$T$-equivariant disc potential and SYZ mirror construction
Yoosik Kim Siu-Cheong Lau Xiao Zheng

	


 Department of Mathematics, Brandeis University, 415 South Street
Waltham, MA 02453, USA & 
Center of Mathematical Sciences and Applications, Harvard University, 20 Garden Street, Cambridge, MA 02138, USA 
yoosik@brandeis.edu, yoosik@cmsa.fas.harvard.edu


Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215, USA
lau@math.bu.edu


Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215, USA
xiaoz259@bu.edu
	
	
		We develop a G-equivariant Lagrangian Floer theory and obtain a curved A_∞ algebra, and in particular a G-equivariant disc potential.  We construct a Morse model, which counts pearly trees in the Borel construction L_G.  When applied to a smooth moment map fiber of a semi-Fano toric manifold, our construction recovers the T-equivariant toric Landau-Ginzburg mirror of Givental. We also study the ^1-equivariant Floer theory of a typical singular SYZ fiber (i.e. a pinched torus) and compute its ^1-equivariant disc potential via the gluing technique developed in <cit.>. 
	
	
	T-equivariant disc potential and SYZ mirror construction
    Xiao Zheng
    December 30, 2023
========================================================



§ INTRODUCTION

	
	
 	Teleman <cit.> conjectured that the mirror of a Hamiltonian action of a compact Lie group G on a symplectic manifold X is a holomorphic map from the mirror X^∨ of X to the space of conjugacy classes in  G_^∨.  
 	Moreover, the mirrors of the symplectic quotients X_λ G for λ∈𝔤^*/Ad^*_G are closely related to the fibers this map.  This agrees with the work of Hori-Vafa <cit.> on mirror symmetry via gauged linear sigma models.

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

	
	For instance, let us consider the well-known Landau-Ginzburg mirror of a compact Fano toric d-fold X <cit.>. It is given by a pair ((^×)^d,W), where the superpotential W:(^×)^d→ is a holomorphic function of the form
	
    W(z)=∑_i=1^m q^α_iz⃗^v_i,
        
	where v_1,…,v_m∈N≅^d are primitive generators of the one-dimensional cones of the fan Σ defining X, z⃗^v_i denote the corresponding Laurent monomials, and q^α_1,…,q^α_m are Kähler parameters associated to effective curves classes 
	α_i∈ H_2(X;). 
	
 Givental <cit.> showed that the equivariant quantum cohomology QH_T^∙(X) of X is mirror to the equivariant superpotential
	
    W_λ(z)=W(z)+∑_i=1^d λ_i logz_i,
   
	where λ_1,…,λ_d are the equivariant parameters for the torus T acting on X. Iritani <cit.> further generalized this to a mirror correspondence between big equivariant quantum cohomology and a universal Landau-Ginzburg potential which has both equivariant and bulk-deformation parameters.
	
	
	Cho-Oh <cit.> (in the Fano case) and Fukaya-Oh-Ohta-Ono <cit.> (in the general compact non-Fano setting) used Lagrangian Floer theory to define the disc potential of a smooth fiber of the toric moment map and showed that it coincides with the Landau-Ginzburg superpotential W. From this perspective, W is viewed as the generating function of open Gromov-Witten invariants. Each term in W corresponds to a holomorphic disc of Maslov index 2 bounded by the torus fiber. This gives an SYZ interpretation of the Landau-Ginzburg mirrors <cit.>. 
	
	It is a natural and interesting question that whether the W_λ can also be interpreted as a generating function of equivariant disc counts, to which we give an affirmative answer to in this paper.
	
	 
		Let X be a semi-projective and semi-Fano toric manifold with _ X=d, and let T⊂ (^1)^d be the subtorus determined by the integral basis u_1,…,u_ℓ∈N≅^d. The T-equivariant disc potential (Definition <ref>) W^L_T of a regular fiber L of the toric moment map is given by
		
    W^L_T= ∑_i=1^m  exp(g_i(q̌(q))) ^ω(β_i)exp (v_i · (x_1,…,x_d))+∑_j=1^ℓ (u_j · (x_1,…,x_d)) λ_j

		where β_i are the basic disc classes bounded by the toric fiber, g_i(q̌(q)) is given by the inverse mirror map in equation (<ref>), q^α = ^ω(α) are the Kähler parameters, and  is the formal Novikov variable.
	
	
	In the Fano case, we have g_i=0. By taking ℓ=d and u_1,…,u_d to be the standard basis of 𝔱^d:=Lie(T), and setting x_i = log z_i for i=1,…,d, the above expression for W^L_T equals to W_λ.  
	
	
	
	
	 
	
	 In relation with the Teleman's conjecture, the T-equivariant disc potential for a Lagrangian submanifold L with nonnegative minimal Maslov index takes the form 
	
    W^L_T= W(b) + ∑_i=1^ℓ h_i(b) λ_i,

	for degree one weak bounding cochains b. W^L_T can be interpreted as follows:  
	The equivariant term 
    Φ(b):=(h_1(b),…,h_ℓ(b))
 
	defines a map from the formal deformation space of L to  (𝔱_^ℓ)^*, assuming its convergence over .
	The non-equivariant term W can be viewed as a family of potentials on the fibers of Φ.  It gives a mirror family of the symplectic quotient X_λ T.
	
	For instance, let us take X=^d.  The non-equivariant part is simply W=z_1+…+z_d.  Suppose X_λ T^ℓ is a semi-Fano toric manifold.  A fiber of Φ is given by z⃗^u_j = c_j for some constants c_j and j=1,…,ℓ.  Then, the restriction of W to the fibers give a mirror family of X_λ T^ℓ.  
	
	
		The terms h_i(b) λ_i can be understood as obstruction terms for a homotopy equivalence between the equivariant Floer theory of L ⊂ X and Floer theory of the quotient L/T ⊂ X_λ T^ℓ.  Daemi-Fukaya <cit.> asserted that this homotopy equivalence can be constructed by Lagrangian correspondence.  In our setting, we can compute the mirror equations h_i(b)=0 explicitly.
		
		The mirror map c(λ) is crucial to precisely identify which fiber in the mirror family corresponds to X_λ T^ℓ.  By the beautiful work of Woodward-Xu <cit.>, the mirror map can be understood as the change from the gauged Floer theory of X to the Fukaya category (X_λ T^ℓ) downstairs.  Gauged Floer theory is formulated in terms of vortex equations.  It would be very interesting to investigate the relation with our formulation.
	
	
	Equivariant Lagrangian Floer theory has seen substantial recent developments.  
	In the exact setting and G=/2, Seidel-Smith <cit.> provided an approach to understanding G-equivariant Floer theory by combining Lagrangian Floer theory and family Morse theory <cit.> on EG→ BG.  Viterbo <cit.> illustrated some related ideas for G=^1.
	Hendricks-Lipshitz-Sarkar <cit.> developed a homotopy coherent method to build up a G-equivariant Floer theory. Daemi-Fukaya <cit.> used G-equivariant Kuranishi structure <cit.> to tackle the G-equivariant transversality problem, and make a formulation using differential forms.  Bao-Honda <cit.> defined equivariant Lagrangian Floer cohomology for finite group action via semi-global Kuranishi structures. Mirror Lagrangian objects of equivariant vector bundles was studied by Lekili-Pascaleff <cit.>. There are many interesting works related to this subject such as <cit.>. 
	
	In Section <ref>, we construct a Morse model for G-equivariant Lagrangian Floer theory by counting pearly trees  <cit.> in the Borel construction L_G, and study the associated equivariant disc potential.  The key ingredient is a collection of Morse models on the finite dimensional approximations of L_G satisfying certain compatibility conditions (Proposition <ref>).
	This is closest to the approach of Seidel-Smith, although the Lagrangians under our consideration are not exact. The Lagrangians of interest in this paper bound non-constant pseudo-holomorphic discs of non-negative Maslov index. These are known as quantum corrections in mirror symmetry, which can have non-trivial contributions to the equivariant parameters in the disc potential. Since there are only finitely many generators for each finite dimensional approximation of L_G, it better serves for computations and for the SYZ mirror construction.
		

	
	In order for the equivariant paramaeters appearing in W_λ to manifest. In Section <ref>  we adapt the homotopy unit construction developed by Fukaya-Oh-Ohta-Ono <cit.> to construct a curved A_∞-algebra 
	
    (C_Morse^∙(L;Λ_0)⊗_Λ_0 H^∙_G(;Λ_0), {^G_k}_k≥ 0),
 
	which satisfies (Theorem <ref>)
	
    _k^G(X_1,…,x⊗λ, …, X_k) = λ·^G_k(X_1,…,x⊗ 1,…,X_k),

	under the assumption that L has minimal Maslov index zero and G is a product of unitary groups. 
	Here Λ_0 is the Novikov ring
	
    Λ_0={∑_i=0^∞ a_iT^A_i|a_i∈, A_i∈_≥ 0; A_i  increases to  +∞,},
	
	This allows us to view H^∙_G(;Λ_0) as a graded coefficient ring. We note that the terms of the form ^G_k(x_1⊗ 1,… x_k⊗ 1) are series in the equivariant parameters and receive non-trivial contributions from pearly trees in L_G. 
	
	
	  
	
	
	
	
	
	
	





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

	 
		The ^1-equivariant disc potential of the immersed sphere L_0 is
		
    W^L_0_^1 = log (1-uv) = -∑_j=1^∞(uv)^j/j

		where (u,v) ∈Λ_0^2, (uv)>0 are the formal deformation parameters corresponding to the degree one immersed generators of L_0.
	

	Floer theory for immersed Lagrangians was developed by Akaho-Joyce <cit.>, which is in line with the theory of Fukaya-Oh-Ohta-Ono <cit.> for smooth Lagrangians.  In <cit.>, a different method via Legendrian topology is used to develop the Floer theory of immersed Lagrangian surfaces.
	
	In a subsequent work <cit.>, we compute the equivariant disc potential for Lagrangian immersions in toric Calabi-Yau manifolds.  They have very interesting expressions in relation with the Gross-Siebert slab functions <cit.> and mirror maps of toric Calabi-Yau manifolds <cit.>.  It is closely related to the topological vertex <cit.>.

	
	 
	
	toc
	

 §.§ Acknowledgments

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


§ A MORSE MODEL FOR LAGRANGIAN FLOER THEORY
  
	
	

 §.§ The singular chain model

	 


 
Let (X,ω_X) be a symplectic manifold of real dimension 2d. We assume that X is convex at infinity or geometrically bounded if it is non-compact. Choose a compatible almost complex structure J_X. For a closed, connected, and relatively spin Lagrangian submanifold L, Fukaya-Oh-Ohta-Ono constructed a countably generated subcomplex C^∙(L;Λ_0) of the singular chain complex S^∙(L;Λ_0) (regarded as a cochain complex) and an A_∞-algebra (C^∙(L;Λ_0),).



We briefly recall their construction below, which will be used in constructing the equivariant Morse model in Section <ref>. We refer the reader to <cit.> for more details. 

For β∈ H_2(X,L;), we denote by ℳ(β;L) the moduli space of J_X-holomorphic stable maps from a bordered Riemann surface of genus 0 representing β and by ℳ_k+1(β;L) the moduli space with k+1 boundary marked points z_0,…,z_k ordered counter-clockwise. The moduli space ℳ_k+1(β;L) has virtual dimension μ(β)+k-2+d, where μ(β) is the Maslov index of β.  It carries the evaluation map 

    :ℳ_k+1(β;L)→ L^k,    u ↦ (u(z_1), …, u(z_k)).
 
Let H_2^eff(X,L) denote the monoid of effective classes in H_2(X,L;), represented by J_X-holomorphic maps, i.e.

    H_2^eff(X,L)= {β∈ H_2(X,L;) |ℳ(β;L)∅}.




For a k-tuple P⃗=(P_1,…,P_k) of smooth singular chains of L, we denote by ℳ_k+1(β;L;P⃗) the fiber product (in the sense of Kuranishi structures) of ℳ_k+1(β;L) with P_1, …, P_k, i.e.,

    ℳ_k+1(β;L;P⃗)=ℳ_k+1(β;L)×_L^k∏ P_i.


We recall a generation g ∈ℤ, which will be used to determine perturbations of moduli spaces and subsets of singular chains inductively in <cit.>.
As preliminaries, for β∈ H_2^eff(X,L), we first set

    β=sup{n |∃β_1,…,β_n∈ H_2^eff(X,L)∖{β_0} with ∑_i=1^n β_i=β}+ω(β)-1    if ββ_0 ; 
    
    -1    if β=β_0

where β_0 is the constant disc class.
Also, we employ a function 𝔡:{1,…,k}→_≥ 0 to keep track of generations of inputs.
For a pair (𝔡,β) with β∈ H_2^eff(X,L), we define

    (𝔡,β)=max_i∈{1,…,k}{𝔡(i)}+β+k    if  k 0;
    β   if  k=0.
	


For a generation g, we shall inductively choose a countable set 𝒳_g(L) of smooth singular chains on L and a system of multisections _,k,β,P⃗ for ℳ^_k+1(β;L;P⃗) satisfying (𝔡,β)=g. Here the superscript  is written to emphasize the generations of the inputs P_i∈𝒳_(i)(L).
Namely, at each inductive step, new multisections _,k,β,P⃗ are chosen to be transversal to the zero section and extend the multisections previously chosen for the boundary strata ∂ℳ_k+1(β;L;P⃗). (To achieve transversality, the Kuranishi structures for the moduli spaces _k+1(β;L) are chosen to be weakly submersive.) Moreover the zero locus

    ℳ^_k+1(β;L;P⃗)^_,k,β,P⃗:=_,k,β,P⃗^-1(0)

is triangulated extending the triangulation on its boundary strata. The new simplices in the triangulation are then regarded as elements of 𝒳_g(L).  Additional singular simplices are added to 𝒳_g(L) so that 

    C_(g)(L;Λ_0)=⊕_g'≤ gΛ_0 ·𝒳_g'(L)

remains a subcomplex of S^∙(L;Λ_0) isomorphic on cohomology.


Let C^∙(L;Λ_0)=lim_→ C_(g)(L;Λ_0). The A_∞-map _k:C^∙(L;Λ_0)^⊗ k→ C^∙(L;Λ_0) is defined by 

    _k(P_1,…,P_k)=∑_β∈ H_2^eff(X,L)^ω_X(β)_k,β(P_1,…,P_k)

where

    _k,β(P_1,…,P_k)=
    
    0    
    
    (-1)^d ∂ P    
    
    (_0)_*(ℳ_k+1(β;L;P⃗)^)





Here, ∂ is the coboundary map on C^∙(L;Λ_0). The map
_k,β is of degree 2-k-μ(β).






We note that it is only possible to choose finitely many multisections at once while still having them being sufficiently close to the original Kuranishi maps. To deal with this technical difficulty, Fukaya-Oh-Ohta-Ono introduced the notion of A_n,K-structure associated to L using moduli space of pseudo-holomorphic discs with bounded energy and number of marked points in <cit.>. The A_∞-structure  on C^∙(L;Λ_0) is obtained from the A_n,K-structures via homological techniques. 



In order to define a -graded Floer cohomology, the conventional grading of the map 𝔪_k is 2-k. This means aside from the case of graded Lagrangian submanifolds, one has to define the Novikov ring Λ_0 using an extra grading parameter in order to compensate for the Maslov indices. In this paper, we work on the chain level and do not follow this convention. The grading of 𝔪̃_k,β is crucial in understanding the vanishing of certain terms when doing computations with the Morse model.


The singular chain model (C^∙(L;Λ_0),) constructed in <cit.> does not have a strict unit in general. It was shown that the fundamental cycle e of L is a homotopy unit.  We briefly describe key properties and ideas below, and refer the readers to <cit.> for the precise definition of a homotopy unit and to <cit.> for details of the homotopy unit construction.

The constructed A_∞-algebra (C^∙(L;Λ_0),) will be enlarged to a unital A_∞-algebra (C^∙(L;Λ_0)^+,^+) that is homotopy equivalent to C^∙(L;Λ_0).
At first, we enlarge the chain complex C^∙(L;Λ_0) by adding a generator e^+ of degree 0 serving as the strict unit and a generator f of degree (-1) serving as a homotopy between e and e^+, that is,

    C^∙(L;Λ_0)^+=C^∙(L;Λ_0)⊕Λ_0 ·e^+ ⊕Λ_0 ·f.

Then the A_∞-maps ^+ = {^+_k }_k≥ 0 are defined with the following properties


  * The restriction of ^+ to C^∙(L;Λ_0) agrees with ,

  * e^+ is the strict unit, i.e.


  * ^+_2(e^+,x)=(-1)^|x|^+_2(x,e^+)=x for x∈ C^∙(L;Λ_0)^+,

  * ^+_k(…,e^+,…)=0 for k 2.


  * f is a homotopy between e and e^+.




























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

A construction and properties of the homotopy f are in order. 


Let a⃗= (a_1,…,a_|a⃗|) be an ordered subset of {1,…,k} satisfying a_1< … < a_|a⃗|. 
For P⃗=(P_1,…,P_k-|a⃗|) with P_i∈𝒳_(i)(L), let P⃗^+ be the k-tuple obtained by inserting e into the a⃗-th places of P⃗. We  assume L∈𝒳_0(L). We set ^+(i)=g_i^+ if P⃗^+=(P_1^+,…,P_k^+) where P^+_i∈𝒳_g^+_i(L).
Consider the forgetful map

    _a⃗:ℳ_k+1(β;L;P⃗^+)→ℳ_k+1-|a⃗|(β;L;P⃗),

which forgets the a⃗-th marked points and then stabilizes.

We choose a perturbation on [0,1]^|a⃗|×_k+1(β;L;P⃗^+) as follows: 
For a splitting a⃗^1∐a⃗^2=a⃗ of a⃗, we denote by P⃗' the (k-|a⃗^1|)-tuple given by removing e from the a⃗^1-th places of P⃗^+.  Let (t_1,…,t_|a⃗|) be coordinates on [0,1]^|a⃗|. If t_i=0 for all i∈a⃗^1, we take the perturbation _^+,k,β,P⃗^+ transversal to the zero section obtained by just inserting e into the a_1,…,a_|a⃗^1|-th place. If t_i=1 for all i∈a⃗^1, we take the perturbation pulled back via the map _a⃗^1:ℳ_k+1(β;L;P⃗^+)→ℳ_k+1-|a⃗^1|(β;L;P⃗') in (<ref>). 

