We study Euler characteristics of moduli spaces of stable representations of
m-Kronecker quivers for m>>0.Comment: submitted versio

Okada, So

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So Okada

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On Euler characteristics for large Kronecker quivers

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C. R. Acad. Sci. Paris, Ser. I 350 (2012) 273–276Contents lists available at SciVerse ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comAlgebraic GeometryOn Euler characteristics for large Kronecker quiversSur la caractéristique d’Euler de l’espace des représentations stables d’un grand carquoisde KroneckerSo OkadaResearch Institute for Mathematical Sciences, Kyoto University, 606-8502, Japana r t i c l e i n f o a b s t r a c tArticle history:Received 18 December 2011Accepted after revision 20 February 2012Available online 2 March 2012Presented by the Editorial BoardWe study Euler characteristics of moduli spaces of stable representations of m-Kroneckerquivers for m  0. In particular, we study an asymptotic log formula of Euler characteristicsand a normalized asymptotic log formula of Euler characteristic, motivated by so-calledDouglas conjecture.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous étudions la caractéristique d’Euler des espaces de modules de représentations stablesdes m-carquois de Kronecker pour m grand. En particulier, nous étudions une formulelog asymptotique pour la caractéristique d’Euler et une formule asymptotique normalisée,motivées par la conjecture de Douglas.© 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionFor each positive integer m, let K m be the m-Kronecker quiver which consists of two vertices and m arrows from oneto the other. For generic non-trivial stability conditions [1] on the category of representations of K m and moduli spaces ofstable representations M(K m(a,b)) of coprime dimension vectors (a,b) [5], we study Euler characteristics χ(K m(a,b)).We give some more details in the later section and we go on as follows. Notice that for the Euler form 〈·,·〉 and asymplectic form {·,·}, which is an anti-symmetrization of the Euler form, we may take a non-trivial stability conditionon the category of representations of K m such that for representations E, F of K m and the slope function μ, we haveμ(E) > μ(F ) if and only if {E, F } > 0.For objects to study in terms of wall-crossings, stability conditions such that the positivity of the difference of slopescoincides with that of symplectic forms on the Grothendieck group have been commonly called Denef’s stability conditionsin physics [2]. We employ these special stability conditions and the terminology.Euler characteristics χ(K m(a,b)) have been studied extensively. In particular, formulas of Kontsevich–Soibelman andReineke [6,10,12] have been known. In this article, we would like to study quantitative questions for m  0.To analyze further, for each coprime a,b and m > 0, let us define the bipartite quiver Q m(a,b) which consists of asource vertices and b terminal vertices with m arrows from each source vertex to each terminal vertex. On representationsof Q m(a,b), we have Denef’s stability conditions (see Section 2).E-mail address: okada@kurims.kyoto-u.ac.jp.1631-073X/$ – see front matter © 2012 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2012.02.008274 S. Okada / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 273–276We denote M(Q m(a,b)) to be the moduli space of stable representations of dimension vectors being one on each vertexof Q m(a,b) and χ(Q m(a,b)) to be the corresponding Euler characteristic. We have the following:Theorem 1. For each coprime a,b, and m  0, we haveχ(Q 1(a,b)) ∼ a!b!ma+b−1χ(K m(a,b)).We would like to mention that in Theorem 1, Euler characteristics in the left-hand and right-hand sides are discussed interms of blackhole counting in supergravity [7] and Witten index in superstring theory [3] (see also [15]).Key tools to obtain Theorem 1 are the recently obtained formula in Theorem 3 on χ(K m(a,b)) by Manschot, Piolineand Sen [7] (MPS formula for short, see also [8,9,14])1 and our Lemma 2.1. We realize that by taking m to be a variable,MPS formula provides the polynomial expansion of χ(K m(a,b)) whose coefficients involve Euler characteristics of bipartitequivers such as Q 1(a,b). Indeed, we are dealing with nothing but the first-order approximation of χ(K m(a,b)) for m  0.By Theorem 1, to compute χ(Q 1(a,b)), we can take the advantage of χ(K m(a,b)). Since the explicit formula ofχ(K m(a,a + 1)) has been provided in [16], we can obtain χ(Q 1(a,a + 1)) as in Corollary 5. Let us mention that for thecases of a = 1 and arbitrary b, we see that Stirling formula explains Theorem 1.Douglas has conjectured the following [4,11,16]. For coprime a,b  0 such that ba ∼ r and each m, we havethat ln(χ(Km(a,b)))a is a continuous function of r. In particular, the conjecture gives an asymptotic closed formula ofln(χ(K m(a,b))). Allowing m to be large, we have the following:Corollary 2. For each coprime a,b and m  0, we haveln(χ(K m(a,b))) ∼ (a + b − 1) ln(m).In particular, for a,b  0 such that ba ∼ r and large enough m depending on a,b, we haveln(χ(K m(a,b)))a∼ (1 + r) ln(m).2. ProofsLet us expand and introduce notions. For each a, let a denote a partition of a such that for non-negative integers al ofl  1, we have∑l lal = a. We put Sa =∑al for our convenience. When a1 = a, we simply write a for a. For a quiver Qand representations E, F of Q , on the Grothendieck group of the category of representations of Q , let 〈E, F 〉Q be the Eulerform and {E, F }Q be the symplectic form 〈F , E〉Q − 〈E, F 〉Q . For a dimension vector d, we call a partition d1, . . . ,ds of dsuch that∑sp=1 dp = d and {∑bp=1 dp,d}Q > 0 for each b = 1, . . . , s − 1 to be admissible.For each m > 0 and partitions a,b of a and b, we define the bipartite quiver Q m(a,b) as follows. It consists of Sa sourcevertices such that for each l, we have al vertices v; for our convenience, we say al is the label of v and we put w(v) = l. Itconsists of Sb terminal vertices with labels and w(·) being defined by the same manner. We put mw(v)w(v ′) arrows fromeach source vertex v to each terminal vertex v ′ .Let us explain Denef’s stability conditions in use. For the m-Kronecker quiver K m , the source vertex (1,0), and theterminal vertex (0,1), the slope function μ satisfies μ(1,0) > μ(0,1). For Q m(a,b) and vertices v and v ′ with the labelsbeing al and bl′ , central chargesZ(v)w(v) andZ(v ′)w(v ′) coincide with those of the vertices (1,0) and (0,1).We write (a,b) for the dimension vector which has one on each vertex of the quiver Q m(a,b). We let M(Q m(a,b)) tobe the moduli space of stable representations of the dimension vector (a,b) of Q m(a,b). We denote P (Q m(a,b), y) to bethe Poincaré polynomial and we put χ(Q m(a,b)) = P (Q m(a,b),1). For the m-Kronecker quiver K m , we have the followingMPS formula by specializing the Poincaré polynomial formula in [7, Appendix D]:Theorem 3 (MPS formula). For each coprime a,b and m > 0, we haveχ(K m(a,b)) = ∑a,bχ(Q m(a,b)) · ∏l1al!(−1)al(l−1)l2al·∏l1bl!(−1)bl(l−1)l2bl.Notice that M(Q m(a,b)) is a non-trivial smooth projective variety, since we have stable representations including oneswith invertible maps on every arrow. We have the following:1 In [7], they give their formula in terms of Poincaré polynomials for Denef’s stabilities on quivers without oriented loops. We use its Euler characteristicversion on Kronecker quivers. In [13], their formula has been motivically generalized and, for complete bipartite quivers and Euler characteristics, identifiedwith a degeneration formula of Gromow–Witten theory.S. Okada / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 273–276 275Lemma 2.1.