BOSONIZATION AND DUALITY IN 2+1-DIMENSIONS: APPLICATIONS IN GAUGED MASSIVE THIRRING MODEL Subir Ghosh 1 Physics Department, 9 Dinabandhu Andrews College, Calcutta 700084, 9 India. 9 1 n a J 3 1 Abstract: 1 v 1 5 Bosonizationof thegauged, massive Thirring modelin 2+1-dimensionsproduces a Maxwell- 0 1 Chern-Simonsgaugetheory, coupledtoadynamical, massivevectorfield. ExploitingtheMaster 0 Lagrangian formalism, two dual theories are constructed, one of them being a gauge theory. 9 The full two-point functions of both the interacting fields are computed in the path integral 9 / quantization scheme. Furthermore, some new dual models, derived from the original master h t lagrangian and valid for different regimes of coupling parameters, are constructed and analysed - p in details. e h : v i X r a 1Email address: ¡subir@boson.bose.res.in¿ 1 Introduction: The notion of duality has played a key role in the common goal of unification in physics. Broadly, two particular models are said to be dual to each other if in some sense they are equivalent to each other as regards to, e.g. effective action, Green’s function, etc.. One way of obtaining a pair of dual systems is to consider the so called Master lagrangian [1] of a model of larger phase space, from which each of the dual models are derivable. The presence of the Master lagrangian assures the equivalence of the said (dual) models. The other more direct way is to derive one model from another one - such as Bosonization (in 1+1-dimensions), where a fermionic model, e.g. the Massive Thirring Model (MTM) was shown to be dual to the (bosonic) sine-Gordon model [2]. Recently there has been a marriage of sorts between these two approaches in higher (2+1) dimensions. Early works [3] in 2+1-dimensional bosonization were not completely satisfactory, asthefermion-bosonmappingwasplaguedwithnon-locality. However, theprogrammereceived an impetus from later works [4], where a local bosonic action for the MTM is obtained at the cost of truncating the Seelay expansion of the fermion determinant at one loop. The small parameter in the expansion is the inverse of fermion mass. In a recent paper [5], we have been able to bosonize the Gauged Massive Thirring Model (GMTM). The spectrum of the bosonic model, a gauge theory, was obtained in a classical setup [5]. The present work is devoted to a quantum analysis of the bosonized GMTM. Exact two-point functions of the GMTM are derived. The concept of duality is utilised in two ways: On the one hand to generate the full propagators of GMTM and on the other hand, to establish the equivalence between several (manifestly) gauge invariant models with non-invariant ones. The latter phenomenon is reminiscent of the duality between abelian self-dual and Maxwell- Chern-Simons models [6], and non-abelian self-dual and Yang-Mills-Chern-Simons models [7]. The rest of this section is devoted to a brief elaboration of the computational scheme adopted. In section II, the GMTM