Finally, we take a perturbation _^+,k,β,P⃗^+^+ on [0,1]^|a⃗|×_k+1(β;L;P⃗^+) transversal to the zero section interpolating between them. 

The maps ^+_k,β for (k,β)( (1,β_0)) with inputs f inserted into a⃗-th place of P⃗ are defined by

    (_0)_*(([0,1]^|a⃗|×_k+1(β;L;P⃗^+))^_^+,k,β,P⃗^+^+/∼)

where ∼ is the equivalence relation collapsing fibers of _a⃗, see <cit.>. Notice that the zero locus of a pullback multisection is a degenerate singular chain which becomes zero in the quotient.


We also set ^+_1,β_0(f)=e^+-e
and therefore

    ^+_1(f)=e^+-e+h,

where

    h=∑_ββ_0T^ω_X(β) (_0)_* (([0,1]×_2(β;L;e))^^+/∼).

Note that h≡ 0 Λ_+ · C^∙(L;Λ_0) and the terms 

    ∑_μ(β)≥ 2 (_0)_* (([0,1]×_2(β;L;e))^^+/∼)

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


	
	
	
	
	
	

	
	
	
	
	
	
	
	
	
	
	
	
	
	
	

	 
	
	
	
	
	
	
	
 	


    
    
    
    
    
     
    
    
    
    
    
    
    
    
    
    
    
    
    
        
        
        
    
    
    
    
    
    
     
     
    

	
		
	
	
	
	
	
	
	
	
	
	
    
    
    
    
    
    
	
	
	
	
	
	
	
	

	

	
	
	
	
	

	

	
	
   	
    


	



 §.§ Morse homology = singular homology: an alternative approach


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

A Morse model of Lagrangian Floer theory was constructed in <cit.> based on their singular chain model.  

We follow their construction to construct a Morse model in Section <ref>, with a modification that we add certain degenerate chains as summands in the definition of unstable chains Δ_p of critical points p and the chains of forward orbits G(τ) of singular chains τ. 
The main reason for the modification is that there are unwanted degenerate chains appeared in ∂Δ_p and ∂ G(τ), and we `contract them away' by adding the cones over the unwanted terms.
This modification enables us to realize a Morse complex as a singular chain complex, which will be explained in this section. 




Let f:L→ be a Morse function. Let V be a negative pseudo-gradient vector field for f, i.e. df(V)|_p≤ 0 and the equality holds if and only if  p∈Crit(f). For each  p∈Crit(f), V coincides with the negative gradient vector field for the Euclidean metric on a Morse chart of p. Let Φ_t be the flow of V. For each p∈Crit(f), we denote by W^s(f;p) and W^u(f;p) its stable and unstable manifolds respectively. Namely, 

    W^s(f;p)={q∈ L |lim_t→ +∞Φ_t(q)=p}   W^u(f;p)={q∈ L |lim_t→ -∞Φ_t(q)=p}.





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

    |p|:=d-ind(p),
 
where ind(p) is the Morse index of p. Then
|p|=W^s(f;p)=codim   W^u(f;p). 
Let C^∙(f;Λ_0) be the cochain complex 

    C^∙(f;Λ_0)=⊕_p∈Crit(f)Λ_0 · p,
 
with grading given by |p|.

We assume that V satisfies the Smale condition and call such a pair (f,V) Morse-Smale. For p,q∈(f), let ℳ(p,q) be the moduli space of flow lines from p to q. The Smale condition implies 

    ℳ(p,q)=|q|-|p|-1.

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

    ℳ(p,q)_k=⋃_r_0=p,r_1,…,r_k,r_k+1=q  distinctℳ(r_0,r_1)×…×ℳ(r_k,r_k+1)

and

    W^u(f;p)_k=⋃_r_0=p,r_1,…,r_k  distinctℳ(r_0,r_1)×…×ℳ(r_k-1,r_k)× W^u(f;r_k).

In particular, we have

    ∂ℳ(p,q)=⋃_r(p,r)×(r,q)   ∂W^u(f;p)=⋃_rℳ(p,r)× W^u(f;r).




In <cit.>, a (non-unital) A_∞-algebra structure was constructed on C^∙(f;Λ_0) assuming L has a triangulation whose simplices are the closures of W^u(f;p). 
To establish an isomorphism between Morse and singular cohomology, we have to associate to each critical point p of degree |p| a singular chain Δ_p∈ S^|p|(L;). A natural candidate for Δ_p is a fundamental chain for W^u(f;p). 
If we make such a natural choice, several issues occur. 

Suppose that we choose such a natrual chain Δ_p. From (<ref>), one can see that in general W^u(f;p) has boundary components of the form ℳ(p,r)×W^u(f;r) with |r|≥ |p|+2. Since ℳ(p,r)≥ 1, and the image of ℳ(p,r)×W^u(f;r) in L is supported on W^u(f;r), the facet of Δ_p representing ℳ(p,r)×W^u(f;r) is a degenerate chain on L. Thus, for the assignment p↦Δ_p to be a chain map, one should deal with degenerate chains.

On the other hand, 𝒳_g(L) are constructed including degenerate chains since the _k product of degenerate chains may no longer be a degenerate chain. 



We overcome this difficulty by adding certain degenerate chains to the fundamental chains (of the compactified unstable submanifolds) and define a map ι:C^∙(f;)→ S^∙(L;) by ι(p)=Δ_p accordingly such that it is a chain map which induces an isomorphism on cohomology, see Figure <ref>. 

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

    d_p∈ C_d-|p|^□(W^u(f;p);)    m_p,q∈ C_|q|-|p|-1^□(ℳ(p,q);)
 
representing the fundamental classes of W^u(f;p) and ℳ(p,q), respectively. Their boundaries are given as 

    ∂ d_p=∑_r (-1)^|r|-|p|-1 m_p,r× d_r   ∂ m_p,q=∑_r (-1)^|r|-|p|-1 m_p,r× m_r,q.


To regard d_p as simplicial singular chains, we choose triangulations for (the domain of) d_p and m_p,q inductively on ind(p)=d-|p|. For any critical point p with ind(p)=0, d_p and m_p,q are 0-simplices. Suppose that we have chosen triangulations for d_p and m_p,q for any p with ind(p)<ℓ. For ind(p)=ℓ, m_p,q with |q|=|p|+1 are 0-simplices. Suppose we have chosen triangulation for m_p,q for |q|-|p|=1,…,m', m'<m. For |q|-|p|=m, we first triangulate ∂ m_p,q as follows: each m_p,r× m_r,q in ∂ m_p,q is a product of simplicial complexes. We choose linear orders on vertices of m_p,r and m_r,q (compatible with their orientations). Then there exists a unique triangulation m_p,r⊗ m_r,q of m_p,r× m_r,q such that the vertices of m_p,r⊗ m_r,q are pairs (x,y), where x is a vertex of m_p,r and y is a vertex of m_r,q, and an n-simplex in m_p,r⊗ m_r,q is defined by a set {(x_0,y_0),…,(x_n,y_n))} such that x_0≤…≤ x_n, y_0≤…≤ y_n, {x_0,…,x_n} defines a simplex τ_1 on m_p,r and {y_0,…,y_n} defines a simplex τ_2 on m_r,q with τ_1+ τ_2≥ n. Notice that this is the standard product structure in the category of simplicial sets. We then triangulate m_p,q extending the triangulation on ∂ m_p,q. Finally, we triangulate d_p extending the triangulation on ∂ d_p given by m_p,r⊗ d_r. 



For each m_p,r× d_r in the boundary of d_p, let Cm_p,r
be the (simplicial) cone over m_p,r with vertices

    {vertices of  Cm_p,r}={vertices of  m_p,r}{*},
  
and simplices

    {n -simplices of  Cm_p,r}   ={n -simplices of  m_p,r}
       {cones of  n-1 -simplices of  m_p,r with the vertex {* }},

for n≥ 1. In particular, this means {*} is the vertex of the cone. As a singular chain on L, the map  Cm_p,r→ L is defined by composing the contraction Cm_p,r→ m_p,r with the map m_p,r→ L. Note that for two simplicial complexes A and B, we have C(A⊗ B)=CA⊗ CB.

We construct Δ_p inductively on ind(p) as follows: For ind(p)=0, we set Δ_p=d_p. Suppose we have constructed Δ_p for ind(p)<ℓ. For ind(p)=ℓ, we define

    d_p^=d_p+∑_m_p,r⊗ d_r  m_p,r⊗Δ_r,

and set 

    Δ_p=d_p^+∑_m_p,r⊗Δ_r Cm_p,r⊗Δ_r.

By construction, we have

    ∂Δ_p=∑_|r|=|p|+1 m_p,r×Δ_r.



 
We define a map 

    ι:C^∙(f;)→ S^∙(L;)   ,

where Δ_p is as defined in (<ref>). 
Then the map ι is a chain map which induces an isomorphism on cohomology H^∙(C^∙(f;);δ)≅ H^∙(L;) where δ denotes the Morse differential.



The approach in <cit.> is to mod out by degenerate chains. 
If we were to consider S^∙(L;) modded out by the subcomplex of degenerate chains, a proof can be found therein. Since we do not wish to mod out by degenerate chains, we would have to modify their arguments. 


Before proving Theorem <ref>, let us introduce some terminologies.
We say a smooth singular chain is generic if each face of every simplex intersects the stable submanifolds of the critical points of f transversely. Let C^∙_gen(L;) be the subcomplex of S^∙(L;) generated by the generic singular chains. It is easy to see that C^∙_gen(L;) and S^∙(L;) are isomorphic on cohomology.

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

    ℳ(τ,q)_k=⋃_j=0^k ⋃_p_1,…, p_j,q  distinct(τ(k-j),p_1)×(p_1,p_2)×…×(p_j-1,p_j)×(p_j,q).

For k=1, we have

    ∂ℳ(τ,q)=(∂τ,q)⋃_p q (-1)^τ+ind(q)(τ,p)×(p,q).
 

We define the forward orbit ℱ(τ) of τ to be the set
ℱ(τ):=[0,∞)×τ
with the map 

    e:ℱ(τ)→ L

where Φ_t is the flow of the negative pseudo-gradient vector field V. 
The orbit ℱ(τ) has a natural compactification to a smooth manifold with corners ℱ(τ), whose codimension k strata are

    ℱ(τ)_k=ℱ(τ(k))⋃_j=1^k⋃_r_1,…, r_j,q  distinct(τ(k-j),r_1)×(r_1,r_2)×…×(r_j-1,r_j)× W^u(r_j).

When k=1, we have

    ∂ℱ(τ)= -τ∪ -ℱ(∂τ) ∪⋃_p∈(f)(τ,p)× W^u(p).


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


	The assertion that ι is a chain map follows from (<ref>). To show that ι induces an isomorphism on cohomology, we modify the proof in <cit.>. 
	
	
	
	
	We have a chain map A:C^∙_gen(L;)→ C^∙(f;) defined by
	
    A(τ)=∑_ind(q)=τ♯ℳ(τ,q)· q.

 In particular, ℳ(τ,p) is empty when τ is a degenerate simplex. Clearly, A∘ι=id:C^∙(f;)→ C^∙(f;). 
 
We define a homotopy G:C^∙_gen(L;)→ S^∙-1(L;) as follows. Let m_τ,q be a smooth cubical singular chain in C^□_τ-ind(q)((τ,q);) representing the fundamental class of (τ,q)  satisfying 

    ∂ m_τ,q= m_∂τ,q+∑_p q (-1)^τ+ind(q) m_τ,p× m_p,q.

We again triangulate m_τ,q inductively on the dimension to regard them as simplicial singular chains. We also choose smooth singular chains F(τ) representing the fundamental class of ℱ(τ) satisfying 

    ∂ F(τ)=-τ-F(∂τ)+∑_r m_τ,r× d_r.

We then construct G(τ) inductively on τ by first replacing m_τ,r× d_r with m_τ,r×Δ_r on ∂ F(τ) and then gluing Cm_τ,r×Δ_r to the boundary of the resulting chain. This ensures that G is a chain homotopy between ι∘ A and the identity, i.e., G satisfies

    Δ_A(τ)-τ= ∂ G(τ)+G(∂τ).

as desired.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
	
	
	
	




 §.§ Morse model with a strict unit



In this section, by adapting the homotopy unit construction in <cit.> (see also Charest-Woodward <cit.>) and using Theorem <ref>, we will explain how to construct a unital A_∞-algebra on a pearl complex. 




For simplicity, we will always assume f has a unique maximum point , so that Δ_=e is the fundamental cycle. 
Let

    CF^∙(L;Λ_0)=C^∙(f;Λ_0)⊕Λ_0 ·⊕Λ_0 ·

with ||=0 and ||=-1. 
The superscripts ▿, ▾ and ▾ are borrowed from <cit.>.  We extend the Morse differential δ to  CF^∙(L;Λ_0) by setting 

    δ()=0,   δ()=(-1)^d (-).


We now construct a unital A_∞-algebra structure on CF^∙(L;Λ_0). Let

    𝒳_-1(L)={Δ_p | p∈(f)}.






For g≥ 0, suppose 𝒳_g'(L) had been constructed for all generations g' (<g). The perturbations  _,ℓ,β,P⃗ and ^+_^+,k,β,P⃗^+ in (<ref>) and (<ref>) can be chosen to have the following properties: Let τ be any face of either a simplex in the triangulation of
	
    (_0)_*(ℳ_ℓ+1(β;L;P_1,…,P_ℓ)^_,ℓ,β,P⃗)

	or a simplex in the triangulation of 
	
    (_0)_*([0,1]^|a⃗|×ℳ_k+1(β;L;P⃗^+)^^+_^+,k,β,P⃗^+) .




  * τ is transversal to the stable submanifold W^s(f;p) for all p∈(f)











  * For each τ of dimension at most d, there exists at most one critical point p(τ)∈(f) such that W^s(f;p(τ)) is of complementary dimension to τ and  intersects τ at a unique point.



Denote by 𝒳^∘_g(L) the set of those singular simplices τ. Define Π(τ)=ϵΔ_p(τ) where ϵ=± 1 is given by

    τ×_L W^s(f;p(τ))=ϵ W^u(f;p(τ)) W^s(f;p(τ))

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

We further add the simplices G(τ) (defined in the proof of Theorem <ref>) for every τ∈𝒳^∘_g(L) to 𝒳_g(L). 
For chains of the form G(τ), we put G(G(τ))=0. The maps  Π: C^∙(L;Λ_0)→ C^∙_(-1)(L;Λ_0) and G:C^∙(L;Λ_0)→ C^∙-1(L;Λ_0) obey 

    Π(τ)-τ= (∂ G(τ))+G(∂τ)),

for τ∈ C^∙(L;Λ_0).

Set 

    C^∙_(-1)(L;Λ_0)^+=C^∙_(-1)(L;Λ_0)⊕Λ_0 ·e^+ ⊕Λ_0 ·f.

We recall that e^+ is the unit and f is a homotopy between e^+ and the fundamental class e. We extend the coboundary operator ∂ to C^∙(L;Λ_0)^+ by setting 

    ∂e^+=0, 
    ∂f=(-1)^d(e^+ -e).
 
We then extend Π and G to maps Π: C^∙(L;Λ_0)^+→ C^∙_(-1)(L;Λ_0)^+ and G:C^∙(L;Λ_0)^+→ C^∙-1(L;Λ_0)^+ respectively by setting

    Π(e^+)=e,  Π(f)=0 
     
    G(e^+)=-f,   G(f)=0.





so that (<ref>) holds for any τ∈ C^∙(L;Λ_0)^+.

A homological perturbation can be applied to reduce the A_∞-algebra structure on  C^∙(L;Λ_0)^+ in (<ref>) to that on C^∙_(-1)(L;Λ_0)^+ in (<ref>). We identify the latter with CF^∙(L;Λ_0) via the map ι:CF^∙(L;Λ_0)→ C^∙(L;Λ_0)^+ defined by (<ref>) and the following assigments

    ι(p)=Δ_p,   ι()=e^+,   ι()=f.

The resulting unital A_∞-algebra is denoted by (CF^∙(L;Λ_0),).
By following <cit.>, we obtain


	The Morse model (CF^∙(L;Λ_0),) is a unital A_∞-algebra. 
	Moreover, the Morse model (CF^∙(L;Λ_0),) and the singular model (C^∙(L;Λ_0)^+,^+) in (<ref>) are unital homotopy equivalent.


We explicitly describe the A_∞-structure maps of the Morse model  (CF^∙(L;Λ_0),). 

The maps _k are given in terms of maps _Γ and _Γ associated to decorated planar rooted trees.
Let us fix a labeling {β_0,β_1,…} of elements of H_2^eff(X,L) where β_0 is the constant disc class.



	A decorated planar rooted tree is a quintuple Γ=(T,i,v_0,V_tad,η) consisting of 
	
		
  * T is a tree;
		
  * i:T→ D^2 is an embedding into the unit disc; 
		
  * v_0 is the root vertex and ι(v_0)∈∂ D^2;
		
  * V_tad is the set of interior vertices with valency 1;
		
  * η:V(Γ)_int→_≥ 0.
	
	where V(Γ) is the set of vertices; V(Γ)_ext=i^-1(∂ D^2) is the set of exterior vertices and V(Γ)_int=V(Γ)∖ V(Γ)_ext is the set of interior vertices. For k ≥ 0, denote by Γ_k+1 the set of isotopy classes represented by Γ=(T,ι,v_0,η) with |V(Γ)_ext|=k+1 and η(v)>0 if the valency ℓ(v) of v is 1 or 2. In other words, the elements of Γ_k+1 are stable.




We denote by Γ^0∈Γ_2 the unique tree with no interior vertices.
For this tree Γ_0, set

    _Γ_0: CF^∙(L;Λ_0)→ CF^∙(L;Λ_0)    
    _Γ_0:CF^∙(L;Λ_0)→ C^∙(L;Λ_0)^+

where ι is the inclusion map in (<ref>). 





