Non-Abelian Chern-Simons models with discrete gauge groups on a lattice

We construct the local Hamiltonian description of the Chern-Simons theory with discrete non-Abelian gauge group on a lattice. We show that the theory is fully determined by the phase factors associated with gauge transformations and classify all possible non-equivalent phase factors. We also construct the gauge invariant electric field operators that move fluxons around and create/anihilate them. We compute the resulting braiding properties of the fluxons. We apply our general results to the simplest class of non-Abelian groups, dihedral groups D_n.

Realization of a quantum computer would be very important for the solution of hard problems in classical mathematics 1 and even more important for the solution of many quantum problems in field theory, quantum chemistry, material physics, etc. 2,3,4 However, despite a large effort of many groups the realization remains far away because of the conflicting requirements imposed by scalability and decoupling from the environment. This dichotomy can be naturally resolved by the error free computation in the topologically protected space 5,6 if the physical systems realizing such spaces are identified and implemented. Though difficult, this seems more realistic than attempts to decouple simple physical systems implementing individual bits from the environment. Thus, the challenge of the error free quantum computation resulted in a surge of interest to physical systems and mathematical models that were considered very exotic before.
The main idea of the whole approach, due to Kitaev 9 , is that the conditions of long decoherence rate and scalability can be in principle satisfied if elementary bits are represented by anyons, the particles that indergo nontrivial transformations when moved adiabatically around each other (braided) 9,10,11 and that can not be distinguished by local operators. One of the most famous examples of such excitations is provided by the fractional Quantum Hall Effect 17,18 . The difficult part is, of course, to identify a realistic physical system that has such excitations and allows their manipulations. To make it more manageable, this problem should be separated into different layers. The bottom layer is the physical system itself, the second is the theoretical model that identifies the low energy processes, the third is the mathematical model that starts with the most relevant low energy degrees of freedom and gives the properties of anyons while the fourth deals with construction of the set of rules on how to move the anyons in order to achieve a set of universal quantum gates (further lies the layer of algorithms and so on). Ideally, the study of each layer should provide the one below it a few simplest alternatives and the one above the resulting properties of the remaining low energy degrees of freedom.
In this paper we focus on the third layer: we discuss a particular set of mathematical models that provides anyon excitations, namely the Chern Simons gauge theories with the discrete gauge groups. Generally, an appealing realization of the anyons is provided by the fluxes in non-Abelian gauge theories. 10,11 . The idea is to encode individual bits in the value of fluxes belonging to the same conjugacy class of the gauge group: such fluxes can not be distinguished locally because they are transformed one into another by a global gauge transformation and would be even completely indistiguishable in the absence of other fluxes in the system. Alternatively, one can protect anyons from the adverse effect of the local operators by spreading the fluxes over a large region of space. In this case one does not need to employ a non-Abelian group: the individual bits can be even encoded by the presence/absence of flux in a given area, for instance a large hole in the lattice. Such models 12,13,14,15,16 are much easier to implement in solid state devices but they do not provide a large set of manipulation that would be sufficient for universal quantum computation. Thus, these models provide a perfect quantum memory but not a quantum processor. On the other hand, the difficulty with the flux representation is that universal computation can be achieved only by large non-Abelian groups (such as A 5 ) that are rather difficult to implement, or if one adds charge motion to the allowed set of manipulations. Because the charges form a non-trivial representation of the local gauge group, it is difficult to protect their coherence in the physical system which makes the latter alternative also difficult to realize in physical systems. The last alternative is to realize a Chern-Simons model where on the top of the conjugacy transformations characteristic of non-Abelian theories, the fluxes acquire non-trivial phase factors when moved around each other. We explore this possibility in this paper.
Chern-Simons theories with finite non-Abelian gauge groups have been extensively studied for a continuous 2 + 1 dimensional space-time. Unlike continous gaugegroup, discrete groups on a continous space allow nontrivial fluxes only around cycles which cannot be contracted to a single point. So in a path-integral approach, and by contrast to the case of continuous groups, the integration domain is restricted to gauge-field configurations for which the local flux density vanishes everywhere. Such path integrals were introduced, analyzed, and classified in the original paper by Dijkgraaf and Witten 20 . They showed that for a given finite gauge group G, all possible Chern-Simons actions in 2 + 1 dimensions are in one to one correspondence with elements in the third cohomology group H 3 (G, U (1)) of G with coefficients in U (1). They also provided a description in terms of a 2 + 1 lattice gauge theory, where space-time is tiled by tetrahedra whose internal states are described by just three independent group elements (g, h, k) because fluxes through all triangular plaquettes vanish. Elements of H 3 (G, U (1)) are then identified with functions α(h, k, l) that play the role of the elementary action for a tetrahedron. This description turns out to be rather cumbersome because α(h, k, l) does not have specific symmetry properties, so that the definition of the total action requires to choose an ordering for all the lattice sites, which cannot be done in a natural way. As a result, it seems very difficult to take the limit of a continuous time that is necessary for our purposes because physical implementations always require an explicit Hamiltonian form.
In principle, the knowledge of an elementary action α in H 3 (G, U (1)) allows to derive all braiding properties of anyonic excitations (i.e. local charges and fluxes). This has been done directly from the original path-integral formulation 21,22 , using as an intermediate step the representation theory of the so-called quasi-quantum double associated to the group G 23 . This mathematical struture is completely defined by the group G and a choice of any element α in H 3 (G, U (1)). Using this later formalism, which bypasses the need to construct microscopic Hamiltonians, many detailed descriptions of anyonic properties for various finite groups, such as direct products of cyclic groups, or dihedral groups, have been given by de Wild Propitius 24 . Unfortunately, these general results do not provide a recipe how to contruct a microscopic Hamiltonian with prescribed braiding properties.
