Crystal melting on toric surfaces
LLPTENS 09/38IHES/P/09/53HWM–09–10EMPG–09–18
Crystal melting on toric surfaces
Michele Cirafici a , Amir-Kian Kashani-Poor b , Richard J. Szabo c a Centro de An´alise Matem´atica, Geometria e Sistemas DinˆamicosDepartamento de Matem´aticaInstituto Superior T´ecnicoAv. Rovisco Pais, 1049-001 Lisboa, Portugal b Laboratoire de Physique Th´eorique de l’ ´Ecole Normale Sup´erieure,24 rue Lhomond, 75231 Paris, France c Department of MathematicsHeriot-Watt UniversityColin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, U.K.and Maxwell Institute for Mathematical Sciences, Edinburgh, U.K.
Abstract
We study the relationship between the statistical mechanics of crystal meltingand instanton counting in N = 4 supersymmetric U (1) gauge theory on toric sur-faces. We argue that, in contrast to their six-dimensional cousins, the two problemsare related but not identical. We develop a vertex formalism for the crystal par-tition function, which calculates a generating function for the dimension 0 and 1subschemes of the toric surface, and describe the modifications required to obtainthe corresponding gauge theory partition function. a r X i v : . [ h e p - t h ] D ec Introduction
The problem of computing instanton contributions to the partition functions of four-dimensional supersymmetric gauge theories has a multitude of applications in field theory,string theory, and black hole physics. The algebraic geometry of the corresponding modulispaces has spawned much interest in mathematics. In this paper, we will consider anew related counting problem in the maximally supersymmetric case. Our approach ismotivated by the six-dimensional cousin of the four-dimensional problem.The vertex formalism [1, 2] allows for the computation of the topological string partitionfunction on arbitrary toric (hence non-compact) Calabi-Yau threefolds. In ref. [3], thisformalism was recast into an intuitive counting prescription for plane partitions (three-dimensional Young diagrams), and shown to have the interpretation of a simple statisticalmechanics model of crystal melting. This reformulation was taken as the starting pointfor relating topological string theory on toric Calabi-Yau manifolds to a six-dimensionalmaximally supersymmetric U (1) gauge theory in ref. [4] (for a recent discussion of in-stanton partition functions in various dimensions, see ref. [5]). In as far as the partitionfunction of the gauge theory is the generating function for Donaldson-Thomas invariants,this relationship was proven in refs. [6, 7].One now observes that in all examples which have been computed thus far, the U (1)partition function of N = 4 Vafa-Witten twisted gauge theory in four dimensions [8] hasas prefactor the Euler function ˆ η ( q ) = q − / η ( q ), the generating function for ordinarypartitions (Young tableaux), raised to the power of the Euler characteristic of the under-lying four-manifold. This suggests that the N = 4 theory might be the four-dimensionalanalogue of the six-dimensional gauge theory underlying Donaldson-Thomas invariants,with its partition function computable from a melting crystal prescription.Instantons of four-dimensional U ( N ) gauge theories on toric surfaces (not necessarilyCalabi-Yau) have been studied in refs. [9, 10, 11, 12, 13, 14, 15, 16]. U (1) instantonsarise as building blocks in these works. In ref. [17], Nakajima studied U ( N ) instantonson ALE spaces and showed (see Theorem 3.2 of that paper) that at the fixed pointsof an appropriately lifted toric action, they decompose into a sum of U (1) instantons(see also refs. [14, 16] for related results). Based on this result, the authors of ref. [10]employed a localization calculation on the explicit ADHM instanton moduli space toargue that in the case of N = 4 Vafa-Witten twisted gauge theories on ALE spaces, the U ( N ) partition function simply factorizes into N powers of the U (1) partition function– the rigorous argument for the factorization of the combinatorial problem was providedin ref. [18]. They also indicate a heuristic argument as to why this factorization shouldhold in general, which is supported by calculations in a two-dimensional reduction ofthe four-dimensional gauge theory on Hirzebruch-Jung spaces [11, 12]. In this paper, the U (1) case will be the focus of attention.In the following, we shall say that an enumerative problem has a melting crystal de-scription if it can be recast in terms of a box counting prescription, analogous to thatof ref. [3]. We shall see that in passing from the study of the Hilbert scheme of points,which features prominently in instanton calculations, to the Hilbert scheme of curves, we This is in contrast to the six-dimensional case where factorization does not generically hold. Seeref. [19] for an explicit analysis of the Coulomb phase of the six-dimensional U ( N ) gauge theory and itsrelationship to U (1) instantons. N = 4 theory on Hirzebruch-Jung surfaces proposedin refs. [10, 11].The organisation of this paper is as follows. In Section 2, we define the enumerativeproblems which we will address, together with the underlying physical motivation, andintroduce the corresponding generating functions. We proceed to compute the weightsthat enter in the definition of these generating function in Section 3. In Section 4, weset up the vertex formalism to compute the crystal melting partition function, and workout the explicit examples of the complex projective plane and Hirzebruch-Jung surfaces.Finally, we describe how these partition functions must be modified to arrive at theinstanton partition function of gauge theory in Section 5. Four appendices at the endof the paper provide calculational details and background material. In Appendix A, wecompute the Euler characteristics of torus invariant subschemes of a toric surface directlyin ˇCech cohomology. We collect the facts we will need about toric surfaces in generaland Hirzebruch-Jung surfaces in particular in Appendix B. Appendix C contains a briefreview of characteristic classes of coherent sheaves. In Appendix D, we illustrate thefactorization of the Hilbert scheme of curves into a divisorial and a punctual part, aresult which plays a central role in the computations of this paper, for the example of theprojective plane.We have made an effort to include many intermediate steps and explanatory notes through-out, in the hope of rendering the exposition more accessible to the casual reader.Unless otherwise noted, all schemes are defined over the field C . In six dimensions, the counting of closed 0 and 1 dimensional subschemes of a projec-tive scheme X is closely related to a gauge theory problem on X . The crystal meltingprescription of ref. [3], in hindsight, is most intuitive in this setting, as the boxes out ofwhich the crystal is built correspond to sections of the structure sheaf O Y of the corre-sponding closed subscheme Y . In this section, we will explain the parametrization ofsubschemes and the gauge theory problem in turn. We then discuss why they coincide insix dimensions, and how they are related in four dimensions.
Grothendieck proved that the Hilbert functor of a projective scheme X is representable bya projective scheme called the Hilbert scheme Hilb XP ( t ) of X . The closed points of Hilb XP ( t ) correspond to closed subschemes Y of X with Hilbert polynomial P XY ( t ) = P ( t ). Recallthat upon fixing a very ample line bundle L , i.e. an embedding of X into projective space, This hindsight is based on the proof of the equivalence of the generating functions for Donaldson-Thomas and Gromov-Witten invariants on toric threefolds [6, 7]. P XY ( t ) ∈ Q [ t ] is defined by the function P XY ( t ) = χ (cid:0) O Y ⊗ L ⊗ t (cid:1) = (cid:88) i ≥ ( − i dim C H i ( X, O Y ⊗ L ⊗ t )for sufficiently large t . The constant term of the Hilbert polynomial is hence the Eulercharacteristic of the subscheme Y . When X = CP r is projective space, the leading termis given by P CP r Y ( t ) = dn ! t n + . . . , with n the dimension of Y and d its degree.One can further stratify the Hilbert scheme, e.g. by considering the Hilbert scheme ofcurves with fixed homology class β ∈ H ( X, Z ). In the case X = CP reviewed inAppendix D, the homology class is determined by the degree of the curve, and can hencebe read off from the Hilbert polynomial. For space curves, i.e. X = CP , this stratificationhas been studied in ref. [21]. For generic smooth projective surfaces, it is also used inref. [22].In general, Hilbert schemes are very complicated objects. The Hilbert scheme of pointshowever, for which the Hilbert polynomial is a constant n , is well understood. It hasalready made various appearances in the physics literature (see e.g. [23]). It is commonlydenoted X [ n ] := Hilb Xn . The Hilbert-Chow morphism X [ n ] −→ S n X , with S n X the n -th symmetric product of the scheme X , reflects the intuition that awayfrom the locus at which points approach each other, the moduli space of n points on X is simply given by n copies of X modulo permutations.The Hilbert scheme of curves generally exhibits much richer structure. On a smoothprojective surface X , this structure simplifies: codimension 1 subschemes factorize intodivisors, i.e. the multiples of integral codimension 1 subschemes, with multiplicity givenby their degree, and sums of free and embedded points (see ref. [24, p. 514], and alsoref. [25, Section 3]). We will denote the Hilbert scheme of subschemes Y of X with β = [ Y ] ∈ H ( X, Z ) and n = χ ( O Y ) as I n ( X, β ). Given a Y ∈ I n ( X, β ), β ∈ H ( X, Z )depends solely on the divisorial part D of Y . The contribution of D to n is given by n β = − β · ( β + K X ), with K X the canonical class of X (see Section 3.1). n − n β is dueto the free and embedded points Y of Y . Thus, I n ( X, β ) ∼ = I n β ( X, β ) × X [ n − n β ] . (2.1)To gain some intuition, we illustrate the factorization (2.1) of the Hilbert scheme at thelevel of the underlying topological spaces explicitly for the toric surface X = CP inAppendix D. Let us now consider the two factors contributing to eq. (2.1). The Hilbertscheme of points X [ n ] on a smooth projective surface X is non-singular and of dimension2 n [24, Theorem 2.4]. As for the moduli space of divisors I n β ( X, β ), on a smooth projectivesurface X of any dimension with H ( X, O X ) = 0, it is a projective space. It follows This definition makes sense because the right-hand side of this equation is a polynomial for sufficientlylarge t , see e.g. [20, Theorem 7.5]. We thank Richard Thomas for explanations concerning this point. I n ( X, β ) is non-singular [24, Corollary 2.7]. This will allow us to define generatingfunctions involving integrals over the fundamental classes [ I n ( X, β )]. Contrary to thesix-dimensional case, recourse to virtual classes, as introduced on projective surfaces inref. [22], will not be required.