For each k ≥ 0, Γ_k+1 contains a unique element that has a single interior vertex v, which is denoted by Γ_k+1.

We then define

    _Γ_k+1=Π∘^+_k,β_η(v), 
    _Γ_k+1=G∘^+_k,β_η(v).





For general Γ, we cut it at the vertex v closest to v_0 so that Γ is decomposed into Γ^(1),…,Γ^(ℓ) and an interval adjacent to v_0 in the counter-clockwise order. We inductively define

    _Γ=Π∘^+_ℓ,β_η(v)∘ (_Γ^(1)⊗…⊗_Γ^(ℓ))
    _Γ=G∘^+_ℓ,β_η(v)∘ (_Γ^(1)⊗…⊗_Γ^(ℓ)).


Finally, we define the A_∞-map _k:CF^∙(L;Λ_0)^⊗ k→ CF^∙(L;Λ_0) by 




    _k=∑_Γ∈Γ_k+1^ω_X(Γ)_Γ,

where ω_X(Γ)=∑_v∈ V_intω_X(β_η(v)).








The maps _Γ restricted to C^∙(f;Λ_0) are given by counting pearly trees as depicted in Figure <ref> (This observation was made in <cit.>. See also <cit.> for a description of _Γ with general inputs in terms of pearly trees).  
For a decorated tree Γ∈Γ_k+1, the exterior vertices v_0,…,v_k are labeled respecting the counter-clockwise orientation. 
Each edge is oriented in the direction from the k input vertices v_1,…,v_k to the root vertex v_0. 
We denote by e_i the edge attached to v_i and by v^±(e) the vertices such that e is the edge from v^-(e) to v^+(e). For p_1,…,p_k,q∈Crit(f), consider the moduli space ℳ_Γ(f;p_1,…,p_k,q) consisting of the following configurations:

	
  * for each interior vertex v, a bordered stable map u_v representing the class β_η(v) with ℓ(v) boundary marked points. We denote by p(e,v) the marked point corresponding to the edge e attached to v;
	
  * for i=1,…,k, the input edge e_i corresponds to a flow line γ_i from p_i to u_v^+(e_i)(p(e_i,v^+(e_i));
	
  * the output edge e_0 corresponds to a flow line γ_0 from u_v^-(e_0)(p(e_0,v^-(e_0))) to q;
	
  * an interior edge e corresponds to a flow line γ_e from u_v^-(e)(p(e,v^-(e))) to u_v^+(e)(p(e,v^+(e))).


The virtual dimension of ℳ_Γ(f;p_1,…,p_k,q) is given by

    ℳ_Γ(f;p_1,…,p_k,q)=k-2+μ(Γ)-∑_i=1^k |p_i|+|q|,

where μ(Γ)=∑_v∈ V_intμ(β_η(v)).
The map 𝔪_Γ:C^∙(f;Λ_0)^⊗ k→ C^∙(f;Λ_0) is given by

    𝔪_Γ(p_1,…,p_k)=∑_q∈Crit(f)♯ℳ_Γ(f;p_1,…,p_k,q)· q,

where ♯ℳ_Γ(f;p_1,…,p_k,q) is the signed count of the moduli space of virtual dimension 0.











 §.§ Disc potentials in Lagrangian Floer theory
 

In this section, we recall the disc potential of (CF^∙(L;Λ_0),) introduced by Fukaya-Oh-Ohta-Ono <cit.>.


We denote by Λ_+⊂Λ_0 the maximal ideal

    Λ_+={∑_i=0^∞ a_iT^A_i|a_i∈, A_i∈_> 0; A_i  increases to  +∞,}.


Let (A,) be an A_∞-algebra over Λ_0 with the strict unit e_A,
and set 

    A_+={x∈ A | x≡ 0 Λ_+· A }.

The weak Maurer-Cartan equation for an element b∈ A_+ is given by

    ^b_0(1)=_0(1)+_1(b)+_2(b,b)+…∈Λ_0 · e_A.

The condition that b∈ A_+ ensures the convergence of ^b_0(1). A solution b of (<ref>) is called a weak bounding cochain. We denote by 

    ℳ𝒞(A)= {b∈ A^odd_+ |_0^b(1)∈Λ_0· e_A },

the space of weak Maurer-Cartan elements. 
We say an A_∞-algebra A is weakly unobstructed if MC(A) is nonempty, in which case we have (^b_1)^2=0 for any b∈ℳ𝒞(A), thus defining a cohomology theory H^∙(A,^b_1). 

Let us put 

    e^b:=1+b+b⊗ b+….

For an element b∈ℳ𝒞(A), we can define a deformation ^b of the A_∞-structure  by

    ^b_k(x_1,…,x_k)=(e^b,x_1,e^b,x_2,e^b,…,e^b,x_k,e^b),    x_1,…,x_k∈ A.



Now, for (A,)=(CF^∙(L;Λ_0),), we denote by ℳ𝒞(L) the space of odd degree 
weak bounding cochains. In general, one should also consider gauge equivalences between weak Maurer Cartan solutions. However, we omit them here since they will trivial in our examples. 

We say that the deformation of L by b (or simply (L,b)) is unobstructed (resp. weakly unobstructed), if 𝔪^b(1)=0 (resp. b∈ℳ𝒞(L)). In particular, if b=0, we will simply say L is unobstructed.


The following lemma concerns the weakly unobstructedness of (CF^∙(L;Λ_0),). This technique of finding weak bounding cochains in the presence of a homotopy unit was introduced in <cit.> and <cit.>.




Let b∈ CF^1(L;Λ_+). Suppose _0^b=W(b) and the minimal Maslov index of L is nonnegative. Then there exists b^+∈ CF^odd(L;Λ_+) such that _0^b^+(1)=W^(b), i.e., (CF^∙(L;Λ_0),) is weakly unobstructed. In particular, if the minimal Maslov index of L is at least two, then, W^(b)=W(b).



By (<ref>), we have

    _1()=- + Π(h),

where 

    Π(h)=Π(∑_ββ_0
    μ(β)=0^ω(β) (_0)_*(([0,1]×_2(β;L;e))^^+/∼))

is a Λ_+ multiple of . The vanishing of the contribution of higher Maslov index discs follows from the fact that the output singular chains are of degrees at most (-2), whose projection to CF^∙(L;Λ_0) vanishes. Hence we can write (<ref>) as

    _1()=-(1-h),

for some h∈Λ_+. 	

Let us tentatively take b'=b+. We have

    _0^b'(1)=_0(1)+_1(b+)+_2(b+,b+)+…=W(b)+∑_μ(β)=0_1,β^b().

The second equality is due to the vanishing of the _k maps with more than one  as an input since the outputs are of degrees at most (-2). Also, notice that the terms _1,β^b() are of degree 0 and therefore are multiples of .

Using the notation of (<ref>), we can write

    ∑_μ(β)=0_1,β^b()=-(1-h(b))=-(1-h)+∑_μ(β) = 0
    ℓ_1 + ℓ_2 ≥ 1_1+ℓ_1+ℓ_2,β(b^⊗ℓ_1, , b^⊗ℓ_2).

Since there do not exist any gradient flow lines from b to the maximum , the contribution of the terms

    ∑_ℓ_1+ℓ_2≥ 1_1+ℓ_1+ℓ_2,β_0(b^⊗ℓ_1,,b^⊗ℓ_2)

are zero and hence h(b)∈Λ_+.

Now, set b^+=b+W/1-h(b). We have 

    _0^b^+(1)=W(b)+W/1-h(b)(-(1-h(b)))=W(b)/1-h(b)=:W^(b),

which yields b^+∈ℳ𝒞(L).

If the minimal Maslov index of L is at least two, then h(b)=0 (see (<ref>)) and hence W^(b)=W(b).
	

Let ℳ𝒞^(L) the space of elements b∈ CF^1(L;Λ_+) which satisfy

    ^b_0(1)=W(b) for some  W(b)∈Λ_0.

By the above lemma, for each such b, there exists b^+∈ CF^odd(L;Λ_+) such that _0^b^+(1)=W^(b). Then, W^(b) can be viewed as a function on ℳ𝒞^(L).

 
We will call W^L:=W^:ℳ𝒞^(L)→Λ_0 the disc potential of L. 










§ A MORSE MODEL FOR EQUIVARIANT LAGRANGIAN FLOER THEORY
  

There are several approaches to G-equivariant Lagrangian Floer theory for a pair of G-invariant Lagrangians in the existing literature.  For G=_2, Seidel-Smith <cit.> used Floer homology coupled with Morse theory on EG to define G-equivariant Lagrangian Floer homology of exact Lagrangians. 
In <cit.>,  Hendricks-Lipshitz-Sarkar used a homotopy theoretic method to define G-equivariant Lagrangian Floer homology for a compact Lie group  G. Daemi-Fukaya <cit.> defined an equivariant de Rham model using G-equivariant Kuranishi structures developed in <cit.>. Bao-Honda <cit.> defined equivariant Lagrangian Floer cohomology in the case of finite group action via semi-global Kuranishi structures.

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

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



 §.§ The equivariant Morse model


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

    EG(0) = G [hookrightarrow]rd    EG(1) [hookrightarrow]rd    EG(2) [hookrightarrow]rd   ⋯
    
    BG(0) = [hookrightarrow]r    BG(1) [hookrightarrow]r    BG(2) [hookrightarrow]r   ⋯

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


We also consider the cotangent bundles T^*EG(N) and T^*BG(N) equipped with the canonical symplectic forms.
Every symplectic G-action on a symplectic manifold can be naturally lifted to a Hamiltonian G-action  on its cotangent bundle. 
We choose a moment map μ_N:T^*EG(N)→𝔤^* for the Hamiltonian G-action on T^*EG(N) lifted from the G-action on EG(N) where 𝔤^* is the dual Lie algebra of G. 

Since G acts on T^*EG(N) freely, we have a canonical isomorphism

    T^*EG(N) G:=μ_N^-1(0)/G≅ T^*BG(N),

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

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


Let X(N)=X×_G μ_N^-1(0) and set X_G=lim_→X(N). Since G acts freely on μ_N^-1(0), there is a canonical map π:X(N)→ T^*BG(N), which is a fiber bundle with the fiber X. 
Let ι:X→ X(N) be inclusion  of a fiber. By the construction, X(N) is endowed with a symplectic form ω_X(N) and a compatible almost complex structure J_X(N) satisfying 

    ω_X(N)|_X(N-1)=ω_X(N-1),   ι^*ω_X(N)=ω_X,

and 

    J_X(N)|_X(N-1)=J_X(N-1),    Dι∘ J_X=J_X(N)∘ Dι,    Dπ∘ J_X(N)=J_T^*BG(N)∘ Dπ.


Let L(N)=L× _G EG(N) for N = 0, 1, …, which are finite dimensional approximations of the Borel construction L_G=L× _G EG. We note that L(N)→ BG(N) is a fiber bundle with the fiber L. 
Regarding BG(N) as the zero section of T^*BG(N), L(N) is a Lagrangian submanifold of X(N) for each N.
The diagram (<ref>) induces a commutative diagram

    L(0)=L [hookrightarrow]rd    L(1) [hookrightarrow]rd    L(2) [hookrightarrow]rd   ⋯
    
    X(0)=X [hookrightarrow]r    X(1) [hookrightarrow]r    X(2) [hookrightarrow]r   ⋯

where the vertical arrows are embeddings of Lagrangian submanifolds. 

Now, we study the disc moduli bounded by L(N). The following proposition tells us that the image of any stable J_X(N)-holomorphic map must be fully contained in a fiber of the map π. 


	
	The induced map ι_*:H_2(X,L;)→ H_2(X(N),L(N);) restricts to a bijection 
	
    ι_*:H_2^eff(X,L) H_2^eff(X(N),L(N)).




	Suppose that there is a J_X(N)-holomorphic map u:(Σ,∂Σ)→ (X(N),L(N)). 
	
    (Σ, ∂Σ) [r,  "u"] [rd, "π∘ u" ']       (X(N), L(N)) [d, "π"]  
         (T^*BG(N), BG(N))

Then the composition π∘ u is a J_T^*BG(N)-holomorphic disc in T^*BG(N) bounded by the zero section BG(N). Since BG(N) is an exact Lagrangian of T^*BG(N), it does not bound any non-constant pseudo-holomorphic discs. It implies that π∘ u is necessarily constant and (u) is contained in a fiber of π over BG(N).


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


	The Maslov index of β∈ H_2^eff(X,L) is equal to the Maslov index of ι_*β∈ H_2^eff(X(N),L(N)). 


By abuse of notation, we denote the disc classes β∈  H_2^eff(X,L) and ι_*β∈ H_2^eff(X(N),L(N)) both by β. We also denote their Maslov index by μ(β) and symplectic area by ω(β).


	
	A J_X(N)-holomorphic disc u:(Σ,∂Σ)→ (X(N),L(N)) is regular if it is regular as a disc in the corresponding fiber of π.
	



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

	where ∂̅ is the linearized Cauchy-Riemann opeartor at u. Pulling the following exact sequences
	
    0→ TX→ TX(N)→ TT^*BG(N)→ 0

	and
	
    0→ TL→ TL(N)→ TBG(N)→ 0
	
	via u, we choose splittings
	
    E=E_1⊕ E_2:=u^*TX⊕ (π∘ u)^*T(T^*BG(N))

	and
	
    F=F_1⊕ F_2:= (∂ u)^* TL⊕ (π∘∂ u)^* TBG(N),

	where F_i⊂ E_1 and F_2⊂ E_2 are subbundles. This gives a decomposition of (<ref>)
	
    A^0(E_1,F_1)⊕ A^0(E_2,F_2) A^1(E_1)⊕ A^1(E_2).

	By Proposition <ref>, π∘ u is a constant map and hence ∂̅_2 is surjective. Therefore, ∂̅ is surjective if and only if ∂̅_1 is surjective. 







	


Our heuristic definition of a G-equivariant Lagrangian Floer theory (CF_G^∙(L;Λ_0),^G) is the (ordinary) Lagrangian Floer theory of the Lagrangian submanifold L_G⊂ X_G. However, since L_G is infinite dimensional, we resort to using finite dimensional approximations {(CF^∙(L(N);Λ_0), {_k^N})}_N∈ associated to a sequence of Morse-Smale pairs {(f_N,𝒱_N)}_N∈ on {L(N)}_N∈. Clearly, arbitrary choices of {(f_N,𝒱_N)}_N∈ and Kuranishi perturbations would not suffice for this purpose. We begin by specifying our choices of Morse-Smale pairs.


 
We call a sequence of Morse-Smale pairs {(f_N,𝒱_N)}_N∈ admissible if it satisfies the following:	
	
  * 
		For each N∈, there is an inclusion of critical point sets (f_N)⊂(f_N+1). Under this identification, we have 
		
			
  * For p∈(f_N),  W^u(f_N+1;p)×_L(N+1)L(N)=W^u(f_N;p). 
			
  * For p∈(f_N), the image of W^s(f_N;p) in L(N)  coincides with W^s(f_N+1;p) in L(N+1).
			
  * For q∈ L(N+1)∖ L(N), we have Φ_t(q)∉ L(N) for all t≥ 0. This implies 
			
    W^u(f_N+1;p)×_L(N+1)L(N)=∅

			for p∈(f_N+1)∖(f_N).
			
			
		
  *  
		For each ℓ≥ 0, there exists an integer N(ℓ)>0 such that |p|>ℓ for all N≥ N(ℓ) and p∈(f_N)∖(f_N-1).	
		
		
  * The Morse function f_N has a unique maximum _L(N) and the inclusion (f_N)⊂(f_N+1) identifies _L(N) with _L(N+1). This allows us to identify CF^∙(L(N);Λ_0) with a subcomplex of CF^∙(L(N+1);Λ_0). We will denote _L(N), _L(N) and _L(N) simply by , , and . 			
	


We explain below how such a sequence of Morse-Smale pairs {(f_N,𝒱_N)}_N∈ can be produced. This is inspired by the family Morse theory of Hutchings <cit.>. 

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



We choose E(U(k)) to be the infinite Stiefel manifold V_k(^∞)=lim_→ V_k(^k+N) and B(U(k)) to be the infinite Grassmannian Gr(k,^∞)=lim_→ Gr(k,^k+N). 
We have an embedding of Gr(k,^∞) into the skew-Hermitian matrices on ^∞ by identifying V∈ Gr(k,^∞) with the orthogonal projection P_V onto V. Let A be the diagonal matrix with entries {1,2,3,…}. The map 

    f_Gr(k,^∞):Gr(k,^∞)→,    f_Gr(k,^∞)(V)= - ((AP_V)),

is a perfect Morse function (see <cit.>). We note that f_Gr(k,^∞) has the following properties:

	
  * The restriction 
	
    f_Gr(k,^k+N)=f_Gr(k,^∞)|_Gr(k,^k+N)

	of f_Gr(k,^∞) to each finite dimensional stratum is again a perfect Morse function.
	
	
  * For all p∈ Gr(k,^k+N) and q∈ Gr(k,^k+N+1)∖ Gr(k,^k+N), we have 
	
    f_Gr(k,^k+N+1)(q)≤ f_Gr(k,^k+N)(p).

	
	
  * If a critical point p of f_Gr(k,^∞) is contained in Gr(k,^∞)∖ Gr(k,^k+N), then |p|> 2N.



We fix an embedding G→ U(k) of Lie groups for some k, and choose   EG=V_k(^∞) and BG=V_k(^∞)/G. This gives a fibration π:BG→ Gr(k,^∞) with fibers the homogeneous space U(k)/G. Let {U_i}_i∈ I be an open cover such that each U_i is contractible, and if p∈(f_Gr(k,^∞)) and p∉ U_i, then p∉U_i. Let {ρ_i}_i∈ I be a partition of unity subordinate to {U_i}_i∈ I. Notice that ρ_i are constant near the critical points of f_Gr(k,^∞) due to our second assumption on the open cover. 

Let f_U(k)/G: U(k)/G→ be a Morse function and define F_U(k)/G:BG→ by

    F_U(k)/G = ∑_i∈ Iρ_i f_U(k)/G.

