# Hofer geometry via toric degeneration

Yusuke Kawamoto

###### Abstract

The main theme of this paper is to use toric degeneration to produce distinct
homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We
focus on the (complex $n$-dimensional) quadric hypersurface and the del Pezzo
surfaces, and study two classes of distinguished Lagrangian submanifolds that
appear naturally in a toric degeneration, namely the Lagrangian torus which is
the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres
that appear as vanishing cycles. For the quadrics, we prove that the group of
Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and
derive some superheaviness results. Along the way, we show that the toric
degeneration is compatible with the Biran decomposition. This implies that for
$n=2$, the Lagrangian fiber torus (Gelfand–Zeitlin torus) is Hamiltonian
isotopic to the Chekanov torus, which answers a question of Y. Kim. We give
applications to $C^{0}$-symplectic topology which includes the
Entov–Polterovich–Py question for the quadric hypersurface. We also prove
analogous results for the del Pezzo surfaces.

###### Contents

  1. 1 Introduction and overview of the results
    1. 1.1 Hofer geometry
    2. 1.2 Main results for quadrics
    3. 1.3 Lagrangians in quadric hypersurfaces
    4. 1.4 Main results for the del Pezzo surfaces
    5. 1.5 Applications
    6. 1.6 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Entov–Polterovich quasimorphisms and (super)heaviness
    2. 2.2 Biran decomposition
    3. 2.3 Various ways to see quadrics
    4. 2.4 Gelfand–Zeitlin system for quadrics
    5. 2.5 Convention
  3. 3 Proofs of Theorem B (1), (2)
    1. 3.1 Toric degeneration
    2. 3.2 Proof
  4. 4 Proofs of Theorem A, B (3)
    1. 4.1 Proof–Part 1
    2. 4.2 The Laurent and Novikov fields
    3. 4.3 Proof–Part 2
    4. 4.4 Proof of Theorem A
  5. 5 Proof of Theorem C
  6. 6 Proofs of Theorem D and Applications

##  1 Introduction and overview of the results

###  1.1 Hofer geometry

The main theme of this paper is to use toric degeneration to produce distinct
homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. The
set of Hamiltonian diffeomorphisms of a closed symplectic manifold
$X=(X,\omega)$, denoted by $\mathrm{Ham}(X)$ (as well as its universal cover
$\widetilde{\mathrm{Ham}}(X)$) forms a group and moreover, it has a remarkable
bi-invariant metric called the Hofer metric [Hof93]. The geometry of
$\mathrm{Ham}(X)$ (and $\widetilde{\mathrm{Ham}}(X)$) with respect to the
Hofer metric has been an active and an important research topic, e.g. [Pol01],
which is now called the Hofer geometry. One of the first important questions
in this topic was the so-called Hofer diameter conjecture which conjectures
that the diameter of $\mathrm{Ham}(X)$ (and $\widetilde{\mathrm{Ham}}(X)$)
with respect to the Hofer metric is infinity. Many fragmented cases were
solved by various symplectic geometers before a major step forward was made by
Entov–Polterovich in [EP03], where they introduced the method of
quasimorphisms to the study of Hofer geometry. This provided a systematic way
to solve the Hofer diameter conjecture for a sufficiently large class of
symplectic manifolds by means of Floer theory. However, beyond the Hofer
diameter conjecture, not much is known about the Hofer geometry. For example,
consider the following question, known as the Kapovich–Polterovich question:

######  Question 1.1.1 (Kapovich–Polterovich question111The original question
was for the case $X=S^{2}$.).

For a closed symplectic manifold $X$, is the group $\mathrm{Ham}(X)$ quasi-
isometric to the real line $\mathbb{R}$ with respect to the Hofer metric?

A negative answer to the Kapovich–Polterovich question will immediately solve
the Hofer diameter conjecture, but extremely little is known about it. At the
time of writing, the only two cases for which Question 1.1.1 has been answered
(in the negative) are the cases of $X=S^{2}\times S^{2}$ Fukaya–Oh–Ohta–Ono
[FOOO19] (see also [EliPol]), and $X=S^{2}$, which was done in 2021,
independently by Cristofaro-Gardiner–Humilière–Seyfaddini [CGHS] and
Polterovich–Shelukhin [PS]. Other than that, even for the weaker version of
Question 1.1.1 where $\mathrm{Ham}(X)$ is replaced by
$\widetilde{\mathrm{Ham}}(X)$, the question remains widely open.

This paper attempts to use toric degeneration to obtain new insights about the
Hofer geometry, especially in (real) dimension higher than $2$. We focus on
the quadric hypersurface

$Q^{n}:=\\{[z_{0}:z_{1}:\cdots:z_{n+1}]\in\mathbb{C}P^{n+1}|z_{0}^{2}+z_{1}^{2}+\cdots+z_{n+1}^{2}=0\\}$

and the symplectic del Pezzo surfaces, and study two distinguished classes of
Lagrangian submanifolds that appear naturally in a toric degeneration, namely
the Lagrangian torus and the Lagrangian spheres that appear as the monotone
fiber of a Lagrangian torus fibration and vanishing cycles, respectively.

###  1.2 Main results for quadrics

We first state the results for the quadric hypersurfaces.

######  Theorem A.

The two Entov–Polterovich type homogeneous quasimorphisms

$\zeta_{\pm}:\widetilde{\mathrm{Ham}}(Q^{n})\to\mathbb{R}$

are distinct, i.e.

$\zeta_{+}\neq\zeta_{-}.$

This immediately answers the $\widetilde{\mathrm{Ham}}(X)$ version of Question
1.1.1 in the negative for $X=Q^{n}$. Theorem A generalizes results of
Eliashberg–Polterovich from [EliPol] and the author from [Kaw22A], where the
$n=2$-case and the $n=4$-case were proven, respectively.

###  1.3 Lagrangians in quadric hypersurfaces

In the proof of Theorem A, producing disjoint Lagrangian submanifolds is a key
step and this is where the toric degeneration plays a crucial role. Recall
that roughly speaking, a toric degeneration is a way to degenerate a
symplectic manifold $X$ into a toric variety $X_{0}$ which allows one to study
the symplectic geometry of $X$ via the toric geometry of $X_{0}$, see Section
3.1. There are two distinguished subsets for toric degenerations. The first is
the monotone Lagrangian fiber torus; given a toric degeneration, one obtains a
Lagrangian torus fibration for $X$ and there is a unique fiber for which the
Lagrangian fiber torus becomes monotone. The second is the vanishing locus,
which is the set obtained by tracing back the points that get crushed into the
singular locus of the toric variety $X_{0}$ in the process of the toric
degeneration. Note that the vanishing locus is not necessarily a geometrically
nice set, e.g. a submanifold, even though in practice it often turns out to be
a Lagrangian cell-complex.

As for the quadric hypersurface $Q^{n}$, the toric degeneration

$\mathfrak{X}_{t}:=\\{(z,t)\in\mathbb{C}P^{n+1}\times\mathbb{C}:z_{0}^{2}+z_{1}^{2}+z_{2}^{2}+t(z_{3}^{2}+\cdots+z_{n+1}^{2})=0\\}$
(1.3.1)

was introduced by Nishinou–Nohara–Ueda in [NNU] which is compatible with the
celebrated Gelfand–Zeitlin (GZ) system on $Q^{n}$. The GZ system defines a
Lagrangian torus fibration on $Q^{n}$ and there is one monotone torus fiber
which we call the GZ torus and denote it by $T^{n}_{\text{GZ}}$. The real
Lagrangian sphere

$S^{n}_{\text{van}}:=\\{z\in\mathbb{C}P^{n+1}:z_{0}^{2}+\cdots+z_{n}^{2}+z_{n+1}^{2}=0,\
z_{0},\cdots,z_{n}\in\mathbb{R},\ z_{n+1}\in i\mathbb{R}\\}$

appears as a vanishing locus for this toric degeneration 1.3.1 and thus, we
denote it by $S^{n}_{\text{van}}$.

The $n=2$ case deserves special attention, as it is well-known that $Q^{2}$ is
symplectomorphic to the monotone $S^{2}\times S^{2}$. Through this
symplectomorphism, we can see the monotone GZ fiber torus $T^{2}_{\text{GZ}}$
as a monotone Lagrangian torus in $S^{2}\times S^{2}$. In [KimA], Yoosik Kim
computed the superpotential for the monotone GZ fiber torus
$T^{n}_{\text{GZ}}$ and found that for $n=2$, it agrees with the
superpotential for the celebrated Chekanov torus in $S^{2}\times S^{2}$. This
lead him to ask the following question:

######  Question 1.3.1.

([KimA, Section 4.2]) In the monotone $S^{2}\times S^{2}$, is the monotone GZ
fiber torus $T^{2}_{\text{GZ}}$ Hamiltonian isotopic to the Chekanov torus
$T^{2}_{\text{Ch}}$?

We study some geometric properties of the distinguished monotone Lagrangian
submanifolds $T^{n}_{\text{GZ}}$ and $S^{n}_{\text{van}}$ in $Q^{n}$. Our
second main result which is the following.

######  Theorem B.

The monotone Lagrangian torus fiber $T^{n}_{\text{GZ}}$ in $Q^{n}$ satisfies
the following properties:

  1. 1.

Consider the polarization $(X,\Sigma)=(Q^{n},Q^{n-1})$. The monotone GZ fiber
$T^{n}_{\text{GZ}}$ in $Q^{n}$ coincides with the distinguished monotone
Lagrangian torus obtained from the Biran circle bundle construction from
$T^{n-1}_{\text{GZ}}$ in $Q^{n-1}$, i.e.

$T^{n}_{\text{GZ}}=\widetilde{T^{n-1}_{\text{GZ}}}.$

  2. 2.

It is disjoint to the real Lagrangian sphere $S^{n}_{\text{van}}$ which is a
vanishing locus of the toric degeneration, i.e.

$S^{n}_{\text{van}}\cap T^{n}_{\text{GZ}}=\emptyset.$

  3. 3.

The Gelfand–Zeitlin torus $T^{n}_{\text{GZ}}$ is $\zeta_{+}$-superheavy and
the vanishing cycle $S^{n}_{\text{van}}$ is $\zeta_{-}$-superheavy.

The third assertion of Theorem B generalizes a result of
Eliashberg–Polterovich from [EliPol] where the $n=2$ case was proven.

Remark 1.3.2.

  1. 1.

For details about polarizations and the Biran circle bundle construction, see
Section 2.2.

  2. 2.

In [Kaw22A], it was proven that $\zeta_{+}\neq\zeta_{-}$ for $n=2,4$ but
nothing about the superheaviness was proved. The new input is the use of some
results from the mirror symmetry literature, e.g. the Auroux–Kontsevich–Seidel
theorem, combined with the comparison technique of spectral invariants with
different coefficients developed in [Kaw22A], see Sections 4.1 and 4.2.

  3. 3.

Theorem B has other applications; it plays an important role in [KawA].

The $n=2$ case of the first assertion in Theorem B has the following
corollary, which answers the aforementioned Question 1.3.1 of Yoosik Kim.

######  Corollary 1.3.3.

In the monotone $S^{2}\times S^{2}$, the monotone GZ fiber $T^{2}_{\text{GZ}}$
is Hamiltonian isotopic to the Chekanov torus $T^{2}_{\text{Ch}}$.

Remark 1.3.4. The answer to Kim’s question 1.3.1, namely Corollary 1.3.3, was
very recently obtained by Kim himself in [KimB], by a different approach.

###  1.4 Main results for the del Pezzo surfaces

In this section, we state the analogous results for the del Pezzo surfaces, as
for the quadrics. Recall that the del Pezzo surfaces are smooth (complex) two-
dimensional Fano varieties, equipped with a monotone symplectic form. They are
classified and consist of $\mathbb{C}P^{2}$, $S^{2}\times S^{2}(=Q^{2})$, and
$\mathbb{D}_{k}:=\mathbb{C}P^{2}\\#k\overline{\mathbb{C}P^{2}},\ 1\leqslant
k\leqslant 8$.

######  Theorem C.

  1. 1.

For $\mathbb{D}_{2}$, there exist two distinct Entov–Polterovich type
homogeneous quasimorphisms

$\zeta_{j}:\widetilde{\mathrm{Ham}}(\mathbb{D}_{2})\to\mathbb{R},\ j=1,2,$
$\zeta_{1}\neq\zeta_{2}.$

  2. 2.

For $\mathbb{D}_{k},\ k=3,4$, there exist three distinct Entov–Polterovich
type homogeneous quasimorphisms

$\zeta_{j}:\widetilde{\mathrm{Ham}}(\mathbb{D}_{k})\to\mathbb{R},\ j=1,2,3,$
$\zeta_{i}\neq\zeta_{j},\text{ if $i\neq j$}.$

Theorem C leads to the following, which answers the Kapovich–Polterovich
question in the negative.

