Instantons, Quivers and Noncommutative Donaldson-Thomas Theory
DDAMTP–2010–124HWM–10–35EMPG–10–25
Instantons, Quiversand Noncommutative Donaldson–Thomas Theory
Michele Cirafici ( a ) , Annamaria Sinkovics ( b ) and 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
Email: [email protected] ( b ) Department of Applied Mathematics and Theoretical PhysicsCentre for Mathematical Sciences, University of CambridgeWilberforce Road, Cambridge CB3 0WA, UK
Email:
[email protected] ( c ) Department of MathematicsHeriot–Watt UniversityColin Maclaurin Building, Riccarton, Edinburgh EH14 4AS, UK and
Maxwell Institute for Mathematical Sciences, Edinburgh, UK
Email:
Abstract
We construct noncommutative Donaldson–Thomas invariants associated with abelian orb-ifold singularities by analysing the instanton contributions to a six-dimensional topological gaugetheory. The noncommutative deformation of this gauge theory localizes on noncommutative in-stantons which can be classified in terms of three-dimensional Young diagrams with a colouringof boxes according to the orbifold group. We construct a moduli space for these gauge field con-figurations which allows us to compute its virtual numbers via the counting of representationsof a quiver with relations. The quiver encodes the instanton dynamics of the noncommutativegauge theory, and is associated to the geometry of the singularity via the generalized McKay cor-respondence. The index of BPS states which compute the noncommutative Donaldson–Thomasinvariants is realized via topological quantum mechanics based on the quiver data. We illustratethese constructions with several explicit examples, involving also higher rank Coulomb branchinvariants and geometries with compact divisors, and connect our approach with other ones inthe literature. a r X i v : . [ h e p - t h ] A ug ontents C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Noncommutative instantons on C / Z . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Coloured instanton partition functions . . . . . . . . . . . . . . . . . . . . . . . . . . 13 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.5 Quiver quantum mechanics on C / Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.6 Pair invariants for quivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.7 Quiver moduli spaces for Donaldson–Thomas data . . . . . . . . . . . . . . . . . . . 385.8 Noncommutative Donaldson–Thomas invariants of type N . . . . . . . . . . . . . . . 395.9 Stability conditions and BPS invariants . . . . . . . . . . . . . . . . . . . . . . . . . 42 C / Z × Z A -fibred threefolds 50 C / Z C / Z orbifold 66 B.1 Line bundle cohomology of divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80B.2 Cohomology of sheaves of differential forms . . . . . . . . . . . . . . . . . . . . . . . 80B.3 Cohomology of ideal sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81B.4 Hypercohomology of torsion sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
C Beilinson monad construction 84
C.1 Beilinson spectral sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84C.2 Generalized ADHM complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Topological string theory on a smooth, six-dimensional toric Calabi–Yau manifold is dual to aclassical statistical mechanics which describes the melting process of a three-dimensional crystal.This duality was originally exhibited in a few examples in [1, 2] and subsequently extended tomore general (non-compact) toric Calabi–Yau threefolds in e.g. [3, 4, 5, 6, 7, 8]. As the temper-ature is increased the crystal melts and certain atomic configurations are removed. The atomicconfigurations correspond in the dual picture to BPS states that are geometrically enumerated byDonaldson–Thomas invariants, which are invariant under deformations of the background. In turn,these configurations are identified in the physical Type IIA string theory with stable bound statesthat a single D6 brane filling the whole Calabi–Yau manifold can form with a gas of D0 and D2branes.As the physical moduli are continuously varied this picture gets modified. Stable states maybecome unstable and decay into more elementary constituents or new physical states can appear inthe spectrum. This type of behaviour is at the core of the solution of N = 2 supersymmetric Yang–Mills theory in four dimensions which was proposed in [9] and adapted to the supergravity settingin [10]. In Calabi–Yau compactifications it is only for a special region of the moduli space thatthe stable objects are enumerated via the Donaldson–Thomas invariants computed by topologicalstring theory.As one moves around the moduli space, certain states can become lighter and different configura-tions become energetically favoured over others. The moduli space can be divided into chambers,each one with a physically distinct spectrum of stable BPS states. As the physical moduli are movedfrom one chamber to another, crossing a so-called wall of marginal stability, the index counting BPSstates jumps according to a wall-crossing formula. There is surmounting evidence that this wall-crossing formula is precisely the one found recently by Kontsevich and Soibelman [11] in developingtheir theory of generalized Donaldson–Thomas invariants. This issue was extensively investigatedin the context of gauge theory in [12, 13, 14] and further in the context of refined/motivic invariantsin [15, 16].The usual Donaldson–Thomas invariants, at least as they are commonly encountered in the contextof topological string theory, are virtual numbers of the moduli space of ideal sheaves with trivialdeterminant. Since ideal sheaves are trivially stable, a generalized theory of Donaldson–Thomasinvariants is needed to fully account for wall-crossing phenomena. This theory is naturally rootedin the formalism of derived categories with the appropriate stability conditions, which is widelybelieved to be the correct framework for addressing questions concerning D branes on Calabi–Yaumanifolds [17]. The same sort of constructions have been pursued also by Joyce and Song in theless general but sometimes more concrete framework of abelian categories [18].2n many cases these constructions have been used to solve for the physical spectrum of BPS states.This is the case for the class of examples of local threefolds without compact four-cycles where thechamber structure of the moduli space has been explicitly constructed in [19, 20] and has founda clear physical interpretation in [21] via a lift to M-theory. Here the partition function of BPSstates at a generic point of the moduli space is seen as receiving competing contributions fromboth M2 and anti-M2 branes. In a certain region of the moduli space the anti-M2 brane statesare all unstable and the partition function of BPS states is purely holomorphic. This is the regionaround the large radius point described by the topological string partition function Z top ( q, Q ), withthe parameter q weighting D0 branes and the parameters Q weighting D2 branes. All the otherregions can be reached by crossing walls of marginal stability and using the Kontsevich–Soibelmanwall-crossing formula.In another region of the moduli space the BPS state partition function has the form Z BPS ( q, Q ) = Z top ( q, Q ) Z top ( q, Q − ) . (1.1)This region corresponds to the noncommutative crepant resolution of a toric singularity where theBPS states are computed by noncommutative Donaldson–Thomas invariants. Ooguri and Yamazakishowed in [22] that these invariants count cyclic modules of a certain quiver which arises in a low-energy approximation of the theory governing a gas of D0 and D2 branes near the singularity inthe sense of Douglas and Moore [23].The quiver diagram is obtained from the toric diagram via a T-duality transformation along the T fibers of the toric threefold. After the duality transformation the D0–D2 system becomes anintricate configuration of D2 and NS5 branes. This configuration has a low-energy description interms of a quiver with a superpotential. Adding the D6 brane modifies the quiver through theaddition of a new vertex and a single arrow from the new vertex to an arbitrary reference vertexof the old quiver. This quiver construction identifies a new kind of melting crystal [22]. Thezero temperature configuration is obtained starting from the reference vertex and consists of layersof coloured atoms, with each colour associated with a different node of the original quiver (thenew vertex only labels the colour of the atom which sits at the top of the pyramid). Each layerrepresents a module of the path algebra of the quiver. Equivalently, in the first layer one draws anumber of atoms corresponding to the nodes of the quiver that can be reached from the referencenode in precisely one step. In the second layer one consider paths of the quiver consisting of twoarrows, and so on. The general picture is obtained similarly, though some further combinatorialcomplications arise from the relations of the quiver, or equivalently the F-term constraints derivedfrom the superpotential.BPS states are counted by removing atomic configurations according to a certain rule which roughlystates that the crystal melts starting from its peak; equivalently, if an atom is removed then soare all the atoms above it. This implies that the complement of the atomic configuration removedis algebraically an ideal in a certain algebra; typically the ideals are generated by monomialsin edge variables associated to the pertinent quiver (before adding the D6 brane node). In thisway one computes the index of BPS states in the region of the moduli space corresponding to anoncommutative deformation of the toric variety, the so-called noncommutative crepant resolution.It was proven by Van den Bergh [24] that the path algebra of a certain quiver with relationsassociated with the toric singularity is a crepant resolution of that singularity. The counting ofBPS states in this region was introduced by Szendr˝oi for the conifold [25], and by Mozgovoy andReineke [26] for (essentially) generic singularities.This picture was further enriched in [27] where Aganagic and Schaeffer study generic toric Calabi–Yau threefolds, possibly with four-cycles. Their picture is general enough to include walls of thesecond kind which are elegantly described via mutations of the low-energy quiver. In particular,3hey offer a clear picture of the relation between melting crystal configurations and D brane charges,thus resolving the apparent mismatch between the number of natural parameters associated toatomic colourings and the number of parameters in the topological string amplitude.In this paper we take a rather different approach, which is somewhat less ambitious. The centralidea is to use a D brane worldvolume perspective and try to understand how much of these proper-ties can be captured by a study of the worldvolume gauge theory, and modifications thereof. Thisapproach was successful in the case of smooth toric threefolds in the topological string chamber [2].The gauge theory in question is the topological twist of six-dimensional N = 2 supersymmetricYang–Mills theory studied in [28, 29, 30, 31, 32] and the relevant BPS configurations are identifiedwith generalized instantons, solutions of the Donaldson–Uhlenbeck–Yau equations. A noncommu-tative deformation of the worldvolume gauge theory provides a natural compactification of theinstanton moduli space and its virtual numbers can be evaluated via equivariant localization, re-producing the partition function for Donaldson–Thomas invariants [33, 34, 35]. Associated withthis noncommutative gauge theory is a quantum mechanics which describes the dynamics of thecollective coordinates on the instanton moduli space [36, 37]. We utilise and adapt to our problemthe techniques of equivariant localization pioneered by Nekrasov in the context of Seiberg–Wittentheory [38, 39, 40]; see e.g. [41] for a review geared at the context of the present paper.We study this gauge theory on orbifolds of the form C / Γ, which we interpret as quotient stacks[ C / Γ], where Γ is a finite subgroup of SL (3 , C ). The topological gauge theory localizes by con-struction on Γ-equivariant instanton configurations and thus poses a novel enumerative problem.This enumerative problem, which reduces to counting the virtual numbers of the moduli space ofΓ-equivariant ideal sheaves, is precisely equivalent to the study of noncommutative Donaldson–Thomas invariants via a quiver gauge theory. Indeed, the local structure of the instanton modulispace on quotient stacks can be encoded in a quiver, which is a modification of the McKay quiverassociated to the singularity and appears to be the same as the quiver used by Ooguri and Yamazakiin [22].Geometrically this problem reduces to the counting of Γ-equivariant closed subschemes of C , aproblem which can be greatly simplified by using equivariant localization techniques with respectto the natural toric action on C . These techniques are only available when the toric action iscompatible with the orbifold action, i.e. when the orbifold group is a subgroup of the torus group.In particular this is true for abelian orbifolds that respect the Calabi–Yau conditions, which is thecase we will focus on in this paper.We use this formalism to compute noncommutative Donaldson–Thomas invariants and assemblethem into partition functions where the formal variables have a specific form which is derived fromthe instanton action. Much in the same way as in [27], the counting variables are not all independentbut related to geometrical quantities, via their relation to the D brane charges in their work, andvia their relation with the instanton action in ours. Our construction of the instanton modulispace computes the instanton partition function in a clear and self-consistent way with as manyparameters as are present in the formalism based on topological string theory. We also constructCoulomb branch invariants associated to arbitrary numbers of D6–D4–D2–D0 branes on thesenoncommutative crepant resolutions. We elucidate our formalism with plenty of examples.This paper is written in a expository way, surveying various known mathematical results, andcomparing them with our gauge theory calculations; it is organised as follows. In Section 2 wedescribe the pertinent gauge theory, and the enumerative geometry problem it is supposed toaddress. In Section 3 we construct its noncommutative instanton contributions, and describe howthe worldvolume gauge theory partition function naturally organises itself into a generating functionfor coloured three-dimensional Young diagrams associated to the Γ-invariant closed subschemes of C . In Section 4 we propose our construction of the instanton moduli space for C / Γ orbifolds; the4onstruction is inspired by the old construction of Kronheimer and Nakajima [42] of the modulivariety of instantons on ALE spaces. In Section 5 we reformulate this construction in terms of atopological quiver quantum mechanics; we explain how the instanton moduli space is characterizedby the quiver, and how the torus fixed points and local characters are computed. Sections 6–9analyse explicit examples with many detailed calculations, comparing our results with the existingliterature. Section 10 summarises and discusses some open technical aspects of our analysis. Wehave included three appendices at the end of the paper containing some of the more technicalconcepts and computations which are used in the main text.
In the large radius limit the existence of bound states of lower-dimensional branes with N D6 braneswrapping a Calabi–Yau threefold X can be addressed directly from the study of the Dirac–Born–Infeld theory defined on the D6 worldvolume. This gauge theory is automatically twisted since on aCalabi–Yau manifold one can identify spinor bundles with bundles of differential forms. In the caseof a local threefold this is approximated in the low-energy limit by an ordinary supersymmetricgauge theory. This gauge theory is the topologically twisted version of six-dimensional N = 2Yang–Mills theory with gauge group U ( N ). On an arbitrary K¨ahler threefold X the bosonic partof the action has the form S = 12 (cid:90) X Tr (cid:16) d A Φ ∧ ∗ d A Φ + (cid:2) Φ , Φ (cid:3) + (cid:12)(cid:12) F , A + ∂ † A ρ (cid:12)(cid:12) + (cid:12)(cid:12) F , A (cid:12)(cid:12) (cid:17) + 12 (cid:90) X Tr (cid:16) F A ∧ F A ∧ ω + ϑ F A ∧ F A ∧ F A (cid:17) , (2.1)where Tr denotes the trace in the fundamental representation of U ( N ), d A = d + i [ A, − ] is thegauge-covariant derivative, Φ is a complex adjoint scalar field, and F A = d A + A ∧ A is the gaugefield strength. Here ρ is a (3 , ∗ is the Hodge duality operator with respect to the K¨ahlermetric of X , ω is the background K¨ahler two-form of X , and ϑ is the six-dimensional theta-anglewhich is identified with the topological string coupling g s .Since the gauge theory is cohomological, its quantum partition function and supersymmetric ob-servables localize onto the moduli space of solutions of the generalized instanton equations F , A = ∂ † A ρ ,F , A ∧ ω ∧ ω + (cid:2) ρ ∧ , ρ (cid:3) = l ω ∧ ω ∧ ω , d A Φ = 0 , (2.2)where the constant l is related to the magnetic charge of the gauge bundle. For a Calabi–Yau back-ground we can consider minima where ρ = 0. On a smooth toric threefold the partition functionof this gauge theory can be evaluated via equivariant localization techniques. The moduli space ofsolutions of the first-order equations (2.2) is desingularized by adding appropriate point-like con-figurations. Since the gauge theory is cohomological every physical observable can be expressed interms of intersection integrals over the moduli space. These integrals can be accordingly computedvia the localization formula.For N = 1 the problem is mathematically well-formulated, and the resulting virtual numbers arethe Donaldson–Thomas invariants which count stable BPS bound states of D2 and D0 braneswith a single D6 brane. For N > X . It is an interesting and ambitious project to understand howmuch of this picture can be captured in terms of gauge theory variables or modifications thereof.Such modifications can include turning on a noncommutative deformation of X via a nontrivial B -field, or including nonlinear higher-derivative corrections to the gauge theory action and henceto the equations (2.2). It is likely that a mixture of these ingredients and string theory effectsshould capture the enumerative problem of stable BPS states at least in some chambers.In this paper we will study this gauge theory on orbifolds of the form C / Γ and we shall propose thatworking equivariantly on C with respect to the linear action of the finite group Γ ⊂ SL (3 , C ), orequivalently working on the quotient stack [ C / Γ], captures the enumerative problem correspond-ing to the noncommutative Donaldson–Thomas invariants. The mathematical intuition behind thisperspective comes from the work of Bryan and Young [47] which studies deformation invariantscounting ideal sheaves of zero-dimensional Γ-equivariant subschemes on C , or equivalently prop-erly supported substacks of [ C / Γ]. These invariants correspond precisely to the noncommutativeDonaldson–Thomas invariants [18].In the following we will interpret the “gauge theory” living in the orbifold phase as a theory ofΓ-equivariant sheaves on C . Some technical aspects of the description of the gauge theory in thissense are briefly discussed in Appendix A. This will allow us to define a quiver which describes thedynamics of the instanton collective coordinates. The study of the representation theory of thisquiver will then yield the noncommutative Donaldson–Thomas invariants. In this section we will study the noncommutative deformation of the gauge theory introduced inSection 2 for U (1) gauge group; this corresponds to subjecting the D6–D2–D0 system to a largeNeveu–Schwarz B -field. The idea is that string theory effects will resolve the orbifold singularity C / Γ and should make the gauge theory well-behaved; see Appendix A for some details. We areinterested in the region of the K¨ahler moduli space where the resolution is still small, for examplewhen the classical volume of the cycles is still vanishing, while the quantum volume as measured bythe B -field is non-zero but small. Even if the B -field is vanishingly small, since the classical volumeof the cycles are zero, the gauge theory sits in the deep noncommutative regime of the K¨ahlermoduli space. We address this issue in more detail in Section 5.9. The BPS state counting in termsof D6–D0 bound states involves fractional branes which can carry both D0 and D2 charge, wherethe D2 charge originates in the large radius limit from D2 branes wrapped on two-cycles whichvanish at the orbifold point. In the more general BPS state counting problem that we considerlater on, fractional branes can also come from wrapped D4 branes or bound states of these D2 andD4 branes. C We begin by reviewing the construction of contributions from noncommutative instantons to thepartition function of the six-dimensional U (1) gauge theory on X = C , following [2, 37]. Inthis case there is a single patch in the geometry and only six-dimensional point-like instantonscontribute. In particular, there are no contributions from four-dimensional instantons stretchedover two-spheres, since there are no non-trivial two-cycles in the geometry.6o compute the partition function for this gauge theory we need to use localization and understandthe moduli space of solutions to the equations (2.2). To resolve short-distance singularities of themoduli space, and to find explicit instanton solutions, we use a noncommutative deformation ofthe gauge theory [2]. The coordinates ( x i ) of C ∼ = R thus satisfy the Heisenberg algebra[ x i , x j ] = i θ ij , i, j = 1 , . . . , , (3.1)where θ = ( θ ij ) = θ − θ θ − θ θ − θ (3.2)is a constant matrix with θ α > α = 1 , ,
3. We change gauge theory variables to the covariantcoordinates X i = x i + i θ ij A j , (3.3)and introduce complex combinations Z α = √ θ α ( X α − + i X α ) for α = 1 , , (cid:2) Z α , Z β (cid:3) + (cid:88) γ =1 (cid:15) αβγ (cid:2) Z † γ , ρ (cid:3) = 0 , (cid:88) α =1 (cid:2) Z α , Z † α (cid:3) + (cid:2) ρ , ρ † (cid:3) = 3 , (cid:2) Z α , Φ (cid:3) = 0 (3.4)for α, β = 1 , ,
3. For the remainder of this section we set the (3 , ρ to zero, as we workon a Calabi–Yau geometry. We now introduce another deformation which regulates the infraredsingularities of the instanton moduli space, by turning on the Ω-background with equivariant pa-rameters (cid:15) , (cid:15) , (cid:15) which parametrize the natural scaling action of the three-torus T on C . Thisdeformation changes the last equation of (3.4) to (cid:2) Z α , Φ (cid:3) = (cid:15) α Z α . (3.5)The set of equations (3.4) can be solved by harmonic oscillator algebra. We represent the fields asoperators on a three-particle quantum mechanical Fock space H , which is the unique irreduciblemodule over the Heisenberg algebra, with the usual creation and annihilation operators a † α =( x α − − i x α ) / √ θ α , a α = ( x α − + i x α ) / √ θ α for α = 1 , , | n , n , n (cid:105) = (cid:81) α ( a † α ) n α / √ n α ! | , , (cid:105) with n α ∈ N . The vacuum solution is then given by Z α = a α and Φ = (cid:88) α =1 (cid:15) α a † α a α . (3.6)Other solutions are found with the solution generating technique [2, 37]. The idea is that one canuse the partial isometry U n on H obeying U † n U n = 1 − (cid:88) n + n + n 00 0 a (2) α a (0) α . (3.23)The first instanton equation in (3.4) then yields a (0) α a (1) β = a (0) β a (1) α ,a (1) α a (2) β = a (1) β a (2) α ,a (2) α a (0) β = a (2) β a (0) α (3.24)9or α, β = 1 , , 3. From the second instanton equation in (3.4) we get the relations a (1) α a (1) † β = a (0) † β a (0) α ,a (2) α a (2) † β = a (1) † β a (1) α ,a (0) α a (0) † β = a (2) † β a (2) α (3.25)for α (cid:54) = β , while for α = β we have a (1) α a (1) † α − a (0) † α a (0) α = P (0) ,a (2) α a (2) † α − a (1) † α a (1) α = P (1) ,a (0) α a (0) † α − a (2) † α a (2) α = P (2) , (3.26)where P ( r ) : H → H ( r ) for r = 0 , , a ( r )1 = ∞ (cid:88) k =0 (cid:88) n + n + n = r +3 k √ n | n − , n , n (cid:105)(cid:104) n , n , n | ,a ( r ) † = ∞ (cid:88) k =0 (cid:88) n + n + n = r − k √ n + 1 | n + 1 , n , n (cid:105)(cid:104) n , n , n | (3.27)for r = 0 , , 2, and analogously for the operators a ( r )2 , a ( r ) † and a ( r )3 , a ( r ) † . The vacuum solution forthe scalar field Φ is now Φ = (cid:88) α =1 (cid:15) α N (0) α (cid:15) α N (1) α 00 0 (cid:15) α N (2) α (3.28)where the number operators N ( r ) α count states in twisted sectors as N ( r ) α = a ( r ) † α a ( r ) α = ∞ (cid:88) k =0 (cid:88) n + n + n = r +3 k n α | n , n , n (cid:105)(cid:104) n , n , n | . (3.29)We now look for the most general solution of the instanton equations (cid:2) Z α , Z β (cid:3) = 0 , (cid:88) α =1 (cid:2) Z α , Z † α (cid:3) = 3 P (0) P (1) 00 0 P (2) , (cid:2) Z α , Φ (cid:3) = (cid:15) α Z α , (3.30)where as before α, β = 1 , , P ( r ) are the projectors for the twisted sectors. To construct thesesolutions we use the partial isometry operators U n from Section 3.1 and split them into twisted10ectors as U n U † n = P (0) P (1) 00 0 P (2) ,U † n U n = P (0) − P (0) n P (1) − P (1) n 00 0 P (2) − P (2) n , (3.31)where P ( r ) n projects onto states with particle number N < n in the sector H ( r ) with P ( r ) n = ∞ (cid:88) k =0 (cid:88) n + n + n = r +3 k 00 0 (cid:15) α N (2) α U † n . (3.36)A more general solution is of the formΦ = U n (cid:80) α =1 (cid:15) α N (0) α Φ ( N (2) ) Φ ( N (1) )Φ ( N (2) ) (cid:80) α =1 (cid:15) α N (1) α Φ ( N (0) )Φ ( N (1) ) Φ ( N (0) ) (cid:80) α =1 (cid:15) α N (2) α U † n . (3.37)11owever, the extra functions here will play no role in the computation of the gauge theory partitionfunction, because Tr H (Φ) = (cid:88) r =0 3 (cid:88) α =1 (cid:15) α Tr H (cid:16) N ( r ) α (cid:0) P ( r ) − P ( r ) n (cid:1)(cid:17) . (3.38)The Γ-equivariant character for the vacuum solution of the noncommutative gauge theory is givenby Char Γ ∅ ( t ) = Tr H ( e t Φ ) = ∞ (cid:88) k =0 2 (cid:88) r =0 (cid:88) n + n + n = r +3 k e t (cid:80) α (cid:15) α n α =: (cid:88) r =0 Char Γ r ( t ) , (3.39)which splits into the twisted sectors. Projecting onto the Γ-invariant sector corresponding to thetrivial orbifold group representation givesChar Γ0 ( t ) = ∞ (cid:88) k =0 (cid:88) n + n + n =3 k e t (cid:80) α (cid:15) α n α = 3 + 2 cosh( (cid:15) − (cid:15) ) t + 2 cosh( (cid:15) − (cid:15) ) t + 2 cosh( (cid:15) − (cid:15) ) t (1 − e (cid:15) t ) (1 − e (cid:15) t ) (1 − e (cid:15) t ) (3.40)where we have used (cid:15) + (cid:15) + (cid:15) = 0. Taking the general solution including the partial isometry U n ,the Γ-invariant character is given byChar Γ π ( t ) = Char Γ0 ( t ) − ∞ (cid:88) k =0 (cid:88) ( n ,n ,n ) ∈ πn + n + n =3 k e t (cid:15) ( n − t (cid:15) ( n − t (cid:15) ( n − , (3.41)where Char Γ0 ( t ) is the orbifold vacuum contribution (3.40) and π is a plane partition. The sumin (3.41) corresponds to Z -invariant zero-dimensional subschemes Y ⊂ C for which Z actstrivially on H ( O Y ). In the following we denote them by π . Thus ( n , n , n ) ∈ π if and only if( n , n , n ) ∈ π and n + n + n ≡ E (3) I is the coefficient of t in the expansion of the function E Γ I ( t ) = 1Char Γ0 ( t ) Tr H (0) I ( e t Φ ) (3.42)for the orbifold gauge theory around t = 0. Expanding we obtain E (3) I = 3 (cid:15) (cid:15) (cid:15) ∞ (cid:88) k =0 (cid:88) ( n ,n ,n ) ∈ πn + n + n =3 k (cid:15) (cid:15) (cid:15) (cid:88) ( n ,n ,n ) ∈ π (cid:15) (cid:15) (cid:15) | π | . (3.43)Hence the weight of an instanton in the sector corresponding to the trivial representation of Z is e ϑ | π | . (3.44)By including the other two twisted orbifold sectors, the instanton contributions are characterizedby plane partitions π together with a 3-colouring π = π (cid:116) π (cid:116) π , where ( n , n , n ) ∈ π r if andonly if n + n + n ≡ r mod 3. The colours of the boxes are in bijection with the set of irreduciblerepresentations of the orbifold group Γ = Z . 