# Hofer–Zehnder conjecture for toric manifolds

Shaoyun Bai  Columbia University, 2990 Broadway, New York, NY 10027, USA
<EMAIL_ADDRESS>and  Guangbo Xu  Department of Mathematics, Rutgers
University, Hill Center–Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ
08854-8019, USA<EMAIL_ADDRESS>
###### Abstract.

We prove that for any compact toric symplectic manifold, if a Hamiltonian
diffeomorphism admits more fixed points, counted homologically, than the total
Betti number, then it has infinitely many simple periodic points. This
provides a vast generalization of Franks’ famous two or infinity dichotomy for
periodic orbits of area-preserving diffeomorphisms on the two-sphere, and
establishes a conjecture attributed to Hofer–Zehnder in the case of toric
manifolds. The key novelty is the application of gauged linear sigma model and
its bulk deformations to the study of Hamiltonian dynamics of symplectic
quotients.

The second author is supported by NSF DMS-2345030.

###### Contents

  1. 1 Introduction
  2. 2 Geometric Preliminaries
  3. 3 Algebraic preliminaries
  4. 4 Vortex Hamiltonian Floer theory
  5. 5 Local Floer theory
  6. 6 Boundary depth
  7. 7 ${\mathbb{Z}}/p$-equivariant vortex Floer theory
  8. 8 Open string theory I. Quasimap Floer theory
  9. 9 Open string theory II. Closed-open maps

##  1\. Introduction

###  1.1. Result

The main theorem of this paper is concerned with the existence of infinitely
many periodic points of Hamiltonian diffeomorphisms of symplectic manifolds.

######  Theorem A.

Let $(X^{2n},\omega)$ be a compact toric symplectic manifold. Suppose
$\phi:X\to X$ is a Hamiltonian diffeomorphism whose set of fixed points ${\rm
Fix}(\phi)$ is finite. For any $x\in{\rm Fix}(\phi)$, denote by ${\it HF}^{\rm
loc}(\phi,x;{\mathbb{Q}})$ the local Floer homology group of the fixed point
$x$ over the rationals. If the inequality

$N(\phi,{\mathbb{Q}}):=\sum_{x\in{\rm Fix}(\phi)}\dim_{{\mathbb{Q}}}{\it
HF}^{\rm
loc}(\phi,x;{\mathbb{Q}})>\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$
(1.1)

holds, then $\phi$ has infinitely many simple periodic points.

Here a point $x\in X$ is called a simple periodic point (of period $k$) if
$\phi^{k}(x)=x$ for some positive integer $k$ and $\phi^{l}(x)\neq x$ for all
$l<k$. The number $N(\phi,{\mathbb{Q}})$ can be viewed as a quantity which
measures the number of fixed points of a generic small Hamiltonian
perturbation of $\phi$. In particular, the following statement holds because
$\dim_{{\mathbb{Q}}}{\it HF}^{\rm loc}(\phi,x;{\mathbb{Q}})=1$ if $x$ is
nondegenerate.

######  Corollary 1.1.

If all the fixed points of $\phi$ are nondegenerate, i.e., for any $x\in{\rm
Fix}(\phi)$, we have $\det(D\phi_{x}-id)\neq 0$, and the inequality

$\\#{\rm Fix}(\phi)>\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$

holds, then $\phi$ has infinitely many simple periodic points.

Theorem A is a high-dimensional generalization of a famous result due to
Franks [Fra92, Fra96]: any area-preserving homeomorphism of $S^{2}$ has either
two or infinitely many simple periodic points. Just as in Franks’ theorem, the
assumption (1.1) is necessary. Indeed, for a toric manifold $(X,\omega)$ with
a Hamiltonian $T^{n}$-action, a generic element of the torus $T^{n}$, which
can be regarded as a higher-dimensional analogue of an irrational rotation on
$S^{2}$, has exactly $\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$
many simple periodic points given by the fixed points of the $T^{n}$-action.
In fact, using the Anosov–Katok method, the paper [LRS22] constructed a
Hamiltonian diffeomorphism on $X$ with exactly
$1+\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$ ergodic measures,
which come from the measure induced by the volume form $\omega^{n}$ and the
Dirac measures supported at the toric fixed points.

Theorem A broadens the scope of the recent investigation of generalizations of
Franks’ dichotomy to higher dimensional symplectic manifolds initiated by
Shelukhin [She22], who proved a similar result on the existence of infinitely
many periodic orbits of Hamiltonian diffeomorphisms defined over monotone
symplectic manifolds with semisimple quantum homology. Our main theorem
resolves a visionary conjecture set forth by Hofer and Zehnder [HZ11, Page
263], which asserts the existence of infinitely many simple periodic points of
a Hamiltonian diffeomorphism $\phi$ of a compact symplectic manifold if
$\\#{\rm Fix}(\phi)$ exceeds the lower bound provided by the Arnold
conjecture, in the case of toric manifolds.

###  1.2. Context from Hamiltonian dynamics and symplectic geometry

As mentioned in the statement of the Hofer–Zehnder conjecture, one rigidity
aspect of Hamiltonian diffeomorphisms is governed by the Arnold conjecture
[Arn65] (see various proofs in [CZ83] [Flo89] [Ono95] [HS95] [FO99] [LT98]
[AB21] [BX22a]), which implies that the inequality
$N(\phi,{\mathbb{Q}})\geq\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$
is always true. Notice that one does not detect simple periodic points of
higher periods by applying the Arnold conjecture to $\phi^{k}$, as fixed
points of $\phi$ are automatically fixed points of $\phi^{k}$.

On the other hand, Hamiltonian diffeomorphisms “tend” to have infinitely many
simple periodic points. The Conley conjecture asserts the infinitude of the
number of simple periodic points for all Hamiltonian diffeomorphisms on a
certain class of symplectic manifolds (originally only conjectured for torus
[Con]). This conjecture has been proved for wider and wider ranges of
manifolds, see [SZ92][FH03][Hin09][Gin10][Hei12][GG09, GG12, GG19a] and the
survey article [GG19b]. However, the Conley conjecture does not hold for all
symplectic manifolds, as easily seen from the case of irrational rotations on
$S^{2}$ (or general toric manifolds).

While Conley conjecture is an unconditional statement about Hamiltonian
diffeomorphisms on certain symplectic manifolds, the Hofer–Zehnder conjecture
observes a simple condition on the Hamiltonian diffeomorphism responsible for
the infinitude of periodic points: the number of fixed points (counted
homologically) is strictly greater than the “Arnold lower bound”. A broader
interpretation of this condition is the existence of “unnecessary” fixed
points, such as non-contractible ones. Distinguished from the intensive
research activities surrounding the Conley conjecture, the Hofer–Zehnder
conjecture has only been understood in limited cases. Except for the cases
covered in [She22] (including projective spaces, Grassmannians, and their
monotone products), the Hofer–Zehnder conjecture is known for weighted
projective spaces [All23], some cases related to non-contractible orbits
([Gür13][GG16][Ori17, Ori19][Sug21]) or hyperbolic fixed points ([GG14]). Our
Theorem A covers a large family of new instances of the original homological
version of the Hofer–Zehnder conjecture. Very interestingly, our proof
actually indicates a surprising connection between mirror symmetry and
Hamiltonian dynamics, see the discussions in Section 1.3.

The holomorphic curve method, most notably the package of Floer homology, has
been a dominant tool in the study of Hamiltonian dynamics. The developments of
Conley conjecture and Hofer–Zehnder conjecture have shown that holomorphic
curves have more subtle influence on Hamiltonian dynamics than being merely a
tool. For example, Conley conjecture is true for Calabi–Yau or negatively
monotone manifolds. These manifolds do not have “very many” holomorphic
curves. Recent progress [CGG20][She21][CGG22] further reveals a close
connection between the failure of Conley conjecture and the effect of
holomorphic curves (which is also related to the Chance–McDuff conjecture).
The semisimplicity condition in the proof of the Hofer–Zehnder conjecture
[She22] can also be viewed as a characterization of the abundance of
holomorphic curves. Note that toric manifolds underlies rational algebraic
varieties, so they provide examples demonstrating this phenomenon. It will be
very interesting to have a more precise and systematic formulation of such a
mechanism for general symplectic manifolds.

###  1.3. Key ingredient: GLSM

While inspired by the method of Shelukhin [She22], our resolution of the
Hofer–Zehnder conjecture for toric symplectic manifolds is largely based on
introducing a new player in Hamiltonian dynamics: _gauged linear sigma model
(GLSM)_.

Gauged linear sigma model was originally introduced by Witten [Wit93] in the
physics context in much greater generality. In the current situation, the
basic usage of the GLSM is to replace holomorphic curves in a toric manifold
$X$ by certain gauge-theoretic objects called vortices. This is possible
because $X$ is the symplectic quotient (or GIT quotient) of a vector space
$V\cong{\mathbb{C}}^{N}$ by a torus $K\cong(S^{1})^{N-n}$ with a moment map
$\mu$. In this situation, a vortex over any Riemann surface $\Sigma$ consists
of a principal $K$-bundle $P\to\Sigma$, a connection $A\in{\mathcal{A}}(P)$,
and a section $u$ of the associated vector bundle $P(V):=(P\times V)/K$,
solving the vortex equation

$\displaystyle\ \overline{\partial}_{A}u=0,\ $ $\displaystyle\
*F_{A}+\mu(u)=0.$

Mathematically, the general symplectic vortex equation was firstly introduced
by Cieliebak–Gaio–Salamon [CGS00] and Mundet [Mun99, Mun03], with many related
technical works by Cieliebak–Gaio–Mundet–Salamon [CGMS02], Ott [Ott14],
Mundet–Tian [MT09], the second author [Xu13], Zilter [Zil09, Zil05, Zil14],
Venugopalan [Ven15], etc. In particular, one can use Hamiltonian perturbed
vortex equation over surfaces with cylindrical ends to develop the _vortex
Floer theory_ (see Frauenfelder [Fra04a, Fra04b] and the second author
[Xu16]).

Many aspects of ordinary Hamiltonian Floer theory has a counterpart in vortex
Hamiltonian Floer theory, including continuation maps and energy filtration.
Accordingly, recent advances on quantitative Floer theory, especially the
theory of persistence modules [PS16, UZ16], can be adapted to the vortex
context. For readers who are not familiar with this variant of Floer theory,
just keep in mind that the chain complex underlying the vortex Hamiltonian
Floer homology is still freely generated by $1$-periodic orbits of the given
Hamiltonian diffeomorphism, and the differentials are defined by counting
solutions to Hamiltonian-perturbed vortex equations instead of Floer
equations, provided that all the $1$-periodic orbits are nondegenerate. For
the general isolated degenerate case, the theory of _local Floer homology_
carries over to the vortex context without much difficulty, therefore all of
our results hold in such a generality.

One remarkable feature of the vortex Floer theory is that we can define Floer
theories over _integers_ in our setting. Indeed, as the target space $V$ is a
symplectic vector space, the Uhlenbeck–Gromov–Floer compactification of moduli
spaces of solutions to vortex equations do not acquire adding configurations
with sphere bubbles, much as in the case of symplectically aspherical
manifolds. Except for simplifying the technical arguments for achieving
transversality, the ability of reducing to characteristic $p$ allows us to
extend the scope of applicability of _symplectic Smith theory_ [Sei15, SZ21]
beyond the exact or semi-positive setting.

With the above explanation, our main results concerning the structural aspects
of (filtered) vortex Hamiltonian Floer theory can be summarized as follows.
Given a commutative ring $R$, let $\Lambda=\Lambda_{R}$ be the upward Novikov
ring

$\Lambda_{R}=\Big{\\{}\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\ |\
g_{i}\in{\mathbb{R}},\ a_{i}\in R,\ \lim_{i\to\infty}g_{i}=+\infty\Big{\\}}.$

Denote by ${\bf z}_{1},\dots,{\bf z}_{N}$ the $K$-equivariant degree $2$
cohomology classes dual to the coordinate hyperplanes $V_{1},\cdots,V_{N}$ in
$V$.

######  Theorem B.

There exists a bulk-deformation of the form

${\mathfrak{b}}=\sum_{j=1}^{N}\log c_{j}\cdot{\bf z}_{j}\ {\rm where}\
c_{j}\in{\mathbb{Z}}[{\bf i}]$ (1.2)

satisfying the following properties.

  1. (1)

The ${\mathfrak{b}}$-deformed vortex quantum homology algebra ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{Q}}})$ is
semisimple over $\Lambda_{\overline{\mathbb{Q}}}$, with the number of
idempotent summands, all of which are 1-dimensional, equal to
$\sum_{i=0}^{2n}\dim_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$.

  2. (2)

The operator ${\mathbb{E}}_{\mathfrak{b}}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{Q}}})\to{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{Q}}})$ given by the
quantum multiplication with the equivariant first Chern class has distinct
nonzero eigenvalues.

######  Remark 1.2.

The reader may wonder about the legitimacy of taking the logarithm of elements
of ${\mathbb{Z}}[{\bf i}]$. In reality, we will take the exponential of the
intersection number between the bulk ${\mathfrak{b}}$ and Riemann surfaces to
deform Floer-theoretic operations in the spirit of the divisor axiom in
Gromov–Witten theory, and we take the above formal expression for the sake of
conciseness.

We explain the central ingredient in the proof of Theorem B: the GLSM version
of the _closed string mirror symmetry_. To find such a bulk and calculate the
quantum homology ring, we develop _closed-open field theory_ , in particular,
the closed-open map

${\rm CO}^{\mathfrak{b}}:{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{Q}}})\to{\it
HH}^{\bullet}({\mathcal{F}}_{\mathfrak{b}}^{K}(V),{\mathcal{F}}_{\mathfrak{b}}^{K}(V))$

where ${\mathcal{F}}_{\mathfrak{b}}^{K}(V)$ is an equivariant version of the
Fukaya category: its objects are roughly Lagrangians in the toric manifold $X$
and its structural maps are defined via equivariant counts of holomorphic
disks upstairs in $V$. Hence we reduce the calculation of the quantum homology
algebra to the determination of Hochschild cohomology by showing that the
closed-open map is a unital ring isomorphism. Moreover, the structural
coefficients of the $A_{\infty}$ operations are governed by the mirror
superpotential, which is a Laurent polynomial
$W:({\mathbb{C}}^{*})^{N}\to{\mathbb{C}}$. The Hochschild cohomology can be
accordingly computed as the Jacobian ring of $W$, and the bulk-deformations
can be understood as taking certain unfoldings of $W$ by adding elements from
the Jacobi ring. It is well-known that a generic unfolding of $W$ only has
nondegenerate critical points, and such unfolding can be realized by adjusting
the bulk-deformation, implying that the quantum homology is generically
semisimple. As for the statement on the first Chern class, it is the
application of the folklore principle, usually attributed to
Auroux–Kontsevich–Seidel (see [Aur07, Section 6] and [She16, Lemma 2.7]), that
generalized eigenvalues of quantum multiplication with the first Chern class
have a one-to-one correspondence with critical values of the mirror
superpotential .

Such a closed-string mirror symmetry statement has been established using
ordinary pseudo-holomorphic curves and Floer theory. For general toric
manifolds, the symplectic version of the mirror superpotential is defined by
counting stable pseudoholomorphic disks, usually having infinitely many terms
(see [CO06, FOOO10, FOOO11, FOOO16]). To Morsify such a superpotential, one
usually needs very general bulk deformations. This in turn demonstrates
another advantage of the GLSM: the mirror superpotential takes a rather simple
form. As shown by Woodward [Woo11], the mirror superpotential in GLSM agrees
with the mirror superpotential given by Givental [Giv95] and Hori–Vafa [HV00].
One can Morsify this superpotential (called the Givental–Hori–Vafa potential)
by only using “small” bulk deformation, i.e., divisor classes.

###  1.4. Proof of the main theorem

Once the crucial Theorem B is established, the rest of the proof can be
streamlined in the same way as [She22]. Many key arguments are in fact
algebraic while other geometric arguments need nontrivial, but straightforward
extensions to the vortex setting.

The first step is to take mod $p$ reductions of the vortex quantum homology
algebra. Notice that as the coefficients of ${\mathfrak{b}}$ are integral, the
mod $p$ deformed counts also define a vortex Hamiltonian Floer homology ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}})$, where
$\overline{\mathbb{F}}_{p}$ is the algebraic closure of
${\mathbb{F}}_{p}\cong{\mathbb{Z}}/p{\mathbb{Z}}$. By a purely algebraic
argument (see Theorem 3.9), one obtains the following corollary of Theorem B.

######  Corollary 1.3.

There exists $p_{0}>0$ such that for all primes $p\geq p_{0}$, the vortex
quantum homology algebra ${\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}})$ is
semisimple with the number of idempotent summands, all of which are
1-dimensional, equal to $\sum_{i=1}^{2n}{\rm
dim}_{\mathbb{Q}}H_{i}(X;{\mathbb{Q}})$.

We also need two quantitative results about the filtered theories in finite
characteristics. Recall that to each Floer–Novikov complex such as the bulk-
deformed vortex Floer complex, one can associate a barcode. In our case one
needs the general formulation by Usher–Zhang [UZ16]. We consider two
associated numerical invariants: the boundary depth, which is the length of
the longest finite bar, and the total bar length, which is the sum of lengths
of all finite bars.

######  Theorem C.

Let ${\mathfrak{b}}$ be a bulk-deformation satisfying Theorem B and $p_{0}$ be
the one from Corollary 1.3. Then there exists $C>0$ satisfying the following
condition. Let ${\it
VCF}_{\bullet}^{\mathfrak{b}}(H;\Lambda_{\overline{\mathbb{F}}_{p}})$ be the
${\mathfrak{b}}$-deformed filtered vortex Floer chain complex associated to a
nondegenerate Hamiltonian $H$ on $X$111In fact the chain complex depends on an
“admissible lift” upstairs as well as an almost complex structure. But the
boundary depth only depends on the Hamiltonian downstairs. and let
$\beta_{(p)}^{\mathfrak{b}}(H)$ be its boundary depth, then for all $p\geq
p_{0}$,

$\beta_{(p)}^{\mathfrak{b}}(H)\leq C.$ (1.3)

The barcodes (with bottleneck distance) have a Lipschitz dependence on
Hamiltonian diffeomorphisms (with the Hofer metric). Hence the above uniform
bound on boundary depth extends to all Hamiltonians on the toric manifold $X$.
On the other hand, the total bar length, denoted by
$\tau^{\mathfrak{b}}_{(p)}(-)$ in characteristic $p$, can be extended to
Hamiltonian diffeomorphisms with isolated fixed points. In particular, the
barcode of a possibly degenerate Hamiltonian diffeomorphism $\phi$ with
isolated fixed points is still finite, and the number of bar ends agrees with
the homological counts of fixed points $N(\phi;\overline{\mathbb{F}}_{p})$
(see Theorem 5.9).

The last key input is about the growth of total bar length of prime iterations
of Hamiltonian diffeomorphisms. Suppose $\phi:X\to X$ is a Hamiltonian
diffeomorphism such that all prime iterations of $\phi$ have isolated fixed
points. If $\phi$ is the time-1 map of a Hamiltonian $H:S^{1}\times
X\to{\mathbb{R}}$, then the $p$-fold iteration $\phi^{p}$ is the time-1 map of
$H^{(p)}:=H_{pt}$.

######  Theorem D.

For any bulk ${\mathfrak{b}}$ for the form (1.2) and any odd prime $p$, we
have the inequality

$\tau_{(p)}^{\mathfrak{b}}(H^{(p)})\geq p\cdot\tau_{(p)}^{\mathfrak{b}}(H).$
(1.4)

######  Remark 1.4.

The above equality should also hold for $p=2$, but we do not give a full
treatment for this case because this would introduce extra notations in the
discussion of equivariant Floer theory. In fact, for the proof, we do not need
the $p=2$ version of (1.4) anyway.

With the above technical ingredients, establishing the Hofer–Zehnder
conjecture for toric manifolds is a matter of elementary arguments.

