aa r X i v : . [ h e p - t h ] J u l Wall Crossing, Quivers and Crystals
Mina Aganagic , , and Kevin Schaeffer Center for Theoretical Physics, Department of Physics,University of California, Berkeley, CA 94720, USA Department of Mathematics,University of California, Berkeley, CA 94720, USA Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa City, Chiba 277-8568, Japan
Abstract
We study the spectrum of BPS D-branes on a Calabi-Yau manifold using the 0 + 1 di-mensional quiver gauge theory that describes the dynamics of the branes at low energies.The results of Kontsevich and Soibelman [1] predict how the degeneracies change. Weargue that Seiberg dualities of the quiver gauge theories, which change the basis of BPSstates, correspond to crossing the “walls of the second kind” in [1]. There is a large classof examples, including local del Pezzo surfaces, where the BPS degeneracies of quiverscorresponding to one D6 brane bound to arbitrary numbers of D4, D2 and D0 branes arecounted by melting crystal configurations. We show that the melting crystals that ariseare a discretization of the Calabi-Yau geometry. The shape of the crystal is determinedby the Calabi-Yau geometry and the background B-field, and its microscopic structure bythe quiver Q . We prove that the BPS degeneracies computed from Q and Q ′ are relatedby the Kontsevich Soibelman formula, using a geometric realization of the Seiberg dualityin the crystal. We also show that, in the limit of infinite B-field, the combinatorics ofcrystals arising from the quivers becomes that of the topological vertex. We thus re-derivethe Gromov-Witten/Donaldson-Thomas correspondence.1 . Introduction There has been remarkable recent progress in understanding the spectra of BPS statesof N = 2 theories in 4 dimensions, driven in part by the mathematical conjectures ofKontsevich and Soibelman [1]. KS conjectured how the degeneracies of BPS states changeas we cross walls of marginal stability. In some cases, we have a physical understandingof why the conjectures of [1] are true. In particular, for BPS states in a gauge theory, theresults of [1] have been explained in [2,3], and from a different perspective in [4] (see alsothe very recent [5]) .In this paper we do three things. First, we give the physical explanation for the “wallsof the second kind” in [1]. Second, we provide further evidence that the conjectures of [1]are true – by proving that the spectrum of BPS bound states of a D6 brane with D4, D2and D0 branes wrapping toric Calabi-Yau manifolds satisfies them, for certain walls. Thespectrum here is in general very complicated (far more so than in the examples studied sofar, corresponding either to four dimensional gauge theories [10], or Calabi-Yau manifoldswithout compact four cycles, where the generating function of the spectrum is computablein closed form [11,12,13,14].). Finally, we use this to shed new light on a relation betweenfamiliar objects: topological strings, Calabi-Yau crystals, and BPS D-branes. These werestudied previously in [15,16,17,18,19]. There are two distinct phenomena that affect the BPS spectrum. One, which wasmost studied in the literature, e.g. in [20], corresponds to crossing a wall of marginalstability where central charges of a pair of states align. There, the degeneracies of boundstates of the pair change. When the BPS states are described in a quiver gauge theory, wecross the wall by varying the Fayet-Iliopoulos terms. We interpret the “walls of the secondkind” of [1], as a kind of Seiberg duality, that changes the basis of BPS branes and the splitof the spectrum into the branes and the anti-branes. In the quiver gauge theory, this canbe made precise. The basis of the BPS branes is provided by the nodes of the quiver. Thebranes are described as linear combinations of these with positive coefficients which arethe ranks of the quiver gauge groups. The walls of the second kind correspond to varyingthe gauge couplings g − of the nodes. When one of them passes through zero, we needto change the description to a new quiver, related to the original one by Seiberg duality See [6] for a matrix model perspective, and [7,8,9] for more mathematical progress. Y . Y is related by mirror symmetry to a toricCalabi-Yau X , which maps the D3 branes to D6, D4, D2 and D0 branes, as we review insection 5. Recently, [25] gave a remarkably simple way of computing degeneracies of BPS boundstates of one D6 brane on a toric Calabi-Yau X , with D4 branes, D2 branes and D0 branes,generalizing the earlier work of [26] to essentially arbitrary toric Calabi-Yau singularities.Adding a D6 brane corresponds to extending the D4-D2-D0 quivers by a node of rank 1[14], in a manner which we make precise. While in principle BPS degeneracies of a quiverare computable for any given choice of ranks as an Euler characteristic of the moduli space,the direct computations become cumbersome as ranks increase. Instead, as we review insection 4, [25] give a combinatorial way to compute the degeneracies for any ranks, bycounting perfect matchings of certain dimer models on a plane. (This can be rephrasedin terms of counting melting crystals, as we will explain shortly.) The relevant dimers arethe lift to R of dimers on T that correspond to D4-D2-D0 quivers in [29,39].We show that crossing the walls of the second kind that take a quiver Q to its Seibergdual Q ′ corresponds to a simple geometric transition in the dimer. In the case of the dimeron T , this was known from [29], and what we have here is a simple lift of this from thedimers on T to the dimers on the plane. Since perfect matchings of the planar dimercount BPS states, this gives a geometric description for how the spectrum jumps. Weshow (in section 4) that this can be used to prove the KS wall crossing formula in thecontext of the quivers of [25].In section 6 we give another example of wall crossing of the second kind correspondingto varying the B-field on X . Since shifting the B-field brings us back to the same pointin the moduli space of X , this should leave the spectrum of BPS states invariant. Indeed, These were studied extensively in [27,28,29,21,30,31,22,23,24,32,33,34,35,36,37,38].
3e show that if one turns on compact B -field, B ∈ H cmpct ( X, Z ), this corresponds to asequence of Seiberg dualities of the quiver, but in the end the quiver comes back to itself.Shifting by a B -field in B ∈ H ( X, Z ) /H cmpct ( X, Z ) also corresponds to a change of basisof branes and Seiberg duality, but this time the quiver does not come back to itself. Instead,this change of basis permutes the quivers describing D6 branes with different amounts ofnoncompact D4 brane charge. There is an intriguing connection between the dimers that appear in [25] and an earlierappearance of dimers in the context of the closed topological string, in [15]. As explainedin [15,40,25,14] there is a close relation between dimer models in the plane, with suitableboundary conditions, and three dimensional melting crystals. We show that the meltingcrystals of [25] have a beautifully simple geometric description: the crystal sites are adiscretization of the Calabi-Yau geometry. The shape of the crystal is determined by thegeometry of the Calabi-Yau base, which is a singular cone at the point in the moduli spacewhere the quiver is defined. The crystal sites are the integral points of the Calabi-Yau.The precise microscopic structure of the crystal depends on the quiver (and changes underSeiberg dualities). The refinement comes from the fact that we are not counting boundstates of the D6 brane with D0 branes, but the more general bound states with D4, D2and D0 branes, corresponding to splitting of the D0 branes into fractional branes. Weshow that, increasing the B-field by D ∈ H ( X, Z ), D6 brane bound states are countedby the crystal that takes the shape of the Calabi-Yau X with Kahler class D , while themicroscopic structure of the crystal does not change.The relation of topological string amplitudes on X with certain melting crystals wasobserved in [15]. The Calabi-Yau crystals that arise in our paper are the same as those in[15,16], but only in the limit of infinite D , where the microscopic structure of the crystalis lost. The crystals in [15] were interpreted in [16] as counting bound states of a D6 braneon X , formulated as the Witten index of a non-commutative N = 2 SYM on X . Theobservations of [16] are known as the Gromov-Witten/Donaldson-Thomas correspondence[17,18]. We thus provide a new derivation of the Gromov-Witten/Donaldson-Thomascorrespondence from the perspective of the 0 + 1 dimensional quiver quantum mechanics,but only in the limit of large D . This is in agreement with [20], which pointed out that thecorrespondence of [16] can hold only in the limit of infinite B-field (this was also verifiedin [13] when X has no compact 4-cycles). This is described in section 7.4 . Walls of the Second Kind and Seiberg Duality Consider BPS states from D-branes wrapping cycles in a Calabi-Yau Y . For definite-ness, take IIB string theory, so that the BPS particles are labeled by charge∆ ∈ H ( Y, Z ) . The mass of the BPS state and the supersymmetry it preserves are determined by thecentral charge, Z (∆) = Z ∆ Ω , where Ω is the (3 ,
0) form on Y . Near the singularities in Y , the D-branes can be describedby quiver gauge theories in 0 + 1 dimensions. The nodes of the quiver correspond to abasis of H ( Y, Z ). Any bound state of the branes with charge ∆ can be written as∆ = X α N α ∆ α , N α ≥ . (2 . G ∆ = Y α U ( N α )gauge theory. The quiver is a good description when all the central charges of the nodesare nearly aligned. By choosing an overall phase of Ω, we can write Z (∆ α ) = ig α + θ α , corresponding to all Z ’s being close to imaginary. Above, θ α is the Fayet-Iliopoulos pa-rameter, and g α is the gauge coupling in the quantum mechanics. The BPS degeneracy ofa state ∆ is determined by the Witten indexΩ Q (∆) = Tr ∆ ,Q ( − F , (2 . G ∆ . Thus, the quiver gauge theory provides both a basisof D-branes, and a means to compute the degeneracies. From the space time perspective, we are computing − F ( − F . The F factor serves toabsorb the contribution of zero modes on R , . This reduces then to Tr( − F where one tracesinternal degrees of freedom only [20].
5t walls in the moduli space, the spectrum of BPS states can change. One kind ofwall is a wall of marginal stability, where central charges of two states, for example of twonodes, align. At the wall, even though (2.2) is an index, it can jump, since the one particlestates it is counting can split, or pair up. In the quiver, since we are near the intersection ofwalls anyhow, crossing this wall corresponds to some combination of FI parameters passingthrough zero.Before we go on, note that the quiver only describes the states with non-negative N α .This is just as well, since the central charges of anti-branes − ∆ α would be anti-parallel tothe rest of the states in the quiver. The states with some of the N α ’s negative and somepositive do not form bound states, and the quiver does not miss anything . Moreover, thisremains true even when the central charges of ∆ α are not nearly aligned, as long as they allremain in the upper half of the complex plane – simply because there are no walls where∆ β and − ∆ α can align. The fact that only the states in the upper half of the complex Z − plane bind, implies that the spectrum can change as a node ∆ ∗ leaves the upper halfof the complex plane, and correspondingly − ∆ ∗ enters it. The node ∆ ∗ leaves the upperhalf of the complex plane when g − ∗ passes through zero. This wall, in real codimensionone of the moduli space, was called the “wall of the second kind” in [1].On the wall, g − ∗ = 0, the gauge coupling of the node is infinitely strong. Even thoughin 0 + 1 dimensions the gauge fields have no local dynamics, the coupling enters the actionas the coefficient of the kinetic terms for the fields on the brane, e.g., g ∗ R | ∂φ | . We cancontinue past infinite coupling since clearly nothing special happens to the D-branes, aslong as we stay away from Z (∆ ∗ ) = 0, but we need to change description. To make g ∗ positive, we need to flip ∆ ∗ → ∆ ′∗ = − ∆ ∗ . (2 . ∗ is not the onlynode that changes. Were we to complete the circle around Z (∆ ∗ ) = 0, all of the nodeswould have changed due to monodromy, which maps ∆ to ∆ ± (∆ ◦ ∆ ∗ )∆ ∗ , for any ∆,with the sign depending on the path. Going halfway around the circle, there are partialmonodromies, which are Seiberg dualities [21]. On the other side of the wall, the theoryis described by a dual gauge theory, based on { ∆ ′ α } and a new quiver Q ′ . There are of course the bound states with all N α negative, but these are just CPT conjugatesof the states at hand. Q and Q ′ . In addition tothe one based on Q and G ∆ , it may have another one, in terms of Q ′ with gauge group G ′ ∆ = Y α U ( N ′ α )where ∆ = X α N ′ α ∆ ′ α , (2 . N ′ α are non-negative. Crossing the “wall of the second kind,” the spectrum can changebecause the spaces of BPS bound states one can obtain from Q and Q ′ (by varying FIparameters, while staying in the upper half of the complex plane) are different. In theterminology of KS, this is a change of t -structure. Kontsevich and Soibelman [1] predict how the degeneracies Ω Q (∆) change as thecentral charges Z are varied in some way. For each charge ∆ ∈ H ( Y, Z ), associate anoperator e ∆ , satisfying [ e ∆ , e ∆ ′ ] = ¯ h ( − ∆ ◦ ∆ ′ ∆ ◦ ∆ ′ e ∆+∆ ′ (2 . e ∆ e ∆ ′ = ( − ∆ ◦ ∆ ′ e ∆+∆ ′ , where ∆ ◦ ∆ ′ is the intersection product on H ( Y ), and an operator A ∆ = exp( 1¯ h ∞ X n =1 e n ∆ n ) . Consider the product A Q = → Y ∆ A Ω Q (∆)∆ (2 . Z (∆) . The KS conjecture states In general, the degeneracies Ω Q (∆) in the KS formula may differ from the physical BPSdegeneracies by a ∆-dependent sign. This additional sign factor will be important in section 4,when we use the KS formula to compute wall crossing for quiver gauge theories. .Consider what happens as we cross the wall where the coupling g ∗ of node ∆ ∗ flipssign. Near the wall, the chamber corresponding to the quiver is as in the figure below. Aswe cross the wall, the state ∆ ∗ leaves the upper half plane from the left, and the state − ∆ ∗ enters it from the right, so that the new degeneracies should satisfy A ′ Q = A − ∗ A Q A − ∆ ∗ . (2 . Fig. 1.
