Orbifold Reduction and 2d (0,2) Gauge Theories

We introduce Orbifold Reduction, a new method for generating $2d$ $(0,2)$ gauge theories associated to D1-branes probing singular toric Calabi-Yau 4-folds starting from $4d$ $\mathcal{N}=1$ gauge theories on D3-branes probing toric Calabi-Yau 3-folds. The new procedure generalizes dimensional reduction and orbifolding. In terms of T-dual configurations, it generates brane brick models starting from brane tilings. Orbifold reduction provides an agile approach for generating $2d$ $(0,2)$ theories with a brane realization. We present three practical applications of the new algorithm: the connection between $4d$ Seiberg duality and $2d$ triality, a combinatorial method for generating theories related by triality and a $2d$ $(0,2)$ generalization of the Klebanov-Witten mass deformation.

1 Introduction D-branes probing singular Calabi-Yau (CY) manifolds provide a fruitful framework for engineering quantum field theories in various dimensions. These setups often give rise to new perspectives and powerful tools for understanding the dynamics of the corresponding gauge theories. A general program for studying the 2d (0, 2) gauge theories that live on the worldvolume of D1-branes probing singular toric CY 4-folds was recently initiated. 1 In [7], a systematic procedure for obtaining the 2d gauge theories on D1-branes probing generic toric CY 4-folds was developed and the general properties of the theories arising from these setups were established. A new class of Type IIA brane configurations, denoted brane brick models, was introduced in [8]. Two of their most remarkable features are that they fully encode the gauge theories on the D1-branes and streamline the connection to the probed CY 4-folds. Brane brick models are related to D1-branes at singularities by T-duality. In [9], a new order-3 IR equivalence among 2d (0, 2) theories, called triality, was discovered. The brane brick model realization of triality was investigated in [10]. In [11], following [12,13], mirror symmetry was used to refine our understanding of the correspondence between D1-brane at singularities, brane brick models and 2d gauge theories. That work also explained how triality is realized in terms of geometric transitions in the mirror geometry.
In this paper we will introduce orbifold reduction, a new method for generating 2d (0, 2) gauge theories associated to D1-branes probing singular toric Calabi-Yau 4-folds starting from 4d N = 1 gauge theories on D3-branes probing toric Calabi-Yau 3-folds. This procedure generalizes dimensional reduction and orbifolding. Orbifold reduction allows us to generate the gauge theories for D1-branes probing complicated CY 4-folds with little effort. This feature makes it a powerful new addition to the toolkit for studying 2d (0, 2) theories in terms of D-branes and we consequently expect it will find several interesting applications. This paper is organized as follows. In section §2, we review 2d (0, 2) theories, D1-branes over toric CY 4-folds and brane brick models. In section §3, we introduce orbifold reduction. Section §4 contains explicit examples illustrating the construction. In order to demonstrate the usefulness of orbifold reduction we then present three possible applications. In §5, we show how it generates 2d triality duals starting from 4d Seiberg dual theories. A distinctive feature of this approach is that both the 2d theories and their 4d parents are realized in terms of D-branes at singularities. In §6, we show how the combinatorics of orbifold reduction leads to non-trivial triality duals.
Finally, in section §7, we use orbifold reduction to construct an explicit example of a theory admitting a 2d (0, 2) generalization of the Klebanov-Witten mass deformation. We conclude in section §8. Additional examples are presented in appendix §A.
2 2d (0,2) Theories, Toric CY 4 's and Brane Brick Models The study of the 2d (0, 2) gauge theories that arise on the worldvolume of D1-branes probing toric CY 4-folds was developed in [7,8,10,11], to which we refer the reader for details. The probed CY 4-fold arises as the classical mesonic moduli space of the gauge theory on the D1-branes.
Brane brick models are Type IIA brane configurations that are related to D1-branes at toric singularities by T-duality. They were introduced in [7,8] and they considerably simplify the connection between gauge theory and the probed CY 4-fold.
It is convenient to represent a brane brick model by its "skeleton" on T 3 . A brane brick model fully encodes a 2d (0, 2) gauge theory following the dictionary in Table 2. Bricks correspond to U (N ) gauge groups. 2 There are two types of faces: oriented and unoriented faces correspond to chiral and Fermi fields, respectively. 3 We will identify chiral and Fermi faces by coloring them grey and red, respectively. Every edge in the brane brick model is attached to a single face corresponding to a Fermi field and a collection of faces corresponding to chiral fields. The chiral faces attached to an edge form a holomorphic monomial product that corresponds to either a J-or E-term that is associated to the Fermi field attached to the same edge. 4 Fermi faces are always 4-sided. This follows from the special structure of J-and E-terms in toric theories, which are always of the form with J ± ji and E ± ij holomorphic monomials in chiral fields [7].

