa=1}$ after identifying generators
of $CW(\Sigma,q_{i},\alpha_{\rho(i)})$ with Hamiltonian chords from $q_{i}$ to
$\alpha_{\rho(i)}$.

Since $\hbar=0$, the only maps $u$ contributing to $\mathcal{F}_{0}$ are such
that $\chi(u)=\kappa$. The domain is then $\kappa$ pseudoholomorphic disks,
each of which is counted in the map $\mathcal{F}_{0,\kappa=1}$. Given a
generator

$\bm{x}=\\{x_{1\rho(1)},\dots,x_{\kappa\rho(\kappa)}\\}\in
CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c},$

where $x_{i\rho(i)}\in
CF(\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma),N^{*}\alpha_{\rho(i)})$,

$\mathcal{F}_{0}(\bm{x})=\otimes_{i=1}^{\kappa}\mathcal{F}_{0,\kappa=1}(x_{i\rho(i)})=\otimes_{i=1}^{\kappa}[\mathcal{L}(x_{i\rho(i)})]\in\otimes_{i=1}^{\kappa}BSk_{1}(\Sigma,q_{i},\alpha_{\rho(i)},\ast)_{c}.$

We construct an inverse

$\mathcal{F}_{0}^{-1}:BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)_{c}|_{\hbar=0}\longrightarrow
CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$

given by

$\bm{\gamma}=\\{\gamma_{1\rho(1)},\dots,\gamma_{\kappa\rho(\kappa)}\\}\mapsto\\{\mathcal{F}_{0,\kappa=1}^{-1}(\gamma_{1\rho(1)}),\dots,\mathcal{F}_{0,\kappa=1}^{-1}(\gamma_{\kappa\rho(\kappa)})\\}.$

Thus $\mathcal{F}_{0}$ is an isomorphism. ∎

We use Lemma 4.10 and reintroduce $\hbar$. It suffices to show that
$\mathcal{F}$ is a bijection. We repeat the argument in [HTY ̵a].

###### Proof of Theorem 1.1.

Injectivity of $\mathcal{F}$: Suppose that there exists $\bm{a}\neq 0$ such
that $\mathcal{F}(\bm{a})=0$. We can write $\bm{a}=\sum_{i\geq
0}\hbar^{i}\bm{a}_{i}$, where $\bm{a}_{i}\in
CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}|_{\hbar=0}$.
Since the codomain $BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$ has no
$\hbar$-torsion, it follows that $\bm{a}_{0}\neq 0$. Then setting $\hbar=0$,
we have $\mathcal{F}(\bm{a}_{0})=\mathcal{F}(\bm{a})=0$. This implies that
$\mathcal{F}_{0}(\bm{a}_{0})=0$, and thus $\bm{a}_{0}=0$, a contradiction.
Therefore, $\mathcal{F}$ is injective.

Surjectivity of $\mathcal{F}$: Let $\bm{b}\in
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$. By Lemma 4.10, there exists
$\bm{a}_{0}$ such that $\mathcal{F}(\bm{a}_{0})\equiv\bm{b}\text{ (mod
}\hbar)$. Let

$\bm{b_{1}}=\frac{\bm{b}-\mathcal{F}(\bm{a}_{0})}{\hbar}|_{\hbar=0}.$

Then there exists an $\bm{a}_{1}$ such that
$\mathcal{F}(\bm{a}_{1})\equiv\bm{b}_{1}\text{ (mod }\hbar)$. Repeating this
procedure, we get $\mathcal{F}(\sum_{i\geq 0}\hbar^{i}\bm{a}_{i})=\bm{b}$.
Thus $\mathcal{F}$ is surjective. ∎

##  5\. Geometric realization of
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
as a module over the braid skein algebra

###  5.1. Algebraic action

We describe the (right) braid skein-module structure on
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$. Specifically, we give the
action of the braid skein algebra $BSk_{\kappa}(\Sigma,\bm{q},\ast)$ on our
proposed module $BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$.

Let $[\gamma_{1}]$ be an element of the braid skein algebra
$BSk_{\kappa}(\Sigma,\bm{q},\ast)$ and $[\gamma_{2}]$ be an element of
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$. Then $[\gamma_{1}]$ and
$[\gamma_{2}]$ represent $c$-deformed homotopy classes of paths in
$\Omega(\text{UConf}_{\kappa}((\Sigma\setminus\\{\ast\\}),\bm{q})$ and
$\Omega(\text{UConf}_{\kappa}((\Sigma\setminus\\{\ast\\}),\bm{q},\bm{\alpha})$,
respectively. Suppose $\gamma_{i}$ is a representative for $[\gamma_{i}]$,
where $\gamma_{i}$ is an element of
$\Omega(\text{UConf}_{\kappa}((\Sigma\setminus\ast))\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]]$ satisfying the appropriate boundary conditions
for the configuration space.

######  Proposition 5.1.

$BSk_{\kappa}(\Sigma,\bm{q},\ast)$ acts on
$BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$ by the map

$\rho:BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)\otimes
BSk_{\kappa}(\Sigma,\bm{q},\ast)\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$

given by

$\rho([\gamma_{2}]_{c},[\gamma_{1}]_{c})\mapsto[\gamma_{1}\gamma_{2}]_{c}.$

###### Proof.

This follows directly from the product defined on the braid skein algebra. ∎

Notation: We will denote the product as a left multiplication to be more
compatible with the composition of braids. Specifically, given $\gamma_{1}\in
BSk_{\kappa}(\Sigma,\bm{q},\ast),\ \gamma_{2}\in
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$, we write
$\gamma_{1}\gamma_{2}:=\rho(\gamma_{2},\gamma_{1})$.

###  5.2. Geometric action

We make use of the bijection between elements in wrapped HDHF and
$\kappa$-tuples of perturbed geodesics. Consider the Hamiltonian
$H_{V}(t,q,p)=\frac{1}{2}|p|^{2}+V(t,q)$ and its Fenchel dual
$L_{V}(t,q,v)=\frac{1}{2}|v|^{2}-V(t,q)$. Then, as in Section 3, we consider
the Morse function $\mathcal{A}_{V}$ on the path space
$\Omega^{1,2}(\Sigma,q,q^{\prime})$ defined by

$\mathcal{A}_{V}(\gamma)=\int_{0}^{1}L_{V}(t,\gamma(t),\dot{\gamma}(t))dt.$

We generalize this to $\kappa$ strands by considering the path spaces

$\Omega_{\rho}(\Sigma,\bm{q},\bm{q}):=\prod_{i=1}^{\kappa}\Omega(\Sigma,q_{i},q_{\rho(i)}),\
\ \
\Omega^{1,2}_{\rho}(\Sigma,\bm{q},\bm{q}):=\prod_{i=1}^{\kappa}\Omega^{1,2}(\Sigma,q_{i},q_{\rho(i)}),$
$\Omega^{1,2}(\Sigma,\bm{q},\bm{q}):=\bigsqcup_{\rho\in
S_{\kappa}}\Omega^{1,2}_{\rho}(\Sigma,\bm{q},\bm{q}),$

where $\rho\in S_{\kappa}$ is a permutation. Then given
$\bm{\gamma}\in\Omega^{1,2}_{\rho}(\Sigma,\bm{q},\bm{q})$, we define

$\mathcal{A}_{V}(\bm{\gamma})=\sum_{i}\mathcal{A}_{V}(\gamma_{i}).$

For generic $V$, the action functional $\mathcal{A}_{V}$ on
$\Omega^{1,2}_{\rho}(\Sigma,\bm{q},\bm{q})$ is a Morse function which
satisfies the Palais-Smale condition.

