Exact Results in Quiver Quantum Mechanics and BPS Bound State Counting

We exactly evaluate the partition function (index) of N=4 supersymmetric quiver quantum mechanics in the Higgs phase by using the localization techniques. We show that the path integral is localized at the fixed points, which are obtained by solving the BRST equations, and D-term and F-term conditions. We turn on background gauge fields of R-symmetries for the chiral multiplets corresponding to the arrows between quiver nodes, but the partition function does not depend on these R-charges. We give explicit examples of the quiver theory including a non-coprime dimension vector. The partition functions completely agree with the mathematical formulae of the Poincare polynomials (chi_y-genus) and the wall crossing for the quiver moduli spaces . We also discuss exact computation of the expectation values of supersymmetric (Q-closed) Wilson loops in the quiver theory.


Introduction
A count of the number of the BPS bound states (index) in supersymmetric theory, including supergravity and superstring theory, is important problem to understand quantum properties of the space-time. For example, the counting of the BPS bound states on D-branes makes relation to the Bekenstein-Hawking entropy of a black hole [1]. Denef proposed that the BPS bound states (multi-centered black holes) in Calabi-Yau compactifications of string theory is interpreted by a suitable quiver quantum mechanics and discussed the correspondence between the bound states of the wrapped D-branes around the Calabi-Yau manifold (Higgs picture) and the multi-centered bound states in the supergravity (Coulomb picture) [2]. These correspondences have been checked for many cases so far and imply that string theory gives the proper description of quantum gravity.
The quantity of the number of the BPS bound states (index) is also interesting mathematically, since it relates to the topological invariants of the manifolds. Therefore, a lot of interesting formulae, which compute the number of the BPS bound states or topological invariants, have been derived. In mathematics, topological properties of the quiver moduli spaces are investigated in [3,4] for example, and they come to fruition of the so-called "wall crossing formula" by Joyce and Song [5], and Kontsevich and Soibelman [6]. The wall crossing formula is physically interpreted and rederived in the Coulomb branch of the quiver quantum mechanics [7,8]. There are many important developments around the wall crossing formula [9][10][11][12][13][14][15][16][17][18][19][20][21][22]. (See also the review of the developments [23].) In this paper, we would like to evaluate the partition function of the quiver quantum mechanics in the Higgs phase by using the localization method. Recently, the index in two-dimensional supersymmetric gauge theory is computed by the localization and Jeffrey-Kirwan residue operation [24,25], and it has been applied to one-dimensional quiver models via a dimensional reduction [26,27]. Here, we derive the index of the supersymmetric quiver quantum mechanics from the beginning in the unpolished way. We reproduce the Poincaré polynomials and wall crossing formalae for the quiver moduli spaces even when the corresponding quiver diagrams only include non-Abelian nodes. Our concrete constructions of the quiver quantum mechanics might also be useful to understand more general matrix quantum mechanics embodying M-theory [28,29].
The localization, which we utilize throughout the paper, reduces the infinitely many dimensional path integral of the partition function to a finite dimensional contour integral with respect to the eigenvalues of an adjoint scalar field. The localization is powerful tool to understand exactly the non-perturbative effects like dualities in the supersymmetric gauge theory. It can be used in order to derive the non-perturbative (instanton) cor-rections in N = 2 supersymmetric Yang-Mills theory [30,31]. In this calculation, the localization fixed points are classified by using the Young diagrams. The fixed point sets in our results are also interpreted in terms of the attractive combinatorial objects.
The organization of this paper is as follows: In section 2, we construct the supersymmetric quantum mechanics with four supercharges by the dimensional reduction from four dimensional N = 1 supersymmetric gauge theory. We construct the BRST charge from a linear combination of the four supercharges and redefine the fields of the theory. For the localization to work, we introduce background gauge fields of R-symmetries, which are also necessary to obtain the partition function in the form of the refined index. We also introduce the physical observables which can be exactly evaluated by the localization. In section 3, we generalize the formulation to the general quiver gauge theory. In section 4, we derive the exact formulae for the partition function and observable of the quiver quantum mechanics by using the localization. We also introduce the equivalent T -character, which is a convenient tool to calculate the residue integrals. In section 5, we give some examples of the quiver quantum mechanics by restricting the gauge groups to the Abelian ones and compute the partition functions and the expectation values of the Wilson loop operators. We find that the dependences of the partition functions on the background gauge fields of the R-symmetries for the chiral multiplets surprisingly disappear, and our results perfectly agree with the Poincaré polynomials of the Higgs branch moduli spaces and the wall crossing formalae. On the other hand, the expectation values of the Wilson loop operators depend on those background gauge fields and become rather complicated polynomials. In section 6, we treat more general cases, which include the non-Abelian gauge groups. By choosing appropriate fixed points carefully, we find that our partition functions again agree with the Poincaré polynomials and the wall crossing formalae. Especially, we find that when the ranks of the gauge groups are non-coprime, the fixed points of the vector multiplets do contribute. The last section is devoted to a summary and future problems. In the appendix, we give a few examples of the wall crossing formalae in a special case.

