MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence
aa r X i v : . [ m a t h . AG ] O c t MPS DEGENERATION FORMULA FOR QUIVER MODULIAND REFINED GW/KRONECKER CORRESPONDENCE
M. REINEKE, J. STOPPA AND T. WEIST
Abstract.
Motivated by string-theoretic arguments Manschot, Pioline andSen discovered a new remarkable formula for the Poincar´e polynomial of asmooth compact moduli space of stable quiver representations which effec-tively reduces to the abelian case (i.e. thin dimension vectors). We first provea motivic generalization of this formula, valid for arbitrary quivers, dimen-sion vectors and stabilities. In the case of complete bipartite quivers we usethe refined GW/Kronecker correspondence between Euler characteristics ofquiver moduli and Gromov-Witten invariants to identify the MPS formula forEuler characteristics with a standard degeneration formula in Gromov-Wittentheory. Finally we combine the MPS formula with localization techniques,obtaining a new formula for quiver Euler characteristics as a sum over trees,and constructing many examples of explicit correspondences between quiverrepresentations and tropical curves. Introduction
In [12], J. Manschot, B. Pioline and A. Sen derive a remarkable formula for thePoincar´e polynomial of a smooth compact moduli space of stable quiver rep-resentations (called MPS degeneration formula in the following), motivated bystring-theoretic techniques (more precisely an interpretation relating quiver mod-uli to multi-centered black hole solutions to N = 2 supergravity). In contrastto the previously available formulae [13] expressing the Poincar´e polynomial ex-plicitly using (a resolution of) a Harder-Narasimhan type recursion, the MPSdegeneration formula expresses it as a summation over Poincar´e polynomials ofmoduli spaces of several other quivers, but only involving very special (thin, i.e.type one) dimension vectors (in the language of [12] the index of certain non-abelian quivers without oriented loops can be reduced to the abelian case, by aphysical argument which allows trading Bose-Fermi statistics with its classicallimit, Maxwell-Boltzmann statistics). One immediate advantage is that the MPSformula specializes to a similar formula for the Euler characteristics, which isnot possible for the Harder-Narasimhan recursion. Surprisingly, the derivation ofthe MPS formula in [12, Appendix D] relies completely on the resolved Harder-Narasimhan recursion of [13]. One should note however that even in the veryspecial cases when Euler characteristics of quiver moduli were already known,the MPS degeneration formula derives these numbers in a highly nontrivial way. Date : October 24, 2011.
As a first result of the present work we prove a motivic generalization of theMPS degeneration formula (Theorem 3.5) which is meaningful for arbitrary quiv-ers, dimension vectors and stabilities. Essentially, the motivic MPS formula ex-presses the motive of the quotient stack of the locus of semistable representationsby the base change group in terms of similar motives for thin dimension vectorsof a covering quiver, as an identity in a suitably localized Grothendieck ring ofvarieties. The proof essentially proceeds along the lines of [12, Appendix D], butavoids the resolution formula for the Harder-Narasimhan recursion, and clarifiesthe role of symmetric function identities implicit in [12]. Specialization of themotivic identity to Poincar´e polynomials recovers a generalization of the formulaof [12], which is now shown to hold for arbitrary quivers, arbitrary stabilitiesand coprime dimension vectors. We also derive a dual MPS degeneration for-mula (Corollary 3.9), which has the advantage of reducing to a smaller coveringquiver, at the expense of having more general dimension vectors.In Section 4 we take up a second line of investigation, connected with theso-called GW/Kronecker correspondence based on [4], [5] or more precisely itsrefinement described in [16]. A typical result of this type states that the Eu-ler characteristic of certain moduli spaces of representations for suitable quivers(e.g. generalized Kronecker quivers) can be computed alternatively as a Gromov-Witten invariant (on a weighted projective plane). In particular this is the casefor coprime dimension vectors of complete bipartite quivers, to which we restrictthroughout Section 4. Writing down the MPS degeneration formula for quiverEuler characteristics in this context one notices a striking similarity with thedegeneration formulae which are commonly used in Gromov-Witten theory, ex-pressing a given Gromov-Witten invariant in terms of relative invariants, withtangency conditions along divisors. Theorem 4.1 puts this intuition on firmground: at least for coprime dimension vectors of bipartite quivers, the MPSformula is indeed completely equivalent to a much more standard degenerationformula in Gromov-Witten theory. The proof hinges on the equality of certainEuler characteristics with tropical counts (Proposition 4.3).Combining localization techniques with the MPS formula leads to the remark-able conclusion that the Euler characteristic of moduli spaces of stable represen-tations is obtained in a purely combinatorial way. Indeed, since the dimensionvectors considered after applying the MPS formula are of type one, the mod-uli spaces corresponding to torus fixed representations are just isolated points,so that every such moduli space corresponds to a tree with a fixed number of(weighted) points. We describe this method for bipartite quivers in Section 5(see especially Corollary 5.3), but it could easily be transferred to general quiverswithout oriented cycles.In Section 6 we analyse the identity of Euler characteristics with tropicalcounts found in Proposition 4.3 from this point of view. On the one hand with a pair of weight vectors ( w ( k ) , w ( k )) we can associate a tropical curvecount N trop [( w ( k ) , w ( k ))], which effectively counts suitable trees. On the otherhand with the same weight vector we can associate a quiver Euler characteristic χ ( M Θ l − st( k ,k ) ( N )), which by the above argument (MPS plus localization) is alsoenumerating certain trees.Thus one would expect to be able to find an explicit way of assigning a quiverlocalization data to one of our tropical curves, and vice versa. The analogybetween quiver localization data and tropical curves was already pointed out in[17], but the MPS formula makes it even stronger. Notice that both tropicalcurves and localization data naturally carry multiplicities: for curves this is thestandard tropical multiplicity (recalled in Section 4), while in the case of quivermoduli spaces, a fixed tree can be coloured in different ways to obtain a numberof torus fixed points. The first natural guess is that the number of colourings andthe multiplicity of some corresponding tropical curve coincide. Unfortunatelythis doesn’t work, simply because in general the numbers of underlying curvesand trees (forgetting the multiplicity) are different.The next more promising attempt is described in Section 6. On both sidesthere is a way to construct new combinatorial data recursively. On the one handwe show that our tropical curves of prescribed slope can be obtained by glueingsmaller ones in a unique way (at least when the set of prescribed, unboundedincoming edges is chosen generically). This construction also gives a recursiveformula for the tropical counts, see Theorem 6.4. On the other hand we havea similar construction for quiver localization data. Starting with a number ofsemistable tuples, i.e. tuples consisting of a tree and a dimension vector such thatthe corresponding moduli space of semistables is not empty, we can glue themin a similar way to obtain a localization data with greater dimension vector, seeTheorem 6.6.In many cases these recursive constructions lead to a direct correspondencebetween tropical curves and quiver localization data. In Section 6.3 we describetwo such families of examples in detail. We do not know at the moment a methodwhich gives a concrete geometric correspondence in full generality. Acknowledgements.
We are grateful to So Okada for drawing our attention tothe formula of Manschot, Pioline and Sen. This research was partially supportedby Trinity College, Cambridge. Part of this work was carried out at the IsaacNewton Institute for Mathematical Sciences, Cambridge and at the HausdorffCenter for Mathematics, Bonn.2.
Recollections and notation
Quivers.
Let Q be a quiver with vertices Q and arrows Q denoted by α : i → j . We denote by Λ = Z Q the free abelian group over Q and by Λ + = N Q the set of dimension vectors written as d = P i ∈ Q d i i . Define ′ Λ + := Λ + \{ } . M. REINEKE, J. STOPPA AND T. WEIST
There exists a bilinear form on Λ, called the Euler form, given by h d, e i = X i ∈ Q d i e i − X α : i → j d i e j . We denote its antisymmetrization by { d, e } = h d, e i − h e, d i . A representation X of Q of dimension d ∈ Λ + is given by complex vector spaces X i of dimension d i forevery i ∈ Q and by linear maps X α : X i → X j for every arrow α : i → j ∈ Q .A vertex q is called a sink (resp. source) if there does not exists an arrow startingat (resp. terminating at) q . A quiver is bipartite if Q = I ∪ J with sources I and sinks J . In the following, we denote by Q ( I ) ∪ Q ( J ) the decomposition intosources and sinks.For a fixed vertex q ∈ Q we denote by N q the set of neighbours of q and for V ⊂ Q define N V := ∪ q ∈ V N q .For a representation X of the quiver Q we denote by dim X ∈ Λ + its dimensionvector. Moreover we choose a level l : Q → N + on the set of vertices. Define twolinear forms Θ , κ ∈ Hom( Z Q , Z ) by Θ( d ) = P q ∈ Q Θ q d q , κ ( d ) = P q ∈ Q l ( q ) d q and a slope function µ : N Q → Q by µ ( d ) = Θ( d ) κ ( d ) . For µ ∈ Q we denote by ′ Λ + µ ⊂ ′ Λ + be the set of dimension vectors of slope µ and define Λ + µ = ′ Λ + µ ∪ { } . This is a subsemigroup of Λ + .For a representation X of the quiver Q we define µ ( X ) := µ (dim X ). Therepresentation X is called (semi-)stable if the slope (weakly) decreases on propernon-zero subrepresentations. Fixing a slope function as above, we denote by R Θ − sst d ( Q ) the set of semistable points and by R Θ − st d ( Q ) the set of stable pointsin the affine variety R d ( Q ) := ⊕ α : i → j Hom( C d i , C d j ) of representations of dimen-sion d ∈ N Q . There exist moduli spaces M Θ − st d ( Q ) (resp. M Θ − sst d ( Q )) of stable(resp. semistable) representations parametrizing isomorphism classes of stable(resp. polystable) representations ([7]). If Q is acyclic and M Θ − st d ( Q ) is non-empty, it is a smooth irreducible variety of dimension 1 − h d, d i . Moreover it isprojective if semistability and stability coincide.Fixing a quiver Q and a dimension vector d ∈ N Q such that there exists a(semi-)stable representation for this tuple we call this tuple (semi-)stable.2.2. The tropical vertex.
