Phases of N=2 Necklace Quivers

We classify the phases of N=2 elliptic models in terms of their global properties i.e. the spectrum of line operators. We show the agreement between the field theory and the M-theory analysis and how the phases form orbits under the action of the S-duality group which corresponds to the mapping class group of the Riemann surface in M-theory.


Introduction
In this note, we study the charge lattices of mutually local bound states of Wilson and 't Hooft lines for N = 2 elliptic models, corresponding to chains of A N−1 gauge groups connected by bifundamental hypermultiplets [1]. We first study the problem in a field theory description by considering the models in the N = 1 formalism. Then we reproduce the results in M-theory, where the models are obtained by wrapping an M5-brane N times on a punctured torus. The charges of the line operators become homologies of closed curves and the lattices are reproduced in terms of the fundamental group of the surface. The geometric description is useful for understanding the action of the S-duality group on the lattices in terms of the mapping class group of the punctured torus.
The phases of N = 4 super Yang-Mills (sym) can be classified in terms of the 't Hooft classification of the possible vacua [2][3][4]. The analysis can be further extended to the cases with N < 4 by adding a supersymmetry-breaking mass deformation [5] . The classification boils down to determining the maximal charge lattice of mutually local bound states of electric Wilson lines (w lines) and magnetic 't Hooft lines (h lines) (see [6] for a precise definition of these operators). The charges are taken with respect to the center of the gauge group and the mutual locality constraints correspond to a generalized Dirac-Schwinger-Zwanziger (dsz) quantization condition. In recent years, this subject has returned to the spotlight of interest due to the discovery of the relation between these lattices and the global properties of the gauge group [7,8]. The gauge group of a quantum field theory is fixed when the gauge algebra is supplemented with additional data such as the charge lattices discussed above.
In the four-dimensional A N−1 N = 4 sym theory, each lattice corresponds to a phase of the SL(2, Z) S-duality group, thus realizing a representation that is in general reducible. In other words, the lattices can be organized in (disjoint) orbits under S-duality. This problem has been reformulated in M-theory in [9]: in this language, the gauge theory lives on M5-branes wrapping the M-theory torus N times, and the bound states are M2-lines wrapping the covering geometry. The problem of computing the possible lattices on the field theory side is translated into the study of the intersections of the closed M2-lines. Indeed, by associating the homologies of these curves to the charges of the lines in field theory, one obtains the dsz quantization condition and recovers the expected charge spectrum.
A similar situation is expected in four-dimensional N = 2 gauge theories arising from wrapping M5-branes on Riemann surfaces. So far, only the case of non-Lagrangian class S theories [10] has been discussed in the literature [11][12][13][14]. These theories can be regarded as the low-energy description of the dynamics of N M5-branes compactified on genus g Riemann surfaces with r punctures, Σ g,r . The case of r = 0 has been reformulated in [14] in terms of the homologies of closed lines on the Riemann surface, while case with punctures has not been fully explored yet. A systematic analysis of the punctured case can however be initiated on a simpler, Lagrangian class of N = 2 gauge theories. It corresponds to the so-called elliptic models of [1], N = 2 Lagrangian gauge theories with product gauge group on a necklace quiver. It is natural to expect that this generalization will lead to a classification of the phases similar to the one discussed in N = 4 sym. This intuition comes from the fact that the case with one puncture corresponds to the N = 2 * theory studied in [5], where it was observed that all the phases present in N = 4 persist after the mass deformation is switched on.
Motivated by this analogy, in this paper we study the phases of the N = 2 elliptic models. In the first part of our analysis, in section two, we study the problem in a purely N = 1 field-theoretical approach. We consider a general quiver with r nodes and compute the charge lattices of the bound states of Wilson-'t Hooft (wh) lines by imposing a generalized dsz condition. The presence of bifundamental hypermultiplets connecting the nodes of the quiver imposes additional constraints on the allowed 2r-dimensional lattices. We show that the possible lattices are actually two dimensional and -as expected -coincide with the ones obtained in N = 4 sym. The second part of the analysis, presented in section three, focuses on the M-theory description . In this picture, we have a genus one Riemann surface with r punctures, Σ 1,r . We show that the analysis of the homologies of closed M2-lines in this geometry reproduces the field theory results. As already observed in [9], also in this case the quantum constraint imposed on the field theory side (the dsz condition) is a classical phenomenon in the geometric description.
