Confinement On the Moose Lattice

In this work we present a new class of N=1 supersymmetric confining gauge theories, with strikingly simple infrared theories that descend from intricate interconnected networks of product gauge groups. A diagram of the gauge groups and the charged matter content of the ultraviolet theory has the structure of a triangular lattice, with $SU(N)$ or $SU(3 N)$ gauge groups at each of the vertices, connected by bifundamental chiral superfields. This structure admits a $U(1)_R$ conserving superpotential with marginal trilinear operators. With the introduction of this superpotential, the $SU(3N)$ and $SU(N)$ gauge groups confine: in the far infrared limit of the supersymmetric theory, the relevant degrees of freedom are gauge invariant"mesons"and"baryons."In this paper we show how the properties of the infrared degrees of freedom depend on the topology and shape of the moose/quiver ``lattice'' of the original gauge theory. We investigate various deformations of the theory, and propose some phenomenological applications for BSM models.


Introduction
Strongly coupled gauge theories are highly relevant to our understanding of the universe, but challenging to analyze. Taking the strong nuclear force (QCD) as an example, perturbation theory is incapable of deriving the masses and interactions of the various hadrons in the Standard Model from high-energy observables. A first-principles calculation in the strongly coupled regime requires nonperturbative methods, such as lattice QCD. Aspects of supersymmetry make this problem more tractable: with a sufficiently high degree of symmetry, it may be possible to constrain the form of the low energy theory to the point that its constituents and interactions can be more precisely identified.
In this article we investigate product gauge groups of the form (SU (3N ) × SU (N ) 3 ) k in N = 1 supersymmetry (SUSY), with a set of chiral bifundamental "quark" matter fields arranged so that a moose diagram of the theory forms a triangular lattice. With the appropriate superpotential, the theory exhibits two stages of confinement, leading to an infrared theory of "baryons" and "mesons" that is surprisingly simple.
The ultraviolet phase of our theory is constructed from three ingredients: the SU (3N ) and SU (N ) gauge groups; chiral matter superfields, transforming in the bifundamental representations of SU (3N ) × SU (N ) or SU (N ) × SU (N ); and a chiral superpotential W , which includes the marginal gauge invariant trace operators of the form W ⊃ λTr (q 1 q 2 q 3 ) for three bifundamentals q i . To analyze the low energy theory we rely on the Seiberg dualities for supersymmetric QCD (SQCD) [1,2].
For the initial discussion, we restrict our attention to the cases where there is some high-energy scale M where all of the gauge couplings are perturbatively small, i.e. g 3N (µ), g N (µ) O(1) at µ = M . As the SU (3N ) and SU (N ) gauge couplings run in opposite directions (i.e. their NSVZ β functions [3] have opposite signs), this situation is not entirely generic. We refer to this µ ∼ M regime as the ultraviolet theory (UV), even though the SU (N ) gauge groups become strongly coupled in the extreme ultraviolet limit µ M . This theory exhibits confinement with chiral symmetry breaking at a scale characterized by Λ 3N , where the one-loop β(g 3N ) function diverges. By describing the theory as confining, we mean that the µ Λ 3N perturbatively coupled (SU (3N )×SU (N ) 3 ) k theory with the trilinear superpotential W is Seiberg-dual to a theory of singlet "baryons" and SU (N ) × SU (N ) bifundamental "mesons" at scales µ Λ 3N . Some of the global symmetries of the UV theory are spontaneously broken by the O(Λ 3N ) expectation values of the baryon operators.
A subset of the SU (N )×SU (N ) bifundamentals acquire vectorlike masses m i of order O(λ i Λ 3N ), inherited from the λ i Tr (q 1 q 2 q 3 ) operators in the UV theory. Like the λ i = 0 UV superpotential terms, these vectorlike masses lift some of the flat directions that would otherwise be included in the moduli space. At scales µ < m i , these massive degrees of freedom can be integrated out. This changes the sign of the β(g N ) function, so that the SU (N ) gauge groups become strongly coupled at some new Λ N < m i . In the far infrared (IR) theory µ Λ N , the only light degrees of freedom are composite baryons and mesons, which can be mapped onto the set of gauge invariant operators of the original UV theory. Despite the formidable complexity of the original ultraviolet theory, this phase of the low energy effective theory is remarkably simple. Sections 2 and 4 present our main results, following the evolution of the theory from µ ∼ M down to µ Λ N step by step, tracking the degrees of freedom and their global symmetries in each regime. Section 2 focuses on the aspects of the calculation that are the least dependent on the actual shape of the "moose lattice," and the easiest to generalize. Details about the boundary conditions become much more important in the infrared limit of the theory, as we show in Section 4. For example, if the moose lattice is given periodic boundary conditions, the SU (N ) groups do not confine, but instead have an unbroken Coulomb phase in the IR. Section 4 explores several such variations on the boundary conditions. As an interlude between Sections 2 and 4, Section 3 follows the global symmetries of the theory from the UV to the IR.
Our focus in this work is restricted to four-dimensional spacetime, and the so-called moose lattice is simply a way to keep track of the gauge groups and matter fields -however, it is highly suggestive of a geometrical interpretation, consistent with the deconstruction [4][5][6][7] of a six-dimensional spacetime with two compact dimensions, as we discuss in our concluding remarks. This view is reinforced by the emergence of some bulk-like and brane-like features in the infrared theory.
In Section 1.1 we provide a review of the familiar Seiberg dualities and confinement in SQCD. Section 1.2 reviews some especially relevant literature on confinement in SU (N ) product gauge theories [8][9][10].

Review of Confinement in N = 1 Theories
Supersymmetry (SUSY) ameliorates some of the challenges of strongly coupled theories, making it possible to derive some infrared properties of a theory exactly. The conjectural Seiberg dualities [1,2] are central to this effort. Given a gauge group such as SU (N ) with some set of matter fields, one can sometimes identify a dual theory with a different gauge group, new matter fields, and possibly some superpotential that describes the interactions of the dual matter fields. In describing the theories as "dual," we mean that the two ultraviolet theories flow to the same infrared behavior, not that this duality is exact at all energy scales. Seiberg dualities have been identified not only for SU (N ) gauge groups, but also Sp(2N ), SO(N ), and the exceptional Lie groups.
A number of SUSY gauge theories have been shown to confine: that is, rather than being dual to another gauge theory, the dual theory has no gauge interactions. Well-known examples include SU (N c ) with F = N c or F = N c + 1 pairs of chiral superfields in the fundamental ( ) and antifundamental ( ) representations of the gauge group (a.k.a. "SQCD"), or SU (N c ) with one field in the antisymmetric ( ) representation and an appropriate number of fundamentals and antifundamentals [11][12][13].
In some cases the infrared theory has a "quantum deformed moduli space," where the classical constraint equations are modified by some terms that depend on the gauge couplings. We refer to these theories as "qdms-confining." The canonical example is the F = N c case of SQCD, where the infrared behavior is described by the gauge invariant operators where Q Nc ≡ det Q is the completely antisymmetric product of i = 1, 2, . . . N c distinct fields Q i , each in the fundamental representation of SU (N c ). Classically, M and BB would obey the constraint equation det(QQ) = det Q det Q, but this classical constraint is modified quantum mechanically [14][15][16] by a term proportional to the holomorphic scale Λ, A second class of theories, known as "s-confining" [17,18], confine smoothly without necessarily breaking any of the global symmetries, with the classical constraint equations enforced by a dynamically generated superpotential. For example, the infrared limit of SQCD with F = N + 1 flavors is described by the gauge invariants

4)
M ij = (QQ) ij ≡ Q α i Q α j , where k refer to SU (N c ) gauge indices, i and j to SU (F ) flavor indices, and is the completely antisymmetric tensor. The constraint equations between M , B and B are not modified quantum mechanically. Instead, the infrared theory has a dynamically generated superpotential [19] W = 1 Λ 2N −1 BM B − det M , (1.5) which enforces the classical constraint equations It is easy to show that this W has R charge +2 under any conserved U (1) R . The origin of moduli space is now a viable solution to the constraint equations, permitting confinement without chiral symmetry breaking.
The Seiberg dualities for SQCD survive a number of nontrivial consistency checks: the dimensionality of the moduli spaces of the UV and IR theories match, the two theories share the same set of global symmetries, and all of the 't Hooft anomaly matching conditions are satisfied. In Appendix A we demonstrate this for some F = N examples chosen to highlight some subtleties associated with anomaly matching on the quantum deformed superpotential.

