\typearea

Floer cohomologies of non-torus fibers
of the Gelfand-Cetlin system

Yuichi Nohara and Kazushi Ueda

Abstract

The Gelfand-Cetlin system has non-torus Lagrangian fibers on some of the boundary strata of the moment polytope. We compute Floer cohomologies of such non-torus Lagrangian fibers in the cases of the 3-dimensional full flag manifold and the Grassmannian of 2-planes in a 4-space.

1 Introduction

Let $P$ be a parabolic subgroup of $\operatorname{GL}(n,\mathbb{C})$ and $F:=\operatorname{GL}(n,\mathbb{C})/P$ be the associated flag manifold. The Gelfand-Cetlin system, introduced by Guillemin and Sternberg [GS83], is a completely integrable system

\Phi:F\longrightarrow\mathbb{R}^{(\dim_{\mathbb{R}}F)/2},

i.e., a set of functionally independent and Poisson commuting functions. The image $\Delta=\Phi(F)$ is a convex polytope called the Gelfand-Cetlin polytope, and $\Phi$ gives a Lagrangian torus fibration structure over the interior $\operatorname{Int}\Delta$ of $\Delta$ . Unlike the case of toric manifolds where the fibers over the relative interior of a $d$ -dimensional face of the moment polytope are $d$ -dimensional isotropic tori, the Gelfand-Cetlin system has non-torus Lagrangian fibers over the relative interiors of some of the faces of $\Delta$ .

Let $(X,\omega)$ be a compact toric manifold of $\dim_{\mathbb{C}}X=N$ , and $\Phi:X\to\mathbb{R}^{N}$ be the toric moment map with the moment polytope $\Delta=\Phi(X)$ . For an interior point ${\boldsymbol{u}}\in\operatorname{Int}\Delta$ , let $L({\boldsymbol{u}})$ denote the Lagrangian torus fiber $\Phi^{-1}({\boldsymbol{u}})$ . Lagrangian intersection Floer theory endows the cohomology group $H^{*}(L({\boldsymbol{u}});\Lambda_{0})$ over the Novikov ring

\Lambda_{0}:=\left\{\left.\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}\,\right|\,a_{i}\in\mathbb{C},\ \lambda_{i}\geq 0,\ \lim_{i\to\infty}\lambda_{i}=\infty\right\}

with a structure $\{\mathfrak{m}_{k}\}_{k\geq 0}$ of a unital filtered $A_{\infty}$ -algebra [FOOO09]. Let $\Lambda$ and $\Lambda_{+}$ be the quotient field and the maximal ideal of the local ring $\Lambda_{0}$ respectively. An odd-degree element $b\in H^{\mathrm{odd}}(L({\boldsymbol{u}});\Lambda_{0})$ is said to be a bounding cochain if it satisfies the Maurer-Cartan equation

\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b^{\otimes k})=0.

(1.1)

A solution $b\in H^{\mathrm{odd}}(L({\boldsymbol{u}});\Lambda_{0})$ to the weak Maurer-Cartan equation

\displaystyle\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b^{\otimes k})\equiv 0\mod\Lambda_{0}\,\mathbf{e}_{0}

(1.2)

is called a weak bounding cochain, where $\mathbf{e}_{0}$ is the unit in $H^{*}(L({\boldsymbol{u}});\Lambda_{0})$ . The set of weak bounding cochains will be denoted by $\widehat{\mathcal{M}}_{\mathrm{weak}}(L({\boldsymbol{u}}))$ . The potential function is a map $\mathfrak{PO}\colon\widehat{\mathcal{M}}_{\mathrm{weak}}(L({\boldsymbol{u}}))\to\Lambda_{0}$ defined by

\displaystyle\sum_{k=0}^{\infty}\mathfrak{m}_{k}(b,\ldots,b)=\mathfrak{PO}(b)\mathbf{e}_{0}.

(1.3)

A weak bounding cochain gives a deformed filtered $A_{\infty}$ -algebra whose $A_{\infty}$ -operations are given by

\displaystyle\mathfrak{m}^{b}_{k}(x_{1},\ldots,x_{k})=\sum_{m_{0}=0}^{\infty}\cdots\sum_{m_{k}=0}^{\infty}\mathfrak{m}_{m_{0}+\cdots+m_{k}+k}(b^{\otimes m_{0}}\otimes x_{1}\otimes b^{\otimes m_{1}}\otimes\cdots\otimes x_{k}\otimes b^{\otimes m_{k}}).

(1.4)

The weak Maurer-Cartan equation implies that $\mathfrak{m}_{1}^{b}$ squares to zero, and the deformed Floer cohomology is defined by

\displaystyle\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b),(L({\boldsymbol{u}}),b);\Lambda_{0})=\left.\operatorname{Ker}(\mathfrak{m}_{1}^{b})\right/\operatorname{Im}(\mathfrak{m}_{1}^{b}).

(1.5)

More generally, one can deform the Floer differential $\mathfrak{m}_{1}$ by

\delta_{b_{0},b_{1}}(x)=\sum_{k_{0},k_{1}\geq 0}\mathfrak{m}_{k_{0}+k_{1}+1}(\underbrace{b_{0},\dots,b_{0}}_{k_{0}},x,\underbrace{b_{1},\dots,b_{1}}_{k_{1}})

(1.6)

for a pair $(b_{0},b_{1})$ of weak bounding cochains with $\mathfrak{PO}(b_{0})=\mathfrak{PO}(b_{1})$ . The Floer cohomology of the pair $((L({\boldsymbol{u}}),b_{0}),(L({\boldsymbol{u}}),b_{1}))$ is defined by

\displaystyle\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b_{0}),(L({\boldsymbol{u}}),b_{1});\Lambda_{0})=\left.\operatorname{Ker}(\delta_{b_{0},b_{1}})\right/\operatorname{Im}(\delta_{b_{0},b_{1}}).

(1.7)

If the toric manifold $X$ is Fano, then the following hold [FOOO10]:

•

$H^{1}(L({\boldsymbol{u}});\Lambda_{0})$ is contained in $\widehat{\mathcal{M}}_{\mathrm{weak}}(L({\boldsymbol{u}}))$ .

•

The potential function $\mathfrak{PO}$ on

\bigcup_{{\boldsymbol{u}}\in\operatorname{Int}\Delta}H^{1}(L({\boldsymbol{u}});\Lambda_{0}/2\pi\sqrt{-1}\mathbb{Z})\cong\operatorname{Int}\Delta\times(\Lambda_{0}/2\pi\sqrt{-1}\mathbb{Z})^{N}

(1.8)

can be considered as a Laurent polynomial, which can be identified with the superpotential of the Landau-Ginzburg mirror of $X$ .

•

Each critical point of $\mathfrak{PO}$ corresponds to a pair $({\boldsymbol{u}},b)$ such that the deformed Floer cohomology $\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b),(L({\boldsymbol{u}}),b);\Lambda)$ over the Novikov field $\Lambda$ is non-trivial.
•

If the deformed Floer cohomology group over the Novikov field is non-trivial, then it is isomorphic to the classical cohomology group;

$\displaystyle\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b),(L({\boldsymbol{u}}),b);\Lambda)\cong H^{*}(T^{N};\Lambda).$ (1.9)
•

The quantum cohomology ring $\mathop{QH}\nolimits(X;\Lambda)$ is isomorphic to the Jacobi ring $\operatorname{Jac}(\mathfrak{PO})$ of the potential function.

In particular, the number of pairs $(L({\boldsymbol{u}}),b)$ with nontrivial Floer cohomology coincides with $\operatorname{rank}\mathop{QH}\nolimits(X;\Lambda)=\operatorname{rank}H^{*}(X;\Lambda).$

Nishinou and the authors [NNU10] introduced the notion of a toric degeneration of an integrable system, and used it to compute the potential function of Lagrangian torus fibers of the Gelfand-Cetlin system. The resulting potential function can be considered as a Laurent polynomial just as in the toric Fano case, which can be identified with the superpotential of the Landau-Ginzburg mirror of the flag manifold given in [Giv97, BCFKvS00]. In contrast to the toric case, the rank of $H^{*}(F;\Lambda)$ is greater in general than the rank of the Jacobi ring $\operatorname{Jac}(\mathfrak{PO})$ , and hence than the number of Lagrangian torus fibers with non-trivial Floer cohomology. In the case of the 3-dimensional flag manifold $\operatorname{Fl}(3)$ , the potential function has six critical points, which is equal to the rank of $H^{*}(\operatorname{Fl}(3);\Lambda)$ . Similarly, the potential function for the Grassmannian $\operatorname{Gr}(2,5)$ of 2-planes in $\mathbb{C}^{5}$ has ten critical points, which is equal to the rank of $H^{*}(\operatorname{Gr}(2,5);\Lambda)$ . On the other hand, the number of critical points of the potential function for the Grassmannian $\operatorname{Gr}(2,4)$ of 2-planes in $\mathbb{C}^{4}$ is four, which is less than the rank of $H^{*}(\operatorname{Gr}(2,4);\Lambda)$ , which is six.

In this paper, we study non-torus Lagrangian fibers of the Gelfand-Cetlin system over the boundary of the Gelfand-Cetlin polytope in the cases of $\operatorname{Fl}(3)$ and $\operatorname{Gr}(2,4)$ . The main results are the following:

Theorem 1.1.

Let $\Phi\colon\operatorname{Fl}(3)\to\mathbb{R}^{3}$ be the Gelfand-Cetlin system with the Gelfand-Cetlin polytope $\Delta=\Phi(\operatorname{Fl}(3))$ .

1.

There exists a vertex ${\boldsymbol{u}}_{0}$ of $\Delta$ such that a fiber $L({\boldsymbol{u}})=\Phi^{-1}({\boldsymbol{u}})$ over a boundary point ${\boldsymbol{u}}\in\partial\Delta$ is a Lagrangian submanifold if and only if ${\boldsymbol{u}}={\boldsymbol{u}}_{0}$ .
2.

The Lagrangian fiber $L({\boldsymbol{u}}_{0})$ is diffeomorphic to $\operatorname{SU}(2)\cong S^{3}$ .
3.

The Floer cohomology of $L({\boldsymbol{u}}_{0})$ over the Novikov field $\Lambda$ is trivial;

$\mathop{H\!F}\nolimits(L({\boldsymbol{u}}_{0}),L({\boldsymbol{u}}_{0});\Lambda)=0.$ (1.10)

Theorem 1.2.

Let $\Phi\colon\operatorname{Gr}(2,4)\to\mathbb{R}^{4}$ be the Gelfand-Cetlin system with the Gelfand-Cetlin polytope $\Delta=\Phi(\operatorname{Gr}(2,4))$ .

1.

There exists an edge of $\Delta$ such that a fiber $L({\boldsymbol{u}})=\Phi^{-1}({\boldsymbol{u}})$ over ${\boldsymbol{u}}\in\partial\Delta$ is a Lagrangian submanifold if and only if ${\boldsymbol{u}}$ is in the relative interior of the edge.
2.

The Lagrangian fiber $L({\boldsymbol{u}})$ over any point ${\boldsymbol{u}}$ in the relative interior of the edge is diffeomorphic to $\operatorname{U}(2)\cong S^{1}\times S^{3}$ .
3.

$H^{1}(L({\boldsymbol{u}});\Lambda_{0})$ is contained in $\widehat{\mathcal{M}}_{\mathrm{weak}}(L({\boldsymbol{u}}))$ .
4.

The potential function is identically zero on $H^{1}(L({\boldsymbol{u}});\Lambda_{0})$ .
5.

The Floer cohomology $\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b),(L({\boldsymbol{u}}),b);\Lambda)$ of a Lagrangian $\operatorname{U}(2)$ -fiber $L({\boldsymbol{u}})$ over the Novikov field $\Lambda$ is non-trivial if and only if ${\boldsymbol{u}}$ is the barycenter ${\boldsymbol{u}}_{0}$ of the edge and $b=\pm\pi\sqrt{-1}/2\,\mathbf{e}_{1}$ , where $\mathbf{e}_{1}$ is a generator of $H^{1}(L({\boldsymbol{u}});\mathbb{Z})\cong\mathbb{Z}$ .

If the deformed Floer cohomology group over the Novikov field is non-trivial, then it is isomorphic to the classical cohomology group;

\displaystyle\mathop{H\!F}\nolimits((L({\boldsymbol{u}}_{0}),\pm\pi\sqrt{-1}/2\,\mathbf{e}_{1}),(L({\boldsymbol{u}}_{0}),\pm\pi\sqrt{-1}/2\,\mathbf{e}_{1});\Lambda)\cong H^{*}(S^{1}\times S^{3};\Lambda).

(1.11)

The Floer cohomology of the pair $((L({\boldsymbol{u}}_{0}),\pi\sqrt{-1}/2\,\mathbf{e}_{1}),(L({\boldsymbol{u}}_{0}),-\pi\sqrt{-1}/2\,\mathbf{e}_{1}))$ is trivial;

\displaystyle\mathop{H\!F}\nolimits((L({\boldsymbol{u}}_{0}),\pi\sqrt{-1}/2\,\mathbf{e}_{1}),(L({\boldsymbol{u}}_{0}),\pi\sqrt{-1}/2\,\mathbf{e}_{1});\Lambda)=0.

(1.12)

More precise statements, which describe the Floer cohomology groups over the Novikov ring $\Lambda_{0}$ , are given in Theorem 4.8, Theorem 4.16, and Theorem 4.20.

A symplectic manifold $(X,\omega)$ is monotone if the cohomology class $[\omega]$ is positively proportional to the first Chern class;

\displaystyle\exists\lambda>0\quad[\omega]=\lambda c_{1}(X).

(1.13)

The quantum cohomology ring of a monotone symplectic manifold does not have any convergence issue, and hence is defined over $\mathbb{C}$ . A Lagrangian submanifold $L$ is monotone if the symplectic area of a disk bounded by $L$ is positively proportional to the Maslov index;

\displaystyle\exists\lambda>0\quad\forall\beta\in\pi_{2}(M,L)\quad\beta\cap\omega=\lambda\mu(\beta).

(1.14)

The $A_{\infty}$ -operations on the Lagrangian intersection Floer complex of a monotone Lagrangian submanifold is defined over $\mathbb{C}$ . The minimal Maslov number of oriented monotone Lagrangian submanifold is greater than or equal to 2, so that the obstruction class $\mathfrak{m}_{0}(1)$ can be written as $\mathfrak{m}_{0}(1)=\mathfrak{m}_{0}(L)\,\mathbf{e}_{0},$ where $\mathfrak{m}_{0}(L)\in\mathbb{C}$ is the count of Maslov index 2 disks bounded by $L$ , weighted by their symplectic areas and holonomies of a flat $U(1)$ -bundle on $L$ along the boundaries of the disks. The monotone Fukaya category is defined as the direct sum

\displaystyle\mathcal{F}(X):=\bigoplus_{\lambda\in\mathbb{C}}\mathcal{F}(X;\lambda),

(1.15)

where $\mathcal{F}(X;\lambda)$ is an $A_{\infty}$ -category over $\mathbb{C}$ whose objects are monotone Lagrangian submanifolds, equipped with flat $U(1)$ -bundles, satisfying $\mathfrak{m}_{0}(L)=\lambda$ . For any monotone Lagrangian submanifold $L$ , there is a natural ring homomorphism

\displaystyle\mathop{QH}\nolimits(X)\to\mathop{H\!F}\nolimits(L,L),

(1.16)

which is known by Auroux [Aur07], Kontsevich, and Seidel to send $c_{1}(X)\in\mathop{QH}\nolimits(X)$ to $\mathfrak{m}_{0}(1)\in\mathop{H\!F}\nolimits(L,L)$ . It follows that $\mathcal{F}(X;\lambda)$ is trivial unless $\lambda$ is an eigenvalue of the quantum cup product by $c_{1}(X)$ .

Now consider the case when $X=\operatorname{Gr}(2,4)$ , which can be written as a quadric hypersurface

\displaystyle X=\left\{[z_{0}:\cdots:z_{5}]\in\mathbb{P}^{5}\mathrel{}\middle|\mathrel{}z_{0}^{2}=z_{1}^{2}+\cdots+z_{5}^{2}\right\}.

