3D Seiberg-like Dualities and M2 Branes

We investigate features of duality in three dimensional N=2 Chern-Simons matter theories conjectured to describe M2 branes at toric Calabi Yau four-fold singularities. For 3D theories with non-chiral 4D parents we propone a Seiberg-like duality which turns out to be a toric duality. For theories with chiral 4D parents we discuss the conditions under which that Seiberg-like duality leads to toric duality. We comment on such duality in 3D theories without 4D parents.


Introduction
Different descriptions of the same physical phenomenon usually provide a better understanding of the phenomenon itself. AdS/CFT correspondence and Seiberg duality are two famous examples. In the AdS 5 /CFT 4 case it happens that to a single geometry correspond different UV field theory descriptions. This phenomenon was called Toric Duality in [1], analyzed in [2,3] and identified as a Seiberg duality in [4,5]. Due to the difficulties to understand the field theory living on M2 branes the AdS 4 /CFT 3 correspondence was less mastered. An important step in this direction has been the realization of the importance of Chern-Simons interactions [6] and the subsequent construction of N = 8 Chern-Simons matter field theories in [7,10,8,9,10,11,12,13]. A breakthrough toward the explicit realization of the AdS 4 /CFT 3 correspondence was done in [14] where the authors propose U (N ) k × U (N ) −k CS matter theories, with N = 6 supersymmetry, to be the low energy theories of N M2-branes at the C 4 /Z k singularities. Afterwards, this construction has been extended to many others CS matter theories with a lower amount of supersymmetries [15,16,17,18,19,20,21,22,23,24,25,26,27].
A large and interesting, but still very peculiar, class of AdS 4 /CFT 3 pairs is realized by M2 branes at Calabi Yau four-fold toric singularities [19,20,21,26,27,28,29]. The low energy theories are proposed to be a special kind of N = 2 Chern-Simons matter theories. It was soon realized [21] that, as in the AdS 5 /CFT 4 case, different N = 2 Chern-Simons matter theories can be associated with the same Calabi Yau fourfold geometry. The phenomenon of toric duality reappears in the AdS 4 /CFT 3 correspondence, but in a much general context. Indeed, contrary to the four dimensional case, in three dimensions one find models with different numbers of gauge group factors to describe the same IR physics (for example mirror symmetry pairs [30]). In the literature some very specific pairs of dual field theories were constructed. A step was done in [31,32] where a sort of Seiberg like duality for three dimensional Chern-Simons matter theory was proposed 1 . In the context of M2 branes at singularities, we can divide the set of dualities in the ones that change and in the ones that do not change the number of gauge group factors. In this paper we will call the second type of duality Seiberg-like toric duality. N = 2 Chern-Simons matter theories for M2 branes at singularities are typically described by a quiver with an assignment of Chern-Simons levels k i and a superpotential in a way similar to the gauge theories for D3 branes at Calabi Yau three-fold singularities. Indeed a class of N = 2 three dimensional theories can be simply obtained from four dimensional N = 1 quivers with superpotential in the following way: rewrite the theory in three dimensions, change the SU (N ) gauge factors to U (N ) factors, disregard the super Yang-Mills actions and add a super Chern-Simons term for every factors. We will say that these three dimensional theories have a four dimensional parent. Viceversa we will call theories without four dimensional parents the three dimensional theories that cannot be obtained in the way just explained [27].
In this paper we investigate Seiberg-like toric dualities for (2+1) dimensional N = 2 Chern-Simons matter theories associated with M2 branes at Calabi Yau four-fold toric singularities. Using a generalization of the forward algorithm for D3 branes [26] we analyze a particular branch of the moduli space that is supposed to reproduce the transverse fourfold Calabi Yau singularity. We identify a set of Seiberg-like toric dualities for three dimensional Chern-Simons quiver theories.
For theories with four dimensional parents one could try to simply extend the four dimensional Seiberg duality to the three dimensional case. In fact, three dimensional theories share the same Master Spaces [35,36] of their four dimensional parents. In the map between four and three dimensions a direction of the Master Space become a direction of the physical Calabi Yau four-fold. Unfortunately, it turns out that an arbitrary assignment of 2+1 dimensional Chern-Simons levels does not in general commute with 3+1 dimensional Seiberg duality [21]. In fact it was shown in [37] that the Master Space for four dimensional Seiberg dual theories are not in general isomorphic. Actually it seems that three dimensional CS theories with chiral four dimensional parents do not admit a simple generalization of the three dimensional SQCD Seiberg duality as it happens in the four dimensional case.
Here, we first analyze non chiral three dimensional CS theories with (3+1)d parents. Using a type IIB brane realization, we propose a Seiberg like duality, with a precise prescription for the transformation of the CS levels and the gauge groups factors. We then check that this proposed Seiberg like duality is indeed a toric duality, namely that the two dual theories are associated with M2 branes probing the same Calabi Yau four-fold singularity.
We try to simply extend to chiral CS theories with (3+1)d parents the rules that we have found for the non chiral theories. For chiral four dimensional theories the Master Space is not isomorphic among Seiberg dual phases [37]. This fact presumably puts constraints on the duality transformations for the three dimensional case. We find indeed difficulties for a straightforward realization of Seiberg like toric dualities for 2+1 dimensional Chern-Simons matter theories with four dimensional chiral parents.
However, by analyzing several examples, we find a rule for the assignments of the CS levels such that toric duality still holds among Seiberg like dual phases.
We finally give some examples of Chern-Simons theories without four dimensional parents. In this case there is no immediate insight from the four dimensions, but we show that the duality proposed for the chiral theories works also for theories without a 3+1 parents.
Our analysis is a first step to the study of Seiberg-like toric dualities in the context of M2 branes. We tried to use the intuition from the non-chiral case and to leave as arbitrary as possible the values of the Chern-Simons levels. It is reasonable that more general transformation rules exist. Moreover it would be nice to investigate more general families of toric dualities, like the ones changing the number of gauge group factors and the large limit for Chern-Simons levels. We leave these topics for future investigations.
As we were finishing this paper, we were informed of [54,55] which discuss related topics.