Superpotential Deformations
Each of the models described so far has been derived from a UV theory with no superpotential. In the case of s-confinement, a superpotential is dynamically generated for the IR theory, which respects all the global symmetries and enforces the classical constraints between the operators. For qdms confinement, there is no dynamically generated superpotential: the quantum modified constraint can only be implemented in a superpotential by the use of Lagrange multipliers.
In the SQCD example, one could perturb the theory by including the gauge-invariant superpotential operators This W explicitly breaks the SU (F ) × SU (F ) r symmetry, as well as U (1) R . Note that if any of the mass terms m ij are larger than Λ, then it is no longer appropriate to treat the problem as F = N SQCD. After integrating out the heavier quarks with m ij Λ, the remaining F < N SQCD theory does not confine: its Seiberg dual has an SU (N − F ) gauge group. The infrared theory thus bears no resemblance to the qdms-confining version of SQCD. If we are to treat the superpotential Eq. (1.7) as a small perturbation, it should be the case that m ij Λ. In this case the global symmetries are still approximately conserved, and the infrared effective theory developed in Section 1.1 is still applicable at scales large compared to m ij (and small compared to Λ).
As defined in Eq. (1.1), M , B and B have mass dimensions 2, N and N , respectively. When matching superpotential terms it can be more convenient to normalize these by factors of Λ to give the operators canonical mass dimension, 2 so that a generic symmetry-violating superpotential for SQCD includes For W ≈ 0 to be a good approximation in the near ultraviolet as well as the infrared, the mass scales associated with the irrelevant operators should satisfy M Λ, so that all of the global symmetries are approximately conserved above and below the scale Λ. For G global to be approximately conserved below Λ, it must also be the case that A ij Λ Λ 2 . In the N = 2 special case, the same should be true for M B Λ and M B Λ.
Solving the equations of motion for M ij , B and B, we find that the F 2 + 1 light degrees of freedom do not remain massless, but instead acquire some potential that lifts various directions of the moduli space. In the F = N case the addition of A ij , β, andβ is sufficient to completely break the global symmetry group SU (F ) × SU (F ) r × U (1) B × U (1) R . The quadratic terms M 2 , M B, M B and BB determine the location of the global minimum of the potential on the moduli space, up to corrections from further irrelevant operators that may be included in the superpotential. Figure 1: The moose diagram for the qdms-confining SU (N ) k model [9], showing the k gauge groups G i = SU (N ) i and the global SU (N ) × SU (N ) r symmetry. Each Q j transforms as a bifundamental (N , N ) under the adjacent G j × G j+1 . The quark charges under the global U (1) B symmetry are indicated in the lower row. Arrows pointing into (out of) a group G j indicate that a quark transforms in the (anti)fundamental representation of that group.

Confinement in Linear Moose Theories
Each of the complicated product gauge group models considered in this paper uses a collection of alternating SU (N )×SU (M )×SU (N )×SU (M )×. . . gauge groups as a building block. Dubbed the "linear moose" model [20], the matter content of this theory consists of one chiral bifundamental quark for each adjacent pair of SU (N )×SU (M ) or SU (M )×SU (N ) groups. This type of structure appears in the k site deconstruction of a five-dimensional theory [4][5][6][7], which in the presence of a Z 2 orbifold produces an SU (N ) k chiral gauge theory.
For the "even" linear moose with equal numbers of SU (N ) and SU (M ) gauge groups, (SU (N )× SU (M )) k , the anomaly matching conditions are saturated, indicating that even in a nonsupersymmetric theory the confinement can proceed without breaking chiral symmetry [20], while for the "odd" linear moose (SU (N )×SU (M )) k ×SU (N ) the chiral symmetry is necessarily spontaneously broken.
Additional information about the low energy behavior can be extracted from supersymmetric moose theories [8-10, 13, 21-23]. For example, with N = 1 supersymmetry, it is often possible to derive the exact form of the chiral superpotential. If an N = 1 theory can be shown to be the limit of N = 2 supersymmetry [2,21,[24][25][26][27], the Kähler potential may be similarly constrained based on the form of the holomorphic prepotential.
In the special case N = M , the Seiberg duality for F = N SQCD can be used to quantify aspects of the infrared theory for the supersymmetric linear mooses [9]. The moose (a.k.a "quiver" [28]) diagram for this theory is shown in Figure 1, together with the matter superfield charge assignments under the global symmetries. Below, we summarize the method and results of Ref. [9], which are utilized several times in Section 2.
Given some large hierarchy between confinement scales, e.g. Λ 1 Λ 2 . . ., the gauge groups G 2,3,... in Figure 1 can be treated as global symmetries in the regime where only G 1 is strongly coupled. In this limit the degrees of freedom at intermediate scales Λ 2,3,... µ Λ 1 include the baryonic det Q 0 and det Q 1 operators, as well as the mesonic (Q 0 Q 1 ) that transforms as the bifundamental of SU (N ) × G 2 . The baryon and meson operators satisfy the usual quantummodified constraint, with b = 2N . If G 2 were not gauged, then det Q 0 , det Q 1 and (Q 0 Q 1 ) would form the set of gauge-invariant operators that describe the flat directions on the moduli space [29]. Next, at µ ∼ Λ 2 , the group G 2 = SU (N ) 2 becomes strongly coupled, thus lifting the pseudo-flat (Q 0 Q 1 ) directions and selecting the det Q 0 det Q 1 = Λ b vacuum. As the light degrees of freedom (Q 0 Q 1 ) and Q 2 resemble F = N SQCD, confinement of G 2 produces the gauge-invariant composite operators det(Q 0 Q 1 ), det Q 2 , and (Q 0 Q 1 Q 2 ). However, given the constraint equation Eq. (1.10), the baryonic operator det(Q 0 Q 1 ) is not an independent degree of freedom, but is redundant with det Q 0 and det Q 1 .
Continuing in this manner for G 3,4,... , and replacing the redundant operators det(Q 0 Q 1 ) and det(Q 0 Q 1 Q 2 . . .) where possible, the constraint equations have the form [9] det 11) where in our shorthand Q N j ≡ det Q j . Given k copies of the gauge group SU (N ), the moduli space is spanned by the reduced set of gauge invariant operators (Q 0 Q 1 . . . Q k ) ij and Q N 0,1,...,k , where the mesonic operator (Q 0 . . . Q k ) is a bifundamental of the SU (N ) × SU (N ) r flavor symmetry, with the single constraint equation (1.12) Following Ref. [9], "neighbor contraction" indicates the replacement The sum in Eq. (1.12) includes all possible contractions.
Here we see explicitly the difference between "even" and "odd" moose theories identified in Ref. [20]. If k is even, then the product (Q N 0 . . . Q N k ) includes an odd number of terms, and the moduli space includes the origin, (Q 0 Q 1 . . . Q k ) ij = 0, Q N j = 0, thus permitting confinement without chiral symmetry breaking. If instead k is odd, then the product (Q N 0 . . . Q N k ) can be fully contracted, and the constraint equation includes the constant term Λ b 1 Λ b 3 . . . Λ b k , thus forcing at least a subset of the operators to acquire nonzero expectation values.
Even though this analysis used the Λ 1 Λ 2 . . . Λ k hierarchy as a simplification, the same conclusion Eq. (1.12) is reached in any other ordering of scales [9]. Furthermore, the model survives a number of consistency checks, including the Λ j → 0 limit where any of the gauge groups is replaced by a global symmetry; the addition of mass terms, where possible; or spontaneous symmetry breaking, where an SU (N ) j is higgsed to one of its subgroups.

