The Volume of the Quiver Vortex Moduli Space
TThe Volume of the Quiver Vortex Moduli Space
Kazutoshi Ohta ∗ and Norisuke Sakai † Institute of Physics, Meiji Gakuin University, Yokohama, Kanagawa 244-8539, Japan Department of Physics, and Research and Education Center for Natural Sciences, KeioUniversity, 4-1-1 Hiyoshi, Yokohama, Kanagawa 223-8521, Japan, and iTHEMS, RIKEN, 2-1Hirasawa, Wako, Saitama 351-0198, Japan
Abstract
We study the moduli space volume of BPS vortices in quiver gauge theories on com-pact Riemann surfaces. The existence of BPS vortices imposes constraints on the quivergauge theories. We show that the moduli space volume is given by a vev of a suitable co-homological operator (volume operator) in a supersymmetric quiver gauge theory, whereBPS equations of the vortices are embedded. In the supersymmetric gauge theory, themoduli space volume is exactly evaluated as a contour integral by using the localization.Graph theory is useful to construct the supersymmetric quiver gauge theory and to derivethe volume formula. The contour integral formula of the volume (generalization of theJeffrey-Kirwan residue formula) leads to the Bradlow bounds (upper bounds on the vor-ticity by the area of the Riemann surface divided by the intrinsic size of the vortex). Wegive some examples of various quiver gauge theories and discuss properties of the modulispace volume in these theories. Our formula are applied to the volume of the vortex mod-uli space in the gauged non-linear sigma model with C P N target space, which is obtainedby a strong coupling limit of a parent quiver gauge theory. We also discuss a non-Abeliangeneralization of the quiver gauge theory and “Abelianization” of the volume formula. ∗ [email protected] † [email protected] a r X i v : . [ h e p - t h ] O c t Introduction
Vortices are co-dimension two solitons and play an important role for non-perturbativeeffects in gauge theories. In particular, the Bogomol’nyi-Prasad-Sommerfield (BPS) vor-tices appear as solutions to the BPS differential equations [1,2] which minimize the energyof the Yang-Mills-Higgs system in three spacetime dimensions.If the vortex equations are considered on compact Riemann surfaces Σ h with the genus h , the number of the vortices (vorticity) is restricted by an upper bound which is givenby the finite area of Σ h divided by the intrinsic size of the vortex. This bound is calledthe Bradlow bound [3, 4].Parameters of the vortex solutions are called moduli, and thier space is called themoduli space. (See for review [5, 6].) The structure of the moduli space is important tounderstand properties of the vortices themselves. The volume of the moduli space appearsin the thermodynamics of the vortices [5, 7–9]. Since the thermodynamical partitionfunction is proportional to the volume of the vortex moduli space, we can derive the freeenergy or equation of state from the volume. Although an integration of the volume formon the moduli space should give the volume of the moduli space, it is generally difficultto know the geometry of the moduli space, including the K¨ahler metric, except for somespecial cases [10]. (See also [5] for details.) On the other hand, the volume of the modulispace can be evaluated exactly without a detailed knowledge of the metric.There are various way to obtain the volume of the moduli space. One way is totake advantage of the property of the moduli space as a K¨ahler manifold [5, 8]. Toevaluate the volume by using the properties of the K¨ahler manifold, we need to know atopological structure of the moduli space like cohomologies or boundary divisors. Theother way is to embed the BPS equation into supersymmetric gauge theory and to utilizethe “localization” [11–15]. The localization method gives the volume as simple contourintegrals even without knowing the geometry of the moduli space. In this sense, thelocalization method is universal and can be applied to any kind of the BPS equations inprinciple. In previous works [13–15], the volume of the moduli space of the vortex witha single U ( N c ) gauge group and N f matters in the fundamental representation has beenevaluated.There have been a number of studies of vortices in gauge theories on curved manifolds,namely gauged nonlinear sigma models (GNLSM) [16–21]. The GNLSM can be obtainedif one considers a product of two gauge groups and matters charged under both of these2auge groups and takes a strong coupling limit of one of the gauge groups. Before takingthe limit, we have linear gauged sigma model with a product of gauge groups and can beconsidered as a parent theory of GNLSM. Vortices in such theories with product gaugegroups have also been studied before [22]. Gauge theories with a product of gauge groupsand matters in bi-fundamental representations between two gauge groups are called quivergauge theories. If we take a decoupling limit of a gauge group in the quiver gauge theory,where a gauge coupling constant goes to zero, the decoupled gauge group behaves asa global symmetry for the matters. So we can obtain the matters in the fundamentalrepresentation from the quiver gauge theory. Thus, the quiver gauge theory includesvarious types of the gauge theory in a very general form. Once a general formula for thevolume of the vortex moduli space for the quiver gauge theory is derived, the BPS vortexequations with various kinds of matters or target space can be obtained. This is a strongmotivation to consider the quiver gauge theory.The quiver gauge theory can be realized by using the graph theory. The BPS vorticesin the supersymmetric gauge theory on the graph has been studied in [23] inspired by the“deconstruction”. In addition, the quiver gauge theory naturally appears in the D-branesystem of superstring theory. Open strings between D-branes give the gauge fields andbi-fundamental matters in the quiver gauge theory. Some of the quiver gauge theoriescan be realized as an effective theory on the D-branes at a tip of an orbifold. The volumeof the vortex moduli space of quiver gauge theory can play an important role for non-perturbative effects in superstring theory.The purpose of our paper is to obtain a formula for the volume of the moduli spaceof BPS vortices in quiver gauge theories on compact Riemann surfaces. We find thatthe graph theory in mathematical literature is useful to describe the quiver gauge theory.The gauge groups and bi-fundamental matters are expressed in terms of a directed graph(quiver diagram), which consists of the vertices and arrows (edges) connecting betweenthe vertices. Each vertex represents a factor of the product gauge group, and each arrowgives the bi-fundamental matter, which transforms as fundamental and anti-fundamentalrepresentation for the gauge group at the source and target vertex of the arrow, respec-tively. Connection of vertices by edges in the graph is represented by a matrix calledincidence matrix, which appears frequently in our construction of volume formulas forBPS vortices. Because of a zero left eigenvector of incidence matrix in a generic quivergauge theories, the existence of BPS vortices imposes a stringent constraint on possiblequiver gauge theories. We find two alternative solutions to the constraint. (i) All gauge3roups have a common gauge coupling (universal coupling case). (ii) There is a gaugegroup whose gauge coupling vanishes (decoupled vertex case).Embedding the vortex system into the supersymmetric quiver gauge theory, we definethe supersymmetric transformation for the fields by a supercharge Q . Vacuum expecta-tion values (vevs) of the cohomological operators, which is Q -closed but not Q -exact, isindependent of the gauge coupling constants of the supersymmetric quiver gauge theory,since the action is Q -exact. So we can control the gauge coupling constants of the super-symmetric gauge theory without changing the vevs of the cohomological operators. If wetake the controllable gauge coupling constants to the same value as the physical couplingconstants in the BPS equations to evaluate the volume of the moduli space, the pathintegral is localized at the solution to the BPS equations. At the fixed points of the BPSsolution, the matter (Higgs) fields take non-trivial value and the supersymmetric quivergauge theory is in the Higgs branch. In the Higgs branch, we can show that the volumeof the vortex moduli space is given by the vev of a suitable cohomological operator calledthe volume operator.On the other hand, if we tune the controllable coupling constants to special values, thevevs of the Higgs fields vanish at the fixed points and the supersymmetric quiver gaugetheory is in the Coulomb branch. In the Coulomb branch, the evaluation of the vev of thevolume operator reduces to simple contour integrals. Using the coupling independence ofthe vev, we expect that the contour integrals also give the volume of the vortex modulispace as well as in the Higgs branch. Thus we obtain the contour integral formula of thevolume of the vortex moduli space in the quiver gauge theory. As concrete examples toapply the contour integral formula for the volume of the vortex moduli space, we considervarious quiver gauge theory with Abelian vertices. We discuss the Abelian quiver gaugetheory with two or three vertices. For the integral to converge, we need a suitable choice ofcontours, which reproduces exactly the Bradlow bounds. The derivation of the Bradlowbounds from the contour integral can be regarded as a generalization of the Jeffrey-Kirwan (JK) residue formula [25]. A similar connection between the Bradlow boundsand the JK residue formula is also considered and utilized in the calculation of the indexon S × Σ h [26, 27]. In some examples of the quiver gauge theory with multiple Abelianvertices, the moduli space becomes non-compact. So we need to introduce regularizationparameters, which can be regarded as the twisted mass of the matters. After taking zerolimits of the regularization parameters, we can see the divergences of the volume of themoduli space corresponding to the non-compactness of the moduli space.4e also apply the contour integral formula of the volume to a quiver gauge theorycorresponding to the parent gauged linear sigma model of Abelian GNLSM with C P N target space with n flavors of charge scalar fields. When restricted to N = n = 1, ourresult agrees with the previous result in [21], which uses an entirely different method.We can also take a strong coupling limit of one of the gauge couplings, which gives thevolume of the vortex moduli space of the GNLSM. Moreover, our contour integral formulaprovides a new results for the moduli space volume of the BPS vortex in the GNLSM withthe target space C P N and its parent GLSM with an arbitrary number n of charged scalarfields.The localization method can be extended to the case of non-Abelian quiver gauge the-ories. Since the non-Abelian gauge groups reduce to a product of U (1)’s in the Coulombbranch, the contour integral is expressed in terms of the Cartan part of the non-Abeliangauge groups, and the non-Abelian vertices in the quiver graph decompose into Abelianvertices. This “Abelianization” [28, 29] occurs in the localization formula and the quivergraph, because of the decomposition of the non-Abelian vertices into Abelian vertices inthe quiver graph. Even with the Abelianization in the localization formula, the explicitevaluation of the contour integral becomes complicated due to the Vandermonde deter-minant which characterizes the non-Abelian case. However, our formula gives in principlethe volume of the vortex moduli space in any non-Abelian quiver gauge theories. Thenon-Abelian generalization of the volume of the vortex moduli space in the GNLSM isalso discussed.The organization of this paper is as follows. In Sect. 2, basics of the quiver gaugetheory and graph theory are explained, and the BPS vortex equations are derived. InSect. 3, the BPS vortex system is embedded into a supersymmetric quiver gauge theory,and the volume operator (a cohomological operator) is introduced to obtain the volumeof the vortex moduli space in the Higgs branch. The contour integral formula for themoduli space volume is obtained from localization in the Coulomb branch. In Sect. 4, wegive various examples of the quiver gauge theory up to three Abelian vertices. Modulispace volumes in Abelian quiver gauge theories are evaluated explicitly by performing thecontour integral. In Sect. 5, the vortex moduli space of a gauged linear sigma model (theparent theory of GNLSM) is obtained. That of the GNLSM is also obtained by takingthe strong coupling limit. In Sect. 6, the contour integral formula is generalized to thenon-Abelian cases, and the “Abelianization” of the non-Abelian quiver diagram is found.The last Sect. 7 is devoted to conclusion and discussions.5 v v v e e e e e Figure 1: An example of the quiver diagram. s ( e ) t ( e ) e Figure 2: A part of the quiver diagram of two vertices connected with an edge. The sourceand target of the arrow (edge) e are denoted by s ( e ) and t ( e ), respectively. A quiver diagram is expressed by a directed graphΓ= (