######  Theorem D (Kapovich–Polterovich question).

The group $\mathrm{Ham}(\mathbb{D}_{k})$ is not quasi-isometric to the real
line $\mathbb{R}$ with respect to the Hofer metric when $k=3,4$.

Remark 1.4.1. As mentioned earlier, to the author’s knowledge, the
Kapovich–Polterovich question has been so far answered in the negative only
for $S^{2}\times S^{2}$ by Fukaya–Oh–Ohta–Ono [FOOO19] (also, see another
approach by Eliashberg–Polterovich [EliPol]), and for $S^{2}$ by Cristofaro-
Gardiner–Humilière–Seyfaddini [CGHS] and Polterovich–Shelukhin [PS].

###  1.5 Applications

We discuss some applications of Theorems A, B, C. The following question of
Entov–Polterovich–Py from 2012 has been an important open question, which is
referred to as the “Quasimorphism question” in the monograph of
McDuff–Salamon:

######  Question 1.5.1 ([EPP12], [MS98, Chapter 14, Problem 23 “Quasimorphism
question”]).

  1. 1.

Does there exist a nonzero homogeneous quasimorphism

$\mu:\mathrm{Ham}(S^{2})\to\mathbb{R}$

that is continuous with respect to the $C^{0}$-topology on
$\mathrm{Ham}(S^{2})$?

  2. 2.

If yes, can it be made Lipschitz with respect to the Hofer metric?

Remark 1.5.2.

  1. 1.

The $C^{0}$-topology on $\mathrm{Ham}(X)$ is induced by the $C^{0}$-metric of
Hamiltonian diffeomorphisms $\phi,\psi\in\mathrm{Ham}(M,\omega)$, which is
defined by

$d_{C^{0}}(\phi,\psi):=\max_{x\in M}d(\phi(x),\psi(x)),$

where $d$ denotes the distance on $X$ induced by a fixed Riemannian metric on
$X$. Different choices of Riemannian metrics on $X$ will induce equivalent
$C^{0}$-topology.

  2. 2.

One point of Question 1.5.1 is that currently the relation between the Hofer
metric and the $C^{0}$-metric is not well understood. See, for example, [JS]
for the latest progress on this topic.

The following extended version was considered in [Kaw22A] before the final
resolution of the original question by Cristofaro-
Gardiner–Humilière–Mak–Seyfaddini–Smith in [CGHMSS]:

######  Question 1.5.3 (Extended Entov–Polterovich–Py question).

Does there exist a closed symplectic manifold $(X,\omega)$ which admits a non-
trivial homogeneous quasimorphism on $\mathrm{Ham}(X,\omega)$ which is
$C^{0}$-continuous? If yes, can it be Hofer Lipschitz continuous?

In [Kaw22A], a positive answer to Question 1.5.3 was provided for $Q^{n},\
n=2,4$. By using Theorems B and C, we manage to generalize this to $Q^{n}$ for
all $n$, and to some del Pezzo surfaces.

######  Theorem E.

For $X=Q^{n}$ and $X=\mathbb{D}_{k},\ k=2,3,4$, there exist non-trivial
homogeneous quasimorphisms

$\mu:\mathrm{Ham}(X)\to\mathbb{R}$

that are both Hofer Lipschitz and $C^{0}$ continuous.

Remark 1.5.4.

  1. 1.

We can actually construct two distinct quasimorphisms with the relevant
continuity properties in Theorem E for $\mathbb{D}_{k},\ k=3,4$.

  2. 2.

It is also known that one can construct quasimorphisms for $\mathbb{C}P^{n}$
that have continuity properties as in Question 1.5.3 by [KS].

The argument of the proof of Theorem E actually shows that the asymptotic
spectral norm $\overline{\gamma}$

$\overline{\gamma}:\mathrm{Ham}(X)\to\mathbb{R}$
$\overline{\gamma}(\phi):=\lim_{k\to+\infty}\frac{\gamma(\phi^{k})}{k}$

is $C^{0}$-continuous for $X=Q^{n}$ and $X=\mathbb{D}_{k},\ k=2,3,4$, even
though the $C^{0}$-continuity of the spectral norm $\gamma$ for none of these
symplectic manifolds is confirmed at the time of writing. Note that the
$C^{0}$-continuity of the spectral norm $\gamma$ is currently proven for the
closed surfaces [Sey13], symplectic aspherical manifolds [BHS21],
$\mathbb{C}P^{n}$ [Sh22], and negative monotone symplectic manifolds [Kaw22B].

######  Theorem F.

Let $(X,\omega)$ be a monotone symplectic manifold whose quantum cohomology
ring $QH(X;\Lambda)$ is semi-simple. Then,

$\overline{\gamma}:\mathrm{Ham}(X)\to\mathbb{R}$

is $C^{0}$-continuous. Moreover, for $Q^{n}$ and $\mathbb{D}_{k},\ k=2,3,4$,
$\overline{\gamma}$ is non-trivial, i.e.

$\overline{\gamma}\neq 0.$

###  1.6 Acknowledgements

I thank Yuhan Sun for a stimulating discussion and for sharing his insight
with me. I thank Sobhan Seyfaddini for useful comments. This work was carried
out in several different institutions; École Normale Supérieure, Paris while
the author was a PhD student, Université de Montréal while the author was a
CRM-postdoctoral fellow at Centre de Recherches Mathématiques (CRM) and ETH
Zürich while the author was a Hermann-Weyl-Instructor at the Institute for
Mathematical Research (FIM). The author thanks all institutions for their
great hospitality.

##  2 Preliminaries

We will review results that we will use in the proofs. A special emphasis is
put on comparing the conventions in [OU16] and other works, as we will use the
convention from [OU16] in the proofs. We will assume that symplectic manifolds
are all closed and monotone.

###  2.1 Entov–Polterovich quasimorphisms and (super)heaviness

It is well-known that on a symplectic manifold $X$, for a non-degenerate
Hamiltonian $H:=\\{H_{t}:X\to\mathbb{R}\\}_{t\in[0,1]}$ and a choice of a nice
coefficient field $\Lambda^{\downarrow}$, which will be either the downward
Laurent coefficients $\Lambda_{\text{Lau}}^{\downarrow}$

$\Lambda_{\text{Lau}}^{\downarrow}:=\\{\sum_{k\leqslant
k_{0}}b_{k}t^{k}:k_{0}\in\mathbb{Z},b_{k}\in\mathbb{C}\\},$

or the downward Novikov coefficients $\Lambda_{\text{Nov}}^{\downarrow}$

$\Lambda_{\text{Nov}}^{\downarrow}:=\\{\sum_{j=1}^{\infty}a_{j}T^{\lambda_{j}}:a_{j}\in\mathbb{C},\lambda_{j}\in\mathbb{R},\lim_{j\to-\infty}\lambda_{j}=+\infty\\},$

one can construct a filtered Floer homology group
$\\{HF^{\tau}(H):=HF^{\tau}(H;\Lambda^{\downarrow})\\}_{\tau\in\mathbb{R}}$
where for two numbers $\tau<\tau^{\prime}$, the groups
$HF^{\tau}(H;\Lambda^{\downarrow})$ and
$HF^{\tau^{\prime}}(H;\Lambda^{\downarrow})$ are related by a map induced by
the inclusion map on the chain level:

$i_{\tau,\tau^{\prime}}:HF^{\tau}(H;\Lambda^{\downarrow})\longrightarrow
HF^{\tau^{\prime}}(H;\Lambda^{\downarrow}),$

and especially we have

$i_{\tau}:HF^{\tau}(H;\Lambda^{\downarrow})\longrightarrow
HF(H;\Lambda^{\downarrow}),$

where $HF(H;\Lambda^{\downarrow})$ is the Floer homology. There is a canonical
ring isomorphism called the Piunikhin–Salamon–Schwarz (PSS)-map [PSS96],
[MS04]

$PSS_{H;\Lambda}:QH(X;\Lambda)\to HF(H;\Lambda^{\downarrow}),$

where $QH(X;\Lambda)$ denotes the quantum cohomology ring of $X$ with
$\Lambda$-coefficients, i.e.

$QH(X;\Lambda):=H^{\ast}(X;\mathbb{C})\otimes\Lambda.$

Here, $\Lambda$ is either the Laurent coefficients $\Lambda_{\text{Lau}}$

$\Lambda_{\text{Lau}}:=\\{\sum_{k\geqslant
k_{0}}b_{k}t^{k}:k_{0}\in\mathbb{Z},b_{k}\in\mathbb{C}\\},$

or the Novikov coefficients $\Lambda_{\text{Nov}}$

$\Lambda_{\text{Nov}}:=\\{\sum_{j=1}^{\infty}a_{j}T^{\lambda_{j}}:a_{j}\in\mathbb{C},\lambda_{j}\in\mathbb{R},\lim_{j\to+\infty}\lambda_{j}=+\infty\\}$

chosen accordingly to the set-up of the Floer homology.

Remark 2.1.1.

  1. 1.

The choice of the coefficient fields will eventually become very important,
see Section 4.2. If we do not precise the choice of it and denote it by
$\Lambda$, it means that the argument/result holds for both
$\Lambda_{\text{Lau}}$ and $\Lambda_{\text{Nov}}$.

  2. 2.

Nevertheless, it might be helpful to keep in mind that $\Lambda_{\text{Lau}}$
can be embedded to $\Lambda_{\text{Nov}}$ by the inclusion given by $t\mapsto
T^{\lambda_{0}}$, where $\lambda_{0}$ is a positive generator of
$\langle\omega,\pi_{2}(X)\rangle$.

  3. 3.

The formal variable $t$ in $\Lambda_{\text{Lau}}$ should be seen as
representing the sphere with the minimal possible positive area
$\lambda_{0}>0$ (and thus, the minimal possible Maslov index as well, by
monotonicity).

The spectral invariants, which were introduced by Schwarz [Sch00] and
developed by Oh [Oh05] following the idea of Viterbo [Vit92], are real numbers
$\\{c(H,a)\in\mathbb{R}\\}$ associated to a pair of a Hamiltonian $H$ and a
class $a\in QH(X;\Lambda)$ in the following way:

$c(H,a):=\inf\\{\tau\in\mathbb{R}:PSS_{H;\Lambda}(a)\in\mathrm{Im}(i_{\tau})\\}.$

Remark 2.1.2. Although the Floer homology is only defined for a non-degenerate
Hamiltonian $H$, the spectral invariants can be defined for any Hamiltonian by
using the following so-called Hofer continuity property:

$c(H,a)-c(G,a)\leqslant\int_{0}^{1}\left(H_{t}(x)-G_{t}(x)\right)dt$

for any $a\in QH(X),\ H$ and $G$.

The spectral norm is

$\begin{gathered}\gamma:\mathrm{Ham}(X)\longrightarrow\mathbb{R}\\\
\phi\mapsto\gamma(\phi):=\inf_{\phi_{H}=\phi}\left(c(H,1_{X})+c(\overline{H},1_{X})\right)\end{gathered}$
(2.1.1)

which defines a non-degenerate metric
$d_{\gamma}(\phi,\phi^{\prime}):=\gamma(\phi^{-1}\circ\phi^{\prime})$ on
$\mathrm{Ham}(X)$. Recall that $\overline{H}(t,x):=-H(t,(\phi_{H}^{t})(x))$
and it generates the Hamiltonian path $t\mapsto(\phi_{H}^{t})^{-1}$.

Entov–Polterovich constructed a special map called the quasimorphism on
$\widetilde{\mathrm{Ham}}(X)$ for under some assumptions via spectral
invariants, which we will briefly review. Recall that a quasimorphism $\mu$ on
a group $G$ is a map to the real line $\mathbb{R}$ that satisfies the
following two properties:

  1. 1.

There exists a constant $C>0$ such that

$|\mu(f\cdot g)-\mu(f)-\mu(g)|<C$

for any $f,g\in G$.

  2. 2.

For any $k\in\mathbb{Z}$ and $f\in G$, we have

$\mu(f^{k})=k\cdot\mu(f).$

######  Theorem 2.1.3 ([EP03]).

Suppose $QH(X;\Lambda)$ has a field factor, i.e.

$QH(X;\Lambda)=Q\oplus A$

where $Q$ is a field and $A$ is some algebra. Decompose the unit $1_{X}$ of
$QH(X;\Lambda)$ with respect to this split, i.e.

$1_{X}=e+a.$

Then, the asymptotic spectral invariant of $\widetilde{\phi}$ with respect to
$e$ defines a quasimorphism, i.e.

$\begin{gathered}\zeta_{e}:\widetilde{\mathrm{Ham}}(X)\longrightarrow\mathbb{R}\\\
\zeta_{e}(\widetilde{\phi}):=\lim_{k\to+\infty}\frac{c(H^{\\#k},e)}{k}\end{gathered}$
(2.1.2)

where $H$ is any mean-normalized Hamiltonian such that the path
$t\mapsto\phi_{H}^{t}$ represents the class $\widetilde{\phi}$ in
$\widetilde{\mathrm{Ham}}(X)$.