Modified Boundary Conditions
A closely related class of product gauge groups was shown in Ref. [10] to s-confine. In this model the N copies of Q 0 in Figure 1 were replaced by four quarks Q and one two-component antisymmetric tensor A ( in Young tableaux notation). This product gauge group confines while dynamically generating a superpotential, with the same "even/odd" behavior identified in Ref. [20] based on the number of gauged SU (N ) groups.
In another variation of the linear SU (N ) k theory, the linear moose of Ref. [9] is modified by gauging the diagonal SU (N ) subgroup of the global SU (N ) × SU (N ) r , so that the moose diagram forms a closed ring [8]. To analyze this theory it is easiest to begin with the limit where this gauged G 0 ⊂ SU (N ) × SU (N ) r is weakly coupled, with Λ 0 Λ 1,2,...,k . If G 0 were not gauged, then the mesonic operator M ij = (Q 0 . . . Q k ) ij would be gauge-invariant. With G 0 gauged, this bifundamental of SU (N ) × SU (N ) r decomposes into irreducible representations of G 0 = SU (N ): a singlet M 0 , and an adjoint M Ad , defined as The moduli space is spanned by the gauge invariant operators Q N j ; M 0 ; and powers of M m Ad for 2 ≤ m ≤ N − 1, with the upper limit on m due to the chiral ring being finitely generated [30,31]. With the adjoint M Ad the only G 0 -charged degree of freedom, it is not possible to completely higgs the gauge group. By giving M Ad an arbitrary expectation value, SU (N ) can be broken into any of its rank N − 1 subgroups, leaving at least an unbroken gauged U (1) N −1 Coulomb phase.
For the SU (N ) k+1 ring moose it is possible to identify a holomorphic prepotential that specifies both W and the Kähler potential. For example, in the Λ 0 Λ 1...k limit, the chiral matter field M Ad can be combined with the vector gauge superfield λ 0 and the antichiral M † Ad into an N = 2 supermultiplet, all of which transform in the adjoint representation of G 0 . In Ref. [8], the prepotential hyperelliptic curve is found by reducing the SU (N ) k product group down to SU (N ) i × SU (N ) j , where Λ i and Λ j are the two smallest holomorphic scales, in analogy with the SU (2) × SU (2) theory in Ref. [2].
Appealing to a geometric interpretation of this theory, we refer to these variants of the SU (N ) k linear moose as having different boundary conditions. In Ref. [9], the linear moose terminates with (anti)fundamental quarks with SU (N ) global symmetries at each end; in Ref. [10], the matter content is altered on one of the boundaries to include the antisymmetric representation; and in Ref. [8] the boundaries are made periodic, turning the moose line into a moose ring. In Section 4 we investigate similar variations to the boundary conditions on the moose lattice.

Confinement on the Moose Lattice
The discretization of Euclidean space in n ≥ 2 dimensions is complicated by the fact that there are multiple lattice arrangements that can be said to contain only nearest-neighbor interactions. This ambiguity is not present for the n = 1 dimensional lattice, where each vertex has precisely two neighbors (or one, if the vertex is on the boundary of the lattice). Two dimensional space, on the other hand, can be tiled by squares, triangles, hexagons, or any variety of non-regular polygons. Our decision to present a triangular (rather than rectangular) lattice is motivated by the fact that the triangular plaquette permits gauge-invariant marginal operators in the superpotential, so that all of the important mass scales in the problem are dynamically generated. This setup leads to a relatively simple analysis, where the confinement proceeds in two stages, at Λ 3N and Λ N .
From Section 2.1 to Section 2.4 we track the evolution of the theory from the ultraviolet (µ ∼ M ) to the infrared (µ Λ N ). This stage of the analysis can be completed without specifying the precise shape of the moose lattice, but for concreteness we will periodically refer to an [SU (3N ) × SU (N ) 3 ] k example with k = 3 × 3 as an illustration. The shape and topology of the moose lattice become much more important in Section 4, where we analyze the different kinds of behavior that can emerge in the far infrared limit of the theory.

The Weakly Coupled Regime
Our discussion begins at the ultraviolet scale µ ∼ M where we take all of the gauge couplings to be perturbatively small, i.e. g i O(1) for all of the SU (3N ) and SU (N ) gauge groups. A moose diagram of one example is shown in Figure 2, with non-Abelian groups SU (N c ) represented by circles, and chiral superfields represented by lines.
The "unit cell" of the lattice consists of a central SU (3N ) gauge group surrounded by six SU (N ) groups; three pairs of bifundamental quarks Q + Q, respectively in the fundamental and antifundamental representations of SU (3N ); and a bifundamental q for each neighboring SU (N )× SU (N ) pair on the boundary of the hexagonal unit cell, for a total of six. Throughout this paper we will borrow lattice-related terms to describe the various components of the model, so that each gauge group is located at a "site"/"node"/"vertex"; the bifundamentals are "links" or edges; and the trilinear gauge invariants Tr(q 1 q 2 q 3 ) will be referred to as "plaquette" operators. Figure 2 shows a k = 3 × 3 example of the [SU (3N ) × SU (N ) 3 ] k model, with nine copies of the unit cell arranged in a parallelogram. Each SU (N c ) group on the boundary of the lattice is a global symmetry. At these sites the cubic anomaly coefficients are nonzero, so these SU (N ) groups cannot be gauged without introducing additional matter fields.
It is not necessary to include an integer number of unit cells in the lattice. Although Figure 2 shows an example where every SU (3N ) group is gauged, and the lattice boundary passes only through SU (N ) sites, we could just as well have routed the lattice boundary through some of the SU (3N ) groups instead. Periodic boundary conditions are more restrictive. A periodic direction should include an integer number of unit cells, so that the gauged SU (3N ) and SU (N ) sites are all anomaly-free.
At the ultraviolet scale M , each of the gauge groups is taken to be weakly coupled. For the SU (3N ) groups this is a natural assumption: the matter content of Figure 2 supplies each SU (3N ) group with F = 3N pairs of (Q+Q) quarks, and SQCD with F = N c exhibits asymptotic freedom. Given a coupling g 3N (M ) defined at the short-distance scale M , the gauge coupling at another scale µ is given by the NSVZ β function, and β < 0 for F = N c . Dimensional transmutation defines the holomorphic scale Λ 3N in terms of the gauge coupling g and the CP violating Yang-Mills phase θ YM , which indicates the scale where the SU (N c ) gauge group becomes strongly coupled, i.e. g(Λ) → ∞.
4π is perturbatively small, then there will be a hierarchy Λ 3N M between the two scales.
In a theory with multiple unit cells, the scales Λ Each SU (N ) node, on the other hand, has 3N + N + N pairs of quarks in the N and N representations, for a total of F = 5N c flavors. In this case, b = −2N , the β(g N ) function is positive, and there is no asymptotic freedom. However, if we start with a weakly coupled g N (M ) 1 at µ = M , the SU (N ) groups remain weakly coupled for all µ < M down to the phase transition at µ ∼ Λ 3N . In this work we do not explore the µ M limit of the theory, where the SU (N ) become strongly coupled.
The final addition to the theory is a chiral superpotential W , composed of the marginal trace operators W ⊃ λTr (Q 1 Q 2 Q 3 ) that encompass each triangular plaquette. Eventually we anticipate introducing a full set of symmetry-violating marginal and irrelevant operators into W , but at this stage of the discussion we restrict W to include only the U (1) R preserving operators. For Figure 2 this includes the plaquettes surrounded by SU (3N ) × SU (N ) × SU (N ) groups, but not the SU (N ) × SU (N ) × SU (N ) plaquettes.
The moose notation is particularly helpful when constructing gauge invariant operators: one simply follows the arrows on the diagram so as to form closed loops. For example, within the bulk of the lattice, the next set of trace operators are the dimension-6, irrelevant operators W ⊃ α(Q 1 Q 2 . . . Q 6 )/M 3 . Another type of gauge invariant operator is formed by open "Wilson lines," which start and end on the boundaries of the lattice. These operators transform as bifundamentals under the SU (N f ) L × SU (N f ) R global symmetries associated with their endpoints, and have the form where i, j are indices for the global symmetries SU (N f ) L,R , and the repeated n 1...k indices correspond to the gauged SU (N c ) groups. Simple examples on Figure 2 include the straight leftto-right lines, which pass either through alternating SU (N ) and SU (3N ) nodes, or exclusively though SU (N ) nodes. In addition to the analogous straight lines along the ϕ = 120 • and ϕ = 240 • directions (where we define ϕ = 0 as left to right in the page), generic Wilson line gauge invariants can include any number of 60 • corners. Most of these operators have no relevance in the infrared theory: as we show in Section 2.2, the degrees of freedom associated with the 60 • corners become massive, so that the low energy theory is dominated by the straight Wilson lines that pass through SU (3N ) sites.
An altogether different set of gauge invariant "baryon" operators is generated by the completely antisymmetric products where q and q are any of the SU (N ) × SU (N ) bifundamentals, and Q a,b,c and Q a,b,c are SU (3N ) × SU (N ) bifundamentals in the (3N, N) and (3N, N) representations, respectively. In the N = 3 special case, q N and q N are marginal operators.
By invoking M as the mass scale associated with the irrelevant operators, we complete the promise made earlier in this section, that the theory is weakly coupled at scales µ M . This statement now applies to the superpotential couplings as well as the gauge interactions, as long as the dimensionless parameters α, λ, etc. are all O(1).
Theories of gravity are generally expected to violate global symmetries [32][33][34][35][36][37], so the flavor symmetry violating superpotential Eq. (2.3) and its baryon number violating cousin Eq. (2.4) may be generated by Planck-scale effects, i.e. with M → M p . A lower scale M S < M p may be appropriate if this theory is to be embedded within string theory, any other N > 1 version of supersymmetry, or any more than four continuous spacetime dimensions.

