Scattering diagrams, theta functions, and refined tropical curve counts
SSCATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICALCURVE COUNTS
TRAVIS MANDEL
Abstract.
Working over various monoid-graded Lie algebras and in arbitrary dimension, we ex-press scattering diagrams and theta functions in terms of counts of tropical curves/disks, weightedby multiplicities given in terms of iterated Lie brackets. Over the tropical vertex group, alreadyimportant in the Gross-Siebert mirror symmetry program, our tropical curve counts give descen-dant log Gromov-Witten invariants. Upcoming work will use this to prove the Gross-Hacking-KeelFrobenius structure conjecture for cluster varieties. The non-degeneracy of the trace-pairing for thisconjecture is also proven here. Working over the quantum torus algebra yields theta functions forquantum cluster varieties, and our tropical description sets up for geometric interpretations of these.As an immediate application, we prove the quantum Frobenius conjecture of [FG09]. We also provea refined version of the [CPS] result on consistency of theta functions.
Contents
1. Introduction 12. Scattering diagrams and theta functions 33. Tropical curves and the main results 164. Cluster varieties and Frobenius maps 31References 351.
Introduction
Mirror symmetry predicts a close relationship between certain counts of holomorphic disks in onespace and the data of certain sections of vector bundles on a mirror space. Motivated by this, variouscombinations of Gross, Hacking, Keel, Kontsevich, and Siebert [CPS, GHK15, GHKK18, GHS] havedefined canonical “theta functions” in terms of combinatorial objects called scattering diagrams andbroken lines which, at least heuristically, capture the data of mirror holomorphic disk counts. In thisarticle, building off ideas from [GPS10], we show how the scattering diagrams and theta functions(along with certain refinements!) can be expressed in terms of certain counts of tropical curves andtropical disks, cf. Corollary 3.8 and Theorem 3.9. The original motivation is that these tropical curvecounts can be related to holomorphic curve counts [Mik05, NS06, Gro18, MRa], and this should leadto proofs of the expected mirror symmetry correspondences. Indeed, upcoming work of the author
The author was supported by the Center of Excellence Grant “Centre for Quantum Geometry of Moduli Spaces”from the Danish National Research Foundation (DNRF95), and later by the National Science Foundation RTG GrantDMS-1246989, and then by the Starter Grant “Categorified Donaldson-Thomas Theory” no. 759967 of the EuropeanResearch Council. a r X i v : . [ m a t h . QA ] S e p TRAVIS MANDEL will use this to prove that the Frobenius structure conjecture of Gross-Hacking-Keel [GHK15, arXivv1, § § § ψ -class conditions (the refineddescendant multiplicities of [BS] are a symmetrization of ours). Various Block-G¨ottsche invariants(with slightly different conditions imposed than in our setup) have been interpreted in terms of refinedSeveri degrees [BG16, GS14], and a motivic interpretation has been investigated in [NPS16].The description of the two-dimensional scattering diagrams in terms of Block-G¨ottsche invariantswas previously found in [FS15, Corollary 4.8], where they noted a relationship to Poincar´e polynomialsof certain moduli of quiver representations (refining the results of [GP10] in terms of the correspondingEuler characteristics). It follows from the ideas of [Bri17] that this Poincar´e polynomial descriptionholds in higher dimensions as well. This will be explained in upcoming joint work of the author withM.-W. Cheung [CM], where we will also express the tropical multiplicities appearing here in terms ofcertain moduli of composition series. New Donaldson-Thomas/Gromov-Witten correspondence theo-rems (and quantum refinements) will follow from comparing these equivalent descriptions of scatteringdiagrams.Alternatively, these Block-G¨ottsche invariants for two-dimensional scattering diagrams have beenrelated to certain higher-genus invariants with lambda classes by Bousseau [Bouc], and to countsof real curves by Mikhalkin [Mik16]. Bousseau has used his correspondence result and the tropicaldescription of two-dimensional quantum scattering diagrams to express these scattering diagrams,hence the mirror construction of [GHK15], in terms of these higher-genus invariants, cf. [Boub, Boua].Future work of the author with Bousseau will use our results to prove a refined version of the Frobeniusstructure conjecture in dimension 2, relating quantum theta functions to these higher-genus invariants.Other upcoming work of the author will extend Mikhalkin’s ideas to more general conditions andhigher dimensions. This will relate our quantum tropical counts (those appearing in the quantumtorus algebra cases of Corollary 3.8 and Theorem 3.9) to certain signed counts of holomorphic diskswith boundary on the positive real locus, the power of the quantum parameter q giving certain areasof the disks. The goal here is a quantum version of the Frobenius structure conjecture for open stringsin the presence of a B -field. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 3
As outlined above, the primary and motivating value of our main results is that the expressions interms of tropical curve counts will lead to nice geometric interpretations. That said, we do presenttwo direct applications of the tropical description to understanding properties of the theta functions.First, in § § This conjecture describes the behavior of quantum theta functionsat roots of unity under the action of the quantum Frobenius map. Outline of the paper. In § § § Theorem 2.16 , andit is an important step towards proving the Frobenius structure conjecture. Degree 0 terms in prod-ucts of theta functions (i.e., the traces) can be understood in terms of tropical curves, as opposed tojust tropical disks, so the correspondence to holomorphic curves is far better understood (although itshould be possible to express the tropical disk counts in terms of the punctured invariants of [GS18]).In § § Proposition 3.5 . Our main results relatingscattering diagrams to tropical curve counts are
Theorem 3.7 and
Corollary 3.8 , and then thedescription of theta functions in terms of tropical curve counts is
Theorem 3.9 . We prove theseresults and the refined [CPS] result (
Theorem 2.14 ) in § § § § § Theorems 4.2 and 4.3 ,respectively).
Acknowledgements.
The author thanks Tom Bridgeland, Man-Wai Cheung, Mark Gross, SeanKeel, Greg Muller, Helge Ruddat, Bernd Siebert, and Jacopo Stoppa for helpful discussions.2.
Scattering diagrams and theta functions
Notation.
Given a finite-rank lattice L , we write L Q := L ⊗ Q and L R := L ⊗ R . We denote the dualpairing between L and Hom( L, Z ) by (cid:104)· , ·(cid:105) . Fock and Goncharov’s Frobenius conjectures should not be confused with the quite different Frobenius structure conjecture of Gross-Hacking-Keel. For quantum cluster varieties from surfaces, the quantum Frobenius conjecture is [AK17, Thm. 1.2.6], althoughthe canonical bases there are not yet known to equal to theta bases.
TRAVIS MANDEL
Scattering diagrams.
Let N denote a lattice of finite rank r , and let M denote the dual latticeHom( N, Z ). Fix a strictly convex rational polyhedral cone σ N ⊕ ⊂ N R . Let N ⊕ := σ N ⊕ ∩ N , and let N + := N ⊕ \{ } . Let g := (cid:76) n ∈ N + g n be a Lie algebra graded by N + , meaning that [ g n , g n ] ⊆ g n + n .For each k ∈ Z ≥ , let kN + := { n + . . . + n k ∈ N | n i ∈ N + for each i = 1 , . . . , k } . (1)Let g ≥ k := (cid:77) n ∈ kN + g n . Note that g ≥ k is a Lie subalgebra of g . Let g k denote the nilpotent Lie algebra g / g ≥ k , and let (cid:98) g := lim ←− g k . We have corresponding Lie groups G := exp g , G k := exp g k , and (cid:98) G := exp (cid:98) g = lim ←− G k .Let σ P denote a convex (but not necessarily strictly convex) rational polyhedral cone in N R con-taining σ N ⊕ , and let P := σ P ∩ N . Let A be a P -graded algebra over A (the degree 0 part) on which g acts via A -derivations respecting the grading, so g n · A p ⊂ A n + p . It will occasionally be useful tothink of g ⊕ A as a Lie algebra under the bracket[ g + a , g + a ] = [ g , g ] + g · a − g · a + [ a , a ] . (2)Let A ≥ k := (cid:76) n ∈ kN + A n , A k := A/A ≥ k , and let (cid:98) A := lim ←− A k denote the N + -adic completion of A .Then (cid:98) G inherits an action on (cid:98) A , (cid:98) G → Aut( (cid:98) A ).For any sublattice L ⊂ N or subspace L ⊂ N R , let g L := (cid:76) n ∈ L ∩ N + g n be the corresponding subLie algebra of g , and let (cid:98) g L denote the ( L ∩ N + )-adic completion. Similarly, define A L := (cid:76) n ∈ L ∩ P A n and let (cid:98) A L be its ( L ∩ N + )-adic completion.Fix a saturated sublattice K ⊂ N such that [ g , g K ] = 0 and g · A K = 0, i.e., such that g K is centralin g , and the action of g on A is via A K -derivations. Let π K : N → N := N/K denote the projection, and let M be the dual lattice to N , canonically identified with K ⊥ ∩ M ⊂ M .We assume that P + K is a lattice, i.e., that P := π K ( P ) is a lattice, and we fix a piecewise-linearsection ϕ : P → P of π K | P such that P = ϕ ( P ) + ( K ∩ P ). For each p ∈ P , we designate a specialelement z ϕ ( p ) ∈ A ϕ ( p ) . We assume ϕ (0) = 0, and z = 1. In our examples it will typically be obviousfrom the notation what these elements z ϕ ( p ) are.For each n ∈ N + , we have a Lie subalgebra g (cid:107) n := (cid:98) g Z n ⊂ (cid:98) g . For n ∈ N + and m ∈ n ⊥ \ { } , let g (cid:107) n,m denote the sub Lie algebra of g (cid:107) n consisting of those g suchthat [ g, g m ⊥ ] = 0 and g · A m ⊥ = 0. For each ( n , m ) and ( n , m ) with n i ∈ N + and nonzero m i ∈ n ⊥ i for i = 1 ,
2, we require that [ g (cid:107) n ,m , g (cid:107) n ,m ] ⊂ g (cid:107) n + n ,µ (( n ,m ) , ( n ,m )) , (3)where µ (( n , m ) , ( n , m )) := (cid:104) n , m (cid:105) m − (cid:104) n , m (cid:105) m ∈ ( n + n ) ⊥ . (4) The reader can safely take K = 0 (so N = N ) and ignore this extra generality, but in certain applications it isuseful to view A K as the coefficient ring for A . Similarly, a reader who is interested only in the scattering diagram, notin broken lines and theta functions, may take A = 0 and P = N , so then all conditions on A and P become trivial. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 5
This setup is motivated by the following examples, which will be built upon throughout the paper.
Examples 2.1.
For use in these examples, we fix a commutative ring R .(i) The tropical vertex group:
Let Θ K ( R [ N ⊕ ]) denote the module of log derivations of R [ N ⊕ ]over R [ K ∩ N ⊕ ]: Θ K ( R [ N ⊕ ]) := R [ N ⊕ ] ⊗ Z M with action on R [ N ⊕ ] defined by f ⊗ m ( z n ) := f (cid:104) n, m (cid:105) z n . We write f ⊗ m as f ∂ m . Θ K ( R [ N ⊕ ]) forms a Lie algebra with bracket [ a, b ] := ab − ba , wheremultiplication means composition of derivations. In particular, one computes[ z n ∂ m , z n ∂ m ] = z n + n ∂ µ (( n ,m ) , ( n ,m )) (5) for µ as defined in (4). Let h := (cid:77) n ∈ N + h n , where h n is the submodule of Θ K ( R [ N ⊕ ]) spanned by elements of the form z n ∂ m with (cid:104) n, m (cid:105) =0. One easily checks that h is closed under the bracket and hence is a Lie subalgebra, clearlygraded by N + . We take g := h . The corresponding pronilpotent group (cid:98) G = (cid:98) H constructedfrom this g as above is called the tropical vertex group.For the algebra A , we take A := R [ P ], so (cid:98) A =: R (cid:74) N ⊕ (cid:75) P is the corresponding Laurentpolynomial ring. One checks that an element of the form exp(log( f ) ∂ m ) ∈ (cid:98) G acts on amonomial z p ∈ (cid:98) A via exp(log( f ) ∂ m ) · z p = z p f (cid:104) p,m (cid:105) . Note that g (cid:107) n,m is generated by z n ∂ m . Condition (3) now follows from (5).(ii) Poisson torus algebras:
This is a special case of the tropical vertex group example and isparticularly important for cluster algebras. For this and Example (iii) below, we assume N is equipped with a Q -valued skew-symmetric form ω = {· , ·} . Each g will be skew-symmetricwith respect to ω in the sense that if { n , n } = 0, then [ g n , g n ] = 0. Similarly, the actionson A will be skew-symmetric, meaning that g n · A n = 0 whenever { n , n } = 0. Note thatthese skew-symmetry conditions imply that g (cid:107) n = g (cid:107) n, { n, ·} , and that Condition (3) also follows.For simplicity, we also assume in this example that either {· , ·} is Z -valued, or that R contains a copy of Q (in which case we identify Θ K ( R [ N ⊕ ]) with R [ N ⊕ ] ⊗ Q M Q ). We definea map ω : N → M Q via ω ( n ) = { n, ·} . A natural choice for K in this and the next exampleis K := ker( ω ).Now, let h be the Lie algebra of the previous example. The elements of the form z n ∂ ω ( n ) (6) generate a Lie subalgebra g ω ⊂ h which we take as our g . We denote the correspondingprounipotent Lie group (cid:98) G by (cid:98) G ω . A and (cid:98) A are as before, and the action of g ω on them is viarestriction from that of h . TRAVIS MANDEL
With this setup, (cid:98) A = R (cid:74) N ⊕ (cid:75) P forms a Poisson algebra with Poisson bracket defined by[ z p , z p ] := { p , p } z p + p . (7) Then ι : z e i ∂ ω ( n ) (cid:55)→ z e i identifies g ω (respectively, (cid:98) g ω ) with the R -span (respectively, thetopological R -span ) of the elements z n ∈ R (cid:74) N ⊕ (cid:75) P with n ∈ N + , the Lie bracket beingidentified with the Poisson bracket. The action of g ω on A is then just the restriction to g ω ofthe adjoint action of A on itself (with the Poisson bracket as the Lie bracket), and similarlyfor the action of (cid:98) g ω on (cid:98) A .(iii) Quantum torus algebras:
The previous example admits a quantization (important forquantum cluster algebras) as follows: Fix some D ∈ Q > such that D {· , ·} is Z -valued. Forany a ∈ D Z ≥ , we have a corresponding “quantum number”[ a ] q := q a − q − a ∈ R [ q ± /D ] . Note that lim q /D → a ] q q − q − = a . Define R q ⊂ R ( q /D ) by adjoining [ a ] − q to R [ q ± /D ] for each a ∈ D Z > .Now, let A = R q [ P ] be the quantum torus algebra: R q [ P ] := R q [ z p | p ∈ P ] / (cid:104) z p z p = q { p ,p } z p + p (cid:105) . The N + -adic completion is (cid:98) A =: R q (cid:74) N ⊕ (cid:75) P . R q [ P ] forms a Lie algebra under the usual commutator, which one easily checks is given by[ z p , z p ] = [ { p , p } ] q z p + p . We take g = g ωq to be the sub-Lie algebra R q [ N + ], spanned over R q by z n with n ∈ N + . Theaction of g ωq on A is just the restriction of the adjoint action. One checks that this specializes to the previous example in the q /D (cid:55)→ z p (cid:55)→ z p for A and z n q − q − (cid:55)→ z n for g .The N + -adic completion of g is (cid:98) g ωq = R q (cid:74) N ⊕ (cid:75) , and exponentiation yields (cid:98) G := (cid:98) G ωq in themultiplicative group of R q (cid:74) N ⊕ (cid:75) ⊂ (cid:98) A . The action of this (cid:98) G on (cid:98) A is then via conjugation, g · a = gag − .(iv) Hall algebras:
The Hall algebra scattering diagrams of [Bri17] provide additional interestingexamples which further refine the Poisson and quantum torus algebra examples above. How-ever, the Hall algebra does not satisfy the condition of (3). To apply the results of this paperthen, including the crucial refined [CPS] result (Theorem 2.14), one must mod out by an idealin order to obtain a skew-symmetric Lie algebra. This setup will not be further discussedhere, but it will be investigated in [CM].