###### Proof of Theorem A.

Let $\phi:X\to X$ be a Hamiltonian diffeomorphism satisfying (1.1). By
Proposition 5.2, one can replace the ordinary local Floer homology by the
(bulk-deformed) local vortex Floer homology. One also knows from Proposition
4.8 and Theorem 4.12 that the total rank of rational homology of $X$ agrees
with the rank of the bulk-deformed vortex Floer homology of $V$. Hence (1.1)
can be rewritten as

$N(\phi,{\mathbb{Q}})=\sum_{x\in{\rm Fix}(\phi)}{\rm
dim}_{\overline{\mathbb{Q}}}{\it VHF}^{\rm loc}(\phi,x;{\mathbb{Q}})>{\rm
dim}_{\Lambda_{\mathbb{Q}}}{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\mathbb{Q}}).$

Because the (local) vortex Floer homology are defined over the integers (see
Section 5), by the universal coefficient theorem, for $p$ sufficiently large,

$\sum_{x\in{\rm Fix}(\phi)}\dim_{\overline{\mathbb{F}}_{p}}{\it VHF}^{\rm
loc}(\phi,x;\overline{\mathbb{F}}_{p})>{\rm
dim}_{\Lambda_{\overline{\mathbb{F}}_{p}}}{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}}).$

Suppose on the contrary that $\phi$ only has only finitely many periodic
points. Then for any sufficiently large prime $p$, for all $k\geq 1$, ${\rm
Fix}(\phi^{p^{k}})={\rm Fix}(\phi)$. Then by Theorem 5.1, one has

$\sum_{x\in{\rm Fix}(\phi^{p^{k}})}{\rm dim}_{\overline{\mathbb{F}}_{p}}{\it
VHF}^{\rm loc}(\phi^{p^{k}},x;\overline{\mathbb{F}}_{p})=\sum_{x\in{\rm
Fix}(\phi)}{\rm dim}_{\overline{\mathbb{F}}_{p}}{\it VHF}^{\rm
loc}(\phi,x;\overline{\mathbb{F}}_{p})>{\rm
dim}_{\Lambda_{\overline{\mathbb{F}}_{p}}}{\it
VHF}_{\bullet}^{\mathfrak{b}}(V;\Lambda_{\overline{\mathbb{F}}_{p}}).$

Consider the barcode of $\phi^{p^{k}}$ coming from the bulk-deformed vortex
Floer theory (over the Novikov field $\Lambda_{\overline{\mathbb{F}}_{p}}$).
The above implies that the number of finite bars is positive and independent
of the iteration $p^{k}$. The uniform bound on the boundary depth (length of
the longest finite bar) given by Theorem C implies that the total bar length
$\tau^{\mathfrak{b}}_{(p)}(\phi^{p^{k}})$ is uniformly bounded.

On the other hand, by Theorem D, for any $k\geq 1$, the total bar length grows
as

$\tau^{\mathfrak{b}}_{(p)}(\phi^{p^{k}})\geq
p^{k}\cdot\tau^{\mathfrak{b}}_{(p)}(\phi)\geq Cp^{k}>0.$

This is a contradiction. Hence $\phi$ must have infinitely many periodic
points. ∎

Because the above argument works for any $p\geq p_{0}$, we know that the
number of periodic points of $\phi$ grows like $\frac{k}{\log(k)}$ as
$k\to\infty$, as a result of the prime number theorem.

######  Remark 1.5.

Arguments of the above form first appeared in [She22, Section 8], which we
reproduce in our context for completeness. As noted above, Shelukhin’s result
on the Hofer–Zehnder conjecture relies on the assumptions that the ambient
symplectic manifold is monotone and that the quantum homology is semisimple,
which respectively account for the inequalities (1.4) (the monotonicity
condition allows one to define Floer theory integrally using classical
methods) and (1.3) (which will be discussed in more detail in the body part of
this paper). For general toric symplectic manifolds, traditional Hamiltonian
Floer homology is only defined over the rationals, which sets difficulties for
establishing symplectic Smith-type inequalities. Moreover, the quantum
homology of toric symplectic manifolds fails to be semisimple in general,
which is already the case even for Fano/monotone toric manifolds [OT09].

###  1.5. Outlook and speculations

We are very surprised to find out that classical considerations from mirror
symmetry can be quite useful for investigations in Hamiltonian dynamics. We
expect such a connection could open up new avenues for future research. As
mentioned above, GLSM can more generally be used to study symplectic topology
and Hamiltonian dynamics of other symplectic/GIT quotients or complete
intersections in them, the latter of which requires studing the gauged Witten
equation (see [TX18a, TX18b]) with Hamiltonian perturbations. It is
conceivable that one could resolve the Hofer–Zehnder conjecture for a broader
class of symplectic quotients, provided that certain form of closed string
mirror symmetry can be established.

On a different note, except for deploying tools like GLSM, there are some
recent advances [BX22b, BX22a] on defining Hamiltonian Floer theory over
integers for general symplectic manifolds. The methods from _loc. cit._ are
general enough for us to expect that a version of symplectic Smith-type
inequality should hold using such a theory. Deriving dynamical applications
using such a toolkit, including proving the Hofer–Zehnder conjecture in more
general settings, is another topic for future research.

###  1.6. Outline of the paper

The following provides an outline of this paper.

  * •

Basic notions related to toric manifolds are recalled in Section 2, which also
includes an introduction to the symplectic vortex equations arising from GLSM.

  * •

In Section 3, various algebraic preliminaries relevant for our purpose,
including semisimple algebras over Novikov rings defined over fields with
possibly positive characteristics, abstract setups for filtered Floer
theories, persistence modules, and $A_{\infty}$ algebras and their Hochschild
cohomology, are recalled systematically.

  * •

A (filtered) Hamiltonian Floer theory package in the vortex setting is
recorded in Section 4. Most notably, we introduce bulk deformations in vortex
Hamiltonian Floer theory which allow us to incorporate ideas from generic
semisimplicity of quantum homology to derive applications in quantitative
symplectic topology.

  * •

In Section 5, we introduce local Floer theory in the vortex setting in order
to establish Theorem A for Hamiltonian diffeomorphisms with isolated but
degenerate fixed points.

  * •

The main purpose of Section 6 is to prove Theorem 6.2 = Theorem C, which
ensures a uniform upper bound on the boundary depth of the bulk-deformed
vortex Hamiltonian Floer persistence module of any Hamiltonian diffeomorphism
provided that the bulk-deformed vortex quantum homology is semisimple.

  * •

In Section 7, we develop ${\mathbb{Z}}/p$-equivariant vortex Hamiltonian Floer
theory by adapting the work [Sei15, SZ21] in the GLSM setting. Theorem D =
Theorem 7.1 is proven as a consequence by appealing to the work of Shelukhin
[She22].

  * •

We turn our attention to Lagrangian Floer theory in Section 8. The key result
is to demonstrate the existence of a “convenient” bulk deformation (cf.
Definition 8.20) whose associated Fukaya category (in the GLSM setting) takes
a very simple form, such that its Hochschild cohomology is a semisimple
algebra.

  * •

Lastly, in Section 9, Theorem 9.1 = Theorem B is proven by showing that the
closed-open string map is a unital ring isomorphism.

### Acknowledgements

: We thank Marcelo Atallah, Hiroshi Iritani, Han Lou, Egor Shelukhin, Nick
Sheridan, Michael Usher, and Chris Woodward for useful discussions and email
correspondences. The first-named author is grateful to the Simons Center for
Geometry and Physics for its warm hospitality during Spring 2023.

##  2\. Geometric Preliminaries

We recall basic notions about toric symplectic manifolds and symplectic vortex
equations.

###  2.1. Toric manifolds as symplectic quotients

We recall the notion of symplectic reduction/quotients. Let $K$ be a compact
Lie group with Lie algebra ${\mathfrak{k}}$. Let $(V,\omega_{V})$ be a
symplectic manifold. A smooth $K$-action on $V$ is called a Hamiltonian action
if there exists a moment map

$\mu:V\to{\mathfrak{k}}^{*}$

satisfying

  1. (1)

$\mu$ is $K$-equivariant (with respect to the co-adjoint action on
${\mathfrak{k}}^{*}$).

  2. (2)

For each $\xi\in{\mathfrak{k}}$, let the infinitesimal action of $\xi$ be
${\mathcal{X}}_{\xi}$. Then

$\omega_{V}({\mathcal{X}}_{\xi},\cdot)=d\langle\mu,\xi\rangle.$

It follows that the level set $\mu^{-1}(0)$ is $K$-invariant. Define the
symplectic reduction of $V$ (with respect to the $K$-action and the moment
map) to be

$X:=\mu^{-1}(0)/K.$

We always assume that $0$ is a regular value of $\mu$ and the $K$-action on
$\mu^{-1}(0)$ is free. This assumption implies that $X$ is a smooth manifold.
In this case, $X$ carries a canonically induced symplectic form $\omega_{X}$.

When $V$ has a $K$-invariant integrable almost complex structure $J_{V}$, the
$K$-action can be extended to its complexification $K^{\mathbb{C}}$ as a
holomorphic action. When this is the case, (under certain extra conditions),
the Kempf–Ness theorem says that the symplectic reduction can be identified
with the geometric invariant theory (GIT) quotient.

####  2.1.1. Compact symplectic toric manifolds

Symplectic toric manifolds can be realized as symplectic quotients of a vector
space. We provide a minimal description of symplectic toric manifolds
necessary for this paper. An $2n$-dimensional compact symplectic toric
manifold $X$ is described by a convex polytope $P\subset{\mathbb{R}}^{n}$
satisfying the following conditions.

  1. (1)

For each face $\partial_{j}P$ of $P$, there are ${\bf
v}_{j}\in{\mathbb{Z}}^{n}$ and $\lambda_{j}\in{\mathbb{R}}$ such that ${\bf
v}_{j}$ is an inward normal vector and the face is defined by

$\partial_{j}P=\\{{\bf u}\in{\mathbb{R}}^{n}\ |\ \langle{\bf v}_{j},{\bf
u}\rangle=\lambda_{j}\\}.$

  2. (2)

For each vertex of $P$, the normal vectors ${\bf v}_{j_{1}},\ldots,{\bf
v}_{j_{n}}$ of all adjacent faces form a ${\mathbb{Z}}$-basis of
${\mathbb{Z}}^{n}$.

In this paper, denote by $N$ the number of faces of $P$. We can realize $X$ as
the symplectic quotient of ${\mathbb{C}}^{N}$ (with the standard symplectic
form) by the $N-n$ dimensional torus $K=T^{N-n}=(S^{1})^{N-n}$. The collection
of vectors ${\bf v}_{1},\ldots,{\bf v}_{N}$ defines a linear map

$\tilde{\pi}_{K}:{\mathbb{R}}^{N}\to{\mathbb{R}}^{n}$

which sends ${\mathbb{Z}}^{N}$ onto ${\mathbb{Z}}^{n}$. Hence it induces a
surjective group homomorphism

$\pi_{K}:T^{N}\to T^{n}.$

Let $K$ be the kernel of $\pi_{P}$. Hence $K$ acts on ${\mathbb{C}}^{N}$ as a
subgroup of $T^{N}$. Notice that for the standard $\widehat{K}=T^{N}$-action,
the moment map can be written as

$\widehat{\mu}(x_{1},\ldots,x_{N})=\left(\pi|x_{1}|^{2}-\lambda_{1},\ldots,\pi|x_{N}|^{2}-\lambda_{N}\right)\in{\mathbb{R}}^{N}\cong\widehat{\mathfrak{k}}^{*}.$

Then the moment map of the $K$-action is simply the composition

$\mu=(d\pi_{K})^{*}\circ\widehat{\mu}:V\to{\mathfrak{k}}^{*}.$

On the other hand, there is a residual torus action on $X$ by the quotient
$T^{n}$ which is the torus action usually appear in the discussion of toric
manifolds. The associated moment map is denoted by

$\pi_{X}:X\to{\mathbb{R}}^{n}$ (2.1)

whose range is actually the moment polytope $P$.

Notice that if one translates the moment polytope $P$ in ${\mathbb{R}}^{N}$ by
a vector in ${\mathbb{R}}^{n}$, then it does not change the moment map $\mu$
and hence the symplectic form on $X$.

###  2.2. Symplectic vortex equation

The symplectic vortex equation was originally introduced by
Cieliebak–Gaio–Salamon [CGS00] and Mundet [Mun99]. It is a generalization of
the pseudoholomorphic curve equation to the equivariant setting. Here we
briefly recall its setup and some basic analytical result.

####  2.2.1. Gauged maps and vortex equation

Let $V$ be the complex vector space acted on by the complex torus
$G=K^{\mathbb{C}}$ with moment map $\mu$ under the $K$-action and symplectic
quotient the toric manifold $X$. Let $\Sigma$ be a Riemann surface. A gauged
map from $\Sigma$ to $V$ is a triple ${\mathfrak{u}}=(P,A,u)$ where
$P\to\Sigma$ is a principal $K$-bundle, $A\in{\mathcal{A}}(P)$ is a connection
on $P$, and $u$ is a section of the associated vector bundle
$P(V):=P\times_{K}V$. The group of gauge transformations ${\mathcal{G}}(P)$,
which, in the abelian case, is the group of smooth maps

$g:\Sigma\to K,$

which acts on gauged maps by

$g^{*}{\mathfrak{u}}=g^{*}(P,A,u)=(P,g^{*}A,g^{*}u)=(P,A+g^{-1}dg,g^{-1}u).$

We need three quantities to define the vortex equation. First the covariant
derivative of $u$ is a section

$d_{A}u\in\Omega^{1}(P,u^{*}TV)$

which descends to an element in $\Omega^{1}(\Sigma,u^{*}TV/K)$. There are also
the curvature and the moment potential

$\displaystyle\ F_{A}\in\Omega^{2}(\Sigma,{\rm ad}P),\ $ $\displaystyle\
\mu(u)\in\Omega^{0}(\Sigma,{\rm ad}P^{*}).$

By choosing an invariant inner product on the Lie algebra ${\mathfrak{k}}$ one
can identify ${\rm ad}P\cong{\rm ad}P^{*}$; by choosing a volume form
$\nu_{\Sigma}$ one can identify $\Omega^{2}\cong\Omega^{0}$. The gauged map
${\mathfrak{u}}$ is called a vortex if

$\displaystyle\ \overline{\partial}_{A}u=0,\ $ $\displaystyle\
*F_{A}+\mu(u)=0.$ (2.2)

Here $\overline{\partial}_{A}u$ is the $(0,1)$-part of the covariant
derivative $d_{A}u$. Both equations are invariant under gauge transformations.
The energy of a vortex is defined to be

$E({\mathfrak{u}})=\frac{1}{2}\int_{\Sigma}\left(\|d_{A}u\|^{2}+\|F_{A}\|^{2}+\|\mu(u)\|^{2}\right)\nu_{\Sigma}.$

Analogous to pseudoholomorphic curves, vortices satisfy an energy identity.
Suppose $\Sigma$ is closed. Then each gauged map ${\mathfrak{u}}$ represents
an equivariant homology class $[{\mathfrak{u}}]\in H_{2}^{K}(V;{\mathbb{Z}})$
defined as follows. The section $u:\Sigma\to P(V)$ can be identified with a
$K$-equivariant map $\tilde{u}:P\to V$. Let $EK\to BK$ be the universal
$K$-bundle. The classifying map of $P\to\Sigma$ is a map $\iota:\Sigma\to BK$
which is covered by a bundle map $\tilde{\iota}:P\to EK$. Then the equivariant
map $(\tilde{\iota},\tilde{u}):P\to EK\times V$ descends to a continuous map
from $\Sigma$ to $(EK\times V)/K$, which represents a class
$[{\mathfrak{u}}]\in H_{2}^{K}(V;{\mathbb{Z}})$. In the toric case, this class
is just the degree of the principal bundle $P\to\Sigma$. Then for any gauged
map ${\mathfrak{u}}=(P,A,u)$, one has

$E({\mathfrak{u}})=\langle\omega^{K},[{\mathfrak{u}}]\rangle+\|\overline{\partial}_{A}u\|_{L^{2}(\Sigma)}^{2}+\|*F_{A}+\mu(u)\|_{L^{2}(\Sigma)}^{2}.$

Here $\omega^{K}\in H_{K}^{2}(V;{\mathbb{R}})$ is the equivariant class
represented by the equivariant 2-form $\omega-\mu$ (see[CGS00, Proposition
3.1] and [Mun03, Lemma 14]).

######  Remark 2.1.

An important feature of the symplectic vortex equation in the toric setting is
that no bubbling happens as the space $V$ is symplectically aspherical. In
general, energy concentration could cause bubbling of holomorphic spheres as
shown in [Mun99, Mun03, CGMS02, Ott14].

One can introduce Hamiltonian perturbations. Given a 1-form

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

with coefficients in the space of $K$-invariant smooth functions on $V$, we
can define a family of Hamiltonian vector fields

$X_{\mathcal{H}}\in\Gamma(\Sigma\times V,\pi_{\Sigma}^{*}T^{*}\Sigma\otimes
TV)$

which is $K$-invariant, where $\pi_{\Sigma}:\Sigma\times V\to\Sigma$ is the
projection to the first factor. Hence for any principal $K$-bundle
$\pi_{P}:P\to\Sigma$, the vector field $X_{\mathcal{H}}$ induces a section on
the total space of the vector bundle $\pi_{P(V)}:P(V)\to\Sigma$

$X_{\mathcal{H}}\in\Gamma(P(V),\pi_{P(V)}^{*}T^{*}\Sigma\otimes P(TV)),$

where $P(TV):=P\times_{K}u^{*}TV$. The perturbed symplectic vortex equation is

$\displaystyle\ \overline{\partial}_{A,{\mathcal{H}}}u=0,\ $ $\displaystyle\
*F_{A}+\mu(u)=0.$ (2.3)

where

$\overline{\partial}_{A,{\mathcal{H}}}u=(d_{A}u)^{0,1}+(X_{\mathcal{H}}(u))^{0,1}.$

For our applications, ${\mathcal{H}}$ is obtained by extending the pullback of
Hamiltonian connections in
$\Omega^{1}(\Sigma,C^{\infty}(X))=\Omega^{1}(\Sigma,C^{\infty}(\mu^{-1}(0)/K))$.

####  2.2.2. Compactness

Although in aspherical targets vortices cannot bubble off holomorphic spheres,
in general holomorphic curves can bubble off. It is the case when one
considers Lagrangian boundary conditions. Let $L\subset V$ be a $K$-invariant
Lagrangian submanifold. One can impose the Lagrangian boundary condition for
gauged maps ${\mathfrak{u}}=(P,A,u)$ from $\Sigma$ to $V$ with
$u|_{\partial\Sigma}\subset P(L)$. Given a sequence of solutions
${\mathfrak{u}}_{i}$ to the vortex equation on $\Sigma$ subject to the
Lagrangian boundary condition, if $u_{i}$ has uniformly bounded image and
$E({\mathfrak{u}}_{i})$ is uniformly bounded, the energy density could blow up
near a boundary point. The boundedness of the images of $u_{i}$ implies that
the curvatures $F_{A_{i}}$ do not blowup. Moreover, if one scales by the rate
of energy concentration, the sequence of connections $A_{i}$ converge
subsequentially (up to gauge transformation) to a flat connection. All
Hamiltonian perturbations and variations of almost complex structures will
also be scaled off. Hence a subsequece can bubble off a (stable) holomorphic
disk in $V$ with boundary in $L$ with respect to a fixed almost complex
structure. See details in [WX17a].

##  3\. Algebraic preliminaries

###  3.1. Novikov rings

We set up the notations for our coefficient rings. In this paper, $R$ always
denotes a commutative ring with a unit, hence comes with a canonical ring map

${\mathbb{Z}}\to R.$

Let $\Lambda=\Lambda_{R}$ be the (upward) Novikov ring

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

The valuation on ${\mathfrak{v}}:\Lambda\to{\mathbb{R}}\cup\\{+\infty\\}$ is
defined by

${\mathfrak{v}}\left(\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\right)=\inf\big{\\{}g_{i}\
|\ a_{i}\neq 0\big{\\}}\ \ \ {\rm and}\ \ \ {\mathfrak{v}}(0)=+\infty.$

We will also need the following version

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

which also comes with a valuation by restricting the above valuation. When $R$
is a field, $\Lambda_{R}$ is also a field, and it is the field of fraction of
$\Lambda_{0,R}$.