12 .3 Coloured instanton partition functions More general abelian orbifolds C / Γ can be treated in exactly the same way and yield the samequalitative behaviours. Let Γ ⊂ T be a finite abelian group acting linearly on C with weights r , r , r and with trivial determinant, r + r + r ≡ . (3.45)The set of irreducible representations of Γ forms a group (cid:98) Γ ∼ = Γ under tensor product; we use anadditive notation for the group operation on weights r , a multiplicative notation for correspondingcharacters χ r : Γ → C , and tensor product for representations ρ r .The Fock space of the noncommutative gauge theory is a Γ-module which admits an isotopicaldecomposition into irreducible representations as H = (cid:77) r ∈ (cid:98) Γ H ( r ) , (3.46)where H ( r ) = (cid:16) | Γ | (cid:88) g ∈ Γ χ r ( g ) g − (cid:17) · C (cid:2) a † , a † , a † (cid:3) | , , (cid:105) = span C (cid:8) | n , n , n (cid:105) | n r + n r + n r ≡ r (cid:9) . (3.47)The covariant coordinates correspondingly decompose into operators Z α = (cid:77) r ∈ (cid:98) Γ Z ( r ) α with Z ( r ) α : H ( r ) −→ H ( r + r α ) (3.48)for α = 1 , , 3. The instanton equations (3.4) then yield Z ( r + r β ) α Z ( r ) β = Z ( r + r α ) β Z ( r ) α (3.49)for α, β = 1 , , r ∈ (cid:98) Γ, and (cid:88) α =1 (cid:0) Z ( r − r α ) α Z ( r − r α ) α † − Z ( r ) α † Z ( r ) α (cid:1) = 3 P ( r ) (3.50)where P ( r ) is the projection onto the isotopical component H ( r ) . The partial isometries U n fromSection 3.1 are accordingly decomposed as U n U † n = 1 and U † n U n = 1 − (cid:77) r ∈ (cid:98) Γ P ( r ) P k r ( n ) P ( r ) , (3.51)where P k r ( n ) is a projection operator of finite rank k r ( n ), the number of states of H ( r ) with N < n . The corresponding noncommutative instantons are labelled by (cid:98) Γ-coloured plane partitions π = (cid:70) r ∈ (cid:98) Γ π r , where ( n , n , n ) ∈ π r if and only if n r + n r + n r ≡ r .The orbifold field theory by construction naturally only keeps contributions from Γ-invariant in-stanton configurations, obtained by projection onto the trivial representation r = 0 of the orbifoldgroup as in the example of Section 3.2. However, in what follows we would like to weigh thecoloured instantons by a set of variables ( p r ) r ∈ (cid:98) Γ indexed by the irreducible representations of theorbifold group Γ. For this, rather than using the descent formula (3.12) from localization, we13ill define the instanton action of the D6 brane gauge theory on C / Γ via the Wess–Zuminocoupling of constant Ramond–Ramond fields C dual to fractional D0 branes (instantons); this en-ables the proper incorporation and weighting of the twisted sectors r (cid:54) = 0 in (3.46) to match withstring theory expectations. Such fields decompose into twisted sectors of the closed string orbifoldas [23, 48, 49, 50] C = (cid:77) r ∈ (cid:98) Γ C ( r ) . (3.52)In Appendix A we justify somewhat the formulation of the gauge theory in this way.Let ρ denote the representation of the orbifold group on (3.46). For the example Γ = Z consideredin Section 3.2, the three-dimensional regular representation ρ ( g ) αβ = ζ α δ αβ naturally correspondsto a superposition of fractional instantons [49]. The corresponding Γ-equivariant Chern characteris given by [50] ch Γ ( F A ) = (cid:77) r ∈ (cid:98) Γ Tr H ( r ) (cid:0) ρ ( r ) exp( − F ( r ) A / π i ) (cid:1) , (3.53)where F ( r ) A = [ X, X ] (cid:12)(cid:12) H ( r ) ∈ End C (cid:0) H ( r ) (cid:1) are the diagonal components in the decomposition of thefield strength F A on the orbifold Hilbert space (3.46) of the noncommutative gauge theory.Then the instanton action is defined by the anomalous coupling to the D6 brane as [23, 48, 50] − ϑ π (cid:90) X F A ∧ F A ∧ F A := ϑ (cid:90) C | Γ | (cid:88) r ∈ (cid:98) Γ C ( r ) Tr H ( r ) (cid:0) ρ ( r ) exp( − F ( r ) A / π i ) (cid:1) , (3.54)which for the linear orbifold group actions on C that we consider in this paper can be expressedas − ϑ π (cid:90) X F A ∧ F A ∧ F A = ϑ π | Γ | (cid:88) r ∈ (cid:98) Γ C ( r ) χ ρ ( r ) Tr H ( r ) (cid:0) F ( r ) A ∧ F ( r ) A ∧ F ( r ) A (cid:1) , (3.55)where χ ρ : Γ → C is the character of the representation ρ . The number of instantons of colour r ∈ (cid:98) Γ is π Tr H ( r ) I (cid:0) F ( r ) A ∧ F ( r ) A ∧ F ( r ) A (cid:1) = | π r | , and by defining ξ r := C ( r ) χ ρ ( r ) / | Γ | the fractionalinstanton action becomes − ϑ (cid:90) X F A ∧ F A ∧ F A = i ϑ (cid:88) r ∈ (cid:98) Γ ξ r | π r | . (3.56)The weighting variables are thus related to D0 brane charges through p r = e i ϑ ξ r , and the instantonpart of the orbifold gauge theory partition function on X = C / Γ with this definition takes theform K DT C / Γ = (cid:88) π χ T ( N π ) (cid:89) r ∈ (cid:98) Γ p | π r | r (3.57)where the sum runs through (cid:98) Γ-coloured plane partitions π = (cid:70) r ∈ (cid:98) Γ π r . By rescaling p r → p p r wemay also include a factor p | π | in (3.57) weighing the total charge | π | = (cid:80) r ∈ (cid:98) Γ | π r | of the collectionof fractional branes.One of the goals of subsequent sections will be to properly define and explicitly compute thegauge theory fluctuation determinants χ T ( N π ) appearing in (3.57) which determine a non-trivialmeasure on the moduli space of noncommutative instantons; we shall find that these are exactlythe noncommutative Donaldson–Thomas invariants. For this, we will properly define the instantonmoduli space (at toric fixed points) in terms of a particular quiver with relations. The idea is that14he effective dynamics of the noncommutative gauge theory (at fixed points) can be encoded ina quiver diagram. Each irreducible representation in (cid:98) Γ is associated to a vertex of the quiver; inthe context of the noncommutative gauge theory, these vertices label isotopical components of theHilbert space (3.46)–(3.47). The links of the quiver diagram specify the bifundamental matter fieldcontent, which arises from the representation of the covariant coordinates (3.48) on the orbifoldHilbert space. The holomorphic conditions (3.49) yield a set of relations for the quiver.A general representation of this quiver corresponds to k r = | π r | noncommutative instantons ofcolour r ∈ (cid:98) Γ. A link joining representations r and r (cid:48) has multiplicity | π r | | π r (cid:48) | in the noncommutativegauge theory. There are no moduli associated to a single instanton of colour r ( k r = 1, k r (cid:48) = 0 for r (cid:48) (cid:54) = r ). But for instantons in the regular representation of the orbifold group Γ, the Γ-orbit is theregular representation on the coordinates ( z , z , z ) ∈ C and so such instantons have a non-trivialpositional moduli space. This will generically require us to impose appropriate stability conditionson the instanton moduli space, and to thereby restrict to fractional D0 branes whose orbits underthe action of the complexified gauge group are closed. In [37] we proposed an ADHM-type formalism to deal with Donaldson–Thomas invariants on C which is directly derived from an analysis of the moduli space of torsion free sheaves. Thegeneralised ADHM equations arise from internal consistency conditions when parametrizing themoduli space of framed torsion free sheaves on P via the Beilinson spectral sequence. Underfavourable conditions the spectral sequence degenerates to a four-term complex whose cohomologyis a generic torsion free sheaf E . The purpose of this section is to extend this construction togeometries which have the form of a crepant resolution of a toric orbifold singularity. We willdo so via a generalisation of the Kronheimer–Nakajima construction [42]. The result will be aconstruction of the instanton moduli space via the representation theory of a quiver with relationswhich governs the generalised ADHM data, much in the spirit of [37]. In Section 5 we will argue thatfor certain choices of the stability parameter of the quiver, one is working on the noncommutativecrepant resolution of the singularity. In the framework of [48], the vacua of the worldvolume quivergauge theory of D0 branes on C / Γ in the different phases corresponding to different choices ofFayet–Iliopoulos (i.e. noncommutativity) parameters all lead to moduli spaces that are simply thegeometric phases of the resolutions X of C / Γ. Before considering orbifolds let us recall briefly the situation for the affine Calabi–Yau space C .Here one would like to construct the instanton moduli space on C , or better a moduli space ofsheaves on the compactification P . In [37] we constructed a model for the moduli space of framedsheaves M N,k (cid:0) P (cid:1) = E = torsion free sheaf on P rank( E ) = N , c ( E ) = 0ch ( E ) = 0 , ch ( E ) = − k E (cid:12)(cid:12) ℘ ∞ ∼ = O ⊕ N℘ ∞ (cid:46) isomorphisms (4.1)where ℘ ∞ is the plane at infinity. Since C ∼ = P / P , in projective coordinates one has explicitly ℘ ∞ = [0 : z : z : z ] ∼ = P . The strategy to construct this moduli space is to adapt Beilin- Actually, as in [42], the construction of the instanton moduli space does not need the assumption that the geometryis toric; our construction should hold whenever the tautological bundles generate the whole derived category (or evenjust the K-theory for some partial results). This assumption is however widely applied in the following sections, forexample in order to use the localization formula. P , via the Beilinson spectralsequence.The construction starts from a sheaf E on P and considers the canonical projections P × P p (cid:124) (cid:124) (cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121)(cid:121) p (cid:35) (cid:35) (cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70)(cid:70) P P . (4.2)The spectral sequence descends from the Fourier–Mukai transform R • p ∗ (cid:0) p ∗ E ⊗ C • (cid:1) , (4.3)where C p := (cid:86) − p (cid:0) O P ( − (cid:2) Q ∨ (cid:1) (4.4)are the terms in the Koszul complex0 −→ (cid:86) (cid:0) O P ( − (cid:2) Q ∨ (cid:1) −→ (cid:86) (cid:0) O P ( − (cid:2) Q ∨ (cid:1) −→ O P ( − (cid:2) Q ∨ −→−→ O P × P −→ O ∆ −→ Q ∨ = Ω P ⊗ O P (1); here ∆ ∼ = P is the diagonal in P × P . Thespectral sequence then has first term E p,q = F p ⊗ H q (cid:0) P , E ⊗ F p (cid:1) (4.6)where C p = F p (cid:2) F p , and it converges to E p,q ∞ = (cid:40) E ( − r ) , if p + q = 0 , , otherwise . (4.7)To obtain a concrete model we were forced in [37] to impose an additional condition on the classof sheaves considered, namely H (cid:0) P , E ( − (cid:1) = 0. This condition is of course restrictive andwas introduced only as a matter of convenience. However, it excludes certain configurations ofsheaves that we are interested in, namely the ideal sheaves of points. Fortunately, this conditioncan be traded for a different one which still has the virtue of collapsing the spectral sequence to afour-term complex and includes the relevant ideal sheaves of points in the parametrization of themoduli space. By imposing H (cid:0) P , E ( − (cid:1) = 0 instead, the spectral sequence degenerates to thecomplex 0 (cid:47) (cid:47) V ⊗ O P ( − a (cid:47) (cid:47) V ⊕ ⊗ O P ( − b (cid:47) (cid:47) b (cid:47) (cid:47) ( V ⊕ ⊕ W ) ⊗ O P c (cid:47) (cid:47) V ⊗ O P (1) (cid:47) (cid:47) , (4.8)where V and W are complex vector spaces with dim C ( V ) = k and dim C ( W ) = N , whileim( a ) = ker( b ) and E = ker( c ) (cid:14) im( b ) . (4.9)This complex is just the dual of the complex considered in [37]. The precise form of the morphisms isnot important for the purposes of this section; they can essentially be read off from [37, Section 4.6].Indeed the choice of complex or its dual is somewhat irrelevant for the construction that follows.In particular, the two moduli spaces are isomorphic, so that we can freely borrow results from [37].We refer the reader to Appendix B and Appendix C for explicit proofs of these claims. In thefollowing we will simply use the complex (4.8). We are grateful to B. Szendr˝oi for pointing this out to us. .2 Intersection theory We begin our study of the instanton moduli space on C / Γ by reviewing some facts from the workof Ito and Nakajima [51] that will be useful in what follows. Their paper is an attempt to extend theMcKay correspondence to three-dimensional orbifolds of the form C / Γ, where Γ ⊂ SL (3 , C ), andtheir natural smooth crepant Calabi–Yau resolutions given by the Hilbert–Chow morphism π : X −→ C (cid:14) Γ (4.10)for the Γ-Hilbert scheme X = Hilb Γ ( C ) consisting of Γ-invariant zero-dimensional subschemes Y of C of length | Γ | such that H ( O Y ) is the regular representation of Γ. Roughly speaking, the McKaycorrespondence in this setting is the statement that any well-posed question about the geometry ofthe resolution X should have a Γ-equivariant answer on C . We will mostly use geometrical notionsin this section; in Section 5 we will comment on the description of the McKay correspondence interms of derived categories. This correspondence has been studied from a physical point of viewanalogous to ours in e.g. [52, 53, 54]; see also [55] for a related description.Consider the universal scheme Z ⊂ X × C with correspondence diagram Z q (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) q (cid:32) (cid:32) (cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64)(cid:64) X C (4.11)and define the tautological bundle R := q ∗ O Z . (4.12)Under the action of Γ on Z , the bundle R transforms in the regular representation. Its fibresare the | Γ | -dimensional vector spaces C [ z , z , z ] /I ∼ = H ( O Y ) for the regular representation of Γ,where I ⊂ C [ z , z , z ] is a Γ-invariant ideal corresponding to a zero-dimensional subscheme Y of C of length | Γ | . Multiplication on the fibres C [ z , z , z ] /I by the coordinates z α of C induces a Γ-equivariant homomorphism B : R → Q ⊗ R , with B ∧ B = 0 as an element of Hom Γ ( R , (cid:86) Q ⊗ R ).Here Q is the fundamental three-dimensional representation of Γ ⊂ SL (3 , C ). If we denote theorbifold action by ( z , z , z ) (cid:55)→ ( r · z , r · z , r · z ), with ρ r α the irreducible one-dimensionalrepresentation of Γ with weight r α , then Q = ρ r ⊕ ρ r ⊕ ρ r . This defines a (cid:98) Γ-colouring N → Γthrough the identification (cid:98) Γ ∼ = Γ by( n , n , n ) (cid:55)−→ ρ ⊗ n r ⊗ ρ ⊗ n r ⊗ ρ ⊗ n r , (4.13)which coincides with the colourings considered in Section 3. We will represent the element B ∈ Q ⊗ End Γ ( R ) by a triple of endomorphisms B = ( B , B , B ), with B ∧ B = (cid:80) α<β [ B α , B β ].The decomposition of the regular representation induces a decomposition of the tautological bundleinto irreducible representations R = (cid:77) r ∈ (cid:98) Γ R r ⊗ ρ r , (4.14)where { ρ r } r ∈ (cid:98) Γ is the set of irreducible representations; we denote the trivial representation by ρ .The tautological bundles R r = Hom Γ ( ρ r , R ) form an integral basis for the Grothendieck group K ( X ) of vector bundles on X , where the bundle corresponding to the trivial representation is the X is crepant if the canonical bundles are isomorphic, K X ∼ = π ∗ ( K C / Γ ); this constraint is required to obtain aCalabi–Yau structure on X from that of C / Γ. R ∼ = O X . Note in particular that they are not line bundles in general, since theirrank depends on the dimension of the irreducible representations of Γ; however, for Γ abelian, thecase considered in this paper, they are always line bundles. Similarly, one can introduce a dual basis S r of K c ( X ), the Grothendieck group of coherent sheaveson π − (0), or equivalently the Grothendieck group of bounded complexes of vector bundles over X which are exact outside the exceptional locus π − (0). The map between the two descriptions isobtained by sending a coherent sheaf on π − (0) to the complex of vector bundles which is a locallyfree resolution of it. The dual basis of K c ( X ) is S r : R ∨ r (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (2) rs R ∨ s (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (1) rs R ∨ s (cid:47) (cid:47) R ∨ r (4.15)where the arrows arise from the decomposition, according to (4.14), of the maps (cid:86) i Q ⊗ R B ∧ −−→ (cid:86) i +1 Q ⊗ R , and (cid:86) i Q ⊗ ρ r = (cid:77) s ∈ (cid:98) Γ a ( i ) sr ρ s with a ( i ) sr = dim C Hom Γ (cid:0) ρ s , (cid:86) i Q ⊗ ρ r (cid:1) . (4.16)Note that the determinant representation (cid:86) Q is trivial as a Γ-module since we assume that Γ is asubgroup of SL (3 , C ), hence (cid:86) Q ⊗ R ∼ = R and a (3) rs = δ rs . This also implies that (cid:86) Q ∼ = Q ∨ andtherefore a (2) sr = a (1) rs . These multiplicities can be computed explicitly from the decompositions Q ⊗ ρ r = ( ρ r ⊕ ρ r ⊕ ρ r ) ⊗ ρ r = ρ r + r ⊕ ρ r + r ⊕ ρ r + r , (4.17)which comparing with (4.16) gives a (1) rs = δ r,s + r + δ r,s + r + δ r,s + r and a (2) rs = δ r,s − r + δ r,s − r + δ r,s − r . (4.18)The purpose of this definition is to relate the representation theory of Γ with geometry. In thecase of the familiar McKay correspondence with ADE singularities, this is just the statement thatthe representation theory of the discrete orbifold group contains all the information about theintersection matrix of the ADE singularity. In higher dimensions there is no such simple anddirect statement, and the tensor product decomposition (4.16) into irreducible representations ofthe discrete group Γ enters rather more indirectly in the basis of K c ( X ).To see this in more detail, define the collection of dual complexes {S ∨ r } r ∈ (cid:98) Γ by S ∨ r : − (cid:104) R r (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (1) rs R s (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (2) rs R s (cid:47) (cid:47) R r (cid:105) . (4.19)On K c ( X ) we can define a perfect pairing( S , T ) = (cid:104) Θ S , T (cid:105) = (cid:90) X ch(Θ S ⊗ T ) ∧ Todd( X ) , (4.20)where (cid:104)− , −(cid:105) is the dual pairing between K ( X ) and K c ( X ), and Θ : K c ( X ) → K ( X ) is the mapwhich sends a complex of vector bundles to the corresponding element in K ( X ) (the alternatingsum of the elements of the complex). For exampleΘ S ∨ r = (cid:88) s ∈ (cid:98) Γ (cid:0) − δ rs + a (2) rs − a (1) rs + δ sr (cid:1) R s = (cid:88) s ∈ (cid:98) Γ (cid:0) a (2) rs − a (1) rs (cid:1) R s . (4.21) In the case of kleininan singularities the ranks are the entries of the vector which generates the kernel of the affineCartan matrix. Geometrically this is related to the annihilator of the quadratic form which gives the intersectionpairing. 18t follows that (cid:0) S ∨ r , S s (cid:1) = (cid:104) Θ S ∨ r , S s (cid:105) = (cid:88) q ∈ (cid:98) Γ (cid:0) a (2) rq − a (1) rq (cid:1) (cid:104)R q , S s (cid:105) = a (2) rs − a (1) rs , (4.22)where we have used the fact that {R s } s ∈ (cid:98) Γ and {S r } r ∈ (cid:98) Γ are dual bases of K ( X ) and K c ( X ). Thisresult underlies the relation between the tensor product decomposition (4.16) and the intersectiontheory of X . These two bases correspond, via the McKay correspondence, with two bases of Γ-equivariantcoherent sheaves on C , as shown in [51]. The Grothendieck groups of Γ-equivariant sheaveson C , K Γ ( C ) and K c Γ ( C ) (with coherent sheaves of compact support), have respective bases { ρ r ⊗ O C } r ∈ (cid:98) Γ and { ρ r ⊗ O } r ∈ (cid:98) Γ where O is the skyscraper sheaf at the origin; the latter basisis naturally identified as the set of fractional 0 branes. All of these groups are isomorphic to therepresentation ring R (Γ) of the orbifold group Γ. This correspondence will be used in the stabilityanalysis of Section 5.9. The main ingredient in the construction of the instanton moduli space is Beilinson’s theorem whichcan be used to parametrize the moduli space via a spectral sequence. A requisite technical ingredientis the resolution of the diagonal sheaf O ∆ of X × X , or better of its compactification. We willcollectively use the notation ∆ for the diagonal of a generic variety to simplify our notation; it willbe clear from the context which variety we are considering. Recall that to construct the Beilinsonspectral sequence on C one needs the identity p ∗ (cid:0) p ∗ E ⊗ O ∆ (cid:1) = p ∗ (cid:0) p ∗ E (cid:12)(cid:12) ∆ (cid:1) = E (4.23)where ∆ ∼ = P ⊂ P × P is the diagonal. This fact was extensively used in [37] to construct explicitlythe instanton moduli space on C . To proceed with the same construction for the crepant resolution X of an orbifold singularity C / Γ, we need two ingredients: a resolution of the diagonal sheaf O ∆ and a compactification of X (in the same way as P is a compactification of C ). We will proceedin two steps: first we find a resolution of the diagonal sheaf of X × X , and then we extend it to itscompactification.A resolution of the diagonal sheaf on X × X is obtained by generalising the argument of [56] tothe higher-dimensional case to get (cid:47) (cid:47) (cid:0) R (cid:2) R ∨ ⊗ (cid:86) Q ∨ (cid:1) Γ B ∧ (cid:47) (cid:47) (cid:0) R (cid:2) R ∨ ⊗ (cid:86) Q ∨ (cid:1) Γ B ∧ (cid:47) (cid:47) B ∧ (cid:47) (cid:47) (cid:0) R (cid:2) R ∨ ⊗ Q ∨ (cid:1) Γ B ∧ (cid:47) (cid:47) (cid:0) R (cid:2) R ∨ (cid:1) Γ Tr (cid:47) (cid:47) O ∆ . (4.24)To unpack this complex a bit, one has to write explicitly the decomposition of the tautologicalbundles from (4.14), use the tensor product decomposition of the representations (4.16), and then For example, in complex dimension two the pairing ( S ∨ r , S s ) would give the extended Cartan matrix of the ADEsingularity. In this exact sequence we explicitly use the Calabi–Yau condition to write the isomorphisms (cid:86) Q ∼ = C and (cid:86) Q ∼ = Q ∨ . (cid:16) R (cid:2) R ∨ ⊗ (cid:86) i Q ∨ (cid:17) Γ = (cid:16) (cid:77) r,s,q ∈ (cid:98) Γ R r ⊗ ρ r (cid:2) a ( i ) qs R ∨ s ⊗ ρ ∨ q (cid:17) Γ = (cid:77) r,s,q ∈ (cid:98) Γ R r (cid:2) a ( i ) sq R ∨ s ⊗ (cid:0) ρ r ⊗ ρ ∨ s (cid:1) Γ = (cid:77) r,s ∈ (cid:98) Γ R r (cid:2) a ( i ) rs R ∨ s , (4.25)where in the last step we used Schur’s lemma. Therefore every term of the resolution (4.24) canbe written as C i = (cid:76) r,s ∈ (cid:98) Γ R r (cid:2) a ( i ) rs R ∨ s .Now let us proceed to the compactification. The idea is that one can mimic directly the com-pactification of C into P by adding a boundary divisor. For P one uses the Koszul resolution(4.5) of the diagonal. We would like to think of the compactification of X to X as obtained bycompactifying C / Γ to P / Γ and then resolving the singularity at the origin, leaving untouchedthe divisor at infinity. This procedure is a generalisation of [42, 56]. It corresponds to an orbifoldcompactification X = X (cid:116) ℘ ∞ and we can regard ℘ ∞ ∼ = P / Γ. More precisely, we glue objects in X with Γ-invariant objects on ℘ ∞ to obtain something defined globally. Then, in a neighbourhood ofthe boundary divisor ℘ ∞ , X looks like P / Γ and the gluing is compatible with the orbifold actionof Γ on ℘ ∞ . We can then glue the resolution of the diagonal of X × X with the Koszul resolutionof the diagonal in P / Γ × P / Γ (i.e., the Koszul resolution of the diagonal of P where the sheavescarry an appropriate Γ-module structure) to get a globally defined resolution on X × X . Indeed,given the nature of the tautological bundles, the complex (4.24) on X × X has a natural projectionto C / Γ × C / Γ.Consider the Koszul resolution on P × P given in (4.5). The sheaf Q (which we would like to glueto Q , the trivial bundle on X which carries the fundamental representation of Γ) is defined with asuitable Γ-module structure through the dual Euler sequence0 (cid:47) (cid:47) O P ( − (cid:47) (cid:47) O ⊕ P (cid:47) (cid:47) Q (cid:47) (cid:47) , (4.26)where we regard O ⊕ P ∼ = ( Q ⊕ ρ ) ⊗ O P , i.e. we consider O ⊕ P as a trivial bundle where each factorcarries an action of Γ. The first three factors are collectively taken care of by the fundamentalrepresentation Q (which acts on C ), while the action of Γ on the fourth factor (which corresponds tothe projective coordinate z centred at the plane at infinity) is trivial. Equivalently, we projectivizethe action of Γ by letting it act trivially on the fourth coordinate z of P [ z : z : z : z ]corresponding to the patch at infinity. This induces a natural Γ-module structure on the sheaf Q via the last morphism in (4.26). We can finally glue together the resolution of the diagonal on X and on P to get a resolution of the diagonal of X × X given by0 (cid:47) (cid:47) (cid:0) R ( − ℘ ∞ ) (cid:2) R ∨ ⊗ (cid:86) Q ∨ (cid:1) Γ (cid:47) (cid:47) (cid:0) R ( − ℘ ∞ ) (cid:2) R ∨ ⊗ (cid:86) Q ∨ (cid:1) Γ (cid:47) (cid:47) (cid:47) (cid:47) (cid:0) R ( − ℘ ∞ ) (cid:2) R ∨ ⊗ Q ∨ (cid:1) Γ (cid:47) (cid:47) (cid:0) R (cid:2) R ∨ (cid:1) Γ (cid:47) (cid:47) O ∆ . (4.27) Consider the Fourier–Mukai transform (4.3) of a torsion free coherent sheaf E on X , where now C • denotes the resolution (4.27). Then by Beilinson’s theorem a sheaf E ( − l ) := E ⊗ O X ( − l ℘ ∞ ) on X is parametrized by a spectral sequence whose first term is E p,q = (cid:16) R ( p ) ⊗ H q (cid:0) X , E ( − l ) ⊗ R ∨ ⊗ (cid:86) − p Q ∨ (cid:1)(cid:17) Γ , (4.28)20here our conventions are p ≤ 0. We will argue that all homological algebra based on this spectralsequence reduce to the familiar case of P .The reason for this is that thanks to the tensor product decomposition (4.16) all the relevantcohomology groups are of the form H • (cid:0) X, E ( − l ) ⊗ R r (cid:1) , where R r is a line bundle (which restrictsto a trivial bundle at infinity, up to the Γ-module structure). Therefore the relevant complex vectorspaces are of the form V = H (cid:0) X , E ( − l ) ⊗ R ∨ (cid:1) = H (cid:16) X , E ( − l ) ⊗ (cid:76) r ∈ (cid:98) Γ R ∨ r ⊗ ρ ∨ r (cid:17) = (cid:77) r ∈ (cid:98) Γ V r ⊗ ρ ∨ r , (4.29)where V r are finite-dimensional vector spaces with trivial Γ-action. Recall that the sheaves (cid:86) − p Q ∨ , p ≤ − p P ( − p ) on P near infinity. Therefore, in a neighbourhood of ℘ ∞ one has E ( − l ) ⊗ R ∨ ⊗ (cid:86) − p Q ∨ ∼ = E ( − l ) ⊗ (cid:77) r ∈ (cid:98) Γ R ∨ r ⊗ ρ ∨ r ⊗ (cid:86) − p Q ∨ ∼ = (cid:77) r,s ∈ (cid:98) Γ a ( − p ) sr ρ ∨ s ⊗ (cid:0) E ( − l ) ⊗ Ω − p P ( − p ) ⊗ R ∨ r (cid:1) . (4.30)Next we have to impose boundary conditions on E . This condition is the same as the framingcondition on P . In particular, the tautological bundles R r only play a passive role since they aretrivial at infinity and their only function is to label a representation ρ r . Therefore we will againimpose that E (cid:12)(cid:12) ℘ ∞ is trivial on a line (cid:96) ∞ ⊂ ℘ ∞ with (cid:96) ∞ ∼ = P , which means that the associatedgauge connection is flat. A choice of different boundary conditions corresponds to using particularconfigurations of D branes as boundary conditions; although this might be related to the noncom-mutative topological vertex formalism of [57], or the orbifold topological vertex formalism of [58],we will leave such investigations for future work.The sheaf cohomology groups are now the same as in the P case (see Appendix B and Appendix C)up to multiplicity factors and the Γ-module structure. We can therefore jump directly to the con-clusion that we can represent the sheaf E as the single non-trivial cohomology of the complex0 (cid:47) (cid:47) (cid:0) V ⊗ R ( − (cid:1) Γ a (cid:47) (cid:47) (cid:0) V ⊗ (cid:86) Q ∨ ⊗ R ( − (cid:1) Γ b (cid:47) (cid:47) b (cid:47) (cid:47) (cid:0) ( V ⊗ Q ∨ ⊕ W ) ⊗ R (cid:1) Γ c (cid:47) (cid:47) (cid:0) V ⊗ R (1) (cid:1) Γ (cid:47) (cid:47) , (4.31)where the complex vector spaces appearing here are finite-dimensional. All the differentials involvedare exactly the same as in the P case, but decomposed equivariantly according to the Γ-modulestructure. The original sheaf E is recovered as in (4.9), while the vector space V is given in(4.29) with dim C V = k . The virtual bundle defined by the cohomology of the complex (4.31) is arepresentative (in equivariant K-theory) of the isomorphism class of the universal sheaf associatedto the (fine) instanton moduli space.The vector space W = H (cid:0) P , ker( b ) (cid:12)(cid:12) (cid:96) ∞ (cid:1) = (cid:77) r ∈ (cid:98) Γ W r ⊗ ρ ∨ r (4.32)parametrizes trivializations of the sheaf E , with dim C W = N . Asymptotically the associated gaugeconnection is flat. The spaces X that we are considering all have the form of a crepant resolution21f an orbifold singularity C / Γ. By [59, Theorem 8.2.3], these resolutions have a unique Ricci-flatALE metric asymptotic to the flat geometry C / Γ. As a smooth manifold, X looks like the Lensspace S / Γ at infinity. As such, the flat connections are labelled by representations of the orbifoldgroup; since π ( S ) = 0 we have π ( S / Γ) = π (Γ) = Γ where in our case Γ is a finite abelian group.Conjugacy classes of homomorphisms from this fundamental group to the gauge group U ( N ) aretherefore in correspondence with gauge equivalence classes of flat connections. This classification ismanifest in the decomposition into irreducible Γ-modules of the fibre at infinity (4.32). At infinity,the gauge sheaf is asymptotically in a representation ρ of the orbifold group Γ and the dimensionsdim C W r = N r label the multiplicities of the decomposition of ρ into irreducible representations,with the constraint N = (cid:88) r ∈ (cid:98) Γ N r . (4.33)From a string theory perspective the dimension k r of the vector space V r corresponds to the numberof fractional D0 branes which transform in the representation ρ r of Γ. The vector space W representsthe D6 branes, to which the D0 branes are bound. Let us make a heuristic check. Since V ⊗ (cid:86) i Q ∨ = (cid:77) r ∈ (cid:98) Γ V r ⊗ ρ ∨ r ⊗ (cid:86) i Q ∨ = (cid:77) r,s ∈ (cid:98) Γ a ( i ) rs V s ⊗ ρ ∨ r (4.34)we can write the complex (4.31) as0 (cid:47) (cid:47) (cid:77) r ∈ (cid:98) Γ V r ⊗ R r ( − a (cid:47) (cid:47) (cid:77) r,s ∈ (cid:98) Γ a (2) rs V s ⊗ R r (1) b (cid:47) (cid:47) b (cid:47) (cid:47) (cid:77) r,s ∈ (cid:98) Γ (cid:0) a (1) rs V s ⊗ R r (cid:1) ⊕ (cid:77) r ∈ (cid:98) Γ W r ⊗ R r c (cid:47) (cid:47) (cid:77) r,s ∈ (cid:98) Γ V r ⊗ R s (1) (cid:47) (cid:47) . (4.35)Let us consider the case of a single D0 brane, k r = 1 for some r ∈ (cid:98) Γ while k r (cid:48) = 0 for all r (cid:48) (cid:54) = r .Then the complex0 (cid:47) (cid:47) R r ( − a (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (2) sr R s ( − b (cid:47) (cid:47) b (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ (cid:0) a (1) sr R s (cid:1) ⊕ (cid:77) r ∈ (cid:98) Γ W r ⊗ R r c (cid:47) (cid:47) R r (1) (cid:47) (cid:47) N = dim C ( W ) D6 branes. Assume now that there areno D6 branes, N = 0, and ignore the boundary divisor ℘ ∞ (as here we are only interested in alocal argument, that can be set up in a neighbourhood of the exceptional locus). Then (4.36)becomes 0 (cid:47) (cid:47) R r (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (2) sr R s (cid:47) (cid:47) (cid:77) s ∈ (cid:98) Γ a (1) sr R s (cid:47) (cid:47) R r (cid:47) (cid:47) , (4.37)which is precisely the object that we called S r in Section 4.2 that denotes an element of a basis of K c ( X ). In other words, this is a sheaf supported on a cycle in the exceptional locus, precisely thebehaviour we would expect from a fractional brane. This is not quite true, as fractional branes in the derived category also have a shift in degree. 