$T$-equivariant disc potential and SYZ mirror construction

TL;DR

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Contribution

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Abstract

We develop a $G$-equivariant Lagrangian Floer theory and obtain a curved $A_\infty$ algebra, and in particular a $G$-equivariant disc potential. We construct a Morse model, which counts pearly trees in the Borel construction $L_G$. When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the $T$-equivariant toric Landau-Ginzburg mirror of Givental. We also study the $\bS^1$-equivariant Floer theory of a typical singular SYZ fiber (i.e. a pinched torus) and compute its $\bS^1$-equivariant disc potential via the gluing technique developed in \cite{CHL18,HKL}.

Figures

Equations

$$W(z)=\sum _{i=1}^m{q}^{{\alpha }_i}{\vec{z}}^{v_i},$$

$$W_{\lambda }(z)=W(z)+\sum _{i=1}^d{\lambda }_i\log z_i,$$

$$W_T^L=\sum _{i=1}^m\exp (g_i(\check{q}(q))){\mathbf{T}}^{\omega ({\beta }_i)}\exp (v_i\cdot (x_1,\dots ,x_d))+\sum _{j=1}^{\ell }(u_j\cdot (x_1,\dots ,x_d)){\lambda }_j$$

$$W_T^L=W(b)+\sum _{i=1}^{\ell }{h}_i(b){\lambda }_i,$$

$$\Phi (b):=(h_1(b),\dots ,h_{\ell }(b))$$

$$\left(C_{\text{Morse}}^{\bullet }(L;{\Lambda }_0){\otimes }_{{\Lambda }_0}{H}_G^{\bullet }(\mathrm{pt};{\Lambda }_0),\{{\mathfrak{m}}_k^G{\}}_{k\ge 0}\right),$$

$${\mathfrak{m}}_k^G(X_1,\dots ,x\otimes \lambda ,\dots ,X_k)=\lambda \cdot {\mathfrak{m}}_k^G(X_1,\dots ,x\otimes 1,\dots ,X_k),$$

$${\Lambda }_0=\left\{\sum _{i=0}^{\infty }{a}_i{T}^{A_i}|a_i\in \mathbb{C},A_i\in {\mathbb{R}}_{\ge 0};A_i\text{ increases to }+\infty ,\right\},$$

$$W_{{\mathbb{S}}^1}^{L_0}=\log (1-uv)=-\sum _{j=1}^{\infty }\frac{(uv{)}^j}{j}$$

$$\mathrm{ev}:{\mathcal{M}}_{k+1}(\beta ;L)\to L^k,u\mapsto (u(z_1),\dots ,u(z_k)).$$

$$H_2^{\mathrm{eff}}(X,L)=\left\{\beta \in H_2(X,L;\mathbb{Z})\mid \mathcal{M}(\beta ;L)\ne \varnothing \right\}.$$

$${\mathcal{M}}_{k+1}(\beta ;L;\vec{P})={\mathcal{M}}_{k+1}(\beta ;L){\times }_{L^k}\prod P_i.$$

$$\Vert \beta \Vert =\{\begin{array}{ll}\sup \{n\mid \exists {\beta }_1,\dots ,{\beta }_n\in H_2^{\mathrm{eff}}(X,L)\setminus \{{\beta }_0\}\text{ with }{\sum }_{i=1}^n{\beta }_i=\beta \}+\lfloor \omega (\beta )\rfloor -1& \text{if }\beta \ne {\beta }_0;\\ -1& \text{if }\beta ={\beta }_0\end{array}$$

$$\Vert (\mathfrak{d},\beta )\Vert =\{\begin{array}{ll}{\max }_{i\in \{1,\dots ,k\}}\{\mathfrak{d}(i)\}+\Vert \beta \Vert +k& \text{if }k\ne 0;\\ \Vert \beta \Vert & \text{if }k=0.\end{array}$$

$${\mathcal{M}}_{k+1}^{\mathfrak{d}}(\beta ;L;\vec{P}{)}^{{\mathfrak{s}}_{\mathfrak{d},k,\beta ,\vec{P}}}:={\mathfrak{s}}_{\mathfrak{d},k,\beta ,\vec{P}}^{-1}(0)$$

$$C_{(g)}(L;{\Lambda }_0)=\underset{g^{\prime }\le g}{⨁}{\Lambda }_0\cdot {\mathcal{X}}_{g^{\prime }}(L)$$

$${\tilde{\mathfrak{m}}}_k(P_1,\dots ,P_k)=\sum _{\beta \in H_2^{\mathrm{eff}}(X,L)}{\mathbf{T}}^{{\omega }_X(\beta )}{\tilde{\mathfrak{m}}}_{k,\beta }(P_1,\dots ,P_k)$$

$$\tilde{\mathfrak{m}}_{k,\beta}(P_{1},\ldots,P_{k})=\begin{cases}0&\mbox{for $(k,\beta)\neq(0,\beta_{0})$}\\ (-1)^{d}\partial P&\mbox{for $(k,\beta)\neq(1,\beta_{0})$}\\ (\mathrm{ev}_{0})_{*}\left(\mathcal{M}_{k+1}(\beta;L;\vec{P})^{\mathfrak{s}}\right)&\mbox{otherwise.}\end{cases}$$

$$C^{\bullet }(L;{\Lambda }_0{)}^{+}=C^{\bullet }(L;{\Lambda }_0)\oplus {\Lambda }_0\cdot {\mathbf{e}}^{+}\oplus {\Lambda }_0\cdot \mathbf{f}.$$

$${\mathfrak{forget}}_{\vec{a}}:{\mathcal{M}}_{k+1}(\beta ;L;{\vec{P}}^{+})\to {\mathcal{M}}_{k+1-|\vec{a}|}(\beta ;L;\vec{P}),$$

$$({\mathrm{ev}}_0{)}_{\ast }\left({\left([0,1{]}^{|\vec{a}|}\times {\mathcal{M}}_{k+1}(\beta ;L;{\vec{P}}^{+})\right)}^{{\mathfrak{s}}_{{\mathfrak{d}}^{+},k,\beta ,{\vec{P}}^{+}}^{+}}/\sim \right)$$

$${\tilde{\mathfrak{m}}}_1^{+}(\mathbf{f})={\mathbf{e}}^{+}-\mathbf{e}+\mathbf{h},$$

$$\mathbf{h}=\sum _{\beta \ne {\beta }_0}{\mathbf{T}}^{{\omega }_X(\beta )}({\mathrm{ev}}_0{)}_{\ast }\left(([0,1]\times {\mathcal{M}}_2(\beta ;L;\mathbf{e}){)}^{{\mathfrak{s}}^{+}}/\sim \right).$$

$$\sum _{\mu (\beta )\ge 2}({\mathrm{ev}}_0{)}_{\ast }\left(([0,1]\times {\mathcal{M}}_2(\beta ;L;\mathbf{e}){)}^{{\mathfrak{s}}^{+}}/\sim \right)$$

$$W^s(f;p)=\left\{q\in L\mid \underset{t\to +\infty }{\lim }{\Phi }_t(q)=p\right\}\text{mbox}and{W}^u(f;p)=\left\{q\in L\mid \underset{t\to -\infty }{\lim }{\Phi }_t(q)=p\right\}.$$

$$|p|:=d-\mathrm{ind}(p),$$

$$C^{\bullet }(f;{\Lambda }_0)=\underset{p\in \mathrm{Crit}(f)}{⨁}{\Lambda }_0\cdot p,$$

$$\dim \mathcal{M}(p,q)=|q|-|p|-1.$$

$$\overline{\mathcal{M}}(p,q{)}_k=\bigcup _{r_0=p,r_1,\dots ,r_k,r_{k+1}=q\text{ distinct}}\mathcal{M}(r_0,r_1)\times \dots \times \mathcal{M}(r_k,r_{k+1})$$

$${\overline{W}}^u(f;p{)}_k=\bigcup _{r_0=p,r_1,\dots ,r_k\text{ distinct}}\mathcal{M}(r_0,r_1)\times \dots \times \mathcal{M}(r_{k-1},r_k)\times W^u(f;r_k).$$

Full text

1. 1 Introduction
2. 2 A Morse model for Lagrangian Floer theory
3. 3 A Morse model for equivariant Lagrangian Floer theory
4. 4 $T$-equivariant disc potentials of toric manifolds
5. 5 $\mathbb{S}^{1}$-equivariant disc potential for the immersed two-sphere

1. Introduction

Teleman [Tel14] conjectured that the mirror of a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $X$ is a holomorphic map from the mirror $X^{\vee}$ of $X$ to the space of conjugacy classes in $G_{\mathbb{C}}^{\vee}$. Moreover, the mirrors of the symplectic quotients $X/\kern-5.0pt/_{\lambda}G$ for $\lambda\in\mathfrak{g}^{*}/\mathrm{Ad}^{*}_{G}$ are closely related to the fibers this map. This agrees with the work of Hori-Vafa [HV00] on mirror symmetry via gauged linear sigma models.

An important step towards understanding this conjecture is to construct a such map. In this paper, we develop an equivariant version of the SYZ mirror construction using equivariant Lagrangian Floer theory. For a $G$-inavariant Lagranguan submanifold of a Hamiltonian $G$-manifold, its $G$-equivariant disc potential gives a fibration of the mirror family over $H^{2}(BG;\mathbb{C})$. For the concrete computations, we focus on the case when $G=T$ is a compact torus, and study the $T$-equivariant Lagrangian Floer theory of torus fibers of an SYZ fibration.

For instance, let us consider the well-known Landau-Ginzburg mirror of a compact Fano toric $d$-fold $X$ [HV00, Giv95, Giv98, LLY99]. It is given by a pair $((\mathbb{C}^{\times})^{d},W)$, where the superpotential $W:(\mathbb{C}^{\times})^{d}\to\mathbb{C}$ is a holomorphic function of the form

[TABLE]

where $v_{1},\ldots,v_{m}\in\bm{N}\cong\mathbb{Z}^{d}$ are primitive generators of the one-dimensional cones of the fan $\Sigma$ defining $X$, $\vec{z}^{v_{i}}$ denote the corresponding Laurent monomials, and $q^{\alpha_{1}},\ldots,q^{\alpha_{m}}$ are Kähler parameters associated to effective curves classes $\alpha_{i}\in H_{2}(X;\mathbb{Z})$.

Givental [Giv98] showed that the equivariant quantum cohomology $QH_{T}^{\bullet}(X)$ of $X$ is mirror to the equivariant superpotential

[TABLE]

where $\lambda_{1},\ldots,\lambda_{d}$ are the equivariant parameters for the torus $T$ acting on $X$. Iritani [Iri17b, Iri17a] further generalized this to a mirror correspondence between big equivariant quantum cohomology and a universal Landau-Ginzburg potential which has both equivariant and bulk-deformation parameters.

Cho-Oh [CO06] (in the Fano case) and Fukaya-Oh-Ohta-Ono [FOOO10] (in the general compact non-Fano setting) used Lagrangian Floer theory to define the disc potential of a smooth fiber of the toric moment map and showed that it coincides with the Landau-Ginzburg superpotential $W$. From this perspective, $W$ is viewed as the generating function of open Gromov-Witten invariants. Each term in $W$ corresponds to a holomorphic disc of Maslov index $2$ bounded by the torus fiber. This gives an SYZ interpretation of the Landau-Ginzburg mirrors [Aur07, CL10].

It is a natural and interesting question that whether the $W_{\lambda}$ can also be interpreted as a generating function of equivariant disc counts, to which we give an affirmative answer to in this paper.

Theorem 1.1** (Corolloary 4.10).**

Let $X$ be a semi-projective and semi-Fano toric manifold with $\dim_{\mathbb{C}}X=d$, and let $T\subset(\mathbb{S}^{1})^{d}$ be the subtorus determined by the integral basis $u_{1},\ldots,u_{\ell}\in\bm{N}\cong\mathbb{Z}^{d}$. The $T$-equivariant disc potential (Definition 3.11) $W^{L}_{T}$ of a regular fiber $L$ of the toric moment map is given by

[TABLE]

where $\beta_{i}$ are the basic disc classes bounded by the toric fiber, $g_{i}(\check{q}(q))$ is given by the inverse mirror map in equation (4.6), $q^{\alpha}=\mathbf{T}^{\omega(\alpha)}$ are the Kähler parameters, and $\mathbf{T}$ is the formal Novikov variable.

In the Fano case, we have $g_{i}=0$. By taking ${\ell}=d$ and $u_{1},\ldots,u_{d}$ to be the standard basis of $\mathfrak{t}^{d}:=\mathrm{Lie}(T)$, and setting $x_{i}=\log z_{i}$ for $i=1,\ldots,d$, the above expression for $W^{L}_{T}$ equals to $W_{\lambda}$.

In relation with the Teleman’s conjecture, the $T$-equivariant disc potential for a Lagrangian submanifold $L$ with nonnegative minimal Maslov index takes the form

[TABLE]

for degree one weak bounding cochains $b$. $W^{L}_{T}$ can be interpreted as follows: The equivariant term

[TABLE]

defines a map from the formal deformation space of $L$ to $(\mathfrak{t}_{\mathbb{C}}^{\ell})^{*}$, assuming its convergence over $\mathbb{C}$. The non-equivariant term $W$ can be viewed as a family of potentials on the fibers of $\Phi$. It gives a mirror family of the symplectic quotient $X/\kern-5.0pt/_{\lambda}T$.

For instance, let us take $X=\mathbb{C}^{d}$. The non-equivariant part is simply $W=z_{1}+\ldots+z_{d}$. Suppose $X/\kern-5.0pt/_{\lambda}T^{\ell}$ is a semi-Fano toric manifold. A fiber of $\Phi$ is given by $\vec{z}^{u_{j}}=c_{j}$ for some constants $c_{j}$ and $j=1,\ldots,{\ell}$. Then, the restriction of $W$ to the fibers give a mirror family of $X/\kern-5.0pt/_{\lambda}T^{\ell}$.

Remark 1.2**.**

The terms $h_{i}(b)\lambda_{i}$ can be understood as obstruction terms for a homotopy equivalence between the equivariant Floer theory of $L\subset X$ and Floer theory of the quotient $L/T\subset X/\kern-5.0pt/_{\lambda}T^{\ell}$. Daemi-Fukaya [DF17] asserted that this homotopy equivalence can be constructed by Lagrangian correspondence. In our setting, we can compute the mirror equations $h_{i}(b)=0$ explicitly.

The mirror map $c(\lambda)$ is crucial to precisely identify which fiber in the mirror family corresponds to $X/\kern-5.0pt/_{\lambda}T^{\ell}$. By the beautiful work of Woodward-Xu [WX18], the mirror map can be understood as the change from the gauged Floer theory of $X$ to the Fukaya category $\mathrm{Fuk}(X/\kern-5.0pt/_{\lambda}T^{\ell})$ downstairs. Gauged Floer theory is formulated in terms of vortex equations. It would be very interesting to investigate the relation with our formulation.

Equivariant Lagrangian Floer theory has seen substantial recent developments. In the exact setting and $G=\mathbb{Z}/2\mathbb{Z}$, Seidel-Smith [SS10] provided an approach to understanding $G$-equivariant Floer theory by combining Lagrangian Floer theory and family Morse theory [Hut08] on $EG\to BG$. Viterbo [Vit99] illustrated some related ideas for $G=\mathbb{S}^{1}$. Hendricks-Lipshitz-Sarkar [HLS16a, HLS16b] developed a homotopy coherent method to build up a $G$-equivariant Floer theory. Daemi-Fukaya [DF17] used G-equivariant Kuranishi structure [Fuk17] to tackle the G-equivariant transversality problem, and make a formulation using differential forms. Bao-Honda [BH18] defined equivariant Lagrangian Floer cohomology for finite group action via semi-global Kuranishi structures. Mirror Lagrangian objects of equivariant vector bundles was studied by Lekili-Pascaleff [LP16]. There are many interesting works related to this subject such as [Sei].

In Section 3, we construct a Morse model for $G$-equivariant Lagrangian Floer theory by counting pearly trees [BC07, FOOO09a] in the Borel construction $L_{G}$, and study the associated equivariant disc potential. The key ingredient is a collection of Morse models on the finite dimensional approximations of $L_{G}$ satisfying certain compatibility conditions (Proposition 3.6). This is closest to the approach of Seidel-Smith, although the Lagrangians under our consideration are not exact. The Lagrangians of interest in this paper bound non-constant pseudo-holomorphic discs of non-negative Maslov index. These are known as quantum corrections in mirror symmetry, which can have non-trivial contributions to the equivariant parameters in the disc potential. Since there are only finitely many generators for each finite dimensional approximation of $L_{G}$, it better serves for computations and for the SYZ mirror construction.

In order for the equivariant paramaeters appearing in $W_{\lambda}$ to manifest. In Section 3.2 we adapt the homotopy unit construction developed by Fukaya-Oh-Ohta-Ono [FOOO09b, Chapter 7] to construct a curved $A_{\infty}$-algebra

[TABLE]

which satisfies (Theorem 3.7)

[TABLE]

under the assumption that $L$ has minimal Maslov index zero and $G$ is a product of unitary groups. Here $\Lambda_{0}$ is the Novikov ring

[TABLE]

This allows us to view $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$ as a graded coefficient ring. We note that the terms of the form $\mathfrak{m}^{G}_{k}(x_{1}\otimes 1,\ldots x_{k}\otimes 1)$ are series in the equivariant parameters and receive non-trivial contributions from pearly trees in $L_{G}$.

In Section 5, we study the $\mathbb{S}^{1}$-equivariant Floer theory of the immersed two-sphere with a single nodal point. This is also known as the pinched two-torus and is the most common singular fiber appearing in an SYZ fibration. Even although it does not bound any non-constant holomorphic discs of Maslov index zero, it still has a non-trivial equivariant disc potential from the contribution of constant polygons at the nodal point. As the corresponding moduli spaces have non-trivial obstructions, the gluing technique via the isomorphism in the Fukaya category between smooth and pinched tori in [HKL18] is crucial in the computation of the explicit expression of the equivariant disc potential.

Theorem 1.3** (Theorem 5.7).**

The $\mathbb{S}^{1}$-equivariant disc potential of the immersed sphere $L_{0}$ is

[TABLE]

where $(u,v)\in\Lambda_{0}^{2}$, $\mathrm{val}(uv)>0$ are the formal deformation parameters corresponding to the degree one immersed generators of $L_{0}$.

Floer theory for immersed Lagrangians was developed by Akaho-Joyce [AJ10], which is in line with the theory of Fukaya-Oh-Ohta-Ono [FOOO09b] for smooth Lagrangians. In [DRET18], a different method via Legendrian topology is used to develop the Floer theory of immersed Lagrangian surfaces.

In a subsequent work [HKLZ19], we compute the equivariant disc potential for Lagrangian immersions in toric Calabi-Yau manifolds. They have very interesting expressions in relation with the Gross-Siebert slab functions [GS11] and mirror maps of toric Calabi-Yau manifolds [CLL12, CCLT]. It is closely related to the topological vertex [KL01, GZ02, FL13, FLT, FLZ16].

Acknowledgments

The first and second named authors express their deep gratitude to Cheol-Hyun Cho for explaining his point of view of equivariant Floer theory using Cartan model. The second named author also thanks Ben Webster and Kevin Costello for the discussion on Teleman’s conjecture at Perimeter Institute.

2. A Morse model for Lagrangian Floer theory

2.1. The singular chain model

Let $(X,\omega_{X})$ be a symplectic manifold of real dimension $2d$. We assume that $X$ is convex at infinity or geometrically bounded if it is non-compact. Choose a compatible almost complex structure $J_{X}$. For a closed, connected, and relatively spin Lagrangian submanifold $L$, Fukaya-Oh-Ohta-Ono constructed a countably generated subcomplex $C^{\bullet}(L;\Lambda_{0})$ of the singular chain complex $S^{\bullet}(L;\Lambda_{0})$ (regarded as a cochain complex) and an $A_{\infty}$-algebra $(C^{\bullet}(L;\Lambda_{0}),\tilde{\mathfrak{m}})$. We briefly recall their construction below, which will be used in constructing the equivariant Morse model in Section 3. We refer the reader to [FOOO09b] for more details.

For $\beta\in H_{2}(X,L;\mathbb{Z})$, we denote by $\mathcal{M}(\beta;L)$ the moduli space of $J_{X}$-holomorphic stable maps from a bordered Riemann surface of genus [math] representing $\beta$ and by $\mathcal{M}_{k+1}(\beta;L)$ the moduli space with $k+1$ boundary marked points $z_{0},\ldots,z_{k}$ ordered counter-clockwise. The moduli space $\mathcal{M}_{k+1}(\beta;L)$ has virtual dimension $\mu(\beta)+k-2+d$, where $\mu(\beta)$ is the Maslov index of $\beta$. It carries the evaluation map

[TABLE]

Let $H_{2}^{\mathrm{eff}}(X,L)$ denote the monoid of effective classes in $H_{2}(X,L;\mathbb{Z})$, represented by $J_{X}$-holomorphic maps, i.e.