M2 branes and N = 2 Chern Simons theories
As discussed in the introduction, supersymmetric Chern Simons theories coupled to matter fields are good candidates to describe the low energy dynamics of M2 branes [6,14]. In this paper we are interested in M2 branes at Calabi Yau four fold toric conical singularities. The authors of [20] proposed that the field theories living on these M2 branes are (2+ 1) dimensional N = 2 Chern Simons theories with gauge group G i=1 U i (N ) with bifundamental and adjoint matter fields. The Lagrangian in N = 2 superspace notation is: where V a are the vector superfields and X ab are bifundamental chiral superfields. The superpotential W (X ab ) satisfies the toricity conditions: every field appears just two times: one time with plus sign and the other time with minus sign. Since these theories are conjectured to be dual to M theory on AdS 4 ×SE 7 , where SE 7 is a seven dimensional Sasaki Einstein manifold, the moduli space of these theories must contain a branch isomorphic to the four-fold Calabi Yau real cone over SE 7 : M 4 = C(SE 7 ). To study the moduli space we need to find the vanishing conditions for the scalar potential. The scalar potential is: , σ a and D a are scalar components of the vector superfield V a , and with abuse of notation X ab is the lowest scalar component of the chiral superfield X ab . The moduli space is the zero locus of the scalar potential and it is given by the equations: In [20] it was shown that if The equation fixing the field σ leaves a Z k action with gcd({k α }) = k by which we need to quotient to obtain the moduli space. In the following we will take gcd({k α }) = 1. Summarizing, the branch of the moduli space we just analyzed is: where H is the (C * ) G−2 kernel of Irr F ♭ is a G + 2 dimensional toric Calabi Yau cone [35,36] and the vectors of charges in H are traceless by construction; it implies that M 4 is a four dimensional Calabi Yau cone and it is understood as the transverse space to the M 2 branes. Following the same procedure in the non abelian case it is possible to see that the moduli space contains the N -times symmetric product of M 4 , and it is interpreted as the the transverse space to a set of N BPS M2 branes.
It is quite generic that a specific Calabi Yau four-fold is a branch of the moduli space of apparently completely different N = 2 Chern-Simons theories. This fact it is called toric duality. We want to systematically study M 4 for some set of Chern Simons theories and see if it is possible to find examples of toric dual pairs. To do this we will use the algorithm proposed in [26].

