Abelianization of BPS Quivers
and the Refined Higgs Index
Seung-Joo Leeemail@example.com, Zhao-Long Wang†††firstname.lastname@example.org, and Piljin Yi‡‡‡email@example.com
School of Physics, Korea Institute for Advanced Study, Seoul 130-722, Korea
[0.2cm] Institute of Modern Physics, Northwest University, Xian 710069, China
We count Higgs “phase” BPS states of general non-Abelian quiver, possibly with loops, by mapping the problem to its Abelian, or toric, counterpart and imposing Weyl invariance later. Precise Higgs index computation is particularly important for quivers with superpotentials; the Coulomb “phase” index is recently shown to miss important BPS states, dubbed intrinsic Higgs states or quiver invariants. We demonstrate how the refined Higgs index is naturally decomposed to a sum over partitions of the charge. We conjecture, and show in simple cases, that this decomposition expresses the Higgs index as a sum over a set of partition-induced Abelian quivers of the same total charge but generically of smaller rank. Unlike the previous approach inspired by a similar decomposition of the Coulomb index, our formulae compute the quiver invariants directly, and thus offer a self-complete routine for counting BPS states.
- 1 Quivers and Indices
- 2 How to Compute Higgs Index
- 3 Examples with a Loop: Triangular Quivers
- 4 Abelianized Higgs Index is a Partition Sum
- A The Ambient D-Term Variety and Maximal Reduced Quivers
1 Quivers and Indices
The low energy dynamics of BPS particles or BPS black holes in four dimensions are most succinctly captured by quiver dynamics, which originate from wrapped D-brane picture of such particles [1, 2] compactified on a Calabi-Yau 3-fold, where particle-like BPS states arise from D3-branes wrapped on special Lagrangian 3-cycles. When the 3-cycle has topology of , the low energy dynamics of wrapped D3-branes would be gauged quantum mechanics with four supercharges. In the phase where the symmetry is broken to , the triplet eigenvalues of the Cartan vector multiplets encode the position of BPS particles along the noncompact , while the residual Weyl group shuffles these identical particles. When more than one 3-cycles are involved, each wrapped by D3-branes as well, we find additional chiral multiplets, in bi-fundamentals, arising from open strings between each pair of D3’s. The number of such chiral fields is identified with the intersection number.
The quiver dynamics itself can be further approximated by integrating out either vector multiplets or chiral multiplets. The two such descriptions are called Higgs and Coulomb “phase” descriptions, respectively. The word “phase” here is very misleading, although it is used conventionally, as the quiver dynamics in question is an one-dimensional system and thus the vacuum expectation values do not imply superselection sectors. It merely refers to particular integrating out procedure, which may or may not be reliable depending on the massgap, although Supersymmetry tends to protect quantities like index further. When both sides are reliable, we expect the computed indices from the two sides to agree with each other. This is the case, as far as we know, when the quiver has no loop [1, 3].
Generally speaking, the Higgs description, better suited for large Fayet-Iliopoulos (FI) constants, ’s, is more reliable as the massive vector multiplet fields being integrated out tend to have uniformly large mass of order , justifying the procedure. The Coulomb description, suitable for small ’s, provides a more intuitive picture of the wall-crossing via its multi-center picture; The positions of charge centers are encoded in the Cartan part of the vector multiplets. While physically more appealing, this latter Coulomb description turns out to involve various subtleties. Identification of the correct index theorem was rigorously argued only very recently , and does not follow from naive truncation to classically flat part of collective coordinates. The derivation has to invoke localization that breaks the natural four supercharges of the low energy dynamics down to one. Another important subtlety arises for cases with the so-called scaling regime, where one finds classical multi-center solutions with mutual distances arbitrarily small. For the latter class, for which the relevant quiver dynamics must have at least one closed loop, the naive massgap for the chiral multiplets fails.
Nevertheless, in the absence of scaling regimes (thus, in the absence of loop), the Coulomb description is very useful as it can be derived as the BPS soliton or black hole dynamics from the underlying theory, and it shows a clear intuitive picture of wall-crossing phenomena as multi-particle bound state physics [5, 6, 7, 8, 9, 10]. This Coulomb description has been derived, ab initio, for Seiberg-Witten dyons [4, 11], i.e., from the field theory itself as low energy dynamics of UV-incomplete solitons; As long as we can ensure the individual constituent particles are actually present in the spectrum, the low energy interaction among them are reliable when we stay very close to the relevant marginal stability wall. Regardless of how we view such multi-particle dynamics, a rather complete derivation of the Coulomb index had emerged very recently [4, 12, 13], which was then shown  to be equivalent among themselves and to the Kontsevich-Soibelman conjecture . We will briefly revisit this Coulomb index in section 4.
