$\mathfrak{cont}:{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{{\mathbb{Z}}[{\bf
i}]})\to{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{{\mathbb{Z}}[{\bf
i}]})$

which is canonical up to chain homotopy. Hence there is a
${\mathbb{Z}}_{2}$-graded $\Lambda_{{\mathbb{Z}}[{\bf i}]}$-module ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf i}]})$, called the
${\mathfrak{b}}$-deformed vortex Floer homology, with canonical isomorphisms

${\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{{\mathbb{Z}}[{\bf
i}]})\cong{\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf
i}]})$

for all regular admissible bulk-avoiding pairs $(\widehat{H},\widehat{J})$.

  3. (3)

There is a linear isomorphism

${\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf i}]})\cong
H_{\bullet}(X;\Lambda_{{\mathbb{Z}}[{\bf i}]}).$

####  4.2.3. Poincaré duality

In Morse–Floer theory one can define the Poincaré duality on the chain-level
by “reversing” the Morse function or the symplectic action functional. We
recall this construction in the setting of vortex Floer theory. If
$\widehat{H}:S^{1}\times V\to{\mathbb{R}}$ is a $K$-invariant Hamiltonian,
define $\widehat{H}^{\rm op}:S^{1}\times V\to{\mathbb{R}}$ by

$\widehat{H}^{\rm op}(t,v)=-\widehat{H}(-t,v).$

Then similar to the case of the ordinary Floer homology (see [MS04, Section
12.3]), there is a one-to-one correspondence between ${\rm
crit}{\mathcal{A}}_{H}$ and ${\rm crit}{\mathcal{A}}_{H^{\rm op}}$. More
precisely, if ${\mathfrak{x}}=(\widehat{x},\eta)\in L^{K}(V)$ is an
equivariant 1-periodic orbit, then

${\mathfrak{x}}^{\rm op}:=(\widehat{x}^{\rm op},\eta^{\rm op})\ {\rm where}\
\widehat{x}^{\rm op}(t)=\widehat{x}(-t),\ \eta^{\rm op}(t)=-\eta(-t)$

solves

$\frac{d}{dt}\widehat{x}^{\rm op}(t)+{\mathcal{X}}_{\eta^{\rm
op}(t)}(\widehat{x}^{\rm op}(t))-X_{\widehat{H}^{\rm op}}(\widehat{x}^{\rm
op}(t))=0$

and hence is an equivariant 1-periodic orbits for $H^{\rm op}$. The map
${\mathfrak{x}}\mapsto{\mathfrak{x}}^{\rm op}$ induces a one-to-one
correspondence

${\rm crit}{\mathcal{A}}_{H}\cong{\rm crit}{\mathcal{A}}_{H^{\rm op}}$

with critical values and Conley–Zehnder indices reversed.

Similarly, if $\widehat{J}_{t}$ is an $S^{1}$-family of $K$-invariant almost
complex structures on $V$, then define

$(\widehat{J}^{\rm op})_{t}=\widehat{J}_{-t}.$

One can verify easily that if $(\widehat{H},\widehat{J})$ is admissible, so is
$(\widehat{H}^{\rm op},\widehat{J}^{\rm op})$.

Now we define a Poincaré pairing on the vortex Floer homology. Let
$(\widehat{H}_{1},\widehat{J}_{1})$ and $(\widehat{H}_{2},\widehat{J}_{2})$ be
two regular bulk-avoiding admissible pairs on $V$. Consider the genus zero
curve with two incoming cylindrical ends, denoted by $\Sigma_{\supset}$.
Choose an area form with cylindrical ends on $\Sigma_{\supset}$. Define a
$K$-invariant Hamiltonian perturbation $\widehat{H}_{\supset}$ on
$\Sigma_{\supset}$ which is equal to $\widehat{H}_{1}dt$ on the first
cylindrical end and which is equal to $\widehat{H}_{2}^{\rm op}dt$ on the
second cylindrical end. Choose a domain-dependent $K$-invariant almost complex
structure $\widehat{J}_{\supset}$ which agrees on $\widehat{J}_{1}$ on the
first cylindrical end and which is equal to $\widehat{J}_{2}^{\rm op}$ on the
second cylindrical end. Consider the $\widehat{H}_{\supset}$-perturbed
symplectic vortex equation on $\Sigma_{\supset}$ with respect to the family of
almost complex structures $\widehat{J}_{\supset}$. Finite energy solutions
converge to critical points of ${\mathcal{A}}_{H_{1}}$ resp.
${\mathcal{A}}_{H_{2}^{\rm op}}$ at the two cylindrical ends. Then given
${\mathfrak{x}}\in{\rm crit}{\mathcal{A}}_{H_{1}}$ and ${\mathfrak{y}}^{\rm
op}\in{\rm crit}{\mathcal{A}}_{H_{2}^{\rm op}}\cong{\rm
crit}{\mathcal{A}}_{H_{2}}$, one can obtain a well-defined count

${\mathfrak{n}}_{\supset}^{\mathfrak{b}}({\mathfrak{x}},{\mathfrak{y}})\in{\mathbb{Z}}$

by looking at rigid solutions. Define a bilinear pairing

$\langle\cdot,\cdot\rangle^{\mathfrak{b}}:{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{R}^{\Gamma})\otimes{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2}^{\rm op},\widehat{J}_{2}^{\rm
op};\Lambda_{R}^{\Gamma})\to R$

by

$\langle\sum_{i=1}^{\infty}a_{i}{\mathfrak{x}}_{i},\sum_{j=1}^{\infty}b_{j}{\mathfrak{y}}_{j}^{\rm
op}\rangle^{\mathfrak{b}}:=\sum_{i,j}a_{i}b_{j}{\mathfrak{n}}_{\supset}({\mathfrak{x}}_{i},{\mathfrak{y}}_{j}^{\rm
op}).$

An argument via energy inequality shows that the above form is finite and
well-defined; by considering 1-dimensional moduli spaces one can show that the
above pairing descends to homology

$\langle\cdot,\cdot\rangle^{\mathfrak{b}}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{R}^{\Gamma})\otimes{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{R}^{\Gamma})\to
R.$

One can also show that the pairing is compatible with respect to the
continuation map. Hence it induces a pairing

$\langle\cdot,\cdot\rangle^{\mathfrak{b}}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})\otimes{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})\to R.$

Now we specialize to the case when
$\widehat{H}_{1}=\widehat{H}_{2}=\widehat{H}$ and
$\widehat{J}_{1}=\widehat{J}_{2}=\widehat{J}$. In this case the pairing takes
a simple form on the chain level. Indeed, if we choose $\widehat{H}_{\supset}$
and $\widehat{J}_{\supset}$ to be the trivial ones, then the countings
$n_{\supset}^{\mathfrak{b}}({\mathfrak{x}},{\mathfrak{y}}^{\rm op})$ is 1 if
${\mathfrak{x}}={\mathfrak{y}}$ and zero otherwise. Then if

$\displaystyle\ \alpha=\sum a_{i}{\mathfrak{x}}_{i}\in{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{R}^{\Gamma}),\
$ $\displaystyle\ \beta=\sum b_{j}{\mathfrak{x}}_{j}^{\rm op}\in{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{\rm op},\widehat{J}^{\rm
op};\Lambda_{R}^{\Gamma})$

one has

$\langle\alpha,\beta\rangle^{\mathfrak{b}}=\sum_{i}a_{i}b_{i}\in R.$

This sum is finite as ${\mathcal{A}}_{H}({\mathfrak{x}}_{i})\to-\infty$ and
${\mathcal{A}}_{H^{\rm op}}({\mathfrak{x}}_{j}^{\rm
op})=-{\mathcal{A}}_{H}({\mathfrak{x}}_{j})\to-\infty$.

####  4.2.4. Pair-of-pants products

A TQFT type construction allows us to define a multiplicative structure on the
vortex Floer homology. In particular, using any volume form on the pair-of-
pants with cylindrical ends, one can define the pair-of-pants product

$\ast_{\mathfrak{b}}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})\otimes{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})\to{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})[n]$

which is associative. Here $2n={\rm dim}X$. The details were given in [WX17b].

There is also an identity element in the vortex Floer homology. Fix a regular
bulk-avoiding admissible pair $(\widehat{H},\widehat{J})$. Consider a once-
punctured sphere $\Sigma_{\rm cigar}$ which is biholomorphic to the complex
plane. View the puncture as an output. Equip $\Sigma_{\rm cigar}$ with a
cylindrical volume form $\nu_{\rm cigar}$ so that one has has the isometric
identification

${\mathbb{C}}\setminus B_{1}\cong[0,+\infty)\times S^{1}.$

Turn on the Hamiltonian perturbation on this cylindrical end, meaning that one
has a Hamiltonian perturbation

${\mathcal{H}}\in\Omega^{1}(\Sigma_{\rm cigar},C^{\infty}_{c}(V)^{K})\ {\rm
s.t.}\ {\mathcal{H}}|_{[S,+\infty)\times S^{1}}=H_{t}dt\ {\rm for}\ S\gg 0.$

Choose a domain-dependent $K$-invariant $\omega_{V}$-compatible almost complex
structure ${\mathcal{J}}$ parametrized by $z\in\Sigma_{\rm cigar}$ such that
over the cylindrical end it agrees with $J_{t}$. Consider the Hamiltonian
perturbed symplectic vortex equation

$\displaystyle\ \overline{\partial}_{A,{\mathcal{H}}}u=0,\ $ $\displaystyle\
F_{A}+\mu(u)\nu^{\rm cigar}=0.$

Each finite energy solution ${\mathfrak{u}}=(A,u)$ converges to an equivariant
1-periodic orbit and hence represents an element ${\mathfrak{x}}\in{\rm
crit}{\mathcal{A}}_{H}$. Hence for each ${\mathfrak{x}}$ there is a moduli
space

${\mathcal{M}}^{\rm cigar}({\mathfrak{x}}).$

Elements in this moduli space have a uniform energy bound by
$-{\mathcal{A}}_{H}({\mathfrak{x}})+C$ where $C$ depends on the perturbation
data on the cigar which is uniformly bounded. The virtual dimension is $n-{\rm
CZ}({\mathfrak{x}})$. Counting elements (with signs) of index zero moduli
spaces ${\mathcal{M}}^{\rm cigar}({\mathfrak{x}})$ defines an element

${\bm{1}}_{{\mathfrak{b}},\widehat{H}}^{{\rm
GLSM}}=\sum_{\mathfrak{x}}{\mathfrak{n}}_{\rm
cigar}^{\mathfrak{b}}({\mathfrak{x}}){\mathfrak{x}}\in{\it
VCF}_{n}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{R}^{\Gamma}).$

Standard TQFT argument shows that ${\bm{1}}_{{\mathfrak{b}},\widehat{H}}^{{\rm
GLSM}}$ is closed, induces a well-defined element in ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})$, and is the
multiplicative identity of ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})$. Denote this element by

${\bm{1}}_{\mathfrak{b}}^{{\rm GLSM}}\in{\it
VHF}_{n}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma}).$ (4.15)

######  Lemma 4.13.

The element ${\bm{1}}_{\mathfrak{b}}^{\rm GLSM}$ is nonzero.

###### Proof.

In the undeformed case this was proved using the closed-open map in [WX17b]
and the fact that some Lagrangian Floer theory is nontrivial. Here as we know
that the algebra ${\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}^{\Gamma})$
is nonzero (see Lemma 4.8) for any ring $R$, one must have
${\bm{1}}_{\mathfrak{b}}^{\rm GLSM}\neq 0$. ∎

######  Lemma 4.14.

One has

$\langle\alpha,\beta\rangle^{\mathfrak{b}}\neq
0\Longrightarrow\langle\alpha\ast_{\mathfrak{b}}\beta,{\bm{1}}_{\mathfrak{b}}^{\rm
GLSM}\rangle^{\mathfrak{b}}\neq 0.$

###### Proof.

This theorem follows from the standard TQFT and cobordism argument. See Figure
1. The details are left to the reader.

Figure 1.

∎

Before we end this part, we state a major step towards our proof of the
Hofer–Zehnder conjecture.

######  Theorem E.

There exists a bulk-deformation ${\mathfrak{b}}$ of the form

${\mathfrak{b}}=\sum_{j=1}^{N}\log c_{j}V_{j}$

with $c_{j}\in{\mathbb{Z}}[{\bf i}]$ such that the algebra ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{Q}}})$ is
semisimple in the sense of Definition 3.6.

The proof occupies Section 8 and Section 9, using the closed-open string map
in the vortex setting.

###  4.3. Bulk-deformed spectral invariants, persistence modules, and barcodes

We fit the bulk-deformed vortex Floer theory into the abstract packages
developed by Usher etc. Let ${\mathfrak{b}}$ be a bulk-deformation of the form
(4.13).

######  Proposition 4.15.

Given a regular bulk-avoiding pair $(\widehat{H},\widehat{J})$, the quadruple

${\mathfrak{c}}^{\mathfrak{b}}(\widehat{H},\widehat{J}):=(P_{H},{\mathcal{A}}_{H},{\rm
CZ}_{(2)},n^{\mathfrak{b}})$

is a ${\mathbb{Z}}_{2}$-graded Floer–Novikov package over $R$ (see Definition
3.14).

###### Proof.

Straightforward. ∎

Next we consider the quantitative dependence of the vortex Floer chain complex
on the Hamiltonian. We restrict to the case where $R={\mathbb{K}}$ is a field.
The vortex Floer chain complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}^{\Gamma})$
is the associated Floer–Novikov complex.

######  Proposition 4.16.

Given two regular bulk-avoiding pairs $(\widehat{H}_{1},\widehat{J}_{1})$ and
$(\widehat{H}_{2},\widehat{J}_{2})$, the quasi-equivalence distance (see
Definition 3.19) between ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{\mathbb{K}}^{\Gamma})$
and ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{\mathbb{K}}^{\Gamma})$
is no greater than the Hofer distance between the induced Hamiltonians
$H_{1},H_{2}$ downstairs, i.e.

$d_{Q}\Big{(}{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{\mathbb{K}}^{\Gamma}),{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{\mathbb{K}}^{\Gamma})\Big{)}\leq\max\Big{\\{}\int_{0}^{1}\max_{X}(H_{2}-H_{1})dt,\
\int_{0}^{1}\max_{X}(H_{1}-H_{2})dt\Big{\\}}.$

###### Proof.

This follows from the quantitative analysis of the continuation map. As the
bulk ${\mathfrak{b}}$ and the coefficient field are fixed, we drop it from
notations. To show that the complex only depends on the induced Hamiltonian
downstairs (measured by quasiequivalence distance), we need to introduce the
parameter $\lambda$ (see (4.8)). For each regular bulk-avoiding triple
$(\lambda,\widehat{H},\widehat{J})$, there is a Floer–Novikov package
${\mathfrak{c}}^{\lambda}(\widehat{H},\widehat{J})$ defined from
(${\mathfrak{b}}$-deformed) counts of solutions to (4.8). Denote the
associated Floer–Novikov complex by ${\it
VCF}_{\bullet}^{\lambda}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}^{\Gamma})$
with valuation denoted by $\ell^{\lambda}$.

######  Lemma 4.17.

The quasi-equivalence distance between ${\it
VCF}_{\bullet}^{\lambda_{1}}(\widehat{H}_{1},\widehat{J}_{1})$ and ${\it
VCF}_{\bullet}^{\lambda_{2}}(\widehat{H}_{2},\widehat{J}_{2})$ is bounded by

$\widehat{d}_{\rm
Hofer}(\widehat{H}_{1},\widehat{H}_{2}):=\max\left\\{\int_{0}^{1}\max_{V}(\widehat{H}_{2}-\widehat{H}_{1})dt,\
\int_{0}^{1}\max_{V}(\widehat{H}_{1}-\widehat{H}_{2})\right\\}.$

###### Proof.

Indeed, this follows from the energy calculation for the continuation maps
(see Proposition 4.7). One can construct chain homotopy equivalences $\Phi$,
$\Psi$ between these two complexes and maps $K_{1}$, $K_{2}$ as in the diagram

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0.0pt\hbox{{}{}{{}}{{}{}{}}{}}}}\ignorespaces{}\ignorespaces{}{}{}{{}{}}\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\raise
0.0pt\hbox{{}{}{}{{}}{{}{}{}\lx@xy@spline@}{}}}}\ignorespaces{}\ignorespaces\hbox{\hbox{\kern
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The first item of Definition 3.19 follows directly from (4.12). Using the same
method, the second item of Definition 3.19 can be verified for the maps
$K_{1}$, $K_{2}$. ∎

We fix the two regular bulk-avoiding pairs
$(\widehat{H}_{\pm},\widehat{J}_{\pm})$. For each $\epsilon>0$, one can find a
$K$-invariant cut-off function $\rho_{\epsilon}:V\to[0,1]$ supported near
$\mu^{-1}(0)$ such that if we define
$\widehat{H}_{\pm}^{\epsilon}:=\rho_{\epsilon}\widehat{H}_{\pm}$, then

$\widehat{d}_{\rm
Hofer}(\widehat{H}_{-}^{\epsilon},\widehat{H}_{+}^{\epsilon})\leq d_{\rm
Hofer}(H_{-},H_{+})+\epsilon.$

Hence in view of Lemma 4.17 above, we only need to prove the following.

######  Lemma 4.18.

Suppose $(\widehat{H}_{\pm},\widehat{J}_{\pm})$ are two regular bulk-avoiding
pairs such that $\widehat{H}_{+}$ and $\widehat{H}_{-}$ descend to the same
Hamiltonian $H$ downstairs. Then

$d_{Q}({\it VCF}_{\bullet}(\widehat{H}_{-},\widehat{J}_{-}),{\it
VCF}_{\bullet}(\widehat{H}_{+},\widehat{J}_{+}))=0.$

###### Proof.

We prove that the quasi-equivalence distance is less than $\epsilon$ for all
$\epsilon>0$. Notice that the potential failure of this assertion comes from
the difference between $\widehat{H}_{-}$ and $\widehat{H}_{+}$ which is a
priori large outside $\mu^{-1}(0)$. We use the adiabatic limit argument to
push solutions contributing to the continuation maps near the level set
$\mu^{-1}(0)$.

Choose a sequence $\lambda_{i}\to\infty$. For each $\lambda_{i}$, one can
choose a $\lambda_{i}$-admissible lift $\widehat{H}_{\pm}^{\lambda_{i}}$ of
$H$. As the admissible condition is only about the infinitesimal behaviors of
the lifts $\widehat{H}_{\pm}^{\lambda_{i}}$ near lifts of 1-periodic orbits of
$H$, we may require that

$\|\widehat{H}_{\pm}^{\lambda_{i}}-\widehat{H}_{\pm}\|_{C^{0}}\leq\epsilon.$

Hence by Lemma 4.17, one only needs to consider the quasi-equivalence

$d_{Q}\left({\it
VCF}_{\bullet}^{\lambda_{i}}(\widehat{H}_{-}^{\lambda_{i}},\widehat{J}_{-}^{\lambda_{i}}),{\it
VCF}_{\bullet}^{\lambda_{i}}(\widehat{H}_{+}^{\lambda_{i}},\widehat{J}_{+}^{\lambda_{i}})\right).$

We claim that the above sequence (in $i$) converges to zero.