An Algorithm to compute M 4
Let us review the algorithm proposed in [26] to compute M 4 . We consider an N = 2 Chern-Simons theory described in the previous section with gauge group U (1) G , and with the following constraints on the Chern-Simons levels: To compute M 4 we need three matrices: the incidence matrix d, the perfect matching matrix P , and the Chern-Simons levels matrix C. d contains the charges of the chiral fields under the gauge group U (1) factors of the theory, and can be easily obtained from the quiver. P is a map between the gauge linear sigma model variables and the chiral fields in the Chern-Simons theory. It can be obtained from the superpotential of the theory and we refer the reader to [26,27] for explanations. Summarizing, the determination of the field theory contains the three matrices d, P , C. They are defined respectively by the gauge group representations of the chiral fields, the chiral fields interactions and the Chern Simons levels. Once we get these three matrices we can obtain the toric diagram of M 4 . From P and d we compute the matrix Q. It is the matrix of charges of the gauge linear sigma model variables under the U (1) G gauge group: d = Q · P T . From Q and C we construct the charge matrix Q D = ker(C) · Q. We denote with K ≡ ker(C). From P we get the charge matrix Q F : Q F = ker(P T ). Once we have Q D and Q F we combine them in the total charge matrix Q t : The toric diagram of M 4 is given by the kernel of Q t : where the columns of G t are the vectors defining the toric diagram of M 4 . Note that, as pointed out in [26], we have to find the integer kernel, that we denote ker * , and not the nullspace of the charge matrix. Each row of G t is reduced to a basis over the integer for every choice of the CS levels. We will see this algorithm at work in the following sections.

Non-Chiral Theories
We consider non-chiral 3D N = 2 CS matter quiver gauge theories which are the three dimensional analog of the four dimensional L aba theories [38,39,40]. and will be denoted as L aba {k i } . We say that a theory is non chiral if for every pair of gauge group factors U (N ) i and U (N ) i+1 the number of bifundamental fields in the representation (N i ,N i+1 ) is the same as the number of bifundamental fields in the conjugate representation (N i , N i+1 ). Otherwise the theory is chiral. The quiver for the L aba {k i } is in figure 1, with gauge groups i U (N ) k i . We label the nodes from left to right. The action is b−a 2 a and S CS (k i , V i ) is the first term in (2.1).

Brane construction
3D gauge theories can be engineered in type IIB string theory as D3 branes suspended among five branes [41]. For 3d CS theories the setup includes (p, q)5 branes [42,43].
Here we construct N = 2 three dimensional L aba {k i } CS theories, in analogy with the 4D construction [44].
As an example we show in figure 2 the realization of the L 121 {k 1 ,k 2 ,k 3 } theory. The generalization to the L aba {k i } is straightforward. We have the brane content resumed in Table 1. The NS branes and the corresponding D5 branes get deformed in (1, p i ) five branes at angles tan θ i ≃ p i , obtaining • N D3 brane along 012 6 • (1, p 1 ) brane along 012 [3,7]     • (1, p 3 ) brane along 012 [3,7] θ 3 89 Since the (1, p 2 ) and the (1, p 3 ) branes are parallel in the 89 direction there is a massless adjoint field on the node 2. This brane system gives the three dimensional L 121 {k 1 ,k 2 ,k 3 } CS theory. The Chern-Simons levels are associated with the relative angle of the branes in the [3,7] directions, i.e.
automatically satisfying (2.3). The gauge groups are all U (N ). Similar brane configurations have been studied in [22,24,25] for N = 3 and/or non toric theories.