Wall of the Second Kind. As we cross this wall, the central charge of ∆ ∗ leaves the upperhalf plane. Crossing the wall as above, we generally do not come back to the same point in themoduli space, so A Q and A ′ Q are not themselves equal, not even after a change of basis.However, the spectrum should be determined uniquely by the point in the moduli spacewe are at. Going around a closed loop in the moduli space, the spectrum of BPS statesmust come back to itself, up to a monodromy that relabels the branes. In particular, itshould not matter which way we go around the singularity. As a simple example, consider the quiver Q with two nodes, ∆ and ∆ ∗ , and onearrow, corresponding to the intersection number ∆ ◦ ∆ ∗ = 1. With the central charges asin figure 2, there are only three BPS states, and the operator A Q is simply [1] A Q = A ∆ A ∆ +∆ ∗ A ∆ ∗ . (2 . More precisely, [1] restrict to a “strict” wedge, meaning one that subtends an angle less than180 ◦ . This simply tells us how to define A Q exactly on the wall, when states ∆ α and − ∆ α bothhave central charges on the real line. ig. 2. Rotating the central charge of ∆ ∗ counterclockwise (above) and clockwise (below). Notethat in both cases we cross a wall of the first kind and a wall of the second kind. Rotating the central charge of ∆ ∗ counterclockwise, eventually Z (∆) and Z (∆ ∗ ) align,and we cross a wall of the first kind. The bound state ∆ + ∆ ∗ decays, and the productbecomes A Q = A ∆ ∗ A ∆ . Continuing past this, eventually the gauge coupling of ∆ ∗ becomes negative, so ∆ ∗ leavesthe upper half of the complex plane, and − ∆ ∗ enters it. To get a good description, weneed to change the quiver from Q , to Q ′ with nodes ∆ , − ∆ ∗ . This corresponds to Seibergduality on the node ∆ ∗ (This quiver and its Seiberg dualities were studied in detail in[21].) with A Q ′ = A − ∗ A Q A − ∆ ∗ = A ∆ A − ∆ ∗ . Now, we could have reached the same point in the moduli space by starting with (2.8) androtating Z (∆ ∗ ) clockwise instead. Then ∆ ∗ leaves the upper half of the complex plane, − ∆ ∗ enters it. We again need to dualize node ∆ ∗ , but now we get the basis ∆ + ∆ ∗ , − ∆ ∗ corresponding to quiver Q ′′ . Moreover, A Q ′′ = A − ∆ ∗ A Q A − ∗ = A − ∆ ∗ A ∆ A ∆ +∆ ∗ . We used the pentagon identity A ∆ A ∆ +∆ ∗ A ∆ ∗ = A ∆ ∗ A ∆ which holds for any two states∆ , ∆ ∗ with intersection number +1 [1].
9o get this to correspond to the same point in the moduli space as Q ′ above, we need tokeep rotating Z (∆ ∗ ), until A Q ′′ becomes A Q ′′ = A ∆ +∆ ∗ A − ∆ ∗ . Since Q ′ and Q ′′ now correspond to two quivers describing physics at exactly the samepoint in the moduli space, they are of course equivalent. The non-trivial relation between A Q ′′ and A Q ′ is a consequece of the fact that to relate them, we need to go once aroundthe Z (∆ ∗ ) = 0. In doing so, there is a monodromy acting on the cycles that maps∆ → ∆ + (∆ ◦ ∆ ∗ )∆ ∗ , looping counter-clockwise, which is precisely how these are related. As an aside, notethat, taking for example the state ∆ + ∆ ∗ of quiver Q , depending on which way we goaround the singularity there are two different interpretations for its fate. Along the pathcorresponding to Q ′ , the state decays into ∆ and ∆ ∗ on the wall of marginal stabilitywhere ∆ ∗ and ∆ align. From the perspective of the split attractor flows [42], the flowcorresponding to ∆ + ∆ ∗ splits on this wall into a flow corresponding to ∆ ∗ , which crashesat Z (∆ ∗ ) = 0, and an honest black hole attractor corresponding to ∆ . Along the pathcorresponding to Q ′′ , the state crosses no walls, but the monodromy changes ∆ + ∆ ∗ to∆ . Following either flow, the attractor point is the same, as expected from [42].
3. Quivers from Calabi-Yau Threefolds
In this section we will review (following [24]) the quiver gauge theories Q that arisefor certain choices of a Calabi-Yau Y , and its moduli. One reason we choose these theoriesis that one has a very direct, geometric understanding of what happens to Q as the centralcharges are varied, and the theory undergoes Seiberg dualities. The second reason is thatthe choice we make implies that the quiver theory has extra symmetries – Q are the socalled toric quiver gauge theories of [29,39] (see [44] for an excellent review). The torussymmetries will allow us to extract very precise information about the quantum BPSspectra of Q , in the next section. The presence of the extra symmetries is related to The change of basis of BPS states was also recently discussed in [43], from the attractorviewpoint. Y is mirror to a toricCalabi-Yau X .Consider a Calabi-Yau Y given by W ( e x , e y ) = w uv = w. (3 . Y as a fibration over the w plane. At a generic point in the w -plane, the fiberis a product of a cylinder uv = w, and a Riemann surface W ( e x , e y ) = w . Over specialpoints, the fiber degenerates. Over a point q with w ( q ) = 0, the S of the cylinder pinches.The 1-cycles of the Riemann surface degenerate over critical points of W ( e x , e y ), p α : ∂ x W = 0 = ∂ y W, α = 1 , . . . , r (3 . w ( p α ) = w α . For each p α , a path in the w -plane connecting it with q , together withan S × S fiber over it, gives an S which we will denote∆ α , α = 1 , . . . , r Above, one of the S ’s corresponds to the cylinder, while the other corresponds to the 1-cycle of the Riemann surface pinching at p α ). The three-cycles ∆ α provide a basis for thecompact homology of Y . We can associate a quiver to the above singularity by consideringD3 branes wrapping the cycles ∆ α . Since the ∆ α ’s are spheres, the D-branes wrappingthem have no massless adjoint matter. For a collection of∆ = X α N α ∆ α , N α ≥ . (3 . G = Y α U ( N α ) (3 . α corresponding to node α of the quiver. Moreover, for ev-ery point of intersection of ∆ α and ∆ β we get a massless chiral bifundamental in either( N α , ¯ N β ), or ( N β , ¯ N α ) representation of the gauge group. Pairs of these can get mass, sothe net number of chiral multiplets going from node α to node β is n αβ = ∆ α ◦ ∆ β .
11s we vary the moduli of Y , the locations of critical points p α in the w plane change.If the critical point p α crosses the cycle ∆ β , due to monodromy, the homology of the cycle∆ β changes to ∆ ′ β ∆ ′ β = ∆ β ± (∆ β ◦ ∆ α )∆ α , (3 . β stands for the original homology class of the cycle (see figure 3). Fig. 3.
Picard lefschetz monodromy as α passes through the ∆ β cycle. The fact that the homology classes of the cycles change implies that the quiver changes.For example, the intersection numbers n αβ change, and with them the number of arrowsconnecting the two nodes of the quiver. Consider varying the moduli so that the gaugecoupling of the nodes ∆ ∗ becomes negative. In the process, p ∗ crosses the cycles ∆ β i , sothat the basis of branes changes to ∆ ′∗ = − ∆ ∗ ∆ ′ β j = ∆ β j ± n β j ∗ ∆ ∗ ∆ ′ γ k = ∆ γ k (3 . W W Fig. 4.
The ∆ ∗ cycle is shown shrinking and then growing in the opposite direction. In theprocess, ∆ ∗ crosses the { ∆ β i } . n ′∗ β k = − n ∗ β k n ′ β k γ j = n β k γ j ± n β j ∗ n ∗ γ k n ′ γ j ∗ = − n γ j α . (3 . N α of the branes on thenode α have to change to N α ′ , so that∆ = X α N α ∆ α = X α N ′ α ∆ ′ α , in order to be consistent with (3.5).Note that (3.6)(3.7) are exactly the same changes of basis as on p. 134 of [1]. Thetransformation is a Seiberg duality of the quiver gauge theory [45][30]when either n ∗ β k > , n ∗ γ j ≤ , for all β k , γ j , or with the direction of inequalities reversed – depending on the sign in(3.6). The choice of sign determines which way around the singularity we go. The newquiver Q ′ is obtained from Q by (i) reversing the arrows beginning or ending on the node ∗ we dualized. The reversed arrows correspond to new chiral fields associated with the dualnode, all of whose intersection numbers have flipped signs. (ii) The original bifundamentalstransforming under the node ∗ are confined in the bifundamental mesons that no longertransform under gauge transformations on node ∗ . (iii) There is an additional gaugeinvariant coupling of the mesons to the new bifundamentals charged under the dual node.These can however pair up with the existing bifundamentals of opposite orientations anddisappear, since only the net intersection number, counted with signs is invariant. The neteffect of (ii) and (iii) is to give a mass to all but n β k γ j ± n β j ∗ n ∗ γ k bifundamentals betweenthe node β k and γ j . When this is not the case, one should still get a dual description, though it may not correspondto the quiver gauge theory one would naively obtain. For example due to the presence of tachyons,the intersection numbers from the geometry need not count chiral multiplets. See, e.g. [30][36]. .1. The mirror of P × P example Consider for example, W ( e x , e y ) = e x + z t e − x + e y + z s e − y + 1 , (3 . z t = e − t , z s = e − s . There are four critical points in the W plane, w α = ± √ z t ± √ z s + 1. W Fig. 5.
The W plane for the mirror of P × P . The four corresponding S ’s are drawn in Figure 5, in the limit z s , z t → . The intersection numbers of the cycles were determined in [46]. The quiver Q that resultshas four nodes, connected by arrows n = n = n = n = 2 , n = 4 . The theory also has a superpotential, computed by the topological A-model on a disk, W = X i,j,a,b =1 , ǫ ij ǫ ab (cid:0) Tr B a A i D jb + Tr ˜ A i ˜ B a D jb (cid:1) . where the fields are labeled in Figure 7. As we vary the complex structure moduli of Y ,the gauge coupling of one of the nodes, say node 2 can become negative. This can beachieved by sending z s , z t → ∞ , z s /z t = e − T fixed. As we vary z ’s the cycles deform as in the Figure 6[24], corresponding to ∆ ′ = ∆ + 2∆ , ∆ ′ = − ∆ , ∆ ′ = ∆ , ∆ ′ = ∆ . (3 . W Fig. 6.
