I generalize the deconstruction lattice formulation of Endres and Kaplan to two-dimensional super-QCD with eight supercharges, denoted by (4,4), and bifundamental matter. I specialize to a particularly interesting (4,4) gauge theory, with gauge group 𝑈(𝑁𝑐)×𝑈(𝑁𝑓), and 𝑈(𝑁𝑓) being weakly gauged. It describes the infrared limit of the D1/D5 brane system, which has been studied extensively as an example of the AdS3/CFT2 correspondence. The construction here preserves two supercharges exactly and has a lattice structure quite similar to that which has previously appeared in the deconstruction approach, that is, site, link, and diagonal fields with both the Bose and Fermi statistics. I remark on possible applications of the lattice theory that would test the AdS3/CFT2 correspondence, particularly one that would exploit the recent worldsheet instanton analysis of Chen and Tong.

1. Introduction

Supersymmetric large 𝑁𝑐 gauge theory seems to afford a window on quantum gravity, through the AdS/CFT correspondence [14]. Recent formulations of lattice supersymmetry give some hope that we may be able to study these ideas on the lattice. In particular, to what extent does the correspondence hold at intermediate 𝑁𝑐, at finite temperature, and for non-BPS quantities?

Many promising lattice formulations of supersymmetric field theories occur in two dimensions (2D). (For an extensive list of references on lattice formulations of supersymmetric field theories, both old and new, see [5, 6].) In some cases, convincing perturbative arguments can be made that the correct continuum limit is obtained without fine tuning [716]. (Another interesting approach, involving noncommutativity on the scale of the lattice, deserves further study of its quantum continuum limit [17, 18].) In other cases, super-renormalizability implies that fine-tuning a small set of one-loop diagrams allows one to obtain the desired continuum limit in perturbation theory, as in [19]. (A 3D analogue is described in [20].)

Broadly speaking, it is the softer ultraviolet (UV) divergences in 2D that generically make it easier to obtain the desired continuum limit in perturbation theory. Whether or not this property holds nonperturbatively is an open question, which at this point can only be answered empirically. In this regard, it is important to note that 2D field theories are more practical to study numerically; a small computer cluster can obtain reasonably accurate results. For some 2D examples, Monte Carlo simulation results have provided information on nonperturbative renormalization. For example, recent simulations of 2D supersymmetric theories that preserve a nilpotent subalgebra seem entirely consistent with continuum expectations [2126]. In this author's opinion, the encouraging results in 2D suggest that it is time to look for interesting applications of the lattice supersymmetry ideas that have been developed thus far.

A well-known example of the AdS/CFT correspondence occurs in the Type IIB superstring, at the intersection of D1 and D5 branes, with four of the directions of the D5 brane wrapped on, say, a torus 𝑇4. The IR limit of the world-volume intersection theory is a 2D (4,4) supersymmetric gauge theory. It can be understood as the dimensional reduction of a 4d 𝒩=2 super-QCD [2729] where 𝑁𝑓 flavors of matter are contained in hypermultiplets, and transform in the fundamental representation of the 𝑈(𝑁𝑐) gauge group. These flavors are minimally coupled, so that there would be a 𝑈(𝑁𝑓) flavor symmetry. In actuality, the 𝑈(𝑁𝑓) symmetry is weakly gauged, and the flavors are bifundamentals of 𝑈(𝑁𝑐)×𝑈(𝑁𝑓).

In this paper, the (2,2) supersymmetric formulation of Endres and Kaplan (EK) [30] will be generalized to (4,4) theories that describe the world-volume gauge theory of the D1/D5 brane system. That work is based on the orbifold approach to lattice supersymmetry, which has been reviewed in [6]. It will be seen that, in the gauge sector, a slight modification of the original (4,4) pure SYM construction of Cohen et al. (CKKU) [10] is required. EK have shown, in general terms, how to construct (2,2) theories with bifundamental matter under a quiver gauge group 𝑈(𝑁)𝑚 (“Example 2” at the end of their paper). The target theory for the D1/D5 system which is aimed at here is (4,4) 2D super-QCD with gauge group 𝑈(𝑁𝑐)×𝑈(𝑁𝑓). In this context, we want the flexibility to choose 𝑁𝑐 and 𝑁𝑓 independently. Also, as already been mentioned, the flavor group 𝑈(𝑁𝑓) should be weakly gauged; that is, the corresponding gauge couplings should satisfy 𝑔𝑁𝑓𝑔𝑁𝑐 for all scales of interest. It will be seen below that it is not difficult to modify EK's technique to obtain a quiver of two factors, 𝑈(𝑁𝑐) and 𝑈(𝑁𝑓), with 𝑁𝑐𝑁𝑓. The nontrivial task is to devise a trick to make the 𝑈(𝑁𝑓) weakly gauged in the EK construction with bifundamental matter. The trick that is used is quite standard and has an interesting interpretation, as will be described below. With these various generalizations of the EK formulation, the lattice theory described here is specially tailored to describe the D1/D5 world-volume theory. Having at hand a fully latticed description of this theory, one can contemplate various nonperturbative studies that would be of interest, both numerical (Monte Carlo simulations) and analytical (strong coupling expansions).