22t is natural now to interpret (4.36) as a bound state of the D0 brane, wrapping a vanishing cycle,with the D6 branes. And in full generality, we regard (4.31) as a bound state of a number of D0and D6 branes, which is precisely what we wanted to obtain. We can identify N = dim C W withthe number of D6 branes, which we will mostly assume to be just one in order to have a U (1)gauge theory. Note, however, that this is not necessary at this stage. Moreover, we will identify k r = dim C V r with the fractional instanton charge associated to the representation ρ ∨ r , which wewill see later on represents the number of boxes in a three-dimensional Young diagram of a givencolour as in Section 3. In Section 4.4 we have explicitly constructed the moduli space of framed instantons on X withfixed topological charges. This moduli space is parametrized via the Beilinson spectral sequenceby a sequence of linear maps between vector spaces. The maps are implicitly determined by thehomological algebra and are nothing else than a particular equivariant decomposition of the mapsalready derived in full generality in [37]. While it is in principle possible to derive them in a closedform, we will refrain from doing so and consider a simpler case, which is the only one where theanalysis can be practically carried on. This is the case where the gauge theory is abelian and N = dim C W = 1, or the gauge theory is non-abelian but in its Coulomb phase with the U ( N )gauge symmetry broken down to the maximal torus U (1) N . In both instances the intersectionindices of the moduli space can be computed directly via equivariant localization. For the moregeneral non-abelian case we have in principle an ADHM-like parametrization of the moduli space,but the analysis is complicated by the lack of suitable techniques and a poor understanding ofstability issues for generic torsion free coherent sheaves.In the U (1) case most of the fields that enter in the generalized ADHM parametrization derivedin [37] can be set to zero or traded for stability conditions, and only the “center of mass” coordinatesremain with [ B , B ] = 0 , [ B , B ] = 0 and [ B , B ] = 0 . (4.38)The decomposition of these maps according to the Γ-action can be neatly summarized in a quiverdiagram as we will explain in the ensuing sections. For the moment we will just remark that themain difference between the orbifold geometries and the flat case studied in [37] is that the choice of atrivialization at infinity carries also the information about inequivalent boundary conditions. Theseare reflected in the form of the universal sheaf on the instanton moduli space via the dependenceon the framing vector N = ( N r ) r ∈ (cid:98) Γ . If one restricts to the U (1) gauge theory then only one of thedimensions N r can be non-vanishing and precisely equal to one by (4.33). The non-abelian gaugetheory, already in its Coulomb branch, offers a combinatorially non-trivial host of possibilities.We therefore decompose the linear maps B ∈ Hom Γ ( V, Q ⊗ V ) with V = (cid:76) r ∈ (cid:98) Γ V r ⊗ ρ ∨ r as B = (cid:77) r ∈ (cid:98) Γ ( B r , B r , B r ) (4.39)where B rα : V r → V r + r α . Then the orbifold generalization of the ADHM equations (4.38) is B r + r α β B rα = B r + r β α B rβ , r ∈ (cid:98) Γ , (4.40)where α, β = 1 , , Q = ρ r ⊕ ρ r ⊕ ρ r . In the U (1) gauge theory there are in principle severalmoduli spaces, characterising instanton configurations with different asymptotics.We will argue that although these configurations are physically distinct, the relevant moduli spacesare isomorphic. Indeed, given a gauge field configuration one can always change its asymptotic23ehaviour by tensoring its gauge bundle with a tautological bundle. Tautological bundles canbe thought of as line bundles whose gauge field is the “elementary” configuration asymptotic toa particular irreducible representation of the orbifold group Γ. This heuristic picture is literallytrue since they form a basis of the topological K-theory group of X . Tensoring with line bundlesthus establishes an isomorphism between different moduli spaces of U (1) instantons with fixedboundary conditions. In particular the local structure of the moduli space is unchanged and so isthe contribution of an instanton configuration to the index of BPS states. What changes is only theform of the instanton action (see Section 4.7 for details). Because of this, when discussing partitionfunctions of BPS states in the U (1) gauge theory we will only consider a certain boundary condition,namely N = 1 corresponding to trivial representations at infinity, with the understanding that theanalysis for different boundary conditions is qualitatively similar. Γ -Hilbert scheme Let us now turn to the cohomology of X . We will review the construction of [60, 61] which givestwo bases of H ( X, Z ) and H ( X, Z ) dual to the bases of exceptional surfaces and curves in theresolution X . The algorithm is combinatorial and allows a direct construction of these bases startingfrom the basis of tautological bundles, labelled by the irreducible representations of the orbifoldgroup Γ. This will enable us to evaluate the integrals in the instanton action which involve theK¨ahler form ω of X . It is accomplished by expanding ω in the basis of H ( X, Z ) and ω ∧ ω in thebasis of H ( X, Z ) given by the tautological bundles.One starts from the triangulation Σ of the toric resolution X = Hilb Γ ( C ) and associates to each linein the diagram the Γ-invariant ratio of monomials which parametrizes that curve. This naturallyassociates to each line a character of the orbifold group; here and in the following we will use thesame symbol for a representation and its character. Then one can associate a character to eachvertex of (the interior of) Σ. Only the following cases are possible: • A vertex v of valency 3. This vertex defines an exceptional projective plane P . All threelines meeting at v are marked by the same character ρ r . Then mark the vertex v with thecharacter ρ m = ρ r ⊗ ρ r . • A vertex v of valency 4. This vertex defines an exceptional Hirzebruch surface F r . Two linesare marked with ρ r and two with ρ s . Then mark the vertex v with the character ρ m = ρ r ⊗ ρ s . • A vertex v of valency 5 or 6 (excluding the case where three straight lines meet at a point).This vertex defines an exceptional Hirzebruch surface F r blown up in one or two pointsrespectively. There are two uniquely determined characters ρ r and ρ s which each mark two lines. The remaining line or two lines are marked with distinct characters. Then mark thevertex v with the character ρ m = ρ r ⊗ ρ s . • A vertex v of valency 6 at the intersection of three straight lines. This vertex defines anexceptional del Pezzo surface d P of degree six. The straight lines are marked by threecharacters ρ r , ρ s and ρ q . Then the monomials defining the pair of morphisms d P → P lie inuniquely determined character spaces ρ l and ρ m obeying the relation ρ l ⊗ ρ m = ρ r ⊗ ρ s ⊗ ρ q .Then mark the vertex v with both characters ρ l and ρ m . We will not need this case explicitlyin this paper.Once Σ is “decorated” by the characters of Γ in this way, several geometrical properties are deter-mined combinatorially. As a start one can show that every non-trivial character of Γ appears inthe toric fan Σ precisely once as either: 24i) marking a line; (ii) marking a vertex; or(iii) the second character ρ l marking the intersection of three straight lines (we will not requirethis case).To anticipate where this is all going, characters along lines will correspond to curves while char-acters on vertices will correspond to surfaces, in a precise sense. Indeed one can prove [61] thatthe first Chern classes c ( R r ) associated with a character of the form (i) and (iii) form a basisof H ( X, Z ).The above decoration encodes the following relations between tautological bundles in the Picardgroup Pic( X ): • If ρ m = ρ r ⊗ ρ r marks a vertex of valency 3, then R m = R r ⊗ R r . • If ρ m = ρ r ⊗ ρ s marks a vertex of valency 4, then R m = R r ⊗ R s . • If ρ m = ρ r ⊗ ρ s marks a vertex of valency 5 or 6, then R m = R r ⊗ R s . • If ρ l and ρ m obeying ρ l ⊗ ρ m = ρ r ⊗ ρ s ⊗ ρ q mark the intersection point v of three straightlines, then R l ⊗ R m = R r ⊗ R s ⊗ R q .Given all of the above ingredients, one can find virtual bundles whose second Chern classes form abasis of H ( X, Z ) dual to the basis of H ( X, Z ) defined by the compact exceptional surfaces. Thevirtual bundles V m are defined as follows:(a) For each relation R m = R r ⊗ R r arising from a vertex of valency 3, define V m = ( R r ⊕ R r ) (cid:9) ( R m ⊕ O X ).(b) For each relation R m = R r ⊗ R s arising from a vertex of valency 4, define V m = ( R r ⊕ R s ) (cid:9) ( R m ⊕ O X ).(c) For each relation R m = R r ⊗ R s arising from a vertex of valency 5 or 6, define V m =( R r ⊕ R s ) (cid:9) ( R m ⊕ O X ).(d) For each relation R l ⊗ R m = R r ⊗ R s ⊗ R q arising from a vertex where three straight linesintersect, define V m = ( R r ⊕ R s ⊕ R q ) (cid:9) ( R l ⊕ R m ⊕ O X ).This completes the characterization of the cohomology of the Γ-Hilbert scheme in terms of thetautological bundles. We can now put together the results we have obtained so far to compute the instanton action inthe general case. Recall that for the U (1) gauge theory this action has the generic form S inst = g s (cid:90) X F A ∧ F A ∧ F A + 12 (cid:90) X ω ∧ F A ∧ F A + (cid:90) X ω ∧ ω ∧ F A . (4.41)Here ω is the K¨ahler form supported in X = X \ ℘ ∞ , and in the following we will understand (4.41)in terms of intersection indices of both compact and non-compact divisors of X and X .Every term in (4.41) can be computed from the knowledge of the Chern character ch( E ) of thesheaf E given as the only non-vanishing cohomology of the complex (4.31). This yieldsch( E ) = − ch (cid:16)(cid:0) V ⊗ R ( − (cid:1) Γ (cid:17) + ch (cid:16)(cid:0) V ⊗ (cid:86) Q ∨ ⊗ R ( − (cid:1) Γ (cid:17) This case also allows for a line, not necessarily straight, passing through several vertices, and thus stretches a bitthe meaning of “precisely once”. See [60, 61] for a discussion of this point. ch (cid:16)(cid:0) ( V ⊗ Q ∨ ⊕ W ) ⊗ R (cid:1) Γ (cid:17) + ch (cid:16)(cid:0) V ⊗ R (1) (cid:1) Γ (cid:17) . (4.42)From this equation we get the instanton numbers c ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) c ( R r )+ (cid:16) δ rs c (cid:0) O X ( − (cid:1) − a (2) rs c (cid:0) O X ( − − δ rs c (cid:0) O X (2) (cid:1)(cid:17) k s (cid:19) , ch ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) ch ( R r ) (4.43)+ c ( R r ) ∧ (cid:16) δ rs c (cid:0) O X ( − (cid:1) − a (2) rs c (cid:0) O X ( − − δ rs c (cid:0) O X (1) (cid:1)(cid:17) k s + (cid:16) δ rs ch (cid:0) O X ( − (cid:1) − a (2) rs ch (cid:0) O X ( − (cid:1) − δ rs ch (cid:0) O X (1) (cid:1)(cid:17) k s (cid:19) , ch ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) ch ( R r )+ c ( R r ) ∧ (cid:16) δ rs ch (cid:0) O X ( − (cid:1) − a (2) rs ch (cid:0) O X ( − (cid:1) − δ rs ch (cid:0) O X (1) (cid:1)(cid:17) k s + ch ( R r ) ∧ (cid:16) δ rs c (cid:0) O X ( − (cid:1) − a (2) rs c (cid:0) O X ( − (cid:1) − δ rs c (cid:0) O X (1) (cid:1)(cid:17) k s + (cid:16) δ rs ch (cid:0) O X ( − (cid:1) − a (2) rs ch (cid:0) O X ( − (cid:1) − δ rs ch (cid:0) O X (1) (cid:1)(cid:17) k s (cid:19) which give an unambiguous instanton action upon integration. Note that c ( R ) = c ( O X ) = 0.From the behaviour of the Chern characters under tensor product we deduce c (cid:0) O X ( − (cid:1) = − c (cid:0) O X (1) (cid:1) ,c (cid:0) O X (2) (cid:1) = 2 c (cid:0) O X (1) (cid:1) , ch (cid:0) O X ( − (cid:1) = ch (cid:0) O X (1) (cid:1) , ch (cid:0) O X (2) (cid:1) = 4 ch (cid:0) O X (1) (cid:1) , ch (cid:0) O X ( − (cid:1) = − ch (cid:0) O X (1) (cid:1) , ch (cid:0) O X (2) (cid:1) = 8 ch (cid:0) O X (1) (cid:1) . (4.44)Together with (4.18) this allows us to simplify (4.43) as c ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) c ( R r ) , ch ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) ch ( R r )+ c (cid:0) O X (1) (cid:1) ∧ c ( R r ) (cid:0) a (2) rs − δ rs (cid:1) k s (cid:19) , (4.45)ch ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) ch ( R r ) + c (cid:0) O X (1) (cid:1) ∧ ch ( R r ) (cid:0) a (2) rs − δ rs (cid:1) k s − c ( R r ) ∧ ch (cid:0) O X (1) (cid:1) (cid:0) a (2) rs − δ rs (cid:1) k s − k s δ rs ch (cid:0) O X (1) (cid:1)(cid:19) . c ( R n ) ∈ H ( X, Z ) and c ( V m ) ∈ H ( X, Z ), we may expand ω = (cid:88) n ∈ (cid:98) Γ ϕ n c ( R n ) and ω ∧ ω = (cid:88) m ∈ (cid:98) Γ ς m c ( V m ) , (4.46)where the parameters ϕ n and ς m play the role of chemical potentials for the D4 and D2 branes.Strictly speaking the K¨ahler class is not an integral class, but there is evidence that it is quantisedin topological string theory [2]. It would be interesting to make this identification more precise bycomputing the D brane charges in our formalism and make a connection with [27].In any case, the resulting integrals over X compute intersection indices among the various compactand non-compact divisors in the geometry, which in general depend on the details of the particularorbifold in question. In particular, the integral of ch (cid:0) O X (1) (cid:1) computes the triple intersection ofthe divisor ℘ ∞ ∼ = P / Γ at infinity. Since three planes P intersect at a point in P and we evaluateorbifold integrals by pullback (see Appendix A), it is given by (cid:90) X ch (cid:0) O X (1) (cid:1) = 16 (cid:90) X c (cid:0) O X (1) (cid:1) ∧ c (cid:0) O X (1) (cid:1) ∧ c (cid:0) O X (1) (cid:1) = 16 | Γ | . (4.47)We have now arrived to the final form of the instanton action given by (cid:90) X ω ∧ ω ∧ c ( E ) = − (cid:88) m,r,s ∈ (cid:98) Γ ς m (cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) (cid:90) X c ( V m ) ∧ c ( R r ) , (4.48) (cid:90) X ω ∧ ch ( E ) = − (cid:88) n,r,s ∈ (cid:98) Γ ϕ n (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) (cid:90) X c ( R n ) ∧ ch ( R r ) (4.49)+ (cid:0) a (2) rs − δ rs (cid:1) k s (cid:90) X c ( R n ) ∧ c (cid:0) O X (1) (cid:1) ∧ c ( R r ) (cid:19) , (cid:90) X ch ( E ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) k s (cid:17) (cid:90) X ch ( R r ) − k s | Γ | δ rs + (cid:0) a (2) rs − δ rs (cid:1) k s (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r ) − (cid:0) a (2) rs − δ rs (cid:1) k s (cid:90) X c ( R r ) ∧ ch (cid:0) O X (cid:0) (cid:1)(cid:19) . (4.50)Note that the “regular” instanton configurations are of the form k r = k for all r ∈ (cid:98) Γ and can benaturally associated with the regular representation (cid:76) r ∈ (cid:98) Γ ρ r of the orbifold group Γ, or equivalentlywith the tautological bundle (4.12). If we consider those instantons which asymptote to the trivialrepresentation at infinity, so that only the framing integer N is non-zero, then by (4.18) and c ( R ) = 0 the integrals involving c ( E ) in (4.48) and ch ( E ) in (4.49) vanish identically, while theinstanton charge (cid:82) X ch ( E ) = k is integer-valued. The instanton moduli space is characterized by a set of equations which generalize the usual ADHMformalism and arise from the degeneration of the Beilinson spectral sequence. But the spectralsequence actually comes with more information. The parametrization of the moduli space of torsion27ree sheaves can be realized via an appropriate quiver, whose vertices correspond to the Γ-moduledecomposition of the vector spaces V and W representing certain cohomology groups in (4.29) and(4.32). The arrows of the quiver are the elements in the decomposition of the differentials whichenter in the spectral sequence, according to their Γ-module structure. We will argue that thisquiver is precisely the framed McKay quiver associated with the orbifold group Γ. We will see thatthe effective action studied in [22] as a low-energy limit of the theory of D0 and D2 branes in thebackground of a single D6 brane is recovered geometrically from the matrix quantum mechanicswhich governs the measure of the instanton moduli space. We will start with a quick review of some facts concerning quivers and their representations, referingthe reader to the reviews [62, 63, 64] for further details. A quiver Q is a directed graph constructedfrom a set of vertices Q and a set of arrows Q connecting the vertices. This information is encodedin maps t , h : Q ⇒ Q that identify the tail and head vertices of each arrow, respectively. A path p in the quiver from a vertex v to the vertex w is a composition of arrows p = a · · · a k such that h ( a m ) = t ( a m +1 ) for 1 ≤ m < k , and t ( p ) := t ( a ) = v while h ( p ) := h ( a k ) = w ; in this case wesay that the path p has length k . In particular each vertex v has associated to it a trivial path e v of length zero, which starts and ends at the same vertex t ( e v ) = h ( e v ) = v . This should not beconfused with a loop, which is a non-trivial path from a vertex to itself of length one.The collection of paths form an associative noncommutative algebra, the path algebra C Q of thequiver Q , with the product of two paths defined by concatenation if the paths compose and zerootherwise. It is graded by path length. The elements e v for v ∈ Q are orthogonal idempotents inthis algebra, i.e. e v e w = δ v,w e v , such that (cid:80) v ∈ Q e v is the identity element of C Q . A relation r of the quiver is a C -linear combination of paths in C Q with the same head and tail vertices, andlength at least two. A bounded quiver ( Q , R ) is a quiver together with a finite set R of relations;they determine an ideal (cid:104) R (cid:105) in the path algebra C Q .We are interested in representations of the quiver Q . They form a category which is equivalent to C Q –mod, the category of finitely-generated left C Q -modules (or equivalently right C Q -modules).For any left C Q -module V , we can form the complex vector spaces V v = e v V for v ∈ Q ofdimension k v . We think of each vector space as living on a vertex of the quiver. The arrows v → w of the quiver induce linear transformations between the vector spaces V v → V w . If the quiver hasrelations R , we furthermore require that the linear maps be compatible with the relations, i.e. thesum of compositions of linear maps corresponding to the relations gives the zero map, and similarlyfor the factor path algebra A = C Q / (cid:104) R (cid:105) . The category of representations of a quiver with relations,Rep C ( Q , R ), is equivalent to the category A –mod of left A -modules.To each vertex v ∈ Q we can associate a one-dimensional simple module D v as the representationwhere V v = C and V w = 0 for all w (cid:54) = v . In the string theory setting these modules correspond tofractional branes. Furthermore we can define P v = e v A , the subspace of the path algebra generatedby all paths that begin at the vertex v . The usefulness of the modules P v is that they are projectiveobjects in the category A –mod which can be used to construct projective resolutions of the simplemodules D v through . . . (cid:47) (cid:47) (cid:77) w ∈ Q d pw,v P w (cid:47) (cid:47) . . . (cid:47) (cid:47) (cid:77) w ∈ Q d w,v P w (cid:47) (cid:47) P v (cid:47) (cid:47) D v (cid:47) (cid:47) d pw,v = dim C Ext p A ( D v , D w ) . (5.2)28hen a quiver has an underlying geometrical interpretation, perhaps via an equivalence of de-rived categories, it is often easier to rephrase the geometrical computations in this algebraic fash-ion. On each node v ∈ Q of the quiver there is a natural GL ( k v , C )-action by basis change automor-phisms. We are thus naturally led to consider the moduli space of isomorphism classes of quiverrepresentations, by factoring the action of the group G k = (cid:89) v ∈ Q GL ( k v , C ) , (5.3)where k := ( k v ) v ∈ Q is the dimension vector characterising the quiver representation. However thedirect quotient is rather badly behaved. The usual strategy in algebraic geometry is to resort togeometric invariant theory. This produces a smooth quotient at the price of having to discard certainorbits of the complexified G k -action. One restricts the quotient to only stable representations thatare defined in a purely algebraic manner via the slope stability parameter [65], which is given forany representation V = (cid:76) v ∈ Q V v with dimension vector k as θ ( V ) = θ ( k ) = θ · k dim C V (5.4)where θ ∈ R Q and dim C V = (cid:80) v ∈ Q k v . A representation V is θ -stable (resp. θ -semistable) if forany proper subrepresentation V (cid:48) ⊂ V one has θ ( V (cid:48) ) < θ ( V ) (resp. θ ( V (cid:48) ) ≤ θ ( V )). Then the modulispace of θ -stable representations is well-behaved and fine (and for generic values of the stabilityparameters θ we do not have to distinguish between stable and semistable representations).The other notion we need is that of a framing of a quiver Q . There are several (equivalent) notionsof framing of quiver representations; here we will follow the treatment of Joyce and Song [18,Section 7.4]. This operation consists in defining a new quiver Q f whose vertex set is doubledcompared to that of Q , i.e. Q f = Q (cid:116) Q . To each vertex v ∈ Q of the original quiver, therecorresponds a new vertex v (cid:48) of Q f (the double of v ) and an additional arrow a I : v (cid:48) → v . Similarlythe representation theory of Q f now involves two sets of vector spaces V v and W v together withadditional maps I v : W v → V v , and we can introduce the notion of framed representations. Thestability of framed representations is essentially the same as stability of the representations before the framing. This defines the moduli spaces of representations of framed quivers Rep θ (cid:0) Q f , k , N (cid:1) with fixed dimension vectors k and N . We will now rephrase the construction of the instanton moduli space in terms of an auxilliaryquiver, the McKay quiver, derived from the representation theory data for the action of the orbifoldgroup Γ. Recall that the problem we are studying has a double life: the representation theory ofΓ-equivariant O C -modules on C and the geometry of the crepant resolution of C / Γ given bythe Γ-Hilbert scheme Hilb Γ ( C ) which parametrizes Γ-invariant configurations of D0 branes. Weshall present a quiver which encodes this construction and has the generalized ADHM equationsas relations.We proceed from the point of view of representation theory. To begin with, we consider all theirreducible representations of Γ ⊂ SL (3 , C ). To each of these representations we associate a tau-tological bundle. We construct the quiver Q by declaring that each node represents a different When a bounded derived category of quiver representations is identified with a bounded derived category ofcoherent sheaves, the simple modules D v correspond to the basis of compactly supported sheaves S v while therepresentations P v which enter in the projective resolutions are related to the basis of tautological bundles R v , whichindeed give locally free resolutions of the sheaves S v . Q = (cid:98) Γ. Two nodes are connected by a number a (1) sr of arrowsgoing from s to r determined by the tensor product decomposition (4.16) for i = 1. Note thatin general the matrix a (1) sr does not have any particular symmetry property (in contrast to thefamiliar case of instantons on ALE spaces where it would be symmetric). The resulting quiver isknown as the bounded McKay quiver ( Q , R ) and it is associated with an ideal of relations (cid:104) R (cid:105) inthe corresponding path algebra C Q .In concrete applications one is interested in the representations of this quiver, which are obtainedby associating with every vertex r a k r -dimensional complex vector space V r and a linear map,represented by a k r × k s matrix, to every arrow from V s to V r . Then the relations between thearrows of the quiver induce equivalence relations between the morphisms of the representations.They can be compactly rewritten in terms of the Γ-equivariant linear map B ∈ Hom Γ ( V, Q ⊗ V )introduced in Section 4.4, and assume the simple form given in (4.40).Associated with this quiver is its moduli space of representations Rep ( Q , R ). This is not quite theend of the story, as there is a natural GL ( V r , C )-action on each vector space V r which lifts to thelinear maps B rα for α = 1 , , r ∈ (cid:98) Γ as B rα (cid:55)−→ g r + r α B rα g r with g r ∈ GL ( V r , C ) . (5.5)Therefore the relevant moduli space is actually the quotient in geometric invariant theory of Rep ( Q , R ) by this group action of (cid:81) r ∈ (cid:98) Γ GL ( V r , C ). The representation theory of quivers with dimension vectors k = (1 , . . . , 1) and dim C W = 1 isintimately related to the smooth geometry of toric varieties. Many toric varieties can be realizedas moduli spaces of representations of a quiver [66]. In many cases, and in particular for thequivers we will consider in this paper, this moduli space of representations, constrained by anappropriate ideal of relations, is a smooth crepant resolution of a toric singularity for generic valuesof the stability parameters. In particular, a moduli space of representations of the bounded McKayquiver with fixed dimension vector k = (1 , . . . , 1) is precisely isomorphic to the natural crepantresolution of an abelian orbifold singularity C / Γ provided by the Γ-Hilbert scheme Hilb Γ ( C ) [51];these quivers correspond to regular instantons. This holds in the chamber of moduli space in whichthe stability parameter θ ρ is positive (see e.g. [63, Remark 4.20] for the definition of this stabilityparameter).Under certain circumstances the path algebra A of the quiver itself can be regarded as anotherdesingularization, the noncommutative crepant resolution of the singularity [24]. In this case A represents a noncommutative deformation of a variety which contains the coordinate algebra of thesingularity as its center. Moreover, the noncommutative space dual to A “knows” about all theother projective crepant resolutions, in the sense that there exists a derived equivalence betweenthe corresponding bounded derived categories of A -modules and of coherent sheaves [67]. Forexample, if the bounded derived category D ( X ) of coherent sheaves on a crepant resolution X ofthe singularity is generated by a tilting object T , then setting A = End D ( X ) ( T ) induces a derivedequivalence D ( X ) ∼ = D ( A ) and A is a noncommutative crepant resolution of its center, with A –mod the category of coherent sheaves on the noncommutative scheme Spec( A ). We explain thisequivalence in more detail in Section 5.9.The path algebra of the McKay quiver for abelian orbifold singularities C / Γ gives a noncommuta-tive crepant resolution. The path algebra of the bounded McKay quiver is isomorphic to the skew30roup algebra [68, Proposition 2.8], which is the standard noncommutative crepant resolution ofthe Γ-invariant ring as in [24]. Moreover, there exists a derived equivalence between the corre-sponding bounded derived categories of A -modules and of coherent sheaves; this is a special caseof [69].Consider for example the orbifold C / Z with the diagonal action [70], as in Section 3.2. Therelevant quiver is v • (cid:120) (cid:120) (cid:0) (cid:0) (cid:123) (cid:123) v • (cid:47) (cid:47) (cid:45) (cid:45) (cid:49) (cid:49) • v (cid:94) (cid:94) (cid:102) (cid:102) (cid:99) (cid:99) (5.6)with weights r α = 1 for α = 1 , , 3, i.e. in this case Q = ρ ⊕ ρ ⊕ ρ . The maps b rα : D r → D r + r α mod 3 corresponding to the arrows of the quiver are of the form b α : D −→ D , b α : D −→ D , b α : D −→ D (5.7)for α = 1 , , 3. The relations are obtained by unpacking the generic form b r + r α β b rα = b r + r β α b rβ ;explicitly we find b b = b b , b b = b b , b b = b b , b b = b b , b b = b b , b b = b b , b b = b b , b b = b b , b b = b b . (5.8)Consider now the associated path algebra A and its center Z ( A ). As a ring, Z ( A ) is generated byelements of the form x αβγ = b α b β b γ with α ≤ β ≤ γ . (5.9)If we choose coordinates ( z , z , z ) on C on which the orbifold group Γ = Z acts as in (3.21),then we can explicitly map these generators to the Γ-invariant elements of the polynomial algebra C [ z , z , z ] as x αβγ = z α z β z γ . This means thatSpec Z ( A ) = C (cid:14) Z , (5.10)and hence the path algebra A is a noncommutative resolution of the C / Z singularity which is seenas its center. By the McKay correspondence there is a bounded derived equivalence D ( A ) ∼ = D ( X ),where the local del Pezzo surface X = O P ( − 3) of degree zero is the unique crepant Calabi–Yauresolution of C / Z obtained by blowing up the singular point at the origin of C to a projectiveplane P [54]. C In this section we describe the quantum mechanics which govern the dynamics of the instantoncollective coordinates. It arises as the dimensional reduction of the noncommutative D6 branegauge theory of Section 3 to the D0 branes. This model is topological and exactly solvable; thestudy of these types of quantum theories was pioneered in [71, 72]. This theory is in its simplest31orm on the affine Calabi–Yau space C ; we begin by briefly reviewing this model following [37].In this case it is based on two vector spaces V and W , of complex dimensions dim C V = k anddim C W = N , introduced in (4.29) and (4.32). In the D brane picture V is spanned by the gas of k D0 branes, while W represents the Chan–Paton bundle on the N (spectator) D6 branes. In thisdescription we fix the topological sector and restrict attention to instantons of charge k .The fields of the quiver are given by X i = ( B , B , B , ϕ, I ) and Ψ i = ( ψ , ψ , ψ , ξ, (cid:37) ) . (5.11)The matrices B α arise from 0–0 strings and represent the position of the coincident D0 branes insidethe D6 branes. The field I describes open strings stretching from the D6 branes to the D0 branes; itcharacterizes the size and orientation of the D0 branes inside the D6 branes, and is required to makethe system supersymmetric. In the noncommutative gauge theory the field ϕ is the dimensionalreduction of the (3 , ρ . Consistently with this open string interpretation we can regardthese fields as linear maps( B , B , B , ϕ ) ∈ Hom C ( V, V ) and I ∈ Hom C ( W, V ) . (5.12)The fields B α and ϕ all lie in the adjoint representation of U ( k ) where k is the number of D0 branes(or the instanton number). On the other hand, I is a U ( k ) × U ( N ) bifundamental field where N is the number of D6 branes (or the rank of the six-dimensional gauge theory). Under the fullsymmetry group U ( k ) × U ( N ) × T the transformation rules are B α (cid:55)−→ e − i (cid:15) α g U ( k ) B α g † U ( k ) ,ϕ (cid:55)−→ g U ( k ) ϕ g † U ( k ) ,I (cid:55)−→ g U ( k ) I g † U ( N ) , (5.13)where in the transformation of ϕ we have imposed the Calabi–Yau condition (cid:15) + (cid:15) + (cid:15) = 0. Thecorresponding BRST transformations read Q B α = ψ α and Q ψ α = [ φ, B α ] − (cid:15) α B α , Q ϕ = ξ and Q ξ = [ φ, ϕ ] , Q I = (cid:37) and Q (cid:37) = φ I − I a , (5.14)where φ is the generator of U ( k ) gauge transformations and a = diag( a , . . . , a N ) is a backgroundfield which parametrizes an element of the Cartan subalgebra u (1) ⊕ N ; in the noncommutative gaugetheory a plays the role of the vev of the Higgs field Φ.The information about the bosonic field content can be summarized in the quiver diagram V • B (cid:53) (cid:53) B (cid:24) (cid:24) B (cid:70) (cid:70) ϕ (cid:105) (cid:105) • W I (cid:111) (cid:111) (5.15)The relationship between the collective coordinates and quivers will also hold for more generalgeometries. The quiver quantum mechanics on C is characterized by the bosonic field equa-32ions E α : [ B α , B β ] + (cid:88) γ =1 (cid:15) αβγ (cid:2) B † γ , ϕ (cid:3) = 0 , E λ : (cid:88) α =1 (cid:2) B α , B † α (cid:3) + (cid:2) ϕ , ϕ † (cid:3) + I I † = λ , E I : I † ϕ = 0 , (5.16)where λ > (cid:126)χ, (cid:126)H ), whichcontain the antighost and auxilliary fields (cid:126)χ = ( χ , χ , χ , χ λ , ζ ) and (cid:126)H = ( H , H , H , H λ , h ). Sincethe auxilliary fields are determined by the equations (cid:126) E , they carry the same quantum numbers.This implies that the antighost fields are defined as maps( χ , χ , χ , χ λ ) ∈ Hom C ( V, V ) and ζ ∈ Hom C ( V, W ) , (5.17)and that the BRST transformations associated with the new fields are Q χ α = H α and Q H α = [ φ, χ α ] + (cid:15) α χ α , Q χ λ = H λ and Q H λ = [ φ, χ λ ] , Q ζ = h and Q h = a ζ − ζ φ . (5.18)To these fields we add the gauge multiplet ( φ, φ, η ) to close the algebra Q φ = 0 , Q φ = η and Q η = (cid:2) φ , φ (cid:3) . (5.19)The action that corresponds to this system of fields and equations is given by S = Q Tr (cid:16) (cid:88) α =1 χ † α ( H α − E α ) + χ λ ( H λ − E λ ) + ζ † ( h − E I )+ (cid:88) α =1 ψ α (cid:2) φ , B † α (cid:3) + ξ (cid:2) φ , ϕ † (cid:3) + (cid:37) φ I † + η (cid:2) φ , φ (cid:3) + h.c. (cid:17) . (5.20)This action is topological and the path integral localizes onto the fixed points of the BRST charge Q given by the equations( B α ) ab ( φ a − φ b − (cid:15) α ) = 0 and I a,l ( φ a − a l ) = 0 . (5.21)Solutions to these equations can be completely characterized by N -vectors of plane partitions (cid:126)π = ( π , . . . , π N ) with | (cid:126)π | = (cid:80) l | π l | = k boxes [37].The fluctuation determinants can be evaluated via standard techniques [71, 72, 37]. However, forpractical purposes it is more efficient to construct a local model of the instanton moduli spacearound each fixed point (cid:126)π of the T -action via the instanton deformation complex [37]Hom C ( V (cid:126)π , V (cid:126)π ) σ (cid:47) (cid:47) Hom C ( V (cid:126)π , V (cid:126)π ⊗ Q ) ⊕ Hom C ( W (cid:126)π , V (cid:126)π ) ⊕ Hom C ( V (cid:126)π , V (cid:126)π ⊗ (cid:86) Q ) τ (cid:47) (cid:47) Hom C ( V (cid:126)π , V (cid:126)π ⊗ (cid:86) Q ) ⊕ Hom C ( V (cid:126)π , W (cid:126)π ⊗ (cid:86) Q ) (5.22)33here the module Q ∼ = C is