In the physics literature, instantons are finite-action minima of a quantum theory. In thecase of pure Yang-Mills theory in four dimensions, they are given by (anti)self-dual connec-tions with appropriate boundary conditions at infinity. Vafa and Witten [8] demonstratedthat N = 4 supersymmetric Yang-Mills theory can be modified, following Witten’s pre-scription of topological twisting, such that its partition function computes, in favorablecircumstances, the Euler characteristic of the moduli space of instantons. This is guar-anteed if certain vanishing theorems for the geometry of the underlying four-manifold X and gauge bundle E → X are met [8, Section 2.4], e.g. if X is a compact K¨ahler manifoldof positive curvature and the structure group of E is SU (2). The twisted theory can beencountered in the wild (i.e. it can describe physical systems) in two situations; when themanifold X on which the gauge theory is defined is hyperk¨ahler, so that the twisted theorycoincides with a subsector of the physical theory, or in certain string theory embeddings,in which the twisting is induced by the background.For the partition function to be well-defined, a smooth, compact instanton moduli spaceis required. One path towards this end is a compactification given by embedding bundleswith irreducible anti-self-dual connection into the space of semistable sheaves. This isthe Gieseker compactification. A semistable sheaf F is in particular torsion-free. Awayfrom a codimension 2 locus, such sheaves are locally free, i.e. vector bundles. Hence,intuitively, in passing from bundles to torsion-free sheaves on surfaces, we are addingpointlike structures. The precise formulation of this statement is eq. (2.2) below.In this paper, we will solely consider the gauge group U (1). This simple case alreadymerits study for two main reasons:1. In six dimensions, this is the gauge group that has been related to the topologicalvertex formalism.2. In the case of ALE spaces X , ref. [10] performs the calculation of the partitionfunction Z ALEU ( N ) ( X ) based on the description of the U ( N ) instanton moduli spaceprovided by ref. [28], and demonstrates that one has the factorization relation Z ALEU ( N ) ( X ) = (cid:0) Z ALEU (1) ( X ) (cid:1) N . The factorization follows from a localization argument which reduces the calculationto the fixed points of an appropriately chosen toric action, and the demonstrationin ref. [17] that the instanton bundles on ALE spaces factorize at the fixed points. The inclusion of torsion-free sheaves in the study of gauge theory instantons was initiated in thephysics literature in ref. [26]. For a relation to non-commutative geometry, see ref. [27]. In fact, ref. [10] argues heuristically that this relation should hold in general, as the gauge symmetryof the U ( N ) theory can be broken to the maximal torus U (1) N by giving vevs to the scalars, and thepartition function is independent of these vevs. This argument is heuristic as the independence of thepartition function from the scalar vevs needs to be established carefully. Proposition 2.1. ([29, Proposition 2.2.6]) If H ( X, R ) = 0 and L is a line bundle overthe surface X , then for any 2-form ω representing c ( L ) there is a unique gauge equivalenceclass of connections with curvature − π i ω . Hence, in particular, in mapping line bundles to connections, we can choose the harmonicrepresentative of c ( L ) with respect to a chosen metric on X . The space of harmonic2-forms H ( X ) on X has a decomposition into subspaces H ± ( X ) of self-dual and anti-self-dual 2-forms, H ( X ) = H + ( X ) ⊕ H − ( X ) . If the intersection matrix on H ( X, R ) is well-defined (a condition we must impose fornon-compact X ) and negative definite, as will be the case for our main example, theHirzebruch-Jung surfaces, it follows that H + ( X ) = 0, and hence every holomorphic linebundle on such a surface admits an anti-self-dual connection. Since exact forms cannotbe anti-self-dual, this connection is unique.Note finally that even in the case of U (1) gauge theory, where the instanton moduli spaceis a lattice and in no need of regularization, we continue to consider the larger space oftorsion-free sheaves, in accord with points 1 and 2 above. On a toric Calabi-Yau threefold X , the problem that Maulik, Nekrasov, Okounkov, andPandharipande address in ref. [6] is the counting of subschemes Y of compact supportwith no component of codimension 1, and with holomorphic Euler characteristic n andsecond homology class β , n = χ ( O Y ) , β = [ Y ] ∈ H ( X, Z ) . They denote the corresponding moduli space of ideal sheaves by I n ( X, β ). They thencompute the generating function Z DT ( X ; q, w ) = (cid:88) β ∈ H ( X, Z ) (cid:88) n ∈ Z ˜ N n,β q n w β for the Donaldson-Thomas invariants˜ N n,β = (cid:90) [ I n ( X,β )] vir , the lengths of the 0 dimensional virtual fundamental cycles. On a projective scheme X , I n ( X, β ) is the moduli space parametrizing isomorphism classes of torsion-free sheavesof rank 1 with trivial determinant, where the singularity sets of the torsion-free sheavesconstitute the subschemes being counted. The argument is the following (see e.g. ref. [30]). A torsion-free sheaf T injects into its double dual.The determinant sheaf of a rank r torsion-free sheaf is defined to bedet T = (cid:0)(cid:86) r T (cid:1) ∗∗ . I n ( X, β )introduced above, now for X a surface. Alternatively, we can drop the trivial determinantcondition, thus enlarging the space to involve factors of line bundles L , given by the doubleduals of torsion-free sheaves T , such that T = L ⊗ I Y . (2.2)Again dimension 1 subschemes come into play, by their relation to effective divisors,however now only up to linear equivalence. Indeed, the decomposition (2.2) correspondsto the gauge theory problem outlined in Section 2.2.In the case of surfaces, we will hence be computing two different generating functions,defined as follows. Crystal melting. Z cm ( X ; q, w ) = (cid:88) β ∈ H ( X, Z ) (cid:88) n ∈ Z N cmn,β q n w β , (2.3)where n = χ ( O Y ), β = [ Y ] ∈ H ( X, Z ) and N cmn,β = (cid:90) I n ( X,β ) e (cid:0) T I n ( X, β ) (cid:1) . (2.4)Here and below, e ( E ) denotes the Euler class of the bundle E . As announced above, norecourse to virtual fundamental classes is taken in these definitions. This is a line bundle, as the double dual of a sheaf is reflexive (i.e. isomorphic to its double dual),and reflexive sheaves of rank 1 are locally free. Rank 1 torsion-free sheaves with trivial determinanthence possess an injection into the structure sheaf, i.e. they are ideal sheaves. Note that the distinctionbetween trivializable determinant and trivial determinant is important here. The singularity set S ( T )of a torsion-free sheaf T occurs in codimension 2 or higher, hence the corresponding subscheme has nocomponent in codimension 0 or 1. This argument can be summarized in the exact sequence of sheaves0 (cid:47) (cid:47) T (cid:47) (cid:47) T ∗∗∼ = (cid:15) (cid:15) (cid:47) (cid:47) S ( T ) (cid:47) (cid:47) O X where the vertical isomorphism is a fixed trivialization.Conversely, the ideal sheaf I Y of a proper closed subscheme Y of a noetherian integral scheme X isa coherent sheaf of rank 1. As a subsheaf of the structure sheaf O X , it is torsion-free by the integralityassumption on X . If Y has no support in codimension 1, then the determinant of I Y is trivial. auge theory. Z gt ( X ; q, w ) = (cid:88) β ∈ H ( X, Z ) (cid:88) n ∈ Q N gtn,β q − n w β , (2.5)where n = ch ( T ), β = ch ( T ) ∈ H ( X, Z ) and N gtn,β = (cid:90) M X ( β,n ) e (cid:0) T M X ( β, n ) (cid:1) , (2.6)with M X ( β, n ) the moduli space parametrizing isomorphism classes of torsion-free sheaves T of the given Chern character. Note that for torsion-free sheaves on non-compact spaces,the second Chern characteristic class n can be fractional; we therefore take the summationin (2.5) over Q , with N gtn,β = 0 away from a fixed common denominator of n , dependingon X .Explaining the various ingredients in these formulae, and interpreting and evaluating theintegrals (2.4) and (2.6), will occupy the rest of this paper. Following the six-dimensional discussion of ref. [6], we organize the crystal melting count-ing problem (2.3) on a toric surface X in terms of the holomorphic Euler characteristic χ ( O Y ) of the subschemes Y , which serves as a weight in the generating function, andtheir second homology class [ Y ] ∈ H ( X, Z ). For the gauge theory partition