On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincar\'e polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package"CoulombHiggs.m"implementing the Coulomb and Higgs branch formulae.

Abstract: In previous work we have shown that the equivariant index of multi-centered N = 2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -the Coulomb branch formulafor computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincaré polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae. arXiv:1302.5498v2 [hep-th] 30

Introduction
BPS states in N = 2 supersymmetric string vacua offer a rich playground for exploring the microscopic properties of black holes in quantum gravity. Indeed, such states have dual descriptions as black hole solutions in supergravity at strong coupling, and brane configurations at weak coupling. The Witten index (more precisely, the second helicity supertrace [1,2]) remains unchanged upon varying the string coupling. Computing the index Ω(γ) in both regimes as function of the electromagnetic charges γ can provide non-trivial tests of the equivalence between the microscopic and gravitational descriptions expected in any theory of quantum gravity [3]. This comparison however is complicated by the fact that on the macroscopic side, contributions to the index Ω(γ) originate not only from single-centered black holes with charge γ, but also from multi-centered black holes with constituents carrying charges {α i } such that γ = α i [4,5,6,7]. The total index carried by single and multi-centered black holes is the quantity that should be compared with the index on the microscopic side. In contrast, the gravitational path integral in the near horizon geometry of each black hole gives information only about the index Ω S (α) associated with single-centered black holes [8,9]. Thus it is necessary to express the contribution from multi-centered black holes in terms of the index Ω S (α i ) associated to each center. For this it suffices to compute the index 1 g Coulomb ({α i }, {c i }) of the supersymmetric quantum mechanics of n centers carrying charges {α i } interacting by Coulomb and Lorentz forces (and other forces related by supersymmetry). This 'Coulomb index' was computed in [10,11], leading to a general prescription -the Coulomb branch formula -for expressing the total index Ω(γ) in terms of the single-centered BPS invariants Ω S (α i ). Unlike these single-centered BPS invariants, the Coulomb index g Coulomb ({α i }, {c i }) depends on the moduli at infinity -indeed, this dependence is responsible for jumps of the total index Ω(γ) across walls of marginal stability, providing a physically transparent derivation of the wall-crossing formulae known in the mathematical literature [12,10,13].
An important technical tool in [10,11] was to consider a 'refined' version of these indices, which computes the trace Tr (−1) F y 2J 3 , where J 3 is the angular momentum generator with respect to a fixed z-axis, and the prime denotes the removal of fermion zero mode contributions before taking the trace. Unlike the helicity supertrace Tr (−1) F , the refined index is not a protected quantity in full-fledged string theory and cannot be directly compared with the microscopic results. This generalization was nonetheless necessary, as one could use localization with respect to rotations along the z axis to compute the refined Coulomb index g Coulomb ({α i }, {c i }; y) for any number of centers n, and then take the limit y → 1 at the end to recover the result for Tr (−1) F . Although the description of the fixed points of this action is straightforward, their enumeration requires solving a set of n − 1 algebraic equations in n − 1 real variables, which quickly becomes unpractical as the number of centers increases. The first goal of this paper is to remove this bottleneck and give a completely algorithmic way of computing the Coulomb index g Coulomb ({α i }; {c i }; y), and hence the total refined index Ω(γ; y). 2 Using the connection between multi-centered black hole quantum mechanics and quiver quantum mechanics described in [14] this leads to an explicit expression for the Poincaré polynomial of quiver moduli spaces. The second goal is to establish the equivalence of this Coulomb branch formula with Reineke's formula [15] and its generalizations [16,12] for the Poincaré polynomial of quiver moduli spaces for general quivers with no oriented loop.

Coulomb index from localization
Before explaining our new algorithm, let us briefly review the prescription of [11] for computing the refined Coulomb index g Coulomb ({α i }, {c i }; y). Let Γ be the charge lattice of electromagnetic charges, equipped with the Dirac-Schwinger-Zwanziger (DSZ) symplectic product ·, · ∈ Z. Consider a multi-centered black hole configuration where each center carries charge α i (i = 1 . . . n), with DSZ products α ij ≡ α i , α j between the charges. For fixed values of the moduli at infinity, encoded in the FI parameters c i , n-centered configurations are parametrized by a 2n − 2-dimensional phase space M n ({α ij }, {c i }) [4,5]. The coordinates of M n are the locations of n centers r i up to overall translations, subject to n − 1 constraints 3 ∀i = 1 . . . n , The space M n admits a symplectic form [5,17,18], such that the action of SO (3) rotations on M n is generated by the moment map 2) equal to the angular momentum carried by the configuration. We denote by g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) = Tr (−y) 2J 3 (1.3) the refined index associated to this multi-centered configuration assuming that all the centers are distinguishible from each other, and that each center carries no intrinsic degeneracy. Mathematically, this is the equivariant Dirac index of the symplectic space M n ({α ij }, {c i }) [11,19]. We refer to (1.3) as the Coulomb index of multi-centered black holes with charges {α i }. This in turn can be used to compute the index associated with a general multi-centered black hole configurations in terms of indices Ω S (α i ) carried by individual centers following the procedure described in [11,14]. When there exists an ordering of the charges {α i } such that i ≤ j if and only if α ij ≥ 0, and for generic values of the parameters c i away from walls of marginal stability, the symplectic space M n is compact, and the Coulomb index can be computed by localization with respect to rotations around a fixed axis, using the Atiyah-Bott Lefschetz fixed point 3 The constraint (1.1) for i = n follows from the sum of others using the fact that n i=1 c i = 0. For a general multi-centered black hole system there are additional constraints besides (1.1) coming from the requirement of the regularity of the metric and other fields. However when the central charges of the constituents nearly align, which is the the limit in which the quiver quantum mechanics becomes a good description, these additional constraints are expected to be satisfied automatically [14]. Throughout this paper we shall be working in this limit. formula [10,11]. The configurations which stay invariant under such rotations are collinear configurations, corresponding to solutions of (1.1) lying along the z axis. The result is expressed as (1.4) where the sum runs over solutions to the equations The z 1 = 0 condition fixes the translational zero-mode. The sign ± in (1.4) is given by the sign of the Hessian det(∂ 2 W/∂z i ∂z j ) of the superpotential 6) whose critical points reproduce the conditions (1.5).
For charges {α i } such that no such ordering exists, the space M n may be non-compact, due to the possibility of 'scaling solutions', i.e. a subset of the centers approaching each other at arbitrary short distances [20,7]. In that case, we continue to define the Coulomb index g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) by the localization formula (1.4), although the result can no longer be interpreted directly as the refined index associated to the multi-centered black hole configuration (in particular, it may not be a symmetric Laurent polynomial). Nevertheless it can be used to construct such a refined index following the procedure described in [11,14] and reviewed in §3.1. When the FI parameters c i sit on a wall of marginal stability, the space M n is also non-compact due to the possibility of separating the centers into two or more clusters at arbitrarily large distances, and we do not assign a value to g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) in such cases.