Here ρ_i and f_U(k)/G are understood as defined on the local trivializations U(k)/G× U_i. 
For each N∈, the restriction of F_U(k)/G,N:=F_U(k)/G|_BG(N) to generic fibers of BG(N)→ Gr(k,^k+N) is a Morse function. In particular, the restriction of F_U(k)/G,N to each fiber over a critical point of f_Gr(k,^k+N) agrees with f_U(k)/G.

We set 

    f_BG(N)=π^*f_Gr(k,^k+N)+ϵ F_U(k)/G,N,   ϵ>0.

Since Gr(k,^k+N) is compact, for sufficiently small ϵ, f_BG(N) has the critical point set 

    (f_BG(N))=  (f_U(k)/G) ×(f_Gr(k,^k+N)) .

Moreover, f_BG(N) is a Morse function. The non-degeneracy of critical points follows from the fact that ρ_i are constant near the critical points of f_Gr(k,^∞). 

Now, let V_Gr(k,^∞) be a negative pseudo-gradient vector field for f_Gr(k,^∞) such that (f_Gr(k,^∞),V_Gr(k,^∞)) is a Morse-Smale pair, and let V_U(k)/G be a fiberwise negative pseudo-gradient for F_U(k)/G. We set

    V_BG= V_U(k)/G+HV_Gr(k,^∞),
 
where H denotes the horizontal lift with respect to an auxiliary connection for the fiber bundle BG→ Gr(k,^∞). For each N∈, we put V_BG(N)=V_BG|_BG(N). Then, for a generic choice of V_U(k)/G, the pairs {(f_BG(N),V_BG(N))}_N∈ are Morse-Smale.

Finally, let f:L→ be a Morse function. By iterating the construction in the above paragraphs, we obtain Morse-Smale pairs {(f_N,𝒱_N)}_N∈ with 

    (f_N)=(f)×(f_BG(N)).

It follows from the properties of f_Gr(k,^∞) that {(f_N,𝒱_N)}_N∈ is an admissible collection in the sense of Definition <ref>.


We now turn to the choice of Kuranishi structures on the moduli spaces _k+1(β;L(N)). Let us first recall that for an element p=[Σ,z⃗,u]∈_k+1(β;L), a Kuranishi chart around u is a quintuple (V_p,E_p,Γ_p,ψ_p,s_p), where Γ_p is the finite automorphism group of (Σ,z⃗,u) acting on the vector bundle E_p → V_p, V_p is a smooth manifold with corners parameterizing smooth stable maps close to u , s_p=∂̅_J_X is a Γ-equivariant smooth section of E_p → V_p, and ψ_p is a homeomorphism from s^-1_p{0}/Γ_p to a neighborhood of p.  

We fix weakly submersive Kuranishi structures for the disc moduli _k+1(β;L) following <cit.>. For N≥ 1, 
since the holomorphic discs with boundary on L(N) are contained in the fibers of X(N)→ T^*BG(N) (over BG(N)), there are no obstruction in the base direction, and therefore we can choose Kuranishi structures to be of the following form: For every N≥ 1 and q∈ BG(N), we choose a contractible neighborhood U_q of q in BG(N) and trivializations X × U_q and L × U_q  compatible with the diagram (<ref>). This gives an trivialization _k+1(β;L)× U_q of _k+1(β;L(N)). Then, for an element p in a fiber of _k+1(β;L)× U_q→ U_q, we can take the (weakly submersive) Kuranishi chart around p to be 

    (V_p× U^k+1_q,E_p⊕ T_qBG(N)^k,Γ_p,(ψ'_p,id),∂̅_J_X(N)),

where ψ'_p is the pullback of ψ_p via the projection s^-1_p(0)/Γ_p× U_q→  s^-1_p(0)/Γ_p 
Then we have 

    _k+1(β;L(N-1))=_k+1(β;L(N))×_L(N) L(N-1),
 
as Kuranishi structures. Moreover, we can orient the moduli spaces using a G-invariant relative spin structure such that the orientations are compatible with the restriction above.



Let {(CF^∙(L(N);Λ_0), {_k^N})}_N∈ be Morse models associated to an admissible collection of Morse-Smale pairs {(f_N,𝒱_N)}_N∈. Let ι_N-1^N:CF^∙(L(N-1);Λ_0)→ CF^∙(L(N);Λ_0) be the  map induced by inclusion of critical points (f_N-1)⊂(f_N). We denote by {_k^N,+} the A_∞-structure maps on the singular chain models  C^∙(L(N);Λ_0)^+ and by {^N_Γ} the maps associated to the decorated planar rooted trees in the definition of {𝔪^N_k}.

The next key proposition enables us to define the A_∞-structure maps of G-equivariant Lagrangian Floer theory.


	
	The singular chain models (C^∙(L(N);Λ_0)^+,^N,+) can be constructed such that the resulting unital Morse models (CF^∙(L(N);Λ_0),^N) satisfy the following property:
	
    ι_N-1^N(𝔪^N-1_Γ(p_1,…,p_k))=^N_Γ(p_1,…,p_k)|_L(N-1)

for p_1,…,p_k∈ CF^∙(L(N-1);Λ_0) and Γ∈Γ_k+1. Here, the (RHS) means that we only consider the outputs of 𝔪^N_Γ in CF^∙(L(N-1);Λ_0).


	
	We proceed by induction on N. For N=0, the statement of the proposition is void. Suppose we have constructed A_∞-algebras (CF^∙(L(N');Λ_0),^N') for all N' with N' < N satisfying (<ref>). We now construct (CF^∙(L(N);Λ_0),^N) satisfying (<ref>) as follows:
	
	Let 𝒳_g'(L(N))⊂𝒳_g(L(N)) be the subset
	
    𝒳_g'(L(N))={P∈𝒳_g(L(N)) | P×_L(N) L(N-1)∅}.
 
	Inductively on g≥ -1, we construct the set of singular simplices 𝒳_g(L), and choose perturbations _,β,ℓ,P⃗_N, (,β)=g, and _^+,k,β,P⃗^+_N^+, (^+,β)=g with following additional properties 
	
	
		
  * There exists a bijection r_g:𝒳_g'(L(N))→𝒳_g(L(N-1)).
		
		
  * For P⃗_N=(P_1,N,…,P_ℓ,N), P_i,N∈𝒳_(i)'(L(N)), and  (i)≤ g, we have
		
    ℳ_ℓ+1(β;L(N);P⃗_N)×_L(N) L(N-1)=ℳ_ℓ+1(β;L(N-1);P_1,N-1,…,P_ℓ,N-1)

		as Kuranishi structures, where P_i,N-1=r_(i)(P_i,N)∈𝒳_(i)(L(N-1)).
		
		
  * For P⃗_N^+=(P_1,N^+,…,P_k,N^+), P_i,N^+∈𝒳_^+(i)'(L(N)) with ^+(i)≤ g, we have
		
    ( [0,1]^|a⃗|×ℳ_k+1(β;L(N);P⃗_N^+)) ×_L(N) L(N-1) 
       =[0,1]^|a⃗|×ℳ_k+1(β;L(N-1);P_1,N-1^+,…,P_k,N-1^+)

		as Kuranishi structures, where P_i,N-1^+=r_(i)(P_i,N^+)∈𝒳_(i)(L(N-1)). In particular, if P_i,N^+=Δ__L(N), then P_i,N-1^+=Δ__L(N-1).
		
		
  * Let P_i,N∈𝒳_(i)'(L(N)) for i=1,…,ℓ. Let _,β,ℓ,P⃗_N-1 be the perturbation chosen at the previous inductive step on N for ℳ_ℓ+1(β;L(N-1);P⃗_N). If (,β)=g, then for each p∈(f_N-1), there is a one-to-one correspondence between the following sets of simplices. The first set is the set of simplices in the triangulation of  
		
    (_0)_*(ℳ_ℓ+1(β;L(N-1);P⃗_N-1)^_,β,ℓ,P⃗_N-1)

		which are of complementary dimension to the stable submanifold W^s(f_N-1;p) in L(N-1) and intersect W^s(f_N-1;p) at a unique point.
		The other set is the set of simplices in the triangulation of 
		
    (_0)_*(ℳ_ℓ+1(β;L(N);P⃗_N)^_,β,ℓ,P⃗_N)

		which are of complementary dimension to the stable submanifold W^s(f_N;p) in L(N) and 
		intersect W^s(f_N;p) at a unique point. Moreover, the intersection points for the corresponding simplices have the same orientation.
		
		
  * The analogue of (4) holds for the perturbation _^+,k,β,P⃗^+_N^+ and the simplices in the triangulation of 
		
    (_0)_*(([0,1]^|a⃗|×ℳ_k+1(β;L(N);P⃗_N^+))^_^+,k,β,P⃗^+_N^+).

	
	These properties together would imply that the resulting Morse model (CF^∙(L(N);Λ_0),^N) satisfies property (<ref>).	

	Denote by Δ_p,N the unstable chain of p∈(f_N) defined as in (<ref>). For g=-1, we have 
	
    𝒳_-1'(L(N))={Δ_p,N| p∈(f_N-1)},

	by Definition <ref> <ref>. We define r_-1:𝒳_-1'(L(N))→𝒳_-1(L(N-1)) by r_-1(Δ_p,N)=Δ_p,N-1.
	
	Let a⃗={a_1,…,a_|a⃗|}⊂{1,…,k}, a_1< … < a_|a⃗|. For p_1,…,p_k-|a⃗|∈(f_N-1), let Δ⃗_N=(Δ_p_1,N,…,Δ_p_k-|a⃗|,N), and let Δ⃗_N^+ be the k-tuple obtained by inserting e=Δ_ into the a_1,…,a_|a⃗|-th places of Δ⃗_N.
	Since any holomorphic disc is contained in a fiber of X(N)→ T^*BG(N) over BG(N) by Proposition <ref>, Definition <ref> <ref> implies 
	
    ℳ_k+1-|a⃗|(β;L(N);Δ⃗_N)×_L(N) L(N-1)=ℳ_k+1(β;L(N-1);Δ⃗_N-1),

	and
	
    ( [0,1]^|a⃗|×ℳ_k+1(β;L(N);Δ⃗_N^+))×_L(N) L(N-1)=[0,1]^|a⃗|×ℳ_k+1(β;L(N-1);Δ⃗_N-1^+),

	as compact subsets of ℳ_k+1-|a⃗|(β;L(N);Δ⃗_N) and [0,1]^|a⃗|×ℳ_k+1(β;L(N);Δ⃗_N^+), respectively. The restriction of the natural Kuranishi structure for the fiber product ℳ_k+1-|a⃗|(β;L(N)) (resp. [0,1]^|a⃗|×ℳ_k+1(β;L(N))) to the (LHS) of (<ref>) (resp. (<ref>)) agrees with the Kuranishi structure of the (RHS) of (<ref>) (resp. (<ref>)). This establishes properties (1)-(3). Notice that properties (4), (5) are void for g=-1. 
	
	Now, suppose we have constructed singular simplices and chosen perturbations satisfying (1)-(5) for g'<g. Let P_i,N∈𝒳_(i)'(L(N)) for i=1,…,ℓ. Suppose (,β)=g. Then (i)<g, and by inductive hypothesis on g'<g, we have  
	
    ℳ_ℓ+1(β;L(N);P⃗_N)×_L(N) L(N-1)=ℳ_ℓ+1(β;L(N-1);P⃗_N-1)

	as Kuranishi structures. We choose perturbation _,β,ℓ,P⃗_N to extend the perturbation _,β,ℓ,P⃗_N-1 and the perturbation for the boundary strata ∂ℳ_ℓ+1(β;L(N);P⃗_N). Note that _,β,ℓ,P⃗_N-1 is compatible with the perturbation for ∂ℳ_ℓ+1(β;L(N);P⃗_N) since the latter is extended from the perturbation for ∂ℳ_ℓ+1(β;L(N);P⃗_N)×_L(N) L(N-1) at previous inductive steps on g. 	
	
	We triangulate  ℳ_ℓ+1(β;L(N);P⃗_N)^_,β,ℓ,P⃗_N extending the triangulation on
 ℳ_ℓ+1(β;L(N-1);P⃗_N-1)^_,β,ℓ,P⃗_N-1 and triangulation on the boundary strata. We can choose the perturbation _,β,ℓ,P⃗_N such that any simplex in this triangulation away from ℳ_ℓ+1(β;L(N-1);P⃗_N-1)^_,β,ℓ,P⃗_N-1
	is transverse to the stable submanifolds of f_N (via the evaluation map _0). Suppose τ^N is a simplex in this triangulation and τ^N×_L(N) L(N-1)∅. By subdividing if necessary, we can assume that τ^N has a face τ^N-1 in ℳ_ℓ+1(β;L(N-1);P⃗_N-1)^_,β,ℓ,P⃗_N-1 and no other faces of τ^N of dimension greater or equal to τ^N-1 are in  ℳ_ℓ+1(β;L(N-1);P⃗_N-1)^_,β,ℓ,P⃗_N-1. Notice that τ^N is not transverse to all stable submanifolds of f_N since the stable submanifolds remain unchanged by Definition <ref> <ref> but the ambient space L(N) is of higher dimension than L(N-1). We can however move the vertices of τ^N so that it is transverse to the stable submanifolds of f_N. We choose the perturbation _^+,k,β,P⃗^+_N^+ and the triangulation for ([0,1]^|a⃗|×ℳ_k+1(β;L(N);P⃗_N^+))^_^+,k,β,P⃗^+_N^+ in a similar manner. 
	
	Finally, we define r_g:𝒳_g'(L(N))→𝒳_g(L(N-1)) by r_g(τ^N)=τ^N-1, and r_g(G(τ^N))=G(τ^N-1).


We are now ready to define a Morse model (CF^∙_G(L;Λ_0), {^G_k }) for the G-equivariant Lagrangian Floer theory of L. 
The equivariant Floer complex CF^∙_G(L;Λ_0) is defined to be the direct limit

    CF^∙_G(L;Λ_0)=lim_→ CF^∙(L(N);Λ_0).

In order to define the A_∞-structure maps {^G_k} on CF^∙_G(L;Λ_0), we first define the maps  {𝔪^G_Γ} for Γ∈Γ_k+1 and k≥ 0 as follows. 
For p_1,…,p_k∈  CF^∙_G(L;Λ_0), there exists a sufficiently large N_0 ∈ℕ such that p_1,…,p_k∈ CF^∙(L(N_0);Λ_0)^+. Setting ℓ=∑_i=1^k |p_i|+2-k-μ(Γ), we define 𝔪^G_Γ by 

    𝔪^G_Γ(p_1,…,p_k)=^N(ℓ)_Γ(p_1,…,p_k)

where N(ℓ) (≥ N_0) comes from Definition <ref> <ref>.
Because of Proposition <ref>, (<ref>) is defined independent of the choice of N(ℓ). Finally, we define the maps 𝔪^G_k:CF_G^∙(L;Λ_0)^⊗ k→ CF_G^∙(L;Λ_0) by

    𝔪^G_k,β=∑_Γ∈Γ_k+1,β=∑β_η(v)^ω(β)𝔪^G_Γ,


    𝔪^G_k=∑_β𝔪^G_k,β.


The pair (CF^∙_G(L;Λ_0), 𝔪^G= {^G_k}_k≥ 0) is an A_∞-algebra with a strict unit . For fixed inputs the A_∞-identity can be checked on (CF^∙(L(N);Λ_0),^N) for sufficiently large N.  

We will refer to the pair (CF^∙_G(L;Λ_0), 𝔪^G) from (<ref>) and (<ref>) as the Morse model for the G-equivariant Lagrangian Floer theory (G-equivariant Morse model) of L.




 §.§ Equivariant parameters as homotopy partial units



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

    W_λ=W+∑_i=1^d λ_i x_i,

as the disc potential of the T-equivariant Lagrangian Floer theory of L constructed in Section <ref>. The terms λ_1,…,λ_d are the equivariant parameters generating the cohomology ring

    H^∙_T(pt)=H^∙(BT;)=[λ_1,…,λ_d].


Since λ_1,…,λ_d have cohomological degree 2, the expression (<ref>) of W_λ suggests that the boundary deformations of curvature _0^T(1) are, a priori, obstructed. For this reason, we shall construct in this section an alternative model which is homotopy equivalent to our equivariant Morse model and define its equivariant disc potential. Our construction in this section replies on the existence of a well-behaved perfect Morse function on BG (e.g. (<ref>) for G=U(k)). For this reason, we will temporarily restrict the discussion to the case when G is a product of unitary groups. 

To begin with, we fix a Morse function f on L with a unique maximum point _L. Following the proof of Proposition <ref>, we can choose an admissible sequence of Morse-Smale pairs {(f_N,𝒱_N)}_N∈ such that f_N is of the form

    f_N=π_N^*φ_N+ϕ_N

where 


  * φ_N is a perfect Morse function on BG(N). 

  * ϕ_N is a (generically) fiberwise Morse function for L(N)→ BG(N).  In particular the restriction of ϕ_N to each fiber over a critical point of φ_N agrees with ϵ f for some ϵ>0. 

Since the choice (<ref>) satisfies (<ref>), we have

    C^∙(f_N;Λ_0)=C^∙(f;Λ_0)⊗_Λ_0 H^∙(BG(N);Λ_0).

We then enlarge C^∙(f_N;Λ_0) fiberwise over (φ_N) to

    CF^∙(L(N);Λ_0)^†=CF^∙(L;Λ_0)⊗_Λ_0 H^∙(BG(N);Λ_0),

where

    CF^∙(L;Λ_0)=C^∙(f;Λ_0)⊕Λ_0 ·_L ⊕Λ_0 ·_L

as in (<ref>). 









We shall produce a unital A_∞-algebra structure 𝔪^G,† on

    CF^∙_G(L;Λ_0)^†=lim_→ CF^∙(L(N);Λ_0)^†=CF^∙(L;Λ_0)⊗_Λ_0 H^∙_G(pt;Λ_0).

over the graded coefficient ring H^∙_G(pt;Λ_0) via finite dimensional approximations.  
Set


    C^∙(L(N);Λ_0)^†=C^∙(L(N);Λ_0)⊕(⊕_λ∈ (φ_N) (Λ_0 ·⊕Λ_0 ·)).