We set up the moduli spaces for the continuation maps. Choose a cut-off
function $\chi:{\mathbb{R}}\to[0,1]$ which is non-decreasing, equals zero on
$(-\infty,0]$, and equals $1$ on $[1,+\infty]$. Consider the equation with

$\widehat{H}_{s,t}^{\lambda_{i}}=(1-\chi(s))\widehat{H}_{-}^{\lambda_{i}}+\chi(s)\widehat{H}_{+}^{\lambda_{i}}.$

We claim that, for all $\epsilon>0$, there exists $i_{\epsilon}>0$ such that
when $i\geq i_{\epsilon}$, for all finite energy solutions to (4.9), if the
limit at $\pm\infty$ is ${\mathfrak{x}}_{\pm}$, then one has

${\mathcal{A}}_{H}({\mathfrak{x}}_{+,i})-{\mathcal{A}}_{H}({\mathfrak{x}}_{-,i})\leq\epsilon.$

This would establish item (1) of Definition 3.19.

Suppose on the contrary that this is not true. Then there exist $\delta>0$, a
subsequence (still indexed by $i$), a sequence of solutions
${\mathfrak{u}}_{i}=(u_{i},\xi_{i},\eta_{i})$ to the equation connecting
${\mathfrak{x}}_{-,i}$ and ${\mathfrak{x}}_{+,i}$ such that

${\mathcal{A}}_{H}({\mathfrak{x}}_{+,i})-{\mathcal{A}}_{H}({\mathfrak{x}}_{-,i})\geq\delta>0.$

By the energy identity (4.11), one has a uniform bound which is independent of
$\lambda_{i}$:

$E_{\lambda_{i}}({\mathfrak{u}}_{i})={\mathcal{A}}_{H}({\mathfrak{x}}_{-,i})-{\mathcal{A}}_{H}({\mathfrak{x}}_{+,i})-\int_{[0,1]\times
S^{1}}\partial_{s}\widehat{H}_{s,t}^{\lambda_{i}}(u)dsdt\leq C.$

Now one can apply the adiabatic limit argument. Notice that although we cannot
guarantee the convergence of $\widehat{H}_{s,t}^{\lambda_{i}}$, but we may
require that $\widehat{J}_{s,t}^{\lambda_{i}}$ converges in sufficiently high
order to a fixed almost complex structure $\widehat{J}$ outside a compact
subset of $V$. In the $\lambda_{i}\to\infty$ limit, a priori there are three
types of bubbles (see [GS05, Section 11]): holomorphic spheres in $V$,
holomorphic spheres in $X$, and affine vortices, which are solutions to the
vortex equation over ${\mathbb{C}}$ (without Hamiltonian term). The three kind
of bubbles can be classified by the rate of energy concentration compared to
the rate of the divergence $\lambda_{i}\to\infty$. As there is a lower bound
on the energy of these bubbles, the uniform bound on energy implies that,
after passing to a subsequence (still indexed by $i$), except near a finite
subset $Z\subset[0,1]\times S^{1}=:Q$ at which bubbling could occur, the
energy density

$|\partial_{s}u_{i}+{\mathcal{X}}_{\xi_{i}}(u_{i})|^{2}+\lambda_{i}^{2}|\mu(u_{i})|^{2}$

stays bounded. In particular, the map $u_{i}|_{Q}$ stays arbitrarily close to
$\mu^{-1}(0)$ except near $Z$ as $i\to\infty$. More precisely, for any $r>0$,
there exists $i_{r}>0$ such that for all $i\geq i_{r}$,

$\sup_{z\in[0,1]\times S^{1}\setminus B_{r}(Z)}|\mu(u_{i}(z))|\leq r.$ (4.16)

Then one has

${\mathcal{A}}_{H}({\mathfrak{x}}_{+,i})-{\mathcal{A}}_{H}({\mathfrak{x}}_{-,i})\leq\int_{Q}|\partial_{s}\widehat{H}_{s,t}^{\lambda_{i}}(u_{i})|dsdt\\\
\leq\int_{Q\setminus
B_{r}(Z)}|\partial_{s}\widehat{H}_{s,t}^{\lambda_{i}}(u_{i})|dsdt+\int_{B_{r}(Z)}|\partial_{s}\widehat{H}_{s,t}^{\lambda_{i}}(u_{i})|dsdt.$

As $\widehat{H}_{-}^{\lambda_{i}}=\widehat{H}_{+}^{\lambda_{i}}$ on
$\mu^{-1}(0)$, the first item is bounded by $Cr$; the second term is bounded
by $C{\rm Area}(B_{r}(Z))$ which can be arbitrarily small. This contradicts
the assumption that
${\mathcal{A}}_{H}({\mathfrak{x}}_{+,i})-{\mathcal{A}}_{H}({\mathfrak{x}}_{-,i})\geq\delta>0$.

Therefore, we established item (1) of Definition 3.19. The case of item (2) is
similar and hence omitted. ∎

Now the proof of Proposition 4.16 is complete. ∎

####  4.3.1. Spectral invariants

Spectral numbers of Hamiltonian diffeomorphisms were introduced by Oh [Oh99],
Schwarz [Sch00] and enhanced by Entov–Polterovich [EP03, EP06, EP09]. In
[WX17b] Wu and the second author constructed the analogue in the vortex Floer
theory.

By Theorem 3.16 and Proposition 4.15, one can define the spectral numbers

$\rho^{\mathfrak{b}}(\alpha;\widehat{H},\widehat{J}):=\rho_{{\mathfrak{c}}^{\mathfrak{b}}(\widehat{H},\widehat{J})}(\alpha)\in{\mathbb{R}}\cup\\{-\infty\\},\
\forall\alpha\in{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf i}]}^{\Gamma}).$

One can establish the following properties of these spectral numbers, which
were proved in [WX17b] in the undeformed (${\mathfrak{b}}=0$) case.

######  Theorem 4.19.

(cf. [WX17b, Proposition 3.6]) The spectral numbers
$\rho^{\mathfrak{b}}(\alpha;\widehat{H},\widehat{J})$ have the following
properties.

  1. (1)

(Independence of lifting and almost complex structure)The number
$\rho^{\mathfrak{b}}(\alpha;\widehat{H},\widehat{J})$ only depends on the
induced Hamiltonian $H$ downstairs. Denote this number by

$c^{\mathfrak{b}}(\alpha,H)\in{\mathbb{R}}.$

  2. (2)

(Homogeneity) Given $\alpha\in{\it VHF}(V;\Lambda_{{\mathbb{Z}}[{\bf
i}]}^{\Gamma})$ and $\lambda\in\Lambda_{{\mathbb{Z}}[{\bf i}]}^{\Gamma}$, for
any $H$, one has

$c^{\mathfrak{b}}(\lambda\alpha,H)=c^{\mathfrak{b}}(\alpha,H)-{\mathfrak{v}}(\lambda).$

One uses this formula to extend the spectral numbers to classes in

${\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf i}]})={\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{{\mathbb{Z}}[{\bf
i}]}^{\Gamma})\otimes_{\Lambda_{{\mathbb{Z}}[{\bf
i}]}^{\Gamma}}\Lambda_{{\mathbb{Z}}[{\bf i}]}.$

  3. (3)

(Lipschitz continuity) Given any two nondegenerate Hamiltonians $H_{1},H_{2}$
downstairs, one has

$\int_{S^{1}}\min_{X}(H_{1}-H_{2})dt\leq
c^{\mathfrak{b}}(\alpha,H_{1})-c^{\mathfrak{b}}(\alpha,H_{2})\leq\int_{S^{1}}\max_{X}(H_{1}-H_{2})dt.$

This implies that $c^{\mathfrak{b}}(\alpha,H)$ is defined for all
Hamiltonians.

  4. (4)

(Invariance) $c^{\mathfrak{b}}(\alpha,H)$ only depends on the homotopy class
of the Hamiltonian path $\tilde{\phi}_{H}$ on $X$. Let ${\rm Ham}(X)$ be the
group of Hamiltonian diffeomorphisms on $X$ and let ${\rm H}{\widetilde{\rm
am}}(X)\to{\rm Ham}(X)$ be the covering of homotopy classes of Hamiltonian
isotopies on $X$. Then we can define

$c^{\mathfrak{b}}(\alpha,\tilde{\phi})\in{\mathbb{R}}\cup\\{-\infty\\}\
\forall\alpha\in{\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R}),\
\tilde{\phi}\in{\rm H}{\widetilde{\rm am}}(X).$

  5. (5)

(Triangle inequality) For any $\alpha_{1},\alpha_{2}\in{\it
VHF}(V;\Lambda_{R})$ and $\tilde{\phi}_{1},\tilde{\phi}_{2}\in{\rm
H}{\widetilde{\rm am}}(X)$ one has

$c^{\mathfrak{b}}(\alpha_{1}\ast\alpha_{2},\tilde{\phi}_{1}\tilde{\phi}_{2})\leq
c^{\mathfrak{b}}(\alpha_{1},\tilde{\phi}_{1})+c^{\mathfrak{b}}(\alpha_{2},\tilde{\phi}_{2}).$

######  Definition 4.20.

The valuation of a class $\alpha\in{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{R})$ is defined to be

${\mathcal{A}}^{\mathfrak{b}}(\alpha):=c^{\mathfrak{b}}(\alpha,\tilde{\rm
Id})\in{\mathbb{R}}\cup\\{+\infty\\}.$

####  4.3.2. Poincaré duality

One useful property of the spectral numbers is related to the Poincaré duality
map.

######  Proposition 4.21.

Let ${\mathbb{K}}$ be a field.

  1. (1)

For any $\alpha\in{\it VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\mathbb{K}})$
and $\tilde{\phi}\in{\rm H}{\widetilde{\rm am}}(X)$, there holds

$c^{\mathfrak{b}}(\alpha,\tilde{\phi})=-\inf\Big{\\{}c^{\mathfrak{b}}(\beta,\tilde{\phi}^{-1})\
|\ \langle\alpha,\beta\rangle^{\mathfrak{b}}\neq 0\Big{\\}}.$

  2. (2)

If $\langle\alpha,\beta\rangle^{\mathfrak{b}}\neq 0$, then

${\mathcal{A}}^{\mathfrak{b}}(\alpha)+{\mathcal{A}}^{\mathfrak{b}}(\beta)\geq
0.$

###### Proof.

Notice that one only needs to prove this proposition for coefficient field
being $\Lambda_{\mathbb{K}}^{\Gamma}$. In the case of ordinary Hamiltonian
Floer theory, the proof of (1) uses the PSS map and the correspondence between
the pairing $\langle\cdot,\cdot\rangle$ and the intersection pairing on the
singular homology of the manifold (see [EP03][Ost06][FOOO19]). It was pointed
in [Ush10] that (1) holds for abstract filtered Floer–Novikov complexes. As
the complex ${\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}^{\Gamma})$
is an abstract filtered Floer–Novikov complex over
$\Lambda_{\mathbb{K}}^{\Gamma}$ (see Proposition 4.15), (1) follows. For (2),
take $\tilde{\phi}={\rm Id}$. Then

${\mathcal{A}}^{\mathfrak{b}}(\alpha)=c^{\mathfrak{b}}(\alpha,{\rm
Id})=-\inf\Big{\\{}{\mathcal{A}}^{\mathfrak{b}}(\beta)\ |\
\langle\alpha,\beta\rangle^{\mathfrak{b}}\neq 0\Big{\\}}.$

Hence if $\langle\alpha,\beta\rangle^{\mathfrak{b}}\neq 0$,
${\mathcal{A}}^{\mathfrak{b}}(\beta)\geq-{\mathcal{A}}^{\mathfrak{b}}(\alpha)$.
∎

####  4.3.3. Persistence modules and barcodes

Recall that (see Subsection 3.4) to any filtered Floer–Novikov complex
$CF_{\bullet}({\mathfrak{c}})$ over the Novikov field
$\Lambda_{\mathbb{K}}^{\Gamma}$ one can associate a persistence module
${\bm{V}}({\mathfrak{c}})$. In particular, for each regular bulk-avoiding
admissible pair $(\widehat{H},\widehat{J})$, the bulk-deformed vortex Floer
complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}^{\Gamma})$
gives a persistence module, denoted by

${\bm{V}}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}^{\Gamma}).$

We omit the dependence on the bulk deformation ${\mathfrak{b}}$ most of the
time. One can check easily that we can extend the coefficient field to the
universal Novikov field $\Lambda_{\mathbb{K}}$, obtaining a persistence module
${\bm{V}}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}})$ with

$V^{s}(\widehat{H},\widehat{J};\Lambda_{\mathbb{K}}):=HF_{\bullet}^{\leq
s}(\widehat{H},\widehat{J};\Lambda_{0,{\mathbb{K}}}^{\Gamma})\otimes_{\Lambda_{0,{\mathbb{K}}}^{\Gamma}}\Lambda_{0,{\mathbb{K}}}.$

When the ground field ${\mathbb{K}}$ is clear from the context, we often
abbreviate this persistence module by ${\bm{V}}(\widehat{H},\widehat{J})$. One
can prove, using the continuation map, that up to isomorphism, this
persistence module is independent of the choice of the almost complex
structure $\widehat{J}$. Hence denote the persistence module by
${\bm{V}}(\widehat{H})$. One can also use the same idea of proving Proposition
4.16 that, for different lifts $\widehat{H}_{1},\widehat{H}_{2}$ of the same
Hamiltonian $H$ downstairs, the interleaving distance between
${\bm{V}}(\widehat{H}_{1})$ and ${\bm{V}}(\widehat{H}_{2})$ is zero. By
identifying persistence modules with zero interleaving distance, the
persistence module only depends on the Hamiltonian path $\tilde{\phi}\in{\rm
H}{\widetilde{\rm am}}(X)$ generated by $H$. Hence we loosely denote the
object by ${\bm{V}}(\tilde{\phi})$.

Recall also that to any Floer–Novikov complex one can associate a barcode (and
hence a reduced barcode). The reduced barcode corresponding to a regular bulk-
avoiding admissible pair $(\widehat{H},\widehat{J})$ is denoted by
${\mathcal{B}}(\widehat{H},\widehat{J})$. One can prove that (similar to the
case of ordinary Floer barcodes, see [Ush13, Proposition 5.3]) the reduced
barcode only depends on the time-1 map $\phi=\phi_{H}$ on the toric manifold
$X$. Hence we also denote it by ${\mathcal{B}}(\phi)$.

##  5\. Local Floer theory

To extend the Hofer–Zehnder conjecture to degenerate Hamiltonian
diffeomorphisms, one needs to have a good notion of counts of fixed points.
Following [She22], we will use the rank of a local version of the vortex Floer
homology (with bulk deformation), which is ultimately isomorphic to the local
Floer homology in the classical sense, to define a homological count of fixed
points. This section can be skipped at first reading, especially if the reader
is mainly interested in the nondegenerate case. The following statements will
be proved in this section.

######  Theorem 5.1.

Let ${\mathbb{K}}$ be a field. Let $\phi:X\to X$ be a Hamiltonian
diffeomorphism and $p\in X$ be an isolated fixed point. Then there is a
${\mathbb{Z}}_{2}$-graded ${\mathbb{K}}$-vector space

${\it VHF}^{\rm loc}(\phi,p;{\mathbb{K}})$

satisfying the following properties.

  1. (1)

If $p$ is a nondegenerate fixed point, then ${\it VHF}^{\rm
loc}(\phi,p;{\mathbb{K}})$ has rank $1$ graded by the Conley–Zehnder index of
$p$ (modulo $2$).

  2. (2)

If $\phi^{s}$, $s\in[0,1]$ is a smooth family of Hamiltonian diffeomorphisms
such that $p$ is a uniformly isolated fixed point, i.e., there exists an open
neighborhood of $p$ in which $\phi_{s}$ has $p$ as the only fixed point for
all $s$. Then ${\it VHF}^{\rm loc}(\phi^{0},p;{\mathbb{K}})\cong{\it VHF}^{\rm
loc}(\phi^{1},p;{\mathbb{K}})$.

  3. (3)

If $\phi^{\prime}$ is a generic $C^{2}$ small perturbation of $\phi$ supported
near $p$, then the number of fixed points of $\phi^{\prime}$ near $p$ is at
least ${\rm rank}{\it VHF}^{\rm loc}(\phi,p;{\mathbb{K}})$.

  4. (4)

If $\phi^{k}$ is an admissible iteration of $\phi$ at $p$, meaning that
$\lambda^{k}\neq 1$ for all eigenvalues $\lambda\neq 1$ of $D\phi_{x}$, which
implies that $p$ is also an isolated fixed point of $\phi^{k}$, then

${\rm rank}{\it VHF}^{\rm loc}(\phi,p;{\mathbb{K}})={\rm rank}{\it VHF}^{\rm
loc}(\phi^{k},p;{\mathbb{K}})$

The homology group ${\it VHF}^{\rm loc}(\phi,p;{\mathbb{K}})$ is constructed
via generating Hamiltonians of $\phi$. For a Hamiltonian $H_{t}:S^{1}\times
X\to{\mathbb{R}}$ with $x:S^{1}\to X$ being an isolated $1$-periodic orbit, we
will define the local Floer homology group

${\it VHF}^{\rm loc}(H,x;{\mathbb{K}}).$

It turns out that the vortex version of local Floer homology is in fact
isomorphic to the classical one.

######  Proposition 5.2.

There is an isomorphism

${\it VHF}^{\rm loc}(H,x;{\mathbb{K}})\cong HF^{\rm loc}(H,x;{\mathbb{K}}).$

As we will use bulk-deformed vortex Floer theory, the right notion of local
Floer homology may _a priori_ depend on the bulk deformation, when the fixed
point is contained in the bulk divisor. However, we will prove (see
Proposition 5.7) that the bulk-deformed local vortex Floer homology is (non-
canonically) isomorphic to the undeformed one. Hence the homological count we
defined here does not see the effect of bulk deformation _a posteri_.

Now we use the rank of local Floer homology to define the so-called
homological count of the number of fixed points.

######  Definition 5.3.

Let ${\mathbb{K}}$ be a field. Given a Hamiltonian diffeomorphism $\phi$ on
$X$ with only isolated fixed points, the homological count (over coefficient
field ${\mathbb{K}}$) of the number of fixed points of $\phi$ is

$N(\phi,{\mathbb{K}}):=\sum_{p\in{\rm Fix}\phi}{\rm dim}_{\mathbb{K}}{\it
VHF}^{\rm loc}(\phi,p;{\mathbb{K}})=\sum_{p\in{\rm Fix}\phi}{\rm
dim}_{\mathbb{K}}{\it HF}^{\rm loc}(\phi,p;{\mathbb{K}}).$ (5.1)

By definition, when $\phi$ is nondegenerate, we have

$N(\phi,{\mathbb{K}})=\\#{\rm Fix}(\phi).$

On the other hand, if $\phi$ be a Hamiltonian diffeomorphism on $X$ with only
isolated fixed points, then for $\phi^{k}$ being an admissible iteration at
all the fixed points of $\phi$ with all fixed points isolated, we know that

$N(\phi,{\mathbb{K}})\leq N(\phi^{k},{\mathbb{K}}).$

###  5.1. Local Morse and Floer homology

####  5.1.1. Local Morse homology

We follow the treatment of local Morse and Floer homology by Ginzburg [Gin10].
First, let $M$ be a smooth manifold and $f:M\to{\mathbb{R}}$ be a smooth
function. Suppose $x$ is an isolated (but not necessarily nondegenerate)
critical point. Then for any coefficient ring $R$, there is an invariant, the
local Morse homology

$HM^{\rm loc}(f,x)$

defined by taking the homology of a Morse-type complex over $R$ of a small
generic perturbation of $f$ in a sufficiently small neighborhood of $x$, which
only takes into account critical points and gradient trajectories (for a
generic Riemannian metric) contained in that neighborhood. We recall the
details of the construction as this is the prototype of the local version of
the (vortex) Floer homology.