Definition 2.2.
For the above data, a wall in N R over g is a triple ( m d , d , g d ) such that: By the topological R -span of a set S in the N + -adic completion (cid:98) A of A , we mean the set of all possibly-infinitesums of elements in S with coefficients in R such that, for each k >
0, all but finitely many terms vanish modulo A ≥ k . Technically, making g ωq → g ω into a well-defined Lie algebra homomorphism requires more care with the coefficientsin g ωq . In § A , but not for g , so we do not take the time to make thisprecise. Actually, our proof of Theorem 3.9 does not use (3) and so applies more generally, but without Theorem 2.14, thetafunctions become less meaningful.
CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 7 • m d is an element of M (which we recall is identified with K ⊥ ∩ M ), determined up to positivescaling (we could require m d to be primitive, but it will often be convenient to allow it to benon-primitive). • d is a closed, convex (but not necessarily strictly convex), rational-polyhedral, codimension-oneaffine cone in N R which is parallel to m ⊥ d . This is called the support of the wall. • g d ∈ g (cid:107) n d ,m d for some primitive n d ∈ m ⊥ d ∩ N . − n d is called the direction of the wall.A scattering diagram D over g is a set of walls over g such that for each k >
0, there are only finitelymany ( m d , d , g d ) ∈ D with g d not projecting to 0 in g k .A wall with direction − n d is called incoming if it is closed under addition by n d . Otherwise, thewall is called outgoing. Note that, given d , the additional data of m d is equivalent to choosing a sideof d to be the positive side of the wall (i.e., the side where m d is positive).We will sometimes denote a wall ( m d , d , g d ) by just d . Denote Supp( D ) := (cid:83) d ∈ D d , andJoints( D ) := (cid:91) d ∈ D ∂ d ∪ (cid:91) d , d ∈ D dim d ∩ d = r − d ∩ d . Remark . We briefly discuss how our definition of a scattering diagram relatesto other definitions which have appeared in the literature.(i) In practice, walls of scattering diagrams are closed under addition by K R . Thus, it is rea-sonable (though more notationally cumbersome for our purposes here) to view the scatteringdiagram as living in N R , replacing each d above with π K ( d ) and viewing m ⊥ d as living in N R instead of N R . This is essentially the approach implicitly used in [GPS10] and [GHK15]. Themodifications for this viewpoint are fairly straightforward: The direction of a wall is then − π K ( n d ) instead of − n d , and incoming walls are then closed under addition by π K ( n d ). Inthe definition of broken lines in Def. 2.10 below, the only modification is that Q should livein N R instead of N R , and γ (cid:48) ( t ) should be − π K ( v i ) in place of − v i . Similarly, when using thisviewpoint, our counts of tropical curves and tropical disks in N R can be replaced with theanalogous counts in N R obtained by applying π K to each value of the tropical degree and toeach incidence condition.(ii) In some setups, e.g., the Hall algebra setup of [Bri17], it is more natural to view the walls ofthe scattering diagram as living in M R , with d being parallel to n ⊥ d . These cases come witha skew-symmetric form {· , ·} on N and a map ω : N → M as mentioned in Example 2.1(ii),and broken lines have γ (cid:48) ( t ) = − ω ( v i ) in place of − v i . These scattering diagrams in M R yieldscattering diagrams in N R as in our setup by taking ω − of the supports of the walls. If g isskew-symmetric with respect to {· , ·} and we take K = ker ω , then ω ( N R ) is identified with N R , and so intersecting the walls in M R with ω ( N R ) recovers the viewpoint of (i) above.Note that for each k >
0, a scattering diagram D over g induces a finite scattering diagram D k over g k with walls corresponding to the d ∈ D for which g d is nontrivial in g k .Consider a smooth immersion γ : [0 , → N R \ Joints( D ) with endpoints not in Supp( D ) which istransverse to each wall of D it crosses. Let ( m d i , d i , g d i ), i = 1 , . . . , s , denote the walls of D k crossedby γ , and say they are crossed at times 0 < t ≤ . . . ≤ t s <
1, respectively (if t i = t i +1 , then therequirement that each g d is in g (cid:107) n d ,m d implies that the ordering of these two walls does not affect (8) TRAVIS MANDEL and therefore does not matter). Define θ d i := exp( g d i ) sgn (cid:104)− γ (cid:48) ( t i ) ,m d i (cid:105) ∈ G k . (8)Let θ kγ, D := θ d s · · · θ d ∈ G k , and define the path-ordered product: θ γ, D := lim ←− k θ kγ, D ∈ (cid:98) G. Definition 2.4.
Two scattering diagrams D and D (cid:48) are equivalent if θ γ, D = θ γ, D (cid:48) for each smoothimmersion γ as above. D is consistent if each θ γ, D depends only on the endpoints of γ . Examples 2.5. (i) Replacing a wall ( m d , d , g d ) ∈ D with the wall ( − m d , d , − g d ) produces an equivalent scatteringdiagram.(ii) Consider a collection of walls { ( m d , d , g d i ∈ g (cid:107) n d ,m d ) ∈ D | i ∈ S } , where S is some countableindex set and n d , m d , and d are independent of i . Replacing this collection of walls with asingle wall ( m d , d , (cid:80) i ∈ S g d i ) produces an equivalent scattering diagram.(iii) Replacing a wall ( m d , d , g d ) ∈ D with a pair of walls ( m d , d i , g d ), i = 1 ,
2, such that d ∪ d = d and codim( d ∩ d ) = 2 produces an equivalent scattering diagram.The following theorem on scattering diagrams is fundamental to the theory. The two-dimensionaltropical vertex group cases were first proved in [KS06]. The tropical vertex cases for higher-dimensionalspaces (including more general affine manifolds than just N R ) were proved in [GS11], and the resultfor more general g follows from [KS14, Thm. 2.1.6] (cf. [GHKK18, Thm. 1.21]). Alternatively, wenote that the result follows from the construction of D ∞ k in § § Theorem 2.6.
Let g be an N + -graded Lie algebra, and let D in be a finite scattering diagram over g whose only walls have full affine hyperplanes as their supports. Then there is a unique-up-to-equivalence scattering diagram D , also denoted Scat( D in ) , such that D is consistent, D ⊃ D in , and D \ D in consists only of outgoing walls. We next give several important examples of initial scattering diagrams D in . For a more specificexample of a possible D in and the corresponding Scat( D in ), cf. Example 2.11. Examples 2.7.
We present some important examples of initial scattering diagrams which will beexamined more in §
4. These examples build off those of Examples 2.1. First though, we fix someadditional data:We fix a multiset (i.e., a set possibly with repetition) E := { e i } i ∈ I of vectors in N , indexed over afinite set I . Let F be a subset of I such that E I \ F := { e i } i ∈ I \ F ⊂ N ⊕ (typically, one would be given N , E , I , and F , and would then choose N ⊕ ⊂ N to contain E ).For the skew-symmetric examples, we also we fix numbers { d i ∈ Q > } i ∈ I and define a bilinear form( · , · ) on N satisfying ( e i , e j ) = d j { e i , e j } . We require that ( n , n ) ∈ Z whenever n , n ∈ N with at least one of n or n being in N ⊕ .The form ( · , · ) determines maps π , π : N ⊕ → M , π ( n ) := ( n, · ) and π ( n ) := ( · , n ). In all ourskew-symmetric examples, we will have g (cid:107) n = g (cid:107) n,π ( n ) . A natural choice for K is K := ker( π ), whichif E spans N Q is the same as ker( ω ). CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 9 (i) Take g = h as in Example 2.1(i). In addition to E , suppose we are given a multiset U = { u i } i ∈ I \ F , this time with vectors u i ∈ M \ { } , such that (cid:104) e i , u i (cid:105) = 0 for each i ∈ I \ F . Thenwe take the initial scattering diagram to be D in := { ( u i , u ⊥ i , log(1 + z e i ) ∂ u i ) | i ∈ I \ F } . The wall-crossing automorphism for crossing from the side of ( u i , u ⊥ i , log(1 + z e i ) ∂ u i ) contain-ing some p ∈ P to the other side then acts by z p (cid:55)→ z p (1 + z e i ) |(cid:104) p,u i (cid:105)| . (9) Such walls are commonly (e.g., in [GPS10] and [GHKK18]) denoted as simply ( u ⊥ i , (1+ z e i ) | u i | ).(ii) Now take g = g ω ⊂ h as in Example 2.1(ii). We take D in to be the special case of D in fromthe previous example in which u i is taken to be − π ( e i ) = d i ω ( e i ) for each i ∈ I \ F .Using the embedding ι : z n ∂ ω ( n ) (cid:55)→ z n of (cid:98) g ω into the Poisson algebra (cid:98) A = R (cid:74) N ⊕ (cid:75) P , theinitial scattering functions log(1 + z e i ) ∂ d i ω ( e i ) become dilogarithms: ι (cid:0) log(1 + z e i ) ∂ d i ω ( e i ) (cid:1) = ι (cid:32) ∞ (cid:88) k =1 ( − k +1 k d i z ke i ∂ ω ( ke i ) (cid:33) = d i ∞ (cid:88) k =1 ( − k +1 z ke i k = − d i Li ( − z e i ) , (10) where Li is the dilogarithm function defined byLi ( x ) := ∞ (cid:88) k =1 x k k . Thus, we can write the initial scattering diagram as D in = { ( ω ( e i ) , ω ( e i ) ⊥ , − d i Li ( − z e i )) | i ∈ I \ F } . (11)(iii) Consider the quantization g = g ωq as in Example 2.1(iii), that is, (cid:98) g = R q (cid:74) N + (cid:75) ⊂ R q (cid:74) N ⊕ (cid:75) P = (cid:98) A . Similarly to in the previous example, we take the initial scattering diagram to be D in := { ( ω ( e i ) , ω ( e i ) ⊥ , − Li ( − z e i ; q /d i ) } , (12) where the scattering functions are now defined in terms of quantum dilogarithms:Li ( x ; q ) := ∞ (cid:88) k =1 x k k [ k ] q . Here, we use our notation [ k ] q = q k − q − k , so [ k ] q /di = [ k/d i ] q . Note that the q (cid:55)→ ( x ; q ) is Li ( x ) (with xq − q − mapping to x ), so this D in does indeed specialize to the onefrom the previous example in the q /D (cid:55)→ z n q − q − mapping to z n ). LetΨ q /di ( z e i ) := exp( − Li ( − z e i ; q /d i )) = ∞ (cid:89) k =1
11 + q (2 k − /d i z e i ∈ (cid:98) G. Then for any p ∈ P , crossing a wall as above from the side containing p to the other side actson z p viaΨ q /di ( z e i ) sgn { e i ,p } · z p = Ψ q /di ( z e i ) sgn { e i ,p } z p Ψ q /di ( z e i ) − sgn { e i ,p } = z p d i |{ e i ,p }| (cid:89) k =1 (1 + q sgn( { e i ,p } )(2 k − /d i z e i ) . (13)Given an N + -graded Lie algebra g as above and any commutative, associative algebra T , we canobtain another N + -graded Lie algebra g ⊗ T with bracket defined by [ g ⊗ t , g ⊗ t ] := [ g , g ] g ⊗ ( t t )(when it is possibly not clear from context, we will use subscripts after brackets to indicate the Liealgebra in which the bracket is performed). We will denote elements g ⊗ t as simply tg . We denote N + -adic completion of g ⊗ T by g (cid:98) ⊗ T , and we similarly denote the corresponding Lie group as G (cid:98) ⊗ T .These act on the algebra A (cid:98) ⊗ T obtained by taking the N + -adic completion of A ⊗ T . Here, the actionof g (cid:98) ⊗ T on A (cid:98) ⊗ T is given by ( tg ) · a = ( g · a ) ⊗ t , also denoted t ( g · a ). We will often use this constructionto adjoin nilpotent elements. The following lemma is straightforward. Lemma 2.8.
Let T be a commutative, associative algebra with t ∈ T , t = 0 . Let g ∈ (cid:98) g , a ∈ (cid:98) A . Then exp( tg ) · a = a + t ( g · a ) . Here, · on the left-hand side is the action of G (cid:98) ⊗ T on A (cid:98) ⊗ T , while · on the right-hand side is the actionof g on (cid:98) A . In § D in ) from D in will depend on repeatedly applying the followingcomputation: Lemma 2.9.
Suppose we have an N + -graded Lie algebra g and a commutative associative algebra T with t , t ∈ T , t = t = 0 . Fix n , n ∈ N + , and fix primitive m , m ∈ M such that (cid:104) n i , m i (cid:105) = 0 for i = 1 , . Also, fix some g i ∈ g n i for i = 1 , . Let D in := { ( m , m ⊥ , t g ) , ( m , m ⊥ , t g ) } be a scattering diagram over g ⊗ T . Then Scat( D in ) = D in ∪ { ( m , d , g ) } , where m := µ (( n , m ) , ( n , m )) , d := ( m ⊥ ∩ m ⊥ ) + R ≤ ( n + n ) , and g := t t [ g , g ] (cid:98) g . Proof.
First, recall from (4) that µ (( n , m ) , ( n , m )) := (cid:104) n , m (cid:105) m − (cid:104) n , m (cid:105) m . One easily checksnow that ( m ∩ m ) ⊥ ⊂ m ⊥ and n + n ∈ m ⊥ , so m ⊥ ⊃ ( m ⊥ ∩ m ⊥ ) + R ( n + n ) . Hence, m ⊥ does contain d . Lemma 2.9 in the cases where dim N = 2 and g = h is essentially [GPS10, Lemma 1.9]. In the cases with dim N = 2and g = g ωq , it is [FS15, Lemma 4.3]. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 11
Now, let γ be a path as in the figure to the right, goingfrom the region with m , m < m > m <
0, then to m , m >
0, then to m < m >
0, and then back to m , m <
0. Then θ γ = exp( t g ) exp( t g ) exp( − t g ) exp( − t g )= [exp( t g ) , exp( t g )] G (cid:98) ⊗ T , where [ a, b ] G (cid:98) ⊗ T := aba − b − for any a, b ∈ G (cid:98) ⊗ T . γ m m − + − +We claim that [exp( t g ) , exp( t g )] G (cid:98) ⊗ T = exp([ t g , t g ] g (cid:98) ⊗ T ) . (14)Indeed, the Baker-Campbell-Hausdorff formula tells us that for any x, y ∈ (cid:98) g , we havelog(exp( t x ) exp( t y )) = t x + t y + 12 t t [ x, y ] , and using this, we compute:log([exp( t g ) , exp( t g )]) = log(exp( t g ) exp( t g ) exp( − t g ) exp( − t g ))= log(exp(log(exp( t g ) exp( t g ))) exp(log(exp( − t g ) exp( − t g )))= log(exp( t g + t g + 12 t t [ g , g ]) exp( − t g − t g + 12 t t [ g , g ]))= ( t g + t g + 12 t t [ g , g ]) + ( − t g − t g + 12 t t [ g , g ])+ 12 [ t g + t g + 12 t t [ g , g ] , − t g − t g + 12 t t [ g , g ]]= [ t g , t g ] . Thus, θ γ = exp( t t [ g , g ]) = exp( g ). Since g = t t [ g , g ] is in g n + n and commutes with both t g and t g , we just have to check that crossing d along γ induces the scattering automorphism g − . That is, we just have to check that (cid:104)− γ (cid:48) ( t ) , m (cid:105) <
0, where t is the time at which γ passes d .Suppose (cid:104) n , m (cid:105) ≥ (cid:104) n , m (cid:105) ≥
0. Then (cid:104) n , m (cid:105) ≥
0, and when γ passes through d , itcomes from the side of d which contains − n . Hence, (cid:104)− γ (cid:48) ( t ) , m (cid:105) ≤
0, as desired. The cases whereone or both of (cid:104) n , m (cid:105) and (cid:104) n , m (cid:105) are negative are similarly checked. (cid:3) Broken lines and theta functions.
Fix a consistent scattering diagram D over g , with (cid:98) g actingon (cid:98) A as in § p ∈ P , we have designated an element z ϕ ( p ) ∈ A ϕ ( p ) . Definition 2.10.
Let p ∈ P \ { } , Q ∈ N R \ Supp( D ). A broken line γ with ends ( p, Q ) is the dataof a continuous map γ : ( −∞ , → N R \ Joints( D ), values −∞ < t ≤ t ≤ . . . ≤ t (cid:96) = 0, and for each i = 0 , . . . , (cid:96) , an associated homogeneous element a i ∈ A v i for some v i ∈ P \ { } , such that:(i) γ (0) = Q .(ii) For i = 1 . . . , (cid:96) , γ (cid:48) ( t ) = − v i for all t ∈ ( t i − , t i ). Similarly, γ (cid:48) ( t ) = − v for all t ∈ ( −∞ , t ).(iii) a = z ϕ ( p ) . (iv) For i = 0 , . . . , (cid:96) − γ ( t i ) ∈ Supp( D ). Let g i := (cid:89) ( m d , d ,g d ) ∈ Dd (cid:51) γ ( t i ) exp( g d ) sgn( (cid:104) v i ,m d (cid:105) ) ∈ (cid:98) G. I.e., g i is the (cid:15) → θ γ | ( ti − (cid:15),ti + (cid:15) ) defined in (8) (usinga smoothing of γ ). Then a i +1 is a homogeneous term of g i · a i .We will call v i +1 − v i ∈ N ⊕ a bend of γ . We assume all bends are nonzero, so we cannot get newbroken lines by just inserting new values of t as trivial bends. A straight broken line is a broken linewith no bends. By the type of a broken line γ as above, we mean the data of the elements a i ∈ A v i , i = 0 , . . . , (cid:96) .Fix a generic point Q ∈ N R \ Supp( D ). For any p ∈ P \ { } , we define a theta function ϑ p,Q := (cid:88) Ends( γ )=( p,Q ) a γ ∈ (cid:98) A. (15)Here, the sum is over all broken lines γ with ends ( p, Q ), and a γ denotes the homogeneous elementof (cid:98) A attached to the final straight segment of γ . That this is well-defined will be proven shortly. Forthe case p = 0, we define ϑ ,Q = 1. d d Q d Figure 2.1.
Example 2.11.
Let N = Z , equipped with the standard skew-symmetric form, and consider thequantum torus algebra setup as in Example 2.1(iii). Consider the scattering diagram D in with walls d := ( e ∗ , ( e ∗ ) ⊥ , − Li ( − z e ; q )) and d := ( − e ∗ , ( − e ∗ ) ⊥ , − Li ( − z e ; q )). Then D := Scat( D in ) con-sists of one additional wall d := ( e ∗ − e ∗ , ( e ∗ − e ∗ ) ⊥ ∩ R ≤ , − Li ( − ( q − q − ) z e + e ; q )). The supports ofthese walls are illustrated in Figure 2.1 as solid lines. The consistency can be written as the expressionΨ q ( z e )Ψ q ( z e ) = Ψ q ( z e )Ψ q (( q − q − ) z e + e )Ψ q ( z e ) (the two sides of this equation correspondingto the two paths from the bottom-right quadrant to the top-left), which is a modified version of the CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 13 quantum pentagon identity of [FK94]. The dashed lines in Figure 2.1 are the broken lines for ϑ e ,Q .See (13) for the formula used for computing the wall-crossings. From bottom to top, the final mono-mials attached to these broken lines are z e , ( q − q − ) z e + e , and ( q − q − ) z e + e , so ϑ e ,Q is thesum of these three terms.We will now prove several facts about these theta functions, beginning with showing that they arewell-defined. Given n ∈ N ⊕ , let d ( n ) ∈ Z ≥ (16)denote the largest number k such that n ∈ kN + (as defined in (1)), taking d (0) to be 0. Note that d ( n + n ) ≥ d ( n ) + d ( n ) for all n , n ∈ N ⊕ . Now, note that for a ∈ (cid:98) A p and g ∈ g n ,exp( g ) · a ∈ a + (cid:77) k ∈ Z > (cid:98) A p + kn . (17)That is, exp( g ) · a is equal to a plus terms of degree equal to p plus a positive multiple of n . Hence,for any broken line γ , we always have d ( v i +1 ) > d ( v i ) (notation as in Definition 2.10). That is, bendsalways increase d of the degree of the elements attached to the straight segments of γ , so a brokenline γ with ends ( p, Q ) contributing a γ ∈ A v γ to (15) has at most d ( v γ − ϕ ( p )) bends. Recall fromDefinition 2.2 the requirement that for each k > g d projects to 0 in g k for all but finitely manywalls d ∈ D . It now follows that for each k >
0, there are only finitely many broken lines γ withEnds( γ ) = ( p, Q ) such that the projection of a γ to g k is non-trivial. Hence, (15) is indeed well-defined.Furthermore, since bends shift the degree of the attached element by an element of N + , we seethat the term in (15) of minimal degree is the one associated to the unbroken line, i.e., z ϕ ( p ) . That is, ϑ p,Q ∈ z ϕ ( p ) + (cid:98) A ϕ ( p )+ N + , (18)where (cid:98) A ϕ ( p )+ N + is the ideal of (cid:98) A consisting of the topological span of terms with grading equal to ϕ ( p ) + n for some n ∈ N + . Let P ◦ ⊂ P be the subset consisting of the elements p such that az ϕ ( p ) (cid:54) = 0 for any nonzero a ∈ (cid:98) A K . It followsfrom (18) that the set { ϑ p,Q ∈ (cid:98) A | p ∈ P ◦ } (with fixed Q ) is linearly independent over (cid:98) A K .Recall that P = ϕ ( P ) + K ∩ P . We will frequently want to make the following assumptions: Assumptions . (i) P ◦ = P (e.g., (cid:98) A is an integral domain and each z ϕ ( p ) is nonzero).(ii) For every p ∈ P , A ϕ ( p )+ P ∩ K = z ϕ ( p ) A K .These assumptions are indeed satisfied in Examples 2.1(i)-(iii).Assumption 2.12(ii) implies that (cid:98) A is topologically spanned over (cid:98) A K by { z ϕ ( p ) | p ∈ P } . It followsfrom this and (18) that { ϑ p,Q | p ∈ P } spans (cid:98) A topologically over (cid:98) A K . In summary, we have thefollowing: Proposition 2.13.
For fixed generic Q ∈ N R \ Supp( D ) and any p ∈ P , (15) gives a well-definedelement ϑ p,Q ∈ z ϕ ( p ) + (cid:98) A p + N + ⊂ (cid:98) A . Under Assumptions 2.12, the theta functions Θ Q := { ϑ p,Q | p ∈ P } form an additive topological basis for (cid:98) A over (cid:98) A K , hence also (at least topologically) span the subalgebra A Θ ,Q ⊂ (cid:98) A generated by Θ Q . The following is a fundamental feature of theta functions. Theorem 2.14 (Refined [CPS] result) . Consider D = Scat( D in ) as in Theorem 2.6. Fix two genericpoints Q , Q ∈ N R \ Supp( D ) . Let γ be a smooth path in N R \ Joints( D ) from Q to Q . Then forany p ∈ P , ϑ p,Q = θ γ, D ( ϑ p,Q ) . When working over the module of log derivations as in Example 2.1(i) or (ii) (but for more generalconsistent scattering structures on more general integral affine manifolds than just N R ), Theorem 2.14is due to [CPS] (their Lemmas 4.7 and 4.9). The author imagines that the arguments of [CPS] can begeneralized to any g and (cid:98) A as above, but in § (cid:98) A K -algebra, A Θ ,Q is independent of thechoice of Q (although the embedding into (cid:98) A does depend on Q ). We will denote this abstract algebraby A Θ , and we let ϑ p ∈ A Θ denote the element ϑ p,Q ∈ A Θ ,Q under this identification A Θ ,Q ∼ = A Θ .Under Assumptions 2.12, one sees that A Θ and the theta functions are determined by the structureconstants α ( p , . . . , p s ; p ) ∈ (cid:98) A K , p , . . . , p s , p ∈ P , defined by ϑ p · · · ϑ p s = (cid:88) p ∈ P : z p (cid:54) =0 α ( p , . . . , p s ; p ) ϑ p . Even when Assumptions 2.12 do not hold, each generic Q ∈ N R \ Supp( D ) determines an embedding A Θ ∼ = A Θ ,Q ⊂ (cid:98) A , hence a P -grading on A Θ , and we define α Q ( p , . . . , p s ; p ) ∈ A p (19)to be the degree p part of ϑ p ,Q · · · ϑ p s ,Q . The next proposition (generalizing [GHKK18, Prop. 6.4(3)]and following the same argument) tells us how to compute the α ’s. Proposition 2.15.