I will now summarize the remainder of this paper. (i)Section 2 (Summary of the Target Theory). Here I describe the continuum 𝑈(𝑁𝑐)×𝑈(𝑁𝑓) (4,4) gauge theory that is aimed at, using the language of 4d 𝒩=1 superfields.(ii)Section 3 (Lattice Construction). First I explain in general terms how a (4,4) theory with gauge group 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)𝑛 and bifundamental matter is obtained. The 𝑈(𝑁𝑓)𝑛 quiver is introduced in order to obtain a weakly coupled diagonal subgroup, 𝑈(𝑁𝑓)diag. It will be explained how this can be interpreted in terms of a deconstructed third dimension. The 𝑈(𝑁𝑐) gauge multiplet does not propagate in this direction but is stuck to the 2D subspace. It is interesting that this mimics what occurs in the D1/D5 system, where 𝑈(𝑁𝑐) gauge fields are stuck to the D1 brane, and 𝑔𝑁𝑓𝑔𝑁𝑐 is due to “volume suppression”. In the 4d 𝒩=1 language, the lowest components of hypermultiplets are 𝑆𝑂(1,3)×𝑈(1)𝑅 neutral and form a doublet of 𝑆𝑈(2)𝑅. It follows that in the conventions of CKKU, the hypermultiplets would have fractional 𝑁-ality with respect to the 𝑍𝑁×𝑍𝑁 that is used to define the lattice theory. This unacceptable situation calls for a minor modification of the choice of global charges used in the 𝑍𝑁×𝑍𝑁 orbifold, relative to CKKU [10]. Finally, I describe how to break to 𝑈(𝑁𝑓)diag, and the conditions that must be satisfied for the Kaluza-Klein (KK) states of the corresponding third dimension to decouple from the 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag effective theory.(iii)Section 4 (Application). Here I describe a simple study of the characteristics of instantons in the sector with unit first Chern class. The distribution of these characteristics—instanton size and orientation—has been shown recently to have AdS3×𝑆3 geometry [31]. This support for the AdS3/CFT2 correspondence would be interesting to study on the lattice where we can access intermediate 𝑁𝑐 and finite temperature. In this regard, the recent results of Rey and Hikida [32] provide continuum results that could be compared to.(iv)Section 5 (Conclusions). Here I summarize this work and outline research that is in progress.(v)Appendices. Some technical details and lengthy formulae have been separated from the main text.

The purpose of this paper is to give a brief outline of the lattice construction and its potential applications. A more thorough discussion of details associated with the lattice system (superspace description, renormalization, etc.), as well as intensive studies of the possible applications mentioned in Section 4, is left to future work.

2. Summary of the Target Theory

The 2D theory is most easily obtained from a dimensional reduction of the 4d theory written in 𝒩=1 superspace. (See [33] for a review of this formalism.) The 𝑈(𝑁𝑐)  𝒩=2 vector multiplet is written in terms of an 𝒩=1 vector superfield 𝑉 and an adjoint 𝑁=1 chiral superfield Φ. For 𝑈(𝑁𝑓) the notation will be Φ𝑉,. The action is compactly described in terms of the (real) Kähler potential 𝐾 and the (holomorphic) superpotential 𝑊.

For the gauge multiplet, we have 𝐾gge=1𝑔2Φ𝐴𝑒𝑉𝐴𝐵Φ𝐵+1̃𝑔2Φ𝑀𝑒𝑉𝑀𝑁Φ𝑁,𝑊gge=14𝑔2𝑊𝛼𝐴𝑊𝐴𝛼+14̃𝑔2𝑊𝛼𝑀𝑊𝑀𝛼,(2.1) written in terms of the usual chiral field strength spinor superfields 𝑊𝛼(𝑉) and 𝑊𝛼(𝑉) and adjoint representation matrices 𝑡𝐴𝐶𝐵 and 𝑡𝐿𝑁𝑀, for 𝑈(𝑁𝑐) and 𝑈(𝑁𝑓), respectively.

The hypermultiplet is written in terms of two chiral multiplets, denoted by 𝑄 and 𝑄. The 𝑈(𝑁𝑐)×𝑈(𝑁𝑓) representations for these superfields are 𝑄𝑎𝑚=𝑁𝑐,𝑁𝑓,𝑄𝑚𝑎=𝑁𝑐,𝑁𝑓.(2.2) The indices range according to 𝑎=1,,𝑁𝑐;  𝑚=1,,𝑁𝑓. It will be convenient to regard 𝑄 as an 𝑁𝑐×𝑁𝑓 matrix, and 𝑄 and an 𝑁𝑓×𝑁𝑐 matrix. Correspondingly, 𝑄 will be an 𝑁𝑓×𝑁𝑐 matrix and 𝑄 will be an 𝑁𝑐×𝑁𝑓 matrix. We can then write the Kähler potential as 𝐾mat=Tr𝑄𝑒𝑉𝑄𝑒𝑉𝑄+Tr𝑒𝑉𝑄𝑒𝑉=𝑄𝑚𝑎𝑒𝑉𝑎𝑏𝑄𝑏𝑛𝑒𝑉𝑛𝑚+𝑄𝑎𝑚𝑒𝑉𝑚𝑛𝑄𝑛𝑏𝑒𝑉𝑏𝑎.(2.3) Note that 𝑉 is expressed in terms of 𝑈(𝑁𝑐) fundamental representation generators 𝑡𝐴𝑎𝑏; a similar statement holds for 𝑉, except that the group is 𝑈(𝑁𝑓). The normalization convention that is assumed in the following is defined by Tr𝑡𝐴𝑡𝐵=(1/2)𝛿𝐴𝐵 for the fundamental representation. In the second step of (2.3), the indices have been written explicitly in order to make the matrix notation clear. Below, such details will be left implicit.

The superpotential is the minimal one, which preserves 𝑈(1)𝑅: 𝑊mat=2Tr𝑄Φ𝑄Φ2Tr𝑄𝑄.(2.4) Here, Φ and Φ are expressed in terms of 𝑈(𝑁𝑐) and 𝑈(𝑁𝑓) fundamental representation generators, respectively. It is easy to check gauge invariance, which acts holomorphically on the chiral superfields: 𝑄𝑒Λ𝑄𝑒Λ,Λ𝑄𝑒𝑄𝑒Λ,𝑒𝑉𝑒Λ𝑒𝑉𝑒Λ𝑉,𝑒𝑒Λ𝑒𝑉𝑒Λ,Φ𝑒ΛΦ𝑒Λ,ΛΦ𝑒Φ𝑒Λ,(2.5) where Λ and Λ are chiral superfields valued in the Lie algebras of 𝑈(𝑁𝑐) and 𝑈(𝑁𝑓), respectively.