Deformation of the cycles as z s , z t → ∞ . Since ∆ is deformed, the good cycle in thislimit is now ∆ ′ . This implies that the non-vanishing intersection numbers are now n ′ = n ′ = n ′ = n ′ = 2 . This results in a new quiver, Q ′ , given in the Figure 7. The superpotential of the theoryalso changes, and becomes W = X i,j,α,β ǫ ij ǫ αβ Tr B α A i B ′ β A ′ j . The dual theory is obtained by Seiberg duality – instead of ˜ A i , ˜ B α we introduce two new pairsof fields A ′ i , B ′ α , with opposite orientation. The original fields are confined in mesons M αi = ˜ B α ˜ A i , and the superpotential becomes W = X i,j,α,β ǫ ij ǫ αβ (cid:0) Tr B α A i D jβ + Tr M iα D jβ + Tr B ′ α A ′ i M jβ (cid:1) . The second term above makes both M and D massive, and they can be integrated out. Thisresults in the effective superpotential we wrote. Q Q'
Fig. 7.
The Quivers for P × P . The Q phase corresponds to z s , z t → Q ′ phasecorresponds to z s , s t → ∞ . The two quiver gauge theories above are examples of the “toric” gauge theories[29]([44] contains an excellent summary). The quiver Q of a toric gauge theory can berepresented as a periodic quiver on a T torus. The periodic quiver gives a tiling ofthe torus which turns out to encode not just the quiver, but also the superpotential W .The terms in the superpotential correspond to the plaquettes on the torus defined by thequiver, whose boundaries are the bifundamental matter fields and where the orientationof the boundary of a plaquette determines the sign of the term. This implies, for example,that a given matter field enters precisely two superpotential terms, with opposite signs .The dual graph, with faces and nodes exchanged, is per definition a bipartite graph on thetorus. The bi-coloring of the nodes is determined by the sign of the corresponding super-potential term. Moreover, the edges of the dual graph connect pairs of nodes of differentcolors.The structure is in part a consequence of mirror symmetry [39], and the fact that Y is fibered by three-tori T . If we view the T as an S fibration over T , where the S fiber corresponds to the uv = w cylinder, and the T is mapped out by the phases of x and y coordinates. Consider a D3 brane, wrapping a generic T fiber (this is mirrorto a D0 brane on X ), and let it approach w = 0 in the base. There, the S fibrationdegenerates over a graph on the T cut out by the Riemann surface W ( e x , e y ) = 0. In thenice cases, corresponding to the quiver being toric, the fibration is such that this sectionsthe D3 brane out into plaquetes, which are the plaquettes of the bipartite graph [47][39].In particular, T = X α ∆ α Any additional coefficients can be set to 1 by a field redefinition. U (1) r quiver gauge theory is the mirror manifold X , since mirror symmetry maps D3 brane on T to D0 brane on X .The toric quiver gauge theories have a global T = U (1) × U (1) R symmetry. The U (1) symmetry is inherited from the U (1) symmetry of the Calabi-Yau,and leaves the superpotential invariant. The U (1) R is an R -symmetry, under which thesuperpotential is homegenous, of degree 2.There is a simple geometric relation between bipartite graphs of a pair of dual quivers[29]. The transformation is local, acting only on the face of the bipartite graph we dualize.We replace the face corresponding to the dualized node of the quiver (in the present case,this is node 2), with a dual face. The dual face is the copy of the original, but with the colorsof all the vertices reversed. For this to fit into the original bipartite graph consistently, weadd a link connecting each original vertex bonding the face to its dual of opposite color.The new links that appear in this way correspond to mesons. Finally, we can erase linkscorresponding to massive fields. The result is the bipartite graph corresponding to thedual gauge theory. An example of this, relating the dimers of quivers Q and Q ′ is in thefigure 8. Q Q' Fig. 8.
The dimer models on T for Q and Q ′ are related by dualizing face 2.
4. BPS Degeneracies and Wall Crossing from Crystals and Dimers
There is a combinatorial way to compute the BPS degeneracies of toric quiver theories[25,14,48] in terms of enumerating certain melting crystal configurations, or equivalently[15,25], by counting dimer configurations. In the language of dimers, Seiberg dualitybecomes geometric. We will use this to prove that the BPS degeneracies of of two Seibergdual quivers satisfy the relations (2.7) for a certain infinite class of states. (As we will seein the next section, these will turn out to be bound states of a single D6 brane bound toD4, D2 and D0 brane wrapping cycles in a mirror toric Calabi-Yau X .)17 .1. BPS states of Quivers and Melting Crystals The BPS degeneracy of a state ∆ is the Witten indexΩ Q (∆) = T r Q, ∆ ( − F of the quiver Q with gauge group G ∆ (3.4). It can be computed as the Euler character ofthe moduli space of the quiver, defined by setting F- and D-terms to zero, and dividing bythe gauge group. In practice, this is doable for any fixed ∆, but it quickly gets cumbersome.There is a combinatorial way to compute the degeneracies Ω Q (∆), for any ∆ , for atoric gauge theory, using the torus T symmetry and localization. The price to pay is thatthe results correspond to degeneracies not of the quiver Q alone, but of its extension byadding one extra node of rank 1. In the figure below we show the extension of the twodual quivers we considered above by an node α = 0 and an arrow. (cid:5) (cid:0) Q Q'
Fig. 9.
The extended quivers for P × P . The extended quiver has the same superpotential as before, since there are no gaugeinvariant operators we can add. We will return to the physical meaning of this extensionin a moment , but for now we simply explain the statement [25,14].Recall that a quiver defines a path algebra A , whose elements are all paths on thequiver obtained by joining arrows in the obvious way, where we consider equivalent twopaths related by F-term constraints. Since we allow paths on the quiver Q that windaround the torus arbitrary numbers of times, consider the lift to a periodic quiver on theplane, Q R . Let A be the subspace of A , corresponding to paths starting on node 0. This was discussed in [12] in the case of the conifold, and in [14] for toric Calabi-Yau withoutcompact 4 cycles. T -charge of the arrows in the quiver assigns T charge to paths in A . A theorem of[49,25] states that any two elements of A with the same T charge are equivalent moduloF-terms. The set of T charges of endpoints of A is a three-dimensional crystal C , whichis a cone (see Figure 10). Keeping track of only the U (1) charge, C projects down to thetwo-dimensional planar quiver we started with. The U (1) R charge defines height of thenodes, making the crystal three dimensional. Fig. 10
The crystal for local P × P . Starting from C one can explicitly construct T -invariant solutions to F-term equations.A melting of the crystal is an ideal C ∆ such that if a path p is in C ∆ , than pa is also in C ∆ for any path a in A . C ∆ is obtained from C by removing N α sites corresponding tonode α where ∆ = P α N α ∆ α . The melting crystal configurations C ∆ are in one-to-onecorrespondence with T -fixed solutions to F-term equations corresponding to quiver Q ,with gauge group G ∆ . We will choose the FI parameters θ α in such a way that every It is crucial for the structure of the crystal that there exist a U (1) R symmetry so that everypath has positive R-charge. One choice is the U (1) R symmetry that can be geometrically realizedin the dimer model as an “isoradial embedding” of the bipartite graph in the plane [44]. To sketch this, consider the finite set of sites we removed to get C ∆ from C . The set of sitescorresponding to node α give vector spaces V α of Chan-Paton factors of rank dim( V α ) = N α .The algebra A is represented on this by matrices, with non-zero entries corresponding to paths inthe crystal. By construction, these satisfy F -term constraints. The vector spaces V α come withthe grading by the torus T charge assigned to them by paths in A . The solutions to the F-termequations we obtained above are fixed under the torus action that transforms the V α and elements θ > θ α =0 <
0. Physically, this means that the central charges of the nodes ∆ α are all roughly aligned, and at an angle with the central charge of the node ∆ . Then,setting D-terms to zero and dividing by the gauge group is equivalent to dividing bythe complexified gauge group. Doing so, the fixed points in the quiver moduli space areisolated, and counting them reduces to enumerating crystal configurations C ∆ .Counting T -fixed points in the classical moduli space gives T r ∆ ( − F up to a sign( − d (∆) , corresponding to the fermion number of the fixed point (which determineswhether the BPS multiplet is bosonic or a fermionic). Thus, the BPS degeneracies ofthe quiver are obtained by counting melting crystal configurations, up to signΩ Q (∆) = X C ∆ ( − d (∆) = ( − d (∆) χ ( M ∆ Q ) , where the sum is over crystals C ∆ with charge ∆ nodes removed, and where we have alsowritten the degeneracies in terms of the euler characteristic of the quiver moduli space M ∆ Q with fixed ranks given by ∆. The sign is computed by d (∆) = X α N α + X α → β N α N β (4 . Q connecting nodes α , and β ,including node 0 . This counts the dimension of the tangent space to the fixed point set.Finally, let us define a generating function for the degeneracies Z Q ( q ) = X ∆ , C ∆ ( − d (∆) q ∆ (4 . q ∆ is the chemical potential, induced by giving weight q α to node α , i.e. q ∆ = Y α> q N α α . We have seen that the degeneracies of N = 1 BPS states of a toric quiver Q extendedby a node, can be computed by counting crystal configurations. Consider now two toric of A by their corresponding weights. We still need to impose D-term constraints, and divide bythe subgroup of the gauge group G ∆ that is preserved by the solution. Since the nodes of V α allcarry different T charge, G ∆ is broken to the maximal abelian subgroup. We thank D. Jafferisfor discussions and explanations of this point. Q , Q ′ related by dualizing a node ∆ ∗ . We will prove that the degeneracies computedby the corresponding crystals satisfy the wall crossing formulas of [1].The wall crossing formula (2.7), predicts that degeneracies corresponding to two quiv-ers Q and Q ′ are related by A Q ′ = A − ∗ A Q A − ∆ ∗ , (4 . e ′ ∆ ∗ = e − ∆ ∗ e ′ ∆ βj = e (∆ βj + n βj ∗ ∆ ∗ ) e ′ ∆ γk = e ∆ γk (4 . n β j ∗ >
0, and n γ k ∗ <
0. (In (4.3) we made a particular choice of the route aroundthe singularity. The other choice is related to this by monodromy, which does not affectthe degeneracies, but relabels the charges.) In the above, A Q contains the informationabout the BPS states with any N , and not just N = 1 states that are counted by thecrystal. To apply this to the present context, consider a truncation of the algebra (2.5)to operators e ∆ with N = 0. Denote by A (0) Q the restriction of A Q operator product tostates with vanishing N . This can be implemented by setting e ∆ = 0. Then, (4.3) readsthe same, just restricted to A (0) Q .Next, restrict to operators e ∆ , with N = 0 ,
1. (Note that this implies that any twooperators with N = 1 commute.) By our choice of the FI parameters, the central chargesof all the states with N = 0 are approximately aligned, and the central charges of all thestates with N = 1 are also aligned, but at an angle to the former. To this order, A Q reduces to a product A (0) Q A (1) Q . Then, (4.3) implies that A (1) Q transforms by conjugationwith A − ∆ ∗ : A (1) Q ′ = A − − ∆ ∗ A (1) Q A − ∆ ∗ . (4 . A (1) Q ′ and A (1) Q , we in addition need to redefine the variables using (4.4). Notethat, in addition to crossing the wall of the second kind, corresponding to a Seiberg duality,we have adjusted the FI terms so that the computation of [25] applies. In other words, allthe central charges of all the ∆ ′ α =0 are aligned, and at an angle to ∆ ′ . Note that this implies that | Z (∆ α ) | ≪ | Z (∆ ) | , an assumption that we will justify in thenext section, when we identify node 0 with a D6 brane wrapping X . N = 1, and using that conjugation by A ∆ acts as A ∆ e ∆ ′ A − = (1 − e ∆ ) ∆ ◦ ∆ ′ e ∆ ′ for any two ∆ , ∆ ′ , we can equivalently rewrite (4.5) as X ∆ χ ( M ∆ Q ′ ) e ∆ = X ∆ χ ( M ∆ Q ) (1 − e − ∆ ∗ ) ∆ ∗ ◦ ∆ e ∆ . Note that we have written this KS formula in terms of the (unsigned) moduli space eulercharacteristics rather than the true (signed) BPS invariants. As noted in section 2, this isbecause KS naturally counts the euler characteristic moduli spaces without the additionalsign. However, this sign is naturally restored in the change of variables below. The sumis over all ∆ with N = 1. Finally, let us write how the partition function transforms.Writing e ∆ → ( − d (∆) q ∆ , and noting that e ∆ e ∆ ′ = ( − ∆ ◦ ∆ ′ e ∆+∆ ′ we can write Z Q ′ ( q ′ ) = X ∆ Ω Q ′ (∆) q ′ ∆ = X ∆ Ω Q (∆) (1 − ( − ∆ ∗ ◦ ∆ q − ∗ ) ∆ ∗ ◦ ∆ q ∆ , (4 . q ′∗ = ( q ∗ ) − q ′ β j = q β j ( q ∗ ) n βj ∗ q ′ γ k = q γ k . (4 . N = 1. When we consider passing through several walls of the secondkind this will end up involving general decays, because the degeneracies of decay productsat one wall can jump on the next. This gives us an explicit prediction for the degeneraciescomputed from one crystal in terms of the other, which one can check term by term. Wecan however do better, and prove that the BPS degeneracies of the two quivers are indeedrelated by (4.5). The proof is elementary, using the dimer point of view on the crystals.22 .3. Dimers and Wall Crossing Recall that Seiberg duality relating quivers Q and Q ′ has a simple geometric realizationin terms of bipartite graphs on T . This will allow us to give a geometric proof of the wallcrossing formula (4.6) in terms of dimers. But, for this we need to translate the countingof BPS states in the quiver from the language of crystals, which we used so far, to dimers.Consider the bipartite graph dual to the periodic quiver Q T on the torus. The liftof this to the covering space is a bipartite graph in the plane. The path set A gives riseto a canonical perfect matching of the bipartite graph. Consider the paths in A that lieon the surface of the crystal. These correspond to short paths in the planar quiver Q R ,paths containing no loops. These short paths define a set of paths on the bipartite graph,where they pick out a subset of edges crossed by them. The edges in the complement ofthis define a perfect matching m of the bipartite graph [25,14].The finite melting crystal configurations are in one to one correspondence with perfectmatchings m which agree with m outside of a finite domain. Removing an atom in thecrystal is rotating the dimers around the corresponding face in the bipartite graph. Thedifference m − m of the two perfect matchings defines closed level sets on the bipartitegraph. We can use them to define a height function whose value is 0 at infinity, andincreases by ± m correspondingto the two quivers Q and Q ′ .
