The log-open correspondence for two-component Looijenga pairs
Yannik Schuler



	A two-component Looijenga pair is a rational smooth projective surface with an anticanonical divisor consisting of two transversally intersecting curves. We establish an all-genus correspondence between the logarithmic Gromov–Witten theory of a two-component Looijenga pair and open Gromov–Witten theory of a toric Calabi–Yau threefold geometrically engineered from the surface geometry. This settles a conjecture of Bousseau, Brini and van Garrel in the case of two boundary components. We also explain how the correspondence implies BPS integrality for the logarithmic invariants and provides a new means for computing them via the topological vertex method.


Iterative Cluster Harvesting for Wafer Map Defect Patterns
    Alina Pleli*, Simon Baeuerle*, Michel Janus, Jonas Barth, Ralf Mikut, Hendrik P. A. Lensch
	A. Pleli and Hendrik P. A. Lensch are with the department of Computer Graphics, University of Tuebingen, D-72076 Tuebingen, Germany. (e-mail: alina.pleli@de.bosch.com) (*Alina Pleli and Simon Baeuerle are co-first authors.) (corresponding author: S. Baeuerle.)
	S. Baeuerle and R. Mikut are with the Institute for Automation and Applied Informatics (IAI), Karlsruhe	Institute of Technology (KIT), D-76344 Eggenstein-Leopoldshafen, Germany.
	A. Pleli, S. Baeuerle, M. Janus and J. Barth are with the Robert Bosch GmbH, D-72762 Reutlingen, Germany.

    April 29, 2024
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§ INTRODUCTION

In <cit.> Bousseau, Brini and van Garrel discovered a surprising relationship between two at first sight quite different curve counting theories:

	
  * [sec: log GW setup]Logarithmic Gromov–Witten theory of a Looijenga pair
	
    (S    D_1+… +D_l)

	which is a rational smooth projective surfaces S together with an anticanonical singular nodal curve D_1+… + D_l. To this datum one can associate a Gromov–Witten invariant
	
    g,ĉ,S    D_1+… +D_l(-1)^g _g Π_i=1^l-1ev_i^ (pt)∈

	enumerating genus g, class  stable logarithmic maps to (S    D_1+… +D_l) with maximum tangency along each irreducible component D_i. Additionally, there are l-1 interior markings with a point condition and an insertion of the top Chern class of the Hodge bundle. The tangency order of the markings along D_1,…,D_l is recorded in a matrix ĉ.
	
	

	
	
  * [sec: open GW setup]Open Gromov–Witten theory of a toric triple
	
    (X,(L_i,f_i)_i=1^k)

	consisting of the toric Calabi–Yau threefold X and a collection of framed Aganagic–Vafa Lagrangian submanifolds (L_i,f_i). To this data one can associate open Gromov–Witten invariants
	
    g,((w_1), …, (w_k)),'X,(L_i,f_i)_i=1^k∈

	enumerating genus g, class ' stable maps to X from Riemann surfaces with k boundaries, each wrapping one of the Lagrangian submanifolds L_i exactly w_i times.	


It was conjectured in <cit.> that starting from a Looijenga pair one can geometrically engineer a toric triple so that the above two curve counts turn out to be essentially equal.

A
	
	<cit.> Let (S    D_1+… +D_l) be a Looijenga pair and  an effective curve class in S satisfying some technical conditions. Then there exists a toric triple (X,(L_i,𝖿_i)_i=1^l-1) and an effective curve class β' in X such that
	
    ∑_g≥ 0ħ^2g-2 g,𝐜,'X, (L_i,f_i)_i=1^l-1 = (∏_i=1^l-1(-1)^D_i ·/D_i ·)  (-1)^D_l · +1/2 sin(D_l ·)ħ2 ∑_g≥ 0ħ^2g-1 g,ĉ,S    D_1+…+D_l(-1)^g _g  Π_i=1^l-1ev_i(pt)

	where 𝐜 is obtained from the contact datum ĉ by deleting the l-1 interior markings and the marking with tangency along D_l.


For l=1 the right-hand side of (<ref>) reduces to the generating series of ordinary Gromov–Witten invariants of X in which case the conjecture has already been proven in <cit.>. In this paper we investigate the case of two-component Looijenga pairs (S    D_1+D_2) and as another application of the main result in <cit.> we will establish <Ref> for these geometries.



 §.§ Logarithmic-Open Correspondence


Consider a two-component Looijenga pair (S  D_1+D_2) together with an effective curve class  in S. We will impose
⋆
	
	(S  D_1+D_2) and  satisfy
	
		
  * D_i · >0, i∈{1,2},
		
  * D_2 · D_2≥ 0,
		
  * (S D_1) deforms into a pair () with  a smooth projective toric surface and  a toric hypersurface.[We expand on the last condition in <Ref>.]
		

Moreover, we denote by ĉ the following contact datum along D_1+D_2

    ĉ = (-0.7ex13.2em0    ⋯    0     c_1    [4.5em]⋯^=  𝐜    c_n     0
    
    				0    [4.5em]⋯_m times    0     0    …    0     D_2 ·)

where we assume n≥ m and c_i>0 for all i∈{1,…,n}. This means there are m interior markings, n markings tangent to D_1 and an additional marking with maximum tangency along D_2. We set

    g,ĉ,S  D_1+D_2 (-1)^g _g  Π_i ev_i=1^n(pt)∫_[_g,ĉ,(S  D_1 + D_2)]^ (-1)^g _g  ∏_i=1^nev_i^(pt) .

Especially, note that we impose a point condition at all m interior markings. Our main result is the following logarithmic to open correspondence.

B  (<Ref>)
	Assuming assumption: star there is a toric triple (X,L,𝖿) (see <Ref>) and a curve class ^' in X satisfying
	
    ∑_g≥ 0ħ^2g-2 g,𝐜,^'X,L,f = (-1)^D_1 ·/m!  ∏_i=0^m-1 c_n-i (-1)^D_2 · +1/2 sin(D_2 ·)ħ2 ∑_g≥ 0ħ^2g-1 g,ĉ,S  D_1+D_2(-1)^g _g  Π_i=1^l ev_i(pt) .



The specialisation of the above correspondence to the case m=n=1 immediately gives

C
	<Ref> holds for all two-component Looijenga pairs (l=2) satisfying <Ref>. 





 §.§ Topological vertex

Computing Gromov–Witten invariants of logarithmic Calabi–Yau surfaces is usually a rather tedious endeavour. In higher genus and the presence of a _g insertion the only general tools available are scattering diagrams and tropical correspondence theorems <cit.>.

As an application of <Ref> we obtain a new — highly efficient — means for computing logarithmic Gromov Witten invariants of Looijenga pairs. Since open Gromov–Witten invariants can be computed using the topological vertex <cit.>, the same method can be used to determine the opposite side of our correspondence theorem.

D (<Ref>)
	The logarithmic Gromov–Witten invariants (<ref>) of a two-component Looijenga pair satisfying <Ref> are computed by the topological vertex.




 §.§ BPS integrality

In general, Gromov–Witten invariants are rational numbers. However, in many cases they are expected to exhibit underlying integral BPS type invariants. For open Gromov–Witten invariants this behaviour was first observed by Labastida–Mariño–Ooguri–Vafa <cit.>, was studied extensively in explicit examples by Luo–Zhu <cit.> and got recently proven by Yu <cit.> for general toric targets. Hence, as a corollary of our main result  we find that logarithmic invariants feature the same property.

E (<Ref>) 
	The logarithmic Gromov–Witten invariants (<ref>) of a two-component Looijenga pair satisfying <Ref> exhibit BPS integrality.


In <Ref> we spell this statement out in more detail. We conjecture that the above observation holds without imposing <Ref> as well (<Ref>).



 §.§ Strategy

The proof of <Ref> splits into several steps which partly have already been carried out elsewhere:

    (S   D_1+D_2)[rr,leftrightarrow,"<cit.>"]       (_S(-D_2)   D_1)[out=240,in=300,loop,looseness=4,swap,"<Ref>"] [rr,leftrightarrow, "<Ref>"]       _S(-D_2)|_S ∖ D_1|[rr,leftrightarrow, "<cit.>"]       (X,L,𝖿)

In <cit.> van Garrel, Nabijou and the author prove a comparison statement between the Gromov–Witten theory of (S  D_1+D_2) and (_S(-D_2)    D_1). Together with a result of Fang and Liu <cit.> expressing the open Gromov–Witten invariants of a toric triple (X,L,𝖿) as descendant invariants of X it is therefore sufficient to prove a formula for relative Gromov–Witten invariants of (_S(-D_2)    D_1) in terms of descendant invariants of X=_S(-D_2)|_S ∖ D_1 in order to establish <Ref>. We will prove such a relation in <Ref> in case there are no interior markings (m=0). Later in <Ref> we will reduce the general case (m≥ 0) to the one proven earlier.



 §.§ Context & Prospects




  §.§.§ Stable maps to Looijenga pairs

The main motivation for this work clearly comes from the conjectures put forward by Bousseau, Brini and van Garrel. <Ref> was formulated in <cit.> motivated by direct calculations of both sides of the correspondence. <Ref> covers all two-component Looijenga pairs considered in loc. cit. but also previously unknown cases. Most notably, our correspondence theorem also applies to the Looijenga pairs dP_1(0,4), dP_2(0,3), dP_3(0,2) for which <Ref> was established by proving an intricate identity of q-hypergeometric functions <cit.>. We illustrate how our main result can be used to reprove these cases in <Ref>.

