Exact duality and local dynamics in SU(N) lattice gauge theory

We construct exact duality transformations in pure SU(N) Hamiltonian lattice gauge theory in (2+1) dimension. This duality is obtained by making a series of iterative canonical transformations on the SU(N) electric vector fields and their conjugate magnetic vector potentials on the four links around every plaquette. The resulting dual description is in terms of the magnetic scalar fields or plaquette flux loops and their conjugate electric scalar potentials. Under SU(N) gauge transformations they both transform like adjoint matter fields. The dual Hamiltonian describes the nonlocal self-interactions of these plaquette flux loops in terms of the electric scalar potentials and with inverted coupling. We show that these nonlocal loop interactions can be made local and converted into minimal couplings by introducing SU(N) auxiliary gauge fields along with new plaquette constraints. The matter fields can be included through minimal coupling. The techniques can be easily generalized to (3+1) dimensions.


I. INTRODUCTION
In the past few decades there have been numerous approaches to dualize gauge theories to obtain their dynamics in terms of the dual potentials [1][2][3]. Many of these attempts are partly inspired by the success of the dual formulation of Abelian lattice gauge theories where duality transformations have led to interesting confining and nonconfining phases in terms of the magnetic monopoles [3]. It is widely believed that color confinement and nonperturbative vacuum structure can also be better understood within the dual framework with inverted couplings [1,2,4,5]. In the recent past, the quest for quantum simulation of non-Abelian lattice gauge theory Hamiltonians using trapped ion or ultra cold atomic gases and optical lattices are important and exciting developments [6][7][8]. The present work with local dual interactions and inverted coupling provides an alternative framework for these quantum simulations in the magnetic basis [6,8]. In fact, dual Hamiltonian formulations and the corresponding magnetic basis are of importance for quantum simulations of gauge theories as they are expected to be more cost efficient for the Hilbert space truncation processes in the weak coupling continuum limit [8]. For this reason in the last few years there has been a surge in the search for the dual representation of various Abelian and non-Abelian lattice models and their application to quantum computations [6,8,9]. The exact duality transformations also naturally lead us to the construct the dual magnetic disorder operators [9,10], which in turn, have been used in Z N and SUðNÞ toric code models to construct anyonic states for topological quantum computations [11].
All duality approaches in the past focus on solving the Abelian or non-Abelian Gauss law constraints to write the electric fields in terms of the dual electric potentials. In Abelian gauge theories such solutions are simple and lead to interesting dynamics [3,8]. However, in non-Abelian cases the duality attempts have not been very successful. Various solutions of non-Abelian Gauss laws lead to the dual descriptions of dynamics, which are involved [1,2,5] and often nonlocal [9] with difficult physical interpretations. These nonlocal interactions also make them computationally unwieldy. Further, many of these duality techniques are tailor made for the SU (2) gauge group [5] and their generalizations to SU(3) and higher SU(N) groups are far from clear. In this paper, using a Hamiltonian approach in (2 þ 1) dimension, we illustrate how to evade the above difficulties and transit from the original SU(N) Kogut-Susskind electric vector field and magnetic vector potential description [see (1)] to the (dual) magnetic scalar field and electric scalar potential description [see (49)]. The dual formulation is also a loop formulation as the dual operators involved are untraced Wilson loops over plaquettes or equivalently the magnetic fields (see Fig. 1) and their conjugate electric scalar potentials. Under SU(N) gauge transformations they both transform like adjoint matter fields. We find that the nonlocal loop-loop interactions, described by electric scalar potentials, can be made local and converted into minimal coupling by introducing auxiliary SU(N) gauge fields through additional plaquette constraints [see (41)]. This should be contrasted with the original interactions which are in terms of the magnetic vector potential holonomies around the plaquettes [see (1)]. This duality between the original plaquette link interactions and the minimal coupling interactions describing loops in (2 þ 1) dimension is a novel feature of the present study. In our previous work [9] we have constructed duality transformations that explicitly solved the SU(N) Gauss law constraints at every lattice site. The dual theory in this case was a SU(N) spin model without any gauge degrees of freedom. The above solutions of SU(N) Gauss law constraints are essentially nonlocal relations between the SU(N) Kogut-Susskind electric fields and the dual electric scalar potentials leading to a nonlocal dual Hamiltonian [9]. These nonlocality issues in the dual formulations have been recently discussed in the context of quantum simulations in the magnetic basis (see Bauer et al. in [6,8]). In the present work we take a different route and define the dual SU(N) electric scalar potentials without solving the Gauss law constraints. We construct SU(N) magnetic scalar or plaquette fields and their conjugate electric scalar potentials [2] by making a series of iterative canonical transformations on the original electric vector fields and their conjugate magnetic vector potentials [12]. These canonical transformations are designed to produce local plaquette loop holonomies (physical magnetic fields) by gluing together its four link holonomies (gluons). This framework is pictorially illustrated in Figs. 1 and 4. Following this process we find the following: (1) The Kogut-Susskind noninteracting electric field g 2 ⃗ E 2 terms dualize to loop interaction terms. As expected, these loop interactions are described by minimal coupling between SU(N) electric scalar potentials and the corresponding auxiliary gauge fields.