(1.17)

The real locus $X_{\mathbb{R}}$ is a monotone Lagrangian sphere, which is the vanishing cycle along a degeneration into a nodal quadric and split-generates the nilpotent summand $D^{\pi}\mathcal{F}(X;0)$ of the monotone Fukaya category [Smi12, Lemma 4.6]. The Floer cohomology $\mathop{H\!F}\nolimits(X_{\mathbb{R}},X_{\mathbb{R}})$ is semisimple, and carries a formal $A_{\infty}$ -structure [Smi12, Lemma 4.7]. It follows that $D^{\pi}\mathcal{F}(X;0)$ is equivalent to the direct sum of two copies of the derived category $D^{b}(\mathbb{C})$ of $\mathbb{C}$ -vector spaces. On the other hand, $(L({\boldsymbol{u}}_{0}),\pm\pi\sqrt{-1}/2\,\mathbf{e}_{1})$ are also objects of the nilpotent summand $D^{\pi}\mathcal{F}(X;0)$ of the monotone Fukaya category, which are non-zero by (1.11). Since $(L({\boldsymbol{u}}_{0}),\pm\sqrt{-1}/2\,\mathbf{e}_{1})$ is a pair of orthogonal non-zero objects in a triangulated category equivalent to $D^{b}(\mathbb{C})\oplus D^{b}(\mathbb{C})$ , they split-generate the whole category:

Corollary 1.3.

The pair $(L({\boldsymbol{u}}_{0}),\pm\pi\sqrt{-1}/2\,\mathbf{e}_{1})$ split-generate $D^{\pi}\mathcal{F}(\operatorname{Gr}(2,4);0)$ .

Acknowledgment: We thank Hiroshi Ohta, Kaoru Ono, and Yoshihiro Ohnita for useful conversations. Y. N. is supported by Grant-in-Aid for Young Scientists (No.23740055). K. U. is supported by Grant-in-Aid for Young Scientists (No.24740043).

2 Non-torus fibers of the Gelfand-Cetlin system

2.1 Flag manifolds

For a sequence $0=n_{0}<n_{1}<\dots<n_{r}<n_{r+1}=n$ of integers, let $F=F(n_{1},\dots,n_{r},n)$ be the flag manifold consisting of flags

0\subset V_{1}\subset\dots\subset V_{r}\subset\mathbb{C}^{n},\quad\dim V_{i}=n_{i}

of $\mathbb{C}^{n}$ . We write the full flag manifold and the Grassmannian as $\operatorname{Fl}(n)=F(1,2,\dots,n)$ and $\operatorname{Gr}(k,n)=F(k,n)$ respectively. The complex dimension of $F(n_{1},\dots,n_{r},n)$ is given by

N=N(n_{1},\dots,n_{r},n):=\dim_{\mathbb{C}}F(n_{1},\dots,n_{r},n)=\sum_{i=1}^{r}(n_{i}-n_{i-1})(n-n_{i}).

Let $P=P(n_{1},\dots,n_{r},n)\subset\operatorname{GL}(n,\mathbb{C})$ be the stabilizer subgroup of the standard flag $(V_{i}=\langle e_{1},\dots,e_{n_{i}}\rangle)_{i=1}^{r},$ where $\{e_{i}\}_{i=1}^{n}$ is the standard basis of $\mathbb{C}^{n}$ . The intersection of $P$ and $\operatorname{U}(n)$ is $\operatorname{U}(k_{1})\times\dots\times\operatorname{U}(k_{r+1})$ for $k_{i}=n_{i}-n_{i-1}$ , and $F$ is written as

F=\operatorname{GL}(n,\mathbb{C})/P=\operatorname{U}(n)/(\operatorname{U}(k_{1})\times\dots\times\operatorname{U}(k_{r+1})).

We take a $\operatorname{U}(n)$ -invariant inner product $\langle x,y\rangle=\operatorname{tr}xy^{*}$ on the Lie algebra $\mathfrak{u}(n)$ of $\operatorname{U}(n)$ , and identify the dual vector space $\mathfrak{u}(n)^{*}$ of $\mathfrak{u}(n)$ with the space $\sqrt{-1}\mathfrak{u}(n)$ of Hermitian matrices. For ${\boldsymbol{\lambda}}=\mathrm{diag}\,(\lambda_{1},\dots,\lambda_{n})\in\sqrt{-1}\mathfrak{u}(n)$ with

\underbrace{\lambda_{1}=\dots=\lambda_{n_{1}}}_{k_{1}}>\underbrace{\lambda_{n_{1}+1}=\dots=\lambda_{n_{2}}}_{k_{2}}>\dots>\underbrace{\lambda_{n_{r}+1}=\dots=\lambda_{n}}_{k_{r+1}},

(2.1)

the flag manifold $F$ is identified with the adjoint orbit $\mathcal{O}_{{\boldsymbol{\lambda}}}\subset\sqrt{-1}\mathfrak{u}(n)$ of ${\boldsymbol{\lambda}}$ . Note that $\mathcal{O}_{{\boldsymbol{\lambda}}}$ consists of Hermitian matrices with fixed eigenvalues $\lambda_{1},\dots,\lambda_{n}$ . Let

\omega(\operatorname{ad}_{\xi}(x),\operatorname{ad}_{\eta}(x))=\frac{1}{2\pi}\langle x,[\xi,\eta]\rangle,\quad\xi,\eta\in\mathfrak{u}(n)

be the (normalized) Kostant-Kirillov form on $\mathcal{O}_{{\boldsymbol{\lambda}}}$ .

For each $i=1,\dots,r$ , we set $\mathbb{P}_{i}:=\mathbb{P}\bigl{(}\bigwedge^{n_{i}}\mathbb{C}^{n}\bigr{)}\cong\mathbb{P}^{\binom{n}{n_{i}}-1}$ . Then the Plücker embedding is given by

\iota:F\hookrightarrow\prod_{i=1}^{r}\mathbb{P}_{i},\quad(0\subset V_{1}\subset\dots\subset V_{r}\subset\mathbb{C}^{n})\mapsto(\textstyle{\bigwedge^{n_{1}}}V_{1},\dots,\textstyle{\bigwedge^{n_{r}}}V_{r}).

Let $\omega_{\mathbb{P}_{i}}$ be the Fubini-Study form on $\mathbb{P}_{i}$ normalized in such a way that it represents the first Chern class $c_{1}(\mathcal{O}(1))$ of the hyperplane bundle. Then the Kostant-Kirillov form $\omega$ and the first Chern form $c_{1}(F)$ of $F$ are given by

\displaystyle\omega

\displaystyle=\sum_{i=1}^{r}(\lambda_{n_{i}}-\lambda_{n_{i+1}})\omega_{\mathbb{P}_{i}}

and

\displaystyle c_{1}(F)

\displaystyle=\sum_{i=1}^{r}(n_{i+1}-n_{i-1})\omega_{\mathbb{P}_{i}}

respectively.

Example 2.1.

The 3-dimensional full flag manifold $\operatorname{Fl}(3)$ is embedded into

\mathbb{P}_{1}\times\mathbb{P}_{2}=\mathbb{P}(\mathbb{C}^{3})\times\mathbb{P}(\textstyle{\bigwedge^{2}\mathbb{C}^{3}})\cong\mathbb{P}^{2}\times\mathbb{P}^{2}

as a hypersurface. The image of $\operatorname{Fl}(3)$ is given by the Plücker relation

Z_{1}Z_{23}+Z_{2}Z_{31}+Z_{3}Z_{12}=0,

where $[Z_{1}:Z_{2}:Z_{3}]$ and $[Z_{23}:Z_{31}:Z_{12}]$ are the Plücker coordinates on $\mathbb{P}_{1}$ and $\mathbb{P}_{2}$ respectively.

Example 2.2.

The Grassmannian $\operatorname{Gr}(2,4)$ of 2-plans in $\mathbb{C}^{4}$ is embedded into $\mathbb{P}(\bigwedge^{2}\mathbb{C}^{4})\cong\mathbb{P}^{5}$ as a hypersurface. The Plücker relation is given by

Z_{12}Z_{34}-Z_{13}Z_{24}+Z_{14}Z_{23}=0,

where $[Z_{12}:Z_{13}:Z_{14}:Z_{23}:Z_{24}:Z_{34}]$ is the Plücker coordinates.

2.2 The Gelfand-Cetlin system

For $x\in\mathcal{O}_{{\boldsymbol{\lambda}}}$ and $k=1,\dots,n-1$ , let $x^{(k)}$ denote the upper-left $k\times k$ submatrix of $x$ . Since $x^{(k)}$ is also a Hermitian matrix, it has real eigenvalues $\lambda^{(k)}_{1}(x)\geq\lambda^{(k)}_{2}(x)\geq\dots\geq\lambda^{(k)}_{k}(x)$ . By taking the eigenvalues for all $k=1,\dots,n-1$ , we obtain a set $(\lambda^{(k)}_{i})_{1\leq i\leq k\leq n-1}$ of $n(n-1)/2$ functions, which satisfy the inequalities

\begin{aligned} \hbox to16.49995pt{$\hfill\lambda_{1}\hfill$}&&&&\hbox to16.49995pt{$\hfill\lambda_{2}\hfill$}&&&&\hbox to16.49995pt{$\hfill\lambda_{3}\hfill$}&&\cdots&&\hbox to16.49995pt{$\hfill\lambda_{n-1}\hfill$}&&&&\hbox to16.49995pt{$\hfill\lambda_{n}\hfill$}\\ &\rotatebox[origin={c}]{315.0}{$\geq$}&&\rotatebox[origin={c}]{45.0}{$\geq$}&&\rotatebox[origin={c}]{315.0}{$\geq$}&&\rotatebox[origin={c}]{45.0}{$\geq$}&&&&&&\rotatebox[origin={c}]{315.0}{$\geq$}&&\rotatebox[origin={c}]{45.0}{$\geq$}&\\ &&\hbox to16.49995pt{$\lambda_{1}^{(n-1)}$}&&&&\hbox to16.49995pt{$\lambda_{2}^{(n-1)}$}&&&&&&&&\hbox to16.49995pt{$\lambda_{n-1}^{(n-1)}$}&&\\ &&&\rotatebox[origin={c}]{315.0}{$\geq$}&&\rotatebox[origin={c}]{45.0}{$\geq$}&&&&&&&&\rotatebox[origin={c}]{45.0}{$\geq$}&&&\\ &&&&\hbox to16.49995pt{$\lambda_{1}^{(n-2)}$}&&&&&&&&\hbox to16.49995pt{$\lambda_{n-2}^{(n-2)}$}&&&&\\ &&&&&\rotatebox[origin={c}]{315.0}{$\geq$}&&&&&&\rotatebox[origin={c}]{45.0}{$\geq$}&&&&&\\ &&&&&&\rotatebox[origin={c}]{315.0}{$\cdots$}&&&&\rotatebox[origin={c}]{45.0}{$\cdots$}&&&&&&\\ &&&&&&&\rotatebox[origin={c}]{315.0}{$\geq$}&&\rotatebox[origin={c}]{45.0}{$\geq$}&&&&&&&\\ &&&&&&&&\hbox to16.49995pt{$\lambda_{1}^{(1)}$}&&&&&&&&&\end{aligned}.

(2.2)

It follows that the number of non-constant $\lambda^{(k)}_{i}$ coincides with $N=\dim_{\mathbb{C}}F$ . Let $I=I(n_{1},\dots,n_{r},n)$ denotes the set of pairs $(i,k)$ such that $\lambda_{i}^{(k)}$ is non-constant. Then the Gelfand-Cetlin system is defined by

\Phi=(\lambda^{(k)}_{i})_{(i,k)\in I}:F(n_{1},\dots,n_{r},n)\longrightarrow\mathbb{R}^{N(n_{1},\dots,n_{r},n)}.

Proposition 2.3 (Guillemin and Sternberg [GS83]).

The map $\Phi$ is a completely integrable system on $(F(n_{1},\dots,n_{r},n),\omega)$ . The functions $\lambda_{i}^{(k)}$ are action variables, and the image $\Delta=\Phi(F)$ is a convex polytope defined by (2.2). The fiber $L({\boldsymbol{u}})=\Phi^{-1}({\boldsymbol{u}})$ over each interior point ${\boldsymbol{u}}\in\operatorname{Int}\Delta$ is a Lagrangian torus.

The image $\Delta\subset\mathbb{R}^{N(n_{1},\dots,n_{r},n)}$ is called the Gelfand-Cetlin polytope. The Gelfand-Cetlin system is not smooth on the locus where $\lambda_{k}^{(i)}=\lambda_{k}^{(i+1)}$ for some $(i,k)$ , or equivalently, where the Gelfand-Cetlin pattern (2.2) contains a set of equalities of the form

\begin{aligned} &&\hbox to16.49995pt{$\hfill{\lambda_{k+1}^{(i+1)}}\hfill$}&&\\ &\rotatebox[origin={c}]{45.0}{$=$}&&\rotatebox[origin={c}]{315.0}{$=$}&&\phantom{\rotatebox[origin={c}]{315.0}{$\geq$}}&\\ \hbox to16.49995pt{$\hfill{\lambda_{k}^{(i)}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{\lambda_{k}^{(i+1)}}\hfill$}&&\\ &\rotatebox[origin={c}]{315.0}{$=$}&&\rotatebox[origin={c}]{45.0}{$=$}&&&\\ &&\hbox to16.49995pt{$\hfill{\lambda_{k-1}^{(i)}}\hfill$}&&&&\end{aligned}.

The image of such loci are faces of $\Delta$ of codimension greater than two where $\Delta$ does not satisfy the Delzant condition. Away from such faces, each fiber $\Phi^{-1}({\boldsymbol{u}})$ of $\Phi$ is an isotropic torus whose dimension is that of the face of $\Delta$ containing ${\boldsymbol{u}}$ in its relative interior.

2.3 The case of $\operatorname{Fl}(3)$

Refer to caption — Figure 2.1: The Gelfand-Cetlin polytope for $\operatorname{Fl}(3)$

After a translation by a scalar matrix, we may assume that $\operatorname{Fl}(3)$ is identified with the adjoint orbit of ${\boldsymbol{\lambda}}=\operatorname{diag}(\lambda_{1},0,-\lambda_{2})$ for $\lambda_{1},\lambda_{2}>0$ . Then the Gelfand-Cetlin polytope $\Delta$ consists of $(u_{1},u_{2},u_{3})\in\mathbb{R}^{3}$ satisfying

$\displaystyle$		$\displaystyle$		$\displaystyle$			(2.3)
	$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$	$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$
$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$
$\displaystyle$

as shown in Figure 2.1. The non-smooth locus of $\Phi$ is the fiber $L_{0}=\Phi^{-1}({\boldsymbol{0}})$ over the vertex ${\boldsymbol{0}}=(0,0,0)\in\Delta$ where four edges intersect.

Definition 2.4 (Evans and Lekili [EL, Definition 1.1.1]).

Let $K$ be a compact connected Lie group. A Lagrangian submanifold $L$ in a Kähler manifold $X$ is said to be $K$ -homogeneous if $K$ acts holomorphically on $X$ in such a way that $L$ is a $K$ -orbit.

Proposition 2.5.

The fiber $L_{0}=\Phi^{-1}({\boldsymbol{0}})$ is a Lagrangian 3-sphere given by

L_{0}=\left\{\left.\begin{pmatrix}0&0&z_{1}\\ 0&0&z_{2}\\ \overline{z}_{1}&\overline{z}_{2}&\lambda_{1}-\lambda_{2}\end{pmatrix}\in\sqrt{-1}\mathfrak{u}(3)\right||z_{1}|^{2}+|z_{2}|^{2}=\lambda_{1}\lambda_{2}\right\},

which is $K$ -homogeneous for

K=\left\{\left.\begin{pmatrix}a_{1}&-\overline{a}_{2}&0\\ a_{2}&\overline{a}_{1}&0\\ 0&0&1\end{pmatrix}\right||a_{1}|^{2}+|a_{2}|^{2}=1\right\}\cong\operatorname{SU}(2).

Proof.

Suppose that $x\in L_{0}$ . Then $\lambda_{1}^{(2)}(x)=\lambda_{2}^{(2)}(x)=0$ implies that $x^{(2)}=0$ and thus $x$ has the form

x=\begin{pmatrix}0&0&z_{1}\\ 0&0&z_{2}\\ \overline{z}_{1}&\overline{z}_{2}&x_{33}\end{pmatrix}

for some $z_{1},z_{2}\in\mathbb{C}$ and $x_{33}\in\mathbb{R}$ . Since

\operatorname{det}(\lambda-x)=\lambda\left(\lambda^{2}-x_{33}\lambda-(|z_{1}|^{2}+|z_{2}|^{2})\right)=0

has solutions $\lambda=\lambda_{1},0,-\lambda_{2}$ , we have $x_{33}=\lambda_{1}-\lambda_{2}$ and $|z_{1}|^{2}+|z_{2}|^{2}=\lambda_{1}\lambda_{2}$ . Hence the fiber $L_{0}$ is the $K$ -orbit of

\begin{pmatrix}0&0&\sqrt{\lambda_{1}\lambda_{2}}\\ 0&0&0\\ \sqrt{\lambda_{1}\lambda_{2}}&0&\lambda_{1}-\lambda_{2}\end{pmatrix}=\operatorname{Ad}_{g_{0}}\begin{pmatrix}\lambda_{1}&0&0\\ 0&0&0\\ 0&0&-\lambda_{2}\end{pmatrix}\in\mathcal{O}_{{\boldsymbol{\lambda}}},

where

g_{0}=\begin{pmatrix}\sqrt{\lambda_{2}/(\lambda_{1}+\lambda_{2})}&0&-\sqrt{\lambda_{1}/(\lambda_{1}+\lambda_{2})}\\ 0&1&0\\ \sqrt{\lambda_{1}/(\lambda_{1}+\lambda_{2})}&0&\sqrt{\lambda_{2}/(\lambda_{1}+\lambda_{2})}\end{pmatrix}\in\operatorname{SU}(3).

