A Deconstruction Lattice Description of the D1/D5 Brane World-Volume Gauge Theory

Abstract

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(𝑁_𝑐)×𝑈(𝑁_𝑓), and 𝑈(𝑁_𝑓) 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 𝑁_𝑐 gauge theory seems to afford a window on quantum gravity, through the AdS/CFT correspondence [1–4]. 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 𝑁_𝑐, 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 [7–16]. (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 [21–26]. 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 [27–29] where 𝑁_𝑓 flavors of matter are contained in hypermultiplets, and transform in the fundamental representation of the 𝑈(𝑁_𝑐) gauge group. These flavors are minimally coupled, so that there would be a 𝑈(𝑁_𝑓) flavor symmetry. In actuality, the 𝑈(𝑁_𝑓) symmetry is weakly gauged, and the flavors are bifundamentals of 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓).

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(𝑁_𝑐)×𝑈(𝑁_𝑓). In this context, we want the flexibility to choose 𝑁_𝑐 and 𝑁_𝑓 independently. Also, as already been mentioned, the flavor group 𝑈(𝑁_𝑓) should be weakly gauged; that is, the corresponding gauge couplings should satisfy 𝑔_𝑁𝑓≪𝑔_𝑁𝑐 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(𝑁_𝑐) and 𝑈(𝑁_𝑓), with 𝑁_𝑐≠𝑁_𝑓. The nontrivial task is to devise a trick to make the 𝑈(𝑁_𝑓) 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(𝑁_𝑐)×𝑈(𝑁_𝑓) (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(𝑁_𝑐)×𝑈(𝑁_𝑓)^𝑛 and bifundamental matter is obtained. The 𝑈(𝑁_𝑓)^𝑛 quiver is introduced in order to obtain a weakly coupled diagonal subgroup, 𝑈(𝑁_𝑓)_diag. It will be explained how this can be interpreted in terms of a deconstructed third dimension. The 𝑈(𝑁_𝑐) 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(𝑁_𝑐) gauge fields are stuck to the D1 brane, and 𝑔_𝑁𝑓≪𝑔_𝑁𝑐 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 𝑍_𝑁×𝑍_𝑁 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 𝑍_𝑁×𝑍_𝑁 orbifold, relative to CKKU [10]. Finally, I describe how to break to 𝑈(𝑁_𝑓)_diag, and the conditions that must be satisfied for the Kaluza-Klein (KK) states of the corresponding third dimension to decouple from the 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)_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 𝑁_𝑐 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(𝑁_𝑐)  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(𝑁_𝑓) the notation will be 𝑉,Φ. The action is compactly described in terms of the (real) Kähler potential 𝐾 and the (holomorphic) superpotential 𝑊.

For the gauge multiplet, we have 𝐾_gge=1𝑔²Φ^†𝐴𝑒^𝑉_𝐴𝐵Φ_𝐵+1𝑔̃²Φ^†𝑀𝑒𝑉_𝑀𝑁𝑊_ggeΦ_𝑁,=14𝑔²𝑊^𝛼𝐴𝑊^𝐴𝛼+14𝑔̃²𝑊^𝛼𝑀𝑊^𝑀𝛼,(2.1) written in terms of the usual chiral field strength spinor superfields 𝑊^𝛼(𝑉) and 𝑊^𝛼(𝑉) and adjoint representation matrices 𝑡^𝐴𝐶𝐵 and 𝑡^𝐿𝑁𝑀, for 𝑈(𝑁_𝑐) and 𝑈(𝑁_𝑓), respectively.