A new algorithm for computing the Coulomb index
Except in very special cases, it is usually impossible to find all solutions of (1.5) explicitly. This is also unnecessary, since the contribution of a given solution of (1.5) to the total index (1.4) depends only on the ordering of the centers, via the angular momentum J 3 = 1 2 i<j α ij sign(z j − z i ) along the z-axis and the sign of the Hessian W . For a small number of centers, it is possible to find approximate solutions numerically, and determine both J 3 and the sign of W , however this becomes quickly unpractical as the number of centers grows. Moreover, the brute force enumeration of solutions of (1.5) does not take into account the fact that there can be cancellations between different solutions with the same ordering. To exploit this fact, it is useful to associate a permutation σ to each solution to (1.5), such that i < j iff z σ(i) < z σ(j) . Definingα i = α σ(i) , x i = z σ(i) andc i = c σ(i) , solutions of (1.5) correspond to critical points of in the physical region 0 ≡ x 1 < x 2 < x 3 · · · < x n . (1.8) Reorganizing (1.4) as a sum over all permutations σ of 1, 2, . . . n, we obtain (1.9) where s(σ) is the sum of the sign of the Hessian of (1.7) over each critical point in the physical region. In particular, s(σ) is insensitive to pairs of solutions of (1.5) with the same ordering, which may appear under small perturbations of the parameters α ij and c i , as long as we stay away from walls of marginal stability in the space of FI parameters {c i } and away from certain 'scaling walls' in the space of DSZ products {α ij } described in more detail below.
The first aim of this paper is to develop an explicit algorithm for computing s(σ) and hence the Coulomb index (1.9) for generic DSZ products {α ij } and FI parameters {c i }. This is achieved in §2, where we prove an inductive formula (2.20) for the indexed number s(σ) = F ({α 1 , · · ·α n }, {c 1 , · · ·c n }) of critical points of the superpotential (1.7), by exploiting its robustness under changes of the DSZ productsα ij . For quivers without oriented loops and generic products, we arrive at the following completely explicit result: (1.10) where Θ is the Heaviside function and the sum runs over all permutations σ of 1, 2, · · · n. If some of the α ij 's vanish then we need to deform them away from zero such that the deformed quiver continues to satisfy the no loop condition, compute the result using (1.10) and then take the limit where the deformations are taken back to zero.
As described in (2.20), in the presence of loops the expression for s(σ) picks up additional contributions ∆F A given in (2.32), which depend on the index F with fewer centers and another auxiliary quantity G(α 1 , · · ·α n ). The latter counts the (indexed) number 4 of collinear scaling solutions, i.e. solutions of (1.5) with c i = 0 which may arise when the total angular momentum 1 2 i<jα ij vanishes. We find that this 'scaling index' can itself be computed inductively using (2.38). These formulae hold when the DSZ productsα ij are generic, but we show that even when this is not the case, all the relevant physical quantities can be computed in terms of limits of these formulae.

Quiver quantum mechanics and pure-Higgs invariants
While the refined index is not protected in full-fledged string theory, it is protected in the context of N = 4 supersymmetric quiver quantum mechanics, which describes the dynamics of certain multi-black hole bound states around special loci in moduli space where the central charges of the constituents become nearly aligned [5,7]. Thus, by considering a black hole whose dynamics in some region of the moduli space is described by a specific quiver quantum mechanics, we can use our general Coulomb branch formula for multi-black hole bound states to parametrize the refined index in the corresponding quiver quantum mechanics in terms of single-centered BPS invariants [14]. For quivers without oriented loops, the single-centered BPS invariants are trivial, and the Coulomb branch formula is completely explicit. We can then use this to establish the equivalence of the Coulomb branch formula for such quivers with explicit formulae for the cohomology of the moduli space of stable quiver representations known in the mathematical literature [15,16,21,22].
Before explaining our results, let us briefly review the relation between quiver quantum mechanics and multi-centered black holes [14]. N = 4 supersymmetric quiver quantum mechanics can be obtained by dimensionally reducing an N = 1 supersymmetric gauge theory in 3+1 dimensions 5 -containing vector multiplets in the adjoint representation of the gauge group G = i=1...K U (N i ) and chiral multiplets in bi-fundamental representations of U (N i ) × U (N j ) -down to 0+1 dimensions. The scalars coming from the chiral multiplets are called the Higgs variables and those from the vector multiplets are called the Coulomb variables. The vacuum moduli space of the Higgs variables at zero values of the Coulomb variables is equivalent to the moduli space M of stable quiver representations (in short quiver moduli space), where the stability condition is determined by the FI parameters ζ 1 , . . . ζ K for each U (N i ) factor. BPS states are in one-to-one correspondence with cohomology classes in H * (M; Z), and the angular momentum is identified with the component J 3 = (p − d)/2 of the Lefschetz SU (2) action on the total cohomology H * (M; Z). Thus the refined index is given by the 'Poincaré-Laurent polynomial' 6 11) where b p (M) are the Betti numbers and d is the complex dimension of M. But the same spectrum may also be calculated by first integrating out all the Higgs variables and considering the effective theory for the Coulomb variables. The latter turns out to be given by 5 As an aside it should be noted that N = 4 supersymmetric quiver quantum mechanics is also useful in computing the spectrum of BPS states in N = 2 supersymmetric gauge theories in 3+1 dimensions [23,19]. Indeed, the BPS spectrum of many gauge theories can be understood in the language of quiver representations [24]. For example in the context of N = 2 supersymmetric pure SU(2) gauge theory, the role of single-centered 'black holes' is played by the monopole and the dyon in Seiberg-Witten theory, which are stable for all values of the moduli and whose bound states generate the complete BPS spectrum. 6 We use this terminology since Q(M; y) differs from the usual Poincaré polynomial p b p y p by a y → −y transformation and a multiplicative factor of (−y) −d .
the same quantum mechanical system as that of multi-centered black holes in N = 2 supersymmetric string theory, upon identifying the charge vector γ with the dimension vector (N 1 , . . . N K ) [5], and the DSZ product γ ij between the basis vectors γ i = (0, . . . , 1, . . . , 0), where the only non-vanishing entry occurs in position i, with the number of arrows from the i-th to the j-th node of the quiver. The Coulomb branch formula of [11] can thus be used to express the Poincaré-Laurent polynomial (1.11) in terms of certain 'single-centered BPS invariants' Ω S (α), which are (conjecturally) independent of the FI parameters and of the fugacity parameter y [14]. In simplest cases, Ω S (γ) enters just as an additive constant in Q(M; y) = Ω(γ), and can be identified as the Lefschetz singlet contribution to the total cohomology H * (M; Z) [25,26,14,27]. In general however, the single-centered BPS invariants Ω S (α) enter in Q(M; y) in a more complicated fashion. It is a very interesting open problem to identify the corresponding 'absolutely stable' classes in H * (M; Z).
For quivers without oriented loops, the only non-vanishing single-centered BPS invariants are those associated to the basis vectors γ i , and for such vectors Ω S (γ i ) takes the value 1. In that case, the Coulomb branch formula gives a completely explicit result for Q(M; y), which can be compared to known results in the mathematical literature. In particular, for primitive dimension vector γ (i.e. such that all N i are coprime), Reineke's formula [15] gives another completely explicit result for Q(M; y). For Abelian quivers (i.e such that all N i are one), the Coulomb branch formula equates the Poincaré-Laurent polynomial Q(M; y) with the Coulomb index g Coulomb ({γ 1 , · · · γ n }; {ζ 1 , · · · ζ n }; y) given in (1.10). In §4.2 we show the equivalence of the Reineke's formula and Coulomb branch results for Abelian quivers, generalizing previous arguments given in the context of wall-crossing [10,28]. For non-Abelian quivers with primitive dimension vector, the equivalence of the Reineke's formula and Coulomb branch results can be reduced to the Abelian case, by using the Abelianization property satisfied by Reineke's formula [28,21] (see (4.18)

below).
For non-Abelian quivers with non-primitive vector, Reineke's formula no longer computes the Poincaré-Laurent polynomial Q(M; y) of the quiver moduli space M (which is singular due to marginal bound states), but rather the 'stack invariant' G Higgs . It is conjectured in the mathematical literature that a bone-fide Poincaré-Laurent polynomial Q(M; y) may be reconstructed from the stack invariants G Higgs [16,12] (see Eq. (4.1) below), but it is unclear in general how to construct a smooth moduli space M whose cohomology would agree with G Higgs . At any rate, using a further property of Abelian stack invariants g Higgs established in [22] (see (4.22) below), we prove that the Coulomb branch formula agrees with the Poincaré-Laurent polynomial Q(M; y) computed from Reineke's formula for the stack invariants. The result can be written as 12) where µ(m) is the Möbius function, {( γ j ) k ( ) j } denotes that we have k ( ) j nodes each carrying charge γ j for ≥ 1 and 1 ≤ j ≤ n, and it is understood that in computing g Coulomb whenever some α ij vanishes we need to deform it away from that value to produce an Abelian quiver without loop, and then use (1.10) for computing it.
For brevity we shall henceforth refer to Reineke's formula and its generalizations as the Higgs branch formulae, since they compute directly the cohomology of the Higgs branch moduli space. For quivers with loops, the Higgs branch formula will refer to the result of any computation of the cohomology of the Higgs branch moduli space, although no general formula is available in such cases.