a carrier space for the torus action with weights t − α = e − i (cid:15) α . Thecharacter of this complex at a fixed point (cid:126)π is given byChar (cid:126)π ( t , t , t ) = W ∨ (cid:126)π ⊗ V (cid:126)π − V ∨ (cid:126)π ⊗ W (cid:126)π + (1 − t ) (1 − t ) (1 − t ) V ∨ (cid:126)π ⊗ V (cid:126)π , (5.23)where we have used the Calabi–Yau condition to set t t t = 1. From this character one cancompute the fluctuation determinant around an instanton solution by the standard conversionformula n (cid:88) i =1 n i e w i −→ n (cid:89) i =1 w n i i , (5.24)where w i = w i ( (cid:15) , (cid:15) , (cid:15) ) are the weights of the toric action on the instanton moduli space. Asexplained in [37], the equivariant index (5.23) in this way computes the ratio of fluctuation de-terminants in the noncommutative gauge theory on C . Below we use this rule to compute theinstanton measure in the noncommutative gauge theory on C / Γ. C / ΓLet us consider now the orbifold case. As we have seen from the study of the Beilinson spectralsequence the structure of the moduli space can be roughly speaking obtained from the instantonmoduli space on C by decomposing each morphism equivariantly according to the Γ-action. Thisperspective has an obvious extension to the instanton quantum mechanics as now each of the fieldsinvolved in the multiplets should be regarded as an equivariant morphism which can be decomposedanalogously to the linear maps (4.39). The relevant bosonic fields and their equations of motionscan be conveniently rephrased in terms of an auxilliary quiver which is essentially the McKay quiverintroduced in Section 5.2, up to some modifications which we will now explain.A first rather irrelevant modification is the addition of the fields ϕ r for r ∈ (cid:98) Γ which play the roleof extra arrows. However, as discussed in [37] we are only interested in those field configurationson which ϕ vanishes identically. This corresponds to adding the new fields on the quiver with newrelations in the path algebra that sets the fields to zero.A somewhat different role is played by the field I . It corresponds to the addition of a vectorspace W r for every vector space V r and a set of linear maps I r : W r → V r for each r ∈ (cid:98) Γ. Thisoperation corresponds precisely to the framing of the quiver discussed in Section 5.1, and indeed inour setting it corresponds to the framing of the moduli space of torsion free sheaves. Typically theframing operation in the usual ADHM formalism involves further sets of fields J r , K r : V r → W r , asreviewed for example in [62]. However, for the U (1) gauge theory or the non-abelian gauge theoryin its Coulomb branch the fields J, K can also be set to zero and substituted by suitable stabilityconditions [37].When the gauge theory is considered on orbifolds C / Γ the construction of Section 5.4 requiresmodification. The orbifold quantum mechanics is constructed to count Γ-equivariant coherentsheaves of compact support on C . One could in principle extend our analysis to the whole systemof quiver quantum mechanics fields. The resulting topological field theory is defined by an actionwhich localizes on the relations of the McKay quiver and is invariant under a set of Γ-equivariantBRST transformations. It is a fairly easy exercise to obtain the explicit formulas, but we refrainfrom writing them down.The quantum mechanics is constructed in essentially the same way as the one for C , but instead ofstarting from the generalized ADHM quiver (5.15) and the associated equations (5.16), one beginswith a modified McKay quiver associated with the singularity and a different set of equations. Nowthe bosonic field content is made up of equivariant matrices( B , B , B , ϕ ) ∈ Hom Γ ( V, V ) and I ∈ Hom Γ ( W, V ) . (5.25)34f we decompose the vector spaces V and W as Γ-modules into irreducible representations r ∈ (cid:98) Γ asin (4.29) and (4.32), then the non-vanishing isotopical components of these fields are maps B rα : V r −→ V r + r α ,ϕ r : V r −→ V r + r + r + r ∼ = V r ,I r : V r −→ V r , (5.26)where as before we have parametrized the fundamental representation of the orbifold group as Q = ρ r ⊕ ρ r ⊕ ρ r and used the fact that the determinant representation is trivial due to theCalabi–Yau condition. These maps uniquely determine the generalized framed McKay quiver interms of the decomposition of the fundamental representation into irreducible Γ-modules.The BRST transformations respect the Γ-module structure and are given by Q B rα = ψ rα and Q ψ rα = [ φ, B rα ] − (cid:15) α B rα , Q ϕ r = ξ r and Q ξ r = [ φ, ϕ r ] , Q I r = (cid:37) r and Q (cid:37) r = φ I r − I r a r , (5.27)where now the vector a r collects all the Higgs field eigenvalues a l associated with the irreduciblerepresentation ρ r . We will discuss their role more thoroughly in Section 5.8. The bosonic equationsof motion change as well into a set of matrix equations labelled by the irreducible representations r ∈ (cid:98) Γ as E rα : B r + r β α B rβ − B r + r α β B rα + (cid:88) γ =1 (cid:15) αβγ (cid:16)(cid:0) B r − r γ γ (cid:1) † ϕ r − ϕ r − r γ (cid:0) B r − r γ γ (cid:1) † (cid:17) = 0 , E rλ : (cid:88) α =1 (cid:16) B r − r α α (cid:0) B r − r α α (cid:1) † − (cid:0) B rα (cid:1) † B rα (cid:17) + (cid:2) ϕ r , ( ϕ r ) † (cid:3) + I r ( I r ) † = λ r , E rI : ( I r ) † ϕ r = 0 , (5.28)where λ r > {E rα , E rI } arises as an ideal of relations in the path algebra ofthe generalized quiver.The multiplets of antighost and auxilliary fields can be added in a similar way. The Γ-modulestructure of the auxilliary fields is dictated by the equations (5.28). The resulting antighost fieldsdecompose as equivariant maps( χ , χ , χ , χ λ ) ∈ Hom Γ ( V, V ) and ζ ∈ Hom Γ ( V, W ) , (5.29)and the BRST transformations close upon adding Q χ rα = H rα and Q H rα = [ φ, χ rα ] + (cid:15) α χ rα , Q χ rλ = H rλ and Q H rλ = [ φ, χ rλ ] , Q ζ r = h r and Q h r = a r ζ r − ζ r φ (5.30)together with the gauge multiplet.The partition function of the topological quantum mechanics can be computed by considering theequivariant version of the instanton deformation complex and using the localization formula as35xplained in Section 5.4. In this case the complex isHom Γ ( V (cid:126)π , V (cid:126)π ) σ (cid:47) (cid:47) Hom Γ ( V (cid:126)π , V (cid:126)π ⊗ Q ) ⊕ Hom Γ ( W (cid:126)π , V (cid:126)π ) ⊕ Hom Γ ( V (cid:126)π , V (cid:126)π ⊗ (cid:86) Q ) τ (cid:47) (cid:47) Hom Γ ( V (cid:126)π , V (cid:126)π ⊗ (cid:86) Q ) ⊕ Hom Γ ( V (cid:126)π , W (cid:126)π ⊗ (cid:86) Q ) (5.31)where we decompose the morphisms of (5.22) according to the Γ-action as dictated by the Beilinsonspectral sequence. The character at a fixed point is now the Γ-invariant part of the character forthe complex on C , i.e.Char Γ (cid:126)π ( t , t , t ) = (cid:0) W ∨ (cid:126)π ⊗ V (cid:126)π − V ∨ (cid:126)π ⊗ W (cid:126)π + (1 − t ) (1 − t ) (1 − t ) V ∨ (cid:126)π ⊗ V (cid:126)π (cid:1) Γ . (5.32)Let us clarify the meaning of the formula (5.32). Consider the rank one case N = 1. Using theCalabi–Yau condition t t t = 1 to eliminate the toric weight t = ( t t ) − , we regard the characteras an element in the virtual representation ring R ( T ) ∼ = Z [ t ± , t ± ] of the torus group T . Theinclusion Γ (cid:44) → T defines a restriction map R ( T ) → R (Γ) ∼ = Z (cid:2) ρ ± r , ρ ± r , ρ ± r (cid:3)(cid:14) (cid:104) ρ r ρ r ρ r − (cid:105) by ( t , t ) (cid:55)→ ( ρ r , ρ r ). Hence by substituting t α = ρ r α we regard the index as an element ofthe representation ring R (Γ) of the orbifold group, and compute (5.32) by composing with theprojection R (Γ) → Z onto the trivial representation.In the process of computing the character we identify the dimensions k r = dim C ( V r ) (cid:126)π with thenumber of boxes in a plane partition which transform in the irreducible representations of theorbifold group Γ; in particular for N = 1 one has k r = | π r | (5.33)as in Section 3.3. The integer k r can be identified with the number of fractional branes associatedto the representation ρ r , which in our formalism is identified with the instanton number. At thefixed points the instanton configurations are parametrized by (cid:98) Γ-coloured plane partitions and thecharacter (5.32) is expressed entirely in terms of combinatorial data.Modulo the issue of stability, which we will discuss momentarily, our quiver seems to reproduceprecisely the quiver used in [22] to compute noncommutative Donaldson–Thomas invariants. Thisquiver was obtained through the low-energy effective field theory of a system of D branes compact-ified on a local Calabi–Yau threefold. However, our perspective partly clarifies its origin. Since inthat case we are dealing with the U (1) gauge theory, our framing only involves a single vector space W r since dim C W is always equal to the rank of the gauge theory. Moreover the pertinent space W r is W , the vector space attached to V , which in turn is labelled by the trivial representationor the trivial line bundle R = O X on the D6 brane. This corresponds to a choice of boundaryconditions on the gauge fields of our D6 brane gauge theory. Incidentally this also explains thephysical meaning of choosing a different reference vertex in the formalism of [22]: it correspondsto BPS configurations whose asymptotic profile at infinity sits in a non-trivial representation ρ r ofthe orbifold group Γ. A particular class of quivers with superpotentials is deeply related to the geometry of Calabi–Yaumanifolds. This relation is at the core of the definition of noncommutative Donaldson–Thomasinvariants given by Szendr˝oi via the counting of cyclic modules of the conifold quiver [25]. Based36n his work, Joyce and Song gave a fully general definition of pair invariants associated to quiverswith superpotentials [18]; they are essentially weighted Euler characteristics of the moduli space offramed quiver representations. Our instanton quivers also come associated with an ideal of rela-tions on the corresponding path algebra, generated by the generalized ADHM equations, which canbe easily ascribed to cyclic derivatives of a superpotential. Representations of the framed McKayquiver are precisely the data that define generalised instantons on C / Γ; this is the main link be-tween our construction of the instanton quantum mechanics and Joyce–Song pair invariants.First let us review some facts from [18] to see explicitly how our construction fits into their moregeneral framework; afterwards we will freely borrow from their results. Given a quiver Q , definea stability condition on quiver representations as follows. Consider two sequences θ ∈ R Q and µ ∈ (0 , ∞ ) Q . Then slope stability can be defined on the category of representations of the quiver,considering only non-zero objects to have a well-defined stability condition. For any non-trivialquiver representation V of dimension vector k = ( k v ) v ∈ Q , the slope stability parameter µ isdefined as µ ( V ) = µ ( k ) = θ · kµ · k . (5.34)This definition generalizes the usual θ -stability parameter (5.4), to which it reduces for the partic-ular sequence µ v = 1 for all v ∈ Q .Consider now the moduli space of representations Rep ( Q , k ) associated with a superpotential W ∈ C Q (cid:14) [ C Q , C Q ], which gives a two-sided ideal of relations (cid:104) R (cid:105) in the path algebra C Q by takingcyclic derivatives ∂ a W for a ∈ Q . Let V k ( A ) be the G k -invariant closed subscheme of Rep ( Q , k )cut out by the equations ∂ a W = 0 (here A = C Q / (cid:104) R (cid:105) is the factor path algebra). This allows us todefine the framed quiver moduli space as the quotient stack M ( Q f , W ; k , N ) = (cid:104)(cid:0) V k ( A ) × Hom C ( W, V ) (cid:1) (cid:46) G k (cid:105) . (5.35)In particular one can define the moduli space of µ -stable framed quiver representations of type ( k , N ), which is a fine moduli space and an open substack M µ ( Q f , W ; k , N ) ⊂ M ( Q f , W ; k , N ).Then following Behrend [73], noncommutative Donaldson–Thomas invariants associated with thismoduli space are defined as the weighted topological Euler characteristics NC µ ( k , N ) = χ (cid:0) M µ ( Q f , W ; k , N ) , ν (cid:1) = (cid:88) n ∈ Z n χ (cid:0) ν − ( n ) (cid:1) , (5.36)where ν : M µ ( Q f , W ; k , N ) → Z is a G k -invariant constructible function related to the Eulercharacteristic of the Milnor fibre of the superpotential W ; at any smooth point V ∈ V k ( A ) onehas ν ( V ) = ( − D where D = dim C M µ ( Q f , W ; k , N ). In particular, this definition makes senseat µ = 0 for which µ -stability coincides with θ -stability at θ = (0 , . . . , A –mod 0-semistable. The new invariants then reproduce precisely the ones introducedby Szendr˝oi for the conifold. In the case of orbifold singularities they enumerate Γ-equivariantsheaves on C via the McKay correspondence; for ideal sheaves they coincide with the orbifoldDonaldson–Thomas invariants defined in [47].In many cases these invariants can be related to the quiver generalized Donaldson–Thomas invari-ants DT µ ( k ) ∈ Q defined by Joyce and Song in [18] via a certain infinite-dimensional Lie algebramorphism acting on the moduli stack of left A -modules. From these invariants one defines thequiver BPS invariants (cid:99) DT µ ( k ) ∈ Q as (cid:99) DT µ ( k ) = (cid:88) m ≥ m | k M¨o( m ) m DT µ ( k /m ) (5.37)37here M¨o : N → Q is the M¨obius function. In special cases these invariants count BPS states,and they generalize the integer Gopakumar–Vafa invariants of Calabi–Yau threefolds [74]. In thegeneral case they are conjectured to be integer-valued. By the M¨obius inversion formula, thisexpression has an inverse given by DT µ ( k ) = (cid:88) m ≥ m | k m (cid:99) DT µ ( k /m ) . (5.38)Noncommutative Donaldson–Thomas invariants are related to the quiver generalized Donaldson–Thomas invariants by [18, Theorem 7.23] NC µ ( k , N ) = ∞ (cid:88) m =1 (cid:88) k ,..., k m (cid:54) = k + ··· + k m = k , µ ( k i )= µ ( k ) ( − m m ! (5.39) × m (cid:89) i =1 (cid:16) ( − k i · N − ¯ χ ( k + ··· + k i − , k i ) (cid:0) k i · N − ¯ χ ( k + · · · + k i − , k i ) (cid:1) DT µ ( k i ) (cid:17) where ¯ χ ( k , k (cid:48) ) = χ ( V, V (cid:48) ) − χ ( V (cid:48) , V ) = (cid:88) a ∈ Q (cid:0) k h ( a ) k (cid:48) t ( a ) − k t ( a ) k (cid:48) h ( a ) (cid:1) (5.40)is the antisymmetrization of the Euler form χ ( V, V (cid:48) ) = (cid:80) p ≥ dim C Ext p A ( V, V (cid:48) ) for V, V (cid:48) ∈ A –mod.Because of the condition on the partitions of k , the sum over m in (5.39) contains only a finitenumber of non-zero terms. In the case of semi-small crepant resolutions, the Euler forms ¯ χ vanishand this relation yields a useful relationship between the corresponding partition functions1 + (cid:88) k : µ ( k )= µ NC µ ( k , N ) p k = exp (cid:16) − (cid:88) k : µ ( k )= µ ( − k · N ( k · N ) DT µ ( k ) p k (cid:17) (5.41)where p k := (cid:81) v ∈ Q p k v v . To see how the Joyce–Song construction is connected with our perspective, let us start by reviewingthe definition of the instanton moduli space on C put forward in [37]. Consider the two vectorspaces V and W , of dimensions dim C V = k and dim C W = N , in the quantum mechanics ofthe gas of D0 branes and the D6 branes on which the gauge theory lives. The moduli space ofrepresentations of the quiver (5.15) is given by M ( k, N ) = Hom C ( V, Q ⊗ V ) ⊕ Hom C ( V, (cid:86) Q ⊗ V ) ⊕ Hom C ( W, V ) , (5.42)on top of which we have the natural action of the complexified gauge group GL ( k, C ). We call anelement ( B, ϕ, I ) of M ( k, N ) a Donaldson–Thomas datum. We define a complex “moment map” µ C = ( E α , E I ) given collectively by the ADHM type equations of the matrix quantum mechanicsin (5.16).The instanton moduli space is defined via θ -stability as the geometric invariant theory quotient M θ ( k, N ) = µ − C (0) (cid:14)(cid:14) θ GL ( k, C ) . (5.43)In the case of a single D6 brane N = 1, we can now proceed to define a Donaldson–Thomasinvariant in the gauge theory formalism as the Euler characteristic (3.17) of the obstruction bundle38ver the moduli space; we regard this number as the gauge theory realization of Behrend’s localweighted Euler characteristic (5.36). Each invariant can be evaluated by using the localizationformula with respect to the natural lift of the toric action on C to the moduli space. Fixed pointsof the toric action are in natural correspondence with certain chains of maps which are classifiedby plane partitions. The local structure of the instanton moduli space around each fixed point iscompletely characterized by the equivariant index of the complex (5.22) generated by the derivative τ of the moment map modulo linearized complex gauge transformations σ .This construction can be generalized to C / Γ orbifolds by considering Γ-equivariant morphisms,as dictated by the instanton deformation complex. The Donaldson–Thomas data now decomposeaccordingly as M Γ ( k , N ) = Hom Γ ( V, Q ⊗ V ) ⊕ Hom Γ ( V, (cid:86) Q ⊗ V ) ⊕ Hom Γ ( W, V ) . (5.44)We use the Γ-equivariant decomposition of the ADHM equations (5.28) to define “moment maps” µ Γ C = ( E rα , E rI ) which correspond to the ideal of relations in the instanton quiver path algebra. Theseequations define a subvariety ( µ Γ C ) − (0) ⊂ Hom Γ ( V, Q ⊗ V ) ⊕ Hom Γ ( V, (cid:86) Q ⊗ V ). This allows usto define the Donaldson–Thomas quiver moduli space as the quotient stack M Γ ( k , N ) = (cid:104)(cid:0) ( µ Γ C ) − (0) × Hom Γ ( W, V ) (cid:1) (cid:46) G k (cid:105) . (5.45)We regard this stack as a moduli space of stable framed representations in the sense of [18, Sec-tion 7.4]. Geometrically this moduli scheme is an Artin stack over C and has certain nice prop-erties which allow us to define enumerative invariants. We postpone a discussion of this to Sec-tion 5.9.From our analysis of the noncommutative gauge theory in Section 3 it follows that the instantonmoduli space is constructed as the moduli space of C but with a decomposition of the instan-ton equations according to the equivariant structure lifted from the orbifold action. Hence theΓ-invariant fixed point set of the instanton moduli space (5.43) for N = 1 admits a decomposi-tion M θ ( k, Γ = (cid:71) k : | k | = k M Γ ( k , , (5.46)where | k | := dim C ( V ) = (cid:80) v ∈ Q k v . It is straightforward to generalize this to the Coulomb branchof the non-abelian gauge theory where the vector space W is no longer one-dimensional and de-composes according to the Γ-action as in (4.32); we discuss this in Section 5.8.Since the natural action of the torus group T commutes with the action of the orbifold groupΓ on C it lifts naturally to the moduli space M Γ ( k , N ). The transformations of the fields arethe usual ones and we can work equivariantly with respect to the toric action. This allows usto immediately classify the torus fixed points that enter into the localization formula which willbe used to compute generating functions of BPS states, since they are precisely the instantonconfigurations on C which are fixed by the toric action and are invariant under the Γ-action.Again we can express fixed points of the toric action via plane partitions, where now each boxcarries an additional colour associated with the Γ-action. By (5.46) and since the orbifold actioncommutes with the torus action on C , the T fixed points on M Γ ( k , N ) coincide with the T fixedpoints of M θ ( k, N ) which are also invariant under the orbifold action. We will substantiate thesearguments further in Section 5.9. N In the following we will propose a physical interpretation of the framed pair invariants NC µ ( k , N ) for µ = 0 as noncommutative Coulomb branch invariants of Donaldson–Thomas type in U ( N ) gauge39heory. In the instanton quiver formalism the framing operation has a clear physical meaning. Itrepresents the choice of boundary conditions for the gauge field living on the worldvolume of thestack of D6 branes. Asymptotically the gauge connection is flat, and flat connections on resolvedgeometries of abelian orbifolds C / Γ are classified by the irreducible representations of the orbifoldgroup Γ.Based on these definitions and on the identification of the relevant quiver as the instanton quiverintroduced in Section 5.5, we can now construct a partition function for the noncommutativeinvariants. It follows from the local character of the instanton moduli space given by (5.32).Neglecting the Γ-action, the two vector spaces V and W can be decomposed at a fixed point (cid:126)π = ( π , . . . , π N ) of the U (1) N × T action on the instanton moduli space as [37] V (cid:126)π = N (cid:88) l =1 e l (cid:88) ( n ,n ,n ) ∈ π l t n − t n − t n − and W (cid:126)π = N (cid:88) l =1 e l . (5.47)Here we view the spaces as U (1) N × T representations regarded as polynomials in t α = e i (cid:15) α , α = 1 , , e l = e i a l , l = 1 , . . . , N , with the sum over boxes of π l for each l corresponding to the T -character on C [ B , B , B ] /I l , i.e. the decomposition of H ( O Z l ) as a T -representation, where Z l is the T -fixed subscheme of C corresponding to the three-dimensional Young diagram π l .Let us now consider the analogous decompositions for the resolved geometry. We can furtherdecompose the vector spaces according to the Γ-action as in (4.29) and (4.32). Recall that thedecomposition of W corresponds to imposing boundary conditions at infinity, which are classifiedby irreducible representations of the orbifold group Γ. In this context each U (1) factor in theCoulomb phase is associated with a vacuum expectation value of the Higgs field a l which correspondsto a certain irreducible representation of Γ. Even if the maximal symmetry breaking pattern U ( N ) → U (1) N is fixed, one still has to specify in which superselection sector one is working.This sector is characterized by choosing which of the eigenvalues a l are in a particular irreduciblerepresentation of Γ. The number of eigenvalues of the Higgs field in the representation ρ ∨ r isprecisely N r = dim C W r .Similarly the dimensions k r = dim C V r give the instanton number of a multi-instanton configurationtransforming in the representation ρ ∨ r . However their T -module decompositions are somewhatcomplicated to compute directly since they do not correspond in a simple way to plane partitions.It follows from (5.47) that each partition carries an action of Γ on its own, but this action is “offset”by the prefactor e l . On the other hand we can also write the decomposition V (cid:126)π = N (cid:77) l =1 (cid:77) r ∈ (cid:98) Γ (cid:0) E l ⊗ ρ ∨ b ( l ) (cid:1) ⊗ (cid:0) P l,r ⊗ ρ ∨ r (cid:1) = N (cid:77) l =1 (cid:77) r ∈ (cid:98) Γ (cid:0) E l ⊗ P l,r (cid:1) ⊗ ρ ∨ r + b ( l ) (5.48)where E l is the module generated by e l , and we have introduced the boundary function b ( l ) which toeach sector l corresponding to a module E l associates the weight of the corresponding representationof Γ; if the vacuum expectation value e l transforms in the irreducible representation ρ s , then b ( l ) = s . Here P l,r is a module which corresponds to the Γ-module decomposition of the sum H ( O Z l ) = (cid:80) ( n ,n ,n ) ∈ π l t n − t n − t n − . Recall that each fixed point is characterized by a vectorof partitions (cid:126)π . Each entry in this vector can be decomposed according to the Γ-action, takingfurther into account the transformation properties of the Higgs field vacuum expectation values e l .In our decomposition (5.48) we have factorized this contribution explicitly so that now dim C P l,r isthe number of boxes in the plane partition at position l of the fixed point vector (cid:126)π = ( π , . . . , π N )which transform in the representation ρ ∨ r , a number which we will call | π l,r | . This should not beconfused with the physical instanton configuration which transforms in the representation ρ ∨ r ⊗ ρ ∨ b ( l ) .The two concepts only differ by the Γ-action. 40n the other hand the module V r contains all the instanton configurations transforming in therepresentation r ∈ (cid:98) Γ (for fixed topological charge k r = dim C V r ). The two parametrizations arerelated by ( V r ) (cid:126)π = N (cid:77) l =1 E l ⊗ P l,r − b ( l ) . (5.49)The instanton numbers are thus expressed via k r = N (cid:88) l =1 | π l,r − b ( l ) | . (5.50)This parametrization is useful for computing explicitly the local contribution of an instanton. Tocompute the Γ-invariant projection of the character (5.32), we write (5.23) asChar (cid:126)π ( t , t , t ) = T + (cid:126)π + T − (cid:126)π (5.51)where T + (cid:126)π = W ∨ (cid:126)π ⊗ V (cid:126)π − V ∨ (cid:126)π ⊗ V (cid:126)π (1 − t ) (1 − t ) t t , T − (cid:126)π = − V ∨ (cid:126)π ⊗ W (cid:126)π + V ∨ (cid:126)π ⊗ V (cid:126)π (1 − t ) (1 − t ) , (5.52)and we have used the Calabi–Yau condition t t t = 1. This splitting has the property (cid:0) T + (cid:126)π (cid:1) ∨ = −T − (cid:126)π , (5.53)where the dual involution acts on the weights as ( t α , e l ) ∨ = ( t − α , e − l ). As in [37], it follows thatone need only consider the partial character T + (cid:126)π , and from (5.24) the contribution of an instantonto the gauge theory fluctuation determinant is given by χ T ( N (cid:126)π ) = ( − K ( (cid:126)π ; N ) with K ( (cid:126)π ; N ) = (cid:0) T + (cid:126)π (cid:1) Γ (cid:12)(cid:12)(cid:12) t α = e l =1 . (5.54)In particular, we need only compute the value of the equivariant index (cid:0) T + (cid:126)π (cid:1) Γ modulo 2.Let us first consider the term (cid:0) W ∨ (cid:126)π ⊗ V (cid:126)π (cid:1) Γ = (cid:16) N (cid:77) l =1 (cid:77) r,s ∈ (cid:98) Γ E l ⊗ P l,r ⊗ ρ ∨ r + b ( l ) ⊗ W ∨ s ⊗ ρ s (cid:17) Γ (5.55)= N (cid:77) l =1 (cid:77) r,s ∈ (cid:98) Γ E l ⊗ P l,r ⊗ W ∨ s ⊗ (cid:0) ρ ∨ r + b ( l ) ⊗ ρ s (cid:1) Γ = N (cid:77) l =1 (cid:77) r ∈ (cid:98) Γ E l ⊗ P l,r ⊗ W ∨ r + b ( l ) . The other terms involve the T weights t and t . As explained in Section 5.5, the weights t α shouldbe properly regarded as the Γ-modules t α (cid:55)→ ρ r α , where r α for α = 1 , , C . We thus find (cid:16) V ∨ (cid:126)π ⊗ V (cid:126)π t t (cid:17) Γ = N (cid:77) l,l (cid:48) =1 (cid:77) r,s ∈ (cid:98) Γ E ∨ l ⊗ P ∨ l,r ⊗ E l (cid:48) ⊗ P l (cid:48) ,s ⊗ (cid:0) ρ r + b ( l ) ⊗ ρ ∨ s + b ( l (cid:48) ) ⊗ ρ ∨ r ⊗ ρ ∨ r (cid:1) Γ = N (cid:77) l,l (cid:48) =1 (cid:77) r ∈ (cid:98) Γ E ∨ l ⊗ P ∨ l,r ⊗ E l (cid:48) ⊗ P l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − r − r , (5.56)41nd similarly (cid:16) V ∨ (cid:126)π ⊗ V (cid:126)π t α (cid:17) Γ = N (cid:77) l,l (cid:48) =1 (cid:77) r ∈ (cid:98) Γ E ∨ l ⊗ P ∨ l,r ⊗ E l (cid:48) ⊗ P l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − r α . (5.57)Therefore the projection of the partial character T + (cid:126)π , evaluated at ( (cid:15) , (cid:15) , (cid:15) , a ) = 0, onto the trivialrepresentation of the orbifold group Γ gives K ( (cid:126)π ; N ) = N (cid:88) l =1 (cid:88) r ∈ (cid:98) Γ | π l,r | N r + b ( l ) − N (cid:88) l,l (cid:48) =1 (cid:88) r ∈ (cid:98) Γ | π l,r | (cid:16) | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − r − r | − | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − r |− | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − r | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) | (cid:17) . (5.58)The fixed point values of the instanton action (4.41) in these new variables can be written as (cid:90) X ω ∧ ω ∧ c ( E (cid:126)π ) = − (cid:88) m,r,s ∈ (cid:98) Γ ς m (cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:17) (cid:90) X c ( V m ) ∧ c ( R r ) , (5.59) (cid:90) X ω ∧ ch ( E (cid:126)π ) = − (cid:88) n,r,s ∈ (cid:98) Γ ϕ n (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:17) (cid:90) X c ( R n ) ∧ ch ( R r )+ (cid:0) a (2) rs − δ rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:90) X c ( R n ) ∧ c (cid:0) O X (1) (cid:1) ∧ c ( R r ) (cid:19) , (5.60) (cid:90) X ch ( E (cid:126)π ) = − (cid:88) r,s ∈ (cid:98) Γ (cid:18)(cid:16) N s δ rs − (cid:0) a (2) rs − a (1) rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:17) (cid:90) X ch ( R r )+ (cid:0) a (2) rs − δ rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r )+ (cid:0) a (2) rs − δ rs (cid:1) N (cid:88) l =1 | π l,s − b ( l ) | (cid:90) X c ( R r ) ∧ ch (cid:0) O X (1) (cid:1)(cid:19) + 1 | Γ | (cid:88) s ∈ (cid:98) Γ N (cid:88) l =1 | π l,s − b ( l ) | . (5.61)Note that the choice of boundary condition enters not only explicitly in the dimensions N r , but alsoimplicitly in the plane partitions. Finally, the partition function for noncommutative Donaldson–Thomas invariants of type N is in full generality given by Z C / Γ ( N ) = (cid:88) (cid:126)π ( − K ( (cid:126)π ; N ) e − S inst [ (cid:126)π ; N ] . (5.62) The problem we are facing now is that the concepts we have used so far (e.g. coherent sheaves,tautological bundles, Beilinson’s theorem) are all large radius concepts. We would like however touse our construction to investigate Donaldson–Thomas invariants near the singular orbifold point42nd argue that this can be done via the McKay correspondence by a suitable choice of the stabilityconditions. To do so we argue that the parameter which identifies the region of the moduli spaceof BPS states we are working in is the stability parameter which enters in the construction of thequiver varieties. While this resembles closely the rigorous mathematical construction of generalisedDonaldson–Thomas invariants presented in e.g. [19, 20, 26], our setting is quite different. Wewill argue that a particular choice of the stability parameter defines the noncommutative crepantresolution. Physically this corresponds to a highly non-geometrical limit in the noncommutativegauge theory, i.e. when the classical volume of all the cycles goes to zero, while their quantum volumes, as measured by the B -field, is still non-vanishing (though very small).Before exposing our arguments let us comment on the physical picture that we expect. It is knownthat the string theory linear sigma-model can be used to investigate the whole extended Calabi–Yaumoduli space, including in principle topology changing transitions. These ideas can be made ratherprecise via the study of the phase structure of the sigma-model [75, 76]. While generically the phaseshave a geometric description, this is not at all necessary and certain abstract “non-geometries” canbe investigated as well. String theory implies that the same point of view, while less studied inthis language, should apply also to the description in terms of weakly coupled solitons [77]. Indeedthere is by now evidence that the appropriate description for BPS solitons given by the derivedcategory behaves consistently with this picture.Still it should be possible to explore at least parts of the whole moduli space of stable BPS solitonsby studying directly the Dirac–Born–Infeld theory at the quantum level. More precisely we expectto be able to describe those chambers in the moduli space which can be accessed from a regimewhere worldsheet instanton corrections are negligible (and of course string theory loops as well).In this regime the theory describing a system of branes is far from