χ(Q m(a,b)) = mSa+Sb−1χ(Q 1(a,b)).Proof. We consider the Poincaré polynomial P (Q m(a,b), y) with Reineke’s formula [10, Corollary 6.8]. For the dimen-sion vector (a,b), we take an admissible partition d1, . . . ,ds and (−1)s−1 y2∑kl∑v→v′ dlv dkv′ . We notice that {·,·}Q m(a,b) =m{·,·}Q 1(a,b) . The set of admissible partitions is invariant under choices of m. For each admissible partition, the power of yabove is the m times of that for P (Q 1(a,b), y). We have that P (Q 1(a,b), y) is a non-zero polynomial. Ignoring an overallfactor of a power of y and writing y2 as q for simplicity, for some non-trivial and non-negative integers αi and βi , we haveP (Q 1(a,b),q) = (q − 1)1−Sa−Sb (∑i0 αi(q − 1)Sa+Sb−1qβi ). For admissible partitions, the second factor is the sum of termsabove. So we have P (Q m(a,b),q) = (q − 1)1−Sa−Sb (∑i0 αi(qm − 1)Sa+Sb−1qmβi ). We give a proof of Theorem 1.Proof. By Lemma 2.1, χ(Q m(a,b)) carries the highest power of m among χ(Q m(a,b)) in Theorem 3. We give a proof of Corollary 2.Proof. When a + b = 1, M(K m(a,b)) is a point. For a + b = 1 and large enough m so that∣∣∣∣ ln(χ(Q 1(a,b))a!b! )(a + b − 1) ln(m)∣∣∣∣ 	 1,the first assertion follows. For the second assertion, with ai,bi,mi such thatbiai→ r, 1ai → 0, andln(χ(K mi (ai ,bi)))ln(mi)(ai+bi−1) → 1 fori → ∞, we use a standard argument. Let us compute χ(Q 1(a,a + 1)) as in the introduction. From [16], we recall the following:Theorem 4. (See [16, Theorem 6.6].)χ(K m(a,a + 1)) = m(a + 1)((m − 1)a + m)((m − 1)2a + (m − 1)ma).By Theorem 1, we have the following:Corollary 5.χ(Q 1(a,a + 1)) = limm→∞χ(K m(a,a + 1))a!(a + 1)!m2a= (a + 1)!(a + 1)−2+a.Remark 1. With the formula of χ(K m(2,2a + 1)) in [10], Manschot has provedχ(Q 1(2,2a + 1)) = (2a + 1)!a!2 .This sequence and the one in Corollary 5 coincide with A002457 and A066319 at oeis.org.AcknowledgementsThe author is supported by JSPS Grant-in-Aid #23740012 and GCOE Program at Kyoto University. The author thanksRIMS and IHES for their excellent research environment. He thanks Professors Dimofte, Hosomichi, I. Kimura, Kontsevich,Manschot, Nakajima, Pioline, Stoppa, Weist, and Zagier for their comments, discussions, or seminar related to the paper.References[1] T. Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2) 166 (2) (2007) 317–345.[2] F. Denef, Supergravity flows and D-brane stability, J. High Energy Phys. 0008 (2000) 50, 40 pp.[3] F. Denef, Quantum quivers and Hall/hole halos, J. High Energy Phys. 0210 (2002) 023, 42 pp.[4] M. Gross, R. Pandharipande, Quivers, curves, and the tropical vertex, Port. Math. 67 (2) (2010) 211–259.[5] A.D. King, Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (180) (1994) 515–530.[6] M. Kontsevich, Y. Soibelman, Stability structures, motivic Donaldson–Thomas invariants and cluster transformations, arXiv:0811.2435.[7] J. Manschot, B. Pioline, A. Sen, Wall-crossing from Boltzmann black hole halos, arXiv:1011.1258.276 S. Okada / C. R. Acad. Sci. Paris, Ser. I 350 (2012) 273–276[8] J. Manschot, B. Pioline, A. Sen, A fixed point formula for the index of multi-centered N = 2 black holes, arXiv:1103.1887.[9] B. Pioline, Four ways across the wall, arXiv:1103.0261.[10] M. Reineke, The Harder–Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2) (2003) 349–368.[11] M. Reineke, Moduli of representations of quivers, in: Trends in Representation Theory of Algebras and Related Topics, in: EMS Ser. Congr. Rep., Eur.Math. Soc., Zürich, 2008, pp. 589–637.[12] M. Reineke, Poisson automorphisms and quiver moduli, J. Inst. Math. Jussieu 9 (3) (2010) 653–667.[13] M. Reineke, J. Stoppa, T. Weist, MPS degeneration formula for quiver moduli and refined GW/Kronecker correspondence, arXiv:1110.4847.[14] A. Sen, Equivalence of three wall crossing formulae, arXiv:1112.2515.[15] J. Stoppa, Universal covers and the GW/Kronecker correspondence, Commun. Number Theory Phys. 5 (2) (2011) 353–396.[16] T. Weist, Localization in quiver moduli spaces, arXiv:0903.5442.

On Euler characteristics for large Kronecker quivers

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