The hypermultiplet is written in terms of two chiral multiplets, denoted by 𝑄 and 𝑄. The 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓) representations for these superfields are 𝑄_𝑎𝑚=𝑁_𝑐,𝑁_𝑓,𝑄^𝑚𝑎=𝑁_𝑐,𝑁_𝑓.(2.2) The indices range according to 𝑎=1,…,𝑁_𝑐;  𝑚=1,…,𝑁_𝑓. It will be convenient to regard 𝑄 as an 𝑁_𝑐×𝑁_𝑓 matrix, and 𝑄 and an 𝑁_𝑓×𝑁_𝑐 matrix. Correspondingly, 𝑄^† will be an 𝑁_𝑓×𝑁_𝑐 matrix and 𝑄^† will be an 𝑁_𝑐×𝑁_𝑓 matrix. We can then write the Kähler potential as 𝐾_mat=Tr𝑄^†𝑒^𝑉𝑄𝑒⁻𝑉+Tr𝑄^†𝑒𝑉𝑄𝑒^−𝑉=𝑄^†_𝑚𝑎𝑒^𝑉_𝑎𝑏𝑄_𝑏𝑛𝑒⁻𝑉_𝑛𝑚+𝑄^†_𝑎𝑚𝑒𝑉_𝑚𝑛𝑄_𝑛𝑏𝑒^−𝑉_𝑏𝑎.(2.3) Note that 𝑉 is expressed in terms of 𝑈(𝑁_𝑐) fundamental representation generators 𝑡^𝐴𝑎𝑏; a similar statement holds for 𝑉, except that the group is 𝑈(𝑁_𝑓). The normalization convention that is assumed in the following is defined by Tr𝑡^𝐴𝑡^𝐵=(1/2)𝛿^𝐴𝐵 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: 𝑊_mat=√2Tr𝑄Φ𝑄−√2Tr𝑄Φ𝑄.(2.4) Here, Φ and Φ are expressed in terms of 𝑈(𝑁_𝑐) and 𝑈(𝑁_𝑓) fundamental representation generators, respectively. It is easy to check gauge invariance, which acts holomorphically on the chiral superfields: 𝑄⟶𝑒^Λ𝑄𝑒⁻Λ,𝑄⟶𝑒Λ𝑄𝑒_−Λ,𝑒^𝑉⟶𝑒^−Λ†𝑒^𝑉𝑒^−Λ,𝑒𝑉⟶𝑒⁻Λ^†𝑒𝑉𝑒⁻Φ⟶𝑒_ΛΛ,Φ𝑒^−Λ,Φ⟶𝑒ΛΦ𝑒⁻Λ,(2.5) where Λ and Λ are chiral superfields valued in the Lie algebras of 𝑈(𝑁_𝑐) and 𝑈(𝑁_𝑓), respectively.

The action is given by a Grassmann integral over superspace coordinates 𝜃^𝛼,𝜃_̇𝛼: 𝑆=𝑑⁴𝑥𝑑⁴𝜃𝐾_gge+𝐾_mat+𝑑²𝜃𝑊_gge+𝑊_mat+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 [8–11]. 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(𝑁_𝑐)×𝑈(𝑁_𝑓). 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(𝑁_𝑓) weakly gauged relative to 𝑈(𝑁_𝑐). This will be addressed through extending to a quiver gauge theory 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)^𝑛 and then breaking 𝑈(𝑁_𝑓)^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((𝑁_𝑐+𝑛𝑁_𝑓)𝑁²) and fundamental matter. This theory is described by 𝐾_gge=1𝑔²Φ^†𝐴𝑒^𝑉_𝐴𝐵Φ_𝐵,𝑊_gge=14𝑔²𝐾_mat𝑊^𝛼𝐴𝑊^𝐴𝛼,=𝑄^†𝑎𝑒^𝑉_𝑎𝑏𝑄_𝑏+𝑄^𝑎𝑒^−𝑉_𝑎𝑏𝑄^†𝑏,𝑊_mat=√2𝑄^𝑎Φ_𝑎𝑏𝑄_𝑏.(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 𝑄,𝑄. 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 𝑍_𝑛+1 symmetry of the mother theory, to break the gauge group according to 𝑈𝑁_𝑐+𝑛𝑁_𝑓𝑁²⟶𝑈𝑁_𝑐𝑁²×𝑈𝑁_𝑓𝑁²^𝑛.(3.2) Then we orbifold by a 𝑍_𝑁×𝑍_𝑁 symmetry of the mother theory to break the gauge group further, according to 𝑈𝑁_𝑐𝑁²×𝑈𝑁_𝑓𝑁²^𝑛⟶𝑈𝑁_𝑐^𝑁2×𝑈𝑁_𝑓^𝑛𝑁2.(3.3)