function,the weight originates in the action, which for anti-self-dual connections in four dimen-sions evaluates to the Chern class of the vector bundle (locally free sheaf) via Chern-Weiltheory. When we compactify the space of gauge connections by including pointlike in-stantons, it is natural to retain the Chern class as weight, as in (2.5). In this section, wewill compute these two weights and find that on Calabi-Yau surfaces, they are equal up tosign, at least for torically invariant Y (this qualification arises due to the non-compactnessof X , see point 3. in Subsection 3.2). Based on the factorization (2.1), we can calculate the Euler characteristic of Y by addingthe contributions from the divisorial and punctual parts, D and Y , of Y : the Eulercharacteristic of a 0 dimensional scheme enumerates its global sections, χ ( O Y ) = h ( Y , O Y ) , while the Euler characteristic of a divisor D on a surface X , as is reviewed in AppendixC, is given by χ ( O D ) = − D · ( D + K X ) , (3.1)with K X the canonical class of the surface. The right-hand side of eq. (3.1) clearly onlydepends on the class of the divisors up to linear equivalence. We will denote this class bysquare brackets [ − ] below. Altogether, χ ( O Y ) = − D · ( D + K X ) + h ( Y , O Y ) . (3.2)7he derivation of formula (3.1) in Appendix C relies on the application of the Hirzebruch-Riemann-Roch theorem, valid for X projective. When relaxing the compactness condi-tion, the terms on the left- and right-hand side of eq. (3.1) remain well-defined for D adivisor with compact support. By calculating the Euler characteristic directly in ˇCechcohomology in Appendix A, following ref. [6], we will see that at least in the case oftorically invariant subschemes, eq. (3.1) remains valid for D of compact support also ona non-compact toric surface X .To explicitly determine the Euler characteristic of a given divisor on a toric surface X , webenefit from the property that the Chow ring A ( X ) is generated by the classes of toricallyinvariant divisors D i (this is in fact true in arbitrary dimensions). We will enumerate the D i via i = 0 , . . . , n + 1, reserving the indices i = 0 and i = n + 1 for non-compact toricdivisors if these are present, otherwise setting D = D n +1 = 0. This notation allows forthe simultaneous treatment of compact and non-compact toric surfaces.Expanding [ D ] in classes [ D i ] generated by compactly supported divisors,[ D ] = n (cid:88) i =1 λ i [ D i ] , with λ i non-negative integers, and with the intersection matrix as given in eq. (B.2) ofAppendix B, the calculation of the Euler characteristic in Appendix A yields χ ( O D ) = n (cid:88) i =1 (cid:16) a i λ i ( λ i − λ i − λ i λ i +1 (cid:17) , (3.3)where the a i denote the negative self-intersection numbers a i = − D i , and λ n +1 = 0 isintroduced for notational convenience.To compare with eq. (3.1), note that the total Chern class of a non-singular toric variety X is given by c t ( X ) = n +1 (cid:89) i =0 (1 + [ D i ]) . It follows that K X = − n +1 (cid:88) i =0 [ D i ] , and hence χ ( O D ) = − n (cid:88) i,j =1 λ i ( λ j − D i · D j + 12 n (cid:88) i =1 λ i D i · ( D + D n +1 ) . (3.4)Borrowing the result (5.5) from Section 5, in which [ D ] and [ D n +1 ] are expressed as linearcombinations of [ D ] , . . . , [ D n ], we conclude that eq. (3.1), though derived for compact X , reproduces the ˇCech cohomology result (3.3) for toric divisors on non-compact X aswell. In the gauge theory setup, we wish to weigh all sheaves via the degree of their secondChern character. This is the natural extension of the notion of instanton number beyond8ocally free sheaves (i.e. vector bundles), see Appendix C. The Chern character satisfiesthe multiplicative property ch(
E ⊗ F ) = ch( E ) · ch( F ) . For the ideal sheaf I Z of a cycle Z , an application of the Grothendieck-Riemann-Rochtheorem yields ch( I Z ) = 1 − η Z , with η Z the class of the cycle (see e.g. ref. [31, p. 159]). Due to the relation ch ( L ) =rk ( L ) = 1 for a line bundle L , we havech ( L ⊗ I Z ) = ch ( L ) − η Z . We can therefore consider the two factors contributing to the weight of a given torsion-freesheaf separately.As we show in Lemma C.1 of Appendix C, deg( η Z ) = χ ( O Z ).For line bundles, we can work in the more familiar cohomological setup. On a compactsurface X , the instanton number evaluates to the intersection pairing of the correspondingdivisors, 12 (cid:90) X c (cid:0) O X ( D ) (cid:1) ∧ c (cid:0) O X ( D ) (cid:1) = 12 D · D . (3.5)For a divisor D with compact support, this relation in fact continues to hold on arbitrarytoric manifolds. The argument consists of three parts:1. Also on non-compact manifolds, the first Chern class c ( O X ( D )) of the line bundleassociated to a divisor and the closed Poincar´e dual η D of the support of the divisorare cohomologous, c ( O X ( D )) ∼ η D .
2. Since the support | D | is compact, we can replace the closed Poincar´e dual by thecompact Poincar´e dual. By localization of this class (in the sense of e.g. ref. [32]),we know that we can choose its support to be contained in an arbitrary open setcontaining | D | , such that the integral in eq. (3.5) is well-defined.3. We verify the relation (3.5) by performing the calculation on a toric compactification¯ X of X for which the compactification divisor does not intersect the image of thesupport of D . This property guarantees that the evaluations of the integrals over X and ¯ X coincide. An example of such a compactification, which always exists forsmooth toric surfaces, is given in Figure 1.The toric assumption can be replaced by the requirement that a compactification withthe requisite properties exists. Recall that the closed Poincar´e dual is integrated against forms of compact support, in contrast tothe compact Poincar´e dual which itself has compact support (see e.g. ref. [32, pp. 51–53]). Hence, theclosed Poincar´e dual η Σ of a cycle Σ satisfies (cid:90) Σ ψ = (cid:90) X η Σ ∧ ψ for any ψ ∈ H ∗ c ( X, R ) (defining both sides to vanish if the form degrees are not appropriate). Byrestricting the integration to the support of ψ , the argument establishing c ( O X ( D )) ∼ η D on compactmanifolds (see e.g. ref. [33, p. 143]) goes through in the non-compact case. A toric compactification of the resolved A singularity (in blue) to P × P blown up at twopoints. We thus arrive at ch (cid:0) O X ( D ) ⊗ I Z (cid:1) = D · D − χ ( O Z ) . For K X = 0 this agrees with eq. (3.2) up to sign, as announced at the beginning of thissection.It turns out that all non-compact divisors in the geometries that we will consider arelinearly equivalent to divisors with compact support. This thus allows us to computethe instanton number on the full Picard group. Note that the instanton numbers of linebundles associated to prime divisors with non-compact support are no longer necessarilyintegral. In the previous sections, we have introduced two closely related enumerative problems:counting subschemes vs. counting torsion-free sheaves on a toric surface. The line bundlefactor in eq. (2.2) requires invoking linear equivalence between toric divisors, and hencecannot be straightforwardly implemented within a vertex formalism that essentially onlyallows for nearest neighbor interactions (in terms of the 2-cones of the toric fan, or thevertices of the dual web diagram). The problem of counting, in an appropriate sense,the 0 and 1 dimensional compactly supported subschemes of a toric surface does howeverhave a melting crystal implementation, as we demonstrate in this section.The factorization (2.1) of the Hilbert scheme discussed in Section 2.1 results in a factor-ization of the partition function Z cm defined in eq. (2.3), Z cm ( X ; q, w ) = (cid:88) β ∈ H ( X, Z ) (cid:90) I nβ ( X,β ) e (cid:0) T I n β ( X, β ) (cid:1) q n β w β (cid:88) n ≥ (cid:90) X [ n ] e (cid:0) T X [ n ] (cid:1) q n . (4.1) We say ‘essentially’ as even the vertex formalism in six dimensions requires identifying homologouscurve classes by hand. In fact, when H ( X, O X ) = H ( X, O X ) = 0, Pic ( X ) ∼ = H ( X, Z ), so ‘linearlyequivalent’ and ‘homologous’ are the same notion on smooth compact toric surfaces (this remains true inany dimension: all cohomology classes on toric manifolds are analytic, hence of pure type ( p, p )). Evenso, in Z cm , all curves are counted and only the weight w β invokes homological equivalence, whereas in Z gt , the enumeration itself proceeds over equivalence classes.