First Stage of Confinement
Approaching the scales µ → Λ (i) 3N , the SU (3N ) i gauge groups become strongly coupled. To understand the behavior of the theory in the infrared, µ Λ 3N , we refer to the Seiberg duality for F = N c = 3N SQCD, an S-duality that relates the weakly-coupled gauge theory at µ Λ 3N to a theory of gauge invariant mesons and baryons at µ Λ 3N . Everything we need to know about the µ Λ 3N regime of the theory can be deduced from studying a single unit cell of the lattice.
Thanks to the positive sign in the β(µ) function for N c = N with F = 5N , the SU (N ) gauge coupling g N -already perturbative at µ = M -becomes even smaller as µ decreases from M towards Λ 3N . As far as the SU (3N ) node is concerned, there are 3N flavors of quarks Q and antiquarks Q, with opposite charges under a U (1) B baryon number, and transforming under approximate SU (3N ) Q and SU (3N ) Q flavor symmetries. By gauging the SU (N ) subgroups, these putative global SU (3N ) Q,Q flavor symmetries are explicitly broken, but at µ Λ 3N this effect is a small perturbation.
The gauge groups and chiral matter associated with the unit cell are shown in Figure 3. The superpotential includes six U (1) R conserving plaquette operators per unit cell, where the trace over gauge indices is implied. Each λ c=1...6 is a dimensionless complex parameter. Under the simplest U (1) R charge assignment, each SU (3N )-charged Q i and Q i is neutral, while the SU (N ) × SU (N ) bifundamentals q i have R charges of +2.
Following Eq. (1.1), the Seiberg dual of the F = N c SQCD is described by F 2 meson operators and two baryon operators, with one constraint equation: , for indices m, n = 1, 2, 3 and i, j = 1, 2, . . . , N . The determinant det M is a shorthand for the completely antisymmetric product of 3N (QQ) operators, which could also be expressed in terms of determinants of the nine distinct SU (N ) charged mesons, det M mn .
2 Triangular/Hexagonal Unit Cell: shows the theory at scales µ < λΛ 3N , after integrating out the vectorlike quarks. In this infrared limit, the only light SU (N )-charged matter fields are the mesons that pass through the center of the unit cell.
In the g N → 0 limit, where the specified SU (N ) node becomes a global symmetry, M ab , B and B all represent flat directions on the moduli space subject to the constraint Eq. (2.6). At symmetryenhanced points on the moduli space, the SU 3N . For gauged SU (N ), many of these flat directions are lifted. The true moduli space is spanned by gauge-invariant operators [29], and the M mn are not gauge invariant: they transform as bifundamentals of SU (N ) × SU (N ). Each gauged SU (N ) introduces a D-term potential for the mesons M mn , so that the vacuum of the theory lies on the BB = −Λ b 3N branch, with spontaneously broken baryon number and unbroken SU (N ) symmetries.
One linear combination of the B and B scalars, the "B + B" direction that changes the value of BB , acquires an O(Λ 3N ) mass. The other linear combination, the "B − B" or "tan β" direction tangential to the BB = −Λ b 3N flat direction, remains massless. This flat direction is lifted if U (1) B is explicitly broken; for example, by the gauge invariant irrelevant operators, which induce small tadpole operators and even smaller baryon mass terms into the superpotential.
Here we have rendered B and B as operators with canonical mass dimension +1, by extracting the appropriate powers of the confinement scale Λ 3N : Applying the same (Q a Q b ) → Λ 3N M ab mapping to the plaquette superpotential Eq. (2.6), we see each of the a = b mesons acquires a vectorlike mass pairing with one of the edge quarks q i : where m a = λ a Λ 3N . Figure 3 illustrates the transition. The middle diagram shows the nine M ab mesons together with the six q c in one unit cell. This moose diagram describes the theory at the intermediate scales m a < µ < Λ 3N . All of the mesons shown in Figure 3 are neutral under the spontaneously broken U (1) B . However, as we show in Section 3, there are a number of unbroken U (1) global symmetries under which B and B are neutral, and the mesons M ab are charged. Integrating Out: On the BB = −Λ b 3N , M ab = 0 branch of the vacuum, all of the M ab degrees of freedom correspond to approximately flat directions on the moduli space, at least if we ignore the D-term potential from the weakly gauged SU (N ) groups. As can be seen from the supersymmetric Lagrangian, which includes terms of the form L ⊃ |∂W/∂Φ| 2 for each of the superfields Φ, many of the otherwise-flat directions are lifted by Eq. (2.9). To pick one example, the m 1 term in W contributes two terms to the so-called F -term potential,

Integrating Out Heavy Mesons and Quarks
In the absence of any other q 1 dependent terms in the superpotential, the scalar potential is minimized at the vacuum solution M 31 = q 1 = 0. Not counting the dimension-6 irrelevant operators, the only other q 1 dependent term in the superpotential comes from the triangular plaquette operator involving q 1 and the q b and q c from the adjacent unit cells, W ∼ λ bc (q 1 q b q c ). Together with the m 1 mass term, the vacuum solution nominally shifts to However, confinement on the b and c unit cells sets q b = q c = 0, so that the minimum of the scalar potential remains at M 31 = 0. In principle the dimension-6 operators do have the potential to shift M ab and q i away from the origin of moduli space; however, thanks to the powers of M 3 in the denominator of such operators, the resulting shift is small enough that it can be safely ignored.
Matching Holomorphic Scales: Before we move on to the strongly coupled regime of the SU (N ) gauge theory, let us take a moment to study the transition at µ ∼ λΛ 3N . This is where the sign of the SU (N ) β function changes, which is what causes SU (N ) to become strongly coupled at µ Λ 3N in the first place. By matching the gauge coupling at the scales µ = m c , the holomorphic scale Λ N for the F = N theory can be derived from g N (µ = M ). Specifically, it is the holomorphic gauge coupling τ that we match at each threshold, We begin with the values of g N and θ YM evaluated at µ = M , and define a Λ F =5N M holomorphic scale: Although the CP -odd θ YM parameter is invariant under the RG evolution, it can acquire threshold corrections at µ ∼ m, so we specify Allowing the four superpotential coupling constants λ a=1,2,3,4 to acquire distinct values, there are generally four distinct mass thresholds, m a > m b > m c > m d . Matching τ (µ = m a ) between the F = 5N and F = 4N theories, with b = −2N and b = −N , respectively, we find (2.14) Applying the same matching procedure at m b,c,d , we find So, the θ YM phase in the F = N theory is given by Inspecting the real part of Eq. (2.15), we find . (2.17) As expected, Λ N for the F = N phase of the theory is exponentially small compared to Λ 3N , which is itself exponentially smaller than M . If any of the λ i are much smaller than For simplicity we have taken λ a,b,c,d ≈ λ to be nearly equal, while also assuming an approximately uniform value of Λ 3N for all of the adjacent unit cells. After the last vectorlike pairs are integrated out, the slope has changed from b = −2N to b = +2N , and g N runs to strong coupling as µ → Λ N . In this graphic we follow two different SU (N ) groups. If their mass thresholds are identical then the ratio between the g 2 N couplings remains fixed, but for generic λ i Λ (j) 3N this is not so.
is suppressed by a further factor of √ λ i . Note that the ultimate expression for Λ N is unaffected by changes to the assumed ordering, λ a > λ b > λ c > λ d .
For simplicity, we assumed in Eq. (2.17) that the two unit cells that border the SU (N ) node have equal values of Λ 3N . Very little changes when this assumption is relaxed. If instead the SU (3N ) L and SU (3N ) R gauge groups have couplings g 3N,L = g 3N,R at µ = M , Eq. (2.17) generalizes to Even if for some reason there is a large hierarchy between Λ 3N,L and Λ 3N,R , it is still true that the SU (N ) remains weakly coupled until after both of its neighboring SU (3N ) gauge groups have confined. Once the first of the strongly coupled SU (3N ) groups confines at (for example) Λ 3N,L , only two pairs of the bifundamental fields acquire O(Λ 3N,L ) masses. After these are integrated out, the F = 5N effective flavors of SU (N ) are reduced to F = 3N ; the coefficient of the one-loop β function switches from b = −2N to b = 0; and the gauge coupling g N (µ) remains fixed at a perturbatively small value. Only after the remaining SU (3N ) R group confines does the β(g N ) function become negative.
After integrating out the vectorlike pairs of quarks q and mesons M ab , the SU (N ) gauge groups become strongly coupled in the infrared. At this stage of the calculation, Λ N,F =N is the only relevant version of the SU (N ) holomorphic scale, so for the remainder of this section we take Λ b N to refer exclusively to where θ F =N and |Λ N,F =N | are given in Eq. (2.16) and Eq. (2.17).
In Figure 4 we show the running gauge couplings for an SU (3N ) and two adjacent SU (N ) groups. At µ = M the various g 2 i are taken to be of the same magnitude. In this example we take the simplifying limit where the neighboring Λ 3N are similar in size, as are the λ i superpotential coupling constants, so that the transition between F = 5N and F = N for the SU (N ) groups occurs sharply at µ ≈ λΛ 3N .
With b = 3N c − F = 6N for the SU (3N ) gauge group, g 3N runs relatively quickly towards strong coupling with decreasing µ < M , while the SU (N ) more gradually become more weakly coupled. For g N , the sharp transition from F = 5N to F = N at µ = λΛ flips the sign of b, implying that only at µ ≈ (λΛ 3N ) 2 /M has g N (µ) returned to its initial value at µ = M . Thus, Λ N (λΛ 3N ) 2 /M is generally much smaller than Λ 3N and M . If Figure 4 were drawn to scale, the red lines corresponding to the different g N would form mirror images in the vicinity of µ ∼ λΛ 3N , and Λ N would be much further to the left on the plot. (2.20) With this substitution, the moduli space in the limit of weakly coupled SU (N ) is given by of the analysis we may restrict our attention to a single string of unit cells, as follows:

Second Stage of Confinement
2 Triangular/Hexagonal Unit Cell: (2.24) Following Ref. [9], the term "neighbor contractions" refers to the replacement of (B j−1 B j ) by a factor of the holomorphic scale −Λ b N associated with the SU (N ) node that connects the M where Λ b j=1,2,3 refer to the jth gauged SU (N ) group, which lies on the border between the jth and (j + 1)th unit cells. The constant term Λ 1 Λ 3 removes the origin from the moduli space: either det M a or some product of baryon operators must acquire an expectation value. If k is odd, then the product (B 1 . . . B k ) cannot be fully contracted: instead, every term in the sum contains an odd number of B j factors. In this case, M a = B This is the stage of the calculation where the boundary conditions, i.e. the shape and topology of the moose lattice, begin to have significant effects on the low energy theory. In this section we have taken the SU (N ) nodes at the boundary of the lattice to be conserved global symmetries.
This kind of boundary is the easiest to analyze: unit cells can be added to or deleted from the k = 3 × 3 example without any change to the analysis above. With different boundary conditions come significantly altered infrared behaviors. In one example, diagonal subgroups of the global SU (N ) ×SU (N ) r groups can be gauged: depending on the resulting topology of the moose lattice, this alteration may prevent the SU (N ) gauge groups from confining. We explore these kinds of possibilities in Section 4.
Symmetry-Breaking Superpotentials: At the corners where the lattice is only one unit cell wide (in the ϕ = 120 • direction), our imposition of SU (N ) ×SU (N ) r global symmetry conservation is important for the consistency of the analysis. These cells admit the gauge invariant mass terms of the form Q 2 Q 2 , e.g. W ⊃ α ij M (Q 2 Q 2 ) ij . Unless the dimensionless α ij is exponentially small, α Λ 3N /M , then these quarks should be integrated out. The theory left behind, SU (3N ) SQCD with F < N c , is not described by the methods of Section 2.2.
Where the moose lattice is wider (m > 1), the analogous gauge invariant operators are irrelevant, with mass dimension 2m > 3. In the m = 2 case, the perturbation to the SU (3N ) theory is small. After SU (3N ) × SU (3N ) confinement, the operator W ∼ α(Q

Conclusion
It is not unseemly to pause here in a spirit of celebration at the simplicity of the low-energy theory. In the µ ∼ M theory depicted in Figure 2, we started with 108 matter fields; nine SU (3N ) and sixteen SU (N ) gauge groups; a superpotential consisting of 62 plaquette operators; and a global symmetry group that includes 22 copies of SU (N ) and a similar, as-yet-uncounted number of U (1) symmetries. In the µ < Λ N limit shown in Figure 5, on the other hand, there are 11 mesons, each transforming as a bifundamental of a global SU (N ) × SU (N ), and a collection of light baryon operators associated with the spontaneously broken U (1) symmetries, subject to constraint equations of the form Eq. (2.24) and Eq. (2.27).

Tracking the Global Symmetries
There are several good reasons why we should keep track of the global symmetries. Matching the anomaly coefficients of the global symmetries in the UV and IR limits provides a consistency check for the Seiberg duality, for example. We may also want to embed the Standard Model within the moose lattice, or to test whether the moose theory descends from some higher dimensional QFT; tracking the global symmetries is important to either effort. A number of the (approximate) global symmetries are spontaneously broken during the various stages of confinement, generating (pseudo-) Nambu-Goldstone bosons and their superpartners. For phenomenological applications these details are important: the approximately massless degrees of freedom may be desirable, e.g. for QCD axion models, or they may be harmful, if for example their presence can be ruled For the moose lattice, the global symmetries can be split into two types. The first class consists of "localized" U (1) symmetries that are associated with a single unit cell. These symmetries have mixed SU (N c ) 2 U (1) anomalies that are cancelled using only the fields from that unit cell. Due to this independence from the neighboring cells and the lattice boundary, we associate these symmetries with the interior "bulk" of the moose lattice.
The second type of global symmetry is associated with the lattice boundary. For these symmetries, the gauge anomalies generated by the matter fields on one boundary are cancelled by matter fields on another part of the boundary, flowing through some set of charged matter fields in the lattice bulk that generally span multiple unit cells.

Global Symmetries In the Bulk
A complete accounting of the anomaly-free global symmetries depends on the shape of the moose lattice. However, it is possible to identify some non-R U (1) symmetries that are properties of the unit cell: that is, where the only matter fields with U (1) charges are the 12 bifundamentals shown in Figure 3.
A simple counting exercise shows why this should be possible. Starting with 12 U (1) phases (one for each matter field), six of the linear combinations are broken by the R-conserving plaquette superpotential; W of Eq. new non-R U (1) symmetries, which are anomaly-free and not broken by the plaquette superpotential.
To make this explicit, consider the following U (1) a × U (1) b charge assignment for the quarks in Figure 6: Here we use a new notation,x(Φ), to concisely report the U (1) x charge of a superfield Φ, or the charge of its scalar component if the U (1) is an R symmetry. Each unit cell has its own U (1) a × U (1) b global symmetry, acting only on the quarks associated with that cell. Transformations of this kind can be called "localized," to distinguish them from global symmetries like U (1) B that act on matter fields from multiple unit cells. This is distinct from (but reminiscent of) a truly local (a.k.a gauged) symmetry.
After SU (3N ) confinement, Eq. It is possible to gauge U (1) a ×U (1) b or any of its U (1) subgroups without adding any additional matter fields, thanks to the cancellation of all of the mixed gauge anomalies involving U (1) a,b and the various gauged SU (N c ) groups. In this case the localized nature of U (1) a × U (1) b in the moose lattice is now highly reminiscent of a gauged U (1) × U (1) symmetry from a 6d spacetime, where the discretization of two compact dimensions causes the local transformation to be realized as a separate U (1) a × U (1) b gauge group for each unit cell.
Plaquette Operators and R: If U (1) a and U (1) b are not gauged, then the superpotential Eq. (2.5) may be expanded to include trace operators from the SU (N ) × SU (N ) × SU (N ) plaquettes. For any three mutually adjacent unit cells r, s, t, these plaquette operators take the form W ⊃ Tr q   So, the addition of the (q 1 q 3 q 5 ) and (q 2 q 4 q 6 ) type plaquettes to W has replaced the k independent localized U (1) a × U (1) b with a single global U (1) a × U (1) b . This picture further reinforces the notion that the moose lattice reconstructs two extra dimensions: either U (1) a × U (1) b is locally conserved, Eq.