(2) The Kogut-Susskind interacting magnetic field 1=g 2 Tr (U plaquette þ H:c:) terms dualize to the noninteracting magnetic fields terms. They create and annihilate single plaquette loops [9,13]. Thus under duality the roles of interacting and noninteracting terms get interchanged resulting in the inversion of coupling constant (g 2 → 1=g 2 ) as expected.
The plan of the paper is as follows: In Sec. II we start with Kogut-Susskind Hamiltonian formulation. This section is added for the sake of completeness and to set up the notations. In Sec. III we discuss the canonical transformations which take us from link description to the plaquette loop description by joining the four links of every plaquette. To make the presentation simple, we first discuss how to join two link holonomies by making a single canonical transformation. In Sec. III A we iterate this step on a simple 2 × 2 plaquette lattice and define four new plaquette loop holonomies (magnetic fields) and their conjugate electric scalar potentials. In Sec. III B we directly generalize these results to N × N plaquette lattice and define N 2 new plaquette holonomies. All technical issues  (28) and (29). As expected, the total number of new configurations (¼ N 2 þ NðN þ 1Þ þ N ¼ 2NðN þ 1Þ) in (b) match with the total number of initial link configurations in (a). and details involved in performing canonical transformations are worked out in Appendix A. In Sec. IV we discuss the dual loop dynamics in (2 þ 1) dimensions in terms of magnetic scalar fields and their conjugate electric scalar potentials. The nonlocality and rotational symmetry problems and their resolutions are discussed. We also compare our SU(N) duality results with the U(1) lattice gauge theory duality results. This simple comparison provides better understanding of the non-Abelian duality relations between electric fields and the electric scalar potentials. We end the paper with a summary and a brief discussion about the future problems.
The notations used are as follows: The lattice sites and links will be denoted by ⃗ n ¼ ðm; nÞ and ð⃗ n;îÞ, respectively, with m; n ¼ 0; 1; 2; …; N and i ¼ 1, 2. We use roman and calligraphic fonts to denote the SU(N) conjugate field operators in the electric (before duality) and the magnetic (after duality) descriptions, respectively. This is clearly illustrated in Table I.

III. CANONICAL TRANSFORMATIONS: LINKS TO LOOPS AND STRINGS
In this section, using canonical transformations, we transit from the Kogut-Susskind link electric field representation to its dual plaquette magnetic field representation in SU(N) lattice gauge theory. These transformations are used to write the Hamiltonian in (1) in its dual form (49). This duality is achieved by canonical gluing the four links around every plaquette on the lattice to define plaquette loop or magnetic operators and their conjugate electric scalar potentials. This is pictorially shown in Figs. 4(a) and 4(b). Note that no attempt is made to solve the SU(N) Gauss laws explicitly to obtain this dual magnetic description. As the above canonical transformation procedure is iterative, we start with gluing two link holonomies and define their electric fields. We then generalize this canonical transformation procedure to 2 × 2 plaquette lattice (see Sec. III A) and then to N × N plaquette lattice (see Sec. III B), respectively. In what follows, we will construct only left (right) plaquette and string electric fields through canonical transformations. Their right (left) electric fields can then be easily obtained using the parallel transport relations (5) with Uð⃗ n;îÞ replaced by the corresponding plaquette or string holonomies. We use calligraphic symbols to denote the new field operators obtained after every canonical transformation.
