Fukaya Algebra over $\mathbb{Z}$ : I

Mohamad Rabah mohamad.rabah@stonybrook.edu

Abstract.

Given a closed connected relatively-spin Lagrangian submanifold of a closed symplectic manifold, we associate to it a curved, gapped, filtered $A_{n,K}$ -algebra over the Novikov ring with integer coefficients.

1. Introduction

Let $(X,\omega)$ be a closed symplectic manifold and let $\Delta\subset X$ be a closed, connected, relatively-spin Lagrangian. Moreover, we fix $(f,g)$ a Morse-Smale pair on $\Delta$ and denote by $\operatorname{Crit}(f)$ the (finite) set of critical points of $f$ . We denote by $\Lambda:=\Lambda^{\mathbb{Z}}_{0,nov}$ the Novikov ring over $\mathbb{Z}$ . Namely,

\Lambda:=\{\Sigma_{i}a_{i}T^{\lambda_{i}}e^{n_{i}}:a_{i},n_{i}\in\mathbb{Z},0\leq\lambda_{1}\leq\lambda_{2}\leq\dots,\lim_{i}\lambda_{i}=\infty\}.

where $T,e$ are formal variables of degrees $0,2$ respectively.
Let $\Pi(\Delta):=\pi_{2}(X,\Delta)/\sim$ where $\alpha\sim\beta$ if and only if $\omega(\alpha)=\omega(\beta)$ and $\mu(\alpha)=\mu(\beta)$ where $\mu:\pi_{2}(X,\Delta)\rightarrow 2\mathbb{Z}$ is the Maslov index of $\Delta$ . We denote by $\operatorname{CM}(f;\mathbb{Z}):=\mathbb{Z}\langle\operatorname{Crit}(f)\rangle$ the Morse complex on $\Delta$ generated by $\operatorname{Crit}(f)$ . In this paper, we prove the following theorem.

Theorem 1.1.

The Morse complex $\operatorname{CM}(f;\Lambda):=\operatorname{CM}(f;\mathbb{Z})\widehat{\otimes}_{\mathbb{Z}}\Lambda$ carries the structure of a curved, gapped, filtered $A_{n,K}$ -algebra $\{m_{k}\}_{k\geq 0}$ . Namely,

(1.1)

m_{k}=\Sigma_{\beta\in\Pi(X,\Delta)}m_{k,\beta}T^{\omega(\beta)}e^{\frac{\mu(\beta)}{2}}

where $m_{k}:\operatorname{CM}(f;\Lambda)^{\otimes k}\rightarrow\operatorname{CM}(f;\Lambda)$ satisfies the $A_{n,K}$ -axioms and $\widehat{\otimes}$ is the completion of the tensor product with respect to the $T$ -adic topology.

1.1. Filtered $A_{n,K}$ -algebra

[FOOO09] We give the monoid $\{(\omega(\beta),\mu(\beta))\}\times\mathbb{N}$ a partial order $\precsim$ as follows:
Given $\beta\in\Pi(\Delta)$ we set

||\beta||:=\sup\{n:\beta=\beta_{1}+\dots+\beta_{n}\}+\lceil\omega(\beta)\rceil-1

where $\beta_{i}\neq 0$ and each can be represented by a pseudo-holomorphic curve.

||\beta||=-1

if $\beta=0\in\Pi(\Delta)$ .
We say $(\beta_{1},k_{1})\prec(\beta_{2},k_{2})$ if and only if

||\beta_{1}||+k_{1}<||\beta_{2}||+k_{2}

||\beta_{1}||+k_{1}=||\beta_{2}||+k_{2}\text{ and }||\beta_{1}||<||\beta_{2}||

We write $(\beta_{1},k_{1})\sim(\beta_{2},k_{2})$ if

||\beta_{1}||+k_{1}=||\beta_{2}||+k_{2}\text{ and }||\beta_{1}||=||\beta_{2}||

An $A_{n,K}$ -algebra structure on $\operatorname{CM}(f;\Lambda)$ is a collection of $\mathbb{Z}$ -linear maps $m_{k,\beta}:\operatorname{CM}(f;\Lambda)^{\otimes k}\rightarrow\operatorname{CM}(f;\Lambda)$ for every $(\beta,k)\precsim(n,K)$ satisfying:

\Sigma_{\beta_{1}+\beta_{2}=\beta,k_{1}+k_{2}=k+1}\Sigma_{i}(-1)^{*_{i}}m_{k_{2},\beta_{2}}(x_{0},\dots,x_{i-1},m_{k_{1},\beta_{1}}(x_{i},\dots,x_{i+k_{1}}),x_{i+k_{1}+1},\dots,x_{k})=0

where $*_{i}=\Sigma_{j=1}^{i}|x_{j}|^{\prime}$ and $|x_{j}|^{\prime}=|x_{j}|-1$ .

Acknowledgements. I am grateful to my Ph.D. advisor, Kenji Fukaya, for a great number of enlightening meetings through out the years and his generosity in sharing his ideas. Equally, I am also thankful to Mark McLean for his generosity in his time and ideas. Also I would like to thank Jiaji Cai, Shuhao Li, Yao Xiao and Guangbo Xu for many fruitful discussions.

2. FOP Perturbations

In this section and following [BX22b], [BX22a], we use the language of derived orbifold charts. In order to define the notion of an FOP-perturbation we need the notion of normally complex derived orbifold charts and the notion of straightening on derived orbifold charts. We recall all the necessary results in this section that will be used later in the proof of the main theorem.

Definition 2.1.

A derived orbifold chart is a tuple $(\mathcal{U},\mathcal{E},s)$ such that $\mathcal{E}\rightarrow\mathcal{U}$ is an orbibundle and $s$ is a continuous section.

We introduce the following notations. Suppose that $(\Gamma,U,E)$ is an orbibundle chart of a orbibudle $\mathcal{E}\rightarrow\mathcal{U}$ , for any subgroup $G\subset\Gamma$ we denote by $U^{G}\subset U$ the $G$ -fixed locus of the induced $G$ -action and we denote by $NU^{G}\rightarrow U^{G}$ the normal bundle of $U^{G}\subset U$ . Moreover, over $U^{G}$ we can decompose $E|_{U^{G}}=\dot{E}^{G}\oplus\check{E}^{G}$ as direct sum of trivial and non-trivial $G$ -repsentations.

Definition 2.2.

[BX22b], [BX22a] A derived orbifold chart is said to be normally complex if and only if for each bundle chart $(\Gamma,E,U)$ and for each subgroup $G\leq\Gamma$ , we have a $G$ -invariant complex structure $I_{G}$ on $NU^{G}$ and $G$ -invariant complex structure $J_{G}$ on the non-trivial $G$ sub-representations of $E$ , satisfying compatibility conditions as in [BX22a]. Namely, within the same chart and for any pair of subgroups $H\subset G\subset\Gamma$ , we have an $H$ -equivariant inclusions

NU^{H}|_{U^{G}}\subset NU^{G}

\check{E}^{H}|_{U^{G}}\subset\check{E}^{G}

We require that the above inclusions to be complex linear with respect to the complex structures above and to be compatible with chart embeddings.

From now on and in order avoid clutter, we assume that all the derived orbifold charts in this section are normally complex.

Definition 2.3.

[BX22b], [BX22a] A straightening of a derived orbifold chart $(\mathcal{U},\mathcal{E},s)$ is a choice of a Riemannian metric on $\mathcal{U}$ and a connection on $\mathcal{E}$ satisfying the following conditions:

(1)

for each chart $(\Gamma,U,E)$ the pullback metric $g_{U}$ satisfies the following condition. Given $G\subset\Gamma$ , we require that on a neighborhood of $U^{G}$ the ambient metric $g_{U}$ agrees with the bundle metric on $NU^{G}$ given by $g_{U}$ via the exponential map along the normal directions.
(2)

for each chart $(\Gamma,U,E)$ we require that the pullback connection $\nabla^{E}$ on $E$ satisfies the following condition. Given $G\subset\Gamma$ and using the exponential map given by $g_{U}$ , we identify a neighborhood of $U^{G}$ by a neighborhood of the zero-section of $NU^{G}$ . Then, after also identifying $E|_{NU^{G}}$ with the pullback of $E|_{U^{G}}$ by $NU^{G}\rightarrow U^{G}$ and the parallet transport along normal geodesics using $\nabla^{E}$ , the connection $\nabla^{E}$ agrees with pullback connection of the restriction $\nabla^{E}$ to $U^{G}$ .

In order to define the notion of an FOP section/perturbation, we need to introduce more notations. Following [BX22b], [BX22a], suppose that $B$ is a smooth manifold and $G$ is a finite group acting trivially on $B$ . Let $\pi_{V}:V\rightarrow B$ and $\pi_{W}:W\rightarrow B$ be two smooth complex $G$ -vectorbundle and fix a positive integer $d$ . We have an induced vectorbundle $\operatorname{Poly}^{G}_{d}(V,W)\rightarrow B$ whose fibre over $b\in B$ is the vectorspace $\operatorname{Poly}^{G}_{d}(V_{b},W_{b})$ of all $G$ -equivariant complex polynomial maps $V_{b}\rightarrow W_{b}$ of degree at most $d$ . Suppose that $V$ is also equipped with a bundle metric and denote by $V_{\epsilon}$ a $G$ -invariant disk subbundle of $V$ . Consider $s:V_{\epsilon}\rightarrow\pi_{V}^{*}W$ a smooth $G$ -equivariant section.

Definition 2.4.

[BX22b], [BX22a]

(1)

$s$ is said to be a normally complex polynomial section of degree at most $d$ if the restriction of $s$ to each fibre $V_{b}\cap V_{\epsilon}$ is an element of $\operatorname{Poly}^{G}_{d}(V_{b},W_{b})$ .
(2)

$s$ is said to be an FOP section of degree at most $d$ if $\forall(b,v)\in V_{\epsilon}$ we can find a neighborhood of $(b,v)$ and a bundle map $f:V_{\epsilon}\rightarrow\operatorname{Poly}^{G}_{d}(V,W)$ such that $\forall(b^{\prime},v^{\prime})$ in such neighborhood, we have

$s(b^{\prime},v^{\prime})=f(b,v^{\prime})(v^{\prime})$

Definition 2.5.

[BX22b], [BX22a] Given $(\mathcal{U},\mathcal{E},s)$ a straightened normally complex derived orbifold chart. We say:

(1)

$s$ is an FOP section if for every bundle chart $(\Gamma,U,E)$ where $s:U\rightarrow E$ is a $\Gamma$ -equivariant continuous section and after identifying a tubular neighborhood of $U^{\Gamma}$ with a disk subbundle $N_{\epsilon}U^{\Gamma}$ of $NU^{\Gamma}\rightarrow U^{\Gamma}$ using the straightened structure and identifying

$E|_{N_{\epsilon}U^{\Gamma}}\cong\pi^{*}_{NU^{\Gamma}}\dot{E}^{\Gamma}\oplus\pi^{*}_{NU^{\Gamma}}\check{E}^{\Gamma}$

we can write $s|_{N_{\epsilon}U^{\Gamma}}=(\dot{s},\check{s})$ . We require $\check{s}$ to be an FOP section as in the above definition.

(2)

Suppose that $s$ is an FOP section. We say $s$ is strongly transverse at $x\in\mathcal{U}$ if the following condition hold. If $(\Gamma,U,E)$ is an orbibundle chart centered at $x$ and after identifying $U$ with a disk subbundle $N_{\epsilon}U^{\Gamma}$ of the normal bundle $NU^{\Gamma}\rightarrow U^{\Gamma}$ and decomposing

E|_{N_{\epsilon}U^{\Gamma}}\cong\pi^{*}_{NU^{\Gamma}}\dot{E}^{\Gamma}\oplus\pi^{*}_{NU^{\Gamma}}\check{E}^{\Gamma}

as above and after writing $s=(\dot{s},\check{s})$ . We consider a bundle map $f:N_{\epsilon}U^{\Gamma}\rightarrow\operatorname{Poly}^{\Gamma}_{d}(NU^{\Gamma},\check{E}^{\Gamma})$ lifting $\check{s}$ and require the induced bundle map

(\dot{s},\operatorname{graph}(f)):N_{\epsilon}U^{\Gamma}\rightarrow\dot{E}^{\Gamma}\oplus(NU^{\Gamma}\oplus\operatorname{Poly}^{G}_{d}(NU^{\Gamma},\check{E}^{\Gamma}))

to be transverse to the subbundle $\{0\}\times\mathcal{Z}^{\Gamma}_{d}(NU^{\Gamma},\check{E}^{\Gamma})$ with respect to the canonical Whitney stratification.

Remark 2.6.

The definition of $\mathcal{Z}(V,W)$ and the notion of canonical Whitney stratification is as [BX22b], [BX22a] since we will not use these constructions in this paper we don’t give their definitions here.

Lemma 2.7.

[BX22b], [BX22a] Given $(\mathcal{U},\mathcal{E},s)$ a derived orbifold chart, we can find a straightening in a neighborhood of $s^{-1}(0)$ .

Lemma 2.8.

[BX22a] Suppose that $(\mathcal{U}^{\prime},\mathcal{E^{\prime}},s^{\prime})$ has an open embedding to $(\mathcal{U},\mathcal{E},s)$ . Then, we can find a compatible straightening on both of $(\mathcal{U}^{\prime},\mathcal{E^{\prime}},s^{\prime}),(\mathcal{U},\mathcal{E},s)$ . Namely, the pull-back of the straightening on $(\mathcal{U},\mathcal{E},s)$ agrees with that of $(\mathcal{U}^{\prime},\mathcal{E^{\prime}},s^{\prime})$ .

Lemma 2.9.

[BX22b] Given $(\mathcal{U},\mathcal{E},s)$ a normally complex derived orbifold chart and $\epsilon>0$ . There exists a strongly transverse FOP section $s_{\epsilon}$ such that $||s-s_{\epsilon}||<\epsilon$ .

We will also need a relative-version of this statement in the proof of the main theorem.

Lemma 2.10.

[BX22b], [BX22a] Suppose that $\mathcal{E}\rightarrow\mathcal{U}$ is an orbibundle and $C\subset\mathcal{U}$ is a closed set. Let $s_{0}:O\rightarrow\mathcal{E}$ be a strongly transverse FOP-section defined on an open neighborhood of $C$ . Then, there exists a strongly transverse FOP-section $s:\mathcal{U}\rightarrow\mathcal{E}$ that agrees with $s_{0}$ on, a possibly smaller neighborhood, of $C$ .

Moreover, as we will see later in the inductive step of the proof of the main theorem, we need the following lemmas.

Lemma 2.11.

[BX22b] Suppose that $(\mathcal{U},\mathcal{E},s)$ is a derived orbifold chart where $s$ is a strongly transverse FOP-section and let $\pi:\mathcal{F}\rightarrow\mathcal{U}$ be a complex orbibundle. Denote by $\tau:\mathcal{F}\rightarrow\pi^{*}\mathcal{F}$ the tautological section. Then, $\pi^{*}s\oplus\tau:\mathcal{F}\rightarrow\pi^{*}\mathcal{E}\oplus\pi^{*}\mathcal{F}$ is a strongly transverse FOP-section.

Lemma 2.12.

[BX22a] Given two derived orbifold charts $(\mathcal{U}_{i},\mathcal{E}_{i},s_{i})$ such that both $s_{i}$ are strongly transverse FOP sections, then $s_{1}\oplus s_{2}:\mathcal{U}_{1}\times\mathcal{U}_{2}\rightarrow\mathcal{E}_{1}\oplus\mathcal{E}_{2}$ is a strongly transverse FOP section.

Lemma 2.13.

[BX22b] Suppose that $(\mathcal{U},\mathcal{E},s)$ is a derived orbifold chart such that $s$ is a strongly transverse FOP section. Denote by $\mathcal{U}_{free}\subset\mathcal{U}$ the open and dense subset (as all orbifolds are effective) of all points with trivial isotropy groups. Then, $s^{-1}(0)\cap\mathcal{U}_{free}\hookrightarrow\mathcal{U}$ defines a pseudo-cycle of real dimension $\dim_{\mathbb{R}}\mathcal{U}-\dim_{\mathbb{R}}\mathcal{E}$ and hence an element in $H_{*}(\mathcal{U},\mathbb{Z})$ independent from the choice of straightening.

3. Global Kuranishi Charts

The notion of Global Kuranishi charts was first introduced in [AMS21].

Definition 3.1.

