Relative quantum cohomology

We establish a system of PDE, called open WDVV, that constrains the bulk-deformed superpotential and associated open Gromov-Witten invariants of a Lagrangian submanifold $L \subset X$ with a bounding chain. Simultaneously, we define the quantum cohomology algebra of $X$ relative to $L$ and prove its associativity. We also define the relative quantum connection and prove it is flat. A wall-crossing formula is derived that allows the interchange of point-like boundary constraints and certain interior constraints in open Gromov-Witten invariants. Another result is a vanishing theorem for open Gromov-Witten invariants of homologically non-trivial Lagrangians with more than one point-like boundary constraint. In this case, the open Gromov-Witten invariants with one point-like boundary constraint are shown to recover certain closed invariants. From open WDVV and the wall-crossing formula, a system of recursive relations is derived that entirely determines the open Gromov-Witten invariants of $(X,L) = (\mathbb{C}P^n, \mathbb{R}P^n)$ with $n$ odd, defined in previous work of the authors. Thus, we obtain explicit formulas for enumerative invariants defined using the Fukaya-Oh-Ohta-Ono theory of bounding chains.

Overview. Let (X, ω) be a symplectic manifold with dim R X = 2n, and let L ⊂ X be a Lagrangian submanifold. We assume L admits a relative spin structure and fix one, so the standard theory of orientations for moduli spaces of J-holomorphic disks [8,27] applies. A bounding chain for L is a solution of the Maurer-Cartan equation in the Fukaya A ∞ algebra of L. Bounding chains provide a systematic method to compensate for the disk bubbling phenomenon that generally spoils the invariance of counts of J-holomorphic curves with boundary. There is a natural equivalence relation on bounding chains known as gauge equivalence. The superpotential of L is a function on the space of bounding chains modulo gauge equivalence. The superpotential counts J-holomorphic disks in X with boundary in L constrained to pass through the given bounding chain. Cycles on X give rise to deformations of the Fukaya A ∞ algebra, bounding chains, and superpotential of L, known as bulk deformations. The deformed superpotential counts J-holomorphic disks in X with boundary on L with the interior constrained to pass through the cycles in X and the boundary constrained to pass through the deformed bounding chain. If one can invariantly parameterize the space of bounding chains modulo gauge equivalence, the superpotential becomes a generating function for open Gromov-Witten invariants [7,33]. Thus, the superpotential is an analog in open Gromov-Witten theory of the genus zero closed Gromov-Witten potential in closed Gromov-Witten theory. Indeed, the closed Gromov-Witten potential is a generating function for genus zero closed Gromov-Witten invariants, which count J-holomorphic spheres in X.
To invariantly count J-holomorphic disks with boundary contractible in L, it is natural to define an enhanced superpotential that includes contributions from J-holomorphic spheres as well as disks [7,20,25]. The sphere contributions compensate for the phenomenon of the boundary of the disk collapsing to a point.
Simultaneously, for a Lagrangian submanifold equipped with a bounding chain for bulk deformations in a Frobenius subalgebra U ⊂ QH(X), we define the relative quantum cohomology algebra QH U (X, L) and prove its associativity. See Theorem 7. Denoting by Q the relevant Novikov ring, we have a long exact sequence of Q-modules, [1] e e L L L L L L L L L L where the top arrow is an algebra homomorphism. The algebra structure on QH(X, L) is given by counting both J-holomorphic spheres in X and J-holomorphic disks in X with boundary in L. When L is a real cohomology sphere, one may consider U = QH(X) and find a bounding chain for all associated bulk-deformations as shown in Theorem 8. A typical situation in which it is useful to consider U ⊂ QH(X) a proper subalgebra is when an anti-symplectic involution fixing L is used to construct the bounding chain as in Theorem 9.
To naturally integrate both disk and sphere contributions in the enhanced superpotential, the open WDVV system, and the relative quantum product, we introduce a cone complex C(i). We define a relative potential Ψ ∈ C(i) and a tensor potential ‫נ‬ : C(i) → C(i), which combine both the enhanced superpotential and the closed Gromov-Witten potential. The open WDVV equations and associativity of relative quantum cohomology follow from a commutation relation for partial derivatives of the tensor potential, which holds up to chain homotopy. See Theorem 5. We interpret the commutation relation for partial derivatives of ‫נ‬ as the flatness of the relative quantum connection in Corollary 1.7.
The chain homotopy underlying the open WDVV equations and associativity is constructed from operators on the Fukaya A ∞ algebra of L associated to moduli spaces of J-holomorphic disks with three marked points constrained to a geodesic. These moduli spaces come in three different families with either one, two, or three, of the marked points on the geodesic being in the interior of the disk, while the rest are on the boundary. In addition, the construction of the chain homotopy uses a family of J-holomorphic sphere moduli spaces with the cross ratio of four marked points constrained to be real. Bubbling in Gromov converging sequences of J-holomorphic disks with three marked points constrained to a geodesic gives rise to A ∞ type relations for the geodesic operators. The Maurer-Cartan equation satisfied by the bounding chain cancels the terms of the geodesic A ∞ relations corresponding to many types of bubbling. The remaining types of bubbling give rise to the open WDVV equations. Figure 1 shows the two types of bubbling of J-holomorphic disks with three interior marked points on a geodesic that contribute to the open WDVV equation. In the bubbling depicted on the left, the components of the Gromov limit are a disk without any geodesic constraints and a sphere. This type of bubbling gives rise to products of the enhanced superpotential and the closed Gromov-Witten potential in the open WDVV equations (15)- (16). In the bubbling depicted on the right, one of the two disk components of the limit open stable map retains the geodesic constraint. Namely, two of the interior marked points and the node must lie on the geodesic. A priori, it is not clear how to interpret the disk component with the geodesic constraint in terms of the enhanced superpotential, which counts disks without a geodesic constraint. Remarkably, the Maurer-Cartan equation and the interaction of the geodesic constraint with the unit of the Fukaya A ∞ algebra of L nonetheless allow the open WDVV equations to capture this type of bubbling with quadratic expressions in the enhanced superpotential. Again using the tensor potential ‫,נ‬ we obtain a wall-crossing formula for open Gromov-Witten invariants that allows one to exchange a boundary constraint for a certain type of interior constraint. See Theorem 6. Furthermore, from the proof that ‫נ‬ is a chain map, we derive a vanishing theorem for open Gromov-Witten invariants with more than one boundary constraint in the case that [L] = 0 ∈ H n (X). In this case, we show the open Gromov-Witten invariants with one boundary constraint recover certain closed invariants. See Theorem 2.
We apply the open WDVV equations and the wall-crossing formula to calculate the superpotential open Gromov-Witten invariants of [33] for (X, L) = (CP n , RP n ) with n odd. See Corollary 1.9 and sample values in Section 1.3.11. When n = 3, it is shown in [33] that the superpotential invariants recover Welschinger's invariants [35]. Thus, our calculations recover those of [2,3]. For arbitrary odd n, interior constraints restricted to odd powers of ω, and no boundary constraints, it is shown in [33] that the superpotential invariants recover the invariants of Georgieva [12]. Thus, our calculations recover those of [13,14]. When at least one interior constraint is an odd power of ω, the invariants of Georgieva vanish. However, our calculations show that the superpotential invariants do not vanish. Also, the superpotential invariants for arbitrary odd n allow for boundary constraints that behave like real point constraints for Welschinger's invariants in dimension 3. Point-like boundary constraints are not allowed in other constructions for n > 3. Our calculations show the superpotential invariants with point-like boundary constraints are non-trivial for n > 3.
The open WDVV equations of the present work are an extension of the equations introduced in [29] in the real setting for n = 2. See also [19] and [4]. We establish the open WDVV equations without regard to real structure and in any dimension. A preliminary version of our results appeared in [34]. A discussion of the formal properties of open WDVV in arbitrary dimension appeared in [1]. The real WDVV equations of [13] can be obtained from open WDVV by setting certain parameters to zero. Recently, the preprint [5] has appeared, which obtains some of our results in the real setting when n = 3 by different methods.