Seiberg-like duality
As in the four dimensional case [45], we argue that electric magnetic duality corresponds to the exchange of two orthogonal (1, p i ) and (1, p i+1 ) branes. During this process, |p i −p i+1 | = |k i | D3 branes are created [42]. Observe that, since the p i and p i+1 of the dualized gauge group are exchanged, this gives non trivial transformations also for the CS level of the neighbour nodes.
From the brane picture (see figure 3) we obtain the rules for a Seiberg like duality on a node without adjoint fields in the L aba {k i } quiver gauge theories. Duality on the i-th node gives and the field content and the superpotential changes as in 4D Seiberg duality. In the following we will verify that this is indeed a toric duality by computing and comparing the branch M 4 of the moduli space (i.e. the toric diagram) of the two dual descriptions.
We observe that the Seiberg like duality (3.4) modifies the rank of the dualized gauge group, introducing fractional branes. This is a novelty of this 3d duality with respect to the 4d case (see also [31,32,46]).
The k fractional D3 branes are stuck between the five branes, so there is no moduli space associated with their motion. This is as discussed in [31], and the same field theory argument can be repeated here. The moduli space of the magnetic description is then the N symmetric product of the abelian moduli space.
The fractional branes can break supersymmetry as a consequence of the s-rule [41]. Indeed it was suggested in [47,48,43] that for U (l) k YM-CS theories supersymmetry is broken if l > |k|. We notice from (3.4) that in the moduli space of the magnetic description, there is a pure U (k) −k YM-CS theory. Thus the bound is satisfied and supersymmetry is unbroken. However, if we perform multiple dualities we can realize configurations with several fractional branes on different nodes. At every duality we have to control via s-rule that supersymmetry is not broken. We leave a more thorough study of these issues related to fractional branes for future investigation.
Finally, the duality proposed maps a theory with a weak coupling limit to a strongly coupled theory. Indeed if we define the i-th 't Hooft coupling as λ i = N/k i , the original theory is weakly coupled for k i >> N . In this limit the dual theory is strongly coupled since the i-th dual 't Hooft coupling isλ The L 121 {k 1 ,k 2 ,k 3 } is the first example that we study. The quiver is given in Figure 4. The toric diagram that encodes the information about the classical mesonic moduli space is computed with the techniques explained in section 2. We extract the incidence matrix d G,i , where G = 1, . . . , N g runs over the labels of the gauge groups, and i runs over the fields (i = 1, . . . , 7 for the L 121 {k 1 ,k 2 ,k 3 } , six bifundamental and one adjoint) The matrix of the perfect matchings P α,i is computed from the determinant of the Kastelein matrix If we order the fields in the determinant of (3.6) we can build the matrix P i,α where α = 1, . . . , c is the number of perfect matchings, that corresponds to the number of monomials of the det Kas. In this matrix we have 1 if the i-th field appears in the α-th element of the determinant, 0 otherwise The matrix Q that represents the charge matrix for the GLSM fields is obtained from the relation The contribution of the D-terms to the moduli space is given by quotienting by the G − 2 FI parameters induced by the CS couplings. These FI parameters are in the integer kernel of the matrix of the CS level The F -term equations are encoded in the matrix Q F = Ker(P ). The Toric diagram is the kernel of the matrix obtained by combining Q D = K · Q and Q F . Acting with an SL(4, Z) transformation the toric diagram reads This system of vectors is co-spatial. This is a CY condition that guarantees that the toric diagram lives on a three dimensional hypersurface in Z 4 . The last three rows of (3.10) defines the toric diagram for the three dimensional Chern-Simons L 121 {k 1 ,k 2 ,k 3 } toric quiver gauge theory. Note that the toric diagram of the (3+1)d parent is recovered by setting to zero the row with the CS levels.
We perform the Seiberg-like duality (3.4) on node 2. The resulting theory is shown in figure 4(b). The L 121 four dimensional parent theory has only one toric phase. The dual theory is in the same phase, thanks to the mapping among the nodes (1 → 3, 2 → 1, 3 → 2), see Figure 4. In the L 121 {k 1 ,k 2 ,k 3 } we should also properly map the CS levels in the two dual descriptions. The transformation rules (3.4) change the CS level as in figure 4. Then we apply the same mapping we used for the gauge groups. After these steps the resulting K matrix is where withk i we denote the CS level of the i-th node in the dual phase. Concerning the field content and the superpotential, the dual theory is in the same phase than the starting theory. Thus we use the same matrices P, d, Q for the computation of the moduli space. The toric diagram is then computed with the usual algorithm. Up to an SL(4, Z) transformation it coincides with the same as the one computed in the original theory (3.10). In this example we have shown that the Seiberg like duality (3.4) is a toric duality. Observe that the non trivial transformation on the CS levels of (3.4) are necessary for the equivalence of the moduli spaces of the two phases.