We briefly review the definition of one of the toolswe shall use, the tropical vertex group, following [4, Section 0].We fix nonnegative integers l , l ≥ R as the formal power series ring R = Q [[ s , . . . , s l , t , . . . , t l ]], with maximal ideal m . Let B be the R -algebra B = Q [ x ± , y ± ][[ s , . . . , s l , t , . . . , t l ]] = Q [ x ± , y ± ] b ⊗ R (a suitable completion of the tensor product). For ( a, b ) ∈ Z and a series f ∈ x a y b Q [ x a y b ] b ⊗ m we consider the R -linear automorphism of B defined by θ ( a,b ) ,f : (cid:26) x xf − b y yf a . Notice that these automorphisms respect the symplectic form dxx ∧ dyy . Definition 2.1.
The tropical vertex group V R ⊂ Aut R ( B ) is defined as the com-pletion with respect to m of the subgroup of Aut R ( B ) generated by all elements θ ( a,b ) ,f as above. We recall that by [8] (see also [4, Theorem 1.3]) there exists a unique infiniteordered product factorization in V R of the form θ (1 , , Q k (1+ s k x ) θ (0 , , Q l (1+ t l y ) = Y b/a decreasing θ ( a,b ) ,f ( a,b ) , the product ranging over all coprime pairs ( a, b ) ∈ N .3. Motivic MPS formula
Some symmetric function identities.
We start with some preliminarieson symmetric functions following [11]. Partitions λ of n are written λ ⊢ n . Witha partition λ ⊢ n we associate a multiplicity vector m ∗ ( λ ), where m i ( λ ) is themultiplicity of the part i in λ . Conversely with a vector m ∗ = ( m l ∈ N ) l ≥ weassociate the partition λ ( m ∗ ) = (1 m m . . . ). This induces a bijection betweenpartitions of n and the set of multiplicity vectors m ∗ such that P l lm l = n . Inthis case, we also write m ∗ ⊢ n .Denote by P Q the ring of symmetric functions with rational coefficients in vari-ables x i for i ≥
1. We consider the so-called principal specialization map p : P Q → Q ( q ) given by x i q i − for all i ≥ e n = P i <...
Lemma 3.1. (1)
The previous identity can be rewritten as e n = X m ∗ ⊢ n Y l m l ! (cid:18) ( − l − l (cid:19) m l p λ ( m ∗ ) . (2) Conversely, we have: p n = ( − n − n X λ ⊢ n ( − l ( λ ) − l ( λ ) l ( λ )! Q l m l ( λ )! e λ . Proof.
The first identity follows from the definitions. For the second we uselog(1 + x ) = P k ≥ − k − k x k to calculatelog E ( t ) = log(1 + X n ≥ e n t n ) = X k ≥ ( − k − k ( X n ≥ e n t n ) k == X k ≥ ( − k − k X n ,...,n k ≥ e n . . . e n k t n + ... + n k == X k ≥ ( − k − k X λ : l ( λ )= k k ! Q l m l ( λ )! e λ t | λ | == X n ≥ X λ ⊢ n ( − l ( λ ) − l ( λ ) l ( λ )! Q l m l ( λ )! e λ t n , where we have used the fact that the number of rearrangements of a partition λ is l ( λ )! Q l m l ( λ )! . Differentiating and using P ( − t ) = ddt (log E ( t )), the lemma follows. (cid:3) For partitions λ , µ of n , denote by L µλ the number of functions f : { , . . . , l ( µ ) } → N such that λ i = P j : f ( j )= i µ j . Then e λ = P µ ε µ z − µ L µλ p µ by [11, I,(6.11)]. Lemma 3.2.
We have e λ = X m ∗ ⊢ n ( X m ∗∗ Y l ≥ m l ! Q j m jl ! ) Y l ≥ m l ! (cid:18) ( − l − l (cid:19) m l p λ ( m ∗ ) , where the inner sum ranges over all tuples ( m jl ) j,l ≥ such that P j m jl = m l forall l ≥ and P l lm jl = λ j for all j ≥ .Proof. Assume µ = (1 m m . . . ). To a function f as above, we associate the sets I jl = { i ∈ { , . . . , m l } : f ( m + . . . + m l − + i ) = j } for j, l ≥
1. Then, for all l ≥
1, the set { , . . . , m l } is the disjoint union of the I jl . Defining m jl as the cardinality of I jl , we thus have P j m jl = m l for all l ≥ P l lm jl = λj by definition of f . This establishes a bijection between theset of functions f which is counted by L µλ and the set of pairs ( m ∗∗ , I ∗∗ ) which iscounted by the inner sum. (cid:3) Under the principal specialization map, e n maps to q ( n ) (1 − q )(1 − q ) ... (1 − q n ) , and p n maps to − q n ) ([11, I,2. Example 4.]). Using the first identity of Lemma 3.1,this yields the identity q ( n )(1 − q ) . . . (1 − q n ) = X m ∗ ⊢ n Y l ≥ m l ! (cid:18) ( − l − l (1 − q l ) (cid:19) m l . Replacing q by q − and multiplying by an appropriate power of q , this is equiv-alent to Lemma 3.3.
We have q ( n )( q n − . . . ( q n − q n − ) = X m ∗ ⊢ n Y l ≥ m l ! (cid:18) ( − l − l [ l ] q (cid:19) m l q − P l m l , where [ l ] q = ( q l − / ( q − . The motivic MPS formula.
We assume throughout that an arbitraryfinite quiver Q and a stability Θ for Q are given and we fix a vertex i ∈ Q .We introduce a new (levelled) quiver b Q by replacing the vertex i by vertices i k,l for k, l ≥
1, thus b Q = Q \ { i } ∪ { i k,l : k, l ≥ } , with vertex i k,l being of level l . The arrows in b Q are given by the following rules: • all arrows α : j → k in Q which are not incident with i induce an arrow α : j → k in b Q , • all arrows α : i → j (resp. α : j → i ) in Q for j = i induce arrows α p : i k,l → j (resp. α p : j → i k,l ) for k, l ≥ p = 1 , . . . , l in b Q , • all loops α : i → i in Q induce arrows α p,q : i k,l → i k ′ ,l ′ for k, l, k ′ , l ′ ≥ p = 1 , . . . , l , q = 1 , . . . , l ′ in b Q .Given a dimension vector d for Q and a multiplicity vector m ∗ ⊢ d i (that is, P l lm l = d i ) as above, we define a dimension vector b d ( m ∗ ) for b Q by b d ( m ∗ ) j = d j for j = i in Q and b d ( m ∗ ) i k,l = (cid:26) , k ≤ m l , k > m l . We have G b d ( m ∗ ) ≃ Y j = i GL d j ( C ) × G P l m l m , where G m = C ∗ denotes the multiplicative group of the field C . We choose anarbitrary basis of C d i indexed by vectors v k,l,p for l ≥
1, 1 ≤ k ≤ m l and 1 ≤ p ≤ l (this is possible since P l lm l = d i ). Then the group G b d ( m ∗ ) embeds into G d byletting the ( k, l )-th component of G m scale the vectors v k,l,p for p = 1 , . . . , l simultaneously, for all l ≥
1, 1 ≤ k ≤ m l .We define a stability b Θ for b Q by b Θ j = Θ j for all j = i in Q and b Θ i k,l = l Θ i forall k, l ≥
1. The associated slope function is denoted by b µ . The following lemmais easily verified by working through the definitions of b Q , b d and b µ : M. REINEKE, J. STOPPA AND T. WEIST
Lemma 3.4.