In many cases we can restrict to a Novikov ring of series $\sum
a_{i}T^{g_{i}}$ where $g_{i}$ are restricted to a finitely generated additive
group $\Gamma\subsetneq{\mathbb{R}}$. In this paper $\Gamma$ is fixed and
actually determined by the GIT presentation of a toric manifold. Indeed, the
discrete monoid $\Gamma$ associated with the toric manifold $X(\Sigma)$ is
defined to be the image of effective $1$-cycles in ${\mathbb{R}}$ defined from
pairing with the cohomology class represented by the symplectic form, see
Section 2.1.1. Denote

$\Lambda_{R}^{\Gamma}:=\Big{\\{}\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\in\Lambda_{R}\
|\ g_{i}\in\Gamma\Big{\\}}$

and

$\Lambda_{0,R}^{\Gamma}:=\Lambda_{0,R}\cap\Lambda_{R}^{\Gamma}.$

However $\Lambda_{R}^{\Gamma}$ does not enjoy certain algebraic properties of
$\Lambda_{R}$. For example, when $R={\mathbb{K}}$ is an algebraically closed
field, $\Lambda_{\mathbb{K}}$ is algebraically closed but
$\Lambda_{\mathbb{K}}^{\Gamma}$ is not.

####  3.1.1. Modules and algebras over Novikov rings

######  Definition 3.1.

A non-Archimedean normed free module over $\Lambda_{R}$ is a pair $(C,\ell)$
where $C$ is a free $\Lambda_{R}$-module endowed with a function
$\ell:C\to{\mathbb{R}}\cup\\{-\infty\\}$ satisfying

  1. (1)

(Nondegeneracy) $\ell(x)=-\infty$ if and only if $x=0$.

  2. (2)

(Homogeneity) For all $\lambda\in\Lambda_{R}$ and $x\in C$, $\ell(\lambda
x)=\ell(x)-{\mathfrak{v}}(\lambda)$.

  3. (3)

(Subadditivity) For $x,y\in C$, $\ell(x+y)\leq\max\\{\ell(x),\ell(y)\\}$; if
$\ell(x)\neq\ell(y)$, then $\ell(x+y)=\max\\{\ell(x),\ell(y)\\}$.222The second
part is a consequence of the first, see [UZ16, Proposition 2.3].

Now suppose ${\mathbb{K}}$ is an $R$-module which is also a field. Then one
can extend the function $\ell$ to $C\otimes_{\Lambda_{R}}\Lambda_{\mathbb{K}}$
via (Homogeneity). The then obtained pair is a non-Archimedean normed vector
space in the sense of [UZ16, Definition 2.2] (except that the coefficient
field was $\Lambda_{\mathbb{K}}^{\Gamma}$ rather than $\Lambda_{\mathbb{K}}$).

We also need to consider multiplicative structures compatible with the non-
Archimedean norm.

######  Definition 3.2.

A non-Archimedean normed algebra over $\Lambda_{R}$ is a non-Archimedean
normed free module $(C,\ell)$ together with a $\Lambda_{R}$-algebra structure
satisfying the

  * •

(Triangle inequality) For all $x,y\in C$,

$\ell(xy)\leq\ell(x)+\ell(y).$

####  3.1.2. Specific coefficients and mod $p$ reductions

In this paper we need to use certain non-traditional coefficient ring or
fields. Here we briefly summarize them and set up the notations. First, let
$\overline{\mathbb{Q}}$ be the algebraic closure of ${\mathbb{Q}}$, which is
viewed as a subfield of ${\mathbb{C}}$. Inside $\overline{\mathbb{Q}}$ there
is the subring of algebraic integers $\overline{\mathbb{Z}}$, which is the set
of algebraic numbers which are solutions to monic polynomials with integer
coefficients. Further, in characteristic $p$ (where $p$ is an odd prime), let
${\mathbb{F}}_{p}\cong{\mathbb{Z}}/p{\mathbb{Z}}$ be the smallest field with
characteristic $p$. Let $\overline{\mathbb{F}}_{p}$ be the algebraic closure
of ${\mathbb{F}}_{p}$, which is only well-defined up to isomorphism of field
extensions.

Notice that the notion of non-Archimedean normed algebras can be transferred
between different coefficient rings via tensor products. A crucial feature of
the geometric construction of this paper is that, as long as one has a
counting theory over ${\mathbb{Z}}$ (or $\overline{\mathbb{Z}}$), then it
automatically induces a theory over any ring $R$ (or
$\overline{\mathbb{Z}}$-algebra). In particular, one needs to perform the “mod
$p$ reduction” which is roughly associated to the ring map
${\mathbb{Z}}\to{\mathbb{F}}_{p}$. In our situation, one needs the
corresponding extension to the algebraic closure of ${\mathbb{F}}_{p}$.

######  Lemma 3.3.

For each prime $p$, there exists a unital ring map

$\overline{\pi}_{p}:\overline{\mathbb{Z}}\to\overline{\mathbb{F}}_{p}.$

###### Proof.

(Following the mathoverflow post [(2̆3])
$p\in{\mathbb{Z}}\subset\overline{\mathbb{Z}}$ is not an invertible element.
Hence there exists a maximal ideal
${\mathfrak{m}}\subset\overline{\mathbb{Z}}$ containing $p$. Consider the
quotient field $\overline{\mathbb{Z}}/{\mathfrak{m}}$, which is an extension
of ${\mathbb{F}}_{p}$. We prove that it is an algebraic closure of
${\mathbb{F}}_{p}$. First, each element
$x\in\overline{\mathbb{Z}}/{\mathfrak{m}}$ has a lift
$\tilde{x}\in\overline{\mathbb{Z}}$ which is the solution to an algebraic
equation with integer coefficients. Hence $x$ is an algebraic element over
${\mathbb{F}}_{p}$. Therefore $\overline{\mathbb{Z}}/{\mathfrak{m}}$ is
algebraic over ${\mathbb{F}}_{p}$. Second, for any monic polynomial $f$ with
${\mathbb{F}}_{p}$-coefficients, one can find a monic integral lift
$\tilde{f}$ and all roots of $\tilde{f}$ are in $\overline{\mathbb{Z}}$. Hence
$f$ has roots in the field $\overline{\mathbb{Z}}/{\mathfrak{m}}$. Therefore
$\overline{\mathbb{Z}}/{\mathfrak{m}}$ is algebraically closed. Hence
$\overline{\mathbb{Z}}/{\mathfrak{m}}$ is an algebraic closure of
${\mathbb{F}}_{p}$. Therefore,
$\overline{\mathbb{Z}}/{\mathfrak{m}}\cong\overline{\mathbb{F}}_{p}$ as
fields. Precomposing with
$\overline{\mathbb{Z}}\to\overline{\mathbb{Z}}/{\mathfrak{m}}$ one obtains the
desired ring map. ∎

######  Remark 3.4.

The map $\overline{\pi}_{p}$ is not unique. But we fix one for each prime $p$.

The “mod $p$ reduction” also makes sense for any rational number as long as
$p$ is greater than the denominator. We need the following lemma to extend
this simple fact to the algebraic closure.

######  Lemma 3.5.

For each $y\in\overline{\mathbb{Q}}$, there exists
$m\in{\mathbb{Z}}\setminus\\{0\\}$ such that $my\in\overline{\mathbb{Z}}$.

###### Proof.

If $y\in\overline{\mathbb{Q}}$ is an algebraic number, then it is a root of a
polynomial with integer coefficients $P=a_{k}u^{k}+\cdots+a_{1}u+a_{0}$ with
$a_{k}\neq 0$. Then

$0=a_{k}^{k-1}P(y)=(a_{k}y)^{k}+a_{k-1}(a_{k}y)^{k-1}+a_{k-2}a_{k}(a_{k}y)^{k-2}+\cdots+a_{1}a_{k}^{k-2}(a_{k}u)+a_{0}a_{k}^{k-1}.$

So $a_{k}y$ is a root of a monic polynomial with integer coefficients, hence
is in $\overline{\mathbb{Z}}$. ∎

###  3.2. Semisimple algebras over Novikov fields

In this paper we use a more restrictive notion of semisimplicity of algebras
over Novikov fields.

######  Definition 3.6.

Let ${\mathbb{F}}$ be a field. A unital ${\mathbb{F}}$-algebra $(A,\ast)$ is
called semisimple if it splits as a direct sum of rings

$A=F_{1}\oplus\cdots\oplus F_{k}$

where $F_{i}\cong{\mathbb{F}}$ as a ring. Each summand $F_{i}$ is called an
idempotent summand of $A$ and the splitting is called the idempotent
splitting.333It is easy to see that if $A$ is semisimple, then the idempotent
splitting is unique up to permuting idempotent summands.

######  Remark 3.7.

In many papers such as [EP03, She22, FOOO10], the meaning of semisimplicity is
more general: for example, each summand $F_{i}$ is allowed to be a finite
extension of the field ${\mathbb{F}}$. The number of idempotent summands also
depends on the choice of the field. In our situation, one can achieve the
above stronger semisimplicity of a version of quantum cohomology algebra by
turning on bulk deformations and taking a sufficiently large field.

Suppose $A$ is semisimple. Then for each idempotent summand $F_{i}$, there is
a unique generator $e_{i}\in F_{i}$ such that $e_{i}\ast e_{i}=e_{i}$. We call
$e_{i}$ the idempotent generator. Then $(e_{1},\ldots,e_{k})$ is a basis of
$A$. Given any element $\alpha=\lambda_{1}e_{1}+\cdots+\lambda_{k}e_{k}$, one
can see that the linear map

$\alpha\ast:A\to A$

has eigenspace decomposition $F_{1}\oplus\cdots\oplus F_{k}$ with eigenvalues
$\lambda_{1},\ldots,\lambda_{k}$. The following statement shows that the
converse also holds under additional assumptions.

######  Lemma 3.8.

Let $A$ be a $k$-dimensional commutative unital ${\mathbb{F}}$-algebra and
$\alpha\in A$. Suppose $\alpha\ast:A\to A$ has $k$ distinct _nonzero_
eigenvalues $\lambda_{1},\ldots,\lambda_{k}$. Then $A$ is semisimple.

###### Proof.

Let $(\varepsilon_{1},\ldots,\varepsilon_{k})$ be an eigen-basis of
$\alpha\ast$. Write

$\alpha=\sum_{i=1}^{k}\mu_{i}\varepsilon_{i}.$

Then we see

$\alpha\ast(\varepsilon_{i}\ast\varepsilon_{j})=\lambda_{i}\varepsilon_{i}\ast\varepsilon_{j}=\lambda_{j}\varepsilon_{i}\ast\varepsilon_{j}.$

As $\lambda_{i}$ are all distinct, one has
$\varepsilon_{i}\ast\varepsilon_{j}=0$ whenever $i\neq j$. Then one obtains

$\alpha\ast\varepsilon_{i}=\mu_{i}\varepsilon_{i}\ast\varepsilon_{i}=\lambda_{i}\varepsilon_{i}.$

As $\lambda_{i}\neq 0$, one can see $\mu_{i}\neq 0$. Define
$e_{i}=\lambda_{i}^{-1}\mu_{i}\varepsilon_{i}$. Then

$e_{i}\ast
e_{i}=(\lambda_{i}^{-1}\mu_{i})^{2}\varepsilon_{i}\ast\varepsilon_{i}=\lambda_{i}^{-1}\mu_{i}\varepsilon_{i}=e_{i}.$

Hence $A$ is semisimple. ∎

####  3.2.1. Semi-simplicity and different characteristics

Here we prove a useful algebraic fact which allows us to derive semi-
simplicity in finite characteristics from semi-simplicity in characteristic
zero. We set up the problem as follows. Let $(A,\ell)$ be a non-Archimedean
normed (free) algebra over the Novikov ring $\Lambda_{\overline{\mathbb{Z}}}$.
Denote

$A_{(0)}:=A\otimes_{\Lambda_{\overline{\mathbb{Z}}}}\Lambda_{\overline{\mathbb{Q}}}$

and for each prime $p$

$A_{(p)}:=A\otimes_{\Lambda_{\overline{\mathbb{Z}}}}\Lambda_{\overline{\mathbb{F}}_{p}}.$

Denote the induced valuations by

$\displaystyle\ \ell_{0}:A_{(0)}\to{\mathbb{R}}\cup\\{-\infty\\},\ $
$\displaystyle\ \ell_{p}:A_{(p)}\to{\mathbb{R}}\cup\\{-\infty\\}.$

Moreover, let ${\mathcal{U}}\in A$ be a distinguished nonzero element (which
will be the first Chern class in quantum homology in our later discussions),
and let ${\mathcal{U}}_{(0)}\in A_{(0)}$, ${\mathcal{U}}_{(p)}\in A_{(p)}$ be
the corresponding induced element. They induce linear operators

$E_{(m)}:A_{(m)}\to A_{(m)},\ x\mapsto{\mathcal{U}}_{(m)}\ast x,\ m=0,p.$

######  Theorem 3.9.

Suppose $A_{(0)}$ is semisimple over $\Lambda_{\overline{\mathbb{Q}}}$ and all
eigenvalues of $E_{(0)}$ are nonzero and distinct. Then there exist $p_{0}>0$
and $C>0$ such that for all prime $p\geq p_{0}$, the following conditions
hold.

  1. (1)

$A_{(p)}$ is semisimple over $\Lambda_{\overline{\mathbb{F}}_{p}}$.

  2. (2)

If $e_{1,p},\ldots,e_{m,p}$ are idempotent generators of $A_{(p)}$, then

$\ell(e_{l,p})\leq C.$

###### Proof of Theorem 3.9 (1).

Consider the operator $E:A\to A$ and its characteristic polynomial $f_{E}$.
Notice that $\Lambda_{\overline{\mathbb{Q}}}$ is the field of fraction of
$\Lambda_{\overline{\mathbb{Z}}}$. Hence $f_{E}$ has $m$ distinct roots in
$\Lambda_{\overline{\mathbb{Q}}}$ and so the discriminant of $f_{E}$, denoted
by $D(f_{E})\in\Lambda_{\overline{\mathbb{Z}}}$, is nonzero. Hence for
sufficiently large prime $p$, the discriminant of $f_{E_{(p)}}$, which is the
mod $p$ reduction of $D(f_{E})$, is nonzero. It follows that $E_{(p)}$ also
has $m$ distinct eigenvalues. Moreover, as all eigenvalues of $E_{(0)}$ are
nonzero, $f_{E}(0)\neq 0$. Hence $f_{E_{(p)}}(0)\neq 0$ when $p$ is
sufficiently large. Hence $E_{(p)}$ is invertible and has no zero eigenvalue.
By Lemma 3.8, $A_{(p)}$ is semisimple for sufficiently large $p$. ∎

####  3.2.2. Proof of Theorem 3.9 (2)

To prove the quantitative statement of Theorem 3.9, we introduce the notion of
truncation. First, given an element

$\lambda=\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\in\Lambda_{\overline{\mathbb{Q}}},$

and $Z\in{\mathbb{R}}$, define the $Z$-truncation of $\lambda$ to be the
element

$\lambda^{Z}:=\sum_{g_{i}\leq Z}a_{i}T^{g_{i}},$

which has only finitely many terms. Then it follows easily

${\mathfrak{v}}(\lambda-\lambda^{Z})\geq Z.$ (3.1)

For an element in a module over $\Lambda_{\overline{\mathbb{Q}}}$, its
truncations are not canonically defined. We fix, throughout the proof, a basis
${\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{m}$, of the
$\Lambda_{\overline{\mathbb{Z}}}$-module $A$. Without loss of generality, we
can choose the basis such that

$\ell({\mathfrak{x}}_{1})=\cdots=\ell({\mathfrak{x}}_{m})=0.$

By abuse of notations, denote the induced basis of $A_{(0)}$ and $A_{(p)}$
still by ${\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{m}$. Then for each
$\alpha\in A_{(0)}$, we can write

$\alpha=\sum_{j=1}^{m}\alpha_{j}{\mathfrak{x}}_{j}$

where $\alpha_{j}\in\Lambda_{\overline{\mathbb{Q}}}$. Then define the
$Z$-truncation

$\alpha^{Z}=\sum_{j=1}^{B}\alpha_{j}^{Z}{\mathfrak{x}}_{j}.$

Then by (3.1) we have the estimate

$\ell_{0}(\alpha-\alpha^{Z})=\ell_{0}\left(\sum_{l=1}^{m}(\alpha_{l}-\alpha_{l}^{Z}){\mathfrak{x}}_{l}\right)\leq\max_{1\leq
l\leq m}\ell_{0}\left((\alpha_{l}-\alpha_{l}^{Z}){\mathfrak{x}}_{l}\right)\\\
=\max_{1\leq l\leq
m}\left(\ell_{0}({\mathfrak{x}}_{l})-{\mathfrak{v}}(\alpha_{l}-\alpha_{l}^{Z})\right)\leq\max_{1\leq
j\leq m}\ell({\mathfrak{x}}_{j})-Z=-Z.$ (3.2)

Running convention. Within this proof, $Z$ is a large real number which can be
fixed from the beginning. The lower bound of $p_{0}$ which is valid for the
statement of Theorem 3.9 depends on the choice of $Z$. The letter $C>0$
denotes a real number which is independent of $Z$ and $p\geq p_{0}$ but whose
value is allowed to change from line to line.

######  Lemma 3.10.

Suppose $e_{1,(0)},\dots,e_{m,(0)}$ constitute the idempotent generators of
$A_{(0)}$, with valuations $\lambda_{1,(0)},\dots,\lambda_{m,(0)}$. Then for
$Z$ sufficiently large, $e_{1,(0)}^{Z},\dots,e_{m,(0)}^{Z}$ form a basis of
$A_{(0)}$, and $\lambda_{1,(0)}^{Z},\dots,\lambda_{m,(0)}^{Z}$ are all nonzero
and distinct.

###### Proof.

With respect to the basis
$({\mathfrak{x}}_{1,(0)},\ldots,{\mathfrak{x}}_{m,(0)})$ of $A_{(0)}$, we
identify $e_{l,(0)}$ with its coordinate vector in
$(\Lambda_{\overline{\mathbb{Q}}})^{m}$. Then the $m\times m$ matrix with
columns $e_{l,(0)}$ is invertible with a nonzero determinant. Then when $Z$ is
sufficiently large, the corresponding determinant with $e_{l,(0)}$ replaced by
$e_{l,(0)}^{Z}$ is also nonzero. On the other hand, as all $\lambda_{l,(0)}$
are nonzero, $\lambda_{l,(0)}^{Z}\neq 0$ when $Z$ is large. ∎

We would like to construct, for large primes $p$, eigenvectors and eigenvalues
of ${\mathbb{E}}_{(p)}$ over the field $\Lambda_{\overline{\mathbb{F}}_{p}}$.
The basic idea is to take some truncation $e_{l,(0)}^{Z}$ of the idempotent
generators and their mod $p$ reductions as an appropriate eigenbasis and then
to apply certain corrections.