The critical points of $\mathcal{A}_{V}$ are exactly the $\kappa$-tuples of
perturbed geodesics which are in bijection with the elements of our wrapped
HDHF.

We define the path space $\Omega^{1,2}(\Sigma,\bm{q},\bm{\alpha})$ in a
similar manner, the only difference being the end points of the paths are
confined to $\alpha_{i}$ instead of $q^{\prime}_{i}$.

Let $\bm{y}\in CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)$ and $\bm{x}\in
CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})$.
We want to define the product $\bm{y}\cdot\bm{x}$. On one hand, this has
already been done in the framework of HDHF ([CHT20]), where a $\mu^{2}$ map
gives us the product $\bm{y}\cdot\bm{x}=\mu^{2}(\bm{x},\bm{y})\in
CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})$.
We aim to give a more geometrically intuitive interpretation of the product
using the bijection with perturbed geodesics, making it more compatible with
the algebraic action given in Section 5.1.

We define a similar Morse function where the endpoint of the curve is allowed
to move along our curves $\alpha_{1},\dots,\alpha_{\kappa}$. With this setup,
the geometric action of $CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}$ on
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
is given by translating elements to $V$-perturbed geodesics viewed as paths,
concatenating the paths, then performing the Morse gradient flow discussed
above. The result is a $V$-perturbed geodesic from
$\sqcup_{i=1}^{\kappa}q_{i}$ to $\sqcup_{i=1}^{\kappa}\alpha_{i}$, which is
then identified with an element of
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$.

When performing the Morse gradient flow after concatenation, we must consider
passings through the marked point as well as any crossings of the perturbed
geodesics. Similarly as before, to account for the marked point, we have the
homotopy $H$ from the concatenated curves $\gamma_{1}\gamma_{2}$ to the
resulting geodesic $\gamma$ and we impose on this the $c$-deformed homotopy
relation $[\gamma]=c^{2\langle H,\ast\rangle}[\gamma_{1}\gamma_{2}]$. We
account for any crossings that occur on $\Sigma$ during our flow by applying
the HOMFLY skein relation at all crossings. We can view each $\kappa$-tuple of
perturbed geodesics as a braid in $[0,1]\times\Sigma$ by mapping
$\gamma(t)\mapsto(t,\gamma(t))$. Then each crossing will either be a positive
crossing $\sigma_{i}$ or a negative crossing $\sigma_{i}^{-1}$, and our HOMFLY
skein relation is $\sigma_{i}-\sigma_{i}^{-1}=\hbar e$, where $e$ is the
resolution of the crossing. We call any time a positive crossing changes to a
negative crossing (or vice versa) a switching. In the Morse theory view, this
results in a bifurcated trajectory; one trajectory is a continuation of the
switching and the other is a resolution of the crossing with a factor of
$\hbar$. As a braid, we get a sum of the same braid with the crossing reversed
and a resolved crossing with an extra $\hbar$ factor. Using this relation, we
can resolve all of the crossings into $\mathbb{Z}[\hbar,c^{\pm 1}]$-linear
combinations of braids which are tuples of $V$-perturbed geodesics.

###  5.3. Equivalence of DAHA modules

Our main goal of this section is to show that the following diagram commutes:

(5.1) ${CW(\sqcup_{i}T^{*}_{q_{i}}\Sigma,\sqcup_{i}N^{*}\alpha_{i})_{c}\otimes
CW(\sqcup_{i}T^{*}_{q_{i}}\Sigma)_{c}}$${BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)\otimes(H_{\kappa}(\Sigma,\bm{q})\otimes\mathbb{Z}[[\hbar]])}$${CW(\sqcup_{i}T^{*}_{q_{i}}\Sigma,\sqcup_{i}N^{*}\alpha_{i})_{c}}$${BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)}$$\scriptstyle{\mathcal{F}_{1}\otimes\mathcal{F}_{2}}$$\scriptstyle{\mu^{2}}$$\scriptstyle{\rho}$$\scriptstyle{\mathcal{F}_{1}}$

We recall each map of the diagram:

The top maps

$\mathcal{F}_{1}:CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$

and

$\mathcal{F}_{2}:CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}\longrightarrow
H_{\kappa}(\Sigma,\bm{q})\otimes\mathbb{Z}[[\hbar]]$

are the isomorphisms defined in Sections 4.3 and 4.2, respectively.

The left map

$\mu^{2}:CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\otimes
CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)_{c}\longrightarrow
CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$

is the $A_{\infty}$-map defined by Equation (2.2) with $m=2$. It is given by a
count of holomorphic maps which project onto a thrice-punctured disk
satisfying the usual boundary conditions; see [CHT20] for more details.

The right map

$\rho:BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)\otimes(H_{\kappa}(\Sigma,\bm{q})\otimes\mathbb{Z}[[\hbar]])\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$

is given in Section 5.1 after identifying
$H_{\kappa}(\Sigma,\bm{q})\otimes\mathbb{Z}[[\hbar]]$ with
$BSk_{\kappa}(\Sigma,\bm{q},\ast)$ .

The bottom map is again the evaluation map from Section 4.3.

######  Lemma 5.2.

Diagram 5.1 commutes.

Before we prove the lemma, we introduce a moduli space of holomorphic curves
similar to that of [HTY ̵a, Section 6.2].