We consider any two adjacent conjugate pairs: ðE a AE ð1Þ; U αβ ð1ÞÞ and ðE a AE ð2Þ; U αβ ð2ÞÞ. They are the two conjugate pairs located on the links ð⃗ n;1Þ and ð⃗ n þ1;1Þ, respectively, as shown in Fig. 3 The transformations (9) are illustrated in Fig. 3. They are canonical as the two new conjugate pairs ðE a AE ð12Þ; U αβ ð12ÞÞ and ðE a AE ð2Þ; U αβ ð2ÞÞ also follow the standard canonical commutation relations: Note that the two new holonomies U αβ ð12Þ and U αβ ð2Þ trivially commute with each other and we have added E a − ð1Þ to define E a þ ð2Þ in (9) The two new conjugate pairs commute with each other and are therefore mutually independent. They are on the same footing as the original two Kogut-Susskind pairs. Note that, in this simplest two link case, if we identify the two end points ð⃗ nÞ and ð⃗ n þ1Þ in Fig. 3 then Uð12Þ transforms like a magnetic flux loop. We can now follow the classification shown in Table I by identifying ðE AE ð12Þ; Uð12ÞÞ with ðE AE ð⃗ nÞ; Wð⃗ nÞÞ and ðE þ ð2Þ; Uð2ÞÞ with the string pair ðE þ ð⃗ n þ1;1Þ; Uð⃗ n þ1;1ÞÞ. We further note that the electric field E AE ð1Þ of the link holonomy U αβ ð1Þ, which is canonically transformed into U αβ ð12Þ, appears in both the final electric fields. This aspect is clearly shown in Fig. 3. This simple fact will lead to the nonlocal duality relations [see (14) and (20a), (20b)], which are obtained after iterating (9)  in turn, will lead to nonlocal dual or loop dynamics [see (39)]. As mentioned earlier, having defined the left electric fields in (9), the right electric fields get fixed by the parallel transport along the new links In the above equations R a b are the SOðN 2 − 1Þ rotation operators and satisfy The new left and right electric fields also satisfy the SU(N) Lie algebra, they commute with each other and their magnitudes are equal. In summary, in this section we have converted the shorter flux line U αβ ð1Þ into longer flux line U αβ ð12Þ using (9). This simple canonical transformation will now be iterated over the entire lattice to convert all horizontal links into local plaquettes starting from the top. This in turn will define the holonomy around a plaquette or the magnetic fields as the fundamental variables in the dual theory [15]. We first generalize the canonical transformations (9) to 2 × 2 plaquette lattice in Sec. III A and then discuss the general N × N plaquette case in Sec. III B.

A. ð2 × 2Þ plaquette lattice
This simple case is illustrated in Fig. 4 and in Table I. The initial 12 Kogut-Susskind link conjugate pairs ðEðm; n;îÞ; Uðm; n;îÞÞ are shown in Fig. 4(a) or in Table I[A]. The final four (physical) plaquette conjugate pairs ðEðm; nÞ; Wðm; nÞÞ and the remaining eight (unphysical) string conjugate pairs ðEðm; n;îÞ; Uðm; n;îÞÞ are shown in Fig. 4(b) or in Tables I [B] and I[C], respectively. As is clear from the figure, we have converted the four Kogut-Susskind horizontal link holonomies and their electric fields at ðm ¼ 0; 1; n ¼ 1; 2Þ into the four plaquette holonomies and their electric fields. The 12 canonical transformations leading to the configurations in Fig. 4 Fig. 4(a) are systematically worked out in Appendix A. In the next section the end results of the above canonical transformations are written down. They have exact duality interpretation.