Brane Brick Model Gauge Theory
Brick Gauge group Oriented face between bricks Chiral field in the bifundamental representation i and j of nodes i and j (adjoint for i = j) Unoriented square face between Fermi field in the bifundamental representation bricks i and j of nodes i and j (adjoint for i = j) Edge Plaquette encoding a monomial in a J-or E-term Table 2. Dictionary between brane brick models and 2d (0, 2) gauge theories.
Brane brick models are in one-to-one correspondence with periodic quivers on T 3 [2,7,8]. The two types of objects are related by graph dualization as shown in Figure  1. Then, periodic quivers also uniquely define a 2d (0, 2) gauge theory. In particular, Jand E-terms correspond to minimal plaquettes. A plaquette is a gauge invariant closed loop in the quiver consisting of an oriented path of chiral fields and a single Fermi field. Most of our discussion in this paper will be phrased in terms of periodic quivers. This choice is motivated by the relative simplicity of the resulting figures. Constructing the corresponding brane brick models is straightforward.

Orbifold Reduction
In this section we introduce orbifold reduction. It is a natural generalization of dimensional reduction and orbifolding, so we review them first.

Dimensional Reduction
Let us start with the dimensional reduction of general 4d N = 1 theories down to 2d (2, 2) theories. Under dimensional reduction, the 4d vector V i and chiral X ij multi-  plets become 2d (2, 2) vector and chiral multiplets, respectively. In terms of 2d (0, 2) multiplets, we have: The J-and E-terms of the 2d theory are where W is the superpotential of the 4d theory.