Global Symmetries From the Boundary
The U (1) a,b type symmetries are distinct from U (1) R and the baryon number U (1) B that is spontaneously broken in the BB = 0 vacuum. Indeed, if we restrict our view to a single unit cell, we find that both U (1) B and U (1) R have nonzero mixed SU (N ) 2 U (1) anomaly coefficients. This means that the anomaly-free versions of U (1) B and U (1) R on the moose lattice must involve matter fields from multiple unit cells. These are the simplest examples of global symmetries associated with the boundaries of the lattice: with the correct charge assignment for a conserved U (1) B , all of the mixed SU (N ) 2 U (1) B anomalies cancel for the gauged SU (N ) groups, but not for the SU (N ) ,r on the boundary of the moose lattice.
In this section we describe a systematic method for enumerating the other boundary-associated global U (1) symmetries. Unlike Section 3.1, we restrict our attention to U (1) rotations that leave Eq. (3.4) as well as Eq. (2.5) invariant. A relatively simple charge basis can be constructed from strings of adjacent quarks with alternating ±1 charges, traversing the bulk of the moose lattice in a zigzag pattern. All plaquette operators are neutral under such a charge assignment. For the mixed SU (N c ) 2 U (1) anomalies to cancel for all gauged SU (N c ), this type of global symmetry needs to involve adjacent rows of unit cells, as shown in Figure 7. The result is a chevron-like charge assignment. The only nonzero SU (N ) 2 U (1) anomaly coefficients involve the global symmetries at the boundaries of the lattice, pairing two SU (N ) groups with their SU (N ) r counterparts on the opposite edge.
There are U (1) symmetries of this type along each of the ϕ = 0, 120 • , 240 • directions. In the (arbitrarily chosen) moose lattice of Figure 7, there are 3 + 5 + 5 = 13 distinct chevron charge assignments. This accounting includes the single-row zigzag versions that run along the edges of the moose lattice: for example, the dashed ϕ = 120 • line in Figure 7, but pushed all the way to the right edge of the lattice. In the k = 3 × 3 model of Figure 2, there are 4 + 4 + 6 = 14 of these U (1) symmetries. More generally, for an arbitrarily shaped moose, the number of chevron global symmetries parallel to ϕ depends on the number of distinct rows of unit cells, i.e.: Because the number of these U (1) symmetries is proportional to the number of rows in the moose lattice, it scales with the length of the perimeter of the lattice, rather than as the area (number of unit cells) of the lattice. These chevron U (1) symmetries can be thought of as partially localized, in analogy with Section 3.1. Rather than being confined to a single unit cell, these U (1)s act on the quarks within a specific horizontal band (or a band parallel to ϕ = ±120 • ) while ignoring the rest of the moose lattice. Unlike the U (1) a ×U (1) b symmetries, these chevron U (1)s mix with the SU (3N ) baryon number: that is, for any participating unit cell, the operators B = (Q N 1 Q N 2 Q N 3 ) and B = (Q

Global symmetry example
where as in Figure 7 we use blue and red to indicate ±1 charges for the quarks.  . (3.8) In this example we added a third row of cells to illustrate the alternating pattern. Note that the charge assignment is symmetric with respect to shifts in the ϕ direction, and that the only . . = 0, and the global symmetry group need not be broken. Most moose lattices will include some odd-length rows in one direction or another, forcing some spontaneous symmetry breaking, but an all-even moose lattice can be constructed from doubly periodic boundary conditions. As we discuss in Section 4, periodic boundary conditions lead to Coulomb phases rather than confinement.

Sources of Explicit Symmetry Breaking
Exactly conserved U (1) symmetries can pose phenomenological problems, especially in contexts (such as spontaneous symmetry breaking) where they correspond to exactly massless particles. For this reason alone, we should parameterize the sources of U (1) breaking which could introduce mass terms for otherwise massless Nambu-Goldstone bosons.
In this context, despite the fact that Λ N is much smaller than Λ 3N , it is still (by definition) large compared to the ultimate IR limit of the theory, µ Λ N . In particular, we have assumed that SUSY is preserved during the SU (N ) confinements. If SUSY is broken (which it must be at some point, if the theory is to describe anything resembling our universe) then the scale of soft SUSY breaking should satisfy m s Λ N . Though we can be glad to ignore any degrees of freedom with O(Λ N ) masses, this does not necessarily extend to the approximate global symmetries that are explicitly broken by their mixed anomalies with SU (N ) gauge groups. Many of these anomalous U (1) A are broken spontaneously by the BB vevs from SU (3N ) confinement at a much higher scale, Λ 3N , where U (1) A can be treated as approximately conserved. The ratio between Λ N and Λ 3N suppresses the particle masses introduced by the triangle anomaly: indeed, this is exactly the setup of a typical axion model [38][39][40][41][42][43][44][45], where the axion mass m a is related to f A , the scale of spontaneous U (1) A breaking, and Λ QCD , the source of explicit U (1) A violation, via In our examples f A is proportional to Λ 3N , while Λ N stands in for Λ QCD , i.e. m a ∼ Λ 2 N /Λ 3N . Incidentally, the fact that all of these mass scales are dynamically generated makes the moose lattice an ideal playground for model building. For example, Refs. [46,47] use similar features in simpler SUSY product gauge theories to construct composite QCD axion models.
Global symmetries can also be broken by superpotential operators. We have seen this when adding plaquette operators to the superpotential; for example, the localized U (1) a × U (1) b global symmetries of Eq.
where the Wilson line runs through k different unit cells. The jth unit cell in the product should be adjacent to its (j ± 1)th neighbors, though the path through the lattice does not need to be in a straight line. We write Eq.  The other type of symmetry violation comes from baryonic superpotential operators, e.g.
Each unit cell supplies a B and B type operator, as well as six q N i = det q i from the SU (N )×SU (N ) bifundamentals. Applying the same mapping between canonically normalized degrees of freedom across the SU (3N ) transition, In principle any of the conserved U (1) symmetries can be gauged, as long as the anomaly cancellation conditions are satisfied. Take U (1) B of Eq. (3.7) for example. With an integer number of unit cells included in the moose lattice, the U (1) 3 B and mixed gauge anomalies all cancel. From the perspective of the 4d theory, gauging U (1) B is straightforward. Unlike U (1) a × U (1) b , however, there is no local U (1) B conservation in the cells of the moose lattice: a U (1) B rotation acting only on the Q i and Q i of a single unit cell has nonzero SU (N ) 2 U (1) anomaly coefficients. For this 4d product gauge theory to descend from a higher dimensional theory with a locally conserved U (1) B , the local U (1) transformations acting on the compact space need to be spontaneously broken as part of the 6d → 4d discretization scheme.

Boundary Conditions
For most of Section 2, the shape of the moose lattice mattered very little to the analysis. In Section 2.4, where the SU (N ) groups become strongly coupled, we did make the assumption that the moose lattice was not periodic. Our discussion of global symmetries in Section 3 is more directly dependent on the shape of the lattice boundary, but our methods remain generic enough that we could switch between the k = 3 × 3 example and a k = 5 + 4 or k = 5 + 4 + 3 version with impunity, as in (3.7) and (3.8). Every example was constructed in the same basic way: by connecting an integer number of unit cells, gauging the SU (N ) nodes that connect adjacent unit cells, while leaving the SU (N ) nodes on the boundary of the moose lattice as global symmetries.
By altering the boundary of the moose lattice, we can construct several new types of gauge theories from this template, often by finding ways to gauge the SU (N ) boundary nodes. For example, we can add new matter fields charged under a single SU (N ) so as to cancel its cubic SU (N ) 3 anomaly; we can add bifundamentals of SU (N ) × SU (N ) r to gauge a pair of boundary nodes; or, we could gauge the diagonal subgroup SU (N ) d ⊂ SU (N ) × SU (N ) r of two boundary nodes. None of these perturbations have much effect on SU (3N ) confinement, but can considerably alter the behavior of the SU (N ) m gauge theory.