A Mathematica package for quiver invariants
It should be clear from the above that the Coulomb branch formula produces a parametrization of the Poincaré-Laurent polynomial of any quiver in terms of single-centered BPS invariants in a completely combinatoric way. However, even for moderately complicated quivers the necessary computations quickly become tiresome, and are best implemented on a computer. We have implemented the Coulomb branch formula as well as Reineke's formula for stack invariants (and many related other routines) in a mathematica package "CoulombHiggs" available from arXiv and described in Appendix A, which we hope will facilitate studies of single-centered BPS invariants. This package has been successfully tested on the examples investigated in [14] and many more.

A formula for the Coulomb index of multi-centered black holes
In this section, we establish a recursive algorithm for computing the Coulomb index g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) for general charge configurations α i , away from the walls of marginal stability in the space of FI parameters c i . We start in §2.1 with charge configurations for which there exists a possible ordering of the α i 's such that and with all DSZ products α ij non-zero. It follows from the discussion in §1 that this corresponds to an Abelian quiver without any oriented loop. For such systems we obtain the simple formula (2.9) for the indexed number s(σ) that enters (1.9). In §2.2 we show that for the purpose of computing the total Coulomb index, the same result can be used even when some of the α ij 's vanish. In §2.3, we turn to general multi-centered black hole configurations for which the charges do not admit an ordering satisfying (2.1), and establish an inductive formulae for computing s(σ). This is given by During this analysis we also derive a similar formula for the coefficient s(σ) for collinear scaling solutions [20,7] for which all the FI parameters vanish and the α i 's satisfy i<j α σ(i)σ(j) = 0. The corresponding inductive formula for s(σ), called G(α σ(1) , · · · α σ(n) ), is given in (2.38). Combining these results and summing over all permutations yields a general algorithm for g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) with non vanishing DSZ products, as summarized in (2.33).
2.1 Abelian quivers with no oriented loops and all α ij = 0 We start by considering the case of charge configurations which admit an ordering such that α ij > 0 for i < j. A special case arises if all α i are positive linear combinations of two charge vectors γ 1 , γ 2 with γ 1 , γ 2 > 0, as is the case for multi-centered configurations relevant for wall-crossing [10]. Defining we can express (1.7) as Each y i takes value from 0 to ∞, but by an appropriate coordinate transformation u i = f (y i ) we can bring the range to 0 ≤ u i < 1 for each i. We can regard the space spanned by the u i 's a unit box. Our goal is to examine the condition under which W has an extremum with respect to the y i 's in the interior of this box, ı.e. there is a solution to the equation ∂W/∂y k = 0 for every k.
To this aim, let us now consider the following deformation of theα ij 's: In the limit λ → 0, onlyα i,i+1 's remain non-zero and (2.4) takes the simple form Thus the set of equations ∂W/∂y k = 0 has a solution in the range 0 < y k < ∞ if and only if Furthermore the sign of the Hessian of W at this solution is easily determined to be These results can be summarized by saying that for quivers without loops the coefficient s(σ) of y i<jα ij associated with a given permutation {α 1 , · · ·α n } is given, for λ = 0, by where Θ(x) is the Heaviside function. Now consider changing λ from 0 to 1. During this deformation new extrema can appear and disappear in pairs in the interior of the box, but since they are weighted by the sign of the Hessian, the result is unaffected. The other possibility is that new extrema can emerge from (or disappear into) the boundary of the box. This happens when a subset of the y i 's approach 0 and/or infinity. Since ∂W/∂y k approaches the constantd k as y k → ∞ irrespective of the values of the other y i 's, it is clear that away from the marginal stability wallsd k = 0, none of the y i 's can approach infinity. Thus the only possible boundary component where extrema of W can appear or disappear is where a subset of the y i 's vanish.
Let us suppose that as λ is increased from 0 to 1, such a phenomenon takes place at some value λ = λ c . If a new extremum appears at λ c then for λ slightly above λ c , there will be an extremum of W where a subset of the y i 's are small, corresponding to a subset of the centers being close to each other. If on the other hand an extremum disappears as λ approaches λ c from below, then such a configuration exists for λ slightly below λ c . These correspond to onset or disappearance of scaling solutions [20,7], with λ = λ c being the point at which the scaling solution becomes collinear. Now if the quiver corresponding to the original α ij 's had no oriented loop then neither does the quiver associated with the deformed α ij 's, since the signs of all the α ij 's are preserved under the deformation (2.5). This implies that the deformed quiver cannot have a scaling configuration, and hence, as we deform λ from 0 to 1, no extremum of W can emerge from or disappear into the boundary. Thus (2.9) gives the correct contribution to s(σ) even at λ = 1. Using (1.9) we now get (2.10) 2.2 Abelian quivers with no oriented loops but some α ij = 0 We now turn to the case of quiver without oriented loops, but for which some of the α ij 's vanish. For this we first deform all the vanishing α ij 's to non-zero values in such a way that the deformed quiver does not have any oriented loop. To see that this is always possible, let us carry out the deformation one link at a time. We begin with the original quiver without oriented loop and make one of the vanishing α ij 's non-zero. If this leads to a quiver with an oriented loop, then there exists some component C of the original quiver, which, together with the new link, gives rise to a quiver with oriented loop. If so let us flip the sign of α ij of the deformed link. In this case C together with the added link no longer forms an oriented loop. Suppose there were another component C of the original quiver, which combined with the new link would now form an oriented loop. Then C + C would form an oriented loop in the original quiver, whih contradicts our assumption. Thus, by choosing the sign of α ij of the deformed link we can ensure that the new quiver also does not have any oriented loop. We can now repeat the argument and show that all the vanishing α ij 's can be made non-zero and for appropriate choice of sign of the deformed α ij 's the new quiver does not have any oriented loop. Thus we can compute its index by our earlier formula (2.10).
We shall now argue that the index of the original quiver can be obtained by taking the limit of the index of the deformed quiver in which the deformation parameters go to zero. For this we shall work with the total index (1.9) rather than a given permutation. We shall use the original charges α i and use their locations z i -related to the x i by x i = z σ(i) -as the independent variables. The z i 's satisfy (1.5).
Let us now consider the effect of taking α pq to 0 for some specific p, q. For any extremum of W at which the locations of α p and α q remain at finite separation, this limit has no drastic effect and the contribution to the index from such extrema at α pq = 0 is the same as what we get by taking the α pq → 0 limit. Thus we only have to examine the fate of the critical points for which the locations of α p and α q approach each other in the α pq → 0 limit, as generically such critical points will disappear for α pq = 0. For such solutions we can replace z q by z p in (1.5) except in the α pq /|z p − z q | terms, and express (1.5) as ( 2.11) By adding and subtracting the last two equations we get (2.12) The first set of equations and the second equation together correspond to the equilibrium configuration of n − 1 charges in which the charges α p and α q have merged to form a charge α p + α q and the corresponding FI parameters have been added. The last equation can be interpreted as an equation for z q − z p . The existence of a solution to this equation requires When this condition is satisfied then z q − z p is of order α pq for small α pq . On the other hand when α pq = 0, the last equation in (2.12) generically has no solution since the left hand side of the equation becomes independent of z q and all the z i 's for i = q are already fixed by the other equations. This shows that the critical points of W associated with solutions to (2.12) disappear at α pq = 0. This could give rise to a discontinuity in the index at α pq = 0. The important point to note however is that even if (2.13) is satisfied, the solutions to the last equation in (2.12) always occur in pairs, related by a reversal of the sign of z p − z q . The exponent of y in (1.4) remains unchanged under this exchange in the α pq → 0 limit since this exchange only flips the sign of the coefficient of α pq in the exponent. Finally it is easy to check that s(σ) changes sign under this exchange. Thus the contribution from this pair of solutions cancel and we get a smooth α pq → 0 limit. Repeating this analysis for the other α pq 's we see that the index associated with the original quiver can be obtained as the limit of the index associated with the deformed quiver. Note that the above argument breaks down if the solution to the equations in the first two lines of (2.12) automatically satisfy This happens for example when α p and α q are parallel so that α ip /α iq = |α p |/|α q | = c p /c q . In this case at α pq = 0 there is a solution to (1.5) at z p = z q , obtained by solving the equations in the first two lines of (2.12). Now consider the case when α pq is deformed away from 0. In this case in order to look for a solution to the last equation in (2.12) where z p and z q are close to each other we can no longer set z p = z q in the regular terms from the beginning, but must keep terms of order (z p − z q ) in the last equation. If we call this term A(z p − z q ) for some constant A then we can express this equation as This equation is no longer invariant under a change of sign of z p − z q , and in fact has a solution only for one sign of z p − z q irrespective of the sign of α pq . In the α pq → 0 limit this solution smoothly continues to the solution with z p = z q at α pq = 0. Thus we again see that the α pq → 0 limit is smooth, and agrees with the result for α pq = 0.