Dimensional Reduction of Toric Theories
Let us now focus on the class of 4d N = 1 theories that arise on D3-branes probing toric CY 3 singularities. Such theories are fully encoded by brane tilings, which are bipartite graphs on T 2 (see [14,15] for details). GLSM fields in the toric description of the CY 3 , namely points in its toric diagram, admit a combinatorial implementation as perfect matchings of the brane tiling. A perfect matching p α is a collection of edges in the tiling such that every node is the endpoint of exactly one edge in p α . Given the map between brane tilings and 4d gauge theories, we can regard perfect matchings as collections of chiral fields. In general, more than one perfect matching is associated to a given point in the toric diagram. This will become important later when we discuss orbifold reduction. For details on how to determine perfect matchings and connect them to toric diagrams we refer the reader to [14] (see also [16] for a more modern perspective).
Brane tilings are in one-to-one correspondence with periodic quivers on T 2 via graph dualization. These periodic quivers are equivalent to brane tilings, so they fully specify the corresponding gauge theories. In particular, every plaquette in the periodic quiver corresponds to a term in the superpotential.
The dimensional reduction of a 4d N = 1 theory associated to a toric CY 3 gives rise to a 2d (2, 2) theory corresponding to the toric CY 4 = CY 3 × C. A systematic lifting algorithm for constructing the periodic quiver on T 3 associated to CY 3 × C starting from the periodic quiver on T 2 for CY 3 was introduced in [7]. Let us refer to the new periodic direction of T 3 as the vertical direction. The lift is achieved by providing chiral and Fermi fields with vertical shifts. The vertical shifts for chiral fields are measured between the tail and the head of arrows. For Fermi fields, we use the same prescription, with the orientation dictated by the corresponding 4d chiral field. The procedure has a beautiful combinatorial implementation in terms of perfect matchings of the original brane tiling. For any perfect matching p 0 , the periodic quiver for the dimensionally reduced theory is achieved by introducing the following vertical shifts for the different types of matter fields: where 1 is the periodicity of the vertical direction. As it is standard in the study of brane tilings [15], let us introduce χ ij , p 0 , which is defined to be equal to 1 if χ ij ∈ p 0 and 0 if χ ij / ∈ p 0 . The vertical shifts can then be compactly written as Figure 2 illustrates this procedure with the dimensional reduction of phase 1 of dP 3 [15,17]. Let us explain how to interpret this kind of figure. In order to avoid clutter, throughout this paper we will represent periodic quivers in terms of three figures. Each of them contains different types of fields between layers: chirals coming from 4d vector multiplets in green, chirals coming from 4d chirals in blue and Fermis coming from 4d chirals in magenta. The combination of the three figures should be regarded as a single periodic quiver. In addition, for clarity, we will frequently present a region of the quiver that is larger than a unit cell, which is easy to determine from the node labels. Any perfect matching can be used to perform the dimensional reduction and the final result is independent of this choice. The 2d toric diagram of the CY 3 becomes the 3d toric diagram of the CY 4 = CY 3 × C by adding an extra point that represents the C factor, as shown in Figure 3. This additional point can be regarded as arising from p 0 . It can be moved to any position on a plane parallel to the 2d toric diagram by an SL(3, Z) transformation, which is another indication of the freedom in choosing p 0 . dP 3 dP 3 ⇥ C Figure 3. The toric diagrams for dP 3 and dP 3 × C.
If we dimensionally reduce two 4d theories connected by Seiberg duality, we obtain a pair of 2d (2, 2) theories related by the duality of [18]. While we are not going to use it in this paper, it is worth mentioning that dimensional reduction also has a beautiful implementation as a lift of brane tilings into brane brick models [8].

Orbifolding
In terms of brane brick models and periodic quivers, orbifolds are constructed by enlarging the unit cell [7,8]. The details of the geometric action of the orbifold group are encoded in the periodic identifications on T 3 .
The lifting algorithm of the previous section can be extended to generate the gauge theories for certain (CY 3 × C)/Z k orbifolds. 5 The process is very simple: 1. Stack k copies of the gauge nodes in the T 2 periodic quiver along the vertical direction. We set the distance between consecutive layers equal to 1.
2. Choose a perfect matching p 0 .
3. For each of the k layers, introduce matter fields with vertical shifts given by (3.3).
The effect on the toric diagram is to expand the point associated to p 0 into a line of length k, as shown in Figure 4. Once again, any perfect matching can be used in this construction. However, unlike in dimensional reduction, perfect matchings associated to different points in the toric diagram give rise to non-SL(3, Z) equivalent geometries. In other words, they correspond to different actions of the orbifold group.