Reflective Boundaries
The reader has probably noticed that the moose lattices depicted in Figures 2 and 5 have notches missing from the edges. Considering the matter further, the reader may have decided that the missing SU (N ) i × SU (N ) j bifundamentals have a minimal impact on the theory after all: if SU (N ) i,j are global symmetries, each of these edge fields is just N 2 chiral fields with no gauge charges. Naturally, if the edge fields are charged under some gauged U (1), or if they are coupled to the other quarks in the superpotential, they are not entirely irrelevant, but they are not especially interesting either.
If the addition of edge quarks allows the boundary SU (N ) nodes to be gauged, their impact on the theory can be much more interesting. Take for example the modified k = 3 × 3 theory shown in Figure 8. Here we have added eight bifundamentals to fill in the notches; however, these edge quarks have the opposite SU (N ) × SU (N ) r charges, i.e. in the (N, N) representation rather than (N, N), or vice versa. The arrows on the moose diagram for these edge quarks appear to be pointing in the wrong way: that is, unlike every other matter field in the moose lattice, the arrows of the new edge quark point in the ϕ = ±60 • , 180 • directions.
From the perspective of the new gauged boundary nodes, there are 3N fundamentals (e.g. from an SU (3N ) node) and 3N antifundamentals (from the three adjacent SU (N ) nodes). This is F = 3N c SQCD, for which the NSVZ β function vanishes, and asymptotic freedom is lost. With our working assumption that g N (M ) O(1) is perturbatively small, the edge SU (N ) gauge couplings begin to run below µ = λ i Λ 3N , where the vectorlike mass terms for the mesons and q i quarks become relevant. The right panel of Figure 8 shows the theory after these fields have been integrated out. This is the same scale at which the β function for the SU (N ) nodes in the bulk switches sign. If the edge and bulk SU (N ) groups have similarly strong couplings at M , i.e. g (i) N (M ), then the running of the different gauge couplings will tend to make the bulk SU (N ) groups more weakly coupled than the edge nodes at µ ∼ Λ 3N , so that the edge nodes are the first to confine.
Previously, the µ < λΛ 3N phase of the k = 3 × 3 theory involved eleven disjointed strings of SU (N ) charged mesons, as depicted in the right panel of Figure 5: three sets each in the ϕ = 0 • , 240 • directions, and five in the ϕ = 120 • direction. With the wrong-way edge quarks and gauged SU (N ) nodes on the boundaries, previously decoupled SU (N ) m sets are joined together. In Figure 8 there are now only three sets of gauge invariant Wilson line operators. One is a simple example of the form Eq. (3.10), running from corner to corner along the ϕ = 120 • direction. The other two appear to bounce off of the lattice boundaries: one originates and terminates in the lower left corner, the other begins and ends in the upper right corner. This is why we refer to the boundary conditions as "reflective." Aside from this detail, the SU (N ) m−1 groups confine in the manner described in Section 2.4 and Ref. [9], now with m = 3 or m = 10. In the far infrared limit the degrees of freedom Although we have added eight edge quarks to the k = 3 × 3 model, the reflective theory has fewer global U (1) symmetries. There is no gauge invariant plaquette superpotential associated with the edge quarks χ i , because products of the form (χ i q a q b ) are charged under the boundary SU (N ) nodes. However, by gauging two SU (N ) nodes for each new edge quark χ, some of the U (1) global symmetries acquire SU (N ) 2 U (1) anomalies that cannot be canceled by assigning χ a charge under U (1).
Coulomb Phases: In the reflective k = 3 × 3 example, all of the IR mesons transform as the bifundamental of a different global SU (N ) × SU (N ) r . This feature is not generic, and is not even generic for k = n × n parallelogram arrangements. In the k = 4 × 4 parallelogram of Figure 9, for example, some of the strings of connected SU (N ) gauge groups form closed loops, with no SU (N ) ,r endpoints on the lattice boundary. This arrangement is studied in Ref. [8], which we review in Section 1.2. These closed loop product groups appear especially frequently in the examples with periodic boundary conditions. At a generic point on the moduli space, each SU (N ) 1 × SU (N ) 2 × . . . SU (N ) m gauge theory is spontaneously broken to U (1) N −1 , a Coulomb phase with N − 1 massless photons. There are two disjoint closed loops of this form in the k = 4 × 4 theory, providing a total of 2(N − 1) unbroken U (1) gauge groups at an arbitrary point on the moduli space. As noted in Ref. [2], for theories in the Coulomb phase one can describe the Lagrangian using a holomorphic prepotential, borrowing methods from N = 2 supersymmetry. The hyperelliptic curves for the SU (N ) m theory are given in Ref. [8].

Cylindrical Moose
All of the examples discussed so far have involved topologically trivial moose lattices, aside from the possibility floated in Section 4.1 of connecting non-adjacent SU (N ) ×SU (N ) r boundary nodes. By making one of the dimensions in the moose lattice periodic, we can construct cylindrical lattices with S 1 topology.
Periodic lattices have additional discrete symmetries associated with reflections or translations. If the coupling constants λ i , g N and g 3N also respect these symmetries, e.g. g . . along the periodic direction, then the discrete symmetries should be manifest in the low energy degrees of freedom. After SU (3N ) confinement and the subsequent integrating-out of the massive quarks, the theory includes three sets of SU (N ) 4 ring product gauge theories, with charged mesons from the 1 → 1, 3 → 3 and 5 → 5 rows. These SU (N ) groups do not confine, but each have an unbroken U (1) N −1 at a generic point on the moduli space. Following Ref. [8], the gauge-invariant degrees of freedom can be written in terms of the M Ad operators of Eq. (1.13),

Symmetric Cylindrical Moose
where M At k = N this u k is related to the other degrees of freedom by a classical constraint, This classical constraint receives quantum corrections of the form B , including a complete set of nearest neighbor replacements as in Eq. (1.12) [8,48].
At scales µ where the SU (N ) groups are weakly coupled (e.g. µ ∼ λΛ 3N ), an arbitrary point on the moduli spaces sees the gauge invariant operators u k acquire expectation values that spontaneously break SU (N ) 4 to its maximal Abelian subgroup, U (1) N −1 . Ref. [8] uses methods from N = 2 supersymmetry to derive the kinetic terms of the supersymmetric Lagrangian in the infrared limit of the theory.
Along the ϕ = 240 • direction we find four copies of SU (N ) 2 gauge theories with open boundaries, as in Figure 1 or Ref. [9]. As described in Section 2.4, these SU (N ) groups confine, so that in the far infrared the theory is described by baryons B Each discrete symmetry of the lattice is only a symmetry of the particle model if the coupling constants cooperate. Unequal values of λ i between different unit cells, for example, tend to break the translation or reflection symmetries.

Alternative Cylinder Boundaries
In Figure 10 we chose an example where the aperiodic boundaries (on the top and bottom rows) respected the shift symmetry in the ϕ = 0 • direction. A more generic example can have cylindrical topology without this "azimuthal" symmetry. For example, one could add or delete unit cells from the aperiodic edges. The periodic edge can be twisted, for example by matching the "1" and "3" on the right edge of Figure 10 to the "3" and "5" nodes (respectively) on the left edge. Other modifications preserve the shift symmetry: one could add wrong-way quark fields to the cylinder ends to recreate the reflective boundary conditions of Section 4.1.
Reflective: If the 4×3 example of Figure 10 is given reflective boundary conditions, it is possible for all 16 of the boundary SU (N ) nodes to be gauged. After SU (3N ) confinement, the charged matter content is given by: Barbershop: The shift symmetry of the cylinder can be broken by adding an apparent rotation or twist to the periodic moose lattice. In Figure 10, the moose lattice is exactly periodic in the ϕ = 0 • direction. We could instead have rolled up the lattice vertically in the page, rather than horizontally: for example, . (4.5) The seven numbered edge nodes are gauged, so that the lattice is approximately periodic in the ϕ ≈ 90 • direction. Unlike the Figure 10  following → 6 → 4 → 2 → r through a total of 11 gauged SU (N ) sites. The ϕ = 120 • direction hosts two open strings with SU (N ) 5 gauge groups: one passes through → 7 → 3 → r , the other through SU (N ) 5 and SU (N ) 1 . Like the stripes on a barbershop pole, the gauge invariant line operators in the ϕ = ±120 • directions wrap around the S 1 direction while traveling horizontally along the cylinder. The number of distinct line operators depends on the size of the cylinder, and on the degree to which it is twisted.
In the example of (4.4), the combination of periodic and reflective boundaries removed all of the global SU (N ) ,r symmetries, ensuring that the IR limit of the theory exhibits a Coulomb phase for each of the ϕ. With the twisted cylindrical moose lattice of (4.5), we encounter the opposite behavior: there are no closed SU (N ) rings or Coulomb phases in any direction, but instead all of the SU (N ) sites confine as in Section 2.4.
Möbius: As a final S 1 related example, we can further modify the k = 4 × 3 cylinder by adding a 180 • twist about the central row, to construct a Möbius strip rather than a simple cylinder. Taking reflective boundary conditions on the top and bottom edges (or rather, the single "top = bottom" edge), the moose diagram after SU (3N ) confinement takes the form:  . (4.6) Compared to (4.4), there are even fewer ϕ = ±120 • distinct gauge invariant line operators: one along 0 → 6 → 0, another following 2 → 4 → 2, each of which encounters 16 gauged SU (N ) groups. Also, the "5" and "1" ϕ = 0 meson lines now join together into a single 1 → 5 → 1. So, the SU (N ) charged mesons can be organized into four sets of SU (N ) m rings, with m = 4, 8 for ϕ = 0, and m = 16, 16 for ϕ = ±120 • . The infrared theory exhibits a Coulomb phase with just four copies of U (1) N −1 , rather than the U (1) 7(N −1) of (4.4).