Next we see that $L_{0}$ is Lagrangian. Since $K$ acts transitively on $L_{0}$ , the tangent space $T_{x}L_{0}$ is spanned by infinitesimal actions $\operatorname{ad}_{\xi}(x)$ of $\xi\in\mathfrak{k}$ , where

\mathfrak{k}=\biggl{\{}\xi=\begin{pmatrix}\xi^{(2)}&0\\ 0&0\end{pmatrix}\in\mathfrak{u}(3)\biggm{|}\xi^{(2)}\in\mathfrak{su}(2)\biggr{\}}\cong\mathfrak{su}(2)

is the Lie algebra of $K$ . Since $x^{(2)}=0$ for $x\in L_{0}$ , we have

\omega(\operatorname{ad}_{\xi}(x),\operatorname{ad}_{\eta}(x))=\frac{\sqrt{-1}}{2\pi}\operatorname{tr}\Bigl{(}x^{(2)}[\xi^{(2)},\eta^{(2)}]\Bigr{)}=0

for any $\xi,\eta\in\mathfrak{k}$ . ∎

Let $\iota:\operatorname{Fl}(3)\to\mathbb{P}_{1}\times\mathbb{P}_{2}=\mathbb{P}(\mathbb{C}^{3})\times\mathbb{P}(\bigwedge^{2}\mathbb{C}^{3})$ be the Plücker embedding and $([Z_{1}:Z_{2}:Z_{3}],[Z_{23}:Z_{31}:Z_{12}])$ be the Plücker coordinates. The Kostant-Kirillov form is given by

\omega=\lambda_{1}\omega_{\mathbb{P}_{1}}+\lambda_{2}\omega_{\mathbb{P}_{2}}.

Since the Lagrangian fiber $L_{0}$ as a submanifold in $\operatorname{SU}(3)/T$ consists of

\begin{pmatrix}a_{1}&-\overline{a}_{2}&0\\ a_{2}&\overline{a}_{1}&0\\ 0&0&1\end{pmatrix}g_{0}=\frac{1}{\sqrt{\lambda_{1}+\lambda_{2}}}\begin{pmatrix}\sqrt{\lambda_{2}}a_{1}&-\sqrt{\lambda_{1}+\lambda_{2}}\overline{a}_{2}&-\sqrt{\lambda_{1}}a_{1}\\ \sqrt{\lambda_{2}}a_{2}&\sqrt{\lambda_{1}+\lambda_{2}}\overline{a}_{1}&-\sqrt{\lambda_{1}}a_{2}\\ \sqrt{\lambda_{1}}&0&\sqrt{\lambda_{2}}\end{pmatrix}\mod T

with $|a_{1}|^{2}+|a_{2}|^{2}=1$ , the image $\iota(L_{0})$ is given by

\iota(L_{0})=\Biggl{\{}\Biggm{(}\left[a_{1}:a_{2}:\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}\right],\left[\overline{a}_{1}:\overline{a}_{2}:-\sqrt{\frac{\lambda_{2}}{\lambda_{1}}}\right]\Biggm{)}\,\Biggm{|}\,|a_{1}|^{2}+|a_{2}|^{2}=1\Biggr{\}}.

(2.4)

Define an anti-holomorphic involution $\tau$ on $\operatorname{Fl}(3)$ by

\displaystyle\tau\left([Z_{1}:Z_{2}:Z_{3}],[Z_{23}:Z_{31}:Z_{12}]\right)

\displaystyle=\left(\left[{\overline{Z}}_{23}:{\overline{Z}}_{31}:-\frac{\lambda_{1}}{\lambda_{2}}{\overline{Z}}_{12}\right],\left[\overline{Z}_{1}:\overline{Z}_{2}:-\frac{\lambda_{2}}{\lambda_{1}}\overline{Z}_{3}\right]\right).

(2.5)

Proposition 2.6.

The Lagrangian $L_{0}$ is the fixed point set of $\tau$ .

One can easily see that $\tau$ is an anti-symplectic involution if and only if $\lambda_{1}=\lambda_{2}$ .

2.4 The case of $\operatorname{Gr}(2,4)$

For $k<n$ , let ${\widetilde{V}}(k,n)$ be the space of $n\times k$ matrices of rank $k$ , and set

V(k,n)=\{Z\in{\widetilde{V}}(k,n)\mid Z^{*}Z=I_{k}\}.

Then the Grassmannian $\operatorname{Gr}(k,n)$ is given by

\operatorname{Gr}(k,n)={\widetilde{V}}(k,n)/\operatorname{GL}(k,\mathbb{C})=V(k,n)/\operatorname{U}(k).

We first consider the Gelfand-Cetlin system on $\operatorname{Gr}(n,2n)$ for general $n$ . Fix $\lambda>0$ and identify $\operatorname{Gr}(n,2n)$ with the adjoint orbit $\mathcal{O}_{{\boldsymbol{\lambda}}}$ of

\displaystyle{\boldsymbol{\lambda}}

\displaystyle=\operatorname{diag}(\underbrace{\lambda,\dots,\lambda}_{n},\underbrace{-\lambda,\dots,-\lambda}_{n}).

The orbit $\mathcal{O}_{{\boldsymbol{\lambda}}}$ consists of matrices of the form $2\lambda ZZ^{*}-\lambda I_{2n}$ for $Z\in V(n,2n)$ . The Gelfand-Cetlin polytope $\Delta$ of $\operatorname{Gr}(n,2n)$ consists of ${\boldsymbol{u}}=(u_{i}^{(k)})_{(i,k)\in I}\in\mathbb{R}^{n^{2}}$ satisfying

\begin{aligned} &&&&&&\hbox to16.49995pt{$\hfill{u_{n}^{(2n-1)}}\hfill$}\\ &&&&&\rotatebox[origin={c}]{45.0}{$\geq$}&&\rotatebox[origin={c}]{315.0}{$\geq$}\\ \hbox to16.49995pt{$\hfill{\lambda}\hfill$}&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\cdots$}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\cdots$}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{-\lambda}\hfill$}\\ &\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\geq$}}\hfill$}&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\geq$}}\hfill$}&&&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\geq$}}\hfill$}&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\geq$}}\hfill$}\\ &&\hbox to16.49995pt{$\hfill{u_{1}^{(n)}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{\cdots}\hfill$}&&&&\hbox to16.49995pt{$\hfill{u_{n}^{(n)}}\hfill$}\\ &&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\geq$}}\hfill$}&&&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\geq$}}\hfill$}\\ &&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\cdots$}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\cdots$}}\hfill$}\\ &&&&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{315.0}{$\geq$}}\hfill$}&&\hbox to16.49995pt{$\hfill{\rotatebox[origin={c}]{45.0}{$\geq$}}\hfill$}\\ &&&&&&\hbox to16.49995pt{$\hfill{u_{1}^{(1)}}\hfill$}\end{aligned}.

For $-\lambda<t<\lambda$ , let $L_{t}=\Phi^{-1}(t,\dots,t)$ be the fiber over the boundary point $u_{1}^{(1)}=\dots=u^{(2n-1)}_{n}=t$ of $\Delta$ .

Proposition 2.7.

The fiber $L_{t}$ is a Lagrangian submanifold given by

L_{t}=\biggl{\{}\begin{pmatrix}tI_{n}&\sqrt{\lambda^{2}-t^{2}}A^{*}\\ \sqrt{\lambda^{2}-t^{2}}A&-tI_{n}\end{pmatrix}\in\sqrt{-1}\mathfrak{u}(2n)\biggm{|}A\in\operatorname{U}(n)\biggr{\}}\cong\operatorname{U}(n),

which is $K$ -homogeneous for

K=\left\{\left.\begin{pmatrix}P&0\\ 0&I_{n}\end{pmatrix}\in\operatorname{U}(2n)\,\right|\,P\in\operatorname{U}(n)\right\}\cong\operatorname{U}(n).

Proof.

We write $x\in\mathcal{O}_{{\boldsymbol{\lambda}}}$ as

x=2\lambda ZZ^{*}-\lambda I_{2n}=\lambda\begin{pmatrix}2Z_{1}Z_{1}^{*}-I_{n}&2Z_{1}Z_{2}^{*}\\ 2Z_{2}Z_{1}^{*}&2Z_{2}Z_{2}^{*}-I_{n}\end{pmatrix}

for $n\times n$ matrices $Z_{1}$ , $Z_{2}$ with

Z=\begin{pmatrix}Z_{1}\\ Z_{2}\end{pmatrix}\in V(n,2n).

Suppose that $x\in L_{t}$ , or equivalently, $\lambda_{1}^{(n)}(x)=\dots=\lambda_{n}^{(n)}(x)=t$ . Then the upper-left $n\times n$ block of $x$ satisfies

x^{(n)}=2\lambda Z_{1}Z_{1}^{*}-\lambda I_{n}=tI_{n},

which means that $Z_{1}\in\sqrt{(\lambda+t)/2\lambda}\operatorname{U}(n)$ . After the right $\operatorname{U}(n)$ -action on $V(n,2n)$ , we may assume that $Z_{1}=\sqrt{(\lambda+t)/2\lambda}I_{n}$ . Then the condition $Z^{*}Z=I_{n}$ implies that

Z_{2}^{*}Z_{2}=I_{n}-\frac{\lambda+t}{2\lambda}I_{n}=\frac{\lambda-t}{2\lambda}I_{n}.

Hence $Z$ has the form

Z=\begin{pmatrix}\sqrt{(\lambda+t)/2\lambda}I_{n}\\ \sqrt{(\lambda-t)/2\lambda}A\end{pmatrix}\in V(n,2n)

(2.6)

for some $A\in\operatorname{U}(n)$ , which shows that

x=2\lambda ZZ^{*}-\lambda I_{2n}=\begin{pmatrix}tI_{n}&\sqrt{\lambda^{2}-t^{2}}A^{*}\\ \sqrt{\lambda^{2}-t^{2}}A&-tI_{n}\end{pmatrix}.

The $K$ -homogeneity is obvious from this expression. Since the tangent space $T_{x}L_{t}$ is spanned by the infinitesimal action of the Lie algebra $\mathfrak{k}$ of $K$ , and $x^{(n)}=tI_{n}$ is a scalar matrix, we have

\omega_{x}(\operatorname{ad}_{\xi}(x),\operatorname{ad}_{\eta}(x))=\frac{1}{2\pi}\operatorname{tr}x^{(n)}[\xi^{(n)},\eta^{(n)}]=0

for

\xi=\begin{pmatrix}\xi^{(n)}&\\ &0\end{pmatrix},\ \eta=\begin{pmatrix}\eta^{(n)}&\\ &0\end{pmatrix}\in\mathfrak{k},

which shows that $L_{t}$ is Lagrangian. ∎

Corollary 2.8.

For $t\neq 0$ , the fiber $L_{t}$ is displaceable, i.e., there exists a Hamiltonian diffeomorphism $\varphi$ on $\operatorname{Gr}(n,2n)$ such that $\varphi(L_{t})\cap L_{t}=\emptyset$ .

Proof.

One has $g(L_{t})=L_{-t}$ for $g=\begin{pmatrix}0&-I_{n}\\ I_{n}&0\end{pmatrix}\in\operatorname{U}(2n).$ ∎

In the rest of this subsection, we restrict ourselves to the case of $\operatorname{Gr}(2,4)$ . We write $(u_{1},u_{2},u_{3},u_{4})=(u_{2}^{(3)},u_{1}^{(2)},u_{2}^{(2)},u_{1}^{(1)})$ for simplicity. Figure 2.2 shows the projection

\Delta\longrightarrow[-\lambda,\lambda],\quad{\boldsymbol{u}}=(u_{1},u_{2},u_{3},u_{4})\longmapsto u_{1}.

The non-smooth locus of $\Phi$ is the inverse image of the edge of $\Delta$ defined by $u_{1}=\dots=u_{4}$ . The fiber $L_{t}$ over $(t,t,t,t)\in\partial\Delta$ is a Lagrangian submanifold consists of $2\lambda ZZ^{*}-\lambda I_{2n}$ with

Z=\frac{1}{\sqrt{2\lambda}}\begin{pmatrix}\sqrt{\lambda+t}I_{2}\\ \sqrt{\lambda-t}A\end{pmatrix}\mod\operatorname{U}(2)

for $A\in\operatorname{U}(2)$ . We identify $\operatorname{U}(2)$ with $\operatorname{U}(1)\times\operatorname{SU}(2)\cong S^{1}\times S^{3}$ by

\operatorname{U}(1)\times\operatorname{SU}(2)\longrightarrow\operatorname{U}(2),\quad\left(a_{0},\begin{pmatrix}a_{1}&-\overline{a}_{2}\\ a_{2}&\overline{a}_{1}\end{pmatrix}\right)\longmapsto\begin{pmatrix}a_{0}&0\\ 0&1\end{pmatrix}\begin{pmatrix}a_{1}&-\overline{a}_{2}\\ a_{2}&\overline{a}_{1}\end{pmatrix}.

Then the image of $L_{t}$ under the Plücker embedding $\iota:\operatorname{Gr}(2,4)\to\mathbb{P}(\bigwedge^{2}\mathbb{C}^{4})\cong\mathbb{P}^{5}$ is given by

\iota(L_{t})=\Biggl{\{}\biggl{[}\sqrt{\frac{\lambda+t}{\lambda-t}}:-a_{0}\overline{a}_{2}:\overline{a}_{1}:-a_{0}a_{1}:-a_{2}:\sqrt{\frac{\lambda-t}{\lambda+t}}a_{0}\biggr{]}\Biggm{|}|a_{0}|^{2}=|a_{1}|^{2}+|a_{2}|^{2}=1\Biggr{\}}.

This expression implies the following.

Proposition 2.9.

For each $t\in(-\lambda,\lambda)$ , we define an anti-holomorphic involution $\tau_{t}$ on $\operatorname{Gr}(2,4)$ defined by

\tau_{t}([Z_{12}:Z_{13}:Z_{14}:Z_{23}:Z_{24}:Z_{34}])=\biggl{[}\frac{\lambda+t}{\lambda-t}\overline{Z}_{34}:\overline{Z}_{24}:-\overline{Z}_{23}:-\overline{Z}_{14}:\overline{Z}_{13}:\frac{\lambda-t}{\lambda+t}\overline{Z}_{12}\biggr{]}

(2.7)

Then $L_{t}$ is the fixed point set of $\tau_{t}$ .

Remark 2.10.

The map $\tau_{0}$ for $t=0$ is an anti-symplectic involution as well, and satisfies $\tau_{0}(L_{t})=L_{-t}$ for each $t\in(-\lambda,\lambda)$ .

2.5 The case of $\operatorname{Gr}(2,5)$

We fix $\lambda>0$ and identify $\operatorname{Gr}(2,5)$ with the adjoint orbit $\mathcal{O}_{{\boldsymbol{\lambda}}}$ of $\operatorname{diag}(\lambda,\lambda,0,0,0)\in\sqrt{-1}\mathfrak{u}(5)$ . The Gelfand-Cetlin polytope $\Delta$ is defined by

$\displaystyle$		$\displaystyle$					(2.8)
	$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$
$\displaystyle$		$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$	$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$
$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$\geq$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$\geq$}$
$\displaystyle$

We first consider the fiber $L_{1}(s_{1},s_{2},t)$ over a boundary point given by

\begin{aligned} \hbox to16.49995pt{$\hfill{\lambda}\hfill$}&&&&\hbox to16.49995pt{$\hfill{s_{2}}\hfill$}\\ &\rotatebox[origin={c}]{315.0}{$>$}&&\rotatebox[origin={c}]{45.0}{$>$}&&\rotatebox[origin={c}]{315.0}{$>$}\\ &&\hbox to16.49995pt{$\hfill{s_{1}}\hfill$}&&&&\hbox to16.49995pt{$\hfill{t}\hfill$}&&&&\hbox to16.49995pt{$\hfill{0}\hfill$}\\ &&&\rotatebox[origin={c}]{315.0}{$>$}&&\rotatebox[origin={c}]{45.0}{$=$}&&\rotatebox[origin={c}]{315.0}{$=$}&&\rotatebox[origin={c}]{45.0}{$>$}\\ &&&&\hbox to16.49995pt{$\hfill{t}\hfill$}&&&&\hbox to16.49995pt{$\hfill{t}\hfill$}\\ &&&&&\rotatebox[origin={c}]{315.0}{$=$}&&\rotatebox[origin={c}]{45.0}{$=$}\\ &&&&&&\hbox to16.49995pt{$\hfill{t}\hfill$}\end{aligned}.