V, E ), which consists of a set ofvertices V and a set of directed edges (arrows) E . We denote elements of V and E by v and e , respectively. We depict an example of the quiver diagram in Fig. 1.We denote the total number of the vertices and edges by n E and n V , respectively.Each directed edge e ∈ E connects from a source vertex s ( e ) ∈ V to a target vertex t ( e ) ∈ V (see Fig. 2), i.e. each directed edge is specified by an ordered pair of two vertices e = ( s ( e ) , t ( e )).To describe the quiver diagram, it is useful to introduce language of graph theory. Thegraph theory describes a structure of the (quiver) graph in terms of elements of matrices.In graph theory, there are various kinds of the matrices or objects which describe andmanipulate the graph structure, but we here introduce some of them only, which will beused to construct the quiver gauge theory.First of all, we introduce the incidence matrix. The incidence matrix L maps from V v v v v e e e e e Figure 1: An example of the quiver diagram. s ( e ) t ( e ) e Figure 2: A part of the quiver diagram of two vertices connected with an edge. The sourceand target of the arrow (edge) e are denoted by s ( e ) and t ( e ), respectively. A quiver diagram is expressed by a directed graph Γ = (
V, E ), which consists of a set ofvertices V and a set of directed edges (arrows) E . We denote elements of V and E by v and e , respectively. We depict an example of the quiver diagram in Fig. 1.We denote the total number of the vertices and edges by n E and n V , respectively.Each directed edge e ∈ E connects from a source vertex s ( e ) ∈ V to a target vertex t ( e ) ∈ V (see Fig. 2), i.e. each directed edge is specified by an ordered pair of two vertices e = ( s ( e ) , t ( e )).To describe the quiver diagram, it is useful to introduce language of graph theory. Thegraph theory describes a structure of the (quiver) graph in terms of elements of matrices.In graph theory, there are various kinds of the matrices or objects which describe andmanipulate the graph structure, but we here introduce some of them only, which will beused to construct the quiver gauge theory.First of all, we introduce the incidence matrix. The incidence matrix L maps from V s ( e ) t ( e ) e Figure 2: A part of the quiver diagram of two vertices connected with an edge. The sourceand target of the arrow (edge) e are denoted by s ( e ) and t ( e ), respectively. A quiver diagram is expressed by a directed graphΓ= (
V, E ), which consists of a set ofvertices V and a set of directed edges (arrows) E . We denote elements of V and E by v and e , respectively. We depict an example of the quiver diagram in Fig. 1.We denote the total number of the vertices and edges by n E and n V , respectively.Each directed edge e ∈ E connects from a source vertex s ( e ) ∈ V to a target vertex t ( e ) ∈ V (see Fig. 2), i.e. each directed edge is specified by an ordered pair of two vertices e = ( s ( e ) , t ( e )).To describe the quiver diagram, it is useful to introduce language of graph theory. Thegraph theory describes a structure of the (quiver) graph in terms of elements of matrices.In graph theory, there are various kinds of the matrices or objects which describe andmanipulate the graph structure, but we here introduce some of them only, which will beused to construct the quiver gauge theory. 6Figure 2: A part of the quiver diagram of two vertices connected with an edge. The sourceand target of the arrow (edge) e are denoted by s ( e ) and t ( e ), respectively. A quiver diagram is expressed by a directed graph Γ = (
V, E ), which consists of a set ofvertices V and a set of directed edges (arrows) E . We denote elements of V and E by v and e , respectively. We depict an example of the quiver diagram in Fig. 1.We denote the total number of the vertices and edges by n E and n V , respectively.Each directed edge e ∈ E connects from a source vertex s ( e ) ∈ V to a target vertex t ( e ) ∈ V (see Fig. 2), i.e. each directed edge is specified by an ordered pair of two vertices e = ( s ( e ) , t ( e )).To describe the quiver diagram, it is useful to introduce language of graph theory. Thegraph theory describes a structure of the (quiver) graph in terms of elements of matrices.In graph theory, there are various kinds of the matrices or objects which describe andmanipulate the graph structure, but we here introduce some of them only, which will beused to construct the quiver gauge theory. 6irst of all, we introduce the incidence matrix. The incidence matrix L maps from V to E , i.e. n V × n E matrix, whose elements are defined by L ve = +1 if s ( e ) = v − t ( e ) = v . (2.1)For example, if we make the incidence matrix for the graph depicted in Fig. 1, we obtain L = − − − − − (2.2)For generic quiver gauge theory, a sum of the elements in each column vanishes (cid:88) v L ve = 0 , (2.3)namely the multiplication of a vector (1 , · · · ,
1) from the left annihilates the incidencematrix L . We will show that this will give a stringent constraint on quiver gauge theoriesadmitting BPS vortices. Once the incidence matrix is given, we can reproduce the directedgraph (quiver diagram).If we assign variables (cid:126)x = ( x , x , . . . , x n V ) on each vertex, the incidence matrix mul-tipied to the vector becomes a difference operator x v L ve = x s ( e ) − x t ( e ) , (2.4)where the repeated upper and lower indices are summed implicitly. This property will beimportant to our formulation in the following.Secondly, let us consider the Laplacian matrix ∆ defined by∆ vv (cid:48) = deg( v ) if v = v (cid:48) − A vv (cid:48) if v (cid:54) = v (cid:48) , (2.5)where deg( v ) represents the number of the edges which connect to the vertex v and A vv (cid:48) isthe number of edges from v to v (cid:48) . ( A vv (cid:48) is also called the adjacency matrix.) The Laplacianmatrix is also constructed from a square of the incidence matrix, i.e. ∆ ≡ LL T . Hence7he Laplacian always has at least one zero eigenvalue with the eigenvector proportionalto (1 , · · · , − − − − − − − − . (2.6)We can notice that the Laplacian matrix is a generalization of the Cartan matrix in theLie algebra.If we assign variables (cid:126)x = ( x , x , . . . , x n V ) on each vertex, we can see that an innerproduct with the Laplacian matrix reduces (cid:126)x ∆ (cid:126)x T = (cid:88) e ∈ E ( x s ( e ) − x t ( e ) ) , (2.7)which is a second order difference operator between vertices. This is a reason why ∆ iscalled the Laplacian on the graph. The Laplacian matrix does not preserve the orientationof the edges. So the Laplacian matrix cannot reproduce the whole structure of the quiverdiagram including the orientation of the edges. The quiver gauge theory is defined via a quiver diagram. Unitary groups U ( N v ) areassigned to each vertex v , where N v is a rank of the unitary group. The quiver gaugetheory has a gauge symmetry of a product group (cid:81) v ∈ V U ( N v ) with the gauge couplings g v . The bi-fundamental matters (scalar fields) H e are associated with each edges e andrepresented by ( N s ( e ) , ¯ N t ( e ) ), namely N s ( e ) × N t ( e ) complex matrices.We first consider 2 + 1D quiver Yang-Mills-Higgs theory on M = R t × Σ h . The metricon M is given by ds = − dt + 2 g z ¯ z dz ⊗ d ¯ z. (2.8)On the Riemann surface Σ h , there exists a volume form ω = √ gdz ∧ d ¯ z . An area of theRiemann surface A is given by an integral of the volume form A = (cid:90) Σ h ω. (2.9)8or each gauge vertex v , there is gauge vector 1-form field A v (3) = A v dt + A vz dz + A v ¯ z d ¯ z .The field strength is given by F v (3) = dA v (3) + iA v (3) ∧ A v (3) . (2.10)On the other hand, on each edge e , we can assign a covariant derivative of the scalar field H e d (3) A H e = d (3) H e + iA s ( e ) H e − iH e A t ( e ) (2.11)The action is written in terms of the quiver diagram by S (3) = − (cid:90) R t × Σ h (cid:34) (cid:88) v ∈ V Tr v (cid:26) g v F v (3) ∧ ∗ F v (3) + g v (cid:16) ζ v N v − (cid:88) e : s ( e )= v H e ¯ H e + (cid:88) e : t ( e )= v ¯ H e H e (cid:17) dt ∧ ω (cid:27) + (cid:88) e ∈ E Tr s ( e ) d (3) A H e ∧ ∗ d (3) A ¯ H e (cid:35) , (2.12)where Tr v stands for a trace over the rank N v gauge group at the vertex v , and the sum (cid:80) e : s ( e )= v ( (cid:80) e : t ( e )= v ) is taken over edges whose sources (targets) are given by v .Taking a static configuration and A = 0 gauge, the gauge vector field reduces to(1,0)-form A v = A vz dz and (0,1)-form ¯ A v = A ¯ z d ¯ z on Σ h , where the field strength is givenby F v = ∂ ¯ A v + ¯ ∂A v + i ( A v ∧ ¯ A v + ¯ A v ∧ A v ) . (2.13)Introducing a “metric” G vv (cid:48) = 1 g v δ vv (cid:48) , G ee (cid:48) = δ ee (cid:48) , (2.14)on v and e , respectively, to raise and lower the indices, the energy is given by E = (cid:90) Σ h Tr (cid:34) µ v ∧ ∗ µ v + 12 ν e ∧ ∗ ¯ ν e + g v ζ v F v (cid:35) ≥ πg v ζ v k v , (2.15)discarding the total divergence, where Tr is taken over suitable size of each term (gaugegroups), and magnetic flux (first Chern class) for each gauge vertices is defined as12 π (cid:90) Σ h Tr F v = k v ∈ Z , (2.16)9nd we have defined moment maps as follows µ v = F v − g v (cid:16) ζ v N v − (cid:88) e : s ( e )= v H e ¯ H e + (cid:88) e : t ( e )= v ¯ H e H e (cid:17) ω, (2.17) ν e = 2 ∂ A ¯ H e , (2.18)¯ ν e = 2 ¯ ∂ A H e , (2.19)where N v stands for a N v × N v unit matrix, and ∂ A ¯ H e ≡ ∂ ¯ H e − i ¯ H e A s ( e ) + iA t ( e ) ¯ H e , ¯ ∂ A H e ≡ ¯ ∂H e + i ¯ A s ( e ) H e − iH e ¯ A t ( e ) . (2.20) µ v , ν e and ¯ ν e are called the moment maps. The energy is saturated at a solution to theso-called BPS equations µ v = ν e = ¯ ν e = 0 . (2.21)We call the solution of the above differential equations on Σ h as the BPS vortex in thequiver gauge theory.Provided g v (cid:54) = 0, we can take a linear combination of the moment map µ v weightedby 1 /g v to obtain 0 = (cid:88) v ∈ V µ v g v = (cid:88) v ∈ V (cid:18) F v g v − ζ v N v ω (cid:19) , (2.22)because of the zero vector for incidence matrix in generic quiver gauge theory in Eq. (2.3).If we take the trace and integral over Σ h , we find (cid:88) v ∈ V (cid:18) πk v g v − N v ζ v A (cid:19) = 0 . (2.23)We will see that the integral formula for the volume of the moduli space gives a constraintidentical to (2.23). Since k v are integer valued, this condition cannot be satisfied for thegeneric g v and ζ v . This means that the BPS vortices (solution) with k v (cid:54) = 0 cannot existon Σ h for the generic g v and ζ v . In fact, Eq. (2.23) gives a stringent restriction nolt onlyon parameters such as g v and ζ v of the theory and also on allowed vorticity k v of BPSvortices.First, the FI parameters ζ v of the theory need to satisfy (cid:88) v ∈ V N v ζ v = 0 , (2.24)in order for vacuum ( k v = 0 for all v ) to exists.In order to allow BPS states with nonzero vorticity, two types of solutions are available10i) Universal coupling: g = g = · · · = g. (2.25)The local constraint (2.22) at each point in space reduces in this case to (cid:88) v ∈ V F v = 0 . (2.26)Therefore the vorticity of N v − g v = n v g with n v ∈ Z + . This solution allows not all but multiple of n v vorticity foreach gauge group v .In sect.4 we consider the case of universal coupling.