To summarize, in spite of a rather large body of theoretical knowledge, we still do not know which Chern-Simons models with a finite gauge group can in principle be realized in a physical system with a local Hamiltonian, that is which one can appear as low energy theory. To answer this question is the purpose of the present paper.
The main ingredients of our construction are the following. We shall be mostly interested in a Hilbert space spanned by dilute classical configurations of fluxes because it is only these configurations that are relevant for quantum computation that involves flux braiding and fusion. Furthermore, we expect that a finite spacial separation between fluxons may facilitate various manipulations and measurements. Notice in this respect that in theories with discrete group symmetry in a continuous space 20,21,22 , non-trivial fluxes appear only thanks to a non-trivial topology of the ambiant space 25 and thus are restricted to large well separated holes in a flat 2D space. The second feature of our construction is that gauge generators are modified by phase-factors which depend on the actual flux configuration in a local way. This is a natural generalization of the procedure we have used recently for the construction of Chern-Simons models with Z Z N symmetry group 19,27 . Note that in the most general situations, the representation of the local gauge group in a discrete Chern-Simons theory becomes projective 20,21,22,23,24 . This would be inappropriate for a robust implementation because projective representations lead to degenerate multiplets that are strongly coupled to local operators, and therefore become very fragile in the presence of external noise. We shall therefore restrict ourselves to the models were no projective representations occur. In practice, they are associated with the nontrivial elements of the group H 2 (G, U (1)), and it turns out that for some interesting classes of groups such as the dihedral groups D N with N odd, H 2 (G, U (1)) is trivial 24 , so this restriction is not too important. As we show in Section III, these assumptions allow us to find all the possible phase factors associated with the gauge transformations in terms of homomorphisms from the subgroups of G that leave invariant a fixed element of G under conjugacy into U (1). The last step is to construct a set of local gauge-invariant operators corresponding to the following elementary processes: moving a single fluxon, creating or annihilating a pair of fluxons with vacuum quantum numbers, and branching processes were two nearby fluxons merge into a single one (or conversely). We shall see that the requirement of local gauge invariance leads to a relatively mild constraint for the possible phase-factors, leaving a rather large set of solutions.
The main result of this work is twofold. First, we provide an explicit computation of holonomy properties of fluxes in a set a models based on dihedral groups. Of special interest is the simplest of them, D 3 which is also the permutation group of 3 elements S 3 . This group belongs to a class which is in principle capable of universal quantum computation 11 . This part is therefore connected to the upper layer (designing a set of univeral gates from properties of anyons) in the classification outlined above. But our construction of a local Hamiltonian version for a Chern-Simons model on a lattice provides some guidelines for possible desirable physical implementations.
The plan of the paper is the following. Section II is mostly pedagogical, providing the motivation for our general construction through the simplest example of a Chern-Simons gauge theory, namely the non-compact U (1) model. In Section III we formulate general conditions on local Chern-Simons phase factors that satisfy gauge invariance conditions. In Section IV we discuss the construction of the electric field operator and derive condition on the phase factor that allows one a gauge invariant fluxon dynamics. In Section V we discuss the fluxon braiding properties and derive the Chern-Simons phase factors associated with the braiding. In Section VI we apply our results to the simplest non-Abelian groups D n . Finally, Section VII gives conclusions. Some technical points relevant for the general construction have been relegated to Appendix A, and Appendix B discusses some interesting features of the torus geometry. Although this geometry is not easy to implement in experiments, it is conceptually very interesting since it is the simplest example of a two-dimensional space without boundary and with topologically non-trivial closed loops.

II. OVERVIEW ON ABELIAN CHERN-SIMONS MODELS
To motivate the construction of the present paper, it is useful to discuss some important features of Chern-Simons gauge theories in the simpler case of Abelian groups. For this purpose, we shall consider here as an illustration the model based on the continuous Abelian group with one generator in its non-compact version. Of course, our main purpose here is to address finite groups, but as we shall discuss, this non-compact U (1) model contains already the key ingredients. On a 2 + 1 dimensional space-time, it is defined from the following Lagrangian density: x is the local magnetic field (a pseudo-scalar in 2 + 1 dimensions) and dots stand for time-derivatives. We have used the gauge in which the time component A 0 of the vector potential is zero. Because of this, we shall consider only invariance under time-independent gauge transformations in this discussion. These are defined as usual by A ρ → A ρ + ∂ ρ f , where f (x, y) is any time-independent smooth scalar function of spacial position. Under such a gauge transformation, the action associated to the system evolution on a two-dimensional space manifold M during the time interval [t 1 , t 2 ] varies by the amount ∆S where: Because f is time-independent, the integrand is a total time-derivative, so we may write ∆S = ν(I(A 2 , f ) − I(A 1 , f )), where A i denotes the field configuration at time t i , i = 1, 2, and: In the case where M has no boundary (and in particular no hole), we may integrate by parts and get: When ν = 0, this modifies in a non-trivial way the behavior of the corresponding quantized model under time-independent gauge transformations. One way to see this is to consider the time-evolution of the system's wave-functional Ψ(A, t). In a path-integral approach, the probability amplitude to go from an initial configuration A 1 (r) at time t 1 to a final A 2 (r) at time t 2 is given by: where fields A(r, t) are required to satisfy boundary conditions A(r, t j ) = A j (r) for j = 1, 2. After the gaugetransformation A ′ = A + ∇f , and using the above expression for ∆S, we see that the probability amplitude connecting the transformed field configurations A ′ 1 and A ′ 2 is: It is therefore natural to define the gauge-transformed wave-functionalΨ by: This definition ensures that Ψ(A, t) andΨ(A, t) evolve according to the same set of probability amplitudes. In a Hamiltonian formulation, we associate to any classical field configuration A(r) a basis state |A . The gauge-transformation corresponding to f is now represented by a unitary operator U (f ) defined by: The presence of the phase-factor is one of the essential features of the Chern-Simons term (i.e. the term proportional to ν) added to the action. Note that when f varies, the family of operators U (f ) gives rise to a representation of the full group of local gauge transformations. Indeed, at the classical level, the composition law in this group is given by the addition of the associated f functions, and because I(A, f ) given in Eq. (4) is itself gauge-invariant, we have U (f )U (g) = U (f + g). It is interesting to give an explicit expression for U (f ). It reads: where Π x and Π y are the canonically conjugated variables to A x and A y . Note that this no longer holds in the case of a manifold M with a boundary, as will be discussed in a few paragraphs. In the Hamiltonian quantization, a very important role is played by the gauge-invariant electric operators E x and E y . In the absence of Chern-Simons term, they are simply equal to Π x and Π y . When ν = 0, because of Eq. (8), the transformation law for Π x and Π y becomes: To compensate for this new gauge sensitivity of conjugated momenta, the gauge-invariant electric field becomes: Any classical gauge-invariant Lagrangian gives rise, after Legendre transformation, to a Hamiltonian which is a functional of E x , E y , and B fields. If we add the Chern-Simons term to the original Lagrangian and perform the Legendre transformation, we get a new Hamiltonian which is expressed in terms of the new gaugeinvariant E x , E y , and B fields through the same functional as without the Chern-Simons term. For the special example of the Maxwell-Chern-Simons Lagrangian (1), this functional reads: But although the Chern-Simons term preserves the Hamiltonian functional, it does modify the dynamical properties of the system through a modification of the basic commutation rules between E x and E y . More precisely, we have: So it appears that finding the appropriate deformations of electric field operators plays a crucial role in constructing the Hamiltonian version of a Chern-Simons theory.