First, choose a small neighborhood $U$ of $p$ which contains no other critical
points of $f$. Fix a reference Riemannian metric $g_{U}$ on $U$ to measure
relevant norms. Let $U^{\prime}$ be a smaller neighborhood of $p$ whose
closure is contained in $U$. Let $f_{1}$ be an $\epsilon$-small perturbation
supported in $U^{\prime}$, i.e.

$\displaystyle\ {\rm supp}(f_{1}-f)\subset U^{\prime},\ $ $\displaystyle\
\|f_{1}-f\|_{C^{2}}<\epsilon.$

A generic such perturbation $f_{1}$ is Morse inside $U$ and ${\rm
crit}(f_{1}|_{U})$ is contained in $U^{\prime}$. Then consider the Morse
complex of $f_{1}:U\to{\mathbb{R}}$, which is freely generated by critical
points of $f_{1}|_{U}$ over $R$, graded by the Morse index. To define the
differential, consider an arbitrary Riemannian metric $g_{1}$ on $U$.
Consider, for each two critical points $p_{1},q_{1}$ of $f_{1}$ the moduli
space of negative gradient trajectories of $f_{1}$ (with respect to $g_{1}$)
that connect $p_{1}$ and $q_{1}$. Then by a compactness argument, for
$\epsilon$ sufficiently small, all trajectories connecting critical points of
$f_{1}|_{U}$ must stay in $U^{\prime}$. Then by a small perturbation of the
Riemannian metric $g_{1}$ to achieve transversality, one can count rigid
negative gradient trajectories over ${\mathbb{Z}}_{2}$; choosing an
orientation of $U$ and orientations on unstable manifolds of critical points
of $f_{1}|_{U}$ one can define integral counts. Hence one obtains a chain
complex whose homology is defined to be the local Morse homology $HM^{\rm
loc}(f,p)$.

Using continuation map one can prove that the local Morse homology is an
invariant, which only depends on the infinitesimal behavior of $f$ at $p$.
Indeed, fix $U$, $U^{\prime}$, $g_{U}$ as above. Let $f_{2}$ be another
$\epsilon$-small perturbation supported in $U^{\prime}$. Let $g_{2}$ be
another Riemannian metric on $U$ for which the local Morse complex of
$(f_{2},g_{2})$ is defined. Choose a homotopy $(f_{\chi(s)},g_{\chi(s)})$
between $(f_{1},g_{1})$ and $(f_{2},g_{2})$ using a fixed cut-off function
$\chi:(-\infty,+\infty)\to[1,2]$. By possibly shrinking the value of
$\epsilon$, one can show that in this case all solutions to the continuation
equation

$\dot{x}(s)+\nabla^{g_{\chi(s)}}f_{\chi(s)}(x(s))=0$

are contained in $U^{\prime}$. By slightly perturbing the data
$(f_{\chi(s)},g_{\chi(s)})$ one can achieve transversality for the moduli
spaces of this equation and hence define a continuation map. The same type of
argument can be applied to show that the continuation map is uniquely
determined up to chain homotopy. As a result, in the same vein of classical
arguments in Floer theory, one can prove that the local Morse homology
$HM^{\rm loc}(f,x)$ is independent of the small perturbation of the function
and the Riemannian metric. One can also see that the homology is neither
dependent on the neighborhoods $U$, $U^{\prime}$ nor the reference metric
$g_{U}$. Hence $HM^{\rm loc}(f,x)$ is an invariant of the germ of $f$ at $x$.

####  5.1.2. Local Floer homology

One can similarly define local homology groups in the Floer setting which
admits extensions to the vortex setting. Note that the explicit construction
depends on the 1-periodic family of Hamiltonians but we will prove that it
only depends on the time-1 map. Let $(M,\omega)$ be a symplectically
aspherical manifold and $H$ be a 1-periodic family of Hamiltonians on $M$ with
time-1 map $\varphi_{H}:M\to M$. Choose a reference Riemannian metric $g_{M}$
which induces a distance function $d_{M}$ on $M$.

######  Lemma 5.4.

If $q\in{\rm Fix}(\varphi_{H})$ is an isolated fixed point corresponding to a
1-periodic orbit $x:S^{1}\to M$, then there exists $r>0$ such that, for all
smooth loop $y:S^{1}\to M$ with

$\sup_{t\in S^{1}}d_{M}(y(t),x(t))<r,$ (5.2)

if $y$ is a 1-periodic orbit of $H$, then $y\equiv x$.

###### Proof.

This follows from the definition of isolated fixed point. ∎

We say that a loop $y:S^{1}\to M$ is $r$-close to $x$ if $y(t)$ satisfies
(5.2). We may choose $r$ smaller than the injectivity radius of $g_{M}$.

Now we review the definition of the local Floer homology. Fix $r$ as in Lemma
5.4. Let $R$ be a coefficient ring such as ${\mathbb{Z}}_{2}$ or
${\mathbb{Z}}$. The local Floer homology $HF^{\rm loc}(H,x)$ is a
${\mathbb{Z}}_{2}$-graded $R$-module. To define it, consider a perturbation
$H_{1,t}:S^{1}\times M\to{\mathbb{R}}$ satisfying

  1. (1)

For all $t\in S^{1}$, ${\rm supp}(H_{1,t}-H_{t})\subset B_{r/2}(x(t))$.

  2. (2)

$\|H_{1}-H\|_{C^{2}(S^{1}\times M)}\leq\delta$.

We call such perturbations $\delta$-small perturbations.

######  Lemma 5.5.

For any $\rho>0$, there exists $\delta>0$ such that for all $\delta$-small
perturbation $H_{1}$, if $y$ is a 1-periodic orbit of $H_{1}$ which is
$r$-close to $x$, then $y$ is $\rho$-close to $x$.

###### Proof.

Suppose this is not true, then there exist $\rho>0$, a sequence $\delta_{i}\to
0$ and a sequence of $\delta_{i}$-small perturbations $H_{i}$, and a sequence
of 1-periodic orbits $y_{i}$ of $H_{i}$ with
$d_{M}(y_{i}(t_{i}),x(t_{i}))\geq\rho$ for some $t_{i}\in S^{1}$. Then
$H_{i}\to H$ in $C^{2}$. By choosing a subsequence, one may assume that
$y_{i}$ converges to a 1-periodic orbit $y_{\infty}$ of $H$. As $y_{i}(0)$ is
$r$-close to $x(0)$, one can see that $y_{\infty}(0)$ is $r$-close to $x(0)$,
hence is a fixed point of $\varphi$ which is $r$-close to $x(0)$. As $x(0)$ is
an isolated fixed point, $y_{\infty}(0)=x(0)$ and hence $y_{\infty}(t)\equiv
x(t)$. This contradicts the assumption. ∎

The following construction is analogous to the Morse case. For each loop
$\overline{x}:S^{1}\to M$ which is $\frac{r}{2}$-close to $x$, one can define
the action functional

${\mathcal{A}}_{H_{1}}^{\rm loc}(\overline{x})=-\int_{[0,1]\times
S^{1}}u^{*}\omega+\int_{S^{1}}H_{1}(\overline{x}(t))dt.$

Here $u:[0,1]\times S^{1}\to M$ is a “small” cobordism connecting
$\overline{x}$ and $x$ whose homotopy class is canonical as $\overline{x}$ is
sufficiently close to $x$. Then for each pair of critical points $x_{1},y_{1}$
of ${\mathcal{A}}_{H_{1}}^{\rm loc}$, the difference

${\mathcal{A}}_{H_{1}}^{\rm loc}(x_{1})-{\mathcal{A}}_{H_{1}}^{\rm
loc}(y_{1})$

is sufficiently small; this is needed to run the compactness argument.

Consider the Floer complex generated over $R$ by critical points of
${\mathcal{A}}_{H_{1}}^{\rm loc}$, graded by the Conley–Zehnder index modulo
$2$. Notice that for any two generators $x_{1},y_{1}$, there is a canonical
homotopy class of (short) cylinders connecting them. To define the
differential, choose a 1-periodic family of $\omega$-compatible almost complex
structures $J_{1}$ and consider Floer trajectories (using $J_{1}$) connecting
generators in the canonical homotopy class. Then the energy identity for Floer
trajectories guarantees that when $\rho$ and $\delta$ are sufficiently small,
the total energy of Floer trajectories can be arbitrarily small. More
precisely, we consider Floer trajectories $u:{\mathbb{R}}\times S^{1}\to M$
for $(H_{1},J_{1})$ satisfying

$\sup_{s\in{\mathbb{R}}}\sup_{t\in S^{1}}d_{M}(u(s,t),x(t))<r.$

The smallness of total energy guarantee that the above supremum can be
arbitrarily small. One can hence guarantee compactness (up to breaking) of
such Floer trajectories which are sufficiently close to $x(t)$ and define a
chain complex. Coherent orientations can also be chosen if one would like to
define the complex over ${\mathbb{Z}}$.

One can prove using the continuation map that the local Floer homology is
independent of the pair $(H_{1},J_{1})$. We omit the details. We denote the
local Floer homology defined in this way as $HF^{\rm loc}(H,x;R)$. Moreover,
we can prove that the local Floer homology only depends on the fixed point $q$
and the time-1 map $\phi\in{\rm Ham}(X)$. Hence we denote the local Floer
homology by

$HF^{\rm loc}(\phi,q;R).$

Among various properties of the local Floer homology we only recall the
following one.

######  Proposition 5.6.

[GG10, Theorem 1.1] Let ${\mathbb{K}}$ be a field. If $q$ is an isolated fixed
point of $\phi$, and $\phi^{k}$ is an iteration admissible at $p$, then

${\rm rank}_{\mathbb{K}}{\it HF}^{\rm loc}(\phi^{k},q;{\mathbb{K}})={\rm
rank}_{\mathbb{K}}{\it HF}^{\rm loc}(\phi,q;{\mathbb{K}}).$

###  5.2. Local vortex Floer homology with bulk

We adapt the definition of local Floer homology in the vortex setting,
possibly with bulk deformations, and establish analogues of the statements in
Proposition 5.6 as listed in Theorem 5.1.

Let ${\mathfrak{b}}$ be a small bulk deformation. Let $\phi:X\to X$ be a
Hamiltonian diffeomorphism and $q\in X$ be an isolated fixed point. We would
like to define a local invariant

${\it VHF}^{\mathfrak{b}}_{\rm loc}(\phi,q;{\mathbb{K}}).$

Indeed, let $H$ be a 1-periodic family of Hamiltonian on $X$ generating the
Hamiltonian isotopy $\phi_{t}$ with $\phi_{1}=\phi$. Let $x(t)=\phi_{t}(p)$ be
the corresponding 1-periodic orbit of $H$. Notice that even if $x(t)$ is
nondegenerate, it may intersect the bulk divisor $D\subset X$. Choose a small
perturbation $H_{1}$ of $H$ supported near $x(t)$ such that all nearby
1-periodic orbits are nondegenerate and are disjoint from the bulk divisor
$D$.

Let $\widehat{H}$ be a $K$-invariant lift of $H$ and $\widehat{H}_{1}$ be a
$K$-invariant admissible lift of $H_{1}$. Then the 1-periodic orbit $x(t)$
lifts to a gauge equivalence class of equivariant 1-periodic orbits. Let
${\mathfrak{x}}(t)=(x(t),\eta(t))$ be a representative. There are also gauge
equivalence classes of equivariant 1-periodic orbits of $H_{1}$ which are near
${\mathfrak{x}}$. Indeed, fixing the $L_{0}K$-orbit of ${\mathfrak{x}}(t)$,
there are well-defined $L_{0}K$-orbits of equivariant 1-periodic orbits which
are nearby. Then for each pair of nearby equivariant 1-periodic orbits
${\mathfrak{x}}_{1}$, ${\mathfrak{y}}_{1}$ of $\widehat{H}_{1}$, there is a
canonical homotopy class of (small) cylinders connecting ${\mathfrak{x}}_{1}$
and ${\mathfrak{y}}_{1}$. Consider the moduli space of solutions to the vortex
equation over the cylinder connecting ${\mathfrak{x}}_{1}$ and
${\mathfrak{y}}_{1}$. The energy of these solutions is

${\mathcal{A}}_{H_{1}}({\mathfrak{x}}_{1})-{\mathcal{A}}_{H_{2}}({\mathfrak{y}}_{1})$

which are arbitrarily small. Then similar to the case of ordinary local Floer
homology, these moduli spaces can be used to define a chain complex over any
coefficient filed ${\mathbb{K}}$. As the orbits are disjoint from $D$, one can
also use topological intersection numbers with the bulk divisor and associated
weighted counts to define the bulk-deformed version. Denote the resulting
homology by

${\it VHF}^{\mathfrak{b}}_{\rm loc}(H,x;{\mathbb{K}}).$

The continuation argument shows that the homology is independent of the data
$(\widehat{H}_{1},\widehat{J}_{1})$. On the other hand, a priori the homology
depends on the bulk ${\mathfrak{b}}$. When ${\mathfrak{b}}=0$, denote this
homology by ${\it VHF}_{\rm loc}(H,x;{\mathbb{K}})$.

######  Proposition 5.7.

One has

${\it VHF}^{\mathfrak{b}}_{\rm loc}(H,x;{\mathbb{K}})\cong{\it VHF}^{\rm
loc}(H,x;{\mathbb{K}}).$ (5.3)

###### Proof.

First, suppose $x$ does not intersect the bulk divisor $D\subset X$. Then for
a small perturbation of $H$, all cylinders contributing to the definition of
the local Floer homology have zero topological intersection number with the
divisor upstairs. Hence (5.3) holds in this case.

Now suppose $x$ intersects the bulk divisor $D$. One can find a loop of
Hamiltonians $\psi(t)$ supported near $x(t)$ such that $y(t):=\psi(t)(x(t))$
is disjoint from $D$. Moreover, define

$y(t)=(\psi(t)\phi(t)\psi(0)^{-1})(\psi(0)(q))=(\psi(t)\phi(t)\psi(0)^{-1})(y(0)),$

then $y(t)$ is a 1-periodic orbit of the Hamiltonian isotopy
$\psi(t)\phi(t)\psi(0)^{-1}$. Let the generating Hamiltonian function of this
new family be $G$, which can be made sufficiently close to $H$. Then $y(t)$ is
also an isolated 1-periodic orbit of $G$. Then a generic perturbation of $G$
also serves as a perturbation of $H$. Hence

${\it VHF}^{\mathfrak{b}}_{\rm loc}(H,x;{\mathbb{K}})\cong{\it
VHF}^{\mathfrak{b}}_{\rm loc}(G,y;{\mathbb{K}}).$

However, as $y$ is disjoint from $D$, the right hand side is isomorphic to
${\it VHF}^{\rm loc}(G,y;{\mathbb{K}})$, which is also isomorphic to ${\it
VHF}^{\rm loc}(H,x;{\mathbb{K}})$. ∎

Now we prove that the local vortex Floer homology is isomorphic to the local
Floer homology inside the symplectic quotient.

###### Proof of Proposition 5.2.

It follows from the adiabatic limit argument in the same spirit as in [DS94,
GS05] and [SX14]. Let $H_{1}$ be a nondegenerate Hamiltonian on $X$ which is
arbitrarily close to $H$. Let $\widehat{H}_{1}$ be an admissible lift and
$\widehat{J}_{1}$ is a generic time-dependent almost complex structure.
Consider the local vortex Floer homology defined by critical points of
${\mathcal{A}}_{H_{1}}$ which are close to the fixed point $x\in{\rm
Fix}(\varphi_{H})$ whose differential counts rigid solutions to the equation
(4.8) (with $(\widehat{H},\widehat{J})$ replaced by
$(\widehat{H}_{1},\widehat{J}_{1})$). Using continuation maps we can show that
the resulting homology is independent of $\lambda$. Moreover, the energy of
relevant solutions can be arbitrarily small. Then consider the
$\lambda\to\infty$ limit. For any sequence $\lambda_{i}\to\infty$ and any
sequence of solutions to (4.8) for $\lambda=\lambda_{i}$ which contributes to
the local vortex Floer differential, there is an upper bound of the energy of
these solutions. Then by the adiabatic limit compactness theorem (see
[GS05][WX17a] in similar settings) a subsequence converges to a possibly
broken ordinary Floer trajectory inside $X$ modulo bubbling. As there is a
lower bound for the energy of bubbles, we can choose the perturbation $H_{1}$
sufficiently close to $H$ so that bubbles can be ruled out. Moreover, we may
assume that the pair $(H_{1},J_{1})$ on the symplectic quotient $X$ induced
from the pair $(\widehat{H}_{1},\widehat{J}_{1})$ makes the local Floer
complex well-defined (i.e. moduli spaces are transverse). Then if we are
considering the zero-dimensional moduli spaces, then the possible limits must
be unbroken trajectories in $X$.

Now we claim that for $\lambda$ sufficiently large, there is an orientation-
preserving bijection between index zero solutions to (4.8) (modulo gauge
transformation) and index zero solutions to the ordinary Floer equation in
$X$. Indeed, using the same kind of estimates as in [DS94][GS05] (and the much
simpler case in [SX14]) one can construct a gluing map from the limiting
moduli space to the vortex moduli space with sufficiently large parameter
$\lambda$. The compactness result explained above shows that the gluing map is
surjective, while via the implicit function theorem one can show that the
gluing map is injective. The fact that the gluing map preserves orientation
follows from the explicit comparison of the linearized Fredholm operators
(they differ by, roughly speaking, an invertible operator). ∎

In view of Proposition 5.2 and the properties of local Floer homology as
proved in, e.g., [Gin10], the assertions in Theorem 5.1 are straightforward.
The following is also immediate.

######  Corollary 5.8.

The local vortex Floer homology has the following properties.

  1. (1)

(Up to isomorphism) ${\it VHF}^{\rm loc}(H,x;{\mathbb{K}})$ only depends on
the fixed point $q$ and the time-1 map $\phi\in{\rm Ham}(X)$. Hence we denote
the (bulk-deformed) local vortex Floer homology by

${\it VHF}_{\rm loc}^{\mathfrak{b}}(\phi,q;{\mathbb{K}}).$

  2. (2)

If $\phi^{k}$ is an admissible iteration of $\phi$ at $q$, then

${\it VHF}_{\rm loc}^{\mathfrak{b}}(\phi,q;{\mathbb{K}})\cong{\it VHF}_{\rm
loc}^{\mathfrak{b}}(\phi^{k},q;{\mathbb{K}}).$

###  5.3. Barcodes of degenerate Hamiltonians

Recall that one can associate to each nondegenerate Hamiltonian on a closed
symplectic manifold a (finite) barcode. As this association is Lipschitz
continuous with respect to the bottleneck distance for barcodes and Hofer
metric for Hamiltonians, we hope one can define barcodes for all Hamiltonians
using this Lipschitz continuity. However, the bottleneck distance is not
complete. Therefore, a priori, the barcode for a general Hamiltonian only
exists in the completion.