The M-theory analysis has the advantage of giving a simple realization for the action of the S-duality group, corresponding to the mapping class group of the punctured Riemann surface Mod(Σ 1,g ) [1] (see also [15,16] for related discussions). In section four we study the action of this group on the geometric side and translate its action on the charges of the bound states of line operators. The net effect is that a part of the S-duality group, generating an SL(2, Z) subgroup, acts on the lattices as in the case of N = 4 sym, while the rest of the action leaves the lattices invariant.
An explicit example, namely the one of the quiver A 1 ⊕ A 1 ⊕ A 1 is discussed in section five and further directions are discussed in section six.

Global properties of elliptic models
In this section, we study the global properties of an infinite class of N = 2 gauge theories with r gauge groups. These gauge theories can be represented conveniently via a quiver diagram. One can associate each gauge group to a node and place the nodes on a circle. Each pair of consecutive nodes is connected by two arrows with opposite orientations. These arrows represent a pair of bifundamental N = 1 chiral fields X l,l+1 and X l+1,l , i.e. the N = 2 hypermultiplets. There is also an N = 1 adjoint field X l,l associated to each node, corresponding to the N = 2 vector multiplets. In Figure 1, an example of such a quiver with r = 4 is shown. The matter fields interact through a superpotential where the sum is cyclic (the label l = r + 1 is identified with l = 1) and the coupling is fixed by supersymmetry. We consider the case in which each gauge component  has algebra A N−1 and the full gauge group has the form where Z N is diagonally embedded [17][18][19]. In the infrared (ir), the overall U(1) In N = 4 sym, the spectra of line operators are determined by imposing this condition on the charges. Here this is not enough: the conditions must be supplemented by some information on the structure of the quiver because the bifundamental matter is not compatible with some of the lattices that solve the dsz quantization. In order to construct the lattices, we can set up the problem as follows. Consider a bifundamental field X l,l+1 charged under the l-th and the l + 1-st group: this corresponds to a line operator where e l = −e l+1 = 1 and all other charges are set to zero. Imposing the dsz condition between X l,l+1 and a generic line we find Applying this constraint to the rest of the quiver, we find m l = m mod N for each value of l. This is the first simplification and we can now express the charge of a line operator as A second simplification is possible because by linearity, the existence of two lines with charges e 1 and e 2 implies the existence of a line with charge e 1 = e 1 + e 2 , e 2 = 0. Let l be the line l = (e 1 , e 2 , 0, . . . ; 0). In the theory, there is always the line l X 1,2 = (1, −1, 0, . . . ; 0) that has the same charge as the bifundamental field X 1,2 . This means that by linearity, the charge l + e 2 l X 1,2 = (e 1 + e 2 , 0, . . . ; 0) is also allowed. In general, if there is a line (e 1 , . . . , e r ; m), there is also a line (∑ r e r , 0, . . . , 0; m) and we can use this line as a representative for the whole family. We conclude that a generic line belongs to a family parametrized by a pair of integer charges, l O = (e; m) where e is the sum of the electric charges and m is the unique magnetic charge. The dsz condition in Eq. (2.7) becomes a condition on the charges (e; m) and (e , m ), viz. em − me = 0 mod N. (2.8) We have just reformulated the lattice (Z N ) r × (Z N ) r as a Z N × Z N lattice. A twodimensional lattice is generated by two non-negative integer vectors (k, 0) and (i, k ), where kk = N and 0 ≤ i < k. Once these two integers are specified, the global gauge group is fixed. We denote the gauge group by where the choice of k fixes the quotient Z k and the integer i is the electric charge of the line with the lowest possible non-vanishing magnetic charge m = N/k. This shows that the lattice structure of the N = 2 elliptic models is identical to the one of N = 4 sym. For N = 4 sym, the different possible lattices for a given algebra A N−1 can be arranged into representations of the SL(2, Z) symmetry acting on the gauge coupling. In the next section, we will derive the lattices from the M-theory description and study the action of the S-duality on the geometry. After that, we will translate this action into the field theory language and study its effect on the charge lattices.