being a local quantum fieldtheory: the full Dirac–Born–Infeld action involves an infinite series of higher derivative corrections,the presence of the B -field induces a noncommutative deformation of the theory, and the otherRamond–Ramond fields yield even more subtle effects which have not been completely understood.Although the situation is unclear it is reasonable to suggest [78] that the full non-linear instantonequations derived in [79] might be already able to capture the chamber structure of the BPS solitonmoduli space, at least for local threefolds. These equations arise as BPS conditions on the D braneembedding from κ -symmetry of the full twisted Dirac–Born–Infeld action.Therefore we do not expect to be able to carry out a fully rigorous derivation of the chamberstructure by studying just the topological Yang–Mills theory living on the D6 branes. We will needto make further assumptions. We know however that string theory defined on the backgrounds weare considering has a certain non-trivial behaviour that is believed to hold at the quantum level:the derived McKay correspondence.The analysis of BPS states on a Calabi–Yau manifold should be ultimately rephrased in terms ofderived categories. Although most of our construction uses abelian categories, it is useful to keep inmind where we stand in the categorical landscape. The category we wish to study is the boundedderived category of coherent sheaves on X , D ( X ). This is generically a very difficult problem.However for toric Calabi–Yau threefolds we can have substantial simplification by using alternativedescriptions. If the Calabi–Yau space is a crepant resolution of an orbifold singularity, then twoalternative models are available. One can characterize the derived category D ( X ) via an equivalentcategory, the derived category of representations of a certain quiver; or one can use the McKaycorrespondence at the derived level and deal with the category D Γ ( C ), the derived category ofΓ-equivariant sheaves on C . We will now review these equivalences.The variety X is the “natural” Calabi–Yau crepant resolution of the singular orbifold C / Γ given bythe Γ-Hilbert scheme Hilb Γ ( C ). The McKay correspondence is established as a derived equivalencebetween D ( X ) and D Γ ( C ) [69]. This equivalence descends to K-theory as follows: there exists a43atural basis { ρ r ⊗ O C } r ∈ (cid:98) Γ of the Grothendieck group of Γ-equivariant sheaves on C , K Γ ( C ),and a natural basis { ρ r ⊗ O } r ∈ (cid:98) Γ of its restriction to coherent sheaves with compact support, K c Γ ( C ), as discussed in Section 4.2. These bases give an explicit ring isomorphism of K Γ ( C ) and K c Γ ( C ) with the representation ring R (Γ). On the other hand the equivalence between the derivedcategories give an explicit isomorphism between these groups and the K-theory groups of X , K ( X )and K c ( X ). Under this isomorphism the two basis sets are mapped respectively to the basis oftautological bundles {R r } r ∈ (cid:98) Γ and of fractional branes {S r } r ∈ (cid:98) Γ introduced in Section 4.2.The Γ-Hilbert scheme Hilb Γ ( C ) can be equivalently realized as the fine moduli space of represen-tations of the McKay quiver. This result underlies another derived equivalence, between D ( X ) andthe bounded derived category of finitely-generated left modules over a certain algebra. To properlydefine this category we need to introduce the notion of a tilting object . Geometrically a coherentsheaf T is said to be tilting if the following three conditions hold: • T can be decomposed as a sum of simple sheaves T = F ⊕ · · · ⊕ F n ; • Ext p O X ( F i , F j ) = 0 for all p > i, j = 1 , . . . , n ; and • T generates D ( X ).Physically the morphisms of the derived category which combine the direct summands F i of T togenerate the whole of D ( X ) correspond to combining D branes via tachyon condensation.Consider now the endomorphism algebra of T , A = End D ( X ) ( T ). This is a noncommutative algebrawith respect to the composition of morphisms. In practice its elements can be seen as matriceswhose entries are elements of the spaces of morphisms Hom O X ( F i , F j ). The tilting object T inducesa derived equivalence through the adjoint functors R Hom O X ( T , − ) : D ( X ) −→ D ( A op ) , T L ⊗ A − : D ( A op ) −→ D ( X ) (5.63)where D ( A ) denotes the bounded derived category of finitely-generated left A -modules and A op isthe opposite algebra of A . The algebra A can be identified with the path algebra of a quiver withrelations C Q / (cid:104) R (cid:105) ; the idempotent elements e i ∈ A are the projections T → F i , with Hom O X ( F i , F j )the space of paths from node j to node i and Ext O X ( F i , F j ) the space of independent relationsimposed on these paths. The moduli space of θ -stable representations of this quiver with dimensionvector k = (1 , . . . , 1) is isomorphic to the crepant resolution X = Hilb Γ ( C ).In our case the quiver Q is the McKay quiver, whose vertices are labelled by the irreducible rep-resentations of the orbifold group Γ and whose arrow set Q is dictated by the decomposition intoirreducible representations of the tensor products Q ⊗ ρ r . In this case the tilting bundle deter-mining the equivalence between the derived categories is the sum of the tautological bundles over X [69, 80, 70] (see also [63, Remark 7.17]), T = (cid:77) r ∈ (cid:98) Γ R r , (5.64)where only Hom O X ( R r , R s ) can be non-trivial. Note that the sheaves in the tilting set are theprojective objects P r = R r , not the fractional branes D r = S r . The relationship between the twosets of D branes is given by projective resolutions (5.1).Our gauge theory construction naturally produces 0-semistable objects of the category A –mod.It is vastly based on the McKay correspondence for threefolds. We have used the Γ-equivariantgeometry of C to describe our instanton moduli space. This is in perfect harmony with the44act, discussed in Section 5.6, that the orbifold Donaldson–Thomas invariants of [47] counting Γ-equivariant ideal sheaves on C coincide exactly with the noncommutative invariants NC µ =0 ( k ) [18,Section 7.4].Let us now turn to the construction of the partition function for these invariants. We have explainedhow to use the geometry of Hilb Γ ( C ) to evaluate explicitly the instanton action in terms of largeradius data. By using the fact that the set {R r } r ∈ (cid:98) Γ generates the topological K-theory group K ( X )and therefore has a direct relation with the homology H • ( X ), we found precise combinations of thebundles R r which correspond to divisors and curves in the resolved geometry. On the other hand,since the object T in (5.64) is a tilting generator, it is by definition constant and well-defined overthe entire K¨ahler moduli space (though the way in which the derived category is generated changesas we move around the moduli space).It is tempting to speculate that our instanton action makes sense also at any point of the modulispace. The only change in the partition function is eventually encoded in the chemical potentials ϕ n and ς m which specify the strengths of the couplings between D branes, and of course in thefact that for given topological instanton charges k the moduli space might be empty. The lattercondition is however automatically taken care of by the measure on the instanton moduli space, atleast in those regions of the K¨ahler moduli space where we can compute it explicitly. Therefore theproblem is reduced to computing the instanton measure for any value of the stability parameterand not just for µ = 0 as we have done above. Note that by working with the tilting set (5.64)we bypass the question of what are the stable fractional branes in each region of the moduli space,or equivalently what happens to the basis of coherent sheaves S r supported on the exceptional setwhen we blow down the exceptional cycles. This is somewhat in line with the proposal of [70] thatthe D branes in the tilting set are everywhere Π-stable over the whole K¨ahler moduli space. C / Z × Z Our first example will be the resolution of C / Z × Z , where the action of Z × Z = { , g , g , g } on C is given by g · ( z , z , z ) = ( − z , − z , z ) ,g · ( z , z , z ) = ( − z , z , − z ) ,g · ( z , z , z ) = ( z , − z , − z ) . (6.1)This singular orbifold has a fan Σ ⊂ Z generated by the lattice vectors D = (0 , , D = (0 , , D = (2 , , C , i.e. D α = { ( z , z , z ) ∈ C | z α = 0 } .One can resolve the singularity in several ways, corresponding to the distinct possible triangulationsof the toric diagram. Here we only consider the symmetric resolution given by X = Hilb Z × Z ( C ),which has the geometry of the closed topological vertex [81, 82], whose fan is depicted in Figure 1.This resolution has three non-compact divisors; we will denote by E αβ the divisor whose vector liesbetween D α and D β . They all have the topology of C × P . Finally there are three compact curvesgiven by the intersections C α = E γα · E αβ . This geometry does not have any compact divisors, asall the compact holomorphic submanifolds are curves, which have codimension two.45 D E D E E Figure 1: Toric fan for the closed topological vertex geometry.The linear equivalences between the non-compact divisors are2 D + E + E ∼ , D + E + E ∼ , D + E + E ∼ . (6.2)The non-vanishing triple intersections are E · D · E = 1 ,E · E · E = 1 ,E · E · D = 1 ,E · D · E = 1 . (6.3)In particular D , D and D generate the K¨ahler cone, while C , C and C are the dual generatorsof the Mori cone with respect to the intersection pairing. This means that the tautological bundlesare R α = O X ( D α ), whose first Chern classes form a basis of H ( X, Z ) with (cid:90) C α c ( R β ) = δ αβ (6.4)for α, β = 1 , , 3. Upon including the trivial bundle R = O X , which generates H ( X, Z ), thesebundles form a canonical integral basis of K ( X ).Let us now turn to the representation theory data. The orbifold group is Γ = Z × Z and it actson C with weights r = (1 , , r = (1 , , 1) and r = r + r = (0 , , ρ r where ρ is the trivial representation, ρ and ρ correspond to the weights r and r , and ρ = ρ ⊗ ρ corresponds to the weight r . The tensor product decomposition of thedefining representation Q gives a matrix (cid:0) a (1) rs (cid:1) = . (6.5) Elements of the K¨ahler cone are cohomology classes η ∈ H ( X, Q ) such that (cid:82) C η ≥ C ∈ H ( X, Q ) (and similarly for η ∧ r on higher-dimensional subvarieties). The Mori cone consists of linearcombinations of compact algebraic cycles with non-negative coefficients and is generated by the exceptional curves. (cid:86) Q ∼ = Q ∨ , one has a (2) rs = a (1) sr and in this particularcase the intersection product vanishes identically, ( S ∨ r , S s ) = 0. This reflects the fact that theresolved geometry has no compact divisors. The quiver constructed from representation theory isthus v • (cid:28) (cid:28) (cid:6) (cid:6) (cid:8) (cid:8) v • (cid:72) (cid:72) (cid:115) (cid:115) (cid:36) (cid:36) v • (cid:56) (cid:56) (cid:51) (cid:51) (cid:47) (cid:47) • v (cid:104) (cid:104) (cid:100) (cid:100) (cid:109) (cid:109) (6.6) To evaluate the partition function we will choose the boundary condition N = (1 , , , 0) corre-sponding to U (1) gauge field configurations that are trivial at infinity. We begin by computing theaction given by (4.48)–(4.50). Since there are no compact four-cycles, we cannot wrap compactD4 branes anywhere and the integral (cid:82) X ω ∧ ω ∧ c ( E ) in (4.48) must vanish identically. Indeed thisis the case, since the first Chern class c ( E ) itself is zero. Because of our boundary condition, thereare no terms proportional to N r since c ( R ) = 0. Moreover the intersection matrix a (2) rs − a (1) rs vanishes, since a (1) rs is symmetric.Now let us turn to the integral (cid:82) X ω ∧ ch ( E ) involving the second Chern class. The first line onthe right-hand side of (4.49) vanishes for the same reasons as above. The remaining term from(4.49) is (cid:90) X ω ∧ ch ( E ) = − (cid:88) n,r,s =0 (cid:16) a (2) rs − δ rs (cid:17) k s ϕ n (cid:90) X c ( R n ) ∧ c ( R r ) ∧ c (cid:0) O X (1) (cid:1) , (6.7)where we recall that c ( R ) = 0. This integral computes a triple intersection number which weevaluate explicitly below.The remaining integral is (4.50), which by the same reasoning as above reduces to (cid:90) X ch ( E ) = − (cid:88) r,s =0 (cid:18)(cid:16) a (2) rs − δ rs (cid:17) k s (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r ) − k s | Γ | δ rs − (cid:16) a (2) rs − δ rs (cid:17) k s (cid:90) X c ( R r ) ∧ ch (cid:0) O X (cid:0) (cid:1)(cid:19) . (6.8)Recall that in our case | Γ | = 4. To evaluate this integral, we note that the integrals on the right-hand side of (6.8) measure, in various forms, all the triple intersections of the non-compact divisorsinvolving the divisor ℘ ∞ at infinity at least once. To evaluate these integrals, we assume that thedivisor at infinity has no intersection with the compact curves that resolve the singularity. Then wecan evaluate the intersection numbers as if they were effectively taken in P (and take care of theorbifold action only when evaluating the pullback by dividing by the order of the orbifold group),i.e. upon compactification the divisors D α are topologically P . Therefore all the intersectionproducts involve three divisors in P , which intersect at a point. By symmetry we can identifytwo independent integrals, those involving the triple intersection of two non-compact divisors withthe divisor at infinity, say D · D · ℘ ∞ , and those involving the self-intersection of a non-compactdivisor, say D · D · ℘ ∞ . We will parametrize these integrals with two integers, b and a .47e can now evaluate the integral (6.7) to be (cid:90) X ω ∧ ch ( E ) = − (cid:88) n,r,s =0 (cid:16) a (2) rs − δ rs (cid:17) k s ϕ n | Γ | I nr (6.9)where we have introduced the intersection matrix( I nr ) = a b b b a b b b a . (6.10)The zeroes come from the fact that R is the trivial bundle with vanishing Chern classes, while theremaining entries come from the intersection products. We finally obtain (cid:90) X ω ∧ ch ( E ) = − (cid:16) b (cid:0) ϕ ( k + k − k − k ) + ϕ ( k − k + k − k )+ ϕ ( k − k − k + k ) (cid:1) + a (cid:0) ϕ ( k + k + k − k )+ ϕ ( k + k − k + k ) + ϕ ( k − k + k + k ) (cid:1)(cid:17) . (6.11)Now let us consider the last term in the instanton action (cid:90) X ch ( E ) = − (cid:88) r,s =0 (cid:18)(cid:16) a (2) rs − δ rs (cid:17) k s γ r | Γ | − (cid:16) a (2) rs − δ rs (cid:17) k s α r | Γ | − δ rs k s | Γ | (cid:19) , (6.12)where we have introduced the vectors ( γ r ) = (0 , c, c, c ) and ( α r ) = (0 , a, a, a ). The integer c is thetriple intersection product D α · ℘ ∞ · ℘ ∞ . The factors of come from the expansion of the Cherncharacter ch . We arrive finally at (cid:90) X ch ( E ) = − ( a − c ) (3 k − k − k − k ) + ( k + k + k + k ) , (6.13)where again we have used | Γ | = 4.Let us now compute the values of the triple intersection numbers a , b and c appearing above. Weknow the intersections between the non-compact divisors in X (when they make sense, i.e. whenthey involve at least some compact curve), and we are modelling the behaviour of X at infinity as P / Γ. We will momentarily ignore the orbifold action. In the compactified geometry the divisors D α look like the compact divisor of P at infinity. Therefore two of them intersect with the divisorat infinity as three ordinary planes P inside P , i.e. at a point. Thus we conclude b = c = 1.However a counts the self-intersection of a divisor with ℘ ∞ . Let us call this compactified divisor˜ D . If our space X were an ordinary P then we could conclude ˜ D · ˜ D · ℘ ∞ = 1; the usual argumentwould be that one can consider a generic intersection with another divisor ˜ D · ˜ D (cid:48) · ℘ ∞ and then“transport” ˜ D (cid:48) back to ˜ D to compute the intersection product. However this “transport” is notpermitted in our case since our variety looks like P only at infinity, and one cannot “transport” adivisor without intersecting the compact curves in the exceptional locus. Therefore to compute thenumber a the heuristic “transporting” argument is not sufficient. On the other hand we know fromthe toric diagram (see Figure 1) that the non-compact divisors D α give always zero intersectionwhenever their self-intersection appears in a triple intersection product, i.e. D α · D α · E αβ = 0 forevery choice of α and β . Therefore we take a = D α · D α · ℘ ∞ = 0.To compute the index of BPS states we need to compute the Z × Z -invariant part of the charac-ter (5.23). We rewrite it as in (5.51)–(5.52) and decompose the vector space V at a fixed point π V π = V ⊕ V ⊕ V ⊕ V , where each subspace V r is associated to the group element representedby ρ r on C . Note that each element is nilpotent. We can write the partial character T + π as T + π = ( V ⊕ V ⊕ V ⊕ V ) − (cid:16) t t − t − t + 1 (cid:17) ( V ⊕ V ⊕ V ⊕ V ) ⊗ (cid:0) V ∨ ⊕ V ∨ ⊕ V ∨ ⊕ V ∨ (cid:1) , (6.14)which upon substituting t α (cid:55)→ ρ r α gives (cid:0) T + π (cid:1) Z × Z = | π | + | π | + | π | (6.15)and hence χ T ( N π ) = ( − | π | + | π | + | π | . (6.16)Combining all of these ingredients together, we can write the partition function of orbifold Donaldson–Thomas invariants as Z C / Z × Z = (cid:88) π χ T ( N π ) e − g s (cid:82) X ch ( E π ) e − (cid:82) X ω ∧ ch ( E π ) = (cid:88) π ( − | π | + | π | + | π | q (3 | π |−| π |−| π |−| π | )+ ( | π | + | π | + | π | + | π | ) × Q | π | + | π |−| π |−| π | Q | π |−| π | + | π |−| π | Q | π |−| π |−| π | + | π | (6.17)where we have introduced the weighting variables q = e − g s and Q α = e − ϕ α for α = 1 , , Now we compare our construction with the available literature. In [47, Definition 1.3], adapted toour case, we learn of a combinatorial partition function K Z × Z = (cid:88) π p | π | p | π | p | π | p | π | (6.18)in formal variables p r which enumerates Z × Z -coloured three-dimensional Young diagrams. In [47,Theorem A.3] it is proven that this partition function is related to the Donaldson–Thomas partitionfunction of the quotient stack [ C / Z × Z ] through K DT C / Z × Z ( p , p , p , p ) = K Z × Z ( p , − p , − p , − p ) . (6.19)This formula only depends on the four variables p = p p p p , p , p and p , and can be writtenas K DT C / Z × Z = (cid:88) π ( − | π | + | π | + | π | p | π | p | π | p | π | p | π | . (6.20)A simple computation shows that after the change of variables p = q / Q Q Q ,p = q − / Q − Q − ,p = q − / Q − Q − ,p = q − / Q − Q − , (6.21)our partition function (6.17) coincides with (6.20). While our original variables seem somewhatapt to an interpretation in terms of D brane charges, the physical meaning of this redefinition is49nclear. The D brane charge corresponding to each configuration represented by a plane partitionis however expected to be a rather non-trivial function of the D2 and D0 charges [22].In this case the partition function has an explicit description as a product of generalized MacMahonfunctions, which generate weighted plane partitions, given by [47, Theorem 1.5] K Z × Z = M ( p ) (cid:102) M ( p p , p ) (cid:102) M ( p p , p ) (cid:102) M ( p p , p ) (cid:102) M ( − p , p ) (cid:102) M ( − p , p ) (cid:102) M ( − p , p ) (cid:102) M ( − p p p , p ) , (6.22)where M ( x, q ) = ∞ (cid:89) n =1 (cid:0) − x q n (cid:1) − n and (cid:102) M ( x, q ) = M ( x, q ) M ( x − , q ) (6.23)with M ( q ) = M (1 , q ). Moreover, since in this case X → C / Γ is a semi-small resolution, i.e. it con-tains no compact four-cycles, by [47, Proposition A.7] the Donaldson–Thomas partition functionsof [ C / Z × Z ] and its natural crepant resolution X = Hilb Z × Z ( C ) are related through K DT C / Z × Z ( p , p , p , p ) = M ( − p ) − K top X ( p, p , p , p ) K top X ( p, p − , p − , p − ) (6.24)where the topological string partition function K top X ( p, p , p , p ) = M ( − p ) M ( p p , − p ) M ( p p , − p ) M ( p p , − p ) M ( p , − p ) M ( p , − p ) M ( p , − p ) M ( p p p , − p ) (6.25)is computed via the topological vertex formalism [82]. Here the variables p , p and p correspondto the basis of curve classes (D2 branes) in X and p to the Euler number (D0 branes). In thisway the gauge theory we have constructed on [ C / Z × Z ] realizes the anticipated wall-crossingbehaviour of the BPS partition function (1.1), connecting in this case the orbifold point with thelarge radius point in the K¨ahler moduli space. This partially justifies some of our arguments fromSection 5.9. A -fibred threefolds Next we will consider another set of examples of semi-small crepant resolutions, this time ob-tained as fibrations of hyper-K¨ahler ALE spaces over the complex plane [83, 84]. These ALEspaces are obtained by blowing up an abelian quotient singularity of the form C / Z n . The result-ing smooth geometry is then trivially fibred over the complex plane to obtain a threefold. Thetheory of Donaldson–Thomas invariants and their wall crossings on these geometries was studiedin [85].We are interested in the local surfaces which are semi-small crepant resolutions of the form X → C / Γ × C , where the orbifold action is g · ( z , z , z ) = ( ζ z , ζ − z , z ) (7.1)with g a generator of Γ = Z n and ζ an n -th root of unity. The resolved Calabi–Yau geometry istherefore a (trivial) fibration of a resolved A n − singularity over the affine line. These A -singularitiesare abelian. The resolved geometry is toric and is in particular a small resolution of the singularity,i.e. a birational morphism such that the exceptional locus consists of curves. The correspondingtoric diagram is obtained by subdividing the long edge of the toric diagram for C into n partsof equal length via the insertion of n − C / Γ. A choice oflattice vectors generating the toric fan is given by D = (1 , , , D = (0 , , , . . . , D n = ( − n + 1 , n, , (7.2)with the linear equivalences among toric divisors D − D − · · · − ( n − D n ∼ D + 2 D + · · · + n D n ∼ . (7.3)The intersection matrix C = ( C rs ) of the exceptional curves D , . . . , D n − is minus the Cartanmatrix of the A n − Lie algebra. The set of generators of the cohomology groups is given in termsof the set of tautological bundles, which in this case are simply the line bundles correspondingto the divisors, plus the trivial bundle. Each tautological bundle R r corresponds naturally to anirreducible representation ρ r which labels the monodromy of its canonical connection at infinity;equivalently it can be read off from the orbifold action on the monomials which are dual to thedivisors. The K¨ahler cone generators are given by the first Chern classes of the tautological bundles, e r = c ( R r ) = (cid:80) s ( C − ) rs D s , and they are dual to the Mori cone generators with respect to theintersection pairing D r · e s = δ rs .The whole geometric structure is encoded in the affine extension of the Cartan matrix of A n − .All of the geometry can be rephrased in terms of the representation theory of the A n − Lie algebraand its affine extension via the McKay correspondence. The representation theory data can also becompactly encoded in the McKay quiver associated with the singularity, whose arrow structure isdictated by the decomposition into irreducible representations of the tensor product Q ⊗ ρ r , where Q is the fundamental representation of Γ ⊂ SU (2) ⊂ SU (3). This construction closely parallelsthe construction of the McKay quiver for threefolds.These sets of data can be used to study the A -fibred singularities. They can be realized via therepresentation theory of a certain quiver, which is a modification of the McKay quiver associatedwith the singularity. This quiver is obtained from the usual McKay quiver by adding a set of loops,arrows from each vertex to itself. Small crepant resolutions of these singularities have an alternativedescription as the moduli space of representations of the modified McKay quivers. The pathalgebras of these quivers are noncommutative crepant resolutions of the fibred singularities.For example, consider the C / Z × C orbifold with weights r = 1, r = 2, r = 0. Its toric fan isdepicted in Figure 2. The regular representation is now Q = ρ ⊕ ρ ⊕ ρ and the tensor productFigure 2: Toric fan for the A -fibration.decomposition gives symmetric matrices (cid:0) a (1) rs (cid:1) = (cid:0) a (2) rs (cid:1) = . (7.4)51he associated quiver is v • (cid:119) (cid:119) (cid:39) (cid:39) (cid:25) (cid:25) v • (cid:49) (cid:49) (cid:55) (cid:55) (cid:54) (cid:54) • v (cid:103) (cid:103) (cid:113) (cid:113) (cid:118) (cid:118) (7.5) Let us start with the U (1) gauge theory. The combinations a (1) rs − a (2) rs vanish identically, andthe boundary condition is imposed by choosing N = (1 , , . . . , , , . . . , n − 1. Let us evaluate thisintegral. It measures the triple intersection between two divisors, corresponding to two compactcurves, with the divisor ℘ ∞ at infinity. The geometry we are considering is a trivial fibration of an A n − singularity over an affine line. Therefore all the intersection numbers are essentially given bythe intersections of curves in the exceptional locus of the blown up singularity in C . However theseintersections are still “fibred” over the affine line. For example, if two exceptional curves intersect ata point in the ALE geometry, then their intersection in the full Calabi–Yau threefold has the formpt × C . In the full Calabi–Yau geometry the exceptional curves are actually non-compact divisorsof the form P × C . The intersection with the boundary divisor is only due to the non-compactfactor C .We can therefore write the action as (cid:90) X ω ∧ ch ( E ) = − n − (cid:88) m,r,s =0 ϕ m (cid:16) a (2) rs − δ rs (cid:17) k s a | Γ | ˜ C mr , (7.6)where the constant a parametrizes the intersections with ℘ ∞ and˜ C = (cid:18) C − (cid:19) (7.7)with C the intersection matrix (minus the A n − Cartan matrix). The rest of the instanton action(4.50) reads as in (6.8). The integrals involved have the form (cid:16) (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r ) (cid:17) r =0 , ,...,n − = a | Γ | (0 , − , − , . . . , − , (cid:16) (cid:90) X c ( R r ) ∧ ch (cid:0) O X (1) (cid:1) (cid:17) r =0 , ,...,n − = 12 | Γ | (0 , b, b, . . . , b ) , (7.8)where we have parametrized the intersection indices of non-compact divisors of the form pt × C and P × C with the divisor at infinity by two integers a and b (whose precise values are not importantat the moment).To compute the instanton measure we consider the Γ-invariant part of the partial character T + π = V π − V π ⊗ V ∨ π (1 − t ) (1 − t ) t t . (7.9)52or the orbifold action (7.1) the terms proportional to V π ⊗ V ∨ π cancel pairwise upon substituting t = ζ , t = ζ − , and therefore (cid:0) T + π (cid:1) Γ = | π | . (7.10)The instanton partition function for the rank one invariants is thus Z C / Z n × C = (cid:88) π ( − | π | e − S inst [ π ] . (7.11)The orbifold group Γ = Z n has n irreducible representations. Therefore it is natural to parametrizethe partition function of orbifold invariants as K DT C / Z n × C = (cid:88) π ( − | π | p | π | p | π | · · · p | π n − | n − , (7.12)which precisely coincides with the result of [47]. This partition function can also be written in aproduct form [47] K DT C / Z n × C = M ( − p ) n (cid:89) 2) ( | π | + | π | )) Q − a ( | π |−| π | )1 Q − a ( | π |−| π | )2 (7.16)where as before we have introduced the weighting variables q = e − g s and Q r = e − ϕ r for r = 1 , p = q (8 a +2 b +2) Q − a/ Q − a/ ,p = q − (4 a + b − Q a/ ,p = q − (4 a + b − Q a/ , (7.17)with p = p p p = q . 