We have also seen that such deformations are strongly constrained by the requirement of invariance under local gauge transformations. This discussion shows that most of the relevant information is implicitely encoded in the additional phase-factor I(A, f ) involved in quantum gauge transformations, as in Eqs. (6) and (7).
Let us now briefly discuss what happens when the model is defined on a two-dimensional space manifold M with a boundary ∂M . Using Stokes' formula, we may recast the phase factor I(A, f ) attached to a gaugetransformation as: In this situation, the phase factor I(A, f ) is no longer gauge-invariant, and this implies that two gauge transformations attached to functions f and g do not commute because: In more mathematical terms, this reflects the fact that the phase-factor I(A, f ), as used in Eqs. (6) and (7) defines only a projective representation of the classical gauge group, that is: As first shown by Witten 26 , this may be understood in terms of a chiral matter field attached to the boundary of M . An explicit example of boundary degrees of freedom induced by a Chern-Simons term has been discussed recently in the case of a Z Z 2 model on a triangular lattice 27 .
To close this preliminary section, it is useful to discuss the case of a finite cylic group Z Z N . In the U (1) case, for a pair of points r and r ′ , we have a natural group element defined by exp(i 2π Φ0 r ′ r A.dl) where the integral is taken along the segment joining r and r ′ , and Φ 0 is the flux quantum in the model. For a finite group G, the notion of a Lie algebra is no longer available, so it is natural to define the model on a lattice. In a classical gauge theory, each oriented link ij carries a group element g ij ∈ G. We have the important constraint g ij g ji = e, where e is the neutral element of the group G. In the quantum version, the Hilbert space attached to link ij is the finite dimensional space generated by the orthogonal basis |g ij where g ij runs over all elements of G. For a lattice, the corresponding Hilbert space is obtained by taking the tensor product of all these finite dimensional spaces associated to links. In the Z Z N model, g ij becomes an integer modulo N , p ij . The connection with the continuous case is obtained through the identification On each link ij, we introduce the unitary operator π + ij which sends |p ij into |p ij + 1 . In the absence of a Chern-Simons term, the generator of the gauge transformation based at site i (which turns where the product involves all the nearest neighbors of site i. By analogy with the continuous case, the presence of a Chern-Simons term is manifested by an additional phase-factor whose precise value depends on the lattice geometry and is to some extent arbitrary, since fluxes are defined on plaquettes, not on lattice sites. On a square lattice, a natural choice is to define U i according to 19 : where L(i) is the oriented loop defined by the outer boundary of the four elementary plaquettes adjacent to site i. This expression has exactly the same structure as Eq. (8), but somehow, the local magnetic field at site i is replaced by a smeared field on the smallest available loop centered around i. It has been shown 19 that a consistent quantum theory can be constructed only when ν/ is an integer multiple of N/π.

III. GENERATORS OF LOCAL GAUGE TRANSFORMATIONS
As discussed in the previous section, the most important step is to construct a non-trivial phase factor which appears in the definition of unitary operators associated to local gauge transformations, generalizing Eq. (7). For this, let us first define the operator L ij (g) which is the left multiplication of g ij by g, namely: L ij (g)|g ij = |gg ij . For any site i and group element g, we choose the generator of a local gauge transformation based at i to be of the following form: where j denotes any nearest neighbor of i and Φ(i, r) is the flux around any of the four square plaquettes, centered at r, adjacent to i. Here, and in all this paper, we shall focus on the square lattice geometry, to simplify the presentation. But adaptations of the basic construction to other lattices are clearly possible. Since we are dealing with a non-Abelian group, we have to specify an origin in order to define these fluxes, and it is natural to choose the same site i, which is expressed through the notation Φ(i, r). Since we wish U i (g) to be unitary, we require |χ Φ (g)| = 1. It is clear from this construction that two generators U i (g) and U j (h) based on different sites commute, since the phase factors χ Φ (g) are gauge invariant. This form is a simple generalization of the lattice Chern-Simons models for the cyclic groups Z Z N discussed in the previous section. In this example, for a square plaquette ijkl, the flux Φ is equal to p ij +p jk +p kl +p li modulo N , and g is simply any integer modulo N . Eq. (13) above may be interpreted as: This is a well defined function for Φ and g modulo N only if ν/ is an integer multiple of 2N/π. We have not succeeded to cast odd integer multiples of N/π for ν/ in the framework of the general construction to be presented now. This is not too surprising since these models were obtained by imposing special periodicity conditions on an infinite-dimensional Hilbert space where p ij can take any integer value 19 . Our goal here is not to write down all possible Chern-Simons theories with a finite group, but to easily construct a large number of them, therefore allowing for non-trivial phase-factors when two localized flux excitations are exchanged. As discussed in the Introduction, a very desirable property, at least for the sake of finding possible physical implementations, is that these deformed generators define a unitary representation of the group G. So we wish to impose: or equivalently: To solve these equations let us first choose a group element Φ. Let us denote by H Φ the stabilizor of Φ under the operation of conjugacy, namely h belongs to H Φ whenever hΦh −1 = Φ. This notion is useful to describe the elements of the conjugacy class of Φ. Indeed, we note that ghΦ(gh) −1 = gΦg −1 if h belongs to H Φ . Therefore, the elements in the conjugacy class of Φ are in one to one correspondence with the left cosets of the form gH Φ . Let us pick one representative g n in each of these cosets.