The action is given by a Grassmann integral over superspace coordinates 𝜃𝛼,𝜃̇𝛼: 𝑑𝑆=4𝑥𝑑4𝜃𝐾gge+𝐾mat+𝑑2𝜃𝑊gge+𝑊mat+h.c..(2.6)

3. Lattice Construction

The EK approach [30] includes matter, in a generalization of earlier work by Kaplan et al., especially CKKU [811]. The Kaplan et al. “deconstruction lattice” approach was an outgrowth of dimensional deconstruction [34, 35]. In “Example 2” given by EK, quiver gauge theories with bifundamental matter were formulated. In this section, I generalize EK’s quiver construction to the case of (4,4) 2D super-QCD with bifundamental matter that is charged under a gauge group 𝑈(𝑁𝑐)×𝑈(𝑁𝑓). A minor modification of the (4,4) setup of CKKU [10] will prove necessary, due to the R-charges of the hypermultiplets that are being added to the theory. The other difficulty will be that we need to have 𝑈(𝑁𝑓) weakly gauged relative to 𝑈(𝑁𝑐). This will be addressed through extending to a quiver gauge theory 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)𝑛 and then breaking 𝑈(𝑁𝑓)𝑛 to its diagonal subgroup.

3.1. Outline

In the present theory, we begin with a matrix model that is the zero-dimensional (0d) reduction (the 0d reduction is obtained by treating all fields as independent of space-time coordinates) of 4d 𝒩=2 super-QCD with gauge group 𝑈((𝑁𝑐+𝑛𝑁𝑓)𝑁2) and fundamental matter. This theory is described by 𝐾gge=1𝑔2Φ𝐴𝑒𝑉𝐴𝐵Φ𝐵,𝑊gge=14𝑔2𝑊𝛼𝐴𝑊𝐴𝛼,𝐾mat=𝑄𝑎𝑒𝑉𝑎𝑏𝑄𝑏+𝑄𝑎𝑒𝑉𝑎𝑏𝑄𝑏,𝑊mat=2𝑄𝑎Φ𝑎𝑏𝑄𝑏.(3.1) Here, indices 𝐴,𝐵 correspond to the adjoint representation, whereas the indices 𝑎,𝑏 correspond to the fundamental representation. We will absorb the overall space-time volume 𝑉4=𝑑4𝑥 associated with the 4d 0d reduction into a redefinition of the coupling constant 𝑔2 and the matter fields 𝑄𝑄,. The resulting 0d theory (fixed to Wess-Zumino gauge) will be referred to as the mother theory, following Kaplan et al.

The next step is to perform an orbifold projection on the mother theory, in order to reduce it to the daughter theory. This “orbifolding” proceeds in two steps. First we orbifold by a 𝑍𝑛+1 symmetry of the mother theory, to break the gauge group according to 𝑈𝑁𝑐+𝑛𝑁𝑓𝑁2𝑁𝑈𝑐𝑁2𝑁×𝑈𝑓𝑁2𝑛.(3.2) Then we orbifold by a 𝑍𝑁×𝑍𝑁 symmetry of the mother theory to break the gauge group further, according to 𝑈𝑁𝑐𝑁2𝑁×𝑈𝑓𝑁2𝑛𝑁𝑈𝑐𝑁2𝑁×𝑈𝑓𝑛𝑁2.(3.3)

It is at this point that the trick to get a weakly gauged 𝑈(𝑁𝑓) comes in. At the final stage of the orbifolding—the RHS of (3.3)—the gauge coupling is universal, with its strength determined by the single coupling 𝑔2 that appears in the original 0d matrix model (3.1) and the overall lattice spacing that is determined by the choice of vacuum (dynamical lattice spacing) for the deconstruction—what was called the 𝑎-configuration in [5]. However, we now “Higgs” the subgroup 𝑈(𝑁𝑓)𝑛𝑁2 on the RHS of (3.3) with universal vacuum expectation values in bifundamental matter of this group, to break to the diagonal subgroup: 𝑈𝑁𝑓𝑛𝑁2𝑁𝑈𝑓𝑁2diag.(3.4) Then the coupling for the diagonal group is ̃𝑔2diag=𝑔2𝑛.(3.5) For large 𝑛 we obtain the desired result—a weakly gauged flavor group.

An alternative picture of this trick is the following. We may regard the factor 𝑛 as counting sites in a third dimension that has been deconstructed. Only the fields with 𝑈(𝑁𝑓) charge propagate in this third dimension. The 𝑈(𝑁𝑐) vector multiplet is stuck to the 2D subspace. It is very interesting that this mimics what happens in the D1/D5 brane system. There, the flavored fields propagate throughout the torus 𝑇4, since they correspond to strings that have one end on the D5 brane that wraps 𝑇4. The D1 branes are stuck at a point in 𝑇4, and so the purely colored fields do not propagate in the 𝑇4 direction. The difference here is that, to simplify the lattice construction, we have only a line interval in the extra dimension. It would be interesting to generalize the present construction to a 𝑈((𝑁𝑐+𝑛4𝑁𝑓)𝑁2) mother theory and to make a deconstructed 𝑇4 appear in the theory. (The more exotic case of a K3 manifold in the extra four dimensions could also be attempted.)

From this perspective we see that it is necessary to keep the third dimension small so that we never see the effects of the KK states. That is, we want only the 𝑈(𝑁𝑓)𝑁2diag states to be light enough to play a role at the scales that we study. In fact, this is exactly what happens in the D1/D5 system. Dimensional reduction of the D5 theory to the 2D intersection gives a volume suppression: 𝑔2D5reduc.𝑔2D14𝑠𝑉4,(3.6) where 𝑉4 is the volume of the torus 𝑇4 and 𝑠 is the string length. For 𝑉44𝑠, the 2D 𝑈(𝑁𝑓) is weakly gauged, and the KK states are supermassive on the scale (recall that in 2D, [𝑔D1]=1 and that this is the scale of non-KK modes) 𝑔D1.

In the discussion of Section 3.5 below, details associated with decoupling the KK states along the third dimension will be addressed.