N = 4 U (N ) supersymmetric quantum mechanics
We first describe an N = 4 U(N) supersymmetric quantum mechanics, which is obtained from the dimensional reduction of four dimensional N = 1 U(N) supersymmetric gauge theory to one dimension. Four dimensional supersymmetric theory originally has U(1) R-symmetry. After the dimensional reduction, a part of the Lorentz symmetry becomes the R-symmetry of the reduced theory. Our supersymmetric matrix quantum mechanics possesses the SU(2) J × U(1) R global R-symmetries [26]. We derive the supersymmetric matrix quantum mechanics explicitly for a vector multiplet part and a chiral multiplet part separately by the dimensional reduction. We follow the conventions used in [32] in what follows.

Vector multiplet
In one dimensional N = 4 supersymmetric theory, a vector multiplet is composed of a gauge field A 0 , three real scalars X i (i = 1, 2, 3), two complex fermions λ α (α = 1, 2), and an auxiliary real scalar D. All fields are in the adjoint representation of U(N). The representations of SU(2) J and U(1) R charges of the vector multiplet are summarized in Table 1. Using these fields, the action is given by where g is the gauge coupling, ζ is the Fayet-Iliopoulos (FI) parameter, and is the covariant derivative. The action (2.1) is invariant under the following supersymmetric transformations: where ξ α represents the supersymmetric parameters, and in terms of the supercharges Q α , the supersymmetric variation is given by To apply the localization method to the theory, we now introduce a linear combination of the supercharges by We call this the BRST charge conventionally in the following. We also should consider the Euclidean theory by the Wick rotation t → −iτ and define linear combinations of the bosonic fields by and for the corresponding fermionic fields, in order to make the localization work well. This operation is usually called "topological twist", but this is nothing but the redefinitions of the supercharges and fields and does not change the original theory in one dimension.
Under the introduced BRST symmetry, the fields are transformed by (2.8) The BRST transformations are nilpotent up to the time translation and (complexified) gauge transformation. The Euclidean action of the theory (2.1) is written as a Q-exact form: where µ R = 1 2 [Z,Z] − g 2 ζ is a (real) moment map constraint which contains the original D-term constraint and describes the moduli space of the vacua. After integrating out the auxiliary field Y R , we obtain the Euclidean action of the original matrix quantum mechanics.
The field redefinitions (topological twist) spoil the original R-symmetries, but the theory is still invariant under the following twisted "R-transformation" U(1) ′ J , which acts on the fields by with an R-transformation parameter θ J . To obtain the refined index, we need a "gauging" of this global R-symmetry, which modifies the moduli space of the theory by induced mass terms. Under the gauged U(1) ′ J symmetry with a constant background A J = ǫ, the τderivatives of Z and λ z are modified into This is known to the Ω-background [30,31]. Thus, we obtain the modified BRST transformations: (2.12) The BRST transformations are nilpotent up to the time translation, gauge transformation including the gauged U(1) ′ J transformation. The action of the modified theory is obtained by replacing simply Q with Q ǫ in (2.9).

Chiral multiplet
Let us now construct the theory which includes chiral multiplets. The chiral multiplet is composed of a complex scalar q, two complex fermions ψ α , and an auxiliary complex scalar F . The chiral multiplet can be taken in an arbitrary representation of U(N). Here, we take it in the fundamental representation for simplicity. (A generalization is straightforward.) The representations under SU(2) J and charges under U(1) R of the chiral multiplet are summarized in Table 2. The action is given by This action is invariant under the following supersymmetric transformations: (2.14) After the Wick rotation, we define the bosonic fields: (2.15) and the fermionic fields: These fields transform under the BRST symmetry by (2.17) Using the BRST charge, the Euclidean action can be written as the Q-exact form: are (complex) moment map constraints associated with the F-term constraints and W (q) is the superpotential. By including the chiral multiplet, the real moment map (D-term constraints) also are modified to After these redefinitions of the fields in the chiral multiplet, the theory possesses the following twisted R-transformations where θ J,R are the R-transformation parameters. As similar as the previous section, we gauge these R-symmetries in the constant backgrounds A J = ǫ and A R =ǫ. The BRST transformations are deformed by Here, we have used the same symbol Q ǫ as that in (2.12), but Q ǫ is regarded as including the whole gauged U(1) ′ J × U(1) ′ R symmetries in the following. Thus, the BRST transformations are now nilpotent up to the time translation, gauge transformation including gauged U(1) ′ J × U(1) ′ R transformations. To summarize, the total action including the chiral multiples is written by a sum of S V and S C , which are Q ǫ -exact. So we can apply the localization arguments (coupling independence) to our models with respect to the BRST charge Q ǫ .