Via the above embedding, we have a G b d ( m ∗ ) -equivariant isomor-phism between R d ( Q ) and R b d ( m ∗ ) ( b Q ) . Furthermore, we have b µ ( b d ( m ∗ )) = µ ( d ) .Proof. (cid:3) Our motivic version of the MPS formula is an identity in a suitably localizedGrothendieck ring of varieties; we refer to [1] for an introduction to this topicsuitable for our purposes. Let K (Var / C ) be the free abelian group generatedby representatives [ X ] of all isomorphism classes of complex varieties X , modulothe relation [ X ] = [ A ] + [ U ] if A is isomorphic to a closed subvariety of X , withcomplement isomorphic to U . Multiplication is given by [ X ] · [ Y ] = [ X × Y ].Denote by [ L ] the class of the affine line. We work in the localization K = ( K (Var / C ) ⊗ Q )[[ L ] − , ([ L ] n − − : n ≥ . Theorem 3.5.
For arbitrary Q , d , Θ and i as above, the following identity holdsin K : [ L ]( di ) [ R sst d ( Q )][ G d ] = X m ∗ ⊢ d i Y l ≥ m l ! (cid:18) ( − l − l [ P l − ] (cid:19) m l [ R sst b d ( m ∗ ) ( b Q )][ G b d ( m ∗ ) ] . Proof.
We start the proof by translating the identity of Lemma 3.3 into the ring K . We note the following identities:[GL n ( C )] = n − Y i =0 ([ L ] n − [ L ] i ) , [ G m ] = [ L ] − , [ P n − ] = ([ L ] n − / ([ L ] −
1) = [ n ] [ L ] . Then the above identity translates into[ L ]( n )[GL n ( C )] = X m ∗ ⊢ n Y l ≥ m l ! (cid:18) ( − l − l [ P l − ] (cid:19) m l G P l m l m ] . Replacing n by d i , multiplying by [ R d ( Q )] / Q j = i [GL d j ( C )] and using the aboveidentifications and Lemma 3.4, this yields the MPS formula for trivial stabilityΘ = 0: [ L ]( di ) [ R d ( Q )][ G d ] = X m ∗ ⊢ d i Y l ≥ m l ! (cid:18) ( − l − l [ P l − ] (cid:19) m l [ R b d ( m ∗ ) ( b Q )][ G b d ( m ∗ ) ] . Now we make use of the Harder-Narasimhan stratification of R d ( Q ) con-structed in [13]: we fix a decomposition d = d + . . . + d s into non-zero dimensionvectors such that µ ( d ) > . . . > µ ( d s ), which we call a HN type for d , denoted by d ∗ = ( d , . . . , d s ) | = d . Denote by R d ∗ d ( Q ) the set of all representations M ∈ R d ( Q )such that in the Harder-Narasimhan filtration 0 = M ⊂ M ⊂ . . . ⊂ M s = M of M , the dimension vector of M i /M i − equals d i for all i = 1 , . . . , s . By [13], wehave R d ∗ d ( Q ) ≃ G d × P d ∗ V d ∗ , where P d ∗ is a parabolic subgroup of G d with Levi isomorphic to Q sk =1 G d k , and V d ∗ is a vector bundle over Q sk =1 R sst d k ( Q ) of rank r d ∗ = P k Corollary 3.6. We have [ L ]( di ) [ R sst d ( Q )][ G d ] = X λ ⊢ d i ε λ z − λ Q j [ P λ j − ] [ R sst b d ( m ∗ ( λ )) ( b Q )][ G b d ( m ∗ ( λ )) ] . There is a well-defined ring homomorphism π : K → Q ( t ) mapping the class of asmooth projective variety X to its Poincar´e polynomial P ( X, t ) = P i dim H i ( X, Q ) t i in singular cohomology. In the case where the dimension vector d is Θ-coprime,that is, µ ( e ) = µ ( d ) for all non-zero dimension vectors e < d , the moduli space M sst d ( Q ) = R sst d ( Q ) /G d ( C ) is a smooth variety, and we have P ( M sst d ( Q ) , t ) = ( t − · π ([ R sst d ( Q )] / [ G d ])by [2, Theorem 2.5]. Specialization of the motivic MPS formula to this case yieldsa formula for its Poincar´e polynomial, and in particular for its Euler characteris-tic: Corollary 3.7. If d is Θ -coprime, we have t d i ( d i − P ( M sst d ( Q ) , t ) = X m ∗ ⊢ d i Y l ≥ m l ! (cid:18) ( − l − l [ l ] t (cid:19) m l P ( M sst b d ( m ∗ ) ( b Q ) , t ) and χ ( M sst d ( Q )) = X m ∗ ⊢ d i Y l ≥ m l ! (cid:18) ( − l − l (cid:19) m l χ ( M sst b d ( m ∗ ) ( b Q )) . Dual MPS formula. We define a quiver ˇ Q as the “level one part” of b Q ,thus ˇ Q = Q \ { i } ∪ { i k : k ≥ } , and the arrows in ˇ Q are given by the followingrules: • all arrows α : j → k in Q which are not incident with i induce an arrow α : j → k in ˇ Q , • all arrows α : i → j (resp. α : j → i ) in Q for j = i induce arrows α : i k → j (resp. α : j → i k ) for k ≥ Q , • all loops α : i → i in Q induce arrows α : i k → i k ′ for k, k ′ ≥ Q . Given a dimension vector d for Q and a partition λ ⊢ d i , we define a dimensionvector ˇ d ( λ ) for ˇ Q by ˇ d ( λ ) j = d j for j = i in Q and ˇ d ( λ ) i k = λ k for k ≥ i k for k ≥ Q yields:[ L ] P j ( λj ) [ R sstˇ d ( λ ) ( ˇ Q )][ G ˇ d ( λ ) ] = X ( m j ∗ ⊢ λ j ) j Y j ≥ Y l ≥ m jl ! (cid:18) ( − l − l [ P l − ] (cid:19) m jl [ R sst b d ( P j m j ∗ ) ( b Q )][ G b d ( P j m j ∗ ) ] == X m ∗ ⊢ d i ( X m ∗∗ Y l ≥ m l ! Q j m jl ! ) Y l ≥ m l ! (cid:18) ( − l − l [ P l − ] (cid:19) m l [ R sst b d ( m ∗ ) ( b Q )][ G b d ( m ∗ ) ] , where the inner sum runs over all tuples ( m jl ) j,l ≥ such that m l = P j m jl for all l and λ j = P l lm jl for all j . Comparison with Lemma 3.2 yields: Proposition 3.8. There is a well-defined map P Q → K of Q -vector spaces suchthat e λ [ L ] P j ( λj ) [ R sstˇ d ( λ ) ( ˇ Q )][ G ˇ d ( λ ) ] , p λ Y j P λ j − ] · [ R sst b d ( m ∗ ( λ )) ( b Q )][ G b d ( m ∗ ( λ )) ] . It might be interesting to ask whether the images of other bases of the ring ofsymmetric functions (monomial symmetric functions, complete symmetric func-tions, Schur functions, ...) have a natural interpretation in terms of motives ofquiver moduli.We can now map the second identity of Lemma 3.1 to K to get the following dualversion of the MPS formula: Corollary 3.9. For given d , denote by b d the dimension vector for b Q with asingle entry on level d i . Then P d i − ] · [ R sst b d ( b Q )][ G b d ( b Q )] = ( − d i − d i X λ ⊢ d i ( − l ( λ ) − ( l ( λ ) − Q l m l ( λ )! [ L ] P j ( λj ) [ R sstˇ d ( λ ) ( ˇ Q )][ G ˇ d ( λ ) ] . The MPS formula as a degeneration formula in Gromov-Wittentheory In the rest of the paper for every bipartite quiver Q we consider the linearform Θ ∈ N Q defined by Θ i = 1 for every i ∈ Q ( I ) and Θ j = 0 for every j ∈ Q ( J ). Additionally fixing a level l : Q → N + , we define the linear form Θ l by(Θ l ) q = l ( q )Θ q and consider the slope µ = Θ l /κ where κ is defined as in Section2. Note that for the trivial level structure, i.e. l ( q ) = 1 for every q ∈ Q , we haveΘ l = Θ and κ = dim.In this section we specialize the MPS formula to Euler characteristics, and atthe same time we restrict to a special class of quivers. These are the completebipartite quivers K ( l , l ) of [16] Section 5, defined by the vertices K ( l , l ) = { i , . . . , i l } ∪ { j , . . . , j l } and the arrows K ( l , l ) = { α k,l : i k → j l | k ∈ { , . . . l } , l ∈ { , . . . l }} . A dimension vector for K ( l , l ) is uniquely determined by a pair of ordered partitions ( P , P ) = ( l X i =1 p i , l X j =1 p j ) . We assume throughout this section that the sizes | P | , | P | are coprime . We fixthe trivial level structure given by l ( q ) = 1 for all q ∈ K ( l , l ) . We denote by M Θ − st ( P , P ) = M Θ − st( P , P ) ( K ( l , l ))the moduli space of stable representations with respect to this choice.The MPS formula in this context can be expressed uniformly for all l , l andall dimension vectors by introducing an infinite quiver N with a suitable levelstructure. We define its vertices by N = { i ( w,m ) | ( w, m ) ∈ N } ∪ { j ( w,m ) | ( w, m ) ∈ N } , and the arrows by N = { α , . . . , α w · w ′ : i ( w,m ) → j ( w ′ ,m ′ ) , ∀ w, w ′ , m, m ′ ∈ N } . The level function is given by l ( q ( w,m ) ) = w, ∀ q ∈ { i, j } , m ∈ N and we fix the linear form Θ l . A refinement of ( P , P ) is a pair of sets of integers( k , k ) = ( { k wi } , { k wj } )such that for i = 1 , . . . , l and j = 1 , . . . , l we have p i = X w wk wi , p j = X w wk wj . We will denote refinements by ( k , k ) ⊢ ( P , P ). The number of entries ofweight w in k i is defined by m w ( k i ) = l i X j =1 k iwj . A fixed refinement ( k , k ) induces