Let $M$ be any metrizable space. A Global Kuranishi chart of $M$ is a tuple $(G,\mathcal{T},Ob,s)$ where

(1)

$G$ is a compact Lie group.
(2)

$\mathcal{T}$ is a smooth $G$ -manifold, with boundaries and corners such that $G$ acts on $\mathcal{T}$ by finite stabilizers.
(3)

$Ob\rightarrow\mathcal{T}$ is a smooth $G$ -vectorbundle.
(4)

$s:\mathcal{T}\rightarrow Ob$ is a continuous section.

such that $s^{-1}(0)/G\cong M$ .

Example 3.2.

Any smooth manifold $M$ can be given a Global Kuranishi representation. Namely $(\{1\},M,TM,0_{TM})$ where $0_{TM}$ is zero-section of $TM$ and $\{1\}$ is the trivial group.

Different Global Kuranishi chart representation of the same space are related by sequence of moves, which we describe below, that preserves the virtual fundamental class [AMS21], [AMS24].

(1)

(Germ Equivalence) If $s^{-1}(0)\subset U\subset\mathcal{T}$ where $U$ is a $G$ -invariant open set, then $(G,\mathcal{T},Ob,s)\sim(G,U,Ob|_{U},s|_{u})$ .
(2)

(Stabilization) If $p:W\rightarrow\mathcal{T}$ is a $G$ -vectorbundle, then $(G,\mathcal{T},Ob,s)\sim(G,W,p^{*}Ob\oplus p^{*}W,p^{*}s\oplus\tau)$ , where $\tau$ is the tautological section of $p^{*}W\rightarrow W$ .
(3)

(Group Enlargement) If $G^{\prime}$ is a compact Lie group and $q:P\rightarrow\mathcal{T}$ is a $G$ -equivariant principal, then $(G,\mathcal{T},Ob,s)\sim(G\times G^{\prime},P,q^{*}Ob,q^{*}s)$ .

Definition 3.3.

A Global Kuranishi chart $(G,\mathcal{T},Ob,s)$ is said to be normally complex if, possibly after stabilization by a free $G$ -representation, the associated derived orbifold chart $(\mathcal{T}/G,Ob/G,s/G)$ is normally complex.

4. Global Kuranishi Charts for J-Disks

From now on, we fix $J$ an $\omega$ -compatible almost complex structure on $X$ .

Definition 4.1.

Let $\beta\in\pi_{2}(X,\Delta)$ and $k,l\in\mathbb{N}$ . We denote by $\mathfrak{M}_{k+1,l}(\beta)=\{u:(\Sigma,\partial\Sigma;\bar{z},\bar{w})\rightarrow(X,\Delta):\bar{\partial}_{J}u=0,u(\partial\Sigma)\subset\Delta$ , $[u]=\beta$ }, where $\Sigma$ is a genus-zero, nodal, bordered Riemann surface with connected boundary. $\bar{z}$ is a tuple of cyclically ordered $(k+1)$ -boundary marked points and $\bar{w}$ is a tuple of $l$ -interior marked points.

We call elements of $\mathfrak{M}_{k+1,l}(\beta)$ $J$ -disks of degree $\beta$ . Fix $\beta\in\pi_{2}(X,\Delta)$ and $k,l\in\mathbb{N}$ .

Lemma 4.2.

The moduli sapce $\mathfrak{M}_{k+1,l}(\beta)$ can be given a normally complex Global Kuranishi chart representation together with a submersive map onto $\Delta^{k+1}$ extending the usual evaluation map.

The proof of the above lemma is long and requires constructions similar to that in [AMS21], [AMS24], [BX22a], [BPX24].

4.1. Preliminary Constructions

Note that, non-degeneracy of $\omega$ is an open condition and since we can approximate $[\omega]\in H^{2}(X;\mathbb{R})$ in the $C^{\infty}$ -norm with elements in $H^{2}(X;\mathbb{Q})$ , it follows upon scaling that we can assume without loss of generality (and abuse of notation) that $[\omega]\in H^{2}(X;\mathbb{Z})$

Definition 4.3.

Let $(\Sigma,\partial\Sigma)$ be a genus zero bordered Riemann surface and $L\rightarrow\Sigma$ be a complex line bundle. $L$ is said to be holomorphic if $L_{|(\Sigma\setminus\partial\Sigma)}\rightarrow\Sigma\setminus\partial\Sigma$ is a holomorphic line bundle.

Definition 4.4.

In the setting of the above definition, $L_{\mathbb{R}}\rightarrow\partial\Sigma$ is said to be totally real line bundle of $L$ if $L_{\mathbb{R}}\rightarrow\partial\Sigma$ is a real line bundle such that, $L_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}=L_{|\partial\Sigma}$ .

Lemma 4.5.

There exists $\Omega\in\Omega^{2}(X)$ so that $[\Omega]\in H^{2}(X,\Delta;\mathbb{Z})\mapsto[\omega]\in H^{2}(X;\mathbb{Z})$ under the pull-back by the quotient map.

Proof.

Upon setting $g=\omega(.\ ,J.)$ as a Riemannian metric on $X$ , it follows by Weinstein’s neighborhood theorem, that $\exists\epsilon>0$ so that $\omega=d\lambda$ on $\mathbf{D}_{\epsilon}(\Delta)$ , a disc bundle of radius $\epsilon$ over $\Delta$ , where $\lambda\in\Omega(X)$ . Choosing $\phi:X\longrightarrow[0,1]$ smooth bump where $\phi\equiv 0$ on $\mathbf{D}_{\frac{\epsilon}{3}}(\Delta)$ and $\phi\equiv 1$ on $\mathbf{D}_{\frac{2\epsilon}{3}}(\Delta)$ and upon setting $\Omega=d(\phi\lambda)$ on $\mathbf{D}_{\frac{\epsilon}{3}}(\Delta)$ and $\Omega=\omega$ on $X\setminus\mathbf{D}_{\frac{2\epsilon}{3}}(\Delta)$ , the result follows. ∎

Lemma 4.6.

There exists a Hermitian line bundle $(L,\nabla)\rightarrow X$ of curvature form $F_{\nabla}=-2\pi i\Omega$ together with a trivialization $\tau:L_{|U}\cong U\times\mathbb{C}$ , where $U$ is an open neighborhood of $\Delta$ .

Proof.

Let $N:=X\setminus\Delta$ and note that $N$ is a non-compact smooth manifold. We denote by $S$ the group of isomorphism classes of pairs $(L,\tau)$ where $L\rightarrow N$ is a complex line bundle and $\tau:L_{|U}\cong U\times\mathbb{C}$ a trivialization at infinity. That is, $\tau$ a trivialization outside a compact set of N, that is $U\subset N$ is an open set where $N\setminus U$ is compact. The group structure on $S$ is given by tensor product. Arguing by compactly-supported Cech cohomology, we have $S\cong H^{2}_{c}(N;\mathbb{Z})$ given by relative first Chern class. By the construction of $\Omega$ in the proof of Lemma 1, it follows that, $\Omega\in\Omega_{c}^{2}(N)$ such that $[\Omega]\in H^{2}_{c}(N;\mathbb{Z})$ . Hence $\exists(L,\tau)\in S$ such that $c_{1}(L,\tau)=[\Omega]$ .
Now let $\nabla$ be any Hermitian connection on $L$ which is trivial with respect to $\tau$ at infinity. Then, $\frac{i}{2\pi}F_{\nabla}=\nabla+d\theta$ where $\theta\in\Omega^{1}_{c}(N)$ . Setting $\nabla^{{}^{\prime}}:=\nabla+i\theta$ , we get $F_{\nabla^{{}^{\prime}}}=-2\pi i\Omega$ . Now the result follows after extending $L$ over $\Delta$ . ∎

Let $u:(\Sigma,\partial\Sigma)\longrightarrow(X,\Delta)$ with $[u]\neq 0\in H_{2}(X,\Delta;\mathbb{Z})$ be a non-constant smooth map and denote by $L_{u}:=u^{*}L$ . In addition, consider the double of $(\Sigma,\partial\Sigma)$ , namely let $\tilde{\Sigma}=\Sigma\cup_{\partial\Sigma}\bar{\Sigma}$ where $\bar{\Sigma}$ is the complex conjugate. Thus, $\tilde{\Sigma}$ is a genus zero orientable nodal surface. Moreover, from this construction we have an anti-holomorphic involution $\iota:\tilde{\Sigma}\longrightarrow\tilde{\Sigma}$ with non-empty fixed locus, Fix $(\iota)\cong\partial\Sigma$ . Similarly, upon pulling-back the trivialization $\tau$ by $u$ , we consider $\mathcal{L}\rightarrow\tilde{\Sigma}$ the double of $L_{u}$ .
We note the following easy lemma.

Lemma 4.7.

Up to biholomorphism, there exists a unique anti-holomorphic involution on $\mathbb{CP}^{1}$ .

Proof.

For existence, consider the conjugation on affine coordinate $z\in\mathbb{C}\subset\mathbb{CP}^{1}$ given by $z\rightarrow\frac{1}{\bar{z}}$ . For uniqueness, let $\iota$ be an anti-holomorphic involution on $\mathbb{CP}^{1}$ and consider $\bar{\iota}:\mathbb{CP}^{1}\longrightarrow\mathbb{CP}^{1}$ . Then $\bar{\iota}$ is a degree 1 holomorphic map and thus a bihilomorphism. ∎

Remark 4.8.

In fact, using brute-force calculations one can see that in affine coordinates, anti-holomorphic involutions on $\mathbb{CP}^{1}$ are of the form $z\mapsto\frac{1}{\bar{z}}$ or $z\mapsto\frac{-1}{\bar{z}}$ , distinguished by whether it has a non-empty fixed locus or not.

Lemma 4.9.

$\mathcal{L}\rightarrow\tilde{\Sigma}$ is a holomorphic line bundle of $\langle c_{1}(\mathcal{L}),[\tilde{\Sigma}]\rangle=2(u^{*}\Omega)[\Sigma,\partial\Sigma]$ .

Proof.

Note that, $\mathcal{L}$ is a Hermitian line bundle over $\tilde{\Sigma}$ and thus, $\mathcal{L}$ is a holomorphic line bundle, as $H^{0,2}(\tilde{\Sigma})={0}$ , with a unique holomorphic structure since $\tilde{\Sigma}$ is simply-connected.
To prove the second claim, we give two proofs:
(1) Note that, we can represent the relative first Chern number of $\langle c_{1}(L_{u},\tau),[\Sigma,\partial\Sigma]\rangle$ with the signed count of number of zeros of a section of $L_{u}$ transverse to the zero-section and being non-zero on the boundary. Now upon doubling this section to a section of $\mathcal{L}$ , we get the second claim.
(2) Alternatively, we calculate as follows on each disk component of $\Sigma$ :
Denote by $U=\mathbb{D}(1+\epsilon)\setminus\mathbb{D}(1-\epsilon)$ and $F_{h_{i}}$ the Chern curvature of $L_{i}=(L,h_{i})$ for $i=1,2$ . For $v_{i}\in\Gamma(L_{i})$ we have $h_{i}(v,v)=e^{\psi_{i}}||v||^{2}$ with $F_{h_{i}}=-\partial\bar{\partial}\psi_{i}$ for $i=1,2$ . On $U$ , $h_{2}=e^{\phi}h_{1}$ and $v_{2}=z^{k}v_{1}$ where $k:=2(u^{*}\Omega)[\mathbb{D},S^{1}]$ and hence,

h_{2}(v_{2},v_{2})=e^{\psi_{2}}||v_{2}||^{2}=e^{\psi_{2}}z^{2k}||v_{1}||^{2}=e^{\psi_{2}+k\log|z|^{2}}||v_{1}||^{2}=e^{\psi_{2}-\psi_{1}+k\log|z|^{2}}h_{1}(v_{1},v_{1})

and thus, $\phi=\psi_{2}-\psi_{1}+k\log|z|^{2}$ .
The total curvature of $g$ is

\int_{S}F_{g}=\int_{\mathbb{D}(1-\epsilon)}F_{h_{1}}+\int_{U}F_{g}+\int_{\mathbb{D}(1-\epsilon)}F_{h_{2}}=-\int_{S^{1}_{-}}\bar{\partial}\psi_{1}+\int_{U}F_{g}+\int_{S^{1}_{+}}\bar{\partial}\psi_{2}

using Stoke’s Theorem and the $S^{1}_{+}$ and $S^{1}_{-}$ denotes the boundary circles of the upper and lower $\mathbb{D}(1-\epsilon)$ respectively. Using Stoke’s Theorem one more time we have

\int_{U}F_{g}=-\int_{S^{1}_{+}}\bar{\partial}\log(\phi_{1}h_{1}+\phi_{2}h_{2})+\int_{S^{1}_{-}}\bar{\partial}\log(\phi_{1}h_{1}+\phi_{2}h_{2})

Noting that, on $S^{1}_{+}$ we have

-\partial\bar{\partial}\log h_{2}=-\partial\bar{\partial}\psi_{2}+k\log|z|^{2}

it follows that,

\int_{S^{1}_{+}}\bar{\partial}\log(\phi_{1}h_{1}+\phi_{2}h_{2})=-\int_{S^{1}_{+}}\bar{\partial}\psi_{2}-k\int_{S^{1}_{+}}\bar{\partial}\log|z|^{2}

Thus by symmetry, it follows that,

\int_{S}F_{g}=-2k\int_{S^{1}_{+}}\bar{\partial}\log(r)=-2k\int_{S^{1}}\bar{\partial}\log(r)

as $\log(r)$ is a harmonic function and thus the above integral is independent of the choice of the radius of $S^{1}_{+}$ . Thus, to find the total curvature of $g$ we use $z=re^{i\theta}$ , $\bar{z}=re^{-i\theta}$ , $d\bar{z}=e^{-i\theta}dr-ire^{i\theta}d\theta$ and $\partial_{\bar{z}}=\partial_{r}\frac{\partial r}{\partial\bar{z}}+\partial_{\theta}\frac{\partial\theta}{\partial\bar{z}}$ . Thus,

\bar{\partial}\log(r)=\frac{\partial\log(r)}{\partial\bar{z}}d\bar{z}=\frac{\partial\log(r)}{\partial r}\frac{\partial r}{\partial\bar{z}}(-ire^{i\theta})d\theta

as $dr=0$ on the unit circle. Noting that, $r^{2}=z\bar{z}$ , it follows that, $2r\frac{\partial r}{\partial\bar{z}}=z$ and hence, $\frac{\partial r}{\partial\bar{z}}=\frac{z}{2r}=\frac{1}{2}e^{-i\theta}$ . Thus,

\bar{\partial}\log(r)=\frac{1}{2r}e^{i\theta}(-ire^{-i\theta})d\theta=-\frac{i}{2}d\theta

and hence,

-2k\int_{S^{1}}\bar{\partial}\log(r)=2k\pi i

By Chern-Weil Formula, the result follows. ∎

Lemma 4.10.

There exists a lift $\tilde{\iota}:\mathcal{L}\rightarrow\mathcal{L}$ of $\iota:\tilde{\Sigma}\rightarrow\tilde{\Sigma}$ . Moreover any two lifts differ by a multiple of a non-zero complex number $\alpha\in\mathbb{C}^{*}$ of norm 1.

Proof.

For existence, we consider $\tilde{\iota}(x,v)=(\iota(x),\bar{v})$ . Now let $\tilde{\iota}_{i}$ for $i=0,1$ be two lifts of $\iota$ and consider $\tilde{\iota}_{1}\circ\tilde{\iota}_{2}(x,v)=(x,g(v))$ where $g:\tilde{\Sigma}\rightarrow\operatorname{GL}_{1}(\mathbb{C})$ as $\tilde{\iota}_{1}\circ\tilde{\iota}_{2}$ is holomorphic. By compactness of $\tilde{\Sigma}$ , it follows that any lift of $\iota$ to an anti-holomorphic on $\mathcal{L}$ is of the form $(x,v)\rightarrow(\iota(x),\alpha\bar{v})$ , where $\alpha$ is a non-zero complex number. Moreover, as $\tilde{\iota}_{i}$ are involutions, it follows that, $\alpha$ is of norm 1, as desired. ∎

Lemma 4.11.

We have a 1-1 correspondence, given by $s\rightarrow\tilde{\iota}\circ s\circ\iota$ on the space of local sections, inducing an isomorphism on $H^{i}(\Sigma,\partial\Sigma;L,L_{|\partial\Sigma})\cong H^{i}(\tilde{\Sigma};\mathcal{L})^{\iota,\tilde{\iota}}$ as $\mathbb{R}$ -vectorspaces and hence, $\dim_{\mathbb{R}}H^{i}(\Sigma,\partial\Sigma;L,L_{|\partial\Sigma})=\dim_{\mathbb{C}}H^{i}(\tilde{\Sigma};\mathcal{L})$ .