Proposition 2.11.

The fiber $L_{1}(s_{1},s_{2},t)$ is a Lagrangian submanifold diffeomorphic to $\operatorname{U}(2)\times T^{2}\cong S^{3}\times T^{3}$ . Moreover, $L_{1}(s_{1},s_{2},t)$ is $K$ -homogeneous for

K=\left\{\left.\begin{pmatrix}P\\ &e^{\sqrt{-1}\theta_{1}}\\ &&e^{\sqrt{-1}\theta_{2}}\\ &&&1\end{pmatrix}\in\operatorname{U}(5)\right|P\in\operatorname{U}(2),\,\theta_{1},\theta_{2}\in\mathbb{R}\right\}\cong\operatorname{U}(2)\times T^{2}.

Proof.

Note that $\mathcal{O}_{{\boldsymbol{\lambda}}}$ consists of matrices of the form

\displaystyle x=\lambda ZZ^{*}=\lambda(z_{i}\overline{z}_{j}+w_{i}\overline{w}_{j})_{1\leq i,j\leq 5}

(2.9)

for

Z=\begin{pmatrix}z_{1}&w_{1}\\ z_{2}&w_{2}\\ z_{3}&w_{3}\\ z_{4}&w_{4}\\ z_{5}&w_{5}\end{pmatrix}\in V(2,5),

i.e.,

\displaystyle\sum_{i=1}^{5}|z_{i}|^{2}=\sum_{i=1}^{5}|w_{i}|^{2}=1,\quad\sum_{i=1}^{5}z_{i}\overline{w}_{i}=0.

(2.10)

Since the upper-left $2\times 2$ submatrix of $x=\lambda(z_{i}\overline{z}_{j}+w_{i}\overline{w}_{j})\in L_{1}(s_{1},s_{2},t)$ satisfies

\displaystyle x^{(2)}=\lambda\begin{pmatrix}|z_{1}|^{2}+|w_{1}|^{2}&z_{1}\overline{z}_{2}+w_{1}\overline{w}_{2}\\ z_{2}\overline{z}_{1}+w_{2}\overline{w}_{1}&|z_{2}|^{2}+|w_{2}|^{2}\end{pmatrix}=\begin{pmatrix}t&0\\ 0&t\end{pmatrix},

(2.11)

we have

\displaystyle\sqrt{\frac{\lambda}{t}}\begin{pmatrix}z_{1}&w_{1}\\ z_{2}&w_{2}\end{pmatrix}\in\operatorname{U}(2),

(2.12)

and in particular, $|z_{1}|^{2}+|z_{2}|^{2}=|w_{1}|^{2}+|w_{2}|^{2}=t/\lambda$ . Then the condition (2.10) implies

$\displaystyle\|z_{3}\|^{2}+\|z_{4}\|^{2}+\|z_{5}\|^{2}$	$\displaystyle=(\lambda-t)/\lambda,$	(2.13)
$\displaystyle\|w_{3}\|^{2}+\|w_{4}\|^{2}+\|w_{5}\|^{2}$	$\displaystyle=(\lambda-t)/\lambda,$	(2.14)
$\displaystyle z_{3}\overline{w}_{3}+z_{4}\overline{w}_{4}+z_{5}\overline{w}_{5}$	$\displaystyle=0.$	(2.15)

On the other hand, the conditions $\operatorname{tr}x^{(3)}=s_{1}+t$ , $\operatorname{tr}x^{(4)}=\lambda+s_{2}$ imply

$\displaystyle\|z_{3}\|^{2}+\|w_{3}\|^{2}$	$\displaystyle=(s_{1}-t)/\lambda,$	(2.16)
$\displaystyle\|z_{4}\|^{2}+\|w_{4}\|^{2}$	$\displaystyle=(\lambda-s_{1}+s_{2}-t)/\lambda,$	(2.17)
$\displaystyle\|z_{5}\|^{2}+\|w_{5}\|^{2}$	$\displaystyle=(\lambda-s_{2})/\lambda.$	(2.18)

After the right $\operatorname{SU}(2)$ -action on $(z,w)$ , we may assume that $(z_{5},w_{5})=\bigl{(}\sqrt{(\lambda-s_{2})/\lambda},0\bigr{)}$ . Then (2.13), (2.14), and (2.15) become

	$\displaystyle\|z_{3}\|^{2}+\|z_{4}\|^{2}$	$\displaystyle=(s_{2}-t)/\lambda,$
	$\displaystyle\|w_{3}\|^{2}+\|w_{4}\|^{2}$	$\displaystyle=(\lambda-t)/\lambda,$
	$\displaystyle z_{3}\overline{w}_{3}+z_{4}\overline{w}_{4}$	$\displaystyle=0,$

which mean that the $2\times 2$ submatrix $(z_{i},w_{i})_{i=3,4}$ has the form

\begin{pmatrix}z_{3}&w_{3}\\ z_{4}&w_{4}\end{pmatrix}=\begin{pmatrix}\sqrt{(s_{2}-t)/\lambda}\,a&-\sqrt{(\lambda-t)/\lambda}\,\overline{b}c\\ \sqrt{(s_{2}-t)/\lambda}\,b&\sqrt{(\lambda-t)/\lambda}\,\overline{a}c\end{pmatrix}

for some

\begin{pmatrix}a&-\overline{b}\\ b&\overline{a}\end{pmatrix}\in\operatorname{SU}(2),\quad c\in\operatorname{U}(1).

Combining this with (2.16) and (2.17) we have

|a|^{2}=\frac{\lambda-s_{1}}{\lambda-s_{2}},\quad|b|^{2}=\frac{s_{1}-s_{2}}{\lambda-s_{2}},

and hence

\begin{pmatrix}z_{3}&w_{3}\\ z_{4}&w_{4}\end{pmatrix}=\frac{1}{\sqrt{\lambda(\lambda-s_{2})}}\begin{pmatrix}\sqrt{(s_{2}-t)(\lambda-s_{1})}\,e^{\sqrt{-1}\theta_{1}}&-\sqrt{(\lambda-t)(s_{1}-s_{2})}\,e^{-\sqrt{-1}\theta_{2}}c\\ \sqrt{(s_{2}-t)(s_{1}-s_{2})}\,e^{\sqrt{-1}\theta_{2}}&\sqrt{(\lambda-t)(\lambda-s_{1})}\,e^{-\sqrt{-1}\theta_{1}}c\end{pmatrix}

for some $\theta_{1},\theta_{2}\in\mathbb{R}$ . After the action of

\left\{\left.\begin{pmatrix}1&0\\ 0&e^{\sqrt{-1}\varphi}\end{pmatrix}\in\operatorname{U}(2)\right|\varphi\in\mathbb{R}\right\}\cong\operatorname{U}(1)

from the right, we may assume that

\begin{pmatrix}z_{3}&w_{3}\\ z_{4}&w_{4}\end{pmatrix}=\frac{1}{\sqrt{\lambda(\lambda-s_{2})}}\begin{pmatrix}\sqrt{(s_{2}-t)(\lambda-s_{1})}\,e^{\sqrt{-1}\theta_{1}}&-\sqrt{(\lambda-t)(s_{1}-s_{2})}\,e^{\sqrt{-1}\theta_{1}}\\ \sqrt{(s_{2}-t)(s_{1}-s_{2})}\,e^{\sqrt{-1}\theta_{2}}&\sqrt{(\lambda-t)(\lambda-s_{1})}\,e^{\sqrt{-1}\theta_{2}}\end{pmatrix}.

Therefore $Z=(z_{i},w_{i})_{i}$ is normalized as

\begin{pmatrix}z_{1}&w_{1}\\ \vdots&\vdots\\ z_{5}&w_{5}\\ \end{pmatrix}=\begin{pmatrix}z_{1}&w_{1}\\ z_{2}&w_{2}\\ \sqrt{(s_{2}-t)(\lambda-s_{1})/\lambda(\lambda-s_{2})}\,e^{\sqrt{-1}\theta_{1}}&-\sqrt{(\lambda-t)(s_{1}-s_{2})/\lambda(\lambda-s_{2})}\,e^{\sqrt{-1}\theta_{1}}\\ \sqrt{(s_{2}-t)(s_{1}-s_{2})/\lambda(\lambda-s_{2})}\,e^{\sqrt{-1}\theta_{2}}&\sqrt{(\lambda-t)(\lambda-s_{1})/\lambda(\lambda-s_{2})}\,e^{\sqrt{-1}\theta_{2}}\\ \sqrt{(\lambda-s_{2})/\lambda}&0\end{pmatrix}

with (2.12), which implies that $L_{1}(s_{1},s_{2},t)$ is a $K$ -orbit and diffeomorphic to $\operatorname{U}(2)\times T^{2}$ .

The assertion that $L_{1}(s_{1},s_{2},t)$ is Lagrangian follows from the $K$ -homogeneity as in the cases of $\operatorname{Fl}(3)$ and $\operatorname{Gr}(n,2n)$ . ∎

Next we consider the fiber $L_{2}(s_{1},s_{2},t)$ over

\begin{aligned} \hbox to16.49995pt{$\hfill{\lambda}\hfill$}&&&&\hbox to16.49995pt{$\hfill{t}\hfill$}\\ &\rotatebox[origin={c}]{315.0}{$>$}&&\rotatebox[origin={c}]{45.0}{$=$}&&\rotatebox[origin={c}]{315.0}{$=$}\\ &&\hbox to16.49995pt{$\hfill{t}\hfill$}&&&&\hbox to16.49995pt{$\hfill{t}\hfill$}&&&&\hbox to16.49995pt{$\hfill{0}\hfill$}\\ &&&\rotatebox[origin={c}]{315.0}{$=$}&&\rotatebox[origin={c}]{45.0}{$=$}&&\rotatebox[origin={c}]{315.0}{$>$}&&\rotatebox[origin={c}]{45.0}{$>$}\\ &&&&\hbox to16.49995pt{$\hfill{t}\hfill$}&&&&\hbox to16.49995pt{$\hfill{s_{1}}\hfill$}\\ &&&&&\rotatebox[origin={c}]{315.0}{$>$}&&\rotatebox[origin={c}]{45.0}{$>$}\\ &&&&&&\hbox to16.49995pt{$\hfill{s_{2}}\hfill$}\end{aligned}.

Suppose that $x=\lambda(z_{i}\overline{z}_{j}+w_{i}\overline{w}_{j})_{1\leq i,j\leq 5}\in L_{2}(s_{1},s_{2},t)$ . The condition that $x^{(3)}=\lambda(z_{i}\overline{z}_{j}+w_{i}\overline{w}_{j})_{1\leq i,j\leq 3}$ has eigenvalues $t,t,0$ is equivalent to

$\displaystyle\|z_{1}\|^{2}+\|z_{2}\|^{2}+\|z_{3}\|^{2}$	$\displaystyle=t/\lambda,$	(2.19)
$\displaystyle\|w_{1}\|^{2}+\|w_{2}\|^{2}+\|w_{3}\|^{2}$	$\displaystyle=t/\lambda,$	(2.20)
$\displaystyle z_{1}\overline{w}_{1}+z_{2}\overline{w}_{2}+z_{3}\overline{w}_{3}$	$\displaystyle=0,$	(2.21)

and hence

\sqrt{\frac{\lambda}{\lambda-t}}\begin{pmatrix}z_{4}&w_{4}\\ z_{5}&w_{5}\end{pmatrix}\in U(2).

On the other hand, the conditions $x^{(1)}=s_{2}$ , $\operatorname{tr}x^{(2)}=t+s_{1}$ , and $\operatorname{tr}x^{(3)}=2t$ imply

	$\displaystyle\|z_{1}\|^{2}+\|w_{1}\|^{2}$	$\displaystyle=s_{2}/\lambda,$
	$\displaystyle\|z_{2}\|^{2}+\|w_{2}\|^{2}$	$\displaystyle=(t-s_{2}+s_{1})/\lambda,$
	$\displaystyle\|z_{3}\|^{2}+\|w_{3}\|^{2}$	$\displaystyle=(t-s_{1})/\lambda.$

Then we have the following.

Proposition 2.12.

The fiber $L_{2}(s_{1},s_{2},t)$ is a $\operatorname{U}(2)\times T^{2}$ -homogeneous Lagrangian submanifold diffeomorphic to $\operatorname{U}(2)\times T^{2}\cong S^{3}\times T^{3}$ . Moreover, the fibers $L_{1}(s_{1},s_{2},t)$ and $L_{2}(s_{1},s_{2},t)$ satisfy

g(L_{2}(s_{1},s_{2},t))=L_{1}(\lambda-s_{1},\lambda-s_{2},\lambda-t)

for

g=\begin{pmatrix}0&&1\\ &\rotatebox[origin={c}]{45.0}{$\cdots$}&\\ 1&&0\end{pmatrix}\in\operatorname{U}(5).

In particular, $L_{1}(s_{1},s_{2},t)$ and $L_{2}(s_{1},s_{2},t)$ are displaceable.

The Hamiltonian isotopy invariance of the Floer cohomology over the Novikov field [FOOO09, Theorem G] implies the following.

Corollary 2.13.

For $i=1,2$ , we have

\mathop{H\!F}\nolimits((L_{i}(s_{1},s_{2},t),b),(L_{i}(s_{1},s_{2},t),b);\Lambda)=0

for any weak bounding cochain $b$ .

Remark 2.14.

Other boundary fibers have lower dimensions. For example, the fiber over

$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$>$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$=$}$		$\displaystyle\rotatebox[origin={c}]{315.0}{$=$}$
$\displaystyle$		$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$=$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$=$}$		$\displaystyle\rotatebox[origin={c}]{315.0}{$=$}$	$\displaystyle\rotatebox[origin={c}]{45.0}{$>$}$
$\displaystyle$		$\displaystyle$
	$\displaystyle\rotatebox[origin={c}]{315.0}{$=$}$		$\displaystyle\rotatebox[origin={c}]{45.0}{$=$}$
$\displaystyle$

consists of

\begin{pmatrix}\sqrt{t/\lambda}&0\\ 0&\sqrt{t/\lambda}\\ 0&0\\ z_{4}&w_{4}\\ z_{5}&w_{5}\end{pmatrix}\mod\operatorname{U}(2)

with

\begin{pmatrix}z_{4}&w_{4}\\ z_{5}&w_{5}\end{pmatrix}\in\sqrt{(\lambda-t)/\lambda}\operatorname{U}(2),

which means that the fiber is diffeomorphic to $\operatorname{U}(2)$ .

3 Critical points of the potential function

Let $\Phi:F=F(n_{1},\dots,n_{r},n)\to\Delta$ be the Gelfand-Cetlin system on the flag manifold, and $\{\theta^{(k)}_{i}\}_{(i,k)\in I}$ be the angle variables dual to the action variables $\{\lambda_{i}^{(k)}\}_{(i,k)\in I}$ . For each ${\boldsymbol{u}}=(u_{k}^{(i)})_{(i,k)\in I}\in\operatorname{Int}\Delta$ , we identify $H^{1}(L({\boldsymbol{u}});\Lambda_{0})$ with $\Lambda_{0}^{N}$ by

b=\sum_{(i,k)\in I}x^{(k)}_{i}d\theta^{(k)}_{i}\in H^{1}(L({\boldsymbol{u}});\Lambda_{0})\longleftrightarrow{\boldsymbol{x}}=(x^{(k)}_{i})_{(i,k)\in I}\in\Lambda_{0}^{N},

and set

	$\displaystyle y_{i}^{(k)}$	$\displaystyle=e^{x_{i}^{(k)}}T^{u_{i}^{(k)}},\qquad(i,k)\in I,$
	$\displaystyle Q_{j}$	$\displaystyle=T^{\lambda_{n_{j}}},\qquad j=1,\dots,r+1.$

Theorem 3.1 ([NNU10, Theorem 10.1]).