Let $T_{2}:=D_{3}$ be our $A_{\infty}$-base where
$\partial_{i}T_{2}=\partial_{i}D_{3}$. Let $\mathcal{T}_{2}$ be the moduli
space of $T_{2}$ modulo automorphisms, and choose representatives $T_{2}$ of
equivalence classes in a smooth manner. Let $\pi_{T^{*}\Sigma}$ be the
projection $T_{2}\times T^{*}\Sigma\to T^{*}\Sigma$ and choose a sufficiently
generic consistent collection of compatible almost complex structures such
that they are close to a split almost complex structure projecting
holomorphically to $T_{2}$, as in Section 2.1. Perturb the 0-section near the
$\alpha_{i}$ and let $\bm{x}$ be the tuple of intersections
$\alpha_{i}\cap\phi_{H^{\prime}}(\Sigma)$ corresponding to the bottom
generators, where $\phi_{H^{\prime}}(\Sigma)$ is the perturbed 0-section. We
denote by $\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
the moduli space of maps

$u:(\dot{F},j)\longrightarrow(T_{2}\times T^{*}\Sigma,J_{T_{2}}),$

where $(F,j)$ is a compact Riemann surface with boundary, $\bm{p}_{0}$,
$\bm{p}_{1}$, $\bm{p}_{2}$, $\bm{p}_{3}$ are disjoint tuples of boundary
punctures of $F$ and $\dot{F}=F\setminus\cup_{i}\ \bm{p}_{i}$, satisfying:

$\begin{cases}du\circ j=J_{T_{2}}\circ du;\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\phi^{2}_{H_{V}}(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)\text{ if
}\pi_{T_{2}}\circ u(z)\subset\partial_{0}T_{2};\\\ \text{each component of
}\partial\dot{F}\text{ that projects to }\partial_{0}T_{2}\text{ maps to a
distinct }\phi^{2}_{H_{V}}(T^{*}_{q_{i}}\Sigma);\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\phi^{1}_{H_{V}}(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma)\text{ if
}\pi_{T_{2}}\circ u(z)\subset\partial_{1}T_{2};\\\ \text{each component of
}\partial\dot{F}\text{ that projects to }\partial_{1}T_{2}\text{ maps to a
distinct }\phi^{1}_{H_{V}}(T^{*}_{q_{i}}\Sigma);\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i}\text{ if }\pi_{T_{2}}\circ
u(z)\subset\partial_{2}T_{2};\\\ \text{each component of
}\partial\dot{F}\text{ that projects to }\partial_{2}T_{2}\text{ maps to a
distinct }N^{*}\alpha_{i};\\\ \pi_{T^{*}\Sigma}\circ
u(z)\in\Sigma\times\\{0\\}\subset T^{*}\Sigma\text{ if }\pi_{T_{2}}\circ
u(z)\subset\partial_{3}T_{2};\\\ \pi_{T^{*}\Sigma}\circ u\text{ tends to
}\bm{q}^{\prime\prime},\ \bm{y}^{\prime},\ \bm{y},\ \bm{x}\text{ as
}s_{0},s_{1},s_{2},s_{3}\to+\infty;\\\ \pi_{T_{1}}\circ u\text{ is a
}\kappa\text{-fold branched cover of a fixed }T_{2}\in\mathcal{T}_{2}.\\\
\end{cases}$

In simpler terms, we look at the moduli space of holomorphic curves between
the Lagrangians involved and the zero section of $T^{*}\Sigma$ in the
framework of HDHF.

Figure 7. The $A_{\infty}$-base $T_{2}$ with boundary conditions

######  Lemma 5.3.

There exists a sufficiently generic consistent collection of almost complex
structures such that the moduli space
$\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$ is of
dimension 1 and is transversely cut out for all $\bm{x},\ \bm{y},\
\bm{y}^{\prime}$ and $\bm{q^{\prime\prime}}$. Moreover,
$\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$ admits a
compactification
$\overline{\mathcal{H}}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
such that its boundary
$\partial\overline{\mathcal{H}}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
is of dimension 0 and contains discrete broken or nodal curves.

###### Proof.

This is identical to Lemma 6.4 in [HTY ̵a]. ∎

As in Section 3.3, we define a map from
$\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})\times[0,1]\longrightarrow(\Sigma)^{\kappa}$
by

(5.2) $\gamma(u)(t)=(\pi_{T^{*}\Sigma}\circ u)\circ(\pi_{\Sigma}\circ
u)^{-1}\circ\tau(t),$

where $\tau:[0,1]\to\partial_{3}T_{2}$ parametrizes the boundary arc from
$p_{0}$ to $p_{3}$. Let

$\mathcal{H}_{0}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})=\\{u\in\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})~{}|~{}\gamma(u)(t)\in\text{UConf}_{\kappa}(\Sigma\setminus\\{\ast\\})\text{
for all }t\\}.$

As before, we define the evaluation map

$\mathcal{G}:\mathcal{H}_{0}(\bm{q^{\prime\prime},y^{\prime},y,x})\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast),$
$u\mapsto(-1)^{\natural(u)}\cdot c^{2\langle
u,\ast\rangle}\cdot\hbar^{\kappa-\chi(u)}\cdot[\gamma(u)].$

###### Proof of Lemma 5.2.

We analyze the boundary of the index 1 moduli space
$\overline{\mathcal{H}}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
by considering the possible degenerations. Let
$\overline{\mathcal{H}}^{\chi}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
be the subset of
$\overline{\mathcal{H}}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
consisting of maps with $\chi(u)=\chi$. For a generic $u$,
$u\in\mathcal{H}_{0}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$.
However, for a 1-parameter family
$u_{t}\in\mathcal{H}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$,
$\gamma(u_{t})$ may intersect the marked point $\ast$ at some $t\in(0,1)$.
However, since we are taking $c$-deformed homotopy classes, we are guaranteed
that $\mathcal{G}(u_{0})=\mathcal{G}(u_{1})$. Hence we will not worry about
intersections with $\ast$. All codimension-1 degenerations occur in the
$A_{\infty}$-base direction, giving us a nice characterization of the possible
breakings. The three types of boundary degenerations are:

  1. (1)

$\bigsqcup_{\bm{y}^{\prime\prime},\chi^{\prime}+\chi^{\prime\prime}-\kappa=\chi}\mathcal{M}^{\text{ind}=0,\chi^{\prime}}(\bm{y}^{\prime},\
\bm{y},\
\bm{y}^{\prime\prime})\times\mathcal{H}^{\text{ind}=0,\chi^{\prime\prime}}(\bm{q}^{\prime\prime},\
\bm{y}^{\prime\prime},\ \bm{x})$;

  2. (2)

$\bigsqcup_{\bm{q}^{\prime},\chi^{\prime}+\chi^{\prime\prime}-\kappa=\chi}\mathcal{H}^{\text{ind}=0,\chi^{\prime}}(\bm{q^{\prime\prime}},\
\bm{y}^{\prime},\
\bm{q}^{\prime})\times\mathcal{H}^{\text{ind}=0,\chi^{\prime\prime}}(\bm{q}^{\prime},\
\bm{y},\ \bm{x})$;

  3. (3)

the set
$\partial_{n}\overline{\mathcal{H}}^{\text{ind}=1,\chi}(\bm{q}^{\prime\prime},\bm{y}^{\prime},\bm{y},\bm{x})$
with a nodal degeneration along $\Sigma$.

The first type is shown on the left-hand side of Figure 8 and contributes
$\mathcal{F}_{1}(\mu^{2}(\bm{y,\ y^{\prime}}))$.