Proof.

We argue using Cech cohomology construction. Indeed, let $\mathcal{A}:=\{U_{i}\}$ be an acyclic cover for the sheaf of local holomorphic sections of $L$ with boundary values in $L_{\partial\Sigma}\cong L_{\mathbb{R}}$ on $\Sigma$ , where is a totally real sub-bundle of $L$ , that is $L_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}\cong L$ . We double $\mathcal{A}$ to $\tilde{A}:=\{\tilde{U}_{i}\}$ where $\tilde{U}_{i}:=U_{i}\sqcup\iota(U_{i})$ if $U_{i}\cap\partial\Sigma=\phi$ and $\tilde{U}_{i}:=U_{i}\cup_{U_{i}\cap\partial\Sigma}\iota(U_{i})$ if $U_{i}\cap\partial\Sigma\neq\phi$ . Upon further restrictions of $A$ , we can assume that $\tilde{A}$ is a acyclic for the sheaf of local holomorphic sections of $\mathcal{L}$ on $\tilde{\Sigma}$ . Now for $U\subseteq\Sigma$ open set and $s\in\Gamma(U;L,L_{\partial\Sigma})$ , $\sigma(s):=\tilde{\iota}\circ s\circ\iota$ is a local section $\mathcal{L}$ on $U$ . On the other hand, for a local section $t$ of $\mathcal{L}$ , $t=\sigma(t)$ if and only if $t\in\Gamma(V;L,L_{\partial\Sigma})\subseteq\Gamma(V;\mathcal{L})$ where $V$ is an open set of $\Sigma$ . Hence, we get an identification between the $\sigma-$ fixed locus of the Cech complex of $\mathcal{L}$ given by $\tilde{A}$ with the Cech complex of $L$ given by $A$ . ∎

4.2. Space of Framings

Now consider, $(\mathbb{CP}^{n},\omega_{FS},J_{std})$ with its Fubini-Study symplectic form and its standard integrable almost complex structure.

Definition 4.12.

Denote by $\mathcal{F}^{d}(\beta)$ the set of all $\phi:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{n},\mathbb{RP}^{n})$ of fixed degree $\beta\in\pi_{2}(\mathbb{CP}^{n},\mathbb{RP}^{n})$ with $(k+1)$ -boundary marked points and $l$ -interior marked points, where $\phi$ is of regularity $W^{k,2}$ satisfying $\bar{\partial}_{J_{std}}\phi=0$ , and hence $\phi$ is smooth by elliptic regularity. We require $(\Sigma,\partial\Sigma)$ to be a genus-zero bordered nodal Riemann surface with $(k+1)$ -boundary marked points and $l$ -interior marked points. We also require that $\phi$ is automorphism-free, that is $\forall\psi:\Sigma_{i}\rightarrow\Sigma_{i}$ biholomorphism of an irreducible component $\phi_{i}\circ\psi\neq\phi_{i}$ where $\phi_{i}:=\phi|_{\Sigma_{i}}$ , and $H^{1}(\Sigma,\partial\Sigma;\phi^{*}\mathcal{O}(1),\phi_{\partial\Sigma}^{*}\mathcal{O}(1)_{\mathbb{R}})=0$ .

Remark 4.13.

We note that the domains of elements of $\mathcal{F}^{d}(\beta)$ are not fixed and hence, the condition $H^{1}(\Sigma,\partial\Sigma;\phi^{*}\mathcal{O}(1),\phi_{\partial\Sigma}^{*}\mathcal{O}(1)_{\mathbb{R}})=0$ is not sufficient for having $\mathcal{F}^{d}(\beta)$ a smooth manifold.

Definition 4.14.

Let $(L_{u},L_{|\mathbb{R}})\rightarrow(\Sigma,\partial\Sigma)$ be a Hermitian line bundle with positive curvature on each unstable irreducible component of $\Sigma$ . Let $F:=\{f_{0},\dots,f_{d}\}$ be an $\mathbb{R}$ -basis of $H^{0}(\Sigma,\partial\Sigma;L,L_{|\mathbb{R}})$ and define a holomorphic embedding $\phi_{F}:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{n},\mathbb{RP}^{n})$ by $\phi_{F}(z):=[f_{i}(z)]$ .

Remark 4.15.

Here we abuse notation and note that $L_{u}$ is a high tensor-power of the line bundle in Lemma 1 twisted by the effective divisor corresponding to nodal points on the normalization.

Lemma 4.16.

$\mathcal{F}^{d}(\beta)$ is a smooth manifold with corners.

Proof.

We note that $\mathcal{M}_{k+1,l}(\mathbb{CP}^{n},\mathbb{RP}^{n};\beta,J_{std})$ the Gromov compactified moduli space of $J_{std}$ -holomorphic disks of degree $\beta$ is a smooth orbifold with corners where the orbifold isotropy groups are the same as the automorphism groups of the underlying curve [Zer17]. On the other hand, $\mathcal{F}^{d}(\beta)\subset\mathcal{M}_{k+1,l}(\mathbb{CP}^{n},\mathbb{RP}^{n};\beta,J_{std})$ is the isotropy-free part of $\mathcal{M}_{k+1,l}(\mathbb{CP}^{n},\mathbb{RP}^{n};\beta,J_{std})$ and hence the result. ∎

Corollary 4.17.

The setting is as in Definition 4.14, then $\phi_{F}$ is an element of $\mathcal{F}^{d}(\beta)$ .

4.3. Proof of Lemma 4.2

4.3.1. Gromov’s Graph Trick

Remark 4.18.

Notice that, $\mathcal{F}^{d}(\beta)$ is a smooth manifold with corners and not necessarily an algebraic space. For this reason, we apply Gromov’s graph trick on $(X\times\mathbb{CP}^{d},\Delta\times\mathbb{RP}^{d})$ , where $d+1=\dim_{\mathbb{R}}H^{0}(L_{u},L_{|\mathbb{R}})$

Let $(E,\nabla)\rightarrow X\times\mathbb{CP}^{d}$ be a Hermitian vectorbundle and denote by

(4.1)

J_{X\times\mathbb{CP}^{d}}:=p_{X}^{*}J+p_{\mathbb{CP}^{d}}^{*}J_{std}

the induced almost-complex structure on $X\times\mathbb{CP}^{d}$ , where

(4.2)

p_{X}:X\times\mathbb{CP}^{d}\rightarrow X

(4.3)

p_{\mathbb{CP}^{d}}:X\times\mathbb{CP}^{d}\rightarrow\mathbb{CP}^{d}

are the natural projection maps. Consider the shearing map

(4.4)

\Psi:E\oplus p_{X}^{*}TX\oplus p_{\mathbb{CP}^{d}}^{*}T\mathbb{CP}^{d}\rightarrow p_{X}^{*}TX\oplus p_{\mathbb{CP}^{d}}^{*}T\mathbb{CP}^{d}

that satisfies:

(4.5)

J_{X\times\mathbb{CP}^{d}}(\Psi(e,h))=-\Psi(e,J_{X\times\mathbb{CP}^{d}}(h))

(4.6)

\Psi(e,\Psi(e,h))=0

Using the Hermitian connection $\nabla$ on $E$ , we define $\Phi:T^{v}E\oplus T^{h}E\rightarrow T^{v}E\oplus T^{h}E$ fibre-wise by:

(4.7)

\Phi(v,h):=(v,h+\Psi(e,h))

$\forall e\in E$ . We set the $\Psi$ -sheared almost-complex structure on the total space of $E$ by

(4.8)

J_{\Psi}:=\Phi^{-1}\circ J_{X\times\mathbb{CP}^{d}}\oplus i\circ\Phi

where $i$ denotes the fibre-wise multiplication by $\sqrt{-1}$ on $E$ . Thus, we have

(4.9)

J_{\Psi}(v,h)=(iv,J_{X\times\mathbb{CP}^{d}}h+2\Psi(e,J_{X\times\mathbb{CP}^{d}}h))

$\forall e\in E$ .
Now for $\tilde{u}:\Sigma\rightarrow E$ a smooth map, we write $\tilde{u}=(u,s)$ where $u:\Sigma\rightarrow X\times\mathbb{CP}^{d}$ is a smooth map and $s\in C^{\infty}(u^{*}E)$ a smooth section. Using the splitting of $TE=T^{v}E\oplus T^{h}E$ induced by $\nabla$ as above we have

(4.10)

d\tilde{u}=(du,u^{*}\nabla s)

Then the $(0,1)$ -part of $d\tilde{u}$ with respect to $J_{\Psi}$ is

(4.11)

(\frac{1}{2}(du+J_{\Psi}\circ du\circ j),u^{*}\nabla^{0,1}s)

From this, the following lemma follows.

Lemma 4.19.

Let $\tilde{u}:\Sigma\rightarrow E$ be a smooth map and write $\tilde{u}=(u,s)$ as above. Then, $\tilde{u}$ is $J_{\Psi}$ -holomorphic if and only if

(4.12)

\bar{\partial}_{J_{X\times\mathbb{CP}^{d}}}u+\Psi(s,J_{X\times\mathbb{CP}^{d}}\circ du\circ j)=0

(4.13)

(u^{*}\nabla)^{0,1}s=0

and thus $s\in H^{0}(u^{*}E)$ .

From now we set $E$ and $\Psi$ as follows.

Definition 4.20.

We define,

(4.14)

E:=p_{X}^{*}TX\otimes_{\mathbb{C}}p_{\mathbb{CP}^{d}}^{*}(\Lambda^{0,1}\mathbb{CP}^{d}\otimes_{\mathbb{C}}\mathcal{O}_{\mathbb{CP}^{d}}(l)\otimes_{\mathbb{C}}\overline{H^{0}(\mathbb{CP}^{d},\mathcal{O}_{\mathbb{CP}^{d}}(l))})

and using the trivialization $L_{\Delta}\cong L_{|\mathbb{R}}$ and trivialization $\mathcal{O}_{\mathbb{CP}^{d}}(l)_{\mathbb{RP}^{d}}\cong\mathcal{O}_{\mathbb{CP}^{d}}(l)_{|\mathbb{R}}$ , we get an induced real sub-bundle $E_{\Delta\times\mathbb{RP}^{d}}\cong E_{|\mathbb{R}}$ , where

(4.15)

E_{|\mathbb{R}}:=p_{X}^{*}T\Delta\otimes_{\mathbb{R}}p_{\mathbb{CP}^{d}}^{*}(T^{*}\mathbb{RP}^{d}\otimes_{\mathbb{R}}\mathcal{O}_{\mathbb{CP}^{d}}(l)_{|\mathbb{R}}\otimes_{\mathbb{R}}H^{0}(\mathbb{CP}^{d},\mathbb{RP}^{d};\mathcal{O}_{\mathbb{CP}^{d}}(l),\mathcal{O}_{\mathbb{CP}^{d}}(l)_{|\mathbb{R}})

where the Hermitian connection on $E$ is the one induced after fixing a Hermitian connection on $(X,J)$ and the Chern connection on $\mathcal{O}(l)\rightarrow\mathbb{CP}^{d}$ and $\mathcal{O}_{\mathbb{CP}^{d}}(l)_{\mathbb{R}}\rightarrow\mathbb{RP}^{d}$ is the unique $\mathbb{R}$ -line bundle that is fixed by the conjugation action on $\mathcal{O}_{\mathbb{CP}^{d}}(l)$ .

Definition 4.21.

We define $\Psi:E\oplus p_{X}^{*}TX\oplus p_{\mathbb{CP}^{d}}^{*}T\mathbb{CP}^{d}\rightarrow p_{X}^{*}TX\oplus p_{\mathbb{CP}^{d}}^{*}T\mathbb{CP}^{d}$

(4.16)

\Psi(\eta,v_{1},v_{2})=(\langle\eta\rangle v_{2},0)

where $\langle.,.\rangle:\mathcal{O}(l)\otimes\overline{H^{0}(\mathbb{CP}^{d},\mathcal{O}(l))}\rightarrow\mathbb{C}$ is induced Hermitian pairing from the $l^{th}$ -tensor power of the Fubini-Study metric on $\mathcal{O}(1)\rightarrow\mathbb{CP}^{d}$ .

Remark 4.22.

In this setting, the $\Psi$ -sheared almost-complex structure on $E$ is given fibre-wise by:

(4.17)

(J_{\Psi})_{(e,x,p)}(e^{\prime},v_{1},v_{2})=(\sqrt{-1}e^{\prime},J_{x}v_{1}+\langle e\rangle v_{2},(J_{std})_{p}v_{2})

and thus using equations (4.5) and (4.6), it follows that, $E_{\mathbb{R}}$ is indeed a totally-real sub-bundle of $E$ for $J_{\psi}$ .

Using the above remark and the above lemma, it follows that,

Corollary 4.23.

Let $\tilde{u}:(\Sigma,\partial\Sigma)\rightarrow(E,E_{\mathbb{R}})$ be a smooth map with real-boundary condition and write $\tilde{u}=(u,s)$ as above. Then, $\tilde{u}$ is $J_{\Psi}$ -holomorphic if and only if on each spherical component of $\Sigma$ we have

(4.18)

\bar{\partial}_{J_{X\times\mathbb{CP}^{d}}}u+\Psi(s,J_{X\times\mathbb{CP}^{d}}\circ du\circ j)=0

(4.19)

s\in H^{0}(u^{*}E)

and on each disk component we have

(4.20)

\bar{\partial}_{J_{X\times\mathbb{CP}^{d}}}u+\Psi(s,J_{X\times\mathbb{CP}^{d}}\circ du\circ j)=0

(4.21)

u(\partial\Sigma)\subseteq E_{\mathbb{R}}

(4.22)

s\in H^{0}(u^{*}E,u^{*}E_{\mathbb{R}})

Remark 4.24.

In the case when $u=v\times\phi$ where $v:(\Sigma,\partial\Sigma)\rightarrow(X,\Delta)$ a smooth map (or of $W^{k,p}-$ regularity as we always have elliptic regularity) and $\phi:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ a $J_{std}$ -holomorphic curve, equations (4.18) and (4.20) are written as

(4.23)

\bar{\partial}_{J}v+\langle\eta\rangle\circ d\phi=0

Lemma 4.25.

(Equivariant Hormander’s Theorem)
Fix $p,q$ positive integers and let $(Y,J_{Y},\omega_{Y})$ be a closed Kahler manifold and $(E,\nabla,h)\rightarrow Y$ be a holomorphic Hermitian vectorbundle of curvature form $iR\in C^{\infty}(\bigwedge^{2}T^{*}Y\otimes Herm(E,E))$ where $Herm(E,E)$ is the bundle of Hermitian endomorphisms on $E$ . Denote by $\bigwedge_{\omega}:\bigwedge^{p,q}T^{*}Y\otimes Herm(E,E)\rightarrow\bigwedge^{p-1,q-1}T^{*}Y\otimes Herm(E,E)$ to be the metric adjoint of the operator $\omega\wedge.$ . Moreover, we assume that there is a finite group $G$ that acts holomorphically on $Y$ and $E$ and isometrically on $TY$ and $E$ (and hence preserving $iR$ and $\bigwedge_{\omega}$ ).
If the commutator of the above two operators, $[iR,\bigwedge_{\omega}]:\bigwedge^{p,q}T^{*}Y\otimes Herm(E,E)\rightarrow\bigwedge^{p,q}T^{*}Y\otimes Herm(E,E)$ is positive-definite on each fibre and if we denote by $c:=||A||$ the operator norm of $A$ then, forall $g\in L^{2}(\bigwedge^{p,q}T^{*}Y\otimes E)$ such that $\bar{\partial}g=0$ and $g$ is $G$ -invariant, there exists $f\in L^{2}(\bigwedge^{p,q}T^{*}Y\otimes E)$ which is $G$ -invariant satisfying $\bar{\partial}f=g$ and $||f||_{2}\leq c||g||_{2}$ .

Proof.