By Lemma 3.5, for each $Z\in{\mathbb{R}}$, there exists $m^{Z}\in{\mathbb{Z}}$
such that

$\displaystyle\ m^{Z}\lambda_{l,(0)}^{Z}\in\Lambda_{\overline{\mathbb{Z}}},\ $
$\displaystyle m^{Z}e_{l,(0)}^{Z}\in A_{\overline{\mathbb{Z}}}.$

This allows us to define the “mod $p$ reduction” of $\lambda_{l,(0)}^{Z}$ and
$e_{l,(0)}^{Z}$ as follows. Fixing $m^{Z}$, by choosing a sufficiently large
$p$ so that it cannot divide $m^{Z}$, the quantity $m^{Z}$ has a nonzero
reduction $[m^{Z}]_{p}\in{\mathbb{F}}_{p}$. Moreover,
$m^{Z}\lambda_{l,(0)}^{Z}$ has a mod $p$ reduction
$[m^{Z}\lambda_{l,(0)}^{Z}]_{p}\in\Lambda_{\overline{\mathbb{F}}_{p}}$ and
$m^{Z}e_{l,(0)}^{Z}$ has a mod $p$ reduction $[m^{Z}e_{l,(0)}^{Z}]_{p}\in
A_{(p)}$ (defined via the integral basis
${\mathfrak{x}}_{1},\ldots,{\mathfrak{x}}_{m}$). Then define

$\displaystyle\
\lambda_{l,(p)}^{Z}:=[m^{Z}]_{p}^{-1}[m^{Z}\lambda_{l,(0)}^{Z}]_{p},\ $
$\displaystyle\ e_{l,(p)}^{Z}:=[m^{Z}]_{p}^{-1}[m^{Z}e_{l,(0)}^{Z}]_{p}.$

######  Lemma 3.11.

There exists $C>0$ such that for any sufficiently large $Z$, upon choosing
$m^{Z}$ as above, there exists $p^{Z}>0$ such that whenever $p\geq p^{Z}$,
$e_{l,(p)}^{Z}$ is a basis of $A_{(p)}$ and all $\lambda_{l,(p)}^{Z}$ are
nonzero and distinct. Moreover, for some constant $C>0$ one has

$\displaystyle\ \ell_{p}(e_{l,(p)}^{Z})\geq-C,\ $ $\displaystyle\
{\mathfrak{v}}(\lambda_{l,(p)}^{Z})\leq C.$

Moreover, for all $k\neq l$

${\mathfrak{v}}(\lambda_{l,(p)}^{Z}-\lambda_{k,(p)}^{Z})\leq C.$

###### Proof.

Straightforward. ∎

######  Proposition 3.12.

There exists $C>0$ such that given any sufficiently large $Z$, for all
sufficiently large prime $p$, there exist eigenvectors $\varepsilon_{l,(p)}$
of $E_{(p)}$ with corresponding distinct eigenvalues
$\lambda_{l,(p)}\in\Lambda_{\overline{\mathbb{F}}_{p}}$ such that

$\ell_{p}(e_{l,(p)}^{Z}-\varepsilon_{l,(p)})\leq-Z+C$

and

${\mathfrak{v}}(\lambda_{l,(p)}^{Z}-\lambda_{l,(p)})\geq Z-C.$

###### Proof.

In $A_{(0)}$, one has

$(m^{Z})^{-1}E_{(0)}(m^{Z}e_{l,(0)})=((m^{Z})^{-1}\lambda_{l,(0)})(m^{Z}e_{l,(0)}).$

Using (3.2), it follows that

$\begin{split}&\
\ell_{0}\Big{(}(m^{Z})^{-1}E_{(0)}(m^{Z}e_{l,(0)}^{Z})-((m^{Z})^{-1}\lambda_{l,(0)}^{Z})(m^{Z}e_{l,(0)}^{Z})\Big{)}\\\
=&\
\ell_{0}\Big{(}E_{(0)}(e_{l,(0)}^{Z})-\lambda_{l,(0)}^{Z}e_{l,(0)}^{Z}\Big{)}\\\
=&\
\ell_{0}\Big{(}E_{(0)}\big{(}e_{l,(0)}^{Z}-e_{l,(0)}\big{)}+(\lambda_{l,(0)}-\lambda_{l,(0)}^{Z})e_{l,(0)}+\lambda_{l,(0)}^{Z}(e_{l,(0)}-e_{l,(0)}^{Z})\Big{)}\\\
\leq&\ \max\Big{\\{}\ell_{0}(c_{1,(0)}\ast(e_{l,(0)}^{Z}-e_{l,(0)})),\
\ell_{0}((\lambda_{l,(0)}-\lambda_{l,(0)}^{Z})e_{l,(0)}),\
\ell_{0}(\lambda_{l,(0)}^{Z}(e_{l,(0)}-e_{l,(0)}^{Z}))\Big{\\}}\\\ \leq&\
\max\Big{\\{}\ell_{0}(c_{1,(0)})+C-Z,\ \ell_{0}(e_{l,(0)})-Z,\
-{\mathfrak{v}}(\lambda_{l,(0)})+C-Z\Big{\\}}\\\ \leq&\ C-Z.\end{split}$ (3.3)

Now take a sufficiently large $p$. Notice that the left hand side of (3.3) is
the valuation of an integral element, hence descends to
$\overline{\mathbb{F}}_{p}$. Then (3.3) implies

$\ell_{p}\left(E_{(p)}(e_{l,(p)}^{Z})-\lambda_{l,(p)}^{Z}e_{l,(p)}^{Z}\right)\leq
C-Z.$ (3.4)

We would like to correct $e_{l,(p)}^{Z}$ by adding higher order terms.

With respect to the basis $(e_{1,(p)}^{Z},\ldots,e_{m,(p)}^{Z})$, let the
matrix of $E_{(p)}$ be $T_{(p)}$. For a matrix $S=(S_{ij})$ with entries in
$\Lambda_{\mathbb{K}}$, we write

${\mathfrak{v}}(S)=\min_{i,j}{\mathfrak{v}}(S_{ij})\in{\mathbb{R}}.$

######  Lemma 3.13.

One has

${\mathfrak{v}}\left(T_{(p)}-\left[\begin{array}[]{cccc}\lambda_{1,(p)}^{Z}&0&\cdots&0\\\
0&\lambda_{2,(p)}^{Z}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda_{m,(p)}^{Z}\end{array}\right]\right)\geq Z-C.$

###### Proof.

We can write

$E_{(p)}(e_{l,(p)}^{Z})-\lambda_{l,(p)}^{Z}e_{l,(p)}^{Z}=\sum_{l=1}^{m}a_{l}e_{l,(p)}^{Z}.$

By (3.4) and the subadditivity property of the function $\ell$ (see Definition
3.1), one has ${\mathfrak{v}}(a_{l})\geq Z-C$ for some appropriate $C$. ∎

For each $l$, consider the following equation for

$\displaystyle\ x=\sum_{k\neq l}x_{k}e_{k,(p)}^{Z},\ $ $\displaystyle\
\delta\in\Lambda_{\overline{\mathbb{F}}_{p}}$

which is

$E_{(p)}\big{(}e_{l,(p)}^{Z}+x\big{)}=\big{(}\lambda_{l,(p)}^{Z}+\delta\big{)}\big{(}e_{l,(p)}^{Z}+x\big{)}$

which is equivalent to

$E_{(p)}\left(\sum_{k\neq l}x_{k}e_{k,(p)}^{Z}\right)-\sum_{k\neq
l}\lambda_{l,(p)}^{Z}x_{k}e_{k,(p)}^{Z}-\delta e_{l,(p)}^{Z}=\delta\sum_{k\neq
l}x_{k}e_{k,(p)}^{Z}+\rho_{l}.$

Here $\rho_{l}$ is the error term with ${\mathfrak{v}}(\rho_{l})\leq-Z+C$ by
(3.4). To simplify notations, assume $l=1$. Then using the basis
$e_{1,(p)}^{Z},\ldots,e_{m,(p)}^{Z}$, this equation is equivalent to the
linear system

$\left(\left[\begin{array}[]{cccc}1&0&\cdots&0\\\
&&T_{(p)}^{\prime}&\end{array}\right]-\left[\begin{array}[]{cccc}0&0&\cdots&0\\\
0&\lambda_{1,(p)}^{Z}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&\lambda_{1,(p)}^{Z}\end{array}\right]\right)\left[\begin{array}[]{c}\delta\\\
x_{2}\\\ \vdots\\\
x_{m}\end{array}\right]=Q(\delta,x_{2},\ldots,x_{m})+\rho_{1}.$

Here the left hand side is linear and $Q$ is quadratic. Let the matrix on the
left hand side be $F_{1}$. Lemma 3.13 implies that $F_{1}$ is invertible with

${\mathfrak{v}}(F_{1})\leq C$

where $C$ is independent of $Z$ and $p$. Then one can use an iteration
argument to solve the equation term by term. The correction term has valuation
at least $Z-C$ for some constant $C$. So the theorem follows. ∎

We continue the proof of (2) of Theorem 3.9. By the proof of Lemma 3.8, each
idempotent generator of $A_{(p)}$ is a multiple of $\varepsilon_{l,(p)}$.
Indeed, if

$\varepsilon_{l,(p)}\ast\varepsilon_{l,(p)}=\mu_{l}\varepsilon_{l,(p)}$

then the corresponding idempotent generator is

$e_{l,(p)}=\mu_{l}^{-1}\varepsilon_{l,(p)}.$

So we need to estimate the valuation of $\mu_{l}$. In characteristic zero one
has

$e_{l,(0)}\ast e_{l,(0)}=e_{l,(0)}.$

Taking truncation at $Z$ one has

$\ell_{0}\big{(}e_{l,(0)}^{Z}\ast e_{l,(0)}^{Z}-e_{l,(0)}^{Z}\big{)}\leq C-Z.$

Taking mod $p$ reduction, one obtains

$\ell_{p}\big{(}e_{l,(p)}^{Z}\ast e_{l,(p)}^{Z}-e_{l,(p)}^{Z}\big{)}\leq C-Z.$

Then

$\ell_{p}\big{(}\mu_{l}\varepsilon_{l,(p)}-e_{l,(p)}^{Z}\big{)}=\ell_{p}\big{(}\varepsilon_{l,(p)}\ast\varepsilon_{l,(p)}-e_{l,(p)}^{Z}\ast
e_{l,(p)}^{Z}+e_{l,(p)}^{Z}\ast e_{l,(p)}^{Z}-e_{l,(p)}^{Z}\big{)}\\\
\leq\max\Big{\\{}\ell_{p}(\varepsilon_{l,(p)}-e_{l,(p)}^{Z})+\ell_{p}(\varepsilon_{l,(p)}+e_{l,(p)}^{Z}),C-Z\Big{\\}}\leq
C-Z.$

As we have $\ell_{p}(e_{l,(p)}^{Z})\geq-C$, it follows that
$\ell_{p}(\mu_{l}\varepsilon_{l,(p)})=\ell_{p}(e_{l,(p)}^{Z})=\ell_{p}(\varepsilon_{l,(p)})$.
Hence ${\mathfrak{v}}(\mu_{l})=0$ and hence

$\ell_{p}(e_{l,(p)})=\ell_{p}(\mu_{l}^{-1}\varepsilon_{l,(p)})=\ell_{p}(\varepsilon_{l,(p)})+{\mathfrak{v}}(\mu_{l})=\ell_{p}(e_{l,(p)}^{Z})=\ell_{0}(e_{l,(0)})$

which is independent of $p$. This finishes the proof of Theorem 3.9.

###  3.3. Floer–Novikov complexes

Let $\Gamma\subsetneq{\mathbb{R}}$ be a proper additive subgroup.

######  Definition 3.14 (Floer–Novikov complex).

(cf. [Ush08, Definition 1.1]) A ${\mathbb{Z}}_{2}$-graded Floer–Novikov
package over a commutative unital ring $R$ consists of data

${\mathfrak{c}}=\Big{(}P,{\mathcal{A}},gr,n_{R}\Big{)}$

where

  1. (1)

$P$ is a $\Gamma$-torsor with $\underline{\smash{P}}:=P/\Gamma$ finite.

  2. (2)

${\mathcal{A}}:P\to{\mathbb{R}}$ is the “action functional” and
$gr:P\to{\mathbb{Z}}_{2}$ is the “grading.”

  3. (3)

For $p\in P$ and $g\in\Gamma$, one has

$\displaystyle\ {\mathcal{A}}(gp)={\mathcal{A}}(p)-g,\ $ $\displaystyle\
gr(gp)=gr(p)$

  4. (4)

$n_{R}:P\times P\to R$ is a function such that

     * •

$n_{R}(p,q)\neq 0\Longrightarrow gr(p)=gr(q)+1,\
{\mathcal{A}}(p)>{\mathcal{A}}(q)$;

     * •

for all $p\in P$ and $C\in{\mathbb{R}}$, the set

$\\{q\in P\ |\ n_{R}(p,q)\neq 0,\ {\mathcal{A}}(q)\geq C\\}$

is finite;

     * •

for any $g\in\Gamma$, we have $n_{R}(gp,gq)=n_{R}(p,q)$;

     * •

the $\Lambda_{R}$-linear map $\partial$ defined in (3.5) satisfies
$\partial^{2}=0$.

Given a Floer–Novikov package one can construct the associated Floer chain
complex. First, define

$CF_{\bullet}({\mathfrak{c}})=\Big{\\{}\sum_{p\in P}a_{p}p\ |\ a_{p}\in R,\
\forall C\in{\mathbb{R}},\\#\\{p\in P\ |\ a_{p}\neq 0,\ {\mathcal{A}}(p)\geq
C\\}<\infty\Big{\\}}$

which is ${\mathbb{Z}}_{2}$-graded. The $\Lambda_{R}^{\Gamma}$-module
structure is induced from the $\Gamma$-action on $P$. Define the differential

$\partial:CF_{\bullet}({\mathfrak{c}})\to CF_{\bullet-1}({\mathfrak{c}})$

by

$\partial\left(\sum_{p\in P}a_{p}p\right)=\sum_{q\in P}\left(\sum_{p\in
P}a_{p}n_{R}(p,q)\right)q.$ (3.5)

We also define the function

$\ell:CF_{\bullet}({\mathfrak{c}})\to{\mathbb{R}}\cup\\{-\infty\\},\
\ell\left(\sum_{p\in P}a_{p}p\right)=\sup\big{\\{}{\mathcal{A}}(p)\ |\
a_{p}\neq 0\Big{\\}}.$ (3.6)

Given a Floer–Novikov package ${\mathfrak{c}}$ over $R$, if
$\iota:R\to\widetilde{R}$ is a ring map, then one can extend ${\mathfrak{c}}$
to a Floer–Novikov package ${\mathfrak{c}}\otimes_{R}\widetilde{R}$ by simply
defining $n_{\widetilde{R}}:=\iota\circ n:P\times P\to\widetilde{R}$

######  Proposition 3.15.

If $R={\mathbb{K}}$ is a field, the triple
$(CF_{\bullet}({\mathfrak{c}}),\partial,\ell)$ is a Floer-type complex over
$\Lambda_{\mathbb{K}}^{\Gamma}$ in the sense of [UZ16, Definition 4.1].

###### Proof.

It follows directly from the definitions of Floer-type complexes. The proof
serves rather as a brief clarification about this concept. First, for each
$k\in{\mathbb{Z}}_{2}$, the pair
$(CF_{k}({\mathfrak{c}}),\ell|_{CF_{k}({\mathfrak{c}})})$ is a non-Archimedean
normed vector space over $\Lambda_{\mathbb{K}}^{\Gamma}$ (see [UZ16,
Definition 2.2]. In addition it is an orthogonalizable
$\Lambda_{\mathbb{K}}^{\Gamma}$-space (see [UZ16, Definition 2.7]. The last
requirement for being a Floer-type complex is the inequality

$\ell(\partial(x))\leq\ell(x)\ \forall x\in CF_{\bullet}({\mathfrak{c}}),$

which is a consequence of the property of the function $n_{\mathbb{K}}$ in the
data ${\mathfrak{c}}$. ∎

####  3.3.1. Spectral invariants

Following Usher [Ush08], one can also define spectral invariants in an
abstract way. First, define the “energy filtration” on the complex
$CF_{\bullet}({\mathfrak{c}})$: for each $\tau\in{\mathbb{R}}$, define

$CF_{\bullet}^{\leq\tau}({\mathfrak{c}}):=\left\\{\sum_{p\in P}a_{p}p\in
CF_{\bullet}({\mathfrak{c}})\ |\ a_{p}\neq
0\Longrightarrow{\mathcal{A}}(p)\leq\tau\right\\}.$

Then since the differential decreases the action, it is a subcomplex with
homology

$HF_{\bullet}^{\leq\tau}({\mathfrak{c}})$

and natural maps when $\tau\leq\kappa$

$\iota^{\tau,\kappa}:HF_{\bullet}^{\leq\tau}({\mathfrak{c}})\to
HF_{\bullet}^{\leq\kappa}({\mathfrak{c}}).$ (3.7)

For $\alpha\in HF_{\bullet}({\mathfrak{c}})$, define

$\rho(\alpha):=\inf\Big{\\{}\tau\in{\mathbb{R}}\ |\ \alpha\in{\rm
Im}\left(\iota^{\tau}:HF_{\bullet}^{\leq\tau}({\mathfrak{c}})\to
HF_{\bullet}({\mathfrak{c}})\right)\Big{\\}}\in{\mathbb{R}}\cup\\{-\infty\\}$

######  Theorem 3.16.

[Ush08, Theorem 1.3, 1.4] Given a Floer–Novikov package ${\mathfrak{c}}$ (over
a Noetherian ring $R$) and $\alpha\in HF({\mathfrak{c}})\setminus\\{0\\}$,
$\rho(\alpha)>-\infty$ and $\alpha\in{\rm Im}(\iota_{\rho(\alpha)})$.

####  3.3.2. Boundary depth

######  Definition 3.17.

[Ush11a] Let ${\mathfrak{c}}$ be a Floer–Novikov package and let
$CF^{\leq\lambda}({\mathfrak{c}})$ be the associated filtered Floer–Novikov
complex over $\Lambda_{\mathbb{K}}^{\Gamma}$. Then the boundary depth of the
filtered complex is the infimum of $\beta>0$ such that for all
$\lambda\in{\mathbb{R}}$

$CF^{\leq\lambda}({\mathfrak{c}})\cap{\rm
Im}\partial\subset\partial(CF^{\leq\lambda+\beta}({\mathfrak{c}})).$

######  Theorem 3.18.

[Ush08, Theorem 1.3] Given a Floer–Novikov package ${\mathfrak{c}}$, the
boundary depth of the associated Floer–Novikov complex is finite.

####  3.3.3. Quasiequivalence distance

We rephrase the notion of quasiequivalences between Floer–Novikov complexes,
which was originally introduced in [UZ16] for the more general situation of
Floer-type complexes.

######  Definition 3.19.

(cf. [UZ16, Definition 1.3]) Let
$(CF_{\bullet}({\mathfrak{c}}_{i}),\partial_{i})$, $i=1,2$, be two
Floer–Novikov complexes associated to Floer–Novikov data ${\mathfrak{c}}_{i}$
over a field ${\mathbb{K}}$. Let $\ell_{i}$ be the valuation function on the
two complexes defined by (3.6). Let $\delta\geq 0$. A
$\delta$-quasiequivalence between $CF_{\bullet}({\mathfrak{c}}_{1})$ and
$CF_{\bullet}({\mathfrak{c}}_{2})$ is a quadruple $(\Phi,\Psi,K_{C},K_{D})$
where

  1. (1)

$\Phi:CF_{\bullet}({\mathfrak{c}}_{1})\to CF_{\bullet}({\mathfrak{c}}_{2})$
and $\Psi:CF_{\bullet}({\mathfrak{c}}_{2})\to
CF_{\bullet}({\mathfrak{c}}_{1})$ are chain maps with

$\displaystyle\ \ell_{2}(\Phi(x_{1}))\leq\ell_{1}(x_{1})+\delta,\ $
$\displaystyle\ \ell_{1}(\Psi(x_{2}))\leq\ell_{2}(x_{2})+\delta$

for all $x_{1}\in CF_{\bullet}({\mathfrak{c}}_{1})$ and $x_{2}\in
CF_{\bullet}({\mathfrak{c}}_{2})$.