(ii) Decoupled vertex:Another solution for the quiver gauge theories admitting BPS vortices is the casewhen there is at least one decoupled vertex g v (cid:48) = 0. The decoupled vertex gives onlya global symmetry and no BPS condition arises for the v (cid:48) vertex. The incidencematrix no longer posseses a zero vector, and the constraint (2.22) is absent. Hencewe can have arbitrary coupling and FI parameters for other gauge groups, providedthere is a decoupled vertex in the quiver diagram. We will consider such a case insect.5. We would like to consider the volume of the moduli space of the quiver BPS vortices,which are the solutions to the equations (2.21). It is useful to embed the system of thequiver BPS vortices into a supersymmetric quiver gauge theory, whose partition functionis localized at the BPS solution.The BPS vortex solution to the quiver BPS equations (2.21) involves the given gaugecoupling constants g v as parameters. On the other hand, the embeded supersymmetric11auge theory has a gauge coupling constants g ,v . The coupling constants g ,v appear asoverall constants of the action of the supersymmetric quiver gauge theory and we will seethat the partition function and vevs are independent of them, thanks to the localizationtheorem. Therefore we can choose g ,v differently from the “physical” gauge coupling g v in the BPS vortex. We will find that the coupling constants g ,v in the supersymmetricquiver gauge theory are controllable parameters which interpolate between the Higgs andCoulomb branch picture.In the following subsections, we concentrate on the quiver gauge theory having onlythe Abelian vertices for a while, since it is sufficient to see the localization theorem andderivation of the volume of the moduli space. We will consider non-Abelian quiver gaugetheories later. We will find that they can be treated by means of a decomposition of thenon-Abelian vertices into the Abelian vertices. Let us consider the supersymmetric quiver gauge theory which contains the Abelian ver-tices only, i.e. the total gauge group is G = U (1) n V .On each vertex, there exist bosonic scalar fields φ v , gauge vector fields A v , ¯ A v and aux-iliary fields Y v , which are 0-forms, (1,0)-forms, (0,1)-forms and 2-forms on Σ h , respectively.There also exist their superpartner fermions η v , λ v , ¯ λ v and χ v , which are Grassmann-valued 0-forms, (1,0)-forms, (0,1)-forms and 2-forms on Σ h , respectively. These bosonsand fermions form vector multiplets of the Abelian gauge theory on each vertex.The supersymmetric transformations between the vector multiplets are given by Qφ v = 0 ,Q ¯ φ v = 2 η v , Qη v = 0 ,QA v = λ v , Qλ v = − ∂φ v ,Q ¯ A v = ¯ λ v , Q ¯ λ v = − ¯ ∂φ v ,QY v = 0 , Qχ v = Y v , (3.1)where ¯ φ v is a complex conjugate of φ v . We note here that if we apply the Q transforma-tions twice on the fields, it generates a gauge transformation with a gauge parameter φ v ,i.e. Q = δ φ v .We also have chiral superfields on each edge. The chiral superfield consists of a complexscalar field H e and its fermionic partner ψ e of 0-form, and an auxiliary field ¯ T e and itsfermionic partner ¯ ρ e of (0,1)-form, on the edge e . The chiral fields generally transform in12he bi-fundamental representation, which means that they possess positive charges of agauge group at the source of the edge s ( e ) and negative charges at the target t ( e ), for theAbelian gauge theory.For those chiral superfields, the supersymmetric transformations are given by QH e = ψ e , Qψ e = iφ v L ( H ) ve ,Q ¯ T e = iφ v L ( ¯ ρ ) ve , Q ¯ ρ e = ¯ T e , (3.2)which also satisfy Q = δ φ v . Here L ( H ) ve is defined from the incidence matrix L ve as L ( H ) ve ≡ L ve H e = + H e if v = s ( e ) − H e if v = t ( e )0 others , (3.3)and similarly L ( ¯ ρ ) ve ≡ L ve ¯ ρ e , where we do not take the sum for the repeated index e .For their complex conjugate fields, which contain 0-form ( ¯ H e , ¯ ψ e ), and (1,0)-form( T e , ρ e ), we have Q ¯ H e = ¯ ψ e , Q ¯ ψ e = − iL T ( ¯ H ) ev φ v ,QT e = − iL T ( ρ ) ev φ v , Qρ e = T e , (3.4)where we defined L T ( ¯ H ) ev ≡ ¯ H e L T ev and L T ( ρ ) ev ≡ ρ e L T ev , without summing over therepeated edge index e , by using the transpose of the incidence matrix.For later convenience, we introduce a norm between forms α and β on Σ h (cid:104) α, β (cid:105) ≡ (cid:90) Σ h α ∧ ∗ ¯ β. (3.5)Using this norm, the action for the vector multiplets on v ∈ V is written as a Q -exactform S V = Q Ξ V , (3.6)where Ξ V = − (cid:2) (cid:104) λ v , ∂φ v (cid:105) + (cid:104) ¯ λ v , ¯ ∂φ v (cid:105) + (cid:104) χ v , Y v − µ v (cid:105) (cid:3) . (3.7)One of the moment maps appears in the action (3.6); µ v = F v − g ,v (cid:16) ζ v − (cid:88) e : s ( e )= v H e ¯ H e + (cid:88) e : t ( e )= v ¯ H e H e (cid:17) ω, (3.8)13hich is the same as the moment map (2.17) for the original vortex system if we replacethe coupling constants g ,v with g v . We however need to distinguish between the momentmaps (3.8) in the supersymmetric action and in the original one of the quiver vortex(2.17), since the solution includes the different coupling constants.In the Q -exact action (3.6), the repeated lower-upper indices are summed implicitlyand the raising or lowering of the indices, such as φ v = G ,vv (cid:48) φ v (cid:48) , is given by a “metric”on the vertices G ,vv (cid:48) = 1 g ,v δ vv (cid:48) , (3.9)which contains the gauge couplings g ,v in contrast to the metric (2.14). In this sense, theaction (3.6) has the gauge couplings g ,v as an overall factor 1 /g ,v .Using the metric G ,vv (cid:48) , we can rewrite the moment map (3.8) as µ v = F v − (cid:16) g ,v ζ v − L ( H ) ve ¯ H e (cid:17) ω, (3.10)where L ( H ) ve = G vv (cid:48) δ ee (cid:48) L ( H ) v (cid:48) e (cid:48) = g ,v δ vv (cid:48) δ ee (cid:48) L ( H ) v (cid:48) e (cid:48) , so L ( H ) ve contains the couplingconstants g ,v unlike L ( H ) ve .For the chiral superfields, we can construct a Q -exact action given by S C = Q Ξ C , (3.11)whereΞ C = 12 (cid:20) (cid:104) ψ e , iφ v L ( H ) ve (cid:105) − (cid:104) ¯ ψ e , iL T ( ¯ H ) ev φ v (cid:105) − (cid:104) ρ e , T e − ν e (cid:105) − (cid:104) ¯ ρ e , ¯ T e − ν e (cid:105) (cid:21) . (3.12)The residual moment maps appear in the chiral superfields action (3.11); ν e = 2 ∂ A ¯ H e , (3.13)¯ ν e = 2 ¯ ∂ A H e , (3.14)where ∂ A ¯ H e = ∂ ¯ H e − iL T ( ¯ H ) ev A v , ¯ ∂ A H e = ¯ ∂H e + i ¯ A v L ( H ) ve , (3.15)for the Abelian theory. The moment maps (3.13) and (3.14) are the same as the originalmoment maps (2.18) and (2.19) since they do not depend on the gauge couplings.14he raising or lowering of the indices of the edge e is just given by δ ee (cid:48) , thus we cansee the Q -exact action (3.11) does not contain any coupling constant g ,v .The total supersymmetric action is given by the sum of the vector and chiral multipletparts S = S V + S C . (3.16)By definition, the total action is also written in a Q -exact form S = Q (Ξ V + Ξ C ) . (3.17)If we rescale the total action like S → tS, (3.18)the partition function or the vev of the supersymmetric operator O , which satisfies Q O =0, is independent of t , since the derivative with respect to t reduces to the vev of the Q -exact operator and vanishes, i.e. ∂∂t (cid:104)O(cid:105) t = − (cid:104)O S (cid:105) t = − (cid:104) Q ( O Ξ) (cid:105) t = 0 , (3.19)where (cid:104)· · · (cid:105) t stands for the vev with the rescaled action tS . Note that we also find by asimilar argument that the partition function or the vev of the supersymmetric operatoris independent of the gauge coupling constants g ,v in S V .If we extract the bosonic part of the action from S V and S C , we obtain S V | B = (cid:104) ∂φ v , ∂φ v (cid:105) + (cid:104) ¯ ∂φ v , ¯ ∂φ v (cid:105) − (cid:104) Y v , Y v (cid:105) + 2 (cid:104) Y v , µ v (cid:105) ,S C | B = 12 (cid:20) (cid:104) φ v L ( H ) ve , φ v L ( H ) ve (cid:105) + (cid:104) L T ( ¯ H ) ev φ v , L T ( ¯ H ) ev φ v (cid:105) − (cid:104) T e , T e (cid:105) + (cid:104) T e , ν e (cid:105) + (cid:104) ν e , T e (cid:105) (cid:21) . (3.20)After integrating out the auxiliary fields Y v and T e , we find S V | B = (cid:104) ∂φ v , ∂φ v (cid:105) + (cid:104) ¯ ∂φ v , ¯ ∂φ v (cid:105) + (cid:104) µ ,v , µ v (cid:105) ,S C | B = 12 (cid:20) (cid:104) φ v L ( H ) ve , φ v L ( H ) ve (cid:105) + (cid:104) L T ( ¯ H ) ev φ v , L T ( ¯ H ) ev φ v (cid:105) + (cid:104) ν e , ν e (cid:105) (cid:21) . (3.21)From the coupling independence, the path integral is localized at the fixed points whichare determined by the equations µ v = ν e = ¯ ν e = 0 , (3.22) ∂φ v = ¯ ∂φ v = 0 , (3.23) φ v L ( H ) ve = L T ( ¯ H ) ev φ v = 0 , (3.24)15he equations in the first line (3.22) are the BPS equation for the quiver vortex at thegauge coupling g ,v . The second line (3.23) and third line (3.24) show that the scalarfields φ v take constant values on Σ h , and φ v and L ( H ) ve are “orthogonal” with eachother, respectively, at the fixed points. The orthogonality conditions (3.24) are solvedeither by (cid:104) φ v (cid:105) = 0 and (cid:104) H e (cid:105) (cid:54) = 0 (the Higgs branch point) or by (cid:104) φ v (cid:105) (cid:54) = 0 and (cid:104) H e (cid:105) = 0(Coulomb branch point), leading to two distinct branches. Eqs. (3.24) can also containspecial solutions in mixed branches ( (cid:104) φ v (cid:105) (cid:54) = 0 and (cid:104) H e (cid:105) (cid:54) = 0), where (cid:104) φ v (cid:105) is proportional tothe vector annihilated by the incidence matrix. This mixed branch will be closely relatedto the constraints (2.22) in the derivation of the volume formula.Finally, we here write down the fermionic part of the action S V | F = 2 (cid:104) λ v , ∂η v (cid:105) + 2 (cid:104) ¯ λ v , ¯ ∂η v (cid:105) − (cid:28) χ v , ∂ ¯ λ v + ¯ ∂λ v + 12 (cid:0) ψ e L T ( ¯ H ) ev + L ( H ) ve ¯ ψ e (cid:1)(cid:29) ,S C | F = i (cid:104) ψ e , η v L ( H ) ve (cid:105) − i (cid:104) ¯ ψ e , L T ( ¯ H ) ev η v (cid:105) + i (cid:104) ψ e , φ v L ( ψ ) ve (cid:105) − i (cid:104) ¯ ψ e , L T ( ¯ ψ ) ev φ v (cid:105) + i (cid:104) ρ e , L T ( ρ ) ev ¯ φ v (cid:105) − (cid:104) ρ e , ∂ A ¯ ψ e − iL T ( ¯ H ) ev λ v (cid:105)− i (cid:104) ¯ ρ e , ¯ φ v L ( ¯ ρ ) ve (cid:105) − (cid:104) ¯ ρ e , ¯ ∂ A ψ e + i ¯ λ v L ( H ) ve (cid:105) , (3.25)for later discussions. Firstly, we consider the localization of the supersymmetric gauge theory in the Higgsbranch, where the scalar fields φ v vanish and the Higgs scalar H e and gauge fields A v takenon-vanishing values in general.Using the coupling independence of the supersymmetric theory, we can choose thecontrollable couplings to be g ,v = g v , where g v are the coupling constants appearedin the BPS quiver vortex equation which we would like to consider. After choosingthe couplings in the Higgs branch, the fixed point equations (3.22) reduce to the BPSequations for the quiver vortex. Thus solutions to the localization fixed point equationsare given by configurations of the quiver vortex.We denote one of the quiver vortex solutions by ˆ A v , ˆ¯ A v , ˆ H e and ˆ¯ H e . In the Higgs16ranch, the fields are expanded around this solution (fixed point) as A v = ˆ A v + √ t ˜ A v , ¯ A v = ˆ¯ A v + √ t ˜¯ A v ,H e = ˆ H e + √ t ˜ H e , ¯ H e = ˆ¯ H e + √ t ˜¯ H e . (3.26)Other bosonic and fermionic fields are expanded around vanishing backgrounds. We justrescale these fields like φ v → φ v / √ t .We now introduce Faddeev-Popov ghosts c v and ¯ c v and Nakanishi-Lautrup (NL) field B v on the vertices to fix the gauge. The BRST transformation, which is nilpotent δ B = 0,is given by δ B ¯ c v = 2 B v ,δ B c v = δ B B v = 0 , (3.27)for the ghosts and NL fields, δ B ˜ A v = − ∂c v ,δ B ˜¯ A v = − ¯ ∂c v ,δ B ˜ H e = ic v L ( ˆ H ) ve ,δ B ˜¯ H e = − iL T ( ˆ¯ H ) ev c v , (3.28)for the fluctuations of the bosonic fields, and similarly for the fermions.To be compatible with the supersymmetric transformation, we need to choose a gaugefixing function by f v = ∂ † ˜ A v + ¯ ∂ † ˜¯ A v + i (cid:16) ˜ H e L T ( ˆ¯ H ) ev − L ( ˆ H ) ve ˜¯ H e (cid:17) − B v , (3.29)where we have introduced co-differentials ∂ † ≡ −∗ ¯ ∂ ∗ , ¯ ∂ † ≡ −∗ ∂ ∗ , (3.30)which give the divergences of the gauge field.The gauge fixing action is given by a δ B -exact form S GF+FP = δ B (cid:104) ¯ c v , f v (cid:105) = 2 (cid:104) B v , f v (cid:105) + (cid:104) ∂c v , ∂c v (cid:105) + (cid:10) ¯ ∂c v , ¯ ∂c v (cid:11) + 12 (cid:68) c v L ( ˆ H ) ve , c v L ( ˆ H ) ve (cid:69) + 12 (cid:68) L T ( ˆ¯ H ) ev c v , L T ( ˆ¯ H ) ev c v (cid:69) . (3.31)17o the gauge fixed total action is given by S (cid:48) = S V + S C + S GF+FP . (3.32)Precisely speaking, the supersymmetric gauge fixing term is written in terms of a linearcombination of Q and δ B ( Q B ≡ Q + δ B ). We can show that Q B is nilpotent and thetotal actions S (cid:48) including the gauge fixing term is written as a Q B -exact form [15]. Thelocalization works for the nilpotent operator Q B . However, this δ B -exact gauge fixingterm is sufficient for our later discussions.It is useful to intruduce combined vector notations (cid:126) B ≡ ( ˜¯ H e , ˜ A v ) T , (cid:126) Y ≡ ( B v , T e / √ , Y v ) T (3.33)for bosonic fields, and (cid:126) F ≡ ( ¯ ψ e , λ v ) T , (cid:126) X ≡ ( η v , ρ e / √ , χ v ) T (3.34)for fermionic fields. Thus we can regard η v as a superpartner of the Nakanishi-Lautrupfield B v , and the degrees of the freedom between the bosons and fermions are balancedwith each other under the Q B -symmetry.Let us now rescale the gauge fixed total action by an overall parameter t like S (cid:48) → tS (cid:48) .Using the vector notation, the rescaled action reduces to t S (cid:48) | B = − (cid:68) (cid:126) Y T , (cid:126) Y (cid:69) + (cid:68) ( ˆ D H (cid:126) B ) T , (cid:126) Y (cid:69) + (cid:68) ( ˆ D † H (cid:126) Y ) T , (cid:126) B (cid:69) + O (1 / √ t ) , (3.35)for bosons, and t S (cid:48) | F = (cid:68) ( ˆ D H (cid:126) F ) T , (cid:126) X (cid:69) − (cid:68) ( ˆ D † H (cid:126) X ) T , (cid:126) F (cid:69) + O (1 / √ t ) , (3.36)for fermions. We here denote the quadratic terms explicitly and cubic or higher orderterms are represented by O (1 / √ t ), which vanish in the t → ∞ limit. We have alsodefined a first order differential operator byˆ D H = − iL ( ˆ H ) ve ∂ † √ ∂ ˆ A − i √ L T ( ˆ¯ H ) ev L ( ˆ H ) ve ∂ . (3.37)Since we can take the t → ∞ limit (WKB or 1-loop approximation) thanks to thecoupling independence of the supersymmetric theory, the above quadratic part of theaction is sufficient to perform the path integral and reproduce the exact results.18t is easy to integrate out all the fluctuations in the quadratic terms (3.35) and (3.36),except for zero modes (the kernel of the operator ˆ D H ). After integrating out all thenon-zero modes of the fluctuations, we obtain a 1-loop determinant(1-loop det) = det (cid:48) ˆ D † H ˆ D H det (cid:48) ˆ D † H ˆ D H = 1 , (3.38)where det (cid:48) stands for the determinants except for the zero modes. The determinants ofthe denominator and numerator are canceled with each other between contributions fromthe bosons and fermions, respectively.After integrating out the non-zero modes, there exist the residual integrals over thezero modes. The bosonic zero modes satisfyˆ D H (cid:126) B = 0 , (3.39)which is a linearized equation of the BPS quiver vortex. So the bosonic zero modes spanthe cotangent space of the vortex moduli space and we finddim ker ˆ D H = dim M k , (3.40)where M k is the moduli space of the quiver vortex of k -flux sector given by (2.16). Theresidual integrals over the bosonic zero modes simply reduce to the integrals over themoduli space of the vortex and just give the volume of the moduli space, which is ourpurpose.On the other hand, there also exists a residual integral over the fermionic zero modes,which satisfy ˆ D H (cid:126) F = 0 . (3.41)This means that the path integral should vanish due to Grassmann integrals of the zeromodes. Thus we need to insert an appropriate supersymmetric operator in order tocompensate the fermionic zero modes. If we consider the vev of this supersymmetricoperartor, we can obtain the volume of the vortex moduli space from the integral of thebosonic zero modes. Q -cohomological Volume Operator In order to compensate the fermionic zero modes, we now introduce an operator whichcontains the fermion as bi-linear terms. It also must be Q -closed (but not Q -exact triv-ially) to preserve the localization arguments (supersymmetry).19o construct the non-trivial Q -closed operator, we first define the following n -formoperator O n by O ≡ W ( φ ) , O ≡ ∂W ( φ ) ∂φ v ( λ v + ¯ λ v ) , O ≡ ∂W ( φ ) ∂φ v F v − ∂ W ( φ ) ∂φ v ∂φ v (cid:48) λ v ∧ ¯ λ v (cid:48) , (3.42)through an arbitrary function W ( φ ) of φ v . These operators obey the so-called descentequations; Q O = 0 ,Q O = − d O ,Q O = d O . (3.43)Thus a possible non-trivial Q -closed operator can be constructed from an integral of theabove 2-form operator I = (cid:90) Σ h O , (3.44)since the Riemann surface does not have the boundary.Choosing W ( φ v ) = ( φ v ) in particular and adding some Q -exact terms, we find anoperator I V ( g v ) = (cid:90) Σ h (cid:20) φ v µ v ( g v ) − λ v ∧ ¯ λ v + i ψ e ¯ ψ e ω (cid:21) , (3.45)is still Q -closed, where µ v ( g v ) is one of moment maps at the coupling g v given by µ v ( g v ) = F v − (cid:0) g v ζ v − L ( H ) ve ¯ H e (cid:1) ω. (3.46)In the Higgs branch, where the coupling constants are tuned to be g ,v = g v , let usconsider a vev of an exponential of the operator I V ( g v ) (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g v k v , (3.47)where a parameter β is introduced. The vev in the path integral around the Higgs branchbackground is denoted as (cid:104)· · · (cid:105) g ,v = g v k v with turning the coupling constants to be g ,v = g v and fixing the magnetic flux (vorticity) as k v . The above vev can be evaluated at the fixedpoints because of the localization in the Higgs branch, since the operator e iβ I V ( g v ) alsobelongs to the Q -cohomological operator, and does not spoil the localization argument. The raising and lowering of the indices v are also done by the metric G vv (cid:48) = g v δ vv (cid:48) . Since we would like to see the volume of the vortex moduli space with the given magnetic flux k v ,topological sectors of the magnetic flux is not summed in our path integral. µ v ( g v ) = 0.Thus the vev (3.47) reduces to (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g v k v = (cid:68) e − iβ (cid:82) Σ h ( λ v ∧ ¯ λ v − i ψ e ¯ ψ e ω ) (cid:69) g ,v = g v k v . (3.48)The fermion bi-linears just compensate the fermionic zero modes as expected. Since thenumber of the fermionic zero modes is equal to the complex dimension of the modulispace, the vev of (3.48) is proportional to β dim C M kv after integrating overall fermioniczero modes.The residual integral over the bosonic zero modes reduces to the integration over themoduli parameters of the solution to the quiver BPS equations, and gives the volume ofthe moduli space. We finally find (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g v k v = N H β dim C M kv Vol( M k v ) , (3.49)up to a numerical factor N H which depends on a definition of the path integral measurein the Higgs branch. Thus the Q -cohomological operator e iβ I V ( g v ) measures the volumeof the moduli space in the path integral.Unfortunately the evaluation of the volume operator e iβ I V ( g v ) in the Higgs branch isdifficult in general, since we do not have a precise knowledge on the metric of the modulispace. If we however evaluate the same operator in the Coulomb branch, then we will seethe path integral reduces to a simple contour integral. Using the coupling independence ofthe supersymmtric theory, we can evaluate the volume of the moduli space in the Coulombbranch at the different coupling constants.In the following, we will consider the Coulomb branch localization. In the Coulomb branch, we tune the controllable coupling constants into special values g ,v → g c,v , which satisfy g c,v = 4 πk v ζ v A , (3.50)i.e. the coupling constants are ajusted to be just at the Bradlow bound for the givenparameters k v , ζ v and A .Since the vevs (backgounds) of the Higgs fields should vanish in the Coulomb branch,the solution of the gauge fields a v and ¯ a v to the moment map (3.10) at the critical couplings21 c,v is given by F v = ∂ ¯ a v + ¯ ∂a v = g c,v ζ v ω. (3.51)Using this solution, we can expand the gauge fields around the backgrounds a v and ¯ a v as A v = a v + 1 √ t ˜ A v , ¯ A v = ¯ a v + 1 √ t ˜¯ A v . (3.52)The scalar fields have vevs (backgrounds) in the Coulomb branch and take constantvalues φ v on Σ h as a consequence of the fixed point equation (3.23), so we can expandthe scalar fields as φ v = φ v + 1 √ t ˜ φ v , ¯ φ = ¯ φ v + 1 √ t ˜¯ φ v . (3.53)Using the index theorem in the Coulomb branch background, we expect that thereexist fermionic zero modes. The number of the fermionic zero modes is determined bythe Betti numbers of the Riemann surface Σ h . First of all, there is one 0-form zero modeon each vertex because of dim H = 1. These are the zero modes of η v , so we denote η v . Secondly, we have one 2-form zero mode χ v , related to dim H = 1, for each χ v . Weexpand these fermionic fields as η v = η v + 1 √ t ˜ η v , χ v = χ v + 1 √ t ˜ χ v . (3.54)There are also 1-form zero modes ( λ v , ¯ λ v ) on Σ h . The (1,0)- and (0,1)-form can beexpanded by cohomology basis γ l ∈ H (1 , and ¯ γ l ∈ H (0 , ( l = 1 , · · · , h ), respectively, cor-responding to each cycle of the Riemann surface Σ h . The cohomology bases are orthogonalwith each other like (cid:104) γ l , γ l (cid:48) (cid:105) = δ ll (cid:48) . (3.55)Thus λ v and ¯ λ v are expanded as λ v = λ v + 1 √ t ˜ λ v , ¯ λ v = ¯ λ v + 1 √ t ˜¯ λ v , (3.56)where the zero modes are also expanded by the bases γ l and ¯ γ l λ v = h (cid:88) l =1 λ v ,l γ l , ¯ λ v = h (cid:88) l =1 ¯ λ v ,l ¯ γ l , (3.57)with Grassmann-valued coefficients λ v ,l and ¯ λ v ,l .22ther fields are just rescaled by 1 / √ t as fluctuations, like H e → H e / √ t . (We omit tilde on these fluctuations expanding around zero.)We again rescale the whole action by S → tS and expand it around the backgroundin the Coulomb branch. The rescaled action becomes tS = (cid:104) ∂ ˜ φ v , ∂ ˜ φ v (cid:105) + (cid:104) ¯ ∂ ˜ φ v , ¯ ∂ ˜ φ v (cid:105) − (cid:104) Y v , Y v (cid:105) + 2 (cid:104) Y v , ∂ ˜¯ A v + ¯ ∂ ˜ A v (cid:105)− (cid:68) ˜ χ v , ∂ ˜¯ λ v + ¯ ∂ ˜ λ v (cid:69) − (cid:68) ˜ η v , ∂ † ˜ λ v + ¯ ∂ † ˜¯ λ v (cid:69) + (cid:68) ( ˆ D C (cid:126) V ) T , (cid:126) V (cid:69) + O (1 / √ t ) , (3.58)up to the quadratic order of the fluctuations. Here we have introduced a vector notation (cid:126) V ≡ ( H e , ψ e , ¯ ρ e / √ T (3.59)and a differential operator (supermatrix)ˆ D C ≡ ∂ † a ¯ ∂ a + | φ v L ve | − L T ev ( iη v + ∗ χ v ) − i √ L T ev λ v ( iη v − ∗ χ v ) L ve i ¯ φ v L ve −√ ∂ † a i √ λ v L ve √ ∂ a − iφ v L ve , (3.60)which is given by the zero modes and incidence matrix L ve (charges of the bi-fundamentalmatters). The first order differential operators ¯ ∂ a and ¯ ∂ † a in ˆ D C are covariant derivativesfor the charged fields in the backgrounds of the gauge fields a v and ¯ a v and acting on (cid:126) V ;e.g. ¯ ∂ a H e = ¯ ∂H e + i ¯ a v L ( H ) ve . (3.61)In the Coulomb branch, we simply choose a Coulomb gauge by a gauge fixing function f v = ∂ † ˜ A v + ¯ ∂ † ˜¯ A v − B v . (3.62)Then, the gauge fixing term and the action for the FP ghosts is given by S GF+FP = δ B (cid:104) ¯ c v , f v (cid:105) = 2 (cid:104) B v , f v (cid:105) + (cid:104) ∂c v , ∂c v (cid:105) + (cid:10) ¯ ∂c v , ¯ ∂c v (cid:11) . (3.63)Using the rescaled action with gauge fixing S (cid:48) → tS (cid:48) = tS + tS GF+FP , (3.64)23e can perform the path integral by the exact Gaussian integral (WKB approximation).Then we obtain only a 1-loop determinant as an exact result of the residual zero modeintegral 1Sdet ˆ D C , (3.65)where Sdet ˆ D C stands for a superdeterminant of ˆ D C , since the Gaussian integrals arecanceled with each other between pairs; ( φ v , ¯ φ v ) ↔ ( c v , ¯ c v ), ( A v , ¯ A v ) ↔ (˜ λ v , ˜¯ λ v ), and( Y v , B v ) ↔ ( ˜ χ v , ˜ η v ).Now if we introduce blocks of the supermatrix differential operator byˆ D C = (cid:32) A BC D (cid:33) , (3.66)where A ≡ ∂ † a ¯ ∂ a + | φ v L ve | ,B ≡ (cid:16) − L T ev ( iη v + ∗ χ v ) − i √ L T ev λ v (cid:17) ,C ≡ (cid:32) ( iη v − ∗ χ v ) L ve i √ λ v L ve (cid:33) ,D ≡ (cid:32) i ¯ φ v L ve −√ ∂ † a √ ∂ a − iφ v L ve (cid:33) , (3.67)then the superdeterminant of ˆ D C can be expressed by1Sdet ˆ D C = det D det A e
Tr log(1 − X ) , (3.68)where X = D − CA − B .Firstly, the ratio of det A and det D are canceled with each other, except for the zeromodes. The number of the zero modes of H e and ψ e is the same, since both are the (0,0)-form fields. On the other hand, the number of the zero modes of H e and ¯ ρ e is different,since ¯ ρ e is the (0,1)-form field, whereas H e is the (0,0)-form field. The difference of thenumber of zero modes is given by the Hirzebruch-Riemann-Roch theoremind ¯ ∂ a = dim H (0 , − dim H (0 , = k v L ve + 12 χ h , (3.69)depending on the charges L ve of the fields H e and ¯ ρ e , background flux k v and Eulercharacteristic χ h on Σ h . Thus we can evaluate explicitly the ratio of the determinant bydet D det A = 1 (cid:81) e ∈ E ( − iφ v L ve ) k v L ve + χ h . (3.70)24econdly, we can evaluate the exponent in (3.68) at the 1-loop level, then we getTr log(1 − X ) (cid:39) − i (cid:88) e ∈ E Tr 1(2 ¯ ∂ † a ¯ ∂ a + | φ v L ve | ) × (cid:8) ( η v L ve )( − iφ v L ve )( ∗ χ v L ve ) + ( λ v L ve )( i ¯ φ v L ve )(¯ λ v L ve ) (cid:9) = − i π (cid:88) e ∈ E (cid:26) ( η v L ve ) 1 i ¯ φ v L ve ( ∗ χ v L ve ) + ( λ v L ve ) 1 − iφ v L ve (¯ λ v L ve ) (cid:27) , (3.71)where we have used the heat kernel to evaluate the above infinite dimensional trace.Let us now consider the vev of the volume operator e iβ I V ( g v ) in the Coulomb branch.The controllable gauge coupling g ,v is now tuned to the critical value g c,v , which saturatesthe Bradlow bound, in the Coulomb branch. The Coulomb branch solution satisfies µ v ( g c,v ) = 0 but not µ v ( g v ) = 0. Indeed, using the Coulomb branch solution (3.51) and (cid:104) H e (cid:105) = (cid:104) ¯ H e (cid:105) = 0, we find µ v ( g v ) = (cid:18) πk v A − g v ζ v (cid:19) ω. (3.72)Thus we have I V ( g v ) = − (cid:88) v ∈ V (cid:40) πφ v (cid:18) ζ v A π − k v g v (cid:19) + 1 g v h (cid:88) l =1 λ v ,l ¯ λ v ,l (cid:41) . (3.73)Now let us consider the vev of the volume operator (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v k v , (3.74)in the Coulomb branch by tuning the controllable parameter as g ,v = g c,v and fixing themagnetic flux as k v . After integrating out all non-zero modes and including all 1-loopcorrections, we obtain an integral over zero modes; (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v k v = N C (cid:90) (cid:89) v ∈ V (cid:40) dφ v π d ¯ φ v π dη v d ∗ χ v h (cid:89) l =1 dλ v ,l d ¯ λ v ,l (cid:41) (cid:81) e ∈ E ( − iφ v L ve ) k v L ve + χ h × exp (cid:34) − πiβ (cid:88) v ∈ V φ v B v + η v M vv (cid:48) ∗ χ v (cid:48) − i h (cid:88) l =1 λ v ,l Ω vv (cid:48) ¯ λ v (cid:48) ,l (cid:35) , (3.75)where N C is an irrelevant numerical constant depending on the path integral measure of25he non-zero modes, and we have defined B v ≡ ζ v A π − k v g v , (3.76) M vv (cid:48) ≡ πi (cid:88) e ∈ E L ve i ¯ φ v (cid:48)(cid:48) L v (cid:48)(cid:48) e L T ev (cid:48) , (3.77)Ω vv (cid:48) ≡ βg v δ vv (cid:48) + 12 π (cid:88) e ∈ E L ve − iφ v (cid:48)(cid:48) L v (cid:48)(cid:48) e