Plaquette, strings, and duality
We first describe the new plaquette sector. The four plaquette fluxes shown in Fig. 4
After duality, the fundamental conjugate pairs describing the dynamics (see Sec. IV) are the magnetic scalar fields and their conjugate electric scalar potentials ðEðm; nÞ; Wðm; nÞÞ. Under SU(N) gauge transformations (7) they transform as adjoint scalar matter fields Wð⃗ nÞ → Λð⃗ nÞWð⃗ nÞΛ † ð⃗ nÞ; We now describe the remaining eight unphysical string sector shown in Fig. 4(b) and Table I[C]. As the iterative canonical transformations preserve the total number of degrees of freedom, these eight strings are the leftover degrees of freedom after defining the four dual plaquette holonomies in (13). They are unphysical and can be completely gauged away as is clear from Fig. 4(b). However we retain them to keep the dual loop dynamics simple and local (see Sec. IV). The two horizontal and six vertical string holonomies are Uðm;n;2Þ ¼ Uðm;n;2Þ; m ¼ 0;1;2; n¼ 0;1: ð19bÞ The corresponding conjugate electric fields are (see FIG. 5. Nonlocal parallel transports S and S 0 required for defining plaquette and string electric field operators in (15a), (15b), (15c) and (21a), (21b), (21c), respectively.
Again like the parallel transport S j ðm; nÞ; ðm; n ¼ 0; 1Þ, the parallel transports S 0 j ðm; nÞ in (20b) are required to construct the new string electric fields (see Fig. 5 The parallel transports S 0 j ðm; nÞ in (21a) In (22) In (23) the conjugate pairs ðE AE ð⃗ n;î ¼ 1Þ; Uð⃗ n;î ¼ 1ÞÞ in (23) exist only when ⃗ n ¼ ðm; 0Þ. Note that this asymmetry in the string holonomy sector is due to the special choice of canonical transformations in Appendix A, which converts all Kogut-Susskind horizontal link holonomies at ðm; n > 0Þ into plaquette loops. Their absence also leads to nonlocal loop-loop interactions. This is because the nearest neighbour electric scalar potentials Eð⃗ nÞ and Eð⃗ n þîÞ cannot be coupled minimally in the horizontal directions (see Sec. IV). In Sec. IVA we will reintroduce the horizontal holonomies through new plaquette constraints (41) and recover the rotational symmetry as well as locality of the original Hamiltonian (1).

B. ðN × NÞ plaquette lattice
We now generalize the dual relations obtained in the previous section to N × N lattice. There are N 2 horizontal links at ðm; n > 0Þ as shown in Fig. 1. Using (9) we canonically transform them into plaquettes in the clockwise direction as shown in Fig. 6(b). This canonical gluing starts from the top left column and goes from the top to the bottom and then repeated iteratively in the adjacent right columns. As each plaquette formation requires three canonical transformations (see Appendix A), we need 3N 2 canonical transformations to cover the entire lattice. At the end we construct (i) N 2 plaquettes pairs: ðEð⃗ nÞ; Wð⃗ nÞÞ, (ii) NðN þ 1Þ vertical strings pairs: ðEð⃗ n;2Þ; Wð⃗ n;2ÞÞ, and (iii) N horizontal string pairs: ðEðm; 0;1Þ; Wðm; 0;1ÞÞ. These dual configurations with their left, right electric fields are shown in Fig. 6(b).

(a):
S j ðm; nÞ ≡ Uðm; n;2ÞUðm; n þ 1;1Þ Y j−1 k¼nþ1 Uðm þ 1; k;2Þ: In (30) j ≥ n þ 2 and S nþ1 ðm; nÞ ≡ Uðm; n;2Þ Uðm; n þ 1;1Þ. The nonlocal parallel transport operators S j ðm; nÞ encode the cumulative effects of all 3N 2 canonical transformations over the entire lattice. As mentioned in the previous section, they are necessary for SU(N) gauge covariance of (30). The asymmetry in the shape of the S j ðm; nÞ is because of the choice of iterative canonical transformations. In this work we started at the left top corner and proceeded toward the bottom in the first column and then moved to the adjacent right column. We know that Wðm; nÞ; n ¼ ðN − 1Þ; ðN − 2Þ; Á Á Á 0 are created sequentially by absorbing Uðm; n þ 1;1Þ at (N − n)th step starting from the top. Therefore its electric field must contain all (N − n) Kogut Susskind electric fields on the horizontal links above it. They are located at different points and are parallel transported to ðm; nÞ via path S to maintain gauge covariance of (29). The plaquette canonical commutation relations (16) and (17) discussed in the previous section on the simple 2 × 2 lattice remain valid.