There is certainly the question whether the techniques that went into the proof of <Ref> can be generalised to cover Looijenga pairs with l>2 boundary components as well. The most difficult part in this endeavour will most likely be to find an appropriate generalisation of <cit.>. Moreover, in the case l≥ 3 the construction of the toric triple proposed in <cit.> involves a flop which numerically does the right job but needs to be understood better geometrically when trying to generalise the techniques of this paper to cover this case as well.



  §.§.§ Open/Closed duality of Liu–Yu

Liu and Yu <cit.> prove a correspondence between genus zero open Gromov–Witten invariants of a toric threefold and local Gromov–Witten invariants of some Calabi–Yau fourfold constructed from the threefold. If we combine <Ref> with <cit.> we obtain a similar relation between open/local Gromov–Witten invariants of

    (X,L,f)[rr,leftrightarrow]       _S(-D_1) ⊕_S(-D_2) .

It is a combinatorial exercise to check that starting from the toric triple (X,L,f) as we define in <Ref> the fourfold constructed in <cit.> is indeed deformation equivalent (in the sense of <Ref>) to the right-hand side of (<ref>).[The key observation is that the toric divisor D we are deleting in <Ref> is reinserted in Liu–Yu's construction <cit.> of the fourfold as the divisor associated to the ray b_R+1_≥ 0.] It should, however, be stressed that the correspondence of Liu and Yu holds in a more general context than just local surfaces.





  §.§.§ BPS invariants

Motivated by <Ref> we conjecture BPS integrality for Gromov–Witten invariants of type (<ref>) for two-component logarithmic Calabi–Yau surfaces not necessarily satisfying <Ref> (<Ref>). This predication overlaps with a conjecture of Bousseau <cit.> in some special cases. So one may wonder about a simultaneous generalisation of both conjectures. Especially, Bousseau's conjecture also covers logarithmic Calabi–Yau surfaces with more than two components. A geometric proof of <Ref> in the case of l>2 components in combination with LMOV integrality <cit.> might allow to make progress towards this direction.




 §.§ Acknowledgements

First and foremost I would like to thank my supervisor Andrea Brini for suggesting this project, innumerous discussions and his continuous support. Special thanks are also due to Cristina Manolache for clarifying discussions concerning key steps in the proof of our main result. I also benefited from discussions with Alberto Cobos Rábano, Michel van Garrel, Samuel Johnston, Navid Nabijou, Dhruv Ranganathan, Ajith Urundolil Kumaran and Song Yu which I am highly grateful for.

Parts of this work have been carried out during research visits at the University of Cambridge. I thank the university and St John's College for hospitality and financial support.



§ STATEMENT OF THE MAIN RESULT




 §.§ Logarithmic Gromov–Witten theory



	
	An l-component Looijenga pair (S  D_1 + … +D_l) is a rational smooth projective surface S together with an anticanonical singular nodal curve D_1 + … + D_l with l irreducible components D_1,…,D_l.

Given a two-component Looijenga pair (S  D_1 + D_2), we denote by

    _g,ĉ,(S  D_1 + D_2)

the moduli stack of genus g, class  stable logarithmic maps to (S  D_1 + D_2) with markings whose tangency along D_1 + D_2 is encoded in ĉ <cit.>. We fix this tangency data to be

    ĉ = (-0.9em13.2em0    ⋯    0     c_1    …    c_n     0
    
    				0    [4.5em]⋯_m times    0     0    …    0     D_2 ·)

for some c_1,…,c_n> 0 with ∑_i c_i = D_1 ·. This means there are m interior markings (no tangency), n markings with tangency along D_1 and a marking having maximum tangency with D_2. The virtual dimension of the moduli stack is g+m+n and so we define

    g,ĉ,S  D_1+D_2(-1)^g _g  ∫_[_g,ĉ,(S  D_1 + D_2)]^ (-1)^g _g  ∏_i=1^n ev_i^ (pt)

where _g is the top Chern class of the Hodge bundle.

The ultimate goal of this section is to relate the above invariants to the ones of an open subset of (a deformation of) the threefold Y _S(-D_2). Let us recall the main result of <cit.> which serves as a first step towards this direction. Writing  for the projection Y→ S, we set D ^-1( D_1) and define

    g,𝐜̃,Y  D∫_[_g,𝐜̃,(Y  D)]^∏_i=1^n ev_i^(^pt)

where 𝐜̃ is obtained from ĉ by deleting the marking with tangency along D_2.


	
	<cit.> Suppose D_2 · D_2 ≥ 0 and  is so that D_1·≥ 0 and  D_2 · > 0. Then 
	
    ∑_g≥ 0ħ^2g-2 g,𝐜̃,Y  D = (-1)^D_2 ·+1/2 sin(D_2 ·)ħ2∑_g≥ 0ħ^2g-1 g,ĉ,S  D_1+D_2(-1)^g _g   .





 §.§ Open Gromov–Witten theory


Open Gromov–Witten invariants are a delicate topic. We will work in the framework of Fang and Liu <cit.> which builds on ideas of <cit.>. We keep details to a minimum since at no point we will be working with open Gromov–Witten invariants directly. We refer the interested reader to <cit.> for more details.



  §.§.§ Preliminaries on toric Calabi–Yau threefolds


Let  be a smooth toric Calabi–Yau threefold. We will write T̂≅^3 for its dense torus and T≅^2 for its Calabi–Yau torus. The latter is defined as the kernel T= of the character  associated to the induced T̂-action on K_≅_.

If  can be realised as a symplectic quotient, which is the case when  is semiprojective <cit.>, Aganagic and Vafa <cit.> construct a certain class of Lagrangian submanifolds L in . Relevant for us here is the fact that these Lagrangian submanifolds are invariant under the action of the maximal compact subgroup T_R↪ T and are homeomorphic to R^2× S^1. Moreover, these Lagrangian submanifolds intersect a unique one dimensional torus orbit closure of  in a circle S^1. For short we will call a Lagrangian submanifold L of the type considered by Aganagic and Vafa an outer brane if it intersects a non-compact one dimensional torus orbit closure.

Let us write  for the moment map

    : ⟶Lie(T_R)^

and fix an isomorphism Lie(T_R)^≅R^2. We call the image of the union of all one dimensional torus orbit closures under  the toric diagram of . It is an embedded trivalent graph since there are always three T̂-preserved one dimensional strata meeting in a torus fixed point (see <Ref>). Now let L be an outer brane in . By our earlier discussion, its image (L) under the moment must be a point on a non-compact edge ℓ. Let us write q for the unique torus fixed point contained in ℓ and denote by

    u,v,w∈^2_T(pt,Z) ⊂Lie(T_R)^


the weights of the induced T action on the tangent spaces of three torus orbit closures at this point. Here, u = c^T_1(T_q ℓ) and v is chosen so that u∧v≥ 0 as indicated in <Ref>. A framing of L is the choice of an element f∈^2_T(pt,Z) satisfying[This definition agrees with the one in <cit.> and differs from <cit.> by a minus sign.]

    u∧f = u∧v .

Equivalently, we have

    f = v - f u

for some f∈Z. Moreover, as an element in ^2_T(pt,Z) one may view f as a character T→. This defines a one dimensional subtorus T_ff↪T which we will refer to as the framing subtorus. Finally, let us denote by |_f=0 the restriction map

    _*^T(pt,Z) ⟶_*^T_f(pt,Z)

as this morphism essentially sets the weight f equal to zero.


	
	A toric triple (,L,f) consists of
	
		
  * a smooth semiprojective toric Calabi–Yau threefold ,
		
  * an outer brane L,
		
  * a framing f.
	


	
	To cover certain interesting cases it will actually be necessary to relax the above definition slightly. Concretely, we would like to include the case where  is not semiprojective in our discussion as well. In this case we replace the second point in the above definition with the choice of some non-compact one dimensional torus orbit closure ℓ.




  §.§.§ Open Gromov–Witten invariants
 Given a toric triple (,L,f) and a collection 𝐜=(c_1,…,c_n) of positive integers we denote by

    _g,𝐜,^'(,L)

the moduli space parametrising genus g stable maps f:(C,∂ C) → (,L) whose domain C is a Riemann surface with n labelled boundary components ∂ C = ∂ C_1 ⊔…⊔∂ C_n such that for all i∈{1,…,n}

    f_ [∂ C_i] = c_i [S^1]∈_1(L,Z)    and    f_ [C] = ^' + ∑_i c_i [B] ∈_2(,L,Z) .

Here, ^' a curve class in X and B is the unique T_R-preserved disk stretching out from q to the circle S^1 in which L and ℓ intersect. The virtual dimension of this moduli problem is zero. We refer to <cit.> for details.

Moduli spaces parametrising stable maps with boundaries are (if they actually exist) usually notoriously hard to work with. In our case there is however a way out. Since the Calabi–Yau torus T_R leaves L invariant the T_R-action on  lifts to an action on _g,𝐜,^'(,L). Then as opposed to its ambient space the (anticipated) fixed locus

    _g,𝐜,^'(,L)^T_R

is a compact complex orbifold. So as in <cit.> we define open Gromov–Witten invariants via virtual localisation:

    g,𝐜,^',L,f∫_[_g,𝐜,^'(,L)^T_R]^_T_R1/e^T_R (N^) |_f=0∈ .

The above integral produces a rational function in the two equivariant parameters of homogenous degree zero. Only the restriction to the framing subtorus yields a rational number. This is how definition (<ref>) depends on the choice of framing f.

We recall a result of Fang and Liu which allows us to express open Gromov–Witten invariants of (,L,f) in terms of descendant invariants of .