L T ev (cid:48) . (3.78)The integral over ¯ φ v , η v and ∗ χ v in (3.75) can be factorized and irrelevant for thevolume of the vortex moduli space, since it does not contain any coupling or parameterlike g v , k v , χ h and β . So we can renormalize the overall constant by N (cid:48) C ≡ N C (cid:90) (cid:89) v ∈ V (cid:26) d ¯ φ v π dη v d ∗ χ v (cid:27) e η v M vv (cid:48) ∗ χ v (cid:48) = N C (cid:90) (cid:89) v ∈ V d ¯ φ v π det M. (3.79)Note here that det M is a function of ¯ φ v , but degenerated for a generic graph since M contains zero eigenvalues. So we need a suitable but irrelevant regularization to define N (cid:48) C .Using the irrelevant overall constant N (cid:48) C , the vev of the volume operator (3.75) reducesto (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v k v = N (cid:48) C (cid:90) (cid:89) v ∈ V (cid:40) dφ v π h (cid:89) l =1 dλ v ,l d ¯ λ v ,l (cid:41) (cid:81) e ∈ E ( − iφ v L ve ) k v L ve + χ h × exp (cid:34) − πiβ (cid:88) v ∈ V φ v B v − i h (cid:88) l =1 λ v ,l Ω vv (cid:48) ¯ λ v (cid:48) ,l (cid:35) = N (cid:48) C (cid:90) (cid:89) v ∈ V dφ v π (det Ω) h (cid:81) e ∈ E ( − iφ v L ve ) k v L ve + χ h e − πiβ (cid:80) v ∈ V φ v B v , (3.80)after integrating out the zero modes λ v and ¯ λ v with a suitable measure.Thus we finally can express the volume of the vortex moduli space as simple line(contour) integrals over φ v , without any explicit information on the metric of the modulispace. In order to evaluate the integral (3.80), we need to choose suitable integral pathof φ v , which determines the condition for the Bradlow bounds and wall crossing. We willsee this phenomenon for concrete examples in the following sections.26 (1) U (1) ... N f H e Figure 3: The quiver diagram of two Abelian vertices with N f flavors. In this section, we apply the integral formula (3.80) for the volume of the vortex modulispace to some Abelian quiver gauge theory. We consider the universal coupling case,although we will keep unconstrained gauge couplings in many places. All the computationsshould be useful in other cases ( g v ! = g v ! ) as well, and the universal coupling case isobtained by taking the limit g v → g at the end. We first start with a quiver which has only two vertices with Abelian gauge groups. Thereexist N f edges (arrows) from one vertex to the other. The quiver diagram is depicted inFig. 3.Each edge corresponds to the bi-fundamental matters. So we have N f kinds of thematters (Higgs fields). For the Abelian theory, this means that the matter has a positivecharge under one U (1) gauge group and a negative charge under the other U (1) . Theincidence matrix L ve is a 2 × N f matrix and represents the charges of the matters by L = N f ! " · · · − − · · · − & . (4.1)27Figure 3: The quiver diagram of two Abelian vertices with N f flavors. In this section, we apply the integral formula (3.80) for the volume of the vortex modulispace to some Abelian quiver gauge theory. We consider the universal coupling case,although we will keep unconstrained gauge couplings in many places. All the computationsshould be useful in other cases ( g v (cid:54) = g v (cid:48) ) as well, and the universal coupling case isobtained by taking the limit g v → g at the end. We first start with a quiver which has only two vertices with Abelian gauge groups. Thereexist N f edges (arrows) from one vertex to the other. The quiver diagram is depicted inFig. 3.Each edge corresponds to the bi-fundamental matters. So we have N f kinds of thematters (Higgs fields). For the Abelian theory, this means that the matter has a positivecharge under one U (1) gauge group and a negative charge under the other U (1) . Theincidence matrix L ve is a 2 × N f matrix and represents the charges of the matters by L = N f (cid:122) (cid:125)(cid:124) (cid:123)(cid:32) · · · − − · · · − (cid:33) . (4.1)27n this model, the BPS vortex equation (moment maps) becomes µ = F − g ζ − N f (cid:88) e =1 H e ¯ H e ω = 0 ,µ = F − g ζ + N f (cid:88) e =1 ¯ H e H e ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.2)Let us consider linear combinations of the moment maps1 g µ + 1 g µ = 1 g F + 1 g F − (cid:0) ζ + ζ (cid:1) ω = 0 , (4.3)1 g µ − g µ = 1 g F − g F − ζ − ζ − N f (cid:88) e =1 H e ¯ H e ω = 0 . (4.4)Integrating (4.3) on Σ h , we find k g + k g = ζ + ζ π A . (4.5)However this equation can not be satisfied for generic value of g v and ζ v since the magneticfluxes k v are integer valued. If ζ + ζ = 0, there exist the vacuum ( k = k = 0) at least,but no BPS vortices is allowed for the generic couplings. If the gauge couplings of two U (1)’s coincide with each other g = g , there are infinitely many BPS vortices when ζ + ζ = 0 and k + k = 0. In this case, U (1) of the difference of the generators in U (1) and U (2) ; A (cid:48) = 12 ( A − A ) (4.6)is isomorphic to a single U (1) theory with N f flavors, and the moment map (4.4) isequivalent to the BPS vortex equation of N f flavors with the flux ( k − k ) / k andFI parameter ( ζ − ζ ) / ζ .On the other hand, integrating (4.4) on Σ h , we get ζ − ζ π A − k g + k g = 12 π N f (cid:88) e =1 (cid:90) Σ h H e ¯ H e ω ≥ . (4.7)28his is a Bradlow bound for the relative charges of the vortex. If g = g , then we need toset ζ + ζ = 0 and k + k = 0 and (4.7) reduces to the Bradlow bound for the Abeliantheory with the single U (1) ζ A π − k g ≥ . (4.8)So there exists an upper bound for the vorticity k on Σ h with the finite area A .Applying the formula (3.80), we obtain (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v k ,k = N (cid:48) C (cid:90) dφ π dφ π (det Ω) h ( − i ( φ − φ )) N f ( k − k + χ h ) e − πiβ ( φ B + φ B ) , (4.9)where det Ω = det (cid:32) βg + π N f − i ( φ − φ ) − π N f − i ( φ − φ ) − π N f − i ( φ − φ ) βg + π N f − i ( φ − φ ) (cid:33) = βg g (cid:18) β + g + g π N f − i ( φ − φ ) (cid:19) . (4.10)Now changing the variables to φ c ≡
12 ( φ + φ ) , ˆ φ ≡ φ − φ , ˆ k ≡ k − k , (4.11)we can write (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v ˆ k = N (cid:48) C (cid:90) dφ c π d ˆ φ π (cid:16) βg g (cid:17) h (cid:16) β + g + g π N f − i ˆ φ (cid:17) h (cid:16) − i ˆ φ (cid:17) N f ( ˆ k + χ h ) e − πiβ ( φ c B c + ˆ φ ˆ B )= N (cid:48) C (cid:90) dφ c π (cid:18) βg g (cid:19) h e − πiβφ c B c (cid:90) d ˆ φ π (cid:16) β + g + g π N f − i ˆ φ (cid:17) h (cid:16) − i ˆ φ (cid:17) N f ( ˆ k + χ h ) e − πiβ ˆ φ ˆ B , (4.12)where B c ≡
12 ( B + B ) = 12 (cid:18) ζ + ζ π A − k g − k g (cid:19) , ˆ B ≡ B − B = ζ − ζ π A − k g + k g . (4.13)29he former integral in (4.12) gives N (cid:48) C (cid:90) dφ c π (cid:18) βg g (cid:19) h e − πiβφ c B c = N (cid:48) C π (cid:18) βg g (cid:19) h δ (2 βB c ) , (4.14)which gives a constraint B c = 0 as we found from (4.3).The latter integral in (4.12); (cid:90) d ˆ φ π (cid:16) β + g + g π N f − i ˆ φ (cid:17) h (cid:16) − i ˆ φ (cid:17) N f ( ˆ k + χ h ) e − πiβ ˆ φ ˆ B , (4.15)is nothing but the integral expression for the volume of the vortex moduli space in U (1)gauge theory with N f flavors [13, 15] up to a redefinition of the parameter β .To evaluate the integral (4.15), we introduce a small twisted mass. Turning on thetwisted mass (cid:15) e for H e , the supersymmetric transformations are modified; e.g. QH e = ψ e , Qψ e = iφ v L ( H ) ve − (cid:15) e H e , (4.16)where we do not sum the repeated index e . This modification by the twisted mass alsomodifies the cohomological volume operator into I (cid:15)V ( g v ) = (cid:90) Σ h (cid:34) φ v µ v ( g v ) + i (cid:88) e ∈ E (cid:15) e H e ¯ H e − λ v ∧ ¯ λ v + i ψ e ¯ ψ e ω (cid:35) . (4.17)Indeed, we can shift the integral path above the real axis without any divergencesfrom the integral of the matter fields, then the integral contour should be closed on thelower half plane (Fig. 4(a)) if ˆ B >
B < φ = − i(cid:15) if ˆ B >
0. So the integral gives a non-vanishing value. This is related to the Bradlow bound condition (4.7). Evaluating the δ ( B c ) diverges at B c = 0, but we absorb and regularize this divergence with the degenerate normal-ization N (cid:48) C at the same time. So we expect a finite constraint B c = 0 from this part. B >
B < φ . The pole exists at ˆ φ = − i(cid:15) . For convergence ofthe integral, we should choose closed circle on the lower half plane (a) if ˆ B >
B <
0. The contour (a) includes the pole inside.integral (4.15), we obtain (cid:90) d ˆ φ π (cid:16) β + g + g π N f − i ˆ φ (cid:17) h e − πiβ ˆ φ ˆ B (cid:16) − i ˆ φ + (cid:15) (cid:17) N f ( ˆ k + χ h ) = h (cid:88) l =0 (cid:18) hl (cid:19) β l (cid:18) g + g π N f (cid:19) h − l × (cid:90) d ˆ φ π e − πiβ ˆ φ ˆ B (cid:16) − i ˆ φ + (cid:15) (cid:17) N f ( ˆ k + χ h ) + h − l = β d h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) g + g π N f (cid:19) h − l (2 π ˆ B ) d − l e − (cid:15)πβ ˆ B ( d − l )! , (4.18)where d ≡ ˆ kN f + ( N f − − h ). So we find, in the (cid:15) → β d h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) g + g π N f (cid:19) h − l (2 π ˆ B ) d − l ( d − l )! (4.19)This is the volume of the moduli space of the Abelian vortex with N f flavor on Σ h . Thedimension of the moduli space is expressed in the power of β , i.e. d = ˆ kN f +( N f − − h ).31 (1) U (1) H H Figure 5: The quiver diagram of two Abelian vertices with a loop.On the other hand, the integral vanish if ˆ
B < φ = 0 is not enclosedinside the contour. This means that there is no BPS vortex solution for ˆ B < B c = 0 and ˆ B > , (4.20)are satisfied, and takes a value of (4.18). The quiver vortex could exist on Σ h with chargeswhich satisfy the condition (4.20). We now consider a model with two vertices, i.e. G = U (1) × U (1) quiver gauge theory.In contrast with the prior model, we have only two matters with opposite charges. Twoedge arrows makes a loop between two vertices. The quiver diagram is depicted in Fig. 5.The incidence matrix is given by L = ! − − " (4.21)The associated moment map is given by µ = F − g ζ − H ¯ H + ¯ H H $ ω = 0 ,µ = F − g ζ + ¯ H H − H ¯ H $ ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.22)32Figure 5: The quiver diagram of two Abelian vertices with a loop.On the other hand, the integral vanish if ˆ B < φ = 0 is not enclosedinside the contour. This means that there is no BPS vortex solution for ˆ B < B c = 0 and ˆ B > , (4.20)are satisfied, and takes a value of (4.18). The quiver vortex could exist on Σ h with chargeswhich satisfy the condition (4.20). We now consider a model with two vertices, i.e. G = U (1) × U (1) quiver gauge theory.In contrast with the prior model, we have only two matters with opposite charges. Twoedge arrows makes a loop between two vertices. The quiver diagram is depicted in Fig. 5.The incidence matrix is given by L = (cid:32) − − (cid:33) (4.21)The associated moment map is given by µ = F − g (cid:0) ζ − H ¯ H + ¯ H H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ + ¯ H H − H ¯ H (cid:1) ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.22)32imilar to the prior model, we can consider sum and difference of the moment maps.1 g µ + 1 g µ = 1 g F + 1 g F − (cid:0) ζ + ζ (cid:1) ω = 0 , (4.23)1 g µ − g µ = 1 g F − g F − (cid:0) ζ − ζ − H ¯ H + 2 ¯ H H (cid:1) ω = 0 . (4.24)From the sum (4.23), we have B c = 12 ( B + B ) = 0 , (4.25)as a constraint for the couplings and FI parameters. From the difference (4.24), we obtainˆ B = B − B = 12 π (cid:90) Σ h H ¯ H ω − π (cid:90) Σ h H ¯ H ω. (4.26)So ˆ B can take any positive and negative values. Thus the moduli space of this modelshould be non-compact since there are infinitely many combinations of the vev of H and H , which give the same difference ˆ B . In particular, if we consider the vacuum( k = k = 0), the vev of H and H is given by the difference of the FI parameters | H | − | H | = ζ − ζ , (4.27)which represents a non-compact moduli space (hyperbolic plane).Applying the integral formula (3.80), we obtain (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v ˆ k = ( − χ h − ˆ k N (cid:48) C (cid:18) βg g (cid:19) h (cid:90) dφ c π e − πiβφ c B c (cid:90) d ˆ φ π e − πiβ ˆ φ ˆ B (cid:16) − i ˆ φ (cid:17) χ h . (4.28)The former integral gives the constraint B c = 0 as expected, but the integral does notdepend on the magnetic flux ˆ k except for the overall sign. And the integral of ˆ φ is highlydegenerated and the choice of the contour is not well-defined.This is because the poles associated with H (cid:54) = 0 (ˆ k >
0) and H (cid:54) = 0 (ˆ k <