Geometry
In this section we rederive the field theory results obtained in the last section via M-theory. The M-theory description of the elliptic models has been originally discussed in [1] as an uplift of the type iia description. The latter consists of a stack of N D4-branes extended along x 0123 and wrapping the compact direction x 6 . There are also r parallel NS5-branes, extended along x 012345 , placed at the positions p l = x 6 l . The lift to M-theory happens along the coordinate x 10 . The N D4-branes branes by themselves would become an M5-brane wrapping N times the two compact directions x 6 and x 10 , while the NS branes lift to M5-branes at fixed positions in x 6 and x 10 . Together, the geometric picture consists of the N-cover of Σ 1,r , a genus one Riemann surface with r punctures. We refer to this covering geometry as Σ N 1,r . By ordering the punctures, one can interpret the distance between two consecutive punctures along x 6 and x 10 as the holomorphic gauge coupling of a node of the quiver of the four-dimensional theory: where the periodicity in the coordinates x 6 and x 10 is respectively 2πL and θR.
In the previous section we have studied the global properties by supplementing the theory with additional data, the charges of the line operators. A w line or a h line is represented in the geometric picture by an M2-brane extended in x 0 (the time direction), x 4 (a direction perpendicular to the M5-brane) and wrapping a geodesic on the Riemann surface Σ N 1,r . Such M2-branes appear as lines on Σ N 1,r and we refer to them as M2-lines. An M2-line extended in x 10 and at fixed x 6 passing between two punctures P l and P l+1 corresponds to a Bogomol'nyi-Prasad-Sommerfield (bps) state with electric charge e l = 1, while any M2-line extended in x 6 and at fixed x 10 is a state with magnetic charge m = 1. More in general, the charges of the line operators on the field theory side correspond to the homologies of the closed M2-lines on Σ 1,r . Following the analysis of [9,14], we can study the charge lattices in terms of the M2-lines by introducing the notion of the fundamental group. This is the set of homotopy classes of curves, where two closed curves are said to be homotopic if one can be continuously deformed into the other. A possible presentation of the fundamental group of the r-punctured torus π 1 (Σ 1,r ) is obtained in terms of the α and β cycles of the torus, plus a set of r cycles {γ l } r l=1 that go around each puncture P l (see Figure 2), together with the condition that there is a non-contractible line of trivial homology that can be written either as the commutator of α and β or as the product of the γ l : This relation can be used to rewrite γ r as a function of the other generators: so that π 1 (Σ 1,r ) is the free group of r + 1 generators, endowed with the symplectic structure i (·, ·) describing the intersection of two curves, which in this basis reads: There is an alternative basis for the free group which is convenient for our problem. Consider a set of r α-cycles α l defined as (see Figure 2) We can invert the relation and write to show that the fundamental group can be recast in the form with the symplectic structure The homology of a curve C can be expressed in terms of either basis as (3.10) which provides the map between the coefficients: e r = e − λ 1 . This reproduces precisely the structure of the charges in the gauge theory. Since there is only one β-cycle 2 , there is only one magnetic charge. The r α-cycles correspond to the r electric charges and the dsz condition is the intersection number between two geodesics on the Riemann surface which only depend on how many times the curve wraps the α and the β cycles, i.e. the sum of the electric charges and the unique magnetic charge.
Now that the geometric structure of the problem is set up, we have to consider the multiple cover of the M-theory torus by the M5-brane to reproduce the stack of N D4-branes in the type iia description and ultimately the non-Abelian SU(N) l gauge factors on the field theory side. By studying the intersection of the cycles introduced above in the covering geometry and their projection to the field theoretical charges, we will be able to construct the lattices via the geometric analysis.