It is at this point that the trick to get a weakly gauged 𝑈(𝑁_𝑓) 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(𝑁_𝑓)^𝑛𝑁2 on the RHS of (3.3) with universal vacuum expectation values in bifundamental matter of this group, to break to the diagonal subgroup: 𝑈𝑁_𝑓^𝑛𝑁2⟶𝑈𝑁_𝑓^𝑁2_diag.(3.4) Then the coupling for the diagonal group is 𝑔̃^2diag=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(𝑁_𝑓) charge propagate in this third dimension. The 𝑈(𝑁_𝑐) 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((𝑁_𝑐+𝑛⁴𝑁_𝑓)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(𝑁_𝑓)^𝑁2_diag 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: 𝑔²_D5reduc.𝑔²_D1≈ℓ^4𝑠𝑉₄,(3.6) where 𝑉₄ is the volume of the torus 𝑇⁴ and ℓ_𝑠 is the string length. For 𝑉₄≫ℓ^4𝑠, the 2D 𝑈(𝑁_𝑓) is weakly gauged, and the KK states are supermassive on the scale (recall that in 2D, [𝑔_D1]=1 and that this is the scale of non-KK modes) 𝑔_D1.

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): 𝑉=𝑣_𝜇,𝜆,𝜆,𝐷,Φ=(𝜙,𝜓,𝐺),𝑄=(𝑄,𝜒,𝐹),𝑄=𝑄,𝜒,𝐹.(3.7) The result, after euclideanization, is: 𝑆_gge=12𝑔²Tr𝑣_𝜇,𝑣_𝜈𝑣_𝜇,𝑣_𝜈+2𝑔²Tr𝑣_𝜇,𝜙𝑣_𝜇,𝜙^†+1𝑔²+2Tr𝐷²+2𝐷𝜙,𝜙^†𝑔²Tr𝐺^†𝐺+2𝑔²TrΨ𝑣_𝜇𝛾_𝜇,Ψ+2√2𝑖𝑔²STr𝜆𝜓,𝜙^†−𝜙,𝜓𝜆,(3.8)_mat=−𝑄^†𝑣_𝜇𝑣_𝜇𝑄+𝐹^†𝐹+𝑄^†𝐷𝑄−𝑄𝑣_𝜇𝑣_𝜇𝑄^†+𝐹𝐹^†+−𝑄𝐷𝑄^†+√2𝐹𝜙𝑄+𝑄𝐺𝑄+𝑄𝜙𝐹+h.c.+𝑖Λ√𝑣_𝜇𝛾_𝜇Λ−√2𝜒𝜙𝜒+𝜒𝜙^†𝜒+𝜒𝜓𝑄+𝑄𝜓𝜒+𝑄^†𝜓𝜒+𝜒𝜓𝑄^†2𝑄^†𝜆𝜒−𝜒𝜆𝑄−𝜒𝜆𝑄^†+𝑄𝜆𝜒.(3.9) Here, the following notations are used (𝛼=1,2):Ψ=𝜆^𝛼𝜓^̇𝛼,Ψ=𝜓^𝛼,𝜆_̇𝛼,Λ=𝜒^𝛼𝜒^̇𝛼,𝛾_𝜇Λ=𝜒^𝛼,𝜒_̇𝛼,=0𝜎^𝜇𝜎_𝜇0,𝜎^𝜇=𝜎⃗,𝑖,𝜎_𝜇=−𝜎⃗,𝑖,Tr𝑇^𝐴𝑇^𝐵=12𝛿^𝐴𝐵.(3.10) It is not difficult to relate (3.8) to the mother theory action of CKKU. The translation is 𝑧₁=1√2𝑣¹+𝑖𝑣₂,𝑧₂=1√2𝑣³Ψ=𝜉₂+𝑖𝑣₄,𝑧₃=𝑖𝜙^†,,𝜉₁,𝜉₃,𝜆,Ψ=𝜓₁,−𝜓₂,𝜒,−𝜓₃.(3.11) Note that 𝜆,  𝜒 here are not the two-component fermions 𝜆_𝛼,𝜒_𝛼 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=Σ_1,2=14𝑖𝛾¹,𝛾₂,𝑞₂=Σ_3,4=14𝑖𝛾³𝑞₃,𝛾₄,=−12𝑄^𝑅,𝑞₄=−𝑇^3𝑅.(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 𝑄_𝑅 is normalized such that gluinos (denoted 𝜆_𝛼,𝜓^̇𝛼 in (3.8)–(3.10)) have 𝑄_𝑅=1. 𝑇^3𝑅=(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,𝜓^̇1𝜓^̇2,Ψ₁,Ψ₂,Ψ₃,Ψ₄=𝜓¹,𝜓²,𝜆_̇1𝜆_̇2(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 𝑄^†𝜆𝜒=𝑄^†𝜆^𝛼𝜒_𝛼=𝑄^†𝜖¹²𝜆₂𝜒₁+𝜖²¹𝜆₁𝜒₂=𝑄^†Ψ₂Λ₁−Ψ₁Λ₂.(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.