Here, the singular chains in χ_-1(L(N))⊂ C^∙(L(N);Λ_0) are the unstable chains of the critical points of f_N. We denote by  the unstable chain of the critical point (_L,λ). The generators  and  are of degrees ||=|| and ||=||-1. We note that  and  are of even degrees while  are of odd degrees.

By using the idea of the homotopy unit construction in Section <ref>, we construct an A_∞-structure ^N,† on C^∙(L(N);Λ_0)^† satisfying the following properties:







  * The restriction of ^N,† to C^∙(L(N);Λ_0)^+ agrees with ^N,+. In particular, this means if λ∈(φ_N) is the maximum point, then =e is the homotopy unit, =e^+ is the strict unit, and =f is the homotopy between them.


  * For λ_i,λ_j∈(φ_N), let λ_k=λ_i∪λ_j where ∪ denotes the cup product of H^∙(BG(N)). Then

    ^N,†_2(_i,_j)= _k= ^N,†_2(_j,_i),
       ^N,†_2(_i,_j)=_k=^N,†_2(_j,_i),
       ^N,†_2(_i,_j)=_k=-^N,†_2(_j,_i).


  * Let λ∈(φ_N) and let P∈ C^∙(L(N);Λ_0)^† be a singular chain. Then we have

    ^N,†_2,β(,P)=
    
    0     
    ^N,†_2,β_0(,P)     ,
    
       ^N,†_2,β(P,)=
    
    0     
    ^N,†_2,β_0 (P,)     .











  * For k 2,

    ^N,†_k(…,,…)=0.



  *  is a homotopy between  and  in the following sense.


Let a⃗=(a_1,…,a_|a⃗|) be an ordered subset of {1,…,k} satisfying a_1< … < a_|a⃗|.
For P⃗=(P_1,…,P_k-|a⃗|) a (k-|a⃗|)-tuple of singular chains,
let P⃗^† be the k-tuple obtained by inserting  into the a⃗-th places of P⃗ (by an abuse of notation, 's inserted can be distinct). We choose a perturbation on [0,1]^|a⃗|×_k+1(β;L(N);P⃗^†) as follows: For a splitting a⃗^1∐a⃗^2=a⃗ of a⃗, 
we denote by P⃗' the (k-|a⃗^1|)-tuple given by removing  from the a⃗^1-th places of P⃗^†. Let (t_1,…,t_|a⃗|) be coordinates on [0,1]^|a⃗|. If t_i=0 for all i∈a⃗^1, we consider a perturbation chosen in Proposition <ref> on the moduli space obtained by just inserting  into the a⃗^1-th place. On the other hand, If t_i=1 for all i∈a⃗^1, we take a perturbation pulled back via _a⃗_1: _k+1(β;L(N);P⃗^†)→_k+1-|a⃗^1|(β;L(N);P⃗'). (We note that the unstable submanifold W^u(f_N,) is the restriction of the fiber bundle L_N→ BG(N) over W^u(φ_N,λ)⊂ BG(N), it therefore makes sense to pullback perturbations.) Finally, we take a perturbation ^† on [0,1]^|a⃗|×_k+1(β;L(N);P⃗^†) interpolating between them. We note that the unstable submanifold W^u(f_N,) is the restriction of the fiber bundle L_N→ BG(N) over the unstable submanifold  W^u(φ_N,λ) in BG(N). Inserting  imposes no constraint in the fiber direction. Therefore it makes sense to pullback perturbations from  _k+1-|a⃗^1|(β;L(N);P⃗') to _k+1(β;L(N);P⃗^†).


The ^N,†_k,β maps for (k,β)  (1,β_0) with inputs  inserted into a⃗-th place of P⃗ are defined by

    (_0)_*(([0,1]^|a⃗|×_k+1(β;L(N);P⃗^†))^^†/∼)

where ∼ is again the equivalence relation collapsing fibers of _a⃗_1. The zero locus of a pullback multisection is a degenerate singular chain which becomes zero in the quotient. 


For (1,β_0), we set ^N,†_1,β_0()=-. Moreover, 

    ^N,†_1()=-+h_λ,

where

    h_λ=∑_ββ_0T^ω_X(β) (_0)_* (([0,1]×_2(β;L(N);))^^†/∼).

Note that h_λ≡ 0 Λ_+· C^∙(L(N);Λ_0)^†.

Now, the homological perturbation as in Theorem <ref> is applied to obtain a Morse model (CF^∙(L(N);Λ_0)^†,^N,†) from the singular chain model (C^∙(L(N);Λ_0)^†,^N,†). 
We finally obtain an A_∞-algebra structure 𝔪^G,† on CF^∙_G(L;Λ_0)^† defined in (<ref>). 


For any λ∈ H^∙_G(pt;Λ_0), we put 

    =_L⊗λ,  =_L⊗λ,   =_L⊗λ.

In particular, for λ=1, we denote

    =_L⊗ 1,  =_L⊗ 1,   =_L⊗ 1.

By our construction, the A_∞-algebra (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) has the following properties: 


  * The restriction of 𝔪^G,† to CF^∙_G(L;Λ_0) coincides with 𝔪^G. In particular,  is the homotopy unit,  is the strict unit, and  is the homotopy between them.


  * For X∈ CF^∙_G(L;Λ_0)^†, let us write X=∑ a_ij x_i⊗λ_j and put

    λ· X=   ∑ a_ij x_i⊗ (λ∪λ_j),
    
    X·λ=   ∑ a_ij x_i⊗ (λ_j∪λ).

Then we have

    𝔪^G,†_2(,X)=λ· X=(-1)^|X|X·λ=(-1)^|X|𝔪^G,†_2(X,)



  * For k  2,

    𝔪^G,†_k(…,,…)=0.





Since the properties (<ref>) and (<ref>) that  satisfies are similar to those of a strict unit, we call  a partial unit and  a homotopy partial unit.















When the minimal Maslov index of L is nonnegative, the elements  and  satisfy

    𝔪^G,†_1()=-(1-h),

for some h∈Λ_+, and

    𝔪^G,†_1()=𝔪^G,†_1(𝔪^G,†_2(,))=-(1-h),

where the last inequality follows from the A_∞-identity.

The equation (<ref>) implies [] = []/(1-h) ∈ HF^0_G(L;Λ_0) (in the weakly unobstructed case so that the equivariant Floer cohomology is well-defined). Since h∈Λ_+,  is a cohomological unit. This is important, for instance when we consider isomorphisms of objects in the Fukaya category.

The following theorem shows that (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) can be defined over the graded coefficient ring H^∙_G(pt;Λ_0), namely

    (CF^∙_G(L;Λ_0)^†, 𝔪^G,†)=(CF^∙(L;H^∙_G(pt;Λ_0)),^G,†).

It is important to point out that the A_∞-structure of the (RHS) is not determined by (CF^∙(L;Λ_0),) since the equivariant parameters can receive nontrivial contributions from the pearly trees in L_G. 










 
Assume that L has non-negative minimal Maslov index. Let X_1,…,X_k∈ CF^∙_G(L;Λ_0)^†. We write X_ℓ=∑ a_ij x_i⊗λ_j for ℓ∈{1,…,k}. Then, we have

    𝔪^G,†_k(X_1,…,X_k)=(-1)^ℓ∑ 
    a_ijλ_j·𝔪^G,†_k(X_1,…,X_ℓ-1,x_i⊗ 1,X_ℓ+1,…,X_k).
	



From (<ref>), it follows that 

    𝔪^G,†_k(X_1,…,X_ℓ-1,x_i⊗λ_j,X_ℓ+1,…,X_k)=𝔪^G,†_k(X_1,…,X_ℓ-1,𝔪^G,†_2(,x_i⊗ 1),X_ℓ+1,…,X_k).

Then, by using (<ref>) and A_∞-identity, one obtains

    𝔪^G,†_k(X_1,…,X_ℓ-1,𝔪^G,†_2(,x_i⊗ 1),X_ℓ+1,…,X_k)
       =(-1)^ℓ-1𝔪^G,†_k(𝔪^G,†_2(,X_1)…,X_ℓ-1,x_i⊗ 1,X_ℓ+1,…,X_k)
       =(-1)^ℓ𝔪^G,†_2(,𝔪^G,†_k(X_1,…,X_ℓ-1,x_i⊗ 1,X_ℓ+1,…,X_k))
       =(-1)^ℓλ_j·𝔪^G,†_k(X_1,…,X_ℓ-1,x_i⊗ 1,X_ℓ+1,…,X_k)

where the last equality follows from (<ref>).


In summary, we have produced the equivariant Morse model (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) as an A_∞-algebra over H^∙_G(pt;Λ_0).



 §.§ Disc potentials in equivariant Lagrangian Floer theory


We now introduce an equivariant disc potential for the equivariant Morse model (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) over the graded coefficient ring H^∙_G(pt;Λ_0). 

For an element b∈ CF^∙_G(L;Λ_+)^†, we define the equivariant weak Maurer-Cartan equation by

    𝔪^G,†,b_0(1)=𝔪^G,†_0(1)+𝔪^G,†_1(b)+𝔪^G,†_2(b,b)+…∈ H^∙_G(pt;Λ_0) ·.

We denote by ℳ𝒞_G(L) the space of odd degree solutions of (<ref>). An element b∈ℳ𝒞_G(L) is called a weak Maurer-Cartan element over H^∙_G(pt;Λ_0). We say that (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) is weakly unobstructed if ℳ𝒞_G(L) is nonempty.














	

The deformation of (L,G) by b (or simply (L,G,b)) is called unobstructed (resp. weakly unobstructed) if 𝔪^G,†,b(1)=0 (resp. b∈ℳ𝒞_G(L)). In particular, if b=0, we simply call (L,G) unobstructed.   






Similar to Lemma <ref>, we have



For b∈ CF^1_G(L;Λ_+), suppose that

    𝔪^G,†,b_0(1)=W(b) +∑_λ =2ϕ_λ(b)

and the minimal Maslov index of L is nonnegative. 
Then there exists b^†∈ CF^odd_G(L;Λ_+)^† such that 

    𝔪^G,†,b^†_0(1)=W^(b)+∑_λ =2ϕ_λ^(b) .
 
In particular, (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) is weakly unobstructed. 

If the minimal Maslov index of L is assumed to be at least two in addition, then W^=W(b) and ϕ_λ^(b)=ϕ_λ(b).



Let us tentatively take b'=b+W(b)+∑_λ =2ϕ_λ(b). We also put b_1=b+W(b),  b_2=W(b)+∑_λ =2ϕ_λ(b), and b_3=b+∑_λ =2ϕ_λ(b). We write 

    𝔪^G,†,b'_0(1)=   ∑_k≥ 0 𝔪^G,†_k(b,…,b) + W(b)𝔪^G,†_1()+ ∑_λ =2ϕ_λ𝔪^G,†_1()
       +∑_k≥ 2( ∑_i=1,2,3𝔪^G,†_k(b_i,…,b_i))-2𝔪^G,†_k(b,…,b)-W(b)^k𝔪^G,†_k(,…,)
       -∑_k≥ 2𝔪^G,†_k(∑_λ =2ϕ_λ(b),…,∑_λ =2ϕ_λ(b)).

It is easy to see that ∑_k≥ 2 W(b)^k𝔪^G,†_k(,…,)=0 since the outputs are of degree at most (-2). On the other hand, by (<ref>) and the proof of Lemma <ref>, we have

    W(b)𝔪^G,†_1()+∑_k≥ 2𝔪^G,†_k(b_1,…,b_1)-𝔪^G,†_k(b,…,b) =W(b)(-(1-h(b)))

for some h(b)∈Λ_+.

Let us now consider the term

    ∑_k≥ 2𝔪^G,†_k(b_3,…,b_3)-𝔪^G,†_k(b,…,b).

By (<ref>), (<ref>) and the A_∞-identity, each term in (<ref>) can be rewritten in the form

    P(λ_1,…,λ_ℓ)·𝔪^G,†_k(…,,…),

where P(λ_1,…,λ_ℓ)∈ H^∙(BG;Λ_0) is a monomial in degree 2 variables and  𝔪^G,†_k(…,,…) has at least one  as an input and remaining inputs are b and . By the degree reason, 𝔪^G,†_k(…,,…)=0 when it has more than one  as inputs. This means (<ref>) equals to

    ∑_k≥ 2, λ =2ϕ_λ(b)𝔪^G,†_k(b,…,b,,b,…,b)=∑_k≥ 2, λ =2 (-1)^δϕ_λ(b)λ·𝔪^G,†_k(b,…,b,,b,…,b),

where δ∈{0,1} depends on the position of . Recall from the proof of Lemma <ref> that 𝔪^G,†_k(b,…,b,,b,…,b) is a Λ_+ multiple of . Since λ·=, 
the (RHS) above is a Λ_+ multiple of . Thus, (<ref>) we have

    ∑_λ =2ϕ_λ𝔪^G,†_1()+∑_k≥ 2𝔪^G,†_k(b_3,…,b_3)-𝔪^G,†_k(b,…,b)=∑_λ =2ϕ_λ(b)(-(1-h_λ(b))),

for some h_λ(b)∈Λ_+. 

Next, we note that the term

    ∑_k≥ 2𝔪^G,†_k(b_2,…,b_2)-𝔪^G,†_k(∑_λ =2ϕ_λ(b),…,∑_λ =2ϕ_λ(b))

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

Finally, let 

    b^†=b+W(b)/1-h(b)+∑_λ =2ϕ_λ(b)/1-h_λ(b).

Then 𝔪^G,†,b^†_0(1) can be expressed as

    𝔪^G,†,b^†_0(1)   =W(b) +∑_λ =2ϕ_λ(b)+W(b)1-h(b)(-(1-h(b)))
       +∑_λ=2ϕ_λ(b)1-h_λ(b)(-(1-h_λ(b)))
       =W(b)1-h(b)+∑_λ =2ϕ_λ(b)1-h_λ(b)=:W^(b)+∑_λ =2ϕ_λ^(b) .

Since both W(b) and ϕ_λ(b) are in Λ_+, we have b^†∈ℳ𝒞_G(L).

If the minimal Maslov index of L is at least two, then h(b)=0, h_λ(b)=0 and therefore W^(b)=W(b), ϕ_λ^(b)=ϕ_λ(b).
	



In the setting of Lemma <ref>, if b∈ C^1(f;Λ_0) and _0^b(1)∈Λ_0 ·, then the equivariant Morse model (CF^∙_G(L;Λ_0)^†, 𝔪^G,†) is weakly unobstructed.



If b∈ C^1(f;Λ_0) and _0^b(1)∈Λ_0 ·, then 𝔪^G,†,b_0(1) is of the form 𝔪^G,†,b_0(1)=W(b) +∑_λ =2ϕ_λ(b).




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

Now, let ℳ𝒞^_G(L) the space of elements b∈ CF^1_G(L;Λ_+) which satisfy

    𝔪^G,†,b_0(1)=W(b) +∑_λ =2ϕ_λ(b)

for some W(b), ϕ_λ(b) ∈Λ. By the Lemma <ref>, for each such b, there exists b^†∈ CF^odd_G(L;Λ_+)^† such that 

    𝔪^G,†,b^†_0(1)=(W^(b)+∑_λ =2ϕ_λ^(b) λ)·,
 
Then, we may view W^L_G(b):=W^(b)+∑_λ =2ϕ_λ^(b) λ as a map from ℳ𝒞^_G(L) to H^∙_G(pt;Λ_0).

 
	We will call W^L_G:=W^:ℳ𝒞^_G(L) → H^∙_G(pt;Λ_0) the equivariant disc potential of (L,G). 










	



§ T-EQUIVARIANT DISC POTENTIALS OF TORIC MANIFOLDS
 
	In this section, we study the equivariant Morse model in the case of a torus T≅ (^1)^ℓ acting on a closed, connected, relative spin Lagrangian submanifold L≅ T× P of product type of a symplectic T-manifold X, such that T acts freely on the first factor of L and trivially on the second factor. In the case L≅ T is a regular moment map fiber of a semi-projective, semi-Fano toric manifold, we recover the equivariant toric superpotential W_λ as the T-equivariant disc potential of L.
	



 §.§ Morse theory on the approximation spaces


  For explicit computations of the equivariant disc potentials, we begin by fixing our choice of admissible Morse-Smale pairs {(f_N,𝒱_N)}_N∈.

		Our choices of the universal bundle ET and the classifying space BT are ET=(^∞)^ℓ and BT=(^∞)^ℓ. We also have the finite dimensional approximations ET(N)= (^2N+1)^ℓ and BT(N)= (^N)^ℓ. Since the T acts trivially on P, we have L_T=T_T× (P× (^∞)^ℓ) and L(N)=T(N)× (P× (^N)^ℓ).
	
	For N≥ 1, let [z_i,0,…,z_i,N] be the homogeneous coordinates on the i-th component of BT(N)=(^N)^ℓ. Let π_N:L(N) → (^N)^ℓ be the projection map. Let {U_j_1… j_ℓ} be the open cover of (^N)^ℓ defined by
	
    U_j_1… j_ℓ={[z_i,0,…,z_i,N]_i=1,…,ℓ∈ (^N)^ℓ| z_i,j_i 0}.