We shall now find all the functions χ gnΦg −1 n (g). First we may specialize Eq. (17) to the case where h belongs to H Φ , giving: In particular, it shows that the function h → χ Φ (h) defines a group homomorphism from H Φ to U (1). Once this homomorphism is known, we can specify completely χ Φ (g) for any group element g once the values χ Φ (g n ) for the coset representatives are known. More explicitely, we have: where h ∈ H Φ . Finally, Eq. (17) yields: Let us now show that for any choice of homomorphism h → χ Φ (h), h ∈ H Φ , and unit complex numbers for χ Φ (g n ), Eqs. (19), (20) reconstruct a function χ Φ (g) which satisfies the condition (17). Any element g ′ in G may be written as g ′ = g n h, with h ∈ H Φ . We have: which is exactly Eq. (17). Note that there are many equivalent ways to choose these functions χ Φ (g). Let us multiply the system wavefunction by a phase-factor of the form i In particular, it is possible to choose the values of ǫ(g n Φg −1 n ) so thatχ Φ (g n ) = 1. Although this does not seem to be required at this stage of the construction, it is also necessary to assume that when the flux Φ is equal to the neutral element e, χ e (g) = 1. This will play an important role later in ensuring that the phase-factor accumulated by the system wave-function as a fluxon winds around another is well defined. Our goal here is to construct local gauge invariant operations for basic fluxon processes: fluxon motion, creation of a pair with vacuum quantum numbers, branching of a single fluxon into two fluxons and their time reversions. These three types of elementary processes can all be derived from a single operation, the electric field operator that at the level of classical configurations, is simply a left multiplication L ij (g) attached to any link ij. To show this, let us consider a pair of two adjacent plaquettes as shown on Fig. 1. We denote by Φ(i, L) (resp. Φ(i, R)) the local flux through the left (resp. right) plaquette, with site i chosen as origin. Similarly, we define fluxes Φ(j, L) and Φ(j, R). Changing the origin from i to j simply conjugates fluxes, according to Φ(j, L(R)) = g ji Φ(i, L(R))g ij . The left multiplication L ij (g) changes g ij into g ′ ij = gg ij . Therefore, it changes simultaneously both fluxes on the left and right plaquettes adjacent to link ij. More specifically, we have: In particular, this implies: Note that transformation laws for fluxes based at site j are slightly more complicated since they read: This asymmetry between i and j arises because g ′ ji = g ji g −1 , so we have: where R ji (h) denotes the right multiplication of g ji by the group element h. In the absence of Chern-Simons term, L ij (g) commutes with all local gauge generators with the exception of U i (h) since: We now apply these general formulas to the elementary processes involving fluxes. Suppose that initially a flux Φ was localized on the left plaquette, and that the right plaquette is fluxless. Applying L ij (g = Φ −1 ) on such configuration gives: which shows that a Φ-fluxon has moved from the left to the right plaquette. A second interesting situation occurs when both plaquettes are initially fluxless. The action of L ij (Φ −1 ) on such state produces a new configuration characterized by: So we have simply created a fluxon and antifluxon pair from the vacuum. Of course, applying L ij (Φ) on the final state annihilates this pair. Finally, a single flux Φ = Φ 2 Φ 1 originaly located on the left plaquette may split into a pair Φ 1 on the left and Φ 2 on the right. This is achieved simply by applying L ij (Φ −1 2 ). In order to incorporate these elementary processes into a Hamiltonian Chern-Simons theory, we have to modify L ij (g) into an electric field operator E ij (g) by introducing phase-factors so that it commutes for all generators U k (h) if k = i and that: As explained in the introduction, we shall need only to construct E ij (g) for special types of configurations for which at least one of the four fluxes Φ(i, L), Φ(i, R), Φ ′ (i, L), Φ ′ (i, R) vanishes. Nevertheless, it is useful to first construct E ij (g) for an arbitrary flux background. The requirement of local gauge-symmetry induces strong constraints on the phase-factors χ Φ (h) as we shall now show. These constraints are less stringent when we restrict the action of E ij (g) to the limited set of configurations just discussed.

B. Construction of gauge-invariant electric field operators
Transition between reference states Generic initial state Final state . The amplitude of the process starting from a generic initial state shown in the lower left can be related to A(ΦL, ΦR, g) using the gauge invariance. This is done in two steps. First, the gauge transformation U ({h}) = U1(h1)U2(h2)U3(h3)U4(h4)Uj (hj) is used to relate the amplitude of the process starting from the generic state to the amplitude of the transition a(ΦL, ΦR, g) between reference state and a special state shown in the middle right. Second, we use gauge transformation W (g) = U4(g)Uj(g)U3(g) to relate the special state to the reference state on the upper right. Site labeling is the same as in Figure 1.