  2. (2)

$K_{i}:CF_{\bullet}({\mathfrak{c}}_{i})\to
CF_{\bullet+1}({\mathfrak{c}}_{i})$, $i=1,2$, obey the homotopy equations

$\displaystyle\ \Psi\circ\Phi-{\rm
Id}_{CF_{\bullet}({\mathfrak{c}}_{1})}=\partial_{1}K_{1}+K_{1}\partial_{1},\ $
$\displaystyle\ \Phi\circ\Psi-{\rm
Id}_{CF_{\bullet}({\mathfrak{c}}_{2})}=\partial_{2}K_{2}+K_{2}\partial_{2}$

and for all $x_{i}\in CF_{\bullet}({\mathfrak{c}}_{i})$, $i=1,2$, one has

$\ell_{i}(K_{i}(x_{i}))\leq\ell_{i}(x_{i})+2\delta.$

The quasiequivalence distance between $CF_{\bullet}({\mathfrak{c}}_{1})$ and
$CF_{\bullet}({\mathfrak{c}}_{2})$, denoted by
$d_{Q}(CF_{\bullet}({\mathfrak{c}}_{1}),CF_{\bullet}({\mathfrak{c}}_{2}))$, is
the infimum of $\delta$ such that there exists a $\delta$-quasiequivalence
between them.

###  3.4. Persistence modules and stability of boundary depth

######  Definition 3.20.

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

  1. (1)

A persistence module ${\bm{V}}$ is a family of ${\mathbb{K}}$-vector spaces

${\bm{V}}=(V^{s})_{s\in{\mathbb{R}}}$

together with linear maps (called the structural maps of ${\bm{V}}$)

$\iota^{s,t}:=\iota^{s,t}_{\bm{V}}:V^{s}\to V^{t}\ \forall s\leq t$

such that $\iota^{s,s}={\rm Id}_{V^{s}}$ for all $s$ and
$\iota^{t,r}\circ\iota^{s,t}=\iota^{s,r}$ for all $s\leq t\leq r$.

  2. (2)

Let ${\bm{V}}$ be a persistence module and $\delta\in{\mathbb{R}}$. The
$\delta$-shift of ${\bm{V}}$ is the persistence module ${\bm{V}}[\delta]$ with
$V[\delta]^{s}=V^{s+\delta}$ and
$\iota[\delta]^{s,t}=\iota^{s+\delta,t+\delta}$.

  3. (3)

Let ${\bm{V}}$ and ${\bm{W}}$ be two persistence modules. A morphism from
${\bm{V}}$ to ${\bm{W}}$ is a collection of linear maps
${\bm{f}}=(f^{s}:V^{s}\to W^{s})_{s\in{\mathbb{R}}}$ such that for all $s\leq
t$ the following diagram commutes.

$\textstyle{V^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{s}}$$\scriptstyle{\iota^{s,t}_{\bm{V}}}$$\textstyle{W^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\iota^{s,t}_{\bm{W}}}$$\textstyle{V^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{t}}$$\textstyle{W^{t}}$

  4. (4)

The direct sum of persistence modules is defined in a natural way.

######  Remark 3.21.

The notion of persistence modules first appeared in topological data analysis
and then was adopted to symplectic topology (see [PS16, UZ16]). In many
papers, notably in the symplectically aspherical or monotone setting such as
[PS16, She22],the Floer persistence modules are usually finite-dimensional and
hence generate barcodes as in the original situation of topological data
analysis. However, in more general situations such as [UZ16] where the Floer
persistence modules are infinite-dimensional, the notion of barcodes becomes
more complicated. See Subsection 3.5 for more details.

####  3.4.1. Interleaving distance

######  Definition 3.22.

Let $\delta\geq 0$. Two persistence modules ${\bm{V}}$, ${\bm{W}}$ are called
$\delta$-interleaved if there are ${\mathbb{K}}$-linear maps

$\displaystyle\ f^{s}:V^{s}\to W^{s+\delta},\ $ $\displaystyle\ g^{s}:W^{s}\to
V^{s+\delta}$

for all $s\in{\mathbb{R}}$ such that for all $s\leq t$ the following diagram
commutes.

$\textstyle{V^{s-\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{s-\delta}}$$\scriptstyle{\iota^{s-\delta,s+\delta}_{\bm{V}}}$$\textstyle{W^{s}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{s}}$$\textstyle{V^{s+\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{s+\delta}}$$\textstyle{W^{s+2\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{V^{t-\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{t-\delta}}$$\textstyle{W^{t}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{t}}$$\scriptstyle{\iota^{t,t+2\delta}_{\bm{W}}}$$\textstyle{V^{t+\delta}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{t+\delta}}$$\textstyle{W^{t+2\delta}}$

Here all vertical arrows are the structural maps in the persistence modules.

Define the interleaving distance between ${\bm{V}}$ and ${\bm{W}}$ to be the
infimum of all $\delta\geq 0$ such that ${\bm{V}}$ and ${\bm{W}}$ are
$\delta$-interleaved; if such $\delta$ does not exist, define the interleaving
distance to be $+\infty$. Here ends this definition.

####  3.4.2. Boundary depth of persistence modules and stability

We observe that one can generalize the notion of boundary depth to persistence
modules.

######  Definition 3.23.

Let ${\bm{V}}$ be a persistence module over ${\mathbb{K}}$. The boundary depth
of ${\bm{V}}$, denoted by $\beta({\bm{V}})$, is the infimum of $\beta>0$ such
that for all $s\in{\mathbb{R}}$, $x\in V^{s}$, if $\iota^{s,t}(x)=0$ for some
$t>s$, then $\iota^{s,s+\beta}(x)=0$.

As we allow persistent modules to be infinite-dimensional, we reprove the
stability result of boundary depth.

######  Proposition 3.24.

Suppose ${\bm{V}}$, ${\bm{W}}$ are $\delta$-interleaved persistence modules.
Suppose ${\bm{V}}$ has finite boundary depth. Then ${\bm{W}}$ has finite
boundary depth and

$\beta({\bm{W}})\leq\beta({\bm{V}})+2\delta.$

###### Proof.

Suppose on the contrary that
$\beta({\bm{W}})\geq\beta({\bm{V}})+2\delta+2\epsilon$ for some $\epsilon>0$.
Then there exist $s\in{\mathbb{R}}$ and $x\in W_{s}$ such that
$\iota^{s,s+\beta({\bm{V}})+2\delta+\epsilon}(x)\neq 0$. Then by the
definition of $\delta$-interleaving, one has $y:=f^{s,s+\delta}(x)\neq 0$ and

$\iota^{s+\delta,s+\delta+\beta({\bm{V}})+\epsilon}(y)\neq 0$

but $y$ cannot survive eventually. This contradicts the definition of
$\beta({\bm{V}})$. ∎

####  3.4.3. Persistence modules associated to filtered Floer–Novikov
complexes

Fix a field ${\mathbb{K}}$. Let ${\mathfrak{c}}$ be a Floer–Novikov package
(see Definition 3.14) and $CF_{\bullet}({\mathfrak{c}})$ be the associated
filtered Floer–Novikov complex. Then the collection of homology groups

$V^{s}({\mathfrak{c}}):=HF_{\bullet}^{\leq
s}({\mathfrak{c}};\Lambda_{\mathbb{K}}^{\Gamma})$

together with the natural maps $\iota^{s,t}$ (cf. Equation (3.7)) is a
persistence module over ${\mathbb{K}}$, denoted by ${\bm{V}}({\mathfrak{c}})$.

It is easy to derive from definitions the following stability results of
persistence modules coming from Floer–Novikov complexes.

######  Proposition 3.25.

Let $CF_{\bullet}({\mathfrak{c}}_{i})$, $i=1,2$ be two Floer–Novikov complexes
over a field ${\mathbb{K}}$ and ${\bm{V}}({\mathfrak{c}}_{i})$ be the
associated persistence module. Then the interleaving distance between
${\bm{V}}({\mathfrak{c}}_{1})$ and ${\bm{V}}({\mathfrak{c}}_{2})$ is no
greater than the quasiequivalence distance between
$CF_{\bullet}({\mathfrak{c}}_{1})$ and $CF_{\bullet}({\mathfrak{c}}_{2})$.

Moreover, the two notions of boundary depths (Definition 3.17 and Definition
3.23) agree.

######  Proposition 3.26.

Let ${\mathfrak{c}}$ be a Floer–Novikov package over ${\mathbb{K}}$. Then the
boundary depth of the filtered Floer–Novikov complex
$CF_{\bullet}({\mathfrak{c}})$ and the boundary depth of the persistence
module ${\bm{V}}({\mathfrak{c}})$ coincide.

###### Proof.

Let $\beta_{1}$ be the boundary depth of $CF_{\bullet}({\mathfrak{c}})$ and
$\beta_{2}$ be the boundary depth of ${\bm{V}}({\mathfrak{c}})$. Suppose
$[x]\in HF^{\leq s}_{\bullet}({\mathfrak{c}})$ which does not survive
eventually. Let $x\in CF^{\leq s}_{\bullet}({\mathfrak{c}})$ be a
representative. Then $x$ is exact. Then by Definition 3.17, for all
$\epsilon>0$, one has

$x\in\partial(CF^{\leq s+\beta_{1}+\epsilon}_{\bullet}({\mathfrak{c}})).$

Hence $\iota^{s,s+\beta_{1}+\epsilon}([x])=0$. As $\epsilon$ is arbitrary,
this implies that $\beta_{2}\leq\beta_{1}$.

On the other hand, for all $s\in{\mathbb{R}}$ and all exact $x\in
CF_{\bullet}^{\leq s}({\mathfrak{c}})$, the class $[x]\in HF_{\bullet}^{\leq
s}({\mathfrak{c}})$ does not survive eventually. Then by Definition 3.23, for
any $\epsilon>0$, one has $\iota^{s,s+\beta_{2}+\epsilon}([x])=0$. This
implies that

$x\in\partial(CF^{\leq s+\beta_{2}+\epsilon}_{\bullet}({\mathfrak{c}})).$

It follows that $\beta_{1}\leq\beta_{2}$. Hence $\beta_{1}=\beta_{2}$. ∎

###  3.5. Barcodes and reduced barcodes

In the symplectically aspherical or monotone case, the notion of barcodes is
the same as the one used in topological data analysis. In more general
situations, Usher–Zhang [UZ16] gave a modification for any Floer-type
complexes (in particular Floer–Novikov complexes) over any Novikov field
$\Lambda_{\mathbb{K}}^{\Gamma}$.

######  Definition 3.27.

(cf. [UZ16, Definition 8.13, 8.14]) Fix a finitely generated subgroup
$\Gamma\subsetneq{\mathbb{R}}$.

  1. (1)

A barcode is a finite multiset $\tilde{\mathcal{B}}$ of elements of
$({\mathbb{R}}/\Gamma)\times(0,+\infty]$. A member of $\tilde{\mathcal{B}}$,
which is usually called a bar, is denoted by $([a],L)$ where
$[a]\in{\mathbb{R}}/\Gamma$ and $L\in(0,+\infty]$.

  2. (2)

A reduced barcode is a finite multiset ${\mathcal{B}}$ of elements of
$(0,+\infty]$. Although ${\mathcal{B}}$ is not a set in general and a member
$L\in{\mathcal{B}}$ may appear multiple times, we still use the same notations
as if $L$ is an element of a set ${\mathcal{B}}$, such as $L\in{\mathcal{B}}$,
without confusion. Let ${\mathcal{B}}_{\rm finite}\subset{\mathcal{B}}$ denote
the submultiset of finite bars, i.e., those with $L<+\infty$. Let
${\mathcal{B}}_{\rm finite}\subset{\mathcal{B}}$ be the submultiset of finite
bars. Notice that a barcode $\tilde{\mathcal{B}}$ induces a reduced barcode
${\mathcal{B}}$ by forgetting the first coordinates.

  3. (3)

The total bar length of a reduced barcode ${\mathcal{B}}$ is

$\tau({\mathcal{B}}):=\sum_{L_{i}\in{\mathcal{B}}_{\rm finite}}L_{i}.$

  4. (4)

The reduced bottleneck distance between two reduced barcodes ${\mathcal{B}}$
and ${\mathcal{B}}^{\prime}$, denoted by
$d_{B}({\mathcal{B}},{\mathcal{B}}^{\prime})$, is the infimum of $\delta>0$
such that, after removing certain submultisets ${\mathcal{B}}_{\rm
short}\subset{\mathcal{B}}$ and ${\mathcal{B}}_{\rm
short}^{\prime}\subset{\mathcal{B}}^{\prime}$ whose members all have length at
most $2\delta$, there is a bijection between
${\mathcal{B}}\setminus{\mathcal{B}}_{\rm short}$ and
${\mathcal{B}}^{\prime}\setminus{\mathcal{B}}_{\rm short}^{\prime}$ such that
the differences of the corresponding bar lengths are all bounded by $\delta$.

The bottleneck distance is symmetric and satisfies the triangle inequality. It
is not a metric in the usual sense as it may take infinite value. Indeed,
$d_{B}({\mathcal{B}},{\mathcal{B}}^{\prime})<\infty$ if and only if
${\mathcal{B}}$ and ${\mathcal{B}}^{\prime}$ has the same number of infinite
bars.

######  Proposition 3.28.

(cf. [RSV21, Proposition 20]) For any $k\geq 0$, the completion of the set of
reduced barcodes having $k$ infinite bars is the set of possibly infinite
reduced barcodes (with $k$ infinite bars) such that for all $\epsilon>0$, the
number of finite bars with length greater than $\epsilon$ is finite.

####  3.5.1. Barcodes associated to Floer–Novikov complexes

Usher–Zhang [UZ16] defined for each ${\mathbb{Z}}_{2}$-graded Floer-type
complexes over $\Lambda_{\mathbb{K}}^{\Gamma}$ and each $k\in{\mathbb{Z}}_{2}$
the associated barcodes (which allows bars of length zero). As Floer–Novikov
complexes are all Floer-type complexes, one has an associated barcode (and
hence a reduced barcode). Let the reduced barcode associated to a
Floer–Novikov complex $CF_{\bullet}({\mathfrak{c}})$ be
${\mathcal{B}}({\mathfrak{c}})$. As the differential strictly decreases the
action, there are no bars of length zero (which was allowed in the abstract
setting of [UZ16]). We do not recall the detail of the definition here.

######  Proposition 3.29.

Let ${\bm{V}}({\mathfrak{c}})$ be the persistence module induced from a
filtered Floer–Novikov complex $CF_{\bullet}({\mathfrak{c}})$ over
$\Lambda_{\mathbb{K}}^{\Gamma}$. Then the boundary depth of
${\bm{V}}({\mathfrak{c}})$ (see Definition 3.17 and Definition 3.23) coincides
with the length of the longest finite bar in ${\mathcal{B}}({\mathfrak{c}})$.
In particular, the boundary depth is zero if and only if
${\mathcal{B}}({\mathfrak{c}})$ has no finite bar.

###### Proof.

It follows from the definitions of boundary depth and barcodes (via singular
value decompositions, see [UZ16]). The details are left to the reader. ∎

####  3.5.2. Stability of barcodes

######  Theorem 3.30.

[UZ16, Theorem 8.17] Let
$(CF_{\bullet}({\mathfrak{c}}_{1}),\partial_{1},\ell_{1})$ and
$(CF_{\bullet}({\mathfrak{c}}_{2}),\partial_{2},\ell_{2})$ be two
Floer–Novikov complexes associated to Floer–Novikov data ${\mathfrak{c}}_{1}$,
${\mathfrak{c}}_{2}$ over a field ${\mathbb{K}}$. Suppose the quasiequivalence
distance between $CF_{\bullet}({\mathfrak{c}}_{1})$ and
$CF_{\bullet}({\mathfrak{c}}_{2})$ is finite. Then

$d_{B}({\mathcal{B}}({\mathfrak{c}}_{1}),{\mathcal{B}}({\mathfrak{c}}_{2}))\leq
2d_{Q}(CF_{\bullet}({\mathfrak{c}}_{1}),CF_{\bullet}({\mathfrak{c}}_{2})).$

###  3.6. $A_{\infty}$ algebras and Hochschild cohomology

Let ${\mathbb{K}}$ be a field of characteristic zero. We recall the notion of
${\mathbb{Z}}_{2}$-graded $A_{\infty}$ algebras over the Novikov field
$\Lambda_{\mathbb{K}}$.

######  Definition 3.31 (Curved $A_{\infty}$ algebra).

  1. (1)

A ${\mathbb{Z}}_{2}$-graded curved $A_{\infty}$ algebra over
$\Lambda_{\mathbb{K}}$ consists of a ${\mathbb{Z}}_{2}$-graded
$\Lambda_{\mathbb{K}}$-vector space ${\mathcal{A}}$ (the degree of a
homogeneous element $a$ is denoted by $|a|$) and for all positive integers
$k\geq 0$ higher composition maps

$m_{k}:{\mathcal{A}}^{\otimes k}\to{\mathcal{A}}\ ({\rm where}\
m_{0}:\Lambda_{\mathbb{K}}\to A)$

(which are $\Lambda_{\mathbb{K}}$-linear and have degree $k\ {\rm mod}\ 2$).
The higher composition maps need to satisfy the following $A_{\infty}$
composition law: for all $k\geq 1$ and
$a_{k},\ldots,a_{1}\in{\mathcal{A}}$,444There are two different conventions:
the variables are either ordered as $a_{1},\ldots,a_{k}$ or ordered as
$a_{k},\ldots,a_{1}$.

$\sum_{i=0}^{k}\sum_{j=0}^{k-i}(-1)^{\maltese_{1}^{j}}m_{k-i+1}\left(a_{k},\ldots,a_{i+j+1},m_{i}(a_{i+j},\ldots,a_{j+1}),a_{j},\ldots,a_{1}\right)=0$

where the symbol $\maltese_{a}^{b}$ for all $a<b$ is defined as

$\maltese_{a}^{b}=\sum_{a\leq i\leq b}\|a_{i}\|\quad{\rm where}\
\|a_{i}\|=|a_{i}|+1.$ (3.8)

  2. (2)

The curvature of a curved $A_{\infty}$ algebra is the element

$m_{0}(1)\in{\mathcal{A}}.$

If $m_{0}=0$, then we say that the $A_{\infty}$ algebra is flat.

  3. (3)

Given a (curved or flat) $A_{\infty}$ algebra ${\mathcal{A}}$, a cohomological
unit is an even element $e\in{\mathcal{A}}$ such that $m_{1}(e)=0$ and that
for all homogeneous $x\in{\mathcal{A}}$

$(-1)^{|x|}m_{2}(e,x)=m_{2}(x,e)=x.$

$e$ is called a strict unit if in addition

$m_{k}(\ldots,e,\ldots)=0\ \forall k\geq 3.$

In these two cases we call $({\mathcal{A}},e)$ a cohomologically unital (resp.
strictly unital) $A_{\infty}$ algebra.

  4. (4)

When ${\mathcal{A}}$ is flat, $A_{\infty}$ composition law implies that
$m_{1}\circ m_{1}=0$. The cohomology algebra of $A$, denoted by
$H^{\bullet}({\mathcal{A}})$, is the ${\mathbb{Z}}_{2}$-graded associative
$\Lambda_{\mathbb{K}}$ algebra whose underlying space is
$H^{\bullet}({\mathcal{A}})={\rm ker}m_{1}/{\rm Im}m_{1}$ and whose
multiplication is induced from $m_{2}$.

Because of bubbling of holomorphic disks, $A_{\infty}$ algebras associated to
a Lagrangian brane is generally curved. There is a way to turn certain curved
$A_{\infty}$ algebras to flat ones.