	
	<cit.> If (,L,f) is a toric triple then
	
    g,𝐜,^',L,f = 
    			∏_i=1^n (-1)^f c_i∏_k=1^c_i-1(f c_i +k)/u· c_i!∫_[_g,n,^'()^T]^_T1/e^T(N^) ∏_i=1^n ev_i^/(uc_i-_i)  |_f=0

	where  is the T-equivariant Poincaré dual of the torus fixed point q contained in the one dimensional torus orbit closure ℓ intersected by L.


	
	In <cit.> Li, Liu, Liu and Zhou generalise the definition of open Gromov–Witten invariants to the case when  is not semiprojective. (This assumption was necessary for the construction of Aganagic–Vafa Lagrangian submanifolds.) In this case the datum of an outer brane gets replaced by the choice of a non-compact one dimensional torus orbit closure ℓ and open Gromov–Witten invariants are defined via formal relative invariants. It is shown in <cit.> that these invariants still satisfy <Ref>.




 §.§ Statement of the main correspondence


Let (S  D_1+D_2) be a two-component Looijenga pair and  an effective curve class in S. We demand that this data satisfies <Ref> which we recall means that

	
  * D_i · >0, i∈{1,2},
	
  * D_2 · D_2≥ 0,
	
  * (S D_1) deforms into a pair () with  a smooth projective toric surface and  a toric hypersurface.


	
	By the last condition we mean that there is a logarithmically smooth morphism of fine saturated logarithmic schemes S→ T with T irreducible and integral together with a line bundle L on S. Moreover, there are regular points t,t^': → T with fibres
	
    S_t = (S D_1)  ,     S_t^' = ()

	on which L restricts to L_t = _S(-D_2) and L_t' = _().


Given () as above we write Y_() and write D for the preimage of  under the projection : Y →. We remark that all compact one dimensional torus orbit closures in Y come from the toric boundary of  embedded via the zero section ↪ Y. This way we may especially view  as a toric curve in Y.


	
	To (S  D_1+D_2) as above we associate the toric triple (,L,f) where
	
    Y ∖ D

	and L⊂⊂ Y is an outer brane intersecting a compact one dimensional torus orbit closure ℓ⊂ Y adjacent to . The action of the Calabi–Yau torus T of X naturally extends to Y. Writing F for the fibre ^-1(p) over the point p = ℓ∩ we set f c_1^T (F) to be the weight of the T-action on F.




We are now able to state our main result. For this we fix c_1,…,c_n > 0 with ∑_i c_i = D_1 · and denote by ĉ the contact datum

    ĉ = (-0.9em13.2em0    ⋯    0     c_1    …    c_n     0
    
    				0    [4.5em]⋯_m times    0     0    …    0     D_2 ·)

along D_1+D_2. We assume m≤ n.

  (<Ref>)
	Under <Ref> we have
	
    ∑_g≥ 0ħ^2g-2 g,𝐜,^',L,f  = (-1)^D_1 ·/m!  ∏_i=0^m-1 c_n-i (-1)^D_2 · +1/2 sin(D_2 ·)ħ2 ∑_g≥ 0ħ^2g-1 g,ĉ,S  D_1+D_2(-1)^g _g  Π_i=1^n ev_i(pt)

	where ^' = ^( - (D_1 ·) [ℓ]) and  is the open inclusion ↪ Y.


See <Ref> for an extended example illustrating <Ref> and an application of the above thm: log open.


	Note that <Ref> depends on the choice which of the two torus orbit closures adjacent to D_1 the Lagrangian submanifold L is intersecting. Hence, a priori, one should expect the right-hand side of (<ref>) to depend on this choice as well and it should come as a surprise that <Ref> actually demonstrates an independence.



	If  as constructed in constr: toric triple turns out to be non-semiprojective the left-hand side of equation (<ref>) can still be defined as described in <Ref>.



	Let us quickly comment on the importance of <Ref>. The first two points guarantee that the moduli stack of relative stable maps to (Y  D) is proper which is required in <Ref>. The last point in <Ref> is necessary for  to be toric which is in turn required in the definition of open Gromov–Witten invariants we use. The condition is rather technical but allows us treat interesting cases such as the example we will discuss in <Ref>.


We observe that in the light of <Ref> and <Ref> it suffices to prove the following prop: log threefold to descedant in order to deduce <Ref>.


	
	Under <Ref> we have
	
    g,𝐜̃,Y    DΠ_i=1^nev_i^(^pt)
       = (-1)^D_1 · m! ( ∏_i=0^m-1c_n-i) ∏_i=1^n (-1)^f c_i∏_k=1^c_i-1(f c_i +k)/u· c_i! ∫_[_g,n,^'()^T]^_T1/e^T(N^) ∏_i=1^n ev_i^/uc_i-_i  |_f=0

	where ^'=^ ( - (D_1 ·) [ℓ]) and  is the T-equivariant Poincaré dual of the torus fixed point q in ℓ∩ X.


We will first prove the prop: log threefold to descedant in the special case m=0 in <Ref> and then deduce the general case from a reduction argument proven in <Ref>. But before we embark on proving <Ref>, let us first argue that it indeed implies <Ref>.


	Comparing formula (<ref>) and (<ref>) we deduce
	
    g,𝐜,^',L,f = (-1)^D_1 ·/m!  ∏_i=0^m-1 c_n-i·g,𝐜̃,Y    DΠ_i=1^nev_i^(^pt) .

	Now by deformation invariance <cit.> (see also <cit.>) the Gromov–Witten invariants of (S  D_1) and () agree. In combination with <Ref> this therefore gives the identity stated in <Ref>.





§ STEP I: NO INTERIOR MARKINGS


This section is devoted to the proof of a special instance of <Ref>.


	
	The statement of <Ref> holds if there are no interior markings (m=0).


We will show this by computing the left-hand side of equation (<ref>) via virtual localisation <cit.>. We start with a careful analysis of the target geometry (Y D) where we adopt the notation introduced in <Ref>. To unload notation we will assume that S is already toric and D_1 is a toric hypersurface throughout this section which means we can take () = (S D_1).




 §.§ The geometric setup


The Calabi–Yau torus T of  also naturally acts on Y as both varieties share the same dense open torus. The compact one dimensional torus orbit closures of Y are given by the irreducible components of the toric boundary of S embedded into Y by the zero section. We write ℓ↪ Y for the component intersected by L. By construction ℓ intersects D_1↪ Y transversely in a torus fixed point which we denote p. We write q for the second torus fixed point of ℓ.



<Ref> shows the image of one dimensional torus orbit closures of Y under the moment map locally around D_1. In this sketch dashed lines correspond to non-compact orbit closures while the dotted lines indicate that the toric boundary of S↪ Y forms a circle. Moreover, the red labels represent the weights with which the torus T is acting on the tangent spaces of the respective torus orbit closures at a fixed point. The observation (see <cit.>) that

    c_1^T(T_pF) = - c_1^T(T_pD_1) = f

will be crucial in our localisation calculation later. Moreover, we also note that the framing factor f introduced in (<ref>) can be identified with the degrees

    f = _S(-D_2)|_ℓ=  - N_ℓY - 1 .

The toric diagram of = Y ∖ D is obtained from the one displayed in <Ref> by erasing the top vertical line. In this process the divisor ℓ gets decompactified and so L is indeed an outer brane.



 §.§ Initialising the localisation

<Ref> involved the choice of a stratum ℓ the outer brane is intersecting. We now do an analogous choice by supporting the cycle whose pushforward gives the Gromov–Witten invariant g,𝐜,Y    DΠ_i=1^nev_i^(^pt) on a substack with evaluations constrained to the fibre F=^-1(p) where  is the projection Y→ S:

    _g,𝐜,(Y  D)|_F[r, hook] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _g,𝐜,(Y  D) [d]
    
    		F^n [r,hook]     D^n .

Pulling back the obstruction theory along the horizontal arrow we obtain

    g,𝐜,Y    DΠ_i=1^nev_i^(^pt) =  [_g,𝐜,(Y  D)|_F]^ .

This choice will turn out rather useful when we will calculate the invariant via virtual localisation. Regarding this we remark that the T-action on Y lifts to an action on _g,𝐜,(Y  D)|_F since D and F are both preserved by T. Especially, this also provides us with an action of the framing subtorus T_f on the moduli stack.

Now equation (<ref>) tells us that the divisor D is pointwise fixed under T_f since the restriction T_f↪T effectively sets the weight f equal to zero. This means we are in a situation where the analysis of Graber and Vakil <cit.> of virtual localisation in the context of stable maps relative a smooth divisor applies. Since their analysis was carried out in the setting of relative stable maps to expanded degenerations as introduced by Li <cit.> we quietly pass to this modification of _g,𝐜,(Y  D) until the end of this section without actually changing our notation. Since eventually we push all cycles forward to a point it does not matter which birational model we choose for our moduli stack <cit.>. See also <cit.> for an analysis of torus localisation in the context of Kim's logarithmic stable maps to expanded degenerations <cit.>.

Following Graber and Vakil <cit.> we split the T_f-fixed locus _g,𝐜,(Y  D)^T_f into two distinguished (not necessarily connected or irreducible) components. First, there is the simple fixed locus

    _g,𝐜,(Y  D)^

whose general points are stable maps with target Y. The complement of this is called composite fixed locus and consequently its general points correspond to maps to non-trivial expanded degenerations of (Y  D). We will denote this component by

    _g,𝐜,(Y  D)^com .