Table 1 

𝑈(1)⁴ charges and 𝑍_𝑁×𝑍_𝑁 orbifold action 𝑁-alities for the gauge multiplet, after modification of (3.24) to accomodate hypermultiplets. The second line connects to the CKKU notation for the fields 𝑧₃,𝜉₂,𝜉₁, and so forth.

Table 2 

𝑈(1)⁴ charges and 𝑍_𝑁×𝑍_𝑁 orbifold action 𝑁-alities for the matter hypermultiplets.

3.3. Orbifolding Details

3.3.1. Projections, Generally

Denote 𝑈((𝑁_𝑐+𝑛𝑁_𝑓)𝑁²) indices collectively by 𝑆≡𝐼𝑚₁𝑚₂,𝐼∈1,…,𝑁_𝑐+𝑛𝑁_𝑓,𝑚₁,𝑚₂∈{0,…,𝑁}.(3.15) The domain of the index 𝐼 should be thought of as follows: 𝐼=+1,𝑁…_𝑓,𝑁_𝑐;𝑁_𝑐+1,…,𝑁_𝑐+𝑁_𝑓;𝑁_𝑐+𝑁_𝑓+1,…,𝑁_𝑐+𝑁_𝑓;…;𝑁_𝑐+(𝑛−1)𝑁_𝑓+1,…,𝑁_𝑐+(𝑛−1)𝑁_𝑓+𝑁_𝑓.(3.16) The interpretation is in terms of a block diagonal matrix, with an 𝑁_𝑐×𝑁_𝑐 block, followed by 𝑛 blocks of size 𝑁_𝑓×𝑁_𝑓. The index 𝑆 then indicates, say, that the entries of the 𝑁_𝑐×𝑁_𝑐 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)𝑁}²=diag𝟏_𝑁𝑐𝑁²,𝟏_𝑁𝑓𝑁²,…,𝟏_𝑁𝑓𝑁²,(3.17) with 𝑛 entries of 𝟏_𝑁𝑓𝑁². Other matrices of this form follow.

Introduce “clock operators” that involve roots of unity 𝜔_𝑘≡exp(2𝜋𝑖/𝑘): 𝑃=diagΩ_𝑁𝟏_𝑁𝑐𝑁²,𝜔_𝑛+1𝟏_𝑁𝑓𝑁²,…,𝜔^𝑛𝑛+1𝟏_𝑁𝑓𝑁²,𝐶_𝑁1,𝑘=diag1,𝜔_𝑁,…,𝜔^𝑁𝑁−1,=𝐶𝟏_𝑘1𝑁⊗Ω_𝑁⊗𝟏_𝑁,𝐶^𝑁2,𝑘=𝟏_𝑘⊗𝟏_𝑁⊗Ω_𝑁,𝐶_2𝑁=diag𝐶^1,𝑁𝑐_𝑁,𝐶^1,𝑁𝑓_𝑁,…,𝐶^1,𝑁𝑓_𝑁,=diag𝐶^2,𝑁𝑐_𝑁,𝐶^2,𝑁𝑓_𝑁,…,𝐶^2,𝑁𝑓_𝑁.(3.18) Orbifold projections for any field 𝒜 are defined by 𝒜≡𝜔^𝑠𝑛+1𝑃𝒜𝑃^†,𝒜≡𝜔^𝑟1_𝑁𝒞^1𝑁𝒜𝒞^𝑁1†,𝒜≡𝜔^𝑟2_𝑁𝒞^2𝑁𝒜𝒞^𝑁2†.(3.19) The charges 𝑠,𝑟₁,𝑟₂ will correspond to, respectively, (𝑛+1)-ality, 𝑁-ality, 𝑁-ality. The origin of the 𝑍_𝑛+1 symmetry in the mother theory—corresponding to the (𝑛+1)-ality—will be discussed shortly. The two 𝑁-alities correspond to a 𝑍_𝑁×𝑍_𝑁 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 𝑍_𝑛+1 projection, 𝒜_𝑠=0⟶⊕⋯𝐚𝐝𝐣⊕𝑈1𝑁,1_𝑐𝑁,…²,,𝐚1𝐝,𝐣…𝑈,1𝑁_𝑓⊕1,𝐚𝐝𝐣𝑈𝑁_𝑓𝑁²,1,…,1𝒜𝑁²_𝑠=1,⟶⊕⋯𝑁_𝑐⊕𝑁²,1𝑁,1_𝑓,𝑁…²,,11,,…𝑁_𝑓,1⊕1,𝑁_𝑓𝑁²,𝑁_𝑓𝑁²,1,…,1𝑁²,𝑁_𝑓𝑁²⊕𝑁_𝑐𝑁²,1,…,1,𝑁_𝑓𝑁²,(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 𝑄⟶𝑒^𝑖𝛼𝑄,𝑄⟶𝑒^−𝑖𝛼𝑄,(3.21) with all other fields neutral and all components in 𝑄,𝑄 transforming identically (it is not an R-symmetry). This is the symmetry that we use for the 𝑍_𝑛+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, 𝑄,𝑄 will be bifundamentals (“links”) of 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)^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 𝑍_𝑛+1 orbifold projection leaves all eight supercharges intact.