(1) The setting is as above. By the usual Hormander’s Theorem, $\exists f\in L^{2}(\bigwedge^{p,q}T^{*}Y\otimes E)$ such that, $\bar{\partial}f=g$ and $||f||_{2}\leq c||g||_{2}$ . Now define $\tilde{f}:=\frac{1}{|G|}\Sigma_{a\in G}a.f$ .
(2) We can argue as in the proof of Lemma 4.11. Using elliptic regularity, we can interpret the result of the usual Hormander’s theorem as $H^{p,q}(Y,E)=0$ . The $G$ -action induces an action on the twisted Doulbeaut complex $(\mathcal{E}^{p,*}(Y,V),\bar{\partial})$ where the invariant part $(\mathcal{E}^{p,*}_{G}(Y,E),\bar{\partial})$ is also a chain complex as $G$ acts holomorphically. Thus $H^{p,q}_{G}(Y,E)\leq H^{p,q}(Y,E)$ is a real-subspace and hence is zero-dimensional by the usual Hormander’s Theorem.
(3) We extend the proof of Hormander’s Theorem to the equivariant case.
Using the metric on $TY$ and $E$ we define an $L^{2}$ -inner-product $(.,.)$ on smooth sections $C^{\infty}(\bigwedge^{p,q}T^{*}Y\otimes E)$ and using $\omega_{Y}$ we define the adjoint $\bar{\partial}^{*}$ of $\bar{\partial}$ . For $h\in C^{\infty}(\bigwedge^{p,q}T^{*}Y\otimes E)$ , we have $||h||_{2}^{2}\leq c(||\bar{\partial}h||^{2}_{2}+||\bar{\partial}^{*}h||^{2}_{2})$ and thus the bilinear pairing $(h_{1},h_{2})_{\mathcal{H}}:=(\bar{\partial}h_{1},\bar{\partial}h_{2})+(\bar{\partial}^{*}h_{1},\bar{\partial}^{*}h_{2})$ defines a metric. Let $\mathcal{H}$ be the Hilbert space completion of $C^{\infty}(\bigwedge^{p,q}T^{*}Y\otimes E)$ with respect to $||.||_{\mathcal{H}}$ . Consider $\psi_{g}(h):=(h,f)_{\mathcal{H}}$ and note that $\psi_{g}\in\mathcal{H}^{*}$ as by Cauchy-Schwartz, we have, $|\psi_{g}(h)|^{2}\leq c||g||^{2}_{\mathcal{H}}||h||_{\mathcal{H}}^{2}=c||g||_{2}^{2}||h||_{\mathcal{H}}^{2}$ . Using Riez-Representation Theorem, it follows that, $\exists v\in\mathcal{H}$ such that, $||v||_{\mathcal{H}}^{2}\leq c||g||_{2}^{2}$ and $(v,h)_{\mathcal{H}}=\psi_{g}(h),\forall h\in\mathcal{H}$ . Noting that $g$ is $G$ -invariant, it follows that, $a.\psi_{g}(h)=\psi_{g}(a.h),\forall a\in G$ and $\forall h\in\mathcal{H}$ and thus $v$ is $G$ -invariant by the non-degeneracy of $(.,.)_{\mathcal{H}}$ . Now setting $f:=\bar{\partial}^{*}v$ solves the equation $\bar{\partial}f=g$ with $f$ being $G$ -invariant. ∎

Definition 4.26.

Let $\mathcal{T}:=\{(u,\Sigma,F,\eta)\}$ where $F$ is an $\mathbb{R}$ -basis of $H^{0}(\Sigma,\partial\Sigma;L,L_{\mathbb{R}})$ and $\eta\in E$ where $E$ is as in Definition 5 and $u:(\Sigma,\partial\Sigma)\rightarrow(X,\Delta)$ is a smooth map satisfying:

(4.24)

\bar{\partial}_{J}u+\langle\eta\rangle\circ d\phi_{F}=0

where $\phi_{F}:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ constructed as in Definition 5, so that the matrix $(\int_{\Sigma}\langle f_{i},f_{j}\rangle)_{i,j}$ is symmetric. Moreover, we give $\mathcal{T}$ a topology induced from the Gromov’s topology on the closures of the images of $u\times\phi_{F}$ in $X\times\mathbb{CP}^{d}$ and the vectorspace topology of $E$ .

Lemma 4.27.

The topology on $\mathcal{T}$ agress with the usual Gromov’s topology.

Proof.

We apply the argument of Lemma 6.14 in [AMS21] after using the structure theorem of [KO00] or [Laz11]. ∎

4.3.2. Smooth Structure on Thickening

Lemma 4.28.

There exists an $l_{0}$ so that for all $l\geq l_{0}$ , the linearization $D_{J_{\Psi}}$ of the $\bar{\partial}_{J_{\Psi}}$ at any point of $\mathcal{T}\hookrightarrow\mathcal{M}(E,J_{\Psi})$ is surjective.

Proof.

We follow the proof of Proposition 6.26 of [AMS21].
Let $p=(u,\Sigma,F,\eta)\in\mathcal{T}$ and assume for a contradiction that, $\operatorname{coker}(D_{J_{\Psi}})\neq\phi$ . We identify $\operatorname{coker}(D_{J_{\Psi}})\cong\ker(D_{J_{\Psi}}^{*})$ where $D_{J_{\Psi}}^{*}$ is the $L^{2}$ -adjoint of $D_{J_{\Psi}}$ and let $e\in\ker(D_{J_{\Psi}}^{*})$ be a non-zero element and assume that the support of $e$ is on a disk-component (as otherwise we argue as in [AMS21] Proposition 6.26). Notice that, if we assume that $e$ is of $W^{k,p}$ -regularity, the equation $D^{*}e=0$ with elliptic regularity implies that $e$ is smooth and thus we assume everything that follows is smooth.
Noting that, $\phi_{F}$ is a regular curve in $(\mathbb{CP}^{d},\mathbb{RP}^{d})$ it follows that $e\in C^{\infty}(u^{*}TX,u^{*}T\Delta)\hookrightarrow C^{\infty}((u\times\phi_{F})^{*}(TX\times T\mathbb{CP}^{d}),(u\times\phi_{F})^{*}(T\Delta\times T\mathbb{RP}^{d}))$ . Moreover, notice that the linearization of $\bar{\partial}_{J_{\Psi}}$ for a fixed complex-structure on the domain $(\Sigma,\partial\Sigma)$ is of the form

(4.25)

D_{J_{\Psi}}=D_{u}\bar{\partial}_{J}+\langle,\rangle\circ d\phi_{F}

Assume that, $u$ is non-constant and has a smooth, non-boundary, non-branched point $z\in\Sigma$ . Now denote by $\tilde{e}\in C^{\infty}(\tilde{\Sigma},u^{*}TX)$ the double of $e$ and fix a smooth $w\in T_{z}\Sigma$ and let $f$ be a complex anti-linear map $T_{\phi_{F}(z)}\mathbb{CP}^{d}\rightarrow T_{u(z)}X$ satisfying

(4.26)

\langle f(d\phi_{F}(j(w))),e(z)\rangle\neq 0

where $j:T\Sigma\rightarrow T\Sigma$ is the complex structure on $\Sigma$ . Using Equivariant Hormander’s Theorem and for $l>>1$ big enough, we can find holomorphic section $s_{l}$ of $\overline{Hom}(\phi_{F}^{*}T\mathbb{CP}^{d},u^{*}TX)\otimes\phi_{F}^{*}\mathcal{O}(l)\rightarrow\tilde{\Sigma}$ which are $\mathbb{Z}/2$ -equivariant with respect to conjugation. In addition, using the proof in [AMS21] Lemma 6.24, we can find holomorphic sections of $\tilde{s}_{l}$ of $\phi_{F}^{*}\mathcal{O}(l)\rightarrow\tilde{\Sigma}$ that converges to $\delta_{z}$ . Using Lemma 4.10, we can lift the conjugation action on $\tilde{\Sigma}$ to a $\mathbb{Z}/2$ -action on $\phi_{F}^{*}\mathcal{O}(l)$ and hence using Equivariant Hormander’s Theorem we deduce that for $l>>1$ and upon restricting the above sections to $\Sigma\subset\tilde{\Sigma}$ we get $\langle s_{l},\tilde{s}_{l}\rangle$ converges to the Dirac delta section $\delta_{f}\in\bigwedge^{0,1}\Sigma\otimes u^{*}TX$ . Now the result follows from the proof of Proposition 6.26 in [AMS21]. Indeed, using equation (25), it follows that, for $l$ large enough, we have,

(4.27)

\langle\langle s_{l}\otimes\tilde{s}_{l}\rangle\circ d\phi_{F}(j(\zeta)),e\rangle\neq 0

where $\zeta$ is a smooth vector field on $\Sigma$ so that $\zeta(z)=w$ . Contradicting the assumption that, $e\in\ker(D_{J_{\Psi}}^{*})$ .
Now for $u$ constant, we note that, $u^{*}TX$ is a trivial holomorphic bundle with a trivial totally-real sub-bundle and thus, upon doubling $u$ to $\tilde{u}$ , we get $\bigwedge^{0,1}\Sigma\otimes u^{*}TX\rightarrow\bigwedge^{0,1}\tilde{\Sigma}\otimes\tilde{u}^{*}TX$ which is injective. Now the result follows by elliptic regularity and Riemann-Roch.
For the general case, we apply the above argument after using the structure theorems of [KO00] or [Laz11]. ∎

Corollary 4.29.

We can further assume that the pertubation term of the equation (4.24) vanishes at the special points, that is, at marked and nodal points.

Proof.

After twisting $E$ with the line bundle corresponding to the effective divisor of the special points on the normalization, it follows by construction that the pertubation term $\eta$ vanishes at these points. Noting that, for $l>>1$ large enough, this twisted bundle is still ample on each component of the normalization, we can argue as above to get a pertubation term with trivial, that is vanishing, matching condition on the nodes. ∎

Lemma 4.30.

$\mathcal{T}$ is a $SO(d+1)$ - topological manifold, with a fibre-wise smooth structure on $\mathcal{T}\rightarrow\mathcal{F}^{d}$ . Moreover, the action has finite stabilizers which can be identified with the automorphism group of the underlying curve.

Proof.

Using Corollary 1 and remark 6 and using the fact that, $H^{1}(E)=0$ , we have a topological open embedding $\mathcal{T}\hookrightarrow\mathcal{M}(E,J_{\Psi})$ the moduli space of $J_{\Psi}$ -holomorphic maps from bordered Riemann surfaces to $E$ with boundary condition in $E_{\mathbb{R}}$ , while equivariance follows from the fact that $SO(d+1)$ -acts freely on $\mathcal{F}^{d}$ and the fact that surjectivity of $D_{v}\bar{\partial}_{J_{\Psi}}$ where $v=(u,\phi_{F})$ is an open condition and thus by Gromov’s Compactness we can choose $l>0$ so that all elements of $\mathcal{T}$ are regular.
Now to get the second claim, we argue as in [AMS21] Lemma 6.4 on the dual graph of the curve $u$ . ∎

Corollary 4.31.

$\mathcal{T}\times V$ can be given an $SO(d+1)$ -equivariant smooth structure, where $V$ is some real $SO(d+1)$ -representation, where such smooth structure is unique up to stable isotopy of smooth $SO(d+1)$ -structures. Moreover upon further stabilization by real $SO(d+1)$ -representation, we can assume that the boundary evaluation map is a submersion.

Proof.

This is an application of Lashoff’s Theorem together with [AMS21] section 4.1, provided that the forgetful map $\mathcal{T}\rightarrow\mathcal{F}^{d}$ is an $SO(d+1)$ -equivariant $C^{1}_{loc}$ -topological submersion. Now consider the forgetful map $(u,\Sigma,F,\eta)\rightarrow\phi_{F}$ . We can identify a neighborhood of $\phi_{F}\in\mathcal{F}^{d}$ by a neighborhood of $\Sigma\in\overline{\mathcal{M}}$ the compactified Deligne-Mumford moduli space of borded Riemann Surfaces as follows:
Fix $U_{0},\dots,U_{s}\subset\mathbb{CP}^{d}$ open of codimension 1, intersecting the image of $\phi$ transversally at distinct points. The existence of such open sets, is done upon noting that $\phi$ is a regular curve and thus we can apply Frobenious Theorem locally on $L_{0},\dots,L_{s}\leq T_{\phi(z_{i})}\mathbb{CP}^{d}$ so that $z_{i}\in\Sigma$ is a point where $\zeta\rightarrow d_{z_{i}}\phi(\zeta)$ is injective and $L_{i}$ is transverse to this map. Now denote by $\mathcal{F}^{d}(L_{*})\subset\mathcal{F}^{d}$ all the elements transverse to $U_{0},\dots,U_{s}$ . Then the map, $\mathcal{F}^{d}(L_{*})\rightarrow\overline{\mathcal{M}}$ sending $\phi^{{}^{\prime}}$ to its domain with $z_{0},\dots,z_{s}$ as extra marked point is the desired local diffeomorphism. Now denote by $\Pi$ the composition of the above map with the forgetful map. For a fixed $(u,\Sigma,F,\eta)$ let $U\subseteq\overline{\mathcal{M}}$ be an open neighborhood so that $\mathcal{F}^{d}(L_{*})\rightarrow\overline{\mathcal{M}}$ is a diffeomorphism onto $U$ and let $K\subset\Pi^{-1}(U)\subseteq\mathcal{T}$ be an open set. Let $g:U\rightarrow g(U)\subseteq\mathcal{F}^{d}$ be a local inverse of $\mathcal{F}^{d}(L_{*})\rightarrow\overline{\mathcal{M}}$ . Upon composing equation (9.7) of [FOOO21] with $g$ , which we denote by $G$ and working on possibly smaller $U$ and $K$ , it follows that, $G\circ\Pi$ is the projection map $U\times K\rightarrow U$ and the restriction $G_{|\{a\}\times K}$ is obtained by Newton-Iterations with initial conditions depending on $K$ and thus depends smoothly on $K$ . Now using equations (9.13) of [FOOO21], we can argue as in [AMS21] Corollary 6.29 to get the first claim.
For the second claim, we stabilize by $\oplus_{i}ev^{*}T\Delta$ , where the sum runs over all marked boundary points and argue as [AMS21] Lemma 4.5. ∎

5. Proof of Lemma 4.2

5.1. Using [AMS21], [BX22a]

Proof.

We set $s(u,\Sigma,F,\eta):=(\eta,(\int_{\Sigma}\langle f_{i},f_{j}\rangle u^{*}\omega)_{i,j})$ and $Ob$ to the direct sum $E$ as in the proof of Corollary 3, the trivial bundle over $\mathcal{T}\times V$ of fibre $(d+1)\times(d+1)$ -symmetric matrices. Then the tuple $(SO(d+1),\mathcal{T}\times V,Ob,(s,0))$ is a Global Kuranishi chart, proving first claim of Lemma 4.2.
In order to show the second claim, let $(u,\Sigma,F,\eta)\in\mathcal{T}$ and note that, each disk element has trivial isotropy group. Thus, it suffice to show the claim, after abuse of notation, assuming $\Sigma$ is a sphere component. Note that, the term $\langle.,.\rangle\circ d\phi_{F}$ in equation (4.25) is a compact operator. Moreover, $D_{u}\bar{\partial}_{J}=O+K$ where $O$ is a complex linear operator and $K$ is a compact operator. Thus, we can define a linear homotopy, $t\rightarrow\ker(O+t(K+\langle,\rangle\circ d\phi_{F}))$ between the vertical tangent space of $\mathcal{T}\rightarrow\mathcal{F}$ and a complex vector space $\ker(O)$ . Then pulling-back the complex structure on $\ker(D_{u}\bar{\partial}_{J})$ by the above homotopy, shows that the vertical tangent can be endowed with a complex structure. Now fix an $SO(d+1)$ -connection on $T\mathcal{T}$ and a non-zero vector $v\in T\mathcal{F}$ . Flowing the above complex structure along the horizontal lift of $v$ , gives the desired claim, after further stabilization $iV$ as in Lashoff’s Theorem. ∎

5.2. Using [AMS24]

In the above proof, we have used the framings, namely the associated holomorphic embeddings $\phi:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ to perturb the $\bar{\partial}$ -equation itself and make its linearization surjective. Following [AMS24], we can also give a construction that proves Lemma 4.2, using the space $\mathcal{F}^{d}$ . We start by recalling the definition of a finite-dimensional approximation scheme from [AMS24].

Definition 5.1.

Let $G$ be a compact Lie group and $\pi:V\rightarrow B$ be a smooth $G$ -vectorbundle. A finite dimensional approximation scheme $(V_{\mu},\lambda_{\mu})_{\mu\in\mathbb{N}}$ for $C^{\infty}_{c}(V)$ the space of compactly-supported smooth sections of $V$ , is a sequence of finite-dimensional $G$ -representations $(V_{\mu})$ together with a sequence of $G$ -equivariant linear maps $\lambda_{\mu}:V_{\mu}\rightarrow C^{\infty}_{c}(V)$ , $\forall\mu\in\mathbb{N}$ satisfying the following conditions:

(1)

$V_{\mu}\leq V_{\mu+1}$ , $\forall\mu\in\mathbb{N}$ is a sub-representation.
(2)

$\lambda_{\mu}|V_{\mu-1}=\lambda_{\mu-1}$ , $\forall\mu\in\mathbb{N}$ .
(3)

$\bigcup_{\mu}\lambda_{\mu}(V_{\mu})\subset C^{\infty}(V)$ is dense with respect to the $C^{1}_{loc}$ -topology.