a dimension vector for N by setting d q ( w,m ) = (cid:26) m = 1 , . . . , m w ( k p ) , m > m w ( k p ) , for q ∈ { i, j } , and p = 1 , q = i, j . With this notation in place, the MPSformula at the level of Euler characteristics can be expressed by χ ( M Θ − st ( P , P )) = X ( k ,k ) ⊢ ( P , P ) χ ( M Θ l − st( k ,k ) ( N )) Y i =1 l i Y j =1 Y w ( − k iw,j ( w − k iw,j ! w k iw,j . (1) The ordered partition ( P , P ) also encodes an a priori very different kindof data, namely the Gromov-Witten invariant N [( P , P )] of [4] Section 0.4.Roughly speaking this is a virtual count of rational curves in the weighted pro-jective plane P ( | P | , | P | , 1) which pass through l j specified distinct points lyingon the distinguished toric divisor D j for j = 1 , 2. We require that these pointsare not fixed by the torus action, that the multiplicities at the points are specifiedby P j , and that the curve touches the remaining toric divisor D out at some pointwhich is also not fixed by the torus. The refined GW/Kronecker correspondenceof [16] (based on [4], [5]) leads to a rather striking consequence ([16] Corollary9.1): N [( P , P )] = χ ( M Θ − st ( P , P )) . The powerful degeneration formula of Gromov-Witten theory ([6], [9], [10])allows one to express N [( P , P )] in terms of certain relative Gromov-Witten in-variants, enumerating rational curves with tangency conditions, as we now brieflydiscuss. Following [4] Section 2.3, we define a weight vector w i as a sequence ofintegers ( w i , . . . , w it i ) with0 < w i ≤ w i ≤ · · · ≤ w it i . The automorphism group Aut( w i ) of a weight vector w i is the subset of the sym-metric group on t i letters which stabilizes w i . A pair of weight vectors ( w , w )encodes a relative Gromov-Witten invariant N rel [( w , w )], virtually enumerat-ing rational curves in P ( | w | , | w | , 1) which are tangent to D i at specified points(not fixed by the torus), with order of tangency specified by w i . The rigorousconstruction of these invariants is carried out in [4] Section 4.4. Let us now fixweight vectors w i with | w i | = | P i | for i = 1 , 2. A set partition I • of w i is adecomposition of the index set I ∪ · · · ∪ I l i = { , . . . , t i } into l i disjoint, possibly empty parts. We say that the set partition I • is compatible with P i (or simply compatible) if for all j we have p ij = X r ∈ I j w ir . The relevant degeneration formula involves the ramification factors R P i | w i = X I • t i Y j =1 ( − w ij − w ij , where we are summing over all compatible set partitions I • . Then Proposition5.3 from [4] yields the equality N [( P , P )] = X ( w , w ) N rel [( w , w )] Y i =1 Q t i j =1 w ij | Aut( w i ) | R P i | w i . (2)We come to the central claim of this section: Theorem 4.1. Given the equality N [( P , P )] = χ ( M Θ − st ( P , P )) (i.e. thecoprime case of the refined GW/Kronecker correspondence), the MPS formula (1) for the Euler characteristics of quiver representations is equivalent to theGromov-Witten degeneration formula (2) . The rest of this section is devoted to a proof of this result. As a first step, tosimplify the comparison, we will rewrite the degeneration formula (2) as a sumover pairs of refinements ( k , k ) rather than pairs of weight vectors ( w , w ).Notice that a fixed refinement k i induces a weight vector w ( k i ) = ( w i , . . . , w it i )of length t i = P w m w ( k i ), by w ij = w for all j = w − X r =1 m r ( k i ) + 1 , . . . , w X r =1 m r ( k i ) . Of course the weight vector w ( k i ) only depends on k i through { m w ( k i ) } w . How-ever we wish to think of w ( k i ) as coming with a distinguished set partition I • ( k i ):the segment of weight w entries in w ( k i ) is partitioned into l i consecutive chunksof size k w , · · · , k wl i , and we declare the indices for the j -th chunk to lie in I j ( k i ). Lemma 4.2. The degeneration formula for Gromov-Witten invariants (2) canbe rewritten as N [( P , P )] = X ( k ,k ) ⊢ ( P , P ) N rel [( w ( k ) , w ( k ))] Y i =1 l i Y j =1 Y w ( − k iw,j ( w − k iw,j ! w k iw,j . Proof. Consider the following operation on a compatible set partition I • : if thereexist a ∈ I p and b ∈ I q with p = q and w i,a = w i,b , then permuting the indices p, q yields a new set partition I ′• which is still compatible. We write [ I • ] forthe equivalence class of set partitions generated by this operation. For a setpartition I • ( k i ) which is induced by a refinement k i , a simple count shows thatthe equivalence class [ I • ( k i )] contains Y w (cid:18) m w ( k i ) k iw, (cid:19)(cid:18) m w ( k i ) − k iw, k iw, (cid:19) . . . (cid:18) m w ( k i ) − P l i − r =1 k iw,r k iw,l i (cid:19) distinct elements.The formula (2) is equivalent to N [( P , P )] = X ( w , w ) X I • X I • N rel [( w , w )] Y i =1 | Aut( w i ) | t i Y j =1 ( − w ij − w ij , where we are summing over all compatible set partitions. But we can enumeratethe data w i , I i • differently: namely, rather than fixing w i and considering alladmissible I i • , we can fix a refinement k i , form the weight vector w ( k i ) andrestrict to the set partitions in the class [ I • ( k i )]. In this case we have | Aut( w ( k i )) | = Y w m w ( k i )! , t i Y j =1 ( − w ij − w ij = l i Y j =1 Y w ( − k iw,j ( w − w k iw,j . Thus the right hand side of the above equation becomes X ( k ,k ) ⊢ ( P , P ) N rel [( w ( k ) , w ( k ))] · Y i =1 l i Y j =1 Y w ( − k iw,j ( w − w k iw,j m w ( k i )! (cid:18) m w ( k i ) − P j − r =1 k iw,r k iw,j (cid:19) . The result follows from the simple calculation1 m w ( k i )! (cid:18) m w ( k i ) k iw, (cid:19)(cid:18) m w ( k i ) − k iw, k iw, (cid:19) . . . (cid:18) m w ( k i ) − P l i − r =1 k iw,r k iw,l i (cid:19) = l i Y j =1 k iw,j ! . (cid:3) Comparing the degeneration formula as rewritten in the Lemma with the MPSformula (1), we see that proving Theorem 4.1 is equivalent to establishing theidentity χ ( M Θ l − st( k ,k ) ( N )) = N rel [( w ( k ) , w ( k ))] Y i =1 l i Y j =1 Y w w k iw,j . (3)In fact the right hand side of (3) has a geometric interpretation as a suitable tropical count . Here we will confine ourselves to the basic notions we need tostate this equivalence, following [4] Section 2.1.Let Γ be a weighted, connected tree with only 1-valent and 3-valent vertices,thought of as a compact topological space in the canonical way. We remove the 1-valent vertices to form the graph Γ. The noncompact edges are called unbounded edges. We denote the induced weight function on the edges of Γ by w Γ . A parametrized rational tropical curve in R is a proper map h : Γ → R such that: • the restriction of h to an edge is an embedding whose image is containedin an affine line of rational slope, and • a balancing condition holds at the vertices. Namely, denoting by m i theprimitive integral vector emanating from the image of a vertex h ( V ) inthe direction of an edge h ( E i ), we require X i =1 w Γ ( E i ) m i = 0 , where we are summing over all the edges which are adjecent to V .A rational tropical curve is the equivalence class of a rational parametrized tropi-cal curve under reparametrizations which respect w Γ . The multiplicity at a vertex V is defined as Mult V ( h ) = w Γ ( E ) w Γ ( E ) | m ∧ m | , where by the balancing condition we can choose E , E to be any two edgesadjecent to V . To total multiplicity of h is then defined asMult( h ) = Y V Mult V ( h ) . Let us write e , e for the versors of R . A pair of weight vectors ( w , w ) encodesa tropical invariant, counting rational tropical curves h which satisfy the followingconditions: • the unbounded edges of Γ are E ij for 1 ≤ i ≤ , ≤ j ≤ t i , plus a sin-gle “outgoing” edge E out . We require that h ( E ij ) is contained