######  Definition 3.32.

Let $({\mathcal{A}},e)$ be a strictly unital $A_{\infty}$ algebra. A weakly
bounding cochain of $({\mathcal{A}},e)$ is an odd element
$b\in{\mathcal{A}}^{\rm odd}$ such that

$m(b):=\sum_{k\geq 0}m_{k}(b,\ldots,b)=W(b)e\ {\rm where}\
W(b)\in\Lambda_{\mathbb{K}}.$

Suppose $b$ is a weakly bounding cochain of $({\mathcal{A}},e)$. Then define
${\mathcal{A}}^{\flat}$ (which depends on the weakly bounding cochain $b$) to
be the flat $A_{\infty}$ algebra whose underlying space is the same as
${\mathcal{A}}$ and whose composition maps $m_{k}^{\flat}$ is defined by

$m_{k}^{\flat}(x_{k},\ldots,x_{1}):=\sum_{l_{0},\ldots,l_{k}\geq
0}m_{k+l_{0}+\cdots+l_{k}}(\underbrace{b,\ldots,b}_{l_{k}},x_{k},\cdots,x_{1},\underbrace{b,\ldots,b}_{l_{0}}).$

######  Lemma 3.33.

${\mathcal{A}}^{\flat}$ is a flat $A_{\infty}$ algebra. ∎

####  3.6.1. Hochschild cohomology for associative algebras

Let $A$ be a ${\mathbb{Z}}_{2}$-graded associative algebra over
${\mathbb{K}}$. Hochschild cohomology ${\it HH}^{\bullet}(A,M)$ can be defined
for all ${\mathbb{Z}}_{2}$-graded bimodules $M$ of $A$. Here we only consider
the case when $M=A$. The Hochschild cochain complex (with coefficients in $A$
itself) is defined by

${\it CC}^{\bullet,n}(A):={\it CC}^{\bullet,n}(A,A):={\rm
Hom}_{\Lambda_{\mathbb{K}}}^{\bullet}(A^{\otimes n},A[n]).$

Here the bullet is the ${\mathbb{Z}}_{2}$-grading on linear maps and $A[n]$ is
the ${\mathbb{Z}}_{2}$-graded vector space $A$ with the
${\mathbb{Z}}_{2}$-grading shifted by $n$ (modulo $2$). Denote the
${\mathbb{Z}}_{2}$-degree of a homogeneous element $\phi\in{\it
CC}^{\bullet,\bullet}(A)$ by $|\phi|\in{\mathbb{Z}}_{2}$ and the reduced
degree by

$\|\phi\|:=|\phi|+1\in{\mathbb{Z}}_{2}.$

A Hochschild cochain is represented by a sequence $\tau=(\tau_{n})_{n\geq 0}$
of such multi-linear maps. The differential $\delta_{{\it CC}}$, which raises
the length grading $n$ by $1$, is defined by

$(\delta_{{\it
CC}}(\tau))(x_{n+1},\ldots,x_{1})=x_{n+1}\tau_{n}(x_{n},\ldots,x_{1})+(-1)^{\|\tau\|\|x_{1}\|}\tau_{n}(x_{n+1},\ldots,x_{2})x_{1}\\\
-\sum_{0\leq
i<n}(-1)^{\|\tau\|+\maltese_{1}^{i}}\tau_{n}(x_{n+1},\ldots,x_{i+2}x_{i+1},x_{i},\ldots,x_{1}).$
(3.9)

The cohomology defined by $\delta_{{\it CC}}$ is called the Hochschild
cohomology of $A$ (with coefficients in $A$). As the simplest example, via a
straightforward calculation one obtains (for $A={\mathbb{K}}$ trivially
graded)

${\it HH}^{\bullet,n}({\mathbb{K}})=\left\\{\begin{array}[]{cc}{\mathbb{K}},&\
n=0\text{ and }n\text{ even},\\\ 0,&\ \text{ otherwise},\end{array}\right.$

where the superscript $n$ comes from the length filtration of Hochschild
cochains.

######  Remark 3.34.

The formula (3.9) differs from the usual version of Hochschild differential,
see for example [Lod13, (1.5.1.1)]. Indeed, suppose $A$ is ungraded, i.e. all
elements are even. Then the ${\mathbb{Z}}_{2}$-grading of a length $n$ cochain
is $n$ mod $2$. In this case (3.9) reduces to

$(\delta_{{\it
CC}}(\tau))(x_{n+1},\ldots,x_{1})=x_{n+1}\tau_{n}(x_{n},\ldots,x_{1})+(-1)^{n+1}\tau_{n}(x_{n+1},\ldots,x_{2})x_{1}\\\
+(-1)^{n+i}\tau_{n}(x_{n+1},\ldots,x_{i+2}x_{i+1},x_{i},\ldots,x_{1}).$

If we replace $A$ by the opposite algebra $A^{\rm op}$ (i.e. the same set with
multiplication reversed), and identify a length $n$ Hochschild cochain $\tau$
on $A$ with $\tau^{\rm op}$ on $A^{\rm op}$ defined by $\tau^{\rm
op}(x_{1},\ldots,x_{n})=\tau(x_{n},\ldots,x_{1})$. Then the above formula
differ from the standard Hochschild differential on $\tau^{\rm op}$ up to a
sign $(-1)^{n+1}$.

####  3.6.2. Hochschild cohomology for $A_{\infty}$ algebras

Now let ${\mathcal{A}}^{\flat}$ be a flat ${\mathcal{A}}_{\infty}$ algebra.
Define the length $n$-part of Hochschild cochain complex of $A^{\flat}$ to be

${\it CC}^{\bullet,n}({\mathcal{A}}^{\flat})={\it
CC}^{\bullet,n}({\mathcal{A}}^{\flat},{\mathcal{A}}^{\flat})={\rm
Hom}_{\Lambda_{\mathbb{K}}}^{\bullet}(({\mathcal{A}}^{\flat})^{\otimes
n},{\mathcal{A}}^{\flat}[n]).$

Here $\bullet$ denotes the ${\mathbb{Z}}_{2}$-grading and
${\mathcal{A}}^{\flat}[n]$ denote the super vector space
${\mathcal{A}}^{\flat}$ with grading shifted by $n$ (mod $2$).

On the Hochschild cochain complex there is the Gerstenhaber product (which is
graded with respect to the reduced grading $\|\cdot\|$) defined by

$(\phi\circ\psi)(x_{s},\ldots,x_{1})=\sum_{i+j+k=s}(-1)^{\|\psi\|\cdot\maltese_{1}^{i}}\phi(x_{s},\ldots,\psi(x_{i+j},\cdots,),x_{i},\ldots,x_{1})$

as well as the Gerstenhaber superbracket

$[\phi,\psi]:=\phi\circ\psi-(-1)^{\|\phi\|\cdot\|\psi\|}\psi\circ\phi.$

Then the $A_{\infty}$-structure on ${\mathcal{A}}^{\flat}$ is equivalent to an
even Hochschild cochain $m^{\flat}$ with $m^{\flat}_{0}=0$ with the
$A_{\infty}$ relation being equivalent to

$[m^{\flat},m^{\flat}]=2m^{\flat}\circ m^{\flat}=0.$

We define the Hochschild differential $\delta_{\it CC}$ on

${\it CC}^{\bullet}({\mathcal{A}}^{\flat})=\prod_{n\geq 0}{\it
CC}^{\bullet,n}({\mathcal{A}}^{\flat})$

by the formula

$\delta_{\it CC}(\phi):=[m^{\flat},\phi].$

Notice that if $m^{\flat}_{k}\neq 0$ only when $k=2$, ${\mathcal{A}}^{\flat}$
is a ${\mathbb{Z}}_{2}$-graded associative algebra with the Hochschild
differential reduces to the differential (3.9). The Hochschild cohomology of
${\mathcal{A}}^{\flat}$ is defined by

${\it HH}^{\bullet}({\mathcal{A}}^{\flat}):={\rm ker}\delta_{\it CC}/{\rm
im}\delta_{\it CC}.$

On ${\it CC}^{\bullet}({\mathcal{A}}^{\flat})$ there is also an $A_{\infty}$
structure whose composition maps start with $\delta_{\it CC}$. We only need
the 2-fold composition map, i.e., the Yoneda product.

######  Definition 3.35.

The Yoneda product on ${\it CC}^{\bullet}({\mathcal{A}}^{\flat})$, denoted by
$\star$, is defined by

$(\phi\star\psi)(a_{k},\ldots,a_{1})\\\
=\sum(-1)^{\clubsuit}m_{k}^{\flat}\left(a_{k},\ldots,\phi_{r}(a_{i+r},\ldots,a_{i+1}),\cdots,\psi_{s}(a_{j+s},\ldots,a_{j+1}),\ldots,a_{1}\right)$
(3.10)

where the sum is taken over all $i,j,r,l$ such that each summand makes sense.
The sign is defined by (see (3.8) for the definition of $\maltese$)

$\clubsuit:=\|\phi\|\cdot\left(\maltese_{1}^{i}+|\psi|\right)+\|\psi\|\cdot\maltese_{1}^{j}.$
(3.11)

As there are many inconsistent conventions in literature (see for example
[Gan12, She20, Mes16]), we verify that the Yoneda product indeed reduces a
product on the cohomology. As recalled from above, the Yoneda product can be
extended to define an $A_{\infty}$ structure on ${\it
CC}^{\bullet}({\mathcal{A}}^{\flat})$, so the induced product on the
Hochschild cohomology ${\it HH}^{\bullet}({\mathcal{A}}^{\flat})$ is
associative.

######  Proposition 3.36.

The map $\star:{\it CC}^{\bullet}({\mathcal{A}}^{\flat})\otimes{\it
CC}^{\bullet}({\mathcal{A}}^{\flat})\to{\it
CC}^{\bullet}({\mathcal{A}}^{\flat})$ is a cochain map of even degree.

###### Proof.

The fact that $\star$ has even degree follows directly from the definition of
the ${\mathbb{Z}}_{2}$-grading on the Hochschild cochain complex and the fact
that $m^{\flat}$ is even. We verify that $\star$ is a chain map. To save
notations, we assume that both $\phi$ and $\psi$ are odd; the general
situation can be verified similarly. Then in this case, we need to prove that

$\begin{split}&\
\big{(}m^{\flat}\circ(\phi\star\psi)\big{)}(a_{k},\ldots,a_{1})+\big{(}(\phi\star\psi)\circ
m^{\flat}\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\big{(}(m^{\flat}\circ\phi-\phi\circ
m^{\flat})\ast\psi\big{)}(a_{k},\ldots,a_{1})\\\ &\
+(-1)^{|\phi|}\big{(}\phi\star(m^{\flat}\circ\psi-\psi\circ
m^{\flat})\big{)}(a_{k},\ldots,a_{1}).\end{split}$ (3.12)

First we compute the left hand side, in which the involved sign $\clubsuit$
(see (3.11)) in $\phi\star\psi$ always vanishes and in which
$\|\phi\star\psi\|=1$. Then

$\begin{split}&\
\big{(}m^{\flat}\circ(\phi\star\psi)\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\maltese_{1}^{i}}m^{\flat}\big{(}a_{k},\ldots,(\phi\star\psi)(a_{i+r},\cdots,a_{i+1}),a_{i},\ldots,a_{1}\big{)}\\\
=&\
\sum(-1)^{\maltese_{1}^{i}}m^{\flat}\big{(}a_{k},\ldots,m^{\flat}(\cdots,\phi(\cdots),\cdots,\psi(\cdots),a_{i},\ldots,a_{1}\big{)}.\end{split}$

Notice that this is a sum of 4-fold compositions using two $m^{\flat}$ with
$\phi$ and $\psi$ such that $\phi$ and $\psi$ are contained in the interior
$m^{\flat}$, which can be abbreviated as

$m^{\flat}(-,m^{\flat}(-,\phi(-),-,\psi(-),-),-).$

We will make similar abbreviations in the following computations; moreover, we
always assume that after the interior $m^{\flat}$ the inputs start with
$a_{i}$. Then similar to above (remember $\clubsuit=0$), one has

$\begin{split}&\ \big{(}(\phi\star\psi)\circ
m^{\flat}\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\maltese_{1}^{i}}(\phi\star\psi)(-,m^{\flat}(-),-)\\\ =&\
\sum(-1)^{\maltese_{1}^{i}}\left(m^{\flat}\big{(}-,\phi(-),-,\psi(-),-,m^{\flat}(-),-\big{)}\right.\\\
&\
\left.+m^{\flat}\big{(}-,\phi(-),-,m^{\flat}(-),-,\psi(-),-\big{)}+m^{\flat}\big{(}-,m^{\flat}(-),-,\phi(-),-,\psi(-),-\big{)}\right)\\\
&\
+\sum(-1)^{\maltese_{1}^{i}}\left(m^{\flat}\big{(}-,\phi(-,m^{\flat}(-),-),-,\psi(-),-\big{)}+m^{\flat}\big{(}-,\phi(-),-,\psi(-,m^{\flat}(-),-),-\big{)}\right).\end{split}$

On the right hand side, the first part is a sum of 4-fold compositions such
that neither $\phi$ nor $\psi$ are contained in the interior $m^{\flat}$ and
the second part is a sum of 4-fold compositions such that either $\phi$ or
$\psi$ contain the interior $m^{\flat}$. Now we can observe that the chain map
property should be a consequence of the $A_{\infty}$ relation $m^{\flat}\circ
m^{\flat}=0$. Notice that to match the signs of the $A_{\infty}$ relation, we
see

$\|\phi(a_{i+r},\cdots,a_{i+1})\|=|\phi|+|a_{i+r}|+\cdots+|a_{i+1}|+r+1=\|\phi\|+\maltese_{i+1}^{i+r}=\maltese_{i+1}^{i+r}$

as $\phi$ is assumed to be odd.

Now compute $(m^{\flat}\circ\phi)\star\psi$. Notice that in this computation,
because $m^{\flat}\circ\phi$ is even and $\psi$ is odd, the sign $\clubsuit$
(see (3.11)) is

$\clubsuit=\|m^{\flat}\circ\phi\|\cdot\big{(}\maltese_{1}^{i}+|\psi|\big{)}+\|\psi\|\cdot\maltese_{1}^{j}=1+\maltese_{1}^{i}.$

Moreover, as $\phi$ is odd, the signs appearing in $m^{\flat}\circ\phi$ (in
fact, all signs in all Gernstenhaber products until the end of the proof)
vanish. Then

$\begin{split}&\
\big{(}(m^{\flat}\circ\phi)\star\psi\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\clubsuit}m^{\flat}\big{(}-,(m^{\flat}\circ\phi)(-),-,\psi(-),-\big{)}\\\
=&\
\sum(-1)^{1+\maltese_{1}^{i}}m^{\flat}\big{(}-,m^{\flat}(-,\phi(-),-),-,\psi(-),-\big{)}.\end{split}$

This is the sum of 4-fold compositions with the interior $m^{\flat}$ only
contains $\phi$. Similarly, for computing $\phi\star(m^{\flat}\circ\psi)$, one
has

$\clubsuit=\|\phi\|\cdot\big{(}\clubsuit_{1}^{i}+|m^{\flat}\circ\psi|\big{)}+\|m^{\flat}\circ\psi\|\cdot\maltese_{1}^{j}=\maltese_{1}^{j}.$

As our running convention is that after the second $m^{\flat}$ the inputs
start with $a_{i}$, this sign is rewritten as $\maltese_{1}^{i}$ below. Then

$\begin{split}&\
(-1)^{|\phi|}\big{(}\phi\star(m^{\flat}\circ\psi)\big{)}(a_{k},\ldots,a_{1})\\\
=&\
\sum(-1)^{1+\clubsuit}m^{\flat}\big{(}-,\phi(-),-,(m^{\flat}\circ\psi)(-),a_{i},\cdots,a_{1}\big{)}\\\
=&\
\sum(-1)^{1+\maltese_{1}^{i}}m^{\flat}\big{(}-,\phi(-),m^{\flat}(-,\psi(-),-),-\big{)}.\end{split}$

Now compute $-(\phi\circ m^{\flat})\star\psi$.

$\begin{split}&\ -\big{(}(\phi\circ
m^{\flat})\star\psi\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\maltese_{1}^{j}}m^{\flat}\big{(}-,(\phi\circ
m^{\flat})(-),a_{j},\cdots,\psi(-),-\big{)}\\\ =&\
\sum(-1)^{\maltese_{1}^{j}+\maltese_{j+1}^{i}}m^{\flat}\big{(}-,\phi(-,m^{\flat}(-),a_{i},\cdots,a_{j+1}),a_{j},\cdots,\psi(-),-\big{)}\\\
=&\
\sum(-1)^{\maltese_{1}^{i}}m^{\flat}(\big{(}-,\phi(-,m^{\flat}(-),-),-,\psi(-),-\big{)}.\end{split}$

Lastly we compute $-(-1)^{|\phi|}\phi\star(\psi\circ
m^{\flat})=\phi\star(\psi\circ m^{\flat})$, which is

$\begin{split}&\ \big{(}\phi\star(\psi\circ
m^{\flat})\big{)}(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\maltese_{1}^{j}}m^{\flat}\big{(}-,\phi(-),-,(\psi\circ
m^{\flat})(-),a_{j},\ldots,a_{1}\big{)}\\\ =&\
\sum(-1)^{\maltese_{1}^{j}+\maltese_{j+1}^{i}}m^{\flat}\big{(}-,\phi(-),-,\psi(-,m^{\flat}(-),a_{i},\ldots,a_{j+1}),a_{j},\ldots,a_{1}\big{)}\\\
=&\
\sum(-1)^{\maltese_{1}^{i}}m^{\flat}\big{(}-,\phi(-),-,\psi(-,m^{\flat}(-),-),-\big{)}.\end{split}$

Gathering all computations together, we see that (3.12) follows from the
$A_{\infty}$ relation $m^{\flat}\circ m^{\flat}=0$. ∎

Therefore the Yoneda product descends to Hochschild cohomology. We still call
the induced one by Yoneda product and denote it by the same symbol $\ast$.

The Yoneda product has a chain-level unit in the strictly unital case.

######  Proposition 3.37.

Suppose ${\mathcal{A}}^{\flat}$ has a strict unit $e$. Then the Hochschild
cochain ${\bm{1}}_{{\mathcal{A}}^{\flat}}$ defined by

${\bm{1}}_{{\mathcal{A}}^{\flat}}(x_{k},\ldots,x_{1})=\left\\{\begin{array}[]{cc}0,&\
k\geq 1,\\\ e,&\ k=0.\end{array}\right.$

is a unit with respect to the Yoneda product.

###### Proof.

By the definition of strict unit and Yoneda product, for any Hochschild
cochain $\phi$, one has

$\begin{split}&\
({\bm{1}}_{{\mathcal{A}}^{\flat}}\star\phi)(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\clubsuit}m^{\flat}\big{(}a_{k},\cdots,e,a_{i},\cdots,\phi(a_{j+l},\cdots,a_{j+1}),\cdots\big{)}\\\
=&\
(-1)^{\maltese_{1}^{k}+|\phi|}m_{2}^{\flat}\big{(}e,\phi(a_{k},\ldots,a_{1})\big{)}\\\
=&\
(-1)^{|\phi(a_{k},\ldots,a_{1})|}m_{2}^{\flat}\big{(}e,\phi(a_{k},\ldots,a_{1})\big{)}\\\
=&\ \phi(a_{k},\ldots,a_{1}).\end{split}$

Similarly

$\begin{split}&\
(\phi\star{\bm{1}}_{{\mathcal{A}}^{\flat}})(a_{k},\ldots,a_{1})\\\ =&\
\sum(-1)^{\clubsuit}m^{\flat}\big{(}a_{k},\cdots,\phi(a_{j+l},\cdots,a_{j+1}),\cdots,e,a_{i},\cdots\big{)}\\\
=&\ (-1)^{\clubsuit}m_{2}^{\flat}\big{(}\phi(a_{k},\ldots,a_{1}),e\big{)}\\\
=&\ m_{2}^{\flat}\big{(}\phi(a_{k},\ldots,a_{1}),e\big{)}\\\ =&\
\phi(a_{k},\ldots,a_{1}).\qed\end{split}$

Finally, we remark that the Yoneda product on ${\it
HH}^{\bullet}({\mathcal{A}}^{\flat})$ is graded commutative. It is compatible
with the Gerstenhaber bracket, which makes ${\it
HH}^{\bullet}({\mathcal{A}}^{\flat})$ into a Gerstenhaber algebra.