Lemma 5.2.

[AMS24] Finite dimensional approximation schemes exists.

Fix $\beta\in\pi_{2}(X,\Delta)$ and consider the universal curve $\pi:\mathcal{C}^{d}(\beta)\rightarrow\mathcal{F}^{d}(\beta)$ . After identifying $\mathcal{C}^{d}$ with $\mathcal{F}^{d}(\beta)$ together with an extra interior marked point, it follows from Lemma 4.16 that, $\mathcal{C}^{d}(\beta)$ is a smooth manifold. Noting that the evaluation map $ev:\mathcal{F}^{d}(\beta)\rightarrow\mathbb{RP}^{d}$ is a submersion, we consider the smooth submanifold $\mathcal{F}:=\{\phi\in\mathcal{F}^{d}(\beta):\phi(z_{0})=[1:0:\dots:0]\}$ and similarly as above, we consider its universal curve $\mathcal{C}\rightarrow\mathcal{F}$ . We have a natural $\operatorname{SO}(d+1)$ -action given by post composition making $\pi:\mathcal{C}^{d}(\beta)\rightarrow\mathcal{F}^{d}(\beta)$ $\operatorname{SO}(d+1)$ -equivariant which induces an $SO(d)$ -equivariant on both of $\mathcal{F},\mathcal{C}$ making $\mathcal{C}\rightarrow\mathcal{F}$ to be $\operatorname{SO}(d)$ -equivariant. In this section we denote by $G_{i}:=\operatorname{SO}(d_{i})$ where $d_{i}$ is some positive integer.
Now denote by $V:=\Omega^{0,1}_{\mathcal{C}/\mathcal{F}}\otimes TX\rightarrow\mathcal{C}\times X$ the $G$ -equivariant vectorbundle and let $(V_{\mu},\lambda_{\mu})_{\mu\in\mathbb{N}}$ of $C^{\infty}_{c}(V)$ .

Definition 5.3.

A consistent domain metric on $\mathcal{F}$ is a fibre-wise metric $\operatorname{SO}(d)$ -invariant on $\mathcal{C}$ .

Lemma 5.4.

$\mathcal{F}$ admits a consistent domain metric.

Proof.

This is an easier case of Lemma 4.17 in [AMS24] as $\operatorname{SO}(d)$ is a compact Lie group. Indeed, upon fixing an $\operatorname{SO}(d)$ -invariant cover of $\mathcal{F}$ formed of trivializing open sets of $\mathcal{C}\rightarrow\mathcal{F}$ and upon fixing a Haar meausre on $\operatorname{SO}(d)$ , we get the result by an averaging argument and partition of unity on such cover. ∎

Definition 5.5.

We define the thickened moduli space $\mathcal{T}(\beta,(V_{\mu},\lambda_{\mu}))$ to be the space of tuples $(F,u,e)$ where $F:=\{f_{0},\dots,f_{d}\}$ is an $\mathbb{R}$ -basis of $H^{0}(L_{u})$ such that $[f_{i}(z_{0})]=[1:0:\dots:0]$ and $e\in V_{\mu}$ for some $\mu$ and $u$ satisfies

(5.1)

\bar{\partial}_{J}u|_{\mathcal{C}^{*}_{\phi}}+(\lambda_{\mu}(e))\circ\Gamma_{u}=0

where $\mathcal{C}^{*}_{\phi}$ is the complement of all nodes in the fibre $\pi:\mathcal{C}^{d}(\beta)\rightarrow\mathcal{F}^{d}(\beta)$ over $\phi$ and $\Gamma_{u}:\mathcal{C}_{\phi}\rightarrow\mathcal{C}\times X$ is the graph map.

We give $\mathcal{T}:=\mathcal{T}(\beta,(V_{\mu},\lambda_{\mu}))$ the Gromov-Hausdorff topology on the graphs given by the closure of the image of $\Gamma_{u}$ and the vectorspace topology on $V_{\mu}$ .

Lemma 5.6.

There exists a $\mu\in\mathbb{N}$ so that the linearization of equation (5.1) is surjective at every point of $\mathcal{T}$ .

Proof.

This follows from Gromov’s compactness and openness condition of having the linearization surjective. Indeed noting that the cokernel of the linearization is a finite-dimensional subspace of $C^{\infty}_{c}(V)$ , the result follows from item (3) in the definition (5.1). ∎

Now fix any $\mu\in\mathbb{N}$ as in the above Lemma.

Definition 5.7.

We define the obstruction bunble $Ob:=Ob(\beta,V_{\mu},\lambda_{\mu})$ to be the vectorbundle over $\mathcal{T}$ given by direct sum of the trivial bundle $V_{\mu}$ with the trivial bundle of fibre $\mathfrak{g}$ the Lie algebra of $G$ .

Remark 5.8.

Note that the above lemma only shows that $\mathcal{T}$ is a topological manifold formed by an uncountable union of smooth manifolds. Namely, it doesn’t say that $\mathcal{T}$ has a smooth structure yet.

Now let $u:(\Sigma,\partial\Sigma)\rightarrow(X,\Delta)$ be any smooth map and let $F$ be an $\mathbb{R}$ -basis of $H^{0}(\Sigma,\partial\Sigma;L_{u},L_{u}|_{\partial\Sigma})$ as above and consider the induced holomorphic embedding $\phi_{F}:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ and note that $\phi_{F}\in\mathcal{F}$ . Moreover denote by $H(F):=(\int_{\Sigma}\langle f_{i},f_{j}\rangle)_{i,j}$ the symmetric matrix given by $F$ .

Definition 5.9.

We define the Kuranishi map $s:\mathcal{T}\rightarrow Ob$ to be $s(F,u,e):=(e,\exp^{-1}(H(F)))$ .

where $\exp:\mathfrak{g}\rightarrow G$ is the usual exponential map. Now we give another proof of Lemma 4.2.

Proof.

From Lemma 5.4, it follows that $\mathcal{T}\rightarrow\mathcal{F}$ has a fibre-wise smooth structure. Applying the same argument as in the proof in section 5.1 it follows that after stabilizing $(G,\mathcal{T},Ob,s)$ by a $G$ -representation we find an equivalent smooth Global Kuranishi chart. As for the normal complex structure we argue as in the first proof of Lemma 4.2. ∎

6. Choice of Finite-dimensional Approximation Scheme

For applications we have to compare different Global Kuranishi charts coming from boundary components of moduli spaces of $J$ -disks. For this reason, we choose to take the Global Kuranishi chart representation as in [AMS24] and we have to carefully choose our finite-dimensional approximation schemes. Let $\beta\in\pi_{2}(X,\Delta)$ and write $\beta=\beta_{1}+\dots+\beta_{k}$ where each $\beta_{i}\neq 0$ and can be represented by a $J$ -disk. In the view of Lemma 5.4, we abuse notation and terminology and call such $V_{\mu,\beta}$ a finite dimensional approximation scheme for $J$ -disks of degree $\beta$ .

Lemma 6.1.

There exists a finite-dimensional approximation scheme $V_{\mu,\beta},$ for the moduli space of $J$ -disks of degree $\beta$ so that $V_{\mu,\beta_{1}}\oplus\dots\oplus V_{\mu,\beta_{k}}\leq V_{\mu,\beta}$ .

Proof.

We argue by upward induction on the discrete monoid $\{(\Omega(\beta),\mu(\beta)):\beta\in\pi_{2}(X,\Delta)\}$ , where $\Omega$ is as in Lemma 4.5 and $\mu(\beta)$ is the Maslov index of $\beta$ .
The base case follows from Lemma 5.2 and Lemma 5.4. Now suppose that we have constructed a finite-dimensional approximation scheme $\forall\beta_{1},\dots,\beta_{k-1}$ and let $\beta_{k}$ be such that $\Omega(\beta_{k})>\Omega(\beta_{i})$ . By Lemma 13 and Lemma 14, we can find a finite-dimensional approximation scheme $V^{\prime}_{\mu,\beta_{k}}$ . Set $V_{\mu,\beta_{k}}:=V^{\prime}_{\mu,\beta_{k}}\oplus V_{\mu,\beta_{1}}\oplus\dots\oplus V_{\mu,\beta_{k-1}}$ where the $G$ -action on the factor $V_{\mu,\beta_{1}}\oplus\dots\oplus V_{\mu,\beta_{k-1}}$ is given by the group embedding $G_{1}\times\dots\times G_{k-1}\hookrightarrow G$ as block matrices. ∎

7. Floer-Morse Trajectories

7.1. Morse Trajectories

We start by investigating the usual Morse case and prove results in this setting that we will use in the next subsection. We follow closely [FO97], [FOOO09].

Definition 7.1.

A ribbon tree is a tuple $t:=(T,\iota,rt)$ where

(1)

T is planar, connected cycle-free rooted tree
(2)

$\iota:T\rightarrow\mathbb{D}$ is a topological embedding
(3)

$\iota^{-1}(\partial\mathbb{D})$ is the set of vertices of valency 1
(4)

$rt\in\iota^{-1}(\partial\mathbb{D})$ is a distinguished point called the root of $t$
(5)

We also require stability condition, namely no vertex has valency 2

The choice of root of $T$ is equivalent to a choice of ribbon structure as then we enumerate the vertices and cyclically order the edges by the pull-back by $\iota$ counter-clockwise orientation on $\mathbb{D}\subset\mathbb{C}$ .
We also introduce the following notation and terminology:

Definition 7.2.

Given a ribbon tree $(T,\iota,rt)$ and denote by $C^{i}(T)$ the $i^{th}$ -cell of $T$ for $i=1,2$ .

(1)

We denote by $C^{0}_{ext}(T):=\iota^{-1}(\partial\mathbb{D})$ the set of external vertices
(2)

We denote by $C^{0}_{int}(T):=C^{0}(T)\setminus C^{0}_{ext}(T)$ the set of internal vertices
(3)

We have a distinguished $rt\in C^{0}_{ext}(T)$ called the root of $T$
(4)

We denote by $C^{1}_{ext}(T)$ the set of all edges adjacent to an external vertex
(5)

We denote by $C^{1}_{int}(T):=C^{1}(T)\setminus C^{1}_{ext}(T)$ the set of all internal edges
(6)

Elements of $C^{1}_{ext}(T)$ are called the leaves of $T$

Definition 7.3.

We say two ribbon trees $(T_{i},\iota_{i},rt_{i})$ are equivalent or of the same combinatorial type if and only if $T_{1}\cong T_{2}$ are homeomorphic and $\iota_{1}$ , $\iota_{2}$ are isotopic.

Definition 7.4.

We denote by $G_{k+1}$ the set of isomorphism classes of ribbon trees with $k+1$ -leaves.

In order to give a topology on $G_{k+1}$ , namely the Gromov-Hausdorff topology we give each ribbon tree a metric.

Definition 7.5.

(1)

Let $t=(T,\iota,rt)$ be a ribbon tree, a metric on $t$ is a map $l:C^{1}_{int}(T)\rightarrow(0,\infty)$
(2)

We denote by $Gr_{k+1}(t):=\{l:C^{1}_{int}(T)\rightarrow(0,\infty)\}$ the set of all metrics on $t$
(3)

We denote by $Gr_{k+1}:=\cup_{G_{k+1}}Gr_{k+1}(t)$

We give $Gr_{k+1}$ a sequential topology as follows:
Let $l_{i}\in Gr_{k+1}(t)$ be a sequence and suppose that $l_{i}(e)\rightarrow l_{\infty}(e)$ , $\forall e\in C^{1}_{int}(T)$ . Let $T^{{}^{\prime}}$ be tree constructed from $T$ by shrinking all $e$ so that $l_{\infty}(e)=0$ . Then $l^{{}^{\prime}}:=l_{\infty|T^{\prime}}\in Gr_{k+1}(t^{\prime})$ .
A theorem of Stasheff says that $Gr_{k+1}$ is homeomorphic to $\mathbb{R}^{k-2}$ and $Gr_{k+1}:=\cup_{G_{k+1}}Gr_{k+1}(t)$ is a cellular decomposition. [FOOO09] upgraded this result as follows:

Theorem 7.6.

[FO97][FOOO09] $Gr_{k+1}$ is a smooth manifold with corners of $dim_{\mathbb{R}}=k-2$ and each of its strata $Gr_{k+1}(t)$ is a smooth manifold with corners of $dim_{\mathbb{R}}=k+1-\Sigma_{C^{0}_{int}(T)}|v|$ , where $|v|$ is the valency of $v$ .

Definition 7.7.

Let $x_{0},\dots,x_{k}$ be pairwise distinct critical points of $f$ . We denote by $\mathfrak{M}(t,f;x_{0},x_{1},\dots,x_{k}):=\{(l,u):l\in Gr_{k+1}(t),u:T\rightarrow L\text{ continuous map satisfying }\star\}$

(1)

$\forall e\in C^{1}(T)$ we require $u_{e}$ , the restriction of $u$ to $e$ , to solve the Morse trajectory equation, namely

(7.1) $\frac{d}{dt}u_{e}+\nabla f(u_{e}(t))=0$
(2)

if $e\in C^{1}_{ext}(T)$ doesn’t contain the root, we give coordinates so that $e\cong(-\infty,0]$ and require $u_{e}(-\infty)=x_{i}$ for some $i=1,\dots,k$
(3)

similarly, if $e\in C^{1}_{ext}(T)$ contains the root then we give coordinates so that $e\cong[0,\infty)$ and require $u_{e}(\infty)=x_{0}$

Remark 7.8.

In the above definition we have abused notation and wrote $u_{e}(\pm\infty)$ as solutions of the Morse trajectories have exponential decay convergence near critical points and hence such limit always exists.

Example 7.9.

Let $x_{0},x_{1},x_{2}$ be critical points of $f$ and consider the unique trivalent tree. Then by the Morse-Smale condition, we have that the unstable manifold $W^{-}_{x_{1}}$ and the stable manifold $W^{+}_{x_{0}}$ intersect transversaly, while by the uniqueness part of the fundamental theorem of ODEs, it follows that $W^{-}_{x_{1}}$ and $W^{-}_{x_{2}}$ only intersect if $x_{1}=x_{2}$ in which case their intersection is not transversal. Thus in order to get a space of solutions of the above equations, we will perturb the Morse trajectory equations away from neighborhoods of the external vertices.

Fix $t\in Gr_{k+1}$ and let $x_{0},\dots,x_{k}$ be distinct critical points of $f$ and fix a small enough (for instance pairwise disjoint) neighborhoods $x_{i}\in U_{i}\subset\Delta$ .

Lemma 7.10.

For a generic choice of $\tilde{f}\in C^{\infty}(\Delta)$ so that $\tilde{f}_{|U_{i}}=f_{|U_{i}}$ we have that $\mathfrak{M}(t,\tilde{f};x_{0},x_{1},\dots,x_{k})$ is a smooth manifold.

Proof.