and its bosonized version is stated from previous works [5]. The lagrangian of GMTM, L (ψ,A ,m,e,g) consists of the following parameters: the fermion (ψ) F µ mass m, the gauge field (A ) coupling e and the Thirring coupling g. The bosonized version µ has L (B ,A ,m,e,g) where B is an auxiliary vector field introduced to linearize the Thirring B µ µ µ interaction. The latter model is our Master Lagrangian (ML). Note that the duality between L and L is not exact, for reasons mentioned earlier. F B In section III we derive the dual partition functions Z(A ) and Z(B ), via integrating µ µ out B and A (in the Lorentz gauge) from ML respectively. This produces an exact duality µ µ between Z(A ), a gauge invariant theory, and Z(B ), a non-invariant one. µ µ The dual system helps us to compute the full propagators < A A > and of ML. µ ν µ ν Limiting values of the parameters are studied to show the consistency of the above framework. This is the content of section IV. Section V comprises of some more dual models, one of the pairs (of a dual system) being a gauge theory. All these models are derived from the starting ML, the distinction being the relative strengths of the couplings e and g. Once again Green’s functions of these models are derived. We end the paper with a brief conclusion in section VI. 2 II. Bosonized model Our starting point is the gauged massive Thirring model, g L = ψ¯iγµ(∂ −ieA )ψ −mψ¯ψ + ψ¯γµψψ¯γ ψ. (1) F µ µ µ 2 Here A is an external, abelian gauge field and g is the Thirring coupling constant. The system µ has abelian gauge invariance under, ψ → exp(ieα), A → A +∂ α. µ µ µ Notice that no kinetic terms for A , (e.g. Maxwell or Chern-Simons term), are present in (1) to µ give dynamics to the photon. However this is no restriction since we will study the bosonized version of (1), where the kinetic terms corresponding to A will be automatically generated in µ the Seelay expansion of the fermion determinant. Hence the coefficients of the starting kinetic terms, (if present), will get renormalized anyway after bosonization. We now introduce an auxiliary field B to linearize the Thirring interaction, µ 1 L = ψ¯iγµ[∂ −i(B +eA )]ψ −mψ¯ψ − BµB . (2) F µ µ µ µ 2g The partition function to be evaluated is ¯ Z = D(ψψ)exp(i L ). (3) B Z Z F Unlike in 1+1-dimensions, the fermion determinant is non-local and conventionally one com- putes the one loop expression in a power series with inverse fermion mass, (i.e.1), as the small m parameter. Keeping the m-independent and O(1) terms only we obtain, m a α 1 L = − (B +eA) (B +eA)µν + ǫ (B +eA)µ(B +eA)νλ − B Bµ, (4) B µν µνλ µ 4 2 2g where α = 1/(4π) and a = −1/(24πm). In the present work, we treat the above L in (4) as B our ML and the analysis actually begins here. The ML has an invariance under local gauge transformation of A . The partion function corresponding to (4) in Lorentz gauge is µ a Z = D(A,B)δ(∂ Aµ)expi[− (B +eA) (B +eA)µν Z µ 4 µν α 1 + ǫ (B +eA)µ(B +eA)νλ − B Bµ +J Aµ +L