3.2. Mother Theory

In 𝒩=1 superfield notation, the mother theory is the 0d reduction of (3.1). It is straightforward to work out the 0d reduction of the component field action in the mother theory. I denote component fields (in Wess-Zumino gauge): 𝑣𝑉=𝜇,𝜆,𝐹𝜆,𝐷,Φ=(𝜙,𝜓,𝐺),𝑄=(𝑄,𝜒,𝐹),𝑄=𝑄,𝜒,.(3.7) The result, after euclideanization, is: 𝑆gge=12𝑔2𝑣Tr𝜇,𝑣𝜈𝑣𝜇,𝑣𝜈+2𝑔2𝑣Tr𝜇𝑣,𝜙𝜇,𝜙+1𝑔2𝐷Tr2+2𝐷𝜙,𝜙+2𝑔2Tr𝐺2𝐺+𝑔2TrΨ𝑣𝜇𝛾𝜇+2,Ψ2𝑖𝑔2𝜆Tr𝜓,𝜙𝜙,𝜓𝜆,S(3.8)mat=𝑄𝑣𝜇𝑣𝜇𝑄+𝐹𝐹+𝑄𝐷𝑄𝑄𝑣𝜇𝑣𝜇𝑄+𝐹𝐹𝑄𝑄𝐷+2+𝐹𝜙𝑄+𝑄𝐺𝑄+𝑄𝜙𝐹+h.c.Λ𝑣𝜇𝛾𝜇Λ2𝜒𝜙𝜒+𝜒𝜙𝜒+𝜒𝜓𝑄+𝑄𝜓𝜒+𝑄𝜓𝜒+𝜒𝜓𝑄+𝑖2𝑄𝜆𝜒𝜒𝑄𝜆𝑄𝜒𝜆+𝑄𝜆.𝜒(3.9) Here, the following notations are used (𝛼=1,2):𝜆Ψ=𝛼𝜓̇𝛼,𝜓Ψ=𝛼,𝜆̇𝛼𝜒,Λ=𝛼𝜒̇𝛼,Λ=𝜒𝛼,𝜒̇𝛼,𝛾𝜇=0𝜎𝜇𝜎𝜇0,𝜎𝜇=,𝜎,𝑖𝜎𝜇=𝜎,𝑖,Tr𝑇𝐴𝑇𝐵=12𝛿𝐴𝐵.(3.10) It is not difficult to relate (3.8) to the mother theory action of CKKU. The translation is 𝑧1=12𝑣1+𝑖𝑣2,𝑧2=12𝑣3+𝑖𝑣4,𝑧3=𝑖𝜙,𝜉Ψ=2,𝜉1,𝜉3,,𝜆𝜓Ψ=1,𝜓2,𝜒,𝜓3.(3.11) Note that 𝜆,  𝜒 here are not the two-component fermions 𝜆𝛼,𝜒𝛼 of the 4d notation (3.10). The 𝑈(1)4 subgroup of 𝑆𝑂(6)×𝑆𝑈(2)𝑅 that CKKU chooses for their orbifold procedure is then𝑞1=Σ1,2=1𝛾4𝑖1,𝛾2,𝑞2=Σ3,4=1𝛾4𝑖3,𝛾4,𝑞31=2𝑄𝑅,𝑞4=𝑇3𝑅.(3.12) I have expressed the last two charges in terms of the conventional 𝑆𝑈(2)𝑅×𝑈(1)𝑅  𝑅-symmetry of the 4d 𝒩=2 theory. The 𝑈(1)𝑅 generator 𝑄𝑅 is normalized such that gluinos (denoted 𝜆𝛼,𝜓̇𝛼 in (3.8)–(3.10)) have 𝑄𝑅=1. 𝑇3𝑅=(1/2)𝜎3 is the diagonal generator of 𝑆𝑈(2)𝑅. The charges of all gauge multiplet fields are summarized in Table 1. The charges of all hypermultiplet fields are summarized in Table 2. For the fermions, the notation is related to (3.8)–(3.10) by Ψ1,Ψ2,Ψ3,Ψ4=𝜆1,𝜆2,𝜓̇1𝜓̇2,Ψ1,Ψ2,Ψ3,Ψ4=𝜓1,𝜓2,𝜆̇1𝜆̇2(3.13) and a similar translation for Λ,Λ. The upper placement and lower placement of indices are significant because of the implicit spinor sums that are in (3.8)–(3.9). For example, in the last line of (3.9), one has the term 𝑄𝜆𝜒=𝑄𝜆𝛼𝜒𝛼=𝑄𝜖12𝜆2𝜒1+𝜖21𝜆1𝜒2=𝑄Ψ2Λ1Ψ1Λ2.(3.14) Here, the conventions of [33] have been used: 𝜖21=𝜖12=1,  𝜖12=𝜖21=1. These details were important in writing down the explicit daughter theory action that is given in Appendix B.

3.3. Orbifolding Details
3.3.1. Projections, Generally

Denote 𝑈((𝑁𝑐+𝑛𝑁𝑓)𝑁2) indices collectively by 𝑆𝐼𝑚1𝑚2,𝐼1,,𝑁𝑐+𝑛𝑁𝑓,𝑚1,𝑚2{0,,𝑁}.(3.15) The domain of the index 𝐼 should be thought of as follows: 𝐼=1,,𝑁𝑐;𝑁𝑐+1,,𝑁𝑐+𝑁𝑓;𝑁𝑐+𝑁𝑓𝑁+1,,𝑐+𝑁𝑓+𝑁𝑓𝑁;;𝑐+(𝑛1)𝑁𝑓𝑁+1,,𝑐+(𝑛1)𝑁𝑓+𝑁𝑓.(3.16) The interpretation is in terms of a block diagonal matrix, with an 𝑁𝑐×𝑁𝑐 block, followed by 𝑛 blocks of size 𝑁𝑓×𝑁𝑓. The index 𝑆 then indicates, say, that the entries of the 𝑁𝑐×𝑁𝑐 matrix are themselves 𝑁2×𝑁2 matrices, and so on. In what follows, “diag” will indicate a block diagonal matrix, with only block entries given explicitly. For example the unit matrix in the mother theory is given by 𝟏(𝑁𝑐+𝑛𝑁𝑓)𝑁2𝟏=diag𝑁𝑐𝑁2,𝟏𝑁𝑓𝑁2,,𝟏𝑁𝑓𝑁2,(3.17) with 𝑛 entries of 𝟏𝑁𝑓𝑁2. Other matrices of this form follow.