Background.
1.2.1. Notation. Consider a symplectic manifold (X, ω) of dimension 2n and a connected, Lagrangian submanifold L with relative spin structure s. Let J be an ω-tame almost complex structure on X. Denote by µ : H 2 (X, L) → Z the Maslov index. Denote by A * (L) the ring of differential forms on L with coefficients in R. Let Π be a quotient of H 2 (X, L; Z) by a possibly trivial subgroup S L contained in the kernel of the homomorphism ω⊕µ : H 2 (X, L; Z) → R⊕Z. Thus the homomorphisms ω, µ, descend to Π. Denote by β 0 the zero element of Π. Let : H 2 (X; Z) → Π (1) denote the composition of the natural map H 2 (X; Z) → H 2 (X, L; Z) with the projection H 2 (X, L; Z) → Π.
Gradings on Λ, Λ c are defined by declaring T β to be of degree µ(β). Define ideals Λ + Λ and Λ + c Λ c by For a graded real vector space V , let R[[V ]] denote the ring of formal functions on the completion of V at the origin and let m V ⊂ R[[V ]] denote the unique maximal ideal. More explicitly, let {v i } i∈I V be a homogeneous basis of V, let {v * i } i∈I V be the dual basis of V * , and let t i be a formal variable of degree −|v i |. We will often identify R[ The vector space V will be used to parameterize deformations associated with marked points in the interior of a Riemann surface while S will be used to parameterize deformations associated with marked points on the boundary of a Riemann surface. So, the grading of V is shifted by the real dimension of a Riemann surface and the grading of S is shifted by the dimension of the boundary. Define ideals K V R V and I V Q V by We may drop the subscript V from the notations Q V , R V , I V , K V , in statements that hold for all choices of V. Denote by Γ V ∈ V ⊗ Q V the vector field corresponding to the parity operator P : V → V given by P(v) = (−1) deg v v. That is, Below, we assume that each index set I V for a basis of a vector space V is endowed with an order, and we implicitly use this order in every graded commutative product over i ∈ I V . We denote by I V the same set with the order reversed. We reserve the formal variables {s i } i∈I S for coordinate functions on the vector space S. We abbreviate s k = i∈I S s k i i and k! = i∈I S k i !. )] = β ∈ Π with one boundary component, k + 1 boundary marked points, and l interior marked points. The boundary points are labeled according to their cyclic order. The space M k+1,l (β) carries evaluation maps associated to boundary marked points evb β j : M k+1,l (β) → L, j = 0, . . . , k, and evaluation maps associated to interior marked points evi β j : M k+1,l (β) → X, j = 1, . . . , l.
Let M l+1 (β) be the moduli space of genus zero J-holomorphic stable maps u : Σ → X of degree u * ([Σ]) = β ∈ H 2 (X; Z) with l + 1 marked points. The space M l+1 (β) carries evaluations maps ev β j : M l+1 (β) → X, j = 0, . . . , l. We assume that all J-holomorphic genus zero open stable maps with one boundary component are regular, the moduli spaces M k+1,l (β; J) are smooth orbifolds with corners, and the evaluation maps evb β 0 are proper submersions. Furthermore, we assume that all the moduli spaces M l+1 (β) are smooth orbifolds and ev 0 is a submersion. Examples include (CP n , RP n ) with the standard symplectic and complex structures or, more generally, flag varieties, Grassmannians, and products thereof. See Example 1.5 and Remark 1.6 in [31]. Throughout the paper we fix a connected component J of the space of ω-tame almost complex structures satisfying our assumptions. All almost complex structures are taken from J . The results and arguments of the paper extend to general target manifolds with arbitrary ω-tame almost complex structures if we use the virtual fundamental class techniques of [6,7,[9][10][11]. Alternatively, it should be possible to use the polyfold theory of [15][16][17][18]23]. See Section 1.3.12 for a detailed discussion on regularity assumptions.