21 32
21 33 Fig. 11.
The vacuum perfect matchings, m , for the dimer model associated to the quivers Q and Q ′ of local P × P . Z Q ( q ) = X m ∈D Q ( − d (∆( m )) q ∆( m ) (4 . q ∆( m ) is the weight of the dimer configuration, chosen to agree with the weights ofthe corresponding crystal [15,40]. We assign a fixed weight w ( e ) to every dimer e on theplane, in such a way that the product of weights of edges around a face, correspondingto node α of Q R equals q α . The edges have a natural orientation, from a white vertexsay to black, and contribute to the product by w ( e ) or w ( e ) − depending on whetherorientation of the edge agrees or disagrees with the orientation of the cycle. The weightof the dimer configuration m is a product over the weights of the edges in the dimer, w ( m ) = Q e ∈ m w ( e ), normalized by w ( m ) = Q e ∈ m w ( e ), and q ∆( m ) = w ( m ) w ( m )The sign in (4.8) does not come from the weights (i.e. for a general Q it cannot be absorbedinto the weights), but is added in by hand.The duality transformation relating the two quivers Q and Q ′ has a simple geometricinterpretation in terms of the bipartite graph, corresponding to replacing a face of thenode we dualize, with the dual face. This operation lifts to perfect matchings of the corre-sponding bipartite graphs as well – from the perfect matching of one graph, we can obtaina perfect matching of the dual graph. Were the operation one to one, the degeneraciescomputed from one dimer and its dual would have been the same. The operation is in factalmost one to one, except for an ambiguity in the mapping of certain dimers on the faceswe dualized.Let D Q and D Q ′ be the sets of the perfect matchings of the two dual bipartite graphsin the plane (with the suitable asymptotics). Any two perfect matchings in D Q ′ differing by“configurations of type 1” in figure 12 come from the same configuration in D Q . In addition,any two perfect matchings in D Q containing a “configuration of type 2” correspond to thesame dimer in D ′ Q . 24 ype 2 Type 1Type 0
Fig. 12.
How dimer configurations transform under Seiberg Duality.
This means we can use perfect matchings of D Q to enumerate perfect matchings of D Q ′ , provided we know the numbers m ), m ) of configurations of type 1 and type2 in each perfect matching m of D Q . To count the perfect matchings in D ′ Q , we needto sum over all the perfect matchings of D Q , and compensate for configurations over- orunder-counted. Schematically, in terms of counting dimer configurations without signs(later, we will be precise about the weights), this gives X m ∈D Q ′ ( q ′ ) ∆( m ′ ) = X m ∈D Q (1 + q ∗ ) I ( m ) q ∆( m ) (4 . I ( m ) = m ) − m ) . The factor (1 + q ∗ ) I ( m ) = (1 + q ∗ ) m ) (1 + q ∗ ) m ) appears since in D Q there are too many configurations of type 2, and too few of typeconfigurations of type 1. 25otice that (4.6) and (4.9) are of the same form, provided I ( m ), which we will callthe “dimer intersection number”, depends only on the charge ∆ of the dimer configuration m , and not on m itself, and equals the intersection number of ∆ ∗ and ∆, I ( m ) = ∆ ∗ ◦ ∆ , where ∆ ∗ is the node being dualized. In the next subsection, we will show that this isindeed the case. Using this, we will prove that the degeneracies of two Seiberg dual toricquivers indeed satisfy (4.6). To be able to count the BPS states using dimers, both the original and the dual quiverneed to be toric. The conditions for this were studied in [29]. The Seiberg duality needsto preserve the fact that a D3 brane wrapping the T fiber corresponds to all ranks ofthe quiver being one. (This is required by the stringy derivation of the relation of quiversand dimer models [39], as we reviewed earlier.) For this to be the case, as is easy to seefrom (3.6) and charge conservation of T = P α ∆ α , the node we dualize has to have twoincoming and two outgoing arrows. Under this restriction, the most general face of thedimer model that can be dualized is shown in Figure 13.Now consider some arbitrary matching m corresponding to charge ∆, with∆ = X α N α ∆ α . If we denote the face to be dualized by, ∆ ∗ , then it follows that its intersection with ∆ is∆ ∗ ◦ ∆ = X α N α n ∗ α = N + N − N − N − n ∗ (4 . n ∗ α is non-zero only for faces that share an edge with ∗ , andits value, including the sign can be read off from the bipartite graph. We are using herethe conventions of figure 13. The last term is not geometric, n ∗ is the number of framingnodes from 0 to node ∗ . In our setup so far, this is either 0 or 1.We will use induction to find the dimer intersection number I ( m ), and show it equalsthe physical intersection number (4.10). Starting with some arbitrary perfect matching, weconsider the effect of melting an additional node. We will show that the I ( m ) and ∆ ∗ ◦ ∆26hange in the same way. In the end, we will show that the cannonical perfect matching m also satisfies the relation, so in fact any perfect matching does so as well.In the dimer model, melting a node simply corresponds to exchanging occupied andunoccupied bonds along the perimeter of that face. Further, from the ingoing and outgoingarrows at face *, a bond on edge a corresponds to face 1 being melted and bond c corre-sponds to 3 melted, while b corresponds to 2 unmelted, and d corresponds to 4 unmelted.Now we observe that removing a bond from the perimeter of ∗ always changes the dimerintersection number by +1, since it either removes a type 2 configuration or adds a type 1configuration. Thus we find that melting faces 1 or 3 both change the dimer intersectionnumber by − n ∗ = 0 then there are no “removable” ∗ faces in theinitial dimer configuration so no type 2 configurations appear. Since the vacuum configu-ration can be seen as the complement of those bonds that intersect arrows in the planarquiver, there are also no type 1 configurations. Such a type 1 configuration would haveboth incoming and outgoing arrows present, but such a configuration cannot appear onthe surface of the crystal. One way to understand this is to note that the vacuum dimerconfiguration breaks the rotational and translational symmetry of the bipartite graph byspecifying the tip of the crystal. For any face in the bipartite graph (which must corre-spond to an atom on the surface of the crystal), the direction toward the tip of the crystalis a preferred direction, and the dimer model reflects this preference. However, a type 1configuration has symmetric arrows with no preferred direction. Thus, it cannot exist inthe vacuum configuration. If n ∗ = 1 then there is exactly one “removable” ∗ face in m , which corresponds to a type 2 configuration, giving a dimer intersection number of -1.Combining these results, we find, I ( m ) = − n ∗ + N + N − N − N We can also see this by breaking the planar dimer into a tiling of T dimers, so that thevacuum configuration corresponds to a set of T dimer configurations. These T dimer matchingsare in one-to-one correspondence with points on the toric diagram [29]. It has been conjectured[14] that only matchings corresponding to external points on the toric diagram appear along thesurfaces of m . A general T matching containing a type 1 or type 2 configuration will always bean internal point in the toric diagram, and thus can only appear at the apex of the crystal.
27e will now tie this all together, and show that, explicitly mapping the dimer con-figurations of D Q to configurations in D ′ Q together with their weights , we reproduce thechange of degeneracies (4.6), together with the change of basis (4.7). If we denote thedimer weight for configuration m by w ( m ) and the partition function variables by q ∆ m ,then by definition, q ∆( m ) = w ( m ) w ( m )Now we must decide how our weights transform under the duality. There is a largeredundancy in the weight assignments, since the partition function (normalized by theweight of the canonical perfect matching) does not depend on the weights of the individualedges, but only on gauge invariant information, the products of weights around closedloops. A convenient choice (Figure 13) is one which does not change the weighting ofType 0 configurations, so that the weighting of the vacuum, w ( m ) remains the same. Weaccomplish this by flipping the edge weights across the dualized face and assigning weight1 to the meson edges. If we denote the old variables by { q α } and the new variables by { q ′ α } and take into account the change in orientation of arrows, we find, q ′ = q ( w a w c ) − q ′ = q ( w b w d ) q ′ = q ( w a w c ) − q ′ = q ( w b w d ) q ′∗ = q − ∗ = ( w a w c )( w b w d ) − q ′ γ = q γ (4 . q γ are correspond to nodes with no intersection with node ∗ . Now we can explicitlytransform the dimer configuration, together with its weight, w ( m ) → w ′ ( m ) = w ( m ) ( w b w d + w a w c ) ( w b w d + w a w c ) . We still have leftover arbitrariness in the weights, since nothing depends on w a , w b , w c , w d separately. We can set for example w b w d = 1 , w a w c = q − ∗ , and then we get the correct change of variables for the Seiberg duality correspondingto + sign in (3.6), and going around the singularity clockwise. This is because n ∗ , n ∗ n ∗ , n ∗ negative, and n γ ∗ vanishes. This is also the choice we have beenmaking in this section. Setting w a w c = 1, instead, the change of variables is correct fora Seiberg duality with a − sign in (3.6). The two choices are related by full monodromyaround Z (∆ ∗ ) = 0, which is just the change of variables at hand. Proceeding with the +sign choice, (4.11) is the change of basis in (4.7). In addition, the contribution of dimerconfiguration m ∈ D Q to D Q ′ is obtained by replacing and w ( m ) → w ′ ( m ) = w ( m )(1 + q − ∗ ) − . To find the contribution to the partition function, we still need to divide by the weight ofthe vacuum configuration, so w ( m ) /w ( m ) = q ∆( m ) → w ′ ( m ) /w ′ ( m ) = q ∆( m ) (1 + ( q ∗ ) − ) ∆ ∗ ◦ ∆ . Finally, to compute BPS degeneracies from the dimer configurations, we need to reintro-duce the sign twists, which means replacing q ∆ by ( − d (∆) q ∆ , and sum over all matchings m . This precisely reproduces (4.6) and (4.7), since ( − d (∆ − ∆ ∗ ) = − ( − d (∆) ( − (∆ ∗ ◦ ∆) .Thus we have derived the KS wall crossing formula purely from how perfect matchings inthe dimer model transform. * * Fig. 13.
