The Non-commutative Topological Vertex
Wall Crossing Phenomena
Kentaro Nagao and Masahito Yamazaki
RIMS, Kyoto University, Kyoto 606-8502, Japan
California Institute of Technology, CA 91125, USA
Department of Physics, University of Tokyo, Tokyo 113-0033, Japan
IPMU, University of Tokyo, Chiba 277-8586, Japan
We propose a generalization of the topological vertex, which we call the “non-commutative topological vertex”. This gives open BPS invariants for a NB. removed: an arbitrarytoric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles. The vertex is defined combinatorially using the crystal melting model proposed recently, and depends on the value of closed string moduli at infinity. The vertex in one special chamber gives the same answer as that computed by the ordinary topological vertex. We prove an identify expressing the non-commutative topological vertex of a toric Calabi-Yau manifold as a specialization of the closed BPS partition function of an orbifold of , thus giving a closed expression for our vertex. We also clarify the action of the Weyl group of an affine NB. correctedLie algebra on chambers, and comment on the generalization of our results to the case of refined BPS invariants.
1 Introduction and Summary
Recently, there has been significant progress in the counting problem of BPS states in type IIA string theory on a toric Calabi-Yau 3-fold111See [1, 2, 3, 4, 5, 6, 7]. See also [8, 9, 10, 11, 12, 13, 14] for mathematical discussions. In the literature, the Calabi-Yau manifold (which we denote by ) is assumed to have no compact 4-cycles, and we consider a BPS configuration of D0/D2-branes wrapping compact holomorphic 0/2-cycles, as well as a single D6-brane filling the entire Calabi-Yau manifold. The question is to count the degeneracy of such BPS bound states of D-branes.
One subtlety in this counting problem is the wall crossing phenomena, stating that the degeneracy of BPS bound states depends on the value of moduli at infinity. Indeed, the closed BPS partition function222The upper index stands for ‘closed’.
which is defined in  as the generation function of the degeneracy of D-brane BPS bound states333The definition of the partition function is the same as the partition function in ., depends on maps specifying a chamber in the Kähler moduli space444See Appendix A for details.. What is interesting is that in one special chamber of the Kähler moduli space, the BPS partition function is equivalent the topological string partition function555Actually, the topological string partition function depends on the choice of the resolution of the singular Calabi-Yau manifold . This is related to the choice of the limit, as will be explained in the main text. (up to the change of variables, which we do not explicitly show here for simplicity):
It is natural to expect that similar story should exist for open BPS invariants as well. Namely, we expect to define open version of the BPS partition function666The upper index stands for ‘open’.
depending on maps specifying the chamber in the Kähler moduli space, such that the partition function reduces to the open topological string partition function in a special chamber :
The question is how to define open BPS degeneracies such that the generating function follows the conditions above.
As a guiding principle of our following argument, we use the crystal melting model developed recently in  (see [8, 11] for mathematical discussions). This crystal melting model generalizes the result of  for to an arbitrary toric Calabi-Yau manifold. In the case of , the crystal melting partition function with the boundary conditions specified by three Young diagrams gives the topological vertex  . By using these vertices as a basic building block, we can compute open topological string partition function with non-compact D-branes wrapping Lagrangian 3-cycles of the topology included . In this story, generalization from closed to open topological string partition function corresponds to the change of the boundary condition of the crystal melting model for .
Now the recent result  shows that the closed BPS partition function discussed above can be written as a statistical mechanical partition function of the crystal model. This model applies to any toric Calabi-Yau manifold, and for the BPS partition function coincides with the topological string partition function. Similarly to the case of the topological string story mentioned in the previous paragraph, we hope to define the open version of the BPS invariants by changing the boundary condition of the crystal melting model. The invariants defined in this way will be defined in any chamber in the Kähler moduli space, and reduces to the ordinary topological vertex in a special chamber. We call such a generalization of the topological vertex “the non-commutative topological vertex”777The word ‘non-commutative’ stems from the mathematical terminologies such as “non-commutative crepant resolution”  and “non-commutative Donaldson-Thomas invariant” . The non-commutativity here refers to that of the path algebra of the quiver. The quiver (together with a superpotential) determines a quiver quantum mechanics, which is the low-energy effective theory on the D-brane worldvolume . , NB. addedfollowing “the orbifold topological vertex” named in . NB. I removed the footnote: See  for a related proposal. Q. Our definition include theirs?.