An N-cover of the r-punctured torus Σ N 1,r is a torus with N × r punctures (Riemann-Hurwitz). A given cover is identified by its fundamental group, which is a subgroup of index N of π 1 (Σ 1,r ). These subgroups are classified in terms of maps from π 1 (Σ 1,r ) to the symmetric group of N elements S N and can be always put into the form where γ l,p can be written as: γ l,p = Ad λ l,p γ l = λ l,p γ l λ −1 l,p (3.14) for some λ l,p ∈ π 1 (Σ 1,r ), chosen such that the relation in the presentation of the fundamental group of the cover is equivalent to the relation in the fundamental group of the base: The integers k, k , i satisfy the relations For fixed N there are σ 1 (N) such covers, where σ 1 is the divisor function, i.e. the sum over all the divisors of N: σ 1 (N) = ∑ d|N d. This is to be compared with the results of the previous section: once more we see that the geometric structure precisely reproduces the results of the gauge theory. The cover Σ N 1,r inherits a symplectic form from the base, given by This means that if we take two closed curves C N and C N on Σ N 1,r , their symplectic product, counting how many times the projections of the curves will intersect on the base Σ 1,r is given by We have studied the homologies of the closed curves in the multiple covering space and interpreted these curves as bound states of w lines and h lines on the field theory side. The situation is analogous to the one discussed in [9]. Again, the intersection number of these curves becomes the dsz condition on the field theory side. Note an interesting aspect of this quantization condition derived from M-theory: on the field theory side, the dsz condition for the product of gauge groups in Eq.(2.4) is different from the one derived coming from the intersection of the lines in Eq. (3.18). They become the same if we consider the presence of the hypermultiplets, because this fixes m i = m in Eq.(2.4). This is expected because the presence of the punctures in the geometry translates into the presence of the hypermultiplets in the field theory description.
This concludes our discussion of the derivation of the lattices for the elliptic models from the M-theory description. We have shown how to interpret the charges of the lines in the geometric language and that the study of the intersection numbers of closed curves in the geometry reproduces the field theory constraints imposed by the mutual locality condition.

S-duality
In this section, we discuss the structure of the S-duality group and its action on the lattices.
Let us start by discussing the situation without punctures. This corresponds to the usual N = 4 sym theory and the S-duality group corresponds to the action of the modular group SL(2, Z) on the complex structure of the torus, τ. The generators of this group act as S : τ → −1/τ and T : τ → τ + 1. The action of these generators on a dyon with charge (e, m) is  This action corresponds to an exact duality on the string coupling. When we add the punctures, there still is an SL(2, Z) acting on the string coupling but now we have r components, each with its own gauge group that a priori has an SL(2, Z) symmetry. This means that the full S-duality group must be more intricate. This situation is clarified by the M-theory description.
We have seen that the four-dimensional gauge theory can be regarded as a reduction from six dimensions on the multiple cover of a punctured torus Σ 1,r . If we apply an isomorphism of the torus before the reduction, this will in general lead to a different four-dimensional gauge theory that is related to the previous one by S-duality. In other words, the action of the "symmetries" of the Riemann surface (the mapping class group Mod(Σ 1,r )) will produce all the possible phases of a given necklace quiver gauge theory.
The mapping class group of a punctured Riemann surface Σ g,r is decomposed into the product of the pure mapping class group PMod(Σ g,r ) that leaves each puncture invariant and the permutation group S r acting on the punctures [21]. More precisely, the following is a short exact sequence: It follows that a generating set for Mod(Σ g,r ) is given by a generating set for PMod(Σ g,r ) together with a set of elements in Mod(Σ g,r ) that project to generators of S r , i.e. the r − 1 transpositions of two consecutive punctures. The group PMod(Σ g,r ) is generated by a set of Dehn twists which, for the punctured torus Σ 1,r , are around the cycles α l and β (see Figure 3(a) and 3(b)). They act on the generators of π 1 (Σ 1,g ) as follows: The transposition of two punctures, say P n and P n+1 , corresponds to a Dehn halftwist around a curve that encloses the two punctures (see Figure 3(c)) and acts on the fundamental group as follows: A minimal set of generators for Mod(Σ 1,r ) is given by two elements from PMod(Σ 1,r ), T = T r and S = T β T r T β −1 , together with two from S r acting as follows: T : {α l , β} → {α 1 , . . . , α r , βα −1 r }, (4.7) Observe that ω cyclically permutes all the punctures, it is the generator of the cyclic group Z r = ω|ω r = 1 . This is the symmetry group of the affineÂ r−1 Dynkin diagram, which has the same shape as our necklace quiver. In this sense, we can think of Z r as of a classical symmetry (realized geometrically in type iia), which is enhanced by quantum effects to Mod(Σ 1,r ) (realized geometrically in M-theory). Each closed curve on the cover Σ N 1,r corresponds to a bps line operator in the necklace quiver gauge theory, whose central charge is 3 Z = r ∑ l=1 e l a l + ma D , (4.11) where a l and a D are the integrals of the Seiberg-Witten differential λ around the cycles α l and β: An element M ∈ Mod(Σ 1,r ) acts as a matrix on the vector (a 1 , . . . , a r , a D ):  where e = e 1 + · · · + e r is the total electric charge. Now we can give a physical interpretation for the action of the mapping class group.  • The operators σ 1 and ω do not change e or m but change the distribution of the electric charge among the gauge groups. These transformations map a state in a given phase into another state in the same phase. This corresponds to the intuition that a permutation of the punctures (the NS5-branes) does not change the total number of M2-branes that are reduced to fundamental strings, but only how the F1s are distributed among the stacks of D4-branes.