[TABLE]

For a $k$-tuple $\vec{P}=(P_{1},\ldots,P_{k})$ of smooth singular chains of $L$, we denote by $\mathcal{M}_{k+1}(\beta;L;\vec{P})$ the fiber product (in the sense of Kuranishi structures) of $\mathcal{M}_{k+1}(\beta;L)$ with $P_{1},\dots,P_{k}$, i.e.,

[TABLE]

We recall a generation $g\in\mathbb{Z}$, which will be used to determine perturbations of moduli spaces and subsets of singular chains inductively in [FOOO09b, Section 7]. As preliminaries, for $\beta\in H_{2}^{\mathrm{eff}}(X,L)$, we first set

[TABLE]

where $\beta_{0}$ is the constant disc class. Also, we employ a function $\mathfrak{d}:\{1,\ldots,k\}\to\mathbb{Z}_{\geq 0}$ to keep track of generations of inputs. For a pair $(\mathfrak{d},\beta)$ with $\beta\in H_{2}^{\mathrm{eff}}(X,L)$, we define

[TABLE]

For a generation $g$, we shall inductively choose a countable set $\mathcal{X}_{g}(L)$ of smooth singular chains on $L$ and a system of multisections $\mathfrak{s}_{\mathfrak{d},k,\beta,\vec{P}}$ for $\mathcal{M}^{\mathfrak{d}}_{k+1}(\beta;L;\vec{P})$ satisfying $\left\lVert(\mathfrak{d},\beta)\right\rVert=g$. Here the superscript $\mathfrak{d}$ is written to emphasize the generations of the inputs $P_{i}\in\mathcal{X}_{\mathfrak{d}(i)}(L)$. Namely, at each inductive step, new multisections $\mathfrak{s}_{\mathfrak{d},k,\beta,\vec{P}}$ are chosen to be transversal to the zero section and extend the multisections previously chosen for the boundary strata $\partial\mathcal{M}_{k+1}(\beta;L;\vec{P})$. (To achieve transversality, the Kuranishi structures for the moduli spaces $\mathcal{M}_{k+1}(\beta;L)$ are chosen to be weakly submersive.) Moreover the zero locus

[TABLE]

is triangulated extending the triangulation on its boundary strata. The new simplices in the triangulation are then regarded as elements of $\mathcal{X}_{g}(L)$. Additional singular simplices are added to $\mathcal{X}_{g}(L)$ so that

[TABLE]

remains a subcomplex of $S^{\bullet}(L;\Lambda_{0})$ isomorphic on cohomology.

Let $C^{\bullet}(L;\Lambda_{0})=\lim_{\to}C_{(g)}(L;\Lambda_{0})$. The $A_{\infty}$-map $\tilde{\mathfrak{m}}_{k}:C^{\bullet}(L;\Lambda_{0})^{\otimes k}\to C^{\bullet}(L;\Lambda_{0})$ is defined by

[TABLE]

where

[TABLE]

Here, $\partial$ is the coboundary map on $C^{\bullet}(L;\Lambda_{0})$. The map $\tilde{\mathfrak{m}}_{k,\beta}$ is of degree $2-k-\mu(\beta)$.

Remark 2.1**.**

We note that it is only possible to choose finitely many multisections at once while still having them being sufficiently close to the original Kuranishi maps. To deal with this technical difficulty, Fukaya-Oh-Ohta-Ono introduced the notion of $A_{n,K}$-structure associated to $L$ using moduli space of pseudo-holomorphic discs with bounded energy and number of marked points in [FOOO09b, Chapter 7]. The $A_{\infty}$-structure $\tilde{\mathfrak{m}}$ on $C^{\bullet}(L;\Lambda_{0})$ is obtained from the $A_{n,K}$-structures via homological techniques.

Remark 2.2**.**

In order to define a $\mathbb{Z}$-graded Floer cohomology, the conventional grading of the map $\mathfrak{m}_{k}$ is $2-k$. This means aside from the case of graded Lagrangian submanifolds, one has to define the Novikov ring $\Lambda_{0}$ using an extra grading parameter in order to compensate for the Maslov indices. In this paper, we work on the chain level and do not follow this convention. The grading of $\tilde{\mathfrak{m}}_{k,\beta}$ is crucial in understanding the vanishing of certain terms when doing computations with the Morse model.

The singular chain model $(C^{\bullet}(L;\Lambda_{0}),\tilde{\mathfrak{m}})$ constructed in [FOOO09b] does not have a strict unit in general. It was shown that the fundamental cycle $\bm{e}$ of $L$ is a homotopy unit. We briefly describe key properties and ideas below, and refer the readers to [FOOO09b, Chapter 3.3] for the precise definition of a homotopy unit and to [FOOO09b, Chapter 7.3] for details of the homotopy unit construction.

The constructed $A_{\infty}$-algebra $(C^{\bullet}(L;\Lambda_{0}),\tilde{\mathfrak{m}})$ will be enlarged to a unital $A_{\infty}$-algebra $(C^{\bullet}(L;\Lambda_{0})^{+},\tilde{\mathfrak{m}}^{+})$ that is homotopy equivalent to $C^{\bullet}(L;\Lambda_{0})$. At first, we enlarge the chain complex $C^{\bullet}(L;\Lambda_{0})$ by adding a generator $\bm{e}^{+}$ of degree [math] serving as the strict unit and a generator $\bm{f}$ of degree $(-1)$ serving as a homotopy between $\bm{e}$ and $\bm{e}^{+}$, that is,

[TABLE]

Then the $A_{\infty}$-maps $\tilde{\mathfrak{m}}^{+}=\{\tilde{\mathfrak{m}}^{+}_{k}\}_{k\geq 0}$ are defined with the following properties$\colon$

1. (1)

The restriction of $\tilde{\mathfrak{m}}^{+}$ to $C^{\bullet}(L;\Lambda_{0})$ agrees with $\tilde{\mathfrak{m}}$, 2. (2)

$\bm{e}^{+}$ is the strict unit, i.e.

1. (i)

$\tilde{\mathfrak{m}}^{+}_{2}(\bm{e}^{+},x)=(-1)^{|x|}\tilde{\mathfrak{m}}^{+}_{2}(x,\bm{e}^{+})=x$ for $x\in C^{\bullet}(L;\Lambda_{0})^{+}$, 2. (ii)

$\tilde{\mathfrak{m}}^{+}_{k}(\ldots,\bm{e}^{+},\ldots)=0$ for $k\neq 2$. 3. (3)

$\bm{f}$ is a homotopy between $\bm{e}$ and $\bm{e}^{+}$.

Later on, we will define a disc potential (Definition 2.8), which is to be of the most interest to us in the manuscript, after passing to the enlarged $A_{\infty}$-algebra with a strict unit. To compute such a disc potential from the original singular chain model (before the enlargement), we need to examine the homotopy $\bm{f}$ between $\bm{e}$ and $\bm{e}^{+}$.

A construction and properties of the homotopy $\bm{f}$ are in order. Let $\vec{a}=(a_{1},\ldots,a_{|\vec{a}|})$ be an ordered subset of $\{1,\ldots,k\}$ satisfying $a_{1}<\ldots<a_{|\vec{a}|}$. For $\vec{P}=(P_{1},\ldots,P_{k-|\vec{a}|})$ with $P_{i}\in\mathcal{X}_{\mathfrak{d}(i)}(L)$, let $\vec{P}^{+}$ be the $k$-tuple obtained by inserting $\bm{e}$ into the $\vec{a}$-th places of $\vec{P}$. We assume $L\in\mathcal{X}_{0}(L)$. We set $\mathfrak{d}^{+}(i)=g_{i}^{+}$ if $\vec{P}^{+}=(P_{1}^{+},\ldots,P_{k}^{+})$ where $P^{+}_{i}\in\mathcal{X}_{g^{+}_{i}}(L)$. Consider the forgetful map

[TABLE]

which forgets the $\vec{a}$-th marked points and then stabilizes.

We choose a perturbation on $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L;\vec{P}^{+})$ as follows: For a splitting $\vec{a}^{1}\coprod\vec{a}^{2}=\vec{a}$ of $\vec{a}$, we denote by $\vec{P}^{\prime}$ the $(k-|\vec{a}^{1}|)$-tuple given by removing $\bm{e}$ from the $\vec{a}^{1}$-th places of $\vec{P}^{+}$. Let $(t_{1},\ldots,t_{|\vec{a}|})$ be coordinates on $[0,1]^{|\vec{a}|}$. If $t_{i}=0$ for all $i\in\vec{a}^{1}$, we take the perturbation $\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}}$ transversal to the zero section obtained by just inserting $\bm{e}$ into the $a_{1},\ldots,a_{|\vec{a}^{1}|}$-th place. If $t_{i}=1$ for all $i\in\vec{a}^{1}$, we take the perturbation pulled back via the map $\mathfrak{forget}_{\vec{a}^{1}}:\mathcal{M}_{k+1}(\beta;L;\vec{P}^{+})\to\mathcal{M}_{k+1-|\vec{a}^{1}|}(\beta;L;\vec{P}^{\prime})$ in (2.3). Finally, we take a perturbation $\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}}^{+}$ on $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L;\vec{P}^{+})$ transversal to the zero section interpolating between them.

The maps $\tilde{\mathfrak{m}}^{+}_{k,\beta}$ for $(k,\beta)(\neq(1,\beta_{0}))$ with inputs $\bm{f}$ inserted into $\vec{a}$-th place of $\vec{P}$ are defined by

[TABLE]

where $\sim$ is the equivalence relation collapsing fibers of $\mathfrak{forget}_{\vec{a}}$, see [FOOO09b, Definition 7.3.28]. Notice that the zero locus of a pullback multisection is a degenerate singular chain which becomes zero in the quotient.

We also set $\tilde{\mathfrak{m}}^{+}_{1,\beta_{0}}(\bm{f})=\bm{e}^{+}-\bm{e}$ and therefore

[TABLE]

where

[TABLE]

Note that $\bm{h}\equiv 0\mod\Lambda_{+}\cdot C^{\bullet}(L;\Lambda_{0})$ and the terms

[TABLE]

are of degrees at most $-2$ and therefore are degenerate singular chains.

2.2. Morse homology $=$ singular homology: an alternative approach

Pearl complex was developed by Biran and Cornea [BC07, BC09] for monotone Lagrangians (a similar complex also appeared in an earlier work of Oh [Oh96]). It has many important applications, including the proof of homological mirror symmetry for Fermat Calabi-Yau hypersurfaces by Sheridan [She15].

A Morse model of Lagrangian Floer theory was constructed in [FOOO09a] based on their singular chain model. We follow their construction to construct a Morse model in Section 2.3, with a modification that we add certain degenerate chains as summands in the definition of unstable chains $\Delta_{p}$ of critical points $p$ and the chains of forward orbits $G(\tau)$ of singular chains $\tau$. The main reason for the modification is that there are unwanted degenerate chains appeared in $\partial\Delta_{p}$ and $\partial G(\tau)$, and we ‘contract them away’ by adding the cones over the unwanted terms. This modification enables us to realize a Morse complex as a singular chain complex, which will be explained in this section.

Let $f:L\to\mathbb{R}$ be a Morse function. Let $V$ be a negative pseudo-gradient vector field for $f$, i.e. $df(V)|_{p}\leq 0$ and the equality holds if and only if $p\in\mathrm{Crit}(f)$. For each $p\in\mathrm{Crit}(f)$, $V$ coincides with the negative gradient vector field for the Euclidean metric on a Morse chart of $p$. Let $\Phi_{t}$ be the flow of $V$. For each $p\in\mathrm{Crit}(f)$, we denote by $W^{s}(f;p)$ and $W^{u}(f;p)$ its stable and unstable manifolds respectively. Namely,

[TABLE]

The degree of $p$ is denoted by $|p|$ and defined by

[TABLE]

where $\mathrm{ind}(p)$ is the Morse index of $p$. Then $|p|=\dim{W^{s}(f;p)}=\mathrm{codim}\,W^{u}(f;p)$. Let $C^{\bullet}(f;\Lambda_{0})$ be the cochain complex

[TABLE]

with grading given by $|p|$.

We assume that $V$ satisfies the Smale condition and call such a pair $(f,V)$ Morse-Smale. For $p,q\in\mathrm{Crit}(f)$, let $\mathcal{M}(p,q)$ be the moduli space of flow lines from $p$ to $q$. The Smale condition implies

[TABLE]

The moduli space $\mathcal{M}(p,q)$ and the unstable manifold $W^{u}(f;p)$ have natural compactifications to smooth manifolds with corners $\overline{\mathcal{M}}(p,q)$ and $\overline{W}^{u}(f;p)$ whose $k$ codimensional strata consist of $k$-times broken flow lines

[TABLE]

and

[TABLE]

In particular, we have

[TABLE]

In [FOOO09a], a (non-unital) $A_{\infty}$-algebra structure was constructed on $C^{\bullet}(f;\Lambda_{0})$ assuming $L$ has a triangulation whose simplices are the closures of $W^{u}(f;p)$. To establish an isomorphism between Morse and singular cohomology, we have to associate to each critical point $p$ of degree $|p|$ a singular chain $\Delta_{p}\in S^{|p|}(L;\mathbb{Z})$. A natural candidate for $\Delta_{p}$ is a fundamental chain for $\overline{W}^{u}(f;p)$. If we make such a natural choice, several issues occur.

Suppose that we choose such a natrual chain $\Delta_{p}$. From (2.7), one can see that in general $\overline{W}^{u}(f;p)$ has boundary components of the form $\overline{\mathcal{M}}(p,r)\times\overline{W}^{u}(f;r)$ with $|r|\geq|p|+2$. Since $\dim\overline{\mathcal{M}}(p,r)\geq 1$, and the image of $\overline{\mathcal{M}}(p,r)\times\overline{W}^{u}(f;r)$ in $L$ is supported on $\overline{W}^{u}(f;r)$, the facet of $\Delta_{p}$ representing $\overline{\mathcal{M}}(p,r)\times\overline{W}^{u}(f;r)$ is a degenerate chain on $L$. Thus, for the assignment $p\mapsto\Delta_{p}$ to be a chain map, one should deal with degenerate chains. On the other hand, $\mathcal{X}_{g}(L)$ are constructed including degenerate chains since the $\tilde{\mathfrak{m}}_{k}$ product of degenerate chains may no longer be a degenerate chain.

We overcome this difficulty by adding certain degenerate chains to the fundamental chains (of the compactified unstable submanifolds) and define a map $\iota:C^{\bullet}(f;\mathbb{Z})\to S^{\bullet}(L;\mathbb{Z})$ by $\iota(p)=\Delta_{p}$ accordingly such that it is a chain map which induces an isomorphism on cohomology, see Figure 1.

More details of a modification are in order. We start with smooth cubical singular chains

[TABLE]

representing the fundamental classes of $\overline{W}^{u}(f;p)$ and $\overline{\mathcal{M}}(p,q)$, respectively. Their boundaries are given as

[TABLE]

To regard $d_{p}$ as simplicial singular chains, we choose triangulations for (the domain of) $d_{p}$ and $m_{p,q}$ inductively on $\mathrm{ind}(p)=d-|p|$. For any critical point $p$ with $\mathrm{ind}(p)=0$, $d_{p}$ and $m_{p,q}$ are [math]-simplices. Suppose that we have chosen triangulations for $d_{p}$ and $m_{p,q}$ for any $p$ with $\mathrm{ind}(p)<\ell$. For $\mathrm{ind}(p)=\ell$, $m_{p,q}$ with $|q|=|p|+1$ are [math]-simplices. Suppose we have chosen triangulation for $m_{p,q}$ for $|q|-|p|=1,\ldots,m^{\prime}$, $m^{\prime}<m$. For $|q|-|p|=m$, we first triangulate $\partial m_{p,q}$ as follows: each $m_{p,r}\times m_{r,q}$ in $\partial m_{p,q}$ is a product of simplicial complexes. We choose linear orders on vertices of $m_{p,r}$ and $m_{r,q}$ (compatible with their orientations). Then there exists a unique triangulation $m_{p,r}\otimes m_{r,q}$ of $m_{p,r}\times m_{r,q}$ such that the vertices of $m_{p,r}\otimes m_{r,q}$ are pairs $(x,y)$, where $x$ is a vertex of $m_{p,r}$ and $y$ is a vertex of $m_{r,q}$, and an $n$-simplex in $m_{p,r}\otimes m_{r,q}$ is defined by a set $\{(x_{0},y_{0}),\ldots,(x_{n},y_{n}))\}$ such that $x_{0}\leq\ldots\leq x_{n}$, $y_{0}\leq\ldots\leq y_{n}$, $\{x_{0},\ldots,x_{n}\}$ defines a simplex $\tau_{1}$ on $m_{p,r}$ and $\{y_{0},\ldots,y_{n}\}$ defines a simplex $\tau_{2}$ on $m_{r,q}$ with $\dim\tau_{1}+\dim\tau_{2}\geq n$. Notice that this is the standard product structure in the category of simplicial sets. We then triangulate $m_{p,q}$ extending the triangulation on $\partial m_{p,q}$. Finally, we triangulate $d_{p}$ extending the triangulation on $\partial d_{p}$ given by $m_{p,r}\otimes d_{r}$.

For each $m_{p,r}\times d_{r}$ in the boundary of $d_{p}$, let $Cm_{p,r}$ be the (simplicial) cone over $m_{p,r}$ with vertices

[TABLE]

and simplices

[TABLE]

for $n\geq 1$. In particular, this means $\{*\}$ is the vertex of the cone. As a singular chain on $L$, the map $Cm_{p,r}\to L$ is defined by composing the contraction $Cm_{p,r}\to m_{p,r}$ with the map $m_{p,r}\to L$. Note that for two simplicial complexes $A$ and $B$, we have $C(A\otimes B)=CA\otimes CB$.

We construct $\Delta_{p}$ inductively on $\mathrm{ind}(p)$ as follows: For $\mathrm{ind}(p)=0$, we set $\Delta_{p}=d_{p}$. Suppose we have constructed $\Delta_{p}$ for $\mathrm{ind}(p)<\ell$. For $\mathrm{ind}(p)=\ell$, we define

[TABLE]

and set

[TABLE]

By construction, we have

[TABLE]

Theorem 2.3**.**

We define a map

[TABLE]

where $\Delta_{p}$ is as defined in (2.8). Then the map $\iota$ is a chain map which induces an isomorphism on cohomology $H^{\bullet}(C^{\bullet}(f;\mathbb{Z});\delta)\cong H^{\bullet}(L;\mathbb{Z})$ where $\delta$ denotes the Morse differential.

Remark 2.4**.**

The approach in [HL99] is to mod out by degenerate chains. If we were to consider $S^{\bullet}(L;\mathbb{Z})$ modded out by the subcomplex of degenerate chains, a proof can be found therein. Since we do not wish to mod out by degenerate chains, we would have to modify their arguments.

Before proving Theorem 2.3, let us introduce some terminologies. We say a smooth singular chain is generic if each face of every simplex intersects the stable submanifolds of the critical points of $f$ transversely. Let $C^{\bullet}_{gen}(L;\mathbb{Z})$ be the subcomplex of $S^{\bullet}(L;\mathbb{Z})$ generated by the generic singular chains. It is easy to see that $C^{\bullet}_{gen}(L;\mathbb{Z})$ and $S^{\bullet}(L;\mathbb{Z})$ are isomorphic on cohomology.

For a generic simplex $\tau$, we denote by $\mathcal{M}(\tau,q)$ the moduli space of flow lines from $\tau$ to a critical point $q$. Its dimension is given by $\dim\mathcal{M}(\tau,q)=\dim\tau-\mathrm{ind}(q)$. Let $\tau(j)$ denote the codimension $j$ strata of $\tau$. The moduli space $\mathcal{M}(\tau,q)$ has a natural compactification to a smooth manifold with corners $\overline{\mathcal{M}}(\tau,q)$. Its strata of codimension $k$ are

[TABLE]

For $k=1$, we have

[TABLE]

We define the forward orbit $\mathscr{F}(\tau)$ of $\tau$ to be the set $\mathscr{F}(\tau):=[0,\infty)\times\tau$ with the map

[TABLE]

where $\Phi_{t}$ is the flow of the negative pseudo-gradient vector field $V$. The orbit $\mathscr{F}(\tau)$ has a natural compactification to a smooth manifold with corners $\overline{\mathscr{F}}(\tau)$, whose codimension $k$ strata are

[TABLE]

When $k=1$, we have

[TABLE]

Now, we are ready to prove the main theorem of this section.

Proof of Theorem 2.3.

The assertion that $\iota$ is a chain map follows from (2.9). To show that $\iota$ induces an isomorphism on cohomology, we modify the proof in [HL99]. We have a chain map $A:C^{\bullet}_{gen}(L;\mathbb{Z})\to C^{\bullet}(f;\mathbb{Z})$ defined by

[TABLE]

In particular, $\mathcal{M}(\tau,p)$ is empty when $\tau$ is a degenerate simplex. Clearly, $A\circ\iota=id:C^{\bullet}(f;\mathbb{Z})\to C^{\bullet}(f;\mathbb{Z})$.