For any interior point ${\boldsymbol{u}}\in\operatorname{Int}\Delta$ , we have an inclusion $H^{1}(L({\boldsymbol{u}});\Lambda_{0})\subset\widehat{\mathcal{M}}_{\mathrm{weak}}(L({\boldsymbol{u}}))$ . As a function on

\bigcup_{{\boldsymbol{u}}\in\operatorname{Int}\Delta}H^{1}(L({\boldsymbol{u}});\Lambda_{0})\cong\operatorname{Int}\Delta\times\Lambda_{0}^{N},

the potential function is given by

\mathfrak{PO}({\boldsymbol{u}},{\boldsymbol{x}})=\sum_{(i,k)\in I}\left(\frac{y_{i}^{(k+1)}}{y_{i}^{(k)}}+\frac{y_{i}^{(k)}}{y_{i+1}^{(k+1)}}\right),

where we put $y_{i}^{(k+1)}=Q_{j}$ if $\lambda_{i}^{(k+1)}=\lambda_{n_{j}}$ is a constant function.

Example 3.2.

We identify the 3-dimensional flag manifold $\operatorname{Fl}(3)$ with the adjoint orbit of ${\boldsymbol{\lambda}}=\operatorname{diag}(\lambda_{1},\lambda_{2},\lambda_{3})$ . The potential function is given by

	$\displaystyle\mathfrak{PO}$	$\displaystyle=e^{-x_{1}}T^{-u_{1}+\lambda_{1}}+e^{x_{1}}T^{u_{1}-\lambda_{2}}+e^{-x_{2}}T^{-u_{2}+\lambda_{2}}$
		$\displaystyle\qquad+e^{x_{2}}T^{u_{2}-\lambda_{3}}+e^{x_{1}-x_{3}}T^{u_{1}-u_{3}}+e^{-x_{2}+x_{3}}T^{-u_{2}+u_{3}}$
		$\displaystyle=\frac{Q_{1}}{y_{1}}+\frac{y_{1}}{Q_{2}}+\frac{Q_{2}}{y_{2}}+\frac{y_{2}}{Q_{3}}+\frac{y_{1}}{y_{3}}+\frac{y_{3}}{y_{2}}.$

The potential function $\mathfrak{PO}$ has six critical points given by

	$\displaystyle y_{1}$	$\displaystyle={y_{3}^{2}}/{y_{2}},$
	$\displaystyle y_{2}$	$\displaystyle=\pm\sqrt{Q_{3}(y_{3}+Q_{2})},$
	$\displaystyle y_{3}$	$\displaystyle=\sqrt[3]{Q_{1}Q_{2}Q_{3}},\ e^{2\pi\sqrt{-1}/3}\sqrt[3]{Q_{1}Q_{2}Q_{3}},\ e^{4\pi\sqrt{-1}/3}\sqrt[3]{Q_{1}Q_{2}Q_{3}}.$

It is easy to see that all critical points are non-degenerate and have the same valuation which lies in the interior of the Gelfand-Cetlin polytope. Hence we have as many critical points as $\dim H^{*}(\operatorname{Fl}(3))=6$ in this case. One can show, using the presentation of the quantum cohomology in [GK95, Theorem 1], that the set of eigenvalues of the quantum cup product by $c_{1}(\operatorname{Fl}(3))$ coincides with the set of critical values of the potential function.

The Floer differential $\mathfrak{m}_{1}^{b}$ is trivial for each critical point $({\boldsymbol{u}},{\boldsymbol{x}})$ of $\mathfrak{PO}$ , and the corresponding Floer cohomology is given by

\mathop{H\!F}\nolimits((L({\boldsymbol{u}}),b),(L({\boldsymbol{u}}),b);\Lambda_{0})\cong H^{*}(L({\boldsymbol{u}});\Lambda_{0})\cong H^{*}(T^{3};\Lambda_{0}).

Example 3.3.

We identify $\operatorname{Gr}(2,4)$ with the adjoint orbit of $\operatorname{diag}(2\lambda,2\lambda,0,0)$ . Setting $Q=T^{2\lambda}$ , the potential function is given by

$\displaystyle\mathfrak{PO}$	$\displaystyle=e^{-x_{2}}T^{-u_{2}+2\lambda}+e^{-x_{1}+x_{2}}T^{-u_{1}+u_{2}}+e^{x_{1}-x_{3}}T^{u_{1}-u_{3}}$
	$\displaystyle\qquad+e^{x_{3}}T^{u_{3}}+e^{x_{2}-x_{4}}T^{u_{2}-u_{4}}+e^{-x_{3}+x_{4}}T^{-u_{3}+u_{4}}$
	$\displaystyle=\frac{Q}{y_{2}}+\frac{y_{2}}{y_{1}}+\frac{y_{1}}{y_{3}}+y_{3}+\frac{y_{2}}{y_{4}}+\frac{y_{4}}{y_{3}}.$	(3.1)

This function has four critical points

(y_{1},y_{2},y_{3},y_{4})=\left((-1)^{i}\sqrt[4]{Q^{2}},\sqrt{-1}^{i}\sqrt[4]{\frac{Q^{3}}{4}},\sqrt{-1}^{i}\sqrt[4]{4Q},(-1)^{i}\sqrt[4]{Q^{2}}\right)

for $i=0,1,2,3$ , and the corresponding critical values are

\displaystyle\mathfrak{PO}=4\sqrt{2}\sqrt{-1}^{i}\sqrt[4]{Q}.

(3.2)

Since $\dim H^{*}(\operatorname{Gr}(2,4))=6$ , one has less critical point than $\dim H^{*}(\operatorname{Gr}(2,4))$ . These critical points are non-degenerate and have a common valuation

{\boldsymbol{u}}_{0}=(\lambda,3\lambda/2,\lambda/2,\lambda)\in\operatorname{Int}\Delta.

Hence there exist four weak bounding cochains $b_{0},\dots,b_{3}$ such that

\mathop{H\!F}\nolimits((L({\boldsymbol{u}}_{0}),b_{i}),(L({\boldsymbol{u}}_{0}),b_{i});\Lambda_{0})\cong H^{*}(L({\boldsymbol{u}}_{0});\Lambda_{0})\cong H^{*}(T^{4};\Lambda_{0})

for $i=0,1,2,3$ . One can show, using the presentation of the quantum cohomology in [ST97, Theorem 0.1], that the set eigenvalues of the quantum cup product by $c_{1}(\operatorname{Gr}(2,4))$ consists of the four critical values of the potential function and the zero eigenvalue with multiplicity two.

Example 3.4.

We identify $\operatorname{Gr}(2,5)$ with the adjoint orbit of $\operatorname{diag}(\lambda,\lambda,0,0,0)$ . Since the Gelfand-Cetlin polytope is defined by (2.8), the potential function is given by

\displaystyle\mathfrak{PO}=\frac{Q}{y_{2}}+\frac{y_{2}}{y_{1}}+\frac{y_{1}}{y_{3}}+\frac{y_{2}}{y_{4}}+\frac{y_{4}}{y_{3}}+\frac{y_{3}}{y_{5}}+y_{5}+\frac{y_{4}}{y_{6}}+\frac{y_{6}}{y_{5}}.

(3.3)

This function has ten critical points defined by

y_{6}^{5}=Q^{5},\quad Qy_{4}=y_{6}(y_{6}^{3}-y_{4}^{2}),

and

y_{1}=\frac{Q}{y_{6}},\quad y_{2}=\frac{Q}{y_{5}},\quad y_{3}=\frac{Q}{y_{4}},\quad y_{5}=\frac{y_{6}^{2}}{y_{4}}.

The set

\displaystyle\left\{5(\zeta_{5}^{i}+\zeta_{5}^{j})Q^{1/5}\mathrel{}\middle|\mathrel{}\zeta_{5}=\exp(2\pi\sqrt{-1}/5)\text{ and }0\leq i<j\leq 4\right\}

(3.4)

of critical values of the potential function coincides with the set of eigenvalues of the quantum cup product by $c_{1}(\operatorname{Gr}(2,5))$ .

4 Floer cohomologies of non-torus fibers

We briefly recall the construction of the $A_{\infty}$ structure $\{\mathfrak{m}_{k}\}_{k\geq 0}$ , omitting various technical details. Let $L$ be a spin, oriented, and compact Lagrangian submanifold in a symplectic manifold $(X,\omega)$ . For an almost complex structure $J$ compatible with $\omega$ , let $\mathcal{M}_{k+1}(J,\beta)$ be the moduli space of stable $J$ -holomorphic maps $v:(\Sigma,\partial\Sigma)\to(X,L)$ from a bordered Riemann surface $\Sigma$ in the class $\beta\in\pi_{2}(X,L)$ of genus zero with $(k+1)$ boundary marked points $z_{0},z_{1},\dots,z_{k}\in\partial\Sigma$ . Then $\mathfrak{m}_{k}=\sum_{\beta\in\pi_{2}(X,L)}T^{\beta\cap\omega}\mathfrak{m}_{k,\beta}\colon H^{*}(L;\Lambda_{0})^{\otimes k}\to H^{*}(L;\Lambda_{0})$ is defined by

\mathfrak{m}_{k,\beta}(x_{1},\dots,x_{k})=(\operatorname{ev}_{0})_{*}(\operatorname{ev}_{1}^{*}x_{1}\cup\dots\cup\operatorname{ev}_{k}^{*}x_{k}),

where $\operatorname{ev}_{i}\colon\mathcal{M}_{k+1}(J,\beta)\to L$ , $[v,(z_{0},\dots,z_{k})]\mapsto v(z_{i})$ is the evaluation map at the $i$ th marked point.

4.1 Holomorphic disks in $(\operatorname{Fl}(3),L_{0})$

We identify $\operatorname{Fl}(3)$ with the adjoint orbit of $\operatorname{diag}(\lambda_{1},0,-\lambda_{2})$ for $\lambda_{1},\lambda_{2}>0$ as in Subsection 2.3. Note that the symplectic form and the first Chern class are given by $\omega=\lambda_{1}\omega_{\mathbb{P}_{1}}+\lambda_{2}\omega_{\mathbb{P}_{2}}$ and $c_{1}(\operatorname{Fl}(3))=2(\omega_{\mathbb{P}_{1}}+\omega_{\mathbb{P}_{2}})$ , respectively.

Recall that the homotopy group $\pi_{2}(\operatorname{Fl}(3))\cong\mathbb{Z}^{2}$ is generated by 1-dimensional Schubert varieties $X_{1}$ and $X_{2}$ , which are rational curves of bidegree $(1,0)$ and $(0,1)$ in $\mathbb{P}_{1}\times\mathbb{P}_{2}\cong\mathbb{P}^{2}\times\mathbb{P}^{2}$ , respectively. Since $L_{0}$ is diffeomorphic to $\operatorname{SU}(2)\cong S^{3}$ , we have $\pi_{1}(L_{0})=\pi_{2}(L_{0})=0$ . The long exact sequence of homotopy groups yields

\pi_{2}(\operatorname{Fl}(3),L_{0})\cong\pi_{2}(\operatorname{Fl}(3))\cong\mathbb{Z}^{2}.

Let $\beta_{1}$ , $\beta_{2}$ be generators of $\pi_{2}(\operatorname{Fl}(3),L_{0})$ corresponding to $X_{1}$ and $X_{2}$ , respectively. The symplectic area of $\beta_{i}$ is given by

\beta_{i}\cap\omega=[X_{i}]\cap(\lambda_{1}\omega_{\mathbb{P}_{1}}+\lambda_{2}\omega_{\mathbb{P}_{2}})=\lambda_{i}.

Let $\tau$ be the anti-holomorphic involution on $\operatorname{Fl}(3)$ defined in (2.5). For a holomorphic disk $v:(D^{2},\partial D^{2})\to(\operatorname{Fl}(3),L_{0})$ , we define a new holomorphic disk $\tau_{*}v:(D^{2},\partial D^{2})\to(\operatorname{Fl}(3),L_{0})$ by

\tau_{*}v(z)=\tau(v(\overline{z})).

Since $L_{0}$ is the fixed point set of $\tau$ , one can glue $v$ and $\tau_{*}v$ along the boundary to obtain a holomorphic curve $w=v\#\tau_{*}v:\mathbb{P}^{1}\to\operatorname{Fl}(3)$ . The induced involution on $\pi_{2}(\operatorname{Fl}(3),L_{0})$ , which is also denoted by $\tau_{*}$ , is given by $\tau_{*}\beta_{1}=\beta_{2}$ . If $v$ represents $\beta_{1}$ or $\beta_{2}$ , then $[w]=\beta_{1}+\beta_{2}=[X_{1}]+[X_{2}]$ , i.e., $w$ is a rational curve of bidegree $(1,1)$ .

Let $\mu_{L_{0}}:\pi_{2}(\operatorname{Fl}(3),L_{0})\to\mathbb{Z}$ be the Maslov index. If we assume $\lambda_{1}=\lambda_{2}$ so that $\tau$ is an anti-symplectic involution, then we have

\mu_{L_{0}}(\beta_{i})=\frac{1}{2}(\mu_{L_{0}}(\beta_{i})+\mu_{L_{0}}(\tau_{*}\beta_{i}))=([X_{1}]+[X_{2}])\cap c_{1}(\operatorname{Fl}(3))=4

for $i=1,2$ . Since the symplectic form $\omega$ and the Lagrangian submanifold $L_{0}$ depend continuously on $\lambda_{1},\lambda_{2}>0$ , the Maslov index $\mu_{L_{0}}(\beta_{1})=\mu_{L_{0}}(\beta_{2})=4$ is independent of $\lambda_{1},\lambda_{2}$ .

To describe holomorphic disks with Lagrangian boundary condition, we identify the unit disk $D^{2}$ with the upper half plane $\mathbb{H}=\mathbb{H}_{+}$ .

Proposition 4.1.

Let $w\colon\mathbb{P}^{1}\to\operatorname{Fl}(3)$ be a holomorphic curve of bidegree $(1,1)$ such that $w(\mathbb{R}\cup\{\infty\})\subset L_{0}$ . After the $\operatorname{SU}(2)$ -action, we may assume

w(\infty)=([1:0:\sqrt{\lambda_{1}/\lambda_{2}}],[1:0:-\sqrt{\lambda_{2}/\lambda_{1}}]).

(4.1)

We can write

w(0)=\Bigl{(}\bigl{[}a_{1}:a_{2}:\sqrt{\lambda_{1}/\lambda_{2}}\bigr{]},\bigl{[}\overline{a}_{1}:\overline{a}_{2}:-\sqrt{\lambda_{2}/\lambda_{1}}\bigr{]}\Bigr{)}\in L_{0}

(4.2)

for some $(a_{1},a_{2})\in S^{3}\setminus\{(1,0)\}$ . Then $w$ is given by

w(z)=\Bigl{(}\bigl{[}cz+a_{1}:a_{2}:\sqrt{\lambda_{1}/\lambda_{2}}(cz+1)\bigr{]},\bigl{[}\overline{c}z+\overline{a}_{1}:\overline{a}_{2}:-\sqrt{\lambda_{2}/\lambda_{1}}(\overline{c}z+1)\bigr{]}\Bigr{)}

with $c/\overline{c}=-(a_{1}-1)/(\overline{a}_{1}-1)$ .

Remark 4.2.

After the action of

\{g\in\operatorname{PSL}(2,\mathbb{R})\,|\,g(0)=0,\,g(\infty)=\infty\}\cong\mathbb{R}_{>0}

on $\mathbb{H}$ , we may assume that $|c|=1$ .

Proof.

The assumptions (4.1) and (4.2) implies that $w$ has the form

w(z)=\Bigl{(}\bigl{[}c_{1}z+a_{1}:a_{2}:\sqrt{\lambda_{1}/\lambda_{2}}(c_{1}z+1)\bigr{]},\bigl{[}c_{2}z+\overline{a}_{1}:\overline{a}_{2}:-\sqrt{\lambda_{2}/\lambda_{1}}(c_{2}z+1)\bigr{]}\Bigr{)}

for some $c_{1},c_{2}\in\mathbb{C}^{*}$ . The Plücker relation

	$\displaystyle 0$	$\displaystyle=-(c_{1}z+a_{1})(c_{2}z+\overline{a}_{1})-\|a_{2}\|^{2}+(c_{1}z+1)(c_{2}z+1)$
		$\displaystyle=(c_{1}-a_{1}c_{1}+c_{2}-\overline{a}_{1}c_{2})z$

implies $c_{1}(\overline{a}_{1}-1)+c_{2}(a-1)=0$ . On the other hand, the Lagrangian boundary condition $w(\mathbb{R})\subset L_{0}$ implies that

\frac{c_{1}x+a_{1}}{c_{1}x+1}=\frac{\overline{c_{2}}x+a_{1}}{\overline{c_{2}}x+1},\quad\frac{a_{2}}{c_{1}x+1}=\frac{a_{2}}{\overline{c_{2}}x+1},\quad x\in\mathbb{R},

which means $c_{2}=\overline{c_{1}}$ . ∎

Note that $\arg c$ is determined by $a_{1}$ up to sign, and the sign corresponds to whether $v=w|_{\mathbb{H}}$ represents $\beta_{1}$ or $\beta_{2}$ . Namely any holomorphic disk in the class $\beta_{i}$ satisfying (4.1) and (4.2) is uniquely determined by $(a_{1},a_{2})$ for $i=1,2$ .