####  3.6.3. Clifford algebras

The Lagrangian Floer cohomology ring of a torus is often isomorphic to a
Clifford algebra. Hence the Hochschild cohomology of Clifford algebras are one
of the most important cases related to symplectic geometry and mirror
symmetry. Recall that given a finite-dimensional ${\mathbb{K}}$-vector space
$W$ equipped with a quadratic form $q$, the Clifford algebra $Cl(W,q)$ is the
tensor algebra of $W$ modulo the relation

$w\otimes w^{\prime}+w^{\prime}\otimes w+2q(w,w^{\prime}){\rm Id}=0.$

We only care about the case when $q$ is nondegenerate and the case when
${\mathbb{K}}$ is algebraically closed. In this case, all nondegenerate
quadratic forms are equivalent to the standard one. When $W$ has dimension
$n$, we abbreviate $Cl(W,q)$ by $Cl_{n}$.

######  Proposition 3.38.

For all $n\geq 0$, the Hochschild cohomology of $Cl_{n}$ is

${\it HH}^{k}(Cl_{n},Cl_{n})=\left\\{\begin{array}[]{cc}{\mathbb{K}},&\
k=0,\\\ 0,&\ k\geq 1.\end{array}\right.$

In particular, ${\it HH}^{0}(Cl_{n},Cl_{n})$ is generated by the identity.

###### Proof.

The calculation was provided by Sheridan [She16] and we recall it here. First,
Hochschild cohomology is Morita invariant (see [Lod13, 1.5.6]). Second, there
are only two Morita equivalence classes among Clifford algebras, the even ones
and the odd ones (Bott periodicity). Hence we only need to calculate for $n=0$
and $n=1$. When $n=0$, $Cl_{0}\cong{\mathbb{K}}$, giving ${\it
HH}^{\bullet}({\mathbb{K}},{\mathbb{K}})={\mathbb{K}}$. When $n=1$ 555The
original calculation of [She16] for the $n=1$ case has a mistake resulting in
a false result. We thank Nick Sheridan for clarifying it., the calculation can
be deduced from the more general case of J. Smith [Smi19, Section 5] using
reduced Hochschild cohomology. ∎

When the Floer cohomology algebra of a Lagrangian brane is isomorphic to a
Clifford algebra, the argument via formality shows that the Hochschild
cohomology of the corresponding $A_{\infty}$ algebra is the same as the
Hochschild cohomology of the cohomology algebra. Recall that an $A_{\infty}$
algebra is called formal if it is $A_{\infty}$ quasi-isomorphic to its
cohomology algebra. An associated algebra $A$ is called intrinsically formal
if any ${\mathbb{Z}}_{2}$-graded $A_{\infty}$ algebra whose cohomology algebra
is isomorphic to $A$ is formal. It was shown in [She16, Corollary 6.4] that
$Cl_{n}$ is intrinsically formal. Due to the Morita invariance of Hochschild
cohomology, the following statement is immediate.

######  Corollary 3.39.

If ${\mathcal{A}}^{\flat}$ is a flat $A_{\infty}$ algebra over ${\mathbb{K}}$
whose cohomology algebra is isomorphic to $Cl_{n}$, then

${\it HH}^{\bullet}({\mathcal{A}}^{\flat})={\mathbb{K}}.$

∎

Notice that if in addition ${\mathcal{A}}^{\flat}$ is strictly unital,
$1_{{\mathcal{A}}^{\flat}}\neq 0$ and it generates the Hochschild cohomology.

##  4\. Vortex Hamiltonian Floer theory

We review the construction of vortex Hamiltonian Floer theory developed by the
second author [Xu16] following the proposal of Cieliebak–Gaio–Salamon [CGS00].

###  4.1. Floer chain complexes

####  4.1.1. Equivariant action functional

Our convention for Hamiltonian vector field is fixed as follows. Let
$(M,\omega)$ be a symplectic manifold and $H:M\to{\mathbb{R}}$ be a smooth
function. The associated Hamiltonian vector field $X_{H}$ is specified by

$dH=\omega(X_{H},\cdot).$

We would like to consider the Hamiltonian dynamics upstairs in the gauged
linear sigma model. Let $X$ be the toric manifold we are considering. Let
$H:S^{1}\times X\to{\mathbb{R}}$ be a smooth Hamiltonian function. Let ${\rm
Per}(H)$ be the set of $1$-periodic orbits 666They must be contractible as $X$
is simply connected. of $H$ whose elements are maps $x:S^{1}\to X$. The
Hamiltonian $H$ lifts to a $K$-invariant function on $S^{1}\times\mu^{-1}(0)$.
Choose an arbitrary $K$-invariant extension $\widehat{H}:S^{1}\times
V\to{\mathbb{R}}$ whose support is compact and disjoint from the unstable
locus $V^{\rm us}$ under the $K^{\mathbb{C}}$-action. Consider the set of
equivariant loops

$L^{K}(V):=\Big{\\{}{\mathfrak{x}}=(\widehat{x},\zeta):S^{1}\to
V\times{\mathfrak{k}}.\Big{\\}}$ (4.1)

Here function $\zeta:S^{1}\to{\mathfrak{k}}$ can be viewed as a gauge field on
$S^{1}$. Notice that as $V$ is contractible, the loop $x$ is contractible and
there is only one homotopy class of cappings. The loop group $LK$ acts on the
set of capped equivariant orbits by

$g\cdot{\mathfrak{x}}=(g\cdot\widehat{x},g\cdot\xi)\ {\rm where}\
(g\cdot\widehat{x})(t)=g(t)\widehat{x}(t),\
(g\cdot\xi)(t)=\zeta(t)-\frac{d}{dt}\log g(t).$

Define the action functional

$\widehat{\mathcal{A}}_{H}:L^{K}(V)\to{\mathbb{R}},\
{\mathfrak{x}}\mapsto-\int_{\mathbb{D}}u^{*}\omega_{V}+\int_{S^{1}}\left(\langle\mu(\widehat{x}(t)),\zeta(t)\rangle-\widehat{H}_{t}(\widehat{x}(t))\right)dt$
(4.2)

where $u:{\mathbb{D}}^{2}\to V$ is any capping. Critical points are solutions

$\displaystyle\ \mu(\widehat{x}(t))\equiv 0,\ $ $\displaystyle\
\widehat{x}^{\prime}(t)=X_{\widehat{H}_{t}}(\widehat{x}(t))-{\mathcal{X}}_{\zeta(t)}(\widehat{x}(t)).$
(4.3)

Here ${\mathcal{X}}_{\zeta(t)}$ is the Hamiltonian vector field of the
function $\langle\mu,\zeta(t)\rangle$.

The action functional satisfies the following transformational law with
respect to the loop group action. Indeed, for $g\in LK$ and ${\mathfrak{x}}\in
L^{K}(V)$, one has

$\widehat{\mathcal{A}}_{H}(g{\mathfrak{x}})=-\omega^{K}(g)+{\mathcal{A}}_{H}({\mathfrak{x}}).$
(4.4)

Denote

${\mathfrak{L}}^{K}(V):=L^{K}(V)/{\rm ker}\omega^{K}.$

Its elements are denoted by $[{\mathfrak{x}}]$. Then $\Gamma\cong LK/{\rm
ker}\omega^{K}$ acts on ${\mathfrak{L}}^{K}(V)$. We denote the action by
$g\cdot[{\mathfrak{x}}]$. Then $\widehat{\mathcal{A}}_{H}$ induces a
functional on ${\mathfrak{L}}^{K}(V)$, denoted by

${\mathcal{A}}_{H}:{\mathfrak{L}}^{K}(V)\to{\mathbb{R}}.$

Each critical point of ${\mathcal{A}}_{H}$ is called an equivariant 1-periodic
Hamiltonian orbit.

There is a correspondence between ordinary Hamiltonian orbits downstairs and
equivariant Hamiltonian orbits upstairs. More precisely, let $\widetilde{\rm
Per}(H)$ be the covering of ${\rm Per}(H)$ consisting of equivalence classes
$[u,x]$ of capped 1-periodic orbits of $H$: the equivalence relation
$(u,x)\sim(u^{\prime},x^{\prime})$ is defined by the equality of action
values:

$(u,x)\sim(u^{\prime},x^{\prime})\Longleftrightarrow x=x^{\prime}\in{\rm
Per}(H)\ {\rm and}\
\int_{{\mathbb{D}}^{2}}u^{*}\omega_{X}=\int_{{\mathbb{D}}^{2}}(u^{\prime})^{*}\omega_{X}.$

Then there is a map

$\iota:\widetilde{\rm Per}(H)\to{\rm
crit}{\mathcal{A}}_{H}\subset{\mathfrak{L}}^{K}(V).$ (4.5)

Indeed, suppose $x:S^{1}\to X$ is a contractible 1-periodic orbit of the
Hamiltonian flow of $H$ and $u:{\mathbb{D}}^{2}\to X$ is a capping of $x$.
View $\mu^{-1}(0)\to X$ as a principal $K$-bundle $P$. The Euclidean metric on
$V$ induces a connection on $P$ whose horizontal distribution is the
orthogonal complement of tangent planes of $K$-orbits; equivalently, this
gives a connection 1-form
$\theta\in\Omega^{1}(\mu^{-1}(0))\otimes{\mathfrak{k}}$. The pullback
$u^{*}P\to{\mathbb{D}}^{2}$ is trivial and different trivializations differ by
a smooth map $g:{\mathbb{D}}^{2}\to K$. Any trivialization of this pullback
bundle induces a connection matrix $u^{*}\theta$ whose boundary restriction is
$\zeta(t)dt$. A trivialization also induces a map
$\widehat{u}:{\mathbb{D}}^{2}\to\mu^{-1}(0)$ lifting $u$. Let the boundary
restriction of $\widehat{u}$ be $\widehat{x}$. Then
${\mathfrak{x}}=(\widehat{x},\zeta)$ is an equivariant 1-periodic orbit, well-
defined up to $L_{0}K$-actions. Furthermore, if $u^{\prime}$ is a different
capping with the same resp. different action value, then the correspondence we
just described gives the same resp. different element in
${\mathfrak{L}}^{K}(V)$.

######  Lemma 4.1.

In the toric case the map (4.5) is bijective.

###### Proof.

Given any equivariant Hamiltonian orbit ${\mathfrak{x}}$ upstairs, the map
$\widehat{x}:S^{1}\to\mu^{-1}(0)$ projects down to a 1-periodic orbit
$x:S^{1}\to X$. As $X$ is simply connected, $x$ is contractible. Choose a
capping $u:{\mathbb{D}}^{2}\to X$ and let
${\mathfrak{x}}^{\prime}=(\widehat{x}^{\prime},\zeta^{\prime})$ be equivariant
Hamiltonian orbit lifting $[u,x]$. As the $K$-action on $\mu^{-1}(0)$ is free,
there is a gauge transformation on the circle making
$\widehat{x}^{\prime}=\widehat{x}$. The condition

$\widehat{x}^{\prime}(t)=X_{H_{t}}(\widehat{x}(t))-{\mathcal{X}}_{\zeta(t)}(\widehat{x}(t))$

implies that $\zeta=\zeta^{\prime}$. ∎

######  Definition 4.2.

The Conley–Zehnder index of an equivariant 1-periodic orbit
${\mathfrak{x}}\in{\rm crit}{\mathcal{A}}_{H}$ is the usual Conley–Zehnder
index of the capped 1-periodic orbit
$\iota^{-1}({\mathfrak{x}})\in\widetilde{\rm Per}(H)$, denoted by ${\rm
CZ}({\mathfrak{x}})\in{\mathbb{Z}}$.

####  4.1.2. Floer trajectories

Similar to the standard Hamiltonian Floer theory, one considers the equation
for the gradient flow of the equivariant action functional. Choose a
1-periodic $K$-invariant $\omega_{V}$-compatible almost complex structure
$\widehat{J}_{t}$ on $V$. Formally the negative gradient flow equation of
$\widehat{\mathcal{A}}_{H}$ is the following equation for pairs
$(u,\eta):{\mathbb{R}}\times S^{1}\to V\times{\mathfrak{k}}$

$\displaystyle\
\partial_{s}u+\widehat{J}_{t}\left(\partial_{t}u+{\mathcal{X}}_{\eta}(u)-X_{\widehat{H}_{t}}(u)\right)=0,\
$ $\displaystyle\ \partial_{s}\eta+\mu(u)=0.$

This is in fact the symplectic vortex equation on the cylinder
${\mathbb{R}}\times S^{1}$ for the trivial $K$-bundle and the standard
cylindrical volume form, written in temporal gauge $A=d+\eta dt$. In general,
for $A=d+\xi ds+\eta dt$, the vortex equation (2.2) reads

$\displaystyle\
\partial_{s}u+{\mathcal{X}}_{\xi}(u)+\widehat{J}_{t}\left(\partial_{t}u+{\mathcal{X}}_{\eta}(u)-X_{\widehat{H}_{t}}(u)\right)=0,\
$ $\displaystyle\ \partial_{s}\eta-\partial_{t}\xi+\mu(u)=0.$ (4.6)

It was shown in [Xu16] that any finite energy solution converges up to gauge
transformation to critical points of ${\mathcal{A}}_{H}$.

######  Theorem 4.3.

[Xu16, Theorem 3.1, Corollary 4.3]

  1. (1)

Given a bounded solution ${\mathfrak{u}}=(u,\xi,\eta)$ (i.e. finite energy
solution with $u({\mathbb{R}}\times S^{1})$ bounded) to (4.6), there exist a
gauge equivalent solution, still denoted by $(u,\xi,\eta)$, as well as
equivariant 1-periodic orbits
${\mathfrak{x}}_{\pm}=(\widehat{x}_{\pm},\zeta_{\pm})$ such that uniformly for
$t\in S^{1}$

$\lim_{s\to\pm\infty}(u(s,\cdot),\xi(s,\cdot),\eta(s,\cdot))=(\widehat{x}_{\pm},0,\zeta_{\pm}).$
(4.7)

  2. (2)

If ${\mathfrak{x}}_{\pm}^{\prime}$ are another pair of equivariant 1-periodic
orbits satisfying (4.7) with ${\mathfrak{u}}$ replaced by any gauge equivalent
solution, then there exists $g_{\pm}\in LK$ with $g_{-}g_{+}^{-1}\in L_{0}K$
such that ${\mathfrak{x}}_{\pm}^{\prime}=g_{\pm}{\mathfrak{x}}_{\pm}$.

  3. (3)

If $H$ is a nondegenerate Hamiltonian downstairs, then one can make the
convergence (4.7) exponentially fast by choosing suitable gauge equivalent
solutions. More precisely, there exist $C>0$ and $\delta>0$ such that

$d_{V}(u(s,t),\widehat{x}_{\pm}(t))+|\xi(s,t)|+|\eta(s,t)-\zeta(t)|\leq
Ce^{-\delta|s|}.$

Here $d_{V}$ is the Euclidean distance on $V$. Similar exponential decay
estimates hold for covariant derivatives of arbitrary higher
orders.777Exponential decay type estimates for vortices can also be found in
[Zil09][Ven15][CWW17].

Therefore, one can use a pair of elements ${\mathfrak{x}}_{\pm}\in{\rm
crit}{\mathcal{A}}_{H}\subset{\mathfrak{L}}^{K}(V)$ to label solutions. Let

${\mathcal{M}}({\mathfrak{x}}_{-},{\mathfrak{x}}_{+})$

be the set of gauge equivalence classes of bounded solutions ${\mathfrak{u}}$
to (4.6) modulo the ${\mathbb{R}}$-translation. One has the energy identity
([Xu16, Proposition 3.8])

$E({\mathfrak{u}})={\mathcal{A}}_{H}({\mathfrak{x}}_{-})-{\mathcal{A}}_{H}({\mathfrak{x}}_{+}).$

######  Remark 4.4.

To achieve transversality, one has to avoid certain “bad” $K$-equivariant
lifts of a given Hamiltonian $H$ downstairs and choose almost complex
structures appropriately. In [Xu16, Section 6] the second author used the
notion of admissible almost complex structures and admissible $K$-invariant
lifts of a Hamiltonian downstairs. We briefly recall the precise meanings of
them adapted to the toric case. First, in the stable locus $V^{\rm st}$ there
is the projection $\pi:V^{\rm st}\to X$ which is invariant under the complex
torus $G$. Hence there is a splitting

$TV|_{V^{\rm st}}\cong\pi^{*}TX\oplus({\mathfrak{k}}\otimes{\mathbb{C}}).$

Throughout this paper we fix a $K$-invariant (small) open neighborhood $U$ of
$\mu^{-1}(0)$ and consider only $K$-invariant, $\omega_{V}$-compatible almost
complex structures $\widehat{J}$ on $V$ which agrees with $J_{V}$ outside $U$
(this is necessary to guarantee the $C^{0}$-compactness in [Xu16]). Moreover,
given a nondegenerate Hamiltonian $H:S^{1}\times X\to{\mathbb{R}}$, an
$S^{1}$-family of almost complex structures $\widehat{J}_{t}$ is said to be
admissible with respect to the $H$ downstairs if for any loop
$\widehat{x}:S^{1}\to\mu^{-1}(0)$ that projects to a 1-periodic orbit
downstairs, one imposes some conditions on the $1$-jet of $\widehat{J}_{t}$
along $\widehat{x}$ (see [Xu16, Definition 6.2]. Then the notion of
admissibility of $K$-invariant lifts of $H$ was defined (see [Xu16, Definition
6.5]), which is a condition on the infinitesimal behavior of the lifts
$\widehat{H}_{t}$ along 1-periodic orbits given in terms of the Hessian of the
equivariant action functional.

######  Theorem 4.5.

Given a nondegenerate Hamiltonian $H_{t}$ downstairs, for a generic admissible
pair $(\widehat{H}_{t},\widehat{J}_{t})$, the following is true.

  1. (1)

Each moduli space ${\mathcal{M}}({\mathfrak{x}},{\mathfrak{y}})$ is regular
and has dimension ${\rm CZ}({\mathfrak{x}})-{\rm CZ}({\mathfrak{y}})-1$.

  2. (2)

Moduli spaces with bounded energy are compact up to breaking as in the usual
setting for the Uhlenbeck–Gromov–Floer compactification.

  3. (3)

If ${\rm CZ}({\mathfrak{x}})-{\rm CZ}({\mathfrak{y}})=1$, then the moduli
space consists of finitely many points.

  4. (4)

When ${\rm CZ}({\mathfrak{x}})-{\rm CZ}({\mathfrak{y}})=2$, the compactified
moduli space is a compact 1-dimensional manifold with boundary.

We briefly explain the reason for transversality. Indeed, as the total energy
is finite and the volume on the cylinder is infinite, near infinity any
solution is contained in the neighborhood $U$ of $\mu^{-1}(0)$ fixed in Remark
4.4. Therefore, there is a nonempty open subset of the cylinder whose image is
contained in the free locus of the $K$-action. Then using an equivariant
version of the argument of Floer–Hofer–Salamon [FHS95] one can achieve
transversality by perturbing $\widehat{J}_{t}$ in a neighborhood of
$\mu^{-1}(0)$.

It is a standard procedure to construct a coherent system of orientations on
the moduli spaces (see [FH93]). Then for $R={\mathbb{Z}}$, there is a well-
defined count

$n({\mathfrak{x}},{\mathfrak{y}})\in{\mathbb{Z}}$

which is the signed count of the number of Floer trajectories in
$0$-dimensional components of ${\mathcal{M}}({\mathfrak{x}},{\mathfrak{y}})$.
When $R$ is any commutative ring with a unit,
$n({\mathfrak{x}},{\mathfrak{y}})$ induces an element

$n_{R}({\mathfrak{x}},{\mathfrak{y}})\in R$

by the induced count via the map ${\mathbb{Z}}\to R$.