######  Theorem 5.9.

Let ${\mathbb{K}}$ be a field. Let $\phi\in{\rm Ham}(X)$ be a Hamiltonian
diffeomorphism with isolated fixed points. Let ${\mathcal{B}}(\phi)$ be the (a
priori infinite) reduced barcode of $\phi$ (in coefficient field
$\Lambda_{\mathbb{K}}^{\Gamma}$). Then ${\mathcal{B}}(\phi)$ has finitely many
bars whose number of end points is equal to $N(\phi)$.

######  Corollary 5.10.

The total bar length is defined for all $\phi\in{\rm Ham}(X)$ with isolated
fixed points.

Now we prove Theorem 5.9. Suppose $\phi\in{\rm Ham}(X)$ has only isolated
fixed points. Let $H$ be a Hamiltonian whose time one map is $\phi$. Let
$\widehat{H}$ be any $K$-invariant lift of $H$ and let $\widehat{J}$ be a
$K$-invariant $\omega_{V}$-compatible almost complex structure. Notice that in
general $(\widehat{H},\widehat{J})$ is not an admissible pair so does not have
a vortex Floer complex. However, one can still consider the vortex equation
with the data $(\widehat{H},\widehat{J})$.

######  Lemma 5.11.

There exists $\delta>0$ which only depends on $(\widehat{H},\widehat{J})$
satisfying the following condition. Let $x(t)\neq y(t)$ be two different
1-periodic orbits of $H$ downstairs. Let ${\mathfrak{u}}$ be a possibly broken
solution to (4.6) which connects $x(t)$ and $y(t)$ (without conditions on
capping). Then the energy of ${\mathfrak{u}}$ is at least $\delta$.

###### Proof.

For admissible $(\widehat{H},\widehat{J})$ this statement is proved as [Xu16,
Proposition 5.5] using a compactness argument. Notice that to run the
compactness argument and to have the notion of converging to a 1-periodic
orbit, one does not really need to require that the Hamiltonian is
nondegenerate or the pair $(\widehat{H},\widehat{J})$ is admissible. ∎

######  Corollary 5.12.

The lengths of all bars in ${\mathcal{B}}(\phi)$ are no less than $\delta$.

###### Proof.

Suppose on the contrary ${\mathcal{B}}(\phi)$ has a finite bar whose length is
positive and smaller than $\delta$. Let $(\widehat{H}_{k},\widehat{J}_{k})$ be
a sequence of regular bulk-avoiding pairs such that
$(\widehat{H}_{k},\widehat{J}_{k})$ converges to $(\widehat{H},\widehat{J})$.
Consider the reduced barcode associated to $\phi_{H_{k}}$. By the continuous
dependence of barcodes on the Hamiltonian, for $k$ sufficiently large, there
exists a finite bar in ${\mathcal{B}}(\phi_{H_{k}})$ whose length is between
$\frac{\delta}{2}$ and $\delta-\epsilon$ for some small $\epsilon$. By the
definition of barcodes by Usher–Zhang, there exists a rigid solution
${\mathfrak{u}}_{k}$ to (4.6) with data $(\widehat{H}_{k},\widehat{J}_{k})$
whose energy is between $\frac{\delta}{2}$ and $\delta-\epsilon$. Via the
compactness argument, there is a subsequence, still indexed by $k$, such that
${\mathfrak{u}}_{k}$ converges to a possibly broken trajectory with data
$(\widehat{H},\widehat{J})$ whose total energy is between $\frac{\delta}{2}$
and $\delta-\epsilon$. This contradicts Lemma 5.11. ∎

###### Proof of Theorem 5.9.

Choose a sequence of regular bulk-avoiding admissible pair
$(\widehat{H}_{k},\widehat{J}_{k})$ converging to $(\widehat{H},\widehat{J})$.
Consider the complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{k},\widehat{J}_{k};\Lambda_{\mathbb{K}}^{\Gamma})$.
One can write

$\partial=\partial_{\rm short}+\partial_{\rm long}$

where $\partial_{\rm short}$ counts rigid trajectories whose energy is smaller
than $\delta$ and $\partial_{\rm long}$ counts rigid trajectories whose energy
is bigger than $\delta$. Then $\partial_{\rm short}^{2}=0$ and its homology
coincides with the direct sum of all local vortex Floer homology of $\phi$.
Moreover, one can decompose the reduced barcode of $\phi_{H_{k}}$ as

${\mathcal{B}}(\phi_{H_{k}})={\mathcal{B}}_{\rm
short}(\phi_{H_{k}})\sqcup{\mathcal{B}}_{\rm
long}(\phi_{H_{k}})\sqcup{\mathcal{B}}_{\infty}(\phi_{H_{k}})$

where the first component consists of finite bars of lengths at most $\delta$
and the second component consists of other finite bars. As $\partial_{\rm
short}^{2}=0$, one can also define a barcode ${\mathcal{B}}_{\rm
local}(\phi_{H_{k}})$ by modifying the definition of Usher–Zhang, whose finite
part coincides with ${\mathcal{B}}_{\rm short}(\phi_{H_{k}})$. Then by the
definition,

$\begin{split}N(\phi_{H_{k}})=&\ \\#{\rm End}({\mathcal{B}}_{\rm
short}(\phi_{H_{k}}))+\sum_{x\in{\rm Fix}(\phi)}{\rm dim}{\it VHF}^{\rm
loc}(\phi,x)\\\ =&\ \\#{\rm End}({\mathcal{B}}_{\rm
short}(\phi_{H_{k}}))+\\#{\rm End}({\mathcal{B}}_{\rm
long}(\phi_{H_{k}}))+{\rm dim}{\it VHF}_{\bullet}(V).\end{split}$

As in the limit, all short bars disappear and long bars survive with respect
to the bottleneck distance, the theorem follows. ∎

##  6\. Boundary depth

In this section we prove Theorem C, namely, under the semisimple condition,
the boundary depth of the vortex Floer complex of any Hamiltonian
diffeomorphism is uniformly bounded from above.

###  6.1. Vortex Floer persistence modules

Recall that from Section 3.4.3 we know that any Floer–Novikov complex over a
Novikov field $\Lambda_{\mathbb{K}}^{\Gamma}$ induces a persistence module
over ${\mathbb{K}}$. Given a regular bulk-avoiding admissible pair
$(\widehat{H},\widehat{J})$ and a bulk deformation

${\mathfrak{b}}=\sum_{j=1}^{N}\log c_{j}V_{j}\ {\rm where}\
c_{j}\in{\mathbb{Z}}[{\bf i}],$

the persistence module induced from the complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{\overline{\mathbb{F}}_{p}}^{\Gamma})$
is denoted by

${\bm{V}}_{(p)}(\widehat{H},\widehat{J}).$

Recall that each filtered Floer–Novikov complex has a finite boundary depth
which coincides with the boundary depth of the associated persistence module.
We denote the boundary depth of ${\bm{V}}_{(p)}(\widehat{H},\widehat{J})$ by

$\beta_{(p)}(\widehat{H},\widehat{J})\in[0,+\infty).$

It is also equal to the length of the longest finite bar in the associated
barcode (cf. Proposition 3.29).

######  Proposition 6.1.

Given any two regular bulk-avoiding admissible pairs
$(\widehat{H}_{1},\widehat{J}_{1})$ and $(\widehat{H}_{2},\widehat{J}_{2})$,
for any prime $p$, one has

$|\beta_{(p)}(\widehat{H}_{1},\widehat{J}_{1})-\beta_{(p)}(\widehat{H}_{2},\widehat{J}_{2})|\leq
2d_{\rm Hofer}(H_{1},H_{2}).$ (6.1)

In particular, the boundary depth only depends on the descent Hamiltonian
downstairs.

###### Proof.

This is a consequence of the stability of the persistence module and the
boundary depth. Indeed, Proposition 4.16 implies that the quasi-equivalence
distance between ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{\overline{\mathbb{F}}_{p}})$
and ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{\overline{\mathbb{F}}_{p}})$
is at most equal to the Hofer distance $d_{\rm Hofer}(H_{1},H_{2})$. Using
Theorem 3.30, it implies that the interleaving distance between the two
associated persistence modules is no greater than the same bound. By
Proposition 3.24, one can conclude (6.1). ∎

Using typical arguments, one can also show that the boundary depth only
depends on the induced (nondegenerate) Hamiltonian isotopy $\tilde{\phi}_{H}$
on the toric manifold $X$. Then Proposition 3.24 implies that $\beta_{(p)}$
descends to a Hofer continuous function

$\beta_{(p)}:{\rm H}{\widetilde{\rm am}}(X)\to[0,+\infty).$

Below is the main theorem of this section.

######  Theorem 6.2.

Suppose there exist $p_{0}>0$ and $C_{0}>0$ such that for all prime $p\geq
p_{0}$, the algebra ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}})$ is a
semisimple $\Lambda_{\overline{\mathbb{F}}_{p}}$-algebra with idempotent
generators $e_{l,(p)},\ldots,e_{m,(p)}$ satisfying

$\ell_{p}(e_{l,(p)})\leq C_{0}.$

Then there exists $C>0$ such that for all prime $p\geq p_{0}$ and all
${\tilde{\phi}}\in{\rm H}{\widetilde{\rm am}}(X)$, one has

$\beta_{(p)}(\tilde{\phi})\leq C.$

###  6.2. Action by quantum multiplication

Recall how we define pair-of-pants product on the vortex Floer homology (see
[WX17b]). On the pair-of-pants $\Sigma^{\rm pop}$, equip the two inputs bulk-
avoiding admissible pairs $(\widehat{H}_{1},\widehat{J}_{1})$ and
$(\widehat{H}_{2},\widehat{J}_{2})$ and equip the output another bulk-avoiding
admissible pair $(\widehat{H}_{3},\widehat{J}_{3})$. Extend these data to a
domain-dependent Hamiltonian perturbation and a domain-dependent almost
complex structure on $\Sigma^{\rm pop}$. By counting solutions to the
Hamiltonian perturbed vortex equation on $\Sigma^{\rm pop}$ (with appropriate
weights coming from the bulk deformation ${\mathfrak{b}}$), one can define a
chain map

${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{1},\widehat{J}_{1};\Lambda_{\mathbb{K}})\otimes{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{2},\widehat{J}_{2};\Lambda_{\mathbb{K}})\to{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}_{3},\widehat{J}_{3};\Lambda_{\mathbb{K}}).$

We fix a class $\alpha\in{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\mathbb{K}})$. For each $\delta>0$,
let $\widehat{H}_{\delta}$ be a bulk-avoiding admissible Hamiltonian on $V$
with $\|\widehat{H}_{\delta}\|_{C^{2}}\leq\delta$. We temporarily omit the
dependence on the almost complex structure and the coefficient field from the
notations. For notational simplicity, we also omit the bulk ${\mathfrak{b}}$
in the formulas at the moment. Consider the chain-level map

${\it VCF}_{\bullet}(\widehat{H}_{\delta})\otimes{\it
VCF}_{\bullet}(\widehat{H})\to{\it VCF}_{\bullet}(\widehat{H}).$

By using the energy inequality, one can show that there exists a constant
$C>0$, such that for all $s,\tau\in{\mathbb{R}}$, the above multiplication
induces a bilinear map

${\it VHF}_{\bullet}^{\leq\tau}(\widehat{H}_{\delta})\otimes{\it
VHF}_{\bullet}^{\leq s}(\widehat{H})\to{\it VHF}_{\bullet}^{\leq
s+\tau+C\delta}(\widehat{H}).$ (6.2)

Denote

${\mathcal{A}}^{\mathfrak{b}}(\alpha):=c^{\mathfrak{b}}(\alpha,0)=\lim_{\delta\to
0}c^{\mathfrak{b}}(\alpha,H_{\delta}).$

Then one has the linear map for all $\epsilon>0$, one can choose $\delta$
sufficiently small so that by setting
$\tau={\mathcal{A}}^{\mathfrak{b}}(\alpha)+\delta$ and inserting a
representative of $\alpha$ in ${\it
VHF}_{\bullet}^{\leq\tau}(\widehat{H}_{\delta})$ in (6.2), one obtains a well-
defined map

$m_{\epsilon}(\alpha):{\it VHF}^{\leq s}_{\bullet}(\widehat{H})\to{\it
VHF}_{\bullet}^{\leq
s+{\mathcal{A}}^{\mathfrak{b}}(\alpha)+\epsilon}(\widehat{H}).$

Using the standard argument one can show that this map only depends on the
class $\alpha$. By applying any positive shift, the above operation defines a
family of operations which are recorded in the following statement.

######  Proposition 6.3.

For all $\epsilon>0$, the maps $m_{\epsilon}(\alpha)$ define a morphism of
persistence modules

$m_{\epsilon}(\alpha):{\bm{V}}({\tilde{\phi}})\to{\bm{V}}({\tilde{\phi}})[{\mathcal{A}}(\alpha)+\epsilon]\
\forall\epsilon>0$

satisfying for all $\epsilon<\epsilon^{\prime}$, one has

$m_{\epsilon^{\prime}}(\alpha)={\rm shift}_{\epsilon^{\prime}-\epsilon}\circ
m_{\epsilon}(\alpha).$

######  Definition 6.4.

Given $\alpha\in{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\mathbb{K}})\setminus\\{0\\}$ and
$\epsilon>0$, the persistence module
${\bm{W}}_{\alpha}(\tilde{\phi})_{\epsilon}$ is defined by

$W_{\alpha}(\tilde{\phi})_{\epsilon}^{s}={\rm
Im}\left(m_{\epsilon}(\alpha):{\it VHF}_{\bullet}^{\leq
s-{\mathcal{A}}^{\mathfrak{b}}(\alpha)}(\tilde{H})\to{\it VHF}_{\bullet}^{\leq
s+\epsilon}(\tilde{H})\right)\subset V(\tilde{\phi})^{s+\epsilon}.$

######  Remark 6.5.

Our notion of persistence modules (Definition 3.20) is very different from the
traditionally used ones (see for example [PSS17] where similar operators were
firstly defined for Floer persistence modules in the monotone case); notably
we allow each piece $V^{s}$ of a persistence module ${\bm{V}}$ to be infinite-
dimensional. Hence it is not straightforward, though not necessarily
difficult, to prove that when $\epsilon\to 0$, the above persistence modules
“converges,” giving a limiting object similar to the one used in [She22].
However, we could also carry the $\epsilon$ everywhere as we are doing here.

###  6.3. Proof of Theorem 6.2

We prove Theorem 6.2 following the strategy of [She22]. This theorem is the
consequence of the following three lemmas (Lemma 6.6, Lemma 6.7, and Lemma
6.8).

We first introduce and simplify the notations. As we work with an individual
prime, we drop the dependence on the prime $p$ in most notations. Let
$e_{1},\ldots,e_{m}$ be the idempotent generators of ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}})$. For
each nondegenerate $\tilde{\phi}\in{\rm H}{\widetilde{\rm am}}(X)$, consider
the direct sum persistence module

${\bm{W}}(\tilde{\phi})_{\epsilon}=\bigoplus_{l=1}^{m}{\bm{W}}_{e_{l}}(\tilde{\phi})_{\epsilon}.$

######  Lemma 6.6.

The interleaving distance between ${\bm{V}}(\tilde{\phi})$ and
${\bm{W}}(\tilde{\phi})_{\epsilon}$ is at most $C_{0}+\epsilon$.

For all ${\tilde{\phi}}\in{\rm H}{\widetilde{\rm am}}(X)$, define

$\gamma({\tilde{\phi}}):=\max_{1\leq l\leq
m}\gamma_{e_{l}}({\tilde{\phi}}):=\max_{1\leq l\leq
m}\left(c^{\mathfrak{b}}(e_{l},{\tilde{\phi}})+c^{\mathfrak{b}}(e_{l},{\tilde{\phi}}^{-1})\right).$

Temporarily let ${\rm pr}:{\rm H}{\widetilde{\rm am}}(X)\to{\rm Ham}(X)$ be
the canonical projection. Define for $\phi\in{\rm Ham}(X)$

$\gamma(\phi):=\inf_{{\rm pr}({\tilde{\phi}})=\phi}\gamma({\tilde{\phi}}).$

The following is an analogue of [She22, Proposition 12].

######  Lemma 6.7.

The boundary depth of the persistence module
${\bm{W}}_{e_{l}}({\tilde{\phi}})_{\epsilon}$ is finite. Moreover, given
nondegenerate ${\tilde{\phi}},{\tilde{\psi}}\in{\rm H}{\widetilde{\rm
am}}(X)$, for each $l=1,\ldots,m$, one has

$\left|\beta({\bm{W}}_{e_{l}}({\tilde{\phi}})_{\epsilon})-\beta({\bm{W}}_{e_{l}}({\tilde{\psi}})_{\epsilon})\right|\leq\gamma_{e_{l}}(\tilde{\phi}\tilde{\psi}^{-1}).$
(6.3)

The following is analogous to [Ush11b, Proposition 5.4] and [She22,
Proposition 13].

######  Lemma 6.8.

For all ${\tilde{\phi}}\in{\rm H}{\widetilde{\rm am}}(X)$, one has

$c^{\mathfrak{b}}(e_{l},{\tilde{\phi}})+c^{\mathfrak{b}}(e_{l},{\tilde{\phi}}^{-1})\leq
4C_{0}.$

###### Proof of Theorem 6.2.

As the boundary depth depends continuously on the Hamiltonian isotopy
$\tilde{\phi}$, one only needs to prove the theorem for nondegenerate ones.
First, by Lemma 6.6, the interleaving distance between
${\bm{V}}(\tilde{\phi})$ and ${\bm{W}}(\tilde{\phi})_{\epsilon}$ is bounded by
$C_{0}$. Hence by Proposition 3.24, it suffices to bound the boundary depth of
${\bm{W}}(\tilde{\phi})_{\epsilon}$. As ${\bm{W}}(\tilde{\phi})_{\epsilon}$ is
the direct sum of ${\bm{W}}_{e_{l}}(\tilde{\phi})_{\epsilon}$, it suffices to
bound the boundary depth of ${\bm{W}}_{e_{l}}(\tilde{\phi})_{\epsilon}$ for
all idempotent generators $e_{l}$. Then applying Lemma 6.7, one obtains

$\beta({\bm{W}}_{e_{l}}(\tilde{\phi})_{\epsilon})\leq\gamma_{e_{l}}(\tilde{\phi}\tilde{\psi}^{-1})+\beta({\bm{W}}_{e_{l}}(\tilde{\psi})_{\epsilon})\leq
4C_{0}+\beta({\bm{W}}_{e_{l}}(\tilde{\psi})_{\epsilon})$

where $\tilde{\psi}\in{\rm H}{\widetilde{\rm am}}(X)$ is an arbitrary fixed
nondegenerate Hamiltonian isotopy on $X$. Then the right hand side is finite
and independent of $\tilde{\phi}$. ∎

###  6.4. Proofs of the technical lemmas

In this subsection we drop all dependence on the bulk deformation from
notations.