The effect of Seiberg Duality on face ∗ for a general brane tiling. Note that the dimerweights, w i are flipped on the inner square and are 1 on the new legs.
5. Mirror Symmetry and Quivers
Mirror symmetry provides a powerful perspective on the quivers in section 3 and4. The mirror of the manifold Y of section 3 is a toric Calabi-Yau manifold X . Mirrorsymmetry also maps D3 branes wrapping three-cycles in IIB on Y to D0-D2-D4-D6-braneswrapping holomorphic submanifolds in IIA string on X . In the mirror, many aspects of29he quiver construction become more transparent. In particular, computing the quivergauge theory on the branes becomes a question in the topological B-model on X , with theanswers provided by a vast machinery of the derived category of coherent sheaves on X (see [50,51] for excellent reviews). The goal of this section is to explain what are the BPSD-branes counted by the crystals in section 4. For the “small quivers” of section 2, thisis well understood. The D3 branes on compact three cycles of Y , are mirror to D4,D2,and D0 branes wrapping compact submanifolds of X [52,24]. The specific combinations ofbranes involved correspond to collections of spherical sheaves on the surface S as we willreview. The extended quivers of section 4 correspond to adding a D6 brane wrapping allof X [14]. We will show that, which node of the D4-D2-D0 quiver ends up extended, isdetermined by a choice of a suitable bundle on the D6 brane. These are essentially thetilting line bundles of [53,30,31]. Having understood this, the mirror perspective givesa simple interpretation to some of the Seiberg dualities, as turning on B-field on X . Theeffect on the quiver ends up depending on the class of B in H ( X, Z ) /H cmpct ( X, Z ), as wewill explain in the next section. The Calabi-Yau manifold Y given by (3.1) W ( e x , e y ) = w uv = w is mirror to IIA on X , where X is a toric Calabi-Yau threefold. The monomials w i = e m i x + n i y in W satisfy relations Y i w Q ai i = e − t a for some complex constants t a , and integers Q ai , satisfying P i Q ai = 0, since Y is Calabi-Yau. The mirror X is given in terms of coordinates z i , one for each monomial w i , satisfying X i Q ai | z i | = r a , (5 . z i ∼ z i e iθ a Q ai . (5 . The fact that the D-brane in question is a D6 brane was proposed in [14], however the specificchoices of bundles are very important. r a = Re( t a ), and the imaginary part of t a , gets related to the NS-NS B-field on X . For each a , we get a curve class [ C a ] ∈ H ( X, Z ), whose volume is r a . Dual to thisare divisor classes D a , corresponding to 4-cycles with D a ◦ C b = δ ba . The toric divisors D i ,obtained by setting z i = 0, are given in terms of these by D i = X i Q ai D a By Poincare duality, we can think of D a as spanning H ( X, Z ). Among the toric divisorsare compact ones, D S , which restrict to compact surfaces S in H ( X, Z ) ≈ H cmpct ( X, Z ).For simplicity, we will restrict to the case when X is a local del Pezzo surface, i.e. whenthere is only one compact S . Mirror symmetry maps D3 branes on three-spheres ∆ α in Y to a collection of sheaves E α , supported on S , for α = 1 , . . . , r . Physically, E α are D4 branes wrapping S , with somespecific bundles turned on, giving the branes specific D2 and D0 brane charges. (Moreprecisely, we need to include both the D4 branes and the anti-D4 branes.) The specificbundles turned on correspond to E α being an exceptional collection of spherical sheaves.Spherical means that the sheaf cohomology group Ext kX ( E α , E α ) is the same as H k ( S ), soboth sets of D-branes have no massless adjoint-valued excitations beyond the gauge fields.The collection of sheaves is in addition such that, for α and β distinct, Ext k ( E α , E β ) isnon-zero only for k = 1 ,
2, corresponding to chiral bifundamental matter. The net number of chiral minus anti-chiral multiplets is an index n αβ = X k =0 ( − k dim Ext kX ( E α , E β )On Y , this corresponded to the intersection numbers of cycles n αβ = ∆ α ◦ ∆ β . In fact, n αβ also has geometric interpretation on X . A D-brane corresponding to E α has charge∆ α = ch( E α ) p td ( X ) , By E α we will really mean a sheaf i ∗ E α induced on X from a sheaf E α on S by the embedding i : S → X of S in X . Non-zero
Ext , ( E α , E β ) would have corresponded to the presence of ghosts. n αβ to computing intersectionson X , n αβ = Z X ch( E α ) v ch( E β ) td ( X ) , where ω v denotes ( − n ω for any 2 n -form ω .In this case, we have an additional simplification that the sheaves E α on X corre-spond to line bundles V α on S . This implies [24] Ext kX ( E α , E β ) = Ext kS ( V α , V β ) ⊕ Ext − kS ( V α , V β ) . (5 . S are twisted [54] by K − / S . This accounts for the fact that the theory on the brane is naturally twisted if S is curved. The relation to bundles on S simplifies things, since for a holomorphic vectorbundle Ext kS ( V α , V β ) = H k ( S, V − α ⊗ V β ).The superpotential can also be computed byfairly elementary means, at least if it is cubic [24,37]. For the more general case, see [55].We can then write (see for example [50])∆ α = ch ( V α ) s td ( S ) td ( N ) = ch ( V α ) (cid:16) e ( S )24 (cid:17) . (5 . N is the normal bundle to S in X . We are implicitly using mirror symmetry andPoincare Duality to identify a Chern class on X with a cycle on Y .When we add a D6 brane wrapping all of X , the quiver gets extended by one nodecorresponding to E , a sheaf supported on all of X . The charges of the brane are∆ = ch ( E ) p td ( X ) = ch ( E ) (cid:16) c ( X )24 (cid:17) We want to choose E so that Ext k ( E , E ) is non-vanishing only for k = 0 ,
3, and extend-ing the quiver by this node does not introduce exotic matter in the quiver, which means
Ext k ( E , E α ) is non-vanishing only for k = 1 ,
2. For the quivers in section 4 where n α = 1for only one node α and zero for all the others, there is a natural construction of E . There As explained in [54][50] even though the sheaf E α on X comes from a vector bundle E α on S (see footnote 17), it does not correspond to a D-brane on S with a vector bundle E α . Itcorresponds to a D-brane with a vector bundle V α , where V α = E α ⊗ K − S , where K S is thecanonical line bundle of S .
32s a “duality” that pairs up sheaves E α supported on S , with the dual line bundles L β on X [31,53,30], such that χ ( E α , L β ) = δ βα . Thus, the requisite extension corresponds simplyto choosing E = L α . Fig. 14.
The D6 brane corresponds to a semi-infinite line, ∆ , in the W-plane. In the mirrorthis corresponds to a dual sheaf, E = L α . P × P example Consider local P × P . In this case, Q t = (1 , , , , − Q s = (0 , , , , − | z | + | z | = 2 | z | + r t , | z | + | z | = 2 | z | + r s , and the corresponding curve classes C t and C s generate H ( X ). The two dual divisorclasses, D t and D s , with the property that C t ◦ D t = 1 = C s ◦ D s , and C t ◦ D s = 0 = C s ◦ D t ,can be taken to correspond to divisors D t = D and D s = D on the figure. Fig. 15.
The toric base of local P × P , with the non-compact divisors labeled by D s and D t . There is one compact divisor D S = D with D S = − D s − D t D S restricts to the surface S = P × P .The four three-cycles ∆ α on Y map to four exceptional sheaves supported on X , whosecharges span the compact homology of X . The D4, D2 and the D0 branes correspond toa collection of exceptional sheaves E = O S ( − , − , E = O S ( − , − − , E = O S ( − , − − , E = O S ( − , − − , supported on S . To compute the charges, we use the fact that a D4-brane with a O S ( m, n )line bundle has Chern class D S e nD t + mD s − K S (1 + c ( S )24 ) , which equals ( D S + ( m + 1) C t + ( n + 1) C S + ( m + 1)( n + 1)pt)(1 + c ( S )24 ) . This gives the charges of fractional D-branes are∆ = ( D S − C t − C s + pt) (cid:16) c ( S )24 (cid:17) ∆ = ( − D S + C t ) (cid:16) c ( S )24 (cid:17) , ∆ = ( − D S + C s ) (cid:16) c ( S )24 (cid:17) , ∆ = D S (cid:16) c ( S )24 (cid:17) . (5 . n = n = n = n = 2 , n = 4 , in agreement with mirror symmetry and section 3.Now we add a D6 brane corresponding to the simplest choice, E = O X [ − , We are using that the canonical class K S of the surface S equals the class of the divisor D S restricting to it in a Calabi-Yau, K S = D S and D S · D s = C t , D S · D t = C s , D t · D t · D t = D s · D s · D s = , D t · D t · D s = D t · D s · D s = − . Moreover D S c ( S ) is just the Euler characteristicof S , which is 4 in this case.
34 trivial line bundle on the Calabi-Yau, with charge∆ = − X (1 + 124 c ( X )) . Clearly,
Ext kX ( E , E ) = 0, except for k = 0 ,
3. We can compute the spectrum of bifunda-mentals using Ext kX ( O X , E α ) = H k ( S, E α ) ,Ext kX ( E α , E β ) = Ext k − p + qX ( E α [ p ] , E β [ q ]) , and dim H ( P , O ( m )) = m + 1, dim H ( P , O ( − m )) = − m − , we find that Ext X ( E , E ) = C , while the rest Ext , ’s vanish in this sector, so there is precisely one chiral bifundamentalmultiplet between E and E . n = 1 .
6. Monodromy and B -field One particularly simple example of monodromy in the moduli space corresponds tochanging B by integer values. Shifts of the NS-NS B-field by a two form D ∈ H ( X, Z ) B → B + D, D ∈ H ( X, Z ) (6 . Recall, see footnote 17, that the sheaf E α is really i ∗ E α , inherited from S by the embedding i : S → X . Therefore, what we call Ext kX ( O X , E α ) is Ext kX ( O X , i ∗ E α ) = Ext kS ( i ∗ O X , E α ) = Ext kS ( O S , E α ) = H k ( S, E α ). We thank E. Sharpe for an explanation of this point. The intersection numbers in the D4-D2-D0 quiver follow similarly. Using that
Ext kS ( V α , V β ) = H k ( S, V ∗ α ⊗ V β ), since V α ’s are bundles on S = P × P . This implies that, for exam-ple Ext X ( E , E ) = H ( S, O (1 , C , and Ext X ( E , E ) = H ( S, O (1 , C = Ext X ( E , E ), as claimed.