in a line e ij + R e i for some prescribed versors e ij , and its unbounded direction is − e i , • w Γ ( E ij ) = w ij .Notice that the balancing condition implies that h ( E out ) lies on an affine line withdirection ( | w | , | w | ). The set of such tropical curves h is finite, and it followsfrom the general theory (see e.g. [3], [14]) that when we count curves h takinginto account the multiplicity Mult( h ) we get an integer N trop [( w , w )] whichis independent on the (generic) choice of displacements e ij . The comparisonresult that we need, relating the Gromov-Witten invariants which appear in thedegeneration formula to tropical counts, is then obtained by combining Theorems3.4 and 4.4 in [4]: N trop [( w , w )] = N rel [( w , w )] Y i =1 t i Y j =1 w ij . (4)Thanks to the equivalence (4), Theorem 4.1 follows from the following result: Proposition 4.3. We have an equality of Euler characteristics and tropicalcounts N trop [( w ( k ) , w ( k ))] = χ ( M Θ l − st( k ,k ) ( N )) . (5)We will prove this equality using the scattering diagrams of [4] and Theorem2.1 in [15]. Notice however that in the special case when all the parts of therefinement ( k , k ) equal 1 the corresponding subquiver of N is isomorphic to K ( l , l ), and (5) is an immediate consequence of the refined GW/Kroneckercorrespondence.Let us we denote by Q ⊂ N the subquiver spanned by the support of (thedimension vector induced by) ( k , k ). This is a complete bipartite quiver with t sources and t sinks. For each w , Q contains m w ( k ) sources (respectively m w ( k ) sinks) with level w . We introduce the ring R = C [[ x j ( w ′ ,m ′ ) , y i ( w,m ) | w, w ′ , m, m ′ ∈ N ]] (6)with a Poisson bracket defined by { x j ( w ′ ,m ′ ) , y i ( w,m ) } = { j ( w ′ ,m ′ ) , i ( w,m ) } x j ( w ′ ,m ′ ) · y i ( w,m ) = ww ′ x j ( w ′ ,m ′ ) · y i ( w,m ) . The Kontsevich-Soibelman Poisson automorphisms in this context are defined by T j ( w,m ) ( x j ( w ′ ,m ′ ) ) = x j ( w ′ ,m ′ ) ,T j ( w,m ) ( y i ( w ′ ,m ′ ) ) = y i ( w ′ ,m ′ ) (cid:16) x j ( w,m ) (cid:17) ww ′ , and similarly T i ( w,m ) ( x j ( w ′ ,m ′ ) ) = x j ( w ′ ,m ′ ) (cid:16) y i ( w,m ) (cid:17) − ww ′ ,T i ( w,m ) ( y i ( w ′ ,m ′ ) ) = y i ( w ′ ,m ′ ) . According to Theorem 2.1 in [15], the product of operators Y j ( w,m ) ∈ Q T j ( w,m ) · Y i ( w ′ ,m ′ ) ∈ Q T i ( w ′ ,m ′ ) can be expressed alternatively as a slope-ordered product Q ← µ ∈ Q T µ , acting e.g.on the y variables as T µ ( y i ( w,m ) ) = y i ( w,m ) Y j ( w ′ ,m ′ ) ∈ Q ( Q µ,j ( w ′ ,m ′ ) ) ww ′ , where we have denoted by Q µ,j ( w ′ ,m ′ ) the generating series of Euler characteristicsfor moduli spaces of stable representations of Q with slope µ and a 1-dimensionalframing at j ( w ′ ,m ′ ) . Recall however that we are only interested in the Euler char-acteristic χ ( M Θ l − st( k ,k ) ( N )), i.e. for representations with dimension 1 at each vertex.In this case framed representations coincide with ordinary representations, andwe find that the coefficient of the monomial Y j ( w ′ ,m ′ ) ∈ Q x j ( w ′ ,m ′ ) · Y i ( w,m ) ∈ Q y i ( w,m ) (7)in the series y − i ( w,m ) T µ ( y i ( w,m ) ) is given by w X w ′ w ′ m w ′ ( k ) χ ( M Θ l − st( k ,k ) ( N )) (8)(where we choose µ to be the slope of the dimension vector induced by ( k , k )).On the other hand we can compute the coefficient of (7) in a different way, bysetting up an appropriate scattering diagram in the sense of [4] Definition 1.2.To this end we need to identify T i ( w,m ) , T j ( w ′ ,m ′ ) with operators acting on the ring R ′ = C [ x ± , y ± ][[ ξ j ( w,m ) , η i ( w ′ ,m ′ ) | w, w ′ , m, m ′ ∈ N ]] . This is possible if we set x j ( w,m ) = (cid:16) ξ j ( w,m ) x (cid:17) w ,y i ( w ′ ,m ′ ) = (cid:16) η i ( w ′ ,m ′ ) y (cid:17) w ′ , (9) from which T j ( w,m ) ( ξ j ( w ′ ,m ′ ) x ) = ξ j ( w ′ ,m ′ ) x,T j ( w,m ) ( η i ( w ′ ,m ′ ) y ) = η i ( w ′ ,m ′ ) y (cid:16) (cid:16) ξ j ( w,m ) x (cid:17) w (cid:17) w , respectively T i ( w,m ) ( ξ j ( w ′ ,m ′ ) x ) = ξ j ( w ′ ,m ′ ) x (cid:16) (cid:16) η i ( w,m ) y (cid:17) w (cid:17) − w ,T i ( w,m ) ( η i ( w ′ ,m ′ ) y ) = η i ( w ′ ,m ′ ) y. Then following the notation of [4] Section 0.1, we can make the identification T j ( w,m ) = θ (1 , , (cid:16) (cid:16) ξ j ( w,m ) x (cid:17) w (cid:17) w with a standard element of the tropical vertex group over R ′ , V R ′ , and similarly T i ( w,m ) = θ (0 , , (cid:16) (cid:16) η i ( w,m ) y (cid:17) w (cid:17) w . We are led to consider the saturated scattering diagram S (in the sense of [4]Section 1) for the product Y j ( w,m ) ∈ Q θ (1 , , (1+( ξ j ( w,m ) x ) w ) w · Y i ( w ′ ,m ′ ) ∈ Q θ (0 , , (1+( η i ( w ′ ,m ′ ) y ) w ′ ) w ′ . (10)We only recall briefly that according to the general theory one starts with ageneric configuration of horizontal lines d j ( w,m ) in R (respectively vertical lines d i ( w ′ ,m ′ ) ), with attached weight functions (1 + ( ξ j ( w,m ) x ) w ) w ((1 + ( η i ( w ′ ,m ′ ) y ) w ′ ) w ′ respectively). According to [4] Section 1.2 with a generic path γ : [0 , → R onecan associate an element θ γ ∈ V R ′ . The (essentially unique) saturated scatteringdiagram S is obtained by adding rays to the original configurations of lines tothat for each closed loop the group element θ γ becomes trivial (if at all defined),see [4] Theorem 1.4. The crucial point for us is that, thanks to the identifications(9), S gives an alternative way of computing the ordered product factorization Q ← µ ∈ Q T µ .In fact since we are only interested in the coefficient of the monomial (7), weare allowed to replace the ring R ′ with its truncation C [ x ± , y ± ][[ ξ j ( w,m ) , η i ( w ′ ,m ′ ) | w, w ′ , m, m ′ ∈ N ]] / ( ξ wj ( w,m ) , η w ′ i ( w ′ ,m ′ ) ) , and thus replace the scattering diagram for (10) with the much simpler scatteringdiagram for the product Y j ( w,m ) ∈ Q θ (1 , , w ( ξ j ( w,m ) x ) w · Y i ( w ′ ,m ′ ) ∈ Q θ (0 , , w ′ ( η i ( w ′ ,m ′ ) y ) w ′ . Making the change of variables u j ( w,m ) = ξ wj ( w,m ) , v i ( w ′ ,m ′ ) = η w ′ i ( w ′ ,m ′ ) we can as well consider the scattering diagram e S for the product Y j ( w,m ) ∈ Q θ (1 , , w u j ( w,m ) x w · Y i ( w ′ ,m ′ ) ∈ Q θ (0 , , w ′ v i ( w ′ ,m ′ ) y w ′ . over the ring f R ′ = C [ x ± , y ± ][[ u j ( w,m ) , v i ( w ′ ,m ′ ) | w, w ′ , m, m ′ ∈ N ]] / ( u j ( w,m ) , v i ( w ′ ,m ′ ) ) . According to [4] Theorem 2.4, there is a one to one correspondence between rays of e S and rational tropical curves for which the set E ij is contained in d j ( w,m ) , d i ( w ′ ,m ′ ) (so that the weight of a leg contained in d j ( w,m ) is w , respectively w ′ for d i ( w ′ ,m ′ ) ).What is more, if f is a weight function containing the monomial (7), it must havethe form f = 1 + Mult( h ) Y j ( w ′ ,m ′ ) ∈ Q u j ( w ′ ,m ′ ) x w ′ · Y i ( w,m ) ∈ Q v i ( w,m ) y w where h is the corresponding tropical curve. This is again a consequence of [4]Theorem 2.4. Indeed taking up for a moment the notation of [4] equation (2.1),in our case we have w out = 1 and the term a i ( J ) q Q j ∈ J u ij vanishes except when J = { } and q is one of our w, w ′ , so Y i,J,q ( J )! a i ( J ) q Y j ∈ J u ij z m out = Y j ( w ′ ,m ′ ) ∈ Q u j ( w ′ ,m ′ ) x w ′ · Y i ( w,m ) ∈ Q v i ( w,m ) y w . Notice that the weight vector of h is ( w ( k ) , w ( k )). Therefore the product ofall such weight functions f equalsΦ = 1 + N trop [( w ( k ) , w ( k ))] Y j ( w ′ ,m ′ ) ∈ Q u j ( w ′ ,m ′ ) x w ′ · Y i ( w,m ) ∈ Q v i ( w,m ) y w Thanks to the choice of level structure on N (i.e. l ( i ( w,m ) ) = w, l ( j ( w ′ ,m ′ ) ) = w ′ ),the slope of each ray underlying one of the weight functions f equals µ , theslope of the dimension vector induced by ( k , k ). By the uniqueness