Generic Abelian quivers with all α ij non-zero
We shall now consider a generic multi-centered black hole configuration with all α ij non-zero, but whose associated quiver may possess oriented loops. In this case we need to take into account possible contributions from scaling solutions. Our goal in this section will be to compute g Coulomb for such configurations.

An inductive formula for the index of collinear solutions
We proceed as in §2.1 and consider the deformation (2.5). It is clear that at λ = 0 the contribution from a given permutation σ will be given by (2.9). Thus we need to investigate the total change in this contribution as λ changes from 0 to 1. As discussed in §2.1 these changes could come from values of λ at which a set A of neighbouring centers come close to each other. This can happen if the total angular momentum carried by this set of centers vanish, i,j∈A;i<jα ij = 0 , (2.16) whereα ij denotes the deformedα ij . To see this, note that in the limit where all y i for i, i + 1 ∈ A approach zero, the part of the superpotential (2.3) involving the y i 's becomes quasi-homogeneous, Differentiating with respect to λ and using ∂W/∂y i = 0 implies (2.16). Since the set A must contain at least three elements there are at most (n − 2)(n − 1)/2 possible sets A, given by the (n − 2)(n − 1)/2 possible ways of choosing the beginning and the end of the set.
Correspondingly there are at most (n − 2)(n − 1)/2 possible values λ A of the deformation parameter λ where such collinear scaling configurations can arise. Using (2.5), the condition (2.16) becomes a linear equation in λ A , The index can jump across λ = λ A if λ A lies between 0 and 1. This is so if and only if the left hand side of (2.18) has opposite signs at λ A = 0 and 1, i.e. i∈A,i+1∈Aα If F ({α 1 , · · ·α n }, {c 1 , · · ·c n }) denotes the coefficient s(σ) of y i<jα ij associated with a given permutation, then we have 20) where F 0 is given by (2.9), and ∆F A is the jump across the critical point λ A . Our goal will be to compute the expression for ∆F A . Let us suppose that set A consists of the integers k, k + 1, · · · . We shall examine the configuration close to the critical point by taking r,s k≤r<s≤ α rs = (2.21) for some small number . We now define z s via 22) and use x 1 , · · · x k , x +1 , · · · x n , y, z k+1 , · · · z −1 as independent variables. Then the relevant equations are given by extremizing ( 2.23) We shall be examining an extremum of W for which y is small, of order . In this case the extrema of W with respect to x 1 , · · · x k , x +1 , · · · x n can be obtained by extremizing ( 2.24) The existence of an extremum of W 1 is equivalent to the existence of a collinear configuration with n − + k centers with chargesα 1 , · · ·α k−1 , s=kα s ,α +1 , · · ·α n and FI parame- Extremization of (2.23) with respect to the parameters z s for k + 1 ≤ s ≤ − 1 can be obtained by extremizing (2.25) The existence of an extremum of W 2 is equivalent to the existence of a collinear scaling configuration of − k + 1 centers with chargesα k , · · ·α and zero FI parameters, with the locations of the centers being at z k = 0, z k+1 , · · · , z −1 and z = 1. Finally y can be obtained by extremizing 7 For genericα ij , a solution to this equation with positive y exists only for one choice of sign of . We shall assume that we have taken the sign of to be such that the solution exists. Let η A denote a quantity which takes value 1 (−1) if the solution exists for λ above (below) the critical value λ A given in (2.18).
Let us now compute the Hessian at this critical point. From (2.24)- (2.26) it is clear that the second derivative of W with respect to all the variables except y remain finite, and we have ∂ 2 W/∂y 2 ∼ s,r k≤s<r≤ α sr /y 2 ∼ /y 2 . Since this is large for y ∼ and all other second derivatives of W remain finite, the full determinant will be given by the product of ∂ 2 W/∂y 2 and the determinant of the Hessian involving the rest of the variables. Furthermore we see from (2.23) that ∂ 2 W/∂x i ∂z s goes to zero as y → 0. Thus the Hessian of W with respect to x i 's and z s 's factorizes into the product of the Hessian of W 1 with respect to x 1 , · · · x k , x +1 , · · · x n and the Hessian of W 2 with respect to z k+1 , · · · z −1 . In our notation the sign of the Hessian of W 1 with respect to x 1 , · · · x k , x +1 , · · · x n is given by F ({α 1 , · · ·α k−1 ,α k + · · ·α ,α +1 , · · ·α n }, {c 1 , · · ·c k−1 ,c k + · · ·c ,c +1 , · · ·c n }). Let be the sign of the Hessian of W 2 with respect to z k+1 , · · · z −1 when the corresponding scaling solution exists; otherwise we take G(α k , · · ·α ) = 0. 8 Then we can write 29) where the last factor imposes the constraint (2.19). Now it follows from (2.5) that at λ = λ A + δλ, (2.31) Substituting this into (2.29) we get ∆F A = F ({α 1 , · · ·α k−1 ,α k + · · ·α ,α +1 , · · ·α n }, {c 1 , · · ·c k−1 ,c k + · · ·c ,c +1 , · · ·c n }) Note that a special case of (2.32) is k = 1, = n in which case the F on the right hand side of this equation is F (α 1 + · · ·α n ;c 1 + · · ·c n ) = 1. If we can compute G(α k , · · ·α ), then we can use (2.20) and (2.32) to compute the function F recursively. Once we know how to compute F , the Coulomb index can be computed as g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y) where the sum runs over all permutations σ.