We define a homotopy $G:C^{\bullet}_{gen}(L;\mathbb{Z})\to S^{\bullet-1}(L;\mathbb{Z})$ as follows. Let $m_{\tau,q}$ be a smooth cubical singular chain in $C^{\square}_{\dim\tau-\mathrm{ind}(q)}(\overline{\mathcal{M}}(\tau,q);\mathbb{Z})$ representing the fundamental class of $\overline{\mathcal{M}}(\tau,q)$ satisfying

[TABLE]

We again triangulate $m_{\tau,q}$ inductively on the dimension to regard them as simplicial singular chains. We also choose smooth singular chains $F(\tau)$ representing the fundamental class of $\overline{\mathscr{F}}(\tau)$ satisfying

[TABLE]

We then construct $G(\tau)$ inductively on $\dim\tau$ by first replacing $m_{\tau,r}\times d_{r}$ with $m_{\tau,r}\times\Delta_{r}$ on $\partial F(\tau)$ and then gluing $Cm_{\tau,r}\times\Delta_{r}$ to the boundary of the resulting chain. This ensures that $G$ is a chain homotopy between $\iota\circ A$ and the identity, i.e., $G$ satisfies

[TABLE]

as desired. ∎

2.3. Morse model with a strict unit

In this section, by adapting the homotopy unit construction in [FOOO09b, Chapter 7] (see also Charest-Woodward [CW15]) and using Theorem 2.3, we will explain how to construct a unital $A_{\infty}$-algebra on a pearl complex.

For simplicity, we will always assume $f$ has a unique maximum point $\bm{1}^{\blacktriangledown}$, so that $\Delta_{\bm{1}^{\blacktriangledown}}=\bm{e}$ is the fundamental cycle. Let

[TABLE]

with $|\bm{1}^{\triangledown}|=0$ and |\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}|=-1. The superscripts $\tiny{\triangledown}$, $\blacktriangledown$ and {\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown} are borrowed from [CW15]. We extend the Morse differential $\delta$ to $CF^{\bullet}(L;\Lambda_{0})$ by setting

[TABLE]

We now construct a unital $A_{\infty}$-algebra structure on $CF^{\bullet}(L;\Lambda_{0})$. Let

[TABLE]

For $g\geq 0$, suppose $\mathcal{X}_{g^{\prime}}(L)$ had been constructed for all generations $g^{\prime}(<g)$. The perturbations $\mathfrak{s}_{\mathfrak{d},\ell,\beta,\vec{P}}$ and $\mathfrak{s}^{+}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}}$ in (2.1) and (2.4) can be chosen to have the following properties: Let $\tau$ be any face of either a simplex in the triangulation of

[TABLE]

or a simplex in the triangulation of

[TABLE]

1. (1)

$\tau$ is transversal to the stable submanifold $W^{s}(f;p)$ for all $p\in\mathrm{Crit}(f)$ 2. (2)

For each $\tau$ of dimension at most $d$, there exists at most one critical point $p(\tau)\in\mathrm{Crit}(f)$ such that $W^{s}(f;p(\tau))$ is of complementary dimension to $\tau$ and intersects $\tau$ at a unique point.

Denote by $\mathcal{X}^{\circ}_{g}(L)$ the set of those singular simplices $\tau$. Define $\Pi(\tau)=\epsilon\Delta_{p(\tau)}$ where $\epsilon=\pm 1$ is given by

[TABLE]

if there exists a unique $p(\tau)$ such that $W^{s}(f;p(\tau))$ intersects $\tau$ at a unique point, and $\Pi(\tau)=0$, otherwise. In particular, we have $\Pi(\tau)=0$ whenever $\tau$ is a degenerate simplex.

We further add the simplices $G(\tau)$ (defined in the proof of Theorem 2.3) for every $\tau\in\mathcal{X}^{\circ}_{g}(L)$ to $\mathcal{X}_{g}(L)$. For chains of the form $G(\tau)$, we put $G(G(\tau))=0$. The maps $\Pi:C^{\bullet}(L;\Lambda_{0})\to C^{\bullet}_{(-1)}(L;\Lambda_{0})$ and $G:C^{\bullet}(L;\Lambda_{0})\to C^{\bullet-1}(L;\Lambda_{0})$ obey

[TABLE]

for $\tau\in C^{\bullet}(L;\Lambda_{0})$.

Set

[TABLE]

We recall that $\bm{e}^{+}$ is the unit and $\bm{f}$ is a homotopy between $\bm{e}^{+}$ and the fundamental class $\bm{e}$. We extend the coboundary operator $\partial$ to $C^{\bullet}(L;\Lambda_{0})^{+}$ by setting

[TABLE]

We then extend $\Pi$ and $G$ to maps $\Pi:C^{\bullet}(L;\Lambda_{0})^{+}\to C^{\bullet}_{(-1)}(L;\Lambda_{0})^{+}$ and $G:C^{\bullet}(L;\Lambda_{0})^{+}\to C^{\bullet-1}(L;\Lambda_{0})^{+}$ respectively by setting

[TABLE]

so that (2.12) holds for any $\tau\in C^{\bullet}(L;\Lambda_{0})^{+}$.

A homological perturbation can be applied to reduce the $A_{\infty}$-algebra structure on $C^{\bullet}(L;\Lambda_{0})^{+}$ in (2.2) to that on $C^{\bullet}_{(-1)}(L;\Lambda_{0})^{+}$ in (2.13). We identify the latter with $CF^{\bullet}(L;\Lambda_{0})$ via the map $\iota:CF^{\bullet}(L;\Lambda_{0})\to C^{\bullet}(L;\Lambda_{0})^{+}$ defined by (2.10) and the following assigments$\colon$

[TABLE]

The resulting unital $A_{\infty}$-algebra is denoted by $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$. By following [FOOO09a, Theorem 5.1], we obtain

Theorem 2.5**.**

The Morse model $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$ is a unital $A_{\infty}$-algebra. Moreover, the Morse model $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$ and the singular model $(C^{\bullet}(L;\Lambda_{0})^{+},\tilde{\mathfrak{m}}^{+})$ in (2.2) are unital homotopy equivalent.

We explicitly describe the $A_{\infty}$-structure maps of the Morse model $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$. The maps $\mathfrak{m}_{k}$ are given in terms of maps $\mathfrak{m}_{\Gamma}$ and $\mathfrak{f}_{\Gamma}$ associated to decorated planar rooted trees. Let us fix a labeling $\{\beta_{0},\beta_{1},\ldots\}$ of elements of $H_{2}^{\mathrm{eff}}(X,L)$ where $\beta_{0}$ is the constant disc class.

Definition 2.6** ([FOOO09a]).**

A decorated planar rooted tree is a quintuple $\Gamma=(T,i,v_{0},V_{tad},\eta)$ consisting of

• •

$T$ is a tree;

• •

$i:T\to D^{2}$ is an embedding into the unit disc;

• •

$v_{0}$ is the root vertex and $\iota(v_{0})\in\partial D^{2}$;

• •

$V_{tad}$ is the set of interior vertices with valency $1$;

• •

$\eta:V(\Gamma)_{int}\to\mathbb{Z}_{\geq 0}$.

where $V(\Gamma)$ is the set of vertices; $V(\Gamma)_{ext}=i^{-1}(\partial D^{2})$ is the set of exterior vertices and $V(\Gamma)_{int}=V(\Gamma)\setminus V(\Gamma)_{ext}$ is the set of interior vertices. For $k\geq 0$, denote by $\bm{\Gamma}_{k+1}$ the set of isotopy classes represented by $\Gamma=(T,\iota,v_{0},\eta)$ with $|V(\Gamma)_{ext}|=k+1$ and $\eta(v)>0$ if the valency $\ell(v)$ of $v$ is $1$ or $2$. In other words, the elements of $\bm{\Gamma}_{k+1}$ are stable.

We denote by $\Gamma^{0}\in\bm{\Gamma}_{2}$ the unique tree with no interior vertices. For this tree $\Gamma_{0}$, set

[TABLE]

where $\iota$ is the inclusion map in (2.14). For each $k\geq 0$, $\bm{\Gamma}_{k+1}$ contains a unique element that has a single interior vertex $v$, which is denoted by $\Gamma_{k+1}$. We then define

[TABLE]

For general $\Gamma$, we cut it at the vertex $v$ closest to $v_{0}$ so that $\Gamma$ is decomposed into $\Gamma^{(1)},\ldots,\Gamma^{(\ell)}$ and an interval adjacent to $v_{0}$ in the counter-clockwise order. We inductively define

[TABLE]

Finally, we define the $A_{\infty}$-map $\mathfrak{m}_{k}:CF^{\bullet}(L;\Lambda_{0})^{\otimes k}\to CF^{\bullet}(L;\Lambda_{0})$ by

[TABLE]

where $\omega_{X}(\Gamma)=\sum_{v\in V_{int}}\omega_{X}(\beta_{\eta(v)})$.

The maps $\mathfrak{m}_{\Gamma}$ restricted to $C^{\bullet}(f;\Lambda_{0})$ are given by counting pearly trees as depicted in Figure 2 (This observation was made in [FOOO09a]. See also [CW15] for a description of $\mathfrak{m}_{\Gamma}$ with general inputs in terms of pearly trees). For a decorated tree $\Gamma\in\bm{\Gamma}_{k+1}$, the exterior vertices $v_{0},\ldots,v_{k}$ are labeled respecting the counter-clockwise orientation. Each edge is oriented in the direction from the $k$ input vertices $v_{1},\ldots,v_{k}$ to the root vertex $v_{0}$. We denote by $e_{i}$ the edge attached to $v_{i}$ and by $v^{\pm}(e)$ the vertices such that $e$ is the edge from $v^{-}(e)$ to $v^{+}(e)$. For $p_{1},\ldots,p_{k},q\in\mathrm{Crit}(f)$, consider the moduli space $\mathcal{M}_{\Gamma}(f;p_{1},\ldots,p_{k},q)$ consisting of the following configurations:

• •

for each interior vertex $v$, a bordered stable map $u_{v}$ representing the class $\beta_{\eta(v)}$ with $\ell(v)$ boundary marked points. We denote by $p(e,v)$ the marked point corresponding to the edge $e$ attached to $v$;

• •

for $i=1,\ldots,k$, the input edge $e_{i}$ corresponds to a flow line $\gamma_{i}$ from $p_{i}$ to $u_{v^{+}(e_{i})}(p(e_{i},v^{+}(e_{i}))$;

• •

the output edge $e_{0}$ corresponds to a flow line $\gamma_{0}$ from $u_{v^{-}(e_{0})}(p(e_{0},v^{-}(e_{0})))$ to $q$;

• •

an interior edge $e$ corresponds to a flow line $\gamma_{e}$ from $u_{v^{-}(e)}(p(e,v^{-}(e)))$ to $u_{v^{+}(e)}(p(e,v^{+}(e)))$.

The virtual dimension of $\mathcal{M}_{\Gamma}(f;p_{1},\ldots,p_{k},q)$ is given by

[TABLE]

where $\mu(\Gamma)=\sum_{v\in V_{int}}\mu(\beta_{\eta(v)})$. The map $\mathfrak{m}_{\Gamma}:C^{\bullet}(f;\Lambda_{0})^{\otimes k}\to C^{\bullet}(f;\Lambda_{0})$ is given by

[TABLE]

where $\sharp\mathcal{M}_{\Gamma}(f;p_{1},\ldots,p_{k},q)$ is the signed count of the moduli space of virtual dimension [math].

2.4. Disc potentials in Lagrangian Floer theory

In this section, we recall the disc potential of $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$ introduced by Fukaya-Oh-Ohta-Ono [FOOO09b].

We denote by $\Lambda_{+}\subset\Lambda_{0}$ the maximal ideal

[TABLE]

Let $(A,\mathfrak{m})$ be an $A_{\infty}$-algebra over $\Lambda_{0}$ with the strict unit $e_{A}$, and set

[TABLE]

The weak Maurer-Cartan equation for an element $b\in A_{+}$ is given by

[TABLE]

The condition that $b\in A_{+}$ ensures the convergence of $\mathfrak{m}^{b}_{0}(1)$. A solution $b$ of (2.4) is called a weak bounding cochain. We denote by

[TABLE]

the space of weak Maurer-Cartan elements. We say an $A_{\infty}$-algebra $A$ is weakly unobstructed if $MC(A)$ is nonempty, in which case we have $(\mathfrak{m}^{b}_{1})^{2}=0$ for any $b\in\mathcal{MC}(A)$, thus defining a cohomology theory $H^{\bullet}(A,\mathfrak{m}^{b}_{1})$.

Let us put

[TABLE]

For an element $b\in\mathcal{MC}(A)$, we can define a deformation $\mathfrak{m}^{b}$ of the $A_{\infty}$-structure $\mathfrak{m}$ by

[TABLE]

Now, for $(A,\mathfrak{m})=(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$, we denote by $\mathcal{MC}(L)$ the space of odd degree weak bounding cochains. In general, one should also consider gauge equivalences between weak Maurer Cartan solutions. However, we omit them here since they will trivial in our examples.

We say that the deformation of $L$ by $b$ (or simply $(L,b)$) is unobstructed (resp. weakly unobstructed), if $\mathfrak{m}^{b}(1)=0$ (resp. $b\in\mathcal{MC}(L)$). In particular, if $b=0$, we will simply say $L$ is unobstructed.

The following lemma concerns the weakly unobstructedness of $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$. This technique of finding weak bounding cochains in the presence of a homotopy unit was introduced in [FOOO09b, Chapter 7] and [CW15, Lemma 2.44].

Lemma 2.7**.**

Let $b\in CF^{1}(L;\Lambda_{+})$. Suppose $\mathfrak{m}_{0}^{b}=W(b)\bm{1}^{\blacktriangledown}$ and the minimal Maslov index of $L$ is nonnegative. Then there exists $b^{+}\in CF^{odd}(L;\Lambda_{+})$ such that $\mathfrak{m}_{0}^{b^{+}}(1)=W^{\triangledown}(b)\bm{1}^{\triangledown}$, i.e., $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$ is weakly unobstructed. In particular, if the minimal Maslov index of $L$ is at least two, then, $W^{\triangledown}(b)=W(b)$.

Proof.

By (2.5), we have

[TABLE]

where

[TABLE]

is a $\Lambda_{+}$ multiple of $\bm{1}^{\blacktriangledown}$. The vanishing of the contribution of higher Maslov index discs follows from the fact that the output singular chains are of degrees at most $(-2)$, whose projection to $CF^{\bullet}(L;\Lambda_{0})$ vanishes. Hence we can write (2.17) as

[TABLE]

for some $h\in\Lambda_{+}$.

Let us tentatively take b^{\prime}=b+\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. We have

[TABLE]

The second equality is due to the vanishing of the $\mathfrak{m}_{k}$ maps with more than one \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} as an input since the outputs are of degrees at most $(-2)$. Also, notice that the terms \mathfrak{m}_{1,\beta}^{b}(\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}) are of degree [math] and therefore are multiples of $\bm{1}^{\blacktriangledown}$.

Using the notation of (2.18), we can write

[TABLE]

Since there do not exist any gradient flow lines from $b$ to the maximum $\bm{1}^{\blacktriangledown}$, the contribution of the terms

[TABLE]

are zero and hence $h(b)\in\Lambda_{+}$.

Now, set b^{+}=b+\frac{W}{1-h(b)}\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. We have

[TABLE]

which yields $b^{+}\in\mathcal{MC}(L)$.

If the minimal Maslov index of $L$ is at least two, then $h(b)=0$ (see (2.6)) and hence $W^{\triangledown}(b)=W(b)$. ∎

Let $\mathcal{MC}^{\blacktriangledown}(L)$ the space of elements $b\in CF^{1}(L;\Lambda_{+})$ which satisfy

[TABLE]

By the above lemma, for each such $b$, there exists $b^{+}\in CF^{odd}(L;\Lambda_{+})$ such that $\mathfrak{m}_{0}^{b^{+}}(1)=W^{\triangledown}(b)\bm{1}^{\triangledown}$. Then, $W^{\triangledown}(b)$ can be viewed as a function on $\mathcal{MC}^{\blacktriangledown}(L)$.

Definition 2.8**.**

We will call $W^{L}:=W^{\triangledown}:\mathcal{MC}^{\blacktriangledown}(L)\to\Lambda_{0}$ the disc potential of $L$.

3. A Morse model for equivariant Lagrangian Floer theory

There are several approaches to $G$-equivariant Lagrangian Floer theory for a pair of G-invariant Lagrangians in the existing literature. For $G=\mathbb{Z}_{2}$, Seidel-Smith [SS10] used Floer homology coupled with Morse theory on $EG$ to define $G$-equivariant Lagrangian Floer homology of exact Lagrangians. In [HLS16a, HLS16b], Hendricks-Lipshitz-Sarkar used a homotopy theoretic method to define $G$-equivariant Lagrangian Floer homology for a compact Lie group $G$. Daemi-Fukaya [DF17] defined an equivariant de Rham model using $G$-equivariant Kuranishi structures developed in [FOOO09b, Fuk17]. Bao-Honda [BH18] defined equivariant Lagrangian Floer cohomology in the case of finite group action via semi-global Kuranishi structures.

In this section, we develop a Morse model for the $G$-equivariant Lagrangian Floer theory focusing on a single $G$-invariant Lagrangian. The underlying cochain complexes can be constructed such that each complex is finite dimensional over the cohomology ring of $BG$, and the $A_{\infty}$-maps are given by counting pearly trees in the Borel construction $L_{G}$. This suits better for our purpose of computing disc potentials and constructing SYZ mirrors. The $A_{\infty}$-algebra we construct will be unital. This uses the homotopy unit construction in [FOOO09b], which has also been adapted to the stabilizing divisor perturbation scheme by Charest-Woodward [CW15] for their (non-equivaraint) unital Morse model.

We define the $G$-equivariant Lagrangian Floer theory as the (ordinary) Lagrangian Floer theory of $L_{G}$ as a Lagrangian submanifold of a certain symplectic manifold $X_{G}$. This avoids the issue of equivariant transversality. The Morse model we use makes the theory much more explicit and computable. Since the Lagrangians we study here bounds non-constant pseudo-holomorphic discs, we use the machinery of [FOOO09b] to handle the obstructions. On the other hand, for computing the equivariant disc potential of toric Fano manifolds, virtual technique is not necessary.

3.1. The equivariant Morse model

Let $G$ be a compact Lie group. We begin by choosing smooth finite dimensional approximations for the universal bundle $EG\to BG$ over the classifying space. Namely, we have the follwing diagram

[TABLE]

where $EG(N)$ and $BG(N)$ are compact smooth manifolds, and the horizontal arrows are smooth embeddings. We note that $BG(N)$ are connected for $N\geq 1$ since $EG(N)$ is $(N-1)$-connected.

We also consider the cotangent bundles $T^{*}EG(N)$ and $T^{*}BG(N)$ equipped with the canonical symplectic forms. Every symplectic $G$-action on a symplectic manifold can be naturally lifted to a Hamiltonian $G$-action on its cotangent bundle. We choose a moment map $\mu_{N}:T^{*}EG(N)\to\mathfrak{g}^{*}$ for the Hamiltonian $G$-action on $T^{*}EG(N)$ lifted from the $G$-action on $EG(N)$ where $\mathfrak{g}^{*}$ is the dual Lie algebra of $G$. Since $G$ acts on $T^{*}EG(N)$ freely, we have a canonical isomorphism

[TABLE]

as symplectic manifolds. We denote by $J_{T^{*}BG(N)}$ the almost complex structure induced by a $G$-invariant compatible almost complex structure on $T^{*}EG(N)$. Notice that $J_{T^{*}BG(N)}$ is compatible with the canonical symplectic form on $T^{*}BG(N)$.

Let $(X,\omega_{X})$ be a symplectic $G$-manifold, i.e. $X$ is a symplectic manifold endowed with a $G$-action which preserves $\omega_{X}$. As in Section 2.1, $X$ is assumed to be convex at infinity or geometrically bounded if it is non-compact. Let $L$ be a $G$-invariant, closed, connected, and relatively spin Lagrangian submanifold $X$. We will assume the relative spin structure is preserved by the $G$-action. Let us fix a $G$-invariant almost complex structure $J_{X}$ compatible with $\omega_{X}$. In this section, we will define a Morse model for the $G$-equivariant Lagrangian Floer theory on $L$.

Let $X(N)=X\times_{G}\mu_{N}^{-1}(0)$ and set $X_{G}=\lim_{\to}X(N)$. Since $G$ acts freely on $\mu_{N}^{-1}(0)$, there is a canonical map $\pi:X(N)\to T^{*}BG(N)$, which is a fiber bundle with the fiber $X$. Let $\iota:X\to X(N)$ be inclusion of a fiber. By the construction, $X(N)$ is endowed with a symplectic form $\omega_{X(N)}$ and a compatible almost complex structure $J_{X(N)}$ satisfying

[TABLE]

and

[TABLE]

Let $L(N)=L\times_{G}EG(N)$ for $N=0,1,\dots$, which are finite dimensional approximations of the Borel construction $L_{G}=L\times_{G}EG$. We note that $L(N)\to BG(N)$ is a fiber bundle with the fiber $L$. Regarding $BG(N)$ as the zero section of $T^{*}BG(N)$, $L(N)$ is a Lagrangian submanifold of $X(N)$ for each $N$. The diagram (3.1) induces a commutative diagram

[TABLE]

where the vertical arrows are embeddings of Lagrangian submanifolds.