In this proof we will do two levels of perturbations, one along external edges so that unstable manifolds intersect transversely and the second along internal ones. We start by perturbing along external edges.
Using the ribbon structure on $t$ we enumerate the internal vertices $v_{1},\dots,v_{s}$ which are adjacent to an external edge. Moreover, $\forall v_{i}$ as above we have a cyclic ordering of all the external edges adjacent to it, say $e^{1}_{i},\dots,e^{h_{i}}_{i}$ . Now by Sard’s Theorem we can find $g\in C^{\infty}(\Delta)$ so that $g_{e^{j}_{i}}=f_{e^{j}_{i}}$ on $(-\infty,2]\cup(1,0]$ or on $[0,1)\cup[2,\infty)$ if ${e^{j}_{i}}=rt$ . Moreover $g_{e}=f_{e}$ , $\forall e\in C^{1}_{int}(T)$ , so that after integrating $\nabla g$ , the images of all unstable manifolds have images intersecting transversely at $v_{i}$ . Denote by $\tilde{W}^{-}_{x_{a}}(e^{j}_{i})$ the image of $W^{-}_{x_{a}}$ by the flow of $\nabla g_{|e^{j}_{i}}$ and thus we can assume that $\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i})$ is a smooth submanifold of $L$ with the understanding that if $e^{j}_{i}=rt$ then $\tilde{W}^{-}_{x_{a}}(e^{j}_{i})=W^{+}_{x_{0}}$ .
Now we realize $\mathfrak{M}(t,g;x_{0},x_{1},\dots,x_{k})$ as a fibre product as follows:
$\forall v_{i}$ as above let $e_{i,1},\dots,e_{i,d}$ be the internal edges forming the unique minimal path joining $v_{1}$ to $v_{i}$ . Such path exists as $T$ is connected, of finite type and uniqueness follows from the cycle-free property of $T$ . For each of the $e_{i,j}$ as above, denote by $W_{i,j}=-\nabla f_{|e_{i,j}}=-\nabla g_{|e_{i,j}}$ and consider the smooth map $exp_{i}:\Delta\times Gr_{k+1}(t)\rightarrow\Delta$ given by $exp_{i}(p,l):=exp_{p}(l(e_{i,1})W_{i,1})\circ\dots\circ exp_{p}(l(e_{i,d})W_{i,d})$ . We define $Exp_{g,t}:\Delta\times Gr_{k+1}(t)\rightarrow\Delta^{s}$ by $Exp_{g,t}(p,l):=(exp_{i}(p,l))_{i=1,\dots,s}$ . Then, $\mathfrak{M}(t,g;x_{0},x_{1},\dots,x_{k})=Exp_{g}^{-1}(\prod_{i=1}^{s}\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i}))$ . We will show that for a generic choice of a smooth function on $\Delta$ as in the statement of the Lemma, $Exp$ is a submersion and hence the result.
Now for each $v_{i}$ as above, let $O_{i}$ be an open subset in the $e_{i,1}$ the edge adjacent to $v_{i}$ not intersecting $v_{i}$ . We require that, if $v_{i}=v_{1}$ then $O_{i}=\phi$ . Moreover, we require that each $O_{i}$ is small enough so that, by Poincare Lemma and the non-degeneracy of the metric, the choice of a vectorfield supported on $O_{i}$ is interchangeable with a choice of function supported on $O_{i}$ of gradient vector equal to the vectorfield. Denote by $C^{k}(O_{i})$ the Banach space of $C^{k}$ -functions on $L$ supported on $O_{i}$ and consider the universal moduli space $\mathfrak{M}(t;x_{0},x_{1},\dots,x_{k}):=\cup_{\prod_{i}^{s}C^{k}(O_{i})}\mathfrak{M}(t,g+f_{1}+\dots+f_{s};x_{0},x_{1},\dots,x_{k})$ . Note that the projection map $\mathfrak{M}(t;x_{0},x_{1},\dots,x_{k})\rightarrow\prod_{i}^{m}C^{k}(O_{i})$ is a Fredholm map and thus by Sard-Smale theorem it suffice to show that $\mathfrak{M}(t;x_{0},x_{1},\dots,x_{k})$ is a Banach manifold. We realize $\mathfrak{M}(t;x_{0},x_{1},\dots,x_{k})$ as a fibre product as above, namely $\mathfrak{M}(t;x_{0},x_{1},\dots,x_{k})=\overline{Exp}^{-1}(\prod_{i=1}^{s}\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i}))$ where $\overline{Exp}:\Delta\times Gr_{k+1}(t)\times\prod_{i}^{s}C^{k}(O_{i})\rightarrow\Delta^{s}$ where $\overline{Exp}(p,l;\bar{f}):=Exp_{g+f_{1}+\dots+f_{s},t}(p,l)$ . It suffice to show that $\overline{Exp}$ is a submersion. Now let $(p,l,\bar{f})\in\Delta\times Gr_{k+1}(t)\times\prod C^{k}(O_{i})$ and let $\overline{Exp}(p,l,\bar{f})=(p_{i})_{i=1,\dots,s}$ and fix a vector $(X_{i})\in\oplus_{i}T_{p_{i}}\Delta$ . We want to find $\bar{f}\in\prod C^{k}(O_{i})$ and a vector $V\in T_{p}\Delta$ so that $d_{(p,l,\bar{f})}Exp_{g+f_{1}+\dots+f_{s},t}(V,l)=(X_{i})$ . We give an ordering on $v_{1},\dots,v_{s}$ by $v_{i}\leq v_{j}$ if and only if the minimal path joining $v_{1}$ to $v_{j}$ passes through $v_{i}$ . Notice that the choice of $f_{i}$ does not affect the $X_{j}$ for $j<i$ this is due to the choice of $O_{i}$ . Moreover for $v_{1}$ , we have that $p_{1}=p$ and thus we can take $V=X_{1}$ . Now suppose we have chosen $f_{1},\dots,f_{i}$ so that the first $i^{th}$ -component $d_{(p,l,\bar{f})}Exp_{g+f_{1}+\dots+f_{s},t}(V,l)$ is $X_{1},\dots,X_{i}$ and the $i+1$ -component is $Y_{i}$ then set $f_{j}=0$ for $j>i+1$ and chose $f_{i+1}$ so that the differential becomes $X_{i}-Y_{i}$ . Continuing in this fashion it follows that $\overline{Exp}^{-1}(\prod_{i=1}^{s}\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i}))$ is a smooth Banach manifold. Now as countable intersection of Baire sets is a Baire set, the result follows. ∎

We introduce the following partial order on elements of $G_{k+1}$ :

Definition 7.11.

Let $t,t^{\prime}\in G_{k+1}$ , we write $t>t^{\prime}$ if and only if $t^{\prime}$ is obtained from $t$ by shrinking some interior edge.

Definition 7.12.

For $t\in G_{k+1}$ we denote by $Gr_{k+1}^{+}(t):=\{l:C^{1}_{int}(T)\rightarrow[0,\infty)\}$ .

Thus, from the (sequential) topology given on $Gr_{k+1}$ , it follows that $Gr_{k+1}^{+}(t)=\cup_{t>t^{\prime}}Gr_{k+1}(t^{\prime})$ . Moreover, using the above Theorem, we can identify a neighborhood of $Gr_{k+1}(t^{\prime})\subset Gr_{k+1}(t)$ by an $\epsilon$ -disk bundle. Namely, for $\epsilon>0$ small enough we denote by

\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime})):=\{exp_{a}(tV):a\in Gr_{k+1}(t^{\prime}),V\in\nu_{a}(Gr_{k+1}(t^{\prime})),0\leq t<\epsilon\}

where $\nu_{a}(Gr_{k+1}(t^{\prime}))$ in the fibre of the normal bundle of $Gr_{k+1}(t^{\prime})\subset Gr_{k+1}(t)$ at $a$ . Thus, $Gr_{k+1}(t^{\prime})\subset\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))\subset Gr_{k+1}(t)$ .

Definition 7.13.

For $\delta>0$ , $t\in Gr_{k+1}$ and $x_{0},\dots,x_{k}$ be distinct critical points of $f$ , we denote by $\mathfrak{M}^{\delta}(t,f;x_{0},x_{1},\dots,x_{k})\subset\mathfrak{M}(t,f;x_{0},x_{1},\dots,x_{k})$ where $l(e)<\delta$ for some $e\in C^{1}_{int}(T)$ .

Lemma 7.14.

There exists, $\epsilon,\delta>0$ where $\delta>0$ depends on $\epsilon>0$ and the injectivity radius of $g$ , where if $\tilde{f}\in C^{\infty}(\Delta)$ so that $\mathfrak{M}(t^{\prime},\tilde{f};x_{0},x_{1},\dots,x_{k})$ is a smooth manifold then, $\mathfrak{M}^{\delta}(t,\tilde{f};x_{0},x_{1},\dots,x_{k})$ is also a smooth manifold, with the understanding that $t^{\prime}$ is formed from $t$ after shrinking the internal edge of length less than $\delta$ . Moreover, we have a diffeomorphism $\mathfrak{M}(t^{\prime},\tilde{f};x_{0},x_{1},\dots,x_{k})\times\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))\cong\mathfrak{M}^{\delta}(t,\tilde{f};x_{0},x_{1},\dots,x_{k})$ .

Proof.

The result follows from the implicit function theorem and the proof of the above Lemma and so we follow the notation above.
In order to prove the first statement, let $\epsilon>0$ and $\tilde{f}\in C^{\infty}(\Delta)$ so that $\mathfrak{M}(t^{\prime},\tilde{f};x_{0},x_{1},\dots,x_{k})$ is a smooth manifold. We note that by the choice of $O_{i}$ as above, namely $O_{i}$ is disjoint from interior vertices and only intersects the edge adjacent to $v_{i}$ in its minimal path to $v_{1}$ . It follows that, $Exp_{\tilde{f},t^{\prime}}$ is independent of the normal variable of $\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))$ . In other words, it is independent of the Stasheff deformation parameter in the contracted edge in $t$ forming $t^{\prime}$ . Thus we can extend it smoothly in that direction to a smooth map $\tilde{Exp}_{\tilde{f},t^{\prime}}$ on $\Delta\times\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))$ . As $\tilde{Exp}_{\tilde{f},t^{\prime}}$ is a submersion on $\Delta\times Gr_{k+1}(t^{\prime})$ , it follows that, for $\epsilon>0$ small enough that $\tilde{Exp}_{\tilde{f},t^{\prime}}$ is a submersion on $\Delta\times\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))$ and thus, $\exists\delta>0$ so that $\mathfrak{M}^{\delta}(t,\tilde{f};x_{0},x_{1},\dots,x_{k})$ is also a smooth manifold.
To prove the second statement, we note that $Gr_{k+1}(t^{\prime})\subset Gr_{k+1}^{+}(t)$ and then we compare $Exp_{\tilde{f},t}$ and $Exp_{\tilde{f},t^{\prime}}$ , where $\tilde{f}\in C^{\infty}(\Delta)$ so that $\mathfrak{M}(t^{\prime},\tilde{f};x_{0},x_{1},\dots,x_{k})$ is a smooth manifold. In order to do so, we extend continuously $Exp_{\tilde{f},t}$ to $Exp_{\tilde{f},t}^{+}:\Delta\times Gr_{k+1}^{+}(t)\rightarrow\Delta^{s}$ . As $t>t^{\prime}$ it follows that, $s\leq s^{\prime}$ where $s,s^{\prime}$ is the number of interior vertices, $v_{1},\dots,v_{s};v_{1}^{\prime},\dots,v_{s^{\prime}}^{\prime}$ in $t,t^{\prime}$ respectively adjacent to an external edge. Moreover it follows that we have a map $m:\{1,\dots,s\}\rightarrow\{1,\dots,s^{\prime}\}$ given by $m(i)=j$ if and only if $v_{j}^{\prime}$ is obtained from $v_{i}$ . Similarly, we define an embedding $e_{t^{\prime},t}:\Delta^{s^{\prime}}\rightarrow\Delta^{s}$ given by $e_{t^{\prime}t}(p_{j})_{j=1,\dots,s^{\prime}}=(q_{i})_{i=1,\dots,s}$ where $p_{j}=q_{i}$ if and only if $m(i)=j$ . Then by construction we have

Exp_{\tilde{f},t|\Delta\times Gr_{k+1}(t^{\prime})}^{+}=e_{t^{\prime},t}\circ Exp_{\tilde{f},t^{\prime}}

Once again by our choice of $O_{i}$ , namely each is supported away from any interior vertex and only intersects the edge adjacent to $v_{i}$ in its minimal path to $v_{1}$ , $e_{t^{\prime},t}$ is an embedding and since $Exp_{\tilde{f},t^{\prime}}$ is transverse to the embedding of $\prod_{i=1}^{s^{\prime}}\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i}))$ , it follows by openness of the transversality condition, that $\exists\epsilon>0$ small enough so that $Exp_{\tilde{f},t|L\times\mathbb{D_{\epsilon}}(Gr_{k+1}(t^{\prime}))}^{+}$ is transverse to the embedding of $\prod_{i=1}^{s}\cap_{j=1,a}^{h_{i}}\tilde{W}^{-}_{x_{a}}(e^{j}_{i}))$ . Now identifying $\mathbb{D_{\epsilon}}(Gr_{k+1}(t^{\prime}))\subset Gr_{k+1}(t)$ by metric ribbon trees with one of there interior edges $e$ has $0\leq l(e)<\delta$ , where $\delta>0$ only depends on $\epsilon$ and injectivity radius of $g$ . Now by the implicit function theorem, we get the desired diffeomorphism $\mathfrak{M}(t^{\prime},\tilde{f};x_{0},x_{1},\dots,x_{k})\times\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))\cong\mathfrak{M}^{\delta}(t,\tilde{f};x_{0},x_{1},\dots,x_{k})$ . ∎

This suggests that in order to achieve transversality of the moduli space of Morse trajectories as a $Gr_{k+1}$ -family we have to change the ambient Morse function as the combinatorial type, that is the number of internal vertices, of the underlying tree changes.

Definition 7.15.

We denote by $\widetilde{Gr_{k+1}}:=\{(t,p):t=(T,\iota,rt,l)\in Gr_{k+1},p\in T\}\rightarrow Gr_{k+1}$ the universal family of metric ribbon trees, together with the preojection map $(t,p)\rightarrow t$ .

Definition 7.16.

[CW15] A domain-dependent Morse function associated to $f$ is a map $\tilde{f}:\widetilde{Gr_{k+1}}\rightarrow C^{\infty}(\Delta)$ satisfying the following conditions:

(1)

$\forall(t,p)\in\widetilde{Gr_{k+1}}$ we have $\tilde{f}_{t}(p,.)\equiv f(.)$ in a neighborhood of $\operatorname{Crit}(f)$ .
(2)

$\forall t\in Gr_{k+1}$ we have $\tilde{f}_{t}(s,.)\equiv f(.)$ , $\forall s$ in a small enough neighborhood of all vertices of the underlying tree of $t$ .

Corollary 7.17.

The Morse complex $\operatorname{CM}(f;\mathbb{Z})$ carries a structure of an $A_{\infty}$ -algebra.

7.2. Floer-Morse Trajectories

In order to define what we mean by a Floer-Morse trajectory in $X$ , we need to introduce more notation on ribbon trees. Let $t:=(T,\iota,rt,l)\in Gr_{k+1}$ be a rooted metric ribbon tree. Using the ribbon structure, we have an induced orientation on the edges of $T$ .

Definition 7.18.

We define $head,tail:C^{1}(T)\rightarrow C^{0}(T)$ to be:

(1)

$head(e):=v_{+}$ where $v_{+}$ is the vertex adjacent to $e$ where the orientation on $e$ is pointing outward from $v_{+}$
(2)

$tail(e):=v_{-}$ where $v_{-}$ is the vertex adjacent to $e$ where the orientation on $e$ is pointing inward from $v_{-}$

We now define the domain of Floer-Morse trajectories.

Definition 7.19.

For $t\in Gr_{k+1}$ as above we denote by $D_{t}:=\bigsqcup_{C^{0}_{int}(T)}\Sigma_{v}\sqcup\bigsqcup_{C^{1}(T)}I_{e}/\sim$ the non-locally Euclidean, compact, connected space, where

(1)

for $v\in C^{0}_{int}(T)$ , $(\Sigma_{v},\partial\Sigma_{v},z_{1},\dots,z_{|v|})$ is a bordered genus-zero Riemann surface with cyclical-ordered boundary marked points
(2)
for $e\in C^{1}(T)$ , $I_{e}$ is either:
1. (a)
  
  $[0,l(e)]$ if $e\in C^{1}_{int}(T)$
2. (b)
  
  $(-\infty,0]$ if $e\in C^{1}_{ext}(T)$ not adjacent to the root
3. (c)
  
  $[0,\infty)$ , otherwise

Constructed as follows: using the ribbon structure on $t$ we have a cyclic-ordering of all edges $e_{v}^{1},\dots,e_{v}^{|v|}$ adjacent to $v$ then, we identify:

(1)

$z_{i}\sim l(e_{v}^{i})$ if $e_{v}^{i}\in C^{1}_{int}(T)$ and $head(e_{v}^{i})=v$
(2)

$z_{i}\sim 0$ , otherwise

Now we fix a domain-dependent Morse function $\tilde{f}:\widetilde{Gr_{k+1}}\rightarrow C^{\infty}(\Delta)$ corresponding to $f$ .

Definition 7.20.