Remark 2.1.4. When we consider Entov–Polterovich quasimorphisms, we take the
Laurent coefficients $\Lambda_{\text{Lau}}$ (see Section 4.2) but the
condition in Theorem 2.1.3 is equivalent for $\Lambda_{\text{Lau}}$ and
$\Lambda_{\text{Nov}}$, see [EP08].

Example 2.1.5. As for the quadric hypersurface $Q^{n}$, the quantum cohomology
is semi-simple and with the Laurent coefficients, it splits into a direct sum
of two fields

$QH(Q^{n};\Lambda_{\text{Lau}})=Q_{+}\oplus Q_{-},$ (2.1.3)

where the unit $1_{Q^{n}}$ splits as

$\begin{gathered}1_{Q^{n}}=e_{+}+e_{-},\\\ e_{\pm}:=\frac{1_{Q^{n}}\pm
PD([pt])t}{2}.\end{gathered}$ (2.1.4)

Thus, we get two Entov–Polterovich quasimorphisms

$\zeta_{\pm}:=\zeta_{e_{\pm}}:\widetilde{\mathrm{Ham}}(Q^{n})\longrightarrow\mathbb{R}$
(2.1.5)

which we prove that they are distinct in Theorem A.

Remark 2.1.6. By slight abuse of notation, we will also see $\zeta_{e}$ as a
function on the set of time-independent Hamiltonians:

$\begin{gathered}\zeta_{e}:C^{\infty}(X)\longrightarrow\mathbb{R}\\\
\zeta_{e}(H):=\lim_{k\to+\infty}\frac{c(H^{\\#k},e)}{k}.\end{gathered}$
(2.1.6)

Not only that Entov–Polterovich defined quasimorphisms on
$\widetilde{\mathrm{Ham}}(X)$, they introduced a level of rigidity for subsets
in $X$ called (super)heaviness.

######  Definition 2.1.7 ([EP09],[EP06]).

Take an idempotent $e\in QH(X;\Lambda)$ and denote the asymptotic spectral
invariant with respect to $e$ by $\zeta_{e}$. A subset $S$ of $X$ is called

  1. 1.

heavy if for any time-independent Hamiltonian $H:X\to\mathbb{R}$, we have

$\inf_{x\in S}H(x)\leqslant\zeta_{e}(H),$

  2. 2.

superheavy if for any time-independent Hamiltonian $H:X\to\mathbb{R}$, we have

$\zeta_{e}(H)\leqslant\sup_{x\in S}H(x).$

The following is an easy corollary of the definition of superheaviness which
is useful.

######  Proposition 2.1.8 ([EP09]).

Assume the same condition on $QH(X;\Lambda)$ as in Theorem 2.1.3. Let $S$ be a
subset of $X$ that is $\zeta$-superheavy. For a time-independent Hamiltonian
$H:X\to\mathbb{R}$ whose restriction to $S$ is constant, i.e. $H|_{S}\equiv
r,\ r\in\mathbb{R}$, we have

$\zeta(H)=r.$

We end this section by giving a criterion for heaviness, proved by FOOO (there
are earlier results with less generality, c.f. [Alb05]). Denote the ring
homomorphism called the (length 0) closed-open map, which is a quantum
analogue of the restriction map, by

$\mathcal{CO}^{0}:QH^{\ast}(X;\Lambda)\to HF^{\ast}(L,\rho)$

where $\rho$ is a $\mathbb{C}^{\ast}$-local system, i.e.

$\rho:H_{1}(L;\mathbb{Z})\longrightarrow\mathbb{C}^{\ast}.$

Remark 2.1.9. The original notation used in [FOOO09] for $\mathcal{CO}^{0}$ is
$i^{\ast}_{qm}$.

######  Theorem 2.1.10.

$($[FOOO19, Theorem 1.6]$)$

Assume

$HF(L,\rho)\neq 0$

for some local system $\rho$. If

$\mathcal{CO}^{0}(e)\neq 0$

for an idempotent $e\in QH^{\ast}(X;\Lambda)$, then $L$ is $\zeta_{e}$-heavy.

Remark 2.1.11. When $\zeta_{e}$ is homogeneous, e.g. when $e$ is a unit of a
field factor of $QH^{\ast}(X;\Lambda)$ and $\zeta_{e}$ is an Entov–Polterovich
quasimorphism, then heaviness and superheaviness are equivalent so Theorem
2.1.10 will be good enough to obtain the superheaviness of $L$.

###  2.2 Biran decomposition

In this section, we briefly review the notion of Biran decomposition, which
was established by Biran in [Bir01], while comparing the conventions in
[Bir01] and [OU16]. We emphasis that in this paper, the convention of
Oakley–Usher [OU16] is used.

We start with the Biran’s setting [Bir01]. Let $X$ be a symplectic Kähler
manifold $X=(X,\omega_{\text{Bir}})$ with an integral symplectic form, i.e.
$[\omega_{\text{Bir}}]\in H^{2}(X;\mathbb{Z})$. Consider a (Donaldson) divisor
$\Sigma$ which gives a polarization of degree $k$, i.e. the pair $(X,\Sigma)$
satisfies

$PD([\Sigma])=k[\omega_{\text{Bir}}]\in H^{2}(X;\mathbb{Z}).$ (2.2.1)

The complement $X\backslash\Sigma$ has the structure of a Stein manifold and
thus, one can define the unstable set of the Stein function which is called
the skeleton $\Delta$. Biran proved that the complement of the skeleton
$\Delta$ is symplectomorphic to a certain disk bundle (defined with
appropriate connection/curvature):

$(X\backslash\Delta,\omega_{\text{Bir}})\xrightarrow[]{\simeq}(D\Sigma,\frac{1}{k}\omega_{\text{can};\text{Bir}})$
(2.2.2)

where

$\omega_{\text{can};\text{Bir}}:=\pi^{\ast}(k\omega_{\text{Bir};\Sigma})+d(r^{2}\alpha_{\text{Bir}}),$

$\omega_{\text{Bir};\Sigma}:=i^{\ast}\omega_{\text{Bir}}$ and
$\alpha_{\text{Bir}}$ is the connection $1$-form satisfying

$\int_{\partial D^{2}=\\{|z|=1\\}}\alpha_{\text{Bir}}=1.$

Thus, the area of a fiber disk of the disk bundle
$(D\Sigma,\omega_{\text{can}})$ is

$\int_{D^{2}=\\{|z|<1\\}}d(r^{2}\alpha_{\text{Bir}})=\int_{\partial
D^{2}}r^{2}\alpha_{\text{Bir}}=1.$

Thus, for the Biran decomposition with Biran’s convention
$(D\Sigma,\frac{1}{k}\omega_{\text{can};\text{Bir}})$, the area of the fiber
disk is $\frac{1}{k}$, i.e. the radius of the disk is $1/\sqrt{\pi k}$.

Next, we explain Oakley–Usher’s convention in [OU16]. The symplectic form of
$X$ $\omega_{\text{OU}}$ is scaled so that

$\omega_{\text{OU}}=2\pi\omega_{\mathrm{Bir}}.$ (2.2.3)

Note that unlike $\omega_{\text{Bir}}$, $\omega_{\text{OU}}$ does not
represent an integral cohomology class, i.e. $[\omega_{\text{OU}}]\notin
H^{2}(X;\mathbb{Z})$. Oakley–Usher describes the aforementioned Biran
decomposition from the relation

$\tau\cdot PD([\Sigma])=[\omega_{\text{OU}}]$ (2.2.4)

for $\tau>0$. Note that from the equations 2.2.1, 2.2.4, and 2.2.3, we obtain

$k=\frac{2\pi}{\tau}.$ (2.2.5)

The Biran decomposition with respect to Oakley–Usher’s convention is as
follows:

$(X\backslash\Delta,\omega_{\text{OU}})\xrightarrow[]{\simeq}(D\Sigma,\frac{1}{k}\omega_{\text{can};\text{OU}})$
(2.2.6)

where

$\omega_{\text{can};\text{OU}}:=\pi^{\ast}(k\omega_{\text{OU};\Sigma})+d(r^{2}\alpha_{\text{OU}}),$

$\omega_{\text{OU};\Sigma}:=i^{\ast}\omega_{\text{OU}}$ and
$\alpha_{\text{OU}}$ is the connection $1$-form satisfying

$\alpha_{\text{OU}}=2\pi\alpha_{\text{Bir}}.$

From the equation 2.2.3, we have

$\omega_{\text{can};\text{OU}}=2\pi\omega_{\text{can};\text{Bir}}$ (2.2.7)

and thus, by using 2.2.5, we have

$\frac{1}{k}\omega_{\text{can};\text{OU}}=\frac{2\pi}{k}\omega_{\text{can};\text{Bir}}=\tau\omega_{\text{can};\text{Bir}}$
(2.2.8)

which means that for the Biran decomposition with Oakley–Usher’s convention
$(D\Sigma,\frac{1}{k}\omega_{\text{can};\text{OU}})$, the area of the disk is
$\tau$, i.e. the radius of the fiber disk is $\sqrt{\tau/\pi}$.

We look at the examples we will be focusing on.

Example 2.2.1. For $X=\mathbb{C}P^{n}$, $\omega_{\text{Bir}}$ is the
Fubini–Study form scaled so that

$\int_{\mathbb{C}P^{1}}\omega_{\text{Bir}}=1.$

whereas $\omega_{\text{OU}}$ satisfies

$\int_{\mathbb{C}P^{1}}\omega_{\text{OU}}=2\pi.$

For $X=Q^{n}$, which is the main case of our interest, $\omega_{\text{Bir}}$
and $\omega_{\text{OU}}$ are

$\omega_{\text{Bir}}=i^{\ast}\omega_{\mathbb{C}P^{n+1};\text{Bir}},$
$\omega_{\text{OU}}=i^{\ast}\omega_{\mathbb{C}P^{n+1};\text{OU}},$

Now, we look at some polarizations.

  1. 1.

$(X,\Sigma)=(\mathbb{C}P^{n},\mathbb{C}P^{n-1}),\ (Q^{n},Q^{n-1})$: As the
degree of this polarization is $k=1$, we have

$\tau=2\pi,$

which implies that the fiber disk has radius $\sqrt{2}$.

  2. 2.

$(X,\Sigma)=(\mathbb{C}P^{n},Q^{n-1})$: As the degree of this polarization is
$k=2$, we have

$\tau=2\pi/2=\pi,$

which implies that the fiber disk has radius $1$.

Biran decomposition has been also known as a nice way to construct interesting
Lagrangians, which we call the Biran circle bundle construction and explain
what it is. Let $L$ be a Lagrangian submanifold in $\Sigma$. Consider the
radius $r>0$ circle bundle associated to the disk bundle $D\Sigma$

$D\Sigma_{|u|=r}:=\\{u\in D\Sigma:|u|=r\\}.$

The set

$\widetilde{L}_{r}:=\pi_{|u|=r}^{-1}(L),\
\pi_{|u|=r}:D\Sigma_{|u|=r}\to\Sigma$

defines a Lagrangian submanifold in $D\Sigma$, which is a circle bundle over
$L$. Note that $\pi_{|u|=r}$ denotes the restricted projection
$D\Sigma_{|u|=r}\to\Sigma$. Via the symplectic identification 2.2.2, we can
see $\widetilde{L}_{r}$ as a Lagrangian submanifold in $X\backslash\Sigma$ or
$X$.

When $L$ is a monotone in $\Sigma$, then there is a distinguished radius
$r_{0}>0$ for which the lifted Lagrangian submanifold $\widetilde{L}$ becomes
also monotone in $X$ and according to [BC09, Proposition 6.4.1], it satisfies

$r_{0}^{2}=\frac{2\kappa_{L}}{2\kappa_{L}+1}$ (2.2.9)

where $\kappa_{L}$ is the monotonicity constant for $L$ in $\Sigma$, i.e.
$\omega_{\Sigma}|_{\pi_{2}(\Sigma,L)}=\kappa_{L}\cdot\mu_{L}|_{\pi_{2}(\Sigma,L)}$.
We sometimes call the radius $r_{0}$ the monotone radius as well.

In the following, the lifted Lagrangian submanifold
$\widetilde{L}:=\widetilde{L}_{r_{0}}$ will always be this distinguished
monotone Lagrangian submanifold in $X$.

###  2.3 Various ways to see quadrics

The aim of this section is to see that the quadric hypersurface $Q^{n}$ can be
identified to some coadjoint orbit $\mathcal{O}_{\lambda}$ of the Lie algebra
$\mathfrak{so}(n+1;\mathbb{R})$.