The natural index in field theory is the second helicity trace
where trace is over the one-particle Hilbert space of the given charge , and is the helicity operator. For four-dimensional field theory, there is a natural equivariant extension, called Protected Spin Character (PSC) 
with belonging to symmetry. When we factor out the universal half-hypermultiplet factor in the BPS supermultiplets, these reduce to the more familiar Witten-type indices as
In the low energy description of these BPS objects, we effectively compute the latter, after removing the free center-of-mass part of the low energy dynamics.
where, as the quiver dynamics is a gauged quantum mechanics with four supercharges, rotation generated by ’s is now an R-symmetry of the quiver dynamics while generates the other R-symmetry . For Higgs “phase,” it has been argued that this equivariant index is computed by (shifted) Hirzebruch characters,
where is the complex dimension of the Higgs moduli space of the quiver. The Higgs moduli space is always Kähler, allowing us to use Hodge decomposition. This collapses to, when ,
which is the Euler number times . In terms of the R-charges of the quiver, and .
Actual computation of is available for some subfamilies of quivers. Reineke has given general formulae for the Poincare polynomial of general quivers without loops ; this can be thought of as Higgs counterpart of the Coulomb index computations mentioned above. More interesting are for quivers with loops, which neither of the above can address. The equivariant index of an arbitrary Abelian cyclic quiver with generic superpotential was computed in Refs. , and along the way was found a new class of BPS states [17, 16], called intrinsic Higgs states. They were found to be wall-crossing-safe, invisible from the Coulomb description, and of zero angular momenta. They are typically far more numerous than Coulomb “phase” states, given a quiver with loops; These states are clearly important ingredients in understanding microstates of single-center black holes, but they also appear in some field theory BPS spectra, such as that of theory .
A challenge we wish to face in this note is how to generalize these Higgs index computations to general non-Abelian quivers with superpotentials. For Abelian quivers that have been studied, the index is computable relatively easily because Higgs moduli spaces are embedded in toric varieties. For non-Abelian cases, one encounters more general symplectic quotients by non-Abelian groups and, with superpotentials, has to intersect the zero loci of sections of vector bundles over such varieties. A general procedure that can recast computation of indices on such spaces to a problem in a bigger toric variety is known in the mathematical literature [20, 21, 22], which we will adapt to the problem at hand. This effectively replaces any given non-Abelian quiver by an Abelian one with the same total charge and of the same rank, by splitting each non-Abelian node, say of rank , to Abelian nodes. Section 2 will declare the procedure and section 3 will elaborate with examples.
An interesting corollary of this Abelianization method is that the end results have some similarity to the Coulomb “phase” wall-crossing formulae in Refs. [12, 4]. In the latter, the gauge symmetry is spontaneously broken to the Cartan part, with massless bosons encoding the positions of the particles. The non-Abelian nature of the quiver enters only at the last step, via the Weyl projection, which has been shown to result in a sum over partitions of the charge . Our Higgs “phase” computation of index is very similar in spirit in that we rely on Cartan subalgebra and the Weyl projection in the end. This naturally leads us to suspect that our Abelianization procedure parallels in some sense the index computation on the Coulomb side. In section 4, we elaborate this idea further for simple examples, and offer a conjecture on how the Higgs index can also be naturally written as a sum over partitions of the charge in a manner that parallels the Coulomb index partition sum.
After this work was completed, Ref.  appeared in the arXiv. There an intriguing transformation rule is suggested for the quiver invariants between different (non-Abelian) quivers related by mutation. Our formulae provided in this note should be capable of verifing explicitly non-Abelian examples in their work.
2 How to Compute Higgs Index
As already explained in the previous section, Higgs phase index can be computed as the Euler number of the Higgs moduli space , which, as we will shortly see, is constructed as a complete intersection via F-terms, embedded in the D-term variety . As is well-known, with the aid of adjunction formula, certain invariants of , including its Euler number , are expressible in terms of the ambient space data . In case of Abelian quivers, the corresponding ambient space is a toric variety and hence, one can easily extract relevant invariants in a straight-forward manner, by using simple combinatorial prescriptions from toric geometry. On the other hand, for general quivers with non-Abelian nodes, it is more difficult to deal with the resulting D-term variety.