Example 4.3.

Suppose that $(a_{1},a_{2})=(-1,0)$ . Then $c=\pm\sqrt{-1}$ , and the corresponding holomorphic disks are given by

v_{\pm}(z)=\Bigl{(}\bigl{[}z\pm\sqrt{-1}:0:\sqrt{\frac{\lambda_{1}}{\lambda_{2}}}(z\mp\sqrt{-1})\bigr{]},\bigl{[}z\mp\sqrt{-1}:0:-\sqrt{\frac{\lambda_{2}}{\lambda_{1}}}(z\pm\sqrt{-1})\bigr{]}\Bigr{)}.

It is easy to see that the image $v_{+}(\mathbb{H})$ (resp. $v_{-}(\mathbb{H})$ ) is the inverse image of the edge of $\Delta$ given by $u^{(1)}_{1}=u^{(2)}_{1}$ and $u^{(2)}_{2}=0$ (resp. $u^{(1)}_{1}=u^{(2)}_{2}$ and $u^{(2)}_{1}=0$ ), which is the upper (resp. lower) vertical edge emanating from the vertex ${\boldsymbol{0}}=(0,0,0)$ . The generators $\beta_{1},\beta_{2}$ of $\pi_{2}(\operatorname{Fl}(3),L_{0})$ are represented by $v_{+}$ and $v_{-}$ respectively.

4.2 Floer cohomology of the $\operatorname{SU}(2)$ -fiber in $\operatorname{Fl}(3)$

Let $J$ be the standard complex structure on $\operatorname{Fl}(3)$ . Since the fiber $L_{0}$ is $\operatorname{SU}(2)$ -homogeneous, [EL, Proposition 3.2.1] implies the following.

Proposition 4.4.

Any $J$ -holomorphic disk in $(\operatorname{Fl}(3),L_{0})$ is Fredholm regular. Hence the moduli space $\mathcal{M}_{k+1}^{\mathrm{reg}}(J,\beta)$ of $J$ -holomorphic disks in the class $\beta$ with $k+1$ boundary marked points is a smooth manifold of dimension

	$\displaystyle\dim\mathcal{M}_{k+1}^{\mathrm{reg}}(J,\beta)$	$\displaystyle=\dim L_{0}+\mu_{L_{0}}(\beta)+k+1-3$
		$\displaystyle=\mu_{L_{0}}(\beta)+k+1.$

In particular, we have $\dim\mathcal{M}_{2}(J,\beta_{i})=6$ for $i=1,2$ . Proposition 4.1 implies the following:

Corollary 4.5.

Let $U=\operatorname{SU}(2)\setminus\{1\}\cong\{(a_{1},a_{2})\in S^{3}\,|\,a_{1}\neq 1\}$ . Then $\mathcal{M}_{2}(J,\beta_{i})$ has an open dense subset diffeomorphic to $\operatorname{SU}(2)\times U$ on which the evaluation map is given by

\operatorname{SU}(2)\times U\longrightarrow L_{0}\times L_{0}\cong\operatorname{SU}(2)\times\operatorname{SU}(2),\quad(g_{1},g_{2})\longmapsto(g_{1},g_{1}g_{2}).

In particular, $\operatorname{ev}:\mathcal{M}_{2}(J,\beta_{i})\to L_{0}\times L_{0}$ is generically one-to-one.

Since the minimal Maslov number is $\mu_{L_{0}}(\beta_{1})=\mu_{L_{0}}(\beta_{2})=4$ and

\deg\mathfrak{m}_{1,\beta}(x)=\deg x+1-\mu_{L_{0}}(\beta),\quad x\in H^{*}(L_{0};\Lambda_{0}),

the only nontrivial parts of the Floer differential are

\mathfrak{m}_{1,\beta_{i}}:H^{3}(L_{0})\cong H_{0}(L_{0})\longrightarrow H^{0}(L_{0})\cong H_{3}(L_{0})

for $i=1,2$ . Corollary 4.5 implies that for the class $[p]\in H_{0}(L_{0})$ of a point, we have

\mathfrak{m}_{1,\beta_{i}}([p])={\operatorname{ev}_{0}}_{*}[\mathcal{M}_{2}(J,\beta_{i}){}_{\operatorname{ev}_{1}}\!\times\{p\}]=\pm[L_{0}].

To see the sign, we use a result on the orientation of the moduli spaces of pseudo-holomorphic disks by Fukaya, Oh, Ohta, and Ono [FOOO, Theorem 1.5]. The following statement is a slightly weaker version of the result, which is sufficient for our purpose.

Theorem 4.6.

Let $(X,\omega)$ be a compact symplectic manifold, and $\tau$ an anti-symplectic involution on $X$ whose fixed point set $L=\operatorname{Fix}(\tau)$ is non-empty, compact, connected, and spin. Then $\mathfrak{m}_{k,\beta}$ and $\mathfrak{m}_{k,\tau_{*}\beta}$ satisfy

\mathfrak{m}_{k,\beta}(P_{1},\dots,P_{k})=(-1)^{\epsilon}\mathfrak{m}_{k,\tau_{*}\beta}(P_{k},\dots,P_{1}),

where

\epsilon=\frac{\mu_{L}(\beta)}{2}+k+1+\sum_{1\leq i<j\leq k}(\deg P_{i}-1)(\deg P_{j}-1).

Corollary 4.7.

We have $\mathfrak{m}_{1,\beta_{1}}=\mathfrak{m}_{1,\beta_{2}}$ for general $\lambda_{1},\lambda_{2}>0$ .

Proof.

If $\lambda_{1}=\lambda_{2}$ , then $\tau$ is anti-symplectic, and thus Theorem 4.6 implies

\mathfrak{m}_{1,\beta_{1}}=(-1)^{\mu_{L_{0}}(\beta_{1})/2+2}\mathfrak{m}_{1,\tau_{*}\beta_{1}}=\mathfrak{m}_{1,\beta_{2}}.

(4.3)

Corollary 4.5 implies that $\mathcal{M}_{2}(J,\beta_{i})$ depends continuously on $\lambda_{1},\lambda_{2}$ , and hence its orientation is independent of $\lambda_{1},\lambda_{2}$ . Thus (4.3) holds for general $\lambda_{1},\lambda_{2}$ . ∎

Then we have

\mathfrak{m}_{1}([p])=\sum_{i=1}^{2}\mathfrak{m}_{1,\beta_{i}}([p])T^{\beta_{i}\cap\omega}=\pm(T^{\lambda_{1}}+T^{\lambda_{2}})[L_{0}],

which implies the following.

Theorem 4.8.

The Floer cohomology of $L_{0}$ over the Novikov ring $\Lambda_{0}$ is

\mathop{H\!F}\nolimits(L_{0},L_{0};\Lambda_{0})\cong\Lambda_{0}/T^{\min\{\lambda_{1},\lambda_{2}\}}\Lambda_{0}.

Theorem 1.1 is an immediate consequence of Theorem 4.8.

4.3 Holomorphic disks in $(\operatorname{Gr}(2,4),L_{t})$

We identify $\operatorname{Gr}(2,4)$ with the adjoint orbit of $\operatorname{diag}(\lambda,\lambda,-\lambda,-\lambda)$ for $\lambda>0$ . Note that the Kostant-Kirillov form and the first Chern class are given by

\omega=2\lambda\omega_{\mathrm{FS}},\quad c_{1}(\operatorname{Gr}(2,4))=4\omega_{\mathrm{FS}},

respectively, where $\omega_{\mathrm{FS}}$ is the Fubini-Study form on $\mathbb{P}(\bigwedge^{2}\mathbb{C}^{4})$ .

Recall that $\pi_{2}(\operatorname{Gr}(2,4))\cong\mathbb{Z}$ is generated by a 1-dimensional Schubert variety $X_{1}$ , which is a rational curve of degree one in $\mathbb{P}(\bigwedge^{2}\mathbb{C}^{4})$ . Since $\pi_{1}(\operatorname{Gr}(2,4))=\pi_{2}(L_{t})=0$ and $\pi_{1}(L_{t})\cong\mathbb{Z}$ , the exact sequence

0\longrightarrow\pi_{2}(\operatorname{Gr}(2,4))\longrightarrow\pi_{2}(\operatorname{Gr}(2,4),L_{t})\longrightarrow\pi_{1}(L_{t})\longrightarrow 0

implies that $\pi_{2}(\operatorname{Gr}(2,4),L_{t})\cong\mathbb{Z}^{2}$ . Let $\beta_{1},\beta_{2}$ be generators of $\pi_{2}(\operatorname{Gr}(2,4),L_{t})$ such that $\beta_{1}+\beta_{2}=[X_{1}]\in\pi_{2}(\operatorname{Gr}(2,4))$ .

Example 4.9.

Consider a holomorphic curve $w:\mathbb{P}^{1}\to\operatorname{Gr}(2,4)$ of degree one defined by

w(z)=\left[\sqrt{\frac{\lambda+t}{\lambda-t}}(z-\sqrt{-1}):0:z-\sqrt{-1}:-z-\sqrt{-1}:0:\sqrt{\frac{\lambda-t}{\lambda+t}}(z+\sqrt{-1})\right].

(4.4)

Since $w$ maps $\mathbb{R}\cup\{\infty\}$ to $L_{t}$ , the restrictions

	$\displaystyle v_{+}$	$\displaystyle=w\|_{\mathbb{H}_{+}}:(\mathbb{H}_{+},\partial\mathbb{H}_{+})\longrightarrow(\operatorname{Gr}(2,4),L_{t}),$
	$\displaystyle v_{-}$	$\displaystyle=w\|_{\mathbb{H}_{-}}:(\mathbb{H}_{-},\partial\mathbb{H}_{-})\longrightarrow(\operatorname{Gr}(2,4),L_{t})$

to the upper and lower half planes give holomorphic disks representing $\beta_{1}$ and $\beta_{2}$ . We define $\beta_{1}=[v_{+}]$ and $\beta_{2}=[v_{-}]$ . It is easy to see that the symplectic areas of $v_{\pm}$ are given by

\omega(\beta_{1})=\int_{\mathbb{H}_{+}}v_{+}^{*}\omega=\lambda+t,\quad\omega(\beta_{2})=\int_{\mathbb{H}_{-}}v_{-}^{*}\omega=\lambda-t.

In the case where $t=0$ , the disk $v_{+}$ sends $\sqrt{-1}\in\mathbb{H}$ to $v_{+}(\sqrt{-1})=[0:0:0:-1:0:1]$ , which is in the fiber $\Phi^{-1}({\boldsymbol{u}}_{1})$ over the point ${\boldsymbol{u}}_{1}\in\Delta$ defined by $u_{1}^{(2)}=u_{1}^{(1)}=\lambda$ and $u_{2}^{(2)}=0$ (see Figure 2.2). On the other hand, $v_{-}(-\sqrt{-1})=[1:0:1:0:0:0]$ lies on the fiber over the point ${\boldsymbol{u}}_{2}\in\Delta$ defined by $u_{2}^{(2)}=u_{1}^{(1)}=-\lambda$ and $u_{1}^{(2)}=0$ .

Let $\tau_{t}$ be the anti-holomorphic involution on $\operatorname{Gr}(2,4)$ defined in (2.7). Note that $(\tau_{t})_{*}$ is given by $(\tau_{t})_{*}v(z)=\tau_{t}(v(-\overline{z}))$ for $v:(\mathbb{H},\partial\mathbb{H})\to(\operatorname{Gr}(2,4),L_{t})$ . Since $(\tau_{t})_{*}v_{+}=v_{-}$ , the induced involution on $\pi_{2}(\operatorname{Gr}(2,4),L_{t})$ is given by $(\tau_{t})_{*}\beta_{1}=\beta_{2}$ . Then the Maslov index of $\beta_{i}$ is given by

\mu_{L_{t}}(\beta_{i})=\frac{1}{2}\left(\mu_{L_{t}}(\beta_{i})+\mu_{L_{t}}((\tau_{t})_{*}\beta_{i})\right)=[X_{1}]\cap c_{1}(\operatorname{Gr}(2,4))=4

for $i=1,2$ .

We describe holomorphic curves $w\colon\mathbb{P}^{1}\to\operatorname{Gr}(n,2n)$ of degree one such that $w(\mathbb{R}\cup\{\infty\})$ is contained in the Lagrangian fiber $L_{t}$ . Proposition 4.10 below is taken from [Sot01, Theorem 2.1], which is well-known in control theory (cf. e.g. [Ros70]).

Proposition 4.10.

Suppose that a holomorphic curve $w\colon\mathbb{P}^{1}\to\operatorname{Gr}(k,n)={\widetilde{V}}(k,n)/\operatorname{GL}(k,\mathbb{C})$ of degree $d$ is given by

w\colon z\longmapsto\begin{pmatrix}I_{k}\\ F(z)\end{pmatrix}\mod\operatorname{GL}(k,\mathbb{C})

for a rational function $F(z)$ with values in $(n-k)\times K$ matrices. Then there exist matrix valued polynomials $P(z)$ , $Q(z)$ of size $k\times k$ and $(n-k)\times k$ respectively such that

1.

$F(z)=Q(z)P(z)^{-1}$ , i.e., the curve $w$ is given by

$w\colon z\longmapsto\begin{pmatrix}P(z)\\ Q(z)\end{pmatrix}\mod\operatorname{GL}(k,\mathbb{C}),$
2.

$P(z)$ and $Q(z)$ are coprime in the sense there exist matrix valued polynomials $X(z)$ , $Y(z)$ such that $X(z)P(z)+Y(z)Q(z)=I_{k}$ , and
3.

$\deg(\operatorname{det}P(z))=d$ .

Such $P(z)$ and $Q(z)$ are unique up to multiplication of elements in $\operatorname{GL}(k,\mathbb{C}[z])$ .

Note that (2.6) implies that the $\operatorname{U}(n)$ -fiber $L_{t}\subset\operatorname{Gr}(n,2n)={\widetilde{V}}(n,2n)/\operatorname{GL}(n,\mathbb{C})$ consists of

\begin{pmatrix}I_{n}\\ \sqrt{(\lambda-t)/(\lambda+t)}\,A\end{pmatrix}\mod\operatorname{GL}(n,\mathbb{C})

for $A\in\operatorname{U}(n)$ .

Proposition 4.11.

Let $w\colon\mathbb{P}^{1}\to\operatorname{Gr}(n,2n)$ be a holomorphic curve of degree one such that $w(\mathbb{R}\cup\{\infty\})\subset L_{t}$ , and let $F(z)$ denote the corresponding rational function with values in $n\times n$ matrices. By the $\operatorname{U}(n)$ -action, we assume that

F(\infty)=\sqrt{\frac{\lambda-t}{\lambda+t}}I_{n}\in\sqrt{\frac{\lambda-t}{\lambda+t}}\operatorname{U}(n),

(4.5)

and set

F(0)=\sqrt{\frac{\lambda-t}{\lambda+t}}A

(4.6)

for $A\in\operatorname{U}(n)$ . Then there exist

a=\begin{pmatrix}a_{1}\\ \vdots\\ a_{n}\end{pmatrix}\in S^{2n-1}/S^{1}=\mathbb{P}^{n-1}

and $c\in\mathbb{C}\setminus\mathbb{R}$ such that

A=I_{n}+\left(\frac{c^{2}}{|c|^{2}}-1\right)aa^{*},

and

F(z)=\sqrt{\frac{\lambda-t}{\lambda+t}}\frac{1}{z-\overline{c}}(zI_{n}-\overline{c}A)=\sqrt{\frac{\lambda-t}{\lambda+t}}\left(I_{n}-\frac{c-\overline{c}}{z-\overline{c}}aa^{*}\right).

(4.7)

Proof.