Now, we study the disc moduli bounded by $L(N)$. The following proposition tells us that the image of any stable $J_{X(N)}$-holomorphic map must be fully contained in a fiber of the map $\pi$.

Proposition 3.1** (Effective disc classes).**

The induced map $\iota_{*}:H_{2}(X,L;\mathbb{Z})\to H_{2}(X(N),L(N);\mathbb{Z})$ restricts to a bijection

[TABLE]

Proof.

Suppose that there is a $J_{X(N)}$-holomorphic map $u:(\Sigma,\partial\Sigma)\to(X(N),L(N))$.

[TABLE]

Then the composition $\pi\circ u$ is a $J_{T^{*}BG(N)}$-holomorphic disc in $T^{*}BG(N)$ bounded by the zero section $BG(N)$. Since $BG(N)$ is an exact Lagrangian of $T^{*}BG(N)$, it does not bound any non-constant pseudo-holomorphic discs. It implies that $\pi\circ u$ is necessarily constant and $\mathrm{Im}(u)$ is contained in a fiber of $\pi$ over $BG(N)$. ∎

Proposition 3.1 has the following corollaries that will play an important role in computing the disc potential of $L(N)$ later on.

Corollary 3.2** (Maslov index).**

The Maslov index of $\beta\in H_{2}^{\mathrm{eff}}(X,L)$ is equal to the Maslov index of $\iota_{*}\beta\in H_{2}^{\mathrm{eff}}(X(N),L(N))$.

By abuse of notation, we denote the disc classes $\beta\in H_{2}^{\mathrm{eff}}(X,L)$ and $\iota_{*}\beta\in H_{2}^{\mathrm{eff}}(X(N),L(N))$ both by $\beta$. We also denote their Maslov index by $\mu(\beta)$ and symplectic area by $\omega(\beta)$.

Corollary 3.3** (Regularity).**

A $J_{X(N)}$-holomorphic disc $u:(\Sigma,\partial\Sigma)\to(X(N),L(N))$ is regular if it is regular as a disc in the corresponding fiber of $\pi$.

Proof.

Let $E:=u^{*}TX(N)$ and $F:=(\partial u)^{*}TL(N)$. We denote by $A^{0}(E,F)$ the space of smooth global sections of E with boundary values in $F$ and by $A^{1}(E)$ the space of smooth global $(0,1)$-forms with coefficient in $E$. Consider the two term elliptic complex

[TABLE]

where $\bar{\partial}$ is the linearized Cauchy-Riemann opeartor at $u$. Pulling the following exact sequences

[TABLE]

and

[TABLE]

via $u$, we choose splittings

[TABLE]

and

[TABLE]

where $F_{i}\subset E_{1}$ and $F_{2}\subset E_{2}$ are subbundles. This gives a decomposition of (3.3)

[TABLE]

By Proposition 3.1, $\pi\circ u$ is a constant map and hence $\bar{\partial}_{2}$ is surjective. Therefore, $\bar{\partial}$ is surjective if and only if $\bar{\partial}_{1}$ is surjective. ∎

Our heuristic definition of a $G$-equivariant Lagrangian Floer theory $(CF_{G}^{\bullet}(L;\Lambda_{0}),\mathfrak{m}^{G})$ is the (ordinary) Lagrangian Floer theory of the Lagrangian submanifold $L_{G}\subset X_{G}$. However, since $L_{G}$ is infinite dimensional, we resort to using finite dimensional approximations $\{(CF^{\bullet}(L(N);\Lambda_{0}),\{\mathfrak{m}_{k}^{N}\})\}_{N\in\mathbb{N}}$ associated to a sequence of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ on $\{L(N)\}_{N\in\mathbb{N}}$. Clearly, arbitrary choices of $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ and Kuranishi perturbations would not suffice for this purpose. We begin by specifying our choices of Morse-Smale pairs.

Definition 3.4**.**

We call a sequence of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ admissible if it satisfies the following:

1. (1)

For each $N\in\mathbb{N}$, there is an inclusion of critical point sets $\mathrm{Crit}(f_{N})\subset\mathrm{Crit}(f_{N+1})$. Under this identification, we have

1. (i)

For $p\in\mathrm{Crit}(f_{N})$, $W^{u}(f_{N+1};p)\times_{L(N+1)}L(N)=W^{u}(f_{N};p)$. 2. (ii)

For $p\in\mathrm{Crit}(f_{N})$, the image of $W^{s}(f_{N};p)$ in $L(N)$ coincides with $W^{s}(f_{N+1};p)$ in $L(N+1)$. 3. (iii)

For $q\in L(N+1)\setminus L(N)$, we have $\Phi_{t}(q)\notin L(N)$ for all $t\geq 0$. This implies

[TABLE]

for $p\in\mathrm{Crit}(f_{N+1})\setminus\mathrm{Crit}(f_{N})$. 2. (2)

For each $\ell\geq 0$, there exists an integer $N(\ell)>0$ such that $|p|>\ell$ for all $N\geq N(\ell)$ and $p\in\mathrm{Crit}(f_{N})\setminus\mathrm{Crit}(f_{N-1})$. 3. (3)

The Morse function $f_{N}$ has a unique maximum $\bm{1}^{\blacktriangledown}_{L(N)}$ and the inclusion $\mathrm{Crit}(f_{N})\subset\mathrm{Crit}(f_{N+1})$ identifies $\bm{1}^{\blacktriangledown}_{L(N)}$ with $\bm{1}^{\blacktriangledown}_{L(N+1)}$. This allows us to identify $CF^{\bullet}(L(N);\Lambda_{0})$ with a subcomplex of $CF^{\bullet}(L(N+1);\Lambda_{0})$. We will denote $\bm{1}^{\blacktriangledown}_{L(N)}$, $\bm{1}^{\triangledown}_{L(N)}$ and \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}_{L(N)} simply by $\bm{1}^{\blacktriangledown}$, $\bm{1}^{\triangledown}$, and \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}.

We explain below how such a sequence of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ can be produced. This is inspired by the family Morse theory of Hutchings [Hut08].

Proposition 3.5**.**

A smooth model for $BG=\lim_{\to}BG(N)$ can be chosen such that admissible sequences of Morse-Smale pairs exist.

Proof.

We choose $E(U(k))$ to be the infinite Stiefel manifold $V_{k}(\mathbb{C}^{\infty})=\lim_{\to}V_{k}(\mathbb{C}^{k+N})$ and $B(U(k))$ to be the infinite Grassmannian $Gr(k,\mathbb{C}^{\infty})=\lim_{\to}Gr(k,\mathbb{C}^{k+N})$. We have an embedding of $Gr(k,\mathbb{C}^{\infty})$ into the skew-Hermitian matrices on $\mathbb{C}^{\infty}$ by identifying $V\in Gr(k,\mathbb{C}^{\infty})$ with the orthogonal projection $P_{V}$ onto $V$. Let $A$ be the diagonal matrix with entries $\{1,2,3,\ldots\}$. The map

[TABLE]

is a perfect Morse function (see [Nic94]). We note that $f_{Gr(k,\mathbb{C}^{\infty})}$ has the following properties:

1. (1)

The restriction

[TABLE]

of $f_{Gr(k,\mathbb{C}^{\infty})}$ to each finite dimensional stratum is again a perfect Morse function. 2. (2)

For all $p\in Gr(k,\mathbb{C}^{k+N})$ and $q\in Gr(k,\mathbb{C}^{k+N+1})\setminus Gr(k,\mathbb{C}^{k+N})$, we have

[TABLE] 3. (3)

If a critical point $p$ of $f_{Gr(k,\mathbb{C}^{\infty})}$ is contained in $Gr(k,\mathbb{C}^{\infty})\setminus Gr(k,\mathbb{C}^{k+N})$, then $|p|>2N$.

We fix an embedding $G\to U(k)$ of Lie groups for some $k$, and choose $EG=V_{k}(\mathbb{C}^{\infty})$ and $BG=V_{k}(\mathbb{C}^{\infty})/G$. This gives a fibration $\pi:BG\to Gr(k,\mathbb{C}^{\infty})$ with fibers the homogeneous space $U(k)/G$. Let $\{U_{i}\}_{i\in I}$ be an open cover such that each $U_{i}$ is contractible, and if $p\in\mathrm{Crit}(f_{Gr(k,\mathbb{C}^{\infty})})$ and $p\notin U_{i}$, then $p\notin\overline{U}_{i}$. Let $\{\rho_{i}\}_{i\in I}$ be a partition of unity subordinate to $\{U_{i}\}_{i\in I}$. Notice that $\rho_{i}$ are constant near the critical points of $f_{Gr(k,\mathbb{C}^{\infty})}$ due to our second assumption on the open cover.

Let $f_{U(k)/G}:U(k)/G\to\mathbb{R}$ be a Morse function and define $F_{U(k)/G}:BG\to\mathbb{R}$ by

[TABLE]

Here $\rho_{i}$ and $f_{U(k)/G}$ are understood as defined on the local trivializations $U(k)/G\times U_{i}$. For each $N\in\mathbb{N}$, the restriction of $F_{U(k)/G,N}:=F_{U(k)/G}|_{BG(N)}$ to generic fibers of $BG(N)\to Gr(k,\mathbb{C}^{k+N})$ is a Morse function. In particular, the restriction of $F_{U(k)/G,N}$ to each fiber over a critical point of $f_{Gr(k,\mathbb{C}^{k+N})}$ agrees with $f_{U(k)/G}$.

We set

[TABLE]

Since $Gr(k,\mathbb{C}^{k+N})$ is compact, for sufficiently small $\epsilon$, $f_{BG(N)}$ has the critical point set

[TABLE]

Moreover, $f_{BG(N)}$ is a Morse function. The non-degeneracy of critical points follows from the fact that $\rho_{i}$ are constant near the critical points of $f_{Gr(k,\mathbb{C}^{\infty})}$.

Now, let $V_{Gr(k,\mathbb{C}^{\infty})}$ be a negative pseudo-gradient vector field for $f_{Gr(k,\mathbb{C}^{\infty})}$ such that $(f_{Gr(k,\mathbb{C}^{\infty})},V_{Gr(k,\mathbb{C}^{\infty})})$ is a Morse-Smale pair, and let $V_{U(k)/G}$ be a fiberwise negative pseudo-gradient for $F_{U(k)/G}$. We set

[TABLE]

where $H$ denotes the horizontal lift with respect to an auxiliary connection for the fiber bundle $BG\to Gr(k,\mathbb{C}^{\infty})$. For each $N\in\mathbb{N}$, we put $V_{BG(N)}=V_{BG}|_{BG(N)}$. Then, for a generic choice of $V_{U(k)/G}$, the pairs $\{(f_{BG(N)},V_{BG(N)})\}_{N\in\mathbb{N}}$ are Morse-Smale.

Finally, let $f:L\to\mathbb{R}$ be a Morse function. By iterating the construction in the above paragraphs, we obtain Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ with

[TABLE]

It follows from the properties of $f_{Gr(k,\mathbb{C}^{\infty})}$ that $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ is an admissible collection in the sense of Definition 3.4. ∎

We now turn to the choice of Kuranishi structures on the moduli spaces $\mathcal{M}_{k+1}(\beta;L(N))$. Let us first recall that for an element $p=[\Sigma,\vec{z},u]\in\mathcal{M}_{k+1}(\beta;L)$, a Kuranishi chart around $u$ is a quintuple $(V_{p},E_{p},\Gamma_{p},\psi_{p},s_{p})$, where $\Gamma_{p}$ is the finite automorphism group of $(\Sigma,\vec{z},u)$ acting on the vector bundle $E_{p}\to V_{p}$, $V_{p}$ is a smooth manifold with corners parameterizing smooth stable maps close to $u$ , $s_{p}=\bar{\partial}_{J_{X}}$ is a $\Gamma$-equivariant smooth section of $E_{p}\to V_{p}$, and $\psi_{p}$ is a homeomorphism from $s^{-1}_{p}\{0\}/\Gamma_{p}$ to a neighborhood of $p$.

We fix weakly submersive Kuranishi structures for the disc moduli $\mathcal{M}_{k+1}(\beta;L)$ following [FOOO09b, Chapter  7.1]. For $N\geq 1$, since the holomorphic discs with boundary on $L(N)$ are contained in the fibers of $X(N)\to T^{*}BG(N)$ (over $BG(N)$), there are no obstruction in the base direction, and therefore we can choose Kuranishi structures to be of the following form: For every $N\geq 1$ and $q\in BG(N)$, we choose a contractible neighborhood $U_{q}$ of $q$ in $BG(N)$ and trivializations $X\times U_{q}$ and $L\times U_{q}$ compatible with the diagram (3.2). This gives an trivialization $\mathcal{M}_{k+1}(\beta;L)\times U_{q}$ of $\mathcal{M}_{k+1}(\beta;L(N))$. Then, for an element $p$ in a fiber of $\mathcal{M}_{k+1}(\beta;L)\times U_{q}\to U_{q}$, we can take the (weakly submersive) Kuranishi chart around $p$ to be

[TABLE]

where $\psi^{\prime}_{p}$ is the pullback of $\psi_{p}$ via the projection $s^{-1}_{p}(0)/\Gamma_{p}\times U_{q}\to s^{-1}_{p}(0)/\Gamma_{p}$ Then we have

[TABLE]

as Kuranishi structures. Moreover, we can orient the moduli spaces using a $G$-invariant relative spin structure such that the orientations are compatible with the restriction above.

Let $\{(CF^{\bullet}(L(N);\Lambda_{0}),\{\mathfrak{m}_{k}^{N}\})\}_{N\in\mathbb{N}}$ be Morse models associated to an admissible collection of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$. Let $\iota_{N-1}^{N}:CF^{\bullet}(L(N-1);\Lambda_{0})\to CF^{\bullet}(L(N);\Lambda_{0})$ be the map induced by inclusion of critical points $\mathrm{Crit}(f_{N-1})\subset\mathrm{Crit}(f_{N})$. We denote by $\{\tilde{\mathfrak{m}}_{k}^{N,+}\}$ the $A_{\infty}$-structure maps on the singular chain models $C^{\bullet}(L(N);\Lambda_{0})^{+}$ and by $\{\mathfrak{m}^{N}_{\Gamma}\}$ the maps associated to the decorated planar rooted trees in the definition of $\{\mathfrak{m}^{N}_{k}\}$.

The next key proposition enables us to define the $A_{\infty}$-structure maps of $G$-equivariant Lagrangian Floer theory.

Proposition 3.6**.**

The singular chain models $(C^{\bullet}(L(N);\Lambda_{0})^{+},\tilde{\mathfrak{m}}^{N,+})$ can be constructed such that the resulting unital Morse models $(CF^{\bullet}(L(N);\Lambda_{0}),\mathfrak{m}^{N})$ satisfy the following property:

[TABLE]

for $p_{1},\ldots,p_{k}\in CF^{\bullet}(L(N-1);\Lambda_{0})$ and $\Gamma\in\bm{\Gamma}_{k+1}$. Here, the (RHS) means that we only consider the outputs of $\mathfrak{m}^{N}_{\Gamma}$ in $CF^{\bullet}(L(N-1);\Lambda_{0})$.

Proof.

We proceed by induction on $N$. For $N=0$, the statement of the proposition is void. Suppose we have constructed $A_{\infty}$-algebras $(CF^{\bullet}(L(N^{\prime});\Lambda_{0}),\mathfrak{m}^{N^{\prime}})$ for all $N^{\prime}$ with $N^{\prime}<N$ satisfying (3.6). We now construct $(CF^{\bullet}(L(N);\Lambda_{0}),\mathfrak{m}^{N})$ satisfying (3.6) as follows:

Let $\mathcal{X}_{g}^{\prime}(L(N))\subset\mathcal{X}_{g}(L(N))$ be the subset

[TABLE]

Inductively on $g\geq-1$, we construct the set of singular simplices $\mathcal{X}_{g}(L)$, and choose perturbations $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N}}$, $\left\lVert(\mathfrak{d},\beta)\right\rVert=g$, and $\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}_{N}}^{+}$, $\left\lVert(\mathfrak{d}^{+},\beta)\right\rVert=g$ with following additional properties $\colon$

1. (1)

There exists a bijection $r_{g}:\mathcal{X}_{g}^{\prime}(L(N))\to\mathcal{X}_{g}(L(N-1))$. 2. (2)

For $\vec{P}_{N}=(P_{1,N},\ldots,P_{\ell,N})$, $P_{i,N}\in\mathcal{X}_{\mathfrak{d}(i)}^{\prime}(L(N))$, and $\mathfrak{d}(i)\leq g$, we have

[TABLE]

as Kuranishi structures, where $P_{i,N-1}=r_{\mathfrak{d}(i)}(P_{i,N})\in\mathcal{X}_{\mathfrak{d}(i)}(L(N-1))$. 3. (3)

For $\vec{P}_{N}^{+}=(P_{1,N}^{+},\ldots,P_{k,N}^{+})$, $P_{i,N}^{+}\in\mathcal{X}_{\mathfrak{d}^{+}(i)}^{\prime}(L(N))$ with $\mathfrak{d}^{+}(i)\leq g$, we have

[TABLE]

as Kuranishi structures, where $P_{i,N-1}^{+}=r_{\mathfrak{d}(i)}(P_{i,N}^{+})\in\mathcal{X}_{\mathfrak{d}(i)}(L(N-1))$. In particular, if $P_{i,N}^{+}=\Delta_{\bm{1}^{\blacktriangledown}_{L(N)}}$, then $P_{i,N-1}^{+}=\Delta_{\bm{1}^{\blacktriangledown}_{L(N-1)}}$. 4. (4)

Let $P_{i,N}\in\mathcal{X}_{\mathfrak{d}(i)}^{\prime}(L(N))$ for $i=1,\ldots,\ell$. Let $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}$ be the perturbation chosen at the previous inductive step on $N$ for $\mathcal{M}_{\ell+1}(\beta;L(N-1);\vec{P}_{N})$. If $\left\lVert(\mathfrak{d},\beta)\right\rVert=g$, then for each $p\in\mathrm{Crit}(f_{N-1})$, there is a one-to-one correspondence between the following sets of simplices. The first set is the set of simplices in the triangulation of

[TABLE]

which are of complementary dimension to the stable submanifold $W^{s}(f_{N-1};p)$ in $L(N-1)$ and intersect $W^{s}(f_{N-1};p)$ at a unique point. The other set is the set of simplices in the triangulation of

[TABLE]

which are of complementary dimension to the stable submanifold $W^{s}(f_{N};p)$ in $L(N)$ and intersect $W^{s}(f_{N};p)$ at a unique point. Moreover, the intersection points for the corresponding simplices have the same orientation. 5. (5)

The analogue of (4) holds for the perturbation $\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}_{N}}^{+}$ and the simplices in the triangulation of

[TABLE]

These properties together would imply that the resulting Morse model $(CF^{\bullet}(L(N);\Lambda_{0}),\mathfrak{m}^{N})$ satisfies property (3.6).

Denote by $\Delta_{p,N}$ the unstable chain of $p\in\mathrm{Crit}(f_{N})$ defined as in (2.8). For $g=-1$, we have

[TABLE]

by Definition 3.4 (1). We define $r_{-1}:\mathcal{X}_{-1}^{\prime}(L(N))\to\mathcal{X}_{-1}(L(N-1))$ by $r_{-1}(\Delta_{p,N})=\Delta_{p,N-1}$.

Let $\vec{a}=\{a_{1},\ldots,a_{|\vec{a}|}\}\subset\{1,\ldots,k\}$, $a_{1}<\ldots<a_{|\vec{a}|}$. For $p_{1},\ldots,p_{k-|\vec{a}|}\in\mathrm{Crit}(f_{N-1})$, let $\vec{\Delta}_{N}=(\Delta_{p_{1},N},\ldots,\Delta_{p_{k-|\vec{a}|},N})$, and let $\vec{\Delta}_{N}^{+}$ be the $k$-tuple obtained by inserting $\bm{e}=\Delta_{\bm{1}^{\blacktriangledown}}$ into the $a_{1},\ldots,a_{|\vec{a}|}$-th places of $\vec{\Delta}_{N}$. Since any holomorphic disc is contained in a fiber of $X(N)\to T^{*}BG(N)$ over $BG(N)$ by Proposition 3.1, Definition 3.4 (1) implies

[TABLE]

and

[TABLE]

as compact subsets of $\mathcal{M}_{k+1-|\vec{a}|}(\beta;L(N);\vec{\Delta}_{N})$ and $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L(N);\vec{\Delta}_{N}^{+})$, respectively. The restriction of the natural Kuranishi structure for the fiber product $\mathcal{M}_{k+1-|\vec{a}|}(\beta;L(N))$ (resp. $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L(N))$) to the (LHS) of (3.9) (resp. (3.10)) agrees with the Kuranishi structure of the (RHS) of (3.9) (resp. (3.10)). This establishes properties (1)-(3). Notice that properties (4), (5) are void for $g=-1$.