The upshot of the computational prescription for Higgs phase index is to first “Abelianize” the quiver and to make use of the corresponding “toric” quiver variety as well as a complete intersection therein. One can then apply the usual toric techniques. In this section, we shall briefly describe the index prescription in full generality at the risk of making the presentation abstract; some concrete, illustrative examples will follow in the ensuing section for triangular quivers.
Before we proceed, it is important to note that we work in individual branches of the quiver. In other words, we presume a definite choice of FI parameters . For each given branch, the Higgs vacuum moduli space can be obtained via two steps; first, we perform a symplectic reduction using D-term conditions, then, if a loop is present, further impose F-term conditions. However, as was seen in Ref.  for Abelian quivers, we can make life slightly easier by noticing that F-terms tend to simplify things. To make the long story short, having a nontrivial F-term subvariety inside the D-term variety often demands that some bi-fundamentals associated with certain pairs of nodes should be set to zero. This is done to reduce the number of F-terms, because F-terms tend to kill entire Higgs moduli space. For each branch of the quiver, we end up freezing certain sets of bi-fundamental fields to zero in order to obtain a nontrivial , which “reduces” the quiver to be without loops by removing links. See section 2 of Ref.  or Appendix A here for an elaboration on this phenomenon.
2.1 Abelianization and the Lift
Given a quiver with the gauge group and given a choice of branch (or choice of FI constants), we have where is the moment map from the D-term, with the shift by FI constants understood. Then we obtain the true moduli space by further imposing F-term conditions. The gauge group that actually participates in the quotient is since there is always one overall that acts trivially on all chiral fields. Again, the choice of branch imposes on us to set certain bi-fundamental fields to zero identically, for otherwise is empty.
To such a non-Abelian quiver , we associate an Abelianized quiver , obtained by splitting each of the non-Abelian nodes of , say, of rank , into Abelian nodes, and simply duplicating the arrows as well as the FI constants. See Figs. 2 and 3 for an example. Via this Abelianization, we reduce the gauge multiplets to those associated with the Cartan subgroup , but keep the same bi-fundamental field contents. (Again, acts nontrivially on the chiral fields.) With such an Abelianized quiver , we end up in the territory of toric geometry. Keeping the same FI parameters, we find the D-term induced variety, , and the subvariety obtained by imposing F-term conditions as well. Thus, can be thought of as the Higgs moduli space of in the given branch. Finally, a useful intermediary that will eventually connect the two D-term varieties and is the space
which can be regarded as a bundle over and also a subvariety of .
where is the Weyl group of the gauge group for the non-Abelian quiver, and is the Euler class of , the Whitney sum of line bundles associated with the “off-diagonal” part, , of the gauge group, that is,
Note that denotes the set of roots of while in bold denotes the corresponding vector bundle. This bundle is naturally decomposed as according to the usual decomposition of into the positive and the negative parts. Here, is the (holomorphic) vector bundle that is tangent to the fibre of . We will shortly see how to express in terms of the toric data for .
The nontrivial part of the bridging rule (2.1) is obviously the lift of , denoted by . Lift is defined via the intermediary, , which naturally admits an inclusion and a projection , in such a way that the relation
holds on . While this does not determine the lift uniquely, given , whatever ambiguity there might be is killed by that follows on the right hand side of (2.1). When the cohomology element is a multiplicative class associated with the (holomorphic) tangent bundle , its lift turns out to be given as
In the next section, we will see how this arises for general quiver varieties in the course of evaluating index by directly constructing a lifted bundle over such that .
2.2 Quivers without Loops, the Indices, and the Fans
For quivers without loops, and thus, with no F-terms present, the above prescription applies directly and simply since the Higgs moduli space is a symplectic reduction, , of the flat space of bi-fundamental chiral fields. When the quiver has a loop, the superpotential will complicate the space further via F-term constraints, which we will address in subsection 2.4.
The simplest invariant is the Euler number of ,
that counts the Higgs BPS states when the quiver in question has no loops and thus no F-terms. In terms of the Chern class, , we find
where is the complex dimension of the Higgs moduli space. The extra sign factor in front is there so as to count each hypermultiplet as .