An inductive formula for the index of scaling collinear solutions
We now turn to the computation of G(α 1 , · · ·α m ), the indexed number of critical points of the superpotential in the range z j > z i for j > i. This coincides with (2.25) under the identifications {α 1 , · · ·α m } = {α k , · · ·α }, and obvious redefinitions of z i . The invariance of the superpotential (2.34) under both translation and rescaling of the z i 's, must be 'gauge fixed' before counting critical points. These invariances were fixed by the conditions z 1 = 0, z m = 1 in (2.22); however in order to compute G(α 1 , · · ·α m ) inductively, it will be more convenient to choose a different gauge z 1 = 0, z m−1 = 1.
Let us now consider the deformation so that the deformedα ij 's (calledα ij ) continue to satisfy i<jα ij = 0. In the limit µ → 0, we can treat the m-th center as a probe in the background of other centers. (2.36) Finally the sign of the Hessian associated with the configuration in the µ → 0 limit, after adding up the contribution from all critical points in the range z j > z i for j > i, is ( 2.42) 3. The scaling configuration involves chargesα k , · · ·α m−1 for 2 ≤ k ≤ (m − 3). In this case we require (2.44) 9 Due to scale invariance a configuration where a subset of the centers get infinitely separated from the others is equivalent to a configuration where a subset of the centers come together.
For each of these cases the computation of ∆G B follows the procedure used for computing ∆F A earlier. We shall quote the final results generalizing (2.32): 1. For µ satisfying (2.40) for m > 4 we have The case m = 4 requires special attention and will be discussed later.
2. For µ satisfying (2.42) with k > 2 we have ( 2.46) The case k = 2 requires special treatment and will be discussed below.
3. For µ satisfying (2.44) we have (2.47) The case where the scaling configuration involves chargesα 2 , · · ·α m requires a special treatment. If we naively consider this as a special case of (2.46) above with k = 2 or of (2.45) for m = 4, we would conclude that the jump vanishes since there are no scaling solution with two centers and hence G(α 1 ,α 2 + · · ·α m ) vanishes. However notice that in this case (2.41) with k = 2 or (2.39) for m = 4 implies m j=2α 1j = 0 (2.48) and hence the two centered configuration with one center with chargeα 1 and the other center with chargeα 2 + · · ·α m is on a wall of threshold stability 10 . As such configurations exist for vanishing FI parameters, we need to analyze the situation more carefully by working at µ = µ B + δµ where µ B is the critical value of µ at which eq.(2.41) is satisfied for k = 2. At this point we have In either case we can proceed to analyze the system following a similar kind of analysis used in computing ∆F A . We denote the locations of the centers as z 1 , z i = z 2 + y w i for 2 ≤ i ≤ m with w 2 ≡ 0, w m ≡ 1, and look for solutions with y ∼ δµ. The solution for z 2 and y are found by extremizing with respect to z 2 and y, and the w i 's are given by extremizing with respect to w 3 , · · · w m−1 . The extremization of W 2 with respect to all the w k and the sign of the corresponding Hessian gives G(α 2 , · · ·α m ). On the other hand the extremization with respect to z 2 and y gives identical conditions 11 (2.53) Following a logic similar to that for ∆F A and using (2.49) we find that at this critical value, the scaling index G jumps by = Θ (−α 23 (α 23 +α 34 +α 24 )) × sign(α 24 +α 34 ) G(α 2 , · · ·α 4 ) for m = 4 .

(2.54)
This gives a recursive procedure for calculating the scaling index G(α 1 , · · ·α m ), and therefore the total index F using (2.20), (2.32). The recursion is initialized by the result for three centers, given below.

Quiver invariants
In this section we shall describe how the results of the previous sections can be used to give a complete prescription for computing the Poincaré-Laurent polynomial of quiver moduli spaces. For this we need to briefly review the prescription given in [14].

Quiver Poincaré-Laurent polynomial from Coulomb index: a review
We shall consider a quiver with K nodes with a U (N ) factor at the -th node, γ k arrows from the -th node to the k-th node representing γ k number of (N ,N k ) representations of U (N ) × U (N k ) and FI parameters ζ 1 , · · · ζ K satisfying N ζ = 0. A negative γ k indicates γ k ≡ −γ k number of (N , N k ) representations of U (N ) × U (N k ). Instead of considering one specific quiver at a time it turns out to be more convenient to consider the family of quivers labelled by different ranks {N } and different values of FI parameters {ζ }. For this we assign to each node a basis vector γ = (0, . . . , 0, 1, 0, . . . 0) -with 1 inserted at the -th position -in an abstract vector space Z K , denote by Γ ⊂ Z K the collection of vectors γ = K =1 N γ where N are non-negative integers, and by C γ the hyperplane K =1 N ζ = 0 in the space of real vectors ζ = K =1 ζ γ ∈ R K . We also introduce a symplectic inner product (3.1) between the elements γ = K =1 N γ of Γ. To any vector γ ∈ Γ and ζ ∈ C γ , we associate a quiver Q(γ, ζ) with K nodes, γ k arrows connecting the node to the node k, gauge group U (N 1 ) × U (N 2 ) × · · · U (N K ), and FI parameters {ζ 1 , · · · ζ K }. If some of the N 's vanish we just drop the corresponding nodes.
Let Q(γ; ζ; y) be the Poincaré-Laurent polynomial where d is the complex dimension of the moduli space M of the quiver Q(γ; ζ) and the b p 's are the topological Betti numbers of M. The Coulomb branch formula for Q(γ; ζ; y), which we denote by Q Coulomb (γ; ζ; y), takes the form: ( 3.3) The first line is the standard relation between integer and rational BPS invariants which has appeared in a variety of contexts [29,10,30,19]. µ(m) is the Möbius function, which is 1 (−1) if m is a square-free positive integer with an even (odd) number of prime factors, and 0 if m is not square-free. In the second line, |Aut({α 1 , · · · α n })| is a symmetry factor given by k s k ! if among the set {α i } there are s 1 identical vectorsα 1 , s 2 identical vectorsα 2 etc., and m|α means that m is a common divisor of (n 1 , · · · , n K ) if α = n γ . The sums over n and {α 1 , · · · α n } in the second equation label all possible ways of expressing γ as (unordered) sums of elements α i of Γ. The coefficients c i are determined in terms of the FI parameters ζ i by c i = A i ζ whenever α i = A i γ . From the restrictions i α i = γ and N ζ = 0 it follows that i c i = 0. The Coulomb indices g Coulomb ({α 1 , · · · , α n }; {c 1 , · · · c n }; y) can be computed from (2.33). The functions Ω tot (α; y) are expressed in terms of the single-centered BPS invariants Ω S through Ω tot (α; y) = Ω S (α; y) +   (3.3), n = 1, α 1 = i k i β i , m 1 = 1 in the second equation in (3.3), and m i = k i in the expression for Ω tot ( i k i β i ; y) in eq. (3.4). We fix H({β i }; {k i }; y) by demanding that the net coefficient of the product i Ω S (β i ; y k i ) in the expression for Q Coulomb ( i k i β i ; y) is a Laurent polynomial in y. This of course leaves open the possibility of adding to H a Laurent polynomial. This is resolved by using the minimal modification hypothesis, which requires that H must be symmetric under y → y −1 and vanish as y → ∞ [11]. We determine H({β i }; {m i }; y) iteratively by beginning with the H's with three β i 's and then determining successively the H's with more β i 's.