First of all, as Oakley–Usher it is convenient to see $Q^{n}$ as the following
quotient: first, we have

$\displaystyle\mathbb{C}P^{n}$
$\displaystyle=\mathbb{C}^{n+1}\backslash\\{0\\}/\mathbb{C}^{\ast}$ (2.3.1)
$\displaystyle=\\{z\in\mathbb{C}^{n+1}:|z|=1\\}/S^{1}.$

Then,

$\displaystyle Q^{n}$
$\displaystyle=\\{z\in\mathbb{C}P^{n+1}:z_{0}^{2}+z_{1}^{2}+\cdots+z_{n+1}^{2}=0\\}$
(2.3.2)
$\displaystyle=\\{z\in\mathbb{C}^{n+2}:|z|=2,z_{0}^{2}+z_{1}^{2}+\cdots+z_{n+1}^{2}=0\\}/S^{1}.$

This is the Oakley–Usher way to see $Q^{n}$. Now, by writing $z=x+iy$, we have

$\displaystyle Q^{n}$
$\displaystyle=\\{z\in\mathbb{C}^{n+2}:|z|=2,z_{0}^{2}+z_{1}^{2}+\cdots+z_{n+1}^{2}=0\\}/S^{1}$
(2.3.3) $\displaystyle=\\{z=x+iy\in\mathbb{C}^{n+2}:|x|=|y|=1,x\cdot
y=0\\}/S^{1}.$

Thus, a point in $Q^{n}$ is a orthogonal frame $x,y$ in $\mathbb{R}^{n+2}$
where the rotations are identified, thus this defines a plane in
$\mathbb{R}^{n+2}$. Thus,

$Q^{n}=Gr_{\mathbb{R}}^{+}(2,n+2),$

where $Gr_{\mathbb{R}}^{+}(2,n+2)$ denotes the real oriented Grassmanian. Now,
the group $SO(n+2;\mathbb{R})$ acts transitively on
$Gr_{\mathbb{R}}^{+}(2,n+2)$ and the isotropy subgroup of the plane spanned by
$(1,0,\cdots,0)$ and $(0,1,0,\cdots,0)$ is $SO(2;\mathbb{R})\times
SO(n;\mathbb{R})$, thus,

$Q^{n}=Gr_{\mathbb{R}}(2,n+2)=SO(n+2;\mathbb{R})/\left(SO(2;\mathbb{R})\times
SO(n;\mathbb{R})\right).$ (2.3.4)

Now, $G:=SO(n+2;\mathbb{R})$ also acts transitively on
$\mathfrak{so}(n+2;\mathbb{R})=\\{A:A^{t}=-A\\}$ by the adjoint action. We try
to find a element $\lambda\in\mathfrak{so}(n+2;\mathbb{R})$ so that the
stabilizer of this action equals the stabilizer of the previous action, namely
$SO(2;\mathbb{R})\times SO(n;\mathbb{R})$. In order to do this, we can
consider

$\lambda:=\begin{pmatrix}0&\lambda_{1}&0&\cdots&0\\\
-\lambda_{1}&0&0&\cdots&0\\\ 0&&\ddots&&0\\\ \vdots&&&&\vdots\\\
0&&&&0\end{pmatrix}\in\mathfrak{so}(n+2;\mathbb{R}).$ (2.3.5)

Thus, we now have

$SO(n+2;\mathbb{R})/SO(2;\mathbb{R})\times
SO(n;\mathbb{R})\xrightarrow[]{\simeq}\mathcal{O}_{\lambda}.$ (2.3.6)

From 2.3.4 and 2.3.6, we conclude

$Q^{n}\xrightarrow[]{\simeq}\mathcal{O}_{\lambda}.$ (2.3.7)

To avoid confusion, we will denote this identification by $A$:

$\begin{gathered}A:Q^{n}\xrightarrow[]{\simeq}\mathcal{O}_{\lambda},\\\
z\mapsto A(z).\end{gathered}$ (2.3.8)

The coadjoint orbits are known to possess a canonical symplectic structure
given by the so-called Kirillov–Kostant–Souriau (KKS) form
$\omega_{\text{KKS}}$ which satisfies

$[\omega_{\text{KKS}}]=c_{1}(T\mathcal{O}_{\lambda})$

and thus via 2.3.8, one obtains a KKS form on $Q^{n}$. The symplectic form we
work with in this paper needs to be appropriately scaled, see Section 2.5 for
this matter.

###  2.4 Gelfand–Zeitlin system for quadrics

In this section, we will review the completely integrable system called the
Gelfand–Zeitlin (GZ) system for the quadric $Q^{n}$ via 2.3.8, as explained in
[KimA, Section 2.2].

For any $z\in Q^{n}$, for each $k,\ 2\leqslant k\leqslant n+2$, we take the
left-upper $k\times k$-submatrix which we will denote by $A(z)^{(k)}$. As
$A(z)$ is a skew-symmetric $(n+2)\times(n+2)$ matrix, each $A(z)^{(k)}$ is
also skew-symmetric, and thus its eigenvalues are either all $0$ or
$\pm\sqrt{-1}\nu^{(k)}(z),\ 0\cdots,0$ where $\nu^{(k)}(z)>0$. Now, define the
following map:

$\begin{gathered}\Phi:Q^{n}\longrightarrow\mathbb{R}^{n}\\\
z\mapsto(\lambda_{1}^{(2)}(z),\cdots,\lambda_{1}^{(n+1)}(z)),\end{gathered}$
(2.4.1)

where

$\lambda_{1}^{(k)}(z):=\begin{cases}\nu^{(k+1)}(z)\text{ if either $k\geqslant
2$ or ($k=1$ and $Pf(A(z)^{(2)})\geqslant 0$)}\\\ -\nu^{(2)}(z)\text{ $k=1$
and $Pf(A(z)^{(2)})<0$},\end{cases}$ (2.4.2)

where $Pf$ is the Phaffian.

Guillemin–Sternberg proved that $\Phi$ forms a completely integrable system.

######  Theorem 2.4.1 ([GS83]).

The map $\Phi$ in 2.4.1 is a completely integrable system on $Q^{n}$. In fact,
$\Phi_{k}$ generates a Hamiltonian $S^{1}$-action on
$\Phi^{-1}(\\{w\in\mathbb{R}^{n}:u_{k}\neq 0\\})$.

In [NNU, Proposition 3.1], Nishinou–Nohara–Ueda computed the GZ system
$\\{\lambda_{1}^{(k)}\\}_{2\leqslant k\leqslant n+1}$ for $Q^{n}$

$\lambda_{1}^{(k)}:Q^{n}\longrightarrow\mathbb{R}$

which is as follows:

$\displaystyle\lambda_{1}^{(2)}(z)$
$\displaystyle=\frac{\lambda}{|z|^{2}}i(z_{1}\overline{z_{2}}-\overline{z_{1}}z_{2}),$
(2.4.3) $\displaystyle(\lambda_{1}^{(k)})^{2}(z)$
$\displaystyle=-\sum_{1\leqslant i<j\leqslant
k}\left(\frac{z_{i}\overline{z_{j}}-\overline{z_{i}}z_{j}}{|z|^{2}}\right)^{2}$
$\displaystyle=\left(\frac{\lambda}{|z|^{2}}\right)^{2}((\sum_{j=1}^{k}|z_{j}|^{2})^{2}-|\sum_{j=1}^{k}z_{j}^{2}|^{2}),\
k\geqslant 3,$ $\displaystyle\lambda:$ $\displaystyle=2.$

Remark 2.4.2. In Kim’s paper [KimA], he uses the notation
$\\{u_{k}\\}_{1\leqslant k\leqslant n}$, where the correspondence is given by

$\lambda_{1}^{(k)}=u_{k-1}.$

The monotone GZ fiber is expressed as follows [KimA, Proposition 3.7]:

$T^{n}_{\text{GZ}}:=\Phi_{Q^{n}}^{-1}\left((0,2\cdot\frac{1}{n},2\cdot\frac{2}{n},\cdots,2\cdot\frac{n-1}{n})\right)$
(2.4.4)

where we used the notation

$\begin{gathered}\Phi_{Q^{n}}:Q^{n}\longrightarrow\mathbb{R}^{n}\\\
\Phi_{Q^{n}}(z):=(\lambda_{1}^{(2)},\lambda_{1}^{(3)},\cdots,\lambda_{1}^{(n+1)})(z).\end{gathered}$
(2.4.5)

###  2.5 Convention

In this section, we will remind some conventions that is used in Section 3 and
4, which are taken from [OU16].

For the quadric hypersurface $Q^{n}\subset\mathbb{C}P^{n+1}$, we equip a
symplectic form $\omega$ such that

$\omega=i^{\ast}\omega_{\mathbb{C}P^{n+1}}$

where

$i:Q^{n}\hookrightarrow\mathbb{C}P^{n+1}$

is the inclusion and $\omega_{\mathbb{C}P^{n+1}}$ is the Fubini–Study form
scaled so that

$\int_{\mathbb{C}P^{1}}\omega_{\mathbb{C}P^{n+1}}=2\pi.$

This makes the monotonicity constant to be equal to $\frac{2\pi}{n}$, i.e.

$\omega|_{\pi_{2}(Q^{n})}=\frac{2\pi}{n}\cdot
c_{1}(TQ^{n})|_{\pi_{2}(Q^{n})}.$

In [KimA] and [NNU], they scale the symplectic form on $Q^{n}$ so that the
monotonicity constant becomes $1$. In fact, they use the
Kirillov–Kostant–Souriau (KKS) form $\omega_{\text{KKS}}$ that satisfies

$[\omega_{\text{KKS}}]=c_{1}(TQ^{n}).$

These different choices of the normalization cause some rescaling in the
results in [KimA] and [NNU] and in this paper, we use their results in the
scaled form which amounts to putting $\lambda=2\pi$ instead of $\lambda=n$ in
their results.

##  3 Proofs of Theorem B (1), (2)

###  3.1 Toric degeneration

In this section, as we briefly review the toric degeneration of a symplectic
manifold and of an integrable system. The latter is a special class of a toric
degeneration which was introduced by Nishinou–Nohara–Ueda in [NNU10] that make
the integrable structure on a symplectic manifold and the natural toric
integrable structure on its degenerated toric variety compatible. We study the
particular case of the quadric hypersurface $Q^{n}$ and we specify the toric
degeneration for the quadric hypersurface $Q^{n}$ that we will use in this
paper. We define two distinguished Lagrangians in $Q^{n}$, namely the monotone
GZ torus $T^{n}_{\text{GZ}}$ and the vanishing Lagrangian sphere
$S^{n}_{\text{van}}$. We also discuss its superpotential, which was computed
by Y. Kim in [KimA].

######  Definition 3.1.1 ([HK15, Definition 2],[Eva, Section 1]).

A toric degeneration of a symplectic manifold $X$ is a flat family
$\pi:\mathfrak{X}\to\mathbb{C}$ whose fibers
$\\{X_{t}:=\pi^{-1}(t)\\}_{t\in\mathbb{C}}$ satisfy the following properties:

  1. 1.

For $t\neq 0$, the fiber $X_{t}$ is smooth and $X_{1}$ is isomorphic to $X$.

  2. 2.

For $t=0$, the fiber $X_{0}$ is a toric variety that is not smooth.

  3. 3.

The fibres $X_{t}$ are projective subvarieties of the same projective space,
i.e. there is a morphism $f:\mathfrak{X}\to\mathbb{C}P^{N}$ such that for
every $t\in\mathbb{C}$, $f_{t}:=f|_{X_{t}}:X_{t}\to\mathbb{C}P^{N}$ is an
embedding.

Remark 3.1.2. In fact, it turns out that all the smooth fibers are
symplectomorphic to each other; see [Eva, Lemma 1.1].

We provide an example of a toric degeneration for the quadric hypersurface
$Q^{n}$ considered by Nishinou–Nohara–Ueda in [NNU], which we will be using
throughout the paper.

Example 3.1.3. Consider the following:

$\mathfrak{X}:=\\{(z,t)\in\mathbb{C}P^{n+1}\times\mathbb{C}|z_{0}^{2}+z_{1}^{2}+z_{2}^{2}+t(z_{3}^{2}+\cdots+z_{n+1}^{2})=0\\}.$
(3.1.1)

We define

$\begin{gathered}\pi:\mathfrak{X}\xrightarrow[]{}\mathbb{C}\\\ (z,t)\mapsto
t.\end{gathered}$ (3.1.2)

Then for $t=1$, the fiber $X_{1}$ is isomorphic to $Q^{n}$ and for $t=0$, the
fiber $X_{0}$ is isomorphic to the weighted projective space
$\mathbb{C}P(1,1,2,\cdots,2)$, which is an orbifold whose singular locus
$\mathbb{C}P(1,1,2,\cdots,2)^{\text{sing}}$ is

$\mathbb{C}P(1,1,2,\cdots,2)^{\text{sing}}=\\{[0:0:0:z_{3}:\cdots:z_{n+1}]\in\mathbb{C}P(1,1,2,\cdots,2)\\}.$

We now define a toric degeneration of a completely integrable system.