35n a B-field is the same as turning on lower dimensional brane charges. The inducedcharges are given by ∆ → ∆ e D , (6 . ∈ H ∗ ( X, Z ), as shifting the B -field is the same as shiftingthe field strength F on the brane by D . The state thus does not come back to itself. Sincewe come back to the same point in the moduli space, we expect the partition function tochange as Z ( q ) = X ∆ Ω(∆) q ∆ → Z ′ ( q ) = X ∆ Ω(∆ e D ) q ∆ (6 . e D at the samepoint in the moduli space. Z and Z ′ are of course equivalent, up to a change of variables.This looping around the moduli space can be represented in terms of crossing a sequenceof walls of the “second kind” and of Seiberg dualities.In the present context, there is a subtlety in the realization of (6.3), due to thefact that the Calabi-Yau is non-compact, and that the quiver does not describe all thepossible D-branes on X . Start with an extended quiver describing bound states of a D6brane, say corresponding to O X , with compact D4-D2-D0 branes. Shifting B-field as in(6.1) corresponds to adding non-compact D4 brane charge to the D6 brane, if the divisorcorresponding to D is non-compact. In this case, we get (6.3) only after summing thecontributions of different quivers in which the ∆ node corresponds to O X ( D ), for different D ∈ H ( X, Z ). If however, we consider shifts by two forms with compact support, D ∈ H cmpct ( X, Z ) , then this is a symmetry of the fixed quiver gauge theory as well. This is because theD6-D4-D2-D0 quiver describes all the BPS states corresponding to a D6 brane bound tocompact D-branes on X . Thus, the different D6-D4-D2-D0 quiver gauge theories we canget on X , at the same point in the moduli space, are classified by D ∈ H ( X, Z ) /H cmpct ( X, Z ) . As an aside, note that if we consider only the D4-D2-D0 quiver, as in section 3, having nonon-compact charge to begin with, any shift of B-field (6.1) is a symmetry.36 .1. The P × P example Consider the effect of changing the B-field on the quiver Q in Figure 5. B → B + n t D t + n s D s , n s , n t ∈ Z . Since H cmpct ( X, Z ) is generated by the divisor D S D S = − D t − D s . shifts by ( n s , n t ) = (2 n, n ) for n an integer, should be a symmetry both of the quiver andof the spectrum. Thus, the different quiver gauge theories one can get by shifting B-fieldare classified by ( n t , n s ) modulo shifts by (2 , B -field by ( n t , n s ) = (1 , Y ,shifting B by D t corresponds to shifting z t , z s → z t e πi , z s . This deforms the 3-cycles ∆ α as in the Figure 16 to ∆ ′ α , related to the good basis westarted out with by ∆ → ∆ ′ = − ∆ ∆ → ∆ ′ = ∆ + 2∆ ∆ → ∆ ′ = − ∆ ∆ → ∆ ′ = ∆ + 2∆ (6 . and ∆ is necessary for their orientations to agree with ∆ ′ and ∆ ′ . W Fig. 16.
The W-Plane after adding one unit of B-field, (1,0), for local P × P . The deformedcycles { ∆ i e D } are shown in black. while the original cycles are shown in blue. X , ∆ are sheaves on X . The shift of B-field corresponds to tensoringwith a line bundle of first chern class D t . The chern classes change by∆ → ∆ ′ = ∆ e D t , where above describes how components of ∆ change in a fixed basis for H ∗ ( X ). To beginwith, for example, ∆ , and ∆ correspond to O S ( − , −
2) and O S ( − , − −
1] respectively.After we change the B-field, they pick up charge, and get mapped to O S ( − , −
1) and O D ( − , − − − ∆ and − ∆ , respectively, in agreement with(6.4). Similarly, ∆ , corresponding to O S ( − , − −
1] gets mapped to O S ( − , − ′ equals∆ ′ = − ( D S + C t − C s − pt)(1 − c (S)24 ) = ∆ + 2∆ , in agreement with (6.4).In the quiver, the shift of the B -field by D t corresponds to a sequence of two Seibergdualities, where we dualize first the node ∆ , (as we did in the last section) and then thenode ∆ (Figure 17). It is easy to see from this that the D4-D2-D0 quiver is invariantunder the shift of the B-field, although the D-branes on the nodes get replaced by a linearcombinations of the ones that we had started out with, given by (6.4). == Fig. 17.
Seiberg Duality corresponding to a shift in the B-field (1,0) for local P × P . Theunframed quiver on top is invariant up to a permutation of the nodes, while the framed quiverdevelops new framing arrows. Now consider the theory with the D6 brane, by including the node ∆ , which has thebundle O X [ − E ′ = O X ( D t )[ − D t . Since no other nodes carrythis charge, E ′ becomes the new framing node, with charge ∆ ′ induced by the B-field∆ = − X → ∆ ′ = − X e D t The extended quiver, with the ∆ node included also transforms by two Seiberg dualities,Figure 17. The quiver in this case is not the same as before, since ∆ α have differentintersection numbers with ∆ ′ than with ∆ . In particular n ′ = ∆ ′ ◦ ∆ = 2 , n ′ = ∆ ◦ ∆ ′ = 1 . This can also be verified directly from sheaf cohomology, showing that for example
Ext ( E ′ , E ) = H S ( O ( − , − W = X i,j =1 , ǫ ij Tr p A i q j + X i,j,a,b =1 , ǫ ij ǫ ab (cid:0) Tr A i B a D jb + Tr ˜ A i ˜ B a D jb (cid:1) . In the previous section, we showed that dualizing a node ∆ ∗ in the quiver is realized asa geometric transition in the dimer model counting the BPS degeneracies, where the facecorresponding to ∆ ∗ is dualized. We showed that this implies the [1] wall crossing formula A (1) Q ′ = A − − ∆ ∗ A (1) Q A − ∆ ∗ , (6 . A (1) Q ′ = A − − ∆ A − − ∆ A (1) Q A − ∆ A − ∆ . In the dimer model, this is a change of the boundary conditions at infinity, replacing thetop of the cone by an edge two sites long (see Figure 18 and 19). It is easy to see that thisis consistent with the crystal one would have derived from the quiver Q ′ by localization.
21 32
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21 333 The relation of Seiberg duality, wall crossing and crystals was first noted in [12]. That workwas one inspiration for this research. ig. 18. The vacuum dimer configuration for P × P grows an edge corresponding to a shift inthe B-field by (1,0). Fig. 19.
The apex of the P × P crystal grows an edge corresponding to a shift in the B-fieldby (1,0). Repeating this m times, shifting B to B + mD t we end up with a quiver Q m with n = m + 1, and n = m . This corresponds to growing a ridge in the crystal, m + 1atoms long as in [26,12]. In the next section, we will give a physical interpretation to thisobservation.More generally, all the possible inequivalent quivers one can obtain in this way cor-respond to shifts of B-field by the inequivalent choices in H ( X, Z ) /H cmpct ( X, Z ). Forexample, it is easy to show (either by Seiberg duality, or direct computation) that shift-ing the B-field by − D t , replaces the node E by O X ( − D t )[ − n = 1 is nonzero for arrows beginning or ending on∆ . Similarly, replacing E by O X ( − D s )[ − n = 1. Finally, we get only n = 1, by taking E = O X ( − D t − D s )[ −
21 34 032 41 0 32 41 0 (a) (b) (c) ig. 20. Framings for local P × P , corresponding to negative shifts of the B-field that replace O X [ −
1] by (a) O X ( − D t )[1], (b) O X ( − D s )[1], and (c) O X ( − D t − D s )[1]. Now consider shifts of the B-field by D = − D S ∈ H cmpct ( X, Z ), B → B − D S . This is a symmetry of the quiver. It can be implemented by changing the B-field by D t , D s , D t and D s , corresponding to a sequence of four Seiberg dualities on the quiver,which in the end leaves the quiver invariant, except that the charges of the nodes of thequiver change. This can also be seen from the perspective of the Kontsevich-Soibelmanalgebra. This corresponds to conjugation of A (1) Q by A − ∆ A − ∆ , A − ∆ ′ A − ∆ ′ , A − ∆ ′′ A − ∆ ′′ ,and A − ∆ ′′′ A − ∆ ′′′ , respectively, where the four ∆ ’s, for example, correspond to the four∆ nodes in the four quivers we get along the way. This can be rewritten as conjugationby M = M ∆ M ∆ ′ M ∆ ′′ M ∆ ′′′ , (6 . M ∆ = A − ∆ A ∆ .M ∆ is a monodromy operator, implementing the shift of the charge M ∆ : e ∆ ′ → ( − ∆ · ∆ ′ e (∆ ′ − (∆ ′ · ∆)∆) . If we denote by M : e ∆ → e f (∆) , the net effect on the partition function is that M : X ∆ Ω Q (∆) q ∆ → X ∆ Ω Q (∆) q f (∆) = X ∆ Ω Q ( f − (∆)) q ∆ , where we changed variables in the last line. It can be shown that f − (∆) is exactly whatis needed for this to correspond to turning on the B-field, − D S , namely, f (∆) = ∆ e D S . The order in which we cross the walls matters, only in so much that the description issimplest for one particular order, and for that ordering we can describe this by Seiberg duality.Otherwise, this does not have a simple interpretation in the gauge theory. The end result, however,is independent of the order. Since M − e ∆ ′ M ∆ = (1 − e − ∆ ) − ∆ · ∆ ′ (1 − e ∆ ) ∆ · ∆ ′ e ∆ ′ . . Geometry of the D6 brane bound states In this section, we begin by considering a fixed state, a D6 brane bound to D4 branes,D2 branes and D0 branes of charge ∆, and ask how its degeneracies change as we shift theB-field by D ∈ H ( X, Z ). Turning on a B-field is the same as turning on lower dimensionalbrane charges on the D6 brane. The induced charges are given by∆ → ∆ e D , (7 . B -field is the same as shifting the field strength F on the brane by D . Whenwe come back to the same point in the moduli space, the states have been re-shuffled by(7.1). So the degeneracies of any one state will change (at the same time, the spectrum asa whole is invariant, as we emphasized in the previous section). The degeneracies, and howthey change, can be found from the quivers and crystals describing the brane. A state ofarbitrary charge ∆, in general, corresponds to a complicated configuration of the meltingcrystal C .It turns out that the vacuum of the crystal C , and certain subset of its states have abeautifully simple geometric description, closely related to the geometry of the Calabi-Yau.We will spend some time explaining this, and then use this result to show that, in the limitof the large B-field, D → ∞ counting of quiver degeneracies reduces to the Donaldson-Thomas theory as formulatedin [16,17,18], in terms of a counting ideal sheaves on X . The latter was discovered bytrying to give a physical interpretation to the combinatorics of the topological vertex [56]and Gromov-Witten theory. In this way, we will be able to derive the famous Donaldson-Thomas/Gromov-Witten correspondence directly from considering quiver representationsin this limit. The proposal that the DT/GW correspondence holds in the large B-field limitwas put forward in [20] using split attractors and verified in [13] for Calabi-Yau manifoldswithout compact 4-cycles, using M-theory. To begin with, consider the D6 brane itself, the sheaf O X [ − − X = ∆ . C is a cone in the lattice Λ = Z ≥ , whose geometry is closely related tothe geometry of the Calabi-Yau. Namely, consider the intersection of C with a sublatticeΛ ⊂ Λ corresponding to points of color α , where α is the framing node. The subset oflattice points C = C ∩ Λ correspond to holomorphic functions on X .This can be seen as follows [38,49,44]. Recall that the cone C is generated by the T weights of the paths in A , starting at the framing node. The subcone C corresponds topaths ending on node α . Because n α = 1, such paths can be viewed as starting andending at the node of color α . These correspond to single trace operators Tr O in thequiver, where we take the ranks to infinity, so that there are no relations between thetraces. The chiral operators O are generated by a set of monomials M i corresponding tothe shortest loops in the quiver, beginning and ending on node α . The monomials M i allcommute, since two paths in the quiver of the same R -charge and the same endpoints areequivalent in the path algebra. Thus, we can simply write O as O = M n M n . . . M n k k .Since the order is irrelevant, the space of operators Tr O is the same as considering allgauge invariant chiral operators in the abelian quiver theory corresponding to a single D0brane on X , described in the quiver by taking all ranks to be 1. Since the moduli space ofa D0 brane is X , the later is the space of holomorphic functions on X .The space of holomorphic functions on X is also a lattice, closely related to thegeometry of X . As we reviewed in section 5, X comes with a set of coordinates z i , satisfying (5.1) X i Q ai | z i | = r a , (7 . X as a fibration over the toric baseobtained by forgetting the phases of z i and using | z i | as coordinates. Consider the integralpoints in the base, where N i = | z i | , N i ∈ Z ≥ . For each such point, we get a monomial in z i Y i z N i i which is a function on X , since X i Q ai N i = 0 . (7 . r a = 0 in (5.1), as we are starting out with singular X . This is because thequiver we started out with has just one framing node, and so C was a cone with an apex43t the origin. Since the Calabi-Yau is three dimensional, the space of all such monomialsis a three dimensional lattice, the singular cone C .Now consider what happens to the D6 brane as we increase the B -field B → B + D. Increasing the B-field is the same as turning on flux on the D6 brane, tensoring O X [ − D . The D6 brane becomes O X ( D )[ − − X e D . Here, we assume D = X a n a D a , n a ≥ . D a generate the Kahler cone. We claim that the state ∆ corresponds to the deformedcrystal C ( D ) whose lattice sites C ( D ) = C ( D ) ∩ Λ are holomorphic sections of an O ( D )bundle over X instead. This corresponding to solving X i Q ai N i = n a . (7 . C is a discretized version of the base of Calabi-Yau, modifying C as in (7.5) modifiesthe moduli. Fig. 21.