As we did in (<ref>) we form fibre products

    _g,𝐜,(Y  D)|_F^  _g,𝐜,(Y  D)^×_D^n F^n ,_g,𝐜,(Y  D)|_F^com  _g,𝐜,(Y  D)^com×_D^n F^n  .

Then virtual localisation <cit.> gives

    g,𝐜,Y    DΠ_i=1^nev_i^(^pt) =     _T_f1/e^T_f(N^_)∩ [_g,𝐜,(Y  D)|_F^]^
    
    		+     _T_f1/e^T_f(N^_com)∩ [_g,𝐜,(Y  D)|_F^com]^ .
 
We will inspect the contribution of the simple and composite fixed locus to the Gromov–Witten invariant individually in the next two subsections.



 §.§ The simple fixed locus

Closely following <cit.>, we start with an analysis of the simple fixed locus. By definition the image of a relative stable map f:C→ Y parametrised by

    _g,𝐜,(Y  D)|_F^

intersects D transversely, ie. f^ D = ∑_i=1^n c_i x_i as Cartier divisors where x_1,…,x_n denote the markings. If we write

    C_1,…,C_n

for the irreducible components containing these markings then since f:C→ Y lies in the T_f-fixed locus and by assumption none of the components C_i get contracted there must be a (non-trivial) lift of the action T_f on C_i (possibly after passing to some cover of T_f) making f|_C_i T_f-equivariant. As a consequence, for all i∈{1,…,n} both C_i and its image under f must be rational curves. Moreover, writing q_i for the second T_f-fixed point of the image rational curve, f|_C_i is the unique degree c_i cover maximally ramified over f(x_i) and q_i.

Now since we confined the evaluation morphisms to the fibre F⊂ Y, we deduce that the image of each component C_i is contained in ^-1(ℓ) where  is the projection Y→ S and ℓ is as in <Ref>. However, since the pullback of the line bundle _S(-D_2) under ∘ f is negative we deduce that f has to factor through the zero section S↪ Y. Hence, the image of each irreducible component C_i has to be the curve ℓ already. This also means that q_i=q for all i∈{1,…,n}.

Now if we write C' ⊔_i=1^n C_i for the connected components of the partial normalisation of C along the nodes y_i∈ C_i mapping to q we observe that f|_C' is a genus g, class

    ^' = ^( - (D_1 ·) [ℓ])

T_f-fixed stable map to = Y ∖ D with markings y_1,…,y_n confined to q. Hence, forming the fibre square

    _g,n,^'() |_q[r,hook] [d][rd,phantom,"□",start anchor=center,end anchor=center]    _g,n,^'() [d,"Π_iev_y_i"] 
    
    	q [r,hook]    ^n

we can summarise the discussion up to now as follows.


	We have
	
    _g,𝐜,(Y  D)|_F^ = [ _g,n,^'() |_q^T_f / ∏_i=1^nZ_c_i ] .



Using this description of the simple fixed locus we are now able to determine its contribution to the Gromov–Witten invariant.


	
	We have
	
    _T_f1/e^T_f(N^_)∩ [_g,𝐜,(Y  D)|_F^]^_T_f    = (-1)^D_1 ·∏_i=1^n (-1)^f c_i∏_k=1^c_i-1(f c_i +k)/u· c_i!
       ×∫_[_g,n,^'()^T_f]^_T_f1/e^T_f(N^) ∏_i=1^n ev_i^/uc_i-_i .

	where  is the T_f-equivariant Poincaré dual of q.



	With our characterisation (<ref>) of the simple fixed locus it remains to analyse the difference between N^_ and the virtual normal bundle N^ of the fixed locus
	
    _g,n,^'()^T_f↪_g,n,^'() .
	
	First of all note that we get an overall factor
	
    ∏_i=1^n 1/c_i

	from the quotient. Moreover, partially normalising along the nodes y_1,…,y_n connecting C' with C_1,…,C_n we see that each degree c_i cover
	
    f_i  f|_C_i   :   C_i →ℓ

	contributes with factors
	
    1/e^T_f(^∙(C_i , f_i^ T^log_ℓ|p)^mov)    = 1/c_i !(u/c_i)^-c_i
    1/e^T_f(^∙(C_i , f_i^ N_ℓS)^mov)    = 1/(-(f+1)c_i)!(u/c_i)^(f+1) c_i
    1/e^T_𝖿( ^∙(C_i, f_i^  O_S(-D_2) )^mov)    = (-1)^fc_i + 1  (-fc_i-1)! (u/c_i)^-f c_i-1

	where f = _S(-D_2)|_ℓ= -  N_ℓY - 1 < 0. Note that if we multiply all these factors together we indeed obtain the top line on the right-hand side of (<ref>). Regarding the second line let us only remark that the insertions
	
    uc_i-_i

	for i∈{1,…,n} come from smoothings of the nodes y_1,…,y_n.





 §.§ Vanishing of composite contributions


We observe that the integral on the right-hand side of equation (<ref>) is the result of computing the one in (<ref>) via T-virtual localisation. Hence, to prove <Ref> it suffices to show


	
	We have
	
    _T_f1/e^T_f(N^_com)∩ [_g,𝐜,(Y  D)|_F^com]^ = 0 .


The proof of this prop: composite contrib vanishes necessitates a careful analysis of the composite fixed locus for which we will closely follow <cit.>.



  §.§.§ The setup

We first set the notation required in this section. Denote by

    Z _D( _D ⊕ N_DY)

the projective completion of N_DY and write D_0 and D_∞ for its zero and infinity section. Note that Z is a line bundle over

    P _D_1( _D_1⊕ N_D_1S) .

More precisely, we have Z≅_P(-2 f) where f is a fibre of P → D_1. 

By definition, the composite fixed locus parametrises relative stable maps whose target is an expanded degeneration of (Y  D) which is the result of gluing (Y  D) and multiple copies of (Z  D_0 + D_∞) to form an accordion. The composite locus decomposes into a disjoint union of components labelled by certain decorated graphs indicating how a relative stable map distributes over the expanded target.

 
	A splitting type Γ is the data of
	
		
  * a bipartite graph with vertices V(Γ), edges E(Γ) and legs L(Γ);
		
		
  * an assignment of a curve class _v to every vertex v∈ V(Γ);
		
		
  * a genus label g_v∈Z_≥ 0 for every vertex v∈ V(Γ);
		
		
  * a contact order d_e ∈Z_> 0 assigned to every edge e∈ E(Γ);
		
		
  * a labelling {1,…,n}≅ L(Γ).
	
	This data has to be compatible with (g,𝐜,) in the usual way. Vertices are separated into Y- and Z-vertices and we will write Γ_Y and Γ_Z for the type of relative stable maps (with possibly disconnected domain) to (Y  D), respectively (Z  D_0+D_∞), obtained by cutting Γ along its edges. All legs are adjacent to Z-vertices.


With this notation the composite locus can be written as a union of disconnected components labelled by splitting types: 

    _g,𝐜,(Y  D)^com = _Γ _Γ .

To characterise the component labelled by a fixed splitting type Γ, let us denote by 

    _Γ_Y(Y  D)^

the simple T_f-fixed locus of relative stable maps of type Γ_Y and write

    _Γ_Z(Z  D_0+D_∞)^

for the moduli stack of type Γ_Z relative stable maps to non-rigid (Z  D_0+D_∞) and form a substack with constrained evaluations at marking legs

    _Γ_Z(Z  D_0+D_∞)|_F^[r, hook] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Z(Z  D_0+D_∞)^[d, "Π_i ev_i"]
    
    		F^n [r, hook]     D_∞^n .

The two moduli stacks support evaluation morphisms to D ≅ D_0 associated to the edges of Γ. Let 
Δ denote the diagonal D^E(Γ)↪ D^E(Γ)× D^E(Γ) and form the fibre product

    _Γ[r, hook] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Y(Y  D)^×_Γ_Z(Z  D_0+D_∞)|_F^[d]
    
    		D^E(Γ)[r, hook, "Δ"]     D^E(Γ)× D^E(Γ) .

Then the connected component _Γ of the composite fixed locus is the quotient of _Γ by (d_e) and moreover, by <cit.>, the Gysin pullback

    [_Γ]^Δ^! ( [_Γ_Y(Y  D)^]^× [_Γ_Z(Z  D_0+D_∞)|_F^]^)

pushes forward to [_Γ]^ under the quotient morphism. This means we get

    _T_f1/e^T_f(N^_com)∩ [_g,𝐜,(Y  D)|_F^com]^_T_f = ∑_Γ1/|(d_e)| _T_f1/e^T_f(N^_Γ)∩ [_Γ]^ .





  §.§.§ First vanishing: restricted curve classes


To prove <Ref> we will show that each term in the sum (<ref>) vanishes in several steps. We first recall a lemma from <cit.>.

	
	<cit.>
	If there is a Z-vertex v∈ V(Γ) such that _v is not a multiple of a fibre of P→ D_1, then
	
    _T_f1/e^T_f(N^_Γ)∩ [_Γ]^ = 0 .




	The above lemma was proven in <cit.> under the assumption that D_1· = 0 but it is straight forward to generalise the proof to the case at hand where D_1· >0. The key observation is that after localising the left-hand side of (<ref>) with respect to T one can identify a trivial factor in the obstruction bundle guaranteeing the vanishing.





  §.§.§ Second vanishing: star shaped graphs


From now on we may assume that every Z-vertex v is decorated with curve class _v which is a multiple of a fibre of P→ D_1.

The following results are inspired by vanishing arguments appearing in the localisation calculations <cit.>. We will present more geometric versions of these arguments here.