	Set U_j_1… j_ℓ=π_N^-1(U_j_1… j_ℓ)≅ L× (^N)^ℓ. We will be working with the atlas {U_j_1… j_ℓ} for L(N) with local coordinates
	
    ((θ_1^(j_1… j_ℓ),…,θ_ℓ^(j_1… j_ℓ), p⃗^(j_1… j_ℓ)),(z_i,0^(j_1… j_ℓ),…,z_i,j_i^(j_1… j_ℓ)=1,…,z_i,N^(j_1… j_ℓ))_i=1,…,ℓ)

	on U_j_1… j_ℓ, where θ_i^(j_1… j_ℓ)∈ [0,2π) are angular coordinates on (^1)^ℓ, p⃗^(j_1… j_ℓ) is any coordinate system on P, and the term under “  " is omitted. Set ϑ_i,j^(j_1… j_ℓ)=Arg(z^(j_1… j_ℓ)_i,j). The transition map U_j_1… j_ℓ to  U_j'_1… j'_ℓ is given by
	
    ((θ_1^(j_1… j_ℓ),…,θ_ℓ^(j_1… j_ℓ),p⃗^(j_1… j_ℓ)),(z_i,0^(j_1… j_ℓ),…,z_i,j_i^(j_1… j_ℓ)=1,…,z_i,N^(j_1… j_ℓ)))↦
    ((θ_1^(j_1… j_ℓ)+ϑ_1,j'_1^(j_1… j_ℓ),…,θ_ℓ^(j_1… j_ℓ)+ϑ_ℓ,j'_ℓ^(j_1… j_ℓ),p⃗^(j_1… j_ℓ)),(z_i,0^(j_1… j_ℓ)z_i,j'_i^(j_N… j_ℓ),…,z_i,j'_i^(j_1… j_ℓ)z_i,j'_i^(j_1… j_d)=1, …,z_i,j_i^(j_1… j_ℓ)z_i,j'_i^(j_1… j_ℓ)…,z_i,N^(j_1… j_ℓ)z_i,j'_i^(j_1… j_ℓ))).

		
	We fix the inclusion (^N)^ℓ (^N+1)^ℓ to be
	
    ([z_i,0,…,z_i,N])↦([z_i,0,…,z_i,N,0]).

	Which in turn, fixes the inclusions X(N) X(N+1) and L(N) L(N+1).
	
	For U(1)=^1, the perfect Morse function (<ref>) on the infinite complex Grassmanian specializes to 
	
    f_^∞: ^∞→,    f_^∞([z_0,z_1,…]) = -∑_k=1^∞ k|z_k|^2∑_k=0^∞ |z_k|^2.

	From which we obtain a perfect Morse function on BT
	
    φ: (^∞)^ℓ→,   φ([z_i,0,z_i,1,…])=-∑_i=1^ℓ∑_k=1^∞ k|z_i,k|^2∑_k=0^∞ |z_i,k|^2.

	The critical points of φ are of the form
	
    (φ)= { ([0,…,0,z_i,j_i 0,0,…]_i=1,…,ℓ) | (j_1,…, j_ℓ)∈_≥ 0^ℓ}.

	with degrees given by ∑_i=1^ℓ 2j_i. We denote the degree 2 critical points with j_i=1, and j_k=0 for k i by λ_i.  We also denote the critical point where φ attains the maximum by 1_BT, i.e.
	
    1_BT=([z_i,0 0,0,0,…]_i=1,…,ℓ).

	
	We set φ_N=φ|_BT(N). Note that φ_N is a perfect Morse function on (^N)^ℓ. We will abuse notation and denote the degree two critical points and the maximum of φ_N again by 1_BT and λ_i, respectively. Note that we have
	
    H^∙(BT(N);)=H^∙((^N)^ℓ;)=[λ_1,…,λ_ℓ]/⟨λ_1^N+1,…,λ_ℓ^N+1⟩,

	and
	
    H^∙_T(;)=[λ_1,…,λ_ℓ].

	
	On the other hand, let f_P be a Morse function on P with a unique maximum 1_P, and let  f_T: (^1)^ℓ→ be the perfect Morse function 
	
    f_T(θ_1,…,θ_ℓ)=∑_i=1^ℓcos(θ_i).

    The critical points of f_T are of the form
    
    (f_T)={(θ_1,…,θ_ℓ)∈ L |θ_i=0  or θ_i=π}.
   
    Let f=(f_T,f_P) be the Morse function on L. We denote by 1_L the critical point where f attains the maximum, 
    i.e. 1_L=((0,…,0),1_P). We also denote by X_1,…,X_ℓ the degree 1 critical points, and X_i∧ X_j, 1≤ i<j≤ℓ the degree 2 critical points of f of the following forms
    
    X_i=((0,…,θ_i=π,…,0),1_P),

    and
    
    X_i∧ X_j=((0,…,θ_i=π,…,θ_j=π,…,0),1_P).


	Let {D_j} be the open cover of ^∞ given by
	
    D_j={(z_0^(j),…,z_N^(j))∈ U_j||z_k^(j)|<2  for all  k}.

	Let {ρ^(j)} be a smooth partition of unity subordinate to {D_j}, and denote by ρ^(j)_i the post-composition of ρ^(j) with the projection from L_T to the i-th component of (^∞)^ℓ. We define ρ^(j_1… j_ℓ): L_T→ by 
	
    ρ^(j_1… j_ℓ)=∏_i=1^ℓρ^(j_i)_i.

	Let ρ^(j_1… j_ℓ)_N=ρ^(j_1… j_ℓ)|_BT(N), then {ρ^(j_1… j_ℓ)_N} is a smooth partition of unity on L(N).
	
	
	Let ϕ^(j_1… j_ℓ)_N:U_j_1… j_ℓ→ be the fiber-wise Morse function
	
    ϕ^(j_1… j_ℓ)_N((θ_1^(j_1… j_ℓ),…,θ^(j_1… j_ℓ)_ℓ,p⃗^(j_1… j_ℓ)),(z_i,0^(j_1… j_ℓ),…,z_i,N^(j_1… j_ℓ)))=f(θ_1^(j_1… j_ℓ),…,θ^(j_1… j_ℓ)_ℓ,p⃗^(j_1… j_ℓ)),

	and set
	
    ϕ_N=∑_(j_1… j_ℓ)ρ^(j_1… j_ℓ)_N  ϕ^(j_1… j_ℓ)_N.

	Notice that ϕ_N=ϕ^(j_1… j_ℓ)_N for some (j_1… j_ℓ) in a neighborhood of π_N^-1(λ) for  λ∈(φ_N). Then, the function f_N:L(N)→ defined by
	
    f_N:=ϵϕ_N+π_N^*φ_N,   ϵ>0,

	is a Morse function for sufficiently small ϵ. f_N has the critical point set
	
    Crit(f_N)={(p,λ)|p∈Crit(ϕ_N|_π_N^-1(λ)), 
    	λ∈Crit(φ_N)}=Crit(f)×Crit(φ_N).

	
	The degree of (p,λ)∈Crit(f_N) is given by |(p,λ)|=|p|+|λ|,
	where |p| and |λ| are the degrees of p and λ as critical points of f and φ_N, respectively.
	
	Let V_P be a negative pseudo-gradient for f_P. Let V_N be the vector field given by
	
    V_N|_U_j_1… j_ℓ=   4∑_i=1^ℓ∑_j j_i (j-j_i)|z_i,j^(j_1… j_ℓ)|^2∂∂|z_i,j^(j_1… j_ℓ)|^2
       +∑_i=1^ℓ(ρ^(j_i)_N,isin(θ_i^(j_1… j_ℓ))+∑_j j_iρ^(j)_N,isin(θ_i^(j_1… j_ℓ)+ϑ_i,j^(j_1… j_ℓ))) ∂∂θ_i^(j_1… j_ℓ).

	Here ρ^(j)_N,i=ρ^(j)_i|_BT(N). Then
	
    𝒱_N=V_N+V_P

	is a negative pseudo-gradient for f_N. For N=0, we set f_0=f, and 𝒱_0=∑_i=1^ℓsin(θ_i) ∂∂θ_i+V_P.
	
	 
	For generic choices of V_P, (f_N,𝒱_N) is a Morse-Smale pair. Moreover, the sequence {(f_N,𝒱_N) }_N∈ is admissible in the sense of Definition <ref>.
	
	
	
	For p,q∈Crit(f_N), let ℳ(p,q) be the moduli space of gradient flow lines from p to q, modulo reparametrizations.   Let (C^∙(f_N;),δ) to be the Morse cochain complex
	
    C^∙(f_N;):=⊕_p∈Crit(f_N)· p

	We have
	
    C^∙(f_N;)=C^∙(f;)⊗[λ_1,…,λ_ℓ]/⟨λ_1^N+1,…,λ_ℓ^N+1⟩.

   For convenience, we will write the elements of (f_N) as p⊗λ, where p∈(f) and λ∈(φ_N). The following result will be used in the computation of the equivariant disc potentials.
	
	
	
		
		For N≥ 1 and X_1,…,X_ℓ, we have
		
    δ(X_i⊗1_BT)=1_L⊗λ_i.

		In particular, there exists a unique gradient flow line from X_i⊗1_BT to 1_L⊗λ_i.
	
	
	
	The possible outputs of δ(X_i⊗1_BT) are of degree 2 critical points of the form (X_j∧ X_k)⊗1_BT, and 1_L⊗λ_j. It is easy to see that ℳ(X_i⊗1_BT, (X_j∧ X_k)⊗1_BT)=∅ if both j i and k i, and ℳ(X_i⊗1_BT, (X_j∧ X_k)⊗1_BT) consists of two points of opposite orientation if either j=i or k=i. 
		
	Suppose Φ:→ L(N) is a flow line from X_i⊗1_BT to 1_L⊗λ_j. Its projection π_N∘Φ:→ (^N)^ℓ is a flow line for the vector field V on (^N)^ℓ given by 
		
    V|_U_j_1… j_ℓ=4∑_i=1^ℓ∑_j j_i (j-j_i)|z_i,j^(j_1… j_d)|^2 ∂∂ |z_i,j^(j_1… j_ℓ)|^2

		from 1_BT→λ_j, whose image is contained in U_0… 0. 
		This means the image of Φ is contained in Ũ_0… 0, and we can therefore write 
		
    Φ(t)=((θ_1^(0…0)(t),…,θ_ℓ^(0… 0)(t)),(z_μ,1^(0…0)(t),…,z_μ,N^(0…0)(t))_μ=1,…,ℓ),

		in term of the coordinates on Ũ_0… 0. For Φ(t) to have the correct asymptotics, we must have z_μ,ν^(0… 0)(t)=0 for μ i and ν 1; θ^(0… 0)_μ(t)=0 for μ i;  z_ℓ,1^(0… 0)(t)=e^2t+iϑ where ϑ∈ [0,2π);  θ_i^(0… 0)(t) satisfies 
		
    lim_t→ -∞θ_i^(0… 0)(t)=π,

		and
		
    lim_t→ +∞θ_i^(0… 0)(t)+δ_ijϑ=0,

		in addition to 
		
    d θ_i^(0… 0)(t)dt=a(t)sin(θ_i^(0… 0)(t))+b(t) sin(θ_i^(0… 0)(t)+ϑ).

		Here a(t)=ρ^(0)_1|_U_0(e^2t+iϑ) and b(t)=ρ^(1)_1|_U_0(e^2t+iϑ), and {ρ^(0)_1, ρ^(1)_1} is the partition of unity for ^1. 
		
		For existence of a flow line when i=j, we note that the flow line with θ_i^(0… 0)(t)=π and ϑ=0 has the desired asymptotics. As for uniqueness, assume without loss of generality that Φ(0) is in a neighborhood of X_i⊗1_BT such that ρ^(0)_N(t)=1 and ρ^(1)_N(t)=0. In this neighborhood, we have explicitly 
		
    θ_i^(0… 0)(t)=2^-1(e^-t(θ_i^(0… 0)(0)2)).

		It is easy to see that 
		
    lim_t→ -∞θ_i^(0… 0)(t)=
    		π   
    
    		0






		Solving for 
    d θ_i^(0… 0)(t)dt=0
 gives
		
    tanθ_i^(0… 0)(t)=b(t)sin(ϑ)a(t)+b(t)cos(ϑ).

		This means we have
		
    lim_t→ +∞θ_i^(0… 0)(t)+δ_ijϑ= 
    		
    		0    
    π





	
	
	


 §.§ Computing T-equivariant disc potentials


	By applying the construction in Section <ref> and <ref> with the choice of Morse-Smale pairs {(f_N,𝒱_N)} made in Section <ref>, one obtains the equivariant Morse model (CF^∙_T (L;Λ_0)^†, 𝔪^T,†) 
    associated to the pair (L,T). Note that we have
    
    CF^∙_T (L;Λ_0)^†= CF^∙(L;Λ_0)⊗_ Λ_0Λ_0[λ_1,…,λ_ℓ],

    where 
    
    CF^∙(L;Λ_0)=C^∙(f;Λ_0)⊕Λ_0 ·_L ⊕Λ_0 ·_L

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

	For simplicity of notations, we will suppress “" and denote the unique maximum _L on L by 1^_L. Let { X_1,…,X_ℓ,Y_1,…,Y_ν} be a basis of C^1(f;) with {X_i} and {Y_j} degree one critical points of f_T and f_P respectively. We will abuse notations and write X_i:=X_i⊗1_BT and Y_i:=Y_i⊗1_BT. We also put  λ_i:=1_L⊗λ_i, and 1:=1_L⊗1_BT.

	
	Let b=∑_i=1^ℓ x_iX_i+∑_i=1^ν y_i Y_i, (x_i, y_i)∈Λ_+^2. 
We consider the boundary deformation of 𝔪^T,†_0 by b

    𝔪^T,†, b_0 (1) =  𝔪^T, b_0(1) = 𝔪^T_0 (1) +𝔪^T_1(b)+𝔪^T_2(b,b)+⋯.

	The first equality above follows  from the fact that the restriction of  𝔪^T, † to CF^∙_T(L;Λ_0) agrees with 𝔪^T.
	
	We compute the obstruction 𝔪_0^T,b(1) by counting pearly trees in L_T ⊂ X_T[To be more precise, we should perform the calculation on the finite dimensional approximations L(N)⊂ X(N) (see (<ref>), (<ref>) and (<ref>)). By abuse of notations, the A_∞-maps on L_T are regarded as the A_∞-maps on the approximation spaces.]. Since m^T_k,β(b,…,b) is of degree 2-μ(β), the outputs of 𝔪_0^T,b(1) have degree either 0 or 2 depending on the Maslov indices of the contributing disc classes. 
	The possible outputs are of the following forms
	
		
  * (Degree zero) 1, 
		
  * (Degree two) p⊗_BT, where p∈(f) is a degree 2 critical point,
		
  * (Degree two) λ_i. 
	
    Notice that all the critical points above are contained in (f_1). Thus, 𝔪^T, b_0(1) can be computed by counting pearly trees in L(1).
		
 
	Suppose T≅ (𝕊^1)^ℓ acts on (X, ω_X) preserving ω_X. 
	Let L⊂ X be a T-invariant closed Lagrangian submanifold of product type  L≅ T × P such that T acts freely on (𝕊^1)^ℓ  and trivially on P. Suppose the minimal Maslov index of L is nonnegative, then
	
    𝔪_0^T,b(1) = _0^b(1) ⊗1_BT + ∑_i=1^ℓ h_i(x⃗,y⃗)  λ_i
		
where  is the A_∞-structure on  CF^∙(L;Λ_0), and h_i(x⃗,y⃗) ∈Λ_0. Moreover, if _0^b(1) = W(x⃗,y⃗)  _L for some W(x⃗,y⃗)∈Λ_+, then (CF^∙_T (L;Λ_0)^†, 𝔪^T,†) is weakly unobstructed.		
	


	The first two types of outputs are contributed by pearly trees contained in the fiber L over the critical point _BT, and coincide with _0^b(1) by Proposition <ref>. Thus, we have the expression (<ref>). The last assertion follows from Corollary <ref>, namely, if _0^b(1) =W(x⃗,y⃗)_L, then	
	
    𝔪_0^T,b(1) = W(x⃗,y⃗) 1 +∑_i=1^ℓ h_i(x⃗,y⃗)λ_i.

	
	
	In the following, we compute _0^b(1) and h_i(x⃗,y⃗) explicitly under additional assumptions. 
	We begin by simplifying h_i(x⃗,y⃗) under the condition that the minimal Maslov index of L is at least 2.
	
		 
		In the situation of Proposition <ref>, if in addition, the minimal Maslov index of L is at least 2, then h_i(x⃗,y⃗) =x_i in (<ref>).

	
	
	The terms h_i(x⃗,y⃗) are contributed by disc classes of Maslov index 0 by degree reason. 
    Since L has minimal Maslov index at least two, the only contribution comes from the trivial disc class β_0, hence the computation of the terms ^T_k(b,…,b) reduces to ^T_Γ(b,…,b), where Γ∈Γ_k+1 is a stable planar rooted tree with all interior vertices decorated by β_0. In other words, the configurations we are counting are Morse flow trees in L_1.
	
	 By Proposition <ref>, we have ^T_1(X_i) = λ_i. We will show momentarily that the coefficients of λ_i in the terms 𝔪_k^T(X_ν_1,…,X_ν_k) are zero for k≥ 2 and (ν_1,…,ν_k) ∈{1,…,ℓ}^k. First, let us consider the case when ν_j i for some j. In the proof of Proposition <ref>, we have showed that there exists no flow line from X_i_j to λ_i. This means for generic small perturbations, the moduli spaces _Γ(f_1;X_ν_1,…,X_ν_k;λ_i) are empty. 
	 
     Now, we consider ^T_k(X_i,…,X_i), k≥ 2. Since rotations on the i-th circle factor of L commute with the structure group of L_1→^1, we have a global ^1-action on L_1 rotating the i-th circle factor of the fibers L. To achieve transversality for the moduli spaces _Γ(f_1;X_i,…,X_i;λ_i), we can perturb the the flow lines from the first k-1 inputs using the ^1-action. We have the unique flow line γ from X_i to λ_i as in Proposition <ref>. For a perturbed flow line from the first k-1 inputs to intersect with γ (in order to form a flow tree), its projection in BT(1) must coincide with π(γ). However, over π(γ), the flow lines from the first k-1 inputs are simply the rotations of γ by ^1, which do not intersect with γ. Thus, _Γ(f_1;X_i,…,X_i;λ_i) are empty for generic small perturbations.
     
     Similarly, since the fiber bundle P_T=P× BT is trivial, there exists no flow line from Y_j to λ_i for any i,j. Therefore the coefficients of λ_i in the terms ^T_k(b,…,Y_j,…,b) are zero. 
		
	
		Thus, in the setting of Lemma <ref>, we have
		
    𝔪_0^T,b(1) = _0^b (1) ⊗_BT + ∑_i=1^ℓ x^i λ_i.