Let $F(z)=Q(z)P(z)^{-1}$ be the factorization given in Proposition 4.10. Then the assumptions (4.5), (4.6), and $\deg(\operatorname{det}P(z))=1$ imply that $F(z)$ has the form

F(z)=\sqrt{\frac{\lambda-t}{\lambda+t}}\frac{1}{z-\overline{c}}(zI_{n}-\overline{c}A)

for some $c\in\mathbb{C}$ . The Lagrangian boundary condition $w(\mathbb{R}\cup\{\infty\})\subset L_{t}$ implies that

\frac{1}{x-\overline{c}}(xI_{n}-\overline{c}A)\in\operatorname{U}(n)

for any $x\in\mathbb{R}$ , which means $\overline{c}A+cA^{*}=(c+\overline{c})I_{n}$ , or equivalently, $\overline{c}A-\operatorname{Re}(c)I_{n}$ is skew-hermitian. Hence $\overline{c}A-\operatorname{Re}(c)I_{n}$ has pure imaginary eigenvalues $\sqrt{-1}\alpha_{1},\dots,\sqrt{-1}\alpha_{n}$ , and can be diagonalized by some $g\in\operatorname{U}(n)$ ;

g^{*}(\overline{c}A-\operatorname{Re}(c)I_{n})g=\operatorname{diag}(\sqrt{-1}\alpha_{1},\dots,\sqrt{-1}\alpha_{n}).

Since

g^{*}Ag=\operatorname{diag}\left(\frac{\operatorname{Re}(c)+\sqrt{-1}\alpha_{1}}{\overline{c}},\dots,\frac{\operatorname{Re}(c)+\sqrt{-1}\alpha_{n}}{\overline{c}}\right)\in\operatorname{U}(n)

has eigenvalues of unit norm, we have $\alpha_{i}=\pm\operatorname{Im}(c)$ for $i=1,\dots,n$ . After the action of a permutation matrix, we may assume that $g^{*}Ag$ has the form

g^{*}Ag=\operatorname{diag}(\underbrace{c/\overline{c},\dots,c/\overline{c}}_{k},\underbrace{1,\dots,1}_{n-k})=:C

(4.8)

for some $k$ . Then $F(z)$ is given by

F(z)=\sqrt{\frac{\lambda-t}{\lambda+t}}\frac{1}{z-\overline{c}}g(zI_{n}-\overline{c}C)g^{*}=\sqrt{\frac{\lambda-t}{\lambda+t}}g\operatorname{diag}\left(\frac{z-c}{z-\overline{c}},\dots,\frac{z-c}{z-\overline{c}},1,\dots,1\right)g^{*}

In particular, we have

\operatorname{det}F(z)=\left(\frac{\lambda-t}{\lambda+t}\right)^{n/2}\left(\frac{z-c}{z-\overline{c}}\right)^{k}.

The condition $\deg(\operatorname{det}P(z))=1$ implies that $k=1$ , i.e.,

C=\operatorname{diag}(c/\overline{c},1,\dots,1)=(c/\overline{c}-1)E_{11}+I_{n},

where $E_{11}=\operatorname{diag}(1,0,\dots,0)\in\mathfrak{gl}(n,\mathbb{C})$ . Let $a\in S^{2n-1}\subset\mathbb{C}^{n}$ be the first column of $g$ . Then we have

A=g\left(\left(\frac{c^{2}}{|c|^{2}}-1\right)E_{11}+I_{n}\right)g^{*}=\left(\frac{c^{2}}{|c|^{2}}-1\right)aa^{*}+I_{n},

which proves the proposition. ∎

Remark 4.12.

1.

The equation (4.8) (with $k=1$ ) implies that $\operatorname{det}A=c/\overline{c}=c^{2}/|c|^{2}$ .
2.

After the $\mathbb{R}_{>0}$ -action on the domain, we may assume that $|c|=1$ .

We now assume that $n=2$ . The sign of $\operatorname{Im}(c)=\operatorname{Im}\sqrt{\operatorname{det}A}$ corresponds to the homotopy class of the holomorphic disk $v=w|_{\mathbb{H}}$ . The curve $w$ corresponding to $a=[1:0]$ and $c=-\sqrt{-1}$ coincides with (4.4), and hence $w|_{\mathbb{H}}=v_{+}$ represents $\beta_{1}$ . Thus $v=w|_{\mathbb{H}}$ represents $\beta_{1}$ (resp. $\beta_{2}$ ) when $\operatorname{Im}(c)=\operatorname{Im}\sqrt{\operatorname{det}A}<0$ (resp. $\operatorname{Im}(c)>0$ ).

4.4 Floer cohomologies of the $\operatorname{U}(2)$ -fibers in $\operatorname{Gr}(2,4)$

Since the minimal Maslov number of the $U(2)$ -fiber $L_{t}$ is $\mu_{L_{t}}(\beta_{i})=4$ , we have the following by degree reason.

Lemma 4.13.

The potential function $\mathfrak{PO}\colon H^{1}(L_{t};\Lambda_{0})\to\Lambda_{0}$ for $L_{t}$ is trivial:

\mathfrak{PO}\equiv 0.

The cohomology of $L_{t}\cong S^{1}\times S^{3}$ is given by

H^{*}(L_{t})\cong H^{*}(S^{1})\otimes H^{*}(S^{3}).

Let $\mathbf{e}_{1}\in H^{1}(L_{t};\mathbb{Z})\cong H^{1}(S^{1};\mathbb{Z})$ and $\mathbf{e}_{3}\in H^{3}(L_{t};\mathbb{Z})\cong H^{3}(S^{3};\mathbb{Z})$ be the generators, and write $b=x\mathbf{e}_{1}\in H^{1}(L_{t};\Lambda_{0})$ . Since $\deg\mathfrak{m}_{1,\beta}^{b}=1-\mu_{L_{t}}(\beta)$ and the minimal Maslov number is four, the only nontrivial parts of the Floer differential $\mathfrak{m}_{1}^{b}$ are

	$\displaystyle\mathfrak{m}_{1,\beta_{i}}^{b}$	$\displaystyle:H^{4}(L_{t})\cong H^{1}(S^{1})\otimes H^{3}(S^{3})\longrightarrow H^{1}(L_{t})\cong H^{1}(S^{1}),$
	$\displaystyle\mathfrak{m}_{1,\beta_{i}}^{b}$	$\displaystyle:H^{3}(L_{t})\cong H^{3}(S^{3})\longrightarrow H^{0}(L_{t})\cong\Lambda_{0}$

for $i=1,2$ .

Since $(\operatorname{Gr}(2,4),L_{t})$ is $\operatorname{U}(2)$ -homogeneous, any $J$ -holomorphic disk is Fredholm regular for the standard complex structure $J$ by [EL, Proposition 3.2.1]. Hence one has $\dim\mathcal{M}_{2}(J,\beta_{i})=7$ for $i=1,2$ . Now Proposition 4.11 implies the following:

Corollary 4.14.

Define $f\colon(0,2\pi)\times\mathbb{P}^{1}\to\operatorname{U}(2)$ by $f(\theta,a)=(e^{\sqrt{-1}\theta}-1)aa^{*}+I_{2}$ . For $i=1,2$ , the moduli space $\mathcal{M}_{2}(J,\beta_{i})$ has an open dense subset diffeomorphic to $\operatorname{U}(2)\times(0,2\pi)\times\mathbb{P}^{1}$ such that the evaluation map is given by

\operatorname{U}(2)\times(0,2\pi)\times\mathbb{P}^{1}\longrightarrow L_{t}\times L_{t}\cong\operatorname{U}(2)\times\operatorname{U}(2),\quad(g,\theta,a)\longmapsto(g,g\cdot f(\theta,a)).

Note that $e^{\sqrt{-1}\theta}=\operatorname{det}f(\theta,a)$ is related to $c\in S^{1}$ in Proposition 4.11 by $c=\exp(\sqrt{-1}(\theta/2+\pi))$ or $c=\exp(\sqrt{-1}\theta/2)$ corresponding to $i=1,2$ .

Next we consider $\mathcal{M}_{k+l+2}(J,\beta_{i})$ . For a rational curve $w\colon\mathbb{P}^{1}\to\operatorname{Gr}(2,4)$ given by (4.7), the composition $\operatorname{det}\circ w|_{\partial\mathbb{H}}\colon\partial\mathbb{H}=\mathbb{R}\to L_{t}\cong\operatorname{U}(2)\to S^{1}$ is given by

x\longmapsto\frac{x-c}{x-\overline{c}}.

Hence each boundary point $x\in\partial\mathbb{H}$ is determined by the argument of $\operatorname{det}w(x)=(x-c)/(x-\overline{c})$ . Fixing the $0$ -th and $(k+1)$ -st boundary marked points, we have the following.

Corollary 4.15.

The moduli space $\mathcal{M}_{k+l+2}(J,\beta_{i})$ has an open dense subset diffeomorphic to

\biggl{\{}(g,\theta,a,(t_{i}),(s_{j}))\in\operatorname{U}(2)\times(0,2\pi)\times\mathbb{P}^{1}\times\mathbb{R}^{k}\times\mathbb{R}^{l}\biggm{|}\begin{array}[]{c}0<t_{1}<\dots<t_{k}<\theta,\\ \theta<s_{1}<\dots<s_{l}<2\pi\end{array}\biggr{\}}

on which the evaluation maps $\operatorname{ev}\colon\mathcal{M}_{k+l+2}(J,\beta_{i})\to L_{t}\cong\operatorname{U}(2)$ satisfy

(\operatorname{ev}_{0},\operatorname{ev}_{k+1})\colon(g,\theta,a,(t_{i}),(s_{j}))\longmapsto(g,g\cdot f(\theta,a))

and

\operatorname{det}\operatorname{ev}_{i}(g,\theta,a,(t_{i}),(s_{j}))=\begin{cases}e^{\sqrt{-1}t_{i}}\operatorname{det}g,&i=1,\dots,k,\\ e^{\sqrt{-1}\theta}\operatorname{det}g,&i=k+1,\\ e^{\sqrt{-1}s_{i-k-1}}\operatorname{det}g,&i=k+2,\dots,k+l+2.\end{cases}

Theorem 4.16.

For $b=x\mathbf{e}_{1}\in H^{1}(L_{0};\Lambda_{0}/2\pi\sqrt{-1}\mathbb{Z})\cong\Lambda_{0}/2\pi\sqrt{-1}\mathbb{Z}$ , the deformed Floer differential $\mathfrak{m}_{1}^{b}$ is given by

	$\displaystyle\mathfrak{m}_{1}^{b}(\mathbf{e}_{3})$	$\displaystyle=e^{x}T^{\lambda+t}+e^{-x}T^{\lambda-t},$		(4.9)
	$\displaystyle\mathfrak{m}_{1}^{b}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})$	$\displaystyle=(e^{x}T^{\lambda+t}+e^{-x}T^{\lambda-t})\mathbf{e}_{1}.$		(4.10)

Hence the Floer cohomology of $(L_{t},b)$ is

\mathop{H\!F}\nolimits((L_{t},b),(L_{t},b);\Lambda_{0})\cong\begin{cases}H^{*}(L_{0};\Lambda_{0})\quad&\text{if $t=0$ and $x=\pm\pi\sqrt{-1}/2$},\\ (\Lambda_{0}/T^{\min\{\lambda-t,\lambda+t\}}\Lambda_{0})^{2}&\text{otherwise}.\end{cases}

The Floer cohomology over the Novikov field is given by

\mathop{H\!F}\nolimits((L_{t},b),(L_{t},b);\Lambda)\cong\begin{cases}H^{*}(L_{0};\Lambda)\quad&\text{if $t=0$ and $x=\pm\pi\sqrt{-1}/2$},\\ 0&\text{otherwise}.\end{cases}

Recall that $\mathbf{e}_{1},\mathbf{e}_{3}\in H^{*}(\operatorname{U}(2))$ are given by

\mathbf{e}_{1}=\frac{1}{2\pi\sqrt{-1}}\operatorname{tr}(g^{-1}dg)=\frac{1}{2\pi\sqrt{-1}}d\log(\operatorname{det}g),\quad\mathbf{e}_{3}=\frac{1}{24\pi^{2}}\operatorname{tr}\left[(g^{-1}dg)^{3}\right],

where $g^{-1}dg$ is the left-invariant Maurer-Cartan form on $\operatorname{U}(2)$ .

Lemma 4.17.

For $f(\theta,a)=(e^{\sqrt{-1}\theta}-1)aa^{*}+I_{2}$ , we have

	$\displaystyle f^{*}\mathbf{e}_{1}$	$\displaystyle=\frac{1}{2\pi}\operatorname{tr}(f^{-1}df)=\frac{d\theta}{2\pi},$		(4.11)
	$\displaystyle f^{*}\mathbf{e}_{3}$	$\displaystyle=\frac{1}{24\pi^{2}}\operatorname{tr}(f^{-1}df)^{3}=(1-\cos\theta)\frac{d\theta}{2\pi}\wedge\omega_{\mathbb{P}^{1}},$		(4.12)

where $\omega_{\mathbb{P}^{1}}$ is the Fubini-Study form on $\mathbb{P}^{1}$ normalized in such a way that

\int_{\mathbb{P}^{1}}\omega_{\mathbb{P}^{1}}=1.

Proof.

The first assertion (4.11) follows from $\operatorname{det}f=e^{\sqrt{-1}\theta}$ . Since $f$ is $\operatorname{SU}(2)$ -equivariant with respect to the natural action on $\mathbb{P}^{1}$ and the adjoint action on $\operatorname{U}(2)$ , it suffices to show (4.12) at $a=[1:0]\in\mathbb{P}^{1}$ . A direct calculation gives

f^{-1}df=\begin{pmatrix}\sqrt{-1}d\theta&-(e^{-\sqrt{-1}\theta}-1)d\overline{a}_{2}\\ (e^{\sqrt{-1}\theta}-1)da_{2}&0\end{pmatrix},

so that

\operatorname{tr}(f^{-1}df)^{3}=3(2-e^{\sqrt{-1}\theta}-e^{-\sqrt{-1}\theta})\sqrt{-1}d\theta\wedge da_{2}\wedge d\overline{a}_{2}

at $a=[1:0]$ . On the other hand, the Fubini-Study form on $\mathbb{P}^{1}$ is given by

\omega_{\mathbb{P}^{1}}=\frac{\sqrt{-1}}{2\pi}da_{2}\wedge d\overline{a}_{2}

at $a=[1:0]$ , which proves (4.12). ∎

Proof of Theorem 4.16.

Note that for $m:\operatorname{U}(2)\times\operatorname{U}(2)\to\operatorname{U}(2)$ , $(g_{1},g_{2})\mapsto g_{1}g_{2}$ , we have $m^{*}\mathbf{e}_{i}=\pi_{1}^{*}\mathbf{e}_{i}+\pi_{2}^{*}\mathbf{e}_{i}$ for $i=1,3$ , where $\pi_{1},\pi_{2}\colon\operatorname{U}(2)\times\operatorname{U}(2)\to\operatorname{U}(2)$ are the projections to the first and the second factors. Then $\operatorname{ev}_{j}^{*}\mathbf{e}_{i}$ are given by

	$\displaystyle\operatorname{ev}_{i}^{*}\mathbf{e}_{1}$	$\displaystyle=\frac{1}{2\pi}dt_{i}+g^{*}\mathbf{e}_{1},\quad i=1,\dots,k,$
	$\displaystyle\operatorname{ev}_{k+1+i}^{*}\mathbf{e}_{1}$	$\displaystyle=\frac{1}{2\pi}dt_{i}+g^{*}\mathbf{e}_{1},\quad i=1,\dots,l,$
	$\displaystyle\operatorname{ev}_{k+1}^{*}\mathbf{e}_{3}$	$\displaystyle=f^{}\mathbf{e}_{3}+g^{}\mathbf{e}_{3}=(1-\cos\theta)\frac{d\theta}{2\pi}\wedge\omega_{\mathbb{P}^{1}}+g^{*}\mathbf{e}_{3},$

where $g^{*}\mathbf{e}_{i}$ is the pull-back of $\mathbf{e}_{i}$ by the projection

\operatorname{U}(2)\times(0,2\pi)\times\mathbb{P}^{1}\longrightarrow\operatorname{U}(2),\quad(g,\theta,a)\longmapsto g

to the first factor. For $\theta\in(0,2\pi)$ , set

	$\displaystyle D_{1}(\theta)$	$\displaystyle=\{(t_{1},\dots,t_{k})\in\mathbb{R}^{k}\mid 0<t_{1}<\dots<t_{k}<\theta\},$
	$\displaystyle D_{2}(\theta)$	$\displaystyle=\{(s_{1},\dots,s_{l})\in\mathbb{R}^{l}\mid\theta<s_{1}<\dots<s_{l}<2\pi\}.$

Taking a suitable orientation on $\mathcal{M}_{k+l+2}(\beta_{1},J)$ , we have from Corollary 4.15 that

	$\displaystyle\mathfrak{m}_{k+l+1,\beta_{1}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{3},\underbrace{b,\dots,b}_{l})$
	$\displaystyle\quad=\int_{(0,2\pi)\times\mathbb{P}^{1}}\left(\int_{D_{1}(\theta)}\left(\frac{x}{2\pi}\right)^{k}dt_{1}\wedge\dots\wedge dt_{k}\right)\left(\int_{D_{2}(\theta)}\left(\frac{x}{2\pi}\right)^{l}ds_{1}\wedge\dots\wedge ds_{l}\right)(1-\cos\theta)\frac{d\theta}{2\pi}\wedge\omega_{\mathbb{P}^{1}}$
	$\displaystyle\quad=\int_{(0,2\pi)}\frac{1}{k!}\left(\frac{\theta}{2\pi}\cdot x\right)^{k}\frac{1}{l!}\left(\left(1-\frac{\theta}{2\pi}\right)x\right)^{l}(1-\cos\theta)\frac{d\theta}{2\pi}.$		(4.13)

Hence

	$\displaystyle\mathfrak{m}_{1,\beta_{1}}^{b}(\mathbf{e}_{3})$	$\displaystyle=\int_{0}^{2\pi}\sum_{k,l\geq 0}\frac{1}{k!}\left(\frac{\theta}{2\pi}\cdot x\right)^{k}\frac{1}{l!}\left(\left(1-\frac{\theta}{2\pi}\right)x\right)^{l}(1-\cos\theta)\frac{d\theta}{2\pi}$
		$\displaystyle=\int_{0}^{2\pi}e^{(\theta/2\pi)x}e^{(1-\theta/2\pi)x}(1-\cos\theta)\frac{d\theta}{2\pi}$
		$\displaystyle=\int_{0}^{2\pi}e^{x}(1-\cos\theta)\frac{d\theta}{2\pi}$
		$\displaystyle=e^{x}.$

The same argument as the proof of Corollary 4.7 gives

	$\displaystyle\mathfrak{m}_{k+l+1,\beta_{2}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{3},\underbrace{b,\dots,b}_{l})$	$\displaystyle=(-1)^{k+l}\mathfrak{m}_{k+l+1,\beta_{1}}(\underbrace{b,\dots,b}_{l},\mathbf{e}_{3},\underbrace{b,\dots,b}_{k})$
		$\displaystyle=\mathfrak{m}_{k+l+1,\beta_{1}}(\underbrace{-b,\dots,-b}_{l},\mathbf{e}_{3},\underbrace{-b,\dots,-b}_{k}),$

so that

\mathfrak{m}_{1,\beta_{2}}^{b}(\mathbf{e}_{3})=e^{-x}.