Having discussed the plaquette loop or magnetic field sector, we now discuss the remaining string sector. The N horizontal and NðN þ 1Þ vertical strings are related to old link variables as and S 0 0 ðm; 0Þ ¼ 1. The canonical transformations ensure that all NðN þ 1Þ vertical string pairs ðE a þ ð⃗ n;2Þ; U αβ ð⃗ n;2ÞÞ and N horizontal string pairs ðE a þ ðm; 0;1Þ; U αβ ðm; 0;1ÞÞ satisfy the standard canonical commutation relations.
As before, under SU(N) gauge transformations (7) the plaquette conjugate pairs transform as adjoint matter (18) and the string conjugate pairs transform as gauge fields (23).

IV. SU(N) DUAL DYNAMICS
The Kogut-Susskind Hamiltonian (1) can now be rewritten in terms of the dual plaquette and string operators as [16] H ¼ X ⃗n g 2 Tr ð∇ 2 ðUÞEð ⃗nÞÞ 2 þðE þ ð ⃗n;2Þ − ∇ 1 ðSÞEð ⃗nÞÞ 2 This dual or loop description is invariant under SU(N) gauge transformations (18), (23) and simple to interpret as follows: The original nontrivial four link interaction term in (1), which dominates near the g 2 → 0 continuum limit, is now a simple noninteracting magnetic field term . This is one of the expected outcomes of duality transformations. On the other hand, the original noninteracting electric field terms in (1) now describe the interactions in terms of the adjoint electric scalar potentials. Note that the dual interaction in the y direction in (39) are the minimal coupling terms between electric scalar potentials and the string fields in the y direction. The immediate problem we face with the above dual description is the nonlocal and asymmetric dynamics due to the presence of Sð⃗ n;1Þ in (37c). The underlying reason for this nonlocality and asymmetry is simply the absence of the horizontal holonomies that have been canonically transformed into Wðm; nÞ as shown in Fig. 4(b). The asymmetric Gauss law constraints associated with the SU(N) gauge invariance (18) and (23) are The above constraints directly follow from the new configurations in Fig. 4(b). As shown in Appendix B the new Gauss law constraints (40) reduce to the old symmetric Gauss law constraints (8) when the canonical relations are used and thus confirming (29) and (32a), (32b). The next section addresses and solves the asymmetry and nonlocality issues by introducing new plaquette constraints.

A. Plaquette constraints
Having obtained the dual magnetic field description in terms of the physical conjugate loop pairs ðEð⃗ nÞ; Wð⃗ nÞÞ, we resolve the above asymmetry and nonlocality problems by reintroduction of horizontal link holonomies Uð⃗ n;1Þ through the local plaquette constraints: Uð⃗ n;2ÞUð⃗ n þ2;1ÞU † ð⃗ n þ1;2ÞU † ð⃗ n;1Þ ¼ Wð⃗ nÞ: Note that the constraints (41) imposed on the dual theory are consistent with the dual gauge transformations (18) and (23). They physically mean that the newly created gauge invariant Wilson loops with gauge fields ðUð⃗ n;1Þ; Uð⃗ n;2ÞÞ do not lead to any additional physical degrees of freedom. The motivation for introducing (41) is that on the constrained surface Uð⃗ n;1Þ ¼ Sð⃗ n;1Þ: Now the nonlocal inverse relation (37c) takes the local form and we write E þ ð⃗ n;îÞ ¼ δ i2 E þ ð⃗ n;îÞ þ ϵ ij ∇ j ðUÞEð⃗ nÞ: In (43) i, j ¼ 1, 2. The plaquette constraints (41) must commute with the Hamiltonian H in (39). It is clear that the magnetic part, H M ∼ Tr Wð⃗ nÞÞ, commutes with (41) as W αβ ð⃗ nÞ and U αβ ð⃗ n;îÞ are mutually independent and commuting dual degrees of freedom. It is easy to see that the constraints (41) will commute with the electric part H E ðH E ∼ ⃗ E 2 ð⃗ n;1Þ þ ⃗ E 2 ð⃗ n;2ÞÞ also if the electric fields E a þ ð⃗ n;1Þ and E a þ ð⃗ n;2Þ defined by (37b) and (37c) rotate both sides of (41) covariantly. We therefore introduce electric fields E þ ð⃗ n;1Þ, which are conjugate to auxiliary gauge fields Uð⃗ n;1Þ and write E þ ð⃗ n;îÞ ¼ E þ ð⃗ n;îÞ þ ϵ ij ∇ j ðUÞEð⃗ nÞ: In (44) the covariant derivatives are defined as As mentioned before the parallel transports in (45a) and (45b) are also consistent with SU(N) gauge covariance. This provides an additional cross check for the validity of the SU(N) canonical or duality transformations. At this stage it is interesting as well as illustrative to compare Eq. (44) with the corresponding equation in U (1) or ZðNÞ lattice gauge theories [9]. In U(1) case the Gauss law constraints in (2 þ 1) dimension are where ∇ i is the simple difference operator in i ¼ 1, 2 directions. The obvious solutions defining the Abelian electric scalar potentials in (2 þ 1) dimension are Eð⃗ n;îÞ ¼ ϵ ij ∇ j Eð⃗ nÞ: The SU(N) electric scalar potentials defining Eq. (44) are obvious generalizations of the corresponding Abelian equation (47) with the ordinary difference operators replaced by the SU(N) covariant difference operators. Note that instead of directly solving (46) to obtain dual electric potential Eð⃗ nÞ in (47), we can also use the present canonical transformation route to reach the same result. In U(1) case the parallel transports in (5) and (38a), (38b) are simple Abelian phase factors and cancel out. Thus there are no strings or link gauge fields and we recover (47) without any nonlocality or asymmetry problems. The SU(N) Gauss law constraints ðE a − ð⃗ n;îÞ þ E a þ ð⃗ n;îÞÞ ¼ 0 are now symmetric as shown in Fig. 10. Under SU(N) gauge transformations all electric fields appearing in (48) transform like adjoint matter fields.
The new Hamiltonian that commutes with the constraints (41) written in terms of the dual operators is The dual Hamiltonian (49) can also be interpreted as the loop Hamiltonian. Its physical interpretation is very simple. The second interacting term in (1) dualizes to the noninteracting magnetic field term in (49). It creates and annihilates the single plaquette loops. This is most transparent in the prepotential operator language [9,13]. The first original noninteracting electric field term in (1) dualizes to the loop-loop interaction term in (49). These SU(N) loop interactions are through minimal couplings of the loop electric scalar potential to the gauge fields. This duality between interacting and noninteracting terms leads to inversion of the coupling constant: g 2 → 1 g 2 . Note that the physical degrees of freedom are associated only with the SU(N) magnetic fields and their conjugate electric potentials ðEð⃗ nÞ; Wð⃗ nÞÞ. The auxiliary string sector ðEð⃗ n;îÞ; Uðm; ⃗ n;îÞÞ with the new constraints (41) makes the dual description local as well as simple and rotationally covariant.
We again emphasize that the N 2 horizontal strings Uðm; n > 0;1Þ can be removed using the N 2 constraints (41). As a result their N 2 conjugate electric fields Eðm; n > 0;1Þ can be put equal to zero without loss of any generality. We thus recover the nonlocal Hamiltonian (39), which in turn is exactly equivalent to the  due to the canonical transformations. In fact, at this stage we can also remove the vertical strings completely. Such SU(N) canonical or duality transformations leading to dual SU(N) spin model without any gauge or string degrees of freedom have been studied in the past [9]. They lead to nonlocal dynamics. In the present framework, with all interactions local and proportional to g 2 , the dual Hamiltonian see (49) can be used to set up a weak coupling perturbation theory near the continuum g 2 → 0 limit. The matter fields can be coupled to the SU(N) gauge fields Uð⃗ n;îÞ through minimal coupling so that the SU(N) gauge invariance (18) and (23) remains intact.