For p , . . . , p s , p ∈ P and generic Q ∈ N R \ Supp( D ) , α Q ( p , . . . , p s ; p ) = (cid:88) γ ,...,γ s Ends( γ i )=( p i ,Q ) ,i =1 ,...,sπ K ( v γ + ... + v γs )= p a γ · · · a γ s , (20) where the sum is over all ordered s -tuples of broken lines ( γ i ) i =1 ,...,s with Ends( γ i ) = ( p i , Q ) , and a γ i ∈ A v γi is the element attached to the final straight segment of γ i .Now suppose Assumptions 2.12 hold. Then α ( p , . . . , p s ; 0) = α Q ( p , . . . , p s ; 0) for each generic Q ∈ N R \ Supp( D ) . More generally, for any p ∈ P , we have α ( p , . . . , p s ; p ) z ϕ ( p ) = (cid:88) (cid:96) ∈ Z ≥ (cid:88) γ ,...,γ s Ends( γ i )=( ϕ ( p i ) ,Q (cid:96) ) ,i =1 ,...,sπ K ( v γ + ... + v γs )= pd ( v γ + ... + v γs − ϕ ( p ))= (cid:96) a γ · · · a γ s ∈ (cid:98) A K , (21) where d is as in (16) and Q (cid:96) shares a maximal cell of D (cid:96) with ϕ ( p ) . When working over h , chambers of the scattering diagram give charts for the mirror manifold, and path-orderedproducts give the transition functions. In this context, Theorem 2.14 can roughly be interpreted as saying that thelocally defined theta functions ϑ p,Q patch together correctly to form global functions on the mirror. We note that α Q ( p , . . . , p s ; p ) is independent of Q if p = 0. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 15
Proof.
This first claim is straightforward from the definitions, with the finiteness of the sum in (20)following from the well-definedness of ϑ p i ,Q ∈ (cid:98) A (Prop. 2.13).For the second claim, we observe that the straight broken line with attached element z ϕ ( p ) is theonly broken line γ over D (cid:96) with π K ( v γ ) = p and end at a point Q (cid:96) which shares a maximal cell of D (cid:96) with ϕ ( p ). To see this, note that if we start at Q (cid:96) and move in the v γ -direction, then we will never hit awall of D (cid:96) and so γ cannot contain any bends. Hence, the only q ∈ P such that ϑ q,Q (cid:96) has a z ϕ ( p ) -termis q = p . On the other hand, (18) says that ϑ p,Q equals z ϕ ( p ) plus higher degree terms. Thus, for any f ∈ (cid:98) A , the z ϕ ( p ) -coefficient of f expanded in the topological (cid:98) A K -module basis { z ϕ ( n ) } n ∈ P of (cid:98) A mustagree, modulo the topological span of A ≥ (cid:96) , with the ϑ p,Q (cid:96) -coefficient of f expanded in the topologicalbasis { ϑ p,Q (cid:96) } p ∈ P . The claim now follows from considering the case f = ϑ p ,Q (cid:96) · · · ϑ p s ,Q (cid:96) . (cid:3) A non-degenerate trace pairing.
The Frobenius Structure Conjecture of [GHK15, arXiv v1, § Y, D ). More precisely, the algebra has an additive (topological) basis of“theta functions,” and the multiplication rule is determined by a “trace” function which is definedin terms of certain descendant log Gromov-Witten invariants of (
Y, D ). In this subsection we willconsider a certain trace function on A Θ and prove that it is non-degenerate, hence is sufficient tocompletely determine the structure constants for the theta function multiplication. Upcoming workof the author will then use Theorem 3.9 and some tropical correspondence results to prove that thistrace really is given by the desired GW invariants, and the combination of these results will prove theFrobenius structure conjecture (at least for cluster varieties).Viewing (cid:98) A as a topological P -graded (cid:98) A K -algebra, we have a map of (cid:98) A K -modulesTr : (cid:98) A → (cid:98) A K taking an element f ∈ (cid:98) A to its degree 0 part (using the P -grading). Since we assumed that g · A K = 0,all wall-crossing automorphisms act trivially on (cid:98) A K , and so Tr induces a map Tr : A Θ → (cid:98) A K as well(no dependence on Q ). Tr also induces an “s-point function”Tr s : (cid:98) A ⊗ s → (cid:98) A K , f ⊗ · · · ⊗ f s (cid:55)→ Tr( f · · · f s ) , and similarly for Tr s : A ⊗ s Θ → (cid:98) A K for each s ≥
1. The following theorem implies that these uniquelydetermines A Θ and the theta functions. Theorem 2.16.
Assume that A is an integral domain and that z ϕ ( p ) is nonzero for each p ∈ P . Themap Tr ∨ : A Θ → Hom (cid:98) A K ( A Θ , (cid:98) A K ) , a (cid:55)→ [ b (cid:55)→ Tr( ab )] is injective. Hence, given the topological (cid:98) A K -module structure on A Θ , the (cid:98) A K -algebra structure (i.e.,the multiplication rule) is uniquely determined by Tr and Tr . In particular, if Assumption 2.12 holds,then all the structure constants α K ( p , . . . , p s ; p ) are determined by those of the form α ( p , p ; 0) and α ( p , p , p ; 0) . An algebro-geometric proof for a version of Theorem 2.16 in the two-dimensional tropical vertex group situationhas previously been found by Gross-Hacking-Keel [GHK]. In fact, the same proof shows the strong statement that the similarly defined map Tr ∨ : (cid:98) A → Hom (cid:98) A K ( A Θ ,Q , (cid:98) A K )is injective for each Q . Proof.
To prove that Tr ∨ is injective, we will show that for any f ∈ (cid:98) A , there exists some p ∈ P suchthat Tr( f ϑ p ) (cid:54) = 0. Pick some generic Q ∈ N R \ Supp( D ) so we can view A Θ as a topological P -graded A -algebra A Θ ,Q . For nonzero f ∈ A Θ ,Q , choose p ∈ P such that d ( p ), as defined in (16), is assmall as possible subject to the condition that the degree p part of f , denoted f p , is nonzero. Let p = π K ( p ). By (18), ϑ − p = z ϕ ( − p ) +[terms with higher d ]. So the degree p + ϕ ( − p ) part of f is f p z ϕ ( − p ) (cid:54) = 0. Since π K ( p + ϕ ( − p )) = 0 ∈ P , degree p + ϕ ( − p ) with respect to the P -gradingimplies degree 0 with respect to the P -grading. Hence, Tr( f ϑ p ) (cid:54) = 0, as desired.For the remaining claims, suppose we want to determine the product of two elements a, b ∈ (cid:98) A . Theabove injectivity implies that it is enough to specify Tr( abc ) = Tr ( ab, c ) for each c ∈ (cid:98) A , and this isequal to Tr ( a, b, c ). The claim about the structure constants then follows because Assumption 2.12implies that the theta functions span (topologically), so knowing the multiplication rule for the thetafunctions determines the whole ring. (cid:3) Remark . Recall that a Frobenius R -algebra is defined to be an R -algebra A ,together with an R -algebra homomorphism Tr : A → R , such that the map Tr ∨ : A → Hom R ( A, R ), a (cid:55)→ [ b (cid:55)→ Tr( ab )], is an isomorphism. This forces A to be finite-dimensional. If we allow Tr ∨ toinstead be just injective, rather than an isomorphism, we could define infinite dimensional Frobeniusalgebras. Such structures appear, for example, in [BSS]. Theorem 2.16 then says that Tr makes A Θ into an infinite dimensional Frobenius (cid:98) A K -algebra.3. Tropical curves and the main results
For use throughout this section, let us fix an initial scattering diagram D in := { ( m d i , d i , g d i ) | i ∈ I } with I a finite index-set ( I here actually corresponds to I \ F in the setup of Examples 2.7) and d i = m ⊥ d i . We can decompose g d i as g d i = (cid:88) j ≥ g ij ∈ g (cid:107) n d i ,m d i (22)with g ij ∈ g jn d i ( j and n d i being multiplied in this subscript). For example, D in could be any ofthe initial scattering diagrams from Examples 2.7. Let D := Scat( D in ) as in Theorem 2.6. We willdescribe D and the associated theta functions in terms of counts of tropical curves and tropical disks.3.1. Tropical curves and tropical disks.Notation 3.1.
For any weighted graph Γ, possibly with some 1-valent vertices removed, we let Γ [0] ,Γ [1] , and Γ [1] ∞ denote the vertices, edges, and non-compact edges, respectively. By “weighted,” wemean that Γ is equipped with a function w : Γ [1] → Z ≥ .Let Γ be a weighted, connected, finite tree without bivalent vertices, and let Γ be the complementof the 1-valent vertices. We mark the non-compact edges via (cid:15) : S ∼ → Γ [1] ∞ for some finite index set S .Given i ∈ S , let E i denote (cid:15) ( i ). Let L be a finite-rank lattice. Definition 3.2.
A parameterized marked tropical curve in L R is the data (Γ , (cid:15) ) as above, along witha proper continuous map h : Γ → L R such that: • For each E ∈ Γ [1] , h | E is an embedding with image contained in an affine line of rational slope. If Γ consists of single edge and no vertices, we view Γ [1] ∞ as including two elements, one for each unboundeddirection. This case without vertices often requires special treatment. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 17 • The following “balancing condition” holds for every V ∈ Γ [0] : For each edge E ∈ Γ [1] con-taining V , let u ( V,E ) ∈ L \ { } denote the primitive integral vector emanating from V in thedirection h ( E ). Then (cid:88) E ∈ Γ [1] E (cid:51) V w ( E ) u ( V,E ) = 0 . (23)Two parameterized marked tropical curves h i : Γ i → L R , i = 1 ,
2, are isomorphic if there is ahomeomorphism φ : Γ → Γ respecting the weights, markings, and maps h i . A (rational) tropicalcurve is an isomorphism class of parameterized marked tropical curves.A tropical disk is defined in nearly the same way, except that Γ is equipped with a marked vertex Q out which is allowed to have any valence (including being univalent or bivalnet). Furthermore, Q out is not required to satisfy the balancing condition.The type of a tropical curve or disk is the data of the weighted marked graph (Γ , (cid:15) ), along with thevectors u ( V,E ) for each V ∈ Γ [0] and E ∈ Γ [1] with E (cid:51) V . If Γ has no vertices, the type includes thedata of the two unbounded directions.For each i ∈ S , let u E i denote the primitive vector pointing in the unbounded direction of h ( E i ).The degree ∆ of a marked tropical curve/disk ( h, Γ , (cid:15) ) is the map ∆ : S → L taking i ∈ S to w ( E i ) u E i ∈ L \ { } .Let B denote a collection { B i ⊂ L R | i ∈ S } of affine subspaces of L R indexed by S , plus an additionalaffine subspace B out if we are considering tropical disks rather than tropical curves. We say that atropical curve ( h, Γ , (cid:15) ) matches the constraints B if h ( E i ) ⊂ B i for each i ∈ S . Similarly for a tropicaldisk with the additional requirement that h ( Q out ) ∈ B out . We call the conditions imposed by B incidence conditions.For s ≥
1, we say that a tropical disk satisfies the ψ -class condition ψ s − Q out ifval( Q out ) ≥ s. Note that we can have s = − Q is univalent.Let T ∆ ( B ) denote the set of tropical curves of degree ∆ satisfying incidence conditions B . Let T (cid:48) ∆ ( B , s −
2) denote the set of tropical disks satisfying incidence conditions B and the ψ -class condition ψ s − Q out .3.1.1. Degrees and incidence conditions coming from scattering diagrams.