The second type is shown on the right-hand side of Figure 8 and contributes
$\rho((\mathcal{F}_{1}\otimes\mathcal{F}_{2})(\bm{y},\ \bm{y}^{\prime}))$.

In fact, all contributions to $\mathcal{F}_{1}(\mu^{2}(\bm{y},\
\bm{y}^{\prime}))$ and $\rho((\mathcal{F}_{1}\otimes\mathcal{F}_{2})(\bm{y},\
\bm{y}^{\prime}))$ come from such degenerations.

The proof of Proposition 6.5 in [HTY ̵a] shows that the total contribution of
the third type over all Euler characteristics $\chi$ is zero. Hence it follows
that $\mathcal{F}_{1}(\mu^{2}(\bm{y},\
\bm{y}^{\prime}))=\rho((\mathcal{F}_{1}\otimes\mathcal{F}_{2})(\bm{y},\
\bm{y}^{\prime}))$ and so the diagram commutes.

∎

Figure 8. $T_{2}$ degenerations

We have thus shown that the map
$\mathcal{F}_{1}:CW(\sqcup_{i=1}^{\kappa}T^{*}_{q_{i}}\Sigma,\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\longrightarrow
BSk_{\kappa}(\Sigma,\bm{q},\bm{\alpha},\ast)$ realizes the wrapped HDHF of
$\kappa$ cotangent fibers and $\kappa$ conormal bundles of simple closed
curves as a DAHA-module.

##  6\. The enhanced polynomial representation

In this section, we specialize to $\Sigma=T^{2}$. We introduce the double
affine Hecke algebra (DAHA) along with its polynomial representation. After
fixing a configuration of points $q_{1},\dots,q_{\kappa}$ and curves
$\alpha_{1},\dots,\alpha_{\kappa}$ in $T^{2}$, we define the enhanced
polynomial representation of DAHA on
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
and prove Theorem 1.3.

###  6.1. Double affine Hecke algebra and its polynomial representation

We briefly review the DAHA and its skein-theoretic realization using braids in
the punctured torus. For more details, refer to [MS21], where these results
are proven and discussed at length.

Viewing $T^{2}$ as a square $I\times I$ with opposite sides identified, we
choose $\ast=(\frac{1}{2},\frac{1}{2})$ and let the $\kappa$ points
$q_{1},\dots,q_{\kappa}$ line up in increasing fashion along the lower part of
the diagonal from $(0,0)$ to $\ast$. We choose a convenient basis for the
braids in $B_{\kappa}(T^{2}\setminus\\{\ast\\},\bm{q})$. Let $x_{i}\
(\text{respectively, }y_{i})$ be the braid which consists of the point $q_{i}$
moving uniformly around the $(-1,0)\ (\text{respectively, }(0,1))$ curve. Let
$\sigma_{i}$ for $1\leq i\leq\kappa-1$ be the braid which locally exchanges
the strings from $q_{i}$ and $q_{i+1}$ in a counterclockwise direction when
looking down onto $T^{2}$, as shown in Figure 7.1(B) below.

(a) Braids $x_{i}$ and $y_{i}$

(b) Braid $\sigma_{i}$

Figure 9. Generators for the braid group on the punctured torus

The key element in this geometric realization of DAHA is the next theorem due
to Morton and Samuelson.

######  Theorem 6.1 (Morton-Samuelson).

The braid skein algebra $BSk_{\kappa}(T^{2},\bm{q},\ast)$ is isomorphic to the
double affine Hecke algebra $\ddot{H}_{\kappa}$.

We now fix the presentation for the skein algebra, and therefore the DAHA,
that we will be using.

######  Theorem 6.2.

The double affine Hecke algebra $\ddot{H}_{\kappa}$ can be presented by the
braids $\sigma_{1},\dots,\sigma_{\kappa-1},x_{1},y_{1}$ with relations:

  1. (1)

$\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i},\ |i-j|>1,$

  2. (2)

$\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1},$

  3. (3)

$\sigma_{i}x_{1}=x_{1}\sigma_{i},\ i>1,$

  4. (4)

$\sigma_{i}y_{1}=y_{1}\sigma_{i},\ i>1,$

  5. (5)

$x_{1}\sigma_{1}x_{1}\sigma_{1}=\sigma_{1}x_{1}\sigma_{1}x_{1},$

  6. (6)

$y_{1}\sigma_{1}y_{1}\sigma_{1}=\sigma_{1}y_{1}\sigma_{1}y_{1},$

  7. (7)

$x_{1}\sigma_{1}y_{1}\sigma_{1}^{-1}=\sigma_{1}y_{1}\sigma_{1}x_{1},$

  8. (8)

$(\sigma_{1}-s)(\sigma_{1}+s^{-1})=0,$

  9. (9)

$x_{1}^{-1}y_{1}x_{1}y_{1}^{-1}=c^{2}\sigma_{1}\sigma_{2}\cdot\cdot\cdot\sigma_{\kappa-1}\sigma_{\kappa-1}\cdot\cdot\cdot\sigma_{2}\sigma_{1}$.

######  Remark 6.3.

The relations stated above are slightly different than those in [MS21].
Specifically, we replace $x_{1}$ with $x_{1}^{-1}$ since the generator $x_{i}$
in [MS21] corresponds to a loop based at $q_{1}$ in the $(1,0)$ direction
whereas our generator goes in the $(-1,0)$ direction. The relations have been
adjusted with this in mind.

Although $\ddot{H}_{\kappa}$ can be generated by
$\sigma_{1},\cdots,\sigma_{\kappa-1},\ x_{1},$ and $y_{1}$, it will be
convenient to make explicit the expression for the braids $x_{i}$ and $y_{i}$.
Using the relations $\sigma_{i}x_{i}\sigma_{i}=x_{i+1}$ and
$\sigma_{i}y_{i}\sigma_{i}=y_{i+1}$, it follows that
$x_{i}=\sigma_{i-1}\cdots\sigma_{1}x_{1}\sigma_{1}\cdots\sigma_{i-1}$ and
$y_{i}=\sigma_{i-1}\cdots\sigma_{1}y_{1}\sigma_{1}\cdots\sigma_{i-1}$.

The DAHA $\ddot{H}_{\kappa}$ has a representation on $\mathbb{Z}[[s]][c^{\pm
1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}]$ called the polynomial
representation. Note that in this paper we are concerned with the polynomial
representation of the DAHA $\ddot{H}_{\kappa}$ instead of the more common
spherical DAHA; we will use the presentation given in [Eno09].