1.2.4.
Operations. We encode the geometry of the moduli spaces of open stable maps in operations q k, l : The push-forward (evb β 0 ) * is defined by integration over the fiber; it is well-defined because evb β 0 is a proper submersion. Intuitively, the η i should be thought of as interior constraints, while α j are boundary constraints. Then the output is a cochain on L that is "Poincaré dual" to the image of the boundaries of disks that satisfy the given constraints.
We define similar operations using moduli spaces of stable maps, as follows. Recall that the relative spin structure s on L determines a class w s ∈ H 2 (X; Z/2Z) such that w 2 (T L) = i * w s . By abuse of notation we think of w s as acting on H 2 (X; Z). Set The sign (−1) ws(β) is designed to balance out the sign of gluing spheres as explained in [31,Lemma 2.12]. Below, we use the same notation for the linear extensions of these operations to spaces of differential forms with larger coefficient rings.
is called a bounding pair if dγ = 0, |γ| = 2, |b| = 1, and there exists c ∈ K V such that |c| = 2 and the Maurer-Cartan equation holds, In this case, we call b a bounding chain. Let W ⊂ H * (X, L; R) be a graded vector subspace. A bounding pair over W is a bounding pair A bounding chain is point-like if the vector space S is one-dimensional and L b = s, where s is the single coordinate on S.
The definition of a bounding chain is due to Fukaya-Oh-Ohta-Ono [8]. The notion of a point-like bounding chain is due to [33]. In Remark 4.17 we explain why generically, all open Gromov-Witten invariants can be obtained from point-like bounding chains.
1.2.6. Gromov-Witten potential. Define a bilinear form on A * (X) by The pairing ·, · X descends to the Poincaré pairing on cohomology, for which we use the same notation. Let U ⊂ H * (X; R) be a linear subspace, and let γ U ∈ I U A * (X; Q U ) satisfy dγ U = 0 and [γ U ] = Γ U .
Consider the formal function Writing where GW β (η 1 , . . . , η N ) denotes the closed Gromov-Witten invariant. The sign (−1) ws(β) compensates for the sign in equation (2). The gradient of Φ U with respect to ·, · X is given by It is well-known that q ∅,l (γ ⊗l U ) is closed and that Φ U only depends on the cohomology class of γ U . 1.2.7. Quantum cohomology. The big quantum product It is well known to be commutative and associative. See Remark 3.6. Moreover, the Poincaré pairing makes (H * (X; Q U ), U ) a Frobenius algebra, We denote this Frobenius algebra by QH U (X).
1.3.1. Relative potential. The usual superpotential of a Lagrangian submanifold L ⊂ X does not give invariant counts of J-holomorphic disks in X with boundary contractible in L and no boundary constraints. The lack of invariance stems from the possibility of the boundary of such disks collapsing to a point in a 1 parameter family. In order to formulate the open WDVV equations, we define an enhanced superpotential that gives invariant counts of all types of J-holomorphic disks in X with boundary in L. Invariance is achieved by including certain contributions from J-holomorphic spheres that cancel the boundary collapse phenomenon. We begin by defining a relative potential that counts both J-holomorphic disks and J-holomorphic spheres. The natural home for the relative potential is the following cone complex. Let W be a graded real vector space and consider the map of complexes of Q W modules where R W [−n] is equipped with the trivial differential. The cone C(i) is the complex with underlying graded Q W module A * (X; Q W ) ⊕ R W [−n − 1] and differential In Section 4.3, we show that d C ψ(γ, b) = 0. Thus, we define the relative potential Ψ(γ, b) to be the cohomology class of ψ(γ, b). Definition 2.23 gives a notion of gauge-equivalence between a bounding pair (γ, b) with respect to J and a bounding pair (γ , b ) with respect to J . We prove the following.
The relative potential and the closed potential. Let denote the natural map. Let and suppose (γ W , b) is a bounding pair over W. Denote by ρ * : Q U → Q W the induced map of rings and let π : H * (C(i)) → H * (X, Q W ) be the natural map. We show in Lemma 5.9 that That is, the relative potential Ψ(γ W , b) lifts the gradient of the closed Gromov-Witten potential ∇Φ U to the cohomology of the cone complex H * (C(i)).
1.3.3. Enhanced superpotential. From the long exact sequence of the cone, i v v n n n n n n n n n n n n we obtain an exact sequence, We choose P : H * (C(i)) −→ Coker i, a left inverse to the mapx from the exact sequence (8) satisfying natural conditions detailed in Section 4.4. The choice of P is equivalent to the choice of a left inverse to the mapȳ : Coker i R → H * (X, L; R) induced by the map y from the long exact sequence i R y y r r r r r r r r r r R[−n].
So, equation (12) holds in R W with Ω replaced by Ω and OGW replaced by OGW . Lemma 4.13 shows that Ω and therefore the invariants OGW are independent of the choice of P. Thus, the choice of P only influences the invariants OGW β,k for k = 0 and β ∈ Im .
Remark 1.3. Lemma 4.14 shows the superpotential Ω coincides with superpotential defined in [33]. For (γ, b) a bounding pair of the type considered in [33], the open Gromov-Witten invariants OGW defined here coincide with those defined there.
1.3.5. Open WDVV equations. Recall the map ρ from (7). To formulate the open WDVV equations, we need the following two assumptions: More explicitly, assumption (A.1) means that U is a subalgebra with respect to the big quantum product U , and the restriction of the Poincaré pairing to U is non-degenerate. All point-like bounding chains satisfy assumption (A.2). Both assumptions are satisfied in the cases discussed in [33] as explained in Section 1.3.9 below. The map P R from (9) determines a complement W = Ker(P R | W ) ⊆ W to the image of the map y from the exact sequence (10).
In particular, ρ| W is injective. Choose index sets I W ⊆ I U , a basis ∆ i ∈ U, i ∈ I U , and a basis Γ i ∈ W , i ∈ I W , such that ρ(Γ i ) = ∆ i . By abuse of notation, denote by ∂ i : and let (g ij ) be the inverse matrix to (g ij ), which exists by assumption (A.1). Abbreviate Φ = Φ U ∈ Q U and Ω = Ω(γ W , b) ∈ R W . Let ρ * : Q U → Q W denote the induced ring homomorphism as in Section 1.3.2. We are now ready to formulate the open WDVV equations.
1.3.6. The tensor potential and the relative quantum connection. To prove Theorem 3, we construct an invariant ‫נ‬ ∈ End(C(i)) called the tensor potential. The tensor potential ‫נ‬ is closely related to the total derivative of the relative potential ψ. The derivative of the tensor potential is the connection 1-form of the relative quantum connection. In greater detail, let (γ, b) ∈ I W A * (X, L; Q W ) ⊕ K W A * (L; R W ) be a bounding pair. We define the tensor potential 1 The open WDVV equations and in fact the closed WDVV equations as well are a consequence of the following theorem. The notation ∂ u ‫נ‬ is explained in detail in Section 4.2.