	
	Unless every vertex in Γ_Z carries exactly one leg we have
	
    [_Γ_Z(Z  D_0+D_∞)|_F^]^ = 0 .




	Clearly, every Z-vertex v has to have at least one leg for the moduli stack to be non-empty since D_∞·_v>0. So suppose there is some vertex with at least two legs. Since the curve class carried by this vertex is a fibre class and the restriction of the line bundle Z→ P to a fibre trivialises, the product of the evaluation morphisms of these two markings factors through the diagonal D_∞→ D_∞^2. Hence, we have the following commuting diagram with Cartesian squares:
	
    _Γ_Z(Z  D_0+D_∞)|_F^[r,hook] [d,"g"] [rd,phantom,"□",start anchor=center,end anchor=center]   _Γ_Z(Z  D_0+D_∞)^[d] 
    
    		F^n-1[r,hook,""] [d, "f"] [rd,phantom,"□",start anchor=center,end anchor=center]     D_∞^n-1[d] 
    
    		F^n[r,hook,""]     D_∞^n .

	Note that the excess intersection bundle
	
    E = f^ N_F^n D_∞^n/ N_F^n-1 D_∞^n-1

	is trivial as is every vector bundle on F^n-1≅n-1. Moreover, note that since T_f is acting trivially on D_∞ and F, this bundle is also T_f-equivariantly trivial. Therefore, by the excess intersection formula <cit.> we have
	
    [_Γ_Z(Z  D_0+D_∞)|_F^]^ = ^! [_Γ_Z(Z  D_0+D_∞)^]^ = c_1^T_f (g^ E) ∩^![_Γ_Z(Z  D_0+D_∞)^]^ = 0 .



As another consequence of our assumption that all curve classes of Z-vertices are multiples of fibre classes the evaluation morphism associated to edges

    _Γ_Z(Z  D_0+D_∞)|_F^⟶ D_0^E(Γ)

factors through F^E(Γ)↪ D_0^E(Γ). If we combine this with the fact that the evaluation morphism

    _Γ_Y(Y  D)^→ D^E(Γ)

factors through D^E(Γ)_1 ↪ D^E(Γ) we may restrict the evaluations to the intersection point p = F ∩ D_1. We get the following commuting diagram with Cartesian squares:

    _Γ_Y(Y  D)|_F^ × _Γ_Z(^1  0+∞)^[d] [r] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Y(Y  D)^×_Γ_Z(Z  D_0+D_∞)|_F^[d] 
    
    	p [d] [r] [rd,phantom,"□",start anchor=center,end anchor=center]     (D_1)^E(Γ)× F^E(Γ)[d] 
    
    	D^E(Γ)[r, "    Δ"]     D^E(Γ)× D^E(Γ) .

Especially, note that we have a fibre square

    _Γ_Z(^1  0+∞)^[d] [r,hook] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Z(Z  D_0+D_∞)|_F^[d] 
    
    	p [r,hook,"ξ"]      F^E(Γ) .

First, we record the following vanishing.


	
	We have [_Γ]^ = 0 unless every Z-vertex has a single adjacent edge.



	The prop: composite unique D0 leg can be proven with an argument similar to the one in the proof of <Ref> using diagram (<ref>). We leave this to the reader.


Moreover, comparing our definition of _Γ in (<ref>) with diagram (<ref>) we learn that

    _Γ≅_Γ_Y(Y  D)|_F^ × _Γ_Z(^1  0+∞)^ .

On the level of virtual fundamental classes we find the following identity.


	
	If there is a Z-vertex v with g_v>0 the cycle [_Γ]^ vanishes. Otherwise, the cycle equates to
	
    [_Γ_Y(Y  D)^|_F]^× [_Γ_Z(^1  0+∞)^]^ .




	Let ξ be as in (<ref>). Then since all squares in (<ref>) are Cartesian we have
	
    [_Γ]^    = Δ^! ( [_Γ_Y(Y  D)^]^× [_Γ_Z(Z  D_0+D_∞)|_F^]^) 
        = [_Γ_Y(Y  D)|_F^]^×ξ^![_Γ_Z(Z  D_0+D_∞)|_F^]^

	by <cit.>. Comparing the usual perfect obstruction theory of _Γ_Z(^1  0+∞)^ with the one pulled back from _Γ_Z(Z  D_0+D_∞)|_F^ we find
	
    ξ^![_Γ_Z(Z  D_0+D_∞)|_F^]^ = _top^2 ∩ [_Γ_Z(^1  0+∞)^]^

	where _top is the top Chern class of the Hodge bundle. The statement of the prop: higher genus vanishing then follows from the fact that _top^2 = 0 if any of the vertices in Γ_Z carries a non-zero genus label and _top^2 = 1 otherwise.




  §.§.§ Final vanishing


Having proven prop: composite unique Dinfty legprop: higher genus vanishing we are finally able to deduce <Ref>.


	According to <Ref> and prop: composite unique D0 leg the only splitting types Γ whose associated summand on the left-hand side of (<ref>) may be non-zero consist of Z-vertices with exactly one adjacent edge and leg. Moreover, by <Ref> every such vertex has to carry a vanishing genus label. Therefore, the virtual dimension of _Γ_Z(^1  0+∞)^ is 
	
    ∑_v∈ V(Γ_Z) (2g_v-2) + |E(Γ)| + |L(Γ)| - 1 = -1

	and so the cycle [_Γ]^ vanishes by <Ref>.






§ STEP II: REDUCING INTERIOR TO CONTACT INSERTIONS




 §.§ The comparison statement
 Throughout this section we will assume that (S  D_1+D_2) is a logarithmic Calabi–Yau surface with boundary the union of two transversally intersecting smooth irreducible curves D_1,D_2. Moreover, we fix an effective curve class  in S and assume 


  * D_i · >0, i∈{1,2},

  * D_2 · D_2≥ 0.

Especially, in this section we do not necessarily assume (S D_1) to deform to a toric pair. We set Y_S (-D_2) and write D for the preimage of D_1 under the projection morphism. 

As before we fix a contact datum 𝐜=(c_1,…,c_n) with D where c_i>0 for all i∈{1,…,n}. We set

    g,𝐜,Y  D∏_i=1^n ev_i^(^pt) = ∫_[_g,𝐜,(Y  D)]^∏_i=1^n ev_i^(^pt) .

Now denote by 

    𝐜̃=(0,…,0,c_1,…,c_n)

the result of adding m≤ n interior markings. Similarly to (<ref>) we define g,𝐜̃,Y  D∏_i=1^n ev_i^(^pt)∈ by imposing point conditions at the first n markings. Note that this especially means that there is a point condition at all m interior markings. We have the following comparison statement which reduces computations to the case without any interior markings.


	
	We have
	
    g,𝐜̃,Y  DΠ_i=1^nev_i^(^pt) = m! ( ∏_i=0^m-1c_n-i) ·g,𝐜,Y  DΠ_i=1^n ev_i^(^pt) .



Together with the special case m=0 of <Ref>, which we have already proven in the last section, the above reduction result implies the general case m≥ 0 of <Ref> as well. The rest of this section is hence dedicated to the proof of <Ref>.




 §.§ Degeneration

To prove the main result of this section we will use a degeneration argument similar to <cit.>. Since by now this approach has become a well-established technique <cit.> we economise on details.

Recall the definition of the line bundle Z→ P from equation (<ref>) and (<ref>) (where we identify S=). We defined Z to be the projective completion of N_D Y whose zero and infinity section we denote by D_0 and D_∞ respectively. Let us consider the degeneration to the normal cone of D in Y:

    ⟶1 .

 is a line bundle over the degeneration to the normal cone of D_1 in S. While the general fibre of  is Y, the special fibre _0 = Y ⊔_D Z is the result of gluing Y along D to Z along the zero section D_0. Now let us write  for the closure of D× (1 - {0}) in  and equip  with the divisorial logarithmic structure with respect to Y+Z+. The resulting logarithmic scheme has general fibre (Y  D) and its special fibre is obtained by gluing

    (Y  D)     and     (Z  D_0+D_∞) .

Finally, for F a fibre of D→ D_1 we denote by  the closure of F × (1-{0}) in . The intersection _0 of  with the special fibre _0 is a fibre of D_∞→ D_1.




 §.§ Decomposition

Composition with the blowup morphism → Y×1 produces a pushforward

    : _g,𝐜̃,(_0) ⟶_g,𝐜̃,(Y  D)

and by the conservation of number principle we have

    [_g,𝐜̃,(Y  D)|_F]^ = (_F_0)_ [_g,𝐜̃,(_0)|_F_0]^

where we introduced the fibre products

    _g,𝐜̃,(Y  D)|_F[r, hook] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _g,𝐜̃,(Y  D) [d, "Π_i=1^nev_i"]       _g,𝐜̃,(_0)|__0[r, hook] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _g,𝐜̃,(_0) [d, "Π_i=1^nev_i"]
    
    		F^n [r,hook]     Y^n       _0^n [r,hook]    _0^n .

and pulled back the respective virtual fundamental classes along horizontal arrows. The pushforward of the left-hand side of (<ref>) to _g,𝐜̃,(Y  D) gives

    Π_i=1^nev_i^ (π^pt) ∩ [_g,𝐜̃,(Y  D)]^

and so pushing forward to a point returns the Gromov–Witten invariant we are after:

    g,𝐜̃,Y  DΠ_i=1^nev_i^ (π^pt) .