We will see that this expectation is indeed true. We adopt the NB. Removed: combinatorialdefinition proposed by one of the authors in the mathematical literature [14, 21]. Our non-commutative topological vertex is defined for a Calabi-Yau manifold without compact 4-cycles, and a set of representations assigned to external legs of the toric diagram. As in the case of topological vertex, encodes the boundary condition of the D4-branes wrapping Lagrangian 3-cycles. We propose our vertex as the building block of open BPS invariants. Here by an open BPS invariant we mean a degeneracy counting the number of BPS bound states of D0/D2/D6-branes wrapping holomorphic 0/2/6-cycles, as well as D2-branes wrapping disks whose boundaries are on D4-branes wrapping non-compact Lagrangian 3-cycles.
We can provide several consistency checks of our proposal (see section 3.4 for more details). First, our vertex by definition reduces to the closed BPS invariant when all the representations are trivial. Second, our vertex shows a wall crossing phenomena as we change the closed string Kähler moduli, and the vertex coincides with the topological vertex computation in the chamber where the closed BPS partition function reduces to the closed topological string partition function. Third, the wall crossing factor is independent of the boundary conditions on D-branes, and is therefore the partition function factorizes into the closed string contribution and the open string contribution, as expected from  and the generalization of .
Given a combinatorial definition of the new vertex, the next question is whether we can compute it, writing it in a closed expression. We show that the answer is affirmative, by showing the following statement. For a Calabi-Yau manifold , the non-commutative topological vertex is equivalent to the closed BPS partition function for an orbifold of , under a suitable identification of variables explained in the main text888More precisely, we need to specify the resolution of and . We also need to impose the condition that two of the representations are trivial. See the discussions in the main text.:
We will give an explicit algorithm to determine and , starting from the data on the open side. Since the infinite-product expression for is already known [13, 6], this gives a closed infinite-product expression for our vertex.
The organization of this paper is as follows. We begin in section 2 with a brief summary of the closed BPS invariants and their wall crossings, and their relation with the topological string theory. In section 3 we define our new vertex using the crystal melting model. We also perform several consistency checks of our proposal. Section 4 contains our main result (1.3), which shows the equivalence of our new vertex with a closed BPS partition function under suitable parameter identifications. We give an explicit algorithm for constructing closed BPS partition function starting from our vertex. In Section 5 we treat several examples in order to illustrate our general results. Section 6 is devoted to discussions. We also include Appendices A-C for mathematical proofs and notations.
2 Closed BPS Invariants
Before discussing the open BPS invariants, we summarize in this section the definition and the properties of the closed BPS invariants.
Throughout this paper, we concentrate on the case of the so-called generalized conifolds. The reason for this is that wall crossing phenomena is understood well only in cases without compact 4-cycles, which means is either a generalized conifold or 999See  for the proof of this statement.. NB. Removed: The case of is similar.
By suitable transformation, we can assume that the toric diagram of a generalized conifold is a trapezoid with height 1, with NB. correctedlength edge at the top and at the bottom (see Figure 1)101010The Calabi-Yau manifold is determined by and as .. If we denote by the sum of the length of the edges on the top and the bottom of the trapezoid, this geometry has independent compact ’s. We label them by , borrowing the language of the root lattice of algebra.
The language of the root lattice will be used extensively throughout this paper111111The root lattice of is exploited in [13, 6, 14]. See also Appendix A.. We can also make more ’s by combining them. For example, combining all the ’s between -th and -th (assume ), we have another which we denote by
This corresponds to a positive root of .