Theorem 5. For all formal vector fields
Since H * (C(i)) is a free Q W module, it can be viewed as the formal sections of a vector bundle over W. The tensor potential ‫נ‬ induces a map ‫נּ‬ : H * (C(i)) → H * (C(i)). We define the relative quantum connection ∇ on H * (C(i)) by A straightforward calculation using Theorem 5 gives the following.  Geometrically, we can understand Theorem 6 as follows. Poincaré-Lefschetz duality gives an isomorphism Under this isomorphism, Γ ∈ H n+1 (X, L; R) corresponds to the class in H n−1 (X \ L; R) of a small (n−1)-dimensional sphere Σ linked with L. As we shrink Σ, it converges to a point in L. Thus, at an intuitive level, it makes sense that the interior constraint Γ can be swapped with a point boundary constraint. To see why Theorem 6 is called the wall-crossing formula, consider the following scenario. Let M be an (n − 1)-dimensional manifold, and let g : [0, 1] × M → X be a map transverse to L such that g −1 (L) is a single point in (t L , m L ) ∈ (0, 1) × M. Let g t : M → X be given by g t (p) = g(t, p). For i = 0, 1, let η i 0 ∈ H n+1 (X, L; R) be the class corresponding to g i * ([M ]) ∈ H n−1 (X \ L, R) under Poincaré-Lefschetz duality. Then, it is easy to see that η 1 0 − η 0 0 = Γ . So, OGW β,k (η 1 0 , η 1 , . . . , η l ) − OGW β,k (η 0 0 , η 1 , . . . , η l ) = OGW β,k (Γ , η 1 , . . . , η l ). On the other hand, roughly speaking, the invariant OGW β,k (η i 0 , η 1 , . . . , η l ) counts disks of degree β with boundary constrained to pass through k points in L and interior constrained to pass through g i (M ) as well as Poincaré duals of η 1 , . . . , η l . To compare the invariants for i = 0, 1, consider the one dimensional family of interior constraints g t (M ). At times t = t L , the invariant is constant. At time t L , the interior constraint g t L (M ) intersects L at the unique point g t L (m L ). As t → t L from below, the interior intersection points with g t (M ) of a subset B of the disks being counted limit to a boundary point. These disks are no longer counted for t > t L . So, the number of disks in B is OGW β,k (η 1 0 , η 1 , . . . , η l ) − OGW β,k (η 0 0 , η 1 , . . . , η l ). On the other hand, at t = t L , the boundaries of the disks in B pass through k + 1 points in L, one of which is g t L (m L ), and the interiors of these disks pass through Poincaré duals to η 1 , . . . , η l . So, the number of disks in B is OGW β,k+1 (η 1 , . . . , η l ).
The map ‫נ‬ : C(i) → C(i) induces a map‫נ‬ : C(i) → C(i), which in turn induces a map ‫נּ‬ : H * ( C(i)) → H * ( C(i)). Since C(i) is the cone of the map i : A * (X; Q W ) → Q W , we have a canonical isomorphism H * (X, L; Q W ) H * ( C(i)).
We define the relative quantum cohomology as a vector space by QH U (X, L) = W ⊗ Q W , and we define the relative quantum product 2 . A more explicit formula for ‫מ‬ is given in Lemma 5.5.
Theorem 7. The relative quantum product ‫מ‬ is graded commutative, associative, and depends only on the gauge-equivalence class of the bounding pair (γ W , b). The following is an immediate consequence of Theorem 2 of [33]. See Section 5.4 of [33] for further details. The open Gromov-Witten invariants OGW associated with such (γ W , b) coincide with those of [33].
A real setting is a quadruple (X, L, ω, φ) where φ : X → X is an anti-symplectic involution such that L ⊂ Fix(φ). Whenever we discuss a real setting, we fix a connected subset J φ ⊂ J consisting of J ∈ J such that φ * J = −J. All almost complex structures of a real setting are taken from J φ . If we use virtual fundamental class techniques, we can treat any ω-tame almost complex structure J satisfying φ * J = −J. In addition, whenever we discuss a real setting, we take S L ⊂ H 2 (X, L; Z) with Im(Id +φ * ) ⊂ S L , so φ * acts on Π L = H 2 (X, L; Z)/S L as − Id . Also, the formal variables t j have even degree. Given a real setting, let H even φ (X) (resp. H even φ (X, L)) denote the direct sum over k of the (−1) k -eigenspaces of φ * acting on H 2k (X; R) (resp. H 2k (X, L; R)). Extend the action of φ * to Λ, Q, R, C, and D, by taking Elements a ∈ Λ, Q, R, C, D, and pairs thereof are called real if The main ingredient in the following is Theorem 3 of [33]. See Appendix A for details.
Theorem 9. Suppose (X, L, ω, φ) is a real setting, s is a spin structure, and n ≡ 1 (mod 4). Moreover, (c) There exists a unique up to gauge equivalence real bounding pair (γ W , b) over W such that b is point-like.
1.3.10. Special case: projective space. Consider the special case (X, L) = (CP n , RP n ) with ω = ω F S the Fubini-Study form, J = J 0 the standard complex structure, and n odd. We normalize ω so that CP 1 ω = 1. Take Π = H 2 (X, L; Z) and identify Π with Z in such a way that β ∈ Π with ω(β) ≥ 0 are identified with non-negative integers. Similarly, identify H 2 (X; Z) with Z in such a way thatβ ∈ H 2 (X; Z) with ω(β) ≥ 0 are identified with nonnegative integers. So, the map : H 2 (X; Z) → Π is given by multiplication by 2. By Theorem 8, our results apply with W = H * (X, L; R). Write Γ j = [ω j ] ∈ H * (X, L; R) and ∆ j = [ω j ] ∈ H * (X; R). We take the map P R of (9) to be the unique left inverse ofȳ such that W = Ker(P R | W ) = Span{Γ j } n j=0 and this determines the map P. The following is a consequence of Corollary 1.6, the Kontsevich-Manin axioms [22], and analogous axioms for open Gromov-Witten invariants given by Proposition 4.19. The proof is given in Section 6.
Moreover, the recursion process readily implies Corollary 6.2, which says the invariants are rational numbers with denominator a power of 2. The denominators arise from the divisor axiom.
1.3.11. Sample values for projective space. We continue with the setting and notation of the preceding section. Below, we write OGW n β,k for invariants of (CP n , RP n ).  Table 1. Sample values with only boundary constraints The invariants OGW 3 β,k coincide with the analogous invariants of Welschinger [35] up to a factor of ±2 1−l by Theorem 5 of [33]. We have verified this for small values of n, l, β, by comparing the tables in [2,3] with computer calculations based on Theorem 10.
On the other hand, we are not aware of a definition of open Gromov-Witten invariants generalizing Welschinger's invariants with k > 0 real point constraints in dimensions n > 3 besides the invariants OGW β,k . In Table 1, we present the results of computer calculations based on Theorem 10, which show these invariants are non-trivial.
For i 1 , . . . , i l , odd, the invariants OGW n β,0 (Γ i 1 , . . . , Γ i l ) coincide with the analogous invariants of Georgieva [12] up to a factor of 2 1−l by Theorem 6 of [33]. We have verified this for small values of n, l, β, by comparing the tables in [13] with computer calculations based on Theorem 10.
On the other hand, if one or more of i 1 , . . . , i l is even, the invariants of [12] vanish, while the invariants OGW n β,0 (Γ i 1 , . . . , Γ i l ) are often non-vanishing. See Tables 2 and 3, which present results of computer computations based on Theorem 10.  Table 2.
). The value of l 1 is determined by β and l 2 .  Table 3. Values of OGW 7 β,0 (Γ ⊗l 1 2 ⊗ Γ ⊗l 2 6 ). The value of l 1 is determined by β and l 2 . Here we chose to take no constraints in Γ 4 .
The reliance on general bounding chains is the main difference between the invariants OGW β,k and the invariants of Welschinger and Georgieva. In the situations where the invariants OGW β,k coincide with Welschinger's and Georgieva's invariants, bounding chains become explicit: either zero or an n-form with integral s. However, the general construction of bounding chains in Theorems 2 and 3 of [33], upon which we rely in Theorems 8 and 9, uses an inductive argument based on the obstruction theory of [8]. It is difficult to give an explicit description of the resulting bounding chain. Nonetheless, the results of this paper allow explicit calculations of the invariants OGW β,k .
Write n = 2r + 1. To illustrate the geometric significance of the relative cohomology H * (X, L) and the wall-crossing formula, we consider the real analog of the classical result that there are r +1 complex lines in CP n through 4 generic complex subspaces of dimension r. Real lines correspond to conjugate pairs of holomorphic disks of degree 1. When β = 1, it is not hard to see that the invariants OGW β,k enumerate disks of degree 1; the bounding chain plays a role only when β > 1. Recall that the class ∆ r+1 = [ω r+1 ] ∈ H n+1 (CP n ) is Poincaré dual to the class of an r plane in H n−1 (CP n ). However, the class Γ r+1 = [ω r+1 ] ∈ H n+1 (CP n , RP n ) is not Poincaré dual to the class of an r plane in H n−1 (CP n \ RP n ). Rather, the classes are Poincaré dual to two distinct classes of r planes in H n−1 (CP n \ RP n ). We have The first two values are stated in Theorem 10 and the third is a consequence of equation (a) and the divisor axiom. Applying the wall-crossing formula of Theorem 6, we obtain OGW n 1,0 (Γ , Γ ) = 2, OGW n 1,0 (Γ r+1 , Γ ) = 0. Thus, it follows by multi-linearity that OGW n 1,0 (λ ± , λ ± ) = 0, OGW n 1,0 (λ ± , λ ∓ ) = −1 When n ≡ 1 (mod 4) the classes λ ± are anti-conjugate, so the Poincaré dual of a conjugate pair of complex r planes is 2Γ r+1 = λ + + λ − . Thus, an invariant count of real lines through two conjugate pairs of complex r planes is half of the corresponding disk count: As expected, this agrees with the complex count mod 2. When n ≡ 3 (mod 4), the classes λ ± are conjugation invariant, so the Poincaré dual of a conjugate pair of complex r planes may be either 2λ + or 2λ − . Thus, there are four possible invariant counts of real lines through two conjugate pairs of complex r planes: Again, these invariants agree with the complex count mod 2. In [21, Example 12], Kollár constructs a generic configuration of two conjugate pairs of complex r planes of the same class, such that there is no real line that intersects them. This shows that the vanishing invariant is optimal for such pairs. However, for conjugate pairs of complex r planes of different classes, we obtain a positive lower bound of 2.
1.3.12. Regularity assumptions. We proceed with the regularity assumptions set in [31], namely, that moduli spaces are smooth orbifolds with corners and the evaluation maps at the zero point are proper submersions. To that we add in Section 3 the assumption that the zero evaluation maps remain submersions after restricting to a subspace of open stable maps where certain marked points are constrained to lie on a geodesic of the hyperbolic metric of the disk. In [31, Example 1.5 and Remark 1.6] we show that the regularity assumptions hold for homogeneous spaces. The additional assumption concerning moduli spaces of open stable maps with geodesic constraints on marked points holds for homogeneous spaces as well. Indeed, suppose J is integrable and suppose there exists a Lie group G X that acts transitively on X by J-holomorphic diffeomorphisms. Furthermore, suppose there exists a subgroup G L ⊂ G X that preserves L and acts transitively on L. Let M k,l;a,b (β) ⊂ M k,l (β) be a moduli space with a geodesic constraint, as defined in Section 3. Then G L acts on M k,l;a,b (β) as well, and the evaluation maps are equivariant. Since G L acts transitively on L, we see that evb 0 remains a submersion after restricting to M k,l;a,b (β).
In particular, (CP n , RP n ) with the standard symplectic and complex structures, or more generally, Grassmannians, flag varieties and products thereof, satisfy our regularity assumptions. Using the theory of the virtual fundamental class from [6,7,[9][10][11] or [15][16][17][18]23], our results extend to general target manifolds. 2. Background 2.1. Integration properties. In the following, M, N and P are orbifolds with corners. We follow the conventions of [32] concerning smooth maps and orientations of orbifolds with corners except that here we require by definition that a proper submersion satisfies the additional property of strong smoothness. Let f : M → N be a proper submersion with fiber dimension rel dim f = r, and let Υ be a graded-commutative algebra over R. Denote by the push-forward of forms along f , that is, integration over the fiber, as defined in [32]. We will need the following properties of f * proved in [32]. In particular, property (a) of Proposition 2.1 is immediate from Definition 4.14 of [32] while the remaining properties of f * are given in Theorem 1. Then be a pull-back diagram of smooth maps, where g is a proper submersion. Let α ∈ A * (P ). Then q * p * α = f * g * α. Then where ∂M is understood as the fiberwise boundary with respect to f.
The sign arises from the possibly non-trivial grading of the coefficient ring. See [31,Remark 2.3] for an extended discussion.
The following result is Lemma 5.4 of [30]. . . , w l ), with z j ∈ ∂Σ, w j ∈ int(Σ), distinct. The labeling of the marked points z j respects the cyclic order given by the orientation of ∂Σ induced by the complex orientation of Σ. Stability means that if Σ i is an irreducible component of Σ, then either u| Σ i is nonconstant or it satisfies the following requirement: If Σ i is a sphere, the number of marked points and nodal points on Σ i is at least 3; if Σ i is a disk, the number of marked and nodal boundary points plus twice the number of marked and nodal interior points is at least 3. An open stable map is called irreducible if its domain consists of a single irreducible component. An isomorphism of open stable maps (Σ, u, z, w) and (Σ , u , z , w ) is a homeomorphism θ : Σ → Σ , biholomorphic on each irreducible component, such that Denote by M k+1,l (β) = M k+1,l (β; J) the moduli space of J-holomorphic genus zero open stable maps to (X, L) of degree β with one boundary component, k + 1 boundary marked points, and l internal marked points. Denote by the evaluation maps given by evb β j ((Σ, u, z, w)) = u(z j ) and evi β j ((Σ, u, z, w)) = u(w j ). We may omit the superscript β when the omission does not create ambiguity.
is a chain map.
The following lemmas are well known.
is a chain map.
Recall the notion of bounding pairs from Definition 1.1.
Definition 2.23. We say a bounding pair (γ, b) with respect to J is gauge-equivalent to a bounding pair (γ , b ) with respect to J , if there existγ ∈ IA * (X; Q) andb ∈ KA * (L; R) such that In this case, we say that (γ,b) is a pseudo-isotopy from (γ, b) to (γ , b ) and write (γ, b) ∼ (γ , b ).