We combine this with the decomposition formula <cit.>

    [_g,𝐜̃,(_0)|_F_0]^ = ∑_m_/ |  | _ [_(_0)|_F_0]^

where  denotes the inclusion

    _(_0)|_F_0↪_g,𝐜̃,(_0)|_F_0

and the sum runs over all rigid tropical types  of tropical maps to the polyhedral complex

    [baseline=(current  bounding  box.center)]
    		 (-3,0) nodeΣ  :;
    		
    		 (-2,0) – (0,0);
    		
    		
    		[->] (0,0) – (3,0);
    		
    		
    		[fill=black] (-2,0) circle[radius=2pt];
    		 (-2,0) node[below]_Y;
    		
    		[fill=black] (0,0) circle[radius=2pt];
    		 (0,0) node[below]_Z;

which is the fibre over 1∈R_≥ 0 of the tropicalisation Σ() →Σ (1) =  R_≥ 0. 

By the discussion in <cit.>  can only be rigid if all vertices of the underlying graph map to either one of the vertices _X or _Z. Hence,  is uniquely reconstructed from the data of its underlying decorated bipartite graph which is exactly the data of a splitting type as introduced in <Ref>. Hence, to be closer to the notation used in <Ref> let us write Γ for the splitting type underlying  and Γ_Y, Γ_Z for the type of stable logarithmic maps to (Y  D), respectively (Z  D_0+D_∞), obtained by cutting Γ along edges.

We remark that only such rigid tropical types can contribute non-trivially to the sum (<ref>) for which all marking legs are attached to a Z-vertex. Indeed, contact markings necessarily have to be adjacent to a Z-vertex and if an interior marking is carried by a Y-vertex then _(_0)|_F_0 is empty (where we crucially use our assumption that m≤ n). Moreover, the factor m_ in the decomposition formula (<ref>) equates to lcm(d_e) where d_e denotes contact order of an edge e.

Since we are in the special setting where Σ is of dimension one the discussion in <cit.> applies. There is a morphism  with target the fibre product

    _(_0)|_F_0[r, ""]    _Γ[r] [d] [rd,phantom,"□"]    _Γ_Y(Y  D)×_Γ_Z(Z  D_0 + D_∞)|_F_0[d] 
        D_0^E(Γ)[r, "Δ"]     D_0^E(Γ)× D_0^E(Γ)

which is finite and satisfies

    _ [_(_0)|_F_0]^ = Π_e   d_e/lcm(d_e) Δ^! (
    	 [_Γ_Y(Y  D)]^× [_Γ_Z(Z  D_0 + D_∞)|_F_0]^) .

There is a gluing morphism : _Γ→_g,m+n,(Y) such that

    _(_0)|_F_0[r, ""] [d, ""]    _Γ[d,""] 
    _g,𝐜̃,(_0)|_F_0[r,"∘_F_0"]    _g,m+n,(Y)

commutes where  denotes the forgetful morphism

    : _g,𝐜̃,(Y  D)|_F⟶_g,m+n,(Y) .

To summarize the discussion so far we combine equation (<ref>) and (<ref>) and use the fact that diagram (<ref>) commutes to deduce

    _ [_g, 𝐜̃ ,(Y  D)|_F]^ = ∑_ΓΠ_e   d_e/|Γ|_ [_Γ]^

where we introduced the notation

    [_Γ]^Δ^! (
    	[_Γ_Y(Y  D)]^× [_Γ_Z(Z  D_0 + D_∞)|_F_0]^) .





 §.§ Vanishing

In this section we will argue that for most splitting types Γ the cycle [_Γ]^ vanishes. We note that our setup is almost the same as in <Ref>. The only notable differences are as follows.

	
  * _Γ_Y(Y  D) parametrise logarithmic stable maps to (Y  D). In <Ref> we considered the simple fixed locus of this moduli stack under some torus action.
	
	
  * _Γ_Z(Z  D_0 + D_∞) parametrises logarithmic stable maps to (Z  D_0 + D_∞) while in <Ref> the target was non-rigid (Z  D_0 + D_∞).
	
	
  * Γ_Z contains interior markings.

We note that all arguments in <Ref> are essentially insensitive to first two differences which allows us to prove the vanishing of [_Γ]^ in similar steps. Only the third point requires some attention.



  §.§.§ First vanishing: fibre classes

We start with the analogue of <Ref>.


	
	We have [_Γ_Z(Z  D_0 + D_∞)|_F_0]^=0 unless Γ_Z only consists of vertices decorated with a curve class which is a multiple of a fibre class.

	
	Let us write _Z for the overall curve class carried by Γ_Z and suppose its projection p__Z along the morphism p:P→ D_1 is non-zero. In other words we assume p__Z = d[D_1] for some d>0. We claim that in this case
	
    [_Γ_Z(Z  D_0 + D_∞)]^=0 .

	To prove this we use an argument from <cit.>. Set k=m+n+|E(Γ)| and consider the morphism
	
    _Γ_Z(Z  D_0 + D_∞) ⟶_g_Z,k, d[D_1] (D_0)

	obtained by forgetting the logarithmic structure and composing with the projection Z → D_0. Write _2(P)^+ and _2(D_1)^+ for the semigroups of effective curve classes in P, respectively D_1 and denote by M_g,n,^(log) the Artin stack of prestable (logarithmic) curves with irreducible components weighted by an element in a semigroup . We factor the projection morphism through
	
    _Γ_Z(Z  D_0 + D_∞) [r, "u"] [dr]    M_Γ_Z[r] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _g_Z,k, d[D_1] (D_0) [d] 
       M_g_Z,k,_2(P)^+^log[r, "v"]    M_g_Z,k,_2(D_1)^+

	Then by <cit.> we have[In the statement of <cit.> van Garrel, Graber and Ruddat assume a positivity property for the logarithmic tangent sheaf of the target which guarantees that u is smooth. In our case at hand this positivity condition is in general not satisfied and u is only virtually smooth in the sense of <cit.>. The latter is however sufficient for our purposes thanks to <cit.>.]
	
    [_Γ_Z(Z  D_0 + D_∞)]^ = u^!_E__Γ_Z(Z  D_0 + D_∞)/M_Γ_Z v^! [_g_Z,k, d[D_1] (D_0)]^ .

	Our claim that [_Γ_Z(Z  D_0 + D_∞)]^=0 directly follows from the fact that
	
    [_g_Z,k, d[D_1] (D_0)]^ = 0

	since D_0 ≅_^1(-2) is a (non-compact) K3 surface and d>0.





  §.§.§ Second vanishing: star shaped graphs
 In this section we apply the same arguments we used in <Ref> in order to constrain the shape of Γ to star shaped graphs.


	We introduce the following notation for labelling sets:
	
		
  * Interior markings with a point condition I_pt{1,…,m};
		
  * Contact markings with a point condition J_pt{m+1,…,n};
		
  * Contact markings without an insertion J_1{n+1,…,m+n}.
	


We may assume that all Z-vertices of Γ are decorated with a curve class which is a multiple of a fibre class. This has the following consequence.

	
	We have [_Γ_Z(Z  D_0 + D_∞)|_F_0]^ = 0 unless every Z-vertex
	
		
  * either carries two legs, one labelled in I_pt and one in J_1;
		
		
  * or carries a single leg with label in J_pt.
	



	It suffices to show that every Z-vertex is carrying at most one leg with label in I_pt∪ J_pt. Indeed, in this case the number of vertices in Γ_Z is bounded from below by |I_pt| + |J_pt| = n. On the other hand it is also bounded from above by the number of contact markings n. We deduce that there have to be exactly n vertices each carrying one of the legs labelled by J_1∪ J_pt. Since we assumed that every vertex carries at most one leg with label in I_pt∪ J_pt, the m interior markings are necessarily distributed among the vertices carrying a leg labelled in J_1.
	
	So suppose there is a Z-vertex with more than one leg with label in I_pt∪ J_pt. Since the evaluation morphism of an interior marking has target Z let us first define a substack with confined evaluations:
	
    _Γ_Z_Γ_Z(Z  D_0 + D_∞) ×_Z^n D_∞^n .

	As every Z-vertex is decorated with a curve class which is a multiple of a fibre class and we assume there is a vertex with two legs labelled in I_pt∪ J_pt the resulting evaluation morphism
	
    _Γ_Z⟶ D_∞^n

	factors through D_∞^n-1→ D_∞^n. We obtain the following commuting diagram with Cartesian squares:
	
    _Γ_Z(Z  D_0 + D_∞)|_F_0[r] [d] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Z[d] [r] [rdd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Z(Z  D_0 + D_∞) [dd,"Π_i=1^n ev_i"]
    F_0^n-1[d] [r] [rd,phantom,"□",start anchor=center,end anchor=center]     D_∞^n-1[d]    
    F_0^n[r]     D_∞^n [r]     Z^n .

	The vanishing of [_Γ_Z(Z  D_0 + D_∞)|_F_0]^ then immediately follows from an excess bundle calculation as the one we did in <Ref>.