We now construct the electric field operators E ij (g) attached to links. In the absence of Chern-Simons term the electric field operator is equivalent to the left multiplication of the link variable: In this case, E ij (g) commutes with all local gauge genera-tors with the exception of U i (h), and Eq. (35) is satisfied. We have also the group property, namely: The Chern-Simons term gives phase factors χ Φ (g) to the gauge generators, so we expect some phase factor, Υ ij , to appear in the electric field operators as well: E = L ij Υ ij . The gauge invariance condition allows us to relate the phase factors associated with different field configurations to each other. Specifically, we introduce the reference states (shown on the top of Fig. 2 ) in which only two bonds carry non-trivial group elements. We define the transition amplitude induced by the electric field between these reference states by A(Φ L , Φ R , g). In order to find the phases Υ ij for arbitrary field configuration we first transform both the initial and final state by . This relates the amplitude of the generic process to the amplitude, a(Φ L , Φ R , g), of the process that starts with the reference state but leads to the special final state shown in Fig. 2 middle right. Collecting the phase factors associated with the gauge transformation U ({h}) before and after the electric field moved the flux we get where Φ(i, L) denotes the flux in the left plaquette counted from site i, Φ(j, R) denotes flux in the right plaquette counted from site j, and prime refers to the flux configuration after the action of the electric field. Finally, we employ the gauge transformation W (g) = U 4 (g)U j (g)U 3 (g) to relate this special final state to the reference state. The phase factor associated with this gauge transformation is χ ΦLg (g)χ g −1 ΦR (g) 2 so In order to express the phase factors, Υ ij , through the initial field configuration we relate the parameters, h k , of the gauge transformations to the bond variables g k by h 1 = g 1i , h 2 = g 2i , h 3 = g 3j g ji , h 4 = g 4j g ji , h j = g ji and the fluxes in the left and right plaquettes before and after electric field operator has changed them. Before the electric field operator has acted the fluxes were Combining the preceding equations and using the relation (17) a few times, we get the final expression for the phase-factor, where we have used the definition g ′ ij = gg ij (g ′ ji = g ji g −1 ) to make it more symmetric.
, and U j (h j ) follows directly from this construction. It can also be checked directly from (40), using the condition (17) on the elementary phase-factors χ Φ (g). Note that sites i and j play different roles, which is expected because E ij (g) acts by left multiplication on g ij whereas E ji (g) acts by right multiplication on the same quantity.
Although the electric field operator commutes with U 1 (h 1 ), U 2 (h 2 ), U 3 (h 3 ), U 4 (h 4 ), and U j (h j ), it does not necessarily commute with U i (h) even if hgh −1 = g. In fact, the requirement of this commutation leads to an important constraint on possible choices of phases χ Φ (g). The appearance of the new constraints becomes clear when one considers a special field configuration shown in Fig. 3. Two identical field configurations shown on the top and bottom left of this figure can be obtained by two different gauge transformations from the reference state if both Φ L and Φ R commute with h: in one case one applies gauge transformation only at site i , in the other one applies gauge transformation on all sites except i. Provided that the resulting states are the same, i.e. gh = hg, the total phase factor Υ ij obtained by these two different ways should be the same.
The phase factors associated with these gaugetransformations are the following Putting all these factors together we conclude that the gauge invariance of the electric field operator implies that This condition can be further simplified by using the main phase factor equation (17). We start by noting that because h commutes with Φ L , Φ R , and g, χ ΦR,L (h)χ ΦR,L (h −1 ) ≡ 1 and χ gΦR,L (h)χ gΦR,L (h −1 ) ≡ 1. This gives Furthermore, combining the identities χ gΦL (h) = χ gΦL (ghg −1 ) = χ ΦLg (g)χ ΦLg (h)χ gΦL (g −1 ) and 1 = χ gΦL (gg −1 ) = χ ΦLg (g)χ gΦL (g −1 ) we get This reduces the condition (42) to a much simpler final form We emphasize that constraint (45) on the phase factors has to be satisfied only for fluxes satisfying the condi- Although we have derived this condition imposing only the gauge invariance of the electrical field acting on a very special field configuration, a more lengthy analysis shows that it is sufficient to ensure that in a general case The details of the proof are presented in Appendix A. Unlike Eq. (17), the constraint (45) relates functions χ Φ (g) and χ Φ ′ (g) for Φ and Φ ′ belonging to different conjugacy classes. As shown in section VI B this constraint strongly reduces the number of possible Chern-Simons theories. Note that if both χ   (17) and (45), their product χ is also a solution. So there is a natural group structure on the set of possible Chern-Simons models based on the group G, which is transparent in the path-integral description: this means that the sum of two Chern-Simons action is also a valid Chern-Simons action.
Is this construction also compatible with the group property (37)? From Eq. (45), we obtain: where: It does not seem that β(Φ L , Φ R , g) are always equal to unity for any choice of χ Φ (g) that in turns determines the amplitudes A(s) (see Appendix A). But this is not really a problem because this group property does not play much role in the construction of gauge-invariant Hamiltonians. We now specialize the most general constraint arising from local gauge-invariance to the various physical processes which are required for fluxon dynamics. For the single fluxon moving operation, we have Φ L = Φ, Φ R = e, and g = Φ −1 , so the condition Eq. (45) is always satisfied. For the pair creation process, we have Φ L = Φ R = e, and g = Φ −1 . The constraint becomes: Finally, let us consider the splitting of an isolated fluxon into two nearby ones. This is described by Φ L = Φ 2 Φ 1 , Φ R = e, and g = Φ −1 2 . We need then to impose: (49) It is clear that condition (48) is a special case of the stronger condition (49). Furthermore, multiplying the conditions (49) for pairs of fluxes (Φ L , Φ R ) and (gΦ L , Φ R g −1 ) we get the most general condition (45); this shows that constraint (49) is necessary and sufficient condition for the gauge invariant definition of electric field operator acting on any flux configuration.