Geodesic conditions
3.1. Geodesic operators. Let a, e ∈ Z ≥0 such that a + e = 3, and let M k+1,l;a,e (β) ⊂ M k+1,l (β) be the closure of the subspace consisting of one-component maps such that a of the boundary points and the first e of the interior points lie on a common geodesic in the domain with respect to the hyperbolic metric. When we need to specify which of the boundary points are taken to lie on a geodesic, we add their labels as sub-indices to a, in which case the order of the indices indicates the order in which the points appear on the geodesic. If not indicated explicitly, the points are assumed to appear according to their labeling order. For example, M k+1,l;2 0,k ,1 (β) is the space of stable disks with k + 1 boundary and l marked points, such that the first interior point lies on the geodesic between the zeroth and last boundary points. In M k+1,l;1 0 ,2 (β), the geodesic starts at the zeroth boundary point and passes through the first and second interior points, in that order. As mentioned in Section 1.3.12, we assume that evb 0 | M k+1,l;a,e (β) is a proper submersion.
To determine the orientation on M k+1,l;a,e (β), it is useful to identify it with a fiber product of oriented orbifolds, as follows. Denote by v 1 , v 2 , v 3 ∈ {z 0 , . . . , z k , w 1 , . . . , w l } the marked points that lie on the geodesic, labeled according to the order in which they appear on the geodesic. Given a nodal Riemann surface with boundary Σ with complex structure j, denote by Σ a copy of Σ with the opposite complex structure −j. For a point v ∈ Σ, letv ∈ Σ denote the corresponding point. The complex double Σ C = Σ ∂Σ Σ is a closed nodal Riemann surface, so it is possible to define the cross ratio of four points on Σ C as in [24,Appendix D.4]. We define On the irreducible locus of M k+1,l (β), the domain Σ can be identified with the upper half plane and χ has the explicit formula Note that the second marked point on the geodesic is necessarily an interior point, sov 2 = v 2 , and thus χ is well defined. Then the condition that v 1 , v 2 , v 3 , lie on a geodesic is equivalent to the condition χ(u, z, w) ∈ [0, 1]. Thus, we have and the fiber product identification determines orientation, as in [32]. Denote by q β k,l;χ = q β k,l;a,e : A * (L; R) ⊗k ⊗ A * (X; Q) ⊗l −→ A * (L; R) the operators defined analogously to q β k, l with M k+1,l;a,e (β) in place of M k+1,l (β). Explicitly, (j + 1)(|α j | + 1) + 1.
Again, specifying boundary points and the order of the points on the geodesic can be done by adding a sub-index to a and e. Lastly, consider the moduli space M l+1 (β) of spheres with l + 1 marked points w 0 , . . . , w l . Let Let B be a list of indices and let η = (η j ) j∈B ⊂ A * (X; Q). For a sublist I ⊂ B, denote by η I the list (η j ) j∈I . For a partition I J of B into two ordered sub-lists, define sgn(σ η I J ) by the equation where the wedge products are taken in the order of the respective lists. Explicitly, sgn(σ η I J ) ≡ i∈I,j∈J j<i |η i | · |η j | (mod 2).
The proof is similar to that of Proposition 3.5 in [31].