Let $\bar{x}=(x_{0},\dots,x_{k})\in\operatorname{Crit}(f)^{k+1}$ be a tuple critical points of $f$ and $\beta_{v}\in\pi_{2}(X,\Delta;\mathbb{Z})$ be relative spherical classes indexed by $C^{0}_{int}(T)$ .
A Floer-Morse trajectory associated to $(\bar{x},(\beta_{v})_{C^{0}_{int}(T)},t)$ is the collection of the following data:

(1)

$\forall v\in C^{0}_{int}(T)$ we have an element $p_{v}=(F_{v},u_{v}:(\Sigma_{v},\partial\Sigma_{v},z_{1},\dots,z_{|v|})\rightarrow(X,\Delta),e_{v})\in\mathcal{T}_{|v|}(\beta_{v})$
(2)
$\forall e\in C^{1}(T)$ , we have $(u_{e},I_{e},0,0)$ where $u_{e}$ satisfies the Morse trajectory equation on $I_{e}$

$\frac{d}{ds}u_{e}(s)+\nabla^{g}\tilde{f}_{t}(u_{e}(s),s)=0$

where
1. (a)
  
  if $e\in C^{1}_{int}(T)$ then, $I_{e}=[0,l(e)]$
2. (b)
  
  if $e\in C^{1}_{ext}(T)$ not adjacent to the root then, $I_{e}=(-\infty,0]$ and $u_{e}(-\infty)=x_{i}$ for some $i=1,\dots,k$
3. (c)
  
  if $e\in C^{1}_{ext}(T)$ is adjacent to the root then, $I_{e}=[0,\infty)$ and $u_{e}(\infty)=x_{0}$
(3)
We also require the matching conditions. Namely, $\forall v\in C^{0}(T)$ and using the cyclic ordering on the edges $e_{v}^{1},\dots,e_{v}^{|v|}$ adjacent to $v$ , we require that
1. (a)
  
  $u_{v}(z_{i})=u_{e_{v}^{i}}(0)$ , if $head(e_{v}^{i})=v$ or $tail(e_{v}^{i})=v$ and $e_{v}^{i}\in C^{1}_{ext}(T)$
2. (b)
  
  otherwise, $u_{v}(z_{i})=u_{e_{v}^{i}}(l(e_{v}^{i}))$

We denote by $\mathcal{T}(t,\tilde{f}_{t},\bar{x},(\beta_{v})_{v\in C^{0}_{int}(T)},g,J)$ the moduli space of all such maps given the Gromov-Hausdorff topology on the closure of the images of $(u_{e},u_{v},\phi)$ and the vectorspace topology.

Definition 7.21.

Using the same notation as in the above definition, we denote by $\mathcal{T}(t,\tilde{f}_{t},\bar{x},\beta,g,J):=\bigcup_{(\beta_{i})_{i}\in\pi_{2}(X,L;\mathbb{Z})^{|C^{0}_{int}(T)|}:\Sigma_{i}\beta_{i}=\beta}\mathcal{T}(t,\tilde{f}_{t},\bar{x},(\beta_{v})_{v\in C^{0}_{int}(T)},g,J)$ , the moduli space of perturbed Floer-Morse trajectories of type $(t,\bar{x},\beta)$ .

Definition 7.22.

We denote by $\mathcal{T}(\tilde{f},\bar{x},\beta;g,J):=\{(t,p):t\in Gr_{k+1},p\in\mathcal{T}(t,\tilde{f}_{t},\bar{x},\beta,g,J)\}$ the moduli space of all Floer-Morse trajectories of type $(\bar{x},\beta)$ .

Remark 7.23.

(1)

Note that, by Gromov’s Compactness we have, the set of $\beta_{i}\in\pi_{2}(X,L;\mathbb{Z})$ realized by $J$ -holomorphic maps so that $\omega(\beta_{i})\leq\omega(\beta)$ , is finite. Thus the $\mathcal{T}(t,\tilde{f}_{t},\bar{x},\beta,g,J)$ is a finite union.
(2)

We do not require that $\beta_{v}\neq 0\in\pi_{2}(X,L;\mathbb{Z})$ for (sequential) compactness reasons. Namely, as such configuration may occur after bubbling off a constant $J$ -disk along a Morse trajectory.

Proposition 7.24.

Using the notation of the above definition, for a generic choice of $\tilde{f}_{t}\in C^{\infty}(\Delta)$ with $Crit(f)=Crit(\tilde{f}_{t})$ , we have $\mathcal{T}(t,\tilde{f}_{t},\bar{x},\beta,g,J)$ is a smooth manifold with corners.

Proof.

From the first part of the above remark, it suffice to show the statement for $\mathcal{T}(t,f,\bar{x},(\beta_{v})_{v\in C^{0}_{int}(T)},g,J)$ . Noting that, for $\beta_{v}\neq 0\in\pi_{2}(X,\Delta)$ , $\mathcal{T}_{|v|}(\beta_{v})$ is a smooth manifold with $ev:\mathcal{T}_{|v|}(\beta_{v})\rightarrow\Delta^{|v|}$ a smooth submersion, the result follows from the Morse-Smale condition on $f$ , provided that $\beta_{v}\neq 0$ , $\forall v\in C_{int}^{0}(T)$ . In order to deal with the case as mentioned in the second part of the above remark, we argue as follows. $\forall v\in C^{0}_{int}(T)$ such that $\beta_{v}=0$ , denote by $T_{v}$ , the maximal sub-tree containing $v$ where all its interior vertices have $\beta=0$ . Namely, consider the connected component of $(\bigcup_{v:\beta_{v}=0}\cup_{j=1}^{|v|}e_{j}^{v})\setminus\cup_{v:\beta_{v}\neq 0}v$ where $e_{j}^{v}$ are all the edges adjacent to $v$ . Such sub-tree exists, as $T$ is connected and has finitely many interior vertices and uniqueness follows from the cycle-free condition on $T$ . Lemma 7.10, tells us that for a generic choice of $\tilde{f}\in C^{\infty}(\Delta)$ so that $Crit(f)=Crit(\tilde{f})$ , the flow of $-\nabla\tilde{f}$ along the edges of all such sub-trees intersect transversely at their interior vertices. Now arguing as in the proof of Lemma 7.10, we can express $\mathcal{T}(t,f,\bar{x},(\beta_{v})_{v\in C^{0}_{int}(T)},g,J)$ as a fibre product by $Exp$ and $ev$ which is smooth as $ev$ is a submersion. ∎

Proposition 7.25.

There exists, $\epsilon,\delta>0$ where $\delta$ depends on $\epsilon$ and the injectivity radius of $g$ , where if $\tilde{f}_{t^{\prime}}\in C^{\infty}(\Delta)$ so that $\mathcal{T}(t^{\prime},\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)$ is a smooth manifold with corners then, $\mathcal{T}^{\delta}(t,\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)$ is also a smooth manifold with corners, with the understanding that $t^{\prime}$ is formed out from $t$ after shrinking the internal edge of length less than $\delta$ . Moreover, we have a diffeomorphism $\mathcal{T}(t^{\prime},\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)\times\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))\cong\mathcal{T}(t,\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)$ .

Proof.

Similarly as above, it suffice to prove the result on one of the componenets of $\mathcal{T}(t,\tilde{f},\bar{x},\beta,g,J)$ as then we can take the minimum of all such $\epsilon,\delta$ .
If the shrinked edge contains a vertex $v$ such that $\beta_{v}\neq 0$ then the first statement follows from the submersion property of the evaluation map corresponding to $v$ . So suppose that the shrinked edge has both $\beta_{v_{+}},\beta_{v_{-}}=0$ , where $v_{+},v_{-}$ are the head and tail of such edge. Denote by $T_{e}$ the maximal sub-tree containing $v_{+},v_{-}$ in $T$ . Then Lemma 7.14 tells us that $\exists\delta>0$ so that, if $\mathfrak{M}(T_{e}^{\prime},f,\bar{x}^{\prime})$ is a smooth manifold with corners so is $\mathfrak{M}^{\delta}(T_{e},f,\bar{x}^{\prime})$ , with the understanding that $T_{e}^{\prime}$ is the tree formed out of $T_{e}$ upon shrinking edge $e$ . Now writing $\mathcal{T}^{\delta}(t,\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)$ and $\mathcal{T}(t^{\prime},\tilde{f}_{t^{\prime}},\bar{x},\beta,g,J)$ as $ev^{-1}(\mathfrak{M}^{\delta}(T_{e},f,\bar{x}^{\prime},g)\times M_{1}\times\dots\times M_{l})$ and $ev^{-1}(\mathfrak{M}(T_{e}^{\prime},f,\bar{x}^{\prime};g)\times M_{1}\times\dots\times M_{l})$ respectively and noting that $ev$ is a submersion and each $M_{i}$ is smooth, the first statement follows. As in the proof of Lemma 7.14, the second statement follows from the implicit function theorem. ∎

Corollary 7.26.

For a generic choice of $\tilde{f}:\widetilde{Gr_{k+1}}\rightarrow C^{\infty}(\Delta)$ , the thickened moduli space $\mathcal{T}(\tilde{f},\bar{x},\beta;g,J)$ is a smooth manifold with corners. Moreover, the forgetful map $\mathcal{T}(\tilde{f},\bar{x},\beta;g,J)\rightarrow Gr_{k+1}$ is a smooth submersion.

Proof.

We argue by upward induction on the cell-structure $Gr_{k+1}=\bigcup_{G_{k+1}}Gr_{k+1}(t)$ . Let $t=(T,\iota,rt,l)\in Gr_{k+1}$ and assume that we have an $\tilde{f}:\bigcup_{t^{\prime}<t}Gr_{k+1}(t^{\prime})\rightarrow C^{\infty}(\Delta)$ so that $\forall t^{\prime\prime}\in\bigcup_{t^{\prime}<t}Gr_{k+1}(t^{\prime})$ , $\mathcal{T}(t^{\prime\prime},\tilde{f}(t^{\prime\prime}),\bar{x},\beta,g,J)$ is a smooth manifold with corners. Noting that the above union is a finite one, indeed we only have finitely many interior edges in $t$ , and using Proposition , we can find $\epsilon,\delta>0$ so that $\forall t^{\prime\prime\prime}\in\cup_{t^{\prime}<t}\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))$ , $\mathcal{T}^{\delta}(t^{\prime\prime\prime},\tilde{f}(t^{\prime\prime\prime}),\bar{x},\beta,g,J)$ is a smooth manifold with corners. Moreover, by Proposition 2.1 and for $\epsilon^{\prime}>0$ small enough, we can find a generic $f\in C^{\infty}(\Delta)$ so that $||f-\tilde{f}||_{C^{\infty}}<\epsilon^{\prime}$ , $\mathcal{T}(t^{\prime\prime\prime},f,\bar{x},\beta,g,J)$ is a smooth manifold with corners, $\forall t^{\prime\prime\prime}\in\cup_{t^{\prime}<t}\mathbb{D}_{\epsilon}(Gr_{k+1}(t^{\prime}))$ . Now using the implicit function theorem, it follows that for $(t,u)\in\mathcal{T}(f,\bar{x},\beta;g,J)$ , $V_{\epsilon}\times U$ , where $V_{\epsilon}$ is a neighborhood of zero in the normal fibre of the $Gr_{k+1}(t)\subset Gr_{k+1}$ and $u\in U\subset\mathcal{T}(t,\tilde{f}(t^{\prime\prime}),\bar{x},\beta,g,J)$ is a smooth coordinate chart, forms an atlas of $\mathcal{T}(f,\bar{x},\beta;g,J)$ .
As for the base case, we note that the minimal cell in $Gr_{k+1}$ is a point corresponding to the unique tree with $1$ -interior vertex and $(k+1)$ -external edges and thus the claim follows from the Morse-Smale condition.
As for the second claim, using Proposition 7.23, locally $\mathcal{T}(f,\bar{x},\beta;g,J)\rightarrow Gr_{k+1}$ is a projection map and thus it is a submersion. ∎

8. Global Kuranishi Chart for the Morse Model

In the above section we have constructed a thickening of the moduli space of Floer-Morse trajectories. We still have to find a global obstruction bundle over the thickening and a group of symmetry on it where the stabilizer group of each point of the thickening is isomorphic to the automorphism-group of the underlying map. In order to do so we have introduce a new type of framing. Indeed, consider a non-perturbed Floer-Morse trajectory $u$ of combinatorial type $t\in Gr_{k+1}$ and denote by $\Sigma:=\vee_{C^{0}_{int}(t)}\Sigma_{v}$ the nodal genus-zero bordered Riemann surface formed after shrinking all the edges of the domain of $u$ . Let $(L,\Omega)\rightarrow X$ be as above and define $L_{u}\rightarrow\Sigma$ by

(8.1)

L_{u}:=\sqcup_{C^{0}_{int}(t)}u_{v}^{*}L/\sim

where if $v_{1},v_{2}$ are two vertices of the same edge $e$ , then we identify the fibre $L_{u_{e}(0)}\sim L_{u_{e}(l(e))}$ by parallel transport along $u_{e}$ .

Definition 8.1.

A framing of $u$ is a choice of an $\mathbb{R}$ -basis of $H^{0}(\Sigma,\partial\Sigma;L_{u},L_{u|\partial\Sigma})$ of holomorphic section of $L_{u}$ with real boundary conditions.

Let $F:=\{f_{0},\dots,f_{d}\}$ be an $\mathbb{R}$ -basis of $H^{0}(\Sigma,\partial\Sigma;L_{u},L_{u|\partial\Sigma})$ such that $[f_{i}(z_{0})]=[1:0:\dots:0]$ and consider the induced embedding $\phi_{F}:(\Sigma,\partial\Sigma)\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ and note that $\phi_{F}\in\mathcal{F}^{d}(\beta)$ . This produces a map $p:\mathcal{T}(f,\bar{x},\beta;g,J)\rightarrow\mathcal{F}$ . Let $V_{\mu,\beta}$ be a finite-dimensional approximation of $J$ -discs of degree $\beta$ as in the above section.

Definition 8.2.

We define the obstruction bundle $Ob(\bar{x},\beta)$ over $\mathcal{T}(f,\bar{x},\beta;g,J)$ to be the trivial bundle of fibre $\mathfrak{g}\oplus V_{\mu,\beta}$ .

Now consider a perturbed Morse-Floer trajectory of type $(t,\bar{x},\beta)$ . For each vertex $v$ we have a tuple $(F_{v},u_{v},e_{v})$ . For simplicity of the discussion, assume that $t$ has only two internal vertices $v_{1},v_{2}$ and write $\phi_{F_{v_{i}}}=[f_{i,0}(z):\dots:f_{i,d_{i}}(z)]$ , where $i=1,2$ . We glue the maps $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}:\Sigma_{v_{1}}\vee\Sigma_{v_{2}}\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ by setting $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}(z)=[f_{1,0}(z):\dots:f_{1,d_{1}}(z):0:\dots:0]$ , $\forall z\in\Sigma_{v_{1}}$ and $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}(z)=[a_{0}f_{2,0}(z):\dots:a_{d_{1}}f_{2,0}(z):f_{2,1}(z):\dots:f_{2,d_{2}}(z)]$ , $\forall z\in\Sigma_{v_{2}}$ , where $a_{i}\in\mathbb{C}$ are chosen so that $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}$ is a continuous map. Then, $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}:\Sigma_{v_{1}}\vee\Sigma_{v_{2}}\rightarrow(\mathbb{CP}^{d},\mathbb{RP}^{d})$ is a holomorphic embedding and hence upon pulling-back $\mathcal{O}(1)$ by $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}$ , it follows that we can represent $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}$ uniquely up to multiplication by $\mathbb{R}^{*}$ by a basis of $H^{0}(\Sigma,\partial\Sigma;L_{u},L_{u|\partial\Sigma})$ . In particular, $\phi_{F_{v_{1}}}\vee\phi_{F_{v_{2}}}\in\mathcal{F}^{d}(\beta)$ . Denote by $F$ the associated basis for the glued framing $\phi_{F_{1}}\vee\phi_{F_{2}}$ and let $H(F)$ be its associated symmetric matrix. Moreover, from our choice of $V_{\mu,\beta}$ , we have the vector $(e_{v})_{C^{0}_{int}(t)}\in V_{\mu,\beta}$ .

Definition 8.3.

We define the Kuranishi map $s:\mathcal{T}(f,\bar{x},\beta;g,J)\rightarrow Ob(\bar{x},\beta)$ by

(8.2)

s(t,p):=((e_{v})_{C^{0}_{int}(t)},exp^{-1}(H(F)))

Theorem 8.4.

The tuple $(SO(d),\mathcal{T}(f,\bar{x},\beta;g,J),Ob(\bar{x},\beta),s)$ is a smooth Global Kuranishi chart for the moduli space of unperturbed Floer-Morse trajectories.

Proof.