3.
H is expected to be independent of the FI parameters and hence can be calculated for any value of these parameters.

4.
After determining H({β i }; {k i }; y) in this way, we set Ω S (γ ; y) = 1 for 1 ≤ ≤ K. For all other charge vectors β, Ω S (β; y) are fixed integers, independent of y and of the FI parameters, which are left undetermined by the Coulomb branch analysis. Since these unknown constants, as well as the quivers, are labelled by the vectors α ∈ Γ, there is one 12 unknown constant for each quiver. This can be fixed e.g. by computing the Euler character of the quiver moduli space for any convenient value of the FI parameters.
As a special case of our result we can consider the case of a general Abelian quiver. This corresponds to γ = N γ with N = 0 or 1. As a result γ, as well as the α i 's appearing on the right hand side of (3.3), are primitive vectors and the α i 's are all distinct. Thus (3.3), (3.4) simplifies to Ω tot (α i ; y) , (3.5) where H({β i }; y) ≡ H({β i }; {1, 1, · · · 1}; y). The functions H({β i }; y) are determined by requiring that they vanish as y → 0, ∞, are invariant under y → y −1 , and that the coefficient of i Ω S (α i ; y) in the expression for Q Coulomb (γ; ζ; y) is a positive integer for each set {α i }.

Coulomb index for non-generic charges
The formulae (3.3), (3.4) are completely explicit provided we have an explicit algorithm for computing g Coulomb ({α 1 , · · · α n }; {c 1 , · · · c n }; y). We have given such an algorithm in the previous sections for generic α ij 's, e.g. all α ij 's non-zero and no ordered subsetᾱ 1 , · · ·ᾱ s of the α i 's satisfying 1≤k< ≤sᾱ k = 0. We have also assumed that the FI parameters stay away from the walls of marginal and threshold stability so that e.g. the quantities k i=1 c σ(i) appearing in (2.10) never vanish. However we need g Coulomb for non-generic α ij 's and c i 's as well. These come from two sources. First of all the γ k 's of the original quiver themselves may be non-generic with some γ k 's vanishing or satisfying special relations. Second, even if the original γ k 's are generic, in the argument of g Coulomb we may have parallel α i 's. For these the corresponding α ij 's will vanish. Also when the total dimension vector {N 1 , · · · N K } is non-primitive, we shall encounter g Coulomb in (3.3) for which the FI parameters sit on the threshold stability walls. In all such cases we need to evaluate g Coulomb by first deforming the α ij 's and/or c i 's to generic values and then taking the limit back to the original configuration. The goal of this section will be to determine a prescription for such deformations. We shall first describe the prescription and then justify it.
1. To deal with the first problem we deform the γ k 's to where 1 is a small positive number and ξ k 's are random numbers between −1 and 1 satisfying ξ k = −ξ k . This will make all the γ k generic. 13 2. At this stage in any given term in the sum in (3.3), all the α ij 's are generic except for subsets of α i 's which are all parallel and/or equal. In computing g Coulomb ({α i }, {c i }; y) for a set of α i 's we consider an arbitrary ordering 14 of all the α i 's and deform them by where 2 is a small positive number that is parametrically smaller than the previous paramater 1 , β ij 's for i < j are randomly chosen positive numbers between 0 and 1 with β ji = −β ij , and f i 's are random numbers satisfying i f i = 0. Under such a deformation any subset of the {α i }'s which are parallel and/or equal to each other get deformed in such a way that the corresponding subquiver does not contain any oriented loop. Also the FI parameters move away from threshold stability walls even if the undeformed configuration sits on such a wall.
3. At the end of the second step the α ij 's and c i 's are generic and can be used to compute g Coulomb . In particular the s(σ) factors in g Coulomb are computed using the deformed α ij 's and c i 's. However in computing the y i<jα ij factors in g Coulomb we use the undeformed α ij 's, since at the end of the computation we are in any case supposed to take the α ij 's to their undeformed values. This g Coulomb is then used to compute Q Coulomb (γ; ζ; y) via (3.3).
In order to prove the validity of the procedure we need to argue that the deformed result reduces to the undeformed one in the limit when the deformations are switched off. First note that there is a qualitative difference between the deformations generated by 1 and those generated by 2 . For the latter the deformation of the α ij 's and c i 's we use is specific to the α i 's and c i 's which appear in the argument of a given g Coulomb . As a result we need to establish that each g Coulomb returns to its undeformed value upon switching off this deformation. On the other hand, the deformation generated by 1 can be carried out for the full index Q Coulomb (γ; ζ; y) as it applies to the whole family of quivers and does not refer to any specific multi-centered black hole configuration. We shall indeed argue that while the individual g Coulomb 's in the 1 deformed system do not necessarily reduce to the undeformed result, the total index does.
Let us begin with the 2 deformation. This contains two parts: deformation of the α ij 's and deformation of the c i 's. First consider the effect of deforming the c i 's. If the initial configuration is away from the walls of marginal and threshold stability then this deformation has no effect. However when γ = i α i is not primitve, it could happen that the set {α 1 , · · · α n } can be divided into two or more sets such that the sum of the α i 's in each set is parallel to the total charge γ. In this case the sum of the c i 's in each set vanishes and the FI parameters sit on the wall of threshold stability. The deformation of the c i 's given in (3.8) is needed to move away from this wall and make g Coulomb well defined, but the result does not depend on how we deform the c i 's.
Next we turn the effect of the 2 deformation on the α ij 's. We begin with the deformed system and take the α ij 's one by one back to their values after the first deformation. During this process g Coulomb can jump if two or more centers come together during the deformation. Using the analysis of §2.2 we know that the possible jumps in g Coulomb could arise if during the deformation a subset A of the centers can come in the collinear scaling configuration by having i<j;i,j∈A α ij = 0. Since at this stage we have already carried out the 1 deformation making the γ k 's generic; the possible subsets where this could happen will only involve the centers carrying equal or parallel charges at the end of the first deformation. However this is ruled out by the fact that the second deformation has been choosen so that any subquiver, containing equal or parallel charges at the end of the first deformation, remains free from oriented loops. This shows that we do not encounter any collinear scaling solutions during the second deformation and hence there is no jump in the index g Coulomb during this deformation.
Finally we turn to the 1 deformation. To deal with this case we note that the analysis of §2.2, showing that the refined index of a quiver changes continuously under de-formations of the α ij 's, breaks down on a subspace on which i,j∈A;i<jα ij = 0 for some subset A. Around this subspace the Coulomb index computed for a given set of charges will depend on the sign of the deformation parameters since a particular collinear solution may exist for one sign of the deformation, but as the deformation parameter approaches zero the centers in the subset A come close together and for the opposite sign of the deformation parameter the solution ceases to exist. However we shall now argue that Q Coulomb (γ; ζ; y) computed from (3.4) remains independent of the sign of the deformation. For this suppose that we have three centers carrying charges α 1 , α 2 and α 3 such that α 12 + α 23 + α 13 = 0, α 12 , α 23 > 0. In that case for the permutation (123) we can have a collinear scaling solution. Now in the deformed system whether this permutation contributes or not depends on whether α 12 , α 23 and α 31 form an oriented triangle or not, and this in turn will depend on the details of the deformation. The difference in g Coulomb ({α 1 , α 2 , α 3 }; {c 1 , c 2 , c 3 }; y) that we shall get between these two cases is, up to a sign, given by (y − y −1 ) −2 since y i<j α ij = 1. In the expression for Q Coulomb (α 1 + α 2 + α 3 ; ζ; y) there will be a term proportional to g Coulomb ({α 1 , α 2 , α 3 }; {c 1 , c 2 , c 3 }; y)Ω S (α 1 )Ω S (α 2 )Ω S (α 3 ) and the ambiguity in g Coulomb described above will lead to an ambiguity proportional to (y − y −1 ) −2 Ω S (α 1 )Ω S (α 2 )Ω S (α 3 ) in the expression for Q Coulomb (α 1 + α 2 + α 3 ; ζ; y). Now the expression for Q Coulomb (α 1 + α 2 + α 3 ; ζ; y) will also contain a term proportional to H({α 1 , α 2 , α 3 }; y)Ω S (α 1 )Ω S (α 2 )Ω S (α 3 ), and the function H({α 1 , α 2 , α 3 }; y) is determined by requiring that H vanishes as y → 0, ∞ and that the net coefficient of Ω S (α 1 )Ω S (α 2 )Ω S (α 3 ) is a polynomial in y, y −1 . Since (y − y −1 ) −2 → 0 as y → 0, ∞ we see that the ambiguity in g Coulomb introduced above is absorbed completely into the function H and does not lead to any ambiguity in the expression for Q Coulomb (α 1 , α 2 , α 3 ; ζ; y). This argument can be easily generalized to argue that all the ambiguities in g Coulomb can be absorbed into the functions H and the expression for Q Coulomb (γ; ζ; y) is independent of the choice of the deformation of the γ k 's for general vector γ.
Using the analysis above we can also answer the question: how small should 1 and 2 be for the procedure described above to hold? The general rule is that if α ij denotes the deformed α ij , then during the deformation we should not have i<j;i,j∈A α ij = 0 for any ordered subset A of the centers. This means that we should not encounter any collinear scaling configurations during the deformation except possibly at the beginning. 15 In [14] we also proposed a formula for the Dolbeault polynomial where h p,q are the Hodge numbers of M. The formula took the same form as (3.3), (3.4), with the only difference that Ω S was allowed to depend on t, and in (3.3), (3.4), all factors of Ω S (α; y m ) were replaced by Ω S (α; y m ; t m ). Eventually we drop the y-dependence of Ω S , but they continue to depend on t, giving a t-dependent formula for Q(γ; ζ; y, t).