Introduce “clock operators” that involve roots of unity 𝜔𝑘exp(2𝜋𝑖/𝑘): 𝟏𝑃=diag𝑁𝑐𝑁2,𝜔𝑛+1𝟏𝑁𝑓𝑁2,,𝜔𝑛𝑛+1𝟏𝑁𝑓𝑁2,Ω𝑁=diag1,𝜔𝑁,,𝜔𝑁𝑁1,𝐶𝑁1,𝑘=𝟏𝑘Ω𝑁𝟏𝑁,𝐶𝑁2,𝑘=𝟏𝑘𝟏𝑁Ω𝑁,𝐶1𝑁𝐶=diag1,𝑁𝑐𝑁,𝐶1,𝑁𝑓𝑁,,𝐶1,𝑁𝑓𝑁,𝐶2𝑁𝐶=diag2,𝑁𝑐𝑁,𝐶2,𝑁𝑓𝑁,,𝐶2,𝑁𝑓𝑁.(3.18) Orbifold projections for any field 𝒜 are defined by 𝒜𝜔𝑠𝑛+1𝑃𝒜𝑃,𝒜𝜔𝑟1𝑁𝒞1𝑁𝒜𝒞𝑁1,𝒜𝜔𝑟2𝑁𝒞2𝑁𝒜𝒞𝑁2.(3.19) The charges 𝑠,𝑟1,𝑟2 will correspond to, respectively, (𝑛+1)-ality, 𝑁-ality, 𝑁-ality. The origin of the 𝑍𝑛+1 symmetry in the mother theory—corresponding to the (𝑛+1)-ality—will be discussed shortly. The two 𝑁-alities correspond to a 𝑍𝑁×𝑍𝑁 subgroup of the 𝑆𝑂(6)×𝑆𝑈(2)𝑅 symmetry of the mother theory.

To understand the effect of (3.19), it is best to look at it in stages. For the 𝑍𝑛+1 projection, 𝒜𝑠=0𝑁𝐚𝐝𝐣𝑈𝑐𝑁2𝑁,1,,11,𝐚𝐝𝐣𝑈𝑓𝑁2𝑁,1,,11,1,,𝐚𝐝𝐣𝑈𝑓𝑁2,𝒜𝑠=1𝑁𝑐𝑁2,𝑁𝑓𝑁2,1,,11,𝑁𝑓𝑁2,𝑁𝑓𝑁2,1,,11,1,,1,𝑁𝑓𝑁2,𝑁𝑓𝑁2𝑁𝑐𝑁2,1,,1,𝑁𝑓𝑁2,(3.20) and 𝑠=1 is conjugate to the latter. This yields “sites” and “links” of the quiver gauge theory (3.2). This structure will persist in the lattice theory and its continuum limit.

The minimal coupling superpotential of the mother theory has the 𝑈(1) global symmetry 𝑄𝑒𝑖𝛼𝑄,𝑄𝑒𝑖𝛼𝑄,(3.21) with all other fields neutral and all components in 𝑄𝑄, transforming identically (it is not an R-symmetry). This is the symmetry that we use for the 𝑍𝑛+1 orbifold. That is, we assign 𝑠=1 to all components of 𝑄, 𝑠=1 to all components of 𝑄, and 𝑠=0 to all components of 𝑉,Φ. In this way, 𝑄𝑄, will be bifundamentals (“links”) of 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)𝑛 quiver gauge theory, whereas 𝑉,Φ will be adjoints (“sites”). The notion of “links” and “sites” used here is distinct from that associated with the 2D lattice that is described next. Note that because (3.21) is not an R-symmetry (i.e., it commutes with the supercharges of the mother theory), the 𝑍𝑛+1 orbifold projection leaves all eight supercharges intact.

The 𝑍𝑁×𝑍𝑁 projections involve clock matrices 𝒞𝑁1,2, which only act on the “site” indices 𝑚1,𝑚2 of (3.15). They have the usual effect of the deconstruction lattice formulation. Label any of the fields in the decomposition (3.20) by 𝒜𝑚1,𝑚2;𝑛1,𝑛2, ignoring indices of the surviving gauge group. Then the surviving components of 𝒜𝑚1,𝑚2;𝑛1,𝑛2 after the 𝑍𝑁×𝑍𝑁 orbifold are those that satisfy 𝑚1𝑛1+𝑟1=0mod𝑁,𝑚2𝑛2+𝑟2=0mod𝑁.(3.22) This yields site, horizontal link, vertical link, and diagonal link interpretations, depending on 𝑟1,𝑟2. The fields are then labeled by site indices 𝑚(𝑚1,𝑚2). Next I discuss particulars with respect to the various fields of the mother theory.

3.3.2. Daughter Theory Gauge Action

A problem arises for the construction of CKKU [10] when we include hypermultiplets. The scalar components are neutral with respect to the 𝑆𝑂(6) global symmetry of the mother theory, which decomposes to 𝑆𝑂(4)×𝑈(1)𝑅 in the 4d theory. In the notation of CKKU, 𝑞1=𝑞2=𝑞3=0. On the other hand these scalars transform as doublets (𝑄,𝑄) under 𝑆𝑈(2)𝑅, and as a consequence 𝑞4=1/2 for 𝑄𝑄,. The 𝑁-alities defined by CKKU are 𝑟1=𝑞1+𝑞4,𝑟2=𝑞2+𝑞4,(3.23) which would lead to half-integral 𝑟1,𝑟2 for the scalars 𝑄𝑄,. Nonintegral 𝑁-alities do not make sense in the lattice interpretation of the orbifolded theory. Therefore we must modify the 𝑁-ality assignments of CKKU in order to include hypermultiplets. It will be seen that this is easily accomplished. The lattice that is obtained is quite similar to the one of CKKU. It is of particular importance that two supercharges are preserved exactly.