53 .3 Coulomb branch invariants Let us now turn to the non-abelian gauge theory. We consider the U ( N ) gauge theory wherethe gauge symmetry is broken to U (1) N according to the pattern dictated by the framing vector N = ( N , N , . . . , N n − ). From (5.59)–(5.61) the instanton action now has the form (cid:90) X ω ∧ ω ∧ c ( E ) = 0 , (cid:90) X ω ∧ ch ( E ) = − n − (cid:88) m,r,s =0 ϕ m (cid:16) a (2) rs − δ rs (cid:17) N (cid:88) l =1 | π l,s − b ( l ) | a | Γ | ˜ C mr , (7.18) (cid:90) X ch ( E ) = − n − (cid:88) r,s =0 N (cid:88) l =1 | π l,s − b ( l ) | (cid:18) (cid:16) a (2) rs − δ rs (cid:17) (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r )+ (cid:16) a (2) rs − δ rs (cid:17) (cid:90) X c ( R r ) ∧ ch (cid:0) O X (1) (cid:1)(cid:19) − δ rs | Γ | , up to constant terms which are proportional to N r but independent of the instanton numbers; suchterms can be safely ignored and absorbed into the normalization of the partition function. Theintegrals appearing here are given by (7.8).Similarly we have to compute the instanton measure from (5.58). The second set of sums vanishesidentically because of the choice of orbifold action (7.1), and we are left with K ( (cid:126)π ; N ) = N (cid:88) l =1 n − (cid:88) r =0 | π l,r | N r + b ( l ) . (7.19)We can therefore write down the partition function for noncommutative Donaldson–Thomas in-variants of type N in the concise form Z C / Z n × C ( N ) = (cid:88) (cid:126)π ( − (cid:80) Nl =1 (cid:80) n − r =0 | π l,r | N r + b ( l ) × q (cid:82) X ch ( E (cid:126)π ) n (cid:89) m =1 Q (cid:80) n − r,s =0 ( a (2) rs − δ rs ) (cid:80) Nl =1 | π l,s − b ( l ) | a | Γ | ˜ C mr m (7.20)where Q m = e − ϕ m .To clarify the content of this formula, let us return to our particular example C / Z × C . Forconcreteness, let us choose the rank N = 5 with the boundary condition N = (2 , , b ( l ), which to an index l = 1 , . . . , N associates the index of the irreduciblerepresentation associated with the Higgs field vacuum expectation value e l , this implies the assign-ments b (1) = 0 = ⇒ e ←→ ρ ,b (2) = 0 = ⇒ e ←→ ρ ,b (3) = 1 = ⇒ e ←→ ρ ,b (4) = 1 = ⇒ e ←→ ρ ,b (5) = 2 = ⇒ e ←→ ρ . (7.21) These factors are irrelevant for the computation of the invariants. On the other hand, they would be crucial forestablishing the modular properties of the partition function. (cid:88) l =1 | π l, − b ( l ) | = | π , | + | π , | + | π , | + | π , | + | π , | , (cid:88) l =1 | π l, − b ( l ) | = | π , | + | π , | + | π , | + | π , | + | π , | , (cid:88) l =1 | π l, − b ( l ) | = | π , | + | π , | + | π , | + | π , | + | π , | , (7.22)the instanton action can be written as (again up to irrelevant constant terms) (cid:90) X ω ∧ ch ( E (cid:126)π ) = a (cid:16) ϕ (cid:0) | π , | − | π , | + | π , | − | π , | + | π , | − | π , | + | π , | − | π , | + | π , | − | π , | (cid:1) + ϕ (cid:0) | π , | − | π , | + | π , | − | π , | + | π , | − | π , | + | π , | − | π , | + | π , | − | π , | (cid:1)(cid:17) (7.23)and (cid:90) X ch ( E (cid:126)π ) = 16 (cid:16) (8 a + 2 b + 2) (cid:0) | π , | + | π , | + | π , | + | π , | + | π , | (cid:1) − (4 a + b − (cid:0) | π , | + | π , | + | π , | + | π , | + | π , | + | π , | + | π , | + | π , | + | π , | + | π , | (cid:1)(cid:17) . (7.24)Similarly the instanton measure (7.19) in this case becomes K (cid:0) (cid:126)π ; N = (2 , , (cid:1) = | π , | + | π , | + | π , | + | π , | + | π , | , (7.25)where we have dropped the even parity terms which do not affect the alternating sign of thefluctuation determinant. By using the change of variables (7.17) of the U (1) gauge theory we canwrite the partition function as Z C / Z × C (cid:0) N = (2 , , (cid:1) = (cid:88) π ( − | π , | p | π , | p | π , | p | π , | (cid:88) π ( − | π , | p | π , | p | π , | p | π , | × (cid:88) π ( − | π , | p | π , | p | π , | p | π , | (cid:88) π ( − | π , | p | π , | p | π , | p | π , | × (cid:88) π ( − | π , | p | π , | p | π , | p | π , | . (7.26)By repeatedly applying (7.14) this partition function can be expressed in closed form as a productof generalized MacMahon functions. For example the sum (cid:88) π ( − | π , | p | π , | p | π , | p | π , | = (cid:88) π ( − | π , | p | π , | ( − p ) | π , | ( − p ) | π , | (7.27)is equal to (7.14) upon redefining p → p , p → − p and p → − p . One then finds Z C / Z × C (cid:0) N = (2 , , (cid:1) = M ( − q ) (cid:102) M ( p , − q ) (cid:102) M ( p , − q ) (cid:102) M ( − p , − q ) × (cid:102) M ( p p , − q ) (cid:102) M ( − p p , − q ) (cid:102) M ( − p p , − q ) . (7.28)55y expanding these functions one obtains expressions for the noncommutative Donaldson–Thomasinvariants NC ( k , N ) of type N = (2 , , 1) for the orbifold geometry C / Z × C . These invariants,although numerically different from those of the U (1) gauge theory, can also be derived fromrank one quiver generalized Donaldson–Thomas invariants via the formula (5.41). By taking thelogarithm of the partition function (7.28) we obtain the free energy F C / Z × C (cid:0) N = (2 , , (cid:1) = ∞ (cid:88) n,l =1 ( − n l nl (cid:16) p n l p n l p n l + 3 p n l p ( n +1) l p n l + 3 p n l p ( n − l p n l + 2( − l p n l p ( n +1) l p ( n +1) l + 2( − l p n l p ( n − l p ( n − l + 4( − l p n l p n l p ( n +1) l + 4( − l p n l p n l p ( n − l (7.29)+ 2( − l p ( n +1) l p n l p ( n +1) l + 2( − l p ( n − l p n l p ( n − l + 3 p ( n +1) l p n l p n l + 3 p ( n − l p n l p n l + p ( n +1) l p ( n +1) l p n l + p ( n − l p ( n − l p n l (cid:17) . By combining terms one finds that this expression indeed fits the pattern of (5.41). For exam-ple ∞ (cid:88) n,l =1 ( − n l nl p n l p ( n − l p n l + ∞ (cid:88) n,l =1 ( − n l + l nl p ( n +1) l p n l p ( n +1) l = ∞ (cid:88) n,l =1 ( − n l n + 2 n − l p n l p ( n − l p n l (7.30)= ∞ (cid:88) n,l =1 ( − n l +2( n − l + n l (cid:0) n l + 2( n − l + n l (cid:1) (cid:18) l (cid:19) p n l p ( n − l p n l , which implies DT ( k ) = − l for k = ( n l, n l − l, n l ) with n, l ≥ . (7.31)Proceeding in a similar way for the remaining terms, we obtain the non-vanishing rank one quiverinvariants DT ( k ) = − l for k = ( n l, n l + l, n l ) , n ≥ , l ≥ , ( n l − l, n l, n l ) , n, l ≥ , ( n l + l, n l, n l ) , n ≥ , l ≥ , ( n l − l, n l − l, n l ) , n, l ≥ , ( n l + l, n l + l, n l ) , n ≥ , l ≥ , (7.32)and DT (cid:0) k = ( k, k, k ) (cid:1) = − (cid:88) l ≥ l | k l for k ≥ . (7.33)Comparing with (5.38) we obtain the non-vanishing integer BPS invariants (cid:99) DT ( k ) = − k = ( n, n − , n ) , n ≥ , ( n, n + 1 , n ) , n ≥ , ( n − , n, n ) , n ≥ , ( n + 1 , n, n ) , n ≥ , ( n − , n − , n ) , n ≥ , ( n + 1 , n + 1 , n ) , n ≥ , (7.34)56nd (cid:99) DT (cid:0) k = ( k, k, k ) (cid:1) = − k ≥ . (7.35)These invariants agree with those obtained in [18] from the rank one partition function (7.14).We can similarly generalize our arguments to get a compact solution for the noncommutativeDonaldson–Thomas invariants for any A -fibred threefold of the form C / Z n × C and any boundarycondition fixed by a framing vector N . The partition function then assumes the form Z C / Z n × C ( N ) = (cid:88) (cid:126)π ( − (cid:80) Nl =1 (cid:80) n − r =0 | π l,r | N r + b ( l ) p (cid:80) Nl =1 | π l, − b ( l ) | · · · p (cid:80) Nl =1 | π l,n − − b ( l ) | n − . (7.36)As we did above, for a given fixed framing vector N it is possible to express this formula in aclosed form as a product of MacMahon functions M ( q ) and (cid:102) M ( x, q ). The corresponding quiverinvariants, independent of N and N , can again be computed explicitly from the formulas (5.41)and (5.38). The integer BPS invariants in this case are computed in [18, Section 7.5.4] from therank one partition function (7.13). We will now describe the wall-crossing formula for Coulomb branch invariants. Although the wallcontributions are all contained in the quiver BPS invariants (cid:99) DT µ ( k ), which are unchanged by wall-crossing in this case [18], it is interesting to examine if the noncommutative invariants NC µ ( k , N )have wall-crossings of their own. Here we consider only very particular walls of stability. In general,there will be walls corresponding to separated D6 branes colliding and forming a bound state; thesewalls are not included in our analysis below, and to get them one should use a non-primitive wall-crossing formula. On the other hand, since the D6 branes are well-separated in the Coulomb branch,it is reasonable to expect that the walls affecting D2–D0 bound states are reached before the wallscorresponding to D6 bound states.The large radius partition function for Coulomb branch invariants can be computed from the U ( N )instanton contributions to the noncommutative gauge theory on the ALE resolution X of the A n − -fibration [37]. It is given by a simple modification of the partition function for topological stringtheory on X as K DT X ( q, p , . . . , p n − ; N ) = M (cid:0) ( − N q (cid:1) n N (cid:89) 1. One then finds the non-abelian wall-crossing formula Z C / Z n × C (cid:0) q, p , . . . , p n − ; N = ( N, , . . . , (cid:1) (7.38)= M (cid:0) ( − N q (cid:1) − n N K DT X ( q, p , . . . , p n − ; N ) K DT X ( q, p − , . . . , p − n − ; N ) . The wall-crossing factor M (cid:0) ( − N q (cid:1) − n N K DT X ( q, p − , . . . , p − n − ; N ) in (7.38) describes the crossingof an infinite number of walls of marginal stability, separating different chambers in the K¨ahlermoduli space, in going from the orbifold point to the large radius point. To identify the individualwalls, let us first recall the situation in the U (1) gauge theory as studied in [85] (see also [87]).In this context the walls are determined by the affine Lie algebra structure associated with the57cKay quiver. If we denote by θ a θ -stability parameter for the (unframed) quiver, then the wallsof marginal stability are determined by W ˆ k = (cid:8) θ ∈ R n (cid:12)(cid:12) θ · ˆ k = 0 , ˆ k ∈ ˆ∆ + (cid:9) (7.39)where ˆ∆ + is the set of affine positive roots. These walls connect different chambers. Of particularrelevance among them is the wall determined by the imaginary root ˆ k im of the affine A n − Liealgebra, which corresponds to the regular representation of Γ = Z n and separates the Donaldson–Thomas chamber from the Pandharipande–Thomas chamber. Our wall-crossing formulas do notinclude this wall.Fix a real positive root k and consider stability parameters on both sides of the associated wall, θ + k and θ − k . Then our vector space V can be identified with the unique θ k -stable module over thepath algebra of dimension vector k constructed in [85]. Therefore the wall-crossing formula of [85,Theorem 4.15] is given by Z θ − k C / Z n × C ( p ) = (cid:0) − ( − k p k (cid:1) − k Z θ + k C / Z n × C ( p ) , (7.40)where the instanton charge k = | π | is singled out by the framing condition. It follows from (7.38)that the proof of [85, Theorem 4.15] can be adapted to our more general situation to give Z θ − k C / Z n × C (cid:0) p ; N = ( N, , . . . , (cid:1) = (cid:0) − ( − k N p k (cid:1) − k N Z θ + k C / Z n × C (cid:0) p ; N = ( N, , . . . , (cid:1) , (7.41)which establishes the wall-crossing formula for noncommutative Donaldson–Thomas invariants oftype N = ( N, , . . . , U (1) sector the wall-crossing formula is the sameup to a redefinition of the assignments of parameters to irreducible representations and some signs.Thus each U (1) factor jumps separately while the walls are the same for every sector. We donot know how to extend these considerations to generic framing vectors N corresponding to moregeneral boundary conditions on the Higgs fields. C / Z Our next example is the C / Z orbifold with weights r = r = r = 1 that was studied inSection 3.2 and Section 5.3; its unique Calabi–Yau crepant resolution given by the Z -Hilbertscheme X = Hilb Z ( C ) is the total space of the fibration O P ( − → P , also known as the localdel Pezzo surface of degree zero. This geometry has one compact divisor, the base E ∼ = P of thefibration, and three rational curves which are homologous. Its fan Σ ⊂ Z is generated by thevectors D = ( − , , , D = (1 , , 1) and D = (0 , , , (8.1)and the toric diagram for the resolved geometry is depicted in Figure 3. The linear equivalencesbetween the divisors are given by D α ∼ D β and 3 D α + E ∼ α, β = 1 , , . (8.2)The Mori cone has a single generator C = D · E = D · E = D · E (8.3)58 D D E Figure 3: Toric fan for the local del Pezzo surface of degree zero.dual to the generator of the K¨ahler cone which is given by any of the linearly equivalent non-compactdivisors D α . In particular E · E · E = 9.The basis of tautological bundles is indexed by the irreducible representations ρ r with r = 0 , , Z on the coordinates of C , is Q = ρ ⊕ ρ ⊕ ρ . The tensor product decompositions (4.16) for i = 1 , (cid:0) a (1) rs (cid:1) = and (cid:0) a (2) rs (cid:1) = (8.4)with a (2) rs = a (1) sr . The associated quiver is given in (5.6).If we write the original coordinates of C as ( z , z , z ), then the rational curves are locally describedby invariant ratios of monomials of the form z /z and cyclic permutations thereof. Therefore theycorrespond to the character ρ , and a generator of H ( X, Z ) dual to the curve class C is givenby c ( R ). In the fan Σ there are now three toric curves intersecting in a vertex of valence 3 andtherefore, following the decoration procedure described in Section 4.6, we associate the character ρ = ρ ⊗ ρ to the vertex. This gives one relation in the Picard group Pic( X ), namely R = R ⊗R .In particular the second Chern class of V = ( R ⊕ R ) (cid:9) ( R ⊕ O X ) generates H ( X, Z ) and isdual to the exceptional divisor with (cid:90) P c ( V ) = 1 . (8.5)In this case we have c ( R ) = 2 c ( R ) and c ( V ) = c ( R ) ∧ c ( R ), with R α = O X ( α D ) where D is one of the linearly equivalent divisors D α which is dual to the class c ( R ) correspondingto P . We begin again with the U (1) gauge theory. As explained in Section 4.5 we need only considerthe boundary condition given by the framing vector N = (1 , , H ( X, Z ) and H ( X, Z ) we can write ω = ϕ c ( R ) and ω ∧ ω = ς c ( V ) = ς c ( R ) ∧ c ( R ) . (8.6)59rom the general form of the instanton action (4.48) we have (cid:90) X ω ∧ ω ∧ c ( E ) = − ς (cid:88) r,s =0 (cid:16) a (1) rs − a (2) rs (cid:17) k s (cid:90) X c ( V ) ∧ c ( R r )= ς (3 k − k + 3 k ) (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) . (8.7)On the other hand since E · E · E = 9 and E ∼ − D , we see that (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · D = − . (8.8)Similarly, from (4.49) one has (cid:90) X ω ∧ ch ( E ) = − ϕ (cid:88) r,s =0 (cid:18) (cid:16) a (1) rs − a (2) rs (cid:17) k s (cid:90) X c ( R ) ∧ ch ( R r )+ (cid:16) a (2) rs − δ rs (cid:17) k s (cid:90) X c ( R ) ∧ c ( R r ) c (cid:0) O X (1) (cid:1)(cid:19) = − ϕ (9 k − k + 3 k ) − ϕ (6 k − k − k ) (cid:90) X c ( R ) ∧ c ( R ) ∧ c (cid:0) O X (1) (cid:1) . (8.9)The last term of the instanton action (4.50) reads (cid:90) X ch ( E ) = − (cid:88) r,s =0 (cid:18) (cid:16) a (1) rs − a (2) rs (cid:17) k s (cid:90) X ch ( R r ) − (cid:16) a (2) rs − δ rs (cid:17) k s (cid:90) X c ( R r ) ∧ ch (cid:0) O X (1) (cid:1) + (cid:16) a (2) rs − δ rs (cid:17) k s (cid:90) X ch ( R r ) ∧ c (cid:0) O X (1) (cid:1) − δ rs k s (cid:19) (8.10)= − (7 k − k + k ) + (6 k − k − k ) (cid:90) X c ( R ) ∧ c (cid:0) O X (1) (cid:1) ∧ c (cid:0) O X (1) (cid:1) − (12 k − k − k ) (cid:90) X c ( R ) ∧ c ( R ) ∧ c (cid:0) O X (1) (cid:1) + ( k + k + k ) . To evaluate the intersection indices arising here we work as follows. This geometry contains acompact divisor. By using linear equivalence, it is possible to rephrase the analysis of Section 7.2in such a way that only compact cycles enter, the divisor and rational curves. Therefore all theintegrals involving c ( O X (1)) and c ( R ) compute the intersection indices between elements of theexceptional locus with the divisor at infinity. We assume that the compactification divisor ℘ ∞ is chosen so that these intersections vanish. Under this assumption, the instanton action finallybecomes (cid:90) X ch ( E ) = − (5 k − k − k ) , (cid:90) X ω ∧ ch ( E ) = − ϕ (3 k − k + k ) , (cid:90) X ω ∧ ω ∧ c ( E ) = − ς ( k − k + k ) . (8.11)We compute the instanton measure by taking the Z -invariant part of the character (5.23). Forthis, let us decompose V π = V ⊕ V ⊕ V at a fixed point π according to the Z -action as before,60here now each term is of order three. The explicit form of the partial character (7.9) is now T + π = ( V ⊕ V ⊕ V ) (8.12) − (cid:16) t t − t − t + 1 (cid:17) (cid:0) V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ (cid:1) . It is easy to find the invariant part by substituting t α = ζ to get the virtual dimension (mod-ulo 2) (cid:0) T + π (cid:1) Z = vdim C (cid:0) V − ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ )+ 2( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ) − ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ) (cid:1) = | π | + | π | + | π | | π | + | π | | π | + | π | | π | , (8.13)where we recall that the fixed points are classified by coloured partitions π = π (cid:116) π (cid:116) π . This resultagrees with that quoted in [47, Remark A.5], and it shows that the equivariant Euler characteristicof the obstruction bundle on the quiver variety at a fixed point π is given by χ T ( N π ) = ( − | π | + | π | + | π | | π | + | π | | π | + | π | | π | . (8.14)We can construct a partition function for these Euler characteristics. As we did in Section 6.2, weare led to define K DT C / Z = (cid:88) π ( − | π | + | π | + | π | | π | + | π | | π | + | π | | π | p | π | p | π | p | π | p | π | . (8.15)Here p is a formal parameter that weighs the number of boxes in a plane partition π , while theother formal variables p r , r = 0 , , k = ( k , k , k ). The first few terms of this partition function can be calculated explicitly tobe K DT C / Z = 1 + p p + 3 p p p + 3 p p p − p p p p + 9 p p p p + p p p − p p p p − p p p p + 9 p p p p − p p p p − p p p p + 15 p p p p + 21 p p p p + 3 p p p p + · · · = (cid:16) − p p p p ) + 12( − p p p p ) + · · · (cid:17) × (cid:16) p p + 3 p p p + 3 p p p + p (9 p p p + p p )+ p (9 p p p − p p p )+ p (15 p p p + 9 p p p + 3 p p p ) + · · · (cid:17) . (8.16)In this expression we recognise a factor of the MacMahon function raised to the power of thetopological Euler characteristic χ ( X ) = χ ( P ) = 3 of the target space. Indeed the generatingfunction of Z -invariant holomorphic polynomials decomposes as13 (cid:88) r =0 − ζ r z ) (1 − ζ r z )(1 − ζ r z ) (8.17)= 1(1 − z z ) (1 − z z ) (1 − z ) + 1(1 − z z ) (1 − z z ) (1 − z ) + 1(1 − z z ) (1 − z z ) (1 − z )61nto three invariant copies of the generating function for C . We thus expect the topological stringpartition function to contain the factor M ( x ) = 1 + 3 x + 12 x + 37 x + 111 x + 303 x + 804 x + · · · (8.18)with x = − q = − p p p p , corresponding to contributions from “regular” instantons (see Sec-tion 4.7).We will now compare the combinatorial partition function (8.15)–(8.16) with our BPS partitionfunction Z C / Z = (cid:88) π ( − | π | + | π | + | π | | π | + | π | | π | + | π | | π | q | π |− (7 | π |− | π | + | π | ) × Q (3 | π |− | π | + | π | ) U | π |− | π | + | π | , (8.19)where q = e − g s , Q = e − ϕ and U = e − ς . A quick computation shows that the two partitionfunctions are related by the change of variables p = q / ,p = q − / Q / U ,p = q / Q − U − ,p = q − / Q / U . (8.20)This BPS partition function contains contributions from non-vanishing D4 brane charge and hasan expansion Z C / Z = − q / U Q / − q / U Q + q / U Q / − q / U Q / + 3 q Q + 9 q / U Q / + 9 q / U Q + 3 q / U Q / + 21 q − q / U Q / − q + 3 q / U Q / − U Q / q / + 1 + U Q / q / + · · · = M ( − q ) (cid:16) U Q / q / + 3 q / U Q / + 3 q / U Q / + 9 q / U Q + q / U Q / + 9 q / U Q / − q / U Q + 15 q / U Q / + 9 q + 3 q Q + · · · (cid:17) . (8.21) We will now describe the noncommutative Donaldson–Thomas invariants NC ( k , N ) of type N inthe present case where the Calabi–Yau threefold has compact four-cycles. Although explicit closed(product) formulas are no longer available, it is possible to compute the invariants order by order asabove for a fixed boundary condition labelled by the framing vector N . In this case the instantonmeasure (5.58) is given by K ( (cid:126)π ; N ) = N (cid:88) l =1 2 (cid:88) r =0 | π l,r | N r + b ( l ) + N (cid:88) l,l (cid:48) =1 2 (cid:88) r =0 | π l,r | (cid:16) | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) | (cid:17) (8.22)where we have dropped irrelevant even parity terms. The rank N action is obtained from the U (1) action by writing the instanton charges as in (5.50). Therefore the partition function ofnoncommutative Donaldson–Thomas invariants of type N is Z C / Z ( N ) = (cid:88) (cid:126)π ( − (cid:80) Nl =1 (cid:80) r =0 | π l,r | ( N r + b ( l ) + (cid:80) Nl (cid:48) =1 ( | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) | )) q (cid:80) Nl =1 ( − | π l, − b ( l ) | +10 | π l, − b ( l ) | + | π l, − b ( l ) | ) (8.23) × Q (cid:80) Nl =1 (3 | π l, − b ( l ) |− | π l, − b ( l ) | + | π l, − b ( l ) | ) U (cid:80) Nl =1 ( | π l, − b ( l ) |− | π l, − b ( l ) | + | π l, − b ( l ) | ) . These invariants, although related to the noncommutative Donaldson–Thomas invariants, appearto be new. They differ from the definitions of [18] by the parameters involved. In our formulationthe invariants of [18] have the form K DT C / Z ( N ) = (cid:88) (cid:126)π ( − (cid:80) Nl =1 (cid:80) r =0 | π l,r | ( N r + b ( l ) + (cid:80) Nl (cid:48) =1 ( | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) | )) × ˜ p (cid:80) Nl =1 | π l, − b ( l ) | ˜ p (cid:80) Nl =1 | π l, − b ( l ) | ˜ p (cid:80) Nl =1 | π l, − b ( l ) | , (8.24)where ˜ p r = p p r = q / p r for r = 0 , , 2. Our invariants are numerically different but are related viathe change of variables (8.20) which allows one set of invariants to be expressed uniquely via theother set. Our formulation seems however to be more physically motivated. The Coulomb branchinvariants are also related to the quiver generalized Donaldson–Thomas invariants, and hence to thequiver BPS invariants, via the formula (5.39). However, in this case the relative Euler form (5.40)is non-zero and hence explicit infinite product forms for the partition functions are not available,thus making the explicit determination of these invariants somewhat more involved. We will now discuss the relationship between the orbifold and large radius phases of the local P geometry. The large radius partition function for rank one BPS states is that of topological stringtheory on X = K P = O P ( − K top X = (cid:88) λ ,λ ,λ ( − Q ) (cid:80) α | λ α | q − (cid:80) α κ λα C ∅ λ (cid:48) λ ( q ) C ∅ λ (cid:48) λ ( q ) C ∅ λ (cid:48) λ ( q ) (8.25)where the sum runs over ordinary partitions λ = ( λ i ) (Young tableaux) labelling the three internallines of the toric diagram dual to the fan of Figure 3, with conjugate partitions λ (cid:48) and κ λ := | λ | + (cid:80) i λ i ( λ i − i ), Q = e − t with t = (cid:82) P ω the K¨ahler parameter corresponding to the hyperplaneclass in P , and q = e − g s . In the melting crystal formulation, the topological vertex can beexpressed as a sum over plane partitions π which asymptote to boundary partitions ( λ , λ , λ )along the three coordinate axes as [1] C λ λ λ ( q ) = M ( q ) − q ( (cid:107) λ (cid:48) (cid:107) + (cid:107) λ (cid:48) (cid:107) + (cid:107) λ (cid:48) (cid:107) ) (cid:88) π : ∂π =( λ ,λ ,λ ) q | π | , (8.26)where (cid:107) λ (cid:107) := (cid:80) i λ i and in this expression | π | denotes the renormalized volume of the infinitethree-dimensional Young diagram π . We can rewrite this expansion in terms of Schur functionsas K top X = (cid:88) λ ,λ ,λ ( − Q ) (cid:80) α | λ α | q − (cid:80) α κ λα / × s λ (cid:48) ( q ρ ) s λ (cid:48) ( q ρ ) s λ (cid:48) ( q ρ ) s λ (cid:48) ( q ρ + λ (cid:48) ) s λ (cid:48) ( q ρ + λ (cid:48) ) s λ (cid:48) ( q ρ + λ (cid:48) ) (8.27)where q λ + ρ := ( q λ i − i +1 / ). In the non-abelian gauge theory, the contributions from noncommutative U ( N ) instantons in the Coulomb branch can be computed following [37] and yield the rank N BPSpartition function K DT X ( q, Q ; N ) = K top X (cid:0) ( − N +1 q , Q (cid:1) N . (8.28)63he orbifold phase is recovered by blowing down the compact divisor P , i.e. by formally setting Q → U = 1 in the orbifold partition functions. In contrast to our previous examples, here thereis no closed form for the partition function either at large radius or in the noncommutative crepantresolution chamber, and the locations of the walls of marginal stability are not known. However,wall-crossing is always described by the Kontsevich–Soibelman formula, and in the present casesince the D6 brane charge is unity one can use the semi-primitive wall-crossing formula. Note thatfor Calabi–Yau threefolds with compact four-cycles, a simple formula such as (1.1) connecting theorbifold point to the large radius point is not anticipated. In the present case, this is because thegeometry contains a divisor which lies over the singular point of C / Z , namely the base P of thefibration, i.e. in this case the crepant resolution π : X → C / Z is not semi-small. Hence theconditions of the crepant resolution conjecture of [47, 58] are not met; the essence of the problemis that the additional non-vanishing homology group H ( X ) introduces more variables into thecounting problem on X than is dictated by the classical McKay correspondence. Moreover, sincein this case ¯ χ (cid:54) = 0, the quiver BPS invariants (cid:99) DT µ ( k ) vary by wall-crossing formulas under changesof stability condition in the derived category D ( X ). To illustrate the moduli space phase structureof marginally stable D brane states on the Calabi–Yau ALE space X = O P ( − F g in the coordinates computed on the saddle point. This prescription can besummarized by the rule (cid:101) F g = F g + Γ g (∆ , ∂ i · · · ∂ i n F r 0. To reach this point one considers the A-model in thelarge radius phase where classical geometry is a good concept. Then one looks at the B-model onthe mirror manifold (see [49, Section 6] for a review ). The mirror is described by a family ofelliptic curves, as prescribed by the rules of local mirror symmetry. The moduli space of this family In [49] the orbifold point is incorrectly set at t (cid:54) = 0. 64s one-dimensional and can be regarded as a projective line P with three punctures at z = 0 , , ∞ .The point z = 0 is the large volume point where the exceptional divisor in the mirror A-model hasinfinite size, the point z = ∞ is the orbifold point with no blow-up, and z = 1 is the conifold pointwhere the underlying worldsheet conformal field theory is singular. The B-model is solved by athree-dimensional vector of periods that satisfies the Picard–Fuchs equation, which can be solvedin each of the three neighbourhoods around z = 0 , , ∞ . Then in each of these neighbourhoods wehave a set of good coordinates that solve the Picard–Fuchs equation.The orbifold mirror map sends linear combinations of these solutions to a basis for the orbifoldcohomology of [ C / Z ]. Thus if we know the topological string free energy in the large radiusneighbourhood F ∞ g then we can recover it around the orbifold point F orb g by using (8.29) as F orb g = F ∞ g + Γ g (cid:0) ∆ , ∂ i · · · ∂ i n F ∞ r 3) are determined entirely by those of the exceptional divisor E = P .Indeed, in [91] it is demonstrated that holomorphic objects near the orbifold point come fromrepresentations of the Beilinson quiver, or equivalently from large volume gauge sheaves. Usingthese facts and local mirror symmetry, it should be possible to map orbifold and large radius phaseobjects into one another, along the lines of [27].In order to make sense of the (argument of the) central charge of the non-compact D6 branes, oneneeds to consider the local P geometry as a large radius limit of a compact Calabi–Yau space.According to [78], the (conjecturally) proper limit involves an extra parameter: a component of the B -field normal to the base survives the local limit. This parameter plays a crucial role in stabilityand wall-crossing analyses. This would also explain why the large radius Donaldson–Thomas theorybehaves in such a complicated manner when trying to approach the orbifold phase, even thoughslope stability is trivial for ideal sheaves: the new stability condition crucially involves this extraparameter. Furthermore, there are various terms in the Dirac–Born–Infeld action which correspondto the possible deformations of the D brane inside the Calabi–Yau manifold. In the local limit,which corresponds to zooming in on a neighbourhood of P in X , many of these terms should bedropped since the brane in the local geometry has much fewer allowed deformations. However, bythe above arguments, the result of the local limit is not the six-dimensional Yang–Mills theory wehave been considering. C / Z orbifold Our final example is the orbifold C / Z with weights r = 1, r = 2, r = 3. It has a toric diagramgiven by three lattice vectors D = ( − , − , , D = (2 , − , 1) and D = ( − , , 1) (9.1)which represent an isolated quotient singularity. We consider the crepant resolution given by theHilbert scheme X = Hilb Z ( C ) which is obtained by adding the vectors D = (0 , − , , D = (1 , − , , D = ( − , , , D = (0 , , 1) (9.2)and the appropriate triangulation shown in Figure 4. This is only one of the five distinct possiblecrepant resolutions. 66 D D D D D D Figure 4: Toric fan for the resolution of the C / Z singularity used in the main text.The resolved geometry has six non-compact divisors D i with i = 1 , . . . , D with the topology of a Hirzebruch surface blown up at one point, together with the linearequivalences 6 D + D + 2 D + 4 D ∼ , D + D + 2 D + D ∼ , D + D + D ∼ , (9.3)which we use to solve for the divisors D ∼ − D − D + D − D ,D ∼ − D + D − D ,D ∼ D − D − D + D . (9.4)The non-vanishing intersection numbers between three distinct divisors can be read off from thetoric diagram by checking whether or not the three divisors lie at the corners of a basic triangle.This gives D · D · D = 1 , D · D · D = 1 , D · D · D = 1 ,D · D · D = 1 , D · D · D = 1 , D · D · D = 1 , (9.5)and the triple intersection numbers of each divisor can be found from these integers by using linearequivalence; for example D · D · D = D · ( − D − D ) · ( − D − D − D ) = 7 . (9.6)The Mori cone is generated by the four compact curves C n , n = 1 , , , D , D , D and − D + D + D − D . Consequentlythe tautological bundles corresponding to the six one-dimesional irreducible representations ρ r ,67 = 0 , , , , , Z are R = O X , R = O X ( D ) , R = O X ( D ) , R = O X ( D ) , R = O X ( − D + D + D − D ) , R = R ⊗ R = O X ( D + D ) . (9.7)The corresponding decoration of the toric fan is depicted in Figure 5. This decoration was also ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ Figure 5: Decorated fan for the resolved C / Z geometry.obtained in [93], which contains a study of the Gromov–Witten theory of the symmetric resolution X as well as a description of its mirror manifold.From the decoration we immediately obtain the generators of the cohomology groups given by H ( X, Z ) = Z (cid:10) c ( R ) , c ( R ) , c ( R ) , c ( R ) (cid:11) and H ( X, Z ) = Z (cid:10) c ( V ) (cid:11) (9.8)where V = ( R ⊕ R ) (cid:9) ( R ⊕ O X ) . (9.9)Let us compute c ( V ) more explicitly. First of all the first Chern class of V vanishes c ( V ) = c ( R ) + c ( R ) − c ( R ) = 0 (9.10)due to R = R ⊗ R and the additivity of the first Chern class under tensor product. Therefore c ( V ) = − ch( V ) which simplifies the computation. By the additivity of the Chern characterch ( V ) = ch ( R ) + ch ( R ) − ch ( R )= (cid:0) c ( R ) ∧ c ( R ) + c ( R ) ∧ c ( R ) − c ( R ) ∧ c ( R ) (cid:1) = (cid:0) c ( R ) ∧ c ( R ) + c ( R ) ∧ c ( R ) − ( c ( R ) + c ( R )) ∧ ( c ( R ) + c ( R )) (cid:1) = − c ( R ) ∧ c ( R ) , (9.11)which implies c ( V ) = c ( R ) ∧ c ( R ) . (9.12)68rom the representation theory data we can construct the matrices a (1) rs and a (2) rs = a (1) sr . They aregiven by the tensor product decompositions of Q = ρ ⊕ ρ ⊕ ρ ; explicitly (cid:0) a (1) rs (cid:1) = and (cid:0) a (2) rs (cid:1) = . (9.13)The associated quiver is v • (cid:36) (cid:36) (cid:28) (cid:28) (cid:12) (cid:12) v • (cid:51) (cid:51) (cid:45) (cid:45) (cid:42) (cid:42) v • (cid:8) (cid:8) (cid:120) (cid:120) (cid:113) (cid:113) v • (cid:72) (cid:72) (cid:56) (cid:56) (cid:49) (cid:49) v • (cid:115) (cid:115) (cid:109) (cid:109) (cid:106) (cid:106) v • (cid:76) (cid:76) (cid:100) (cid:100) (cid:92) (cid:92) (9.14) We now compute the instanton action with boundary condition N = (1 , , , , , (cid:90) X ω ∧ ω ∧ c ( E ) = − ς (cid:88) r,s =0 (cid:16) a (1) rs − a (2) rs (cid:17) k s (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R r ) . (9.15)To evaluate this contribution we need the integrals (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = 0 , (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · D = 0 , (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · D = D · D · (cid:0) − ( D + 2 D + D ) (cid:1) = − , (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · D = D · D · (cid:0) − ( D + D ) (cid:1) = − , (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · ( − D + D + D − D ) = − − = − , (cid:90) X c ( R ) ∧ c ( R ) ∧ c ( R ) = D · D · ( D + D ) = − , (9.16)and finally (cid:90) X ω ∧ ω ∧ c ( E ) = ς (cid:16) k − k − k − k + 3 k + 8 k (cid:17) . (9.17)69onsider now the second Chern character (4.49) given by (cid:90) X ω ∧ ch ( E ) = − (cid:88) n =1 5 (cid:88) r,s =0 ϕ n k s (cid:18) (cid:16) a (1) rs − a (2) rs (cid:17) (cid:90) X c ( R n ) ∧ ch ( R r ) (9.18)+ (cid:16) a (2) rs − δ rs (cid:17) (cid:90) X c ( R n ) ∧ c (cid:0) O X (1) (cid:1) ∧ c ( R r ) (cid:19) . To evaluate the first term one has to compute the triple intersection numbers between the divisors.For example the triple intersection (cid:90) X c ( R ) ∧ ch ( R ) = D · D · D (9.19)is linearly equivalent to D · ( D + D + D + D ) · (cid:0) − ( D + D ) (cid:1) = − D · D · D = − . (9.20)Laborious manipulations give altogether (cid:16) (cid:90) X c ( R n ) ∧ ch ( R r ) (cid:17) n,r =0 = − 00 0 0 − − − − − − − − − − − . (9.21)To evaluate the intersections with the boundary divisor ℘ ∞ we proceed as follows. As a start byusing linear equivalence we can write D ∼ D − D + D + 2 D , (9.22)so that all the intersection products involve either the original non-compact divisors D , D and D of C , or the compact divisor D