Physical observables
If an operator O is Q ǫ -closed such that Q ǫ O = 0, but not Q ǫ -exact such that O = Q ǫ O ′ , the vacuum expectation value (vev) of the operator O can be evaluated exactly by the localization without changing the coupling independent property. So the operator O belonging to the Q ǫ -cohomology (equivariant cohomology) becomes a non-trivial physical observable. We here discuss the possible observables in the matrix quantum mechanics.
First of all, noting that a combination of the fields is Q ǫ -closed itself because of the BRST transformations (2.12), we find that any gauge invariant function of Φ becomes the physical observable. A possible gauge invariant function made of Φ is a supersymmetric Wilson (Polyakov) loop operator along the Euclidean "time" direction: where P stands for the path ordered product and the trace is taken over the representation R. We can evaluate the vev of W R (Φ) exactly by using the localization in principle. Secondly, another interesting physical observable is obtained from the dimensional reduction of the supersymmetric Chern-Simons (CS) action in three dimensions or BF action in two dimensions, which explains why lower dimensional gauge theories are exactly solvable [33][34][35][36]. In one dimensional model, dimensionally reduced CS type operator becomes We can check this operator is Q ǫ -closed from the BRST transformations (2.12). We can also define a similar Q ǫ -closed operator from the chiral multiplet as follows: Using the similar arguments in [33], we expect that the vev of the Q ǫ -closed operator gives a partition function of a bosonic (non-supersymmetric) matrix quantum mechanics, since where S MQM = dτ Tr Z D τ Z +Z[σ, Z] + iǫZZ +qD τ q +qσq + irǫqq . (2.28) Here we have used the coupling independence of the Q ǫ -exact action and performed the Gaussian integrals of the fermions. We will discuss the exact vevs of the above physical observables later.

Quiver quantum mechanics
Let us now apply the general formulation of the matrix quantum mechanics in the previous section to the quiver quantum mechanics. The quiver gauge theory has a gauge symmetry of a product of gauge groups G = v G v and contains chiral matter multiplets represented between two gauge groups. We assume all gauge groups are unitary groups of rank N v , that is, G v = U(N v ). The structure of the quiver gauge theory is represented by the socalled quiver diagram depicted in Figure 1. A quiver diagram is composed of nodes v and arrows a whose ends are attached on the nodes. A node v represents a vector multiplet in the adjoint representation of U(N v ). An arrow, whose direction is v → w, represents a chiral multiplet in the bi-fundamental representation ( ,¯ ) of U(N v ) × U(N w ). If ends of an arrow are attached on the same node, the chiral multiplet is in the adjoint representation. Figure 1: A generic quiver diagram with 6 nodes.
Following the representations of the quiver quantum mechanics, we see the BRST transformations for the vector multiplet at the node v are where Φ v ≡ σ v + iA v , and the BRST transformations for the chiral multiplet along the arrow a : v → w are If there is no superpotential, the U(1) ′ R symmetry exists independently in each chiral multiplet. Thus, ǫ a is interpreted as a constant U(1) ′ R gauge field which is coupled to the chiral multiplet labeled by a with unit charge.
For later convenience, we introduce a vector notation of the fields. For bosonic and fermionic fields in the vector multiplet at each node v, we define and Here "·" denotes an inner product of the vectors of the fields with a suitable norm. The moment map constraints are now given by where • denotes all nodes which are connected to v by arrows in the quiver diagram, and the FI parameters . If there is no closed oriented loop in the quiver diagram, the corresponding quiver gauge theory cannot possess the superpotential by the gauge invariance. The physical observables discussed in the previous section are generalized to the quiver theory. We can naively define a Q ǫ -closed Wilson loop operator as a product of Wilson loop operators with respect to each node v: However, as we will see, the above Wilson loop generally contains the diagonal U(1) part of the quiver group, which associated with the center of mass integral coordinate. If the Wilson loop has an overall factor like e i dτ φc , where φ c is the center of mass coordinate, the vev vanishes after integrating over the center of mass coordinate. So we should use the center of mass free Wilson loop operators, which are written typically in terms of the "relative" coordinates associated with the arrows a : v → w where R v and R w are suitably chosen to remove the center of mass coordinate. Then, we can evaluate the non-vanishing vev for the Wilson loop operator.
In addition, the sum of the CS-type action, where is Q ǫ -closed in the quiver quantum mechanics. We can derive the partition function of a bosonic quiver quantum mechanics as the vev of the above CS -type Q ǫ -closed operator.

Exact partition function of quiver quantum mechanics
Using the formulation and field redefinitions in the previous sections, we derive a generic formula for the partition function of the quiver quantum mechanics exactly by the localization method. In order to find the relation between the exact partition function and (refined) index of the BPS states, we assume that the Euclidean time direction is compact as τ ∼ τ + β in the following calculations.