The 𝑍_𝑁×𝑍_𝑁 projections involve clock matrices 𝒞^𝑁1,2, 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 𝒜_{𝑚1,𝑚2;𝑛1,𝑛2}, ignoring indices of the surviving gauge group. Then the surviving components of 𝒜_{𝑚1,𝑚2;𝑛1,𝑛2} after the 𝑍_𝑁×𝑍_𝑁 orbifold are those that satisfy 𝑚₁−𝑛₁+𝑟₁=0mod𝑁,𝑚₂−𝑛₂+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 𝑄,𝑄. 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 𝑄,𝑄. 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=𝑞₁−𝑞₃,𝑟₂=𝑞₂−𝑞₃.(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: 𝑧𝜉₃:site⟶(−)diagonallink,₂𝜉:−𝑒₁̂link⟶𝑒₂̂link,₁𝜓𝜒::dia−g𝑒o₂̂nallilninkk⟶⟶𝑒s₁̂itlein,₃k,: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) 𝑧^†3,𝑚,𝑧_3,𝑚⟶𝑧^†3,𝑚𝑧_3,𝑚+𝑒̂₁₊𝑒₂̂−𝑧_3,𝑚𝑧^†3,𝑚−𝑒̂₁₋𝑒₂̂,(3.26) to take into account that 𝑧₃,𝑧^†3 are now −/+ diagonal link fields. The fermion action is still of the form 𝑆_𝐹,𝑔=2√2𝑔²_𝑚TrΔ_𝑚𝜆,𝑧^†𝑎,𝜓_𝑎−Δ^𝑚𝜒,𝑧^†𝑎,𝜉_𝑎+𝜖^𝑎𝑏𝑐Δ_𝑚𝜓_𝑎,𝑧_𝑏,𝜉_𝑐,(3.27) with Δ(𝐴,𝐵,𝐶)=𝐴𝐵𝐶−𝐴𝐶𝐵 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 𝑍_𝑁×𝑍_𝑁 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 𝑣_𝜇 in terms of 𝑧_𝑖,𝑧^†𝑖, 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(𝑁_𝑓)_diag subgroup of 𝑈(𝑁_𝑓)^𝑛 survives at the scale 𝑔_𝑐=𝑔𝑎 of the 𝑈(𝑁_𝑐) gauge theory, we only require the application of the deconstruction idea to the 𝑈(𝑁_𝑓)^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(𝑁_𝑓)₁×⋯×𝑈(𝑁_𝑓)_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 𝐾_mat=^𝑛−1_𝑖=1𝑊_matTr𝑄^†𝑖𝑒^𝑉𝑖𝑄_𝑖𝑒^{−𝑉𝑖+1}+𝑄^†𝑖𝑒^𝑉𝑖+1𝑄_𝑖𝑒^−𝑉𝑖,=√2^𝑛−1_𝑖=1Tr𝑄_𝑖Φ_𝑖𝑄_𝑖−𝑄_𝑖Φ_𝑖+1𝑄_𝑖.(3.28) Formally, this is quite similar to the quiver theory studied in [36]. I do not write 𝐾_gge,𝑊_gge since it is just an 𝑛-fold replication of terms of the form (3.1). Holomorphic gauge invariance is given by 𝑄Φ_𝑖𝑖⟶𝑒^Λ𝑖𝑄_𝑖𝑒^−Λ𝑖+1,𝑄_𝑖⟶𝑒^Λ𝑖+1𝑄_𝑖𝑒^−Λ𝑖,⟶𝑒^Λ𝑖Φ_𝑖𝑒^−Λ𝑖,𝑒^𝑉𝑖⟶𝑒^−Λ†𝑖𝑒^𝑉𝑖𝑒^−Λ𝑖.(3.29)