Suppose that we have a Calabi-Yau manifold without compact 4-cycles. We also consider a single D6-brane filling the entire and D0/D2-branes wrapping compact holomorphic 0/2-cycles specified by and , respectively. We can then define the BPS degeneracy counting BPS degeneracy of D-branes121212More precisely, this BPS degeneracy is defined by the second helicity supertrace.. The closed BPS partition function is then defined by
The closed BPS partition function for generalized conifolds is studied in [13, 6]. To describe the results, let us first specify the resolution (crepant resolution131313Crepant resolution is a resolution such that , where and are canonical bundles of and .) of 141414This is not essential, since by varying the value of the Kähler moduli we can go to the geometry with other choices of resolutions. We just need to specify an arbitrary resolution in order to begin the discussion. See Appendix A for more about this.. Each of the ’s are either -curve or -curve. In the language of the toric diagram, this is to specify the triangulation of the toric diagram. We specify this choice by a map
In the following we sometimes write instead of . When (), the -th triangle from the left has one of its edges on the top (bottom) edge of the trapezoid. This means that the -th is a -curve (-curve) when (). By definition, we have .
For example, in the case of Suspended Pinched Point () whose toric diagram is shown in Figure 2, and there are 3 difference choice of resolutions. This is represented by
where is the genus 0 Gopakumar-Vafa (GV) invariant151515Higher genus GV invariants vanish for generalized conifolds.. For the 2-cycle , the explicit form of depends on and is given by
By CPT invariance in five dimensions , we have We also have , where the Euler character for a toric Calabi-Yau manifold is the same as twice the area of the toric diagram.
where the central charge is given by
Here denotes (up to proportionality constants) the central charge of the D0 brane, and following  we choose the complexified Kähler moduli to be real. Also, the notation means the B-field flux through the cycle 161616This was written in ..
Now suppose that is positive171717Under this condition we are discussing only half of chambers of the Kähler moduli space, which lie between the Donaldson-Thomas chamber and the non-commutative Donaldson-Thomas chamber. The other half arises when is negative.. From (2.7) and (2), it follows that the wall crossing occurs when the integer part of the value of the B-field through the cycle change. For the cycle , this is given by
Since there are NB. corrected ’s in , there are NB. corrected such parameters.
just as advertised in (1.1).
For concreteness, let us discuss an example. We use the example of the Suspended Pinched Point () using the triangulation in (2.5). In this example, the topological string partition function is
where is the MacMahon function
The BPS partition function is given by
The parameters specify the chamber, but as we can see from the definition they are not completely independent parameters. Since we only have NB. corrected real parameters , it is likely that this parametrization is redundant. Indeed, as explained in the Appendix A we can specify the chamber by a map , which is specified by NB. corrected half-integers, NB. corrected, satisfying one constraint
This means we can indeed parametrize the chamber by NB. corrected independent (half-)integers, which is what we expected. As discussed in Appendix A, is an element of the Weyl group of NB. corrected.
3 The Noncommutative Topological Vertex
In this section we give a general definition of the non-commutative topological vertex using the crystal melting model. This definition is equivalent to the one given in  using the dimer model181818See Appendix A of  for the equivalence between crystal melting model and the dimer model.. NB. addedSee  for more conceptual definition in terms of Bridgeland’s stability conditions and moduli spaces.
To define our vertex, we need the following set of data:
As already explained in section 2, this gives a triangulation of the toric diagram, or equivalently the choice of the resolution of the Calabi-Yau manifold.
A map . As explained in Appendix A in the case of closed BPS invariants, and specify the chamber structure of the open BPS invariants.
A set of Young diagrams , assigned to external legs of the -web. This specifies the boundary condition of the non-compact D-branes ending on the -web. We denote by NB. corrected the Young diagrams for the top and the bottom edges of the trapezoid, and by the remaining two. We sometimes write , where NB. corrected and . In the example shown in Figure 3, there are five external legs and we have five representations.
For later purposes, we combine into a single representation by
In other words, we choose such that L-quotients of give . By abuse of notation, we use the same symbol for a set of representations as well as a single representation define above.
Given and , we define the non-commutative topological vertex
In the following we drop the subscript BPS for simplicity.
Before going into the general definition, we first illustrate our idea using simple example of the resolved conifold.
3.1 Example: Resolved Conifold
In this example there is only one and the BPS partition function depends on a single positive integer . In the language of , NB. I change the notation:
We fix to be
Without losing generality we concentrate on , since corresponds to a flopped geometry, where is replaced by (see Appendix A).
The ground state crystal for is shown in Figure 4. This crystal, sometimes called a pyramid, consists of infinite layers of atoms, the color alternating between black and white NB. added([8, 9]). In the -th chamber there are atoms on the top.