Localization
As we have seen in section 2, the reduced supersymmetric Yang-Mills action is written as an exact form of a part of the supercharges. If we denote by B I and F I the vectors of the bosonic and fermionic fields, respectively, we find that a Q-exact supersymmetric Yang-Mills action takes the following form generically: where we have introduced the different coupling constants t and t ′ for the Gaussian and constraint part. A metric g IJ gives a norm on the field variables and χ i are superpartners of the auxiliary fields Y i , which will be the moment map constraints µ i = 0. Note that there exists a Q-closed combination of the fields Φ in addition to B I and F I . We can show that the partition function for the generic Q-exact action (4.1) 2) or the vev of the Q-closed operator is independent of the couplings t and t ′ . So we can take the limit of t, t ′ → ∞ (weak coupling limit) without changing the value of the partition function or vev. If we first take the limit of t → ∞, we find that the path integral becomes WKB exact and localizes at the fixed points QF I = QB I = 0, since the action with the coupling t is essentially Gaussian. The Gaussian integral also induces Jacobians (1-loop determinants) to the measure, which are given by super determinants (super Hessian) of the BRST transformations evaluated at the fixed points [36][37][38][39][40][41] The dependence of the coupling t disappears from the 1-loop determinants as expected, because of the cancellation between the bosons and fermions.
On the other hand, if we take the limit of t ′ → ∞, the action with the coupling t ′ imposes delta-functional constrains µ i = 0 in the path integral after integrating out the auxiliary field Y i . These constraints work in any coupling region because of the coupling independence. So we always should take the moment map constraints µ i = 0 into account in addition to the fixed points.
To summarize, the partition function or vev of theory with Q-exact action is given by a summation over a finite set of the fixed points on the moment map constraints (vacuum moduli space). We obtain the Duistmaat-Heckman localization formula for supersymmetric Yang-Mills theory where ∆ gh (Φ * ) is the 1-loop determinant from the Faddeev-Popov ghosts, which we will discuss in the next subsection. Our residual task is to find the explicit 1-loop determinants and solutions of the fixed points of the quiver quantum mechanics.

Gauge fixing
Originally, the quiver quantum mechanics has the gauge symmetry of v U(N v ), but the existence of the real adjoint scalar fields σ v enhances the gauge symmetry to the complexified one, namely v GL(N v , C). We can regard a Q ǫ -closed combination of the gauge field and adjoint scalar factor. Under these complexified gauge symmetries, the D-term constraints (µ R,v = 0) become redundant, since there is an isomorphism in the vacuum moduli space given by the moment map quotient space .
So we can utilize these complexified gauge symmetries instead of taking D-term constraints into account directly.
Using the complexified gauge symmetries, we can choose the gauge condition so that the off-diagonal components of Φ v to be zero as similar as the matrix models, After imposing the gauge condition, we still have GL(1, C) N gauge symmetries. To fix the residual gauge degrees of freedom, we choose for the Cartan (diagonal) part of the each gauge field. In other words, the imaginary part of Φ v is chosen to be zero. At this moment, the total number of the gauge conditions is 2(N 2 −N)+N = 2N 2 −N. Then, we still have N gauge degrees of freedom, since GL(N, C) possesses 2N 2 degrees of freedom. However, as we explained above, since this residual gauge symmetry is equivalent to treat the Abelian part of the D-term conditions, we can fix it by imposing the real moment map constraints on the fixed point sets after the gauge fixing above. We will see that the solutions to the D-term constraints correspond to a choice of integral contour over Φ v . Then the residual gauge degrees of freedom are fixed by the choice of the integral contour. This is the essential reason why the wall crossing phenomena occurs in the quiver matrix model. Each solution of the D-term condition (stability condition) relates to the choice of the contour and the value of the partition function (index) or vev in the quiver quantum mechanics jumps by each choice of the contour. We will see this relation in more concrete examples later.
At least, in the above gauge choice, we obtain the gauge fixing action with the Faddeev- for each U(N v ) node. This gives the 1-loop determinant for the ghosts are the diagonal components of Φ v , and we have used the infinite product representation of the hyperbolic sine function and zeta-function regularization for the infinite product.

Partition function
Now, let us find the exact partition function of the quiver quantum mechanics. Under the diagonal gauge, the BRST fixed point equation becomes that is, only the constant modes of σ v (Φ v ) survive in the path integral. These constant eigenvalues of Φ v are denoted by φ 1 , . . . , φ Nv in what follows. Using the super determinant formula (4.4), we find that the 1-loop determinant for a vector multiplet, including the gauge fixing ghost contribution, with a node v is given by , (4.14) in terms of the constant eigenvalues of Φ v at the fixed points. Similarly, the 1-loop determinant for a chiral multiplet with an arrow a : v → w becomes . (4.16) We finally obtain an integral formula for the partition function of the generic quiver quantum mechanics where the 1/N v ! factor comes from the order of the Weyl group. We give two comments on the formula of the partition function. Firstly, since the integrand of the partition function depends only on the relative variables one trivial integration is left, which leads to the infrared divergence. This is due to the center of mass motion of the system [2]. We will ignore this divergence in the subsequent sections. Secondly, we note on the contour integrals over the constant modes φ v i . This means that there still exists the gauge degrees of freedom as we have discussed above. The integral formula contains poles in the denominator of the integrand. Choosing a suitable contour, some poles in the denominator are picked up as the residue integral, which correspond to the BRST fixed points on the moment map constraints.
Thus, we find the localization formula for the partition function of the quiver quantum mechanics: where φ * stands for the fixed point sets which satisfy the BRST fixed point equation and the moment map constraints, and the 1-loop determinants are evaluated at the fixed points in a sense of the residue integral. Similarly, the vev of the Q ǫ -closed operator is also evaluated exactly by the localization: We apply the localization formula to some concrete examples in the subsequent sections.
Before closing this section, we make a comment. The integrand of the formula (4.17) has infinitely many poles due to the trigonometric property of the determinant. These infinite poles are related to the large gauge transformations