L 222 {k i }
The second example is the L 222 {k i } theory. The main difference is that this theory has two phases with a different matter content and superpotential (see Figure 5), obtained by dualizing node 2. These phases are dual for the (3+1)d parents theory. Here we show that the same holds in three dimensions with the Seiberg like duality (3.4). The P, d, Q and K N2+|k2|,−k2 matrices for the first phase are The resulting toric diagram is By duality on node 2 we obtain the inequivalent phase of L 222 {k i } , see figure 5. The toric diagram is computed with new d, P , Q and K matrices which is equivalent to (3.14).

The general L aba {k i }
In the previous section we have seen two simple examples. In this section we consider the generic case of non-chiral N = 2 toric three dimensional CS quiver gauge theories L aba {k i } singularities. Four dimensional theories based on these singularities share the same toric diagram among the different Seiberg dual phases. Here we show that two theories that are related by Seiberg like duality in three dimension (3.4) share the same toric diagram. Our argument is based on the algorithm [28] that extracts toric data of the CY four-fold by using brane tiling.

Toric diagrams from bipartite graphs
Let us remind the reader that to every quiver describing a four dimensional conformal field theory on D3 branes at Calabi Yau three-fold singularities it is possible to associate a bipartite diagram drawn on a torus. It is called tiling or dimer [49,50], and it encodes all the informations in the quiver and in the superpotential. To every face in the dimer we can associate a gauge group factor, to every edge a bifundamental field and to every node a term in the superpotential. A similar tiling can be associated with three dimensional Chern-Simons matter theories living on M2 branes probing a Calabi Yau four-fold singularity simply adding a flux of Chern-Simons charge. The CS levels are described as a conserved flow on the quiver, or equivalently on the dimer. To every edge we associate a flux of Chern-Simons charge and the CS level of the gauge group is the sum of these contributions taken with sign depending on the orientation of the arrow.
In this section we review the proposal of [28] for the computation of the moduli space of three dimensional CS toric quiver gauge theories. This method furnishes the toric data from the bipartite graphs associated with the quiver model and with the CS levels.
One has to choose a set of paths (p 1 , . . . , p 4 )on the dimer. The paths p 1 and p 2 correspond to the α and β cycles of the torus described by the dimer. The path p 4 is a paths encircling one of the vertices. One can also associate mesonic operators to these paths. These operators correspond to the product of the corresponding bifundamentals along the paths. For example the operator associated on the p 4 path is a term of the superpotential. The moduli space of the three dimensional theory requires also the definition of the path p 3 . This is a product of paths corresponding to a closed flow of CS charges along the quiver. We choose p 3 in the tiling of the L aba singularity by taking a minimal closed path connecting the bifundamentals from the first node to the a + b-th node of the quiver. Then one associates the CS charge n i=1 k i (3.18) to each bifundamental in this closed loop, with n = 1, . . . , a + b. The last charge is zero since it corresponds to sum of all the CS levels in the theory. This conserved CS charges flow is represented on the dimer by a set of oriented arrows connecting the faces of the dimer, the gauge groups. In [28] it has been shown that the toric polytope of CY 4 is given by the convex hull of all lattice point where D α are the perfect matchings. The operation ., . in (3.19) is the signed intersection number of the perfect matching D α with the path p i . Note that v 4 is always 1, since there is always only one perfect matching connected with the node encircled by p 4 .