In the modification, we want to leave the 𝑁-alities of link bosons 𝑧1,𝑧2 unchanged, since these must ultimately get a vacuum expectation value that links neighboring sites. (Actually, it is an interesting question whether or not a dynamical lattice spacing can be associated with, say, diagonal link bosons. I will not pursue this here.) We must choose 𝑟1,𝑟2 such that all fields have integer 𝑁-ality. Also we would like to preserve two supercharges, as in the pure gauge construction of CKKU. According to the CKKU rubric, we therefore must choose 𝑟1,𝑟2 such that two components of the fermions are neutral. Here we choose to keep Ψ4=𝜆 neutral, as in the CKKU construction (other choices are of course possible, but lead to similar results, due to the symmetries of the mother theory). Then it is easy to show (cf. Appendix A) that the unique choice that satisfies all of our requirements is 𝑟1=𝑞1𝑞3,𝑟2=𝑞2𝑞3.(3.24) In addition to Ψ4, the fermion component Ψ3=𝜒 is 𝑟1,𝑟2 neutral. The charges for the vector multiplet are summarized in Table 1.

Relative to the formulation of CKKU, only the following five fields of the gauge multiplet change their nature: 𝑧3𝜉:site()diagonallink,2:̂𝑒1link̂𝑒2𝜉link,1:̂𝑒2link̂𝑒1𝜓link,𝜒:diagonallinksite,3:site()diagonallink.(3.25) This merely leads to modest changes in the site labels for the daughter theory action of CKKU, their (1.2) and (1.4). These changes are all obvious from the 𝑟1,𝑟2 assignments of Table 1. For instance, in their bosonic action one replaces (CKKU uses the notation 𝑧 while 𝑧 is used here) 𝑧3,𝑚,𝑧3,𝑚𝑧3,𝑚𝑧3,𝑚+̂𝑒1+̂𝑒2𝑧3,𝑚𝑧3,𝑚̂𝑒1̂𝑒2,(3.26) to take into account that 𝑧3,𝑧3 are now /+ diagonal link fields. The fermion action is still of the form 𝑆𝐹,𝑔=22𝑔2𝑚ΔTr𝑚𝜆,𝑧𝑎,𝜓𝑎Δ𝑚𝜒,𝑧𝑎,𝜉𝑎+𝜖𝑎𝑏𝑐Δ𝑚𝜓𝑎,𝑧𝑏,𝜉𝑐,(3.27) with Δ(𝐴,𝐵,𝐶)=𝐴𝐵𝐶𝐴𝐶𝐵 and site labels assigned according to the nature of the fields that appear.

Referring to Table 1, we note that there is a twofold degeneracy for the 𝑟1,𝑟2 charges among the fermions. The reason for this is that the orbifold charges (3.24) do not involve the 𝑆𝑈(2)𝑅 diagonal generator 𝑞4. Thus the 𝑆𝑈(2)𝑅 symmetry of the mother theory is preserved, unlike that which occurs in the CKKU construction. Since the (4,4) gluinos fall into doublets of 𝑆𝑈(2)𝑅, we are guaranteed to have the twofold degeneracy with respect to 𝑟1,𝑟2.

Note also that the 𝑟1,𝑟2 neutral fermions are those that have 𝑞1=𝑞2=𝑞3. It follows that the two supercharges that are preserved in the daughter theory are those that have 𝑞1=𝑞2=𝑞3.

3.3.3. Daughter Theory Matter Action

Having explained how the gauge action is modified, we next turn to the matter action. The daughter theory is obtained in a simple application of the orbifold procedure to the mother theory (3.9), as determined by the 𝑟1,𝑟2 assignments that appear in Table 2. Due to the CKKU calculus, we are assured to obtain the correct classical continuum limit, just as in the EK examples.

We have already seen from the discussion of the daughter theory gauge action that there are two supercharges that are neutral with respect to the 𝑍𝑁×𝑍𝑁 charges 𝑟1,𝑟2. This symmetry of the matter mother theory action will be an exact supersymmetry of the matter daughter theory as well. Upon inspection of Table 2, one sees that the 𝑟1,𝑟2 neutral fermions are once again those that have 𝑞1=𝑞2=𝑞3. It follows that the two supercharges that are preserved in the daughter theory matter action are those that have 𝑞1=𝑞2=𝑞3. It is no accident that this is identical to what occurs in the daughter theory gauge action: the supercharges are inherited from the mother theory. This illustrates the usefulness of the orbifold technique of CKKU.

Straightforward manipulations yield the daughter theory matter action. One merely writes out the fermion components in (3.9) explicitly, reexpresses 𝑣𝜇 in terms of 𝑧𝑖,𝑧𝑖, and adds site labels as determined by the 𝑟1,𝑟2 charges given in Table 2. Because the result is somewhat lengthy, it has been relegated to Appendix B.

3.4. Higgsing Details

To “Higgs” the theory, such that only the 𝑈(𝑁𝑓)diag subgroup of 𝑈(𝑁𝑓)𝑛 survives at the scale 𝑔𝑐=𝑔𝑎 of the 𝑈(𝑁𝑐) gauge theory, we only require the application of the deconstruction idea to the 𝑈(𝑁𝑓)𝑛 quiver. This 1d quiver is similar to that considered in [34], in that it is an extra dimensional interval (in this case a third dimension), 𝑈(𝑁𝑓)1××𝑈(𝑁𝑓)𝑛, and it is not necessary to rework all the details.