Fixed points and character
After taking the diagonal gauge, the gauge symmetry of the quiver theory breaks to the product of the Abelian gauge groups, which is a maximal torus group T . So we have to find the fixed point set of the BRST equations Q ǫ F = Q ǫ B = 0 and the moment map constraints µ R = µ C =μ C = 0 with respect to the maximal torus. This situation is the same as in the Atiyah-Bott-Berline-Vergne localization formula. The 1-loop determinants correspond to the equivariant Euler class of the localization formula. The fixed point set of the torus T is classified by combinatorial objects such as partitions or Young diagrams, similar to the Nekrasov formula of the instanton counting [30,31]. Once we find the complete set of the fixed points, we should evaluate the 1-loop determinants at each fixed point in the sense of the residue integral (4.17). However, in the instanton calculus, there exists more convenient way to evaluate the 1-loop determinant at the fixed points, that is, the evaluation of the equivariant T -character [42,43].
The character maps the products in the 1-loop determinant of the partition function to summations of the polynomials. As a result, cancellation of the poles and zeros in the residue integral is suitably managed. We schematically give a correspondence between the 1-loop determinants of the partition function and T -character as follows: Using the mapping rules, we find that the T -character for the quiver quantum mechanics is given by where we have defined and t ≡ e iβǫ , x a ≡ e iβǫa . (4.23) We here have inserted "−1" in the square bracket of (4.21) in order to remove a singularity which corresponds to the infrared divergence due to the center of mass motion. Strictly speaking, the value of the partition function obtained from the above T -character is slightly different from the original residue integral (4.18), but the essential part of the Higgs branch index still holds as we will see. We denote the partition function evaluated from the T -character byẐ, which represents a contribution from the relative coordinates, to be distinguished. Then, the original partition function Z is related toẐ by where C is a renormalized constant of the divergence coming from the integral over the center of mass motion. We set C = 1 in the following. Note here that the exponent is the dimension of the Higgs branch in the relative coordinates. We need to classify the whole set of the fixed points and evaluate the T -character at the fixed points, but it is difficult to explain for the general quiver model. In the following section, we give some concrete examples. We will explain how to classify the fixed points and evaluate the T -character and partition function for more explicit models.

Abelian nodes
In the following two sections, we give several examples of the quiver quantum mechanics and compute the partition functions and the vevs of the physical observables exactly using the localization formula. Here, we treat the theories with no closed oriented loop in the quiver diagrams for simplicity, but the generalization to the case with closed oriented loops will be straightforward. In this section, we consider the Abelian quiver quantum mechanics.

Two nodes
We first start with a simplest example which consists of two Abelian nodes and k arrows between them, which is depicted in Fig. 2. In the Higgs picture, this theory describes the BPS bound states of two D-branes which are wrapped around two distinct cycles in a Calabi-Yau three manifold (CY 3 ) and intersect transversely in k points [2]. The adjoint scalars in the two vector multiplets X i v (v = 1, 2) represent the positions of the two D-branes in the non-compact space and the chiral multiplet represents the open string localized at the intersection point, whose ends are attached on the different D-branes. In the Coulomb picture, this theory describes the BPS bound states of two particles with mutually nonlocal charges γ 1 and γ 2 in a four dimensional N = 2 supergravity theory which arises in the low energy limit of type II string theory compactified on CY 3 [2]. The number of the chiral multiplets k corresponds to the Dirac-Schwinger-Zwanziger (DSZ) product of the two charges γ 1 , γ 2 . where h a is an integral constant. Forcing the periodic boundary condition q a (τ + β) = q a (τ ), we see that h l = 0 except for only one label a (l = a) by setting This represents the infinitely many poles of the hyperbolic sine function. But, as we have explained in section 4.3, we only pick up a pole with n = 0. This fixed point is denoted by Φ * a . The non-vanishing complex scalar q a at the fixed point is also the solution to the D-term conditions, where the FI parameters satisfy the constraint θ(N) = ζ 1 + ζ 2 = 0. The D-term equations mean that there exist the fixed points (|q a | = 0) if ζ ≡ ζ 1 = −ζ 2 > 0, but not if ζ < 0. The partition function vanishes for the later case. This is nothing but the wall crossing formula for the quiver quantum mechanics [2,44]. Let us evaluate the partition function for the case of ζ > 0. The solution (5.2) means that where u 1 = e βφ 1 = V 1 , u 2 = e βφ 2 = V 2 , and x a = e iǫa .
The T -character of N = (1, 1) with k arrows becomes 1 at each fixed point labelled by a = 1, . . . , k. Using the mapping rules between the character and 1-loop determinants in the partition function, we obtain a contribution from each fixed point to the partition function The total partition function is given by a summation over the fixed point set where all x a (ǫ a ) dependences surprisingly disappear. The polynomial ofẐ k 1,1 in t is the Poincaré polynomial of CP k−1 , which is the Higgs branch moduli. To compare it with the known results in [2,8], setting t 1/2 = −y, the original partition function becomes This result agrees with them.
We can also evaluate the vev of the supersymmetric Wilson loop. The insertion of the supersymmetric Wilson loop in the path integral does not violate the above localization arguments. The Wilson loop operator takes a value at the localization fixed point. At the fixed point, we find a center of mass independent Wilson loop by using the solution to (5.5). Thus, we obtain the vev of the Wilson loop by (5.12) The dependence of x a (ǫ a ) does not disappear in contrast to the partition function.