0) aremerged. To avoid the degeneration, we modify the model by introducing twisted masses(Ω-backgrounds) for the matter H and H . The introduction of the twisted masseschanges the supersymmetric transformations to Qψ = i ( φ − φ + i(cid:15) ) H = i ( ˆ φ + i(cid:15) ) H ,Qψ = i ( φ − φ + i(cid:15) ) H = i ( − ˆ φ + i(cid:15) ) H , (4.29)33here (cid:15) and (cid:15) are real and positive parameters. The fixed point equation Qψ = Qψ = 0means that H (cid:54) = 0 (or H (cid:54) = 0) contributes near the pole at ˆ φ + i(cid:15) ≈ − ˆ φ + i(cid:15) ≈ B > H (cid:54) = 0, or ˆ B < H (cid:54) = 0. Turning onthe twisted mass does not admit the mixed branch H (cid:54) = 0 and H (cid:54) = 0.The integral formula for the volume is also modified by the twisted mass into (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v ˆ k = N (cid:48) C (cid:90) dφ c π e − πiβφ c B c × (cid:90) d ˆ φ π (det Ω) h e − πiβ ˆ φ ˆ B (cid:16) − i ˆ φ + (cid:15) (cid:17) ˆ k + χ h (cid:16) i ˆ φ + (cid:15) (cid:17) − ˆ k + χ h , (4.30)where det Ω = βg g (cid:18) β + g + g π − i ˆ φ + (cid:15) + g + g π i ˆ φ + (cid:15) (cid:19) = βg g β + g + g π (cid:15) + (cid:15) (cid:16) − i ˆ φ + (cid:15) (cid:17) (cid:16) i ˆ φ + (cid:15) (cid:17) . (4.31)The former integral gives the constraint B c = 0 again. If ˆ B >
0, we need to choose thecontour on the lower half plane. Then we obtain the volume of the moduli space as (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v ˆ k = N (cid:48)(cid:48) C β ˆ k (cid:15) + (cid:15) ) − h − ˆ k h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) g + g π (cid:19) h − l (2 π ˆ B ) ˆ k − l (ˆ k − l )! , (4.32)and ˆ k should be positive, where N (cid:48)(cid:48) C ≡ N (cid:48) C (cid:18) βg g (cid:19) h (cid:90) dφ c π e − πiβφ c B c , (4.33)includes the irrelevant constants and constraint. If ˆ B <
0, we need to choose the contouron the upper-half plane and get (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v ˆ k = N (cid:48)(cid:48) C β − ˆ k (cid:15) + (cid:15) ) − h +ˆ k h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) g + g π (cid:19) h − l ( − π ˆ B ) − ˆ k − l ( − ˆ k − l )! , (4.34)and ˆ k should be negative. 34 (1) U (1) U (1) H H Figure 6: The quiver diagram of three Abelian vertices of the non-unidirectional chain.After regularizing the volume by introducing the twisted masses ! and ! , we find thevolume is proportional to ( ! + ! ) h + | ˆ k |− . So the volume diverges in the limit of ! → ! → h = 0 and ˆ k = 0. This reflects that fact that the moduli space of thevacuum on S is non-compact. The regularization causes the separation of the branch ofthe moduli space. Each branch contributes to the volumes as Abelian BPS vortices of N f = 1. We can see this results from the equation (4.26) if the moduli space is separatedby two branches of ˆ B >
0, ˆ k > H " = 0, or ˆ B <
0, ˆ k < H " = 0 Non-unidirectional chain
We next consider a quiver diagram with three Abelian vertices. The first example is twomatter fields (edges) between three vertices. Orientations of the edges are from the secondto the first and from the second to the third; i.e. the two arrows are emitted from thesecond vertex and oriented in opposite directions to each other. The quiver diagram isdepicted in Fig. 6.The incidence matrix is given by L = − − , (4.35)35Figure 6: The quiver diagram of three Abelian vertices of the non-unidirectional chain.After regularizing the volume by introducing the twisted masses (cid:15) and (cid:15) , we find thevolume is proportional to ( (cid:15) + (cid:15) ) h + | ˆ k |− . So the volume diverges in the limit of (cid:15) → (cid:15) → h = 0 and ˆ k = 0. This reflects that fact that the moduli space of thevacuum on S is non-compact. The regularization causes the separation of the branch ofthe moduli space. Each branch contributes to the volumes as Abelian BPS vortices of N f = 1. We can see this results from the equation (4.26) if the moduli space is separatedby two branches of ˆ B >
0, ˆ k > H (cid:54) = 0, or ˆ B <
0, ˆ k < H (cid:54) = 0 Non-unidirectional chain
We next consider a quiver diagram with three Abelian vertices. The first example is twomatter fields (edges) between three vertices. Orientations of the edges are from the secondto the first and from the second to the third; i.e. the two arrows are emitted from thesecond vertex and oriented in opposite directions to each other. The quiver diagram isdepicted in Fig. 6.The incidence matrix is given by L = − − , (4.35)35nd the moment maps (BPS equations) are µ = F − g (cid:0) ζ + ¯ H H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ − H ¯ H − H ¯ H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ + ¯ H H (cid:1) ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.36)Integrating µ v on Σ h , we find a constraint and the Bradlow bounds B + B + B = 0 ,B ≤ , B ≥ , B ≤ . (4.37)The volume of the moduli space is expressed by an integral over φ , φ and φ (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π dφ π dφ π J (cid:15) ( φ , φ , φ ) , (4.38)where the integrand J (cid:15) ( φ , φ , φ ) is a rational function of φ , φ and φ with poles.Introducing notations φ vv (cid:48) ≡ φ v − φ v (cid:48) , k vv (cid:48) ≡ k v − k v (cid:48) , (4.39)the integrand is given by J (cid:15) ( φ , φ , φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h , (4.40)for this model after turning on the twisted masses (cid:15) and (cid:15) for each edge, wheredet Ω = βg g g (cid:32) β + β π (cid:18) g + g − iφ + (cid:15) + g + g − iφ + (cid:15) (cid:19) + 1(2 π ) g g + g g + g g ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) (cid:33) . (4.41)Integrating φ and φ first, we obtain (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π Res φ = φ + i(cid:15) Res φ = φ + i(cid:15) J (cid:15) ( φ , φ , φ ) , (4.42)and the condition B < B < k + χ h > k + χ h >
0) is needed tocontain poles inside the contour and get non-vanishing value. The final integral depends36 (1) U (1) U (1) H H Figure 7: The quiver diagram of three Abelian vertices of the unidirectional chain (orientedarrows).only on φ such as e − πiβφ ( B + B + B ) , which reduces to the constraint B + B + B = 0as expected. (So we also have B > h = 0), we find ! e iβ I !V ( g v ) " g ,v = g c,v k ,k ,k = N " C dφ π e − πiβφ ( B + B + B ) ( − πB ) k k ! ( − πB ) k k ! e − πβ ( $ B + $ B ) , (4.43)which is finite in the limit of e → N f = 1.For higher genus case h ≥
0, we can also perform the integral in the similar way byexpanding (det Ω) h . Unidirectional chain
Next we consider a quiver chain with three vertices and oriented (unidirectional) arrows.The quiver diagram is depicted in Fig. 7.The incidence matrix and associated moment maps (BPS vortex equations) are givenby L = − − , (4.44)and µ = F − g * ζ − H ¯ H + ω = 0 ,µ = F − g * ζ + ¯ H H − H ¯ H + ω = 0 ,µ = F − g * ζ + ¯ H H + ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.45)37Figure 7: The quiver diagram of three Abelian vertices of the unidirectional chain (orientedarrows).only on φ such as e − πiβφ ( B + B + B ) , which reduces to the constraint B + B + B = 0as expected. (So we also have B > h = 0), we find (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π e − πiβφ ( B + B + B ) ( − πB ) k k ! ( − πB ) k k ! e − πβ ( (cid:15) B + (cid:15) B ) , (4.43)which is finite in the limit of (cid:15) e → N f = 1.For higher genus case h ≥
0, we can also perform the integral in the similar way byexpanding (det Ω) h . Unidirectional chain
Next we consider a quiver chain with three vertices and oriented (unidirectional) arrows.The quiver diagram is depicted in Fig. 7.The incidence matrix and associated moment maps (BPS vortex equations) are givenby L = − − , (4.44)37nd µ = F − g (cid:0) ζ − H ¯ H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ + ¯ H H − H ¯ H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ + ¯ H H (cid:1) ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.45)Expected constraint and Bradlow bounds from the moment maps are B + B + B = 0 ,B ≥ , B ≤ . (4.46)The volume of the moduli space is given by (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π Res φ = φ − i(cid:15) Res φ = φ + i(cid:15) J (cid:15) ( φ , φ , φ ) , (4.47)where J (cid:15) ( φ , φ , φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h , (4.48)anddet Ω = βg g g (cid:32) β + β π (cid:18) g + g − iφ + (cid:15) + g + g − iφ + (cid:15) (cid:19) + 1(2 π ) g g + g g + g g ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) (cid:33) . (4.49)Only the difference from the previous case, we obtain the bounds and constraint as B > B < B + B + B = 0. For the sphere ( h = 0), we also find the volumeis proportional to a product of two finite moduli space of vortex and anti-vortex as(2 πB ) k k ! ( − πB ) k k ! . (4.50)This means that total moduli space is compact and determined by the vortex from µ = 0and anti-vortex from µ = 0 with N f = 1. As a result, the moduli space determined from µ = 0 still remains finite. 38 (1) U (1) U (1) H H H Figure 8: The quiver diagram of three Abelian vertices with a non-unidirectional loop.The moment maps (BPS vortex equations) are µ = F − g ! ζ − H ¯ H − H ¯ H " ω = 0 ,µ = F − g ! ζ − H ¯ H + ¯ H H " ω = 0 ,µ = F − g ! ζ + ¯ H H + ¯ H H " ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.52)Integrating µ v on Σ h , we get a constraint and bounds B + B + B = 0 ,B ≥ , B ≤ ,B − B ≥ . (4.53)The integral formula of the volume is given by e iβ I !V ( g v ) $ g ,v = g c,v k ,k ,k = N ! C % dφ π dφ π dφ π J " ( φ ,φ ,φ ) , (4.54)where J " ( φ ,φ ,φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + ’ ) k + χ h ( − iφ + ’ ) k + χ h ( − iφ + ’ ) k + χ h ,φ vv ! ≡ φ v − φ v ! ,k vv ! ≡ k v − k v ! , (4.55)39Figure 8: The quiver diagram of three Abelian vertices with a non-unidirectional loop. Non-unidirectional closed loop
There are arrows (edges) between the vertices, but two arrows are emitted from the firstvertex U (1) and one arrow is started from U (1) and ended at U (1) . So there is a loopwithout one way (unidirectional) arrows. We depicted the quiver diagram in Fig. 8.The incidence matrix is given by L = − − − . (4.51)The moment maps (BPS vortex equations) are µ = F − g (cid:0) ζ − H ¯ H − H ¯ H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ − H ¯ H + ¯ H H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ + ¯ H H + ¯ H H (cid:1) ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.52)Integrating µ v on Σ h , we get a constraint and bounds B + B + B = 0 ,B ≥ , B ≤ ,B − B ≥ . (4.53)39he integral formula of the volume is given by (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π dφ π dφ π J (cid:15) ( φ , φ , φ ) , (4.54)where J (cid:15) ( φ , φ , φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h ,φ vv (cid:48) ≡ φ v − φ v (cid:48) ,k vv (cid:48) ≡ k v − k v (cid:48) , (4.55)anddet Ω = βg g g (cid:32) β + β π (cid:18) g + g − iφ + (cid:15) + g + g − iφ + (cid:15) + g + g − iφ + (cid:15) (cid:19) + g g + g g + g g (2 π ) − iφ + (cid:15) + (cid:15) + (cid:15) ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) (cid:33) . (4.56)Integrating φ and φ of (4.54) first in turn, we obtain (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π (cid:104) Res φ = φ − i(cid:15) Res φ = φ + i(cid:15) + Res φ = φ − i ( (cid:15) − (cid:15) ) Res φ = φ + i(cid:15) + Res φ = φ − i(cid:15) Res φ = φ + i(cid:15) + Res φ = φ − i ( (cid:15) − (cid:15) ) Res φ = φ + i(cid:15) (cid:105) J (cid:15) ( φ , φ , φ ) . (4.57)To pick up the residues of the poles inside the contour and obtain a non-vanishing volume,we need to assume that B > B <
0, which agrees with the Bradlow bound. In thefinal integral of φ , the integrand depends only on φ through the factor e − πiβφ ( B + B + B ) .So the integral by φ gives the constraint B + B + B = 0.For simplicity, let us consider a vacuum on the sphere ( k v = 0 and h = 0). In thiscase, we can ignore the contribution from det Ω. The volume of the moduli space isproportional to e − β ( (cid:15) ζ +( (cid:15) − (cid:15) ) ζ ) A / (cid:15) + (cid:15) − (cid:15) (cid:16) − e β ( (cid:15) + (cid:15) − (cid:15) ) ζ A / (cid:17) , (4.58)which takes a finite value − βζ A / (cid:15) e →
0. So we can see the moduli spaceof vacua is compact at least. 40 (1) U (1) U (1) H H H Figure 9: The quiver diagram of three Abelian vertices with a unidirectional loop.The associated moment maps (BPS vortex equations) are µ = F − g ! ζ − H ¯ H + ¯ H H " ω = 0 ,µ = F − g ! ζ − H ¯ H + ¯ H H " ω = 0 ,µ = F − g ! ζ − H ¯ H + ¯ H H " ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.60)Each moment map µ v contains both of positive and negative charge matters. So themoduli space of vacua at least is non-compact.The volume of the moduli space is given by the following integral e iβ I !V ( g v ) $ g ,v = g c,v k ,k ,k = N ! C % dφ π dφ π dφ π J " ( φ ,φ ,φ ) , (4.61)where J " ( φ ,φ ,φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + ’ ) k + χ h ( − iφ + ’ ) k + χ h ( − iφ + ’ ) k + χ h , (4.62)anddet Ω = βg g g & β + β π ’ g + g − iφ + ’ + g + g − iφ + ’ + g + g − iφ + ’ ( + g g + g g + g g (2 π ) ’ + ’ + ’ ( − iφ + ’ ) ( − iφ + ’ ) ( − iφ + ’ ) ) . (4.63)41Figure 9: The quiver diagram of three Abelian vertices with a unidirectional loop. Unidirectional closed loop