Seiberg-like duality on L aba {k i } and toric duality
In subsection (3.1) we argued that the action of Seiberg duality on the field content and on the superpotential is the same as in four dimensions. The only difference is the change of the CS levels associated with the groups involved in the duality (3.4).  In an L aba theory, if duality is performed on node N 2 , the levels become The action of Seiberg duality (3.4) not only modifies the dimer as in the 4d, but also the CS levels of the gauge groups. This changes the path p 3 in the dual description. Many SL(4, Z) equivalent choices are possible. Among them we select the p 3 path as in figure  6. We associate a charge l 1 to Y 12 and l 2 to Y 23 , but with the opposite arrows that before. The values of the charges l 1 and l 2 are derived from (3.20) and are The field X 34 is not involved in this duality and it contributes to the CS flow with the same charge k 1 + k 2 + k 3 in both phases. We claim that the two theories share the same toric diagram. For the proof of this relation it is useful to distinguish two sector of fields from which all the perfect matching are built. In Figure 7 we separated these two sectors for the electric and magnetic phase of an L aaa theory (the same distinction is possible in a generic L aba theory). Every perfect matching is built by choosing in these sets only one field associated with each vertex. For example in Figure 7(a) every perfect matching is a set of blue lines chosen such that every vertex is involved only once.
The paths p 1 and p 2 are shown in figure 8 for the electric and the magnetic phase. The intersection numbers p 1 , D α and p 2 , D α give the same bidimensional toric diagram of the associated four dimensional L aba theories, up to an overall SL(3, Z) translation. This is shown by mapping the perfect matching in the two description. This mapping is done with  If duality is performed on node N i , the field X i−1,i , X i,i−1 , X i,i+1 and X i+1,i are respectively mapped in the fields Y i,i+1 , Y i+1,i , Y i−1,i and Y i.i−1 of the dual theory. This prescription gives a 1 − 1 map of each point of the 2d toric diagrams of the two dual theories. The whole diagram for the three dimensional L aba theory is obtained by considering the intersection numbers p 3 , D α , which give the third component of the vectors v α . The path p 3 corresponds to the flux of CS charges from the group N 1 to the group N g , where the last arrow is omitted since it carries zero charge. In the dual theory the path p 3 changes as explained above, and as we show in Figure 8 for duality on a node labelled by 2. Note that only the arrows connected with the dualized gauge group change.
With this choice of p 3 and using the basis of perfect matching we prescribed, the intersection numbers p 3 , D α coincide in the two phases for every point of the 2d toric diagram. This is shown by associating the relevant part of the path p 3 to each sector of perfect matching as in Figure 9. The arrow that carries charge k 1 in the electric theory corresponds to the arrow with charge −k 1 in the magnetic theory. Its contribution to the Thus the 3d toric diagrams of the two dual theories are the same. We conclude that the action of three dimensional Seiberg like duality (3.4) in the L aba {k i } theories implies toric duality.

Chiral theories
In this section we study Seiberg like duality for N = 2 three dimensional Chern-Simons theories with four dimensional parent chiral theories, which as such suffer from anomalies. The anomaly free condition imposes constraints on the rank distribution. In 3d there are no local gauge anomalies. Nevertheless we work with all U (N ) gauge groups such that the moduli space is the N symmetric product of the abelian one. Moreover for three dimensional Chern-Simons chiral theories we do not have a brane construction as simple as for the non-chiral case, and were not able to deduce the duality from the brane picture.
However, using what we learnt from the non-chiral case, we infer that at least a subset of the possible three dimensional Seiberg-like toric dualities acts on the field content and on the superpotential as it does in 4d and moreover recombines the Chern-Simons levels in a similar way as in (3.4). As a matter of fact, a straightforward extension of the rule (3.4) does not seem to work in the chiral case. This could be related to the fact that the Master Spaces of the four dimensional dual Seiberg parents are not isomorphic [37]. For this reason we restrict ourself to the case where the CS level of the group which undergoes duality is set to zero. We assume that the other CS levels are unchanged, and no fractional branes are created. This could be suggested by the parity anomaly matching argument (see appendix A). We also set to zero the CS level of those gauge groups that after duality have the same interactions with the rest of the quiver as the dualized gauge group.
By direct inspection we find that under these assumptions on the CS levels also for chiral 3D CS theories Seiberg like duality leads to toric duality. For F 0 we can take milder assumptions. Indeed we find that a generalization of the rule (3.4) to the chiral case still gives toric duality for this theory.