n nodes
Let us now generalize the above result to the Abelian quiver with n-nodes. We assume that the direction of the arrows between two nodes is identical. Then the quiver structure is specified by an antisymmetric matrix K ij = −K ji (i, j = 1, . . . , n), where |K ij | represents the number of the arrows from node i to j if K ij > 0, or the arrows from node j to i if K ij < 0. We also assume that the arrows always go from lower to higher node, namely K ij > 0 for i < j, to avoid the oriented loops (superpotentials). This system represents the bound states of n distinguishable particles, and will give basic building blocks of the Manschot-Pioline-Sen (MPS) formula [8].
The gauge group of the model is G = n r=1 U(1) r and there are n integral variables φ i . Except for the center of mass, there are n − 1 independent relative variables and those are fixed by demanding n − 1 independent BRST fixed point equations or equivalently u i u −1 j x ij a ij = 1, for i < j, (5.14) where n − 1 combinations of two nodes are chosen. A set of the indices (i, j) which appear in the above (n − 1) equations is denoted by I. From the BRST fixed point equations, we find that q ij a ij can get non-zero vev in the Higgs branch, where q ij a ij is a scalar component of the chiral multiplet, corresponding to the a ij -th arrow between the i-th and j-th node. The D-term equations with a constraint Once the fixed point I is found, we obtain the T -character where p ij is a monomial of x ij a ij 's and satisfies u i u −1 j p ij = 1 for (i, j) ∈ I. Therefore, a contribution to the partition function at the fixed point is given by We finally obtain the partition function where p 13 = x 12 a 12 x 23 a 23 . The D-term equations become |q 12 a 12 | 2 = ζ 1 , |q 23 a 23 | 2 − |q 12 a 12 | 2 = ζ 2 , −|q 23 a 23 | 2 = ζ 3 . , there is a solution if ζ 1 > 0, ζ 2 < 0 and ζ 3 < 0 (ζ 1 > 0, ζ 2 > 0 and ζ 3 < 0). Thus, in a chamber of ζ 2 > 0 (ζ 2 < 0), I 1 and I 3 (I 1 and I 2 ) are chosen for the fixed points.
At the end of this section, we give an expression of the Wilson loop operator vev. The center of mass free Wilson loop operators are given by a polynomial of u i u −1 j associated with each arrow (link). At the fixed point, u i u −1 j = 1/x ij a ij for (i, j) ∈ I and u i u −1 j = 1/p ij for (i, j) ∈ I. If the polynomial evaluated at the fixed points I is denoted by P I (x ij a ij , p ij ), the vev of the Wilson loop is given by The expression is rather complicated, but can be evaluated exactly, in principle.

Non-Abelian nodes
In this section, we consider the quiver quantum mechanics including the non-Abelian nodes.