3.3.2.
Linearity. The following is a direct analog of the linearity properties of the usual q operators. The signs reflect the fact that the map of shifted complexes q k,l;χ : A * (L; R) [1] ⊗k ⊗ A * (X; Q) ⊗l −→ A * (L; R) [1] has degree 0 (mod 2).

3.3.3.
Unit on the geodesic. The followings lemmas concern geodesic operators where the unit is fed to one of the inputs constrained to the geodesic. They have no direct analog for the usual q operators. Lemma 3.9.
The proofs of this lemma and the next follow after Lemma 3.11.
Proof of Lemma 3.10. Let evi j , evb j , be the evaluation maps on M k,l (β), and set Let p i : M k+1,l;χ i (β) → M k,l (β) be the forgetful map from Lemma 3.11 with v 2 , v 3 , taken to be w 1 , w 2 , for the first equation, or w 1 , z 0 , for the second equation. Varying i, we get a diffeomorphism onto an open dense subset of the irreducible stratum: See Figure 6.
Let evi j , evb j , be the evaluation maps on M k,l (β), and let evi i j , evb i j , be the evaluation maps on M k+1,l;χ i (β). In particular, Then, by Proposition 2.1(c),

3.3.4.
Reversing the geodesic. We consider the effect on the geodesic operators of reversing the order of the marked points constrained to the geodesic.
The proof is similar to that of Proposition 3.6 in [31].

3.4.
Deformed q operators. Let γ ∈ IA * (X; Q) such that |γ| = 2 and dγ = 0. Let b ∈ KA * (L; R) such that |b| = 1. Define For k ≥ 0, define Define also For j > 0, the deformed geodesic q operators are given by   (17) we have defined the operator Lemma 4.1. For any η ∈ A * (X; Q W ), we have . By Lemmas 2.12 and 2.11, This proves Proof. For any space, denote by pt the map from it to a point. By Proposition 2.1 (c), we have Proof of Theorem 2. By assumption, L i * η = 1. Therefore, by Corollary 4.2, we have Lemma 4.4. The map ‫נ‬ is a chain map.
Proof of Theorem 4. The first part is given by Lemma 4.4. The second part is given by Lemma 4.7.

4.2.
Flatness relation. The objective of this section is to prove Theorem 5. We begin by explaining the notation in greater detail.
Let W and S be graded real vector spaces. In this section, it will be useful to note that elements of W and S define derivations on , which in turn induces a derivation on A * (X; Q W ) = A * (X; R) ⊗ Q W . In addition, the derivation on Q W extends trivially to R W . In total, we get For s ∈ S, the derivation ∂ s extends trivially from R[[S]] to R W , and acts trivially on A * (X; Q W ). This defines It is immediate from definition that ∂ u , ∂ s , are chain maps, and so descend to cohomology. Furthermore, for u ∈ Q W ⊗ W of the form u = r ⊗ w, we define ∂ u = r · ∂ w , and similarly for u ∈ Q W ⊗ S. Moreover, since ∂ s for s ∈ S acts as zero on the first component of C(i), we can in fact define ∂ u for u ∈ R W ⊗ S, and it is a chain map. In other words, there are chain map derivations Finally, for a chain map the derivative operator ∂ u Θ is defined by The four summands of the second component of H uv correspond to the four pictures in Figure 7. We show that The heart of the proof is based on an analysis of the boundaries of the moduli spaces of stable disks with geodesic constraints shown in Figure 7 using the geodesic structure equations. In preparation for this analysis, we expand both sides of equation (29) into their constituent parts. Let (η, ξ) ∈ C(i). By Lemma 2.16 the derivative ‫נ‬ operator is given by To compute the left-hand side of (29), first calculate Symmetrizing, we get Compute the right hand side of equation (29): Equality in the first component of (29) follows from Proposition 3.5. Equality in the second component reads To show this, we use the deformed versions of the geodesic structure equations. Figure 8 depicts the boundary components of the moduli spaces A,B,C,D, in Figure 7 that contribute to our calculation. First, keeping in mind line D of Figure 8, consider the contribution from Proposition 3.4: Lemma 3.7 gives |q γ ∅,3;(0,1,2,3)∈(1,∞) (∂ u γ, η, ∂ v γ)| ≡ |u| + |η| + |v| + 1 (mod 2), so Using the Maurer-Cartan equation, q γ,b 0,0 = c · 1, we can apply the unit property, Lemma 3.13, to get Multiply the equation by (−1) 1+|η|+|u|+|v|+|v||η| to get ). The summands of equation (32) correspond to the pictures in line D of Figure 8.
Lastly, keeping in mind line A of Figure 8, compute the contribution of Proposition 3.1, again using q γ,b 0,0 = c · 1 and Lemma 3.13: Pairing with ∂ u b and using the cyclic properties of Lemmas 2.8 and 3.15 together with the degree properties of Lemmas 2.9 and 3.7, we get Apply Lemma 3.12 to the last summand of the right hand side of the preceding equation and multiply the entire equation by (−1) (|η|+1)(|u|+|v|+1)+|u||v| to get The summands of equation (34) correspond to the pictures in line A of Figure 8.