Coulomb/Higgs equivalence for quivers without loops
For quivers without oriented loops, quiver invariants can be computed using the Harder-Narasimhan recursion, or equivalently using Reineke's solution to this recursion [15]. In this section we shall show that for such quivers the Coulomb branch formula (3.3) agrees with Reineke's formula, both for Abelian and non-Abelian quivers.

Reineke's formula for quivers without loops
As reviewed in [14], the Poincaré-Laurent polynomial of a quiver Q without oriented loops can be computed using the Harder-Narasimhan recursion. Henceforth we shall denote the Poincaré-Laurent polynomial computed by this method by Q Higgs ( N γ ; ζ; y), to distinguish it from the Coulomb branch formula (3.3). The expression for Q Higgs takes the form [12], [16,Theorem 6.8] The sum over { M (i) } runs over all ordered partitions of N ≡ (N 1 , · · · N K ) into parallel vectors M (1) , · · · M ( ) , and G Higgs ({M (i) 1 , · · · M (i) K }; {γ 1 , · · · γ K }; {ζ 1 , · · · ζ K }; y) is the 'stack invariant' associated to the quiver with dimension vector M (i) . If M (i) is primitive then the quiver moduli space is smooth and G Higgs is just its Poincaré-Laurent polynomial, however in general it is a rational function of y, not necessarily invariant under y → 1/y. In all cases however, we assume that it is given by Reineke's formula [15], 16 where the sum over partitions in (4.2) runs over all ordered partitions of the vector (M 1 , · · · M L ) into non-zero vectors {(N a 1 , · · · N a L ), a = 1, . . . , s} for s = 1, . . . , for all b between 1 and s − It is worth noting that (4.1) can be inverted to express the stack invariants in terms of the rational Poincaré-Laurent polynomials [16,Theorem 6.8]: Higgs (α i ; ζ; y) , (4.5) where again the sum over {α i } runs over all ordered partition of N γ into parallel vectors α 1 , α 2 , · · · α k . We shall refer to the formula (4.1) (or equivalently (4.5)) and (4.2) as the Higgs branch formula for the Poincaré-Laurent polynomial Q(γ; ζ; y). Our goal will be to show the equality of this Higgs branch result with the Coulomb branch formula (3.3).
If N γ is a primitive vector, ı.e. the N 's have no common factor other than unity, then (4.1) reduces to a simple form In this case G Higgs is a symmetric Laurent polynomial since Q Higgs is.
• For each such breakup of {σ(1), · · · σ(K)} into increasing subsequences, we can associate a partition of γ 1 + · · · γ K into vectors β (a) as follows: (4.11) This generates an ordered partition of γ but does not necessarily satisfy (4.9). It is easy to show that with this choice the power of y in y i<j γ σ(i)σ(j) in (1.9) matches the power of y inside the sum in (4.8) For illustration we can consider the permutation σ(1234) = (3412 The above analysis shows how to each permutation σ we can associate a set of ordered partitions of γ 1 , · · · γ K into vectors β (1) , · · · β (s) . The converse is also true -all ordered partitions of the vectors (γ 1 , . . . , γ K ), before imposing (4.9), are in one to one correspondence with the set of all increasing subsequences of all the permutations of (12 . . . n) via the rule (4.11). For a given permutation we shall call an increasing subsequence maximal if it is not possible to build bigger increasing subsequences involving the elements of the subsequence. In the example described above the maximal increasing subsequences are {{34}, {12}}. The complete contribution to the coefficient of y i<j γ σ(i)σ(j) in the Reineke formula can be generated by beginning with the maximal increasing subsequences associated with the permutation σ and combining them with the contribution from other increasing subsequences associated with the same permutation, but we must be careful to pick only those partitions which satisfy (4.9). As shown in [10], a given permutation contributes if and only if its maximal increasing subsequences generate a partition via (4.11) satisfying (4.9), and none of the other (non-maximal) increasing subsequences generate an allowed partition. 17 The sign of the contribution is given by (−1) s−1 where s is the number of maximal increasing subsequences.

(4.16)
For charge configurations relevant for wall-crossing, where all the γ 's lie in a plane and the ζ 's are determined in terms of γ ij , this equivalence was proved by induction in [28]. Here we have shown that it holds for any Abelian quiver without oriented loop.

Non-Abelian quivers with primitive dimension vector and without loops
For non-Abelian quivers without loops we still can set Ω tot (α; y) = Ω S (α; y). We also have Ω S (α; y) = 0 unless α = γ , and Ω S (γ ; y) = 1. Thus Ω S (α i /m i ; y m i ) is non-vanishing (and equal to 1) only when α i /m i is a basis vector. Furthermore for primitive charge vector, Q Coulomb (γ; ζ; y) is equal toQ Coulomb (γ; ζ; y). In that case we can express (3.3) as The argument of g Coulomb given above corresponds to a choice of {α 1 , · · · α n } where we have k ( ) j copies of the vector γ j for = 1, 2, 3, · · · , j = 1, · · · K, and the FI parameter associated with each such copy is ζ j . It is understood that if some of the α ij 's appearing in the argument of g Coulomb vanish then we must deform them in a way so that the deformed quiver does not contain any oriented loop, compute g Coulomb and then take the deformations back to zero. The i k ( ) i ! factor in the denominator represents |Aut({α 1 , α 2 , · · · , α n })|. On the other hand, it was shown in [10,21]  • The vertices ofQ are obtained by replacing the vertex i of Q by a collection i ,k of vertices with k = 1, . . . k for any in the partition N i = k , and keeping all the other vertices j = i of Q; • Each arrow j → k in Q with j, k = i induces an arrow j → k inQ; inQ for each , k; • The dimension at each node i ,k inQ is equal to one, hence justifying the name; the dimensions at the other nodes j = i are the same as the dimensions N j in the original quiver Q; • The FI parameter at each node i ,k inQ is equal to ζ i ; the FI parameters at the other nodes j = i are the same as the FI parameters ζ j in the original quiver Q.
Thus for example on the right hand side of (4.18) the set {N j } will include all the N j 's except N i and k number of 1's, the set {γ j } will include all the γ j 's except γ i , k 1 copy of γ i , k 2 copy of 2γ i , etc., and the set {ζ i } will include all the ζ j 's except ζ i , k 1 copies of ζ i , k 2 copies of 2ζ i etc. By successive application of this formula we can express the stack invariant of any non-Abelian quiver in terms of that of a family of Abelian quivers: (4.19) Using (4.6) we can replace the left hand side by Q Higgs K i=1 N i γ i ; ζ; y . The resulting equation is identical to (4.17) with Q Coulomb replaced by Q Higgs and g Coulomb replaced by g Higgs . Using the equivalence of g Coulomb and g Higgs proven in (4.16), we get Q Higgs N γ ; ζ; y = Q Coulomb N γ ; ζ; y . (4.20) Thus the Higgs and Coulomb branch formulae are equivalent for non-Abelian quivers as well, so long as the dimension vector {N i } is primitive.