F 0{k i }
Here we study the (2+1)d CS chiral theory whose (3+1)d parent is F 0 . In (3+1)d there are two dual toric phases of F 0 , denoted as F I 0 and F II 0 . In (2+1)d the two phases for arbitrary choices of the CS levels do not have the same moduli space. Nevertheless it is possible to find assignments for the Chern-Simons levels such that the two phases have the same toric diagram. The quiver representing the F I 0 phase is in figure 10(a). The superpotential is The incidence matrix and the matrix of perfect matchings are  The second phase F II 0 is obtained by dualizing node 2. The dual superpotential is The matrices d, P and Q are determined from the quiver and the superpotential

Two families
The F 0{k i } theories turn out to be a special case. Indeed one can single out two different possibilities: in the first one we put to zero just the CS level associated with the group 2; while in the second case we can fix to zero just the Chern-Simons level of the group 4 and transform the CS levels as in the non-chiral case. In the first case we choose the CS levels as (k 1 , k 2 , k 3 , k 4 ) = (k, 0, p, −k − p). The CS level matrix for both phases is: and the toric diagram is given by: is equivalent to the one above.
In the second case we choose the CS levels as (k 1 , k 2 , k 3 , k 4 ) = (k, p, −k − p, 0). The phase F II 0 is computed by dualizing the node 2. We observe that by applying the rules (3.4) the CS levels of F II 0 are (k + p, −p, −k, 0). The C matrices for the two phases are: The toric diagram for the first phase is: while the toric diagram for the second phase is: And they are equivalent. The F 0 theory seems to be the only case where the assumptions we gave at the beginning of this section can be relaxed. In the following examples we will just apply those basic rules.

dP 1{k i }
Here we study the (2+1)d CS chiral theory whose (3+1)d parent is dP 1 . In (3+1)d dP 1 has only one phase. After Seiberg duality on node 2 the theory has a self similar structure and it is described by the same quiver. The only difference is that we have to change the labels of the groups as (1 → 2, 2 → 4, 3 → 1, 4 → 3). The matrix d, P and Q are unchanged by duality. The CS levels in the C matrix change as the labels of the gauge groups do. We give in figure 11 the two phases.
We take the assumption described in the introduction of section 4. Hence we choose the CS level of the group that undergoes duality to be 0. With this choice the two phases of the 2 + 1 dimensional theory (see figure 11) have the same toric diagram.
The superpotential is The d, P, Q matrices are Following the relabeling of the gauge groups, the C matrix in the two phases are The toric diagram for the first theory is given by the matrix G The toric diagram of the dual theory, up to SL(4, Z) transformation, is This shows that the two systems of vectors give the same toric diagram and that the two theories have the same abelian moduli space also in the 2 + 1 dimensions, provided k 2 = 0.

dP 2{k i }
We analyze the (2+1)d CS chiral theory with dP 2 as (3+1)d parents. The 4D theory has two inequivalent phases. The two phases are connected by duality on node 5 and are reported in figure 12.
The constraint on the CS levels explained in the introduction of this section imposes k 2 = 0 and k 5 = 0. Under this assumption the two phases have the same toric diagram also for the (2+1)d CS theory. The superpotential for the two phases are We compute the toric diagrams G They result the same.

dP 3{k i }
Here we study dP 3{k i } . This theory has four phases in four dimensions, with superpotentials Phases (II, III, IV) are computed from phases (I, II, III) by dualizing nodes (6,4,1) respectively. The quivers associated with each phase are given in Figure 13. We now This is the last chiral theory we analyze. In this case, after duality, there is not an identification between the gauge group that undergoes duality with other groups. This implies that the assumptions of section 4 impose only k g = 0, where k g is the CS level of the dualized gauge group. As for the case of dP 1 and all the Y p,p−1 theories, the Y 3,2 theory is self similar under four dimensional Seiberg duality. We can evaluate M 4 for one phase and then the toric diagram associated with a dual phase is given by an appropriate change of the D-term modding matrix. We fix the conventions on the groups by giving the tiling of the two dual phases, see Figure 14. The two phases are connected by duality on node 5, so we set k 5 = 0. The CS matrices for the two dual phases are: The toric diagrams are encoded in the G t matrices.