10e begin by considering the contribution of the free and embedded 0 dimensional sub-schemes. The corresponding generating function for smooth projective surfaces has beencalculated by G¨ottsche in ref. [34]. We reproduce his result in the case of toric sur-faces via a localization calculation, which then permits an extension to the non-compactcase. This calculation will also be relevant in the gauge theory context of Section 5. Wenext apply a similar localization argument to the divisorial contribution to the partitionfunction. Finally, we show how the computation of Z cm can be encapsulated in a smallset of diagrammatic rules, and illustrate these in the examples of projective space andHirzebruch-Jung spaces. Localization arguments lie at the heart of the calculations inthis section. For X a smooth projective surface, the moduli space X [ n ] of 0 dimensional subschemes oflength n is non-singular of dimension 2 n [24, Theorem 2.4]. The generating function weare after was computed by G¨ottsche for smooth projective surfaces as [34] (cid:88) n ≥ χ (cid:0) X [ n ] (cid:1) q n = (cid:0) ˆ η ( q ) − (cid:1) χ ( X ) , (4.2)with ˆ η ( q ) = ∞ (cid:89) k =1 (cid:0) − q k (cid:1) the generating function of partitions. The computation of ref. [34] does not require a torusaction on the surface. If such an action exists, i.e. in the case of a toric surface X , we canreproduce G¨ottsche’s formula (4.2) by a localization computation (see also ref. [35] andref. [18, Appendix A]). As the Hilbert scheme X [ n ] is smooth, we can use conventionalAtiyah-Bott localization [36] in equivariant cohomology. We quote here the more generalintegration formula of Edidin and Graham [37] in equivariant Chow theory; this is theframework that generalizes beyond smooth varieties. Theorem 4.1. ([37, Proposition 5]) Let M be a smooth and complete scheme with theaction of a torus T = ( C ∗ ) k . Denote the fixed point locus of the T -action by M T , withembedding i : M T (cid:44) → M .
Let a ∈ A ( M ) descend from an equivariant class α ∈ A T ( M ) , i.e. a = i ∗ α . Then deg( a ) = (cid:88) F ⊂ M T π F ∗ (cid:16) i ∗ F αe T ( N F M ) (cid:17) , (4.3) where the sum runs through the connected components of the fixed point locus, N F M denotes the normal bundle over F in M , i F the embedding of F into M , and π F theprojection of F to a point. e T ( N F M ) in formula (4.3) denotes the T -equivariant Euler class, which is indeed invert-ible in A T ∗ ( F ) ⊗ Q [ t ,...,t k ] Q [ t , . . . , t k ] m , where Q [ t , . . . , t k ] m is the localization of the ring11 [ t , . . . , t k ] at the maximal ideal m spanned by the generators t , . . . , t k of the equivariantring of T .When the set of T -fixed points is a union of isolated points, the tangent bundle to each F is trivial, and thus e T ( N F M ) = e T ( T M | F ) . With a = e ( T M ), we have α = e T ( T M ), and hence i ∗ F α = e T ( T M | F ) cancels the de-nominator in the integrand on the right-hand side of the localization formula (4.3). Inthis case, the integral (4.3) can simply be evaluated by counting the fixed points of the T -action on M .Note that characteristic classes of equivariant vector bundles can be extended to equiv-ariant classes. Theorem 4.1 hence applies to our case of interest, with M = X [ n ] and a = e ( T X [ n ] ) the Euler class of the tangent bundle of the Hilbert scheme X [ n ] .The fixed points of the torus action on X [ n ] parametrize the torically invariant 0 dimen-sional subschemes of X with holomorphic Euler characteristic n . They can be enumeratedby considering an affine patch C [ x, y ] around each set theoretic fixed point, and ideals I ⊂ C [ x, y ] generated by monomials x m y n giving rise to non-reduced schemes with sup-port at this point. The ideals I are in one-to-one correspondence with Young tableaux π I , as illustrated in Figure 2. The boxes of the Young tableau map to a basis of globalFigure 2: The Young tableau encoding the ideal generated by the monomials x m i y n i , i = 1 , . . . ,
4, with( m i , n i ) the coordinates of the shaded boxes. sections of the corresponding 0 dimensional subscheme Y . Its Euler characteristic is henceequal to χ ( O Y ) = h ( Y, O Y ) = dim C (cid:0) C [ x, y ] (cid:14) I (cid:1) = | π I | , the number of boxes in the Young tableau π I . The contribution to the partition functionper geometric fixed point is hence (cid:88) π q | π | = ˆ η ( q ) − . The toric fixed points correspond to the maximal cones of the toric fan of X . Since theEuler characteristic χ ( X ) of a toric manifold X is given by the number of maximal conesof X , this reproduces the formula (4.2).The application of standard theorems is complicated when the surface X is non-compact.We will proceed by applying the localization formula to a toric compactification ¯ X of X ,12nd then restrict to the fixed points lying in X . This procedure is clearly independent ofthe choice of compactification. We now want to apply Theorem 4.1 to the integral in eq. (4.1) over [ I n β ( X, β )]. For X asmooth projective surface, this class exists as the corresponding Hilbert scheme of curvesis smooth, as argued in Section 2.1. For X non-compact, we will again consider a toriccompactification ¯ X of X , as illustrated in Figure 1. This compactification is obtained bygluing in a set of torically invariant divisors which have vanishing intersection with thecompactly supported divisors of X . For β the class of such a divisor, it follows that I n β ( X, β ) ∼ = I n β ( ¯ X, β ) , as for D such that [ D ] = β , all divisors linearly equivalent to D will lie within X . Weconclude that for β the class of a compactly supported divisor, I n β ( X, β ) is smooth on asmooth quasi-projective toric surface as well.
Above, we considered ideal sheaves corresponding to 0 dimensional subschemes. Thecentral property in that analysis, that ideal sheaves invariant under the torus action aremonomial, i.e. locally generated by monomials, holds for ideal sheaves of any dimension.We will describe such an ideal sheaf I by specifying it locally on the torically invariantopen sets U i of the surface X , I i = I ( U i ) ⊂ C [ x, y ], such that restrictions to overlapscoincide. The monomial ideals I i are in a one-to-one relation to Young tableaux which indistinction to the 0 dimensional case may be infinite, i.e. the generators do not necessarilyinclude monomials of the form x m or y n .The factorization (2.1) of the Hilbert scheme into a divisorial and a punctual part isimmediate when restricting to the toric fixed points: the possible associated primes to I i are ( x ), ( y ), and ( x, y ) (see e.g. ref. [38] for an explanation of this notion), the latterimplying the existence of an embedded point. It is easy to see that all Young diagramsother than hook diagrams correspond to closed subschemes with embedded points. Thedecomposition { infinite Young tableau } ←→ (cid:0) N ∪ { } (cid:1) × { finite Young tableau } illustrated in Figure 3 hence corresponds to the decomposition of the fixed point into aneffective divisor, and free and embedded point contributions (free in case λ i = λ i +1 = 0).We now consider the two factors contributing to Z cm as given in eq. (4.1) in turn. Points.
The torically invariant ideal sheaves of points were discussed in Section 4.1.They are in one-to-one correspondence with tuples of finite Young tableaux π i , one tableauassigned to each toric fixed point of the surface, which we recall correspond to 2-conesof the toric fan. The Euler characteristic of the corresponding subscheme Y is givenby χ ( O Y ) = (cid:88) | π i | . Decomposition of a subscheme into a reduced and a 0 dimensional component.
Below we will identify toric fixed points by the two bounding 1-cones, and whence usethe notation π i,i +1 for the corresponding Young tableau, as e.g. in Figure 3. Divisors.