Let us consider one more case of the three vertices. In contrast with the previous case,all arrows are aligned in one direction (unidirectional) on the loop. The quiver diagramis depicted in Fig. 9.The incidence matrix is given by L = − − − . (4.59)The associated moment maps (BPS vortex equations) are µ = F − g (cid:0) ζ − H ¯ H + ¯ H H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ − H ¯ H + ¯ H H (cid:1) ω = 0 ,µ = F − g (cid:0) ζ − H ¯ H + ¯ H H (cid:1) ω = 0 ,ν e = 2 ∂ A ¯ H e = 0 , ¯ ν e = 2 ¯ ∂ A H e = 0 . (4.60)Each moment map µ v contains both of positive and negative charge matters. So themoduli space of vacua at least is non-compact.The volume of the moduli space is given by the following integral (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:90) dφ π dφ π dφ π J (cid:15) ( φ , φ , φ ) , (4.61)41here J (cid:15) ( φ , φ , φ ) ≡ (det Ω) h e − πiβ ( φ B + φ B + φ B )( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h ( − iφ + (cid:15) ) k + χ h , (4.62)anddet Ω = βg g g (cid:32) β + β π (cid:18) g + g − iφ + (cid:15) + g + g − iφ + (cid:15) + g + g − iφ + (cid:15) (cid:19) + g g + g g + g g (2 π ) (cid:15) + (cid:15) + (cid:15) ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) ( − iφ + (cid:15) ) (cid:33) . (4.63)Thus the volume of the moduli space is given by residues of J (cid:15) (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = N (cid:48) C (cid:82) dφ π Res φ = φ − i ( (cid:15) + (cid:15) ) Res φ = φ − i(cid:15) J (cid:15) ( φ , φ , φ ) if B > B > N (cid:48) C (cid:82) dφ π Res φ = φ + i(cid:15) Res φ = φ − i(cid:15) J (cid:15) ( φ , φ , φ ) if B > B < N (cid:48) C (cid:82) dφ π Res φ = φ − i ( (cid:15) + (cid:15) ) Res φ = φ + i(cid:15) J (cid:15) ( φ , φ , φ ) if B < B > N (cid:48) C (cid:82) dφ π Res φ = φ + i(cid:15) Res φ = φ + i(cid:15) J (cid:15) ( φ , φ , φ ) if B < B < . (4.64)In particular, if we consider the vacuum on the sphere ( k v = 0 and h = 0), we obtain (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k ,k = −N (cid:48) C (cid:90) dφ π e − iβφ ( ζ + ζ + ζ ) A / e − β ( ( (cid:15) + (cid:15) ) ζ + (cid:15) ζ ) A / (cid:15) + (cid:15) + (cid:15) , (4.65)if B > B >
0. Including all other cases, each volume is proportional to 1 / ( (cid:15) + (cid:15) + (cid:15) ) and diverges in the limit of (cid:15) e →
0. This means that the volume of the modulispace of vacua is non-compact.Finally, we would like to comment on an interesting fact. Each residues in (4.64)diverges in the limit of (cid:15) e → N (cid:48) C (cid:90) dφ π e − iβφ ( ζ + ζ + ζ ) A / e β ( (cid:15) ζ − (cid:15) ζ ) A / (cid:16) e β ( (cid:15) + (cid:15) + (cid:15) ) ζ A / − (cid:17) (cid:15) + (cid:15) + (cid:15) , (4.66)which is finite in the limit of (cid:15) e →
0. This is similar to the previous case of the com-pact moduli space. The contributions of the divergence from each region seem to becomplementary to each other. 42
Application to Vortex in Gauged Non-linear SigmaModel
In this section, we would like to consider vortices in gauged non-linear sigma model on ageneric Riemann surface Σ h with a genus h . If we assume that a target space is a K¨ahlermanifold X , the (anti-)BPS vortex equation is defined by µ = F − g (cid:0) ζ + || Z || (cid:1) = 0 ,ν i = 2 ∂ A ¯ Z i = 0 , ¯ ν i = 2 ¯ ∂ A Z i = 0 , (5.1)where Z i are the (inhomogeneous) coordinates of X and || Z || is a positive definite map-ping function from X to R (moment map), which is invariant under a part of U (1)isometries of X . The U (1) gauge symmetry is regarded as a gauging of the U (1) isometryof X , under which the moment map || Z || is invariant. For later convenience, we hereconsider the anti-BPS equation; i.e. the flux k = π (cid:82) Σ h F and FI parameter ζ should benegative. We will consider only the case of X = C P N as an example in the following.According to [21], the above vortex system in gauged non-linear sigma model can beobtained in a strong coupling limit of a certain gauged linear sigma model (GLSM). TheGLSM contains two U (1) gauge groups and two kinds of matter (Higgs) fields, whose totalnumber is N + 1 ( N + 1 arrows in total). n of the matter fields are charged with respectto both U (1) and denoted as H e ( e = 1 , , . . . , n ). The other N − n + 1 matter fields havepositive charges only on one U (1) and are denoted as H e (cid:48) ( e (cid:48) = n + 1 , n + 2 , . . . , N + 1).We call this model as a parent GLSM following [21].The parent GLSM is expressed in terms of the quiver gauge theory. The generic quivergauge theory contains only the bi-fundamental matters (charged under two U (1) vertices),and no fields in the fundamental representation. However, we can introduce a decoupledvertex, which is defined as a vertex with decoupled gauge fields; i.e. the gauge couplingon that vertex is taken in the weak coupling limit. Since the decoupled vertex U ( N )stands for a global U ( N ) symmetry instead of a local symmetry, a matter associated withan arrow between a U ( N c ) vertex and a decoupled vertex becomes N flavors of fields inthe fundamental representation of U ( N c ) (charged fields if N c = 1). We will denote thedecoupled vertex in terms of box vertex.The quiver diagram of the parent GLSM is depicted in Fig. 10. There is n arrows43 (1) U (1) ... H e · · · H e ! Figure 10: The quiver diagram with two Abelian vertices which is a parent of the gaugednon-linear σ -model. The box stands for a decoupled vertex.the moment maps (BPS vortex equations) for the parent GLSM as µ = F − g ! ζ − n " e =1 H e ¯ H e − N +1 " e ! = n +1 H e ! ¯ H e ! ω = 0 ,µ = F − g ! ζ + n " e =1 ¯ H e H e ω = 0 ,ν e = 2 $ ∂ − iA + iA % ¯ H e = 0 , ¯ ν e = 2 $ ¯ ∂ + i ¯ A − i ¯ A % H e = 0 ,ν e ! = 2 $ ∂ − iA % ¯ H e ! = 0 , ¯ ν e ! = 2 $ ¯ ∂ + i ¯ A % H e ! = 0 . (5.2)Integrating µ v on Σ h , we can see Bradlow bounds B ≥ , B ≤ . (5.3)In addition to the above, if we consider a linear combination µ + µ , then we particularlyhave B + B ≥ . (5.4)In contrast with the generic quiver model in the previous section, there is no constrainton B v ’s, reflecting the existence of the decoupled vertex.In the parent GLSM, we have two gauge couplings g and g associated with two U (1)’s. If we take a strong coupling limit of one gauge coupling g → ∞ , the matter fields44Figure 10: The quiver diagram with two Abelian vertices which is a parent of the gaugednon-linear σ -model. The box stands for a decoupled vertex.between two U (1) vertices, which correspond to H e . We also have N − n + 1 arrows fromone U (1) to the fixed vertex, which represent the matter fields H e (cid:48) .Taking care with the charges (representations) of the matter fields, we can write downthe moment maps (BPS vortex equations) for the parent GLSM as µ = F − g (cid:32) ζ − n (cid:88) e =1 H e ¯ H e − N +1 (cid:88) e (cid:48) = n +1 H e (cid:48) ¯ H e (cid:48) (cid:33) ω = 0 ,µ = F − g (cid:32) ζ + n (cid:88) e =1 ¯ H e H e (cid:33) ω = 0 ,ν e = 2 (cid:0) ∂ − iA + iA (cid:1) ¯ H e = 0 , ¯ ν e = 2 (cid:0) ¯ ∂ + i ¯ A − i ¯ A (cid:1) H e = 0 ,ν e (cid:48) = 2 (cid:0) ∂ − iA (cid:1) ¯ H e (cid:48) = 0 , ¯ ν e (cid:48) = 2 (cid:0) ¯ ∂ + i ¯ A (cid:1) H e (cid:48) = 0 . (5.2)Integrating µ v on Σ h , we can see Bradlow bounds B ≥ , B ≤ . (5.3)In addition to the above, if we consider a linear combination µ + µ , then we particularlyhave B + B ≥ . (5.4)In contrast with the generic quiver model in the previous section, there is no constrainton B v ’s, reflecting the existence of the decoupled vertex.44n the parent GLSM, we have two gauge couplings g and g associated with two U (1)’s. If we take a strong coupling limit of one gauge coupling g → ∞ , the matter fieldsare captured on a constraint n (cid:88) e =1 H e ¯ H e + N +1 (cid:88) e (cid:48) = n +1 H e (cid:48) ¯ H e (cid:48) = ζ . (5.5)Using this constraint and quotient by U (1) gauge symmetry, we can regard H e as a setof the inhomogeneous coordinate of C P N . Thus we expect that the moment maps µ = ν e = ¯ ν e = 0 ( e = 1 , . . . , n ) , (5.6)express the BPS equation (5.1) of the (anti-)vortex of the gauged non-linear sigma modelwith the target C P N in the strong coupling limit g → ∞ .We are interested in the volume of the moduli space of the vortex in the gaugednon-linear sigma model with the target C P N . However we first would like to derive thevolume of the moduli space of the parent quiver theory by using the integral formulain the Coulomb branch, since we can obtain the non-linear sigma model in the strongcoupling limit.The incidence matrix of the parent quiver theory is given by L = n (cid:122) (cid:125)(cid:124) (cid:123) N − n +1 (cid:122) (cid:125)(cid:124) (cid:123)(cid:18) (cid:19) · · · · · · − · · · − · · · , (5.7)The N − n + 1 right-most columns represent the matters (arrows) from one U (1) vertexto the decoupled vertex and contains only the positive charge +1. This point is ratherspecial than the usual incidence matrix of the oriented graph.Using the generic integral formula for the quiver theory, the volume of the modulispace of the vortices in the parent GLSM is given by (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k = N (cid:48) C (cid:90) dφ π dφ π J (cid:15) ( φ , φ ) . (5.8)Turning on twisted masses (cid:15) and (cid:15) (cid:48) for H e and H e (cid:48) , respectively, the integrand becomes J (cid:15) ( φ , φ ) = (det Ω (cid:15) ) h e − πiβ ( φ B + φ B )( − iφ + (cid:15) ) n ( k + χ h )( − iφ + (cid:15) (cid:48) ) ( N − n +1) ( k + χ h ) , (5.9)45heredet Ω (cid:15) = 1 g g (cid:18) β + β π (cid:18) n ( g + g ) − iφ + (cid:15) + ( N − n + 1) g − iφ + (cid:15) (cid:48) (cid:19) + 1(2 π ) n ( N − n + 1) g g ( − iφ + (cid:15) )( − iφ + (cid:15) (cid:48) ) (cid:19) = (cid:18) βg + 12 π n − iφ + (cid:15) (cid:19) (cid:18) βg + βg + 12 π N − n + 1 − iφ + (cid:15) (cid:48) (cid:19) − β g . (5.10)Here we rearranged det Ω (cid:15) in the final form for later convenience.We first expand (det Ω (cid:15) ) h by using the binomial theorem as(det Ω (cid:15) ) h = h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) − β g (cid:19) l (cid:18) βg + 12 π n − iφ + (cid:15) (cid:19) h − l (cid:18) βg + βg + 12 π N − n + 1 − iφ + (cid:15) (cid:48) (cid:19) h − l = h (cid:88) l =0 (cid:18) hl (cid:19) (cid:18) − β g (cid:19) l h (cid:88) j = l (cid:18) h − lj − l (cid:19) (cid:18) βg (cid:19) j − l (cid:18) π n − iφ + (cid:15) (cid:19) h − j × h (cid:88) j = l (cid:18) h − lj − l (cid:19) (cid:18) βg + βg (cid:19) j − l (cid:18) π N − n + 1 − iφ + (cid:15) (cid:48) (cid:19) h − j . (5.11)Then the integrand J (cid:15) ( φ , φ ) also can be expanded as follows J (cid:15) ( φ , φ ) = h (cid:88) l =0 h !( h − l )!( − l l ! (cid:18) βg (cid:19) l × h (cid:88) j = l (cid:0) n π (cid:1) h − j (cid:16) βg (cid:17) j − l ( j − l )!( h − j )! 1( − iφ + (cid:15) ) d − j +1 × h (cid:88) j = l (cid:0) N − n +12 π (cid:1) h − j (cid:16) βg + βg (cid:17) j − l ( j − l )!( h − j )! 1( − iφ + (cid:15) (cid:48) ) d − j +1 × e − πiβ ( φ B + φ B ) , (5.12)where we have defined d ≡ nk + ( n − − h ) and d ≡ ( N − n + 1) k + ( N − n )(1 − h ).Using this expansion, we can integrate φ and φ in turn. The volume is expressed in46erms of the residues for each term (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k = N (cid:48) C Res φ = − i(cid:15) (cid:48) Res φ = φ + i(cid:15) J (cid:15) ( φ , φ )= N (cid:48) C (2 πβ ) d + d (2 π ) h h (cid:88) l =0 h !( h − l )!( − l l ! (cid:18) g (cid:19) l × h (cid:88) j = l n h − j (cid:16) g (cid:17) j − l ( − B ) d − j ( j − l )!( h − j )!( k − j )! × h (cid:88) j = l ( N − n + 1) h − j (cid:16) g + g (cid:17) j − l ( B + B ) d − j ( j − l )!( h − j )!( d − j )! × e − π(cid:15)B − π ( (cid:15) (cid:48) − (cid:15) ) B . (5.13)We need to require at least d = nk + ( n − − h ) ≥ d = ( N − n + 1) k + ( N − n )(1 − h ) ≥ . (5.14)So we have k ≥ max (cid:26) , ( n − h − n (cid:27) and k ≥ max (cid:26) , ( N − n )( h − N − n + 1 (cid:27) . (5.15)In integrating φ , we get a bound B < , (5.16)to include the pole at φ = φ + i(cid:15) inside the contour. The integral of φ also gives B + B > , (5.17)as a consequence of the pole at φ = − i(cid:15) . These agree with the Bradlow bounds.The volume is always finite in the limit of (cid:15) → (cid:15) (cid:48) → (cid:15) = (cid:15) (cid:48) = 0 gives the moduli space volume of BPSvortices in the U (1) × U (1) GLSM with n bifundamental and N − n + 1 fundamentalscalar fields. When restricted to N = 1 and n = 1 (corresponding to X = C P case), ourGLSM reduces to that studied in [21]. The results precisely agree with each other pro-vided different conventions are appropriately translated. Our field theoretical derivation47s based on a scheme that is entirely different from that in [21]. Moreover, our resultsinclude the more general cases for generic N and n (corresponding to X = C P N casewith n flavors of charged scalars) and are obtained without any concrete knowledge of themoduli space including the metric.The dimension of the moduli space is given by the overall power of β . So we can seethe dimension of the total moduli space is d + d , which is the sum of the dimensionof the moduli space of the Abelian vortex with n and N − n + 1 flavors. And also thevolume of the moduli space (5.13) is an almost direct product of each Abelian vortexmoduli space with n and N − n + 1 matters, except for the combinatorial factor and sums.We found the volume of the vortex moduli space of the parent model. As we explainedabove, we can expect that the volume of the vortex moduli space of the non-linear sigmamodel with the target space C P N can be obtained by the strong coupling limit of g → ∞ .In this limit, we find B → ζ A π . (5.18)Thus we obtain the volume of the vortex moduli space of the U (1) GNLSM with thetarget space C P N and with n flavors aslim (cid:15),(cid:15) (cid:48) → g →∞ (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v k ,k = N (cid:48) C (2 πβ ) d + d (2 π ) h h (cid:88) l =0 h !