Orbifold Reduction
Orbifold reduction is a simple generalization of the orbifolding procedure discussed in the previous section and can be summarized as follows: 1. Stack k copies of the gauge nodes in the T 2 periodic quiver along the vertical direction. The distance between consecutive layers is set to 1.  Let us denote k + and k − the number of plus and minus signs in s, respectively. Orbifold reduction generates a gauge theory that corresponds to a CY 4 whose toric diagram is obtained by expanding the point associated to p 0 into a line of length k, with k + points above the original 2d toric diagram and k − points below it. This generalizes the transformations of toric diagrams discussed in the two previous sections. Figure 5 shows an example. Clearly s and −s give rise to the same theory, since their periodic quivers are simply related by a reflection along the vertical direction. Furthermore, s = (+, . . . , +) corre-sponds to a (CY 3 × C)/Z k orbifold. In particular s = (+) corresponds to dimensional reduction.
We previously saw that any perfect matching can be used as p 0 for both dimensional reduction and orbifolding. However, this is no longer the case for orbifold reduction. In particular, certain combinations of s and p 0 can result in theories with non-vanishing non-abelian anomalies. Hence, it is always necessary to verify at an initial stage that we pick a p 0 that, in combination with s, does not give rise to anomalies. In order to determine whether a given p 0 is consistent with any s, it is sufficient to check that it does not lead to anomalies for a layer between (+, −) signs (layers between equal signs are automatically free of anomalies, as in dimensional reduction and orbifolding). Our study suggests that it is always possible to pick a p 0 such that any point in a 2d toric diagram can be consistently lifted. It would be interesting to prove this fact. Denoting the numbers of gauge groups, chirals and Fermis in the orbifold reduced theory and G (4d) and N (4d) χ the numbers of gauge groups and chirals in the 4d parent, we have (3.4) A convenient notation for specifying a 2d theory T 2 obtained by orbifold reduction is which emphasizes the necessary data for performing orbifold reduction: a toric 4d parent theory T 4 , a sign vector s and a perfect matching p 0 . In this paper, we will not consider pairs of theories that differ only by the choice of p 0 . For simplicity, we will thus omit p 0 from the label of T 2 .

Examples
Below we illustrate orbifold reduction with two simple examples. They are the two inequivalent k = 2 models obtained from the conifold: C (+,+) and C (+,−) . Additional examples are presented in the appendix.
4.1 C (+,+) Figure 6 shows the toric diagram and periodic quiver for the conifold C. Each of the four points in the toric diagram corresponds to a single perfect matching. The four points are equivalent, so any choice of p 0 to be used in orbifold reduction generates the same theory. Performing orbifold reduction with s = (+, +), we obtain a 2d theory that we call C (+,+) . Its periodic quiver is shown in Figure 7.
In all the examples we consider in this paper, it is straightforward to identify the p 0 that was used from the periodic quiver, as we now explain. The quiver on any of the layers is almost identical to the periodic quiver for the 4d parent. The only difference is given by the Fermi fields (red edges) which, according to (3.3), correspond precisely to the chiral fields contained in p 0 . Figure 2 gives a good idea of how this works.
From the periodic quiver, we read the following J-and E-terms: Computing the classical mesonic moduli space for this theory, we obtain the toric diagram that is expected from the general discussion in section §3.3, which is shown in Figure 8.

C (+,−)
Let us start again from the conifold and perform orbifold reduction with s = (+, −). We refer to this theory as C (+,−) . Figure 9 shows its periodic quiver. Te corresponding J-and E-terms are: The classical mesonic moduli space for this theory corresponds to the toric diagram shown in Figure 8, as expected.

Triality from Seiberg Duality
Orbifold reduction of Seiberg dual toric theories generically leads to different 2d (0, 2) theories associated to the same CY 4-fold. Following [10,11], we expect such theories to be related by triality (namely, either a single triality transformation or a sequence of them). This approach to triality is in the same general spirit of other constructions that derive it from Seiberg duality, such as [19].
Let us illustrate this idea with an explicit example. Consider the complex cone over F 0 or, for brevity, just F 0 from now on. Its toric diagram is shown in Figure 11. There are two toric phases for F 0 , which are related by Seiberg duality. They have been extensively studied in the literature (see e.g. [15,17,20,21]). Their periodic quivers are shown in Figure 12. Starting from the two theories, we will perform orbifold reductions that lift the center point in the toric diagram with s = (+, −). The toric diagram is then transformed into the one for Q 1,1,1 /Z 2 , as shown in Figure 13. This geometry has several toric phases connected by triality, whose study was initiated in [10]. 6 Some of these phases were analyzed using mirror symmetry in [11].