As reviewed in Appendix B, the torically invariant divisors are in one-to-one correspondence with the 1-cones of the toric fan. Labeling the compactly supportedtorically invariant divisors as D i , i = 1 , . . . , n (i.e. disregarding the two outermost 1-conesin the case of non-compact surfaces), a general effective divisor D kept fixed by the toricaction is parametrized by n non-negative integers λ i , D = n (cid:88) i =1 λ i D i . The Euler characteristic of D is then computed via (3.4), χ ( O D ) = n (cid:88) i =1 (cid:16) a i λ i ( λ i − λ i − λ i λ i +1 (cid:17) . The self-intersection numbers are here denoted D i = − a i . We have furthermore set λ n +1 = λ in the compact case and λ n +1 = 0 in the non-compact case. We discusshow to determine the intersection matrix of the compactly supported prime divisors inAppendix B. To emphasize the similarity with melting crystal combinatorics in six dimensions [3], wecan express the Euler characteristic χ ( O Y ) = χ ( O Y ) + χ ( O D ) in terms of the infiniteYoung tableaux ˜ π i,i +1 as χ ( O Y ) = n (cid:88) i =0 | ˜ π i,i +1 | + n (cid:88) i =1 (cid:16) a i λ i ( λ i − λ i (cid:17) , where the box count of the infinite Young tableaux is defined as (see Figure 3) | ˜ π i,i +1 | := (cid:16) (cid:88) ( I,J ) ∈ ˜ π i,i +1 ∩ [0 , ,...,N ] (cid:17) − ( N + 1) λ i − ( N + 1) λ i +1 , N (cid:29) (cid:16) (cid:88) ( I,J ) ∈ π i,i +1 (cid:17) − λ i λ i +1 . π i,i +1 and λ i . Note that for the Calabi-Yau case, the self-intersection numbers are all given by a i = 2 and the Euler characteristic simplifies to χ ( O Y ) = n (cid:88) i =0 | ˜ π i,i +1 | + n (cid:88) i =1 λ i . We can now easily summarize the computation of the partition function Z cm ( X ; q, w ) interms of a simple set of vertex rules:1. Draw the dual web diagram of the toric fan. 2-cones are dual to vertices, and 1-conesare dual to legs.2. Each vertex i carries two positive integer labels λ i and λ i +1 (“one-dimensional Youngtableaux”), one assigned to each emanating leg, and contributes a vertex factor V λ i ,λ i +1 ( q ) = 1ˆ η ( q ) q − λ i λ i +1 to the partition function.3. Vertices are glued along legs carrying the same integer label λ i with a gluing factor G λ i ( q, w i ) = q a i λi ( λi − + λ i w λ i i , where the self-intersection numbers − a i are determined graphically as describedbelow. w i labels the homology class of the curve corresponding to the leg alongwhich the vertices are glued.4. Multiplying the vertex and gluing factors together, summing the λ i on internal legsover all non-negative integers while setting those on external legs to zero then yieldsthe melting crystal partition function Z cm ( X ; q, w ).We can determine the self-intersection numbers − a i graphically as follows. Recall thatthey are given by the relation a i v i = v i − + v i +1 between the generator v i of the 1-cone associated to D i and those of the two neighboring1-cones, see Figure 4. Note that since the surface X is non-singular by assumption, onehas v i − × v i = v i × v i +1 = 1 , where the × product computes the volume of the cell spanned by the generators. Hence a i = v i − × v i +1 . An example of a curve with self-intersection number − C at the origin). On the right-hand side, we have indicated the dual web diagram with the analogueof the framing vectors of the six-dimensional vertex formalism [1, 2]. The toric fan and web diagram of the projective space CP are depicted in Figure 5. Theself-intersection numbers − a i of the torically invariant divisors are a i = −
1. The partitionfunction is thus Z cm ( CP ; q, w ) = ∞ (cid:88) λ ,λ ,λ =0 η ( q ) q − λ λ q − λ + λ w λ η ( q ) q − λ λ q − λ + λ w λ × η ( q ) q − λ λ q − λ + λ w λ = 1ˆ η ( q ) ∞ (cid:88) λ ,λ ,λ =0 q − ( λ + λ + λ ) + ( λ + λ + λ ) w λ + λ + λ , with w labeling the hyperplane class.Figure 5: The toric fan for CP , and the corresponding web diagram, with the legs of the verticeslabelled. As a check, we can use this formula to extract the Euler characteristic χ (cid:0) I n β ( X, d ) (cid:1) ofthe moduli space of degree d divisorial curves on X = CP . By our result for the partitionfunction, it is given by the number of ways to obtain d as the sum of three non-negativeintegers, d = λ + λ + λ . As | I n β ( X, d ) | = P ( H ( X, O X ( D )) for a choice of divisor D with [ D ] = β , (see e.g. [33,p. 137]), and χ ( CP n ) = n + 1, this is indeed the correct result.16 .4.2 Hirzebruch-Jung surfaces A choice of 1-cones describing the Hirzebruch-Jung surfaces Y p,q is given in Appendix B.The corresponding fan for the example Y , = A is depicted in Figure 6, together withthe dual web diagram.Figure 6: The toric fan for A , and the corresponding web diagram, with the legs of the vertices labelled. The vertex rules yield the partition function Z cm ( Y p,q ; q, w ) = ∞ (cid:88) λ ,...,λ n =0 η ( q ) q − λ λ q a λ + λ (1 − a i ) w λ η ( q ) q − λ λ · · ·× q a n λ n + λ n (1 − a n ) w λ n n η ( q ) q − λ n λ n +1 = 1ˆ η ( q ) n +1 ∞ (cid:88) λ ,...,λ n =0 q λ · Cλ − λ · Ce − λ − λ n w λ , where we have defined e := (1 , . . . , λ := ( λ , . . . , λ n ), and w λ := w λ · · · w λ n n . The neg-ative of C is the intersection matrix (B.2) of the compact divisors given in Appendix B,where we also review how to determine the self-intersection numbers − a i for these sur-faces.ALE spaces have vanishing canonical class. By specializing the above formula to thiscase, with all a i = 2, we observe the ensuing simplification to Z cm ( A n ; q, w ) = 1ˆ η ( q ) n +1 ∞ (cid:88) λ ,...,λ n =0 q λ · Cλ w λ . On non-compact surfaces, the vertex rules from the previous section only capture partof the complete gauge theory partition function. To obtain the full partition function,we need to include contributions from both negative and non-compact divisors. In thecompact case, linear equivalence will furthermore identify divisors in the gauge theorythat correspond to distinct fixed points of the torus action, changing the combinatoricsof the problem.The factorization (2.2) of rank 1 torsion-free sheaves T , T = L ⊗ I Z , M X ( β, n ) = Pic Xβ × X [ n − n β ] , with the Picard group Pic Xβ parametrizing line bundles which contribute n β = − β · ( β + K X ) to the Euler characteristic of the torsion-free sheaf. It follows that the generatingfunction (2.5) for the counting problem splits into a discrete and a continuous part, Z gt ( X ; q, w ) = (cid:88) β ∈ H ( X, Z ) (cid:88) L∈ Pic Xβ q − n β w β (cid:88) n ≥ (cid:90) X [ n ] e (cid:0) T X [ n ] (cid:1) q n . For the continuous part, we need to count 0 dimensional subschemes. These contributeidentically to Z gt and Z cm , by the factor 1ˆ η ( q ) χ ( X ) , as determined in Section 4.1.It remains to enumerate the holomorphic line bundles L ∈
Pic Xβ . In fact, on a toricmanifold, the Picard group is spanned by the classes of torically invariant divisors. Ourtask will be to determine an integral generating set among these. We do this for each ofthe examples considered in Section 4.4 in turn. The homology of CP is spanned by the hyperplane class, with self-intersection number 1.Holomorphic line bundles on CP hence permit self-dual, but not anti-self-dual connec-tions. Of course, the two conditions are interchanged upon reversing the orientation ofthe surface. Let us proceed to determine the gauge theory partition function, in the sensedeveloped in Section 2.2, upon replacing anti-self-duality by self-duality.The three torically invariant divisors of the complex projective plane are linearly equiv-alent (the Picard group of complex projective space in any dimension is spanned by thehyperplane divisor). In contrast to the non-compact examples discussed below, we obtainthe gauge theory partition function from the melting crystal partition function simply bydropping the sum over equivalent bundles and the restriction to effective divisors, and bytaking into account the change in weight due to K CP (cid:54) = 0. One thereby finds Z gt ( CP ; q, w ) = 1ˆ η ( q ) ∞ (cid:88) u = −∞ q − u w u . When including non-compact prime divisors, the full set of divisors associated to the 1-cones of the toric fan of a Y p,q space become linearly dependent. We will now determine an Note the deceptive similarities to the case of counting subschemes. The sum over torically invariantdivisors there was due to localization. Furthermore, we will start off here by considering two additionaltoric divisors (those of non-compact support), but taking linear equivalence into account will result inthe same number of summations as previously. A n , though they are of course encompassed by the subsequent treatmentof general Hirzebruch-Jung surfaces Y p,q , upon setting ( p, q ) = ( n + 1 , n ). ALE spaces.