One then gives an expectation value to the (𝑁^𝑓(𝑖),𝑁^𝑓(𝑖+1)),  𝑖=1,…,𝑛−1, bifundamentals and their conjugates: ⟨𝑄_𝑖⟩=𝑄_𝑖=1√2𝑎₃.(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 𝑧_1,𝐦,𝑧_2,𝐦 that transform as adjoints of the 𝑈(𝑁_𝑓)^𝑛 group, excepting the combination corresponding to 𝑈(𝑁_𝑓)_diag) ^𝑛−1_𝑖=1𝑔²4𝑎²³𝐴^{𝜇,𝑚𝑖+1}−𝐴^{𝑖𝜇,𝑚}²,(3.31) where a contraction over the 4d index 𝜇 and 𝑈(𝑁_𝑓) index 𝑚=1,…,𝑁^2𝑓 is implied. The scaling 𝐴→𝑔̂𝐴 has been performed to make the kinetic terms for gauge bosons canonical. Here, 𝑔̂=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(𝑁_𝑓)_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 𝑀^2𝑛=𝑔̂²𝑎²³sin²𝑗𝜋𝑛,𝑗=0,…,𝑛−1.(3.32) The radius 𝑅 of this third deconstructed dimension and the KK mass scale 𝑀 are therefore 𝑅≈𝑛𝑎³𝑔̂,𝑀=𝑅𝜋.(3.33) The effective gauge coupling of the 𝑈(𝑁_𝑓)_diag theory is given by (3.5).

3.5. Decoupling KK States

The condition that the KK states decouple from the 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)_diag gauge theory is just 𝑀≫𝑔_𝑐=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(𝑁_𝑐)×𝑈(𝑁_𝑓)_diag gauge theory: 𝑅≡𝑎. This translates into 𝑛=𝑎𝑔̂𝑎₃=𝑔_𝑐𝑎²𝑎₃.(3.34) Thus as we take the continuum limit 𝑎→0 in the 2D 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)_diag gauge theory, with 𝑛,𝑔_𝑐 held fixed, we have the scaling 𝑎₃∼𝑎². This would decouple the effects of the 𝑈(𝑁_𝑓)^𝑛 quiver at the UV scale of the 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)_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 𝑔_𝑐𝑅 fixed and small. This should also decouple the KK states before important 𝑈(𝑁_𝑐)×𝑈(𝑁_𝑓)_diag physics sets in. This translates into 𝑎₃𝑎=𝑓𝑛,𝑓≪1.(3.35) Holding the factors 𝑓,𝑛 fixed, we see that a scaling 𝑎₃∼𝑎 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 𝑁_𝑐, given the current fashion for applying AdS/QCD ideas to real-world QCD, where 𝑁_𝑐=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(𝑁_𝑐)×𝑈(𝑁_𝑓)^𝑛 quiver theory. Next, I showed how to treat the 𝑈(𝑁_𝑓)^𝑛 quiver as a deconstructed third dimension and how to obtain a weakly coupled 2D remnant 𝑈(𝑁_𝑓)_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.