(+,−)
Let us first consider the s = (+, −) orbifold reduction of phase I of F 0 using a p 0 associated to the central point of the toric diagram. We refer to the resulting theory as F I 0 (+,−) and present its periodic quiver in Figure 14. As explained earlier, our choice of p 0 can be immediately identified from this figure. The J-and E-terms are: This theory corresponds to Q 1,1,1 /Z 2 . In fact, this is phase C in the classification of [10]. The ease with which orbifold reduction generates a phase for such a complicated geometry is truly remarkable.

(+,−)
Next, let us construct a second theory for Q 1,1,1 /Z 2 , by performing the orbifold reduction of phase II of F 0 with s = (+, −) and a p 0 associated to the central point in the toric diagram. We call this theory F II 0 (+,−) and present its periodic quiver in Figure  15. The J-and E-terms are: This is a new theory that has not appeared in the existing partial survey of phases of Q 1,1,1 /Z 2 of [10]. It is possible to show that it is indeed related by triality to the other phases. In particular, it can be obtained from F I 0 (+,−) (phase C) by consecutive triality transformations on nodes 1 and 4 and relabeling of nodes. 7 The two theories considered in this section demonstrate one of the salient features of orbifold reduction: the simplicity with which it generates 2d (0, 2) gauge theories for relatively complicated CY 4-folds. They also show that, in general, not all toric phases of a CY 4 can be derived by orbifold reduction. In particular, only phases whose field contents are related to 4d parents by (3.4) can be constructed this way.

Triality from Combinatorics in Multi-Layered Orbifold Reduction
Things become very interesting when the number of layers k is increased. Whenever we have a pair of orbifold reduced theories obtained by lifting the same point in the toric diagram and whose sign vectors are related by permutations but not by an overall sign flip, we expect them to be different gauge theories that correspond to the same CY 4 , and hence to be related by triality. This phenomenon first arises for k = 4, for which s = (+, +, −, −) and s = (+, −, +, −) produce different gauge theories. Below we consider two explicit examples based on the conifold: C (+,+,−,−) and C (+,−,+,−) . We have explicitly verified that classical mesonic moduli spaces for both of them correspond to the toric diagram shown in Figure 16.  The periodic quiver for this theory is shown in Figure 17. The J-and E-terms are:  The periodic quiver for this theory is shown in Figure 18. The J-and E-terms are: It is possible to show that the two theories discussed above are related by triality, as expected. Explicitly: starting from C (+,+,−,−) , performing a triality transformation on node 7, relabeling nodes according to (1,2,3,4,5,6,7,8) → (2,1,4,3,6,5,7,8) and charge conjugating all fields, we obtain precisely the C (+,−,+,−) theory.
The field content and types of interactions terms in the two theories are very similar. The action of the permutation of the sign vectors as a rearrangement of layers along the vertical direction of the periodic quiver is reminiscent of the motion of impurities in Y p,q and L a,b,a quivers due to Seiberg duality [23][24][25][26]. It would be interesting to explore this connection in further detail.