Consider the vectors (1 ,
0) and (0 ,
1) in the toric fan of the resolvedgeometry of A n = C / Z n +1 introduced in Appendix B.2. They correspond to the twoprincipal divisors div (cid:0) χ (1 , (cid:1) = D − D − D − . . . − n D n +1 , div (cid:0) χ (0 , (cid:1) = D + 2 D + . . . + ( n + 1) D n +1 , (5.1)where we have labelled toric divisors in anti-clockwise order. The two divisors corre-sponding to the outermost 1-cones, D and D n +1 , have non-compact support. Based onthe relations of linear equivalence induced by eq. (5.1), we now demonstrate that theclasses e i := − n (cid:88) j =1 (cid:0) C − (cid:1) ij [ D j ] , i = 1 , . . . , n , (5.2)with − C the intersection matrix of the compact divisors as given in eq. (B.2) of Ap-pendix B, constitute an integral generating set for the Picard group A ( X ). As theentries of C − are fractional, we need to demonstrate both that the elements e i are gener-ators, and that they are integral linear combinations of the toric divisors (including [ D ]and [ D n +1 ]). Both properties follow upon providing the following recursive presentationof the e i (for n >
1; the case n = 1 is trivial, with a single generator [ D ] = [ D ]). It iseasy to verify that e = [ D ] and e n = [ D n +1 ]. For i = 2 , . . . , n −
1, the e i satisfy e i = e i − − e n − [ D i ] − . . . − [ D n +1 ] . It follows that { e i } represents an extension of the set of non-compact torically invariantdivisors { [ D ] , [ D n +1 ] } to an integral generating set for the Picard group.Parametrizing the class of a divisor D = D u in terms of the generators e i ,[ D u ] = n (cid:88) i =1 u i e i with u = ( u , . . . , u n ) ∈ Z n , we can now compute its second Chern character. Note thatthe Chern classes corresponding to divisor classes are integral, irrespective of the supportof the divisor. At the level of Chern classes, we can hence invoke the presentation of e = [ D ] and e n = [ D n +1 ] in eq. (5.2) to solve for e and e n , and the right-hand side,despite the appearance of the fraction n +1 , maps into integral cohomology. With theintersection pairing (3.5), we thus arrive atch (cid:0) O X ( D u ) (cid:1) = u · C − u . (5.3) See e.g. ref. [39] for the notation χ u , which assigns a function to the lattice vector u . Such a generating set is of course not unique. Our choice provides a dual set, via the intersectionproduct linearly extended to non-compact divisors, to the compactly supported divisors D , . . . , D n , andas such corresponds to the basis of bundles constructed by Kronheimer and Nakajima in ref. [28]. eneral Y p,q spaces. With the parametrization of the toric fan given in Appendix B,the principal divisors corresponding to the lattice vectors (1 ,
0) and (0 ,
1) arediv (cid:0) χ (1 , (cid:1) = n +1 (cid:88) i =0 x i D i , div (cid:0) χ (0 , (cid:1) = n +1 (cid:88) i =0 y i D i , where we have introduced the notation v i = ( x i , y i ) for the generator of the i th v = (1 , D ] = 1 y n +1 n (cid:88) i =1 ( x n +1 y i − x i y n +1 ) [ D i ] , [ D n +1 ] = − y n +1 n (cid:88) i =1 y i [ D i ] . By invoking eq. (B.3) from Appendix B and the relation x i − y i +1 − x i +1 y i − = a i , which follows from eq. (B.3) and the fact that the resolved geometry is non-singular (i.e.the 2-cones have volume 1), we can easily verify that in A ( Y p,q ) ⊗ Q one has[ D ] = − n (cid:88) i =1 (cid:0) C − (cid:1) i [ D i ] , (5.4)[ D n +1 ] = − n (cid:88) i =1 (cid:0) C − (cid:1) n i [ D i ] . (5.5)We can now complete the set { [ D ] , [ D n +1 ] } to an integral generating set for A ( Y p,q ) bysetting e = [ D ], e n = [ D n +1 ] and defining e i , i = 2 , . . . , n − e i = e i − − n (cid:88) j = i c ij [ D i ] − c in +1 e n , where c ii = 1 ,c ij = a j − c ij − − , j = i + 1 , . . . , n ,c in +1 = − c in − + a n c in . The generators so defined satisfy e i = − n (cid:88) j =1 (cid:0) C − (cid:1) ij [ D j ] . D u ] = n (cid:88) i =1 u i e i , the second Chern character as given in (5.3).Combining the contributions from the line bundles and the ideal sheaves of points, weobtain the partition function Z gt ( Y p,q ; q, v ) = 1ˆ η ( q ) χ ( Y p,q ) (cid:88) u ∈ Z n q − u · C − u w u with w u := w u · · · w u n n . This coincides with the results obtained in ref. [10]. Acknowledgements
We would like to thank Ugo Bruzzo, Francesco Fucito, Elizabeth Gasparim, Antony Ma-ciocia, Jose F. Morales, Rubik Poghossian, Alessandro Tanzini and Constantin Telemanfor discussions. AK would like to thank Pierre Cartier, Ofer Gabber, Nicol`o Sibilla, and es-pecially Ilya Shapiro for patient explanations. MC is partially supported by the Funda¸c˜aopara a Ciˆencia e a Tecnologia (FCT/Portugal), and by the Center for Mathematical Anal-ysis, Geometry and Dynamical Systems. AK is supported in part by l’Agence Nationalede la Recherche under the grant ANR-BLAN06-3-137168. RJS is partially supported bygrant ST/G000514/1 “String Theory Scotland” from the UK Science and TechnologyFacilities Council.
A Euler characteristic of torus invariant subschemes
In this appendix, we will calculate the Euler characteristic χ ( O Y ) of torically invariantsubschemes Y of a non-compact toric surface X using ˇCech cohomology. We will computethe cohomology with respect to the canonical torically invariant open cover { U i } of X ,where each U i corresponds to a maximal cone. We choose the index i = 0 , . . . , n toenumerate consecutive maximal cones in anti-clockwise order. The collection of sets thusdefined has the properties U i ∩ U j (cid:40) = ( C ∗ ) , j (cid:54) = i ± , ⊃ ( C ∗ ) , j = i ± , (A.1)and U i ∩ U j ∩ U k = ( C ∗ ) for any i, j, k . Our strategy to compute χ ( O Y ) is as follows. Given V = b (cid:91) i = a U i , (A.2)21et A V = O Y ( U a − ∪ V ∪ U b +1 ) (cid:12)(cid:12) V be the space of global sections of O Y ( V ) that lift to O Y ( U a − ) and O Y ( U b +1 ), and de-fine χ V = dim C ( A V ) − ˇ h (cid:0) O Y (cid:12)(cid:12) U a − ∪ V ∪ U b +1 (cid:1) . Note that we avoid the use of ˇ h ( O Y | U a − ∪ V ∪ U b +1 ) in place of dim C ( A V ), as the formercould be infinite. We will compute χ V for V = U i , and given two adjacent such sets, suchas V in eq. (A.2) and W = c (cid:91) i = b +1 U i , together with the integers χ V and χ W , we will determine χ V ∪ W . Applying this gluingoperation a finite number of times will yield χ ( O Y ).The monomial generators of O Y ( U i ) are, as explained in Section 4.1, in one-to-one corre-spondence with the boxes of a possibly infinite Young tableau π , such that the numberof boxes in the k -th row and column of π stabilize at large k . We will denote these stablevalues as λ i +1 and λ i , respectively. The boxes with coordinates ( k, l ), k ≥ λ i , l ≥ λ i +1 correspond to sections that restrict to 0 outside of U i , and therefore can be attributed tothe punctual factor in the decomposition (2.1). Each such box contributes a summand of1 to χ ( O Y ). In the following, we can hence restrict attention to subschemes Y withoutsuch free or embedded points, corresponding to hook Young tableaux.We turn to the calculation of χ U i . We can choose coordinates in the three neighboringpatches U i − , U i , U i +1 such that O Y ( U i − ) = C (cid:2) x , x a i y (cid:3) (cid:14) (cid:0) ( x ) λ i − ( x a i y ) λ i (cid:1) , O Y ( U i ) = C [ x, y ] (cid:14) (cid:0) x λ i +1 y λ i (cid:1) , O Y ( U i +1 ) = C (cid:2) x y a i +1 , y (cid:3) (cid:14) (cid:0) ( x y a i +1 ) λ i +1 ( y ) λ i +2 (cid:1) , together with O Y ( U i − ,i ) = C (cid:2) x , x , y (cid:3) (cid:14) (cid:0) y λ i (cid:1) , O Y ( U i,i +1 ) = C (cid:2) x , y , y (cid:3) (cid:14) (cid:0) x λ i +1 (cid:1) , and O Y ( U i − ,i,i +1 ) = 0 . We have introduced the notation U i,...,j = U i ∩ . . . ∩ U j . The negative self-intersectionnumbers a i are discussed in Section 4 and Appendix B. A moment’s thought yieldsdim C ( A U i ) = λ i − (cid:88) s =0 a i s (cid:88) r =0 λ i +1 − (cid:88) s =0 a i +1 s (cid:88) r =0 − λ i +1 λ i = a i ( λ i − λ i λ i + a i +1 ( λ i +1 − λ i +1 λ i +1 − λ i λ i +1 , and ˇ H (cid:0) O Y (cid:12)(cid:12) U i − ∪ U i ∪ U i +1 (cid:1) = 0 , χ U i = dim C ( A U i ) . Next, we turn to the compution of χ V ∪ W given χ V and χ W , using the notation for V and W introduced above. Note that by eq. (A.1), V ∩ W = U b ∩ U b +1 . If we consider the sum χ V + χ W as an approximation to χ V ∪ W , then we make the following mistakes: • We count generators of A V and A W that have the same restriction to V ∩ W twice. • We count generators of A V | V ∩ W that do not lift to A W , and likewise elements of A W | V ∩ W that do not lift to A V . • We do not subtract new contributions to ˇ H , i.e. generators in O Y ( V ∩ W ) that arenot exact.Now consider the space B V,W = O Y ( U b ∪ U b +1 ) | U b ∩ U b +1 . Generators of this space eitherlift to elements in both A V and A W , or in either A V or A W but not both, or in neither.By the following lemma, the generators of B V,W are hence in one-to-one correspondencewith the elements over-counted above.
Lemma A.1.