of orderedproduct factorizations in V e R ′ , we have that the coefficient of the monomial (7)in the series y − i ( w,m ) T µ ( y i ( w,m ) ) equals the nontrivial coefficient of the action of θ Φ on v i ( w,m ) y w . Namely we have θ Φ ( v i ( w,m ) y w ) = θ Φ ( η wi ( w,m ) y w ) = η wi ( w,m ) y w · Φ w P w ′ w ′ m w ′ ( k ) , which when expanded contains as the only nontrivial coefficient w X w ′ w ′ m w ′ ( k ) N trop [( w ( k ) , w ( k ))] . (11)Our claim (5) follows by comparing (8), (11). Remark. One can show (arguing by induction on | p ij | ) that the MPS formulaand the coprime case of the refined GW/Kronecker correspondence imply theequality (5). Euler characteristic via counting trees In this section we continue the investigation of the MPS formula (4). Combinedwith localization techniques it implies that to calculate the Euler characteristicof moduli spaces it suffices to count trees. For a quiver Q we denote by ˜ Q itsuniversal cover given by the vertex set˜ Q = { ( q, w ) | q ∈ Q , w ∈ W ( Q ) } and the arrow set˜ Q = { α ( q,w ) : ( q, w ) → ( q ′ , wα ) | α : q → q ′ ∈ Q } . Here W ( Q ) denotes the set of words of Q , see [18, Section 3.4] for a precisedefinition. Recall the localization theorem [18, Corollary 3.14]: Theorem 5.1. We have χ ( M Θ − st d ( Q )) = X ˜ d χ ( M ˜Θ − st˜ d ( ˜ Q )) , where ˜ d ranges over all equivalence classes being compatible with d , and the slopefunction considered on ˜ Q is the one induced by the slope function fixed on Q , i.e.we define the corresponding linear form ˜Θ by ˜Θ q ′ = Θ q for all q ∈ Q and for all q ′ ∈ ˜ Q corresponding to q . We call a tuple ( Q , d ) consisting of a finite subquiver Q of ˜ Q and a dimensionvector d ∈ N Q localization data if M ˜Θ − st d ( Q ) = ∅ .Even if the following machinery applies in a more general setting, we concen-trate on the quivers K ( l , l ) and N respectively. We fix a partition ( P , P )with | P | = d, | P | = e and a refinement ( k , k ). We consider the quiver N ( k , k ) := supp( k , k ) consisting of the full subquiver of N with verticessupp( k , k ) = { q ∈ N | ( k , k ) q = 0 } . Since we have ( k , k ) q = 1 forall q ∈ N ( k , k ) , every localization data ( T, ( k , k )) defines a subtree T of N ( k , k ), i.e. a subquiver without cycles with at most one arrow between eachtwo vertices.In the following, fixing a subtree T we denote by β the dimension vector and,moreover, we denote by ( d, e ) the dimension type, i.e. d := X i ∈ T ( I ) β i l ( i ) and e := X j ∈ T ( J ) β j l ( j ) Remark 5.2. Since for the dimension vectors β ∈ N T we are mostly interestedin we have β q = 1 for all q ∈ T and since T is a tree, there exists only one stablerepresentation X up to isomorphism. In particular, we can assume that X α = 1for all α ∈ T . Moreover, we have M ˜Θ l − st β ( T ) = { pt } .Since, in general, every connected tree with n vertices has n − N ( k , k ) has P i =1 P w P l i j =1 k iw,j − T ( k , k )be the set of connected subtrees of N ( k , k ). For a tree T ∈ T ( k , k ) and asubset I ′ ( T ( I ) we define σ I ′ ( T ) = P j ∈ N I ′ l ( j ). Then ( T, β ) with β q = 1 for all q ∈ T is a localization data if and only if σ I ′ ( T ) > ed | I ′ | for all ∅ 6 = I ′ ( T ( I )where | I ′ | := P i ∈ I ′ l ( i ). Moreover since we have χ ( M ˜Θ l − st β ( T )) = 1 it suffices tocount such subtrees in order to calculate the Euler characteristic. More preciselydefining w ( T ) = (cid:26) σ I ′ ( T ) > ed | I ′ | for all ∅ 6 = I ′ ( T ( I )0 otherwise (cid:27) we obtain the following: Corollary 5.3. We have χ ( M Θ l − st( k ,k ) ( N )) = P T ∈ T ( k ,k ) w ( T ) . Notice that fixing a tree T ∈ T ( k , k ) with w ( T ) = 1 we can also forget aboutthe colouring (but fix the level structure), and ask for the number of differentembeddings into N ( k , k ).6. A connection between localization data and tropical curves In this section we connect a recursive construction of tropical curves to a similarconstruction of localization data. This gives a possible recipe to obtain a directcorrespondence between rational tropical curves and quiver localization data, assuggested by the equality of the respective counts Proposition 4.3. We work outthis correspondence in some examples in the following sections. In every case thisconstruction can be used to compute the number of tropical curves and, therefore,the Euler characteristic of the corresponding moduli spaces recursively.6.1. Recursive construction of curves. The aim of this section is to con-struct tropical curves recursively. Therefore we first show that by choosing thelines h ( E ij ) in a suitable (but generic) way we can remove the last edge E ,t ,effectively decomposing one of our tropical curves into smaller ones. Moreoverthis construction works in the other direction as well. In particular we show thatevery tropical curve is obtained by glueing smaller ones.In the following we denote the coordinates of a vector x ∈ R by ( x , x ).Let h : Γ → R be a connected parametrized rational tropical curve with un-bounded edges E ij of weights w ij and E out for 1 ≤ i ≤ ≤ j ≤ t i . Let h ( E ij ) be contained in the line e ij + R e i . For every unbounded edge E thereexists a unique vertex V such that E is adjacent to V . We denote this vertexby V ( E ). For every compact edge E there exist two vertices V ( E ) and V ( E )which are adjacent to E where we assume that h ( V ( E )) > h ( V ( E )) . For afixed tropical curve h : Γ → R we have h ( V ( E )) = h ( V ( E )) + λw Γ ( E ) m forsome λ ∈ R where m denotes the primitive vector emanating from h ( V ( E )) inthe direction of h ( V ( E )). The slope w Γ ( E ) m w Γ ( E ) m of h ( E ) is abbreviated to µ ( E ) inthe following. Note that it is important that the weight of E is taken into account.By the methods of [3], see also [4, Proposition 2.7], we have that N trop [ w , w ]does not depend on the (general) choice of the vectors e ij . So we always assumethat e ,j > e ,j − and e ,j > e ,j − . Let F , . . . , F n be the edges such that there exists a point ( x i , x i ) ∈ h ( F i )satisfying e ,t − < x i < e ,t and, moreover, h ( V ( E )) ≥ e ,t . Moreover, let h ( F i ) ⊂ R f ,i + f ,i =: f i and denote by s i,j ∈ R , 1 ≤ i < j ≤ n , the intersectionpoint of f i and f j if there exists one. Again, by the methods of [3] we may assumethat s i,j < e ,t (by moving e ,t ). This means that all intersection points of anytwo affine lines f i and f j lie on the left hand side of the line e ,t + R e . Inparticular we may assume that the edges F i are slope ordered, i.e. µ ( F i ) ≤ µ ( F j )for i < j .In the following, we say that a tropical curve satisfying these conditions is slopeordered. Then we have the following lemma: Lemma 6.1. We can choose the lines h ( E ij ) in such a way that every tropicalcurve is slope ordered. In the following we assume that we have chosen the lines in this way and we alsocall such an arrangement of lines slope ordered. Let µ ( F i ) = e i d i with e i , d i ∈ N and d i > 0. Clearly the vertex V ( E ,t ) must be adjacent to F and G := E ,t and induces an edge G with µ ( G ) = e + w ,t d such that e + w ,t d > e d . Withthis notation in place, the following lemma can be proved by induction: Lemma 6.2. Let h : Γ → R be a slope ordered tropical curve. Then there existedges G , . . . , G n and vertices V , . . . , V n such that V i is adjacent to F k , G k − and G k and such that µ ( G k ) = w ,t + P ki =1 e i P ki =1 d i > e k +1 d k +1 for k = 1 , . . . , n − . Moreover we have G n = E out . Remark 6.3. Note that for w ,t = 1 these conditions are part of the glueingconditions of [18, Section 4.3]. The only missing property is the fourth one.Fixing ( d, e ) ∈ N and w ∈ N we call