######  Definition 6.4.

The polynomial representation of $\ddot{H}_{\kappa}$ on
$\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}]$ is
defined by the following:

$\displaystyle x_{i}$ $\displaystyle\mapsto X_{i},$ $\displaystyle\sigma_{i}$
$\displaystyle\mapsto
s\tau_{i}+\frac{s-s^{-1}}{X_{i}X_{i+1}^{-1}-1}(\tau_{i}-1),$ $\displaystyle
y_{1}$
$\displaystyle\mapsto\sigma_{1}^{-1}\cdots\sigma_{\kappa-1}^{-1}\omega,$

where $\tau_{i}$ permutes $X_{i}$ and $X_{i+1}$ and for any
$f\in\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}]$,

$(\omega
f)(X_{1},\cdots,X_{\kappa})=f(c^{2}X_{\kappa},X_{1},\cdots,X_{\kappa-1}).$

Denote the action above by

$p:\ddot{H}_{\kappa}\times\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm
1},\cdots,X_{\kappa}^{\pm 1}]\longrightarrow\mathbb{Z}[[s]][c^{\pm
1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}].$

######  Remark 6.5.

Observe that in the definition of the DAHA, the variable $s$ does not appear
on its own. The HOMFLY skein relation in the definition of the braid skein
algebra uses $s-s^{-1}$; in the presentation of Theorem 6.2, expanding the
relation (8) gives the term $s-s^{-1}$.

For our purposes, we let $\hbar=s-s^{-1}$ and change the coefficient ring from
$\mathbb{Z}[[s]][c^{\pm 1}]$ to $\mathbb{Z}[[\hbar]][c^{\pm 1}]$ when we are
not dealing with the polynomial representation.

###  6.2. The enhanced polynomial representation

In this section, we compute the DAHA-module
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)$ for the configuration of points
$q_{1},\dots,q_{\kappa}$ as in Section 6.1 and simple closed curves
$\alpha_{1},\dots,\alpha_{\kappa}$ similar to Section 2.4.

Viewing $T^{2}$ as $I\times I$ with opposite sides identified, let
$q_{i}=(\frac{i}{2(\kappa+1)},\frac{i}{2(\kappa+1)})$ and
$\alpha_{i}=\\{\frac{1}{2}+\frac{i}{2\kappa+2}\\}\times I$. Moreover, let
$\ast=(\frac{1}{2},\frac{1}{2})$. Choose a perturbation term $V(t,q)$ such
that $q^{\prime}_{i}=\phi^{1}_{H_{V}}(T^{*}_{q_{i}}T^{2})\cap T^{2}$ is to the
left of $q_{i}$ when viewed on $I\times I$ and
$|q_{i}-q^{\prime}_{i}|>|q_{j}-q^{\prime}_{j}|$ whenever $i<j$.

We introduce an additional parameter $d$ which keeps track of the ends of the
braids sliding along the $\alpha_{i}$. Specifically, consider the projection

$\pi_{\alpha}:\alpha_{1}\times\cdots\times\alpha_{\kappa}\longrightarrow
T^{\kappa}$

of the $\alpha_{i}$ to $T^{\kappa}=(S^{1})^{\kappa}$ given by dropping the $x$
coordinate for each $\alpha_{i}$. Let

$\Delta=\\{(x_{1},\dots,x_{\kappa})~{}|~{}x_{i}=x_{j}\text{ for some }i\neq
j\\}$

be the big diagonal in $T^{\kappa}$. The parameter $d$ counts signed
intersections with $\Delta$ as the braids are isotoped.

######  Definition 6.6.

The $d$-deformed braid skein group
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$ is the free
$\mathbb{Z}[[\hbar]][c^{\pm 1},d^{\pm 1}]$-module generated by elements of the
braid skein group $BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)$ subject to the
ends-slide relation:

Let $[\gamma]_{c,d}$ be the equivalence class of the braid $\gamma$ in
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$ which keeps track of the
parameters $c$ and $d$.

We now expand our wrapped HDHF group with parameter $c$ to one with parameters
$c$ and $d$.

######  Definition 6.7.

The wrapped HDHF group of $\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2}$ and
$\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i}$ with parameters $c$ and $d$ is

$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c,d}:=CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}\otimes\mathbb{Z}[d^{\pm
1}].$

A slight enhancement of Theorem 1.1 which keeps track of intersections with
$\Delta\subset T^{\kappa}$ viewed as a subset of $T^{2}\times\\{1\\}$ gives:

Theorem 1.1’.
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c,d}\simeq
BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$

Under the configuration of points $q_{i}$ and curves $\alpha_{i}$ described at
the start of this subsection, we choose a specific presentation for
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$ (and
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c,d}$).

Consider a $\kappa$-tuple of perturbed geodesics
$\bm{\gamma}=\\{\gamma_{1},\dots,\gamma_{\kappa}\\}$ viewed as a braid in
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$, where each $\gamma_{i}$ is
a perturbed geodesic from $q_{i}$ to $\alpha_{\sigma(i)}$ for some permutation
$\sigma\in S_{\kappa}$. Let $\alpha=\\{\frac{1}{2}\\}\times S^{1}\in T^{2}$ be
a simple closed curve between the set of points $q_{i}$ and curves
$\alpha_{i}$. We define the signed intersection number
$n_{i}=\langle\gamma_{i},\alpha\rangle$ of each perturbed geodesic with
$\alpha$. The sign of $n_{i}$ is set to be positive if $\gamma_{i}$ is in the
$(-1,0)$ direction and negative if it is in the $(1,0)$ direction. This
intersection number is similar to the process described in the proof of Lemma
2.9.

The following is a slight enhancement of Lemma 1.2:

Lemma 1.2’.
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}\simeq(\mathbb{Z}[a_{1}^{\pm
1},\dots,a_{\kappa}^{\pm
1}]\otimes\mathbb{Z}[S_{\kappa}])\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]]\otimes\mathbb{Z}[d^{\pm 1}].$

We denote the module with this presentation by $PR_{\kappa}$.

###### Proof.

Every element in $BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$ can be
identified with a $\mathbb{Z}[[\hbar]][c^{\pm 1},d^{\pm 1}]$-linear
combination of $\kappa$-tuples of perturbed geodesics viewed as a braid. This
is done by homotoping the strands to perturbed geodesics while keeping track
of intersections within the strands, with the marked point, and with the big
diagonal $\Delta$ of $T^{\kappa}$.

Let

$f:BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}\longrightarrow(\mathbb{Z}[a_{1}^{\pm
1},\dots,a_{\kappa}^{\pm
1}]\otimes\mathbb{Z}[S_{\kappa}])\otimes\mathbb{Z}[c^{\pm
1}]\otimes\mathbb{Z}[[\hbar]]\otimes\mathbb{Z}[d^{\pm 1}]$

be the $\mathbb{Z}[[\hbar]][c^{\pm 1},d^{\pm 1}]$-linear map which sends

$\bm{\gamma}=\\{\gamma_{1},\dots,\gamma_{\kappa}\\}\mapsto(a_{1}^{n_{1}}\cdots
a_{\kappa}^{n_{\kappa}},\sigma),$

where $\bm{\gamma}$ is a $\kappa$-tuple of perturbed geodesics and
$n_{i}=\langle\gamma_{i},\alpha\rangle$ is the signed intersection number
described above.