V. SUMMARY AND DISCUSSION
In this work we have constructed the canonical transformations in SU(N) lattice gauge theory that lead to local dual Hamiltonian with minimal interactions between dual electric scalar potentials and the auxiliary gauge fields. This result is easy to understand as under gauge transformations the magnetic or plaquette loop fields as well as their conjugate electric scalar potentials transform like SU(N) adjoint matter fields. The transformations convert the plaquette interaction terms into the pure noninteracting magnetic field terms and the pure noninteracting electric field terms into the electric scalar potential minimal coupling interaction terms. These results are important as the plaquette interaction terms involving four links, which dominate near the continuum g 2 → 0 limit, have been completely simplified. In the past, even in the simple SU(2) lattice gauge theory case, these plaquette interactions become extremely complicated in the loop Hilbert space [5,13]. Therefore, it will be interesting to develop a systematic weak coupling loop perturbation theory in the g 2 → 0 continuum limit with the dual Hamiltonian (49).
In the context of quantum simulations of non-Abelian lattice gauge theories [6][7][8], the problem with the electric basis [7,13], is the complicated matrix elements of the magnetic field terms H M in (1). On the other hand, in the magnetic basis the electric field terms H E in (1) becomes complicated and nonlocal [6,8,9]. The present work provides a magnetic basis without the above nonlocality problem and therefore may be better suited for quantum simulations near the continuum limit.
In (3 þ 1) dimension these canonical transformation can be carried out on every ðXZÞ and ðYZÞ plane similar to the present (2 þ 1) dimensional case. We thus convert all X, Y links at z > 0 into ðXZÞ, ðYZÞ plaquettes, respectively, and Z links into the unphysical strings. Now the dual formulation will have nonlocality in both the electric and magnetic field parts of the Hamiltonian. The nonlocality in the electric field part, like in (2 þ 1) dimension, will be due to gauge invariance, whereas the absence of (XY) plaquette will introduce nonlocality in the magnetic part. This nonlocal dynamics can again be made local by FIG. 10. The new symmetric Gauss Laws E a − ð⃗ nÞþ E a þ ð⃗ nÞ þ P 2 i¼1 ðE a þ ð⃗ n;îÞ þ E a − ð⃗ n;îÞÞ ¼ 0. There are six electric fields at each site ⃗ n. The two plaquette electric fields or electric scalar potentials are shown by dark bullets • and the four string electric fields are shown by gray bullets •.
introducing new plaquette constraints. The work in these directions is in progress and will be reported elsewhere.

APPENDIX A: CANONICAL TRANSFORMATION ON A 2 × 2 PLAQUETTE LATTICE
In this appendix, we will explicitly work out canonical transformations (14), (20a), and (20b) for the simple 2 × 2 plaquette lattice. Starting from the top-left plaquette, we make canonical transformations over four plaquettes in the following four steps I, II, III and IV to construct the plaquettes Wð0; 1Þ; Wð0; 0Þ; Wð1; 1Þ, and Wð1; 0Þ, respectively (see Fig. 11). Each of these four steps involves three gluings of Kogut-Susskind link holonomies through canonical transformations illustrated in Fig. 3.
The corresponding electric fields are Now we notice that in step I, we have traded off Uð0; 2;1Þ into the plaquette Wð0; 1Þ so its electric field E − ð1; 2;1Þ appears in all the four new plaquette and string electric fields with appropriate parallel transports (A7) and (A14). Now we will perform steps II, III, and IV using Eqs. (A19) and (A20). So we have two dual holonomies as required and two intermediate holonomies that will be used for canonical transformation in steps II and III. Electric fields for these holonomies are, see Fig. 12.

Construction of Wð1; 1Þ
In the third step, we start with four holonomies Uð1; 1;2Þ, Uð1; 2;1Þ, Uð2; 1;2Þ, and Uð1; 1;1Þ of top right plaquette, see Fig. 11(II), and performing canonical transformations similar to previous step II we convert them into one plaquette, two strings, and one intermediary holonomy: The above holonomies are shown in Fig. 11(III) and their electric fields are given by which is (20b) for m ¼ 2, n ¼ 1, and (A37) is used for canonical transformations in step IV.