###### Proof of Lemma 6.6.

We construct maps between persistence modules
${\bm{f}}_{\epsilon}:{\bm{V}}(\tilde{\phi})\to{\bm{W}}(\tilde{\phi})_{\epsilon}[C_{0}]$
and
${\bm{g}}_{\epsilon}:{\bm{W}}(\tilde{\phi})_{\epsilon}\to{\bm{V}}(\tilde{\phi})[C_{0}]$
as follows. For $s\in{\mathbb{R}}$, define

$f_{\epsilon}^{s}:V(\tilde{\phi})^{s}\to\bigoplus_{l=1}^{m}W_{e_{l}}(\tilde{\phi})^{s+C_{0}}_{\epsilon}$

to be the composition of

$\displaystyle\
V(\tilde{\phi})^{s}\to\bigoplus_{l=1}^{m}W_{e_{l}}(\tilde{\phi})_{\epsilon}^{s+{\mathcal{A}}(e_{l})},\
$ $\displaystyle\ \alpha\mapsto(e_{1}\ast\alpha,\ldots,e_{m}\ast\alpha)$

and the natural map

$\bigoplus_{l=1}^{m}W_{e_{l}}(\tilde{\phi})_{\epsilon}^{s+{\mathcal{A}}(e_{l})}\to\bigoplus_{l=1}^{m}W_{e_{l}}(\tilde{\phi})_{\epsilon}^{s+C_{0}}.$

Define

$\displaystyle\
g_{+\epsilon}^{s}:\bigoplus_{l=1}^{m}W_{e_{l}}(\tilde{\phi})_{\epsilon}^{s}\to
V(\tilde{\phi})^{s+C_{0}},\ $ $\displaystyle\
(\alpha_{1},\ldots,\alpha_{m})\mapsto\iota^{s+\epsilon,s+C_{0}}(\alpha_{1}+\cdots+\alpha_{m}).$

It is straightforward to check, using the fact that
$e_{1}+\cdots+e_{m}={\bm{1}}^{{\rm GLSM}}_{\mathfrak{b}}$ and that $e_{l}$ are
idempotent generators, that ${\bm{f}}_{\epsilon}$, ${\bm{g}}_{\epsilon}$
provide $C_{0}$-interleaving between ${\bm{V}}(\tilde{\phi})$ and
${\bm{W}}(\tilde{\phi})_{\epsilon}$. ∎

###### Proof of Lemma 6.7.

The detailed proof would be almost identical to the part of the proof of
[She22, Proposition 12] corresponding to this lemma. Hence we only briefly
sketch it. First we show the finiteness of the boundary depth. The boundary
depth of ${\bm{V}}(\tilde{\phi})$ is finite because it coincides of the
boundary depth of the associated Floer–Novikov complex (see Proposition 3.26).
Hence by Lemma 6.6 and Proposition 3.24, ${\bm{W}}(\tilde{\phi})_{\epsilon}$
has finite boundary depth. Therefore each summand
${\bm{W}}_{e_{l}}(\tilde{\phi})_{\epsilon}$ has finite boundary depth.

Now we prove the inequality (6.3). Let $F,G$ be Hamiltonians downstairs with
time-1 maps $\tilde{\phi}$ and $\tilde{\psi}$ respectively. Choose bulk-
avoiding admissible lifts $\widehat{F}$, $\widehat{G}$ upstairs and let
$(\widehat{F},\widehat{J}_{F})$, $(\widehat{G},\widehat{J}_{G})$ be regular
pairs. Let $\ell_{F}$ resp. $\ell_{G}$ be the non-Archimedean valuation on the
complex ${\it VCF}_{\bullet}(\widehat{F},\widehat{J}_{F})$ resp. ${\it
VCF}_{\bullet}(\widehat{G},\widehat{J}_{G})$. Let
$\Delta_{\widehat{F},\widehat{G}}=G\\#\overline{F}$ be the difference
Hamiltonian upstairs with descent difference Hamiltonian $\Delta_{F,G}$
downstairs. Let $\widehat{J}_{F,G}$ be an admissible almost complex structure
so that the pair $(\Delta_{\widehat{F},\widehat{G}},\widehat{J}_{F,G})$ is
regular. One can obtain a pair
$(\Delta_{\widehat{G},\widehat{F}},\widehat{J}_{G,F})$ with the roles of
$\widehat{F}$ and $\widehat{G}$ reversed.

Now fix $\epsilon>0$. Choose a cycle
$c_{\widehat{F},\widehat{G},\epsilon}\in{\it
VCF}_{\bullet}(\Delta_{\widehat{F},\widehat{G}},\widehat{J}_{F,G})$
representing $e_{l}$ such that

$\ell(c_{\widehat{F},\widehat{G},\epsilon})\leq
c(e_{l},\Delta_{\widehat{F},\widehat{G}})+\epsilon.$

We also choose a cycle $c_{\widehat{G},\widehat{F},\epsilon}\in{\it
VCF}_{\bullet}(\Delta_{\widehat{G},\widehat{F}},\widehat{J}_{G,F})$
representing $e_{l}$ with

$\ell(c_{\widehat{G},\widehat{F},\epsilon})\leq
c(e_{l},\Delta_{\widehat{G},\widehat{F}})+\epsilon.$

Now after choosing perturbation data on the pair-of-pants, one can define a
chain map

$C_{\widehat{F},\widehat{G},\epsilon}:{\it
VCF}_{\bullet}(\widehat{F},\widehat{J}_{F})\to{\it
VCF}_{\bullet}(\widehat{G},\widehat{J}_{G}),\ x\mapsto
c_{\widehat{F},\widehat{G},\epsilon}\ast x.$

satisfying

$\ell_{G}(C_{\widehat{F},\widehat{G},\epsilon}(x))\leq
c(e_{l},\Delta_{\widehat{F},\widehat{G}})+\ell_{F}(x)+2\epsilon.$

Similarly, by using the cycle $c_{\widehat{G},\widehat{F},\epsilon}$ one can
also define a chain map

$C_{\widehat{G},\widehat{F},\epsilon}:{\it
VCF}_{\bullet}(\widehat{G},\widehat{J}_{G})\to{\it
VCF}_{\bullet}(\widehat{F},\widehat{J}_{F})$

satisfying

$\ell_{F}(C_{\widehat{G},\widehat{F},\epsilon}(y))\leq
c(e_{l},\Delta_{G,F})+\ell_{G}(y)+2\epsilon.$

The lemma will follow from the following claim.

Claim. $C_{\widehat{F},\widehat{G},\epsilon}$ and
$C_{\widehat{G},\widehat{F},\epsilon}$ induce a
$\frac{1}{2}\gamma_{e_{l}}(\tilde{\phi}\tilde{\psi}^{-1})+4\epsilon$-interleaving
between ${\bm{W}}_{e_{l}}(\widehat{F})_{\epsilon}$ and
${\bm{W}}_{e_{l}}(\widehat{G})_{\epsilon}$.

The detailed proof would also be almost identical to that of [She22] except
for notations. We omit the details. ∎

######  Remark 6.9.

As one can infer from the above proof, the inequality “$\beta\leq\gamma$” is a
consequence of studying filtered continuations maps in terms of taking the
pair-of-pants product with the filtered continuation elements, which in
particular does not depend on the semi-simplicity assumption.

###### Proof of Lemma 6.8.

Using Proposition 4.21, Lemma 4.14 and the triangle inequality for spectral
invariants, one has

$\begin{split}-c(e_{l},{\tilde{\phi}}^{-1})=&\
\inf\Big{\\{}c(\alpha,{\tilde{\phi}})\ |\ \langle e_{l},\alpha\rangle\neq
0\Big{\\}}\\\ \geq&\
-{\mathcal{A}}(e_{l})+\inf\Big{\\{}c(e_{l}\ast\alpha,{\tilde{\phi}})\ |\
\langle e_{l}\ast\alpha,{\bm{1}}^{\rm GLSM}\rangle\neq 0\Big{\\}}\\\ \geq&\
-{\mathcal{A}}(e_{l})+\inf\Big{\\{}c(e_{l},{\tilde{\phi}})-{\mathcal{A}}((e_{l}\ast\alpha)^{-1})\
|\ \langle e_{l}\ast\alpha,{\bm{1}}^{\rm GLSM}\rangle\neq 0\Big{\\}}\\\ \geq&\
-{\mathcal{A}}(e_{l})+c(e_{l},{\tilde{\phi}})+\inf\Big{\\{}-{\mathcal{A}}(e_{l}\ast\alpha)-{\mathcal{A}}((e_{l}\ast\alpha)^{-1})\
|\ e_{l}\ast\alpha\neq 0\Big{\\}}\\\ &\
+\inf\Big{\\{}{\mathcal{A}}(e_{l}\ast\alpha)\ |\ \langle
e_{l}\ast\alpha,{\bm{1}}^{\rm GLSM}\rangle\neq 0\Big{\\}}\end{split}$

Here the quantum product and the Poincaré pairing are both the bulk-deformed
versions. Notice that as $e_{l}$ is an idempotent generator,
$e_{l}\ast\alpha=\lambda(\alpha)e_{l}$ and
$(e_{l}\ast\alpha)^{-1}=\lambda(\alpha)^{-1}e_{l}$. Hence

${\mathcal{A}}(e_{l}\ast\alpha)+{\mathcal{A}}((e_{l}\ast\alpha)^{-1})=2{\mathcal{A}}(e_{l})-{\mathfrak{v}}(\lambda(\alpha))-{\mathfrak{v}}(\lambda(\alpha)^{-1})=2{\mathcal{A}}(e_{l})$

which is uniformly bounded. Moreover, by Proposition 4.21

$\inf\Big{\\{}{\mathcal{A}}(e_{l}\ast\alpha)\ |\ \langle
e_{l}\ast\alpha,{\bm{1}}^{\rm GLSM}\rangle\neq
0\Big{\\}}\geq-{\mathcal{A}}({\bm{1}}^{\rm GLSM}).$

Therefore

$c(e_{l},{\tilde{\phi}})+c(e_{l},{\tilde{\phi}}^{-1})\leq
3{\mathcal{A}}(e_{l})+{\mathcal{A}}({\bm{1}}^{\rm GLSM}).$

Lemma 6.8 follows by using the assumption ${\mathcal{A}}(e_{l})\leq C_{0}$ and
noticing

${\mathcal{A}}({\bm{1}}^{\rm
GLSM})={\mathcal{A}}(e_{1}+\cdots+e_{m})\leq\max_{1\leq l\leq
m}{\mathcal{A}}(e_{l})\leq C_{0}.\qed$

######  Remark 6.10.

The above argument crucially relies on the semi-simplicity assumption, which
allows us to take advantage of the feature that any nonzero element in a field
summand of the quantum homology is invertible. Note that such a phenomenon is
ultimately due to the abundance of rational curves in toric manifolds.

##  7\. ${\mathbb{Z}}/p$-equivariant vortex Floer theory

Following [Sei15, SZ21], we develop ${\mathbb{Z}}/p$-equivariant Hamiltonian
Floer theory in the vortex setting. Using equivariant pair of pants
operations, we show that the following analogue of [She22, Theorem D] about
the total bar length holds in our setting.

######  Theorem 7.1.

Let $\phi$ be a Hamiltonian diffeomorphism on the toric symplectic manifold
$(X,\omega)$ with lift $\tilde{\phi}\in{\rm H}{\widetilde{\rm am}}(X)$. Then
for any odd prime $p$, if ${\rm Fix}(\phi)$ and ${\rm Fix}(\phi^{p})$ are
finite, then

$\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi}^{p})\geq
p\cdot\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi})$ (7.1)

Here we work over $\Lambda_{\overline{\mathbb{F}}_{p}}$, which is omitted from
the notations above. Given the arguments from [She22, Section 6], the only
missing ingredient for establishing Theorem 7.1 is the package of
${\mathbb{Z}}/p$ Borel equivariant vortex Floer theory with bulk deformation.
As demonstrated in other parts of the paper, one salient feature of vortex
Floer theory is the absence of sphere bubbles due to the contractibility of
symplectic vector space, which allows us to achieve transversality in many
settings by only perturbing the almost complex structure. Specializing to the
Borel equivariant theory, except for the necessity to deal with the symplectic
vortex equations and the appearance of Novikov coefficients, our theory is
quite similar to the exact setting as from the original reference [Sei15,
SZ21], at least for bulk-avoiding Hamiltonians, which suffice for our purpose
via a limiting argument. Therefore, unless there is anything special in our
situation, we will be brief and refer the reader to the original references
for full proofs.

In this section the bulk deformation ${\mathfrak{b}}$ is fixed. All curve
counts are weighted by the bulk term. We often drop it in order to shorten the
notations.

###  7.1. The Borel construction

We take the following model of $E{\mathbb{Z}}/p$: the ambient space is

$S^{\infty}:=\\{(z_{0},z_{1},\dots)\ |\ z_{k}\in{\mathbb{C}}\text{ for
}k\in{\mathbb{Z}}_{\geq 0},\sum|z_{k}|^{2}=1,\text{ only finitely many
}z_{k}\text{'s are nonzero}\\},$

and the group ${\mathbb{Z}}/p$ freely acts on $S^{\infty}$ by multiplying each
coordinate by $p$-th roots of unity. The quotient space of $S^{\infty}$ under
this ${\mathbb{Z}}/p$-action is a model for the classifying space
$B{\mathbb{Z}}/p$. The group cohomology of ${\mathbb{Z}}/p$ over
${\mathbb{F}}_{p}$ is recovered as the (graded-commutative) cohomology ring

${\it H}^{*}(B{\mathbb{Z}}/p;{\mathbb{F}}_{p})={\mathbb{F}}_{p}\llbracket
u\rrbracket\langle\theta\rangle,\mathrm{deg}(u)=2\text{ and
}\mathrm{deg}(\theta)=1.$

For $\epsilon>0$ sufficiently small, $E{\mathbb{Z}}/p$ admits a
${\mathbb{Z}}/p$-invariant Morse function

$\tilde{F}(z)=\sum k|z_{k}|^{2}+\epsilon\sum\mathrm{re}(z_{k}^{p})$

obtained by perturbing the standard Morse–Bott function $\sum k|z_{k}|^{2}$ on
$S^{\infty}$ along the critical submanifolds. The function $\tilde{F}(z)$ has
the following properties:

  1. (1)

defining the map

$\displaystyle\tilde{\tau}:S^{\infty}$ $\displaystyle\to S^{\infty}$ (7.2)
$\displaystyle(z_{0},z_{1},\dots)$
$\displaystyle\mapsto(0,z_{0},z_{1},\dots),$

then we have $\tilde{F}\circ\tilde{\tau}=\tilde{F}+1$;

  2. (2)

for any $l\in{\mathbb{Z}}_{\geq 0}$, the critical points of $\tilde{F}$
obtained from perturbing the critical submanifold $\\{|z_{l}|=1\\}$ of $\sum
k|z_{k}|^{2}$ can be indexed by

$Z_{2l}^{0},\dots,Z_{2l}^{p-1},\text{ and
},Z_{2l+1}^{0},\dots,Z_{2l+1}^{p-1},$

where each $Z_{2l}^{i}$ has Morse index $2l$ and each $Z_{2l+1}^{i}$ has Morse
index $2l+1$;

  3. (3)

the sets $\\{Z_{2l}^{0},\dots,Z_{2l}^{p-1}\\}$ and
$\\{Z_{2l+1}^{0},\dots,Z_{2l+1}^{p-1}\\}$ respectively form an
${\mathbb{Z}}/p$-orbit of the ${\mathbb{Z}}/p$-action on $S^{\infty}$;

  4. (4)

there exists a ${\mathbb{Z}}/p$-equivariant Riemannian metric $\tilde{g}$ on
$S^{\infty}$ such that $(\tilde{f},\tilde{g})$ is Morse–Smale, and the
differential on the corresponding Morse cochain complex is

$\displaystyle Z_{2l}^{m}$ $\displaystyle\mapsto Z_{2l+1}^{m}-Z_{2l+1}^{m+1},$
$\displaystyle Z_{2l+1}^{m}$ $\displaystyle\mapsto
Z_{2l+2}^{0}+\cdots+Z_{2l+2}^{p-1},$

where the index $m\in{\mathbb{Z}}/p$ is read cyclically.

###  7.2. The Tate construction

Next, we review the Tate construction for cyclic groups of prime order. Let
$R$ be a unital commutative ring which is an ${\mathbb{F}}_{p}$-algebra (later
$R$ will become $\overline{\mathbb{F}}_{p}$). Suppose
$(\hat{C}_{\bullet},d_{\hat{C}})$ is a ${\mathbb{Z}}_{2}$-graded chain complex
defined over the Novikov ring $\Lambda_{0,R}$. Note that $\Lambda_{0,R}$ is a
module over $\Lambda_{0,{\mathbb{F}}_{p}}$. Introduce the graded field

${\mathcal{K}}={\mathbb{F}}_{p}[u^{-1},u\rrbracket,\ \mathrm{deg}(u)=2.$

Then the ${\mathbb{Z}}/p$-equivariant Tate complex

$C_{\mathrm{Tate}}({\mathbb{Z}}/p,\hat{C}_{\bullet}^{\otimes p})$

is a module over $\Lambda_{0,\mathcal{K}}\langle\theta\rangle$ where
$\mathrm{deg}(\theta)=1,\theta^{2}=0$, explicitly given by

$\hat{C}_{\bullet}^{\otimes
p}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}}\langle\theta\rangle.$

The differential $d_{\mathrm{Tate}}$ is
$\Lambda_{0,R}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}}$-linear,
such that for $x_{0}\otimes\cdots\otimes x_{p-1}\in\hat{C}_{\bullet}^{\otimes
p}$, we have

$\displaystyle d_{\mathrm{Tate}}(x_{0}\otimes\cdots\otimes x_{p-1})$
$\displaystyle=d_{\hat{C}}^{\otimes p}(x_{0}\otimes\cdots\otimes
x_{p-1})+\theta(id-\zeta)(x_{0}\otimes\cdots\otimes x_{p-1}),$ $\displaystyle
d_{\mathrm{Tate}}(\theta(x_{0}\otimes\cdots\otimes x_{p-1}))$
$\displaystyle=-\theta d_{\hat{C}}^{\otimes p}(x_{0}\otimes\cdots\otimes
x_{p-1})+u(id+\zeta+\cdots+\zeta^{p-1})(x_{0}\otimes\cdots\otimes x_{p-1}),$

in which the $\zeta$ is the automorphism on $\hat{C}_{\bullet}^{\otimes p}$
defined by

$x_{0}\otimes\cdots\otimes
x_{p-1}\mapsto(-1)^{|x_{p-1}|(|x_{0}|+\cdots+|x_{p-2}|)}x_{p-1}\otimes
x_{0}\otimes\cdots\otimes x_{p-2}.$

In other words, the Tate complex
$(C_{\mathrm{Tate}}({\mathbb{Z}}/p,\hat{C}_{\bullet}^{\otimes
p}),d_{\mathrm{Tate}})$ is obtained from the ${\mathbb{Z}}/p$ group cohomology
of the chain complex $(\hat{C}_{\bullet}^{\otimes p},d_{\hat{C}}^{\otimes p})$
by inverting the equivariant parameter $u$. Denote the homology of the Tate
complex by

$H_{\rm Tate}({\mathbb{Z}}/p,\hat{C}_{\bullet}^{\otimes p}).$

The following algebraic statement will be used in establishing the
localization result proved later.