######  Definition 3.1.4 ([NNU10, Definition 1.1]).

Let $X^{2n}=(X,\omega)$ be a symplectic manifold and $\Phi:X\to\mathbb{R}^{n}$
a completely integrable system with respect to $\omega$. A toric degeneration
of the completely integrable system is a toric degeneration of $X$

$\pi:\mathfrak{X}\longrightarrow\mathbb{C}$

with the following data:

  * •

a (piecewise smooth) path $\gamma:[0,1]\to\mathbb{C},\ \gamma(0)=1,\
\gamma(1)=0$,

  * •

a continuous map

$\widetilde{\Phi}:\mathfrak{X}|_{\gamma([0,1])}\to\mathbb{R}^{n},$

  * •

a flow $\phi_{t}$ on $\mathfrak{X}|_{\gamma([0,1])}$ which is defined away
from the singular loci $\bigcup_{t\in[0,1]}X_{t}^{\text{sing}}$ where
$X_{t}:=\pi^{-1}(\gamma(t))\\}_{t\in[0,1]}$,

satisfy the following properties:

  1. 1.

For each $t$, $\Phi_{t}:=\widetilde{\Phi}|_{X_{t}}:X_{t}\to\mathbb{R}^{n}$
defines a completely integrable system where for $t=0$, it coincides with the
toric system $\Phi_{0}:X_{0}\to\mathbb{R}^{n}$ and for $t=1$, it coincides
with $\Phi:X\to\mathbb{R}^{n}$.

  2. 2.

Away from the singular loci, the flow $\phi_{t}$ restricts to a
symplectomorphism between $X_{s}$ and $X_{s-t}$ that preserves the integrable
system:

${(X_{t},\omega_{t})}$${(X_{t-s},\omega_{t-s})}$${\mathbb{R}^{n}}$.$\scriptstyle{\Phi_{t}}$$\scriptstyle{\phi_{t}}$$\scriptstyle{\Phi_{t-s}}$
(3.1.3)

We now would like to construct a toric degeneration for the GZ system on
$Q^{n}$. The method of Nishinou–Nohara–Ueda from [NNU10] and Harada–Kaveh from
[HK15] which is based on the gradient-Hamiltonian vector field (due to Ruan
[Rua01]) allows us to do that starting from the toric degeneration of $Q^{n}$
in Example 3.1.1, which is of our main interest.

We roughly review Ruan’s idea. Let $\mathfrak{X}$ be an algebraic variety
equipped with a Kähler form $\widetilde{\omega}$. Let
$\pi:\mathfrak{X}\to\mathbb{C}$ be a morphism (i.e. an algebraic map) and
$\nabla\mathrm{Re}(\pi)$ be the gradient vector field on (the smooth locus of)
$\mathfrak{X}$ with respect to to the Kähler form $\widetilde{\omega}$ where
$\mathrm{Re}(\pi)$ is the real part of the holomorphic function $\pi$. Now,
the gradient-Hamiltonian vector field $V$ on the smooth locus of
$\mathfrak{X}$ is defined by

$V:=-\frac{\nabla\mathrm{Re}(\pi)}{|\nabla\mathrm{Re}(\pi)|^{2}}.$ (3.1.4)

From the normalization, it follows that

$V(\mathrm{Re}(\pi))=-1.$ (3.1.5)

Although the flow of the gradient-Hamiltonian vector field $V$ is not complete
due to not being defined on the singular loci, Harada–Kaveh [HK15] proved that
one can extend the flow continuously on the whole $\mathfrak{X}$.

By using this gradient-Hamiltonian flow applied to the toric degeneration of
$Q^{n}$ in Example 3.1.1, Nishinou–Nohara–Ueda obtained a toric degeneration
for the GZ system on $Q^{n}$.

######  Proposition 3.1.5 ([NNU, Proposition 3.1], [KimA, Proposition 2.6]).

The toric degeneration $\pi:\mathfrak{X}\to\mathbb{C}$ in Example 3.1.1
together with the gradient-Hamiltonian flow $\phi_{t}$ of $\pi$ defines a
toric degeneration of the GZ system on $Q^{n}$ (i.e. one takes the path
$\gamma(t):=1-t$ and $\Phi_{t}:=\Phi_{0}\circ\phi_{t}$.)

The compatibility between the GZ system on $X=Q^{n}$ and the toric system on
$X_{0}=\mathbb{C}P(1,1,2,\cdots,2)$ implies that the monotone GZ torus is sent
to a toric torus through the toric degeneration with the gradient-Hamiltonian
flow, i.e. the monotone GZ torus fiber $T^{n}_{\text{GZ}}=\Phi^{-1}(x_{0})$
satisfies

$\phi_{1}(T^{n}_{\text{GZ}})=\Phi_{0}^{-1}(x_{0}).$ (3.1.6)

The map $\phi_{1}$ allows us to define the vanishing loci.

######  Definition 3.1.6.

Let $S$ be a subset of the singular locus of $X_{0}$, i.e. $S\subset
X_{0}^{\text{sing}}$. The vanishing locus of $S$ is the set
$\phi_{1}^{-1}(S)$.

Now, we get back to the toric degeneration 3.1.1, 3.1.5 for $Q^{n}$ which is
of our specific interest.

######  Proposition 3.1.7.

The Lagrangian sphere

$S^{n}=\\{x\in\mathbb{C}P^{n+1}:x_{0}^{2}+x_{1}^{2}+\cdots+x_{n-1}^{2}=x_{n}^{2},\
x_{j}\in\mathbb{R}\\}$

is the vanishing locus of the set

$\\{[0:0:0:x_{3}:\cdots:x_{n}:ix_{n+1}]\in\mathbb{C}P^{n+1}:x_{j}\in\mathbb{R},\
\forall 3\leqslant j\leqslant n+1\\}.$

Remark 3.1.8. Because of Proposition 3.1.7, we will denote the sphere $S^{n}$
by $S^{n}_{\text{van}}$.

###### Proof of Proposition 3.1.7.

Consider the following anti-symplectic involution:

$\begin{gathered}\tau:\mathbb{C}P^{n+1}\to\mathbb{C}P^{n+1}\\\
[z_{0}:z_{1}:\cdots:z_{n}:z_{n+1}]\mapsto[\overline{z_{0}}:\overline{z_{1}}:\cdots:\overline{z_{n}}:-\overline{z_{n+1}}].\end{gathered}$
(3.1.7)

This restricts to an anti-symplectic involution on $X_{t}$ for each $t$:

$\begin{gathered}\tau_{t}:X_{t}\to X_{t}\\\
[z_{0}:z_{1}:\cdots:z_{n}:z_{n+1}]\mapsto[\overline{z_{0}}:\overline{z_{1}}:\cdots:\overline{z_{n}}:-\overline{z_{n+1}}].\end{gathered}$
(3.1.8)

As the Kähler form $\widetilde{\omega}$ on $\mathfrak{X}$ and the gradient-
Hamiltonian vector field $V$ are both preserved by the anti-symplectic
involution $\tau$,

$\mathrm{Fix}(\tau_{1})=X\cap\mathrm{Fix}(\tau)=\\{x\in\mathbb{C}P^{n+1}:x_{0}^{2}+x_{1}^{2}+\cdots+x_{n-1}^{2}+(ix_{n})^{2}=0,\
x_{j}\in\mathbb{R}\\}$

is sent to

$\begin{gathered}\mathrm{Fix}(\tau_{1-t})=X_{1-t}\cap\mathrm{Fix}(\tau)\\\
=\\{x\in\mathbb{C}P^{n+1}:x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+(1-t)(x_{3}^{2}+\cdots+x_{n-1}^{2}+(ix_{n})^{2})=0,\
x_{j}\in\mathbb{R}\\}\end{gathered}$ (3.1.9)

by $\phi_{t}$:

$\phi_{t}(\mathrm{Fix}(\tau_{1}))=\mathrm{Fix}(\tau_{1-t}).$

Thus, the $t=1$ case will give us that

$\mathrm{Fix}(\tau_{0})=\phi_{1}(\mathrm{Fix}(\tau_{1}))$

where

$\mathrm{Fix}(\tau_{1})=\\{x\in\mathbb{C}P^{n+1}:x_{0}^{2}+x_{1}^{2}+\cdots+x_{n-1}^{2}=x_{n}^{2}\\}=S^{n}$

is the Lagrangian sphere in $Q^{n}$ that we are interested in, and

$\mathrm{Fix}(\tau_{0})=\\{[0:0:0:z_{3}:\cdots:z_{n+1}]\in\mathbb{C}P^{n+1}\\}\subset
X_{0}^{\text{sing}}.$

Thus, $S^{n}$ is the vanishing locus of the singular locus

$\\{[0:0:0:x_{3}:\cdots:x_{n}:ix_{n+1}]\in\mathbb{C}P^{n+1}:x_{j}\in\mathbb{R},\
\forall 3\leqslant j\leqslant n+1\\}.$

∎

Nishinou–Nohara–Ueda applied toric degenerations of GZ systems to compute the
superpotential of the monotone Lagrangian torus fiber for some toric
degenerations in [NNU10, NNU], which enlarged the cases where one can compute
the superpotential. This idea inspired FOOO and Y. Kim to compute the
superpotential of the Chekanov torus in $S^{2}\times S^{2}\simeq Q^{2}$
[FOOO12] and the GZ torus $T^{n}_{\text{GZ}}$ in $Q^{n}$ [KimA], respectively.
We now summarize Y. Kim’s work from [KimA].

According to Y. Kim, the superpotential of $T^{n}_{\text{GZ}}$ takes the
following form:

$W_{T^{n}_{\text{GZ}}}(z)=\frac{1}{z_{n}}+\frac{z_{n}}{z_{n-1}}+\cdots+\frac{z_{2}}{z_{1}}+2z_{2}+z_{1}z_{2}.$
(3.1.10)

This has $n$ different (non-degenerate) critical points:

$(z_{1},z_{2},\cdots,z_{n})=(1,\xi^{-(n-1)},\xi^{-(n-2)},\cdots,\xi^{-2},\xi^{-1})$
(3.1.11)

where $\xi^{n}=4$, and the critical values are as follows:

$W_{T^{n}_{\text{GZ}}}(z)=n\cdot\xi.$ (3.1.12)

This implies that there are $n$ different local systems
$\rho_{j}:H_{1}(T^{n}_{\text{GZ}};\mathbb{Z})\to\mathbb{C}^{\ast},\
j=1,\cdots,n$ so that the Floer homology of $T^{n}_{\text{GZ}}$ with respect
to these local systems are non-zero:

$HF(T^{n}_{\text{GZ}},\rho_{j})\neq 0.$ (3.1.13)

Remark 3.1.9. As Y. Kim points out in [KimA], FOOO’s superpotential for the
Chekanov torus in $S^{2}\times S^{2}\simeq Q^{2}$ [FOOO12] coincides with the
$n=2$ case of his superpotential 3.1.10 for $T^{n}_{\text{GZ}}$. This made his
ask whether the two dimensional GZ torus $T^{2}_{\text{GZ}}$ is Hamiltonian
isotopic to the Chekanov torus (Question 1.3.1). We will answer this in the
positive in Corollary 1.3.3.

###  3.2 Proof

In this section, we prove the main results, namely Theorems A and B, and
Corollary 1.3.1. We first look at how Corollary 1.3.3 follows from Theorem B.

###### Proof of Corollary 1.3.3.

First of all, note that $Q^{2}$ is symplectomorphic to the monotone
$S^{2}\times S^{2}$ (with appropriate normalization), i.e.

$\Phi:Q^{2}\xrightarrow[]{\simeq}S^{2}\times S^{2},$

and through this identification, the polarization $(Q^{2},Q^{1})$ gets
translated to the polarization $(S^{2}\times S^{2},\Delta)$ where $\Delta$
denotes the diagonal sphere

$\Delta:=\\{(x,x)\in S^{2}\times S^{2}\\},$

i.e.

$\Phi:(Q^{2},Q^{1})\xrightarrow[]{\simeq}(S^{2}\times S^{2},\Delta).$ (3.2.1)

Now, our aim is to prove that $\Phi(T^{2}_{\text{GZ}})$ is Hamiltonian
isotopic to the Chekanov torus. From Theorem B, we know that
$T^{2}_{\text{GZ}}$ in $Q^{2}$ is equal to the Biran circle fibration of
$T^{1}_{\text{GZ}}$ in $Q^{1}$, i.e.