Now, suppose we have constructed singular simplices and chosen perturbations satisfying (1)-(5) for $g^{\prime}<g$. Let $P_{i,N}\in\mathcal{X}_{\mathfrak{d}(i)}^{\prime}(L(N))$ for $i=1,\ldots,\ell$. Suppose $\left\lVert(\mathfrak{d},\beta)\right\rVert=g$. Then $\mathfrak{d}(i)<g$, and by inductive hypothesis on $g^{\prime}<g$, we have

[TABLE]

as Kuranishi structures. We choose perturbation $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N}}$ to extend the perturbation $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}$ and the perturbation for the boundary strata $\partial\mathcal{M}_{\ell+1}(\beta;L(N);\vec{P}_{N})$. Note that $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}$ is compatible with the perturbation for $\partial\mathcal{M}_{\ell+1}(\beta;L(N);\vec{P}_{N})$ since the latter is extended from the perturbation for $\partial\mathcal{M}_{\ell+1}(\beta;L(N);\vec{P}_{N})\times_{L(N)}L(N-1)$ at previous inductive steps on $g$.

We triangulate $\mathcal{M}_{\ell+1}(\beta;L(N);\vec{P}_{N})^{\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N}}}$ extending the triangulation on

$\mathcal{M}_{\ell+1}(\beta;L(N-1);\vec{P}_{N-1})^{\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}}$ and triangulation on the boundary strata. We can choose the perturbation $\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N}}$ such that any simplex in this triangulation away from $\mathcal{M}_{\ell+1}(\beta;L(N-1);\vec{P}_{N-1})^{\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}}$ is transverse to the stable submanifolds of $f_{N}$ (via the evaluation map $\mathrm{ev}_{0}$). Suppose $\tau^{N}$ is a simplex in this triangulation and $\tau^{N}\times_{L(N)}L(N-1)\neq\emptyset$. By subdividing if necessary, we can assume that $\tau^{N}$ has a face $\tau^{N-1}$ in $\mathcal{M}_{\ell+1}(\beta;L(N-1);\vec{P}_{N-1})^{\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}}$ and no other faces of $\tau^{N}$ of dimension greater or equal to $\dim\tau^{N-1}$ are in $\mathcal{M}_{\ell+1}(\beta;L(N-1);\vec{P}_{N-1})^{\mathfrak{s}_{\mathfrak{d},\beta,\ell,\vec{P}_{N-1}}}$. Notice that $\tau^{N}$ is not transverse to all stable submanifolds of $f_{N}$ since the stable submanifolds remain unchanged by Definition 3.4 (1) but the ambient space $L(N)$ is of higher dimension than $L(N-1)$. We can however move the vertices of $\tau^{N}$ so that it is transverse to the stable submanifolds of $f_{N}$. We choose the perturbation $\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}_{N}}^{+}$ and the triangulation for $\left([0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L(N);\vec{P}_{N}^{+})\right)^{\mathfrak{s}_{\mathfrak{d}^{+},k,\beta,\vec{P}^{+}_{N}}^{+}}$ in a similar manner.

Finally, we define $r_{g}:\mathcal{X}_{g}^{\prime}(L(N))\to\mathcal{X}_{g}(L(N-1))$ by $r_{g}(\tau^{N})=\tau^{N-1}$, and $r_{g}(G(\tau^{N}))=G(\tau^{N-1})$. ∎

We are now ready to define a Morse model $(CF^{\bullet}_{G}(L;\Lambda_{0}),\{\mathfrak{m}^{G}_{k}\})$ for the $G$-equivariant Lagrangian Floer theory of $L$. The equivariant Floer complex $CF^{\bullet}_{G}(L;\Lambda_{0})$ is defined to be the direct limit

[TABLE]

In order to define the $A_{\infty}$-structure maps $\{\mathfrak{m}^{G}_{k}\}$ on $CF^{\bullet}_{G}(L;\Lambda_{0})$, we first define the maps $\{\mathfrak{m}^{G}_{\Gamma}\}$ for $\Gamma\in\bm{\Gamma}_{k+1}$ and $k\geq 0$ as follows. For $p_{1},\ldots,p_{k}\in CF^{\bullet}_{G}(L;\Lambda_{0})$, there exists a sufficiently large $N_{0}\in\mathbb{N}$ such that $p_{1},\ldots,p_{k}\in CF^{\bullet}(L(N_{0});\Lambda_{0})^{+}$. Setting $\ell=\sum_{i=1}^{k}|p_{i}|+2-k-\mu(\Gamma)$, we define $\mathfrak{m}^{G}_{\Gamma}$ by

[TABLE]

where $N(\ell)(\geq N_{0})$ comes from Definition 3.4 (2). Because of Proposition 3.6, (3.12) is defined independent of the choice of $N(\ell)$. Finally, we define the maps $\mathfrak{m}^{G}_{k}:CF_{G}^{\bullet}(L;\Lambda_{0})^{\otimes k}\to CF_{G}^{\bullet}(L;\Lambda_{0})$ by

[TABLE]

[TABLE]

The pair $(CF^{\bullet}_{G}(L;\Lambda_{0}),\mathfrak{m}^{G}=\{\mathfrak{m}^{G}_{k}\}_{k\geq 0})$ is an $A_{\infty}$-algebra with a strict unit $\bm{1}^{\triangledown}$. For fixed inputs the $A_{\infty}$-identity can be checked on $(CF^{\bullet}(L(N);\Lambda_{0}),\mathfrak{m}^{N})$ for sufficiently large $N$.

We will refer to the pair $(CF^{\bullet}_{G}(L;\Lambda_{0}),\mathfrak{m}^{G})$ from (3.11) and (3.14) as the Morse model for the $G$-equivariant Lagrangian Floer theory ($G$-equivariant Morse model) of $L$.

3.2. Equivariant parameters as homotopy partial units

Consider a Lagrangian toric fiber $L$ of a compact semi-Fano toric manifold of the complex $d$ dimension. One of the main goals of this paper is to understand the Givental’s equivariant toric superpotential

[TABLE]

as the disc potential of the $T$-equivariant Lagrangian Floer theory of $L$ constructed in Section 3.1. The terms $\lambda_{1},\ldots,\lambda_{d}$ are the equivariant parameters generating the cohomology ring

[TABLE]

Since $\lambda_{1},\ldots,\lambda_{d}$ have cohomological degree $2$, the expression (3.15) of $W_{\lambda}$ suggests that the boundary deformations of curvature $\mathfrak{m}_{0}^{T}(1)$ are, a priori, obstructed. For this reason, we shall construct in this section an alternative model which is homotopy equivalent to our equivariant Morse model and define its equivariant disc potential. Our construction in this section replies on the existence of a well-behaved perfect Morse function on $BG$ (e.g. (3.4) for $G=U(k)$). For this reason, we will temporarily restrict the discussion to the case when $G$ is a product of unitary groups.

To begin with, we fix a Morse function $f$ on $L$ with a unique maximum point $\bm{1}^{\blacktriangledown}_{L}$. Following the proof of Proposition 3.5, we can choose an admissible sequence of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ such that $f_{N}$ is of the form

[TABLE]

where

• •

$\varphi_{N}$ is a perfect Morse function on $BG(N)$.

• •

$\phi_{N}$ is a (generically) fiberwise Morse function for $L(N)\to BG(N)$. In particular the restriction of $\phi_{N}$ to each fiber over a critical point of $\varphi_{N}$ agrees with $\epsilon f$ for some $\epsilon>0$.

Since the choice (3.16) satisfies (3.5), we have

[TABLE]

We then enlarge $C^{\bullet}(f_{N};\Lambda_{0})$ fiberwise over $\mathrm{Crit}(\varphi_{N})$ to

[TABLE]

where

[TABLE]

as in (2.11).

We shall produce a unital $A_{\infty}$-algebra structure $\mathfrak{m}^{G,\dagger}$ on

[TABLE]

over the graded coefficient ring $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$ via finite dimensional approximations. Set

[TABLE]

Here, the singular chains in $\chi_{-1}(L(N))\subset C^{\bullet}(L(N);\Lambda_{0})$ are the unstable chains of the critical points of $f_{N}$. We denote by $\bm{\lambda}^{\blacktriangledown}$ the unstable chain of the critical point $(\bm{1}^{\blacktriangledown}_{L},\lambda)$. The generators $\bm{\lambda}^{\triangledown}$ and \bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} are of degrees $|\bm{\lambda}^{\triangledown}|=|\bm{\lambda}^{\blacktriangledown}|$ and |\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}|=|\bm{\lambda}^{\blacktriangledown}|-1. We note that $\bm{\lambda}^{\blacktriangledown}$ and $\bm{\lambda}^{\triangledown}$ are of even degrees while \bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} are of odd degrees.

By using the idea of the homotopy unit construction in Section 2.3, we construct an $A_{\infty}$-structure $\tilde{\mathfrak{m}}^{N,\dagger}$ on $C^{\bullet}(L(N);\Lambda_{0})^{\dagger}$ satisfying the following properties:

1. (1)

The restriction of $\tilde{\mathfrak{m}}^{N,\dagger}$ to $C^{\bullet}(L(N);\Lambda_{0})^{+}$ agrees with $\tilde{\mathfrak{m}}^{N,+}$. In particular, this means if $\lambda\in\mathrm{Crit}(\varphi_{N})$ is the maximum point, then $\bm{\lambda}^{\blacktriangledown}=\bm{e}$ is the homotopy unit, $\bm{\lambda}^{\triangledown}=\bm{e}^{+}$ is the strict unit, and \bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}=\bm{f} is the homotopy between them. 2. (2)

For $\lambda_{i},\lambda_{j}\in\mathrm{Crit}(\varphi_{N})$, let $\lambda_{k}=\lambda_{i}\cup\lambda_{j}$ where $\cup$ denotes the cup product of $H^{\bullet}(BG(N))$. Then

[TABLE] 3. (3)

Let $\lambda\in\mathrm{Crit}(\varphi_{N})$ and let $P\in C^{\bullet}(L(N);\Lambda_{0})^{\dagger}$ be a singular chain. Then we have

[TABLE] 4. (4)

For $k\neq 2$,

[TABLE] 5. (5)

\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} is a homotopy between $\bm{\lambda}^{\blacktriangledown}$ and $\bm{\lambda}^{\triangledown}$ in the following sense.

Let $\vec{a}=(a_{1},\ldots,a_{|\vec{a}|})$ be an ordered subset of $\{1,\ldots,k\}$ satisfying $a_{1}<\ldots<a_{|\vec{a}|}$. For $\vec{P}=(P_{1},\ldots,P_{k-|\vec{a}|})$ a $(k-|\vec{a}|)$-tuple of singular chains, let $\vec{P}^{\dagger}$ be the $k$-tuple obtained by inserting $\bm{\lambda}^{\blacktriangledown}$ into the $\vec{a}$-th places of $\vec{P}$ (by an abuse of notation, $\bm{\lambda}^{\blacktriangledown}$’s inserted can be distinct). We choose a perturbation on $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L(N);\vec{P}^{\dagger})$ as follows: For a splitting $\vec{a}^{1}\coprod\vec{a}^{2}=\vec{a}$ of $\vec{a}$, we denote by $\vec{P}^{\prime}$ the $(k-|\vec{a}^{1}|)$-tuple given by removing $\bm{\lambda}^{\blacktriangledown}$ from the $\vec{a}^{1}$-th places of $\vec{P}^{\dagger}$. Let $(t_{1},\ldots,t_{|\vec{a}|})$ be coordinates on $[0,1]^{|\vec{a}|}$. If $t_{i}=0$ for all $i\in\vec{a}^{1}$, we consider a perturbation chosen in Proposition 3.6 on the moduli space obtained by just inserting $\bm{\lambda}^{\blacktriangledown}$ into the $\vec{a}^{1}$-th place. On the other hand, If $t_{i}=1$ for all $i\in\vec{a}^{1}$, we take a perturbation pulled back via $\mathfrak{forget}_{\vec{a}_{1}}:\mathcal{M}_{k+1}(\beta;L(N);\vec{P}^{\dagger})\to\mathcal{M}_{k+1-|\vec{a}^{1}|}(\beta;L(N);\vec{P}^{\prime})$. (We note that the unstable submanifold $W^{u}(f_{N},\bm{\lambda}^{\blacktriangledown})$ is the restriction of the fiber bundle $L_{N}\to BG(N)$ over $W^{u}(\varphi_{N},\lambda)\subset BG(N)$, it therefore makes sense to pullback perturbations.) Finally, we take a perturbation $\mathfrak{s}^{\dagger}$ on $[0,1]^{|\vec{a}|}\times\mathcal{M}_{k+1}(\beta;L(N);\vec{P}^{\dagger})$ interpolating between them. We note that the unstable submanifold $W^{u}(f_{N},\bm{\lambda}^{\blacktriangledown})$ is the restriction of the fiber bundle $L_{N}\to BG(N)$ over the unstable submanifold $W^{u}(\varphi_{N},\lambda)$ in $BG(N)$. Inserting $\bm{\lambda}^{\blacktriangledown}$ imposes no constraint in the fiber direction. Therefore it makes sense to pullback perturbations from $\mathcal{M}_{k+1-|\vec{a}^{1}|}(\beta;L(N);\vec{P}^{\prime})$ to $\mathcal{M}_{k+1}(\beta;L(N);\vec{P}^{\dagger})$.

The $\tilde{\mathfrak{m}}^{N,\dagger}_{k,\beta}$ maps for $(k,\beta)\neq(1,\beta_{0})$ with inputs \bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} inserted into $\vec{a}$-th place of $\vec{P}$ are defined by

[TABLE]

where $\sim$ is again the equivalence relation collapsing fibers of $\mathfrak{forget}_{\vec{a}_{1}}$. The zero locus of a pullback multisection is a degenerate singular chain which becomes zero in the quotient.

For $(1,\beta_{0})$, we set \tilde{\mathfrak{m}}^{N,\dagger}_{1,\beta_{0}}(\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}})=\bm{\lambda}^{\triangledown}-\bm{\lambda}^{\blacktriangledown}. Moreover,

[TABLE]

where

[TABLE]

Note that $\bm{h}_{\lambda}\equiv 0\mod\Lambda_{+}\cdot C^{\bullet}(L(N);\Lambda_{0})^{\dagger}$.

Now, the homological perturbation as in Theorem 2.5 is applied to obtain a Morse model $(CF^{\bullet}(L(N);\Lambda_{0})^{\dagger},\mathfrak{m}^{N,\dagger})$ from the singular chain model $(C^{\bullet}(L(N);\Lambda_{0})^{\dagger},\tilde{\mathfrak{m}}^{N,\dagger})$. We finally obtain an $A_{\infty}$-algebra structure $\mathfrak{m}^{G,\dagger}$ on $CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger}$ defined in (3.17).

For any $\lambda\in H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$, we put

[TABLE]

In particular, for $\lambda=1$, we denote

[TABLE]

By our construction, the $A_{\infty}$-algebra $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ has the following properties:

1. (1)

The restriction of $\mathfrak{m}^{G,\dagger}$ to $CF^{\bullet}_{G}(L;\Lambda_{0})$ coincides with $\mathfrak{m}^{G}$. In particular, $\bm{1}^{\blacktriangledown}$ is the homotopy unit, $\bm{1}^{\triangledown}$ is the strict unit, and \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} is the homotopy between them. 2. (2)

For $X\in CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger}$, let us write $X=\sum a_{ij}x_{i}\otimes\lambda_{j}$ and put

[TABLE]

Then we have

[TABLE] 3. (3)

For $k\neq 2$,

[TABLE]

Since the properties (3.20) and (3.21) that $\bm{\lambda}^{\triangledown}$ satisfies are similar to those of a strict unit, we call $\bm{\lambda}^{\triangledown}$ a partial unit and $\bm{\lambda}^{\blacktriangledown}$ a homotopy partial unit.

When the minimal Maslov index of $L$ is nonnegative, the elements \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} and \bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} satisfy

[TABLE]

for some $h\in\Lambda_{+}$, and

[TABLE]

where the last inequality follows from the $A_{\infty}$-identity.

The equation $\eqref{eq:coh_unit}$ implies $[\bm{1}^{\blacktriangledown}]=[\bm{1}^{\triangledown}]/(1-h)\in HF^{0}_{G}(L;\Lambda_{0})$ (in the weakly unobstructed case so that the equivariant Floer cohomology is well-defined). Since $h\in\Lambda_{+}$, $\bm{1}^{\blacktriangledown}$ is a cohomological unit. This is important, for instance when we consider isomorphisms of objects in the Fukaya category.

The following theorem shows that $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ can be defined over the graded coefficient ring $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$, namely

[TABLE]

It is important to point out that the $A_{\infty}$-structure of the (RHS) is not determined by $(CF^{\bullet}(L;\Lambda_{0}),\mathfrak{m})$ since the equivariant parameters can receive nontrivial contributions from the pearly trees in $L_{G}$.

Theorem 3.7**.**

Assume that $L$ has non-negative minimal Maslov index. Let $X_{1},\ldots,X_{k}\in CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger}$. We write $X_{\ell}=\sum a_{ij}x_{i}\otimes\lambda_{j}$ for $\ell\in\{1,\ldots,k\}$. Then, we have

[TABLE]

Proof.

From (3.20), it follows that

[TABLE]

Then, by using (3.21) and $A_{\infty}$-identity, one obtains

[TABLE]

where the last equality follows from (3.20). ∎

In summary, we have produced the equivariant Morse model $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ as an $A_{\infty}$-algebra over $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$.

3.3. Disc potentials in equivariant Lagrangian Floer theory

We now introduce an equivariant disc potential for the equivariant Morse model $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ over the graded coefficient ring $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$.

For an element $b\in CF^{\bullet}_{G}(L;\Lambda_{+})^{\dagger}$, we define the equivariant weak Maurer-Cartan equation by

[TABLE]

We denote by $\mathcal{MC}_{G}(L)$ the space of odd degree solutions of (3.25). An element $b\in\mathcal{MC}_{G}(L)$ is called a weak Maurer-Cartan element over $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$. We say that $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ is weakly unobstructed if $\mathcal{MC}_{G}(L)$ is nonempty.

Definition 3.8**.**

The deformation of $(L,G)$ by $b$ (or simply $(L,G,b)$) is called unobstructed (resp. weakly unobstructed) if $\mathfrak{m}^{G,\dagger,b}(1)=0$ (resp. $b\in\mathcal{MC}_{G}(L)$). In particular, if $b=0$, we simply call $(L,G)$ unobstructed.

Similar to Lemma 2.7, we have

Lemma 3.9**.**

For $b\in CF^{1}_{G}(L;\Lambda_{+})$, suppose that

[TABLE]

and the minimal Maslov index of L is nonnegative. Then there exists $b^{\dagger}\in CF^{odd}_{G}(L;\Lambda_{+})^{\dagger}$ such that

[TABLE]

In particular, $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ is weakly unobstructed.

If the minimal Maslov index of $L$ is assumed to be at least two in addition, then $W^{\triangledown}=W(b)$ and $\phi_{\lambda}^{\triangledown}(b)=\phi_{\lambda}(b)$.

Proof.

Let us tentatively take b^{\prime}=b+W(b)\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}+\sum_{\deg\lambda=2}\phi_{\lambda}(b)\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. We also put b_{1}=b+W(b)\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}, b_{2}=W(b)\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}+\sum_{\deg\lambda=2}\phi_{\lambda}(b)\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}, and b_{3}=b+\sum_{\deg\lambda=2}\phi_{\lambda}(b)\bm{\lambda}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. We write

[TABLE]

It is easy to see that \sum_{k\geq 2}W(b)^{k}\mathfrak{m}^{G,\dagger}_{k}(\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}},\ldots,\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}})=0 since the outputs are of degree at most $(-2)$. On the other hand, by (3.22) and the proof of Lemma 2.7, we have

[TABLE]

for some $h(b)\in\Lambda_{+}$.

Let us now consider the term

[TABLE]

By (3.20), (3.21) and the $A_{\infty}$-identity, each term in (3.26) can be rewritten in the form

[TABLE]

where $P(\lambda_{1},\ldots,\lambda_{\ell})\in H^{\bullet}(BG;\Lambda_{0})$ is a monomial in degree $2$ variables and \mathfrak{m}^{G,\dagger}_{k}(\ldots,\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}},\ldots) has at least one \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} as an input and remaining inputs are $b$ and \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. By the degree reason, \mathfrak{m}^{G,\dagger}_{k}(\ldots,\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}},\ldots)=0 when it has more than one \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}} as inputs. This means (3.26) equals to

[TABLE]

where $\delta\in\{0,1\}$ depends on the position of \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}. Recall from the proof of Lemma 2.7 that \mathfrak{m}^{G,\dagger}_{k}(b,\ldots,b,\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}},b,\ldots,b) is a $\Lambda_{+}$ multiple of $\bm{1}^{\blacktriangledown}$. Since $\lambda\cdot\bm{1}^{\blacktriangledown}=\bm{\lambda}^{\blacktriangledown}$, the (RHS) above is a $\Lambda_{+}$ multiple of $\bm{\lambda}^{\blacktriangledown}$. Thus, (3.23) we have

[TABLE]

for some $h_{\lambda}(b)\in\Lambda_{+}$.

Next, we note that the term

[TABLE]

vanishes for the same reason we explain in the above paragraph.

Finally, let

[TABLE]

Then $\mathfrak{m}^{G,\dagger,b^{\dagger}}_{0}(1)$ can be expressed as

[TABLE]

Since both $W(b)$ and $\phi_{\lambda}(b)$ are in $\Lambda_{+}$, we have $b^{\dagger}\in\mathcal{MC}_{G}(L)$.