2d (0,2) Klebanov-Witten Deformations
In their seminal paper [27], Klebanov and Witten (KW) introduced a mass deformation that connects the 4d N = 2 gauge theory on D3-branes probing C 2 /Z 2 ×C to the N = 1 gauge theory associated to the conifold. This type of deformation has been generalized to a wide class of 4d theories on D3-branes probing toric CY 3 singularities and studied in detail in terms of brane tilings [28]. The new examples include starting points with only N = 1 SUSY.
It is natural to ask whether 2d analogues of KW deformations exist: namely mass deformations connecting the gauge theories on D1-branes probing two different CY 4 geometries. 8 First, it is easy to verify that the dimensional reductions of 4d KW-type deformations work. Such deformations, connect pairs of theories with at least (2, 2) SUSY. A more interesting question is whether (0, 2) KW deformations exist. Below we present an explicit example in which both the initial and final theories have (0, 2) SUSY. It would certainly be interesting to carry out a systematic investigation of (0, 2) KW deformations. We leave this question for future work.
In order to visualize the relation between the matter contents of both theories, it is convenient to consider their standard, instead of periodic quivers. These quivers are shown in Figure 20. The two gauge theories are connected by turning on appropriate mass terms for the (Λ 12 , Y 12 ) and (Λ 34 , Y 34 ) chiral-Fermi pairs and integrating them out. While the process is rather straightforward, we consider it is instructive to go over the computation in detail. Let us first consider the effect that integrating out these fields has on the quiver. As shown in Figure 20, the final quiver agrees with the one for Q 1,1,1 , which was introduced in [7]. Figure 19. Transition between the toric diagrams for C (+,−) and Q 1,1,1 . This transformation of the quiver is promising. However, it is crucial to determine whether the J-and E-terms become those of Q 1,1,1 . The mass terms correspond to adding a term in linear in Y 12 to the E-term for Λ 12 and a term linear in Y 34 to the E-term for Λ 34 as follows: 9 J E Λ 12 : where we have indicated the new terms in blue. The relative minus sign is crucial for the deformation to have the desired effect. It is reminiscent of analogous relative signs between pairs of mass terms that are necessary in the original KW deformation [27] and its generalizations to other 4d theories [28]. Λ 12 and Λ 34 are integrated out and, consequently, the corresponding rows in (7.1) disappear. In addition, Y 12 and Y 34 are replaced by Interestingly, whenever these two chiral fields appear in a surviving J-or E-term, they do so in pairs. This fact will be important for many cancellations, as we now show. Using (7.1) and (7.2), we obtain For each Λ i 34 , the terms shown in green are identical to the E-terms of another Λ j 34 and hence vanish on the moduli space. The pairs of terms shown in red vanish directly.
Taking these cancellations into account, we get which are precisely the J-and E-terms for Q 1,1,1 [7]. We conclude that the mass deformation we introduced in (7.1) transforms C (+,−) into Q 1,1,1 , providing an explicit example of a 2d (0, 2) KW deformation.

Conclusions
We introduced orbifold reduction, a novel method for generating 2d (0, 2) gauge theories associated to D1-branes probing toric CY 4-folds starting from 4d N = 1 gauge theories on D3-branes probing toric CY 3-folds. This procedure generalizes dimensional reduction and orbifolding. Orbifold reduction generates the periodic quiver on T 3 that encodes a 2d theory starting from the periodic quiver on T 2 for a 4d one. Equivalently, it generates brane brick models from brane tilings. Orbifold reduction allows us to construct gauge theories for D1-branes probing toric CY 4-folds almost effortlessly. It is thus an ideal tool for generating new examples, which can in turn be used for studying both the D-brane configurations and the dynamics of the gauge theories.
In order to illustrate the usefulness of orbifold reduction we presented three applications. We first showed how it connects 4d Seiberg duality to 2d triality. Next, we discussed how theories related by permutations of the sign vectors are automatically triality duals. The periodic quivers for such theories differ by a reorganization of layers that is similar to the motion of impurities in certain 4d toric theories due to Seiberg duality. Finally, we exploited orbifold reduction to construct an explicit example of a 2d (0, 2) KW deformation.
Our work suggests various interesting directions for future investigation. Let us mention a couple of them. It would be interesting to perform a systematic study of 2d KW deformations determining, among other things, the effect of the deformations on phase boundaries (the brane brick model analogues of zig-zag paths for brane tilings [8]) and establishing whether there is a general criterion that identifies theories with such deformations. Finally, it would be interesting to investigate whether a generalization of orbifold reduction similarly simplifies the construction of 0d N = 1 matrix models arising on the worldvolume of D(-1)-branes probing toric CY 5-folds. Understanding these theories is certainly a natural next step after brane tilings and brane brick models. Moreover, recent studies based on mirror symmetry suggest that these matrix models exhibit a quadrality symmetry [11,29], making the development of new tools for studying such theories even more timely.