The closed BPS partition function is defined by removing a finite set of atoms from the crystal. When we do this, we follow the melting rule [3, 11] such that whenever an atom is removed from the crystal, we remove all the atom above it. In other words, since an atom is in one-to-one correspondence with an F-term equivalent class of paths starting from a fixed node of the quiver diagram , for an arrow and and an atom we can define . The melting rule then says
We then define the partition function by summing over such :
where and are the number of white and black atoms in , respectively.
The weights () assigned to white (black) atoms in the -th chamber are determined as follows. We can slice the crystal by the plane, and each slice is specified by an integer (see Figure 5). We choose so that atoms on the top of the crystal is located at .
The weight depends on the chamber and is given by NB. corrected
when is odd, and
when is even. For example, when , and NB. corrected. The change of variables arises from the Seiberg duality on the quiver quantum mechanics , geometrically mutations in the derived category of coherent sheaves [2, 12], or in more combinatorial language the dimer shuffling . The parameters defined here are related to the D0/D2 chemical potentials introduced in section 2 by191919 The equation (3.5) is the same for odd and even if we suitably exchange the two nodes of the quiver diagram. The relation (3.6) can also be written as when is even, and and exchanged when odd. This coincides with the expression in .
Now let us discuss the open case. When non-trivial representations are assigned to each of the four external legs of the -web, the only thing we need to do is to change the ground state of the crystal.
The crystal has four ridges, corresponding to four external legs of the -web. When we assign a representation, we remove the atoms with the shape of the Young diagram. More precisely, we remove the atoms with the shape of the Young diagram in the asymptotic direction of the -web, as well as all the atoms above them, so that the melting rule is satisfied. See Figure 6 for an example.
The partition function is defined in exactly the same way by (3.3), and the result is denoted by .
Several comments are now in order.
NB. AddedFirst, let us explain the origin of the name “the non-commutative topological vertex”. Recall that, in commutative case, topological vertex is defined for . For a general affine toric Calabi-Yau manifold , we divide the polygon into triangles and assign a topological vertex to each trivalent vertex of the dual graph. We can get the topological string partition function for the smooth toric Calabi-Yau manifold by gluing them with propagators. Similarly, NB. added assume that a polygon is divided into trapezoids. Then we can assign a non-commutative topological vertex to each vertex of the dual graph and glue them by propagators. The BPS partition function defined in this way NB. changed is related to the topological string partition function via wall-crossing202020Given a devision of a polygon into trapezoids, we get a partial resolution of and a non-commutative algebra over the partial resolution, which is derived equivalent to . The BPS partition function given by gluing non-commutative topological vertices counts torus invariants -modules.. In [20, 24] they study the case when a polygon is divided into (not necessary minimal) triangles.
NB. Removed: First, let us emphasize the difference between ordinary (‘commutative’) topological vertex and our non-commutative topological vertex. In commutative case, topological vertex is defined only for . For a more general toric Calabi-Yau manifold, we cut the -web into pieces, assign a topological vertex to each trivalent vertex and we glue them by propagators. For example, for the resolved conifold we need to glue two vertices. However, our NCTV is defined for any toric Calabi-Yau manifold, and thus we do not need to glue them. In this sense, our vertex is not a ‘vertex’.
Second, it is possible to give more geometric definition of the vertex NB. Added(see ). For the closed BPS invariants, the crystal arises as a torus fixed point of the moduli space of the modules of the path algebra quiver (under suitable -stability conditions). The moduli space is the vacuum moduli space of the quiver quantum mechanics arising as the low-energy effective theory of D-branes . The similar story exists in our case. Namely, the crystal is in one to one correspondence with the fixed point of the moduli space arising from a quiver diagram. For example, for conifold with \yng(1,1), the quiver is given in Figure 7.
Third, in the case of , our vertex reproduces the topological vertex of by definition.
3.2 General Definition from Crystal Melting
We next give a general definition of the vertex. Readers not interested in the details of the definition of the non-commutative topological vertex can skip this section on first reading.
Given a boundary condition specified by and , we would like to construct a ground state of the crystal, and determine the weights assigned to the atoms of the crystal.