See Figure 4 for the full diagram showing the complete action of PMod(Σ 1,3 ) on the three lattices of the A 1 ⊕ A 1 ⊕ A 1 necklace quiver.

Further directions
In this paper we have studied the global properties of N = 2 necklace quiver gauge theories with r nodes. They can be understood in terms of the charge lattices of mutually local bound states of w lines and h lines. We find that they can be formulated as two-dimensional lattices which correspond to the ones obtained in N = 4 sym. We have reinterpreted the analysis in a geometric language by studying the uplift of this system to M-theory. In this picture, the problem reduces to studying the homologies of closed M2-lines on the N-cover of a torus with r punctures. We have reproduced the field theory results by introducing the notion of the fundamental group. Finally, we have shown how to connect different lattices by S-duality, corresponding to the action of the mapping class group of the Riemann surface. The latter is decomposed into the combined action of the SL(2, Z) symmetry on the torus and of the permutation and shift symmetries of the punctures. Only the generators of SL(2, Z) act non-trivially on the lattices, which can be organized into separate orbits of the S-duality group, just as in N = 4 sym.
Our geometrical analysis can also be useful for class S theories. These theories are constructed by gluing fundamental N = 2 T N blocks, with SU(N) 3 global symmetry. In the M-theory description, these blocks represent spheres with three punctures and the gluing operation corresponds to the gauging of the global symmetries. The four-dimensional theories are in general non-Lagrangian and are obtained by a partially twisted compactification of the Riemann surface obtained by gluing T N blocks. The M-theory description has been used in [14] to derive the global properties of these four-dimensional gauge theories, but the analysis was restricted to the case of compact Riemann surfaces. Here, we have considered the presence of punctures in similar geometries. It would be interesting to generalize our current understanding to the case of class S theories with generic punctures.
Another interesting line of research consists in studying N = 1 theories. One can indeed generalize the analysis to N = 1 theories with an M-theory origin. It can be done by giving some masses to the adjoints in the elliptic models (e.g. by embedding the construction in a fluxtrap background [22][23][24]), adding fluxes (see e.g. [25,26]) or by looking at some generalizations of the class S theories, like the Sicilian theories [27] and the class S k theories [28]. In these cases, the possible lattices have to coincide with the ones studied here. This can be verified by reproducing our N = 1 field theory analysis of section 2. As already observed there, this result is also expected from the brane description: there is a U(1) symmetry, namely the center of mass of the stack of branes on which the gauge theory lives, that decouples in the ir. The different consistent factorizations of this U(1) symmetry correspond to the various theories associated to the same algebra [20].
Another extension of our discussion regards the classification of the lattices for theories with real gauge groups, corresponding to the presence of orientifold fixed points in the M-theory picture. This requires taking into account the effect of these fixed points in the fundamental group. In the N = 1 case this analysis may have interesting consequences on the structure of the S-duality group.
It follows that they leave the metric on the moduli space ds 2 = Im[da D,l da l ] invariant.
The central charge of a bps object in the theory can be written in terms of the a l and a D,l as Z = e l a l + m l a D,l . (A.9) Taken separately, the integrals along the paths β l diverge but the divergence coming from the puncture P l appears with opposite signs in a D,l−1 and a D,l . This means that the central charge is finite if and only if all the coefficients m l are equal. The result is that we can interpret the configuration in terms of m l = m M2-lines of finite length wrapping the cycle β. In the type iia reduction this corresponds to having the same number of D2-branes between each pair of NS5s, i.e. the magnetic charge of a bps state must be the same for each of the gauge components. Once more we see a field-theoretical quantum condition resulting from a classical condition in M-theory. Since the only magnetic component remaining is m l = m we can rewrite the central charge as in Eq. (4.11) where a D = ∑ l a D,l .