V. GENERAL EXPRESSION FOR HOLONOMY OF FLUXONS
Let us consider two isolated fluxes carrying group elements g 1 and g 2 , and move the first one counterclockwise around the second one, as shown on Fig. 4. This can be done by successive applications of local gauge-invariant electric field operators as discussed in the previous section. Although we wish to work in the gauge-invariant subspace, it is very convenient to use special configurations of link variables to illustrate what happens in such a process. We simply have to project all special states on the gauge invariant subspace, which is straightforward since the fluxon moving operator commutes with this projector. The initial fluxes are described by two vertical strings of links carrying the group elements g 1 and g 2 , see Fig. 4. When several other fluxes are present, besides the two to be exchanged, it is necessary to choose the location of the strings in such a way that no other flux is present in the vertical band delimited by the strings attached to the two fluxons. During the exchange process, the first fluxon collides with the second string, and after this event, it no longer carries the group element g 1 , but its conjugate g ′ 1 = g −1 2 g 1 g 2 . After the process is completed, the first flux has recovered its original position, but the configuration of group elements has changed. If we measure the second flux using a path starting at point O shown on Fig. 4 we find that it has also changed into g ′ 2 = g −1 2 g −1 1 g 2 g 1 g 2 . The final state can be reduced to its template state built from two strings carrying groups elements g ′ 1 and g ′ 2 by the gauge transformation i U i (h i ) where h i is locally constant in the two following regions: the core region inside the circuit followed by the first fluxon (h i = h core ), the intermediate region delimited by the two initial vertical strings and the upper part of the circuit (h i = h int ). Note that because we do not wish to modify external fluxes, we cannot perform gauge transformations in the bulk, outside of these regions.
Group elements h core and h int have to satisfy the following conditions, which may be obtained readily upon inspection of Fig. 4 h These equations are mutually compatible, and we get h int = g −1 1 g −1 2 g 1 g 2 . Since fluxes are present in the core region, they will contribute a phase-factor f (g 1 , g 2 ) when the gauge transformation from the template to the actual final state is performed. The final result may be summarized as follows: The new phase-factor f (g 1 , g 2 ) appears not to depend on the detailed path taken by the first fluxon, but just on the fact that it winds exactly once around the second. In this sense, our construction really implements a topological theory.

A. General properties of dihedral groups
The dihedral groups D N are among the most natural to consider, since they contain a normal cyclic group Z Z N which is simply extended by a Z Z 2 factor to form a semidirect product. In this respect, one may view this family as the most weakly non-Abelian groups. D N can be described as the isometry group of a planar regular polygon with N vertices. The Z Z N subgroup corresponds to rotations with angles multiples of 2π/N . We shall denote by C the generator of this subgroup, so C may be identified with the 2π/N rotation. D N contains also N reflections, of the form τ C n . The two elements C and τ generate D N , and they are subjected to the following minimal set of relations: This last relation shows that indeed D N is non-Abelian. The next useful information about these groups is the list of conjugacy classes. If N = 2M + 1, D N contains M + 2 classes which are: As shown in section III, in order to construct possible phase factors associated to gauge transformations, we need to know the stabilizors of group elements for the conjugacy operation of D N acting on itself. Here is a list of these stabilizors: For N odd: Finally, we need to choose homomorphisms from these stabilizors into U (1). In the case of a cyclic group Z Z N generated by C, homomorphisms χ are completely determined by χ(C), so that χ(C p ) = χ(C) p , with the constraint: χ(C) N = 1. For the group D N itself, we have: χ(τ r C p ) = χ(τ ) r χ(C) p , with the following constraints: These are direct consequences of generator relations (54),(55),(56). Again, the parity of N is relevant. For N odd, χ(C) = 1, which leaves only two homomorphisms from D N into U (1). For N even, χ(C) = ±1, and there are four such homomorphisms. The last possible stabilizor to consider is the four element subgroup of D 2M , S = {e, C M , τ C p , τ C p+M }. This abelian group has four possible homomorphisms into U (1), which are characterized as follows: (61) B. Classification of possible models

N odd
Let us first consider conjugacy classes of the form {C p , C −p }. Since the stabilizor of C p for the conjugacy action of D N is Z Z N , we have: where ω N p = 1. Choosing χ C P (τ ) = 1, we have for fluxes in the cyclic group generated by C: For the remaining conjugacy class, we have χ τ (τ ) = η, with η = ±1. Choosing χ τ (C p ) = 1, and using Eqs. (19) and (20), we obtain: All these possible phase-factors satisfy the following property: so that Eq. (48) is always satisfied. So no new constraint is imposed by the requirement to create or annihilate a pair of fluxons. What about the stronger condition Eq. (49)? Its form suggests that we should first determine pairs of fluxes (Φ 1 , Φ 2 ) such that their stabilizors H Φ1 and H Φ1 have a non-trivial intersection. This occurs if both Φ 1 and Φ 2 are in the Z Z N normal subgroup generated by C, or if Φ 1 = Φ 2 = τ C p . The second case simply implies χ τ (τ ) 2 = 1, which is not a new condition. In the first case, choosing h in Z Z N as well shows that χ Φ (h) 4 is a homomorphism from Z Z N to U (1) with respect to both Φ and h. This is satisfied in particular if χ Φ (h) itself is a group homomorphism in both arguments. This sufficient (but possibly not necessary) condition simplifies algebraic considerations; it can be also justified from physical argument that the theory should allow for the sites with different number of neighbours, Z, which would change Replacing this the constraint on χ Φ (h) 4 by the constraint on χ Φ (h), we get Therefore, the class of possible phase-factors (which is stable under multiplication) is isomorphic to the group Z Z N × Z Z 2 . This group of phase-factors is identical to the group of possible Chern-Simons actions since in this case, H 3 (D N , U (1)) = Z Z N × Z Z 2 24 . Very likely, the coincidence of these two results can be traced to the absence of projective representations for D N for N odd 24 : as explained above, the projective representations are not allowed in our construction but are allowed in the classification of all possible Chern-Simons actions 24 .