which by assumption has no intersection with ℘ ∞ . Furthermorewe can argue by symmetry that the intersection indices can be parametrized by two numbers ℘ ∞ · D i · D i = a for i = 1 , , ,℘ ∞ · D i · D j = b for i (cid:54) = j = 1 , , . (9.23)The intersection matrix can be therefore parametrized as (cid:16) (cid:90) X c ( R n ) ∧ c (cid:0) O X (1) (cid:1) ∧ c ( R r ) (cid:17) n,r =0 = 1 | Γ | a b b − a + 3 b b b a b a a + b b b a a − b a + b − a + 3 b a a − b a − b a − b (9.24)where here | Γ | = 6.As in Section 6.2 we can set b = 1 and a = 0. We will however for the time being keep bothparameters arbitrary. Under these conditions the second Chern character term in the instantonaction (4.49) becomes (cid:90) X ω ∧ ch ( E ) = − (cid:16) ϕ ( k − k − k + k ) + 5 ϕ (3 k + k − k − k − k + 2 k )70 ϕ (7 k + 2 k − k − k − k + 5 k )+ ϕ (10 k + 3 k − k − k − k + 7 k ) (cid:17) + a (cid:16) − ϕ (4 k + 3 k + k − k − k )+ ϕ ( − k − k + 2 k + 7 k + 26 k − k )+ ϕ ( − k − k + 2 k − k + 2 k + k )+ ϕ (2 k + 5 k − k − k − k + 2 k ) (cid:17) + b (cid:16) ϕ ( − k + k − k + k − k + k )+ ϕ ( − k − k + k + 2 k + 7 k + k )+ 2 ϕ (5 k + 9 k − k − k − k + 3 k )+ ϕ (3 k + k − k − k + 3 k ) (cid:17) . (9.25)Finally we are left with the last part of the instanton action (4.50). We have to evaluate the inte-grals involving the Chern classes. Fortunately we have already computed most of the intersectionproducts. In vector notation we have (cid:16) (cid:90) X ch ( R r ) (cid:17) r =0 = (cid:16) (cid:90) X c ( R r ) ∧ c ( R r ) ∧ c ( R r ) (cid:17) r =0 = 16 (cid:16) , , , − , − , − (cid:17) , (cid:16) (cid:90) X c (cid:0) O X (1) (cid:1) ∧ ch ( R r ) (cid:17) r =0 = (cid:16) (cid:90) X c (cid:0) O X (1) (cid:1) ∧ c ( R r ) ∧ c ( R r ) (cid:17) r =0 = 12 | Γ | (cid:0) , a, a, a, a − b, a + 2 b (cid:1) . (9.26)The remaining integrals involve the double intersection of the divisor at infinity. Arguing again bysymmetry we can parametrize these integrals with a single number (cid:16) (cid:90) X c ( R r ) ∧ ch (cid:0) O X (1) (cid:1) (cid:17) r =0 = (cid:16) (cid:90) X c ( R r ) ∧ c (cid:0) O X (1) (cid:1) ∧ c (cid:0) O X (1) (cid:1) (cid:17) r =0 = 12 | Γ | (cid:0) , c, c, c, c, c (cid:1) , (9.27)where we have expressed D in terms of D , D and D by using linear equivalence, and taken theintersection with D to be zero so that D · ℘ ∞ · ℘ ∞ = ( D − D + D + 2 D ) · ℘ ∞ · ℘ ∞ = − c + c + 2 c = c . (9.28)Finally the last term in the instanton action is (cid:90) X ch ( E ) = 112 (cid:16) − (9 k + 5 k − k − k − k + 4 k )+ a ( − k − k + k + 24 k − k ) + 2 b (3 k + 3 k − k − k + 7 k )+ c (4 k − k − k ) + 2( k + k + k + k + k + k ) (cid:17) . (9.29)We now need to compute the Z -invariant part of the character (5.23). For this, we decompose thevector space V at a fixed point π according to the Z -action as V π = V ⊕ V ⊕ V ⊕ V ⊕ V ⊕ V . (9.30)71he explicit form of the partial character (7.9) is now T + π = ( V ⊕ V ⊕ V ⊕ V ⊕ V ⊕ V ) − (cid:16) t t − t − t + 1 (cid:17) (cid:0) V ⊕ V ⊕ V ⊕ V ⊕ V ⊕ V (cid:1) ⊗ (cid:0) V ∨ ⊕ V ∨ ⊕ V ∨ ⊕ V ∨ ⊕ V ∨ ⊕ V ∨ (cid:1) . (9.31)The invariant part is given by substituting t α = ζ α with ζ = 1 to get (cid:0) T + π (cid:1) Z = vdim C (cid:0) V − ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ )+ ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ )+ ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ) − ( V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ⊕ V ⊗ V ∨ ) (cid:1) = −| π | − | π | − | π | − | π | − | π | + | π | (cid:0) | π | + | π | + | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | (cid:1) + | π | | π |− (cid:0) | π | | π | + | π | | π | + | π | | π | (cid:1) , (9.32)where we have introduced a Z -colouring of the partitions π = π (cid:116) π (cid:116) π (cid:116) π (cid:116) π (cid:116) π . Thereforethe equivariant Euler characteristic of the obstruction bundle on the quiver variety at a fixed point π is χ T ( N π ) = ( − K ( π ) , (9.33)where the phase factor is K ( π ) = | π | + | π | + | π | + | π | + | π | + | π | (cid:0) | π | + | π | + | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | (cid:1) + | π | (cid:0) | π | + | π | (cid:1) + | π | | π | (9.34)since the even parity terms do not contribute.We will now evaluate the partition function K DT C / Z = (cid:88) π ( − K ( π ) p | π | p | π | p | π | p | π | p | π | p | π | p | π | (9.35)= 1 − p p + p (cid:0) − p p − p p − p p (cid:1) + p (cid:0) − p p + 2 p p p + p p p + p p p − p p p (cid:1) + p (cid:0) p p − p p p − p p p − p p p − p p p − p p p p + 4 p p p p + p p p p + p p p p − p p p p (cid:1) + p (cid:0) − p p − p p p − p p p − p p p p + 2 p p p p − p p p − p p p p − p p p p + 2 p p p p + 2 p p p p + 3 p p p p + p p p p p + p p p p + p p p p p − p p p p p (cid:1) + p (cid:0) − p p − p p p − p p p − p p p p + p p p p − p p p p + p p p + p p p p − p p p p − p p p p p + 2 p p p p + p p p p − p p p p − p p p p − p p p p + 2 p p p p p − p p p p p − p p p p − p p p p p + p p p p p − p p p p p + 4 p p p p p + 6 p p p p p − p p p p p p (cid:1) + · · · . Again one observes the factorization K DT C / Z = (cid:16) − p p p p p p p + 33 p p p p p p p − p p p p p p p + · · · (cid:17) × (cid:16) − p p + (cid:0) − p p − p p − p p (cid:1) p + (cid:0) − p p + 2 p p p + p p p + p p p − p p p (cid:1) p (9.36)+ (cid:0) p p − p p p − p p p − p p p − p p p − p p p p + 4 p p p p p p p p + p p p p − p p p p (cid:1) p + · · · (cid:17) , where M ( x ) = 1 + 6 x + 33 x + 146 x + · · · (9.37)with x = − q = − p p p p p p p . This factor is again expected to appear in the large radiusregime as the contribution from degree zero curve classes (“regular D0 branes”).On the other hand we can use our computation of the instanton action to write down the partitionfunction in a more “physical” form. According to our discussion in Section 6.2, we set a = 0 and b = c = 1. The instanton partition function has the form Z C / Z = (cid:88) π ( − K ( π ) e − g s (cid:82) X ch ( E π ) e − (cid:82) X ω ∧ ch ( E π ) e − (cid:82) X ω ∧ ω ∧ c ( E π ) (9.38)where the universal sheaf is evaluated on the fixed points π (i.e. k r = | π r | ). Introducing the K¨ahlerparameters U = e − ς and Q n = e − ϕ n for n = 1 , , , 4, we can then write Z C / Z = (cid:88) π ( − K ( π ) q I ( π ) Q B ( π )1 Q B ( π )2 Q B ( π )3 Q B ( π )4 U A ( π ) (9.39)where from our computations above we have I ( π ) = (cid:0) | π | + 3 | π | + 4 | π | + 10 | π | − | π | + 9 | π | (cid:1) , B ( π ) = (cid:0) | π | + 5 | π | − | π | − | π | − | π | + | π | (cid:1) , B ( π ) = (cid:0) | π | − | π | − | π | + | π | (cid:1) , B ( π ) = (cid:0) | π | − | π | − | π | + | π | + 2 | π | (cid:1) , B ( π ) = (cid:0) | π | − | π | − | π | − | π | + 21 | π | + 4 | π | (cid:1) , A ( π ) = (cid:0) | π | − | π | − | π | − | π | + 3 | π | + 8 | π | (cid:1) . (9.40)In these new variables the partition function has an expansion Z C / Z = · · · − Q Q q / Q − Q / q / U / Q / √ Q √ Q + Q / Q / q / Q / + 4 Q / q / √ U Q / Q / Q / + Q Q q / + Q q / Q + 2 Q q / U / Q / Q Q − √ U Q / q / Q / Q + q / U / (cid:112) Q (cid:112) Q Q / Q / − √ Q Q / q / √ Q − Q / q U / Q / √ Q Q / − Q / q / U / Q / Q / Q / − √ Q Q / Q / √ q + Q / √ q Q / √ Q − √ Q Q / √ Q q / √ U − q / √ U √ Q Q / Q √ Q − √ Q q / U / Q / Q / Q / − q + 1 q / U / Q / √ Q Q + 1 + q / √ UQ / Q Q √ Q + 1 √ q U / Q Q / Q √ Q − q / U / Q / Q Q √ Q + √ qU / √ Q √ Q Q / Q / + 2 q / U / Q / √ Q Q / Q / − √ Q √ Q q / √ U Q / − q / U / √ Q √ Q Q / Q / q / U / Q / Q / Q / Q / − q / U / √ Q Q / Q / Q − √ q Q / Q / √ U Q / − √ q √ Q √ U √ Q Q / + 4 U / Q / Q Q / Q / + 2 q / Q √ Q Q / − √ q Q / √ U Q / − q / U / √ Q Q Q / − q / U / Q / Q Q Q / − √ q U / √ Q Q / Q Q / + 6 q / U / Q Q Q Q / − √ q Q / U / √ Q Q / Q / − √ qU / √ Q √ Q Q / Q / − q / √ Q Q Q / Q + q √ Q U / Q / Q / Q − qU / √ Q Q / Q / Q + 3 q / U Q / Q / Q / Q − q / U / Q / Q / Q / Q / − q / √ Q U / √ Q Q / Q / − q / √ Q U / Q / Q / Q / + 2 √ qU / Q / Q Q / Q / − q / Q / √ U Q √ Q Q / + q / U / √ Q Q / Q Q / + 2 q / U / √ Q Q Q / Q − q / Q / √ U Q Q / Q / − q / U / Q / Q / Q / Q / + q U / Q / Q Q / Q / + q / Q / Q Q / Q − q / Q / U / Q / Q Q / − q / U / Q Q / Q Q / − q / √ Q U / Q / Q Q / + 2 q / U / Q Q / Q + · · · . (9.41) Finally, the change of variables between the two partition functions reads p = q / ,p = q / Q / Q / Q / Q / U / ,p = q / Q / Q − / U − / ,p = q / Q − / Q − / Q − / Q − / U − / ,p = q / Q − / Q − / Q − / Q − / U − / ,p = q − / Q − / Q / Q / U / ,p = q / Q / Q / Q / Q / U / . (9.42) Finally we can present the partition function for the Coulomb branch invariants. In this case theinstanton measure (5.58) has the form K ( (cid:126)π ; N ) = N (cid:88) l =1 5 (cid:88) r =0 | π l,r | N r + b ( l ) + N (cid:88) l,l (cid:48) =1 5 (cid:88) r =0 | π l,r | (cid:16) − | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | + | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) − | − | π l (cid:48) ,r + b ( l ) − b ( l (cid:48) ) | (cid:17) . (9.43)74e can thus write the partition function for noncommutative Donaldson–Thomas invariants oftype N as K DT C / Z ( N ) = (cid:88) (cid:126)π ( − K ( (cid:126)π ; N ) p (cid:80) Nl =1 | π l, − b ( l ) | p (cid:80) Nl =1 | π l, − b ( l ) | p (cid:80) Nl =1 | π l, − b ( l ) | × p (cid:80) Nl =1 | π l, − b ( l ) | p (cid:80) Nl =1 | π l, − b ( l ) | p (cid:80) Nl =1 | π l, − b ( l ) | . (9.44)In physical variables given by the transformation (9.42), this partition function becomes Z C / Z ( N ) = (cid:88) (cid:126)π ( − K ( (cid:126)π ; N ) q I ( (cid:126)π ; N ) Q B ( (cid:126)π ; N )1 Q B ( (cid:126)π ; N )2 Q B ( (cid:126)π ; N )3 Q B ( (cid:126)π ; N )4 U A ( (cid:126)π ; N ) (9.45)where I ( (cid:126)π ; N ) = 112 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | + 3 | π l, − b ( l ) | + 4 | π l, − b ( l ) | + 10 | π l, − b ( l ) | − | π l, − b ( l ) | + 9 | π l, − b ( l ) | (cid:17) , B ( (cid:126)π ; N ) = 16 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | + 5 | π l, − b ( l ) | − | π l, − b ( l ) |− | π l, − b ( l ) | − | π l, − b ( l ) | + | π l, − b ( l ) | (cid:17) , B ( (cid:126)π ; N ) = 12 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | − | π l, − b ( l ) | − | π l, − b ( l ) | + | π l, − b ( l ) | (cid:17) , B ( (cid:126)π ; N ) = 13 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | − | π l, − b ( l ) | − | π l, − b ( l ) | + | π l, − b ( l ) | + 2 | π l, − b ( l ) | (cid:17) , B ( (cid:126)π ; N ) = 16 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | − | π l, − b ( l ) | − | π l, − b ( l ) |− | π l, − b ( l ) | + 21 | π l, − b ( l ) | + 4 | π l, − b ( l ) | (cid:17) , A ( (cid:126)π ; N ) = 16 N (cid:88) l =1 (cid:16) | π l, − b ( l ) | − | π l, − b ( l ) | − | π l, − b ( l ) |− | π l, − b ( l ) | + 3 | π l, − b ( l ) | + 8 | π l, − b ( l ) | (cid:17) . (9.46) 10 Discussion In this paper we have taken a gauge theory approach to the study of noncommutative Donaldson–Thomas invariants defined on noncommutative crepant resolutions of orbifold singularities. Thisgauge theory is defined on geometries of the form C / Γ, by which we really mean a gauge theoryon C whose observables are Γ-equivariant. In this gauge theory one can study a moduli space ofΓ-equivariant instantons, and define an enumerative problem associated with this moduli space.We demonstrated that this moduli space may also be identified with the moduli space of repre-sentations of a certain quiver. The structure of the quiver is dictated by the singularity via theMcKay correspondence. A certain topological matrix quantum mechanics based on the quiver datacan be used to study the local structure of the moduli space and hence to compute its virtualnumbers. 75ur analysis however leaves many open issues. In particular it would be desirable to develop directlya connection with physical states in string theory in order to obtain actual partition functions forthe invariants, where the counting parameters are the D brane charges as in [27]. Furthermore ourapproach seems to be limited to resolutions of abelian orbifolds and do not include more generalsingularities, such as the conifold. While it does not seem impossible that our approach can begeneralized to other singularities, major technical problems arise, such as how to impose boundaryconditions at infinity.Another technical point concerns the condition we have to impose in deriving a compact parametriza-tion of the moduli space, thus excluding certain classes of sheaves. While it seems that this conditionis not restrictive for the purpose of enumeration of the BPS states considered in this paper, as it issatisfied for ideal sheaves, it certainly is for more generic physical states. It would be desirable tohave more control over the full moduli space; we expect this issue to became critical when studyinghigher rank invariants beyond the Coulomb phase of the non-abelian gauge theory.It would also be interesting to study the wall-crossing behaviour of generalized Donaldson–Thomasinvariants from the D brane perspective and across different phases. It is natural to expect that thegauge theory analysis could at least capture the qualitative behaviour. In the favourable cases wherethe set of tautological bundles is also a set of generators for the derived category, the enumerativeproblem is already well-posed and what remains to be done is to evaluate the virtual numbers fordifferent values of the stability parameter, and use the tilting set to obtain the proper D branecharges in each phase. It would be interesting to clarify how variations of the slope or θ -stabilityparameters of Section 5.6, and hence the crossing of chambers, could be understood purely fromthe gauge theory perspective as modifications of the D-term conditions.It would also be interesting to investigate more deeply the connection to the crystal melting pic-ture, for example by exploring the crystal picture in the framework of [26] specifically for orbifoldsingularities. The melting rules could be understood as a counting of coloured plane partitionswhen an atom is removed. One could then also explore the high-temperature limit of such a crystaland the algebraic curve to which it is related; one may further investigate the boundary conditionsat infinity in this way along the lines of [27].Another point which we have left basically untouched is the study of higher rank invariants ofnoncommutative crepant resolutions. This is a formidable problem plagued by technical and con-ceptual difficulties. We were able to restrict ourselves to the Coulomb branch of the non-abeliangauge theory where torus equivariant localization is still a viable approach. One however couldhope that, similarly to what happens in Seiberg–Witten theory, by combining the Coulomb branchresult with an appropriate modular behaviour of the amplitudes one could derive non-abelian in-variants in full generality. Equivalently this problem could be solved via a wall-crossing analysis,where different BPS states corresponding to bound states centred around well-separated D6 branescould coalesce together and form a higher rank stable state.Finally one cannot help noticing how many concepts and techniques that have entered the presentwork seem also to appear in the study of D branes probing the singularity from a low-energy per-spective, in the approach pioneered in [94]. Recent papers have focused on the role that noncommu-tative crepant resolutions have in the properties of low-energy effective gauge theories [95, 96, 97].It would be interesting to understand this correspondence further. Acknowledgments We thank D.-E. Diaconescu and B. Szendr˝oi for helpful discussions, and A. Craw for comments onthe manuscript. Preliminary results concerning the C / Z orbifold were presented by M.C. at the“Fourth Regional Meeting in String Theory” in Patras, Greece in June 2007. The work of M.C. was76upported in part by the Funda¸c˜ao para a Ciˆencia e Tecnologia (FCT/Portugal). M.C. is grateful tothe Institute des Hautes ´Etudes Scientifiques (IHES) for the warm hospitality and support duringthe final stages of this work. The work of R.J.S. was supported in part by grant ST/G000514/1“String Theory Scotland” from the UK Science and Technology Facilities Council. A Gauge theory on quotient stacks The invariant way to describe orbifolds independently of a particular presentation is through thelanguage of Deligne–Mumford stacks. For global orbifolds, obtained as the quotient of a smoothmanifold M by the action of a group Γ, the relevant stacks are called quotient stacks. In thisappendix we collect some properties of quotient stacks, focusing mostly on physical perspectivesand streamlining most of the mathematical technicalities. In particular, we use these notionsto present some heuristic justification for our definition of the gauge theory in Section 3.3 andSection 4. The point of view adapted here closely follows the approach of [98].Let M be a smooth manifold and Γ a finite group acting properly on M ; in the main text weconsider the case M = C and Γ ⊂ SL (3 , C ) abelian. The quotient stack is denoted X = [ M/ Γ], asopposed to the quotient space M/ Γ. One can think of [ M/ Γ] as something similar to M/ Γ awayfrom the orbifold singularities, but with extra structure at the singularities. The intuition is thatany point x ∈ X comes with a finite group Γ x , the stabilizer subgroup of Γ whose elements areregarded as “automorphisms of x ”.In contrast to an ordinary space, the points of a stack do not form a set but objects in a category;the morphisms in fact are all invertible, hence the category is a groupoid, in this case the actiongroupoid Γ × M ⇒ M whose objects are the points of M and whose morphisms are given by theactions of elements of Γ on M . This implies that a single point can have non-trivial automorphisms(as it happens with the moduli space of sigma-model maps from a worldsheet into a Calabi–Yauthreefold), or two distinct points can be isomorphic. The category of points of [ M/ Γ] consists ofthe orbits of points of M under the action of Γ, and there is a one-to-one correspondence betweenisomorphism classes of points of [ M/ Γ] and points of M/ Γ. In other words, the coarse modulispace of the stack X is the underlying singular variety M/ Γ. Thus [ M/ Γ] is similar to M/ Γ awayfrom the singularities, but at the fixed points of the Γ-action on M the stack [ M/ Γ] has non-trivialautomorphisms.An important property of stacks is that a sheaf defined on [ M/ Γ] is the same thing as a Γ-equivariantsheaf on M . This feature extends to all the objects that derive from sheaves, for example differentialforms, spinors, and functions can all be regarded as sections of sheaves and so on; in particular adifferential form on [ M/ Γ] is a Γ-invariant form on M . Moreover, one can also prove that [ M/ Γ]is smooth; an appropriate definition of smoothness is provided by the smooth “orbifold atlas”( M, f ) for the stack, where f : M → [ M/ Γ] is the canonical surjection (which is a principalΓ-bundle).When one constructs the usual string theory sigma-model on a target space M , the spectrum ofmassless closed string modes is given by the cohomology of M ; this is because massless modes areassociated with constant (zero momentum) maps from the worldsheet to the target space. Howeverwhen one considers a sigma-model of maps into a stack [ M/ Γ] the situation is a bit different becauseof the non-trivial automorphisms at the fixed points of the Γ-action. For generic points of [ M/ Γ]the only automorphism is the identity and one basically recovers the quotient space M/ Γ. But atthe fixed points there are non-trivial automorphisms, and so the appropriate cohomology is notjust the cohomology of the target [ M/ Γ] but an appropriate generalization that keeps track ofthe automorphisms of cohomology classes. This is called the orbifold cohomology; it contains acombination of geometric and representation theory data.77hese notions motivate the definition of the inertia stack I [ M/ Γ] associated to the quotient stack[ M/ Γ], defined as the substack of points ( x, g ) ∈ X × Γ with g ∈ Γ x . It has the structure of adisjoint union of orbifolds I [ M/ Γ] ∼ = (cid:71) [ g ] (cid:2) M g (cid:14) C ( g ) (cid:3) , (A.1)where [ g ] denotes the conjugacy class of an element g ∈ Γ, C ( g ) is the centralizer of g in G , and M g is the submanifold of points of M which are invariant under the action of g . Twist fields livein the cohomology of this auxilliary stack, i.e. we define the orbifold cohomology H • orb ([ M/ Γ]) of[ M/ Γ] as the ordinary cohomology of I [ M/ Γ]. As a particular case, the quotient stack [pt / Γ], with pt some fixed point with trivial Γ-action, isthe classifying stack of Γ, denoted B Γ; it is a moduli space for principal Γ-bundles. If Γ acts freelyon a manifold M then H • orb ([ M/ Γ]) ∼ = H • ( M ) Γ . In the more general case H • ( M ) Γ is a subspace of H • orb ([ M/ Γ]) and its orthogonal complement is the space of twisted sectors.Assume now that X is a crepant resolution of the underlying singular space M/ Γ of a stack X .The crepant resolution conjecture [90] tells us that the Gromov–Witten theories of X and of X are equivalent. This means, in particular, that the cohomology groups H • orb ( X ) and H • ( X ) areisomorphic, and there is a prescription which takes the Gromov–Witten prepotential of X , F X , tothe Gromov–Witten prepotential of the orbifold X , F X , via a non-trivial transformation.In the example considered in Section 8, wherein X = [ C / Z ] and X = O P ( − Z = { , ζ, ζ } where ζ = e π i / ; it acts on C as in (3.21). The action of the torus T liftsboth to X and X ; we work equivariantly with respect to this toric action.The inertia stack I X has three components corresponding to the three elements 1 , ζ, ζ . Eachcomponent is contractible. A basis for the equivariant orbifold cohomology H • orb , T ( X ) is given bythe classes { O , O ζ , O ζ } corresponding to the elements 1 , ζ, ζ , with O ∈ H ( X ), O ζ ∈ H ( X )and O ζ ∈ H ( X ). Then the genus zero free energy for the Gromov–Witten series on X is givenby F X = (cid:88) n ,n ,n ≥ (cid:104) O n O n ζ O n ζ (cid:105) x n n ! x n ζ n ! x n ζ n != 13 x + 13 x x ζ x ζ + (cid:88) m,n> m +2 n ≡ (cid:104) O mζ O nζ (cid:105) x mζ m ! x nζ n ! . (A.2)The invariants here are defined via pullback through the evaluation map ev that computes the valueof the sigma-model string field embedding the curve in the Calabi–Yau target space. Since we workat genus zero, the unstable terms, with less than three operator insertions, drop out. The countingof “divisorial classes” corresponds to the n = 0 sector of this series and should be compared withthe free energy (8.31) for g = 0.The localization calculation is completely determined by what happens at [0 / Z ] ∈ [ C / Z ] where0 ∈ C is fixed by the Z -action. The component of the moduli space of sigma-model fields withtarget [ C / Z ] that parametrizes constant maps with image [0 / Z ] ∈ [ C / Z ] is then identifiedwith the moduli space of twisted stable maps to the orbifold B Z = [0 / Z ], or equivalently themoduli spaces of admissible Z -covers of genus zero curves. This is analogous to what happens To be precise there is also a shift in the degree of the cohomology that is called “age”; we will ignore the age inorder to simplify our presentation a bit. 78n the Gromov–Witten theory of a local Calabi–Yau threefold, where the relevant maps are thosewhich cover the base of the threefold, i.e. the sigma-model fields factor through the zero section.See [100, Section 4.3] and [101] for a discussion of this point, and [102] for the extension to highergenus invariants.Now let us see how this discussion can be used to model our gauge theory formulation. The factthat Gromov–Witten theory is defined through the quotient stack [ C / Z ] suggests that one shouldconsider objects on C that are invariant under the Z -action. For example, the Z -invariant partof the character on C is computed in Section 3.2, see (3.42). By expanding (3.42) as a powerseries in t around t = 0 and taking the coefficient proportional to t one obtains the third Cherncharacter. If one applies the same procedure to the first and second Chern characters one finds anon-vanishing result. The interpretation of this within the context of the localization calculationis as follows.To have e.g. a non-vanishing second Chern class, one needs a non-trivial toric four-cycle D in thebackground. According to our discussion above, the gauge theory should really be formulated onthe quotient stack [ C / Z ] and the localization calculation reduces the target to the classifyingstack [0 / Z ] = B Z . The cohomology of the classifying space B Z can be computed from H odd ( B Z n , Z ) = 0 and H even ( B Z n , Z ) = Z n . (A.3)This means that there are effectively four-cycles in the orbifold geometry. As they are purelytorsion, they cannot be modelled at the level of differential forms. In what follows we describea gauge theory realization of the non-trivial orbifold cohomology classes { O , O ζ , O ζ } that aroseabove in the Gromov–Witten theory.The computation in Gromov–Witten theory involves insertions of a local operator, which de-fines the invariants through equivariant localization on the moduli space of stable maps intothe stack X . Such insertions correspond to what on a generic threefold X one would reallycall (primary) descendent invariants. If one takes them literally as descendent fields, then theGromov–Witten/Donaldson–Thomas correspondence of [34] says that on the gauge theory sideone should consider Donaldson–Thomas (primary) descendent invariants. On the topologicalstring theory side one takes a basis γ , . . . , γ m of H • ( X, Q ) and defines the primary descendentfields by integrating products of ev ∗ ( γ l i ). On the gauge theory side one considers the operatorch ( γ ) : H • ( M , Q ) → H • +2 − l ( M , Q ) where γ ∈ H l ( X, Z ); roughly speaking it is given by integrat-ing the Chern character of the universal sheaf E over γ on X and the virtual fundamental class ofthe rank one instanton moduli space M . Then one defines the descendent invariants of the gaugetheory as the integrals ( − r (cid:90) M ch ( γ l ) ∧ · · · ∧ ch ( γ l r ) . (A.4)The reduced partition function equals the Gromov–Witten partition function with the usual changeof variables, up to normalization [34].The operators ch ( γ ) are equivalent to insertions of an F A ∧ F A term integrated over some appropri-ate homology class (dual to γ ). To see this, recall that the toric action on X lifts to the instantonmoduli space M . Hence one should compute the Chern character of the universal sheaf at a fixedpoint of the toric action on M × X through localization. But by definition the universal sheaf E restricted to a point of the moduli space gives precisely the ideal sheaf I on the target space X , i.e. E | I× X ∼ = I where I is now a fixed point of the toric action on M . Thus heuristically the descendentinvariants are recovered by D2 charge insertions of F A ∧ F A which is equivalent to the expansionof the term exp (cid:0) − (cid:82) D F A ∧ F A (cid:1) in the gauge theory path integral. This way of interpretingthe second Chern character partly justifies the approach to the gauge theory undertaken in thispaper. 