Moreover, by our assumption that all Z-vertices carry curve classes which are multiples of a fibre class the evaluation morphism associated to edges

    _Γ_Z(Z  D_0 + D_∞)|_F_0⟶ D_0^E(Γ)

factors through F^E(Γ)↪ D_0^E(Γ) where we write F for the image of _0 under Z→ D_0. We combine this observation with the fact that stable maps to Y factor through the zero section S↪ Y in order to constrain the image of Z-vertices to the fibre of Z→ D_0 over the point p = F ∩ D_1. Indeed, as we just argued the evaluation morphism

    _Γ_Y(Y  D) ⟶ D^E(Γ)

factors through D_1^E(Γ)↪ D^E(Γ). Writing _Γ_Y(Y  D)|_F for the fibre product _Γ_Y(Y  D)×_D^E(Γ) F^E(Γ) we find that

    _Γ_Y(Y  D)|_F × _Γ_Z(^1  0+∞)|_∞[d] [r] [rd,phantom,"□",start anchor=center,end anchor=center]    _Γ_Y(Y  D) ×_Γ_Z(Z  D_0+D_∞)|__0[d] 
    
    		p [d] [r] [rd,phantom,"□",start anchor=center,end anchor=center]     D_1^E(Γ)× F^E(Γ)[d] 
    
    		D_0^E(Γ)[r, "    Δ"]     D_0^E(Γ)× D_0^E(Γ) .

commutes and both squares are Cartesian. An excess intersection argument as in <Ref> gives


	
	We have [_Γ]^=0 if there is a Z-vertex with more than one edge.

Since all squares in (<ref>) are Cartesian, a comparison with (<ref>) gives

    _Γ≅_Γ_Y(Y  D)|_F × _Γ_Z(^1  0+∞)|_∞ .

The following prop: higher genus vanishing 2 comparing the virtual fundamental classes of the above moduli stacks can be proven with the same arguments we used to show <Ref>.


	
	If Γ contains a Z-vertex v with g_v>0 the cycle [_Γ]^ vanishes. Otherwise, it equates to
	
		      [_Γ_Y(Y D)|_F]^ ×[_Γ_Z(^1 0+∞)|_∞]^ .      	


Let us summarise the results of this section.


	
	The cycle [_Γ]^ vanishes unless the rigid tropical type  associated to Γ is of the form
	
    [baseline=(current  bounding  box.center),scale=0.85]
    			[fill=black] (-2,2) circle[radius=2pt];
    			 (-2,2) node[below]v_0;
    			
    			 (-2,2) – (0,1);
    			[fill=black] (0,1) circle[radius=2pt];		
    			
    			[->] (0,1) – (3,1);
    			
    			 (0.5,1.44) node⋮;
    			
    			 (-2,2) – (0,1.75);
    			[fill=black] (0,1.75) circle[radius=2pt];		
    			
    			[->] (0,1.75) – (3,1.75);
    			
    			 (-2,2) – (0,2.25);
    			[fill=black] (0,2.25) circle[radius=2pt];		
    			
    			[->] (0,2.25) – (3,2.25);
    			[->] (0,2.25) – (0,2.6);
    			
    			 (0.5,2.69) node⋮;
    			
    			 (-2,2) – (0,3);
    			[fill=black] (0,3) circle[radius=2pt];		
    			
    			[->] (0,3) – (3,3);
    			[->] (0,3) – (0,3.35);
    			
    			
    			
    			[->] (0.5,0.75) – (0.5,0.25);
    			
    			
    			 (-2,0) – (0,0);
    			
    			[->] (0,0) – (3,0);
    			
    			[fill=black] (-2,0) circle[radius=2pt];
    			 (-2,0) node[below]_Y;
    			
    			[fill=black] (0,0) circle[radius=2pt];
    			 (0,0) node[below]_Z;
    			
    			 [decorate,decoration=brace,amplitude=5pt,mirror]
    			(3.2,0.9) – (3.2,1.85) node[midway,xshift=2em]J_pt-legs;
    			 [decorate,decoration=brace,amplitude=5pt,mirror]
    			(3.2,2.15) – (3.2,3.1) node[midway,xshift=2em]J_1-legs;

	where v_0 is decorated with g_v_0=g, _v_0= and all other vertices are decorated with multiples of fibre classes. Moreover, in this case 
	
		      [_Γ]^ = [_Γ_Y(Y D)|_F]^ ×[_Γ_Z(^1 0+∞)|_∞]^ .        	






 §.§ Analysis of remaining terms

Now our discussion will deviate from the analysis in <Ref> because as opposed to <Ref> we will show that all remaining contributions characterised in <Ref> do contribute to the Gromov–Witten invariant and will exactly yield the right-hand side of equation (<ref>).

Let Γ be as in <Ref>. We fix a labelling of the markings of _Γ_Y(Y  D)|_F by fixing a bijection {1,…,n}≅ E(Γ) where we impose that the ith edge is adjacent to the Z-vertex carrying the (m+i)th leg. With this assignment we can naturally identify Γ_Y with the data (g,𝐜,).

Our first goal is to simplify

    _ [_Γ]^ = _( [_g,𝐜,(Y  D)|_F]^× [_Γ_Z(^1  0+∞)|_∞]^) .

Under  all images of stable maps which lie in a fibre of P→ D_1 get contracted to a point. Hence, the morphism factors into

    _g,𝐜,(Y  D)|_F × _Γ_Z(^1  0+∞)|_∞[r, "Id×"] [rr, start anchor=[yshift=0.4em]south east,  end anchor=[yshift=0.4em]south west, swap, bend right = 10, ""]    _g,𝐜,(Y  D)|_F × (_0,3)^m[r, ""]    _g,m+n,(Y)

where  is the forgetful morphism which only remembers the stabilised domain curve and  is a morphism which forgets the logarithmic structure and attaches a contracted rational component at the last m markings.

Consequently, a Z-vertex carrying a leg labelled with i+m ∈ J_1 contributes with a factor

    [_0,((c_i,0),(0,c_i)),c_i(^1  0+∞)|_∞]^ =  [_0,((c_i,0),(0,c_i)),c_i(^1  0+∞)]^ = 1/c_i

where the denominator is due to the degree c_i automorphism group acting on the unique degree c_i cover ^1 →^1 fully ramified over 0 and infinity. Similarly, a Z-vertex with leg labelled by i+m ∈  J_pt contributes

    [_0,(0,(c_i,0),(0,c_i)),c_i(^1  0+∞)|_∞]^ = ( ev_1^(pt)∩ [_0,(0,(c_i,0),(0,c_i)),c_i(^1  0+∞)]^) = 1

because the additional marking kills the automorphism group acting on the domain. We obtain

    _ [_Γ]^ = (∏_i=1^n-m1c_i) ·_([_g, 𝐜 ,(Y  D)|_F]^× [_0,3]^× m) .


Note that a splitting type Γ as in <Ref> is uniquely determined by the bijection I_pt≅ J_1 it induces. This means there are exactly m! such splitting types each one contributing with (<ref>) to the overall Gromov–Witten invariant. Thus, plugging the last equation into (<ref>) we find

    _ [_g, 𝐜̃ ,(Y  D)|_F]^ = m! (∏_i=0^m-1c_n-i) ·_([_g, 𝐜 ,(Y  D)|_F]^× [_0,3]^× m) .

Pushing the identity forward to a point yields formula (<ref>) which closes the proof of <Ref>.




	


§ APPLICATIONS





 §.§ Topological Vertex


Open Gromov–Witten invariants can be computed efficiently using the topological vertex method <cit.>. Hence, formula (<ref>) equating logarithmic Gromov–Witten invariants — which are usually rather cumbersome to compute — against open ones presents a new venue for calculating the former.



  §.§.§ The general recipe

We refer to <cit.> or <cit.> for a general summary of the topological vertex method which allows to determine the open Gromov–Witten invariants of a toric triple just from the information of its toric diagram. Let (X,L,f) be the outcome of <Ref> for some two-component Looijenga pair (Y D_1 + D_2). We will write

    _(X,L,f)(q,Q)

for the generating series of disconnected open Gromov–Witten invariants of (X,L,f) in representation basis where the winding around L is encoded in a partition . In the special case of a single boundary wrapping the Lagrangian submanifold L we can extract the connected invariant defined in (<ref>) via <cit.>[We corrected <cit.> by an overall factor -. See <cit.> for a discussion of this factor.]

    ∑_'∑_g≥ 0g,(c),'X,L,f ħ^2g-1  Q^' = -∑_k=0^c (-1)^k/c_(c-k,1^k)(X,L,f)/_∅(X,L,f)

under the change of variables q=^ħ where we fixed a square root  of -1.



As we discussed in <Ref>, the toric diagram of (X,L,f) will have a shape as displayed in <Ref> where we labelled the compact one dimensional torus orbit closures by ℓ_1,…,ℓ_k starting from the stratum ℓ_0 intersected by L and assigned partitions _1,…,_k to all edges. Then every edge ℓ_i, i∈{0,…,k}, contributes with a factor

    (-1)^( N_ℓ_i) · |_i| q^( N_ℓ_i + 1)·__i/2

to (<ref>) where _∑_j _j (_j - 2j +j). Each trivalent vertex corresponding to a torus fixed point ℓ_i∩ℓ_i+1 contributes

    __i+1,_i^,∅ s__i+1(q^+_i)   s__i^(q^) = q^-__i/2 s__i+1(q^) s__i(q^+_i+1) = q^__i+1/2∑_ s__i^(q^)   s__i+1^(q^)

where by convention _0= and _k+1=∅ and s_/(q^+) denotes the skew Schur function s_/(x_1,x_2,…) evaluated at x_j = -j + 1/2 -_j. As an application of <cit.> we obtain

	
	For (X,L,f) the outcome of <Ref> we have
	
    _(X,L,f)(q,Q) = ∑__1,…,_k Q^∑_i=1^k |_i| · [ℓ_i] ∏_i=0^k (-1)^( N_ℓ_i) · |_i| q^( N_ℓ_i + 1) ·__i/2__i+1,_i^,∅ .