Hence we have

\mathfrak{m}_{1}^{b}(\mathbf{e}_{3})=\sum_{i=1}^{2}\mathfrak{m}_{1,\beta_{i}}^{b}(\mathbf{e}_{3})T^{\beta_{i}\cap\omega}=e^{x}T^{\lambda+t}+e^{-x}T^{\lambda-t}.

Next we compute $\mathfrak{m}_{1}^{b}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})\in H^{1}(L_{0})$ . Note that

\operatorname{ev}_{k+1}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})=(g^{*}\mathbf{e}_{1}+f^{*}\mathbf{e}_{1})\otimes(g^{*}\mathbf{e}_{3}+f^{*}\mathbf{e}_{3})=g^{*}\mathbf{e}_{1}\otimes f^{*}\mathbf{e}_{3}+\dots.

Since only the term $g^{*}\mathbf{e}_{1}\otimes f^{*}\mathbf{e}_{3}$ contribute to $\mathfrak{m}_{k+l+1,\beta_{i}}(b,\dots,b,\mathbf{e}_{1}\otimes\mathbf{e}_{3},b,\dots,b)$ by degree reason, we have

\mathfrak{m}_{k+l+1,\beta_{i}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{1}\otimes\mathbf{e}_{3},\underbrace{b,\dots,b}_{l})=\mathfrak{m}_{k+l+1,\beta_{i}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{1},\underbrace{b,\dots,b}_{l})g^{*}\mathbf{e}_{1}.

Hence we obtain

	$\displaystyle\mathfrak{m}_{1}^{b}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})$	$\displaystyle=\sum_{i=1}^{2}\mathfrak{m}_{1,\beta_{i}}^{b}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})T^{\beta_{i}\cap\omega}$
		$\displaystyle=\sum_{i=1}^{2}\mathfrak{m}_{1,\beta_{i}}^{b}(\mathbf{e}_{1})T^{\beta_{i}\cap\omega}\mathbf{e}_{1}$
		$\displaystyle=(e^{x}T^{\lambda+t}+e^{-x}T^{\lambda-t})\mathbf{e}_{1}.$

∎

Remark 4.18.

Iriyeh, Sakai, and Tasaki [IST13] computed Floer cohomologies $\mathop{H\!F}\nolimits(L,L^{\prime};\mathbb{Z}/2\mathbb{Z})$ of real forms in a compact Hermitian symmetric space, i.e., fixed point sets $L=\operatorname{Fix}(\tau)$ , $L^{\prime}=\operatorname{Fix}(\tau^{\prime})$ of anti-holomorphic and anti-symplectic involutions $\tau$ , $\tau^{\prime}$ . In particular, the Floer cohomology of the $\operatorname{U}(2)$ -fiber $L_{0}=\operatorname{Fix}(\tau_{0})$ with coefficients in $\mathbb{Z}/2\mathbb{Z}$ is given by

\mathop{H\!F}\nolimits(L_{0},L_{0};\mathbb{Z}/2\mathbb{Z})\cong H^{*}(L_{0};\mathbb{Z}/2\mathbb{Z})\cong(\mathbb{Z}/2\mathbb{Z})^{4}.

On the other hand, (4.9) and (4.10) implies that

\mathop{H\!F}\nolimits(L_{0},L_{0};\Lambda^{\mathbb{Z}}_{0})\cong(\Lambda_{0}^{\mathbb{Z}}/2T^{\lambda}\Lambda_{0}^{\mathbb{Z}})^{2},

where

\Lambda_{0}^{\mathbb{Z}}=\left\{\left.\sum_{i=1}^{\infty}a_{i}T^{\lambda_{i}}\,\right|\,a_{i}\in\mathbb{Z},\ \lambda_{i}\geq 0,\ \lim_{i\to\infty}\lambda_{i}=\infty\right\}

is the Novikov ring over $\mathbb{Z}$ .

Remark 4.19.

Here we consider a Lagrangian $\operatorname{U}(n)$ -fiber $L_{t}$ in $\operatorname{Gr}(n,2n)$ for general $n$ . The one-parameter subgroup $g_{\theta}=\exp(\theta\xi)$ of $\operatorname{U}(2n)$ given by

\xi=\begin{pmatrix}0&-E_{11}\\ E_{11}&0\end{pmatrix}\in\mathfrak{u}(2n)

sends

x=\left(\begin{array}[]{ccc|ccc}t&&&\overline{x}^{1}_{1}&\dots&\overline{x}^{n}_{1}\\ &\ddots&&\vdots&&\vdots\\ &&t&\overline{x}^{1}_{n}&\dots&\overline{x}^{n}_{n}\\ \hline\cr x^{1}_{1}&\dots&x^{1}_{n}&-t&&\\ \vdots&&\vdots&&\ddots&\\ x^{n}_{1}&\dots&x^{n}_{n}&&&-t\end{array}\right)\in L_{t}

to $\operatorname{Ad}_{g_{\theta}}(x)\in\mathcal{O}_{{\boldsymbol{\lambda}}}$ whose upper-left $n\times n$ block is given by

(\operatorname{Ad}_{g_{\theta}}(x))^{(n)}=\begin{pmatrix}t(1-2\sin^{2}\theta)-(x^{1}_{1}+\overline{x}^{1}_{1})\sin\theta\cos\theta&-x^{1}_{2}\sin\theta&\dots&-x^{1}_{n}\sin\theta\\ -\overline{x}^{1}_{n}\sin\theta&t\\ \vdots&&\ddots\\ -\overline{x}^{1}_{n}\sin\theta&&&t\end{pmatrix}.

If $\operatorname{Ad}_{g_{\theta}}(x)$ is still in $L_{t}$ , i.e., $(g_{\theta}xg_{\theta}^{*})^{(n)}=tI_{n}$ , then we have $x^{1}_{2}=\dots=x^{1}_{n}=0$ and $\operatorname{Re}x^{1}_{1}=-t\tan\theta$ . Since $|\operatorname{Re}x^{1}_{1}|\leq\sqrt{\lambda^{2}-t^{2}}$ , one has $g_{\theta}(L_{t})\cap L_{t}=\emptyset$ if

|\theta|>\arctan\sqrt{\frac{\lambda^{2}-t^{2}}{t^{2}}}.

Note that the moment map $\mu:\mathcal{O}_{{\boldsymbol{\lambda}}}\to\mathfrak{u}(2n)$ of the $\operatorname{U}(2n)$ -action is given by $\mu(x)=(\sqrt{-1}/2\pi)x$ in our setting. Hence the Hamiltonian of $g_{\theta}$ is given by

H(x)=\frac{\sqrt{-1}}{2\pi}\langle x,\xi\rangle.

Since $\max_{\mathcal{O}_{\lambda}}H=\lambda/\pi$ and $\min_{\mathcal{O}_{\lambda}}H=-\lambda/\pi$ , the norm of $g_{\theta}$ is given by

\int_{0}^{\theta}\Bigl{(}\max_{\mathcal{O}_{\lambda}}H-\min_{\mathcal{O}_{\lambda}}H\Bigr{)}d\theta=\frac{2\lambda}{\pi}\theta.

Hence the displacement energy of $L_{t}$ is bounded from above by

h(t)=\frac{2\lambda}{\pi}\arctan\sqrt{\frac{\lambda^{2}-t^{2}}{t^{2}}}.

Note that $h(t)$ is a concave function on $[-\lambda,\lambda]$ such that $h(\pm\lambda)=0$ , $h(0)=\lambda$ , and $h(t)>\min\{\lambda-t,\lambda+t\}$ for $t\neq 0,\pm\lambda$ .

Theorem 4.20.

The Floer cohomology of the pair $(L_{0},\pi\sqrt{-1}/2\mathbf{e}_{1})$ , $(L_{0},-\pi\sqrt{-1}/2\mathbf{e}_{1})$ is given by

\mathop{H\!F}\nolimits((L_{0},\pm\pi\sqrt{-1}/2\mathbf{e}_{1}),(L_{0},\mp\pi\sqrt{-1}/2\mathbf{e}_{1});\Lambda_{0})\cong(\Lambda_{0}/T^{\lambda}\Lambda_{0})^{2}.

In particular, the Floer cohomology over the Novikov field is trivial;

\mathop{H\!F}\nolimits((L_{0},\pm\pi\sqrt{-1}/2\mathbf{e}_{1}),(L_{0},\mp\pi\sqrt{-1}/2\mathbf{e}_{1});\Lambda)=0.

Proof.

For $b=\sqrt{-1}\pi/2\mathbf{e}_{1}\in H^{1}(L_{0};\Lambda_{0})$ , we have from (4.13) that

	$\displaystyle\mathfrak{m}_{k+l+1,\beta_{i}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{3},\underbrace{-b,\dots,-b}_{l})$
	$\displaystyle\quad=\int_{(0,2\pi)}\frac{1}{k!}\left(\frac{\sqrt{-1}}{4}\theta\right)^{k}\frac{1}{l!}\left(\frac{\sqrt{-1}}{4}\theta-\frac{\pi\sqrt{-1}}{2}\right)^{l}(1-\cos\theta)\frac{d\theta}{2\pi}.$

Hence the Floer differential is given by

	$\displaystyle\delta_{b,-b}(\mathbf{e}_{3})$	$\displaystyle=\sum_{i=1,2}\sum_{k,l\geq 0}\mathfrak{m}_{k+l+1,\beta_{i}}(\underbrace{b,\dots,b}_{k},\mathbf{e}_{3},\underbrace{-b,\dots,-b}_{l})T^{\beta_{i}\cap\omega}$
		$\displaystyle=2T^{\lambda}\int_{0}^{2\pi}\sum_{k,l\geq 0}\frac{1}{k!}\left(\frac{\sqrt{-1}}{4}\theta\right)^{k}\frac{1}{l!}\left(\sqrt{-1}\left(\frac{\theta}{4}-\frac{\pi}{2}\right)\right)^{l}(1-\cos\theta)\frac{d\theta}{2\pi}$
		$\displaystyle=2T^{\lambda}\int_{0}^{2\pi}e^{\sqrt{-1}(\theta/2-\pi/2)}(1-\cos\theta)\frac{d\theta}{2\pi}$
		$\displaystyle=\frac{16}{3\pi}T^{\lambda}.$

Similarly we have

\delta_{b,-b}(\mathbf{e}_{1}\otimes\mathbf{e}_{3})=\frac{32}{3\pi}T^{\lambda}\mathbf{e}_{1},

and consequently,

\mathop{H\!F}\nolimits((L_{0},\pi\sqrt{-1}/2\mathbf{e}_{1}),(L_{0},-\pi\sqrt{-1}/2\mathbf{e}_{1});\Lambda_{0})\cong(\Lambda_{0}/T^{\lambda}\Lambda_{0})^{2}.

The computation of $\mathop{H\!F}\nolimits((L_{0},-\pi\sqrt{-1}/2\mathbf{e}_{1}),(L_{0},\pi\sqrt{-1}/2\mathbf{e}_{1});\Lambda_{0})$ is completely parallel. ∎

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Yuichi Nohara

Faculty of Education, Kagawa University, Saiwai-cho 1-1, Takamatsu, Kagawa, 760-8522, Japan.

e-mail address : nohara@ed.kagawa-u.ac.jp

Kazushi Ueda

Department of Mathematics, Graduate School of Science, Osaka University, Machikaneyama 1-1, Toyonaka, Osaka, 560-0043, Japan.

e-mail address : kazushi@math.sci.osaka-u.ac.jp

Floer cohomologies of non-torus fibers of the Gelfand-Cetlin system

Abstract

1 Introduction

Theorem 1.1.

Theorem 1.2.

Corollary 1.3.

2 Non-torus fibers of the Gelfand-Cetlin system

2.1 Flag manifolds

Example 2.1.

Example 2.2.

2.2 The Gelfand-Cetlin system

Proposition 2.3 (Guillemin and Sternberg [GS83]).

2.3 The case of Fl⁡(3)\operatorname{Fl}(3)

Definition 2.4 (Evans and Lekili [EL, Definition 1.1.1]).

Proposition 2.5.

Proof.

Proposition 2.6.

2.4 The case of Gr⁡(2,4)\operatorname{Gr}(2,4)

Proposition 2.7.

Proof.

Corollary 2.8.

Proof.

Proposition 2.9.

Remark 2.10.

2.5 The case of Gr⁡(2,5)\operatorname{Gr}(2,5)

Proposition 2.11.

Proof.

Proposition 2.12.

Corollary 2.13.

Remark 2.14.

3 Critical points of the potential function

Theorem 3.1 ([NNU10, Theorem 10.1]).

Example 3.2.

Example 3.3.

Example 3.4.

4 Floer cohomologies of non-torus fibers

4.1 Holomorphic disks in (Fl⁡(3),L0)(\operatorname{Fl}(3),L_{0})

Proposition 4.1.

Remark 4.2.

Proof.

Example 4.3.

4.2 Floer cohomology of the SU⁡(2)\operatorname{SU}(2)-fiber in Fl⁡(3)\operatorname{Fl}(3)

Proposition 4.4.

Corollary 4.5.

Theorem 4.6.

Corollary 4.7.

Proof.

Theorem 4.8.

4.3 Holomorphic disks in (Gr⁡(2,4),Lt)(\operatorname{Gr}(2,4),L_{t})

Example 4.9.

Proposition 4.10.

Proposition 4.11.

Proof.

Remark 4.12.

4.4 Floer cohomologies of the U⁡(2)\operatorname{U}(2)-fibers in Gr⁡(2,4)\operatorname{Gr}(2,4)

Lemma 4.13.

Corollary 4.14.

Corollary 4.15.

Theorem 4.16.

Lemma 4.17.

Proof.

Proof of Theorem 4.16.

Remark 4.18.

Remark 4.19.

Theorem 4.20.

Proof.

References

Floer cohomologies of non-torus fibers
of the Gelfand-Cetlin system

2.3 The case of $\operatorname{Fl}(3)$

2.4 The case of $\operatorname{Gr}(2,4)$

2.5 The case of $\operatorname{Gr}(2,5)$

4.1 Holomorphic disks in $(\operatorname{Fl}(3),L_{0})$

4.2 Floer cohomology of the $\operatorname{SU}(2)$ -fiber in $\operatorname{Fl}(3)$

4.3 Holomorphic disks in $(\operatorname{Gr}(2,4),L_{t})$

4.4 Floer cohomologies of the $\operatorname{U}(2)$ -fibers in $\operatorname{Gr}(2,4)$