a tuple ( d i , e i ) i =1 ,...,n of pairs of naturalnumbers satisfying w + n X i =1 e i = e, n X i =1 d i = d, e i d i ≤ e i +1 d i +1 and w + P ki =1 e i P ki =1 d i > e k +1 d k +1 (12)a w -admissible decomposition of ( d, e ). Obviously every slope ordered tropicalcurve defines a w t -admissible decomposition of ( d, e ).We call a set partition I • as introduced in Section 4 proper if all parts arenot empty. Fix a weight vector ( w , w ) with d = P t j =1 w j and e = P t j =1 w j satisfying w ij ≤ w i ( j +1) . Every w ,t -admissible decomposition of ( d, e ) definestwo ordered partitions of d and e − w ,t respectively. Then every tuple of setpartitions of { , . . . , t } and { , . . . , t − } respectively which is compatible with( d i , e i ) i =1 ,...,n defines n tuples of weight vectors ( w ( i ) , w ( i ) ) with d i = P j w ( i ) j and e i = P j w ( i ) j .Now let h i : Γ i → R , i = 1 , . . . , n , be tropical curves with unbounded edgescorresponding to the weights ( w ( i ) , w ( i ) ) and with outgoing edges E i, out . More-over let h ( E ,t ) ⊆ e ,t + R e be the embedding of E ,t . We may assume that for all intersection points s ij of the affine lines f i = R f ,i + f ,i containing h ( E i, out )we have s ij < e ,t . We may assume that f ,i ≥ 0. In particular, we have µ ( E i, out ) ≤ µ ( E i +1 , out ).Therefore, these curves recursively define a tropical curve h : Γ → R in thefollowing way: h ( E , out ) and h ( E ,t ) have a unique intersection point s . Thusto Γ we add a vertex V with h ( V ) = s and an unbounded edge F with adja-cent vertex V setting h ( F ) = { h ( V ) + R ≥ (( d , e ) + (0 , w ,t )) } . Additionallywe bind E , out by V and so we modify its image in an appropriate way. Ingeneral, since ( d i , e i ) i is a w t -admissible decomposition, there exists an inter-section point s j +1 of h ( F j ) and h ( E j +1 , out ). Thus we add a vertex V j +1 with h ( V j +1 ) = s j +1 and an unbounded edge F j +1 with adjacent vertex V j +1 setting h ( F j +1 ) = { h ( V j +1 ) + R ≥ ( P j +1 i =1 ( d i , e i ) + (0 , w ,t )) } . As above we bind E j +1 , out and F j by V j +1 and we again modify their images appropriately.Considering all the w t -admissible decompositions and all the sets of tropicalcurves (embedded in the chosen line arrangement) as above at once we can assumethat s ij < e ,t for all possible intersection points. So we have the following: Theorem 6.4. N trop ( w , w ) = X ( d i ,e i ) i X I • n Y i =1 N trop [ w ( i ) , w ( i ) ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n Y k =1 ( e k k − X i =1 d i − d k ( k − X i =1 e i + w ,t )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) where we first sum over all w t -admissible decompositions of ( d, e ) and then overall proper set partitions I • which are compatible with the partitions ( d, e − w ,t ) =( P ni =1 d i , P ni =1 e i ) .Proof. We just need to determine the multiplicities of the vertices where theoriginal curves are glued. For their multiplicities we getMult V k ( h ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) P k − i =1 d i d k P k − i =1 e i + w ,t e k (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) for k = 1 , . . . , n . (cid:3) Recursive construction of localization data. On the quiver side wehave a similar construction: let ( Q , β ) , . . . , ( Q n , β n ) be semistable consisting ofdisjoint subquivers of N and a dimension vector β i ∈ N ( Q i ) of type one (namely( β i ) q = 1 for all q ∈ ( Q i ) ), of dimension type ( d i , e i ), i.e. we have X q ∈Q i ( I ) l ( q ) = d i and X q ∈Q i ( J ) l ( q ) = e i , and M Θ l − sst β i ( Q i ) = ∅ . Moreover let the tuple ( d i , e i ) i =1 ,...,n be a w -admissible de-composition of ( d, e ) where e := w + P ni =1 e i and d := P ni =1 d i . Consider the tuple ( Q , β ) consisting of the quiver Q defined by the vertices Q = S nj =1 ( Q j ) ∪ { q } with l ( q ) = w and the arrows Q = S nj =1 ( Q j ) ∪ { α : i → q | i ∈ Q j ( I ) , j =1 , . . . , n } and the dimension vector β obtained by setting β q = 1. Lemma 6.5. Let ( d i , e i ) i be a w -admissible decomposition of ( d, e ) and let I ( { , . . . , n } . Then we have (1) P i ∈ I e i P i ∈ I d i ≤ e n d n (2) w + P i ∈ I e i P i ∈ I d i > w + P ni =1 e i P ni =1 d i Proof. Let I max ∈ I the largest number in I . We proceed by induction on n .If n ∈ I , the first inequality is equivalent to P i ∈ I \{ n } e i P i ∈ I \{ n } d i ≤ e n d n . By induction hypothesis we have P i ∈ I \{ n } e i P i ∈ I \{ n } d i ≤ e ( I \{ n } ) max d ( I \{ n } ) max ≤ e n − d n − ≤ e n d n . If n / ∈ I , we can apply the induction hypothesis.In order to prove the second inequality, we first assume that n / ∈ I . Then weproceed by induction on n . It is easy to check that w + P n − i =1 e i P n − i =1 d i > w + P ni =1 e i P ni =1 d i ⇔ w + P n − i =1 e i P n − i =1 d i > e n d n . Moreover by the induction hypothesis we have w + P i ∈ I e i P i ∈ I d i > w + P n − i =1 e i P n − i =1 d i . If n ∈ I , by the first statement we have P i ∈ I ′ e i P i ∈ I ′ d i ≤ e n d n where I ′ := { , . . . , n }\ I .Since the second inequality of the statement is equivalent to w + P i ∈ I e i P i ∈ I d i > P i ∈ I ′ e i P i ∈ I ′ d i , it suffices to show that w + P i ∈ I e i P i ∈ I d i > e n d n . Since this is equivalent to w + P i ∈ I \{ n } e i P i ∈ I \{ n } d i > e n d n , we can apply the induction hypothesis using w + P n − i =1 e i P n − i =1 d i > e n d n . (cid:3) Theorem 6.6. The tuple ( Q , β ) is stable. In particular, every stable torus fixedpoint of this quiver defines a stable torus fixed point of N of type ( d, e ) .Proof. Since we have β q = 1 for all q ∈ Q , we consider the representation X defined by X α = 1 for all α ∈ Q . Let I k ⊆ Q k ( I ) be arbitrary subsets with t k := P i ∈ I k l ( i ) and let s k := σ I k such that I k = Q ( I ) k for at least one k and I k = ∅ for at least one k . We have to show w + n X i =1 s i > w + P ni =1 e i P ni =1 d i ( n X i =1 t i ) . Since the tuples we started with are semistable, we have w + n X i =1 s i ≥ w + n X i =1 e i d i t i . Thus it suffices to show w > n X j =1 t j d j ( w + P ni =1 e i ) − e j ( P ni =1 d i ) d j ( P ni =1 d i ) . Therefore, if k j := d j ( w + P ni =1 e i ) − e j ( P ni =1 d i ) d j ( P ni =1 d i ) > , we can assume that t j = d j , and otherwise we can assume that t j = 0. Let I ⊆ { , . . . , n } such that j ∈ I ⇔ k j > 0. Then we have to show that w + P i ∈ I e i P i ∈ I d i > w + P ni =1 e i P ni =1 d i what follows by the preceding lemma. Note that if I = { , . . . , n } , i.e. k j > ≤ j ≤ n , from w = P nj =1 d j k j it follows that w > P nj =1 t j k j if t j < d j for atleast one j . (cid:3) Remark 6.7. • It would be interesting to know if every localization data can be obtainedby this construction. We conjecture that this is true, but it seems moredifficult to prove this than the tropical analogue. • The main goal we have in mind is to construct a direct correspondencebetween tropical curves and localization data. Given a tropical curveof slope ( d, e ), say with multiplicity m , there should be m localizationdata of dimension type ( d, e ) corresponding to this tropical curve. We seethe two recursive constructions which we have described as a step in thisdirection: they give us a way to glue smaller objects in order to build morecomplicated ones. Moreover on both sides we have the same numerical conditions. So starting with smaller objects for which a correspondenceis known this should give a correspondence between the glued objects. • In some cases such a correspondence is obtained immediately: assumethat we have decomposed ( d, e ) = ( d s , e s ) + ( d ′ , e ′ ) as in [18, Section 4.3].Then