Appendices

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=⁴_𝑖=1𝑐_1𝑖𝑞_𝑖,𝑟₂=⁴_𝑖=1𝑐_2𝑖𝑞_𝑖.(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+𝑐₁₃𝑞₃+𝑞₄,𝑟₂=𝑞₂+𝑞₄+𝑐₂₃𝑞₃+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, 𝑆_mat=𝑆₁+𝑆₂+𝑆₃,(B.1) where 𝑆₁=−𝑄^†𝑚𝑧^𝑖,𝑚𝑧^{†𝑖,𝑚+}𝑒_𝑖̂+𝑧^†𝑖,𝑚𝑧_𝑖,𝑚−𝑒_𝑖̂𝑄^𝑚−𝑄_𝑚𝑧^𝑖,𝑚𝑧^{†𝑖,𝑚+}𝑒_𝑖̂+𝑧^†𝑖,𝑚𝑧_𝑖,𝑚−𝑒_𝑖̂𝑄^†𝑚+𝐹^†𝑚𝐹_𝑚+𝑒̂₁₊𝑒₂̂+𝐹_𝑚𝐹^†𝑚−𝑒̂₁₋𝑒₂̂+𝑄^†𝑚𝐷_𝑚𝑄_𝑚−𝑄_𝑚𝐷_𝑚𝑄^†𝑚+√2𝐹_𝑚𝜙_𝑚−𝑒̂₁₋𝑒₂̂𝑄_𝑚+𝑄_𝑚𝐺_𝑚𝑄_𝑚+𝑄_𝑚𝜙_𝑚𝐹_𝑚+𝑒̂₁₊𝑒₂̂+𝑄^†𝑚𝜙^†𝑚𝐹^†𝑚−𝑒̂₁₋𝑒₂̂+𝑄^†𝑚𝐺^†𝑚𝑄^†𝑚+𝐹^†𝑚𝜙^†𝑚+𝑒̂₁₊𝑒₂̂𝑄^†𝑚𝑆₂,=√2Λ_1,𝑚𝑧^†1,𝑚−𝑒₂̂+Λ_4,𝑚−𝑒̂₁₋𝑒₂̂+𝑧_2,𝑚−𝑒₂̂Λ_3,𝑚Λ_2,𝑚𝑧^1,𝑚−𝑒₁̂Λ_3,𝑚−𝑧^†2,𝑚−𝑒₁̂−Λ_4,𝑚−𝑒̂₁₋𝑒₂̂Λ_3,𝑚𝑧^†1,𝑚+𝑒̂₁₊𝑒₂̂Λ_2,𝑚+𝑒₂̂−𝑧^†2,𝑚+𝑒̂₁₊𝑒₂̂−Λ_1,𝑚+𝑒₁̂𝑆Λ_4,𝑚𝑧^1,𝑚Λ_1,𝑚+𝑒₁̂+𝑧_2,𝑚Λ_2,𝑚+𝑒₂̂₃,=−√2Λ_1,𝑚𝜙_𝑚−𝑒₂̂Λ_1,𝑚+𝑒₁̂+Λ_2,𝑚𝜙_𝑚−𝑒₁̂Λ_2,𝑚+𝑒₂̂+Λ_3,𝑚𝜙^†𝑚+𝑒̂₁₊𝑒₂̂Λ_3,𝑚+Λ_4,𝑚𝜙^†𝑚Λ_4,𝑚−𝑒̂₁₋𝑒₂̂−Λ_1,𝑚Ψ_2,𝑚−𝑒₂̂−Λ_2,𝑚Ψ_1,𝑚−𝑒₁̂𝑄^𝑚++𝑄^𝑚Ψ_1,𝑚Λ_1,𝑚+𝑒₁̂+Ψ_2,𝑚Λ_2,𝑚+𝑒₂̂−𝑄^†𝑚Ψ^4,𝑚Λ_3,𝑚−Ψ_3,𝑚Λ_4,𝑚−𝑒̂₁₋𝑒₂̂+𝑖√Λ_3,𝑚Ψ_3,𝑚+𝑒̂₁₊𝑒₂̂+Λ_4,𝑚Ψ_4,𝑚𝑄^†𝑚2𝑄^†𝑚Ψ^2,𝑚Λ_1,𝑚+𝑒₁̂−Ψ_1,𝑚Λ_2,𝑚+𝑒₂̂−Λ_3,𝑚Ψ_4,𝑚+𝑒̂₁₊𝑒₂̂−Λ_4,𝑚Ψ_3,𝑚𝑄^𝑚−Λ_1,𝑚Ψ_1,𝑚−𝑒₂̂+Λ_2,𝑚Ψ_2,𝑚−𝑒₁̂𝑄^†𝑚+𝑄_𝑚Ψ_3,𝑚Λ_3,𝑚+Ψ_4,𝑚Λ_4,𝑚−𝑒̂₁₋𝑒₂̂.(B.2) Here, the site indices 𝑚 are implicitly summed.

Acknowledgment

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