		
		In <cit.>, it was shown that the moment map fibers of a compact semi-Fano toric manifold are weakly unobstructed in the de Rham model.  The following is the analogous statement in the Morse model. 
		
		
		
			Let L be a regular moment map fiber of a compact semi-Fano toric manifold X. Then _0^b(1)∈Λ_0 ·_L.
		
	
		
		The only possible outputs of _0^b(1) are multiples of _L and degree two critical points of f (which are of the form X_i∧ X_j), contributed by Maslov index zero and two disc classes, respectively. Since X is semi-Fano, L has minimal Maslov index 2. This means only Morse flow trees contribute to the degree two critical points. We have _1(X_i)=0, since f is a perfect Morse function on L. For _k(X_ν_1,…,X_ν_k) and k≥ 2, if there are two repeated inputs X_ν_i=X_ν_j, then by perturbing unstable hypertori of X_ν_i using the ^1-action rotating the ν_i-th circle factor of L, the perturbed hypertori do not intersect and hence _k(X_ν_1,…,X_ν_k)=0. In the case of distinct inputs, we have 
		
    ∑_σ∈ S_k_k(X_ν_σ(1),,…,X_ν_σ(k))=0,    k≥2,

		due to the orientations on the corresponding moduli spaces.
		
		
		
		
			
		Before proceeding, a remark is in order addressing the perturbations used in the proof of Lemma <ref>, and <ref>. We do not perturb the input singular chains when choosing perturbation for a fiber product _k+1(β;L(N);P⃗) since the singular chains are fixed during the inductive construction and doing so would destroy the A_∞-structure. On the other hand, since the Kuranishi structure on the disc moduli are weakly submersive, we can realize the perturbation of input singular chains by perturbing the respective evaluation maps.
				
		
		Combining the above lemmas, to show that the equivariant toric superpotentials coincide with the equivariant disc potentials, it remains to compute the (non-equivariant) disc potential.
		
		Recall that the holomorphic disc classes are generated by the basic disc classes β_i for 1≤ i≤ m <cit.>.  Moreover, by <cit.>, a stable disc class of Maslov index two must be of the form β_i+α for some effective curve class α with c_1(α)=0.  Let n_1(β_i+α) be the degree of the virtual fundamental class _*[_1(β_i+α)].  In the Morse model, it is given by the intersection number of _*[_1(β_i+α)] with the maximal point _L.

	
		
		Let X be a compact semi-Fano toric manifolds with _ X=d, and let v_1,…,v_m∈^d be the primitive generators of the one-dimensional cones of the fan Σ defining X. Let L⊂ X be a regular moment map fiber. Then, 
		
    𝔪^T,b_0(1) = W^L (x_1,…,x_d) 1+∑_i=1^d x_i λ_i=(W^L (x_1,…,x_d)+∑_i=1^d λ_i x_i ):=W^L_T(x_1,…,x_d)  ,

		where
		
    W^L (x_1,…,x_d) = ∑_i=1^m (∑_α: c_1(α)=0 n_1(β_i + α) ^ω_X(α)) ^ω_X(β_i)exp (v_i · (x_1,…,x_d)).

		
		In particular in the Fano case,	
		
    𝔪^T,b_0=(W^toric(e^x_1,…,e^x_d) +∑_i=1^d λ_i x_i ),

		where W^toric is the superpotential of Givental and Hori-Vafa <cit.>.
	
	
	
	Recall from Proposition <ref> that W^L·_L=^b_0(1). Since L has minimal Maslov index 2, by degree reason, a stable rooted tree Γ∈Γ_k+1 contributing to the term _k(b,…,b), k≥ 1, must have exactly one interior vertex decorated by a Maslov index 2 disc class and the remaining interior vertices are decorated by the constant disc class β_0. 
	
	Let us abuse notation and denote by X_j the unstable hypertorus of the degree one critical point X_j of the perfect Morse function f=∑_i=1^dcos(θ_i) on (^1)^d. We can choose an identification L≅ (^1)^d such that the boundaries of the finitely many holomorphic discs in basic disc classes β_1,…,β_m, passing through the maximal point _L do not intersect with X_μ∩ X_ν for any μ≠ν. This means the only configurations Γ∈Γ_k+1, contributing to _k(b,…,b) have exactly one interior vertex, which are decorated by a Maslov index 2 disc class β_i+α. It remains to consider contributions from the moduli spaces of the form _k+1(β_i+α;L;X_μ_1,…,X_μ_k 	), (μ_1,…,μ_k)∈{1,…,d}^k.	
	
	We note that _k+1(β_i+α;L;X_μ_1,…,X_μ_k)^=∅ for generic perturbations  unless ∂β_i intersects all the input hypertori X_μ. Therefore, it suffices to assume that is the case. Let us denote by r_j the number of times X_j appears in the inputs, and by m_j is the intersection multiplicity of ∂β_i⋔ X_j. In order to obtain the expression (<ref>), we choose the following Kuranishi structure and perturbation for  _k+1(β_i+α;L;X_μ_1,…,X_μ_k).
	
	Let ={_p} be the multisection chosen for _1(β_i+α;L), which gives rise to the open Gromov-Witten invaraint n_1(β_i+α). (We note that n_1(β_i+α) is in fact independent of the choice of a transversal multisection due to the absence of disc bubbles.) For k≥ 1, let :_k+1(β_i+α;L)→_1(β_i+α;L) be the forgetful map which forgets the k input marked points and stabilizes. To relate the fiber product _k+1(β_i+α;L;X_μ_1,…,X_μ_k) with n_1(β_i+α), we first pullback the Kuranishi structure and multisection from _1(β_i+α;L) to _k+1(β_i+α;L) via . Let p^+=[Σ,z⃗,u]∈_k+1(β_i+α;L) such that (p^+)=p∈_1(β_i+α;L). Let V^+_p^+, E^+_p^+, and _p^+^+:V^+_p^+→ (E^+_p^+)^ℓ/S_ℓ be pullback of the Kuranishi neighborhood, obstruction bundle, and multisection at p via  (see <cit.> for the precise definitions). To make the Kuranishi structure on _k+1(β_i+α;L) weakly submersive, we enlarge Kuranishi neighborhood V_p^+ and the obstruction bundle E_p^+ at p^+ to be of the form
	
	    
  * E_p^+=E^+_p^+⊕_i=1,…,k T_u(z_i)L.
		
    
  * V_p^+=V^+_p^+× U_p^+, where U_p^+⊂⊕_i=1,…,k T_u(z_i) is an open subset containing 0.
	
Then, ^+_p^+ naturally induces a multisection '_p^+: V_p^+→ (E_p^+)^ℓ/S_ℓ which is characterized as follows. Let _p^+:V_p^+→ (E_p^+)^ℓ be the lift of _p^+ and write

    _p^+=((s^1_p^+,1,s^2,1_p^+,…,s^2,k_p^+),…,(s^1_p^+,ℓ,s^2,1_p^+,…,s^2,k_p^+) ).

Here, the restriction of (s^1_p^+,1,…,s^1_p^+,ℓ):V_p^+→ (E^+_p^+)^ℓ to V^+_p^+×{0} agrees with the lift of _p^+^+ and is constant along the U_p^+-direction; (s^2,1_p^+,…,s^2,k_p^+):V_p^+→⊕_i=1,…,k T_u(z_i) is the tautological section given by the embedding U_p^+⊂⊕_i=1,…,k T_u(z_i).

Now, let v_1,…,v_k be distinct, generic, small vectors in the direction of ∂β_i. (Since TL is trivial, we can view v_1,…,v_k as constant vector fields on L.)
We define a new multisection '={'_p^+} by 

    '_p^+:V_p^+→ E_p^+^ℓ· k!,
 

    '_p^+=((s^1_p^+,1,s^2,1_p^++v_σ(1),…,s^2,k_p^++v_σ(k)),…,(s^1_p^+,ℓ,s^2,1_p^++v_σ(1),…,s^2,k_p^++v_σ(k)))_σ∈ S_k,

and '_p^+='_p^+/S_ℓ· k!. Then, ' induces a transversal multisection on the fiber product _k+1(β_i+α;L;X_μ_1,…,X_μ_k) whose zero locus is given by 

    _k+1(β_i+α;L)^'×_L^k∏_ν=1,… k X_μ_ν=∐_1,…,∏_j=1^d m_j^r_j^-1(0)

weighted by 1/k!. Its intersection with _L gives the coefficient
	n_1(β_i+α)∏_j=1^d m_j^r_j/k!. (See Figure <ref> for an illustration of such perturbations.) Summing over all the possible inputs of k hypertori and the k!/∏_j=1^d (r_j!) arrangements among them, we obtain the term n_1(β_i+α) exp (v_i · (x_1,…,x_d)).
	



	
		For the above theorem, we have chosen the perturbations for the fiber products to be the average of all configurations of the perturbed unstable hypertori of X_i.  Indeed, we can choose other perturbations, which gives different expressions of the (non-equivariant) disc potential.
		
		Take ^1 as an example.  Let L be the equator.  Let us perturb X_1 in the counterclockwise order with respect to the left hemisphere.  Then the non-equivariant disc potential will read
		
    W^L = ^A/2( 1/1-x_1 +(1-x_1) )

		instead of the well-known expression ^A/2( e^x_1 + e^-x_1).
	

		

	In the Fano case, n_1(β_i+α)=0 whenever α≠0 by dimension reason.  Moreover n_1(β_i)=1 <cit.>.  More generally, the non-equivariant disc potential for compact semi-Fano toric manifold has been computed by <cit.> using Seidel representations.  The coefficients are given by the inverse mirror map.  We recall it in the following theorem.
	
	 
		Let X be a compact semi-Fano toric manifold. Let v_1,…,v_m be the primitive generators of the one dimensional cones of the fan defining X. We denote by D_i the toric prime divisor corresponding to v_i. Then
		
    ∑_α: c_1(α)=0 n_1(β_i + α) ^ω_X(α) = exp(g_i(q̌(q))),

		where
		
    g_i(q̌):=∑_C(-1)^(D_i· C)(-(D_i· C)-1)!/∏_j≠ i (D_j· C)!q̌^C,   q̌^C:=^ω_X(C) ,

		and the summation is over all effective curve classes C∈ H_2^eff(X) satisfying
		
    -K_X· C=0, D_i· C<0  and  D_j· C ≥ 0  for all  j≠ i

		and q̌=q̌(q) is the inverse of the mirror map q=q(q̌).
	

	A similar result also holds for toric semi-Fano Gorenstein orbifold <cit.>.  However we stick with the manifold case for simplicity.  Moreover, compactness of X can be replaced by requiring X to be semi-projective, so that the disc moduli spaces are compact.  A toric manifold X is said to be semi-projective if it has a torus-fixed point, and the natural map X→H^0(X,𝒪_X) is projective <cit.>. By combining Theorem <ref> and <ref>, we get the following:
	
	 
		For a toric fiber of a semi-projective and semi-Fano toric manifold, the equivariant disc potential equals to
		
    W_T^L = ∑_i=1^m  exp(g_i(q̌(q))) ^ω_X(β_i)exp (v_i · (x_1,…,x_d))+∑_i=1^d λ_i x_i,

		where g_i(q̌(q)) is given by the inverse mirror map in equation (<ref>).
	


	









§ ^1-EQUIVARIANT DISC POTENTIAL FOR THE IMMERSED TWO-SPHERE
 

In this section, we study the ^1-equivaraint Floer theory of an immersed Lagrangian two-sphere L_0 which has a single nodal point. This will serve as a prototypical example for the authors' subsequent work <cit.> with Hong on the ^1-equivariant Floer theory of immersed SYZ fibers in tori Calabi-Yau manifolds.

 

	

 §.§ Equivariant Floer theory for Lagrangian immersions

	
	In this section we describe a generalization of the equivariant Morse model to immersed Lagrangians with clean self-intersections. 
	
	In <cit.>, Akaho and Joyce developed Lagrangian Floer theory for immersed Lagrangians L with transverse self-intersection (which is a nodal point), generalizing the work of Fukaya-Oh-Ohta-Ono <cit.> in the embedded case.
	Their construction can be further generalized to the case where L is an immersed Lagrangian with clean self-intersection as we describe below. See <cit.> for the development of Floer theory of clean intersections in different settings. 
	
Let (X,ω_X) be as in Section <ref> and let L be a closed, connected, relatively spin, immersed Lagrangian submanifold with clean self-intersections. We denote by ι:L̃→ X an immersion with image L and by  the self-intersection. As in Akaho-Joyce <cit.>, the inverse image of  under the immersion ι is assumed to be the disjoint union ^-∐^+⊂L̃ of two branches, each of which is diffeomorphic to .  Then the following fiber product

    L̃×_L L̃=∐_i=-1,0,1 R_i,

consists of the diagonal component R_0, and the two immersed sectors
	
    R_1={(p_-, p_+)∈L̃×L̃| p_-∈^-, p_+∈^+, ι(p_-)=ι(p_+)  },
    
    	R_-1={(p_+, p_-)∈L̃×L̃| p_+∈^+, p_-∈^-, ι(p_+)=ι(p_-) },

	 We have canonical isomorphisms R_0≅L̃ and R_-1≅ R_1≅. We also have the involution σ:R_-1∐ R_1→ R_-1∐ R_1 swapping the two immersed sectors, i.e, σ(p_-,p_+)=(p_+,p_-).
	 
	 	Let J_X be a compatible almost complex structure of (X,ω_X). For a map α:{0,…, k}→{-1,0,1}, we consider quintuples (Σ,z⃗,u,ũ,l) where
	 
	 	
  * Σ is a prestable genus 0 bordered Riemann surface,
	 	
  * z⃗=(z_0,…,z_k) are distinct counter-clockwise ordered smooth points on ∂Σ, 
	 	
  * u:(Σ,∂Σ)→ (X,L) is a J-holomorphic map with (Σ,z⃗,u) stable,
	 	
  * ũ:^1 ∖{ζ_i:=l^-1(z_i):α(i) 0}→L̃ is a local lift of u|_∂Σ, i.e., 
	 	
    ι∘ũ=u∘ l (lim_θ→ 0^-ũ(e^𝐢θζ_i),lim_θ→ 0^+ũ(e^𝐢θζ_i))∈ R_α(i),

	 	where α(i) 0 and 𝐢:=√(-1). 
	 	
	 	
	 	
	 	
  * l:^1 →∂Σ is an orientation preserving continuous map (unique up to a reparametrization) characterized by that the inverse image of a smooth point is a point and the inverse image of a singular point consists of two points. 
	 
	 
	 
	 
	 
	 
	 
	 
	 
	 
	 Let [Σ,z⃗,u,ũ,l] be the equivalence class of (Σ,z⃗,u,ũ,l) given by the automorphisms. 
	 For β∈ H_2(X,L;) and α a map as described above, we denote by _k+1(α,β) the moduli space of (equivalence classes of) such quintuples [Σ,z⃗,u,ũ,l] satisfying u_*([Σ])=β. The moduli spaces come with the following evaluation maps _i:_k+1(α,β) →L̃×_L L̃ defined by
	 
    _i([Σ,z⃗,u,ũ,l])=ũ(z_i)∈ R_0    α(i)=0 
    (lim_θ→ 0^-ũ(e^𝐢θζ_i),lim_θ→ 0^+ũ(e^𝐢θζ_i))∈ R_α(i)   α(i) 0,

	 at the input marked points i=1,…, k, and 
	 
    _0([Σ,z⃗,u,ũ,l])=ũ(z_0)∈ R_0    α(0)=0 
    σ(lim_θ→ 0^-ũ(e^𝐢θζ_0),lim_θ→ 0^+ũ(e^𝐢θζ_0))∈ R_-α(0)   α(0) 0,

	 at the output marked point. 
	 
	 For the convenience of writing, we will call an element of _k+1(α,β) a stable polygon if α(i) 0 for some i∈{0,…,k}. In this case, the corners of a polygon are the boundary marked points z_i with α(i) 0. If α(i)=0 for all i, we will simply refer to an element of _k+1(α,β) as a stable disc. Note that α dictates the branch jumps at the boundary marked points.

	
	The Kuranishi structures on _k+1(α,β;L) are chosen to be weakly submersive. 
For P⃗=(P_1,…, P_k), P_1,…,P_k∈ S^∙(L̃×_LL̃;), we denote by _k+1(α,β,P⃗) the fiber product 
	
    _k+1(α,β,P⃗)=_k+1(α,β,L) ×_(L̃×_L L̃)^kP⃗

	in the sense of Kuranishi structures. We can then construct an A_∞-structure  on a countably generated subcomplex C^∙(L;Λ_0)⊂ S^∙(L̃×_LL̃;Λ_0) as in Section <ref>. 
    We define the maps _k,β:C^∙(L;Λ_0)^⊗ k→ C^∙(L;Λ_0)⊂ S^∙(L̃×_LL̃;Λ_0) by
	
    _1,β_0(P)=(-1)^n ∂ P,

	and by
	
    _k,(α,β)(P_1,…,P_k)=(_0)_*(ℳ_k+1(α,β;P⃗)^),

	
    _k,β(P_1,…,P_k)=∑_α_k,(α,β)(P_1,…,P_k)

	for (k,β) (1,β_0). Notice that _k,(α,β)(P_1,…,P_k)=∅ unless P_i is a singular chain on R_α(i). The A_∞-structure map _k is given by
	
    _k(P_1,…,P_k)=∑_β^ω_X(β)_k,β(P_1,…,P_k).

	
	Similarly, we have a unital A_∞-algebra (C^∙(L;Λ_0)^+, ^+) that is homotopy equivalent to (C^∙(L;Λ_0), ) obtained via the homotopy unit construction.
	
	
	We choose for each connected component _i⊂, a base point p_i, and a path of Lagrangian subspaces in T_p_iX connecting dι (T_p_i,-L) and dι (T_p_i,+L), ι(p_i,-)=ι(p_i,+)=p_i. Then, the dimensions of the moduli spaces and the grading of singular chains on the immersed sectors depend on the choices of paths in the same manner as in <cit.>. On the other hand, for our application in Section <ref>, the immersed Lagrangian will be graded by a holomorphic volume form and these will be determined uniquely once paths of graded Lagrangian subspaces are chosen.  
	