Let w := ( w i ) i ∈ I be a tupleof weight vectors w i := ( w i , . . . , w il i ) with 0 < w i ≤ . . . ≤ w il i , w ij ∈ Z . For Σ l i denoting the groupof permutations of { , . . . , l i } , let Aut( w ) ⊂ (cid:89) i ∈ I Σ l i be the group of automorphisms of the second indices of the weights w i which act trivially on w .Recall the lattice N = N/K = π K ( N ) from § N R . Let ∆ w denote the tropical curve degree ∆ w : I w → N \ { } Higher-valence conditions as a tropical analog of ψ -class conditions first appeared in [Mik07], with proofs of variousdescendant correspondence theorems appearing in [MR09, Gro10, Ove15, Gro18, MRa]. The last two of these apply inparticular to the tropical curve counts which appear here when working over h . with I w := { ( i, j ) | i ∈ I, j = 1 , . . . , l i } ∪ { out } , ∆(( i, j )) = w ij n d i , and ∆(out) := − n out , where n out := (cid:88) i,j w ij n d i . Here, out is the label for an unbounded edge E out . We will typically write E ( i,j ) as simply E ij .Now let p := ( p , . . . , p s ) denote an s -tuple of vectors in P \ { } , s ≥
1. We let ∆ w , p denote thetropical disk degree ∆ w , p : I w , p → N \ { } with I w , p := { ( i, j ) | i ∈ I, j = 1 , . . . , l i } ∪ { , . . . , s } , ∆(( i, j )) := w ij n d i , and ∆( k ) = ϕ ( p k ) for k = 1 , . . . , s .Given n ∈ N R \ { } , let B w ,n denote the incidence conditions { B ij | ( i, j ) ∈ I w } ∪ { B out } with each B ij a generic translate of d i , and with B out a generic translate of the span of n and n out = (cid:80) i,j w ij n d i .Similarly, given a generic point Q ∈ N R \ Supp( D ), we define the incidence conditions B w , p ,Q as follows: take B ij ’s as before, take B k := N R for each k = 1 , . . . , s (i.e., there are no incidenceconditions on the E k ’s), and after fixing the generic B ij ’s, take B out to be a single point rQ for r (cid:29) r sufficiently large relative to the distance of the B ij ’s from the origin).With these conditions, T ∆ w ( B w ,n ) and T (cid:48) ∆ w , p ( B w , p ,Q , s −
2) are finite, so we can “count” theelements of these sets after assigning appropriate multiplicities to each tropical curve.We note that for generic incidence conditions, every vertex of the tropical curves/disks in thesesets will be trivalent except for possibly Q out , which will be s -valent. Furthermore, for the tropicaldisks, each of the s components of Γ \ Q out will consist of exactly one of the edges of the form E k , k = 1 , . . . , s .3.1.2. Multiplicities.
We next define the multiplicities of the tropical curves/disks in T ∆ w ( B w ,n ) and T (cid:48) ∆ w , p ( B w , p ,Q , s − ∈ T ∆ w ( B w ,n ) for some n ∈ N R \ { } . By thinking of the edges E ij ∈ Γ [1] ∞ asbeing incoming edges and the edge E out ∈ Γ [1] ∞ as being an outgoing edge, we obtain a flow on Γ. Foreach E ∈ Γ [1] , let u E ∈ N be the primitive vector tangent to E pointing in the opposite directionof the flow of Γ. To each incoming edge E ij we associate the element m d i ∈ M and the element g iw ij ∈ g w ij u i ,m d i from the expansion of g d i given in (22).We now use the flow to recursively associate, up to signs, an element m E ∈ M ⊂ M and an element g E ∈ g (cid:107) w ( E ) u E ,m E to every edge E ∈ Γ [1] as follows: Suppose two edges E and E flow into a commonvertex V , with edge E flowing out of V , such that E and E have associated elements m E , m E ∈ M and g E ∈ g n ,m E , g E ∈ g n ,m E for n = w ( E ) u E and n = w ( E ) u E . Then we define m E := µ (( n , m E ) , ( n , m E )) ∈ M (24)for µ as defined in (4), and we define g E := [ g E , g E ] ∈ g n + n ,m E , (25)where the containment in g n + n ,m E utilizes Condition (3). Note that reordering E and E willchange the signs of both m E and g E above. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 19
This flow process determines (up to simultaneously changing both signs) elements m Γ := m E out ∈ M and g Γ := g E out ∈ g n out ,m E out (26)associated to the outgoing edge of Γ.Now, suppose n = ϕ ( p ) for some p ∈ P . If n / ∈ m ⊥ Γ , we defineMult(Γ) := sgn (cid:104) n, m Γ (cid:105) ( g Γ · z n ) ∈ A. (27)where · is the action of g on A . If n is in m ⊥ Γ , we take Mult(Γ) := 0 (which in practice is typicallyequal to g Γ · z n in this case anyway). Note that the factor sgn (cid:104) n, m Γ (cid:105) makes up for the ambiguity inthe ordering of the edges E and E above.Now suppose Γ ∈ T (cid:48) ∆ w , p ( B w , p ,Q , s − Q out being the sink. We again associate g iw ij ∈ g and m d i ∈ M to E ij foreach ( i, j ), and we associate z ϕ ( p i ) ∈ A to E k ∈ Γ [1] ∞ for k = 1 , . . . , s .We now recursively assign to every edge E either elements m E and g E as before, or an element a E ∈ A w ( E ) u E . When two edges with associated elements of M and g flow into a vertex, outgoingelements in M and g are determined as before. On the other hand, if E , E flow into a vertex,and E has associated elements m E ∈ u ⊥ E and g E ∈ g w ( E ) u E ,m E , while E has associated element a E ∈ A w ( E ) u E , we associate to the outgoing edge E the element a E := sgn (cid:104) u E , m E (cid:105) ( g E · a E ) ∈ A w ( E ) u E . (28)We note that g E · a E above may be viewed as a bracket [ g E , a E ] as in (2), so (28) is indeed analogousto (25).Now, for k = 1 , . . . , s , let E out ,k denote the edge of Γ containing Q out which is in the same connectedcomponent of Γ \ Q out as E k . ThenMult(Γ) := a E out , a E out , · · · a E out ,s ∈ A n out ⊂ A, (29)where now, n out = (cid:80) (cid:96) ∈ I w , p ∆ w , p ( (cid:96) ) = (cid:80) ( i,j ) w ij n d i + (cid:80) k ϕ ( p k ). Remark . The sign issues in the multiplicitydefinitions above can be simplified when g is skew-symmetric—there is a canonical choice of orderingfor the commutators and an easy way to find each m E when using this choice. Recall that we call g skew-symmetric if there is a skew-symmetric form ω = {· , ·} on N such that [ g n , g n ] = 0 whenever { n , n } = 0. Using Example 2.5(i), we assume that m d i = ω ( n d i ) for every wall d i ∈ D in . Then,when choosing an ordering for a commutator as in (25) above, pick [ g E , g E ] if { u E , u E } ≥ m E is always givenby ω ( w ( E ) u E ). Hence, the factor sgn (cid:104) n, m Γ (cid:105) of (27) is simply sgn( { u E , n } ). Similarly, the factorsgn( (cid:104) u E , m E (cid:105) ) from (28) is simply sgn( { u E , u E } ).We now explain how Mult(Γ) can be computed more simply in the cases of Examples 2.1(i)-(iii). Examples 3.4. (i) An alternative approach to scattering diagrams over h as in Example 2.1(i) (cf. [GPS10]) is toattach not an element of (cid:98) h to each wall, but rather, an element of R (cid:74) N ⊕ (cid:75) . In this perspective,a wall is expressed as ( d , f d ), f d ∈ R (cid:74) N ⊕ (cid:75) . Letting m d be either primitive element of M whichvanishes on d , the wall ( d , f d ) would in our approach be written as ( m d , d , f d ∂ m d ). Now, suppose that the elements of h attached to the initial edges E ij have the form g iw ij = a iw ij z w ij e i ∂ m di for some constants a iw ij ∈ R . Let a w := (cid:81) i,j a iw ij . If Γ ∈ T ∆ w ( B w ,n ), then g Γ = a w z n out ∂ m Γ , and so if n ∈ ϕ ( P ), Mult(Γ) = a w |(cid:104) n, m Γ (cid:105)| z n + n out . (30) Note that the computation of (30) does not actually require knowing the sign of m Γ .Similarly, for Γ ∈ T (cid:48) ∆ w , p ( B w , p ,Q , s − g E · a E as in (28) without worrying about signs: the product in (29) will bean element of the form a w kz n out (31) for some k ∈ Z , and the correct sign choices will result in k being non-negative.It follows from upcoming joint work of the author and H. Ruddat that the factor |(cid:104) n, m Γ (cid:105)| in (30), and the factor | k | of (31) (in the case n out = 0 so we have honest tropical curves) arethe correct multiplicities for counting tropical curves if one wishes for the counts to equal theappropriate corresponding Gromov-Witten invariants. Furthermore, the factors a w are relatedto counts of multiple covers of certain ( − g = g ω ⊂ h as in Example 2.1(ii), the multiplicity formula simplifies further.For each vertex V (cid:54) = Q out with edges E , E containing V , n := w ( E ) u E and n := w ( E ) u E , define Mult( V ) := |{ n , n }| . For Γ ∈ T ∆ w ( B w ,n ), using Remark 3.3, one finds thatMult(Γ) = a w (cid:89) V ∈ Γ [0] Mult( V ) |{ n out , n }| z n + n out . For Γ ∈ T (cid:48) ∆ w , p ( B w , p ,Q , s − Q out ) = 1. ThenMult(Γ) = a w (cid:89) V ∈ Γ [0] Mult( V ) z n out . (iii) Similarly for the quantization g = g ωq : For V (cid:54) = Q out , take Mult q ( V ) := [ |{ n , n }| ] q , where n and n are weighted tangent vectors of edges containing V , and we recall [ a ] q denotes q a − q − a .Then for Γ ∈ T ∆ w ( B w ,n ) we haveMult(Γ) = a w (cid:89) V ∈ Γ [0] Mult q ( V ) [ |{ n out , n }| ] q z n . For Γ ∈ T (cid:48) ∆ w , p ( B w , p ,Q , s − q ( Q out ) = q (cid:80) { n i ,n j } CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 21 where the sum is over all pairs i, j ∈ { , . . . , s } with i < j , and n k := w ( E out ,k ) u E out ,k .Equivalently, Mult q ( Q out ) is determined by z n z n · · · z n s = Mult q ( Q out ) z n out . ThenMult(Γ) = a w (cid:89) V ∈ Γ [0] Mult q ( V ) z n out . (32) We note that (after removing the a w -factors and z n or z n out factors) these quantum multiplic-ities extend the Block-G¨ottsche multiplicities of [BG16] to allow for these higher-dimensionalcases with ψ -class conditions.With these multiplicities, we can defineN trop w ( p ) := (cid:88) Γ ∈ T ∆ w ( B w ,ϕ ( p ) ) Mult(Γ)for each p ∈ P \ { } , and N trop w , p ( Q ) := (cid:88) Γ ∈ T (cid:48) ∆ w , p ( B w , p ,Q ,s − Mult(Γ)for each generic Q ∈ N R \ Supp( D ). Also, for n ∈ N + primitive, w ∈ W ( n ), m ∈ n ⊥ ∩ M , and n ∈ N R \ { } , we define N trop w ( n ; m ) := (cid:88) Γ ∈ T ∆ w ( B w ,n ) m Γ ∈ R m sgn( m Γ /m ) g Γ . Here m Γ and g Γ are given as in (26), and sgn( m Γ /m ) is defined to be +1 if m Γ is a positive multipleof m and − Proposition 3.5.
The quantities N trop w ( n ; m ) and N trop w ( p ) do not depend on the choices of genericrepresentatives of the incidence conditions B . For fixed Q , N trop w , p ( Q ) does not depend on the genericchoices of representatives for the conditions { B ij } ij . If n out := (cid:80) ( i,j ) w ij n d i + (cid:80) k ϕ ( p k ) is containedin K , then N trop w , p ( Q ) is also independent of the generic choice of Q .Proof. The invariance of N trop w ( n ; m ), N trop w ( p ), and N trop w , p ( Q ) (for fixed Q ) will follow as immediatecorollaries of Theorem 3.7, Corollary 3.8, and Theorem 3.9, respectively. The final statement willfollow once we prove Theorem 2.14 since all wall-crossings act trivially on A K . (cid:3) Remark . An earlier version of this paper (arXiv v3) claimed a direct proof of Proposition 3.5 ratherthan realizing it as a corollary of the results below. However, that argument had a flaw, namely, theclaim of Footnote 11 in that version is nontrivial, and in fact is false without our Condition (3) whichwas not present in that version. However, the key ideas of that argument, plus a proof of the flawedfootnote for some cases, will still be used in § n ∈ N , let W p ( n ) be the set of weight vectors w such that n out := (cid:88) i,j w ij n d i + s (cid:88) k =1 ϕ ( p k ) = n. We will write just W ( n ) for the cases where p is empty (i.e., when considering tropical curves in T ∆ w ( B w ,n ) for some n ), so W ( n ) is the set of weight vectors such that n out := (cid:80) i,j w ij n d i = n .We are now ready to state the main theorems. Theorem 3.7.
For n ∈ N + primitive and m ∈ n ⊥ ∩ M , let D ( n, m ) be the set of walls in D withdirection − n and support parallel to m , i.e., walls ( m d , d , g d ) with m d ∈ Q m and g d ∈ g (cid:107) n . By applyingthe equivalence of Example 2.5(i), assume that each m d here is in fact a positive rational multiple of m . Let n ∈ N R \ m ⊥ . Then (cid:88) d ∈ D ( n,m ) g d = (cid:88) k> w ∈W ( kn ) N trop w ( n ; m ) | Aut( w ) | . (33)If every wall in D ( n, m ) has the same support, then the sum on the left-hand side of (33) appearswhen combining the walls into a single wall via the equivalence of Example 2.5(ii). This is themotivation for considering such an expression.From the definition of the multiplicity of tropical curves in T ∆ w ( B w ,n ), we immediately obtain thefollowing as a corollary of Theorem 3.7. Corollary 3.8.