$T^{2}_{\text{GZ}}=\widetilde{T^{1}_{\text{GZ}}}.$

The circle $T^{1}_{\text{GZ}}$ in $Q^{1}$ is the equatorial circle
$T^{1}_{\text{eq}}$ via the identification between $Q^{1}$ and $S^{2}$, i.e.

$\Phi(T^{1}_{\text{GZ}})=T^{1}_{\text{eq}}.$

In [OU16], it was proven that for the polarization $(S^{2}\times
S^{2},\Delta)$, the Lagrangian torus obtained as the Biran circle bundle of
the equatorial circle in $\Delta\simeq S^{2}$, i.e.
$\widetilde{T^{1}_{\text{eq}}}$ (which is denoted by $T_{\text{BC}}$ in the
paper), is Hamiltonian isotopic to the Chekanov torus (and to the FOOO torus,
Entov–Polterovich torus, Albers–Frauenfelder torus). From the equivalence of
polarizations 3.2.1, we have

$\displaystyle\Phi(T^{2}_{\text{GZ}})$
$\displaystyle=\Phi(\widetilde{T^{1}_{\text{GZ}}})$ (3.2.2)
$\displaystyle=\widetilde{\Phi(T^{1}_{\text{GZ}})}$
$\displaystyle=\widetilde{T^{1}_{\text{eq}}}=T_{\text{BC}}.$

Thus, we conclude that $\Phi(T^{2}_{\text{GZ}})$ is Hamiltonian isotopic to
the Chekanov torus in $S^{2}\times S^{2}$. ∎

We now prove the first assertion of Theorem B.

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

Recall from Section 2.4 that the monotone GZ torus $T^{n}_{\text{GZ}}$ is
defined as

$T^{n}_{\text{GZ}}:=\Phi_{Q^{n}}^{-1}\left((0,2\cdot\frac{1}{n},2\cdot\frac{2}{n},\cdots,2\cdot\frac{n-1}{n})\right)$
(3.2.3)

where $\Phi_{Q^{n}}$ is

$\begin{gathered}\Phi_{Q^{n}}:Q^{n}\longrightarrow\mathbb{R}^{n}\\\
\Phi_{Q^{n}}(z):=(\lambda_{1}^{(2)},\lambda_{1}^{(3)},\cdots,\lambda_{1}^{(n+1)})(z),\end{gathered}$
(3.2.4)

with $\lambda_{1}^{(k)}:Q^{n}\longrightarrow\mathbb{R}$ such that

$\displaystyle\lambda_{1}^{(2)}(z)$
$\displaystyle=\frac{\lambda}{|z|^{2}}i(z_{1}\overline{z_{2}}-\overline{z_{1}}z_{2}),$
(3.2.5) $\displaystyle(\lambda_{1}^{(k)})^{2}(z)$
$\displaystyle=-\sum_{1\leqslant i<j\leqslant
k}\left(\frac{z_{i}\overline{z_{j}}-\overline{z_{i}}z_{j}}{|z|^{2}}\right)^{2}$
$\displaystyle=\left(\frac{\lambda}{|z|^{2}}\right)^{2}((\sum_{j=1}^{k}|z_{j}|^{2})^{2}-|\sum_{j=1}^{k}z_{j}^{2}|^{2}),\
k\geqslant 3,$ $\displaystyle\lambda:$ $\displaystyle=2.$

From now on, we will be looking at the GZ systems on $Q^{n}$ and its
submanifold $Q^{n-1}=\\{z_{n+1}=0\\}$ so we will introduce the notation

$\lambda_{Q^{n}}^{(k)}:Q^{n}\longrightarrow\mathbb{R}$

to denote the GZ system for $Q^{n}$ and

$\lambda_{Q^{n-1}}^{(k)}:Q^{n-1}\longrightarrow\mathbb{R}$

to denote the GZ system for $Q^{n-1}$.

Now, we prove that the monotone GZ fiber in $Q^{n}$ is equal to the monotone
Biran circle fibration of the monotone GZ fiber in $Q^{n-1}$ for the
polarization, i.e.

$T^{n}_{\text{GZ}}=\widetilde{T^{n-1}_{\text{GZ}}}.$

Note that apriori, the two geometric constructions, namely the toric
degeneration and the Biran decomposition has nothing to do with each other.

The Biran decomposition associated to the polarization $(Q^{n},Q^{n-1})$ is
expressed in terms of coordinate as follows [OU16, Section 4.2]:

$\begin{gathered}DQ^{n-1}\xrightarrow[]{\simeq}Q^{n}\backslash\Delta\\\
(w,\zeta)\mapsto[z_{0}:\cdots:z_{n+1}]=\left[\left(1-\frac{|\zeta|^{2}}{4}\right)w-\frac{\zeta^{2}\overline{w}}{4}:\left(1-\frac{|\zeta|^{2}}{4}\right)^{1/2}\zeta\right],\end{gathered}$
(3.2.6)

First, we focus on the Hamiltonian $\lambda_{Q^{n}}^{(n+1)}$. By using

$\sum_{j=0}^{n+1}z_{j}^{2}=0,$

the expression of $\lambda_{1}^{(n+1)}$ can we rewritten as follows:

$\displaystyle(\lambda_{Q^{n}}^{(n+1)})^{2}$
$\displaystyle=\left(\frac{\lambda}{|z|^{2}}\right)^{2}((\sum_{j=0}^{n}|z_{j}|^{2})^{2}-|\sum_{j=0}^{n}z_{j}^{2}|^{2})$
(3.2.7)
$\displaystyle=\left(\frac{\lambda}{|z|^{2}}\right)^{2}((|z|^{2}-|z_{n+1}|^{2})^{2}-|-z_{n+1}^{2}|^{2})$
$\displaystyle=\left(\frac{\lambda}{|z|^{2}}\right)^{2}(|z|^{4}-2|z|^{2}|z_{n+1}|^{2})$
$\displaystyle=\lambda^{2}(1-2\frac{|z_{n+1}|^{2}}{|z|^{2}}),$

thus, we have

$\lambda_{Q^{n}}^{(n+1)}(z)=\lambda(1-2\frac{|z_{n+1}|^{2}}{|z|^{2}})^{1/2}.$

By using that $|z|^{2}=2$ and that the symplectomorphism 3.2.6, we further
have

$\displaystyle\lambda_{Q^{n}}^{(n+1)}(z)$
$\displaystyle=\lambda(1-2\frac{|z_{n+1}|^{2}}{|z|^{2}})^{1/2}$ (3.2.8)
$\displaystyle=\lambda(1-2\cdot\frac{1}{2}\left(1-\frac{|\zeta|^{2}}{4}\right)|\zeta|^{2})^{1/2}$
$\displaystyle=\lambda(1-|\zeta|^{2}+\frac{|\zeta|^{4}}{4})^{1/2}$
$\displaystyle=\lambda(1-\frac{|\zeta|^{2}}{2}).$

From 2.4.4, the monotone GZ fiber satisfies

$\lambda_{Q^{n}}^{(n+1)}(z)=2\cdot\frac{n-1}{n}$

which is equivalent to

$\lambda(1-\frac{|\zeta|^{2}}{2})=2\cdot\frac{n-1}{n}$

which is (as $\lambda=2$)

$\frac{|\zeta|^{2}}{2}=\frac{1}{n}.$ (3.2.9)

We will see that this is precisely the monotone radius of the Biran circle
bundle construction for $(Q^{n},Q^{n-1})$. Indeed, by [OU16, Section 4.4] (see
also [BC09, Proposition 6.4.1]), the monotone radius $r_{0}$ satisfies

$\frac{r_{0}^{2}}{2}=\frac{\kappa_{Q^{n}}}{2}=\frac{2/n}{2}=\frac{1}{n}$

where $\kappa_{Q^{n}}$ is the monotonicity constant for $Q^{n}$ (see our
convention in Section 2.5). Thus,

$\displaystyle\frac{r_{0}^{2}}{2}$
$\displaystyle=\frac{2\kappa_{L}}{2\kappa_{L}+1}$ (3.2.10)
$\displaystyle=\frac{2\cdot 1/2(n-1)}{2\cdot 1/2(n-1)+1}$
$\displaystyle=\frac{1}{n-1}\cdot\frac{n-1}{n}$ $\displaystyle=\frac{1}{n}.$

Now, we shift our focus to $\lambda_{Q^{n}}^{(k)}$ where $2\leqslant
k\leqslant n$. By using the symplectomorphism 3.2.6, we rewrite the
Hamiltonians $\lambda_{Q^{n}}^{(k)}$ as follows:

$\displaystyle\lambda_{Q^{n}}^{(k)}(z)$
$\displaystyle=-\frac{\lambda}{|z|^{2}}\left(\sum_{1\leqslant i<j\leqslant
k}(z_{i}\overline{z_{j}}-\overline{z_{i}}z_{j})^{2}\right)^{1/2}$ (3.2.11)
$\displaystyle=-\frac{\lambda}{|z|^{2}}\left(\sum_{1\leqslant i<j\leqslant
k}(2i\mathrm{Im}(z_{i}\overline{z_{j}}))^{2}\right)^{1/2}$
$\displaystyle=-\lambda\left(\sum_{1\leqslant i<j\leqslant
k}(i\mathrm{Im}(z_{i}\overline{z_{j}}))^{2}\right)^{1/2}$
$\displaystyle=-\lambda\left(\sum_{1\leqslant i<j\leqslant
k}(\left(1-\frac{|\zeta|^{2}}{2}\right)i\mathrm{Im}(w_{i}\overline{w_{j}}))^{2}\right)^{1/2}$
$\displaystyle=-\lambda\left(1-\frac{|\zeta|^{2}}{2}\right)\left(\sum_{1\leqslant
i<j\leqslant k}(i\mathrm{Im}(w_{i}\overline{w_{j}}))^{2}\right)^{1/2}$
$\displaystyle=\left(1-\frac{|\zeta|^{2}}{2}\right)\lambda_{Q^{n-1}}^{(k)}(w).$

By using 3.2.9, we have

$\displaystyle\lambda_{Q^{n}}^{(k)}(z)$
$\displaystyle=\left(1-\frac{|\zeta|^{2}}{2}\right)\lambda_{Q^{n-1}}^{(k)}(w)$
(3.2.12) $\displaystyle=\left(1-\frac{1}{n}\right)\lambda_{Q^{n-1}}^{(k)}(w)$
$\displaystyle=\frac{n-1}{n}\cdot\lambda_{Q^{n-1}}^{(k)}(w).$

Finally, we compare the monotone GZ tori $T^{n}_{\text{GZ}}$ and
$T^{n-1}_{\text{GZ}}$ by using 3.2.9 and 3.2.12. For $2\leqslant k\leqslant
n$, by 2.4.4 the monotone GZ torus $T^{n}_{\text{GZ}}$ satisfies

$\lambda_{Q^{n}}^{(k)}(z)=2\cdot\frac{k-2}{n}$ (3.2.13)

which, according to 3.2.12, is equivalent to

$\lambda_{Q^{n-1}}^{(k)}(w)=2\cdot\frac{k-2}{n-1}$ (3.2.14)

which is nothing but the description of $T^{n-1}_{\text{GZ}}$. This implies
that $T^{n}_{\text{GZ}}$ is the monotone Biran circle fibration (i.e. the
Biran circle fibration with radius as in 3.2.9) over $T^{n-1}_{\text{GZ}}$,
namely

$T^{n}_{\text{GZ}}=\widetilde{T^{n-1}_{\text{GZ}}}.$

This completes the proof of the first assertion of Theorem B.

∎

We now prove the second assertion of Theorem B.

###### Proof of Theorem B (2).

From Proposition 3.1.7, we know that the vanishing sphere $S^{n}_{\text{van}}$
in $X=X_{1}$ gets mapped to the singular locus of $X_{0}$ by $\phi_{1}$, i.e.

$\phi_{1}(S^{n}_{\text{van}})\subset X_{0}^{\text{sing}}.$ (3.2.15)

On the other hand, we have seen in 3.1.6 that from Proposition 3.1.5, we get

$\phi_{1}(T^{n}_{\text{GZ}})=\Phi_{0}^{-1}(x_{0}),$ (3.2.16)

which implies

$\phi_{1}(T^{n}_{\text{GZ}})\subset X_{0}\backslash X_{0}^{\text{sing}}.$
(3.2.17)

The properties 3.2.15 and 3.2.17 imply

$S^{n}_{\text{van}}\cap T^{n}_{\text{GZ}}=\emptyset.$

∎

##  4 Proofs of Theorem A, B (3)

###  4.1 Proof–Part 1

###### Proof of Theorem B (3).

We use the following theorem due to Auroux–Kontsevich–Seidel.

######  Theorem 4.1.1 ([Aur07, Section 6],[Sh16, Lemma 2.7, Proposition 2.9]).

Let $X$ be a closed monotone symplectic manifold. The (length 0) closed-open
map

$\mathcal{CO}^{0}:QH(X;\Lambda_{\text{Nov}})\longrightarrow HF(L,\rho)$

has the following properties:

  1. 1.