Elements in B V,W that lift neither to elements in A V nor in A W lie in ˇ H ( V ∪ W ) .Proof. A preimage r under the ˇCech differential δ of a monomial element in ˇ C ( V ∪ W )of the form s ij = (cid:40) a if { i, j } = { b, b + 1 } ,0 otherwisemust satisfy r i − r j (cid:12)(cid:12) U i,j = (cid:40) ± a if { i, j } = { b, b + 1 } ,0 otherwise .Such an element exists if and only if a lifts to an element in A V or A W .Finally, the number of generators of B V,W already entered into our computation of thedimension of A U i above, dim C ( B V,W ) = λ b +1 − (cid:88) s =0 a b +1 s (cid:88) r =0 . Combining all of these observations, we arrive at the desired formula for the holomorphicEuler characteristic of Y , χ ( O Y ) = n (cid:88) i =1 (cid:16) a i λ i ( λ i − λ i − λ i λ i +1 (cid:17) . Note that with very little extra effort, this calculation can be modified to encompass thecase of compact X as well. 23 Toric surfaces
B.1 General non-singular toric surfaces
A non-singular toric surface is determined by a sequence of integral vectors v i in Z ,taken in counter-clockwise order, that generate the 1-cones of the toric fan. For compactsurfaces, we will enumerate these v through v n . Any two adjacent vectors span a 2-conein this case, and by non-singularity, generate the lattice. For non-compact surfaces, twoof the 2-cones have only one neighbor. For this case, we denote the outer-most vectorsby v and v n +1 , giving rise to a total of n + 2 1-cones.The coordinate transformation between neighboring 2-cones is of the form( x, y ) (cid:55)−→ (cid:0) x , x a y (cid:1) . For each 2-cone, generated by integral vectors v i and v i +1 , one determines the integer a = a i by considering the generator v i +1 of the neighboring cone, counter-clockwise,which satisfies v i +1 = − v i − + a i v i . (B.1)The torically invariant prime divisors of a toric manifold are in one-to-one correspondencewith the integral generators of the 1-cones. In the conventions introduced above, thenumber of compactly supported divisors is n , both in the compact and the non-compactcase. The intersection matrix of the compactly supported divisors is determined as follows.Two divisors whose associated 1-cones span a 2-cone of the fan intersect transversally,while all others are disjoint. The negative self-intersection number of the divisor D i associated to the generator v i is given by the constant a i in eq. (B.1), D i = − a i (recallthat we are excluding i = 0 and i = n + 1; indeed, due to the non-compact support of theassociated divisors, the naive intersection number here is not defined). The intersectionmatrix − C for the compactly supported toric divisors therefore has the form C = a − . . . − a − . . . − −
10 0 . . . − a n . (B.2) B.2 Hirzebruch-Jung surfaces
Hirzebruch-Jung spaces X = Y p,q are non-compact toric surfaces, parametrized by twopositive coprime integers p and q with p > q . They are defined as the resolutions of A p,q quotients, i.e. the quotients of C by the action of the cyclic group Z p generated byΓ = (cid:18) ξ ξ q (cid:19) , where ξ = e π i /p . The toric fan for the singular space is given by the two 1-conesgenerated by the integral vectors v = (1 ,
0) and v n +1 = ( − q, p ) respectively, and the24-cone generated by the pair. The resolution is obtained via subdivision with 1-conesgenerated by integral vectors v , . . . , v n such that v i − + v i +1 = a i v i (B.3)for i = 1 , . . . , n . The integers a i can be read off from the continued fraction expansion of p/q , pq = a − a − ... a n − − an . With these entries, C of eq. (B.2) is positive definite, the intersection matrix hence neg-ative definite.Topologically, ALE spaces are resolutions of A n singularities. These are the Hirzebruch-Jung spaces Y n +1 ,n . For these surfaces, a i = 2 for i = 1 , . . . , n , and a choice of integralvectors generating the 1-cones of the toric fan is given by v = (1 , , v = (0 , , . . . , v n +1 = ( − n, n + 1) . We have depicted the fan for the surface A in Figure 7.Figure 7: The toric fan for A , with the torically invariant divisors indicated. C Characteristic classes of coherent sheaves
In physics, we are most familiar with characteristic classes assigned to vector bundleson a manifold X , taking values in H ∗ ( X, Z ) ⊗ Q (the sheaf cohomology of the locallyconstant sheaf); we often deal with the image in de Rham cohomology. Chern classesform basic building blocks for all characteristic classes. The total Chern class satisfies themultiplicative property c t ( E ) = c t ( E (cid:48) ) · c t ( E (cid:48)(cid:48) ) (C.1)whenever the bundles E , E (cid:48) , and E (cid:48)(cid:48) fit into an exact sequence0 (cid:47) (cid:47) E (cid:48) (cid:47) (cid:47) E (cid:47) (cid:47) E (cid:48)(cid:48) (cid:47) (cid:47) . The product in eq. (C.1) is the cup product or the wedge product, respectively.A refined version of characteristic classes takes values in the Chow ring A ∗ ( X ) ⊗ Q of X .One path between the two definitions is via the splitting principle and the relation of line25undles to divisors. We take the image of an irreducible closed subscheme Y in the Chowring to be the underlying closed subset endowed with the reduced induced structure, Y red ,with multiplicity given by the length of the local ring O y,Y at the generic point y of Y red (we will unravel this perhaps unfamiliar sounding definition below in the simple case of a0 dimensional subscheme).The Grothendieck group K ( X ) of a scheme X is the free abelian group generated by thecoherent sheaves on X , modulo the relation F −F (cid:48) −F (cid:48)(cid:48) whenever these sheaves fit into anexact sequence. As coherent sheaves on well-behaved schemes allow for a finite locally freeresolution, the property (C.1) allows for the extension of the definition of characteristicclasses to all of K ( X ).For a complete scheme X of dimension n , there is a degree map A n ( X ) → Z , givenby deg( (cid:80) n i [ Y i ]) = (cid:80) n i , for [ Y i ] integral. The degree of an irreducible 0 dimensionalsubscheme Y in A n ( X ) is hence its multiplicity, as defined above.With these definitions, we can derive the Euler characteristic (3.1) of the structure sheafof a subscheme Y as follows. The ideal sheaf I Y associated to Y is defined via the exactsequence 0 (cid:47) (cid:47) I Y (cid:47) (cid:47) O X (cid:47) (cid:47) O Y (cid:47) (cid:47) . By additivity of the Euler characteristic, one has χ ( O Y ) = χ ( O X ) − χ ( I Y ) . (C.2)If we consider I Y as an abstract sheaf, forgetting about its embedding into O X , then itis isomorphic to an invertible sheaf on X , i.e. a line bundle. This is simply the familiarcorrespondence between divisors D = [ Y ] and line bundles O X ( D ), O X ( − D ) = I Y . To calculate χ ( O Y ) using eq. (C.2), we invoke the Hirzebruch-Riemann-Roch theorem tocompute χ ( O X ) and χ ( O X ( − D )). It states that the Euler characteristic of a locally freesheaf E of rank r on a nonsingular projective variety X of dimension n is given by χ ( E ) = deg (cid:0) ch( E ) · td ( X ) (cid:1) n , where ( − ) n denotes the component of degree n in the Chow ring A ∗ ( X ) ⊗ Q and a dotdenotes intersection product. When X is a surface, the Todd class is given bytd ( X ) = 1 − K X + (cid:0) K X + c ( X ) (cid:1) , where K X = − c ( X ) is the canonical divisor. Since ch( O X ) = 1 one finds χ ( O X ) = (cid:0) K X + c ( X ) (cid:1) . For a line bundle O X ( − D ), one has ch( O X ( − D )) = exp( − D ). Therefore, χ (cid:0) O X ( − D ) (cid:1) = D · ( D + K X ) + (cid:0) K X + c ( X ) (cid:1) . Collecting these results, we arrive at formula (3.1).Finally, we present a simple application of the definition of multiplicity presented aboveto the case of zero dimensional subschemes.26 emma C.1.
The multiplicity of an irreducible zero dimensional subscheme Y of ascheme X of finite type over an algebraically closed field k is given by dim k ( H ( Y, O Y )) .Proof. The question is local, so we can assume X affine with coordinate ring A , and Y = Spec A/I . The generic point y of Y red is √ I (this is prime by the irreducibilityassumption). The local ring at the generic point is O y,Y = ( A/I ) √ I = A/I . The secondequality follows from dim
A/I = 0. As the length of
A/I as a module over k is equal toits dimension as a vector space, the lemma follows. D The Hilbert scheme of CP P r ) is easily obtained. For a hypersurface Y of degree d , whose Hilbert polynomialis given by P P r Y ( t ) = (cid:18) r + tr (cid:19) − (cid:18) r + t − dr (cid:19) , it is given by projective space P N , with N = h ( P r , O P r ( − d H )) − H the hyperplanedivisor of P r . Consider in particular the case r = 2. As defined above, the hypersurfacescan be non-reduced and reducible, but they cannot include embedded points (as principalideals do not possess embedded components). Including such points increases the Eulercharacteristic of the subscheme. Hence the constant term of P P Y ( t ), P P Y (0) = 3 d − d , is a lower bound for the Euler characteristic of a degree d subscheme of P . In fact, thisfollows from a corollary of Hartshorne [40]: Corollary D.1.