A well-known equivariant version of the Euler number is the refined Euler character, available upon the Hodge decomposition as
which reduces to the Euler number when . Recall that is computed via the class (see for instance Ref. )
where and are the multiplicative classes associated, respectively, with and . The Abelianization asserts that this quantity can be computed as
We will show examples of these equivariant and non-equivariant indices in the next section.
Alternatively, we can directly consider the topological class associated with the refined Higgs index
which will be more useful in interpreting the Abelianization physically in section 4. For this, it is convenient to view the factor as a (trivial) multiplicative class associated with the constant function , whereby we find is directly computed by another multiplicative class associated with the function
where has been replaced by . In other words,
with eigen-forms of the curvature of the holomorphic tangent bundle . Again, we have
in the lifted form.
To understand the origin of the contribution from in Eq. (2.4) (and consequently in Eqs. (2.6), (2.14) and (2.19)), it is useful to consider how the lift of the tangent bundle is related to the tangent bundle of the Abelianized variety . The relevant exact sequence for general quiver can be written as
with , where ’s are copies of the trivial line bundle, call it , over . The label serves as a reminder how the corresponding Cartan generator determines the map . In turn, the latter is a sum of line bundles, , where belongs to the collection of one-dimensional cones, , in the “fan,” , for the Abelianized toric variety .
For any multiplicative class , then, we have
As the Euler sequence for the Abelianized D-term variety is given by
we conclude that
Given the toric data for , this carries all the information required to express the (equivariant) index as the integral of a specific cohomology class over .
To evaluate this integral, we still need to determine the various intersection numbers, which can only be understood through the complete fan structure for . First of all, associated with the Abelianized quiver is a corresponding “charge matrix”, denoted by , each row of which lists the charges of the bi-fundamental fields under each gauge group. Note that the row and the column indices for the matrix range over the regions and , respectively. The charge matrix itself has a certain amount of information on the toric variety . For instance, any multiplicative class of the tangent bundle , say, the Chern class of , can be expressed explicitly in terms of ,
where form a basis of the , which turns out to be of rank . However, the charge matrix does not uniquely determine the fan; in toric terms, its rows correspond to the linear relations of the rays in the fan, but the incidence information for higher-dimensional cones is missing.
For the rest of this subsection, we summarize how the complete fan structure for the toric variety is determined from the given quiver data, and also present the recipe for the intersection numbers. The technical details will not be needed in reading the rest of this paper and the way we state the procedures here is by no means pedagogical. Interested readers are kindly referred to the excellent texts [31, 32, 33, 34] for a more complete review.
It turns out that the charge matrix, when equipped with values assigned to the quiver nodes (that is, a choice of -stability criterion), does determine the fan completely; the notion of stability of reduced quivers can be defined accordingly, from which the fan structure is determined . Practically, however, the procedure illustrated in Ref.  can be more accessible, which goes as follows: an index set is defined as
and by collecting the maximal elements of the complement inside the power set of , one obtains the Stanley-Reisner ideal , from which the corresponding fan is constructed as#1#1#1By abuse of notation we denote the cone simply by the set, , of its generators.
Now given the charge matrix and the fan for the toric variety , the intersection numbers defined as
are determined by simultaneously solving the linear equations of the following form,
where denote the ray corresponding to the -th column of , and , the collection of -dimensional cones for .
2.3 An Illustration: Grassmannian
As an illustration, let us consider the quiver with gauge group , the two nodes for which are linked by arrows. The Higgs moduli space is the Grassmannian, whose indices are of course well-known already. Nevertheless, let us proceed to compute its topological invariants following the Abelianization procedure. The Abelianized variety consists of copies of projective spaces and we denote by the Kähler class of each copy. The intersection structure is simple;
is the only nonvanishing intersection number.
From the Euler sequence (2.22) for , or that for each of the projective spaces
where the products run over the range . The factors associated with can be read off from the off-diagonal parts of . The ’s are associated with , under which the off-diagonal parts are labeled by a pair of ordered indices, , which have the charge of the form
So, , which leads to
These combined, Eq. (2.6) (and Eq. (2.19), respectively) reproduces the (refined) Higgs index of the Grassmannian faithfully. We will come back to this example in section 4, and try to give the resulting index formula a little more physical interpretation.