######  Lemma 7.2.

[She22, Lemma 21] Denote the homology of $(\hat{C}_{\bullet},d_{\hat{C}})$ by
$\hat{H}_{\bullet}$. The $p$-th power map

$\displaystyle\hat{C}_{\bullet}$ $\displaystyle\to\hat{C}_{\bullet}^{\otimes
p}$ (7.3) $\displaystyle x$ $\displaystyle\mapsto x\otimes\cdots\otimes x$

induces an isomorphism of
$\Lambda_{0,R}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}}$-modules

$r_{p}^{*}(\hat{H}_{\bullet}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}})\to
H_{\mathrm{Tate}}({\mathbb{Z}}/p,\hat{C}_{\bullet}^{\otimes p})$

where $r_{p}$ is the operator on
$\Lambda_{0,R}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}}$
defined by mapping the Novikov variable $T$ to $T^{1/p}$.

This is referred to as the quasi-Frobenius isomorphism in [She22, Section 7].

###  7.3. ${\mathbb{Z}}/p$-equivariant vortex Floer theory

Given a 1-periodic Hamiltonian $H_{t}$ on $X$, its $p$-th iteration is the
family $H^{(p)}_{t}:=H_{pt}$ If $\phi:X\to X$ is the time-1 map of $H$, then
the time-1 map of $H^{(p)}$ is the iteration $\phi^{p}$. Following [Sei15,
SZ21], we define the ${\mathbb{Z}}/p$-equivariant vortex Hamiltonian Floer
homology for $H^{(p)}$ by using the family Floer homology coming from the
Borel construction. For all the Floer-theoretic constructions involving moduli
spaces, we always assume that the Hamiltonians involved in the discussion are
nondegenerate.

Recall that the toric divisors of $X$ are given by $D_{1},\cdots,D_{N}$, which
are obtained as the symplectic quotient of the coordinate hyperplanes
$V_{1},\cdots,V_{N}$ in the symplectic vector space $V$. As in the definition
of bulk-deformed Floer homology, we assume that the Hamiltonian $H$ is bulk-
avoiding; in particular, for any odd prime $p$, $1$-periodic orbits of $H$ and
$H^{(p)}$ are disjoint from $V_{1}\cup\cdots\cup V_{N}$. We also assume that
both $H$ and $H^{(p)}$ are nondegenerate. Let $\widehat{H}$ be an admissible
lift of $H$ and $\widehat{H}^{(p)}$ an admissible lift of $H^{(p)}$ (see
Remark 4.4). Let $\widehat{J}^{(p)}=\\{\widehat{J}_{t}^{(p)}\\}_{t\in S^{1}}$
be a $1$-periodic family of compatible almost complex structures on $V$ such
that the pair $(\widehat{H}^{(p)},\widehat{J}^{(p)})$ is admissible and the
Floer chain complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$
has a well-defined bulk-deformed differential $\partial^{(p)}_{\mathfrak{b}}$,
where ${\mathfrak{b}}=\sum_{i=1}^{N}\log c_{i}\ V_{i}$ is a chosen bulk in
which $c_{i}\in{\mathbb{Z}}[{\bf i}]$. Note that we work over $\Lambda_{0,R}$
instead of $\Lambda_{R}$, which does not introduce any further difficulty due
to the fact that $\partial^{(p)}_{\mathfrak{b}}$ preserves the energy
filtration on ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$.

To define equivariant differentials, we include more parameters from the Borel
construction. We choose an $S^{\infty}=E{\mathbb{Z}}/p$ family of time-
dependent compatible _admissible_ almost complex structures

$\widehat{J}_{\infty}^{(p)}=\\{\widehat{J}^{(p)}_{t,z}\\}_{t\in S^{1},z\in
S^{\infty}}$

satisfying the following requirements:

  1. (1)

Near each critical point $Z^{0}_{i},i\in{\mathbb{Z}}_{\geq 0}$ of the Morse
function $\tilde{F}(z)$ on $S^{\infty}$, we have
$\widehat{J}^{(p)}_{t,z}=\widehat{J}^{(p)}_{t}$;

  2. (2)

Regard ${\mathbb{Z}}/p\subset S^{1}$. For any $m\in{\mathbb{Z}}/p$ and $z\in
S^{\infty}$, there holds the equivariance relation

$\widehat{J}^{(p)}_{t-m,z}=\widehat{J}^{(p)}_{t,m\cdot z};$

  3. (3)

$\widehat{J}^{(p)}_{t,z}$ is invariant under the translation (7.2). Namely

$\widehat{J}^{(p)}_{t,\tilde{\tau}(z)}=\widehat{J}^{(p)}_{t,z}.$

After making such a choice, we can write down the following version of
parametrized vortex Floer equation. Let
${\mathfrak{x}}_{\pm}=(x_{\pm},\eta_{\pm})\in{\rm
crit}{\mathcal{A}}_{H^{(p)}}$ be a pair of equivariant $1$-periodic orbits of
$H^{(p)}$ (which do not depend on the lift $\widehat{H}^{(p)}$). Given
$i\in{\mathbb{Z}}_{\geq 0},m\in{\mathbb{Z}}/p$ and $\alpha\in\\{0,1\\}$, the
moduli space

${\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$

consists of gauge equivalence classes of pairs of smooth maps (the gauge
transformations act on the $(u,\phi,\psi)$-component)

$(u,\phi,\psi):{\mathbb{R}}_{s}\times S^{1}_{t}\to
V\times{\mathfrak{k}}\times{\mathfrak{k}},\quad\quad\quad
w:{\mathbb{R}}_{s}\to S^{\infty}$

which satisfy the equations and asymptotic conditions

$\left\\{\begin{array}[]{llll}\displaystyle\partial_{s}u+{\mathcal{X}}_{\phi}(u)+\widehat{J}_{w(s),t}^{(p)}(\partial_{t}u+{\mathcal{X}}_{\psi}(u)-X_{\widehat{H}_{t}^{(p)}}(u))=0,&\
\partial_{s}\psi-\partial_{t}\phi+\mu(u)=0,\\\
\partial_{s}w(s)-\nabla\tilde{F}(w)=0,\\\
\displaystyle\lim_{s\to-\infty}(u(s,\cdot),\phi(s,\cdot),\psi(s,\cdot),w(s))=(x_{-},0,\eta_{-},Z_{\alpha}^{0}),\\\
\displaystyle\lim_{s\to+\infty}(u(s,\cdot),\phi(s,\cdot),\psi(s,\cdot),w(s))=(x_{+},0,\eta_{+},Z_{i}^{m}),\end{array}\right.$
(7.4)

modulo the ${\mathbb{R}}$-translation action given by

$(u(s,\cdot),\phi(s,\cdot),\psi(s,\cdot),w(s))\mapsto(u(s+r,\cdot),\phi(s+r,\cdot),\psi(s+r,\cdot),w(s+r)),\quad\quad\quad
r\in{\mathbb{R}}.$

Because of the absence of sphere bubbles, as the capped orbits impose an upper
bound on energy, the moduli space
${\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$ admits a
Uhlenbeck–Gromov–Floer compactification
$\overline{{\mathcal{M}}}{}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$
by adding equivalence classes of solutions the above coupled equations defined
over broken configurations. On the other hand, for a generic choice of
$\\{J^{(p)}_{t,z}\\}_{t\in S^{1},z\in S^{\infty}}$, the moduli space
${\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$ is
transversely cut out, and the dimension of the moduli space satisfies

$\dim{\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})={\rm
CZ}({\mathfrak{x}}_{-})-{\rm CZ}({\mathfrak{x}}_{+})+i-\alpha-1.$

For a more detailed discussion of these facts, the reader may consult [Sei15,
Section 4], [SZ21, Section 6], whose arguments apply to our case after using
the setup from [Xu16, Section 6].

After achieving transversality, for each triple $i\in{\mathbb{Z}}_{\geq
0},m\in{\mathbb{Z}}/p$ and $\alpha\in\\{0,1\\}$, we can define a
$\Lambda_{0,R}$-linear map $\partial_{\alpha,{\mathfrak{b}}}^{i,m}$ on ${\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$ of the form

$\partial_{\alpha,{\mathfrak{b}}}^{i,m}({\mathfrak{x}})=\sum_{{\mathfrak{y}}\atop{\rm
CZ}({\mathfrak{x}})-{\rm
CZ}({\mathfrak{y}})+i-\alpha=1}\left(\sum_{[({\mathfrak{u}},w)]\in{{\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}},{\mathfrak{y}})}}\epsilon([({\mathfrak{u}},w)])\exp\left(\sum_{i=1}^{N}\log
c_{i}\ [{\mathfrak{u}}]\cap V_{i}\right)\right){\mathfrak{y}},$

where $\epsilon([({\mathfrak{u}},w)])\in\\{\pm 1\\}$ is the sign of the rigid
solution $[{\mathfrak{u}}]$, which is well-defined due to the existence of
coherent orientations, and $[{\mathfrak{u}}]\cap V_{i}$ is defined as before,
coming from the topological intersection number. We further introduce the
notation

$\partial_{\alpha,{\mathfrak{b}}}^{i}=\partial_{\alpha,{\mathfrak{b}}}^{i,0}+\cdots+\partial_{\alpha,{\mathfrak{b}}}^{i,p-1}.$

######  Definition 7.3.

The ${\mathbb{Z}}/p$-equivariant ${\mathfrak{b}}$-deformed vortex Floer chain
complex

${\it
VCF}_{\bullet}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R})$

is the ${\mathbb{Z}}_{2}$-graded $\Lambda_{0,R}$-module given by

${\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})\llbracket
u\rrbracket\langle\theta\rangle,\mathrm{deg}(u)=2,\mathrm{deg}(\theta)=1$

with $\Lambda_{0,R}\llbracket u\rrbracket$-linear differential

$\displaystyle\partial^{(p)}_{eq,{\mathfrak{b}}}({\mathfrak{x}}\otimes 1)$
$\displaystyle=\sum_{i\geq
0}\partial_{0,{\mathfrak{b}}}^{2i}({\mathfrak{x}})\otimes u^{i}+\sum_{i\geq
0}\partial_{0,{\mathfrak{b}}}^{2i+1}({\mathfrak{x}})\otimes u^{i}\theta,$
$\displaystyle\partial^{(p)}_{eq,{\mathfrak{b}}}({\mathfrak{x}}\otimes\theta)$
$\displaystyle=\sum_{i\geq
0}\partial_{1,{\mathfrak{b}}}^{2i+1}({\mathfrak{x}})\otimes
u^{i}\theta+\sum_{i\geq
1}\partial_{1,{\mathfrak{b}}}^{2i}({\mathfrak{x}})\otimes u^{i}.$

The statement that $(\partial^{(p)}_{eq,{\mathfrak{b}}})^{2}=0$ follows from
the signed count of boundaries of the compactified $1$-dimensional moduli
spaces
$\overline{{\mathcal{M}}}{}^{i.m}_{\alpha}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$.
The differential is well-defined over $\Lambda_{R}^{0}$ because we only
perturb the almost complex structure to achieve transversality. By
continuation map considerations, the resulting homology group

${\it
VHF}_{\bullet}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}_{\infty}^{(p)};\Lambda_{0,R})$

is independent of the choice of $\widehat{J}_{\infty}^{(p)}$, and it is a
module over $\Lambda_{0,R}\llbracket u\rrbracket\langle\theta\rangle$. By
inverting $u$, we can define

${\it
VCF}_{\mathrm{Tate}}(\widehat{H}^{(p)},\widehat{J}_{\infty}^{(p)};\Lambda_{0,R})={\it
VCF}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R})[u^{-1},u\rrbracket\langle\theta\rangle$

for which the differential is the $\Lambda_{0,R}[u^{-1},u\rrbracket$-linear
extension of $\partial^{(p)}_{eq,{\mathfrak{b}}}$. The homology group is
written as

${\it VHF}_{\mathrm{Tate}}(\widehat{H}^{(p)};\Lambda_{0,R}),$

which is a module over
$\Lambda^{0,R}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda^{0,{\mathcal{K}}}\langle\theta\rangle$.

Here is some explanation of the definition of the equivariant differential. By
definition, the leading order term $\partial_{0,{\mathfrak{b}}}^{0}$ agrees
with the differential $\partial^{(p)}_{\mathfrak{b}}$ on the complex ${\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$, so does
$\partial^{1}_{1,{\mathfrak{b}}}$. The space of equivariant loops $L^{K}(V)$
admits an $S^{1}$-action by shifting the domain parameter, and the natural
inclusion ${\mathbb{Z}}/p\subset S^{1}$ defines a ${\mathbb{Z}}/p$-action on
$L^{K}(V)$ such that the action functional ${\mathcal{A}}_{H^{(p)}}$ is
invariant under such an action. More concretely, the reparametrization

${\mathfrak{x}}(t)=(x(t),\eta(t))\mapsto(x(t+\frac{1}{p}),\eta(t+\frac{1}{p}))$

generates a ${\mathbb{Z}}/p$-action on the Floer homology

$R_{1/p}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})\to{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})$

which is realized by the composition

${\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})\xrightarrow[\text{pullback}]{\sim}{\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\cdot-1/p};\Lambda_{0,R})\xrightarrow{\text{continuation}}{\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$

after passing to homology. Here $\widehat{J}_{\cdot-\frac{1}{p}}^{(p)}$ is the
$S^{1}$-family of almost complex structures whose value at moment $t$ is
$\widehat{J}_{t-\frac{1}{p}}^{(p)}$. The action $R_{1/p}$ generates a
${\mathbb{Z}}/p$-action on homology; we denote

$R_{m/p}:=(R_{1/p})^{m}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})\to{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R}).$

Then the map $\partial^{1}_{0,{\mathfrak{b}}}$ descends to

$id-R_{1/p}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})\to{\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})$

on homology, while the map $\partial^{2}_{1,{\mathfrak{b}}}$ descends to
$id+R_{1/p}+\cdots+R_{(p-1)/p}$. The higher order terms encodes the chain
homotopies realizing relations of the form $(R_{1/p})^{p}=id$ on homology, and
higher homotopy relations.

Finally, we observe that the degree filtration on the chain complex ${\it
VCF}_{\bullet}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}_{\infty}^{(p)};\Lambda_{R}^{0})$
induced from variables $u$ and $\theta$ is preserved by the equivariant
differential $\partial^{(p)}_{eq,{\mathfrak{b}}}$, and such a filtration is
complete and exhaustive. Therefore, we have a spectral sequence converging to
${\it VHF}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)};\Lambda_{0,R})$, whose first
page can be identified with ${\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H}^{(p)};\Lambda_{0,R})\llbracket
u\rrbracket\langle\theta\rangle$. The same holds for the Tate version, which
inverts the variable $u$.

###  7.4. Equivariant $p$-legged pants operations

In this subsection, we define equivariant “$p$-legged” pants operations on
vortex Hamiltonian Floer theory, which generalizes the constructions from
[Sei15, SZ21] to our situation. We use the homological convention, so the
roles of the positive and negative cylindrical ends are the opposite of those
from _loc. cit._. We will continue the setup from the previous subsection, and
keep using the notations $H$, $\widehat{H}$, $H^{(p)}$, $\widehat{H}^{(p)}$,
and $\widehat{J}_{t,z}^{(p)}$. Furthermore, we choose a $1$-parameter family
of compatible almost complex structures $\widehat{J}$ on $V$ such that
$(\widehat{H},\widehat{J})$ is regular and the Floer chain complex ${\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{0,R})$ is well-
defined.

The equivariant pants operation is defined over a particularly designed
domain. Let $\pi:S_{\mathcal{P}}\to{\mathbb{R}}\times S^{1}$ be the $p$-fold
branched cover with unique branch point $(0,0)\in{\mathbb{R}}\times S^{1}$
whose ramification point has maximal ramification order. Then
$S_{\mathcal{P}}$ has $p+1$ punctures, regarded as $p$ negative ends and one
positive ends. Suppose $S_{\mathcal{P}}$ are equipped with cylindrical ends

$\epsilon_{i}^{-}:(-\infty,-1]\times S^{1}\to
S_{\mathcal{P}},\quad\epsilon_{i}^{+}:[1,\infty)\times S^{1}_{p}\to
S_{\mathcal{P}},\quad i\in{\mathbb{Z}}/p,$

subject to the conditions

$\displaystyle\pi(\epsilon_{i}^{-}(s,t))$ $\displaystyle=(s,t),\quad
m\cdot(\epsilon_{i}^{-}(s,t))=\epsilon_{i+m}^{-}(s,t)$
$\displaystyle\pi(\epsilon_{i}^{+}(s,t))$ $\displaystyle=(s,t),\quad
m\cdot(\epsilon_{i}^{+}(s,t))=\epsilon_{i+m}^{+}(s,t)=\epsilon_{i}^{+}(s,t+m),\quad\text{
for }m\in{\mathbb{Z}}/p,$

where $S^{1}_{p}:={\mathbb{R}}/p{\mathbb{Z}}$ is the $p$-fold cover of
$S^{1}={\mathbb{R}}/{\mathbb{Z}}$. Note that all $\epsilon_{i}^{+}$ are
obtained from shifting from each other by certain $m\in{\mathbb{Z}}/p$.