The basic idea is the same as in the conifold example. First, the closed string BPS partition function is equivalent to the statistical partition function of crystal melting. The ground state crystal can be sliced by an infinite number of parallel planes parametrized by integers , just as in Figure 5. On each slice, there are infinitely many atoms, labeled by integers . Therefore, the atoms in the crystal are label by .
Let us show this in the example of the Suspended Pinched Point. The crystal in Figure 8 clearly shows this structure.
Another way of explaining this is to construct a crystal starting from a bipartite graph on , shown in Figure 9212121This is a universal cover of the bipartite graph on , which appears in the study of four-dimensional quiver gauge theories. See [26, 27, 28] for original references, and [29, 30] for reviews.. In this example, the bipartite graph consists of hexagons and squares, and periodically changes its shape along the horizontal directions.
Now the atoms of the crystal are located at the centers of the faces of the bipartite graph. and it thus follows we can slice the crystal along the horizontal axis. Each slice consist of an infinite number of atoms labeled by two integers , since there are two directions, the horizontal direction and the perpendicular direction to the paper222222In general the bipartite graph is determined by . A hexagon (a square) corresponds to -curve (-curve). In other words, the th polygon is a hexagon (square) if ..
Now consider the open case. In this case, we construct a new ground state by removing atoms from the closed ground state. By the melting rule, the atoms removed from the -th plane should be labeled by , where is a Young diagram. Depending on the representations on external legs, increases or decreases as we change . Thus the ground state crystal for open BPS invariants are determined by such a sequence of Young diagrams , called transitions below. In the following we make this idea more rigorous.
Let us begin with some notations. Let and be two Young diagrams. We say if the row lengths satisfy
and if the column lengths satisfy
We define a transition of Young diagrams of type as a map from the set of integers to the set of Young diagrams such that
for and for ,
Then as shown in  there is a transition of Young diagrams of type such that for any transition of Young diagrams of type and for any we have . The transition is the sequence of transitions discussed above.
For a transition of Young diagram of type , the ground state crystal can be defined by
where denotes the atom at position .
Having defined the ground state crystal, the partition functions is defined again as the sum over a subset of satisfying the following two conditions:232323It is straightforward to show that a subset satisfies the two conditions if and only if for a transition of Young diagram of type .
is finite set, and
satisfies the melting rule (3.2). In other words, if and for an arrow , then .242424 We can also explicitly write down the melting rule using the coordinates . Let us write when there is a path (a composition of arrows) such that . The partial order is then generated by
For a crystal , we define the weight by the number of atoms with the color contained in :
Also, for , we put
where we define for periodically,
We then define the vertex by
The parameters defined here are related to the D0/D2 chemical potentials introduced in section 2 by
NB. added the whole section
Let us recall the meaning of the refined BPS counting, first in the closed case. When the type IIA brane configuration is lifted to M-theory  and when we use the 4d/5d correspondence [34, 35], the D0/D2-branes are lifted to spinning M2-branes in , which has spin under the little group in 5d, namely . The ordinary BPS invariant is defined as an index; it keeps only the spin, while taking an alternate sum over the spin. The refined closed BPS invariants is defined by taking both spins into account.
The situation changes slightly when we consider open refined BPS invariants. The D4-branes wrapping Lagrangians, when included, are mapped to M5-branes on . This means that is broken to , and we have only one spin. However, there is an R-symmetry for supersymmetry in three dimensions, and in the definition of the ordinary open BPS invariants we keep only one linear combination of the two, while tracing out the other combination . The refined open BPS invariants studied here takes boths of the two charges into account.
In the language of crystal melting used in this paper, the open refined BPS invariants are defined simply by modifying the definition of the weights. Here, we explain how to modify the weights in the case of .
For an integer with
We also define the weights by
when , and
when . We then define the refined vertex by
By definition, the refined vertex reduces to the unrefined vertex by setting . The reader can refer to  for the definition of weights in general cases .
3.4 Consistency Checks of Our Proposal
In Appendix C we gave a purely combinatorial definition of the non-commutative topological vertex. We now claim that is captures open BPS invariants in the following sense.