Bµ]. (5) µνλ µ µ µ 2 2g Thesourcecurrent J isconserved (∂ Jµ = 0),butthereisnosuchrestrictiononthesourcecur- µ µ rent L . The classical analysis including the spectrum of this model has already been discussed µ in [5]. At present we are interested in the quantum Green Functions (GF) < A (x)A (y) > µ ν and of the model. This will be achieved by the introduction of two dual µ ν lagrangians, obtained by selective integration of the A and B fields respectively. µ µ 3 III. Dual lagrangians Our objective is to integrate out A and B separately from the ML. Since A is a gauge µ µ µ field, the computations have been carried out in a particular (Lorentz) gauge, so that ∂ Aµ = o. µ The effective action is rewritten as, a α 1 Z = D(B)expi[− B Bµν + ǫ BµBνλ − B Bµ +B Lµ] Z 4 µν 2 µνλ 2g µ µ D(A)expi[e2AµD Aν +eH Aµ] (6) Z µν µ where, a∂2 a D = e2[ g +(ζ − )∂ ∂ +(−α)ǫ ∂λ] µν µν µ ν µνλ 2 2 = e2(Pg +Q∂ ∂ +Rǫ ∂λ), µν µ ν µνλ H = ea∂νB +eαǫ Bνλ +J . µ νµ µνλ µ ζ is the gauge parameter. Let us define, 1 D−1 = [pg +q∂ ∂ +rǫ ∂λ]. (7) µν e2 µν µ ν µνλ Demanding DµνD−1 = gµ, we solve for p, q and r and get νλ λ P 2a∂2 e2p = = , P2 +R2∂2 a2(∂2)2 +4α2∂2 R 4α e2r = − = , P2 +R2∂2 a2(∂2)2 +4α2∂2 R2 −QP 4α2 −a(2ζ −a)∂2 e2q = = . (P2 +R2∂2)(P +Q∂2) ζ∂2(a2(∂2)2 +4α2∂2) After integration of A we are left with, µ a α 1 Z (B ) = D(B )expi [(− B Bµν + ǫ BµBνλ − B Bµ +B Lµ) B µ Z µ Z 4 µν 2 µνλ 2g µ µ i +(− )(HµD(−1)Hν)]. (8) 4 µν After simplification we arrive at i 2 1 2a 1 Z(B ) = D(B)exp(− )[ B Bµ +4B ( Jµ −Lµ)+ J Jµ µ Z 4 g µ µ e e2 µa∂2 +4α2 4α 1 − J ǫ ∂νJλ] (9) µ µνλ e2 ∂2(a2∂2 +4α2) 4 Even though the source free expression of Z(B ) looks trivial with only a contact term in B , µ µ the Green functions are quite involved, as we will demonstrate in the next section. In a similar way, we now carry out the B integration. No gauge fixing is required here. µ The starting partion function is reexpressed as ae2 αe2 Z (A ) = D(A)δ(∂ Aµ)expi[− AµνA + ǫ AµAνλ +J Aµ] B µ Z µ 4 µν 2 µνλ µ a α 1 ae D(B)expi[− B Bµν+ ǫ BµBνλ− B Bµ− B Aµν+αeǫ BµAνλ+L Bµ]. (10) Z 4 µν 2 µνλ 2g µ 2 µν µνλ µ Once again we define the B -part as µ D(B)expi[BµK Bν +h Bµ] Z µν µ a∂2 1 a K = [( − )g +(− )∂ ∂ +(−α)ǫ ∂λ] µν µν µ ν µνλ 2 2g 2 = Pg +Q∂ ∂ +Rǫ ∂λ (11) µν µ ν µνλ h = ea∂νA +eαǫ Aνλ +L . (12) µ νµ µνλ µ Proceeding exactly as in the previous case, the required inverse operator is, K−1 = [pg +q∂ ∂ +rǫ ∂λ]. (13) µν µν µ ν µνλ P 2g(ag∂2 −1) p = = , P2 +R2∂2 (ag∂2 −1)2 +4α2g2∂2 R2 −QP) 2g2(4α2g +a2g∂2) q = = − (P2 +R2∂2)(P2 +Q∂2) (ag∂2 −1)2 +4α2g2∂2 R 4αg2 r = − = . P2 +R2∂2 (ag∂2 −1)2 +4α2g2∂2 The integration of B leads to µ ae2 αe2 Z (A ) = D(A )δ(∂ Aµ)expi[− AµνA + ǫ AµAνλ +J Aµ] B µ Z µ µ 4 µν 2 µνλ µ i +(− )(hµK(−1)hν)]. (14) 4 µν The final result is the gauge invariant action, 4α2g −a+a2g∂2 Z(A ) = Dδ(∂ Aµ)exp(−i/4)[2e2A (∂2gµν −∂µ∂ν)A µ Z µ µ(ag∂2 −1)2 +4α2g2∂2 ν {−4α+8g(1−α)∂2(2α2g −a+a2g∂2)} +e2A ǫµνλ∂ A µ ν λ (ag∂2 −1)2 +4α2g2∂2 5 4α2g −a+a2g∂2 +4egA (∂2gµν −∂µ∂ν)L µ ν (ag∂2 −1)2 +4α2g2∂2 1 −8αegLµǫ ∂νAλ µνλ (ag∂2 −1)2 +4α2g2∂2 1 −2g2Lµ∂ (4α2g −a+a2g∂2) ∂ Lν µ ν (ag∂2 −1)2 +4α2g2∂2 1 1 −4αg2L ǫµνλ∂ L +2gL (ag∂2−1) Lµ−4J Aµ]. µ ν λ µ µ (ag∂2 −1)2 +4α2g2∂2 (ag∂2 −1)2 +4α2g2∂2 (15) The ∂.A terms in (6) have been displayed to show the manifest gauge invariance but they are dropped later since Lorentz gauge has been adopted. Thus we have established an exact duality between the complicated gauge invariant effective action in (15) and the more simpler looking one in (9). It should be kept in mind that the duality regarding the original fermion model in (1) and its bosonized version in (5),(as mentioned before), was not exact. On the other hand, the just exposed duality is exact since they both originate from the same master lagrangian in (5) and no further approximation are involved so far. The next task is to compute the A and µ B GF’s by exploiting the latter duality. µ IV. Green’s functions The sources present in (5) allow us to formally define the GF’s, 1 δ2Z ( )2 B ≡< Aµ(x)Aν(y) >, (16) i δJ (y)δJ (x) ν µ 1 δ2Z ( )2 B ≡< Bµ(x)Bν(y) > . (17) i δL (y)δL (x) ν µ But thanks to the duality we can replace Z by Z (A ) or Z (B ). Using Z (B ) in (10) we B B µ B µ B µ find, i 2α e2 < Aµ(x)Aν(y) >=< Bµ(x)Bν(y) > + (agµνδ(x−y)+ ǫµνλ∂ δ(x−y)). (18) λ a2∂2 +4α2 ∂2 In a similar way, performing the variation (1)2δ2Z (Aµ)/(δLµ(x)δLν(y)) on (17), we obtain i B the relation, (4α2g −a+a2g∂2) = [eg ∂2A (x) µ ν µ (ag∂2 −1)2 +4α2g2∂2 1 (4α2g −a+a2g∂2) −2αeg ǫ ∂αAβ(x)][eg ∂2A (y) µαβ ν (ag∂2 −1)2 +4α2g2∂2 (ag∂2 −1)2 +4α2g2∂2 1 −2αeg ǫ ∂ρAσ(y)] νρσ (ag∂2 −1)2 +4α2g2∂2 (4α2g −a+a2g∂2) 1 +i[−g2 ∂ ∂ δ(x−y)+2αg2 ǫ ∂λδ(x−y) µ ν µνλ (ag∂2 −1)2 +4α2g2∂2 (ag∂2 −1)2 +4α2g2∂2 6 1 +g(ag∂2 −1) g δ(x−y)]. (19) µν (ag∂2 −1)2 +4α2g2∂2 From the above coupled Green functions, we can derive Green function identities containing exclusively A or B fields, which are given below in a compact form. µ µ e2 < A (x)A (y) >= e2g2Mµλ(x)Mνσ(y) < A (x)A (y) > µ ν λ σ a +i[{ +g(ag∂2 −1)φ}gµνδ(x−y) a2∂2 +4α2 2α −g2(4α2g −a+a2g∂2)φ∂µ∂νδ(x−y)+{ +2αg2φ}ǫµνλ∂ δ(x−y)], (20) λ ∂2(a2∂2 +4α2) = g2Mµλ(x)Mνσ(y) µ ν λ σ ag2∂2 m2 +i[{ ( 1 +m2)+g(ag∂2 −1)φ}gµνδ(x−y) a2∂2 +4α2 ∂2 3 ag2 m2 −{ ( 1 +m2)+g2(4α2g −a+a2g∂2)φ}∂µ∂νδ(x−y) a2∂2 +4α2 ∂2 3 2αg2 m2 +{ ( 1 +m2)+2αg2φ}ǫµνλ∂ δ(x−y)]. (21) (a2∂2 +4α2) ∂2 3 λ The operator Mµλ(x) acts on x and is defined below, Mµλ = m gµλ +m ∂µ∂λ +m ǫµνλ∂ 1 2 3 ν = (4α2g −a+a2g∂2)φ[∂2gµλ −∂µ∂λ]−2αφǫµνλ∂ , ν φ = [(ag∂2 −1)2 +4α2g2∂2]−1. (22) For the sake of comparison, we also write down the propagators for the Maxwell- Chern-Simons model, −a L = A Aµν +αǫµνλA ∂ A +J Aµ, ∂.J = 0, (23) MCS µν µ ν λ µ 4 and the Maxwell-Chern-Simons-Proca model, −a 1 L = B Bµν +αǫµνλB ∂ B − B Bµ +L Bµ. (24) MCSP µν µ ν λ µ µ 4 2g The propagators are respectively, a 2α < Aµ(x)Aν(y) > = i( gµνδ(x−y)+ ǫµνλ∂ δ(x−y)), (25) MCS λ a2∂2 +4α2 ∂2(a2∂2 +4α2) i < Bµ(x)Bν(y) > = [2g(ag∂2 −1)gµνδ(x−y)−2g2(4α2g +a2g∂2 −a)∂µ∂νδ(x−y) MCSP 4φ +4αg2ǫµνλ∂ δ(x−y)], (26) λ It is imperative