For $c_{1}:=c_{1}(TX)$,

$\mathcal{CO}^{0}(c_{1})=W_{L}(\rho)\cdot 1_{(L,\rho)}$

where $W_{L}(\rho)$ is the value of the superpotential of the Lagrangian $L$
equipped with a local system $\rho$.

  2. 2.

Consider the map

$c_{1}\ast-:QH(X;\Lambda_{\text{Nov}})\longrightarrow
QH(X;\Lambda_{\text{Nov}})$ (4.1.1)

and split $QH(X;\Lambda_{\text{Nov}})$ into generalized eigenspaces with
respect to $c_{1}\ast-$:

$QH(X;\Lambda_{\text{Nov}})=\bigoplus_{w}QH(X;\Lambda_{\text{Nov}})_{w}$
(4.1.2)

where $w$ is an eigenvalue of $c_{1}\ast-$. The map

$\mathcal{CO}^{0}:QH(X;\Lambda_{\text{Nov}})_{w}\longrightarrow HF(L,\rho)$

is zero if $w\neq W_{L}(\rho)$ and is a unital homomorphism if
$w=W_{L}(\rho)$.

Remark 4.1.2. Note that in Theorem 4.1.1, it is important that we take the
universal Novikov field

$\Lambda_{\text{Nov}}:=\\{\sum_{j=1}^{\infty}a_{j}T^{\lambda_{j}}:a_{j}\in\mathbb{C},\lambda_{j}\in\mathbb{R},\lim_{j\to+\infty}\lambda_{j}=+\infty\\}$

for the quantum cohomology instead of the field of the Laurent series

$\Lambda_{\text{Lau}}:=\\{\sum_{k\geqslant
k_{0}}b_{k}t^{k}:k_{0}\in\mathbb{Z},b_{k}\in\mathbb{C}\\},$

as $\Lambda_{\text{Nov}}$ is algebraically closed and $\Lambda_{\text{Lau}}$
is not.

Now, we apply Theorem 4.1.1 to $Q^{n}$. The eigenvalues of the map

$c_{1}\ast-:QH(Q^{n};\Lambda_{\text{Nov}})\longrightarrow
QH(Q^{n};\Lambda_{\text{Nov}})$

are $0$ and $n\cdot\xi_{j},\ j=1,\cdots,n$ where
$\\{\xi_{j}\\}_{j=1,\cdots,n}$ are solutions to $\xi^{n}=4$ (see [Sh16,
Corollary 1.14]). Thus, $QH(X;\Lambda_{\text{Nov}})$ splits into a direct sum
of generalized eigenspaces with respect to $c_{1}\ast-$ as follows:

$QH(Q^{n};\Lambda_{\text{Nov}})=\bigoplus_{1\leqslant j\leqslant
n}QH(X;\Lambda_{\text{Nov}})_{n\cdot\xi_{j}}\oplus
QH(X;\Lambda_{\text{Nov}})_{0}.$ (4.1.3)

We decompose the unit $1_{Q^{n}}$ with respect to this split:

$1_{Q^{n}}=\sum_{1\leqslant j\leqslant n}e_{n\cdot\xi_{j}}+e_{0}.$ (4.1.4)

In view of the superpotential computation for $T^{n}_{\text{GZ}}$ of Y. Kim
3.1.11, 3.1.12, Theorem 4.1.1 implies that

$\begin{gathered}\mathcal{CO}^{0}:QH(X;\Lambda_{\text{Nov}})_{n\cdot\xi_{i}}\longrightarrow
HF(T^{n}_{\text{GZ}},\rho_{j})\\\
\mathcal{CO}^{0}(e_{n\cdot\xi_{i}})=\delta_{i,j}\cdot
1_{(T^{n}_{\text{GZ}},\rho_{j})},\end{gathered}$ (4.1.5)

where

$\delta_{i,j}=\begin{cases}1,\ \text{if $i=j$}\\\ 0,\ \text{if $i\neq
j$},\end{cases}$ (4.1.6)

and

$\begin{gathered}\mathcal{CO}^{0}:QH(X;\Lambda_{\text{Nov}})_{0}\longrightarrow
HF(S^{n}_{\text{van}})\\\
\mathcal{CO}^{0}(e_{0})=1_{S^{n}_{\text{van}}}.\end{gathered}$ (4.1.7)

The heaviness criterion Theorem 2.1.10 applied to 4.1.5 and 4.1.7 imply that
$T^{n}_{\text{GZ}}$ is $\zeta_{e_{n\cdot\xi_{j}}}$-superheavy for all
$1\leqslant j\leqslant n$ and $S^{n}_{\text{van}}$ is $\zeta_{e_{0}}$-heavy.
Here, we have used that $QH(X;\Lambda_{\text{Nov}})_{n\cdot\xi_{j}}$ is a
field for any $1\leqslant j\leqslant n$ and $QH(X;\Lambda_{\text{Nov}})_{0}$
is not, if $n$ is even; see Section 4.2 for this in more detail.

###  4.2 The Laurent and Novikov fields

The aim of this section is to clarify the relation between spectral invariants
defined with respect to different coefficient fields. Note that
$\zeta_{\pm}:\widetilde{\mathrm{Ham}}(Q^{n})\to\mathbb{R}$ are defined with
the Laurent coefficients

$\Lambda_{\text{Lau}}=\\{\sum_{k\geqslant
k_{0}}b_{k}t^{k}:k_{0}\in\mathbb{Z},b_{k}\in\mathbb{C}\\},$

while we have worked entirely with the Novikov coefficients

$\Lambda_{\text{Nov}}=\\{\sum_{j=1}^{\infty}a_{j}T^{\lambda_{j}}:a_{j}\in\mathbb{C},\lambda_{j}\in\mathbb{R},\lim_{j\to+\infty}\lambda_{j}=+\infty\\}$

in Section 4.1 (see Remark 4.1.1). Recall that $\Lambda_{\text{Lau}}$ can be
embedded to $\Lambda_{\text{Nov}}$ by the inclusion given by $t\mapsto
T^{\lambda_{0}}$, and this inclusion extends to

$i:QH(X;\Lambda_{\text{Lau}})\hookrightarrow QH(X;\Lambda_{\text{Nov}}).$

This subtlety of the choice of the coefficient field, which was analyzed in
[Kaw22A, Section 4.2, 4.5], might seem technical but turns out to be very
useful and important. To summarize the points from [Kaw22A, Section 4.2, 4.5],
to work with spectral invariants, e.g. Entov–Polterovich quasimorphisms, it is
more convenient to work with the Laurent coefficient while Lagrangian Floer
theory is more suited to work with the universal Novikov field, e.g. Theorem
4.1.1.

We will focus on the case of $Q^{n}$. With the Laurent coefficients, the
quantum cohomology splits into a direct sum of two fields

$QH(Q^{n};\Lambda_{\text{Lau}})=Q_{+}\oplus Q_{-},$ (4.2.1)

where the unit $1_{Q^{n}}$ splits as

$\begin{gathered}1_{Q^{n}}=e_{+}+e_{-},\\\ e_{\pm}:=\frac{1_{Q^{n}}\pm
PD([pt])t}{2}.\end{gathered}$ (4.2.2)

However, when we consider $QH(Q^{n};\Lambda_{\text{Nov}})$, $Q_{+}$ and
$Q_{-}$ further splits into finer fields; $Q_{+}$ splits as a direct sum of
$n$ fields

$Q_{+}=\bigoplus_{1\leqslant i\leqslant n}Q_{+,i},$

and $Q_{-}$ splits as

$Q_{-}=\begin{cases}Q_{-},\text{ if $n$ is odd}\\\ Q_{-,1}\oplus
Q_{-,2},\text{ if $n$ is even}.\end{cases}$ (4.2.3)

Thus, we have

$QH(Q^{n};\Lambda_{\text{Nov}})=\begin{cases}\left(\bigoplus_{1\leqslant
i\leqslant n}Q_{+,i}\right)\oplus Q_{-},\text{ if $n$ is odd}\\\
\left(\bigoplus_{1\leqslant i\leqslant
n}Q_{+,i}\right)\oplus\left(\bigoplus_{j=1,2}Q_{-,j}\right),\text{ if $n$ is
even}\end{cases}$ (4.2.4)

where the unit $1_{Q^{n}}$ splits as

$1_{Q^{n}}=\begin{cases}\sum_{1\leqslant i\leqslant n}e_{+,i}+e_{-},\text{ if
$n$ is odd}\\\ \sum_{1\leqslant i\leqslant
n}e_{+,i}+\sum_{j=1,2}e_{-,j},\text{ if $n$ is even}.\end{cases}$ (4.2.5)

$\begin{gathered}1_{Q^{n}}=\sum_{1\leqslant i\leqslant
n}e_{+,i}+\sum_{j=1,2}e_{-,j},\\\ i(e_{+})=\sum_{1\leqslant i\leqslant
n}e_{+,i},\\\ i(e_{-})=\sum_{j=1,2}e_{-,j}.\end{gathered}$ (4.2.6)

Remark 4.2.1. When $n$ is even, the idempotents $e_{-,j},\ j=1,2$ are given by

$e_{-,j}:=\frac{1}{2}\left(1_{Q^{n}}\pm PD([S^{n}])\cdot
T^{\lambda_{0}/2}-PD([pt])\cdot T^{\lambda_{0}}\right)$ (4.2.7)

where $[S^{n}]$ is the homology class represented by the vanishing sphere
$S^{n}_{\text{van}}$. The precise expression for $\\{e_{+,i}\\}$ will be
omitted but can be obtained similarly.

In Section 4.1, we have considered yet another split of
$QH(Q^{n};\Lambda_{\text{Nov}})$, namely the eigenvalue decomposition 4.1.3:

$QH(Q^{n};\Lambda_{\text{Nov}})=\bigoplus_{1\leqslant j\leqslant
n}QH(X;\Lambda_{\text{Nov}})_{n\cdot\xi_{j}}\oplus
QH(X;\Lambda_{\text{Nov}})_{0},$ (4.2.8)

and the unit $1_{Q^{n}}$ is decomposed as

$1_{Q^{n}}=\sum_{1\leqslant j\leqslant n}e_{n\cdot\xi_{j}}+e_{0}.$ (4.2.9)

One can check that $c_{1}\ast e_{+,i}=n\cdot\xi_{i}\cdot e_{+,i}$ and
$c_{1}\ast e_{-,j}=0$ and thus, the relation between the splits 4.2.1, 4.2.4
and 4.2.8 is

$\begin{gathered}e_{n\cdot\xi_{i}}=e_{+,i},\\\ e_{+}=\sum_{1\leqslant
i\leqslant n}e_{+,i},\end{gathered}$ (4.2.10)

and

$\begin{gathered}e_{-}=e_{0},\\\ e_{0}=\sum_{j=1,2}e_{-,j},\text{ if $n$ is
even}\end{gathered}$ (4.2.11)

which means that $QH(X;\Lambda_{\text{Nov}})_{n\cdot\xi_{j}}$ is a field for
any $1\leqslant j\leqslant n$ and $QH(X;\Lambda_{\text{Nov}})_{0}$ is not (it
is a direct sum of two fields). All these are not trivial but easy to see from
[Sh16, Section 7.4].

In [Kaw22A, Lemma 31, 32] (see also [Kaw22A, Proof of Theorem 6, Remark 44]),
the author studied the relation between spectral invariants of a class seen as
elements of quantum cohomology with different coefficient fields and the lemma
implies the following.

######  Theorem 4.2.2.

We have the following relation between asymptotic spectral invariants:

$\begin{gathered}\zeta_{e_{+}}=\zeta_{n\cdot\xi_{i}}=\zeta_{e_{+,i}},\
i=1,\cdots,n,\\\ \zeta_{e_{-}}=\zeta_{e_{0}}=\zeta_{e_{-,j}},\
j=1,2.\end{gathered}$ (4.2.12)

###  4.3 Proof–Part 2

In this section, we will combine results from Sections 4.1 and 4.2 to complete
the proof of Theorem B.

In Section 4.1, we have shown that $T^{n}_{\text{GZ}}$ is
$\zeta_{e_{n\cdot\xi_{j}}}$-superheavy for any $1\leqslant j\leqslant n$ and
$S^{n}_{\text{van}}$ is $\zeta_{e_{0}}$-heavy and thus, Theorem 4.2.2 implies
that $T^{n}_{\text{GZ}}$ is $\zeta_{e_{+}}$-superheavy and
$S^{n}_{\text{van}}$ is $\zeta_{e_{-}}$-heavy. As $e_{-}$ is a unit of a field
factor of $QH(X,\Lambda_{\text{Lau}})$, $\zeta_{e_{-}}$ is homogeneous and
thus $S^{n}_{\text{van}}$ is actually $\zeta_{e_{-}}$-superheavy. We have
completed the proof of Theorem B. ∎

###  4.4 Proof of Theorem A

We prove Theorem A by using Theorem B.