Abelian and non-Abelian nodes: the Hall halo
We firstly consider the quiver quantum mechanics which is represented by two nodes of U(1) 1 × U(N) 2 with k arrows in Fig. 3. This case can be regarded as a charge k magnetic 1 N k Figure 3: monopole surrounded by N mutually non-interacting electrons with charge one in the Coulomb picture [2]. If we first assume that Z 2 andZ 2 are diagonal, then the BRST fixed point equations (3.1) say that Z 2 =Z 2 = 0 because of the periodicity Z 2 (τ + β) = Z 2 (τ ) andZ 2 (τ + β) = Z 2 (τ ). Then we can repeat the similar arguments in the Abelian case, and we find the fixed point equations . . .
where n i ∈ Z. The N indices a i must be chosen from different integers within k. Otherwise, the partition function vanishes because of the numerator of the 1-loop determinant for the vector multiplet. Then, the fixed points are classified by a set of the indices (a 1 , a 2 , . . . , a N ). We can assume that a 1 < a 2 < · · · < a N by using the Weyl permutations. For this ordering, the total number of the fixed points is k N . These fixed points are denoted by Φ * a 1 a 2 ···a N . Since [Z 2 ,Z 2 ] = 0 in this case, the D-term equations are expressed only by the Higgs in terms of the N absolute values of q i a i , associated with the BRST fixed points. Under the constraint of the FI parameter θ(N) = ζ 1 + Nζ 2 = 0, the D-term equations have a solution |q 1 Using u 1 = e βφ 1 and u 2,i = e βφ 2,i , the BRST fixed point equations give Thus, we obtain The fixed point data gives the T -character The corresponding determinant in the partition function at each fixed point is Investigating various values of N and k, we find that the total relative partition function is given by the t-binomial coefficient where the N! factor comes from the Weyl permutations. This is the Poincaré polynomial of the Grassmannian Gr (N, k), that is the moduli space of the Higgs branch. This also agrees with [2]. So far we have assumed that Z 2 =Z 2 = 0. One may wonder whether there is fixed point for the case of [Z 2 ,Z 2 ] = 0 when ζ v = 0. This case corresponds to the poles in (4.17) coming from the 1-loop determinant of the vector multiplet. If there is a contribution from these poles, the classification of the fixed points becomes more complicated due to off-diagonal components of Z 2 andZ 2 . The fixed points for the non-commutative Z 2 and Z 2 are classified as similar as the Nekrasov formula by using the Young diagrams [30,31], but the partition function should vanish in this branch if these poles are rejected by the F-term and D-term conditions.
Let us now discuss the possibility of the branch with [Z 2 ,Z 2 ] = 0 in our quiver model. First of all, from the BRST equation, requiring the periodicity of Z 2 andZ 2 , we obtain The above equations have generally blockwise solutions up to the Weyl permutations 10) where (L (ns) 0 ) j,j = j − 1 for j = 1, · · · , n s , and (L (ns) − ) j+1,j = z s,j and (L (ns) + ) j,j+1 =z s,j for j = 1, · · · , n − 1, otherwise zero. Here n s 's satisfy d s=1 n s = N, then the fixed points are classified in terms of the partitions of N, namely the Young diagrams of N boxes.
Secondly, we have to find solutions which satisfy the other BRST equations for the chiral multiplet, F-term and D-term conditions at the same time. Solving the F-term condition Z 1 q a − q a Z 2 = 0, (6.12) and BRST equations ∂ τ q a + φ 1 q a − q a Φ 2 + iǫ a q a = 0, (6.13) we find that Z 1 = 0 and α s = φ 1 + iǫ as (s = 1, · · · , d), and also each q as has a highest component only, namely q as = (h as , 0, . . . , 0). For the above solutions, however, there is no solution to the D-term equations. Indeed, for each n-dimensional block, we see that which does not have any solution for both ζ 2 > 0 and ζ 2 < 0. Thus we can conclude that there is no Nakajima-Nekrasov type fixed point in our quiver matrix model. This is a consequence of the orthogonality between the Higgs branch (Z = 0 and q = 0) and the Coulomb branch (Z = 0 and q = 0) from the F-term condition 2 . The Wilson loop for the non-Abelian part is obtained by a trace over a representation R. In the diagonal gauge, it is written in terms of a symmetric polynomial of u 2,i , that is, a Schur polynomial associated with the representation (Young diagram) R. If we denote the polynomial by s R (u 2,1 , u 2,1 , . . . , u 2,N ), the center of mass motion free Wilson loop is expressed by u −d 1 s R (u 2,1 , u 2,1 , . . . , u 2,N ), where d is the degree of the polynomial. At the fixed point, the Wilson loop becomes simply s R (x a 1 , x a 2 , . . . , x a N ) in N variables chosen from k. Therefore, we find the Wilson loop in the representation R is evaluated by This results might be interesting from a point of the view of the physics and the symmetric polynomial of t and x a i 's, but we do not pursue the Wilson loops anymore. We will concentrate on the partition function only in the following.

Coprime dimension vector
We generalize our arguments to the non-Abelian nodes. The classification of the fixed points in quiver theory only with non-Abelian nodes is rather complicated. So we here explain only a few examples.
We first start with two nodes with a coprime dimension vector, that is U(2) 1 × U(3) 2 with k arrows. There are five integral variables of φ 1,i (i = 1, 2) and φ 2,i ′ (i ′ = 1, 2, 3). Except for the center of mass, we can give four fixed point equations to solve in φ 1,i and φ 2,i ′ . For the exponent variables, a possible set of the BRST fixed point equations is The corresponding squarks q a 11 , q b 12 , q c 21 , and q d 23 could have the vevs as a consequence of the D-term equations. The D-term equation becomes If ζ > 0, there exist 4! k 4 + 3 · 3! k 3 + 2! k 2 = k(k − 1) 3 fixed points in total by choosing appropriate a, b, c, d from integers within k. The fixed point set of this type is denoted by Φ * [ ab c d ], which reflects the structure of the BRST fixed points (6.16) in a 2 × 3 matrix of the labels. We can also find that there are other 5 fixed point sets of the similar structure like Φ * [ ab cd ], Φ * [ ab cd ], Φ * [ a b cd ], etc. This classification of the fixed points is already known in the mathematical literature [4]. These fixed points are depicted in the diagram: All of the fixed points of this kind will give the same contribution to the T -character and partition function. So we here evaluate the contribution from the fixed point set of Φ * [ ab c d ] only as follows.
Solving (6.16), we find where the remainingũ 3 corresponds to the degree of the center of mass. Using this solution, we obtain the T -character at the fixed point Then the contribution to the partition function from this fixed point is given by In this quiver gauge theory, it is important to notice that there is yet an another kind of the fixed points. To find it, we go back to the original expression of the partition function, .