Dualities for CS theories without 4d parents
Three dimensional CS theories with four dimensional parents are a subset of all the possible 3d CS theories [27]. For CS theories without four dimensional parents we miss in principle the intuition from the 4d Seiberg duality. In this short section we see that we can still describe a subset of 3d CS theories with the same mesonic moduli space if we just apply the rules we learnt in the previous sections.
We study a case associated with Q 111 . We show that by performing a Seiberg-like duality and by setting the CS level of the dualized gauge group to zero, the toric diagrams of the two models coincide.

Example
The theory is described by the quiver given in Figure 15a. It is a generalization of the C(Q 1,1,1 ) with arbitrary CS levels [27]. The superpotential is The toric diagram for k 4 = 0 is given by Seiberg duality on node N 4 gives the superpotential 32 X 23 X 31 + X 24 X 43 X 32 − X 24 X

X
(1) 32 43 X 31 − X 14 X 43 X (2) 31 (5.3) and the theory is described by the quiver given in Figure 15b. The toric diagram in this case is and it is equivalent to the toric diagram for the first phase.

Conclusion
In this paper we have some advances towards the understanding of toric duality for M2 branes. Generalizing the work of [32,31], we proposed a Seiberg-like duality for non-chiral three dimensional CS matter theories and we verified that the mesonic moduli space of dual theories is indeed the same four-fold Calabi Yau probed by the M2 branes. In the chiral case and in the case in which the three dimensional theories do not have a four dimensional parent, the situation is more complicated. However, fixing to zero the value of some of the Chern-Simons levels, we were able to realize toric dual pairs.
We have just analyzed the mesonic moduli space, it would be important to study the complete moduli space, including baryonic operators.
For the non-chiral case the two main limitations are the lack of understanding of the transformation rule for the rank of the gauge groups and the fact that we forced to zero some of the k i . It is reasonable that there exist some more general and precise transformation rules and we would like to investigate them.
We concentrated on Seiberg-like transformations, but it is well known that in three dimension there exist duality maps that change the number of gauge group factors. It would be interesting to systematically study these more general transformations.
A lot of possible directions and generalizations are opening up and after these first steps we hope to step up.

A. Parity anomaly
We briefly review parity anomaly for 3D gauge theories [51,52] and the parity anomaly matching agument [33]. In three dimensions there are no local gauge anomalies. However gauge invariance can require the introduction of a classical Chern-Simons term, which breaks parity. This is referred to as parity anomaly.
For abelian theories with multiple U (1)'s, there is a parity anomaly if Here (q f ) i is the charge of the fermion f under the U (1) i . We work in a basis where all the charges are integers.

Parity anomaly matching
In the context of dualities in 4D gauge theory, a relevant tool have been the t'Hooft anomaly matching between the electric and the magnetic description. Having some global symmetries, we suppose that they are gauged and we compute their anomaly. The result of this computation should be equal in the two dual descriptions. The same technique can be used here for the parity anomaly. We suppose we gauge the global U (1)'s of the theory, and we compute their parity anomaly both in the electric and in the magnetic description. The two computations should match. The parity anomaly matching is much weaker then the t'Hooft one. Indeed in 4D the precise anomalies associated with gauging global symmetries must match. In 3D there is a weaker Z 2 type condition.
For the L aba {k i } theories, parity anomaly matching is obeyed by the Seiberg like duality we propose.
In chiral theories, parity anomaly matching can be non trivial between dual phases if fractional branes are introduced by the duality.
As an example we analyze the toric chiral (2+1)d CS theory which has as (3+1)d parents dP 2 . We consider the parity anomaly associated to the two U (1) flavour symmetries F 1 and F 2 . The charges of the chiral fields of the theory under these symmetries can be derived from [37]. The electric theory has equal ranks n. In the magnetic description we set rank n + k for the dualized gauge group (number 5) and n for the others. The integer k counts the number of fractional branes introduced in the duality. The parity anomaly matrices are One can see that in the electric description there are no parity anomalies. In the magnetic description the off diagonal components of A mag can instead lead to parity anomaly if kn is odd. If we set k = 0 the electric and magnetic theories satisfies the parity anomaly matching for the two flavour symmetries.