If the minimal Maslov index of $L$ is at least two, then $h(b)=0$, $h_{\lambda}(b)=0$ and therefore $W^{\triangledown}(b)=W(b)$, $\phi_{\lambda}^{\triangledown}(b)=\phi_{\lambda}(b)$. ∎

Corollary 3.10**.**

In the setting of Lemma 3.9, if $b\in C^{1}(f;\Lambda_{0})$ and $\mathfrak{m}_{0}^{b}(1)\in\Lambda_{0}\cdot\bm{1}^{\blacktriangledown}$, then the equivariant Morse model $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ is weakly unobstructed.

Proof.

If $b\in C^{1}(f;\Lambda_{0})$ and $\mathfrak{m}_{0}^{b}(1)\in\Lambda_{0}\cdot\bm{1}^{\blacktriangledown}$, then $\mathfrak{m}^{G,\dagger,b}_{0}(1)$ is of the form $\mathfrak{m}^{G,\dagger,b}_{0}(1)=W(b)\bm{1}^{\blacktriangledown}+\sum_{\deg\lambda=2}\phi_{\lambda}(b)\bm{\lambda}^{\blacktriangledown}$. ∎

We remark that even if the minimal Maslov index of $L$ is $2$ and $(L,b)$ is unobstructed, one can only expect $(L,G,b)$ to be weakly unobstructed in general. This is due to the possibility of the constant disc class contributing to degree $2$ equivariant parameters.

Now, let $\mathcal{MC}^{\blacktriangledown}_{G}(L)$ the space of elements $b\in CF^{1}_{G}(L;\Lambda_{+})$ which satisfy

[TABLE]

for some $W(b),\phi_{\lambda}(b)\in\Lambda$. By the Lemma 3.9, for each such $b$, there exists $b^{\dagger}\in CF^{odd}_{G}(L;\Lambda_{+})^{\dagger}$ such that

[TABLE]

Then, we may view $W^{L}_{G}(b):=W^{\triangledown}(b)+\sum_{\deg\lambda=2}\phi_{\lambda}^{\triangledown}(b)\lambda$ as a map from $\mathcal{MC}^{\blacktriangledown}_{G}(L)$ to $H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$.

Definition 3.11**.**

We will call $W^{L}_{G}:=W^{\triangledown}:\mathcal{MC}^{\blacktriangledown}_{G}(L)\to H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$ the equivariant disc potential of $(L,G)$.

4. $T$-equivariant disc potentials of toric manifolds

In this section, we study the equivariant Morse model in the case of a torus $T\cong(\mathbb{S}^{1})^{\ell}$ acting on a closed, connected, relative spin Lagrangian submanifold $L\cong T\times P$ of product type of a symplectic $T$-manifold $X$, such that $T$ acts freely on the first factor of $L$ and trivially on the second factor. In the case $L\cong T$ is a regular moment map fiber of a semi-projective, semi-Fano toric manifold, we recover the equivariant toric superpotential $W_{\lambda}$ as the $T$-equivariant disc potential of $L$.

4.1. Morse theory on the approximation spaces

For explicit computations of the equivariant disc potentials, we begin by fixing our choice of admissible Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$.

Our choices of the universal bundle $ET$ and the classifying space $BT$ are $ET=(\mathbb{S}^{\infty})^{\ell}$ and $BT=(\mathbb{C}\mathbb{P}^{\infty})^{\ell}$. We also have the finite dimensional approximations $ET(N)=(\mathbb{S}^{2N+1})^{\ell}$ and $BT(N)=(\mathbb{C}\mathbb{P}^{N})^{\ell}$. Since the $T$ acts trivially on $P$, we have $L_{T}=T_{T}\times(P\times(\mathbb{C}\mathbb{P}^{\infty})^{\ell})$ and $L(N)=T(N)\times(P\times(\mathbb{C}\mathbb{P}^{N})^{\ell}).$

For $N\geq 1$, let $[z_{i,0},\ldots,z_{i,N}]$ be the homogeneous coordinates on the $i$-th component of $BT(N)=(\mathbb{C}\mathbb{P}^{N})^{\ell}$. Let $\pi_{N}:L(N)\to(\mathbb{C}\mathbb{P}^{N})^{\ell}$ be the projection map. Let $\{U_{j_{1}\ldots j_{\ell}}\}$ be the open cover of $(\mathbb{C}\mathbb{P}^{N})^{\ell}$ defined by

[TABLE]

Set $\widetilde{U}_{j_{1}\ldots j_{\ell}}=\pi_{N}^{-1}(U_{j_{1}\ldots j_{\ell}})\cong L\times(\mathbb{C}^{N})^{\ell}$. We will be working with the atlas $\{\widetilde{U}_{j_{1}\ldots j_{\ell}}\}$ for $L(N)$ with local coordinates

[TABLE]

on $\widetilde{U}_{j_{1}\ldots j_{\ell}}$, where $\theta_{i}^{(j_{1}\ldots j_{\ell})}\in[0,2\pi)$ are angular coordinates on $(\mathbb{S}^{1})^{\ell}$, $\vec{p}^{(j_{1}\ldots j_{\ell})}$ is any coordinate system on $P$, and the term under $``\widehat{\quad}"$ is omitted. Set $\vartheta_{i,j}^{(j_{1}\ldots j_{\ell})}=\mathrm{Arg}\left(z^{(j_{1}\ldots j_{\ell})}_{i,j}\right)$. The transition map $\widetilde{U}_{j_{1}\ldots j_{\ell}}$ to $\widetilde{U}_{j^{\prime}_{1}\ldots j^{\prime}_{\ell}}$ is given by

[TABLE]

We fix the inclusion $(\mathbb{C}\mathbb{P}^{N})^{\ell}\hookrightarrow(\mathbb{C}\mathbb{P}^{N+1})^{\ell}$ to be

[TABLE]

Which in turn, fixes the inclusions $X(N)\hookrightarrow X(N+1)$ and $L(N)\hookrightarrow L(N+1)$.

For $U(1)=\mathbb{S}^{1}$, the perfect Morse function (3.4) on the infinite complex Grassmanian specializes to

[TABLE]

From which we obtain a perfect Morse function on $BT$

[TABLE]

The critical points of $\varphi$ are of the form

[TABLE]

with degrees given by $\sum_{i=1}^{\ell}2j_{i}$. We denote the degree $2$ critical points with $j_{i}=1$, and $j_{k}=0$ for $k\neq i$ by $\lambda_{i}$. We also denote the critical point where $\varphi$ attains the maximum by $\bm{1}_{BT}$, i.e.

[TABLE]

We set $\varphi_{N}=\varphi|_{BT(N)}$. Note that $\varphi_{N}$ is a perfect Morse function on $(\mathbb{C}\mathbb{P}^{N})^{\ell}$. We will abuse notation and denote the degree two critical points and the maximum of $\varphi_{N}$ again by $\bm{1}_{BT}$ and $\lambda_{i}$, respectively. Note that we have

[TABLE]

and

[TABLE]

On the other hand, let $f_{P}$ be a Morse function on $P$ with a unique maximum $\bm{1}_{P}$, and let $f_{T}:(\mathbb{S}^{1})^{\ell}\to\mathbb{R}$ be the perfect Morse function

[TABLE]

The critical points of $f_{T}$ are of the form

[TABLE]

Let $f=(f_{T},f_{P})$ be the Morse function on $L$. We denote by $\bm{1}_{L}$ the critical point where $f$ attains the maximum, i.e. $\bm{1}_{L}=((0,\ldots,0),\bm{1}_{P})$. We also denote by $X_{1},\ldots,X_{\ell}$ the degree $1$ critical points, and $X_{i}\wedge X_{j}$, $1\leq i<j\leq\ell$ the degree $2$ critical points of $f$ of the following forms

[TABLE]

and

[TABLE]

Let $\{D_{j}\}$ be the open cover of $\mathbb{C}\mathbb{P}^{\infty}$ given by

[TABLE]

Let $\{\rho^{(j)}\}$ be a smooth partition of unity subordinate to $\{D_{j}\}$, and denote by $\rho^{(j)}_{i}$ the post-composition of $\rho^{(j)}$ with the projection from $L_{T}$ to the $i$-th component of $(\mathbb{C}\mathbb{P}^{\infty})^{\ell}$. We define $\rho^{(j_{1}\ldots j_{\ell})}:L_{T}\to\mathbb{R}$ by

[TABLE]

Let $\rho^{(j_{1}\ldots j_{\ell})}_{N}=\rho^{(j_{1}\ldots j_{\ell})}|_{BT(N)}$, then $\{\rho^{(j_{1}\ldots j_{\ell})}_{N}\}$ is a smooth partition of unity on $L(N)$.

Let $\phi^{(j_{1}\ldots j_{\ell})}_{N}:\widetilde{U}_{j_{1}\ldots j_{\ell}}\to\mathbb{R}$ be the fiber-wise Morse function

[TABLE]

and set

[TABLE]

Notice that $\phi_{N}=\phi^{(j_{1}\ldots j_{\ell})}_{N}$ for some $(j_{1}\ldots j_{\ell})$ in a neighborhood of $\pi_{N}^{-1}(\lambda)$ for $\lambda\in\mathrm{Crit}(\varphi_{N})$. Then, the function $f_{N}:L(N)\to\mathbb{R}$ defined by

[TABLE]

is a Morse function for sufficiently small $\epsilon$. $f_{N}$ has the critical point set

[TABLE]

The degree of $(p,\lambda)\in\mathrm{Crit}(f_{N})$ is given by $|(p,\lambda)|=|p|+|\lambda|$, where $|p|$ and $|\lambda|$ are the degrees of $p$ and $\lambda$ as critical points of $f$ and $\varphi_{N}$, respectively.

Let $V_{P}$ be a negative pseudo-gradient for $f_{P}$. Let $V_{N}$ be the vector field given by

[TABLE]

Here $\rho^{(j)}_{N,i}=\rho^{(j)}_{i}|_{BT(N)}$. Then

[TABLE]

is a negative pseudo-gradient for $f_{N}$. For $N=0$, we set $f_{0}=f$, and $\mathscr{V}_{0}=\sum_{i=1}^{\ell}\sin(\theta_{i})\cfrac{\partial}{\partial\theta_{i}}+V_{P}$.

Proposition 4.1**.**

For generic choices of $V_{P}$, $(f_{N},\mathscr{V}_{N})$ is a Morse-Smale pair. Moreover, the sequence $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ is admissible in the sense of Definition 3.4.

For $p,q\in\mathrm{Crit}(f_{N})$, let $\mathcal{M}(p,q)$ be the moduli space of gradient flow lines from $p$ to $q$, modulo reparametrizations. Let $(C^{\bullet}(f_{N};\mathbb{Z}),\delta)$ to be the Morse cochain complex

[TABLE]

We have

[TABLE]

For convenience, we will write the elements of $\mathrm{Crit}(f_{N})$ as $p\otimes\lambda$, where $p\in\mathrm{Crit}(f)$ and $\lambda\in\mathrm{Crit}(\varphi_{N})$. The following result will be used in the computation of the equivariant disc potentials.

Proposition 4.2**.**

For $N\geq 1$ and $X_{1},\ldots,X_{\ell}$, we have

[TABLE]

In particular, there exists a unique gradient flow line from $X_{i}\otimes\bm{1}_{BT}$ to $\bm{1}_{L}\otimes\lambda_{i}$.

Proof.

The possible outputs of $\delta(X_{i}\otimes\bm{1}_{BT})$ are of degree $2$ critical points of the form $(X_{j}\wedge X_{k})\otimes\bm{1}_{BT}$, and $\bm{1}_{L}\otimes\lambda_{j}$. It is easy to see that $\mathcal{M}(X_{i}\otimes\bm{1}_{BT},(X_{j}\wedge X_{k})\otimes\bm{1}_{BT})=\emptyset$ if both $j\neq i$ and $k\neq i$, and $\mathcal{M}(X_{i}\otimes\bm{1}_{BT},(X_{j}\wedge X_{k})\otimes\bm{1}_{BT})$ consists of two points of opposite orientation if either $j=i$ or $k=i$.

Suppose $\Phi:\mathbb{R}\to L(N)$ is a flow line from $X_{i}\otimes\bm{1}_{BT}$ to $\bm{1}_{L}\otimes\lambda_{j}$. Its projection $\pi_{N}\circ\Phi:\mathbb{R}\to(\mathbb{C}\mathbb{P}^{N})^{\ell}$ is a flow line for the vector field $V$ on $(\mathbb{C}\mathbb{P}^{N})^{\ell}$ given by

[TABLE]

from $\bm{1}_{BT}\to\lambda_{j}$, whose image is contained in $U_{0\ldots 0}$. This means the image of $\Phi$ is contained in $\tilde{U}_{0\ldots 0}$, and we can therefore write

[TABLE]

in term of the coordinates on $\tilde{U}_{0\ldots 0}$. For $\Phi(t)$ to have the correct asymptotics, we must have $z_{\mu,\nu}^{(0\ldots 0)}(t)=0$ for $\mu\neq i$ and $\nu\neq 1$; $\theta^{(0\ldots 0)}_{\mu}(t)=0$ for $\mu\neq i$; $z_{\ell,1}^{(0\ldots 0)}(t)=e^{2t+i\vartheta}$ where $\vartheta\in[0,2\pi)$; $\theta_{i}^{(0\ldots 0)}(t)$ satisfies

[TABLE]

and

[TABLE]

in addition to

[TABLE]

Here $a(t)=\rho^{(0)}_{1}|_{U_{0}}\left(e^{2t+i\vartheta}\right)$ and $b(t)=\rho^{(1)}_{1}|_{U_{0}}\left(e^{2t+i\vartheta}\right)$, and $\{\rho^{(0)}_{1},\rho^{(1)}_{1}\}$ is the partition of unity for $\mathbb{C}\mathbb{P}^{1}$.

For existence of a flow line when $i=j$, we note that the flow line with $\theta_{i}^{(0\ldots 0)}(t)=\pi$ and $\vartheta=0$ has the desired asymptotics. As for uniqueness, assume without loss of generality that $\Phi(0)$ is in a neighborhood of $X_{i}\otimes\bm{1}_{BT}$ such that $\rho^{(0)}_{N}(t)=1$ and $\rho^{(1)}_{N}(t)=0$. In this neighborhood, we have explicitly

[TABLE]

It is easy to see that

[TABLE]

Solving for

[TABLE]

gives

[TABLE]

This means we have

[TABLE]

∎

4.2. Computing $T$-equivariant disc potentials

By applying the construction in Section 3.1 and 3.2 with the choice of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}$ made in Section 4.1, one obtains the equivariant Morse model $(CF^{\bullet}_{T}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{T,\dagger})$ associated to the pair $(L,T)$. Note that we have

[TABLE]

where

[TABLE]

is the unital Morse complex associated to $f:L\to\mathbb{R}$. In this section, we compute the $T$-equivariant disc potential of $L$ assuming the minimal Maslov index of $L$ is nonnegative.

For simplicity of notations, we will suppress $``\blacktriangledown"$ and denote the unique maximum $\bm{1}^{\blacktriangledown}_{L}$ on $L$ by $\bm{1}^{\vphantom{\blacktriangledown}}_{L}$. Let $\{X_{1},\ldots,X_{\ell},Y_{1},\ldots,Y_{\nu}\}$ be a basis of $C^{1}(f;\mathbb{Q})$ with $\{X_{i}\}$ and $\{Y_{j}\}$ degree one critical points of $f_{T}$ and $f_{P}$ respectively. We will abuse notations and write $X_{i}:=X_{i}\otimes\bm{1}_{BT}$ and $Y_{i}:=Y_{i}\otimes\bm{1}_{BT}$. We also put $\bm{\lambda}_{i}:=\bm{1}_{L}\otimes\lambda_{i}$, and $\bm{1}:=\bm{1}_{L}\otimes\bm{1}_{BT}$.

Let $b=\sum_{i=1}^{\ell}x_{i}X_{i}+\sum_{i=1}^{\nu}y_{i}Y_{i}$, $(x_{i},y_{i})\in\Lambda_{+}^{2}$. We consider the boundary deformation of $\mathfrak{m}^{T,\dagger}_{0}$ by $b$

[TABLE]

The first equality above follows from the fact that the restriction of $\mathfrak{m}^{T,\dagger}$ to $CF^{\bullet}_{T}(L;\Lambda_{0})$ agrees with $\mathfrak{m}^{T}$.

We compute the obstruction $\mathfrak{m}_{0}^{T,b}(1)$ by counting pearly trees in $L_{T}\subset X_{T}$111To be more precise, we should perform the calculation on the finite dimensional approximations $L(N)\subset X(N)$ (see (3.12), (3.13) and (3.14)). By abuse of notations, the $A_{\infty}$-maps on $L_{T}$ are regarded as the $A_{\infty}$-maps on the approximation spaces.. Since $m^{T}_{k,\beta}(b,\ldots,b)$ is of degree $2-\mu(\beta)$, the outputs of $\mathfrak{m}_{0}^{T,b}(1)$ have degree either [math] or $2$ depending on the Maslov indices of the contributing disc classes. The possible outputs are of the following forms$\colon$

1. (1)

(Degree zero) $\bm{1}$, 2. (2)

(Degree two) $p\otimes\mathbf{1}_{BT}$, where $p\in\mathrm{Crit}(f)$ is a degree $2$ critical point, 3. (3)

(Degree two) $\bm{\lambda}_{i}$.

Notice that all the critical points above are contained in $\mathrm{Crit}(f_{1})$. Thus, $\mathfrak{m}^{T,b}_{0}(1)$ can be computed by counting pearly trees in $L(1)$.

Proposition 4.3**.**

Suppose $T\cong(\mathbb{S}^{1})^{\ell}$ acts on $(X,\omega_{X})$ preserving $\omega_{X}$. Let $L\subset X$ be a $T$-invariant closed Lagrangian submanifold of product type $L\cong T\times P$ such that $T$ acts freely on $(\mathbb{S}^{1})^{\ell}$ and trivially on $P$. Suppose the minimal Maslov index of $L$ is nonnegative, then

[TABLE]

where $\mathfrak{m}$ is the $A_{\infty}$-structure on $CF^{\bullet}(L;\Lambda_{0})$, and $h_{i}(\vec{x},\vec{y})\in\Lambda_{0}$. Moreover, if $\mathfrak{m}_{0}^{b}(1)=W(\vec{x},\vec{y})\mathbf{1}_{L}$ for some $W(\vec{x},\vec{y})\in\Lambda_{+}$, then $(CF^{\bullet}_{T}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{T,\dagger})$ is weakly unobstructed.

Proof.

The first two types of outputs are contributed by pearly trees contained in the fiber $L$ over the critical point $\mathbf{1}_{BT}$, and coincide with $\mathfrak{m}_{0}^{b}(1)$ by Proposition 3.6. Thus, we have the expression (4.3). The last assertion follows from Corollary 3.10, namely, if $\mathfrak{m}_{0}^{b}(1)=W(\vec{x},\vec{y})\mathbf{1}_{L}$, then

[TABLE]

∎

In the following, we compute $\mathfrak{m}_{0}^{b}(1)$ and $h_{i}(\vec{x},\vec{y})$ explicitly under additional assumptions. We begin by simplifying $h_{i}(\vec{x},\vec{y})$ under the condition that the minimal Maslov index of $L$ is at least $2$.

Lemma 4.4**.**

In the situation of Proposition 4.3, if in addition, the minimal Maslov index of $L$ is at least $2$, then $h_{i}(\vec{x},\vec{y})=x_{i}$ in (4.3).

Proof.

The terms $h_{i}(\vec{x},\vec{y})$ are contributed by disc classes of Maslov index [math] by degree reason. Since $L$ has minimal Maslov index at least two, the only contribution comes from the trivial disc class $\beta_{0}$, hence the computation of the terms $\mathfrak{m}^{T}_{k}(b,\ldots,b)$ reduces to $\mathfrak{m}^{T}_{\Gamma}(b,\ldots,b)$, where $\Gamma\in\bm{\Gamma}_{k+1}$ is a stable planar rooted tree with all interior vertices decorated by $\beta_{0}$. In other words, the configurations we are counting are Morse flow trees in $L_{1}$.

By Proposition 4.2, we have $\mathfrak{m}^{T}_{1}(X_{i})=\bm{\lambda}_{i}$. We will show momentarily that the coefficients of $\bm{\lambda}_{i}$ in the terms $\mathfrak{m}_{k}^{T}(X_{\nu_{1}},\ldots,X_{\nu_{k}})$ are zero for $k\geq 2$ and $(\nu_{1},\ldots,\nu_{k})\in\{1,\ldots,\ell\}^{k}$. First, let us consider the case when $\nu_{j}\neq i$ for some $j$. In the proof of Proposition 4.2, we have showed that there exists no flow line from $X_{i_{j}}$ to $\bm{\lambda}_{i}$. This means for generic small perturbations, the moduli spaces $\mathcal{M}_{\Gamma}(f_{1};X_{\nu_{1}},\ldots,X_{\nu_{k}};\bm{\lambda_{i}})$ are empty.