Non-coprime dimension vector
Next we treat the case of N = (2, 2), which has a common divisor among the dimensions. The first type of the BRST fixed points comes from poles of the chiral multiplet (6.40) which can be solved by The corresponding vev of the squarks q a 11 , q b 12 , and q c 21 satisfies the D-term equation if a = b and a = c. Thus, the above solution is valid for the localization fixed point, which is denoted by Φ * [ ab c ] and depicted by a diagram: There are k(k − 1) 2 fixed points of this kind in total.
The T -character at this fixed point reduces to So the 1-loop determinant at the fixed point is Another kind of the fixed point is given by which are solved by V 2 =ũ 2 (t −1 + 1). (6.47) The first equation in (6.45) and solution (6.46) and (6.47) means no longer Z 1 = Z 2 = 0, but they contains off-diagonal elements The F-term equation Z 1 q a − q a Z 2 = 0 is satisfied if q a 11 z 1 − q a 22 z 2 = 0. And the D-term equations become where we have slightly modified the FI parameters by introducing a parameter δ > 0. The trace condition on the FI parameters is the same as θ(N) = 2ζ 1 + 2ζ 2 = 0 after introducing δ, but partially they satisfy (ζ 1 + δ) This δ corresponds to the concept of the θ-stability. Indeed, the above D-term equations have a solution |z 1 | = |z 2 | = 2g 2 δ and |q a 11 | = |q a 22 | = √ ζ if ζ ≡ ζ 1 = −ζ 2 > 0. In the limit of δ → 0, the vevs of Z v vanish, but the Higgs vevs are still kept, and so the value of the partition function in the Higgs phase holds in this limit since it is independent of δ.
In this way, we obtain the extra k fixed points, which are denoted by Φ * [ a a ] and a diagram: In the case of the coprime dimension vector, the Nakajima-Nekrasov type fixed points Z v = 0 are rejected by the F-term and D-term equations. It is subtle, but we should take into account the Nakajima-Nekrasov type fixed points in the present non-coprime case. Using the solution (6.46) and (6.47), we find the T -character x l x a , (6.50) and the 1-loop determinant at the fixed point Note here that the contribution to the index from each fixed point is − 1 2 up to the Weyl factor. We can also find additional three contributions from the fixed points of the same type, such that Φ * [ a a ] exchanging the role of Z v 's by the Weyl permutations. Combining all of contributions from the fixed points, we obtain the total partition functionẐ We again can see all x a dependences disappear in the final results. As mentioned above, in the limit of t → 1, the former term in (6.52) contributes to the index by +1, while the latter term contributes by − 1 2 . Thus we find the index (Euler characteristic) for this model lim which takes a value in a half integer. We finally give our results of the partition function for smaller k: 55) Z 3 2,2 = t −5/2 1 + t 1 + t 2 + t 4 2 + 3t + 2t 2 , (6.56) Z 4 2,2 = t −9/2 1 + t (1 + t 2 + t 4 + t 6 )(2 + 4t + 5t 2 + 4t 3 + 2t 4 ), (6.57) Z 5 2,2 = t −13/2 1 + t 1 + t 2 + t 4 + t 6 + t 8 2 + 4t + 6t 2 + 7t 3 + 6t 4 + 4t 5 + 2t 6 . (6.58) In this case, we cannot compare our results with Reineke's formula because Reineke's formula is valid only when the dimension vector is coprime [3]. But we can compare them with the wall crossing formula. In fact, our results agree with the special case of the wall crossing formulae (A.5)-(A.9) by setting t 1/2 = −y up to an overall factor 1/2. 3

Conclusion and Discussion
In this paper, we have derived the exact partition functions and expectation values of the Wilson loop operators of the quiver quantum mechanics in the Higgs phase by the localization techniques. We have considered several examples of the quiver quantum mechanics which include only Abelian nodes in section 5, a non-Abelian and an Abelian node in section 6.1, and only non-Abelian nodes in section 6.2 and 6.3. We have found that those partition functions give the (refined) Witten index of the quiver theory physically, or Poincaré polynomial of the quiver moduli mathematically. Our results also agree with ones from the wall crossing formula.
To regularize the path integral of the matrix quantum mechanics, we have gauged the R-symmetries for both vector multiplet and chiral multiplet. The background gauge fields of the R-symmetries lift up the flat direction and the localization fixed points become isolated. In terms of the residue integral, this means that all poles are decomposed into the simple poles. We have considered the chiral multiplets with the general R-charges, but the all final results are independent of these R-charges, except for the R-charge of the vector multiplet, and thus its background gauge field becomes the refined parameter of the index. This disappearance of the R-charges from the index is a significant property in our results although the vevs of Wilson loops depend on those R-charges. As we have seen in the explicit examples, it is very subtle task to scrutinize the D-term and F-term conditions in the non-Abelian cases. However, if we fail to take the correct fixed points into account, the R-charge dependence never disappears. Thus, we can use this property for a criterion of correctness of the partition functions.
We have considered the localization in the Higgs branch by solving directly the linear (quiver) sigma model. On the other hand, the quiver theory goes to the Coulomb phase in the IR limit g s ≫ 1. The effective theory in the Coulomb phase can be written in terms of a non-linear sigma model of the Abelian vector multiplets, that describe more directly charged BPS particles in N = 2 supergravity. The localization in the Coulomb branch has been discussed already in [7,8], but we would like to revisit the Coulomb branch localization explicitly from a physical (matrix model) point of view, since we believe that it is a key to understand the correspondence between the gauge theory and gravity via the localization. Deeper understandings of the correspondence may make clear how to emerge the supergravity or M-theory from the matrix quantum mechanics, proposed in [28,29]. ∆Ω ref (2γ 1 + 3γ 2 , y)| k=3 = 1 y 6 (1 + y 2 + 3y 4 + 3y 6 + 3y 8 + y 10 + y 12 ), (A