The domain-dependent almost complex structure needs to have particular
symmetry. We consider almost complex structures $\widehat{J}^{+}_{\infty}$ on
$V$ parametrized by $z\in S^{\infty}$, $t\in S^{1}$, and $s\geq-1$, such that:

  1. (1)

for $s\geq 2$ and $z\in S^{\infty}$, we have
$\widehat{J}^{+}_{s,t,z}=\widehat{J}^{(p)}_{t,z}$;

  2. (2)

for any $m\in{\mathbb{Z}}/p$ and $z\in S^{\infty}$, there holds the
equivariance relation

$\widehat{J}^{+}_{s,t-\frac{m}{p},z}=\widehat{J}^{+}_{s,t,m\cdot z};$

  3. (3)

$\widehat{J}^{+}_{s,t,z}$ is invariant under the translation:

$\widehat{J}^{+}_{s,t,\tilde{\tau}(z)}=\widehat{J}^{+}_{s,t,z}.$

Given such a choice, we further look at almost complex structures
$\widehat{J}^{-,i}_{\infty}$ parametrized by with $s\leq 1$, $t\in S^{1}$,
$z\in S^{\infty}$, and indexed by $i\in{\mathbb{Z}}/p$ (the label of negative
ends) satisfying:

  1. (1)

for $s\leq-2$ and any $w\in S^{\infty}$, we have
$\widehat{J}^{-,i}_{s,t,z}=\widehat{J}_{t}$ for any $i\in{\mathbb{Z}}/p$;

  2. (2)

for any $i\in{\mathbb{Z}}/p$ and $w\in S^{\infty}$, we have the equality
$\widehat{J}^{-,i}_{s,t,z}=\widehat{J}^{+}_{s,t,z}$ hold for $-1\leq s\leq 1$;

  3. (3)

for any $m,i\in{\mathbb{Z}}/p$ and $z\in S^{\infty}$, there holds the
equivariance relation

$\widehat{J}^{-,i}_{s,t-\frac{m}{p},z}=\widehat{J}^{-,i+m}_{s,t,z};$

  4. (4)

$\widehat{J}^{-,i}_{s,t,z}$ is invariant under the translation:

$\widehat{J}^{-,i}_{s,t,\tilde{\tau}(z)}=\widehat{J}^{-,i}_{s,t,z}.$

If $w:{\mathbb{R}}\to S^{\infty}$ is a parametrized negative gradient flow
line of $\tilde{F}$, the above data specify a family of almost complex
structures $\\{\widehat{J}^{\mathcal{P}}_{v,w}\\}_{v\in S_{\mathcal{P}}}$
given by:

  1. (1)

$\widehat{J}^{\mathcal{P}}_{v,w}=\pi^{*}\widehat{J}^{-,i}_{s,t,w(s)}=\pi^{*}J^{+}_{s,t,w(s)}$
for $v\in\pi^{-1}([-1,1]\times S^{1})$ and $\pi(v)=(s,t)$;

  2. (2)

over the negative ends,
$\widehat{J}^{\mathcal{P}}_{v,w}=\pi^{*}\widehat{J}^{-,i}_{s,t,w(s)}$ if
$v=\epsilon_{i}^{-}(s,t)$ for all $i=0,1,\dots,p-1$;

  3. (3)

over the positive end,
$\widehat{J}^{\mathcal{P}}_{v,w}=\widehat{J}^{+}_{s,t,m\cdot w(s)}$ for all
$m\in{\mathbb{Z}}/p$ and $z=\epsilon_{m}^{+}(s,t)$.

We need to further introduce a Hamiltonian perturbation term

$\widehat{\mathcal{H}}^{\mathcal{P}}\in\Omega^{1}(S_{\mathcal{P}},C^{\infty}(V)^{K})$

satisfying the following conditions.

  1. (1)

For any $i\in{\mathbb{Z}}/p$, we have
$\widehat{\mathcal{H}}^{\mathcal{P}}(\epsilon_{i}^{-}(s,t))=\widehat{H}_{t}\otimes
dt$;

  2. (2)

On the positive end, for any $i\in{\mathbb{Z}}/p$, there holds
$\widehat{\mathcal{H}}^{\mathcal{P}}(\epsilon_{i}^{+}(s,t))=\widehat{H}_{t+i}^{(p)}\otimes
dt$;

  3. (3)

The ${\mathbb{Z}}/p$-equivariance condition
$\widehat{\mathcal{H}}^{\mathcal{P}}(m\cdot
v)=\widehat{\mathcal{H}}^{\mathcal{P}}(v)$ holds;

  4. (4)

Let ${\mathcal{H}}^{\mathcal{P}}\in\Omega^{1}(S_{\mathcal{P}},C^{\infty}(X))$
be the induced Hamiltonian perturbation term on $X$. Then the curvature of the
Hamiltonian connection ${\mathcal{H}}^{\mathcal{P}}$ on $S_{\mathcal{P}}$ is
$0$.

Consider moduli spaces of perturbed vortex equation over the surface
$S_{\mathcal{P}}$. Let $P\to S_{\mathcal{P}}$ be the trivial $K$-bundle. Given
${\mathfrak{x}}_{+}=(x_{+},\eta_{+})\in{\rm crit}{\mathcal{A}}_{H^{(p)}}$ and
${\mathfrak{x}}_{0}=(x_{0},\eta_{0}),\dots,{\mathfrak{x}}_{p-1}=(x_{p-1},\eta_{p-1})\in{\rm
crit}{\mathcal{A}}_{H}$, for any $i\in{\mathbb{Z}}_{\geq
0},m\in{\mathbb{Z}}/p$ and $\alpha\in\\{0,1\\}$ we can introduce the moduli
space

${\mathcal{M}}^{i.m}_{{\mathcal{P}},\alpha}({\mathfrak{x}}_{0},\dots,{\mathfrak{x}}_{p-1};{\mathfrak{x}}_{+})$

which parametrizes gauge equivalence classes of pairs

$(u,A)\in C^{\infty}(S_{\mathcal{P}},V)\times{\mathcal{A}}(P),\quad\quad\quad
w:{\mathbb{R}}_{s}\to S^{\infty}$

which satisfy the equations and asymptotic conditions

$\left\\{\begin{array}[]{llll}\displaystyle\overline{\partial}_{A,\widehat{\mathcal{H}}^{\mathcal{P}},\widehat{J}^{\mathcal{P}}_{v,w}}u=0,\quad\quad\quad*F_{A}+\mu(u)=0,\\\
w^{\prime}(s)-\nabla\tilde{F}(w)=0,\\\
\displaystyle\lim_{s\to-\infty}(u(\epsilon_{j}^{-}(s,\cdot)),A(\epsilon_{j}^{-}(s,\cdot)),w(\epsilon_{j}^{-}(s,\cdot)))=(x_{j},0,\eta_{j},Z_{\alpha}^{0}),\quad\forall
j\in{\mathbb{Z}}/p,\\\
\displaystyle\lim_{s\to\infty}(u(\epsilon_{0}^{+}(s,\cdot)),A(\epsilon_{0}^{+}(s,\cdot)),w(\epsilon_{0}^{+}(s,\cdot)))=(x_{+},0,\eta_{+},Z_{i}^{m}).\end{array}\right.$

As expected, the moduli space
${\mathcal{M}}^{i.m}_{{\mathcal{P}},\alpha}({\mathfrak{x}}_{0},\dots,{\mathfrak{x}}_{p-1};{\mathfrak{x}}_{+})$
admits an Uhlenbeck–Gromov–Floer compactification
$\overline{{\mathcal{M}}}{}^{i.m}_{{\mathcal{P}},\alpha}({\mathfrak{x}}_{0},\dots,{\mathfrak{x}}_{p-1};{\mathfrak{x}}_{+})$,
whose detailed description can be found in [Sei15, Section (4c)]. For a
generic choice of almost complex structures and Hamiltonian connections, the
moduli space
${\mathcal{M}}^{i.m}_{{\mathcal{P}},\alpha}({\mathfrak{x}}_{0},\dots,{\mathfrak{x}}_{p-1};{\mathfrak{x}}_{+})$
is cut out transversely, whose dimension is given by

${\rm CZ}({\mathfrak{x}}_{0})+\cdots+{\rm CZ}({\mathfrak{x}}_{p-1})-{\rm
CZ}({\mathfrak{x}}_{+})+i-\alpha.$

We define the pants operations using the above moduli spaces. For each
$i\in{\mathbb{Z}}_{\geq 0}$, $m\in{\mathbb{Z}}/p$, and $\alpha\in\\{0,1\\}$,
define

$\displaystyle{\mathcal{P}}^{i,m}_{\alpha,{\mathfrak{b}}}:{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p}$
$\displaystyle\to{\it
VCF}_{\bullet}(\widehat{H}^{(p)},\widehat{J}^{(p)};\Lambda_{0,R})$
$\displaystyle{\mathfrak{x}}_{0}\otimes\cdots\otimes{\mathfrak{x}}_{p-1}\mapsto\sum_{{\mathfrak{x}}\atop{\rm
CZ}({\mathfrak{x}}_{0})+\cdots+{\rm CZ}({\mathfrak{x}}_{p-1})-{\rm
CZ}({\mathfrak{x}})+i-\alpha=0}$
$\displaystyle\left(\sum_{[({\mathfrak{u}},w)]\in{{\mathcal{M}}^{i.m}_{\alpha}({\mathfrak{x}}_{0},\dots,{\mathfrak{x}}_{p-1};{\mathfrak{x}})}}\epsilon([({\mathfrak{u}},w)])\exp\left(\sum_{i=1}^{N}\log
c_{i}\ [{\mathfrak{u}}]\cap V_{i}\right)\right){\mathfrak{x}},$

where a discussion on the sign $\epsilon([({\mathfrak{u}},w)])$ can be found
in [SZ21, Appendix A].

######  Definition 7.4.

Let
${\mathcal{P}}^{i}_{\alpha,{\mathfrak{b}}}={\mathcal{P}}^{i,0}_{\alpha,{\mathfrak{b}}}+\cdots+{\mathcal{P}}^{i,p-1}_{\alpha,{\mathfrak{b}}}$.
The ${\mathbb{Z}}/p$-equivariant product is defined to be

$\displaystyle{\mathcal{P}}:{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p}\llbracket
u\rrbracket\langle\theta\rangle$ $\displaystyle\to{\it
VCF}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R})$
$\displaystyle{\mathcal{P}}(-\otimes 1)$ $\displaystyle=\sum_{i\geq
0}{\mathcal{P}}^{2i}_{0,{\mathfrak{b}}}\otimes u^{i}+\sum_{i\geq
0}{\mathcal{P}}^{2i+1}_{0,{\mathfrak{b}}}\otimes u^{i}\theta,$
$\displaystyle{\mathcal{P}}(-\otimes\theta)$ $\displaystyle=\sum_{i\geq
1}{\mathcal{P}}^{2i}_{1,{\mathfrak{b}}}\otimes u^{i}+\sum_{i\geq
0}{\mathcal{P}}^{2i+1}_{1,{\mathfrak{b}}}\otimes u^{i}\theta.$

We can apply the Tate construction to the complex ${\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})$, obtaining the Tate
complex

$(C_{\mathrm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes
p}),\partial_{\mathrm{Tate}}).$

By inverting $u$, the ${\mathbb{Z}}/p$-equivariant product ${\mathcal{P}}$
induces a map

$C_{\mathrm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p})\to{\it
VCF}_{\mathrm{Tate}}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R}),$

which is also denoted by ${\mathcal{P}}$. By [SZ21, Section 8.3],
${\mathcal{P}}$ defines a chain map on the Tate chain complexes. In fact, the
chain map property holds without inverting $u$, but we will not need such a
statement.

We also need to define a ${\mathbb{Z}}/p$-equivariant coproduct operation

${\mathcal{C}}:{\it
VCF}^{{\mathbb{Z}}/p}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R})\to{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p}\llbracket
u\rrbracket\langle\theta\rangle$

by counting solutions to Hamiltonian-perturbed vortex equations defined over
the Riemann surface $S_{\mathcal{C}}$ which is a $p$-fold branched cover of
${\mathbb{R}}\times S^{1}$ with unique branch point
$(0,0)\in{\mathbb{R}}\times S^{1}$ whose ramification point has maximal
ramification order, coupled with negative gradient trajectory equations of
$\tilde{F}:S^{\infty}\to{\mathbb{R}}$, but this time $S_{\mathcal{C}}$ has a
single negative cylindrical end and $p$ positive cylindrical ends. By
inverting the $u$-variable, ${\mathcal{C}}$ induces a map

${\it
VCF}_{\mathrm{Tate}}(\widehat{H}^{(p)},\widehat{J}^{(p)}_{\infty};\Lambda_{0,R})\to
C_{\mathrm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p}).$

###  7.5. Local theories and equivariant localization

In this subsection, we sketch the ingredients necessary for the proof of the
following statement:

######  Theorem 7.5.

The equivariant ${\mathbb{Z}}/p$-equivariant product on homology

${\mathcal{P}}:{\it H}_{\rm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes
p})\to{\it
VHF}_{\mathrm{Tate}}(\widehat{H}^{(p)},\widehat{J}_{\infty}^{(p)};\Lambda_{0,R})$
(7.5)

is an isomorphism.

First, one needs an equivariant version of the local vortex Floer theory as
discussed in Section 5. Namely, if $x$ is a $1$-periodic orbit of the
Hamiltonian $H$, and if we denote by $x^{(p)}$ the $p$-th iteration of $x$,
which is necessarily a $1$-periodic orbit of $H^{(p)}$, then there is a well-
defined (bulk-deformed) ${\mathbb{Z}}/p$-equivariant local Floer homology
group

${\it VHF}^{{\mathbb{Z}}/p}_{\rm loc}(H^{(p)},x^{(p)};\Lambda_{0,R}),$

which is defined by looking at contributions to
$\partial^{(p)}_{eq,{\mathfrak{b}}}$ from solutions to (7.4) which are
contained in a $C^{2}$-small neighborhood of the equivariant lift of
$x^{(p)}$. By inverting $u$, the Tate version is denoted by ${\it VHF}_{\rm
Tate}^{\rm loc}(H^{(p)},x^{(p)};\Lambda_{0,R})$. One can similarly define the
local version of the ${\mathbb{Z}}/p$-equivariant product and coproduct
operation localized near $x$

$\displaystyle{\mathcal{P}}_{x}^{\rm loc}:H_{\rm{Tate}}({\mathbb{Z}}/p,{\it
VCF}^{\rm loc}_{\bullet}(H,x;\Lambda_{0,R})^{\otimes p})\to{\it
VHF}_{\rm{Tate}}^{\rm loc}(H^{(p)},x^{(p)};\Lambda_{0,R}),$
$\displaystyle{\mathcal{C}}_{x}^{\rm loc}:{\it VHF}_{\rm{Tate}}^{\rm
loc}(H^{(p)},x^{(p)};\Lambda_{R}^{0})\to H_{\rm{Tate}}({\mathbb{Z}}/p,{\it
VCF}^{\rm loc}(H,x;\Lambda_{0,R})^{\otimes p}).$

Note that just as in the non-equivariant setting, equivariant local Floer
theories can be defined for isolated but not necessarily nondegenerate
iterations.

Second, the main result of [SZ21, Section 10] shows that if $x$ and $x^{(p)}$
are nondegenerate, the composition satisfies

${\mathcal{C}}_{x}^{\rm loc}\circ{\mathcal{P}}_{x}^{\rm
loc}=(-1)^{n}u^{n(p-1)}\cdot id,$

which is an isomorphism as $u$ is invertible in the ground ring of the Tate
version, thus ${\mathcal{P}}_{x}^{\rm loc}$ is an isomorphism by rank
considerations. The proof in _loc. cit._ goes through an auxiliary operation
${\mathcal{Z}}_{x}^{\rm loc}$ satisfying ${\mathcal{C}}_{x}^{\rm
loc}\circ{\mathcal{P}}_{x}^{\rm loc}={\mathcal{Z}}_{x}^{\rm loc}$, which can
be defined in our setting following [SZ21, Definition 10.1]. On the other
hand, the calculation ${\mathcal{Z}}_{x}^{\rm loc}=(-1)^{n}u^{n(p-1)}$ is
based on reducing to the case of local Morse theory by a deformation argument,
which is also legitimate in the vortex setting. Then by virtue of the proof of
Proposition 5.2, when the Hamiltonian $H$ is a $C^{2}$-small Morse function,
we can match the upstairs and downstairs moduli spaces, so that the
calculation also works in our setting. Note that by Proposition 5.7, the
presence of bulk ${\mathfrak{b}}$ does not affect the argument.

Finally, we can write

${\mathcal{P}}=\sum_{x}{\mathcal{P}}_{x}^{\rm
loc}+O(T^{\delta}),\quad\quad\quad\delta>0,$

in which $x$ ranges over all $1$-periodic orbits of $H$ and $O(T^{\delta})$
denotes an operation with positive valuation. Because the local operations
${\mathcal{P}}_{x}^{\rm loc}$ are isomorphisms, and the contributions of the
simple (i.e., non-iterated) $1$-periodic orbits of $H^{(p)}$ to the Tate
construction are trivial, we see that ${\mathcal{P}}$ is an isomorphism over
$\Lambda_{0,R}[u^{-1},u\rrbracket=\Lambda_{0,R}\otimes_{\Lambda_{0,{\mathbb{F}}_{p}}}\Lambda_{0,\mathcal{K}}$.
This finishes the sketch of the proof of the equivariant localization
isomorphism.

###  7.6. Growth of total bar length

After demonstrating the existence of the equivariant Hamiltonian Floer package
in the vortex setting, we are in the right position to prove the inequality of
total bar length.

###### Proof of Theorem 7.1.

With equivariant Hamiltonian Floer theory and local Floer homology in our
hands, the arguments from [She22, Section 7] can easily be adapted to the
current situation without much modification. Consequently, we will only
provide a sketch of the proof, and refer the reader to _loc. cit._ for
complete arguments.

Firstly, we recall the following alternative characterization of the total bar
length. Given a field ${\mathbb{K}}$, if we define the vortex Hamiltonian
Floer homology over the Novikov ring

$\Lambda_{0,\mathbb{K}}:=\Big{\\{}\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\ |\
g_{i}\in{\mathbb{R}}_{\geq 0},\ a_{i}\in{\mathbb{K}},\
\lim_{i\to\infty}g_{i}=+\infty\Big{\\}}$

instead of its field of fractions $\Lambda_{\mathbb{K}}$, then finite bars are
reflected as nontrivial _torsion components_ , which is the language used in
[FOOO13]. If we denote the direct sum of the torsion components of the Floer
homology ${\it
VHF}_{\bullet}^{\mathfrak{b}}(\tilde{\phi};\Lambda_{0,\mathbb{K}})$ by

$\Lambda_{0,\mathbb{K}}/T^{g_{1}}\Lambda_{0,\mathbb{K}}\oplus\cdots\oplus\Lambda_{0,\mathbb{K}}/T^{g_{s}}\Lambda_{0,\mathbb{K}},\quad\quad\quad\text{
with }\ g_{1}\geq\cdots\geq g_{s}\geq 0,$ (7.6)

then we can write the total bar length of $\tilde{\phi}$ over
$\Lambda_{\mathbb{K}}$ as

$\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi},\Lambda_{\mathbb{K}})=g_{1}+\cdots+g_{s},$

and the boundary depth is given by $g_{1}$, c.f. [She22, Section 4.4.4]. Note
that these torsion exponents correspond to _verbose bar-length spectrum_ in
the sense of [UZ16], which means that $g_{i}$ can be $0$, due to that fact
that the Floer differential in our discussion may not strictly decrease the
energy.

The claim is more easily to be proved when $\tilde{\phi}^{p}$ is nondegenerate
and bulk-avoiding. We choose a generating Hamiltonian $H$ for the Hamiltonian
isotopy $\tilde{\phi}$. The comparison between
$\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi}^{p})$ and
$\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi})$ is established in the following
three steps.

  1. (1)

Using the quasi-Frobenius isomorphism from Lemma 7.2, one can show that the
total bar length of

$(C_{\mathrm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes
p}),\partial_{\mathrm{Tate}}),$

i.e., the sum of torsion exponents of the homology group ${\it
H}_{\rm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p})$, is equal
to $p$ times the quantity $\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi})$.

  2. (2)

By appealing to the isomorphism in Equation 7.5 and an application of the
homological perturbation lemma, it is shown in [She22, Section 7.3.1] the
total boundary depth, i.e., the sum of torsion exponents of ${\it
H}_{\rm{Tate}}({\mathbb{Z}}/p,{\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{0,R})^{\otimes p})$ agrees
with that of ${\it VHF}_{\mathrm{Tate}}(\widehat{H}^{(p)};\Lambda_{0,R})$.