( h − l )!( − l l ! × h (cid:88) j = l n h − j (cid:16) g (cid:17) j ( − B ) d − j ( j − l )!( h − j )!( k − j )! × h (cid:88) j = l ( N − n + 1) h − j (cid:16) g (cid:17) j (cid:16) B + ζ A π (cid:17) d − j ( j − l )!( h − j )!( d − j )! . (5.19) We now generalize the Abelian quiver gauge theory to quiver gauge theory with non-Abelian vertices. There exist U ( N ) non-Abelian gauge groups on each vertex. So we48ave quiver gauge theory with a gauge symmetry G = (cid:81) v ∈ V U ( N v ), where N v is a rankof gauge group on a vertex v ∈ V .There are also directed arrows (edges) connecting between vertices, which representsmatter fields in bi-fundamental representations. If we pick up one edge e ∈ E , the matterfield transform as a fundamental representation under the gauge group U ( N s ( e ) ) at thesource of the edge and anti-fundamental representation under U ( N t ( e ) ) at the target ofthe edge.The BPS quiver vortex equations in this non-Abelian theory are given by µ v = ν e = ¯ ν e = 0 , (6.1)where the moment maps are defined before by Eqs. (2.17)-(2.19).Next, we consider an embedding of the above BPS vortex system into a supersym-metric gauge theory. To define the supersymmetric gauge theory, we introduce vectormultiplets and chiral multiplets. The vector multiplets exist on each vertex v and contain0-form scalar fields Φ v , 1-form vector fields ( A v , ¯ A v ), 2-form auxiliary fields Y v , and theirfermionic super partners η v , ( λ v , ¯ λ v ), χ v . All fields are N v × N v matrices and belong tothe adjoint representation.The supersymmetry transformations of the vector multiplets are given by Q Φ v = 0 ,Q ¯Φ v = 2 η v , Qη v = i [Φ v , ¯Φ v ] ,QA v = λ v , Qλ v = − ∂ A Φ v ,Q ¯ A v = ¯ λ v , Q ¯ λ v = − ¯ ∂ A Φ v ,QY v = i [Φ v , χ v ] , Qχ v = Y v , (6.2)where ∂ A Φ v ≡ ∂ Φ v + i [ A v , Φ v ] and ¯ ∂ A Φ v ≡ ¯ ∂ Φ v + i [ ¯ A v , Φ v ]. We can see Q = δ Φ , whichis a gauge transformation with respect to a parameter Φ v .The chiral multiplets correspond to each arrow on the graph. We can devide the chiralmultiplets into two sets. One of the sets contains N s ( e ) × N t ( e ) matrix of 0-form bosonsand fermions, which we denote by ( H e , ψ e ), and (0 , T e , ¯ ρ e ). For this part of thechiral multiplets, we can define the supersymmetry by QH e = ψ e , Qψ e = i Φ v · L ( H ) ve ,Q ¯ T e = i Φ v · L ( ¯ ρ ) ve , Q ¯ ρ e = ¯ T e , (6.3)where Φ v · L ( H ) ve ≡ Φ s ( e ) H e − H e Φ t ( e ) , Φ v · L ( ¯ ρ ) ve ≡ Φ s ( e ) ¯ ρ e − ¯ ρ e Φ t ( e ) , (6.4)49.e. L ( H ) ve is a non-Abelian generalization of the (covariant) incidence matrix, and Φ v is acting on the bi-fundamental representation H e in a suitable way. Again we can see Q = δ Φ .Another set of the chiral multiplets is a conjugate of the above. We have N t ( e ) × N s ( e ) H e , ¯ ψ e ) and (1 , T e , ρ e ) Q ¯ H e = ¯ ψ e , Q ¯ ψ e = − iL T ( ¯ H ) ev · Φ v ,QT e = − iL T ( ρ ) ev · Φ v , Qρ e = T e , (6.5)where L T ( ¯ H ) ev · Φ v ≡ ¯ H e Φ s ( e ) − Φ t ( e ) ¯ H e ,L T ( ρ ) ev · Φ v ≡ ρ e Φ s ( e ) − Φ t ( e ) ρ e . (6.6)Using these multiplets and supersymmetry transformations, we can define the super-symmetric action in a Q -exact form S = Q Ξ V + Q Ξ C , (6.7)where Ξ V = Tr (cid:20) (cid:104) λ v , ∂ A Φ v (cid:105) + (cid:104) ¯ λ v , ¯ ∂ Φ v (cid:105) + (cid:104) η v , i v , ¯Φ v ] (cid:105) + (cid:104) χ v , Y v − µ v (cid:105) (cid:21) , (6.8)Ξ C = 12 Tr (cid:20) (cid:104) ψ e , i Φ v · L ( H ) ve (cid:105) − (cid:104) ¯ ψ e , iL T ( ¯ H ) ev · ¯Φ v (cid:105) (6.9) − (cid:104) ρ e , T e − ν e (cid:105) − (cid:104) ¯ ρ e , ¯ T e − ν e (cid:105) (cid:21) . (6.10)The action contains the moment maps µ v = F v − g ,v (cid:16) ζ v N v − (cid:88) e : s ( e )= v H e ¯ H e + (cid:88) e : t ( e )= v ¯ H e H e (cid:17) ω, (6.11) ν e = 2 ∂ A ¯ H e , (6.12)¯ ν e = 2 ¯ ∂ A H e . (6.13)The moment maps in the supersymmetric action have the same form as the momentmaps of the BPS vortex equations (2.17)-(2.19), but are written in terms of different(controllable) coupling g ,v . 50ince the action is written in the Q -exact form, we can show that the path integral isindependent of an overall coupling t of an action rescaling S → tS. (6.14)So we can evaluate exactly the path integral of the supersymmetric theory by the WKB(1-loop) approximation in the limit of t → ∞ .In the 1-loop approximation, the path integral is localized at fixed points which aredetermined by the following equations µ v = ν e = ¯ ν e = 0 , (6.15) ∂ A Φ v = ¯ ∂ A Φ v = 0 , (6.16)[Φ v , ¯Φ v ] = 0 , (6.17)Φ v · L ( H ) ve = L T ( ¯ H ) ev · Φ v = 0 . (6.18)If we exclude the possibility of a mixed branch, Eqs. (6.15)-(6.18) have two kind ofsolutions; the Higgs branch (cid:104) H e (cid:105) (cid:54) = 0 and (cid:104) Φ v (cid:105) = 0, or the Coulomb branch (cid:104) H e (cid:105) = 0 and (cid:104) Φ v (cid:105) (cid:54) = 0.In the Higgs brach, the solution to the fixed point equation is generically given by thevortex solution at the coupling g ,v . On the other hand, a solution in the Coulomb branchbreaks the gauge symmetry at each vertex to U (1) N v and Φ v are diagonalized intoΦ v = diag( φ v, , φ v, , . . . , φ v,N v ) , (6.19)where φ v,a ( a = 1 , . . . , N v ) are constant on Σ h .If we tune the controllable coupling to g ,v = g v in the Higgs branch, the path integralis localized at the solution to the original BPS equations and reduces to an integral overthe vortex moduli space. However, the path integral itself vanishes due to the existenceof the fermion zero mode.To save this, we need to insert a compensator of the fermion zero modes (volumeoperator) e iβ I V ( g v ) = exp (cid:26) iβ (cid:90) Σ h Tr (cid:20) Φ v µ v ( g v ) − λ v ∧ ¯ λ v + i ψ e ¯ ψ e ω (cid:21)(cid:27) . (6.20)So we expect that the vev of the volume operator in the Higgs branch gives the volumeof the moduli space of the BPS vortex by turning the coupling g ,v → g v .It is difficult to evaluate the above vev in the Higgs branch since we do not know themetric of the moduli space. So we next try to evaluate the vev in the Coulomb branch51icture. Using the coupling independence, we can adjust the controllable coupling g ,v toa critical value g c,v , without changing the vev of the volume operator.At the critical coupling g c,v in the Coulomb branch, after fixing a suitable gauge, thepath integral reduces to contour integrals (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v (cid:126)k v = N (cid:48) C (cid:90) (cid:89) v ∈ V N v (cid:89) a =1 dφ v,a π (cid:81) v ∈ V (cid:81) a
2) and one edge between them. We assume N = 3 and N = 2; i.e. G = U (3) × U (2) quiver gauge theory. (See the upper in Fig. 11.)In the original quiver diagram, there is one arrow between two vertices. So the inci-dence matrix is expressed by 2 × × L ve = (cid:32) − (cid:33) −→ ˜ L ˜ v ˜ e = − − − − − − , (6.27)where ˜ L is the expanded incidence matrix, and ˜ v and ˜ e are the indices of decomposedvertices and edges.Using these expanded vertices, edges and incidence matrix, we can express simply theintegral formula (6.21) as well as the integral formula of the quiver gauge theory with theAbelian vertices (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v (cid:126)k v = N (cid:48) C (cid:90) (cid:89) ˜ v ∈ ˜ V dφ ˜ v π (cid:81) v ∈ V (cid:81) a
1) (2 , ,
2) (2 , , Figure 11: An Abelian decomposition of U (3) × U (2) non-Abelian graph. U (3) and U (2)non-Abelian vertices are decomposed into three Abelian vertices ˜ v = (1 , , (1 , , (1 , v = (2 , , (2 , U (3) × U (2) non-Abelian graph. U (3) and U (2)non-Abelian vertices are decomposed into three Abelian vertices ˜ v = (1 , , (1 , , (1 , v = (2 , , (2 , ˜ v ˜ v (cid:48) ≡ βg v δ ˜ v ˜ v (cid:48) + 12 π (cid:88) ˜ e ∈ ˜ E ˜ L ˜ v ˜ e − iφ ˜ v (cid:48)(cid:48) ˜ L ˜ v (cid:48)(cid:48) ˜ e ˜ L T ˜ e ˜ v (cid:48) , (6.29)and we have set the gauge coupling to be the same as the original non-Abelian verticeslike g ˜ v =( v,a ) = g v .In addition to the Abelian decomposition of the edges, we can consider extra edgesinside the original non-Abelian vertices, which we have depicted in Fig. 11 as the dashedarrows. These extra edges form a directed complete graph in each non-Abelian vertexand are regarded as reproducing the Vandermonde determinant in the numerator of theintegral formula, such that (cid:89) v ∈ V (cid:89) a
1) (2 , ,
2) (2 , ,N c ) (2 ,N f ) Figure 12: The quiver diagram of two non-Abelian vertices, which has U ( N c ) × U ( N f ) symmetry. U ( N f ) vertices will be decoupled and becomes a global symmetry.56Figure 12: The quiver diagram of two non-Abelian vertices, which has U ( N c ) × U ( N f ) symmetry. U ( N f ) vertices will be decoupled and becomes a global symmetry.56he integral formula of this quiver gauge theory is given by (cid:10) e iβ I V ( g v ) (cid:11) g ,v = g c,v (cid:126)k ,(cid:126)k = N (cid:48) C (cid:90) N c (cid:89) a =1 dφ ,a π N f (cid:89) i =1 dφ ,i π (cid:81) a
Using the above observations, let us consider a non-Abelian generalization of the vortexin gauged non-linear sigma model discussed in Sec. 5.We first start with a quiver diagram of three non-Abelian vertices. There are twoarrows from first to second and from first to third vertex. There exist bi-fundamentalmatters associated with the arrows. One is denoted by H , which is N × N matrix, andanother is denoted by H (cid:48) , which is N × N matrix. The quiver diagram is depicted inFig. 13If we consider a decoupling limit of gauge coupling of the third vertex by taking g →
0, the third vertex is decoupled and gives N flavors of U ( N ) gauge theory at thefirst vertex. 57 ( N ) U ( N ) U ( N ) HH ! Figure 13: The quiver diagram with three non-Abelian vertices which is a parent of thegauged non-linear sigma model. The third vertex will be decoupled by taking g → g → ∞ , we get the non-linear sigmamodel.After decoupling the third vertex, we obtain the moment maps of the parent GLSMof gauged non-linear sigma model as µ = F − g ! ζ N − H ¯ H − H ! ¯ H ! " ω = 0 ,µ = F − g ! ζ N + ¯ HH " ω = 0 ,ν = 2 ! ∂ ¯ H − i ¯ HA + iA ¯ H " = 0 , ¯ ν = 2 ! ¯ ∂H + i ¯ A H − iH ¯ A " = 0 ,ν ! = 2 ! ∂ ¯ H ! − i ¯ H ! A " = 0 , ¯ ν ! = 2 ! ¯ ∂H ! + i ¯ A H ! " = 0 . (6.34)Furthermore, if we take the strong coupling limit of the first vertex g → ∞ , we get aconstraint H ¯ H + H ! ¯ H ! = ζ N . (6.35) H and H ! of a solution to the constraint parametrizes the Grassmann coset moduli spaceof vacua M Gr = U ( N + N ) U ( N ) × U ( N + N − N ) . (6.36)So we expect that the moment map equation µ = ν = ¯ ν = 0 represents the vortexequation with the target manifold M Gr . 58Figure 13: The quiver diagram with three non-Abelian vertices which is a parent of thegauged non-linear sigma model. The third vertex will be decoupled by taking g → g → ∞ , we get the non-linear sigmamodel.After decoupling the third vertex, we obtain the moment maps of the parent GLSMof gauged non-linear sigma model as µ = F − g (cid:0) ζ N − H ¯ H − H (cid:48) ¯ H (cid:48) (cid:1) ω = 0 ,µ = F − g (cid:0) ζ N + ¯ HH (cid:1) ω = 0 ,ν = 2 (cid:0) ∂ ¯ H − i ¯ HA + iA ¯ H (cid:1) = 0 , ¯ ν = 2 (cid:0) ¯ ∂H + i ¯ A H − iH ¯ A (cid:1) = 0 ,ν (cid:48) = 2 (cid:0) ∂ ¯ H (cid:48) − i ¯ H (cid:48) A (cid:1) = 0 , ¯ ν (cid:48) = 2 (cid:0) ¯ ∂H (cid:48) + i ¯ A H (cid:48) (cid:1) = 0 . (6.34)Furthermore, if we take the strong coupling limit of the first vertex g → ∞ , we get aconstraint H ¯ H + H (cid:48) ¯ H (cid:48) = ζ N . (6.35) H and H (cid:48) of a solution to the constraint parametrizes the Grassmann coset moduli spaceof vacua M Gr = U ( N + N ) U ( N ) × U ( N + N − N ) . (6.36)So we expect that the moment map equation µ = ν = ¯ ν = 0 represents the vortexequation with the target manifold M Gr . 58he volume of the moduli space of the vortex in the parent model is given by thefollowing integral formula after decoupling the third vertex (cid:10) e iβ I (cid:15)V ( g v ) (cid:11) g ,v = g c,v (cid:126)k ,(cid:126)k = N (cid:48) C (cid:90) N (cid:89) i =1 dφ ,i π N (cid:89) a =1 dφ ,a π (cid:89) i N. S. would like to thank J. M. Speight for giving a stimulating seminar at Keio University.This work is supported in part by Grant-in-Aid for Scientific Research (KAKENHI) (B)Grant Number 17K05422 (K. O.) and 18H01217 (N. S.).60 eferences [1] E. B. Bogomolny, Sov. J. Nucl. Phys. (1976) 449 [Yad. Fiz. (1976) 861];[2] M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. (1975) 760.[3] S. B. Bradlow, Commun. 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