For primitive n ∈ N + and any p ∈ P , let g n,p := (cid:88) ( m d , d ,g d ) ∈ D n d = n sgn( p, m d ) g d · z ϕ ( p ) . The sum here is over all walls of D with direction − n . I.e., exp( g n,p ) is the image of z ϕ ( p ) under theautomorphism associated with crossing these walls while moving in the direction − ϕ ( p ) . Then g n,p = (cid:88) k> w ∈W ( kn ) N trop w ( p ) | Aut( w ) | . Now recall the structure constants α Q ( p , . . . , p s ; p ) of (19). Theorem 3.9 (Main Theorem) . For p = ( p , . . . , p s ) an s -tuple of elements of P \ { } , s ≥ , p ∈ P ,and Q ∈ N R generic, we have α Q ( p , . . . , p s ; p ) = (cid:88) r ∈ K ∩ P (cid:88) w ∈W p ( ϕ ( p )+ r ) N trop w , p ( Q ) | Aut( w ) | . (34) Remark . We note that each of the above results can be restated using tropicalribbons in place of tropical curves/disks. By a tropical ribbon, we mean the data of a tropical curve ordisk, plus the additional data of a cyclic ordering of the edges at each vertex (cf. the ribbon trees andribbon graphs of [GS16, Abo09, Sla11] for applications of such objects in related contexts). Let us view g as part of the commutator Lie algebra of some associative algebra U g (e.g., the universal envelopingalgebra of g ). Then, in the definition of the multiplicities in § g E , g E ]of (25), we take the product g E g E if the ordering E , E , E agrees with the ribbon structure at V and the product − g E g E if it does not (keeping m E defined as in (24)). Similarly for (28), viewing g ⊕ A now as part of the commutator Lie algebra of some associative algebra U g ⊕ A . That is, we take a E there to be sgn( (cid:104) u E , m E (cid:105) ) g E a E if the ribbon structure agrees with the ordering E , E , E , and − sgn( (cid:104) u E , m E (cid:105) ) a E g E otherwise, where the products here are in U g ⊕ A . The ribbon structure at Q is taken to be the one induced by the ordering of the theta functions being multiplied. It is then clearthat the multiplicity of a tropical curve as in § CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 23 perspective because the ribbon multiplicities will have a more natural geometric interpretation thanthe tropical multiplicities.3.2.
Factored, perturbed, and asymptotic scattering diagrams.
Factoring and perturbing the initial scattering diagram.
To prove Theorems 3.7 and 3.9, weextend and build off ideas from [GPS10, § § Definition 3.11.
For any scattering diagram D , the asymptotic scattering diagram D as of D isdefined as follows: Every wall ( m d , n + d , g d ) ∈ D , with d denoting a rational polyhedral cone (apexat the origin) and n ∈ N R translating this cone, is replaced by the wall ( m d , d , g d ).Note that Scat( D as ) = (Scat( D )) as . We will use the technique from [GPS10] in which one factors an initial scattering diagram D in , deformsthe factored scattering diagram by moving the supports of the initial walls, constructs Scat of thedeformed scattering diagram, and then takes the asymptotic scattering diagram to obtain Scat( D in ).Let T denote the commutative polynomial ring Z [ t i | i ∈ I ], and let T k := T / (cid:104) t k +1 i | i ∈ I (cid:105) . Let D in ,T k and D in ,T be the initial scattering diagrams over g ⊗ T k and g ⊗ T , respectively, given by replacingeach g d i = (cid:80) j ≥ g ij from D in with g (cid:48) d i := (cid:80) j ≥ t ji g ij .We will show that Theorems 3.7 and 3.9 hold for D T k := Scat( D in ,T k ) for all k , hence for D T :=Scat( D in ,T ). Taking t i = 1 for each i then recovers the theorems for D = Scat( D in ).We have an inclusion of commutative rings T k (cid:44) → T (cid:48) k := Z [ u ij | i ∈ I, ≤ j ≤ k ] / (cid:104) u ij | i ∈ I, ≤ j ≤ k (cid:105) t i (cid:55)→ k (cid:88) j =1 u ij . Using this inclusion to work in g ⊗ T (cid:48) k , we have g (cid:48) d i = k (cid:88) w =1 t ji g ij = k (cid:88) w =1 (cid:88) J = w w ! g iw u iJ , (35)where the second sum is over all subsets J ⊂ { , . . . , k } of size w , and u iJ := (cid:89) j ∈ J u ij . Applying the equivalence from Examples 2.5(ii) in reverse and then perturbing the walls (i.e., trans-lating the walls by some generic amount), we obtain a scattering diagram D k := { ( m d i , d iwJ , w ! g iw u iJ ) | ≤ w ≤ k, J ⊂ { , . . . , k } , J = w } , (36)where d iwJ is some generic translation of d i = m ⊥ d i . Note that Scat( D k ) as = D in ,T k .It shall be useful for us to work over a new commutative ring (cid:101) T k , defined by (cid:101) T k := Z [ u iJ | i ∈ I, J ⊂ { , . . . , k } ] / (cid:104) u iJ u iJ | J ∩ J (cid:54) = ∅(cid:105) . Note that we have a surjective homomorphisms π : (cid:101) T k → T (cid:48) k , u iJ (cid:55)→ u iJ . (37) Let D k denote the initial scattering diagram over g ⊗ (cid:101) T k defined as in (36), but with the factors u iJ replaced by u iJ , i.e., D k := { ( m d i , d iwJ , w ! g iw u iJ ) | ≤ w ≤ k, J ⊂ { , . . . , k } , J = w } . (38)3.2.2. Constructing the consistent scattering diagram D ∞ k . As in [GPS10, § D k , D k , D k , . . . , D k I − k =: D ∞ k = Scat( D k ). Assume inductivelythat:(a) Each wall in D ik is of the form ( m d , d , g d u J d ), where g d ∈ g n d for some n d ∈ N + , J d is acollection of pairwise-disjoint subsets of I × { , . . . , k } of the form ( i, J ) for various i ∈ I and J ⊂ { , . . . , k } , and u J d := (cid:89) ( i,J ) ∈ J d u iJ . (39)(b) There is no set W of walls in D ik of cardinality ≥ (cid:84) d ∈ W d has codimension ≤ u J d u J d (cid:54) = 0 for each pair of distinct walls d , d ∈ W .These conditions clearly hold for D k . To get D lk from D l − k , consider each pair d , d ∈ D l − k whichsatisfies:(i) { d , d } (cid:42) D l − k ,(ii) d ∩ d (cid:54) = ∅ has codimension 2 and is not contained in the boundary of either d or d ,(iii) u J d u J d (cid:54) = 0.Given such a pair, Lemma 2.9 says that adding the following new wall will result in consistency aroundthe joint d ∩ d (i.e., path-ordered products around this joint will be trivial): d ( d , d ) := ( µ (( n d , m d ) , ( n d , m d )) , ( d ∩ d ) + R ≤ ( n d + n d ) , [ g d u J d , g d u J d ]) . (40)We now define D lk := D l − k ∪ { d ( d , d ) | d , d satisfying (i)-(iii) above } . Definition 3.12. If d = d ( d , d ), define Parents( d ) := { d , d } , and if d ∈ D k , define Parents( d ) := ∅ .Recursively define Ancestors( d ) by Ancestors( d ) := { d } ∪ (cid:83) d (cid:48) ∈ Parents( d ) Ancestors( d (cid:48) ). DefineLeaves( d ) := { d (cid:48) ∈ Ancestors( d ) | d (cid:48) is the support of a wall in D k } . It is clear that D lk satisfies inductive hypothesis (a). For hypothesis (b), suppose we do have sucha bad set of walls W . Since the products u d u d are nonzero for each d , d ∈ W , the sets Leaves( d )for d ∈ W must be pairwise disjoint. Thus, slightly shifting the initial walls’ supports will shift thewalls in W independently, and so we can avoid having this bad set W by choosing the walls d iwJ moregenerically.For each J = { ( i, J ij ) ⊂ I × { , . . . , k }} ij , let I J := (cid:91) ( i,J ) ∈ J ( i, J ) ⊂ I × { , . . . , k } . (41)Since the cardinality of I J d for the new walls increases with each step and is bounded by k I , wesee that the process stabilizes with the scattering diagram D k I − k . Furthermore, since the wall-crossing automorphisms θ d and θ d commute for u J d u J d = 0, and since Lemma 2.9 and (40) ensure CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 25 consistency around all other joints, D k I − k is consistent. Thus, we have D k I − k = Scat( D k ), asdesired. Hence, for D ∞ k := D k I − k , we have D T k = π ∗ ( D ∞ k ) as , where the π ∗ means that we apply the homomorphism π : (cid:101) T k → T (cid:48) k of (37) to each g d .3.2.3. The tropical description of D ∞ k . We will continue to use J to denote collections of pairwise-disjoint sets J = { ( i, J ij ) ⊂ I × { , . . . , k }} ij as in inductive hypothesis (a) of § u J as in (39) and I J as in (41).Now, as in § w := ( w i ) i ∈ I , w i := ( w i , . . . , w il i ) with 0 < w i ≤ . . . ≤ w il i .Let J w denote the set of all possible collections J as above, subject to the requirement that J ij = w ij .Note that each J ∈ J w corresponds to a set D k, J = { d iw ij J ij } ij of walls of D k , and two choices of J correspond to the same D k, J exactly if they are related by an element of Aut( w ). Let D ∞ k, J denotethe set of walls in D ∞ k whose leaves are precisely the walls of D k, J . Note that, for J ∈ J w and( m d , d , g d u J ) ∈ D ∞ k, J , we must have g d ∈ g n out where n out := (cid:80) i,j w ij n d i .We will write B w ,n, J and B w , p ,Q, J to indicate that we are choosing the representatives of theincidence conditions B w ,n and B w , p ,Q so that B ij = d iw ij J ij . Recall that for n ∈ N R \ { } , thecondition B out from B w ,n is a generic translate of the R -span of n and n out . In particular, if n / ∈ m ⊥ d (cid:51) n out , then B out ∩ d is a ray or a line. The following is essentially a generalization of[GPS10, Thm 2.4] (which used g = h and dimension 2) to higher dimensions and more general g (thetwo-dimensional case over g ωq is [FS15, Lemmas 4.5-4.6]). Lemma 3.13.
For every wall ( m d , d , g d u J ) ∈ D ∞ k, J , there exists a unique tropical curve h : Γ → N R in T ∆ w ( B w ,n , J ) with h ( E out ) ⊂ d , where n is any element of N R \ m ⊥ d . Furthermore, h ( ∂E out ) ∈ ∂ d ,and up to an equivalence as in Example 2.5(i) (plus possibly a positive re-scaling of m d ), we have m d = m Γ ∈ M and g d := g Γ (cid:89) ij ( w ij !)(42) for m Γ and g Γ as defined in (26) .Proof. The proof of [GPS10, Theorem 2.4] is easily modified to prove this Lemma. The idea is toconstruct the tropical curve by starting with the ray d ∩ B out and considering the endpoint p ∈ d ∩ d ,where { d , d } = Parents( d ). The resulting stratum is given weight | n d | (the index of n d , i.e., n d equals | n d | > g d ∈ g n d . From p , extend the tropical curve in thedirections n d and n d with weights | n d | and | n d | , respectively, until reaching the boundaries of thewalls d and d . The balancing condition at p follows easily from (40) and the fact that commutatorsin g respect the N + -grading. The process is repeated for each of these branches, and continues untilevery branch extends to infinity in some leaf. This gives the desired tropical curve. The formulas for g d , and m d follow easily from (40) and the definitions of g Γ and m Γ , noting that the (cid:81) w ij ! factorappears because of the fact that g iw is multiplied by w ! in the definition of D k in (38), and similarlyfor the u J factor. (cid:3) Proofs of the main theorems.
We will need a certain formula for relating the t - and u -variables. For a weight vector w as above, let | w i | := (cid:80) l i j =1 w ij , and let t w = (cid:81) i,j t w ij i = (cid:81) i t | w i | i .Note that t | w i | = | w i | ! (cid:88) J i ⊂{ ,...,k } J i = | w i | u iJ i . (43)Given J ∈ J w , let J i = (cid:83) j J ij . Note that given the sets J i , there are (cid:81) i | w i | ! (cid:81) j w ij ! possible refinementsinto the sets J ij . We thus find (cid:88) J ∈ J w u I J = (cid:89) i ∈ I l i (cid:89) j =1 (cid:18) w ij ! (cid:19) (cid:88) J i ⊂{ ,...,k } J i = | w i | | w i | ! u iJ i . (44)Now combining (43) and (44) yields t w = (cid:88) J ∈ J w u I J (cid:89) i,j w ij ! . (45)3.3.1. Proof of Theorem 3.7.
Fix n , m , and n as in the statement of the theorem. Let D ∞ k ( n, m ) bethe set of walls in D ∞ k of the form ( m d , d , g d ) with m d ∈ Z m and g d ∈ g (cid:107) n .Recall that each J ∈ J w determines a set of walls D k, J , and in the reverse direction, each D k, J determines an orbit of Aut( w ) in J w . Similarly, u J uniquely determines and is determined by an orbitof Aut( w ) in J w . We see that the sum from the left-hand side of (33), with D ( n, m ) there replacedby D ∞ k ( n, m ), is equal to (cid:88) k> w ∈W ( kn ) (cid:88) J ∈ J w / Aut( w ) (cid:88) d ∈ D ∞ k, J m d ∈ R m sgn( m d /m ) g d u J . (46)Applying Lemma 3.13, this becomes (cid:88) k> w ∈W ( kn ) (cid:88) J ∈ J w / Aut( w ) (cid:88) Γ ∈ T ∆ w ( B w ,n , J ) m Γ ∈ R m sgn( m Γ /m ) g Γ (cid:89) ( i,j ) ∈ I J ( w ij !) u J . (47)Now, note that for each w , ( D k ) as is symmetric with respect to permuting the elements of J w , i.e.,for J , J ∈ J w , swapping the supports of d iw J and d iw J in (38) does not affect ( D k ) as . Hence, thesum in the large parentheses of (46) must be independent of J , so we can write T ∆ w ( B w ,n ) in placeof T ∆ w ( B w ,n , J ) in (47). Now, pulling the quotient by Aut( w ) into the sum, applying π : (cid:101) T k → T (cid:48) k as in (37), and utilizing (45), the expression (47) becomes (cid:88) k> w ∈W ( kn ) (cid:88) Γ ∈ T ∆ w ( B w ,n ) m Γ ∈ R m (cid:18) | Aut( w ) | (cid:19) sgn( m Γ /m ) g Γ t w . The claim follows. (cid:3) Here it is important that we are using u J instead of u J . We note that this step is really what gives us the invarianceof N trop w ( n , m ) in Proposition 3.5. CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 27
Proof of Theorem 3.9.