to check the consistency of the above procedure through a qualitative analysis of (20) and (21), by considering limiting values of the parameters involved, specifically the fermion mass m and the Thirring coupling g. 7 The bosonization result was obtained in the large m limit. In keeping conformity with that, it is natural to keep only O(m−1) terms and hence a2 ≈ 0. Indeed, this choice is by no means mandatory, since the bosonized ML is an independent field theory by itself. First we consider the large g limit and keep terms of O(a,g−1) only in the Green’s functions. ae2 a ∂µ∂. ∂ν∂. < Aµ(x)Aν(y) >= e2( −1) < A(x)[ A(y) 2α2g 2α2g ∂2 ∂2 ∂ν∂. ∂µ∂. +Aµ(x) A(y)+ A(x)Aν(y)] > ∂2 ∂2 e2 ǫνρσ∂ ǫµρσ∂ ∂µ∂. ǫνρσ∂ + < [−Aµ(x) ρA (y)−Aν(y) ρA (x)+ A(x) ρb (y) σ σ σ 2αg ∂2 ∂2 ∂2 ∂2 ǫµρσ∂ ∂ν∂. ρ + A (x) A(y)] > σ ∂2 ∂2 a 1 a ∂µ∂ν 1 +i[( − )gµνδ(x−y)−( +g) δ(x−y)+ ǫµνλ∂ δ(x−y). (27) λ 4α2 4α2g∂2 4α2 ∂2 α∂2 a a ∂µ∂. < Bµ(x)Bν(y) >= ( −1) < B(x) 2α2g 2α2g ∂2 ∂ν∂. ∂ν∂. ∂µ∂. [ B(y)+Bµ(x) B(y)+ B(x)Bν(y)] > ∂2 ∂2 ∂2 1 ǫνρσ∂ ǫµρσ∂ ∂µ∂. ǫνρσ∂ + < [−Bµ(x) ρB (y)−Bν(y) ρB (x)+ B(x) ρB (y) σ σ σ 2αg ∂2 ∂2 ∂2 ∂2 ǫµρσ∂ ∂ν∂. ρ + B (x) B(y)] > σ ∂2 ∂2 a 1 a ∂µ∂ν 1 a +i[( − )gµνδ(x−y)−( +g) δ(x−y)+ (1+ )ǫµνλ∂ δ(x−y)]. (28) λ 2α2 4α2g∂2 2α2 ∂2 2α∂2 2α2g If we consider g-independent terms only, the A and B propagators become identical, but for µ µ anoverall factorof 1 inthec-number expressions. Thisis alsothesame astheL propagator 2 MCS in (25). This is consistent with the fact that in the master lagrangian in (5), the difference between A and B disappears as g → ∞ and the model tends to L in the composite field µ µ MCS (B +eA ). µ µ In case of small g, we have φ ≈ 1, m ≈ −2α and m ≈ −2α. This reproduces the following 1 3 approximate propagators, < Bµ(x)Bν(y) >≈ −igµνδ(x−y), (29) a 2α e2 < Aµ(x)Aν(y) >≈ i[ gµνδ(x−y)+ ǫµνλ∂ δ(x−y). (30) λ a2∂2 +4α2 ∂2(a2∂2 +4α2) Notice that the A -propagator in (29) is identical to the MCS propagator in (25). This be- µ haviour is justified in the limit of small g where the mass term for B in (4) becomes infinite µ and hence the B field effectively decouples. This is also the reason for trivial form of the B µ µ propagator in (29). V: New dualities 8 We now consider two different ML’s, which can be thought of as large or small e-the electromagnetic coupling limit of the original ML in (4) considered in the beginning. (i). Small e limit: This procedure produces a different ML where we drop O(e2) terms from starting ML in (4) and integrate B first. The resulting partition function is µ ae Z = D(A)δ(∂.A)expi(J Aµ) D(B)[− B Aµν +eαǫ BµAνλ +J Aµ 1 Z µ Z 2 µν µνλ µ a α 1 − B Bµν + ǫ BµBνλ − B Bµ +B Lµ]. (31) µν µνλ µ µ 4 2 2g with ∂.J = 0. The result is i Z (A ) = D(A)δ(∂.A)exp(− )[−4J Aµ +e2A {2a2g(∂2)2(ag∂2 −1) 1 µ Z 4 µ µ +4α2g∂2(ag∂2 +1)φ}(∂2gµν −∂µ∂ν)A ν +e2A {4g∂2(αa2g∂2 −2a−4α2g)}φǫµνλ∂ A µ ν λ +4eg(A {4α2g −a+a2g∂2}φ(∂2gµν −∂µ∂ )A µ ν ν 1 −8αegLµǫ ∂νAλ µνλ (ag∂2 −1)2 +4α2g2∂2 1 −2g2Lµ∂ (4α2g −a+a2g∂2) ∂ Lν µ ν (ag∂2 −1)2 +4α2g2∂2 1 1 −4αg2L ǫµνλ∂ L +2gL (ag∂2 −1) Lµ]. (32) µ ν λ µ (ag∂2 −1)2 +4α2g2∂2 (ag∂2 −1)2 +4α2g2∂2 In case of A integration, the gauge fixing is really not necessary since A appears only linearly µ µ in the action. The A equation of motion produces the relation µ −ea∂νB +eαǫ Bνλ +J = 0. (33) νµ µνλ µ Substituting the above relation in the action, the result is a 1 1 Z (B ) = D(B)expi[− B Bµν − B Bµ − B Jµ +B Lm]. (34) 1 µ Z 2 µν 2g µ 2e µ µ Without the sources, this is the Proca Model. However, the Green’s functions are, 4e2 < Aµ(x)Aν(y) >= e2g2Mµλ(x)Mνσ(y) < A (x)A (y) > λ σ +i{g(ag∂2)φgµνδ(x−y)−g2(4α2g −a+ag∂2)φ∂µ∂νδ(x−y)+2αg2φǫµνλ∂ δ(x−y)}, (35) λ where the operator Mµν(x) is the same as in (22). The B Green function is obtained from the µ above one by replacing A by B /(2e). The reason is that the variation by −iδ/δJ reproduces µ µ µ A and −B /(2e) when operated upon Z and Z (B ) respectively. µ µ 1 1 µ (ii) Large e limit: We drop the e-independent quadratic B-terms from the original ML. ae2 αe2 Z = D(A)δ(∂ Aµ)expi[− AµνA + ǫ AµAνλ +J Aµ] 2 Z µ 4 µν 2 µνλ µ 9 1 ae D(B)expi[− B Bµ − B Aµν +αeǫ BµAνλ +L Bµ]. (36) Z 2g µ 2 µν µνλ µ After B integration, the result is µ a a2g Z (A ) = D(A)δ(∂ Aµ)exp(−i)[e2A ( + ∂2−2α2g)(∂2gµν−∂µ∂ν)A +2αage2A ∂2ǫµνλ∂ A 2 µ Z µ µ 2 2 ν µ ν λ g +eagL ∂ Aνµ + L Lµ +2αegLµǫµνλ∂ A ]. (37) µ ν µ ν λ 2 Integrating out the A field in the Lorentz gauge gives, µ a 1 1 a 1 Z (B ) = D(B)exp(−i)[ (B ∂2Bµ−B ∂µ∂νB )+ B Bµ+B ( Jµ−Lµ)+ J Jµ 2 µ Z 2 µ µ ν 2g µ µ e 2e2 µa∂2 +4α2 α 1 − J ǫ ∂νJλ]. (38) µ µνλ e2 ∂2(a2∂2 +4α2) Without the sources, this is the Chern-Simons-Proca model. The Green’s functions in this case are 1 < Aµ(x)Aν(y) >= Nµβ(x)Nνλ(y) < A (x)A (y) > β λ e2 i a 2α + {(g + )gµνδ(x−y)− ǫµνλ∂ δ(x−y)}, (39) λ e2 a2∂2 +4α2 ∂2(a2∂2 +4α2) 1 < Bµ(x)Bν(y) >= Nµβ(x)Nνλ(y) β λ e2 1 a 2α +i{ggµνδ(x−y)+ Nµβ(x)Nνλ(y)( g δ(x−y)+ ǫ ∂σδ(x−y))}, βλ βλσ e2 a2∂2 +4α2 ∂2(a2∂2 +4α2) (40) The operator Nµν is defined as Nµν = eag∂2gµν −eag∂µ∂ν −2αegǫµνλ∂ . λ VI: Conclusions Bosonized version of the gauged, massive Thirring model is analysed exhaustively at the level of effective action. Duality invariance is used to construct exactly equivalent gauge invariant and non-invariant models. The correlation functions of the bosonized models are studied. Validity of this new scheme to achieve this is tested in different coupling regimes. Depending on the relative strengths of the electromagnetic and Thirring couplings, some new dual models, including their propagators, are also presented. Acknowledgements It is a pleasure to thank Dr. Rabin Banerjee for a number of helpful discussions. I am also grateful to Professor C. K. Majumdar, Director, S. N. Bose National Centre for Basic Sciences, Calcutta, for allowing me to use the Institute facilities. 10

Bosonization and Duality in 2+1-Dimensions: Applications in Gauged Massive Thirring Model

Similar Bosonization and Duality in 2+1-Dimensions: Applications in Gauged Massive Thirring Model