Consider a generalized conifold (a toric Calabi-Yau manifold without compact 4-cycles) with representations assigned to each leg of the -web. Each representation specifies a boundary condition on the non-compact D4-brane wrapping Lagrangian 3-cycle of topology .
In the absence of D4-branes, we are counting particles of D0/D2-branes wrapping 0/2-cycles, which makes a bound state with a single D6-brane filling the entire Calabi-Yau manifold. When the D4-branes are included, D2-branes can wrap disks ending on the worldvolume of D4-branes. The degeneracy of such D-brane configurations is what we mean by the open BPS degeneracies. Note that supersymmetry is broken by half due to the inclusion of D4-branes; our counting of BPS particles makes sense because we are counting BPS states in lower dimensions, where the minimal amount of supersymmetry is lower.
We can provide several consistency checks of our proposal. First, our vertex by definition reduces to the closed BPS invariant when all the representations are trivial:
The second consistency check comes from the wall crossing phenomena. As shown in , the vertex goes through a series of wall crossings as we move around the closed string moduli space (Kähler moduli of the Calabi-Yau manifold), just as in the case of closed invariants. It was also shown in  that in the chamber where the closed BPS partition function reduces to the closed topological string partition function, our vertex gives the same answer as that computed from the topological vertex (in the standard framing):
The third consistency check comes from the fact that the wall crossing factor is independent of representations. In other words,
does not depend on [14, 21]262626To be exact, we have to normalize the generating function by a monomial. See [21, Corollary 3.21] for the precise statement.. This means that the open BPS partition function, which is defined by the sum over representations, takes a factorized form
Since takes an infinite product form as explained in section 2 and also takes the infinite product form , itself should take an infinite product form, which is consistent with a suitable generalization of the argument of .
Using (3.8), we can compute our vertex using the ordinary topological vertex formalism. In the next section, we give a yet another way of computing the non-commutative topological vertex. The advantage of our approach is that the final expression manifestly takes a simple infinite product form, and we do not have to worry about the summation of Schur functions.
4 The Closed Expression for the Vertex
In this section we give a closed expression for our non-commutative topological vertex. We do this by proving a curious identity stating the equivalence of our vertex for a toric Calabi-Yau manifold with a closed BPS partition function for an orbifold of 272727This is reminiscent of story of , where the ‘bubbling geometry’ is constructed for given a toric Calabi-Yau manifold such that the open+closed topological string partition function on is equivalent to closed topological string partition function on . However, our story is different in that the vertex computes only a part of the full open BPS partition function; the partition function itself is given by summation of our vertices over representations.. For another method using vertex operators, see [21, 38].
Start from a non-commutative topological vertex for a generalized conifold , which has compact ’s. As we discussed above, for the definition of the vertex we need (1) for a choice of the crepant resolution of , (2) a map specifying the chamber together with (3) a set of representations , and the resulting vertex is denoted by . In the following we consider the special case .
Choose an and consider the orbifold of . We choose the orbifold action such that when the toric diagram of is a trapezoid with a top and the bottom edge of length and respectively, has length and . We also choose map
where . NB. Modified: See Appendix C for the proof of this statement. The Appendix also discusses See Appendix C for an explicit method for choosing such satisfying (4.1) as well as generalization of (4.2) to the case of refined BPS invariants.
Since the infinite-product expression for closed BPS partition function for a generalized conifold is already known (section 2), we have a closed expression of our vertex when .
Let us illustrate the above procedure by several examples.
First, we begin with the non-commutative topological vertex for . Since there is no wall crossing phenomena involved in this case282828There is one wall between and , however we do not discuss such a wall since we are specializing to the case ., the vertex should coincide with the ordinary topological vertex, thus providing a useful consistency check of our proposal. For , we have .
Take with representation with at one leg. The above-mentioned procedure gives , and thus . The method in Appendix C gives
in the case of odd. NB. Modified (for odd; the case of even is similar)
and by solving for
Substituting this into the closed BPS partition function
we have NB. corrected
5.2 Resolved Conifold
Now let us discuss the next simplest example, the resolved conifold.
NB. In this case, and we choose a triangulation given by . The -th chamber is given by .
Consider the representation , with NB. changed . In this case, the method in Appendix C gives NB. Modified
Then we have NB. Modified