Fix r ∈ K ∩ P and let n = ϕ ( p ) + r . We wish to describe the degree n part (for the P -grading) of ϑ p ,Q · · · ϑ p s ,Q := ϑ D ∞ k p ,Q · · · ϑ D ∞ k p s ,Q , in terms of tropical curve counts (using z p i ⊗ ∈ A ⊗ (cid:101) T K as our initial monomials). We can assume that Q is far enough from the origin forthe degree n part of the product over D ∞ k to agree with that over ( D ∞ k ) as = Scat k ( D in ).Specifically, we want to show that the degree n part of ϑ p ,Q · · · ϑ p s ,Q ∈ A ⊗ (cid:101) T k is (cid:88) w ∈W p ( n ) N trop w , p ( Q ) | Aut( w ) | (cid:88) J ∈ J w u J (cid:89) i,j w ij ! . (48)Then applying π and using (45), this becomes (cid:88) w ∈W p ( n ) N trop w , p ( Q ) | Aut( w ) | t w , and setting t i = 1 for each i yields the desired result.Consider a collection of broken lines { γ k } k for D ∞ k contributing non-trivially to ϑ p ,Q · · · ϑ p s ,Q asin (20). For any wall d ∈ D ∞ k along which some γ k bends at a point Q d , we glue to γ k the truncation h Q d at Q d of a tropical curve from Lemma 3.13, a so-called Maslov index 0 tropical disk. Note that h Q d together with γ k (weighted by the indexes of the degrees of the attached elements of g and A )satisfies the balancing condition at Q d , so repeating this for every bend of γ k results in a tropical disk h γ k with 1-valent point at Q . One then concatenates these tropical disks h γ k at Q for each k = 1 , . . . , s to obtain a tropical disk in T (cid:48) ∆ w , p ( B w , p ,Q, J , s −
2) for some w and J ∈ J w . By design (and usingLemma 2.8 and (42)), the corresponding product of final elements a γ · · · a γ s as in (20) is preciselyMult( h γ i ), times a factor of (cid:81) ij w ij ! as in Lemma 3.13. See Figure 3.2 for an example.Now, for each w ∈ W p ( n ) and each J ∈ J w , we can apply the above computation to all broken linesfor D ∞ k whose corresponding tropical disk lives in T (cid:48) ∆ w , p ( B w , p ,Q, J , s − J ∈ J w and applying the same tricks as in the above proof of Theorem 3.7 (e.g., noting that for each w , theresult must be symmetric with respect to permuting the J ’s in J w ), one finds that the sum of thefinal monomials of all the relevant broken lines indeed yields (48). (cid:3) See Figure 3.2 for an example of the above construction with k = 1. We note that the complexityof the scattering diagram does increase quickly as soon as the k in (cid:101) T k is increased. See [GPS10, Figure1.3] for an illustration when k = 2.3.3.3. Proof of Theorem 2.14 (the refined [CPS] result).
We want to show that ϑ p,Q for different genericvalues of Q are related by path-ordered products. Note that it suffices to prove this for the scatteringdiagrams D ∞ k as described in Lemma 3.13. Recall from the proof of Theorem 3.9 in § ϑ p,Q (for the scattering diagram D ∞ k ) correspond bijectively totropical disks in T (cid:48) ∆ w ,p ( B w ,p,Q, J , −
1) for some w and some Aut( w )-orbit of J in J w . Furthermore,for a broken line γ and corresponding tropical disk h γ , we have a γ = Mult( h γ ). As we move Q , thereare two issues that could result in changes to the types of broken lines contributing to ϑ p,Q . Themost obvious is that Q may move across a wall d of D ∞ k , resulting in the possible gluing or losing ofa Maslov index 0 tropical disk associated to the wall. By Lemma 3.13 and Lemma 2.8, the resultingchanges to the tropical disk counts are exactly accounted for by the wall-crossing automorphisms.There is one other way that moving Q might affect the types of tropical disks being enumerated.Namely, as we translate Q and correspondingly deform Γ ∈ T (cid:48) ∆ w ,p ( B w ,p,Q, J , − d d Q d Figure 3.2.
A consistent scattering diagram over g ωq ⊗ (cid:101) T with g d = u z e , g d = u z e , and g d = [ g d , g d ] = u u ( q − q − ) z e + e . The solid lines (both boldand unbold) are the supports of the walls. The bold dashed lines are a pair of brokenlines (one without any bends) contributing to the product ϑ e ,Q ϑ e ,Q . The bold lines(dashed and undashed) form the tropical disk Γ (which has weight w = ((1) , (1)))corresponding to this pair of broken lines. The contribution of this pair of brokenlines to the product is given by [ z e , [ g d , g d ]] z e = u u q ( q − q − ) z e +2 e . Ifwe view Γ as being in T (cid:48) ∆ w , p ( B w , p ,Q , s −
2) (using d and d as the incidence condi-tions for the legs they contain), then the corresponding contribution to N trop w , p ( Q ) isMult(Γ) = [ z e , [ z e , z e ]] z e = q ( q − q − ) z e +2 e . Using Example 3.4(iii), the coef-ficient q ( q − q − ) can be computed as a product of vertex-multiplicities: Mult q ( Q ) = q { e + e ,e } = q , while the other two vertices, moving from top-right to bottom left,have multiplicities [ |{ e , e }| ] q = [1] q = q − q − and [ |{ e , e + e }| ] q = [1] q = q − q − .collapse to have length 0, resulting in a 4-valent vertex. Let Q be a point for which some Γ in T (cid:48) ∆ w ,p ( B w ,p,Q, J , −
1) has a 4-valent vertex, and assume Q is generic among such points. Then thereis some neighborhood U of Q and some affine hyperplane H containing Q such that, for each Q ∈ H ∩ U , there is a unique tropical disk of the same type as Γ in T (cid:48) ∆ w ,p ( B w ,p,Q, J , − E , E , E be the edges flowing into the 4-valent vertex V for Γ as above, and let E be theoutward-flowing edge, flowing towards Q out , cf. Figure 3.3(b). For Q in one component of U \ ( H ∩ U ),there is exactly one way to extend the 4-valent vertex into a compact edge to yield a tropical diskΓ ∈ T (cid:48) ∆ w ,p ( B w ,p,Q, J , − E and E meeting first, cf. Figure 3.3(a). For Q on the otherside of U \ ( H ∩ U ), there are either one or two ways to insert a compact edge yielding tropical disksin T (cid:48) ∆ w ,p ( B w ,p,Q, J , − E and E meeting first, or with E and E meeting first, cf.Figure 3.3(c)(d). We wish to show first that if one of the two tropical curve types of Figure 3.3(c)(d)does not occur, then the tropical multiplicity associated to that type is 0. We will then show thatthe multiplicity associated to the tropical curve type of Figure 3.3(a) is the sum of the multiplicitiesassociated to the tropical curve types of Figure 3.3(c)(d). See Figure 3.4 for an illustration of howthis wall-crossing the tropical moduli space arises as a result of a broken line crossing a joint of D ∞ k . We note this strategy for proving invariance of tropical counts was first employed in the genus 0 cases of [GM07].
CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 29 E E E E E E E E E E E E E E E E (a) (b) (c) (d) Figure 3.3.
Tropical wall crossing. Locally in the space of choices for the incidenceconditions, there is a codimensions one “wall” of non-generic choices resulting in a4-valent vertex as in (b). One one side of this wall, there is a single tropical curvetype (a) satisfying the deformed conditions. On the other side, there are up to twotypes (c) and (d).The situation in which one of the two types from Figure 3.3(c)(d), say, the type from (d), doesnot occur arises under the following circumstances: consider the four tropical disk-types associated tothe connected components of the tropical curve Figure 3.3(b) with its vertex removed. The incidenceconditions on these components force each E i to live in some affine space B i . Then the tropical curvetype from (d) does not occur if either B and B are not transverse, or B and B are not transverse.For i = 1 , ,
3, if E i has an element g E i ∈ g (cid:107) n Ei ,m Ei associated to it in the definition of Mult(Γ),then B i = m ⊥ E i , while if E i has an element of A associated to it, then B i is all of N R . So the spaces B and B will automatically be transverse if either E or E has an element of A associated to it.On the other hand, if B and B are each associated with elements of g (cid:107) n Ei ,m Ei , then B and B willonly fail to be transverse if m E and m E are parallel. But then n E and n E are both containedin m ⊥ E = m ⊥ E , and so since g E i ∈ g (cid:107) n Ei ,m Ei , we have [ g E , g E ] = 0. So then this missing type hasmultiplicity 0 and does not affect the counts.Now suppose that B and B fail to be transverse. Let E denote the compact edge in Figure 3.3(d).As above, it must be the case that B has an element of g n E ,m E associated to it, not an element of A , and so B is parallel to m ⊥ E . On the other hand, B is parallel to R n E , and so non-transversailtymeans n E ∈ m ⊥ E . But then the balancing condition forces n E ∈ m E ⊥ as well. Since g E ∈ g (cid:107) n E ,m E ,we now have g E · a g E = 0. Thus, these missing tropical disk types have multiplicity 0, as desired.So now we may indeed assume that each of the 3 possible tropical types of Figure 3.3(a,c,d) occursnear the wall. For convenience, let us now view the elements attached to the edges of Γ not as livingin g or A , but instead as living in g ⊕ A , always denoting the element associated to an edge E by g E .For the side of H associated to (a), the element g E ∈ g ⊕ A is, up to sign, given by [ g E , [ g E , g E ]].For the other side of the wall, the g E ’s corresponding to the two possible types Figure 3.3 (c) and (d)are, up to signs, [[ g E , g E ] , g E ] and [ g E , [ g E , g E ]], respectively. So equality of the tropical countson the two sides of H comes down to checking that ± [ g E , [ g E , g E ]] = ± [[ g E , g E ] , g E ] ± [ g E , [ g E , g E ]] , (49) Q Q Q QE E E E E E E E E E E E E E E E (a) (b) (c) (d) Figure 3.4.
Broken lines (the dashed segments) moving past a joint of a scatteringdiagram (the solid rays), and the corresponding transition in tropical disk types.When Q moves onto a certain local hyperplane H (the opaque dotted ray), brokenlines γ ending at Q collide with a joint (b, top), resulting in a tropical disk witha 4-valent vertex (b, bottom). For Q on one side of H , there is one possibility forthe additional straight segment of γ (a, top), resulting in one tropical disk type (a,bottom). On the other side of H , there are up to two possibilities for the new edgeof the broken line (c, top) and (d, top), resulting in two corresponding tropical disktypes (c, bottom) and (d, bottom), respectively. Note that the bottom row herecorresponds to the tropical disks of Figure 3.3.where the signs have yet to be addressed. If we can show that the signs of nonzero terms in (49) areeither all positive or all negative, then the equality follows from the Jacobi identity. We may of courseassume that the terms of (49) are not all 0, since this case is trivial.Note that the signs in (49) are independent of the specific choice of g and A , instead being deter-mined entirely by the vectors m E and n E associated to the edges. Thus, it suffices to check the caseof the tropical vertex group h as in Example 2.1(i). In this case, Theorem 2.14 is known to hold by[CPS], so all the signs of nonzero terms in (49) must be the same.Now, in the tropical vertex group setting, for i , i , i the distinct elements of { , , } in someorder, we have that [ g E i , [ g E i , g E i ]] is nonzero if and only if B i and B i are transverse and B i and B are transverse (here we use the assumption that not all terms of (49) are 0 to ensure thatthe vanishing of powers of the u ij ’s does not cause 0 multiplicity). Furthermore, as we saw in ourtransversality arguments above, non-transversality of B i and B i or of B i and B forces the bracketto be 0 for any choice of g and A . Thus, the signs of all nonzero terms in (49) agreeing in the tropicalvertex group setting is sufficient. This completes the proof. (cid:3) CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 31
We note that the above proof used the fact that Theorem 2.14 is known to hold over the tropicalvertex group, but this can be avoided, either by tediously checking the signs of (49) in several differentcases, or by using the results of [MRb] to relate the multiplicities in the tropical vertex group settingto tropical Gromov-Witten counts that are known to be invariant.4.
Cluster varieties and Frobenius maps
In this section we briefly explain how to get the initial scattering diagrams used for constructingtheta functions on cluster varieties, including both the classical and quantum versions. We then useTheorem 3.9 to prove Fock and Goncharov’s conjectures [FG09, §
4] on symmetries of theta functionswith respect to certain Frobenius automorphisms (not to be confused with Gross-Hacking-Keel’sFrobenius structure conjecture).4.1.