N even
The conjugacy classes of the form {C p , C −p } behave in the same way as for N odd. So writing N = 2M , we have: The conjugacy class {C M } is special since its stabilizor is D N itself. As discussed in Section VI A above, there are four homomorphisms form D N into U (1) that we denote by: Let us now turn to χ τ (g) with the corresponding stabilizor equal to {e, C M , τ, τ C M }. As seen in section VI A, the four possible homomorphisms may be written as: From this, we derive the expression of χ τ (g), in the following form (0 ≤ p ≤ M − 1): Furthermore, because C p τ C −p = τ C −2p , Eq. (20) implies: The last conjugacy class to consider contains τ C, with the stabilizor {e, C M , τ C, τ C 1+M }. In this case, we may set (0 ≤ p ≤ M − 1): Here again, the constraint Eq. (48) is always satisfied. To impose Eq. (49), we have to consider pairs of fluxes (Φ 1 , Φ 2 ) such that H Φ1 ∩ H Φ2 is non-trivial. As before, choosing Φ 1 and Φ 2 in the Z Z N subgroup imposes ω p = ω p , with ω N = 1. A new constraint arises by choosing Φ 1 = C p and Φ 2 = τ C p ′ . In this case, C M belongs to their common stabilizor. Eq. (49) implies: (93) But using Eq. (17) this yields: Therefore we have the constraint: ζ 0 = ζ 1 = 1, which enables us to simplify drastically the above expression for phase-factors: and But Eq. (93) now implies that χ C p (C M ) = 1 for any p, which is satisfied only when ω M = 1. Specializing to p = M , we see that the common stabilizor contains now two more elements, namely τ C p ′ and τ C p ′ +M . Eq. (93) now requires that: It is then easy to check that considering the common stabilizor of τ C p and τ C p ′ , which may contain two or four elements, does not bring any new constraint. Finally, ω belongs to the Z Z M group and among the three binary variables η 0 , η 1 , andω, only two are independent. Therefore, the set all all possible phase-factors for the D 2M group is identical to Z Z M × Z Z 2 × Z Z 2 . This contains only half of H 3 (D 2M , U (1)) which is equal to Z Z 2M × Z Z 2 × Z Z 2 24 . But D 2M admits non trivial projective representations since H 2 (D 2M , U (1)) = Z Z 2 which cannot appear in our construction. The important result is that in spite of this restriction, we get a non-trivial subset of the possible theories also in this case.

C. Holonomy properties
Using the general expression (53), and the above description of possible phase-factors, we may compute the adiabatic phase induced by a process where a fluxon g 1 winds once around another fluxon g 2 . The results are listed in the last column of table I, and they are valid for both parities of N .  I: Adiabatic phase f (g1, g2) generated in the process where fluxon g1 winds once around a fluxon g2. Values of phase-factors χ g ′ 1 (g ′ 1 ) and χ g ′ 2 (g ′ 1 ) are given for dihedral groups DN with odd N . For even N , expressions for χ g ′ 1 (g ′ 1 ) and χ g ′ 2 (g ′ 1 ) are slightly more complicated, but interestingly, the main result for f (g1, g2) is the same as for odd N , namely it involves only the complex number ω.

VII. CONCLUSION
Generally, in order to appear as a low energy sector of some physical Hamiltonian, the Chern-Simons gauge theory has to involve gauge transformations that depend only on a local flux configuration. Furthermore, to be interesting from the view point of quantum computation, the theory should allow for a local gauge invariant electric field operator that moves a flux or fuses two fluxes together. Here we have analyzed non-Abelian gauge models that satisfy these general conditions; our main result is the equation (49) for the phase factor χ associated with a gauge transformation. Furthermore, we have computed the flux braiding properties for a given phase factor that satisfies these conditions. Finally, we have applied our general results to the simplest class of non-Abelian groups, dihedral groups D n . The fluxon braiding properties in these groups are summarized in Table I. Inspection of the Table I shows that even for the smallest groups, Chern-Simons term modifies braiding properties in a very non-trivial manner and gives phases to the braiding that were trivial in the absence of the Chern-Simons term. In the scheme 11 where the pair of two compensating fluxes (τ C p , τ C p ) are used to encode one bit, the transformations allowed in the absence of Chern-Simons term are limited to conjugation. In the presence of Chern-Simons term the braiding of such bit with the controlled flux results in a richer set of transformations that involve both conjugation by group elements and phase factors (see Table I) but does not change the state of the flux as it should. We hope that this will make it possible to construct the universal quantum computation with the simplest group D 3 that does not involve operations that are difficult to protect (such as charge doublets).
The implementation of the microscopic Hamiltonians discussed in this paper in a realistic physical system is a challenging problem. The straightforward implementation would mean that the dominant term in the microscopic Hamiltonian is H = −t i,g U i (g) so that all low energy states are gauge invariant. This is not easy to realize in a physical system because operator U (g) involves a significant number of surrounding bonds. We hope, however, this can be achieved by a mapping to the appropriate spin model as was the case for Abelian Chern-Simons theories; this is the subject of the future research. Acknowledgments We are thankful to M. Görbig and R. Moessner for useful discussions. LI is thankful to LPTHE, Jussieu for their hospitality while BD has enjoyed the hospitality of the Physics Department at Rutgers University. This work was made possible by support from NATO CLG grant 979979, NSF DMR 0210575. In the absence of additional constraints on the gauge transformation phase factors, χ Φ (g), the definition of the electric field operator implies that it is gauge invariant under gauge transformations on all sites except i while the the gauge transformation on the latter gives . This involves gauge transformations on all grey (blue) sites. The gauge invariance implies that the total phase factor accumulated when moving around the diagram is unity which implies the condition on the ratio A(s)/A(h(s)). As in text, s stands for a triple (ΦL, ΦR, g), and h(s) is obtained by conjugating the components of s by h.