In case the toric triple is a strip geometry the above sum of partitions can be carried out explicitly to yield a closed form solution for _(X,L,f) <cit.>. Plugged into (<ref>) and (<ref>) this can be used to give a closed form solution for the logarithmic Gromov–Witten invariants of (S D_1+D_2) and will be illustrated in a concrete example in the following.





  §.§.§ Extended example: dP3(0,2)


Let us consider (dP_3  D_1+D_2) where dP_3 is a del Pezzo surface which is the result of blowing up three points in ^2. This Looijenga pair was denoted by dP_3(0,2) in <cit.>. If we write E_1,E_2,E_3 for exceptional divisors we take D_1∈ |H-E_1| and D_2 ∈ |2H-E_2-E_3| smooth with transverse intersections. We use the following notation for a curve class in dP_3:

    = d_0 (H-E_1-E_2-E_3) + d_1 E_1 +d_2 E_2 + d_3 E_3 .

It was conjectured in <cit.> and later proven in <cit.> that the maximum contact logarithmic Gromov–Witten invariants of this geometry have the following closed form solution.



	
	<cit.> For all effective curve classes  in dP_3 with D_i · >0, i∈{1,2}, and contact data ĉ = [     0 D_1 ·     0;     0     0 D_2 · ] we have
	
    ∑_g≥ 0ħ^2g-1 g,ĉ,dP_3  D_1+D_2(-1)^g _g  ev_1(pt)
       = [d_1]_q [d_2+d_3]_q/[d_0]_q [d_1 + d_2 + d_3 - d_0]_qd_1d_0-d_3_q d_1d_0-d_2_q d_0d_1_q d_1+d_2+d_3-d_0d_1_q

	as formal Laurent series in ħ under the identification q=^ħ.

In this thm: dP3 0 2 we use the notation

    [n]_q  q^n/2-q^-n/2 ,   nm_q ∏_k=1^m[n-m+k]_q/[k]_q    0≤ m ≤ n, 
    
    		0    otherwise.

The statement was proven in <cit.> by extracting the Gromov–Witten invariants from a quantum scattering diagram of a toric model of (dP_3  D_1+D_2). The outcome of this calculation is a convoluted sum of products of q-binomial coefficients which with a lot of effort can be simplified to yield the above formula. We will give a new proof of this identity using the topological vertex method.


We start by applying <Ref> to (dP_3  D_1+D_2). For this let us write p_1,p_2,p_3 for the three points in ^2 whose blow up gives dP_3. Then D_1 is the strict transform of a line through p_1 and D_2 the strict transform of a conic through p_2 and p_3. We note that we can always choose the ^2-action on ^2 in such a way that D_1 is a toric hypersurface and p_1 and p_2 are torus fixed points. Now with this choice of ^2-action the blowups at p_1 and p_2 are toric as opposed to the one at p_3 since all points are assumed to be in general position. However, there is a deformation of dP_3 to the blow up of ^2 in p_1,p_2 and a torus fixed point of E_2 leaving D_1 invariant. We denote this deformation by dP^'_3 which is indeed toric as required. We display its fan in <Ref> where we labelled each ray with its curve class. 

Since dP^'_3 is toric with D_1 a toric hypersurface, we can choose

    X = _dP^'_3 (-D_2) |_dP^'_3 ∖ D_1 .

The toric diagram of (X,L,f) is displayed in <Ref> where we chose L to intersect the torus invariant curve in class H-E_2-E_3. We note that this particular toric triple has already been studied by Bousseau, Brini and van Garrel in <cit.>. Hence, from here on we basically follow their calculation.

Observe that each compact edge in the toric diagram corresponds to a ray in the fan. Hence, we can read off the degree of the normal bundles of ℓ_0,…,ℓ_3 from <Ref> to be -1, -1, -2 and -1 respectively. Thus, by <Ref> we have

    _(X,L,f) = ∑__1,_2,_3 
    _1,_2 (-1)^||+|_1|+|_3|  s_^(q^)   s__1^(q^+)    s__1_1(q^)    s__2_1(q^)    s__2_2(q^)    s__3_2(q^)   s__3^(q^)   Q^∑_i=1^3 |_i| · [ℓ_i] .

This can now be plugged into (<ref>) and simplified using the techniques developed in <cit.>. These steps have already been carried out in <cit.> and result in

    ∑_'∑_g≥ 0g,(c),'X,L,f ħ^2g-1  Q^'
       = ∑_c_1,c_2,c_3 (-1)^c+c_1+c_3+11/c [c_1]_qcc_1 -c_2_q cc_2 -c_3_q c+c_3-1c_3_q c_1c_q   Q^∑_i=1^3 c_i [ℓ_i] .

Extracting the coefficient of Q^ι^( - d_1 [ℓ_0]) from the above formula we get

    ∑_g≥ 0ħ^2g-1 g,(d_1),ι^( - d_1 [ℓ_0])X,L,f
        =   (-1)^d_1+d_2+d_3+1 [d_1]_q/d_1 [d_0]_q [d_1 + d_2 + d_3 - d_0]_qd_1d_0-d_3_q d_1d_0-d_2_q d_0d_1_q d_1+d_2+d_3-d_0d_1_q

which can be plugged into (<ref>) to yield the formula stated in <Ref>.


We stress that the above calculation is substantially easier than the one in <cit.> using scattering diagrams. This hopefully illustrates the usefulness of our main result <Ref>.




 §.§ BPS integrality


It is a general feature rather than a coincidence that the all-genus generating series of logarithmic Gromov–Witten invariants lifts to a rational function in ^ħ as we saw for instance in formula (<ref>). This property is usually called BPS integrality.



  §.§.§ LMOV invariants

In the context of open Gromov–Witten invariants of toric Calabi–Yau threefolds BPS integrality was first systematically studied by Labastida–Marino–Ooguri–Vafa <cit.>. Just recently Yu gave a proof of this phenomenon by carefully analysing the combinatorics of the topological vertex <cit.>.

To review Yu's result, let (X,L,f) be a toric triple. For a curve class ^' in X and a winding profile 𝐜=(c_1,…,c_n) along L we will write

    k    (𝐜,^')

for k∈ if k divides c_i for all i∈{1,…,n} and there is a curve class ^'' satisfying k^''=^'.

Now for every tuple (𝐜,^') we define 

    LMOV_𝐜,^'(ħ) ∈ħ^2

by demanding that these formal power series satisfy

    ∑_g≥ 0ħ^2g-2+n g,𝐜,^'X,L,f∏_i=1^n 2 sinħ c_i2/c_i·∑_k|(𝐜,^') k^n-1(2 sinħ k2)^-2 LMOV_𝐜/k,^'/k(kħ) .

Writing  for the Möbius function one can check that

    LMOV_𝐜,^'(ħ) =  (2 sinħ2)^2 ∏_i=1^n c_i/2 sinħ c_i2 ·∑_k | (𝐜,^')(k)/k∑_g≥ 0 (kħ)^2g-2+n g,𝐜/k,^'/kX,L,f

is indeed a solution to the above defining equation.


	
	<cit.> LMOV_𝐜,^' lifts to a Laurent polynomial in q=^ħ with integer coefficients.





  §.§.§ BPS integrality of logarithmic invariants

Now let (S D_1+D_2) be a logarithmic Calabi–Yau surface with boundary the union of two transversally intersecting smooth curves D_1,D_2. To a tuple (ĉ,), where  is an effective curve class in S satisfying D_1 ·≥ 0 and D_2 · > 0 and ĉ is a contact condition of the form

    ĉ = (-0.9em13.2em0    ⋯    0     c_1    …    c_n     0
    
    				0    [4.5em]⋯_m times    0     0    …    0     D_2 ·)

for some c_1,…,c_n >0 and m≤ n, we associate a formal power series

    BPS_ĉ,(ħ) ∈ħ^2

solving the recursive system

    (-1)^K_S · +1 ∑_g≥ 0ħ^2g-1+n g,ĉ,S  D_1+D_2(-1)^g _g  Π_i=1^n ev_i(pt)
       (∏_i=1^n-m2 sinħ c_i2/c_i) (1/m!∏_i=n-m+1^n 2 sinħ c_i2) (2 sinħ (D_2 ·)2) ∑_k | (ĉ,) k^n-1 (2 sinħ k2)^-2 BPS_ĉ/k,/k (k ħ)  .

Again, one can check that

    BPS_ĉ,   = (2 sinħ2)^2 (∏_i=1^n-mc_i/2 sinħ c_i2) (1/m!∏_i=n-m+1^n1/2 sinħ c_i2) 1/2 sinħ (D_2 ·)2
       ×∑_k | (ĉ,)(k)   k^m-1  (-1)^K_S · /k +1∑_g≥ 0 (kħ)^2g-1+n g,ĉ/k,/kS  D_1+D_2(-1)^g _g  Π_i=1^n ev_i(pt)

is indeed a solution to this defining equation. Motivated by the last section we expect the following.


	
	BPS_ĉ, lifts to a Laurent polynomial in q=^ħ with integer coefficients.


Evidence for the conj: BPS integrality comes from the following immediate corollary of Yu's LMOV integrality (<Ref>) and our logarithmic-open correspondence (<Ref>).


	
	<Ref> holds if (S  D_1 + D_2) satisfies <Ref>. Especially, if (X,L,f) is the toric triple from <Ref> we have
	
		       BPS_ĉ, = LMOV_𝐜,^' .        


Note that the pattern in which the sine factors enter the right-hand side of equation (<ref>) naturally suggests a generalisation to the case where we permit more than one point of contact with D_2. It remains to be seen whether BPS integrality persists in this case as well.