It remains to check that the isotropy group of a point $(t,p)\in s^{-1}(0)$ agrees with the automorphism group of the underlying Floer-Morse trajectory. As all the Morse flow-lines have prescribed initial value, it follows that any automorphism of the underlying Floer-Morse trajectory comes from the automorphism group of the maps associated to the vertices. Let $(f_{v})_{C^{0}_{int}(t)}$ be a collection of $f_{v}:\Sigma_{v}\rightarrow\Sigma_{v}$ domain automorphism satisfying $u_{v}\circ f_{v}=u_{v}$ . Then $u_{v}^{*}\Omega=f_{v}^{*}u^{*}\Omega$ and hence $\phi_{F_{v}}\circ f_{v}=\phi_{g_{v}F_{v}}$ for some $g_{v}\in SO(d_{v})$ . Thus using the fixed ordering of the internal vertices, it follows that, $g:=\operatorname{diag}(g_{v})_{C^{0}_{int}(t)}\in SO(d)$ will fix $(t,p)$ . On the other hand, if $g\in SO(d)$ fixes $(t,p)$ then $g\circ\vee\phi_{F_{v}}=\vee\phi_{F_{v}}$ . Now by construction of $\vee\phi_{F_{v}}$ and using the ordering on the internal vertices to induct on them, it follows that $g$ is a block matrix $g=\operatorname{diag}(g_{v})$ and hence each $g_{v}\in SO(d_{v})$ , such that $g_{v}$ fixes the corresponding element $u_{v}$ of $\mathfrak{M}(\beta_{v})\subset\mathcal{T}_{|v|}(\beta_{v})$ , namely the corresponding unperturbed $J$ -disk at vertex $v$ . Using Lemma 4.2, we can find $f_{v}:\Sigma_{v}\rightarrow\Sigma_{v}$ such that, $u_{v}\circ f_{v}=u_{v}$ and $\phi_{F_{v}}\circ f_{v}=\phi_{g_{v}F_{v}}$ . ∎

8.1. Boundary Components of the Thickening

We study the codimension 1 corner of $\mathcal{T}(f,\bar{x},\beta,g,J)$ .
Consider a Floer-Gromov convergent sequence of elements of $(t_{n},p_{n})\in\mathcal{T}(f,\bar{x},\beta,g,J)$ . We have the following 3 case:

(1)

Bubbling at an internal vertex.
(2)

Shrinking of a finite length Morse edge to zero.
(3)

Breaking of a finite length Morse edge into two infinite ones.

Consider case 1 and denote by $v$ the internal vertex where we have bubbling. Recall that the smooth structure on $\mathcal{T}_{|v|}(\beta_{v})$ is an extension of the $C^{1}_{loc}$ -structure given by gluing (perturbed) $J$ -disks. On the other hand, for such configuration, we can also find another sequence $(t^{\prime}_{n},q_{n})\in\mathcal{T}(f,\bar{x},\beta,g,J)$ where $t_{n}\leq t^{\prime}_{n}$ that converges to the same limit. Namely, separate the bubbled off pseudo-holomorphic map by a finite length Morse edge that converges to zero as in case 2. Now using the appendix of [CW15] or [Sch16], we get that the normal fibre in the $C^{1}$ -sense of this limit point inside $\mathcal{T}(f,\bar{x},\beta,g,J)$ is identified by $\mathbb{R}$ . Therefore, limit points as in cases 1 and 2 are actually interior points of $\mathcal{T}(f,\bar{x},\beta,g,J)$ .

Corollary 8.5.

For a generic choice of $f:\widetilde{Gr_{k+1}}\rightarrow C^{\infty}(\Delta)$ , the thickened moduli space $\mathcal{T}(f,\bar{x},\beta;g,J)$ is a smooth manifold with corners, which admits a bordification by adding a disk-bundle over $\bigcup\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ with $\beta=\beta_{1}+\beta_{2}$ , $\bar{x}_{1}\cup\bar{x}_{2}=\bar{x}\cup\{y\}$ for some $y\in Crit(f)$ such that $\mu(\bar{x}_{1},\beta_{1})+\mu(\bar{x}_{2},\beta_{2})=\mu(\bar{x},\beta)-1$ , where $\mu(\bar{x},\beta)$ is the expected dimension of $\mathcal{T}(f,\bar{x},\beta;g,J)$ .

Proof.

We construct such $f$ by upward induction over $\omega(\beta)$ the energy and over $k$ the number of external vertices. Indeed suppose that we have constructed $f^{\prime}$ on lower dimensional strata $Gr_{k_{1}+1}\times Gr_{k_{2}+1}$ of the compactification of $Gr_{k+1}$ . Noting that, the perturbation of $f^{\prime}$ is compactly-supported and away from the vertices, we can use Whitney’s Extension theorem to extend $f^{\prime}$ in a small neighborhood of $Gr_{k_{1}+1}\times Gr_{k_{2}+1}$ . Thus the Banach space of all $f:\overline{Gr}_{k+1}\rightarrow C^{k}(\Delta)$ such that $f\equiv f^{\prime}$ on $Gr_{k_{1}+1}\times Gr_{k_{2}+1}$ is non-empty. Now arguing as in the proof of Lemma 7.10, we get the inductive step.
The base case is precisely Corollary 7.5.
Now given $((t_{1},p_{1}),(t_{2},p_{2}))\in\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ . Standard gluing results of Morse flow-lines provides us a neighborhood $((t_{1},p_{1}),(t_{2},p_{2}))\in U\subset\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ together with an embedding $U\times[R,\infty)\rightarrow\mathcal{T}(f,\bar{x},\beta;g,J)$ . On the other hand, the failure of $\bigcup\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ being the codimension 1 corner, comes from the perturbation factors of the equations on the vertices. Indeed consider the normal bundle of the image of the embedding above inside the codimension 1 boundary of $\mathcal{T}(f,\bar{x},\beta;g,J)$ , its fibre over a point $p\in\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ is $V_{\mu,\beta}/(V_{\mu,\beta_{1})}\oplus V_{\mu,\beta_{2}})$ . Thus, after stabilizing $\mathcal{T}(f,\bar{x}_{1},\beta_{1};g,J)\times\mathcal{T}(f,\bar{x}_{2},\beta_{2};g,J)$ by the trivial bundle $V_{\mu,\beta}/(V_{\mu,\beta_{1})}\oplus V_{\mu,\beta_{2}})$ and group enlarging $G_{1}\times G_{2}\hookrightarrow G$ , we get the desired result. ∎

Proposition 8.6.

The Global Kuranishi chart $(SO(d),\mathcal{T}(f,\bar{x},\beta;g,J),Ob(\bar{x},\beta),s)$ is stably normally complex. Moreover, the normal complex structure is compatible with that of its boundary components.

Proof.

As all Morse flow-line have prescribed initial values, it follows that, the non-trivial sub-representations of $T\mathcal{T}(f,\bar{x},\beta;g,J)$ at any point $(t,p)$ comes from $\bigoplus_{C^{0}_{int}(t)}\ker(D\bar{\partial}_{J}+K)_{v}$ the kernel of the linearization of the perturbed $\bar{\partial}_{J}$ -equation at sphere components. Thus the result follows from the proof of Lemma 4.2. We show that the (stable) normal complex structure on $\mathcal{T}(f,\bar{x},\beta;g,J)$ is compatible with that of its codimesnion 1 corner. To this end, we argue by upward induction over $\omega(\beta)$ the energy and over $k$ the number of external vertices. The base case follows from the above. As for the inductive step and simplicity of notation, suppose that we have a once broken tree and denote by $u,v$ the underlying Floer-Morse trajectories. Denote by $D_{u}$ and $D_{v}$ the linearization of these maps. Noting that both $D_{u}$ and $D_{v}$ are surjective, it suffice to study the non-trivial sub-representations of $\ker(D_{u})$ and $\ker(D_{v})$ , which we denote by $\tilde{\ker(D_{u})}$ and $\tilde{\ker(D_{v})}$ respectively. From the proof of Theorem 1, we can build a homotopy $H_{u}$ between $\tilde{\ker(D_{u})}$ and a complex vectorspace $W_{u}$ and a homotopy $H_{v}$ between $\tilde{\ker(D_{v})}$ and a complex vectorspace $W_{v}$ . Using the same notation as in the above proof, we consider nearby curves, namely $u\#_{r}v$ images of the embedding $U\times[R,\infty)\rightarrow\mathcal{T}(f,\bar{x},\beta;g,J)$ . As automorphisms of the Floer-Morse trajectory $u\#_{r}v$ comes from the sphere components, it follows that, the non-trivial sub-representation of $\ker(D_{u\#_{r}v})$ is decomposed into $\tilde{\ker(D_{u\#_{r}v})}=\tilde{\ker(D_{u})}\oplus\tilde{\ker(D_{v})}$ and hence $r$ -independent. Thus $H_{u}\oplus H_{v}$ is a homotopy between $\tilde{\ker(D_{u\#_{r}v})}$ and a complex vectorspace. Now the inductive step follows from the fact that any continuous map defined on a subcomplex of a the stratification of a manifold with corners can be extended to an additional cell. ∎

9. Proof of Theorem 1.1

Proof.

Given $\beta\in\pi_{2}(X,\Delta)$ and a pure tensor $x_{1}\otimes\dots\otimes x_{k}$ of critical points of $f$ we consider the associated derived orbifold chart $(\mathcal{U},\mathcal{E},s)$ of $\mathcal{T}(f,\bar{x},\beta;g,J)$ , where $\bar{x}=(x_{0},x_{1},\dots,x_{k})$ . We set

m_{k,\beta}(x_{1}\otimes\dots\otimes X_{k}):=\Sigma_{x_{0}\in Crit(f)}(-1)^{*}|s_{\epsilon}^{-1}(0)\cap\mathcal{U}_{free}|x_{0}

where the sum is taken over all $x_{0}\in Crit(f)$ such that $\mu(\bar{x},\beta)=0$ and $||s-s_{\epsilon}||_{C^{0}}<\epsilon$ is an FOP perturbation of $s$ . We set

m_{k}:=\Sigma_{\beta}m_{k,\beta}T^{\omega(\beta)}e^{\frac{\mu(\beta)}{2}}

and extend $\Lambda$ -linearily. Now the proof follows from the codimension 1 boundary description of $\mathcal{T}(\tilde{f},\bar{x},\beta;g,J)$ , provided that we have chosen our FOP perturbations consistently. In order to do so we argue by upward induction over $\omega(\beta)$ the energy and $k$ the number of leaves.
The base case follows from Lemma 2.9 and Lemma 2.13. As for the inductive step, we have to consider $\mathcal{T}(\tilde{f},\bar{x},\beta,:g,J)$ where the expected dimension $\mu(\bar{x},\beta)=1$ and construct FOP-sections on its associated derived orbifold chart, which we denote by $(\mathcal{U},\mathcal{E},s)$ , that agrees with the one already constructed on its codimesion 1 corners. Indeed, consider a component of the codimesion 1 corner of $\mathcal{T}(\tilde{f},\bar{x},\beta,:g,J)$ , this can be given a Global Kuranishi chart

(G_{1}\times G_{2},\mathcal{T}(\tilde{f},\bar{x}_{1},\beta_{1},:g,J)\times\mathcal{T}(\tilde{f},\bar{x}_{2},\beta_{2},:g,J),Ob(\bar{x_{1}},\beta_{1})\oplus Ob(\bar{x_{2}},\beta_{2}),s_{1}\oplus s_{2})

where by the inductive hypothesis we are assuming that both $s_{1}/G_{1},s_{2}/G_{2}$ are strongly transverse FOP-sections. By Lemma 2.12, it follows that the associated orbibundle section of $s_{1}\oplus s_{2}$ is strongly transverse. Now after group enlargement by $G_{1}\times G_{2}\hookrightarrow G$ and stabilization, we get an equivalent Global Kuranishi chart, where we denote its associated derived orbifold chart by $(\mathcal{U}^{\prime},\mathcal{E}^{\prime},s^{\prime})$ together with an open embedding of $(\mathcal{U}^{\prime},\mathcal{E}^{\prime})$ into $(\mathcal{U},\mathcal{E})$ . Now the inductive step follows from Lemma 2.10, Lemma 2.11 and Lemma 2.13. ∎

References

[AMS21] Mohammed Abouzaid, Mark McLean, and Ivan Smith. Complex Cobordism, Hamiltonian loops and Global Kuranishi Charts. arXiv preprint arXiv:2110.14320, 2021.
[AMS24] Mohammed Abouzaid, Mark McLean, and Ivan Smith. Gromov-Witten Invariants in Complex and Morava-Local K-Theories. Geometric and Functional Analysis, pages 1–87, 2024.
[BPX24] Shaoyun Bai, Daniel Pomerleano, and Guangbo Xu. Cohomological splitting over rationally connected bases. arXiv preprint arXiv:2406.00931, 2024.
[BX22a] Shaoyun Bai and Guangbo Xu. Arnold Conjecture over Integers. arXiv preprint arXiv:2209.08599, 2022.
[BX22b] Shaoyun Bai and Guangbo Xu. An integral euler cycle in normally complex orbifolds and z-valued gromov-witten type invariants. arXiv preprint arXiv:2201.02688, 2022.
[CW15] François Charest and Chris T Woodward. Floer Theory and Flips. arXiv preprint arXiv:1508.01573, 2015.
[FO97] Kenji Fukaya and Yong-Geun Oh. Zero-loop Open Strings in the cotangent bundle and Morse homotopy. Asian Journal of Mathematics, 1(1):96–180, 1997.
[FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono. Lagrangian Intersection Floer Theory: Anomaly and Obstruction (Part 1), volume 46.1 of AMS/IP Studies in Advanced Mathematics. American Mathematical Society / International Press, 2009.
[FOOO21] K Fukaya, YG Oh, H Ohta, and K Ono. Construction of Kuranishi structures on the moduli spaces of pseudo holomorphic disks: I, to appear in Surveys in Differential Geometry. arXiv preprint arXiv:1710.01459, 2021.
[KO00] Daesung Kwon and Yong-Geun Oh. Structure of the Image of (pseudo)-holomorphic Discs with Totally Real Boundary Condition. Communications in Analysis and Geometry, 8(1):31–82, 2000.
[Laz11] Laurent Lazzarini. Relative Frames on J-holomorphic Curves. Journal of Fixed Point Theory and Applications, 9:213–256, 2011.
[Raba] Mohamad Rabah. Fukaya Algebra over Z: II. Manuscript in preparation.
[Rabb] Mohamad Rabah. Lagrangian Floer Theory over Z. Manuscript in preparation.
[Sch16] Felix Schmäschke. Floer Homology of Lagrangians in Clean Intersection. arXiv preprint arXiv:1606.05327, 2016.
[Zer17] Amitai Netser Zernik. Moduli of Open Stable Maps to a Homogeneous Space. arXiv preprint arXiv:1709.07402, 2017.

Fukaya Algebra over ℤ\mathbb{Z}: I

Abstract.

1. Introduction

Theorem 1.1.

1.1. Filtered An,KA_{n,K}-algebra

2. FOP Perturbations

Definition 2.1.

Definition 2.2.

Definition 2.3.

Definition 2.4.

Definition 2.5.

Remark 2.6.

Lemma 2.7.

Lemma 2.8.

Lemma 2.9.

Lemma 2.10.

Lemma 2.11.

Lemma 2.12.

Lemma 2.13.

3. Global Kuranishi Charts

Definition 3.1.

Example 3.2.

Definition 3.3.

4. Global Kuranishi Charts for J-Disks

Definition 4.1.

Lemma 4.2.

4.1. Preliminary Constructions

Definition 4.3.

Definition 4.4.

Lemma 4.5.

Proof.

Lemma 4.6.

Proof.

Lemma 4.7.

Proof.

Remark 4.8.

Lemma 4.9.

Proof.

Lemma 4.10.

Proof.

Lemma 4.11.

Proof.

4.2. Space of Framings

Definition 4.12.

Remark 4.13.

Definition 4.14.

Remark 4.15.

Lemma 4.16.

Proof.

Corollary 4.17.

4.3. Proof of Lemma 4.2

4.3.1. Gromov’s Graph Trick

Remark 4.18.

Lemma 4.19.

Definition 4.20.

Definition 4.21.

Remark 4.22.

Corollary 4.23.

Remark 4.24.

Lemma 4.25.

Proof.

Definition 4.26.

Lemma 4.27.

Proof.

4.3.2. Smooth Structure on Thickening

Lemma 4.28.

Proof.

Corollary 4.29.

Proof.

Lemma 4.30.

Proof.

Corollary 4.31.

Proof.

5. Proof of Lemma 4.2

5.1. Using [AMS21], [BX22a]

Proof.

5.2. Using [AMS24]

Definition 5.1.

Lemma 5.2.

Definition 5.3.

Fukaya Algebra over $\mathbb{Z}$ : I

1.1. Filtered $A_{n,K}$ -algebra