Seeds.
As in [FG09, § S = { L, I, E := { e i } i ∈ I , F, {· , ·} , { d i } i ∈ I } , (50)where L is a finitely generated free Abelian group, I is a finite index set, E is a basis for N indexedby I , F is a subset of I , {· , ·} is a skew-symmetric Q -valued bilinear form, and the d i ’s are positiverational numbers such that d j { e i , e j } is in Z whenever i and j are not both in F . One considers thebilinear form ( · , · ) defined by ( e i , e j ) := d j { e i , e j } . One calls e i a frozen vector if i ∈ F . We let π and π be the maps L → L ∗ defined by n (cid:55)→ ( n, · ) and n (cid:55)→ ( · , n ), respectively. The reader should notice the resemblance of this setup to that of Examples2.7. If the seed S is not clear from context, we may write the data with subscripts S to clarify, e.g., S = { L S , I S , E S = { e S,i } , F S , {· , ·} S , { d S,i }} .Given S as above, the Langland’s dual seed S ∨ has the same L , I , E , and F as S , but {· , ·} isreplaced with the form {· , ·} ∨ defined by { e i , e j } ∨ := d i d j { e i , e j } , and for each i ∈ I , d i is replacedby d ∨ i := d i . The main effect of this is that the form ( · , · ) ∨ defined by ( e i , e j ) ∨ = d ∨ j { e i , e j } ∨ is thenegative transpose of ( · , · ), so π ∨ = − π and π ∨ = − π .We refer to [FG09, § A and X associated to the seed S . For thequantum version X q of the X -space, cf. [FG09, § A q of A , cf. [BZ05](alternatively, the reader may confer v2 of this article on arXiv).Fix a seed S as in (50). In the construction of the theta functions used in [GHKK18], one worksnot with S , but with the seed S prin defined as follows: • L S prin := L ⊕ L ∗ . • I S prin is the disjoint union of two copies of I . We will call them I and I to distinguishbetween them. • E S prin := { ( e i , | i ∈ I } ∪ { (0 , e ∗ i ) | i ∈ I }• F S prin := F ∪ I , where F is F viewed as a subset of I . • { ( n , m ) , ( n , m ) } S prin := { n , n } + m ( n ) − m ( n ). • The d i ’s are the same as before (viewing i in I or I as an element of I ). The initial cluster scattering diagrams.
The theta functions in [GHKK18] are constructedfirst for A prin , and then certain restrictions of subsets of these theta functions yield the theta functionson A and X (cf. their Section 7.2). We will briefly give the initial scattering diagrams for directlyconstructing theta functions for X and (if a “compatible pair” exists) for A . Theta functions for A prin can then be constructed by applying the A -case to the seed S prin . Similarly, we will give the initialscattering diagrams for constructing the quantum theta functions on X q and A q .4.2.1. Theta functions on X and X q . The initial scattering diagram for constructing theta functionson X is defined using Example 2.7(ii) in the obvious way. That is, we take N = L with E , I , F , {· , ·} , and { d i } as for the seed S . Then, using the equivalence of Example 2.5(i), the resulting initialscattering diagram is D X in := { ( π ( e i ) , π ( e i ) ⊥ , log(1 + z e i ) ∂ π ( e i ) ) } i ∈ I \ F . We note a couple alternative ways to express this. In terms of the Langland’s dual seed S ∨ and usingthe dilogarithm description of (10), and applying the equivalence of Example 2.5(i) again, we canwrite the above scattering diagram as D X in = { ( π ∨ ( e i ) , π ∨ ( e i ) ⊥ , − d i Li ( − z e i )) } i ∈ I \ F . On the other hand, in terms of the version of scattering diagrams sketched in Remark 2.3(ii), wewould write D X in as { ( e i , e ⊥ i , log(1 + z e i ) ∂ π ( e i ) ) } i ∈ I \ F .Similarly, the initial scattering diagram for the quantization X q is given as in (12) by D X q in := { ( π ∨ ( e i ) , π ∨ ( e i ) ⊥ , − Li ( − z e i ; q /d i ) } . where we recall that Li ( x ; q ) := (cid:80) ∞ k =1 x k k [ k ] q and [ k ] q := q k − q − k . We note that the construction ofthis quantum initial scattering diagram was outlined in [GHKK18, arXiv v1, Construction 1.31].4.2.2. Theta functions on A and A prin , and on A q and A prin q . To construct the initial scatteringdiagram for A , we will use what [BZ05] calls a compatible pair, i.e., a skew-symmetric bilinear formΛ on L ∗ such that Λ( π ( e i ) , · ) = d i e i for each i ∈ I \ F .(The other part of the “pair” is the data of the matrix B for ( · , · ) with respect to the basis E ). Onesees that the existence of such a Λ is equivalent to the condition that the restriction of p to the spanof { e i } i ∈ I \ F is injective (called the “Injectivity Assumption” in [GHKK18, § S prin because ( · , · ) prin is unimodular.We now fix such a Λ, assuming one exists. We then apply Example 2.7(ii) to the data N = L ∗ S , I = I S , F = F S , E = { π ( e S,i ) } i ∈ I S , ω = Λ, and d i = d S,i for each i ∈ I . This yields the desiredinitial scattering diagram: D A in = { e i , e ⊥ i , log(1 + z π ( e i ) ) ∂ e i } . Similarly, the initial quantum scattering diagram is obtained by applying Example 2.7(iii) to thisdata, thus yielding D A q in = { e i , e ⊥ i , − Li ( − z π ( e i ) ; q /d i ) } , In fact, since the theta functions are, in general, formal, they are more accurately defined only on various formalversions of these spaces. We will ignore this issue here as it does not matter for our purposes.
CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 33
Here, − Li ( − z π ( e i ) ; q /d i ) lives in the completion of the quantum torus algebra g Λ q associated to L ∗ and Λ via the construction in Example 2.1(iii).The initial scattering diagrams for A prin and A prin q are constructed in the same way but using S prin in place of S .4.3. The Frobenius maps.
Prior to the definition of the theta functions in [GHKK18], [FG09, §
4] predicted their existence and conjectured several properties they should satisfy. Among theseproperties are certain symmetries under a (quantum) Frobenius automorphism, predicted there fortheta functions on the X -space, but proven here to also hold for the A -spaces.First, we will need the following, which is little more than a restatement of [GHKK18, Thm 1.13]. Theorem 4.1 ([GHKK18], Thm 1.13) . Let D in be an initial scattering diagram over a Poisson torusalgebra as in (11) (this includes each D X in and D A in of § D := Scat( D in ) . Then D is equivalentto a scattering diagram D (cid:48) such that, for any wall d ∈ D (cid:48) , and for any u ∈ P , crossing from the sideof d containing u to the side not containing u acts on z u via z u (cid:55)→ z u (1 + z n ) cd i |{ n,u }| (51) for some n ∈ N + and some positive integer c . Consequently, every theta function constructed frombroken lines for D has non-negative integer coefficients. In particular, the integrality allows us to consider the coefficients modulo a prime p . In [FG09, § X -space cases of the following theorem, whichthey called the Frobenius Conjecture: Theorem 4.2 (Frobenius Conjecture, classical version) . Consider D in as in (11) and D = Scat( D in ) .For any prime p and any u ∈ P , the theta functions constructed from D satisfy ϑ pu ≡ ϑ pu (mod p ) . Proof.
We work with a representative D (cid:48) of the equivalence class of D as in (51). Consider brokenlines with attached monomials az v and az pv ( a ∈ Z , v ∈ P ) crossing a wall of D (cid:48) with associatedwall-crossing automorphism ν . By (51), ν ( az v ) = az v (1 + z n ) k for some n ∈ N + , k ∈ Z ≥ . Similarly, ν ( az pv ) = az pv (1+ z n ) pk . By the freshman’s dream and Fermat’s little theorem, we see that ν ( az pv ) ≡ ν ( az v ) p (mod p ). It follows that the broken lines contributing to ϑ pu,Q in characteristic p are the sameas the broken lines contributing to ϑ u,Q in characteristic p , except that the attached monomials forbroken lines contributing to ϑ pu,Q are the p -th powers of the corresponding attached monomials for ϑ u,Q . The result now follows by applying the freshman’s dream to ϑ pu . (cid:3) [FG09] also predicted the following quantum version of the Frobenius Conjecture, their Conjecture4.8.6. First we introduce some notation. Denote by ϑ u,Q ( z n ) = (cid:80) c n z n ∈ (cid:98) A = R q (cid:74) N ⊕ (cid:75) P theLaurent series expansion of ϑ u,Q in terms of monomials z n , n ∈ P . Then for k ∈ Z > , denote ϑ u,Q ( z kn ) := (cid:80) c n z kn , the series obtained by multiplying each exponent by k . When we want tospecify that we are taking a certain limit for q , we will write this value in the subscript, as in ϑ u,Q,q . Theorem 4.3 (Frobenius Conjecture, quantum version) . Consider theta functions with respect to D = Scat( D in ) for D in as in (12) (so this includes D in equal to any D X q in or D A q in ). Suppose q andeach q /d i are primitive k -th roots of unity for a positive odd integer k . Then for any u ∈ P , we have ϑ ku,Q,q ( z n ) = ϑ u,Q, ( z kn ) The map ϑ u,Q,q ( z n ) (cid:55)→ ϑ u,Q, ( z kn ) is what [FG09] calls the quantum Frobenius map. The case ofquantum cluster varieties from surfaces is [AK17, Theorem 1.2.6], assuming that their canonical basesturn out to equal the theta bases. Since we do not have a version of (51) in the quantum setting, themethods from the proof of Theorem 4.2 will not be useful here. Instead, we make use of Theorem 3.9. Proof.
Consider a tropical disk making a nonzero contribution to (34) for ϑ ku,Q . I.e., we consider atropical disk Γ contributing to some N trop w , p ( Q ) in (cid:88) w ∈W p ( ku ) N trop w , p ( Q ) | Aut( w ) | . Let w (Γ) denote the corresponding weight vector. Using the description of Mult(Γ) given in (32), wesee that the contribution of Γ is z n out times (cid:89) V ∈ Γ [0] \ Q [Mult Γ ( V )] q (cid:89) w ij ∈ w (Γ) ( − w ij − w ij [ w ij /d i ] q | Aut( w (Γ)) | . (52)Here, each factor ( − wij − w ij [ w ij /d i ] q , which we will denote as R w ij ,d i ; q , arises as the z w ij e i -coefficient in thequantum dilogarithm − Li ( − z e i ; q /d i ), so this product is the factor called a w in (32).The initial segment of the broken line corresponding to Γ has weight a multiple of k . We show byinduction that the same is true for every edge of Γ. Let S be a maximal subset of Γ \ Q out such thateach edge E ∈ S has weight a multiple of k and the closure of Γ \ S in Γ is connected. Suppose S isnot all of Γ \ Q out . Then S is a union of trees that each contain exactly 1 univalent vertex, with theremainder of the vertices being trivalent. To see this, note that there are no bivalent vertices in thesetrees because if two edges containing a vertex have weights a multiple of k , then the third does too.Also, if there were more than one univalent vertex, then the closure of Γ \ S would not be connected.On the other hand, the vertex of a component of S whose distance from Q out is minimal must beunivalant.Now, the number of vertices of S is equal to the number of undbounded edges in S . Since S contains the unbounded edge corresponding to the initial direction of the broken line, this means thatΓ has more vertices of multiplicity a multiple of k than there are elements of w (Γ) that are a multipleof k . But for ζ a primitive k -th root of unity, lim q → ζ [ a ] q [ b ] q = 0 if a is a multiple of k and b is not, andthe limit equals a finite nonzero number (see below) if both a and b are multiples of k . Hence, thecontribution of such a curve would be 0. So every edge of Γ must have been weight a multiple of k .We now see that a tropical curve contributes to ϑ ku,Q,q if and only if it can be obtained by takinga tropical curve contributing to ϑ u,Q, and multiplying each weight by k . This multiplication ofeach weight by k takes each vertex multiplicity [ a ] q to [ k a ] q , and each R w ij ,d i ; q = ( − wij − w ij [ w ij /d i ] q to R kw ij ,d i ; q = ( − kwij − kw ij [ kw ij /d i ] q . The number of trivalent vertices of Γ is the same as the number of weights w ij in w (Γ), so we can pair the trivalent vertices up with the w ij ’s and compute, for ζ a primitive k -th root of unity,lim q → ζ [ k a ] q R kw ij ,d i ; q = lim q → ζ ( q k a − q − k a )( − kw ij − kw ij ( q kw ij /d i − q − kw ij /d i )= ( − kw ij − kw ij lim q → ζ q kw ij /d i − k a ( q k a − q kw ij /d i − . CATTERING DIAGRAMS, THETA FUNCTIONS, AND REFINED TROPICAL CURVE COUNTS 35
Since q /d i was also assumed to be a primitive k -th root of unity, lim q → ζ q kw ij /d i − k a = 1. Using thisand L’Hospital’s rule, the above now further simplifies to( − kw ij − kw ij lim q → ζ k aq k a − (2 kw ij /d i ) q kw ij /d i − = ad i ( − kw ij − w ij = ad i ( − w ij − w ij , where the last equality used the assumption that k is odd. This is equal to lim q → [ a ] q R w ij ,d i ; q , andthe result follows from applying this to every such vertex-weight pair. (cid:3) References [Abo09] M. Abouzaid,
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