Non-primitive dimension vector
We now turn to the case of a non-Abelian quiver with non-primitive dimension vector. In this case the Coulomb branch formula is still given by (3.3) although Q Coulomb andQ Coulomb are no longer identical. On the other hand the Higgs branch formula is given by (4.1) (or equivalently (4.5)) and (4.2). Our goal in this section will be to prove the equality of Q Coulomb and Q Higgs . When the dimension vector {N i } i=1...K ≡ N is not primitive, the Abelianization formula (4.19) is still known to hold, and can be used to express G Higgs in terms of g Higgs However some of the Abelian stack invariants g Higgs ({γ i }; {ζ i }; y) appearing on the r.h.s. have stability conditions {ζ} lying on walls of threshold stability. This happens when the set {γ i } can be divided into several subsets A 1 , A 2 , · · · such that i∈Asγ i N γ for each s. As a result g Higgs ({γ i }; {ζ i }; y) can no longer be equated to g Coulomb ({γ i }; {ζ i }; y) (see footnote 18). Let {ζ i } be a sufficiently generic perturbation of the FI parameters away from the wall and sufficiently small so that no other walls of marginal stability are crossed. In this case g Higgs ({γ i }; {ζ i } ; y) can again be equated to g Coulomb ({γ i }; {ζ i } ; y) and furthermore will be independent of the deformation. Now the HN recursion method [31,15] relates g Higgs ({γ i }; {ζ i } ; y) and g Higgs ({γ i }; {ζ i }; y) as follows: g Higgs (α 1 , · · · α n ; c 1 , · · · c n ; y) = k≥1 (y −1 − y) 1−k {As} i∈As c i =0 ∀ s; ∪ k s=1 As={1,2,···n} k s=1 g Higgs ({α i ; i ∈ A s }; {c i ; i ∈ A s }; y) (4.21) where the sum over {A s } runs over all unordered partitions of the integers 1, 2, · · · n into sets A 1 , A 2 , · · · A k subject to the conditions indicated above. The symbol {c i } denotes a generic deformation in which the FI parameters associated with each of the n nodes are deformed independently so that the configuration moves away from the wall of threshold stability, subject to the constraint that the sum of the FI parameters carried by all the nodes must vanish. It was shown in Ref. [22] that the combinatorics of the summation in (4.21) can be summarized by an equality of generating functions, which in our notation reads (4.26) Let us now define G Coulomb by a formula analogous to (4.5) withQ Higgs replaced byQ Coulomb on the right hand side: G Coulomb ({N 1 , · · · N K }; {γ 1 , · · · γ K }; {ζ 1 , · · · ζ K }; y) Coulomb (α i ; ζ; y) . (4.27) Since the right hand side matches the right hand side of (4.25), it follows that G Higgs and G Coulomb are identical, and therefore the Higgs and Coulomb branch computations Q Higgs and Q Coulomb are equivalent, even for non-primitive dimension vector. Given the equivalence of Q Higgs (defined by the Harder-Narasimhan recursion (4.1), (4.2)) and Q Coulomb (defined by the Abelianization formula (4.26)), and given the equivalence of the Abelian indices g Coulomb and g Higgs for deformed FI parameters, we conclude that the Poincaré-Laurent polynomial Q Higgs of a quiver without loop satisfies, for arbitrary dimension vector, the Abelianization formula: Q Higgs (γ; ζ; y) = m|γ µ(m) m y − y −1 y m − y −mQ Higgs (γ/m; ζ; y m ) Q Higgs (γ; ζ; y) = (4.30) This can be regarded as the main result of this subsection.

A. The Mathematica package "CoulombHiggs"
To facilitate further investigation, we provide a Mathematica package allowing to compute the Coulomb index g Coulomb for multi-centered black holes and the Poincaré-Laurent polynomial Q Coulomb (γ; ζ; y) and Q Higgs (γ; ζ; y) of quiver moduli spaces using the Coulomb branch and Higgs branch formulae. We also provide three example files where this package is used to evaluate quiver invariants for the Kronecker quiver (Kronecker.nb), for non-Abelian 3-node quivers (Threenode.nb) and for several 4 and 5-node Abelian quivers considered in [14] (Multinode.nb). The validity of the algorithm for the F and G indices is tested in a fourth file CoulombIndexCheck.nb. All these files are included in the "source" of this paper available from arXiv and can be obtained from the second name authors' webpage.
Assuming that the file CoulombHiggs.m is present in the user's Mathematica Application directory, the package is loaded by entering CoulombHiggs v 2.0 -A package for evaluating quiver invariants using the Coulomb and Higgs branch formulae.
If the file CoulombHiggs.m has not yet been copied in the user's Mathematica Application directory but is in the same directory as the notebook, evaluate instead The first main routine is CoulombBranchFormula, whose basic usage is illustrated below: This routine computes the Dolbeault polynomial (3.9) of the quiver moduli space, expressed in terms of the single-centered indices. The first argument corresponds to the matrix of DSZ products α ij (an antisymmetric matrix of integers), the second to the FI parameters ζ i (a vector of rational numbers), the third to the dimension vector N i (a vector of integers). The variables y and t are fugacities conjugate to the sum of the Dolbeault degrees p + q (i.e. the angular momentum) and to the difference of the Dolbeault degrees p − q, respectively. The Poincaré-Laurent polynomial is obtained by setting t = 1. For generic superpotential, the single-centered indices Ω S (γ, y, t) are conjectured to be independent of y. In the above example, the Dolbeault polynomial of the moduli space of a three-node Abelian cyclic quiver with 4 arrows between each subsequent node is expressed in terms of the single-centered index Ω S (γ 1 + γ 2 + γ 3 , y, t). The second main routine is HiggsBranchFormula, which computes the Poincaré-Laurent polynomial using the Higgs branch formula (4.1) (which is only valid for quivers without oriented loop, but the routine works irrespective of this assumption). The arguments are the same as for CoulombBranchFormula: The above command computes the Poincaré-Laurent polynomial for the Kronecker quiver with 3 arrows, FI parameters (1/2, −1/2), dimension vector (2,2). The package allows for much more, as documented below. Inline documentation can be obtained by typing e.g. • y: fugacity conjugate to the sum of Dolbeault degrees p + q (i.e. angular momentum); • t: fugacity conjugate to the difference of Dolbeault degrees p − q; 20 Note the following changes in v2.0: the fugacity y is no longer a parameter of CoulombBranchFormula and QuiverBranchFormula, and the former computes the Dolbeault polynomial in terms of Ω S (α i , t), rather than expressing the Poincaré polynomial in terms of Ω S (α i ). Other changes are highlighted by margin notes below.