In terms of 𝒩=1 superfields, the quiver is described by the 0d reduction of the theory with 𝐾mat=𝑛1𝑖=1𝑄Tr𝑖𝑒𝑉𝑖𝑄𝑖𝑒𝑉𝑖+1+𝑄𝑖𝑒𝑉𝑖+1𝑄𝑖𝑒𝑉𝑖,𝑊mat=2𝑛1𝑖=1𝑄Tr𝑖Φ𝑖𝑄𝑖𝑄𝑖Φ𝑖+1𝑄𝑖.(3.28) Formally, this is quite similar to the quiver theory studied in [36]. I do not write 𝐾gge,𝑊gge since it is just an 𝑛-fold replication of terms of the form (3.1). Holomorphic gauge invariance is given by 𝑄𝑖𝑒Λ𝑖𝑄𝑖𝑒Λ𝑖+1,𝑄𝑖𝑒Λ𝑖+1𝑄𝑖𝑒Λ𝑖,Φ𝑖𝑒Λ𝑖Φ𝑖𝑒Λ𝑖,𝑒𝑉𝑖𝑒Λ𝑖𝑒𝑉𝑖𝑒Λ𝑖.(3.29)

One then gives an expectation value to the (𝑁𝑓(𝑖),𝑁𝑓(𝑖+1)),  𝑖=1,,𝑛1, bifundamentals and their conjugates: 𝑄𝑖𝑄=𝑖=12𝑎3.(3.30) Then, for instance, the quadratic terms in the 2D Lagrangian for the gauge bosons are (here I am hiding all the details of the 2D lattice theory and just emphasizing the quiver in the third dimension. The modes of the lattice theory that are getting mass here are just the 𝑧1,𝐦,𝑧2,𝐦 that transform as adjoints of the 𝑈(𝑁𝑓)𝑛 group, excepting the combination corresponding to 𝑈(𝑁𝑓)diag) 𝑛1𝑖=1𝑔24𝑎23𝐴𝜇,𝑚𝑖+1𝐴𝑖𝜇,𝑚2,(3.31) where a contraction over the 4d index 𝜇 and 𝑈(𝑁𝑓) index 𝑚=1,,𝑁2𝑓 is implied. The scaling 𝐴̂𝑔𝐴 has been performed to make the kinetic terms for gauge bosons canonical. Here, ̂𝑔=𝑔𝑎2 is the dimensionless coupling, that is, the coupling of the matrix model expressed in units of the 2D lattice. It follows immediately from the considerations of [34] that only 𝑈(𝑁𝑓)diag has a massless gauge boson. All other modes are quanta with configuration energies of order 1/(𝑛𝑎3), corresponding to discrete momenta in the third dimension. To be precise, the spectrum is 𝑀2𝑛=̂𝑔2𝑎23sin2𝑗𝜋𝑛,𝑗=0,,𝑛1.(3.32) The radius 𝑅 of this third deconstructed dimension and the KK mass scale 𝑀 are therefore 𝑅𝑛𝑎3𝜋̂𝑔,𝑀=𝑅.(3.33) The effective gauge coupling of the 𝑈(𝑁𝑓)diag theory is given by (3.5).

3.5. Decoupling KK States

The condition that the KK states decouple from the 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag gauge theory is just 𝑀𝑔𝑐=𝑔𝑎. Various realizations of this could be imagined. A strong one is that we set the KK scale at the UV cutoff of the 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag gauge theory: 𝑅𝑎. This translates into 𝑛=𝑎̂𝑔𝑎3=𝑔𝑐𝑎2𝑎3.(3.34) Thus as we take the continuum limit 𝑎0 in the 2D 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag gauge theory, with 𝑛,𝑔𝑐 held fixed, we have the scaling 𝑎3𝑎2. This would decouple the effects of the 𝑈(𝑁𝑓)𝑛 quiver at the UV scale of the 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag gauge theory, and just represents a slightly different UV completion that should not have physical consequences—based on universality arguments.

A less aggressive prescription is to take 𝑔𝑐𝑅 fixed and small. This should also decouple the KK states before important 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)diag physics sets in. This translates into 𝑎3𝑎=𝑓𝑛,𝑓1.(3.35) Holding the factors 𝑓,𝑛 fixed, we see that a scaling 𝑎3𝑎 is prescribed as the continuum limit is taken.

4. Application

Here I mention one possible application of the lattice theory. Recently, Chen and Tong have studied the D1/D5 effective worldsheet instanton partition function on the Higgs branch. In the gauge theory one looks at the distribution of instanton size 𝜌 and orientational modes Ω, where the latter are points on 𝑆3. Indeed, it is found that the distribution has the AdS3×𝑆3 geometry in the sector with first Chern class 𝑘=1, that is, a unit of winding in the 𝑈(1)diag of the color group.

In a numerical study of this phenomenon, one would build up a histogram in the 𝑘=1 topological sector. Twisted boundary conditions could be imposed to force nontrivial topology for the gauge fields. The histogram would count configurations with a given instanton size 𝜌 and orientation Ω. If the weight is identical to the AdS3×𝑆3 density, it would provide evidence of the correspondence. In particular, it is interesting to explore the correspondence for intermediate values of 𝑁𝑐, given the current fashion for applying AdS/QCD ideas to real-world QCD, where 𝑁𝑐=3.

It would also be interesting to explore the correspondence at finite temperature, since continuum methods start to break down if the temperature is too far from zero. The recent results of Rey and Hikida for small 't Hooft coupling and finite temperature [32] provide continuum results that could be compared to. Finally, one would like to study correlation functions that are not BPS saturated. Again, continuum methods are generally unreliable in that case.

5. Conclusions

In this paper I have generalized the EK construction to 2D (4,4) gauge theories. I have specialized to a 𝑈(𝑁𝑐)×𝑈(𝑁𝑓)𝑛 quiver theory. Next, I showed how to treat the 𝑈(𝑁𝑓)𝑛 quiver as a deconstructed third dimension and how to obtain a weakly coupled 2D remnant 𝑈(𝑁𝑓)diag, mimicking what really happens in the D1/D5 brane system. I described a simple test of AdS3/𝐶𝐹𝑇2 that could be conducted numerically. It is worth noting that it should be straightforward to include the Fayet-Iliopoulos (FI) terms in the mother theory, and thus in the lattice theory, indeed, this has already been illustrated by EK in their “Example 2.”

Work in progress includes a careful study of renormalization in the lattice theory, the number of counterterms that need to be fine-tuned, their exact calculation in perturbation theory (the lattice theory is super-renormalizable since the coupling has positive mass dimension), and a numerical study of the correspondence. Renormalization of the theory, such as has been studied in [37], is certainly a pressing question in the presence of matter. It remains to be seen the extent to which complex phase problems of the pure gauge lattice theory [38, 39] persist once matter is introduced. If FI terms are introduced and the theory is studied on the Higgs branch, the complex phase may be less of a problem.