2.4 Loops, the Superpotential, and the Normal Bundle
Computation of any multiplicative class for embedded in is straightforward as long as we understand the normal bundle of this embedding. The general rule states that
For the problem at hand, we are interested in for the unrefined index, and in either or for the refined one.#2#2#2The class of a bundle denotes the class of the dual bundle. For Abelian quivers this general formula has been used very fruitfully in Refs. [17, 16, 30], where a new class of BPS states, intrinsic Higgs states, was discovered. For non-Abelian quivers, this is again lifted to the Abelianized form,
so it remains to understand how the normal bundle of in is lifted to a bundle over in .
For quivers with a loop, , the superpotential generates a F-term constraint
for each chiral multiplet in the quiver and defines the embedding of in . Note that, with a generic choice of superpotential and a generic choice of FI constants, the D-term and the F-term constraints are independent. When we Abelianize to , we are removing non-Cartan part of the D-term constraints from the data but leave the chiral field contents and the superpotential thereof intact. This shows that, generically the fibre of coincides with that of , i.e., the normal bundle of the Abelianized Higgs moduli space embedded into its D-term ambient . This is in contrast with how fibre of is a sum of the fibre of and that of .
In fact, a natural and simple lift of the normal bundle and its topological classes dictates
In other words,
computes the -integral of the multiplicative class via its Abelianized D-term variety and the embedded . Again, we will be working with toric varieties, so all quantities here can be straightforwardly read-off from .
3 Examples with a Loop: Triangular Quivers
Having seen the general prescription for computing the Higgs phase index, we shall illustrate it, in this section, with simplest examples with an oriented loop: the triangular quivers.
3.1 A Simplest Non-Abelian Triangular Quiver
Let us consider the quiver with the adjacency matrix
and the dimension vector , as depicted in Fig. 2.
The computation of is illustrated in the branch where and so that the single bi-fundamental field from node to node gets a zero VEV. Firstly, by imposing D-terms, one is led to the ambient variety . The vacuum moduli space is then embedded in through the F-term, defined as a section of the rank-2 vector bundle associated with the vanishing -bi-fundamental field under , where the subscripts for gauge groups label the nodes.
Now, we shall apply the general prescription of section 2 and move towards the territory of line bundles on toric geometry, as opposed to that of vector bundles on Grassmannian geometry. Upon Abelianizing the quiver, the rank-2 node gives rise to two Abelian nodes. Thus, the resulting quiver is described by the following adjacency matrix
with the dimension vector , as depicted in Fig. 3.
It is easy to see that the corresponding D-term variety is . Note that the piece of the original D-term variety has led to the last two factors of upon Abelianization.
By taking the multiplicative class in Eq. (2.48) to be the Chern class , one is thus led to the following expression for as an integral over ,
where , and denote the Kähler classes of the three factors, respectively, and in the last step, via the trivial intersection structure of , the integration of the cohomology class has switched to a contour integral around the origin.#3#3#3Note that we could have had replacing in Eq. (3.7) to conform with the convention used in subsection 3.2 for a general triangular quiver. Under such a choice, would not lie in the Kähler cone and Eq. (3.1) should get an extra sign factor due to the negative intersection. Note that the factor is implicit in each of the contour integral measures. It is straightforward to evaluate Eq. (3.1) and we obtain .
As for the computation of the refined Euler character, we again apply Eq. (2.48), now with ,
where the four factors in the integrand are written in turn as
Similarly to the unrefined case, we are led to a straightforward contour integral and thereby obtain
which gives the refined Higgs index
As desired, for , the refined Euler character (3.11) does reduce to the Euler number .
3.2 General Triangular Quivers
Let us now consider triangular quivers in full generality (see Fig. 4).
In the previous example, the Abelian ambient variety was a product of projective spaces and hence, the integration (3.7) of a cohomology class, for instance, turned into the contour integral (3.1) in a trivial manner. In general, complications may arise due to the non-trivial intersection structure of . We are still in the territory of toric geometry, however, and topological invariants can be obtained by some simple combinatorics. Let us work in the branch where the fields that transform as under vanish simultaneously.
The D-term ambient space is a toric quiver variety and hence, can be completely described by its fan, which itself is determined by the charge matrix together with values on the nodes (or the -stability criterion). Amongst the Abelian groups, that is, for , respectively, for , one can ignore an overall and we choose to take
with the last Abelian factor quotiented from .
Then, the matrix , in an appropriate arrow ordering, can be written as follows:
where and are row vectors of length with and in all directions, respectively, and similarly,