we have | d s e − de s | = 1. In particular, the preceding methods givea one-to-one correspondence in this case. Indeed, we can understand thevertex corresponding to the last leg as the glueing vertex. • Unfortunately, the construction does not always give a canonical corre-spondence. Consider the data11 mmmmm ( ( QQQQQ o o / / mmmmm ( ( QQQQQ fixed colouring. Glueing an additional sink, we get the data111 @ @ (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2) mmmmm ( ( QQQQQ Y Y o o / / mmmmm ( ( QQQQQ , 4) and one of slope (0 , • Notice that Corollary 6.4 also gives a recursive formula for the Euler char-acteristic of moduli spaces. Indeed, even for w -admissible decompositionsof ( d, e ) involving non-primitive vectors ( d i , e i ) one ends up with primitivevectors after finitely many steps.6.3. Examples and discussion. In this section we discuss two examples inwhich we obtain a direct correspondence between tropical curves and localizationdata by using the methods described above.6.3.1. The case (2 , n + 1) . We consider the example of 2 n + 1 points in theprojective plane, i.e. ( P , P ) = (2 , n +1 ). There exist two refined partitionswhich are the partition itself and ( k , k ) = (1 + 1 , n +1 ). In the first case, the only tree to consider is ji nnnnnnnnnnnnnn . . . . . . . . . i n +1 h h RRRRRRRRRRRRRR with l ( j ) = 2 and l ( i k ) = 1. In the second case, the only tree to consider is j j i > > }}}}}} . . . i n +1 b b EEEEEEE < < yyyyyyy . . . i n +1 c c FFFFFFF Now it is easy to check that we have 2 n +1 different embeddings (or colourings)in the first case and (cid:18) n + 1 n (cid:19)(cid:18) n + 1 n (cid:19) different embeddings in the second case. Thus by the MPS formula for the Eulercharacteristic we get: χ ( M Θ − st (2 , n +1 )) = 12 (cid:18) n + 1 n (cid:19)(cid:18) n + 1 n (cid:19) − 14 2 n +1 . (13)Following the construction of the last section, we have to decompose the vector(2 , n + 1) into a 1-admissible tuple ( d i , e i ). The only two possibilities are( d , e ) = ( d , e ) = (1 , n ) and ( d , e ) = (2 , n ) , and, moreover, the only 1-admissible decomposition of (2 , n ) is ( d , e ) = (2 , n − , n ) and glue them in i n +1 . In thesecond case assume that we have already constructed the curves correspondingto (2 , n − , n + 1) fromsuch a curve is to glue twice a curve of slope (0 , 1) to it. If m is the multiplicityof the tropical curve of slope (2 , n − m . On the quiver side this means that we have to construct four localizationdata of type (2 , n + 1) from every localization data of type (2 , n − j j i > > }}}}}} . . . i n ` ` AAAAAA > > }}}}}} . . . i n − c c FFFFFFF Considering the construction of the last section we can construct a stable dataof type (2 , n ) starting with this one. This leads to two semistable tuples bydeleting one of the two new arrows. In short, we just glue the vertex i n to one of the sinks. By the last section we now have to consider the following tuple i n +1 FFFFFFF { { xxxxxxx j j i > > }}}}}} . . . i n c c FFFFFFFF ; ; xxxxxxxx . . . i n − c c FFFFFFF i n i i SSSSSSSSSSSSSSSS But now it is easy to check that we have two possibilities to obtain a localizationdata from this. On the curve side we consider a line arrangement of the followingshape: . . . . . .with 2 n + 1 vertical legs. In order to determine the corresponding tropical curves,we first consider the tropical curve of weight one for the partition (1 , n ) (herefor n = 3), i.e.: (cid:0)(cid:0)✁✁✁✂✂✂✂✂✍ For general n we have (cid:0) nn (cid:1) possibilities to embed the curves corresponding to thepartition (1 , n ) into the upper row of the line arrangement above and anothercurve of the same slope into the lower row. Then we can glue these two curvesas described in the last section. For the other refined partition which is thepartition itself we obviously get one curve of weight 2 n +1 . To sum up, we get N trop (2 , n +1 ) = ( (cid:0) nn (cid:1) + 4 (cid:0) nn − (cid:1)(cid:0) n − n − (cid:1) ) − n +1 which is easily seen to be thesame as the expression (13). For n = 1 we get the following localization data andthe following curves of multiplicity four, one and one respectively: i i j j j (cid:0)(cid:0)(cid:0) (cid:0)(cid:0)✁✁✁ ✡✡✡✡✡✣ i k ✟✟✟✟✟✟✯❍❍❍❍❍❍❥ i k ✟✟✟✟✟✟✯❍❍❍❍❍❍❥ j l j l j l with l , l ∈ { , } and k , k ∈ { , } and l = 3 which are four localizationdata. For the curves i i j j j (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✁✁✁✡✡✡✣ i i j j j (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)✁✁✁✁✁✁✁✁✁✡✡✡✣ we get the same quiver coloured by k = 1 , k = 2 , l = 2 , l = 3 , l = 1and k = 1 , k = 2 , l = 1 , l = 3 , l = 2 respectively.6.3.2. The case ( d, d + 1) . We consider the dimension vector ( d, d + 1) concen-trating on the trivial refinement (1 d , d +1 ). The 1-admissible decompositions of( d, d + 1) are given by ( d, d + 1) = (0 , 1) + P ni =1 ( d i , d i ) for some n ≤ d . More-over every slope-ordered tropical curve of slope ( d, d ) is obtained by a tropicalcurve of slope ( d − , d ) glued with one of slope (1 , d i − , d i ) we have to glue them in a certainway in order to get new tropical curves/localization data. If the multiplicities ofthese tropical curves are m , . . . , m n the multiplicity of the new curve is easilydetermined to be n Y i =1 m i d i . On the quiver side this means that we have to construct Π ni =1 d i new localizationdata from those of type ( d i − , d i ) i . By the results of [18, Section 6.2] we know that every source of a localization data of dimension type ( d, d + 1) of type onehas exactly two neighbours and, therefore, is obtained by glueing the followingdata 11 mmmmm ( ( QQQQQ n -tuple of localization data ( Q i , β i ) of type( d i − , d i ) with a fixed embedding into N . Let R i ⊂ Q i ( I ) × Q i ( J ) be the arrowsof Q i . By [18, Section 6.2] it is known that every connected subdata of dimensiontype ( d i , d i +1) is a localization data of this dimension type. So we may restrict tothe case n = 1. We are interested in certain semistable subtuples of type ( d i , d i )such that every source has at most two neighbours. Fixed such a subtuple thereis exactly one possibility to glue an additional sink in order to get a localizationdata of type ( d i , d i + 1) which is a subdata of the one constructed in Theorem 6.6.We proceed as follows: Let Q ′ ( I ) := Q ( I ) ∪ { i d } and Q ′ ( J ) = Q ( J ). Now thereare several possibilities for the arrows. Initially, we consider R ′ := R ∪ { ( i d , j ) } for some j ∈ Q ′ ( J ). Note that this gives us d choices. Secondly, we consider thearrows given by R ′ := ( R ∪ { ( i d , j ) , ( i d , j ) } ) \{ ( i, j k ) } for some j , j ∈ Q ′ ( J ) with j = j , i ∈ N j k (with i = i d ) for k ∈ { , } . Thisgives 2 (cid:0) d (cid:1) choices. Note that d + 2 (cid:0) d (cid:1) = d . Theorem 6.8. By this construction we get Q ni =1 d i localization data of type ( d i , d i + 1) starting with localization data of type ( d i − , d i ) for i = 1 , . . . , n .Moreover every localization data is obtained in this way.Proof. Consider a localization data of type ( d, d + 1). By deleting the vertex j d +1 (including the corresponding arrows) we get n semistable subdata of type( d i , d i ) for i = 1 , . . . , n . Let q ( i ) max ∈ Q i ( I ) be the source with the maximalindex. If | N q ( i ) max | = 1 we also delete this vertex and get a localization data oftype ( d i − , d i ). If | N q ( i ) max | = 2 there exists exactly one source q ( i ) ∈ Q i ( I )such that | N q ( i ) | = 1. After deleting q ( i ) max , there exists one possibility to obtaina localization data by adding an extra arrow ( q ( i ) , j ) where j ∈ N q ( i ) max . 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