####  4.1.3. Floer homology

We first define the Floer chain group for a smaller Novikov ring. Recall that
one has the finitely generated abelian group

$\Gamma:=LK/{\rm ker}\omega^{K}$

which naturally embeds into ${\mathbb{R}}$. For any commutative ring $R$,
introduce

$\Lambda_{R}^{\Gamma}:=\Big{\\{}\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\in\Lambda_{R}\
|\ g_{i}\in\Gamma\Big{\\}}.$

We define the Floer chain group ${\it VCF}_{\bullet}(\widehat{H})$ to be the
“downward” completion:

${\it
VCF}_{\bullet}(\widehat{H};\Lambda_{R}^{\Gamma})=\Big{\\{}\sum_{i=1}^{\infty}b_{i}{\mathfrak{x}}_{i}\
|\ b_{i}\in R,\ {\mathfrak{x}}_{i}\in{\rm crit}{\mathcal{A}}_{H},\
\lim_{i\to\infty}{\mathcal{A}}_{H}({\mathfrak{x}}_{i})=-\infty\Big{\\}}.$

It is graded by the Conley–Zehnder index (modulo 2). The
$\Lambda_{R}^{\Gamma}$-module structure is defined by

$\Big{(}\sum_{i=1}^{\infty}a_{i}T^{g_{i}}\Big{)}\Big{(}\sum_{j=1}^{\infty}b_{j}{\mathfrak{x}}_{j}\Big{)}=\sum_{i,j=1}^{\infty}a_{i}b_{j}(g_{i}\cdot{\mathfrak{x}}_{j}).$

By (4.4), the right hand side is in ${\it
VCF}_{\bullet}(\widehat{H};\Lambda_{R}^{\Gamma})$ and this is a well-defined
action. Define

${\it VCF}_{\bullet}(\widehat{H};\Lambda_{R}):={\it
VCF}_{\bullet}(\widehat{H};\Lambda_{R}^{\Gamma})\otimes_{\Lambda_{R}^{\Gamma}}\Lambda_{R}.$

The Floer differential $\partial_{\widehat{J}}:{\it
VCF}_{\bullet}(\widehat{H};\Lambda_{R}^{\Gamma})\to{\it
VCF}_{\bullet-1}(\widehat{H};\Lambda_{R}^{\Gamma})$ is defined by the counts
$n_{R}({\mathfrak{x}},{\mathfrak{y}})$. More precisely, on generators,

$\partial_{\widehat{J}}{\mathfrak{x}}=\sum_{{\mathfrak{y}}}n_{R}({\mathfrak{x}},{\mathfrak{y}}){\mathfrak{y}}.$

One has $\partial_{\widehat{J}}^{2}=0$, resulting in the vortex Floer homology

${\it VHF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{R}^{\Gamma}).$

Notice that the differential $\partial_{\widehat{J}}$ decreases the action.

####  4.1.4. Adiabatic limit

The adiabatic limit argument allows us to relate the gauged linear sigma model
with holomorphic curves in the symplectic quotient. While we do not need a
complete analysis of such a correspondence, we do need to consider the family
of vortex equations related to the adiabatic limit argument. Indeed, if on the
infinite cylinder we choose, instead of the standard area form $dsdt$, a
rescaled one $\lambda^{2}dsdt$, then the corresponding vortex Floer equation
reads

$\displaystyle\
\partial_{s}u+{\mathcal{X}}_{\xi}(u)+\widehat{J}_{t}\left(\partial_{t}u+{\mathcal{X}}_{\eta}(u)-X_{\widehat{H}_{t}}(u)\right)=0,$
$\displaystyle\ \partial_{s}\eta-\partial_{t}\xi+\lambda^{2}\mu(u)=0.$ (4.8)

One can define a vortex Floer chain complex for the triple
$(\lambda,\widehat{H},\widehat{J})$ in completely the same way as the
$\lambda=1$ case, once transversality holds, which can be achieved via
perturbation. We denote the vortex Floer chain complex by ${\it
VCF}_{\bullet}^{\lambda}(\widehat{H},\widehat{J};\Lambda_{R}^{\Gamma})$. The
corresponding homology is denoted by

${\it VHF}^{\lambda}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{R}^{\Gamma}).$

There are a few subtleties. First, given a nondegenerate Hamiltonian $H$
downstairs on $X$, the notion of admissible almost complex structures ([Xu16,
Definition 6.2]) is independent of $\lambda$; the notion of admissible lifts,
however, depends on $\lambda$.

######  Definition 4.6.

A triple $(\lambda,\widehat{H},\widehat{J})$ is called a regular triple if

  1. (1)

The descent Hamiltonian $H$ on $X$ is nondegenerate.

  2. (2)

$(\widehat{H},\widehat{J})$ is admissible with respect to $H$.

  3. (3)

Moduli spaces of gauge equivalence classes of finite energy solutions to (4.8)
are all regular.

####  4.1.5. Continuation map

Given two regular triples
$(\lambda_{\pm},\widehat{H}_{\pm},\widehat{J}_{\pm})$, one compares the two
associated vortex Floer complexes via continuation maps. By an interpolation
between these two triples, we mean a triple
$(\lambda_{s},\widehat{H}_{s},\widehat{J}_{s})$ where
$\lambda_{s}\in{\mathbb{R}}_{+}$ is a smooth function in $s\in{\mathbb{R}}$
which agrees with $\lambda_{\pm}$ near $\pm\infty$, $\widehat{H}_{s}$ is a
smooth family of $K$-invariant compactly supported functions parametrized by
$(s,t)\in{\mathbb{R}}\times S^{1}$ which agrees with $\widehat{H}_{\pm}$ near
$\pm\infty$, and $\widehat{J}_{s}$ is a smooth family of $K$-invariant
$\omega_{V}$-compatible almost complex structures parametrized by
$(s,t)\in{\mathbb{R}}\times S^{1}$ which agrees with $\widehat{J}_{\pm}$ near
$\pm\infty$.

Choosing a generic interpolation
$(\lambda_{s},\widehat{H}_{s},\widehat{J}_{s})$, by considering moduli spaces
of gauge equivalence classes of solutions to the equation

$\displaystyle\
\partial_{s}u+{\mathcal{X}}_{\xi}+\widehat{J}_{s,t}\left(\partial_{t}u+{\mathcal{X}}_{\eta}-X_{\widehat{H}_{s,t}}(u)\right)=0,\
$ $\displaystyle\ \partial_{s}\eta-\partial_{t}\xi+\lambda_{s}^{2}\mu(u)=0,$
(4.9)

one can define a continuation map

$\mathfrak{cont}:{\it
VCF}_{\bullet}^{\lambda_{-}}(\widehat{H}_{-},\widehat{J}_{-};\Lambda_{R}^{\Gamma})\to{\it
VCF}_{\bullet}^{\lambda_{+}}(\widehat{H}_{+},\widehat{J}_{+};\Lambda_{R}^{\Gamma})$

completely analogous to the case of classical Hamiltonian Floer theory. The
map $\mathfrak{cont}$ is a chain homotopy equivalence, inducing an isomorphism
on Floer homology

${\it
VHF}_{\bullet}^{\lambda_{-}}(\widehat{H}_{-},\widehat{J}_{-};\Lambda_{R}^{\Gamma})\cong{\it
VHF}_{\bullet}^{\lambda_{+}}(\widehat{H}_{+},\widehat{J}_{+};\Lambda_{R}^{\Gamma}).$

Completely analogous to the classical situation, these isomorphisms are
natural, hence the resulting homology groups define a common object called the
vortex Hamiltonian Floer homology of $V$, denoted by

${\it VHF}_{\bullet}(V;\Lambda_{R}^{\Gamma}).$ (4.10)

Define

${\it VHF}_{\bullet}(V;\Lambda_{R}):={\it
VHF}_{\bullet}(V;\Lambda_{R}^{\Gamma})\otimes_{\Lambda_{R}^{\Gamma}}\Lambda_{R}.$

In order to consider effects on the filtered theories, one needs to estimate
the energy of solutions contributing to the continuation maps.

######  Proposition 4.7.

Given any solution ${\mathfrak{u}}=(u,\xi,\eta)$ to (4.9) which converges to
${\mathfrak{x}}_{\pm}\in{\rm crit}{\mathcal{A}}_{H_{\pm}}$ at $\pm\infty$, one
has

$\int_{{\mathbb{R}}\times
S^{1}}\Big{(}|\partial_{s}u+{\mathcal{X}}_{\xi}(u)|^{2}+\lambda_{s}^{2}|\mu(u)|^{2}\Big{)}dsdt={\mathcal{A}}_{H_{-}}({\mathfrak{x}}_{-})-{\mathcal{A}}_{H_{+}}({\mathfrak{x}}_{+})-\int_{{\mathbb{R}}\times
S^{1}}\frac{\partial\widehat{H}_{s,t}}{\partial s}(u)dsdt.$ (4.11)

In particular, if
$\widehat{H}_{s,t}=(1-\chi(s))\widehat{H}_{-}+\chi(s)\widehat{H}_{+}$ for some
non-decreasing function $\chi:{\mathbb{R}}\to[0,1]$, then one has

${\mathcal{A}}_{H_{+}}({\mathfrak{x}}_{+})\leq{\mathcal{A}}_{H_{-}}({\mathfrak{x}}_{-})+\int_{0}^{1}\max_{V}\left(\widehat{H}_{-}-\widehat{H}_{+}\right)dt.$
(4.12)

###### Proof.

When $\lambda_{s}$ is a constant, (4.11) is [Xu16, Proposition 7.5]. The
general case is the same as the area form on the domain does not affect the
topological nature of the energy. As the left-hand-side of (4.11) is
nonnegative, (4.12) follows. ∎

####  4.1.6. Computation of ${\it VHF}$

It is expected that the vortex Floer homology is isomorphic to the Hamiltonian
Floer homology of the symplectic quotient, and hence its singular homology (in
appropriate coefficients). However, such a calculation relies involved
technical constructions. The Piunikhin–Salamon–Schwarz (PSS) approach forces
one to deal with multiple covers of equivariant Floer cylinders with $H\equiv
0$ which may have negative equivariant Chern number. The adiabatic limit
approach (similar to [GS05]) requires the study of affine vortices for a
general toric manifold. In particular, for general symplectic quotients both
approaches require the use of the virtual technique.

However, in the toric case, even without having the PSS map, it is rather easy
to compute the rank of ${\it VHF}(V)$ as one can find a perfect Morse
function.

######  Proposition 4.8.

For any commutative ring $R$, as $\Lambda_{R}^{\Gamma}$-modules, ${\it
VHF}_{\bullet}(V;\Lambda_{R}^{\Gamma})$ is isomorphic to
$H_{\bullet}(X;\Lambda_{R}^{\Gamma})$ (with the reduced
${\mathbb{Z}}_{2}$-grading) up to a degree shifting.

###### Proof.

Recall that the $2n$-dimensional toric manifold $X$ carries a Hamiltonian
$T^{n}$-action. For a generic circle $S^{1}\subset T^{n}$, the induced moment
map $f:X\to{\mathbb{R}}$ is a perfect Morse function whose critical points are
the toric fixed points. In particular, the Morse indices are all even. Then
for $\epsilon$ small, $\epsilon f$ is a nondegenerate time-independent
Hamiltonian. After a small perturbation and $K$-invariant lift to $V$, the
corresponding vortex Floer chain complex has no two generators with adjacent
degrees. Hence the ${\it VHF}_{\bullet}(V;\Lambda_{R}^{\Gamma})$ has the same
rank as $H_{\bullet}(X;\Lambda_{R}^{\Gamma})$. Lastly, the usual normalization
of the Conley–Zehnder index is taken in such a way that if $x$ is a critical
point of $\epsilon f$ viewed as a 1-periodic orbit with a constant capping,
then

${\rm CZ}(x)=n-{\rm index}_{f}(x)$

where $2n={\rm dim}X$ and ${\rm index}_{f}(x)$ is the Morse index of $x$ (see
[MS04, (12.1.7)]). ∎

###  4.2. Small bulk deformations

Here we define a family of deformations of the vortex Floer homology
parametrized by “small” bulk deformations. Recall that the toric manifold $X$
has $N$ toric divisors $D_{j}$ corresponding to the $N$ faces of the moment
polytope. These divisors are GIT quotients of the coordinate hyperplanes

$V_{j}=\\{(x_{1},\ldots,x_{N})\in V\ |\ x_{j}=0\\}.$

Introduce a small bulk deformation of the form

${\mathfrak{b}}=\sum_{j=1}^{N}c_{j}V_{j}\ {\rm where}\ c_{j}\in\Lambda_{0,R}.$
(4.13)

The ${\mathfrak{b}}$-deformed vortex Floer complex is the complex generated by
equivariant 1-periodic orbits upstairs whose differential counts gauge
equivalence classes of solutions to the vortex equation in a different way:
for each rigid (modulo gauge transformation) solution
${\mathfrak{u}}=(u,\xi,\eta)$, we weight the count by the factor

$\exp\left(\sum_{j=1}^{N}c_{j}(u\cap V_{j})\right)\in\Lambda_{R}$

where $u\cap V_{j}$ is the intersection number between the cylinder $u$ and
the divisor $V_{j}$. Formally, this count coincides with the count of
solutions on the cylinder with markings mapped to $V_{j}$.

######  Remark 4.9.

The use of bulk deformations in Lagrangian Floer theory was invented by
Fukaya–Oh–Ohta–Ono [FOOO11, FOOO16] which resembles the notion of big quantum
cohomology in Gromov–Witten theory. Bulk deformations are adapted to
Hamiltonian Floer theory in [Ush11b] and [FOOO19]. In gauged linear sigma
model it was discussed in [Woo11]. The term “small” used here comes from the
terminology in Gromov–Witten theory where small means deforming Gromov–Witten
invariants by divisor classes and “big” means deforming by classes with
arbitrary degrees.

####  4.2.1. Bulk-avoiding Hamiltonians

One can only have a well-defined topological intersection number between Floer
cylinders and the divisors if periodic orbits do not intersect these toric
divisors. We introduce the following type of Hamiltonians on the toric
manifold.

######  Definition 4.10 (Bulk-avoiding Hamiltonians).

  1. (1)

A Hamiltonian $H$ on the toric manifold $X$ is called bulk-avoiding if all
1-periodic orbits of $nH$ for all $n\geq 1$ do not intersect the divisor
$D_{1}\cup\cdots\cup D_{N}$.

  2. (2)

Denote by

${\mathcal{H}}_{K}^{**}(V)\subset{\mathcal{H}}_{K}^{*}(V)$

the space of admissible $K$-invariant Hamiltonians on $V$ whose reductions are
bulk-avoiding.

  3. (3)

A bulk-avoiding admissible pair is an admissible pair
$(\widehat{H},\widehat{J})$ such that $\widehat{H}$ descends to a bulk-
avoiding Hamiltonian downstairs.

It is easy to see that a $C^{2}$-small perturbation of any Hamiltonian is
bulk-avoiding. Now we can define the topological intersection numbers. Let
${\mathfrak{u}}=(u,\xi,\eta)$ be a solution to (4.6) which converges to
equivariant 1-periodic orbits ${\mathfrak{x}}$ resp. ${\mathfrak{y}}$ at
$-\infty$ resp. $+\infty$. Then a generic compactly supported perturbation
$\tilde{u}$ intersects transversely with $V_{j}$. Define

$[{\mathfrak{u}}]\cap V_{j}=\tilde{u}\cap V_{j}\in{\mathbb{Z}}$

which counts transverse intersection points with signs. Notice that this
number is well-defined: first, if $\tilde{u}^{\prime}$ is another
perturbation, then $\tilde{u}\cap V_{j}=\tilde{u}^{\prime}\cap V_{j}$; second,
if ${\mathfrak{u}}^{\prime}=(u^{\prime},\xi^{\prime},\eta^{\prime})$ is gauge
equivalent to ${\mathfrak{u}}$ via a gauge transformation $g$, then
$\tilde{u}^{\prime}:=g\tilde{u}$ is a perturbation of $u^{\prime}$. As $V_{j}$
is $K$-invariant, $\tilde{u}^{\prime}$ still intersect transversely with
$V_{j}$ and the intersection number is the same.

####  4.2.2. Bulk-deformed vortex Floer complex

For our application, we only consider small bulk deformations of the form

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

Here ${\bf i}=\sqrt{-1}$ and one can regard ${\mathbb{Z}}[{\bf
i}]\subset{\mathbb{C}}$. The weighted counts eventually only depend on $c_{j}$
so we allow $c_{j}$ to be zero and the ambiguity of taking logarithm does not
affect further discussions. Consider the vortex Floer chain complex

${\it VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{{\mathbb{Z}}[{\bf
i}]}).$

Due to the special behavior of the bulk ${\mathfrak{b}}$, the weighted counts
of cylinders are still integral. Define the bulk-deformed vortex differential

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

by

$\partial^{\mathfrak{b}}({\mathfrak{x}})=\sum_{{\mathfrak{y}}\atop{\rm
CZ}({\mathfrak{x}})-{\rm
CZ}({\mathfrak{y}})=1}\left(\sum_{[{\mathfrak{u}}]\in{\mathcal{M}}^{\rm
cyl}({\mathfrak{x}},{\mathfrak{y}})}\epsilon([{\mathfrak{u}}])\exp\left(\sum_{j=1}^{N}\log
c_{j}\ [{\mathfrak{u}}]\cap V_{j}\right)\right){\mathfrak{y}}.$ (4.14)

Here $\epsilon([{\mathfrak{u}}])\in\\{\pm 1\\}$ is the sign of the rigid
solution $[{\mathfrak{u}}]$. In particular, when ${\mathfrak{b}}=0$, the above
coincides with the original differential map $\partial$.

######  Lemma 4.11.

$\partial_{\mathfrak{b}}$ is a legitimate linear map and
$(\partial^{\mathfrak{b}})^{2}=0$.

###### Proof.

First, as $c_{j}\in{\mathbb{Z}}[{\bf i}]$, the weights

$\exp\left(\sum_{j=1}^{N}\log c_{j}\ [{\mathfrak{u}}]\cap
V_{j}\right)=\prod_{j=1}^{N}c_{j}^{[{\mathfrak{u}}]\cap
V_{j}}\in{\mathbb{Z}}[{\bf i}].$

Hence the coefficients on the right hand side of (4.14) are still in
${\mathbb{Z}}[{\bf i}]$. Second, by Gromov compactness, the sum (4.14) is
still in the module ${\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{{\mathbb{Z}}[{\bf i}]})$.
Hence $\partial^{\mathfrak{b}}$ is a well-defined linear map. To prove that
its square is zero, consider for each ${\mathfrak{x}}$ and ${\mathfrak{z}}$
with Conley–Zehnder indices differing by $2$ and consider the $1$-dimensional
components of the moduli space $\overline{\mathcal{M}}{}^{\rm
cyl}({\mathfrak{x}},{\mathfrak{z}})$. It can be further decomposed into
connected components. Within each connected components, the topological
intersection number for each cylinder with each $V_{j}$ is a constant.
Moreover, for the concatenation of two cylinders $[{\mathfrak{u}}_{1}]$ and
$[{\mathfrak{u}}_{2}]$ which is in the boundary of such a component, this
intersection number with $V_{j}$ is equal to the sum $[{\mathfrak{u}}_{1}]\cap
V_{j}+[{\mathfrak{u}}_{2}]\cap V_{j}$. It follows that
$(\partial^{\mathfrak{b}})^{2}=0$. ∎

Hence for each regular admissible bulk-avoiding pair
$(\widehat{H},\widehat{J})$, one can define the ${\mathfrak{b}}$-deformed
vortex Floer homology by

${\it
VHF}_{\bullet}^{\mathfrak{b}}(\widehat{H},\widehat{J};\Lambda_{{\mathbb{Z}}[{\bf
i}]}):={\rm ker}\partial^{\mathfrak{b}}/{\rm im}\partial^{\mathfrak{b}}.$

Below we summarize its properties.

######  Theorem 4.12 (Properties of bulk-deformed vortex Floer complex).

  1. (1)

For each regular bulk-avoiding admissible pair $(\widehat{H},\widehat{J})$,
the complex ${\it
VCF}_{\bullet}(\widehat{H},\widehat{J};\Lambda_{{\mathbb{Z}}[{\bf i}]})$ with
differential $\partial^{\mathfrak{b}}$ is a ${\mathbb{Z}}_{2}$-graded filtered
Floer–Novikov complex (see Definition (3.14)).

  2. (2)

For each two regular admissible bulk-avoiding pairs
$(\widehat{H}_{1},\widehat{J}_{1})$ and $(\widehat{H}_{2},\widehat{J}_{2})$,
there is a continuation map