Integral points in the toric base of local P correspond to holomorphic sections of O ( D ). The cases D ∈ H cmpct ( X, Z ) and D ∈ H ( X, Z ) /H cmpct ( X, Z ), should be discussedseparately, since they differ in character. When D = D ∈ H ( X, Z ) /H cmpct ( X, Z ) , (7 . O X ( D )[ − − X ) e D = ∆ C ( D ) ⊂ Λ associated with thisnew quiver. We are growing the edges of the crystal, corresponding to increasing thelengths of curves, but no faces open up. In other words, shifts of the B-field by non-compact divisors (7.6) correspond to purely non-normalizable deformations of the crystaland the Calabi-Yau. Next, consider shifting the B-field by D , such that D − D ∈ H cmpct ( X, Z ) , (7 . D − D = 0 corresponds to shifting the normalizable moduli of the Calabi-Yau,opening up faces in the toric base, and deforming C without changing its asymptotics.This adds compact D4 brane charge to the D6 brane, and this can always be described inthe quiver we had started with. To verify that the configuration of the crystal correspondsto O X ( D )[ − , we need to express its Chern character∆ = − X e D in terms of the Chern characters of the nodes of the quiver∆ = ∆ + X α N α ∆ α , where ∆ corresponds to the D6 brane node,∆ = − X e D . Note that ranks N α depend both on D and D . The set of N α ’s we obtain in this waycorresponds to how many atoms of color α we need to remove to describe O X ( D )[ − Consider the local P × P . In this case, Q = (1 , , , , − Q = (0 , , , , −
2) sothe “pure” D6 brane O X [ −
1] on X corresponds to the set of points N + N = 2 N , N + N = 2 N ,
45n Λ . The cone direction here is parameterized by N . At fixed N , we get a square with(2 N + 1) × (2 N + 1) integral points. It is easy to see that this agrees with we get byconsidering either of the two quivers in section 3, and a subset of points corresponding tonode α = 3, in this case. Of course, the finer structure of the crystal C is different in thetwo cases, but the geometry of C is the same. Fig. 22
The crystal for local P × P . Take now O X ( mD t )[ − Q m in the previoussection. The corresponding crystal C gets deformed to C ( mD t ) by replacing the apex ofthe cone with an edge m + 1 sites long. The crystal sites of C ( mD t ) are given by N + N = 2 N + m, N + N = 2 N , (7 . C t length m + 1. Add to this − n units of thecompact D4 brane charge, by taking D = ( m + 2 n ) D t + 2 nD s instead. From what we hadsaid above, we expect a face of ( m + 2 n + 1) × (2 n + 1) nodes of color ∆ to open up atthe appex N + N = 2 N + m + 2 n, N + N = 2 N + 2 n. (7 .
111 111111 111
Fig. 23.
The crystal for local P × P with compact B-field. Now we will show that this precisely describes the state O X ( D )[ − − X (1 + c ( X )24 ) e ( m +2 n ) D t +2 nD s . in terms of the crystal associated with the quiver Q m . The quiver has the D6 brane node O X ( mD t )[ −
1] of charge ∆ = − X (1 + c ( X )24 ) e mD t . The difference of these charges∆ − ∆ = nD + ( n + m ) nC s + n C t − ( 43 n + 12 mn − n )pt , (7 . C corresponding to the quiver Q m is n − X i =0 (2 i + 1 + m )(2 i + 1)∆ + (2 i + 1)(2 i + 2 + m )∆ +(2 i + 2)( m + 2 i + 1)∆ + (2 i + 2)(2 i + 2 + m )∆ . (7 . = ( D S − C t − C s + pt)(1 + 124 c ( S ))∆ = ( − D S + C s )(1 + 124 c ( S ))∆ = ( − D S + C t )(1 + 124 c ( S ))∆ = D S (1 + 124 c ( S )) . Adding up (7.11) and using that, for a divisor D S which restricts to a surface S in theCalabi-Yau c ( X ) D S = ( c ( S ) − c ( S ) ) D S (7 . c ( S ) D S = χ ( S )pt , c ( S ) D = (12 − χ ( S ))pt , we recover (7.10). Above χ ( S ) is the euler characteristic of S . Here, S = P × P , and χ ( S ) = 4.While we did the explicit computation for the quiver Q m , we could have just as wellused the dual quiver Q ′ m , obtained by dualizing node 2. This changes the microscopics ofthe crystal, so C ( D ) changes, however the shape of the crystal and C ( D ) stays the same.This had better be the case, as C does not know about the full quiver, but only about asubset of its nodes that are untouched by the quiver mutation. Explicitly, the charges ofthe nodes of Q m and Q ′ m are related by∆ ′ = ∆ + 2∆ ∆ ′ = − ∆ ∆ ′ = ∆ ∆ ′ = ∆ while ∆ = ∆ ′ . Using this to rewrite (7.11), we get n − X i =0 (2 i + 1 + m )(2 i + 1)∆ ′ + (2 i + 2)(2 i + 1 + m )∆ ′ +(2 i + 2)(2 i + 2 + m )∆ ′ + (2 i + 3)(2 i + 2 + m )∆ ′ . It is easy to see that this counts the nodes in the cut-off top of the crystal based on thequiver Q ′ m . 48n the next subsection, we will consider meltings of C ( D ). This counts a subset ofstates of the original crystal C , which naturally should correspond to bound states ofD6 brane with an O X ( D ) bundle on it, with lower dimensional branes. As an aside,note that, while this problem can be phrased in terms of counting a subset of statesof the original quiver, we can try to define it more directly as well, in terms of a D4-D2-D0 quiver, with a framing node corresponding to O X ( D )[ − Ext , ( E α , E β ), which correspond to ghosts, not chiralmultiplets [36,57]. Take a D6 brane − X , with any number of i. D4 branes wrapping surfaces S in H ( X, Z ), ii. D2 branes wrapping curves C in H ( X, Z ) and iii. D0 branes where werequire D S , C to be positive , and the number of D0 branes to be non-negative.Changing the B-field by D (7.1), gives this state a simple geometric interpretation interms of removing i. faces corresponding to toric divisors D S , restricting to S , ii. edgescorresponding to C and iii. vertices of C ( D ). This holds for any D large enough that the“edges”, “faces” and “vertices” of C ( D ) have an unambiguous meaning., and the positivityconstraint comes from the fact that we can only remove, but not add sites along the edges,faces and vertices. Fig. 24
Melting a face (a) corresponds to adding compact D4 brane charge, melting an edge (b)adds D2 brane charge, and melting a node in C ( D ) adds D0 brane charge. As we discussed above, the D6 brane O X ( D )[ −
1] in the background of B-field D isdescribed by the crystal C ( D ) such that C ( D ) are integral points in the Calabi-Yau withKahler class D . Adding to this a D4 brane on a divisor D S corresponds to changing the More precisely, we require C to be in the Mori Cone of X, and we require S to be a veryample divisor so that S · D > To be completely clear, arbitrary meltings of the crystal correspond to a larger set of chargesthat do not obey this, but the other states do not have such an intuitive description. O X ( D − D S )[ − C ( D ) to C ( D − D S ). Thissimply changes the Kahler class of the Calabi-Yau base by D S . From the perspective ofthe crystal C ( D ) , C ( D − D S ) is obtained by removing sites along the face corresponding to D S . If D S is a compact divisor, we remove a finite number of sites. Consistency requiresthat the charge carried by these be the charge of the D4 brane on D S in this background.In other words, if S is the surface the divisor D S restricts to, the charge should be that of O S ( D ), obtained from the pure D4 brane O S on D S by shifting the B-field by D .Explicitly, the crystals C ( D ) and C ( D − D S ) carry the charges of the D6 branescorresponding to O X ( D )[ −
1] and O X ( D − D S )[ − , − X e D (1 + 124 c ( X ))and − X e D − D S (1 + 124 c ( X )) . Thus, the sites that we remove in going from one crystal to the other must carry thedifference of the charges. This is D S (1 − D S + 13! D S + 124 c ( X )) e D = D S (1 + 124 c ( S )) e D − K S which equals the charge of a D4 brane on S in the background B-field O S ( D ). We usedhere the fact that one can think of the sheaf supported on a surface S as a bundle on S twisted by − / K S of the surface, and that K S = D S , on aCalabi-Yau.Similarly, a D2 brane wrapping a curve C in X corresponding to O C ( D ) is obtainedby removing an edge in the crystal along C . To see this, suppose that C lies on theintersection of two divisors C = D S D T . Then, analogously to what we had done for theD6 branes and the D4 branes, we can express the D2 brane as a difference of the D4 braneson D S when we change the background from D to D − D T . The corresponding chargesare D S e D − K S (1 + 124 c ( S )) (7 . D S e D − D T − K S (1 + 124 c ( S )) . (7 . C (cid:16) −
12 ( D T + D S ) (cid:17) e D = C (cid:0) c ( C ) (cid:1) e D , O C ( D ) supported on C in the background B-field D .We could also remove n edges along C in the crystal, but then we have the choiceof melting additional edges along the face or down the side of the crystal. The differencein charges corresponds exactly to n O C ( D ) + k [ pt ] for some k determined by the crystalstructure, where the additional D0 charge arises from the fact that in a general crystal,the lengths of multiple melted edges will not all be the same. In our local P × P example, consider X with the B-field shifted by D = ( m +2 n ) D t + 2 nD s . Removing a face worth of sites from C (and the relevant sites from itsrefinement C ), (2 n + 1 + m )(2 n + 1)∆ + (2 n + 1)(2 n + 2 + m )∆ +(2 n + 2)( m + 2 n + 1)∆ + (2 n + 2)(2 n + 2 + m )∆ (7 . D S + ( m + 2 n + 1) C s + (2 n + 1) C t + (cid:16) (2 n + 1 + m )(2 n + 1) + 16 (cid:17) ptwhich is the same as D S (1 + 124 c ( S )) e D − K S , the chern character of O S ( D ), where S = P × P , and K S = − D s − D t = D S .Similarly, removing an edge, corresponding to C t say, we remove sites of net charge(2 n + 1 + m )∆ + (2 n + 2 + m )∆ + ( m + 2 n + 1)∆ + (2 n + 2 + m )∆ Adding this up, the net charge is C t + (2 n + 1 + m )pt = C t (1 + 12 c ( C t )) e D corresponding to O C t ( D ).To summarize, the states that have a simple geometric description are those we obtainfrom “pure” D6, D4, D2 (and of course D0 branes) O X , O S and O C by turning on a B-field B → B + D, in other words O X ( D ), O S ( D ) and O C ( D ).51 .5. The large B -field limit and DT/GW correspondence Now consider C ( D ) in the limit of large D → ∞ , or more precisely n a → ∞ (7 . D = X a n a D a , n a ≥ . (7 . X , where the lengths of all edges in C ( D ) go to infinity.When we consider meltings of this crystal, we need to specify what we keep fixed inthis limit. Without doing anything, working in terms of the fixed basis of H ∗ ( X, Z ),corresponding to X , D a , C a and pt the states corresponding to melting of C ( D ) would allhave infinite charges. Moreover, because the edges of the crystal are infinitely long, thecounting problem itself is not well defined. Fig. 25