Here the notations are those of Fig. 2, where state |Ψ corresponds to the initial state on the figure. The computation leading to this equation is depicted on Fig 5. The question we have to address now is whether it is possible to choose amplitudes A(Φ L , Φ R , g) so that the function α(Φ L , Φ R , g|h) is equal to unity. To simplify notations, let us denote triples (Φ L , Φ R , g) by a single label s. The group G acts on these triples according to: (A4) Let us also introduce the function f (s|h) defined as: Let us now show that when these two conditions are satisfied, we can always reconstruct a system of amplitudes solving Eq. (A9). From the form of this equation, we see that the various orbits in S for the action of G remain uncoupled. Let us then concentrate on one orbit generated by a fixed given element s. Elements of this orbit are in one to one correspondence with the left cosets gH s of G, since gh(s) = g(s) when h ∈ H s . Let us choose in each coset a representative element g n . Our problem is to find amplitudes A(g n (s)) knowing A(s). Let us define: From Eq. (A13), we have A(g(s)) = A(s)f (s|g) for any g in G, since g may always be written as g n h with h ∈ H s . Now Eq. (A9) is an immediate consequence of the condition (A11). To conclude, when conditions (A10) and (A11) are satisfied, solutions of Eq. (A9) can be constructed independently on each orbit of S for the action of G. On a given orbit, the solution is unique, up to the choice of A(s) for one particular element of this orbit.

APPENDIX B: STRUCTURE OF THE GROUND STATE ON A TORUS
The distinguishing feature of a torus is the appearance of non-trivial closed loops that are classified according to their winding numbers associated with two fundamental cycles γ x and γ y , chosen with a common origin O. As a consequence, even if local fluxes vanish on all plaquettes, the fluxes Φ x and Φ y associated to γ x and γ y may not vanish. These global degrees of freedom are the origin of the topological degeneracies exhibited by a large class of gauge-invariant models on a closed space with non-trivial topology 28 . How are these degeneracies affected by the presence of a Chern-Simons term? Precise formulas for the Hilbert-space dimension of the pure Chern-Simons theory with a finite gauge group G have been given 20 in terms of the action via functions α(h, k, l) in H 3 (G, U (1)). In this paper we consider a much larger Hilbert space induced by all allowed classical gauge configurations, |{g ij } . We may expect to recover the pure Chern-Simons Hilbert-space (which dimension is independent of the system size) by projecting the full Hilbert-space onto the gauge-invariant groundstate of some local gauge-invariant Hamiltonian. This goes beyond the scope of the present paper because our goal here is classification and basic properties of such local gauge invariant Hamiltonian. However, it is instructive to understand how, in the present formalism, a Chern-Simons term affects the gauge dynamics in the topological fluxless sector on a torus.  Starting from the trivial configuration g ij = e, we may induce a new fluxless state in the bulk but with a nontrivial Φ x along γ x by creating a string of parallel horizontal links carrying each the group element Φ x as shown on Fig. 6. This string may be viewed as the result of a succession of elementary processes discussed in section IV: first the creation of a Φ x , Φ −1 x fluxon pair, then the motion of one of them along γ y and finally annihilation of the two fluxons. As we have seen, each of these processes may be described by a gauge-invariant operator. Let us denote by C x (Φ x ) the product of all these operators involved in the creation of the string. Similarly, we define C y (Φ y ). More generally, we may start from a state already characterized by a pair of fluxes (Φ x , Φ y ). Since γ x and γ y commute up to homotopy, we require that Φ x and Φ y commute in G. The string picture allows us to construct operators C x (g x ) and C y (g y ) such that: Denoting by |Φ x , Φ y any state with the pair of fluxes (Φ x , Φ y ), we show below that C x (g x ) and C y (g y ) do not commute, but their actions on a state |Φ x , Φ y are related by phase-factors: C y (g y )C x (g x )|Φ x , Φ y = λ(Φ x , Φ y |g x , g y ) × C x (g x )C y (g y )|Φ x , Φ y (B3) λ(Φ x , Φ y |g x , g y ) = χ gx (g y ) 2 χ Φ −1 x gxΦx (g y ) 2 χ g −1 y (g x ) 2 χ Φ −1 y g −1 y Φy (g x ) 2 (B4) Note that in order for all four states |Φ x , Φ y , |g x Φ x , Φ y , |Φ x , g y Φ y , |g x Φ x , g y Φ y to be defined, the following constraints have to be imposed: This relation (B4) is reminiscent of the phase factors involved in permuting electrical field operators attached to adjacent links for Abelian models with a Chern-Simons term 19,27 .
Let us now briefly explain how to derive the relations (B3) and (B4) expressing the commutation rules between C x (g x ) and C y (g y ). The first step is to choose template states for a pair of fluxes (Φ x , Φ y ) along elementary cycles γ x and γ y . These are depicted on Fig. 6. Let us now compare the action of C x (g x )C y (g y ) and C y (g y )C x (g x ) on this template |Φ x , Φ y . The situation is illustrated on Fig. 7. Note that we do not need explicitely the values of the fluxon hopping amplitudes A(Φ L , Φ R , g), but simply the fact they are local and that fluxon moving operators commute with gauge transformation. The value of λ(Φ x , Φ y |g x , g y ) obtained in this way (see Eq. (B4)) is valid not only for our template state |Φ x , Φ y but for any state deduced from it by a gauge transformation.