A generalization of the model computation in Section 2.4 shows us that $f$ is
surjective. That is, we can construct a $\kappa$-tuple of perturbed geodesics
with the right permutation and intersections with $\alpha$.

Let $\bm{y}\in BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$ and suppose
$f(\bm{y})=0$. By an argument similar to the discussion at the end of Section
5.2, we can identify $\bm{y}$ with a $\mathbb{Z}[[\hbar]][c^{\pm 1},d^{\pm
1}]$-linear combination of distinct $\kappa$-tuples of perturbed geodesics
$\bm{\gamma}_{i}$:

$\bm{y}=\sum_{i=1}^{n}g_{i}\bm{\gamma}_{i},$

where $g_{i}\in\mathbb{Z}[[\hbar]][c^{\pm 1},d^{\pm 1}]$. Since $f(\bm{y})=0$,
it follows that $\sum_{i=1}^{n}g_{i}f(\bm{\gamma}_{i})=0$. Since the perturbed
geodesics $\bm{\gamma}_{i}$ are distinct, they belong to different homotopy
classes and so their images under $f$ are linearly independent. It follows
that $\bm{y}=0$ and thus $f$ is injective. ∎

Consider an element $(1,\sigma)\in PR_{\kappa}$. This element is represented
by $V$-perturbed geodesics from each $q_{i}$ to $\alpha_{\sigma(i)}$ which do
not intersect each other or the curve $\alpha$. Let
$\sigma_{i}\in\ddot{H}_{\kappa}$, viewed as a braid consisting of strands
$q_{i}\mapsto q^{\prime}_{i+1}$, $q_{i+1}\mapsto q^{\prime}_{i}$ and
$q_{j}\mapsto q^{\prime}_{j}$ for $j\neq i,\ i+1$, where by $a\mapsto b$ we
mean ”from $a$ to $b$”. Our choice of perturbation term $V(t,q)$ guarantees
that the strand from $q_{i}$ crosses over the strand from $q_{i+1}$ when
projected down from $T^{2}\times[0,1]\to T^{2}$.

######  Lemma 6.8.

In the situation above, if $\sigma(i)<\sigma(i+1)$, then the concatenation of
the braid $\sigma_{i}$ and the geodesics representing $(1,\sigma)$ can be
isotoped to geodesics $d^{-1}(1,\sigma_{i}\sigma)$. (In the expression
$\sigma_{i}\sigma$, $\sigma_{i}$ is viewed as an element of $S_{\kappa}$ under
the projection $B_{\kappa}(T^{2},\bm{q})\to S_{\kappa}$.)

On the other hand, if $\sigma(i)>\sigma(i+1)$, then the concatenation of the
braid $\sigma_{i}$ and the geodesics representing $(1,\sigma)$ is equivalent
to a linear combination of geodesics $\hbar(1,\sigma)+d(1,\sigma_{i}\sigma)$.

###### Proof.

Suppose $\sigma(i)<\sigma(i+1)$. (See the left-hand side of Figure 10.) Slide
the end of the strand going to $\alpha_{\sigma(i+1)}$ along
$\alpha_{\sigma(i+1)}$ past the strand going to $\alpha_{\sigma(i)}$. Due to
the arrangement of the $\alpha_{j}$, this creates a crossing in which the
sliding strand crosses over the other, picking up a factor of $d^{-1}$. The
result is a braid which has a positive crossing at the bottom (by this we mean
at a lower $t$-coordinate where the braid is in $T^{2}\times[0,1]$ with
coordinates $(q,t)$) due to the $\sigma_{i}$ and a negative crossing at the
top. Thus we can isotope the two strands apart, arriving at the set of
geodesics representing $d^{-1}(1,\sigma_{i}\sigma)$.

Suppose on the other hand that $\sigma(i)>\sigma(i+1)$. (See the right-hand
side of Figure 10.) Resolve the $\sigma_{i}$ braid as $\sigma_{i}^{-1}+\hbar$
to get two braids: $\sigma_{i}^{-1}\cdot(1,\sigma)+\hbar(1,\sigma)$. For the
$\sigma_{i}^{-1}\cdot(1,\sigma)$ braid, slide the strand along
$\alpha_{\sigma(i)}$. This creates a positive crossing and picks up a factor
of $d$. With this positive crossing, we can isotope the strands apart to get a
braid $d(1,\sigma_{i}\sigma)$. Therefore,
$\rho((1,\sigma),\sigma_{i})=\hbar(1,\sigma)+d(1,\sigma_{i}\sigma)$. ∎

Figure 10. The compositions $\sigma_{2}(1,(12))$ and $\sigma_{2}(1,(123))$. On
the left, we have the case where $\sigma(i)<\sigma(i+1)$ and sliding the ends
of the strands creates a crossing of the dotted purple arcs which allows us to
separate the strands. On the right, we have the case where
$\sigma(i)>\sigma(i+1)$ and we see that the strands are linked after sliding
the ends of the strands across each other.

Notation: Recall that
$CW(\sqcup_{i=1}^{\kappa}T_{q_{i}}^{*}T^{2},\sqcup_{i=1}^{\kappa}N^{*}\alpha_{i})_{c}$
is a right DAHA-module. It follows that there is a right $\ddot{H}_{\kappa}$
action on $PR_{\kappa}$. It is often convenient to identify elements of
$PR_{\kappa}$ with $\kappa$-tuples of paths which are elements of
$BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)$. Since composition of paths is
written as a left multiplication, we choose to adopt the notation for a left
module. Let $h\in\ddot{H}_{\kappa}$ and $x\in PR_{\kappa}$, then we denote
$\rho(x,h)=h\cdot x$, where $\rho$ is as in Section 5.1. After identifying
$h_{1},h_{2}\in\ddot{H}_{\kappa}$ with braids $\gamma_{1},\gamma_{2}$ we have
that $\rho(x,\tilde{\rho}(h_{2},h_{1}))=(\gamma_{1}\gamma_{2})\cdot
x=\gamma_{1}\cdot(\gamma_{2}\cdot x)$, a left module structure, where
$\tilde{\rho}:BSk_{\kappa}(T^{2},\bm{q},\bm{q},\ast)\otimes
BSk_{\kappa}(T^{2},\bm{q},\bm{q},\ast)\longrightarrow
BSk_{\kappa}(T^{2},\bm{q},\bm{q},\ast)$ is the product on the braid skein
algebra.