Toroidal Moose
Finally, let us discuss the toroidal topologies (T 2 = S 1 × S 1 ) that arise when periodic boundary conditions are imposed on all edges of the moose lattice. By construction, these geometries have no SU (N ) sites that are not gauged. A generic periodic flat torus can be represented by a k = m × n parallelogram, with some scheme for matching the nodes on opposite edges. That is, the boundaries can be twisted in the manner of (4.5), while preserving the two-dimensional shift symmetries of the lattice. As a first concrete example, we return to the k = 4 × 3 parallelogram, depicted in Figure 11 before and after SU (3N ) confinement. With this particular choice for the periodic boundaries, Using the technology from Section 2, it is straightforward to follow the theory from µ ∼ M towards the infrared, past SU (3N ) confinement and the generation of masses for the vectorlike pairs of mesons and quarks. The remaining light SU (N ) charged degrees of freedom are shown in the right side of Figure 11. The ϕ = 0 mesons form four sets of SU (N ) 3 , following j → j for j = 1, 3, 5, 7. Similarly, the ϕ = 240 • mesons provide another three SU (N ) 4 rings, with j → j for j = 1 , 3 , 5 . Lastly, the ϕ = 120 • mesons form a single closed loop encompassing an SU (N ) 12 gauge group, following 0 → 6 → 4 → 4 → 2 → 2 → 0.
The doubly periodic lattice has a two dimensional shift symmetry, spanned by unit cell translations in the ϕ = 0 • and ϕ = 240 • directions. Respectively, these operations cyclically permute the sets of ϕ = 240 • and ϕ = 0 • mesons. The single ϕ = 120 • line transforms as the identity under both kinds of translation. The moduli space is spanned by the u k type operators of Eq. Either type of edge can be twisted, by acting on the node labels with cyclic permutations. As a simple example, we could shift the bottom row of labels four spaces to the right, 0 → 4 , so that the top and bottom rows now match along the ϕ = 90 • vertical direction. The k = 3 × 4 version is shown in Figure 12, after SU (3N ) confinement. In this example the lattice is symmetric with respect to reflections about the vertical axis: this was not true of Figure 11. Now, both sets of ϕ = ±120 • mesons form rings of SU (N ) 12 . The ϕ = 120 • example follows 0 → 4 → 2 → 6 → 4 → 2 → 0 ; for ϕ = 240 • , the line operator follows 1 → 3 → 5 → 1 instead. The ϕ = 0 • operators are not impacted by the twist: they still form four sets of SU (N ) 3 rings. At a generic point on the moduli space, the SU (N ) groups are spontaneously broken to U (1) 6(N −1) .
In the present work we are content to restrict ourselves to T 2 and S 1 topologies for the moose lattice. More complicated topologies can of course be constructed by folding and connecting lattices of different shapes, but the methods for determining the low energy behavior remain the same.

Conclusion
This paper is dedicated to the N = 1 supersymmetric triangular moose lattice in four spacetime dimensions, with the [SU (3N ) × SU (N )] k style product gauge group. Assuming that there is a high energy scale M at which the coupling constants are perturbatively small, we have shown that the SU (3N ) gauge groups confine, and that some of the SU (N ) charged quarks and mesons subsequently acquire vectorlike masses. Depending on the lattice boundary conditions, the SU (N ) gauge groups may either confine or form Coulomb phases.
Aspects of the infrared theory are highly suggestive of a higher dimensional interpretation, where the effectively two-dimensional moose lattice is associated with two compact extra dimensions. Some of the degrees of freedom, such as the baryon operators, are localized to specific segments of the moose lattice. Each gauged SU (3N ), for example, is associated with the B and B operators defined in Eq. (2.6), as if B and B were composite fields propagating in the bulk of the extra-dimensional theory. As we show in Section 3.1, some global symmetries can act in the same way, especially if some subgroup of the U (1) a × U (1) b is gauged. Other degrees of freedom, like the M of Eq. (2.23), are associated more closely with the boundaries of the lattice, as are the chevron-type global U (1) symmetries of Section 3.2.
Especially for the examples with periodic boundary conditions presented in Section 4, we see that the meson and baryon operators in the farthest infrared limit possess shift symmetries that recall a discretized version of translation invariance. An approximate version of this translation invariance appears in the bulk of the moose lattice even in non-periodic topologies. All of these signs point towards a geometric physical interpretation of the moose lattice gauge theory, providing a clear direction for future research on this topic.
The Coulomb phases associated with the periodic or reflective boundary conditions provide another area for future study. Aside from noting that Ref. [8] provides expressions for the holomorphic prepotential of the SU (N ) m ring theories, we have not yet taken advantage of the approximate N = 2 supersymmetry to constrain the Lagrangian for the infrared degrees of freedom.
The structure of the model also provides many opportunities for model building. The strong CP problem, and the associated axion quality problem, supply one such well-motivated target. The QCD axion provides an elegant mechanism to explain the otherwise confoundingly tiny value of the SU (3) c CP violating θ parameter, where an approximate U (1) PQ with nonzero SU (3) 2 c U (1) PQ anomaly coefficient is spontaneously broken at a high scale, f a Λ QCD . Nonperturbative QCD effects generate a potential for the pseudo-Nambu-Goldstone boson of U (1) PQ that sets the effective value of θ to zero.
This mechanism requires that U (1) PQ should be classically conserved to an extremely high degree, broken only by QCD effects. However, gravitational effects are generally expected to break global symmetries [32][33][34][35][36][37], and even relatively tiny perturbations to the axion potential can ruin the solution to the strong CP problem. A successful "high quality" axion model protects U (1) PQ against these gravitational intrusions by ensuring that all PQ-charged gauge-invariant operators permitted in the Lagrangian are sufficiently suppressed [46,47,[49][50][51][52][53][54][55][56]. Especially for large moose lattices, or for the "barbershop" arrangements of Section 4.2.2, it can be relatively easy to embed SU (3) c and a high quality QCD axion within this model. Indeed, compared to the relative simplicity of Ref. [47], invoking a whole moose lattice for the sole purpose of dealing with the axion quality problem may be seen as overly aggressive.
The automatically generated hierarchy between the scales Λ N and Λ 3N depicted in Figure 4 is another feature of the moose lattice that has possible model building applications. Due to the sign change in the β(g N ) function induced by SU (3N ) confinement, the scale Λ N is suppressed by a factor of Λ 2 3N /M . This inverse relationship, reminiscent of the seesaw mechanism for neutrinos, allows for an unusually small Λ N even if the SU (N ) and SU (3N ) couplings are of similar size at µ ∼ M .
Finally, it may be worthwhile to generalize the two-dimensional triangular lattice beyond the [SU (3N ) × SU (N ) 3 ] k paradigm to include higher dimensions and alternative lattice arrangements.

Acknowledgements
I am grateful to Patrick Draper, Arvind Rajaraman, Yuri Shirman, and Tim M. P. Tait for several conversations during the development of this paper, and to Carlos Blanco, Aaron Friedman, Robert McGehee, and Pavel Maksimov for their patience at the social occasions where I have presented the principal results. Special thanks go to Patrick Draper for helpful feedback on this manuscript. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. This work was partially supported by a grant from the Simons Foundation. Some of this work was supported by NSF Grant No. PHY-1620638 and the Chair's Dissertation Fellowship from the Department of Physics & Astronomy at UC Irvine.

A Global Symmetries and 't Hooft Anomaly Matching
For the infrared theory of M , B and B to be dual to the SU (N ) gauge theory of Q and Q, it must satisfy a number of nontrivial constraints that arise from the global symmetries of the theory. At every point on the moduli space, the 't Hooft trace anomalies of the preserved global symmetries should match between the two theories. For qdms-confinement, we show that the full set of anomaly matching conditions is not satisfied at the origin of moduli space: but, the anomaly coefficients do match everywhere on the quantum-deformed moduli space Eq. (1.3), for those global symmetries which are not spontaneously broken. This provides a separate confirmation that the origin should be excised from the qdms-confined theories. In the case of s-confinement, the moduli space includes the origin, so the full set of 't Hooft anomaly coefficients must match in the infrared theory.
In SQCD with N colors and F flavors, the global symmetry group is In addition to G global , there is also an approximate symmetry U (1) A , which has a nonzero SU (N ) 2 -U (1) A anomaly coefficient. It is explicitly broken by SU (N ) instantons, at a scale characterized by Λ. As the F = N model features prominently in the product gauge groups introduced in this paper, let us take a moment to explore its infrared effective theory in more detail. For this special case the U (1) R symmetry can be defined such that the scalar parts of the Q and Q supermultiplets have zero R charge, as shown in Table 1. Using the canonical normalization for U (1) R , the fermionic components of Q and Q (and B, B and M ) have R charges −1. This definition of U (1) R is not unique: as in any theory with multiple conserved U (1) charges, it is possible to define a U (1) R or a U (1) B out of linear combinations of U (1) R and U (1) B .
To demonstrate the use of the anomaly matching conditions in the presence of spontaneous symmetry breaking, consider the U (1) 3 R cubic anomaly coefficient: where we have naively evaluated the anomaly coefficient A IR at the origin of the moduli space. The two A do not match: this is because we have overcounted the IR degrees of freedom by neglecting the quantum modified constraint, Eq. (1.3). On the BB = −Λ b branch of the moduli space,