Finally, it is of some interest to work out a superfield description of the daughter theory in this model. This would be useful in a super-Feynman diagram perturbative analysis, as well as for understanding the renormalizations to the tree-level action that are possible.


A. Uniqueness of 𝑟1,𝑟2 with Conditions Imposed

The conditions that we will impose are the following. (i)The link bosons 𝑧1,𝑧2 should have (𝑟1,𝑟2)=(1,0) and (0,1), respectively. (ii)The fermion component Ψ4 should have (𝑟1,𝑟2)=(0,0). (iii)At least one other fermion component in Ψ,Ψ should have (𝑟1,𝑟2)=(0,0). (iv)All fields should have integral values of 𝑟1,𝑟2.

It is completely general to write 𝑟1=4𝑖=1𝑐1𝑖𝑞𝑖,𝑟2=4𝑖=1𝑐2𝑖𝑞𝑖.(A.1) Condition (i) yields immediately 𝑐11=𝑐22=1,  𝑐21=𝑐12=0. Condition (ii) gives 𝑐14=𝑐13+1,𝑐24=𝑐23+1. Thus the charges reduce to 𝑟1=𝑞1+𝑞4+𝑐13𝑞3+𝑞4,𝑟2=𝑞2+𝑞4+𝑐23𝑞3+𝑞4.(A.2) It is easy to see from Table 2 that the components of the matter fermions Λ,Λ have (𝑞1+𝑞4)=±1/2,  (𝑞2+𝑞4)=±1/2,  (𝑞3+𝑞4)=±1/2. It follows that we must take 𝑐13 and 𝑐23 to be odd integers, in order to satisfy condition (iv). The remaining condition (iii) then has a unique solution, as can be checked from Table 1. It is 𝑐13=𝑐23=1 which gives (3.24).

B. Daughter Theory Matter Action

The action can be expressed as three terms, 𝑆mat=𝑆1+𝑆2+𝑆3,(B.1) where 𝑆1=𝑄𝑚𝑧𝑖,𝑚𝑧𝑖,𝑚+̂𝑒𝑖+𝑧𝑖,𝑚𝑧𝑖,𝑚̂𝑒𝑖𝑄𝑚𝑄𝑚𝑧𝑖,𝑚𝑧𝑖,𝑚+̂𝑒𝑖+𝑧𝑖,𝑚𝑧𝑖,𝑚̂𝑒𝑖𝑄𝑚+𝐹𝑚𝐹𝑚+̂𝑒1+̂𝑒2+𝐹𝑚𝐹𝑚̂𝑒1̂𝑒2+𝑄𝑚𝐷𝑚𝑄𝑚𝑄𝑚𝐷𝑚𝑄𝑚+2𝐹𝑚𝜙𝑚̂𝑒1̂𝑒2𝑄𝑚+𝑄𝑚𝐺𝑚𝑄𝑚+𝑄𝑚𝜙𝑚𝐹𝑚+̂𝑒1+̂𝑒2+𝑄𝑚𝜙𝑚𝐹𝑚̂𝑒1̂𝑒2+𝑄𝑚𝐺𝑚𝑄𝑚+𝐹𝑚𝜙𝑚+̂𝑒1+̂𝑒2𝑄𝑚,𝑆2=2Λ1,𝑚𝑧1,𝑚̂𝑒2Λ4,𝑚̂𝑒1̂𝑒2+𝑧2,𝑚̂𝑒2Λ3,𝑚+Λ2,𝑚𝑧1,𝑚̂𝑒1Λ3,𝑚𝑧2,𝑚̂𝑒1Λ4,𝑚̂𝑒1̂𝑒2Λ3,𝑚𝑧1,𝑚+̂𝑒1+̂𝑒2Λ2,𝑚+̂𝑒2𝑧2,𝑚+̂𝑒1+̂𝑒2Λ1,𝑚+̂𝑒1Λ4,𝑚𝑧1,𝑚Λ1,𝑚+̂𝑒1+𝑧2,𝑚Λ2,𝑚+̂𝑒2,𝑆3=2Λ1,𝑚𝜙𝑚̂𝑒2Λ1,𝑚+̂𝑒1+Λ2,𝑚𝜙𝑚̂𝑒1Λ2,𝑚+̂𝑒2+Λ3,𝑚𝜙𝑚+̂𝑒1+̂𝑒2Λ3,𝑚+Λ4,𝑚𝜙𝑚Λ4,𝑚̂𝑒1̂𝑒2Λ1,𝑚Ψ2,𝑚̂𝑒2Λ2,𝑚Ψ1,𝑚̂𝑒1𝑄𝑚+𝑄𝑚Ψ1,𝑚Λ1,𝑚+̂𝑒1+Ψ2,𝑚Λ2,𝑚+̂𝑒2𝑄𝑚Ψ4,𝑚Λ3,𝑚Ψ3,𝑚Λ4,𝑚̂𝑒1̂𝑒2+Λ3,𝑚Ψ3,𝑚+̂𝑒1+̂𝑒2+Λ4,𝑚Ψ4,𝑚𝑄𝑚+𝑖2𝑄𝑚Ψ2,𝑚Λ1,𝑚+̂𝑒1Ψ1,𝑚Λ2,𝑚+̂𝑒2Λ3,𝑚Ψ4,𝑚+̂𝑒1+̂𝑒2Λ4,𝑚Ψ3,𝑚𝑄𝑚Λ1,𝑚Ψ1,𝑚̂𝑒2+Λ2,𝑚Ψ2,𝑚̂𝑒1𝑄𝑚+𝑄𝑚Ψ3,𝑚Λ3,𝑚+Ψ4,𝑚Λ4,𝑚̂𝑒1̂𝑒2.(B.2) Here, the site indices 𝑚 are implicitly summed.


This work was supported by the U.S. Department of Energy under Grants No. DE-FG02-94ER-40823 and DE-FG02-08ER-41575.