79 Sheaf cohomology B.1 Line bundle cohomology of divisors In this appendix we collect and prove some results concerning the cohomology of sheaves on theprojective space P . We begin by quoting the elementary results that will be exploited in thefollowing. We havedim C H (cid:0) P , O P ( − r ) (cid:1) = (cid:18) − r − r (cid:19) dim C H (cid:0) P , O P ( − r ) (cid:1) = 0 = dim C H (cid:0) P , O P ( − r ) (cid:1) , dim C H (cid:0) P , O P ( − r ) (cid:1) = (cid:18) r − r − (cid:19) , (B.1)and therefore H (cid:0) P , O P ( − r ) (cid:1) = 0 for r > ,H (cid:0) P , O P ( − r ) (cid:1) = 0 = H (cid:0) P , O P ( − r ) (cid:1) ,H (cid:0) P , O P ( − r ) (cid:1) = 0 for r < . (B.2)These results and others that we use throughout this appendix can be found in [103]. B.2 Cohomology of sheaves of differential forms The following lemma computes the strongest bounds on the vanishing cohomology groups forsheaves of differential forms that we were able to find. Its proof will repeatedly make use of theEuler sequences for differential forms on P obtained via truncation of the Koszul complex (4.5).They are given by 0 (cid:47) (cid:47) Ω P (cid:47) (cid:47) O P ( − ⊕ (cid:47) (cid:47) O P (cid:47) (cid:47) , (B.3)0 (cid:47) (cid:47) Ω P (cid:47) (cid:47) O P ( − ⊕ (cid:47) (cid:47) Ω P (cid:47) (cid:47) , (B.4)0 (cid:47) (cid:47) Ω P (cid:47) (cid:47) O P ( − ⊕ (cid:47) (cid:47) Ω P (cid:47) (cid:47) . (B.5) Lemma B.6 One has the vanishing results H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > − ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > − ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r (cid:54) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 = H (cid:0) P , Ω P ( − r ) (cid:1) ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r (cid:54) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < . (B.7)80 roof : From the Euler sequence for one-forms (B.3) we obtain the short exact sequence0 (cid:47) (cid:47) Ω P ( − r ) (cid:47) (cid:47) O P ( − r − ⊕ (cid:47) (cid:47) O P ( − r ) (cid:47) (cid:47) . (B.8)Applying the snake lemma to write the associated long exact sequence in cohomology, togetherwith (B.2) we easily conclude H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > − ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < . (B.9)Since Ω P = O P ( − (cid:47) (cid:47) O P ( − − r ) (cid:47) (cid:47) O P ( − − r ) ⊕ (cid:47) (cid:47) Ω P ( − r ) (cid:47) (cid:47) . (B.10)Using the associated long exact cohomology sequence and the vanishing results (B.2) we thusfind H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > − ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < . (B.11)From the Euler sequence for two-forms (B.4) we obtain the final short exact sequence0 (cid:47) (cid:47) Ω P ( − r ) (cid:47) (cid:47) O P ( − r − ⊕ (cid:47) (cid:47) Ω P ( − r ) (cid:47) (cid:47) . (B.12)From the corresponding long exact sequence in cohomology and (B.2), this in particular im-plies H (cid:0) P , Ω P ( − r ) (cid:1) = H (cid:0) P , Ω P ( − r ) (cid:1) , (B.13)but since the left-hand side vanishes for r > r < r = 0. In this case we thus conclude H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r > − ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r (cid:54) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r (cid:54) = 0 ,H (cid:0) P , Ω P ( − r ) (cid:1) = 0 for r < . (B.14)Putting everything together we arrive finally at (B.7). (cid:4) B.3 Cohomology of ideal sheaves We will now study the cohomology of the ideal sheaves of a point. Consider the short exact sequenceof sheaves 0 (cid:47) (cid:47) I (cid:47) (cid:47) O P (cid:47) (cid:47) O z (cid:47) (cid:47) , (B.15)where O z is the skyscraper sheaf of a point z ∈ C which is not torsion free.81 emma B.16 The cohomology of the ideal sheaf I is given by H (cid:0) P , I ( − r ) (cid:1) = 0 for r > ,H (cid:0) P , I ( − r ) (cid:1) = C for r > ,H (cid:0) P , I ( − r ) (cid:1) = 0 ,H (cid:0) P , I ( − r ) (cid:1) = 0 for r < . (B.17) Proof : Take the tensor product of the exact sequence (B.15) with O P ( − r ) to get0 (cid:47) (cid:47) I ( − r ) (cid:47) (cid:47) O P ( − r ) (cid:47) (cid:47) O P ( − r ) ⊗ O z (cid:47) (cid:47) , (B.18)since O P ( − r ) is locally free and hence Tor O P ( O P ( − r ) , O z ) = 0. The skyscraper sheaf is acyclic, H (cid:0) P , O z (cid:1) = C and H n (cid:0) P , O z (cid:1) = 0 for n (cid:54) = 0 , (B.19)and this property is unaltered by twisting. Therefore writing the associated long exact sequence incohomology and applying (B.2) yields (B.17). (cid:4) The next lemma is needed in the proof of Lemma B.21 below, but we defer its proof to Sec-tion B.4. Lemma B.20 The sheaves Ω m P ( k ) ⊗ O z are acyclic. Lemma B.21 The groups H n ( P , Ω m P ( m ) ⊗ I ( − for m = 1 , and H n ( P , Ω P (1) ⊗ I ( − arenon-zero only for n = 1 , while H n ( P , Ω P (2) ⊗ I ( − are non-zero for both n = 1 , . Proof : From (B.15) we get the exact sequence of sheaves0 (cid:47) (cid:47) Ω m P ( r ) ⊗ I (cid:47) (cid:47) Ω m P ( r ) (cid:47) (cid:47) Ω m P ( r ) ⊗ O z (cid:47) (cid:47) , (B.22)where again Tor O P (Ω m P ( r ) , O z ) = 0 since Ω m P ( r ) is locally free and hence flat. We are interestedin computing the sheaf cohomology groups H • (cid:0) P , I ( − ⊗ Ω P (1) (cid:1) = H • (cid:0) P , I ⊗ Ω P (cid:1) ,H • (cid:0) P , I ( − ⊗ Ω P (1) (cid:1) = H • (cid:0) P , I ⊗ Ω P ( − (cid:1) , (B.23)which come from taking r = 0 and r = − r = 0 , − 1, we thus consider the longexact cohomology sequence which, by Lemma B.20 and Lemma B.6, yields H (cid:0) P , Ω P (1) ⊗ I ( r − (cid:1) = 0 ,H (cid:0) P , Ω P (1) ⊗ I ( r − (cid:1) (cid:54) = 0 ,H (cid:0) P , Ω P (1) ⊗ I ( r − (cid:1) = 0 ,H (cid:0) P , Ω P (1) ⊗ I ( r − (cid:1) = 0 . (B.24)Next we compute H • (cid:0) P , I ( − ⊗ Ω P (2) (cid:1) = H • (cid:0) P , I ⊗ Ω P (1) (cid:1) ,H • (cid:0) P , I ( − ⊗ Ω P (2) (cid:1) = H • (cid:0) P , I ⊗ Ω P (cid:1) , (B.25)82hich correspond respectively to r = 1 and r = 0. For r = 0 , 1, we therefore consider the longexact cohomology sequence, which as above gives H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) (cid:54) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) (cid:54) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) = 0 (B.26)for r = 0, while H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) (cid:54) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) = 0 ,H (cid:0) P , Ω P (2) ⊗ I ( − (cid:1) = 0 (B.27)for r = 1. (cid:4) B.4 Hypercohomology of torsion sheaves We now prove Lemma B.20. We want to compute the cohomology groups H • ( P , Ω m P ( k ) ⊗ O z ) ofthe skyscraper sheaf. They are equal to Ext •O P (Ω m P ( k ) ∨ , O z ). The sheaf Ω m P ( k ) ∨ is the holomorphictangent bundle of P for m = 1 and k = 0. In particular it is locally free and therefore has a triviallocally free resolution 0 (cid:47) (cid:47) Ω m P ( k ) ∨ = (cid:47) (cid:47) Ω m P ( k ) ∨ (cid:47) (cid:47) . (B.28)The strategy is to compute first local Ext sheaves and then global Ext groups using the local-to-global spectral sequence.Given this trivial locally free resolution, local Ext sheaves are defined as the cohomology sheavesof the complex 0 (cid:47) (cid:47) Hom (cid:0) Ω m P ( k ) ∨ , O z (cid:1) (cid:47) (cid:47) n O P (cid:0) Ω m P ( k ) ∨ , O z (cid:1) = (cid:26) Hom (cid:0) Ω m P ( k ) ∨ , O z (cid:1) = Ω m P ( k ) ⊗ O z , n = 0 , , n = 1 , , . (B.30)Global Ext groups are now computed via the local-to-global spectral sequence E p,q = H p (cid:0) P , Ext q O P ( E , F ) (cid:1) = ⇒ Ext p + q O P ( E , F ) . (B.31)However, the spectral sequence in the present case is trivial since the local Ext sheaves have supportonly on points; in particular E p,q = 0 for p > 0. ThereforeExt n O P (cid:0) Ω m P ( k ) ∨ , O z (cid:1) = (cid:26) H (cid:0) P , Ω m P ( k ) ⊗ O z (cid:1) , n = 0 , , n = 1 , , . (B.32)The result for n = 0 is tautological, but the conclusion we are interested in is that all highercohomology groups vanish, H n> (cid:0) P , Ω m P ( k ) ⊗ O z (cid:1) = 0 , (B.33)83s is required by Lemma B.20. Note that these Ext groups compute B-model open string spectrabetween the D0 branes O z and other branes.We can check this result also directly with the Serre dual. Consider the cohomology groupExt n O P ( O z , Ω m P ( k ) ∨ ). On P we have the locally free resolution of the skyscraper sheaf O z givenby0 (cid:47) (cid:47) O P ( − (cid:47) (cid:47) O P ( − ⊕ (cid:47) (cid:47) O P ( − ⊕ (cid:47) (cid:47) O P (cid:47) (cid:47) O z (cid:47) (cid:47) . (B.34)Local Ext sheaves are now given by the cohomology sheaves of the complexHom (cid:0) O P , Ω m P ( k ) ∨ (cid:1) (cid:47) (cid:47) Hom (cid:0) O P ( − ⊕ , Ω m P ( k ) ∨ (cid:1) (cid:47) (cid:47) Hom (cid:0) O P ( − ⊕ , Ω m P ( k ) ∨ (cid:1) (cid:47) (cid:47) Hom (cid:0) O P ( − , Ω m P ( k ) ∨ (cid:1) . (B.35)Arguing as in [104, Section 2.4], the cohomology sheaf of this complex isExt n O P (cid:0) O z , Ω m P ( k ) ∨ (cid:1) = (cid:26) , n = 0 , , , F , n = 3 (B.36)where F is a certain sheaf that arises as the cohomology sheaf at the right-most position in thecomplex (it is essentially O z ⊗ Ω m P ( k ) ∨ ); in particular it has zero-dimensional support. It followsthat the local-to-global spectral sequence is trivial andExt n O P (cid:0) O z , Ω m P ( k ) ∨ (cid:1) = (cid:26) , n = 0 , , ,H ( P , F ) , n = 3 . (B.37)This result is consistent with Serre duality between coherent sheaves which in the present caseimplies [103] Ext n O P ( E , F ) = Ext − n O P (cid:0) F , E ⊗ O P ( − (cid:1) (B.38)where O P ( − ∼ = Ω P is the canonical line bundle on P . C Beilinson monad construction C.1 Beilinson spectral sequence In this appendix we derive the monad parametrization (4.8)–(4.9) of sheaves in the moduli space(4.1). The Beilinson theorem implies that for any coherent sheaf E on P there is a spectralsequence E p,qs with E -term (4.6) which converges to (4.7) for each fixed integer r ≥ 0, where E ( − r ) = E ⊗ O P O P ( − r ). Explicitly, the first term is given by E p,q = H q (cid:0) P , E ( − r ) ⊗ Ω − p P ( − p ) (cid:1) ⊗ O P ( p ) (C.1)84or p ≤ 0. The E -complexes of the spectral sequence can be summarized in the diagram E − , 31 d (cid:47) (cid:47) E − , 31 d (cid:47) (cid:47) E − , 31 d (cid:47) (cid:47) E , E − , 21 d (cid:47) (cid:47) E − , 21 d (cid:47) (cid:47) E − , 21 d (cid:47) (cid:47) E , E − , 11 d (cid:47) (cid:47) E − , 11 d (cid:47) (cid:47) E − , 11 d (cid:47) (cid:47) E , E − , 01 d (cid:47) (cid:47) E − , 01 d (cid:47) (cid:47) E − , 01 d (cid:47) (cid:47) E , (cid:47) (cid:47) (cid:79) (cid:79) p q (C.2)where all other entries are zero for dimensional reasons and the only nonvanishing differentiald : E p,q −→ E p +1 ,q (C.3)is determined by the morphisms in the Koszul complex (4.5).For explicit computations we will again exploit the Euler sequences (B.3)–(B.5) together with0 (cid:47) (cid:47) O P ( − × z (cid:47) (cid:47) O P z =0 (cid:47) (cid:47) O ℘ ∞ (cid:47) (cid:47) . (C.4)This sequence defines the plane at infinity ℘ ∞ = [0 : z : z : z ] ∼ = P . We take the tensor productof the sequence (C.4) with E ( − r ) to get0 (cid:47) (cid:47) E ( − r − (cid:47) (cid:47) E ( − r ) (cid:47) (cid:47) E ( − r ) (cid:12)(cid:12) ℘ ∞ (cid:47) (cid:47) , (C.5)where we have used the fact that O P is a locally free sheaf to set Tor O P ( E ( − r ) | ℘ ∞ , O ℘ ∞ ) = 0.The following result is proven in [37]. Lemma C.6 For [ E ] ∈ M N,k (cid:0) P (cid:1) one has the vanishing results H (cid:0) P , E ( − r ) (cid:12)(cid:12) ℘ ∞ (cid:1) = 0 for r ≥ ,H (cid:0) P , (Ω P (1) ⊗ E ( − r )) (cid:12)(cid:12) ℘ ∞ (cid:1) = 0 for r ≥ ,H (cid:0) P , E ( − r ) (cid:1) = 0 for r ≥ ,H (cid:0) P , Ω P (2) ⊗ E ( − r ) (cid:1) = 0 for r ≥ ,H (cid:0) P , E ( − r ) (cid:12)(cid:12) ℘ ∞ (cid:1) = 0 for r ≤ ,H (cid:0) P , (Ω P (1) ⊗ E ( − r )) (cid:12)(cid:12) ℘ ∞ (cid:1) = 0 for r ≤ ,H (cid:0) P , E ( − r ) (cid:1) = 0 for r ≤ ,H (cid:0) P , Ω P (1) ⊗ E ( − r ) (cid:1) = 0 for r ≤ ,H (cid:0) P , Ω P (2) ⊗ E ( − r ) (cid:1) = 0 for r ≤ . (C.7)85emma C.6 imples that for r = 1 , H and H vanish, and hence thefirst and last rows of the Beilinson spectral sequence (C.2) are 0. We will now impose the additionalcondition H (cid:0) P , E ( − (cid:1) = 0 . (C.8)Then the following result implies the vanishing of the second row of the spectral sequence (C.2). Lemma C.9 For [ E ] ∈ M N,k (cid:0) P (cid:1) satisfying H (cid:0) P , E ( − (cid:1) = 0 one has the vanishing results H (cid:0) P , E ( − (cid:1) = H (cid:0) P , E ( − ⊗ Ω P (1) (cid:1) = H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) = 0 . (C.10) Proof : The long exact sequence in cohomology induced by the short exact sequence of sheaves(C.5) for r = 1 contains H (cid:0) P , E ( − (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:12)(cid:12) ℘ ∞ (cid:1) , (C.11)and therefore H (cid:0) P , E ( − (cid:1) = 0 by (C.8) and Lemma C.6. Using (B.4) we consider now theshort exact sequence0 (cid:47) (cid:47) E ( − r ) ⊗ Ω P (cid:47) (cid:47) E ( − r − ⊕ (cid:47) (cid:47) E ( − r ) ⊗ Ω P (cid:47) (cid:47) . (C.12)For r = 0 the corresponding long exact cohomology sequence contains H (cid:0) P , E ( − (cid:1) ⊕ (cid:47) (cid:47) H (cid:0) P , E ( − ⊗ Ω P (1) (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) , (C.13)and therefore H (cid:0) P , E ( − ⊗ Ω P (1) (cid:1) = 0 by (C.8) and Lemma C.6. Taking the tensor productof (B.10) for r = − E ( − s ) gives0 (cid:47) (cid:47) E ( − s − (cid:47) (cid:47) E ( − s − ⊕ (cid:47) (cid:47) E ( − s ) ⊗ Ω P (2) (cid:47) (cid:47) . (C.14)The associated long exact sequence in cohomology for s = 1 contains H (cid:0) P , E ( − (cid:1) ⊕ (cid:47) (cid:47) H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) , (C.15)and finally we conclude H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) = 0 by (C.8) and Lemma C.6. (cid:4) By Lemma C.6, Lemma C.9 and Beilinson’s theorem it follows that the cohomology of the differ-ential complex (C.2) reduces to0 0 0 00 0 0 0 E − , ∞ = 0 E − , ∞ = 0 E − , ∞ = E ( − E , ∞ = 00 0 0 0 (cid:47) (cid:47) (cid:79) (cid:79) p q (C.16)86 .2 Generalized ADHM complex The following result is proven in [37]. Lemma C.17 For [ E ] ∈ M N,k (cid:0) P (cid:1) one has the Euler characters χ (cid:0) E ( − (cid:1) = − k ,χ (cid:0) E ( − ⊗ Ω P (2) (cid:1) = − k ,χ (cid:0) E ( − ⊗ Ω P (1) (cid:1) = − k − N ,χ (cid:0) E ( − (cid:1) = − k . (C.18)The reduction of the Beilinson spectral sequence from Section C.1 is equivalent to the complex0 (cid:47) (cid:47) V ⊗ O P ( − a (cid:47) (cid:47) B ⊗ O P ( − b (cid:47) (cid:47) C ⊗ O P ( − c (cid:47) (cid:47) D ⊗ O P (cid:47) (cid:47) , (C.19)where we have defined the complex vector spaces V = H (cid:0) P , E ( − (cid:1) ∼ = C k ,B = H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) ∼ = C k ,C = H (cid:0) P , E ( − ⊗ Ω P (1) (cid:1) ∼ = C k + N ,D = H (cid:0) P , E ( − (cid:1) ∼ = C k , (C.20)whose (stable) dimensions are computed by Lemma C.6, Lemma C.9 and Lemma C.17. There aresome natural identifications. From the short exact sequence of sheaves (C.4) we have H (cid:0) P , E ( − r ) (cid:12)(cid:12) ℘ ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − r − (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − r ) (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − r ) (cid:12)(cid:12) ℘ ∞ (cid:1) . (C.21)For r = 1, the first term vanishes by Lemma C.6, while dim C H (cid:0) P , E ( − (cid:12)(cid:12) ℘ ∞ (cid:1) is the instantoncharge at infinity which we assume to vanish. It follows that D ∼ = V .Consider now the vector space B in (C.20). The exact sequence (C.14) gives H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) ⊕ (cid:47) (cid:47) H (cid:0) P , E ( − ⊗ Ω P (2) (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) . (C.22)The first term vanishes by Lemma C.6. Consider the cohomology sequence (C.21) for r = 2; the firstterm vanishes by Lemma C.6, while from [105, p. 16] we have the vector space isomorphism H (cid:0) P , E ( − (cid:12)(cid:12) ℘ ∞ (cid:1) = H (cid:0) P , E ( − (cid:12)(cid:12) ℘ ∞ (cid:1) (C.23)whose dimension is again the instanton charge at infinity which we assume to vanish. There-fore H (cid:0) P , E ( − (cid:1) ∼ = H (cid:0) P , E ( − (cid:1) = V . Finally, the group H (cid:0) P , E ( − (cid:1) fits in the exactsequence H (cid:0) P , E ( − (cid:12)(cid:12) ℘ ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) (cid:47) (cid:47) H (cid:0) P , E ( − (cid:1) (C.24)87btained from (C.14). However, the first term again vanishes due to the vacuum configuration atinfinity, while the third term is zero due to (C.8), and therefore H (cid:0) P , E ( − (cid:1) = 0 . (C.25)We have just shown that the long exact sequence (C.22) reduces to the short exact sequence0 (cid:47) (cid:47) V (cid:47) (cid:47) V ⊕ (cid:47) (cid:47) B (cid:47) (cid:47) . (C.26)Since a short exact sequence of vector spaces is always split we conclude that B = V ⊕ V ⊕ V .Finally, we determine the vector space C in (C.20). Since we really want the sheaf E , we twist thecomplex by O P (1) to get0 (cid:47) (cid:47) V ⊗ O P ( − a (cid:47) (cid:47) V ⊕ ⊗ O P ( − b (cid:47) (cid:47) C ⊗ O P c (cid:47) (cid:47) V ⊗ O P (1) (cid:47) (cid:47) , (C.27)where im( a ) = ker( b ) and E = ker( c ) (cid:14) im( b ) . (C.28)The restriction of this complex to a line (cid:96) ∞ ∼ = P at infinity reads0 (cid:47) (cid:47) V ⊗ O P ( − (cid:12)(cid:12) (cid:96) ∞ a ∞ (cid:47) (cid:47) V ⊕ ⊗ O P ( − (cid:12)(cid:12) (cid:96) ∞ b ∞ (cid:47) (cid:47) C ⊗ O P (cid:12)(cid:12) (cid:96) ∞ c ∞ (cid:47) (cid:47) V ⊗ O P (1) (cid:12)(cid:12) (cid:96) ∞ (cid:47) (cid:47) . (C.29)To this complex we can associate the short exact sequences0 (cid:47) (cid:47) ker( a ∞ ) (cid:47) (cid:47) V ⊗ O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:47) (cid:47) im( a ∞ ) (cid:47) (cid:47) , (C.30)0 (cid:47) (cid:47) im( a ∞ ) (cid:47) (cid:47) V ⊕ ⊗ O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:47) (cid:47) im( b ∞ ) (cid:47) (cid:47) , (C.31)0 (cid:47) (cid:47) ker( c ∞ ) (cid:47) (cid:47) C ⊗ O P (cid:12)(cid:12) (cid:96) ∞ c ∞ (cid:47) (cid:47) V ⊗ O P (1) (cid:12)(cid:12) (cid:96) ∞ (cid:47) (cid:47) , (C.32)0 (cid:47) (cid:47) im( b ∞ ) (cid:47) (cid:47) ker( c ∞ ) (cid:47) (cid:47) E (cid:12)(cid:12) (cid:96) ∞ (cid:47) (cid:47) , (C.33)where we have used the isomorphisms ker( b ∞ ) ∼ = im( a ∞ ) and ker( c ∞ ) (cid:14) im( b ∞ ) ∼ = E (cid:12)(cid:12) (cid:96) ∞ .Since the morphism a ∞ is injective, from (C.30) it follows that H (cid:0) P , im( a ∞ ) (cid:1) = 0 ,V ⊗ H (cid:0) P , O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:1) ∼ = V ∼ = H (cid:0) P , im( a ∞ ) (cid:1) . (C.34)The long exact sequence in cohomology associated with (C.31) is0 (cid:47) (cid:47) H (cid:0) P , im( a ∞ ) (cid:1) (cid:47) (cid:47) V ⊕ ⊗ H (cid:0) P , O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , im( b ∞ ) (cid:1) (cid:47) (cid:47) H (cid:0) P , im( a ∞ ) (cid:1) (cid:47) (cid:47) V ⊕ ⊗ H (cid:0) P , O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , im( b ∞ ) (cid:1) (cid:47) (cid:47) , (C.35)88hich using H (cid:0) P , O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:1) ∼ = 0 ∼ = H (cid:0) P , O P ( − (cid:12)(cid:12) (cid:96) ∞ (cid:1) implies H (cid:0) P , im( a ∞ ) (cid:1) = 0 = H (cid:0) P , im( b ∞ ) (cid:1) and H (cid:0) P , im( b ∞ ) (cid:1) ∼ = H (cid:0) P , im( a ∞ ) (cid:1) . (C.36)From (C.33) we have0 (cid:47) (cid:47) H (cid:0) P , im( b ∞ ) (cid:1) (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) H (cid:0) P , E (cid:12)(cid:12) (cid:96) ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , im( b ∞ ) (cid:1) (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) H (cid:0) P , E (cid:12)(cid:12) (cid:96) ∞ (cid:1) (cid:47) (cid:47) , (C.37)where due to our boundary condition we have H ( P , E| (cid:96) ∞ ) = 0.As in (4.32), we define the framing vector space W = H (cid:0) P , E (cid:12)(cid:12) (cid:96) ∞ (cid:1) . (C.38)If we put together (C.34), (C.36) and (C.37), then we find that H (cid:0) P , ker( c ∞ ) (cid:1) = 0 (C.39)and that the sequence0 (cid:47) (cid:47) V (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) W (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) ∼ = V ⊕ W . (C.41)Finally, from (C.32) we have0 (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) C ⊗ H (cid:0) P , O P (cid:12)(cid:12) (cid:96) ∞ (cid:1) c ∞ (cid:47) (cid:47) V ⊗ H (cid:0) P , O P (1) (cid:12)(cid:12) (cid:96) ∞ (cid:1) (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) C ⊗ H (cid:0) P , O P (cid:12)(cid:12) (cid:96) ∞ (cid:1) c ∞ (cid:47) (cid:47) V ⊗ H (cid:0) P , O P (1) (cid:12)(cid:12) ℘ ∞ (cid:1) (cid:47) (cid:47) (cid:47) (cid:47) H (cid:0) P , ker( c ∞ ) (cid:1) (cid:47) (cid:47) C c ∞ (cid:47) (cid:47) V ⊕ V (cid:47) (cid:47) , (C.43)and therefore C ∼ = H (cid:0) P , ker( c ∞ ) (cid:1) ⊕ V ⊕ V ∼ = V ⊕ V ⊕ V ⊕ W . (C.44)Putting everything together, we have reduced the complex (C.19) to (4.8)–(4.9). References [1] A. Okounkov, N. Reshetikhin and C. Vafa, “Quantum Calabi–Yau and classical crystals,”Progr. Math. (2006) 597 [arXiv:hep-th/0309208].[2] A. Iqbal, N. A. Nekrasov, A. Okounkov and C. Vafa, “Quantum foam and topological strings,”J. High Energy Phys. (2008) 011 [arXiv:hep-th/0312022.].893] N. Saulina and C. Vafa, “D-branes as defects in the Calabi–Yau crystal,” [arXiv:hep-th/0404246].[4] R. Dijkgraaf, A. Sinkovics and M. Temurhan, “Universal correlators from geometry,” J. HighEnergy Phys. (2004) 012 [arXiv:hep-th/0406247].[5] T. Okuda, “Derivation of Calabi–Yau crystals from Chern–Simons gauge theory,” J. HighEnergy Phys. (2005) 047 [arXiv:hep-th/0409270].[6] N. Halmagyi, A. Sinkovics and P. Sulkowski, “Knot invariants and Calabi–Yau crystals,” J.High Energy Phys. (2006) 040 [arXiv:hep-th/0506230].[7] P. Sulkowski, “Crystal model for the closed topological vertex geometry,” J. High Energy Phys. (2006) 030 [arXiv:hep-th/0606055].[8] J. J. Heckman and C. Vafa, “Crystal melting and black holes,” J. High Energy Phys. (2007)011 [arXiv:hep-th/0610005].[9] N. Seiberg and E. Witten, “Monopole condensation and confinement in N = 2 supersymmetricYang–Mills theory,” Nucl. Phys. B (1994) 19 [Erratum-ibid. B (1994) 485] [arXiv:hep-th/9407087].[10] F. Denef and G. W. Moore, “Split states, entropy enigmas, holes and halos,” arXiv:hep-th/0702146.[11] M. Kontsevich and Y. Soibelman, “Stability structures, motivic Donaldson–Thomas invariantsand cluster transformations,” arXiv:0811.2435 [math.AG].[12] D. Gaiotto, G. W. Moore and A. Neitzke, “Framed BPS states,” arXiv:1006.0146 [hep-th].[13] D. Gaiotto, G. W. Moore and A. Neitzke, “Wall-crossing, Hitchin systems, and the WKBapproximation,” arXiv:0907.3987 [hep-th].[14] D. Gaiotto, G. W. Moore and A. Neitzke, “Four-dimensional wall-crossing via three-dimensional field theory,” Commun. Math. Phys. (2010) 163 [arXiv:0807.4723 [hep-th]].[15] T. Dimofte, S. Gukov and Y. Soibelman, “Quantum wall-crossing in N = 2 gauge theories,”Lett. Math. Phys. (2011) 1 [arXiv:0912.1346 [hep-th]].[16] T. Dimofte and S. Gukov, “Refined, motivic, and quantum,” Lett. Math. Phys. (2010) 1[arXiv:0904.1420 [hep-th]].[17] P. S. Aspinwall, “D branes on Calabi–Yau manifolds,” in: Recent Trends in String Theory ,ed. J. M. Maldacena (World Scientific, 2005), p. 1 [arXiv:hep-th/0403166].[18] D. Joyce and Y. Song “A theory of generalized Donaldson–Thomas invariants,”arXiv:0810.5645 [math.AG].[19] K. Nagao, H. Nakajima, “Counting invariants of perverse coherent sheaves and its wall-crossing,” [arXiv:0809.2992 [math.AG]].[20] K. Nagao, “Derived categories of small toric Calabi–Yau 3-folds and counting invariants,”[arXiv:0809.2994 [math.AG]].[21] M. Aganagic, H. Ooguri, C. Vafa and M. Yamazaki, “Wall-crossing and M-theory,”arXiv:0908.1194 [hep-th].[22] H. Ooguri and M. Yamazaki, “Crystal melting and toric Calabi–Yau manifolds,” Commun.Math. Phys. (2009) 179 [arXiv:0811.2801 [hep-th]].[23] M. R. Douglas and G. W. Moore, “D branes, quivers, and ALE instantons,” arXiv:hep-th/9603167. 9024] M. Van den Bergh, “Noncommutative crepant resolutions,” arXiv:math/0211064 [math.RA];“Three-dimensional flops and noncommutative rings,” Duke Math. J. (2004) 423[arXiv:math/0207170 [math.AG]].[25] B. Szendr˝oi, “Noncommutative Donaldson–Thomas theory and the conifold,” Geom. Topol. (2008) 1171 [arXiv:0705.3419 [math.AG]].[26] S. Mozgovoy and M. Reineke, “On the noncommutative Donaldson–Thomas invariants arisingfrom brane tilings,” Adv. Math. (2010) 1521 [arXiv:0809.0117 [math.AG]].[27] M. Aganagic and K. Schaeffer, “Wall-crossing, quivers and crystals,” arXiv:1006.2113 [hep-th].[28] B. S. Acharya, M. O’Loughlin and B. J. Spence, “Higher-dimensional analogues of Donaldson–Witten theory,” Nucl. Phys. B (1997) 657 [arXiv:hep-th/9705138].[29] R. Dijkgraaf and G. W. Moore, “Balanced topological field theories,” Commun. Math. Phys. (1997) 411 [arXiv:hep-th/9608169].[30] M. Blau and G. Thompson, “Euclidean SYM theories by time reduction and special holonomymanifolds,” Phys. Lett. B (1997) 242 [arXiv:hep-th/9706225].[31] C. Hofman and J.-S. Park, “Cohomological Yang–Mills theories on K¨ahler 3-folds,” Nucl. Phys.B (2001) 133 [arXiv:hep-th/0010103].[32] L. Baulieu, H. Kanno and I. M. Singer, “Special quantum field theories in eight and otherdimensions,” Commun. Math. Phys. (1998) 149 [arXiv:hep-th/9704167].[33] D. Maulik, N. A. Nekrasov, A. Okounkov and R. Pandharipande, “Gromov–Witten theoryand Donaldson–Thomas theory I,” Compos. Math. (2006) 1263 [arXiv:math/0312059[math.AG]].[34] D. Maulik, N. A. Nekrasov, A. Okounkov and R. Pandharipande, “Gromov–Witten theoryand Donaldson–Thomas theory II,” Compos. Math. (2006) 1286 [arXiv:math/0406092[math.AG]].[35] D. Maulik, A. Oblomkov, A. Okounkov and R. Pandharipande, “Gromov–Witten/Donaldson–Thomas correspondence for toric 3-folds,” arXiv:0809.3976 [math.AG].[36] D. L. Jafferis, “Topological quiver matrix models and quantum foam,” arXiv:0705.2250 [hep-th].[37] M. Cirafici, A. Sinkovics and R. J. Szabo, “Cohomological gauge theory, quiver matrix modelsand Donaldson–Thomas theory,” Nucl. Phys. B (2009) 452 [arXiv:0803.4188 [hep-th]].[38] N. A. Nekrasov, “Seiberg–Witten prepotential from instanton counting,” Adv. Theor. Math.Phys. (2004) 831 [arXiv:hep-th/0206161].[39] N. A. Nekrasov and A. Okounkov, “Seiberg–Witten theory and random partitions,” Progr.Math. (2006) 525 [arXiv:hep-th/0306238].[40] N. A. Nekrasov, “Localizing gauge theories,” in: , ed. J.-C. Zambrini (World Scientific, 2005), p. 644.[41] R. J. Szabo, “Instantons, topological strings and enumerative geometry,” Adv. Math. Phys. (2010) 107857 [arXiv:0912.1509 [hep-th]].[42] P. B. Kronheimer and H. Nakajima, “Yang–Mills instantons on ALE gravitational instantons,”Math. Ann. (1990) 263.[43] H. Awata and H. Kanno, “Quiver matrix model and topological partition function in sixdimensions,” J. High Energy Phys. (2009) 076 [arXiv:0905.0184 [hep-th]].9144] H. Liu, “M-theory and the Coulomb phase of higher rank DT invariants,” J. High EnergyPhys. (2010) 024 [arXiv:1004.1812 [hep-th]].[45] J. Stoppa, “D0–D6 states counting and GW invariants,” arXiv:0912.2923 [math.AG].[46] D.-E. Diaconescu, “Moduli of ADHM sheaves and local Donaldson–Thomas theory,”arXiv:0801.0820 [math.AG].[47] B. Young and J. Bryan, “Generating functions for coloured 3 D Young diagrams and theDonaldson–Thomas invariants of orbifolds,” Duke Math. J. (2010) 115 [arXiv:0802.3948[math.CO]].[48] M. R. Douglas, B. R. Greene and D. R. Morrison, “Orbifold resolution by D branes,” Nucl.Phys. B (1997) 84 [arXiv:hep-th/9704151].[49] D.-E. Diaconescu and J. Gomis, “Fractional branes and boundary states in orbifold theories,”J. High Energy Phys. (2000) 001 [arXiv:hep-th/9906242].[50] R. J. Szabo and A. Valentino, “Ramond–Ramond fields, fractional branes and orbifold differ-ential K-theory,” Commun. Math. Phys. (2010) 647 [arXiv:0710.2773 [hep-th]].[51] Y. Ito and H. Nakajima, “McKay correspondence and Hilbert schemes in dimension three,”Topology (2000) 1155 [arXiv:math/9803120 [math.AG]].[52] A. Tomasiello, “D-branes on Calabi–Yau manifolds and helices,” J. High Energy Phys. (2001) 008 [arXiv:hep-th/0010217].[53] P. Mayr, “Phases of supersymmetric D-branes on K¨ahler manifolds and the McKay correspon-dence,” J. High Energy Phys. (2001) 018 [arXiv:hep-th/0010223].[54] B. Ezhuthachan, S. Govindarajan and T. Jayaraman, “Fractional two-branes, toric orbifoldsand the quantum McKay correspondence,” J. High Energy Phys. (2006) 032 [arXiv:hep-th/0606154].[55] A. Degeratu, “Geometrical McKay correspondence for isolated singularities,”arXiv:math/0302068 [math.DG].[56] H. Nakajima, “Sheaves on ALE spaces and quiver varieties,” Moscow Math. J. (2007) 699.[57] K. Nagao and M. Yamazaki, “The noncommutative topological vertex and wall-crossing phe-nomena,” arXiv:0910.5479 [hep-th].[58] J. Bryan, C. Cadman and B. Young, “The orbifold topological vertex,” arXiv:1008.4205[math.AG].[59] D. Joyce, Compact Manifolds with Special Holonomy (Oxford University Press, 2000).[60] A. Craw and M. Reid, “How to Calculate A –Hilb C ,” Semin. Congr. Soc. Math. France (2002) 129 [arXiv:math/9909085 [math.AG]].[61] A. Craw, “An explicit construction of the McKay correspondence for A –Hilb C ,” J. Algebra (2005) 682 [arXiv:math/0010053 [math.AG]].[62] V. Ginzburg, “Lectures on Nakajima’s quiver variaties,” arXiv:0905.0686 [math.RT].[63] A. Craw, “Quiver representations in toric geometry,” arXiv:0807.2191 [math.AG].[64] M. Reineke, “Moduli of representations of quivers,” arXiv:0802.2147 [math.RT].[65] A. King, “Moduli of representations of finite dimensional algebras,” Quart. J. Math. Oxford (1994) 515. 9266] A. Craw and G. G. Smith, “Projective toric varieties as fine moduli spaces of quiver represen-tations,” Amer. J. Math. (2008) 1509 [arXiv:math/0608183 [math.AG]].[67] A. Craw and A. Ishii, “Flops of G –Hilb and equivalences of derived categories by variation ofGIT quotient,” Duke Math. J. (2004) 259 [arXiv:math/0211360 [math.AG]].[68] A. Craw, D. Maclagan and R. R. Thomas, “Moduli of McKay quiver representations II.Gr¨obner basis techniques,” J. Algebra (2007) 514 [arXiv:math/0611840 [math.AG]].[69] T. Bridgeland, A. King and M. Reid, “Mukai implies McKay: The McKay correspondence asan equivalence of derived categories,” J. Amer. Math. Soc. (2001) 535 [arXiv:math/9908027[math.AG]].[70] P. S. Aspinwall, “D branes on toric Calabi–Yau varieties,” arXiv:0806.2612 [hep-th].[71] G. W. Moore, N. A. Nekrasov and S. Shatashvili, “D particle bound states and generalizedinstantons,” Commun. Math. Phys. (2000) 77 [arXiv:hep-th/9803265].[72] G. W. Moore, N. A. Nekrasov and S. Shatashvili, “Integrating over Higgs branches,” Commun.Math. Phys. (2000) 97 [arXiv:hep-th/9712241].[73] K. Behrend, “Donaldson–Thomas invariants via microlocal geometry,”, Ann. Math. (2009)1307 [arXiv:math/0507523 [math.AG]].[74] R. Gopakumar and C. Vafa, “M-theory and topological strings II,” arXiv:hep-th/9812127.[75] E. Witten, “Phases of N = 2 theories in two dimensions,” Nucl. Phys. B (1993) 159[arXiv:hep-th/9301042].[76] P. S. Aspinwall, B. R. Greene and D. R. Morrison, “Calabi–Yau moduli space, mirror manifoldsand spacetime topology change in string theory,” Nucl. Phys. B (1994) 414 [arXiv:hep-th/9309097].[77] B. R. Greene, “D brane topology changing transitions,” Nucl. Phys. B (1998) 284[arXiv:hep-th/9711124].[78] D. L. Jafferis and G. W. Moore, “Wall-crossing in local Calabi–Yau manifolds,”arXiv:0810.4909 [hep-th].[79] M. Mari˜no, R. Minasian, G. W. Moore and A. Strominger, “Nonlinear instantons from super-symmetric p -branes,” J. High Energy Phys. (2000) 005 [arXiv:hep-th/9911206].[80] A. Ishii and K. Ueda, “Dimer models and the special McKay correspondence,” arXiv:0905.0059[math.AG]; “Dimer models and exceptional collections,” arXiv:0911.4529 [math.AG].[81] J. Bryan and D. Karp, “The closed topological vertex via the Cremona transform,” J. AlgebraicGeom. (2005) 529 [arXiv:math/0311208 [math.AG]].[82] D. Karp, C.-C. M. Liu and M. Mari˜no, “The local Gromov–Witten invariants of configurationsof rational curves,” Geom. Topol. (2006) 115 [arXiv:math/0506488 [math.AG]].[83] F. Cachazo, S. Katz and C. Vafa, “Geometric transitions and N = 1 quiver theories,”arXiv:hep-th/0108120.[84] B. Szendr˝oi, “Sheaves on fibred threefolds and quiver sheaves,” Commun. Math. Phys. (2008) 627 [arXiv:math/0506301 [math.AG]].[85] A. Gholampour and Y. Jiang, “Counting invariants for the ADE McKay quivers,”arXiv:0910.5551 [math.AG].[86] J. Bryan and A. Gholampour, “The quantum McKay correspondence for polyhedral singular-ities,” Invent. Math. (2009) 655 [arXiv:0803.3766 [math.AG]].9387] P. Sulkowski, “Wall-crossing, free fermions and crystal melting,” Commun. Math. Phys. (2011) 517 [arXiv:0910.5485 [hep-th]].[88] M. Aganagic, A. Klemm, M. Mari˜no and C. Vafa, “The topological vertex,” Commun. Math.Phys. (2005) 425 [arXiv:hep-th/0305132].[89] M. Aganagic, V. Bouchard and A. Klemm, “Topological strings and (almost) modular forms,”Commun. Math. Phys. (2008) 771 [arXiv:hep-th/0607100].[90] J. Bryan and T. Graber, “The crepant resolution conjecture,” Proc. Symp. Pure Math. (2009) 23 [arXiv:math/0610129 [math.AG]].[91] M. R. Douglas, B. Fiol and C. Romelsberger, “The spectrum of BPS branes on a noncompactCalabi–Yau,” J. High Energy Phys. (2005) 057 [arXiv:hep-th/0003263].[92] P. S. Aspinwall and I. V. Melnikov, “D branes on vanishing del Pezzo surfaces,” J. High EnergyPhys. (2004) 042 [arXiv:hep-th/0405134].[93] S. L. Cacciatori and M. Compagnoni, “D branes on C Part I: Prepotential and GW invari-ants,” Adv. Theor. Math. Phys. (2009) 1371 [arXiv:0806.2372 [hep-th]].[94] D. Berenstein and R. G. Leigh, “Resolution of stringy singularities by noncommutative alge-bras,” J. High Energy Phys. (2001) 030 [arXiv:hep-th/0105229].[95] C. Beil and D. Berenstein, “Geometric aspects of dibaryon operators,” arXiv:0811.1819 [hep-th].[96] C. Beil, “The noncommutative geometry of square superpotential algebras,” arXiv:0811.2439[math.AG].[97] R. Eager, “Brane tilings and noncommutative geometry,” J. High Energy Phys. (2011) 026[arXiv:1003.2862 [hep-th]].[98] E. R. Sharpe, “String orbifolds and quotient stacks,” Nucl. Phys. B (2002) 445 [arXiv:hep-th/0102211].[99] T. Coates, A. Corti, H. Iritani and H. Tseng, “Computing genus zero twisted Gromov–Witteninvariants,” Duke Math. J. (2009) 377 [arXiv:math/0702234 [math.AG]].[100] H. Tseng, “Orbifold quantum Riemann–Roch, Lefschetz and Serre,” Geom. Topol. (2010)1 [arXiv:math/0506111 [math.AG]].[101] T. J. Jarvis and T. Kimura, “Orbifold quantum cohomology of the classifying space of a finitegroup,” Contemp. Math. (2002) 123 [arXiv:math/0112037 [math.AG]].[102] V. Bouchard and R. Cavalieri, “On the mathematics and physics of high genus invariants of[ C / Z ],” Adv. Theor. Math. Phys. (2009) 695 [arXiv:0709.3805 [math.AG]].[103] C. Okonek, M. Schneider and H. Spindler, Vector Bundles on Complex Projective Spaces (Birkh¨auser, 1980).[104] E. R. Sharpe, “Lectures on D branes and sheaves,” arXiv:hep-th/0307245.[105] H. Nakajima,