4.4.
Enhanced superpotential. We begin by giving a full account of the conditions imposed on the map P discussed in Section 1.3.3. There is a natural map of complexes A(X, L) → C(i) given by η → (η, 0). Denote by a : H * (X, L; R) → H * (C(i)) the induced map on cohomology.
Consider the commutative diagram of long exact sequences, i w w n n n n n n n n n n n n w w n n n n n n n n n n n n where a is injective by the five lemma. Observe that for this diagram to commute, we need the map x to be given at chain level by r → (0, −r). There is a canonical chain map denote the induced map on cohomology. Let q : R W → R W /Q W denote the quotient map, and letq : Coker i → R W /Q W be the induced map. We obtain the following diagram with exact rows and columns. 0 0 We choose P : H * (C(i)) −→ Coker i, a left inverse to the mapx from the diagram (37) satisfying the following two conditions. The first condition is thatq Note that if [L] = 0, thenq is an isomorphism, so this determines P completely. Condition (38) and the exactness of diagram (37) imply that there exists a unique P Q : H * (X, L; Q W ) → Coker i Q such that the following diagram commutes.
The second condition is that there exists P R : H * (X, L; R) → Coker i R , such that Recall the exact sequence (10). Denote byȳ : Coker i R → H * (X, L; R) the induced map. Proof. By the exactness of diagram (37), the mapā is injective. So, to prove P Q •ȳ Q = Id, it suffices to prove thatā • P Q •ȳ Q =ā. By commutativity of diagrams (39) and (37), we obtain To see P R •ȳ = Id, observe thatȳ Q =ȳ ⊗ Id Q . and the splitting lemma implies that H * (C(i)) Imx ⊕ a(Ker l Q ).
Take P to be the unique left inverse ofx such that Ker P = a(Ker l Q ). Diagram (37) and the above splittings imply condition (38) and l Q = P Q . Condition (40) follows.
Following [33], we write Ω = (−1) n q γ,b −1,0 . Remark 4.15. In [33] we defined By Lemma 4.14 and the condition q(Ω) =q(Ω), this definition coincides with the definition of equation (13)   Proof. By Lemma 4.9, Lemma 4.14 and the Maurer-Cartan equation (3) give Let U ⊂ H * (X; R) be a subspace, and let (γ W , b) be a bounding pair over W := ρ −1 (U ) ⊂ H * (X, L). Recall that in Section 1.3.7 we define Γ := y(1). By abuse of notation, denote by ∂ : R W → R W the derivation corresponding to Γ . Similar axioms for the invariants OGW were proved in [33]. In the following, we assume that the subspace W ⊂ H * (X, L; R) and the bounding pair (γ W , b) are as in Theorem 8 or Theorem 9. In particular, writing W i ⊂ W for the degree i homogeneous part, by Lemma 5.11 of [33], there is a natural inclusion So, for A ∈ W 2 , and β ∈ H 2 (X, L; Z), the pairing β A is well-defined.

5.2.
Relative quantum product. Recall that in Section 1.3.8 we defined Q W = Q W ⊗ Λc Λ and As noted, there is a natural isomorphism H * (X, L; Q W ) H * ( C(i)). In particular, we can think of QH U (X, L) as a subspace of H * ( C(i)).
π u u l l l l l l l l l l l l l H * (X; Q W ) By Lemma 5.2, we have Tensoring with Q W , we get the required result.
Proof. For any f ∈ Q W , by definition of the graded Lie bracket, On the other hand, since the canonical connection on affine space is symmetric, we have Taking f = q γ W ,b −1,0 , and using also Lemma 2.16, we see that the right-hand side in Lemma 5.5 is graded symmetric in u, v, which implies part (a). Keeping in mind the shifted grading of R[−n − 1] in the definition of C(i), part (b) follows from Lemmas 2.15 and 2.9.
Proof of Theorem 7. Graded commutativity and associativity are given by Lemmas 5.6(a) and 5.7, respectively. Invariance under gauge equivalence follows from Lemma 5.3(a).

Open WDVV.
In the following, we make some calculations that will be useful in the proof of Theorem 3. As in Section 1.3.5, abbreviate Lemma 5.8. Let r ∈ Coker(i).
Recall the definitions of W and V from equations (48) and (52) respectively.
(55) Thus, since Imx ⊂ V, it suffices to show that V ∩ Ker P = a(W ⊗ Q W ).
By Lemma 4.11, we have Ker P = a(Ker P Q ). Thus, On the other hand, . Therefore, V ∩ Ker P = a(W ⊗ Q W ) and the lemma follows.
Lemma 5.12. Let u ∈ W and v ∈ W ⊕ S, let v W be the projection of v to W, and let u = ρ(u),v = ρ(v W ). We have Proof. By Lemmas 5.2 and 5.11, we obtain a unique decomposition To compute r, recall that P •x = Id and Υ m ∈ Ker P . By Lemma 4.9, we have To find r m , take l ∈ I U and compute On the other hand, So, m∈I W r m g ml = ρ * (∂v∂ū∂ l Φ).
To get the open WDVV equation (14), apply P .
Remark 5.13. From equation (57) we can obtain the standard WDVV equation for closed genus zero Gromov-Witten invariants by applying π and pairing via , X with Γ e .
Proof of Corollary 1.6. Use Lemma 4.16 as follows. If [L] = 0, then c = ∂ s Ω and the equations follow. If [L] = 0, then c = ∂ s Ω. Let z : R W /Q W → R W be the unique right-inverse of q such that D • z = 0. In particular, z(Ω) = Ω. Observe that if u ∈ W , then for all g ∈ R W /Q W we have ∂ u (z(g)) = z(∂ u g). The desired equation then follows from applying z to equation (14).