Now, we consider $\mathfrak{m}^{T}_{k}(X_{i},\ldots,X_{i})$, $k\geq 2$. Since rotations on the $i$-th circle factor of $L$ commute with the structure group of $L_{1}\to\mathbb{C}\mathbb{P}^{1}$, we have a global $\mathbb{S}^{1}$-action on $L_{1}$ rotating the $i$-th circle factor of the fibers $L$. To achieve transversality for the moduli spaces $\mathcal{M}_{\Gamma}(f_{1};X_{i},\ldots,X_{i};\bm{\lambda_{i}})$, we can perturb the the flow lines from the first $k-1$ inputs using the $\mathbb{S}^{1}$-action. We have the unique flow line $\gamma$ from $X_{i}$ to $\bm{\lambda}_{i}$ as in Proposition 4.2. For a perturbed flow line from the first $k-1$ inputs to intersect with $\gamma$ (in order to form a flow tree), its projection in $BT(1)$ must coincide with $\pi(\gamma)$. However, over $\pi(\gamma)$, the flow lines from the first $k-1$ inputs are simply the rotations of $\gamma$ by $\mathbb{S}^{1}$, which do not intersect with $\gamma$. Thus, $\mathcal{M}_{\Gamma}(f_{1};X_{i},\ldots,X_{i};\bm{\lambda_{i}})$ are empty for generic small perturbations.

Similarly, since the fiber bundle $P_{T}=P\times BT$ is trivial, there exists no flow line from $Y_{j}$ to $\bm{\lambda}_{i}$ for any $i,j$. Therefore the coefficients of $\bm{\lambda}_{i}$ in the terms $\mathfrak{m}^{T}_{k}(b,\ldots,Y_{j},\ldots,b)$ are zero. ∎

Thus, in the setting of Lemma 4.4, we have

[TABLE]

In [FOOO10], it was shown that the moment map fibers of a compact semi-Fano toric manifold are weakly unobstructed in the de Rham model. The following is the analogous statement in the Morse model.

Lemma 4.5**.**

Let $L$ be a regular moment map fiber of a compact semi-Fano toric manifold $X$. Then $\mathfrak{m}_{0}^{b}(1)\in\Lambda_{0}\cdot\mathbf{1}_{L}$.

Proof.

The only possible outputs of $\mathfrak{m}_{0}^{b}(1)$ are multiples of $\mathbf{1}_{L}$ and degree two critical points of $f$ (which are of the form $X_{i}\wedge X_{j}$), contributed by Maslov index zero and two disc classes, respectively. Since $X$ is semi-Fano, $L$ has minimal Maslov index $2$. This means only Morse flow trees contribute to the degree two critical points. We have $\mathfrak{m}_{1}(X_{i})=0$, since $f$ is a perfect Morse function on $L$. For $\mathfrak{m}_{k}(X_{\nu_{1}},\ldots,X_{\nu_{k}})$ and $k\geq 2$, if there are two repeated inputs $X_{\nu_{i}}=X_{\nu_{j}}$, then by perturbing unstable hypertori of $X_{\nu_{i}}$ using the $\mathbb{S}^{1}$-action rotating the $\nu_{i}$-th circle factor of $L$, the perturbed hypertori do not intersect and hence $\mathfrak{m}_{k}(X_{\nu_{1}},\ldots,X_{\nu_{k}})=0$. In the case of distinct inputs, we have

[TABLE]

due to the orientations on the corresponding moduli spaces.

∎

Remark 4.6**.**

Before proceeding, a remark is in order addressing the perturbations used in the proof of Lemma 4.4, and 4.5. We do not perturb the input singular chains when choosing perturbation for a fiber product $\mathcal{M}_{k+1}(\beta;L(N);\vec{P})$ since the singular chains are fixed during the inductive construction and doing so would destroy the $A_{\infty}$-structure. On the other hand, since the Kuranishi structure on the disc moduli are weakly submersive, we can realize the perturbation of input singular chains by perturbing the respective evaluation maps.

Combining the above lemmas, to show that the equivariant toric superpotentials coincide with the equivariant disc potentials, it remains to compute the (non-equivariant) disc potential.

Recall that the holomorphic disc classes are generated by the basic disc classes $\beta_{i}$ for $1\leq i\leq m$ [CO06]. Moreover, by [FOOO10], a stable disc class of Maslov index two must be of the form $\beta_{i}+\alpha$ for some effective curve class $\alpha$ with $c_{1}(\alpha)=0$. Let $n_{1}(\beta_{i}+\alpha)$ be the degree of the virtual fundamental class $\mathrm{ev}_{*}[\mathcal{M}_{1}(\beta_{i}+\alpha)]$. In the Morse model, it is given by the intersection number of $\mathrm{ev}_{*}[\mathcal{M}_{1}(\beta_{i}+\alpha)]$ with the maximal point $\mathbf{1}_{L}$.

Theorem 4.7** (Equivariant toric superpotential).**

Let $X$ be a compact semi-Fano toric manifolds with $\dim_{\mathbb{C}}X=d$, and let $v_{1},\ldots,v_{m}\in\mathbb{Z}^{d}$ be the primitive generators of the one-dimensional cones of the fan $\Sigma$ defining $X$. Let $L\subset X$ be a regular moment map fiber. Then,

[TABLE]

where

[TABLE]

In particular in the Fano case,

[TABLE]

where $W^{\mathrm{toric}}$ is the superpotential of Givental and Hori-Vafa [Giv98, HV00, HKK*+*03].

Proof.

Recall from Proposition 4.3 that $W^{L}\cdot\mathbf{1}_{L}=\mathfrak{m}^{b}_{0}(1)$. Since $L$ has minimal Maslov index $2$, by degree reason, a stable rooted tree $\Gamma\in\bm{\Gamma}_{k+1}$ contributing to the term $\mathfrak{m}_{k}(b,\ldots,b)$, $k\geq 1$, must have exactly one interior vertex decorated by a Maslov index $2$ disc class and the remaining interior vertices are decorated by the constant disc class $\beta_{0}$.

Let us abuse notation and denote by $X_{j}$ the unstable hypertorus of the degree one critical point $X_{j}$ of the perfect Morse function $f=\sum_{i=1}^{d}\cos(\theta_{i})$ on $(\mathbb{S}^{1})^{d}$. We can choose an identification $L\cong(\mathbb{S}^{1})^{d}$ such that the boundaries of the finitely many holomorphic discs in basic disc classes $\beta_{1},\ldots,\beta_{m}$, passing through the maximal point $\mathbf{1}_{L}$ do not intersect with $X_{\mu}\cap X_{\nu}$ for any $\mu\not=\nu$. This means the only configurations $\Gamma\in\bm{\Gamma}_{k+1}$, contributing to $\mathfrak{m}_{k}(b,\ldots,b)$ have exactly one interior vertex, which are decorated by a Maslov index $2$ disc class $\beta_{i}+\alpha$. It remains to consider contributions from the moduli spaces of the form $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L;X_{\mu_{1}},\ldots,X_{\mu_{k}})$, $(\mu_{1},\ldots,\mu_{k})\in\{1,\ldots,d\}^{k}$.

We note that $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L;X_{\mu_{1}},\ldots,X_{\mu_{k}})^{\mathfrak{s}}=\emptyset$ for generic perturbations $\mathfrak{s}$ unless $\partial\beta_{i}$ intersects all the input hypertori $X_{\mu}$. Therefore, it suffices to assume that is the case. Let us denote by $r_{j}$ the number of times $X_{j}$ appears in the inputs, and by $m_{j}$ is the intersection multiplicity of $\partial\beta_{i}\pitchfork X_{j}$. In order to obtain the expression (4.5), we choose the following Kuranishi structure and perturbation for $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L;X_{\mu_{1}},\ldots,X_{\mu_{k}})$.

Let $\mathfrak{s}=\{\mathfrak{s}_{p}\}$ be the multisection chosen for $\mathcal{M}_{1}(\beta_{i}+\alpha;L)$, which gives rise to the open Gromov-Witten invaraint $n_{1}(\beta_{i}+\alpha)$. (We note that $n_{1}(\beta_{i}+\alpha)$ is in fact independent of the choice of a transversal multisection due to the absence of disc bubbles.) For $k\geq 1$, let $\mathfrak{forget}:\mathcal{M}_{k+1}(\beta_{i}+\alpha;L)\to\mathcal{M}_{1}(\beta_{i}+\alpha;L)$ be the forgetful map which forgets the $k$ input marked points and stabilizes. To relate the fiber product $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L;X_{\mu_{1}},\ldots,X_{\mu_{k}})$ with $n_{1}(\beta_{i}+\alpha)$, we first pullback the Kuranishi structure and multisection from $\mathcal{M}_{1}(\beta_{i}+\alpha;L)$ to $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L)$ via $\mathfrak{forget}$. Let $p^{+}=[\Sigma,\vec{z},u]\in\mathcal{M}_{k+1}(\beta_{i}+\alpha;L)$ such that $\mathfrak{forget}(p^{+})=p\in\mathcal{M}_{1}(\beta_{i}+\alpha;L)$. Let $V^{+}_{p^{+}}$, $E^{+}_{p^{+}}$, and $\mathfrak{s}_{p^{+}}^{+}:V^{+}_{p^{+}}\to(E^{+}_{p^{+}})^{\ell}/S_{\ell}$ be pullback of the Kuranishi neighborhood, obstruction bundle, and multisection at $p$ via $\mathfrak{forget}$ (see [FOOO09b, Lemma 7.3.8] for the precise definitions). To make the Kuranishi structure on $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L)$ weakly submersive, we enlarge Kuranishi neighborhood $V_{p^{+}}$ and the obstruction bundle $E_{p^{+}}$ at $p^{+}$ to be of the form

1. (1)

$E_{p^{+}}=E^{+}_{p^{+}}\bigoplus_{i=1,\ldots,k}T_{u(z_{i})}L$. 2. (2)

$V_{p^{+}}=V^{+}_{p^{+}}\times U_{p^{+}}$, where $U_{p^{+}}\subset\bigoplus_{i=1,\ldots,k}T_{u(z_{i})}$ is an open subset containing [math].

Then, $\mathfrak{s}^{+}_{p^{+}}$ naturally induces a multisection $\mathfrak{s}^{\prime}_{p^{+}}:V_{p^{+}}\to(E_{p^{+}})^{\ell}/S_{\ell}$ which is characterized as follows. Let $\widetilde{\mathfrak{s}_{p^{+}}}:V_{p^{+}}\to(E_{p^{+}})^{\ell}$ be the lift of $\mathfrak{s}_{p^{+}}$ and write

[TABLE]

Here, the restriction of $(s^{1}_{p^{+},1},\ldots,s^{1}_{p^{+},\ell}):V_{p^{+}}\to(E^{+}_{p^{+}})^{\ell}$ to $V^{+}_{p^{+}}\times\{0\}$ agrees with the lift of $\mathfrak{s}_{p^{+}}^{+}$ and is constant along the $U_{p^{+}}$-direction; $(s^{2,1}_{p^{+}},\ldots,s^{2,k}_{p^{+}}):V_{p^{+}}\to\bigoplus_{i=1,\ldots,k}T_{u(z_{i})}$ is the tautological section given by the embedding $U_{p^{+}}\subset\bigoplus_{i=1,\ldots,k}T_{u(z_{i})}$.

Now, let $v_{1},\ldots,v_{k}$ be distinct, generic, small vectors in the direction of $\partial\beta_{i}$. (Since $TL$ is trivial, we can view $v_{1},\ldots,v_{k}$ as constant vector fields on $L$.) We define a new multisection $\mathfrak{s}^{\prime}=\{\mathfrak{s}^{\prime}_{p^{+}}\}$ by

[TABLE]

[TABLE]

and $\mathfrak{s}^{\prime}_{p^{+}}=\widetilde{\mathfrak{s}^{\prime}_{p^{+}}}/S_{\ell\cdot k!}$. Then, $\mathfrak{s}^{\prime}$ induces a transversal multisection on the fiber product $\mathcal{M}_{k+1}(\beta_{i}+\alpha;L;X_{\mu_{1}},\ldots,X_{\mu_{k}})$ whose zero locus is given by

[TABLE]

weighted by $1/k!$. Its intersection with $\mathbf{1}_{L}$ gives the coefficient $\frac{n_{1}(\beta_{i}+\alpha)\prod_{j=1}^{d}m_{j}^{r_{j}}}{k!}$. (See Figure 3 for an illustration of such perturbations.) Summing over all the possible inputs of $k$ hypertori and the $\frac{k!}{\prod_{j=1}^{d}(r_{j}!)}$ arrangements among them, we obtain the term $n_{1}(\beta_{i}+\alpha)\exp(v_{i}\cdot(x_{1},\ldots,x_{d}))$. ∎

Remark 4.8**.**

For the above theorem, we have chosen the perturbations for the fiber products to be the average of all configurations of the perturbed unstable hypertori of $X_{i}$. Indeed, we can choose other perturbations, which gives different expressions of the (non-equivariant) disc potential.

Take $\mathbb{P}^{1}$ as an example. Let $L$ be the equator. Let us perturb $X_{1}$ in the counterclockwise order with respect to the left hemisphere. Then the non-equivariant disc potential will read

[TABLE]

instead of the well-known expression $\mathbf{T}^{A/2}\left(e^{x_{1}}+e^{-x_{1}}\right)$.

In the Fano case, $n_{1}(\beta_{i}+\alpha)=0$ whenever $\alpha\not=0$ by dimension reason. Moreover $n_{1}(\beta_{i})=1$ [CO06]. More generally, the non-equivariant disc potential for compact semi-Fano toric manifold has been computed by [CLLT17] using Seidel representations. The coefficients are given by the inverse mirror map. We recall it in the following theorem.

Theorem 4.9** ([CLLT17]).**

Let $X$ be a compact semi-Fano toric manifold. Let $v_{1},\ldots,v_{m}$ be the primitive generators of the one dimensional cones of the fan defining $X$. We denote by $D_{i}$ the toric prime divisor corresponding to $v_{i}$. Then

[TABLE]

where

[TABLE]

and the summation is over all effective curve classes $C\in H_{2}^{\mathrm{eff}}(X)$ satisfying

[TABLE]

and $\check{q}=\check{q}(q)$ is the inverse of the mirror map $q=q(\check{q})$.

A similar result also holds for toric semi-Fano Gorenstein orbifold [CCL*+*19]. However we stick with the manifold case for simplicity. Moreover, compactness of $X$ can be replaced by requiring $X$ to be semi-projective, so that the disc moduli spaces are compact. A toric manifold $X$ is said to be semi-projective if it has a torus-fixed point, and the natural map $X\to\operatorname{Spec}{H^{0}(X,\mathcal{O}_{X})}$ is projective [CLS11, Section 7.2]. By combining Theorem 4.7 and 4.9, we get the following:

Corollary 4.10**.**

For a toric fiber of a semi-projective and semi-Fano toric manifold, the equivariant disc potential equals to

[TABLE]

where $g_{i}(\check{q}(q))$ is given by the inverse mirror map in equation (4.6).

5. $\mathbb{S}^{1}$-equivariant disc potential for the immersed two-sphere

In this section, we study the $\mathbb{S}^{1}$-equivaraint Floer theory of an immersed Lagrangian two-sphere $L_{0}$ which has a single nodal point. This will serve as a prototypical example for the authors’ subsequent work [HKLZ19] with Hong on the $\mathbb{S}^{1}$-equivariant Floer theory of immersed SYZ fibers in tori Calabi-Yau manifolds.

5.1. Equivariant Floer theory for Lagrangian immersions

In this section we describe a generalization of the equivariant Morse model to immersed Lagrangians with clean self-intersections.

In [AJ10], Akaho and Joyce developed Lagrangian Floer theory for immersed Lagrangians $L$ with transverse self-intersection (which is a nodal point), generalizing the work of Fukaya-Oh-Ohta-Ono [FOOO09b] in the embedded case. Their construction can be further generalized to the case where $L$ is an immersed Lagrangian with clean self-intersection as we describe below. See [Sch16, CW15, Fuk18] for the development of Floer theory of clean intersections in different settings.

Let $(X,\omega_{X})$ be as in Section 2.1 and let $L$ be a closed, connected, relatively spin, immersed Lagrangian submanifold with clean self-intersections. We denote by $\iota:\tilde{L}\to X$ an immersion with image $L$ and by $\mathcal{I}$ the self-intersection. As in Akaho-Joyce [AJ10], the inverse image of $\mathcal{I}$ under the immersion $\iota$ is assumed to be the disjoint union $\mathcal{I}^{-}\coprod\mathcal{I}^{+}\subset\tilde{L}$ of two branches, each of which is diffeomorphic to $\mathcal{I}$. Then the following fiber product

[TABLE]

consists of the diagonal component $R_{0}$, and the two immersed sectors

[TABLE]

We have canonical isomorphisms $R_{0}\cong\tilde{L}$ and $R_{-1}\cong R_{1}\cong\mathcal{I}$. We also have the involution $\sigma:R_{-1}\coprod R_{1}\to R_{-1}\coprod R_{1}$ swapping the two immersed sectors, i.e, $\sigma(p_{-},p_{+})=(p_{+},p_{-})$.

Let $J_{X}$ be a compatible almost complex structure of $(X,\omega_{X})$. For a map $\alpha:\{0,\ldots,k\}\to\{-1,0,1\}$, we consider quintuples $(\Sigma,\vec{z},u,\tilde{u},l)$ where

• •

$\Sigma$ is a prestable genus [math] bordered Riemann surface,

• •

$\vec{z}=(z_{0},\ldots,z_{k})$ are distinct counter-clockwise ordered smooth points on $\partial\Sigma$,

• •

$u:(\Sigma,\partial\Sigma)\to(X,L)$ is a $J$-holomorphic map with $(\Sigma,\vec{z},u)$ stable,

• •

$\tilde{u}:\mathbb{S}^{1}\setminus\{\zeta_{i}:=l^{-1}(z_{i}):\alpha(i)\neq 0\}\to\tilde{L}$ is a local lift of $u|_{\partial\Sigma}$, i.e.,

[TABLE]

where $\alpha(i)\neq 0$ and $\mathbf{i}:=\sqrt{-1}$.

• •

$l:\mathbb{S}^{1}\to\partial\Sigma$ is an orientation preserving continuous map (unique up to a reparametrization) characterized by that the inverse image of a smooth point is a point and the inverse image of a singular point consists of two points.

Let $[\Sigma,\vec{z},u,\tilde{u},l]$ be the equivalence class of $(\Sigma,\vec{z},u,\tilde{u},l)$ given by the automorphisms. For $\beta\in H_{2}(X,L;\mathbb{Z})$ and $\alpha$ a map as described above, we denote by $\mathcal{M}_{k+1}(\alpha,\beta)$ the moduli space of (equivalence classes of) such quintuples $[\Sigma,\vec{z},u,\tilde{u},l]$ satisfying $u_{*}([\Sigma])=\beta$. The moduli spaces come with the following evaluation maps $\mathrm{ev}_{i}:\mathcal{M}_{k+1}(\alpha,\beta)\to\tilde{L}\times_{L}\tilde{L}$ defined by

[TABLE]

at the input marked points $i=1,\ldots,k$, and

[TABLE]

at the output marked point.

For the convenience of writing, we will call an element of $\mathcal{M}_{k+1}(\alpha,\beta)$ a stable polygon if $\alpha(i)\neq 0$ for some $i\in\{0,\ldots,k\}$. In this case, the corners of a polygon are the boundary marked points $z_{i}$ with $\alpha(i)\neq 0$. If $\alpha(i)=0$ for all $i$, we will simply refer to an element of $\mathcal{M}_{k+1}(\alpha,\beta)$ as a stable disc. Note that $\alpha$ dictates the branch jumps at the boundary marked points.

The Kuranishi structures on $\mathcal{M}_{k+1}(\alpha,\beta;L)$ are chosen to be weakly submersive. For $\vec{P}=(P_{1},\ldots,P_{k}),P_{1},\ldots,P_{k}\in S^{\bullet}(\tilde{L}\times_{L}\tilde{L};\mathbb{Q})$, we denote by $\mathcal{M}_{k+1}(\alpha,\beta,\vec{P})$ the fiber product

[TABLE]

in the sense of Kuranishi structures. We can then construct an $A_{\infty}$-structure $\tilde{\mathfrak{m}}$ on a countably generated subcomplex $C^{\bullet}(L;\Lambda_{0})\subset S^{\bullet}(\tilde{L}\times_{L}\tilde{L};\Lambda_{0})$ as in Section 2.1. We define the maps $\tilde{\mathfrak{m}}_{k,\beta}:C^{\bullet}(L;\Lambda_{0})^{\otimes k}\to C^{\bullet}(L;\Lambda_{0})\subset S^{\bullet}(\tilde{L}\times_{L}\tilde{L};\Lambda_{0})$ by

[TABLE]

and by

[TABLE]

[TABLE]

for $(k,\beta)\neq(1,\beta_{0})$. Notice that $\mathfrak{m}_{k,(\alpha,\beta)}(P_{1},\ldots,P_{k})=\emptyset$ unless $P_{i}$ is a singular chain on $R_{\alpha(i)}$. The $A_{\infty}$-structure map $\tilde{\mathfrak{m}}_{k}$ is given by

[TABLE]

Similarly, we have a unital $A_{\infty}$-algebra $(C^{\bullet}(L;\Lambda_{0})^{+},\tilde{\mathfrak{m}}^{+})$ that is homotopy equivalent to $(C^{\bullet}(L;\Lambda_{0}),\tilde{\mathfrak{m}})$ obtained via the homotopy unit construction.