Melting the local P × P crystal with a large compact B-field turned on. To remedy this, consider changing the basis to X ′ = X e D , D ′ a = D a e D , C ′ a = C a e D , and counting states whose charges in terms X ′ , D ′ a , C ′ a and pt are finite, i.e. states of theform X ′ + k a D ′ a + ℓ a C ′ a + m pt (7 . k, ℓ and m are all finite. In terms of the quiver we started with, this is implementedby taking a limit in the weight space, where the weights of all but one node go to zero, q α → , α = 1 , . . . k − . k Y α =1 q α = q. Since the sum of the charges of the nodes is one unit of D0 brane charge, states weightedby q m carry m units of D0 brane charge.Taking the limits (7.16) and (7.19) together, makes the state counting well defined. Inthe limit, the only local excitations of the crystal C ( D ) that survive are those where equalnumbers of nodes of all colors are excited – corresponding to removing pure D0 branes,weighted by q m . One can think of this as a kind of a phase transition in the crystal C ( D )where excitations of individual nodes freeze out, and one can only remove atoms in groupsweighted by q . In this limit, C ( D ) becomes C ( D ). Moreover, the crystal can melt onlyfrom the vertices, and near each vertex [16], the crystal C ( D ) looks like a copy of the C crystal in [15]. In C we only have D0 branes to begin with, so the crystals C and C forthis coincide. Note that, while we were originally counting signed partitions, with signs( − d (∆) from (4.1), in the large D limit the signs become simply ( − m = ( − m , up toan overall sign, since we are restricting to configurations where all ranks are equal.But, in addition, some large excitations survive as well. There are edge excitationswhich carry finite number of D2 brane charge ℓ a in (7.18), and also excitations along facescarrying D4 brane charges k a . We have shown in the previous section that, even at finite D , excitations along an edge P a ℓ a C a in H ( X, Z ) carry charge X a ℓ a C ′ a = ( X a ℓ a C a ) e D . Similarly, removing the face in C ( D ) in the class P k D a corresponds to adding charge X a k a D ′ a = ( X a k a D a ) e D to the D6 brane − X ′ . We also showed that adding the D4 brane is the same as shifting D by P a k a D a . This clearly does not affect the degeneracies of D0-D2-D6 branes in the53imit where we take D to infinity – it only shifts what we mean by D , in agreement with[59][20].The relation of topological string amplitudes on X with certain melting crystals wasobserved in [15]. The combinatorics of the topological vertex [56] and the A-model topo-logical string on X is the same [15] as the combinatorics of C crystals glued togetherover the edges between C patches in X . In [16] a physical explanation of this was pro-posed, by relating the crystals to D6 brane bound states. Starting with the crystal C ( D )corresponding to integral points in the base of a toric Calabi-Yau X , [16] showed that inthe limit where one takes the Kahler class D of X to infinity, the crystal degenerates to C crystals glued together over long legs, exactly as in [15]. On the other hand, it wasshown that in the same limit, the crystal counts bound states of a D6 brane on X , withD2 and D0 branes (in the language of sheaves, these are ideal sheaves on X ). The countin [16] was formulated in terms of the maximally supersymmetric SYM on X , topologi-cally twisted, and non-commutative. The conjecture of [16] relating the topological stringon X to counting bound states of a single D6 brane on X with D0 and D2 branes, forany Calabi-Yau X , is known as the Gromov-Witten/Donaldson-Thomas correspondence[17,18]. Recently, it was proven for toric threefolds by [60].We have shown that the bound states of a D6 brane on X , with D4, D2 and D0 branesdescribed by a quiver Q and in the background of B-field D , are counted by crystals C ( D ),at any D . The crystal roughly corresponds to integral points in the base of the Calabi-Yau with Kahler class D , though the precise microscopic details depend on Q . In the limitof infinite D , the microscopic structure is lost, and C ( D ) becomes the same as the crystal C ( D ) – and hence the same as the crystal in [15][16]. Thus, the count of the D6 branebound states from the six dimensional perspective of [16] and the 0 + 1 dimensional quiverquantum mechanics of D6 branes bound to D4-D2 and D0 branes, agree – but only in thislimit. We have thus re-derived the Gromov-Witten/Donaldson-Thomas correspondenceof [16,17,18] from the quiver perspective.More than that, we provided an answer to the question raised in [61]: what is thecrystal C ( D ) in [16] is counting at finite D ? The crystal C ( D ), or more precisely itsrefinement C ( D ), is counting Donaldson-Thomas invariants defined as the Witten indicesof the quiver quantum mechanics describing one D6 brane on X , bound to D4, D2 and D0branes in the background B field D . This is in accord with [20], which pointed out that the correspondence of [16] can hold onlyin the limit of infinite B-field. One should be able to understand this a consequence of essentiallyinfinite non-commutativity turned on in [16]. . Acknowledgments We would like to thank T. Dimofte, Y. Nakayama, N. Reshetikhin, E. Sharpe, C. Vafaand M. Yamazaki for very helpful discussions. We are especially grateful to D. Jafferis,A. Hanany, R. Kenyon, H. Ooguri and Y. Soibelman for explanations of their work. Thiswork was supported by the Berkeley Center for Theoretical Physics, by the National ScienceFoundation (award number 0855653), by the Institute for the Physics and Mathematics ofthe Universe, and by the US Department of Energy under Contract DE-AC02-05CH11231.
Appendix A. The Conifold
We will show how we can use Seiberg duality and dimer mapping explained in section4, to reproduce the results of [26,62] on the partition function of the conifold. Recall thatthe conifold has only two nodes, which makes this model especially simple. The conifoldquiver is shown for chamber n (with n framing nodes) in Figure 26 while the conifold dimeris shown in Figure 27. (n)(n-1) 12 0
Fig. 26.
The conifold quiver for chamber n.
Starting from the configuration shown in Figure 27.a, we can perform Seiberg Dualityon Node 1. Seiberg Duality results in a dual face for 1, as explained in section 4. However,the resulting brane tiling has two-valent vertices, which correspond to mass terms in thesuperpotential. Integrating out results in the quiver shown in Figure 27.c. This trans-formation takes the quiver from chamber n to chamber n + 1. As an explicit example ofthe techniques in section 4, we will derive the exact wall crossing formula, along with thechange of variables from the dimer model.The exact partition function for the conifold is known [26,62]. In chamber n, it isgiven by, Z ( n, q , q ) = M (1 , − q q ) Y k ≥ (cid:0) q k ( − q ) k − (cid:1) k + n − Y k ≥ n (cid:0) q k ( − q ) k +1 (cid:1) k − n +1 M ( x, q ) is the MacMahon function, M ( x, q ) ≡ ∞ Y m =1 (cid:16) − xq m (cid:17) m
222 22111 1 1 111 2 2222 222 22111 1 (a) (b) (c)
Fig. 27.
The effect of Seiberg Duality on the conifold dimer. In the second step, we haveintegrated out the fields with a mass term in the superpotential coming from the 2-valent vertices.
Now consider crossing the wall from chamber n to chamber n+1. This gives, Z ( n + 1) = Z ( n, q , q )(1 + q ) h q ,q ∆ i = Z ( n, q , q )(1 + q ) − n since for the conifold, every melting configuration must have the same intersection number,∆ ◦ ∆ = − n . By a simple change of variables q = − Q − q = − Q Q we find, Z ( n + 1) = M (1 , − Q Q ) Y k ≥ (cid:0) Q k ( − Q ) k − (cid:1) k + n +1 Y k ≥ n +1 (cid:0) Q k ( − Q ) k +1 (cid:1) k − n which agrees with the general formula for the conifold partition function in chamber (n+1). Note that in Section 4 we only considered dualizing on nodes with n ∗ = 1 or 0 framingarrows. However, the conifold is simple enough to explicitly check that the dimer intersectionnumber equals the quiver intersection number for arbitrary n ∗ . This change of variables includes additional minus signs relative to (4.7). This is because inthis infinite product form, we have implicitly absorbed signs in the { q i } . These signs flip (fromthe ( − d (∆) factors) when we cross the wall from chamber n to chamber n + 1. ppendix B. The local P example As another example, consider the toric quiver corresponding to X = O ( − → P .The D4-D2-D0 quiver has three nodes, corresponding to ˜ E = O S ( − E = O S ( − − E = O S ( − − O S ( n ) is D S e nD t − K S (1 + 124 c ( S )) (B.1)where D S is the divisor corresponding to the surface, and D t generates the Kahler class, so D S = − D t . Also, D t D S = C t and per definition D t C t = 1 . Thus, (B.1) can be rewrittenas D S + ( n + 32 ) C t + 12 ( n + 1)( n + 2) pt −
18 pt (B.2)We can write ˜∆ = D S + pt + ( − C t + 14 pt)˜∆ = D S + 2 C t + ( − C t + 14 pt)˜∆ = − D S − C t + ( 32 C t −
14 pt) . This has n = n = 3, n = 6, where n ij is the number of arrows from node i to node j . Add to this the D6 brane, corresponding to O X [ − = − X (1 + c ( X )24 ) , Since c ( X ) · D S = ( c ( S ) − c ( S )) D S = −
6, it is easy to see that n = 1 and n = n = 0. The intersection numbers can be computed by setting up the usual Ext machinery, and then reducing this to cohomology calculations on S = P . For exam-ple, Ext X ( O X [ − , O S ( − Ext X ( O X , O S ( − H ( P , O S ( − C , while all theother Ext groups vanish in this sector. This is not a toric quiver yet, we need to dualize node ˜∆ . We get,∆ = ˜∆ = D S + pt + ( − C t + 14 pt)∆ = ˜∆ + 3 ˜∆ = − D S − C t + 2( 32 C t −
14 pt)∆ = − ˜∆ = D S + C t − ( 32 C t −
14 pt) . The last step follows from dim H ( P k , O ( m )) = (cid:0) m + kk (cid:1) , dim H k ( P k , O ( m )) = (cid:0) − m − − k − m − (cid:1) , anddimH n ( P k , O ( m )) = 0, for n = 0 , k , see e.g. [50]. = ˜∆ , unchanged. The new bundles are given by, E = O S ( − E = O S ( − − E = e O S . This has n = n = n = 3, n = 1, and the superpotential W = X i,j,k =1 ǫ ijk Tr A i B j C k where n ij is the number of arrows from node i to node j . The resulting quiver is in figure28. Q Fig. 28
The quiver for the orbifold phase of local P . The crystal C corresponding to this quiver is on the left in the figure 29. The corre-sponding crystal C is on the right. This corresponds to a set of points N + N + N = 3 N , in agreement with the fact that for local P , Q = (1 , , , − Fig. 29.
The full crystal, C for local P is shown on the left, while the subcrystal, C correspondingto holomorphic functions is shown on the right. D = − nD S = 3 nD t , the lattice C ( D ) becomes (see figure30) N + N + N = 3 N + 3 n. Getting C ( D ) from C corresponds to removing sites n − X i =0 (3 i + 1)(3 i + 2)2 ∆ + (3 i + 2)(3 i + 3)2 ∆ + (3 i + 3)(3 i + 4)2 ∆ Fig. 30.
The full crystal, C ( D ), and the subcrystal, C ( D ) for local P after turning on a largeB-field. Summing up the charges, we find nD S + 32 n C t + ( 32 n − n )ptThis accounts for the charge the D6 brane picks up by putting it in the background B-field,which takes it to O X ( D )[ − − X e D (1 + c ( X )24 ) , Namely, it is easy to see that this agrees with the difference∆ − ∆ = nD S − n D S D S + n D S D S D S + n C ( X ) D S D S D S = − C t , D S D S D S = 9 , Similarly, the face of the crystal carries charge(3 n + 1)(3 n + 2)2 ∆ + (3 n + 2)(3 n + 3)2 ∆ + (3 n + 3)(3 n + 4)2 ∆ which equals D S + (3 n + 32 ) C t + ( (3 n + 1)(3 n + 2)2 + 14 )pt . From above, we see that this is the charge of O S ( D ) , as we claimed in the text. Similarly, the charge of an edge is(3 n + 1)∆ + (3 n + 2)∆ + (3 n + 3)∆ . This equals C t + (3 n + 1)pt , the charge of O C t ( D ) . eferences [1] M. Kontsevich and Y. 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