Let f:L̃×_L L̃→ be a Morse function with a unique maximum point  on the diagonal component L̃. Let C^∙(f;Λ_0) be the cochain complex generated by the critical points of f

    C^∙(f;Λ_0)=⊕_p∈(f)Λ_0 · p,

and set

    CF^∙(L;Λ_0)=C^∙(f;Λ_0)⊕Λ_0 ·⊕Λ_0 ·.

Let

    𝒳_-1(L)={Δ_p | p∈(f)},

where Δ_p is defined by (<ref>). Let C^∙_(-1)(L;Λ_0)⊂ S^∙(L̃×_LL̃;Λ_0) be the subcomplex generated elements in 𝒳_-1(L), and set

    C^∙_(-1)(L;Λ_0)^+=C^∙_(-1)(L;Λ_0)⊕Λ_0  ·e^+ ⊕Λ_0 ·f.

We again identify CF^∙(L;Λ_0) with C^∙_(-1)(L;Λ_0)^+ via the assignment p↦Δ_p and equip CF^∙(L;Λ_0) with a unital A_∞-structure  via homological peturbation as in Section <ref>. 

The A_∞-maps _k counts stable pearly trees with interior vertices decorated by stable polygons (α,β). The each edge of a tree correspond to a negative gradient flow line of the Morse function f (in a connected component of L̃×_LL̃). A flow line connected to the i^th input marked point of a polygon (α,β) must be contained R_α(i), for otherwise the pearly tree does not exist. See Figure <ref> for some (non-) examples.

For a pair (L_1,L_2) of closed, connected, relatively spin, embedded Lagrangian submanifolds intersecting cleanly, the union L=L_1∪ L_2 is a immersed Lagrangian with clean self-intersections with L̃=L_1∐ L_2. We choose the splitting ι^-1()=^-∐^+ so that ^-⊂ L_1 and ^+⊂ L_2.

We can define a pearl complex (CF^∙(L_1,L_2;Λ_0),_1^L_1,L_2) for the Lagrangian intersection Floer theory of (L_1,L_2) as follows: 

    CF^∙(L_1,L_2;Λ_0)=⊕_p∈(f|_R_1) Λ_0 · p

is the subcomplex of CF^∙(L;Λ_0) generated by critical points of f in R_1, and _1^L_1,L_2 counts stable pearly trees Γ∈Γ_2 with input and output vertices in R_1. 








For the equivariant Morse model, we consider a G-equivariant Lagrangian immersion ι: L̃→ X with image L=ι(L̃)⊂ X a G-invariant immersed Lagrangian L.  
We can choose an admissible collection of Morse-Smale pairs {(f_N,𝒱_N)}_N∈ on the finite dimensional approximations of L̃_G ×_L_GL̃_G and proceed to define A_∞-algebras (CF^∙_G(L;Λ_0),^G) and (CF^∙_G(L;Λ_0)^†,^G,†) following Section <ref> and <ref>. This gives rise to interesting equivariant disc potentials even in the case when L is unobstructed as we will see in Section <ref>.
	
	
	
	

	





 §.§ ^1-equivariant disc potential via the gluing method

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

 
	Let L_1,L_2 be two graded Lagrangians in the Fukaya category that cleanly intersect with each other, and α∈ CF^0(L_1,L_2).  Suppose that CF^j(L_1,L_2)=0 for all j<0.  For any b'_i ∈ CF^1(L_i,Λ_+) and w_i ∈Λ_0, we put b_i=b'_i+w_i·_L_i. Then ,
	
    𝔪^b_1', b_2'_1(α) = 𝔪^b_1, b_2_1(α).

	
	Similarly, if further CF^j(L_2,L_1)=0 for all j<0 and β∈ CF^0(L_2,L_1), then 
	
    𝔪^b_1',b_2',b_1'_2(β,α)    = 𝔪^b_1, b_2,b_1_2(β,α), 
    𝔪^b_2',b_1',b_2'_2(α,β)    = 𝔪^b_2, b_1,b_2_2(α,β).




For (<ref>), since the degree of _L_i is (-1), the extra terms

    𝔪^b_1', b_2'_1(α) - 𝔪^b_1, b_2_1(α)=∑_k≥ 2 
     k_1+k_2+1=k_k(b_1',…,b_1'_k_1,α,b_2',…,b_2'_k_2)-_k(b_1,…,b_1_k_1,α,b_2,…,b_2_k_2)
	
have degrees 2-k + (-1)· (k-1) = 3-2k < 0  in CF^∙(L_1,L_2) and hence must vanish. Similarly for (<ref>), the extra terms in 	

    𝔪^b_1',b_2',b_1'_2(β,α) - 𝔪^b_1, b_2,b_1_2(β,α)∈ CF^∙(L_1;Λ_0)

and 

    𝔪^b_2',b_1',b_2'_2(α,β) - 𝔪^b_2, b_1,b_2_2(α,β)∈ CF^∙(L_2;Λ_0)

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


In the setting of Section <ref>, the above lemma has a direct G-equivariant analog, by replacing 𝔪_k with 𝔪^G_k and setting w_i = w_i^0 + ∑_λ=2ϕ_λ·λ∈ H^∙_G(;Λ_0), as a consequence of Proposition <ref>.  

An isomorphism between (L_1,b_1) and (L_2,b_2), where b_i are weak bounding cochains, consists of α∈ CF^0(L_1,L_2) and β∈ CF^0(L_2,L_1) satisfying 

    𝔪^b_1, b_2_1(α) = 𝔪^b_2, b_1_1(β)= 0, 
       𝔪^b_1, b_2,b_1_2(β,α) =  _L_1 + 𝔪^b_1_1 (γ_1), 
       𝔪^b_2, b_1,b_2_2(α,β) =  _L_2 + 𝔪^b_2_1 (γ_2),

for some 
γ_i ∈ CF^∙(L_i;Λ_0). Since 𝔪^b_i_1(_L_i) = _L_i -  (1-  h(b_i) )_L_i for some h(b_i) ∈Λ_+, by Lemma <ref>, it suffices to check that 

    𝔪^b_1', b_2'_1(α)=0,   𝔪^b_2', b_1'_1(β)=0,   𝔪^b_1', b_2',b_1'_2(β,α) = _L_1,   𝔪^b_2', b_1',b_2'_2(α,β) = _L_2.
 


Similarly, an isomorphism between (L_1,G,b_1) and (L_2,G,b_2) consists of α∈ CF^0_G(L_1,L_2) and β∈ CF^0_G(L_2,L_1) such that the obvious analog of equations (<ref>) hold. 




Let us now consider the following immersed two-sphere in 

    X := { (a, b, z) ∈^2 ×^×| ab = 1 + z }.
 
We equip X with the symplectic form inherited from the standard symplectic form on ^3.
Let Π X →^× be the projection to the z-component. 
Regarding 𝕊^1 as the unit circle in the complex plane , the space X has a fiberwise Hamiltonian 𝕊^1-action given by
η· (a, b, z) ↦(η· a, η^-1· b, z).
One obtains the special Lagrangian torus fibration 

    (|a|^2-|b|^2, |z|)
 with respect to the holomorphic volume form 

    Ω_X = da ∧ dbz.

See <cit.>.
In particular, a singular fiber which will be denoted by L_0 occurs over the point (|a|^2-|b|^2 = 0, |z| = 1) is an immersed two-sphere which intersects itself transversally at the point (0, 0, -1). 




Let ι: L_0=^2→ X be a Lagrangian immersion with image the immersed two-sphere L_0.  In this case, the self-intersection r∈ L_0 is a single nodal point fixed by the 𝕊^1-action, and the two branches of the immersed loci are given by U=(p,q), V=(q,p) ∈^2×^2, p q, and ι(p)=ι(q)=r.  










We have finite dimensional approximations of the fiber product 

    (^2)_^1×_(L_0)_^1 (^2)_^1≅^∞∐ (^2)_^1∐^∞ .

Let f_∞: (^2)_^1×_(L_0)_^1 (^2)_^1→ be a Morse function such that its restriction to each of the ^∞-components is the perfect Morse function φ:B^1=^∞→ (see (<ref>)), and the restriction to the diagonal component (^2)_^1 is the sum of a horizontal lift of the perfect Morse function φ and function on the total space (^2)_^1→^∞ which restricts to a perfect Morse function f_^2:^2 → on generic fibers. We require (f_^2) to be away from p,q∈^2.

We will denote the critical points of f_∞|_(^2)_^1 by _^2⊗λ^ℓ and _^2⊗λ^ℓ, ℓ≥ 0, where _^2 and _^2 are the maximal and minimal critical points of ϕ, respectively. We will also denote the critical points of f_∞ on the non-diagonal components by U ⊗λ^ℓ and V ⊗λ^ℓ, ℓ≥ 0, respectively.




Following Section <ref> and <ref>, we obtain an A_∞-algebra (CF_^1^∙(L_0;Λ_0)^†,^^1,†) associated to (L_0,^1). We have

    CF_^1^∙(L_0;Λ_0)^†=CF^∙(L_0;Λ_0)⊗_Λ_0Λ_0[λ],

where CF^∙(L_0;Λ_0) is the unital Morse complex associated to f_∞. 
By abuse of notations, degree one immersed generators (graded with respect to the ^1-invariant holomorphic volume form Ω_X) will simply be denoted by U:=U⊗ 1, and V:=V⊗ 1.
We now consider the formal deformations b_0=uU+vV of (L_0,^1) for (u,v)∈Λ^2_0 with (uv) >0.


	The formal deformations of (L_0,^1) by  b_0=uU+vV are weakly unobstructed, i.e., 𝔪_0^^1,†, b_0(1)∈ H^∙_^1(;Λ_0)·.



	Since L_0 has minimal Maslov index two, 𝔪_0^^1,†, b_0(1) can be expressed as 
	
    𝔪_0^^1,†, b_0(1) =𝔪_0^b_0(1) ⊗ 1+ W(u,v) ·_^2⊗λ

	by the same argument in the proof Proposition <ref>.
	Although L_0 does not bound any non-constant holomorphic discs, extra care is necessary for 𝔪_0^b_0(1) as constant polygons might bubble off at the immersed loci. 
	We consider the anti-symplectic involution σ X → X given by (a,b, z) ↦ (b, a, z), which preserves L_0 and swaps two immersed generators U and V.
	In <cit.>, by taking orientations respecting the involution, the constant polygon contributions are canceled out. 
	Thus (L_0,b_0=uU+vV) is unobstructed, that is, 𝔪_0^b_0(1)=0. Therefore, 
	
    𝔪_0^^1,†, b_0(1) = W(u,v) ·_^2⊗λ.

	
	


More precisely, the _^2⊗λ appeared above is _^2⊗λ= in the notation of Section <ref>.  By Lemma <ref>, the above b_0 can be modified (by adding terms involving ) to b_0^† such that 𝔪_0^^1,†, b_0^†(1)= W^(u,v) ·.  By abuse of notations, we shall simply replace b_0 with b_0^† and denote _^2 by _^2, =_^2⊗ 1 by , and W^ by W.


	
	This line of thoughts is similar to the work in <cit.>.
	In this example L_0 does not bound non-constant holomorphic discs by maximal principle.  However, there are still highly interesting non-trivial contributions coming from constant polygons with corners at U and V.  Note that for each constant polygon, U and V corners must occur in pairs (in order to return to the original branch).  This is the reason that the formal deformations of (L_0,^1) by b_0=uU+vV are convergent for (u,v)∈Λ_0^2 with (uv) >0. 
	
	


In order to compute W(u,v), we compare (L_0,^1) with the ^1-equivariant Floer theory of Chekanov and Clifford tori L_1 and L_2 respectively.  
For the special Lagrangian torus fibration (<ref>), the Lagrangian torus over a point in the chamber |z| < 1 (resp. |z| > 1) is called a Chekanov (resp. Clifford) torus.
Since any pair of distinct fibers in (<ref>) does not intersect, they cannot be isomorphic. 
We apply Lagrangian isotopies to Chekanov torus and Clifford torus without intersecting the wall z = 0 to make them intersect as in Figure <ref> (a). 
Those isotoped Lagrangians are still -graded and called Chekanov torus and Clifford torus respectively. 




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

We equip L_1 and L_2, with the non-trivial spin structure along the ^1-orbit direction.  This will introduce extra systematic contributions to the orientation bundle of the moduli space of strips bounded by L_1 and L_2 and other Lagrangians.  The reason for doing this will be explained below.

We fix a perfect Morse function on each L_i for i=1,2, such that the (compactified) unstable submanifolds of the degree one generators X_i,Y_i are the hypertori chosen in <cit.>.  The unstable submanifold of X_i is transverse to the ^1-orbits in L_i.  We also have a Morse function on (L_i)_^1 as in Section <ref>, whose critical points are labeled by η⊗λ^ℓ, ℓ≥ 0 where η is one of the following

    _L_i,   X_i,    Y_i,    X_i∧ Y_i.
 

We consider the formal deformations of  L_i for i=1,2, by b_i=x_i X_i+y_i Y_i, x_i,y_i∈Λ_+. (Here X_i:=X_i⊗ 1 and  Y_i:=Y_i⊗ 1, by abuse of notations.) By Proposition <ref>, the equivariant disc potential of (L_i,^1) is given by

    W^L_i_^1 =  λ x_i.

Since L_i does not bound any non-constant holomorphic discs (particularly Maslov index zero discs), we have W^L^(i)_^1 =(W^L^(i)_^1)^.

Each pair of Lagrangians L_i and L_j, 0≤ i<j≤ 2, intersect cleanly at two circles C^L_i,L_j_α and C^L_i,L_j_β (see Figure <ref>) on which ^1 acts freely. We fix a perfect Morse function on their clean intersections and denote by CF^∙(L_i,L_j), and CF^∙(L_j,L_i) the corresponding Lagrangian Floer intersection complexes. 

Let 

    α_ij∈ CF^0(L_i,L_j),   β_ji∈ CF^0(L_j,L_i)

be the generators corresponding to the maximal critical points on C^L_i,L_j_α and C^L_i,L_j_β, respectively. Then, in the non-equivariant setting, we have the following gluing formula between the formal deformation space of the Lagrangians:

<cit.>
The three formally deformed Lagrangians (L_0, b_0), (L_1, b_1) and (L_2, b_2) are isomorphic, with the pairwise isomorphisms provided by (<ref>), if and only if
	
    uv = 1-exp (x_i)    , 
    
    	u = exp (y_1), 
    
    	v = exp (-y_2).

(The minus sign in front of exp(x_i) is due to the non-trivial spin structure we chose for L_i.)


We now turn to the gluing formula in the equivariant setting. For i,j∈{0,1,2} and i j, let CF^∙_^1(L_i,L_j) be the ^1-equivariant Lagrangian Floer intersection complex with 

    CF^∙_^1(L_i,L_j)=CF^∙(L_i,L_j)⊗_Λ_0Λ_0[λ].



(L_0,^1,b_0), (L_1,^1,b_1) and (L_2,^1,b_2) are isomorphic, with the pairwise isomorphisms provided by 

    α_ij⊗ 1 ∈ CF^0_^1(L_i,L_j),   β_ji⊗ 1 ∈ CF^0_^1(L_j,L_i),

if and only if (<ref>) holds.



This is simply due to degree reason. Since the equivariant parameter λ has  degree two, we have
	
    (_1^^1)^b_i,b_j(α_ij⊗ 1)= _1^b_i,b_j(α_ij), 
    
    (_1^^1)^b_j,b_i(β_ji⊗ 1)= _1^b_j,b_i(β_ji), 
    
    (_2^^1)^b_i,b_j,b_i(β_ji⊗ 1, α_ij⊗ 1)=_2^b_i,b_j,b_i(β_ji, α_ij), 
    
    (_2^^1)^b_j,b_i,b_j(α_ij⊗ 1, β_ij⊗ 1)=_2^b_j,b_i,b_j(α_ij, β_ji) .

This implies that the gluing formula remains the same in the equivariant setting. Note that we have implicitly identified CF^∙(L_i;Λ_0) with CF^∙(L_i;Λ_0)⊗ 1⊂ CF^∙_^1(L_i;Λ_0) in the above equations.


Since the disc potentials are compatible with the gluing formulas (see <cit.>), we can compute W^L_0_^1 using the above lemma and the fact that W^L_1_^1 = λ x_1.

 
	The ^1-equivariant potential m of the immersed two-sphere L_0 is given by
	
    W^L_0_^1 =λ W(u,v)=λlog (1-uv) = -λ∑_j=1^∞(uv)^j/j.




By the A_∞-identity, we have

    ((_1^^1)^b_1,b_0)^2(α_10⊗ 1)=λ W(u,v) (_2^^1)^b_0,b_0,b_1(α_10⊗ 1,_^2⊗ 1)- λ x_1 (_2^^1)^b_0,b_1,b_1(_L_1⊗ 1,α_10⊗ 1).
	
(<ref>) forces the (LHS) to vanish, and thus
	
    λ W(u,v) (_2^^1)^b_0,b_0,b_1(α_10⊗ 1,_^2⊗ 1)=λ x_1 (_2^^1)^b_0,b_1,b_1(_L_1⊗ 1,α_10⊗ 1).

	On the other hand, we have
	
    (_2^^1)^b_0,b_0,b_1(α_10⊗ 1,_^2⊗ 1)
    	= (_2^^1)^b_0,b_1,b_1(_L_1⊗ 1,α_10⊗ 1) = α_10⊗ 1.

	Therefore, 
	
    W(u,v) = x_1.

	Since exp (x_1) = 1-uv, we get
	
    W(u,v)=   log (1-uv) = - ∑_j=1^∞(uv)^j/j.




	Note that if we did not take the non-trivial spin structure, then the gluing formula reads exp (x_1) = -1+uv.  However, the LHS lies in 1 + Λ_+, while the RHS lies in -1 + Λ_+. This means the formally deformed Lagrangians (L_1,^1,b_1) equipped with the trivial spin structure cannot be isomorphic to (L_0,^1,b).  This is reason why we equipped L_1 and L_2 with the non-trivial spin structure in the first place.












	

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