  3. (3)

Using [She22, Proposition 17, Lemma 18], one can prove that the total boundary
depth of ${\it VHF}_{\mathrm{Tate}}(\widehat{H}^{(p)};\Lambda_{0,R})$ is
bounded from above by $\tau_{(p)}^{\mathfrak{b}}(\tilde{\phi}^{p})$, which is
a reminiscent of the Borel spectral sequence in the context of filtered Floer
theory.

Finally, to establish Theorem 7.1 for $\tilde{\phi}$ and $\tilde{\phi}^{p}$
which are not necessarily bulk-avoiding and may admit isolated degenerate
fixed points, an approximation argument and multiple applications of the
homological perturbation lemma as in [She22, Section 7.4] suffice. ∎

######  Remark 7.6.

To prove Theorem 7.1 for degenerate Hamiltonian diffeomorphisms with isolated
fixed points assuming the corresponding result for nondegenerate ones, one can
alternatively use the following more elementary argument. Suppose $H$ is not
necessarily bulk-avoiding and may have isolated but degenerate fixed points
and periodic points of period $p$. Let $H_{i}$ be a sequence of nondegenerate
and bulk-avoiding Hamiltonians on $X$ which converges to $H$ under
$C^{2}$-norm. We can choose the perturbations $H_{i}$ to be supported in an
arbitrarily small neighborhood of the one period orbits of $H$ and $H^{(p)}$,
over which the perturbation is modeled on $\epsilon_{i}f$ where $f$ is a Morse
function. Then the above implies that

$\tau_{(p)}^{\mathfrak{b}}(H_{i}^{(p)})\geq
p\tau_{(p)}^{\mathfrak{b}}(H_{i}).$

Notice that the reduced barcode of $H_{i}$ resp. $H_{i}^{(p)}$ is a Cauchy
sequence with respect to the bottleneck distance with a uniformly bounded
number of bars. Our choice of perturbation also guarantees a uniform upper
bound for the short bars. Moreover, we know that the barcode of the limit $H$
resp. $H^{(p)}$ is finite. Hence the total bar length of $H_{i}$ resp.
$H_{i}^{(p)}$ converges to that of $H$ resp. $H^{(p)}$, which implies the
desired result.

##  8\. Open string theory I. Quasimap Floer theory

In this section we recall the construction of quasimap Lagrangian Floer theory
developed by Woodward [Woo11]. The basic idea agrees with the philosophy of
gauged linear sigma model [Wit93]: one replaces the count of holomorphic
curves in the toric manifold $X$ by an equivariant count of holomorphic curves
upstairs. There are two significant consequences: first, one can achieve
transversality of moduli spaces at a very low cost; second, the counts of
curves are all integer-valued.

We use the Morse–Bott model for Lagrangians Floer theory to construct open-
string theories and closed-open maps. We extend the use of domain-dependent
perturbations for bulk-deformed vortex Floer cohomology to the open-string
situation. We first need to fix certain notions and notations to describe the
combinatorial data of various moduli spaces.

###  8.1. Trees and treed disks

We first set up the convention for the notion of trees used in this paper.

######  Convention 8.1 (Convention for trees).

A tree, usually denoted by $\Gamma$, consists of a nonempty set of vertices
$V_{\Gamma}$ and a nonempty set of edges $E_{\Gamma}$. The set of vertices is
decomposed into the set of finite vertices and the set of vertices at
infinity, and the decomposition is denote by

$V_{\Gamma}=V_{\Gamma}^{\rm finite}\sqcup V_{\Gamma}^{\infty}.$

We always assume

  1. (1)

$V_{\Gamma}^{\infty}$ contains a distinguished vertex $v_{\rm root}$ called
the root.

  2. (2)

The valence (degree) of any $v\in V_{\Gamma}^{\infty}$ is either one or two.

The set $V_{\Gamma}$ is partially ordered in the following way: we denote by
$v_{\alpha}\succ v_{\beta}$ if $v_{\alpha}$ and $v_{\beta}$ are adjacent and
$v_{\beta}$ is closer to the root. In this way vertices at infinities are
either incoming (called inputs) or outgoing (called outputs); in particular
the output $v_{\rm root}$ is outgoing.

Edges are decomposed into four groups: the set of finite edges
$E_{\Gamma}^{\rm finite}$ consisting of edges connecting two finite vertices,
the set of incoming semi-infinite edges $E_{\Gamma}^{\rm in}$ consisting of
edges connecting $v_{\alpha}\in V_{\Gamma}^{\infty}$ with $v_{\beta}\in
V_{\Gamma}^{\rm finite}$ with $v_{\alpha}\succ v_{\beta}$, the set of outgoing
semi-infinite edges $E_{\Gamma}^{\rm out}$ consisting edges connecting
$v_{\alpha}\in V_{\Gamma}^{\rm finite}$ and $v_{\beta}\in V_{\Gamma}^{\infty}$
with $v_{\alpha}\succ v_{\beta}$, and the set of infinite edges
$E_{\Gamma}^{\infty}$ connecting two vertices at infinity. We also call
incoming resp. outgoing semi-infinite edges inputs resp. outputs.

A tree $\Gamma$ is called unbroken if all vertices $v\in V_{\Gamma}^{\infty}$
has valence $1$. A vertex $v\in V_{\Gamma}^{\infty}$ of valence $2$ is called
a breaking of the tree $\Gamma$. Breakings separate $\Gamma$ into unbroken
components.

A ribbon tree is a tree $\Gamma$ together with an isotopy class of embeddings
$\Gamma\hookrightarrow{\mathbb{R}}^{2}$. Equivalently, it means for each
vertex $v\in V_{\Gamma}$ the adjacent edges are cyclically ordered. As
$\Gamma$ is rooted, it follows that all incoming edges are strictly ordered.

A ribbon tree is stable if the valence of each finite vertex is at least
three.

####  8.1.1. Metric ribbon trees

A metric on a ribbon tree $\Gamma$ is a function

${\bm{l}}:E_{\Gamma}^{\rm finite}\to[0,+\infty).$

The underlying decomposition

$E_{\Gamma}^{{\rm finite}}=E_{\Gamma}^{{\rm finite},0}\sqcup E_{\Gamma}^{{\rm
finite},+}={\bm{l}}^{-1}(\\{0\\})\sqcup{\bm{l}}^{-1}((0,+\infty))$

is called a metric type, denoted by $[{\bm{l}}]$. We often call the pair
$(\Gamma,[{\bm{l}}])$ a domain type. A metric ribbon tree of type
$(\Gamma,[{\bm{l}}])$ is a pair $(\Gamma,{\bm{l}})$ such that ${\bm{l}}$ has
the metric type $[{\bm{l}}]$.

As in [Woo11, Section 3.3], one needs to work with unstable trees. We hence
replace the usual stability condition by another minimality condition. We say
that a metric ribbon trees $(\Gamma,{\bm{l}})$ (resp. domain type
$(\Gamma,[{\bm{l}}])$) is minimal if it has no finite edges of length zero or
infinite edges. Hence for each domain type $\Gamma$, there is a canonical
minimal one $\Gamma^{\rm min}$ obtained from $\Gamma$ by shrinking edges
violating the minimality condition.

We define perturbations over the universal trees. Consider a minimal domain
type $\Gamma=(\Gamma,[{\bm{l}}])$ (which is not necessarily stable). Then
there is a moduli space of metric trees of type $\Gamma$, denoted by
$\mathcal{MT}_{\Gamma}$, which is homeomorphic to
$(0,+\infty)^{\\#E_{\underline{\smash{\Gamma}}}^{{\rm finite},+}}$, whose
elements parametrize the lengths of finite edges with positive lengths. There
is also a universal tree

$\mathcal{UT}_{\Gamma}\to\mathcal{MT}_{\Gamma}$

whose fiber over a point $p\in\mathcal{MT}_{\Gamma}$ is homeomorphic to a
metric tree representing $p$ (the infinities of semi-infinite or infinite
edges are regarded as points in the metric tree).

The above moduli spaces have natural compactifications. In fact, we can define
a partial order among all minimal domain types. We say that a minimal domain
type $\Gamma$ degenerates to another minimal domain type $\Pi$, denoted by
$\Pi\preceq\Gamma$, if $\Pi$ is obtained from $\Gamma$ by composing the
following types of operations

  1. (1)

Shrinking the length of a finite edge in $\Gamma$ to zero and collapse this
edge.

  2. (2)

Breaking a finite edge of positive length to a pair of semi-infinite edges
joined at a new vertex at infinity.

Notice that if $\Pi\preceq\Gamma$, then there is a canonical surjective map
$\rho:V_{\Gamma}^{\rm finite}\to V_{\Pi}^{\rm finite}$. Then
$\mathcal{MT}_{\Gamma}$ has the natural compactification

$\overline{\mathcal{MT}}_{\Gamma}:=\bigsqcup_{\Pi\preceq\Gamma}\mathcal{MT}_{\Pi}.$

The universal tree is also extended to the compactification, which is denoted
by

$\overline{\mathcal{UT}}_{\Gamma}\to\overline{\mathcal{MT}}_{\Gamma}.$

There is a special closed subset $\overline{\mathcal{UT}}{}_{\Gamma}^{\rm
node}\subset\overline{\mathcal{UT}}_{\Gamma}$ corresponding to infinities or
vertices. Notice that the complement of
$\overline{\mathcal{UT}}{}_{\Gamma}^{\rm node}$ inside the interior
$\mathcal{UT}_{\Gamma}$ is a smooth manifold.

####  8.1.2. Treed disks

######  Definition 8.2.

Given a domain type $\Gamma=(\Gamma,[{\bm{l}}])$. A treed disk of type
$\Gamma$, denoted by $C=S\cup T$, is the configuration given by the union of
disk components $S_{\alpha}\cong{\mathbb{D}}$ for all vertices $v_{\alpha}\in
V_{\Gamma}^{\rm finite}$, a metric ${\bm{l}}$ on $\Gamma$ of type $[{\bm{l}}]$
and an interval $I_{e}$ of length ${\bm{l}}(e)$ for each finite edge $e\in
E_{\Gamma}^{\rm finite}$. The notion of isomorphisms between treed disks is
standard and omitted.

###  8.2. Quasimap Floer theory for Lagrangians

We recall the quasimap Floer theory developed by Woodward [Woo11]. Let ${\bf
u}\in{\rm Int}P\subset{\mathbb{R}}^{n}$ be an interior point of the moment
polytope $P$ of the toric manifold $X$. Recall that the number of faces $N$ of
$P$ coincides with the dimension of $V$. Let $L=L({\bf u})\subset X$ be the
torus fiber over ${\bf u}$. Let $\widehat{L}=\widehat{L}({\bf
u})\subset\mu^{-1}(0)\subset V$ be the lift of $L({\bf u})$, which is a
$K$-invariant Lagrangian torus in $V$. Explicitly, we have

$\widehat{L}=\prod_{i=1}^{N}\Big{\\{}z_{i}\in{\mathbb{C}}\ |\
|z_{i}|^{2}=\tau_{i}\Big{\\}}$

where $\tau_{i}$ are determined by ${\bf u}$ and the constant term in the
moment map $\mu$.

A holomorphic quasidisk is an ordinary holomorphic map
$u:({\mathbb{D}},\partial{\mathbb{D}})\to(V,\widehat{L})$ (with respect to the
standard complex structure $\widehat{J}_{V}$). Two holomorphic quasidisks $u$
and $u^{\prime}$ are $K$-equivalent if there exists $g\in K$ such that
$gu=u^{\prime}$. Each $K$-equivalence class of holomorphic quasidisks
represents a disk class

$\beta\in H_{2}(V,\widehat{L})/K\cong H_{2}(V,\widehat{L}).$

Each such class has a well-defined energy

$\omega(\beta)=\omega_{V}(\beta)\in{\mathbb{R}}$

and a well-defined Maslov index

$i(\beta)\in 2{\mathbb{Z}}.$

Given $k$ and $\beta\in H_{2}(V,\widehat{L})$, let ${\mathcal{M}}_{k+1}^{\rm
disk}(\beta)$ be the moduli space of $K$-equivalence classes of holomorphic
quasidisks of class $\beta$ with $k+1$ boundary marked points, and let
$\overline{\mathcal{M}}{}_{k+1}^{\rm disk}(\beta)$ be its compactification.
Notice that as $V$ is aspherical, configurations in
$\overline{\mathcal{M}}{}_{k+1}^{\rm disk}(\beta)$ have only disks bubbles but
not sphere bubbles. The evaluation of a $K$-equivalence class of quasidisks at
the last boundary marked point is well-defined as a point in the quotient
Lagrangian $L\subset X$. Hence there is a continuous map

${\rm ev}:\overline{\mathcal{M}}{}_{k+1}^{\rm disk}(\beta)\to L.$

######  Theorem 8.3 (Blaschke product).

Let $u:{\mathbb{D}}^{2}\to V$ be a holomorphic quasidisk. Then there exist
$\theta_{1},\ldots,\theta_{N}\in[0,2\pi)$ and
$(a_{i,k})_{k=1,\ldots,d_{i}}\subset{\mathbb{D}}^{2}\subset{\mathbb{C}}$ for
$i=1,\ldots,N$ such that

$u(z)=\left(\sqrt{\tau_{1}}e^{i\theta_{1}}\prod_{k=1}^{d_{1}}\frac{z-a_{1,k}}{1-\overline{a_{1,k}}z},\ldots,\sqrt{\tau_{N}}e^{i\theta_{N}}\prod_{k=1}^{d_{N}}\frac{z-a_{N,k}}{1-\overline{a_{N,k}}z}\right).$
(8.1)

Moreover, the Maslov index of $u$ is $2(d_{1}+\cdots+d_{N})$.

In particular, there are $N$ “basic” Maslov two disk classes
$\beta_{1},\ldots,\beta_{N}\in H_{2}(V,L)$ where each $\beta_{i}$ is
represented by a quasidisk given as above with $d_{j}=\delta_{ij}$. These
Maslov two classes form a basis of $H_{2}(V,\widehat{L})$.

######  Theorem 8.4.

The moduli space ${\mathcal{M}}_{k+1}^{\rm disk}(\beta)$ is regular of
dimension $n+2i(\beta)+k-2$ and the evaluation map

${\rm ev}:{\mathcal{M}}_{k+1}^{\rm disk}(\beta)\to L$

is a smooth submersion.

###### Proof.

See [CO06, Section 6]. ∎

A consequence is that each stratum of the compactification
$\overline{\mathcal{M}}{}_{k+1}^{\rm disk}(\beta)$ is regular. To be more
precise, let $\Gamma$ denote a ribbon tree representing the combinatorial type
of a nodal disk (with $k$ inputs and $1$ output) with each vertex labelled by
a disk class whose sum is equal to $\beta$. Then there is a stratum
${\mathcal{M}}_{\Gamma}^{\rm disk}\subset\overline{\mathcal{M}}{}_{k+1}^{\rm
disk}(\beta)$.

######  Corollary 8.5.

Each stratum ${\mathcal{M}}{}_{\Gamma}^{\rm
disk}\subset\overline{\mathcal{M}}_{k+1}(\beta)$ is regular and the evaluation
map ${\rm ev}:{\mathcal{M}}_{\Gamma}^{\rm disk}\to L$ is a submersion.

###### Proof.

See [Woo11, Corollary 6.2]. ∎

####  8.2.1. Treed holomorphic quasimaps

The idea of treed holomorphic disks goes back to Cornea–Lalonde [CL05, CL06].
We recall the adaptation by Woodward [Woo11] in order to define the quasimap
$A_{\infty}$ algebras. Throughout our discussion, we fix a smooth perfect
Morse function $f_{L}:L\to{\mathbb{R}}$ defined over the Lagrangian torus
$L\subset X$, which has exactly $2^{n}$ critical points.

Given a treed disk $C=S\cup T$ of type $\Gamma$, suppose we have a domain-
dependent perturbation $f$ of the Morse function $f_{L}:L\to{\mathbb{R}}$
parametrized by points $t$ on the tree part $T$, a treed holomorphic quasimap
on $C$ is a collection of objects

$\Big{(}(u_{v})_{v\in V_{\Gamma}^{\rm finite}},(x_{e})_{e\in
E_{\Gamma}}\Big{)}$

where for each finite vertex $v\in V_{\Gamma}^{\rm finite}$, we assign a
smooth map $u_{v}:S_{v}\to V$ satisfying

$\overline{\partial}u_{v}=0,\ u_{v}(\partial S_{v})\subset\widehat{L},$

$x_{e}:I_{e}\to L$ is a smooth map satisfying

$x_{e}^{\prime}(t)+\nabla f(x_{e}(t))=0;$

moreover, the matching condition requires 1) for each node joining a boundary
point $z$ of some surface component $S_{v}$ and a finite end of an edge $e$,
the value of $x_{e}(z)$ lies in the same $K$-orbit as the value of $u_{v}(z)$;
2) for each infinite vertex $v\in V_{\Gamma}^{\infty}$ joining two
(semi-)infinite edges $e_{1}$ and $e_{2}$, the limits of $x_{e_{1}}$ and
$x_{e_{2}}$ at the corresponding infinities agree. Here to ensure the
convergence of the maps $x_{e}$, we require that the perturbation $f$ is
supported away from the infinities.

Two treed holomorphic quasimaps are regarded as equivalent if after
identifying domains, the maps on corresponding surfaces parts are
$K$-equivalent (recall $K$ is the gauge group).

To define the $A_{\infty}$ structure (or other structures) one would like to
regularize the moduli spaces of equivalence classes of treed holomorphic
quasimaps and their boundaries. One first needs to use coherent systems of
perturbations to describe such moduli spaces.

####  8.2.2. Perturbations for the $A_{\infty}$ algebra

To achieve transversality relevant for defining the $A_{\infty}$ algebra, we
only need to perturb the Morse function on edges. Hence for a given minimal
metric type $\Gamma$, a domain-dependent perturbation can be viewed as a map

$P_{\Gamma}:\overline{\mathcal{UT}}_{\Gamma}\to C^{\infty}(L).$

We require any such perturbation to vanish near infinities, i.e., vanish near
the closed subset

$\overline{\mathcal{UT}}{}_{\Gamma}^{\infty}\subset\overline{\mathcal{UT}}_{\Gamma}$

corresponding to positions of vertices at infinity. Notice that if $\Gamma$ is
not necessarily stable, a perturbation $P_{\Gamma^{\rm min}}$ for the minimal
form is enough to determine the treed holomorphic map on any treed disks $C$
of type $\Gamma$. Indeed, on any infinite edges of $C$ (if any) the negative
gradient flow equation is taken for the unperturbed Morse function $f_{L}$.

In order to establish the $A_{\infty}$ relation, we also need to require that,
if $\Gamma$ degenerates to $\Pi$, then the restriction of $P_{\Gamma}$ to the
stratum $\overline{\mathcal{UT}}_{\Pi}\subset\overline{\mathcal{UT}}_{\Gamma}$
must agree with the perturbation $P_{\Pi}$ which have been chosen for the
minimal domain type $\Pi$. Hence we need to construct a coherent system of
perturbations indexed for all minimal domain types $\Gamma$. To use the
Sard–Smale theorem to prove that generic perturbations are regular, we also
need to specify the neighborhood of
$\overline{\mathcal{UT}}{}_{\Gamma}^{\infty}$ where we require the