######  Corollary 6.9.

Let $\sigma_{i}\in\ddot{H}_{\kappa}$ and $(1,\sigma)\in PR_{\kappa}$. Then,
setting $d=s$,

$\sigma_{i}\cdot((1,\sigma)+(1,\sigma_{i}\sigma))=s((1,\sigma)+(1,\sigma_{i}\sigma)).$

###### Proof.

Suppose $\sigma(i)<\sigma(i+1)$. Then
$\sigma_{i}\cdot(1,\sigma)=s^{-1}(1,\sigma_{i}\sigma)$ by Lemma 6.8. On the
other hand, $(\sigma_{i}\sigma)(i)>(\sigma_{i}\sigma)(i+1)$, so
$\sigma_{i}\cdot(1,\sigma_{i}\sigma)=\hbar(1,\sigma_{i}\sigma)+s(1,\sigma)$.
Adding the terms, expanding $\hbar=s-s^{-1}$, and canceling the
$s^{-1}(1,\sigma_{i}\sigma)$ gives the result.

The case $\sigma(i)>\sigma(i+1)$ follows immediately by letting
$\sigma=\sigma_{i}\sigma$. ∎

######  Definition 6.10.

Let $(\bm{a},\sigma)=(a_{1}^{n_{1}}\dotsm
a_{\kappa}^{n_{\kappa}},\sigma)\in\mathbb{Z}[a_{1}^{\pm
1},\dots,a_{\kappa}^{\pm 1}]\times S_{\kappa}$ be an element of $PR_{\kappa}$.
The enhanced polynomial representation of the DAHA $\ddot{H}_{\kappa}$ (with
presentation given in Theorem 6.2) on $PR_{\kappa}$ is defined on generators
as follows:

(1) $x_{i}\cdot(\bm{a},\sigma)=(a_{i}\cdot\bm{a},\sigma),$

(2) $\sigma_{i}\cdot(1,\sigma)=\begin{cases}d^{-1}(1,\sigma_{i}\sigma)&\text{
if }\sigma(i)<\sigma(i+1)\\\ d(1,\sigma_{i}\sigma)+\hbar(1,\sigma)&\text{ if
}\sigma(i)>\sigma(i+1)\par\end{cases},$

(3)
$y_{1}\cdot(\bm{a},\sigma)=c^{2n_{1}}\tau_{\kappa}^{-1}\cdot(\bm{a}_{\tau_{\kappa}},\tau_{\kappa}\sigma)$,

where $\tau_{\kappa}=\sigma_{\kappa-1}\cdots\sigma_{1}$ and
$\bm{a}_{\tau_{\kappa}}=a_{\kappa}^{n_{1}}a_{1}^{n_{2}}\cdots
a_{\kappa-1}^{n_{\kappa}}$.

(2) in Definition 6.10 only defines the action of $\sigma_{i}$ on an element
$(1,\sigma)$, but we can extend this to an action on $(\bm{a},\sigma)$ by
using the action of $x_{i}$ along with the relations of $\ddot{H}_{\kappa}$.
Since $\sigma_{i}x_{i}=x_{i+1}\sigma_{i}^{-1}$ and
$\sigma_{i}-\sigma_{i}^{-1}=\hbar$, it follows that
$\sigma_{i}x_{i}=x_{i+1}(\sigma_{i}-\hbar)=x_{i+1}\sigma_{i}-\hbar x_{i+1}$.
Similarly, $\sigma_{i}x_{i+1}=x_{i}\sigma_{i}+\hbar x_{i+1}$. If $j\neq i,\
i+1$, then $\sigma_{i}x_{j}=x_{j}\sigma_{i}$. Thus we are able to express any
product of $\sigma_{i}$ and $x_{j}$ as an expression where all the $x_{j}$ are
in front of the $\sigma_{i}$.

Since $(\bm{a},\sigma)=(x_{1}^{n_{1}}\cdots
x_{\kappa}^{n_{\kappa}})\cdot(1,\sigma)$, it follows that
$\sigma_{i}\cdot(\bm{a},\sigma)=(\sigma_{i}\cdot x_{1}^{n_{1}}\cdots
x_{\kappa}^{n_{\kappa}})\cdot(1,\sigma)$.

######  Claim 6.11.

We can express $\sigma_{i}\cdot x_{1}^{n_{1}}\cdots x_{\kappa}^{n_{\kappa}}$
as

$f(x_{1},\cdots,x_{\kappa})\sigma_{i}+g(x_{1},\cdots,x_{\kappa}),$

where $f,g\in\mathbb{Z}[\hbar,x_{1}^{\pm 1},\cdots,x_{\kappa}^{\pm 1}]$.

###### Proof.

This follows from repeated applications of the relations discussed above. ∎

The action of $\sigma_{i}$ on a general element $(\bm{a},\sigma)$ then follows
from the claim above and Definition 6.10.

Similarly, we can compute $y_{i}\cdot(\bm{a},\sigma)$ by using the relation
$y_{i}=\sigma_{i-1}y_{i-1}\sigma_{i-1}$ repeatedly to reduce to an expression
of transpositions and $y_{1}$.

######  Proposition 6.12.

The action defined above is the one given by

$\rho_{d}:BSk_{\kappa}(T^{2},\bm{q},\bm{q},\ast)\otimes
BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}\longrightarrow
BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d},$

where
$\rho_{d}([\gamma_{1}]_{c},[\gamma_{2}]_{c,d})=[\gamma_{1}\gamma_{2}]_{c,d}.$

###### Proof.

It suffices to verify (1), (2), and (3) in Definition 6.10.

(1) is immediate from the definitions.

(2) follows from Lemma 6.8.

(3) is easiest seen on the universal cover of $T^{2}$; see Figure 11 below. We
slide the endpoint of the strand going from $q_{1}$ to $\alpha_{\sigma(i)}$ in
$y_{1}\cdot(\bm{a},\sigma)$ down along $\alpha_{\sigma(i)}$. We can homotope
the strand until it looks like the right side of Figure 11 without creating
any crossings with other strands or the marked point. Thus the two braids are
the same element in $BSk_{\kappa}(T^{2},\bm{q},\bm{\alpha},\ast)_{d}$. Next,
we can pull the strand down across the marked point, picking up a factor of
$c^{2}$ for each marked point we cross in this direction. The resulting braid
is equivalent to
$c^{2n_{1}}(\tau_{\kappa})^{-1}\cdot(\bm{a}_{\tau_{\kappa}},\tau_{\kappa}\sigma)$.
This gives (3). ∎

Figure 11. The action of $y_{1}$ on $(a_{1},(13))$.

######  Example 6.13.

Suppose $\kappa=2$. We compute $(\sigma_{1}\cdot
y_{1})\cdot(a_{1}^{2}a_{2}^{-1},\sigma_{1})$. First of all,

$y_{1}\cdot(a_{1}^{2}a_{2}^{-1},\sigma_{1})=c^{4}\sigma_{1}^{-1}(a_{1}^{-1}a_{2}^{2},e).$

Then $\sigma_{1}\cdot
c^{4}\sigma_{1}^{-1}(a_{1}^{-1}a_{2}^{2},e)=c^{4}(a_{1}^{-1}a_{2}^{2},e)$.