###### Proof of Theorem A.

From Theorem B, the Lagrangian sphere $S^{n}_{\text{van}}$ and the monotone GZ
torus $T^{n}_{\text{GZ}}$ are $e_{-}$-superheavy and $e_{+}$-superheavy,
respectively. Also by Theorem B, $S^{n}_{\text{van}}$ and $T^{n}_{\text{GZ}}$
are disjoint so one can take a Hamiltonian $H$ on $Q^{n}$ such that it is
time-independent and its restriction to $S^{n}_{\text{van}}$ and
$T^{n}_{\text{GZ}}$ are $0$ and $1$, respectively, i.e.

$H|_{S^{n}_{\text{van}}}\equiv 0,\ H|_{T^{n}_{\text{GZ}}}\equiv 1.$

By Proposition 2.1.8, we have

$\zeta_{-}(H)=0,\ \zeta_{+}(H)=1,$

which implies

$\zeta_{-}\neq\zeta_{+}.$

We have proven Theorem A. ∎

##  5 Proof of Theorem C

In this section, we prove Theorem C. Unlike the case of the quadrics, the
relevant geometry of the toric degeneration (,i.e. the geometry of the
vanishing cycles) was already worked out by Y. Sun in [Sun20], so we focus
only on how to use Y. Sun’s results to obtain Theorem C.

###### Proof of Theorem C.

We start by summarizing Y. Sun’s result. In [Sun20], Y. Sun considered toric
degenerations for the del Pezzo surfaces to compute the superpotentials of the
Lagrangian tori therein. Y. Sun’s superpotentials (for the monotone torus
fiber without bulk-deformation) have non-degenerate critical points and thus,
the monotone Lagrangian torus fiber has non-zero self-Floer homology. Y. Sun
also classifies the singularities that appear in each toric degeneration, and
the result is as follows:

  1. 1.

For $\mathbb{D}_{2}$, there is one $A_{1}$-singularity ([Sun20, Section 3.4]).

  2. 2.

For $\mathbb{D}_{3}$, there is one $A_{1}$-singularity and one
$A_{2}$-singularity ([Sun20, Appendix B, case of $X_{3}$]).

  3. 3.

For $\mathbb{D}_{4}$, there is one $A_{1}$-singularity and one
$A_{2}$-singularity ([Sun20, Appendix B, case of $X_{6}$]).

Remark 5.0.1.

  1. 1.

Y. Sun considers $\mathbb{D}_{5}$ as well, but we will not deal with this, as
$QH(\mathbb{D}_{5})$ is not semi-simple.

  2. 2.

For $\mathbb{D}_{3}$ and $\mathbb{D}_{4}$, there are several toric
degenerations; see [Sun20, Appendix B, cases of $X_{3},X_{4},X_{5}$], [Sun20,
Appendix B, case of $X_{6},X_{7}$], respectively.

  3. 3.

One difference between the case of the quadrics and the case of del Pezzo
surfaces is that in the former, the singular locus was not a discrete set,
while in the latter the singular locus is discrete.

By looking at the vanishing cycles of the $A_{m}$-singularities, one obtains a
chain of Lagrangian spheres (i.e. an $A_{m}$-configuration), and in particular
we have the following:

  1. 1.

In $\mathbb{D}_{2}$, there is one Lagrangian sphere $S_{1}$ that is disjoint
from the monotone Lagrangian torus.

  2. 2.

In $\mathbb{D}_{3}$, there is one Lagrangian sphere $S_{1}$ (from the
$A_{1}$-singularity) and one Lagrangian sphere $S_{2}$ (from the
$A_{2}$-singularity) that are both disjoint to each other and also from the
monotone Lagrangian torus.

  3. 3.

In $\mathbb{D}_{4}$, there is one Lagrangian sphere $S_{1}$ (from the
$A_{1}$-singularity) and one Lagrangian sphere $S_{2}$ (from the
$A_{2}$-singularity) that are both disjoint to each other and also from the
monotone Lagrangian torus.

Remark 5.0.2. The properties of the spectral invariants of the Lagrangian
spheres forming an $A_{m}$ configuration is studied in [KawB]. The relation
between the spheres and the corresponding idempotents is also explained in
more detail in [KawB].

Note that all the Lagrangian spheres that appear above have non-zero Floer
homology, c.f. [BM15, Proposition A.2]. Now, using that $QH(\mathbb{D}_{k})$
for $k=2,3,4$ are semi-simple, c.f. [BM04], and by employing an analogous
argument to Section 4, we have the following:

  1. 1.

For $k=2$, there exist two units of field factors of $QH(\mathbb{D}_{2})$
$e_{1},e_{2}$ for which the Lagrangian sphere $S_{1}$ is $e_{1}$-superheavy
and the monotone Lagrangian torus is $e_{2}$-superheavy.

  2. 2.

For each $k=3,4$, there exist three units of field factors of
$QH(\mathbb{D}_{k})$ $e_{1},e_{2},e_{3}$ for which the Lagrangian sphere
$S_{1}$ is $e_{1}$-superheavy, the Lagrangian sphere $S_{2}$ is
$e_{2}$-superheavy, and the monotone Lagrangian torus is $e_{3}$-superheavy.

As all the involved Lagrangians are disjoint, we have

  1. 1.

For $k=2$, $\zeta_{e_{1}}\neq\zeta_{e_{2}}$.

  2. 2.

For $k=3,4$, $\zeta_{e_{i}}\neq\zeta_{e_{j}}$, $1\leqslant i<j\leqslant 3$.

This finishes the proof of Theorem C.

∎

Remark 5.0.3. For the del Pezzo surfaces, unlike the quadrics, we did not have
to deal with the change of coefficient as in Section 4.2. This is because the
minimal Chern number of the del Pezzo surfaces $\mathbb{D}_{k},\ k\geqslant 1$
is $1$, and thus Laurent coefficients and Novikov coefficients will give the
same split in quantum cohomology.

##  6 Proofs of Theorem D and Applications

In this section, we prove Theorems Theorem D, E and F.

We first prove Theorem E.

###### Proof of Theorem E.

First, recall the following result from [Kaw22A]:

######  Theorem 6.0.1 ([Kaw22A, Theorem 22]).

Let $(X,\omega)$ be a monotone symplectic manifold. Assume its quantum
cohomology ring $QH^{\ast}(X;\Lambda)$ is semi-simple i.e.

$QH^{\ast}(X;\Lambda)=Q_{1}\oplus Q_{2}\oplus\cdots\oplus Q_{l}$

for some $l\in\mathbb{N}$ where each $Q_{j}$ is a field. We decompose the unit
$1_{X}\in QH^{\ast}(X;\Lambda)$ into a sum of idempotents with respect to this
split:

$1_{X}=e_{1}+e_{2}+\cdots+e_{l},\ e_{j}\in Q_{j}.$

Then for any $i,j\in\\{1,2,\cdots,l\\},$

$\mu:=\zeta_{e_{i}}-\zeta_{e_{j}}$

descends from $\widetilde{\mathrm{Ham}}(X,\omega)$ to $\mathrm{Ham}(X,\omega)$
and defines a homogeneous quasimorphism on $\mathrm{Ham}(X,\omega)$ which is
$C^{0}$-continuous i.e.

$\mu:(\mathrm{Ham}(X,\omega),d_{C^{0}})\to\mathbb{R}$

is continuous. Moreover, it is Hofer Lipschitz continuous.

Now, when $X=Q^{n}$, we take

$\mu:=\zeta_{+}-\zeta_{-}.$

Theorem 6.0.1 and Theorem A imply Theorem E.

When $X=\mathbb{D}_{k},\ k=2,3,4$, we take

$\mu_{i,j}:=\zeta_{e_{i}}-\zeta_{e_{j}}$

with $i\neq j$, where the Entov–Polterovich quasimorphisms
$\\{\zeta_{e_{i}}\\}$ are taken as in Section 5. Theorem 6.0.1 and Theorem C
imply Theorem E.

∎

Remark 6.0.2. Note that the above quasimorphism extends to the $C^{0}$-closure
of $\mathrm{Ham}(X)$, so $\mu=\zeta_{+}-\zeta_{-}$ and
$\mu_{i,j}=\zeta_{e_{i}}-\zeta_{e_{j}}$ give quasimorphisms on
$\overline{\mathrm{Ham}}^{C^{0}}(Q^{n})$ and
$\overline{\mathrm{Ham}}^{C^{0}}(\mathbb{D}_{k}),\ k=2,3,4$, respectively.

We prove Theorem D.

###### Proof of Theorem D.

When $k=3,4$, in the final step of the proof of Theorem E, we had

$\begin{gathered}\mu_{1,2},\mu_{2,3}:\mathrm{Ham}(\mathbb{D}_{k})\to\mathbb{R}\\\
\mu_{1,2}\neq\mu_{2,3}.\end{gathered}$ (6.0.1)

As $\mu_{1,2}$ and $\mu_{2,3}$ are both Hofer Lipschitz continuous and
linearly independent, we conclude that $\mathrm{Ham}(\mathbb{D}_{k})$ is not
quasi-isometric to the real line $\mathbb{R}$ with respect to the Hofer
metric, which proves Theorem D. ∎

We prove Theorem F.

###### Proof of Theorem F.

We start with the following claim.

######  Claim 6.0.3.

Suppose $QH(X)$ is semi-simple and denote the split of it by

$QH(X;\Lambda)=Q_{1}\oplus Q_{2}\oplus\cdots\oplus Q_{l}.$

For any Hamiltonian $H$, we have

$\zeta_{1_{X}}(H)=\max_{1\leqslant j\leqslant l}\zeta_{e_{j}}(H).$

Claim 6.0.3 implies that for any $\phi\in\mathrm{Ham}(X)$,

$\overline{\gamma}(\phi)=\max_{1\leqslant i,j\leqslant l}\mu_{i,j}(\phi)$
(6.0.2)

where

$\mu_{i,j}(\phi_{H})=\mu_{i,j}(H):=\zeta_{e_{i}}(H)-\zeta_{e_{j}}(H).$

In fact,

$\displaystyle\overline{\gamma}(\phi_{H})$
$\displaystyle=\zeta_{1_{X}}(H)+\zeta_{1_{X}}(\overline{H})$ (6.0.3)
$\displaystyle=\max_{1\leqslant i\leqslant l}\zeta_{e_{i}}(H)+\max_{1\leqslant
j\leqslant l}\zeta_{e_{j}}(\overline{H})$ $\displaystyle=\max_{1\leqslant
i,j\leqslant l}\\{\zeta_{e_{i}}(H)-\zeta_{e_{j}}(H)\\}$
$\displaystyle=\max_{1\leqslant i,j\leqslant l}\mu_{i,j}(\phi).$

By Theorem 6.0.1, we know that for each $i,j$, $\mu_{i,j}$ is
$C^{0}$-continuous and therefore, $\overline{\gamma}$ is $C^{0}$-continuous.
Theorem E implies $\overline{\gamma}\neq 0$ for $X=Q^{n}$ and
$X=\mathbb{D}_{k},\ k=2,3,4$. ∎

We prove Claim 6.0.3.

###### Proof of Claim 6.0.3.

We first prove $\zeta_{1_{X}}(H)\geqslant\max_{1\leqslant j\leqslant
l}\zeta_{e_{j}}(H)$ for any $H$. By the triangle inequality, we get

$c(H,1_{X})+\nu(e_{j})\geqslant c(H,e_{j})$

for any $j$ and Hamiltonian $H$ and thus

$\zeta_{1_{X}}(H)\geqslant\zeta_{e_{j}}(H)$

for any $j$ and Hamiltonian $H$. Therefore,

$\zeta_{1_{X}}(H)\geqslant\max_{1\leqslant j\leqslant l}\zeta_{e_{j}}(H)$

for any Hamiltonian $H$.

Next, we prove $\zeta_{1_{X}}(H)\leqslant\max_{1\leqslant j\leqslant
l}\zeta_{e_{j}}(H)$. A standard property of spectral invariants implies

$c(H,1_{X})\leqslant\max_{1\leqslant j\leqslant l}c(H,e_{j})$

as $1_{X}=e_{1}+e_{2}+\cdots+e_{l}$ and thus

$\zeta_{1_{X}}(H)\leqslant\max_{1\leqslant j\leqslant l}\zeta_{e_{j}}(H)$

for any Hamiltonian $H$. This completes the proof of Claim 6.0.3.

∎

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Yusuke Kawamoto, Institute for Mathematical Research (FIM), Rämistrasse 101,
8092 Zürich Switzerland

E-mail address<EMAIL_ADDRESS><EMAIL_ADDRESS>