Remark 5.1**.**

We choose for each connected component $\mathcal{I}_{i}\subset\mathcal{I}$, a base point $p_{i}$, and a path of Lagrangian subspaces in $T_{p_{i}}X$ connecting $d\iota(T_{p_{i,-}}L)$ and $d\iota(T_{p_{i,+}}L)$, $\iota(p_{i,-})$=$\iota(p_{i,+})=p_{i}$. Then, the dimensions of the moduli spaces and the grading of singular chains on the immersed sectors depend on the choices of paths in the same manner as in [AJ10]. On the other hand, for our application in Section 5.2, the immersed Lagrangian will be graded by a holomorphic volume form and these will be determined uniquely once paths of graded Lagrangian subspaces are chosen.

Let $f:\tilde{L}\times_{L}\tilde{L}\to\mathbb{R}$ be a Morse function with a unique maximum point $\bm{1}^{\blacktriangledown}$ on the diagonal component $\tilde{L}$. Let $C^{\bullet}(f;\Lambda_{0})$ be the cochain complex generated by the critical points of $f$

[TABLE]

and set

[TABLE]

Let

[TABLE]

where $\Delta_{p}$ is defined by (2.8). Let $C^{\bullet}_{(-1)}(L;\Lambda_{0})\subset S^{\bullet}(\tilde{L}\times_{L}\tilde{L};\Lambda_{0})$ be the subcomplex generated elements in $\mathcal{X}_{-1}(L)$, and set

[TABLE]

We again identify $CF^{\bullet}(L;\Lambda_{0})$ with $C^{\bullet}_{(-1)}(L;\Lambda_{0})^{+}$ via the assignment $p\mapsto\Delta_{p}$ and equip $CF^{\bullet}(L;\Lambda_{0})$ with a unital $A_{\infty}$-structure $\mathfrak{m}$ via homological peturbation as in Section 2.3.

The $A_{\infty}$-maps $\mathfrak{m}_{k}$ counts stable pearly trees with interior vertices decorated by stable polygons $(\alpha,\beta)$. The each edge of a tree correspond to a negative gradient flow line of the Morse function $f$ (in a connected component of $\tilde{L}\times_{L}\tilde{L}$). A flow line connected to the $i^{\mathrm{th}}$ input marked point of a polygon $(\alpha,\beta)$ must be contained $R_{\alpha(i)}$, for otherwise the pearly tree does not exist. See Figure 5 for some (non-) examples.

For a pair $(L_{1},L_{2})$ of closed, connected, relatively spin, embedded Lagrangian submanifolds intersecting cleanly, the union $L=L_{1}\cup L_{2}$ is a immersed Lagrangian with clean self-intersections with $\tilde{L}=L_{1}\coprod L_{2}$. We choose the splitting $\iota^{-1}(\mathcal{I})=\mathcal{I}^{-}\coprod\mathcal{I}^{+}$ so that $\mathcal{I}^{-}\subset L_{1}$ and $\mathcal{I}^{+}\subset L_{2}$.

We can define a pearl complex $(CF^{\bullet}(L_{1},L_{2};\Lambda_{0}),\mathfrak{m}_{1}^{L_{1},L_{2}})$ for the Lagrangian intersection Floer theory of $(L_{1},L_{2})$ as follows:

[TABLE]

is the subcomplex of $CF^{\bullet}(L;\Lambda_{0})$ generated by critical points of $f$ in $R_{1}$, and $\mathfrak{m}_{1}^{L_{1},L_{2}}$ counts stable pearly trees $\Gamma\in\bm{\Gamma}_{2}$ with input and output vertices in $R_{1}$.

For the equivariant Morse model, we consider a $G$-equivariant Lagrangian immersion $\iota:\tilde{L}\to X$ with image $L=\iota(\tilde{L})\subset X$ a $G$-invariant immersed Lagrangian $L$. We can choose an admissible collection of Morse-Smale pairs $\{(f_{N},\mathscr{V}_{N})\}_{N\in\mathbb{N}}$ on the finite dimensional approximations of $\tilde{L}_{G}\times_{L_{G}}\tilde{L}_{G}$ and proceed to define $A_{\infty}$-algebras $(CF^{\bullet}_{G}(L;\Lambda_{0}),\mathfrak{m}^{G})$ and $(CF^{\bullet}_{G}(L;\Lambda_{0})^{\dagger},\mathfrak{m}^{G,\dagger})$ following Section 2.3 and 3.2. This gives rise to interesting equivariant disc potentials even in the case when $L$ is unobstructed as we will see in Section 5.2.

5.2. $\mathbb{S}^{1}$-equivariant disc potential via the gluing method

First, we make the following observation which is important for finding isomorphisms between objects in the Fukaya category.

Lemma 5.2**.**

Let $L_{1},L_{2}$ be two graded Lagrangians in the Fukaya category that cleanly intersect with each other, and $\alpha\in CF^{0}(L_{1},L_{2})$. Suppose that $CF^{j}(L_{1},L_{2})=0$ for all $j<0$. For any $b^{\prime}_{i}\in CF^{1}(L_{i},\Lambda_{+})$ and $w_{i}\in\Lambda_{0}$, we put b_{i}=b^{\prime}_{i}+w_{i}\cdot\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}_{L_{i}}. Then ,

[TABLE]

Similarly, if further $CF^{j}(L_{2},L_{1})=0$ for all $j<0$ and $\beta\in CF^{0}(L_{2},L_{1})$, then

[TABLE]

Proof.

For (5.1), since the degree of \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}_{L_{i}} is $(-1)$, the extra terms

[TABLE]

have degrees $2-k+(-1)\cdot(k-1)=3-2k<0$ in $CF^{\bullet}(L_{1},L_{2})$ and hence must vanish. Similarly for (5.2), the extra terms in

[TABLE]

and

[TABLE]

have degrees at most $(-2)$ and hence vanish for degree reason. ∎

In the setting of Section 3.2, the above lemma has a direct $G$-equivariant analog, by replacing $\mathfrak{m}_{k}$ with $\mathfrak{m}^{G}_{k}$ and setting $w_{i}=w_{i}^{0}+\sum_{\deg\lambda=2}\phi_{\lambda}\cdot\lambda\in H^{\bullet}_{G}(\mathrm{pt};\Lambda_{0})$, as a consequence of Proposition 3.7.

An isomorphism between $(L_{1},b_{1})$ and $(L_{2},b_{2})$, where $b_{i}$ are weak bounding cochains, consists of $\alpha\in CF^{0}(L_{1},L_{2})$ and $\beta\in CF^{0}(L_{2},L_{1})$ satisfying

[TABLE]

for some $\gamma_{i}\in CF^{\bullet}(L_{i};\Lambda_{0})$. Since \mathfrak{m}^{b_{i}}_{1}(\bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}_{L_{i}})=\bm{1}^{\triangledown}_{L_{i}}-(1-h(b_{i}))\bm{1}^{\blacktriangledown}_{L_{i}} for some $h(b_{i})\in\Lambda_{+}$, by Lemma 5.2, it suffices to check that

[TABLE]

Similarly, an isomorphism between $(L_{1},G,b_{1})$ and $(L_{2},G,b_{2})$ consists of $\alpha\in CF^{0}_{G}(L_{1},L_{2})$ and $\beta\in CF^{0}_{G}(L_{2},L_{1})$ such that the obvious analog of equations (5.2) hold.

Let us now consider the following immersed two-sphere in

[TABLE]

We equip $X$ with the symplectic form inherited from the standard symplectic form on $\mathbb{C}^{3}$. Let $\Pi\colon X\to\mathbb{C}^{\times}$ be the projection to the $z$-component. Regarding $\mathbb{S}^{1}$ as the unit circle in the complex plane $\mathbb{C}$, the space $X$ has a fiberwise Hamiltonian $\mathbb{S}^{1}$-action given by $\eta\cdot(a,b,z)\mapsto\left(\eta\cdot a,\eta^{-1}\cdot b,z\right).$ One obtains the special Lagrangian torus fibration

[TABLE]

with respect to the holomorphic volume form

[TABLE]

See [HL82, Gol01, Gro01]. In particular, a singular fiber which will be denoted by $L_{0}$ occurs over the point $(|a|^{2}-|b|^{2}=0,|z|=1)$ is an immersed two-sphere which intersects itself transversally at the point $(0,0,-1)$.

Let $\iota:\widetilde{L_{0}}=\mathbb{S}^{2}\to X$ be a Lagrangian immersion with image the immersed two-sphere $L_{0}$. In this case, the self-intersection $r\in L_{0}$ is a single nodal point fixed by the $\mathbb{S}^{1}$-action, and the two branches of the immersed loci are given by $U=(p,q)$, $V=(q,p)\in\mathbb{S}^{2}\times\mathbb{S}^{2}$, $p\neq q$, and $\iota(p)=\iota(q)=r$.

We have finite dimensional approximations of the fiber product

[TABLE]

Let $f_{\infty}:(\mathbb{S}^{2})_{\mathbb{S}^{1}}\times_{(L_{0})_{\mathbb{S}^{1}}}(\mathbb{S}^{2})_{\mathbb{S}^{1}}\to\mathbb{R}$ be a Morse function such that its restriction to each of the $\mathbb{C}\mathbb{P}^{\infty}$-components is the perfect Morse function $\varphi:B\mathbb{S}^{1}=\mathbb{C}\mathbb{P}^{\infty}\to\mathbb{R}$ (see (4.2)), and the restriction to the diagonal component $(\mathbb{S}^{2})_{\mathbb{S}^{1}}$ is the sum of a horizontal lift of the perfect Morse function $\varphi$ and function on the total space $(\mathbb{S}^{2})_{\mathbb{S}^{1}}\to\mathbb{C}\mathbb{P}^{\infty}$ which restricts to a perfect Morse function $f_{\mathbb{S}^{2}}:\mathbb{S}^{2}\to\mathbb{R}$ on generic fibers. We require $\mathrm{Crit}(f_{\mathbb{S}^{2}})$ to be away from $p,q\in\mathbb{S}^{2}$.

We will denote the critical points of $f_{\infty}|_{(\mathbb{S}^{2})_{\mathbb{S}^{1}}}$ by $\mathbf{1}_{\mathbb{S}^{2}}\otimes\lambda^{\ell}$ and $\mathrm{pt}_{\mathbb{S}^{2}}\otimes\lambda^{\ell}$, $\ell\geq 0$, where $\mathbf{1}_{\mathbb{S}^{2}}$ and $\mathrm{pt}_{\mathbb{S}^{2}}$ are the maximal and minimal critical points of $\phi$, respectively. We will also denote the critical points of $f_{\infty}$ on the non-diagonal components by $U\otimes\lambda^{\ell}$ and $V\otimes\lambda^{\ell}$, $\ell\geq 0$, respectively.

Following Section 3.2 and 5.1, we obtain an $A_{\infty}$-algebra $(CF_{\mathbb{S}^{1}}^{\bullet}(L_{0};\Lambda_{0})^{\dagger},\mathfrak{m}^{\mathbb{S}^{1},\dagger})$ associated to $(L_{0},\mathbb{S}^{1})$. We have

[TABLE]

where $CF^{\bullet}(L_{0};\Lambda_{0})$ is the unital Morse complex associated to $f_{\infty}$. By abuse of notations, degree one immersed generators (graded with respect to the $\mathbb{S}^{1}$-invariant holomorphic volume form $\Omega_{X}$) will simply be denoted by $U:=U\otimes 1$, and $V:=V\otimes 1$. We now consider the formal deformations $b_{0}=uU+vV$ of $(L_{0},\mathbb{S}^{1})$ for $(u,v)\in\Lambda^{2}_{0}$ with $\mathrm{val}(uv)>0$.

Lemma 5.3**.**

The formal deformations of $(L_{0},\mathbb{S}^{1})$ by $b_{0}=uU+vV$ are weakly unobstructed, i.e., $\mathfrak{m}_{0}^{\mathbb{S}^{1},\dagger,b_{0}}(1)\in H^{\bullet}_{\mathbb{S}^{1}}(\mathrm{pt};\Lambda_{0})\cdot\mathbf{1}$.

Proof.

Since $L_{0}$ has minimal Maslov index two, $\mathfrak{m}_{0}^{\mathbb{S}^{1},\dagger,b_{0}}(1)$ can be expressed as

[TABLE]

by the same argument in the proof Proposition 4.3. Although $L_{0}$ does not bound any non-constant holomorphic discs, extra care is necessary for $\mathfrak{m}_{0}^{b_{0}}(1)$ as constant polygons might bubble off at the immersed loci. We consider the anti-symplectic involution $\sigma\colon X\to X$ given by $(a,b,z)\mapsto(\overline{b},\overline{a},\overline{z})$, which preserves $L_{0}$ and swaps two immersed generators $U$ and $V$. In [HKL18], by taking orientations respecting the involution, the constant polygon contributions are canceled out. Thus $(L_{0},b_{0}=uU+vV)$ is unobstructed, that is, $\mathfrak{m}_{0}^{b_{0}}(1)=0$. Therefore,

[TABLE]

∎

More precisely, the $\mathbf{1}_{\mathbb{S}^{2}}\otimes\lambda$ appeared above is $\bm{1}^{\blacktriangledown}_{\mathbb{S}^{2}}\otimes\lambda=\bm{\lambda}^{\blacktriangledown}$ in the notation of Section 3.2. By Lemma 3.9, the above $b_{0}$ can be modified (by adding terms involving \bm{1}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}\blacktriangledown}}) to $b_{0}^{\dagger}$ such that $\mathfrak{m}_{0}^{\mathbb{S}^{1},\dagger,b_{0}^{\dagger}}(1)=W^{\triangledown}(u,v)\cdot\bm{\lambda}^{\triangledown}$. By abuse of notations, we shall simply replace $b_{0}$ with $b_{0}^{\dagger}$ and denote $\bm{1}^{\triangledown}_{\mathbb{S}^{2}}$ by $\mathbf{1}_{\mathbb{S}^{2}}$, $\bm{1}^{\triangledown}=\bm{1}^{\triangledown}_{\mathbb{S}^{2}}\otimes 1$ by $\mathbf{1}$, and $W^{\triangledown}$ by $W$.

Remark 5.4**.**

This line of thoughts is similar to the work in [HKL18]. In this example $L_{0}$ does not bound non-constant holomorphic discs by maximal principle. However, there are still highly interesting non-trivial contributions coming from constant polygons with corners at $U$ and $V$. Note that for each constant polygon, $U$ and $V$ corners must occur in pairs (in order to return to the original branch). This is the reason that the formal deformations of $(L_{0},\mathbb{S}^{1})$ by $b_{0}=uU+vV$ are convergent for $(u,v)\in\Lambda_{0}^{2}$ with $\mathrm{val}(uv)>0$.

In order to compute $W(u,v)$, we compare $(L_{0},\mathbb{S}^{1})$ with the $\mathbb{S}^{1}$-equivariant Floer theory of Chekanov and Clifford tori $L_{1}$ and $L_{2}$ respectively. For the special Lagrangian torus fibration (5.4), the Lagrangian torus over a point in the chamber $|z|<1$ (resp. $|z|>1$) is called a Chekanov (resp. Clifford) torus. Since any pair of distinct fibers in (5.4) does not intersect, they cannot be isomorphic. We apply Lagrangian isotopies to Chekanov torus and Clifford torus without intersecting the wall $z=0$ to make them intersect as in Figure 6 (a). Those isotoped Lagrangians are still $\mathbb{Z}$-graded and called Chekanov torus and Clifford torus respectively.

The equivariant theory for the Lagrangian tori $L_{i}$ for $i=1,2$, is understood in the same way as in the toric case in Section 4, except that in this local case they do not bound any non-constant holomorphic disc.

We equip $L_{1}$ and $L_{2}$, with the non-trivial spin structure along the $\mathbb{S}^{1}$-orbit direction. This will introduce extra systematic contributions to the orientation bundle of the moduli space of strips bounded by $L_{1}$ and $L_{2}$ and other Lagrangians. The reason for doing this will be explained below.

We fix a perfect Morse function on each $L_{i}$ for $i=1,2$, such that the (compactified) unstable submanifolds of the degree one generators $X_{i},Y_{i}$ are the hypertori chosen in [HKL18, Section 3.3]. The unstable submanifold of $X_{i}$ is transverse to the $\mathbb{S}^{1}$-orbits in $L_{i}$. We also have a Morse function on $(L_{i})_{\mathbb{S}^{1}}$ as in Section 4, whose critical points are labeled by $\eta\otimes\lambda^{\ell}$, $\ell\geq 0$ where $\eta$ is one of the following

[TABLE]

We consider the formal deformations of $L_{i}$ for $i=1,2$, by $b_{i}=x_{i}X_{i}+y_{i}Y_{i}$, $x_{i},y_{i}\in\Lambda_{+}$. (Here $X_{i}:=X_{i}\otimes 1$ and $Y_{i}:=Y_{i}\otimes 1$, by abuse of notations.) By Proposition 4.3, the equivariant disc potential of $(L_{i},\mathbb{S}^{1})$ is given by

[TABLE]

Since $L_{i}$ does not bound any non-constant holomorphic discs (particularly Maslov index zero discs), we have $W^{L^{(i)}}_{\mathbb{S}^{1}}=(W^{L^{(i)}}_{\mathbb{S}^{1}})^{\triangledown}$.

Each pair of Lagrangians $L_{i}$ and $L_{j}$, $0\leq i<j\leq 2$, intersect cleanly at two circles $C^{L_{i},L_{j}}_{\alpha}$ and $C^{L_{i},L_{j}}_{\beta}$ (see Figure 6) on which $\mathbb{S}^{1}$ acts freely. We fix a perfect Morse function on their clean intersections and denote by $CF^{\bullet}(L_{i},L_{j}),$ and $CF^{\bullet}(L_{j},L_{i})$ the corresponding Lagrangian Floer intersection complexes.

Let

[TABLE]

be the generators corresponding to the maximal critical points on $C^{L_{i},L_{j}}_{\alpha}$ and $C^{L_{i},L_{j}}_{\beta}$, respectively. Then, in the non-equivariant setting, we have the following gluing formula between the formal deformation space of the Lagrangians:

Lemma 5.5**.**

[HKL18, Theorem 3.7]** The three formally deformed Lagrangians $(L_{0},b_{0})$, $(L_{1},b_{1})$ and $(L_{2},b_{2})$ are isomorphic, with the pairwise isomorphisms provided by (5.5), if and only if

[TABLE]

(The minus sign in front of $\exp(x_{i})$ is due to the non-trivial spin structure we chose for $L_{i}$.)

We now turn to the gluing formula in the equivariant setting. For $i,j\in\{0,1,2\}$ and $i\neq j$, let $CF^{\bullet}_{\mathbb{S}^{1}}(L_{i},L_{j})$ be the $\mathbb{S}^{1}$-equivariant Lagrangian Floer intersection complex with

[TABLE]

Lemma 5.6**.**

$(L_{0},\mathbb{S}^{1},b_{0})$, $(L_{1},\mathbb{S}^{1},b_{1})$ and $(L_{2},\mathbb{S}^{1},b_{2})$ are isomorphic, with the pairwise isomorphisms provided by

[TABLE]

if and only if $\eqref{equ_isorel}$ holds.

Proof.

This is simply due to degree reason. Since the equivariant parameter $\lambda$ has degree two, we have

[TABLE]

This implies that the gluing formula remains the same in the equivariant setting. Note that we have implicitly identified $CF^{\bullet}(L_{i};\Lambda_{0})$ with $CF^{\bullet}(L_{i};\Lambda_{0})\otimes 1\subset CF^{\bullet}_{\mathbb{S}^{1}}(L_{i};\Lambda_{0})$ in the above equations. ∎

Since the disc potentials are compatible with the gluing formulas (see [CHL18, Theorem 4.7]), we can compute $W^{L_{0}}_{\mathbb{S}^{1}}$ using the above lemma and the fact that $W^{L_{1}}_{\mathbb{S}^{1}}=\lambda x_{1}$.

Theorem 5.7**.**

The $\mathbb{S}^{1}$-equivariant potential m of the immersed two-sphere $L_{0}$ is given by

[TABLE]

Proof.

By the $A_{\infty}$-identity, we have

[TABLE]

(5.6) forces the (LHS) to vanish, and thus

[TABLE]

On the other hand, we have

[TABLE]

Therefore,

[TABLE]

Since $\exp(x_{1})=1-uv$, we get

[TABLE]

∎

Remark 5.8**.**

Note that if we did not take the non-trivial spin structure, then the gluing formula reads $\exp(x_{1})=-1+uv$. However, the LHS lies in $1+\Lambda_{+}$, while the RHS lies in $-1+\Lambda_{+}$. This means the formally deformed Lagrangians $(L_{1},\mathbb{S}^{1},b_{1})$ equipped with the trivial spin structure cannot be isomorphic to $(L_{0},\mathbb{S}^{1},b)$. This is reason why we equipped $L_{1}$ and $L_{2}$ with the non-trivial spin structure in the first place.