Computations for projective space
The objective of this section is to describe a recursive process for the computation of our invariants for (X, L) = (CP n , RP n ) n odd. As in Section 1.3.10, take ω = ω F S the Fubini-Study form, J = J 0 the standard complex structure, and Π = H 2 (X, L; Z). Equip (X, L) with a relative spin structure. Thus, Theorem 8 holds, and we have a bounding pair (γ, b) over W = H * (X, L; R) with b point-like. Abbreviate Γ j = [ω j ] ∈ H * (X, L; R) and Γ j = [ω j ] ∈ H * (X; R). Observe that [L] = 0 ∈ H n (X; Z). So, together with Γ = y(1), the classes Γ j form a basis of H * (X, L; R). Denote by t j ∈ R[[W ]] the coordinates on W corresponding to Γ j for j = 0, . . . , n, . Let s ∈ S be the coordinate on S with L b = s. Identify H 2 (X, L; Z) with Z so that the non-negative integers correspond to classes β ∈ H 2 (X, L; Z) with ω(β) ≥ 0. By Lemma 4.11, choose P : H * (C(i)) → Coker i to be the unique left inverse tox satisfying conditions (38) and (40) such that P R (Γ j ) = 0 for j = 0, . . . , n. Thus, by definition (48) we have W = Span{Γ j } n j=0 .
Proof. For any space, denote by pt the map from it to a point. Recall Remark 4.15. Since γ ∈ I W A * (X, L; Q W ) and b ∈ K W A * (L; R W ), Lemmas 2.8, 2.11, and Proposition 2.1 (c), imply that the coefficient of T 1 in −∂ 2 s Ω| s=t j =0 is (∂ 2 s q β 1 ;γ,b −1,0 )| s=t j =0 =(∂ 2 s q β 1 ;0,b −1,0 )| s=t j =0 = q β 1 1,0 (∂ s b), ∂ s b =(−1) n+1 pt * ((evb 0 ) * (evb * Let Y = L × L \ and let Z = (evb 0 × evb 1 ) −1 (Y ) ⊂ M 2,0 (1). It is easy to see that g = (evb 0 × evb 1 )| Z : Z → Y is a covering map. Indeed, the fiber of g over a point (x 1 , x 2 ) ∈ Y can be identified with the pair of oriented lines in RP n passing through x 1 and x 2 . We claim that for an appropriate choice of relative P in structure, the degree of g is −2. Indeed, the two points in the preimage of a point in Y are conjugate disks of degree 1 with two marked points. Proposition 5.1 of [28] shows that the covering transformation that interchanges these two disks is orientation preserving, so the degree is ±2. Finally, Lemma 2.10 of [28] shows that one can choose the relative spin structure on L so as to make the degree −2.
For i = 1, 2, let p i : L × L → L be the projections. Since Y ⊂ L × L and Z ⊂ M 2,0 (β) are open dense subsets, using that b is point-like, we obtain Combining this calculation with equation (59) we obtain the desired result.
Proof of Theorem 10. Use the axioms of GW given in [22,Section 2] and [24,Chapter 7], and the axioms of OGW given in Proposition 4.19. As in Remark 4.15, the invariants OGW are given by derivatives of Ω. Note that, with the identification H 2 (CP n ; Z) Z, we have w s (β) ≡ n + 1 2 ·β (mod 2).
As shown in Lemma 5.11 of [33], the natural map H 2 (X, L; R) → H * (X, L; R) is an isomorphism. So, the pairing of Γ 1 = [ω] ∈ H * (X, L; R) with β ∈ Π = H 2 (X, L; Z) Z is well-defined and the result is We use this implicitly each time we invoke the divisor axiom (47) in the following argument.
This recovers the second recursion.
Proof of Corollary 1.9. By Theorem 6 invariants with interior constraints in Γ are computable in terms of invariants with interior constraints of the form Γ j = [ω j ]. Further, by the unit (45) and divisor (47) axioms, we may assume that |Γ j | > 2. It follows from the degree axiom (44) that for any β there are only finitely many values of k, l, for which there may be nonzero invariants with constraints of the above type. Thus, we give a process for computing OGW β,k (Γ i 1 , . . . , Γ i l ) which is inductive on (β, k, l) with respect to the lexicographical order on Z ⊕3 ≥0 . For β = 0, all values are given by the zero axiom (46). For (β, k, l) with β = 1 and l ≤ 1, all possible values have been computed explicitly in Theorem 10. Indeed, assume for convenience that interior constraints are written in ascending degree order. By the degree axiom (44), β = 1 =⇒ n − 3 + n + 1 + k + 2l = kn + l j=1 |Γ i l | =⇒ 0 = (k − 2)(n − 1) + l j=1 (|Γ i l | − 2).
Since |Γ i j | > 2, equality cannot occur when k > 2. For k = 2, equality holds if and only if l = 0, for k = 1 if and only if l = 1 and |Γ i 1 | = n + 1, and for k = 0 if and only if l = 1 and |Γ i 1 | = 2n.
In the following, we often use the zero axiom (46) without mention to deduce the vanishing of open Gromov-Witten invariants with β = 0. For this purpose it is important that W is closed under the cup product so that for A 1 , A 2 ∈ W , we have P R (A 1 A 2 ) = 0.
Consider a triple (β, k, l) with l ≥ 2. By Theorem 10(a) we can express the invariant as a combination of invariants that either have degree smaller than β, or have at most the same amount of interior constraints as the original invariant but with a smaller minimal degree. Proceed to reduce the degree of the smallest constraint until you arrive at a divisor, then eliminate this constraint by the divisor axiom (47). In the process, summands of degree β do not increase the value of k. Thus, the invariant is reduced to invariants with data of smaller lexicographical order, known by induction.
Consider a triple (β, k, l) with l ≤ 1. For β = 0, 1, the values have been computed above. For β > 1, the degree axiom (44) implies that k ≥ 2. Using Theorem 10(b), express the required invariant as sums of invariants that are either of smaller degree or have equal degree and less boundary marked points. Either way, we get invariants with data of smaller lexicographical order, known by induction. Corollary 6.2. All the invariants of (CP n , RP n ) are of the form m 2 r with m, r ∈ Z. Proof. This is immediate from the recursive process noting that the initial conditions are integer, and the only contribution to the denominators comes from the divisor axiom. Therefore, the denominators consist of powers of 2.
Appendix A. The real setting The objective of this section is to prove Theorem 9. In particular, we operate under the assumptions of Theorem 9 throughout the section.
For any nodal Riemann surface with boundary Σ, denote by Σ be the conjugate surface, as in Section 3.1. Denote by ψ Σ : Σ → Σ the anti-holomorphic map given by the identity Corollary A.3. Let U = H even φ (X). Then U ⊗ Q U ⊂ QH U (X) is a Frobenius subalgebra.