Let k be a field, r > an integer and p ∈ Q [ z ] a numerical polynomial.Then a necessary and sufficient condition that p be the Hilbert polynomial of a properclosed subscheme of P rk is that when p is written in the form p ( z ) = ∞ (cid:88) t =0 (cid:20)(cid:18) z + tt + 1 (cid:19) − (cid:18) z + t − m t t + 1 (cid:19)(cid:21) , one has m ≥ m ≥ . . . ≥ m r − ≥ and m r = m r +1 = . . . = 0 . In the case of interest, p ( z ) = m z + 2 m + m − m , from which our claim follows.This observation already suggests the factorization of the Hilbert scheme of curves on P as Hilb P d t + n = Hilb P d t + n d × (cid:0) P (cid:1) [ n − n d ] , for n ≥ n d = d − d . It follows from Corollary D.1 that the left-hand side is empty for n < n d . 27t the level of the underlying topological spaces, this decomposition follows easily. With S = C [ x , x , x ] the homogeneous coordinate ring of P , the subschemes of P are inone-to-one correspondence with homogeneous ideals I of S , via the map I (cid:55)→ Proj
S/I .Since S is a unique factorization domain, we can decompose I uniquely into irreducibleideals I = (cid:16) (cid:89) i I i (cid:17) (cid:16) (cid:89) j I j (cid:17) , where the I i are generated by one element and correspond to subschemes of dimension 1,and the I j are generated by more than one element and correspond to subschemes ofdimension 0. References [1] M. Aganagic, A. Klemm, M. Marino, and C. Vafa, “The topological vertex,”
Commun. Math. Phys. (2005) 425–478, arXiv:hep-th/0305132 .[2] J. Li, C.-C. M. Liu, K. Liu, and J. Zhou, “A mathematical theory of the topologicalvertex,”
Geom. Topol. (2009) no. 1, 527–621, arXiv:math/0408426 [math.AG] .[3] A. Okounkov, N. Reshetikhin, and C. Vafa, “Quantum Calabi-Yau and classicalcrystals,” Progr. Math. (2006) 597–618, arXiv:hep-th/0309208 .[4] A. Iqbal, N. Nekrasov, A. Okounkov, and C. Vafa, “Quantum foam and topologicalstrings,”
JHEP (2008) 011, arXiv:hep-th/0312022 .[5] N. Nekrasov, “Instanton partition functions and M-theory,” Japan J. Math. (2009) 63–93.[6] D. Maulik, N. Nekrasov, A. Okounkov, and R. Pandharipande, “Gromov-Wittentheory and Donaldson-Thomas theory. I,” Compos. Math. (2006) no. 5,1263–1285, arXiv:math/0312059 [math.AG] .[7] D. Maulik, A. Oblomkov, A. Okounkov, and R. Pandharipande,“Gromov-Witten/Donaldson-Thomas correspondence for toric 3-folds,” arXiv:0809.3976 [math.AG] .[8] C. Vafa and E. Witten, “A strong coupling test of S -duality,” Nucl. Phys.
B431 (1994) 3–77, arXiv:hep-th/9408074 .[9] N. A. Nekrasov, “Localizing gauge theories,”. Prepared for 14th InternationalCongress on Mathematical Physics (ICMP 2003), Lisbon, Portugal, 28 Jul - 2 Aug2003.[10] F. Fucito, J. F. Morales, and R. Poghossian, “Instanton on toric singularities andblack hole countings,”
JHEP (2006) 073, arXiv:hep-th/0610154 .[11] L. Griguolo, D. Seminara, R. J. Szabo, and A. Tanzini, “Black holes, instantoncounting on toric singularities and q -deformed two-dimensional Yang-Mills theory,” Nucl. Phys.
B772 (2007) 1–24, arXiv:hep-th/0610155 .[12] G. Bonelli and A. Tanzini, “Topological gauge theories on local spaces and blackhole entropy countings,”
Adv. Theor. Math. Phys. (2008) no. 6, 1429–1446, arXiv:0706.2633 [hep-th] . 2813] R. Dijkgraaf and P. Sulkowski, “Instantons on ALE spaces and orbifold partitions,” JHEP (2008) 013, arXiv:0712.1427 [hep-th] .[14] E. Gasparim and C.-C. M. Liu, “The Nekrasov conjecture for toric surfaces,” arXiv:0808.0884 [math.AG] .[15] M. Kool, “Euler characteristics of moduli spaces of torsion free sheaves on toricsurfaces,” arXiv:0906.3393 [math.AG] .[16] U. Bruzzo, R. Poghossian, and A. Tanzini, “Poincar´e polynomial of moduli spacesof framed sheaves on (stacky) Hirzebruch surfaces,” arXiv:0909.1458 [math.AG] .[17] H. Nakajima, “Homology of moduli spaces of instantons on ALE spaces. I,” J. Diff.Geom. (1994) no. 1, 105–127.[18] S. Fujii and S. Minabe, “A combinatorial study on quiver varieties,” arXiv:math/0510455 [math.AG] .[19] M. Cirafici, A. Sinkovics, and R. J. Szabo, “Cohomological gauge theory, quivermatrix models and Donaldson-Thomas theory,” Nucl. Phys.
B809 (2009) 452–518, arXiv:0803.4188 [hep-th] .[20] R. Hartshorne,
Algebraic Geometry . Springer-Verlag, New York, 1977. GraduateTexts in Mathematics, No. 52.[21] M. Martin-Deschamps and D. Perrin, “Sur la classification des courbes gauches,”
Ast´erisque (1990) .[22] M. D¨urr, A. Kabanov, and C. Okonek, “Poincar´e invariants,”
Topology (2007)no. 3, 225–294, arXiv:math/0408131 [math.AG] .[23] R. Dijkgraaf, G. W. Moore, E. P. Verlinde, and H. L. Verlinde, “Elliptic genera ofsymmetric products and second quantized strings,” Commun. Math. Phys. (1997) 197–209, arXiv:hep-th/9608096 .[24] J. Fogarty, “Algebraic families on an algebraic surface,”
Amer. J. Math. (1968)511–521.[25] J. Stoppa and R. P. Thomas, “Hilbert schemes and stable pairs: GIT and derivedcategory wall crossings,” arXiv:0903.1444 [math.AG] .[26] A. Losev, G. W. Moore, N. Nekrasov, and S. Shatashvili, “Four-dimensional avatarsof two-dimensional RCFT,” Nucl. Phys. Proc. Suppl. (1996) 130–145, arXiv:hep-th/9509151 .[27] N. Nekrasov and A. S. Schwarz, “Instantons on noncommutative R**4 and (2,0)superconformal six dimensional theory,” Commun. Math. Phys. (1998)689–703, arXiv:hep-th/9802068 .[28] P. B. Kronheimer and H. Nakajima, “Yang-Mills instantons on ALE gravitationalinstantons,”
Math. Ann. (1990) no. 2, 263–307.[29] S. K. Donaldson and P. B. Kronheimer,
The Geometry of Four-Manifolds . OxfordMathematical Monographs. Oxford University Press, New York, 1990. OxfordScience Publications.[30] C. Okonek, M. Schneider, and H. Spindler,
Vector Bundles on Complex ProjectiveSpaces , vol. 3 of
Progress in Mathematics . Birkh¨auser Boston, Mass., 1980.[31] J. Harris and I. Morrison,
Moduli of Curves , vol. 187 of
Graduate Texts inMathematics . Springer-Verlag, New York, 1998.2932] R. Bott and L. W. Tu,
Differential Forms in Algebraic Topology , vol. 82 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 1982.[33] P. Griffiths and J. Harris,
Principles of Algebraic Geometry . Wiley Classics Library.John Wiley & Sons Inc., New York, 1994. Reprint of the 1978 original.[34] L. G¨ottsche, “The Betti numbers of the Hilbert scheme of points on a smoothprojective surface,”
Math. Ann. (1990) no. 1-3, 193–207.[35] G. Ellingsrud and S. A. Strømme, “On the homology of the Hilbert scheme ofpoints in the plane,”
Invent. Math. (1987) no. 2, 343–352.[36] M. F. Atiyah and R. Bott, “The moment map and equivariant cohomology,” Topology (1984) no. 1, 1–28.[37] D. Edidin and W. Graham, “Localization in equivariant intersection theory and theBott residue formula,” Amer. J. Math. (1998) no. 3, 619–636, arXiv:alg-geom/9508001 .[38] D. Eisenbud and J. Harris,
The Geometry of Schemes , vol. 197 of
Graduate Texts inMathematics . Springer-Verlag, New York, 2000.[39] W. Fulton,
Introduction to Toric Varieties , vol. 131 of
Annals of MathematicsStudies . Princeton University Press, 1997.[40] R. Hartshorne, “Connectedness of the Hilbert scheme,”
Publ. Math. Inst. Hautes´Etudes Sci.29