In order to identify our enhanced polynomial representation with the standard
polynomial representation from Definition 6.4, we must first find a way to
eliminate the $S_{\kappa}$ factor of $PR_{\kappa}$. The standard polynomial
representation acts on $\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm
1},\cdots,X_{\kappa}^{\pm 1}]$, whereas the generators of $PR_{\kappa}$ have
permutations $\sigma\in S_{\kappa}$ associated to them.

We identify $a_{i}$ with $X_{i}$ and substitute $\hbar=s-s^{-1}$ to revert
back to the ring $\mathbb{Z}[s^{\pm 1},c^{\pm 1}]$. We take an average of the
permutations which defines a $\mathbb{Z}[[s]][c^{\pm 1}]$-linear map:

$\hypertarget{identificationmap}{S}:\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm
1},\cdots,X_{\kappa}^{\pm 1}]\longrightarrow PR_{\kappa}$ $X_{1}^{n_{1}}\cdots
X_{\kappa}^{n_{\kappa}}\mapsto\sum_{\sigma\in S_{\kappa}}(a_{1}^{n_{1}}\cdots
a_{\kappa}^{n_{\kappa}},\sigma).$

######  Remark 6.14.

The $\alpha_{i}$ are distinct and this is captured by our permutation term
$\sigma$ in a generator $(\bm{a},\sigma)$. The permutation-averaging can be
thought of as getting rid of this distinctness, resulting in a permutation-
free polynomial determined by $\bm{a}$.

The following theorem is a more precise version of Theorem 1.3.

######  Theorem 6.15.

After setting $d=s$, the standard polynomial representation agrees with the
enhanced polynomial representation composed with the permutation-averaging map
$S$.

More precisely, given an element $h\in\ddot{H}_{\kappa}$ and an element
$f\in\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}]$,

$S(p(h,f))=\rho_{d}(h,S(f)),$

where $p(h,f)$ is the action of the standard polynomial representation and
$\rho_{d}(h,S(f))$ is the action of the enhanced polynomial representation.

###### Proof.

It suffices to show that the equality holds for the generators as in
Definition 6.10:

(1) Let $x_{i}\in\ddot{H}_{\kappa}$ and
$f(X_{1},\cdots,X_{\kappa})\in\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm
1},\cdots,X_{\kappa}^{\pm 1}]$. Then

$\displaystyle S(p(x_{i},f))$ $\displaystyle=S(X_{i}f)=\sum_{\sigma\in
S_{\kappa}}(a_{i}f(a_{1},\dots,a_{\kappa}),\sigma)$
$\displaystyle=\rho_{d}\bigl{(}x_{i},\sum_{\sigma\in
S_{\kappa}}(f(a_{1},\dots,a_{\kappa}),\sigma)\bigr{)}=\rho_{d}(x_{i},S(f)).$

(2) Let $\sigma_{i}\in\ddot{H_{\kappa}}$ and consider
$1\in\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm 1},\cdots,X_{\kappa}^{\pm 1}]$.

Let $A_{\kappa}$ be the alternating group on $\kappa$ elements, i.e. the
subgroup of $S_{\kappa}$ consisting of even permutations. For a transposition
$\sigma_{i}$ and permutation $\sigma\in S_{\kappa}$, either $\sigma\in
A_{\kappa}$ or $\sigma_{i}\sigma\in A_{\kappa}$, so

$\sum_{\sigma\in S_{\kappa}}(1,\sigma)=\sum_{\rho\in
A_{\kappa}}((1,\rho)+(1,\sigma_{i}\rho)).$

Then, using Corollary 6.9,

$\displaystyle S(p(\sigma_{i},1))$ $\displaystyle=S(s)=\sum_{\sigma\in
S_{\kappa}}s(1,\sigma)=\sum_{\rho\in
A_{\kappa}}s((1,\rho)+(1,\sigma_{i}\rho))$ $\displaystyle=\sum_{\rho\in
A_{\kappa}}\rho_{d}\bigl{(}\sigma_{i},(1,\rho)+(1,\sigma_{i}\rho)\bigr{)}=\rho_{d}\bigl{(}\sigma_{i},\sum_{\sigma\in
S_{\kappa}}(1,\sigma)\bigr{)}$ $\displaystyle=\rho_{d}(\sigma_{i},S(1)).$

(3) Let $y_{1}\in\ddot{H}_{\kappa}$ and
$f(X_{1},\cdots,X_{\kappa})=X_{1}^{n_{1}}\cdots
X_{\kappa}^{n_{\kappa}}\in\mathbb{Z}[[s]][c^{\pm 1}][X_{1}^{\pm
1},\cdots,X_{\kappa}^{\pm 1}]$.

Then $\omega(X_{1}^{n_{1}}\cdots
X_{\kappa}^{n_{\kappa}})=c^{2n_{1}}X_{\kappa}^{n_{1}}X_{1}^{n_{2}}\cdots
X_{\kappa-1}^{n_{\kappa}}$.

Taking advantage of the first two parts of the proof, we see that

$\displaystyle S(p(y_{1},f))$
$\displaystyle=S(\tau_{\kappa}^{-1}\omega(f))=S(\tau_{\kappa}^{-1}c^{2n_{1}}X_{\kappa}^{n_{1}}X_{1}^{n_{2}}\cdots
X_{\kappa-1}^{n_{\kappa}})$
$\displaystyle=c^{2n_{1}}\tau_{\kappa}^{-1}S(X_{\kappa}^{n_{1}}X_{1}^{n_{2}}\cdots
X_{\kappa-1}^{n_{\kappa}})=c^{2n_{1}}\tau_{\kappa}^{-1}\sum_{\sigma\in
S_{\kappa}}(a_{\kappa}^{n_{1}}a_{1}^{n_{2}}\cdots
a_{\kappa-1}^{n_{\kappa}},\sigma)$
$\displaystyle=c^{2n_{1}}\tau_{\kappa}^{-1}\sum_{\sigma\in
S_{\kappa}}(a_{\kappa}^{n_{1}}a_{1}^{n_{2}}\cdots
a_{\kappa-1}^{n_{\kappa}},\tau_{\kappa}\sigma)=y_{1}\cdot\sum_{\sigma\in
S_{\kappa}}(f(a_{1},\dots,a_{\kappa}),\sigma)$
$\displaystyle=\rho_{d}(y_{1},S(f)).$

∎

Theorem 6.15 can be restated as the following corollary:

######  Corollary 6.16.

Let $W\subset PR_{\kappa}$ be the submodule generated over
$\mathbb{Z}[[s]][c^{\pm 1}]$ by elements of the form $\sum_{\sigma\in
S_{\kappa}}(\bm{a},\sigma)$. Then the enhanced polynomial representation has a
subrepresentation over $W$ which is isomorphic to the polynomial
representation of the double affine Hecke algebra.

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